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:

  lexicographic combination of reduction pairs:
  
    1. max/plus interpretations on natural numbers:
    
      active#_A(x1) = x1 + 1291
      zeros_A = 0
      mark#_A(x1) = 1291
      cons_A(x1,x2) = 0
      |0|_A = 0
      U96_A(x1) = 0
      mark_A(x1) = x1 + 137
      take_A(x1,x2) = 0
      s_A(x1) = 0
      U131_A(x1,x2,x3,x4) = 0
      isNatIList_A(x1) = 0
      U95_A(x1,x2) = 0
      U121_A(x1,x2) = 0
      U94_A(x1,x2,x3) = 0
      length_A(x1) = 0
      U111_A(x1,x2,x3) = 0
      isNatList_A(x1) = 0
      U93_A(x1,x2,x3) = 0
      U101_A(x1,x2,x3) = 0
      isNatKind_A(x1) = 0
      U92_A(x1,x2,x3) = 0
      U91_A(x1,x2,x3) = 0
      U81_A(x1) = 0
      U71_A(x1) = 0
      isNatIListKind_A(x1) = 0
      U61_A(x1,x2) = 0
      U62_A(x1) = 0
      U51_A(x1,x2) = 0
      U41_A(x1,x2,x3) = 0
      U52_A(x1) = 0
      U31_A(x1,x2) = 0
      isNat_A(x1) = 0
      U21_A(x1,x2) = 0
      U46_A(x1) = 0
      U11_A(x1,x2) = 0
      U45_A(x1,x2) = 0
      tt_A = 0
      U44_A(x1,x2,x3) = 0
      U43_A(x1,x2,x3) = 0
      U42_A(x1,x2,x3) = 0
      U33_A(x1) = 0
      U32_A(x1,x2) = 0
      U23_A(x1) = 0
      U22_A(x1,x2) = 0
      U136_A(x1,x2,x3,x4) = 0
      U135_A(x1,x2,x3,x4) = 0
      U134_A(x1,x2,x3,x4) = 0
      U133_A(x1,x2,x3,x4) = 0
      U132_A(x1,x2,x3,x4) = 0
      U122_A(x1) = 0
      U13_A(x1) = 0
      U12_A(x1,x2) = 0
      U114_A(x1,x2) = 0
      U113_A(x1,x2,x3) = 0
      U112_A(x1,x2,x3) = 0
      U105_A(x1,x2) = 0
      U106_A(x1) = 0
      U104_A(x1,x2,x3) = 0
      U103_A(x1,x2,x3) = 0
      U102_A(x1,x2,x3) = 0
      active_A(x1) = x1 + 137
      nil_A = 0
    
    2. max/plus interpretations on natural numbers:
    
      active#_A(x1) = max{0, x1 - 20}
      zeros_A = 33
      mark#_A(x1) = 13
      cons_A(x1,x2) = 14
      |0|_A = 33
      U96_A(x1) = 9
      mark_A(x1) = max{0, x1 - 14}
      take_A(x1,x2) = 33
      s_A(x1) = 21
      U131_A(x1,x2,x3,x4) = 33
      isNatIList_A(x1) = 33
      U95_A(x1,x2) = 33
      U121_A(x1,x2) = 33
      U94_A(x1,x2,x3) = 33
      length_A(x1) = 33
      U111_A(x1,x2,x3) = 33
      isNatList_A(x1) = 33
      U93_A(x1,x2,x3) = 33
      U101_A(x1,x2,x3) = 33
      isNatKind_A(x1) = 33
      U92_A(x1,x2,x3) = 33
      U91_A(x1,x2,x3) = 33
      U81_A(x1) = 11
      U71_A(x1) = 9
      isNatIListKind_A(x1) = 33
      U61_A(x1,x2) = 33
      U62_A(x1) = 9
      U51_A(x1,x2) = 33
      U41_A(x1,x2,x3) = 33
      U52_A(x1) = 2
      U31_A(x1,x2) = 33
      isNat_A(x1) = 33
      U21_A(x1,x2) = 33
      U46_A(x1) = 9
      U11_A(x1,x2) = 33
      U45_A(x1,x2) = 33
      tt_A = 10
      U44_A(x1,x2,x3) = 33
      U43_A(x1,x2,x3) = 33
      U42_A(x1,x2,x3) = 33
      U33_A(x1) = 9
      U32_A(x1,x2) = 33
      U23_A(x1) = 23
      U22_A(x1,x2) = 33
      U136_A(x1,x2,x3,x4) = 33
      U135_A(x1,x2,x3,x4) = 33
      U134_A(x1,x2,x3,x4) = 33
      U133_A(x1,x2,x3,x4) = 33
      U132_A(x1,x2,x3,x4) = 33
      U122_A(x1) = 2
      U13_A(x1) = 9
      U12_A(x1,x2) = 33
      U114_A(x1,x2) = 33
      U113_A(x1,x2,x3) = 33
      U112_A(x1,x2,x3) = 33
      U105_A(x1,x2) = 33
      U106_A(x1) = 14
      U104_A(x1,x2,x3) = 33
      U103_A(x1,x2,x3) = 33
      U102_A(x1,x2,x3) = 33
      active_A(x1) = max{0, x1 - 14}
      nil_A = 14
    

The next rules are strictly ordered:

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


-- 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)) -> mark#(X)
p47: mark#(length(X)) -> active#(length(mark(X)))
p48: active#(U134(tt(),IL,M,N)) -> mark#(U135(isNat(N),IL,M,N))
p49: mark#(length(X)) -> mark#(X)
p50: mark#(U13(X)) -> mark#(X)
p51: mark#(isNatList(X)) -> active#(isNatList(X))
p52: active#(U135(tt(),IL,M,N)) -> mark#(U136(isNatKind(N),IL,M,N))
p53: mark#(U121(X1,X2)) -> active#(U121(mark(X1),X2))
p54: active#(U136(tt(),IL,M,N)) -> mark#(cons(N,take(M,IL)))
p55: mark#(U121(X1,X2)) -> mark#(X1)
p56: mark#(U122(X)) -> mark#(X)
p57: mark#(U131(X1,X2,X3,X4)) -> active#(U131(mark(X1),X2,X3,X4))
p58: active#(U21(tt(),V1)) -> mark#(U22(isNatKind(V1),V1))
p59: mark#(U131(X1,X2,X3,X4)) -> mark#(X1)
p60: mark#(U132(X1,X2,X3,X4)) -> active#(U132(mark(X1),X2,X3,X4))
p61: active#(U22(tt(),V1)) -> mark#(U23(isNat(V1)))
p62: mark#(U132(X1,X2,X3,X4)) -> mark#(X1)
p63: mark#(U133(X1,X2,X3,X4)) -> active#(U133(mark(X1),X2,X3,X4))
p64: active#(U31(tt(),V)) -> mark#(U32(isNatIListKind(V),V))
p65: mark#(U133(X1,X2,X3,X4)) -> mark#(X1)
p66: mark#(U134(X1,X2,X3,X4)) -> active#(U134(mark(X1),X2,X3,X4))
p67: active#(U32(tt(),V)) -> mark#(U33(isNatList(V)))
p68: mark#(U134(X1,X2,X3,X4)) -> mark#(X1)
p69: mark#(U135(X1,X2,X3,X4)) -> active#(U135(mark(X1),X2,X3,X4))
p70: active#(U41(tt(),V1,V2)) -> mark#(U42(isNatKind(V1),V1,V2))
p71: mark#(U135(X1,X2,X3,X4)) -> mark#(X1)
p72: mark#(U136(X1,X2,X3,X4)) -> active#(U136(mark(X1),X2,X3,X4))
p73: active#(U42(tt(),V1,V2)) -> mark#(U43(isNatIListKind(V2),V1,V2))
p74: mark#(U136(X1,X2,X3,X4)) -> mark#(X1)
p75: mark#(take(X1,X2)) -> active#(take(mark(X1),mark(X2)))
p76: active#(U43(tt(),V1,V2)) -> mark#(U44(isNatIListKind(V2),V1,V2))
p77: mark#(take(X1,X2)) -> mark#(X1)
p78: mark#(take(X1,X2)) -> mark#(X2)
p79: mark#(U21(X1,X2)) -> active#(U21(mark(X1),X2))
p80: active#(U44(tt(),V1,V2)) -> mark#(U45(isNat(V1),V2))
p81: mark#(U21(X1,X2)) -> mark#(X1)
p82: mark#(U22(X1,X2)) -> active#(U22(mark(X1),X2))
p83: active#(U45(tt(),V2)) -> mark#(U46(isNatIList(V2)))
p84: mark#(U22(X1,X2)) -> mark#(X1)
p85: mark#(U23(X)) -> mark#(X)
p86: mark#(U31(X1,X2)) -> active#(U31(mark(X1),X2))
p87: active#(U51(tt(),V2)) -> mark#(U52(isNatIListKind(V2)))
p88: mark#(U31(X1,X2)) -> mark#(X1)
p89: mark#(U32(X1,X2)) -> active#(U32(mark(X1),X2))
p90: active#(U61(tt(),V2)) -> mark#(U62(isNatIListKind(V2)))
p91: mark#(U32(X1,X2)) -> mark#(X1)
p92: mark#(U33(X)) -> mark#(X)
p93: mark#(U41(X1,X2,X3)) -> active#(U41(mark(X1),X2,X3))
p94: active#(U91(tt(),V1,V2)) -> mark#(U92(isNatKind(V1),V1,V2))
p95: mark#(U41(X1,X2,X3)) -> mark#(X1)
p96: mark#(U42(X1,X2,X3)) -> active#(U42(mark(X1),X2,X3))
p97: active#(U92(tt(),V1,V2)) -> mark#(U93(isNatIListKind(V2),V1,V2))
p98: mark#(U42(X1,X2,X3)) -> mark#(X1)
p99: mark#(U43(X1,X2,X3)) -> active#(U43(mark(X1),X2,X3))
p100: active#(U93(tt(),V1,V2)) -> mark#(U94(isNatIListKind(V2),V1,V2))
p101: mark#(U43(X1,X2,X3)) -> mark#(X1)
p102: mark#(U44(X1,X2,X3)) -> active#(U44(mark(X1),X2,X3))
p103: active#(U94(tt(),V1,V2)) -> mark#(U95(isNat(V1),V2))
p104: mark#(U44(X1,X2,X3)) -> mark#(X1)
p105: mark#(U45(X1,X2)) -> active#(U45(mark(X1),X2))
p106: active#(U95(tt(),V2)) -> mark#(U96(isNatList(V2)))
p107: mark#(U45(X1,X2)) -> mark#(X1)
p108: mark#(U46(X)) -> mark#(X)
p109: mark#(U51(X1,X2)) -> active#(U51(mark(X1),X2))
p110: active#(isNat(length(V1))) -> mark#(U11(isNatIListKind(V1),V1))
p111: mark#(U51(X1,X2)) -> mark#(X1)
p112: mark#(U52(X)) -> mark#(X)
p113: mark#(U61(X1,X2)) -> active#(U61(mark(X1),X2))
p114: active#(isNat(s(V1))) -> mark#(U21(isNatKind(V1),V1))
p115: mark#(U61(X1,X2)) -> mark#(X1)
p116: mark#(U62(X)) -> mark#(X)
p117: mark#(U71(X)) -> mark#(X)
p118: mark#(U81(X)) -> mark#(X)
p119: mark#(U91(X1,X2,X3)) -> active#(U91(mark(X1),X2,X3))
p120: active#(isNatIList(V)) -> mark#(U31(isNatIListKind(V),V))
p121: mark#(U91(X1,X2,X3)) -> mark#(X1)
p122: mark#(U92(X1,X2,X3)) -> active#(U92(mark(X1),X2,X3))
p123: active#(isNatIList(cons(V1,V2))) -> mark#(U41(isNatKind(V1),V1,V2))
p124: mark#(U92(X1,X2,X3)) -> mark#(X1)
p125: mark#(U93(X1,X2,X3)) -> active#(U93(mark(X1),X2,X3))
p126: active#(isNatIListKind(cons(V1,V2))) -> mark#(U51(isNatKind(V1),V2))
p127: mark#(U93(X1,X2,X3)) -> mark#(X1)
p128: mark#(U94(X1,X2,X3)) -> active#(U94(mark(X1),X2,X3))
p129: active#(isNatIListKind(take(V1,V2))) -> mark#(U61(isNatKind(V1),V2))
p130: mark#(U94(X1,X2,X3)) -> mark#(X1)
p131: mark#(U95(X1,X2)) -> active#(U95(mark(X1),X2))
p132: active#(isNatKind(length(V1))) -> mark#(U71(isNatIListKind(V1)))
p133: mark#(U95(X1,X2)) -> mark#(X1)
p134: mark#(U96(X)) -> mark#(X)
p135: active#(isNatKind(s(V1))) -> mark#(U81(isNatKind(V1)))
p136: active#(isNatList(cons(V1,V2))) -> mark#(U91(isNatKind(V1),V1,V2))
p137: active#(isNatList(take(V1,V2))) -> mark#(U101(isNatKind(V1),V1,V2))
p138: active#(length(cons(N,L))) -> mark#(U111(isNatList(L),L,N))
p139: active#(take(|0|(),IL)) -> mark#(U121(isNatIList(IL),IL))
p140: 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:

  lexicographic combination of reduction pairs:
  
    1. max/plus interpretations on natural numbers:
    
      active#_A(x1) = max{0, x1 - 90}
      zeros_A = 909
      mark#_A(x1) = max{77, x1 - 90}
      cons_A(x1,x2) = max{909, x1 + 735, x2}
      |0|_A = 63
      U101_A(x1,x2,x3) = max{1161, x1 + 975, x2 + 350, x3 + 535}
      mark_A(x1) = max{174, x1}
      tt_A = 75
      U102_A(x1,x2,x3) = max{1049, x1 + 168, x2 + 253, x3 + 534}
      isNatKind_A(x1) = max{182, x1 + 11}
      U103_A(x1,x2,x3) = max{828, x1 + 91, x2 + 252, x3 + 288}
      isNatIListKind_A(x1) = max{163, x1 - 366}
      U104_A(x1,x2,x3) = max{828, x1 + 654, x2 + 251, x3 + 287}
      U105_A(x1,x2) = max{828, x1 + 213, x2 + 286}
      isNat_A(x1) = max{183, x1 + 37}
      U106_A(x1) = max{287, x1 + 100}
      isNatIList_A(x1) = max{710, x1 + 185}
      U11_A(x1,x2) = max{542, x1 + 19, x2 + 1}
      U12_A(x1,x2) = max{542, x1 + 367, x2}
      U111_A(x1,x2,x3) = max{909, x1, x2, x3 + 735}
      U112_A(x1,x2,x3) = max{908, x1 + 167, x2, x3 + 340}
      U113_A(x1,x2,x3) = max{507, x1, x2, x3 + 339}
      U114_A(x1,x2) = max{506, x1, x2}
      s_A(x1) = max{173, x1}
      length_A(x1) = max{505, x1}
      U13_A(x1) = max{443, x1 + 2}
      isNatList_A(x1) = max{89, x1 - 3}
      U121_A(x1,x2) = max{1164, x1 + 342, x2 + 539}
      U122_A(x1) = max{1079, x1 + 905}
      U131_A(x1,x2,x3,x4) = max{x1 + 278, x2 + 539, x3 + 989, x4 + 1274}
      U132_A(x1,x2,x3,x4) = max{x1 + 905, x2 + 539, x3 + 989, x4 + 1274}
      U133_A(x1,x2,x3,x4) = max{x1 + 169, x2 + 539, x3 + 989, x4 + 1274}
      U134_A(x1,x2,x3,x4) = max{x1 + 356, x2 + 539, x3 + 989, x4 + 1274}
      U135_A(x1,x2,x3,x4) = max{x1 + 247, x2 + 539, x3 + 989, x4 + 1273}
      U136_A(x1,x2,x3,x4) = max{1264, x1 + 1090, x2 + 539, x3 + 989, x4 + 735}
      take_A(x1,x2) = max{1164, x1 + 989, x2 + 539}
      U21_A(x1,x2) = max{209, x1 + 1, x2 + 37}
      U22_A(x1,x2) = max{200, x1 + 26, x2 + 37}
      U23_A(x1) = max{200, x1}
      U31_A(x1,x2) = max{709, x1, x2 + 167}
      U32_A(x1,x2) = max{708, x1 + 533, x2 + 167}
      U33_A(x1) = max{609, x1 + 170}
      U41_A(x1,x2,x3) = max{x1 + 4, x2 + 728, x3 + 185}
      U42_A(x1,x2,x3) = max{x1 + 12, x2 + 727, x3 + 185}
      U43_A(x1,x2,x3) = max{x1 + 550, x2 + 726, x3 + 185}
      U44_A(x1,x2,x3) = max{726, x1 + 551, x2 + 724, x3 + 185}
      U45_A(x1,x2) = max{726, x1 + 1, x2 + 185}
      U46_A(x1) = max{174, x1}
      U51_A(x1,x2) = max{370, x1, x2 - 366}
      U52_A(x1) = max{174, x1}
      U61_A(x1,x2) = max{622, x1, x2 + 173}
      U62_A(x1) = max{174, x1}
      U91_A(x1,x2,x3) = max{733, x1 + 91, x2 + 565, x3 - 3}
      U92_A(x1,x2,x3) = max{x1 + 164, x2 + 338, x3 - 3}
      U93_A(x1,x2,x3) = max{x1 + 1, x2 + 337, x3 - 3}
      U94_A(x1,x2,x3) = max{x1 + 1, x2 + 336, x3 - 3}
      U95_A(x1,x2) = max{336, x1 + 153, x2 - 3}
      U96_A(x1) = max{174, x1}
      U71_A(x1) = max{181, x1 + 7}
      U81_A(x1) = max{183, x1}
      active_A(x1) = max{169, x1}
      nil_A = 541
    
    2. max/plus interpretations on natural numbers:
    
      active#_A(x1) = 0
      zeros_A = 37
      mark#_A(x1) = 0
      cons_A(x1,x2) = 99
      |0|_A = 64
      U101_A(x1,x2,x3) = 70
      mark_A(x1) = 63
      tt_A = 41
      U102_A(x1,x2,x3) = 47
      isNatKind_A(x1) = 76
      U103_A(x1,x2,x3) = 37
      isNatIListKind_A(x1) = 29
      U104_A(x1,x2,x3) = 95
      U105_A(x1,x2) = 85
      isNat_A(x1) = 40
      U106_A(x1) = 62
      isNatIList_A(x1) = 61
      U11_A(x1,x2) = 63
      U12_A(x1,x2) = 42
      U111_A(x1,x2,x3) = 52
      U112_A(x1,x2,x3) = 42
      U113_A(x1,x2,x3) = 42
      U114_A(x1,x2) = 42
      s_A(x1) = 28
      length_A(x1) = 26
      U13_A(x1) = 93
      isNatList_A(x1) = 40
      U121_A(x1,x2) = 42
      U122_A(x1) = 61
      U131_A(x1,x2,x3,x4) = 31
      U132_A(x1,x2,x3,x4) = 42
      U133_A(x1,x2,x3,x4) = 52
      U134_A(x1,x2,x3,x4) = 42
      U135_A(x1,x2,x3,x4) = 40
      U136_A(x1,x2,x3,x4) = 30
      take_A(x1,x2) = 52
      U21_A(x1,x2) = 42
      U22_A(x1,x2) = 94
      U23_A(x1) = 84
      U31_A(x1,x2) = 51
      U32_A(x1,x2) = 42
      U33_A(x1) = 42
      U41_A(x1,x2,x3) = 53
      U42_A(x1,x2,x3) = 62
      U43_A(x1,x2,x3) = 62
      U44_A(x1,x2,x3) = 73
      U45_A(x1,x2) = 50
      U46_A(x1) = 42
      U51_A(x1,x2) = 29
      U52_A(x1) = 21
      U61_A(x1,x2) = 64
      U62_A(x1) = 56
      U91_A(x1,x2,x3) = 30
      U92_A(x1,x2,x3) = 42
      U93_A(x1,x2,x3) = 32
      U94_A(x1,x2,x3) = 52
      U95_A(x1,x2) = 42
      U96_A(x1) = 42
      U71_A(x1) = 62
      U81_A(x1) = 68
      active_A(x1) = 63
      nil_A = 64
    

The next rules are strictly ordered:

  p2, p5, p6, p8, p9, p14, p16, p19, p22, p25, p30, p32, p33, p39, p50, p55, p56, p59, p62, p65, p68, p71, p74, p77, p78, p81, p84, p90, p91, p92, p95, p98, p101, p104, p107, p117, p120, p121, p124, p127, p130, p132, p133

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

  {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}


-- Reduction pair.

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

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

  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:

  lexicographic combination of reduction pairs:
  
    1. max/plus interpretations on natural numbers:
    
      mark#_A(x1) = 141
      U96_A(x1) = 109
      U95_A(x1,x2) = 109
      active#_A(x1) = x1 + 32
      mark_A(x1) = max{57, x1 - 95}
      take_A(x1,x2) = 109
      s_A(x1) = 2
      cons_A(x1,x2) = 95
      U131_A(x1,x2,x3,x4) = 109
      isNatIList_A(x1) = 109
      U94_A(x1,x2,x3) = 109
      |0|_A = 96
      U121_A(x1,x2) = 109
      U93_A(x1,x2,x3) = 109
      length_A(x1) = 109
      U111_A(x1,x2,x3) = 109
      isNatList_A(x1) = 109
      U92_A(x1,x2,x3) = 109
      U101_A(x1,x2,x3) = 109
      isNatKind_A(x1) = 109
      U91_A(x1,x2,x3) = 109
      U81_A(x1) = 109
      U62_A(x1) = 109
      U61_A(x1,x2) = 109
      U52_A(x1) = 109
      U51_A(x1,x2) = 109
      isNatIListKind_A(x1) = 109
      U46_A(x1) = 109
      U45_A(x1,x2) = 109
      U44_A(x1,x2,x3) = 109
      U41_A(x1,x2,x3) = 109
      U43_A(x1,x2,x3) = 109
      isNat_A(x1) = 109
      U21_A(x1,x2) = 109
      U42_A(x1,x2,x3) = 109
      U11_A(x1,x2) = 109
      tt_A = 96
      U32_A(x1,x2) = 109
      U31_A(x1,x2) = 109
      U23_A(x1) = 109
      U22_A(x1,x2) = 109
      U136_A(x1,x2,x3,x4) = 109
      U135_A(x1,x2,x3,x4) = 109
      U134_A(x1,x2,x3,x4) = 109
      U133_A(x1,x2,x3,x4) = 109
      U132_A(x1,x2,x3,x4) = 109
      U33_A(x1) = 109
      U114_A(x1,x2) = 109
      U113_A(x1,x2,x3) = 109
      U112_A(x1,x2,x3) = 109
      U12_A(x1,x2) = 109
      U122_A(x1) = 109
      U105_A(x1,x2) = 109
      U104_A(x1,x2,x3) = 109
      U103_A(x1,x2,x3) = 109
      U102_A(x1,x2,x3) = 109
      active_A(x1) = max{0, x1 - 52}
      zeros_A = 109
      U106_A(x1) = 109
      U13_A(x1) = 109
      nil_A = 109
      U71_A(x1) = 109
    
    2. max/plus interpretations on natural numbers:
    
      mark#_A(x1) = 2
      U96_A(x1) = 35
      U95_A(x1,x2) = 18
      active#_A(x1) = max{0, x1 - 16}
      mark_A(x1) = 28
      take_A(x1,x2) = 18
      s_A(x1) = 29
      cons_A(x1,x2) = 65
      U131_A(x1,x2,x3,x4) = 18
      isNatIList_A(x1) = 18
      U94_A(x1,x2,x3) = 18
      |0|_A = 48
      U121_A(x1,x2) = 18
      U93_A(x1,x2,x3) = 18
      length_A(x1) = 18
      U111_A(x1,x2,x3) = 18
      isNatList_A(x1) = 18
      U92_A(x1,x2,x3) = 18
      U101_A(x1,x2,x3) = 18
      isNatKind_A(x1) = 18
      U91_A(x1,x2,x3) = 18
      U81_A(x1) = 14
      U62_A(x1) = 53
      U61_A(x1,x2) = 18
      U52_A(x1) = 29
      U51_A(x1,x2) = 18
      isNatIListKind_A(x1) = 18
      U46_A(x1) = 29
      U45_A(x1,x2) = 18
      U44_A(x1,x2,x3) = 18
      U41_A(x1,x2,x3) = 18
      U43_A(x1,x2,x3) = 18
      isNat_A(x1) = 18
      U21_A(x1,x2) = 18
      U42_A(x1,x2,x3) = 18
      U11_A(x1,x2) = 18
      tt_A = 29
      U32_A(x1,x2) = 18
      U31_A(x1,x2) = 18
      U23_A(x1) = 16
      U22_A(x1,x2) = 18
      U136_A(x1,x2,x3,x4) = 18
      U135_A(x1,x2,x3,x4) = 18
      U134_A(x1,x2,x3,x4) = 18
      U133_A(x1,x2,x3,x4) = 18
      U132_A(x1,x2,x3,x4) = 18
      U33_A(x1) = 26
      U114_A(x1,x2) = 18
      U113_A(x1,x2,x3) = 18
      U112_A(x1,x2,x3) = 18
      U12_A(x1,x2) = 12
      U122_A(x1) = 12
      U105_A(x1,x2) = 18
      U104_A(x1,x2,x3) = 18
      U103_A(x1,x2,x3) = 18
      U102_A(x1,x2,x3) = 18
      active_A(x1) = 28
      zeros_A = 0
      U106_A(x1) = 29
      U13_A(x1) = 66
      nil_A = 29
      U71_A(x1) = 27
    

The next rules are strictly ordered:

  p75

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


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

  lexicographic combination of reduction pairs:
  
    1. max/plus interpretations on natural numbers:
    
      mark#_A(x1) = 2669
      U96_A(x1) = 345
      U101_A(x1,x2,x3) = 345
      active#_A(x1) = x1 + 2324
      mark_A(x1) = max{165, x1 - 51}
      U103_A(x1,x2,x3) = 345
      tt_A = 345
      U104_A(x1,x2,x3) = 345
      isNatIListKind_A(x1) = 345
      U102_A(x1,x2,x3) = 345
      U105_A(x1,x2) = 345
      isNat_A(x1) = 345
      isNatKind_A(x1) = 345
      U11_A(x1,x2) = 345
      U12_A(x1,x2) = 345
      U111_A(x1,x2,x3) = 345
      U112_A(x1,x2,x3) = 345
      U113_A(x1,x2,x3) = 345
      U114_A(x1,x2) = 345
      s_A(x1) = 345
      length_A(x1) = 345
      U121_A(x1,x2) = 345
      U122_A(x1) = 345
      isNatIList_A(x1) = 345
      U131_A(x1,x2,x3,x4) = 345
      U132_A(x1,x2,x3,x4) = 345
      U133_A(x1,x2,x3,x4) = 345
      U134_A(x1,x2,x3,x4) = 345
      U135_A(x1,x2,x3,x4) = 345
      U136_A(x1,x2,x3,x4) = 345
      cons_A(x1,x2) = 345
      take_A(x1,x2) = 345
      U21_A(x1,x2) = 345
      U22_A(x1,x2) = 345
      isNatList_A(x1) = 345
      U23_A(x1) = 345
      U31_A(x1,x2) = 345
      U32_A(x1,x2) = 345
      U33_A(x1) = 345
      U41_A(x1,x2,x3) = 345
      U42_A(x1,x2,x3) = 345
      U43_A(x1,x2,x3) = 345
      U44_A(x1,x2,x3) = 345
      U45_A(x1,x2) = 345
      U46_A(x1) = 345
      U51_A(x1,x2) = 345
      U52_A(x1) = 345
      U91_A(x1,x2,x3) = 345
      U92_A(x1,x2,x3) = 345
      U93_A(x1,x2,x3) = 345
      U94_A(x1,x2,x3) = 345
      U95_A(x1,x2) = 345
      U61_A(x1,x2) = 345
      U81_A(x1) = 345
      U62_A(x1) = 345
      |0|_A = 345
      active_A(x1) = max{294, x1 - 56}
      zeros_A = 351
      U106_A(x1) = 345
      U13_A(x1) = 345
      nil_A = 345
      U71_A(x1) = 345
    
    2. max/plus interpretations on natural numbers:
    
      mark#_A(x1) = 2
      U96_A(x1) = 55
      U101_A(x1,x2,x3) = 47
      active#_A(x1) = max{0, x1 - 48}
      mark_A(x1) = 93
      U103_A(x1,x2,x3) = 50
      tt_A = 35
      U104_A(x1,x2,x3) = 50
      isNatIListKind_A(x1) = 50
      U102_A(x1,x2,x3) = 14
      U105_A(x1,x2) = 47
      isNat_A(x1) = 50
      isNatKind_A(x1) = 50
      U11_A(x1,x2) = 50
      U12_A(x1,x2) = 45
      U111_A(x1,x2,x3) = 50
      U112_A(x1,x2,x3) = 50
      U113_A(x1,x2,x3) = 50
      U114_A(x1,x2) = 50
      s_A(x1) = 45
      length_A(x1) = 50
      U121_A(x1,x2) = 50
      U122_A(x1) = 45
      isNatIList_A(x1) = 50
      U131_A(x1,x2,x3,x4) = 50
      U132_A(x1,x2,x3,x4) = 50
      U133_A(x1,x2,x3,x4) = 50
      U134_A(x1,x2,x3,x4) = 50
      U135_A(x1,x2,x3,x4) = 50
      U136_A(x1,x2,x3,x4) = 50
      cons_A(x1,x2) = 96
      take_A(x1,x2) = 50
      U21_A(x1,x2) = 50
      U22_A(x1,x2) = 50
      isNatList_A(x1) = 50
      U23_A(x1) = 48
      U31_A(x1,x2) = 50
      U32_A(x1,x2) = 50
      U33_A(x1) = 66
      U41_A(x1,x2,x3) = 50
      U42_A(x1,x2,x3) = 50
      U43_A(x1,x2,x3) = 50
      U44_A(x1,x2,x3) = 50
      U45_A(x1,x2) = 50
      U46_A(x1) = 47
      U51_A(x1,x2) = 50
      U52_A(x1) = 49
      U91_A(x1,x2,x3) = 50
      U92_A(x1,x2,x3) = 50
      U93_A(x1,x2,x3) = 50
      U94_A(x1,x2,x3) = 50
      U95_A(x1,x2) = 50
      U61_A(x1,x2) = 0
      U81_A(x1) = 35
      U62_A(x1) = 53
      |0|_A = 55
      active_A(x1) = 93
      zeros_A = 21
      U106_A(x1) = 53
      U13_A(x1) = 12
      nil_A = 55
      U71_A(x1) = 55
    

The next rules are strictly ordered:

  p2, p4, p14, p80

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


-- Reduction pair.

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

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

  lexicographic combination of reduction pairs:
  
    1. max/plus interpretations on natural numbers:
    
      mark#_A(x1) = max{0, x1 - 56}
      U96_A(x1) = max{225, x1}
      U95_A(x1,x2) = 814
      active#_A(x1) = max{21, x1 - 56}
      mark_A(x1) = max{71, x1}
      take_A(x1,x2) = max{1033, x1 + 5, x2 + 962}
      s_A(x1) = max{82, x1}
      cons_A(x1,x2) = max{896, x1 + 306, x2}
      U131_A(x1,x2,x3,x4) = max{x2 + 962, x3 + 5, x4 + 959}
      isNatIList_A(x1) = x1 + 650
      U94_A(x1,x2,x3) = 814
      |0|_A = 138
      U121_A(x1,x2) = max{x1 + 311, x2 + 382}
      U93_A(x1,x2,x3) = 814
      length_A(x1) = max{139, x1 + 60}
      U111_A(x1,x2,x3) = max{x1 + 141, x2 + 60, x3 + 366}
      isNatList_A(x1) = 814
      U92_A(x1,x2,x3) = 814
      U101_A(x1,x2,x3) = 813
      isNatKind_A(x1) = x1 + 88
      U91_A(x1,x2,x3) = 814
      U81_A(x1) = max{89, x1}
      U62_A(x1) = max{226, x1 + 81}
      U61_A(x1,x2) = max{x1, x2 + 1045}
      U52_A(x1) = max{82, x1}
      U51_A(x1,x2) = max{x1 + 4, x2 + 83}
      U46_A(x1) = max{225, x1}
      U45_A(x1,x2) = max{x1 + 147, x2 + 650}
      isNatIListKind_A(x1) = max{86, x1 + 83}
      U44_A(x1,x2,x3) = max{x1 + 567, x2 + 264, x3 + 650}
      U43_A(x1,x2,x3) = max{x1 + 428, x2 + 346, x3 + 650}
      U41_A(x1,x2,x3) = max{x1 + 147, x2 + 533, x3 + 650}
      U42_A(x1,x2,x3) = max{x1 + 445, x2 + 515, x3 + 650}
      isNat_A(x1) = x1 + 117
      U21_A(x1,x2) = x2 + 117
      U11_A(x1,x2) = x2 + 161
      U32_A(x1,x2) = max{x1 + 84, x2 + 227}
      tt_A = 225
      U31_A(x1,x2) = max{x1, x2 + 228}
      U23_A(x1) = max{116, x1}
      U22_A(x1,x2) = x2 + 117
      U136_A(x1,x2,x3,x4) = max{1034, x1 + 809, x2 + 962, x3 + 5, x4 + 897}
      U135_A(x1,x2,x3,x4) = max{968, x1 + 827, x2 + 962, x3 + 5, x4 + 897}
      U134_A(x1,x2,x3,x4) = max{968, x2 + 962, x3 + 5, x4 + 944}
      U133_A(x1,x2,x3,x4) = max{1025, x2 + 962, x3 + 5, x4 + 944}
      U132_A(x1,x2,x3,x4) = max{x1 + 874, x2 + 962, x3 + 5, x4 + 944}
      U33_A(x1) = max{226, x1 - 732}
      U114_A(x1,x2) = max{138, x1 + 57, x2 + 60}
      U113_A(x1,x2,x3) = max{x1 + 249, x2 + 60, x3 + 366}
      U112_A(x1,x2,x3) = max{x2 + 60, x3 + 366}
      U104_A(x1,x2,x3) = 435
      U122_A(x1) = 294
      U103_A(x1,x2,x3) = 591
      U12_A(x1,x2) = max{x1 + 2, x2 + 83}
      U105_A(x1,x2) = 392
      active_A(x1) = max{81, x1}
      zeros_A = 897
      U102_A(x1,x2,x3) = 812
      U106_A(x1) = 226
      U13_A(x1) = max{226, x1 - 732}
      nil_A = 293
      U71_A(x1) = max{227, x1 - 9}
    
    2. max/plus interpretations on natural numbers:
    
      mark#_A(x1) = 0
      U96_A(x1) = 46
      U95_A(x1,x2) = 49
      active#_A(x1) = 0
      mark_A(x1) = 23
      take_A(x1,x2) = 28
      s_A(x1) = 53
      cons_A(x1,x2) = 14
      U131_A(x1,x2,x3,x4) = 164
      isNatIList_A(x1) = 5
      U94_A(x1,x2,x3) = 24
      |0|_A = 8
      U121_A(x1,x2) = 126
      U93_A(x1,x2,x3) = 6
      length_A(x1) = 51
      U111_A(x1,x2,x3) = 6
      isNatList_A(x1) = 5
      U92_A(x1,x2,x3) = 24
      U101_A(x1,x2,x3) = 22
      isNatKind_A(x1) = 5
      U91_A(x1,x2,x3) = 6
      U81_A(x1) = 55
      U62_A(x1) = 22
      U61_A(x1,x2) = 0
      U52_A(x1) = 0
      U51_A(x1,x2) = 22
      U46_A(x1) = 22
      U45_A(x1,x2) = 24
      isNatIListKind_A(x1) = 5
      U44_A(x1,x2,x3) = 49
      U43_A(x1,x2,x3) = 74
      U41_A(x1,x2,x3) = 22
      U42_A(x1,x2,x3) = 99
      isNat_A(x1) = 5
      U21_A(x1,x2) = 24
      U11_A(x1,x2) = 87
      U32_A(x1,x2) = 28
      tt_A = 10
      U31_A(x1,x2) = 53
      U23_A(x1) = 24
      U22_A(x1,x2) = 24
      U136_A(x1,x2,x3,x4) = 39
      U135_A(x1,x2,x3,x4) = 24
      U134_A(x1,x2,x3,x4) = 49
      U133_A(x1,x2,x3,x4) = 94
      U132_A(x1,x2,x3,x4) = 139
      U33_A(x1) = 3
      U114_A(x1,x2) = 98
      U113_A(x1,x2,x3) = 121
      U112_A(x1,x2,x3) = 5
      U104_A(x1,x2,x3) = 114
      U122_A(x1) = 81
      U103_A(x1,x2,x3) = 24
      U12_A(x1,x2) = 64
      U105_A(x1,x2) = 69
      active_A(x1) = 23
      zeros_A = 0
      U102_A(x1,x2,x3) = 0
      U106_A(x1) = 24
      U13_A(x1) = 59
      nil_A = 36
      U71_A(x1) = 0
    

The next rules are strictly ordered:

  p5, p9, p13, p16, p29, p55, p57, p58, p62, p65, p70, p80, p88, p89, p90

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)) -> active#(U95(mark(X1),X2))
p3: active#(take(s(M),cons(N,IL))) -> mark#(U131(isNatIList(IL),IL,M,N))
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)) -> active#(U92(mark(X1),X2,X3))
p8: mark#(U91(X1,X2,X3)) -> active#(U91(mark(X1),X2,X3))
p9: active#(isNatList(cons(V1,V2))) -> mark#(U91(isNatKind(V1),V1,V2))
p10: mark#(U81(X)) -> mark#(X)
p11: mark#(U61(X1,X2)) -> mark#(X1)
p12: mark#(U52(X)) -> mark#(X)
p13: mark#(U51(X1,X2)) -> active#(U51(mark(X1),X2))
p14: active#(isNatKind(s(V1))) -> mark#(U81(isNatKind(V1)))
p15: mark#(U46(X)) -> mark#(X)
p16: mark#(U45(X1,X2)) -> active#(U45(mark(X1),X2))
p17: active#(isNatIListKind(take(V1,V2))) -> mark#(U61(isNatKind(V1),V2))
p18: mark#(U44(X1,X2,X3)) -> active#(U44(mark(X1),X2,X3))
p19: active#(isNatIListKind(cons(V1,V2))) -> mark#(U51(isNatKind(V1),V2))
p20: mark#(U43(X1,X2,X3)) -> active#(U43(mark(X1),X2,X3))
p21: active#(isNatIList(cons(V1,V2))) -> mark#(U41(isNatKind(V1),V1,V2))
p22: mark#(U42(X1,X2,X3)) -> active#(U42(mark(X1),X2,X3))
p23: active#(isNat(s(V1))) -> mark#(U21(isNatKind(V1),V1))
p24: mark#(U41(X1,X2,X3)) -> active#(U41(mark(X1),X2,X3))
p25: mark#(U32(X1,X2)) -> active#(U32(mark(X1),X2))
p26: active#(U95(tt(),V2)) -> mark#(U96(isNatList(V2)))
p27: mark#(U31(X1,X2)) -> mark#(X1)
p28: mark#(U31(X1,X2)) -> active#(U31(mark(X1),X2))
p29: active#(U94(tt(),V1,V2)) -> mark#(U95(isNat(V1),V2))
p30: mark#(U23(X)) -> mark#(X)
p31: mark#(U22(X1,X2)) -> active#(U22(mark(X1),X2))
p32: active#(U93(tt(),V1,V2)) -> mark#(U94(isNatIListKind(V2),V1,V2))
p33: mark#(U21(X1,X2)) -> active#(U21(mark(X1),X2))
p34: active#(U92(tt(),V1,V2)) -> mark#(U93(isNatIListKind(V2),V1,V2))
p35: mark#(take(X1,X2)) -> active#(take(mark(X1),mark(X2)))
p36: active#(U91(tt(),V1,V2)) -> mark#(U92(isNatKind(V1),V1,V2))
p37: mark#(U136(X1,X2,X3,X4)) -> active#(U136(mark(X1),X2,X3,X4))
p38: active#(U51(tt(),V2)) -> mark#(U52(isNatIListKind(V2)))
p39: mark#(U135(X1,X2,X3,X4)) -> active#(U135(mark(X1),X2,X3,X4))
p40: active#(U45(tt(),V2)) -> mark#(U46(isNatIList(V2)))
p41: mark#(U134(X1,X2,X3,X4)) -> active#(U134(mark(X1),X2,X3,X4))
p42: active#(U44(tt(),V1,V2)) -> mark#(U45(isNat(V1),V2))
p43: mark#(U133(X1,X2,X3,X4)) -> active#(U133(mark(X1),X2,X3,X4))
p44: active#(U43(tt(),V1,V2)) -> mark#(U44(isNatIListKind(V2),V1,V2))
p45: mark#(U132(X1,X2,X3,X4)) -> active#(U132(mark(X1),X2,X3,X4))
p46: active#(U42(tt(),V1,V2)) -> mark#(U43(isNatIListKind(V2),V1,V2))
p47: mark#(U131(X1,X2,X3,X4)) -> active#(U131(mark(X1),X2,X3,X4))
p48: active#(U41(tt(),V1,V2)) -> mark#(U42(isNatKind(V1),V1,V2))
p49: mark#(U121(X1,X2)) -> active#(U121(mark(X1),X2))
p50: mark#(isNatList(X)) -> active#(isNatList(X))
p51: mark#(length(X)) -> active#(length(mark(X)))
p52: active#(U22(tt(),V1)) -> mark#(U23(isNat(V1)))
p53: mark#(s(X)) -> mark#(X)
p54: mark#(U114(X1,X2)) -> active#(U114(mark(X1),X2))
p55: active#(U21(tt(),V1)) -> mark#(U22(isNatKind(V1),V1))
p56: mark#(U113(X1,X2,X3)) -> active#(U113(mark(X1),X2,X3))
p57: active#(U136(tt(),IL,M,N)) -> mark#(cons(N,take(M,IL)))
p58: mark#(U112(X1,X2,X3)) -> active#(U112(mark(X1),X2,X3))
p59: active#(U135(tt(),IL,M,N)) -> mark#(U136(isNatKind(N),IL,M,N))
p60: mark#(U111(X1,X2,X3)) -> active#(U111(mark(X1),X2,X3))
p61: active#(U134(tt(),IL,M,N)) -> mark#(U135(isNat(N),IL,M,N))
p62: mark#(U11(X1,X2)) -> active#(U11(mark(X1),X2))
p63: active#(U133(tt(),IL,M,N)) -> mark#(U134(isNatKind(M),IL,M,N))
p64: mark#(isNatIList(X)) -> active#(isNatIList(X))
p65: active#(U132(tt(),IL,M,N)) -> mark#(U133(isNat(M),IL,M,N))
p66: mark#(isNat(X)) -> active#(isNat(X))
p67: active#(U131(tt(),IL,M,N)) -> mark#(U132(isNatIListKind(IL),IL,M,N))
p68: mark#(U104(X1,X2,X3)) -> active#(U104(mark(X1),X2,X3))
p69: mark#(isNatIListKind(X)) -> active#(isNatIListKind(X))
p70: active#(U114(tt(),L)) -> mark#(s(length(L)))
p71: mark#(U103(X1,X2,X3)) -> active#(U103(mark(X1),X2,X3))
p72: active#(U113(tt(),L,N)) -> mark#(U114(isNatKind(N),L))
p73: mark#(isNatKind(X)) -> active#(isNatKind(X))
p74: active#(U112(tt(),L,N)) -> mark#(U113(isNat(N),L,N))
p75: active#(U111(tt(),L,N)) -> mark#(U112(isNatIListKind(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}


-- Reduction pair.

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

p1: mark#(U96(X)) -> mark#(X)
p2: mark#(isNatKind(X)) -> active#(isNatKind(X))
p3: active#(U111(tt(),L,N)) -> mark#(U112(isNatIListKind(L),L,N))
p4: mark#(U103(X1,X2,X3)) -> active#(U103(mark(X1),X2,X3))
p5: active#(U112(tt(),L,N)) -> mark#(U113(isNat(N),L,N))
p6: mark#(isNatIListKind(X)) -> active#(isNatIListKind(X))
p7: active#(U113(tt(),L,N)) -> mark#(U114(isNatKind(N),L))
p8: mark#(U104(X1,X2,X3)) -> active#(U104(mark(X1),X2,X3))
p9: active#(U114(tt(),L)) -> mark#(s(length(L)))
p10: mark#(isNat(X)) -> active#(isNat(X))
p11: active#(U131(tt(),IL,M,N)) -> mark#(U132(isNatIListKind(IL),IL,M,N))
p12: mark#(isNatIList(X)) -> active#(isNatIList(X))
p13: active#(U132(tt(),IL,M,N)) -> mark#(U133(isNat(M),IL,M,N))
p14: mark#(U11(X1,X2)) -> active#(U11(mark(X1),X2))
p15: active#(U133(tt(),IL,M,N)) -> mark#(U134(isNatKind(M),IL,M,N))
p16: mark#(U111(X1,X2,X3)) -> active#(U111(mark(X1),X2,X3))
p17: active#(U134(tt(),IL,M,N)) -> mark#(U135(isNat(N),IL,M,N))
p18: mark#(U112(X1,X2,X3)) -> active#(U112(mark(X1),X2,X3))
p19: active#(U135(tt(),IL,M,N)) -> mark#(U136(isNatKind(N),IL,M,N))
p20: mark#(U113(X1,X2,X3)) -> active#(U113(mark(X1),X2,X3))
p21: active#(U136(tt(),IL,M,N)) -> mark#(cons(N,take(M,IL)))
p22: mark#(U114(X1,X2)) -> active#(U114(mark(X1),X2))
p23: active#(U21(tt(),V1)) -> mark#(U22(isNatKind(V1),V1))
p24: mark#(s(X)) -> mark#(X)
p25: mark#(length(X)) -> active#(length(mark(X)))
p26: active#(U22(tt(),V1)) -> mark#(U23(isNat(V1)))
p27: mark#(isNatList(X)) -> active#(isNatList(X))
p28: active#(U41(tt(),V1,V2)) -> mark#(U42(isNatKind(V1),V1,V2))
p29: mark#(U121(X1,X2)) -> active#(U121(mark(X1),X2))
p30: active#(U42(tt(),V1,V2)) -> mark#(U43(isNatIListKind(V2),V1,V2))
p31: mark#(U131(X1,X2,X3,X4)) -> active#(U131(mark(X1),X2,X3,X4))
p32: active#(U43(tt(),V1,V2)) -> mark#(U44(isNatIListKind(V2),V1,V2))
p33: mark#(U132(X1,X2,X3,X4)) -> active#(U132(mark(X1),X2,X3,X4))
p34: active#(U44(tt(),V1,V2)) -> mark#(U45(isNat(V1),V2))
p35: mark#(U133(X1,X2,X3,X4)) -> active#(U133(mark(X1),X2,X3,X4))
p36: active#(U45(tt(),V2)) -> mark#(U46(isNatIList(V2)))
p37: mark#(U134(X1,X2,X3,X4)) -> active#(U134(mark(X1),X2,X3,X4))
p38: active#(U51(tt(),V2)) -> mark#(U52(isNatIListKind(V2)))
p39: mark#(U135(X1,X2,X3,X4)) -> active#(U135(mark(X1),X2,X3,X4))
p40: active#(U91(tt(),V1,V2)) -> mark#(U92(isNatKind(V1),V1,V2))
p41: mark#(U136(X1,X2,X3,X4)) -> active#(U136(mark(X1),X2,X3,X4))
p42: active#(U92(tt(),V1,V2)) -> mark#(U93(isNatIListKind(V2),V1,V2))
p43: mark#(take(X1,X2)) -> active#(take(mark(X1),mark(X2)))
p44: active#(U93(tt(),V1,V2)) -> mark#(U94(isNatIListKind(V2),V1,V2))
p45: mark#(U21(X1,X2)) -> active#(U21(mark(X1),X2))
p46: active#(U94(tt(),V1,V2)) -> mark#(U95(isNat(V1),V2))
p47: mark#(U22(X1,X2)) -> active#(U22(mark(X1),X2))
p48: active#(U95(tt(),V2)) -> mark#(U96(isNatList(V2)))
p49: mark#(U23(X)) -> mark#(X)
p50: mark#(U31(X1,X2)) -> active#(U31(mark(X1),X2))
p51: active#(isNat(s(V1))) -> mark#(U21(isNatKind(V1),V1))
p52: mark#(U31(X1,X2)) -> mark#(X1)
p53: mark#(U32(X1,X2)) -> active#(U32(mark(X1),X2))
p54: active#(isNatIList(cons(V1,V2))) -> mark#(U41(isNatKind(V1),V1,V2))
p55: mark#(U41(X1,X2,X3)) -> active#(U41(mark(X1),X2,X3))
p56: active#(isNatIListKind(cons(V1,V2))) -> mark#(U51(isNatKind(V1),V2))
p57: mark#(U42(X1,X2,X3)) -> active#(U42(mark(X1),X2,X3))
p58: active#(isNatIListKind(take(V1,V2))) -> mark#(U61(isNatKind(V1),V2))
p59: mark#(U43(X1,X2,X3)) -> active#(U43(mark(X1),X2,X3))
p60: active#(isNatKind(s(V1))) -> mark#(U81(isNatKind(V1)))
p61: mark#(U44(X1,X2,X3)) -> active#(U44(mark(X1),X2,X3))
p62: active#(isNatList(cons(V1,V2))) -> mark#(U91(isNatKind(V1),V1,V2))
p63: mark#(U45(X1,X2)) -> active#(U45(mark(X1),X2))
p64: active#(length(cons(N,L))) -> mark#(U111(isNatList(L),L,N))
p65: mark#(U46(X)) -> mark#(X)
p66: mark#(U51(X1,X2)) -> active#(U51(mark(X1),X2))
p67: active#(take(s(M),cons(N,IL))) -> mark#(U131(isNatIList(IL),IL,M,N))
p68: mark#(U52(X)) -> mark#(X)
p69: mark#(U61(X1,X2)) -> mark#(X1)
p70: mark#(U81(X)) -> mark#(X)
p71: mark#(U91(X1,X2,X3)) -> active#(U91(mark(X1),X2,X3))
p72: mark#(U92(X1,X2,X3)) -> active#(U92(mark(X1),X2,X3))
p73: mark#(U93(X1,X2,X3)) -> active#(U93(mark(X1),X2,X3))
p74: mark#(U94(X1,X2,X3)) -> active#(U94(mark(X1),X2,X3))
p75: 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 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:

  lexicographic combination of reduction pairs:
  
    1. max/plus interpretations on natural numbers:
    
      mark#_A(x1) = 437
      U96_A(x1) = 206
      isNatKind_A(x1) = 437
      active#_A(x1) = x1
      U111_A(x1,x2,x3) = 437
      tt_A = 130
      U112_A(x1,x2,x3) = 437
      isNatIListKind_A(x1) = 437
      U103_A(x1,x2,x3) = 0
      mark_A(x1) = max{558, x1 + 104}
      U113_A(x1,x2,x3) = 437
      isNat_A(x1) = 437
      U114_A(x1,x2) = 437
      U104_A(x1,x2,x3) = 66
      s_A(x1) = 453
      length_A(x1) = 437
      U131_A(x1,x2,x3,x4) = 437
      U132_A(x1,x2,x3,x4) = 437
      isNatIList_A(x1) = 437
      U133_A(x1,x2,x3,x4) = 437
      U11_A(x1,x2) = 345
      U134_A(x1,x2,x3,x4) = 437
      U135_A(x1,x2,x3,x4) = 437
      U136_A(x1,x2,x3,x4) = 437
      cons_A(x1,x2) = 0
      take_A(x1,x2) = 437
      U21_A(x1,x2) = 437
      U22_A(x1,x2) = 437
      U23_A(x1) = 454
      isNatList_A(x1) = 437
      U41_A(x1,x2,x3) = 437
      U42_A(x1,x2,x3) = 437
      U121_A(x1,x2) = 436
      U43_A(x1,x2,x3) = 437
      U44_A(x1,x2,x3) = 437
      U45_A(x1,x2) = 437
      U46_A(x1) = 17
      U51_A(x1,x2) = 437
      U52_A(x1) = 345
      U91_A(x1,x2,x3) = 437
      U92_A(x1,x2,x3) = 437
      U93_A(x1,x2,x3) = 437
      U94_A(x1,x2,x3) = 437
      U95_A(x1,x2) = 437
      U31_A(x1,x2) = 345
      U32_A(x1,x2) = 314
      U61_A(x1,x2) = 454
      U81_A(x1) = 454
      active_A(x1) = max{558, x1 + 13}
      zeros_A = 118
      |0|_A = 345
      U101_A(x1,x2,x3) = 454
      U102_A(x1,x2,x3) = 66
      U105_A(x1,x2) = 429
      U106_A(x1) = 222
      U12_A(x1,x2) = 454
      U13_A(x1) = 259
      U122_A(x1) = 454
      nil_A = 454
      U33_A(x1) = 222
      U62_A(x1) = 362
      U71_A(x1) = 196
    
    2. max/plus interpretations on natural numbers:
    
      mark#_A(x1) = 0
      U96_A(x1) = 59
      isNatKind_A(x1) = 65
      active#_A(x1) = max{0, x1 - 65}
      U111_A(x1,x2,x3) = 59
      tt_A = 32
      U112_A(x1,x2,x3) = 65
      isNatIListKind_A(x1) = 54
      U103_A(x1,x2,x3) = 48
      mark_A(x1) = 63
      U113_A(x1,x2,x3) = 58
      isNat_A(x1) = 48
      U114_A(x1,x2) = 47
      U104_A(x1,x2,x3) = 66
      s_A(x1) = 60
      length_A(x1) = 65
      U131_A(x1,x2,x3,x4) = 65
      U132_A(x1,x2,x3,x4) = 48
      isNatIList_A(x1) = 60
      U133_A(x1,x2,x3,x4) = 41
      U11_A(x1,x2) = 65
      U134_A(x1,x2,x3,x4) = 25
      U135_A(x1,x2,x3,x4) = 48
      U136_A(x1,x2,x3,x4) = 65
      cons_A(x1,x2) = 7
      take_A(x1,x2) = 65
      U21_A(x1,x2) = 48
      U22_A(x1,x2) = 41
      U23_A(x1) = 0
      isNatList_A(x1) = 65
      U41_A(x1,x2,x3) = 48
      U42_A(x1,x2,x3) = 65
      U121_A(x1,x2) = 66
      U43_A(x1,x2,x3) = 58
      U44_A(x1,x2,x3) = 48
      U45_A(x1,x2) = 41
      U46_A(x1) = 64
      U51_A(x1,x2) = 62
      U52_A(x1) = 55
      U91_A(x1,x2,x3) = 58
      U92_A(x1,x2,x3) = 65
      U93_A(x1,x2,x3) = 62
      U94_A(x1,x2,x3) = 48
      U95_A(x1,x2) = 48
      U31_A(x1,x2) = 48
      U32_A(x1,x2) = 62
      U61_A(x1,x2) = 48
      U81_A(x1) = 60
      active_A(x1) = 63
      zeros_A = 0
      |0|_A = 64
      U101_A(x1,x2,x3) = 64
      U102_A(x1,x2,x3) = 48
      U105_A(x1,x2) = 52
      U106_A(x1) = 64
      U12_A(x1,x2) = 113
      U13_A(x1) = 48
      U122_A(x1) = 48
      nil_A = 67
      U33_A(x1) = 64
      U62_A(x1) = 64
      U71_A(x1) = 51
    

The next rules are strictly ordered:

  p4, p8, p14, p29, p50, p53

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#(isNatKind(X)) -> active#(isNatKind(X))
p3: active#(U111(tt(),L,N)) -> mark#(U112(isNatIListKind(L),L,N))
p4: active#(U112(tt(),L,N)) -> mark#(U113(isNat(N),L,N))
p5: mark#(isNatIListKind(X)) -> active#(isNatIListKind(X))
p6: active#(U113(tt(),L,N)) -> mark#(U114(isNatKind(N),L))
p7: active#(U114(tt(),L)) -> mark#(s(length(L)))
p8: mark#(isNat(X)) -> active#(isNat(X))
p9: active#(U131(tt(),IL,M,N)) -> mark#(U132(isNatIListKind(IL),IL,M,N))
p10: mark#(isNatIList(X)) -> active#(isNatIList(X))
p11: active#(U132(tt(),IL,M,N)) -> mark#(U133(isNat(M),IL,M,N))
p12: active#(U133(tt(),IL,M,N)) -> mark#(U134(isNatKind(M),IL,M,N))
p13: mark#(U111(X1,X2,X3)) -> active#(U111(mark(X1),X2,X3))
p14: active#(U134(tt(),IL,M,N)) -> mark#(U135(isNat(N),IL,M,N))
p15: mark#(U112(X1,X2,X3)) -> active#(U112(mark(X1),X2,X3))
p16: active#(U135(tt(),IL,M,N)) -> mark#(U136(isNatKind(N),IL,M,N))
p17: mark#(U113(X1,X2,X3)) -> active#(U113(mark(X1),X2,X3))
p18: active#(U136(tt(),IL,M,N)) -> mark#(cons(N,take(M,IL)))
p19: mark#(U114(X1,X2)) -> active#(U114(mark(X1),X2))
p20: active#(U21(tt(),V1)) -> mark#(U22(isNatKind(V1),V1))
p21: mark#(s(X)) -> mark#(X)
p22: mark#(length(X)) -> active#(length(mark(X)))
p23: active#(U22(tt(),V1)) -> mark#(U23(isNat(V1)))
p24: mark#(isNatList(X)) -> active#(isNatList(X))
p25: active#(U41(tt(),V1,V2)) -> mark#(U42(isNatKind(V1),V1,V2))
p26: active#(U42(tt(),V1,V2)) -> mark#(U43(isNatIListKind(V2),V1,V2))
p27: mark#(U131(X1,X2,X3,X4)) -> active#(U131(mark(X1),X2,X3,X4))
p28: active#(U43(tt(),V1,V2)) -> mark#(U44(isNatIListKind(V2),V1,V2))
p29: mark#(U132(X1,X2,X3,X4)) -> active#(U132(mark(X1),X2,X3,X4))
p30: active#(U44(tt(),V1,V2)) -> mark#(U45(isNat(V1),V2))
p31: mark#(U133(X1,X2,X3,X4)) -> active#(U133(mark(X1),X2,X3,X4))
p32: active#(U45(tt(),V2)) -> mark#(U46(isNatIList(V2)))
p33: mark#(U134(X1,X2,X3,X4)) -> active#(U134(mark(X1),X2,X3,X4))
p34: active#(U51(tt(),V2)) -> mark#(U52(isNatIListKind(V2)))
p35: mark#(U135(X1,X2,X3,X4)) -> active#(U135(mark(X1),X2,X3,X4))
p36: active#(U91(tt(),V1,V2)) -> mark#(U92(isNatKind(V1),V1,V2))
p37: mark#(U136(X1,X2,X3,X4)) -> active#(U136(mark(X1),X2,X3,X4))
p38: active#(U92(tt(),V1,V2)) -> mark#(U93(isNatIListKind(V2),V1,V2))
p39: mark#(take(X1,X2)) -> active#(take(mark(X1),mark(X2)))
p40: active#(U93(tt(),V1,V2)) -> mark#(U94(isNatIListKind(V2),V1,V2))
p41: mark#(U21(X1,X2)) -> active#(U21(mark(X1),X2))
p42: active#(U94(tt(),V1,V2)) -> mark#(U95(isNat(V1),V2))
p43: mark#(U22(X1,X2)) -> active#(U22(mark(X1),X2))
p44: active#(U95(tt(),V2)) -> mark#(U96(isNatList(V2)))
p45: mark#(U23(X)) -> mark#(X)
p46: active#(isNat(s(V1))) -> mark#(U21(isNatKind(V1),V1))
p47: mark#(U31(X1,X2)) -> mark#(X1)
p48: active#(isNatIList(cons(V1,V2))) -> mark#(U41(isNatKind(V1),V1,V2))
p49: mark#(U41(X1,X2,X3)) -> active#(U41(mark(X1),X2,X3))
p50: active#(isNatIListKind(cons(V1,V2))) -> mark#(U51(isNatKind(V1),V2))
p51: mark#(U42(X1,X2,X3)) -> active#(U42(mark(X1),X2,X3))
p52: active#(isNatIListKind(take(V1,V2))) -> mark#(U61(isNatKind(V1),V2))
p53: mark#(U43(X1,X2,X3)) -> active#(U43(mark(X1),X2,X3))
p54: active#(isNatKind(s(V1))) -> mark#(U81(isNatKind(V1)))
p55: mark#(U44(X1,X2,X3)) -> active#(U44(mark(X1),X2,X3))
p56: active#(isNatList(cons(V1,V2))) -> mark#(U91(isNatKind(V1),V1,V2))
p57: mark#(U45(X1,X2)) -> active#(U45(mark(X1),X2))
p58: active#(length(cons(N,L))) -> mark#(U111(isNatList(L),L,N))
p59: mark#(U46(X)) -> mark#(X)
p60: mark#(U51(X1,X2)) -> active#(U51(mark(X1),X2))
p61: active#(take(s(M),cons(N,IL))) -> mark#(U131(isNatIList(IL),IL,M,N))
p62: mark#(U52(X)) -> mark#(X)
p63: mark#(U61(X1,X2)) -> mark#(X1)
p64: mark#(U81(X)) -> mark#(X)
p65: mark#(U91(X1,X2,X3)) -> active#(U91(mark(X1),X2,X3))
p66: mark#(U92(X1,X2,X3)) -> active#(U92(mark(X1),X2,X3))
p67: mark#(U93(X1,X2,X3)) -> active#(U93(mark(X1),X2,X3))
p68: mark#(U94(X1,X2,X3)) -> active#(U94(mark(X1),X2,X3))
p69: 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, p51, p52, p53, p54, p55, p56, p57, p58, p59, p60, p61, p62, p63, p64, p65, p66, p67, p68, p69}


-- Reduction pair.

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

p1: mark#(U96(X)) -> mark#(X)
p2: mark#(U95(X1,X2)) -> active#(U95(mark(X1),X2))
p3: active#(take(s(M),cons(N,IL))) -> mark#(U131(isNatIList(IL),IL,M,N))
p4: mark#(U94(X1,X2,X3)) -> active#(U94(mark(X1),X2,X3))
p5: active#(length(cons(N,L))) -> mark#(U111(isNatList(L),L,N))
p6: mark#(U93(X1,X2,X3)) -> active#(U93(mark(X1),X2,X3))
p7: active#(isNatList(cons(V1,V2))) -> mark#(U91(isNatKind(V1),V1,V2))
p8: mark#(U92(X1,X2,X3)) -> active#(U92(mark(X1),X2,X3))
p9: active#(isNatKind(s(V1))) -> mark#(U81(isNatKind(V1)))
p10: mark#(U91(X1,X2,X3)) -> active#(U91(mark(X1),X2,X3))
p11: active#(isNatIListKind(take(V1,V2))) -> mark#(U61(isNatKind(V1),V2))
p12: mark#(U81(X)) -> mark#(X)
p13: mark#(U61(X1,X2)) -> mark#(X1)
p14: mark#(U52(X)) -> mark#(X)
p15: mark#(U51(X1,X2)) -> active#(U51(mark(X1),X2))
p16: active#(isNatIListKind(cons(V1,V2))) -> mark#(U51(isNatKind(V1),V2))
p17: mark#(U46(X)) -> mark#(X)
p18: mark#(U45(X1,X2)) -> active#(U45(mark(X1),X2))
p19: active#(isNatIList(cons(V1,V2))) -> mark#(U41(isNatKind(V1),V1,V2))
p20: mark#(U44(X1,X2,X3)) -> active#(U44(mark(X1),X2,X3))
p21: active#(isNat(s(V1))) -> mark#(U21(isNatKind(V1),V1))
p22: mark#(U43(X1,X2,X3)) -> active#(U43(mark(X1),X2,X3))
p23: active#(U95(tt(),V2)) -> mark#(U96(isNatList(V2)))
p24: mark#(U42(X1,X2,X3)) -> active#(U42(mark(X1),X2,X3))
p25: active#(U94(tt(),V1,V2)) -> mark#(U95(isNat(V1),V2))
p26: mark#(U41(X1,X2,X3)) -> active#(U41(mark(X1),X2,X3))
p27: active#(U93(tt(),V1,V2)) -> mark#(U94(isNatIListKind(V2),V1,V2))
p28: mark#(U31(X1,X2)) -> mark#(X1)
p29: mark#(U23(X)) -> mark#(X)
p30: mark#(U22(X1,X2)) -> active#(U22(mark(X1),X2))
p31: active#(U92(tt(),V1,V2)) -> mark#(U93(isNatIListKind(V2),V1,V2))
p32: mark#(U21(X1,X2)) -> active#(U21(mark(X1),X2))
p33: active#(U91(tt(),V1,V2)) -> mark#(U92(isNatKind(V1),V1,V2))
p34: mark#(take(X1,X2)) -> active#(take(mark(X1),mark(X2)))
p35: active#(U51(tt(),V2)) -> mark#(U52(isNatIListKind(V2)))
p36: mark#(U136(X1,X2,X3,X4)) -> active#(U136(mark(X1),X2,X3,X4))
p37: active#(U45(tt(),V2)) -> mark#(U46(isNatIList(V2)))
p38: mark#(U135(X1,X2,X3,X4)) -> active#(U135(mark(X1),X2,X3,X4))
p39: active#(U44(tt(),V1,V2)) -> mark#(U45(isNat(V1),V2))
p40: mark#(U134(X1,X2,X3,X4)) -> active#(U134(mark(X1),X2,X3,X4))
p41: active#(U43(tt(),V1,V2)) -> mark#(U44(isNatIListKind(V2),V1,V2))
p42: mark#(U133(X1,X2,X3,X4)) -> active#(U133(mark(X1),X2,X3,X4))
p43: active#(U42(tt(),V1,V2)) -> mark#(U43(isNatIListKind(V2),V1,V2))
p44: mark#(U132(X1,X2,X3,X4)) -> active#(U132(mark(X1),X2,X3,X4))
p45: active#(U41(tt(),V1,V2)) -> mark#(U42(isNatKind(V1),V1,V2))
p46: mark#(U131(X1,X2,X3,X4)) -> active#(U131(mark(X1),X2,X3,X4))
p47: active#(U22(tt(),V1)) -> mark#(U23(isNat(V1)))
p48: mark#(isNatList(X)) -> active#(isNatList(X))
p49: active#(U21(tt(),V1)) -> mark#(U22(isNatKind(V1),V1))
p50: mark#(length(X)) -> active#(length(mark(X)))
p51: active#(U136(tt(),IL,M,N)) -> mark#(cons(N,take(M,IL)))
p52: mark#(s(X)) -> mark#(X)
p53: mark#(U114(X1,X2)) -> active#(U114(mark(X1),X2))
p54: active#(U135(tt(),IL,M,N)) -> mark#(U136(isNatKind(N),IL,M,N))
p55: mark#(U113(X1,X2,X3)) -> active#(U113(mark(X1),X2,X3))
p56: active#(U134(tt(),IL,M,N)) -> mark#(U135(isNat(N),IL,M,N))
p57: mark#(U112(X1,X2,X3)) -> active#(U112(mark(X1),X2,X3))
p58: active#(U133(tt(),IL,M,N)) -> mark#(U134(isNatKind(M),IL,M,N))
p59: mark#(U111(X1,X2,X3)) -> active#(U111(mark(X1),X2,X3))
p60: active#(U132(tt(),IL,M,N)) -> mark#(U133(isNat(M),IL,M,N))
p61: mark#(isNatIList(X)) -> active#(isNatIList(X))
p62: active#(U131(tt(),IL,M,N)) -> mark#(U132(isNatIListKind(IL),IL,M,N))
p63: mark#(isNat(X)) -> active#(isNat(X))
p64: active#(U114(tt(),L)) -> mark#(s(length(L)))
p65: mark#(isNatIListKind(X)) -> active#(isNatIListKind(X))
p66: active#(U113(tt(),L,N)) -> mark#(U114(isNatKind(N),L))
p67: mark#(isNatKind(X)) -> active#(isNatKind(X))
p68: active#(U112(tt(),L,N)) -> mark#(U113(isNat(N),L,N))
p69: active#(U111(tt(),L,N)) -> mark#(U112(isNatIListKind(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:

  lexicographic combination of reduction pairs:
  
    1. max/plus interpretations on natural numbers:
    
      mark#_A(x1) = max{0, x1 - 1135}
      U96_A(x1) = max{1134, x1 + 1}
      U95_A(x1,x2) = x2 + 4253
      active#_A(x1) = max{0, x1 - 1156}
      mark_A(x1) = max{1130, x1}
      take_A(x1,x2) = 754
      s_A(x1) = max{1129, x1}
      cons_A(x1,x2) = max{1125, x2 + 375}
      U131_A(x1,x2,x3,x4) = 732
      isNatIList_A(x1) = x1 + 2016
      U94_A(x1,x2,x3) = max{x1 + 1167, x3 + 4279}
      length_A(x1) = 1134
      U111_A(x1,x2,x3) = 1134
      isNatList_A(x1) = x1 + 4230
      U93_A(x1,x2,x3) = max{x1 + 2305, x3 + 4301}
      U91_A(x1,x2,x3) = max{4583, x3 + 4345}
      isNatKind_A(x1) = 784
      U92_A(x1,x2,x3) = max{4561, x3 + 4323}
      U81_A(x1) = max{1077, x1}
      isNatIListKind_A(x1) = max{1125, x1 + 897}
      U61_A(x1,x2) = max{1077, x1}
      U52_A(x1) = max{1130, x1}
      U51_A(x1,x2) = max{1126, x2 + 1124}
      U46_A(x1) = max{1136, x1 + 1}
      U45_A(x1,x2) = max{x1 + 1149, x2 + 2280}
      U41_A(x1,x2,x3) = x3 + 2370
      U44_A(x1,x2,x3) = x3 + 2304
      isNat_A(x1) = 1133
      U21_A(x1,x2) = 1133
      U43_A(x1,x2,x3) = x3 + 2326
      tt_A = 9
      U42_A(x1,x2,x3) = x3 + 2348
      U31_A(x1,x2) = x1
      U23_A(x1) = max{337, x1}
      U22_A(x1,x2) = 1133
      U136_A(x1,x2,x3,x4) = 596
      U135_A(x1,x2,x3,x4) = 1129
      U134_A(x1,x2,x3,x4) = 1093
      U133_A(x1,x2,x3,x4) = 1113
      U132_A(x1,x2,x3,x4) = 708
      U114_A(x1,x2) = 1134
      U113_A(x1,x2,x3) = 1134
      U112_A(x1,x2,x3) = 1134
      active_A(x1) = max{1130, x1}
      zeros_A = 233
      |0|_A = 1133
      U101_A(x1,x2,x3) = max{2261, x1 + 345}
      U102_A(x1,x2,x3) = 347
      U103_A(x1,x2,x3) = 1129
      U104_A(x1,x2,x3) = 1128
      U105_A(x1,x2) = 1127
      U106_A(x1) = 1126
      U11_A(x1,x2) = 1123
      U12_A(x1,x2) = 1123
      U13_A(x1) = 1122
      U121_A(x1,x2) = 0
      U122_A(x1) = 1
      nil_A = 233
      U32_A(x1,x2) = 1130
      U33_A(x1) = 1130
      U62_A(x1) = 16
      U71_A(x1) = 1130
    
    2. max/plus interpretations on natural numbers:
    
      mark#_A(x1) = 0
      U96_A(x1) = 140
      U95_A(x1,x2) = 102
      active#_A(x1) = 0
      mark_A(x1) = 94
      take_A(x1,x2) = 120
      s_A(x1) = 40
      cons_A(x1,x2) = 45
      U131_A(x1,x2,x3,x4) = 110
      isNatIList_A(x1) = 102
      U94_A(x1,x2,x3) = 114
      length_A(x1) = 0
      U111_A(x1,x2,x3) = 126
      isNatList_A(x1) = 43
      U93_A(x1,x2,x3) = 126
      U91_A(x1,x2,x3) = 150
      isNatKind_A(x1) = 59
      U92_A(x1,x2,x3) = 138
      U81_A(x1) = 47
      isNatIListKind_A(x1) = 13
      U61_A(x1,x2) = 89
      U52_A(x1) = 91
      U51_A(x1,x2) = 100
      U46_A(x1) = 22
      U45_A(x1,x2) = 62
      U41_A(x1,x2,x3) = 89
      U44_A(x1,x2,x3) = 102
      isNat_A(x1) = 138
      U21_A(x1,x2) = 57
      U43_A(x1,x2,x3) = 114
      tt_A = 100
      U42_A(x1,x2,x3) = 126
      U31_A(x1,x2) = 62
      U23_A(x1) = 54
      U22_A(x1,x2) = 150
      U136_A(x1,x2,x3,x4) = 102
      U135_A(x1,x2,x3,x4) = 102
      U134_A(x1,x2,x3,x4) = 114
      U133_A(x1,x2,x3,x4) = 126
      U132_A(x1,x2,x3,x4) = 91
      U114_A(x1,x2) = 57
      U113_A(x1,x2,x3) = 102
      U112_A(x1,x2,x3) = 114
      active_A(x1) = 94
      zeros_A = 59
      |0|_A = 0
      U101_A(x1,x2,x3) = 141
      U102_A(x1,x2,x3) = 99
      U103_A(x1,x2,x3) = 58
      U104_A(x1,x2,x3) = 58
      U105_A(x1,x2) = 132
      U106_A(x1) = 92
      U11_A(x1,x2) = 98
      U12_A(x1,x2) = 100
      U13_A(x1) = 111
      U121_A(x1,x2) = 101
      U122_A(x1) = 59
      nil_A = 17
      U32_A(x1,x2) = 59
      U33_A(x1) = 142
      U62_A(x1) = 140
      U71_A(x1) = 140
    

The next rules are strictly ordered:

  p2, p4, p6, p7, p8, p10, p11, p16, p17, p18, p20, p22, p23, p24, p25, p26, p27, p31, p33, p37, p39, p41, p43, p45, p48, p61

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: active#(take(s(M),cons(N,IL))) -> mark#(U131(isNatIList(IL),IL,M,N))
p3: active#(length(cons(N,L))) -> mark#(U111(isNatList(L),L,N))
p4: active#(isNatKind(s(V1))) -> mark#(U81(isNatKind(V1)))
p5: mark#(U81(X)) -> mark#(X)
p6: mark#(U61(X1,X2)) -> mark#(X1)
p7: mark#(U52(X)) -> mark#(X)
p8: mark#(U51(X1,X2)) -> active#(U51(mark(X1),X2))
p9: active#(isNatIList(cons(V1,V2))) -> mark#(U41(isNatKind(V1),V1,V2))
p10: active#(isNat(s(V1))) -> mark#(U21(isNatKind(V1),V1))
p11: mark#(U31(X1,X2)) -> mark#(X1)
p12: mark#(U23(X)) -> mark#(X)
p13: mark#(U22(X1,X2)) -> active#(U22(mark(X1),X2))
p14: mark#(U21(X1,X2)) -> active#(U21(mark(X1),X2))
p15: mark#(take(X1,X2)) -> active#(take(mark(X1),mark(X2)))
p16: active#(U51(tt(),V2)) -> mark#(U52(isNatIListKind(V2)))
p17: mark#(U136(X1,X2,X3,X4)) -> active#(U136(mark(X1),X2,X3,X4))
p18: mark#(U135(X1,X2,X3,X4)) -> active#(U135(mark(X1),X2,X3,X4))
p19: mark#(U134(X1,X2,X3,X4)) -> active#(U134(mark(X1),X2,X3,X4))
p20: mark#(U133(X1,X2,X3,X4)) -> active#(U133(mark(X1),X2,X3,X4))
p21: mark#(U132(X1,X2,X3,X4)) -> active#(U132(mark(X1),X2,X3,X4))
p22: mark#(U131(X1,X2,X3,X4)) -> active#(U131(mark(X1),X2,X3,X4))
p23: active#(U22(tt(),V1)) -> mark#(U23(isNat(V1)))
p24: active#(U21(tt(),V1)) -> mark#(U22(isNatKind(V1),V1))
p25: mark#(length(X)) -> active#(length(mark(X)))
p26: active#(U136(tt(),IL,M,N)) -> mark#(cons(N,take(M,IL)))
p27: mark#(s(X)) -> mark#(X)
p28: mark#(U114(X1,X2)) -> active#(U114(mark(X1),X2))
p29: active#(U135(tt(),IL,M,N)) -> mark#(U136(isNatKind(N),IL,M,N))
p30: mark#(U113(X1,X2,X3)) -> active#(U113(mark(X1),X2,X3))
p31: active#(U134(tt(),IL,M,N)) -> mark#(U135(isNat(N),IL,M,N))
p32: mark#(U112(X1,X2,X3)) -> active#(U112(mark(X1),X2,X3))
p33: active#(U133(tt(),IL,M,N)) -> mark#(U134(isNatKind(M),IL,M,N))
p34: mark#(U111(X1,X2,X3)) -> active#(U111(mark(X1),X2,X3))
p35: active#(U132(tt(),IL,M,N)) -> mark#(U133(isNat(M),IL,M,N))
p36: active#(U131(tt(),IL,M,N)) -> mark#(U132(isNatIListKind(IL),IL,M,N))
p37: mark#(isNat(X)) -> active#(isNat(X))
p38: active#(U114(tt(),L)) -> mark#(s(length(L)))
p39: mark#(isNatIListKind(X)) -> active#(isNatIListKind(X))
p40: active#(U113(tt(),L,N)) -> mark#(U114(isNatKind(N),L))
p41: mark#(isNatKind(X)) -> active#(isNatKind(X))
p42: active#(U112(tt(),L,N)) -> mark#(U113(isNat(N),L,N))
p43: active#(U111(tt(),L,N)) -> mark#(U112(isNatIListKind(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}


-- Reduction pair.

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

p1: mark#(U96(X)) -> mark#(X)
p2: mark#(isNatKind(X)) -> active#(isNatKind(X))
p3: active#(U111(tt(),L,N)) -> mark#(U112(isNatIListKind(L),L,N))
p4: mark#(isNatIListKind(X)) -> active#(isNatIListKind(X))
p5: active#(U112(tt(),L,N)) -> mark#(U113(isNat(N),L,N))
p6: mark#(isNat(X)) -> active#(isNat(X))
p7: active#(U113(tt(),L,N)) -> mark#(U114(isNatKind(N),L))
p8: mark#(U111(X1,X2,X3)) -> active#(U111(mark(X1),X2,X3))
p9: active#(U114(tt(),L)) -> mark#(s(length(L)))
p10: mark#(U112(X1,X2,X3)) -> active#(U112(mark(X1),X2,X3))
p11: active#(U131(tt(),IL,M,N)) -> mark#(U132(isNatIListKind(IL),IL,M,N))
p12: mark#(U113(X1,X2,X3)) -> active#(U113(mark(X1),X2,X3))
p13: active#(U132(tt(),IL,M,N)) -> mark#(U133(isNat(M),IL,M,N))
p14: mark#(U114(X1,X2)) -> active#(U114(mark(X1),X2))
p15: active#(U133(tt(),IL,M,N)) -> mark#(U134(isNatKind(M),IL,M,N))
p16: mark#(s(X)) -> mark#(X)
p17: mark#(length(X)) -> active#(length(mark(X)))
p18: active#(U134(tt(),IL,M,N)) -> mark#(U135(isNat(N),IL,M,N))
p19: mark#(U131(X1,X2,X3,X4)) -> active#(U131(mark(X1),X2,X3,X4))
p20: active#(U135(tt(),IL,M,N)) -> mark#(U136(isNatKind(N),IL,M,N))
p21: mark#(U132(X1,X2,X3,X4)) -> active#(U132(mark(X1),X2,X3,X4))
p22: active#(U136(tt(),IL,M,N)) -> mark#(cons(N,take(M,IL)))
p23: mark#(U133(X1,X2,X3,X4)) -> active#(U133(mark(X1),X2,X3,X4))
p24: active#(U21(tt(),V1)) -> mark#(U22(isNatKind(V1),V1))
p25: mark#(U134(X1,X2,X3,X4)) -> active#(U134(mark(X1),X2,X3,X4))
p26: active#(U22(tt(),V1)) -> mark#(U23(isNat(V1)))
p27: mark#(U135(X1,X2,X3,X4)) -> active#(U135(mark(X1),X2,X3,X4))
p28: active#(U51(tt(),V2)) -> mark#(U52(isNatIListKind(V2)))
p29: mark#(U136(X1,X2,X3,X4)) -> active#(U136(mark(X1),X2,X3,X4))
p30: active#(isNat(s(V1))) -> mark#(U21(isNatKind(V1),V1))
p31: mark#(take(X1,X2)) -> active#(take(mark(X1),mark(X2)))
p32: active#(isNatIList(cons(V1,V2))) -> mark#(U41(isNatKind(V1),V1,V2))
p33: mark#(U21(X1,X2)) -> active#(U21(mark(X1),X2))
p34: active#(isNatKind(s(V1))) -> mark#(U81(isNatKind(V1)))
p35: mark#(U22(X1,X2)) -> active#(U22(mark(X1),X2))
p36: active#(length(cons(N,L))) -> mark#(U111(isNatList(L),L,N))
p37: mark#(U23(X)) -> mark#(X)
p38: mark#(U31(X1,X2)) -> mark#(X1)
p39: mark#(U51(X1,X2)) -> active#(U51(mark(X1),X2))
p40: active#(take(s(M),cons(N,IL))) -> mark#(U131(isNatIList(IL),IL,M,N))
p41: mark#(U52(X)) -> mark#(X)
p42: mark#(U61(X1,X2)) -> mark#(X1)
p43: mark#(U81(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:

  lexicographic combination of reduction pairs:
  
    1. max/plus interpretations on natural numbers:
    
      mark#_A(x1) = 0
      U96_A(x1) = 2572
      isNatKind_A(x1) = 2727
      active#_A(x1) = max{0, x1 - 7090}
      U111_A(x1,x2,x3) = 6929
      tt_A = 2572
      U112_A(x1,x2,x3) = 5945
      isNatIListKind_A(x1) = 6081
      U113_A(x1,x2,x3) = 5783
      isNat_A(x1) = 7089
      U114_A(x1,x2) = 5621
      mark_A(x1) = max{2717, x1 + 150}
      s_A(x1) = 3979
      length_A(x1) = 7090
      U131_A(x1,x2,x3,x4) = 6928
      U132_A(x1,x2,x3,x4) = 6766
      U133_A(x1,x2,x3,x4) = 6457
      U134_A(x1,x2,x3,x4) = 5135
      U135_A(x1,x2,x3,x4) = 4974
      U136_A(x1,x2,x3,x4) = 4665
      cons_A(x1,x2) = 4503
      take_A(x1,x2) = 7090
      U21_A(x1,x2) = 6928
      U22_A(x1,x2) = 4231
      U23_A(x1) = 2734
      U51_A(x1,x2) = 5755
      U52_A(x1) = 5594
      isNatIList_A(x1) = 7092
      U41_A(x1,x2,x3) = 5593
      U81_A(x1) = 2572
      isNatList_A(x1) = 12764
      U31_A(x1,x2) = 3406
      U61_A(x1,x2) = 3041
      active_A(x1) = max{2722, x1 - 11}
      zeros_A = 4665
      |0|_A = 2572
      U101_A(x1,x2,x3) = 12603
      U102_A(x1,x2,x3) = 12442
      U103_A(x1,x2,x3) = 12281
      U104_A(x1,x2,x3) = 12119
      U105_A(x1,x2) = 11958
      U106_A(x1) = 7280
      U11_A(x1,x2) = 5453
      U12_A(x1,x2) = 5292
      U13_A(x1) = 5130
      U121_A(x1,x2) = 6929
      U122_A(x1) = 2572
      nil_A = 2572
      U32_A(x1,x2) = 3244
      U33_A(x1) = 2572
      U42_A(x1,x2,x3) = 2735
      U43_A(x1,x2,x3) = 2573
      U44_A(x1,x2,x3) = 2572
      U45_A(x1,x2) = 2572
      U46_A(x1) = 2572
      U62_A(x1) = 2879
      U71_A(x1) = 2572
      U91_A(x1,x2,x3) = 9286
      U92_A(x1,x2,x3) = 9125
      U93_A(x1,x2,x3) = 8961
      U94_A(x1,x2,x3) = 5292
      U95_A(x1,x2) = 5130
    
    2. max/plus interpretations on natural numbers:
    
      mark#_A(x1) = 0
      U96_A(x1) = 53
      isNatKind_A(x1) = 48
      active#_A(x1) = 0
      U111_A(x1,x2,x3) = 84
      tt_A = 33
      U112_A(x1,x2,x3) = 68
      isNatIListKind_A(x1) = 48
      U113_A(x1,x2,x3) = 48
      isNat_A(x1) = 49
      U114_A(x1,x2) = 51
      mark_A(x1) = 131
      s_A(x1) = 19
      length_A(x1) = 50
      U131_A(x1,x2,x3,x4) = 50
      U132_A(x1,x2,x3,x4) = 50
      U133_A(x1,x2,x3,x4) = 50
      U134_A(x1,x2,x3,x4) = 48
      U135_A(x1,x2,x3,x4) = 48
      U136_A(x1,x2,x3,x4) = 48
      cons_A(x1,x2) = 74
      take_A(x1,x2) = 49
      U21_A(x1,x2) = 68
      U22_A(x1,x2) = 48
      U23_A(x1) = 26
      U51_A(x1,x2) = 51
      U52_A(x1) = 51
      isNatIList_A(x1) = 112
      U41_A(x1,x2,x3) = 92
      U81_A(x1) = 168
      isNatList_A(x1) = 51
      U31_A(x1,x2) = 92
      U61_A(x1,x2) = 71
      active_A(x1) = 131
      zeros_A = 51
      |0|_A = 84
      U101_A(x1,x2,x3) = 74
      U102_A(x1,x2,x3) = 54
      U103_A(x1,x2,x3) = 0
      U104_A(x1,x2,x3) = 74
      U105_A(x1,x2) = 54
      U106_A(x1) = 72
      U11_A(x1,x2) = 53
      U12_A(x1,x2) = 53
      U13_A(x1) = 22
      U121_A(x1,x2) = 109
      U122_A(x1) = 96
      nil_A = 74
      U32_A(x1,x2) = 206
      U33_A(x1) = 184
      U42_A(x1,x2,x3) = 72
      U43_A(x1,x2,x3) = 72
      U44_A(x1,x2,x3) = 50
      U45_A(x1,x2) = 28
      U46_A(x1) = 72
      U62_A(x1) = 51
      U71_A(x1) = 151
      U91_A(x1,x2,x3) = 53
      U92_A(x1,x2,x3) = 53
      U93_A(x1,x2,x3) = 53
      U94_A(x1,x2,x3) = 53
      U95_A(x1,x2) = 72
    

The next rules are strictly ordered:

  p32

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#(isNatKind(X)) -> active#(isNatKind(X))
p3: active#(U111(tt(),L,N)) -> mark#(U112(isNatIListKind(L),L,N))
p4: mark#(isNatIListKind(X)) -> active#(isNatIListKind(X))
p5: active#(U112(tt(),L,N)) -> mark#(U113(isNat(N),L,N))
p6: mark#(isNat(X)) -> active#(isNat(X))
p7: active#(U113(tt(),L,N)) -> mark#(U114(isNatKind(N),L))
p8: mark#(U111(X1,X2,X3)) -> active#(U111(mark(X1),X2,X3))
p9: active#(U114(tt(),L)) -> mark#(s(length(L)))
p10: mark#(U112(X1,X2,X3)) -> active#(U112(mark(X1),X2,X3))
p11: active#(U131(tt(),IL,M,N)) -> mark#(U132(isNatIListKind(IL),IL,M,N))
p12: mark#(U113(X1,X2,X3)) -> active#(U113(mark(X1),X2,X3))
p13: active#(U132(tt(),IL,M,N)) -> mark#(U133(isNat(M),IL,M,N))
p14: mark#(U114(X1,X2)) -> active#(U114(mark(X1),X2))
p15: active#(U133(tt(),IL,M,N)) -> mark#(U134(isNatKind(M),IL,M,N))
p16: mark#(s(X)) -> mark#(X)
p17: mark#(length(X)) -> active#(length(mark(X)))
p18: active#(U134(tt(),IL,M,N)) -> mark#(U135(isNat(N),IL,M,N))
p19: mark#(U131(X1,X2,X3,X4)) -> active#(U131(mark(X1),X2,X3,X4))
p20: active#(U135(tt(),IL,M,N)) -> mark#(U136(isNatKind(N),IL,M,N))
p21: mark#(U132(X1,X2,X3,X4)) -> active#(U132(mark(X1),X2,X3,X4))
p22: active#(U136(tt(),IL,M,N)) -> mark#(cons(N,take(M,IL)))
p23: mark#(U133(X1,X2,X3,X4)) -> active#(U133(mark(X1),X2,X3,X4))
p24: active#(U21(tt(),V1)) -> mark#(U22(isNatKind(V1),V1))
p25: mark#(U134(X1,X2,X3,X4)) -> active#(U134(mark(X1),X2,X3,X4))
p26: active#(U22(tt(),V1)) -> mark#(U23(isNat(V1)))
p27: mark#(U135(X1,X2,X3,X4)) -> active#(U135(mark(X1),X2,X3,X4))
p28: active#(U51(tt(),V2)) -> mark#(U52(isNatIListKind(V2)))
p29: mark#(U136(X1,X2,X3,X4)) -> active#(U136(mark(X1),X2,X3,X4))
p30: active#(isNat(s(V1))) -> mark#(U21(isNatKind(V1),V1))
p31: mark#(take(X1,X2)) -> active#(take(mark(X1),mark(X2)))
p32: mark#(U21(X1,X2)) -> active#(U21(mark(X1),X2))
p33: active#(isNatKind(s(V1))) -> mark#(U81(isNatKind(V1)))
p34: mark#(U22(X1,X2)) -> active#(U22(mark(X1),X2))
p35: active#(length(cons(N,L))) -> mark#(U111(isNatList(L),L,N))
p36: mark#(U23(X)) -> mark#(X)
p37: mark#(U31(X1,X2)) -> mark#(X1)
p38: mark#(U51(X1,X2)) -> active#(U51(mark(X1),X2))
p39: active#(take(s(M),cons(N,IL))) -> mark#(U131(isNatIList(IL),IL,M,N))
p40: mark#(U52(X)) -> mark#(X)
p41: mark#(U61(X1,X2)) -> mark#(X1)
p42: mark#(U81(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}


-- Reduction pair.

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

p1: mark#(U96(X)) -> mark#(X)
p2: mark#(U81(X)) -> mark#(X)
p3: mark#(U61(X1,X2)) -> mark#(X1)
p4: mark#(U52(X)) -> mark#(X)
p5: mark#(U51(X1,X2)) -> active#(U51(mark(X1),X2))
p6: active#(take(s(M),cons(N,IL))) -> mark#(U131(isNatIList(IL),IL,M,N))
p7: mark#(U31(X1,X2)) -> mark#(X1)
p8: mark#(U23(X)) -> mark#(X)
p9: mark#(U22(X1,X2)) -> active#(U22(mark(X1),X2))
p10: active#(length(cons(N,L))) -> mark#(U111(isNatList(L),L,N))
p11: mark#(U21(X1,X2)) -> active#(U21(mark(X1),X2))
p12: active#(isNatKind(s(V1))) -> mark#(U81(isNatKind(V1)))
p13: mark#(take(X1,X2)) -> active#(take(mark(X1),mark(X2)))
p14: active#(isNat(s(V1))) -> mark#(U21(isNatKind(V1),V1))
p15: mark#(U136(X1,X2,X3,X4)) -> active#(U136(mark(X1),X2,X3,X4))
p16: active#(U51(tt(),V2)) -> mark#(U52(isNatIListKind(V2)))
p17: mark#(U135(X1,X2,X3,X4)) -> active#(U135(mark(X1),X2,X3,X4))
p18: active#(U22(tt(),V1)) -> mark#(U23(isNat(V1)))
p19: mark#(U134(X1,X2,X3,X4)) -> active#(U134(mark(X1),X2,X3,X4))
p20: active#(U21(tt(),V1)) -> mark#(U22(isNatKind(V1),V1))
p21: mark#(U133(X1,X2,X3,X4)) -> active#(U133(mark(X1),X2,X3,X4))
p22: active#(U136(tt(),IL,M,N)) -> mark#(cons(N,take(M,IL)))
p23: mark#(U132(X1,X2,X3,X4)) -> active#(U132(mark(X1),X2,X3,X4))
p24: active#(U135(tt(),IL,M,N)) -> mark#(U136(isNatKind(N),IL,M,N))
p25: mark#(U131(X1,X2,X3,X4)) -> active#(U131(mark(X1),X2,X3,X4))
p26: active#(U134(tt(),IL,M,N)) -> mark#(U135(isNat(N),IL,M,N))
p27: mark#(length(X)) -> active#(length(mark(X)))
p28: active#(U133(tt(),IL,M,N)) -> mark#(U134(isNatKind(M),IL,M,N))
p29: mark#(s(X)) -> mark#(X)
p30: mark#(U114(X1,X2)) -> active#(U114(mark(X1),X2))
p31: active#(U132(tt(),IL,M,N)) -> mark#(U133(isNat(M),IL,M,N))
p32: mark#(U113(X1,X2,X3)) -> active#(U113(mark(X1),X2,X3))
p33: active#(U131(tt(),IL,M,N)) -> mark#(U132(isNatIListKind(IL),IL,M,N))
p34: mark#(U112(X1,X2,X3)) -> active#(U112(mark(X1),X2,X3))
p35: active#(U114(tt(),L)) -> mark#(s(length(L)))
p36: mark#(U111(X1,X2,X3)) -> active#(U111(mark(X1),X2,X3))
p37: active#(U113(tt(),L,N)) -> mark#(U114(isNatKind(N),L))
p38: mark#(isNat(X)) -> active#(isNat(X))
p39: active#(U112(tt(),L,N)) -> mark#(U113(isNat(N),L,N))
p40: mark#(isNatIListKind(X)) -> active#(isNatIListKind(X))
p41: active#(U111(tt(),L,N)) -> mark#(U112(isNatIListKind(L),L,N))
p42: 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:

  lexicographic combination of reduction pairs:
  
    1. max/plus interpretations on natural numbers:
    
      mark#_A(x1) = 29
      U96_A(x1) = 24
      U81_A(x1) = 24
      U61_A(x1,x2) = 24
      U52_A(x1) = 24
      U51_A(x1,x2) = 24
      active#_A(x1) = x1 + 5
      mark_A(x1) = max{78, x1 - 5}
      take_A(x1,x2) = 24
      s_A(x1) = 21
      cons_A(x1,x2) = 22
      U131_A(x1,x2,x3,x4) = 24
      isNatIList_A(x1) = 24
      U31_A(x1,x2) = 24
      U23_A(x1) = 24
      U22_A(x1,x2) = 24
      length_A(x1) = 24
      U111_A(x1,x2,x3) = 24
      isNatList_A(x1) = 24
      U21_A(x1,x2) = 24
      isNatKind_A(x1) = 24
      isNat_A(x1) = 24
      U136_A(x1,x2,x3,x4) = 24
      tt_A = 24
      isNatIListKind_A(x1) = 24
      U135_A(x1,x2,x3,x4) = 24
      U134_A(x1,x2,x3,x4) = 24
      U133_A(x1,x2,x3,x4) = 24
      U132_A(x1,x2,x3,x4) = 24
      U114_A(x1,x2) = 24
      U113_A(x1,x2,x3) = 24
      U112_A(x1,x2,x3) = 24
      active_A(x1) = x1 + 54
      zeros_A = 24
      |0|_A = 23
      U101_A(x1,x2,x3) = 24
      U102_A(x1,x2,x3) = 24
      U103_A(x1,x2,x3) = 24
      U104_A(x1,x2,x3) = 24
      U105_A(x1,x2) = 24
      U106_A(x1) = 24
      U11_A(x1,x2) = 24
      U12_A(x1,x2) = 24
      U13_A(x1) = 24
      U121_A(x1,x2) = 24
      U122_A(x1) = 24
      nil_A = 24
      U32_A(x1,x2) = 24
      U33_A(x1) = 24
      U41_A(x1,x2,x3) = 24
      U42_A(x1,x2,x3) = 24
      U43_A(x1,x2,x3) = 24
      U44_A(x1,x2,x3) = 24
      U45_A(x1,x2) = 24
      U46_A(x1) = 24
      U62_A(x1) = 24
      U71_A(x1) = 24
      U91_A(x1,x2,x3) = 24
      U92_A(x1,x2,x3) = 24
      U93_A(x1,x2,x3) = 24
      U94_A(x1,x2,x3) = 24
      U95_A(x1,x2) = 24
    
    2. max/plus interpretations on natural numbers:
    
      mark#_A(x1) = 53
      U96_A(x1) = 18
      U81_A(x1) = 9
      U61_A(x1,x2) = 18
      U52_A(x1) = 18
      U51_A(x1,x2) = 18
      active#_A(x1) = x1 + 35
      mark_A(x1) = 7
      take_A(x1,x2) = 18
      s_A(x1) = 21
      cons_A(x1,x2) = 19
      U131_A(x1,x2,x3,x4) = 18
      isNatIList_A(x1) = 2
      U31_A(x1,x2) = 18
      U23_A(x1) = 17
      U22_A(x1,x2) = 18
      length_A(x1) = 18
      U111_A(x1,x2,x3) = 18
      isNatList_A(x1) = 4
      U21_A(x1,x2) = 18
      isNatKind_A(x1) = 18
      isNat_A(x1) = 18
      U136_A(x1,x2,x3,x4) = 18
      tt_A = 8
      isNatIListKind_A(x1) = 17
      U135_A(x1,x2,x3,x4) = 18
      U134_A(x1,x2,x3,x4) = 18
      U133_A(x1,x2,x3,x4) = 18
      U132_A(x1,x2,x3,x4) = 18
      U114_A(x1,x2) = 18
      U113_A(x1,x2,x3) = 18
      U112_A(x1,x2,x3) = 18
      active_A(x1) = max{7, x1 - 11}
      zeros_A = 9
      |0|_A = 36
      U101_A(x1,x2,x3) = 18
      U102_A(x1,x2,x3) = 17
      U103_A(x1,x2,x3) = 18
      U104_A(x1,x2,x3) = 18
      U105_A(x1,x2) = 18
      U106_A(x1) = 18
      U11_A(x1,x2) = 18
      U12_A(x1,x2) = 18
      U13_A(x1) = 18
      U121_A(x1,x2) = 9
      U122_A(x1) = 18
      nil_A = 11
      U32_A(x1,x2) = 0
      U33_A(x1) = 18
      U41_A(x1,x2,x3) = 18
      U42_A(x1,x2,x3) = 10
      U43_A(x1,x2,x3) = 0
      U44_A(x1,x2,x3) = 18
      U45_A(x1,x2) = 18
      U46_A(x1) = 18
      U62_A(x1) = 17
      U71_A(x1) = 18
      U91_A(x1,x2,x3) = 10
      U92_A(x1,x2,x3) = 18
      U93_A(x1,x2,x3) = 18
      U94_A(x1,x2,x3) = 18
      U95_A(x1,x2) = 18
    

The next rules are strictly ordered:

  p40

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#(U81(X)) -> mark#(X)
p3: mark#(U61(X1,X2)) -> mark#(X1)
p4: mark#(U52(X)) -> mark#(X)
p5: mark#(U51(X1,X2)) -> active#(U51(mark(X1),X2))
p6: active#(take(s(M),cons(N,IL))) -> mark#(U131(isNatIList(IL),IL,M,N))
p7: mark#(U31(X1,X2)) -> mark#(X1)
p8: mark#(U23(X)) -> mark#(X)
p9: mark#(U22(X1,X2)) -> active#(U22(mark(X1),X2))
p10: active#(length(cons(N,L))) -> mark#(U111(isNatList(L),L,N))
p11: mark#(U21(X1,X2)) -> active#(U21(mark(X1),X2))
p12: active#(isNatKind(s(V1))) -> mark#(U81(isNatKind(V1)))
p13: mark#(take(X1,X2)) -> active#(take(mark(X1),mark(X2)))
p14: active#(isNat(s(V1))) -> mark#(U21(isNatKind(V1),V1))
p15: mark#(U136(X1,X2,X3,X4)) -> active#(U136(mark(X1),X2,X3,X4))
p16: active#(U51(tt(),V2)) -> mark#(U52(isNatIListKind(V2)))
p17: mark#(U135(X1,X2,X3,X4)) -> active#(U135(mark(X1),X2,X3,X4))
p18: active#(U22(tt(),V1)) -> mark#(U23(isNat(V1)))
p19: mark#(U134(X1,X2,X3,X4)) -> active#(U134(mark(X1),X2,X3,X4))
p20: active#(U21(tt(),V1)) -> mark#(U22(isNatKind(V1),V1))
p21: mark#(U133(X1,X2,X3,X4)) -> active#(U133(mark(X1),X2,X3,X4))
p22: active#(U136(tt(),IL,M,N)) -> mark#(cons(N,take(M,IL)))
p23: mark#(U132(X1,X2,X3,X4)) -> active#(U132(mark(X1),X2,X3,X4))
p24: active#(U135(tt(),IL,M,N)) -> mark#(U136(isNatKind(N),IL,M,N))
p25: mark#(U131(X1,X2,X3,X4)) -> active#(U131(mark(X1),X2,X3,X4))
p26: active#(U134(tt(),IL,M,N)) -> mark#(U135(isNat(N),IL,M,N))
p27: mark#(length(X)) -> active#(length(mark(X)))
p28: active#(U133(tt(),IL,M,N)) -> mark#(U134(isNatKind(M),IL,M,N))
p29: mark#(s(X)) -> mark#(X)
p30: mark#(U114(X1,X2)) -> active#(U114(mark(X1),X2))
p31: active#(U132(tt(),IL,M,N)) -> mark#(U133(isNat(M),IL,M,N))
p32: mark#(U113(X1,X2,X3)) -> active#(U113(mark(X1),X2,X3))
p33: active#(U131(tt(),IL,M,N)) -> mark#(U132(isNatIListKind(IL),IL,M,N))
p34: mark#(U112(X1,X2,X3)) -> active#(U112(mark(X1),X2,X3))
p35: active#(U114(tt(),L)) -> mark#(s(length(L)))
p36: mark#(U111(X1,X2,X3)) -> active#(U111(mark(X1),X2,X3))
p37: active#(U113(tt(),L,N)) -> mark#(U114(isNatKind(N),L))
p38: mark#(isNat(X)) -> active#(isNat(X))
p39: active#(U112(tt(),L,N)) -> mark#(U113(isNat(N),L,N))
p40: active#(U111(tt(),L,N)) -> mark#(U112(isNatIListKind(L),L,N))
p41: 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}


-- Reduction pair.

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

p1: mark#(U96(X)) -> mark#(X)
p2: mark#(isNatKind(X)) -> active#(isNatKind(X))
p3: active#(U111(tt(),L,N)) -> mark#(U112(isNatIListKind(L),L,N))
p4: mark#(isNat(X)) -> active#(isNat(X))
p5: active#(U112(tt(),L,N)) -> mark#(U113(isNat(N),L,N))
p6: mark#(U111(X1,X2,X3)) -> active#(U111(mark(X1),X2,X3))
p7: active#(U113(tt(),L,N)) -> mark#(U114(isNatKind(N),L))
p8: mark#(U112(X1,X2,X3)) -> active#(U112(mark(X1),X2,X3))
p9: active#(U114(tt(),L)) -> mark#(s(length(L)))
p10: mark#(U113(X1,X2,X3)) -> active#(U113(mark(X1),X2,X3))
p11: active#(U131(tt(),IL,M,N)) -> mark#(U132(isNatIListKind(IL),IL,M,N))
p12: mark#(U114(X1,X2)) -> active#(U114(mark(X1),X2))
p13: active#(U132(tt(),IL,M,N)) -> mark#(U133(isNat(M),IL,M,N))
p14: mark#(s(X)) -> mark#(X)
p15: mark#(length(X)) -> active#(length(mark(X)))
p16: active#(U133(tt(),IL,M,N)) -> mark#(U134(isNatKind(M),IL,M,N))
p17: mark#(U131(X1,X2,X3,X4)) -> active#(U131(mark(X1),X2,X3,X4))
p18: active#(U134(tt(),IL,M,N)) -> mark#(U135(isNat(N),IL,M,N))
p19: mark#(U132(X1,X2,X3,X4)) -> active#(U132(mark(X1),X2,X3,X4))
p20: active#(U135(tt(),IL,M,N)) -> mark#(U136(isNatKind(N),IL,M,N))
p21: mark#(U133(X1,X2,X3,X4)) -> active#(U133(mark(X1),X2,X3,X4))
p22: active#(U136(tt(),IL,M,N)) -> mark#(cons(N,take(M,IL)))
p23: mark#(U134(X1,X2,X3,X4)) -> active#(U134(mark(X1),X2,X3,X4))
p24: active#(U21(tt(),V1)) -> mark#(U22(isNatKind(V1),V1))
p25: mark#(U135(X1,X2,X3,X4)) -> active#(U135(mark(X1),X2,X3,X4))
p26: active#(U22(tt(),V1)) -> mark#(U23(isNat(V1)))
p27: mark#(U136(X1,X2,X3,X4)) -> active#(U136(mark(X1),X2,X3,X4))
p28: active#(U51(tt(),V2)) -> mark#(U52(isNatIListKind(V2)))
p29: mark#(take(X1,X2)) -> active#(take(mark(X1),mark(X2)))
p30: active#(isNat(s(V1))) -> mark#(U21(isNatKind(V1),V1))
p31: mark#(U21(X1,X2)) -> active#(U21(mark(X1),X2))
p32: active#(isNatKind(s(V1))) -> mark#(U81(isNatKind(V1)))
p33: mark#(U22(X1,X2)) -> active#(U22(mark(X1),X2))
p34: active#(length(cons(N,L))) -> mark#(U111(isNatList(L),L,N))
p35: mark#(U23(X)) -> mark#(X)
p36: mark#(U31(X1,X2)) -> mark#(X1)
p37: mark#(U51(X1,X2)) -> active#(U51(mark(X1),X2))
p38: active#(take(s(M),cons(N,IL))) -> mark#(U131(isNatIList(IL),IL,M,N))
p39: mark#(U52(X)) -> mark#(X)
p40: mark#(U61(X1,X2)) -> mark#(X1)
p41: mark#(U81(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:

  lexicographic combination of reduction pairs:
  
    1. max/plus interpretations on natural numbers:
    
      mark#_A(x1) = x1
      U96_A(x1) = max{357, x1 + 1}
      isNatKind_A(x1) = x1 + 760
      active#_A(x1) = max{178, x1}
      U111_A(x1,x2,x3) = max{x1 + 592, x2 + 949}
      tt_A = 754
      U112_A(x1,x2,x3) = max{1346, x2 + 949}
      isNatIListKind_A(x1) = x1 + 745
      isNat_A(x1) = x1 + 1168
      U113_A(x1,x2,x3) = max{1346, x2 + 949}
      mark_A(x1) = max{353, x1}
      U114_A(x1,x2) = max{1346, x2 + 949}
      s_A(x1) = max{1345, x1 + 34}
      length_A(x1) = max{1311, x1 + 915}
      U131_A(x1,x2,x3,x4) = max{949, x3 + 477}
      U132_A(x1,x2,x3,x4) = max{949, x3 + 477}
      U133_A(x1,x2,x3,x4) = max{881, x3 + 477}
      U134_A(x1,x2,x3,x4) = max{881, x3 + 477}
      U135_A(x1,x2,x3,x4) = max{880, x3 + 477}
      U136_A(x1,x2,x3,x4) = max{842, x3 + 477}
      cons_A(x1,x2) = max{179, x2 + 34}
      take_A(x1,x2) = max{796, x1 + 443}
      U21_A(x1,x2) = x2 + 1202
      U22_A(x1,x2) = x2 + 1177
      U23_A(x1) = max{1176, x1 + 9}
      U51_A(x1,x2) = x2 + 778
      U52_A(x1) = max{753, x1 + 32}
      U81_A(x1) = max{386, x1 + 33}
      isNatList_A(x1) = max{356, x1 + 318}
      U31_A(x1,x2) = max{x1 + 790, x2 + 1144}
      isNatIList_A(x1) = x1 + 1535
      U61_A(x1,x2) = max{354, x1 + 1}
      active_A(x1) = max{353, x1}
      zeros_A = 179
      |0|_A = 352
      U101_A(x1,x2,x3) = max{352, x1}
      U102_A(x1,x2,x3) = 754
      U103_A(x1,x2,x3) = 754
      U104_A(x1,x2,x3) = 754
      U105_A(x1,x2) = 754
      U106_A(x1) = 754
      U11_A(x1,x2) = max{2479, x2 + 318}
      U12_A(x1,x2) = max{2479, x2 + 318}
      U13_A(x1) = x1
      U121_A(x1,x2) = 796
      U122_A(x1) = 437
      nil_A = 437
      U32_A(x1,x2) = max{752, x1 + 399, x2 + 670}
      U33_A(x1) = max{753, x1 + 352}
      U41_A(x1,x2,x3) = 1568
      U42_A(x1,x2,x3) = 1567
      U43_A(x1,x2,x3) = 1317
      U44_A(x1,x2,x3) = 963
      U45_A(x1,x2) = 962
      U46_A(x1) = 962
      U62_A(x1) = 754
      U71_A(x1) = 754
      U91_A(x1,x2,x3) = max{355, x3 + 351}
      U92_A(x1,x2,x3) = max{354, x3 + 350}
      U93_A(x1,x2,x3) = max{354, x1 - 400, x3 + 350}
      U94_A(x1,x2,x3) = max{354, x1 - 396, x3 + 349}
      U95_A(x1,x2) = max{357, x2 + 348}
    
    2. max/plus interpretations on natural numbers:
    
      mark#_A(x1) = 0
      U96_A(x1) = 16
      isNatKind_A(x1) = 21
      active#_A(x1) = 0
      U111_A(x1,x2,x3) = 29
      tt_A = 1
      U112_A(x1,x2,x3) = 38
      isNatIListKind_A(x1) = 0
      isNat_A(x1) = 16
      U113_A(x1,x2,x3) = 16
      mark_A(x1) = 12
      U114_A(x1,x2) = 97
      s_A(x1) = 75
      length_A(x1) = 28
      U131_A(x1,x2,x3,x4) = 89
      U132_A(x1,x2,x3,x4) = 67
      U133_A(x1,x2,x3,x4) = 45
      U134_A(x1,x2,x3,x4) = 23
      U135_A(x1,x2,x3,x4) = 57
      U136_A(x1,x2,x3,x4) = 35
      cons_A(x1,x2) = 13
      take_A(x1,x2) = 7
      U21_A(x1,x2) = 38
      U22_A(x1,x2) = 16
      U23_A(x1) = 38
      U51_A(x1,x2) = 38
      U52_A(x1) = 16
      U81_A(x1) = 13
      isNatList_A(x1) = 3
      U31_A(x1,x2) = 0
      isNatIList_A(x1) = 11
      U61_A(x1,x2) = 0
      active_A(x1) = 12
      zeros_A = 13
      |0|_A = 87
      U101_A(x1,x2,x3) = 22
      U102_A(x1,x2,x3) = 0
      U103_A(x1,x2,x3) = 13
      U104_A(x1,x2,x3) = 0
      U105_A(x1,x2) = 0
      U106_A(x1) = 0
      U11_A(x1,x2) = 13
      U12_A(x1,x2) = 11
      U13_A(x1) = 11
      U121_A(x1,x2) = 13
      U122_A(x1) = 43
      nil_A = 0
      U32_A(x1,x2) = 13
      U33_A(x1) = 23
      U41_A(x1,x2,x3) = 0
      U42_A(x1,x2,x3) = 22
      U43_A(x1,x2,x3) = 0
      U44_A(x1,x2,x3) = 0
      U45_A(x1,x2) = 11
      U46_A(x1) = 0
      U62_A(x1) = 23
      U71_A(x1) = 13
      U91_A(x1,x2,x3) = 0
      U92_A(x1,x2,x3) = 0
      U93_A(x1,x2,x3) = 0
      U94_A(x1,x2,x3) = 0
      U95_A(x1,x2) = 0
    

The next rules are strictly ordered:

  p1, p14, p24, p28, p32, p35, p36, p39, p40, p41

We remove them from the problem.

-- SCC decomposition.

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

p1: mark#(isNatKind(X)) -> active#(isNatKind(X))
p2: active#(U111(tt(),L,N)) -> mark#(U112(isNatIListKind(L),L,N))
p3: mark#(isNat(X)) -> active#(isNat(X))
p4: active#(U112(tt(),L,N)) -> mark#(U113(isNat(N),L,N))
p5: mark#(U111(X1,X2,X3)) -> active#(U111(mark(X1),X2,X3))
p6: active#(U113(tt(),L,N)) -> mark#(U114(isNatKind(N),L))
p7: mark#(U112(X1,X2,X3)) -> active#(U112(mark(X1),X2,X3))
p8: active#(U114(tt(),L)) -> mark#(s(length(L)))
p9: mark#(U113(X1,X2,X3)) -> active#(U113(mark(X1),X2,X3))
p10: active#(U131(tt(),IL,M,N)) -> mark#(U132(isNatIListKind(IL),IL,M,N))
p11: mark#(U114(X1,X2)) -> active#(U114(mark(X1),X2))
p12: active#(U132(tt(),IL,M,N)) -> mark#(U133(isNat(M),IL,M,N))
p13: mark#(length(X)) -> active#(length(mark(X)))
p14: active#(U133(tt(),IL,M,N)) -> mark#(U134(isNatKind(M),IL,M,N))
p15: mark#(U131(X1,X2,X3,X4)) -> active#(U131(mark(X1),X2,X3,X4))
p16: active#(U134(tt(),IL,M,N)) -> mark#(U135(isNat(N),IL,M,N))
p17: mark#(U132(X1,X2,X3,X4)) -> active#(U132(mark(X1),X2,X3,X4))
p18: active#(U135(tt(),IL,M,N)) -> mark#(U136(isNatKind(N),IL,M,N))
p19: mark#(U133(X1,X2,X3,X4)) -> active#(U133(mark(X1),X2,X3,X4))
p20: active#(U136(tt(),IL,M,N)) -> mark#(cons(N,take(M,IL)))
p21: mark#(U134(X1,X2,X3,X4)) -> active#(U134(mark(X1),X2,X3,X4))
p22: mark#(U135(X1,X2,X3,X4)) -> active#(U135(mark(X1),X2,X3,X4))
p23: active#(U22(tt(),V1)) -> mark#(U23(isNat(V1)))
p24: mark#(U136(X1,X2,X3,X4)) -> active#(U136(mark(X1),X2,X3,X4))
p25: mark#(take(X1,X2)) -> active#(take(mark(X1),mark(X2)))
p26: active#(isNat(s(V1))) -> mark#(U21(isNatKind(V1),V1))
p27: mark#(U21(X1,X2)) -> active#(U21(mark(X1),X2))
p28: mark#(U22(X1,X2)) -> active#(U22(mark(X1),X2))
p29: active#(length(cons(N,L))) -> mark#(U111(isNatList(L),L,N))
p30: mark#(U51(X1,X2)) -> active#(U51(mark(X1),X2))
p31: 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}


-- Reduction pair.

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

p1: mark#(isNatKind(X)) -> active#(isNatKind(X))
p2: active#(take(s(M),cons(N,IL))) -> mark#(U131(isNatIList(IL),IL,M,N))
p3: mark#(U51(X1,X2)) -> active#(U51(mark(X1),X2))
p4: active#(length(cons(N,L))) -> mark#(U111(isNatList(L),L,N))
p5: mark#(U22(X1,X2)) -> active#(U22(mark(X1),X2))
p6: active#(isNat(s(V1))) -> mark#(U21(isNatKind(V1),V1))
p7: mark#(U21(X1,X2)) -> active#(U21(mark(X1),X2))
p8: active#(U22(tt(),V1)) -> mark#(U23(isNat(V1)))
p9: mark#(take(X1,X2)) -> active#(take(mark(X1),mark(X2)))
p10: active#(U136(tt(),IL,M,N)) -> mark#(cons(N,take(M,IL)))
p11: mark#(U136(X1,X2,X3,X4)) -> active#(U136(mark(X1),X2,X3,X4))
p12: active#(U135(tt(),IL,M,N)) -> mark#(U136(isNatKind(N),IL,M,N))
p13: mark#(U135(X1,X2,X3,X4)) -> active#(U135(mark(X1),X2,X3,X4))
p14: active#(U134(tt(),IL,M,N)) -> mark#(U135(isNat(N),IL,M,N))
p15: mark#(U134(X1,X2,X3,X4)) -> active#(U134(mark(X1),X2,X3,X4))
p16: active#(U133(tt(),IL,M,N)) -> mark#(U134(isNatKind(M),IL,M,N))
p17: mark#(U133(X1,X2,X3,X4)) -> active#(U133(mark(X1),X2,X3,X4))
p18: active#(U132(tt(),IL,M,N)) -> mark#(U133(isNat(M),IL,M,N))
p19: mark#(U132(X1,X2,X3,X4)) -> active#(U132(mark(X1),X2,X3,X4))
p20: active#(U131(tt(),IL,M,N)) -> mark#(U132(isNatIListKind(IL),IL,M,N))
p21: mark#(U131(X1,X2,X3,X4)) -> active#(U131(mark(X1),X2,X3,X4))
p22: active#(U114(tt(),L)) -> mark#(s(length(L)))
p23: mark#(length(X)) -> active#(length(mark(X)))
p24: active#(U113(tt(),L,N)) -> mark#(U114(isNatKind(N),L))
p25: mark#(U114(X1,X2)) -> active#(U114(mark(X1),X2))
p26: active#(U112(tt(),L,N)) -> mark#(U113(isNat(N),L,N))
p27: mark#(U113(X1,X2,X3)) -> active#(U113(mark(X1),X2,X3))
p28: active#(U111(tt(),L,N)) -> mark#(U112(isNatIListKind(L),L,N))
p29: mark#(U112(X1,X2,X3)) -> active#(U112(mark(X1),X2,X3))
p30: mark#(U111(X1,X2,X3)) -> active#(U111(mark(X1),X2,X3))
p31: mark#(isNat(X)) -> active#(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

  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:

  lexicographic combination of reduction pairs:
  
    1. max/plus interpretations on natural numbers:
    
      mark#_A(x1) = max{0, x1 - 697}
      isNatKind_A(x1) = 696
      active#_A(x1) = max{0, x1 - 760}
      take_A(x1,x2) = 3947
      s_A(x1) = 91
      cons_A(x1,x2) = 0
      U131_A(x1,x2,x3,x4) = 3859
      isNatIList_A(x1) = 4453
      U51_A(x1,x2) = 724
      mark_A(x1) = x1 + 85
      length_A(x1) = 5911
      U111_A(x1,x2,x3) = 762
      isNatList_A(x1) = 5069
      U22_A(x1,x2) = 786
      isNat_A(x1) = 1886
      U21_A(x1,x2) = 1799
      tt_A = 0
      U23_A(x1) = 699
      U136_A(x1,x2,x3,x4) = 713
      U135_A(x1,x2,x3,x4) = 800
      U134_A(x1,x2,x3,x4) = 2776
      U133_A(x1,x2,x3,x4) = 2928
      U132_A(x1,x2,x3,x4) = 3015
      isNatIListKind_A(x1) = 812
      U114_A(x1,x2) = 248
      U113_A(x1,x2,x3) = 335
      U112_A(x1,x2,x3) = 449
      active_A(x1) = max{9, x1 - 2}
      zeros_A = 603
      |0|_A = 592
      U101_A(x1,x2,x3) = 4981
      U102_A(x1,x2,x3) = 4893
      U103_A(x1,x2,x3) = 4805
      U104_A(x1,x2,x3) = 4717
      U105_A(x1,x2) = 4629
      U106_A(x1) = 176
      U11_A(x1,x2) = 1799
      U12_A(x1,x2) = 900
      U13_A(x1) = 88
      U121_A(x1,x2) = 3860
      U122_A(x1) = 87
      nil_A = 0
      U31_A(x1,x2) = 986
      U32_A(x1,x2) = 176
      U33_A(x1) = 88
      U41_A(x1,x2,x3) = 1423
      U42_A(x1,x2,x3) = 1336
      U43_A(x1,x2,x3) = 1249
      U44_A(x1,x2,x3) = 358
      U45_A(x1,x2) = 261
      U46_A(x1) = 174
      U52_A(x1) = 171
      U61_A(x1,x2) = 724
      U62_A(x1) = 87
      U71_A(x1) = 609
      U81_A(x1) = 97
      U91_A(x1,x2,x3) = 2241
      U92_A(x1,x2,x3) = 2153
      U93_A(x1,x2,x3) = 2065
      U94_A(x1,x2,x3) = 1977
      U95_A(x1,x2) = 176
      U96_A(x1) = 88
    
    2. max/plus interpretations on natural numbers:
    
      mark#_A(x1) = 0
      isNatKind_A(x1) = 40
      active#_A(x1) = 0
      take_A(x1,x2) = 0
      s_A(x1) = 53
      cons_A(x1,x2) = 55
      U131_A(x1,x2,x3,x4) = 49
      isNatIList_A(x1) = 55
      U51_A(x1,x2) = 49
      mark_A(x1) = max{34, x1 + 9}
      length_A(x1) = 52
      U111_A(x1,x2,x3) = 66
      isNatList_A(x1) = 38
      U22_A(x1,x2) = 37
      isNat_A(x1) = 53
      U21_A(x1,x2) = 44
      tt_A = 23
      U23_A(x1) = 40
      U136_A(x1,x2,x3,x4) = 44
      U135_A(x1,x2,x3,x4) = 53
      U134_A(x1,x2,x3,x4) = 4
      U133_A(x1,x2,x3,x4) = 23
      U132_A(x1,x2,x3,x4) = 42
      isNatIListKind_A(x1) = 40
      U114_A(x1,x2) = 49
      U113_A(x1,x2,x3) = 55
      U112_A(x1,x2,x3) = 57
      active_A(x1) = 63
      zeros_A = 0
      |0|_A = 44
      U101_A(x1,x2,x3) = 44
      U102_A(x1,x2,x3) = 27
      U103_A(x1,x2,x3) = 19
      U104_A(x1,x2,x3) = 0
      U105_A(x1,x2) = 95
      U106_A(x1) = 40
      U11_A(x1,x2) = 42
      U12_A(x1,x2) = 44
      U13_A(x1) = 84
      U121_A(x1,x2) = 0
      U122_A(x1) = 42
      nil_A = 53
      U31_A(x1,x2) = 55
      U32_A(x1,x2) = 42
      U33_A(x1) = 82
      U41_A(x1,x2,x3) = 38
      U42_A(x1,x2,x3) = 0
      U43_A(x1,x2,x3) = 44
      U44_A(x1,x2,x3) = 27
      U45_A(x1,x2) = 97
      U46_A(x1) = 40
      U52_A(x1) = 40
      U61_A(x1,x2) = 74
      U62_A(x1) = 55
      U71_A(x1) = 40
      U81_A(x1) = 40
      U91_A(x1,x2,x3) = 44
      U92_A(x1,x2,x3) = 0
      U93_A(x1,x2,x3) = 0
      U94_A(x1,x2,x3) = 44
      U95_A(x1,x2) = 27
      U96_A(x1) = 40
    

The next rules are strictly ordered:

  p2, p3, p4, p5, p6, p7, p8, p9, p11, p12, p13, p14, p15, p16, p17, p18, p19, p20, p21, p23, p28, p30, p31

We remove them from the problem.

-- SCC decomposition.

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

p1: mark#(isNatKind(X)) -> active#(isNatKind(X))
p2: active#(U136(tt(),IL,M,N)) -> mark#(cons(N,take(M,IL)))
p3: active#(U114(tt(),L)) -> mark#(s(length(L)))
p4: active#(U113(tt(),L,N)) -> mark#(U114(isNatKind(N),L))
p5: mark#(U114(X1,X2)) -> active#(U114(mark(X1),X2))
p6: active#(U112(tt(),L,N)) -> mark#(U113(isNat(N),L,N))
p7: mark#(U113(X1,X2,X3)) -> active#(U113(mark(X1),X2,X3))
p8: mark#(U112(X1,X2,X3)) -> active#(U112(mark(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, p6, p7, p8}


-- Reduction pair.

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

p1: mark#(isNatKind(X)) -> active#(isNatKind(X))
p2: active#(U112(tt(),L,N)) -> mark#(U113(isNat(N),L,N))
p3: mark#(U112(X1,X2,X3)) -> active#(U112(mark(X1),X2,X3))
p4: active#(U113(tt(),L,N)) -> mark#(U114(isNatKind(N),L))
p5: mark#(U113(X1,X2,X3)) -> active#(U113(mark(X1),X2,X3))
p6: active#(U114(tt(),L)) -> mark#(s(length(L)))
p7: mark#(U114(X1,X2)) -> active#(U114(mark(X1),X2))
p8: active#(U136(tt(),IL,M,N)) -> mark#(cons(N,take(M,IL)))

and R consists of:

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

  lexicographic combination of reduction pairs:
  
    1. max/plus interpretations on natural numbers:
    
      mark#_A(x1) = x1 + 136
      isNatKind_A(x1) = x1 + 1718
      active#_A(x1) = x1 + 113
      U112_A(x1,x2,x3) = 3515
      tt_A = 20
      U113_A(x1,x2,x3) = 2268
      isNat_A(x1) = x1 + 629
      mark_A(x1) = max{640, x1 + 620}
      U114_A(x1,x2) = 529
      s_A(x1) = 505
      length_A(x1) = 3717
      U136_A(x1,x2,x3,x4) = 1087
      cons_A(x1,x2) = max{346, x2 - 135}
      take_A(x1,x2) = 1198
      active_A(x1) = x1 + 602
      zeros_A = 385
      |0|_A = 39
      U101_A(x1,x2,x3) = 2218
      U102_A(x1,x2,x3) = 844
      U103_A(x1,x2,x3) = 825
      isNatIListKind_A(x1) = max{804, x1 + 789}
      U104_A(x1,x2,x3) = 327
      U105_A(x1,x2) = 57
      U106_A(x1) = 38
      isNatIList_A(x1) = max{1543, x1 + 903}
      U11_A(x1,x2) = 520
      U12_A(x1,x2) = 336
      U111_A(x1,x2,x3) = 3698
      U13_A(x1) = 38
      isNatList_A(x1) = x1 + 1039
      U121_A(x1,x2) = 855
      U122_A(x1) = 39
      nil_A = 0
      U131_A(x1,x2,x3,x4) = 1180
      U132_A(x1,x2,x3,x4) = 1161
      U133_A(x1,x2,x3,x4) = 1143
      U134_A(x1,x2,x3,x4) = 1125
      U135_A(x1,x2,x3,x4) = 1106
      U21_A(x1,x2) = 75
      U22_A(x1,x2) = 56
      U23_A(x1) = 38
      U31_A(x1,x2) = x2 + 688
      U32_A(x1,x2) = 488
      U33_A(x1) = 466
      U41_A(x1,x2,x3) = max{1232, x3 + 749}
      U42_A(x1,x2,x3) = x3 + 730
      U43_A(x1,x2,x3) = x3 + 711
      U44_A(x1,x2,x3) = max{511, x3 + 509}
      U45_A(x1,x2) = max{490, x2 + 488}
      U46_A(x1) = 39
      U51_A(x1,x2) = x2 + 636
      U52_A(x1) = 39
      U61_A(x1,x2) = 346
      U62_A(x1) = 39
      U71_A(x1) = 1109
      U81_A(x1) = 482
      U91_A(x1,x2,x3) = max{902, x3 + 885}
      U92_A(x1,x2,x3) = max{884, x3 + 867}
      U93_A(x1,x2,x3) = x3 + 682
      U94_A(x1,x2,x3) = x3 + 663
      U95_A(x1,x2) = max{642, x2 + 159}
      U96_A(x1) = 39
    
    2. max/plus interpretations on natural numbers:
    
      mark#_A(x1) = max{147, x1 - 11}
      isNatKind_A(x1) = max{174, x1 + 12}
      active#_A(x1) = max{0, x1 - 11}
      U112_A(x1,x2,x3) = 189
      tt_A = 138
      U113_A(x1,x2,x3) = 173
      isNat_A(x1) = max{154, x1 + 128}
      mark_A(x1) = 136
      U114_A(x1,x2) = 157
      s_A(x1) = 16
      length_A(x1) = 132
      U136_A(x1,x2,x3,x4) = 128
      cons_A(x1,x2) = 112
      take_A(x1,x2) = 144
      active_A(x1) = max{21, x1 - 7}
      zeros_A = 140
      |0|_A = 14
      U101_A(x1,x2,x3) = 143
      U102_A(x1,x2,x3) = 142
      U103_A(x1,x2,x3) = 126
      isNatIListKind_A(x1) = 152
      U104_A(x1,x2,x3) = 144
      U105_A(x1,x2) = 142
      U106_A(x1) = 154
      isNatIList_A(x1) = 118
      U11_A(x1,x2) = 44
      U12_A(x1,x2) = 28
      U111_A(x1,x2,x3) = 110
      U13_A(x1) = 154
      isNatList_A(x1) = max{120, x1 - 16}
      U121_A(x1,x2) = 128
      U122_A(x1) = 29
      nil_A = 142
      U131_A(x1,x2,x3,x4) = 14
      U132_A(x1,x2,x3,x4) = 144
      U133_A(x1,x2,x3,x4) = 158
      U134_A(x1,x2,x3,x4) = 142
      U135_A(x1,x2,x3,x4) = 144
      U21_A(x1,x2) = 121
      U22_A(x1,x2) = 293
      U23_A(x1) = 144
      U31_A(x1,x2) = 102
      U32_A(x1,x2) = 170
      U33_A(x1) = 154
      U41_A(x1,x2,x3) = 78
      U42_A(x1,x2,x3) = 62
      U43_A(x1,x2,x3) = 46
      U44_A(x1,x2,x3) = 30
      U45_A(x1,x2) = 14
      U46_A(x1) = 142
      U51_A(x1,x2) = 136
      U52_A(x1) = 120
      U61_A(x1,x2) = 142
      U62_A(x1) = 136
      U71_A(x1) = 128
      U81_A(x1) = 138
      U91_A(x1,x2,x3) = 143
      U92_A(x1,x2,x3) = 129
      U93_A(x1,x2,x3) = 115
      U94_A(x1,x2,x3) = 142
      U95_A(x1,x2) = 126
      U96_A(x1) = 14
    

The next rules are strictly ordered:

  p1, p2, p3, p4, p5, p6, p7, p8

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: 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:

  lexicographic combination of reduction pairs:
  
    1. max/plus interpretations on natural numbers:
    
      U102#_A(x1,x2,x3) = max{0, x2 - 1}
      mark_A(x1) = max{3, x1 + 2}
      active_A(x1) = max{1, x1}
    
    2. max/plus interpretations on natural numbers:
    
      U102#_A(x1,x2,x3) = 0
      mark_A(x1) = 0
      active_A(x1) = 0
    

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: 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)

and R consists of:

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

  lexicographic combination of reduction pairs:
  
    1. max/plus interpretations on natural numbers:
    
      U102#_A(x1,x2,x3) = x1 + 3
      mark_A(x1) = max{2, x1 + 1}
      active_A(x1) = x1 + 1
    
    2. max/plus interpretations on natural numbers:
    
      U102#_A(x1,x2,x3) = 0
      mark_A(x1) = 0
      active_A(x1) = x1
    

The next rules are strictly ordered:

  p1, p3

We remove them from the problem.

-- SCC decomposition.

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

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

and R consists of:

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

  lexicographic combination of reduction pairs:
  
    1. max/plus interpretations on natural numbers:
    
      U102#_A(x1,x2,x3) = x3 + 1
      mark_A(x1) = max{1, x1}
      active_A(x1) = x1 + 1
    
    2. max/plus interpretations on natural numbers:
    
      U102#_A(x1,x2,x3) = 0
      mark_A(x1) = 0
      active_A(x1) = x1
    

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#(X1,X2,mark(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#(X1,X2,mark(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:

  lexicographic combination of reduction pairs:
  
    1. max/plus interpretations on natural numbers:
    
      U102#_A(x1,x2,x3) = x2
      mark_A(x1) = max{2, x1 + 1}
      active_A(x1) = max{2, x1 + 1}
    
    2. max/plus interpretations on natural numbers:
    
      U102#_A(x1,x2,x3) = 0
      mark_A(x1) = 0
      active_A(x1) = 0
    

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#(X1,X2,mark(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,X2,mark(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:

  lexicographic combination of reduction pairs:
  
    1. max/plus interpretations on natural numbers:
    
      U102#_A(x1,x2,x3) = x3
      mark_A(x1) = max{2, x1 + 1}
    
    2. max/plus interpretations on natural numbers:
    
      U102#_A(x1,x2,x3) = 0
      mark_A(x1) = 0
    

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 monotone reduction pair:

  lexicographic combination of reduction pairs:
  
    1. max/plus interpretations on natural numbers:
    
      isNatKind#_A(x1) = x1
      mark_A(x1) = x1 + 1
      active_A(x1) = x1
    
    2. max/plus interpretations on natural numbers:
    
      isNatKind#_A(x1) = x1
      mark_A(x1) = x1 + 1
      active_A(x1) = x1 + 1
    

The next rules are strictly ordered:

  p1, p2

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:

  lexicographic combination of reduction pairs:
  
    1. max/plus interpretations on natural numbers:
    
      U103#_A(x1,x2,x3) = max{0, x2 - 1}
      mark_A(x1) = max{3, x1 + 2}
      active_A(x1) = max{1, x1}
    
    2. max/plus interpretations on natural numbers:
    
      U103#_A(x1,x2,x3) = 0
      mark_A(x1) = 0
      active_A(x1) = 0
    

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: 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)

and R consists of:

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

  lexicographic combination of reduction pairs:
  
    1. max/plus interpretations on natural numbers:
    
      U103#_A(x1,x2,x3) = x1 + 2
      mark_A(x1) = max{2, x1 + 1}
      active_A(x1) = max{2, x1 + 1}
    
    2. max/plus interpretations on natural numbers:
    
      U103#_A(x1,x2,x3) = 0
      mark_A(x1) = 0
      active_A(x1) = 0
    

The next rules are strictly ordered:

  p1, p3

We remove them from the problem.

-- SCC decomposition.

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

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

and R consists of:

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

  lexicographic combination of reduction pairs:
  
    1. max/plus interpretations on natural numbers:
    
      U103#_A(x1,x2,x3) = x2 + 2
      mark_A(x1) = max{2, x1 + 1}
      active_A(x1) = max{2, x1 + 1}
    
    2. max/plus interpretations on natural numbers:
    
      U103#_A(x1,x2,x3) = 0
      mark_A(x1) = 0
      active_A(x1) = 0
    

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: U103#(X1,X2,mark(X3)) -> U103#(X1,X2,X3)
p2: 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 estimated dependency graph contains the following SCCs:

  {p1, p2}


-- Reduction pair.

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

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

  lexicographic combination of reduction pairs:
  
    1. max/plus interpretations on natural numbers:
    
      U103#_A(x1,x2,x3) = x3 + 1
      mark_A(x1) = max{1, x1}
      active_A(x1) = max{2, x1 + 1}
    
    2. max/plus interpretations on natural numbers:
    
      U103#_A(x1,x2,x3) = 0
      mark_A(x1) = 0
      active_A(x1) = 0
    

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#(X1,X2,mark(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,X2,mark(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:

  lexicographic combination of reduction pairs:
  
    1. max/plus interpretations on natural numbers:
    
      U103#_A(x1,x2,x3) = x3
      mark_A(x1) = max{2, x1 + 1}
    
    2. max/plus interpretations on natural numbers:
    
      U103#_A(x1,x2,x3) = 0
      mark_A(x1) = 0
    

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 monotone reduction pair:

  lexicographic combination of reduction pairs:
  
    1. max/plus interpretations on natural numbers:
    
      isNatIListKind#_A(x1) = x1
      mark_A(x1) = x1 + 1
      active_A(x1) = x1
    
    2. max/plus interpretations on natural numbers:
    
      isNatIListKind#_A(x1) = x1
      mark_A(x1) = x1 + 1
      active_A(x1) = x1 + 1
    

The next rules are strictly ordered:

  p1, p2

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:

  lexicographic combination of reduction pairs:
  
    1. max/plus interpretations on natural numbers:
    
      U104#_A(x1,x2,x3) = x3
      mark_A(x1) = max{4, x1 + 3}
      active_A(x1) = max{2, x1 + 1}
    
    2. max/plus interpretations on natural numbers:
    
      U104#_A(x1,x2,x3) = 0
      mark_A(x1) = 0
      active_A(x1) = 0
    

The next rules are strictly ordered:

  p2, 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,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 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,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 set of usable rules consists of

  (no rules)

Take the reduction pair:

  lexicographic combination of reduction pairs:
  
    1. max/plus interpretations on natural numbers:
    
      U104#_A(x1,x2,x3) = max{2, x2}
      mark_A(x1) = max{2, x1 + 1}
      active_A(x1) = max{4, x1 + 3}
    
    2. max/plus interpretations on natural numbers:
    
      U104#_A(x1,x2,x3) = 0
      mark_A(x1) = 0
      active_A(x1) = 0
    

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,mark(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,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:

  lexicographic combination of reduction pairs:
  
    1. max/plus interpretations on natural numbers:
    
      U104#_A(x1,x2,x3) = x1 + 1
      mark_A(x1) = x1
      active_A(x1) = x1 + 1
    
    2. max/plus interpretations on natural numbers:
    
      U104#_A(x1,x2,x3) = 0
      mark_A(x1) = x1
      active_A(x1) = x1
    

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,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}


-- 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)

and R consists of:

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

  lexicographic combination of reduction pairs:
  
    1. max/plus interpretations on natural numbers:
    
      U104#_A(x1,x2,x3) = x2
      mark_A(x1) = max{2, x1 + 1}
    
    2. max/plus interpretations on natural numbers:
    
      U104#_A(x1,x2,x3) = 0
      mark_A(x1) = 0
    

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)

and R consists of:

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

  (no rules)

Take the reduction pair:

  lexicographic combination of reduction pairs:
  
    1. max/plus interpretations on natural numbers:
    
      U104#_A(x1,x2,x3) = x1
      mark_A(x1) = max{2, x1 + 1}
    
    2. max/plus interpretations on natural numbers:
    
      U104#_A(x1,x2,x3) = 0
      mark_A(x1) = 0
    

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:

  lexicographic combination of reduction pairs:
  
    1. max/plus interpretations on natural numbers:
    
      U105#_A(x1,x2) = x2 + 3
      mark_A(x1) = max{2, x1 + 1}
      active_A(x1) = max{1, x1}
    
    2. max/plus interpretations on natural numbers:
    
      U105#_A(x1,x2) = 0
      mark_A(x1) = 0
      active_A(x1) = 0
    

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:

  lexicographic combination of reduction pairs:
  
    1. max/plus interpretations on natural numbers:
    
      U105#_A(x1,x2) = x2 + 2
      mark_A(x1) = max{2, x1 + 1}
      active_A(x1) = max{2, x1 + 1}
    
    2. max/plus interpretations on natural numbers:
    
      U105#_A(x1,x2) = 0
      mark_A(x1) = 0
      active_A(x1) = 0
    

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:

  lexicographic combination of reduction pairs:
  
    1. max/plus interpretations on natural numbers:
    
      U105#_A(x1,x2) = x1
      mark_A(x1) = max{2, x1 + 1}
      active_A(x1) = max{2, x1 + 1}
    
    2. max/plus interpretations on natural numbers:
    
      U105#_A(x1,x2) = 0
      mark_A(x1) = 0
      active_A(x1) = 0
    

The next rules are strictly ordered:

  p1, p2

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 monotone reduction pair:

  lexicographic combination of reduction pairs:
  
    1. max/plus interpretations on natural numbers:
    
      isNat#_A(x1) = x1
      mark_A(x1) = x1 + 1
      active_A(x1) = x1
    
    2. max/plus interpretations on natural numbers:
    
      isNat#_A(x1) = x1
      mark_A(x1) = x1 + 1
      active_A(x1) = x1 + 1
    

The next rules are strictly ordered:

  p1, p2

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 monotone reduction pair:

  lexicographic combination of reduction pairs:
  
    1. max/plus interpretations on natural numbers:
    
      U106#_A(x1) = x1
      mark_A(x1) = x1 + 1
      active_A(x1) = x1
    
    2. max/plus interpretations on natural numbers:
    
      U106#_A(x1) = x1
      mark_A(x1) = x1 + 1
      active_A(x1) = x1 + 1
    

The next rules are strictly ordered:

  p1, p2

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 monotone reduction pair:

  lexicographic combination of reduction pairs:
  
    1. max/plus interpretations on natural numbers:
    
      isNatIList#_A(x1) = x1
      mark_A(x1) = x1 + 1
      active_A(x1) = x1
    
    2. max/plus interpretations on natural numbers:
    
      isNatIList#_A(x1) = x1
      mark_A(x1) = x1 + 1
      active_A(x1) = x1 + 1
    

The next rules are strictly ordered:

  p1, p2

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:

  lexicographic combination of reduction pairs:
  
    1. max/plus interpretations on natural numbers:
    
      U12#_A(x1,x2) = x2 + 2
      mark_A(x1) = max{3, x1 + 2}
      active_A(x1) = max{1, x1}
    
    2. max/plus interpretations on natural numbers:
    
      U12#_A(x1,x2) = 0
      mark_A(x1) = 0
      active_A(x1) = 0
    

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:

  lexicographic combination of reduction pairs:
  
    1. max/plus interpretations on natural numbers:
    
      U12#_A(x1,x2) = x2 + 2
      mark_A(x1) = max{2, x1 + 1}
      active_A(x1) = max{2, x1 + 1}
    
    2. max/plus interpretations on natural numbers:
    
      U12#_A(x1,x2) = 0
      mark_A(x1) = 0
      active_A(x1) = 0
    

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:

  lexicographic combination of reduction pairs:
  
    1. max/plus interpretations on natural numbers:
    
      U12#_A(x1,x2) = x1
      mark_A(x1) = max{2, x1 + 1}
      active_A(x1) = max{2, x1 + 1}
    
    2. max/plus interpretations on natural numbers:
    
      U12#_A(x1,x2) = 0
      mark_A(x1) = 0
      active_A(x1) = 0
    

The next rules are strictly ordered:

  p1, p2

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:

  lexicographic combination of reduction pairs:
  
    1. max/plus interpretations on natural numbers:
    
      U112#_A(x1,x2,x3) = x3
      mark_A(x1) = x1 + 1
      active_A(x1) = max{3, x1 + 2}
    
    2. max/plus interpretations on natural numbers:
    
      U112#_A(x1,x2,x3) = 0
      mark_A(x1) = x1
      active_A(x1) = 0
    

The next rules are strictly ordered:

  p2, 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,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 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,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 set of usable rules consists of

  (no rules)

Take the reduction pair:

  lexicographic combination of reduction pairs:
  
    1. max/plus interpretations on natural numbers:
    
      U112#_A(x1,x2,x3) = max{1, x2}
      mark_A(x1) = max{1, x1}
      active_A(x1) = max{3, x1 + 2}
    
    2. max/plus interpretations on natural numbers:
    
      U112#_A(x1,x2,x3) = 0
      mark_A(x1) = 0
      active_A(x1) = 0
    

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,mark(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,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:

  lexicographic combination of reduction pairs:
  
    1. max/plus interpretations on natural numbers:
    
      U112#_A(x1,x2,x3) = x2 + 2
      mark_A(x1) = max{2, x1 + 1}
      active_A(x1) = max{2, x1 + 1}
    
    2. max/plus interpretations on natural numbers:
    
      U112#_A(x1,x2,x3) = 0
      mark_A(x1) = 0
      active_A(x1) = 0
    

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


-- 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)

and R consists of:

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

  lexicographic combination of reduction pairs:
  
    1. max/plus interpretations on natural numbers:
    
      U112#_A(x1,x2,x3) = x1 + 1
      mark_A(x1) = max{1, x1}
      active_A(x1) = max{2, x1 + 1}
    
    2. max/plus interpretations on natural numbers:
    
      U112#_A(x1,x2,x3) = 0
      mark_A(x1) = 0
      active_A(x1) = 0
    

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)

and R consists of:

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

  (no rules)

Take the reduction pair:

  lexicographic combination of reduction pairs:
  
    1. max/plus interpretations on natural numbers:
    
      U112#_A(x1,x2,x3) = x1
      mark_A(x1) = max{2, x1 + 1}
    
    2. max/plus interpretations on natural numbers:
    
      U112#_A(x1,x2,x3) = 0
      mark_A(x1) = 0
    

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:

  lexicographic combination of reduction pairs:
  
    1. max/plus interpretations on natural numbers:
    
      U113#_A(x1,x2,x3) = x3
      mark_A(x1) = x1 + 1
      active_A(x1) = max{3, x1 + 2}
    
    2. max/plus interpretations on natural numbers:
    
      U113#_A(x1,x2,x3) = 0
      mark_A(x1) = x1
      active_A(x1) = 0
    

The next rules are strictly ordered:

  p2, 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,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 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,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 set of usable rules consists of

  (no rules)

Take the reduction pair:

  lexicographic combination of reduction pairs:
  
    1. max/plus interpretations on natural numbers:
    
      U113#_A(x1,x2,x3) = x2 + 2
      mark_A(x1) = max{2, x1 + 1}
      active_A(x1) = max{4, x1 + 3}
    
    2. max/plus interpretations on natural numbers:
    
      U113#_A(x1,x2,x3) = 0
      mark_A(x1) = 0
      active_A(x1) = 0
    

The next rules are strictly ordered:

  p2, 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#(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}


-- 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)

and R consists of:

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

  lexicographic combination of reduction pairs:
  
    1. max/plus interpretations on natural numbers:
    
      U113#_A(x1,x2,x3) = x1 + 1
      mark_A(x1) = max{1, x1}
      active_A(x1) = max{2, x1 + 1}
    
    2. max/plus interpretations on natural numbers:
    
      U113#_A(x1,x2,x3) = 0
      mark_A(x1) = 0
      active_A(x1) = 0
    

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)

and R consists of:

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

  (no rules)

Take the reduction pair:

  lexicographic combination of reduction pairs:
  
    1. max/plus interpretations on natural numbers:
    
      U113#_A(x1,x2,x3) = x1
      mark_A(x1) = max{2, x1 + 1}
    
    2. max/plus interpretations on natural numbers:
    
      U113#_A(x1,x2,x3) = 0
      mark_A(x1) = 0
    

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:

  lexicographic combination of reduction pairs:
  
    1. max/plus interpretations on natural numbers:
    
      U114#_A(x1,x2) = x2 + 3
      mark_A(x1) = max{2, x1 + 1}
      active_A(x1) = max{1, x1}
    
    2. max/plus interpretations on natural numbers:
    
      U114#_A(x1,x2) = 0
      mark_A(x1) = 0
      active_A(x1) = 0
    

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:

  lexicographic combination of reduction pairs:
  
    1. max/plus interpretations on natural numbers:
    
      U114#_A(x1,x2) = x2 + 2
      mark_A(x1) = max{2, x1 + 1}
      active_A(x1) = max{2, x1 + 1}
    
    2. max/plus interpretations on natural numbers:
    
      U114#_A(x1,x2) = 0
      mark_A(x1) = 0
      active_A(x1) = 0
    

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:

  lexicographic combination of reduction pairs:
  
    1. max/plus interpretations on natural numbers:
    
      U114#_A(x1,x2) = x1
      mark_A(x1) = max{2, x1 + 1}
      active_A(x1) = max{2, x1 + 1}
    
    2. max/plus interpretations on natural numbers:
    
      U114#_A(x1,x2) = 0
      mark_A(x1) = 0
      active_A(x1) = 0
    

The next rules are strictly ordered:

  p1, p2

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 monotone reduction pair:

  lexicographic combination of reduction pairs:
  
    1. max/plus interpretations on natural numbers:
    
      s#_A(x1) = x1
      mark_A(x1) = x1 + 1
      active_A(x1) = x1
    
    2. max/plus interpretations on natural numbers:
    
      s#_A(x1) = x1
      mark_A(x1) = x1 + 1
      active_A(x1) = x1 + 1
    

The next rules are strictly ordered:

  p1, p2

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 monotone reduction pair:

  lexicographic combination of reduction pairs:
  
    1. max/plus interpretations on natural numbers:
    
      length#_A(x1) = x1
      mark_A(x1) = x1 + 1
      active_A(x1) = x1
    
    2. max/plus interpretations on natural numbers:
    
      length#_A(x1) = x1
      mark_A(x1) = x1 + 1
      active_A(x1) = x1 + 1
    

The next rules are strictly ordered:

  p1, p2

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 monotone reduction pair:

  lexicographic combination of reduction pairs:
  
    1. max/plus interpretations on natural numbers:
    
      U13#_A(x1) = x1
      mark_A(x1) = x1 + 1
      active_A(x1) = x1
    
    2. max/plus interpretations on natural numbers:
    
      U13#_A(x1) = x1
      mark_A(x1) = x1 + 1
      active_A(x1) = x1 + 1
    

The next rules are strictly ordered:

  p1, p2

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 monotone reduction pair:

  lexicographic combination of reduction pairs:
  
    1. max/plus interpretations on natural numbers:
    
      isNatList#_A(x1) = x1
      mark_A(x1) = x1 + 1
      active_A(x1) = x1
    
    2. max/plus interpretations on natural numbers:
    
      isNatList#_A(x1) = x1
      mark_A(x1) = x1 + 1
      active_A(x1) = x1 + 1
    

The next rules are strictly ordered:

  p1, p2

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 monotone reduction pair:

  lexicographic combination of reduction pairs:
  
    1. max/plus interpretations on natural numbers:
    
      U122#_A(x1) = x1
      mark_A(x1) = x1 + 1
      active_A(x1) = x1
    
    2. max/plus interpretations on natural numbers:
    
      U122#_A(x1) = x1
      mark_A(x1) = x1 + 1
      active_A(x1) = x1 + 1
    

The next rules are strictly ordered:

  p1, p2

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:

  lexicographic combination of reduction pairs:
  
    1. max/plus interpretations on natural numbers:
    
      U132#_A(x1,x2,x3,x4) = max{0, x2 - 1}
      mark_A(x1) = max{3, x1 + 2}
      active_A(x1) = x1 + 1
    
    2. max/plus interpretations on natural numbers:
    
      U132#_A(x1,x2,x3,x4) = 0
      mark_A(x1) = 0
      active_A(x1) = x1
    

The next rules are strictly ordered:

  p8

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,X3,mark(X4)) -> U132#(X1,X2,X3,X4)
p7: 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 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,X2,mark(X3),X4) -> U132#(X1,X2,X3,X4)
p3: U132#(X1,X2,X3,mark(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:

  lexicographic combination of reduction pairs:
  
    1. max/plus interpretations on natural numbers:
    
      U132#_A(x1,x2,x3,x4) = x1 + 1
      mark_A(x1) = x1 + 1
      active_A(x1) = x1 + 1
    
    2. max/plus interpretations on natural numbers:
    
      U132#_A(x1,x2,x3,x4) = 0
      mark_A(x1) = x1
      active_A(x1) = x1
    

The next rules are strictly ordered:

  p1, p4

We remove them from the problem.

-- SCC decomposition.

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

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


-- Reduction pair.

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

p1: U132#(X1,X2,mark(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,X3,mark(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:

  lexicographic combination of reduction pairs:
  
    1. max/plus interpretations on natural numbers:
    
      U132#_A(x1,x2,x3,x4) = x4 + 3
      mark_A(x1) = x1 + 1
      active_A(x1) = max{2, x1 + 1}
    
    2. max/plus interpretations on natural numbers:
    
      U132#_A(x1,x2,x3,x4) = 0
      mark_A(x1) = x1
      active_A(x1) = 0
    

The next rules are strictly ordered:

  p2, p5

We remove them from the problem.

-- SCC decomposition.

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

p1: U132#(X1,X2,mark(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,X2,mark(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:

  lexicographic combination of reduction pairs:
  
    1. max/plus interpretations on natural numbers:
    
      U132#_A(x1,x2,x3,x4) = x3 + 1
      mark_A(x1) = x1
      active_A(x1) = x1 + 1
    
    2. max/plus interpretations on natural numbers:
    
      U132#_A(x1,x2,x3,x4) = 0
      mark_A(x1) = x1
      active_A(x1) = x1
    

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


-- Reduction pair.

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

p1: U132#(X1,X2,mark(X3),X4) -> U132#(X1,X2,X3,X4)
p2: 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 set of usable rules consists of

  (no rules)

Take the reduction pair:

  lexicographic combination of reduction pairs:
  
    1. max/plus interpretations on natural numbers:
    
      U132#_A(x1,x2,x3,x4) = max{2, x3}
      mark_A(x1) = max{4, x1 + 3}
      active_A(x1) = max{2, x1 + 1}
    
    2. max/plus interpretations on natural numbers:
    
      U132#_A(x1,x2,x3,x4) = 0
      mark_A(x1) = 0
      active_A(x1) = 0
    

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,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}


-- Reduction pair.

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

p1: 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 set of usable rules consists of

  (no rules)

Take the reduction pair:

  lexicographic combination of reduction pairs:
  
    1. max/plus interpretations on natural numbers:
    
      U132#_A(x1,x2,x3,x4) = x2
      active_A(x1) = max{2, x1 + 1}
    
    2. max/plus interpretations on natural numbers:
    
      U132#_A(x1,x2,x3,x4) = 0
      active_A(x1) = 0
    

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:

  lexicographic combination of reduction pairs:
  
    1. max/plus interpretations on natural numbers:
    
      U133#_A(x1,x2,x3,x4) = x2
      mark_A(x1) = x1 + 1
      active_A(x1) = x1 + 2
    
    2. max/plus interpretations on natural numbers:
    
      U133#_A(x1,x2,x3,x4) = 0
      mark_A(x1) = x1
      active_A(x1) = x1
    

The next rules are strictly ordered:

  p4, p8

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#(active(X1),X2,X3,X4) -> U133#(X1,X2,X3,X4)
p5: U133#(X1,X2,X3,mark(X4)) -> U133#(X1,X2,X3,X4)
p6: 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 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,mark(X3),X4) -> U133#(X1,X2,X3,X4)
p3: U133#(X1,X2,X3,mark(X4)) -> U133#(X1,X2,X3,X4)
p4: U133#(active(X1),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 set of usable rules consists of

  (no rules)

Take the reduction pair:

  lexicographic combination of reduction pairs:
  
    1. max/plus interpretations on natural numbers:
    
      U133#_A(x1,x2,x3,x4) = max{2, x3}
      mark_A(x1) = max{2, x1 + 1}
      active_A(x1) = x1 + 3
    
    2. max/plus interpretations on natural numbers:
    
      U133#_A(x1,x2,x3,x4) = 0
      mark_A(x1) = 0
      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: U133#(mark(X1),X2,X3,X4) -> U133#(X1,X2,X3,X4)
p2: U133#(X1,X2,mark(X3),X4) -> U133#(X1,X2,X3,X4)
p3: U133#(X1,X2,X3,mark(X4)) -> U133#(X1,X2,X3,X4)
p4: U133#(active(X1),X2,X3,X4) -> U133#(X1,X2,X3,X4)
p5: 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}


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

  lexicographic combination of reduction pairs:
  
    1. max/plus interpretations on natural numbers:
    
      U133#_A(x1,x2,x3,x4) = x3 + 2
      mark_A(x1) = max{2, x1 + 1}
      active_A(x1) = max{2, x1 + 1}
    
    2. max/plus interpretations on natural numbers:
    
      U133#_A(x1,x2,x3,x4) = 0
      mark_A(x1) = 0
      active_A(x1) = 0
    

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

  lexicographic combination of reduction pairs:
  
    1. max/plus interpretations on natural numbers:
    
      U133#_A(x1,x2,x3,x4) = x1 + 3
      mark_A(x1) = x1
      active_A(x1) = max{2, x1 + 1}
    
    2. max/plus interpretations on natural numbers:
    
      U133#_A(x1,x2,x3,x4) = 0
      mark_A(x1) = x1
      active_A(x1) = 0
    

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


-- 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,X3,mark(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:

  lexicographic combination of reduction pairs:
  
    1. max/plus interpretations on natural numbers:
    
      U133#_A(x1,x2,x3,x4) = max{1, x4}
      mark_A(x1) = x1 + 2
      active_A(x1) = x1 + 1
    
    2. max/plus interpretations on natural numbers:
    
      U133#_A(x1,x2,x3,x4) = 0
      mark_A(x1) = x1
      active_A(x1) = x1
    

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


-- 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)

and R consists of:

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

  lexicographic combination of reduction pairs:
  
    1. max/plus interpretations on natural numbers:
    
      U133#_A(x1,x2,x3,x4) = max{2, x1}
      mark_A(x1) = max{4, x1 + 3}
      active_A(x1) = max{2, x1 + 1}
    
    2. max/plus interpretations on natural numbers:
    
      U133#_A(x1,x2,x3,x4) = 0
      mark_A(x1) = 0
      active_A(x1) = 0
    

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)

and R consists of:

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

  lexicographic combination of reduction pairs:
  
    1. max/plus interpretations on natural numbers:
    
      U133#_A(x1,x2,x3,x4) = x4
      active_A(x1) = max{2, x1 + 1}
    
    2. max/plus interpretations on natural numbers:
    
      U133#_A(x1,x2,x3,x4) = 0
      active_A(x1) = 0
    

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:

  lexicographic combination of reduction pairs:
  
    1. max/plus interpretations on natural numbers:
    
      U134#_A(x1,x2,x3,x4) = max{0, x2 - 1}
      mark_A(x1) = max{3, x1 + 2}
      active_A(x1) = x1 + 1
    
    2. max/plus interpretations on natural numbers:
    
      U134#_A(x1,x2,x3,x4) = 0
      mark_A(x1) = 0
      active_A(x1) = x1
    

The next rules are strictly ordered:

  p8

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,X3,mark(X4)) -> U134#(X1,X2,X3,X4)
p7: 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 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,X2,mark(X3),X4) -> U134#(X1,X2,X3,X4)
p3: U134#(X1,X2,X3,mark(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:

  lexicographic combination of reduction pairs:
  
    1. max/plus interpretations on natural numbers:
    
      U134#_A(x1,x2,x3,x4) = x1 + 1
      mark_A(x1) = x1 + 1
      active_A(x1) = x1 + 1
    
    2. max/plus interpretations on natural numbers:
    
      U134#_A(x1,x2,x3,x4) = 0
      mark_A(x1) = x1
      active_A(x1) = x1
    

The next rules are strictly ordered:

  p1, p4

We remove them from the problem.

-- SCC decomposition.

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

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


-- Reduction pair.

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

p1: U134#(X1,X2,mark(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,X3,mark(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:

  lexicographic combination of reduction pairs:
  
    1. max/plus interpretations on natural numbers:
    
      U134#_A(x1,x2,x3,x4) = x4 + 2
      mark_A(x1) = x1 + 1
      active_A(x1) = max{1, x1}
    
    2. max/plus interpretations on natural numbers:
    
      U134#_A(x1,x2,x3,x4) = 0
      mark_A(x1) = x1
      active_A(x1) = 0
    

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: U134#(X1,X2,mark(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)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> 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,X2,mark(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)
p4: 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:

  lexicographic combination of reduction pairs:
  
    1. max/plus interpretations on natural numbers:
    
      U134#_A(x1,x2,x3,x4) = max{4, x3 + 2}
      mark_A(x1) = max{2, x1 + 1}
      active_A(x1) = max{4, x1 + 3}
    
    2. max/plus interpretations on natural numbers:
    
      U134#_A(x1,x2,x3,x4) = 0
      mark_A(x1) = 0
      active_A(x1) = 0
    

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


-- Reduction pair.

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

p1: U134#(X1,X2,mark(X3),X4) -> U134#(X1,X2,X3,X4)
p2: U134#(X1,X2,X3,active(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 set of usable rules consists of

  (no rules)

Take the reduction pair:

  lexicographic combination of reduction pairs:
  
    1. max/plus interpretations on natural numbers:
    
      U134#_A(x1,x2,x3,x4) = x3
      mark_A(x1) = x1 + 1
      active_A(x1) = 0
    
    2. max/plus interpretations on natural numbers:
    
      U134#_A(x1,x2,x3,x4) = 0
      mark_A(x1) = x1
      active_A(x1) = 0
    

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,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}


-- 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,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 set of usable rules consists of

  (no rules)

Take the reduction pair:

  lexicographic combination of reduction pairs:
  
    1. max/plus interpretations on natural numbers:
    
      U134#_A(x1,x2,x3,x4) = x2
      active_A(x1) = max{2, x1 + 1}
    
    2. max/plus interpretations on natural numbers:
    
      U134#_A(x1,x2,x3,x4) = 0
      active_A(x1) = 0
    

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,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}


-- 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)

and R consists of:

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

  lexicographic combination of reduction pairs:
  
    1. max/plus interpretations on natural numbers:
    
      U134#_A(x1,x2,x3,x4) = x4
      active_A(x1) = max{2, x1 + 1}
    
    2. max/plus interpretations on natural numbers:
    
      U134#_A(x1,x2,x3,x4) = 0
      active_A(x1) = 0
    

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:

  lexicographic combination of reduction pairs:
  
    1. max/plus interpretations on natural numbers:
    
      U135#_A(x1,x2,x3,x4) = max{0, x2 - 1}
      mark_A(x1) = max{3, x1 + 2}
      active_A(x1) = x1 + 1
    
    2. max/plus interpretations on natural numbers:
    
      U135#_A(x1,x2,x3,x4) = 0
      mark_A(x1) = 0
      active_A(x1) = x1
    

The next rules are strictly ordered:

  p8

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,X3,mark(X4)) -> U135#(X1,X2,X3,X4)
p7: 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 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,X2,mark(X3),X4) -> U135#(X1,X2,X3,X4)
p3: U135#(X1,X2,X3,mark(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:

  lexicographic combination of reduction pairs:
  
    1. max/plus interpretations on natural numbers:
    
      U135#_A(x1,x2,x3,x4) = x1 + 1
      mark_A(x1) = x1 + 1
      active_A(x1) = x1 + 1
    
    2. max/plus interpretations on natural numbers:
    
      U135#_A(x1,x2,x3,x4) = 0
      mark_A(x1) = x1
      active_A(x1) = x1
    

The next rules are strictly ordered:

  p1, p4

We remove them from the problem.

-- SCC decomposition.

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

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


-- Reduction pair.

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

p1: U135#(X1,X2,mark(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,X3,mark(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:

  lexicographic combination of reduction pairs:
  
    1. max/plus interpretations on natural numbers:
    
      U135#_A(x1,x2,x3,x4) = x4 + 3
      mark_A(x1) = x1 + 1
      active_A(x1) = max{2, x1 + 1}
    
    2. max/plus interpretations on natural numbers:
    
      U135#_A(x1,x2,x3,x4) = 0
      mark_A(x1) = x1
      active_A(x1) = 0
    

The next rules are strictly ordered:

  p2, p5

We remove them from the problem.

-- SCC decomposition.

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

p1: U135#(X1,X2,mark(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,X2,mark(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:

  lexicographic combination of reduction pairs:
  
    1. max/plus interpretations on natural numbers:
    
      U135#_A(x1,x2,x3,x4) = x3 + 1
      mark_A(x1) = x1
      active_A(x1) = x1 + 1
    
    2. max/plus interpretations on natural numbers:
    
      U135#_A(x1,x2,x3,x4) = 0
      mark_A(x1) = x1
      active_A(x1) = x1
    

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: U135#(X1,X2,mark(X3),X4) -> U135#(X1,X2,X3,X4)
p2: 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}


-- Reduction pair.

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

p1: U135#(X1,X2,mark(X3),X4) -> U135#(X1,X2,X3,X4)
p2: 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 set of usable rules consists of

  (no rules)

Take the reduction pair:

  lexicographic combination of reduction pairs:
  
    1. max/plus interpretations on natural numbers:
    
      U135#_A(x1,x2,x3,x4) = max{2, x3}
      mark_A(x1) = max{4, x1 + 3}
      active_A(x1) = max{2, x1 + 1}
    
    2. max/plus interpretations on natural numbers:
    
      U135#_A(x1,x2,x3,x4) = 0
      mark_A(x1) = 0
      active_A(x1) = 0
    

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,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}


-- Reduction pair.

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

p1: 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 set of usable rules consists of

  (no rules)

Take the reduction pair:

  lexicographic combination of reduction pairs:
  
    1. max/plus interpretations on natural numbers:
    
      U135#_A(x1,x2,x3,x4) = x2
      active_A(x1) = max{2, x1 + 1}
    
    2. max/plus interpretations on natural numbers:
    
      U135#_A(x1,x2,x3,x4) = 0
      active_A(x1) = 0
    

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:

  lexicographic combination of reduction pairs:
  
    1. max/plus interpretations on natural numbers:
    
      U136#_A(x1,x2,x3,x4) = x2
      mark_A(x1) = x1 + 1
      active_A(x1) = x1 + 2
    
    2. max/plus interpretations on natural numbers:
    
      U136#_A(x1,x2,x3,x4) = 0
      mark_A(x1) = x1
      active_A(x1) = x1
    

The next rules are strictly ordered:

  p4, p8

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#(active(X1),X2,X3,X4) -> U136#(X1,X2,X3,X4)
p5: U136#(X1,X2,X3,mark(X4)) -> U136#(X1,X2,X3,X4)
p6: 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 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,mark(X3),X4) -> U136#(X1,X2,X3,X4)
p3: U136#(X1,X2,X3,mark(X4)) -> U136#(X1,X2,X3,X4)
p4: U136#(active(X1),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 set of usable rules consists of

  (no rules)

Take the reduction pair:

  lexicographic combination of reduction pairs:
  
    1. max/plus interpretations on natural numbers:
    
      U136#_A(x1,x2,x3,x4) = max{2, x3}
      mark_A(x1) = max{2, x1 + 1}
      active_A(x1) = x1 + 3
    
    2. max/plus interpretations on natural numbers:
    
      U136#_A(x1,x2,x3,x4) = 0
      mark_A(x1) = 0
      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: U136#(mark(X1),X2,X3,X4) -> U136#(X1,X2,X3,X4)
p2: U136#(X1,X2,mark(X3),X4) -> U136#(X1,X2,X3,X4)
p3: U136#(X1,X2,X3,mark(X4)) -> U136#(X1,X2,X3,X4)
p4: U136#(active(X1),X2,X3,X4) -> U136#(X1,X2,X3,X4)
p5: 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}


-- 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#(active(X1),X2,X3,X4) -> U136#(X1,X2,X3,X4)
p4: U136#(X1,X2,X3,mark(X4)) -> U136#(X1,X2,X3,X4)
p5: 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:

  lexicographic combination of reduction pairs:
  
    1. max/plus interpretations on natural numbers:
    
      U136#_A(x1,x2,x3,x4) = x3 + 2
      mark_A(x1) = max{2, x1 + 1}
      active_A(x1) = max{2, x1 + 1}
    
    2. max/plus interpretations on natural numbers:
    
      U136#_A(x1,x2,x3,x4) = 0
      mark_A(x1) = 0
      active_A(x1) = 0
    

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: 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#(active(X1),X2,X3,X4) -> U136#(X1,X2,X3,X4)
p4: U136#(X1,X2,X3,mark(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#(mark(X1),X2,X3,X4) -> U136#(X1,X2,X3,X4)
p2: U136#(X1,X2,X3,mark(X4)) -> U136#(X1,X2,X3,X4)
p3: U136#(active(X1),X2,X3,X4) -> U136#(X1,X2,X3,X4)
p4: 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:

  lexicographic combination of reduction pairs:
  
    1. max/plus interpretations on natural numbers:
    
      U136#_A(x1,x2,x3,x4) = x1 + 3
      mark_A(x1) = x1
      active_A(x1) = max{2, x1 + 1}
    
    2. max/plus interpretations on natural numbers:
    
      U136#_A(x1,x2,x3,x4) = 0
      mark_A(x1) = x1
      active_A(x1) = 0
    

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: U136#(mark(X1),X2,X3,X4) -> U136#(X1,X2,X3,X4)
p2: U136#(X1,X2,X3,mark(X4)) -> U136#(X1,X2,X3,X4)
p3: 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}


-- 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,X3,mark(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:

  lexicographic combination of reduction pairs:
  
    1. max/plus interpretations on natural numbers:
    
      U136#_A(x1,x2,x3,x4) = max{1, x4}
      mark_A(x1) = x1 + 2
      active_A(x1) = x1 + 1
    
    2. max/plus interpretations on natural numbers:
    
      U136#_A(x1,x2,x3,x4) = 0
      mark_A(x1) = x1
      active_A(x1) = x1
    

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: U136#(mark(X1),X2,X3,X4) -> U136#(X1,X2,X3,X4)
p2: 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}


-- 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)

and R consists of:

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

  lexicographic combination of reduction pairs:
  
    1. max/plus interpretations on natural numbers:
    
      U136#_A(x1,x2,x3,x4) = max{2, x1}
      mark_A(x1) = max{4, x1 + 3}
      active_A(x1) = max{2, x1 + 1}
    
    2. max/plus interpretations on natural numbers:
    
      U136#_A(x1,x2,x3,x4) = 0
      mark_A(x1) = 0
      active_A(x1) = 0
    

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)

and R consists of:

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

  lexicographic combination of reduction pairs:
  
    1. max/plus interpretations on natural numbers:
    
      U136#_A(x1,x2,x3,x4) = x4
      active_A(x1) = max{2, x1 + 1}
    
    2. max/plus interpretations on natural numbers:
    
      U136#_A(x1,x2,x3,x4) = 0
      active_A(x1) = 0
    

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:

  lexicographic combination of reduction pairs:
  
    1. max/plus interpretations on natural numbers:
    
      cons#_A(x1,x2) = x2 + 2
      mark_A(x1) = max{3, x1 + 2}
      active_A(x1) = max{1, x1}
    
    2. max/plus interpretations on natural numbers:
    
      cons#_A(x1,x2) = 0
      mark_A(x1) = 0
      active_A(x1) = 0
    

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:

  lexicographic combination of reduction pairs:
  
    1. max/plus interpretations on natural numbers:
    
      cons#_A(x1,x2) = x2 + 2
      mark_A(x1) = max{2, x1 + 1}
      active_A(x1) = max{2, x1 + 1}
    
    2. max/plus interpretations on natural numbers:
    
      cons#_A(x1,x2) = 0
      mark_A(x1) = 0
      active_A(x1) = 0
    

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:

  lexicographic combination of reduction pairs:
  
    1. max/plus interpretations on natural numbers:
    
      cons#_A(x1,x2) = x1
      mark_A(x1) = max{2, x1 + 1}
      active_A(x1) = max{2, x1 + 1}
    
    2. max/plus interpretations on natural numbers:
    
      cons#_A(x1,x2) = 0
      mark_A(x1) = 0
      active_A(x1) = 0
    

The next rules are strictly ordered:

  p1, p2

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:

  lexicographic combination of reduction pairs:
  
    1. max/plus interpretations on natural numbers:
    
      take#_A(x1,x2) = x2 + 2
      mark_A(x1) = max{3, x1 + 2}
      active_A(x1) = max{1, x1}
    
    2. max/plus interpretations on natural numbers:
    
      take#_A(x1,x2) = 0
      mark_A(x1) = 0
      active_A(x1) = 0
    

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:

  lexicographic combination of reduction pairs:
  
    1. max/plus interpretations on natural numbers:
    
      take#_A(x1,x2) = x2 + 2
      mark_A(x1) = max{2, x1 + 1}
      active_A(x1) = max{2, x1 + 1}
    
    2. max/plus interpretations on natural numbers:
    
      take#_A(x1,x2) = 0
      mark_A(x1) = 0
      active_A(x1) = 0
    

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:

  lexicographic combination of reduction pairs:
  
    1. max/plus interpretations on natural numbers:
    
      take#_A(x1,x2) = x1
      mark_A(x1) = max{2, x1 + 1}
      active_A(x1) = max{2, x1 + 1}
    
    2. max/plus interpretations on natural numbers:
    
      take#_A(x1,x2) = 0
      mark_A(x1) = 0
      active_A(x1) = 0
    

The next rules are strictly ordered:

  p1, p2

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:

  lexicographic combination of reduction pairs:
  
    1. max/plus interpretations on natural numbers:
    
      U22#_A(x1,x2) = x2 + 2
      mark_A(x1) = max{3, x1 + 2}
      active_A(x1) = max{1, x1}
    
    2. max/plus interpretations on natural numbers:
    
      U22#_A(x1,x2) = 0
      mark_A(x1) = 0
      active_A(x1) = 0
    

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:

  lexicographic combination of reduction pairs:
  
    1. max/plus interpretations on natural numbers:
    
      U22#_A(x1,x2) = x2 + 2
      mark_A(x1) = max{2, x1 + 1}
      active_A(x1) = max{2, x1 + 1}
    
    2. max/plus interpretations on natural numbers:
    
      U22#_A(x1,x2) = 0
      mark_A(x1) = 0
      active_A(x1) = 0
    

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:

  lexicographic combination of reduction pairs:
  
    1. max/plus interpretations on natural numbers:
    
      U22#_A(x1,x2) = x1
      mark_A(x1) = max{2, x1 + 1}
      active_A(x1) = max{2, x1 + 1}
    
    2. max/plus interpretations on natural numbers:
    
      U22#_A(x1,x2) = 0
      mark_A(x1) = 0
      active_A(x1) = 0
    

The next rules are strictly ordered:

  p1, p2

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 monotone reduction pair:

  lexicographic combination of reduction pairs:
  
    1. max/plus interpretations on natural numbers:
    
      U23#_A(x1) = x1
      mark_A(x1) = x1 + 1
      active_A(x1) = x1
    
    2. max/plus interpretations on natural numbers:
    
      U23#_A(x1) = x1
      mark_A(x1) = x1 + 1
      active_A(x1) = x1 + 1
    

The next rules are strictly ordered:

  p1, p2

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:

  lexicographic combination of reduction pairs:
  
    1. max/plus interpretations on natural numbers:
    
      U32#_A(x1,x2) = x2 + 2
      mark_A(x1) = max{3, x1 + 2}
      active_A(x1) = max{1, x1}
    
    2. max/plus interpretations on natural numbers:
    
      U32#_A(x1,x2) = 0
      mark_A(x1) = 0
      active_A(x1) = 0
    

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:

  lexicographic combination of reduction pairs:
  
    1. max/plus interpretations on natural numbers:
    
      U32#_A(x1,x2) = x2 + 2
      mark_A(x1) = max{2, x1 + 1}
      active_A(x1) = max{2, x1 + 1}
    
    2. max/plus interpretations on natural numbers:
    
      U32#_A(x1,x2) = 0
      mark_A(x1) = 0
      active_A(x1) = 0
    

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:

  lexicographic combination of reduction pairs:
  
    1. max/plus interpretations on natural numbers:
    
      U32#_A(x1,x2) = x1
      mark_A(x1) = max{2, x1 + 1}
      active_A(x1) = max{2, x1 + 1}
    
    2. max/plus interpretations on natural numbers:
    
      U32#_A(x1,x2) = 0
      mark_A(x1) = 0
      active_A(x1) = 0
    

The next rules are strictly ordered:

  p1, p2

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 monotone reduction pair:

  lexicographic combination of reduction pairs:
  
    1. max/plus interpretations on natural numbers:
    
      U33#_A(x1) = x1
      mark_A(x1) = x1 + 1
      active_A(x1) = x1
    
    2. max/plus interpretations on natural numbers:
    
      U33#_A(x1) = x1
      mark_A(x1) = x1 + 1
      active_A(x1) = x1 + 1
    

The next rules are strictly ordered:

  p1, p2

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:

  lexicographic combination of reduction pairs:
  
    1. max/plus interpretations on natural numbers:
    
      U42#_A(x1,x2,x3) = x3
      mark_A(x1) = max{2, x1 + 1}
      active_A(x1) = x1 + 3
    
    2. max/plus interpretations on natural numbers:
    
      U42#_A(x1,x2,x3) = 0
      mark_A(x1) = 0
      active_A(x1) = x1
    

The next rules are strictly ordered:

  p2, 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,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 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,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 set of usable rules consists of

  (no rules)

Take the reduction pair:

  lexicographic combination of reduction pairs:
  
    1. max/plus interpretations on natural numbers:
    
      U42#_A(x1,x2,x3) = max{2, x2}
      mark_A(x1) = max{2, x1 + 1}
      active_A(x1) = max{4, x1 + 3}
    
    2. max/plus interpretations on natural numbers:
    
      U42#_A(x1,x2,x3) = 0
      mark_A(x1) = 0
      active_A(x1) = 0
    

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,mark(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,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:

  lexicographic combination of reduction pairs:
  
    1. max/plus interpretations on natural numbers:
    
      U42#_A(x1,x2,x3) = x1 + 1
      mark_A(x1) = x1
      active_A(x1) = x1 + 1
    
    2. max/plus interpretations on natural numbers:
    
      U42#_A(x1,x2,x3) = 0
      mark_A(x1) = x1
      active_A(x1) = x1
    

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,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}


-- 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)

and R consists of:

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

  lexicographic combination of reduction pairs:
  
    1. max/plus interpretations on natural numbers:
    
      U42#_A(x1,x2,x3) = x2
      mark_A(x1) = max{2, x1 + 1}
    
    2. max/plus interpretations on natural numbers:
    
      U42#_A(x1,x2,x3) = 0
      mark_A(x1) = 0
    

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)

and R consists of:

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

  (no rules)

Take the reduction pair:

  lexicographic combination of reduction pairs:
  
    1. max/plus interpretations on natural numbers:
    
      U42#_A(x1,x2,x3) = x1
      mark_A(x1) = max{2, x1 + 1}
    
    2. max/plus interpretations on natural numbers:
    
      U42#_A(x1,x2,x3) = 0
      mark_A(x1) = 0
    

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:

  lexicographic combination of reduction pairs:
  
    1. max/plus interpretations on natural numbers:
    
      U43#_A(x1,x2,x3) = x3
      mark_A(x1) = x1 + 1
      active_A(x1) = max{3, x1 + 2}
    
    2. max/plus interpretations on natural numbers:
    
      U43#_A(x1,x2,x3) = 0
      mark_A(x1) = x1
      active_A(x1) = 0
    

The next rules are strictly ordered:

  p2, 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,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 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,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 set of usable rules consists of

  (no rules)

Take the reduction pair:

  lexicographic combination of reduction pairs:
  
    1. max/plus interpretations on natural numbers:
    
      U43#_A(x1,x2,x3) = max{2, x2}
      mark_A(x1) = max{2, x1 + 1}
      active_A(x1) = max{4, x1 + 3}
    
    2. max/plus interpretations on natural numbers:
    
      U43#_A(x1,x2,x3) = 0
      mark_A(x1) = 0
      active_A(x1) = 0
    

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,mark(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,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:

  lexicographic combination of reduction pairs:
  
    1. max/plus interpretations on natural numbers:
    
      U43#_A(x1,x2,x3) = x2 + 2
      mark_A(x1) = max{2, x1 + 1}
      active_A(x1) = max{2, x1 + 1}
    
    2. max/plus interpretations on natural numbers:
    
      U43#_A(x1,x2,x3) = 0
      mark_A(x1) = 0
      active_A(x1) = 0
    

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: U43#(mark(X1),X2,X3) -> U43#(X1,X2,X3)
p2: 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}


-- 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)

and R consists of:

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

  lexicographic combination of reduction pairs:
  
    1. max/plus interpretations on natural numbers:
    
      U43#_A(x1,x2,x3) = x1 + 1
      mark_A(x1) = max{1, x1}
      active_A(x1) = max{2, x1 + 1}
    
    2. max/plus interpretations on natural numbers:
    
      U43#_A(x1,x2,x3) = 0
      mark_A(x1) = 0
      active_A(x1) = 0
    

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)

and R consists of:

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

  (no rules)

Take the reduction pair:

  lexicographic combination of reduction pairs:
  
    1. max/plus interpretations on natural numbers:
    
      U43#_A(x1,x2,x3) = x1
      mark_A(x1) = max{2, x1 + 1}
    
    2. max/plus interpretations on natural numbers:
    
      U43#_A(x1,x2,x3) = 0
      mark_A(x1) = 0
    

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:

  lexicographic combination of reduction pairs:
  
    1. max/plus interpretations on natural numbers:
    
      U44#_A(x1,x2,x3) = x3
      mark_A(x1) = max{2, x1 + 1}
      active_A(x1) = x1 + 3
    
    2. max/plus interpretations on natural numbers:
    
      U44#_A(x1,x2,x3) = 0
      mark_A(x1) = 0
      active_A(x1) = x1
    

The next rules are strictly ordered:

  p2, 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,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 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,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 set of usable rules consists of

  (no rules)

Take the reduction pair:

  lexicographic combination of reduction pairs:
  
    1. max/plus interpretations on natural numbers:
    
      U44#_A(x1,x2,x3) = x2
      mark_A(x1) = x1 + 1
      active_A(x1) = max{2, x1 + 1}
    
    2. max/plus interpretations on natural numbers:
    
      U44#_A(x1,x2,x3) = 0
      mark_A(x1) = x1
      active_A(x1) = 0
    

The next rules are strictly ordered:

  p2, 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#(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}


-- 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)

and R consists of:

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

  lexicographic combination of reduction pairs:
  
    1. max/plus interpretations on natural numbers:
    
      U44#_A(x1,x2,x3) = x1 + 1
      mark_A(x1) = max{1, x1}
      active_A(x1) = max{2, x1 + 1}
    
    2. max/plus interpretations on natural numbers:
    
      U44#_A(x1,x2,x3) = 0
      mark_A(x1) = 0
      active_A(x1) = 0
    

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)

and R consists of:

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

  (no rules)

Take the reduction pair:

  lexicographic combination of reduction pairs:
  
    1. max/plus interpretations on natural numbers:
    
      U44#_A(x1,x2,x3) = x1
      mark_A(x1) = max{2, x1 + 1}
    
    2. max/plus interpretations on natural numbers:
    
      U44#_A(x1,x2,x3) = 0
      mark_A(x1) = 0
    

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:

  lexicographic combination of reduction pairs:
  
    1. max/plus interpretations on natural numbers:
    
      U45#_A(x1,x2) = x2 + 2
      mark_A(x1) = max{3, x1 + 2}
      active_A(x1) = max{1, x1}
    
    2. max/plus interpretations on natural numbers:
    
      U45#_A(x1,x2) = 0
      mark_A(x1) = 0
      active_A(x1) = 0
    

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:

  lexicographic combination of reduction pairs:
  
    1. max/plus interpretations on natural numbers:
    
      U45#_A(x1,x2) = x2 + 2
      mark_A(x1) = max{2, x1 + 1}
      active_A(x1) = max{2, x1 + 1}
    
    2. max/plus interpretations on natural numbers:
    
      U45#_A(x1,x2) = 0
      mark_A(x1) = 0
      active_A(x1) = 0
    

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:

  lexicographic combination of reduction pairs:
  
    1. max/plus interpretations on natural numbers:
    
      U45#_A(x1,x2) = x1
      mark_A(x1) = max{2, x1 + 1}
      active_A(x1) = max{2, x1 + 1}
    
    2. max/plus interpretations on natural numbers:
    
      U45#_A(x1,x2) = 0
      mark_A(x1) = 0
      active_A(x1) = 0
    

The next rules are strictly ordered:

  p1, p2

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 monotone reduction pair:

  lexicographic combination of reduction pairs:
  
    1. max/plus interpretations on natural numbers:
    
      U46#_A(x1) = x1
      mark_A(x1) = x1 + 1
      active_A(x1) = x1
    
    2. max/plus interpretations on natural numbers:
    
      U46#_A(x1) = x1
      mark_A(x1) = x1 + 1
      active_A(x1) = x1 + 1
    

The next rules are strictly ordered:

  p1, p2

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 monotone reduction pair:

  lexicographic combination of reduction pairs:
  
    1. max/plus interpretations on natural numbers:
    
      U52#_A(x1) = x1
      mark_A(x1) = x1 + 1
      active_A(x1) = x1
    
    2. max/plus interpretations on natural numbers:
    
      U52#_A(x1) = x1
      mark_A(x1) = x1 + 1
      active_A(x1) = x1 + 1
    

The next rules are strictly ordered:

  p1, p2

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 monotone reduction pair:

  lexicographic combination of reduction pairs:
  
    1. max/plus interpretations on natural numbers:
    
      U62#_A(x1) = x1
      mark_A(x1) = x1 + 1
      active_A(x1) = x1
    
    2. max/plus interpretations on natural numbers:
    
      U62#_A(x1) = x1
      mark_A(x1) = x1 + 1
      active_A(x1) = x1 + 1
    

The next rules are strictly ordered:

  p1, p2

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:

  lexicographic combination of reduction pairs:
  
    1. max/plus interpretations on natural numbers:
    
      U92#_A(x1,x2,x3) = x3
      mark_A(x1) = max{2, x1 + 1}
      active_A(x1) = x1 + 3
    
    2. max/plus interpretations on natural numbers:
    
      U92#_A(x1,x2,x3) = 0
      mark_A(x1) = 0
      active_A(x1) = x1
    

The next rules are strictly ordered:

  p2, 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,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 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,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 set of usable rules consists of

  (no rules)

Take the reduction pair:

  lexicographic combination of reduction pairs:
  
    1. max/plus interpretations on natural numbers:
    
      U92#_A(x1,x2,x3) = max{1, x2}
      mark_A(x1) = max{1, x1}
      active_A(x1) = max{3, x1 + 2}
    
    2. max/plus interpretations on natural numbers:
    
      U92#_A(x1,x2,x3) = 0
      mark_A(x1) = 0
      active_A(x1) = 0
    

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,mark(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,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:

  lexicographic combination of reduction pairs:
  
    1. max/plus interpretations on natural numbers:
    
      U92#_A(x1,x2,x3) = x2 + 2
      mark_A(x1) = max{2, x1 + 1}
      active_A(x1) = max{2, x1 + 1}
    
    2. max/plus interpretations on natural numbers:
    
      U92#_A(x1,x2,x3) = 0
      mark_A(x1) = 0
      active_A(x1) = 0
    

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: U92#(mark(X1),X2,X3) -> U92#(X1,X2,X3)
p2: 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}


-- 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)

and R consists of:

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

  lexicographic combination of reduction pairs:
  
    1. max/plus interpretations on natural numbers:
    
      U92#_A(x1,x2,x3) = x1 + 1
      mark_A(x1) = max{1, x1}
      active_A(x1) = max{2, x1 + 1}
    
    2. max/plus interpretations on natural numbers:
    
      U92#_A(x1,x2,x3) = 0
      mark_A(x1) = 0
      active_A(x1) = 0
    

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)

and R consists of:

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

  (no rules)

Take the reduction pair:

  lexicographic combination of reduction pairs:
  
    1. max/plus interpretations on natural numbers:
    
      U92#_A(x1,x2,x3) = x1
      mark_A(x1) = max{2, x1 + 1}
    
    2. max/plus interpretations on natural numbers:
    
      U92#_A(x1,x2,x3) = 0
      mark_A(x1) = 0
    

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:

  lexicographic combination of reduction pairs:
  
    1. max/plus interpretations on natural numbers:
    
      U93#_A(x1,x2,x3) = x3
      mark_A(x1) = x1 + 1
      active_A(x1) = max{3, x1 + 2}
    
    2. max/plus interpretations on natural numbers:
    
      U93#_A(x1,x2,x3) = 0
      mark_A(x1) = x1
      active_A(x1) = 0
    

The next rules are strictly ordered:

  p2, 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,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 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,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 set of usable rules consists of

  (no rules)

Take the reduction pair:

  lexicographic combination of reduction pairs:
  
    1. max/plus interpretations on natural numbers:
    
      U93#_A(x1,x2,x3) = x2 + 2
      mark_A(x1) = max{2, x1 + 1}
      active_A(x1) = max{4, x1 + 3}
    
    2. max/plus interpretations on natural numbers:
    
      U93#_A(x1,x2,x3) = 0
      mark_A(x1) = 0
      active_A(x1) = 0
    

The next rules are strictly ordered:

  p2, 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#(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}


-- 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)

and R consists of:

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

  lexicographic combination of reduction pairs:
  
    1. max/plus interpretations on natural numbers:
    
      U93#_A(x1,x2,x3) = x1 + 1
      mark_A(x1) = max{1, x1}
      active_A(x1) = max{2, x1 + 1}
    
    2. max/plus interpretations on natural numbers:
    
      U93#_A(x1,x2,x3) = 0
      mark_A(x1) = 0
      active_A(x1) = 0
    

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)

and R consists of:

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

  (no rules)

Take the reduction pair:

  lexicographic combination of reduction pairs:
  
    1. max/plus interpretations on natural numbers:
    
      U93#_A(x1,x2,x3) = x1
      mark_A(x1) = max{2, x1 + 1}
    
    2. max/plus interpretations on natural numbers:
    
      U93#_A(x1,x2,x3) = 0
      mark_A(x1) = 0
    

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:

  lexicographic combination of reduction pairs:
  
    1. max/plus interpretations on natural numbers:
    
      U94#_A(x1,x2,x3) = x3
      mark_A(x1) = max{4, x1 + 3}
      active_A(x1) = max{2, x1 + 1}
    
    2. max/plus interpretations on natural numbers:
    
      U94#_A(x1,x2,x3) = 0
      mark_A(x1) = 0
      active_A(x1) = 0
    

The next rules are strictly ordered:

  p2, 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,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 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,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 set of usable rules consists of

  (no rules)

Take the reduction pair:

  lexicographic combination of reduction pairs:
  
    1. max/plus interpretations on natural numbers:
    
      U94#_A(x1,x2,x3) = x2
      mark_A(x1) = x1 + 1
      active_A(x1) = max{2, x1 + 1}
    
    2. max/plus interpretations on natural numbers:
    
      U94#_A(x1,x2,x3) = 0
      mark_A(x1) = x1
      active_A(x1) = 0
    

The next rules are strictly ordered:

  p2, 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#(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}


-- 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)

and R consists of:

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

  lexicographic combination of reduction pairs:
  
    1. max/plus interpretations on natural numbers:
    
      U94#_A(x1,x2,x3) = x1 + 1
      mark_A(x1) = max{1, x1}
      active_A(x1) = max{2, x1 + 1}
    
    2. max/plus interpretations on natural numbers:
    
      U94#_A(x1,x2,x3) = 0
      mark_A(x1) = 0
      active_A(x1) = 0
    

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)

and R consists of:

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

  (no rules)

Take the reduction pair:

  lexicographic combination of reduction pairs:
  
    1. max/plus interpretations on natural numbers:
    
      U94#_A(x1,x2,x3) = x1
      mark_A(x1) = max{2, x1 + 1}
    
    2. max/plus interpretations on natural numbers:
    
      U94#_A(x1,x2,x3) = 0
      mark_A(x1) = 0
    

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:

  lexicographic combination of reduction pairs:
  
    1. max/plus interpretations on natural numbers:
    
      U95#_A(x1,x2) = x2 + 3
      mark_A(x1) = max{2, x1 + 1}
      active_A(x1) = max{1, x1}
    
    2. max/plus interpretations on natural numbers:
    
      U95#_A(x1,x2) = 0
      mark_A(x1) = 0
      active_A(x1) = 0
    

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:

  lexicographic combination of reduction pairs:
  
    1. max/plus interpretations on natural numbers:
    
      U95#_A(x1,x2) = x2 + 2
      mark_A(x1) = max{2, x1 + 1}
      active_A(x1) = max{2, x1 + 1}
    
    2. max/plus interpretations on natural numbers:
    
      U95#_A(x1,x2) = 0
      mark_A(x1) = 0
      active_A(x1) = 0
    

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:

  lexicographic combination of reduction pairs:
  
    1. max/plus interpretations on natural numbers:
    
      U95#_A(x1,x2) = x1
      mark_A(x1) = max{2, x1 + 1}
      active_A(x1) = max{2, x1 + 1}
    
    2. max/plus interpretations on natural numbers:
    
      U95#_A(x1,x2) = 0
      mark_A(x1) = 0
      active_A(x1) = 0
    

The next rules are strictly ordered:

  p1, p2

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 monotone reduction pair:

  lexicographic combination of reduction pairs:
  
    1. max/plus interpretations on natural numbers:
    
      U96#_A(x1) = x1
      mark_A(x1) = x1 + 1
      active_A(x1) = x1
    
    2. max/plus interpretations on natural numbers:
    
      U96#_A(x1) = x1
      mark_A(x1) = x1 + 1
      active_A(x1) = x1 + 1
    

The next rules are strictly ordered:

  p1, p2

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:

  lexicographic combination of reduction pairs:
  
    1. max/plus interpretations on natural numbers:
    
      U11#_A(x1,x2) = x2 + 2
      mark_A(x1) = max{3, x1 + 2}
      active_A(x1) = max{1, x1}
    
    2. max/plus interpretations on natural numbers:
    
      U11#_A(x1,x2) = 0
      mark_A(x1) = 0
      active_A(x1) = 0
    

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:

  lexicographic combination of reduction pairs:
  
    1. max/plus interpretations on natural numbers:
    
      U11#_A(x1,x2) = x2 + 2
      mark_A(x1) = max{2, x1 + 1}
      active_A(x1) = max{2, x1 + 1}
    
    2. max/plus interpretations on natural numbers:
    
      U11#_A(x1,x2) = 0
      mark_A(x1) = 0
      active_A(x1) = 0
    

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:

  lexicographic combination of reduction pairs:
  
    1. max/plus interpretations on natural numbers:
    
      U11#_A(x1,x2) = x1
      mark_A(x1) = max{2, x1 + 1}
      active_A(x1) = max{2, x1 + 1}
    
    2. max/plus interpretations on natural numbers:
    
      U11#_A(x1,x2) = 0
      mark_A(x1) = 0
      active_A(x1) = 0
    

The next rules are strictly ordered:

  p1, p2

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:

  lexicographic combination of reduction pairs:
  
    1. max/plus interpretations on natural numbers:
    
      U21#_A(x1,x2) = x2 + 2
      mark_A(x1) = max{3, x1 + 2}
      active_A(x1) = max{1, x1}
    
    2. max/plus interpretations on natural numbers:
    
      U21#_A(x1,x2) = 0
      mark_A(x1) = 0
      active_A(x1) = 0
    

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:

  lexicographic combination of reduction pairs:
  
    1. max/plus interpretations on natural numbers:
    
      U21#_A(x1,x2) = x2 + 2
      mark_A(x1) = max{2, x1 + 1}
      active_A(x1) = max{2, x1 + 1}
    
    2. max/plus interpretations on natural numbers:
    
      U21#_A(x1,x2) = 0
      mark_A(x1) = 0
      active_A(x1) = 0
    

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:

  lexicographic combination of reduction pairs:
  
    1. max/plus interpretations on natural numbers:
    
      U21#_A(x1,x2) = x1
      mark_A(x1) = max{2, x1 + 1}
      active_A(x1) = max{2, x1 + 1}
    
    2. max/plus interpretations on natural numbers:
    
      U21#_A(x1,x2) = 0
      mark_A(x1) = 0
      active_A(x1) = 0
    

The next rules are strictly ordered:

  p1, p2

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:

  lexicographic combination of reduction pairs:
  
    1. max/plus interpretations on natural numbers:
    
      U31#_A(x1,x2) = x2 + 2
      mark_A(x1) = max{3, x1 + 2}
      active_A(x1) = max{1, x1}
    
    2. max/plus interpretations on natural numbers:
    
      U31#_A(x1,x2) = 0
      mark_A(x1) = 0
      active_A(x1) = 0
    

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:

  lexicographic combination of reduction pairs:
  
    1. max/plus interpretations on natural numbers:
    
      U31#_A(x1,x2) = x2 + 2
      mark_A(x1) = max{2, x1 + 1}
      active_A(x1) = max{2, x1 + 1}
    
    2. max/plus interpretations on natural numbers:
    
      U31#_A(x1,x2) = 0
      mark_A(x1) = 0
      active_A(x1) = 0
    

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:

  lexicographic combination of reduction pairs:
  
    1. max/plus interpretations on natural numbers:
    
      U31#_A(x1,x2) = x1
      mark_A(x1) = max{2, x1 + 1}
      active_A(x1) = max{2, x1 + 1}
    
    2. max/plus interpretations on natural numbers:
    
      U31#_A(x1,x2) = 0
      mark_A(x1) = 0
      active_A(x1) = 0
    

The next rules are strictly ordered:

  p1, p2

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:

  lexicographic combination of reduction pairs:
  
    1. max/plus interpretations on natural numbers:
    
      U41#_A(x1,x2,x3) = x3
      mark_A(x1) = x1 + 1
      active_A(x1) = max{3, x1 + 2}
    
    2. max/plus interpretations on natural numbers:
    
      U41#_A(x1,x2,x3) = 0
      mark_A(x1) = x1
      active_A(x1) = 0
    

The next rules are strictly ordered:

  p2, 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,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 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,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 set of usable rules consists of

  (no rules)

Take the reduction pair:

  lexicographic combination of reduction pairs:
  
    1. max/plus interpretations on natural numbers:
    
      U41#_A(x1,x2,x3) = max{2, x2}
      mark_A(x1) = max{2, x1 + 1}
      active_A(x1) = max{4, x1 + 3}
    
    2. max/plus interpretations on natural numbers:
    
      U41#_A(x1,x2,x3) = 0
      mark_A(x1) = 0
      active_A(x1) = 0
    

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,mark(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,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:

  lexicographic combination of reduction pairs:
  
    1. max/plus interpretations on natural numbers:
    
      U41#_A(x1,x2,x3) = x1 + 1
      mark_A(x1) = max{1, x1}
      active_A(x1) = x1 + 1
    
    2. max/plus interpretations on natural numbers:
    
      U41#_A(x1,x2,x3) = 0
      mark_A(x1) = 0
      active_A(x1) = x1
    

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,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}


-- 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)

and R consists of:

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

  lexicographic combination of reduction pairs:
  
    1. max/plus interpretations on natural numbers:
    
      U41#_A(x1,x2,x3) = x2
      mark_A(x1) = max{2, x1 + 1}
    
    2. max/plus interpretations on natural numbers:
    
      U41#_A(x1,x2,x3) = 0
      mark_A(x1) = 0
    

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)

and R consists of:

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

  (no rules)

Take the reduction pair:

  lexicographic combination of reduction pairs:
  
    1. max/plus interpretations on natural numbers:
    
      U41#_A(x1,x2,x3) = x1
      mark_A(x1) = max{2, x1 + 1}
    
    2. max/plus interpretations on natural numbers:
    
      U41#_A(x1,x2,x3) = 0
      mark_A(x1) = 0
    

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:

  lexicographic combination of reduction pairs:
  
    1. max/plus interpretations on natural numbers:
    
      U51#_A(x1,x2) = x2 + 2
      mark_A(x1) = max{3, x1 + 2}
      active_A(x1) = max{1, x1}
    
    2. max/plus interpretations on natural numbers:
    
      U51#_A(x1,x2) = 0
      mark_A(x1) = 0
      active_A(x1) = 0
    

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:

  lexicographic combination of reduction pairs:
  
    1. max/plus interpretations on natural numbers:
    
      U51#_A(x1,x2) = x2 + 2
      mark_A(x1) = max{2, x1 + 1}
      active_A(x1) = max{2, x1 + 1}
    
    2. max/plus interpretations on natural numbers:
    
      U51#_A(x1,x2) = 0
      mark_A(x1) = 0
      active_A(x1) = 0
    

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:

  lexicographic combination of reduction pairs:
  
    1. max/plus interpretations on natural numbers:
    
      U51#_A(x1,x2) = x1
      mark_A(x1) = max{2, x1 + 1}
      active_A(x1) = max{2, x1 + 1}
    
    2. max/plus interpretations on natural numbers:
    
      U51#_A(x1,x2) = 0
      mark_A(x1) = 0
      active_A(x1) = 0
    

The next rules are strictly ordered:

  p1, p2

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:

  lexicographic combination of reduction pairs:
  
    1. max/plus interpretations on natural numbers:
    
      U61#_A(x1,x2) = x2 + 2
      mark_A(x1) = max{3, x1 + 2}
      active_A(x1) = max{1, x1}
    
    2. max/plus interpretations on natural numbers:
    
      U61#_A(x1,x2) = 0
      mark_A(x1) = 0
      active_A(x1) = 0
    

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:

  lexicographic combination of reduction pairs:
  
    1. max/plus interpretations on natural numbers:
    
      U61#_A(x1,x2) = x2 + 2
      mark_A(x1) = max{2, x1 + 1}
      active_A(x1) = max{2, x1 + 1}
    
    2. max/plus interpretations on natural numbers:
    
      U61#_A(x1,x2) = 0
      mark_A(x1) = 0
      active_A(x1) = 0
    

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:

  lexicographic combination of reduction pairs:
  
    1. max/plus interpretations on natural numbers:
    
      U61#_A(x1,x2) = x1
      mark_A(x1) = max{2, x1 + 1}
      active_A(x1) = max{2, x1 + 1}
    
    2. max/plus interpretations on natural numbers:
    
      U61#_A(x1,x2) = 0
      mark_A(x1) = 0
      active_A(x1) = 0
    

The next rules are strictly ordered:

  p1, p2

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 monotone reduction pair:

  lexicographic combination of reduction pairs:
  
    1. max/plus interpretations on natural numbers:
    
      U71#_A(x1) = x1
      mark_A(x1) = x1 + 1
      active_A(x1) = x1
    
    2. max/plus interpretations on natural numbers:
    
      U71#_A(x1) = x1
      mark_A(x1) = x1 + 1
      active_A(x1) = x1 + 1
    

The next rules are strictly ordered:

  p1, p2

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 monotone reduction pair:

  lexicographic combination of reduction pairs:
  
    1. max/plus interpretations on natural numbers:
    
      U81#_A(x1) = x1
      mark_A(x1) = x1 + 1
      active_A(x1) = x1
    
    2. max/plus interpretations on natural numbers:
    
      U81#_A(x1) = x1
      mark_A(x1) = x1 + 1
      active_A(x1) = x1 + 1
    

The next rules are strictly ordered:

  p1, p2

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:

  lexicographic combination of reduction pairs:
  
    1. max/plus interpretations on natural numbers:
    
      U91#_A(x1,x2,x3) = x3
      mark_A(x1) = max{4, x1 + 3}
      active_A(x1) = max{2, x1 + 1}
    
    2. max/plus interpretations on natural numbers:
    
      U91#_A(x1,x2,x3) = 0
      mark_A(x1) = 0
      active_A(x1) = 0
    

The next rules are strictly ordered:

  p2, 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,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 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,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 set of usable rules consists of

  (no rules)

Take the reduction pair:

  lexicographic combination of reduction pairs:
  
    1. max/plus interpretations on natural numbers:
    
      U91#_A(x1,x2,x3) = max{1, x2}
      mark_A(x1) = max{1, x1}
      active_A(x1) = max{3, x1 + 2}
    
    2. max/plus interpretations on natural numbers:
    
      U91#_A(x1,x2,x3) = 0
      mark_A(x1) = 0
      active_A(x1) = 0
    

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,mark(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,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:

  lexicographic combination of reduction pairs:
  
    1. max/plus interpretations on natural numbers:
    
      U91#_A(x1,x2,x3) = x1 + 1
      mark_A(x1) = max{1, x1}
      active_A(x1) = x1 + 1
    
    2. max/plus interpretations on natural numbers:
    
      U91#_A(x1,x2,x3) = 0
      mark_A(x1) = 0
      active_A(x1) = x1
    

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,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}


-- 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)

and R consists of:

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

  lexicographic combination of reduction pairs:
  
    1. max/plus interpretations on natural numbers:
    
      U91#_A(x1,x2,x3) = x2
      mark_A(x1) = max{2, x1 + 1}
    
    2. max/plus interpretations on natural numbers:
    
      U91#_A(x1,x2,x3) = 0
      mark_A(x1) = 0
    

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)

and R consists of:

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

  (no rules)

Take the reduction pair:

  lexicographic combination of reduction pairs:
  
    1. max/plus interpretations on natural numbers:
    
      U91#_A(x1,x2,x3) = x1
      mark_A(x1) = max{2, x1 + 1}
    
    2. max/plus interpretations on natural numbers:
    
      U91#_A(x1,x2,x3) = 0
      mark_A(x1) = 0
    

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:

  lexicographic combination of reduction pairs:
  
    1. max/plus interpretations on natural numbers:
    
      U101#_A(x1,x2,x3) = max{0, x2 - 1}
      mark_A(x1) = max{3, x1 + 2}
      active_A(x1) = max{1, x1}
    
    2. max/plus interpretations on natural numbers:
    
      U101#_A(x1,x2,x3) = 0
      mark_A(x1) = 0
      active_A(x1) = 0
    

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: 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)

and R consists of:

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

  lexicographic combination of reduction pairs:
  
    1. max/plus interpretations on natural numbers:
    
      U101#_A(x1,x2,x3) = x1 + 3
      mark_A(x1) = max{2, x1 + 1}
      active_A(x1) = x1 + 1
    
    2. max/plus interpretations on natural numbers:
    
      U101#_A(x1,x2,x3) = 0
      mark_A(x1) = 0
      active_A(x1) = x1
    

The next rules are strictly ordered:

  p1, p3

We remove them from the problem.

-- SCC decomposition.

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

p1: U101#(X1,X2,mark(X3)) -> U101#(X1,X2,X3)
p2: U101#(X1,active(X2),X3) -> U101#(X1,X2,X3)
p3: 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 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#(X1,X2,mark(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)

and R consists of:

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

  lexicographic combination of reduction pairs:
  
    1. max/plus interpretations on natural numbers:
    
      U101#_A(x1,x2,x3) = x3 + 1
      mark_A(x1) = max{1, x1}
      active_A(x1) = x1 + 1
    
    2. max/plus interpretations on natural numbers:
    
      U101#_A(x1,x2,x3) = 0
      mark_A(x1) = 0
      active_A(x1) = x1
    

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#(X1,X2,mark(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#(X1,X2,mark(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:

  lexicographic combination of reduction pairs:
  
    1. max/plus interpretations on natural numbers:
    
      U101#_A(x1,x2,x3) = x2
      mark_A(x1) = max{2, x1 + 1}
      active_A(x1) = max{2, x1 + 1}
    
    2. max/plus interpretations on natural numbers:
    
      U101#_A(x1,x2,x3) = 0
      mark_A(x1) = 0
      active_A(x1) = 0
    

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#(X1,X2,mark(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,X2,mark(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:

  lexicographic combination of reduction pairs:
  
    1. max/plus interpretations on natural numbers:
    
      U101#_A(x1,x2,x3) = x3
      mark_A(x1) = max{2, x1 + 1}
    
    2. max/plus interpretations on natural numbers:
    
      U101#_A(x1,x2,x3) = 0
      mark_A(x1) = 0
    

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:

  lexicographic combination of reduction pairs:
  
    1. max/plus interpretations on natural numbers:
    
      U111#_A(x1,x2,x3) = x3
      mark_A(x1) = max{4, x1 + 3}
      active_A(x1) = max{2, x1 + 1}
    
    2. max/plus interpretations on natural numbers:
    
      U111#_A(x1,x2,x3) = 0
      mark_A(x1) = 0
      active_A(x1) = 0
    

The next rules are strictly ordered:

  p2, 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,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 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,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 set of usable rules consists of

  (no rules)

Take the reduction pair:

  lexicographic combination of reduction pairs:
  
    1. max/plus interpretations on natural numbers:
    
      U111#_A(x1,x2,x3) = max{1, x2}
      mark_A(x1) = max{1, x1}
      active_A(x1) = max{3, x1 + 2}
    
    2. max/plus interpretations on natural numbers:
    
      U111#_A(x1,x2,x3) = 0
      mark_A(x1) = 0
      active_A(x1) = 0
    

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,mark(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,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:

  lexicographic combination of reduction pairs:
  
    1. max/plus interpretations on natural numbers:
    
      U111#_A(x1,x2,x3) = x1 + 1
      mark_A(x1) = x1
      active_A(x1) = x1 + 1
    
    2. max/plus interpretations on natural numbers:
    
      U111#_A(x1,x2,x3) = 0
      mark_A(x1) = x1
      active_A(x1) = x1
    

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,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}


-- 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)

and R consists of:

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

  lexicographic combination of reduction pairs:
  
    1. max/plus interpretations on natural numbers:
    
      U111#_A(x1,x2,x3) = x2
      mark_A(x1) = max{2, x1 + 1}
    
    2. max/plus interpretations on natural numbers:
    
      U111#_A(x1,x2,x3) = 0
      mark_A(x1) = 0
    

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)

and R consists of:

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

  (no rules)

Take the reduction pair:

  lexicographic combination of reduction pairs:
  
    1. max/plus interpretations on natural numbers:
    
      U111#_A(x1,x2,x3) = x1
      mark_A(x1) = max{2, x1 + 1}
    
    2. max/plus interpretations on natural numbers:
    
      U111#_A(x1,x2,x3) = 0
      mark_A(x1) = 0
    

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:

  lexicographic combination of reduction pairs:
  
    1. max/plus interpretations on natural numbers:
    
      U121#_A(x1,x2) = x2 + 2
      mark_A(x1) = max{3, x1 + 2}
      active_A(x1) = max{1, x1}
    
    2. max/plus interpretations on natural numbers:
    
      U121#_A(x1,x2) = 0
      mark_A(x1) = 0
      active_A(x1) = 0
    

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:

  lexicographic combination of reduction pairs:
  
    1. max/plus interpretations on natural numbers:
    
      U121#_A(x1,x2) = x2 + 2
      mark_A(x1) = max{2, x1 + 1}
      active_A(x1) = max{2, x1 + 1}
    
    2. max/plus interpretations on natural numbers:
    
      U121#_A(x1,x2) = 0
      mark_A(x1) = 0
      active_A(x1) = 0
    

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:

  lexicographic combination of reduction pairs:
  
    1. max/plus interpretations on natural numbers:
    
      U121#_A(x1,x2) = x1
      mark_A(x1) = max{2, x1 + 1}
      active_A(x1) = max{2, x1 + 1}
    
    2. max/plus interpretations on natural numbers:
    
      U121#_A(x1,x2) = 0
      mark_A(x1) = 0
      active_A(x1) = 0
    

The next rules are strictly ordered:

  p1, p2

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:

  lexicographic combination of reduction pairs:
  
    1. max/plus interpretations on natural numbers:
    
      U131#_A(x1,x2,x3,x4) = max{0, x2 - 1}
      mark_A(x1) = max{3, x1 + 2}
      active_A(x1) = x1 + 1
    
    2. max/plus interpretations on natural numbers:
    
      U131#_A(x1,x2,x3,x4) = 0
      mark_A(x1) = 0
      active_A(x1) = x1
    

The next rules are strictly ordered:

  p8

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,X3,mark(X4)) -> U131#(X1,X2,X3,X4)
p7: 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 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,X2,mark(X3),X4) -> U131#(X1,X2,X3,X4)
p3: U131#(X1,X2,X3,mark(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:

  lexicographic combination of reduction pairs:
  
    1. max/plus interpretations on natural numbers:
    
      U131#_A(x1,x2,x3,x4) = x1 + 1
      mark_A(x1) = x1 + 1
      active_A(x1) = x1 + 1
    
    2. max/plus interpretations on natural numbers:
    
      U131#_A(x1,x2,x3,x4) = 0
      mark_A(x1) = x1
      active_A(x1) = x1
    

The next rules are strictly ordered:

  p1, p4

We remove them from the problem.

-- SCC decomposition.

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

p1: U131#(X1,X2,mark(X3),X4) -> U131#(X1,X2,X3,X4)
p2: U131#(X1,X2,X3,mark(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)
p5: 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}


-- Reduction pair.

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

p1: U131#(X1,X2,mark(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,X3,mark(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:

  lexicographic combination of reduction pairs:
  
    1. max/plus interpretations on natural numbers:
    
      U131#_A(x1,x2,x3,x4) = x4 + 2
      mark_A(x1) = x1 + 1
      active_A(x1) = max{1, x1}
    
    2. max/plus interpretations on natural numbers:
    
      U131#_A(x1,x2,x3,x4) = 0
      mark_A(x1) = x1
      active_A(x1) = 0
    

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: U131#(X1,X2,mark(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)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> 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,X2,mark(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)
p4: 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:

  lexicographic combination of reduction pairs:
  
    1. max/plus interpretations on natural numbers:
    
      U131#_A(x1,x2,x3,x4) = max{4, x3 + 2}
      mark_A(x1) = max{2, x1 + 1}
      active_A(x1) = max{4, x1 + 3}
    
    2. max/plus interpretations on natural numbers:
    
      U131#_A(x1,x2,x3,x4) = 0
      mark_A(x1) = 0
      active_A(x1) = 0
    

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: U131#(X1,X2,mark(X3),X4) -> U131#(X1,X2,X3,X4)
p2: U131#(X1,active(X2),X3,X4) -> U131#(X1,X2,X3,X4)
p3: 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}


-- Reduction pair.

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

p1: U131#(X1,X2,mark(X3),X4) -> U131#(X1,X2,X3,X4)
p2: U131#(X1,X2,X3,active(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 set of usable rules consists of

  (no rules)

Take the reduction pair:

  lexicographic combination of reduction pairs:
  
    1. max/plus interpretations on natural numbers:
    
      U131#_A(x1,x2,x3,x4) = x2
      mark_A(x1) = max{2, x1 + 1}
      active_A(x1) = max{2, x1 + 1}
    
    2. max/plus interpretations on natural numbers:
    
      U131#_A(x1,x2,x3,x4) = 0
      mark_A(x1) = 0
      active_A(x1) = 0
    

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: U131#(X1,X2,mark(X3),X4) -> U131#(X1,X2,X3,X4)
p2: 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}


-- Reduction pair.

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

p1: U131#(X1,X2,mark(X3),X4) -> U131#(X1,X2,X3,X4)
p2: 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:

  lexicographic combination of reduction pairs:
  
    1. max/plus interpretations on natural numbers:
    
      U131#_A(x1,x2,x3,x4) = max{2, x3}
      mark_A(x1) = max{4, x1 + 3}
      active_A(x1) = max{2, x1 + 1}
    
    2. max/plus interpretations on natural numbers:
    
      U131#_A(x1,x2,x3,x4) = 0
      mark_A(x1) = 0
      active_A(x1) = 0
    

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)

and R consists of:

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

  lexicographic combination of reduction pairs:
  
    1. max/plus interpretations on natural numbers:
    
      U131#_A(x1,x2,x3,x4) = x4
      active_A(x1) = max{2, x1 + 1}
    
    2. max/plus interpretations on natural numbers:
    
      U131#_A(x1,x2,x3,x4) = 0
      active_A(x1) = 0
    

The next rules are strictly ordered:

  p1

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