YES

We show the termination of the TRS R:

  a__zeros() -> cons(|0|(),zeros())
  a__U11(tt()) -> tt()
  a__U21(tt()) -> tt()
  a__U31(tt()) -> tt()
  a__U41(tt(),V2) -> a__U42(a__isNatIList(V2))
  a__U42(tt()) -> tt()
  a__U51(tt(),V2) -> a__U52(a__isNatList(V2))
  a__U52(tt()) -> tt()
  a__U61(tt(),V2) -> a__U62(a__isNatIList(V2))
  a__U62(tt()) -> tt()
  a__U71(tt(),L,N) -> a__U72(a__isNat(N),L)
  a__U72(tt(),L) -> s(a__length(mark(L)))
  a__U81(tt()) -> nil()
  a__U91(tt(),IL,M,N) -> a__U92(a__isNat(M),IL,M,N)
  a__U92(tt(),IL,M,N) -> a__U93(a__isNat(N),IL,M,N)
  a__U93(tt(),IL,M,N) -> cons(mark(N),take(M,IL))
  a__isNat(|0|()) -> tt()
  a__isNat(length(V1)) -> a__U11(a__isNatList(V1))
  a__isNat(s(V1)) -> a__U21(a__isNat(V1))
  a__isNatIList(V) -> a__U31(a__isNatList(V))
  a__isNatIList(zeros()) -> tt()
  a__isNatIList(cons(V1,V2)) -> a__U41(a__isNat(V1),V2)
  a__isNatList(nil()) -> tt()
  a__isNatList(cons(V1,V2)) -> a__U51(a__isNat(V1),V2)
  a__isNatList(take(V1,V2)) -> a__U61(a__isNat(V1),V2)
  a__length(nil()) -> |0|()
  a__length(cons(N,L)) -> a__U71(a__isNatList(L),L,N)
  a__take(|0|(),IL) -> a__U81(a__isNatIList(IL))
  a__take(s(M),cons(N,IL)) -> a__U91(a__isNatIList(IL),IL,M,N)
  mark(zeros()) -> a__zeros()
  mark(U11(X)) -> a__U11(mark(X))
  mark(U21(X)) -> a__U21(mark(X))
  mark(U31(X)) -> a__U31(mark(X))
  mark(U41(X1,X2)) -> a__U41(mark(X1),X2)
  mark(U42(X)) -> a__U42(mark(X))
  mark(isNatIList(X)) -> a__isNatIList(X)
  mark(U51(X1,X2)) -> a__U51(mark(X1),X2)
  mark(U52(X)) -> a__U52(mark(X))
  mark(isNatList(X)) -> a__isNatList(X)
  mark(U61(X1,X2)) -> a__U61(mark(X1),X2)
  mark(U62(X)) -> a__U62(mark(X))
  mark(U71(X1,X2,X3)) -> a__U71(mark(X1),X2,X3)
  mark(U72(X1,X2)) -> a__U72(mark(X1),X2)
  mark(isNat(X)) -> a__isNat(X)
  mark(length(X)) -> a__length(mark(X))
  mark(U81(X)) -> a__U81(mark(X))
  mark(U91(X1,X2,X3,X4)) -> a__U91(mark(X1),X2,X3,X4)
  mark(U92(X1,X2,X3,X4)) -> a__U92(mark(X1),X2,X3,X4)
  mark(U93(X1,X2,X3,X4)) -> a__U93(mark(X1),X2,X3,X4)
  mark(take(X1,X2)) -> a__take(mark(X1),mark(X2))
  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__U11(X) -> U11(X)
  a__U21(X) -> U21(X)
  a__U31(X) -> U31(X)
  a__U41(X1,X2) -> U41(X1,X2)
  a__U42(X) -> U42(X)
  a__isNatIList(X) -> isNatIList(X)
  a__U51(X1,X2) -> U51(X1,X2)
  a__U52(X) -> U52(X)
  a__isNatList(X) -> isNatList(X)
  a__U61(X1,X2) -> U61(X1,X2)
  a__U62(X) -> U62(X)
  a__U71(X1,X2,X3) -> U71(X1,X2,X3)
  a__U72(X1,X2) -> U72(X1,X2)
  a__isNat(X) -> isNat(X)
  a__length(X) -> length(X)
  a__U81(X) -> U81(X)
  a__U91(X1,X2,X3,X4) -> U91(X1,X2,X3,X4)
  a__U92(X1,X2,X3,X4) -> U92(X1,X2,X3,X4)
  a__U93(X1,X2,X3,X4) -> U93(X1,X2,X3,X4)
  a__take(X1,X2) -> take(X1,X2)

-- SCC decomposition.

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

p1: a__U41#(tt(),V2) -> a__U42#(a__isNatIList(V2))
p2: a__U41#(tt(),V2) -> a__isNatIList#(V2)
p3: a__U51#(tt(),V2) -> a__U52#(a__isNatList(V2))
p4: a__U51#(tt(),V2) -> a__isNatList#(V2)
p5: a__U61#(tt(),V2) -> a__U62#(a__isNatIList(V2))
p6: a__U61#(tt(),V2) -> a__isNatIList#(V2)
p7: a__U71#(tt(),L,N) -> a__U72#(a__isNat(N),L)
p8: a__U71#(tt(),L,N) -> a__isNat#(N)
p9: a__U72#(tt(),L) -> a__length#(mark(L))
p10: a__U72#(tt(),L) -> mark#(L)
p11: a__U91#(tt(),IL,M,N) -> a__U92#(a__isNat(M),IL,M,N)
p12: a__U91#(tt(),IL,M,N) -> a__isNat#(M)
p13: a__U92#(tt(),IL,M,N) -> a__U93#(a__isNat(N),IL,M,N)
p14: a__U92#(tt(),IL,M,N) -> a__isNat#(N)
p15: a__U93#(tt(),IL,M,N) -> mark#(N)
p16: a__isNat#(length(V1)) -> a__U11#(a__isNatList(V1))
p17: a__isNat#(length(V1)) -> a__isNatList#(V1)
p18: a__isNat#(s(V1)) -> a__U21#(a__isNat(V1))
p19: a__isNat#(s(V1)) -> a__isNat#(V1)
p20: a__isNatIList#(V) -> a__U31#(a__isNatList(V))
p21: a__isNatIList#(V) -> a__isNatList#(V)
p22: a__isNatIList#(cons(V1,V2)) -> a__U41#(a__isNat(V1),V2)
p23: a__isNatIList#(cons(V1,V2)) -> a__isNat#(V1)
p24: a__isNatList#(cons(V1,V2)) -> a__U51#(a__isNat(V1),V2)
p25: a__isNatList#(cons(V1,V2)) -> a__isNat#(V1)
p26: a__isNatList#(take(V1,V2)) -> a__U61#(a__isNat(V1),V2)
p27: a__isNatList#(take(V1,V2)) -> a__isNat#(V1)
p28: a__length#(cons(N,L)) -> a__U71#(a__isNatList(L),L,N)
p29: a__length#(cons(N,L)) -> a__isNatList#(L)
p30: a__take#(|0|(),IL) -> a__U81#(a__isNatIList(IL))
p31: a__take#(|0|(),IL) -> a__isNatIList#(IL)
p32: a__take#(s(M),cons(N,IL)) -> a__U91#(a__isNatIList(IL),IL,M,N)
p33: a__take#(s(M),cons(N,IL)) -> a__isNatIList#(IL)
p34: mark#(zeros()) -> a__zeros#()
p35: mark#(U11(X)) -> a__U11#(mark(X))
p36: mark#(U11(X)) -> mark#(X)
p37: mark#(U21(X)) -> a__U21#(mark(X))
p38: mark#(U21(X)) -> mark#(X)
p39: mark#(U31(X)) -> a__U31#(mark(X))
p40: mark#(U31(X)) -> mark#(X)
p41: mark#(U41(X1,X2)) -> a__U41#(mark(X1),X2)
p42: mark#(U41(X1,X2)) -> mark#(X1)
p43: mark#(U42(X)) -> a__U42#(mark(X))
p44: mark#(U42(X)) -> mark#(X)
p45: mark#(isNatIList(X)) -> a__isNatIList#(X)
p46: mark#(U51(X1,X2)) -> a__U51#(mark(X1),X2)
p47: mark#(U51(X1,X2)) -> mark#(X1)
p48: mark#(U52(X)) -> a__U52#(mark(X))
p49: mark#(U52(X)) -> mark#(X)
p50: mark#(isNatList(X)) -> a__isNatList#(X)
p51: mark#(U61(X1,X2)) -> a__U61#(mark(X1),X2)
p52: mark#(U61(X1,X2)) -> mark#(X1)
p53: mark#(U62(X)) -> a__U62#(mark(X))
p54: mark#(U62(X)) -> mark#(X)
p55: mark#(U71(X1,X2,X3)) -> a__U71#(mark(X1),X2,X3)
p56: mark#(U71(X1,X2,X3)) -> mark#(X1)
p57: mark#(U72(X1,X2)) -> a__U72#(mark(X1),X2)
p58: mark#(U72(X1,X2)) -> mark#(X1)
p59: mark#(isNat(X)) -> a__isNat#(X)
p60: mark#(length(X)) -> a__length#(mark(X))
p61: mark#(length(X)) -> mark#(X)
p62: mark#(U81(X)) -> a__U81#(mark(X))
p63: mark#(U81(X)) -> mark#(X)
p64: mark#(U91(X1,X2,X3,X4)) -> a__U91#(mark(X1),X2,X3,X4)
p65: mark#(U91(X1,X2,X3,X4)) -> mark#(X1)
p66: mark#(U92(X1,X2,X3,X4)) -> a__U92#(mark(X1),X2,X3,X4)
p67: mark#(U92(X1,X2,X3,X4)) -> mark#(X1)
p68: mark#(U93(X1,X2,X3,X4)) -> a__U93#(mark(X1),X2,X3,X4)
p69: mark#(U93(X1,X2,X3,X4)) -> mark#(X1)
p70: mark#(take(X1,X2)) -> a__take#(mark(X1),mark(X2))
p71: mark#(take(X1,X2)) -> mark#(X1)
p72: mark#(take(X1,X2)) -> mark#(X2)
p73: mark#(cons(X1,X2)) -> mark#(X1)
p74: mark#(s(X)) -> mark#(X)

and R consists of:

r1: a__zeros() -> cons(|0|(),zeros())
r2: a__U11(tt()) -> tt()
r3: a__U21(tt()) -> tt()
r4: a__U31(tt()) -> tt()
r5: a__U41(tt(),V2) -> a__U42(a__isNatIList(V2))
r6: a__U42(tt()) -> tt()
r7: a__U51(tt(),V2) -> a__U52(a__isNatList(V2))
r8: a__U52(tt()) -> tt()
r9: a__U61(tt(),V2) -> a__U62(a__isNatIList(V2))
r10: a__U62(tt()) -> tt()
r11: a__U71(tt(),L,N) -> a__U72(a__isNat(N),L)
r12: a__U72(tt(),L) -> s(a__length(mark(L)))
r13: a__U81(tt()) -> nil()
r14: a__U91(tt(),IL,M,N) -> a__U92(a__isNat(M),IL,M,N)
r15: a__U92(tt(),IL,M,N) -> a__U93(a__isNat(N),IL,M,N)
r16: a__U93(tt(),IL,M,N) -> cons(mark(N),take(M,IL))
r17: a__isNat(|0|()) -> tt()
r18: a__isNat(length(V1)) -> a__U11(a__isNatList(V1))
r19: a__isNat(s(V1)) -> a__U21(a__isNat(V1))
r20: a__isNatIList(V) -> a__U31(a__isNatList(V))
r21: a__isNatIList(zeros()) -> tt()
r22: a__isNatIList(cons(V1,V2)) -> a__U41(a__isNat(V1),V2)
r23: a__isNatList(nil()) -> tt()
r24: a__isNatList(cons(V1,V2)) -> a__U51(a__isNat(V1),V2)
r25: a__isNatList(take(V1,V2)) -> a__U61(a__isNat(V1),V2)
r26: a__length(nil()) -> |0|()
r27: a__length(cons(N,L)) -> a__U71(a__isNatList(L),L,N)
r28: a__take(|0|(),IL) -> a__U81(a__isNatIList(IL))
r29: a__take(s(M),cons(N,IL)) -> a__U91(a__isNatIList(IL),IL,M,N)
r30: mark(zeros()) -> a__zeros()
r31: mark(U11(X)) -> a__U11(mark(X))
r32: mark(U21(X)) -> a__U21(mark(X))
r33: mark(U31(X)) -> a__U31(mark(X))
r34: mark(U41(X1,X2)) -> a__U41(mark(X1),X2)
r35: mark(U42(X)) -> a__U42(mark(X))
r36: mark(isNatIList(X)) -> a__isNatIList(X)
r37: mark(U51(X1,X2)) -> a__U51(mark(X1),X2)
r38: mark(U52(X)) -> a__U52(mark(X))
r39: mark(isNatList(X)) -> a__isNatList(X)
r40: mark(U61(X1,X2)) -> a__U61(mark(X1),X2)
r41: mark(U62(X)) -> a__U62(mark(X))
r42: mark(U71(X1,X2,X3)) -> a__U71(mark(X1),X2,X3)
r43: mark(U72(X1,X2)) -> a__U72(mark(X1),X2)
r44: mark(isNat(X)) -> a__isNat(X)
r45: mark(length(X)) -> a__length(mark(X))
r46: mark(U81(X)) -> a__U81(mark(X))
r47: mark(U91(X1,X2,X3,X4)) -> a__U91(mark(X1),X2,X3,X4)
r48: mark(U92(X1,X2,X3,X4)) -> a__U92(mark(X1),X2,X3,X4)
r49: mark(U93(X1,X2,X3,X4)) -> a__U93(mark(X1),X2,X3,X4)
r50: mark(take(X1,X2)) -> a__take(mark(X1),mark(X2))
r51: mark(cons(X1,X2)) -> cons(mark(X1),X2)
r52: mark(|0|()) -> |0|()
r53: mark(tt()) -> tt()
r54: mark(s(X)) -> s(mark(X))
r55: mark(nil()) -> nil()
r56: a__zeros() -> zeros()
r57: a__U11(X) -> U11(X)
r58: a__U21(X) -> U21(X)
r59: a__U31(X) -> U31(X)
r60: a__U41(X1,X2) -> U41(X1,X2)
r61: a__U42(X) -> U42(X)
r62: a__isNatIList(X) -> isNatIList(X)
r63: a__U51(X1,X2) -> U51(X1,X2)
r64: a__U52(X) -> U52(X)
r65: a__isNatList(X) -> isNatList(X)
r66: a__U61(X1,X2) -> U61(X1,X2)
r67: a__U62(X) -> U62(X)
r68: a__U71(X1,X2,X3) -> U71(X1,X2,X3)
r69: a__U72(X1,X2) -> U72(X1,X2)
r70: a__isNat(X) -> isNat(X)
r71: a__length(X) -> length(X)
r72: a__U81(X) -> U81(X)
r73: a__U91(X1,X2,X3,X4) -> U91(X1,X2,X3,X4)
r74: a__U92(X1,X2,X3,X4) -> U92(X1,X2,X3,X4)
r75: a__U93(X1,X2,X3,X4) -> U93(X1,X2,X3,X4)
r76: a__take(X1,X2) -> take(X1,X2)

The estimated dependency graph contains the following SCCs:

  {p7, p9, p10, p11, p13, p15, p28, p32, p36, p38, p40, p42, p44, p47, p49, p52, p54, p55, p56, p57, p58, p60, p61, p63, p64, p65, p66, p67, p68, p69, p70, p71, p72, p73, p74}
  {p2, p4, p6, p17, p19, p21, p22, p23, p24, p25, p26, p27}


-- Reduction pair.

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

p1: a__U93#(tt(),IL,M,N) -> mark#(N)
p2: mark#(s(X)) -> mark#(X)
p3: mark#(cons(X1,X2)) -> mark#(X1)
p4: mark#(take(X1,X2)) -> mark#(X2)
p5: mark#(take(X1,X2)) -> mark#(X1)
p6: mark#(take(X1,X2)) -> a__take#(mark(X1),mark(X2))
p7: a__take#(s(M),cons(N,IL)) -> a__U91#(a__isNatIList(IL),IL,M,N)
p8: a__U91#(tt(),IL,M,N) -> a__U92#(a__isNat(M),IL,M,N)
p9: a__U92#(tt(),IL,M,N) -> a__U93#(a__isNat(N),IL,M,N)
p10: mark#(U93(X1,X2,X3,X4)) -> mark#(X1)
p11: mark#(U93(X1,X2,X3,X4)) -> a__U93#(mark(X1),X2,X3,X4)
p12: mark#(U92(X1,X2,X3,X4)) -> mark#(X1)
p13: mark#(U92(X1,X2,X3,X4)) -> a__U92#(mark(X1),X2,X3,X4)
p14: mark#(U91(X1,X2,X3,X4)) -> mark#(X1)
p15: mark#(U91(X1,X2,X3,X4)) -> a__U91#(mark(X1),X2,X3,X4)
p16: mark#(U81(X)) -> mark#(X)
p17: mark#(length(X)) -> mark#(X)
p18: mark#(length(X)) -> a__length#(mark(X))
p19: a__length#(cons(N,L)) -> a__U71#(a__isNatList(L),L,N)
p20: a__U71#(tt(),L,N) -> a__U72#(a__isNat(N),L)
p21: a__U72#(tt(),L) -> mark#(L)
p22: mark#(U72(X1,X2)) -> mark#(X1)
p23: mark#(U72(X1,X2)) -> a__U72#(mark(X1),X2)
p24: a__U72#(tt(),L) -> a__length#(mark(L))
p25: mark#(U71(X1,X2,X3)) -> mark#(X1)
p26: mark#(U71(X1,X2,X3)) -> a__U71#(mark(X1),X2,X3)
p27: mark#(U62(X)) -> mark#(X)
p28: mark#(U61(X1,X2)) -> mark#(X1)
p29: mark#(U52(X)) -> mark#(X)
p30: mark#(U51(X1,X2)) -> mark#(X1)
p31: mark#(U42(X)) -> mark#(X)
p32: mark#(U41(X1,X2)) -> mark#(X1)
p33: mark#(U31(X)) -> mark#(X)
p34: mark#(U21(X)) -> mark#(X)
p35: mark#(U11(X)) -> mark#(X)

and R consists of:

r1: a__zeros() -> cons(|0|(),zeros())
r2: a__U11(tt()) -> tt()
r3: a__U21(tt()) -> tt()
r4: a__U31(tt()) -> tt()
r5: a__U41(tt(),V2) -> a__U42(a__isNatIList(V2))
r6: a__U42(tt()) -> tt()
r7: a__U51(tt(),V2) -> a__U52(a__isNatList(V2))
r8: a__U52(tt()) -> tt()
r9: a__U61(tt(),V2) -> a__U62(a__isNatIList(V2))
r10: a__U62(tt()) -> tt()
r11: a__U71(tt(),L,N) -> a__U72(a__isNat(N),L)
r12: a__U72(tt(),L) -> s(a__length(mark(L)))
r13: a__U81(tt()) -> nil()
r14: a__U91(tt(),IL,M,N) -> a__U92(a__isNat(M),IL,M,N)
r15: a__U92(tt(),IL,M,N) -> a__U93(a__isNat(N),IL,M,N)
r16: a__U93(tt(),IL,M,N) -> cons(mark(N),take(M,IL))
r17: a__isNat(|0|()) -> tt()
r18: a__isNat(length(V1)) -> a__U11(a__isNatList(V1))
r19: a__isNat(s(V1)) -> a__U21(a__isNat(V1))
r20: a__isNatIList(V) -> a__U31(a__isNatList(V))
r21: a__isNatIList(zeros()) -> tt()
r22: a__isNatIList(cons(V1,V2)) -> a__U41(a__isNat(V1),V2)
r23: a__isNatList(nil()) -> tt()
r24: a__isNatList(cons(V1,V2)) -> a__U51(a__isNat(V1),V2)
r25: a__isNatList(take(V1,V2)) -> a__U61(a__isNat(V1),V2)
r26: a__length(nil()) -> |0|()
r27: a__length(cons(N,L)) -> a__U71(a__isNatList(L),L,N)
r28: a__take(|0|(),IL) -> a__U81(a__isNatIList(IL))
r29: a__take(s(M),cons(N,IL)) -> a__U91(a__isNatIList(IL),IL,M,N)
r30: mark(zeros()) -> a__zeros()
r31: mark(U11(X)) -> a__U11(mark(X))
r32: mark(U21(X)) -> a__U21(mark(X))
r33: mark(U31(X)) -> a__U31(mark(X))
r34: mark(U41(X1,X2)) -> a__U41(mark(X1),X2)
r35: mark(U42(X)) -> a__U42(mark(X))
r36: mark(isNatIList(X)) -> a__isNatIList(X)
r37: mark(U51(X1,X2)) -> a__U51(mark(X1),X2)
r38: mark(U52(X)) -> a__U52(mark(X))
r39: mark(isNatList(X)) -> a__isNatList(X)
r40: mark(U61(X1,X2)) -> a__U61(mark(X1),X2)
r41: mark(U62(X)) -> a__U62(mark(X))
r42: mark(U71(X1,X2,X3)) -> a__U71(mark(X1),X2,X3)
r43: mark(U72(X1,X2)) -> a__U72(mark(X1),X2)
r44: mark(isNat(X)) -> a__isNat(X)
r45: mark(length(X)) -> a__length(mark(X))
r46: mark(U81(X)) -> a__U81(mark(X))
r47: mark(U91(X1,X2,X3,X4)) -> a__U91(mark(X1),X2,X3,X4)
r48: mark(U92(X1,X2,X3,X4)) -> a__U92(mark(X1),X2,X3,X4)
r49: mark(U93(X1,X2,X3,X4)) -> a__U93(mark(X1),X2,X3,X4)
r50: mark(take(X1,X2)) -> a__take(mark(X1),mark(X2))
r51: mark(cons(X1,X2)) -> cons(mark(X1),X2)
r52: mark(|0|()) -> |0|()
r53: mark(tt()) -> tt()
r54: mark(s(X)) -> s(mark(X))
r55: mark(nil()) -> nil()
r56: a__zeros() -> zeros()
r57: a__U11(X) -> U11(X)
r58: a__U21(X) -> U21(X)
r59: a__U31(X) -> U31(X)
r60: a__U41(X1,X2) -> U41(X1,X2)
r61: a__U42(X) -> U42(X)
r62: a__isNatIList(X) -> isNatIList(X)
r63: a__U51(X1,X2) -> U51(X1,X2)
r64: a__U52(X) -> U52(X)
r65: a__isNatList(X) -> isNatList(X)
r66: a__U61(X1,X2) -> U61(X1,X2)
r67: a__U62(X) -> U62(X)
r68: a__U71(X1,X2,X3) -> U71(X1,X2,X3)
r69: a__U72(X1,X2) -> U72(X1,X2)
r70: a__isNat(X) -> isNat(X)
r71: a__length(X) -> length(X)
r72: a__U81(X) -> U81(X)
r73: a__U91(X1,X2,X3,X4) -> U91(X1,X2,X3,X4)
r74: a__U92(X1,X2,X3,X4) -> U92(X1,X2,X3,X4)
r75: a__U93(X1,X2,X3,X4) -> U93(X1,X2,X3,X4)
r76: a__take(X1,X2) -> take(X1,X2)

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

Take the reduction pair:

  lexicographic combination of reduction pairs:
  
    1. weighted path order
    
      base order:
    
        max/plus interpretations on natural numbers:
        
          a__U93#_A(x1,x2,x3,x4) = max{x1 - 2, x3 + 3, x4 + 88}
          tt_A = 157
          mark#_A(x1) = x1 + 87
          s_A(x1) = max{246, x1}
          cons_A(x1,x2) = max{400, x1 + 358, x2}
          take_A(x1,x2) = max{472, x1 + 320, x2 + 281}
          a__take#_A(x1,x2) = max{473, x1 + 281, x2 + 205}
          mark_A(x1) = max{40, x1}
          a__U91#_A(x1,x2,x3,x4) = max{x1 + 87, x2 + 5, x3 + 4, x4 + 360}
          a__isNatIList_A(x1) = max{474, x1 + 117}
          a__U92#_A(x1,x2,x3,x4) = max{x3 + 4, x4 + 359}
          a__isNat_A(x1) = x1 + 9
          U93_A(x1,x2,x3,x4) = max{473, x1 + 242, x2 + 281, x3 + 320, x4 + 358}
          U92_A(x1,x2,x3,x4) = max{473, x1 + 279, x2 + 281, x3 + 320, x4 + 358}
          U91_A(x1,x2,x3,x4) = max{681, x1 + 87, x2 + 281, x3 + 320, x4 + 474}
          U81_A(x1) = max{245, x1 + 88}
          length_A(x1) = max{530, x1 + 71}
          a__length#_A(x1) = x1 + 116
          a__U71#_A(x1,x2,x3) = max{x1 + 1, x2 + 116, x3 + 183}
          a__isNatList_A(x1) = max{472, x1 + 11}
          a__U72#_A(x1,x2) = max{157, x1 - 1, x2 + 116}
          U72_A(x1,x2) = max{530, x1 + 88, x2 + 71}
          U71_A(x1,x2,x3) = max{530, x1 + 58, x2 + 71, x3 + 97}
          U62_A(x1) = max{158, x1 + 3}
          U61_A(x1,x2) = max{482, x1 + 273, x2 + 159}
          U52_A(x1) = max{161, x1}
          U51_A(x1,x2) = max{472, x1 + 3, x2 + 11}
          U42_A(x1) = max{41, x1}
          U41_A(x1,x2) = max{474, x1, x2 + 117}
          U31_A(x1) = max{118, x1 + 1}
          U21_A(x1) = x1
          U11_A(x1) = max{158, x1 + 59}
          a__zeros_A = 602
          |0|_A = 243
          zeros_A = 602
          a__U11_A(x1) = max{158, x1 + 59}
          a__U21_A(x1) = max{8, x1}
          a__U31_A(x1) = max{118, x1 + 1}
          a__U41_A(x1,x2) = max{474, x1, x2 + 117}
          a__U42_A(x1) = max{41, x1}
          a__U51_A(x1,x2) = max{472, x1 + 3, x2 + 11}
          a__U52_A(x1) = max{161, x1}
          a__U61_A(x1,x2) = max{482, x1 + 273, x2 + 159}
          a__U62_A(x1) = max{158, x1 + 3}
          a__U71_A(x1,x2,x3) = max{530, x1 + 58, x2 + 71, x3 + 97}
          a__U72_A(x1,x2) = max{530, x1 + 88, x2 + 71}
          a__length_A(x1) = max{530, x1 + 71}
          a__U81_A(x1) = max{245, x1 + 88}
          nil_A = 244
          a__U91_A(x1,x2,x3,x4) = max{681, x1 + 87, x2 + 281, x3 + 320, x4 + 474}
          a__U92_A(x1,x2,x3,x4) = max{473, x1 + 279, x2 + 281, x3 + 320, x4 + 358}
          a__U93_A(x1,x2,x3,x4) = max{473, x1 + 242, x2 + 281, x3 + 320, x4 + 358}
          a__take_A(x1,x2) = max{472, x1 + 320, x2 + 281}
          isNatIList_A(x1) = max{474, x1 + 117}
          isNatList_A(x1) = max{472, x1 + 11}
          isNat_A(x1) = x1 + 9
    
      precedence:
      
        a__U93# = tt = mark# = s = cons = take = a__take# = mark = a__U91# = a__isNatIList = a__U92# = a__isNat = U93 = U92 = U91 = U81 = length = a__length# = a__U71# = a__isNatList = a__U72# = U72 = U71 = U62 = U61 = U52 = U51 = U42 = U41 = U31 = U21 = U11 = a__zeros = |0| = zeros = a__U11 = a__U21 = a__U31 = a__U41 = a__U42 = a__U51 = a__U52 = a__U61 = a__U62 = a__U71 = a__U72 = a__length = a__U81 = nil = a__U91 = a__U92 = a__U93 = a__take = isNatIList = isNatList = isNat
      
      partial status:
    
        pi(a__U93#) = []
        pi(tt) = []
        pi(mark#) = []
        pi(s) = []
        pi(cons) = []
        pi(take) = []
        pi(a__take#) = []
        pi(mark) = []
        pi(a__U91#) = []
        pi(a__isNatIList) = []
        pi(a__U92#) = []
        pi(a__isNat) = []
        pi(U93) = []
        pi(U92) = []
        pi(U91) = []
        pi(U81) = []
        pi(length) = []
        pi(a__length#) = []
        pi(a__U71#) = []
        pi(a__isNatList) = []
        pi(a__U72#) = []
        pi(U72) = []
        pi(U71) = []
        pi(U62) = []
        pi(U61) = []
        pi(U52) = []
        pi(U51) = []
        pi(U42) = []
        pi(U41) = []
        pi(U31) = []
        pi(U21) = []
        pi(U11) = []
        pi(a__zeros) = []
        pi(|0|) = []
        pi(zeros) = []
        pi(a__U11) = []
        pi(a__U21) = []
        pi(a__U31) = []
        pi(a__U41) = []
        pi(a__U42) = []
        pi(a__U51) = []
        pi(a__U52) = []
        pi(a__U61) = []
        pi(a__U62) = []
        pi(a__U71) = []
        pi(a__U72) = []
        pi(a__length) = []
        pi(a__U81) = []
        pi(nil) = []
        pi(a__U91) = []
        pi(a__U92) = []
        pi(a__U93) = []
        pi(a__take) = []
        pi(isNatIList) = []
        pi(isNatList) = []
        pi(isNat) = []
    
    2. weighted path order
    
      base order:
    
        max/plus interpretations on natural numbers:
        
          a__U93#_A(x1,x2,x3,x4) = 74
          tt_A = 95
          mark#_A(x1) = 74
          s_A(x1) = 73
          cons_A(x1,x2) = 102
          take_A(x1,x2) = 88
          a__take#_A(x1,x2) = 73
          mark_A(x1) = 136
          a__U91#_A(x1,x2,x3,x4) = 56
          a__isNatIList_A(x1) = 136
          a__U92#_A(x1,x2,x3,x4) = 55
          a__isNat_A(x1) = 133
          U93_A(x1,x2,x3,x4) = 91
          U92_A(x1,x2,x3,x4) = 74
          U91_A(x1,x2,x3,x4) = 19
          U81_A(x1) = 126
          length_A(x1) = 75
          a__length#_A(x1) = 137
          a__U71#_A(x1,x2,x3) = 137
          a__isNatList_A(x1) = 136
          a__U72#_A(x1,x2) = 137
          U72_A(x1,x2) = 126
          U71_A(x1,x2,x3) = 75
          U62_A(x1) = 73
          U61_A(x1,x2) = 73
          U52_A(x1) = 135
          U51_A(x1,x2) = 136
          U42_A(x1) = 1
          U41_A(x1,x2) = 3
          U31_A(x1) = 127
          U21_A(x1) = 64
          U11_A(x1) = 1
          a__zeros_A = 103
          |0|_A = 0
          zeros_A = 94
          a__U11_A(x1) = 12
          a__U21_A(x1) = 96
          a__U31_A(x1) = 127
          a__U41_A(x1,x2) = 101
          a__U42_A(x1) = 100
          a__U51_A(x1,x2) = 136
          a__U52_A(x1) = 135
          a__U61_A(x1,x2) = 95
          a__U62_A(x1) = 94
          a__U71_A(x1,x2,x3) = 134
          a__U72_A(x1,x2) = 126
          a__length_A(x1) = 135
          a__U81_A(x1) = 136
          nil_A = 94
          a__U91_A(x1,x2,x3,x4) = 136
          a__U92_A(x1,x2,x3,x4) = 136
          a__U93_A(x1,x2,x3,x4) = 102
          a__take_A(x1,x2) = 136
          isNatIList_A(x1) = 136
          isNatList_A(x1) = 126
          isNat_A(x1) = 133
    
      precedence:
      
        U41 = zeros > U61 > U93 > a__U93# = mark# = a__U92# > a__take# = a__U91# > mark > s > a__U62 > a__isNatList > a__U51 > U71 > isNatList > U81 = a__U81 = a__take > U91 = a__U91 > a__U93 > a__U92 > U92 > U62 > a__U61 > nil > a__length# = a__U71# = a__U72# = a__U11 > a__isNat > a__U41 > a__isNatIList > cons = a__U31 > a__U21 > U42 > a__U42 > tt > a__zeros = isNatIList > a__U52 > U52 > |0| > a__length > a__U72 > a__U71 > length > U72 > take = U51 = U31 = U21 > U11 > isNat
      
      partial status:
    
        pi(a__U93#) = []
        pi(tt) = []
        pi(mark#) = []
        pi(s) = []
        pi(cons) = []
        pi(take) = []
        pi(a__take#) = []
        pi(mark) = []
        pi(a__U91#) = []
        pi(a__isNatIList) = []
        pi(a__U92#) = []
        pi(a__isNat) = []
        pi(U93) = []
        pi(U92) = []
        pi(U91) = []
        pi(U81) = []
        pi(length) = []
        pi(a__length#) = []
        pi(a__U71#) = []
        pi(a__isNatList) = []
        pi(a__U72#) = []
        pi(U72) = []
        pi(U71) = []
        pi(U62) = []
        pi(U61) = []
        pi(U52) = []
        pi(U51) = []
        pi(U42) = []
        pi(U41) = []
        pi(U31) = []
        pi(U21) = []
        pi(U11) = []
        pi(a__zeros) = []
        pi(|0|) = []
        pi(zeros) = []
        pi(a__U11) = []
        pi(a__U21) = []
        pi(a__U31) = []
        pi(a__U41) = []
        pi(a__U42) = []
        pi(a__U51) = []
        pi(a__U52) = []
        pi(a__U61) = []
        pi(a__U62) = []
        pi(a__U71) = []
        pi(a__U72) = []
        pi(a__length) = []
        pi(a__U81) = []
        pi(nil) = []
        pi(a__U91) = []
        pi(a__U92) = []
        pi(a__U93) = []
        pi(a__take) = []
        pi(isNatIList) = []
        pi(isNatList) = []
        pi(isNat) = []
    

The next rules are strictly ordered:

  p9, p18, p21, p23, p26

We remove them from the problem.

-- SCC decomposition.

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

p1: a__U93#(tt(),IL,M,N) -> mark#(N)
p2: mark#(s(X)) -> mark#(X)
p3: mark#(cons(X1,X2)) -> mark#(X1)
p4: mark#(take(X1,X2)) -> mark#(X2)
p5: mark#(take(X1,X2)) -> mark#(X1)
p6: mark#(take(X1,X2)) -> a__take#(mark(X1),mark(X2))
p7: a__take#(s(M),cons(N,IL)) -> a__U91#(a__isNatIList(IL),IL,M,N)
p8: a__U91#(tt(),IL,M,N) -> a__U92#(a__isNat(M),IL,M,N)
p9: mark#(U93(X1,X2,X3,X4)) -> mark#(X1)
p10: mark#(U93(X1,X2,X3,X4)) -> a__U93#(mark(X1),X2,X3,X4)
p11: mark#(U92(X1,X2,X3,X4)) -> mark#(X1)
p12: mark#(U92(X1,X2,X3,X4)) -> a__U92#(mark(X1),X2,X3,X4)
p13: mark#(U91(X1,X2,X3,X4)) -> mark#(X1)
p14: mark#(U91(X1,X2,X3,X4)) -> a__U91#(mark(X1),X2,X3,X4)
p15: mark#(U81(X)) -> mark#(X)
p16: mark#(length(X)) -> mark#(X)
p17: a__length#(cons(N,L)) -> a__U71#(a__isNatList(L),L,N)
p18: a__U71#(tt(),L,N) -> a__U72#(a__isNat(N),L)
p19: mark#(U72(X1,X2)) -> mark#(X1)
p20: a__U72#(tt(),L) -> a__length#(mark(L))
p21: mark#(U71(X1,X2,X3)) -> mark#(X1)
p22: mark#(U62(X)) -> mark#(X)
p23: mark#(U61(X1,X2)) -> mark#(X1)
p24: mark#(U52(X)) -> mark#(X)
p25: mark#(U51(X1,X2)) -> mark#(X1)
p26: mark#(U42(X)) -> mark#(X)
p27: mark#(U41(X1,X2)) -> mark#(X1)
p28: mark#(U31(X)) -> mark#(X)
p29: mark#(U21(X)) -> mark#(X)
p30: mark#(U11(X)) -> mark#(X)

and R consists of:

r1: a__zeros() -> cons(|0|(),zeros())
r2: a__U11(tt()) -> tt()
r3: a__U21(tt()) -> tt()
r4: a__U31(tt()) -> tt()
r5: a__U41(tt(),V2) -> a__U42(a__isNatIList(V2))
r6: a__U42(tt()) -> tt()
r7: a__U51(tt(),V2) -> a__U52(a__isNatList(V2))
r8: a__U52(tt()) -> tt()
r9: a__U61(tt(),V2) -> a__U62(a__isNatIList(V2))
r10: a__U62(tt()) -> tt()
r11: a__U71(tt(),L,N) -> a__U72(a__isNat(N),L)
r12: a__U72(tt(),L) -> s(a__length(mark(L)))
r13: a__U81(tt()) -> nil()
r14: a__U91(tt(),IL,M,N) -> a__U92(a__isNat(M),IL,M,N)
r15: a__U92(tt(),IL,M,N) -> a__U93(a__isNat(N),IL,M,N)
r16: a__U93(tt(),IL,M,N) -> cons(mark(N),take(M,IL))
r17: a__isNat(|0|()) -> tt()
r18: a__isNat(length(V1)) -> a__U11(a__isNatList(V1))
r19: a__isNat(s(V1)) -> a__U21(a__isNat(V1))
r20: a__isNatIList(V) -> a__U31(a__isNatList(V))
r21: a__isNatIList(zeros()) -> tt()
r22: a__isNatIList(cons(V1,V2)) -> a__U41(a__isNat(V1),V2)
r23: a__isNatList(nil()) -> tt()
r24: a__isNatList(cons(V1,V2)) -> a__U51(a__isNat(V1),V2)
r25: a__isNatList(take(V1,V2)) -> a__U61(a__isNat(V1),V2)
r26: a__length(nil()) -> |0|()
r27: a__length(cons(N,L)) -> a__U71(a__isNatList(L),L,N)
r28: a__take(|0|(),IL) -> a__U81(a__isNatIList(IL))
r29: a__take(s(M),cons(N,IL)) -> a__U91(a__isNatIList(IL),IL,M,N)
r30: mark(zeros()) -> a__zeros()
r31: mark(U11(X)) -> a__U11(mark(X))
r32: mark(U21(X)) -> a__U21(mark(X))
r33: mark(U31(X)) -> a__U31(mark(X))
r34: mark(U41(X1,X2)) -> a__U41(mark(X1),X2)
r35: mark(U42(X)) -> a__U42(mark(X))
r36: mark(isNatIList(X)) -> a__isNatIList(X)
r37: mark(U51(X1,X2)) -> a__U51(mark(X1),X2)
r38: mark(U52(X)) -> a__U52(mark(X))
r39: mark(isNatList(X)) -> a__isNatList(X)
r40: mark(U61(X1,X2)) -> a__U61(mark(X1),X2)
r41: mark(U62(X)) -> a__U62(mark(X))
r42: mark(U71(X1,X2,X3)) -> a__U71(mark(X1),X2,X3)
r43: mark(U72(X1,X2)) -> a__U72(mark(X1),X2)
r44: mark(isNat(X)) -> a__isNat(X)
r45: mark(length(X)) -> a__length(mark(X))
r46: mark(U81(X)) -> a__U81(mark(X))
r47: mark(U91(X1,X2,X3,X4)) -> a__U91(mark(X1),X2,X3,X4)
r48: mark(U92(X1,X2,X3,X4)) -> a__U92(mark(X1),X2,X3,X4)
r49: mark(U93(X1,X2,X3,X4)) -> a__U93(mark(X1),X2,X3,X4)
r50: mark(take(X1,X2)) -> a__take(mark(X1),mark(X2))
r51: mark(cons(X1,X2)) -> cons(mark(X1),X2)
r52: mark(|0|()) -> |0|()
r53: mark(tt()) -> tt()
r54: mark(s(X)) -> s(mark(X))
r55: mark(nil()) -> nil()
r56: a__zeros() -> zeros()
r57: a__U11(X) -> U11(X)
r58: a__U21(X) -> U21(X)
r59: a__U31(X) -> U31(X)
r60: a__U41(X1,X2) -> U41(X1,X2)
r61: a__U42(X) -> U42(X)
r62: a__isNatIList(X) -> isNatIList(X)
r63: a__U51(X1,X2) -> U51(X1,X2)
r64: a__U52(X) -> U52(X)
r65: a__isNatList(X) -> isNatList(X)
r66: a__U61(X1,X2) -> U61(X1,X2)
r67: a__U62(X) -> U62(X)
r68: a__U71(X1,X2,X3) -> U71(X1,X2,X3)
r69: a__U72(X1,X2) -> U72(X1,X2)
r70: a__isNat(X) -> isNat(X)
r71: a__length(X) -> length(X)
r72: a__U81(X) -> U81(X)
r73: a__U91(X1,X2,X3,X4) -> U91(X1,X2,X3,X4)
r74: a__U92(X1,X2,X3,X4) -> U92(X1,X2,X3,X4)
r75: a__U93(X1,X2,X3,X4) -> U93(X1,X2,X3,X4)
r76: a__take(X1,X2) -> take(X1,X2)

The estimated dependency graph contains the following SCCs:

  {p1, p2, p3, p4, p5, p9, p10, p11, p13, p15, p16, p19, p21, p22, p23, p24, p25, p26, p27, p28, p29, p30}
  {p17, p18, p20}


-- Reduction pair.

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

p1: a__U93#(tt(),IL,M,N) -> mark#(N)
p2: mark#(U11(X)) -> mark#(X)
p3: mark#(U21(X)) -> mark#(X)
p4: mark#(U31(X)) -> mark#(X)
p5: mark#(U41(X1,X2)) -> mark#(X1)
p6: mark#(U42(X)) -> mark#(X)
p7: mark#(U51(X1,X2)) -> mark#(X1)
p8: mark#(U52(X)) -> mark#(X)
p9: mark#(U61(X1,X2)) -> mark#(X1)
p10: mark#(U62(X)) -> mark#(X)
p11: mark#(U71(X1,X2,X3)) -> mark#(X1)
p12: mark#(U72(X1,X2)) -> mark#(X1)
p13: mark#(length(X)) -> mark#(X)
p14: mark#(U81(X)) -> mark#(X)
p15: mark#(U91(X1,X2,X3,X4)) -> mark#(X1)
p16: mark#(U92(X1,X2,X3,X4)) -> mark#(X1)
p17: mark#(U93(X1,X2,X3,X4)) -> a__U93#(mark(X1),X2,X3,X4)
p18: mark#(U93(X1,X2,X3,X4)) -> mark#(X1)
p19: mark#(take(X1,X2)) -> mark#(X1)
p20: mark#(take(X1,X2)) -> mark#(X2)
p21: mark#(cons(X1,X2)) -> mark#(X1)
p22: mark#(s(X)) -> mark#(X)

and R consists of:

r1: a__zeros() -> cons(|0|(),zeros())
r2: a__U11(tt()) -> tt()
r3: a__U21(tt()) -> tt()
r4: a__U31(tt()) -> tt()
r5: a__U41(tt(),V2) -> a__U42(a__isNatIList(V2))
r6: a__U42(tt()) -> tt()
r7: a__U51(tt(),V2) -> a__U52(a__isNatList(V2))
r8: a__U52(tt()) -> tt()
r9: a__U61(tt(),V2) -> a__U62(a__isNatIList(V2))
r10: a__U62(tt()) -> tt()
r11: a__U71(tt(),L,N) -> a__U72(a__isNat(N),L)
r12: a__U72(tt(),L) -> s(a__length(mark(L)))
r13: a__U81(tt()) -> nil()
r14: a__U91(tt(),IL,M,N) -> a__U92(a__isNat(M),IL,M,N)
r15: a__U92(tt(),IL,M,N) -> a__U93(a__isNat(N),IL,M,N)
r16: a__U93(tt(),IL,M,N) -> cons(mark(N),take(M,IL))
r17: a__isNat(|0|()) -> tt()
r18: a__isNat(length(V1)) -> a__U11(a__isNatList(V1))
r19: a__isNat(s(V1)) -> a__U21(a__isNat(V1))
r20: a__isNatIList(V) -> a__U31(a__isNatList(V))
r21: a__isNatIList(zeros()) -> tt()
r22: a__isNatIList(cons(V1,V2)) -> a__U41(a__isNat(V1),V2)
r23: a__isNatList(nil()) -> tt()
r24: a__isNatList(cons(V1,V2)) -> a__U51(a__isNat(V1),V2)
r25: a__isNatList(take(V1,V2)) -> a__U61(a__isNat(V1),V2)
r26: a__length(nil()) -> |0|()
r27: a__length(cons(N,L)) -> a__U71(a__isNatList(L),L,N)
r28: a__take(|0|(),IL) -> a__U81(a__isNatIList(IL))
r29: a__take(s(M),cons(N,IL)) -> a__U91(a__isNatIList(IL),IL,M,N)
r30: mark(zeros()) -> a__zeros()
r31: mark(U11(X)) -> a__U11(mark(X))
r32: mark(U21(X)) -> a__U21(mark(X))
r33: mark(U31(X)) -> a__U31(mark(X))
r34: mark(U41(X1,X2)) -> a__U41(mark(X1),X2)
r35: mark(U42(X)) -> a__U42(mark(X))
r36: mark(isNatIList(X)) -> a__isNatIList(X)
r37: mark(U51(X1,X2)) -> a__U51(mark(X1),X2)
r38: mark(U52(X)) -> a__U52(mark(X))
r39: mark(isNatList(X)) -> a__isNatList(X)
r40: mark(U61(X1,X2)) -> a__U61(mark(X1),X2)
r41: mark(U62(X)) -> a__U62(mark(X))
r42: mark(U71(X1,X2,X3)) -> a__U71(mark(X1),X2,X3)
r43: mark(U72(X1,X2)) -> a__U72(mark(X1),X2)
r44: mark(isNat(X)) -> a__isNat(X)
r45: mark(length(X)) -> a__length(mark(X))
r46: mark(U81(X)) -> a__U81(mark(X))
r47: mark(U91(X1,X2,X3,X4)) -> a__U91(mark(X1),X2,X3,X4)
r48: mark(U92(X1,X2,X3,X4)) -> a__U92(mark(X1),X2,X3,X4)
r49: mark(U93(X1,X2,X3,X4)) -> a__U93(mark(X1),X2,X3,X4)
r50: mark(take(X1,X2)) -> a__take(mark(X1),mark(X2))
r51: mark(cons(X1,X2)) -> cons(mark(X1),X2)
r52: mark(|0|()) -> |0|()
r53: mark(tt()) -> tt()
r54: mark(s(X)) -> s(mark(X))
r55: mark(nil()) -> nil()
r56: a__zeros() -> zeros()
r57: a__U11(X) -> U11(X)
r58: a__U21(X) -> U21(X)
r59: a__U31(X) -> U31(X)
r60: a__U41(X1,X2) -> U41(X1,X2)
r61: a__U42(X) -> U42(X)
r62: a__isNatIList(X) -> isNatIList(X)
r63: a__U51(X1,X2) -> U51(X1,X2)
r64: a__U52(X) -> U52(X)
r65: a__isNatList(X) -> isNatList(X)
r66: a__U61(X1,X2) -> U61(X1,X2)
r67: a__U62(X) -> U62(X)
r68: a__U71(X1,X2,X3) -> U71(X1,X2,X3)
r69: a__U72(X1,X2) -> U72(X1,X2)
r70: a__isNat(X) -> isNat(X)
r71: a__length(X) -> length(X)
r72: a__U81(X) -> U81(X)
r73: a__U91(X1,X2,X3,X4) -> U91(X1,X2,X3,X4)
r74: a__U92(X1,X2,X3,X4) -> U92(X1,X2,X3,X4)
r75: a__U93(X1,X2,X3,X4) -> U93(X1,X2,X3,X4)
r76: a__take(X1,X2) -> take(X1,X2)

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

Take the reduction pair:

  lexicographic combination of reduction pairs:
  
    1. weighted path order
    
      base order:
    
        max/plus interpretations on natural numbers:
        
          a__U93#_A(x1,x2,x3,x4) = max{x1 - 77, x2, x4 + 66}
          tt_A = 0
          mark#_A(x1) = max{65, x1 - 53}
          U11_A(x1) = max{147, x1 + 120}
          U21_A(x1) = max{24, x1}
          U31_A(x1) = max{58, x1 + 24}
          U41_A(x1,x2) = max{77, x1 + 54, x2 - 5}
          U42_A(x1) = max{53, x1}
          U51_A(x1,x2) = max{25, x1 + 1, x2 - 30}
          U52_A(x1) = max{1, x1}
          U61_A(x1,x2) = max{x1 + 41, x2 + 64}
          U62_A(x1) = max{42, x1 + 6}
          U71_A(x1,x2,x3) = max{x1 + 119, x2 + 89, x3 + 148}
          U72_A(x1,x2) = max{147, x1 + 124, x2 + 89}
          length_A(x1) = max{146, x1 + 89}
          U81_A(x1) = max{66, x1}
          U91_A(x1,x2,x3,x4) = max{x1 + 117, x2 + 116, x3 + 75, x4 + 176}
          U92_A(x1,x2,x3,x4) = max{x1 + 64, x2 + 116, x3 + 75, x4 + 176}
          U93_A(x1,x2,x3,x4) = max{176, x1, x2 + 116, x3 + 75, x4 + 142}
          mark_A(x1) = max{23, x1}
          take_A(x1,x2) = max{176, x1 + 75, x2 + 116}
          cons_A(x1,x2) = max{143, x1 + 60, x2}
          s_A(x1) = max{125, x1}
          a__zeros_A = 144
          |0|_A = 22
          zeros_A = 144
          a__U11_A(x1) = max{147, x1 + 120}
          a__U21_A(x1) = max{24, x1}
          a__U31_A(x1) = max{58, x1 + 24}
          a__U41_A(x1,x2) = max{77, x1 + 54, x2 - 5}
          a__U42_A(x1) = max{53, x1}
          a__isNatIList_A(x1) = max{58, x1 - 5}
          a__U51_A(x1,x2) = max{25, x1 + 1, x2 - 30}
          a__U52_A(x1) = max{23, x1}
          a__isNatList_A(x1) = max{25, x1 - 30}
          a__U61_A(x1,x2) = max{x1 + 41, x2 + 64}
          a__U62_A(x1) = max{42, x1 + 6}
          a__U71_A(x1,x2,x3) = max{x1 + 119, x2 + 89, x3 + 148}
          a__U72_A(x1,x2) = max{147, x1 + 124, x2 + 89}
          a__isNat_A(x1) = max{3, x1 + 1}
          a__length_A(x1) = max{146, x1 + 89}
          a__U81_A(x1) = max{66, x1}
          nil_A = 31
          a__U91_A(x1,x2,x3,x4) = max{x1 + 117, x2 + 116, x3 + 75, x4 + 176}
          a__U92_A(x1,x2,x3,x4) = max{x1 + 64, x2 + 116, x3 + 75, x4 + 176}
          a__U93_A(x1,x2,x3,x4) = max{176, x1, x2 + 116, x3 + 75, x4 + 142}
          a__take_A(x1,x2) = max{176, x1 + 75, x2 + 116}
          isNatIList_A(x1) = max{58, x1 - 5}
          isNatList_A(x1) = max{25, x1 - 30}
          isNat_A(x1) = max{2, x1 + 1}
    
      precedence:
      
        mark > a__U81 > a__take > a__zeros > a__U91 > a__U92 > a__U93 > U92 = a__isNat > tt > U93 > cons > a__isNatIList > a__U41 > a__length > a__U71 > U81 = a__U72 > U31 = a__U31 > a__U21 > a__isNatList > isNat > U41 > a__U51 > s > take > zeros > U61 = a__U61 > a__U62 > nil > a__U42 > U21 > mark# = isNatList > length > isNatIList > a__U11 > U72 > U62 > |0| > U11 = a__U52 > a__U93# > U52 = U71 = U91 > U51 > U42
      
      partial status:
    
        pi(a__U93#) = []
        pi(tt) = []
        pi(mark#) = []
        pi(U11) = [1]
        pi(U21) = []
        pi(U31) = []
        pi(U41) = []
        pi(U42) = []
        pi(U51) = [1]
        pi(U52) = []
        pi(U61) = [1]
        pi(U62) = []
        pi(U71) = [1, 2]
        pi(U72) = [2]
        pi(length) = []
        pi(U81) = []
        pi(U91) = [1]
        pi(U92) = []
        pi(U93) = []
        pi(mark) = []
        pi(take) = []
        pi(cons) = []
        pi(s) = []
        pi(a__zeros) = []
        pi(|0|) = []
        pi(zeros) = []
        pi(a__U11) = [1]
        pi(a__U21) = []
        pi(a__U31) = []
        pi(a__U41) = [1]
        pi(a__U42) = []
        pi(a__isNatIList) = []
        pi(a__U51) = []
        pi(a__U52) = []
        pi(a__isNatList) = []
        pi(a__U61) = [1, 2]
        pi(a__U62) = []
        pi(a__U71) = [1, 2]
        pi(a__U72) = []
        pi(a__isNat) = []
        pi(a__length) = []
        pi(a__U81) = []
        pi(nil) = []
        pi(a__U91) = [1, 4]
        pi(a__U92) = []
        pi(a__U93) = []
        pi(a__take) = [2]
        pi(isNatIList) = []
        pi(isNatList) = []
        pi(isNat) = []
    
    2. weighted path order
    
      base order:
    
        max/plus interpretations on natural numbers:
        
          a__U93#_A(x1,x2,x3,x4) = 170
          tt_A = 155
          mark#_A(x1) = 170
          U11_A(x1) = 1
          U21_A(x1) = 6
          U31_A(x1) = 33
          U41_A(x1,x2) = 74
          U42_A(x1) = 74
          U51_A(x1,x2) = 0
          U52_A(x1) = 1
          U61_A(x1,x2) = 166
          U62_A(x1) = 3
          U71_A(x1,x2,x3) = max{x1 + 165, x2 + 242}
          U72_A(x1,x2) = 7
          length_A(x1) = 153
          U81_A(x1) = 78
          U91_A(x1,x2,x3,x4) = 156
          U92_A(x1,x2,x3,x4) = 155
          U93_A(x1,x2,x3,x4) = 147
          mark_A(x1) = 164
          take_A(x1,x2) = 81
          cons_A(x1,x2) = 121
          s_A(x1) = 0
          a__zeros_A = 122
          |0|_A = 5
          zeros_A = 88
          a__U11_A(x1) = 154
          a__U21_A(x1) = 7
          a__U31_A(x1) = 75
          a__U41_A(x1,x2) = max{152, x1}
          a__U42_A(x1) = 153
          a__isNatIList_A(x1) = 120
          a__U51_A(x1,x2) = 164
          a__U52_A(x1) = 164
          a__isNatList_A(x1) = 164
          a__U61_A(x1,x2) = max{x1 + 167, x2 + 167}
          a__U62_A(x1) = 154
          a__U71_A(x1,x2,x3) = max{x1 + 166, x2 + 243}
          a__U72_A(x1,x2) = 163
          a__isNat_A(x1) = 154
          a__length_A(x1) = 166
          a__U81_A(x1) = 78
          nil_A = 0
          a__U91_A(x1,x2,x3,x4) = 168
          a__U92_A(x1,x2,x3,x4) = 155
          a__U93_A(x1,x2,x3,x4) = 148
          a__take_A(x1,x2) = max{165, x2 + 82}
          isNatIList_A(x1) = 119
          isNatList_A(x1) = 74
          isNat_A(x1) = 66
    
      precedence:
      
        mark > tt > U52 > zeros > a__isNatIList > a__isNatList > a__length > a__U42 > a__U41 > a__U51 = a__U52 = a__U61 > a__U31 = a__isNat > isNatList > a__U81 > a__take > a__U91 > a__U62 > U11 = U71 = a__U71 = a__U92 > U61 > a__U93 > length = U92 > nil > |0| > a__zeros > cons > U93 > mark# > s > isNat > U42 = U91 = a__U72 > a__U93# > U41 > a__U21 > U72 > U31 > U51 = U62 > U81 > a__U11 > U21 = take = isNatIList
      
      partial status:
    
        pi(a__U93#) = []
        pi(tt) = []
        pi(mark#) = []
        pi(U11) = []
        pi(U21) = []
        pi(U31) = []
        pi(U41) = []
        pi(U42) = []
        pi(U51) = []
        pi(U52) = []
        pi(U61) = []
        pi(U62) = []
        pi(U71) = []
        pi(U72) = []
        pi(length) = []
        pi(U81) = []
        pi(U91) = []
        pi(U92) = []
        pi(U93) = []
        pi(mark) = []
        pi(take) = []
        pi(cons) = []
        pi(s) = []
        pi(a__zeros) = []
        pi(|0|) = []
        pi(zeros) = []
        pi(a__U11) = []
        pi(a__U21) = []
        pi(a__U31) = []
        pi(a__U41) = []
        pi(a__U42) = []
        pi(a__isNatIList) = []
        pi(a__U51) = []
        pi(a__U52) = []
        pi(a__isNatList) = []
        pi(a__U61) = []
        pi(a__U62) = []
        pi(a__U71) = [2]
        pi(a__U72) = []
        pi(a__isNat) = []
        pi(a__length) = []
        pi(a__U81) = []
        pi(nil) = []
        pi(a__U91) = []
        pi(a__U92) = []
        pi(a__U93) = []
        pi(a__take) = []
        pi(isNatIList) = []
        pi(isNatList) = []
        pi(isNat) = []
    

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: mark#(U11(X)) -> mark#(X)
p2: mark#(U21(X)) -> mark#(X)
p3: mark#(U31(X)) -> mark#(X)
p4: mark#(U41(X1,X2)) -> mark#(X1)
p5: mark#(U42(X)) -> mark#(X)
p6: mark#(U51(X1,X2)) -> mark#(X1)
p7: mark#(U52(X)) -> mark#(X)
p8: mark#(U61(X1,X2)) -> mark#(X1)
p9: mark#(U62(X)) -> mark#(X)
p10: mark#(U71(X1,X2,X3)) -> mark#(X1)
p11: mark#(U72(X1,X2)) -> mark#(X1)
p12: mark#(length(X)) -> mark#(X)
p13: mark#(U81(X)) -> mark#(X)
p14: mark#(U91(X1,X2,X3,X4)) -> mark#(X1)
p15: mark#(U92(X1,X2,X3,X4)) -> mark#(X1)
p16: mark#(U93(X1,X2,X3,X4)) -> a__U93#(mark(X1),X2,X3,X4)
p17: mark#(U93(X1,X2,X3,X4)) -> mark#(X1)
p18: mark#(take(X1,X2)) -> mark#(X1)
p19: mark#(take(X1,X2)) -> mark#(X2)
p20: mark#(cons(X1,X2)) -> mark#(X1)
p21: mark#(s(X)) -> mark#(X)

and R consists of:

r1: a__zeros() -> cons(|0|(),zeros())
r2: a__U11(tt()) -> tt()
r3: a__U21(tt()) -> tt()
r4: a__U31(tt()) -> tt()
r5: a__U41(tt(),V2) -> a__U42(a__isNatIList(V2))
r6: a__U42(tt()) -> tt()
r7: a__U51(tt(),V2) -> a__U52(a__isNatList(V2))
r8: a__U52(tt()) -> tt()
r9: a__U61(tt(),V2) -> a__U62(a__isNatIList(V2))
r10: a__U62(tt()) -> tt()
r11: a__U71(tt(),L,N) -> a__U72(a__isNat(N),L)
r12: a__U72(tt(),L) -> s(a__length(mark(L)))
r13: a__U81(tt()) -> nil()
r14: a__U91(tt(),IL,M,N) -> a__U92(a__isNat(M),IL,M,N)
r15: a__U92(tt(),IL,M,N) -> a__U93(a__isNat(N),IL,M,N)
r16: a__U93(tt(),IL,M,N) -> cons(mark(N),take(M,IL))
r17: a__isNat(|0|()) -> tt()
r18: a__isNat(length(V1)) -> a__U11(a__isNatList(V1))
r19: a__isNat(s(V1)) -> a__U21(a__isNat(V1))
r20: a__isNatIList(V) -> a__U31(a__isNatList(V))
r21: a__isNatIList(zeros()) -> tt()
r22: a__isNatIList(cons(V1,V2)) -> a__U41(a__isNat(V1),V2)
r23: a__isNatList(nil()) -> tt()
r24: a__isNatList(cons(V1,V2)) -> a__U51(a__isNat(V1),V2)
r25: a__isNatList(take(V1,V2)) -> a__U61(a__isNat(V1),V2)
r26: a__length(nil()) -> |0|()
r27: a__length(cons(N,L)) -> a__U71(a__isNatList(L),L,N)
r28: a__take(|0|(),IL) -> a__U81(a__isNatIList(IL))
r29: a__take(s(M),cons(N,IL)) -> a__U91(a__isNatIList(IL),IL,M,N)
r30: mark(zeros()) -> a__zeros()
r31: mark(U11(X)) -> a__U11(mark(X))
r32: mark(U21(X)) -> a__U21(mark(X))
r33: mark(U31(X)) -> a__U31(mark(X))
r34: mark(U41(X1,X2)) -> a__U41(mark(X1),X2)
r35: mark(U42(X)) -> a__U42(mark(X))
r36: mark(isNatIList(X)) -> a__isNatIList(X)
r37: mark(U51(X1,X2)) -> a__U51(mark(X1),X2)
r38: mark(U52(X)) -> a__U52(mark(X))
r39: mark(isNatList(X)) -> a__isNatList(X)
r40: mark(U61(X1,X2)) -> a__U61(mark(X1),X2)
r41: mark(U62(X)) -> a__U62(mark(X))
r42: mark(U71(X1,X2,X3)) -> a__U71(mark(X1),X2,X3)
r43: mark(U72(X1,X2)) -> a__U72(mark(X1),X2)
r44: mark(isNat(X)) -> a__isNat(X)
r45: mark(length(X)) -> a__length(mark(X))
r46: mark(U81(X)) -> a__U81(mark(X))
r47: mark(U91(X1,X2,X3,X4)) -> a__U91(mark(X1),X2,X3,X4)
r48: mark(U92(X1,X2,X3,X4)) -> a__U92(mark(X1),X2,X3,X4)
r49: mark(U93(X1,X2,X3,X4)) -> a__U93(mark(X1),X2,X3,X4)
r50: mark(take(X1,X2)) -> a__take(mark(X1),mark(X2))
r51: mark(cons(X1,X2)) -> cons(mark(X1),X2)
r52: mark(|0|()) -> |0|()
r53: mark(tt()) -> tt()
r54: mark(s(X)) -> s(mark(X))
r55: mark(nil()) -> nil()
r56: a__zeros() -> zeros()
r57: a__U11(X) -> U11(X)
r58: a__U21(X) -> U21(X)
r59: a__U31(X) -> U31(X)
r60: a__U41(X1,X2) -> U41(X1,X2)
r61: a__U42(X) -> U42(X)
r62: a__isNatIList(X) -> isNatIList(X)
r63: a__U51(X1,X2) -> U51(X1,X2)
r64: a__U52(X) -> U52(X)
r65: a__isNatList(X) -> isNatList(X)
r66: a__U61(X1,X2) -> U61(X1,X2)
r67: a__U62(X) -> U62(X)
r68: a__U71(X1,X2,X3) -> U71(X1,X2,X3)
r69: a__U72(X1,X2) -> U72(X1,X2)
r70: a__isNat(X) -> isNat(X)
r71: a__length(X) -> length(X)
r72: a__U81(X) -> U81(X)
r73: a__U91(X1,X2,X3,X4) -> U91(X1,X2,X3,X4)
r74: a__U92(X1,X2,X3,X4) -> U92(X1,X2,X3,X4)
r75: a__U93(X1,X2,X3,X4) -> U93(X1,X2,X3,X4)
r76: a__take(X1,X2) -> take(X1,X2)

The estimated dependency graph contains the following SCCs:

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


-- Reduction pair.

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

p1: mark#(U11(X)) -> mark#(X)
p2: mark#(s(X)) -> mark#(X)
p3: mark#(cons(X1,X2)) -> mark#(X1)
p4: mark#(take(X1,X2)) -> mark#(X2)
p5: mark#(take(X1,X2)) -> mark#(X1)
p6: mark#(U93(X1,X2,X3,X4)) -> mark#(X1)
p7: mark#(U92(X1,X2,X3,X4)) -> mark#(X1)
p8: mark#(U91(X1,X2,X3,X4)) -> mark#(X1)
p9: mark#(U81(X)) -> mark#(X)
p10: mark#(length(X)) -> mark#(X)
p11: mark#(U72(X1,X2)) -> mark#(X1)
p12: mark#(U71(X1,X2,X3)) -> mark#(X1)
p13: mark#(U62(X)) -> mark#(X)
p14: mark#(U61(X1,X2)) -> mark#(X1)
p15: mark#(U52(X)) -> mark#(X)
p16: mark#(U51(X1,X2)) -> mark#(X1)
p17: mark#(U42(X)) -> mark#(X)
p18: mark#(U41(X1,X2)) -> mark#(X1)
p19: mark#(U31(X)) -> mark#(X)
p20: mark#(U21(X)) -> mark#(X)

and R consists of:

r1: a__zeros() -> cons(|0|(),zeros())
r2: a__U11(tt()) -> tt()
r3: a__U21(tt()) -> tt()
r4: a__U31(tt()) -> tt()
r5: a__U41(tt(),V2) -> a__U42(a__isNatIList(V2))
r6: a__U42(tt()) -> tt()
r7: a__U51(tt(),V2) -> a__U52(a__isNatList(V2))
r8: a__U52(tt()) -> tt()
r9: a__U61(tt(),V2) -> a__U62(a__isNatIList(V2))
r10: a__U62(tt()) -> tt()
r11: a__U71(tt(),L,N) -> a__U72(a__isNat(N),L)
r12: a__U72(tt(),L) -> s(a__length(mark(L)))
r13: a__U81(tt()) -> nil()
r14: a__U91(tt(),IL,M,N) -> a__U92(a__isNat(M),IL,M,N)
r15: a__U92(tt(),IL,M,N) -> a__U93(a__isNat(N),IL,M,N)
r16: a__U93(tt(),IL,M,N) -> cons(mark(N),take(M,IL))
r17: a__isNat(|0|()) -> tt()
r18: a__isNat(length(V1)) -> a__U11(a__isNatList(V1))
r19: a__isNat(s(V1)) -> a__U21(a__isNat(V1))
r20: a__isNatIList(V) -> a__U31(a__isNatList(V))
r21: a__isNatIList(zeros()) -> tt()
r22: a__isNatIList(cons(V1,V2)) -> a__U41(a__isNat(V1),V2)
r23: a__isNatList(nil()) -> tt()
r24: a__isNatList(cons(V1,V2)) -> a__U51(a__isNat(V1),V2)
r25: a__isNatList(take(V1,V2)) -> a__U61(a__isNat(V1),V2)
r26: a__length(nil()) -> |0|()
r27: a__length(cons(N,L)) -> a__U71(a__isNatList(L),L,N)
r28: a__take(|0|(),IL) -> a__U81(a__isNatIList(IL))
r29: a__take(s(M),cons(N,IL)) -> a__U91(a__isNatIList(IL),IL,M,N)
r30: mark(zeros()) -> a__zeros()
r31: mark(U11(X)) -> a__U11(mark(X))
r32: mark(U21(X)) -> a__U21(mark(X))
r33: mark(U31(X)) -> a__U31(mark(X))
r34: mark(U41(X1,X2)) -> a__U41(mark(X1),X2)
r35: mark(U42(X)) -> a__U42(mark(X))
r36: mark(isNatIList(X)) -> a__isNatIList(X)
r37: mark(U51(X1,X2)) -> a__U51(mark(X1),X2)
r38: mark(U52(X)) -> a__U52(mark(X))
r39: mark(isNatList(X)) -> a__isNatList(X)
r40: mark(U61(X1,X2)) -> a__U61(mark(X1),X2)
r41: mark(U62(X)) -> a__U62(mark(X))
r42: mark(U71(X1,X2,X3)) -> a__U71(mark(X1),X2,X3)
r43: mark(U72(X1,X2)) -> a__U72(mark(X1),X2)
r44: mark(isNat(X)) -> a__isNat(X)
r45: mark(length(X)) -> a__length(mark(X))
r46: mark(U81(X)) -> a__U81(mark(X))
r47: mark(U91(X1,X2,X3,X4)) -> a__U91(mark(X1),X2,X3,X4)
r48: mark(U92(X1,X2,X3,X4)) -> a__U92(mark(X1),X2,X3,X4)
r49: mark(U93(X1,X2,X3,X4)) -> a__U93(mark(X1),X2,X3,X4)
r50: mark(take(X1,X2)) -> a__take(mark(X1),mark(X2))
r51: mark(cons(X1,X2)) -> cons(mark(X1),X2)
r52: mark(|0|()) -> |0|()
r53: mark(tt()) -> tt()
r54: mark(s(X)) -> s(mark(X))
r55: mark(nil()) -> nil()
r56: a__zeros() -> zeros()
r57: a__U11(X) -> U11(X)
r58: a__U21(X) -> U21(X)
r59: a__U31(X) -> U31(X)
r60: a__U41(X1,X2) -> U41(X1,X2)
r61: a__U42(X) -> U42(X)
r62: a__isNatIList(X) -> isNatIList(X)
r63: a__U51(X1,X2) -> U51(X1,X2)
r64: a__U52(X) -> U52(X)
r65: a__isNatList(X) -> isNatList(X)
r66: a__U61(X1,X2) -> U61(X1,X2)
r67: a__U62(X) -> U62(X)
r68: a__U71(X1,X2,X3) -> U71(X1,X2,X3)
r69: a__U72(X1,X2) -> U72(X1,X2)
r70: a__isNat(X) -> isNat(X)
r71: a__length(X) -> length(X)
r72: a__U81(X) -> U81(X)
r73: a__U91(X1,X2,X3,X4) -> U91(X1,X2,X3,X4)
r74: a__U92(X1,X2,X3,X4) -> U92(X1,X2,X3,X4)
r75: a__U93(X1,X2,X3,X4) -> U93(X1,X2,X3,X4)
r76: a__take(X1,X2) -> take(X1,X2)

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  lexicographic combination of reduction pairs:
  
    1. weighted path order
    
      base order:
    
        max/plus interpretations on natural numbers:
        
          mark#_A(x1) = max{6, x1 + 4}
          U11_A(x1) = max{5, x1 + 4}
          s_A(x1) = max{4, x1 + 3}
          cons_A(x1,x2) = max{x1, x2}
          take_A(x1,x2) = max{x1, x2}
          U93_A(x1,x2,x3,x4) = max{x1, x4}
          U92_A(x1,x2,x3,x4) = max{x1, x4}
          U91_A(x1,x2,x3,x4) = max{x1, x4}
          U81_A(x1) = max{1, x1}
          length_A(x1) = max{1, x1}
          U72_A(x1,x2) = max{x1, x2}
          U71_A(x1,x2,x3) = max{x1, x3}
          U62_A(x1) = max{1, x1}
          U61_A(x1,x2) = max{x1, x2}
          U52_A(x1) = max{1, x1}
          U51_A(x1,x2) = max{x1, x2}
          U42_A(x1) = max{1, x1}
          U41_A(x1,x2) = max{x1, x2}
          U31_A(x1) = max{1, x1}
          U21_A(x1) = max{1, x1}
    
      precedence:
      
        U93 > mark# = U11 = s = cons = take = U92 = U91 = U81 = length = U72 = U71 = U62 = U61 = U52 = U51 = U42 = U41 = U31 = U21
      
      partial status:
    
        pi(mark#) = [1]
        pi(U11) = [1]
        pi(s) = [1]
        pi(cons) = [1, 2]
        pi(take) = [1, 2]
        pi(U93) = [1, 4]
        pi(U92) = [1, 4]
        pi(U91) = [1, 4]
        pi(U81) = [1]
        pi(length) = [1]
        pi(U72) = [1, 2]
        pi(U71) = [1, 3]
        pi(U62) = [1]
        pi(U61) = [1, 2]
        pi(U52) = [1]
        pi(U51) = [1, 2]
        pi(U42) = [1]
        pi(U41) = [1, 2]
        pi(U31) = [1]
        pi(U21) = [1]
    
    2. weighted path order
    
      base order:
    
        max/plus interpretations on natural numbers:
        
          mark#_A(x1) = x1 + 2
          U11_A(x1) = x1
          s_A(x1) = x1
          cons_A(x1,x2) = max{x1, x2}
          take_A(x1,x2) = max{x1, x2}
          U93_A(x1,x2,x3,x4) = max{x1, x4}
          U92_A(x1,x2,x3,x4) = max{x1, x4}
          U91_A(x1,x2,x3,x4) = max{x1, x4}
          U81_A(x1) = x1
          length_A(x1) = x1
          U72_A(x1,x2) = max{x1, x2}
          U71_A(x1,x2,x3) = max{x1, x3}
          U62_A(x1) = x1
          U61_A(x1,x2) = max{x1, x2}
          U52_A(x1) = x1
          U51_A(x1,x2) = max{x1, x2}
          U42_A(x1) = x1
          U41_A(x1,x2) = max{x1, x2}
          U31_A(x1) = x1
          U21_A(x1) = x1
    
      precedence:
      
        mark# = U11 = s = cons = take = U93 = U92 = U91 = U81 = length = U72 = U71 = U62 = U61 = U52 = U51 = U42 = U41 = U31 = U21
      
      partial status:
    
        pi(mark#) = [1]
        pi(U11) = [1]
        pi(s) = [1]
        pi(cons) = [1, 2]
        pi(take) = [1, 2]
        pi(U93) = [1, 4]
        pi(U92) = [1, 4]
        pi(U91) = [1, 4]
        pi(U81) = [1]
        pi(length) = [1]
        pi(U72) = [1, 2]
        pi(U71) = [1, 3]
        pi(U62) = [1]
        pi(U61) = [1, 2]
        pi(U52) = [1]
        pi(U51) = [1, 2]
        pi(U42) = [1]
        pi(U41) = [1, 2]
        pi(U31) = [1]
        pi(U21) = [1]
    

The next rules are strictly ordered:

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

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__length#(cons(N,L)) -> a__U71#(a__isNatList(L),L,N)
p2: a__U71#(tt(),L,N) -> a__U72#(a__isNat(N),L)
p3: a__U72#(tt(),L) -> a__length#(mark(L))

and R consists of:

r1: a__zeros() -> cons(|0|(),zeros())
r2: a__U11(tt()) -> tt()
r3: a__U21(tt()) -> tt()
r4: a__U31(tt()) -> tt()
r5: a__U41(tt(),V2) -> a__U42(a__isNatIList(V2))
r6: a__U42(tt()) -> tt()
r7: a__U51(tt(),V2) -> a__U52(a__isNatList(V2))
r8: a__U52(tt()) -> tt()
r9: a__U61(tt(),V2) -> a__U62(a__isNatIList(V2))
r10: a__U62(tt()) -> tt()
r11: a__U71(tt(),L,N) -> a__U72(a__isNat(N),L)
r12: a__U72(tt(),L) -> s(a__length(mark(L)))
r13: a__U81(tt()) -> nil()
r14: a__U91(tt(),IL,M,N) -> a__U92(a__isNat(M),IL,M,N)
r15: a__U92(tt(),IL,M,N) -> a__U93(a__isNat(N),IL,M,N)
r16: a__U93(tt(),IL,M,N) -> cons(mark(N),take(M,IL))
r17: a__isNat(|0|()) -> tt()
r18: a__isNat(length(V1)) -> a__U11(a__isNatList(V1))
r19: a__isNat(s(V1)) -> a__U21(a__isNat(V1))
r20: a__isNatIList(V) -> a__U31(a__isNatList(V))
r21: a__isNatIList(zeros()) -> tt()
r22: a__isNatIList(cons(V1,V2)) -> a__U41(a__isNat(V1),V2)
r23: a__isNatList(nil()) -> tt()
r24: a__isNatList(cons(V1,V2)) -> a__U51(a__isNat(V1),V2)
r25: a__isNatList(take(V1,V2)) -> a__U61(a__isNat(V1),V2)
r26: a__length(nil()) -> |0|()
r27: a__length(cons(N,L)) -> a__U71(a__isNatList(L),L,N)
r28: a__take(|0|(),IL) -> a__U81(a__isNatIList(IL))
r29: a__take(s(M),cons(N,IL)) -> a__U91(a__isNatIList(IL),IL,M,N)
r30: mark(zeros()) -> a__zeros()
r31: mark(U11(X)) -> a__U11(mark(X))
r32: mark(U21(X)) -> a__U21(mark(X))
r33: mark(U31(X)) -> a__U31(mark(X))
r34: mark(U41(X1,X2)) -> a__U41(mark(X1),X2)
r35: mark(U42(X)) -> a__U42(mark(X))
r36: mark(isNatIList(X)) -> a__isNatIList(X)
r37: mark(U51(X1,X2)) -> a__U51(mark(X1),X2)
r38: mark(U52(X)) -> a__U52(mark(X))
r39: mark(isNatList(X)) -> a__isNatList(X)
r40: mark(U61(X1,X2)) -> a__U61(mark(X1),X2)
r41: mark(U62(X)) -> a__U62(mark(X))
r42: mark(U71(X1,X2,X3)) -> a__U71(mark(X1),X2,X3)
r43: mark(U72(X1,X2)) -> a__U72(mark(X1),X2)
r44: mark(isNat(X)) -> a__isNat(X)
r45: mark(length(X)) -> a__length(mark(X))
r46: mark(U81(X)) -> a__U81(mark(X))
r47: mark(U91(X1,X2,X3,X4)) -> a__U91(mark(X1),X2,X3,X4)
r48: mark(U92(X1,X2,X3,X4)) -> a__U92(mark(X1),X2,X3,X4)
r49: mark(U93(X1,X2,X3,X4)) -> a__U93(mark(X1),X2,X3,X4)
r50: mark(take(X1,X2)) -> a__take(mark(X1),mark(X2))
r51: mark(cons(X1,X2)) -> cons(mark(X1),X2)
r52: mark(|0|()) -> |0|()
r53: mark(tt()) -> tt()
r54: mark(s(X)) -> s(mark(X))
r55: mark(nil()) -> nil()
r56: a__zeros() -> zeros()
r57: a__U11(X) -> U11(X)
r58: a__U21(X) -> U21(X)
r59: a__U31(X) -> U31(X)
r60: a__U41(X1,X2) -> U41(X1,X2)
r61: a__U42(X) -> U42(X)
r62: a__isNatIList(X) -> isNatIList(X)
r63: a__U51(X1,X2) -> U51(X1,X2)
r64: a__U52(X) -> U52(X)
r65: a__isNatList(X) -> isNatList(X)
r66: a__U61(X1,X2) -> U61(X1,X2)
r67: a__U62(X) -> U62(X)
r68: a__U71(X1,X2,X3) -> U71(X1,X2,X3)
r69: a__U72(X1,X2) -> U72(X1,X2)
r70: a__isNat(X) -> isNat(X)
r71: a__length(X) -> length(X)
r72: a__U81(X) -> U81(X)
r73: a__U91(X1,X2,X3,X4) -> U91(X1,X2,X3,X4)
r74: a__U92(X1,X2,X3,X4) -> U92(X1,X2,X3,X4)
r75: a__U93(X1,X2,X3,X4) -> U93(X1,X2,X3,X4)
r76: a__take(X1,X2) -> take(X1,X2)

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

Take the reduction pair:

  lexicographic combination of reduction pairs:
  
    1. weighted path order
    
      base order:
    
        max/plus interpretations on natural numbers:
        
          a__length#_A(x1) = x1 + 6
          cons_A(x1,x2) = max{x1 - 1, x2 + 7}
          a__U71#_A(x1,x2,x3) = max{x1 - 34, x2 + 7, x3 - 2}
          a__isNatList_A(x1) = x1 + 46
          tt_A = 84
          a__U72#_A(x1,x2) = max{21, x1 - 80, x2 + 6}
          a__isNat_A(x1) = x1 + 73
          mark_A(x1) = max{14, x1}
          a__zeros_A = 13
          |0|_A = 12
          zeros_A = 0
          a__U11_A(x1) = max{73, x1 + 15}
          a__U21_A(x1) = max{85, x1}
          a__U31_A(x1) = 85
          a__U41_A(x1,x2) = 86
          a__U42_A(x1) = 85
          a__isNatIList_A(x1) = 86
          a__U51_A(x1,x2) = max{x1 - 84, x2 + 46}
          a__U52_A(x1) = max{13, x1}
          a__U61_A(x1,x2) = max{87, x1 + 15, x2 + 30}
          a__U62_A(x1) = 87
          a__U71_A(x1,x2,x3) = max{x1 - 40, x2 + 7, x3 - 2}
          a__U72_A(x1,x2) = max{44, x1 - 76, x2 + 7}
          s_A(x1) = max{44, x1 + 7}
          a__length_A(x1) = x1
          a__U81_A(x1) = max{58, x1 - 85}
          nil_A = 57
          a__U91_A(x1,x2,x3,x4) = max{59, x1 - 16, x2 + 7, x3 + 50, x4 - 1}
          a__U92_A(x1,x2,x3,x4) = max{67, x1 - 24, x2 + 7, x3 + 50, x4 - 1}
          a__U93_A(x1,x2,x3,x4) = max{66, x2 + 7, x3 + 50, x4 - 1}
          take_A(x1,x2) = max{58, x1 + 43, x2}
          a__take_A(x1,x2) = max{58, x1 + 43, x2}
          U11_A(x1) = max{73, x1 + 15}
          U21_A(x1) = max{85, x1}
          U31_A(x1) = 85
          U41_A(x1,x2) = 86
          U42_A(x1) = 85
          isNatIList_A(x1) = 86
          U51_A(x1,x2) = max{x1 - 84, x2 + 46}
          U52_A(x1) = max{13, x1}
          U61_A(x1,x2) = max{87, x1 + 15, x2 + 30}
          U62_A(x1) = 87
          U71_A(x1,x2,x3) = max{x1 - 40, x2 + 7, x3 - 2}
          U72_A(x1,x2) = max{44, x1 - 76, x2 + 7}
          length_A(x1) = x1
          U81_A(x1) = max{58, x1 - 85}
          U91_A(x1,x2,x3,x4) = max{59, x1 - 16, x2 + 7, x3 + 50, x4 - 1}
          U92_A(x1,x2,x3,x4) = max{67, x1 - 24, x2 + 7, x3 + 50, x4 - 1}
          U93_A(x1,x2,x3,x4) = max{66, x2 + 7, x3 + 50, x4 - 1}
          isNatList_A(x1) = x1 + 46
          isNat_A(x1) = x1 + 73
    
      precedence:
      
        mark = a__U31 = U31 > a__length > |0| = a__isNatIList = a__take > a__U91 > a__isNatList = a__U41 = a__U42 = a__U51 = a__U52 = a__U61 = a__U71 = a__U72 = a__U81 = take = U42 = U51 = U61 = U71 = U72 = U91 > isNatList > a__zeros = a__U92 = a__U93 = U92 > cons > a__U71# > a__U72# > length = U93 > s > a__length# > tt = a__isNat = zeros = a__U11 = a__U21 = a__U62 = nil = U11 = U21 = U41 = isNatIList = U52 = U62 = U81 = isNat
      
      partial status:
    
        pi(a__length#) = [1]
        pi(cons) = [2]
        pi(a__U71#) = []
        pi(a__isNatList) = [1]
        pi(tt) = []
        pi(a__U72#) = []
        pi(a__isNat) = [1]
        pi(mark) = []
        pi(a__zeros) = []
        pi(|0|) = []
        pi(zeros) = []
        pi(a__U11) = []
        pi(a__U21) = []
        pi(a__U31) = []
        pi(a__U41) = []
        pi(a__U42) = []
        pi(a__isNatIList) = []
        pi(a__U51) = [2]
        pi(a__U52) = []
        pi(a__U61) = []
        pi(a__U62) = []
        pi(a__U71) = []
        pi(a__U72) = []
        pi(s) = [1]
        pi(a__length) = []
        pi(a__U81) = []
        pi(nil) = []
        pi(a__U91) = []
        pi(a__U92) = [3]
        pi(a__U93) = []
        pi(take) = []
        pi(a__take) = [1]
        pi(U11) = []
        pi(U21) = []
        pi(U31) = []
        pi(U41) = []
        pi(U42) = []
        pi(isNatIList) = []
        pi(U51) = []
        pi(U52) = []
        pi(U61) = []
        pi(U62) = []
        pi(U71) = []
        pi(U72) = []
        pi(length) = []
        pi(U81) = []
        pi(U91) = []
        pi(U92) = []
        pi(U93) = []
        pi(isNatList) = [1]
        pi(isNat) = []
    
    2. weighted path order
    
      base order:
    
        max/plus interpretations on natural numbers:
        
          a__length#_A(x1) = max{15, x1 + 2}
          cons_A(x1,x2) = x2 + 51
          a__U71#_A(x1,x2,x3) = 32
          a__isNatList_A(x1) = 16
          tt_A = 1
          a__U72#_A(x1,x2) = 20
          a__isNat_A(x1) = 7
          mark_A(x1) = 17
          a__zeros_A = 4
          |0|_A = 0
          zeros_A = 0
          a__U11_A(x1) = 7
          a__U21_A(x1) = 17
          a__U31_A(x1) = 6
          a__U41_A(x1,x2) = 6
          a__U42_A(x1) = 17
          a__isNatIList_A(x1) = 6
          a__U51_A(x1,x2) = 16
          a__U52_A(x1) = 15
          a__U61_A(x1,x2) = 16
          a__U62_A(x1) = 16
          a__U71_A(x1,x2,x3) = 17
          a__U72_A(x1,x2) = 17
          s_A(x1) = 17
          a__length_A(x1) = 17
          a__U81_A(x1) = 17
          nil_A = 18
          a__U91_A(x1,x2,x3,x4) = 17
          a__U92_A(x1,x2,x3,x4) = x3 + 70
          a__U93_A(x1,x2,x3,x4) = 64
          take_A(x1,x2) = 13
          a__take_A(x1,x2) = max{17, x1}
          U11_A(x1) = 7
          U21_A(x1) = 16
          U31_A(x1) = 6
          U41_A(x1,x2) = 5
          U42_A(x1) = 16
          isNatIList_A(x1) = 5
          U51_A(x1,x2) = 15
          U52_A(x1) = 14
          U61_A(x1,x2) = 15
          U62_A(x1) = 15
          U71_A(x1,x2,x3) = 16
          U72_A(x1,x2) = 17
          length_A(x1) = 6
          U81_A(x1) = 18
          U91_A(x1,x2,x3,x4) = 17
          U92_A(x1,x2,x3,x4) = 61
          U93_A(x1,x2,x3,x4) = 61
          isNatList_A(x1) = max{19, x1 + 17}
          isNat_A(x1) = 7
    
      precedence:
      
        a__zeros > a__isNatList = tt > take > a__length# = a__U71# = a__U41 = a__isNatIList = a__U51 = a__U52 = isNatIList = U52 > a__U91 > a__U72# = a__U93 > a__U61 = a__U62 = a__U92 = U51 = U61 > U92 > mark = a__U31 = a__U81 = U81 > a__isNat > isNat > a__U42 = a__take > U41 = U42 = U62 = U91 = U93 > cons > U31 > |0| = a__U71 = a__U72 = a__length = nil = isNatList > zeros = a__U11 = a__U21 = s = U11 = U21 = U71 = U72 = length
      
      partial status:
    
        pi(a__length#) = [1]
        pi(cons) = [2]
        pi(a__U71#) = []
        pi(a__isNatList) = []
        pi(tt) = []
        pi(a__U72#) = []
        pi(a__isNat) = []
        pi(mark) = []
        pi(a__zeros) = []
        pi(|0|) = []
        pi(zeros) = []
        pi(a__U11) = []
        pi(a__U21) = []
        pi(a__U31) = []
        pi(a__U41) = []
        pi(a__U42) = []
        pi(a__isNatIList) = []
        pi(a__U51) = []
        pi(a__U52) = []
        pi(a__U61) = []
        pi(a__U62) = []
        pi(a__U71) = []
        pi(a__U72) = []
        pi(s) = []
        pi(a__length) = []
        pi(a__U81) = []
        pi(nil) = []
        pi(a__U91) = []
        pi(a__U92) = []
        pi(a__U93) = []
        pi(take) = []
        pi(a__take) = []
        pi(U11) = []
        pi(U21) = []
        pi(U31) = []
        pi(U41) = []
        pi(U42) = []
        pi(isNatIList) = []
        pi(U51) = []
        pi(U52) = []
        pi(U61) = []
        pi(U62) = []
        pi(U71) = []
        pi(U72) = []
        pi(length) = []
        pi(U81) = []
        pi(U91) = []
        pi(U92) = []
        pi(U93) = []
        pi(isNatList) = []
        pi(isNat) = []
    

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: a__length#(cons(N,L)) -> a__U71#(a__isNatList(L),L,N)
p2: a__U71#(tt(),L,N) -> a__U72#(a__isNat(N),L)

and R consists of:

r1: a__zeros() -> cons(|0|(),zeros())
r2: a__U11(tt()) -> tt()
r3: a__U21(tt()) -> tt()
r4: a__U31(tt()) -> tt()
r5: a__U41(tt(),V2) -> a__U42(a__isNatIList(V2))
r6: a__U42(tt()) -> tt()
r7: a__U51(tt(),V2) -> a__U52(a__isNatList(V2))
r8: a__U52(tt()) -> tt()
r9: a__U61(tt(),V2) -> a__U62(a__isNatIList(V2))
r10: a__U62(tt()) -> tt()
r11: a__U71(tt(),L,N) -> a__U72(a__isNat(N),L)
r12: a__U72(tt(),L) -> s(a__length(mark(L)))
r13: a__U81(tt()) -> nil()
r14: a__U91(tt(),IL,M,N) -> a__U92(a__isNat(M),IL,M,N)
r15: a__U92(tt(),IL,M,N) -> a__U93(a__isNat(N),IL,M,N)
r16: a__U93(tt(),IL,M,N) -> cons(mark(N),take(M,IL))
r17: a__isNat(|0|()) -> tt()
r18: a__isNat(length(V1)) -> a__U11(a__isNatList(V1))
r19: a__isNat(s(V1)) -> a__U21(a__isNat(V1))
r20: a__isNatIList(V) -> a__U31(a__isNatList(V))
r21: a__isNatIList(zeros()) -> tt()
r22: a__isNatIList(cons(V1,V2)) -> a__U41(a__isNat(V1),V2)
r23: a__isNatList(nil()) -> tt()
r24: a__isNatList(cons(V1,V2)) -> a__U51(a__isNat(V1),V2)
r25: a__isNatList(take(V1,V2)) -> a__U61(a__isNat(V1),V2)
r26: a__length(nil()) -> |0|()
r27: a__length(cons(N,L)) -> a__U71(a__isNatList(L),L,N)
r28: a__take(|0|(),IL) -> a__U81(a__isNatIList(IL))
r29: a__take(s(M),cons(N,IL)) -> a__U91(a__isNatIList(IL),IL,M,N)
r30: mark(zeros()) -> a__zeros()
r31: mark(U11(X)) -> a__U11(mark(X))
r32: mark(U21(X)) -> a__U21(mark(X))
r33: mark(U31(X)) -> a__U31(mark(X))
r34: mark(U41(X1,X2)) -> a__U41(mark(X1),X2)
r35: mark(U42(X)) -> a__U42(mark(X))
r36: mark(isNatIList(X)) -> a__isNatIList(X)
r37: mark(U51(X1,X2)) -> a__U51(mark(X1),X2)
r38: mark(U52(X)) -> a__U52(mark(X))
r39: mark(isNatList(X)) -> a__isNatList(X)
r40: mark(U61(X1,X2)) -> a__U61(mark(X1),X2)
r41: mark(U62(X)) -> a__U62(mark(X))
r42: mark(U71(X1,X2,X3)) -> a__U71(mark(X1),X2,X3)
r43: mark(U72(X1,X2)) -> a__U72(mark(X1),X2)
r44: mark(isNat(X)) -> a__isNat(X)
r45: mark(length(X)) -> a__length(mark(X))
r46: mark(U81(X)) -> a__U81(mark(X))
r47: mark(U91(X1,X2,X3,X4)) -> a__U91(mark(X1),X2,X3,X4)
r48: mark(U92(X1,X2,X3,X4)) -> a__U92(mark(X1),X2,X3,X4)
r49: mark(U93(X1,X2,X3,X4)) -> a__U93(mark(X1),X2,X3,X4)
r50: mark(take(X1,X2)) -> a__take(mark(X1),mark(X2))
r51: mark(cons(X1,X2)) -> cons(mark(X1),X2)
r52: mark(|0|()) -> |0|()
r53: mark(tt()) -> tt()
r54: mark(s(X)) -> s(mark(X))
r55: mark(nil()) -> nil()
r56: a__zeros() -> zeros()
r57: a__U11(X) -> U11(X)
r58: a__U21(X) -> U21(X)
r59: a__U31(X) -> U31(X)
r60: a__U41(X1,X2) -> U41(X1,X2)
r61: a__U42(X) -> U42(X)
r62: a__isNatIList(X) -> isNatIList(X)
r63: a__U51(X1,X2) -> U51(X1,X2)
r64: a__U52(X) -> U52(X)
r65: a__isNatList(X) -> isNatList(X)
r66: a__U61(X1,X2) -> U61(X1,X2)
r67: a__U62(X) -> U62(X)
r68: a__U71(X1,X2,X3) -> U71(X1,X2,X3)
r69: a__U72(X1,X2) -> U72(X1,X2)
r70: a__isNat(X) -> isNat(X)
r71: a__length(X) -> length(X)
r72: a__U81(X) -> U81(X)
r73: a__U91(X1,X2,X3,X4) -> U91(X1,X2,X3,X4)
r74: a__U92(X1,X2,X3,X4) -> U92(X1,X2,X3,X4)
r75: a__U93(X1,X2,X3,X4) -> U93(X1,X2,X3,X4)
r76: a__take(X1,X2) -> take(X1,X2)

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__U41#(tt(),V2) -> a__isNatIList#(V2)
p2: a__isNatIList#(cons(V1,V2)) -> a__isNat#(V1)
p3: a__isNat#(s(V1)) -> a__isNat#(V1)
p4: a__isNat#(length(V1)) -> a__isNatList#(V1)
p5: a__isNatList#(take(V1,V2)) -> a__isNat#(V1)
p6: a__isNatList#(take(V1,V2)) -> a__U61#(a__isNat(V1),V2)
p7: a__U61#(tt(),V2) -> a__isNatIList#(V2)
p8: a__isNatIList#(cons(V1,V2)) -> a__U41#(a__isNat(V1),V2)
p9: a__isNatIList#(V) -> a__isNatList#(V)
p10: a__isNatList#(cons(V1,V2)) -> a__isNat#(V1)
p11: a__isNatList#(cons(V1,V2)) -> a__U51#(a__isNat(V1),V2)
p12: a__U51#(tt(),V2) -> a__isNatList#(V2)

and R consists of:

r1: a__zeros() -> cons(|0|(),zeros())
r2: a__U11(tt()) -> tt()
r3: a__U21(tt()) -> tt()
r4: a__U31(tt()) -> tt()
r5: a__U41(tt(),V2) -> a__U42(a__isNatIList(V2))
r6: a__U42(tt()) -> tt()
r7: a__U51(tt(),V2) -> a__U52(a__isNatList(V2))
r8: a__U52(tt()) -> tt()
r9: a__U61(tt(),V2) -> a__U62(a__isNatIList(V2))
r10: a__U62(tt()) -> tt()
r11: a__U71(tt(),L,N) -> a__U72(a__isNat(N),L)
r12: a__U72(tt(),L) -> s(a__length(mark(L)))
r13: a__U81(tt()) -> nil()
r14: a__U91(tt(),IL,M,N) -> a__U92(a__isNat(M),IL,M,N)
r15: a__U92(tt(),IL,M,N) -> a__U93(a__isNat(N),IL,M,N)
r16: a__U93(tt(),IL,M,N) -> cons(mark(N),take(M,IL))
r17: a__isNat(|0|()) -> tt()
r18: a__isNat(length(V1)) -> a__U11(a__isNatList(V1))
r19: a__isNat(s(V1)) -> a__U21(a__isNat(V1))
r20: a__isNatIList(V) -> a__U31(a__isNatList(V))
r21: a__isNatIList(zeros()) -> tt()
r22: a__isNatIList(cons(V1,V2)) -> a__U41(a__isNat(V1),V2)
r23: a__isNatList(nil()) -> tt()
r24: a__isNatList(cons(V1,V2)) -> a__U51(a__isNat(V1),V2)
r25: a__isNatList(take(V1,V2)) -> a__U61(a__isNat(V1),V2)
r26: a__length(nil()) -> |0|()
r27: a__length(cons(N,L)) -> a__U71(a__isNatList(L),L,N)
r28: a__take(|0|(),IL) -> a__U81(a__isNatIList(IL))
r29: a__take(s(M),cons(N,IL)) -> a__U91(a__isNatIList(IL),IL,M,N)
r30: mark(zeros()) -> a__zeros()
r31: mark(U11(X)) -> a__U11(mark(X))
r32: mark(U21(X)) -> a__U21(mark(X))
r33: mark(U31(X)) -> a__U31(mark(X))
r34: mark(U41(X1,X2)) -> a__U41(mark(X1),X2)
r35: mark(U42(X)) -> a__U42(mark(X))
r36: mark(isNatIList(X)) -> a__isNatIList(X)
r37: mark(U51(X1,X2)) -> a__U51(mark(X1),X2)
r38: mark(U52(X)) -> a__U52(mark(X))
r39: mark(isNatList(X)) -> a__isNatList(X)
r40: mark(U61(X1,X2)) -> a__U61(mark(X1),X2)
r41: mark(U62(X)) -> a__U62(mark(X))
r42: mark(U71(X1,X2,X3)) -> a__U71(mark(X1),X2,X3)
r43: mark(U72(X1,X2)) -> a__U72(mark(X1),X2)
r44: mark(isNat(X)) -> a__isNat(X)
r45: mark(length(X)) -> a__length(mark(X))
r46: mark(U81(X)) -> a__U81(mark(X))
r47: mark(U91(X1,X2,X3,X4)) -> a__U91(mark(X1),X2,X3,X4)
r48: mark(U92(X1,X2,X3,X4)) -> a__U92(mark(X1),X2,X3,X4)
r49: mark(U93(X1,X2,X3,X4)) -> a__U93(mark(X1),X2,X3,X4)
r50: mark(take(X1,X2)) -> a__take(mark(X1),mark(X2))
r51: mark(cons(X1,X2)) -> cons(mark(X1),X2)
r52: mark(|0|()) -> |0|()
r53: mark(tt()) -> tt()
r54: mark(s(X)) -> s(mark(X))
r55: mark(nil()) -> nil()
r56: a__zeros() -> zeros()
r57: a__U11(X) -> U11(X)
r58: a__U21(X) -> U21(X)
r59: a__U31(X) -> U31(X)
r60: a__U41(X1,X2) -> U41(X1,X2)
r61: a__U42(X) -> U42(X)
r62: a__isNatIList(X) -> isNatIList(X)
r63: a__U51(X1,X2) -> U51(X1,X2)
r64: a__U52(X) -> U52(X)
r65: a__isNatList(X) -> isNatList(X)
r66: a__U61(X1,X2) -> U61(X1,X2)
r67: a__U62(X) -> U62(X)
r68: a__U71(X1,X2,X3) -> U71(X1,X2,X3)
r69: a__U72(X1,X2) -> U72(X1,X2)
r70: a__isNat(X) -> isNat(X)
r71: a__length(X) -> length(X)
r72: a__U81(X) -> U81(X)
r73: a__U91(X1,X2,X3,X4) -> U91(X1,X2,X3,X4)
r74: a__U92(X1,X2,X3,X4) -> U92(X1,X2,X3,X4)
r75: a__U93(X1,X2,X3,X4) -> U93(X1,X2,X3,X4)
r76: a__take(X1,X2) -> take(X1,X2)

The set of usable rules consists of

  r2, r3, r4, r5, r6, r7, r8, r9, r10, r17, r18, r19, r20, r21, r22, r23, r24, r25, r57, r58, r59, r60, r61, r62, r63, r64, r65, r66, r67, r70

Take the reduction pair:

  lexicographic combination of reduction pairs:
  
    1. weighted path order
    
      base order:
    
        max/plus interpretations on natural numbers:
        
          a__U41#_A(x1,x2) = max{51, x1 + 4, x2 + 38}
          tt_A = 10
          a__isNatIList#_A(x1) = x1 + 37
          cons_A(x1,x2) = max{x1 + 50, x2 + 17}
          a__isNat#_A(x1) = x1 + 43
          s_A(x1) = x1 + 43
          length_A(x1) = x1 + 14
          a__isNatList#_A(x1) = x1 + 29
          take_A(x1,x2) = max{42, x1 + 40, x2 + 40}
          a__U61#_A(x1,x2) = max{70, x1, x2 + 69}
          a__isNat_A(x1) = x1 + 48
          a__U51#_A(x1,x2) = max{51, x1 + 30, x2 + 29}
          a__U42_A(x1) = max{9, x1 + 8}
          U42_A(x1) = x1
          a__U31_A(x1) = x1 + 3
          a__U41_A(x1,x2) = max{67, x1 + 18, x2 + 34}
          a__isNatIList_A(x1) = max{35, x1 + 17}
          U31_A(x1) = max{2, x1}
          U41_A(x1,x2) = max{x1 + 18, x2 + 34}
          a__U52_A(x1) = max{9, x1 + 2}
          a__U62_A(x1) = x1 + 3
          a__isNatList_A(x1) = max{13, x1 + 5}
          zeros_A = 11
          isNatIList_A(x1) = max{1, x1}
          U52_A(x1) = 0
          U62_A(x1) = max{2, x1 + 1}
          a__U51_A(x1,x2) = max{x1 + 2, x2 + 17}
          a__U61_A(x1,x2) = max{x1 - 8, x2 + 40}
          U51_A(x1,x2) = max{x1 + 1, x2 + 1}
          U61_A(x1,x2) = x2 + 40
          a__U11_A(x1) = max{48, x1 + 37}
          a__U21_A(x1) = max{91, x1 + 3}
          nil_A = 11
          U11_A(x1) = max{2, x1 + 1}
          U21_A(x1) = max{2, x1 + 1}
          isNatList_A(x1) = x1
          |0|_A = 0
          isNat_A(x1) = max{1, x1}
    
      precedence:
      
        a__U51# > U31 = U51 > a__U11 = U21 = isNatList > s > a__isNat# > cons > a__U42 = nil > U42 > take = a__U41 = a__isNatIList = a__isNatList = a__U61 > tt = a__isNat = U41 > U52 > U61 > isNatIList = isNat > a__isNatList# = |0| > a__U61# = a__U31 = a__U62 = zeros = a__U51 > a__U52 = U11 > a__U41# = a__isNatIList# = length = a__U21 > U62
      
      partial status:
    
        pi(a__U41#) = [1, 2]
        pi(tt) = []
        pi(a__isNatIList#) = [1]
        pi(cons) = [1, 2]
        pi(a__isNat#) = [1]
        pi(s) = [1]
        pi(length) = [1]
        pi(a__isNatList#) = [1]
        pi(take) = [1, 2]
        pi(a__U61#) = [1, 2]
        pi(a__isNat) = [1]
        pi(a__U51#) = [2]
        pi(a__U42) = [1]
        pi(U42) = [1]
        pi(a__U31) = []
        pi(a__U41) = [1, 2]
        pi(a__isNatIList) = [1]
        pi(U31) = [1]
        pi(U41) = [1, 2]
        pi(a__U52) = [1]
        pi(a__U62) = [1]
        pi(a__isNatList) = [1]
        pi(zeros) = []
        pi(isNatIList) = [1]
        pi(U52) = []
        pi(U62) = [1]
        pi(a__U51) = []
        pi(a__U61) = []
        pi(U51) = [1, 2]
        pi(U61) = [2]
        pi(a__U11) = [1]
        pi(a__U21) = [1]
        pi(nil) = []
        pi(U11) = [1]
        pi(U21) = [1]
        pi(isNatList) = [1]
        pi(|0|) = []
        pi(isNat) = [1]
    
    2. weighted path order
    
      base order:
    
        max/plus interpretations on natural numbers:
        
          a__U41#_A(x1,x2) = max{13, x1 + 4, x2 - 1}
          tt_A = 12
          a__isNatIList#_A(x1) = max{16, x1 - 4}
          cons_A(x1,x2) = max{x1 + 12, x2 + 12}
          a__isNat#_A(x1) = x1 + 2
          s_A(x1) = x1
          length_A(x1) = x1
          a__isNatList#_A(x1) = max{1, x1 - 5}
          take_A(x1,x2) = max{x1 + 23, x2 + 29}
          a__U61#_A(x1,x2) = max{x1 + 15, x2 + 24}
          a__isNat_A(x1) = max{9, x1 + 3}
          a__U51#_A(x1,x2) = x2 + 2
          a__U42_A(x1) = max{48, x1 + 34}
          U42_A(x1) = max{49, x1 + 35}
          a__U31_A(x1) = 44
          a__U41_A(x1,x2) = max{x1 + 35, x2 + 44}
          a__isNatIList_A(x1) = max{43, x1 + 11}
          U31_A(x1) = max{45, x1 - 1}
          U41_A(x1,x2) = max{x1 + 36, x2 + 45}
          a__U52_A(x1) = max{26, x1 + 16}
          a__U62_A(x1) = max{43, x1}
          a__isNatList_A(x1) = x1 + 12
          zeros_A = 0
          isNatIList_A(x1) = max{44, x1 + 12}
          U52_A(x1) = 27
          U62_A(x1) = max{44, x1 + 1}
          a__U51_A(x1,x2) = 25
          a__U61_A(x1,x2) = 42
          U51_A(x1,x2) = max{26, x1 + 3, x2 - 1}
          U61_A(x1,x2) = max{43, x2 - 1}
          a__U11_A(x1) = max{10, x1 + 1}
          a__U21_A(x1) = max{10, x1 + 1}
          nil_A = 1
          U11_A(x1) = max{11, x1 + 2}
          U21_A(x1) = max{11, x1 + 2}
          isNatList_A(x1) = x1 + 13
          |0|_A = 10
          isNat_A(x1) = max{10, x1 + 4}
    
      precedence:
      
        a__U41# = tt = a__isNat# = s = length > a__isNatIList# = a__isNatList# > cons = a__U61# > take = a__isNat > a__U51# = a__U42 = U42 = a__U31 = a__U41 = a__isNatIList = U31 = U41 = a__U52 = a__U62 = a__isNatList = zeros = isNatIList = U52 = U62 = a__U51 = a__U61 = U51 = U61 = a__U11 = a__U21 = nil = U11 = U21 = isNatList = |0| = isNat
      
      partial status:
    
        pi(a__U41#) = [1]
        pi(tt) = []
        pi(a__isNatIList#) = []
        pi(cons) = []
        pi(a__isNat#) = [1]
        pi(s) = [1]
        pi(length) = [1]
        pi(a__isNatList#) = []
        pi(take) = []
        pi(a__U61#) = [2]
        pi(a__isNat) = [1]
        pi(a__U51#) = [2]
        pi(a__U42) = []
        pi(U42) = []
        pi(a__U31) = []
        pi(a__U41) = [1, 2]
        pi(a__isNatIList) = [1]
        pi(U31) = []
        pi(U41) = [1]
        pi(a__U52) = [1]
        pi(a__U62) = [1]
        pi(a__isNatList) = [1]
        pi(zeros) = []
        pi(isNatIList) = []
        pi(U52) = []
        pi(U62) = []
        pi(a__U51) = []
        pi(a__U61) = []
        pi(U51) = []
        pi(U61) = []
        pi(a__U11) = []
        pi(a__U21) = []
        pi(nil) = []
        pi(U11) = []
        pi(U21) = []
        pi(isNatList) = []
        pi(|0|) = []
        pi(isNat) = []
    

The next rules are strictly ordered:

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

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