YES

We show the termination of the TRS R:

  a__zeros() -> cons(|0|(),zeros())
  a__U101(tt(),V1,V2) -> a__U102(a__isNatKind(V1),V1,V2)
  a__U102(tt(),V1,V2) -> a__U103(a__isNatIListKind(V2),V1,V2)
  a__U103(tt(),V1,V2) -> a__U104(a__isNatIListKind(V2),V1,V2)
  a__U104(tt(),V1,V2) -> a__U105(a__isNat(V1),V2)
  a__U105(tt(),V2) -> a__U106(a__isNatIList(V2))
  a__U106(tt()) -> tt()
  a__U11(tt(),V1) -> a__U12(a__isNatIListKind(V1),V1)
  a__U111(tt(),L,N) -> a__U112(a__isNatIListKind(L),L,N)
  a__U112(tt(),L,N) -> a__U113(a__isNat(N),L,N)
  a__U113(tt(),L,N) -> a__U114(a__isNatKind(N),L)
  a__U114(tt(),L) -> s(a__length(mark(L)))
  a__U12(tt(),V1) -> a__U13(a__isNatList(V1))
  a__U121(tt(),IL) -> a__U122(a__isNatIListKind(IL))
  a__U122(tt()) -> nil()
  a__U13(tt()) -> tt()
  a__U131(tt(),IL,M,N) -> a__U132(a__isNatIListKind(IL),IL,M,N)
  a__U132(tt(),IL,M,N) -> a__U133(a__isNat(M),IL,M,N)
  a__U133(tt(),IL,M,N) -> a__U134(a__isNatKind(M),IL,M,N)
  a__U134(tt(),IL,M,N) -> a__U135(a__isNat(N),IL,M,N)
  a__U135(tt(),IL,M,N) -> a__U136(a__isNatKind(N),IL,M,N)
  a__U136(tt(),IL,M,N) -> cons(mark(N),take(M,IL))
  a__U21(tt(),V1) -> a__U22(a__isNatKind(V1),V1)
  a__U22(tt(),V1) -> a__U23(a__isNat(V1))
  a__U23(tt()) -> tt()
  a__U31(tt(),V) -> a__U32(a__isNatIListKind(V),V)
  a__U32(tt(),V) -> a__U33(a__isNatList(V))
  a__U33(tt()) -> tt()
  a__U41(tt(),V1,V2) -> a__U42(a__isNatKind(V1),V1,V2)
  a__U42(tt(),V1,V2) -> a__U43(a__isNatIListKind(V2),V1,V2)
  a__U43(tt(),V1,V2) -> a__U44(a__isNatIListKind(V2),V1,V2)
  a__U44(tt(),V1,V2) -> a__U45(a__isNat(V1),V2)
  a__U45(tt(),V2) -> a__U46(a__isNatIList(V2))
  a__U46(tt()) -> tt()
  a__U51(tt(),V2) -> a__U52(a__isNatIListKind(V2))
  a__U52(tt()) -> tt()
  a__U61(tt(),V2) -> a__U62(a__isNatIListKind(V2))
  a__U62(tt()) -> tt()
  a__U71(tt()) -> tt()
  a__U81(tt()) -> tt()
  a__U91(tt(),V1,V2) -> a__U92(a__isNatKind(V1),V1,V2)
  a__U92(tt(),V1,V2) -> a__U93(a__isNatIListKind(V2),V1,V2)
  a__U93(tt(),V1,V2) -> a__U94(a__isNatIListKind(V2),V1,V2)
  a__U94(tt(),V1,V2) -> a__U95(a__isNat(V1),V2)
  a__U95(tt(),V2) -> a__U96(a__isNatList(V2))
  a__U96(tt()) -> tt()
  a__isNat(|0|()) -> tt()
  a__isNat(length(V1)) -> a__U11(a__isNatIListKind(V1),V1)
  a__isNat(s(V1)) -> a__U21(a__isNatKind(V1),V1)
  a__isNatIList(V) -> a__U31(a__isNatIListKind(V),V)
  a__isNatIList(zeros()) -> tt()
  a__isNatIList(cons(V1,V2)) -> a__U41(a__isNatKind(V1),V1,V2)
  a__isNatIListKind(nil()) -> tt()
  a__isNatIListKind(zeros()) -> tt()
  a__isNatIListKind(cons(V1,V2)) -> a__U51(a__isNatKind(V1),V2)
  a__isNatIListKind(take(V1,V2)) -> a__U61(a__isNatKind(V1),V2)
  a__isNatKind(|0|()) -> tt()
  a__isNatKind(length(V1)) -> a__U71(a__isNatIListKind(V1))
  a__isNatKind(s(V1)) -> a__U81(a__isNatKind(V1))
  a__isNatList(nil()) -> tt()
  a__isNatList(cons(V1,V2)) -> a__U91(a__isNatKind(V1),V1,V2)
  a__isNatList(take(V1,V2)) -> a__U101(a__isNatKind(V1),V1,V2)
  a__length(nil()) -> |0|()
  a__length(cons(N,L)) -> a__U111(a__isNatList(L),L,N)
  a__take(|0|(),IL) -> a__U121(a__isNatIList(IL),IL)
  a__take(s(M),cons(N,IL)) -> a__U131(a__isNatIList(IL),IL,M,N)
  mark(zeros()) -> a__zeros()
  mark(U101(X1,X2,X3)) -> a__U101(mark(X1),X2,X3)
  mark(U102(X1,X2,X3)) -> a__U102(mark(X1),X2,X3)
  mark(isNatKind(X)) -> a__isNatKind(X)
  mark(U103(X1,X2,X3)) -> a__U103(mark(X1),X2,X3)
  mark(isNatIListKind(X)) -> a__isNatIListKind(X)
  mark(U104(X1,X2,X3)) -> a__U104(mark(X1),X2,X3)
  mark(U105(X1,X2)) -> a__U105(mark(X1),X2)
  mark(isNat(X)) -> a__isNat(X)
  mark(U106(X)) -> a__U106(mark(X))
  mark(isNatIList(X)) -> a__isNatIList(X)
  mark(U11(X1,X2)) -> a__U11(mark(X1),X2)
  mark(U12(X1,X2)) -> a__U12(mark(X1),X2)
  mark(U111(X1,X2,X3)) -> a__U111(mark(X1),X2,X3)
  mark(U112(X1,X2,X3)) -> a__U112(mark(X1),X2,X3)
  mark(U113(X1,X2,X3)) -> a__U113(mark(X1),X2,X3)
  mark(U114(X1,X2)) -> a__U114(mark(X1),X2)
  mark(length(X)) -> a__length(mark(X))
  mark(U13(X)) -> a__U13(mark(X))
  mark(isNatList(X)) -> a__isNatList(X)
  mark(U121(X1,X2)) -> a__U121(mark(X1),X2)
  mark(U122(X)) -> a__U122(mark(X))
  mark(U131(X1,X2,X3,X4)) -> a__U131(mark(X1),X2,X3,X4)
  mark(U132(X1,X2,X3,X4)) -> a__U132(mark(X1),X2,X3,X4)
  mark(U133(X1,X2,X3,X4)) -> a__U133(mark(X1),X2,X3,X4)
  mark(U134(X1,X2,X3,X4)) -> a__U134(mark(X1),X2,X3,X4)
  mark(U135(X1,X2,X3,X4)) -> a__U135(mark(X1),X2,X3,X4)
  mark(U136(X1,X2,X3,X4)) -> a__U136(mark(X1),X2,X3,X4)
  mark(take(X1,X2)) -> a__take(mark(X1),mark(X2))
  mark(U21(X1,X2)) -> a__U21(mark(X1),X2)
  mark(U22(X1,X2)) -> a__U22(mark(X1),X2)
  mark(U23(X)) -> a__U23(mark(X))
  mark(U31(X1,X2)) -> a__U31(mark(X1),X2)
  mark(U32(X1,X2)) -> a__U32(mark(X1),X2)
  mark(U33(X)) -> a__U33(mark(X))
  mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3)
  mark(U42(X1,X2,X3)) -> a__U42(mark(X1),X2,X3)
  mark(U43(X1,X2,X3)) -> a__U43(mark(X1),X2,X3)
  mark(U44(X1,X2,X3)) -> a__U44(mark(X1),X2,X3)
  mark(U45(X1,X2)) -> a__U45(mark(X1),X2)
  mark(U46(X)) -> a__U46(mark(X))
  mark(U51(X1,X2)) -> a__U51(mark(X1),X2)
  mark(U52(X)) -> a__U52(mark(X))
  mark(U61(X1,X2)) -> a__U61(mark(X1),X2)
  mark(U62(X)) -> a__U62(mark(X))
  mark(U71(X)) -> a__U71(mark(X))
  mark(U81(X)) -> a__U81(mark(X))
  mark(U91(X1,X2,X3)) -> a__U91(mark(X1),X2,X3)
  mark(U92(X1,X2,X3)) -> a__U92(mark(X1),X2,X3)
  mark(U93(X1,X2,X3)) -> a__U93(mark(X1),X2,X3)
  mark(U94(X1,X2,X3)) -> a__U94(mark(X1),X2,X3)
  mark(U95(X1,X2)) -> a__U95(mark(X1),X2)
  mark(U96(X)) -> a__U96(mark(X))
  mark(cons(X1,X2)) -> cons(mark(X1),X2)
  mark(|0|()) -> |0|()
  mark(tt()) -> tt()
  mark(s(X)) -> s(mark(X))
  mark(nil()) -> nil()
  a__zeros() -> zeros()
  a__U101(X1,X2,X3) -> U101(X1,X2,X3)
  a__U102(X1,X2,X3) -> U102(X1,X2,X3)
  a__isNatKind(X) -> isNatKind(X)
  a__U103(X1,X2,X3) -> U103(X1,X2,X3)
  a__isNatIListKind(X) -> isNatIListKind(X)
  a__U104(X1,X2,X3) -> U104(X1,X2,X3)
  a__U105(X1,X2) -> U105(X1,X2)
  a__isNat(X) -> isNat(X)
  a__U106(X) -> U106(X)
  a__isNatIList(X) -> isNatIList(X)
  a__U11(X1,X2) -> U11(X1,X2)
  a__U12(X1,X2) -> U12(X1,X2)
  a__U111(X1,X2,X3) -> U111(X1,X2,X3)
  a__U112(X1,X2,X3) -> U112(X1,X2,X3)
  a__U113(X1,X2,X3) -> U113(X1,X2,X3)
  a__U114(X1,X2) -> U114(X1,X2)
  a__length(X) -> length(X)
  a__U13(X) -> U13(X)
  a__isNatList(X) -> isNatList(X)
  a__U121(X1,X2) -> U121(X1,X2)
  a__U122(X) -> U122(X)
  a__U131(X1,X2,X3,X4) -> U131(X1,X2,X3,X4)
  a__U132(X1,X2,X3,X4) -> U132(X1,X2,X3,X4)
  a__U133(X1,X2,X3,X4) -> U133(X1,X2,X3,X4)
  a__U134(X1,X2,X3,X4) -> U134(X1,X2,X3,X4)
  a__U135(X1,X2,X3,X4) -> U135(X1,X2,X3,X4)
  a__U136(X1,X2,X3,X4) -> U136(X1,X2,X3,X4)
  a__take(X1,X2) -> take(X1,X2)
  a__U21(X1,X2) -> U21(X1,X2)
  a__U22(X1,X2) -> U22(X1,X2)
  a__U23(X) -> U23(X)
  a__U31(X1,X2) -> U31(X1,X2)
  a__U32(X1,X2) -> U32(X1,X2)
  a__U33(X) -> U33(X)
  a__U41(X1,X2,X3) -> U41(X1,X2,X3)
  a__U42(X1,X2,X3) -> U42(X1,X2,X3)
  a__U43(X1,X2,X3) -> U43(X1,X2,X3)
  a__U44(X1,X2,X3) -> U44(X1,X2,X3)
  a__U45(X1,X2) -> U45(X1,X2)
  a__U46(X) -> U46(X)
  a__U51(X1,X2) -> U51(X1,X2)
  a__U52(X) -> U52(X)
  a__U61(X1,X2) -> U61(X1,X2)
  a__U62(X) -> U62(X)
  a__U71(X) -> U71(X)
  a__U81(X) -> U81(X)
  a__U91(X1,X2,X3) -> U91(X1,X2,X3)
  a__U92(X1,X2,X3) -> U92(X1,X2,X3)
  a__U93(X1,X2,X3) -> U93(X1,X2,X3)
  a__U94(X1,X2,X3) -> U94(X1,X2,X3)
  a__U95(X1,X2) -> U95(X1,X2)
  a__U96(X) -> U96(X)

-- SCC decomposition.

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

p1: a__U101#(tt(),V1,V2) -> a__U102#(a__isNatKind(V1),V1,V2)
p2: a__U101#(tt(),V1,V2) -> a__isNatKind#(V1)
p3: a__U102#(tt(),V1,V2) -> a__U103#(a__isNatIListKind(V2),V1,V2)
p4: a__U102#(tt(),V1,V2) -> a__isNatIListKind#(V2)
p5: a__U103#(tt(),V1,V2) -> a__U104#(a__isNatIListKind(V2),V1,V2)
p6: a__U103#(tt(),V1,V2) -> a__isNatIListKind#(V2)
p7: a__U104#(tt(),V1,V2) -> a__U105#(a__isNat(V1),V2)
p8: a__U104#(tt(),V1,V2) -> a__isNat#(V1)
p9: a__U105#(tt(),V2) -> a__U106#(a__isNatIList(V2))
p10: a__U105#(tt(),V2) -> a__isNatIList#(V2)
p11: a__U11#(tt(),V1) -> a__U12#(a__isNatIListKind(V1),V1)
p12: a__U11#(tt(),V1) -> a__isNatIListKind#(V1)
p13: a__U111#(tt(),L,N) -> a__U112#(a__isNatIListKind(L),L,N)
p14: a__U111#(tt(),L,N) -> a__isNatIListKind#(L)
p15: a__U112#(tt(),L,N) -> a__U113#(a__isNat(N),L,N)
p16: a__U112#(tt(),L,N) -> a__isNat#(N)
p17: a__U113#(tt(),L,N) -> a__U114#(a__isNatKind(N),L)
p18: a__U113#(tt(),L,N) -> a__isNatKind#(N)
p19: a__U114#(tt(),L) -> a__length#(mark(L))
p20: a__U114#(tt(),L) -> mark#(L)
p21: a__U12#(tt(),V1) -> a__U13#(a__isNatList(V1))
p22: a__U12#(tt(),V1) -> a__isNatList#(V1)
p23: a__U121#(tt(),IL) -> a__U122#(a__isNatIListKind(IL))
p24: a__U121#(tt(),IL) -> a__isNatIListKind#(IL)
p25: a__U131#(tt(),IL,M,N) -> a__U132#(a__isNatIListKind(IL),IL,M,N)
p26: a__U131#(tt(),IL,M,N) -> a__isNatIListKind#(IL)
p27: a__U132#(tt(),IL,M,N) -> a__U133#(a__isNat(M),IL,M,N)
p28: a__U132#(tt(),IL,M,N) -> a__isNat#(M)
p29: a__U133#(tt(),IL,M,N) -> a__U134#(a__isNatKind(M),IL,M,N)
p30: a__U133#(tt(),IL,M,N) -> a__isNatKind#(M)
p31: a__U134#(tt(),IL,M,N) -> a__U135#(a__isNat(N),IL,M,N)
p32: a__U134#(tt(),IL,M,N) -> a__isNat#(N)
p33: a__U135#(tt(),IL,M,N) -> a__U136#(a__isNatKind(N),IL,M,N)
p34: a__U135#(tt(),IL,M,N) -> a__isNatKind#(N)
p35: a__U136#(tt(),IL,M,N) -> mark#(N)
p36: a__U21#(tt(),V1) -> a__U22#(a__isNatKind(V1),V1)
p37: a__U21#(tt(),V1) -> a__isNatKind#(V1)
p38: a__U22#(tt(),V1) -> a__U23#(a__isNat(V1))
p39: a__U22#(tt(),V1) -> a__isNat#(V1)
p40: a__U31#(tt(),V) -> a__U32#(a__isNatIListKind(V),V)
p41: a__U31#(tt(),V) -> a__isNatIListKind#(V)
p42: a__U32#(tt(),V) -> a__U33#(a__isNatList(V))
p43: a__U32#(tt(),V) -> a__isNatList#(V)
p44: a__U41#(tt(),V1,V2) -> a__U42#(a__isNatKind(V1),V1,V2)
p45: a__U41#(tt(),V1,V2) -> a__isNatKind#(V1)
p46: a__U42#(tt(),V1,V2) -> a__U43#(a__isNatIListKind(V2),V1,V2)
p47: a__U42#(tt(),V1,V2) -> a__isNatIListKind#(V2)
p48: a__U43#(tt(),V1,V2) -> a__U44#(a__isNatIListKind(V2),V1,V2)
p49: a__U43#(tt(),V1,V2) -> a__isNatIListKind#(V2)
p50: a__U44#(tt(),V1,V2) -> a__U45#(a__isNat(V1),V2)
p51: a__U44#(tt(),V1,V2) -> a__isNat#(V1)
p52: a__U45#(tt(),V2) -> a__U46#(a__isNatIList(V2))
p53: a__U45#(tt(),V2) -> a__isNatIList#(V2)
p54: a__U51#(tt(),V2) -> a__U52#(a__isNatIListKind(V2))
p55: a__U51#(tt(),V2) -> a__isNatIListKind#(V2)
p56: a__U61#(tt(),V2) -> a__U62#(a__isNatIListKind(V2))
p57: a__U61#(tt(),V2) -> a__isNatIListKind#(V2)
p58: a__U91#(tt(),V1,V2) -> a__U92#(a__isNatKind(V1),V1,V2)
p59: a__U91#(tt(),V1,V2) -> a__isNatKind#(V1)
p60: a__U92#(tt(),V1,V2) -> a__U93#(a__isNatIListKind(V2),V1,V2)
p61: a__U92#(tt(),V1,V2) -> a__isNatIListKind#(V2)
p62: a__U93#(tt(),V1,V2) -> a__U94#(a__isNatIListKind(V2),V1,V2)
p63: a__U93#(tt(),V1,V2) -> a__isNatIListKind#(V2)
p64: a__U94#(tt(),V1,V2) -> a__U95#(a__isNat(V1),V2)
p65: a__U94#(tt(),V1,V2) -> a__isNat#(V1)
p66: a__U95#(tt(),V2) -> a__U96#(a__isNatList(V2))
p67: a__U95#(tt(),V2) -> a__isNatList#(V2)
p68: a__isNat#(length(V1)) -> a__U11#(a__isNatIListKind(V1),V1)
p69: a__isNat#(length(V1)) -> a__isNatIListKind#(V1)
p70: a__isNat#(s(V1)) -> a__U21#(a__isNatKind(V1),V1)
p71: a__isNat#(s(V1)) -> a__isNatKind#(V1)
p72: a__isNatIList#(V) -> a__U31#(a__isNatIListKind(V),V)
p73: a__isNatIList#(V) -> a__isNatIListKind#(V)
p74: a__isNatIList#(cons(V1,V2)) -> a__U41#(a__isNatKind(V1),V1,V2)
p75: a__isNatIList#(cons(V1,V2)) -> a__isNatKind#(V1)
p76: a__isNatIListKind#(cons(V1,V2)) -> a__U51#(a__isNatKind(V1),V2)
p77: a__isNatIListKind#(cons(V1,V2)) -> a__isNatKind#(V1)
p78: a__isNatIListKind#(take(V1,V2)) -> a__U61#(a__isNatKind(V1),V2)
p79: a__isNatIListKind#(take(V1,V2)) -> a__isNatKind#(V1)
p80: a__isNatKind#(length(V1)) -> a__U71#(a__isNatIListKind(V1))
p81: a__isNatKind#(length(V1)) -> a__isNatIListKind#(V1)
p82: a__isNatKind#(s(V1)) -> a__U81#(a__isNatKind(V1))
p83: a__isNatKind#(s(V1)) -> a__isNatKind#(V1)
p84: a__isNatList#(cons(V1,V2)) -> a__U91#(a__isNatKind(V1),V1,V2)
p85: a__isNatList#(cons(V1,V2)) -> a__isNatKind#(V1)
p86: a__isNatList#(take(V1,V2)) -> a__U101#(a__isNatKind(V1),V1,V2)
p87: a__isNatList#(take(V1,V2)) -> a__isNatKind#(V1)
p88: a__length#(cons(N,L)) -> a__U111#(a__isNatList(L),L,N)
p89: a__length#(cons(N,L)) -> a__isNatList#(L)
p90: a__take#(|0|(),IL) -> a__U121#(a__isNatIList(IL),IL)
p91: a__take#(|0|(),IL) -> a__isNatIList#(IL)
p92: a__take#(s(M),cons(N,IL)) -> a__U131#(a__isNatIList(IL),IL,M,N)
p93: a__take#(s(M),cons(N,IL)) -> a__isNatIList#(IL)
p94: mark#(zeros()) -> a__zeros#()
p95: mark#(U101(X1,X2,X3)) -> a__U101#(mark(X1),X2,X3)
p96: mark#(U101(X1,X2,X3)) -> mark#(X1)
p97: mark#(U102(X1,X2,X3)) -> a__U102#(mark(X1),X2,X3)
p98: mark#(U102(X1,X2,X3)) -> mark#(X1)
p99: mark#(isNatKind(X)) -> a__isNatKind#(X)
p100: mark#(U103(X1,X2,X3)) -> a__U103#(mark(X1),X2,X3)
p101: mark#(U103(X1,X2,X3)) -> mark#(X1)
p102: mark#(isNatIListKind(X)) -> a__isNatIListKind#(X)
p103: mark#(U104(X1,X2,X3)) -> a__U104#(mark(X1),X2,X3)
p104: mark#(U104(X1,X2,X3)) -> mark#(X1)
p105: mark#(U105(X1,X2)) -> a__U105#(mark(X1),X2)
p106: mark#(U105(X1,X2)) -> mark#(X1)
p107: mark#(isNat(X)) -> a__isNat#(X)
p108: mark#(U106(X)) -> a__U106#(mark(X))
p109: mark#(U106(X)) -> mark#(X)
p110: mark#(isNatIList(X)) -> a__isNatIList#(X)
p111: mark#(U11(X1,X2)) -> a__U11#(mark(X1),X2)
p112: mark#(U11(X1,X2)) -> mark#(X1)
p113: mark#(U12(X1,X2)) -> a__U12#(mark(X1),X2)
p114: mark#(U12(X1,X2)) -> mark#(X1)
p115: mark#(U111(X1,X2,X3)) -> a__U111#(mark(X1),X2,X3)
p116: mark#(U111(X1,X2,X3)) -> mark#(X1)
p117: mark#(U112(X1,X2,X3)) -> a__U112#(mark(X1),X2,X3)
p118: mark#(U112(X1,X2,X3)) -> mark#(X1)
p119: mark#(U113(X1,X2,X3)) -> a__U113#(mark(X1),X2,X3)
p120: mark#(U113(X1,X2,X3)) -> mark#(X1)
p121: mark#(U114(X1,X2)) -> a__U114#(mark(X1),X2)
p122: mark#(U114(X1,X2)) -> mark#(X1)
p123: mark#(length(X)) -> a__length#(mark(X))
p124: mark#(length(X)) -> mark#(X)
p125: mark#(U13(X)) -> a__U13#(mark(X))
p126: mark#(U13(X)) -> mark#(X)
p127: mark#(isNatList(X)) -> a__isNatList#(X)
p128: mark#(U121(X1,X2)) -> a__U121#(mark(X1),X2)
p129: mark#(U121(X1,X2)) -> mark#(X1)
p130: mark#(U122(X)) -> a__U122#(mark(X))
p131: mark#(U122(X)) -> mark#(X)
p132: mark#(U131(X1,X2,X3,X4)) -> a__U131#(mark(X1),X2,X3,X4)
p133: mark#(U131(X1,X2,X3,X4)) -> mark#(X1)
p134: mark#(U132(X1,X2,X3,X4)) -> a__U132#(mark(X1),X2,X3,X4)
p135: mark#(U132(X1,X2,X3,X4)) -> mark#(X1)
p136: mark#(U133(X1,X2,X3,X4)) -> a__U133#(mark(X1),X2,X3,X4)
p137: mark#(U133(X1,X2,X3,X4)) -> mark#(X1)
p138: mark#(U134(X1,X2,X3,X4)) -> a__U134#(mark(X1),X2,X3,X4)
p139: mark#(U134(X1,X2,X3,X4)) -> mark#(X1)
p140: mark#(U135(X1,X2,X3,X4)) -> a__U135#(mark(X1),X2,X3,X4)
p141: mark#(U135(X1,X2,X3,X4)) -> mark#(X1)
p142: mark#(U136(X1,X2,X3,X4)) -> a__U136#(mark(X1),X2,X3,X4)
p143: mark#(U136(X1,X2,X3,X4)) -> mark#(X1)
p144: mark#(take(X1,X2)) -> a__take#(mark(X1),mark(X2))
p145: mark#(take(X1,X2)) -> mark#(X1)
p146: mark#(take(X1,X2)) -> mark#(X2)
p147: mark#(U21(X1,X2)) -> a__U21#(mark(X1),X2)
p148: mark#(U21(X1,X2)) -> mark#(X1)
p149: mark#(U22(X1,X2)) -> a__U22#(mark(X1),X2)
p150: mark#(U22(X1,X2)) -> mark#(X1)
p151: mark#(U23(X)) -> a__U23#(mark(X))
p152: mark#(U23(X)) -> mark#(X)
p153: mark#(U31(X1,X2)) -> a__U31#(mark(X1),X2)
p154: mark#(U31(X1,X2)) -> mark#(X1)
p155: mark#(U32(X1,X2)) -> a__U32#(mark(X1),X2)
p156: mark#(U32(X1,X2)) -> mark#(X1)
p157: mark#(U33(X)) -> a__U33#(mark(X))
p158: mark#(U33(X)) -> mark#(X)
p159: mark#(U41(X1,X2,X3)) -> a__U41#(mark(X1),X2,X3)
p160: mark#(U41(X1,X2,X3)) -> mark#(X1)
p161: mark#(U42(X1,X2,X3)) -> a__U42#(mark(X1),X2,X3)
p162: mark#(U42(X1,X2,X3)) -> mark#(X1)
p163: mark#(U43(X1,X2,X3)) -> a__U43#(mark(X1),X2,X3)
p164: mark#(U43(X1,X2,X3)) -> mark#(X1)
p165: mark#(U44(X1,X2,X3)) -> a__U44#(mark(X1),X2,X3)
p166: mark#(U44(X1,X2,X3)) -> mark#(X1)
p167: mark#(U45(X1,X2)) -> a__U45#(mark(X1),X2)
p168: mark#(U45(X1,X2)) -> mark#(X1)
p169: mark#(U46(X)) -> a__U46#(mark(X))
p170: mark#(U46(X)) -> mark#(X)
p171: mark#(U51(X1,X2)) -> a__U51#(mark(X1),X2)
p172: mark#(U51(X1,X2)) -> mark#(X1)
p173: mark#(U52(X)) -> a__U52#(mark(X))
p174: mark#(U52(X)) -> mark#(X)
p175: mark#(U61(X1,X2)) -> a__U61#(mark(X1),X2)
p176: mark#(U61(X1,X2)) -> mark#(X1)
p177: mark#(U62(X)) -> a__U62#(mark(X))
p178: mark#(U62(X)) -> mark#(X)
p179: mark#(U71(X)) -> a__U71#(mark(X))
p180: mark#(U71(X)) -> mark#(X)
p181: mark#(U81(X)) -> a__U81#(mark(X))
p182: mark#(U81(X)) -> mark#(X)
p183: mark#(U91(X1,X2,X3)) -> a__U91#(mark(X1),X2,X3)
p184: mark#(U91(X1,X2,X3)) -> mark#(X1)
p185: mark#(U92(X1,X2,X3)) -> a__U92#(mark(X1),X2,X3)
p186: mark#(U92(X1,X2,X3)) -> mark#(X1)
p187: mark#(U93(X1,X2,X3)) -> a__U93#(mark(X1),X2,X3)
p188: mark#(U93(X1,X2,X3)) -> mark#(X1)
p189: mark#(U94(X1,X2,X3)) -> a__U94#(mark(X1),X2,X3)
p190: mark#(U94(X1,X2,X3)) -> mark#(X1)
p191: mark#(U95(X1,X2)) -> a__U95#(mark(X1),X2)
p192: mark#(U95(X1,X2)) -> mark#(X1)
p193: mark#(U96(X)) -> a__U96#(mark(X))
p194: mark#(U96(X)) -> mark#(X)
p195: mark#(cons(X1,X2)) -> mark#(X1)
p196: mark#(s(X)) -> mark#(X)

and R consists of:

r1: a__zeros() -> cons(|0|(),zeros())
r2: a__U101(tt(),V1,V2) -> a__U102(a__isNatKind(V1),V1,V2)
r3: a__U102(tt(),V1,V2) -> a__U103(a__isNatIListKind(V2),V1,V2)
r4: a__U103(tt(),V1,V2) -> a__U104(a__isNatIListKind(V2),V1,V2)
r5: a__U104(tt(),V1,V2) -> a__U105(a__isNat(V1),V2)
r6: a__U105(tt(),V2) -> a__U106(a__isNatIList(V2))
r7: a__U106(tt()) -> tt()
r8: a__U11(tt(),V1) -> a__U12(a__isNatIListKind(V1),V1)
r9: a__U111(tt(),L,N) -> a__U112(a__isNatIListKind(L),L,N)
r10: a__U112(tt(),L,N) -> a__U113(a__isNat(N),L,N)
r11: a__U113(tt(),L,N) -> a__U114(a__isNatKind(N),L)
r12: a__U114(tt(),L) -> s(a__length(mark(L)))
r13: a__U12(tt(),V1) -> a__U13(a__isNatList(V1))
r14: a__U121(tt(),IL) -> a__U122(a__isNatIListKind(IL))
r15: a__U122(tt()) -> nil()
r16: a__U13(tt()) -> tt()
r17: a__U131(tt(),IL,M,N) -> a__U132(a__isNatIListKind(IL),IL,M,N)
r18: a__U132(tt(),IL,M,N) -> a__U133(a__isNat(M),IL,M,N)
r19: a__U133(tt(),IL,M,N) -> a__U134(a__isNatKind(M),IL,M,N)
r20: a__U134(tt(),IL,M,N) -> a__U135(a__isNat(N),IL,M,N)
r21: a__U135(tt(),IL,M,N) -> a__U136(a__isNatKind(N),IL,M,N)
r22: a__U136(tt(),IL,M,N) -> cons(mark(N),take(M,IL))
r23: a__U21(tt(),V1) -> a__U22(a__isNatKind(V1),V1)
r24: a__U22(tt(),V1) -> a__U23(a__isNat(V1))
r25: a__U23(tt()) -> tt()
r26: a__U31(tt(),V) -> a__U32(a__isNatIListKind(V),V)
r27: a__U32(tt(),V) -> a__U33(a__isNatList(V))
r28: a__U33(tt()) -> tt()
r29: a__U41(tt(),V1,V2) -> a__U42(a__isNatKind(V1),V1,V2)
r30: a__U42(tt(),V1,V2) -> a__U43(a__isNatIListKind(V2),V1,V2)
r31: a__U43(tt(),V1,V2) -> a__U44(a__isNatIListKind(V2),V1,V2)
r32: a__U44(tt(),V1,V2) -> a__U45(a__isNat(V1),V2)
r33: a__U45(tt(),V2) -> a__U46(a__isNatIList(V2))
r34: a__U46(tt()) -> tt()
r35: a__U51(tt(),V2) -> a__U52(a__isNatIListKind(V2))
r36: a__U52(tt()) -> tt()
r37: a__U61(tt(),V2) -> a__U62(a__isNatIListKind(V2))
r38: a__U62(tt()) -> tt()
r39: a__U71(tt()) -> tt()
r40: a__U81(tt()) -> tt()
r41: a__U91(tt(),V1,V2) -> a__U92(a__isNatKind(V1),V1,V2)
r42: a__U92(tt(),V1,V2) -> a__U93(a__isNatIListKind(V2),V1,V2)
r43: a__U93(tt(),V1,V2) -> a__U94(a__isNatIListKind(V2),V1,V2)
r44: a__U94(tt(),V1,V2) -> a__U95(a__isNat(V1),V2)
r45: a__U95(tt(),V2) -> a__U96(a__isNatList(V2))
r46: a__U96(tt()) -> tt()
r47: a__isNat(|0|()) -> tt()
r48: a__isNat(length(V1)) -> a__U11(a__isNatIListKind(V1),V1)
r49: a__isNat(s(V1)) -> a__U21(a__isNatKind(V1),V1)
r50: a__isNatIList(V) -> a__U31(a__isNatIListKind(V),V)
r51: a__isNatIList(zeros()) -> tt()
r52: a__isNatIList(cons(V1,V2)) -> a__U41(a__isNatKind(V1),V1,V2)
r53: a__isNatIListKind(nil()) -> tt()
r54: a__isNatIListKind(zeros()) -> tt()
r55: a__isNatIListKind(cons(V1,V2)) -> a__U51(a__isNatKind(V1),V2)
r56: a__isNatIListKind(take(V1,V2)) -> a__U61(a__isNatKind(V1),V2)
r57: a__isNatKind(|0|()) -> tt()
r58: a__isNatKind(length(V1)) -> a__U71(a__isNatIListKind(V1))
r59: a__isNatKind(s(V1)) -> a__U81(a__isNatKind(V1))
r60: a__isNatList(nil()) -> tt()
r61: a__isNatList(cons(V1,V2)) -> a__U91(a__isNatKind(V1),V1,V2)
r62: a__isNatList(take(V1,V2)) -> a__U101(a__isNatKind(V1),V1,V2)
r63: a__length(nil()) -> |0|()
r64: a__length(cons(N,L)) -> a__U111(a__isNatList(L),L,N)
r65: a__take(|0|(),IL) -> a__U121(a__isNatIList(IL),IL)
r66: a__take(s(M),cons(N,IL)) -> a__U131(a__isNatIList(IL),IL,M,N)
r67: mark(zeros()) -> a__zeros()
r68: mark(U101(X1,X2,X3)) -> a__U101(mark(X1),X2,X3)
r69: mark(U102(X1,X2,X3)) -> a__U102(mark(X1),X2,X3)
r70: mark(isNatKind(X)) -> a__isNatKind(X)
r71: mark(U103(X1,X2,X3)) -> a__U103(mark(X1),X2,X3)
r72: mark(isNatIListKind(X)) -> a__isNatIListKind(X)
r73: mark(U104(X1,X2,X3)) -> a__U104(mark(X1),X2,X3)
r74: mark(U105(X1,X2)) -> a__U105(mark(X1),X2)
r75: mark(isNat(X)) -> a__isNat(X)
r76: mark(U106(X)) -> a__U106(mark(X))
r77: mark(isNatIList(X)) -> a__isNatIList(X)
r78: mark(U11(X1,X2)) -> a__U11(mark(X1),X2)
r79: mark(U12(X1,X2)) -> a__U12(mark(X1),X2)
r80: mark(U111(X1,X2,X3)) -> a__U111(mark(X1),X2,X3)
r81: mark(U112(X1,X2,X3)) -> a__U112(mark(X1),X2,X3)
r82: mark(U113(X1,X2,X3)) -> a__U113(mark(X1),X2,X3)
r83: mark(U114(X1,X2)) -> a__U114(mark(X1),X2)
r84: mark(length(X)) -> a__length(mark(X))
r85: mark(U13(X)) -> a__U13(mark(X))
r86: mark(isNatList(X)) -> a__isNatList(X)
r87: mark(U121(X1,X2)) -> a__U121(mark(X1),X2)
r88: mark(U122(X)) -> a__U122(mark(X))
r89: mark(U131(X1,X2,X3,X4)) -> a__U131(mark(X1),X2,X3,X4)
r90: mark(U132(X1,X2,X3,X4)) -> a__U132(mark(X1),X2,X3,X4)
r91: mark(U133(X1,X2,X3,X4)) -> a__U133(mark(X1),X2,X3,X4)
r92: mark(U134(X1,X2,X3,X4)) -> a__U134(mark(X1),X2,X3,X4)
r93: mark(U135(X1,X2,X3,X4)) -> a__U135(mark(X1),X2,X3,X4)
r94: mark(U136(X1,X2,X3,X4)) -> a__U136(mark(X1),X2,X3,X4)
r95: mark(take(X1,X2)) -> a__take(mark(X1),mark(X2))
r96: mark(U21(X1,X2)) -> a__U21(mark(X1),X2)
r97: mark(U22(X1,X2)) -> a__U22(mark(X1),X2)
r98: mark(U23(X)) -> a__U23(mark(X))
r99: mark(U31(X1,X2)) -> a__U31(mark(X1),X2)
r100: mark(U32(X1,X2)) -> a__U32(mark(X1),X2)
r101: mark(U33(X)) -> a__U33(mark(X))
r102: mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3)
r103: mark(U42(X1,X2,X3)) -> a__U42(mark(X1),X2,X3)
r104: mark(U43(X1,X2,X3)) -> a__U43(mark(X1),X2,X3)
r105: mark(U44(X1,X2,X3)) -> a__U44(mark(X1),X2,X3)
r106: mark(U45(X1,X2)) -> a__U45(mark(X1),X2)
r107: mark(U46(X)) -> a__U46(mark(X))
r108: mark(U51(X1,X2)) -> a__U51(mark(X1),X2)
r109: mark(U52(X)) -> a__U52(mark(X))
r110: mark(U61(X1,X2)) -> a__U61(mark(X1),X2)
r111: mark(U62(X)) -> a__U62(mark(X))
r112: mark(U71(X)) -> a__U71(mark(X))
r113: mark(U81(X)) -> a__U81(mark(X))
r114: mark(U91(X1,X2,X3)) -> a__U91(mark(X1),X2,X3)
r115: mark(U92(X1,X2,X3)) -> a__U92(mark(X1),X2,X3)
r116: mark(U93(X1,X2,X3)) -> a__U93(mark(X1),X2,X3)
r117: mark(U94(X1,X2,X3)) -> a__U94(mark(X1),X2,X3)
r118: mark(U95(X1,X2)) -> a__U95(mark(X1),X2)
r119: mark(U96(X)) -> a__U96(mark(X))
r120: mark(cons(X1,X2)) -> cons(mark(X1),X2)
r121: mark(|0|()) -> |0|()
r122: mark(tt()) -> tt()
r123: mark(s(X)) -> s(mark(X))
r124: mark(nil()) -> nil()
r125: a__zeros() -> zeros()
r126: a__U101(X1,X2,X3) -> U101(X1,X2,X3)
r127: a__U102(X1,X2,X3) -> U102(X1,X2,X3)
r128: a__isNatKind(X) -> isNatKind(X)
r129: a__U103(X1,X2,X3) -> U103(X1,X2,X3)
r130: a__isNatIListKind(X) -> isNatIListKind(X)
r131: a__U104(X1,X2,X3) -> U104(X1,X2,X3)
r132: a__U105(X1,X2) -> U105(X1,X2)
r133: a__isNat(X) -> isNat(X)
r134: a__U106(X) -> U106(X)
r135: a__isNatIList(X) -> isNatIList(X)
r136: a__U11(X1,X2) -> U11(X1,X2)
r137: a__U12(X1,X2) -> U12(X1,X2)
r138: a__U111(X1,X2,X3) -> U111(X1,X2,X3)
r139: a__U112(X1,X2,X3) -> U112(X1,X2,X3)
r140: a__U113(X1,X2,X3) -> U113(X1,X2,X3)
r141: a__U114(X1,X2) -> U114(X1,X2)
r142: a__length(X) -> length(X)
r143: a__U13(X) -> U13(X)
r144: a__isNatList(X) -> isNatList(X)
r145: a__U121(X1,X2) -> U121(X1,X2)
r146: a__U122(X) -> U122(X)
r147: a__U131(X1,X2,X3,X4) -> U131(X1,X2,X3,X4)
r148: a__U132(X1,X2,X3,X4) -> U132(X1,X2,X3,X4)
r149: a__U133(X1,X2,X3,X4) -> U133(X1,X2,X3,X4)
r150: a__U134(X1,X2,X3,X4) -> U134(X1,X2,X3,X4)
r151: a__U135(X1,X2,X3,X4) -> U135(X1,X2,X3,X4)
r152: a__U136(X1,X2,X3,X4) -> U136(X1,X2,X3,X4)
r153: a__take(X1,X2) -> take(X1,X2)
r154: a__U21(X1,X2) -> U21(X1,X2)
r155: a__U22(X1,X2) -> U22(X1,X2)
r156: a__U23(X) -> U23(X)
r157: a__U31(X1,X2) -> U31(X1,X2)
r158: a__U32(X1,X2) -> U32(X1,X2)
r159: a__U33(X) -> U33(X)
r160: a__U41(X1,X2,X3) -> U41(X1,X2,X3)
r161: a__U42(X1,X2,X3) -> U42(X1,X2,X3)
r162: a__U43(X1,X2,X3) -> U43(X1,X2,X3)
r163: a__U44(X1,X2,X3) -> U44(X1,X2,X3)
r164: a__U45(X1,X2) -> U45(X1,X2)
r165: a__U46(X) -> U46(X)
r166: a__U51(X1,X2) -> U51(X1,X2)
r167: a__U52(X) -> U52(X)
r168: a__U61(X1,X2) -> U61(X1,X2)
r169: a__U62(X) -> U62(X)
r170: a__U71(X) -> U71(X)
r171: a__U81(X) -> U81(X)
r172: a__U91(X1,X2,X3) -> U91(X1,X2,X3)
r173: a__U92(X1,X2,X3) -> U92(X1,X2,X3)
r174: a__U93(X1,X2,X3) -> U93(X1,X2,X3)
r175: a__U94(X1,X2,X3) -> U94(X1,X2,X3)
r176: a__U95(X1,X2) -> U95(X1,X2)
r177: a__U96(X) -> U96(X)

The estimated dependency graph contains the following SCCs:

  {p13, p15, p17, p19, p20, p25, p27, p29, p31, p33, p35, p88, p92, p96, p98, p101, p104, p106, p109, p112, p114, p115, p116, p117, p118, p119, p120, p121, p122, p123, p124, p126, p129, p131, p132, p133, p134, p135, p136, p137, p138, p139, p140, p141, p142, p143, p144, p145, p146, p148, p150, p152, p154, p156, p158, p160, p162, p164, p166, p168, p170, p172, p174, p176, p178, p180, p182, p184, p186, p188, p190, p192, p194, p195, p196}
  {p1, p3, p5, p7, p8, p10, p11, p22, p36, p39, p40, p43, p44, p46, p48, p50, p51, p53, p58, p60, p62, p64, p65, p67, p68, p70, p72, p74, p84, p86}
  {p55, p57, p76, p77, p78, p79, p81, p83}


-- Reduction pair.

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

p1: a__U136#(tt(),IL,M,N) -> mark#(N)
p2: mark#(s(X)) -> mark#(X)
p3: mark#(cons(X1,X2)) -> mark#(X1)
p4: mark#(U96(X)) -> mark#(X)
p5: mark#(U95(X1,X2)) -> mark#(X1)
p6: mark#(U94(X1,X2,X3)) -> mark#(X1)
p7: mark#(U93(X1,X2,X3)) -> mark#(X1)
p8: mark#(U92(X1,X2,X3)) -> mark#(X1)
p9: mark#(U91(X1,X2,X3)) -> mark#(X1)
p10: mark#(U81(X)) -> mark#(X)
p11: mark#(U71(X)) -> mark#(X)
p12: mark#(U62(X)) -> mark#(X)
p13: mark#(U61(X1,X2)) -> mark#(X1)
p14: mark#(U52(X)) -> mark#(X)
p15: mark#(U51(X1,X2)) -> mark#(X1)
p16: mark#(U46(X)) -> mark#(X)
p17: mark#(U45(X1,X2)) -> mark#(X1)
p18: mark#(U44(X1,X2,X3)) -> mark#(X1)
p19: mark#(U43(X1,X2,X3)) -> mark#(X1)
p20: mark#(U42(X1,X2,X3)) -> mark#(X1)
p21: mark#(U41(X1,X2,X3)) -> mark#(X1)
p22: mark#(U33(X)) -> mark#(X)
p23: mark#(U32(X1,X2)) -> mark#(X1)
p24: mark#(U31(X1,X2)) -> mark#(X1)
p25: mark#(U23(X)) -> mark#(X)
p26: mark#(U22(X1,X2)) -> mark#(X1)
p27: mark#(U21(X1,X2)) -> mark#(X1)
p28: mark#(take(X1,X2)) -> mark#(X2)
p29: mark#(take(X1,X2)) -> mark#(X1)
p30: mark#(take(X1,X2)) -> a__take#(mark(X1),mark(X2))
p31: a__take#(s(M),cons(N,IL)) -> a__U131#(a__isNatIList(IL),IL,M,N)
p32: a__U131#(tt(),IL,M,N) -> a__U132#(a__isNatIListKind(IL),IL,M,N)
p33: a__U132#(tt(),IL,M,N) -> a__U133#(a__isNat(M),IL,M,N)
p34: a__U133#(tt(),IL,M,N) -> a__U134#(a__isNatKind(M),IL,M,N)
p35: a__U134#(tt(),IL,M,N) -> a__U135#(a__isNat(N),IL,M,N)
p36: a__U135#(tt(),IL,M,N) -> a__U136#(a__isNatKind(N),IL,M,N)
p37: mark#(U136(X1,X2,X3,X4)) -> mark#(X1)
p38: mark#(U136(X1,X2,X3,X4)) -> a__U136#(mark(X1),X2,X3,X4)
p39: mark#(U135(X1,X2,X3,X4)) -> mark#(X1)
p40: mark#(U135(X1,X2,X3,X4)) -> a__U135#(mark(X1),X2,X3,X4)
p41: mark#(U134(X1,X2,X3,X4)) -> mark#(X1)
p42: mark#(U134(X1,X2,X3,X4)) -> a__U134#(mark(X1),X2,X3,X4)
p43: mark#(U133(X1,X2,X3,X4)) -> mark#(X1)
p44: mark#(U133(X1,X2,X3,X4)) -> a__U133#(mark(X1),X2,X3,X4)
p45: mark#(U132(X1,X2,X3,X4)) -> mark#(X1)
p46: mark#(U132(X1,X2,X3,X4)) -> a__U132#(mark(X1),X2,X3,X4)
p47: mark#(U131(X1,X2,X3,X4)) -> mark#(X1)
p48: mark#(U131(X1,X2,X3,X4)) -> a__U131#(mark(X1),X2,X3,X4)
p49: mark#(U122(X)) -> mark#(X)
p50: mark#(U121(X1,X2)) -> mark#(X1)
p51: mark#(U13(X)) -> mark#(X)
p52: mark#(length(X)) -> mark#(X)
p53: mark#(length(X)) -> a__length#(mark(X))
p54: a__length#(cons(N,L)) -> a__U111#(a__isNatList(L),L,N)
p55: a__U111#(tt(),L,N) -> a__U112#(a__isNatIListKind(L),L,N)
p56: a__U112#(tt(),L,N) -> a__U113#(a__isNat(N),L,N)
p57: a__U113#(tt(),L,N) -> a__U114#(a__isNatKind(N),L)
p58: a__U114#(tt(),L) -> mark#(L)
p59: mark#(U114(X1,X2)) -> mark#(X1)
p60: mark#(U114(X1,X2)) -> a__U114#(mark(X1),X2)
p61: a__U114#(tt(),L) -> a__length#(mark(L))
p62: mark#(U113(X1,X2,X3)) -> mark#(X1)
p63: mark#(U113(X1,X2,X3)) -> a__U113#(mark(X1),X2,X3)
p64: mark#(U112(X1,X2,X3)) -> mark#(X1)
p65: mark#(U112(X1,X2,X3)) -> a__U112#(mark(X1),X2,X3)
p66: mark#(U111(X1,X2,X3)) -> mark#(X1)
p67: mark#(U111(X1,X2,X3)) -> a__U111#(mark(X1),X2,X3)
p68: mark#(U12(X1,X2)) -> mark#(X1)
p69: mark#(U11(X1,X2)) -> mark#(X1)
p70: mark#(U106(X)) -> mark#(X)
p71: mark#(U105(X1,X2)) -> mark#(X1)
p72: mark#(U104(X1,X2,X3)) -> mark#(X1)
p73: mark#(U103(X1,X2,X3)) -> mark#(X1)
p74: mark#(U102(X1,X2,X3)) -> mark#(X1)
p75: mark#(U101(X1,X2,X3)) -> mark#(X1)

and R consists of:

r1: a__zeros() -> cons(|0|(),zeros())
r2: a__U101(tt(),V1,V2) -> a__U102(a__isNatKind(V1),V1,V2)
r3: a__U102(tt(),V1,V2) -> a__U103(a__isNatIListKind(V2),V1,V2)
r4: a__U103(tt(),V1,V2) -> a__U104(a__isNatIListKind(V2),V1,V2)
r5: a__U104(tt(),V1,V2) -> a__U105(a__isNat(V1),V2)
r6: a__U105(tt(),V2) -> a__U106(a__isNatIList(V2))
r7: a__U106(tt()) -> tt()
r8: a__U11(tt(),V1) -> a__U12(a__isNatIListKind(V1),V1)
r9: a__U111(tt(),L,N) -> a__U112(a__isNatIListKind(L),L,N)
r10: a__U112(tt(),L,N) -> a__U113(a__isNat(N),L,N)
r11: a__U113(tt(),L,N) -> a__U114(a__isNatKind(N),L)
r12: a__U114(tt(),L) -> s(a__length(mark(L)))
r13: a__U12(tt(),V1) -> a__U13(a__isNatList(V1))
r14: a__U121(tt(),IL) -> a__U122(a__isNatIListKind(IL))
r15: a__U122(tt()) -> nil()
r16: a__U13(tt()) -> tt()
r17: a__U131(tt(),IL,M,N) -> a__U132(a__isNatIListKind(IL),IL,M,N)
r18: a__U132(tt(),IL,M,N) -> a__U133(a__isNat(M),IL,M,N)
r19: a__U133(tt(),IL,M,N) -> a__U134(a__isNatKind(M),IL,M,N)
r20: a__U134(tt(),IL,M,N) -> a__U135(a__isNat(N),IL,M,N)
r21: a__U135(tt(),IL,M,N) -> a__U136(a__isNatKind(N),IL,M,N)
r22: a__U136(tt(),IL,M,N) -> cons(mark(N),take(M,IL))
r23: a__U21(tt(),V1) -> a__U22(a__isNatKind(V1),V1)
r24: a__U22(tt(),V1) -> a__U23(a__isNat(V1))
r25: a__U23(tt()) -> tt()
r26: a__U31(tt(),V) -> a__U32(a__isNatIListKind(V),V)
r27: a__U32(tt(),V) -> a__U33(a__isNatList(V))
r28: a__U33(tt()) -> tt()
r29: a__U41(tt(),V1,V2) -> a__U42(a__isNatKind(V1),V1,V2)
r30: a__U42(tt(),V1,V2) -> a__U43(a__isNatIListKind(V2),V1,V2)
r31: a__U43(tt(),V1,V2) -> a__U44(a__isNatIListKind(V2),V1,V2)
r32: a__U44(tt(),V1,V2) -> a__U45(a__isNat(V1),V2)
r33: a__U45(tt(),V2) -> a__U46(a__isNatIList(V2))
r34: a__U46(tt()) -> tt()
r35: a__U51(tt(),V2) -> a__U52(a__isNatIListKind(V2))
r36: a__U52(tt()) -> tt()
r37: a__U61(tt(),V2) -> a__U62(a__isNatIListKind(V2))
r38: a__U62(tt()) -> tt()
r39: a__U71(tt()) -> tt()
r40: a__U81(tt()) -> tt()
r41: a__U91(tt(),V1,V2) -> a__U92(a__isNatKind(V1),V1,V2)
r42: a__U92(tt(),V1,V2) -> a__U93(a__isNatIListKind(V2),V1,V2)
r43: a__U93(tt(),V1,V2) -> a__U94(a__isNatIListKind(V2),V1,V2)
r44: a__U94(tt(),V1,V2) -> a__U95(a__isNat(V1),V2)
r45: a__U95(tt(),V2) -> a__U96(a__isNatList(V2))
r46: a__U96(tt()) -> tt()
r47: a__isNat(|0|()) -> tt()
r48: a__isNat(length(V1)) -> a__U11(a__isNatIListKind(V1),V1)
r49: a__isNat(s(V1)) -> a__U21(a__isNatKind(V1),V1)
r50: a__isNatIList(V) -> a__U31(a__isNatIListKind(V),V)
r51: a__isNatIList(zeros()) -> tt()
r52: a__isNatIList(cons(V1,V2)) -> a__U41(a__isNatKind(V1),V1,V2)
r53: a__isNatIListKind(nil()) -> tt()
r54: a__isNatIListKind(zeros()) -> tt()
r55: a__isNatIListKind(cons(V1,V2)) -> a__U51(a__isNatKind(V1),V2)
r56: a__isNatIListKind(take(V1,V2)) -> a__U61(a__isNatKind(V1),V2)
r57: a__isNatKind(|0|()) -> tt()
r58: a__isNatKind(length(V1)) -> a__U71(a__isNatIListKind(V1))
r59: a__isNatKind(s(V1)) -> a__U81(a__isNatKind(V1))
r60: a__isNatList(nil()) -> tt()
r61: a__isNatList(cons(V1,V2)) -> a__U91(a__isNatKind(V1),V1,V2)
r62: a__isNatList(take(V1,V2)) -> a__U101(a__isNatKind(V1),V1,V2)
r63: a__length(nil()) -> |0|()
r64: a__length(cons(N,L)) -> a__U111(a__isNatList(L),L,N)
r65: a__take(|0|(),IL) -> a__U121(a__isNatIList(IL),IL)
r66: a__take(s(M),cons(N,IL)) -> a__U131(a__isNatIList(IL),IL,M,N)
r67: mark(zeros()) -> a__zeros()
r68: mark(U101(X1,X2,X3)) -> a__U101(mark(X1),X2,X3)
r69: mark(U102(X1,X2,X3)) -> a__U102(mark(X1),X2,X3)
r70: mark(isNatKind(X)) -> a__isNatKind(X)
r71: mark(U103(X1,X2,X3)) -> a__U103(mark(X1),X2,X3)
r72: mark(isNatIListKind(X)) -> a__isNatIListKind(X)
r73: mark(U104(X1,X2,X3)) -> a__U104(mark(X1),X2,X3)
r74: mark(U105(X1,X2)) -> a__U105(mark(X1),X2)
r75: mark(isNat(X)) -> a__isNat(X)
r76: mark(U106(X)) -> a__U106(mark(X))
r77: mark(isNatIList(X)) -> a__isNatIList(X)
r78: mark(U11(X1,X2)) -> a__U11(mark(X1),X2)
r79: mark(U12(X1,X2)) -> a__U12(mark(X1),X2)
r80: mark(U111(X1,X2,X3)) -> a__U111(mark(X1),X2,X3)
r81: mark(U112(X1,X2,X3)) -> a__U112(mark(X1),X2,X3)
r82: mark(U113(X1,X2,X3)) -> a__U113(mark(X1),X2,X3)
r83: mark(U114(X1,X2)) -> a__U114(mark(X1),X2)
r84: mark(length(X)) -> a__length(mark(X))
r85: mark(U13(X)) -> a__U13(mark(X))
r86: mark(isNatList(X)) -> a__isNatList(X)
r87: mark(U121(X1,X2)) -> a__U121(mark(X1),X2)
r88: mark(U122(X)) -> a__U122(mark(X))
r89: mark(U131(X1,X2,X3,X4)) -> a__U131(mark(X1),X2,X3,X4)
r90: mark(U132(X1,X2,X3,X4)) -> a__U132(mark(X1),X2,X3,X4)
r91: mark(U133(X1,X2,X3,X4)) -> a__U133(mark(X1),X2,X3,X4)
r92: mark(U134(X1,X2,X3,X4)) -> a__U134(mark(X1),X2,X3,X4)
r93: mark(U135(X1,X2,X3,X4)) -> a__U135(mark(X1),X2,X3,X4)
r94: mark(U136(X1,X2,X3,X4)) -> a__U136(mark(X1),X2,X3,X4)
r95: mark(take(X1,X2)) -> a__take(mark(X1),mark(X2))
r96: mark(U21(X1,X2)) -> a__U21(mark(X1),X2)
r97: mark(U22(X1,X2)) -> a__U22(mark(X1),X2)
r98: mark(U23(X)) -> a__U23(mark(X))
r99: mark(U31(X1,X2)) -> a__U31(mark(X1),X2)
r100: mark(U32(X1,X2)) -> a__U32(mark(X1),X2)
r101: mark(U33(X)) -> a__U33(mark(X))
r102: mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3)
r103: mark(U42(X1,X2,X3)) -> a__U42(mark(X1),X2,X3)
r104: mark(U43(X1,X2,X3)) -> a__U43(mark(X1),X2,X3)
r105: mark(U44(X1,X2,X3)) -> a__U44(mark(X1),X2,X3)
r106: mark(U45(X1,X2)) -> a__U45(mark(X1),X2)
r107: mark(U46(X)) -> a__U46(mark(X))
r108: mark(U51(X1,X2)) -> a__U51(mark(X1),X2)
r109: mark(U52(X)) -> a__U52(mark(X))
r110: mark(U61(X1,X2)) -> a__U61(mark(X1),X2)
r111: mark(U62(X)) -> a__U62(mark(X))
r112: mark(U71(X)) -> a__U71(mark(X))
r113: mark(U81(X)) -> a__U81(mark(X))
r114: mark(U91(X1,X2,X3)) -> a__U91(mark(X1),X2,X3)
r115: mark(U92(X1,X2,X3)) -> a__U92(mark(X1),X2,X3)
r116: mark(U93(X1,X2,X3)) -> a__U93(mark(X1),X2,X3)
r117: mark(U94(X1,X2,X3)) -> a__U94(mark(X1),X2,X3)
r118: mark(U95(X1,X2)) -> a__U95(mark(X1),X2)
r119: mark(U96(X)) -> a__U96(mark(X))
r120: mark(cons(X1,X2)) -> cons(mark(X1),X2)
r121: mark(|0|()) -> |0|()
r122: mark(tt()) -> tt()
r123: mark(s(X)) -> s(mark(X))
r124: mark(nil()) -> nil()
r125: a__zeros() -> zeros()
r126: a__U101(X1,X2,X3) -> U101(X1,X2,X3)
r127: a__U102(X1,X2,X3) -> U102(X1,X2,X3)
r128: a__isNatKind(X) -> isNatKind(X)
r129: a__U103(X1,X2,X3) -> U103(X1,X2,X3)
r130: a__isNatIListKind(X) -> isNatIListKind(X)
r131: a__U104(X1,X2,X3) -> U104(X1,X2,X3)
r132: a__U105(X1,X2) -> U105(X1,X2)
r133: a__isNat(X) -> isNat(X)
r134: a__U106(X) -> U106(X)
r135: a__isNatIList(X) -> isNatIList(X)
r136: a__U11(X1,X2) -> U11(X1,X2)
r137: a__U12(X1,X2) -> U12(X1,X2)
r138: a__U111(X1,X2,X3) -> U111(X1,X2,X3)
r139: a__U112(X1,X2,X3) -> U112(X1,X2,X3)
r140: a__U113(X1,X2,X3) -> U113(X1,X2,X3)
r141: a__U114(X1,X2) -> U114(X1,X2)
r142: a__length(X) -> length(X)
r143: a__U13(X) -> U13(X)
r144: a__isNatList(X) -> isNatList(X)
r145: a__U121(X1,X2) -> U121(X1,X2)
r146: a__U122(X) -> U122(X)
r147: a__U131(X1,X2,X3,X4) -> U131(X1,X2,X3,X4)
r148: a__U132(X1,X2,X3,X4) -> U132(X1,X2,X3,X4)
r149: a__U133(X1,X2,X3,X4) -> U133(X1,X2,X3,X4)
r150: a__U134(X1,X2,X3,X4) -> U134(X1,X2,X3,X4)
r151: a__U135(X1,X2,X3,X4) -> U135(X1,X2,X3,X4)
r152: a__U136(X1,X2,X3,X4) -> U136(X1,X2,X3,X4)
r153: a__take(X1,X2) -> take(X1,X2)
r154: a__U21(X1,X2) -> U21(X1,X2)
r155: a__U22(X1,X2) -> U22(X1,X2)
r156: a__U23(X) -> U23(X)
r157: a__U31(X1,X2) -> U31(X1,X2)
r158: a__U32(X1,X2) -> U32(X1,X2)
r159: a__U33(X) -> U33(X)
r160: a__U41(X1,X2,X3) -> U41(X1,X2,X3)
r161: a__U42(X1,X2,X3) -> U42(X1,X2,X3)
r162: a__U43(X1,X2,X3) -> U43(X1,X2,X3)
r163: a__U44(X1,X2,X3) -> U44(X1,X2,X3)
r164: a__U45(X1,X2) -> U45(X1,X2)
r165: a__U46(X) -> U46(X)
r166: a__U51(X1,X2) -> U51(X1,X2)
r167: a__U52(X) -> U52(X)
r168: a__U61(X1,X2) -> U61(X1,X2)
r169: a__U62(X) -> U62(X)
r170: a__U71(X) -> U71(X)
r171: a__U81(X) -> U81(X)
r172: a__U91(X1,X2,X3) -> U91(X1,X2,X3)
r173: a__U92(X1,X2,X3) -> U92(X1,X2,X3)
r174: a__U93(X1,X2,X3) -> U93(X1,X2,X3)
r175: a__U94(X1,X2,X3) -> U94(X1,X2,X3)
r176: a__U95(X1,X2) -> U95(X1,X2)
r177: a__U96(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

Take the reduction pair:

  max/plus interpretations on natural numbers:
  
    a__U136#_A(x1,x2,x3,x4) = x4 + 18
    tt_A = 0
    mark#_A(x1) = max{18, x1 + 1}
    s_A(x1) = max{60, x1}
    cons_A(x1,x2) = max{x1, x2}
    U96_A(x1) = max{18, x1}
    U95_A(x1,x2) = max{x1, x2 + 22}
    U94_A(x1,x2,x3) = max{x1, x2 + 9, x3 + 22}
    U93_A(x1,x2,x3) = max{x1 + 11, x2 + 9, x3 + 22}
    U92_A(x1,x2,x3) = max{x1 + 11, x2 + 9, x3 + 22}
    U91_A(x1,x2,x3) = max{x1 + 7, x2 + 18, x3 + 22}
    U81_A(x1) = x1
    U71_A(x1) = x1
    U62_A(x1) = max{1, x1}
    U61_A(x1,x2) = max{11, x1}
    U52_A(x1) = max{11, x1}
    U51_A(x1,x2) = max{11, x1}
    U46_A(x1) = max{21, x1}
    U45_A(x1,x2) = max{x1 + 18, x2 + 24}
    U44_A(x1,x2,x3) = max{x1 + 13, x2 - 1, x3 + 24}
    U43_A(x1,x2,x3) = max{x1 + 2, x2 - 1, x3 + 24}
    U42_A(x1,x2,x3) = max{x1, x2 - 1, x3 + 24}
    U41_A(x1,x2,x3) = max{x1 + 9, x2 + 11, x3 + 24}
    U33_A(x1) = max{2, x1 + 1}
    U32_A(x1,x2) = max{x1, x2 + 23}
    U31_A(x1,x2) = max{24, x1, x2 + 23}
    U23_A(x1) = x1
    U22_A(x1,x2) = max{x1 + 3, x2 - 38}
    U21_A(x1,x2) = max{17, x1 + 1, x2 - 38}
    take_A(x1,x2) = max{x1, x2 + 26}
    a__take#_A(x1,x2) = max{x1 + 1, x2 + 27}
    mark_A(x1) = x1
    a__U131#_A(x1,x2,x3,x4) = max{x2 + 27, x4 + 19}
    a__isNatIList_A(x1) = x1 + 24
    a__U132#_A(x1,x2,x3,x4) = max{x2 + 18, x4 + 19}
    a__isNatIListKind_A(x1) = 11
    a__U133#_A(x1,x2,x3,x4) = max{x2 + 12, x4 + 19}
    a__isNat_A(x1) = max{0, x1 - 38}
    a__U134#_A(x1,x2,x3,x4) = max{x1 + 1, x2 + 12, x4 + 19}
    a__isNatKind_A(x1) = 11
    a__U135#_A(x1,x2,x3,x4) = x4 + 18
    U136_A(x1,x2,x3,x4) = max{x1, x2 + 26, x3, x4 + 18}
    U135_A(x1,x2,x3,x4) = max{x1 + 38, x2 + 26, x3, x4 + 18}
    U134_A(x1,x2,x3,x4) = max{38, x1, x2 + 26, x3, x4 + 19}
    U133_A(x1,x2,x3,x4) = max{x1 + 38, x2 + 26, x3, x4 + 19}
    U132_A(x1,x2,x3,x4) = max{38, x1, x2 + 26, x3, x4 + 26}
    U131_A(x1,x2,x3,x4) = max{38, x1, x2 + 26, x3, x4 + 26}
    U122_A(x1) = x1 + 1
    U121_A(x1,x2) = max{x1, x2 + 18}
    U13_A(x1) = x1 + 1
    length_A(x1) = x1 + 67
    a__length#_A(x1) = x1 + 18
    a__U111#_A(x1,x2,x3) = max{x2 + 18, x3 + 18}
    a__isNatList_A(x1) = x1 + 22
    a__U112#_A(x1,x2,x3) = x2 + 18
    a__U113#_A(x1,x2,x3) = x2 + 18
    a__U114#_A(x1,x2) = x2 + 18
    U114_A(x1,x2) = max{x1 + 51, x2 + 67}
    U113_A(x1,x2,x3) = max{x1 + 57, x2 + 67}
    U112_A(x1,x2,x3) = max{x1 + 56, x2 + 67, x3 + 19}
    U111_A(x1,x2,x3) = max{x1, x2 + 67, x3 + 52}
    U12_A(x1,x2) = max{x1, x2 + 23}
    U11_A(x1,x2) = max{x1, x2 + 23}
    U106_A(x1) = x1 + 1
    U105_A(x1,x2) = max{x1, x2 + 25}
    U104_A(x1,x2,x3) = max{x1, x2 + 21, x3 + 25}
    U103_A(x1,x2,x3) = max{x1, x2 + 21, x3 + 25}
    U102_A(x1,x2,x3) = max{x1 + 19, x2 + 21, x3 + 26}
    U101_A(x1,x2,x3) = max{x1, x2 + 22, x3 + 30}
    a__zeros_A = 0
    |0|_A = 0
    zeros_A = 0
    a__U101_A(x1,x2,x3) = max{x1, x2 + 22, x3 + 30}
    a__U102_A(x1,x2,x3) = max{x1 + 19, x2 + 21, x3 + 26}
    a__U103_A(x1,x2,x3) = max{x1, x2 + 21, x3 + 25}
    a__U104_A(x1,x2,x3) = max{x1, x2 + 21, x3 + 25}
    a__U105_A(x1,x2) = max{x1, x2 + 25}
    a__U106_A(x1) = x1 + 1
    a__U11_A(x1,x2) = max{x1, x2 + 23}
    a__U12_A(x1,x2) = max{x1, x2 + 23}
    a__U111_A(x1,x2,x3) = max{x1, x2 + 67, x3 + 52}
    a__U112_A(x1,x2,x3) = max{x1 + 56, x2 + 67, x3 + 19}
    a__U113_A(x1,x2,x3) = max{x1 + 57, x2 + 67}
    a__U114_A(x1,x2) = max{x1 + 51, x2 + 67}
    a__length_A(x1) = x1 + 67
    a__U13_A(x1) = x1 + 1
    a__U121_A(x1,x2) = max{x1, x2 + 18}
    a__U122_A(x1) = x1 + 1
    nil_A = 0
    a__U131_A(x1,x2,x3,x4) = max{38, x1, x2 + 26, x3, x4 + 26}
    a__U132_A(x1,x2,x3,x4) = max{38, x1, x2 + 26, x3, x4 + 26}
    a__U133_A(x1,x2,x3,x4) = max{x1 + 38, x2 + 26, x3, x4 + 19}
    a__U134_A(x1,x2,x3,x4) = max{38, x1, x2 + 26, x3, x4 + 19}
    a__U135_A(x1,x2,x3,x4) = max{x1 + 38, x2 + 26, x3, x4 + 18}
    a__U136_A(x1,x2,x3,x4) = max{x1, x2 + 26, x3, x4 + 18}
    a__U21_A(x1,x2) = max{17, x1 + 1, x2 - 38}
    a__U22_A(x1,x2) = max{x1 + 3, x2 - 38}
    a__U23_A(x1) = x1
    a__U31_A(x1,x2) = max{24, x1, x2 + 23}
    a__U32_A(x1,x2) = max{x1, x2 + 23}
    a__U33_A(x1) = max{2, x1 + 1}
    a__U41_A(x1,x2,x3) = max{x1 + 9, x2 + 11, x3 + 24}
    a__U42_A(x1,x2,x3) = max{x1, x2 - 1, x3 + 24}
    a__U43_A(x1,x2,x3) = max{x1 + 2, x2 - 1, x3 + 24}
    a__U44_A(x1,x2,x3) = max{x1 + 13, x2 - 1, x3 + 24}
    a__U45_A(x1,x2) = max{x1 + 18, x2 + 24}
    a__U46_A(x1) = max{21, x1}
    a__U51_A(x1,x2) = max{11, x1}
    a__U52_A(x1) = max{11, x1}
    a__U61_A(x1,x2) = max{11, x1}
    a__U62_A(x1) = max{1, x1}
    a__U71_A(x1) = x1
    a__U81_A(x1) = x1
    a__U91_A(x1,x2,x3) = max{x1 + 7, x2 + 18, x3 + 22}
    a__U92_A(x1,x2,x3) = max{x1 + 11, x2 + 9, x3 + 22}
    a__U93_A(x1,x2,x3) = max{x1 + 11, x2 + 9, x3 + 22}
    a__U94_A(x1,x2,x3) = max{x1, x2 + 9, x3 + 22}
    a__U95_A(x1,x2) = max{x1, x2 + 22}
    a__U96_A(x1) = max{18, x1}
    a__take_A(x1,x2) = max{x1, x2 + 26}
    isNatKind_A(x1) = 11
    isNatIListKind_A(x1) = 11
    isNat_A(x1) = max{0, x1 - 38}
    isNatIList_A(x1) = x1 + 24
    isNatList_A(x1) = x1 + 22

The next rules are strictly ordered:

  p7, p8, p9, p17, p18, p19, p21, p28, p35, p38, p39, p40, p43, p44, p46, p52, p53, p59, p60, p62, p63, p64, p65, p67, p74

We remove them from the problem.

-- SCC decomposition.

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

p1: a__U136#(tt(),IL,M,N) -> mark#(N)
p2: mark#(s(X)) -> mark#(X)
p3: mark#(cons(X1,X2)) -> mark#(X1)
p4: mark#(U96(X)) -> mark#(X)
p5: mark#(U95(X1,X2)) -> mark#(X1)
p6: mark#(U94(X1,X2,X3)) -> mark#(X1)
p7: mark#(U81(X)) -> mark#(X)
p8: mark#(U71(X)) -> mark#(X)
p9: mark#(U62(X)) -> mark#(X)
p10: mark#(U61(X1,X2)) -> mark#(X1)
p11: mark#(U52(X)) -> mark#(X)
p12: mark#(U51(X1,X2)) -> mark#(X1)
p13: mark#(U46(X)) -> mark#(X)
p14: mark#(U42(X1,X2,X3)) -> mark#(X1)
p15: mark#(U33(X)) -> mark#(X)
p16: mark#(U32(X1,X2)) -> mark#(X1)
p17: mark#(U31(X1,X2)) -> mark#(X1)
p18: mark#(U23(X)) -> mark#(X)
p19: mark#(U22(X1,X2)) -> mark#(X1)
p20: mark#(U21(X1,X2)) -> mark#(X1)
p21: mark#(take(X1,X2)) -> mark#(X1)
p22: mark#(take(X1,X2)) -> a__take#(mark(X1),mark(X2))
p23: a__take#(s(M),cons(N,IL)) -> a__U131#(a__isNatIList(IL),IL,M,N)
p24: a__U131#(tt(),IL,M,N) -> a__U132#(a__isNatIListKind(IL),IL,M,N)
p25: a__U132#(tt(),IL,M,N) -> a__U133#(a__isNat(M),IL,M,N)
p26: a__U133#(tt(),IL,M,N) -> a__U134#(a__isNatKind(M),IL,M,N)
p27: a__U135#(tt(),IL,M,N) -> a__U136#(a__isNatKind(N),IL,M,N)
p28: mark#(U136(X1,X2,X3,X4)) -> mark#(X1)
p29: mark#(U134(X1,X2,X3,X4)) -> mark#(X1)
p30: mark#(U134(X1,X2,X3,X4)) -> a__U134#(mark(X1),X2,X3,X4)
p31: mark#(U132(X1,X2,X3,X4)) -> mark#(X1)
p32: mark#(U131(X1,X2,X3,X4)) -> mark#(X1)
p33: mark#(U131(X1,X2,X3,X4)) -> a__U131#(mark(X1),X2,X3,X4)
p34: mark#(U122(X)) -> mark#(X)
p35: mark#(U121(X1,X2)) -> mark#(X1)
p36: mark#(U13(X)) -> mark#(X)
p37: a__length#(cons(N,L)) -> a__U111#(a__isNatList(L),L,N)
p38: a__U111#(tt(),L,N) -> a__U112#(a__isNatIListKind(L),L,N)
p39: a__U112#(tt(),L,N) -> a__U113#(a__isNat(N),L,N)
p40: a__U113#(tt(),L,N) -> a__U114#(a__isNatKind(N),L)
p41: a__U114#(tt(),L) -> mark#(L)
p42: a__U114#(tt(),L) -> a__length#(mark(L))
p43: mark#(U111(X1,X2,X3)) -> mark#(X1)
p44: mark#(U12(X1,X2)) -> mark#(X1)
p45: mark#(U11(X1,X2)) -> mark#(X1)
p46: mark#(U106(X)) -> mark#(X)
p47: mark#(U105(X1,X2)) -> mark#(X1)
p48: mark#(U104(X1,X2,X3)) -> mark#(X1)
p49: mark#(U103(X1,X2,X3)) -> mark#(X1)
p50: mark#(U101(X1,X2,X3)) -> mark#(X1)

and R consists of:

r1: a__zeros() -> cons(|0|(),zeros())
r2: a__U101(tt(),V1,V2) -> a__U102(a__isNatKind(V1),V1,V2)
r3: a__U102(tt(),V1,V2) -> a__U103(a__isNatIListKind(V2),V1,V2)
r4: a__U103(tt(),V1,V2) -> a__U104(a__isNatIListKind(V2),V1,V2)
r5: a__U104(tt(),V1,V2) -> a__U105(a__isNat(V1),V2)
r6: a__U105(tt(),V2) -> a__U106(a__isNatIList(V2))
r7: a__U106(tt()) -> tt()
r8: a__U11(tt(),V1) -> a__U12(a__isNatIListKind(V1),V1)
r9: a__U111(tt(),L,N) -> a__U112(a__isNatIListKind(L),L,N)
r10: a__U112(tt(),L,N) -> a__U113(a__isNat(N),L,N)
r11: a__U113(tt(),L,N) -> a__U114(a__isNatKind(N),L)
r12: a__U114(tt(),L) -> s(a__length(mark(L)))
r13: a__U12(tt(),V1) -> a__U13(a__isNatList(V1))
r14: a__U121(tt(),IL) -> a__U122(a__isNatIListKind(IL))
r15: a__U122(tt()) -> nil()
r16: a__U13(tt()) -> tt()
r17: a__U131(tt(),IL,M,N) -> a__U132(a__isNatIListKind(IL),IL,M,N)
r18: a__U132(tt(),IL,M,N) -> a__U133(a__isNat(M),IL,M,N)
r19: a__U133(tt(),IL,M,N) -> a__U134(a__isNatKind(M),IL,M,N)
r20: a__U134(tt(),IL,M,N) -> a__U135(a__isNat(N),IL,M,N)
r21: a__U135(tt(),IL,M,N) -> a__U136(a__isNatKind(N),IL,M,N)
r22: a__U136(tt(),IL,M,N) -> cons(mark(N),take(M,IL))
r23: a__U21(tt(),V1) -> a__U22(a__isNatKind(V1),V1)
r24: a__U22(tt(),V1) -> a__U23(a__isNat(V1))
r25: a__U23(tt()) -> tt()
r26: a__U31(tt(),V) -> a__U32(a__isNatIListKind(V),V)
r27: a__U32(tt(),V) -> a__U33(a__isNatList(V))
r28: a__U33(tt()) -> tt()
r29: a__U41(tt(),V1,V2) -> a__U42(a__isNatKind(V1),V1,V2)
r30: a__U42(tt(),V1,V2) -> a__U43(a__isNatIListKind(V2),V1,V2)
r31: a__U43(tt(),V1,V2) -> a__U44(a__isNatIListKind(V2),V1,V2)
r32: a__U44(tt(),V1,V2) -> a__U45(a__isNat(V1),V2)
r33: a__U45(tt(),V2) -> a__U46(a__isNatIList(V2))
r34: a__U46(tt()) -> tt()
r35: a__U51(tt(),V2) -> a__U52(a__isNatIListKind(V2))
r36: a__U52(tt()) -> tt()
r37: a__U61(tt(),V2) -> a__U62(a__isNatIListKind(V2))
r38: a__U62(tt()) -> tt()
r39: a__U71(tt()) -> tt()
r40: a__U81(tt()) -> tt()
r41: a__U91(tt(),V1,V2) -> a__U92(a__isNatKind(V1),V1,V2)
r42: a__U92(tt(),V1,V2) -> a__U93(a__isNatIListKind(V2),V1,V2)
r43: a__U93(tt(),V1,V2) -> a__U94(a__isNatIListKind(V2),V1,V2)
r44: a__U94(tt(),V1,V2) -> a__U95(a__isNat(V1),V2)
r45: a__U95(tt(),V2) -> a__U96(a__isNatList(V2))
r46: a__U96(tt()) -> tt()
r47: a__isNat(|0|()) -> tt()
r48: a__isNat(length(V1)) -> a__U11(a__isNatIListKind(V1),V1)
r49: a__isNat(s(V1)) -> a__U21(a__isNatKind(V1),V1)
r50: a__isNatIList(V) -> a__U31(a__isNatIListKind(V),V)
r51: a__isNatIList(zeros()) -> tt()
r52: a__isNatIList(cons(V1,V2)) -> a__U41(a__isNatKind(V1),V1,V2)
r53: a__isNatIListKind(nil()) -> tt()
r54: a__isNatIListKind(zeros()) -> tt()
r55: a__isNatIListKind(cons(V1,V2)) -> a__U51(a__isNatKind(V1),V2)
r56: a__isNatIListKind(take(V1,V2)) -> a__U61(a__isNatKind(V1),V2)
r57: a__isNatKind(|0|()) -> tt()
r58: a__isNatKind(length(V1)) -> a__U71(a__isNatIListKind(V1))
r59: a__isNatKind(s(V1)) -> a__U81(a__isNatKind(V1))
r60: a__isNatList(nil()) -> tt()
r61: a__isNatList(cons(V1,V2)) -> a__U91(a__isNatKind(V1),V1,V2)
r62: a__isNatList(take(V1,V2)) -> a__U101(a__isNatKind(V1),V1,V2)
r63: a__length(nil()) -> |0|()
r64: a__length(cons(N,L)) -> a__U111(a__isNatList(L),L,N)
r65: a__take(|0|(),IL) -> a__U121(a__isNatIList(IL),IL)
r66: a__take(s(M),cons(N,IL)) -> a__U131(a__isNatIList(IL),IL,M,N)
r67: mark(zeros()) -> a__zeros()
r68: mark(U101(X1,X2,X3)) -> a__U101(mark(X1),X2,X3)
r69: mark(U102(X1,X2,X3)) -> a__U102(mark(X1),X2,X3)
r70: mark(isNatKind(X)) -> a__isNatKind(X)
r71: mark(U103(X1,X2,X3)) -> a__U103(mark(X1),X2,X3)
r72: mark(isNatIListKind(X)) -> a__isNatIListKind(X)
r73: mark(U104(X1,X2,X3)) -> a__U104(mark(X1),X2,X3)
r74: mark(U105(X1,X2)) -> a__U105(mark(X1),X2)
r75: mark(isNat(X)) -> a__isNat(X)
r76: mark(U106(X)) -> a__U106(mark(X))
r77: mark(isNatIList(X)) -> a__isNatIList(X)
r78: mark(U11(X1,X2)) -> a__U11(mark(X1),X2)
r79: mark(U12(X1,X2)) -> a__U12(mark(X1),X2)
r80: mark(U111(X1,X2,X3)) -> a__U111(mark(X1),X2,X3)
r81: mark(U112(X1,X2,X3)) -> a__U112(mark(X1),X2,X3)
r82: mark(U113(X1,X2,X3)) -> a__U113(mark(X1),X2,X3)
r83: mark(U114(X1,X2)) -> a__U114(mark(X1),X2)
r84: mark(length(X)) -> a__length(mark(X))
r85: mark(U13(X)) -> a__U13(mark(X))
r86: mark(isNatList(X)) -> a__isNatList(X)
r87: mark(U121(X1,X2)) -> a__U121(mark(X1),X2)
r88: mark(U122(X)) -> a__U122(mark(X))
r89: mark(U131(X1,X2,X3,X4)) -> a__U131(mark(X1),X2,X3,X4)
r90: mark(U132(X1,X2,X3,X4)) -> a__U132(mark(X1),X2,X3,X4)
r91: mark(U133(X1,X2,X3,X4)) -> a__U133(mark(X1),X2,X3,X4)
r92: mark(U134(X1,X2,X3,X4)) -> a__U134(mark(X1),X2,X3,X4)
r93: mark(U135(X1,X2,X3,X4)) -> a__U135(mark(X1),X2,X3,X4)
r94: mark(U136(X1,X2,X3,X4)) -> a__U136(mark(X1),X2,X3,X4)
r95: mark(take(X1,X2)) -> a__take(mark(X1),mark(X2))
r96: mark(U21(X1,X2)) -> a__U21(mark(X1),X2)
r97: mark(U22(X1,X2)) -> a__U22(mark(X1),X2)
r98: mark(U23(X)) -> a__U23(mark(X))
r99: mark(U31(X1,X2)) -> a__U31(mark(X1),X2)
r100: mark(U32(X1,X2)) -> a__U32(mark(X1),X2)
r101: mark(U33(X)) -> a__U33(mark(X))
r102: mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3)
r103: mark(U42(X1,X2,X3)) -> a__U42(mark(X1),X2,X3)
r104: mark(U43(X1,X2,X3)) -> a__U43(mark(X1),X2,X3)
r105: mark(U44(X1,X2,X3)) -> a__U44(mark(X1),X2,X3)
r106: mark(U45(X1,X2)) -> a__U45(mark(X1),X2)
r107: mark(U46(X)) -> a__U46(mark(X))
r108: mark(U51(X1,X2)) -> a__U51(mark(X1),X2)
r109: mark(U52(X)) -> a__U52(mark(X))
r110: mark(U61(X1,X2)) -> a__U61(mark(X1),X2)
r111: mark(U62(X)) -> a__U62(mark(X))
r112: mark(U71(X)) -> a__U71(mark(X))
r113: mark(U81(X)) -> a__U81(mark(X))
r114: mark(U91(X1,X2,X3)) -> a__U91(mark(X1),X2,X3)
r115: mark(U92(X1,X2,X3)) -> a__U92(mark(X1),X2,X3)
r116: mark(U93(X1,X2,X3)) -> a__U93(mark(X1),X2,X3)
r117: mark(U94(X1,X2,X3)) -> a__U94(mark(X1),X2,X3)
r118: mark(U95(X1,X2)) -> a__U95(mark(X1),X2)
r119: mark(U96(X)) -> a__U96(mark(X))
r120: mark(cons(X1,X2)) -> cons(mark(X1),X2)
r121: mark(|0|()) -> |0|()
r122: mark(tt()) -> tt()
r123: mark(s(X)) -> s(mark(X))
r124: mark(nil()) -> nil()
r125: a__zeros() -> zeros()
r126: a__U101(X1,X2,X3) -> U101(X1,X2,X3)
r127: a__U102(X1,X2,X3) -> U102(X1,X2,X3)
r128: a__isNatKind(X) -> isNatKind(X)
r129: a__U103(X1,X2,X3) -> U103(X1,X2,X3)
r130: a__isNatIListKind(X) -> isNatIListKind(X)
r131: a__U104(X1,X2,X3) -> U104(X1,X2,X3)
r132: a__U105(X1,X2) -> U105(X1,X2)
r133: a__isNat(X) -> isNat(X)
r134: a__U106(X) -> U106(X)
r135: a__isNatIList(X) -> isNatIList(X)
r136: a__U11(X1,X2) -> U11(X1,X2)
r137: a__U12(X1,X2) -> U12(X1,X2)
r138: a__U111(X1,X2,X3) -> U111(X1,X2,X3)
r139: a__U112(X1,X2,X3) -> U112(X1,X2,X3)
r140: a__U113(X1,X2,X3) -> U113(X1,X2,X3)
r141: a__U114(X1,X2) -> U114(X1,X2)
r142: a__length(X) -> length(X)
r143: a__U13(X) -> U13(X)
r144: a__isNatList(X) -> isNatList(X)
r145: a__U121(X1,X2) -> U121(X1,X2)
r146: a__U122(X) -> U122(X)
r147: a__U131(X1,X2,X3,X4) -> U131(X1,X2,X3,X4)
r148: a__U132(X1,X2,X3,X4) -> U132(X1,X2,X3,X4)
r149: a__U133(X1,X2,X3,X4) -> U133(X1,X2,X3,X4)
r150: a__U134(X1,X2,X3,X4) -> U134(X1,X2,X3,X4)
r151: a__U135(X1,X2,X3,X4) -> U135(X1,X2,X3,X4)
r152: a__U136(X1,X2,X3,X4) -> U136(X1,X2,X3,X4)
r153: a__take(X1,X2) -> take(X1,X2)
r154: a__U21(X1,X2) -> U21(X1,X2)
r155: a__U22(X1,X2) -> U22(X1,X2)
r156: a__U23(X) -> U23(X)
r157: a__U31(X1,X2) -> U31(X1,X2)
r158: a__U32(X1,X2) -> U32(X1,X2)
r159: a__U33(X) -> U33(X)
r160: a__U41(X1,X2,X3) -> U41(X1,X2,X3)
r161: a__U42(X1,X2,X3) -> U42(X1,X2,X3)
r162: a__U43(X1,X2,X3) -> U43(X1,X2,X3)
r163: a__U44(X1,X2,X3) -> U44(X1,X2,X3)
r164: a__U45(X1,X2) -> U45(X1,X2)
r165: a__U46(X) -> U46(X)
r166: a__U51(X1,X2) -> U51(X1,X2)
r167: a__U52(X) -> U52(X)
r168: a__U61(X1,X2) -> U61(X1,X2)
r169: a__U62(X) -> U62(X)
r170: a__U71(X) -> U71(X)
r171: a__U81(X) -> U81(X)
r172: a__U91(X1,X2,X3) -> U91(X1,X2,X3)
r173: a__U92(X1,X2,X3) -> U92(X1,X2,X3)
r174: a__U93(X1,X2,X3) -> U93(X1,X2,X3)
r175: a__U94(X1,X2,X3) -> U94(X1,X2,X3)
r176: a__U95(X1,X2) -> U95(X1,X2)
r177: a__U96(X) -> U96(X)

The estimated dependency graph contains the following SCCs:

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


-- Reduction pair.

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

p1: a__U113#(tt(),L,N) -> a__U114#(a__isNatKind(N),L)
p2: a__U114#(tt(),L) -> a__length#(mark(L))
p3: a__length#(cons(N,L)) -> a__U111#(a__isNatList(L),L,N)
p4: a__U111#(tt(),L,N) -> a__U112#(a__isNatIListKind(L),L,N)
p5: a__U112#(tt(),L,N) -> a__U113#(a__isNat(N),L,N)

and R consists of:

r1: a__zeros() -> cons(|0|(),zeros())
r2: a__U101(tt(),V1,V2) -> a__U102(a__isNatKind(V1),V1,V2)
r3: a__U102(tt(),V1,V2) -> a__U103(a__isNatIListKind(V2),V1,V2)
r4: a__U103(tt(),V1,V2) -> a__U104(a__isNatIListKind(V2),V1,V2)
r5: a__U104(tt(),V1,V2) -> a__U105(a__isNat(V1),V2)
r6: a__U105(tt(),V2) -> a__U106(a__isNatIList(V2))
r7: a__U106(tt()) -> tt()
r8: a__U11(tt(),V1) -> a__U12(a__isNatIListKind(V1),V1)
r9: a__U111(tt(),L,N) -> a__U112(a__isNatIListKind(L),L,N)
r10: a__U112(tt(),L,N) -> a__U113(a__isNat(N),L,N)
r11: a__U113(tt(),L,N) -> a__U114(a__isNatKind(N),L)
r12: a__U114(tt(),L) -> s(a__length(mark(L)))
r13: a__U12(tt(),V1) -> a__U13(a__isNatList(V1))
r14: a__U121(tt(),IL) -> a__U122(a__isNatIListKind(IL))
r15: a__U122(tt()) -> nil()
r16: a__U13(tt()) -> tt()
r17: a__U131(tt(),IL,M,N) -> a__U132(a__isNatIListKind(IL),IL,M,N)
r18: a__U132(tt(),IL,M,N) -> a__U133(a__isNat(M),IL,M,N)
r19: a__U133(tt(),IL,M,N) -> a__U134(a__isNatKind(M),IL,M,N)
r20: a__U134(tt(),IL,M,N) -> a__U135(a__isNat(N),IL,M,N)
r21: a__U135(tt(),IL,M,N) -> a__U136(a__isNatKind(N),IL,M,N)
r22: a__U136(tt(),IL,M,N) -> cons(mark(N),take(M,IL))
r23: a__U21(tt(),V1) -> a__U22(a__isNatKind(V1),V1)
r24: a__U22(tt(),V1) -> a__U23(a__isNat(V1))
r25: a__U23(tt()) -> tt()
r26: a__U31(tt(),V) -> a__U32(a__isNatIListKind(V),V)
r27: a__U32(tt(),V) -> a__U33(a__isNatList(V))
r28: a__U33(tt()) -> tt()
r29: a__U41(tt(),V1,V2) -> a__U42(a__isNatKind(V1),V1,V2)
r30: a__U42(tt(),V1,V2) -> a__U43(a__isNatIListKind(V2),V1,V2)
r31: a__U43(tt(),V1,V2) -> a__U44(a__isNatIListKind(V2),V1,V2)
r32: a__U44(tt(),V1,V2) -> a__U45(a__isNat(V1),V2)
r33: a__U45(tt(),V2) -> a__U46(a__isNatIList(V2))
r34: a__U46(tt()) -> tt()
r35: a__U51(tt(),V2) -> a__U52(a__isNatIListKind(V2))
r36: a__U52(tt()) -> tt()
r37: a__U61(tt(),V2) -> a__U62(a__isNatIListKind(V2))
r38: a__U62(tt()) -> tt()
r39: a__U71(tt()) -> tt()
r40: a__U81(tt()) -> tt()
r41: a__U91(tt(),V1,V2) -> a__U92(a__isNatKind(V1),V1,V2)
r42: a__U92(tt(),V1,V2) -> a__U93(a__isNatIListKind(V2),V1,V2)
r43: a__U93(tt(),V1,V2) -> a__U94(a__isNatIListKind(V2),V1,V2)
r44: a__U94(tt(),V1,V2) -> a__U95(a__isNat(V1),V2)
r45: a__U95(tt(),V2) -> a__U96(a__isNatList(V2))
r46: a__U96(tt()) -> tt()
r47: a__isNat(|0|()) -> tt()
r48: a__isNat(length(V1)) -> a__U11(a__isNatIListKind(V1),V1)
r49: a__isNat(s(V1)) -> a__U21(a__isNatKind(V1),V1)
r50: a__isNatIList(V) -> a__U31(a__isNatIListKind(V),V)
r51: a__isNatIList(zeros()) -> tt()
r52: a__isNatIList(cons(V1,V2)) -> a__U41(a__isNatKind(V1),V1,V2)
r53: a__isNatIListKind(nil()) -> tt()
r54: a__isNatIListKind(zeros()) -> tt()
r55: a__isNatIListKind(cons(V1,V2)) -> a__U51(a__isNatKind(V1),V2)
r56: a__isNatIListKind(take(V1,V2)) -> a__U61(a__isNatKind(V1),V2)
r57: a__isNatKind(|0|()) -> tt()
r58: a__isNatKind(length(V1)) -> a__U71(a__isNatIListKind(V1))
r59: a__isNatKind(s(V1)) -> a__U81(a__isNatKind(V1))
r60: a__isNatList(nil()) -> tt()
r61: a__isNatList(cons(V1,V2)) -> a__U91(a__isNatKind(V1),V1,V2)
r62: a__isNatList(take(V1,V2)) -> a__U101(a__isNatKind(V1),V1,V2)
r63: a__length(nil()) -> |0|()
r64: a__length(cons(N,L)) -> a__U111(a__isNatList(L),L,N)
r65: a__take(|0|(),IL) -> a__U121(a__isNatIList(IL),IL)
r66: a__take(s(M),cons(N,IL)) -> a__U131(a__isNatIList(IL),IL,M,N)
r67: mark(zeros()) -> a__zeros()
r68: mark(U101(X1,X2,X3)) -> a__U101(mark(X1),X2,X3)
r69: mark(U102(X1,X2,X3)) -> a__U102(mark(X1),X2,X3)
r70: mark(isNatKind(X)) -> a__isNatKind(X)
r71: mark(U103(X1,X2,X3)) -> a__U103(mark(X1),X2,X3)
r72: mark(isNatIListKind(X)) -> a__isNatIListKind(X)
r73: mark(U104(X1,X2,X3)) -> a__U104(mark(X1),X2,X3)
r74: mark(U105(X1,X2)) -> a__U105(mark(X1),X2)
r75: mark(isNat(X)) -> a__isNat(X)
r76: mark(U106(X)) -> a__U106(mark(X))
r77: mark(isNatIList(X)) -> a__isNatIList(X)
r78: mark(U11(X1,X2)) -> a__U11(mark(X1),X2)
r79: mark(U12(X1,X2)) -> a__U12(mark(X1),X2)
r80: mark(U111(X1,X2,X3)) -> a__U111(mark(X1),X2,X3)
r81: mark(U112(X1,X2,X3)) -> a__U112(mark(X1),X2,X3)
r82: mark(U113(X1,X2,X3)) -> a__U113(mark(X1),X2,X3)
r83: mark(U114(X1,X2)) -> a__U114(mark(X1),X2)
r84: mark(length(X)) -> a__length(mark(X))
r85: mark(U13(X)) -> a__U13(mark(X))
r86: mark(isNatList(X)) -> a__isNatList(X)
r87: mark(U121(X1,X2)) -> a__U121(mark(X1),X2)
r88: mark(U122(X)) -> a__U122(mark(X))
r89: mark(U131(X1,X2,X3,X4)) -> a__U131(mark(X1),X2,X3,X4)
r90: mark(U132(X1,X2,X3,X4)) -> a__U132(mark(X1),X2,X3,X4)
r91: mark(U133(X1,X2,X3,X4)) -> a__U133(mark(X1),X2,X3,X4)
r92: mark(U134(X1,X2,X3,X4)) -> a__U134(mark(X1),X2,X3,X4)
r93: mark(U135(X1,X2,X3,X4)) -> a__U135(mark(X1),X2,X3,X4)
r94: mark(U136(X1,X2,X3,X4)) -> a__U136(mark(X1),X2,X3,X4)
r95: mark(take(X1,X2)) -> a__take(mark(X1),mark(X2))
r96: mark(U21(X1,X2)) -> a__U21(mark(X1),X2)
r97: mark(U22(X1,X2)) -> a__U22(mark(X1),X2)
r98: mark(U23(X)) -> a__U23(mark(X))
r99: mark(U31(X1,X2)) -> a__U31(mark(X1),X2)
r100: mark(U32(X1,X2)) -> a__U32(mark(X1),X2)
r101: mark(U33(X)) -> a__U33(mark(X))
r102: mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3)
r103: mark(U42(X1,X2,X3)) -> a__U42(mark(X1),X2,X3)
r104: mark(U43(X1,X2,X3)) -> a__U43(mark(X1),X2,X3)
r105: mark(U44(X1,X2,X3)) -> a__U44(mark(X1),X2,X3)
r106: mark(U45(X1,X2)) -> a__U45(mark(X1),X2)
r107: mark(U46(X)) -> a__U46(mark(X))
r108: mark(U51(X1,X2)) -> a__U51(mark(X1),X2)
r109: mark(U52(X)) -> a__U52(mark(X))
r110: mark(U61(X1,X2)) -> a__U61(mark(X1),X2)
r111: mark(U62(X)) -> a__U62(mark(X))
r112: mark(U71(X)) -> a__U71(mark(X))
r113: mark(U81(X)) -> a__U81(mark(X))
r114: mark(U91(X1,X2,X3)) -> a__U91(mark(X1),X2,X3)
r115: mark(U92(X1,X2,X3)) -> a__U92(mark(X1),X2,X3)
r116: mark(U93(X1,X2,X3)) -> a__U93(mark(X1),X2,X3)
r117: mark(U94(X1,X2,X3)) -> a__U94(mark(X1),X2,X3)
r118: mark(U95(X1,X2)) -> a__U95(mark(X1),X2)
r119: mark(U96(X)) -> a__U96(mark(X))
r120: mark(cons(X1,X2)) -> cons(mark(X1),X2)
r121: mark(|0|()) -> |0|()
r122: mark(tt()) -> tt()
r123: mark(s(X)) -> s(mark(X))
r124: mark(nil()) -> nil()
r125: a__zeros() -> zeros()
r126: a__U101(X1,X2,X3) -> U101(X1,X2,X3)
r127: a__U102(X1,X2,X3) -> U102(X1,X2,X3)
r128: a__isNatKind(X) -> isNatKind(X)
r129: a__U103(X1,X2,X3) -> U103(X1,X2,X3)
r130: a__isNatIListKind(X) -> isNatIListKind(X)
r131: a__U104(X1,X2,X3) -> U104(X1,X2,X3)
r132: a__U105(X1,X2) -> U105(X1,X2)
r133: a__isNat(X) -> isNat(X)
r134: a__U106(X) -> U106(X)
r135: a__isNatIList(X) -> isNatIList(X)
r136: a__U11(X1,X2) -> U11(X1,X2)
r137: a__U12(X1,X2) -> U12(X1,X2)
r138: a__U111(X1,X2,X3) -> U111(X1,X2,X3)
r139: a__U112(X1,X2,X3) -> U112(X1,X2,X3)
r140: a__U113(X1,X2,X3) -> U113(X1,X2,X3)
r141: a__U114(X1,X2) -> U114(X1,X2)
r142: a__length(X) -> length(X)
r143: a__U13(X) -> U13(X)
r144: a__isNatList(X) -> isNatList(X)
r145: a__U121(X1,X2) -> U121(X1,X2)
r146: a__U122(X) -> U122(X)
r147: a__U131(X1,X2,X3,X4) -> U131(X1,X2,X3,X4)
r148: a__U132(X1,X2,X3,X4) -> U132(X1,X2,X3,X4)
r149: a__U133(X1,X2,X3,X4) -> U133(X1,X2,X3,X4)
r150: a__U134(X1,X2,X3,X4) -> U134(X1,X2,X3,X4)
r151: a__U135(X1,X2,X3,X4) -> U135(X1,X2,X3,X4)
r152: a__U136(X1,X2,X3,X4) -> U136(X1,X2,X3,X4)
r153: a__take(X1,X2) -> take(X1,X2)
r154: a__U21(X1,X2) -> U21(X1,X2)
r155: a__U22(X1,X2) -> U22(X1,X2)
r156: a__U23(X) -> U23(X)
r157: a__U31(X1,X2) -> U31(X1,X2)
r158: a__U32(X1,X2) -> U32(X1,X2)
r159: a__U33(X) -> U33(X)
r160: a__U41(X1,X2,X3) -> U41(X1,X2,X3)
r161: a__U42(X1,X2,X3) -> U42(X1,X2,X3)
r162: a__U43(X1,X2,X3) -> U43(X1,X2,X3)
r163: a__U44(X1,X2,X3) -> U44(X1,X2,X3)
r164: a__U45(X1,X2) -> U45(X1,X2)
r165: a__U46(X) -> U46(X)
r166: a__U51(X1,X2) -> U51(X1,X2)
r167: a__U52(X) -> U52(X)
r168: a__U61(X1,X2) -> U61(X1,X2)
r169: a__U62(X) -> U62(X)
r170: a__U71(X) -> U71(X)
r171: a__U81(X) -> U81(X)
r172: a__U91(X1,X2,X3) -> U91(X1,X2,X3)
r173: a__U92(X1,X2,X3) -> U92(X1,X2,X3)
r174: a__U93(X1,X2,X3) -> U93(X1,X2,X3)
r175: a__U94(X1,X2,X3) -> U94(X1,X2,X3)
r176: a__U95(X1,X2) -> U95(X1,X2)
r177: a__U96(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

Take the reduction pair:

  max/plus interpretations on natural numbers:
  
    a__U113#_A(x1,x2,x3) = max{233, x1 + 113, x2 + 232}
    tt_A = 113
    a__U114#_A(x1,x2) = max{x1 + 119, x2 + 195}
    a__isNatKind_A(x1) = 113
    a__length#_A(x1) = x1 + 194
    mark_A(x1) = max{38, x1}
    cons_A(x1,x2) = x2 + 38
    a__U111#_A(x1,x2,x3) = max{x1 + 121, x2 + 232}
    a__isNatList_A(x1) = max{53, x1 + 51}
    a__U112#_A(x1,x2,x3) = max{233, x2 + 232}
    a__isNatIListKind_A(x1) = 124
    a__isNat_A(x1) = 119
    a__zeros_A = 38
    |0|_A = 88
    zeros_A = 0
    a__U101_A(x1,x2,x3) = 126
    a__U102_A(x1,x2,x3) = max{126, x1 + 1}
    a__U103_A(x1,x2,x3) = 126
    a__U104_A(x1,x2,x3) = max{126, x1 + 1}
    a__U105_A(x1,x2) = max{115, x1 - 4}
    a__U106_A(x1) = 115
    a__isNatIList_A(x1) = x1 + 113
    a__U11_A(x1,x2) = 119
    a__U12_A(x1,x2) = 113
    a__U111_A(x1,x2,x3) = max{228, x1 + 175, x2 + 227}
    a__U112_A(x1,x2,x3) = max{229, x1 + 153, x2 + 227}
    a__U113_A(x1,x2,x3) = max{266, x1 + 112, x2 + 227}
    a__U114_A(x1,x2) = max{266, x1 + 113, x2 + 227}
    s_A(x1) = max{227, x1 + 38}
    a__length_A(x1) = max{228, x1 + 189}
    a__U13_A(x1) = 113
    a__U121_A(x1,x2) = 62
    a__U122_A(x1) = 62
    nil_A = 62
    a__U131_A(x1,x2,x3,x4) = max{120, x3 - 69}
    a__U132_A(x1,x2,x3,x4) = max{120, x1 - 106, x3 - 69}
    a__U133_A(x1,x2,x3,x4) = max{40, x1 + 1, x3 - 69}
    a__U134_A(x1,x2,x3,x4) = max{113, x3 - 69}
    a__U135_A(x1,x2,x3,x4) = max{113, x3 - 69}
    a__U136_A(x1,x2,x3,x4) = max{21, x1, x3 - 69}
    take_A(x1,x2) = max{75, x1 - 107}
    a__U21_A(x1,x2) = 119
    a__U22_A(x1,x2) = max{119, x1 - 75}
    a__U23_A(x1) = max{39, x1}
    a__U31_A(x1,x2) = x2 + 113
    a__U32_A(x1,x2) = max{x1 - 11, x2 + 103}
    a__U33_A(x1) = max{103, x1 + 50}
    a__U41_A(x1,x2,x3) = x3 + 127
    a__U42_A(x1,x2,x3) = max{x1 + 14, x3 + 126}
    a__U43_A(x1,x2,x3) = max{127, x1, x3 + 126}
    a__U44_A(x1,x2,x3) = max{x1 + 3, x3 + 123}
    a__U45_A(x1,x2) = x2 + 123
    a__U46_A(x1) = 118
    a__U51_A(x1,x2) = x1
    a__U52_A(x1) = 113
    a__U61_A(x1,x2) = max{124, x1}
    a__U62_A(x1) = max{39, x1}
    a__U71_A(x1) = max{113, x1 - 11}
    a__U81_A(x1) = max{39, x1}
    a__U91_A(x1,x2,x3) = x3 + 89
    a__U92_A(x1,x2,x3) = max{x1 - 37, x3 + 89}
    a__U93_A(x1,x2,x3) = max{77, x1 - 35, x3 + 76}
    a__U94_A(x1,x2,x3) = max{76, x3 + 75}
    a__U95_A(x1,x2) = max{68, x1 - 44, x2 + 67}
    a__U96_A(x1) = max{69, x1 + 16}
    a__take_A(x1,x2) = max{75, x1 - 107}
    U101_A(x1,x2,x3) = 126
    U102_A(x1,x2,x3) = max{126, x1 + 1}
    U103_A(x1,x2,x3) = 126
    U104_A(x1,x2,x3) = max{126, x1 + 1}
    U105_A(x1,x2) = max{115, x1 - 4}
    U106_A(x1) = 115
    isNatIList_A(x1) = x1 + 113
    U11_A(x1,x2) = 119
    U12_A(x1,x2) = 113
    U111_A(x1,x2,x3) = max{228, x1 + 175, x2 + 227}
    U112_A(x1,x2,x3) = max{229, x1 + 153, x2 + 227}
    U113_A(x1,x2,x3) = max{266, x1 + 112, x2 + 227}
    U114_A(x1,x2) = max{266, x1 + 113, x2 + 227}
    length_A(x1) = max{228, x1 + 189}
    U13_A(x1) = 113
    U121_A(x1,x2) = 62
    U122_A(x1) = 62
    U131_A(x1,x2,x3,x4) = max{120, x3 - 69}
    U132_A(x1,x2,x3,x4) = max{120, x1 - 106, x3 - 69}
    U133_A(x1,x2,x3,x4) = max{40, x1 + 1, x3 - 69}
    U134_A(x1,x2,x3,x4) = max{113, x3 - 69}
    U135_A(x1,x2,x3,x4) = max{113, x3 - 69}
    U136_A(x1,x2,x3,x4) = max{20, x1, x3 - 69}
    U21_A(x1,x2) = 119
    U22_A(x1,x2) = max{119, x1 - 75}
    U23_A(x1) = max{39, x1}
    U31_A(x1,x2) = x2 + 113
    U32_A(x1,x2) = max{x1 - 11, x2 + 103}
    U33_A(x1) = max{103, x1 + 50}
    U41_A(x1,x2,x3) = x3 + 127
    U42_A(x1,x2,x3) = max{x1 + 14, x3 + 126}
    U43_A(x1,x2,x3) = max{127, x1, x3 + 126}
    U44_A(x1,x2,x3) = max{x1 + 3, x3 + 123}
    U45_A(x1,x2) = x2 + 123
    U46_A(x1) = 118
    U51_A(x1,x2) = x1
    U52_A(x1) = 113
    U61_A(x1,x2) = max{124, x1}
    U62_A(x1) = max{39, x1}
    U71_A(x1) = max{113, x1 - 11}
    U81_A(x1) = max{39, x1}
    U91_A(x1,x2,x3) = x3 + 89
    U92_A(x1,x2,x3) = max{x1 - 37, x3 + 89}
    U93_A(x1,x2,x3) = max{77, x1 - 35, x3 + 76}
    U94_A(x1,x2,x3) = max{76, x3 + 75}
    U95_A(x1,x2) = max{68, x1 - 44, x2 + 67}
    U96_A(x1) = max{69, x1 + 16}
    isNatKind_A(x1) = 113
    isNatIListKind_A(x1) = 124
    isNat_A(x1) = 119
    isNatList_A(x1) = max{53, x1 + 51}

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: a__U114#(tt(),L) -> a__length#(mark(L))
p2: a__length#(cons(N,L)) -> a__U111#(a__isNatList(L),L,N)
p3: a__U111#(tt(),L,N) -> a__U112#(a__isNatIListKind(L),L,N)
p4: a__U112#(tt(),L,N) -> a__U113#(a__isNat(N),L,N)

and R consists of:

r1: a__zeros() -> cons(|0|(),zeros())
r2: a__U101(tt(),V1,V2) -> a__U102(a__isNatKind(V1),V1,V2)
r3: a__U102(tt(),V1,V2) -> a__U103(a__isNatIListKind(V2),V1,V2)
r4: a__U103(tt(),V1,V2) -> a__U104(a__isNatIListKind(V2),V1,V2)
r5: a__U104(tt(),V1,V2) -> a__U105(a__isNat(V1),V2)
r6: a__U105(tt(),V2) -> a__U106(a__isNatIList(V2))
r7: a__U106(tt()) -> tt()
r8: a__U11(tt(),V1) -> a__U12(a__isNatIListKind(V1),V1)
r9: a__U111(tt(),L,N) -> a__U112(a__isNatIListKind(L),L,N)
r10: a__U112(tt(),L,N) -> a__U113(a__isNat(N),L,N)
r11: a__U113(tt(),L,N) -> a__U114(a__isNatKind(N),L)
r12: a__U114(tt(),L) -> s(a__length(mark(L)))
r13: a__U12(tt(),V1) -> a__U13(a__isNatList(V1))
r14: a__U121(tt(),IL) -> a__U122(a__isNatIListKind(IL))
r15: a__U122(tt()) -> nil()
r16: a__U13(tt()) -> tt()
r17: a__U131(tt(),IL,M,N) -> a__U132(a__isNatIListKind(IL),IL,M,N)
r18: a__U132(tt(),IL,M,N) -> a__U133(a__isNat(M),IL,M,N)
r19: a__U133(tt(),IL,M,N) -> a__U134(a__isNatKind(M),IL,M,N)
r20: a__U134(tt(),IL,M,N) -> a__U135(a__isNat(N),IL,M,N)
r21: a__U135(tt(),IL,M,N) -> a__U136(a__isNatKind(N),IL,M,N)
r22: a__U136(tt(),IL,M,N) -> cons(mark(N),take(M,IL))
r23: a__U21(tt(),V1) -> a__U22(a__isNatKind(V1),V1)
r24: a__U22(tt(),V1) -> a__U23(a__isNat(V1))
r25: a__U23(tt()) -> tt()
r26: a__U31(tt(),V) -> a__U32(a__isNatIListKind(V),V)
r27: a__U32(tt(),V) -> a__U33(a__isNatList(V))
r28: a__U33(tt()) -> tt()
r29: a__U41(tt(),V1,V2) -> a__U42(a__isNatKind(V1),V1,V2)
r30: a__U42(tt(),V1,V2) -> a__U43(a__isNatIListKind(V2),V1,V2)
r31: a__U43(tt(),V1,V2) -> a__U44(a__isNatIListKind(V2),V1,V2)
r32: a__U44(tt(),V1,V2) -> a__U45(a__isNat(V1),V2)
r33: a__U45(tt(),V2) -> a__U46(a__isNatIList(V2))
r34: a__U46(tt()) -> tt()
r35: a__U51(tt(),V2) -> a__U52(a__isNatIListKind(V2))
r36: a__U52(tt()) -> tt()
r37: a__U61(tt(),V2) -> a__U62(a__isNatIListKind(V2))
r38: a__U62(tt()) -> tt()
r39: a__U71(tt()) -> tt()
r40: a__U81(tt()) -> tt()
r41: a__U91(tt(),V1,V2) -> a__U92(a__isNatKind(V1),V1,V2)
r42: a__U92(tt(),V1,V2) -> a__U93(a__isNatIListKind(V2),V1,V2)
r43: a__U93(tt(),V1,V2) -> a__U94(a__isNatIListKind(V2),V1,V2)
r44: a__U94(tt(),V1,V2) -> a__U95(a__isNat(V1),V2)
r45: a__U95(tt(),V2) -> a__U96(a__isNatList(V2))
r46: a__U96(tt()) -> tt()
r47: a__isNat(|0|()) -> tt()
r48: a__isNat(length(V1)) -> a__U11(a__isNatIListKind(V1),V1)
r49: a__isNat(s(V1)) -> a__U21(a__isNatKind(V1),V1)
r50: a__isNatIList(V) -> a__U31(a__isNatIListKind(V),V)
r51: a__isNatIList(zeros()) -> tt()
r52: a__isNatIList(cons(V1,V2)) -> a__U41(a__isNatKind(V1),V1,V2)
r53: a__isNatIListKind(nil()) -> tt()
r54: a__isNatIListKind(zeros()) -> tt()
r55: a__isNatIListKind(cons(V1,V2)) -> a__U51(a__isNatKind(V1),V2)
r56: a__isNatIListKind(take(V1,V2)) -> a__U61(a__isNatKind(V1),V2)
r57: a__isNatKind(|0|()) -> tt()
r58: a__isNatKind(length(V1)) -> a__U71(a__isNatIListKind(V1))
r59: a__isNatKind(s(V1)) -> a__U81(a__isNatKind(V1))
r60: a__isNatList(nil()) -> tt()
r61: a__isNatList(cons(V1,V2)) -> a__U91(a__isNatKind(V1),V1,V2)
r62: a__isNatList(take(V1,V2)) -> a__U101(a__isNatKind(V1),V1,V2)
r63: a__length(nil()) -> |0|()
r64: a__length(cons(N,L)) -> a__U111(a__isNatList(L),L,N)
r65: a__take(|0|(),IL) -> a__U121(a__isNatIList(IL),IL)
r66: a__take(s(M),cons(N,IL)) -> a__U131(a__isNatIList(IL),IL,M,N)
r67: mark(zeros()) -> a__zeros()
r68: mark(U101(X1,X2,X3)) -> a__U101(mark(X1),X2,X3)
r69: mark(U102(X1,X2,X3)) -> a__U102(mark(X1),X2,X3)
r70: mark(isNatKind(X)) -> a__isNatKind(X)
r71: mark(U103(X1,X2,X3)) -> a__U103(mark(X1),X2,X3)
r72: mark(isNatIListKind(X)) -> a__isNatIListKind(X)
r73: mark(U104(X1,X2,X3)) -> a__U104(mark(X1),X2,X3)
r74: mark(U105(X1,X2)) -> a__U105(mark(X1),X2)
r75: mark(isNat(X)) -> a__isNat(X)
r76: mark(U106(X)) -> a__U106(mark(X))
r77: mark(isNatIList(X)) -> a__isNatIList(X)
r78: mark(U11(X1,X2)) -> a__U11(mark(X1),X2)
r79: mark(U12(X1,X2)) -> a__U12(mark(X1),X2)
r80: mark(U111(X1,X2,X3)) -> a__U111(mark(X1),X2,X3)
r81: mark(U112(X1,X2,X3)) -> a__U112(mark(X1),X2,X3)
r82: mark(U113(X1,X2,X3)) -> a__U113(mark(X1),X2,X3)
r83: mark(U114(X1,X2)) -> a__U114(mark(X1),X2)
r84: mark(length(X)) -> a__length(mark(X))
r85: mark(U13(X)) -> a__U13(mark(X))
r86: mark(isNatList(X)) -> a__isNatList(X)
r87: mark(U121(X1,X2)) -> a__U121(mark(X1),X2)
r88: mark(U122(X)) -> a__U122(mark(X))
r89: mark(U131(X1,X2,X3,X4)) -> a__U131(mark(X1),X2,X3,X4)
r90: mark(U132(X1,X2,X3,X4)) -> a__U132(mark(X1),X2,X3,X4)
r91: mark(U133(X1,X2,X3,X4)) -> a__U133(mark(X1),X2,X3,X4)
r92: mark(U134(X1,X2,X3,X4)) -> a__U134(mark(X1),X2,X3,X4)
r93: mark(U135(X1,X2,X3,X4)) -> a__U135(mark(X1),X2,X3,X4)
r94: mark(U136(X1,X2,X3,X4)) -> a__U136(mark(X1),X2,X3,X4)
r95: mark(take(X1,X2)) -> a__take(mark(X1),mark(X2))
r96: mark(U21(X1,X2)) -> a__U21(mark(X1),X2)
r97: mark(U22(X1,X2)) -> a__U22(mark(X1),X2)
r98: mark(U23(X)) -> a__U23(mark(X))
r99: mark(U31(X1,X2)) -> a__U31(mark(X1),X2)
r100: mark(U32(X1,X2)) -> a__U32(mark(X1),X2)
r101: mark(U33(X)) -> a__U33(mark(X))
r102: mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3)
r103: mark(U42(X1,X2,X3)) -> a__U42(mark(X1),X2,X3)
r104: mark(U43(X1,X2,X3)) -> a__U43(mark(X1),X2,X3)
r105: mark(U44(X1,X2,X3)) -> a__U44(mark(X1),X2,X3)
r106: mark(U45(X1,X2)) -> a__U45(mark(X1),X2)
r107: mark(U46(X)) -> a__U46(mark(X))
r108: mark(U51(X1,X2)) -> a__U51(mark(X1),X2)
r109: mark(U52(X)) -> a__U52(mark(X))
r110: mark(U61(X1,X2)) -> a__U61(mark(X1),X2)
r111: mark(U62(X)) -> a__U62(mark(X))
r112: mark(U71(X)) -> a__U71(mark(X))
r113: mark(U81(X)) -> a__U81(mark(X))
r114: mark(U91(X1,X2,X3)) -> a__U91(mark(X1),X2,X3)
r115: mark(U92(X1,X2,X3)) -> a__U92(mark(X1),X2,X3)
r116: mark(U93(X1,X2,X3)) -> a__U93(mark(X1),X2,X3)
r117: mark(U94(X1,X2,X3)) -> a__U94(mark(X1),X2,X3)
r118: mark(U95(X1,X2)) -> a__U95(mark(X1),X2)
r119: mark(U96(X)) -> a__U96(mark(X))
r120: mark(cons(X1,X2)) -> cons(mark(X1),X2)
r121: mark(|0|()) -> |0|()
r122: mark(tt()) -> tt()
r123: mark(s(X)) -> s(mark(X))
r124: mark(nil()) -> nil()
r125: a__zeros() -> zeros()
r126: a__U101(X1,X2,X3) -> U101(X1,X2,X3)
r127: a__U102(X1,X2,X3) -> U102(X1,X2,X3)
r128: a__isNatKind(X) -> isNatKind(X)
r129: a__U103(X1,X2,X3) -> U103(X1,X2,X3)
r130: a__isNatIListKind(X) -> isNatIListKind(X)
r131: a__U104(X1,X2,X3) -> U104(X1,X2,X3)
r132: a__U105(X1,X2) -> U105(X1,X2)
r133: a__isNat(X) -> isNat(X)
r134: a__U106(X) -> U106(X)
r135: a__isNatIList(X) -> isNatIList(X)
r136: a__U11(X1,X2) -> U11(X1,X2)
r137: a__U12(X1,X2) -> U12(X1,X2)
r138: a__U111(X1,X2,X3) -> U111(X1,X2,X3)
r139: a__U112(X1,X2,X3) -> U112(X1,X2,X3)
r140: a__U113(X1,X2,X3) -> U113(X1,X2,X3)
r141: a__U114(X1,X2) -> U114(X1,X2)
r142: a__length(X) -> length(X)
r143: a__U13(X) -> U13(X)
r144: a__isNatList(X) -> isNatList(X)
r145: a__U121(X1,X2) -> U121(X1,X2)
r146: a__U122(X) -> U122(X)
r147: a__U131(X1,X2,X3,X4) -> U131(X1,X2,X3,X4)
r148: a__U132(X1,X2,X3,X4) -> U132(X1,X2,X3,X4)
r149: a__U133(X1,X2,X3,X4) -> U133(X1,X2,X3,X4)
r150: a__U134(X1,X2,X3,X4) -> U134(X1,X2,X3,X4)
r151: a__U135(X1,X2,X3,X4) -> U135(X1,X2,X3,X4)
r152: a__U136(X1,X2,X3,X4) -> U136(X1,X2,X3,X4)
r153: a__take(X1,X2) -> take(X1,X2)
r154: a__U21(X1,X2) -> U21(X1,X2)
r155: a__U22(X1,X2) -> U22(X1,X2)
r156: a__U23(X) -> U23(X)
r157: a__U31(X1,X2) -> U31(X1,X2)
r158: a__U32(X1,X2) -> U32(X1,X2)
r159: a__U33(X) -> U33(X)
r160: a__U41(X1,X2,X3) -> U41(X1,X2,X3)
r161: a__U42(X1,X2,X3) -> U42(X1,X2,X3)
r162: a__U43(X1,X2,X3) -> U43(X1,X2,X3)
r163: a__U44(X1,X2,X3) -> U44(X1,X2,X3)
r164: a__U45(X1,X2) -> U45(X1,X2)
r165: a__U46(X) -> U46(X)
r166: a__U51(X1,X2) -> U51(X1,X2)
r167: a__U52(X) -> U52(X)
r168: a__U61(X1,X2) -> U61(X1,X2)
r169: a__U62(X) -> U62(X)
r170: a__U71(X) -> U71(X)
r171: a__U81(X) -> U81(X)
r172: a__U91(X1,X2,X3) -> U91(X1,X2,X3)
r173: a__U92(X1,X2,X3) -> U92(X1,X2,X3)
r174: a__U93(X1,X2,X3) -> U93(X1,X2,X3)
r175: a__U94(X1,X2,X3) -> U94(X1,X2,X3)
r176: a__U95(X1,X2) -> U95(X1,X2)
r177: a__U96(X) -> U96(X)

The estimated dependency graph contains the following SCCs:

  (no SCCs)

-- Reduction pair.

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

p1: mark#(s(X)) -> mark#(X)
p2: mark#(U101(X1,X2,X3)) -> mark#(X1)
p3: mark#(U103(X1,X2,X3)) -> mark#(X1)
p4: mark#(U104(X1,X2,X3)) -> mark#(X1)
p5: mark#(U105(X1,X2)) -> mark#(X1)
p6: mark#(U106(X)) -> mark#(X)
p7: mark#(U11(X1,X2)) -> mark#(X1)
p8: mark#(U12(X1,X2)) -> mark#(X1)
p9: mark#(U111(X1,X2,X3)) -> mark#(X1)
p10: mark#(U13(X)) -> mark#(X)
p11: mark#(U121(X1,X2)) -> mark#(X1)
p12: mark#(U122(X)) -> mark#(X)
p13: mark#(U131(X1,X2,X3,X4)) -> mark#(X1)
p14: mark#(U132(X1,X2,X3,X4)) -> mark#(X1)
p15: mark#(U134(X1,X2,X3,X4)) -> mark#(X1)
p16: mark#(U136(X1,X2,X3,X4)) -> mark#(X1)
p17: mark#(take(X1,X2)) -> mark#(X1)
p18: mark#(U21(X1,X2)) -> mark#(X1)
p19: mark#(U22(X1,X2)) -> mark#(X1)
p20: mark#(U23(X)) -> mark#(X)
p21: mark#(U31(X1,X2)) -> mark#(X1)
p22: mark#(U32(X1,X2)) -> mark#(X1)
p23: mark#(U33(X)) -> mark#(X)
p24: mark#(U42(X1,X2,X3)) -> mark#(X1)
p25: mark#(U46(X)) -> mark#(X)
p26: mark#(U51(X1,X2)) -> mark#(X1)
p27: mark#(U52(X)) -> mark#(X)
p28: mark#(U61(X1,X2)) -> mark#(X1)
p29: mark#(U62(X)) -> mark#(X)
p30: mark#(U71(X)) -> mark#(X)
p31: mark#(U81(X)) -> mark#(X)
p32: mark#(U94(X1,X2,X3)) -> mark#(X1)
p33: mark#(U95(X1,X2)) -> mark#(X1)
p34: mark#(U96(X)) -> mark#(X)
p35: mark#(cons(X1,X2)) -> mark#(X1)

and R consists of:

r1: a__zeros() -> cons(|0|(),zeros())
r2: a__U101(tt(),V1,V2) -> a__U102(a__isNatKind(V1),V1,V2)
r3: a__U102(tt(),V1,V2) -> a__U103(a__isNatIListKind(V2),V1,V2)
r4: a__U103(tt(),V1,V2) -> a__U104(a__isNatIListKind(V2),V1,V2)
r5: a__U104(tt(),V1,V2) -> a__U105(a__isNat(V1),V2)
r6: a__U105(tt(),V2) -> a__U106(a__isNatIList(V2))
r7: a__U106(tt()) -> tt()
r8: a__U11(tt(),V1) -> a__U12(a__isNatIListKind(V1),V1)
r9: a__U111(tt(),L,N) -> a__U112(a__isNatIListKind(L),L,N)
r10: a__U112(tt(),L,N) -> a__U113(a__isNat(N),L,N)
r11: a__U113(tt(),L,N) -> a__U114(a__isNatKind(N),L)
r12: a__U114(tt(),L) -> s(a__length(mark(L)))
r13: a__U12(tt(),V1) -> a__U13(a__isNatList(V1))
r14: a__U121(tt(),IL) -> a__U122(a__isNatIListKind(IL))
r15: a__U122(tt()) -> nil()
r16: a__U13(tt()) -> tt()
r17: a__U131(tt(),IL,M,N) -> a__U132(a__isNatIListKind(IL),IL,M,N)
r18: a__U132(tt(),IL,M,N) -> a__U133(a__isNat(M),IL,M,N)
r19: a__U133(tt(),IL,M,N) -> a__U134(a__isNatKind(M),IL,M,N)
r20: a__U134(tt(),IL,M,N) -> a__U135(a__isNat(N),IL,M,N)
r21: a__U135(tt(),IL,M,N) -> a__U136(a__isNatKind(N),IL,M,N)
r22: a__U136(tt(),IL,M,N) -> cons(mark(N),take(M,IL))
r23: a__U21(tt(),V1) -> a__U22(a__isNatKind(V1),V1)
r24: a__U22(tt(),V1) -> a__U23(a__isNat(V1))
r25: a__U23(tt()) -> tt()
r26: a__U31(tt(),V) -> a__U32(a__isNatIListKind(V),V)
r27: a__U32(tt(),V) -> a__U33(a__isNatList(V))
r28: a__U33(tt()) -> tt()
r29: a__U41(tt(),V1,V2) -> a__U42(a__isNatKind(V1),V1,V2)
r30: a__U42(tt(),V1,V2) -> a__U43(a__isNatIListKind(V2),V1,V2)
r31: a__U43(tt(),V1,V2) -> a__U44(a__isNatIListKind(V2),V1,V2)
r32: a__U44(tt(),V1,V2) -> a__U45(a__isNat(V1),V2)
r33: a__U45(tt(),V2) -> a__U46(a__isNatIList(V2))
r34: a__U46(tt()) -> tt()
r35: a__U51(tt(),V2) -> a__U52(a__isNatIListKind(V2))
r36: a__U52(tt()) -> tt()
r37: a__U61(tt(),V2) -> a__U62(a__isNatIListKind(V2))
r38: a__U62(tt()) -> tt()
r39: a__U71(tt()) -> tt()
r40: a__U81(tt()) -> tt()
r41: a__U91(tt(),V1,V2) -> a__U92(a__isNatKind(V1),V1,V2)
r42: a__U92(tt(),V1,V2) -> a__U93(a__isNatIListKind(V2),V1,V2)
r43: a__U93(tt(),V1,V2) -> a__U94(a__isNatIListKind(V2),V1,V2)
r44: a__U94(tt(),V1,V2) -> a__U95(a__isNat(V1),V2)
r45: a__U95(tt(),V2) -> a__U96(a__isNatList(V2))
r46: a__U96(tt()) -> tt()
r47: a__isNat(|0|()) -> tt()
r48: a__isNat(length(V1)) -> a__U11(a__isNatIListKind(V1),V1)
r49: a__isNat(s(V1)) -> a__U21(a__isNatKind(V1),V1)
r50: a__isNatIList(V) -> a__U31(a__isNatIListKind(V),V)
r51: a__isNatIList(zeros()) -> tt()
r52: a__isNatIList(cons(V1,V2)) -> a__U41(a__isNatKind(V1),V1,V2)
r53: a__isNatIListKind(nil()) -> tt()
r54: a__isNatIListKind(zeros()) -> tt()
r55: a__isNatIListKind(cons(V1,V2)) -> a__U51(a__isNatKind(V1),V2)
r56: a__isNatIListKind(take(V1,V2)) -> a__U61(a__isNatKind(V1),V2)
r57: a__isNatKind(|0|()) -> tt()
r58: a__isNatKind(length(V1)) -> a__U71(a__isNatIListKind(V1))
r59: a__isNatKind(s(V1)) -> a__U81(a__isNatKind(V1))
r60: a__isNatList(nil()) -> tt()
r61: a__isNatList(cons(V1,V2)) -> a__U91(a__isNatKind(V1),V1,V2)
r62: a__isNatList(take(V1,V2)) -> a__U101(a__isNatKind(V1),V1,V2)
r63: a__length(nil()) -> |0|()
r64: a__length(cons(N,L)) -> a__U111(a__isNatList(L),L,N)
r65: a__take(|0|(),IL) -> a__U121(a__isNatIList(IL),IL)
r66: a__take(s(M),cons(N,IL)) -> a__U131(a__isNatIList(IL),IL,M,N)
r67: mark(zeros()) -> a__zeros()
r68: mark(U101(X1,X2,X3)) -> a__U101(mark(X1),X2,X3)
r69: mark(U102(X1,X2,X3)) -> a__U102(mark(X1),X2,X3)
r70: mark(isNatKind(X)) -> a__isNatKind(X)
r71: mark(U103(X1,X2,X3)) -> a__U103(mark(X1),X2,X3)
r72: mark(isNatIListKind(X)) -> a__isNatIListKind(X)
r73: mark(U104(X1,X2,X3)) -> a__U104(mark(X1),X2,X3)
r74: mark(U105(X1,X2)) -> a__U105(mark(X1),X2)
r75: mark(isNat(X)) -> a__isNat(X)
r76: mark(U106(X)) -> a__U106(mark(X))
r77: mark(isNatIList(X)) -> a__isNatIList(X)
r78: mark(U11(X1,X2)) -> a__U11(mark(X1),X2)
r79: mark(U12(X1,X2)) -> a__U12(mark(X1),X2)
r80: mark(U111(X1,X2,X3)) -> a__U111(mark(X1),X2,X3)
r81: mark(U112(X1,X2,X3)) -> a__U112(mark(X1),X2,X3)
r82: mark(U113(X1,X2,X3)) -> a__U113(mark(X1),X2,X3)
r83: mark(U114(X1,X2)) -> a__U114(mark(X1),X2)
r84: mark(length(X)) -> a__length(mark(X))
r85: mark(U13(X)) -> a__U13(mark(X))
r86: mark(isNatList(X)) -> a__isNatList(X)
r87: mark(U121(X1,X2)) -> a__U121(mark(X1),X2)
r88: mark(U122(X)) -> a__U122(mark(X))
r89: mark(U131(X1,X2,X3,X4)) -> a__U131(mark(X1),X2,X3,X4)
r90: mark(U132(X1,X2,X3,X4)) -> a__U132(mark(X1),X2,X3,X4)
r91: mark(U133(X1,X2,X3,X4)) -> a__U133(mark(X1),X2,X3,X4)
r92: mark(U134(X1,X2,X3,X4)) -> a__U134(mark(X1),X2,X3,X4)
r93: mark(U135(X1,X2,X3,X4)) -> a__U135(mark(X1),X2,X3,X4)
r94: mark(U136(X1,X2,X3,X4)) -> a__U136(mark(X1),X2,X3,X4)
r95: mark(take(X1,X2)) -> a__take(mark(X1),mark(X2))
r96: mark(U21(X1,X2)) -> a__U21(mark(X1),X2)
r97: mark(U22(X1,X2)) -> a__U22(mark(X1),X2)
r98: mark(U23(X)) -> a__U23(mark(X))
r99: mark(U31(X1,X2)) -> a__U31(mark(X1),X2)
r100: mark(U32(X1,X2)) -> a__U32(mark(X1),X2)
r101: mark(U33(X)) -> a__U33(mark(X))
r102: mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3)
r103: mark(U42(X1,X2,X3)) -> a__U42(mark(X1),X2,X3)
r104: mark(U43(X1,X2,X3)) -> a__U43(mark(X1),X2,X3)
r105: mark(U44(X1,X2,X3)) -> a__U44(mark(X1),X2,X3)
r106: mark(U45(X1,X2)) -> a__U45(mark(X1),X2)
r107: mark(U46(X)) -> a__U46(mark(X))
r108: mark(U51(X1,X2)) -> a__U51(mark(X1),X2)
r109: mark(U52(X)) -> a__U52(mark(X))
r110: mark(U61(X1,X2)) -> a__U61(mark(X1),X2)
r111: mark(U62(X)) -> a__U62(mark(X))
r112: mark(U71(X)) -> a__U71(mark(X))
r113: mark(U81(X)) -> a__U81(mark(X))
r114: mark(U91(X1,X2,X3)) -> a__U91(mark(X1),X2,X3)
r115: mark(U92(X1,X2,X3)) -> a__U92(mark(X1),X2,X3)
r116: mark(U93(X1,X2,X3)) -> a__U93(mark(X1),X2,X3)
r117: mark(U94(X1,X2,X3)) -> a__U94(mark(X1),X2,X3)
r118: mark(U95(X1,X2)) -> a__U95(mark(X1),X2)
r119: mark(U96(X)) -> a__U96(mark(X))
r120: mark(cons(X1,X2)) -> cons(mark(X1),X2)
r121: mark(|0|()) -> |0|()
r122: mark(tt()) -> tt()
r123: mark(s(X)) -> s(mark(X))
r124: mark(nil()) -> nil()
r125: a__zeros() -> zeros()
r126: a__U101(X1,X2,X3) -> U101(X1,X2,X3)
r127: a__U102(X1,X2,X3) -> U102(X1,X2,X3)
r128: a__isNatKind(X) -> isNatKind(X)
r129: a__U103(X1,X2,X3) -> U103(X1,X2,X3)
r130: a__isNatIListKind(X) -> isNatIListKind(X)
r131: a__U104(X1,X2,X3) -> U104(X1,X2,X3)
r132: a__U105(X1,X2) -> U105(X1,X2)
r133: a__isNat(X) -> isNat(X)
r134: a__U106(X) -> U106(X)
r135: a__isNatIList(X) -> isNatIList(X)
r136: a__U11(X1,X2) -> U11(X1,X2)
r137: a__U12(X1,X2) -> U12(X1,X2)
r138: a__U111(X1,X2,X3) -> U111(X1,X2,X3)
r139: a__U112(X1,X2,X3) -> U112(X1,X2,X3)
r140: a__U113(X1,X2,X3) -> U113(X1,X2,X3)
r141: a__U114(X1,X2) -> U114(X1,X2)
r142: a__length(X) -> length(X)
r143: a__U13(X) -> U13(X)
r144: a__isNatList(X) -> isNatList(X)
r145: a__U121(X1,X2) -> U121(X1,X2)
r146: a__U122(X) -> U122(X)
r147: a__U131(X1,X2,X3,X4) -> U131(X1,X2,X3,X4)
r148: a__U132(X1,X2,X3,X4) -> U132(X1,X2,X3,X4)
r149: a__U133(X1,X2,X3,X4) -> U133(X1,X2,X3,X4)
r150: a__U134(X1,X2,X3,X4) -> U134(X1,X2,X3,X4)
r151: a__U135(X1,X2,X3,X4) -> U135(X1,X2,X3,X4)
r152: a__U136(X1,X2,X3,X4) -> U136(X1,X2,X3,X4)
r153: a__take(X1,X2) -> take(X1,X2)
r154: a__U21(X1,X2) -> U21(X1,X2)
r155: a__U22(X1,X2) -> U22(X1,X2)
r156: a__U23(X) -> U23(X)
r157: a__U31(X1,X2) -> U31(X1,X2)
r158: a__U32(X1,X2) -> U32(X1,X2)
r159: a__U33(X) -> U33(X)
r160: a__U41(X1,X2,X3) -> U41(X1,X2,X3)
r161: a__U42(X1,X2,X3) -> U42(X1,X2,X3)
r162: a__U43(X1,X2,X3) -> U43(X1,X2,X3)
r163: a__U44(X1,X2,X3) -> U44(X1,X2,X3)
r164: a__U45(X1,X2) -> U45(X1,X2)
r165: a__U46(X) -> U46(X)
r166: a__U51(X1,X2) -> U51(X1,X2)
r167: a__U52(X) -> U52(X)
r168: a__U61(X1,X2) -> U61(X1,X2)
r169: a__U62(X) -> U62(X)
r170: a__U71(X) -> U71(X)
r171: a__U81(X) -> U81(X)
r172: a__U91(X1,X2,X3) -> U91(X1,X2,X3)
r173: a__U92(X1,X2,X3) -> U92(X1,X2,X3)
r174: a__U93(X1,X2,X3) -> U93(X1,X2,X3)
r175: a__U94(X1,X2,X3) -> U94(X1,X2,X3)
r176: a__U95(X1,X2) -> U95(X1,X2)
r177: a__U96(X) -> U96(X)

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  max/plus interpretations on natural numbers:
  
    mark#_A(x1) = max{0, x1 - 1}
    s_A(x1) = x1 + 2
    U101_A(x1,x2,x3) = max{x1, x3 + 1}
    U103_A(x1,x2,x3) = max{x1, x2 + 2, x3 + 2}
    U104_A(x1,x2,x3) = max{x1 + 2, x2 + 2, x3 + 2}
    U105_A(x1,x2) = max{x1 + 2, x2 + 2}
    U106_A(x1) = x1
    U11_A(x1,x2) = x1 + 1
    U12_A(x1,x2) = max{x1, x2 + 1}
    U111_A(x1,x2,x3) = max{x1, x2 + 2, x3 + 2}
    U13_A(x1) = x1 + 1
    U121_A(x1,x2) = max{x1 + 1, x2 + 1}
    U122_A(x1) = x1 + 1
    U131_A(x1,x2,x3,x4) = max{x1, x2 + 2, x3 + 2, x4 + 2}
    U132_A(x1,x2,x3,x4) = max{x1 + 2, x2 + 2, x3 + 2, x4 + 2}
    U134_A(x1,x2,x3,x4) = max{x1, x2 + 2, x3 + 2, x4 + 2}
    U136_A(x1,x2,x3,x4) = max{x1 + 2, x2 + 2, x3 + 2, x4 + 2}
    take_A(x1,x2) = max{x1 + 2, x2 + 2}
    U21_A(x1,x2) = max{x1, x2 + 1}
    U22_A(x1,x2) = max{x1, x2 + 1}
    U23_A(x1) = x1
    U31_A(x1,x2) = max{x1, x2 + 1}
    U32_A(x1,x2) = max{x1, x2 + 1}
    U33_A(x1) = x1
    U42_A(x1,x2,x3) = max{x1, x2 + 1, x3 + 1}
    U46_A(x1) = x1
    U51_A(x1,x2) = max{x1 + 2, x2 + 2}
    U52_A(x1) = x1
    U61_A(x1,x2) = max{x1 + 2, x2 + 2}
    U62_A(x1) = x1
    U71_A(x1) = x1
    U81_A(x1) = x1
    U94_A(x1,x2,x3) = max{x1 + 2, x2 + 2, x3 + 2}
    U95_A(x1,x2) = x1 + 1
    U96_A(x1) = x1 + 1
    cons_A(x1,x2) = x1 + 1

The next rules are strictly ordered:

  p1, p4, p5, p14, p16, p17, p26, p28, p32

We remove them from the problem.

-- SCC decomposition.

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

p1: mark#(U101(X1,X2,X3)) -> mark#(X1)
p2: mark#(U103(X1,X2,X3)) -> mark#(X1)
p3: mark#(U106(X)) -> mark#(X)
p4: mark#(U11(X1,X2)) -> mark#(X1)
p5: mark#(U12(X1,X2)) -> mark#(X1)
p6: mark#(U111(X1,X2,X3)) -> mark#(X1)
p7: mark#(U13(X)) -> mark#(X)
p8: mark#(U121(X1,X2)) -> mark#(X1)
p9: mark#(U122(X)) -> mark#(X)
p10: mark#(U131(X1,X2,X3,X4)) -> mark#(X1)
p11: mark#(U134(X1,X2,X3,X4)) -> mark#(X1)
p12: mark#(U21(X1,X2)) -> mark#(X1)
p13: mark#(U22(X1,X2)) -> mark#(X1)
p14: mark#(U23(X)) -> mark#(X)
p15: mark#(U31(X1,X2)) -> mark#(X1)
p16: mark#(U32(X1,X2)) -> mark#(X1)
p17: mark#(U33(X)) -> mark#(X)
p18: mark#(U42(X1,X2,X3)) -> mark#(X1)
p19: mark#(U46(X)) -> mark#(X)
p20: mark#(U52(X)) -> mark#(X)
p21: mark#(U62(X)) -> mark#(X)
p22: mark#(U71(X)) -> mark#(X)
p23: mark#(U81(X)) -> mark#(X)
p24: mark#(U95(X1,X2)) -> mark#(X1)
p25: mark#(U96(X)) -> mark#(X)
p26: mark#(cons(X1,X2)) -> mark#(X1)

and R consists of:

r1: a__zeros() -> cons(|0|(),zeros())
r2: a__U101(tt(),V1,V2) -> a__U102(a__isNatKind(V1),V1,V2)
r3: a__U102(tt(),V1,V2) -> a__U103(a__isNatIListKind(V2),V1,V2)
r4: a__U103(tt(),V1,V2) -> a__U104(a__isNatIListKind(V2),V1,V2)
r5: a__U104(tt(),V1,V2) -> a__U105(a__isNat(V1),V2)
r6: a__U105(tt(),V2) -> a__U106(a__isNatIList(V2))
r7: a__U106(tt()) -> tt()
r8: a__U11(tt(),V1) -> a__U12(a__isNatIListKind(V1),V1)
r9: a__U111(tt(),L,N) -> a__U112(a__isNatIListKind(L),L,N)
r10: a__U112(tt(),L,N) -> a__U113(a__isNat(N),L,N)
r11: a__U113(tt(),L,N) -> a__U114(a__isNatKind(N),L)
r12: a__U114(tt(),L) -> s(a__length(mark(L)))
r13: a__U12(tt(),V1) -> a__U13(a__isNatList(V1))
r14: a__U121(tt(),IL) -> a__U122(a__isNatIListKind(IL))
r15: a__U122(tt()) -> nil()
r16: a__U13(tt()) -> tt()
r17: a__U131(tt(),IL,M,N) -> a__U132(a__isNatIListKind(IL),IL,M,N)
r18: a__U132(tt(),IL,M,N) -> a__U133(a__isNat(M),IL,M,N)
r19: a__U133(tt(),IL,M,N) -> a__U134(a__isNatKind(M),IL,M,N)
r20: a__U134(tt(),IL,M,N) -> a__U135(a__isNat(N),IL,M,N)
r21: a__U135(tt(),IL,M,N) -> a__U136(a__isNatKind(N),IL,M,N)
r22: a__U136(tt(),IL,M,N) -> cons(mark(N),take(M,IL))
r23: a__U21(tt(),V1) -> a__U22(a__isNatKind(V1),V1)
r24: a__U22(tt(),V1) -> a__U23(a__isNat(V1))
r25: a__U23(tt()) -> tt()
r26: a__U31(tt(),V) -> a__U32(a__isNatIListKind(V),V)
r27: a__U32(tt(),V) -> a__U33(a__isNatList(V))
r28: a__U33(tt()) -> tt()
r29: a__U41(tt(),V1,V2) -> a__U42(a__isNatKind(V1),V1,V2)
r30: a__U42(tt(),V1,V2) -> a__U43(a__isNatIListKind(V2),V1,V2)
r31: a__U43(tt(),V1,V2) -> a__U44(a__isNatIListKind(V2),V1,V2)
r32: a__U44(tt(),V1,V2) -> a__U45(a__isNat(V1),V2)
r33: a__U45(tt(),V2) -> a__U46(a__isNatIList(V2))
r34: a__U46(tt()) -> tt()
r35: a__U51(tt(),V2) -> a__U52(a__isNatIListKind(V2))
r36: a__U52(tt()) -> tt()
r37: a__U61(tt(),V2) -> a__U62(a__isNatIListKind(V2))
r38: a__U62(tt()) -> tt()
r39: a__U71(tt()) -> tt()
r40: a__U81(tt()) -> tt()
r41: a__U91(tt(),V1,V2) -> a__U92(a__isNatKind(V1),V1,V2)
r42: a__U92(tt(),V1,V2) -> a__U93(a__isNatIListKind(V2),V1,V2)
r43: a__U93(tt(),V1,V2) -> a__U94(a__isNatIListKind(V2),V1,V2)
r44: a__U94(tt(),V1,V2) -> a__U95(a__isNat(V1),V2)
r45: a__U95(tt(),V2) -> a__U96(a__isNatList(V2))
r46: a__U96(tt()) -> tt()
r47: a__isNat(|0|()) -> tt()
r48: a__isNat(length(V1)) -> a__U11(a__isNatIListKind(V1),V1)
r49: a__isNat(s(V1)) -> a__U21(a__isNatKind(V1),V1)
r50: a__isNatIList(V) -> a__U31(a__isNatIListKind(V),V)
r51: a__isNatIList(zeros()) -> tt()
r52: a__isNatIList(cons(V1,V2)) -> a__U41(a__isNatKind(V1),V1,V2)
r53: a__isNatIListKind(nil()) -> tt()
r54: a__isNatIListKind(zeros()) -> tt()
r55: a__isNatIListKind(cons(V1,V2)) -> a__U51(a__isNatKind(V1),V2)
r56: a__isNatIListKind(take(V1,V2)) -> a__U61(a__isNatKind(V1),V2)
r57: a__isNatKind(|0|()) -> tt()
r58: a__isNatKind(length(V1)) -> a__U71(a__isNatIListKind(V1))
r59: a__isNatKind(s(V1)) -> a__U81(a__isNatKind(V1))
r60: a__isNatList(nil()) -> tt()
r61: a__isNatList(cons(V1,V2)) -> a__U91(a__isNatKind(V1),V1,V2)
r62: a__isNatList(take(V1,V2)) -> a__U101(a__isNatKind(V1),V1,V2)
r63: a__length(nil()) -> |0|()
r64: a__length(cons(N,L)) -> a__U111(a__isNatList(L),L,N)
r65: a__take(|0|(),IL) -> a__U121(a__isNatIList(IL),IL)
r66: a__take(s(M),cons(N,IL)) -> a__U131(a__isNatIList(IL),IL,M,N)
r67: mark(zeros()) -> a__zeros()
r68: mark(U101(X1,X2,X3)) -> a__U101(mark(X1),X2,X3)
r69: mark(U102(X1,X2,X3)) -> a__U102(mark(X1),X2,X3)
r70: mark(isNatKind(X)) -> a__isNatKind(X)
r71: mark(U103(X1,X2,X3)) -> a__U103(mark(X1),X2,X3)
r72: mark(isNatIListKind(X)) -> a__isNatIListKind(X)
r73: mark(U104(X1,X2,X3)) -> a__U104(mark(X1),X2,X3)
r74: mark(U105(X1,X2)) -> a__U105(mark(X1),X2)
r75: mark(isNat(X)) -> a__isNat(X)
r76: mark(U106(X)) -> a__U106(mark(X))
r77: mark(isNatIList(X)) -> a__isNatIList(X)
r78: mark(U11(X1,X2)) -> a__U11(mark(X1),X2)
r79: mark(U12(X1,X2)) -> a__U12(mark(X1),X2)
r80: mark(U111(X1,X2,X3)) -> a__U111(mark(X1),X2,X3)
r81: mark(U112(X1,X2,X3)) -> a__U112(mark(X1),X2,X3)
r82: mark(U113(X1,X2,X3)) -> a__U113(mark(X1),X2,X3)
r83: mark(U114(X1,X2)) -> a__U114(mark(X1),X2)
r84: mark(length(X)) -> a__length(mark(X))
r85: mark(U13(X)) -> a__U13(mark(X))
r86: mark(isNatList(X)) -> a__isNatList(X)
r87: mark(U121(X1,X2)) -> a__U121(mark(X1),X2)
r88: mark(U122(X)) -> a__U122(mark(X))
r89: mark(U131(X1,X2,X3,X4)) -> a__U131(mark(X1),X2,X3,X4)
r90: mark(U132(X1,X2,X3,X4)) -> a__U132(mark(X1),X2,X3,X4)
r91: mark(U133(X1,X2,X3,X4)) -> a__U133(mark(X1),X2,X3,X4)
r92: mark(U134(X1,X2,X3,X4)) -> a__U134(mark(X1),X2,X3,X4)
r93: mark(U135(X1,X2,X3,X4)) -> a__U135(mark(X1),X2,X3,X4)
r94: mark(U136(X1,X2,X3,X4)) -> a__U136(mark(X1),X2,X3,X4)
r95: mark(take(X1,X2)) -> a__take(mark(X1),mark(X2))
r96: mark(U21(X1,X2)) -> a__U21(mark(X1),X2)
r97: mark(U22(X1,X2)) -> a__U22(mark(X1),X2)
r98: mark(U23(X)) -> a__U23(mark(X))
r99: mark(U31(X1,X2)) -> a__U31(mark(X1),X2)
r100: mark(U32(X1,X2)) -> a__U32(mark(X1),X2)
r101: mark(U33(X)) -> a__U33(mark(X))
r102: mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3)
r103: mark(U42(X1,X2,X3)) -> a__U42(mark(X1),X2,X3)
r104: mark(U43(X1,X2,X3)) -> a__U43(mark(X1),X2,X3)
r105: mark(U44(X1,X2,X3)) -> a__U44(mark(X1),X2,X3)
r106: mark(U45(X1,X2)) -> a__U45(mark(X1),X2)
r107: mark(U46(X)) -> a__U46(mark(X))
r108: mark(U51(X1,X2)) -> a__U51(mark(X1),X2)
r109: mark(U52(X)) -> a__U52(mark(X))
r110: mark(U61(X1,X2)) -> a__U61(mark(X1),X2)
r111: mark(U62(X)) -> a__U62(mark(X))
r112: mark(U71(X)) -> a__U71(mark(X))
r113: mark(U81(X)) -> a__U81(mark(X))
r114: mark(U91(X1,X2,X3)) -> a__U91(mark(X1),X2,X3)
r115: mark(U92(X1,X2,X3)) -> a__U92(mark(X1),X2,X3)
r116: mark(U93(X1,X2,X3)) -> a__U93(mark(X1),X2,X3)
r117: mark(U94(X1,X2,X3)) -> a__U94(mark(X1),X2,X3)
r118: mark(U95(X1,X2)) -> a__U95(mark(X1),X2)
r119: mark(U96(X)) -> a__U96(mark(X))
r120: mark(cons(X1,X2)) -> cons(mark(X1),X2)
r121: mark(|0|()) -> |0|()
r122: mark(tt()) -> tt()
r123: mark(s(X)) -> s(mark(X))
r124: mark(nil()) -> nil()
r125: a__zeros() -> zeros()
r126: a__U101(X1,X2,X3) -> U101(X1,X2,X3)
r127: a__U102(X1,X2,X3) -> U102(X1,X2,X3)
r128: a__isNatKind(X) -> isNatKind(X)
r129: a__U103(X1,X2,X3) -> U103(X1,X2,X3)
r130: a__isNatIListKind(X) -> isNatIListKind(X)
r131: a__U104(X1,X2,X3) -> U104(X1,X2,X3)
r132: a__U105(X1,X2) -> U105(X1,X2)
r133: a__isNat(X) -> isNat(X)
r134: a__U106(X) -> U106(X)
r135: a__isNatIList(X) -> isNatIList(X)
r136: a__U11(X1,X2) -> U11(X1,X2)
r137: a__U12(X1,X2) -> U12(X1,X2)
r138: a__U111(X1,X2,X3) -> U111(X1,X2,X3)
r139: a__U112(X1,X2,X3) -> U112(X1,X2,X3)
r140: a__U113(X1,X2,X3) -> U113(X1,X2,X3)
r141: a__U114(X1,X2) -> U114(X1,X2)
r142: a__length(X) -> length(X)
r143: a__U13(X) -> U13(X)
r144: a__isNatList(X) -> isNatList(X)
r145: a__U121(X1,X2) -> U121(X1,X2)
r146: a__U122(X) -> U122(X)
r147: a__U131(X1,X2,X3,X4) -> U131(X1,X2,X3,X4)
r148: a__U132(X1,X2,X3,X4) -> U132(X1,X2,X3,X4)
r149: a__U133(X1,X2,X3,X4) -> U133(X1,X2,X3,X4)
r150: a__U134(X1,X2,X3,X4) -> U134(X1,X2,X3,X4)
r151: a__U135(X1,X2,X3,X4) -> U135(X1,X2,X3,X4)
r152: a__U136(X1,X2,X3,X4) -> U136(X1,X2,X3,X4)
r153: a__take(X1,X2) -> take(X1,X2)
r154: a__U21(X1,X2) -> U21(X1,X2)
r155: a__U22(X1,X2) -> U22(X1,X2)
r156: a__U23(X) -> U23(X)
r157: a__U31(X1,X2) -> U31(X1,X2)
r158: a__U32(X1,X2) -> U32(X1,X2)
r159: a__U33(X) -> U33(X)
r160: a__U41(X1,X2,X3) -> U41(X1,X2,X3)
r161: a__U42(X1,X2,X3) -> U42(X1,X2,X3)
r162: a__U43(X1,X2,X3) -> U43(X1,X2,X3)
r163: a__U44(X1,X2,X3) -> U44(X1,X2,X3)
r164: a__U45(X1,X2) -> U45(X1,X2)
r165: a__U46(X) -> U46(X)
r166: a__U51(X1,X2) -> U51(X1,X2)
r167: a__U52(X) -> U52(X)
r168: a__U61(X1,X2) -> U61(X1,X2)
r169: a__U62(X) -> U62(X)
r170: a__U71(X) -> U71(X)
r171: a__U81(X) -> U81(X)
r172: a__U91(X1,X2,X3) -> U91(X1,X2,X3)
r173: a__U92(X1,X2,X3) -> U92(X1,X2,X3)
r174: a__U93(X1,X2,X3) -> U93(X1,X2,X3)
r175: a__U94(X1,X2,X3) -> U94(X1,X2,X3)
r176: a__U95(X1,X2) -> U95(X1,X2)
r177: a__U96(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}


-- Reduction pair.

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

p1: mark#(U101(X1,X2,X3)) -> mark#(X1)
p2: mark#(cons(X1,X2)) -> mark#(X1)
p3: mark#(U96(X)) -> mark#(X)
p4: mark#(U95(X1,X2)) -> mark#(X1)
p5: mark#(U81(X)) -> mark#(X)
p6: mark#(U71(X)) -> mark#(X)
p7: mark#(U62(X)) -> mark#(X)
p8: mark#(U52(X)) -> mark#(X)
p9: mark#(U46(X)) -> mark#(X)
p10: mark#(U42(X1,X2,X3)) -> mark#(X1)
p11: mark#(U33(X)) -> mark#(X)
p12: mark#(U32(X1,X2)) -> mark#(X1)
p13: mark#(U31(X1,X2)) -> mark#(X1)
p14: mark#(U23(X)) -> mark#(X)
p15: mark#(U22(X1,X2)) -> mark#(X1)
p16: mark#(U21(X1,X2)) -> mark#(X1)
p17: mark#(U134(X1,X2,X3,X4)) -> mark#(X1)
p18: mark#(U131(X1,X2,X3,X4)) -> mark#(X1)
p19: mark#(U122(X)) -> mark#(X)
p20: mark#(U121(X1,X2)) -> mark#(X1)
p21: mark#(U13(X)) -> mark#(X)
p22: mark#(U111(X1,X2,X3)) -> mark#(X1)
p23: mark#(U12(X1,X2)) -> mark#(X1)
p24: mark#(U11(X1,X2)) -> mark#(X1)
p25: mark#(U106(X)) -> mark#(X)
p26: mark#(U103(X1,X2,X3)) -> mark#(X1)

and R consists of:

r1: a__zeros() -> cons(|0|(),zeros())
r2: a__U101(tt(),V1,V2) -> a__U102(a__isNatKind(V1),V1,V2)
r3: a__U102(tt(),V1,V2) -> a__U103(a__isNatIListKind(V2),V1,V2)
r4: a__U103(tt(),V1,V2) -> a__U104(a__isNatIListKind(V2),V1,V2)
r5: a__U104(tt(),V1,V2) -> a__U105(a__isNat(V1),V2)
r6: a__U105(tt(),V2) -> a__U106(a__isNatIList(V2))
r7: a__U106(tt()) -> tt()
r8: a__U11(tt(),V1) -> a__U12(a__isNatIListKind(V1),V1)
r9: a__U111(tt(),L,N) -> a__U112(a__isNatIListKind(L),L,N)
r10: a__U112(tt(),L,N) -> a__U113(a__isNat(N),L,N)
r11: a__U113(tt(),L,N) -> a__U114(a__isNatKind(N),L)
r12: a__U114(tt(),L) -> s(a__length(mark(L)))
r13: a__U12(tt(),V1) -> a__U13(a__isNatList(V1))
r14: a__U121(tt(),IL) -> a__U122(a__isNatIListKind(IL))
r15: a__U122(tt()) -> nil()
r16: a__U13(tt()) -> tt()
r17: a__U131(tt(),IL,M,N) -> a__U132(a__isNatIListKind(IL),IL,M,N)
r18: a__U132(tt(),IL,M,N) -> a__U133(a__isNat(M),IL,M,N)
r19: a__U133(tt(),IL,M,N) -> a__U134(a__isNatKind(M),IL,M,N)
r20: a__U134(tt(),IL,M,N) -> a__U135(a__isNat(N),IL,M,N)
r21: a__U135(tt(),IL,M,N) -> a__U136(a__isNatKind(N),IL,M,N)
r22: a__U136(tt(),IL,M,N) -> cons(mark(N),take(M,IL))
r23: a__U21(tt(),V1) -> a__U22(a__isNatKind(V1),V1)
r24: a__U22(tt(),V1) -> a__U23(a__isNat(V1))
r25: a__U23(tt()) -> tt()
r26: a__U31(tt(),V) -> a__U32(a__isNatIListKind(V),V)
r27: a__U32(tt(),V) -> a__U33(a__isNatList(V))
r28: a__U33(tt()) -> tt()
r29: a__U41(tt(),V1,V2) -> a__U42(a__isNatKind(V1),V1,V2)
r30: a__U42(tt(),V1,V2) -> a__U43(a__isNatIListKind(V2),V1,V2)
r31: a__U43(tt(),V1,V2) -> a__U44(a__isNatIListKind(V2),V1,V2)
r32: a__U44(tt(),V1,V2) -> a__U45(a__isNat(V1),V2)
r33: a__U45(tt(),V2) -> a__U46(a__isNatIList(V2))
r34: a__U46(tt()) -> tt()
r35: a__U51(tt(),V2) -> a__U52(a__isNatIListKind(V2))
r36: a__U52(tt()) -> tt()
r37: a__U61(tt(),V2) -> a__U62(a__isNatIListKind(V2))
r38: a__U62(tt()) -> tt()
r39: a__U71(tt()) -> tt()
r40: a__U81(tt()) -> tt()
r41: a__U91(tt(),V1,V2) -> a__U92(a__isNatKind(V1),V1,V2)
r42: a__U92(tt(),V1,V2) -> a__U93(a__isNatIListKind(V2),V1,V2)
r43: a__U93(tt(),V1,V2) -> a__U94(a__isNatIListKind(V2),V1,V2)
r44: a__U94(tt(),V1,V2) -> a__U95(a__isNat(V1),V2)
r45: a__U95(tt(),V2) -> a__U96(a__isNatList(V2))
r46: a__U96(tt()) -> tt()
r47: a__isNat(|0|()) -> tt()
r48: a__isNat(length(V1)) -> a__U11(a__isNatIListKind(V1),V1)
r49: a__isNat(s(V1)) -> a__U21(a__isNatKind(V1),V1)
r50: a__isNatIList(V) -> a__U31(a__isNatIListKind(V),V)
r51: a__isNatIList(zeros()) -> tt()
r52: a__isNatIList(cons(V1,V2)) -> a__U41(a__isNatKind(V1),V1,V2)
r53: a__isNatIListKind(nil()) -> tt()
r54: a__isNatIListKind(zeros()) -> tt()
r55: a__isNatIListKind(cons(V1,V2)) -> a__U51(a__isNatKind(V1),V2)
r56: a__isNatIListKind(take(V1,V2)) -> a__U61(a__isNatKind(V1),V2)
r57: a__isNatKind(|0|()) -> tt()
r58: a__isNatKind(length(V1)) -> a__U71(a__isNatIListKind(V1))
r59: a__isNatKind(s(V1)) -> a__U81(a__isNatKind(V1))
r60: a__isNatList(nil()) -> tt()
r61: a__isNatList(cons(V1,V2)) -> a__U91(a__isNatKind(V1),V1,V2)
r62: a__isNatList(take(V1,V2)) -> a__U101(a__isNatKind(V1),V1,V2)
r63: a__length(nil()) -> |0|()
r64: a__length(cons(N,L)) -> a__U111(a__isNatList(L),L,N)
r65: a__take(|0|(),IL) -> a__U121(a__isNatIList(IL),IL)
r66: a__take(s(M),cons(N,IL)) -> a__U131(a__isNatIList(IL),IL,M,N)
r67: mark(zeros()) -> a__zeros()
r68: mark(U101(X1,X2,X3)) -> a__U101(mark(X1),X2,X3)
r69: mark(U102(X1,X2,X3)) -> a__U102(mark(X1),X2,X3)
r70: mark(isNatKind(X)) -> a__isNatKind(X)
r71: mark(U103(X1,X2,X3)) -> a__U103(mark(X1),X2,X3)
r72: mark(isNatIListKind(X)) -> a__isNatIListKind(X)
r73: mark(U104(X1,X2,X3)) -> a__U104(mark(X1),X2,X3)
r74: mark(U105(X1,X2)) -> a__U105(mark(X1),X2)
r75: mark(isNat(X)) -> a__isNat(X)
r76: mark(U106(X)) -> a__U106(mark(X))
r77: mark(isNatIList(X)) -> a__isNatIList(X)
r78: mark(U11(X1,X2)) -> a__U11(mark(X1),X2)
r79: mark(U12(X1,X2)) -> a__U12(mark(X1),X2)
r80: mark(U111(X1,X2,X3)) -> a__U111(mark(X1),X2,X3)
r81: mark(U112(X1,X2,X3)) -> a__U112(mark(X1),X2,X3)
r82: mark(U113(X1,X2,X3)) -> a__U113(mark(X1),X2,X3)
r83: mark(U114(X1,X2)) -> a__U114(mark(X1),X2)
r84: mark(length(X)) -> a__length(mark(X))
r85: mark(U13(X)) -> a__U13(mark(X))
r86: mark(isNatList(X)) -> a__isNatList(X)
r87: mark(U121(X1,X2)) -> a__U121(mark(X1),X2)
r88: mark(U122(X)) -> a__U122(mark(X))
r89: mark(U131(X1,X2,X3,X4)) -> a__U131(mark(X1),X2,X3,X4)
r90: mark(U132(X1,X2,X3,X4)) -> a__U132(mark(X1),X2,X3,X4)
r91: mark(U133(X1,X2,X3,X4)) -> a__U133(mark(X1),X2,X3,X4)
r92: mark(U134(X1,X2,X3,X4)) -> a__U134(mark(X1),X2,X3,X4)
r93: mark(U135(X1,X2,X3,X4)) -> a__U135(mark(X1),X2,X3,X4)
r94: mark(U136(X1,X2,X3,X4)) -> a__U136(mark(X1),X2,X3,X4)
r95: mark(take(X1,X2)) -> a__take(mark(X1),mark(X2))
r96: mark(U21(X1,X2)) -> a__U21(mark(X1),X2)
r97: mark(U22(X1,X2)) -> a__U22(mark(X1),X2)
r98: mark(U23(X)) -> a__U23(mark(X))
r99: mark(U31(X1,X2)) -> a__U31(mark(X1),X2)
r100: mark(U32(X1,X2)) -> a__U32(mark(X1),X2)
r101: mark(U33(X)) -> a__U33(mark(X))
r102: mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3)
r103: mark(U42(X1,X2,X3)) -> a__U42(mark(X1),X2,X3)
r104: mark(U43(X1,X2,X3)) -> a__U43(mark(X1),X2,X3)
r105: mark(U44(X1,X2,X3)) -> a__U44(mark(X1),X2,X3)
r106: mark(U45(X1,X2)) -> a__U45(mark(X1),X2)
r107: mark(U46(X)) -> a__U46(mark(X))
r108: mark(U51(X1,X2)) -> a__U51(mark(X1),X2)
r109: mark(U52(X)) -> a__U52(mark(X))
r110: mark(U61(X1,X2)) -> a__U61(mark(X1),X2)
r111: mark(U62(X)) -> a__U62(mark(X))
r112: mark(U71(X)) -> a__U71(mark(X))
r113: mark(U81(X)) -> a__U81(mark(X))
r114: mark(U91(X1,X2,X3)) -> a__U91(mark(X1),X2,X3)
r115: mark(U92(X1,X2,X3)) -> a__U92(mark(X1),X2,X3)
r116: mark(U93(X1,X2,X3)) -> a__U93(mark(X1),X2,X3)
r117: mark(U94(X1,X2,X3)) -> a__U94(mark(X1),X2,X3)
r118: mark(U95(X1,X2)) -> a__U95(mark(X1),X2)
r119: mark(U96(X)) -> a__U96(mark(X))
r120: mark(cons(X1,X2)) -> cons(mark(X1),X2)
r121: mark(|0|()) -> |0|()
r122: mark(tt()) -> tt()
r123: mark(s(X)) -> s(mark(X))
r124: mark(nil()) -> nil()
r125: a__zeros() -> zeros()
r126: a__U101(X1,X2,X3) -> U101(X1,X2,X3)
r127: a__U102(X1,X2,X3) -> U102(X1,X2,X3)
r128: a__isNatKind(X) -> isNatKind(X)
r129: a__U103(X1,X2,X3) -> U103(X1,X2,X3)
r130: a__isNatIListKind(X) -> isNatIListKind(X)
r131: a__U104(X1,X2,X3) -> U104(X1,X2,X3)
r132: a__U105(X1,X2) -> U105(X1,X2)
r133: a__isNat(X) -> isNat(X)
r134: a__U106(X) -> U106(X)
r135: a__isNatIList(X) -> isNatIList(X)
r136: a__U11(X1,X2) -> U11(X1,X2)
r137: a__U12(X1,X2) -> U12(X1,X2)
r138: a__U111(X1,X2,X3) -> U111(X1,X2,X3)
r139: a__U112(X1,X2,X3) -> U112(X1,X2,X3)
r140: a__U113(X1,X2,X3) -> U113(X1,X2,X3)
r141: a__U114(X1,X2) -> U114(X1,X2)
r142: a__length(X) -> length(X)
r143: a__U13(X) -> U13(X)
r144: a__isNatList(X) -> isNatList(X)
r145: a__U121(X1,X2) -> U121(X1,X2)
r146: a__U122(X) -> U122(X)
r147: a__U131(X1,X2,X3,X4) -> U131(X1,X2,X3,X4)
r148: a__U132(X1,X2,X3,X4) -> U132(X1,X2,X3,X4)
r149: a__U133(X1,X2,X3,X4) -> U133(X1,X2,X3,X4)
r150: a__U134(X1,X2,X3,X4) -> U134(X1,X2,X3,X4)
r151: a__U135(X1,X2,X3,X4) -> U135(X1,X2,X3,X4)
r152: a__U136(X1,X2,X3,X4) -> U136(X1,X2,X3,X4)
r153: a__take(X1,X2) -> take(X1,X2)
r154: a__U21(X1,X2) -> U21(X1,X2)
r155: a__U22(X1,X2) -> U22(X1,X2)
r156: a__U23(X) -> U23(X)
r157: a__U31(X1,X2) -> U31(X1,X2)
r158: a__U32(X1,X2) -> U32(X1,X2)
r159: a__U33(X) -> U33(X)
r160: a__U41(X1,X2,X3) -> U41(X1,X2,X3)
r161: a__U42(X1,X2,X3) -> U42(X1,X2,X3)
r162: a__U43(X1,X2,X3) -> U43(X1,X2,X3)
r163: a__U44(X1,X2,X3) -> U44(X1,X2,X3)
r164: a__U45(X1,X2) -> U45(X1,X2)
r165: a__U46(X) -> U46(X)
r166: a__U51(X1,X2) -> U51(X1,X2)
r167: a__U52(X) -> U52(X)
r168: a__U61(X1,X2) -> U61(X1,X2)
r169: a__U62(X) -> U62(X)
r170: a__U71(X) -> U71(X)
r171: a__U81(X) -> U81(X)
r172: a__U91(X1,X2,X3) -> U91(X1,X2,X3)
r173: a__U92(X1,X2,X3) -> U92(X1,X2,X3)
r174: a__U93(X1,X2,X3) -> U93(X1,X2,X3)
r175: a__U94(X1,X2,X3) -> U94(X1,X2,X3)
r176: a__U95(X1,X2) -> U95(X1,X2)
r177: a__U96(X) -> U96(X)

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  max/plus interpretations on natural numbers:
  
    mark#_A(x1) = x1 + 1
    U101_A(x1,x2,x3) = max{x1 + 1, x2, x3}
    cons_A(x1,x2) = max{x1, x2}
    U96_A(x1) = x1
    U95_A(x1,x2) = max{x1, x2}
    U81_A(x1) = x1
    U71_A(x1) = x1
    U62_A(x1) = x1
    U52_A(x1) = x1
    U46_A(x1) = x1
    U42_A(x1,x2,x3) = max{x1, x2, x3}
    U33_A(x1) = x1
    U32_A(x1,x2) = max{x1, x2}
    U31_A(x1,x2) = max{x1, x2}
    U23_A(x1) = x1
    U22_A(x1,x2) = max{x1, x2}
    U21_A(x1,x2) = max{x1, x2}
    U134_A(x1,x2,x3,x4) = max{x1, x2, x3, x4}
    U131_A(x1,x2,x3,x4) = max{x1, x2, x3, x4}
    U122_A(x1) = x1
    U121_A(x1,x2) = max{x1, x2}
    U13_A(x1) = x1
    U111_A(x1,x2,x3) = max{x1, x2, x3}
    U12_A(x1,x2) = max{x1, x2}
    U11_A(x1,x2) = max{x1, x2}
    U106_A(x1) = x1 + 1
    U103_A(x1,x2,x3) = max{x1, x2, x3}

The next rules are strictly ordered:

  p1, p25

We remove them from the problem.

-- SCC decomposition.

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

p1: mark#(cons(X1,X2)) -> mark#(X1)
p2: mark#(U96(X)) -> mark#(X)
p3: mark#(U95(X1,X2)) -> mark#(X1)
p4: mark#(U81(X)) -> mark#(X)
p5: mark#(U71(X)) -> mark#(X)
p6: mark#(U62(X)) -> mark#(X)
p7: mark#(U52(X)) -> mark#(X)
p8: mark#(U46(X)) -> mark#(X)
p9: mark#(U42(X1,X2,X3)) -> mark#(X1)
p10: mark#(U33(X)) -> mark#(X)
p11: mark#(U32(X1,X2)) -> mark#(X1)
p12: mark#(U31(X1,X2)) -> mark#(X1)
p13: mark#(U23(X)) -> mark#(X)
p14: mark#(U22(X1,X2)) -> mark#(X1)
p15: mark#(U21(X1,X2)) -> mark#(X1)
p16: mark#(U134(X1,X2,X3,X4)) -> mark#(X1)
p17: mark#(U131(X1,X2,X3,X4)) -> mark#(X1)
p18: mark#(U122(X)) -> mark#(X)
p19: mark#(U121(X1,X2)) -> mark#(X1)
p20: mark#(U13(X)) -> mark#(X)
p21: mark#(U111(X1,X2,X3)) -> mark#(X1)
p22: mark#(U12(X1,X2)) -> mark#(X1)
p23: mark#(U11(X1,X2)) -> mark#(X1)
p24: mark#(U103(X1,X2,X3)) -> mark#(X1)

and R consists of:

r1: a__zeros() -> cons(|0|(),zeros())
r2: a__U101(tt(),V1,V2) -> a__U102(a__isNatKind(V1),V1,V2)
r3: a__U102(tt(),V1,V2) -> a__U103(a__isNatIListKind(V2),V1,V2)
r4: a__U103(tt(),V1,V2) -> a__U104(a__isNatIListKind(V2),V1,V2)
r5: a__U104(tt(),V1,V2) -> a__U105(a__isNat(V1),V2)
r6: a__U105(tt(),V2) -> a__U106(a__isNatIList(V2))
r7: a__U106(tt()) -> tt()
r8: a__U11(tt(),V1) -> a__U12(a__isNatIListKind(V1),V1)
r9: a__U111(tt(),L,N) -> a__U112(a__isNatIListKind(L),L,N)
r10: a__U112(tt(),L,N) -> a__U113(a__isNat(N),L,N)
r11: a__U113(tt(),L,N) -> a__U114(a__isNatKind(N),L)
r12: a__U114(tt(),L) -> s(a__length(mark(L)))
r13: a__U12(tt(),V1) -> a__U13(a__isNatList(V1))
r14: a__U121(tt(),IL) -> a__U122(a__isNatIListKind(IL))
r15: a__U122(tt()) -> nil()
r16: a__U13(tt()) -> tt()
r17: a__U131(tt(),IL,M,N) -> a__U132(a__isNatIListKind(IL),IL,M,N)
r18: a__U132(tt(),IL,M,N) -> a__U133(a__isNat(M),IL,M,N)
r19: a__U133(tt(),IL,M,N) -> a__U134(a__isNatKind(M),IL,M,N)
r20: a__U134(tt(),IL,M,N) -> a__U135(a__isNat(N),IL,M,N)
r21: a__U135(tt(),IL,M,N) -> a__U136(a__isNatKind(N),IL,M,N)
r22: a__U136(tt(),IL,M,N) -> cons(mark(N),take(M,IL))
r23: a__U21(tt(),V1) -> a__U22(a__isNatKind(V1),V1)
r24: a__U22(tt(),V1) -> a__U23(a__isNat(V1))
r25: a__U23(tt()) -> tt()
r26: a__U31(tt(),V) -> a__U32(a__isNatIListKind(V),V)
r27: a__U32(tt(),V) -> a__U33(a__isNatList(V))
r28: a__U33(tt()) -> tt()
r29: a__U41(tt(),V1,V2) -> a__U42(a__isNatKind(V1),V1,V2)
r30: a__U42(tt(),V1,V2) -> a__U43(a__isNatIListKind(V2),V1,V2)
r31: a__U43(tt(),V1,V2) -> a__U44(a__isNatIListKind(V2),V1,V2)
r32: a__U44(tt(),V1,V2) -> a__U45(a__isNat(V1),V2)
r33: a__U45(tt(),V2) -> a__U46(a__isNatIList(V2))
r34: a__U46(tt()) -> tt()
r35: a__U51(tt(),V2) -> a__U52(a__isNatIListKind(V2))
r36: a__U52(tt()) -> tt()
r37: a__U61(tt(),V2) -> a__U62(a__isNatIListKind(V2))
r38: a__U62(tt()) -> tt()
r39: a__U71(tt()) -> tt()
r40: a__U81(tt()) -> tt()
r41: a__U91(tt(),V1,V2) -> a__U92(a__isNatKind(V1),V1,V2)
r42: a__U92(tt(),V1,V2) -> a__U93(a__isNatIListKind(V2),V1,V2)
r43: a__U93(tt(),V1,V2) -> a__U94(a__isNatIListKind(V2),V1,V2)
r44: a__U94(tt(),V1,V2) -> a__U95(a__isNat(V1),V2)
r45: a__U95(tt(),V2) -> a__U96(a__isNatList(V2))
r46: a__U96(tt()) -> tt()
r47: a__isNat(|0|()) -> tt()
r48: a__isNat(length(V1)) -> a__U11(a__isNatIListKind(V1),V1)
r49: a__isNat(s(V1)) -> a__U21(a__isNatKind(V1),V1)
r50: a__isNatIList(V) -> a__U31(a__isNatIListKind(V),V)
r51: a__isNatIList(zeros()) -> tt()
r52: a__isNatIList(cons(V1,V2)) -> a__U41(a__isNatKind(V1),V1,V2)
r53: a__isNatIListKind(nil()) -> tt()
r54: a__isNatIListKind(zeros()) -> tt()
r55: a__isNatIListKind(cons(V1,V2)) -> a__U51(a__isNatKind(V1),V2)
r56: a__isNatIListKind(take(V1,V2)) -> a__U61(a__isNatKind(V1),V2)
r57: a__isNatKind(|0|()) -> tt()
r58: a__isNatKind(length(V1)) -> a__U71(a__isNatIListKind(V1))
r59: a__isNatKind(s(V1)) -> a__U81(a__isNatKind(V1))
r60: a__isNatList(nil()) -> tt()
r61: a__isNatList(cons(V1,V2)) -> a__U91(a__isNatKind(V1),V1,V2)
r62: a__isNatList(take(V1,V2)) -> a__U101(a__isNatKind(V1),V1,V2)
r63: a__length(nil()) -> |0|()
r64: a__length(cons(N,L)) -> a__U111(a__isNatList(L),L,N)
r65: a__take(|0|(),IL) -> a__U121(a__isNatIList(IL),IL)
r66: a__take(s(M),cons(N,IL)) -> a__U131(a__isNatIList(IL),IL,M,N)
r67: mark(zeros()) -> a__zeros()
r68: mark(U101(X1,X2,X3)) -> a__U101(mark(X1),X2,X3)
r69: mark(U102(X1,X2,X3)) -> a__U102(mark(X1),X2,X3)
r70: mark(isNatKind(X)) -> a__isNatKind(X)
r71: mark(U103(X1,X2,X3)) -> a__U103(mark(X1),X2,X3)
r72: mark(isNatIListKind(X)) -> a__isNatIListKind(X)
r73: mark(U104(X1,X2,X3)) -> a__U104(mark(X1),X2,X3)
r74: mark(U105(X1,X2)) -> a__U105(mark(X1),X2)
r75: mark(isNat(X)) -> a__isNat(X)
r76: mark(U106(X)) -> a__U106(mark(X))
r77: mark(isNatIList(X)) -> a__isNatIList(X)
r78: mark(U11(X1,X2)) -> a__U11(mark(X1),X2)
r79: mark(U12(X1,X2)) -> a__U12(mark(X1),X2)
r80: mark(U111(X1,X2,X3)) -> a__U111(mark(X1),X2,X3)
r81: mark(U112(X1,X2,X3)) -> a__U112(mark(X1),X2,X3)
r82: mark(U113(X1,X2,X3)) -> a__U113(mark(X1),X2,X3)
r83: mark(U114(X1,X2)) -> a__U114(mark(X1),X2)
r84: mark(length(X)) -> a__length(mark(X))
r85: mark(U13(X)) -> a__U13(mark(X))
r86: mark(isNatList(X)) -> a__isNatList(X)
r87: mark(U121(X1,X2)) -> a__U121(mark(X1),X2)
r88: mark(U122(X)) -> a__U122(mark(X))
r89: mark(U131(X1,X2,X3,X4)) -> a__U131(mark(X1),X2,X3,X4)
r90: mark(U132(X1,X2,X3,X4)) -> a__U132(mark(X1),X2,X3,X4)
r91: mark(U133(X1,X2,X3,X4)) -> a__U133(mark(X1),X2,X3,X4)
r92: mark(U134(X1,X2,X3,X4)) -> a__U134(mark(X1),X2,X3,X4)
r93: mark(U135(X1,X2,X3,X4)) -> a__U135(mark(X1),X2,X3,X4)
r94: mark(U136(X1,X2,X3,X4)) -> a__U136(mark(X1),X2,X3,X4)
r95: mark(take(X1,X2)) -> a__take(mark(X1),mark(X2))
r96: mark(U21(X1,X2)) -> a__U21(mark(X1),X2)
r97: mark(U22(X1,X2)) -> a__U22(mark(X1),X2)
r98: mark(U23(X)) -> a__U23(mark(X))
r99: mark(U31(X1,X2)) -> a__U31(mark(X1),X2)
r100: mark(U32(X1,X2)) -> a__U32(mark(X1),X2)
r101: mark(U33(X)) -> a__U33(mark(X))
r102: mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3)
r103: mark(U42(X1,X2,X3)) -> a__U42(mark(X1),X2,X3)
r104: mark(U43(X1,X2,X3)) -> a__U43(mark(X1),X2,X3)
r105: mark(U44(X1,X2,X3)) -> a__U44(mark(X1),X2,X3)
r106: mark(U45(X1,X2)) -> a__U45(mark(X1),X2)
r107: mark(U46(X)) -> a__U46(mark(X))
r108: mark(U51(X1,X2)) -> a__U51(mark(X1),X2)
r109: mark(U52(X)) -> a__U52(mark(X))
r110: mark(U61(X1,X2)) -> a__U61(mark(X1),X2)
r111: mark(U62(X)) -> a__U62(mark(X))
r112: mark(U71(X)) -> a__U71(mark(X))
r113: mark(U81(X)) -> a__U81(mark(X))
r114: mark(U91(X1,X2,X3)) -> a__U91(mark(X1),X2,X3)
r115: mark(U92(X1,X2,X3)) -> a__U92(mark(X1),X2,X3)
r116: mark(U93(X1,X2,X3)) -> a__U93(mark(X1),X2,X3)
r117: mark(U94(X1,X2,X3)) -> a__U94(mark(X1),X2,X3)
r118: mark(U95(X1,X2)) -> a__U95(mark(X1),X2)
r119: mark(U96(X)) -> a__U96(mark(X))
r120: mark(cons(X1,X2)) -> cons(mark(X1),X2)
r121: mark(|0|()) -> |0|()
r122: mark(tt()) -> tt()
r123: mark(s(X)) -> s(mark(X))
r124: mark(nil()) -> nil()
r125: a__zeros() -> zeros()
r126: a__U101(X1,X2,X3) -> U101(X1,X2,X3)
r127: a__U102(X1,X2,X3) -> U102(X1,X2,X3)
r128: a__isNatKind(X) -> isNatKind(X)
r129: a__U103(X1,X2,X3) -> U103(X1,X2,X3)
r130: a__isNatIListKind(X) -> isNatIListKind(X)
r131: a__U104(X1,X2,X3) -> U104(X1,X2,X3)
r132: a__U105(X1,X2) -> U105(X1,X2)
r133: a__isNat(X) -> isNat(X)
r134: a__U106(X) -> U106(X)
r135: a__isNatIList(X) -> isNatIList(X)
r136: a__U11(X1,X2) -> U11(X1,X2)
r137: a__U12(X1,X2) -> U12(X1,X2)
r138: a__U111(X1,X2,X3) -> U111(X1,X2,X3)
r139: a__U112(X1,X2,X3) -> U112(X1,X2,X3)
r140: a__U113(X1,X2,X3) -> U113(X1,X2,X3)
r141: a__U114(X1,X2) -> U114(X1,X2)
r142: a__length(X) -> length(X)
r143: a__U13(X) -> U13(X)
r144: a__isNatList(X) -> isNatList(X)
r145: a__U121(X1,X2) -> U121(X1,X2)
r146: a__U122(X) -> U122(X)
r147: a__U131(X1,X2,X3,X4) -> U131(X1,X2,X3,X4)
r148: a__U132(X1,X2,X3,X4) -> U132(X1,X2,X3,X4)
r149: a__U133(X1,X2,X3,X4) -> U133(X1,X2,X3,X4)
r150: a__U134(X1,X2,X3,X4) -> U134(X1,X2,X3,X4)
r151: a__U135(X1,X2,X3,X4) -> U135(X1,X2,X3,X4)
r152: a__U136(X1,X2,X3,X4) -> U136(X1,X2,X3,X4)
r153: a__take(X1,X2) -> take(X1,X2)
r154: a__U21(X1,X2) -> U21(X1,X2)
r155: a__U22(X1,X2) -> U22(X1,X2)
r156: a__U23(X) -> U23(X)
r157: a__U31(X1,X2) -> U31(X1,X2)
r158: a__U32(X1,X2) -> U32(X1,X2)
r159: a__U33(X) -> U33(X)
r160: a__U41(X1,X2,X3) -> U41(X1,X2,X3)
r161: a__U42(X1,X2,X3) -> U42(X1,X2,X3)
r162: a__U43(X1,X2,X3) -> U43(X1,X2,X3)
r163: a__U44(X1,X2,X3) -> U44(X1,X2,X3)
r164: a__U45(X1,X2) -> U45(X1,X2)
r165: a__U46(X) -> U46(X)
r166: a__U51(X1,X2) -> U51(X1,X2)
r167: a__U52(X) -> U52(X)
r168: a__U61(X1,X2) -> U61(X1,X2)
r169: a__U62(X) -> U62(X)
r170: a__U71(X) -> U71(X)
r171: a__U81(X) -> U81(X)
r172: a__U91(X1,X2,X3) -> U91(X1,X2,X3)
r173: a__U92(X1,X2,X3) -> U92(X1,X2,X3)
r174: a__U93(X1,X2,X3) -> U93(X1,X2,X3)
r175: a__U94(X1,X2,X3) -> U94(X1,X2,X3)
r176: a__U95(X1,X2) -> U95(X1,X2)
r177: a__U96(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}


-- Reduction pair.

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

p1: mark#(cons(X1,X2)) -> mark#(X1)
p2: mark#(U103(X1,X2,X3)) -> mark#(X1)
p3: mark#(U11(X1,X2)) -> mark#(X1)
p4: mark#(U12(X1,X2)) -> mark#(X1)
p5: mark#(U111(X1,X2,X3)) -> mark#(X1)
p6: mark#(U13(X)) -> mark#(X)
p7: mark#(U121(X1,X2)) -> mark#(X1)
p8: mark#(U122(X)) -> mark#(X)
p9: mark#(U131(X1,X2,X3,X4)) -> mark#(X1)
p10: mark#(U134(X1,X2,X3,X4)) -> mark#(X1)
p11: mark#(U21(X1,X2)) -> mark#(X1)
p12: mark#(U22(X1,X2)) -> mark#(X1)
p13: mark#(U23(X)) -> mark#(X)
p14: mark#(U31(X1,X2)) -> mark#(X1)
p15: mark#(U32(X1,X2)) -> mark#(X1)
p16: mark#(U33(X)) -> mark#(X)
p17: mark#(U42(X1,X2,X3)) -> mark#(X1)
p18: mark#(U46(X)) -> mark#(X)
p19: mark#(U52(X)) -> mark#(X)
p20: mark#(U62(X)) -> mark#(X)
p21: mark#(U71(X)) -> mark#(X)
p22: mark#(U81(X)) -> mark#(X)
p23: mark#(U95(X1,X2)) -> mark#(X1)
p24: mark#(U96(X)) -> mark#(X)

and R consists of:

r1: a__zeros() -> cons(|0|(),zeros())
r2: a__U101(tt(),V1,V2) -> a__U102(a__isNatKind(V1),V1,V2)
r3: a__U102(tt(),V1,V2) -> a__U103(a__isNatIListKind(V2),V1,V2)
r4: a__U103(tt(),V1,V2) -> a__U104(a__isNatIListKind(V2),V1,V2)
r5: a__U104(tt(),V1,V2) -> a__U105(a__isNat(V1),V2)
r6: a__U105(tt(),V2) -> a__U106(a__isNatIList(V2))
r7: a__U106(tt()) -> tt()
r8: a__U11(tt(),V1) -> a__U12(a__isNatIListKind(V1),V1)
r9: a__U111(tt(),L,N) -> a__U112(a__isNatIListKind(L),L,N)
r10: a__U112(tt(),L,N) -> a__U113(a__isNat(N),L,N)
r11: a__U113(tt(),L,N) -> a__U114(a__isNatKind(N),L)
r12: a__U114(tt(),L) -> s(a__length(mark(L)))
r13: a__U12(tt(),V1) -> a__U13(a__isNatList(V1))
r14: a__U121(tt(),IL) -> a__U122(a__isNatIListKind(IL))
r15: a__U122(tt()) -> nil()
r16: a__U13(tt()) -> tt()
r17: a__U131(tt(),IL,M,N) -> a__U132(a__isNatIListKind(IL),IL,M,N)
r18: a__U132(tt(),IL,M,N) -> a__U133(a__isNat(M),IL,M,N)
r19: a__U133(tt(),IL,M,N) -> a__U134(a__isNatKind(M),IL,M,N)
r20: a__U134(tt(),IL,M,N) -> a__U135(a__isNat(N),IL,M,N)
r21: a__U135(tt(),IL,M,N) -> a__U136(a__isNatKind(N),IL,M,N)
r22: a__U136(tt(),IL,M,N) -> cons(mark(N),take(M,IL))
r23: a__U21(tt(),V1) -> a__U22(a__isNatKind(V1),V1)
r24: a__U22(tt(),V1) -> a__U23(a__isNat(V1))
r25: a__U23(tt()) -> tt()
r26: a__U31(tt(),V) -> a__U32(a__isNatIListKind(V),V)
r27: a__U32(tt(),V) -> a__U33(a__isNatList(V))
r28: a__U33(tt()) -> tt()
r29: a__U41(tt(),V1,V2) -> a__U42(a__isNatKind(V1),V1,V2)
r30: a__U42(tt(),V1,V2) -> a__U43(a__isNatIListKind(V2),V1,V2)
r31: a__U43(tt(),V1,V2) -> a__U44(a__isNatIListKind(V2),V1,V2)
r32: a__U44(tt(),V1,V2) -> a__U45(a__isNat(V1),V2)
r33: a__U45(tt(),V2) -> a__U46(a__isNatIList(V2))
r34: a__U46(tt()) -> tt()
r35: a__U51(tt(),V2) -> a__U52(a__isNatIListKind(V2))
r36: a__U52(tt()) -> tt()
r37: a__U61(tt(),V2) -> a__U62(a__isNatIListKind(V2))
r38: a__U62(tt()) -> tt()
r39: a__U71(tt()) -> tt()
r40: a__U81(tt()) -> tt()
r41: a__U91(tt(),V1,V2) -> a__U92(a__isNatKind(V1),V1,V2)
r42: a__U92(tt(),V1,V2) -> a__U93(a__isNatIListKind(V2),V1,V2)
r43: a__U93(tt(),V1,V2) -> a__U94(a__isNatIListKind(V2),V1,V2)
r44: a__U94(tt(),V1,V2) -> a__U95(a__isNat(V1),V2)
r45: a__U95(tt(),V2) -> a__U96(a__isNatList(V2))
r46: a__U96(tt()) -> tt()
r47: a__isNat(|0|()) -> tt()
r48: a__isNat(length(V1)) -> a__U11(a__isNatIListKind(V1),V1)
r49: a__isNat(s(V1)) -> a__U21(a__isNatKind(V1),V1)
r50: a__isNatIList(V) -> a__U31(a__isNatIListKind(V),V)
r51: a__isNatIList(zeros()) -> tt()
r52: a__isNatIList(cons(V1,V2)) -> a__U41(a__isNatKind(V1),V1,V2)
r53: a__isNatIListKind(nil()) -> tt()
r54: a__isNatIListKind(zeros()) -> tt()
r55: a__isNatIListKind(cons(V1,V2)) -> a__U51(a__isNatKind(V1),V2)
r56: a__isNatIListKind(take(V1,V2)) -> a__U61(a__isNatKind(V1),V2)
r57: a__isNatKind(|0|()) -> tt()
r58: a__isNatKind(length(V1)) -> a__U71(a__isNatIListKind(V1))
r59: a__isNatKind(s(V1)) -> a__U81(a__isNatKind(V1))
r60: a__isNatList(nil()) -> tt()
r61: a__isNatList(cons(V1,V2)) -> a__U91(a__isNatKind(V1),V1,V2)
r62: a__isNatList(take(V1,V2)) -> a__U101(a__isNatKind(V1),V1,V2)
r63: a__length(nil()) -> |0|()
r64: a__length(cons(N,L)) -> a__U111(a__isNatList(L),L,N)
r65: a__take(|0|(),IL) -> a__U121(a__isNatIList(IL),IL)
r66: a__take(s(M),cons(N,IL)) -> a__U131(a__isNatIList(IL),IL,M,N)
r67: mark(zeros()) -> a__zeros()
r68: mark(U101(X1,X2,X3)) -> a__U101(mark(X1),X2,X3)
r69: mark(U102(X1,X2,X3)) -> a__U102(mark(X1),X2,X3)
r70: mark(isNatKind(X)) -> a__isNatKind(X)
r71: mark(U103(X1,X2,X3)) -> a__U103(mark(X1),X2,X3)
r72: mark(isNatIListKind(X)) -> a__isNatIListKind(X)
r73: mark(U104(X1,X2,X3)) -> a__U104(mark(X1),X2,X3)
r74: mark(U105(X1,X2)) -> a__U105(mark(X1),X2)
r75: mark(isNat(X)) -> a__isNat(X)
r76: mark(U106(X)) -> a__U106(mark(X))
r77: mark(isNatIList(X)) -> a__isNatIList(X)
r78: mark(U11(X1,X2)) -> a__U11(mark(X1),X2)
r79: mark(U12(X1,X2)) -> a__U12(mark(X1),X2)
r80: mark(U111(X1,X2,X3)) -> a__U111(mark(X1),X2,X3)
r81: mark(U112(X1,X2,X3)) -> a__U112(mark(X1),X2,X3)
r82: mark(U113(X1,X2,X3)) -> a__U113(mark(X1),X2,X3)
r83: mark(U114(X1,X2)) -> a__U114(mark(X1),X2)
r84: mark(length(X)) -> a__length(mark(X))
r85: mark(U13(X)) -> a__U13(mark(X))
r86: mark(isNatList(X)) -> a__isNatList(X)
r87: mark(U121(X1,X2)) -> a__U121(mark(X1),X2)
r88: mark(U122(X)) -> a__U122(mark(X))
r89: mark(U131(X1,X2,X3,X4)) -> a__U131(mark(X1),X2,X3,X4)
r90: mark(U132(X1,X2,X3,X4)) -> a__U132(mark(X1),X2,X3,X4)
r91: mark(U133(X1,X2,X3,X4)) -> a__U133(mark(X1),X2,X3,X4)
r92: mark(U134(X1,X2,X3,X4)) -> a__U134(mark(X1),X2,X3,X4)
r93: mark(U135(X1,X2,X3,X4)) -> a__U135(mark(X1),X2,X3,X4)
r94: mark(U136(X1,X2,X3,X4)) -> a__U136(mark(X1),X2,X3,X4)
r95: mark(take(X1,X2)) -> a__take(mark(X1),mark(X2))
r96: mark(U21(X1,X2)) -> a__U21(mark(X1),X2)
r97: mark(U22(X1,X2)) -> a__U22(mark(X1),X2)
r98: mark(U23(X)) -> a__U23(mark(X))
r99: mark(U31(X1,X2)) -> a__U31(mark(X1),X2)
r100: mark(U32(X1,X2)) -> a__U32(mark(X1),X2)
r101: mark(U33(X)) -> a__U33(mark(X))
r102: mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3)
r103: mark(U42(X1,X2,X3)) -> a__U42(mark(X1),X2,X3)
r104: mark(U43(X1,X2,X3)) -> a__U43(mark(X1),X2,X3)
r105: mark(U44(X1,X2,X3)) -> a__U44(mark(X1),X2,X3)
r106: mark(U45(X1,X2)) -> a__U45(mark(X1),X2)
r107: mark(U46(X)) -> a__U46(mark(X))
r108: mark(U51(X1,X2)) -> a__U51(mark(X1),X2)
r109: mark(U52(X)) -> a__U52(mark(X))
r110: mark(U61(X1,X2)) -> a__U61(mark(X1),X2)
r111: mark(U62(X)) -> a__U62(mark(X))
r112: mark(U71(X)) -> a__U71(mark(X))
r113: mark(U81(X)) -> a__U81(mark(X))
r114: mark(U91(X1,X2,X3)) -> a__U91(mark(X1),X2,X3)
r115: mark(U92(X1,X2,X3)) -> a__U92(mark(X1),X2,X3)
r116: mark(U93(X1,X2,X3)) -> a__U93(mark(X1),X2,X3)
r117: mark(U94(X1,X2,X3)) -> a__U94(mark(X1),X2,X3)
r118: mark(U95(X1,X2)) -> a__U95(mark(X1),X2)
r119: mark(U96(X)) -> a__U96(mark(X))
r120: mark(cons(X1,X2)) -> cons(mark(X1),X2)
r121: mark(|0|()) -> |0|()
r122: mark(tt()) -> tt()
r123: mark(s(X)) -> s(mark(X))
r124: mark(nil()) -> nil()
r125: a__zeros() -> zeros()
r126: a__U101(X1,X2,X3) -> U101(X1,X2,X3)
r127: a__U102(X1,X2,X3) -> U102(X1,X2,X3)
r128: a__isNatKind(X) -> isNatKind(X)
r129: a__U103(X1,X2,X3) -> U103(X1,X2,X3)
r130: a__isNatIListKind(X) -> isNatIListKind(X)
r131: a__U104(X1,X2,X3) -> U104(X1,X2,X3)
r132: a__U105(X1,X2) -> U105(X1,X2)
r133: a__isNat(X) -> isNat(X)
r134: a__U106(X) -> U106(X)
r135: a__isNatIList(X) -> isNatIList(X)
r136: a__U11(X1,X2) -> U11(X1,X2)
r137: a__U12(X1,X2) -> U12(X1,X2)
r138: a__U111(X1,X2,X3) -> U111(X1,X2,X3)
r139: a__U112(X1,X2,X3) -> U112(X1,X2,X3)
r140: a__U113(X1,X2,X3) -> U113(X1,X2,X3)
r141: a__U114(X1,X2) -> U114(X1,X2)
r142: a__length(X) -> length(X)
r143: a__U13(X) -> U13(X)
r144: a__isNatList(X) -> isNatList(X)
r145: a__U121(X1,X2) -> U121(X1,X2)
r146: a__U122(X) -> U122(X)
r147: a__U131(X1,X2,X3,X4) -> U131(X1,X2,X3,X4)
r148: a__U132(X1,X2,X3,X4) -> U132(X1,X2,X3,X4)
r149: a__U133(X1,X2,X3,X4) -> U133(X1,X2,X3,X4)
r150: a__U134(X1,X2,X3,X4) -> U134(X1,X2,X3,X4)
r151: a__U135(X1,X2,X3,X4) -> U135(X1,X2,X3,X4)
r152: a__U136(X1,X2,X3,X4) -> U136(X1,X2,X3,X4)
r153: a__take(X1,X2) -> take(X1,X2)
r154: a__U21(X1,X2) -> U21(X1,X2)
r155: a__U22(X1,X2) -> U22(X1,X2)
r156: a__U23(X) -> U23(X)
r157: a__U31(X1,X2) -> U31(X1,X2)
r158: a__U32(X1,X2) -> U32(X1,X2)
r159: a__U33(X) -> U33(X)
r160: a__U41(X1,X2,X3) -> U41(X1,X2,X3)
r161: a__U42(X1,X2,X3) -> U42(X1,X2,X3)
r162: a__U43(X1,X2,X3) -> U43(X1,X2,X3)
r163: a__U44(X1,X2,X3) -> U44(X1,X2,X3)
r164: a__U45(X1,X2) -> U45(X1,X2)
r165: a__U46(X) -> U46(X)
r166: a__U51(X1,X2) -> U51(X1,X2)
r167: a__U52(X) -> U52(X)
r168: a__U61(X1,X2) -> U61(X1,X2)
r169: a__U62(X) -> U62(X)
r170: a__U71(X) -> U71(X)
r171: a__U81(X) -> U81(X)
r172: a__U91(X1,X2,X3) -> U91(X1,X2,X3)
r173: a__U92(X1,X2,X3) -> U92(X1,X2,X3)
r174: a__U93(X1,X2,X3) -> U93(X1,X2,X3)
r175: a__U94(X1,X2,X3) -> U94(X1,X2,X3)
r176: a__U95(X1,X2) -> U95(X1,X2)
r177: a__U96(X) -> U96(X)

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  max/plus interpretations on natural numbers:
  
    mark#_A(x1) = x1 + 1
    cons_A(x1,x2) = max{x1 + 1, x2}
    U103_A(x1,x2,x3) = max{x1, x2, x3}
    U11_A(x1,x2) = max{x1, x2}
    U12_A(x1,x2) = max{x1, x2}
    U111_A(x1,x2,x3) = max{x1, x2, x3}
    U13_A(x1) = x1
    U121_A(x1,x2) = max{x1, x2}
    U122_A(x1) = x1
    U131_A(x1,x2,x3,x4) = max{x1, x2, x3, x4}
    U134_A(x1,x2,x3,x4) = max{x1, x2, x3, x4}
    U21_A(x1,x2) = max{x1, x2}
    U22_A(x1,x2) = max{x1 + 1, x2}
    U23_A(x1) = x1
    U31_A(x1,x2) = max{x1, x2}
    U32_A(x1,x2) = max{x1, x2}
    U33_A(x1) = x1
    U42_A(x1,x2,x3) = max{x1, x2, x3}
    U46_A(x1) = x1
    U52_A(x1) = x1
    U62_A(x1) = x1
    U71_A(x1) = x1
    U81_A(x1) = x1
    U95_A(x1,x2) = max{x1, x2}
    U96_A(x1) = x1

The next rules are strictly ordered:

  p1, p12

We remove them from the problem.

-- SCC decomposition.

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

p1: mark#(U103(X1,X2,X3)) -> mark#(X1)
p2: mark#(U11(X1,X2)) -> mark#(X1)
p3: mark#(U12(X1,X2)) -> mark#(X1)
p4: mark#(U111(X1,X2,X3)) -> mark#(X1)
p5: mark#(U13(X)) -> mark#(X)
p6: mark#(U121(X1,X2)) -> mark#(X1)
p7: mark#(U122(X)) -> mark#(X)
p8: mark#(U131(X1,X2,X3,X4)) -> mark#(X1)
p9: mark#(U134(X1,X2,X3,X4)) -> mark#(X1)
p10: mark#(U21(X1,X2)) -> mark#(X1)
p11: mark#(U23(X)) -> mark#(X)
p12: mark#(U31(X1,X2)) -> mark#(X1)
p13: mark#(U32(X1,X2)) -> mark#(X1)
p14: mark#(U33(X)) -> mark#(X)
p15: mark#(U42(X1,X2,X3)) -> mark#(X1)
p16: mark#(U46(X)) -> mark#(X)
p17: mark#(U52(X)) -> mark#(X)
p18: mark#(U62(X)) -> mark#(X)
p19: mark#(U71(X)) -> mark#(X)
p20: mark#(U81(X)) -> mark#(X)
p21: mark#(U95(X1,X2)) -> mark#(X1)
p22: mark#(U96(X)) -> mark#(X)

and R consists of:

r1: a__zeros() -> cons(|0|(),zeros())
r2: a__U101(tt(),V1,V2) -> a__U102(a__isNatKind(V1),V1,V2)
r3: a__U102(tt(),V1,V2) -> a__U103(a__isNatIListKind(V2),V1,V2)
r4: a__U103(tt(),V1,V2) -> a__U104(a__isNatIListKind(V2),V1,V2)
r5: a__U104(tt(),V1,V2) -> a__U105(a__isNat(V1),V2)
r6: a__U105(tt(),V2) -> a__U106(a__isNatIList(V2))
r7: a__U106(tt()) -> tt()
r8: a__U11(tt(),V1) -> a__U12(a__isNatIListKind(V1),V1)
r9: a__U111(tt(),L,N) -> a__U112(a__isNatIListKind(L),L,N)
r10: a__U112(tt(),L,N) -> a__U113(a__isNat(N),L,N)
r11: a__U113(tt(),L,N) -> a__U114(a__isNatKind(N),L)
r12: a__U114(tt(),L) -> s(a__length(mark(L)))
r13: a__U12(tt(),V1) -> a__U13(a__isNatList(V1))
r14: a__U121(tt(),IL) -> a__U122(a__isNatIListKind(IL))
r15: a__U122(tt()) -> nil()
r16: a__U13(tt()) -> tt()
r17: a__U131(tt(),IL,M,N) -> a__U132(a__isNatIListKind(IL),IL,M,N)
r18: a__U132(tt(),IL,M,N) -> a__U133(a__isNat(M),IL,M,N)
r19: a__U133(tt(),IL,M,N) -> a__U134(a__isNatKind(M),IL,M,N)
r20: a__U134(tt(),IL,M,N) -> a__U135(a__isNat(N),IL,M,N)
r21: a__U135(tt(),IL,M,N) -> a__U136(a__isNatKind(N),IL,M,N)
r22: a__U136(tt(),IL,M,N) -> cons(mark(N),take(M,IL))
r23: a__U21(tt(),V1) -> a__U22(a__isNatKind(V1),V1)
r24: a__U22(tt(),V1) -> a__U23(a__isNat(V1))
r25: a__U23(tt()) -> tt()
r26: a__U31(tt(),V) -> a__U32(a__isNatIListKind(V),V)
r27: a__U32(tt(),V) -> a__U33(a__isNatList(V))
r28: a__U33(tt()) -> tt()
r29: a__U41(tt(),V1,V2) -> a__U42(a__isNatKind(V1),V1,V2)
r30: a__U42(tt(),V1,V2) -> a__U43(a__isNatIListKind(V2),V1,V2)
r31: a__U43(tt(),V1,V2) -> a__U44(a__isNatIListKind(V2),V1,V2)
r32: a__U44(tt(),V1,V2) -> a__U45(a__isNat(V1),V2)
r33: a__U45(tt(),V2) -> a__U46(a__isNatIList(V2))
r34: a__U46(tt()) -> tt()
r35: a__U51(tt(),V2) -> a__U52(a__isNatIListKind(V2))
r36: a__U52(tt()) -> tt()
r37: a__U61(tt(),V2) -> a__U62(a__isNatIListKind(V2))
r38: a__U62(tt()) -> tt()
r39: a__U71(tt()) -> tt()
r40: a__U81(tt()) -> tt()
r41: a__U91(tt(),V1,V2) -> a__U92(a__isNatKind(V1),V1,V2)
r42: a__U92(tt(),V1,V2) -> a__U93(a__isNatIListKind(V2),V1,V2)
r43: a__U93(tt(),V1,V2) -> a__U94(a__isNatIListKind(V2),V1,V2)
r44: a__U94(tt(),V1,V2) -> a__U95(a__isNat(V1),V2)
r45: a__U95(tt(),V2) -> a__U96(a__isNatList(V2))
r46: a__U96(tt()) -> tt()
r47: a__isNat(|0|()) -> tt()
r48: a__isNat(length(V1)) -> a__U11(a__isNatIListKind(V1),V1)
r49: a__isNat(s(V1)) -> a__U21(a__isNatKind(V1),V1)
r50: a__isNatIList(V) -> a__U31(a__isNatIListKind(V),V)
r51: a__isNatIList(zeros()) -> tt()
r52: a__isNatIList(cons(V1,V2)) -> a__U41(a__isNatKind(V1),V1,V2)
r53: a__isNatIListKind(nil()) -> tt()
r54: a__isNatIListKind(zeros()) -> tt()
r55: a__isNatIListKind(cons(V1,V2)) -> a__U51(a__isNatKind(V1),V2)
r56: a__isNatIListKind(take(V1,V2)) -> a__U61(a__isNatKind(V1),V2)
r57: a__isNatKind(|0|()) -> tt()
r58: a__isNatKind(length(V1)) -> a__U71(a__isNatIListKind(V1))
r59: a__isNatKind(s(V1)) -> a__U81(a__isNatKind(V1))
r60: a__isNatList(nil()) -> tt()
r61: a__isNatList(cons(V1,V2)) -> a__U91(a__isNatKind(V1),V1,V2)
r62: a__isNatList(take(V1,V2)) -> a__U101(a__isNatKind(V1),V1,V2)
r63: a__length(nil()) -> |0|()
r64: a__length(cons(N,L)) -> a__U111(a__isNatList(L),L,N)
r65: a__take(|0|(),IL) -> a__U121(a__isNatIList(IL),IL)
r66: a__take(s(M),cons(N,IL)) -> a__U131(a__isNatIList(IL),IL,M,N)
r67: mark(zeros()) -> a__zeros()
r68: mark(U101(X1,X2,X3)) -> a__U101(mark(X1),X2,X3)
r69: mark(U102(X1,X2,X3)) -> a__U102(mark(X1),X2,X3)
r70: mark(isNatKind(X)) -> a__isNatKind(X)
r71: mark(U103(X1,X2,X3)) -> a__U103(mark(X1),X2,X3)
r72: mark(isNatIListKind(X)) -> a__isNatIListKind(X)
r73: mark(U104(X1,X2,X3)) -> a__U104(mark(X1),X2,X3)
r74: mark(U105(X1,X2)) -> a__U105(mark(X1),X2)
r75: mark(isNat(X)) -> a__isNat(X)
r76: mark(U106(X)) -> a__U106(mark(X))
r77: mark(isNatIList(X)) -> a__isNatIList(X)
r78: mark(U11(X1,X2)) -> a__U11(mark(X1),X2)
r79: mark(U12(X1,X2)) -> a__U12(mark(X1),X2)
r80: mark(U111(X1,X2,X3)) -> a__U111(mark(X1),X2,X3)
r81: mark(U112(X1,X2,X3)) -> a__U112(mark(X1),X2,X3)
r82: mark(U113(X1,X2,X3)) -> a__U113(mark(X1),X2,X3)
r83: mark(U114(X1,X2)) -> a__U114(mark(X1),X2)
r84: mark(length(X)) -> a__length(mark(X))
r85: mark(U13(X)) -> a__U13(mark(X))
r86: mark(isNatList(X)) -> a__isNatList(X)
r87: mark(U121(X1,X2)) -> a__U121(mark(X1),X2)
r88: mark(U122(X)) -> a__U122(mark(X))
r89: mark(U131(X1,X2,X3,X4)) -> a__U131(mark(X1),X2,X3,X4)
r90: mark(U132(X1,X2,X3,X4)) -> a__U132(mark(X1),X2,X3,X4)
r91: mark(U133(X1,X2,X3,X4)) -> a__U133(mark(X1),X2,X3,X4)
r92: mark(U134(X1,X2,X3,X4)) -> a__U134(mark(X1),X2,X3,X4)
r93: mark(U135(X1,X2,X3,X4)) -> a__U135(mark(X1),X2,X3,X4)
r94: mark(U136(X1,X2,X3,X4)) -> a__U136(mark(X1),X2,X3,X4)
r95: mark(take(X1,X2)) -> a__take(mark(X1),mark(X2))
r96: mark(U21(X1,X2)) -> a__U21(mark(X1),X2)
r97: mark(U22(X1,X2)) -> a__U22(mark(X1),X2)
r98: mark(U23(X)) -> a__U23(mark(X))
r99: mark(U31(X1,X2)) -> a__U31(mark(X1),X2)
r100: mark(U32(X1,X2)) -> a__U32(mark(X1),X2)
r101: mark(U33(X)) -> a__U33(mark(X))
r102: mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3)
r103: mark(U42(X1,X2,X3)) -> a__U42(mark(X1),X2,X3)
r104: mark(U43(X1,X2,X3)) -> a__U43(mark(X1),X2,X3)
r105: mark(U44(X1,X2,X3)) -> a__U44(mark(X1),X2,X3)
r106: mark(U45(X1,X2)) -> a__U45(mark(X1),X2)
r107: mark(U46(X)) -> a__U46(mark(X))
r108: mark(U51(X1,X2)) -> a__U51(mark(X1),X2)
r109: mark(U52(X)) -> a__U52(mark(X))
r110: mark(U61(X1,X2)) -> a__U61(mark(X1),X2)
r111: mark(U62(X)) -> a__U62(mark(X))
r112: mark(U71(X)) -> a__U71(mark(X))
r113: mark(U81(X)) -> a__U81(mark(X))
r114: mark(U91(X1,X2,X3)) -> a__U91(mark(X1),X2,X3)
r115: mark(U92(X1,X2,X3)) -> a__U92(mark(X1),X2,X3)
r116: mark(U93(X1,X2,X3)) -> a__U93(mark(X1),X2,X3)
r117: mark(U94(X1,X2,X3)) -> a__U94(mark(X1),X2,X3)
r118: mark(U95(X1,X2)) -> a__U95(mark(X1),X2)
r119: mark(U96(X)) -> a__U96(mark(X))
r120: mark(cons(X1,X2)) -> cons(mark(X1),X2)
r121: mark(|0|()) -> |0|()
r122: mark(tt()) -> tt()
r123: mark(s(X)) -> s(mark(X))
r124: mark(nil()) -> nil()
r125: a__zeros() -> zeros()
r126: a__U101(X1,X2,X3) -> U101(X1,X2,X3)
r127: a__U102(X1,X2,X3) -> U102(X1,X2,X3)
r128: a__isNatKind(X) -> isNatKind(X)
r129: a__U103(X1,X2,X3) -> U103(X1,X2,X3)
r130: a__isNatIListKind(X) -> isNatIListKind(X)
r131: a__U104(X1,X2,X3) -> U104(X1,X2,X3)
r132: a__U105(X1,X2) -> U105(X1,X2)
r133: a__isNat(X) -> isNat(X)
r134: a__U106(X) -> U106(X)
r135: a__isNatIList(X) -> isNatIList(X)
r136: a__U11(X1,X2) -> U11(X1,X2)
r137: a__U12(X1,X2) -> U12(X1,X2)
r138: a__U111(X1,X2,X3) -> U111(X1,X2,X3)
r139: a__U112(X1,X2,X3) -> U112(X1,X2,X3)
r140: a__U113(X1,X2,X3) -> U113(X1,X2,X3)
r141: a__U114(X1,X2) -> U114(X1,X2)
r142: a__length(X) -> length(X)
r143: a__U13(X) -> U13(X)
r144: a__isNatList(X) -> isNatList(X)
r145: a__U121(X1,X2) -> U121(X1,X2)
r146: a__U122(X) -> U122(X)
r147: a__U131(X1,X2,X3,X4) -> U131(X1,X2,X3,X4)
r148: a__U132(X1,X2,X3,X4) -> U132(X1,X2,X3,X4)
r149: a__U133(X1,X2,X3,X4) -> U133(X1,X2,X3,X4)
r150: a__U134(X1,X2,X3,X4) -> U134(X1,X2,X3,X4)
r151: a__U135(X1,X2,X3,X4) -> U135(X1,X2,X3,X4)
r152: a__U136(X1,X2,X3,X4) -> U136(X1,X2,X3,X4)
r153: a__take(X1,X2) -> take(X1,X2)
r154: a__U21(X1,X2) -> U21(X1,X2)
r155: a__U22(X1,X2) -> U22(X1,X2)
r156: a__U23(X) -> U23(X)
r157: a__U31(X1,X2) -> U31(X1,X2)
r158: a__U32(X1,X2) -> U32(X1,X2)
r159: a__U33(X) -> U33(X)
r160: a__U41(X1,X2,X3) -> U41(X1,X2,X3)
r161: a__U42(X1,X2,X3) -> U42(X1,X2,X3)
r162: a__U43(X1,X2,X3) -> U43(X1,X2,X3)
r163: a__U44(X1,X2,X3) -> U44(X1,X2,X3)
r164: a__U45(X1,X2) -> U45(X1,X2)
r165: a__U46(X) -> U46(X)
r166: a__U51(X1,X2) -> U51(X1,X2)
r167: a__U52(X) -> U52(X)
r168: a__U61(X1,X2) -> U61(X1,X2)
r169: a__U62(X) -> U62(X)
r170: a__U71(X) -> U71(X)
r171: a__U81(X) -> U81(X)
r172: a__U91(X1,X2,X3) -> U91(X1,X2,X3)
r173: a__U92(X1,X2,X3) -> U92(X1,X2,X3)
r174: a__U93(X1,X2,X3) -> U93(X1,X2,X3)
r175: a__U94(X1,X2,X3) -> U94(X1,X2,X3)
r176: a__U95(X1,X2) -> U95(X1,X2)
r177: a__U96(X) -> U96(X)

The estimated dependency graph contains the following SCCs:

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


-- Reduction pair.

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

p1: mark#(U103(X1,X2,X3)) -> mark#(X1)
p2: mark#(U96(X)) -> mark#(X)
p3: mark#(U95(X1,X2)) -> mark#(X1)
p4: mark#(U81(X)) -> mark#(X)
p5: mark#(U71(X)) -> mark#(X)
p6: mark#(U62(X)) -> mark#(X)
p7: mark#(U52(X)) -> mark#(X)
p8: mark#(U46(X)) -> mark#(X)
p9: mark#(U42(X1,X2,X3)) -> mark#(X1)
p10: mark#(U33(X)) -> mark#(X)
p11: mark#(U32(X1,X2)) -> mark#(X1)
p12: mark#(U31(X1,X2)) -> mark#(X1)
p13: mark#(U23(X)) -> mark#(X)
p14: mark#(U21(X1,X2)) -> mark#(X1)
p15: mark#(U134(X1,X2,X3,X4)) -> mark#(X1)
p16: mark#(U131(X1,X2,X3,X4)) -> mark#(X1)
p17: mark#(U122(X)) -> mark#(X)
p18: mark#(U121(X1,X2)) -> mark#(X1)
p19: mark#(U13(X)) -> mark#(X)
p20: mark#(U111(X1,X2,X3)) -> mark#(X1)
p21: mark#(U12(X1,X2)) -> mark#(X1)
p22: mark#(U11(X1,X2)) -> mark#(X1)

and R consists of:

r1: a__zeros() -> cons(|0|(),zeros())
r2: a__U101(tt(),V1,V2) -> a__U102(a__isNatKind(V1),V1,V2)
r3: a__U102(tt(),V1,V2) -> a__U103(a__isNatIListKind(V2),V1,V2)
r4: a__U103(tt(),V1,V2) -> a__U104(a__isNatIListKind(V2),V1,V2)
r5: a__U104(tt(),V1,V2) -> a__U105(a__isNat(V1),V2)
r6: a__U105(tt(),V2) -> a__U106(a__isNatIList(V2))
r7: a__U106(tt()) -> tt()
r8: a__U11(tt(),V1) -> a__U12(a__isNatIListKind(V1),V1)
r9: a__U111(tt(),L,N) -> a__U112(a__isNatIListKind(L),L,N)
r10: a__U112(tt(),L,N) -> a__U113(a__isNat(N),L,N)
r11: a__U113(tt(),L,N) -> a__U114(a__isNatKind(N),L)
r12: a__U114(tt(),L) -> s(a__length(mark(L)))
r13: a__U12(tt(),V1) -> a__U13(a__isNatList(V1))
r14: a__U121(tt(),IL) -> a__U122(a__isNatIListKind(IL))
r15: a__U122(tt()) -> nil()
r16: a__U13(tt()) -> tt()
r17: a__U131(tt(),IL,M,N) -> a__U132(a__isNatIListKind(IL),IL,M,N)
r18: a__U132(tt(),IL,M,N) -> a__U133(a__isNat(M),IL,M,N)
r19: a__U133(tt(),IL,M,N) -> a__U134(a__isNatKind(M),IL,M,N)
r20: a__U134(tt(),IL,M,N) -> a__U135(a__isNat(N),IL,M,N)
r21: a__U135(tt(),IL,M,N) -> a__U136(a__isNatKind(N),IL,M,N)
r22: a__U136(tt(),IL,M,N) -> cons(mark(N),take(M,IL))
r23: a__U21(tt(),V1) -> a__U22(a__isNatKind(V1),V1)
r24: a__U22(tt(),V1) -> a__U23(a__isNat(V1))
r25: a__U23(tt()) -> tt()
r26: a__U31(tt(),V) -> a__U32(a__isNatIListKind(V),V)
r27: a__U32(tt(),V) -> a__U33(a__isNatList(V))
r28: a__U33(tt()) -> tt()
r29: a__U41(tt(),V1,V2) -> a__U42(a__isNatKind(V1),V1,V2)
r30: a__U42(tt(),V1,V2) -> a__U43(a__isNatIListKind(V2),V1,V2)
r31: a__U43(tt(),V1,V2) -> a__U44(a__isNatIListKind(V2),V1,V2)
r32: a__U44(tt(),V1,V2) -> a__U45(a__isNat(V1),V2)
r33: a__U45(tt(),V2) -> a__U46(a__isNatIList(V2))
r34: a__U46(tt()) -> tt()
r35: a__U51(tt(),V2) -> a__U52(a__isNatIListKind(V2))
r36: a__U52(tt()) -> tt()
r37: a__U61(tt(),V2) -> a__U62(a__isNatIListKind(V2))
r38: a__U62(tt()) -> tt()
r39: a__U71(tt()) -> tt()
r40: a__U81(tt()) -> tt()
r41: a__U91(tt(),V1,V2) -> a__U92(a__isNatKind(V1),V1,V2)
r42: a__U92(tt(),V1,V2) -> a__U93(a__isNatIListKind(V2),V1,V2)
r43: a__U93(tt(),V1,V2) -> a__U94(a__isNatIListKind(V2),V1,V2)
r44: a__U94(tt(),V1,V2) -> a__U95(a__isNat(V1),V2)
r45: a__U95(tt(),V2) -> a__U96(a__isNatList(V2))
r46: a__U96(tt()) -> tt()
r47: a__isNat(|0|()) -> tt()
r48: a__isNat(length(V1)) -> a__U11(a__isNatIListKind(V1),V1)
r49: a__isNat(s(V1)) -> a__U21(a__isNatKind(V1),V1)
r50: a__isNatIList(V) -> a__U31(a__isNatIListKind(V),V)
r51: a__isNatIList(zeros()) -> tt()
r52: a__isNatIList(cons(V1,V2)) -> a__U41(a__isNatKind(V1),V1,V2)
r53: a__isNatIListKind(nil()) -> tt()
r54: a__isNatIListKind(zeros()) -> tt()
r55: a__isNatIListKind(cons(V1,V2)) -> a__U51(a__isNatKind(V1),V2)
r56: a__isNatIListKind(take(V1,V2)) -> a__U61(a__isNatKind(V1),V2)
r57: a__isNatKind(|0|()) -> tt()
r58: a__isNatKind(length(V1)) -> a__U71(a__isNatIListKind(V1))
r59: a__isNatKind(s(V1)) -> a__U81(a__isNatKind(V1))
r60: a__isNatList(nil()) -> tt()
r61: a__isNatList(cons(V1,V2)) -> a__U91(a__isNatKind(V1),V1,V2)
r62: a__isNatList(take(V1,V2)) -> a__U101(a__isNatKind(V1),V1,V2)
r63: a__length(nil()) -> |0|()
r64: a__length(cons(N,L)) -> a__U111(a__isNatList(L),L,N)
r65: a__take(|0|(),IL) -> a__U121(a__isNatIList(IL),IL)
r66: a__take(s(M),cons(N,IL)) -> a__U131(a__isNatIList(IL),IL,M,N)
r67: mark(zeros()) -> a__zeros()
r68: mark(U101(X1,X2,X3)) -> a__U101(mark(X1),X2,X3)
r69: mark(U102(X1,X2,X3)) -> a__U102(mark(X1),X2,X3)
r70: mark(isNatKind(X)) -> a__isNatKind(X)
r71: mark(U103(X1,X2,X3)) -> a__U103(mark(X1),X2,X3)
r72: mark(isNatIListKind(X)) -> a__isNatIListKind(X)
r73: mark(U104(X1,X2,X3)) -> a__U104(mark(X1),X2,X3)
r74: mark(U105(X1,X2)) -> a__U105(mark(X1),X2)
r75: mark(isNat(X)) -> a__isNat(X)
r76: mark(U106(X)) -> a__U106(mark(X))
r77: mark(isNatIList(X)) -> a__isNatIList(X)
r78: mark(U11(X1,X2)) -> a__U11(mark(X1),X2)
r79: mark(U12(X1,X2)) -> a__U12(mark(X1),X2)
r80: mark(U111(X1,X2,X3)) -> a__U111(mark(X1),X2,X3)
r81: mark(U112(X1,X2,X3)) -> a__U112(mark(X1),X2,X3)
r82: mark(U113(X1,X2,X3)) -> a__U113(mark(X1),X2,X3)
r83: mark(U114(X1,X2)) -> a__U114(mark(X1),X2)
r84: mark(length(X)) -> a__length(mark(X))
r85: mark(U13(X)) -> a__U13(mark(X))
r86: mark(isNatList(X)) -> a__isNatList(X)
r87: mark(U121(X1,X2)) -> a__U121(mark(X1),X2)
r88: mark(U122(X)) -> a__U122(mark(X))
r89: mark(U131(X1,X2,X3,X4)) -> a__U131(mark(X1),X2,X3,X4)
r90: mark(U132(X1,X2,X3,X4)) -> a__U132(mark(X1),X2,X3,X4)
r91: mark(U133(X1,X2,X3,X4)) -> a__U133(mark(X1),X2,X3,X4)
r92: mark(U134(X1,X2,X3,X4)) -> a__U134(mark(X1),X2,X3,X4)
r93: mark(U135(X1,X2,X3,X4)) -> a__U135(mark(X1),X2,X3,X4)
r94: mark(U136(X1,X2,X3,X4)) -> a__U136(mark(X1),X2,X3,X4)
r95: mark(take(X1,X2)) -> a__take(mark(X1),mark(X2))
r96: mark(U21(X1,X2)) -> a__U21(mark(X1),X2)
r97: mark(U22(X1,X2)) -> a__U22(mark(X1),X2)
r98: mark(U23(X)) -> a__U23(mark(X))
r99: mark(U31(X1,X2)) -> a__U31(mark(X1),X2)
r100: mark(U32(X1,X2)) -> a__U32(mark(X1),X2)
r101: mark(U33(X)) -> a__U33(mark(X))
r102: mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3)
r103: mark(U42(X1,X2,X3)) -> a__U42(mark(X1),X2,X3)
r104: mark(U43(X1,X2,X3)) -> a__U43(mark(X1),X2,X3)
r105: mark(U44(X1,X2,X3)) -> a__U44(mark(X1),X2,X3)
r106: mark(U45(X1,X2)) -> a__U45(mark(X1),X2)
r107: mark(U46(X)) -> a__U46(mark(X))
r108: mark(U51(X1,X2)) -> a__U51(mark(X1),X2)
r109: mark(U52(X)) -> a__U52(mark(X))
r110: mark(U61(X1,X2)) -> a__U61(mark(X1),X2)
r111: mark(U62(X)) -> a__U62(mark(X))
r112: mark(U71(X)) -> a__U71(mark(X))
r113: mark(U81(X)) -> a__U81(mark(X))
r114: mark(U91(X1,X2,X3)) -> a__U91(mark(X1),X2,X3)
r115: mark(U92(X1,X2,X3)) -> a__U92(mark(X1),X2,X3)
r116: mark(U93(X1,X2,X3)) -> a__U93(mark(X1),X2,X3)
r117: mark(U94(X1,X2,X3)) -> a__U94(mark(X1),X2,X3)
r118: mark(U95(X1,X2)) -> a__U95(mark(X1),X2)
r119: mark(U96(X)) -> a__U96(mark(X))
r120: mark(cons(X1,X2)) -> cons(mark(X1),X2)
r121: mark(|0|()) -> |0|()
r122: mark(tt()) -> tt()
r123: mark(s(X)) -> s(mark(X))
r124: mark(nil()) -> nil()
r125: a__zeros() -> zeros()
r126: a__U101(X1,X2,X3) -> U101(X1,X2,X3)
r127: a__U102(X1,X2,X3) -> U102(X1,X2,X3)
r128: a__isNatKind(X) -> isNatKind(X)
r129: a__U103(X1,X2,X3) -> U103(X1,X2,X3)
r130: a__isNatIListKind(X) -> isNatIListKind(X)
r131: a__U104(X1,X2,X3) -> U104(X1,X2,X3)
r132: a__U105(X1,X2) -> U105(X1,X2)
r133: a__isNat(X) -> isNat(X)
r134: a__U106(X) -> U106(X)
r135: a__isNatIList(X) -> isNatIList(X)
r136: a__U11(X1,X2) -> U11(X1,X2)
r137: a__U12(X1,X2) -> U12(X1,X2)
r138: a__U111(X1,X2,X3) -> U111(X1,X2,X3)
r139: a__U112(X1,X2,X3) -> U112(X1,X2,X3)
r140: a__U113(X1,X2,X3) -> U113(X1,X2,X3)
r141: a__U114(X1,X2) -> U114(X1,X2)
r142: a__length(X) -> length(X)
r143: a__U13(X) -> U13(X)
r144: a__isNatList(X) -> isNatList(X)
r145: a__U121(X1,X2) -> U121(X1,X2)
r146: a__U122(X) -> U122(X)
r147: a__U131(X1,X2,X3,X4) -> U131(X1,X2,X3,X4)
r148: a__U132(X1,X2,X3,X4) -> U132(X1,X2,X3,X4)
r149: a__U133(X1,X2,X3,X4) -> U133(X1,X2,X3,X4)
r150: a__U134(X1,X2,X3,X4) -> U134(X1,X2,X3,X4)
r151: a__U135(X1,X2,X3,X4) -> U135(X1,X2,X3,X4)
r152: a__U136(X1,X2,X3,X4) -> U136(X1,X2,X3,X4)
r153: a__take(X1,X2) -> take(X1,X2)
r154: a__U21(X1,X2) -> U21(X1,X2)
r155: a__U22(X1,X2) -> U22(X1,X2)
r156: a__U23(X) -> U23(X)
r157: a__U31(X1,X2) -> U31(X1,X2)
r158: a__U32(X1,X2) -> U32(X1,X2)
r159: a__U33(X) -> U33(X)
r160: a__U41(X1,X2,X3) -> U41(X1,X2,X3)
r161: a__U42(X1,X2,X3) -> U42(X1,X2,X3)
r162: a__U43(X1,X2,X3) -> U43(X1,X2,X3)
r163: a__U44(X1,X2,X3) -> U44(X1,X2,X3)
r164: a__U45(X1,X2) -> U45(X1,X2)
r165: a__U46(X) -> U46(X)
r166: a__U51(X1,X2) -> U51(X1,X2)
r167: a__U52(X) -> U52(X)
r168: a__U61(X1,X2) -> U61(X1,X2)
r169: a__U62(X) -> U62(X)
r170: a__U71(X) -> U71(X)
r171: a__U81(X) -> U81(X)
r172: a__U91(X1,X2,X3) -> U91(X1,X2,X3)
r173: a__U92(X1,X2,X3) -> U92(X1,X2,X3)
r174: a__U93(X1,X2,X3) -> U93(X1,X2,X3)
r175: a__U94(X1,X2,X3) -> U94(X1,X2,X3)
r176: a__U95(X1,X2) -> U95(X1,X2)
r177: a__U96(X) -> U96(X)

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  max/plus interpretations on natural numbers:
  
    mark#_A(x1) = max{0, x1 - 1}
    U103_A(x1,x2,x3) = max{x1 + 1, x2 + 1, x3 + 1}
    U96_A(x1) = x1 + 2
    U95_A(x1,x2) = max{x1 + 1, x2 + 1}
    U81_A(x1) = x1 + 1
    U71_A(x1) = x1 + 1
    U62_A(x1) = max{2, x1}
    U52_A(x1) = x1 + 1
    U46_A(x1) = x1
    U42_A(x1,x2,x3) = max{x1 + 1, x2 + 1}
    U33_A(x1) = x1 + 1
    U32_A(x1,x2) = max{x1, x2 + 2}
    U31_A(x1,x2) = max{x1, x2 + 1}
    U23_A(x1) = x1
    U21_A(x1,x2) = max{x1, x2 + 1}
    U134_A(x1,x2,x3,x4) = max{x1, x2 + 2, x3 + 2, x4 + 2}
    U131_A(x1,x2,x3,x4) = max{x1, x2 + 2, x3 + 2, x4 + 2}
    U122_A(x1) = x1
    U121_A(x1,x2) = max{x1, x2 + 1}
    U13_A(x1) = x1
    U111_A(x1,x2,x3) = max{x1, x2 + 2, x3 + 2}
    U12_A(x1,x2) = max{x1, x2 + 2}
    U11_A(x1,x2) = max{x1 + 1, x2 + 1}

The next rules are strictly ordered:

  p2

We remove them from the problem.

-- SCC decomposition.

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

p1: mark#(U103(X1,X2,X3)) -> mark#(X1)
p2: mark#(U95(X1,X2)) -> mark#(X1)
p3: mark#(U81(X)) -> mark#(X)
p4: mark#(U71(X)) -> mark#(X)
p5: mark#(U62(X)) -> mark#(X)
p6: mark#(U52(X)) -> mark#(X)
p7: mark#(U46(X)) -> mark#(X)
p8: mark#(U42(X1,X2,X3)) -> mark#(X1)
p9: mark#(U33(X)) -> mark#(X)
p10: mark#(U32(X1,X2)) -> mark#(X1)
p11: mark#(U31(X1,X2)) -> mark#(X1)
p12: mark#(U23(X)) -> mark#(X)
p13: mark#(U21(X1,X2)) -> mark#(X1)
p14: mark#(U134(X1,X2,X3,X4)) -> mark#(X1)
p15: mark#(U131(X1,X2,X3,X4)) -> mark#(X1)
p16: mark#(U122(X)) -> mark#(X)
p17: mark#(U121(X1,X2)) -> mark#(X1)
p18: mark#(U13(X)) -> mark#(X)
p19: mark#(U111(X1,X2,X3)) -> mark#(X1)
p20: mark#(U12(X1,X2)) -> mark#(X1)
p21: mark#(U11(X1,X2)) -> mark#(X1)

and R consists of:

r1: a__zeros() -> cons(|0|(),zeros())
r2: a__U101(tt(),V1,V2) -> a__U102(a__isNatKind(V1),V1,V2)
r3: a__U102(tt(),V1,V2) -> a__U103(a__isNatIListKind(V2),V1,V2)
r4: a__U103(tt(),V1,V2) -> a__U104(a__isNatIListKind(V2),V1,V2)
r5: a__U104(tt(),V1,V2) -> a__U105(a__isNat(V1),V2)
r6: a__U105(tt(),V2) -> a__U106(a__isNatIList(V2))
r7: a__U106(tt()) -> tt()
r8: a__U11(tt(),V1) -> a__U12(a__isNatIListKind(V1),V1)
r9: a__U111(tt(),L,N) -> a__U112(a__isNatIListKind(L),L,N)
r10: a__U112(tt(),L,N) -> a__U113(a__isNat(N),L,N)
r11: a__U113(tt(),L,N) -> a__U114(a__isNatKind(N),L)
r12: a__U114(tt(),L) -> s(a__length(mark(L)))
r13: a__U12(tt(),V1) -> a__U13(a__isNatList(V1))
r14: a__U121(tt(),IL) -> a__U122(a__isNatIListKind(IL))
r15: a__U122(tt()) -> nil()
r16: a__U13(tt()) -> tt()
r17: a__U131(tt(),IL,M,N) -> a__U132(a__isNatIListKind(IL),IL,M,N)
r18: a__U132(tt(),IL,M,N) -> a__U133(a__isNat(M),IL,M,N)
r19: a__U133(tt(),IL,M,N) -> a__U134(a__isNatKind(M),IL,M,N)
r20: a__U134(tt(),IL,M,N) -> a__U135(a__isNat(N),IL,M,N)
r21: a__U135(tt(),IL,M,N) -> a__U136(a__isNatKind(N),IL,M,N)
r22: a__U136(tt(),IL,M,N) -> cons(mark(N),take(M,IL))
r23: a__U21(tt(),V1) -> a__U22(a__isNatKind(V1),V1)
r24: a__U22(tt(),V1) -> a__U23(a__isNat(V1))
r25: a__U23(tt()) -> tt()
r26: a__U31(tt(),V) -> a__U32(a__isNatIListKind(V),V)
r27: a__U32(tt(),V) -> a__U33(a__isNatList(V))
r28: a__U33(tt()) -> tt()
r29: a__U41(tt(),V1,V2) -> a__U42(a__isNatKind(V1),V1,V2)
r30: a__U42(tt(),V1,V2) -> a__U43(a__isNatIListKind(V2),V1,V2)
r31: a__U43(tt(),V1,V2) -> a__U44(a__isNatIListKind(V2),V1,V2)
r32: a__U44(tt(),V1,V2) -> a__U45(a__isNat(V1),V2)
r33: a__U45(tt(),V2) -> a__U46(a__isNatIList(V2))
r34: a__U46(tt()) -> tt()
r35: a__U51(tt(),V2) -> a__U52(a__isNatIListKind(V2))
r36: a__U52(tt()) -> tt()
r37: a__U61(tt(),V2) -> a__U62(a__isNatIListKind(V2))
r38: a__U62(tt()) -> tt()
r39: a__U71(tt()) -> tt()
r40: a__U81(tt()) -> tt()
r41: a__U91(tt(),V1,V2) -> a__U92(a__isNatKind(V1),V1,V2)
r42: a__U92(tt(),V1,V2) -> a__U93(a__isNatIListKind(V2),V1,V2)
r43: a__U93(tt(),V1,V2) -> a__U94(a__isNatIListKind(V2),V1,V2)
r44: a__U94(tt(),V1,V2) -> a__U95(a__isNat(V1),V2)
r45: a__U95(tt(),V2) -> a__U96(a__isNatList(V2))
r46: a__U96(tt()) -> tt()
r47: a__isNat(|0|()) -> tt()
r48: a__isNat(length(V1)) -> a__U11(a__isNatIListKind(V1),V1)
r49: a__isNat(s(V1)) -> a__U21(a__isNatKind(V1),V1)
r50: a__isNatIList(V) -> a__U31(a__isNatIListKind(V),V)
r51: a__isNatIList(zeros()) -> tt()
r52: a__isNatIList(cons(V1,V2)) -> a__U41(a__isNatKind(V1),V1,V2)
r53: a__isNatIListKind(nil()) -> tt()
r54: a__isNatIListKind(zeros()) -> tt()
r55: a__isNatIListKind(cons(V1,V2)) -> a__U51(a__isNatKind(V1),V2)
r56: a__isNatIListKind(take(V1,V2)) -> a__U61(a__isNatKind(V1),V2)
r57: a__isNatKind(|0|()) -> tt()
r58: a__isNatKind(length(V1)) -> a__U71(a__isNatIListKind(V1))
r59: a__isNatKind(s(V1)) -> a__U81(a__isNatKind(V1))
r60: a__isNatList(nil()) -> tt()
r61: a__isNatList(cons(V1,V2)) -> a__U91(a__isNatKind(V1),V1,V2)
r62: a__isNatList(take(V1,V2)) -> a__U101(a__isNatKind(V1),V1,V2)
r63: a__length(nil()) -> |0|()
r64: a__length(cons(N,L)) -> a__U111(a__isNatList(L),L,N)
r65: a__take(|0|(),IL) -> a__U121(a__isNatIList(IL),IL)
r66: a__take(s(M),cons(N,IL)) -> a__U131(a__isNatIList(IL),IL,M,N)
r67: mark(zeros()) -> a__zeros()
r68: mark(U101(X1,X2,X3)) -> a__U101(mark(X1),X2,X3)
r69: mark(U102(X1,X2,X3)) -> a__U102(mark(X1),X2,X3)
r70: mark(isNatKind(X)) -> a__isNatKind(X)
r71: mark(U103(X1,X2,X3)) -> a__U103(mark(X1),X2,X3)
r72: mark(isNatIListKind(X)) -> a__isNatIListKind(X)
r73: mark(U104(X1,X2,X3)) -> a__U104(mark(X1),X2,X3)
r74: mark(U105(X1,X2)) -> a__U105(mark(X1),X2)
r75: mark(isNat(X)) -> a__isNat(X)
r76: mark(U106(X)) -> a__U106(mark(X))
r77: mark(isNatIList(X)) -> a__isNatIList(X)
r78: mark(U11(X1,X2)) -> a__U11(mark(X1),X2)
r79: mark(U12(X1,X2)) -> a__U12(mark(X1),X2)
r80: mark(U111(X1,X2,X3)) -> a__U111(mark(X1),X2,X3)
r81: mark(U112(X1,X2,X3)) -> a__U112(mark(X1),X2,X3)
r82: mark(U113(X1,X2,X3)) -> a__U113(mark(X1),X2,X3)
r83: mark(U114(X1,X2)) -> a__U114(mark(X1),X2)
r84: mark(length(X)) -> a__length(mark(X))
r85: mark(U13(X)) -> a__U13(mark(X))
r86: mark(isNatList(X)) -> a__isNatList(X)
r87: mark(U121(X1,X2)) -> a__U121(mark(X1),X2)
r88: mark(U122(X)) -> a__U122(mark(X))
r89: mark(U131(X1,X2,X3,X4)) -> a__U131(mark(X1),X2,X3,X4)
r90: mark(U132(X1,X2,X3,X4)) -> a__U132(mark(X1),X2,X3,X4)
r91: mark(U133(X1,X2,X3,X4)) -> a__U133(mark(X1),X2,X3,X4)
r92: mark(U134(X1,X2,X3,X4)) -> a__U134(mark(X1),X2,X3,X4)
r93: mark(U135(X1,X2,X3,X4)) -> a__U135(mark(X1),X2,X3,X4)
r94: mark(U136(X1,X2,X3,X4)) -> a__U136(mark(X1),X2,X3,X4)
r95: mark(take(X1,X2)) -> a__take(mark(X1),mark(X2))
r96: mark(U21(X1,X2)) -> a__U21(mark(X1),X2)
r97: mark(U22(X1,X2)) -> a__U22(mark(X1),X2)
r98: mark(U23(X)) -> a__U23(mark(X))
r99: mark(U31(X1,X2)) -> a__U31(mark(X1),X2)
r100: mark(U32(X1,X2)) -> a__U32(mark(X1),X2)
r101: mark(U33(X)) -> a__U33(mark(X))
r102: mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3)
r103: mark(U42(X1,X2,X3)) -> a__U42(mark(X1),X2,X3)
r104: mark(U43(X1,X2,X3)) -> a__U43(mark(X1),X2,X3)
r105: mark(U44(X1,X2,X3)) -> a__U44(mark(X1),X2,X3)
r106: mark(U45(X1,X2)) -> a__U45(mark(X1),X2)
r107: mark(U46(X)) -> a__U46(mark(X))
r108: mark(U51(X1,X2)) -> a__U51(mark(X1),X2)
r109: mark(U52(X)) -> a__U52(mark(X))
r110: mark(U61(X1,X2)) -> a__U61(mark(X1),X2)
r111: mark(U62(X)) -> a__U62(mark(X))
r112: mark(U71(X)) -> a__U71(mark(X))
r113: mark(U81(X)) -> a__U81(mark(X))
r114: mark(U91(X1,X2,X3)) -> a__U91(mark(X1),X2,X3)
r115: mark(U92(X1,X2,X3)) -> a__U92(mark(X1),X2,X3)
r116: mark(U93(X1,X2,X3)) -> a__U93(mark(X1),X2,X3)
r117: mark(U94(X1,X2,X3)) -> a__U94(mark(X1),X2,X3)
r118: mark(U95(X1,X2)) -> a__U95(mark(X1),X2)
r119: mark(U96(X)) -> a__U96(mark(X))
r120: mark(cons(X1,X2)) -> cons(mark(X1),X2)
r121: mark(|0|()) -> |0|()
r122: mark(tt()) -> tt()
r123: mark(s(X)) -> s(mark(X))
r124: mark(nil()) -> nil()
r125: a__zeros() -> zeros()
r126: a__U101(X1,X2,X3) -> U101(X1,X2,X3)
r127: a__U102(X1,X2,X3) -> U102(X1,X2,X3)
r128: a__isNatKind(X) -> isNatKind(X)
r129: a__U103(X1,X2,X3) -> U103(X1,X2,X3)
r130: a__isNatIListKind(X) -> isNatIListKind(X)
r131: a__U104(X1,X2,X3) -> U104(X1,X2,X3)
r132: a__U105(X1,X2) -> U105(X1,X2)
r133: a__isNat(X) -> isNat(X)
r134: a__U106(X) -> U106(X)
r135: a__isNatIList(X) -> isNatIList(X)
r136: a__U11(X1,X2) -> U11(X1,X2)
r137: a__U12(X1,X2) -> U12(X1,X2)
r138: a__U111(X1,X2,X3) -> U111(X1,X2,X3)
r139: a__U112(X1,X2,X3) -> U112(X1,X2,X3)
r140: a__U113(X1,X2,X3) -> U113(X1,X2,X3)
r141: a__U114(X1,X2) -> U114(X1,X2)
r142: a__length(X) -> length(X)
r143: a__U13(X) -> U13(X)
r144: a__isNatList(X) -> isNatList(X)
r145: a__U121(X1,X2) -> U121(X1,X2)
r146: a__U122(X) -> U122(X)
r147: a__U131(X1,X2,X3,X4) -> U131(X1,X2,X3,X4)
r148: a__U132(X1,X2,X3,X4) -> U132(X1,X2,X3,X4)
r149: a__U133(X1,X2,X3,X4) -> U133(X1,X2,X3,X4)
r150: a__U134(X1,X2,X3,X4) -> U134(X1,X2,X3,X4)
r151: a__U135(X1,X2,X3,X4) -> U135(X1,X2,X3,X4)
r152: a__U136(X1,X2,X3,X4) -> U136(X1,X2,X3,X4)
r153: a__take(X1,X2) -> take(X1,X2)
r154: a__U21(X1,X2) -> U21(X1,X2)
r155: a__U22(X1,X2) -> U22(X1,X2)
r156: a__U23(X) -> U23(X)
r157: a__U31(X1,X2) -> U31(X1,X2)
r158: a__U32(X1,X2) -> U32(X1,X2)
r159: a__U33(X) -> U33(X)
r160: a__U41(X1,X2,X3) -> U41(X1,X2,X3)
r161: a__U42(X1,X2,X3) -> U42(X1,X2,X3)
r162: a__U43(X1,X2,X3) -> U43(X1,X2,X3)
r163: a__U44(X1,X2,X3) -> U44(X1,X2,X3)
r164: a__U45(X1,X2) -> U45(X1,X2)
r165: a__U46(X) -> U46(X)
r166: a__U51(X1,X2) -> U51(X1,X2)
r167: a__U52(X) -> U52(X)
r168: a__U61(X1,X2) -> U61(X1,X2)
r169: a__U62(X) -> U62(X)
r170: a__U71(X) -> U71(X)
r171: a__U81(X) -> U81(X)
r172: a__U91(X1,X2,X3) -> U91(X1,X2,X3)
r173: a__U92(X1,X2,X3) -> U92(X1,X2,X3)
r174: a__U93(X1,X2,X3) -> U93(X1,X2,X3)
r175: a__U94(X1,X2,X3) -> U94(X1,X2,X3)
r176: a__U95(X1,X2) -> U95(X1,X2)
r177: a__U96(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}


-- Reduction pair.

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

p1: mark#(U103(X1,X2,X3)) -> mark#(X1)
p2: mark#(U11(X1,X2)) -> mark#(X1)
p3: mark#(U12(X1,X2)) -> mark#(X1)
p4: mark#(U111(X1,X2,X3)) -> mark#(X1)
p5: mark#(U13(X)) -> mark#(X)
p6: mark#(U121(X1,X2)) -> mark#(X1)
p7: mark#(U122(X)) -> mark#(X)
p8: mark#(U131(X1,X2,X3,X4)) -> mark#(X1)
p9: mark#(U134(X1,X2,X3,X4)) -> mark#(X1)
p10: mark#(U21(X1,X2)) -> mark#(X1)
p11: mark#(U23(X)) -> mark#(X)
p12: mark#(U31(X1,X2)) -> mark#(X1)
p13: mark#(U32(X1,X2)) -> mark#(X1)
p14: mark#(U33(X)) -> mark#(X)
p15: mark#(U42(X1,X2,X3)) -> mark#(X1)
p16: mark#(U46(X)) -> mark#(X)
p17: mark#(U52(X)) -> mark#(X)
p18: mark#(U62(X)) -> mark#(X)
p19: mark#(U71(X)) -> mark#(X)
p20: mark#(U81(X)) -> mark#(X)
p21: mark#(U95(X1,X2)) -> mark#(X1)

and R consists of:

r1: a__zeros() -> cons(|0|(),zeros())
r2: a__U101(tt(),V1,V2) -> a__U102(a__isNatKind(V1),V1,V2)
r3: a__U102(tt(),V1,V2) -> a__U103(a__isNatIListKind(V2),V1,V2)
r4: a__U103(tt(),V1,V2) -> a__U104(a__isNatIListKind(V2),V1,V2)
r5: a__U104(tt(),V1,V2) -> a__U105(a__isNat(V1),V2)
r6: a__U105(tt(),V2) -> a__U106(a__isNatIList(V2))
r7: a__U106(tt()) -> tt()
r8: a__U11(tt(),V1) -> a__U12(a__isNatIListKind(V1),V1)
r9: a__U111(tt(),L,N) -> a__U112(a__isNatIListKind(L),L,N)
r10: a__U112(tt(),L,N) -> a__U113(a__isNat(N),L,N)
r11: a__U113(tt(),L,N) -> a__U114(a__isNatKind(N),L)
r12: a__U114(tt(),L) -> s(a__length(mark(L)))
r13: a__U12(tt(),V1) -> a__U13(a__isNatList(V1))
r14: a__U121(tt(),IL) -> a__U122(a__isNatIListKind(IL))
r15: a__U122(tt()) -> nil()
r16: a__U13(tt()) -> tt()
r17: a__U131(tt(),IL,M,N) -> a__U132(a__isNatIListKind(IL),IL,M,N)
r18: a__U132(tt(),IL,M,N) -> a__U133(a__isNat(M),IL,M,N)
r19: a__U133(tt(),IL,M,N) -> a__U134(a__isNatKind(M),IL,M,N)
r20: a__U134(tt(),IL,M,N) -> a__U135(a__isNat(N),IL,M,N)
r21: a__U135(tt(),IL,M,N) -> a__U136(a__isNatKind(N),IL,M,N)
r22: a__U136(tt(),IL,M,N) -> cons(mark(N),take(M,IL))
r23: a__U21(tt(),V1) -> a__U22(a__isNatKind(V1),V1)
r24: a__U22(tt(),V1) -> a__U23(a__isNat(V1))
r25: a__U23(tt()) -> tt()
r26: a__U31(tt(),V) -> a__U32(a__isNatIListKind(V),V)
r27: a__U32(tt(),V) -> a__U33(a__isNatList(V))
r28: a__U33(tt()) -> tt()
r29: a__U41(tt(),V1,V2) -> a__U42(a__isNatKind(V1),V1,V2)
r30: a__U42(tt(),V1,V2) -> a__U43(a__isNatIListKind(V2),V1,V2)
r31: a__U43(tt(),V1,V2) -> a__U44(a__isNatIListKind(V2),V1,V2)
r32: a__U44(tt(),V1,V2) -> a__U45(a__isNat(V1),V2)
r33: a__U45(tt(),V2) -> a__U46(a__isNatIList(V2))
r34: a__U46(tt()) -> tt()
r35: a__U51(tt(),V2) -> a__U52(a__isNatIListKind(V2))
r36: a__U52(tt()) -> tt()
r37: a__U61(tt(),V2) -> a__U62(a__isNatIListKind(V2))
r38: a__U62(tt()) -> tt()
r39: a__U71(tt()) -> tt()
r40: a__U81(tt()) -> tt()
r41: a__U91(tt(),V1,V2) -> a__U92(a__isNatKind(V1),V1,V2)
r42: a__U92(tt(),V1,V2) -> a__U93(a__isNatIListKind(V2),V1,V2)
r43: a__U93(tt(),V1,V2) -> a__U94(a__isNatIListKind(V2),V1,V2)
r44: a__U94(tt(),V1,V2) -> a__U95(a__isNat(V1),V2)
r45: a__U95(tt(),V2) -> a__U96(a__isNatList(V2))
r46: a__U96(tt()) -> tt()
r47: a__isNat(|0|()) -> tt()
r48: a__isNat(length(V1)) -> a__U11(a__isNatIListKind(V1),V1)
r49: a__isNat(s(V1)) -> a__U21(a__isNatKind(V1),V1)
r50: a__isNatIList(V) -> a__U31(a__isNatIListKind(V),V)
r51: a__isNatIList(zeros()) -> tt()
r52: a__isNatIList(cons(V1,V2)) -> a__U41(a__isNatKind(V1),V1,V2)
r53: a__isNatIListKind(nil()) -> tt()
r54: a__isNatIListKind(zeros()) -> tt()
r55: a__isNatIListKind(cons(V1,V2)) -> a__U51(a__isNatKind(V1),V2)
r56: a__isNatIListKind(take(V1,V2)) -> a__U61(a__isNatKind(V1),V2)
r57: a__isNatKind(|0|()) -> tt()
r58: a__isNatKind(length(V1)) -> a__U71(a__isNatIListKind(V1))
r59: a__isNatKind(s(V1)) -> a__U81(a__isNatKind(V1))
r60: a__isNatList(nil()) -> tt()
r61: a__isNatList(cons(V1,V2)) -> a__U91(a__isNatKind(V1),V1,V2)
r62: a__isNatList(take(V1,V2)) -> a__U101(a__isNatKind(V1),V1,V2)
r63: a__length(nil()) -> |0|()
r64: a__length(cons(N,L)) -> a__U111(a__isNatList(L),L,N)
r65: a__take(|0|(),IL) -> a__U121(a__isNatIList(IL),IL)
r66: a__take(s(M),cons(N,IL)) -> a__U131(a__isNatIList(IL),IL,M,N)
r67: mark(zeros()) -> a__zeros()
r68: mark(U101(X1,X2,X3)) -> a__U101(mark(X1),X2,X3)
r69: mark(U102(X1,X2,X3)) -> a__U102(mark(X1),X2,X3)
r70: mark(isNatKind(X)) -> a__isNatKind(X)
r71: mark(U103(X1,X2,X3)) -> a__U103(mark(X1),X2,X3)
r72: mark(isNatIListKind(X)) -> a__isNatIListKind(X)
r73: mark(U104(X1,X2,X3)) -> a__U104(mark(X1),X2,X3)
r74: mark(U105(X1,X2)) -> a__U105(mark(X1),X2)
r75: mark(isNat(X)) -> a__isNat(X)
r76: mark(U106(X)) -> a__U106(mark(X))
r77: mark(isNatIList(X)) -> a__isNatIList(X)
r78: mark(U11(X1,X2)) -> a__U11(mark(X1),X2)
r79: mark(U12(X1,X2)) -> a__U12(mark(X1),X2)
r80: mark(U111(X1,X2,X3)) -> a__U111(mark(X1),X2,X3)
r81: mark(U112(X1,X2,X3)) -> a__U112(mark(X1),X2,X3)
r82: mark(U113(X1,X2,X3)) -> a__U113(mark(X1),X2,X3)
r83: mark(U114(X1,X2)) -> a__U114(mark(X1),X2)
r84: mark(length(X)) -> a__length(mark(X))
r85: mark(U13(X)) -> a__U13(mark(X))
r86: mark(isNatList(X)) -> a__isNatList(X)
r87: mark(U121(X1,X2)) -> a__U121(mark(X1),X2)
r88: mark(U122(X)) -> a__U122(mark(X))
r89: mark(U131(X1,X2,X3,X4)) -> a__U131(mark(X1),X2,X3,X4)
r90: mark(U132(X1,X2,X3,X4)) -> a__U132(mark(X1),X2,X3,X4)
r91: mark(U133(X1,X2,X3,X4)) -> a__U133(mark(X1),X2,X3,X4)
r92: mark(U134(X1,X2,X3,X4)) -> a__U134(mark(X1),X2,X3,X4)
r93: mark(U135(X1,X2,X3,X4)) -> a__U135(mark(X1),X2,X3,X4)
r94: mark(U136(X1,X2,X3,X4)) -> a__U136(mark(X1),X2,X3,X4)
r95: mark(take(X1,X2)) -> a__take(mark(X1),mark(X2))
r96: mark(U21(X1,X2)) -> a__U21(mark(X1),X2)
r97: mark(U22(X1,X2)) -> a__U22(mark(X1),X2)
r98: mark(U23(X)) -> a__U23(mark(X))
r99: mark(U31(X1,X2)) -> a__U31(mark(X1),X2)
r100: mark(U32(X1,X2)) -> a__U32(mark(X1),X2)
r101: mark(U33(X)) -> a__U33(mark(X))
r102: mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3)
r103: mark(U42(X1,X2,X3)) -> a__U42(mark(X1),X2,X3)
r104: mark(U43(X1,X2,X3)) -> a__U43(mark(X1),X2,X3)
r105: mark(U44(X1,X2,X3)) -> a__U44(mark(X1),X2,X3)
r106: mark(U45(X1,X2)) -> a__U45(mark(X1),X2)
r107: mark(U46(X)) -> a__U46(mark(X))
r108: mark(U51(X1,X2)) -> a__U51(mark(X1),X2)
r109: mark(U52(X)) -> a__U52(mark(X))
r110: mark(U61(X1,X2)) -> a__U61(mark(X1),X2)
r111: mark(U62(X)) -> a__U62(mark(X))
r112: mark(U71(X)) -> a__U71(mark(X))
r113: mark(U81(X)) -> a__U81(mark(X))
r114: mark(U91(X1,X2,X3)) -> a__U91(mark(X1),X2,X3)
r115: mark(U92(X1,X2,X3)) -> a__U92(mark(X1),X2,X3)
r116: mark(U93(X1,X2,X3)) -> a__U93(mark(X1),X2,X3)
r117: mark(U94(X1,X2,X3)) -> a__U94(mark(X1),X2,X3)
r118: mark(U95(X1,X2)) -> a__U95(mark(X1),X2)
r119: mark(U96(X)) -> a__U96(mark(X))
r120: mark(cons(X1,X2)) -> cons(mark(X1),X2)
r121: mark(|0|()) -> |0|()
r122: mark(tt()) -> tt()
r123: mark(s(X)) -> s(mark(X))
r124: mark(nil()) -> nil()
r125: a__zeros() -> zeros()
r126: a__U101(X1,X2,X3) -> U101(X1,X2,X3)
r127: a__U102(X1,X2,X3) -> U102(X1,X2,X3)
r128: a__isNatKind(X) -> isNatKind(X)
r129: a__U103(X1,X2,X3) -> U103(X1,X2,X3)
r130: a__isNatIListKind(X) -> isNatIListKind(X)
r131: a__U104(X1,X2,X3) -> U104(X1,X2,X3)
r132: a__U105(X1,X2) -> U105(X1,X2)
r133: a__isNat(X) -> isNat(X)
r134: a__U106(X) -> U106(X)
r135: a__isNatIList(X) -> isNatIList(X)
r136: a__U11(X1,X2) -> U11(X1,X2)
r137: a__U12(X1,X2) -> U12(X1,X2)
r138: a__U111(X1,X2,X3) -> U111(X1,X2,X3)
r139: a__U112(X1,X2,X3) -> U112(X1,X2,X3)
r140: a__U113(X1,X2,X3) -> U113(X1,X2,X3)
r141: a__U114(X1,X2) -> U114(X1,X2)
r142: a__length(X) -> length(X)
r143: a__U13(X) -> U13(X)
r144: a__isNatList(X) -> isNatList(X)
r145: a__U121(X1,X2) -> U121(X1,X2)
r146: a__U122(X) -> U122(X)
r147: a__U131(X1,X2,X3,X4) -> U131(X1,X2,X3,X4)
r148: a__U132(X1,X2,X3,X4) -> U132(X1,X2,X3,X4)
r149: a__U133(X1,X2,X3,X4) -> U133(X1,X2,X3,X4)
r150: a__U134(X1,X2,X3,X4) -> U134(X1,X2,X3,X4)
r151: a__U135(X1,X2,X3,X4) -> U135(X1,X2,X3,X4)
r152: a__U136(X1,X2,X3,X4) -> U136(X1,X2,X3,X4)
r153: a__take(X1,X2) -> take(X1,X2)
r154: a__U21(X1,X2) -> U21(X1,X2)
r155: a__U22(X1,X2) -> U22(X1,X2)
r156: a__U23(X) -> U23(X)
r157: a__U31(X1,X2) -> U31(X1,X2)
r158: a__U32(X1,X2) -> U32(X1,X2)
r159: a__U33(X) -> U33(X)
r160: a__U41(X1,X2,X3) -> U41(X1,X2,X3)
r161: a__U42(X1,X2,X3) -> U42(X1,X2,X3)
r162: a__U43(X1,X2,X3) -> U43(X1,X2,X3)
r163: a__U44(X1,X2,X3) -> U44(X1,X2,X3)
r164: a__U45(X1,X2) -> U45(X1,X2)
r165: a__U46(X) -> U46(X)
r166: a__U51(X1,X2) -> U51(X1,X2)
r167: a__U52(X) -> U52(X)
r168: a__U61(X1,X2) -> U61(X1,X2)
r169: a__U62(X) -> U62(X)
r170: a__U71(X) -> U71(X)
r171: a__U81(X) -> U81(X)
r172: a__U91(X1,X2,X3) -> U91(X1,X2,X3)
r173: a__U92(X1,X2,X3) -> U92(X1,X2,X3)
r174: a__U93(X1,X2,X3) -> U93(X1,X2,X3)
r175: a__U94(X1,X2,X3) -> U94(X1,X2,X3)
r176: a__U95(X1,X2) -> U95(X1,X2)
r177: a__U96(X) -> U96(X)

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  max/plus interpretations on natural numbers:
  
    mark#_A(x1) = x1 + 1
    U103_A(x1,x2,x3) = max{x1 + 1, x2, x3}
    U11_A(x1,x2) = max{x1, x2}
    U12_A(x1,x2) = max{x1, x2}
    U111_A(x1,x2,x3) = max{x1, x2, x3}
    U13_A(x1) = x1 + 1
    U121_A(x1,x2) = max{x1, x2}
    U122_A(x1) = x1
    U131_A(x1,x2,x3,x4) = max{x1, x2, x3, x4}
    U134_A(x1,x2,x3,x4) = max{x1, x2, x3, x4}
    U21_A(x1,x2) = max{x1, x2}
    U23_A(x1) = x1
    U31_A(x1,x2) = max{x1, x2}
    U32_A(x1,x2) = max{x1 + 1, x2}
    U33_A(x1) = x1
    U42_A(x1,x2,x3) = max{x1, x2, x3}
    U46_A(x1) = x1
    U52_A(x1) = x1
    U62_A(x1) = x1
    U71_A(x1) = x1 + 1
    U81_A(x1) = x1
    U95_A(x1,x2) = max{x1, x2}

The next rules are strictly ordered:

  p1, p5, p13, p19

We remove them from the problem.

-- SCC decomposition.

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

p1: mark#(U11(X1,X2)) -> mark#(X1)
p2: mark#(U12(X1,X2)) -> mark#(X1)
p3: mark#(U111(X1,X2,X3)) -> mark#(X1)
p4: mark#(U121(X1,X2)) -> mark#(X1)
p5: mark#(U122(X)) -> mark#(X)
p6: mark#(U131(X1,X2,X3,X4)) -> mark#(X1)
p7: mark#(U134(X1,X2,X3,X4)) -> mark#(X1)
p8: mark#(U21(X1,X2)) -> mark#(X1)
p9: mark#(U23(X)) -> mark#(X)
p10: mark#(U31(X1,X2)) -> mark#(X1)
p11: mark#(U33(X)) -> mark#(X)
p12: mark#(U42(X1,X2,X3)) -> mark#(X1)
p13: mark#(U46(X)) -> mark#(X)
p14: mark#(U52(X)) -> mark#(X)
p15: mark#(U62(X)) -> mark#(X)
p16: mark#(U81(X)) -> mark#(X)
p17: mark#(U95(X1,X2)) -> mark#(X1)

and R consists of:

r1: a__zeros() -> cons(|0|(),zeros())
r2: a__U101(tt(),V1,V2) -> a__U102(a__isNatKind(V1),V1,V2)
r3: a__U102(tt(),V1,V2) -> a__U103(a__isNatIListKind(V2),V1,V2)
r4: a__U103(tt(),V1,V2) -> a__U104(a__isNatIListKind(V2),V1,V2)
r5: a__U104(tt(),V1,V2) -> a__U105(a__isNat(V1),V2)
r6: a__U105(tt(),V2) -> a__U106(a__isNatIList(V2))
r7: a__U106(tt()) -> tt()
r8: a__U11(tt(),V1) -> a__U12(a__isNatIListKind(V1),V1)
r9: a__U111(tt(),L,N) -> a__U112(a__isNatIListKind(L),L,N)
r10: a__U112(tt(),L,N) -> a__U113(a__isNat(N),L,N)
r11: a__U113(tt(),L,N) -> a__U114(a__isNatKind(N),L)
r12: a__U114(tt(),L) -> s(a__length(mark(L)))
r13: a__U12(tt(),V1) -> a__U13(a__isNatList(V1))
r14: a__U121(tt(),IL) -> a__U122(a__isNatIListKind(IL))
r15: a__U122(tt()) -> nil()
r16: a__U13(tt()) -> tt()
r17: a__U131(tt(),IL,M,N) -> a__U132(a__isNatIListKind(IL),IL,M,N)
r18: a__U132(tt(),IL,M,N) -> a__U133(a__isNat(M),IL,M,N)
r19: a__U133(tt(),IL,M,N) -> a__U134(a__isNatKind(M),IL,M,N)
r20: a__U134(tt(),IL,M,N) -> a__U135(a__isNat(N),IL,M,N)
r21: a__U135(tt(),IL,M,N) -> a__U136(a__isNatKind(N),IL,M,N)
r22: a__U136(tt(),IL,M,N) -> cons(mark(N),take(M,IL))
r23: a__U21(tt(),V1) -> a__U22(a__isNatKind(V1),V1)
r24: a__U22(tt(),V1) -> a__U23(a__isNat(V1))
r25: a__U23(tt()) -> tt()
r26: a__U31(tt(),V) -> a__U32(a__isNatIListKind(V),V)
r27: a__U32(tt(),V) -> a__U33(a__isNatList(V))
r28: a__U33(tt()) -> tt()
r29: a__U41(tt(),V1,V2) -> a__U42(a__isNatKind(V1),V1,V2)
r30: a__U42(tt(),V1,V2) -> a__U43(a__isNatIListKind(V2),V1,V2)
r31: a__U43(tt(),V1,V2) -> a__U44(a__isNatIListKind(V2),V1,V2)
r32: a__U44(tt(),V1,V2) -> a__U45(a__isNat(V1),V2)
r33: a__U45(tt(),V2) -> a__U46(a__isNatIList(V2))
r34: a__U46(tt()) -> tt()
r35: a__U51(tt(),V2) -> a__U52(a__isNatIListKind(V2))
r36: a__U52(tt()) -> tt()
r37: a__U61(tt(),V2) -> a__U62(a__isNatIListKind(V2))
r38: a__U62(tt()) -> tt()
r39: a__U71(tt()) -> tt()
r40: a__U81(tt()) -> tt()
r41: a__U91(tt(),V1,V2) -> a__U92(a__isNatKind(V1),V1,V2)
r42: a__U92(tt(),V1,V2) -> a__U93(a__isNatIListKind(V2),V1,V2)
r43: a__U93(tt(),V1,V2) -> a__U94(a__isNatIListKind(V2),V1,V2)
r44: a__U94(tt(),V1,V2) -> a__U95(a__isNat(V1),V2)
r45: a__U95(tt(),V2) -> a__U96(a__isNatList(V2))
r46: a__U96(tt()) -> tt()
r47: a__isNat(|0|()) -> tt()
r48: a__isNat(length(V1)) -> a__U11(a__isNatIListKind(V1),V1)
r49: a__isNat(s(V1)) -> a__U21(a__isNatKind(V1),V1)
r50: a__isNatIList(V) -> a__U31(a__isNatIListKind(V),V)
r51: a__isNatIList(zeros()) -> tt()
r52: a__isNatIList(cons(V1,V2)) -> a__U41(a__isNatKind(V1),V1,V2)
r53: a__isNatIListKind(nil()) -> tt()
r54: a__isNatIListKind(zeros()) -> tt()
r55: a__isNatIListKind(cons(V1,V2)) -> a__U51(a__isNatKind(V1),V2)
r56: a__isNatIListKind(take(V1,V2)) -> a__U61(a__isNatKind(V1),V2)
r57: a__isNatKind(|0|()) -> tt()
r58: a__isNatKind(length(V1)) -> a__U71(a__isNatIListKind(V1))
r59: a__isNatKind(s(V1)) -> a__U81(a__isNatKind(V1))
r60: a__isNatList(nil()) -> tt()
r61: a__isNatList(cons(V1,V2)) -> a__U91(a__isNatKind(V1),V1,V2)
r62: a__isNatList(take(V1,V2)) -> a__U101(a__isNatKind(V1),V1,V2)
r63: a__length(nil()) -> |0|()
r64: a__length(cons(N,L)) -> a__U111(a__isNatList(L),L,N)
r65: a__take(|0|(),IL) -> a__U121(a__isNatIList(IL),IL)
r66: a__take(s(M),cons(N,IL)) -> a__U131(a__isNatIList(IL),IL,M,N)
r67: mark(zeros()) -> a__zeros()
r68: mark(U101(X1,X2,X3)) -> a__U101(mark(X1),X2,X3)
r69: mark(U102(X1,X2,X3)) -> a__U102(mark(X1),X2,X3)
r70: mark(isNatKind(X)) -> a__isNatKind(X)
r71: mark(U103(X1,X2,X3)) -> a__U103(mark(X1),X2,X3)
r72: mark(isNatIListKind(X)) -> a__isNatIListKind(X)
r73: mark(U104(X1,X2,X3)) -> a__U104(mark(X1),X2,X3)
r74: mark(U105(X1,X2)) -> a__U105(mark(X1),X2)
r75: mark(isNat(X)) -> a__isNat(X)
r76: mark(U106(X)) -> a__U106(mark(X))
r77: mark(isNatIList(X)) -> a__isNatIList(X)
r78: mark(U11(X1,X2)) -> a__U11(mark(X1),X2)
r79: mark(U12(X1,X2)) -> a__U12(mark(X1),X2)
r80: mark(U111(X1,X2,X3)) -> a__U111(mark(X1),X2,X3)
r81: mark(U112(X1,X2,X3)) -> a__U112(mark(X1),X2,X3)
r82: mark(U113(X1,X2,X3)) -> a__U113(mark(X1),X2,X3)
r83: mark(U114(X1,X2)) -> a__U114(mark(X1),X2)
r84: mark(length(X)) -> a__length(mark(X))
r85: mark(U13(X)) -> a__U13(mark(X))
r86: mark(isNatList(X)) -> a__isNatList(X)
r87: mark(U121(X1,X2)) -> a__U121(mark(X1),X2)
r88: mark(U122(X)) -> a__U122(mark(X))
r89: mark(U131(X1,X2,X3,X4)) -> a__U131(mark(X1),X2,X3,X4)
r90: mark(U132(X1,X2,X3,X4)) -> a__U132(mark(X1),X2,X3,X4)
r91: mark(U133(X1,X2,X3,X4)) -> a__U133(mark(X1),X2,X3,X4)
r92: mark(U134(X1,X2,X3,X4)) -> a__U134(mark(X1),X2,X3,X4)
r93: mark(U135(X1,X2,X3,X4)) -> a__U135(mark(X1),X2,X3,X4)
r94: mark(U136(X1,X2,X3,X4)) -> a__U136(mark(X1),X2,X3,X4)
r95: mark(take(X1,X2)) -> a__take(mark(X1),mark(X2))
r96: mark(U21(X1,X2)) -> a__U21(mark(X1),X2)
r97: mark(U22(X1,X2)) -> a__U22(mark(X1),X2)
r98: mark(U23(X)) -> a__U23(mark(X))
r99: mark(U31(X1,X2)) -> a__U31(mark(X1),X2)
r100: mark(U32(X1,X2)) -> a__U32(mark(X1),X2)
r101: mark(U33(X)) -> a__U33(mark(X))
r102: mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3)
r103: mark(U42(X1,X2,X3)) -> a__U42(mark(X1),X2,X3)
r104: mark(U43(X1,X2,X3)) -> a__U43(mark(X1),X2,X3)
r105: mark(U44(X1,X2,X3)) -> a__U44(mark(X1),X2,X3)
r106: mark(U45(X1,X2)) -> a__U45(mark(X1),X2)
r107: mark(U46(X)) -> a__U46(mark(X))
r108: mark(U51(X1,X2)) -> a__U51(mark(X1),X2)
r109: mark(U52(X)) -> a__U52(mark(X))
r110: mark(U61(X1,X2)) -> a__U61(mark(X1),X2)
r111: mark(U62(X)) -> a__U62(mark(X))
r112: mark(U71(X)) -> a__U71(mark(X))
r113: mark(U81(X)) -> a__U81(mark(X))
r114: mark(U91(X1,X2,X3)) -> a__U91(mark(X1),X2,X3)
r115: mark(U92(X1,X2,X3)) -> a__U92(mark(X1),X2,X3)
r116: mark(U93(X1,X2,X3)) -> a__U93(mark(X1),X2,X3)
r117: mark(U94(X1,X2,X3)) -> a__U94(mark(X1),X2,X3)
r118: mark(U95(X1,X2)) -> a__U95(mark(X1),X2)
r119: mark(U96(X)) -> a__U96(mark(X))
r120: mark(cons(X1,X2)) -> cons(mark(X1),X2)
r121: mark(|0|()) -> |0|()
r122: mark(tt()) -> tt()
r123: mark(s(X)) -> s(mark(X))
r124: mark(nil()) -> nil()
r125: a__zeros() -> zeros()
r126: a__U101(X1,X2,X3) -> U101(X1,X2,X3)
r127: a__U102(X1,X2,X3) -> U102(X1,X2,X3)
r128: a__isNatKind(X) -> isNatKind(X)
r129: a__U103(X1,X2,X3) -> U103(X1,X2,X3)
r130: a__isNatIListKind(X) -> isNatIListKind(X)
r131: a__U104(X1,X2,X3) -> U104(X1,X2,X3)
r132: a__U105(X1,X2) -> U105(X1,X2)
r133: a__isNat(X) -> isNat(X)
r134: a__U106(X) -> U106(X)
r135: a__isNatIList(X) -> isNatIList(X)
r136: a__U11(X1,X2) -> U11(X1,X2)
r137: a__U12(X1,X2) -> U12(X1,X2)
r138: a__U111(X1,X2,X3) -> U111(X1,X2,X3)
r139: a__U112(X1,X2,X3) -> U112(X1,X2,X3)
r140: a__U113(X1,X2,X3) -> U113(X1,X2,X3)
r141: a__U114(X1,X2) -> U114(X1,X2)
r142: a__length(X) -> length(X)
r143: a__U13(X) -> U13(X)
r144: a__isNatList(X) -> isNatList(X)
r145: a__U121(X1,X2) -> U121(X1,X2)
r146: a__U122(X) -> U122(X)
r147: a__U131(X1,X2,X3,X4) -> U131(X1,X2,X3,X4)
r148: a__U132(X1,X2,X3,X4) -> U132(X1,X2,X3,X4)
r149: a__U133(X1,X2,X3,X4) -> U133(X1,X2,X3,X4)
r150: a__U134(X1,X2,X3,X4) -> U134(X1,X2,X3,X4)
r151: a__U135(X1,X2,X3,X4) -> U135(X1,X2,X3,X4)
r152: a__U136(X1,X2,X3,X4) -> U136(X1,X2,X3,X4)
r153: a__take(X1,X2) -> take(X1,X2)
r154: a__U21(X1,X2) -> U21(X1,X2)
r155: a__U22(X1,X2) -> U22(X1,X2)
r156: a__U23(X) -> U23(X)
r157: a__U31(X1,X2) -> U31(X1,X2)
r158: a__U32(X1,X2) -> U32(X1,X2)
r159: a__U33(X) -> U33(X)
r160: a__U41(X1,X2,X3) -> U41(X1,X2,X3)
r161: a__U42(X1,X2,X3) -> U42(X1,X2,X3)
r162: a__U43(X1,X2,X3) -> U43(X1,X2,X3)
r163: a__U44(X1,X2,X3) -> U44(X1,X2,X3)
r164: a__U45(X1,X2) -> U45(X1,X2)
r165: a__U46(X) -> U46(X)
r166: a__U51(X1,X2) -> U51(X1,X2)
r167: a__U52(X) -> U52(X)
r168: a__U61(X1,X2) -> U61(X1,X2)
r169: a__U62(X) -> U62(X)
r170: a__U71(X) -> U71(X)
r171: a__U81(X) -> U81(X)
r172: a__U91(X1,X2,X3) -> U91(X1,X2,X3)
r173: a__U92(X1,X2,X3) -> U92(X1,X2,X3)
r174: a__U93(X1,X2,X3) -> U93(X1,X2,X3)
r175: a__U94(X1,X2,X3) -> U94(X1,X2,X3)
r176: a__U95(X1,X2) -> U95(X1,X2)
r177: a__U96(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}


-- Reduction pair.

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

p1: mark#(U11(X1,X2)) -> mark#(X1)
p2: mark#(U95(X1,X2)) -> mark#(X1)
p3: mark#(U81(X)) -> mark#(X)
p4: mark#(U62(X)) -> mark#(X)
p5: mark#(U52(X)) -> mark#(X)
p6: mark#(U46(X)) -> mark#(X)
p7: mark#(U42(X1,X2,X3)) -> mark#(X1)
p8: mark#(U33(X)) -> mark#(X)
p9: mark#(U31(X1,X2)) -> mark#(X1)
p10: mark#(U23(X)) -> mark#(X)
p11: mark#(U21(X1,X2)) -> mark#(X1)
p12: mark#(U134(X1,X2,X3,X4)) -> mark#(X1)
p13: mark#(U131(X1,X2,X3,X4)) -> mark#(X1)
p14: mark#(U122(X)) -> mark#(X)
p15: mark#(U121(X1,X2)) -> mark#(X1)
p16: mark#(U111(X1,X2,X3)) -> mark#(X1)
p17: mark#(U12(X1,X2)) -> mark#(X1)

and R consists of:

r1: a__zeros() -> cons(|0|(),zeros())
r2: a__U101(tt(),V1,V2) -> a__U102(a__isNatKind(V1),V1,V2)
r3: a__U102(tt(),V1,V2) -> a__U103(a__isNatIListKind(V2),V1,V2)
r4: a__U103(tt(),V1,V2) -> a__U104(a__isNatIListKind(V2),V1,V2)
r5: a__U104(tt(),V1,V2) -> a__U105(a__isNat(V1),V2)
r6: a__U105(tt(),V2) -> a__U106(a__isNatIList(V2))
r7: a__U106(tt()) -> tt()
r8: a__U11(tt(),V1) -> a__U12(a__isNatIListKind(V1),V1)
r9: a__U111(tt(),L,N) -> a__U112(a__isNatIListKind(L),L,N)
r10: a__U112(tt(),L,N) -> a__U113(a__isNat(N),L,N)
r11: a__U113(tt(),L,N) -> a__U114(a__isNatKind(N),L)
r12: a__U114(tt(),L) -> s(a__length(mark(L)))
r13: a__U12(tt(),V1) -> a__U13(a__isNatList(V1))
r14: a__U121(tt(),IL) -> a__U122(a__isNatIListKind(IL))
r15: a__U122(tt()) -> nil()
r16: a__U13(tt()) -> tt()
r17: a__U131(tt(),IL,M,N) -> a__U132(a__isNatIListKind(IL),IL,M,N)
r18: a__U132(tt(),IL,M,N) -> a__U133(a__isNat(M),IL,M,N)
r19: a__U133(tt(),IL,M,N) -> a__U134(a__isNatKind(M),IL,M,N)
r20: a__U134(tt(),IL,M,N) -> a__U135(a__isNat(N),IL,M,N)
r21: a__U135(tt(),IL,M,N) -> a__U136(a__isNatKind(N),IL,M,N)
r22: a__U136(tt(),IL,M,N) -> cons(mark(N),take(M,IL))
r23: a__U21(tt(),V1) -> a__U22(a__isNatKind(V1),V1)
r24: a__U22(tt(),V1) -> a__U23(a__isNat(V1))
r25: a__U23(tt()) -> tt()
r26: a__U31(tt(),V) -> a__U32(a__isNatIListKind(V),V)
r27: a__U32(tt(),V) -> a__U33(a__isNatList(V))
r28: a__U33(tt()) -> tt()
r29: a__U41(tt(),V1,V2) -> a__U42(a__isNatKind(V1),V1,V2)
r30: a__U42(tt(),V1,V2) -> a__U43(a__isNatIListKind(V2),V1,V2)
r31: a__U43(tt(),V1,V2) -> a__U44(a__isNatIListKind(V2),V1,V2)
r32: a__U44(tt(),V1,V2) -> a__U45(a__isNat(V1),V2)
r33: a__U45(tt(),V2) -> a__U46(a__isNatIList(V2))
r34: a__U46(tt()) -> tt()
r35: a__U51(tt(),V2) -> a__U52(a__isNatIListKind(V2))
r36: a__U52(tt()) -> tt()
r37: a__U61(tt(),V2) -> a__U62(a__isNatIListKind(V2))
r38: a__U62(tt()) -> tt()
r39: a__U71(tt()) -> tt()
r40: a__U81(tt()) -> tt()
r41: a__U91(tt(),V1,V2) -> a__U92(a__isNatKind(V1),V1,V2)
r42: a__U92(tt(),V1,V2) -> a__U93(a__isNatIListKind(V2),V1,V2)
r43: a__U93(tt(),V1,V2) -> a__U94(a__isNatIListKind(V2),V1,V2)
r44: a__U94(tt(),V1,V2) -> a__U95(a__isNat(V1),V2)
r45: a__U95(tt(),V2) -> a__U96(a__isNatList(V2))
r46: a__U96(tt()) -> tt()
r47: a__isNat(|0|()) -> tt()
r48: a__isNat(length(V1)) -> a__U11(a__isNatIListKind(V1),V1)
r49: a__isNat(s(V1)) -> a__U21(a__isNatKind(V1),V1)
r50: a__isNatIList(V) -> a__U31(a__isNatIListKind(V),V)
r51: a__isNatIList(zeros()) -> tt()
r52: a__isNatIList(cons(V1,V2)) -> a__U41(a__isNatKind(V1),V1,V2)
r53: a__isNatIListKind(nil()) -> tt()
r54: a__isNatIListKind(zeros()) -> tt()
r55: a__isNatIListKind(cons(V1,V2)) -> a__U51(a__isNatKind(V1),V2)
r56: a__isNatIListKind(take(V1,V2)) -> a__U61(a__isNatKind(V1),V2)
r57: a__isNatKind(|0|()) -> tt()
r58: a__isNatKind(length(V1)) -> a__U71(a__isNatIListKind(V1))
r59: a__isNatKind(s(V1)) -> a__U81(a__isNatKind(V1))
r60: a__isNatList(nil()) -> tt()
r61: a__isNatList(cons(V1,V2)) -> a__U91(a__isNatKind(V1),V1,V2)
r62: a__isNatList(take(V1,V2)) -> a__U101(a__isNatKind(V1),V1,V2)
r63: a__length(nil()) -> |0|()
r64: a__length(cons(N,L)) -> a__U111(a__isNatList(L),L,N)
r65: a__take(|0|(),IL) -> a__U121(a__isNatIList(IL),IL)
r66: a__take(s(M),cons(N,IL)) -> a__U131(a__isNatIList(IL),IL,M,N)
r67: mark(zeros()) -> a__zeros()
r68: mark(U101(X1,X2,X3)) -> a__U101(mark(X1),X2,X3)
r69: mark(U102(X1,X2,X3)) -> a__U102(mark(X1),X2,X3)
r70: mark(isNatKind(X)) -> a__isNatKind(X)
r71: mark(U103(X1,X2,X3)) -> a__U103(mark(X1),X2,X3)
r72: mark(isNatIListKind(X)) -> a__isNatIListKind(X)
r73: mark(U104(X1,X2,X3)) -> a__U104(mark(X1),X2,X3)
r74: mark(U105(X1,X2)) -> a__U105(mark(X1),X2)
r75: mark(isNat(X)) -> a__isNat(X)
r76: mark(U106(X)) -> a__U106(mark(X))
r77: mark(isNatIList(X)) -> a__isNatIList(X)
r78: mark(U11(X1,X2)) -> a__U11(mark(X1),X2)
r79: mark(U12(X1,X2)) -> a__U12(mark(X1),X2)
r80: mark(U111(X1,X2,X3)) -> a__U111(mark(X1),X2,X3)
r81: mark(U112(X1,X2,X3)) -> a__U112(mark(X1),X2,X3)
r82: mark(U113(X1,X2,X3)) -> a__U113(mark(X1),X2,X3)
r83: mark(U114(X1,X2)) -> a__U114(mark(X1),X2)
r84: mark(length(X)) -> a__length(mark(X))
r85: mark(U13(X)) -> a__U13(mark(X))
r86: mark(isNatList(X)) -> a__isNatList(X)
r87: mark(U121(X1,X2)) -> a__U121(mark(X1),X2)
r88: mark(U122(X)) -> a__U122(mark(X))
r89: mark(U131(X1,X2,X3,X4)) -> a__U131(mark(X1),X2,X3,X4)
r90: mark(U132(X1,X2,X3,X4)) -> a__U132(mark(X1),X2,X3,X4)
r91: mark(U133(X1,X2,X3,X4)) -> a__U133(mark(X1),X2,X3,X4)
r92: mark(U134(X1,X2,X3,X4)) -> a__U134(mark(X1),X2,X3,X4)
r93: mark(U135(X1,X2,X3,X4)) -> a__U135(mark(X1),X2,X3,X4)
r94: mark(U136(X1,X2,X3,X4)) -> a__U136(mark(X1),X2,X3,X4)
r95: mark(take(X1,X2)) -> a__take(mark(X1),mark(X2))
r96: mark(U21(X1,X2)) -> a__U21(mark(X1),X2)
r97: mark(U22(X1,X2)) -> a__U22(mark(X1),X2)
r98: mark(U23(X)) -> a__U23(mark(X))
r99: mark(U31(X1,X2)) -> a__U31(mark(X1),X2)
r100: mark(U32(X1,X2)) -> a__U32(mark(X1),X2)
r101: mark(U33(X)) -> a__U33(mark(X))
r102: mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3)
r103: mark(U42(X1,X2,X3)) -> a__U42(mark(X1),X2,X3)
r104: mark(U43(X1,X2,X3)) -> a__U43(mark(X1),X2,X3)
r105: mark(U44(X1,X2,X3)) -> a__U44(mark(X1),X2,X3)
r106: mark(U45(X1,X2)) -> a__U45(mark(X1),X2)
r107: mark(U46(X)) -> a__U46(mark(X))
r108: mark(U51(X1,X2)) -> a__U51(mark(X1),X2)
r109: mark(U52(X)) -> a__U52(mark(X))
r110: mark(U61(X1,X2)) -> a__U61(mark(X1),X2)
r111: mark(U62(X)) -> a__U62(mark(X))
r112: mark(U71(X)) -> a__U71(mark(X))
r113: mark(U81(X)) -> a__U81(mark(X))
r114: mark(U91(X1,X2,X3)) -> a__U91(mark(X1),X2,X3)
r115: mark(U92(X1,X2,X3)) -> a__U92(mark(X1),X2,X3)
r116: mark(U93(X1,X2,X3)) -> a__U93(mark(X1),X2,X3)
r117: mark(U94(X1,X2,X3)) -> a__U94(mark(X1),X2,X3)
r118: mark(U95(X1,X2)) -> a__U95(mark(X1),X2)
r119: mark(U96(X)) -> a__U96(mark(X))
r120: mark(cons(X1,X2)) -> cons(mark(X1),X2)
r121: mark(|0|()) -> |0|()
r122: mark(tt()) -> tt()
r123: mark(s(X)) -> s(mark(X))
r124: mark(nil()) -> nil()
r125: a__zeros() -> zeros()
r126: a__U101(X1,X2,X3) -> U101(X1,X2,X3)
r127: a__U102(X1,X2,X3) -> U102(X1,X2,X3)
r128: a__isNatKind(X) -> isNatKind(X)
r129: a__U103(X1,X2,X3) -> U103(X1,X2,X3)
r130: a__isNatIListKind(X) -> isNatIListKind(X)
r131: a__U104(X1,X2,X3) -> U104(X1,X2,X3)
r132: a__U105(X1,X2) -> U105(X1,X2)
r133: a__isNat(X) -> isNat(X)
r134: a__U106(X) -> U106(X)
r135: a__isNatIList(X) -> isNatIList(X)
r136: a__U11(X1,X2) -> U11(X1,X2)
r137: a__U12(X1,X2) -> U12(X1,X2)
r138: a__U111(X1,X2,X3) -> U111(X1,X2,X3)
r139: a__U112(X1,X2,X3) -> U112(X1,X2,X3)
r140: a__U113(X1,X2,X3) -> U113(X1,X2,X3)
r141: a__U114(X1,X2) -> U114(X1,X2)
r142: a__length(X) -> length(X)
r143: a__U13(X) -> U13(X)
r144: a__isNatList(X) -> isNatList(X)
r145: a__U121(X1,X2) -> U121(X1,X2)
r146: a__U122(X) -> U122(X)
r147: a__U131(X1,X2,X3,X4) -> U131(X1,X2,X3,X4)
r148: a__U132(X1,X2,X3,X4) -> U132(X1,X2,X3,X4)
r149: a__U133(X1,X2,X3,X4) -> U133(X1,X2,X3,X4)
r150: a__U134(X1,X2,X3,X4) -> U134(X1,X2,X3,X4)
r151: a__U135(X1,X2,X3,X4) -> U135(X1,X2,X3,X4)
r152: a__U136(X1,X2,X3,X4) -> U136(X1,X2,X3,X4)
r153: a__take(X1,X2) -> take(X1,X2)
r154: a__U21(X1,X2) -> U21(X1,X2)
r155: a__U22(X1,X2) -> U22(X1,X2)
r156: a__U23(X) -> U23(X)
r157: a__U31(X1,X2) -> U31(X1,X2)
r158: a__U32(X1,X2) -> U32(X1,X2)
r159: a__U33(X) -> U33(X)
r160: a__U41(X1,X2,X3) -> U41(X1,X2,X3)
r161: a__U42(X1,X2,X3) -> U42(X1,X2,X3)
r162: a__U43(X1,X2,X3) -> U43(X1,X2,X3)
r163: a__U44(X1,X2,X3) -> U44(X1,X2,X3)
r164: a__U45(X1,X2) -> U45(X1,X2)
r165: a__U46(X) -> U46(X)
r166: a__U51(X1,X2) -> U51(X1,X2)
r167: a__U52(X) -> U52(X)
r168: a__U61(X1,X2) -> U61(X1,X2)
r169: a__U62(X) -> U62(X)
r170: a__U71(X) -> U71(X)
r171: a__U81(X) -> U81(X)
r172: a__U91(X1,X2,X3) -> U91(X1,X2,X3)
r173: a__U92(X1,X2,X3) -> U92(X1,X2,X3)
r174: a__U93(X1,X2,X3) -> U93(X1,X2,X3)
r175: a__U94(X1,X2,X3) -> U94(X1,X2,X3)
r176: a__U95(X1,X2) -> U95(X1,X2)
r177: a__U96(X) -> U96(X)

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  max/plus interpretations on natural numbers:
  
    mark#_A(x1) = x1 + 1
    U11_A(x1,x2) = max{x1, x2}
    U95_A(x1,x2) = max{x1, x2}
    U81_A(x1) = x1
    U62_A(x1) = x1
    U52_A(x1) = x1 + 1
    U46_A(x1) = x1
    U42_A(x1,x2,x3) = max{x1, x2, x3}
    U33_A(x1) = x1
    U31_A(x1,x2) = max{x1, x2}
    U23_A(x1) = x1 + 1
    U21_A(x1,x2) = max{x1, x2}
    U134_A(x1,x2,x3,x4) = max{x1, x2, x3, x4}
    U131_A(x1,x2,x3,x4) = max{x1, x2, x3, x4}
    U122_A(x1) = x1
    U121_A(x1,x2) = max{x1, x2}
    U111_A(x1,x2,x3) = max{x1, x2, x3}
    U12_A(x1,x2) = max{x1, x2}

The next rules are strictly ordered:

  p5, p10

We remove them from the problem.

-- SCC decomposition.

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

p1: mark#(U11(X1,X2)) -> mark#(X1)
p2: mark#(U95(X1,X2)) -> mark#(X1)
p3: mark#(U81(X)) -> mark#(X)
p4: mark#(U62(X)) -> mark#(X)
p5: mark#(U46(X)) -> mark#(X)
p6: mark#(U42(X1,X2,X3)) -> mark#(X1)
p7: mark#(U33(X)) -> mark#(X)
p8: mark#(U31(X1,X2)) -> mark#(X1)
p9: mark#(U21(X1,X2)) -> mark#(X1)
p10: mark#(U134(X1,X2,X3,X4)) -> mark#(X1)
p11: mark#(U131(X1,X2,X3,X4)) -> mark#(X1)
p12: mark#(U122(X)) -> mark#(X)
p13: mark#(U121(X1,X2)) -> mark#(X1)
p14: mark#(U111(X1,X2,X3)) -> mark#(X1)
p15: mark#(U12(X1,X2)) -> mark#(X1)

and R consists of:

r1: a__zeros() -> cons(|0|(),zeros())
r2: a__U101(tt(),V1,V2) -> a__U102(a__isNatKind(V1),V1,V2)
r3: a__U102(tt(),V1,V2) -> a__U103(a__isNatIListKind(V2),V1,V2)
r4: a__U103(tt(),V1,V2) -> a__U104(a__isNatIListKind(V2),V1,V2)
r5: a__U104(tt(),V1,V2) -> a__U105(a__isNat(V1),V2)
r6: a__U105(tt(),V2) -> a__U106(a__isNatIList(V2))
r7: a__U106(tt()) -> tt()
r8: a__U11(tt(),V1) -> a__U12(a__isNatIListKind(V1),V1)
r9: a__U111(tt(),L,N) -> a__U112(a__isNatIListKind(L),L,N)
r10: a__U112(tt(),L,N) -> a__U113(a__isNat(N),L,N)
r11: a__U113(tt(),L,N) -> a__U114(a__isNatKind(N),L)
r12: a__U114(tt(),L) -> s(a__length(mark(L)))
r13: a__U12(tt(),V1) -> a__U13(a__isNatList(V1))
r14: a__U121(tt(),IL) -> a__U122(a__isNatIListKind(IL))
r15: a__U122(tt()) -> nil()
r16: a__U13(tt()) -> tt()
r17: a__U131(tt(),IL,M,N) -> a__U132(a__isNatIListKind(IL),IL,M,N)
r18: a__U132(tt(),IL,M,N) -> a__U133(a__isNat(M),IL,M,N)
r19: a__U133(tt(),IL,M,N) -> a__U134(a__isNatKind(M),IL,M,N)
r20: a__U134(tt(),IL,M,N) -> a__U135(a__isNat(N),IL,M,N)
r21: a__U135(tt(),IL,M,N) -> a__U136(a__isNatKind(N),IL,M,N)
r22: a__U136(tt(),IL,M,N) -> cons(mark(N),take(M,IL))
r23: a__U21(tt(),V1) -> a__U22(a__isNatKind(V1),V1)
r24: a__U22(tt(),V1) -> a__U23(a__isNat(V1))
r25: a__U23(tt()) -> tt()
r26: a__U31(tt(),V) -> a__U32(a__isNatIListKind(V),V)
r27: a__U32(tt(),V) -> a__U33(a__isNatList(V))
r28: a__U33(tt()) -> tt()
r29: a__U41(tt(),V1,V2) -> a__U42(a__isNatKind(V1),V1,V2)
r30: a__U42(tt(),V1,V2) -> a__U43(a__isNatIListKind(V2),V1,V2)
r31: a__U43(tt(),V1,V2) -> a__U44(a__isNatIListKind(V2),V1,V2)
r32: a__U44(tt(),V1,V2) -> a__U45(a__isNat(V1),V2)
r33: a__U45(tt(),V2) -> a__U46(a__isNatIList(V2))
r34: a__U46(tt()) -> tt()
r35: a__U51(tt(),V2) -> a__U52(a__isNatIListKind(V2))
r36: a__U52(tt()) -> tt()
r37: a__U61(tt(),V2) -> a__U62(a__isNatIListKind(V2))
r38: a__U62(tt()) -> tt()
r39: a__U71(tt()) -> tt()
r40: a__U81(tt()) -> tt()
r41: a__U91(tt(),V1,V2) -> a__U92(a__isNatKind(V1),V1,V2)
r42: a__U92(tt(),V1,V2) -> a__U93(a__isNatIListKind(V2),V1,V2)
r43: a__U93(tt(),V1,V2) -> a__U94(a__isNatIListKind(V2),V1,V2)
r44: a__U94(tt(),V1,V2) -> a__U95(a__isNat(V1),V2)
r45: a__U95(tt(),V2) -> a__U96(a__isNatList(V2))
r46: a__U96(tt()) -> tt()
r47: a__isNat(|0|()) -> tt()
r48: a__isNat(length(V1)) -> a__U11(a__isNatIListKind(V1),V1)
r49: a__isNat(s(V1)) -> a__U21(a__isNatKind(V1),V1)
r50: a__isNatIList(V) -> a__U31(a__isNatIListKind(V),V)
r51: a__isNatIList(zeros()) -> tt()
r52: a__isNatIList(cons(V1,V2)) -> a__U41(a__isNatKind(V1),V1,V2)
r53: a__isNatIListKind(nil()) -> tt()
r54: a__isNatIListKind(zeros()) -> tt()
r55: a__isNatIListKind(cons(V1,V2)) -> a__U51(a__isNatKind(V1),V2)
r56: a__isNatIListKind(take(V1,V2)) -> a__U61(a__isNatKind(V1),V2)
r57: a__isNatKind(|0|()) -> tt()
r58: a__isNatKind(length(V1)) -> a__U71(a__isNatIListKind(V1))
r59: a__isNatKind(s(V1)) -> a__U81(a__isNatKind(V1))
r60: a__isNatList(nil()) -> tt()
r61: a__isNatList(cons(V1,V2)) -> a__U91(a__isNatKind(V1),V1,V2)
r62: a__isNatList(take(V1,V2)) -> a__U101(a__isNatKind(V1),V1,V2)
r63: a__length(nil()) -> |0|()
r64: a__length(cons(N,L)) -> a__U111(a__isNatList(L),L,N)
r65: a__take(|0|(),IL) -> a__U121(a__isNatIList(IL),IL)
r66: a__take(s(M),cons(N,IL)) -> a__U131(a__isNatIList(IL),IL,M,N)
r67: mark(zeros()) -> a__zeros()
r68: mark(U101(X1,X2,X3)) -> a__U101(mark(X1),X2,X3)
r69: mark(U102(X1,X2,X3)) -> a__U102(mark(X1),X2,X3)
r70: mark(isNatKind(X)) -> a__isNatKind(X)
r71: mark(U103(X1,X2,X3)) -> a__U103(mark(X1),X2,X3)
r72: mark(isNatIListKind(X)) -> a__isNatIListKind(X)
r73: mark(U104(X1,X2,X3)) -> a__U104(mark(X1),X2,X3)
r74: mark(U105(X1,X2)) -> a__U105(mark(X1),X2)
r75: mark(isNat(X)) -> a__isNat(X)
r76: mark(U106(X)) -> a__U106(mark(X))
r77: mark(isNatIList(X)) -> a__isNatIList(X)
r78: mark(U11(X1,X2)) -> a__U11(mark(X1),X2)
r79: mark(U12(X1,X2)) -> a__U12(mark(X1),X2)
r80: mark(U111(X1,X2,X3)) -> a__U111(mark(X1),X2,X3)
r81: mark(U112(X1,X2,X3)) -> a__U112(mark(X1),X2,X3)
r82: mark(U113(X1,X2,X3)) -> a__U113(mark(X1),X2,X3)
r83: mark(U114(X1,X2)) -> a__U114(mark(X1),X2)
r84: mark(length(X)) -> a__length(mark(X))
r85: mark(U13(X)) -> a__U13(mark(X))
r86: mark(isNatList(X)) -> a__isNatList(X)
r87: mark(U121(X1,X2)) -> a__U121(mark(X1),X2)
r88: mark(U122(X)) -> a__U122(mark(X))
r89: mark(U131(X1,X2,X3,X4)) -> a__U131(mark(X1),X2,X3,X4)
r90: mark(U132(X1,X2,X3,X4)) -> a__U132(mark(X1),X2,X3,X4)
r91: mark(U133(X1,X2,X3,X4)) -> a__U133(mark(X1),X2,X3,X4)
r92: mark(U134(X1,X2,X3,X4)) -> a__U134(mark(X1),X2,X3,X4)
r93: mark(U135(X1,X2,X3,X4)) -> a__U135(mark(X1),X2,X3,X4)
r94: mark(U136(X1,X2,X3,X4)) -> a__U136(mark(X1),X2,X3,X4)
r95: mark(take(X1,X2)) -> a__take(mark(X1),mark(X2))
r96: mark(U21(X1,X2)) -> a__U21(mark(X1),X2)
r97: mark(U22(X1,X2)) -> a__U22(mark(X1),X2)
r98: mark(U23(X)) -> a__U23(mark(X))
r99: mark(U31(X1,X2)) -> a__U31(mark(X1),X2)
r100: mark(U32(X1,X2)) -> a__U32(mark(X1),X2)
r101: mark(U33(X)) -> a__U33(mark(X))
r102: mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3)
r103: mark(U42(X1,X2,X3)) -> a__U42(mark(X1),X2,X3)
r104: mark(U43(X1,X2,X3)) -> a__U43(mark(X1),X2,X3)
r105: mark(U44(X1,X2,X3)) -> a__U44(mark(X1),X2,X3)
r106: mark(U45(X1,X2)) -> a__U45(mark(X1),X2)
r107: mark(U46(X)) -> a__U46(mark(X))
r108: mark(U51(X1,X2)) -> a__U51(mark(X1),X2)
r109: mark(U52(X)) -> a__U52(mark(X))
r110: mark(U61(X1,X2)) -> a__U61(mark(X1),X2)
r111: mark(U62(X)) -> a__U62(mark(X))
r112: mark(U71(X)) -> a__U71(mark(X))
r113: mark(U81(X)) -> a__U81(mark(X))
r114: mark(U91(X1,X2,X3)) -> a__U91(mark(X1),X2,X3)
r115: mark(U92(X1,X2,X3)) -> a__U92(mark(X1),X2,X3)
r116: mark(U93(X1,X2,X3)) -> a__U93(mark(X1),X2,X3)
r117: mark(U94(X1,X2,X3)) -> a__U94(mark(X1),X2,X3)
r118: mark(U95(X1,X2)) -> a__U95(mark(X1),X2)
r119: mark(U96(X)) -> a__U96(mark(X))
r120: mark(cons(X1,X2)) -> cons(mark(X1),X2)
r121: mark(|0|()) -> |0|()
r122: mark(tt()) -> tt()
r123: mark(s(X)) -> s(mark(X))
r124: mark(nil()) -> nil()
r125: a__zeros() -> zeros()
r126: a__U101(X1,X2,X3) -> U101(X1,X2,X3)
r127: a__U102(X1,X2,X3) -> U102(X1,X2,X3)
r128: a__isNatKind(X) -> isNatKind(X)
r129: a__U103(X1,X2,X3) -> U103(X1,X2,X3)
r130: a__isNatIListKind(X) -> isNatIListKind(X)
r131: a__U104(X1,X2,X3) -> U104(X1,X2,X3)
r132: a__U105(X1,X2) -> U105(X1,X2)
r133: a__isNat(X) -> isNat(X)
r134: a__U106(X) -> U106(X)
r135: a__isNatIList(X) -> isNatIList(X)
r136: a__U11(X1,X2) -> U11(X1,X2)
r137: a__U12(X1,X2) -> U12(X1,X2)
r138: a__U111(X1,X2,X3) -> U111(X1,X2,X3)
r139: a__U112(X1,X2,X3) -> U112(X1,X2,X3)
r140: a__U113(X1,X2,X3) -> U113(X1,X2,X3)
r141: a__U114(X1,X2) -> U114(X1,X2)
r142: a__length(X) -> length(X)
r143: a__U13(X) -> U13(X)
r144: a__isNatList(X) -> isNatList(X)
r145: a__U121(X1,X2) -> U121(X1,X2)
r146: a__U122(X) -> U122(X)
r147: a__U131(X1,X2,X3,X4) -> U131(X1,X2,X3,X4)
r148: a__U132(X1,X2,X3,X4) -> U132(X1,X2,X3,X4)
r149: a__U133(X1,X2,X3,X4) -> U133(X1,X2,X3,X4)
r150: a__U134(X1,X2,X3,X4) -> U134(X1,X2,X3,X4)
r151: a__U135(X1,X2,X3,X4) -> U135(X1,X2,X3,X4)
r152: a__U136(X1,X2,X3,X4) -> U136(X1,X2,X3,X4)
r153: a__take(X1,X2) -> take(X1,X2)
r154: a__U21(X1,X2) -> U21(X1,X2)
r155: a__U22(X1,X2) -> U22(X1,X2)
r156: a__U23(X) -> U23(X)
r157: a__U31(X1,X2) -> U31(X1,X2)
r158: a__U32(X1,X2) -> U32(X1,X2)
r159: a__U33(X) -> U33(X)
r160: a__U41(X1,X2,X3) -> U41(X1,X2,X3)
r161: a__U42(X1,X2,X3) -> U42(X1,X2,X3)
r162: a__U43(X1,X2,X3) -> U43(X1,X2,X3)
r163: a__U44(X1,X2,X3) -> U44(X1,X2,X3)
r164: a__U45(X1,X2) -> U45(X1,X2)
r165: a__U46(X) -> U46(X)
r166: a__U51(X1,X2) -> U51(X1,X2)
r167: a__U52(X) -> U52(X)
r168: a__U61(X1,X2) -> U61(X1,X2)
r169: a__U62(X) -> U62(X)
r170: a__U71(X) -> U71(X)
r171: a__U81(X) -> U81(X)
r172: a__U91(X1,X2,X3) -> U91(X1,X2,X3)
r173: a__U92(X1,X2,X3) -> U92(X1,X2,X3)
r174: a__U93(X1,X2,X3) -> U93(X1,X2,X3)
r175: a__U94(X1,X2,X3) -> U94(X1,X2,X3)
r176: a__U95(X1,X2) -> U95(X1,X2)
r177: a__U96(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}


-- Reduction pair.

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

p1: mark#(U11(X1,X2)) -> mark#(X1)
p2: mark#(U12(X1,X2)) -> mark#(X1)
p3: mark#(U111(X1,X2,X3)) -> mark#(X1)
p4: mark#(U121(X1,X2)) -> mark#(X1)
p5: mark#(U122(X)) -> mark#(X)
p6: mark#(U131(X1,X2,X3,X4)) -> mark#(X1)
p7: mark#(U134(X1,X2,X3,X4)) -> mark#(X1)
p8: mark#(U21(X1,X2)) -> mark#(X1)
p9: mark#(U31(X1,X2)) -> mark#(X1)
p10: mark#(U33(X)) -> mark#(X)
p11: mark#(U42(X1,X2,X3)) -> mark#(X1)
p12: mark#(U46(X)) -> mark#(X)
p13: mark#(U62(X)) -> mark#(X)
p14: mark#(U81(X)) -> mark#(X)
p15: mark#(U95(X1,X2)) -> mark#(X1)

and R consists of:

r1: a__zeros() -> cons(|0|(),zeros())
r2: a__U101(tt(),V1,V2) -> a__U102(a__isNatKind(V1),V1,V2)
r3: a__U102(tt(),V1,V2) -> a__U103(a__isNatIListKind(V2),V1,V2)
r4: a__U103(tt(),V1,V2) -> a__U104(a__isNatIListKind(V2),V1,V2)
r5: a__U104(tt(),V1,V2) -> a__U105(a__isNat(V1),V2)
r6: a__U105(tt(),V2) -> a__U106(a__isNatIList(V2))
r7: a__U106(tt()) -> tt()
r8: a__U11(tt(),V1) -> a__U12(a__isNatIListKind(V1),V1)
r9: a__U111(tt(),L,N) -> a__U112(a__isNatIListKind(L),L,N)
r10: a__U112(tt(),L,N) -> a__U113(a__isNat(N),L,N)
r11: a__U113(tt(),L,N) -> a__U114(a__isNatKind(N),L)
r12: a__U114(tt(),L) -> s(a__length(mark(L)))
r13: a__U12(tt(),V1) -> a__U13(a__isNatList(V1))
r14: a__U121(tt(),IL) -> a__U122(a__isNatIListKind(IL))
r15: a__U122(tt()) -> nil()
r16: a__U13(tt()) -> tt()
r17: a__U131(tt(),IL,M,N) -> a__U132(a__isNatIListKind(IL),IL,M,N)
r18: a__U132(tt(),IL,M,N) -> a__U133(a__isNat(M),IL,M,N)
r19: a__U133(tt(),IL,M,N) -> a__U134(a__isNatKind(M),IL,M,N)
r20: a__U134(tt(),IL,M,N) -> a__U135(a__isNat(N),IL,M,N)
r21: a__U135(tt(),IL,M,N) -> a__U136(a__isNatKind(N),IL,M,N)
r22: a__U136(tt(),IL,M,N) -> cons(mark(N),take(M,IL))
r23: a__U21(tt(),V1) -> a__U22(a__isNatKind(V1),V1)
r24: a__U22(tt(),V1) -> a__U23(a__isNat(V1))
r25: a__U23(tt()) -> tt()
r26: a__U31(tt(),V) -> a__U32(a__isNatIListKind(V),V)
r27: a__U32(tt(),V) -> a__U33(a__isNatList(V))
r28: a__U33(tt()) -> tt()
r29: a__U41(tt(),V1,V2) -> a__U42(a__isNatKind(V1),V1,V2)
r30: a__U42(tt(),V1,V2) -> a__U43(a__isNatIListKind(V2),V1,V2)
r31: a__U43(tt(),V1,V2) -> a__U44(a__isNatIListKind(V2),V1,V2)
r32: a__U44(tt(),V1,V2) -> a__U45(a__isNat(V1),V2)
r33: a__U45(tt(),V2) -> a__U46(a__isNatIList(V2))
r34: a__U46(tt()) -> tt()
r35: a__U51(tt(),V2) -> a__U52(a__isNatIListKind(V2))
r36: a__U52(tt()) -> tt()
r37: a__U61(tt(),V2) -> a__U62(a__isNatIListKind(V2))
r38: a__U62(tt()) -> tt()
r39: a__U71(tt()) -> tt()
r40: a__U81(tt()) -> tt()
r41: a__U91(tt(),V1,V2) -> a__U92(a__isNatKind(V1),V1,V2)
r42: a__U92(tt(),V1,V2) -> a__U93(a__isNatIListKind(V2),V1,V2)
r43: a__U93(tt(),V1,V2) -> a__U94(a__isNatIListKind(V2),V1,V2)
r44: a__U94(tt(),V1,V2) -> a__U95(a__isNat(V1),V2)
r45: a__U95(tt(),V2) -> a__U96(a__isNatList(V2))
r46: a__U96(tt()) -> tt()
r47: a__isNat(|0|()) -> tt()
r48: a__isNat(length(V1)) -> a__U11(a__isNatIListKind(V1),V1)
r49: a__isNat(s(V1)) -> a__U21(a__isNatKind(V1),V1)
r50: a__isNatIList(V) -> a__U31(a__isNatIListKind(V),V)
r51: a__isNatIList(zeros()) -> tt()
r52: a__isNatIList(cons(V1,V2)) -> a__U41(a__isNatKind(V1),V1,V2)
r53: a__isNatIListKind(nil()) -> tt()
r54: a__isNatIListKind(zeros()) -> tt()
r55: a__isNatIListKind(cons(V1,V2)) -> a__U51(a__isNatKind(V1),V2)
r56: a__isNatIListKind(take(V1,V2)) -> a__U61(a__isNatKind(V1),V2)
r57: a__isNatKind(|0|()) -> tt()
r58: a__isNatKind(length(V1)) -> a__U71(a__isNatIListKind(V1))
r59: a__isNatKind(s(V1)) -> a__U81(a__isNatKind(V1))
r60: a__isNatList(nil()) -> tt()
r61: a__isNatList(cons(V1,V2)) -> a__U91(a__isNatKind(V1),V1,V2)
r62: a__isNatList(take(V1,V2)) -> a__U101(a__isNatKind(V1),V1,V2)
r63: a__length(nil()) -> |0|()
r64: a__length(cons(N,L)) -> a__U111(a__isNatList(L),L,N)
r65: a__take(|0|(),IL) -> a__U121(a__isNatIList(IL),IL)
r66: a__take(s(M),cons(N,IL)) -> a__U131(a__isNatIList(IL),IL,M,N)
r67: mark(zeros()) -> a__zeros()
r68: mark(U101(X1,X2,X3)) -> a__U101(mark(X1),X2,X3)
r69: mark(U102(X1,X2,X3)) -> a__U102(mark(X1),X2,X3)
r70: mark(isNatKind(X)) -> a__isNatKind(X)
r71: mark(U103(X1,X2,X3)) -> a__U103(mark(X1),X2,X3)
r72: mark(isNatIListKind(X)) -> a__isNatIListKind(X)
r73: mark(U104(X1,X2,X3)) -> a__U104(mark(X1),X2,X3)
r74: mark(U105(X1,X2)) -> a__U105(mark(X1),X2)
r75: mark(isNat(X)) -> a__isNat(X)
r76: mark(U106(X)) -> a__U106(mark(X))
r77: mark(isNatIList(X)) -> a__isNatIList(X)
r78: mark(U11(X1,X2)) -> a__U11(mark(X1),X2)
r79: mark(U12(X1,X2)) -> a__U12(mark(X1),X2)
r80: mark(U111(X1,X2,X3)) -> a__U111(mark(X1),X2,X3)
r81: mark(U112(X1,X2,X3)) -> a__U112(mark(X1),X2,X3)
r82: mark(U113(X1,X2,X3)) -> a__U113(mark(X1),X2,X3)
r83: mark(U114(X1,X2)) -> a__U114(mark(X1),X2)
r84: mark(length(X)) -> a__length(mark(X))
r85: mark(U13(X)) -> a__U13(mark(X))
r86: mark(isNatList(X)) -> a__isNatList(X)
r87: mark(U121(X1,X2)) -> a__U121(mark(X1),X2)
r88: mark(U122(X)) -> a__U122(mark(X))
r89: mark(U131(X1,X2,X3,X4)) -> a__U131(mark(X1),X2,X3,X4)
r90: mark(U132(X1,X2,X3,X4)) -> a__U132(mark(X1),X2,X3,X4)
r91: mark(U133(X1,X2,X3,X4)) -> a__U133(mark(X1),X2,X3,X4)
r92: mark(U134(X1,X2,X3,X4)) -> a__U134(mark(X1),X2,X3,X4)
r93: mark(U135(X1,X2,X3,X4)) -> a__U135(mark(X1),X2,X3,X4)
r94: mark(U136(X1,X2,X3,X4)) -> a__U136(mark(X1),X2,X3,X4)
r95: mark(take(X1,X2)) -> a__take(mark(X1),mark(X2))
r96: mark(U21(X1,X2)) -> a__U21(mark(X1),X2)
r97: mark(U22(X1,X2)) -> a__U22(mark(X1),X2)
r98: mark(U23(X)) -> a__U23(mark(X))
r99: mark(U31(X1,X2)) -> a__U31(mark(X1),X2)
r100: mark(U32(X1,X2)) -> a__U32(mark(X1),X2)
r101: mark(U33(X)) -> a__U33(mark(X))
r102: mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3)
r103: mark(U42(X1,X2,X3)) -> a__U42(mark(X1),X2,X3)
r104: mark(U43(X1,X2,X3)) -> a__U43(mark(X1),X2,X3)
r105: mark(U44(X1,X2,X3)) -> a__U44(mark(X1),X2,X3)
r106: mark(U45(X1,X2)) -> a__U45(mark(X1),X2)
r107: mark(U46(X)) -> a__U46(mark(X))
r108: mark(U51(X1,X2)) -> a__U51(mark(X1),X2)
r109: mark(U52(X)) -> a__U52(mark(X))
r110: mark(U61(X1,X2)) -> a__U61(mark(X1),X2)
r111: mark(U62(X)) -> a__U62(mark(X))
r112: mark(U71(X)) -> a__U71(mark(X))
r113: mark(U81(X)) -> a__U81(mark(X))
r114: mark(U91(X1,X2,X3)) -> a__U91(mark(X1),X2,X3)
r115: mark(U92(X1,X2,X3)) -> a__U92(mark(X1),X2,X3)
r116: mark(U93(X1,X2,X3)) -> a__U93(mark(X1),X2,X3)
r117: mark(U94(X1,X2,X3)) -> a__U94(mark(X1),X2,X3)
r118: mark(U95(X1,X2)) -> a__U95(mark(X1),X2)
r119: mark(U96(X)) -> a__U96(mark(X))
r120: mark(cons(X1,X2)) -> cons(mark(X1),X2)
r121: mark(|0|()) -> |0|()
r122: mark(tt()) -> tt()
r123: mark(s(X)) -> s(mark(X))
r124: mark(nil()) -> nil()
r125: a__zeros() -> zeros()
r126: a__U101(X1,X2,X3) -> U101(X1,X2,X3)
r127: a__U102(X1,X2,X3) -> U102(X1,X2,X3)
r128: a__isNatKind(X) -> isNatKind(X)
r129: a__U103(X1,X2,X3) -> U103(X1,X2,X3)
r130: a__isNatIListKind(X) -> isNatIListKind(X)
r131: a__U104(X1,X2,X3) -> U104(X1,X2,X3)
r132: a__U105(X1,X2) -> U105(X1,X2)
r133: a__isNat(X) -> isNat(X)
r134: a__U106(X) -> U106(X)
r135: a__isNatIList(X) -> isNatIList(X)
r136: a__U11(X1,X2) -> U11(X1,X2)
r137: a__U12(X1,X2) -> U12(X1,X2)
r138: a__U111(X1,X2,X3) -> U111(X1,X2,X3)
r139: a__U112(X1,X2,X3) -> U112(X1,X2,X3)
r140: a__U113(X1,X2,X3) -> U113(X1,X2,X3)
r141: a__U114(X1,X2) -> U114(X1,X2)
r142: a__length(X) -> length(X)
r143: a__U13(X) -> U13(X)
r144: a__isNatList(X) -> isNatList(X)
r145: a__U121(X1,X2) -> U121(X1,X2)
r146: a__U122(X) -> U122(X)
r147: a__U131(X1,X2,X3,X4) -> U131(X1,X2,X3,X4)
r148: a__U132(X1,X2,X3,X4) -> U132(X1,X2,X3,X4)
r149: a__U133(X1,X2,X3,X4) -> U133(X1,X2,X3,X4)
r150: a__U134(X1,X2,X3,X4) -> U134(X1,X2,X3,X4)
r151: a__U135(X1,X2,X3,X4) -> U135(X1,X2,X3,X4)
r152: a__U136(X1,X2,X3,X4) -> U136(X1,X2,X3,X4)
r153: a__take(X1,X2) -> take(X1,X2)
r154: a__U21(X1,X2) -> U21(X1,X2)
r155: a__U22(X1,X2) -> U22(X1,X2)
r156: a__U23(X) -> U23(X)
r157: a__U31(X1,X2) -> U31(X1,X2)
r158: a__U32(X1,X2) -> U32(X1,X2)
r159: a__U33(X) -> U33(X)
r160: a__U41(X1,X2,X3) -> U41(X1,X2,X3)
r161: a__U42(X1,X2,X3) -> U42(X1,X2,X3)
r162: a__U43(X1,X2,X3) -> U43(X1,X2,X3)
r163: a__U44(X1,X2,X3) -> U44(X1,X2,X3)
r164: a__U45(X1,X2) -> U45(X1,X2)
r165: a__U46(X) -> U46(X)
r166: a__U51(X1,X2) -> U51(X1,X2)
r167: a__U52(X) -> U52(X)
r168: a__U61(X1,X2) -> U61(X1,X2)
r169: a__U62(X) -> U62(X)
r170: a__U71(X) -> U71(X)
r171: a__U81(X) -> U81(X)
r172: a__U91(X1,X2,X3) -> U91(X1,X2,X3)
r173: a__U92(X1,X2,X3) -> U92(X1,X2,X3)
r174: a__U93(X1,X2,X3) -> U93(X1,X2,X3)
r175: a__U94(X1,X2,X3) -> U94(X1,X2,X3)
r176: a__U95(X1,X2) -> U95(X1,X2)
r177: a__U96(X) -> U96(X)

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  max/plus interpretations on natural numbers:
  
    mark#_A(x1) = x1 + 1
    U11_A(x1,x2) = max{x1, x2}
    U12_A(x1,x2) = max{x1, x2}
    U111_A(x1,x2,x3) = max{x1, x2, x3}
    U121_A(x1,x2) = max{x1, x2}
    U122_A(x1) = x1
    U131_A(x1,x2,x3,x4) = max{x1, x2, x3, x4}
    U134_A(x1,x2,x3,x4) = max{x1, x2, x3, x4}
    U21_A(x1,x2) = max{x1, x2}
    U31_A(x1,x2) = max{x1, x2}
    U33_A(x1) = x1
    U42_A(x1,x2,x3) = max{x1, x2, x3}
    U46_A(x1) = x1
    U62_A(x1) = x1
    U81_A(x1) = x1
    U95_A(x1,x2) = max{x1 + 1, x2}

The next rules are strictly ordered:

  p15

We remove them from the problem.

-- SCC decomposition.

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

p1: mark#(U11(X1,X2)) -> mark#(X1)
p2: mark#(U12(X1,X2)) -> mark#(X1)
p3: mark#(U111(X1,X2,X3)) -> mark#(X1)
p4: mark#(U121(X1,X2)) -> mark#(X1)
p5: mark#(U122(X)) -> mark#(X)
p6: mark#(U131(X1,X2,X3,X4)) -> mark#(X1)
p7: mark#(U134(X1,X2,X3,X4)) -> mark#(X1)
p8: mark#(U21(X1,X2)) -> mark#(X1)
p9: mark#(U31(X1,X2)) -> mark#(X1)
p10: mark#(U33(X)) -> mark#(X)
p11: mark#(U42(X1,X2,X3)) -> mark#(X1)
p12: mark#(U46(X)) -> mark#(X)
p13: mark#(U62(X)) -> mark#(X)
p14: mark#(U81(X)) -> mark#(X)

and R consists of:

r1: a__zeros() -> cons(|0|(),zeros())
r2: a__U101(tt(),V1,V2) -> a__U102(a__isNatKind(V1),V1,V2)
r3: a__U102(tt(),V1,V2) -> a__U103(a__isNatIListKind(V2),V1,V2)
r4: a__U103(tt(),V1,V2) -> a__U104(a__isNatIListKind(V2),V1,V2)
r5: a__U104(tt(),V1,V2) -> a__U105(a__isNat(V1),V2)
r6: a__U105(tt(),V2) -> a__U106(a__isNatIList(V2))
r7: a__U106(tt()) -> tt()
r8: a__U11(tt(),V1) -> a__U12(a__isNatIListKind(V1),V1)
r9: a__U111(tt(),L,N) -> a__U112(a__isNatIListKind(L),L,N)
r10: a__U112(tt(),L,N) -> a__U113(a__isNat(N),L,N)
r11: a__U113(tt(),L,N) -> a__U114(a__isNatKind(N),L)
r12: a__U114(tt(),L) -> s(a__length(mark(L)))
r13: a__U12(tt(),V1) -> a__U13(a__isNatList(V1))
r14: a__U121(tt(),IL) -> a__U122(a__isNatIListKind(IL))
r15: a__U122(tt()) -> nil()
r16: a__U13(tt()) -> tt()
r17: a__U131(tt(),IL,M,N) -> a__U132(a__isNatIListKind(IL),IL,M,N)
r18: a__U132(tt(),IL,M,N) -> a__U133(a__isNat(M),IL,M,N)
r19: a__U133(tt(),IL,M,N) -> a__U134(a__isNatKind(M),IL,M,N)
r20: a__U134(tt(),IL,M,N) -> a__U135(a__isNat(N),IL,M,N)
r21: a__U135(tt(),IL,M,N) -> a__U136(a__isNatKind(N),IL,M,N)
r22: a__U136(tt(),IL,M,N) -> cons(mark(N),take(M,IL))
r23: a__U21(tt(),V1) -> a__U22(a__isNatKind(V1),V1)
r24: a__U22(tt(),V1) -> a__U23(a__isNat(V1))
r25: a__U23(tt()) -> tt()
r26: a__U31(tt(),V) -> a__U32(a__isNatIListKind(V),V)
r27: a__U32(tt(),V) -> a__U33(a__isNatList(V))
r28: a__U33(tt()) -> tt()
r29: a__U41(tt(),V1,V2) -> a__U42(a__isNatKind(V1),V1,V2)
r30: a__U42(tt(),V1,V2) -> a__U43(a__isNatIListKind(V2),V1,V2)
r31: a__U43(tt(),V1,V2) -> a__U44(a__isNatIListKind(V2),V1,V2)
r32: a__U44(tt(),V1,V2) -> a__U45(a__isNat(V1),V2)
r33: a__U45(tt(),V2) -> a__U46(a__isNatIList(V2))
r34: a__U46(tt()) -> tt()
r35: a__U51(tt(),V2) -> a__U52(a__isNatIListKind(V2))
r36: a__U52(tt()) -> tt()
r37: a__U61(tt(),V2) -> a__U62(a__isNatIListKind(V2))
r38: a__U62(tt()) -> tt()
r39: a__U71(tt()) -> tt()
r40: a__U81(tt()) -> tt()
r41: a__U91(tt(),V1,V2) -> a__U92(a__isNatKind(V1),V1,V2)
r42: a__U92(tt(),V1,V2) -> a__U93(a__isNatIListKind(V2),V1,V2)
r43: a__U93(tt(),V1,V2) -> a__U94(a__isNatIListKind(V2),V1,V2)
r44: a__U94(tt(),V1,V2) -> a__U95(a__isNat(V1),V2)
r45: a__U95(tt(),V2) -> a__U96(a__isNatList(V2))
r46: a__U96(tt()) -> tt()
r47: a__isNat(|0|()) -> tt()
r48: a__isNat(length(V1)) -> a__U11(a__isNatIListKind(V1),V1)
r49: a__isNat(s(V1)) -> a__U21(a__isNatKind(V1),V1)
r50: a__isNatIList(V) -> a__U31(a__isNatIListKind(V),V)
r51: a__isNatIList(zeros()) -> tt()
r52: a__isNatIList(cons(V1,V2)) -> a__U41(a__isNatKind(V1),V1,V2)
r53: a__isNatIListKind(nil()) -> tt()
r54: a__isNatIListKind(zeros()) -> tt()
r55: a__isNatIListKind(cons(V1,V2)) -> a__U51(a__isNatKind(V1),V2)
r56: a__isNatIListKind(take(V1,V2)) -> a__U61(a__isNatKind(V1),V2)
r57: a__isNatKind(|0|()) -> tt()
r58: a__isNatKind(length(V1)) -> a__U71(a__isNatIListKind(V1))
r59: a__isNatKind(s(V1)) -> a__U81(a__isNatKind(V1))
r60: a__isNatList(nil()) -> tt()
r61: a__isNatList(cons(V1,V2)) -> a__U91(a__isNatKind(V1),V1,V2)
r62: a__isNatList(take(V1,V2)) -> a__U101(a__isNatKind(V1),V1,V2)
r63: a__length(nil()) -> |0|()
r64: a__length(cons(N,L)) -> a__U111(a__isNatList(L),L,N)
r65: a__take(|0|(),IL) -> a__U121(a__isNatIList(IL),IL)
r66: a__take(s(M),cons(N,IL)) -> a__U131(a__isNatIList(IL),IL,M,N)
r67: mark(zeros()) -> a__zeros()
r68: mark(U101(X1,X2,X3)) -> a__U101(mark(X1),X2,X3)
r69: mark(U102(X1,X2,X3)) -> a__U102(mark(X1),X2,X3)
r70: mark(isNatKind(X)) -> a__isNatKind(X)
r71: mark(U103(X1,X2,X3)) -> a__U103(mark(X1),X2,X3)
r72: mark(isNatIListKind(X)) -> a__isNatIListKind(X)
r73: mark(U104(X1,X2,X3)) -> a__U104(mark(X1),X2,X3)
r74: mark(U105(X1,X2)) -> a__U105(mark(X1),X2)
r75: mark(isNat(X)) -> a__isNat(X)
r76: mark(U106(X)) -> a__U106(mark(X))
r77: mark(isNatIList(X)) -> a__isNatIList(X)
r78: mark(U11(X1,X2)) -> a__U11(mark(X1),X2)
r79: mark(U12(X1,X2)) -> a__U12(mark(X1),X2)
r80: mark(U111(X1,X2,X3)) -> a__U111(mark(X1),X2,X3)
r81: mark(U112(X1,X2,X3)) -> a__U112(mark(X1),X2,X3)
r82: mark(U113(X1,X2,X3)) -> a__U113(mark(X1),X2,X3)
r83: mark(U114(X1,X2)) -> a__U114(mark(X1),X2)
r84: mark(length(X)) -> a__length(mark(X))
r85: mark(U13(X)) -> a__U13(mark(X))
r86: mark(isNatList(X)) -> a__isNatList(X)
r87: mark(U121(X1,X2)) -> a__U121(mark(X1),X2)
r88: mark(U122(X)) -> a__U122(mark(X))
r89: mark(U131(X1,X2,X3,X4)) -> a__U131(mark(X1),X2,X3,X4)
r90: mark(U132(X1,X2,X3,X4)) -> a__U132(mark(X1),X2,X3,X4)
r91: mark(U133(X1,X2,X3,X4)) -> a__U133(mark(X1),X2,X3,X4)
r92: mark(U134(X1,X2,X3,X4)) -> a__U134(mark(X1),X2,X3,X4)
r93: mark(U135(X1,X2,X3,X4)) -> a__U135(mark(X1),X2,X3,X4)
r94: mark(U136(X1,X2,X3,X4)) -> a__U136(mark(X1),X2,X3,X4)
r95: mark(take(X1,X2)) -> a__take(mark(X1),mark(X2))
r96: mark(U21(X1,X2)) -> a__U21(mark(X1),X2)
r97: mark(U22(X1,X2)) -> a__U22(mark(X1),X2)
r98: mark(U23(X)) -> a__U23(mark(X))
r99: mark(U31(X1,X2)) -> a__U31(mark(X1),X2)
r100: mark(U32(X1,X2)) -> a__U32(mark(X1),X2)
r101: mark(U33(X)) -> a__U33(mark(X))
r102: mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3)
r103: mark(U42(X1,X2,X3)) -> a__U42(mark(X1),X2,X3)
r104: mark(U43(X1,X2,X3)) -> a__U43(mark(X1),X2,X3)
r105: mark(U44(X1,X2,X3)) -> a__U44(mark(X1),X2,X3)
r106: mark(U45(X1,X2)) -> a__U45(mark(X1),X2)
r107: mark(U46(X)) -> a__U46(mark(X))
r108: mark(U51(X1,X2)) -> a__U51(mark(X1),X2)
r109: mark(U52(X)) -> a__U52(mark(X))
r110: mark(U61(X1,X2)) -> a__U61(mark(X1),X2)
r111: mark(U62(X)) -> a__U62(mark(X))
r112: mark(U71(X)) -> a__U71(mark(X))
r113: mark(U81(X)) -> a__U81(mark(X))
r114: mark(U91(X1,X2,X3)) -> a__U91(mark(X1),X2,X3)
r115: mark(U92(X1,X2,X3)) -> a__U92(mark(X1),X2,X3)
r116: mark(U93(X1,X2,X3)) -> a__U93(mark(X1),X2,X3)
r117: mark(U94(X1,X2,X3)) -> a__U94(mark(X1),X2,X3)
r118: mark(U95(X1,X2)) -> a__U95(mark(X1),X2)
r119: mark(U96(X)) -> a__U96(mark(X))
r120: mark(cons(X1,X2)) -> cons(mark(X1),X2)
r121: mark(|0|()) -> |0|()
r122: mark(tt()) -> tt()
r123: mark(s(X)) -> s(mark(X))
r124: mark(nil()) -> nil()
r125: a__zeros() -> zeros()
r126: a__U101(X1,X2,X3) -> U101(X1,X2,X3)
r127: a__U102(X1,X2,X3) -> U102(X1,X2,X3)
r128: a__isNatKind(X) -> isNatKind(X)
r129: a__U103(X1,X2,X3) -> U103(X1,X2,X3)
r130: a__isNatIListKind(X) -> isNatIListKind(X)
r131: a__U104(X1,X2,X3) -> U104(X1,X2,X3)
r132: a__U105(X1,X2) -> U105(X1,X2)
r133: a__isNat(X) -> isNat(X)
r134: a__U106(X) -> U106(X)
r135: a__isNatIList(X) -> isNatIList(X)
r136: a__U11(X1,X2) -> U11(X1,X2)
r137: a__U12(X1,X2) -> U12(X1,X2)
r138: a__U111(X1,X2,X3) -> U111(X1,X2,X3)
r139: a__U112(X1,X2,X3) -> U112(X1,X2,X3)
r140: a__U113(X1,X2,X3) -> U113(X1,X2,X3)
r141: a__U114(X1,X2) -> U114(X1,X2)
r142: a__length(X) -> length(X)
r143: a__U13(X) -> U13(X)
r144: a__isNatList(X) -> isNatList(X)
r145: a__U121(X1,X2) -> U121(X1,X2)
r146: a__U122(X) -> U122(X)
r147: a__U131(X1,X2,X3,X4) -> U131(X1,X2,X3,X4)
r148: a__U132(X1,X2,X3,X4) -> U132(X1,X2,X3,X4)
r149: a__U133(X1,X2,X3,X4) -> U133(X1,X2,X3,X4)
r150: a__U134(X1,X2,X3,X4) -> U134(X1,X2,X3,X4)
r151: a__U135(X1,X2,X3,X4) -> U135(X1,X2,X3,X4)
r152: a__U136(X1,X2,X3,X4) -> U136(X1,X2,X3,X4)
r153: a__take(X1,X2) -> take(X1,X2)
r154: a__U21(X1,X2) -> U21(X1,X2)
r155: a__U22(X1,X2) -> U22(X1,X2)
r156: a__U23(X) -> U23(X)
r157: a__U31(X1,X2) -> U31(X1,X2)
r158: a__U32(X1,X2) -> U32(X1,X2)
r159: a__U33(X) -> U33(X)
r160: a__U41(X1,X2,X3) -> U41(X1,X2,X3)
r161: a__U42(X1,X2,X3) -> U42(X1,X2,X3)
r162: a__U43(X1,X2,X3) -> U43(X1,X2,X3)
r163: a__U44(X1,X2,X3) -> U44(X1,X2,X3)
r164: a__U45(X1,X2) -> U45(X1,X2)
r165: a__U46(X) -> U46(X)
r166: a__U51(X1,X2) -> U51(X1,X2)
r167: a__U52(X) -> U52(X)
r168: a__U61(X1,X2) -> U61(X1,X2)
r169: a__U62(X) -> U62(X)
r170: a__U71(X) -> U71(X)
r171: a__U81(X) -> U81(X)
r172: a__U91(X1,X2,X3) -> U91(X1,X2,X3)
r173: a__U92(X1,X2,X3) -> U92(X1,X2,X3)
r174: a__U93(X1,X2,X3) -> U93(X1,X2,X3)
r175: a__U94(X1,X2,X3) -> U94(X1,X2,X3)
r176: a__U95(X1,X2) -> U95(X1,X2)
r177: a__U96(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}


-- Reduction pair.

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

p1: mark#(U11(X1,X2)) -> mark#(X1)
p2: mark#(U81(X)) -> mark#(X)
p3: mark#(U62(X)) -> mark#(X)
p4: mark#(U46(X)) -> mark#(X)
p5: mark#(U42(X1,X2,X3)) -> mark#(X1)
p6: mark#(U33(X)) -> mark#(X)
p7: mark#(U31(X1,X2)) -> mark#(X1)
p8: mark#(U21(X1,X2)) -> mark#(X1)
p9: mark#(U134(X1,X2,X3,X4)) -> mark#(X1)
p10: mark#(U131(X1,X2,X3,X4)) -> mark#(X1)
p11: mark#(U122(X)) -> mark#(X)
p12: mark#(U121(X1,X2)) -> mark#(X1)
p13: mark#(U111(X1,X2,X3)) -> mark#(X1)
p14: mark#(U12(X1,X2)) -> mark#(X1)

and R consists of:

r1: a__zeros() -> cons(|0|(),zeros())
r2: a__U101(tt(),V1,V2) -> a__U102(a__isNatKind(V1),V1,V2)
r3: a__U102(tt(),V1,V2) -> a__U103(a__isNatIListKind(V2),V1,V2)
r4: a__U103(tt(),V1,V2) -> a__U104(a__isNatIListKind(V2),V1,V2)
r5: a__U104(tt(),V1,V2) -> a__U105(a__isNat(V1),V2)
r6: a__U105(tt(),V2) -> a__U106(a__isNatIList(V2))
r7: a__U106(tt()) -> tt()
r8: a__U11(tt(),V1) -> a__U12(a__isNatIListKind(V1),V1)
r9: a__U111(tt(),L,N) -> a__U112(a__isNatIListKind(L),L,N)
r10: a__U112(tt(),L,N) -> a__U113(a__isNat(N),L,N)
r11: a__U113(tt(),L,N) -> a__U114(a__isNatKind(N),L)
r12: a__U114(tt(),L) -> s(a__length(mark(L)))
r13: a__U12(tt(),V1) -> a__U13(a__isNatList(V1))
r14: a__U121(tt(),IL) -> a__U122(a__isNatIListKind(IL))
r15: a__U122(tt()) -> nil()
r16: a__U13(tt()) -> tt()
r17: a__U131(tt(),IL,M,N) -> a__U132(a__isNatIListKind(IL),IL,M,N)
r18: a__U132(tt(),IL,M,N) -> a__U133(a__isNat(M),IL,M,N)
r19: a__U133(tt(),IL,M,N) -> a__U134(a__isNatKind(M),IL,M,N)
r20: a__U134(tt(),IL,M,N) -> a__U135(a__isNat(N),IL,M,N)
r21: a__U135(tt(),IL,M,N) -> a__U136(a__isNatKind(N),IL,M,N)
r22: a__U136(tt(),IL,M,N) -> cons(mark(N),take(M,IL))
r23: a__U21(tt(),V1) -> a__U22(a__isNatKind(V1),V1)
r24: a__U22(tt(),V1) -> a__U23(a__isNat(V1))
r25: a__U23(tt()) -> tt()
r26: a__U31(tt(),V) -> a__U32(a__isNatIListKind(V),V)
r27: a__U32(tt(),V) -> a__U33(a__isNatList(V))
r28: a__U33(tt()) -> tt()
r29: a__U41(tt(),V1,V2) -> a__U42(a__isNatKind(V1),V1,V2)
r30: a__U42(tt(),V1,V2) -> a__U43(a__isNatIListKind(V2),V1,V2)
r31: a__U43(tt(),V1,V2) -> a__U44(a__isNatIListKind(V2),V1,V2)
r32: a__U44(tt(),V1,V2) -> a__U45(a__isNat(V1),V2)
r33: a__U45(tt(),V2) -> a__U46(a__isNatIList(V2))
r34: a__U46(tt()) -> tt()
r35: a__U51(tt(),V2) -> a__U52(a__isNatIListKind(V2))
r36: a__U52(tt()) -> tt()
r37: a__U61(tt(),V2) -> a__U62(a__isNatIListKind(V2))
r38: a__U62(tt()) -> tt()
r39: a__U71(tt()) -> tt()
r40: a__U81(tt()) -> tt()
r41: a__U91(tt(),V1,V2) -> a__U92(a__isNatKind(V1),V1,V2)
r42: a__U92(tt(),V1,V2) -> a__U93(a__isNatIListKind(V2),V1,V2)
r43: a__U93(tt(),V1,V2) -> a__U94(a__isNatIListKind(V2),V1,V2)
r44: a__U94(tt(),V1,V2) -> a__U95(a__isNat(V1),V2)
r45: a__U95(tt(),V2) -> a__U96(a__isNatList(V2))
r46: a__U96(tt()) -> tt()
r47: a__isNat(|0|()) -> tt()
r48: a__isNat(length(V1)) -> a__U11(a__isNatIListKind(V1),V1)
r49: a__isNat(s(V1)) -> a__U21(a__isNatKind(V1),V1)
r50: a__isNatIList(V) -> a__U31(a__isNatIListKind(V),V)
r51: a__isNatIList(zeros()) -> tt()
r52: a__isNatIList(cons(V1,V2)) -> a__U41(a__isNatKind(V1),V1,V2)
r53: a__isNatIListKind(nil()) -> tt()
r54: a__isNatIListKind(zeros()) -> tt()
r55: a__isNatIListKind(cons(V1,V2)) -> a__U51(a__isNatKind(V1),V2)
r56: a__isNatIListKind(take(V1,V2)) -> a__U61(a__isNatKind(V1),V2)
r57: a__isNatKind(|0|()) -> tt()
r58: a__isNatKind(length(V1)) -> a__U71(a__isNatIListKind(V1))
r59: a__isNatKind(s(V1)) -> a__U81(a__isNatKind(V1))
r60: a__isNatList(nil()) -> tt()
r61: a__isNatList(cons(V1,V2)) -> a__U91(a__isNatKind(V1),V1,V2)
r62: a__isNatList(take(V1,V2)) -> a__U101(a__isNatKind(V1),V1,V2)
r63: a__length(nil()) -> |0|()
r64: a__length(cons(N,L)) -> a__U111(a__isNatList(L),L,N)
r65: a__take(|0|(),IL) -> a__U121(a__isNatIList(IL),IL)
r66: a__take(s(M),cons(N,IL)) -> a__U131(a__isNatIList(IL),IL,M,N)
r67: mark(zeros()) -> a__zeros()
r68: mark(U101(X1,X2,X3)) -> a__U101(mark(X1),X2,X3)
r69: mark(U102(X1,X2,X3)) -> a__U102(mark(X1),X2,X3)
r70: mark(isNatKind(X)) -> a__isNatKind(X)
r71: mark(U103(X1,X2,X3)) -> a__U103(mark(X1),X2,X3)
r72: mark(isNatIListKind(X)) -> a__isNatIListKind(X)
r73: mark(U104(X1,X2,X3)) -> a__U104(mark(X1),X2,X3)
r74: mark(U105(X1,X2)) -> a__U105(mark(X1),X2)
r75: mark(isNat(X)) -> a__isNat(X)
r76: mark(U106(X)) -> a__U106(mark(X))
r77: mark(isNatIList(X)) -> a__isNatIList(X)
r78: mark(U11(X1,X2)) -> a__U11(mark(X1),X2)
r79: mark(U12(X1,X2)) -> a__U12(mark(X1),X2)
r80: mark(U111(X1,X2,X3)) -> a__U111(mark(X1),X2,X3)
r81: mark(U112(X1,X2,X3)) -> a__U112(mark(X1),X2,X3)
r82: mark(U113(X1,X2,X3)) -> a__U113(mark(X1),X2,X3)
r83: mark(U114(X1,X2)) -> a__U114(mark(X1),X2)
r84: mark(length(X)) -> a__length(mark(X))
r85: mark(U13(X)) -> a__U13(mark(X))
r86: mark(isNatList(X)) -> a__isNatList(X)
r87: mark(U121(X1,X2)) -> a__U121(mark(X1),X2)
r88: mark(U122(X)) -> a__U122(mark(X))
r89: mark(U131(X1,X2,X3,X4)) -> a__U131(mark(X1),X2,X3,X4)
r90: mark(U132(X1,X2,X3,X4)) -> a__U132(mark(X1),X2,X3,X4)
r91: mark(U133(X1,X2,X3,X4)) -> a__U133(mark(X1),X2,X3,X4)
r92: mark(U134(X1,X2,X3,X4)) -> a__U134(mark(X1),X2,X3,X4)
r93: mark(U135(X1,X2,X3,X4)) -> a__U135(mark(X1),X2,X3,X4)
r94: mark(U136(X1,X2,X3,X4)) -> a__U136(mark(X1),X2,X3,X4)
r95: mark(take(X1,X2)) -> a__take(mark(X1),mark(X2))
r96: mark(U21(X1,X2)) -> a__U21(mark(X1),X2)
r97: mark(U22(X1,X2)) -> a__U22(mark(X1),X2)
r98: mark(U23(X)) -> a__U23(mark(X))
r99: mark(U31(X1,X2)) -> a__U31(mark(X1),X2)
r100: mark(U32(X1,X2)) -> a__U32(mark(X1),X2)
r101: mark(U33(X)) -> a__U33(mark(X))
r102: mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3)
r103: mark(U42(X1,X2,X3)) -> a__U42(mark(X1),X2,X3)
r104: mark(U43(X1,X2,X3)) -> a__U43(mark(X1),X2,X3)
r105: mark(U44(X1,X2,X3)) -> a__U44(mark(X1),X2,X3)
r106: mark(U45(X1,X2)) -> a__U45(mark(X1),X2)
r107: mark(U46(X)) -> a__U46(mark(X))
r108: mark(U51(X1,X2)) -> a__U51(mark(X1),X2)
r109: mark(U52(X)) -> a__U52(mark(X))
r110: mark(U61(X1,X2)) -> a__U61(mark(X1),X2)
r111: mark(U62(X)) -> a__U62(mark(X))
r112: mark(U71(X)) -> a__U71(mark(X))
r113: mark(U81(X)) -> a__U81(mark(X))
r114: mark(U91(X1,X2,X3)) -> a__U91(mark(X1),X2,X3)
r115: mark(U92(X1,X2,X3)) -> a__U92(mark(X1),X2,X3)
r116: mark(U93(X1,X2,X3)) -> a__U93(mark(X1),X2,X3)
r117: mark(U94(X1,X2,X3)) -> a__U94(mark(X1),X2,X3)
r118: mark(U95(X1,X2)) -> a__U95(mark(X1),X2)
r119: mark(U96(X)) -> a__U96(mark(X))
r120: mark(cons(X1,X2)) -> cons(mark(X1),X2)
r121: mark(|0|()) -> |0|()
r122: mark(tt()) -> tt()
r123: mark(s(X)) -> s(mark(X))
r124: mark(nil()) -> nil()
r125: a__zeros() -> zeros()
r126: a__U101(X1,X2,X3) -> U101(X1,X2,X3)
r127: a__U102(X1,X2,X3) -> U102(X1,X2,X3)
r128: a__isNatKind(X) -> isNatKind(X)
r129: a__U103(X1,X2,X3) -> U103(X1,X2,X3)
r130: a__isNatIListKind(X) -> isNatIListKind(X)
r131: a__U104(X1,X2,X3) -> U104(X1,X2,X3)
r132: a__U105(X1,X2) -> U105(X1,X2)
r133: a__isNat(X) -> isNat(X)
r134: a__U106(X) -> U106(X)
r135: a__isNatIList(X) -> isNatIList(X)
r136: a__U11(X1,X2) -> U11(X1,X2)
r137: a__U12(X1,X2) -> U12(X1,X2)
r138: a__U111(X1,X2,X3) -> U111(X1,X2,X3)
r139: a__U112(X1,X2,X3) -> U112(X1,X2,X3)
r140: a__U113(X1,X2,X3) -> U113(X1,X2,X3)
r141: a__U114(X1,X2) -> U114(X1,X2)
r142: a__length(X) -> length(X)
r143: a__U13(X) -> U13(X)
r144: a__isNatList(X) -> isNatList(X)
r145: a__U121(X1,X2) -> U121(X1,X2)
r146: a__U122(X) -> U122(X)
r147: a__U131(X1,X2,X3,X4) -> U131(X1,X2,X3,X4)
r148: a__U132(X1,X2,X3,X4) -> U132(X1,X2,X3,X4)
r149: a__U133(X1,X2,X3,X4) -> U133(X1,X2,X3,X4)
r150: a__U134(X1,X2,X3,X4) -> U134(X1,X2,X3,X4)
r151: a__U135(X1,X2,X3,X4) -> U135(X1,X2,X3,X4)
r152: a__U136(X1,X2,X3,X4) -> U136(X1,X2,X3,X4)
r153: a__take(X1,X2) -> take(X1,X2)
r154: a__U21(X1,X2) -> U21(X1,X2)
r155: a__U22(X1,X2) -> U22(X1,X2)
r156: a__U23(X) -> U23(X)
r157: a__U31(X1,X2) -> U31(X1,X2)
r158: a__U32(X1,X2) -> U32(X1,X2)
r159: a__U33(X) -> U33(X)
r160: a__U41(X1,X2,X3) -> U41(X1,X2,X3)
r161: a__U42(X1,X2,X3) -> U42(X1,X2,X3)
r162: a__U43(X1,X2,X3) -> U43(X1,X2,X3)
r163: a__U44(X1,X2,X3) -> U44(X1,X2,X3)
r164: a__U45(X1,X2) -> U45(X1,X2)
r165: a__U46(X) -> U46(X)
r166: a__U51(X1,X2) -> U51(X1,X2)
r167: a__U52(X) -> U52(X)
r168: a__U61(X1,X2) -> U61(X1,X2)
r169: a__U62(X) -> U62(X)
r170: a__U71(X) -> U71(X)
r171: a__U81(X) -> U81(X)
r172: a__U91(X1,X2,X3) -> U91(X1,X2,X3)
r173: a__U92(X1,X2,X3) -> U92(X1,X2,X3)
r174: a__U93(X1,X2,X3) -> U93(X1,X2,X3)
r175: a__U94(X1,X2,X3) -> U94(X1,X2,X3)
r176: a__U95(X1,X2) -> U95(X1,X2)
r177: a__U96(X) -> U96(X)

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  max/plus interpretations on natural numbers:
  
    mark#_A(x1) = max{1, x1}
    U11_A(x1,x2) = max{x1, x2 + 2}
    U81_A(x1) = x1 + 1
    U62_A(x1) = x1 + 1
    U46_A(x1) = x1 + 1
    U42_A(x1,x2,x3) = max{x1 + 2, x2 + 2, x3 + 2}
    U33_A(x1) = x1
    U31_A(x1,x2) = max{x1, x2 + 2}
    U21_A(x1,x2) = max{x1, x2 + 1}
    U134_A(x1,x2,x3,x4) = max{x1, x2 + 2, x3 + 2, x4 + 2}
    U131_A(x1,x2,x3,x4) = max{x1, x3 + 1}
    U122_A(x1) = x1
    U121_A(x1,x2) = max{x1 + 2, x2 + 2}
    U111_A(x1,x2,x3) = max{x1, x2 + 1, x3 + 1}
    U12_A(x1,x2) = max{x1, x2 + 2}

The next rules are strictly ordered:

  p5, p12

We remove them from the problem.

-- SCC decomposition.

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

p1: mark#(U11(X1,X2)) -> mark#(X1)
p2: mark#(U81(X)) -> mark#(X)
p3: mark#(U62(X)) -> mark#(X)
p4: mark#(U46(X)) -> mark#(X)
p5: mark#(U33(X)) -> mark#(X)
p6: mark#(U31(X1,X2)) -> mark#(X1)
p7: mark#(U21(X1,X2)) -> mark#(X1)
p8: mark#(U134(X1,X2,X3,X4)) -> mark#(X1)
p9: mark#(U131(X1,X2,X3,X4)) -> mark#(X1)
p10: mark#(U122(X)) -> mark#(X)
p11: mark#(U111(X1,X2,X3)) -> mark#(X1)
p12: mark#(U12(X1,X2)) -> mark#(X1)

and R consists of:

r1: a__zeros() -> cons(|0|(),zeros())
r2: a__U101(tt(),V1,V2) -> a__U102(a__isNatKind(V1),V1,V2)
r3: a__U102(tt(),V1,V2) -> a__U103(a__isNatIListKind(V2),V1,V2)
r4: a__U103(tt(),V1,V2) -> a__U104(a__isNatIListKind(V2),V1,V2)
r5: a__U104(tt(),V1,V2) -> a__U105(a__isNat(V1),V2)
r6: a__U105(tt(),V2) -> a__U106(a__isNatIList(V2))
r7: a__U106(tt()) -> tt()
r8: a__U11(tt(),V1) -> a__U12(a__isNatIListKind(V1),V1)
r9: a__U111(tt(),L,N) -> a__U112(a__isNatIListKind(L),L,N)
r10: a__U112(tt(),L,N) -> a__U113(a__isNat(N),L,N)
r11: a__U113(tt(),L,N) -> a__U114(a__isNatKind(N),L)
r12: a__U114(tt(),L) -> s(a__length(mark(L)))
r13: a__U12(tt(),V1) -> a__U13(a__isNatList(V1))
r14: a__U121(tt(),IL) -> a__U122(a__isNatIListKind(IL))
r15: a__U122(tt()) -> nil()
r16: a__U13(tt()) -> tt()
r17: a__U131(tt(),IL,M,N) -> a__U132(a__isNatIListKind(IL),IL,M,N)
r18: a__U132(tt(),IL,M,N) -> a__U133(a__isNat(M),IL,M,N)
r19: a__U133(tt(),IL,M,N) -> a__U134(a__isNatKind(M),IL,M,N)
r20: a__U134(tt(),IL,M,N) -> a__U135(a__isNat(N),IL,M,N)
r21: a__U135(tt(),IL,M,N) -> a__U136(a__isNatKind(N),IL,M,N)
r22: a__U136(tt(),IL,M,N) -> cons(mark(N),take(M,IL))
r23: a__U21(tt(),V1) -> a__U22(a__isNatKind(V1),V1)
r24: a__U22(tt(),V1) -> a__U23(a__isNat(V1))
r25: a__U23(tt()) -> tt()
r26: a__U31(tt(),V) -> a__U32(a__isNatIListKind(V),V)
r27: a__U32(tt(),V) -> a__U33(a__isNatList(V))
r28: a__U33(tt()) -> tt()
r29: a__U41(tt(),V1,V2) -> a__U42(a__isNatKind(V1),V1,V2)
r30: a__U42(tt(),V1,V2) -> a__U43(a__isNatIListKind(V2),V1,V2)
r31: a__U43(tt(),V1,V2) -> a__U44(a__isNatIListKind(V2),V1,V2)
r32: a__U44(tt(),V1,V2) -> a__U45(a__isNat(V1),V2)
r33: a__U45(tt(),V2) -> a__U46(a__isNatIList(V2))
r34: a__U46(tt()) -> tt()
r35: a__U51(tt(),V2) -> a__U52(a__isNatIListKind(V2))
r36: a__U52(tt()) -> tt()
r37: a__U61(tt(),V2) -> a__U62(a__isNatIListKind(V2))
r38: a__U62(tt()) -> tt()
r39: a__U71(tt()) -> tt()
r40: a__U81(tt()) -> tt()
r41: a__U91(tt(),V1,V2) -> a__U92(a__isNatKind(V1),V1,V2)
r42: a__U92(tt(),V1,V2) -> a__U93(a__isNatIListKind(V2),V1,V2)
r43: a__U93(tt(),V1,V2) -> a__U94(a__isNatIListKind(V2),V1,V2)
r44: a__U94(tt(),V1,V2) -> a__U95(a__isNat(V1),V2)
r45: a__U95(tt(),V2) -> a__U96(a__isNatList(V2))
r46: a__U96(tt()) -> tt()
r47: a__isNat(|0|()) -> tt()
r48: a__isNat(length(V1)) -> a__U11(a__isNatIListKind(V1),V1)
r49: a__isNat(s(V1)) -> a__U21(a__isNatKind(V1),V1)
r50: a__isNatIList(V) -> a__U31(a__isNatIListKind(V),V)
r51: a__isNatIList(zeros()) -> tt()
r52: a__isNatIList(cons(V1,V2)) -> a__U41(a__isNatKind(V1),V1,V2)
r53: a__isNatIListKind(nil()) -> tt()
r54: a__isNatIListKind(zeros()) -> tt()
r55: a__isNatIListKind(cons(V1,V2)) -> a__U51(a__isNatKind(V1),V2)
r56: a__isNatIListKind(take(V1,V2)) -> a__U61(a__isNatKind(V1),V2)
r57: a__isNatKind(|0|()) -> tt()
r58: a__isNatKind(length(V1)) -> a__U71(a__isNatIListKind(V1))
r59: a__isNatKind(s(V1)) -> a__U81(a__isNatKind(V1))
r60: a__isNatList(nil()) -> tt()
r61: a__isNatList(cons(V1,V2)) -> a__U91(a__isNatKind(V1),V1,V2)
r62: a__isNatList(take(V1,V2)) -> a__U101(a__isNatKind(V1),V1,V2)
r63: a__length(nil()) -> |0|()
r64: a__length(cons(N,L)) -> a__U111(a__isNatList(L),L,N)
r65: a__take(|0|(),IL) -> a__U121(a__isNatIList(IL),IL)
r66: a__take(s(M),cons(N,IL)) -> a__U131(a__isNatIList(IL),IL,M,N)
r67: mark(zeros()) -> a__zeros()
r68: mark(U101(X1,X2,X3)) -> a__U101(mark(X1),X2,X3)
r69: mark(U102(X1,X2,X3)) -> a__U102(mark(X1),X2,X3)
r70: mark(isNatKind(X)) -> a__isNatKind(X)
r71: mark(U103(X1,X2,X3)) -> a__U103(mark(X1),X2,X3)
r72: mark(isNatIListKind(X)) -> a__isNatIListKind(X)
r73: mark(U104(X1,X2,X3)) -> a__U104(mark(X1),X2,X3)
r74: mark(U105(X1,X2)) -> a__U105(mark(X1),X2)
r75: mark(isNat(X)) -> a__isNat(X)
r76: mark(U106(X)) -> a__U106(mark(X))
r77: mark(isNatIList(X)) -> a__isNatIList(X)
r78: mark(U11(X1,X2)) -> a__U11(mark(X1),X2)
r79: mark(U12(X1,X2)) -> a__U12(mark(X1),X2)
r80: mark(U111(X1,X2,X3)) -> a__U111(mark(X1),X2,X3)
r81: mark(U112(X1,X2,X3)) -> a__U112(mark(X1),X2,X3)
r82: mark(U113(X1,X2,X3)) -> a__U113(mark(X1),X2,X3)
r83: mark(U114(X1,X2)) -> a__U114(mark(X1),X2)
r84: mark(length(X)) -> a__length(mark(X))
r85: mark(U13(X)) -> a__U13(mark(X))
r86: mark(isNatList(X)) -> a__isNatList(X)
r87: mark(U121(X1,X2)) -> a__U121(mark(X1),X2)
r88: mark(U122(X)) -> a__U122(mark(X))
r89: mark(U131(X1,X2,X3,X4)) -> a__U131(mark(X1),X2,X3,X4)
r90: mark(U132(X1,X2,X3,X4)) -> a__U132(mark(X1),X2,X3,X4)
r91: mark(U133(X1,X2,X3,X4)) -> a__U133(mark(X1),X2,X3,X4)
r92: mark(U134(X1,X2,X3,X4)) -> a__U134(mark(X1),X2,X3,X4)
r93: mark(U135(X1,X2,X3,X4)) -> a__U135(mark(X1),X2,X3,X4)
r94: mark(U136(X1,X2,X3,X4)) -> a__U136(mark(X1),X2,X3,X4)
r95: mark(take(X1,X2)) -> a__take(mark(X1),mark(X2))
r96: mark(U21(X1,X2)) -> a__U21(mark(X1),X2)
r97: mark(U22(X1,X2)) -> a__U22(mark(X1),X2)
r98: mark(U23(X)) -> a__U23(mark(X))
r99: mark(U31(X1,X2)) -> a__U31(mark(X1),X2)
r100: mark(U32(X1,X2)) -> a__U32(mark(X1),X2)
r101: mark(U33(X)) -> a__U33(mark(X))
r102: mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3)
r103: mark(U42(X1,X2,X3)) -> a__U42(mark(X1),X2,X3)
r104: mark(U43(X1,X2,X3)) -> a__U43(mark(X1),X2,X3)
r105: mark(U44(X1,X2,X3)) -> a__U44(mark(X1),X2,X3)
r106: mark(U45(X1,X2)) -> a__U45(mark(X1),X2)
r107: mark(U46(X)) -> a__U46(mark(X))
r108: mark(U51(X1,X2)) -> a__U51(mark(X1),X2)
r109: mark(U52(X)) -> a__U52(mark(X))
r110: mark(U61(X1,X2)) -> a__U61(mark(X1),X2)
r111: mark(U62(X)) -> a__U62(mark(X))
r112: mark(U71(X)) -> a__U71(mark(X))
r113: mark(U81(X)) -> a__U81(mark(X))
r114: mark(U91(X1,X2,X3)) -> a__U91(mark(X1),X2,X3)
r115: mark(U92(X1,X2,X3)) -> a__U92(mark(X1),X2,X3)
r116: mark(U93(X1,X2,X3)) -> a__U93(mark(X1),X2,X3)
r117: mark(U94(X1,X2,X3)) -> a__U94(mark(X1),X2,X3)
r118: mark(U95(X1,X2)) -> a__U95(mark(X1),X2)
r119: mark(U96(X)) -> a__U96(mark(X))
r120: mark(cons(X1,X2)) -> cons(mark(X1),X2)
r121: mark(|0|()) -> |0|()
r122: mark(tt()) -> tt()
r123: mark(s(X)) -> s(mark(X))
r124: mark(nil()) -> nil()
r125: a__zeros() -> zeros()
r126: a__U101(X1,X2,X3) -> U101(X1,X2,X3)
r127: a__U102(X1,X2,X3) -> U102(X1,X2,X3)
r128: a__isNatKind(X) -> isNatKind(X)
r129: a__U103(X1,X2,X3) -> U103(X1,X2,X3)
r130: a__isNatIListKind(X) -> isNatIListKind(X)
r131: a__U104(X1,X2,X3) -> U104(X1,X2,X3)
r132: a__U105(X1,X2) -> U105(X1,X2)
r133: a__isNat(X) -> isNat(X)
r134: a__U106(X) -> U106(X)
r135: a__isNatIList(X) -> isNatIList(X)
r136: a__U11(X1,X2) -> U11(X1,X2)
r137: a__U12(X1,X2) -> U12(X1,X2)
r138: a__U111(X1,X2,X3) -> U111(X1,X2,X3)
r139: a__U112(X1,X2,X3) -> U112(X1,X2,X3)
r140: a__U113(X1,X2,X3) -> U113(X1,X2,X3)
r141: a__U114(X1,X2) -> U114(X1,X2)
r142: a__length(X) -> length(X)
r143: a__U13(X) -> U13(X)
r144: a__isNatList(X) -> isNatList(X)
r145: a__U121(X1,X2) -> U121(X1,X2)
r146: a__U122(X) -> U122(X)
r147: a__U131(X1,X2,X3,X4) -> U131(X1,X2,X3,X4)
r148: a__U132(X1,X2,X3,X4) -> U132(X1,X2,X3,X4)
r149: a__U133(X1,X2,X3,X4) -> U133(X1,X2,X3,X4)
r150: a__U134(X1,X2,X3,X4) -> U134(X1,X2,X3,X4)
r151: a__U135(X1,X2,X3,X4) -> U135(X1,X2,X3,X4)
r152: a__U136(X1,X2,X3,X4) -> U136(X1,X2,X3,X4)
r153: a__take(X1,X2) -> take(X1,X2)
r154: a__U21(X1,X2) -> U21(X1,X2)
r155: a__U22(X1,X2) -> U22(X1,X2)
r156: a__U23(X) -> U23(X)
r157: a__U31(X1,X2) -> U31(X1,X2)
r158: a__U32(X1,X2) -> U32(X1,X2)
r159: a__U33(X) -> U33(X)
r160: a__U41(X1,X2,X3) -> U41(X1,X2,X3)
r161: a__U42(X1,X2,X3) -> U42(X1,X2,X3)
r162: a__U43(X1,X2,X3) -> U43(X1,X2,X3)
r163: a__U44(X1,X2,X3) -> U44(X1,X2,X3)
r164: a__U45(X1,X2) -> U45(X1,X2)
r165: a__U46(X) -> U46(X)
r166: a__U51(X1,X2) -> U51(X1,X2)
r167: a__U52(X) -> U52(X)
r168: a__U61(X1,X2) -> U61(X1,X2)
r169: a__U62(X) -> U62(X)
r170: a__U71(X) -> U71(X)
r171: a__U81(X) -> U81(X)
r172: a__U91(X1,X2,X3) -> U91(X1,X2,X3)
r173: a__U92(X1,X2,X3) -> U92(X1,X2,X3)
r174: a__U93(X1,X2,X3) -> U93(X1,X2,X3)
r175: a__U94(X1,X2,X3) -> U94(X1,X2,X3)
r176: a__U95(X1,X2) -> U95(X1,X2)
r177: a__U96(X) -> U96(X)

The estimated dependency graph contains the following SCCs:

  {p1, p2, p3, p4, p5, p6, p7, p8, p9, p10, p11, p12}


-- Reduction pair.

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

p1: mark#(U11(X1,X2)) -> mark#(X1)
p2: mark#(U12(X1,X2)) -> mark#(X1)
p3: mark#(U111(X1,X2,X3)) -> mark#(X1)
p4: mark#(U122(X)) -> mark#(X)
p5: mark#(U131(X1,X2,X3,X4)) -> mark#(X1)
p6: mark#(U134(X1,X2,X3,X4)) -> mark#(X1)
p7: mark#(U21(X1,X2)) -> mark#(X1)
p8: mark#(U31(X1,X2)) -> mark#(X1)
p9: mark#(U33(X)) -> mark#(X)
p10: mark#(U46(X)) -> mark#(X)
p11: mark#(U62(X)) -> mark#(X)
p12: mark#(U81(X)) -> mark#(X)

and R consists of:

r1: a__zeros() -> cons(|0|(),zeros())
r2: a__U101(tt(),V1,V2) -> a__U102(a__isNatKind(V1),V1,V2)
r3: a__U102(tt(),V1,V2) -> a__U103(a__isNatIListKind(V2),V1,V2)
r4: a__U103(tt(),V1,V2) -> a__U104(a__isNatIListKind(V2),V1,V2)
r5: a__U104(tt(),V1,V2) -> a__U105(a__isNat(V1),V2)
r6: a__U105(tt(),V2) -> a__U106(a__isNatIList(V2))
r7: a__U106(tt()) -> tt()
r8: a__U11(tt(),V1) -> a__U12(a__isNatIListKind(V1),V1)
r9: a__U111(tt(),L,N) -> a__U112(a__isNatIListKind(L),L,N)
r10: a__U112(tt(),L,N) -> a__U113(a__isNat(N),L,N)
r11: a__U113(tt(),L,N) -> a__U114(a__isNatKind(N),L)
r12: a__U114(tt(),L) -> s(a__length(mark(L)))
r13: a__U12(tt(),V1) -> a__U13(a__isNatList(V1))
r14: a__U121(tt(),IL) -> a__U122(a__isNatIListKind(IL))
r15: a__U122(tt()) -> nil()
r16: a__U13(tt()) -> tt()
r17: a__U131(tt(),IL,M,N) -> a__U132(a__isNatIListKind(IL),IL,M,N)
r18: a__U132(tt(),IL,M,N) -> a__U133(a__isNat(M),IL,M,N)
r19: a__U133(tt(),IL,M,N) -> a__U134(a__isNatKind(M),IL,M,N)
r20: a__U134(tt(),IL,M,N) -> a__U135(a__isNat(N),IL,M,N)
r21: a__U135(tt(),IL,M,N) -> a__U136(a__isNatKind(N),IL,M,N)
r22: a__U136(tt(),IL,M,N) -> cons(mark(N),take(M,IL))
r23: a__U21(tt(),V1) -> a__U22(a__isNatKind(V1),V1)
r24: a__U22(tt(),V1) -> a__U23(a__isNat(V1))
r25: a__U23(tt()) -> tt()
r26: a__U31(tt(),V) -> a__U32(a__isNatIListKind(V),V)
r27: a__U32(tt(),V) -> a__U33(a__isNatList(V))
r28: a__U33(tt()) -> tt()
r29: a__U41(tt(),V1,V2) -> a__U42(a__isNatKind(V1),V1,V2)
r30: a__U42(tt(),V1,V2) -> a__U43(a__isNatIListKind(V2),V1,V2)
r31: a__U43(tt(),V1,V2) -> a__U44(a__isNatIListKind(V2),V1,V2)
r32: a__U44(tt(),V1,V2) -> a__U45(a__isNat(V1),V2)
r33: a__U45(tt(),V2) -> a__U46(a__isNatIList(V2))
r34: a__U46(tt()) -> tt()
r35: a__U51(tt(),V2) -> a__U52(a__isNatIListKind(V2))
r36: a__U52(tt()) -> tt()
r37: a__U61(tt(),V2) -> a__U62(a__isNatIListKind(V2))
r38: a__U62(tt()) -> tt()
r39: a__U71(tt()) -> tt()
r40: a__U81(tt()) -> tt()
r41: a__U91(tt(),V1,V2) -> a__U92(a__isNatKind(V1),V1,V2)
r42: a__U92(tt(),V1,V2) -> a__U93(a__isNatIListKind(V2),V1,V2)
r43: a__U93(tt(),V1,V2) -> a__U94(a__isNatIListKind(V2),V1,V2)
r44: a__U94(tt(),V1,V2) -> a__U95(a__isNat(V1),V2)
r45: a__U95(tt(),V2) -> a__U96(a__isNatList(V2))
r46: a__U96(tt()) -> tt()
r47: a__isNat(|0|()) -> tt()
r48: a__isNat(length(V1)) -> a__U11(a__isNatIListKind(V1),V1)
r49: a__isNat(s(V1)) -> a__U21(a__isNatKind(V1),V1)
r50: a__isNatIList(V) -> a__U31(a__isNatIListKind(V),V)
r51: a__isNatIList(zeros()) -> tt()
r52: a__isNatIList(cons(V1,V2)) -> a__U41(a__isNatKind(V1),V1,V2)
r53: a__isNatIListKind(nil()) -> tt()
r54: a__isNatIListKind(zeros()) -> tt()
r55: a__isNatIListKind(cons(V1,V2)) -> a__U51(a__isNatKind(V1),V2)
r56: a__isNatIListKind(take(V1,V2)) -> a__U61(a__isNatKind(V1),V2)
r57: a__isNatKind(|0|()) -> tt()
r58: a__isNatKind(length(V1)) -> a__U71(a__isNatIListKind(V1))
r59: a__isNatKind(s(V1)) -> a__U81(a__isNatKind(V1))
r60: a__isNatList(nil()) -> tt()
r61: a__isNatList(cons(V1,V2)) -> a__U91(a__isNatKind(V1),V1,V2)
r62: a__isNatList(take(V1,V2)) -> a__U101(a__isNatKind(V1),V1,V2)
r63: a__length(nil()) -> |0|()
r64: a__length(cons(N,L)) -> a__U111(a__isNatList(L),L,N)
r65: a__take(|0|(),IL) -> a__U121(a__isNatIList(IL),IL)
r66: a__take(s(M),cons(N,IL)) -> a__U131(a__isNatIList(IL),IL,M,N)
r67: mark(zeros()) -> a__zeros()
r68: mark(U101(X1,X2,X3)) -> a__U101(mark(X1),X2,X3)
r69: mark(U102(X1,X2,X3)) -> a__U102(mark(X1),X2,X3)
r70: mark(isNatKind(X)) -> a__isNatKind(X)
r71: mark(U103(X1,X2,X3)) -> a__U103(mark(X1),X2,X3)
r72: mark(isNatIListKind(X)) -> a__isNatIListKind(X)
r73: mark(U104(X1,X2,X3)) -> a__U104(mark(X1),X2,X3)
r74: mark(U105(X1,X2)) -> a__U105(mark(X1),X2)
r75: mark(isNat(X)) -> a__isNat(X)
r76: mark(U106(X)) -> a__U106(mark(X))
r77: mark(isNatIList(X)) -> a__isNatIList(X)
r78: mark(U11(X1,X2)) -> a__U11(mark(X1),X2)
r79: mark(U12(X1,X2)) -> a__U12(mark(X1),X2)
r80: mark(U111(X1,X2,X3)) -> a__U111(mark(X1),X2,X3)
r81: mark(U112(X1,X2,X3)) -> a__U112(mark(X1),X2,X3)
r82: mark(U113(X1,X2,X3)) -> a__U113(mark(X1),X2,X3)
r83: mark(U114(X1,X2)) -> a__U114(mark(X1),X2)
r84: mark(length(X)) -> a__length(mark(X))
r85: mark(U13(X)) -> a__U13(mark(X))
r86: mark(isNatList(X)) -> a__isNatList(X)
r87: mark(U121(X1,X2)) -> a__U121(mark(X1),X2)
r88: mark(U122(X)) -> a__U122(mark(X))
r89: mark(U131(X1,X2,X3,X4)) -> a__U131(mark(X1),X2,X3,X4)
r90: mark(U132(X1,X2,X3,X4)) -> a__U132(mark(X1),X2,X3,X4)
r91: mark(U133(X1,X2,X3,X4)) -> a__U133(mark(X1),X2,X3,X4)
r92: mark(U134(X1,X2,X3,X4)) -> a__U134(mark(X1),X2,X3,X4)
r93: mark(U135(X1,X2,X3,X4)) -> a__U135(mark(X1),X2,X3,X4)
r94: mark(U136(X1,X2,X3,X4)) -> a__U136(mark(X1),X2,X3,X4)
r95: mark(take(X1,X2)) -> a__take(mark(X1),mark(X2))
r96: mark(U21(X1,X2)) -> a__U21(mark(X1),X2)
r97: mark(U22(X1,X2)) -> a__U22(mark(X1),X2)
r98: mark(U23(X)) -> a__U23(mark(X))
r99: mark(U31(X1,X2)) -> a__U31(mark(X1),X2)
r100: mark(U32(X1,X2)) -> a__U32(mark(X1),X2)
r101: mark(U33(X)) -> a__U33(mark(X))
r102: mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3)
r103: mark(U42(X1,X2,X3)) -> a__U42(mark(X1),X2,X3)
r104: mark(U43(X1,X2,X3)) -> a__U43(mark(X1),X2,X3)
r105: mark(U44(X1,X2,X3)) -> a__U44(mark(X1),X2,X3)
r106: mark(U45(X1,X2)) -> a__U45(mark(X1),X2)
r107: mark(U46(X)) -> a__U46(mark(X))
r108: mark(U51(X1,X2)) -> a__U51(mark(X1),X2)
r109: mark(U52(X)) -> a__U52(mark(X))
r110: mark(U61(X1,X2)) -> a__U61(mark(X1),X2)
r111: mark(U62(X)) -> a__U62(mark(X))
r112: mark(U71(X)) -> a__U71(mark(X))
r113: mark(U81(X)) -> a__U81(mark(X))
r114: mark(U91(X1,X2,X3)) -> a__U91(mark(X1),X2,X3)
r115: mark(U92(X1,X2,X3)) -> a__U92(mark(X1),X2,X3)
r116: mark(U93(X1,X2,X3)) -> a__U93(mark(X1),X2,X3)
r117: mark(U94(X1,X2,X3)) -> a__U94(mark(X1),X2,X3)
r118: mark(U95(X1,X2)) -> a__U95(mark(X1),X2)
r119: mark(U96(X)) -> a__U96(mark(X))
r120: mark(cons(X1,X2)) -> cons(mark(X1),X2)
r121: mark(|0|()) -> |0|()
r122: mark(tt()) -> tt()
r123: mark(s(X)) -> s(mark(X))
r124: mark(nil()) -> nil()
r125: a__zeros() -> zeros()
r126: a__U101(X1,X2,X3) -> U101(X1,X2,X3)
r127: a__U102(X1,X2,X3) -> U102(X1,X2,X3)
r128: a__isNatKind(X) -> isNatKind(X)
r129: a__U103(X1,X2,X3) -> U103(X1,X2,X3)
r130: a__isNatIListKind(X) -> isNatIListKind(X)
r131: a__U104(X1,X2,X3) -> U104(X1,X2,X3)
r132: a__U105(X1,X2) -> U105(X1,X2)
r133: a__isNat(X) -> isNat(X)
r134: a__U106(X) -> U106(X)
r135: a__isNatIList(X) -> isNatIList(X)
r136: a__U11(X1,X2) -> U11(X1,X2)
r137: a__U12(X1,X2) -> U12(X1,X2)
r138: a__U111(X1,X2,X3) -> U111(X1,X2,X3)
r139: a__U112(X1,X2,X3) -> U112(X1,X2,X3)
r140: a__U113(X1,X2,X3) -> U113(X1,X2,X3)
r141: a__U114(X1,X2) -> U114(X1,X2)
r142: a__length(X) -> length(X)
r143: a__U13(X) -> U13(X)
r144: a__isNatList(X) -> isNatList(X)
r145: a__U121(X1,X2) -> U121(X1,X2)
r146: a__U122(X) -> U122(X)
r147: a__U131(X1,X2,X3,X4) -> U131(X1,X2,X3,X4)
r148: a__U132(X1,X2,X3,X4) -> U132(X1,X2,X3,X4)
r149: a__U133(X1,X2,X3,X4) -> U133(X1,X2,X3,X4)
r150: a__U134(X1,X2,X3,X4) -> U134(X1,X2,X3,X4)
r151: a__U135(X1,X2,X3,X4) -> U135(X1,X2,X3,X4)
r152: a__U136(X1,X2,X3,X4) -> U136(X1,X2,X3,X4)
r153: a__take(X1,X2) -> take(X1,X2)
r154: a__U21(X1,X2) -> U21(X1,X2)
r155: a__U22(X1,X2) -> U22(X1,X2)
r156: a__U23(X) -> U23(X)
r157: a__U31(X1,X2) -> U31(X1,X2)
r158: a__U32(X1,X2) -> U32(X1,X2)
r159: a__U33(X) -> U33(X)
r160: a__U41(X1,X2,X3) -> U41(X1,X2,X3)
r161: a__U42(X1,X2,X3) -> U42(X1,X2,X3)
r162: a__U43(X1,X2,X3) -> U43(X1,X2,X3)
r163: a__U44(X1,X2,X3) -> U44(X1,X2,X3)
r164: a__U45(X1,X2) -> U45(X1,X2)
r165: a__U46(X) -> U46(X)
r166: a__U51(X1,X2) -> U51(X1,X2)
r167: a__U52(X) -> U52(X)
r168: a__U61(X1,X2) -> U61(X1,X2)
r169: a__U62(X) -> U62(X)
r170: a__U71(X) -> U71(X)
r171: a__U81(X) -> U81(X)
r172: a__U91(X1,X2,X3) -> U91(X1,X2,X3)
r173: a__U92(X1,X2,X3) -> U92(X1,X2,X3)
r174: a__U93(X1,X2,X3) -> U93(X1,X2,X3)
r175: a__U94(X1,X2,X3) -> U94(X1,X2,X3)
r176: a__U95(X1,X2) -> U95(X1,X2)
r177: a__U96(X) -> U96(X)

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  max/plus interpretations on natural numbers:
  
    mark#_A(x1) = max{0, x1 - 1}
    U11_A(x1,x2) = max{x1, x2 + 2}
    U12_A(x1,x2) = x1
    U111_A(x1,x2,x3) = max{x1 + 1, x2 + 1}
    U122_A(x1) = x1
    U131_A(x1,x2,x3,x4) = max{x1, x2 + 2, x3 + 2, x4 + 2}
    U134_A(x1,x2,x3,x4) = max{x1, x3 + 1}
    U21_A(x1,x2) = max{x1 + 2, x2 + 2}
    U31_A(x1,x2) = max{x1, x2 + 1}
    U33_A(x1) = x1
    U46_A(x1) = x1 + 1
    U62_A(x1) = x1 + 1
    U81_A(x1) = x1 + 1

The next rules are strictly ordered:

  p7

We remove them from the problem.

-- SCC decomposition.

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

p1: mark#(U11(X1,X2)) -> mark#(X1)
p2: mark#(U12(X1,X2)) -> mark#(X1)
p3: mark#(U111(X1,X2,X3)) -> mark#(X1)
p4: mark#(U122(X)) -> mark#(X)
p5: mark#(U131(X1,X2,X3,X4)) -> mark#(X1)
p6: mark#(U134(X1,X2,X3,X4)) -> mark#(X1)
p7: mark#(U31(X1,X2)) -> mark#(X1)
p8: mark#(U33(X)) -> mark#(X)
p9: mark#(U46(X)) -> mark#(X)
p10: mark#(U62(X)) -> mark#(X)
p11: mark#(U81(X)) -> mark#(X)

and R consists of:

r1: a__zeros() -> cons(|0|(),zeros())
r2: a__U101(tt(),V1,V2) -> a__U102(a__isNatKind(V1),V1,V2)
r3: a__U102(tt(),V1,V2) -> a__U103(a__isNatIListKind(V2),V1,V2)
r4: a__U103(tt(),V1,V2) -> a__U104(a__isNatIListKind(V2),V1,V2)
r5: a__U104(tt(),V1,V2) -> a__U105(a__isNat(V1),V2)
r6: a__U105(tt(),V2) -> a__U106(a__isNatIList(V2))
r7: a__U106(tt()) -> tt()
r8: a__U11(tt(),V1) -> a__U12(a__isNatIListKind(V1),V1)
r9: a__U111(tt(),L,N) -> a__U112(a__isNatIListKind(L),L,N)
r10: a__U112(tt(),L,N) -> a__U113(a__isNat(N),L,N)
r11: a__U113(tt(),L,N) -> a__U114(a__isNatKind(N),L)
r12: a__U114(tt(),L) -> s(a__length(mark(L)))
r13: a__U12(tt(),V1) -> a__U13(a__isNatList(V1))
r14: a__U121(tt(),IL) -> a__U122(a__isNatIListKind(IL))
r15: a__U122(tt()) -> nil()
r16: a__U13(tt()) -> tt()
r17: a__U131(tt(),IL,M,N) -> a__U132(a__isNatIListKind(IL),IL,M,N)
r18: a__U132(tt(),IL,M,N) -> a__U133(a__isNat(M),IL,M,N)
r19: a__U133(tt(),IL,M,N) -> a__U134(a__isNatKind(M),IL,M,N)
r20: a__U134(tt(),IL,M,N) -> a__U135(a__isNat(N),IL,M,N)
r21: a__U135(tt(),IL,M,N) -> a__U136(a__isNatKind(N),IL,M,N)
r22: a__U136(tt(),IL,M,N) -> cons(mark(N),take(M,IL))
r23: a__U21(tt(),V1) -> a__U22(a__isNatKind(V1),V1)
r24: a__U22(tt(),V1) -> a__U23(a__isNat(V1))
r25: a__U23(tt()) -> tt()
r26: a__U31(tt(),V) -> a__U32(a__isNatIListKind(V),V)
r27: a__U32(tt(),V) -> a__U33(a__isNatList(V))
r28: a__U33(tt()) -> tt()
r29: a__U41(tt(),V1,V2) -> a__U42(a__isNatKind(V1),V1,V2)
r30: a__U42(tt(),V1,V2) -> a__U43(a__isNatIListKind(V2),V1,V2)
r31: a__U43(tt(),V1,V2) -> a__U44(a__isNatIListKind(V2),V1,V2)
r32: a__U44(tt(),V1,V2) -> a__U45(a__isNat(V1),V2)
r33: a__U45(tt(),V2) -> a__U46(a__isNatIList(V2))
r34: a__U46(tt()) -> tt()
r35: a__U51(tt(),V2) -> a__U52(a__isNatIListKind(V2))
r36: a__U52(tt()) -> tt()
r37: a__U61(tt(),V2) -> a__U62(a__isNatIListKind(V2))
r38: a__U62(tt()) -> tt()
r39: a__U71(tt()) -> tt()
r40: a__U81(tt()) -> tt()
r41: a__U91(tt(),V1,V2) -> a__U92(a__isNatKind(V1),V1,V2)
r42: a__U92(tt(),V1,V2) -> a__U93(a__isNatIListKind(V2),V1,V2)
r43: a__U93(tt(),V1,V2) -> a__U94(a__isNatIListKind(V2),V1,V2)
r44: a__U94(tt(),V1,V2) -> a__U95(a__isNat(V1),V2)
r45: a__U95(tt(),V2) -> a__U96(a__isNatList(V2))
r46: a__U96(tt()) -> tt()
r47: a__isNat(|0|()) -> tt()
r48: a__isNat(length(V1)) -> a__U11(a__isNatIListKind(V1),V1)
r49: a__isNat(s(V1)) -> a__U21(a__isNatKind(V1),V1)
r50: a__isNatIList(V) -> a__U31(a__isNatIListKind(V),V)
r51: a__isNatIList(zeros()) -> tt()
r52: a__isNatIList(cons(V1,V2)) -> a__U41(a__isNatKind(V1),V1,V2)
r53: a__isNatIListKind(nil()) -> tt()
r54: a__isNatIListKind(zeros()) -> tt()
r55: a__isNatIListKind(cons(V1,V2)) -> a__U51(a__isNatKind(V1),V2)
r56: a__isNatIListKind(take(V1,V2)) -> a__U61(a__isNatKind(V1),V2)
r57: a__isNatKind(|0|()) -> tt()
r58: a__isNatKind(length(V1)) -> a__U71(a__isNatIListKind(V1))
r59: a__isNatKind(s(V1)) -> a__U81(a__isNatKind(V1))
r60: a__isNatList(nil()) -> tt()
r61: a__isNatList(cons(V1,V2)) -> a__U91(a__isNatKind(V1),V1,V2)
r62: a__isNatList(take(V1,V2)) -> a__U101(a__isNatKind(V1),V1,V2)
r63: a__length(nil()) -> |0|()
r64: a__length(cons(N,L)) -> a__U111(a__isNatList(L),L,N)
r65: a__take(|0|(),IL) -> a__U121(a__isNatIList(IL),IL)
r66: a__take(s(M),cons(N,IL)) -> a__U131(a__isNatIList(IL),IL,M,N)
r67: mark(zeros()) -> a__zeros()
r68: mark(U101(X1,X2,X3)) -> a__U101(mark(X1),X2,X3)
r69: mark(U102(X1,X2,X3)) -> a__U102(mark(X1),X2,X3)
r70: mark(isNatKind(X)) -> a__isNatKind(X)
r71: mark(U103(X1,X2,X3)) -> a__U103(mark(X1),X2,X3)
r72: mark(isNatIListKind(X)) -> a__isNatIListKind(X)
r73: mark(U104(X1,X2,X3)) -> a__U104(mark(X1),X2,X3)
r74: mark(U105(X1,X2)) -> a__U105(mark(X1),X2)
r75: mark(isNat(X)) -> a__isNat(X)
r76: mark(U106(X)) -> a__U106(mark(X))
r77: mark(isNatIList(X)) -> a__isNatIList(X)
r78: mark(U11(X1,X2)) -> a__U11(mark(X1),X2)
r79: mark(U12(X1,X2)) -> a__U12(mark(X1),X2)
r80: mark(U111(X1,X2,X3)) -> a__U111(mark(X1),X2,X3)
r81: mark(U112(X1,X2,X3)) -> a__U112(mark(X1),X2,X3)
r82: mark(U113(X1,X2,X3)) -> a__U113(mark(X1),X2,X3)
r83: mark(U114(X1,X2)) -> a__U114(mark(X1),X2)
r84: mark(length(X)) -> a__length(mark(X))
r85: mark(U13(X)) -> a__U13(mark(X))
r86: mark(isNatList(X)) -> a__isNatList(X)
r87: mark(U121(X1,X2)) -> a__U121(mark(X1),X2)
r88: mark(U122(X)) -> a__U122(mark(X))
r89: mark(U131(X1,X2,X3,X4)) -> a__U131(mark(X1),X2,X3,X4)
r90: mark(U132(X1,X2,X3,X4)) -> a__U132(mark(X1),X2,X3,X4)
r91: mark(U133(X1,X2,X3,X4)) -> a__U133(mark(X1),X2,X3,X4)
r92: mark(U134(X1,X2,X3,X4)) -> a__U134(mark(X1),X2,X3,X4)
r93: mark(U135(X1,X2,X3,X4)) -> a__U135(mark(X1),X2,X3,X4)
r94: mark(U136(X1,X2,X3,X4)) -> a__U136(mark(X1),X2,X3,X4)
r95: mark(take(X1,X2)) -> a__take(mark(X1),mark(X2))
r96: mark(U21(X1,X2)) -> a__U21(mark(X1),X2)
r97: mark(U22(X1,X2)) -> a__U22(mark(X1),X2)
r98: mark(U23(X)) -> a__U23(mark(X))
r99: mark(U31(X1,X2)) -> a__U31(mark(X1),X2)
r100: mark(U32(X1,X2)) -> a__U32(mark(X1),X2)
r101: mark(U33(X)) -> a__U33(mark(X))
r102: mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3)
r103: mark(U42(X1,X2,X3)) -> a__U42(mark(X1),X2,X3)
r104: mark(U43(X1,X2,X3)) -> a__U43(mark(X1),X2,X3)
r105: mark(U44(X1,X2,X3)) -> a__U44(mark(X1),X2,X3)
r106: mark(U45(X1,X2)) -> a__U45(mark(X1),X2)
r107: mark(U46(X)) -> a__U46(mark(X))
r108: mark(U51(X1,X2)) -> a__U51(mark(X1),X2)
r109: mark(U52(X)) -> a__U52(mark(X))
r110: mark(U61(X1,X2)) -> a__U61(mark(X1),X2)
r111: mark(U62(X)) -> a__U62(mark(X))
r112: mark(U71(X)) -> a__U71(mark(X))
r113: mark(U81(X)) -> a__U81(mark(X))
r114: mark(U91(X1,X2,X3)) -> a__U91(mark(X1),X2,X3)
r115: mark(U92(X1,X2,X3)) -> a__U92(mark(X1),X2,X3)
r116: mark(U93(X1,X2,X3)) -> a__U93(mark(X1),X2,X3)
r117: mark(U94(X1,X2,X3)) -> a__U94(mark(X1),X2,X3)
r118: mark(U95(X1,X2)) -> a__U95(mark(X1),X2)
r119: mark(U96(X)) -> a__U96(mark(X))
r120: mark(cons(X1,X2)) -> cons(mark(X1),X2)
r121: mark(|0|()) -> |0|()
r122: mark(tt()) -> tt()
r123: mark(s(X)) -> s(mark(X))
r124: mark(nil()) -> nil()
r125: a__zeros() -> zeros()
r126: a__U101(X1,X2,X3) -> U101(X1,X2,X3)
r127: a__U102(X1,X2,X3) -> U102(X1,X2,X3)
r128: a__isNatKind(X) -> isNatKind(X)
r129: a__U103(X1,X2,X3) -> U103(X1,X2,X3)
r130: a__isNatIListKind(X) -> isNatIListKind(X)
r131: a__U104(X1,X2,X3) -> U104(X1,X2,X3)
r132: a__U105(X1,X2) -> U105(X1,X2)
r133: a__isNat(X) -> isNat(X)
r134: a__U106(X) -> U106(X)
r135: a__isNatIList(X) -> isNatIList(X)
r136: a__U11(X1,X2) -> U11(X1,X2)
r137: a__U12(X1,X2) -> U12(X1,X2)
r138: a__U111(X1,X2,X3) -> U111(X1,X2,X3)
r139: a__U112(X1,X2,X3) -> U112(X1,X2,X3)
r140: a__U113(X1,X2,X3) -> U113(X1,X2,X3)
r141: a__U114(X1,X2) -> U114(X1,X2)
r142: a__length(X) -> length(X)
r143: a__U13(X) -> U13(X)
r144: a__isNatList(X) -> isNatList(X)
r145: a__U121(X1,X2) -> U121(X1,X2)
r146: a__U122(X) -> U122(X)
r147: a__U131(X1,X2,X3,X4) -> U131(X1,X2,X3,X4)
r148: a__U132(X1,X2,X3,X4) -> U132(X1,X2,X3,X4)
r149: a__U133(X1,X2,X3,X4) -> U133(X1,X2,X3,X4)
r150: a__U134(X1,X2,X3,X4) -> U134(X1,X2,X3,X4)
r151: a__U135(X1,X2,X3,X4) -> U135(X1,X2,X3,X4)
r152: a__U136(X1,X2,X3,X4) -> U136(X1,X2,X3,X4)
r153: a__take(X1,X2) -> take(X1,X2)
r154: a__U21(X1,X2) -> U21(X1,X2)
r155: a__U22(X1,X2) -> U22(X1,X2)
r156: a__U23(X) -> U23(X)
r157: a__U31(X1,X2) -> U31(X1,X2)
r158: a__U32(X1,X2) -> U32(X1,X2)
r159: a__U33(X) -> U33(X)
r160: a__U41(X1,X2,X3) -> U41(X1,X2,X3)
r161: a__U42(X1,X2,X3) -> U42(X1,X2,X3)
r162: a__U43(X1,X2,X3) -> U43(X1,X2,X3)
r163: a__U44(X1,X2,X3) -> U44(X1,X2,X3)
r164: a__U45(X1,X2) -> U45(X1,X2)
r165: a__U46(X) -> U46(X)
r166: a__U51(X1,X2) -> U51(X1,X2)
r167: a__U52(X) -> U52(X)
r168: a__U61(X1,X2) -> U61(X1,X2)
r169: a__U62(X) -> U62(X)
r170: a__U71(X) -> U71(X)
r171: a__U81(X) -> U81(X)
r172: a__U91(X1,X2,X3) -> U91(X1,X2,X3)
r173: a__U92(X1,X2,X3) -> U92(X1,X2,X3)
r174: a__U93(X1,X2,X3) -> U93(X1,X2,X3)
r175: a__U94(X1,X2,X3) -> U94(X1,X2,X3)
r176: a__U95(X1,X2) -> U95(X1,X2)
r177: a__U96(X) -> U96(X)

The estimated dependency graph contains the following SCCs:

  {p1, p2, p3, p4, p5, p6, p7, p8, p9, p10, p11}


-- Reduction pair.

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

p1: mark#(U11(X1,X2)) -> mark#(X1)
p2: mark#(U81(X)) -> mark#(X)
p3: mark#(U62(X)) -> mark#(X)
p4: mark#(U46(X)) -> mark#(X)
p5: mark#(U33(X)) -> mark#(X)
p6: mark#(U31(X1,X2)) -> mark#(X1)
p7: mark#(U134(X1,X2,X3,X4)) -> mark#(X1)
p8: mark#(U131(X1,X2,X3,X4)) -> mark#(X1)
p9: mark#(U122(X)) -> mark#(X)
p10: mark#(U111(X1,X2,X3)) -> mark#(X1)
p11: mark#(U12(X1,X2)) -> mark#(X1)

and R consists of:

r1: a__zeros() -> cons(|0|(),zeros())
r2: a__U101(tt(),V1,V2) -> a__U102(a__isNatKind(V1),V1,V2)
r3: a__U102(tt(),V1,V2) -> a__U103(a__isNatIListKind(V2),V1,V2)
r4: a__U103(tt(),V1,V2) -> a__U104(a__isNatIListKind(V2),V1,V2)
r5: a__U104(tt(),V1,V2) -> a__U105(a__isNat(V1),V2)
r6: a__U105(tt(),V2) -> a__U106(a__isNatIList(V2))
r7: a__U106(tt()) -> tt()
r8: a__U11(tt(),V1) -> a__U12(a__isNatIListKind(V1),V1)
r9: a__U111(tt(),L,N) -> a__U112(a__isNatIListKind(L),L,N)
r10: a__U112(tt(),L,N) -> a__U113(a__isNat(N),L,N)
r11: a__U113(tt(),L,N) -> a__U114(a__isNatKind(N),L)
r12: a__U114(tt(),L) -> s(a__length(mark(L)))
r13: a__U12(tt(),V1) -> a__U13(a__isNatList(V1))
r14: a__U121(tt(),IL) -> a__U122(a__isNatIListKind(IL))
r15: a__U122(tt()) -> nil()
r16: a__U13(tt()) -> tt()
r17: a__U131(tt(),IL,M,N) -> a__U132(a__isNatIListKind(IL),IL,M,N)
r18: a__U132(tt(),IL,M,N) -> a__U133(a__isNat(M),IL,M,N)
r19: a__U133(tt(),IL,M,N) -> a__U134(a__isNatKind(M),IL,M,N)
r20: a__U134(tt(),IL,M,N) -> a__U135(a__isNat(N),IL,M,N)
r21: a__U135(tt(),IL,M,N) -> a__U136(a__isNatKind(N),IL,M,N)
r22: a__U136(tt(),IL,M,N) -> cons(mark(N),take(M,IL))
r23: a__U21(tt(),V1) -> a__U22(a__isNatKind(V1),V1)
r24: a__U22(tt(),V1) -> a__U23(a__isNat(V1))
r25: a__U23(tt()) -> tt()
r26: a__U31(tt(),V) -> a__U32(a__isNatIListKind(V),V)
r27: a__U32(tt(),V) -> a__U33(a__isNatList(V))
r28: a__U33(tt()) -> tt()
r29: a__U41(tt(),V1,V2) -> a__U42(a__isNatKind(V1),V1,V2)
r30: a__U42(tt(),V1,V2) -> a__U43(a__isNatIListKind(V2),V1,V2)
r31: a__U43(tt(),V1,V2) -> a__U44(a__isNatIListKind(V2),V1,V2)
r32: a__U44(tt(),V1,V2) -> a__U45(a__isNat(V1),V2)
r33: a__U45(tt(),V2) -> a__U46(a__isNatIList(V2))
r34: a__U46(tt()) -> tt()
r35: a__U51(tt(),V2) -> a__U52(a__isNatIListKind(V2))
r36: a__U52(tt()) -> tt()
r37: a__U61(tt(),V2) -> a__U62(a__isNatIListKind(V2))
r38: a__U62(tt()) -> tt()
r39: a__U71(tt()) -> tt()
r40: a__U81(tt()) -> tt()
r41: a__U91(tt(),V1,V2) -> a__U92(a__isNatKind(V1),V1,V2)
r42: a__U92(tt(),V1,V2) -> a__U93(a__isNatIListKind(V2),V1,V2)
r43: a__U93(tt(),V1,V2) -> a__U94(a__isNatIListKind(V2),V1,V2)
r44: a__U94(tt(),V1,V2) -> a__U95(a__isNat(V1),V2)
r45: a__U95(tt(),V2) -> a__U96(a__isNatList(V2))
r46: a__U96(tt()) -> tt()
r47: a__isNat(|0|()) -> tt()
r48: a__isNat(length(V1)) -> a__U11(a__isNatIListKind(V1),V1)
r49: a__isNat(s(V1)) -> a__U21(a__isNatKind(V1),V1)
r50: a__isNatIList(V) -> a__U31(a__isNatIListKind(V),V)
r51: a__isNatIList(zeros()) -> tt()
r52: a__isNatIList(cons(V1,V2)) -> a__U41(a__isNatKind(V1),V1,V2)
r53: a__isNatIListKind(nil()) -> tt()
r54: a__isNatIListKind(zeros()) -> tt()
r55: a__isNatIListKind(cons(V1,V2)) -> a__U51(a__isNatKind(V1),V2)
r56: a__isNatIListKind(take(V1,V2)) -> a__U61(a__isNatKind(V1),V2)
r57: a__isNatKind(|0|()) -> tt()
r58: a__isNatKind(length(V1)) -> a__U71(a__isNatIListKind(V1))
r59: a__isNatKind(s(V1)) -> a__U81(a__isNatKind(V1))
r60: a__isNatList(nil()) -> tt()
r61: a__isNatList(cons(V1,V2)) -> a__U91(a__isNatKind(V1),V1,V2)
r62: a__isNatList(take(V1,V2)) -> a__U101(a__isNatKind(V1),V1,V2)
r63: a__length(nil()) -> |0|()
r64: a__length(cons(N,L)) -> a__U111(a__isNatList(L),L,N)
r65: a__take(|0|(),IL) -> a__U121(a__isNatIList(IL),IL)
r66: a__take(s(M),cons(N,IL)) -> a__U131(a__isNatIList(IL),IL,M,N)
r67: mark(zeros()) -> a__zeros()
r68: mark(U101(X1,X2,X3)) -> a__U101(mark(X1),X2,X3)
r69: mark(U102(X1,X2,X3)) -> a__U102(mark(X1),X2,X3)
r70: mark(isNatKind(X)) -> a__isNatKind(X)
r71: mark(U103(X1,X2,X3)) -> a__U103(mark(X1),X2,X3)
r72: mark(isNatIListKind(X)) -> a__isNatIListKind(X)
r73: mark(U104(X1,X2,X3)) -> a__U104(mark(X1),X2,X3)
r74: mark(U105(X1,X2)) -> a__U105(mark(X1),X2)
r75: mark(isNat(X)) -> a__isNat(X)
r76: mark(U106(X)) -> a__U106(mark(X))
r77: mark(isNatIList(X)) -> a__isNatIList(X)
r78: mark(U11(X1,X2)) -> a__U11(mark(X1),X2)
r79: mark(U12(X1,X2)) -> a__U12(mark(X1),X2)
r80: mark(U111(X1,X2,X3)) -> a__U111(mark(X1),X2,X3)
r81: mark(U112(X1,X2,X3)) -> a__U112(mark(X1),X2,X3)
r82: mark(U113(X1,X2,X3)) -> a__U113(mark(X1),X2,X3)
r83: mark(U114(X1,X2)) -> a__U114(mark(X1),X2)
r84: mark(length(X)) -> a__length(mark(X))
r85: mark(U13(X)) -> a__U13(mark(X))
r86: mark(isNatList(X)) -> a__isNatList(X)
r87: mark(U121(X1,X2)) -> a__U121(mark(X1),X2)
r88: mark(U122(X)) -> a__U122(mark(X))
r89: mark(U131(X1,X2,X3,X4)) -> a__U131(mark(X1),X2,X3,X4)
r90: mark(U132(X1,X2,X3,X4)) -> a__U132(mark(X1),X2,X3,X4)
r91: mark(U133(X1,X2,X3,X4)) -> a__U133(mark(X1),X2,X3,X4)
r92: mark(U134(X1,X2,X3,X4)) -> a__U134(mark(X1),X2,X3,X4)
r93: mark(U135(X1,X2,X3,X4)) -> a__U135(mark(X1),X2,X3,X4)
r94: mark(U136(X1,X2,X3,X4)) -> a__U136(mark(X1),X2,X3,X4)
r95: mark(take(X1,X2)) -> a__take(mark(X1),mark(X2))
r96: mark(U21(X1,X2)) -> a__U21(mark(X1),X2)
r97: mark(U22(X1,X2)) -> a__U22(mark(X1),X2)
r98: mark(U23(X)) -> a__U23(mark(X))
r99: mark(U31(X1,X2)) -> a__U31(mark(X1),X2)
r100: mark(U32(X1,X2)) -> a__U32(mark(X1),X2)
r101: mark(U33(X)) -> a__U33(mark(X))
r102: mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3)
r103: mark(U42(X1,X2,X3)) -> a__U42(mark(X1),X2,X3)
r104: mark(U43(X1,X2,X3)) -> a__U43(mark(X1),X2,X3)
r105: mark(U44(X1,X2,X3)) -> a__U44(mark(X1),X2,X3)
r106: mark(U45(X1,X2)) -> a__U45(mark(X1),X2)
r107: mark(U46(X)) -> a__U46(mark(X))
r108: mark(U51(X1,X2)) -> a__U51(mark(X1),X2)
r109: mark(U52(X)) -> a__U52(mark(X))
r110: mark(U61(X1,X2)) -> a__U61(mark(X1),X2)
r111: mark(U62(X)) -> a__U62(mark(X))
r112: mark(U71(X)) -> a__U71(mark(X))
r113: mark(U81(X)) -> a__U81(mark(X))
r114: mark(U91(X1,X2,X3)) -> a__U91(mark(X1),X2,X3)
r115: mark(U92(X1,X2,X3)) -> a__U92(mark(X1),X2,X3)
r116: mark(U93(X1,X2,X3)) -> a__U93(mark(X1),X2,X3)
r117: mark(U94(X1,X2,X3)) -> a__U94(mark(X1),X2,X3)
r118: mark(U95(X1,X2)) -> a__U95(mark(X1),X2)
r119: mark(U96(X)) -> a__U96(mark(X))
r120: mark(cons(X1,X2)) -> cons(mark(X1),X2)
r121: mark(|0|()) -> |0|()
r122: mark(tt()) -> tt()
r123: mark(s(X)) -> s(mark(X))
r124: mark(nil()) -> nil()
r125: a__zeros() -> zeros()
r126: a__U101(X1,X2,X3) -> U101(X1,X2,X3)
r127: a__U102(X1,X2,X3) -> U102(X1,X2,X3)
r128: a__isNatKind(X) -> isNatKind(X)
r129: a__U103(X1,X2,X3) -> U103(X1,X2,X3)
r130: a__isNatIListKind(X) -> isNatIListKind(X)
r131: a__U104(X1,X2,X3) -> U104(X1,X2,X3)
r132: a__U105(X1,X2) -> U105(X1,X2)
r133: a__isNat(X) -> isNat(X)
r134: a__U106(X) -> U106(X)
r135: a__isNatIList(X) -> isNatIList(X)
r136: a__U11(X1,X2) -> U11(X1,X2)
r137: a__U12(X1,X2) -> U12(X1,X2)
r138: a__U111(X1,X2,X3) -> U111(X1,X2,X3)
r139: a__U112(X1,X2,X3) -> U112(X1,X2,X3)
r140: a__U113(X1,X2,X3) -> U113(X1,X2,X3)
r141: a__U114(X1,X2) -> U114(X1,X2)
r142: a__length(X) -> length(X)
r143: a__U13(X) -> U13(X)
r144: a__isNatList(X) -> isNatList(X)
r145: a__U121(X1,X2) -> U121(X1,X2)
r146: a__U122(X) -> U122(X)
r147: a__U131(X1,X2,X3,X4) -> U131(X1,X2,X3,X4)
r148: a__U132(X1,X2,X3,X4) -> U132(X1,X2,X3,X4)
r149: a__U133(X1,X2,X3,X4) -> U133(X1,X2,X3,X4)
r150: a__U134(X1,X2,X3,X4) -> U134(X1,X2,X3,X4)
r151: a__U135(X1,X2,X3,X4) -> U135(X1,X2,X3,X4)
r152: a__U136(X1,X2,X3,X4) -> U136(X1,X2,X3,X4)
r153: a__take(X1,X2) -> take(X1,X2)
r154: a__U21(X1,X2) -> U21(X1,X2)
r155: a__U22(X1,X2) -> U22(X1,X2)
r156: a__U23(X) -> U23(X)
r157: a__U31(X1,X2) -> U31(X1,X2)
r158: a__U32(X1,X2) -> U32(X1,X2)
r159: a__U33(X) -> U33(X)
r160: a__U41(X1,X2,X3) -> U41(X1,X2,X3)
r161: a__U42(X1,X2,X3) -> U42(X1,X2,X3)
r162: a__U43(X1,X2,X3) -> U43(X1,X2,X3)
r163: a__U44(X1,X2,X3) -> U44(X1,X2,X3)
r164: a__U45(X1,X2) -> U45(X1,X2)
r165: a__U46(X) -> U46(X)
r166: a__U51(X1,X2) -> U51(X1,X2)
r167: a__U52(X) -> U52(X)
r168: a__U61(X1,X2) -> U61(X1,X2)
r169: a__U62(X) -> U62(X)
r170: a__U71(X) -> U71(X)
r171: a__U81(X) -> U81(X)
r172: a__U91(X1,X2,X3) -> U91(X1,X2,X3)
r173: a__U92(X1,X2,X3) -> U92(X1,X2,X3)
r174: a__U93(X1,X2,X3) -> U93(X1,X2,X3)
r175: a__U94(X1,X2,X3) -> U94(X1,X2,X3)
r176: a__U95(X1,X2) -> U95(X1,X2)
r177: a__U96(X) -> U96(X)

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  max/plus interpretations on natural numbers:
  
    mark#_A(x1) = x1 + 1
    U11_A(x1,x2) = max{x1, x2}
    U81_A(x1) = x1
    U62_A(x1) = x1
    U46_A(x1) = x1 + 1
    U33_A(x1) = x1
    U31_A(x1,x2) = max{x1, x2}
    U134_A(x1,x2,x3,x4) = max{x1, x2, x3, x4}
    U131_A(x1,x2,x3,x4) = max{x1, x2, x3, x4}
    U122_A(x1) = x1
    U111_A(x1,x2,x3) = max{x1, x2, x3}
    U12_A(x1,x2) = max{x1, x2}

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: mark#(U11(X1,X2)) -> mark#(X1)
p2: mark#(U81(X)) -> mark#(X)
p3: mark#(U62(X)) -> mark#(X)
p4: mark#(U33(X)) -> mark#(X)
p5: mark#(U31(X1,X2)) -> mark#(X1)
p6: mark#(U134(X1,X2,X3,X4)) -> mark#(X1)
p7: mark#(U131(X1,X2,X3,X4)) -> mark#(X1)
p8: mark#(U122(X)) -> mark#(X)
p9: mark#(U111(X1,X2,X3)) -> mark#(X1)
p10: mark#(U12(X1,X2)) -> mark#(X1)

and R consists of:

r1: a__zeros() -> cons(|0|(),zeros())
r2: a__U101(tt(),V1,V2) -> a__U102(a__isNatKind(V1),V1,V2)
r3: a__U102(tt(),V1,V2) -> a__U103(a__isNatIListKind(V2),V1,V2)
r4: a__U103(tt(),V1,V2) -> a__U104(a__isNatIListKind(V2),V1,V2)
r5: a__U104(tt(),V1,V2) -> a__U105(a__isNat(V1),V2)
r6: a__U105(tt(),V2) -> a__U106(a__isNatIList(V2))
r7: a__U106(tt()) -> tt()
r8: a__U11(tt(),V1) -> a__U12(a__isNatIListKind(V1),V1)
r9: a__U111(tt(),L,N) -> a__U112(a__isNatIListKind(L),L,N)
r10: a__U112(tt(),L,N) -> a__U113(a__isNat(N),L,N)
r11: a__U113(tt(),L,N) -> a__U114(a__isNatKind(N),L)
r12: a__U114(tt(),L) -> s(a__length(mark(L)))
r13: a__U12(tt(),V1) -> a__U13(a__isNatList(V1))
r14: a__U121(tt(),IL) -> a__U122(a__isNatIListKind(IL))
r15: a__U122(tt()) -> nil()
r16: a__U13(tt()) -> tt()
r17: a__U131(tt(),IL,M,N) -> a__U132(a__isNatIListKind(IL),IL,M,N)
r18: a__U132(tt(),IL,M,N) -> a__U133(a__isNat(M),IL,M,N)
r19: a__U133(tt(),IL,M,N) -> a__U134(a__isNatKind(M),IL,M,N)
r20: a__U134(tt(),IL,M,N) -> a__U135(a__isNat(N),IL,M,N)
r21: a__U135(tt(),IL,M,N) -> a__U136(a__isNatKind(N),IL,M,N)
r22: a__U136(tt(),IL,M,N) -> cons(mark(N),take(M,IL))
r23: a__U21(tt(),V1) -> a__U22(a__isNatKind(V1),V1)
r24: a__U22(tt(),V1) -> a__U23(a__isNat(V1))
r25: a__U23(tt()) -> tt()
r26: a__U31(tt(),V) -> a__U32(a__isNatIListKind(V),V)
r27: a__U32(tt(),V) -> a__U33(a__isNatList(V))
r28: a__U33(tt()) -> tt()
r29: a__U41(tt(),V1,V2) -> a__U42(a__isNatKind(V1),V1,V2)
r30: a__U42(tt(),V1,V2) -> a__U43(a__isNatIListKind(V2),V1,V2)
r31: a__U43(tt(),V1,V2) -> a__U44(a__isNatIListKind(V2),V1,V2)
r32: a__U44(tt(),V1,V2) -> a__U45(a__isNat(V1),V2)
r33: a__U45(tt(),V2) -> a__U46(a__isNatIList(V2))
r34: a__U46(tt()) -> tt()
r35: a__U51(tt(),V2) -> a__U52(a__isNatIListKind(V2))
r36: a__U52(tt()) -> tt()
r37: a__U61(tt(),V2) -> a__U62(a__isNatIListKind(V2))
r38: a__U62(tt()) -> tt()
r39: a__U71(tt()) -> tt()
r40: a__U81(tt()) -> tt()
r41: a__U91(tt(),V1,V2) -> a__U92(a__isNatKind(V1),V1,V2)
r42: a__U92(tt(),V1,V2) -> a__U93(a__isNatIListKind(V2),V1,V2)
r43: a__U93(tt(),V1,V2) -> a__U94(a__isNatIListKind(V2),V1,V2)
r44: a__U94(tt(),V1,V2) -> a__U95(a__isNat(V1),V2)
r45: a__U95(tt(),V2) -> a__U96(a__isNatList(V2))
r46: a__U96(tt()) -> tt()
r47: a__isNat(|0|()) -> tt()
r48: a__isNat(length(V1)) -> a__U11(a__isNatIListKind(V1),V1)
r49: a__isNat(s(V1)) -> a__U21(a__isNatKind(V1),V1)
r50: a__isNatIList(V) -> a__U31(a__isNatIListKind(V),V)
r51: a__isNatIList(zeros()) -> tt()
r52: a__isNatIList(cons(V1,V2)) -> a__U41(a__isNatKind(V1),V1,V2)
r53: a__isNatIListKind(nil()) -> tt()
r54: a__isNatIListKind(zeros()) -> tt()
r55: a__isNatIListKind(cons(V1,V2)) -> a__U51(a__isNatKind(V1),V2)
r56: a__isNatIListKind(take(V1,V2)) -> a__U61(a__isNatKind(V1),V2)
r57: a__isNatKind(|0|()) -> tt()
r58: a__isNatKind(length(V1)) -> a__U71(a__isNatIListKind(V1))
r59: a__isNatKind(s(V1)) -> a__U81(a__isNatKind(V1))
r60: a__isNatList(nil()) -> tt()
r61: a__isNatList(cons(V1,V2)) -> a__U91(a__isNatKind(V1),V1,V2)
r62: a__isNatList(take(V1,V2)) -> a__U101(a__isNatKind(V1),V1,V2)
r63: a__length(nil()) -> |0|()
r64: a__length(cons(N,L)) -> a__U111(a__isNatList(L),L,N)
r65: a__take(|0|(),IL) -> a__U121(a__isNatIList(IL),IL)
r66: a__take(s(M),cons(N,IL)) -> a__U131(a__isNatIList(IL),IL,M,N)
r67: mark(zeros()) -> a__zeros()
r68: mark(U101(X1,X2,X3)) -> a__U101(mark(X1),X2,X3)
r69: mark(U102(X1,X2,X3)) -> a__U102(mark(X1),X2,X3)
r70: mark(isNatKind(X)) -> a__isNatKind(X)
r71: mark(U103(X1,X2,X3)) -> a__U103(mark(X1),X2,X3)
r72: mark(isNatIListKind(X)) -> a__isNatIListKind(X)
r73: mark(U104(X1,X2,X3)) -> a__U104(mark(X1),X2,X3)
r74: mark(U105(X1,X2)) -> a__U105(mark(X1),X2)
r75: mark(isNat(X)) -> a__isNat(X)
r76: mark(U106(X)) -> a__U106(mark(X))
r77: mark(isNatIList(X)) -> a__isNatIList(X)
r78: mark(U11(X1,X2)) -> a__U11(mark(X1),X2)
r79: mark(U12(X1,X2)) -> a__U12(mark(X1),X2)
r80: mark(U111(X1,X2,X3)) -> a__U111(mark(X1),X2,X3)
r81: mark(U112(X1,X2,X3)) -> a__U112(mark(X1),X2,X3)
r82: mark(U113(X1,X2,X3)) -> a__U113(mark(X1),X2,X3)
r83: mark(U114(X1,X2)) -> a__U114(mark(X1),X2)
r84: mark(length(X)) -> a__length(mark(X))
r85: mark(U13(X)) -> a__U13(mark(X))
r86: mark(isNatList(X)) -> a__isNatList(X)
r87: mark(U121(X1,X2)) -> a__U121(mark(X1),X2)
r88: mark(U122(X)) -> a__U122(mark(X))
r89: mark(U131(X1,X2,X3,X4)) -> a__U131(mark(X1),X2,X3,X4)
r90: mark(U132(X1,X2,X3,X4)) -> a__U132(mark(X1),X2,X3,X4)
r91: mark(U133(X1,X2,X3,X4)) -> a__U133(mark(X1),X2,X3,X4)
r92: mark(U134(X1,X2,X3,X4)) -> a__U134(mark(X1),X2,X3,X4)
r93: mark(U135(X1,X2,X3,X4)) -> a__U135(mark(X1),X2,X3,X4)
r94: mark(U136(X1,X2,X3,X4)) -> a__U136(mark(X1),X2,X3,X4)
r95: mark(take(X1,X2)) -> a__take(mark(X1),mark(X2))
r96: mark(U21(X1,X2)) -> a__U21(mark(X1),X2)
r97: mark(U22(X1,X2)) -> a__U22(mark(X1),X2)
r98: mark(U23(X)) -> a__U23(mark(X))
r99: mark(U31(X1,X2)) -> a__U31(mark(X1),X2)
r100: mark(U32(X1,X2)) -> a__U32(mark(X1),X2)
r101: mark(U33(X)) -> a__U33(mark(X))
r102: mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3)
r103: mark(U42(X1,X2,X3)) -> a__U42(mark(X1),X2,X3)
r104: mark(U43(X1,X2,X3)) -> a__U43(mark(X1),X2,X3)
r105: mark(U44(X1,X2,X3)) -> a__U44(mark(X1),X2,X3)
r106: mark(U45(X1,X2)) -> a__U45(mark(X1),X2)
r107: mark(U46(X)) -> a__U46(mark(X))
r108: mark(U51(X1,X2)) -> a__U51(mark(X1),X2)
r109: mark(U52(X)) -> a__U52(mark(X))
r110: mark(U61(X1,X2)) -> a__U61(mark(X1),X2)
r111: mark(U62(X)) -> a__U62(mark(X))
r112: mark(U71(X)) -> a__U71(mark(X))
r113: mark(U81(X)) -> a__U81(mark(X))
r114: mark(U91(X1,X2,X3)) -> a__U91(mark(X1),X2,X3)
r115: mark(U92(X1,X2,X3)) -> a__U92(mark(X1),X2,X3)
r116: mark(U93(X1,X2,X3)) -> a__U93(mark(X1),X2,X3)
r117: mark(U94(X1,X2,X3)) -> a__U94(mark(X1),X2,X3)
r118: mark(U95(X1,X2)) -> a__U95(mark(X1),X2)
r119: mark(U96(X)) -> a__U96(mark(X))
r120: mark(cons(X1,X2)) -> cons(mark(X1),X2)
r121: mark(|0|()) -> |0|()
r122: mark(tt()) -> tt()
r123: mark(s(X)) -> s(mark(X))
r124: mark(nil()) -> nil()
r125: a__zeros() -> zeros()
r126: a__U101(X1,X2,X3) -> U101(X1,X2,X3)
r127: a__U102(X1,X2,X3) -> U102(X1,X2,X3)
r128: a__isNatKind(X) -> isNatKind(X)
r129: a__U103(X1,X2,X3) -> U103(X1,X2,X3)
r130: a__isNatIListKind(X) -> isNatIListKind(X)
r131: a__U104(X1,X2,X3) -> U104(X1,X2,X3)
r132: a__U105(X1,X2) -> U105(X1,X2)
r133: a__isNat(X) -> isNat(X)
r134: a__U106(X) -> U106(X)
r135: a__isNatIList(X) -> isNatIList(X)
r136: a__U11(X1,X2) -> U11(X1,X2)
r137: a__U12(X1,X2) -> U12(X1,X2)
r138: a__U111(X1,X2,X3) -> U111(X1,X2,X3)
r139: a__U112(X1,X2,X3) -> U112(X1,X2,X3)
r140: a__U113(X1,X2,X3) -> U113(X1,X2,X3)
r141: a__U114(X1,X2) -> U114(X1,X2)
r142: a__length(X) -> length(X)
r143: a__U13(X) -> U13(X)
r144: a__isNatList(X) -> isNatList(X)
r145: a__U121(X1,X2) -> U121(X1,X2)
r146: a__U122(X) -> U122(X)
r147: a__U131(X1,X2,X3,X4) -> U131(X1,X2,X3,X4)
r148: a__U132(X1,X2,X3,X4) -> U132(X1,X2,X3,X4)
r149: a__U133(X1,X2,X3,X4) -> U133(X1,X2,X3,X4)
r150: a__U134(X1,X2,X3,X4) -> U134(X1,X2,X3,X4)
r151: a__U135(X1,X2,X3,X4) -> U135(X1,X2,X3,X4)
r152: a__U136(X1,X2,X3,X4) -> U136(X1,X2,X3,X4)
r153: a__take(X1,X2) -> take(X1,X2)
r154: a__U21(X1,X2) -> U21(X1,X2)
r155: a__U22(X1,X2) -> U22(X1,X2)
r156: a__U23(X) -> U23(X)
r157: a__U31(X1,X2) -> U31(X1,X2)
r158: a__U32(X1,X2) -> U32(X1,X2)
r159: a__U33(X) -> U33(X)
r160: a__U41(X1,X2,X3) -> U41(X1,X2,X3)
r161: a__U42(X1,X2,X3) -> U42(X1,X2,X3)
r162: a__U43(X1,X2,X3) -> U43(X1,X2,X3)
r163: a__U44(X1,X2,X3) -> U44(X1,X2,X3)
r164: a__U45(X1,X2) -> U45(X1,X2)
r165: a__U46(X) -> U46(X)
r166: a__U51(X1,X2) -> U51(X1,X2)
r167: a__U52(X) -> U52(X)
r168: a__U61(X1,X2) -> U61(X1,X2)
r169: a__U62(X) -> U62(X)
r170: a__U71(X) -> U71(X)
r171: a__U81(X) -> U81(X)
r172: a__U91(X1,X2,X3) -> U91(X1,X2,X3)
r173: a__U92(X1,X2,X3) -> U92(X1,X2,X3)
r174: a__U93(X1,X2,X3) -> U93(X1,X2,X3)
r175: a__U94(X1,X2,X3) -> U94(X1,X2,X3)
r176: a__U95(X1,X2) -> U95(X1,X2)
r177: a__U96(X) -> U96(X)

The estimated dependency graph contains the following SCCs:

  {p1, p2, p3, p4, p5, p6, p7, p8, p9, p10}


-- Reduction pair.

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

p1: mark#(U11(X1,X2)) -> mark#(X1)
p2: mark#(U12(X1,X2)) -> mark#(X1)
p3: mark#(U111(X1,X2,X3)) -> mark#(X1)
p4: mark#(U122(X)) -> mark#(X)
p5: mark#(U131(X1,X2,X3,X4)) -> mark#(X1)
p6: mark#(U134(X1,X2,X3,X4)) -> mark#(X1)
p7: mark#(U31(X1,X2)) -> mark#(X1)
p8: mark#(U33(X)) -> mark#(X)
p9: mark#(U62(X)) -> mark#(X)
p10: mark#(U81(X)) -> mark#(X)

and R consists of:

r1: a__zeros() -> cons(|0|(),zeros())
r2: a__U101(tt(),V1,V2) -> a__U102(a__isNatKind(V1),V1,V2)
r3: a__U102(tt(),V1,V2) -> a__U103(a__isNatIListKind(V2),V1,V2)
r4: a__U103(tt(),V1,V2) -> a__U104(a__isNatIListKind(V2),V1,V2)
r5: a__U104(tt(),V1,V2) -> a__U105(a__isNat(V1),V2)
r6: a__U105(tt(),V2) -> a__U106(a__isNatIList(V2))
r7: a__U106(tt()) -> tt()
r8: a__U11(tt(),V1) -> a__U12(a__isNatIListKind(V1),V1)
r9: a__U111(tt(),L,N) -> a__U112(a__isNatIListKind(L),L,N)
r10: a__U112(tt(),L,N) -> a__U113(a__isNat(N),L,N)
r11: a__U113(tt(),L,N) -> a__U114(a__isNatKind(N),L)
r12: a__U114(tt(),L) -> s(a__length(mark(L)))
r13: a__U12(tt(),V1) -> a__U13(a__isNatList(V1))
r14: a__U121(tt(),IL) -> a__U122(a__isNatIListKind(IL))
r15: a__U122(tt()) -> nil()
r16: a__U13(tt()) -> tt()
r17: a__U131(tt(),IL,M,N) -> a__U132(a__isNatIListKind(IL),IL,M,N)
r18: a__U132(tt(),IL,M,N) -> a__U133(a__isNat(M),IL,M,N)
r19: a__U133(tt(),IL,M,N) -> a__U134(a__isNatKind(M),IL,M,N)
r20: a__U134(tt(),IL,M,N) -> a__U135(a__isNat(N),IL,M,N)
r21: a__U135(tt(),IL,M,N) -> a__U136(a__isNatKind(N),IL,M,N)
r22: a__U136(tt(),IL,M,N) -> cons(mark(N),take(M,IL))
r23: a__U21(tt(),V1) -> a__U22(a__isNatKind(V1),V1)
r24: a__U22(tt(),V1) -> a__U23(a__isNat(V1))
r25: a__U23(tt()) -> tt()
r26: a__U31(tt(),V) -> a__U32(a__isNatIListKind(V),V)
r27: a__U32(tt(),V) -> a__U33(a__isNatList(V))
r28: a__U33(tt()) -> tt()
r29: a__U41(tt(),V1,V2) -> a__U42(a__isNatKind(V1),V1,V2)
r30: a__U42(tt(),V1,V2) -> a__U43(a__isNatIListKind(V2),V1,V2)
r31: a__U43(tt(),V1,V2) -> a__U44(a__isNatIListKind(V2),V1,V2)
r32: a__U44(tt(),V1,V2) -> a__U45(a__isNat(V1),V2)
r33: a__U45(tt(),V2) -> a__U46(a__isNatIList(V2))
r34: a__U46(tt()) -> tt()
r35: a__U51(tt(),V2) -> a__U52(a__isNatIListKind(V2))
r36: a__U52(tt()) -> tt()
r37: a__U61(tt(),V2) -> a__U62(a__isNatIListKind(V2))
r38: a__U62(tt()) -> tt()
r39: a__U71(tt()) -> tt()
r40: a__U81(tt()) -> tt()
r41: a__U91(tt(),V1,V2) -> a__U92(a__isNatKind(V1),V1,V2)
r42: a__U92(tt(),V1,V2) -> a__U93(a__isNatIListKind(V2),V1,V2)
r43: a__U93(tt(),V1,V2) -> a__U94(a__isNatIListKind(V2),V1,V2)
r44: a__U94(tt(),V1,V2) -> a__U95(a__isNat(V1),V2)
r45: a__U95(tt(),V2) -> a__U96(a__isNatList(V2))
r46: a__U96(tt()) -> tt()
r47: a__isNat(|0|()) -> tt()
r48: a__isNat(length(V1)) -> a__U11(a__isNatIListKind(V1),V1)
r49: a__isNat(s(V1)) -> a__U21(a__isNatKind(V1),V1)
r50: a__isNatIList(V) -> a__U31(a__isNatIListKind(V),V)
r51: a__isNatIList(zeros()) -> tt()
r52: a__isNatIList(cons(V1,V2)) -> a__U41(a__isNatKind(V1),V1,V2)
r53: a__isNatIListKind(nil()) -> tt()
r54: a__isNatIListKind(zeros()) -> tt()
r55: a__isNatIListKind(cons(V1,V2)) -> a__U51(a__isNatKind(V1),V2)
r56: a__isNatIListKind(take(V1,V2)) -> a__U61(a__isNatKind(V1),V2)
r57: a__isNatKind(|0|()) -> tt()
r58: a__isNatKind(length(V1)) -> a__U71(a__isNatIListKind(V1))
r59: a__isNatKind(s(V1)) -> a__U81(a__isNatKind(V1))
r60: a__isNatList(nil()) -> tt()
r61: a__isNatList(cons(V1,V2)) -> a__U91(a__isNatKind(V1),V1,V2)
r62: a__isNatList(take(V1,V2)) -> a__U101(a__isNatKind(V1),V1,V2)
r63: a__length(nil()) -> |0|()
r64: a__length(cons(N,L)) -> a__U111(a__isNatList(L),L,N)
r65: a__take(|0|(),IL) -> a__U121(a__isNatIList(IL),IL)
r66: a__take(s(M),cons(N,IL)) -> a__U131(a__isNatIList(IL),IL,M,N)
r67: mark(zeros()) -> a__zeros()
r68: mark(U101(X1,X2,X3)) -> a__U101(mark(X1),X2,X3)
r69: mark(U102(X1,X2,X3)) -> a__U102(mark(X1),X2,X3)
r70: mark(isNatKind(X)) -> a__isNatKind(X)
r71: mark(U103(X1,X2,X3)) -> a__U103(mark(X1),X2,X3)
r72: mark(isNatIListKind(X)) -> a__isNatIListKind(X)
r73: mark(U104(X1,X2,X3)) -> a__U104(mark(X1),X2,X3)
r74: mark(U105(X1,X2)) -> a__U105(mark(X1),X2)
r75: mark(isNat(X)) -> a__isNat(X)
r76: mark(U106(X)) -> a__U106(mark(X))
r77: mark(isNatIList(X)) -> a__isNatIList(X)
r78: mark(U11(X1,X2)) -> a__U11(mark(X1),X2)
r79: mark(U12(X1,X2)) -> a__U12(mark(X1),X2)
r80: mark(U111(X1,X2,X3)) -> a__U111(mark(X1),X2,X3)
r81: mark(U112(X1,X2,X3)) -> a__U112(mark(X1),X2,X3)
r82: mark(U113(X1,X2,X3)) -> a__U113(mark(X1),X2,X3)
r83: mark(U114(X1,X2)) -> a__U114(mark(X1),X2)
r84: mark(length(X)) -> a__length(mark(X))
r85: mark(U13(X)) -> a__U13(mark(X))
r86: mark(isNatList(X)) -> a__isNatList(X)
r87: mark(U121(X1,X2)) -> a__U121(mark(X1),X2)
r88: mark(U122(X)) -> a__U122(mark(X))
r89: mark(U131(X1,X2,X3,X4)) -> a__U131(mark(X1),X2,X3,X4)
r90: mark(U132(X1,X2,X3,X4)) -> a__U132(mark(X1),X2,X3,X4)
r91: mark(U133(X1,X2,X3,X4)) -> a__U133(mark(X1),X2,X3,X4)
r92: mark(U134(X1,X2,X3,X4)) -> a__U134(mark(X1),X2,X3,X4)
r93: mark(U135(X1,X2,X3,X4)) -> a__U135(mark(X1),X2,X3,X4)
r94: mark(U136(X1,X2,X3,X4)) -> a__U136(mark(X1),X2,X3,X4)
r95: mark(take(X1,X2)) -> a__take(mark(X1),mark(X2))
r96: mark(U21(X1,X2)) -> a__U21(mark(X1),X2)
r97: mark(U22(X1,X2)) -> a__U22(mark(X1),X2)
r98: mark(U23(X)) -> a__U23(mark(X))
r99: mark(U31(X1,X2)) -> a__U31(mark(X1),X2)
r100: mark(U32(X1,X2)) -> a__U32(mark(X1),X2)
r101: mark(U33(X)) -> a__U33(mark(X))
r102: mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3)
r103: mark(U42(X1,X2,X3)) -> a__U42(mark(X1),X2,X3)
r104: mark(U43(X1,X2,X3)) -> a__U43(mark(X1),X2,X3)
r105: mark(U44(X1,X2,X3)) -> a__U44(mark(X1),X2,X3)
r106: mark(U45(X1,X2)) -> a__U45(mark(X1),X2)
r107: mark(U46(X)) -> a__U46(mark(X))
r108: mark(U51(X1,X2)) -> a__U51(mark(X1),X2)
r109: mark(U52(X)) -> a__U52(mark(X))
r110: mark(U61(X1,X2)) -> a__U61(mark(X1),X2)
r111: mark(U62(X)) -> a__U62(mark(X))
r112: mark(U71(X)) -> a__U71(mark(X))
r113: mark(U81(X)) -> a__U81(mark(X))
r114: mark(U91(X1,X2,X3)) -> a__U91(mark(X1),X2,X3)
r115: mark(U92(X1,X2,X3)) -> a__U92(mark(X1),X2,X3)
r116: mark(U93(X1,X2,X3)) -> a__U93(mark(X1),X2,X3)
r117: mark(U94(X1,X2,X3)) -> a__U94(mark(X1),X2,X3)
r118: mark(U95(X1,X2)) -> a__U95(mark(X1),X2)
r119: mark(U96(X)) -> a__U96(mark(X))
r120: mark(cons(X1,X2)) -> cons(mark(X1),X2)
r121: mark(|0|()) -> |0|()
r122: mark(tt()) -> tt()
r123: mark(s(X)) -> s(mark(X))
r124: mark(nil()) -> nil()
r125: a__zeros() -> zeros()
r126: a__U101(X1,X2,X3) -> U101(X1,X2,X3)
r127: a__U102(X1,X2,X3) -> U102(X1,X2,X3)
r128: a__isNatKind(X) -> isNatKind(X)
r129: a__U103(X1,X2,X3) -> U103(X1,X2,X3)
r130: a__isNatIListKind(X) -> isNatIListKind(X)
r131: a__U104(X1,X2,X3) -> U104(X1,X2,X3)
r132: a__U105(X1,X2) -> U105(X1,X2)
r133: a__isNat(X) -> isNat(X)
r134: a__U106(X) -> U106(X)
r135: a__isNatIList(X) -> isNatIList(X)
r136: a__U11(X1,X2) -> U11(X1,X2)
r137: a__U12(X1,X2) -> U12(X1,X2)
r138: a__U111(X1,X2,X3) -> U111(X1,X2,X3)
r139: a__U112(X1,X2,X3) -> U112(X1,X2,X3)
r140: a__U113(X1,X2,X3) -> U113(X1,X2,X3)
r141: a__U114(X1,X2) -> U114(X1,X2)
r142: a__length(X) -> length(X)
r143: a__U13(X) -> U13(X)
r144: a__isNatList(X) -> isNatList(X)
r145: a__U121(X1,X2) -> U121(X1,X2)
r146: a__U122(X) -> U122(X)
r147: a__U131(X1,X2,X3,X4) -> U131(X1,X2,X3,X4)
r148: a__U132(X1,X2,X3,X4) -> U132(X1,X2,X3,X4)
r149: a__U133(X1,X2,X3,X4) -> U133(X1,X2,X3,X4)
r150: a__U134(X1,X2,X3,X4) -> U134(X1,X2,X3,X4)
r151: a__U135(X1,X2,X3,X4) -> U135(X1,X2,X3,X4)
r152: a__U136(X1,X2,X3,X4) -> U136(X1,X2,X3,X4)
r153: a__take(X1,X2) -> take(X1,X2)
r154: a__U21(X1,X2) -> U21(X1,X2)
r155: a__U22(X1,X2) -> U22(X1,X2)
r156: a__U23(X) -> U23(X)
r157: a__U31(X1,X2) -> U31(X1,X2)
r158: a__U32(X1,X2) -> U32(X1,X2)
r159: a__U33(X) -> U33(X)
r160: a__U41(X1,X2,X3) -> U41(X1,X2,X3)
r161: a__U42(X1,X2,X3) -> U42(X1,X2,X3)
r162: a__U43(X1,X2,X3) -> U43(X1,X2,X3)
r163: a__U44(X1,X2,X3) -> U44(X1,X2,X3)
r164: a__U45(X1,X2) -> U45(X1,X2)
r165: a__U46(X) -> U46(X)
r166: a__U51(X1,X2) -> U51(X1,X2)
r167: a__U52(X) -> U52(X)
r168: a__U61(X1,X2) -> U61(X1,X2)
r169: a__U62(X) -> U62(X)
r170: a__U71(X) -> U71(X)
r171: a__U81(X) -> U81(X)
r172: a__U91(X1,X2,X3) -> U91(X1,X2,X3)
r173: a__U92(X1,X2,X3) -> U92(X1,X2,X3)
r174: a__U93(X1,X2,X3) -> U93(X1,X2,X3)
r175: a__U94(X1,X2,X3) -> U94(X1,X2,X3)
r176: a__U95(X1,X2) -> U95(X1,X2)
r177: a__U96(X) -> U96(X)

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  max/plus interpretations on natural numbers:
  
    mark#_A(x1) = max{0, x1 - 1}
    U11_A(x1,x2) = max{x1 + 1, x2 + 1}
    U12_A(x1,x2) = x1 + 1
    U111_A(x1,x2,x3) = max{x1, x2 + 2, x3 + 2}
    U122_A(x1) = x1 + 1
    U131_A(x1,x2,x3,x4) = max{x1, x2 + 2, x3 + 2, x4 + 2}
    U134_A(x1,x2,x3,x4) = max{x1, x3 + 1}
    U31_A(x1,x2) = max{x1 + 2, x2 + 2}
    U33_A(x1) = x1
    U62_A(x1) = x1 + 1
    U81_A(x1) = x1 + 1

The next rules are strictly ordered:

  p7

We remove them from the problem.

-- SCC decomposition.

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

p1: mark#(U11(X1,X2)) -> mark#(X1)
p2: mark#(U12(X1,X2)) -> mark#(X1)
p3: mark#(U111(X1,X2,X3)) -> mark#(X1)
p4: mark#(U122(X)) -> mark#(X)
p5: mark#(U131(X1,X2,X3,X4)) -> mark#(X1)
p6: mark#(U134(X1,X2,X3,X4)) -> mark#(X1)
p7: mark#(U33(X)) -> mark#(X)
p8: mark#(U62(X)) -> mark#(X)
p9: mark#(U81(X)) -> mark#(X)

and R consists of:

r1: a__zeros() -> cons(|0|(),zeros())
r2: a__U101(tt(),V1,V2) -> a__U102(a__isNatKind(V1),V1,V2)
r3: a__U102(tt(),V1,V2) -> a__U103(a__isNatIListKind(V2),V1,V2)
r4: a__U103(tt(),V1,V2) -> a__U104(a__isNatIListKind(V2),V1,V2)
r5: a__U104(tt(),V1,V2) -> a__U105(a__isNat(V1),V2)
r6: a__U105(tt(),V2) -> a__U106(a__isNatIList(V2))
r7: a__U106(tt()) -> tt()
r8: a__U11(tt(),V1) -> a__U12(a__isNatIListKind(V1),V1)
r9: a__U111(tt(),L,N) -> a__U112(a__isNatIListKind(L),L,N)
r10: a__U112(tt(),L,N) -> a__U113(a__isNat(N),L,N)
r11: a__U113(tt(),L,N) -> a__U114(a__isNatKind(N),L)
r12: a__U114(tt(),L) -> s(a__length(mark(L)))
r13: a__U12(tt(),V1) -> a__U13(a__isNatList(V1))
r14: a__U121(tt(),IL) -> a__U122(a__isNatIListKind(IL))
r15: a__U122(tt()) -> nil()
r16: a__U13(tt()) -> tt()
r17: a__U131(tt(),IL,M,N) -> a__U132(a__isNatIListKind(IL),IL,M,N)
r18: a__U132(tt(),IL,M,N) -> a__U133(a__isNat(M),IL,M,N)
r19: a__U133(tt(),IL,M,N) -> a__U134(a__isNatKind(M),IL,M,N)
r20: a__U134(tt(),IL,M,N) -> a__U135(a__isNat(N),IL,M,N)
r21: a__U135(tt(),IL,M,N) -> a__U136(a__isNatKind(N),IL,M,N)
r22: a__U136(tt(),IL,M,N) -> cons(mark(N),take(M,IL))
r23: a__U21(tt(),V1) -> a__U22(a__isNatKind(V1),V1)
r24: a__U22(tt(),V1) -> a__U23(a__isNat(V1))
r25: a__U23(tt()) -> tt()
r26: a__U31(tt(),V) -> a__U32(a__isNatIListKind(V),V)
r27: a__U32(tt(),V) -> a__U33(a__isNatList(V))
r28: a__U33(tt()) -> tt()
r29: a__U41(tt(),V1,V2) -> a__U42(a__isNatKind(V1),V1,V2)
r30: a__U42(tt(),V1,V2) -> a__U43(a__isNatIListKind(V2),V1,V2)
r31: a__U43(tt(),V1,V2) -> a__U44(a__isNatIListKind(V2),V1,V2)
r32: a__U44(tt(),V1,V2) -> a__U45(a__isNat(V1),V2)
r33: a__U45(tt(),V2) -> a__U46(a__isNatIList(V2))
r34: a__U46(tt()) -> tt()
r35: a__U51(tt(),V2) -> a__U52(a__isNatIListKind(V2))
r36: a__U52(tt()) -> tt()
r37: a__U61(tt(),V2) -> a__U62(a__isNatIListKind(V2))
r38: a__U62(tt()) -> tt()
r39: a__U71(tt()) -> tt()
r40: a__U81(tt()) -> tt()
r41: a__U91(tt(),V1,V2) -> a__U92(a__isNatKind(V1),V1,V2)
r42: a__U92(tt(),V1,V2) -> a__U93(a__isNatIListKind(V2),V1,V2)
r43: a__U93(tt(),V1,V2) -> a__U94(a__isNatIListKind(V2),V1,V2)
r44: a__U94(tt(),V1,V2) -> a__U95(a__isNat(V1),V2)
r45: a__U95(tt(),V2) -> a__U96(a__isNatList(V2))
r46: a__U96(tt()) -> tt()
r47: a__isNat(|0|()) -> tt()
r48: a__isNat(length(V1)) -> a__U11(a__isNatIListKind(V1),V1)
r49: a__isNat(s(V1)) -> a__U21(a__isNatKind(V1),V1)
r50: a__isNatIList(V) -> a__U31(a__isNatIListKind(V),V)
r51: a__isNatIList(zeros()) -> tt()
r52: a__isNatIList(cons(V1,V2)) -> a__U41(a__isNatKind(V1),V1,V2)
r53: a__isNatIListKind(nil()) -> tt()
r54: a__isNatIListKind(zeros()) -> tt()
r55: a__isNatIListKind(cons(V1,V2)) -> a__U51(a__isNatKind(V1),V2)
r56: a__isNatIListKind(take(V1,V2)) -> a__U61(a__isNatKind(V1),V2)
r57: a__isNatKind(|0|()) -> tt()
r58: a__isNatKind(length(V1)) -> a__U71(a__isNatIListKind(V1))
r59: a__isNatKind(s(V1)) -> a__U81(a__isNatKind(V1))
r60: a__isNatList(nil()) -> tt()
r61: a__isNatList(cons(V1,V2)) -> a__U91(a__isNatKind(V1),V1,V2)
r62: a__isNatList(take(V1,V2)) -> a__U101(a__isNatKind(V1),V1,V2)
r63: a__length(nil()) -> |0|()
r64: a__length(cons(N,L)) -> a__U111(a__isNatList(L),L,N)
r65: a__take(|0|(),IL) -> a__U121(a__isNatIList(IL),IL)
r66: a__take(s(M),cons(N,IL)) -> a__U131(a__isNatIList(IL),IL,M,N)
r67: mark(zeros()) -> a__zeros()
r68: mark(U101(X1,X2,X3)) -> a__U101(mark(X1),X2,X3)
r69: mark(U102(X1,X2,X3)) -> a__U102(mark(X1),X2,X3)
r70: mark(isNatKind(X)) -> a__isNatKind(X)
r71: mark(U103(X1,X2,X3)) -> a__U103(mark(X1),X2,X3)
r72: mark(isNatIListKind(X)) -> a__isNatIListKind(X)
r73: mark(U104(X1,X2,X3)) -> a__U104(mark(X1),X2,X3)
r74: mark(U105(X1,X2)) -> a__U105(mark(X1),X2)
r75: mark(isNat(X)) -> a__isNat(X)
r76: mark(U106(X)) -> a__U106(mark(X))
r77: mark(isNatIList(X)) -> a__isNatIList(X)
r78: mark(U11(X1,X2)) -> a__U11(mark(X1),X2)
r79: mark(U12(X1,X2)) -> a__U12(mark(X1),X2)
r80: mark(U111(X1,X2,X3)) -> a__U111(mark(X1),X2,X3)
r81: mark(U112(X1,X2,X3)) -> a__U112(mark(X1),X2,X3)
r82: mark(U113(X1,X2,X3)) -> a__U113(mark(X1),X2,X3)
r83: mark(U114(X1,X2)) -> a__U114(mark(X1),X2)
r84: mark(length(X)) -> a__length(mark(X))
r85: mark(U13(X)) -> a__U13(mark(X))
r86: mark(isNatList(X)) -> a__isNatList(X)
r87: mark(U121(X1,X2)) -> a__U121(mark(X1),X2)
r88: mark(U122(X)) -> a__U122(mark(X))
r89: mark(U131(X1,X2,X3,X4)) -> a__U131(mark(X1),X2,X3,X4)
r90: mark(U132(X1,X2,X3,X4)) -> a__U132(mark(X1),X2,X3,X4)
r91: mark(U133(X1,X2,X3,X4)) -> a__U133(mark(X1),X2,X3,X4)
r92: mark(U134(X1,X2,X3,X4)) -> a__U134(mark(X1),X2,X3,X4)
r93: mark(U135(X1,X2,X3,X4)) -> a__U135(mark(X1),X2,X3,X4)
r94: mark(U136(X1,X2,X3,X4)) -> a__U136(mark(X1),X2,X3,X4)
r95: mark(take(X1,X2)) -> a__take(mark(X1),mark(X2))
r96: mark(U21(X1,X2)) -> a__U21(mark(X1),X2)
r97: mark(U22(X1,X2)) -> a__U22(mark(X1),X2)
r98: mark(U23(X)) -> a__U23(mark(X))
r99: mark(U31(X1,X2)) -> a__U31(mark(X1),X2)
r100: mark(U32(X1,X2)) -> a__U32(mark(X1),X2)
r101: mark(U33(X)) -> a__U33(mark(X))
r102: mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3)
r103: mark(U42(X1,X2,X3)) -> a__U42(mark(X1),X2,X3)
r104: mark(U43(X1,X2,X3)) -> a__U43(mark(X1),X2,X3)
r105: mark(U44(X1,X2,X3)) -> a__U44(mark(X1),X2,X3)
r106: mark(U45(X1,X2)) -> a__U45(mark(X1),X2)
r107: mark(U46(X)) -> a__U46(mark(X))
r108: mark(U51(X1,X2)) -> a__U51(mark(X1),X2)
r109: mark(U52(X)) -> a__U52(mark(X))
r110: mark(U61(X1,X2)) -> a__U61(mark(X1),X2)
r111: mark(U62(X)) -> a__U62(mark(X))
r112: mark(U71(X)) -> a__U71(mark(X))
r113: mark(U81(X)) -> a__U81(mark(X))
r114: mark(U91(X1,X2,X3)) -> a__U91(mark(X1),X2,X3)
r115: mark(U92(X1,X2,X3)) -> a__U92(mark(X1),X2,X3)
r116: mark(U93(X1,X2,X3)) -> a__U93(mark(X1),X2,X3)
r117: mark(U94(X1,X2,X3)) -> a__U94(mark(X1),X2,X3)
r118: mark(U95(X1,X2)) -> a__U95(mark(X1),X2)
r119: mark(U96(X)) -> a__U96(mark(X))
r120: mark(cons(X1,X2)) -> cons(mark(X1),X2)
r121: mark(|0|()) -> |0|()
r122: mark(tt()) -> tt()
r123: mark(s(X)) -> s(mark(X))
r124: mark(nil()) -> nil()
r125: a__zeros() -> zeros()
r126: a__U101(X1,X2,X3) -> U101(X1,X2,X3)
r127: a__U102(X1,X2,X3) -> U102(X1,X2,X3)
r128: a__isNatKind(X) -> isNatKind(X)
r129: a__U103(X1,X2,X3) -> U103(X1,X2,X3)
r130: a__isNatIListKind(X) -> isNatIListKind(X)
r131: a__U104(X1,X2,X3) -> U104(X1,X2,X3)
r132: a__U105(X1,X2) -> U105(X1,X2)
r133: a__isNat(X) -> isNat(X)
r134: a__U106(X) -> U106(X)
r135: a__isNatIList(X) -> isNatIList(X)
r136: a__U11(X1,X2) -> U11(X1,X2)
r137: a__U12(X1,X2) -> U12(X1,X2)
r138: a__U111(X1,X2,X3) -> U111(X1,X2,X3)
r139: a__U112(X1,X2,X3) -> U112(X1,X2,X3)
r140: a__U113(X1,X2,X3) -> U113(X1,X2,X3)
r141: a__U114(X1,X2) -> U114(X1,X2)
r142: a__length(X) -> length(X)
r143: a__U13(X) -> U13(X)
r144: a__isNatList(X) -> isNatList(X)
r145: a__U121(X1,X2) -> U121(X1,X2)
r146: a__U122(X) -> U122(X)
r147: a__U131(X1,X2,X3,X4) -> U131(X1,X2,X3,X4)
r148: a__U132(X1,X2,X3,X4) -> U132(X1,X2,X3,X4)
r149: a__U133(X1,X2,X3,X4) -> U133(X1,X2,X3,X4)
r150: a__U134(X1,X2,X3,X4) -> U134(X1,X2,X3,X4)
r151: a__U135(X1,X2,X3,X4) -> U135(X1,X2,X3,X4)
r152: a__U136(X1,X2,X3,X4) -> U136(X1,X2,X3,X4)
r153: a__take(X1,X2) -> take(X1,X2)
r154: a__U21(X1,X2) -> U21(X1,X2)
r155: a__U22(X1,X2) -> U22(X1,X2)
r156: a__U23(X) -> U23(X)
r157: a__U31(X1,X2) -> U31(X1,X2)
r158: a__U32(X1,X2) -> U32(X1,X2)
r159: a__U33(X) -> U33(X)
r160: a__U41(X1,X2,X3) -> U41(X1,X2,X3)
r161: a__U42(X1,X2,X3) -> U42(X1,X2,X3)
r162: a__U43(X1,X2,X3) -> U43(X1,X2,X3)
r163: a__U44(X1,X2,X3) -> U44(X1,X2,X3)
r164: a__U45(X1,X2) -> U45(X1,X2)
r165: a__U46(X) -> U46(X)
r166: a__U51(X1,X2) -> U51(X1,X2)
r167: a__U52(X) -> U52(X)
r168: a__U61(X1,X2) -> U61(X1,X2)
r169: a__U62(X) -> U62(X)
r170: a__U71(X) -> U71(X)
r171: a__U81(X) -> U81(X)
r172: a__U91(X1,X2,X3) -> U91(X1,X2,X3)
r173: a__U92(X1,X2,X3) -> U92(X1,X2,X3)
r174: a__U93(X1,X2,X3) -> U93(X1,X2,X3)
r175: a__U94(X1,X2,X3) -> U94(X1,X2,X3)
r176: a__U95(X1,X2) -> U95(X1,X2)
r177: a__U96(X) -> U96(X)

The estimated dependency graph contains the following SCCs:

  {p1, p2, p3, p4, p5, p6, p7, p8, p9}


-- Reduction pair.

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

p1: mark#(U11(X1,X2)) -> mark#(X1)
p2: mark#(U81(X)) -> mark#(X)
p3: mark#(U62(X)) -> mark#(X)
p4: mark#(U33(X)) -> mark#(X)
p5: mark#(U134(X1,X2,X3,X4)) -> mark#(X1)
p6: mark#(U131(X1,X2,X3,X4)) -> mark#(X1)
p7: mark#(U122(X)) -> mark#(X)
p8: mark#(U111(X1,X2,X3)) -> mark#(X1)
p9: mark#(U12(X1,X2)) -> mark#(X1)

and R consists of:

r1: a__zeros() -> cons(|0|(),zeros())
r2: a__U101(tt(),V1,V2) -> a__U102(a__isNatKind(V1),V1,V2)
r3: a__U102(tt(),V1,V2) -> a__U103(a__isNatIListKind(V2),V1,V2)
r4: a__U103(tt(),V1,V2) -> a__U104(a__isNatIListKind(V2),V1,V2)
r5: a__U104(tt(),V1,V2) -> a__U105(a__isNat(V1),V2)
r6: a__U105(tt(),V2) -> a__U106(a__isNatIList(V2))
r7: a__U106(tt()) -> tt()
r8: a__U11(tt(),V1) -> a__U12(a__isNatIListKind(V1),V1)
r9: a__U111(tt(),L,N) -> a__U112(a__isNatIListKind(L),L,N)
r10: a__U112(tt(),L,N) -> a__U113(a__isNat(N),L,N)
r11: a__U113(tt(),L,N) -> a__U114(a__isNatKind(N),L)
r12: a__U114(tt(),L) -> s(a__length(mark(L)))
r13: a__U12(tt(),V1) -> a__U13(a__isNatList(V1))
r14: a__U121(tt(),IL) -> a__U122(a__isNatIListKind(IL))
r15: a__U122(tt()) -> nil()
r16: a__U13(tt()) -> tt()
r17: a__U131(tt(),IL,M,N) -> a__U132(a__isNatIListKind(IL),IL,M,N)
r18: a__U132(tt(),IL,M,N) -> a__U133(a__isNat(M),IL,M,N)
r19: a__U133(tt(),IL,M,N) -> a__U134(a__isNatKind(M),IL,M,N)
r20: a__U134(tt(),IL,M,N) -> a__U135(a__isNat(N),IL,M,N)
r21: a__U135(tt(),IL,M,N) -> a__U136(a__isNatKind(N),IL,M,N)
r22: a__U136(tt(),IL,M,N) -> cons(mark(N),take(M,IL))
r23: a__U21(tt(),V1) -> a__U22(a__isNatKind(V1),V1)
r24: a__U22(tt(),V1) -> a__U23(a__isNat(V1))
r25: a__U23(tt()) -> tt()
r26: a__U31(tt(),V) -> a__U32(a__isNatIListKind(V),V)
r27: a__U32(tt(),V) -> a__U33(a__isNatList(V))
r28: a__U33(tt()) -> tt()
r29: a__U41(tt(),V1,V2) -> a__U42(a__isNatKind(V1),V1,V2)
r30: a__U42(tt(),V1,V2) -> a__U43(a__isNatIListKind(V2),V1,V2)
r31: a__U43(tt(),V1,V2) -> a__U44(a__isNatIListKind(V2),V1,V2)
r32: a__U44(tt(),V1,V2) -> a__U45(a__isNat(V1),V2)
r33: a__U45(tt(),V2) -> a__U46(a__isNatIList(V2))
r34: a__U46(tt()) -> tt()
r35: a__U51(tt(),V2) -> a__U52(a__isNatIListKind(V2))
r36: a__U52(tt()) -> tt()
r37: a__U61(tt(),V2) -> a__U62(a__isNatIListKind(V2))
r38: a__U62(tt()) -> tt()
r39: a__U71(tt()) -> tt()
r40: a__U81(tt()) -> tt()
r41: a__U91(tt(),V1,V2) -> a__U92(a__isNatKind(V1),V1,V2)
r42: a__U92(tt(),V1,V2) -> a__U93(a__isNatIListKind(V2),V1,V2)
r43: a__U93(tt(),V1,V2) -> a__U94(a__isNatIListKind(V2),V1,V2)
r44: a__U94(tt(),V1,V2) -> a__U95(a__isNat(V1),V2)
r45: a__U95(tt(),V2) -> a__U96(a__isNatList(V2))
r46: a__U96(tt()) -> tt()
r47: a__isNat(|0|()) -> tt()
r48: a__isNat(length(V1)) -> a__U11(a__isNatIListKind(V1),V1)
r49: a__isNat(s(V1)) -> a__U21(a__isNatKind(V1),V1)
r50: a__isNatIList(V) -> a__U31(a__isNatIListKind(V),V)
r51: a__isNatIList(zeros()) -> tt()
r52: a__isNatIList(cons(V1,V2)) -> a__U41(a__isNatKind(V1),V1,V2)
r53: a__isNatIListKind(nil()) -> tt()
r54: a__isNatIListKind(zeros()) -> tt()
r55: a__isNatIListKind(cons(V1,V2)) -> a__U51(a__isNatKind(V1),V2)
r56: a__isNatIListKind(take(V1,V2)) -> a__U61(a__isNatKind(V1),V2)
r57: a__isNatKind(|0|()) -> tt()
r58: a__isNatKind(length(V1)) -> a__U71(a__isNatIListKind(V1))
r59: a__isNatKind(s(V1)) -> a__U81(a__isNatKind(V1))
r60: a__isNatList(nil()) -> tt()
r61: a__isNatList(cons(V1,V2)) -> a__U91(a__isNatKind(V1),V1,V2)
r62: a__isNatList(take(V1,V2)) -> a__U101(a__isNatKind(V1),V1,V2)
r63: a__length(nil()) -> |0|()
r64: a__length(cons(N,L)) -> a__U111(a__isNatList(L),L,N)
r65: a__take(|0|(),IL) -> a__U121(a__isNatIList(IL),IL)
r66: a__take(s(M),cons(N,IL)) -> a__U131(a__isNatIList(IL),IL,M,N)
r67: mark(zeros()) -> a__zeros()
r68: mark(U101(X1,X2,X3)) -> a__U101(mark(X1),X2,X3)
r69: mark(U102(X1,X2,X3)) -> a__U102(mark(X1),X2,X3)
r70: mark(isNatKind(X)) -> a__isNatKind(X)
r71: mark(U103(X1,X2,X3)) -> a__U103(mark(X1),X2,X3)
r72: mark(isNatIListKind(X)) -> a__isNatIListKind(X)
r73: mark(U104(X1,X2,X3)) -> a__U104(mark(X1),X2,X3)
r74: mark(U105(X1,X2)) -> a__U105(mark(X1),X2)
r75: mark(isNat(X)) -> a__isNat(X)
r76: mark(U106(X)) -> a__U106(mark(X))
r77: mark(isNatIList(X)) -> a__isNatIList(X)
r78: mark(U11(X1,X2)) -> a__U11(mark(X1),X2)
r79: mark(U12(X1,X2)) -> a__U12(mark(X1),X2)
r80: mark(U111(X1,X2,X3)) -> a__U111(mark(X1),X2,X3)
r81: mark(U112(X1,X2,X3)) -> a__U112(mark(X1),X2,X3)
r82: mark(U113(X1,X2,X3)) -> a__U113(mark(X1),X2,X3)
r83: mark(U114(X1,X2)) -> a__U114(mark(X1),X2)
r84: mark(length(X)) -> a__length(mark(X))
r85: mark(U13(X)) -> a__U13(mark(X))
r86: mark(isNatList(X)) -> a__isNatList(X)
r87: mark(U121(X1,X2)) -> a__U121(mark(X1),X2)
r88: mark(U122(X)) -> a__U122(mark(X))
r89: mark(U131(X1,X2,X3,X4)) -> a__U131(mark(X1),X2,X3,X4)
r90: mark(U132(X1,X2,X3,X4)) -> a__U132(mark(X1),X2,X3,X4)
r91: mark(U133(X1,X2,X3,X4)) -> a__U133(mark(X1),X2,X3,X4)
r92: mark(U134(X1,X2,X3,X4)) -> a__U134(mark(X1),X2,X3,X4)
r93: mark(U135(X1,X2,X3,X4)) -> a__U135(mark(X1),X2,X3,X4)
r94: mark(U136(X1,X2,X3,X4)) -> a__U136(mark(X1),X2,X3,X4)
r95: mark(take(X1,X2)) -> a__take(mark(X1),mark(X2))
r96: mark(U21(X1,X2)) -> a__U21(mark(X1),X2)
r97: mark(U22(X1,X2)) -> a__U22(mark(X1),X2)
r98: mark(U23(X)) -> a__U23(mark(X))
r99: mark(U31(X1,X2)) -> a__U31(mark(X1),X2)
r100: mark(U32(X1,X2)) -> a__U32(mark(X1),X2)
r101: mark(U33(X)) -> a__U33(mark(X))
r102: mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3)
r103: mark(U42(X1,X2,X3)) -> a__U42(mark(X1),X2,X3)
r104: mark(U43(X1,X2,X3)) -> a__U43(mark(X1),X2,X3)
r105: mark(U44(X1,X2,X3)) -> a__U44(mark(X1),X2,X3)
r106: mark(U45(X1,X2)) -> a__U45(mark(X1),X2)
r107: mark(U46(X)) -> a__U46(mark(X))
r108: mark(U51(X1,X2)) -> a__U51(mark(X1),X2)
r109: mark(U52(X)) -> a__U52(mark(X))
r110: mark(U61(X1,X2)) -> a__U61(mark(X1),X2)
r111: mark(U62(X)) -> a__U62(mark(X))
r112: mark(U71(X)) -> a__U71(mark(X))
r113: mark(U81(X)) -> a__U81(mark(X))
r114: mark(U91(X1,X2,X3)) -> a__U91(mark(X1),X2,X3)
r115: mark(U92(X1,X2,X3)) -> a__U92(mark(X1),X2,X3)
r116: mark(U93(X1,X2,X3)) -> a__U93(mark(X1),X2,X3)
r117: mark(U94(X1,X2,X3)) -> a__U94(mark(X1),X2,X3)
r118: mark(U95(X1,X2)) -> a__U95(mark(X1),X2)
r119: mark(U96(X)) -> a__U96(mark(X))
r120: mark(cons(X1,X2)) -> cons(mark(X1),X2)
r121: mark(|0|()) -> |0|()
r122: mark(tt()) -> tt()
r123: mark(s(X)) -> s(mark(X))
r124: mark(nil()) -> nil()
r125: a__zeros() -> zeros()
r126: a__U101(X1,X2,X3) -> U101(X1,X2,X3)
r127: a__U102(X1,X2,X3) -> U102(X1,X2,X3)
r128: a__isNatKind(X) -> isNatKind(X)
r129: a__U103(X1,X2,X3) -> U103(X1,X2,X3)
r130: a__isNatIListKind(X) -> isNatIListKind(X)
r131: a__U104(X1,X2,X3) -> U104(X1,X2,X3)
r132: a__U105(X1,X2) -> U105(X1,X2)
r133: a__isNat(X) -> isNat(X)
r134: a__U106(X) -> U106(X)
r135: a__isNatIList(X) -> isNatIList(X)
r136: a__U11(X1,X2) -> U11(X1,X2)
r137: a__U12(X1,X2) -> U12(X1,X2)
r138: a__U111(X1,X2,X3) -> U111(X1,X2,X3)
r139: a__U112(X1,X2,X3) -> U112(X1,X2,X3)
r140: a__U113(X1,X2,X3) -> U113(X1,X2,X3)
r141: a__U114(X1,X2) -> U114(X1,X2)
r142: a__length(X) -> length(X)
r143: a__U13(X) -> U13(X)
r144: a__isNatList(X) -> isNatList(X)
r145: a__U121(X1,X2) -> U121(X1,X2)
r146: a__U122(X) -> U122(X)
r147: a__U131(X1,X2,X3,X4) -> U131(X1,X2,X3,X4)
r148: a__U132(X1,X2,X3,X4) -> U132(X1,X2,X3,X4)
r149: a__U133(X1,X2,X3,X4) -> U133(X1,X2,X3,X4)
r150: a__U134(X1,X2,X3,X4) -> U134(X1,X2,X3,X4)
r151: a__U135(X1,X2,X3,X4) -> U135(X1,X2,X3,X4)
r152: a__U136(X1,X2,X3,X4) -> U136(X1,X2,X3,X4)
r153: a__take(X1,X2) -> take(X1,X2)
r154: a__U21(X1,X2) -> U21(X1,X2)
r155: a__U22(X1,X2) -> U22(X1,X2)
r156: a__U23(X) -> U23(X)
r157: a__U31(X1,X2) -> U31(X1,X2)
r158: a__U32(X1,X2) -> U32(X1,X2)
r159: a__U33(X) -> U33(X)
r160: a__U41(X1,X2,X3) -> U41(X1,X2,X3)
r161: a__U42(X1,X2,X3) -> U42(X1,X2,X3)
r162: a__U43(X1,X2,X3) -> U43(X1,X2,X3)
r163: a__U44(X1,X2,X3) -> U44(X1,X2,X3)
r164: a__U45(X1,X2) -> U45(X1,X2)
r165: a__U46(X) -> U46(X)
r166: a__U51(X1,X2) -> U51(X1,X2)
r167: a__U52(X) -> U52(X)
r168: a__U61(X1,X2) -> U61(X1,X2)
r169: a__U62(X) -> U62(X)
r170: a__U71(X) -> U71(X)
r171: a__U81(X) -> U81(X)
r172: a__U91(X1,X2,X3) -> U91(X1,X2,X3)
r173: a__U92(X1,X2,X3) -> U92(X1,X2,X3)
r174: a__U93(X1,X2,X3) -> U93(X1,X2,X3)
r175: a__U94(X1,X2,X3) -> U94(X1,X2,X3)
r176: a__U95(X1,X2) -> U95(X1,X2)
r177: a__U96(X) -> U96(X)

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  max/plus interpretations on natural numbers:
  
    mark#_A(x1) = x1 + 1
    U11_A(x1,x2) = max{x1, x2}
    U81_A(x1) = x1
    U62_A(x1) = x1
    U33_A(x1) = x1 + 1
    U134_A(x1,x2,x3,x4) = max{x1, x2, x3, x4}
    U131_A(x1,x2,x3,x4) = max{x1, x2, x3, x4}
    U122_A(x1) = x1
    U111_A(x1,x2,x3) = max{x1, x2, x3}
    U12_A(x1,x2) = max{x1, x2}

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: mark#(U11(X1,X2)) -> mark#(X1)
p2: mark#(U81(X)) -> mark#(X)
p3: mark#(U62(X)) -> mark#(X)
p4: mark#(U134(X1,X2,X3,X4)) -> mark#(X1)
p5: mark#(U131(X1,X2,X3,X4)) -> mark#(X1)
p6: mark#(U122(X)) -> mark#(X)
p7: mark#(U111(X1,X2,X3)) -> mark#(X1)
p8: mark#(U12(X1,X2)) -> mark#(X1)

and R consists of:

r1: a__zeros() -> cons(|0|(),zeros())
r2: a__U101(tt(),V1,V2) -> a__U102(a__isNatKind(V1),V1,V2)
r3: a__U102(tt(),V1,V2) -> a__U103(a__isNatIListKind(V2),V1,V2)
r4: a__U103(tt(),V1,V2) -> a__U104(a__isNatIListKind(V2),V1,V2)
r5: a__U104(tt(),V1,V2) -> a__U105(a__isNat(V1),V2)
r6: a__U105(tt(),V2) -> a__U106(a__isNatIList(V2))
r7: a__U106(tt()) -> tt()
r8: a__U11(tt(),V1) -> a__U12(a__isNatIListKind(V1),V1)
r9: a__U111(tt(),L,N) -> a__U112(a__isNatIListKind(L),L,N)
r10: a__U112(tt(),L,N) -> a__U113(a__isNat(N),L,N)
r11: a__U113(tt(),L,N) -> a__U114(a__isNatKind(N),L)
r12: a__U114(tt(),L) -> s(a__length(mark(L)))
r13: a__U12(tt(),V1) -> a__U13(a__isNatList(V1))
r14: a__U121(tt(),IL) -> a__U122(a__isNatIListKind(IL))
r15: a__U122(tt()) -> nil()
r16: a__U13(tt()) -> tt()
r17: a__U131(tt(),IL,M,N) -> a__U132(a__isNatIListKind(IL),IL,M,N)
r18: a__U132(tt(),IL,M,N) -> a__U133(a__isNat(M),IL,M,N)
r19: a__U133(tt(),IL,M,N) -> a__U134(a__isNatKind(M),IL,M,N)
r20: a__U134(tt(),IL,M,N) -> a__U135(a__isNat(N),IL,M,N)
r21: a__U135(tt(),IL,M,N) -> a__U136(a__isNatKind(N),IL,M,N)
r22: a__U136(tt(),IL,M,N) -> cons(mark(N),take(M,IL))
r23: a__U21(tt(),V1) -> a__U22(a__isNatKind(V1),V1)
r24: a__U22(tt(),V1) -> a__U23(a__isNat(V1))
r25: a__U23(tt()) -> tt()
r26: a__U31(tt(),V) -> a__U32(a__isNatIListKind(V),V)
r27: a__U32(tt(),V) -> a__U33(a__isNatList(V))
r28: a__U33(tt()) -> tt()
r29: a__U41(tt(),V1,V2) -> a__U42(a__isNatKind(V1),V1,V2)
r30: a__U42(tt(),V1,V2) -> a__U43(a__isNatIListKind(V2),V1,V2)
r31: a__U43(tt(),V1,V2) -> a__U44(a__isNatIListKind(V2),V1,V2)
r32: a__U44(tt(),V1,V2) -> a__U45(a__isNat(V1),V2)
r33: a__U45(tt(),V2) -> a__U46(a__isNatIList(V2))
r34: a__U46(tt()) -> tt()
r35: a__U51(tt(),V2) -> a__U52(a__isNatIListKind(V2))
r36: a__U52(tt()) -> tt()
r37: a__U61(tt(),V2) -> a__U62(a__isNatIListKind(V2))
r38: a__U62(tt()) -> tt()
r39: a__U71(tt()) -> tt()
r40: a__U81(tt()) -> tt()
r41: a__U91(tt(),V1,V2) -> a__U92(a__isNatKind(V1),V1,V2)
r42: a__U92(tt(),V1,V2) -> a__U93(a__isNatIListKind(V2),V1,V2)
r43: a__U93(tt(),V1,V2) -> a__U94(a__isNatIListKind(V2),V1,V2)
r44: a__U94(tt(),V1,V2) -> a__U95(a__isNat(V1),V2)
r45: a__U95(tt(),V2) -> a__U96(a__isNatList(V2))
r46: a__U96(tt()) -> tt()
r47: a__isNat(|0|()) -> tt()
r48: a__isNat(length(V1)) -> a__U11(a__isNatIListKind(V1),V1)
r49: a__isNat(s(V1)) -> a__U21(a__isNatKind(V1),V1)
r50: a__isNatIList(V) -> a__U31(a__isNatIListKind(V),V)
r51: a__isNatIList(zeros()) -> tt()
r52: a__isNatIList(cons(V1,V2)) -> a__U41(a__isNatKind(V1),V1,V2)
r53: a__isNatIListKind(nil()) -> tt()
r54: a__isNatIListKind(zeros()) -> tt()
r55: a__isNatIListKind(cons(V1,V2)) -> a__U51(a__isNatKind(V1),V2)
r56: a__isNatIListKind(take(V1,V2)) -> a__U61(a__isNatKind(V1),V2)
r57: a__isNatKind(|0|()) -> tt()
r58: a__isNatKind(length(V1)) -> a__U71(a__isNatIListKind(V1))
r59: a__isNatKind(s(V1)) -> a__U81(a__isNatKind(V1))
r60: a__isNatList(nil()) -> tt()
r61: a__isNatList(cons(V1,V2)) -> a__U91(a__isNatKind(V1),V1,V2)
r62: a__isNatList(take(V1,V2)) -> a__U101(a__isNatKind(V1),V1,V2)
r63: a__length(nil()) -> |0|()
r64: a__length(cons(N,L)) -> a__U111(a__isNatList(L),L,N)
r65: a__take(|0|(),IL) -> a__U121(a__isNatIList(IL),IL)
r66: a__take(s(M),cons(N,IL)) -> a__U131(a__isNatIList(IL),IL,M,N)
r67: mark(zeros()) -> a__zeros()
r68: mark(U101(X1,X2,X3)) -> a__U101(mark(X1),X2,X3)
r69: mark(U102(X1,X2,X3)) -> a__U102(mark(X1),X2,X3)
r70: mark(isNatKind(X)) -> a__isNatKind(X)
r71: mark(U103(X1,X2,X3)) -> a__U103(mark(X1),X2,X3)
r72: mark(isNatIListKind(X)) -> a__isNatIListKind(X)
r73: mark(U104(X1,X2,X3)) -> a__U104(mark(X1),X2,X3)
r74: mark(U105(X1,X2)) -> a__U105(mark(X1),X2)
r75: mark(isNat(X)) -> a__isNat(X)
r76: mark(U106(X)) -> a__U106(mark(X))
r77: mark(isNatIList(X)) -> a__isNatIList(X)
r78: mark(U11(X1,X2)) -> a__U11(mark(X1),X2)
r79: mark(U12(X1,X2)) -> a__U12(mark(X1),X2)
r80: mark(U111(X1,X2,X3)) -> a__U111(mark(X1),X2,X3)
r81: mark(U112(X1,X2,X3)) -> a__U112(mark(X1),X2,X3)
r82: mark(U113(X1,X2,X3)) -> a__U113(mark(X1),X2,X3)
r83: mark(U114(X1,X2)) -> a__U114(mark(X1),X2)
r84: mark(length(X)) -> a__length(mark(X))
r85: mark(U13(X)) -> a__U13(mark(X))
r86: mark(isNatList(X)) -> a__isNatList(X)
r87: mark(U121(X1,X2)) -> a__U121(mark(X1),X2)
r88: mark(U122(X)) -> a__U122(mark(X))
r89: mark(U131(X1,X2,X3,X4)) -> a__U131(mark(X1),X2,X3,X4)
r90: mark(U132(X1,X2,X3,X4)) -> a__U132(mark(X1),X2,X3,X4)
r91: mark(U133(X1,X2,X3,X4)) -> a__U133(mark(X1),X2,X3,X4)
r92: mark(U134(X1,X2,X3,X4)) -> a__U134(mark(X1),X2,X3,X4)
r93: mark(U135(X1,X2,X3,X4)) -> a__U135(mark(X1),X2,X3,X4)
r94: mark(U136(X1,X2,X3,X4)) -> a__U136(mark(X1),X2,X3,X4)
r95: mark(take(X1,X2)) -> a__take(mark(X1),mark(X2))
r96: mark(U21(X1,X2)) -> a__U21(mark(X1),X2)
r97: mark(U22(X1,X2)) -> a__U22(mark(X1),X2)
r98: mark(U23(X)) -> a__U23(mark(X))
r99: mark(U31(X1,X2)) -> a__U31(mark(X1),X2)
r100: mark(U32(X1,X2)) -> a__U32(mark(X1),X2)
r101: mark(U33(X)) -> a__U33(mark(X))
r102: mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3)
r103: mark(U42(X1,X2,X3)) -> a__U42(mark(X1),X2,X3)
r104: mark(U43(X1,X2,X3)) -> a__U43(mark(X1),X2,X3)
r105: mark(U44(X1,X2,X3)) -> a__U44(mark(X1),X2,X3)
r106: mark(U45(X1,X2)) -> a__U45(mark(X1),X2)
r107: mark(U46(X)) -> a__U46(mark(X))
r108: mark(U51(X1,X2)) -> a__U51(mark(X1),X2)
r109: mark(U52(X)) -> a__U52(mark(X))
r110: mark(U61(X1,X2)) -> a__U61(mark(X1),X2)
r111: mark(U62(X)) -> a__U62(mark(X))
r112: mark(U71(X)) -> a__U71(mark(X))
r113: mark(U81(X)) -> a__U81(mark(X))
r114: mark(U91(X1,X2,X3)) -> a__U91(mark(X1),X2,X3)
r115: mark(U92(X1,X2,X3)) -> a__U92(mark(X1),X2,X3)
r116: mark(U93(X1,X2,X3)) -> a__U93(mark(X1),X2,X3)
r117: mark(U94(X1,X2,X3)) -> a__U94(mark(X1),X2,X3)
r118: mark(U95(X1,X2)) -> a__U95(mark(X1),X2)
r119: mark(U96(X)) -> a__U96(mark(X))
r120: mark(cons(X1,X2)) -> cons(mark(X1),X2)
r121: mark(|0|()) -> |0|()
r122: mark(tt()) -> tt()
r123: mark(s(X)) -> s(mark(X))
r124: mark(nil()) -> nil()
r125: a__zeros() -> zeros()
r126: a__U101(X1,X2,X3) -> U101(X1,X2,X3)
r127: a__U102(X1,X2,X3) -> U102(X1,X2,X3)
r128: a__isNatKind(X) -> isNatKind(X)
r129: a__U103(X1,X2,X3) -> U103(X1,X2,X3)
r130: a__isNatIListKind(X) -> isNatIListKind(X)
r131: a__U104(X1,X2,X3) -> U104(X1,X2,X3)
r132: a__U105(X1,X2) -> U105(X1,X2)
r133: a__isNat(X) -> isNat(X)
r134: a__U106(X) -> U106(X)
r135: a__isNatIList(X) -> isNatIList(X)
r136: a__U11(X1,X2) -> U11(X1,X2)
r137: a__U12(X1,X2) -> U12(X1,X2)
r138: a__U111(X1,X2,X3) -> U111(X1,X2,X3)
r139: a__U112(X1,X2,X3) -> U112(X1,X2,X3)
r140: a__U113(X1,X2,X3) -> U113(X1,X2,X3)
r141: a__U114(X1,X2) -> U114(X1,X2)
r142: a__length(X) -> length(X)
r143: a__U13(X) -> U13(X)
r144: a__isNatList(X) -> isNatList(X)
r145: a__U121(X1,X2) -> U121(X1,X2)
r146: a__U122(X) -> U122(X)
r147: a__U131(X1,X2,X3,X4) -> U131(X1,X2,X3,X4)
r148: a__U132(X1,X2,X3,X4) -> U132(X1,X2,X3,X4)
r149: a__U133(X1,X2,X3,X4) -> U133(X1,X2,X3,X4)
r150: a__U134(X1,X2,X3,X4) -> U134(X1,X2,X3,X4)
r151: a__U135(X1,X2,X3,X4) -> U135(X1,X2,X3,X4)
r152: a__U136(X1,X2,X3,X4) -> U136(X1,X2,X3,X4)
r153: a__take(X1,X2) -> take(X1,X2)
r154: a__U21(X1,X2) -> U21(X1,X2)
r155: a__U22(X1,X2) -> U22(X1,X2)
r156: a__U23(X) -> U23(X)
r157: a__U31(X1,X2) -> U31(X1,X2)
r158: a__U32(X1,X2) -> U32(X1,X2)
r159: a__U33(X) -> U33(X)
r160: a__U41(X1,X2,X3) -> U41(X1,X2,X3)
r161: a__U42(X1,X2,X3) -> U42(X1,X2,X3)
r162: a__U43(X1,X2,X3) -> U43(X1,X2,X3)
r163: a__U44(X1,X2,X3) -> U44(X1,X2,X3)
r164: a__U45(X1,X2) -> U45(X1,X2)
r165: a__U46(X) -> U46(X)
r166: a__U51(X1,X2) -> U51(X1,X2)
r167: a__U52(X) -> U52(X)
r168: a__U61(X1,X2) -> U61(X1,X2)
r169: a__U62(X) -> U62(X)
r170: a__U71(X) -> U71(X)
r171: a__U81(X) -> U81(X)
r172: a__U91(X1,X2,X3) -> U91(X1,X2,X3)
r173: a__U92(X1,X2,X3) -> U92(X1,X2,X3)
r174: a__U93(X1,X2,X3) -> U93(X1,X2,X3)
r175: a__U94(X1,X2,X3) -> U94(X1,X2,X3)
r176: a__U95(X1,X2) -> U95(X1,X2)
r177: a__U96(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#(U11(X1,X2)) -> mark#(X1)
p2: mark#(U12(X1,X2)) -> mark#(X1)
p3: mark#(U111(X1,X2,X3)) -> mark#(X1)
p4: mark#(U122(X)) -> mark#(X)
p5: mark#(U131(X1,X2,X3,X4)) -> mark#(X1)
p6: mark#(U134(X1,X2,X3,X4)) -> mark#(X1)
p7: mark#(U62(X)) -> mark#(X)
p8: mark#(U81(X)) -> mark#(X)

and R consists of:

r1: a__zeros() -> cons(|0|(),zeros())
r2: a__U101(tt(),V1,V2) -> a__U102(a__isNatKind(V1),V1,V2)
r3: a__U102(tt(),V1,V2) -> a__U103(a__isNatIListKind(V2),V1,V2)
r4: a__U103(tt(),V1,V2) -> a__U104(a__isNatIListKind(V2),V1,V2)
r5: a__U104(tt(),V1,V2) -> a__U105(a__isNat(V1),V2)
r6: a__U105(tt(),V2) -> a__U106(a__isNatIList(V2))
r7: a__U106(tt()) -> tt()
r8: a__U11(tt(),V1) -> a__U12(a__isNatIListKind(V1),V1)
r9: a__U111(tt(),L,N) -> a__U112(a__isNatIListKind(L),L,N)
r10: a__U112(tt(),L,N) -> a__U113(a__isNat(N),L,N)
r11: a__U113(tt(),L,N) -> a__U114(a__isNatKind(N),L)
r12: a__U114(tt(),L) -> s(a__length(mark(L)))
r13: a__U12(tt(),V1) -> a__U13(a__isNatList(V1))
r14: a__U121(tt(),IL) -> a__U122(a__isNatIListKind(IL))
r15: a__U122(tt()) -> nil()
r16: a__U13(tt()) -> tt()
r17: a__U131(tt(),IL,M,N) -> a__U132(a__isNatIListKind(IL),IL,M,N)
r18: a__U132(tt(),IL,M,N) -> a__U133(a__isNat(M),IL,M,N)
r19: a__U133(tt(),IL,M,N) -> a__U134(a__isNatKind(M),IL,M,N)
r20: a__U134(tt(),IL,M,N) -> a__U135(a__isNat(N),IL,M,N)
r21: a__U135(tt(),IL,M,N) -> a__U136(a__isNatKind(N),IL,M,N)
r22: a__U136(tt(),IL,M,N) -> cons(mark(N),take(M,IL))
r23: a__U21(tt(),V1) -> a__U22(a__isNatKind(V1),V1)
r24: a__U22(tt(),V1) -> a__U23(a__isNat(V1))
r25: a__U23(tt()) -> tt()
r26: a__U31(tt(),V) -> a__U32(a__isNatIListKind(V),V)
r27: a__U32(tt(),V) -> a__U33(a__isNatList(V))
r28: a__U33(tt()) -> tt()
r29: a__U41(tt(),V1,V2) -> a__U42(a__isNatKind(V1),V1,V2)
r30: a__U42(tt(),V1,V2) -> a__U43(a__isNatIListKind(V2),V1,V2)
r31: a__U43(tt(),V1,V2) -> a__U44(a__isNatIListKind(V2),V1,V2)
r32: a__U44(tt(),V1,V2) -> a__U45(a__isNat(V1),V2)
r33: a__U45(tt(),V2) -> a__U46(a__isNatIList(V2))
r34: a__U46(tt()) -> tt()
r35: a__U51(tt(),V2) -> a__U52(a__isNatIListKind(V2))
r36: a__U52(tt()) -> tt()
r37: a__U61(tt(),V2) -> a__U62(a__isNatIListKind(V2))
r38: a__U62(tt()) -> tt()
r39: a__U71(tt()) -> tt()
r40: a__U81(tt()) -> tt()
r41: a__U91(tt(),V1,V2) -> a__U92(a__isNatKind(V1),V1,V2)
r42: a__U92(tt(),V1,V2) -> a__U93(a__isNatIListKind(V2),V1,V2)
r43: a__U93(tt(),V1,V2) -> a__U94(a__isNatIListKind(V2),V1,V2)
r44: a__U94(tt(),V1,V2) -> a__U95(a__isNat(V1),V2)
r45: a__U95(tt(),V2) -> a__U96(a__isNatList(V2))
r46: a__U96(tt()) -> tt()
r47: a__isNat(|0|()) -> tt()
r48: a__isNat(length(V1)) -> a__U11(a__isNatIListKind(V1),V1)
r49: a__isNat(s(V1)) -> a__U21(a__isNatKind(V1),V1)
r50: a__isNatIList(V) -> a__U31(a__isNatIListKind(V),V)
r51: a__isNatIList(zeros()) -> tt()
r52: a__isNatIList(cons(V1,V2)) -> a__U41(a__isNatKind(V1),V1,V2)
r53: a__isNatIListKind(nil()) -> tt()
r54: a__isNatIListKind(zeros()) -> tt()
r55: a__isNatIListKind(cons(V1,V2)) -> a__U51(a__isNatKind(V1),V2)
r56: a__isNatIListKind(take(V1,V2)) -> a__U61(a__isNatKind(V1),V2)
r57: a__isNatKind(|0|()) -> tt()
r58: a__isNatKind(length(V1)) -> a__U71(a__isNatIListKind(V1))
r59: a__isNatKind(s(V1)) -> a__U81(a__isNatKind(V1))
r60: a__isNatList(nil()) -> tt()
r61: a__isNatList(cons(V1,V2)) -> a__U91(a__isNatKind(V1),V1,V2)
r62: a__isNatList(take(V1,V2)) -> a__U101(a__isNatKind(V1),V1,V2)
r63: a__length(nil()) -> |0|()
r64: a__length(cons(N,L)) -> a__U111(a__isNatList(L),L,N)
r65: a__take(|0|(),IL) -> a__U121(a__isNatIList(IL),IL)
r66: a__take(s(M),cons(N,IL)) -> a__U131(a__isNatIList(IL),IL,M,N)
r67: mark(zeros()) -> a__zeros()
r68: mark(U101(X1,X2,X3)) -> a__U101(mark(X1),X2,X3)
r69: mark(U102(X1,X2,X3)) -> a__U102(mark(X1),X2,X3)
r70: mark(isNatKind(X)) -> a__isNatKind(X)
r71: mark(U103(X1,X2,X3)) -> a__U103(mark(X1),X2,X3)
r72: mark(isNatIListKind(X)) -> a__isNatIListKind(X)
r73: mark(U104(X1,X2,X3)) -> a__U104(mark(X1),X2,X3)
r74: mark(U105(X1,X2)) -> a__U105(mark(X1),X2)
r75: mark(isNat(X)) -> a__isNat(X)
r76: mark(U106(X)) -> a__U106(mark(X))
r77: mark(isNatIList(X)) -> a__isNatIList(X)
r78: mark(U11(X1,X2)) -> a__U11(mark(X1),X2)
r79: mark(U12(X1,X2)) -> a__U12(mark(X1),X2)
r80: mark(U111(X1,X2,X3)) -> a__U111(mark(X1),X2,X3)
r81: mark(U112(X1,X2,X3)) -> a__U112(mark(X1),X2,X3)
r82: mark(U113(X1,X2,X3)) -> a__U113(mark(X1),X2,X3)
r83: mark(U114(X1,X2)) -> a__U114(mark(X1),X2)
r84: mark(length(X)) -> a__length(mark(X))
r85: mark(U13(X)) -> a__U13(mark(X))
r86: mark(isNatList(X)) -> a__isNatList(X)
r87: mark(U121(X1,X2)) -> a__U121(mark(X1),X2)
r88: mark(U122(X)) -> a__U122(mark(X))
r89: mark(U131(X1,X2,X3,X4)) -> a__U131(mark(X1),X2,X3,X4)
r90: mark(U132(X1,X2,X3,X4)) -> a__U132(mark(X1),X2,X3,X4)
r91: mark(U133(X1,X2,X3,X4)) -> a__U133(mark(X1),X2,X3,X4)
r92: mark(U134(X1,X2,X3,X4)) -> a__U134(mark(X1),X2,X3,X4)
r93: mark(U135(X1,X2,X3,X4)) -> a__U135(mark(X1),X2,X3,X4)
r94: mark(U136(X1,X2,X3,X4)) -> a__U136(mark(X1),X2,X3,X4)
r95: mark(take(X1,X2)) -> a__take(mark(X1),mark(X2))
r96: mark(U21(X1,X2)) -> a__U21(mark(X1),X2)
r97: mark(U22(X1,X2)) -> a__U22(mark(X1),X2)
r98: mark(U23(X)) -> a__U23(mark(X))
r99: mark(U31(X1,X2)) -> a__U31(mark(X1),X2)
r100: mark(U32(X1,X2)) -> a__U32(mark(X1),X2)
r101: mark(U33(X)) -> a__U33(mark(X))
r102: mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3)
r103: mark(U42(X1,X2,X3)) -> a__U42(mark(X1),X2,X3)
r104: mark(U43(X1,X2,X3)) -> a__U43(mark(X1),X2,X3)
r105: mark(U44(X1,X2,X3)) -> a__U44(mark(X1),X2,X3)
r106: mark(U45(X1,X2)) -> a__U45(mark(X1),X2)
r107: mark(U46(X)) -> a__U46(mark(X))
r108: mark(U51(X1,X2)) -> a__U51(mark(X1),X2)
r109: mark(U52(X)) -> a__U52(mark(X))
r110: mark(U61(X1,X2)) -> a__U61(mark(X1),X2)
r111: mark(U62(X)) -> a__U62(mark(X))
r112: mark(U71(X)) -> a__U71(mark(X))
r113: mark(U81(X)) -> a__U81(mark(X))
r114: mark(U91(X1,X2,X3)) -> a__U91(mark(X1),X2,X3)
r115: mark(U92(X1,X2,X3)) -> a__U92(mark(X1),X2,X3)
r116: mark(U93(X1,X2,X3)) -> a__U93(mark(X1),X2,X3)
r117: mark(U94(X1,X2,X3)) -> a__U94(mark(X1),X2,X3)
r118: mark(U95(X1,X2)) -> a__U95(mark(X1),X2)
r119: mark(U96(X)) -> a__U96(mark(X))
r120: mark(cons(X1,X2)) -> cons(mark(X1),X2)
r121: mark(|0|()) -> |0|()
r122: mark(tt()) -> tt()
r123: mark(s(X)) -> s(mark(X))
r124: mark(nil()) -> nil()
r125: a__zeros() -> zeros()
r126: a__U101(X1,X2,X3) -> U101(X1,X2,X3)
r127: a__U102(X1,X2,X3) -> U102(X1,X2,X3)
r128: a__isNatKind(X) -> isNatKind(X)
r129: a__U103(X1,X2,X3) -> U103(X1,X2,X3)
r130: a__isNatIListKind(X) -> isNatIListKind(X)
r131: a__U104(X1,X2,X3) -> U104(X1,X2,X3)
r132: a__U105(X1,X2) -> U105(X1,X2)
r133: a__isNat(X) -> isNat(X)
r134: a__U106(X) -> U106(X)
r135: a__isNatIList(X) -> isNatIList(X)
r136: a__U11(X1,X2) -> U11(X1,X2)
r137: a__U12(X1,X2) -> U12(X1,X2)
r138: a__U111(X1,X2,X3) -> U111(X1,X2,X3)
r139: a__U112(X1,X2,X3) -> U112(X1,X2,X3)
r140: a__U113(X1,X2,X3) -> U113(X1,X2,X3)
r141: a__U114(X1,X2) -> U114(X1,X2)
r142: a__length(X) -> length(X)
r143: a__U13(X) -> U13(X)
r144: a__isNatList(X) -> isNatList(X)
r145: a__U121(X1,X2) -> U121(X1,X2)
r146: a__U122(X) -> U122(X)
r147: a__U131(X1,X2,X3,X4) -> U131(X1,X2,X3,X4)
r148: a__U132(X1,X2,X3,X4) -> U132(X1,X2,X3,X4)
r149: a__U133(X1,X2,X3,X4) -> U133(X1,X2,X3,X4)
r150: a__U134(X1,X2,X3,X4) -> U134(X1,X2,X3,X4)
r151: a__U135(X1,X2,X3,X4) -> U135(X1,X2,X3,X4)
r152: a__U136(X1,X2,X3,X4) -> U136(X1,X2,X3,X4)
r153: a__take(X1,X2) -> take(X1,X2)
r154: a__U21(X1,X2) -> U21(X1,X2)
r155: a__U22(X1,X2) -> U22(X1,X2)
r156: a__U23(X) -> U23(X)
r157: a__U31(X1,X2) -> U31(X1,X2)
r158: a__U32(X1,X2) -> U32(X1,X2)
r159: a__U33(X) -> U33(X)
r160: a__U41(X1,X2,X3) -> U41(X1,X2,X3)
r161: a__U42(X1,X2,X3) -> U42(X1,X2,X3)
r162: a__U43(X1,X2,X3) -> U43(X1,X2,X3)
r163: a__U44(X1,X2,X3) -> U44(X1,X2,X3)
r164: a__U45(X1,X2) -> U45(X1,X2)
r165: a__U46(X) -> U46(X)
r166: a__U51(X1,X2) -> U51(X1,X2)
r167: a__U52(X) -> U52(X)
r168: a__U61(X1,X2) -> U61(X1,X2)
r169: a__U62(X) -> U62(X)
r170: a__U71(X) -> U71(X)
r171: a__U81(X) -> U81(X)
r172: a__U91(X1,X2,X3) -> U91(X1,X2,X3)
r173: a__U92(X1,X2,X3) -> U92(X1,X2,X3)
r174: a__U93(X1,X2,X3) -> U93(X1,X2,X3)
r175: a__U94(X1,X2,X3) -> U94(X1,X2,X3)
r176: a__U95(X1,X2) -> U95(X1,X2)
r177: a__U96(X) -> U96(X)

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  max/plus interpretations on natural numbers:
  
    mark#_A(x1) = max{1, x1}
    U11_A(x1,x2) = max{x1 + 1, x2 + 1}
    U12_A(x1,x2) = max{x1, x2 + 2}
    U111_A(x1,x2,x3) = max{x1, x2 + 1}
    U122_A(x1) = x1 + 1
    U131_A(x1,x2,x3,x4) = max{x1, x2 + 1}
    U134_A(x1,x2,x3,x4) = max{x1, x3 + 1}
    U62_A(x1) = x1 + 2
    U81_A(x1) = x1 + 1

The next rules are strictly ordered:

  p7

We remove them from the problem.

-- SCC decomposition.

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

p1: mark#(U11(X1,X2)) -> mark#(X1)
p2: mark#(U12(X1,X2)) -> mark#(X1)
p3: mark#(U111(X1,X2,X3)) -> mark#(X1)
p4: mark#(U122(X)) -> mark#(X)
p5: mark#(U131(X1,X2,X3,X4)) -> mark#(X1)
p6: mark#(U134(X1,X2,X3,X4)) -> mark#(X1)
p7: mark#(U81(X)) -> mark#(X)

and R consists of:

r1: a__zeros() -> cons(|0|(),zeros())
r2: a__U101(tt(),V1,V2) -> a__U102(a__isNatKind(V1),V1,V2)
r3: a__U102(tt(),V1,V2) -> a__U103(a__isNatIListKind(V2),V1,V2)
r4: a__U103(tt(),V1,V2) -> a__U104(a__isNatIListKind(V2),V1,V2)
r5: a__U104(tt(),V1,V2) -> a__U105(a__isNat(V1),V2)
r6: a__U105(tt(),V2) -> a__U106(a__isNatIList(V2))
r7: a__U106(tt()) -> tt()
r8: a__U11(tt(),V1) -> a__U12(a__isNatIListKind(V1),V1)
r9: a__U111(tt(),L,N) -> a__U112(a__isNatIListKind(L),L,N)
r10: a__U112(tt(),L,N) -> a__U113(a__isNat(N),L,N)
r11: a__U113(tt(),L,N) -> a__U114(a__isNatKind(N),L)
r12: a__U114(tt(),L) -> s(a__length(mark(L)))
r13: a__U12(tt(),V1) -> a__U13(a__isNatList(V1))
r14: a__U121(tt(),IL) -> a__U122(a__isNatIListKind(IL))
r15: a__U122(tt()) -> nil()
r16: a__U13(tt()) -> tt()
r17: a__U131(tt(),IL,M,N) -> a__U132(a__isNatIListKind(IL),IL,M,N)
r18: a__U132(tt(),IL,M,N) -> a__U133(a__isNat(M),IL,M,N)
r19: a__U133(tt(),IL,M,N) -> a__U134(a__isNatKind(M),IL,M,N)
r20: a__U134(tt(),IL,M,N) -> a__U135(a__isNat(N),IL,M,N)
r21: a__U135(tt(),IL,M,N) -> a__U136(a__isNatKind(N),IL,M,N)
r22: a__U136(tt(),IL,M,N) -> cons(mark(N),take(M,IL))
r23: a__U21(tt(),V1) -> a__U22(a__isNatKind(V1),V1)
r24: a__U22(tt(),V1) -> a__U23(a__isNat(V1))
r25: a__U23(tt()) -> tt()
r26: a__U31(tt(),V) -> a__U32(a__isNatIListKind(V),V)
r27: a__U32(tt(),V) -> a__U33(a__isNatList(V))
r28: a__U33(tt()) -> tt()
r29: a__U41(tt(),V1,V2) -> a__U42(a__isNatKind(V1),V1,V2)
r30: a__U42(tt(),V1,V2) -> a__U43(a__isNatIListKind(V2),V1,V2)
r31: a__U43(tt(),V1,V2) -> a__U44(a__isNatIListKind(V2),V1,V2)
r32: a__U44(tt(),V1,V2) -> a__U45(a__isNat(V1),V2)
r33: a__U45(tt(),V2) -> a__U46(a__isNatIList(V2))
r34: a__U46(tt()) -> tt()
r35: a__U51(tt(),V2) -> a__U52(a__isNatIListKind(V2))
r36: a__U52(tt()) -> tt()
r37: a__U61(tt(),V2) -> a__U62(a__isNatIListKind(V2))
r38: a__U62(tt()) -> tt()
r39: a__U71(tt()) -> tt()
r40: a__U81(tt()) -> tt()
r41: a__U91(tt(),V1,V2) -> a__U92(a__isNatKind(V1),V1,V2)
r42: a__U92(tt(),V1,V2) -> a__U93(a__isNatIListKind(V2),V1,V2)
r43: a__U93(tt(),V1,V2) -> a__U94(a__isNatIListKind(V2),V1,V2)
r44: a__U94(tt(),V1,V2) -> a__U95(a__isNat(V1),V2)
r45: a__U95(tt(),V2) -> a__U96(a__isNatList(V2))
r46: a__U96(tt()) -> tt()
r47: a__isNat(|0|()) -> tt()
r48: a__isNat(length(V1)) -> a__U11(a__isNatIListKind(V1),V1)
r49: a__isNat(s(V1)) -> a__U21(a__isNatKind(V1),V1)
r50: a__isNatIList(V) -> a__U31(a__isNatIListKind(V),V)
r51: a__isNatIList(zeros()) -> tt()
r52: a__isNatIList(cons(V1,V2)) -> a__U41(a__isNatKind(V1),V1,V2)
r53: a__isNatIListKind(nil()) -> tt()
r54: a__isNatIListKind(zeros()) -> tt()
r55: a__isNatIListKind(cons(V1,V2)) -> a__U51(a__isNatKind(V1),V2)
r56: a__isNatIListKind(take(V1,V2)) -> a__U61(a__isNatKind(V1),V2)
r57: a__isNatKind(|0|()) -> tt()
r58: a__isNatKind(length(V1)) -> a__U71(a__isNatIListKind(V1))
r59: a__isNatKind(s(V1)) -> a__U81(a__isNatKind(V1))
r60: a__isNatList(nil()) -> tt()
r61: a__isNatList(cons(V1,V2)) -> a__U91(a__isNatKind(V1),V1,V2)
r62: a__isNatList(take(V1,V2)) -> a__U101(a__isNatKind(V1),V1,V2)
r63: a__length(nil()) -> |0|()
r64: a__length(cons(N,L)) -> a__U111(a__isNatList(L),L,N)
r65: a__take(|0|(),IL) -> a__U121(a__isNatIList(IL),IL)
r66: a__take(s(M),cons(N,IL)) -> a__U131(a__isNatIList(IL),IL,M,N)
r67: mark(zeros()) -> a__zeros()
r68: mark(U101(X1,X2,X3)) -> a__U101(mark(X1),X2,X3)
r69: mark(U102(X1,X2,X3)) -> a__U102(mark(X1),X2,X3)
r70: mark(isNatKind(X)) -> a__isNatKind(X)
r71: mark(U103(X1,X2,X3)) -> a__U103(mark(X1),X2,X3)
r72: mark(isNatIListKind(X)) -> a__isNatIListKind(X)
r73: mark(U104(X1,X2,X3)) -> a__U104(mark(X1),X2,X3)
r74: mark(U105(X1,X2)) -> a__U105(mark(X1),X2)
r75: mark(isNat(X)) -> a__isNat(X)
r76: mark(U106(X)) -> a__U106(mark(X))
r77: mark(isNatIList(X)) -> a__isNatIList(X)
r78: mark(U11(X1,X2)) -> a__U11(mark(X1),X2)
r79: mark(U12(X1,X2)) -> a__U12(mark(X1),X2)
r80: mark(U111(X1,X2,X3)) -> a__U111(mark(X1),X2,X3)
r81: mark(U112(X1,X2,X3)) -> a__U112(mark(X1),X2,X3)
r82: mark(U113(X1,X2,X3)) -> a__U113(mark(X1),X2,X3)
r83: mark(U114(X1,X2)) -> a__U114(mark(X1),X2)
r84: mark(length(X)) -> a__length(mark(X))
r85: mark(U13(X)) -> a__U13(mark(X))
r86: mark(isNatList(X)) -> a__isNatList(X)
r87: mark(U121(X1,X2)) -> a__U121(mark(X1),X2)
r88: mark(U122(X)) -> a__U122(mark(X))
r89: mark(U131(X1,X2,X3,X4)) -> a__U131(mark(X1),X2,X3,X4)
r90: mark(U132(X1,X2,X3,X4)) -> a__U132(mark(X1),X2,X3,X4)
r91: mark(U133(X1,X2,X3,X4)) -> a__U133(mark(X1),X2,X3,X4)
r92: mark(U134(X1,X2,X3,X4)) -> a__U134(mark(X1),X2,X3,X4)
r93: mark(U135(X1,X2,X3,X4)) -> a__U135(mark(X1),X2,X3,X4)
r94: mark(U136(X1,X2,X3,X4)) -> a__U136(mark(X1),X2,X3,X4)
r95: mark(take(X1,X2)) -> a__take(mark(X1),mark(X2))
r96: mark(U21(X1,X2)) -> a__U21(mark(X1),X2)
r97: mark(U22(X1,X2)) -> a__U22(mark(X1),X2)
r98: mark(U23(X)) -> a__U23(mark(X))
r99: mark(U31(X1,X2)) -> a__U31(mark(X1),X2)
r100: mark(U32(X1,X2)) -> a__U32(mark(X1),X2)
r101: mark(U33(X)) -> a__U33(mark(X))
r102: mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3)
r103: mark(U42(X1,X2,X3)) -> a__U42(mark(X1),X2,X3)
r104: mark(U43(X1,X2,X3)) -> a__U43(mark(X1),X2,X3)
r105: mark(U44(X1,X2,X3)) -> a__U44(mark(X1),X2,X3)
r106: mark(U45(X1,X2)) -> a__U45(mark(X1),X2)
r107: mark(U46(X)) -> a__U46(mark(X))
r108: mark(U51(X1,X2)) -> a__U51(mark(X1),X2)
r109: mark(U52(X)) -> a__U52(mark(X))
r110: mark(U61(X1,X2)) -> a__U61(mark(X1),X2)
r111: mark(U62(X)) -> a__U62(mark(X))
r112: mark(U71(X)) -> a__U71(mark(X))
r113: mark(U81(X)) -> a__U81(mark(X))
r114: mark(U91(X1,X2,X3)) -> a__U91(mark(X1),X2,X3)
r115: mark(U92(X1,X2,X3)) -> a__U92(mark(X1),X2,X3)
r116: mark(U93(X1,X2,X3)) -> a__U93(mark(X1),X2,X3)
r117: mark(U94(X1,X2,X3)) -> a__U94(mark(X1),X2,X3)
r118: mark(U95(X1,X2)) -> a__U95(mark(X1),X2)
r119: mark(U96(X)) -> a__U96(mark(X))
r120: mark(cons(X1,X2)) -> cons(mark(X1),X2)
r121: mark(|0|()) -> |0|()
r122: mark(tt()) -> tt()
r123: mark(s(X)) -> s(mark(X))
r124: mark(nil()) -> nil()
r125: a__zeros() -> zeros()
r126: a__U101(X1,X2,X3) -> U101(X1,X2,X3)
r127: a__U102(X1,X2,X3) -> U102(X1,X2,X3)
r128: a__isNatKind(X) -> isNatKind(X)
r129: a__U103(X1,X2,X3) -> U103(X1,X2,X3)
r130: a__isNatIListKind(X) -> isNatIListKind(X)
r131: a__U104(X1,X2,X3) -> U104(X1,X2,X3)
r132: a__U105(X1,X2) -> U105(X1,X2)
r133: a__isNat(X) -> isNat(X)
r134: a__U106(X) -> U106(X)
r135: a__isNatIList(X) -> isNatIList(X)
r136: a__U11(X1,X2) -> U11(X1,X2)
r137: a__U12(X1,X2) -> U12(X1,X2)
r138: a__U111(X1,X2,X3) -> U111(X1,X2,X3)
r139: a__U112(X1,X2,X3) -> U112(X1,X2,X3)
r140: a__U113(X1,X2,X3) -> U113(X1,X2,X3)
r141: a__U114(X1,X2) -> U114(X1,X2)
r142: a__length(X) -> length(X)
r143: a__U13(X) -> U13(X)
r144: a__isNatList(X) -> isNatList(X)
r145: a__U121(X1,X2) -> U121(X1,X2)
r146: a__U122(X) -> U122(X)
r147: a__U131(X1,X2,X3,X4) -> U131(X1,X2,X3,X4)
r148: a__U132(X1,X2,X3,X4) -> U132(X1,X2,X3,X4)
r149: a__U133(X1,X2,X3,X4) -> U133(X1,X2,X3,X4)
r150: a__U134(X1,X2,X3,X4) -> U134(X1,X2,X3,X4)
r151: a__U135(X1,X2,X3,X4) -> U135(X1,X2,X3,X4)
r152: a__U136(X1,X2,X3,X4) -> U136(X1,X2,X3,X4)
r153: a__take(X1,X2) -> take(X1,X2)
r154: a__U21(X1,X2) -> U21(X1,X2)
r155: a__U22(X1,X2) -> U22(X1,X2)
r156: a__U23(X) -> U23(X)
r157: a__U31(X1,X2) -> U31(X1,X2)
r158: a__U32(X1,X2) -> U32(X1,X2)
r159: a__U33(X) -> U33(X)
r160: a__U41(X1,X2,X3) -> U41(X1,X2,X3)
r161: a__U42(X1,X2,X3) -> U42(X1,X2,X3)
r162: a__U43(X1,X2,X3) -> U43(X1,X2,X3)
r163: a__U44(X1,X2,X3) -> U44(X1,X2,X3)
r164: a__U45(X1,X2) -> U45(X1,X2)
r165: a__U46(X) -> U46(X)
r166: a__U51(X1,X2) -> U51(X1,X2)
r167: a__U52(X) -> U52(X)
r168: a__U61(X1,X2) -> U61(X1,X2)
r169: a__U62(X) -> U62(X)
r170: a__U71(X) -> U71(X)
r171: a__U81(X) -> U81(X)
r172: a__U91(X1,X2,X3) -> U91(X1,X2,X3)
r173: a__U92(X1,X2,X3) -> U92(X1,X2,X3)
r174: a__U93(X1,X2,X3) -> U93(X1,X2,X3)
r175: a__U94(X1,X2,X3) -> U94(X1,X2,X3)
r176: a__U95(X1,X2) -> U95(X1,X2)
r177: a__U96(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: mark#(U11(X1,X2)) -> mark#(X1)
p2: mark#(U81(X)) -> mark#(X)
p3: mark#(U134(X1,X2,X3,X4)) -> mark#(X1)
p4: mark#(U131(X1,X2,X3,X4)) -> mark#(X1)
p5: mark#(U122(X)) -> mark#(X)
p6: mark#(U111(X1,X2,X3)) -> mark#(X1)
p7: mark#(U12(X1,X2)) -> mark#(X1)

and R consists of:

r1: a__zeros() -> cons(|0|(),zeros())
r2: a__U101(tt(),V1,V2) -> a__U102(a__isNatKind(V1),V1,V2)
r3: a__U102(tt(),V1,V2) -> a__U103(a__isNatIListKind(V2),V1,V2)
r4: a__U103(tt(),V1,V2) -> a__U104(a__isNatIListKind(V2),V1,V2)
r5: a__U104(tt(),V1,V2) -> a__U105(a__isNat(V1),V2)
r6: a__U105(tt(),V2) -> a__U106(a__isNatIList(V2))
r7: a__U106(tt()) -> tt()
r8: a__U11(tt(),V1) -> a__U12(a__isNatIListKind(V1),V1)
r9: a__U111(tt(),L,N) -> a__U112(a__isNatIListKind(L),L,N)
r10: a__U112(tt(),L,N) -> a__U113(a__isNat(N),L,N)
r11: a__U113(tt(),L,N) -> a__U114(a__isNatKind(N),L)
r12: a__U114(tt(),L) -> s(a__length(mark(L)))
r13: a__U12(tt(),V1) -> a__U13(a__isNatList(V1))
r14: a__U121(tt(),IL) -> a__U122(a__isNatIListKind(IL))
r15: a__U122(tt()) -> nil()
r16: a__U13(tt()) -> tt()
r17: a__U131(tt(),IL,M,N) -> a__U132(a__isNatIListKind(IL),IL,M,N)
r18: a__U132(tt(),IL,M,N) -> a__U133(a__isNat(M),IL,M,N)
r19: a__U133(tt(),IL,M,N) -> a__U134(a__isNatKind(M),IL,M,N)
r20: a__U134(tt(),IL,M,N) -> a__U135(a__isNat(N),IL,M,N)
r21: a__U135(tt(),IL,M,N) -> a__U136(a__isNatKind(N),IL,M,N)
r22: a__U136(tt(),IL,M,N) -> cons(mark(N),take(M,IL))
r23: a__U21(tt(),V1) -> a__U22(a__isNatKind(V1),V1)
r24: a__U22(tt(),V1) -> a__U23(a__isNat(V1))
r25: a__U23(tt()) -> tt()
r26: a__U31(tt(),V) -> a__U32(a__isNatIListKind(V),V)
r27: a__U32(tt(),V) -> a__U33(a__isNatList(V))
r28: a__U33(tt()) -> tt()
r29: a__U41(tt(),V1,V2) -> a__U42(a__isNatKind(V1),V1,V2)
r30: a__U42(tt(),V1,V2) -> a__U43(a__isNatIListKind(V2),V1,V2)
r31: a__U43(tt(),V1,V2) -> a__U44(a__isNatIListKind(V2),V1,V2)
r32: a__U44(tt(),V1,V2) -> a__U45(a__isNat(V1),V2)
r33: a__U45(tt(),V2) -> a__U46(a__isNatIList(V2))
r34: a__U46(tt()) -> tt()
r35: a__U51(tt(),V2) -> a__U52(a__isNatIListKind(V2))
r36: a__U52(tt()) -> tt()
r37: a__U61(tt(),V2) -> a__U62(a__isNatIListKind(V2))
r38: a__U62(tt()) -> tt()
r39: a__U71(tt()) -> tt()
r40: a__U81(tt()) -> tt()
r41: a__U91(tt(),V1,V2) -> a__U92(a__isNatKind(V1),V1,V2)
r42: a__U92(tt(),V1,V2) -> a__U93(a__isNatIListKind(V2),V1,V2)
r43: a__U93(tt(),V1,V2) -> a__U94(a__isNatIListKind(V2),V1,V2)
r44: a__U94(tt(),V1,V2) -> a__U95(a__isNat(V1),V2)
r45: a__U95(tt(),V2) -> a__U96(a__isNatList(V2))
r46: a__U96(tt()) -> tt()
r47: a__isNat(|0|()) -> tt()
r48: a__isNat(length(V1)) -> a__U11(a__isNatIListKind(V1),V1)
r49: a__isNat(s(V1)) -> a__U21(a__isNatKind(V1),V1)
r50: a__isNatIList(V) -> a__U31(a__isNatIListKind(V),V)
r51: a__isNatIList(zeros()) -> tt()
r52: a__isNatIList(cons(V1,V2)) -> a__U41(a__isNatKind(V1),V1,V2)
r53: a__isNatIListKind(nil()) -> tt()
r54: a__isNatIListKind(zeros()) -> tt()
r55: a__isNatIListKind(cons(V1,V2)) -> a__U51(a__isNatKind(V1),V2)
r56: a__isNatIListKind(take(V1,V2)) -> a__U61(a__isNatKind(V1),V2)
r57: a__isNatKind(|0|()) -> tt()
r58: a__isNatKind(length(V1)) -> a__U71(a__isNatIListKind(V1))
r59: a__isNatKind(s(V1)) -> a__U81(a__isNatKind(V1))
r60: a__isNatList(nil()) -> tt()
r61: a__isNatList(cons(V1,V2)) -> a__U91(a__isNatKind(V1),V1,V2)
r62: a__isNatList(take(V1,V2)) -> a__U101(a__isNatKind(V1),V1,V2)
r63: a__length(nil()) -> |0|()
r64: a__length(cons(N,L)) -> a__U111(a__isNatList(L),L,N)
r65: a__take(|0|(),IL) -> a__U121(a__isNatIList(IL),IL)
r66: a__take(s(M),cons(N,IL)) -> a__U131(a__isNatIList(IL),IL,M,N)
r67: mark(zeros()) -> a__zeros()
r68: mark(U101(X1,X2,X3)) -> a__U101(mark(X1),X2,X3)
r69: mark(U102(X1,X2,X3)) -> a__U102(mark(X1),X2,X3)
r70: mark(isNatKind(X)) -> a__isNatKind(X)
r71: mark(U103(X1,X2,X3)) -> a__U103(mark(X1),X2,X3)
r72: mark(isNatIListKind(X)) -> a__isNatIListKind(X)
r73: mark(U104(X1,X2,X3)) -> a__U104(mark(X1),X2,X3)
r74: mark(U105(X1,X2)) -> a__U105(mark(X1),X2)
r75: mark(isNat(X)) -> a__isNat(X)
r76: mark(U106(X)) -> a__U106(mark(X))
r77: mark(isNatIList(X)) -> a__isNatIList(X)
r78: mark(U11(X1,X2)) -> a__U11(mark(X1),X2)
r79: mark(U12(X1,X2)) -> a__U12(mark(X1),X2)
r80: mark(U111(X1,X2,X3)) -> a__U111(mark(X1),X2,X3)
r81: mark(U112(X1,X2,X3)) -> a__U112(mark(X1),X2,X3)
r82: mark(U113(X1,X2,X3)) -> a__U113(mark(X1),X2,X3)
r83: mark(U114(X1,X2)) -> a__U114(mark(X1),X2)
r84: mark(length(X)) -> a__length(mark(X))
r85: mark(U13(X)) -> a__U13(mark(X))
r86: mark(isNatList(X)) -> a__isNatList(X)
r87: mark(U121(X1,X2)) -> a__U121(mark(X1),X2)
r88: mark(U122(X)) -> a__U122(mark(X))
r89: mark(U131(X1,X2,X3,X4)) -> a__U131(mark(X1),X2,X3,X4)
r90: mark(U132(X1,X2,X3,X4)) -> a__U132(mark(X1),X2,X3,X4)
r91: mark(U133(X1,X2,X3,X4)) -> a__U133(mark(X1),X2,X3,X4)
r92: mark(U134(X1,X2,X3,X4)) -> a__U134(mark(X1),X2,X3,X4)
r93: mark(U135(X1,X2,X3,X4)) -> a__U135(mark(X1),X2,X3,X4)
r94: mark(U136(X1,X2,X3,X4)) -> a__U136(mark(X1),X2,X3,X4)
r95: mark(take(X1,X2)) -> a__take(mark(X1),mark(X2))
r96: mark(U21(X1,X2)) -> a__U21(mark(X1),X2)
r97: mark(U22(X1,X2)) -> a__U22(mark(X1),X2)
r98: mark(U23(X)) -> a__U23(mark(X))
r99: mark(U31(X1,X2)) -> a__U31(mark(X1),X2)
r100: mark(U32(X1,X2)) -> a__U32(mark(X1),X2)
r101: mark(U33(X)) -> a__U33(mark(X))
r102: mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3)
r103: mark(U42(X1,X2,X3)) -> a__U42(mark(X1),X2,X3)
r104: mark(U43(X1,X2,X3)) -> a__U43(mark(X1),X2,X3)
r105: mark(U44(X1,X2,X3)) -> a__U44(mark(X1),X2,X3)
r106: mark(U45(X1,X2)) -> a__U45(mark(X1),X2)
r107: mark(U46(X)) -> a__U46(mark(X))
r108: mark(U51(X1,X2)) -> a__U51(mark(X1),X2)
r109: mark(U52(X)) -> a__U52(mark(X))
r110: mark(U61(X1,X2)) -> a__U61(mark(X1),X2)
r111: mark(U62(X)) -> a__U62(mark(X))
r112: mark(U71(X)) -> a__U71(mark(X))
r113: mark(U81(X)) -> a__U81(mark(X))
r114: mark(U91(X1,X2,X3)) -> a__U91(mark(X1),X2,X3)
r115: mark(U92(X1,X2,X3)) -> a__U92(mark(X1),X2,X3)
r116: mark(U93(X1,X2,X3)) -> a__U93(mark(X1),X2,X3)
r117: mark(U94(X1,X2,X3)) -> a__U94(mark(X1),X2,X3)
r118: mark(U95(X1,X2)) -> a__U95(mark(X1),X2)
r119: mark(U96(X)) -> a__U96(mark(X))
r120: mark(cons(X1,X2)) -> cons(mark(X1),X2)
r121: mark(|0|()) -> |0|()
r122: mark(tt()) -> tt()
r123: mark(s(X)) -> s(mark(X))
r124: mark(nil()) -> nil()
r125: a__zeros() -> zeros()
r126: a__U101(X1,X2,X3) -> U101(X1,X2,X3)
r127: a__U102(X1,X2,X3) -> U102(X1,X2,X3)
r128: a__isNatKind(X) -> isNatKind(X)
r129: a__U103(X1,X2,X3) -> U103(X1,X2,X3)
r130: a__isNatIListKind(X) -> isNatIListKind(X)
r131: a__U104(X1,X2,X3) -> U104(X1,X2,X3)
r132: a__U105(X1,X2) -> U105(X1,X2)
r133: a__isNat(X) -> isNat(X)
r134: a__U106(X) -> U106(X)
r135: a__isNatIList(X) -> isNatIList(X)
r136: a__U11(X1,X2) -> U11(X1,X2)
r137: a__U12(X1,X2) -> U12(X1,X2)
r138: a__U111(X1,X2,X3) -> U111(X1,X2,X3)
r139: a__U112(X1,X2,X3) -> U112(X1,X2,X3)
r140: a__U113(X1,X2,X3) -> U113(X1,X2,X3)
r141: a__U114(X1,X2) -> U114(X1,X2)
r142: a__length(X) -> length(X)
r143: a__U13(X) -> U13(X)
r144: a__isNatList(X) -> isNatList(X)
r145: a__U121(X1,X2) -> U121(X1,X2)
r146: a__U122(X) -> U122(X)
r147: a__U131(X1,X2,X3,X4) -> U131(X1,X2,X3,X4)
r148: a__U132(X1,X2,X3,X4) -> U132(X1,X2,X3,X4)
r149: a__U133(X1,X2,X3,X4) -> U133(X1,X2,X3,X4)
r150: a__U134(X1,X2,X3,X4) -> U134(X1,X2,X3,X4)
r151: a__U135(X1,X2,X3,X4) -> U135(X1,X2,X3,X4)
r152: a__U136(X1,X2,X3,X4) -> U136(X1,X2,X3,X4)
r153: a__take(X1,X2) -> take(X1,X2)
r154: a__U21(X1,X2) -> U21(X1,X2)
r155: a__U22(X1,X2) -> U22(X1,X2)
r156: a__U23(X) -> U23(X)
r157: a__U31(X1,X2) -> U31(X1,X2)
r158: a__U32(X1,X2) -> U32(X1,X2)
r159: a__U33(X) -> U33(X)
r160: a__U41(X1,X2,X3) -> U41(X1,X2,X3)
r161: a__U42(X1,X2,X3) -> U42(X1,X2,X3)
r162: a__U43(X1,X2,X3) -> U43(X1,X2,X3)
r163: a__U44(X1,X2,X3) -> U44(X1,X2,X3)
r164: a__U45(X1,X2) -> U45(X1,X2)
r165: a__U46(X) -> U46(X)
r166: a__U51(X1,X2) -> U51(X1,X2)
r167: a__U52(X) -> U52(X)
r168: a__U61(X1,X2) -> U61(X1,X2)
r169: a__U62(X) -> U62(X)
r170: a__U71(X) -> U71(X)
r171: a__U81(X) -> U81(X)
r172: a__U91(X1,X2,X3) -> U91(X1,X2,X3)
r173: a__U92(X1,X2,X3) -> U92(X1,X2,X3)
r174: a__U93(X1,X2,X3) -> U93(X1,X2,X3)
r175: a__U94(X1,X2,X3) -> U94(X1,X2,X3)
r176: a__U95(X1,X2) -> U95(X1,X2)
r177: a__U96(X) -> U96(X)

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  max/plus interpretations on natural numbers:
  
    mark#_A(x1) = x1 + 1
    U11_A(x1,x2) = max{x1, x2}
    U81_A(x1) = x1
    U134_A(x1,x2,x3,x4) = max{x1, x2, x3, x4}
    U131_A(x1,x2,x3,x4) = max{x1, x2, x3, x4}
    U122_A(x1) = x1
    U111_A(x1,x2,x3) = max{x1, x2, x3}
    U12_A(x1,x2) = max{x1 + 1, x2}

The next rules are strictly ordered:

  p7

We remove them from the problem.

-- SCC decomposition.

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

p1: mark#(U11(X1,X2)) -> mark#(X1)
p2: mark#(U81(X)) -> mark#(X)
p3: mark#(U134(X1,X2,X3,X4)) -> mark#(X1)
p4: mark#(U131(X1,X2,X3,X4)) -> mark#(X1)
p5: mark#(U122(X)) -> mark#(X)
p6: mark#(U111(X1,X2,X3)) -> mark#(X1)

and R consists of:

r1: a__zeros() -> cons(|0|(),zeros())
r2: a__U101(tt(),V1,V2) -> a__U102(a__isNatKind(V1),V1,V2)
r3: a__U102(tt(),V1,V2) -> a__U103(a__isNatIListKind(V2),V1,V2)
r4: a__U103(tt(),V1,V2) -> a__U104(a__isNatIListKind(V2),V1,V2)
r5: a__U104(tt(),V1,V2) -> a__U105(a__isNat(V1),V2)
r6: a__U105(tt(),V2) -> a__U106(a__isNatIList(V2))
r7: a__U106(tt()) -> tt()
r8: a__U11(tt(),V1) -> a__U12(a__isNatIListKind(V1),V1)
r9: a__U111(tt(),L,N) -> a__U112(a__isNatIListKind(L),L,N)
r10: a__U112(tt(),L,N) -> a__U113(a__isNat(N),L,N)
r11: a__U113(tt(),L,N) -> a__U114(a__isNatKind(N),L)
r12: a__U114(tt(),L) -> s(a__length(mark(L)))
r13: a__U12(tt(),V1) -> a__U13(a__isNatList(V1))
r14: a__U121(tt(),IL) -> a__U122(a__isNatIListKind(IL))
r15: a__U122(tt()) -> nil()
r16: a__U13(tt()) -> tt()
r17: a__U131(tt(),IL,M,N) -> a__U132(a__isNatIListKind(IL),IL,M,N)
r18: a__U132(tt(),IL,M,N) -> a__U133(a__isNat(M),IL,M,N)
r19: a__U133(tt(),IL,M,N) -> a__U134(a__isNatKind(M),IL,M,N)
r20: a__U134(tt(),IL,M,N) -> a__U135(a__isNat(N),IL,M,N)
r21: a__U135(tt(),IL,M,N) -> a__U136(a__isNatKind(N),IL,M,N)
r22: a__U136(tt(),IL,M,N) -> cons(mark(N),take(M,IL))
r23: a__U21(tt(),V1) -> a__U22(a__isNatKind(V1),V1)
r24: a__U22(tt(),V1) -> a__U23(a__isNat(V1))
r25: a__U23(tt()) -> tt()
r26: a__U31(tt(),V) -> a__U32(a__isNatIListKind(V),V)
r27: a__U32(tt(),V) -> a__U33(a__isNatList(V))
r28: a__U33(tt()) -> tt()
r29: a__U41(tt(),V1,V2) -> a__U42(a__isNatKind(V1),V1,V2)
r30: a__U42(tt(),V1,V2) -> a__U43(a__isNatIListKind(V2),V1,V2)
r31: a__U43(tt(),V1,V2) -> a__U44(a__isNatIListKind(V2),V1,V2)
r32: a__U44(tt(),V1,V2) -> a__U45(a__isNat(V1),V2)
r33: a__U45(tt(),V2) -> a__U46(a__isNatIList(V2))
r34: a__U46(tt()) -> tt()
r35: a__U51(tt(),V2) -> a__U52(a__isNatIListKind(V2))
r36: a__U52(tt()) -> tt()
r37: a__U61(tt(),V2) -> a__U62(a__isNatIListKind(V2))
r38: a__U62(tt()) -> tt()
r39: a__U71(tt()) -> tt()
r40: a__U81(tt()) -> tt()
r41: a__U91(tt(),V1,V2) -> a__U92(a__isNatKind(V1),V1,V2)
r42: a__U92(tt(),V1,V2) -> a__U93(a__isNatIListKind(V2),V1,V2)
r43: a__U93(tt(),V1,V2) -> a__U94(a__isNatIListKind(V2),V1,V2)
r44: a__U94(tt(),V1,V2) -> a__U95(a__isNat(V1),V2)
r45: a__U95(tt(),V2) -> a__U96(a__isNatList(V2))
r46: a__U96(tt()) -> tt()
r47: a__isNat(|0|()) -> tt()
r48: a__isNat(length(V1)) -> a__U11(a__isNatIListKind(V1),V1)
r49: a__isNat(s(V1)) -> a__U21(a__isNatKind(V1),V1)
r50: a__isNatIList(V) -> a__U31(a__isNatIListKind(V),V)
r51: a__isNatIList(zeros()) -> tt()
r52: a__isNatIList(cons(V1,V2)) -> a__U41(a__isNatKind(V1),V1,V2)
r53: a__isNatIListKind(nil()) -> tt()
r54: a__isNatIListKind(zeros()) -> tt()
r55: a__isNatIListKind(cons(V1,V2)) -> a__U51(a__isNatKind(V1),V2)
r56: a__isNatIListKind(take(V1,V2)) -> a__U61(a__isNatKind(V1),V2)
r57: a__isNatKind(|0|()) -> tt()
r58: a__isNatKind(length(V1)) -> a__U71(a__isNatIListKind(V1))
r59: a__isNatKind(s(V1)) -> a__U81(a__isNatKind(V1))
r60: a__isNatList(nil()) -> tt()
r61: a__isNatList(cons(V1,V2)) -> a__U91(a__isNatKind(V1),V1,V2)
r62: a__isNatList(take(V1,V2)) -> a__U101(a__isNatKind(V1),V1,V2)
r63: a__length(nil()) -> |0|()
r64: a__length(cons(N,L)) -> a__U111(a__isNatList(L),L,N)
r65: a__take(|0|(),IL) -> a__U121(a__isNatIList(IL),IL)
r66: a__take(s(M),cons(N,IL)) -> a__U131(a__isNatIList(IL),IL,M,N)
r67: mark(zeros()) -> a__zeros()
r68: mark(U101(X1,X2,X3)) -> a__U101(mark(X1),X2,X3)
r69: mark(U102(X1,X2,X3)) -> a__U102(mark(X1),X2,X3)
r70: mark(isNatKind(X)) -> a__isNatKind(X)
r71: mark(U103(X1,X2,X3)) -> a__U103(mark(X1),X2,X3)
r72: mark(isNatIListKind(X)) -> a__isNatIListKind(X)
r73: mark(U104(X1,X2,X3)) -> a__U104(mark(X1),X2,X3)
r74: mark(U105(X1,X2)) -> a__U105(mark(X1),X2)
r75: mark(isNat(X)) -> a__isNat(X)
r76: mark(U106(X)) -> a__U106(mark(X))
r77: mark(isNatIList(X)) -> a__isNatIList(X)
r78: mark(U11(X1,X2)) -> a__U11(mark(X1),X2)
r79: mark(U12(X1,X2)) -> a__U12(mark(X1),X2)
r80: mark(U111(X1,X2,X3)) -> a__U111(mark(X1),X2,X3)
r81: mark(U112(X1,X2,X3)) -> a__U112(mark(X1),X2,X3)
r82: mark(U113(X1,X2,X3)) -> a__U113(mark(X1),X2,X3)
r83: mark(U114(X1,X2)) -> a__U114(mark(X1),X2)
r84: mark(length(X)) -> a__length(mark(X))
r85: mark(U13(X)) -> a__U13(mark(X))
r86: mark(isNatList(X)) -> a__isNatList(X)
r87: mark(U121(X1,X2)) -> a__U121(mark(X1),X2)
r88: mark(U122(X)) -> a__U122(mark(X))
r89: mark(U131(X1,X2,X3,X4)) -> a__U131(mark(X1),X2,X3,X4)
r90: mark(U132(X1,X2,X3,X4)) -> a__U132(mark(X1),X2,X3,X4)
r91: mark(U133(X1,X2,X3,X4)) -> a__U133(mark(X1),X2,X3,X4)
r92: mark(U134(X1,X2,X3,X4)) -> a__U134(mark(X1),X2,X3,X4)
r93: mark(U135(X1,X2,X3,X4)) -> a__U135(mark(X1),X2,X3,X4)
r94: mark(U136(X1,X2,X3,X4)) -> a__U136(mark(X1),X2,X3,X4)
r95: mark(take(X1,X2)) -> a__take(mark(X1),mark(X2))
r96: mark(U21(X1,X2)) -> a__U21(mark(X1),X2)
r97: mark(U22(X1,X2)) -> a__U22(mark(X1),X2)
r98: mark(U23(X)) -> a__U23(mark(X))
r99: mark(U31(X1,X2)) -> a__U31(mark(X1),X2)
r100: mark(U32(X1,X2)) -> a__U32(mark(X1),X2)
r101: mark(U33(X)) -> a__U33(mark(X))
r102: mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3)
r103: mark(U42(X1,X2,X3)) -> a__U42(mark(X1),X2,X3)
r104: mark(U43(X1,X2,X3)) -> a__U43(mark(X1),X2,X3)
r105: mark(U44(X1,X2,X3)) -> a__U44(mark(X1),X2,X3)
r106: mark(U45(X1,X2)) -> a__U45(mark(X1),X2)
r107: mark(U46(X)) -> a__U46(mark(X))
r108: mark(U51(X1,X2)) -> a__U51(mark(X1),X2)
r109: mark(U52(X)) -> a__U52(mark(X))
r110: mark(U61(X1,X2)) -> a__U61(mark(X1),X2)
r111: mark(U62(X)) -> a__U62(mark(X))
r112: mark(U71(X)) -> a__U71(mark(X))
r113: mark(U81(X)) -> a__U81(mark(X))
r114: mark(U91(X1,X2,X3)) -> a__U91(mark(X1),X2,X3)
r115: mark(U92(X1,X2,X3)) -> a__U92(mark(X1),X2,X3)
r116: mark(U93(X1,X2,X3)) -> a__U93(mark(X1),X2,X3)
r117: mark(U94(X1,X2,X3)) -> a__U94(mark(X1),X2,X3)
r118: mark(U95(X1,X2)) -> a__U95(mark(X1),X2)
r119: mark(U96(X)) -> a__U96(mark(X))
r120: mark(cons(X1,X2)) -> cons(mark(X1),X2)
r121: mark(|0|()) -> |0|()
r122: mark(tt()) -> tt()
r123: mark(s(X)) -> s(mark(X))
r124: mark(nil()) -> nil()
r125: a__zeros() -> zeros()
r126: a__U101(X1,X2,X3) -> U101(X1,X2,X3)
r127: a__U102(X1,X2,X3) -> U102(X1,X2,X3)
r128: a__isNatKind(X) -> isNatKind(X)
r129: a__U103(X1,X2,X3) -> U103(X1,X2,X3)
r130: a__isNatIListKind(X) -> isNatIListKind(X)
r131: a__U104(X1,X2,X3) -> U104(X1,X2,X3)
r132: a__U105(X1,X2) -> U105(X1,X2)
r133: a__isNat(X) -> isNat(X)
r134: a__U106(X) -> U106(X)
r135: a__isNatIList(X) -> isNatIList(X)
r136: a__U11(X1,X2) -> U11(X1,X2)
r137: a__U12(X1,X2) -> U12(X1,X2)
r138: a__U111(X1,X2,X3) -> U111(X1,X2,X3)
r139: a__U112(X1,X2,X3) -> U112(X1,X2,X3)
r140: a__U113(X1,X2,X3) -> U113(X1,X2,X3)
r141: a__U114(X1,X2) -> U114(X1,X2)
r142: a__length(X) -> length(X)
r143: a__U13(X) -> U13(X)
r144: a__isNatList(X) -> isNatList(X)
r145: a__U121(X1,X2) -> U121(X1,X2)
r146: a__U122(X) -> U122(X)
r147: a__U131(X1,X2,X3,X4) -> U131(X1,X2,X3,X4)
r148: a__U132(X1,X2,X3,X4) -> U132(X1,X2,X3,X4)
r149: a__U133(X1,X2,X3,X4) -> U133(X1,X2,X3,X4)
r150: a__U134(X1,X2,X3,X4) -> U134(X1,X2,X3,X4)
r151: a__U135(X1,X2,X3,X4) -> U135(X1,X2,X3,X4)
r152: a__U136(X1,X2,X3,X4) -> U136(X1,X2,X3,X4)
r153: a__take(X1,X2) -> take(X1,X2)
r154: a__U21(X1,X2) -> U21(X1,X2)
r155: a__U22(X1,X2) -> U22(X1,X2)
r156: a__U23(X) -> U23(X)
r157: a__U31(X1,X2) -> U31(X1,X2)
r158: a__U32(X1,X2) -> U32(X1,X2)
r159: a__U33(X) -> U33(X)
r160: a__U41(X1,X2,X3) -> U41(X1,X2,X3)
r161: a__U42(X1,X2,X3) -> U42(X1,X2,X3)
r162: a__U43(X1,X2,X3) -> U43(X1,X2,X3)
r163: a__U44(X1,X2,X3) -> U44(X1,X2,X3)
r164: a__U45(X1,X2) -> U45(X1,X2)
r165: a__U46(X) -> U46(X)
r166: a__U51(X1,X2) -> U51(X1,X2)
r167: a__U52(X) -> U52(X)
r168: a__U61(X1,X2) -> U61(X1,X2)
r169: a__U62(X) -> U62(X)
r170: a__U71(X) -> U71(X)
r171: a__U81(X) -> U81(X)
r172: a__U91(X1,X2,X3) -> U91(X1,X2,X3)
r173: a__U92(X1,X2,X3) -> U92(X1,X2,X3)
r174: a__U93(X1,X2,X3) -> U93(X1,X2,X3)
r175: a__U94(X1,X2,X3) -> U94(X1,X2,X3)
r176: a__U95(X1,X2) -> U95(X1,X2)
r177: a__U96(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: mark#(U11(X1,X2)) -> mark#(X1)
p2: mark#(U111(X1,X2,X3)) -> mark#(X1)
p3: mark#(U122(X)) -> mark#(X)
p4: mark#(U131(X1,X2,X3,X4)) -> mark#(X1)
p5: mark#(U134(X1,X2,X3,X4)) -> mark#(X1)
p6: mark#(U81(X)) -> mark#(X)

and R consists of:

r1: a__zeros() -> cons(|0|(),zeros())
r2: a__U101(tt(),V1,V2) -> a__U102(a__isNatKind(V1),V1,V2)
r3: a__U102(tt(),V1,V2) -> a__U103(a__isNatIListKind(V2),V1,V2)
r4: a__U103(tt(),V1,V2) -> a__U104(a__isNatIListKind(V2),V1,V2)
r5: a__U104(tt(),V1,V2) -> a__U105(a__isNat(V1),V2)
r6: a__U105(tt(),V2) -> a__U106(a__isNatIList(V2))
r7: a__U106(tt()) -> tt()
r8: a__U11(tt(),V1) -> a__U12(a__isNatIListKind(V1),V1)
r9: a__U111(tt(),L,N) -> a__U112(a__isNatIListKind(L),L,N)
r10: a__U112(tt(),L,N) -> a__U113(a__isNat(N),L,N)
r11: a__U113(tt(),L,N) -> a__U114(a__isNatKind(N),L)
r12: a__U114(tt(),L) -> s(a__length(mark(L)))
r13: a__U12(tt(),V1) -> a__U13(a__isNatList(V1))
r14: a__U121(tt(),IL) -> a__U122(a__isNatIListKind(IL))
r15: a__U122(tt()) -> nil()
r16: a__U13(tt()) -> tt()
r17: a__U131(tt(),IL,M,N) -> a__U132(a__isNatIListKind(IL),IL,M,N)
r18: a__U132(tt(),IL,M,N) -> a__U133(a__isNat(M),IL,M,N)
r19: a__U133(tt(),IL,M,N) -> a__U134(a__isNatKind(M),IL,M,N)
r20: a__U134(tt(),IL,M,N) -> a__U135(a__isNat(N),IL,M,N)
r21: a__U135(tt(),IL,M,N) -> a__U136(a__isNatKind(N),IL,M,N)
r22: a__U136(tt(),IL,M,N) -> cons(mark(N),take(M,IL))
r23: a__U21(tt(),V1) -> a__U22(a__isNatKind(V1),V1)
r24: a__U22(tt(),V1) -> a__U23(a__isNat(V1))
r25: a__U23(tt()) -> tt()
r26: a__U31(tt(),V) -> a__U32(a__isNatIListKind(V),V)
r27: a__U32(tt(),V) -> a__U33(a__isNatList(V))
r28: a__U33(tt()) -> tt()
r29: a__U41(tt(),V1,V2) -> a__U42(a__isNatKind(V1),V1,V2)
r30: a__U42(tt(),V1,V2) -> a__U43(a__isNatIListKind(V2),V1,V2)
r31: a__U43(tt(),V1,V2) -> a__U44(a__isNatIListKind(V2),V1,V2)
r32: a__U44(tt(),V1,V2) -> a__U45(a__isNat(V1),V2)
r33: a__U45(tt(),V2) -> a__U46(a__isNatIList(V2))
r34: a__U46(tt()) -> tt()
r35: a__U51(tt(),V2) -> a__U52(a__isNatIListKind(V2))
r36: a__U52(tt()) -> tt()
r37: a__U61(tt(),V2) -> a__U62(a__isNatIListKind(V2))
r38: a__U62(tt()) -> tt()
r39: a__U71(tt()) -> tt()
r40: a__U81(tt()) -> tt()
r41: a__U91(tt(),V1,V2) -> a__U92(a__isNatKind(V1),V1,V2)
r42: a__U92(tt(),V1,V2) -> a__U93(a__isNatIListKind(V2),V1,V2)
r43: a__U93(tt(),V1,V2) -> a__U94(a__isNatIListKind(V2),V1,V2)
r44: a__U94(tt(),V1,V2) -> a__U95(a__isNat(V1),V2)
r45: a__U95(tt(),V2) -> a__U96(a__isNatList(V2))
r46: a__U96(tt()) -> tt()
r47: a__isNat(|0|()) -> tt()
r48: a__isNat(length(V1)) -> a__U11(a__isNatIListKind(V1),V1)
r49: a__isNat(s(V1)) -> a__U21(a__isNatKind(V1),V1)
r50: a__isNatIList(V) -> a__U31(a__isNatIListKind(V),V)
r51: a__isNatIList(zeros()) -> tt()
r52: a__isNatIList(cons(V1,V2)) -> a__U41(a__isNatKind(V1),V1,V2)
r53: a__isNatIListKind(nil()) -> tt()
r54: a__isNatIListKind(zeros()) -> tt()
r55: a__isNatIListKind(cons(V1,V2)) -> a__U51(a__isNatKind(V1),V2)
r56: a__isNatIListKind(take(V1,V2)) -> a__U61(a__isNatKind(V1),V2)
r57: a__isNatKind(|0|()) -> tt()
r58: a__isNatKind(length(V1)) -> a__U71(a__isNatIListKind(V1))
r59: a__isNatKind(s(V1)) -> a__U81(a__isNatKind(V1))
r60: a__isNatList(nil()) -> tt()
r61: a__isNatList(cons(V1,V2)) -> a__U91(a__isNatKind(V1),V1,V2)
r62: a__isNatList(take(V1,V2)) -> a__U101(a__isNatKind(V1),V1,V2)
r63: a__length(nil()) -> |0|()
r64: a__length(cons(N,L)) -> a__U111(a__isNatList(L),L,N)
r65: a__take(|0|(),IL) -> a__U121(a__isNatIList(IL),IL)
r66: a__take(s(M),cons(N,IL)) -> a__U131(a__isNatIList(IL),IL,M,N)
r67: mark(zeros()) -> a__zeros()
r68: mark(U101(X1,X2,X3)) -> a__U101(mark(X1),X2,X3)
r69: mark(U102(X1,X2,X3)) -> a__U102(mark(X1),X2,X3)
r70: mark(isNatKind(X)) -> a__isNatKind(X)
r71: mark(U103(X1,X2,X3)) -> a__U103(mark(X1),X2,X3)
r72: mark(isNatIListKind(X)) -> a__isNatIListKind(X)
r73: mark(U104(X1,X2,X3)) -> a__U104(mark(X1),X2,X3)
r74: mark(U105(X1,X2)) -> a__U105(mark(X1),X2)
r75: mark(isNat(X)) -> a__isNat(X)
r76: mark(U106(X)) -> a__U106(mark(X))
r77: mark(isNatIList(X)) -> a__isNatIList(X)
r78: mark(U11(X1,X2)) -> a__U11(mark(X1),X2)
r79: mark(U12(X1,X2)) -> a__U12(mark(X1),X2)
r80: mark(U111(X1,X2,X3)) -> a__U111(mark(X1),X2,X3)
r81: mark(U112(X1,X2,X3)) -> a__U112(mark(X1),X2,X3)
r82: mark(U113(X1,X2,X3)) -> a__U113(mark(X1),X2,X3)
r83: mark(U114(X1,X2)) -> a__U114(mark(X1),X2)
r84: mark(length(X)) -> a__length(mark(X))
r85: mark(U13(X)) -> a__U13(mark(X))
r86: mark(isNatList(X)) -> a__isNatList(X)
r87: mark(U121(X1,X2)) -> a__U121(mark(X1),X2)
r88: mark(U122(X)) -> a__U122(mark(X))
r89: mark(U131(X1,X2,X3,X4)) -> a__U131(mark(X1),X2,X3,X4)
r90: mark(U132(X1,X2,X3,X4)) -> a__U132(mark(X1),X2,X3,X4)
r91: mark(U133(X1,X2,X3,X4)) -> a__U133(mark(X1),X2,X3,X4)
r92: mark(U134(X1,X2,X3,X4)) -> a__U134(mark(X1),X2,X3,X4)
r93: mark(U135(X1,X2,X3,X4)) -> a__U135(mark(X1),X2,X3,X4)
r94: mark(U136(X1,X2,X3,X4)) -> a__U136(mark(X1),X2,X3,X4)
r95: mark(take(X1,X2)) -> a__take(mark(X1),mark(X2))
r96: mark(U21(X1,X2)) -> a__U21(mark(X1),X2)
r97: mark(U22(X1,X2)) -> a__U22(mark(X1),X2)
r98: mark(U23(X)) -> a__U23(mark(X))
r99: mark(U31(X1,X2)) -> a__U31(mark(X1),X2)
r100: mark(U32(X1,X2)) -> a__U32(mark(X1),X2)
r101: mark(U33(X)) -> a__U33(mark(X))
r102: mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3)
r103: mark(U42(X1,X2,X3)) -> a__U42(mark(X1),X2,X3)
r104: mark(U43(X1,X2,X3)) -> a__U43(mark(X1),X2,X3)
r105: mark(U44(X1,X2,X3)) -> a__U44(mark(X1),X2,X3)
r106: mark(U45(X1,X2)) -> a__U45(mark(X1),X2)
r107: mark(U46(X)) -> a__U46(mark(X))
r108: mark(U51(X1,X2)) -> a__U51(mark(X1),X2)
r109: mark(U52(X)) -> a__U52(mark(X))
r110: mark(U61(X1,X2)) -> a__U61(mark(X1),X2)
r111: mark(U62(X)) -> a__U62(mark(X))
r112: mark(U71(X)) -> a__U71(mark(X))
r113: mark(U81(X)) -> a__U81(mark(X))
r114: mark(U91(X1,X2,X3)) -> a__U91(mark(X1),X2,X3)
r115: mark(U92(X1,X2,X3)) -> a__U92(mark(X1),X2,X3)
r116: mark(U93(X1,X2,X3)) -> a__U93(mark(X1),X2,X3)
r117: mark(U94(X1,X2,X3)) -> a__U94(mark(X1),X2,X3)
r118: mark(U95(X1,X2)) -> a__U95(mark(X1),X2)
r119: mark(U96(X)) -> a__U96(mark(X))
r120: mark(cons(X1,X2)) -> cons(mark(X1),X2)
r121: mark(|0|()) -> |0|()
r122: mark(tt()) -> tt()
r123: mark(s(X)) -> s(mark(X))
r124: mark(nil()) -> nil()
r125: a__zeros() -> zeros()
r126: a__U101(X1,X2,X3) -> U101(X1,X2,X3)
r127: a__U102(X1,X2,X3) -> U102(X1,X2,X3)
r128: a__isNatKind(X) -> isNatKind(X)
r129: a__U103(X1,X2,X3) -> U103(X1,X2,X3)
r130: a__isNatIListKind(X) -> isNatIListKind(X)
r131: a__U104(X1,X2,X3) -> U104(X1,X2,X3)
r132: a__U105(X1,X2) -> U105(X1,X2)
r133: a__isNat(X) -> isNat(X)
r134: a__U106(X) -> U106(X)
r135: a__isNatIList(X) -> isNatIList(X)
r136: a__U11(X1,X2) -> U11(X1,X2)
r137: a__U12(X1,X2) -> U12(X1,X2)
r138: a__U111(X1,X2,X3) -> U111(X1,X2,X3)
r139: a__U112(X1,X2,X3) -> U112(X1,X2,X3)
r140: a__U113(X1,X2,X3) -> U113(X1,X2,X3)
r141: a__U114(X1,X2) -> U114(X1,X2)
r142: a__length(X) -> length(X)
r143: a__U13(X) -> U13(X)
r144: a__isNatList(X) -> isNatList(X)
r145: a__U121(X1,X2) -> U121(X1,X2)
r146: a__U122(X) -> U122(X)
r147: a__U131(X1,X2,X3,X4) -> U131(X1,X2,X3,X4)
r148: a__U132(X1,X2,X3,X4) -> U132(X1,X2,X3,X4)
r149: a__U133(X1,X2,X3,X4) -> U133(X1,X2,X3,X4)
r150: a__U134(X1,X2,X3,X4) -> U134(X1,X2,X3,X4)
r151: a__U135(X1,X2,X3,X4) -> U135(X1,X2,X3,X4)
r152: a__U136(X1,X2,X3,X4) -> U136(X1,X2,X3,X4)
r153: a__take(X1,X2) -> take(X1,X2)
r154: a__U21(X1,X2) -> U21(X1,X2)
r155: a__U22(X1,X2) -> U22(X1,X2)
r156: a__U23(X) -> U23(X)
r157: a__U31(X1,X2) -> U31(X1,X2)
r158: a__U32(X1,X2) -> U32(X1,X2)
r159: a__U33(X) -> U33(X)
r160: a__U41(X1,X2,X3) -> U41(X1,X2,X3)
r161: a__U42(X1,X2,X3) -> U42(X1,X2,X3)
r162: a__U43(X1,X2,X3) -> U43(X1,X2,X3)
r163: a__U44(X1,X2,X3) -> U44(X1,X2,X3)
r164: a__U45(X1,X2) -> U45(X1,X2)
r165: a__U46(X) -> U46(X)
r166: a__U51(X1,X2) -> U51(X1,X2)
r167: a__U52(X) -> U52(X)
r168: a__U61(X1,X2) -> U61(X1,X2)
r169: a__U62(X) -> U62(X)
r170: a__U71(X) -> U71(X)
r171: a__U81(X) -> U81(X)
r172: a__U91(X1,X2,X3) -> U91(X1,X2,X3)
r173: a__U92(X1,X2,X3) -> U92(X1,X2,X3)
r174: a__U93(X1,X2,X3) -> U93(X1,X2,X3)
r175: a__U94(X1,X2,X3) -> U94(X1,X2,X3)
r176: a__U95(X1,X2) -> U95(X1,X2)
r177: a__U96(X) -> U96(X)

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  max/plus interpretations on natural numbers:
  
    mark#_A(x1) = x1
    U11_A(x1,x2) = max{x1, x2 + 1}
    U111_A(x1,x2,x3) = max{x1, x2 + 1, x3 + 1}
    U122_A(x1) = max{1, x1}
    U131_A(x1,x2,x3,x4) = max{x1, x2}
    U134_A(x1,x2,x3,x4) = max{x1, x3, x4}
    U81_A(x1) = x1 + 1

The next rules are strictly ordered:

  p6

We remove them from the problem.

-- SCC decomposition.

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

p1: mark#(U11(X1,X2)) -> mark#(X1)
p2: mark#(U111(X1,X2,X3)) -> mark#(X1)
p3: mark#(U122(X)) -> mark#(X)
p4: mark#(U131(X1,X2,X3,X4)) -> mark#(X1)
p5: mark#(U134(X1,X2,X3,X4)) -> mark#(X1)

and R consists of:

r1: a__zeros() -> cons(|0|(),zeros())
r2: a__U101(tt(),V1,V2) -> a__U102(a__isNatKind(V1),V1,V2)
r3: a__U102(tt(),V1,V2) -> a__U103(a__isNatIListKind(V2),V1,V2)
r4: a__U103(tt(),V1,V2) -> a__U104(a__isNatIListKind(V2),V1,V2)
r5: a__U104(tt(),V1,V2) -> a__U105(a__isNat(V1),V2)
r6: a__U105(tt(),V2) -> a__U106(a__isNatIList(V2))
r7: a__U106(tt()) -> tt()
r8: a__U11(tt(),V1) -> a__U12(a__isNatIListKind(V1),V1)
r9: a__U111(tt(),L,N) -> a__U112(a__isNatIListKind(L),L,N)
r10: a__U112(tt(),L,N) -> a__U113(a__isNat(N),L,N)
r11: a__U113(tt(),L,N) -> a__U114(a__isNatKind(N),L)
r12: a__U114(tt(),L) -> s(a__length(mark(L)))
r13: a__U12(tt(),V1) -> a__U13(a__isNatList(V1))
r14: a__U121(tt(),IL) -> a__U122(a__isNatIListKind(IL))
r15: a__U122(tt()) -> nil()
r16: a__U13(tt()) -> tt()
r17: a__U131(tt(),IL,M,N) -> a__U132(a__isNatIListKind(IL),IL,M,N)
r18: a__U132(tt(),IL,M,N) -> a__U133(a__isNat(M),IL,M,N)
r19: a__U133(tt(),IL,M,N) -> a__U134(a__isNatKind(M),IL,M,N)
r20: a__U134(tt(),IL,M,N) -> a__U135(a__isNat(N),IL,M,N)
r21: a__U135(tt(),IL,M,N) -> a__U136(a__isNatKind(N),IL,M,N)
r22: a__U136(tt(),IL,M,N) -> cons(mark(N),take(M,IL))
r23: a__U21(tt(),V1) -> a__U22(a__isNatKind(V1),V1)
r24: a__U22(tt(),V1) -> a__U23(a__isNat(V1))
r25: a__U23(tt()) -> tt()
r26: a__U31(tt(),V) -> a__U32(a__isNatIListKind(V),V)
r27: a__U32(tt(),V) -> a__U33(a__isNatList(V))
r28: a__U33(tt()) -> tt()
r29: a__U41(tt(),V1,V2) -> a__U42(a__isNatKind(V1),V1,V2)
r30: a__U42(tt(),V1,V2) -> a__U43(a__isNatIListKind(V2),V1,V2)
r31: a__U43(tt(),V1,V2) -> a__U44(a__isNatIListKind(V2),V1,V2)
r32: a__U44(tt(),V1,V2) -> a__U45(a__isNat(V1),V2)
r33: a__U45(tt(),V2) -> a__U46(a__isNatIList(V2))
r34: a__U46(tt()) -> tt()
r35: a__U51(tt(),V2) -> a__U52(a__isNatIListKind(V2))
r36: a__U52(tt()) -> tt()
r37: a__U61(tt(),V2) -> a__U62(a__isNatIListKind(V2))
r38: a__U62(tt()) -> tt()
r39: a__U71(tt()) -> tt()
r40: a__U81(tt()) -> tt()
r41: a__U91(tt(),V1,V2) -> a__U92(a__isNatKind(V1),V1,V2)
r42: a__U92(tt(),V1,V2) -> a__U93(a__isNatIListKind(V2),V1,V2)
r43: a__U93(tt(),V1,V2) -> a__U94(a__isNatIListKind(V2),V1,V2)
r44: a__U94(tt(),V1,V2) -> a__U95(a__isNat(V1),V2)
r45: a__U95(tt(),V2) -> a__U96(a__isNatList(V2))
r46: a__U96(tt()) -> tt()
r47: a__isNat(|0|()) -> tt()
r48: a__isNat(length(V1)) -> a__U11(a__isNatIListKind(V1),V1)
r49: a__isNat(s(V1)) -> a__U21(a__isNatKind(V1),V1)
r50: a__isNatIList(V) -> a__U31(a__isNatIListKind(V),V)
r51: a__isNatIList(zeros()) -> tt()
r52: a__isNatIList(cons(V1,V2)) -> a__U41(a__isNatKind(V1),V1,V2)
r53: a__isNatIListKind(nil()) -> tt()
r54: a__isNatIListKind(zeros()) -> tt()
r55: a__isNatIListKind(cons(V1,V2)) -> a__U51(a__isNatKind(V1),V2)
r56: a__isNatIListKind(take(V1,V2)) -> a__U61(a__isNatKind(V1),V2)
r57: a__isNatKind(|0|()) -> tt()
r58: a__isNatKind(length(V1)) -> a__U71(a__isNatIListKind(V1))
r59: a__isNatKind(s(V1)) -> a__U81(a__isNatKind(V1))
r60: a__isNatList(nil()) -> tt()
r61: a__isNatList(cons(V1,V2)) -> a__U91(a__isNatKind(V1),V1,V2)
r62: a__isNatList(take(V1,V2)) -> a__U101(a__isNatKind(V1),V1,V2)
r63: a__length(nil()) -> |0|()
r64: a__length(cons(N,L)) -> a__U111(a__isNatList(L),L,N)
r65: a__take(|0|(),IL) -> a__U121(a__isNatIList(IL),IL)
r66: a__take(s(M),cons(N,IL)) -> a__U131(a__isNatIList(IL),IL,M,N)
r67: mark(zeros()) -> a__zeros()
r68: mark(U101(X1,X2,X3)) -> a__U101(mark(X1),X2,X3)
r69: mark(U102(X1,X2,X3)) -> a__U102(mark(X1),X2,X3)
r70: mark(isNatKind(X)) -> a__isNatKind(X)
r71: mark(U103(X1,X2,X3)) -> a__U103(mark(X1),X2,X3)
r72: mark(isNatIListKind(X)) -> a__isNatIListKind(X)
r73: mark(U104(X1,X2,X3)) -> a__U104(mark(X1),X2,X3)
r74: mark(U105(X1,X2)) -> a__U105(mark(X1),X2)
r75: mark(isNat(X)) -> a__isNat(X)
r76: mark(U106(X)) -> a__U106(mark(X))
r77: mark(isNatIList(X)) -> a__isNatIList(X)
r78: mark(U11(X1,X2)) -> a__U11(mark(X1),X2)
r79: mark(U12(X1,X2)) -> a__U12(mark(X1),X2)
r80: mark(U111(X1,X2,X3)) -> a__U111(mark(X1),X2,X3)
r81: mark(U112(X1,X2,X3)) -> a__U112(mark(X1),X2,X3)
r82: mark(U113(X1,X2,X3)) -> a__U113(mark(X1),X2,X3)
r83: mark(U114(X1,X2)) -> a__U114(mark(X1),X2)
r84: mark(length(X)) -> a__length(mark(X))
r85: mark(U13(X)) -> a__U13(mark(X))
r86: mark(isNatList(X)) -> a__isNatList(X)
r87: mark(U121(X1,X2)) -> a__U121(mark(X1),X2)
r88: mark(U122(X)) -> a__U122(mark(X))
r89: mark(U131(X1,X2,X3,X4)) -> a__U131(mark(X1),X2,X3,X4)
r90: mark(U132(X1,X2,X3,X4)) -> a__U132(mark(X1),X2,X3,X4)
r91: mark(U133(X1,X2,X3,X4)) -> a__U133(mark(X1),X2,X3,X4)
r92: mark(U134(X1,X2,X3,X4)) -> a__U134(mark(X1),X2,X3,X4)
r93: mark(U135(X1,X2,X3,X4)) -> a__U135(mark(X1),X2,X3,X4)
r94: mark(U136(X1,X2,X3,X4)) -> a__U136(mark(X1),X2,X3,X4)
r95: mark(take(X1,X2)) -> a__take(mark(X1),mark(X2))
r96: mark(U21(X1,X2)) -> a__U21(mark(X1),X2)
r97: mark(U22(X1,X2)) -> a__U22(mark(X1),X2)
r98: mark(U23(X)) -> a__U23(mark(X))
r99: mark(U31(X1,X2)) -> a__U31(mark(X1),X2)
r100: mark(U32(X1,X2)) -> a__U32(mark(X1),X2)
r101: mark(U33(X)) -> a__U33(mark(X))
r102: mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3)
r103: mark(U42(X1,X2,X3)) -> a__U42(mark(X1),X2,X3)
r104: mark(U43(X1,X2,X3)) -> a__U43(mark(X1),X2,X3)
r105: mark(U44(X1,X2,X3)) -> a__U44(mark(X1),X2,X3)
r106: mark(U45(X1,X2)) -> a__U45(mark(X1),X2)
r107: mark(U46(X)) -> a__U46(mark(X))
r108: mark(U51(X1,X2)) -> a__U51(mark(X1),X2)
r109: mark(U52(X)) -> a__U52(mark(X))
r110: mark(U61(X1,X2)) -> a__U61(mark(X1),X2)
r111: mark(U62(X)) -> a__U62(mark(X))
r112: mark(U71(X)) -> a__U71(mark(X))
r113: mark(U81(X)) -> a__U81(mark(X))
r114: mark(U91(X1,X2,X3)) -> a__U91(mark(X1),X2,X3)
r115: mark(U92(X1,X2,X3)) -> a__U92(mark(X1),X2,X3)
r116: mark(U93(X1,X2,X3)) -> a__U93(mark(X1),X2,X3)
r117: mark(U94(X1,X2,X3)) -> a__U94(mark(X1),X2,X3)
r118: mark(U95(X1,X2)) -> a__U95(mark(X1),X2)
r119: mark(U96(X)) -> a__U96(mark(X))
r120: mark(cons(X1,X2)) -> cons(mark(X1),X2)
r121: mark(|0|()) -> |0|()
r122: mark(tt()) -> tt()
r123: mark(s(X)) -> s(mark(X))
r124: mark(nil()) -> nil()
r125: a__zeros() -> zeros()
r126: a__U101(X1,X2,X3) -> U101(X1,X2,X3)
r127: a__U102(X1,X2,X3) -> U102(X1,X2,X3)
r128: a__isNatKind(X) -> isNatKind(X)
r129: a__U103(X1,X2,X3) -> U103(X1,X2,X3)
r130: a__isNatIListKind(X) -> isNatIListKind(X)
r131: a__U104(X1,X2,X3) -> U104(X1,X2,X3)
r132: a__U105(X1,X2) -> U105(X1,X2)
r133: a__isNat(X) -> isNat(X)
r134: a__U106(X) -> U106(X)
r135: a__isNatIList(X) -> isNatIList(X)
r136: a__U11(X1,X2) -> U11(X1,X2)
r137: a__U12(X1,X2) -> U12(X1,X2)
r138: a__U111(X1,X2,X3) -> U111(X1,X2,X3)
r139: a__U112(X1,X2,X3) -> U112(X1,X2,X3)
r140: a__U113(X1,X2,X3) -> U113(X1,X2,X3)
r141: a__U114(X1,X2) -> U114(X1,X2)
r142: a__length(X) -> length(X)
r143: a__U13(X) -> U13(X)
r144: a__isNatList(X) -> isNatList(X)
r145: a__U121(X1,X2) -> U121(X1,X2)
r146: a__U122(X) -> U122(X)
r147: a__U131(X1,X2,X3,X4) -> U131(X1,X2,X3,X4)
r148: a__U132(X1,X2,X3,X4) -> U132(X1,X2,X3,X4)
r149: a__U133(X1,X2,X3,X4) -> U133(X1,X2,X3,X4)
r150: a__U134(X1,X2,X3,X4) -> U134(X1,X2,X3,X4)
r151: a__U135(X1,X2,X3,X4) -> U135(X1,X2,X3,X4)
r152: a__U136(X1,X2,X3,X4) -> U136(X1,X2,X3,X4)
r153: a__take(X1,X2) -> take(X1,X2)
r154: a__U21(X1,X2) -> U21(X1,X2)
r155: a__U22(X1,X2) -> U22(X1,X2)
r156: a__U23(X) -> U23(X)
r157: a__U31(X1,X2) -> U31(X1,X2)
r158: a__U32(X1,X2) -> U32(X1,X2)
r159: a__U33(X) -> U33(X)
r160: a__U41(X1,X2,X3) -> U41(X1,X2,X3)
r161: a__U42(X1,X2,X3) -> U42(X1,X2,X3)
r162: a__U43(X1,X2,X3) -> U43(X1,X2,X3)
r163: a__U44(X1,X2,X3) -> U44(X1,X2,X3)
r164: a__U45(X1,X2) -> U45(X1,X2)
r165: a__U46(X) -> U46(X)
r166: a__U51(X1,X2) -> U51(X1,X2)
r167: a__U52(X) -> U52(X)
r168: a__U61(X1,X2) -> U61(X1,X2)
r169: a__U62(X) -> U62(X)
r170: a__U71(X) -> U71(X)
r171: a__U81(X) -> U81(X)
r172: a__U91(X1,X2,X3) -> U91(X1,X2,X3)
r173: a__U92(X1,X2,X3) -> U92(X1,X2,X3)
r174: a__U93(X1,X2,X3) -> U93(X1,X2,X3)
r175: a__U94(X1,X2,X3) -> U94(X1,X2,X3)
r176: a__U95(X1,X2) -> U95(X1,X2)
r177: a__U96(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: mark#(U11(X1,X2)) -> mark#(X1)
p2: mark#(U134(X1,X2,X3,X4)) -> mark#(X1)
p3: mark#(U131(X1,X2,X3,X4)) -> mark#(X1)
p4: mark#(U122(X)) -> mark#(X)
p5: mark#(U111(X1,X2,X3)) -> mark#(X1)

and R consists of:

r1: a__zeros() -> cons(|0|(),zeros())
r2: a__U101(tt(),V1,V2) -> a__U102(a__isNatKind(V1),V1,V2)
r3: a__U102(tt(),V1,V2) -> a__U103(a__isNatIListKind(V2),V1,V2)
r4: a__U103(tt(),V1,V2) -> a__U104(a__isNatIListKind(V2),V1,V2)
r5: a__U104(tt(),V1,V2) -> a__U105(a__isNat(V1),V2)
r6: a__U105(tt(),V2) -> a__U106(a__isNatIList(V2))
r7: a__U106(tt()) -> tt()
r8: a__U11(tt(),V1) -> a__U12(a__isNatIListKind(V1),V1)
r9: a__U111(tt(),L,N) -> a__U112(a__isNatIListKind(L),L,N)
r10: a__U112(tt(),L,N) -> a__U113(a__isNat(N),L,N)
r11: a__U113(tt(),L,N) -> a__U114(a__isNatKind(N),L)
r12: a__U114(tt(),L) -> s(a__length(mark(L)))
r13: a__U12(tt(),V1) -> a__U13(a__isNatList(V1))
r14: a__U121(tt(),IL) -> a__U122(a__isNatIListKind(IL))
r15: a__U122(tt()) -> nil()
r16: a__U13(tt()) -> tt()
r17: a__U131(tt(),IL,M,N) -> a__U132(a__isNatIListKind(IL),IL,M,N)
r18: a__U132(tt(),IL,M,N) -> a__U133(a__isNat(M),IL,M,N)
r19: a__U133(tt(),IL,M,N) -> a__U134(a__isNatKind(M),IL,M,N)
r20: a__U134(tt(),IL,M,N) -> a__U135(a__isNat(N),IL,M,N)
r21: a__U135(tt(),IL,M,N) -> a__U136(a__isNatKind(N),IL,M,N)
r22: a__U136(tt(),IL,M,N) -> cons(mark(N),take(M,IL))
r23: a__U21(tt(),V1) -> a__U22(a__isNatKind(V1),V1)
r24: a__U22(tt(),V1) -> a__U23(a__isNat(V1))
r25: a__U23(tt()) -> tt()
r26: a__U31(tt(),V) -> a__U32(a__isNatIListKind(V),V)
r27: a__U32(tt(),V) -> a__U33(a__isNatList(V))
r28: a__U33(tt()) -> tt()
r29: a__U41(tt(),V1,V2) -> a__U42(a__isNatKind(V1),V1,V2)
r30: a__U42(tt(),V1,V2) -> a__U43(a__isNatIListKind(V2),V1,V2)
r31: a__U43(tt(),V1,V2) -> a__U44(a__isNatIListKind(V2),V1,V2)
r32: a__U44(tt(),V1,V2) -> a__U45(a__isNat(V1),V2)
r33: a__U45(tt(),V2) -> a__U46(a__isNatIList(V2))
r34: a__U46(tt()) -> tt()
r35: a__U51(tt(),V2) -> a__U52(a__isNatIListKind(V2))
r36: a__U52(tt()) -> tt()
r37: a__U61(tt(),V2) -> a__U62(a__isNatIListKind(V2))
r38: a__U62(tt()) -> tt()
r39: a__U71(tt()) -> tt()
r40: a__U81(tt()) -> tt()
r41: a__U91(tt(),V1,V2) -> a__U92(a__isNatKind(V1),V1,V2)
r42: a__U92(tt(),V1,V2) -> a__U93(a__isNatIListKind(V2),V1,V2)
r43: a__U93(tt(),V1,V2) -> a__U94(a__isNatIListKind(V2),V1,V2)
r44: a__U94(tt(),V1,V2) -> a__U95(a__isNat(V1),V2)
r45: a__U95(tt(),V2) -> a__U96(a__isNatList(V2))
r46: a__U96(tt()) -> tt()
r47: a__isNat(|0|()) -> tt()
r48: a__isNat(length(V1)) -> a__U11(a__isNatIListKind(V1),V1)
r49: a__isNat(s(V1)) -> a__U21(a__isNatKind(V1),V1)
r50: a__isNatIList(V) -> a__U31(a__isNatIListKind(V),V)
r51: a__isNatIList(zeros()) -> tt()
r52: a__isNatIList(cons(V1,V2)) -> a__U41(a__isNatKind(V1),V1,V2)
r53: a__isNatIListKind(nil()) -> tt()
r54: a__isNatIListKind(zeros()) -> tt()
r55: a__isNatIListKind(cons(V1,V2)) -> a__U51(a__isNatKind(V1),V2)
r56: a__isNatIListKind(take(V1,V2)) -> a__U61(a__isNatKind(V1),V2)
r57: a__isNatKind(|0|()) -> tt()
r58: a__isNatKind(length(V1)) -> a__U71(a__isNatIListKind(V1))
r59: a__isNatKind(s(V1)) -> a__U81(a__isNatKind(V1))
r60: a__isNatList(nil()) -> tt()
r61: a__isNatList(cons(V1,V2)) -> a__U91(a__isNatKind(V1),V1,V2)
r62: a__isNatList(take(V1,V2)) -> a__U101(a__isNatKind(V1),V1,V2)
r63: a__length(nil()) -> |0|()
r64: a__length(cons(N,L)) -> a__U111(a__isNatList(L),L,N)
r65: a__take(|0|(),IL) -> a__U121(a__isNatIList(IL),IL)
r66: a__take(s(M),cons(N,IL)) -> a__U131(a__isNatIList(IL),IL,M,N)
r67: mark(zeros()) -> a__zeros()
r68: mark(U101(X1,X2,X3)) -> a__U101(mark(X1),X2,X3)
r69: mark(U102(X1,X2,X3)) -> a__U102(mark(X1),X2,X3)
r70: mark(isNatKind(X)) -> a__isNatKind(X)
r71: mark(U103(X1,X2,X3)) -> a__U103(mark(X1),X2,X3)
r72: mark(isNatIListKind(X)) -> a__isNatIListKind(X)
r73: mark(U104(X1,X2,X3)) -> a__U104(mark(X1),X2,X3)
r74: mark(U105(X1,X2)) -> a__U105(mark(X1),X2)
r75: mark(isNat(X)) -> a__isNat(X)
r76: mark(U106(X)) -> a__U106(mark(X))
r77: mark(isNatIList(X)) -> a__isNatIList(X)
r78: mark(U11(X1,X2)) -> a__U11(mark(X1),X2)
r79: mark(U12(X1,X2)) -> a__U12(mark(X1),X2)
r80: mark(U111(X1,X2,X3)) -> a__U111(mark(X1),X2,X3)
r81: mark(U112(X1,X2,X3)) -> a__U112(mark(X1),X2,X3)
r82: mark(U113(X1,X2,X3)) -> a__U113(mark(X1),X2,X3)
r83: mark(U114(X1,X2)) -> a__U114(mark(X1),X2)
r84: mark(length(X)) -> a__length(mark(X))
r85: mark(U13(X)) -> a__U13(mark(X))
r86: mark(isNatList(X)) -> a__isNatList(X)
r87: mark(U121(X1,X2)) -> a__U121(mark(X1),X2)
r88: mark(U122(X)) -> a__U122(mark(X))
r89: mark(U131(X1,X2,X3,X4)) -> a__U131(mark(X1),X2,X3,X4)
r90: mark(U132(X1,X2,X3,X4)) -> a__U132(mark(X1),X2,X3,X4)
r91: mark(U133(X1,X2,X3,X4)) -> a__U133(mark(X1),X2,X3,X4)
r92: mark(U134(X1,X2,X3,X4)) -> a__U134(mark(X1),X2,X3,X4)
r93: mark(U135(X1,X2,X3,X4)) -> a__U135(mark(X1),X2,X3,X4)
r94: mark(U136(X1,X2,X3,X4)) -> a__U136(mark(X1),X2,X3,X4)
r95: mark(take(X1,X2)) -> a__take(mark(X1),mark(X2))
r96: mark(U21(X1,X2)) -> a__U21(mark(X1),X2)
r97: mark(U22(X1,X2)) -> a__U22(mark(X1),X2)
r98: mark(U23(X)) -> a__U23(mark(X))
r99: mark(U31(X1,X2)) -> a__U31(mark(X1),X2)
r100: mark(U32(X1,X2)) -> a__U32(mark(X1),X2)
r101: mark(U33(X)) -> a__U33(mark(X))
r102: mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3)
r103: mark(U42(X1,X2,X3)) -> a__U42(mark(X1),X2,X3)
r104: mark(U43(X1,X2,X3)) -> a__U43(mark(X1),X2,X3)
r105: mark(U44(X1,X2,X3)) -> a__U44(mark(X1),X2,X3)
r106: mark(U45(X1,X2)) -> a__U45(mark(X1),X2)
r107: mark(U46(X)) -> a__U46(mark(X))
r108: mark(U51(X1,X2)) -> a__U51(mark(X1),X2)
r109: mark(U52(X)) -> a__U52(mark(X))
r110: mark(U61(X1,X2)) -> a__U61(mark(X1),X2)
r111: mark(U62(X)) -> a__U62(mark(X))
r112: mark(U71(X)) -> a__U71(mark(X))
r113: mark(U81(X)) -> a__U81(mark(X))
r114: mark(U91(X1,X2,X3)) -> a__U91(mark(X1),X2,X3)
r115: mark(U92(X1,X2,X3)) -> a__U92(mark(X1),X2,X3)
r116: mark(U93(X1,X2,X3)) -> a__U93(mark(X1),X2,X3)
r117: mark(U94(X1,X2,X3)) -> a__U94(mark(X1),X2,X3)
r118: mark(U95(X1,X2)) -> a__U95(mark(X1),X2)
r119: mark(U96(X)) -> a__U96(mark(X))
r120: mark(cons(X1,X2)) -> cons(mark(X1),X2)
r121: mark(|0|()) -> |0|()
r122: mark(tt()) -> tt()
r123: mark(s(X)) -> s(mark(X))
r124: mark(nil()) -> nil()
r125: a__zeros() -> zeros()
r126: a__U101(X1,X2,X3) -> U101(X1,X2,X3)
r127: a__U102(X1,X2,X3) -> U102(X1,X2,X3)
r128: a__isNatKind(X) -> isNatKind(X)
r129: a__U103(X1,X2,X3) -> U103(X1,X2,X3)
r130: a__isNatIListKind(X) -> isNatIListKind(X)
r131: a__U104(X1,X2,X3) -> U104(X1,X2,X3)
r132: a__U105(X1,X2) -> U105(X1,X2)
r133: a__isNat(X) -> isNat(X)
r134: a__U106(X) -> U106(X)
r135: a__isNatIList(X) -> isNatIList(X)
r136: a__U11(X1,X2) -> U11(X1,X2)
r137: a__U12(X1,X2) -> U12(X1,X2)
r138: a__U111(X1,X2,X3) -> U111(X1,X2,X3)
r139: a__U112(X1,X2,X3) -> U112(X1,X2,X3)
r140: a__U113(X1,X2,X3) -> U113(X1,X2,X3)
r141: a__U114(X1,X2) -> U114(X1,X2)
r142: a__length(X) -> length(X)
r143: a__U13(X) -> U13(X)
r144: a__isNatList(X) -> isNatList(X)
r145: a__U121(X1,X2) -> U121(X1,X2)
r146: a__U122(X) -> U122(X)
r147: a__U131(X1,X2,X3,X4) -> U131(X1,X2,X3,X4)
r148: a__U132(X1,X2,X3,X4) -> U132(X1,X2,X3,X4)
r149: a__U133(X1,X2,X3,X4) -> U133(X1,X2,X3,X4)
r150: a__U134(X1,X2,X3,X4) -> U134(X1,X2,X3,X4)
r151: a__U135(X1,X2,X3,X4) -> U135(X1,X2,X3,X4)
r152: a__U136(X1,X2,X3,X4) -> U136(X1,X2,X3,X4)
r153: a__take(X1,X2) -> take(X1,X2)
r154: a__U21(X1,X2) -> U21(X1,X2)
r155: a__U22(X1,X2) -> U22(X1,X2)
r156: a__U23(X) -> U23(X)
r157: a__U31(X1,X2) -> U31(X1,X2)
r158: a__U32(X1,X2) -> U32(X1,X2)
r159: a__U33(X) -> U33(X)
r160: a__U41(X1,X2,X3) -> U41(X1,X2,X3)
r161: a__U42(X1,X2,X3) -> U42(X1,X2,X3)
r162: a__U43(X1,X2,X3) -> U43(X1,X2,X3)
r163: a__U44(X1,X2,X3) -> U44(X1,X2,X3)
r164: a__U45(X1,X2) -> U45(X1,X2)
r165: a__U46(X) -> U46(X)
r166: a__U51(X1,X2) -> U51(X1,X2)
r167: a__U52(X) -> U52(X)
r168: a__U61(X1,X2) -> U61(X1,X2)
r169: a__U62(X) -> U62(X)
r170: a__U71(X) -> U71(X)
r171: a__U81(X) -> U81(X)
r172: a__U91(X1,X2,X3) -> U91(X1,X2,X3)
r173: a__U92(X1,X2,X3) -> U92(X1,X2,X3)
r174: a__U93(X1,X2,X3) -> U93(X1,X2,X3)
r175: a__U94(X1,X2,X3) -> U94(X1,X2,X3)
r176: a__U95(X1,X2) -> U95(X1,X2)
r177: a__U96(X) -> U96(X)

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  max/plus interpretations on natural numbers:
  
    mark#_A(x1) = x1 + 1
    U11_A(x1,x2) = max{x1, x2}
    U134_A(x1,x2,x3,x4) = max{x1, x2, x3, x4}
    U131_A(x1,x2,x3,x4) = max{x1 + 1, x2, x3, x4}
    U122_A(x1) = x1
    U111_A(x1,x2,x3) = max{x1, x2, x3}

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: mark#(U11(X1,X2)) -> mark#(X1)
p2: mark#(U134(X1,X2,X3,X4)) -> mark#(X1)
p3: mark#(U122(X)) -> mark#(X)
p4: mark#(U111(X1,X2,X3)) -> mark#(X1)

and R consists of:

r1: a__zeros() -> cons(|0|(),zeros())
r2: a__U101(tt(),V1,V2) -> a__U102(a__isNatKind(V1),V1,V2)
r3: a__U102(tt(),V1,V2) -> a__U103(a__isNatIListKind(V2),V1,V2)
r4: a__U103(tt(),V1,V2) -> a__U104(a__isNatIListKind(V2),V1,V2)
r5: a__U104(tt(),V1,V2) -> a__U105(a__isNat(V1),V2)
r6: a__U105(tt(),V2) -> a__U106(a__isNatIList(V2))
r7: a__U106(tt()) -> tt()
r8: a__U11(tt(),V1) -> a__U12(a__isNatIListKind(V1),V1)
r9: a__U111(tt(),L,N) -> a__U112(a__isNatIListKind(L),L,N)
r10: a__U112(tt(),L,N) -> a__U113(a__isNat(N),L,N)
r11: a__U113(tt(),L,N) -> a__U114(a__isNatKind(N),L)
r12: a__U114(tt(),L) -> s(a__length(mark(L)))
r13: a__U12(tt(),V1) -> a__U13(a__isNatList(V1))
r14: a__U121(tt(),IL) -> a__U122(a__isNatIListKind(IL))
r15: a__U122(tt()) -> nil()
r16: a__U13(tt()) -> tt()
r17: a__U131(tt(),IL,M,N) -> a__U132(a__isNatIListKind(IL),IL,M,N)
r18: a__U132(tt(),IL,M,N) -> a__U133(a__isNat(M),IL,M,N)
r19: a__U133(tt(),IL,M,N) -> a__U134(a__isNatKind(M),IL,M,N)
r20: a__U134(tt(),IL,M,N) -> a__U135(a__isNat(N),IL,M,N)
r21: a__U135(tt(),IL,M,N) -> a__U136(a__isNatKind(N),IL,M,N)
r22: a__U136(tt(),IL,M,N) -> cons(mark(N),take(M,IL))
r23: a__U21(tt(),V1) -> a__U22(a__isNatKind(V1),V1)
r24: a__U22(tt(),V1) -> a__U23(a__isNat(V1))
r25: a__U23(tt()) -> tt()
r26: a__U31(tt(),V) -> a__U32(a__isNatIListKind(V),V)
r27: a__U32(tt(),V) -> a__U33(a__isNatList(V))
r28: a__U33(tt()) -> tt()
r29: a__U41(tt(),V1,V2) -> a__U42(a__isNatKind(V1),V1,V2)
r30: a__U42(tt(),V1,V2) -> a__U43(a__isNatIListKind(V2),V1,V2)
r31: a__U43(tt(),V1,V2) -> a__U44(a__isNatIListKind(V2),V1,V2)
r32: a__U44(tt(),V1,V2) -> a__U45(a__isNat(V1),V2)
r33: a__U45(tt(),V2) -> a__U46(a__isNatIList(V2))
r34: a__U46(tt()) -> tt()
r35: a__U51(tt(),V2) -> a__U52(a__isNatIListKind(V2))
r36: a__U52(tt()) -> tt()
r37: a__U61(tt(),V2) -> a__U62(a__isNatIListKind(V2))
r38: a__U62(tt()) -> tt()
r39: a__U71(tt()) -> tt()
r40: a__U81(tt()) -> tt()
r41: a__U91(tt(),V1,V2) -> a__U92(a__isNatKind(V1),V1,V2)
r42: a__U92(tt(),V1,V2) -> a__U93(a__isNatIListKind(V2),V1,V2)
r43: a__U93(tt(),V1,V2) -> a__U94(a__isNatIListKind(V2),V1,V2)
r44: a__U94(tt(),V1,V2) -> a__U95(a__isNat(V1),V2)
r45: a__U95(tt(),V2) -> a__U96(a__isNatList(V2))
r46: a__U96(tt()) -> tt()
r47: a__isNat(|0|()) -> tt()
r48: a__isNat(length(V1)) -> a__U11(a__isNatIListKind(V1),V1)
r49: a__isNat(s(V1)) -> a__U21(a__isNatKind(V1),V1)
r50: a__isNatIList(V) -> a__U31(a__isNatIListKind(V),V)
r51: a__isNatIList(zeros()) -> tt()
r52: a__isNatIList(cons(V1,V2)) -> a__U41(a__isNatKind(V1),V1,V2)
r53: a__isNatIListKind(nil()) -> tt()
r54: a__isNatIListKind(zeros()) -> tt()
r55: a__isNatIListKind(cons(V1,V2)) -> a__U51(a__isNatKind(V1),V2)
r56: a__isNatIListKind(take(V1,V2)) -> a__U61(a__isNatKind(V1),V2)
r57: a__isNatKind(|0|()) -> tt()
r58: a__isNatKind(length(V1)) -> a__U71(a__isNatIListKind(V1))
r59: a__isNatKind(s(V1)) -> a__U81(a__isNatKind(V1))
r60: a__isNatList(nil()) -> tt()
r61: a__isNatList(cons(V1,V2)) -> a__U91(a__isNatKind(V1),V1,V2)
r62: a__isNatList(take(V1,V2)) -> a__U101(a__isNatKind(V1),V1,V2)
r63: a__length(nil()) -> |0|()
r64: a__length(cons(N,L)) -> a__U111(a__isNatList(L),L,N)
r65: a__take(|0|(),IL) -> a__U121(a__isNatIList(IL),IL)
r66: a__take(s(M),cons(N,IL)) -> a__U131(a__isNatIList(IL),IL,M,N)
r67: mark(zeros()) -> a__zeros()
r68: mark(U101(X1,X2,X3)) -> a__U101(mark(X1),X2,X3)
r69: mark(U102(X1,X2,X3)) -> a__U102(mark(X1),X2,X3)
r70: mark(isNatKind(X)) -> a__isNatKind(X)
r71: mark(U103(X1,X2,X3)) -> a__U103(mark(X1),X2,X3)
r72: mark(isNatIListKind(X)) -> a__isNatIListKind(X)
r73: mark(U104(X1,X2,X3)) -> a__U104(mark(X1),X2,X3)
r74: mark(U105(X1,X2)) -> a__U105(mark(X1),X2)
r75: mark(isNat(X)) -> a__isNat(X)
r76: mark(U106(X)) -> a__U106(mark(X))
r77: mark(isNatIList(X)) -> a__isNatIList(X)
r78: mark(U11(X1,X2)) -> a__U11(mark(X1),X2)
r79: mark(U12(X1,X2)) -> a__U12(mark(X1),X2)
r80: mark(U111(X1,X2,X3)) -> a__U111(mark(X1),X2,X3)
r81: mark(U112(X1,X2,X3)) -> a__U112(mark(X1),X2,X3)
r82: mark(U113(X1,X2,X3)) -> a__U113(mark(X1),X2,X3)
r83: mark(U114(X1,X2)) -> a__U114(mark(X1),X2)
r84: mark(length(X)) -> a__length(mark(X))
r85: mark(U13(X)) -> a__U13(mark(X))
r86: mark(isNatList(X)) -> a__isNatList(X)
r87: mark(U121(X1,X2)) -> a__U121(mark(X1),X2)
r88: mark(U122(X)) -> a__U122(mark(X))
r89: mark(U131(X1,X2,X3,X4)) -> a__U131(mark(X1),X2,X3,X4)
r90: mark(U132(X1,X2,X3,X4)) -> a__U132(mark(X1),X2,X3,X4)
r91: mark(U133(X1,X2,X3,X4)) -> a__U133(mark(X1),X2,X3,X4)
r92: mark(U134(X1,X2,X3,X4)) -> a__U134(mark(X1),X2,X3,X4)
r93: mark(U135(X1,X2,X3,X4)) -> a__U135(mark(X1),X2,X3,X4)
r94: mark(U136(X1,X2,X3,X4)) -> a__U136(mark(X1),X2,X3,X4)
r95: mark(take(X1,X2)) -> a__take(mark(X1),mark(X2))
r96: mark(U21(X1,X2)) -> a__U21(mark(X1),X2)
r97: mark(U22(X1,X2)) -> a__U22(mark(X1),X2)
r98: mark(U23(X)) -> a__U23(mark(X))
r99: mark(U31(X1,X2)) -> a__U31(mark(X1),X2)
r100: mark(U32(X1,X2)) -> a__U32(mark(X1),X2)
r101: mark(U33(X)) -> a__U33(mark(X))
r102: mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3)
r103: mark(U42(X1,X2,X3)) -> a__U42(mark(X1),X2,X3)
r104: mark(U43(X1,X2,X3)) -> a__U43(mark(X1),X2,X3)
r105: mark(U44(X1,X2,X3)) -> a__U44(mark(X1),X2,X3)
r106: mark(U45(X1,X2)) -> a__U45(mark(X1),X2)
r107: mark(U46(X)) -> a__U46(mark(X))
r108: mark(U51(X1,X2)) -> a__U51(mark(X1),X2)
r109: mark(U52(X)) -> a__U52(mark(X))
r110: mark(U61(X1,X2)) -> a__U61(mark(X1),X2)
r111: mark(U62(X)) -> a__U62(mark(X))
r112: mark(U71(X)) -> a__U71(mark(X))
r113: mark(U81(X)) -> a__U81(mark(X))
r114: mark(U91(X1,X2,X3)) -> a__U91(mark(X1),X2,X3)
r115: mark(U92(X1,X2,X3)) -> a__U92(mark(X1),X2,X3)
r116: mark(U93(X1,X2,X3)) -> a__U93(mark(X1),X2,X3)
r117: mark(U94(X1,X2,X3)) -> a__U94(mark(X1),X2,X3)
r118: mark(U95(X1,X2)) -> a__U95(mark(X1),X2)
r119: mark(U96(X)) -> a__U96(mark(X))
r120: mark(cons(X1,X2)) -> cons(mark(X1),X2)
r121: mark(|0|()) -> |0|()
r122: mark(tt()) -> tt()
r123: mark(s(X)) -> s(mark(X))
r124: mark(nil()) -> nil()
r125: a__zeros() -> zeros()
r126: a__U101(X1,X2,X3) -> U101(X1,X2,X3)
r127: a__U102(X1,X2,X3) -> U102(X1,X2,X3)
r128: a__isNatKind(X) -> isNatKind(X)
r129: a__U103(X1,X2,X3) -> U103(X1,X2,X3)
r130: a__isNatIListKind(X) -> isNatIListKind(X)
r131: a__U104(X1,X2,X3) -> U104(X1,X2,X3)
r132: a__U105(X1,X2) -> U105(X1,X2)
r133: a__isNat(X) -> isNat(X)
r134: a__U106(X) -> U106(X)
r135: a__isNatIList(X) -> isNatIList(X)
r136: a__U11(X1,X2) -> U11(X1,X2)
r137: a__U12(X1,X2) -> U12(X1,X2)
r138: a__U111(X1,X2,X3) -> U111(X1,X2,X3)
r139: a__U112(X1,X2,X3) -> U112(X1,X2,X3)
r140: a__U113(X1,X2,X3) -> U113(X1,X2,X3)
r141: a__U114(X1,X2) -> U114(X1,X2)
r142: a__length(X) -> length(X)
r143: a__U13(X) -> U13(X)
r144: a__isNatList(X) -> isNatList(X)
r145: a__U121(X1,X2) -> U121(X1,X2)
r146: a__U122(X) -> U122(X)
r147: a__U131(X1,X2,X3,X4) -> U131(X1,X2,X3,X4)
r148: a__U132(X1,X2,X3,X4) -> U132(X1,X2,X3,X4)
r149: a__U133(X1,X2,X3,X4) -> U133(X1,X2,X3,X4)
r150: a__U134(X1,X2,X3,X4) -> U134(X1,X2,X3,X4)
r151: a__U135(X1,X2,X3,X4) -> U135(X1,X2,X3,X4)
r152: a__U136(X1,X2,X3,X4) -> U136(X1,X2,X3,X4)
r153: a__take(X1,X2) -> take(X1,X2)
r154: a__U21(X1,X2) -> U21(X1,X2)
r155: a__U22(X1,X2) -> U22(X1,X2)
r156: a__U23(X) -> U23(X)
r157: a__U31(X1,X2) -> U31(X1,X2)
r158: a__U32(X1,X2) -> U32(X1,X2)
r159: a__U33(X) -> U33(X)
r160: a__U41(X1,X2,X3) -> U41(X1,X2,X3)
r161: a__U42(X1,X2,X3) -> U42(X1,X2,X3)
r162: a__U43(X1,X2,X3) -> U43(X1,X2,X3)
r163: a__U44(X1,X2,X3) -> U44(X1,X2,X3)
r164: a__U45(X1,X2) -> U45(X1,X2)
r165: a__U46(X) -> U46(X)
r166: a__U51(X1,X2) -> U51(X1,X2)
r167: a__U52(X) -> U52(X)
r168: a__U61(X1,X2) -> U61(X1,X2)
r169: a__U62(X) -> U62(X)
r170: a__U71(X) -> U71(X)
r171: a__U81(X) -> U81(X)
r172: a__U91(X1,X2,X3) -> U91(X1,X2,X3)
r173: a__U92(X1,X2,X3) -> U92(X1,X2,X3)
r174: a__U93(X1,X2,X3) -> U93(X1,X2,X3)
r175: a__U94(X1,X2,X3) -> U94(X1,X2,X3)
r176: a__U95(X1,X2) -> U95(X1,X2)
r177: a__U96(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: mark#(U11(X1,X2)) -> mark#(X1)
p2: mark#(U111(X1,X2,X3)) -> mark#(X1)
p3: mark#(U122(X)) -> mark#(X)
p4: mark#(U134(X1,X2,X3,X4)) -> mark#(X1)

and R consists of:

r1: a__zeros() -> cons(|0|(),zeros())
r2: a__U101(tt(),V1,V2) -> a__U102(a__isNatKind(V1),V1,V2)
r3: a__U102(tt(),V1,V2) -> a__U103(a__isNatIListKind(V2),V1,V2)
r4: a__U103(tt(),V1,V2) -> a__U104(a__isNatIListKind(V2),V1,V2)
r5: a__U104(tt(),V1,V2) -> a__U105(a__isNat(V1),V2)
r6: a__U105(tt(),V2) -> a__U106(a__isNatIList(V2))
r7: a__U106(tt()) -> tt()
r8: a__U11(tt(),V1) -> a__U12(a__isNatIListKind(V1),V1)
r9: a__U111(tt(),L,N) -> a__U112(a__isNatIListKind(L),L,N)
r10: a__U112(tt(),L,N) -> a__U113(a__isNat(N),L,N)
r11: a__U113(tt(),L,N) -> a__U114(a__isNatKind(N),L)
r12: a__U114(tt(),L) -> s(a__length(mark(L)))
r13: a__U12(tt(),V1) -> a__U13(a__isNatList(V1))
r14: a__U121(tt(),IL) -> a__U122(a__isNatIListKind(IL))
r15: a__U122(tt()) -> nil()
r16: a__U13(tt()) -> tt()
r17: a__U131(tt(),IL,M,N) -> a__U132(a__isNatIListKind(IL),IL,M,N)
r18: a__U132(tt(),IL,M,N) -> a__U133(a__isNat(M),IL,M,N)
r19: a__U133(tt(),IL,M,N) -> a__U134(a__isNatKind(M),IL,M,N)
r20: a__U134(tt(),IL,M,N) -> a__U135(a__isNat(N),IL,M,N)
r21: a__U135(tt(),IL,M,N) -> a__U136(a__isNatKind(N),IL,M,N)
r22: a__U136(tt(),IL,M,N) -> cons(mark(N),take(M,IL))
r23: a__U21(tt(),V1) -> a__U22(a__isNatKind(V1),V1)
r24: a__U22(tt(),V1) -> a__U23(a__isNat(V1))
r25: a__U23(tt()) -> tt()
r26: a__U31(tt(),V) -> a__U32(a__isNatIListKind(V),V)
r27: a__U32(tt(),V) -> a__U33(a__isNatList(V))
r28: a__U33(tt()) -> tt()
r29: a__U41(tt(),V1,V2) -> a__U42(a__isNatKind(V1),V1,V2)
r30: a__U42(tt(),V1,V2) -> a__U43(a__isNatIListKind(V2),V1,V2)
r31: a__U43(tt(),V1,V2) -> a__U44(a__isNatIListKind(V2),V1,V2)
r32: a__U44(tt(),V1,V2) -> a__U45(a__isNat(V1),V2)
r33: a__U45(tt(),V2) -> a__U46(a__isNatIList(V2))
r34: a__U46(tt()) -> tt()
r35: a__U51(tt(),V2) -> a__U52(a__isNatIListKind(V2))
r36: a__U52(tt()) -> tt()
r37: a__U61(tt(),V2) -> a__U62(a__isNatIListKind(V2))
r38: a__U62(tt()) -> tt()
r39: a__U71(tt()) -> tt()
r40: a__U81(tt()) -> tt()
r41: a__U91(tt(),V1,V2) -> a__U92(a__isNatKind(V1),V1,V2)
r42: a__U92(tt(),V1,V2) -> a__U93(a__isNatIListKind(V2),V1,V2)
r43: a__U93(tt(),V1,V2) -> a__U94(a__isNatIListKind(V2),V1,V2)
r44: a__U94(tt(),V1,V2) -> a__U95(a__isNat(V1),V2)
r45: a__U95(tt(),V2) -> a__U96(a__isNatList(V2))
r46: a__U96(tt()) -> tt()
r47: a__isNat(|0|()) -> tt()
r48: a__isNat(length(V1)) -> a__U11(a__isNatIListKind(V1),V1)
r49: a__isNat(s(V1)) -> a__U21(a__isNatKind(V1),V1)
r50: a__isNatIList(V) -> a__U31(a__isNatIListKind(V),V)
r51: a__isNatIList(zeros()) -> tt()
r52: a__isNatIList(cons(V1,V2)) -> a__U41(a__isNatKind(V1),V1,V2)
r53: a__isNatIListKind(nil()) -> tt()
r54: a__isNatIListKind(zeros()) -> tt()
r55: a__isNatIListKind(cons(V1,V2)) -> a__U51(a__isNatKind(V1),V2)
r56: a__isNatIListKind(take(V1,V2)) -> a__U61(a__isNatKind(V1),V2)
r57: a__isNatKind(|0|()) -> tt()
r58: a__isNatKind(length(V1)) -> a__U71(a__isNatIListKind(V1))
r59: a__isNatKind(s(V1)) -> a__U81(a__isNatKind(V1))
r60: a__isNatList(nil()) -> tt()
r61: a__isNatList(cons(V1,V2)) -> a__U91(a__isNatKind(V1),V1,V2)
r62: a__isNatList(take(V1,V2)) -> a__U101(a__isNatKind(V1),V1,V2)
r63: a__length(nil()) -> |0|()
r64: a__length(cons(N,L)) -> a__U111(a__isNatList(L),L,N)
r65: a__take(|0|(),IL) -> a__U121(a__isNatIList(IL),IL)
r66: a__take(s(M),cons(N,IL)) -> a__U131(a__isNatIList(IL),IL,M,N)
r67: mark(zeros()) -> a__zeros()
r68: mark(U101(X1,X2,X3)) -> a__U101(mark(X1),X2,X3)
r69: mark(U102(X1,X2,X3)) -> a__U102(mark(X1),X2,X3)
r70: mark(isNatKind(X)) -> a__isNatKind(X)
r71: mark(U103(X1,X2,X3)) -> a__U103(mark(X1),X2,X3)
r72: mark(isNatIListKind(X)) -> a__isNatIListKind(X)
r73: mark(U104(X1,X2,X3)) -> a__U104(mark(X1),X2,X3)
r74: mark(U105(X1,X2)) -> a__U105(mark(X1),X2)
r75: mark(isNat(X)) -> a__isNat(X)
r76: mark(U106(X)) -> a__U106(mark(X))
r77: mark(isNatIList(X)) -> a__isNatIList(X)
r78: mark(U11(X1,X2)) -> a__U11(mark(X1),X2)
r79: mark(U12(X1,X2)) -> a__U12(mark(X1),X2)
r80: mark(U111(X1,X2,X3)) -> a__U111(mark(X1),X2,X3)
r81: mark(U112(X1,X2,X3)) -> a__U112(mark(X1),X2,X3)
r82: mark(U113(X1,X2,X3)) -> a__U113(mark(X1),X2,X3)
r83: mark(U114(X1,X2)) -> a__U114(mark(X1),X2)
r84: mark(length(X)) -> a__length(mark(X))
r85: mark(U13(X)) -> a__U13(mark(X))
r86: mark(isNatList(X)) -> a__isNatList(X)
r87: mark(U121(X1,X2)) -> a__U121(mark(X1),X2)
r88: mark(U122(X)) -> a__U122(mark(X))
r89: mark(U131(X1,X2,X3,X4)) -> a__U131(mark(X1),X2,X3,X4)
r90: mark(U132(X1,X2,X3,X4)) -> a__U132(mark(X1),X2,X3,X4)
r91: mark(U133(X1,X2,X3,X4)) -> a__U133(mark(X1),X2,X3,X4)
r92: mark(U134(X1,X2,X3,X4)) -> a__U134(mark(X1),X2,X3,X4)
r93: mark(U135(X1,X2,X3,X4)) -> a__U135(mark(X1),X2,X3,X4)
r94: mark(U136(X1,X2,X3,X4)) -> a__U136(mark(X1),X2,X3,X4)
r95: mark(take(X1,X2)) -> a__take(mark(X1),mark(X2))
r96: mark(U21(X1,X2)) -> a__U21(mark(X1),X2)
r97: mark(U22(X1,X2)) -> a__U22(mark(X1),X2)
r98: mark(U23(X)) -> a__U23(mark(X))
r99: mark(U31(X1,X2)) -> a__U31(mark(X1),X2)
r100: mark(U32(X1,X2)) -> a__U32(mark(X1),X2)
r101: mark(U33(X)) -> a__U33(mark(X))
r102: mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3)
r103: mark(U42(X1,X2,X3)) -> a__U42(mark(X1),X2,X3)
r104: mark(U43(X1,X2,X3)) -> a__U43(mark(X1),X2,X3)
r105: mark(U44(X1,X2,X3)) -> a__U44(mark(X1),X2,X3)
r106: mark(U45(X1,X2)) -> a__U45(mark(X1),X2)
r107: mark(U46(X)) -> a__U46(mark(X))
r108: mark(U51(X1,X2)) -> a__U51(mark(X1),X2)
r109: mark(U52(X)) -> a__U52(mark(X))
r110: mark(U61(X1,X2)) -> a__U61(mark(X1),X2)
r111: mark(U62(X)) -> a__U62(mark(X))
r112: mark(U71(X)) -> a__U71(mark(X))
r113: mark(U81(X)) -> a__U81(mark(X))
r114: mark(U91(X1,X2,X3)) -> a__U91(mark(X1),X2,X3)
r115: mark(U92(X1,X2,X3)) -> a__U92(mark(X1),X2,X3)
r116: mark(U93(X1,X2,X3)) -> a__U93(mark(X1),X2,X3)
r117: mark(U94(X1,X2,X3)) -> a__U94(mark(X1),X2,X3)
r118: mark(U95(X1,X2)) -> a__U95(mark(X1),X2)
r119: mark(U96(X)) -> a__U96(mark(X))
r120: mark(cons(X1,X2)) -> cons(mark(X1),X2)
r121: mark(|0|()) -> |0|()
r122: mark(tt()) -> tt()
r123: mark(s(X)) -> s(mark(X))
r124: mark(nil()) -> nil()
r125: a__zeros() -> zeros()
r126: a__U101(X1,X2,X3) -> U101(X1,X2,X3)
r127: a__U102(X1,X2,X3) -> U102(X1,X2,X3)
r128: a__isNatKind(X) -> isNatKind(X)
r129: a__U103(X1,X2,X3) -> U103(X1,X2,X3)
r130: a__isNatIListKind(X) -> isNatIListKind(X)
r131: a__U104(X1,X2,X3) -> U104(X1,X2,X3)
r132: a__U105(X1,X2) -> U105(X1,X2)
r133: a__isNat(X) -> isNat(X)
r134: a__U106(X) -> U106(X)
r135: a__isNatIList(X) -> isNatIList(X)
r136: a__U11(X1,X2) -> U11(X1,X2)
r137: a__U12(X1,X2) -> U12(X1,X2)
r138: a__U111(X1,X2,X3) -> U111(X1,X2,X3)
r139: a__U112(X1,X2,X3) -> U112(X1,X2,X3)
r140: a__U113(X1,X2,X3) -> U113(X1,X2,X3)
r141: a__U114(X1,X2) -> U114(X1,X2)
r142: a__length(X) -> length(X)
r143: a__U13(X) -> U13(X)
r144: a__isNatList(X) -> isNatList(X)
r145: a__U121(X1,X2) -> U121(X1,X2)
r146: a__U122(X) -> U122(X)
r147: a__U131(X1,X2,X3,X4) -> U131(X1,X2,X3,X4)
r148: a__U132(X1,X2,X3,X4) -> U132(X1,X2,X3,X4)
r149: a__U133(X1,X2,X3,X4) -> U133(X1,X2,X3,X4)
r150: a__U134(X1,X2,X3,X4) -> U134(X1,X2,X3,X4)
r151: a__U135(X1,X2,X3,X4) -> U135(X1,X2,X3,X4)
r152: a__U136(X1,X2,X3,X4) -> U136(X1,X2,X3,X4)
r153: a__take(X1,X2) -> take(X1,X2)
r154: a__U21(X1,X2) -> U21(X1,X2)
r155: a__U22(X1,X2) -> U22(X1,X2)
r156: a__U23(X) -> U23(X)
r157: a__U31(X1,X2) -> U31(X1,X2)
r158: a__U32(X1,X2) -> U32(X1,X2)
r159: a__U33(X) -> U33(X)
r160: a__U41(X1,X2,X3) -> U41(X1,X2,X3)
r161: a__U42(X1,X2,X3) -> U42(X1,X2,X3)
r162: a__U43(X1,X2,X3) -> U43(X1,X2,X3)
r163: a__U44(X1,X2,X3) -> U44(X1,X2,X3)
r164: a__U45(X1,X2) -> U45(X1,X2)
r165: a__U46(X) -> U46(X)
r166: a__U51(X1,X2) -> U51(X1,X2)
r167: a__U52(X) -> U52(X)
r168: a__U61(X1,X2) -> U61(X1,X2)
r169: a__U62(X) -> U62(X)
r170: a__U71(X) -> U71(X)
r171: a__U81(X) -> U81(X)
r172: a__U91(X1,X2,X3) -> U91(X1,X2,X3)
r173: a__U92(X1,X2,X3) -> U92(X1,X2,X3)
r174: a__U93(X1,X2,X3) -> U93(X1,X2,X3)
r175: a__U94(X1,X2,X3) -> U94(X1,X2,X3)
r176: a__U95(X1,X2) -> U95(X1,X2)
r177: a__U96(X) -> U96(X)

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  max/plus interpretations on natural numbers:
  
    mark#_A(x1) = x1 + 1
    U11_A(x1,x2) = max{x1, x2}
    U111_A(x1,x2,x3) = max{x1, x2, x3}
    U122_A(x1) = x1 + 1
    U134_A(x1,x2,x3,x4) = max{x1, x2, x3, x4}

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: mark#(U11(X1,X2)) -> mark#(X1)
p2: mark#(U111(X1,X2,X3)) -> mark#(X1)
p3: mark#(U134(X1,X2,X3,X4)) -> mark#(X1)

and R consists of:

r1: a__zeros() -> cons(|0|(),zeros())
r2: a__U101(tt(),V1,V2) -> a__U102(a__isNatKind(V1),V1,V2)
r3: a__U102(tt(),V1,V2) -> a__U103(a__isNatIListKind(V2),V1,V2)
r4: a__U103(tt(),V1,V2) -> a__U104(a__isNatIListKind(V2),V1,V2)
r5: a__U104(tt(),V1,V2) -> a__U105(a__isNat(V1),V2)
r6: a__U105(tt(),V2) -> a__U106(a__isNatIList(V2))
r7: a__U106(tt()) -> tt()
r8: a__U11(tt(),V1) -> a__U12(a__isNatIListKind(V1),V1)
r9: a__U111(tt(),L,N) -> a__U112(a__isNatIListKind(L),L,N)
r10: a__U112(tt(),L,N) -> a__U113(a__isNat(N),L,N)
r11: a__U113(tt(),L,N) -> a__U114(a__isNatKind(N),L)
r12: a__U114(tt(),L) -> s(a__length(mark(L)))
r13: a__U12(tt(),V1) -> a__U13(a__isNatList(V1))
r14: a__U121(tt(),IL) -> a__U122(a__isNatIListKind(IL))
r15: a__U122(tt()) -> nil()
r16: a__U13(tt()) -> tt()
r17: a__U131(tt(),IL,M,N) -> a__U132(a__isNatIListKind(IL),IL,M,N)
r18: a__U132(tt(),IL,M,N) -> a__U133(a__isNat(M),IL,M,N)
r19: a__U133(tt(),IL,M,N) -> a__U134(a__isNatKind(M),IL,M,N)
r20: a__U134(tt(),IL,M,N) -> a__U135(a__isNat(N),IL,M,N)
r21: a__U135(tt(),IL,M,N) -> a__U136(a__isNatKind(N),IL,M,N)
r22: a__U136(tt(),IL,M,N) -> cons(mark(N),take(M,IL))
r23: a__U21(tt(),V1) -> a__U22(a__isNatKind(V1),V1)
r24: a__U22(tt(),V1) -> a__U23(a__isNat(V1))
r25: a__U23(tt()) -> tt()
r26: a__U31(tt(),V) -> a__U32(a__isNatIListKind(V),V)
r27: a__U32(tt(),V) -> a__U33(a__isNatList(V))
r28: a__U33(tt()) -> tt()
r29: a__U41(tt(),V1,V2) -> a__U42(a__isNatKind(V1),V1,V2)
r30: a__U42(tt(),V1,V2) -> a__U43(a__isNatIListKind(V2),V1,V2)
r31: a__U43(tt(),V1,V2) -> a__U44(a__isNatIListKind(V2),V1,V2)
r32: a__U44(tt(),V1,V2) -> a__U45(a__isNat(V1),V2)
r33: a__U45(tt(),V2) -> a__U46(a__isNatIList(V2))
r34: a__U46(tt()) -> tt()
r35: a__U51(tt(),V2) -> a__U52(a__isNatIListKind(V2))
r36: a__U52(tt()) -> tt()
r37: a__U61(tt(),V2) -> a__U62(a__isNatIListKind(V2))
r38: a__U62(tt()) -> tt()
r39: a__U71(tt()) -> tt()
r40: a__U81(tt()) -> tt()
r41: a__U91(tt(),V1,V2) -> a__U92(a__isNatKind(V1),V1,V2)
r42: a__U92(tt(),V1,V2) -> a__U93(a__isNatIListKind(V2),V1,V2)
r43: a__U93(tt(),V1,V2) -> a__U94(a__isNatIListKind(V2),V1,V2)
r44: a__U94(tt(),V1,V2) -> a__U95(a__isNat(V1),V2)
r45: a__U95(tt(),V2) -> a__U96(a__isNatList(V2))
r46: a__U96(tt()) -> tt()
r47: a__isNat(|0|()) -> tt()
r48: a__isNat(length(V1)) -> a__U11(a__isNatIListKind(V1),V1)
r49: a__isNat(s(V1)) -> a__U21(a__isNatKind(V1),V1)
r50: a__isNatIList(V) -> a__U31(a__isNatIListKind(V),V)
r51: a__isNatIList(zeros()) -> tt()
r52: a__isNatIList(cons(V1,V2)) -> a__U41(a__isNatKind(V1),V1,V2)
r53: a__isNatIListKind(nil()) -> tt()
r54: a__isNatIListKind(zeros()) -> tt()
r55: a__isNatIListKind(cons(V1,V2)) -> a__U51(a__isNatKind(V1),V2)
r56: a__isNatIListKind(take(V1,V2)) -> a__U61(a__isNatKind(V1),V2)
r57: a__isNatKind(|0|()) -> tt()
r58: a__isNatKind(length(V1)) -> a__U71(a__isNatIListKind(V1))
r59: a__isNatKind(s(V1)) -> a__U81(a__isNatKind(V1))
r60: a__isNatList(nil()) -> tt()
r61: a__isNatList(cons(V1,V2)) -> a__U91(a__isNatKind(V1),V1,V2)
r62: a__isNatList(take(V1,V2)) -> a__U101(a__isNatKind(V1),V1,V2)
r63: a__length(nil()) -> |0|()
r64: a__length(cons(N,L)) -> a__U111(a__isNatList(L),L,N)
r65: a__take(|0|(),IL) -> a__U121(a__isNatIList(IL),IL)
r66: a__take(s(M),cons(N,IL)) -> a__U131(a__isNatIList(IL),IL,M,N)
r67: mark(zeros()) -> a__zeros()
r68: mark(U101(X1,X2,X3)) -> a__U101(mark(X1),X2,X3)
r69: mark(U102(X1,X2,X3)) -> a__U102(mark(X1),X2,X3)
r70: mark(isNatKind(X)) -> a__isNatKind(X)
r71: mark(U103(X1,X2,X3)) -> a__U103(mark(X1),X2,X3)
r72: mark(isNatIListKind(X)) -> a__isNatIListKind(X)
r73: mark(U104(X1,X2,X3)) -> a__U104(mark(X1),X2,X3)
r74: mark(U105(X1,X2)) -> a__U105(mark(X1),X2)
r75: mark(isNat(X)) -> a__isNat(X)
r76: mark(U106(X)) -> a__U106(mark(X))
r77: mark(isNatIList(X)) -> a__isNatIList(X)
r78: mark(U11(X1,X2)) -> a__U11(mark(X1),X2)
r79: mark(U12(X1,X2)) -> a__U12(mark(X1),X2)
r80: mark(U111(X1,X2,X3)) -> a__U111(mark(X1),X2,X3)
r81: mark(U112(X1,X2,X3)) -> a__U112(mark(X1),X2,X3)
r82: mark(U113(X1,X2,X3)) -> a__U113(mark(X1),X2,X3)
r83: mark(U114(X1,X2)) -> a__U114(mark(X1),X2)
r84: mark(length(X)) -> a__length(mark(X))
r85: mark(U13(X)) -> a__U13(mark(X))
r86: mark(isNatList(X)) -> a__isNatList(X)
r87: mark(U121(X1,X2)) -> a__U121(mark(X1),X2)
r88: mark(U122(X)) -> a__U122(mark(X))
r89: mark(U131(X1,X2,X3,X4)) -> a__U131(mark(X1),X2,X3,X4)
r90: mark(U132(X1,X2,X3,X4)) -> a__U132(mark(X1),X2,X3,X4)
r91: mark(U133(X1,X2,X3,X4)) -> a__U133(mark(X1),X2,X3,X4)
r92: mark(U134(X1,X2,X3,X4)) -> a__U134(mark(X1),X2,X3,X4)
r93: mark(U135(X1,X2,X3,X4)) -> a__U135(mark(X1),X2,X3,X4)
r94: mark(U136(X1,X2,X3,X4)) -> a__U136(mark(X1),X2,X3,X4)
r95: mark(take(X1,X2)) -> a__take(mark(X1),mark(X2))
r96: mark(U21(X1,X2)) -> a__U21(mark(X1),X2)
r97: mark(U22(X1,X2)) -> a__U22(mark(X1),X2)
r98: mark(U23(X)) -> a__U23(mark(X))
r99: mark(U31(X1,X2)) -> a__U31(mark(X1),X2)
r100: mark(U32(X1,X2)) -> a__U32(mark(X1),X2)
r101: mark(U33(X)) -> a__U33(mark(X))
r102: mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3)
r103: mark(U42(X1,X2,X3)) -> a__U42(mark(X1),X2,X3)
r104: mark(U43(X1,X2,X3)) -> a__U43(mark(X1),X2,X3)
r105: mark(U44(X1,X2,X3)) -> a__U44(mark(X1),X2,X3)
r106: mark(U45(X1,X2)) -> a__U45(mark(X1),X2)
r107: mark(U46(X)) -> a__U46(mark(X))
r108: mark(U51(X1,X2)) -> a__U51(mark(X1),X2)
r109: mark(U52(X)) -> a__U52(mark(X))
r110: mark(U61(X1,X2)) -> a__U61(mark(X1),X2)
r111: mark(U62(X)) -> a__U62(mark(X))
r112: mark(U71(X)) -> a__U71(mark(X))
r113: mark(U81(X)) -> a__U81(mark(X))
r114: mark(U91(X1,X2,X3)) -> a__U91(mark(X1),X2,X3)
r115: mark(U92(X1,X2,X3)) -> a__U92(mark(X1),X2,X3)
r116: mark(U93(X1,X2,X3)) -> a__U93(mark(X1),X2,X3)
r117: mark(U94(X1,X2,X3)) -> a__U94(mark(X1),X2,X3)
r118: mark(U95(X1,X2)) -> a__U95(mark(X1),X2)
r119: mark(U96(X)) -> a__U96(mark(X))
r120: mark(cons(X1,X2)) -> cons(mark(X1),X2)
r121: mark(|0|()) -> |0|()
r122: mark(tt()) -> tt()
r123: mark(s(X)) -> s(mark(X))
r124: mark(nil()) -> nil()
r125: a__zeros() -> zeros()
r126: a__U101(X1,X2,X3) -> U101(X1,X2,X3)
r127: a__U102(X1,X2,X3) -> U102(X1,X2,X3)
r128: a__isNatKind(X) -> isNatKind(X)
r129: a__U103(X1,X2,X3) -> U103(X1,X2,X3)
r130: a__isNatIListKind(X) -> isNatIListKind(X)
r131: a__U104(X1,X2,X3) -> U104(X1,X2,X3)
r132: a__U105(X1,X2) -> U105(X1,X2)
r133: a__isNat(X) -> isNat(X)
r134: a__U106(X) -> U106(X)
r135: a__isNatIList(X) -> isNatIList(X)
r136: a__U11(X1,X2) -> U11(X1,X2)
r137: a__U12(X1,X2) -> U12(X1,X2)
r138: a__U111(X1,X2,X3) -> U111(X1,X2,X3)
r139: a__U112(X1,X2,X3) -> U112(X1,X2,X3)
r140: a__U113(X1,X2,X3) -> U113(X1,X2,X3)
r141: a__U114(X1,X2) -> U114(X1,X2)
r142: a__length(X) -> length(X)
r143: a__U13(X) -> U13(X)
r144: a__isNatList(X) -> isNatList(X)
r145: a__U121(X1,X2) -> U121(X1,X2)
r146: a__U122(X) -> U122(X)
r147: a__U131(X1,X2,X3,X4) -> U131(X1,X2,X3,X4)
r148: a__U132(X1,X2,X3,X4) -> U132(X1,X2,X3,X4)
r149: a__U133(X1,X2,X3,X4) -> U133(X1,X2,X3,X4)
r150: a__U134(X1,X2,X3,X4) -> U134(X1,X2,X3,X4)
r151: a__U135(X1,X2,X3,X4) -> U135(X1,X2,X3,X4)
r152: a__U136(X1,X2,X3,X4) -> U136(X1,X2,X3,X4)
r153: a__take(X1,X2) -> take(X1,X2)
r154: a__U21(X1,X2) -> U21(X1,X2)
r155: a__U22(X1,X2) -> U22(X1,X2)
r156: a__U23(X) -> U23(X)
r157: a__U31(X1,X2) -> U31(X1,X2)
r158: a__U32(X1,X2) -> U32(X1,X2)
r159: a__U33(X) -> U33(X)
r160: a__U41(X1,X2,X3) -> U41(X1,X2,X3)
r161: a__U42(X1,X2,X3) -> U42(X1,X2,X3)
r162: a__U43(X1,X2,X3) -> U43(X1,X2,X3)
r163: a__U44(X1,X2,X3) -> U44(X1,X2,X3)
r164: a__U45(X1,X2) -> U45(X1,X2)
r165: a__U46(X) -> U46(X)
r166: a__U51(X1,X2) -> U51(X1,X2)
r167: a__U52(X) -> U52(X)
r168: a__U61(X1,X2) -> U61(X1,X2)
r169: a__U62(X) -> U62(X)
r170: a__U71(X) -> U71(X)
r171: a__U81(X) -> U81(X)
r172: a__U91(X1,X2,X3) -> U91(X1,X2,X3)
r173: a__U92(X1,X2,X3) -> U92(X1,X2,X3)
r174: a__U93(X1,X2,X3) -> U93(X1,X2,X3)
r175: a__U94(X1,X2,X3) -> U94(X1,X2,X3)
r176: a__U95(X1,X2) -> U95(X1,X2)
r177: a__U96(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: mark#(U11(X1,X2)) -> mark#(X1)
p2: mark#(U134(X1,X2,X3,X4)) -> mark#(X1)
p3: mark#(U111(X1,X2,X3)) -> mark#(X1)

and R consists of:

r1: a__zeros() -> cons(|0|(),zeros())
r2: a__U101(tt(),V1,V2) -> a__U102(a__isNatKind(V1),V1,V2)
r3: a__U102(tt(),V1,V2) -> a__U103(a__isNatIListKind(V2),V1,V2)
r4: a__U103(tt(),V1,V2) -> a__U104(a__isNatIListKind(V2),V1,V2)
r5: a__U104(tt(),V1,V2) -> a__U105(a__isNat(V1),V2)
r6: a__U105(tt(),V2) -> a__U106(a__isNatIList(V2))
r7: a__U106(tt()) -> tt()
r8: a__U11(tt(),V1) -> a__U12(a__isNatIListKind(V1),V1)
r9: a__U111(tt(),L,N) -> a__U112(a__isNatIListKind(L),L,N)
r10: a__U112(tt(),L,N) -> a__U113(a__isNat(N),L,N)
r11: a__U113(tt(),L,N) -> a__U114(a__isNatKind(N),L)
r12: a__U114(tt(),L) -> s(a__length(mark(L)))
r13: a__U12(tt(),V1) -> a__U13(a__isNatList(V1))
r14: a__U121(tt(),IL) -> a__U122(a__isNatIListKind(IL))
r15: a__U122(tt()) -> nil()
r16: a__U13(tt()) -> tt()
r17: a__U131(tt(),IL,M,N) -> a__U132(a__isNatIListKind(IL),IL,M,N)
r18: a__U132(tt(),IL,M,N) -> a__U133(a__isNat(M),IL,M,N)
r19: a__U133(tt(),IL,M,N) -> a__U134(a__isNatKind(M),IL,M,N)
r20: a__U134(tt(),IL,M,N) -> a__U135(a__isNat(N),IL,M,N)
r21: a__U135(tt(),IL,M,N) -> a__U136(a__isNatKind(N),IL,M,N)
r22: a__U136(tt(),IL,M,N) -> cons(mark(N),take(M,IL))
r23: a__U21(tt(),V1) -> a__U22(a__isNatKind(V1),V1)
r24: a__U22(tt(),V1) -> a__U23(a__isNat(V1))
r25: a__U23(tt()) -> tt()
r26: a__U31(tt(),V) -> a__U32(a__isNatIListKind(V),V)
r27: a__U32(tt(),V) -> a__U33(a__isNatList(V))
r28: a__U33(tt()) -> tt()
r29: a__U41(tt(),V1,V2) -> a__U42(a__isNatKind(V1),V1,V2)
r30: a__U42(tt(),V1,V2) -> a__U43(a__isNatIListKind(V2),V1,V2)
r31: a__U43(tt(),V1,V2) -> a__U44(a__isNatIListKind(V2),V1,V2)
r32: a__U44(tt(),V1,V2) -> a__U45(a__isNat(V1),V2)
r33: a__U45(tt(),V2) -> a__U46(a__isNatIList(V2))
r34: a__U46(tt()) -> tt()
r35: a__U51(tt(),V2) -> a__U52(a__isNatIListKind(V2))
r36: a__U52(tt()) -> tt()
r37: a__U61(tt(),V2) -> a__U62(a__isNatIListKind(V2))
r38: a__U62(tt()) -> tt()
r39: a__U71(tt()) -> tt()
r40: a__U81(tt()) -> tt()
r41: a__U91(tt(),V1,V2) -> a__U92(a__isNatKind(V1),V1,V2)
r42: a__U92(tt(),V1,V2) -> a__U93(a__isNatIListKind(V2),V1,V2)
r43: a__U93(tt(),V1,V2) -> a__U94(a__isNatIListKind(V2),V1,V2)
r44: a__U94(tt(),V1,V2) -> a__U95(a__isNat(V1),V2)
r45: a__U95(tt(),V2) -> a__U96(a__isNatList(V2))
r46: a__U96(tt()) -> tt()
r47: a__isNat(|0|()) -> tt()
r48: a__isNat(length(V1)) -> a__U11(a__isNatIListKind(V1),V1)
r49: a__isNat(s(V1)) -> a__U21(a__isNatKind(V1),V1)
r50: a__isNatIList(V) -> a__U31(a__isNatIListKind(V),V)
r51: a__isNatIList(zeros()) -> tt()
r52: a__isNatIList(cons(V1,V2)) -> a__U41(a__isNatKind(V1),V1,V2)
r53: a__isNatIListKind(nil()) -> tt()
r54: a__isNatIListKind(zeros()) -> tt()
r55: a__isNatIListKind(cons(V1,V2)) -> a__U51(a__isNatKind(V1),V2)
r56: a__isNatIListKind(take(V1,V2)) -> a__U61(a__isNatKind(V1),V2)
r57: a__isNatKind(|0|()) -> tt()
r58: a__isNatKind(length(V1)) -> a__U71(a__isNatIListKind(V1))
r59: a__isNatKind(s(V1)) -> a__U81(a__isNatKind(V1))
r60: a__isNatList(nil()) -> tt()
r61: a__isNatList(cons(V1,V2)) -> a__U91(a__isNatKind(V1),V1,V2)
r62: a__isNatList(take(V1,V2)) -> a__U101(a__isNatKind(V1),V1,V2)
r63: a__length(nil()) -> |0|()
r64: a__length(cons(N,L)) -> a__U111(a__isNatList(L),L,N)
r65: a__take(|0|(),IL) -> a__U121(a__isNatIList(IL),IL)
r66: a__take(s(M),cons(N,IL)) -> a__U131(a__isNatIList(IL),IL,M,N)
r67: mark(zeros()) -> a__zeros()
r68: mark(U101(X1,X2,X3)) -> a__U101(mark(X1),X2,X3)
r69: mark(U102(X1,X2,X3)) -> a__U102(mark(X1),X2,X3)
r70: mark(isNatKind(X)) -> a__isNatKind(X)
r71: mark(U103(X1,X2,X3)) -> a__U103(mark(X1),X2,X3)
r72: mark(isNatIListKind(X)) -> a__isNatIListKind(X)
r73: mark(U104(X1,X2,X3)) -> a__U104(mark(X1),X2,X3)
r74: mark(U105(X1,X2)) -> a__U105(mark(X1),X2)
r75: mark(isNat(X)) -> a__isNat(X)
r76: mark(U106(X)) -> a__U106(mark(X))
r77: mark(isNatIList(X)) -> a__isNatIList(X)
r78: mark(U11(X1,X2)) -> a__U11(mark(X1),X2)
r79: mark(U12(X1,X2)) -> a__U12(mark(X1),X2)
r80: mark(U111(X1,X2,X3)) -> a__U111(mark(X1),X2,X3)
r81: mark(U112(X1,X2,X3)) -> a__U112(mark(X1),X2,X3)
r82: mark(U113(X1,X2,X3)) -> a__U113(mark(X1),X2,X3)
r83: mark(U114(X1,X2)) -> a__U114(mark(X1),X2)
r84: mark(length(X)) -> a__length(mark(X))
r85: mark(U13(X)) -> a__U13(mark(X))
r86: mark(isNatList(X)) -> a__isNatList(X)
r87: mark(U121(X1,X2)) -> a__U121(mark(X1),X2)
r88: mark(U122(X)) -> a__U122(mark(X))
r89: mark(U131(X1,X2,X3,X4)) -> a__U131(mark(X1),X2,X3,X4)
r90: mark(U132(X1,X2,X3,X4)) -> a__U132(mark(X1),X2,X3,X4)
r91: mark(U133(X1,X2,X3,X4)) -> a__U133(mark(X1),X2,X3,X4)
r92: mark(U134(X1,X2,X3,X4)) -> a__U134(mark(X1),X2,X3,X4)
r93: mark(U135(X1,X2,X3,X4)) -> a__U135(mark(X1),X2,X3,X4)
r94: mark(U136(X1,X2,X3,X4)) -> a__U136(mark(X1),X2,X3,X4)
r95: mark(take(X1,X2)) -> a__take(mark(X1),mark(X2))
r96: mark(U21(X1,X2)) -> a__U21(mark(X1),X2)
r97: mark(U22(X1,X2)) -> a__U22(mark(X1),X2)
r98: mark(U23(X)) -> a__U23(mark(X))
r99: mark(U31(X1,X2)) -> a__U31(mark(X1),X2)
r100: mark(U32(X1,X2)) -> a__U32(mark(X1),X2)
r101: mark(U33(X)) -> a__U33(mark(X))
r102: mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3)
r103: mark(U42(X1,X2,X3)) -> a__U42(mark(X1),X2,X3)
r104: mark(U43(X1,X2,X3)) -> a__U43(mark(X1),X2,X3)
r105: mark(U44(X1,X2,X3)) -> a__U44(mark(X1),X2,X3)
r106: mark(U45(X1,X2)) -> a__U45(mark(X1),X2)
r107: mark(U46(X)) -> a__U46(mark(X))
r108: mark(U51(X1,X2)) -> a__U51(mark(X1),X2)
r109: mark(U52(X)) -> a__U52(mark(X))
r110: mark(U61(X1,X2)) -> a__U61(mark(X1),X2)
r111: mark(U62(X)) -> a__U62(mark(X))
r112: mark(U71(X)) -> a__U71(mark(X))
r113: mark(U81(X)) -> a__U81(mark(X))
r114: mark(U91(X1,X2,X3)) -> a__U91(mark(X1),X2,X3)
r115: mark(U92(X1,X2,X3)) -> a__U92(mark(X1),X2,X3)
r116: mark(U93(X1,X2,X3)) -> a__U93(mark(X1),X2,X3)
r117: mark(U94(X1,X2,X3)) -> a__U94(mark(X1),X2,X3)
r118: mark(U95(X1,X2)) -> a__U95(mark(X1),X2)
r119: mark(U96(X)) -> a__U96(mark(X))
r120: mark(cons(X1,X2)) -> cons(mark(X1),X2)
r121: mark(|0|()) -> |0|()
r122: mark(tt()) -> tt()
r123: mark(s(X)) -> s(mark(X))
r124: mark(nil()) -> nil()
r125: a__zeros() -> zeros()
r126: a__U101(X1,X2,X3) -> U101(X1,X2,X3)
r127: a__U102(X1,X2,X3) -> U102(X1,X2,X3)
r128: a__isNatKind(X) -> isNatKind(X)
r129: a__U103(X1,X2,X3) -> U103(X1,X2,X3)
r130: a__isNatIListKind(X) -> isNatIListKind(X)
r131: a__U104(X1,X2,X3) -> U104(X1,X2,X3)
r132: a__U105(X1,X2) -> U105(X1,X2)
r133: a__isNat(X) -> isNat(X)
r134: a__U106(X) -> U106(X)
r135: a__isNatIList(X) -> isNatIList(X)
r136: a__U11(X1,X2) -> U11(X1,X2)
r137: a__U12(X1,X2) -> U12(X1,X2)
r138: a__U111(X1,X2,X3) -> U111(X1,X2,X3)
r139: a__U112(X1,X2,X3) -> U112(X1,X2,X3)
r140: a__U113(X1,X2,X3) -> U113(X1,X2,X3)
r141: a__U114(X1,X2) -> U114(X1,X2)
r142: a__length(X) -> length(X)
r143: a__U13(X) -> U13(X)
r144: a__isNatList(X) -> isNatList(X)
r145: a__U121(X1,X2) -> U121(X1,X2)
r146: a__U122(X) -> U122(X)
r147: a__U131(X1,X2,X3,X4) -> U131(X1,X2,X3,X4)
r148: a__U132(X1,X2,X3,X4) -> U132(X1,X2,X3,X4)
r149: a__U133(X1,X2,X3,X4) -> U133(X1,X2,X3,X4)
r150: a__U134(X1,X2,X3,X4) -> U134(X1,X2,X3,X4)
r151: a__U135(X1,X2,X3,X4) -> U135(X1,X2,X3,X4)
r152: a__U136(X1,X2,X3,X4) -> U136(X1,X2,X3,X4)
r153: a__take(X1,X2) -> take(X1,X2)
r154: a__U21(X1,X2) -> U21(X1,X2)
r155: a__U22(X1,X2) -> U22(X1,X2)
r156: a__U23(X) -> U23(X)
r157: a__U31(X1,X2) -> U31(X1,X2)
r158: a__U32(X1,X2) -> U32(X1,X2)
r159: a__U33(X) -> U33(X)
r160: a__U41(X1,X2,X3) -> U41(X1,X2,X3)
r161: a__U42(X1,X2,X3) -> U42(X1,X2,X3)
r162: a__U43(X1,X2,X3) -> U43(X1,X2,X3)
r163: a__U44(X1,X2,X3) -> U44(X1,X2,X3)
r164: a__U45(X1,X2) -> U45(X1,X2)
r165: a__U46(X) -> U46(X)
r166: a__U51(X1,X2) -> U51(X1,X2)
r167: a__U52(X) -> U52(X)
r168: a__U61(X1,X2) -> U61(X1,X2)
r169: a__U62(X) -> U62(X)
r170: a__U71(X) -> U71(X)
r171: a__U81(X) -> U81(X)
r172: a__U91(X1,X2,X3) -> U91(X1,X2,X3)
r173: a__U92(X1,X2,X3) -> U92(X1,X2,X3)
r174: a__U93(X1,X2,X3) -> U93(X1,X2,X3)
r175: a__U94(X1,X2,X3) -> U94(X1,X2,X3)
r176: a__U95(X1,X2) -> U95(X1,X2)
r177: a__U96(X) -> U96(X)

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  max/plus interpretations on natural numbers:
  
    mark#_A(x1) = max{0, x1 - 1}
    U11_A(x1,x2) = max{x1 + 1, x2 + 1}
    U134_A(x1,x2,x3,x4) = max{x1, x2 + 1, x4 + 1}
    U111_A(x1,x2,x3) = max{x1 + 2, x2 + 2, x3 + 2}

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: mark#(U11(X1,X2)) -> mark#(X1)
p2: mark#(U134(X1,X2,X3,X4)) -> mark#(X1)

and R consists of:

r1: a__zeros() -> cons(|0|(),zeros())
r2: a__U101(tt(),V1,V2) -> a__U102(a__isNatKind(V1),V1,V2)
r3: a__U102(tt(),V1,V2) -> a__U103(a__isNatIListKind(V2),V1,V2)
r4: a__U103(tt(),V1,V2) -> a__U104(a__isNatIListKind(V2),V1,V2)
r5: a__U104(tt(),V1,V2) -> a__U105(a__isNat(V1),V2)
r6: a__U105(tt(),V2) -> a__U106(a__isNatIList(V2))
r7: a__U106(tt()) -> tt()
r8: a__U11(tt(),V1) -> a__U12(a__isNatIListKind(V1),V1)
r9: a__U111(tt(),L,N) -> a__U112(a__isNatIListKind(L),L,N)
r10: a__U112(tt(),L,N) -> a__U113(a__isNat(N),L,N)
r11: a__U113(tt(),L,N) -> a__U114(a__isNatKind(N),L)
r12: a__U114(tt(),L) -> s(a__length(mark(L)))
r13: a__U12(tt(),V1) -> a__U13(a__isNatList(V1))
r14: a__U121(tt(),IL) -> a__U122(a__isNatIListKind(IL))
r15: a__U122(tt()) -> nil()
r16: a__U13(tt()) -> tt()
r17: a__U131(tt(),IL,M,N) -> a__U132(a__isNatIListKind(IL),IL,M,N)
r18: a__U132(tt(),IL,M,N) -> a__U133(a__isNat(M),IL,M,N)
r19: a__U133(tt(),IL,M,N) -> a__U134(a__isNatKind(M),IL,M,N)
r20: a__U134(tt(),IL,M,N) -> a__U135(a__isNat(N),IL,M,N)
r21: a__U135(tt(),IL,M,N) -> a__U136(a__isNatKind(N),IL,M,N)
r22: a__U136(tt(),IL,M,N) -> cons(mark(N),take(M,IL))
r23: a__U21(tt(),V1) -> a__U22(a__isNatKind(V1),V1)
r24: a__U22(tt(),V1) -> a__U23(a__isNat(V1))
r25: a__U23(tt()) -> tt()
r26: a__U31(tt(),V) -> a__U32(a__isNatIListKind(V),V)
r27: a__U32(tt(),V) -> a__U33(a__isNatList(V))
r28: a__U33(tt()) -> tt()
r29: a__U41(tt(),V1,V2) -> a__U42(a__isNatKind(V1),V1,V2)
r30: a__U42(tt(),V1,V2) -> a__U43(a__isNatIListKind(V2),V1,V2)
r31: a__U43(tt(),V1,V2) -> a__U44(a__isNatIListKind(V2),V1,V2)
r32: a__U44(tt(),V1,V2) -> a__U45(a__isNat(V1),V2)
r33: a__U45(tt(),V2) -> a__U46(a__isNatIList(V2))
r34: a__U46(tt()) -> tt()
r35: a__U51(tt(),V2) -> a__U52(a__isNatIListKind(V2))
r36: a__U52(tt()) -> tt()
r37: a__U61(tt(),V2) -> a__U62(a__isNatIListKind(V2))
r38: a__U62(tt()) -> tt()
r39: a__U71(tt()) -> tt()
r40: a__U81(tt()) -> tt()
r41: a__U91(tt(),V1,V2) -> a__U92(a__isNatKind(V1),V1,V2)
r42: a__U92(tt(),V1,V2) -> a__U93(a__isNatIListKind(V2),V1,V2)
r43: a__U93(tt(),V1,V2) -> a__U94(a__isNatIListKind(V2),V1,V2)
r44: a__U94(tt(),V1,V2) -> a__U95(a__isNat(V1),V2)
r45: a__U95(tt(),V2) -> a__U96(a__isNatList(V2))
r46: a__U96(tt()) -> tt()
r47: a__isNat(|0|()) -> tt()
r48: a__isNat(length(V1)) -> a__U11(a__isNatIListKind(V1),V1)
r49: a__isNat(s(V1)) -> a__U21(a__isNatKind(V1),V1)
r50: a__isNatIList(V) -> a__U31(a__isNatIListKind(V),V)
r51: a__isNatIList(zeros()) -> tt()
r52: a__isNatIList(cons(V1,V2)) -> a__U41(a__isNatKind(V1),V1,V2)
r53: a__isNatIListKind(nil()) -> tt()
r54: a__isNatIListKind(zeros()) -> tt()
r55: a__isNatIListKind(cons(V1,V2)) -> a__U51(a__isNatKind(V1),V2)
r56: a__isNatIListKind(take(V1,V2)) -> a__U61(a__isNatKind(V1),V2)
r57: a__isNatKind(|0|()) -> tt()
r58: a__isNatKind(length(V1)) -> a__U71(a__isNatIListKind(V1))
r59: a__isNatKind(s(V1)) -> a__U81(a__isNatKind(V1))
r60: a__isNatList(nil()) -> tt()
r61: a__isNatList(cons(V1,V2)) -> a__U91(a__isNatKind(V1),V1,V2)
r62: a__isNatList(take(V1,V2)) -> a__U101(a__isNatKind(V1),V1,V2)
r63: a__length(nil()) -> |0|()
r64: a__length(cons(N,L)) -> a__U111(a__isNatList(L),L,N)
r65: a__take(|0|(),IL) -> a__U121(a__isNatIList(IL),IL)
r66: a__take(s(M),cons(N,IL)) -> a__U131(a__isNatIList(IL),IL,M,N)
r67: mark(zeros()) -> a__zeros()
r68: mark(U101(X1,X2,X3)) -> a__U101(mark(X1),X2,X3)
r69: mark(U102(X1,X2,X3)) -> a__U102(mark(X1),X2,X3)
r70: mark(isNatKind(X)) -> a__isNatKind(X)
r71: mark(U103(X1,X2,X3)) -> a__U103(mark(X1),X2,X3)
r72: mark(isNatIListKind(X)) -> a__isNatIListKind(X)
r73: mark(U104(X1,X2,X3)) -> a__U104(mark(X1),X2,X3)
r74: mark(U105(X1,X2)) -> a__U105(mark(X1),X2)
r75: mark(isNat(X)) -> a__isNat(X)
r76: mark(U106(X)) -> a__U106(mark(X))
r77: mark(isNatIList(X)) -> a__isNatIList(X)
r78: mark(U11(X1,X2)) -> a__U11(mark(X1),X2)
r79: mark(U12(X1,X2)) -> a__U12(mark(X1),X2)
r80: mark(U111(X1,X2,X3)) -> a__U111(mark(X1),X2,X3)
r81: mark(U112(X1,X2,X3)) -> a__U112(mark(X1),X2,X3)
r82: mark(U113(X1,X2,X3)) -> a__U113(mark(X1),X2,X3)
r83: mark(U114(X1,X2)) -> a__U114(mark(X1),X2)
r84: mark(length(X)) -> a__length(mark(X))
r85: mark(U13(X)) -> a__U13(mark(X))
r86: mark(isNatList(X)) -> a__isNatList(X)
r87: mark(U121(X1,X2)) -> a__U121(mark(X1),X2)
r88: mark(U122(X)) -> a__U122(mark(X))
r89: mark(U131(X1,X2,X3,X4)) -> a__U131(mark(X1),X2,X3,X4)
r90: mark(U132(X1,X2,X3,X4)) -> a__U132(mark(X1),X2,X3,X4)
r91: mark(U133(X1,X2,X3,X4)) -> a__U133(mark(X1),X2,X3,X4)
r92: mark(U134(X1,X2,X3,X4)) -> a__U134(mark(X1),X2,X3,X4)
r93: mark(U135(X1,X2,X3,X4)) -> a__U135(mark(X1),X2,X3,X4)
r94: mark(U136(X1,X2,X3,X4)) -> a__U136(mark(X1),X2,X3,X4)
r95: mark(take(X1,X2)) -> a__take(mark(X1),mark(X2))
r96: mark(U21(X1,X2)) -> a__U21(mark(X1),X2)
r97: mark(U22(X1,X2)) -> a__U22(mark(X1),X2)
r98: mark(U23(X)) -> a__U23(mark(X))
r99: mark(U31(X1,X2)) -> a__U31(mark(X1),X2)
r100: mark(U32(X1,X2)) -> a__U32(mark(X1),X2)
r101: mark(U33(X)) -> a__U33(mark(X))
r102: mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3)
r103: mark(U42(X1,X2,X3)) -> a__U42(mark(X1),X2,X3)
r104: mark(U43(X1,X2,X3)) -> a__U43(mark(X1),X2,X3)
r105: mark(U44(X1,X2,X3)) -> a__U44(mark(X1),X2,X3)
r106: mark(U45(X1,X2)) -> a__U45(mark(X1),X2)
r107: mark(U46(X)) -> a__U46(mark(X))
r108: mark(U51(X1,X2)) -> a__U51(mark(X1),X2)
r109: mark(U52(X)) -> a__U52(mark(X))
r110: mark(U61(X1,X2)) -> a__U61(mark(X1),X2)
r111: mark(U62(X)) -> a__U62(mark(X))
r112: mark(U71(X)) -> a__U71(mark(X))
r113: mark(U81(X)) -> a__U81(mark(X))
r114: mark(U91(X1,X2,X3)) -> a__U91(mark(X1),X2,X3)
r115: mark(U92(X1,X2,X3)) -> a__U92(mark(X1),X2,X3)
r116: mark(U93(X1,X2,X3)) -> a__U93(mark(X1),X2,X3)
r117: mark(U94(X1,X2,X3)) -> a__U94(mark(X1),X2,X3)
r118: mark(U95(X1,X2)) -> a__U95(mark(X1),X2)
r119: mark(U96(X)) -> a__U96(mark(X))
r120: mark(cons(X1,X2)) -> cons(mark(X1),X2)
r121: mark(|0|()) -> |0|()
r122: mark(tt()) -> tt()
r123: mark(s(X)) -> s(mark(X))
r124: mark(nil()) -> nil()
r125: a__zeros() -> zeros()
r126: a__U101(X1,X2,X3) -> U101(X1,X2,X3)
r127: a__U102(X1,X2,X3) -> U102(X1,X2,X3)
r128: a__isNatKind(X) -> isNatKind(X)
r129: a__U103(X1,X2,X3) -> U103(X1,X2,X3)
r130: a__isNatIListKind(X) -> isNatIListKind(X)
r131: a__U104(X1,X2,X3) -> U104(X1,X2,X3)
r132: a__U105(X1,X2) -> U105(X1,X2)
r133: a__isNat(X) -> isNat(X)
r134: a__U106(X) -> U106(X)
r135: a__isNatIList(X) -> isNatIList(X)
r136: a__U11(X1,X2) -> U11(X1,X2)
r137: a__U12(X1,X2) -> U12(X1,X2)
r138: a__U111(X1,X2,X3) -> U111(X1,X2,X3)
r139: a__U112(X1,X2,X3) -> U112(X1,X2,X3)
r140: a__U113(X1,X2,X3) -> U113(X1,X2,X3)
r141: a__U114(X1,X2) -> U114(X1,X2)
r142: a__length(X) -> length(X)
r143: a__U13(X) -> U13(X)
r144: a__isNatList(X) -> isNatList(X)
r145: a__U121(X1,X2) -> U121(X1,X2)
r146: a__U122(X) -> U122(X)
r147: a__U131(X1,X2,X3,X4) -> U131(X1,X2,X3,X4)
r148: a__U132(X1,X2,X3,X4) -> U132(X1,X2,X3,X4)
r149: a__U133(X1,X2,X3,X4) -> U133(X1,X2,X3,X4)
r150: a__U134(X1,X2,X3,X4) -> U134(X1,X2,X3,X4)
r151: a__U135(X1,X2,X3,X4) -> U135(X1,X2,X3,X4)
r152: a__U136(X1,X2,X3,X4) -> U136(X1,X2,X3,X4)
r153: a__take(X1,X2) -> take(X1,X2)
r154: a__U21(X1,X2) -> U21(X1,X2)
r155: a__U22(X1,X2) -> U22(X1,X2)
r156: a__U23(X) -> U23(X)
r157: a__U31(X1,X2) -> U31(X1,X2)
r158: a__U32(X1,X2) -> U32(X1,X2)
r159: a__U33(X) -> U33(X)
r160: a__U41(X1,X2,X3) -> U41(X1,X2,X3)
r161: a__U42(X1,X2,X3) -> U42(X1,X2,X3)
r162: a__U43(X1,X2,X3) -> U43(X1,X2,X3)
r163: a__U44(X1,X2,X3) -> U44(X1,X2,X3)
r164: a__U45(X1,X2) -> U45(X1,X2)
r165: a__U46(X) -> U46(X)
r166: a__U51(X1,X2) -> U51(X1,X2)
r167: a__U52(X) -> U52(X)
r168: a__U61(X1,X2) -> U61(X1,X2)
r169: a__U62(X) -> U62(X)
r170: a__U71(X) -> U71(X)
r171: a__U81(X) -> U81(X)
r172: a__U91(X1,X2,X3) -> U91(X1,X2,X3)
r173: a__U92(X1,X2,X3) -> U92(X1,X2,X3)
r174: a__U93(X1,X2,X3) -> U93(X1,X2,X3)
r175: a__U94(X1,X2,X3) -> U94(X1,X2,X3)
r176: a__U95(X1,X2) -> U95(X1,X2)
r177: a__U96(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: mark#(U11(X1,X2)) -> mark#(X1)
p2: mark#(U134(X1,X2,X3,X4)) -> mark#(X1)

and R consists of:

r1: a__zeros() -> cons(|0|(),zeros())
r2: a__U101(tt(),V1,V2) -> a__U102(a__isNatKind(V1),V1,V2)
r3: a__U102(tt(),V1,V2) -> a__U103(a__isNatIListKind(V2),V1,V2)
r4: a__U103(tt(),V1,V2) -> a__U104(a__isNatIListKind(V2),V1,V2)
r5: a__U104(tt(),V1,V2) -> a__U105(a__isNat(V1),V2)
r6: a__U105(tt(),V2) -> a__U106(a__isNatIList(V2))
r7: a__U106(tt()) -> tt()
r8: a__U11(tt(),V1) -> a__U12(a__isNatIListKind(V1),V1)
r9: a__U111(tt(),L,N) -> a__U112(a__isNatIListKind(L),L,N)
r10: a__U112(tt(),L,N) -> a__U113(a__isNat(N),L,N)
r11: a__U113(tt(),L,N) -> a__U114(a__isNatKind(N),L)
r12: a__U114(tt(),L) -> s(a__length(mark(L)))
r13: a__U12(tt(),V1) -> a__U13(a__isNatList(V1))
r14: a__U121(tt(),IL) -> a__U122(a__isNatIListKind(IL))
r15: a__U122(tt()) -> nil()
r16: a__U13(tt()) -> tt()
r17: a__U131(tt(),IL,M,N) -> a__U132(a__isNatIListKind(IL),IL,M,N)
r18: a__U132(tt(),IL,M,N) -> a__U133(a__isNat(M),IL,M,N)
r19: a__U133(tt(),IL,M,N) -> a__U134(a__isNatKind(M),IL,M,N)
r20: a__U134(tt(),IL,M,N) -> a__U135(a__isNat(N),IL,M,N)
r21: a__U135(tt(),IL,M,N) -> a__U136(a__isNatKind(N),IL,M,N)
r22: a__U136(tt(),IL,M,N) -> cons(mark(N),take(M,IL))
r23: a__U21(tt(),V1) -> a__U22(a__isNatKind(V1),V1)
r24: a__U22(tt(),V1) -> a__U23(a__isNat(V1))
r25: a__U23(tt()) -> tt()
r26: a__U31(tt(),V) -> a__U32(a__isNatIListKind(V),V)
r27: a__U32(tt(),V) -> a__U33(a__isNatList(V))
r28: a__U33(tt()) -> tt()
r29: a__U41(tt(),V1,V2) -> a__U42(a__isNatKind(V1),V1,V2)
r30: a__U42(tt(),V1,V2) -> a__U43(a__isNatIListKind(V2),V1,V2)
r31: a__U43(tt(),V1,V2) -> a__U44(a__isNatIListKind(V2),V1,V2)
r32: a__U44(tt(),V1,V2) -> a__U45(a__isNat(V1),V2)
r33: a__U45(tt(),V2) -> a__U46(a__isNatIList(V2))
r34: a__U46(tt()) -> tt()
r35: a__U51(tt(),V2) -> a__U52(a__isNatIListKind(V2))
r36: a__U52(tt()) -> tt()
r37: a__U61(tt(),V2) -> a__U62(a__isNatIListKind(V2))
r38: a__U62(tt()) -> tt()
r39: a__U71(tt()) -> tt()
r40: a__U81(tt()) -> tt()
r41: a__U91(tt(),V1,V2) -> a__U92(a__isNatKind(V1),V1,V2)
r42: a__U92(tt(),V1,V2) -> a__U93(a__isNatIListKind(V2),V1,V2)
r43: a__U93(tt(),V1,V2) -> a__U94(a__isNatIListKind(V2),V1,V2)
r44: a__U94(tt(),V1,V2) -> a__U95(a__isNat(V1),V2)
r45: a__U95(tt(),V2) -> a__U96(a__isNatList(V2))
r46: a__U96(tt()) -> tt()
r47: a__isNat(|0|()) -> tt()
r48: a__isNat(length(V1)) -> a__U11(a__isNatIListKind(V1),V1)
r49: a__isNat(s(V1)) -> a__U21(a__isNatKind(V1),V1)
r50: a__isNatIList(V) -> a__U31(a__isNatIListKind(V),V)
r51: a__isNatIList(zeros()) -> tt()
r52: a__isNatIList(cons(V1,V2)) -> a__U41(a__isNatKind(V1),V1,V2)
r53: a__isNatIListKind(nil()) -> tt()
r54: a__isNatIListKind(zeros()) -> tt()
r55: a__isNatIListKind(cons(V1,V2)) -> a__U51(a__isNatKind(V1),V2)
r56: a__isNatIListKind(take(V1,V2)) -> a__U61(a__isNatKind(V1),V2)
r57: a__isNatKind(|0|()) -> tt()
r58: a__isNatKind(length(V1)) -> a__U71(a__isNatIListKind(V1))
r59: a__isNatKind(s(V1)) -> a__U81(a__isNatKind(V1))
r60: a__isNatList(nil()) -> tt()
r61: a__isNatList(cons(V1,V2)) -> a__U91(a__isNatKind(V1),V1,V2)
r62: a__isNatList(take(V1,V2)) -> a__U101(a__isNatKind(V1),V1,V2)
r63: a__length(nil()) -> |0|()
r64: a__length(cons(N,L)) -> a__U111(a__isNatList(L),L,N)
r65: a__take(|0|(),IL) -> a__U121(a__isNatIList(IL),IL)
r66: a__take(s(M),cons(N,IL)) -> a__U131(a__isNatIList(IL),IL,M,N)
r67: mark(zeros()) -> a__zeros()
r68: mark(U101(X1,X2,X3)) -> a__U101(mark(X1),X2,X3)
r69: mark(U102(X1,X2,X3)) -> a__U102(mark(X1),X2,X3)
r70: mark(isNatKind(X)) -> a__isNatKind(X)
r71: mark(U103(X1,X2,X3)) -> a__U103(mark(X1),X2,X3)
r72: mark(isNatIListKind(X)) -> a__isNatIListKind(X)
r73: mark(U104(X1,X2,X3)) -> a__U104(mark(X1),X2,X3)
r74: mark(U105(X1,X2)) -> a__U105(mark(X1),X2)
r75: mark(isNat(X)) -> a__isNat(X)
r76: mark(U106(X)) -> a__U106(mark(X))
r77: mark(isNatIList(X)) -> a__isNatIList(X)
r78: mark(U11(X1,X2)) -> a__U11(mark(X1),X2)
r79: mark(U12(X1,X2)) -> a__U12(mark(X1),X2)
r80: mark(U111(X1,X2,X3)) -> a__U111(mark(X1),X2,X3)
r81: mark(U112(X1,X2,X3)) -> a__U112(mark(X1),X2,X3)
r82: mark(U113(X1,X2,X3)) -> a__U113(mark(X1),X2,X3)
r83: mark(U114(X1,X2)) -> a__U114(mark(X1),X2)
r84: mark(length(X)) -> a__length(mark(X))
r85: mark(U13(X)) -> a__U13(mark(X))
r86: mark(isNatList(X)) -> a__isNatList(X)
r87: mark(U121(X1,X2)) -> a__U121(mark(X1),X2)
r88: mark(U122(X)) -> a__U122(mark(X))
r89: mark(U131(X1,X2,X3,X4)) -> a__U131(mark(X1),X2,X3,X4)
r90: mark(U132(X1,X2,X3,X4)) -> a__U132(mark(X1),X2,X3,X4)
r91: mark(U133(X1,X2,X3,X4)) -> a__U133(mark(X1),X2,X3,X4)
r92: mark(U134(X1,X2,X3,X4)) -> a__U134(mark(X1),X2,X3,X4)
r93: mark(U135(X1,X2,X3,X4)) -> a__U135(mark(X1),X2,X3,X4)
r94: mark(U136(X1,X2,X3,X4)) -> a__U136(mark(X1),X2,X3,X4)
r95: mark(take(X1,X2)) -> a__take(mark(X1),mark(X2))
r96: mark(U21(X1,X2)) -> a__U21(mark(X1),X2)
r97: mark(U22(X1,X2)) -> a__U22(mark(X1),X2)
r98: mark(U23(X)) -> a__U23(mark(X))
r99: mark(U31(X1,X2)) -> a__U31(mark(X1),X2)
r100: mark(U32(X1,X2)) -> a__U32(mark(X1),X2)
r101: mark(U33(X)) -> a__U33(mark(X))
r102: mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3)
r103: mark(U42(X1,X2,X3)) -> a__U42(mark(X1),X2,X3)
r104: mark(U43(X1,X2,X3)) -> a__U43(mark(X1),X2,X3)
r105: mark(U44(X1,X2,X3)) -> a__U44(mark(X1),X2,X3)
r106: mark(U45(X1,X2)) -> a__U45(mark(X1),X2)
r107: mark(U46(X)) -> a__U46(mark(X))
r108: mark(U51(X1,X2)) -> a__U51(mark(X1),X2)
r109: mark(U52(X)) -> a__U52(mark(X))
r110: mark(U61(X1,X2)) -> a__U61(mark(X1),X2)
r111: mark(U62(X)) -> a__U62(mark(X))
r112: mark(U71(X)) -> a__U71(mark(X))
r113: mark(U81(X)) -> a__U81(mark(X))
r114: mark(U91(X1,X2,X3)) -> a__U91(mark(X1),X2,X3)
r115: mark(U92(X1,X2,X3)) -> a__U92(mark(X1),X2,X3)
r116: mark(U93(X1,X2,X3)) -> a__U93(mark(X1),X2,X3)
r117: mark(U94(X1,X2,X3)) -> a__U94(mark(X1),X2,X3)
r118: mark(U95(X1,X2)) -> a__U95(mark(X1),X2)
r119: mark(U96(X)) -> a__U96(mark(X))
r120: mark(cons(X1,X2)) -> cons(mark(X1),X2)
r121: mark(|0|()) -> |0|()
r122: mark(tt()) -> tt()
r123: mark(s(X)) -> s(mark(X))
r124: mark(nil()) -> nil()
r125: a__zeros() -> zeros()
r126: a__U101(X1,X2,X3) -> U101(X1,X2,X3)
r127: a__U102(X1,X2,X3) -> U102(X1,X2,X3)
r128: a__isNatKind(X) -> isNatKind(X)
r129: a__U103(X1,X2,X3) -> U103(X1,X2,X3)
r130: a__isNatIListKind(X) -> isNatIListKind(X)
r131: a__U104(X1,X2,X3) -> U104(X1,X2,X3)
r132: a__U105(X1,X2) -> U105(X1,X2)
r133: a__isNat(X) -> isNat(X)
r134: a__U106(X) -> U106(X)
r135: a__isNatIList(X) -> isNatIList(X)
r136: a__U11(X1,X2) -> U11(X1,X2)
r137: a__U12(X1,X2) -> U12(X1,X2)
r138: a__U111(X1,X2,X3) -> U111(X1,X2,X3)
r139: a__U112(X1,X2,X3) -> U112(X1,X2,X3)
r140: a__U113(X1,X2,X3) -> U113(X1,X2,X3)
r141: a__U114(X1,X2) -> U114(X1,X2)
r142: a__length(X) -> length(X)
r143: a__U13(X) -> U13(X)
r144: a__isNatList(X) -> isNatList(X)
r145: a__U121(X1,X2) -> U121(X1,X2)
r146: a__U122(X) -> U122(X)
r147: a__U131(X1,X2,X3,X4) -> U131(X1,X2,X3,X4)
r148: a__U132(X1,X2,X3,X4) -> U132(X1,X2,X3,X4)
r149: a__U133(X1,X2,X3,X4) -> U133(X1,X2,X3,X4)
r150: a__U134(X1,X2,X3,X4) -> U134(X1,X2,X3,X4)
r151: a__U135(X1,X2,X3,X4) -> U135(X1,X2,X3,X4)
r152: a__U136(X1,X2,X3,X4) -> U136(X1,X2,X3,X4)
r153: a__take(X1,X2) -> take(X1,X2)
r154: a__U21(X1,X2) -> U21(X1,X2)
r155: a__U22(X1,X2) -> U22(X1,X2)
r156: a__U23(X) -> U23(X)
r157: a__U31(X1,X2) -> U31(X1,X2)
r158: a__U32(X1,X2) -> U32(X1,X2)
r159: a__U33(X) -> U33(X)
r160: a__U41(X1,X2,X3) -> U41(X1,X2,X3)
r161: a__U42(X1,X2,X3) -> U42(X1,X2,X3)
r162: a__U43(X1,X2,X3) -> U43(X1,X2,X3)
r163: a__U44(X1,X2,X3) -> U44(X1,X2,X3)
r164: a__U45(X1,X2) -> U45(X1,X2)
r165: a__U46(X) -> U46(X)
r166: a__U51(X1,X2) -> U51(X1,X2)
r167: a__U52(X) -> U52(X)
r168: a__U61(X1,X2) -> U61(X1,X2)
r169: a__U62(X) -> U62(X)
r170: a__U71(X) -> U71(X)
r171: a__U81(X) -> U81(X)
r172: a__U91(X1,X2,X3) -> U91(X1,X2,X3)
r173: a__U92(X1,X2,X3) -> U92(X1,X2,X3)
r174: a__U93(X1,X2,X3) -> U93(X1,X2,X3)
r175: a__U94(X1,X2,X3) -> U94(X1,X2,X3)
r176: a__U95(X1,X2) -> U95(X1,X2)
r177: a__U96(X) -> U96(X)

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  max/plus interpretations on natural numbers:
  
    mark#_A(x1) = max{0, x1 - 1}
    U11_A(x1,x2) = x1 + 1
    U134_A(x1,x2,x3,x4) = max{x1 + 2, x2 + 2, x3 + 2, x4 + 2}

The next rules are strictly ordered:

  p2

We remove them from the problem.

-- SCC decomposition.

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

p1: mark#(U11(X1,X2)) -> mark#(X1)

and R consists of:

r1: a__zeros() -> cons(|0|(),zeros())
r2: a__U101(tt(),V1,V2) -> a__U102(a__isNatKind(V1),V1,V2)
r3: a__U102(tt(),V1,V2) -> a__U103(a__isNatIListKind(V2),V1,V2)
r4: a__U103(tt(),V1,V2) -> a__U104(a__isNatIListKind(V2),V1,V2)
r5: a__U104(tt(),V1,V2) -> a__U105(a__isNat(V1),V2)
r6: a__U105(tt(),V2) -> a__U106(a__isNatIList(V2))
r7: a__U106(tt()) -> tt()
r8: a__U11(tt(),V1) -> a__U12(a__isNatIListKind(V1),V1)
r9: a__U111(tt(),L,N) -> a__U112(a__isNatIListKind(L),L,N)
r10: a__U112(tt(),L,N) -> a__U113(a__isNat(N),L,N)
r11: a__U113(tt(),L,N) -> a__U114(a__isNatKind(N),L)
r12: a__U114(tt(),L) -> s(a__length(mark(L)))
r13: a__U12(tt(),V1) -> a__U13(a__isNatList(V1))
r14: a__U121(tt(),IL) -> a__U122(a__isNatIListKind(IL))
r15: a__U122(tt()) -> nil()
r16: a__U13(tt()) -> tt()
r17: a__U131(tt(),IL,M,N) -> a__U132(a__isNatIListKind(IL),IL,M,N)
r18: a__U132(tt(),IL,M,N) -> a__U133(a__isNat(M),IL,M,N)
r19: a__U133(tt(),IL,M,N) -> a__U134(a__isNatKind(M),IL,M,N)
r20: a__U134(tt(),IL,M,N) -> a__U135(a__isNat(N),IL,M,N)
r21: a__U135(tt(),IL,M,N) -> a__U136(a__isNatKind(N),IL,M,N)
r22: a__U136(tt(),IL,M,N) -> cons(mark(N),take(M,IL))
r23: a__U21(tt(),V1) -> a__U22(a__isNatKind(V1),V1)
r24: a__U22(tt(),V1) -> a__U23(a__isNat(V1))
r25: a__U23(tt()) -> tt()
r26: a__U31(tt(),V) -> a__U32(a__isNatIListKind(V),V)
r27: a__U32(tt(),V) -> a__U33(a__isNatList(V))
r28: a__U33(tt()) -> tt()
r29: a__U41(tt(),V1,V2) -> a__U42(a__isNatKind(V1),V1,V2)
r30: a__U42(tt(),V1,V2) -> a__U43(a__isNatIListKind(V2),V1,V2)
r31: a__U43(tt(),V1,V2) -> a__U44(a__isNatIListKind(V2),V1,V2)
r32: a__U44(tt(),V1,V2) -> a__U45(a__isNat(V1),V2)
r33: a__U45(tt(),V2) -> a__U46(a__isNatIList(V2))
r34: a__U46(tt()) -> tt()
r35: a__U51(tt(),V2) -> a__U52(a__isNatIListKind(V2))
r36: a__U52(tt()) -> tt()
r37: a__U61(tt(),V2) -> a__U62(a__isNatIListKind(V2))
r38: a__U62(tt()) -> tt()
r39: a__U71(tt()) -> tt()
r40: a__U81(tt()) -> tt()
r41: a__U91(tt(),V1,V2) -> a__U92(a__isNatKind(V1),V1,V2)
r42: a__U92(tt(),V1,V2) -> a__U93(a__isNatIListKind(V2),V1,V2)
r43: a__U93(tt(),V1,V2) -> a__U94(a__isNatIListKind(V2),V1,V2)
r44: a__U94(tt(),V1,V2) -> a__U95(a__isNat(V1),V2)
r45: a__U95(tt(),V2) -> a__U96(a__isNatList(V2))
r46: a__U96(tt()) -> tt()
r47: a__isNat(|0|()) -> tt()
r48: a__isNat(length(V1)) -> a__U11(a__isNatIListKind(V1),V1)
r49: a__isNat(s(V1)) -> a__U21(a__isNatKind(V1),V1)
r50: a__isNatIList(V) -> a__U31(a__isNatIListKind(V),V)
r51: a__isNatIList(zeros()) -> tt()
r52: a__isNatIList(cons(V1,V2)) -> a__U41(a__isNatKind(V1),V1,V2)
r53: a__isNatIListKind(nil()) -> tt()
r54: a__isNatIListKind(zeros()) -> tt()
r55: a__isNatIListKind(cons(V1,V2)) -> a__U51(a__isNatKind(V1),V2)
r56: a__isNatIListKind(take(V1,V2)) -> a__U61(a__isNatKind(V1),V2)
r57: a__isNatKind(|0|()) -> tt()
r58: a__isNatKind(length(V1)) -> a__U71(a__isNatIListKind(V1))
r59: a__isNatKind(s(V1)) -> a__U81(a__isNatKind(V1))
r60: a__isNatList(nil()) -> tt()
r61: a__isNatList(cons(V1,V2)) -> a__U91(a__isNatKind(V1),V1,V2)
r62: a__isNatList(take(V1,V2)) -> a__U101(a__isNatKind(V1),V1,V2)
r63: a__length(nil()) -> |0|()
r64: a__length(cons(N,L)) -> a__U111(a__isNatList(L),L,N)
r65: a__take(|0|(),IL) -> a__U121(a__isNatIList(IL),IL)
r66: a__take(s(M),cons(N,IL)) -> a__U131(a__isNatIList(IL),IL,M,N)
r67: mark(zeros()) -> a__zeros()
r68: mark(U101(X1,X2,X3)) -> a__U101(mark(X1),X2,X3)
r69: mark(U102(X1,X2,X3)) -> a__U102(mark(X1),X2,X3)
r70: mark(isNatKind(X)) -> a__isNatKind(X)
r71: mark(U103(X1,X2,X3)) -> a__U103(mark(X1),X2,X3)
r72: mark(isNatIListKind(X)) -> a__isNatIListKind(X)
r73: mark(U104(X1,X2,X3)) -> a__U104(mark(X1),X2,X3)
r74: mark(U105(X1,X2)) -> a__U105(mark(X1),X2)
r75: mark(isNat(X)) -> a__isNat(X)
r76: mark(U106(X)) -> a__U106(mark(X))
r77: mark(isNatIList(X)) -> a__isNatIList(X)
r78: mark(U11(X1,X2)) -> a__U11(mark(X1),X2)
r79: mark(U12(X1,X2)) -> a__U12(mark(X1),X2)
r80: mark(U111(X1,X2,X3)) -> a__U111(mark(X1),X2,X3)
r81: mark(U112(X1,X2,X3)) -> a__U112(mark(X1),X2,X3)
r82: mark(U113(X1,X2,X3)) -> a__U113(mark(X1),X2,X3)
r83: mark(U114(X1,X2)) -> a__U114(mark(X1),X2)
r84: mark(length(X)) -> a__length(mark(X))
r85: mark(U13(X)) -> a__U13(mark(X))
r86: mark(isNatList(X)) -> a__isNatList(X)
r87: mark(U121(X1,X2)) -> a__U121(mark(X1),X2)
r88: mark(U122(X)) -> a__U122(mark(X))
r89: mark(U131(X1,X2,X3,X4)) -> a__U131(mark(X1),X2,X3,X4)
r90: mark(U132(X1,X2,X3,X4)) -> a__U132(mark(X1),X2,X3,X4)
r91: mark(U133(X1,X2,X3,X4)) -> a__U133(mark(X1),X2,X3,X4)
r92: mark(U134(X1,X2,X3,X4)) -> a__U134(mark(X1),X2,X3,X4)
r93: mark(U135(X1,X2,X3,X4)) -> a__U135(mark(X1),X2,X3,X4)
r94: mark(U136(X1,X2,X3,X4)) -> a__U136(mark(X1),X2,X3,X4)
r95: mark(take(X1,X2)) -> a__take(mark(X1),mark(X2))
r96: mark(U21(X1,X2)) -> a__U21(mark(X1),X2)
r97: mark(U22(X1,X2)) -> a__U22(mark(X1),X2)
r98: mark(U23(X)) -> a__U23(mark(X))
r99: mark(U31(X1,X2)) -> a__U31(mark(X1),X2)
r100: mark(U32(X1,X2)) -> a__U32(mark(X1),X2)
r101: mark(U33(X)) -> a__U33(mark(X))
r102: mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3)
r103: mark(U42(X1,X2,X3)) -> a__U42(mark(X1),X2,X3)
r104: mark(U43(X1,X2,X3)) -> a__U43(mark(X1),X2,X3)
r105: mark(U44(X1,X2,X3)) -> a__U44(mark(X1),X2,X3)
r106: mark(U45(X1,X2)) -> a__U45(mark(X1),X2)
r107: mark(U46(X)) -> a__U46(mark(X))
r108: mark(U51(X1,X2)) -> a__U51(mark(X1),X2)
r109: mark(U52(X)) -> a__U52(mark(X))
r110: mark(U61(X1,X2)) -> a__U61(mark(X1),X2)
r111: mark(U62(X)) -> a__U62(mark(X))
r112: mark(U71(X)) -> a__U71(mark(X))
r113: mark(U81(X)) -> a__U81(mark(X))
r114: mark(U91(X1,X2,X3)) -> a__U91(mark(X1),X2,X3)
r115: mark(U92(X1,X2,X3)) -> a__U92(mark(X1),X2,X3)
r116: mark(U93(X1,X2,X3)) -> a__U93(mark(X1),X2,X3)
r117: mark(U94(X1,X2,X3)) -> a__U94(mark(X1),X2,X3)
r118: mark(U95(X1,X2)) -> a__U95(mark(X1),X2)
r119: mark(U96(X)) -> a__U96(mark(X))
r120: mark(cons(X1,X2)) -> cons(mark(X1),X2)
r121: mark(|0|()) -> |0|()
r122: mark(tt()) -> tt()
r123: mark(s(X)) -> s(mark(X))
r124: mark(nil()) -> nil()
r125: a__zeros() -> zeros()
r126: a__U101(X1,X2,X3) -> U101(X1,X2,X3)
r127: a__U102(X1,X2,X3) -> U102(X1,X2,X3)
r128: a__isNatKind(X) -> isNatKind(X)
r129: a__U103(X1,X2,X3) -> U103(X1,X2,X3)
r130: a__isNatIListKind(X) -> isNatIListKind(X)
r131: a__U104(X1,X2,X3) -> U104(X1,X2,X3)
r132: a__U105(X1,X2) -> U105(X1,X2)
r133: a__isNat(X) -> isNat(X)
r134: a__U106(X) -> U106(X)
r135: a__isNatIList(X) -> isNatIList(X)
r136: a__U11(X1,X2) -> U11(X1,X2)
r137: a__U12(X1,X2) -> U12(X1,X2)
r138: a__U111(X1,X2,X3) -> U111(X1,X2,X3)
r139: a__U112(X1,X2,X3) -> U112(X1,X2,X3)
r140: a__U113(X1,X2,X3) -> U113(X1,X2,X3)
r141: a__U114(X1,X2) -> U114(X1,X2)
r142: a__length(X) -> length(X)
r143: a__U13(X) -> U13(X)
r144: a__isNatList(X) -> isNatList(X)
r145: a__U121(X1,X2) -> U121(X1,X2)
r146: a__U122(X) -> U122(X)
r147: a__U131(X1,X2,X3,X4) -> U131(X1,X2,X3,X4)
r148: a__U132(X1,X2,X3,X4) -> U132(X1,X2,X3,X4)
r149: a__U133(X1,X2,X3,X4) -> U133(X1,X2,X3,X4)
r150: a__U134(X1,X2,X3,X4) -> U134(X1,X2,X3,X4)
r151: a__U135(X1,X2,X3,X4) -> U135(X1,X2,X3,X4)
r152: a__U136(X1,X2,X3,X4) -> U136(X1,X2,X3,X4)
r153: a__take(X1,X2) -> take(X1,X2)
r154: a__U21(X1,X2) -> U21(X1,X2)
r155: a__U22(X1,X2) -> U22(X1,X2)
r156: a__U23(X) -> U23(X)
r157: a__U31(X1,X2) -> U31(X1,X2)
r158: a__U32(X1,X2) -> U32(X1,X2)
r159: a__U33(X) -> U33(X)
r160: a__U41(X1,X2,X3) -> U41(X1,X2,X3)
r161: a__U42(X1,X2,X3) -> U42(X1,X2,X3)
r162: a__U43(X1,X2,X3) -> U43(X1,X2,X3)
r163: a__U44(X1,X2,X3) -> U44(X1,X2,X3)
r164: a__U45(X1,X2) -> U45(X1,X2)
r165: a__U46(X) -> U46(X)
r166: a__U51(X1,X2) -> U51(X1,X2)
r167: a__U52(X) -> U52(X)
r168: a__U61(X1,X2) -> U61(X1,X2)
r169: a__U62(X) -> U62(X)
r170: a__U71(X) -> U71(X)
r171: a__U81(X) -> U81(X)
r172: a__U91(X1,X2,X3) -> U91(X1,X2,X3)
r173: a__U92(X1,X2,X3) -> U92(X1,X2,X3)
r174: a__U93(X1,X2,X3) -> U93(X1,X2,X3)
r175: a__U94(X1,X2,X3) -> U94(X1,X2,X3)
r176: a__U95(X1,X2) -> U95(X1,X2)
r177: a__U96(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: mark#(U11(X1,X2)) -> mark#(X1)

and R consists of:

r1: a__zeros() -> cons(|0|(),zeros())
r2: a__U101(tt(),V1,V2) -> a__U102(a__isNatKind(V1),V1,V2)
r3: a__U102(tt(),V1,V2) -> a__U103(a__isNatIListKind(V2),V1,V2)
r4: a__U103(tt(),V1,V2) -> a__U104(a__isNatIListKind(V2),V1,V2)
r5: a__U104(tt(),V1,V2) -> a__U105(a__isNat(V1),V2)
r6: a__U105(tt(),V2) -> a__U106(a__isNatIList(V2))
r7: a__U106(tt()) -> tt()
r8: a__U11(tt(),V1) -> a__U12(a__isNatIListKind(V1),V1)
r9: a__U111(tt(),L,N) -> a__U112(a__isNatIListKind(L),L,N)
r10: a__U112(tt(),L,N) -> a__U113(a__isNat(N),L,N)
r11: a__U113(tt(),L,N) -> a__U114(a__isNatKind(N),L)
r12: a__U114(tt(),L) -> s(a__length(mark(L)))
r13: a__U12(tt(),V1) -> a__U13(a__isNatList(V1))
r14: a__U121(tt(),IL) -> a__U122(a__isNatIListKind(IL))
r15: a__U122(tt()) -> nil()
r16: a__U13(tt()) -> tt()
r17: a__U131(tt(),IL,M,N) -> a__U132(a__isNatIListKind(IL),IL,M,N)
r18: a__U132(tt(),IL,M,N) -> a__U133(a__isNat(M),IL,M,N)
r19: a__U133(tt(),IL,M,N) -> a__U134(a__isNatKind(M),IL,M,N)
r20: a__U134(tt(),IL,M,N) -> a__U135(a__isNat(N),IL,M,N)
r21: a__U135(tt(),IL,M,N) -> a__U136(a__isNatKind(N),IL,M,N)
r22: a__U136(tt(),IL,M,N) -> cons(mark(N),take(M,IL))
r23: a__U21(tt(),V1) -> a__U22(a__isNatKind(V1),V1)
r24: a__U22(tt(),V1) -> a__U23(a__isNat(V1))
r25: a__U23(tt()) -> tt()
r26: a__U31(tt(),V) -> a__U32(a__isNatIListKind(V),V)
r27: a__U32(tt(),V) -> a__U33(a__isNatList(V))
r28: a__U33(tt()) -> tt()
r29: a__U41(tt(),V1,V2) -> a__U42(a__isNatKind(V1),V1,V2)
r30: a__U42(tt(),V1,V2) -> a__U43(a__isNatIListKind(V2),V1,V2)
r31: a__U43(tt(),V1,V2) -> a__U44(a__isNatIListKind(V2),V1,V2)
r32: a__U44(tt(),V1,V2) -> a__U45(a__isNat(V1),V2)
r33: a__U45(tt(),V2) -> a__U46(a__isNatIList(V2))
r34: a__U46(tt()) -> tt()
r35: a__U51(tt(),V2) -> a__U52(a__isNatIListKind(V2))
r36: a__U52(tt()) -> tt()
r37: a__U61(tt(),V2) -> a__U62(a__isNatIListKind(V2))
r38: a__U62(tt()) -> tt()
r39: a__U71(tt()) -> tt()
r40: a__U81(tt()) -> tt()
r41: a__U91(tt(),V1,V2) -> a__U92(a__isNatKind(V1),V1,V2)
r42: a__U92(tt(),V1,V2) -> a__U93(a__isNatIListKind(V2),V1,V2)
r43: a__U93(tt(),V1,V2) -> a__U94(a__isNatIListKind(V2),V1,V2)
r44: a__U94(tt(),V1,V2) -> a__U95(a__isNat(V1),V2)
r45: a__U95(tt(),V2) -> a__U96(a__isNatList(V2))
r46: a__U96(tt()) -> tt()
r47: a__isNat(|0|()) -> tt()
r48: a__isNat(length(V1)) -> a__U11(a__isNatIListKind(V1),V1)
r49: a__isNat(s(V1)) -> a__U21(a__isNatKind(V1),V1)
r50: a__isNatIList(V) -> a__U31(a__isNatIListKind(V),V)
r51: a__isNatIList(zeros()) -> tt()
r52: a__isNatIList(cons(V1,V2)) -> a__U41(a__isNatKind(V1),V1,V2)
r53: a__isNatIListKind(nil()) -> tt()
r54: a__isNatIListKind(zeros()) -> tt()
r55: a__isNatIListKind(cons(V1,V2)) -> a__U51(a__isNatKind(V1),V2)
r56: a__isNatIListKind(take(V1,V2)) -> a__U61(a__isNatKind(V1),V2)
r57: a__isNatKind(|0|()) -> tt()
r58: a__isNatKind(length(V1)) -> a__U71(a__isNatIListKind(V1))
r59: a__isNatKind(s(V1)) -> a__U81(a__isNatKind(V1))
r60: a__isNatList(nil()) -> tt()
r61: a__isNatList(cons(V1,V2)) -> a__U91(a__isNatKind(V1),V1,V2)
r62: a__isNatList(take(V1,V2)) -> a__U101(a__isNatKind(V1),V1,V2)
r63: a__length(nil()) -> |0|()
r64: a__length(cons(N,L)) -> a__U111(a__isNatList(L),L,N)
r65: a__take(|0|(),IL) -> a__U121(a__isNatIList(IL),IL)
r66: a__take(s(M),cons(N,IL)) -> a__U131(a__isNatIList(IL),IL,M,N)
r67: mark(zeros()) -> a__zeros()
r68: mark(U101(X1,X2,X3)) -> a__U101(mark(X1),X2,X3)
r69: mark(U102(X1,X2,X3)) -> a__U102(mark(X1),X2,X3)
r70: mark(isNatKind(X)) -> a__isNatKind(X)
r71: mark(U103(X1,X2,X3)) -> a__U103(mark(X1),X2,X3)
r72: mark(isNatIListKind(X)) -> a__isNatIListKind(X)
r73: mark(U104(X1,X2,X3)) -> a__U104(mark(X1),X2,X3)
r74: mark(U105(X1,X2)) -> a__U105(mark(X1),X2)
r75: mark(isNat(X)) -> a__isNat(X)
r76: mark(U106(X)) -> a__U106(mark(X))
r77: mark(isNatIList(X)) -> a__isNatIList(X)
r78: mark(U11(X1,X2)) -> a__U11(mark(X1),X2)
r79: mark(U12(X1,X2)) -> a__U12(mark(X1),X2)
r80: mark(U111(X1,X2,X3)) -> a__U111(mark(X1),X2,X3)
r81: mark(U112(X1,X2,X3)) -> a__U112(mark(X1),X2,X3)
r82: mark(U113(X1,X2,X3)) -> a__U113(mark(X1),X2,X3)
r83: mark(U114(X1,X2)) -> a__U114(mark(X1),X2)
r84: mark(length(X)) -> a__length(mark(X))
r85: mark(U13(X)) -> a__U13(mark(X))
r86: mark(isNatList(X)) -> a__isNatList(X)
r87: mark(U121(X1,X2)) -> a__U121(mark(X1),X2)
r88: mark(U122(X)) -> a__U122(mark(X))
r89: mark(U131(X1,X2,X3,X4)) -> a__U131(mark(X1),X2,X3,X4)
r90: mark(U132(X1,X2,X3,X4)) -> a__U132(mark(X1),X2,X3,X4)
r91: mark(U133(X1,X2,X3,X4)) -> a__U133(mark(X1),X2,X3,X4)
r92: mark(U134(X1,X2,X3,X4)) -> a__U134(mark(X1),X2,X3,X4)
r93: mark(U135(X1,X2,X3,X4)) -> a__U135(mark(X1),X2,X3,X4)
r94: mark(U136(X1,X2,X3,X4)) -> a__U136(mark(X1),X2,X3,X4)
r95: mark(take(X1,X2)) -> a__take(mark(X1),mark(X2))
r96: mark(U21(X1,X2)) -> a__U21(mark(X1),X2)
r97: mark(U22(X1,X2)) -> a__U22(mark(X1),X2)
r98: mark(U23(X)) -> a__U23(mark(X))
r99: mark(U31(X1,X2)) -> a__U31(mark(X1),X2)
r100: mark(U32(X1,X2)) -> a__U32(mark(X1),X2)
r101: mark(U33(X)) -> a__U33(mark(X))
r102: mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3)
r103: mark(U42(X1,X2,X3)) -> a__U42(mark(X1),X2,X3)
r104: mark(U43(X1,X2,X3)) -> a__U43(mark(X1),X2,X3)
r105: mark(U44(X1,X2,X3)) -> a__U44(mark(X1),X2,X3)
r106: mark(U45(X1,X2)) -> a__U45(mark(X1),X2)
r107: mark(U46(X)) -> a__U46(mark(X))
r108: mark(U51(X1,X2)) -> a__U51(mark(X1),X2)
r109: mark(U52(X)) -> a__U52(mark(X))
r110: mark(U61(X1,X2)) -> a__U61(mark(X1),X2)
r111: mark(U62(X)) -> a__U62(mark(X))
r112: mark(U71(X)) -> a__U71(mark(X))
r113: mark(U81(X)) -> a__U81(mark(X))
r114: mark(U91(X1,X2,X3)) -> a__U91(mark(X1),X2,X3)
r115: mark(U92(X1,X2,X3)) -> a__U92(mark(X1),X2,X3)
r116: mark(U93(X1,X2,X3)) -> a__U93(mark(X1),X2,X3)
r117: mark(U94(X1,X2,X3)) -> a__U94(mark(X1),X2,X3)
r118: mark(U95(X1,X2)) -> a__U95(mark(X1),X2)
r119: mark(U96(X)) -> a__U96(mark(X))
r120: mark(cons(X1,X2)) -> cons(mark(X1),X2)
r121: mark(|0|()) -> |0|()
r122: mark(tt()) -> tt()
r123: mark(s(X)) -> s(mark(X))
r124: mark(nil()) -> nil()
r125: a__zeros() -> zeros()
r126: a__U101(X1,X2,X3) -> U101(X1,X2,X3)
r127: a__U102(X1,X2,X3) -> U102(X1,X2,X3)
r128: a__isNatKind(X) -> isNatKind(X)
r129: a__U103(X1,X2,X3) -> U103(X1,X2,X3)
r130: a__isNatIListKind(X) -> isNatIListKind(X)
r131: a__U104(X1,X2,X3) -> U104(X1,X2,X3)
r132: a__U105(X1,X2) -> U105(X1,X2)
r133: a__isNat(X) -> isNat(X)
r134: a__U106(X) -> U106(X)
r135: a__isNatIList(X) -> isNatIList(X)
r136: a__U11(X1,X2) -> U11(X1,X2)
r137: a__U12(X1,X2) -> U12(X1,X2)
r138: a__U111(X1,X2,X3) -> U111(X1,X2,X3)
r139: a__U112(X1,X2,X3) -> U112(X1,X2,X3)
r140: a__U113(X1,X2,X3) -> U113(X1,X2,X3)
r141: a__U114(X1,X2) -> U114(X1,X2)
r142: a__length(X) -> length(X)
r143: a__U13(X) -> U13(X)
r144: a__isNatList(X) -> isNatList(X)
r145: a__U121(X1,X2) -> U121(X1,X2)
r146: a__U122(X) -> U122(X)
r147: a__U131(X1,X2,X3,X4) -> U131(X1,X2,X3,X4)
r148: a__U132(X1,X2,X3,X4) -> U132(X1,X2,X3,X4)
r149: a__U133(X1,X2,X3,X4) -> U133(X1,X2,X3,X4)
r150: a__U134(X1,X2,X3,X4) -> U134(X1,X2,X3,X4)
r151: a__U135(X1,X2,X3,X4) -> U135(X1,X2,X3,X4)
r152: a__U136(X1,X2,X3,X4) -> U136(X1,X2,X3,X4)
r153: a__take(X1,X2) -> take(X1,X2)
r154: a__U21(X1,X2) -> U21(X1,X2)
r155: a__U22(X1,X2) -> U22(X1,X2)
r156: a__U23(X) -> U23(X)
r157: a__U31(X1,X2) -> U31(X1,X2)
r158: a__U32(X1,X2) -> U32(X1,X2)
r159: a__U33(X) -> U33(X)
r160: a__U41(X1,X2,X3) -> U41(X1,X2,X3)
r161: a__U42(X1,X2,X3) -> U42(X1,X2,X3)
r162: a__U43(X1,X2,X3) -> U43(X1,X2,X3)
r163: a__U44(X1,X2,X3) -> U44(X1,X2,X3)
r164: a__U45(X1,X2) -> U45(X1,X2)
r165: a__U46(X) -> U46(X)
r166: a__U51(X1,X2) -> U51(X1,X2)
r167: a__U52(X) -> U52(X)
r168: a__U61(X1,X2) -> U61(X1,X2)
r169: a__U62(X) -> U62(X)
r170: a__U71(X) -> U71(X)
r171: a__U81(X) -> U81(X)
r172: a__U91(X1,X2,X3) -> U91(X1,X2,X3)
r173: a__U92(X1,X2,X3) -> U92(X1,X2,X3)
r174: a__U93(X1,X2,X3) -> U93(X1,X2,X3)
r175: a__U94(X1,X2,X3) -> U94(X1,X2,X3)
r176: a__U95(X1,X2) -> U95(X1,X2)
r177: a__U96(X) -> U96(X)

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  max/plus interpretations on natural numbers:
  
    mark#_A(x1) = x1
    U11_A(x1,x2) = max{x1 + 1, x2 + 1}

The next rules are strictly ordered:

  p1

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

-- Reduction pair.

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

p1: a__U101#(tt(),V1,V2) -> a__U102#(a__isNatKind(V1),V1,V2)
p2: a__U102#(tt(),V1,V2) -> a__U103#(a__isNatIListKind(V2),V1,V2)
p3: a__U103#(tt(),V1,V2) -> a__U104#(a__isNatIListKind(V2),V1,V2)
p4: a__U104#(tt(),V1,V2) -> a__isNat#(V1)
p5: a__isNat#(s(V1)) -> a__U21#(a__isNatKind(V1),V1)
p6: a__U21#(tt(),V1) -> a__U22#(a__isNatKind(V1),V1)
p7: a__U22#(tt(),V1) -> a__isNat#(V1)
p8: a__isNat#(length(V1)) -> a__U11#(a__isNatIListKind(V1),V1)
p9: a__U11#(tt(),V1) -> a__U12#(a__isNatIListKind(V1),V1)
p10: a__U12#(tt(),V1) -> a__isNatList#(V1)
p11: a__isNatList#(take(V1,V2)) -> a__U101#(a__isNatKind(V1),V1,V2)
p12: a__isNatList#(cons(V1,V2)) -> a__U91#(a__isNatKind(V1),V1,V2)
p13: a__U91#(tt(),V1,V2) -> a__U92#(a__isNatKind(V1),V1,V2)
p14: a__U92#(tt(),V1,V2) -> a__U93#(a__isNatIListKind(V2),V1,V2)
p15: a__U93#(tt(),V1,V2) -> a__U94#(a__isNatIListKind(V2),V1,V2)
p16: a__U94#(tt(),V1,V2) -> a__isNat#(V1)
p17: a__U94#(tt(),V1,V2) -> a__U95#(a__isNat(V1),V2)
p18: a__U95#(tt(),V2) -> a__isNatList#(V2)
p19: a__U104#(tt(),V1,V2) -> a__U105#(a__isNat(V1),V2)
p20: a__U105#(tt(),V2) -> a__isNatIList#(V2)
p21: a__isNatIList#(cons(V1,V2)) -> a__U41#(a__isNatKind(V1),V1,V2)
p22: a__U41#(tt(),V1,V2) -> a__U42#(a__isNatKind(V1),V1,V2)
p23: a__U42#(tt(),V1,V2) -> a__U43#(a__isNatIListKind(V2),V1,V2)
p24: a__U43#(tt(),V1,V2) -> a__U44#(a__isNatIListKind(V2),V1,V2)
p25: a__U44#(tt(),V1,V2) -> a__isNat#(V1)
p26: a__U44#(tt(),V1,V2) -> a__U45#(a__isNat(V1),V2)
p27: a__U45#(tt(),V2) -> a__isNatIList#(V2)
p28: a__isNatIList#(V) -> a__U31#(a__isNatIListKind(V),V)
p29: a__U31#(tt(),V) -> a__U32#(a__isNatIListKind(V),V)
p30: a__U32#(tt(),V) -> a__isNatList#(V)

and R consists of:

r1: a__zeros() -> cons(|0|(),zeros())
r2: a__U101(tt(),V1,V2) -> a__U102(a__isNatKind(V1),V1,V2)
r3: a__U102(tt(),V1,V2) -> a__U103(a__isNatIListKind(V2),V1,V2)
r4: a__U103(tt(),V1,V2) -> a__U104(a__isNatIListKind(V2),V1,V2)
r5: a__U104(tt(),V1,V2) -> a__U105(a__isNat(V1),V2)
r6: a__U105(tt(),V2) -> a__U106(a__isNatIList(V2))
r7: a__U106(tt()) -> tt()
r8: a__U11(tt(),V1) -> a__U12(a__isNatIListKind(V1),V1)
r9: a__U111(tt(),L,N) -> a__U112(a__isNatIListKind(L),L,N)
r10: a__U112(tt(),L,N) -> a__U113(a__isNat(N),L,N)
r11: a__U113(tt(),L,N) -> a__U114(a__isNatKind(N),L)
r12: a__U114(tt(),L) -> s(a__length(mark(L)))
r13: a__U12(tt(),V1) -> a__U13(a__isNatList(V1))
r14: a__U121(tt(),IL) -> a__U122(a__isNatIListKind(IL))
r15: a__U122(tt()) -> nil()
r16: a__U13(tt()) -> tt()
r17: a__U131(tt(),IL,M,N) -> a__U132(a__isNatIListKind(IL),IL,M,N)
r18: a__U132(tt(),IL,M,N) -> a__U133(a__isNat(M),IL,M,N)
r19: a__U133(tt(),IL,M,N) -> a__U134(a__isNatKind(M),IL,M,N)
r20: a__U134(tt(),IL,M,N) -> a__U135(a__isNat(N),IL,M,N)
r21: a__U135(tt(),IL,M,N) -> a__U136(a__isNatKind(N),IL,M,N)
r22: a__U136(tt(),IL,M,N) -> cons(mark(N),take(M,IL))
r23: a__U21(tt(),V1) -> a__U22(a__isNatKind(V1),V1)
r24: a__U22(tt(),V1) -> a__U23(a__isNat(V1))
r25: a__U23(tt()) -> tt()
r26: a__U31(tt(),V) -> a__U32(a__isNatIListKind(V),V)
r27: a__U32(tt(),V) -> a__U33(a__isNatList(V))
r28: a__U33(tt()) -> tt()
r29: a__U41(tt(),V1,V2) -> a__U42(a__isNatKind(V1),V1,V2)
r30: a__U42(tt(),V1,V2) -> a__U43(a__isNatIListKind(V2),V1,V2)
r31: a__U43(tt(),V1,V2) -> a__U44(a__isNatIListKind(V2),V1,V2)
r32: a__U44(tt(),V1,V2) -> a__U45(a__isNat(V1),V2)
r33: a__U45(tt(),V2) -> a__U46(a__isNatIList(V2))
r34: a__U46(tt()) -> tt()
r35: a__U51(tt(),V2) -> a__U52(a__isNatIListKind(V2))
r36: a__U52(tt()) -> tt()
r37: a__U61(tt(),V2) -> a__U62(a__isNatIListKind(V2))
r38: a__U62(tt()) -> tt()
r39: a__U71(tt()) -> tt()
r40: a__U81(tt()) -> tt()
r41: a__U91(tt(),V1,V2) -> a__U92(a__isNatKind(V1),V1,V2)
r42: a__U92(tt(),V1,V2) -> a__U93(a__isNatIListKind(V2),V1,V2)
r43: a__U93(tt(),V1,V2) -> a__U94(a__isNatIListKind(V2),V1,V2)
r44: a__U94(tt(),V1,V2) -> a__U95(a__isNat(V1),V2)
r45: a__U95(tt(),V2) -> a__U96(a__isNatList(V2))
r46: a__U96(tt()) -> tt()
r47: a__isNat(|0|()) -> tt()
r48: a__isNat(length(V1)) -> a__U11(a__isNatIListKind(V1),V1)
r49: a__isNat(s(V1)) -> a__U21(a__isNatKind(V1),V1)
r50: a__isNatIList(V) -> a__U31(a__isNatIListKind(V),V)
r51: a__isNatIList(zeros()) -> tt()
r52: a__isNatIList(cons(V1,V2)) -> a__U41(a__isNatKind(V1),V1,V2)
r53: a__isNatIListKind(nil()) -> tt()
r54: a__isNatIListKind(zeros()) -> tt()
r55: a__isNatIListKind(cons(V1,V2)) -> a__U51(a__isNatKind(V1),V2)
r56: a__isNatIListKind(take(V1,V2)) -> a__U61(a__isNatKind(V1),V2)
r57: a__isNatKind(|0|()) -> tt()
r58: a__isNatKind(length(V1)) -> a__U71(a__isNatIListKind(V1))
r59: a__isNatKind(s(V1)) -> a__U81(a__isNatKind(V1))
r60: a__isNatList(nil()) -> tt()
r61: a__isNatList(cons(V1,V2)) -> a__U91(a__isNatKind(V1),V1,V2)
r62: a__isNatList(take(V1,V2)) -> a__U101(a__isNatKind(V1),V1,V2)
r63: a__length(nil()) -> |0|()
r64: a__length(cons(N,L)) -> a__U111(a__isNatList(L),L,N)
r65: a__take(|0|(),IL) -> a__U121(a__isNatIList(IL),IL)
r66: a__take(s(M),cons(N,IL)) -> a__U131(a__isNatIList(IL),IL,M,N)
r67: mark(zeros()) -> a__zeros()
r68: mark(U101(X1,X2,X3)) -> a__U101(mark(X1),X2,X3)
r69: mark(U102(X1,X2,X3)) -> a__U102(mark(X1),X2,X3)
r70: mark(isNatKind(X)) -> a__isNatKind(X)
r71: mark(U103(X1,X2,X3)) -> a__U103(mark(X1),X2,X3)
r72: mark(isNatIListKind(X)) -> a__isNatIListKind(X)
r73: mark(U104(X1,X2,X3)) -> a__U104(mark(X1),X2,X3)
r74: mark(U105(X1,X2)) -> a__U105(mark(X1),X2)
r75: mark(isNat(X)) -> a__isNat(X)
r76: mark(U106(X)) -> a__U106(mark(X))
r77: mark(isNatIList(X)) -> a__isNatIList(X)
r78: mark(U11(X1,X2)) -> a__U11(mark(X1),X2)
r79: mark(U12(X1,X2)) -> a__U12(mark(X1),X2)
r80: mark(U111(X1,X2,X3)) -> a__U111(mark(X1),X2,X3)
r81: mark(U112(X1,X2,X3)) -> a__U112(mark(X1),X2,X3)
r82: mark(U113(X1,X2,X3)) -> a__U113(mark(X1),X2,X3)
r83: mark(U114(X1,X2)) -> a__U114(mark(X1),X2)
r84: mark(length(X)) -> a__length(mark(X))
r85: mark(U13(X)) -> a__U13(mark(X))
r86: mark(isNatList(X)) -> a__isNatList(X)
r87: mark(U121(X1,X2)) -> a__U121(mark(X1),X2)
r88: mark(U122(X)) -> a__U122(mark(X))
r89: mark(U131(X1,X2,X3,X4)) -> a__U131(mark(X1),X2,X3,X4)
r90: mark(U132(X1,X2,X3,X4)) -> a__U132(mark(X1),X2,X3,X4)
r91: mark(U133(X1,X2,X3,X4)) -> a__U133(mark(X1),X2,X3,X4)
r92: mark(U134(X1,X2,X3,X4)) -> a__U134(mark(X1),X2,X3,X4)
r93: mark(U135(X1,X2,X3,X4)) -> a__U135(mark(X1),X2,X3,X4)
r94: mark(U136(X1,X2,X3,X4)) -> a__U136(mark(X1),X2,X3,X4)
r95: mark(take(X1,X2)) -> a__take(mark(X1),mark(X2))
r96: mark(U21(X1,X2)) -> a__U21(mark(X1),X2)
r97: mark(U22(X1,X2)) -> a__U22(mark(X1),X2)
r98: mark(U23(X)) -> a__U23(mark(X))
r99: mark(U31(X1,X2)) -> a__U31(mark(X1),X2)
r100: mark(U32(X1,X2)) -> a__U32(mark(X1),X2)
r101: mark(U33(X)) -> a__U33(mark(X))
r102: mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3)
r103: mark(U42(X1,X2,X3)) -> a__U42(mark(X1),X2,X3)
r104: mark(U43(X1,X2,X3)) -> a__U43(mark(X1),X2,X3)
r105: mark(U44(X1,X2,X3)) -> a__U44(mark(X1),X2,X3)
r106: mark(U45(X1,X2)) -> a__U45(mark(X1),X2)
r107: mark(U46(X)) -> a__U46(mark(X))
r108: mark(U51(X1,X2)) -> a__U51(mark(X1),X2)
r109: mark(U52(X)) -> a__U52(mark(X))
r110: mark(U61(X1,X2)) -> a__U61(mark(X1),X2)
r111: mark(U62(X)) -> a__U62(mark(X))
r112: mark(U71(X)) -> a__U71(mark(X))
r113: mark(U81(X)) -> a__U81(mark(X))
r114: mark(U91(X1,X2,X3)) -> a__U91(mark(X1),X2,X3)
r115: mark(U92(X1,X2,X3)) -> a__U92(mark(X1),X2,X3)
r116: mark(U93(X1,X2,X3)) -> a__U93(mark(X1),X2,X3)
r117: mark(U94(X1,X2,X3)) -> a__U94(mark(X1),X2,X3)
r118: mark(U95(X1,X2)) -> a__U95(mark(X1),X2)
r119: mark(U96(X)) -> a__U96(mark(X))
r120: mark(cons(X1,X2)) -> cons(mark(X1),X2)
r121: mark(|0|()) -> |0|()
r122: mark(tt()) -> tt()
r123: mark(s(X)) -> s(mark(X))
r124: mark(nil()) -> nil()
r125: a__zeros() -> zeros()
r126: a__U101(X1,X2,X3) -> U101(X1,X2,X3)
r127: a__U102(X1,X2,X3) -> U102(X1,X2,X3)
r128: a__isNatKind(X) -> isNatKind(X)
r129: a__U103(X1,X2,X3) -> U103(X1,X2,X3)
r130: a__isNatIListKind(X) -> isNatIListKind(X)
r131: a__U104(X1,X2,X3) -> U104(X1,X2,X3)
r132: a__U105(X1,X2) -> U105(X1,X2)
r133: a__isNat(X) -> isNat(X)
r134: a__U106(X) -> U106(X)
r135: a__isNatIList(X) -> isNatIList(X)
r136: a__U11(X1,X2) -> U11(X1,X2)
r137: a__U12(X1,X2) -> U12(X1,X2)
r138: a__U111(X1,X2,X3) -> U111(X1,X2,X3)
r139: a__U112(X1,X2,X3) -> U112(X1,X2,X3)
r140: a__U113(X1,X2,X3) -> U113(X1,X2,X3)
r141: a__U114(X1,X2) -> U114(X1,X2)
r142: a__length(X) -> length(X)
r143: a__U13(X) -> U13(X)
r144: a__isNatList(X) -> isNatList(X)
r145: a__U121(X1,X2) -> U121(X1,X2)
r146: a__U122(X) -> U122(X)
r147: a__U131(X1,X2,X3,X4) -> U131(X1,X2,X3,X4)
r148: a__U132(X1,X2,X3,X4) -> U132(X1,X2,X3,X4)
r149: a__U133(X1,X2,X3,X4) -> U133(X1,X2,X3,X4)
r150: a__U134(X1,X2,X3,X4) -> U134(X1,X2,X3,X4)
r151: a__U135(X1,X2,X3,X4) -> U135(X1,X2,X3,X4)
r152: a__U136(X1,X2,X3,X4) -> U136(X1,X2,X3,X4)
r153: a__take(X1,X2) -> take(X1,X2)
r154: a__U21(X1,X2) -> U21(X1,X2)
r155: a__U22(X1,X2) -> U22(X1,X2)
r156: a__U23(X) -> U23(X)
r157: a__U31(X1,X2) -> U31(X1,X2)
r158: a__U32(X1,X2) -> U32(X1,X2)
r159: a__U33(X) -> U33(X)
r160: a__U41(X1,X2,X3) -> U41(X1,X2,X3)
r161: a__U42(X1,X2,X3) -> U42(X1,X2,X3)
r162: a__U43(X1,X2,X3) -> U43(X1,X2,X3)
r163: a__U44(X1,X2,X3) -> U44(X1,X2,X3)
r164: a__U45(X1,X2) -> U45(X1,X2)
r165: a__U46(X) -> U46(X)
r166: a__U51(X1,X2) -> U51(X1,X2)
r167: a__U52(X) -> U52(X)
r168: a__U61(X1,X2) -> U61(X1,X2)
r169: a__U62(X) -> U62(X)
r170: a__U71(X) -> U71(X)
r171: a__U81(X) -> U81(X)
r172: a__U91(X1,X2,X3) -> U91(X1,X2,X3)
r173: a__U92(X1,X2,X3) -> U92(X1,X2,X3)
r174: a__U93(X1,X2,X3) -> U93(X1,X2,X3)
r175: a__U94(X1,X2,X3) -> U94(X1,X2,X3)
r176: a__U95(X1,X2) -> U95(X1,X2)
r177: a__U96(X) -> U96(X)

The set of usable rules consists of

  r2, r3, r4, r5, r6, r7, r8, r13, r16, 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, r126, r127, r128, r129, r130, r131, r132, r133, r134, r135, r136, r137, r143, r144, r154, r155, r156, r157, r158, r159, r160, r161, r162, r163, r164, r165, r166, r167, r168, r169, r170, r171, r172, r173, r174, r175, r176, r177

Take the reduction pair:

  max/plus interpretations on natural numbers:
  
    a__U101#_A(x1,x2,x3) = max{200, x2 + 148, x3 + 148}
    tt_A = 28
    a__U102#_A(x1,x2,x3) = max{x1 + 113, x2 + 148, x3 + 148}
    a__isNatKind_A(x1) = max{34, x1 + 27}
    a__U103#_A(x1,x2,x3) = max{x1 + 92, x2 + 120, x3 + 120}
    a__isNatIListKind_A(x1) = max{48, x1 + 27}
    a__U104#_A(x1,x2,x3) = max{x1 + 71, x2 + 27, x3 + 120}
    a__isNat#_A(x1) = max{93, x1 + 27}
    s_A(x1) = x1 + 80
    a__U21#_A(x1,x2) = max{x1 + 72, x2 + 107}
    a__U22#_A(x1,x2) = max{x1 + 65, x2 + 27}
    length_A(x1) = x1 + 49
    a__U11#_A(x1,x2) = x2 + 76
    a__U12#_A(x1,x2) = max{x1 + 27, x2 + 76}
    a__isNatList#_A(x1) = x1 + 76
    take_A(x1,x2) = max{x1 + 73, x2 + 147}
    cons_A(x1,x2) = max{x1 + 57, x2 + 66}
    a__U91#_A(x1,x2,x3) = max{x1 + 102, x2 + 133, x3 + 141}
    a__U92#_A(x1,x2,x3) = max{x1 + 106, x2 + 133, x3 + 141}
    a__U93#_A(x1,x2,x3) = max{x1 + 84, x2 + 112, x3 + 112}
    a__U94#_A(x1,x2,x3) = max{x1 + 64, x2 + 93, x3 + 93}
    a__U95#_A(x1,x2) = max{x1 + 40, x2 + 76}
    a__isNat_A(x1) = max{51, x1 + 27}
    a__U105#_A(x1,x2) = max{x1 - 25, x2 + 120}
    a__isNatIList#_A(x1) = x1 + 120
    a__U41#_A(x1,x2,x3) = max{x1 + 142, x2 + 177, x3 + 184}
    a__U42#_A(x1,x2,x3) = max{x1 + 150, x2 + 176, x3 + 184}
    a__U43#_A(x1,x2,x3) = max{x1 + 127, x2 + 155, x3 + 184}
    a__U44#_A(x1,x2,x3) = max{x1 + 106, x2 + 134, x3 + 134}
    a__U45#_A(x1,x2) = max{x1 + 82, x2 + 134}
    a__U31#_A(x1,x2) = max{98, x2 + 77}
    a__U32#_A(x1,x2) = max{x1 + 50, x2 + 76}
    a__U46_A(x1) = 78
    U46_A(x1) = 73
    a__U45_A(x1,x2) = 78
    a__isNatIList_A(x1) = max{28, x1 + 27}
    U45_A(x1,x2) = 78
    a__U44_A(x1,x2,x3) = 78
    U44_A(x1,x2,x3) = 78
    a__U33_A(x1) = 28
    a__U43_A(x1,x2,x3) = 78
    U33_A(x1) = 0
    U43_A(x1,x2,x3) = 78
    a__U32_A(x1,x2) = 28
    a__isNatList_A(x1) = max{30, x1 + 27}
    a__U42_A(x1,x2,x3) = 78
    U32_A(x1,x2) = 0
    U42_A(x1,x2,x3) = 78
    a__U31_A(x1,x2) = max{28, x1 - 21}
    a__U41_A(x1,x2,x3) = max{77, x1 + 50}
    U31_A(x1,x2) = max{0, x1 - 21}
    U41_A(x1,x2,x3) = max{77, x1 + 50}
    a__U106_A(x1) = 134
    a__U96_A(x1) = 78
    zeros_A = 0
    U106_A(x1) = 134
    isNatIList_A(x1) = max{28, x1 + 27}
    U96_A(x1) = 73
    a__U105_A(x1,x2) = 134
    a__U95_A(x1,x2) = 78
    U105_A(x1,x2) = 134
    U95_A(x1,x2) = 78
    a__U104_A(x1,x2,x3) = x1 + 106
    a__U94_A(x1,x2,x3) = 78
    U104_A(x1,x2,x3) = x1 + 106
    U94_A(x1,x2,x3) = 78
    a__U103_A(x1,x2,x3) = max{x1 + 126, x3 + 154}
    a__U93_A(x1,x2,x3) = 78
    U103_A(x1,x2,x3) = max{x1 + 126, x3 + 154}
    U93_A(x1,x2,x3) = 78
    a__U102_A(x1,x2,x3) = x3 + 174
    a__U92_A(x1,x2,x3) = 78
    U102_A(x1,x2,x3) = x3 + 174
    U92_A(x1,x2,x3) = 78
    a__U101_A(x1,x2,x3) = max{x1 + 66, x3 + 174}
    a__U91_A(x1,x2,x3) = max{86, x1 + 50}
    U101_A(x1,x2,x3) = max{x1 + 66, x3 + 174}
    U91_A(x1,x2,x3) = max{86, x1 + 50}
    a__U13_A(x1) = 53
    a__U23_A(x1) = 84
    nil_A = 0
    U13_A(x1) = 53
    isNatList_A(x1) = max{28, x1 + 27}
    U23_A(x1) = 84
    a__U12_A(x1,x2) = 53
    a__U22_A(x1,x2) = 84
    a__U52_A(x1) = 69
    a__U62_A(x1) = 150
    U12_A(x1,x2) = 52
    U22_A(x1,x2) = 84
    U52_A(x1) = 69
    U62_A(x1) = 150
    a__U11_A(x1,x2) = max{52, x1 + 25}
    a__U21_A(x1,x2) = max{57, x1 + 56}
    a__U51_A(x1,x2) = x1 + 50
    a__U61_A(x1,x2) = max{150, x1 + 66}
    a__U71_A(x1) = max{35, x1 - 1}
    a__U81_A(x1) = max{35, x1 + 1}
    U11_A(x1,x2) = max{26, x1 + 25}
    U21_A(x1,x2) = max{57, x1 + 56}
    U51_A(x1,x2) = x1 + 50
    U61_A(x1,x2) = max{67, x1 + 66}
    U71_A(x1) = max{35, x1 - 1}
    U81_A(x1) = max{2, x1 + 1}
    |0|_A = 0
    isNatKind_A(x1) = max{28, x1 + 27}
    isNatIListKind_A(x1) = max{48, x1 + 27}
    isNat_A(x1) = max{51, x1 + 27}

The next rules are strictly ordered:

  p2, p6, p11, p14, p17, p24, p25, p27, p28

We remove them from the problem.

-- SCC decomposition.

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

p1: a__U101#(tt(),V1,V2) -> a__U102#(a__isNatKind(V1),V1,V2)
p2: a__U103#(tt(),V1,V2) -> a__U104#(a__isNatIListKind(V2),V1,V2)
p3: a__U104#(tt(),V1,V2) -> a__isNat#(V1)
p4: a__isNat#(s(V1)) -> a__U21#(a__isNatKind(V1),V1)
p5: a__U22#(tt(),V1) -> a__isNat#(V1)
p6: a__isNat#(length(V1)) -> a__U11#(a__isNatIListKind(V1),V1)
p7: a__U11#(tt(),V1) -> a__U12#(a__isNatIListKind(V1),V1)
p8: a__U12#(tt(),V1) -> a__isNatList#(V1)
p9: a__isNatList#(cons(V1,V2)) -> a__U91#(a__isNatKind(V1),V1,V2)
p10: a__U91#(tt(),V1,V2) -> a__U92#(a__isNatKind(V1),V1,V2)
p11: a__U93#(tt(),V1,V2) -> a__U94#(a__isNatIListKind(V2),V1,V2)
p12: a__U94#(tt(),V1,V2) -> a__isNat#(V1)
p13: a__U95#(tt(),V2) -> a__isNatList#(V2)
p14: a__U104#(tt(),V1,V2) -> a__U105#(a__isNat(V1),V2)
p15: a__U105#(tt(),V2) -> a__isNatIList#(V2)
p16: a__isNatIList#(cons(V1,V2)) -> a__U41#(a__isNatKind(V1),V1,V2)
p17: a__U41#(tt(),V1,V2) -> a__U42#(a__isNatKind(V1),V1,V2)
p18: a__U42#(tt(),V1,V2) -> a__U43#(a__isNatIListKind(V2),V1,V2)
p19: a__U44#(tt(),V1,V2) -> a__U45#(a__isNat(V1),V2)
p20: a__U31#(tt(),V) -> a__U32#(a__isNatIListKind(V),V)
p21: a__U32#(tt(),V) -> a__isNatList#(V)

and R consists of:

r1: a__zeros() -> cons(|0|(),zeros())
r2: a__U101(tt(),V1,V2) -> a__U102(a__isNatKind(V1),V1,V2)
r3: a__U102(tt(),V1,V2) -> a__U103(a__isNatIListKind(V2),V1,V2)
r4: a__U103(tt(),V1,V2) -> a__U104(a__isNatIListKind(V2),V1,V2)
r5: a__U104(tt(),V1,V2) -> a__U105(a__isNat(V1),V2)
r6: a__U105(tt(),V2) -> a__U106(a__isNatIList(V2))
r7: a__U106(tt()) -> tt()
r8: a__U11(tt(),V1) -> a__U12(a__isNatIListKind(V1),V1)
r9: a__U111(tt(),L,N) -> a__U112(a__isNatIListKind(L),L,N)
r10: a__U112(tt(),L,N) -> a__U113(a__isNat(N),L,N)
r11: a__U113(tt(),L,N) -> a__U114(a__isNatKind(N),L)
r12: a__U114(tt(),L) -> s(a__length(mark(L)))
r13: a__U12(tt(),V1) -> a__U13(a__isNatList(V1))
r14: a__U121(tt(),IL) -> a__U122(a__isNatIListKind(IL))
r15: a__U122(tt()) -> nil()
r16: a__U13(tt()) -> tt()
r17: a__U131(tt(),IL,M,N) -> a__U132(a__isNatIListKind(IL),IL,M,N)
r18: a__U132(tt(),IL,M,N) -> a__U133(a__isNat(M),IL,M,N)
r19: a__U133(tt(),IL,M,N) -> a__U134(a__isNatKind(M),IL,M,N)
r20: a__U134(tt(),IL,M,N) -> a__U135(a__isNat(N),IL,M,N)
r21: a__U135(tt(),IL,M,N) -> a__U136(a__isNatKind(N),IL,M,N)
r22: a__U136(tt(),IL,M,N) -> cons(mark(N),take(M,IL))
r23: a__U21(tt(),V1) -> a__U22(a__isNatKind(V1),V1)
r24: a__U22(tt(),V1) -> a__U23(a__isNat(V1))
r25: a__U23(tt()) -> tt()
r26: a__U31(tt(),V) -> a__U32(a__isNatIListKind(V),V)
r27: a__U32(tt(),V) -> a__U33(a__isNatList(V))
r28: a__U33(tt()) -> tt()
r29: a__U41(tt(),V1,V2) -> a__U42(a__isNatKind(V1),V1,V2)
r30: a__U42(tt(),V1,V2) -> a__U43(a__isNatIListKind(V2),V1,V2)
r31: a__U43(tt(),V1,V2) -> a__U44(a__isNatIListKind(V2),V1,V2)
r32: a__U44(tt(),V1,V2) -> a__U45(a__isNat(V1),V2)
r33: a__U45(tt(),V2) -> a__U46(a__isNatIList(V2))
r34: a__U46(tt()) -> tt()
r35: a__U51(tt(),V2) -> a__U52(a__isNatIListKind(V2))
r36: a__U52(tt()) -> tt()
r37: a__U61(tt(),V2) -> a__U62(a__isNatIListKind(V2))
r38: a__U62(tt()) -> tt()
r39: a__U71(tt()) -> tt()
r40: a__U81(tt()) -> tt()
r41: a__U91(tt(),V1,V2) -> a__U92(a__isNatKind(V1),V1,V2)
r42: a__U92(tt(),V1,V2) -> a__U93(a__isNatIListKind(V2),V1,V2)
r43: a__U93(tt(),V1,V2) -> a__U94(a__isNatIListKind(V2),V1,V2)
r44: a__U94(tt(),V1,V2) -> a__U95(a__isNat(V1),V2)
r45: a__U95(tt(),V2) -> a__U96(a__isNatList(V2))
r46: a__U96(tt()) -> tt()
r47: a__isNat(|0|()) -> tt()
r48: a__isNat(length(V1)) -> a__U11(a__isNatIListKind(V1),V1)
r49: a__isNat(s(V1)) -> a__U21(a__isNatKind(V1),V1)
r50: a__isNatIList(V) -> a__U31(a__isNatIListKind(V),V)
r51: a__isNatIList(zeros()) -> tt()
r52: a__isNatIList(cons(V1,V2)) -> a__U41(a__isNatKind(V1),V1,V2)
r53: a__isNatIListKind(nil()) -> tt()
r54: a__isNatIListKind(zeros()) -> tt()
r55: a__isNatIListKind(cons(V1,V2)) -> a__U51(a__isNatKind(V1),V2)
r56: a__isNatIListKind(take(V1,V2)) -> a__U61(a__isNatKind(V1),V2)
r57: a__isNatKind(|0|()) -> tt()
r58: a__isNatKind(length(V1)) -> a__U71(a__isNatIListKind(V1))
r59: a__isNatKind(s(V1)) -> a__U81(a__isNatKind(V1))
r60: a__isNatList(nil()) -> tt()
r61: a__isNatList(cons(V1,V2)) -> a__U91(a__isNatKind(V1),V1,V2)
r62: a__isNatList(take(V1,V2)) -> a__U101(a__isNatKind(V1),V1,V2)
r63: a__length(nil()) -> |0|()
r64: a__length(cons(N,L)) -> a__U111(a__isNatList(L),L,N)
r65: a__take(|0|(),IL) -> a__U121(a__isNatIList(IL),IL)
r66: a__take(s(M),cons(N,IL)) -> a__U131(a__isNatIList(IL),IL,M,N)
r67: mark(zeros()) -> a__zeros()
r68: mark(U101(X1,X2,X3)) -> a__U101(mark(X1),X2,X3)
r69: mark(U102(X1,X2,X3)) -> a__U102(mark(X1),X2,X3)
r70: mark(isNatKind(X)) -> a__isNatKind(X)
r71: mark(U103(X1,X2,X3)) -> a__U103(mark(X1),X2,X3)
r72: mark(isNatIListKind(X)) -> a__isNatIListKind(X)
r73: mark(U104(X1,X2,X3)) -> a__U104(mark(X1),X2,X3)
r74: mark(U105(X1,X2)) -> a__U105(mark(X1),X2)
r75: mark(isNat(X)) -> a__isNat(X)
r76: mark(U106(X)) -> a__U106(mark(X))
r77: mark(isNatIList(X)) -> a__isNatIList(X)
r78: mark(U11(X1,X2)) -> a__U11(mark(X1),X2)
r79: mark(U12(X1,X2)) -> a__U12(mark(X1),X2)
r80: mark(U111(X1,X2,X3)) -> a__U111(mark(X1),X2,X3)
r81: mark(U112(X1,X2,X3)) -> a__U112(mark(X1),X2,X3)
r82: mark(U113(X1,X2,X3)) -> a__U113(mark(X1),X2,X3)
r83: mark(U114(X1,X2)) -> a__U114(mark(X1),X2)
r84: mark(length(X)) -> a__length(mark(X))
r85: mark(U13(X)) -> a__U13(mark(X))
r86: mark(isNatList(X)) -> a__isNatList(X)
r87: mark(U121(X1,X2)) -> a__U121(mark(X1),X2)
r88: mark(U122(X)) -> a__U122(mark(X))
r89: mark(U131(X1,X2,X3,X4)) -> a__U131(mark(X1),X2,X3,X4)
r90: mark(U132(X1,X2,X3,X4)) -> a__U132(mark(X1),X2,X3,X4)
r91: mark(U133(X1,X2,X3,X4)) -> a__U133(mark(X1),X2,X3,X4)
r92: mark(U134(X1,X2,X3,X4)) -> a__U134(mark(X1),X2,X3,X4)
r93: mark(U135(X1,X2,X3,X4)) -> a__U135(mark(X1),X2,X3,X4)
r94: mark(U136(X1,X2,X3,X4)) -> a__U136(mark(X1),X2,X3,X4)
r95: mark(take(X1,X2)) -> a__take(mark(X1),mark(X2))
r96: mark(U21(X1,X2)) -> a__U21(mark(X1),X2)
r97: mark(U22(X1,X2)) -> a__U22(mark(X1),X2)
r98: mark(U23(X)) -> a__U23(mark(X))
r99: mark(U31(X1,X2)) -> a__U31(mark(X1),X2)
r100: mark(U32(X1,X2)) -> a__U32(mark(X1),X2)
r101: mark(U33(X)) -> a__U33(mark(X))
r102: mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3)
r103: mark(U42(X1,X2,X3)) -> a__U42(mark(X1),X2,X3)
r104: mark(U43(X1,X2,X3)) -> a__U43(mark(X1),X2,X3)
r105: mark(U44(X1,X2,X3)) -> a__U44(mark(X1),X2,X3)
r106: mark(U45(X1,X2)) -> a__U45(mark(X1),X2)
r107: mark(U46(X)) -> a__U46(mark(X))
r108: mark(U51(X1,X2)) -> a__U51(mark(X1),X2)
r109: mark(U52(X)) -> a__U52(mark(X))
r110: mark(U61(X1,X2)) -> a__U61(mark(X1),X2)
r111: mark(U62(X)) -> a__U62(mark(X))
r112: mark(U71(X)) -> a__U71(mark(X))
r113: mark(U81(X)) -> a__U81(mark(X))
r114: mark(U91(X1,X2,X3)) -> a__U91(mark(X1),X2,X3)
r115: mark(U92(X1,X2,X3)) -> a__U92(mark(X1),X2,X3)
r116: mark(U93(X1,X2,X3)) -> a__U93(mark(X1),X2,X3)
r117: mark(U94(X1,X2,X3)) -> a__U94(mark(X1),X2,X3)
r118: mark(U95(X1,X2)) -> a__U95(mark(X1),X2)
r119: mark(U96(X)) -> a__U96(mark(X))
r120: mark(cons(X1,X2)) -> cons(mark(X1),X2)
r121: mark(|0|()) -> |0|()
r122: mark(tt()) -> tt()
r123: mark(s(X)) -> s(mark(X))
r124: mark(nil()) -> nil()
r125: a__zeros() -> zeros()
r126: a__U101(X1,X2,X3) -> U101(X1,X2,X3)
r127: a__U102(X1,X2,X3) -> U102(X1,X2,X3)
r128: a__isNatKind(X) -> isNatKind(X)
r129: a__U103(X1,X2,X3) -> U103(X1,X2,X3)
r130: a__isNatIListKind(X) -> isNatIListKind(X)
r131: a__U104(X1,X2,X3) -> U104(X1,X2,X3)
r132: a__U105(X1,X2) -> U105(X1,X2)
r133: a__isNat(X) -> isNat(X)
r134: a__U106(X) -> U106(X)
r135: a__isNatIList(X) -> isNatIList(X)
r136: a__U11(X1,X2) -> U11(X1,X2)
r137: a__U12(X1,X2) -> U12(X1,X2)
r138: a__U111(X1,X2,X3) -> U111(X1,X2,X3)
r139: a__U112(X1,X2,X3) -> U112(X1,X2,X3)
r140: a__U113(X1,X2,X3) -> U113(X1,X2,X3)
r141: a__U114(X1,X2) -> U114(X1,X2)
r142: a__length(X) -> length(X)
r143: a__U13(X) -> U13(X)
r144: a__isNatList(X) -> isNatList(X)
r145: a__U121(X1,X2) -> U121(X1,X2)
r146: a__U122(X) -> U122(X)
r147: a__U131(X1,X2,X3,X4) -> U131(X1,X2,X3,X4)
r148: a__U132(X1,X2,X3,X4) -> U132(X1,X2,X3,X4)
r149: a__U133(X1,X2,X3,X4) -> U133(X1,X2,X3,X4)
r150: a__U134(X1,X2,X3,X4) -> U134(X1,X2,X3,X4)
r151: a__U135(X1,X2,X3,X4) -> U135(X1,X2,X3,X4)
r152: a__U136(X1,X2,X3,X4) -> U136(X1,X2,X3,X4)
r153: a__take(X1,X2) -> take(X1,X2)
r154: a__U21(X1,X2) -> U21(X1,X2)
r155: a__U22(X1,X2) -> U22(X1,X2)
r156: a__U23(X) -> U23(X)
r157: a__U31(X1,X2) -> U31(X1,X2)
r158: a__U32(X1,X2) -> U32(X1,X2)
r159: a__U33(X) -> U33(X)
r160: a__U41(X1,X2,X3) -> U41(X1,X2,X3)
r161: a__U42(X1,X2,X3) -> U42(X1,X2,X3)
r162: a__U43(X1,X2,X3) -> U43(X1,X2,X3)
r163: a__U44(X1,X2,X3) -> U44(X1,X2,X3)
r164: a__U45(X1,X2) -> U45(X1,X2)
r165: a__U46(X) -> U46(X)
r166: a__U51(X1,X2) -> U51(X1,X2)
r167: a__U52(X) -> U52(X)
r168: a__U61(X1,X2) -> U61(X1,X2)
r169: a__U62(X) -> U62(X)
r170: a__U71(X) -> U71(X)
r171: a__U81(X) -> U81(X)
r172: a__U91(X1,X2,X3) -> U91(X1,X2,X3)
r173: a__U92(X1,X2,X3) -> U92(X1,X2,X3)
r174: a__U93(X1,X2,X3) -> U93(X1,X2,X3)
r175: a__U94(X1,X2,X3) -> U94(X1,X2,X3)
r176: a__U95(X1,X2) -> U95(X1,X2)
r177: a__U96(X) -> U96(X)

The estimated dependency graph contains the following SCCs:

  (no SCCs)

-- Reduction pair.

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

p1: a__isNatIListKind#(take(V1,V2)) -> a__isNatKind#(V1)
p2: a__isNatKind#(s(V1)) -> a__isNatKind#(V1)
p3: a__isNatKind#(length(V1)) -> a__isNatIListKind#(V1)
p4: a__isNatIListKind#(take(V1,V2)) -> a__U61#(a__isNatKind(V1),V2)
p5: a__U61#(tt(),V2) -> a__isNatIListKind#(V2)
p6: a__isNatIListKind#(cons(V1,V2)) -> a__isNatKind#(V1)
p7: a__isNatIListKind#(cons(V1,V2)) -> a__U51#(a__isNatKind(V1),V2)
p8: a__U51#(tt(),V2) -> a__isNatIListKind#(V2)

and R consists of:

r1: a__zeros() -> cons(|0|(),zeros())
r2: a__U101(tt(),V1,V2) -> a__U102(a__isNatKind(V1),V1,V2)
r3: a__U102(tt(),V1,V2) -> a__U103(a__isNatIListKind(V2),V1,V2)
r4: a__U103(tt(),V1,V2) -> a__U104(a__isNatIListKind(V2),V1,V2)
r5: a__U104(tt(),V1,V2) -> a__U105(a__isNat(V1),V2)
r6: a__U105(tt(),V2) -> a__U106(a__isNatIList(V2))
r7: a__U106(tt()) -> tt()
r8: a__U11(tt(),V1) -> a__U12(a__isNatIListKind(V1),V1)
r9: a__U111(tt(),L,N) -> a__U112(a__isNatIListKind(L),L,N)
r10: a__U112(tt(),L,N) -> a__U113(a__isNat(N),L,N)
r11: a__U113(tt(),L,N) -> a__U114(a__isNatKind(N),L)
r12: a__U114(tt(),L) -> s(a__length(mark(L)))
r13: a__U12(tt(),V1) -> a__U13(a__isNatList(V1))
r14: a__U121(tt(),IL) -> a__U122(a__isNatIListKind(IL))
r15: a__U122(tt()) -> nil()
r16: a__U13(tt()) -> tt()
r17: a__U131(tt(),IL,M,N) -> a__U132(a__isNatIListKind(IL),IL,M,N)
r18: a__U132(tt(),IL,M,N) -> a__U133(a__isNat(M),IL,M,N)
r19: a__U133(tt(),IL,M,N) -> a__U134(a__isNatKind(M),IL,M,N)
r20: a__U134(tt(),IL,M,N) -> a__U135(a__isNat(N),IL,M,N)
r21: a__U135(tt(),IL,M,N) -> a__U136(a__isNatKind(N),IL,M,N)
r22: a__U136(tt(),IL,M,N) -> cons(mark(N),take(M,IL))
r23: a__U21(tt(),V1) -> a__U22(a__isNatKind(V1),V1)
r24: a__U22(tt(),V1) -> a__U23(a__isNat(V1))
r25: a__U23(tt()) -> tt()
r26: a__U31(tt(),V) -> a__U32(a__isNatIListKind(V),V)
r27: a__U32(tt(),V) -> a__U33(a__isNatList(V))
r28: a__U33(tt()) -> tt()
r29: a__U41(tt(),V1,V2) -> a__U42(a__isNatKind(V1),V1,V2)
r30: a__U42(tt(),V1,V2) -> a__U43(a__isNatIListKind(V2),V1,V2)
r31: a__U43(tt(),V1,V2) -> a__U44(a__isNatIListKind(V2),V1,V2)
r32: a__U44(tt(),V1,V2) -> a__U45(a__isNat(V1),V2)
r33: a__U45(tt(),V2) -> a__U46(a__isNatIList(V2))
r34: a__U46(tt()) -> tt()
r35: a__U51(tt(),V2) -> a__U52(a__isNatIListKind(V2))
r36: a__U52(tt()) -> tt()
r37: a__U61(tt(),V2) -> a__U62(a__isNatIListKind(V2))
r38: a__U62(tt()) -> tt()
r39: a__U71(tt()) -> tt()
r40: a__U81(tt()) -> tt()
r41: a__U91(tt(),V1,V2) -> a__U92(a__isNatKind(V1),V1,V2)
r42: a__U92(tt(),V1,V2) -> a__U93(a__isNatIListKind(V2),V1,V2)
r43: a__U93(tt(),V1,V2) -> a__U94(a__isNatIListKind(V2),V1,V2)
r44: a__U94(tt(),V1,V2) -> a__U95(a__isNat(V1),V2)
r45: a__U95(tt(),V2) -> a__U96(a__isNatList(V2))
r46: a__U96(tt()) -> tt()
r47: a__isNat(|0|()) -> tt()
r48: a__isNat(length(V1)) -> a__U11(a__isNatIListKind(V1),V1)
r49: a__isNat(s(V1)) -> a__U21(a__isNatKind(V1),V1)
r50: a__isNatIList(V) -> a__U31(a__isNatIListKind(V),V)
r51: a__isNatIList(zeros()) -> tt()
r52: a__isNatIList(cons(V1,V2)) -> a__U41(a__isNatKind(V1),V1,V2)
r53: a__isNatIListKind(nil()) -> tt()
r54: a__isNatIListKind(zeros()) -> tt()
r55: a__isNatIListKind(cons(V1,V2)) -> a__U51(a__isNatKind(V1),V2)
r56: a__isNatIListKind(take(V1,V2)) -> a__U61(a__isNatKind(V1),V2)
r57: a__isNatKind(|0|()) -> tt()
r58: a__isNatKind(length(V1)) -> a__U71(a__isNatIListKind(V1))
r59: a__isNatKind(s(V1)) -> a__U81(a__isNatKind(V1))
r60: a__isNatList(nil()) -> tt()
r61: a__isNatList(cons(V1,V2)) -> a__U91(a__isNatKind(V1),V1,V2)
r62: a__isNatList(take(V1,V2)) -> a__U101(a__isNatKind(V1),V1,V2)
r63: a__length(nil()) -> |0|()
r64: a__length(cons(N,L)) -> a__U111(a__isNatList(L),L,N)
r65: a__take(|0|(),IL) -> a__U121(a__isNatIList(IL),IL)
r66: a__take(s(M),cons(N,IL)) -> a__U131(a__isNatIList(IL),IL,M,N)
r67: mark(zeros()) -> a__zeros()
r68: mark(U101(X1,X2,X3)) -> a__U101(mark(X1),X2,X3)
r69: mark(U102(X1,X2,X3)) -> a__U102(mark(X1),X2,X3)
r70: mark(isNatKind(X)) -> a__isNatKind(X)
r71: mark(U103(X1,X2,X3)) -> a__U103(mark(X1),X2,X3)
r72: mark(isNatIListKind(X)) -> a__isNatIListKind(X)
r73: mark(U104(X1,X2,X3)) -> a__U104(mark(X1),X2,X3)
r74: mark(U105(X1,X2)) -> a__U105(mark(X1),X2)
r75: mark(isNat(X)) -> a__isNat(X)
r76: mark(U106(X)) -> a__U106(mark(X))
r77: mark(isNatIList(X)) -> a__isNatIList(X)
r78: mark(U11(X1,X2)) -> a__U11(mark(X1),X2)
r79: mark(U12(X1,X2)) -> a__U12(mark(X1),X2)
r80: mark(U111(X1,X2,X3)) -> a__U111(mark(X1),X2,X3)
r81: mark(U112(X1,X2,X3)) -> a__U112(mark(X1),X2,X3)
r82: mark(U113(X1,X2,X3)) -> a__U113(mark(X1),X2,X3)
r83: mark(U114(X1,X2)) -> a__U114(mark(X1),X2)
r84: mark(length(X)) -> a__length(mark(X))
r85: mark(U13(X)) -> a__U13(mark(X))
r86: mark(isNatList(X)) -> a__isNatList(X)
r87: mark(U121(X1,X2)) -> a__U121(mark(X1),X2)
r88: mark(U122(X)) -> a__U122(mark(X))
r89: mark(U131(X1,X2,X3,X4)) -> a__U131(mark(X1),X2,X3,X4)
r90: mark(U132(X1,X2,X3,X4)) -> a__U132(mark(X1),X2,X3,X4)
r91: mark(U133(X1,X2,X3,X4)) -> a__U133(mark(X1),X2,X3,X4)
r92: mark(U134(X1,X2,X3,X4)) -> a__U134(mark(X1),X2,X3,X4)
r93: mark(U135(X1,X2,X3,X4)) -> a__U135(mark(X1),X2,X3,X4)
r94: mark(U136(X1,X2,X3,X4)) -> a__U136(mark(X1),X2,X3,X4)
r95: mark(take(X1,X2)) -> a__take(mark(X1),mark(X2))
r96: mark(U21(X1,X2)) -> a__U21(mark(X1),X2)
r97: mark(U22(X1,X2)) -> a__U22(mark(X1),X2)
r98: mark(U23(X)) -> a__U23(mark(X))
r99: mark(U31(X1,X2)) -> a__U31(mark(X1),X2)
r100: mark(U32(X1,X2)) -> a__U32(mark(X1),X2)
r101: mark(U33(X)) -> a__U33(mark(X))
r102: mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3)
r103: mark(U42(X1,X2,X3)) -> a__U42(mark(X1),X2,X3)
r104: mark(U43(X1,X2,X3)) -> a__U43(mark(X1),X2,X3)
r105: mark(U44(X1,X2,X3)) -> a__U44(mark(X1),X2,X3)
r106: mark(U45(X1,X2)) -> a__U45(mark(X1),X2)
r107: mark(U46(X)) -> a__U46(mark(X))
r108: mark(U51(X1,X2)) -> a__U51(mark(X1),X2)
r109: mark(U52(X)) -> a__U52(mark(X))
r110: mark(U61(X1,X2)) -> a__U61(mark(X1),X2)
r111: mark(U62(X)) -> a__U62(mark(X))
r112: mark(U71(X)) -> a__U71(mark(X))
r113: mark(U81(X)) -> a__U81(mark(X))
r114: mark(U91(X1,X2,X3)) -> a__U91(mark(X1),X2,X3)
r115: mark(U92(X1,X2,X3)) -> a__U92(mark(X1),X2,X3)
r116: mark(U93(X1,X2,X3)) -> a__U93(mark(X1),X2,X3)
r117: mark(U94(X1,X2,X3)) -> a__U94(mark(X1),X2,X3)
r118: mark(U95(X1,X2)) -> a__U95(mark(X1),X2)
r119: mark(U96(X)) -> a__U96(mark(X))
r120: mark(cons(X1,X2)) -> cons(mark(X1),X2)
r121: mark(|0|()) -> |0|()
r122: mark(tt()) -> tt()
r123: mark(s(X)) -> s(mark(X))
r124: mark(nil()) -> nil()
r125: a__zeros() -> zeros()
r126: a__U101(X1,X2,X3) -> U101(X1,X2,X3)
r127: a__U102(X1,X2,X3) -> U102(X1,X2,X3)
r128: a__isNatKind(X) -> isNatKind(X)
r129: a__U103(X1,X2,X3) -> U103(X1,X2,X3)
r130: a__isNatIListKind(X) -> isNatIListKind(X)
r131: a__U104(X1,X2,X3) -> U104(X1,X2,X3)
r132: a__U105(X1,X2) -> U105(X1,X2)
r133: a__isNat(X) -> isNat(X)
r134: a__U106(X) -> U106(X)
r135: a__isNatIList(X) -> isNatIList(X)
r136: a__U11(X1,X2) -> U11(X1,X2)
r137: a__U12(X1,X2) -> U12(X1,X2)
r138: a__U111(X1,X2,X3) -> U111(X1,X2,X3)
r139: a__U112(X1,X2,X3) -> U112(X1,X2,X3)
r140: a__U113(X1,X2,X3) -> U113(X1,X2,X3)
r141: a__U114(X1,X2) -> U114(X1,X2)
r142: a__length(X) -> length(X)
r143: a__U13(X) -> U13(X)
r144: a__isNatList(X) -> isNatList(X)
r145: a__U121(X1,X2) -> U121(X1,X2)
r146: a__U122(X) -> U122(X)
r147: a__U131(X1,X2,X3,X4) -> U131(X1,X2,X3,X4)
r148: a__U132(X1,X2,X3,X4) -> U132(X1,X2,X3,X4)
r149: a__U133(X1,X2,X3,X4) -> U133(X1,X2,X3,X4)
r150: a__U134(X1,X2,X3,X4) -> U134(X1,X2,X3,X4)
r151: a__U135(X1,X2,X3,X4) -> U135(X1,X2,X3,X4)
r152: a__U136(X1,X2,X3,X4) -> U136(X1,X2,X3,X4)
r153: a__take(X1,X2) -> take(X1,X2)
r154: a__U21(X1,X2) -> U21(X1,X2)
r155: a__U22(X1,X2) -> U22(X1,X2)
r156: a__U23(X) -> U23(X)
r157: a__U31(X1,X2) -> U31(X1,X2)
r158: a__U32(X1,X2) -> U32(X1,X2)
r159: a__U33(X) -> U33(X)
r160: a__U41(X1,X2,X3) -> U41(X1,X2,X3)
r161: a__U42(X1,X2,X3) -> U42(X1,X2,X3)
r162: a__U43(X1,X2,X3) -> U43(X1,X2,X3)
r163: a__U44(X1,X2,X3) -> U44(X1,X2,X3)
r164: a__U45(X1,X2) -> U45(X1,X2)
r165: a__U46(X) -> U46(X)
r166: a__U51(X1,X2) -> U51(X1,X2)
r167: a__U52(X) -> U52(X)
r168: a__U61(X1,X2) -> U61(X1,X2)
r169: a__U62(X) -> U62(X)
r170: a__U71(X) -> U71(X)
r171: a__U81(X) -> U81(X)
r172: a__U91(X1,X2,X3) -> U91(X1,X2,X3)
r173: a__U92(X1,X2,X3) -> U92(X1,X2,X3)
r174: a__U93(X1,X2,X3) -> U93(X1,X2,X3)
r175: a__U94(X1,X2,X3) -> U94(X1,X2,X3)
r176: a__U95(X1,X2) -> U95(X1,X2)
r177: a__U96(X) -> U96(X)

The set of usable rules consists of

  r35, r36, r37, r38, r39, r40, r53, r54, r55, r56, r57, r58, r59, r128, r130, r166, r167, r168, r169, r170, r171

Take the reduction pair:

  max/plus interpretations on natural numbers:
  
    a__isNatIListKind#_A(x1) = max{10, x1}
    take_A(x1,x2) = max{x1 + 8, x2 + 10}
    a__isNatKind#_A(x1) = x1 + 8
    s_A(x1) = x1 + 2
    length_A(x1) = x1 + 2
    a__U61#_A(x1,x2) = max{x1 + 1, x2 + 10}
    a__isNatKind_A(x1) = max{6, x1 + 5}
    tt_A = 6
    cons_A(x1,x2) = max{x1 + 11, x2 + 11}
    a__U51#_A(x1,x2) = max{x1 + 5, x2 + 11}
    a__U52_A(x1) = 14
    a__U62_A(x1) = 13
    U52_A(x1) = 14
    U62_A(x1) = 13
    a__U51_A(x1,x2) = x1 + 10
    a__isNatIListKind_A(x1) = max{8, x1 + 5}
    a__U61_A(x1,x2) = max{12, x1 + 7}
    U51_A(x1,x2) = x1 + 10
    U61_A(x1,x2) = max{12, x1 + 7}
    a__U71_A(x1) = max{7, x1 - 1}
    a__U81_A(x1) = max{7, x1 + 1}
    nil_A = 0
    zeros_A = 0
    isNatIListKind_A(x1) = max{6, x1 + 5}
    U71_A(x1) = max{7, x1 - 1}
    U81_A(x1) = max{7, x1 + 1}
    |0|_A = 0
    isNatKind_A(x1) = max{6, x1 + 5}

The next rules are strictly ordered:

  p2, p6, p8

We remove them from the problem.

-- SCC decomposition.

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

p1: a__isNatIListKind#(take(V1,V2)) -> a__isNatKind#(V1)
p2: a__isNatKind#(length(V1)) -> a__isNatIListKind#(V1)
p3: a__isNatIListKind#(take(V1,V2)) -> a__U61#(a__isNatKind(V1),V2)
p4: a__U61#(tt(),V2) -> a__isNatIListKind#(V2)
p5: a__isNatIListKind#(cons(V1,V2)) -> a__U51#(a__isNatKind(V1),V2)

and R consists of:

r1: a__zeros() -> cons(|0|(),zeros())
r2: a__U101(tt(),V1,V2) -> a__U102(a__isNatKind(V1),V1,V2)
r3: a__U102(tt(),V1,V2) -> a__U103(a__isNatIListKind(V2),V1,V2)
r4: a__U103(tt(),V1,V2) -> a__U104(a__isNatIListKind(V2),V1,V2)
r5: a__U104(tt(),V1,V2) -> a__U105(a__isNat(V1),V2)
r6: a__U105(tt(),V2) -> a__U106(a__isNatIList(V2))
r7: a__U106(tt()) -> tt()
r8: a__U11(tt(),V1) -> a__U12(a__isNatIListKind(V1),V1)
r9: a__U111(tt(),L,N) -> a__U112(a__isNatIListKind(L),L,N)
r10: a__U112(tt(),L,N) -> a__U113(a__isNat(N),L,N)
r11: a__U113(tt(),L,N) -> a__U114(a__isNatKind(N),L)
r12: a__U114(tt(),L) -> s(a__length(mark(L)))
r13: a__U12(tt(),V1) -> a__U13(a__isNatList(V1))
r14: a__U121(tt(),IL) -> a__U122(a__isNatIListKind(IL))
r15: a__U122(tt()) -> nil()
r16: a__U13(tt()) -> tt()
r17: a__U131(tt(),IL,M,N) -> a__U132(a__isNatIListKind(IL),IL,M,N)
r18: a__U132(tt(),IL,M,N) -> a__U133(a__isNat(M),IL,M,N)
r19: a__U133(tt(),IL,M,N) -> a__U134(a__isNatKind(M),IL,M,N)
r20: a__U134(tt(),IL,M,N) -> a__U135(a__isNat(N),IL,M,N)
r21: a__U135(tt(),IL,M,N) -> a__U136(a__isNatKind(N),IL,M,N)
r22: a__U136(tt(),IL,M,N) -> cons(mark(N),take(M,IL))
r23: a__U21(tt(),V1) -> a__U22(a__isNatKind(V1),V1)
r24: a__U22(tt(),V1) -> a__U23(a__isNat(V1))
r25: a__U23(tt()) -> tt()
r26: a__U31(tt(),V) -> a__U32(a__isNatIListKind(V),V)
r27: a__U32(tt(),V) -> a__U33(a__isNatList(V))
r28: a__U33(tt()) -> tt()
r29: a__U41(tt(),V1,V2) -> a__U42(a__isNatKind(V1),V1,V2)
r30: a__U42(tt(),V1,V2) -> a__U43(a__isNatIListKind(V2),V1,V2)
r31: a__U43(tt(),V1,V2) -> a__U44(a__isNatIListKind(V2),V1,V2)
r32: a__U44(tt(),V1,V2) -> a__U45(a__isNat(V1),V2)
r33: a__U45(tt(),V2) -> a__U46(a__isNatIList(V2))
r34: a__U46(tt()) -> tt()
r35: a__U51(tt(),V2) -> a__U52(a__isNatIListKind(V2))
r36: a__U52(tt()) -> tt()
r37: a__U61(tt(),V2) -> a__U62(a__isNatIListKind(V2))
r38: a__U62(tt()) -> tt()
r39: a__U71(tt()) -> tt()
r40: a__U81(tt()) -> tt()
r41: a__U91(tt(),V1,V2) -> a__U92(a__isNatKind(V1),V1,V2)
r42: a__U92(tt(),V1,V2) -> a__U93(a__isNatIListKind(V2),V1,V2)
r43: a__U93(tt(),V1,V2) -> a__U94(a__isNatIListKind(V2),V1,V2)
r44: a__U94(tt(),V1,V2) -> a__U95(a__isNat(V1),V2)
r45: a__U95(tt(),V2) -> a__U96(a__isNatList(V2))
r46: a__U96(tt()) -> tt()
r47: a__isNat(|0|()) -> tt()
r48: a__isNat(length(V1)) -> a__U11(a__isNatIListKind(V1),V1)
r49: a__isNat(s(V1)) -> a__U21(a__isNatKind(V1),V1)
r50: a__isNatIList(V) -> a__U31(a__isNatIListKind(V),V)
r51: a__isNatIList(zeros()) -> tt()
r52: a__isNatIList(cons(V1,V2)) -> a__U41(a__isNatKind(V1),V1,V2)
r53: a__isNatIListKind(nil()) -> tt()
r54: a__isNatIListKind(zeros()) -> tt()
r55: a__isNatIListKind(cons(V1,V2)) -> a__U51(a__isNatKind(V1),V2)
r56: a__isNatIListKind(take(V1,V2)) -> a__U61(a__isNatKind(V1),V2)
r57: a__isNatKind(|0|()) -> tt()
r58: a__isNatKind(length(V1)) -> a__U71(a__isNatIListKind(V1))
r59: a__isNatKind(s(V1)) -> a__U81(a__isNatKind(V1))
r60: a__isNatList(nil()) -> tt()
r61: a__isNatList(cons(V1,V2)) -> a__U91(a__isNatKind(V1),V1,V2)
r62: a__isNatList(take(V1,V2)) -> a__U101(a__isNatKind(V1),V1,V2)
r63: a__length(nil()) -> |0|()
r64: a__length(cons(N,L)) -> a__U111(a__isNatList(L),L,N)
r65: a__take(|0|(),IL) -> a__U121(a__isNatIList(IL),IL)
r66: a__take(s(M),cons(N,IL)) -> a__U131(a__isNatIList(IL),IL,M,N)
r67: mark(zeros()) -> a__zeros()
r68: mark(U101(X1,X2,X3)) -> a__U101(mark(X1),X2,X3)
r69: mark(U102(X1,X2,X3)) -> a__U102(mark(X1),X2,X3)
r70: mark(isNatKind(X)) -> a__isNatKind(X)
r71: mark(U103(X1,X2,X3)) -> a__U103(mark(X1),X2,X3)
r72: mark(isNatIListKind(X)) -> a__isNatIListKind(X)
r73: mark(U104(X1,X2,X3)) -> a__U104(mark(X1),X2,X3)
r74: mark(U105(X1,X2)) -> a__U105(mark(X1),X2)
r75: mark(isNat(X)) -> a__isNat(X)
r76: mark(U106(X)) -> a__U106(mark(X))
r77: mark(isNatIList(X)) -> a__isNatIList(X)
r78: mark(U11(X1,X2)) -> a__U11(mark(X1),X2)
r79: mark(U12(X1,X2)) -> a__U12(mark(X1),X2)
r80: mark(U111(X1,X2,X3)) -> a__U111(mark(X1),X2,X3)
r81: mark(U112(X1,X2,X3)) -> a__U112(mark(X1),X2,X3)
r82: mark(U113(X1,X2,X3)) -> a__U113(mark(X1),X2,X3)
r83: mark(U114(X1,X2)) -> a__U114(mark(X1),X2)
r84: mark(length(X)) -> a__length(mark(X))
r85: mark(U13(X)) -> a__U13(mark(X))
r86: mark(isNatList(X)) -> a__isNatList(X)
r87: mark(U121(X1,X2)) -> a__U121(mark(X1),X2)
r88: mark(U122(X)) -> a__U122(mark(X))
r89: mark(U131(X1,X2,X3,X4)) -> a__U131(mark(X1),X2,X3,X4)
r90: mark(U132(X1,X2,X3,X4)) -> a__U132(mark(X1),X2,X3,X4)
r91: mark(U133(X1,X2,X3,X4)) -> a__U133(mark(X1),X2,X3,X4)
r92: mark(U134(X1,X2,X3,X4)) -> a__U134(mark(X1),X2,X3,X4)
r93: mark(U135(X1,X2,X3,X4)) -> a__U135(mark(X1),X2,X3,X4)
r94: mark(U136(X1,X2,X3,X4)) -> a__U136(mark(X1),X2,X3,X4)
r95: mark(take(X1,X2)) -> a__take(mark(X1),mark(X2))
r96: mark(U21(X1,X2)) -> a__U21(mark(X1),X2)
r97: mark(U22(X1,X2)) -> a__U22(mark(X1),X2)
r98: mark(U23(X)) -> a__U23(mark(X))
r99: mark(U31(X1,X2)) -> a__U31(mark(X1),X2)
r100: mark(U32(X1,X2)) -> a__U32(mark(X1),X2)
r101: mark(U33(X)) -> a__U33(mark(X))
r102: mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3)
r103: mark(U42(X1,X2,X3)) -> a__U42(mark(X1),X2,X3)
r104: mark(U43(X1,X2,X3)) -> a__U43(mark(X1),X2,X3)
r105: mark(U44(X1,X2,X3)) -> a__U44(mark(X1),X2,X3)
r106: mark(U45(X1,X2)) -> a__U45(mark(X1),X2)
r107: mark(U46(X)) -> a__U46(mark(X))
r108: mark(U51(X1,X2)) -> a__U51(mark(X1),X2)
r109: mark(U52(X)) -> a__U52(mark(X))
r110: mark(U61(X1,X2)) -> a__U61(mark(X1),X2)
r111: mark(U62(X)) -> a__U62(mark(X))
r112: mark(U71(X)) -> a__U71(mark(X))
r113: mark(U81(X)) -> a__U81(mark(X))
r114: mark(U91(X1,X2,X3)) -> a__U91(mark(X1),X2,X3)
r115: mark(U92(X1,X2,X3)) -> a__U92(mark(X1),X2,X3)
r116: mark(U93(X1,X2,X3)) -> a__U93(mark(X1),X2,X3)
r117: mark(U94(X1,X2,X3)) -> a__U94(mark(X1),X2,X3)
r118: mark(U95(X1,X2)) -> a__U95(mark(X1),X2)
r119: mark(U96(X)) -> a__U96(mark(X))
r120: mark(cons(X1,X2)) -> cons(mark(X1),X2)
r121: mark(|0|()) -> |0|()
r122: mark(tt()) -> tt()
r123: mark(s(X)) -> s(mark(X))
r124: mark(nil()) -> nil()
r125: a__zeros() -> zeros()
r126: a__U101(X1,X2,X3) -> U101(X1,X2,X3)
r127: a__U102(X1,X2,X3) -> U102(X1,X2,X3)
r128: a__isNatKind(X) -> isNatKind(X)
r129: a__U103(X1,X2,X3) -> U103(X1,X2,X3)
r130: a__isNatIListKind(X) -> isNatIListKind(X)
r131: a__U104(X1,X2,X3) -> U104(X1,X2,X3)
r132: a__U105(X1,X2) -> U105(X1,X2)
r133: a__isNat(X) -> isNat(X)
r134: a__U106(X) -> U106(X)
r135: a__isNatIList(X) -> isNatIList(X)
r136: a__U11(X1,X2) -> U11(X1,X2)
r137: a__U12(X1,X2) -> U12(X1,X2)
r138: a__U111(X1,X2,X3) -> U111(X1,X2,X3)
r139: a__U112(X1,X2,X3) -> U112(X1,X2,X3)
r140: a__U113(X1,X2,X3) -> U113(X1,X2,X3)
r141: a__U114(X1,X2) -> U114(X1,X2)
r142: a__length(X) -> length(X)
r143: a__U13(X) -> U13(X)
r144: a__isNatList(X) -> isNatList(X)
r145: a__U121(X1,X2) -> U121(X1,X2)
r146: a__U122(X) -> U122(X)
r147: a__U131(X1,X2,X3,X4) -> U131(X1,X2,X3,X4)
r148: a__U132(X1,X2,X3,X4) -> U132(X1,X2,X3,X4)
r149: a__U133(X1,X2,X3,X4) -> U133(X1,X2,X3,X4)
r150: a__U134(X1,X2,X3,X4) -> U134(X1,X2,X3,X4)
r151: a__U135(X1,X2,X3,X4) -> U135(X1,X2,X3,X4)
r152: a__U136(X1,X2,X3,X4) -> U136(X1,X2,X3,X4)
r153: a__take(X1,X2) -> take(X1,X2)
r154: a__U21(X1,X2) -> U21(X1,X2)
r155: a__U22(X1,X2) -> U22(X1,X2)
r156: a__U23(X) -> U23(X)
r157: a__U31(X1,X2) -> U31(X1,X2)
r158: a__U32(X1,X2) -> U32(X1,X2)
r159: a__U33(X) -> U33(X)
r160: a__U41(X1,X2,X3) -> U41(X1,X2,X3)
r161: a__U42(X1,X2,X3) -> U42(X1,X2,X3)
r162: a__U43(X1,X2,X3) -> U43(X1,X2,X3)
r163: a__U44(X1,X2,X3) -> U44(X1,X2,X3)
r164: a__U45(X1,X2) -> U45(X1,X2)
r165: a__U46(X) -> U46(X)
r166: a__U51(X1,X2) -> U51(X1,X2)
r167: a__U52(X) -> U52(X)
r168: a__U61(X1,X2) -> U61(X1,X2)
r169: a__U62(X) -> U62(X)
r170: a__U71(X) -> U71(X)
r171: a__U81(X) -> U81(X)
r172: a__U91(X1,X2,X3) -> U91(X1,X2,X3)
r173: a__U92(X1,X2,X3) -> U92(X1,X2,X3)
r174: a__U93(X1,X2,X3) -> U93(X1,X2,X3)
r175: a__U94(X1,X2,X3) -> U94(X1,X2,X3)
r176: a__U95(X1,X2) -> U95(X1,X2)
r177: a__U96(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: a__isNatIListKind#(take(V1,V2)) -> a__isNatKind#(V1)
p2: a__isNatKind#(length(V1)) -> a__isNatIListKind#(V1)
p3: a__isNatIListKind#(take(V1,V2)) -> a__U61#(a__isNatKind(V1),V2)
p4: a__U61#(tt(),V2) -> a__isNatIListKind#(V2)

and R consists of:

r1: a__zeros() -> cons(|0|(),zeros())
r2: a__U101(tt(),V1,V2) -> a__U102(a__isNatKind(V1),V1,V2)
r3: a__U102(tt(),V1,V2) -> a__U103(a__isNatIListKind(V2),V1,V2)
r4: a__U103(tt(),V1,V2) -> a__U104(a__isNatIListKind(V2),V1,V2)
r5: a__U104(tt(),V1,V2) -> a__U105(a__isNat(V1),V2)
r6: a__U105(tt(),V2) -> a__U106(a__isNatIList(V2))
r7: a__U106(tt()) -> tt()
r8: a__U11(tt(),V1) -> a__U12(a__isNatIListKind(V1),V1)
r9: a__U111(tt(),L,N) -> a__U112(a__isNatIListKind(L),L,N)
r10: a__U112(tt(),L,N) -> a__U113(a__isNat(N),L,N)
r11: a__U113(tt(),L,N) -> a__U114(a__isNatKind(N),L)
r12: a__U114(tt(),L) -> s(a__length(mark(L)))
r13: a__U12(tt(),V1) -> a__U13(a__isNatList(V1))
r14: a__U121(tt(),IL) -> a__U122(a__isNatIListKind(IL))
r15: a__U122(tt()) -> nil()
r16: a__U13(tt()) -> tt()
r17: a__U131(tt(),IL,M,N) -> a__U132(a__isNatIListKind(IL),IL,M,N)
r18: a__U132(tt(),IL,M,N) -> a__U133(a__isNat(M),IL,M,N)
r19: a__U133(tt(),IL,M,N) -> a__U134(a__isNatKind(M),IL,M,N)
r20: a__U134(tt(),IL,M,N) -> a__U135(a__isNat(N),IL,M,N)
r21: a__U135(tt(),IL,M,N) -> a__U136(a__isNatKind(N),IL,M,N)
r22: a__U136(tt(),IL,M,N) -> cons(mark(N),take(M,IL))
r23: a__U21(tt(),V1) -> a__U22(a__isNatKind(V1),V1)
r24: a__U22(tt(),V1) -> a__U23(a__isNat(V1))
r25: a__U23(tt()) -> tt()
r26: a__U31(tt(),V) -> a__U32(a__isNatIListKind(V),V)
r27: a__U32(tt(),V) -> a__U33(a__isNatList(V))
r28: a__U33(tt()) -> tt()
r29: a__U41(tt(),V1,V2) -> a__U42(a__isNatKind(V1),V1,V2)
r30: a__U42(tt(),V1,V2) -> a__U43(a__isNatIListKind(V2),V1,V2)
r31: a__U43(tt(),V1,V2) -> a__U44(a__isNatIListKind(V2),V1,V2)
r32: a__U44(tt(),V1,V2) -> a__U45(a__isNat(V1),V2)
r33: a__U45(tt(),V2) -> a__U46(a__isNatIList(V2))
r34: a__U46(tt()) -> tt()
r35: a__U51(tt(),V2) -> a__U52(a__isNatIListKind(V2))
r36: a__U52(tt()) -> tt()
r37: a__U61(tt(),V2) -> a__U62(a__isNatIListKind(V2))
r38: a__U62(tt()) -> tt()
r39: a__U71(tt()) -> tt()
r40: a__U81(tt()) -> tt()
r41: a__U91(tt(),V1,V2) -> a__U92(a__isNatKind(V1),V1,V2)
r42: a__U92(tt(),V1,V2) -> a__U93(a__isNatIListKind(V2),V1,V2)
r43: a__U93(tt(),V1,V2) -> a__U94(a__isNatIListKind(V2),V1,V2)
r44: a__U94(tt(),V1,V2) -> a__U95(a__isNat(V1),V2)
r45: a__U95(tt(),V2) -> a__U96(a__isNatList(V2))
r46: a__U96(tt()) -> tt()
r47: a__isNat(|0|()) -> tt()
r48: a__isNat(length(V1)) -> a__U11(a__isNatIListKind(V1),V1)
r49: a__isNat(s(V1)) -> a__U21(a__isNatKind(V1),V1)
r50: a__isNatIList(V) -> a__U31(a__isNatIListKind(V),V)
r51: a__isNatIList(zeros()) -> tt()
r52: a__isNatIList(cons(V1,V2)) -> a__U41(a__isNatKind(V1),V1,V2)
r53: a__isNatIListKind(nil()) -> tt()
r54: a__isNatIListKind(zeros()) -> tt()
r55: a__isNatIListKind(cons(V1,V2)) -> a__U51(a__isNatKind(V1),V2)
r56: a__isNatIListKind(take(V1,V2)) -> a__U61(a__isNatKind(V1),V2)
r57: a__isNatKind(|0|()) -> tt()
r58: a__isNatKind(length(V1)) -> a__U71(a__isNatIListKind(V1))
r59: a__isNatKind(s(V1)) -> a__U81(a__isNatKind(V1))
r60: a__isNatList(nil()) -> tt()
r61: a__isNatList(cons(V1,V2)) -> a__U91(a__isNatKind(V1),V1,V2)
r62: a__isNatList(take(V1,V2)) -> a__U101(a__isNatKind(V1),V1,V2)
r63: a__length(nil()) -> |0|()
r64: a__length(cons(N,L)) -> a__U111(a__isNatList(L),L,N)
r65: a__take(|0|(),IL) -> a__U121(a__isNatIList(IL),IL)
r66: a__take(s(M),cons(N,IL)) -> a__U131(a__isNatIList(IL),IL,M,N)
r67: mark(zeros()) -> a__zeros()
r68: mark(U101(X1,X2,X3)) -> a__U101(mark(X1),X2,X3)
r69: mark(U102(X1,X2,X3)) -> a__U102(mark(X1),X2,X3)
r70: mark(isNatKind(X)) -> a__isNatKind(X)
r71: mark(U103(X1,X2,X3)) -> a__U103(mark(X1),X2,X3)
r72: mark(isNatIListKind(X)) -> a__isNatIListKind(X)
r73: mark(U104(X1,X2,X3)) -> a__U104(mark(X1),X2,X3)
r74: mark(U105(X1,X2)) -> a__U105(mark(X1),X2)
r75: mark(isNat(X)) -> a__isNat(X)
r76: mark(U106(X)) -> a__U106(mark(X))
r77: mark(isNatIList(X)) -> a__isNatIList(X)
r78: mark(U11(X1,X2)) -> a__U11(mark(X1),X2)
r79: mark(U12(X1,X2)) -> a__U12(mark(X1),X2)
r80: mark(U111(X1,X2,X3)) -> a__U111(mark(X1),X2,X3)
r81: mark(U112(X1,X2,X3)) -> a__U112(mark(X1),X2,X3)
r82: mark(U113(X1,X2,X3)) -> a__U113(mark(X1),X2,X3)
r83: mark(U114(X1,X2)) -> a__U114(mark(X1),X2)
r84: mark(length(X)) -> a__length(mark(X))
r85: mark(U13(X)) -> a__U13(mark(X))
r86: mark(isNatList(X)) -> a__isNatList(X)
r87: mark(U121(X1,X2)) -> a__U121(mark(X1),X2)
r88: mark(U122(X)) -> a__U122(mark(X))
r89: mark(U131(X1,X2,X3,X4)) -> a__U131(mark(X1),X2,X3,X4)
r90: mark(U132(X1,X2,X3,X4)) -> a__U132(mark(X1),X2,X3,X4)
r91: mark(U133(X1,X2,X3,X4)) -> a__U133(mark(X1),X2,X3,X4)
r92: mark(U134(X1,X2,X3,X4)) -> a__U134(mark(X1),X2,X3,X4)
r93: mark(U135(X1,X2,X3,X4)) -> a__U135(mark(X1),X2,X3,X4)
r94: mark(U136(X1,X2,X3,X4)) -> a__U136(mark(X1),X2,X3,X4)
r95: mark(take(X1,X2)) -> a__take(mark(X1),mark(X2))
r96: mark(U21(X1,X2)) -> a__U21(mark(X1),X2)
r97: mark(U22(X1,X2)) -> a__U22(mark(X1),X2)
r98: mark(U23(X)) -> a__U23(mark(X))
r99: mark(U31(X1,X2)) -> a__U31(mark(X1),X2)
r100: mark(U32(X1,X2)) -> a__U32(mark(X1),X2)
r101: mark(U33(X)) -> a__U33(mark(X))
r102: mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3)
r103: mark(U42(X1,X2,X3)) -> a__U42(mark(X1),X2,X3)
r104: mark(U43(X1,X2,X3)) -> a__U43(mark(X1),X2,X3)
r105: mark(U44(X1,X2,X3)) -> a__U44(mark(X1),X2,X3)
r106: mark(U45(X1,X2)) -> a__U45(mark(X1),X2)
r107: mark(U46(X)) -> a__U46(mark(X))
r108: mark(U51(X1,X2)) -> a__U51(mark(X1),X2)
r109: mark(U52(X)) -> a__U52(mark(X))
r110: mark(U61(X1,X2)) -> a__U61(mark(X1),X2)
r111: mark(U62(X)) -> a__U62(mark(X))
r112: mark(U71(X)) -> a__U71(mark(X))
r113: mark(U81(X)) -> a__U81(mark(X))
r114: mark(U91(X1,X2,X3)) -> a__U91(mark(X1),X2,X3)
r115: mark(U92(X1,X2,X3)) -> a__U92(mark(X1),X2,X3)
r116: mark(U93(X1,X2,X3)) -> a__U93(mark(X1),X2,X3)
r117: mark(U94(X1,X2,X3)) -> a__U94(mark(X1),X2,X3)
r118: mark(U95(X1,X2)) -> a__U95(mark(X1),X2)
r119: mark(U96(X)) -> a__U96(mark(X))
r120: mark(cons(X1,X2)) -> cons(mark(X1),X2)
r121: mark(|0|()) -> |0|()
r122: mark(tt()) -> tt()
r123: mark(s(X)) -> s(mark(X))
r124: mark(nil()) -> nil()
r125: a__zeros() -> zeros()
r126: a__U101(X1,X2,X3) -> U101(X1,X2,X3)
r127: a__U102(X1,X2,X3) -> U102(X1,X2,X3)
r128: a__isNatKind(X) -> isNatKind(X)
r129: a__U103(X1,X2,X3) -> U103(X1,X2,X3)
r130: a__isNatIListKind(X) -> isNatIListKind(X)
r131: a__U104(X1,X2,X3) -> U104(X1,X2,X3)
r132: a__U105(X1,X2) -> U105(X1,X2)
r133: a__isNat(X) -> isNat(X)
r134: a__U106(X) -> U106(X)
r135: a__isNatIList(X) -> isNatIList(X)
r136: a__U11(X1,X2) -> U11(X1,X2)
r137: a__U12(X1,X2) -> U12(X1,X2)
r138: a__U111(X1,X2,X3) -> U111(X1,X2,X3)
r139: a__U112(X1,X2,X3) -> U112(X1,X2,X3)
r140: a__U113(X1,X2,X3) -> U113(X1,X2,X3)
r141: a__U114(X1,X2) -> U114(X1,X2)
r142: a__length(X) -> length(X)
r143: a__U13(X) -> U13(X)
r144: a__isNatList(X) -> isNatList(X)
r145: a__U121(X1,X2) -> U121(X1,X2)
r146: a__U122(X) -> U122(X)
r147: a__U131(X1,X2,X3,X4) -> U131(X1,X2,X3,X4)
r148: a__U132(X1,X2,X3,X4) -> U132(X1,X2,X3,X4)
r149: a__U133(X1,X2,X3,X4) -> U133(X1,X2,X3,X4)
r150: a__U134(X1,X2,X3,X4) -> U134(X1,X2,X3,X4)
r151: a__U135(X1,X2,X3,X4) -> U135(X1,X2,X3,X4)
r152: a__U136(X1,X2,X3,X4) -> U136(X1,X2,X3,X4)
r153: a__take(X1,X2) -> take(X1,X2)
r154: a__U21(X1,X2) -> U21(X1,X2)
r155: a__U22(X1,X2) -> U22(X1,X2)
r156: a__U23(X) -> U23(X)
r157: a__U31(X1,X2) -> U31(X1,X2)
r158: a__U32(X1,X2) -> U32(X1,X2)
r159: a__U33(X) -> U33(X)
r160: a__U41(X1,X2,X3) -> U41(X1,X2,X3)
r161: a__U42(X1,X2,X3) -> U42(X1,X2,X3)
r162: a__U43(X1,X2,X3) -> U43(X1,X2,X3)
r163: a__U44(X1,X2,X3) -> U44(X1,X2,X3)
r164: a__U45(X1,X2) -> U45(X1,X2)
r165: a__U46(X) -> U46(X)
r166: a__U51(X1,X2) -> U51(X1,X2)
r167: a__U52(X) -> U52(X)
r168: a__U61(X1,X2) -> U61(X1,X2)
r169: a__U62(X) -> U62(X)
r170: a__U71(X) -> U71(X)
r171: a__U81(X) -> U81(X)
r172: a__U91(X1,X2,X3) -> U91(X1,X2,X3)
r173: a__U92(X1,X2,X3) -> U92(X1,X2,X3)
r174: a__U93(X1,X2,X3) -> U93(X1,X2,X3)
r175: a__U94(X1,X2,X3) -> U94(X1,X2,X3)
r176: a__U95(X1,X2) -> U95(X1,X2)
r177: a__U96(X) -> U96(X)

The set of usable rules consists of

  r35, r36, r37, r38, r39, r40, r53, r54, r55, r56, r57, r58, r59, r128, r130, r166, r167, r168, r169, r170, r171

Take the reduction pair:

  max/plus interpretations on natural numbers:
  
    a__isNatIListKind#_A(x1) = x1 + 1
    take_A(x1,x2) = max{x1 + 5, x2 + 5}
    a__isNatKind#_A(x1) = max{1, x1 - 5}
    length_A(x1) = x1 + 6
    a__U61#_A(x1,x2) = max{x1 - 3, x2 + 1}
    a__isNatKind_A(x1) = max{4, x1 - 1}
    tt_A = 4
    a__U52_A(x1) = 12
    a__U62_A(x1) = 8
    U52_A(x1) = 12
    U62_A(x1) = 8
    a__U51_A(x1,x2) = x1 + 8
    a__isNatIListKind_A(x1) = max{6, x1 + 3}
    a__U61_A(x1,x2) = max{8, x1 + 3}
    U51_A(x1,x2) = x1 + 8
    U61_A(x1,x2) = max{8, x1 + 3}
    a__U71_A(x1) = max{5, x1 - 1}
    a__U81_A(x1) = max{5, x1 + 4}
    nil_A = 0
    zeros_A = 0
    cons_A(x1,x2) = max{x1 + 9, x2 + 9}
    isNatIListKind_A(x1) = max{4, x1 + 3}
    U71_A(x1) = max{0, x1 - 1}
    U81_A(x1) = max{5, x1 + 4}
    |0|_A = 0
    s_A(x1) = x1 + 9
    isNatKind_A(x1) = max{0, x1 - 1}

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: a__isNatKind#(length(V1)) -> a__isNatIListKind#(V1)
p2: a__U61#(tt(),V2) -> a__isNatIListKind#(V2)

and R consists of:

r1: a__zeros() -> cons(|0|(),zeros())
r2: a__U101(tt(),V1,V2) -> a__U102(a__isNatKind(V1),V1,V2)
r3: a__U102(tt(),V1,V2) -> a__U103(a__isNatIListKind(V2),V1,V2)
r4: a__U103(tt(),V1,V2) -> a__U104(a__isNatIListKind(V2),V1,V2)
r5: a__U104(tt(),V1,V2) -> a__U105(a__isNat(V1),V2)
r6: a__U105(tt(),V2) -> a__U106(a__isNatIList(V2))
r7: a__U106(tt()) -> tt()
r8: a__U11(tt(),V1) -> a__U12(a__isNatIListKind(V1),V1)
r9: a__U111(tt(),L,N) -> a__U112(a__isNatIListKind(L),L,N)
r10: a__U112(tt(),L,N) -> a__U113(a__isNat(N),L,N)
r11: a__U113(tt(),L,N) -> a__U114(a__isNatKind(N),L)
r12: a__U114(tt(),L) -> s(a__length(mark(L)))
r13: a__U12(tt(),V1) -> a__U13(a__isNatList(V1))
r14: a__U121(tt(),IL) -> a__U122(a__isNatIListKind(IL))
r15: a__U122(tt()) -> nil()
r16: a__U13(tt()) -> tt()
r17: a__U131(tt(),IL,M,N) -> a__U132(a__isNatIListKind(IL),IL,M,N)
r18: a__U132(tt(),IL,M,N) -> a__U133(a__isNat(M),IL,M,N)
r19: a__U133(tt(),IL,M,N) -> a__U134(a__isNatKind(M),IL,M,N)
r20: a__U134(tt(),IL,M,N) -> a__U135(a__isNat(N),IL,M,N)
r21: a__U135(tt(),IL,M,N) -> a__U136(a__isNatKind(N),IL,M,N)
r22: a__U136(tt(),IL,M,N) -> cons(mark(N),take(M,IL))
r23: a__U21(tt(),V1) -> a__U22(a__isNatKind(V1),V1)
r24: a__U22(tt(),V1) -> a__U23(a__isNat(V1))
r25: a__U23(tt()) -> tt()
r26: a__U31(tt(),V) -> a__U32(a__isNatIListKind(V),V)
r27: a__U32(tt(),V) -> a__U33(a__isNatList(V))
r28: a__U33(tt()) -> tt()
r29: a__U41(tt(),V1,V2) -> a__U42(a__isNatKind(V1),V1,V2)
r30: a__U42(tt(),V1,V2) -> a__U43(a__isNatIListKind(V2),V1,V2)
r31: a__U43(tt(),V1,V2) -> a__U44(a__isNatIListKind(V2),V1,V2)
r32: a__U44(tt(),V1,V2) -> a__U45(a__isNat(V1),V2)
r33: a__U45(tt(),V2) -> a__U46(a__isNatIList(V2))
r34: a__U46(tt()) -> tt()
r35: a__U51(tt(),V2) -> a__U52(a__isNatIListKind(V2))
r36: a__U52(tt()) -> tt()
r37: a__U61(tt(),V2) -> a__U62(a__isNatIListKind(V2))
r38: a__U62(tt()) -> tt()
r39: a__U71(tt()) -> tt()
r40: a__U81(tt()) -> tt()
r41: a__U91(tt(),V1,V2) -> a__U92(a__isNatKind(V1),V1,V2)
r42: a__U92(tt(),V1,V2) -> a__U93(a__isNatIListKind(V2),V1,V2)
r43: a__U93(tt(),V1,V2) -> a__U94(a__isNatIListKind(V2),V1,V2)
r44: a__U94(tt(),V1,V2) -> a__U95(a__isNat(V1),V2)
r45: a__U95(tt(),V2) -> a__U96(a__isNatList(V2))
r46: a__U96(tt()) -> tt()
r47: a__isNat(|0|()) -> tt()
r48: a__isNat(length(V1)) -> a__U11(a__isNatIListKind(V1),V1)
r49: a__isNat(s(V1)) -> a__U21(a__isNatKind(V1),V1)
r50: a__isNatIList(V) -> a__U31(a__isNatIListKind(V),V)
r51: a__isNatIList(zeros()) -> tt()
r52: a__isNatIList(cons(V1,V2)) -> a__U41(a__isNatKind(V1),V1,V2)
r53: a__isNatIListKind(nil()) -> tt()
r54: a__isNatIListKind(zeros()) -> tt()
r55: a__isNatIListKind(cons(V1,V2)) -> a__U51(a__isNatKind(V1),V2)
r56: a__isNatIListKind(take(V1,V2)) -> a__U61(a__isNatKind(V1),V2)
r57: a__isNatKind(|0|()) -> tt()
r58: a__isNatKind(length(V1)) -> a__U71(a__isNatIListKind(V1))
r59: a__isNatKind(s(V1)) -> a__U81(a__isNatKind(V1))
r60: a__isNatList(nil()) -> tt()
r61: a__isNatList(cons(V1,V2)) -> a__U91(a__isNatKind(V1),V1,V2)
r62: a__isNatList(take(V1,V2)) -> a__U101(a__isNatKind(V1),V1,V2)
r63: a__length(nil()) -> |0|()
r64: a__length(cons(N,L)) -> a__U111(a__isNatList(L),L,N)
r65: a__take(|0|(),IL) -> a__U121(a__isNatIList(IL),IL)
r66: a__take(s(M),cons(N,IL)) -> a__U131(a__isNatIList(IL),IL,M,N)
r67: mark(zeros()) -> a__zeros()
r68: mark(U101(X1,X2,X3)) -> a__U101(mark(X1),X2,X3)
r69: mark(U102(X1,X2,X3)) -> a__U102(mark(X1),X2,X3)
r70: mark(isNatKind(X)) -> a__isNatKind(X)
r71: mark(U103(X1,X2,X3)) -> a__U103(mark(X1),X2,X3)
r72: mark(isNatIListKind(X)) -> a__isNatIListKind(X)
r73: mark(U104(X1,X2,X3)) -> a__U104(mark(X1),X2,X3)
r74: mark(U105(X1,X2)) -> a__U105(mark(X1),X2)
r75: mark(isNat(X)) -> a__isNat(X)
r76: mark(U106(X)) -> a__U106(mark(X))
r77: mark(isNatIList(X)) -> a__isNatIList(X)
r78: mark(U11(X1,X2)) -> a__U11(mark(X1),X2)
r79: mark(U12(X1,X2)) -> a__U12(mark(X1),X2)
r80: mark(U111(X1,X2,X3)) -> a__U111(mark(X1),X2,X3)
r81: mark(U112(X1,X2,X3)) -> a__U112(mark(X1),X2,X3)
r82: mark(U113(X1,X2,X3)) -> a__U113(mark(X1),X2,X3)
r83: mark(U114(X1,X2)) -> a__U114(mark(X1),X2)
r84: mark(length(X)) -> a__length(mark(X))
r85: mark(U13(X)) -> a__U13(mark(X))
r86: mark(isNatList(X)) -> a__isNatList(X)
r87: mark(U121(X1,X2)) -> a__U121(mark(X1),X2)
r88: mark(U122(X)) -> a__U122(mark(X))
r89: mark(U131(X1,X2,X3,X4)) -> a__U131(mark(X1),X2,X3,X4)
r90: mark(U132(X1,X2,X3,X4)) -> a__U132(mark(X1),X2,X3,X4)
r91: mark(U133(X1,X2,X3,X4)) -> a__U133(mark(X1),X2,X3,X4)
r92: mark(U134(X1,X2,X3,X4)) -> a__U134(mark(X1),X2,X3,X4)
r93: mark(U135(X1,X2,X3,X4)) -> a__U135(mark(X1),X2,X3,X4)
r94: mark(U136(X1,X2,X3,X4)) -> a__U136(mark(X1),X2,X3,X4)
r95: mark(take(X1,X2)) -> a__take(mark(X1),mark(X2))
r96: mark(U21(X1,X2)) -> a__U21(mark(X1),X2)
r97: mark(U22(X1,X2)) -> a__U22(mark(X1),X2)
r98: mark(U23(X)) -> a__U23(mark(X))
r99: mark(U31(X1,X2)) -> a__U31(mark(X1),X2)
r100: mark(U32(X1,X2)) -> a__U32(mark(X1),X2)
r101: mark(U33(X)) -> a__U33(mark(X))
r102: mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3)
r103: mark(U42(X1,X2,X3)) -> a__U42(mark(X1),X2,X3)
r104: mark(U43(X1,X2,X3)) -> a__U43(mark(X1),X2,X3)
r105: mark(U44(X1,X2,X3)) -> a__U44(mark(X1),X2,X3)
r106: mark(U45(X1,X2)) -> a__U45(mark(X1),X2)
r107: mark(U46(X)) -> a__U46(mark(X))
r108: mark(U51(X1,X2)) -> a__U51(mark(X1),X2)
r109: mark(U52(X)) -> a__U52(mark(X))
r110: mark(U61(X1,X2)) -> a__U61(mark(X1),X2)
r111: mark(U62(X)) -> a__U62(mark(X))
r112: mark(U71(X)) -> a__U71(mark(X))
r113: mark(U81(X)) -> a__U81(mark(X))
r114: mark(U91(X1,X2,X3)) -> a__U91(mark(X1),X2,X3)
r115: mark(U92(X1,X2,X3)) -> a__U92(mark(X1),X2,X3)
r116: mark(U93(X1,X2,X3)) -> a__U93(mark(X1),X2,X3)
r117: mark(U94(X1,X2,X3)) -> a__U94(mark(X1),X2,X3)
r118: mark(U95(X1,X2)) -> a__U95(mark(X1),X2)
r119: mark(U96(X)) -> a__U96(mark(X))
r120: mark(cons(X1,X2)) -> cons(mark(X1),X2)
r121: mark(|0|()) -> |0|()
r122: mark(tt()) -> tt()
r123: mark(s(X)) -> s(mark(X))
r124: mark(nil()) -> nil()
r125: a__zeros() -> zeros()
r126: a__U101(X1,X2,X3) -> U101(X1,X2,X3)
r127: a__U102(X1,X2,X3) -> U102(X1,X2,X3)
r128: a__isNatKind(X) -> isNatKind(X)
r129: a__U103(X1,X2,X3) -> U103(X1,X2,X3)
r130: a__isNatIListKind(X) -> isNatIListKind(X)
r131: a__U104(X1,X2,X3) -> U104(X1,X2,X3)
r132: a__U105(X1,X2) -> U105(X1,X2)
r133: a__isNat(X) -> isNat(X)
r134: a__U106(X) -> U106(X)
r135: a__isNatIList(X) -> isNatIList(X)
r136: a__U11(X1,X2) -> U11(X1,X2)
r137: a__U12(X1,X2) -> U12(X1,X2)
r138: a__U111(X1,X2,X3) -> U111(X1,X2,X3)
r139: a__U112(X1,X2,X3) -> U112(X1,X2,X3)
r140: a__U113(X1,X2,X3) -> U113(X1,X2,X3)
r141: a__U114(X1,X2) -> U114(X1,X2)
r142: a__length(X) -> length(X)
r143: a__U13(X) -> U13(X)
r144: a__isNatList(X) -> isNatList(X)
r145: a__U121(X1,X2) -> U121(X1,X2)
r146: a__U122(X) -> U122(X)
r147: a__U131(X1,X2,X3,X4) -> U131(X1,X2,X3,X4)
r148: a__U132(X1,X2,X3,X4) -> U132(X1,X2,X3,X4)
r149: a__U133(X1,X2,X3,X4) -> U133(X1,X2,X3,X4)
r150: a__U134(X1,X2,X3,X4) -> U134(X1,X2,X3,X4)
r151: a__U135(X1,X2,X3,X4) -> U135(X1,X2,X3,X4)
r152: a__U136(X1,X2,X3,X4) -> U136(X1,X2,X3,X4)
r153: a__take(X1,X2) -> take(X1,X2)
r154: a__U21(X1,X2) -> U21(X1,X2)
r155: a__U22(X1,X2) -> U22(X1,X2)
r156: a__U23(X) -> U23(X)
r157: a__U31(X1,X2) -> U31(X1,X2)
r158: a__U32(X1,X2) -> U32(X1,X2)
r159: a__U33(X) -> U33(X)
r160: a__U41(X1,X2,X3) -> U41(X1,X2,X3)
r161: a__U42(X1,X2,X3) -> U42(X1,X2,X3)
r162: a__U43(X1,X2,X3) -> U43(X1,X2,X3)
r163: a__U44(X1,X2,X3) -> U44(X1,X2,X3)
r164: a__U45(X1,X2) -> U45(X1,X2)
r165: a__U46(X) -> U46(X)
r166: a__U51(X1,X2) -> U51(X1,X2)
r167: a__U52(X) -> U52(X)
r168: a__U61(X1,X2) -> U61(X1,X2)
r169: a__U62(X) -> U62(X)
r170: a__U71(X) -> U71(X)
r171: a__U81(X) -> U81(X)
r172: a__U91(X1,X2,X3) -> U91(X1,X2,X3)
r173: a__U92(X1,X2,X3) -> U92(X1,X2,X3)
r174: a__U93(X1,X2,X3) -> U93(X1,X2,X3)
r175: a__U94(X1,X2,X3) -> U94(X1,X2,X3)
r176: a__U95(X1,X2) -> U95(X1,X2)
r177: a__U96(X) -> U96(X)

The estimated dependency graph contains the following SCCs:

  (no SCCs)