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:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^2
        order: lexicographic order
        interpretations:
          a__U93#_A(x1,x2,x3,x4) = ((0,0),(1,0)) x4 + (7,15)
          tt_A() = (0,14)
          mark#_A(x1) = ((0,0),(1,0)) x1 + (7,13)
          s_A(x1) = ((1,0),(0,0)) x1 + (0,12)
          cons_A(x1,x2) = ((1,0),(0,0)) x1 + x2
          take_A(x1,x2) = ((1,0),(1,0)) x1 + ((1,0),(1,0)) x2 + (6,10)
          a__take#_A(x1,x2) = ((0,0),(1,0)) x2 + (7,18)
          mark_A(x1) = x1 + (0,11)
          a__U91#_A(x1,x2,x3,x4) = ((0,0),(1,0)) x4 + (7,16)
          a__isNatIList_A(x1) = ((0,0),(1,0)) x1 + (0,18)
          a__U92#_A(x1,x2,x3,x4) = ((0,0),(1,0)) x4 + (7,16)
          a__isNat_A(x1) = ((0,0),(1,0)) x1 + (0,17)
          U93_A(x1,x2,x3,x4) = ((1,0),(0,0)) x1 + ((1,0),(1,0)) x2 + ((1,0),(1,0)) x3 + ((1,0),(0,0)) x4 + (6,10)
          U92_A(x1,x2,x3,x4) = ((1,0),(0,0)) x1 + ((1,0),(1,0)) x2 + ((1,0),(1,0)) x3 + ((1,0),(0,0)) x4 + (6,20)
          U91_A(x1,x2,x3,x4) = ((1,0),(0,0)) x1 + ((1,0),(1,0)) x2 + ((1,0),(1,0)) x3 + ((1,0),(0,0)) x4 + (6,9)
          U81_A(x1) = x1 + (2,1)
          length_A(x1) = ((1,0),(0,0)) x1 + (0,15)
          a__length#_A(x1) = ((0,0),(1,0)) x1 + (7,13)
          a__U71#_A(x1,x2,x3) = ((0,0),(1,0)) x1 + ((0,0),(1,0)) x2 + ((0,0),(1,0)) x3 + (7,13)
          a__isNatList_A(x1) = ((0,0),(1,0)) x1 + (0,17)
          a__U72#_A(x1,x2) = ((0,0),(1,0)) x2 + (7,13)
          U72_A(x1,x2) = ((1,0),(0,0)) x1 + ((1,0),(0,0)) x2 + (0,3)
          U71_A(x1,x2,x3) = ((1,0),(0,0)) x1 + ((1,0),(0,0)) x2 + ((1,0),(0,0)) x3 + (0,10)
          U62_A(x1) = ((1,0),(0,0)) x1 + (0,4)
          U61_A(x1,x2) = ((1,0),(0,0)) x1 + (0,15)
          U52_A(x1) = ((1,0),(0,0)) x1 + (0,3)
          U51_A(x1,x2) = ((1,0),(1,0)) x1 + (0,6)
          U42_A(x1) = ((1,0),(0,0)) x1 + (0,3)
          U41_A(x1,x2) = x1
          U31_A(x1) = x1 + (0,1)
          U21_A(x1) = ((1,0),(0,0)) x1 + (0,15)
          U11_A(x1) = ((1,0),(0,0)) x1 + (0,14)
          a__zeros_A() = (1,13)
          |0|_A() = (0,20)
          zeros_A() = (1,13)
          a__U11_A(x1) = ((1,0),(0,0)) x1 + (0,14)
          a__U21_A(x1) = ((1,0),(0,0)) x1 + (0,15)
          a__U31_A(x1) = x1 + (0,1)
          a__U41_A(x1,x2) = x1
          a__U42_A(x1) = ((1,0),(0,0)) x1 + (0,14)
          a__U51_A(x1,x2) = ((1,0),(1,0)) x1 + (0,17)
          a__U52_A(x1) = ((1,0),(0,0)) x1 + (0,14)
          a__U61_A(x1,x2) = ((1,0),(0,0)) x1 + (0,15)
          a__U62_A(x1) = ((1,0),(0,0)) x1 + (0,14)
          a__U71_A(x1,x2,x3) = ((1,0),(0,0)) x1 + ((1,0),(0,0)) x2 + ((1,0),(0,0)) x3 + (0,14)
          a__U72_A(x1,x2) = ((1,0),(0,0)) x1 + ((1,0),(0,0)) x2 + (0,13)
          a__length_A(x1) = ((1,0),(0,0)) x1 + (0,21)
          a__U81_A(x1) = x1 + (2,1)
          nil_A() = (1,13)
          a__U91_A(x1,x2,x3,x4) = ((1,0),(0,0)) x1 + ((1,0),(1,0)) x2 + ((1,0),(1,0)) x3 + ((1,0),(0,0)) x4 + (6,20)
          a__U92_A(x1,x2,x3,x4) = ((1,0),(0,0)) x1 + ((1,0),(1,0)) x2 + ((1,0),(1,0)) x3 + ((1,0),(0,0)) x4 + (6,20)
          a__U93_A(x1,x2,x3,x4) = ((1,0),(0,0)) x1 + ((1,0),(1,0)) x2 + ((1,0),(1,0)) x3 + ((1,0),(0,0)) x4 + (6,10)
          a__take_A(x1,x2) = ((1,0),(1,0)) x1 + ((1,0),(1,0)) x2 + (6,20)
          isNatIList_A(x1) = ((0,0),(1,0)) x1 + (0,8)
          isNatList_A(x1) = ((0,0),(1,0)) x1 + (0,16)
          isNat_A(x1) = ((0,0),(1,0)) x1 + (0,7)
  
    precedence:
    
      U71 = a__U71 > U52 > mark# = a__length# = a__U71# = a__U72# > a__take# = a__U91# > s = U72 = a__U72 > a__isNatList = isNatList > tt = mark = a__isNatIList = U61 = a__zeros = a__U11 = a__U21 = a__U31 = a__U41 = a__U42 = a__U51 = a__U52 = a__U61 = a__U62 > U93 = U91 = zeros = a__U91 = a__U92 = a__U93 = a__take > take > a__isNat = U11 > cons > |0| = isNat > U42 > length = U41 = a__length = isNatIList > U31 > a__U93# = a__U92# = U92 = U81 = U62 = U51 = U21 = a__U81 = nil
    
    partial status:
  
      pi(a__U93#) = []
      pi(tt) = []
      pi(mark#) = []
      pi(s) = []
      pi(cons) = [2]
      pi(take) = []
      pi(a__take#) = []
      pi(mark) = [1]
      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:

  p6

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


-- 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)) -> a__U71#(mark(X1),X2,X3)
p12: a__U71#(tt(),L,N) -> a__U72#(a__isNat(N),L)
p13: a__U72#(tt(),L) -> a__length#(mark(L))
p14: a__length#(cons(N,L)) -> a__U71#(a__isNatList(L),L,N)
p15: a__U72#(tt(),L) -> mark#(L)
p16: mark#(U71(X1,X2,X3)) -> mark#(X1)
p17: mark#(U72(X1,X2)) -> a__U72#(mark(X1),X2)
p18: mark#(U72(X1,X2)) -> mark#(X1)
p19: mark#(length(X)) -> a__length#(mark(X))
p20: mark#(length(X)) -> mark#(X)
p21: mark#(U81(X)) -> mark#(X)
p22: mark#(U91(X1,X2,X3,X4)) -> a__U91#(mark(X1),X2,X3,X4)
p23: a__U91#(tt(),IL,M,N) -> a__U92#(a__isNat(M),IL,M,N)
p24: a__U92#(tt(),IL,M,N) -> a__U93#(a__isNat(N),IL,M,N)
p25: mark#(U91(X1,X2,X3,X4)) -> mark#(X1)
p26: mark#(U92(X1,X2,X3,X4)) -> a__U92#(mark(X1),X2,X3,X4)
p27: mark#(U92(X1,X2,X3,X4)) -> mark#(X1)
p28: mark#(U93(X1,X2,X3,X4)) -> a__U93#(mark(X1),X2,X3,X4)
p29: mark#(U93(X1,X2,X3,X4)) -> mark#(X1)
p30: mark#(take(X1,X2)) -> mark#(X1)
p31: mark#(take(X1,X2)) -> mark#(X2)
p32: mark#(cons(X1,X2)) -> mark#(X1)
p33: 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:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^2
        order: lexicographic order
        interpretations:
          a__U93#_A(x1,x2,x3,x4) = ((1,0),(1,0)) x4 + (6,7)
          tt_A() = (0,3)
          mark#_A(x1) = ((1,0),(1,0)) x1 + (5,10)
          U11_A(x1) = ((1,0),(0,0)) x1 + (0,3)
          U21_A(x1) = x1
          U31_A(x1) = x1 + (0,1)
          U41_A(x1,x2) = x1 + ((0,0),(1,0)) x2
          U42_A(x1) = ((1,0),(0,0)) x1 + (0,3)
          U51_A(x1,x2) = ((1,0),(0,0)) x1 + ((0,0),(1,0)) x2
          U52_A(x1) = x1
          U61_A(x1,x2) = x1 + ((0,0),(1,0)) x2 + (0,4)
          U62_A(x1) = x1 + (0,4)
          U71_A(x1,x2,x3) = x1 + ((1,0),(0,0)) x2 + ((1,0),(0,0)) x3 + (6,0)
          a__U71#_A(x1,x2,x3) = ((1,0),(0,0)) x1 + ((1,0),(0,0)) x2 + ((1,0),(0,0)) x3 + (7,11)
          mark_A(x1) = ((1,0),(1,1)) x1
          a__U72#_A(x1,x2) = ((1,0),(0,0)) x1 + ((1,0),(0,0)) x2 + (7,11)
          a__isNat_A(x1) = (0,3)
          a__length#_A(x1) = ((1,0),(0,0)) x1 + (7,11)
          cons_A(x1,x2) = ((1,0),(1,0)) x1 + ((1,0),(1,0)) x2
          a__isNatList_A(x1) = ((0,0),(1,0)) x1
          U72_A(x1,x2) = ((1,0),(0,0)) x1 + ((1,0),(0,0)) x2 + (6,1)
          length_A(x1) = ((1,0),(1,1)) x1 + (6,0)
          U81_A(x1) = ((1,0),(0,0)) x1 + (4,9)
          U91_A(x1,x2,x3,x4) = ((1,0),(1,0)) x1 + ((1,0),(1,0)) x2 + ((1,0),(1,0)) x3 + ((1,0),(0,0)) x4 + (7,4)
          a__U91#_A(x1,x2,x3,x4) = ((1,0),(0,0)) x2 + ((1,0),(0,0)) x3 + ((1,0),(0,0)) x4 + (9,5)
          a__U92#_A(x1,x2,x3,x4) = ((0,0),(1,0)) x2 + ((1,0),(0,0)) x3 + ((1,0),(0,0)) x4 + (8,6)
          U92_A(x1,x2,x3,x4) = ((1,0),(1,1)) x1 + ((1,0),(1,0)) x2 + ((1,0),(0,0)) x3 + ((1,0),(1,0)) x4 + (7,0)
          U93_A(x1,x2,x3,x4) = ((1,0),(0,0)) x1 + ((1,0),(1,0)) x2 + ((1,0),(0,0)) x3 + ((1,0),(1,0)) x4 + (7,2)
          take_A(x1,x2) = ((1,0),(0,0)) x1 + ((1,0),(1,0)) x2 + (7,11)
          s_A(x1) = ((1,0),(0,0)) x1 + (0,1)
          a__zeros_A() = (4,4)
          |0|_A() = (0,0)
          zeros_A() = (4,3)
          a__U11_A(x1) = ((1,0),(0,0)) x1 + (0,3)
          a__U21_A(x1) = x1
          a__U31_A(x1) = x1 + (0,1)
          a__U41_A(x1,x2) = x1 + ((0,0),(1,0)) x2
          a__U42_A(x1) = ((1,0),(0,0)) x1 + (0,3)
          a__isNatIList_A(x1) = ((0,0),(1,0)) x1 + (0,3)
          a__U51_A(x1,x2) = ((1,0),(0,0)) x1 + ((0,0),(1,0)) x2
          a__U52_A(x1) = x1
          a__U61_A(x1,x2) = x1 + ((0,0),(1,0)) x2 + (0,4)
          a__U62_A(x1) = x1 + (0,4)
          a__U71_A(x1,x2,x3) = x1 + ((1,0),(0,0)) x2 + ((1,0),(0,0)) x3 + (6,0)
          a__U72_A(x1,x2) = ((1,0),(0,0)) x1 + ((1,0),(0,0)) x2 + (6,2)
          a__length_A(x1) = ((1,0),(1,1)) x1 + (6,0)
          a__U81_A(x1) = ((1,0),(0,0)) x1 + (4,12)
          nil_A() = (3,3)
          a__U91_A(x1,x2,x3,x4) = ((1,0),(1,0)) x1 + ((1,0),(1,0)) x2 + ((1,0),(1,0)) x3 + ((1,0),(1,0)) x4 + (7,10)
          a__U92_A(x1,x2,x3,x4) = ((1,0),(1,1)) x1 + ((1,0),(1,0)) x2 + ((1,0),(1,0)) x3 + ((1,0),(1,0)) x4 + (7,6)
          a__U93_A(x1,x2,x3,x4) = ((1,0),(0,0)) x1 + ((1,0),(1,0)) x2 + ((1,0),(1,0)) x3 + ((1,0),(1,0)) x4 + (7,8)
          a__take_A(x1,x2) = ((1,0),(1,0)) x1 + ((1,0),(1,0)) x2 + (7,11)
          isNatIList_A(x1) = ((0,0),(1,0)) x1 + (0,3)
          isNatList_A(x1) = ((0,0),(1,0)) x1
          isNat_A(x1) = (0,3)
  
    precedence:
    
      U93 > mark# = a__U91# > a__U93# > U72 = a__U72 > a__U71# = a__U72# = a__length# > U31 = U41 = U61 = U71 = mark = a__isNatList = length = U81 = U91 = take = s = a__U31 = a__U41 = a__isNatIList = a__U51 = a__U61 = a__U71 = a__length = a__U81 = a__U91 = a__U92 = a__U93 = a__take = isNatIList > U42 = a__U42 > cons = a__zeros = a__U62 > tt = U21 = U51 = U52 = U62 = a__isNat = a__U92# = |0| = zeros = a__U11 = a__U21 = a__U52 = nil = isNatList = isNat > U11 = U92
    
    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(a__U71#) = []
      pi(mark) = []
      pi(a__U72#) = []
      pi(a__isNat) = []
      pi(a__length#) = []
      pi(cons) = []
      pi(a__isNatList) = []
      pi(U72) = []
      pi(length) = []
      pi(U81) = []
      pi(U91) = []
      pi(a__U91#) = []
      pi(a__U92#) = []
      pi(U92) = []
      pi(U93) = []
      pi(take) = []
      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__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:

  p19

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#(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)) -> a__U71#(mark(X1),X2,X3)
p12: a__U71#(tt(),L,N) -> a__U72#(a__isNat(N),L)
p13: a__U72#(tt(),L) -> a__length#(mark(L))
p14: a__length#(cons(N,L)) -> a__U71#(a__isNatList(L),L,N)
p15: a__U72#(tt(),L) -> mark#(L)
p16: mark#(U71(X1,X2,X3)) -> mark#(X1)
p17: mark#(U72(X1,X2)) -> a__U72#(mark(X1),X2)
p18: mark#(U72(X1,X2)) -> mark#(X1)
p19: mark#(length(X)) -> mark#(X)
p20: mark#(U81(X)) -> mark#(X)
p21: mark#(U91(X1,X2,X3,X4)) -> a__U91#(mark(X1),X2,X3,X4)
p22: a__U91#(tt(),IL,M,N) -> a__U92#(a__isNat(M),IL,M,N)
p23: a__U92#(tt(),IL,M,N) -> a__U93#(a__isNat(N),IL,M,N)
p24: mark#(U91(X1,X2,X3,X4)) -> mark#(X1)
p25: mark#(U92(X1,X2,X3,X4)) -> a__U92#(mark(X1),X2,X3,X4)
p26: mark#(U92(X1,X2,X3,X4)) -> mark#(X1)
p27: mark#(U93(X1,X2,X3,X4)) -> a__U93#(mark(X1),X2,X3,X4)
p28: mark#(U93(X1,X2,X3,X4)) -> mark#(X1)
p29: mark#(take(X1,X2)) -> mark#(X1)
p30: mark#(take(X1,X2)) -> mark#(X2)
p31: mark#(cons(X1,X2)) -> mark#(X1)
p32: 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, p16, p17, p18, p19, p20, p21, p22, p23, p24, p25, p26, p27, p28, p29, p30, p31, p32}


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

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^2
        order: lexicographic order
        interpretations:
          a__U93#_A(x1,x2,x3,x4) = x1 + ((1,0),(1,1)) x3 + ((1,0),(1,1)) x4 + (1,10)
          tt_A() = (7,15)
          mark#_A(x1) = ((1,0),(0,0)) x1 + (5,21)
          s_A(x1) = ((1,0),(0,0)) x1 + (0,17)
          cons_A(x1,x2) = ((1,0),(0,0)) x1 + ((1,0),(1,0)) x2
          take_A(x1,x2) = ((1,0),(0,0)) x1 + ((1,0),(1,1)) x2 + (43,0)
          U93_A(x1,x2,x3,x4) = ((1,0),(0,0)) x1 + ((1,0),(1,0)) x2 + ((1,0),(1,0)) x3 + ((1,0),(0,0)) x4 + (36,9)
          mark_A(x1) = ((1,0),(1,1)) x1
          U92_A(x1,x2,x3,x4) = ((1,0),(1,1)) x1 + ((1,0),(0,0)) x2 + ((1,0),(1,0)) x3 + ((1,0),(1,0)) x4 + (36,22)
          a__U92#_A(x1,x2,x3,x4) = ((0,0),(1,0)) x1 + ((1,0),(1,1)) x2 + ((1,0),(1,1)) x3 + x4 + (9,20)
          a__isNat_A(x1) = (7,16)
          U91_A(x1,x2,x3,x4) = x1 + ((1,0),(1,0)) x2 + ((1,0),(1,0)) x3 + ((1,0),(1,0)) x4 + (36,21)
          a__U91#_A(x1,x2,x3,x4) = x1 + ((1,0),(1,1)) x2 + ((1,0),(1,1)) x3 + ((1,0),(1,1)) x4 + (3,22)
          U81_A(x1) = ((1,0),(0,0)) x1 + (36,3)
          length_A(x1) = ((1,0),(0,0)) x1 + (14,8)
          U72_A(x1,x2) = x1 + ((1,0),(0,0)) x2 + (7,1)
          a__U72#_A(x1,x2) = ((1,0),(0,0)) x2 + (6,17)
          U71_A(x1,x2,x3) = ((1,0),(1,0)) x1 + ((1,0),(0,0)) x2 + ((1,0),(0,0)) x3 + (7,13)
          a__U71#_A(x1,x2,x3) = ((1,0),(0,0)) x2 + (6,17)
          a__length#_A(x1) = ((1,0),(0,0)) x1 + (6,17)
          a__isNatList_A(x1) = ((0,0),(1,0)) x1 + (7,8)
          U62_A(x1) = x1 + (0,6)
          U61_A(x1,x2) = ((1,0),(0,0)) x1 + ((0,0),(1,0)) x2 + (0,17)
          U52_A(x1) = x1
          U51_A(x1,x2) = ((1,0),(1,0)) x1 + ((0,0),(1,0)) x2 + (0,1)
          U42_A(x1) = x1
          U41_A(x1,x2) = ((1,0),(0,0)) x1 + ((0,0),(1,0)) x2 + (0,10)
          U31_A(x1) = x1 + (0,1)
          U21_A(x1) = ((1,0),(0,0)) x1 + (0,15)
          U11_A(x1) = ((1,0),(0,0)) x1 + (0,15)
          a__zeros_A() = (6,7)
          |0|_A() = (0,1)
          zeros_A() = (6,2)
          a__U11_A(x1) = ((1,0),(0,0)) x1 + (0,15)
          a__U21_A(x1) = ((1,0),(0,0)) x1 + (0,15)
          a__U31_A(x1) = x1 + (0,1)
          a__U41_A(x1,x2) = ((1,0),(0,0)) x1 + ((0,0),(1,0)) x2 + (0,10)
          a__U42_A(x1) = x1
          a__isNatIList_A(x1) = ((0,0),(1,0)) x1 + (7,10)
          a__U51_A(x1,x2) = ((1,0),(1,0)) x1 + ((0,0),(1,0)) x2 + (0,1)
          a__U52_A(x1) = x1
          a__U61_A(x1,x2) = ((1,0),(0,0)) x1 + ((0,0),(1,0)) x2 + (0,17)
          a__U62_A(x1) = x1 + (0,6)
          a__U71_A(x1,x2,x3) = ((1,0),(1,0)) x1 + ((1,0),(0,0)) x2 + ((1,0),(0,0)) x3 + (7,13)
          a__U72_A(x1,x2) = x1 + ((1,0),(0,0)) x2 + (7,3)
          a__length_A(x1) = ((1,0),(0,0)) x1 + (14,21)
          a__U81_A(x1) = ((1,0),(0,0)) x1 + (36,3)
          nil_A() = (43,2)
          a__U91_A(x1,x2,x3,x4) = x1 + ((1,0),(1,0)) x2 + ((1,0),(1,0)) x3 + ((1,0),(1,0)) x4 + (36,31)
          a__U92_A(x1,x2,x3,x4) = ((1,0),(1,1)) x1 + ((1,0),(1,0)) x2 + ((1,0),(1,0)) x3 + ((1,0),(1,0)) x4 + (36,23)
          a__U93_A(x1,x2,x3,x4) = ((1,0),(0,0)) x1 + ((1,0),(1,0)) x2 + ((1,0),(1,0)) x3 + ((1,0),(1,0)) x4 + (36,44)
          a__take_A(x1,x2) = ((1,0),(1,0)) x1 + ((1,0),(1,1)) x2 + (43,42)
          isNatIList_A(x1) = ((0,0),(1,0)) x1 + (7,10)
          isNatList_A(x1) = ((0,0),(1,0)) x1 + (7,2)
          isNat_A(x1) = (7,9)
  
    precedence:
    
      mark# = a__U91# = a__U72# = a__U71# = a__length# > length > U21 = a__U21 > U93 > a__isNatIList = isNatIList > mark = a__isNat = U62 = U61 = U31 = a__U11 = a__U31 = a__U41 = a__U42 = a__U61 = a__U62 = a__length = a__U93 = isNat > a__U93# = a__U92# = a__zeros > cons > tt = U71 = a__isNatList = a__U51 = a__U52 = a__U71 > take = U91 = U81 = a__U81 = nil = a__U91 = a__take > s = a__U72 > U72 > U51 = isNatList > U52 > U92 = U11 = zeros = a__U92 > U42 = U41 = |0|
    
    partial status:
  
      pi(a__U93#) = []
      pi(tt) = []
      pi(mark#) = []
      pi(s) = []
      pi(cons) = []
      pi(take) = []
      pi(U93) = []
      pi(mark) = [1]
      pi(U92) = []
      pi(a__U92#) = [4]
      pi(a__isNat) = []
      pi(U91) = []
      pi(a__U91#) = []
      pi(U81) = []
      pi(length) = []
      pi(U72) = []
      pi(a__U72#) = []
      pi(U71) = []
      pi(a__U71#) = []
      pi(a__length#) = []
      pi(a__isNatList) = []
      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__isNatIList) = []
      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) = [2]
      pi(isNatIList) = []
      pi(isNatList) = []
      pi(isNat) = []

The next rules are strictly ordered:

  p10

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


-- Reduction pair.

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

p1: 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)) -> a__U71#(mark(X1),X2,X3)
p12: a__U71#(tt(),L,N) -> a__U72#(a__isNat(N),L)
p13: a__U72#(tt(),L) -> a__length#(mark(L))
p14: a__length#(cons(N,L)) -> a__U71#(a__isNatList(L),L,N)
p15: a__U72#(tt(),L) -> mark#(L)
p16: mark#(U71(X1,X2,X3)) -> mark#(X1)
p17: mark#(U72(X1,X2)) -> a__U72#(mark(X1),X2)
p18: mark#(U72(X1,X2)) -> mark#(X1)
p19: mark#(length(X)) -> mark#(X)
p20: mark#(U81(X)) -> mark#(X)
p21: mark#(U91(X1,X2,X3,X4)) -> mark#(X1)
p22: mark#(U92(X1,X2,X3,X4)) -> mark#(X1)
p23: mark#(U93(X1,X2,X3,X4)) -> a__U93#(mark(X1),X2,X3,X4)
p24: mark#(U93(X1,X2,X3,X4)) -> mark#(X1)
p25: mark#(take(X1,X2)) -> mark#(X1)
p26: mark#(take(X1,X2)) -> mark#(X2)
p27: mark#(cons(X1,X2)) -> mark#(X1)
p28: 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:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^2
        order: lexicographic order
        interpretations:
          a__U93#_A(x1,x2,x3,x4) = ((0,0),(1,0)) x1 + ((0,0),(1,0)) x2 + ((0,0),(1,0)) x4 + (14,46)
          tt_A() = (15,31)
          mark#_A(x1) = ((0,0),(1,0)) x1 + (14,30)
          U11_A(x1) = ((1,0),(1,0)) x1 + (0,16)
          U21_A(x1) = ((1,0),(0,0)) x1 + (0,31)
          U31_A(x1) = x1 + (0,1)
          U41_A(x1,x2) = ((1,0),(0,0)) x1 + ((0,0),(1,0)) x2 + (0,28)
          U42_A(x1) = x1
          U51_A(x1,x2) = ((1,0),(1,0)) x1 + ((0,0),(1,0)) x2
          U52_A(x1) = x1
          U61_A(x1,x2) = ((1,0),(1,1)) x1 + (0,1)
          U62_A(x1) = ((1,0),(0,0)) x1 + (0,47)
          U71_A(x1,x2,x3) = ((1,0),(1,0)) x1 + ((1,0),(1,0)) x2 + ((1,0),(0,0)) x3 + (3,9)
          a__U71#_A(x1,x2,x3) = ((0,0),(1,0)) x1 + ((0,0),(1,0)) x2 + ((0,0),(1,0)) x3 + (14,17)
          mark_A(x1) = x1
          a__U72#_A(x1,x2) = ((0,0),(1,0)) x2 + (14,32)
          a__isNat_A(x1) = (15,31)
          a__length#_A(x1) = ((0,0),(1,0)) x1 + (14,32)
          cons_A(x1,x2) = ((1,0),(1,0)) x1 + ((1,0),(1,0)) x2 + (0,25)
          a__isNatList_A(x1) = ((0,0),(1,0)) x1 + (15,15)
          U72_A(x1,x2) = ((1,0),(0,0)) x1 + ((1,0),(1,0)) x2 + (3,23)
          length_A(x1) = x1 + (18,0)
          U81_A(x1) = ((1,0),(0,0)) x1 + (1,28)
          U91_A(x1,x2,x3,x4) = ((1,0),(1,0)) x1 + ((1,0),(1,0)) x2 + ((1,0),(1,0)) x3 + ((1,0),(1,0)) x4 + (17,45)
          U92_A(x1,x2,x3,x4) = ((1,0),(1,0)) x1 + ((1,0),(1,0)) x2 + ((1,0),(1,0)) x3 + ((1,0),(1,0)) x4 + (17,44)
          U93_A(x1,x2,x3,x4) = ((1,0),(1,0)) x1 + ((1,0),(1,0)) x2 + ((1,0),(1,0)) x3 + ((1,0),(1,0)) x4 + (17,43)
          take_A(x1,x2) = ((1,0),(1,1)) x1 + ((1,0),(1,0)) x2 + (32,57)
          s_A(x1) = ((1,0),(1,0)) x1 + (0,4)
          a__zeros_A() = (4,30)
          |0|_A() = (0,29)
          zeros_A() = (4,30)
          a__U11_A(x1) = ((1,0),(1,0)) x1 + (0,16)
          a__U21_A(x1) = ((1,0),(0,0)) x1 + (0,31)
          a__U31_A(x1) = x1 + (0,1)
          a__U41_A(x1,x2) = ((1,0),(0,0)) x1 + ((0,0),(1,0)) x2 + (0,28)
          a__U42_A(x1) = x1
          a__isNatIList_A(x1) = ((0,0),(1,0)) x1 + (15,28)
          a__U51_A(x1,x2) = ((1,0),(1,0)) x1 + ((0,0),(1,0)) x2
          a__U52_A(x1) = x1
          a__U61_A(x1,x2) = ((1,0),(1,1)) x1 + (0,1)
          a__U62_A(x1) = ((1,0),(0,0)) x1 + (0,47)
          a__U71_A(x1,x2,x3) = ((1,0),(1,0)) x1 + ((1,0),(1,0)) x2 + ((1,0),(0,0)) x3 + (3,9)
          a__U72_A(x1,x2) = ((1,0),(0,0)) x1 + ((1,0),(1,0)) x2 + (3,23)
          a__length_A(x1) = x1 + (18,0)
          a__U81_A(x1) = ((1,0),(0,0)) x1 + (1,28)
          nil_A() = (16,0)
          a__U91_A(x1,x2,x3,x4) = ((1,0),(1,0)) x1 + ((1,0),(1,0)) x2 + ((1,0),(1,0)) x3 + ((1,0),(1,0)) x4 + (17,45)
          a__U92_A(x1,x2,x3,x4) = ((1,0),(1,0)) x1 + ((1,0),(1,0)) x2 + ((1,0),(1,0)) x3 + ((1,0),(1,0)) x4 + (17,44)
          a__U93_A(x1,x2,x3,x4) = ((1,0),(1,0)) x1 + ((1,0),(1,0)) x2 + ((1,0),(1,0)) x3 + ((1,0),(1,0)) x4 + (17,43)
          a__take_A(x1,x2) = ((1,0),(1,1)) x1 + ((1,0),(1,0)) x2 + (32,57)
          isNatIList_A(x1) = ((0,0),(1,0)) x1 + (15,28)
          isNatList_A(x1) = ((0,0),(1,0)) x1 + (15,15)
          isNat_A(x1) = (15,31)
  
    precedence:
    
      a__isNatIList = isNatIList > U31 = U41 = a__U31 = a__U41 > tt = mark# = U42 = U51 = mark = a__isNat = a__isNatList = U92 = take = a__U11 = a__U21 = a__U42 = a__U51 = a__U52 = a__U61 = a__U62 = a__U91 = a__U92 = a__U93 = a__take > a__U93# > U21 > a__U71# = a__U72# = a__length# > U71 = length = a__U71 = a__length > U72 = U91 = a__U72 > U52 > isNat > U62 > |0| > a__zeros > cons > isNatList > U61 > s = zeros > U93 = a__U81 > nil > U81 > U11
    
    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(a__U71#) = []
      pi(mark) = [1]
      pi(a__U72#) = []
      pi(a__isNat) = []
      pi(a__length#) = []
      pi(cons) = []
      pi(a__isNatList) = []
      pi(U72) = []
      pi(length) = []
      pi(U81) = []
      pi(U91) = []
      pi(U92) = []
      pi(U93) = []
      pi(take) = []
      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__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) = [1]
      pi(isNatIList) = []
      pi(isNatList) = []
      pi(isNat) = []

The next rules are strictly ordered:

  p23

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#(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)) -> a__U71#(mark(X1),X2,X3)
p12: a__U71#(tt(),L,N) -> a__U72#(a__isNat(N),L)
p13: a__U72#(tt(),L) -> a__length#(mark(L))
p14: a__length#(cons(N,L)) -> a__U71#(a__isNatList(L),L,N)
p15: a__U72#(tt(),L) -> mark#(L)
p16: mark#(U71(X1,X2,X3)) -> mark#(X1)
p17: mark#(U72(X1,X2)) -> a__U72#(mark(X1),X2)
p18: mark#(U72(X1,X2)) -> mark#(X1)
p19: mark#(length(X)) -> mark#(X)
p20: mark#(U81(X)) -> mark#(X)
p21: mark#(U91(X1,X2,X3,X4)) -> mark#(X1)
p22: mark#(U92(X1,X2,X3,X4)) -> mark#(X1)
p23: mark#(U93(X1,X2,X3,X4)) -> mark#(X1)
p24: mark#(take(X1,X2)) -> mark#(X1)
p25: mark#(take(X1,X2)) -> mark#(X2)
p26: mark#(cons(X1,X2)) -> mark#(X1)
p27: 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:

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


-- 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#(U72(X1,X2)) -> a__U72#(mark(X1),X2)
p13: a__U72#(tt(),L) -> mark#(L)
p14: mark#(U71(X1,X2,X3)) -> mark#(X1)
p15: mark#(U71(X1,X2,X3)) -> a__U71#(mark(X1),X2,X3)
p16: a__U71#(tt(),L,N) -> a__U72#(a__isNat(N),L)
p17: a__U72#(tt(),L) -> a__length#(mark(L))
p18: a__length#(cons(N,L)) -> a__U71#(a__isNatList(L),L,N)
p19: mark#(U62(X)) -> mark#(X)
p20: mark#(U61(X1,X2)) -> mark#(X1)
p21: mark#(U52(X)) -> mark#(X)
p22: mark#(U51(X1,X2)) -> mark#(X1)
p23: mark#(U42(X)) -> mark#(X)
p24: mark#(U41(X1,X2)) -> mark#(X1)
p25: mark#(U31(X)) -> mark#(X)
p26: 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

  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:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^2
        order: lexicographic order
        interpretations:
          mark#_A(x1) = ((0,0),(1,0)) x1 + (0,7)
          U11_A(x1) = x1 + (0,8)
          s_A(x1) = ((1,0),(1,0)) x1 + (0,6)
          cons_A(x1,x2) = ((1,0),(0,0)) x1 + ((1,0),(0,0)) x2 + (0,16)
          take_A(x1,x2) = ((1,0),(0,0)) x1 + x2 + (22,39)
          U93_A(x1,x2,x3,x4) = ((1,0),(0,0)) x1 + ((1,0),(0,0)) x2 + ((1,0),(0,0)) x3 + ((1,0),(0,0)) x4 + (17,6)
          U92_A(x1,x2,x3,x4) = ((1,0),(1,0)) x1 + ((1,0),(0,0)) x2 + ((1,0),(0,0)) x3 + ((1,0),(0,0)) x4 + (17,13)
          U91_A(x1,x2,x3,x4) = ((1,0),(1,0)) x1 + ((1,0),(0,0)) x2 + ((1,0),(0,0)) x3 + ((1,0),(0,0)) x4 + (17,17)
          U81_A(x1) = ((1,0),(0,0)) x1 + (0,6)
          length_A(x1) = ((1,0),(1,1)) x1 + (8,8)
          U72_A(x1,x2) = ((1,0),(0,0)) x1 + ((1,0),(1,0)) x2 + (3,19)
          a__U72#_A(x1,x2) = ((0,0),(1,0)) x1 + ((0,0),(1,0)) x2 + (0,10)
          mark_A(x1) = ((1,0),(1,1)) x1 + (0,18)
          tt_A() = (5,0)
          U71_A(x1,x2,x3) = ((1,0),(0,0)) x1 + ((1,0),(0,0)) x2 + ((1,0),(0,0)) x3 + (3,1)
          a__U71#_A(x1,x2,x3) = ((0,0),(1,0)) x1 + ((0,0),(1,0)) x2 + (0,10)
          a__isNat_A(x1) = ((0,0),(1,0)) x1 + (5,1)
          a__length#_A(x1) = ((0,0),(1,0)) x1 + (0,15)
          a__isNatList_A(x1) = ((0,0),(1,0)) x1 + (5,0)
          U62_A(x1) = ((1,0),(0,0)) x1 + (0,1)
          U61_A(x1,x2) = ((1,0),(0,0)) x1 + (0,17)
          U52_A(x1) = ((1,0),(0,0)) x1
          U51_A(x1,x2) = ((1,0),(0,0)) x1
          U42_A(x1) = ((1,0),(0,0)) x1 + (0,19)
          U41_A(x1,x2) = x1 + (0,20)
          U31_A(x1) = ((1,0),(1,0)) x1 + (0,1)
          U21_A(x1) = ((1,0),(0,0)) x1
          a__zeros_A() = (0,17)
          |0|_A() = (0,22)
          zeros_A() = (0,0)
          a__U11_A(x1) = x1 + (0,8)
          a__U21_A(x1) = ((1,0),(0,0)) x1
          a__U31_A(x1) = ((1,0),(1,0)) x1 + (0,2)
          a__U41_A(x1,x2) = x1 + (0,20)
          a__U42_A(x1) = ((1,0),(0,0)) x1 + (0,19)
          a__isNatIList_A(x1) = ((0,0),(1,0)) x1 + (5,21)
          a__U51_A(x1,x2) = ((1,0),(0,0)) x1
          a__U52_A(x1) = ((1,0),(0,0)) x1
          a__U61_A(x1,x2) = ((1,0),(1,0)) x1 + (0,17)
          a__U62_A(x1) = ((1,0),(0,0)) x1 + (0,19)
          a__U71_A(x1,x2,x3) = ((1,0),(0,0)) x1 + ((1,0),(1,0)) x2 + ((1,0),(0,0)) x3 + (3,22)
          a__U72_A(x1,x2) = ((1,0),(0,0)) x1 + ((1,0),(1,0)) x2 + (3,20)
          a__length_A(x1) = ((1,0),(1,1)) x1 + (8,9)
          a__U81_A(x1) = ((1,0),(0,0)) x1 + (0,23)
          nil_A() = (1,0)
          a__U91_A(x1,x2,x3,x4) = ((1,0),(1,0)) x1 + ((1,0),(0,0)) x2 + ((1,0),(0,0)) x3 + ((1,0),(0,0)) x4 + (17,49)
          a__U92_A(x1,x2,x3,x4) = ((1,0),(1,0)) x1 + ((1,0),(0,0)) x2 + ((1,0),(0,0)) x3 + ((1,0),(0,0)) x4 + (17,48)
          a__U93_A(x1,x2,x3,x4) = ((1,0),(0,0)) x1 + ((1,0),(0,0)) x2 + ((1,0),(0,0)) x3 + ((1,0),(0,0)) x4 + (17,40)
          a__take_A(x1,x2) = ((1,0),(0,0)) x1 + x2 + (22,39)
          isNatIList_A(x1) = ((0,0),(1,0)) x1 + (5,1)
          isNatList_A(x1) = ((0,0),(1,0)) x1 + (5,0)
          isNat_A(x1) = ((0,0),(1,0)) x1 + (5,1)
  
    precedence:
    
      |0| > nil > mark# > a__isNat > U81 > a__U31 > a__isNatList = U61 = a__U61 = isNatList > s = mark = U52 = U42 = a__zeros = zeros = a__U11 = a__U21 = a__U41 = a__U42 = a__isNatIList = a__U51 = a__U52 = a__U62 = a__U72 = isNatIList > U92 = a__U81 = a__U91 = a__U92 = a__U93 = a__take > cons > U31 > U11 > a__U71 = a__length > take = U91 = U71 > length > U41 > U93 = U72 = a__U72# = tt = a__U71# = a__length# = U62 = U51 = U21 = isNat
    
    partial status:
  
      pi(mark#) = []
      pi(U11) = [1]
      pi(s) = []
      pi(cons) = []
      pi(take) = []
      pi(U93) = []
      pi(U92) = []
      pi(U91) = []
      pi(U81) = []
      pi(length) = []
      pi(U72) = []
      pi(a__U72#) = []
      pi(mark) = [1]
      pi(tt) = []
      pi(U71) = []
      pi(a__U71#) = []
      pi(a__isNat) = []
      pi(a__length#) = []
      pi(a__isNatList) = []
      pi(U62) = []
      pi(U61) = []
      pi(U52) = []
      pi(U51) = []
      pi(U42) = []
      pi(U41) = []
      pi(U31) = []
      pi(U21) = []
      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(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:

  p15

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#(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#(U72(X1,X2)) -> a__U72#(mark(X1),X2)
p13: a__U72#(tt(),L) -> mark#(L)
p14: mark#(U71(X1,X2,X3)) -> mark#(X1)
p15: a__U71#(tt(),L,N) -> a__U72#(a__isNat(N),L)
p16: a__U72#(tt(),L) -> a__length#(mark(L))
p17: a__length#(cons(N,L)) -> a__U71#(a__isNatList(L),L,N)
p18: mark#(U62(X)) -> mark#(X)
p19: mark#(U61(X1,X2)) -> mark#(X1)
p20: mark#(U52(X)) -> mark#(X)
p21: mark#(U51(X1,X2)) -> mark#(X1)
p22: mark#(U42(X)) -> mark#(X)
p23: mark#(U41(X1,X2)) -> mark#(X1)
p24: mark#(U31(X)) -> mark#(X)
p25: 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 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}


-- Reduction pair.

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)) -> a__U72#(mark(X1),X2)
p12: a__U72#(tt(),L) -> a__length#(mark(L))
p13: a__length#(cons(N,L)) -> a__U71#(a__isNatList(L),L,N)
p14: a__U71#(tt(),L,N) -> a__U72#(a__isNat(N),L)
p15: a__U72#(tt(),L) -> mark#(L)
p16: mark#(U72(X1,X2)) -> mark#(X1)
p17: mark#(length(X)) -> mark#(X)
p18: mark#(U81(X)) -> mark#(X)
p19: mark#(U91(X1,X2,X3,X4)) -> mark#(X1)
p20: mark#(U92(X1,X2,X3,X4)) -> mark#(X1)
p21: mark#(U93(X1,X2,X3,X4)) -> mark#(X1)
p22: mark#(take(X1,X2)) -> mark#(X1)
p23: mark#(take(X1,X2)) -> mark#(X2)
p24: mark#(cons(X1,X2)) -> mark#(X1)
p25: 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:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^2
        order: lexicographic order
        interpretations:
          mark#_A(x1) = ((1,0),(1,0)) x1 + (0,23)
          U11_A(x1) = x1 + (0,1)
          U21_A(x1) = ((1,0),(0,0)) x1 + (0,24)
          U31_A(x1) = ((1,0),(0,0)) x1 + (0,35)
          U41_A(x1,x2) = ((1,0),(1,1)) x1 + ((0,0),(1,0)) x2 + (0,24)
          U42_A(x1) = ((1,0),(0,0)) x1 + (0,1)
          U51_A(x1,x2) = ((1,0),(1,0)) x1 + (0,35)
          U52_A(x1) = ((1,0),(0,0)) x1 + (0,35)
          U61_A(x1,x2) = ((1,0),(1,1)) x1 + ((0,0),(1,0)) x2 + (0,27)
          U62_A(x1) = x1 + (0,1)
          U71_A(x1,x2,x3) = ((1,0),(0,0)) x1 + ((1,0),(1,0)) x2 + ((1,0),(1,0)) x3 + (26,0)
          U72_A(x1,x2) = ((1,0),(0,0)) x1 + ((1,0),(1,0)) x2 + (26,0)
          a__U72#_A(x1,x2) = ((1,0),(1,1)) x1 + ((1,0),(1,0)) x2 + (1,14)
          mark_A(x1) = x1 + (0,34)
          tt_A() = (15,24)
          a__length#_A(x1) = ((1,0),(1,0)) x1 + (16,53)
          cons_A(x1,x2) = ((1,0),(1,0)) x1 + ((1,0),(1,0)) x2 + (0,52)
          a__U71#_A(x1,x2,x3) = ((1,0),(0,0)) x1 + ((1,0),(1,0)) x2 + ((1,0),(1,0)) x3 + (1,53)
          a__isNatList_A(x1) = ((0,0),(1,0)) x1 + (15,51)
          a__isNat_A(x1) = ((0,0),(1,0)) x1 + (15,24)
          length_A(x1) = ((1,0),(1,0)) x1 + (41,0)
          U81_A(x1) = x1 + (1,0)
          U91_A(x1,x2,x3,x4) = ((1,0),(1,0)) x1 + ((1,0),(1,0)) x2 + ((1,0),(1,0)) x3 + ((1,0),(1,0)) x4 + (1,22)
          U92_A(x1,x2,x3,x4) = ((1,0),(0,0)) x1 + ((1,0),(1,0)) x2 + ((1,0),(1,0)) x3 + ((1,0),(1,0)) x4 + (1,35)
          U93_A(x1,x2,x3,x4) = ((1,0),(0,0)) x1 + ((1,0),(1,0)) x2 + ((1,0),(1,0)) x3 + ((1,0),(1,0)) x4 + (1,68)
          take_A(x1,x2) = ((1,0),(1,0)) x1 + ((1,0),(1,0)) x2 + (16,69)
          s_A(x1) = ((1,0),(0,0)) x1
          a__zeros_A() = (1,54)
          |0|_A() = (0,0)
          zeros_A() = (1,25)
          a__U11_A(x1) = x1 + (0,1)
          a__U21_A(x1) = ((1,0),(0,0)) x1 + (0,24)
          a__U31_A(x1) = ((1,0),(0,0)) x1 + (0,35)
          a__U41_A(x1,x2) = ((1,0),(1,1)) x1 + ((0,0),(1,0)) x2 + (0,24)
          a__U42_A(x1) = ((1,0),(0,0)) x1 + (0,25)
          a__isNatIList_A(x1) = ((0,0),(1,0)) x1 + (15,64)
          a__U51_A(x1,x2) = ((1,0),(1,0)) x1 + (0,35)
          a__U52_A(x1) = ((1,0),(0,0)) x1 + (0,35)
          a__U61_A(x1,x2) = ((1,0),(1,1)) x1 + ((0,0),(1,0)) x2 + (0,27)
          a__U62_A(x1) = x1 + (0,1)
          a__U71_A(x1,x2,x3) = ((1,0),(0,0)) x1 + ((1,0),(1,0)) x2 + ((1,0),(1,0)) x3 + (26,0)
          a__U72_A(x1,x2) = ((1,0),(0,0)) x1 + ((1,0),(1,0)) x2 + (26,0)
          a__length_A(x1) = ((1,0),(1,0)) x1 + (41,0)
          a__U81_A(x1) = x1 + (1,0)
          nil_A() = (1,24)
          a__U91_A(x1,x2,x3,x4) = ((1,0),(1,0)) x1 + ((1,0),(1,0)) x2 + ((1,0),(1,0)) x3 + ((1,0),(1,0)) x4 + (1,55)
          a__U92_A(x1,x2,x3,x4) = ((1,0),(0,0)) x1 + ((1,0),(1,0)) x2 + ((1,0),(1,0)) x3 + ((1,0),(1,0)) x4 + (1,69)
          a__U93_A(x1,x2,x3,x4) = ((1,0),(0,0)) x1 + ((1,0),(1,0)) x2 + ((1,0),(1,0)) x3 + ((1,0),(1,0)) x4 + (1,68)
          a__take_A(x1,x2) = ((1,0),(1,0)) x1 + ((1,0),(1,0)) x2 + (16,103)
          isNatIList_A(x1) = ((0,0),(1,0)) x1 + (15,31)
          isNatList_A(x1) = ((0,0),(1,0)) x1 + (15,51)
          isNat_A(x1) = ((0,0),(1,0)) x1 + (15,0)
  
    precedence:
    
      a__isNat > a__isNatList = U92 = a__length = isNatList > U21 = tt = a__U21 = isNat > isNatIList > mark = a__U61 = a__U62 = a__U81 = a__U92 = a__U93 > U81 > mark# = U41 = a__U41 = a__isNatIList > a__U72# = a__length# = a__U71# = take > a__zeros = a__take > U31 = U42 = U51 = U52 = |0| = zeros = a__U31 = a__U42 = a__U51 = a__U52 > U61 = U71 = U72 = U91 = s = a__U71 = a__U72 = nil = a__U91 > U11 = U62 = cons = length = U93 = a__U11
    
    partial status:
  
      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(a__U72#) = []
      pi(mark) = [1]
      pi(tt) = []
      pi(a__length#) = []
      pi(cons) = []
      pi(a__U71#) = []
      pi(a__isNatList) = []
      pi(a__isNat) = []
      pi(length) = []
      pi(U81) = []
      pi(U91) = []
      pi(U92) = []
      pi(U93) = []
      pi(take) = []
      pi(s) = []
      pi(a__zeros) = []
      pi(|0|) = []
      pi(zeros) = []
      pi(a__U11) = []
      pi(a__U21) = []
      pi(a__U31) = []
      pi(a__U41) = [1]
      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(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:

  p11

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: a__U72#(tt(),L) -> a__length#(mark(L))
p12: a__length#(cons(N,L)) -> a__U71#(a__isNatList(L),L,N)
p13: a__U71#(tt(),L,N) -> a__U72#(a__isNat(N),L)
p14: a__U72#(tt(),L) -> mark#(L)
p15: mark#(U72(X1,X2)) -> mark#(X1)
p16: mark#(length(X)) -> mark#(X)
p17: mark#(U81(X)) -> mark#(X)
p18: mark#(U91(X1,X2,X3,X4)) -> mark#(X1)
p19: mark#(U92(X1,X2,X3,X4)) -> mark#(X1)
p20: mark#(U93(X1,X2,X3,X4)) -> mark#(X1)
p21: mark#(take(X1,X2)) -> mark#(X1)
p22: mark#(take(X1,X2)) -> mark#(X2)
p23: mark#(cons(X1,X2)) -> mark#(X1)
p24: 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:

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


-- Reduction pair.

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

p1: a__U71#(tt(),L,N) -> a__U72#(a__isNat(N),L)
p2: a__U72#(tt(),L) -> a__length#(mark(L))
p3: a__length#(cons(N,L)) -> a__U71#(a__isNatList(L),L,N)

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:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^2
        order: lexicographic order
        interpretations:
          a__U71#_A(x1,x2,x3) = ((0,0),(1,0)) x1 + x2 + (1,1)
          tt_A() = (7,5)
          a__U72#_A(x1,x2) = x2 + (1,6)
          a__isNat_A(x1) = ((1,0),(0,0)) x1 + (2,7)
          a__length#_A(x1) = x1 + (1,4)
          mark_A(x1) = x1
          cons_A(x1,x2) = ((1,0),(1,1)) x2
          a__isNatList_A(x1) = ((1,0),(1,1)) x1 + (2,8)
          a__zeros_A() = (0,6)
          |0|_A() = (8,7)
          zeros_A() = (0,6)
          a__U11_A(x1) = ((1,0),(0,0)) x1 + (0,6)
          a__U21_A(x1) = ((1,0),(0,0)) x1 + (0,6)
          a__U31_A(x1) = ((1,0),(0,0)) x1 + (4,5)
          a__U41_A(x1,x2) = x2 + (7,0)
          a__U42_A(x1) = x1
          a__isNatIList_A(x1) = x1 + (7,0)
          a__U51_A(x1,x2) = ((1,0),(1,0)) x2 + (2,4)
          a__U52_A(x1) = ((1,0),(1,0)) x1 + (0,1)
          a__U61_A(x1,x2) = ((1,0),(1,1)) x1 + ((1,0),(0,0)) x2 + (0,1)
          a__U62_A(x1) = ((1,0),(0,0)) x1 + (0,12)
          a__U71_A(x1,x2,x3) = ((1,0),(1,1)) x2
          a__U72_A(x1,x2) = ((1,0),(1,1)) x2
          s_A(x1) = ((1,0),(1,1)) x1
          a__length_A(x1) = x1
          a__U81_A(x1) = ((1,0),(0,0)) x1 + (1,9)
          nil_A() = (8,8)
          a__U91_A(x1,x2,x3,x4) = ((1,0),(1,1)) x2 + ((1,0),(1,1)) x3 + (0,8)
          a__U92_A(x1,x2,x3,x4) = ((1,0),(1,1)) x2 + ((1,0),(1,1)) x3 + (0,8)
          a__U93_A(x1,x2,x3,x4) = ((1,0),(1,1)) x2 + ((1,0),(1,1)) x3 + (0,8)
          take_A(x1,x2) = x1 + x2 + (0,8)
          a__take_A(x1,x2) = x1 + x2 + (0,8)
          U11_A(x1) = ((1,0),(0,0)) x1 + (0,6)
          U21_A(x1) = ((1,0),(0,0)) x1 + (0,6)
          U31_A(x1) = ((1,0),(0,0)) x1 + (4,5)
          U41_A(x1,x2) = x2 + (7,0)
          U42_A(x1) = x1
          isNatIList_A(x1) = x1 + (7,0)
          U51_A(x1,x2) = ((1,0),(1,0)) x2 + (2,4)
          U52_A(x1) = ((1,0),(1,0)) x1 + (0,1)
          U61_A(x1,x2) = ((1,0),(1,1)) x1 + ((1,0),(0,0)) x2 + (0,1)
          U62_A(x1) = ((1,0),(0,0)) x1 + (0,12)
          U71_A(x1,x2,x3) = ((1,0),(1,1)) x2
          U72_A(x1,x2) = ((1,0),(1,1)) x2
          length_A(x1) = x1
          U81_A(x1) = ((1,0),(0,0)) x1 + (1,9)
          U91_A(x1,x2,x3,x4) = ((1,0),(1,1)) x2 + ((1,0),(1,1)) x3 + (0,8)
          U92_A(x1,x2,x3,x4) = ((1,0),(1,1)) x2 + ((1,0),(1,1)) x3 + (0,8)
          U93_A(x1,x2,x3,x4) = ((1,0),(1,1)) x2 + ((1,0),(1,1)) x3 + (0,8)
          isNatList_A(x1) = ((1,0),(1,1)) x1 + (2,8)
          isNat_A(x1) = ((1,0),(0,0)) x1 + (2,7)
  
    precedence:
    
      a__length# > a__U71# > tt = a__U72# = a__isNat = mark = cons = a__isNatList = a__zeros = |0| = zeros = a__U11 = a__U21 = a__U31 = a__U41 = a__U42 = a__isNatIList = a__U51 = a__U52 = a__U61 = a__U62 = a__U71 = a__U72 = s = a__length = a__U81 = nil = a__U91 = a__U92 = a__U93 = take = a__take = U11 = U21 = U31 = U41 = U42 = isNatIList = U51 = U52 = U61 = U62 = U71 = U72 = length = U81 = U91 = U92 = U93 = isNatList = isNat
    
    partial status:
  
      pi(a__U71#) = [2]
      pi(tt) = []
      pi(a__U72#) = [2]
      pi(a__isNat) = []
      pi(a__length#) = [1]
      pi(mark) = []
      pi(cons) = []
      pi(a__isNatList) = []
      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__U71#(tt(),L,N) -> a__U72#(a__isNat(N),L)
p2: 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 estimated dependency graph contains the following SCCs:

  (no SCCs)

-- 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:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^2
        order: lexicographic order
        interpretations:
          mark#_A(x1) = ((1,0),(0,0)) x1 + (2,2)
          U11_A(x1) = x1 + (1,1)
          s_A(x1) = ((1,0),(1,1)) x1 + (3,3)
          cons_A(x1,x2) = ((1,0),(1,1)) x1 + (3,3)
          take_A(x1,x2) = ((1,0),(1,0)) x1 + ((1,0),(1,0)) x2 + (3,3)
          U93_A(x1,x2,x3,x4) = ((1,0),(1,1)) x1 + x2 + x3 + x4 + (3,3)
          U92_A(x1,x2,x3,x4) = ((1,0),(1,1)) x1 + ((1,0),(1,1)) x2 + x3 + x4 + (3,3)
          U91_A(x1,x2,x3,x4) = x1 + x4 + (3,1)
          U81_A(x1) = ((1,0),(1,1)) x1 + (3,3)
          length_A(x1) = ((1,0),(1,1)) x1 + (3,3)
          U72_A(x1,x2) = ((1,0),(0,0)) x1 + x2 + (3,3)
          U71_A(x1,x2,x3) = x1 + x2 + (3,1)
          U62_A(x1) = x1 + (3,1)
          U61_A(x1,x2) = ((1,0),(0,0)) x1 + (3,3)
          U52_A(x1) = x1 + (3,1)
          U51_A(x1,x2) = x1 + x2 + (3,1)
          U42_A(x1) = ((1,0),(1,0)) x1 + (3,3)
          U41_A(x1,x2) = ((1,0),(1,1)) x1 + (3,1)
          U31_A(x1) = ((1,0),(0,0)) x1 + (2,3)
          U21_A(x1) = ((1,0),(0,0)) x1 + (3,3)
  
    precedence:
    
      mark# = s = take = U93 = U92 = U81 = U72 = U52 = U51 > U11 = cons = U31 > U61 > U91 = length = U62 > U71 > U42 = U41 = U21
    
    partial status:
  
      pi(mark#) = []
      pi(U11) = []
      pi(s) = []
      pi(cons) = [1]
      pi(take) = []
      pi(U93) = []
      pi(U92) = []
      pi(U91) = [4]
      pi(U81) = []
      pi(length) = [1]
      pi(U72) = []
      pi(U71) = [1]
      pi(U62) = [1]
      pi(U61) = []
      pi(U52) = []
      pi(U51) = []
      pi(U42) = []
      pi(U41) = [1]
      pi(U31) = []
      pi(U21) = []

The next rules are strictly ordered:

  p10

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#(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#(U72(X1,X2)) -> mark#(X1)
p11: mark#(U71(X1,X2,X3)) -> mark#(X1)
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#(U42(X)) -> mark#(X)
p17: mark#(U41(X1,X2)) -> mark#(X1)
p18: mark#(U31(X)) -> mark#(X)
p19: 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 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}


-- Reduction pair.

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#(U81(X)) -> mark#(X)
p13: mark#(U91(X1,X2,X3,X4)) -> mark#(X1)
p14: mark#(U92(X1,X2,X3,X4)) -> mark#(X1)
p15: mark#(U93(X1,X2,X3,X4)) -> mark#(X1)
p16: mark#(take(X1,X2)) -> mark#(X1)
p17: mark#(take(X1,X2)) -> mark#(X2)
p18: mark#(cons(X1,X2)) -> mark#(X1)
p19: 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

  (no rules)

Take the reduction pair:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^2
        order: lexicographic order
        interpretations:
          mark#_A(x1) = ((1,0),(0,0)) x1 + (1,2)
          U11_A(x1) = ((1,0),(1,1)) x1 + (2,3)
          U21_A(x1) = x1 + (2,3)
          U31_A(x1) = x1 + (2,1)
          U41_A(x1,x2) = ((1,0),(1,1)) x1 + (2,1)
          U42_A(x1) = ((1,0),(1,1)) x1 + (2,1)
          U51_A(x1,x2) = ((1,0),(1,0)) x1 + x2 + (2,3)
          U52_A(x1) = ((1,0),(1,1)) x1 + (2,3)
          U61_A(x1,x2) = ((1,0),(1,1)) x1 + x2 + (2,3)
          U62_A(x1) = x1 + (2,1)
          U71_A(x1,x2,x3) = ((1,0),(0,0)) x1 + x2 + (2,1)
          U72_A(x1,x2) = ((1,0),(0,0)) x1 + x2 + (2,1)
          U81_A(x1) = x1 + (2,1)
          U91_A(x1,x2,x3,x4) = ((1,0),(1,1)) x1 + x2 + x4 + (2,1)
          U92_A(x1,x2,x3,x4) = x1 + x2 + x3 + (2,1)
          U93_A(x1,x2,x3,x4) = ((1,0),(1,1)) x1 + x2 + x3 + (2,3)
          take_A(x1,x2) = ((1,0),(0,0)) x1 + ((1,0),(0,0)) x2 + (2,3)
          cons_A(x1,x2) = ((1,0),(0,0)) x1 + x2 + (2,1)
          s_A(x1) = ((1,0),(0,0)) x1 + (2,3)
  
    precedence:
    
      U91 = U92 > U31 = U72 = U81 > U41 = U61 = U93 > U11 = U42 > mark# = U21 = U51 = U62 > U52 = U71 = cons > take = s
    
    partial status:
  
      pi(mark#) = []
      pi(U11) = []
      pi(U21) = []
      pi(U31) = [1]
      pi(U41) = [1]
      pi(U42) = [1]
      pi(U51) = [2]
      pi(U52) = [1]
      pi(U61) = [1]
      pi(U62) = [1]
      pi(U71) = []
      pi(U72) = []
      pi(U81) = [1]
      pi(U91) = []
      pi(U92) = []
      pi(U93) = [3]
      pi(take) = []
      pi(cons) = [2]
      pi(s) = []

The next rules are strictly ordered:

  p12

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#(U91(X1,X2,X3,X4)) -> mark#(X1)
p13: mark#(U92(X1,X2,X3,X4)) -> mark#(X1)
p14: mark#(U93(X1,X2,X3,X4)) -> mark#(X1)
p15: mark#(take(X1,X2)) -> mark#(X1)
p16: mark#(take(X1,X2)) -> mark#(X2)
p17: mark#(cons(X1,X2)) -> mark#(X1)
p18: 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, p16, p17, p18}


-- 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#(U72(X1,X2)) -> mark#(X1)
p10: mark#(U71(X1,X2,X3)) -> mark#(X1)
p11: mark#(U62(X)) -> mark#(X)
p12: mark#(U61(X1,X2)) -> mark#(X1)
p13: mark#(U52(X)) -> mark#(X)
p14: mark#(U51(X1,X2)) -> mark#(X1)
p15: mark#(U42(X)) -> mark#(X)
p16: mark#(U41(X1,X2)) -> mark#(X1)
p17: mark#(U31(X)) -> mark#(X)
p18: 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:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^2
        order: lexicographic order
        interpretations:
          mark#_A(x1) = ((1,0),(0,0)) x1 + (2,2)
          U11_A(x1) = ((1,0),(1,1)) x1 + (3,3)
          s_A(x1) = x1 + (3,3)
          cons_A(x1,x2) = ((1,0),(1,0)) x1 + x2 + (3,3)
          take_A(x1,x2) = ((1,0),(0,0)) x1 + ((1,0),(1,1)) x2 + (3,3)
          U93_A(x1,x2,x3,x4) = x1 + x2 + x4 + (1,1)
          U92_A(x1,x2,x3,x4) = x1 + x2 + x3 + x4 + (3,1)
          U91_A(x1,x2,x3,x4) = x1 + x3 + x4 + (3,3)
          U72_A(x1,x2) = x1 + x2 + (3,1)
          U71_A(x1,x2,x3) = ((1,0),(0,0)) x1 + x2 + x3 + (3,1)
          U62_A(x1) = ((1,0),(1,1)) x1 + (3,1)
          U61_A(x1,x2) = ((1,0),(1,1)) x1 + x2 + (3,3)
          U52_A(x1) = x1 + (3,3)
          U51_A(x1,x2) = ((1,0),(0,0)) x1 + x2 + (3,1)
          U42_A(x1) = ((1,0),(1,0)) x1 + (3,3)
          U41_A(x1,x2) = ((1,0),(0,0)) x1 + x2 + (1,1)
          U31_A(x1) = x1 + (3,3)
          U21_A(x1) = ((1,0),(0,0)) x1 + (3,1)
  
    precedence:
    
      U51 > U52 > mark# = U62 > U61 > U92 = U72 = U42 > U11 = take = U93 = U71 > U91 > U41 > s = cons = U31 > U21
    
    partial status:
  
      pi(mark#) = []
      pi(U11) = [1]
      pi(s) = []
      pi(cons) = []
      pi(take) = [2]
      pi(U93) = [2]
      pi(U92) = [1]
      pi(U91) = [4]
      pi(U72) = [2]
      pi(U71) = [3]
      pi(U62) = [1]
      pi(U61) = [1]
      pi(U52) = [1]
      pi(U51) = [2]
      pi(U42) = []
      pi(U41) = [2]
      pi(U31) = [1]
      pi(U21) = []

The next rules are strictly ordered:

  p12

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#(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#(U72(X1,X2)) -> mark#(X1)
p10: mark#(U71(X1,X2,X3)) -> mark#(X1)
p11: mark#(U62(X)) -> mark#(X)
p12: mark#(U52(X)) -> mark#(X)
p13: mark#(U51(X1,X2)) -> mark#(X1)
p14: mark#(U42(X)) -> mark#(X)
p15: mark#(U41(X1,X2)) -> mark#(X1)
p16: mark#(U31(X)) -> mark#(X)
p17: 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 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(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#(U62(X)) -> mark#(X)
p9: mark#(U71(X1,X2,X3)) -> mark#(X1)
p10: mark#(U72(X1,X2)) -> mark#(X1)
p11: mark#(U91(X1,X2,X3,X4)) -> mark#(X1)
p12: mark#(U92(X1,X2,X3,X4)) -> mark#(X1)
p13: mark#(U93(X1,X2,X3,X4)) -> mark#(X1)
p14: mark#(take(X1,X2)) -> mark#(X1)
p15: mark#(take(X1,X2)) -> mark#(X2)
p16: mark#(cons(X1,X2)) -> mark#(X1)
p17: 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

  (no rules)

Take the reduction pair:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^2
        order: lexicographic order
        interpretations:
          mark#_A(x1) = x1 + (1,2)
          U11_A(x1) = ((1,0),(1,1)) x1 + (2,3)
          U21_A(x1) = ((1,0),(1,1)) x1 + (2,1)
          U31_A(x1) = ((1,0),(1,1)) x1 + (2,3)
          U41_A(x1,x2) = ((1,0),(1,1)) x1 + ((0,0),(1,0)) x2 + (2,3)
          U42_A(x1) = x1 + (2,3)
          U51_A(x1,x2) = ((1,0),(1,1)) x1 + ((1,0),(1,1)) x2 + (2,3)
          U52_A(x1) = x1 + (2,1)
          U62_A(x1) = x1 + (2,1)
          U71_A(x1,x2,x3) = x1 + ((1,0),(1,1)) x3 + (2,1)
          U72_A(x1,x2) = x1 + x2 + (2,1)
          U91_A(x1,x2,x3,x4) = ((1,0),(1,1)) x1 + ((0,0),(1,0)) x2 + ((0,0),(1,0)) x3 + ((1,0),(1,1)) x4 + (2,1)
          U92_A(x1,x2,x3,x4) = x1 + x3 + x4 + (2,1)
          U93_A(x1,x2,x3,x4) = x1 + ((1,0),(1,1)) x2 + ((0,0),(1,0)) x3 + ((0,0),(1,0)) x4 + (2,3)
          take_A(x1,x2) = ((1,0),(1,1)) x1 + x2 + (2,3)
          cons_A(x1,x2) = x1 + ((1,0),(1,1)) x2 + (2,2)
          s_A(x1) = x1 + (2,1)
  
    precedence:
    
      U72 > U91 > U11 = take > U52 = U93 > U31 = U41 = U71 > mark# = U42 = U51 = cons > U21 = U92 > U62 = s
    
    partial status:
  
      pi(mark#) = [1]
      pi(U11) = []
      pi(U21) = []
      pi(U31) = []
      pi(U41) = [1]
      pi(U42) = [1]
      pi(U51) = [1, 2]
      pi(U52) = [1]
      pi(U62) = []
      pi(U71) = [1, 3]
      pi(U72) = [1, 2]
      pi(U91) = []
      pi(U92) = [1]
      pi(U93) = [1]
      pi(take) = [1, 2]
      pi(cons) = [1, 2]
      pi(s) = [1]

The next rules are strictly ordered:

  p1

We remove them from the problem.

-- SCC decomposition.

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

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


-- Reduction pair.

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

p1: mark#(U21(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#(U72(X1,X2)) -> mark#(X1)
p10: mark#(U71(X1,X2,X3)) -> mark#(X1)
p11: mark#(U62(X)) -> mark#(X)
p12: mark#(U52(X)) -> mark#(X)
p13: mark#(U51(X1,X2)) -> mark#(X1)
p14: mark#(U42(X)) -> mark#(X)
p15: mark#(U41(X1,X2)) -> mark#(X1)
p16: mark#(U31(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:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^2
        order: lexicographic order
        interpretations:
          mark#_A(x1) = ((1,0),(0,0)) x1 + (2,2)
          U21_A(x1) = ((1,0),(1,1)) x1 + (3,1)
          s_A(x1) = ((1,0),(1,0)) x1 + (3,3)
          cons_A(x1,x2) = ((1,0),(1,0)) x1 + x2 + (3,3)
          take_A(x1,x2) = ((1,0),(1,1)) x1 + ((1,0),(1,1)) x2 + (3,3)
          U93_A(x1,x2,x3,x4) = ((1,0),(1,1)) x1 + x2 + x3 + (3,1)
          U92_A(x1,x2,x3,x4) = x1 + x2 + x3 + x4 + (1,1)
          U91_A(x1,x2,x3,x4) = ((1,0),(1,1)) x1 + x3 + (3,3)
          U72_A(x1,x2) = ((1,0),(1,1)) x1 + x2 + (3,3)
          U71_A(x1,x2,x3) = x1 + x2 + x3 + (1,1)
          U62_A(x1) = ((1,0),(1,1)) x1 + (3,3)
          U52_A(x1) = ((1,0),(0,0)) x1 + (3,1)
          U51_A(x1,x2) = ((1,0),(0,0)) x1 + x2 + (3,3)
          U42_A(x1) = x1 + (3,3)
          U41_A(x1,x2) = x1 + x2 + (1,1)
          U31_A(x1) = ((1,0),(0,0)) x1 + (3,1)
  
    precedence:
    
      U51 > U42 > s = cons = take = U71 > mark# = U91 = U72 = U62 > U52 > U41 > U93 = U92 > U21 = U31
    
    partial status:
  
      pi(mark#) = []
      pi(U21) = []
      pi(s) = []
      pi(cons) = [2]
      pi(take) = [2]
      pi(U93) = [3]
      pi(U92) = [1]
      pi(U91) = [1]
      pi(U72) = [1]
      pi(U71) = [1, 3]
      pi(U62) = []
      pi(U52) = []
      pi(U51) = [2]
      pi(U42) = [1]
      pi(U41) = [1]
      pi(U31) = []

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#(U21(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#(U72(X1,X2)) -> mark#(X1)
p10: mark#(U71(X1,X2,X3)) -> mark#(X1)
p11: mark#(U62(X)) -> mark#(X)
p12: mark#(U52(X)) -> mark#(X)
p13: mark#(U51(X1,X2)) -> mark#(X1)
p14: mark#(U42(X)) -> mark#(X)
p15: mark#(U31(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}


-- Reduction pair.

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

p1: mark#(U21(X)) -> mark#(X)
p2: mark#(U31(X)) -> mark#(X)
p3: mark#(U42(X)) -> mark#(X)
p4: mark#(U51(X1,X2)) -> mark#(X1)
p5: mark#(U52(X)) -> mark#(X)
p6: mark#(U62(X)) -> mark#(X)
p7: mark#(U71(X1,X2,X3)) -> mark#(X1)
p8: mark#(U72(X1,X2)) -> mark#(X1)
p9: mark#(U91(X1,X2,X3,X4)) -> mark#(X1)
p10: mark#(U92(X1,X2,X3,X4)) -> mark#(X1)
p11: mark#(U93(X1,X2,X3,X4)) -> mark#(X1)
p12: mark#(take(X1,X2)) -> mark#(X1)
p13: mark#(take(X1,X2)) -> mark#(X2)
p14: mark#(cons(X1,X2)) -> mark#(X1)
p15: 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

  (no rules)

Take the reduction pair:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^2
        order: lexicographic order
        interpretations:
          mark#_A(x1) = ((1,0),(1,0)) x1
          U21_A(x1) = ((1,0),(1,1)) x1 + (1,1)
          U31_A(x1) = ((1,0),(1,1)) x1 + (1,1)
          U42_A(x1) = ((1,0),(1,1)) x1 + (1,1)
          U51_A(x1,x2) = ((1,0),(1,1)) x1 + (1,1)
          U52_A(x1) = ((1,0),(1,1)) x1 + (1,1)
          U62_A(x1) = ((1,0),(1,1)) x1 + (1,1)
          U71_A(x1,x2,x3) = ((1,0),(1,1)) x1 + x2 + x3 + (1,1)
          U72_A(x1,x2) = ((1,0),(0,0)) x1 + x2 + (1,1)
          U91_A(x1,x2,x3,x4) = ((1,0),(0,0)) x1 + x2 + x3 + x4 + (1,1)
          U92_A(x1,x2,x3,x4) = ((1,0),(0,0)) x1 + x2 + (1,1)
          U93_A(x1,x2,x3,x4) = ((1,0),(0,0)) x1 + x3 + x4 + (1,1)
          take_A(x1,x2) = x1 + x2 + (1,1)
          cons_A(x1,x2) = ((1,0),(1,1)) x1 + x2 + (1,1)
          s_A(x1) = ((1,0),(0,0)) x1 + (1,1)
  
    precedence:
    
      mark# > cons > U93 > U52 = U71 = U72 = U92 > take > U51 = U62 = U91 = s > U21 = U42 > U31
    
    partial status:
  
      pi(mark#) = []
      pi(U21) = []
      pi(U31) = []
      pi(U42) = [1]
      pi(U51) = [1]
      pi(U52) = [1]
      pi(U62) = [1]
      pi(U71) = [3]
      pi(U72) = [2]
      pi(U91) = []
      pi(U92) = []
      pi(U93) = [4]
      pi(take) = [1, 2]
      pi(cons) = [2]
      pi(s) = []

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#(U21(X)) -> mark#(X)
p2: mark#(U42(X)) -> mark#(X)
p3: mark#(U51(X1,X2)) -> mark#(X1)
p4: mark#(U52(X)) -> mark#(X)
p5: mark#(U62(X)) -> mark#(X)
p6: mark#(U71(X1,X2,X3)) -> mark#(X1)
p7: mark#(U72(X1,X2)) -> mark#(X1)
p8: mark#(U91(X1,X2,X3,X4)) -> mark#(X1)
p9: mark#(U92(X1,X2,X3,X4)) -> mark#(X1)
p10: mark#(U93(X1,X2,X3,X4)) -> mark#(X1)
p11: mark#(take(X1,X2)) -> mark#(X1)
p12: mark#(take(X1,X2)) -> mark#(X2)
p13: mark#(cons(X1,X2)) -> mark#(X1)
p14: 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}


-- Reduction pair.

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

p1: mark#(U21(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#(U72(X1,X2)) -> mark#(X1)
p10: mark#(U71(X1,X2,X3)) -> mark#(X1)
p11: mark#(U62(X)) -> mark#(X)
p12: mark#(U52(X)) -> mark#(X)
p13: mark#(U51(X1,X2)) -> mark#(X1)
p14: mark#(U42(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:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^2
        order: lexicographic order
        interpretations:
          mark#_A(x1) = ((0,0),(1,0)) x1 + (2,2)
          U21_A(x1) = ((1,0),(1,1)) x1 + (3,1)
          s_A(x1) = ((1,0),(1,1)) x1 + (3,3)
          cons_A(x1,x2) = x1 + x2 + (3,1)
          take_A(x1,x2) = ((1,0),(1,0)) x1 + ((1,0),(1,1)) x2 + (3,1)
          U93_A(x1,x2,x3,x4) = x1 + x2 + x3 + (3,1)
          U92_A(x1,x2,x3,x4) = ((1,0),(0,0)) x1 + x2 + x3 + x4 + (3,1)
          U91_A(x1,x2,x3,x4) = ((1,0),(0,0)) x1 + x2 + x4 + (3,1)
          U72_A(x1,x2) = x1 + x2 + (3,3)
          U71_A(x1,x2,x3) = x1 + x2 + x3 + (3,1)
          U62_A(x1) = ((1,0),(1,0)) x1 + (3,3)
          U52_A(x1) = x1 + (1,1)
          U51_A(x1,x2) = x1 + (1,3)
          U42_A(x1) = ((1,0),(0,0)) x1 + (1,3)
  
    precedence:
    
      U42 > U62 > U52 > U92 = U51 > U21 = U93 = U71 > mark# = cons = take = U91 = U72 > s
    
    partial status:
  
      pi(mark#) = []
      pi(U21) = [1]
      pi(s) = []
      pi(cons) = [2]
      pi(take) = [2]
      pi(U93) = [3]
      pi(U92) = [3]
      pi(U91) = []
      pi(U72) = [1]
      pi(U71) = []
      pi(U62) = []
      pi(U52) = [1]
      pi(U51) = [1]
      pi(U42) = []

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#(U21(X)) -> mark#(X)
p2: mark#(s(X)) -> mark#(X)
p3: mark#(take(X1,X2)) -> mark#(X2)
p4: mark#(take(X1,X2)) -> mark#(X1)
p5: mark#(U93(X1,X2,X3,X4)) -> mark#(X1)
p6: mark#(U92(X1,X2,X3,X4)) -> mark#(X1)
p7: mark#(U91(X1,X2,X3,X4)) -> mark#(X1)
p8: mark#(U72(X1,X2)) -> mark#(X1)
p9: mark#(U71(X1,X2,X3)) -> mark#(X1)
p10: mark#(U62(X)) -> mark#(X)
p11: mark#(U52(X)) -> mark#(X)
p12: mark#(U51(X1,X2)) -> mark#(X1)
p13: mark#(U42(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}


-- Reduction pair.

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

p1: mark#(U21(X)) -> mark#(X)
p2: mark#(U42(X)) -> mark#(X)
p3: mark#(U51(X1,X2)) -> mark#(X1)
p4: mark#(U52(X)) -> mark#(X)
p5: mark#(U62(X)) -> mark#(X)
p6: mark#(U71(X1,X2,X3)) -> mark#(X1)
p7: mark#(U72(X1,X2)) -> mark#(X1)
p8: mark#(U91(X1,X2,X3,X4)) -> mark#(X1)
p9: mark#(U92(X1,X2,X3,X4)) -> mark#(X1)
p10: mark#(U93(X1,X2,X3,X4)) -> mark#(X1)
p11: mark#(take(X1,X2)) -> mark#(X1)
p12: mark#(take(X1,X2)) -> mark#(X2)
p13: 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

  (no rules)

Take the reduction pair:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^2
        order: lexicographic order
        interpretations:
          mark#_A(x1) = ((1,0),(0,0)) x1 + (2,2)
          U21_A(x1) = ((1,0),(1,1)) x1 + (3,3)
          U42_A(x1) = x1 + (3,1)
          U51_A(x1,x2) = ((1,0),(1,1)) x1 + x2 + (3,3)
          U52_A(x1) = ((1,0),(1,1)) x1 + (3,3)
          U62_A(x1) = ((1,0),(1,1)) x1 + (3,3)
          U71_A(x1,x2,x3) = x1 + (1,1)
          U72_A(x1,x2) = x1 + x2 + (3,1)
          U91_A(x1,x2,x3,x4) = ((1,0),(1,1)) x1 + x2 + x3 + x4 + (3,1)
          U92_A(x1,x2,x3,x4) = ((1,0),(0,0)) x1 + x2 + x3 + x4 + (3,1)
          U93_A(x1,x2,x3,x4) = x1 + x2 + (3,3)
          take_A(x1,x2) = ((1,0),(0,0)) x1 + x2 + (3,1)
          s_A(x1) = ((1,0),(0,0)) x1 + (3,3)
  
    precedence:
    
      U92 > s > U72 > mark# > U93 > take > U71 > U51 > U52 > U21 = U91 > U42 > U62
    
    partial status:
  
      pi(mark#) = []
      pi(U21) = []
      pi(U42) = []
      pi(U51) = [1, 2]
      pi(U52) = []
      pi(U62) = []
      pi(U71) = []
      pi(U72) = [2]
      pi(U91) = [1, 4]
      pi(U92) = [4]
      pi(U93) = []
      pi(take) = [2]
      pi(s) = []

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#(U21(X)) -> mark#(X)
p2: mark#(U42(X)) -> mark#(X)
p3: mark#(U52(X)) -> mark#(X)
p4: mark#(U62(X)) -> mark#(X)
p5: mark#(U71(X1,X2,X3)) -> mark#(X1)
p6: mark#(U72(X1,X2)) -> mark#(X1)
p7: mark#(U91(X1,X2,X3,X4)) -> mark#(X1)
p8: mark#(U92(X1,X2,X3,X4)) -> mark#(X1)
p9: mark#(U93(X1,X2,X3,X4)) -> mark#(X1)
p10: mark#(take(X1,X2)) -> mark#(X1)
p11: mark#(take(X1,X2)) -> mark#(X2)
p12: 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}


-- Reduction pair.

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

p1: mark#(U21(X)) -> mark#(X)
p2: mark#(s(X)) -> mark#(X)
p3: mark#(take(X1,X2)) -> mark#(X2)
p4: mark#(take(X1,X2)) -> mark#(X1)
p5: mark#(U93(X1,X2,X3,X4)) -> mark#(X1)
p6: mark#(U92(X1,X2,X3,X4)) -> mark#(X1)
p7: mark#(U91(X1,X2,X3,X4)) -> mark#(X1)
p8: mark#(U72(X1,X2)) -> mark#(X1)
p9: mark#(U71(X1,X2,X3)) -> mark#(X1)
p10: mark#(U62(X)) -> mark#(X)
p11: mark#(U52(X)) -> mark#(X)
p12: mark#(U42(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:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^2
        order: lexicographic order
        interpretations:
          mark#_A(x1) = x1 + (1,2)
          U21_A(x1) = ((1,0),(1,1)) x1 + (2,3)
          s_A(x1) = ((1,0),(1,1)) x1 + (2,1)
          take_A(x1,x2) = x1 + x2 + (2,1)
          U93_A(x1,x2,x3,x4) = x1 + ((1,0),(1,1)) x2 + ((1,0),(1,1)) x3 + ((0,0),(1,0)) x4 + (2,1)
          U92_A(x1,x2,x3,x4) = ((1,0),(1,1)) x1 + ((0,0),(1,0)) x3 + x4 + (2,1)
          U91_A(x1,x2,x3,x4) = x1 + x3 + x4 + (2,3)
          U72_A(x1,x2) = x1 + x2 + (2,3)
          U71_A(x1,x2,x3) = x1 + ((1,0),(1,1)) x2 + ((1,0),(1,1)) x3 + (2,1)
          U62_A(x1) = x1 + (2,3)
          U52_A(x1) = x1 + (2,1)
          U42_A(x1) = x1 + (2,3)
  
    precedence:
    
      U62 > U52 > take = U42 > U21 > s > U93 > mark# = U92 = U91 = U71 > U72
    
    partial status:
  
      pi(mark#) = [1]
      pi(U21) = [1]
      pi(s) = []
      pi(take) = [1, 2]
      pi(U93) = []
      pi(U92) = [1, 4]
      pi(U91) = [1, 3]
      pi(U72) = [1, 2]
      pi(U71) = [1]
      pi(U62) = [1]
      pi(U52) = []
      pi(U42) = []

The next rules are strictly ordered:

  p10

We remove them from the problem.

-- SCC decomposition.

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

p1: mark#(U21(X)) -> mark#(X)
p2: mark#(s(X)) -> mark#(X)
p3: mark#(take(X1,X2)) -> mark#(X2)
p4: mark#(take(X1,X2)) -> mark#(X1)
p5: mark#(U93(X1,X2,X3,X4)) -> mark#(X1)
p6: mark#(U92(X1,X2,X3,X4)) -> mark#(X1)
p7: mark#(U91(X1,X2,X3,X4)) -> mark#(X1)
p8: mark#(U72(X1,X2)) -> mark#(X1)
p9: mark#(U71(X1,X2,X3)) -> mark#(X1)
p10: mark#(U52(X)) -> mark#(X)
p11: mark#(U42(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}


-- Reduction pair.

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

p1: mark#(U21(X)) -> mark#(X)
p2: mark#(U42(X)) -> mark#(X)
p3: mark#(U52(X)) -> mark#(X)
p4: mark#(U71(X1,X2,X3)) -> mark#(X1)
p5: mark#(U72(X1,X2)) -> mark#(X1)
p6: mark#(U91(X1,X2,X3,X4)) -> mark#(X1)
p7: mark#(U92(X1,X2,X3,X4)) -> mark#(X1)
p8: mark#(U93(X1,X2,X3,X4)) -> mark#(X1)
p9: mark#(take(X1,X2)) -> mark#(X1)
p10: mark#(take(X1,X2)) -> mark#(X2)
p11: 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

  (no rules)

Take the reduction pair:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^2
        order: lexicographic order
        interpretations:
          mark#_A(x1) = ((1,0),(1,1)) x1 + (1,2)
          U21_A(x1) = ((1,0),(1,1)) x1 + (2,3)
          U42_A(x1) = ((1,0),(1,1)) x1 + (2,1)
          U52_A(x1) = ((1,0),(1,1)) x1 + (2,1)
          U71_A(x1,x2,x3) = x1 + x2 + x3 + (2,1)
          U72_A(x1,x2) = x1 + x2 + (2,1)
          U91_A(x1,x2,x3,x4) = x1 + x2 + x3 + x4 + (2,1)
          U92_A(x1,x2,x3,x4) = x1 + x2 + x3 + (2,1)
          U93_A(x1,x2,x3,x4) = x1 + ((1,0),(1,1)) x2 + ((0,0),(1,0)) x3 + ((1,0),(1,1)) x4 + (2,1)
          take_A(x1,x2) = ((1,0),(1,1)) x1 + ((1,0),(1,1)) x2 + (2,3)
          s_A(x1) = ((1,0),(1,1)) x1 + (2,3)
  
    precedence:
    
      take > U21 > s > U42 > U52 > U71 > U91 > U93 > mark# = U72 = U92
    
    partial status:
  
      pi(mark#) = [1]
      pi(U21) = []
      pi(U42) = []
      pi(U52) = []
      pi(U71) = [1, 3]
      pi(U72) = [1, 2]
      pi(U91) = []
      pi(U92) = [1, 3]
      pi(U93) = [1, 4]
      pi(take) = []
      pi(s) = []

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#(U21(X)) -> mark#(X)
p2: mark#(U42(X)) -> mark#(X)
p3: mark#(U52(X)) -> mark#(X)
p4: mark#(U71(X1,X2,X3)) -> mark#(X1)
p5: mark#(U72(X1,X2)) -> mark#(X1)
p6: mark#(U91(X1,X2,X3,X4)) -> mark#(X1)
p7: mark#(U93(X1,X2,X3,X4)) -> mark#(X1)
p8: mark#(take(X1,X2)) -> mark#(X1)
p9: mark#(take(X1,X2)) -> mark#(X2)
p10: 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}


-- Reduction pair.

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

p1: mark#(U21(X)) -> mark#(X)
p2: mark#(s(X)) -> mark#(X)
p3: mark#(take(X1,X2)) -> mark#(X2)
p4: mark#(take(X1,X2)) -> mark#(X1)
p5: mark#(U93(X1,X2,X3,X4)) -> mark#(X1)
p6: mark#(U91(X1,X2,X3,X4)) -> mark#(X1)
p7: mark#(U72(X1,X2)) -> mark#(X1)
p8: mark#(U71(X1,X2,X3)) -> mark#(X1)
p9: mark#(U52(X)) -> mark#(X)
p10: mark#(U42(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:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^2
        order: lexicographic order
        interpretations:
          mark#_A(x1) = x1 + (1,2)
          U21_A(x1) = ((1,0),(1,1)) x1 + (2,3)
          s_A(x1) = ((1,0),(1,1)) x1 + (2,1)
          take_A(x1,x2) = ((1,0),(1,1)) x1 + ((1,0),(1,1)) x2 + (2,3)
          U93_A(x1,x2,x3,x4) = x1 + x3 + (2,1)
          U91_A(x1,x2,x3,x4) = x1 + x2 + x4 + (2,1)
          U72_A(x1,x2) = x1 + x2 + (2,3)
          U71_A(x1,x2,x3) = x1 + ((1,0),(1,1)) x2 + ((0,0),(1,0)) x3 + (2,3)
          U52_A(x1) = ((1,0),(1,1)) x1 + (2,3)
          U42_A(x1) = x1 + (2,3)
  
    precedence:
    
      U42 > U21 > s > take > U93 > U71 > U91 > U52 > U72 > mark#
    
    partial status:
  
      pi(mark#) = [1]
      pi(U21) = []
      pi(s) = []
      pi(take) = [1, 2]
      pi(U93) = []
      pi(U91) = []
      pi(U72) = []
      pi(U71) = []
      pi(U52) = []
      pi(U42) = [1]

The next rules are strictly ordered:

  p10

We remove them from the problem.

-- SCC decomposition.

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

p1: mark#(U21(X)) -> mark#(X)
p2: mark#(s(X)) -> mark#(X)
p3: mark#(take(X1,X2)) -> mark#(X2)
p4: mark#(take(X1,X2)) -> mark#(X1)
p5: mark#(U93(X1,X2,X3,X4)) -> mark#(X1)
p6: mark#(U91(X1,X2,X3,X4)) -> mark#(X1)
p7: mark#(U72(X1,X2)) -> mark#(X1)
p8: mark#(U71(X1,X2,X3)) -> mark#(X1)
p9: mark#(U52(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}


-- Reduction pair.

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

p1: mark#(U21(X)) -> mark#(X)
p2: mark#(U52(X)) -> mark#(X)
p3: mark#(U71(X1,X2,X3)) -> mark#(X1)
p4: mark#(U72(X1,X2)) -> mark#(X1)
p5: mark#(U91(X1,X2,X3,X4)) -> mark#(X1)
p6: mark#(U93(X1,X2,X3,X4)) -> mark#(X1)
p7: mark#(take(X1,X2)) -> mark#(X1)
p8: mark#(take(X1,X2)) -> mark#(X2)
p9: 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

  (no rules)

Take the reduction pair:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^2
        order: lexicographic order
        interpretations:
          mark#_A(x1) = x1 + (2,2)
          U21_A(x1) = ((1,0),(1,1)) x1 + (3,1)
          U52_A(x1) = x1 + (1,1)
          U71_A(x1,x2,x3) = ((1,0),(1,1)) x1 + x3 + (3,3)
          U72_A(x1,x2) = x1 + (1,1)
          U91_A(x1,x2,x3,x4) = x1 + (3,3)
          U93_A(x1,x2,x3,x4) = x1 + x2 + x3 + x4 + (1,1)
          take_A(x1,x2) = ((1,0),(1,1)) x1 + ((1,0),(1,1)) x2 + (3,3)
          s_A(x1) = x1 + (3,3)
  
    precedence:
    
      mark# = U93 = take > s > U21 > U52 > U71 > U72 > U91
    
    partial status:
  
      pi(mark#) = []
      pi(U21) = []
      pi(U52) = []
      pi(U71) = [3]
      pi(U72) = []
      pi(U91) = []
      pi(U93) = []
      pi(take) = []
      pi(s) = [1]

The next rules are strictly ordered:

  p1

We remove them from the problem.

-- SCC decomposition.

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

p1: mark#(U52(X)) -> mark#(X)
p2: mark#(U71(X1,X2,X3)) -> mark#(X1)
p3: mark#(U72(X1,X2)) -> mark#(X1)
p4: mark#(U91(X1,X2,X3,X4)) -> mark#(X1)
p5: mark#(U93(X1,X2,X3,X4)) -> mark#(X1)
p6: mark#(take(X1,X2)) -> mark#(X1)
p7: mark#(take(X1,X2)) -> mark#(X2)
p8: 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}


-- Reduction pair.

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

p1: mark#(U52(X)) -> mark#(X)
p2: mark#(s(X)) -> mark#(X)
p3: mark#(take(X1,X2)) -> mark#(X2)
p4: mark#(take(X1,X2)) -> mark#(X1)
p5: mark#(U93(X1,X2,X3,X4)) -> mark#(X1)
p6: mark#(U91(X1,X2,X3,X4)) -> mark#(X1)
p7: mark#(U72(X1,X2)) -> mark#(X1)
p8: mark#(U71(X1,X2,X3)) -> mark#(X1)

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:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^2
        order: lexicographic order
        interpretations:
          mark#_A(x1) = x1 + (1,2)
          U52_A(x1) = ((1,0),(1,1)) x1 + (2,1)
          s_A(x1) = ((1,0),(1,1)) x1 + (2,3)
          take_A(x1,x2) = x1 + ((1,0),(1,1)) x2 + (2,3)
          U93_A(x1,x2,x3,x4) = x1 + x2 + x3 + x4 + (2,1)
          U91_A(x1,x2,x3,x4) = x1 + x3 + x4 + (2,3)
          U72_A(x1,x2) = ((1,0),(1,1)) x1 + (2,2)
          U71_A(x1,x2,x3) = x1 + x2 + x3 + (2,3)
  
    precedence:
    
      U71 > s > take > U91 > U72 > U52 > mark# = U93
    
    partial status:
  
      pi(mark#) = [1]
      pi(U52) = []
      pi(s) = [1]
      pi(take) = []
      pi(U93) = [1, 4]
      pi(U91) = []
      pi(U72) = []
      pi(U71) = [1, 3]

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#(s(X)) -> mark#(X)
p2: mark#(take(X1,X2)) -> mark#(X2)
p3: mark#(take(X1,X2)) -> mark#(X1)
p4: mark#(U93(X1,X2,X3,X4)) -> mark#(X1)
p5: mark#(U91(X1,X2,X3,X4)) -> mark#(X1)
p6: mark#(U72(X1,X2)) -> mark#(X1)
p7: mark#(U71(X1,X2,X3)) -> mark#(X1)

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}


-- Reduction pair.

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

p1: mark#(s(X)) -> mark#(X)
p2: mark#(U71(X1,X2,X3)) -> mark#(X1)
p3: mark#(U72(X1,X2)) -> mark#(X1)
p4: mark#(U91(X1,X2,X3,X4)) -> mark#(X1)
p5: mark#(U93(X1,X2,X3,X4)) -> mark#(X1)
p6: mark#(take(X1,X2)) -> mark#(X1)
p7: mark#(take(X1,X2)) -> mark#(X2)

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:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^2
        order: lexicographic order
        interpretations:
          mark#_A(x1) = ((1,0),(1,1)) x1 + (1,2)
          s_A(x1) = ((1,0),(1,1)) x1 + (2,1)
          U71_A(x1,x2,x3) = x1 + x2 + x3 + (2,1)
          U72_A(x1,x2) = x1 + (2,1)
          U91_A(x1,x2,x3,x4) = ((1,0),(1,1)) x1 + x2 + x4 + (2,1)
          U93_A(x1,x2,x3,x4) = ((1,0),(1,1)) x1 + ((1,0),(1,1)) x2 + ((1,0),(1,1)) x3 + ((1,0),(1,1)) x4 + (2,1)
          take_A(x1,x2) = x1 + ((1,0),(1,1)) x2 + (2,3)
  
    precedence:
    
      s > U71 > U72 > U91 > take > mark# > U93
    
    partial status:
  
      pi(mark#) = [1]
      pi(s) = []
      pi(U71) = []
      pi(U72) = []
      pi(U91) = []
      pi(U93) = [1, 4]
      pi(take) = [1, 2]

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#(s(X)) -> mark#(X)
p2: mark#(U71(X1,X2,X3)) -> mark#(X1)
p3: mark#(U72(X1,X2)) -> mark#(X1)
p4: mark#(U91(X1,X2,X3,X4)) -> mark#(X1)
p5: mark#(U93(X1,X2,X3,X4)) -> mark#(X1)
p6: mark#(take(X1,X2)) -> mark#(X2)

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}


-- Reduction pair.

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

p1: mark#(s(X)) -> mark#(X)
p2: mark#(take(X1,X2)) -> mark#(X2)
p3: mark#(U93(X1,X2,X3,X4)) -> mark#(X1)
p4: mark#(U91(X1,X2,X3,X4)) -> mark#(X1)
p5: mark#(U72(X1,X2)) -> mark#(X1)
p6: mark#(U71(X1,X2,X3)) -> mark#(X1)

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:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^2
        order: lexicographic order
        interpretations:
          mark#_A(x1) = ((1,0),(1,1)) x1 + (1,2)
          s_A(x1) = x1 + (2,1)
          take_A(x1,x2) = x1 + x2 + (2,1)
          U93_A(x1,x2,x3,x4) = x1 + x2 + x4 + (2,1)
          U91_A(x1,x2,x3,x4) = x1 + x2 + x3 + x4 + (2,1)
          U72_A(x1,x2) = ((1,0),(1,1)) x1 + x2 + (2,3)
          U71_A(x1,x2,x3) = x1 + (2,3)
  
    precedence:
    
      U72 > take > U71 > U91 > mark# = U93 > s
    
    partial status:
  
      pi(mark#) = [1]
      pi(s) = []
      pi(take) = []
      pi(U93) = [1, 4]
      pi(U91) = [1, 4]
      pi(U72) = [1, 2]
      pi(U71) = [1]

The next rules are strictly ordered:

  p4

We remove them from the problem.

-- SCC decomposition.

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

p1: mark#(s(X)) -> mark#(X)
p2: mark#(take(X1,X2)) -> mark#(X2)
p3: mark#(U93(X1,X2,X3,X4)) -> mark#(X1)
p4: mark#(U72(X1,X2)) -> mark#(X1)
p5: mark#(U71(X1,X2,X3)) -> mark#(X1)

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}


-- Reduction pair.

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

p1: mark#(s(X)) -> mark#(X)
p2: mark#(U71(X1,X2,X3)) -> mark#(X1)
p3: mark#(U72(X1,X2)) -> mark#(X1)
p4: mark#(U93(X1,X2,X3,X4)) -> mark#(X1)
p5: mark#(take(X1,X2)) -> mark#(X2)

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:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^2
        order: lexicographic order
        interpretations:
          mark#_A(x1) = x1 + (1,2)
          s_A(x1) = ((1,0),(1,1)) x1 + (2,3)
          U71_A(x1,x2,x3) = ((1,0),(1,1)) x1 + ((0,0),(1,0)) x2 + ((1,0),(1,1)) x3 + (2,3)
          U72_A(x1,x2) = x1 + (2,1)
          U93_A(x1,x2,x3,x4) = ((1,0),(1,1)) x1 + ((1,0),(1,1)) x2 + ((0,0),(1,0)) x3 + ((1,0),(1,1)) x4 + (2,3)
          take_A(x1,x2) = x1 + x2 + (2,3)
  
    precedence:
    
      take > U71 > mark# = U72 = U93 > s
    
    partial status:
  
      pi(mark#) = [1]
      pi(s) = []
      pi(U71) = [1, 3]
      pi(U72) = [1]
      pi(U93) = [1, 4]
      pi(take) = [2]

The next rules are strictly ordered:

  p5

We remove them from the problem.

-- SCC decomposition.

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

p1: mark#(s(X)) -> mark#(X)
p2: mark#(U71(X1,X2,X3)) -> mark#(X1)
p3: mark#(U72(X1,X2)) -> mark#(X1)
p4: mark#(U93(X1,X2,X3,X4)) -> mark#(X1)

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}


-- Reduction pair.

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

p1: mark#(s(X)) -> mark#(X)
p2: mark#(U93(X1,X2,X3,X4)) -> mark#(X1)
p3: mark#(U72(X1,X2)) -> mark#(X1)
p4: mark#(U71(X1,X2,X3)) -> mark#(X1)

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:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^2
        order: lexicographic order
        interpretations:
          mark#_A(x1) = x1 + (1,2)
          s_A(x1) = x1 + (2,1)
          U93_A(x1,x2,x3,x4) = ((1,0),(1,1)) x1 + x4 + (2,3)
          U72_A(x1,x2) = x1 + x2 + (2,1)
          U71_A(x1,x2,x3) = ((1,0),(1,1)) x1 + x2 + ((0,0),(1,0)) x3 + (2,3)
  
    precedence:
    
      U72 > mark# = U93 > U71 > s
    
    partial status:
  
      pi(mark#) = [1]
      pi(s) = []
      pi(U93) = [1, 4]
      pi(U72) = []
      pi(U71) = [1, 2]

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#(U93(X1,X2,X3,X4)) -> mark#(X1)
p2: mark#(U72(X1,X2)) -> mark#(X1)
p3: mark#(U71(X1,X2,X3)) -> mark#(X1)

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}


-- Reduction pair.

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

p1: mark#(U93(X1,X2,X3,X4)) -> mark#(X1)
p2: mark#(U71(X1,X2,X3)) -> mark#(X1)
p3: mark#(U72(X1,X2)) -> mark#(X1)

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:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^2
        order: lexicographic order
        interpretations:
          mark#_A(x1) = ((1,0),(0,0)) x1 + (2,2)
          U93_A(x1,x2,x3,x4) = ((1,0),(1,1)) x1 + x3 + x4 + (3,3)
          U71_A(x1,x2,x3) = ((1,0),(0,0)) x1 + x2 + x3 + (1,1)
          U72_A(x1,x2) = ((1,0),(1,0)) x1 + (3,1)
  
    precedence:
    
      mark# = U72 > U93 > U71
    
    partial status:
  
      pi(mark#) = []
      pi(U93) = []
      pi(U71) = []
      pi(U72) = []

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#(U93(X1,X2,X3,X4)) -> mark#(X1)
p2: mark#(U71(X1,X2,X3)) -> mark#(X1)

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}


-- Reduction pair.

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

p1: mark#(U93(X1,X2,X3,X4)) -> mark#(X1)
p2: mark#(U71(X1,X2,X3)) -> mark#(X1)

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:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^2
        order: lexicographic order
        interpretations:
          mark#_A(x1) = ((1,0),(0,0)) x1 + (2,1)
          U93_A(x1,x2,x3,x4) = ((1,0),(1,1)) x1 + ((0,0),(1,0)) x2 + ((1,0),(1,1)) x3 + ((1,0),(1,1)) x4 + (3,2)
          U71_A(x1,x2,x3) = ((1,0),(0,0)) x1 + x2 + x3 + (1,0)
  
    precedence:
    
      mark# = U93 = U71
    
    partial status:
  
      pi(mark#) = []
      pi(U93) = [4]
      pi(U71) = [3]

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#(U71(X1,X2,X3)) -> mark#(X1)

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}


-- Reduction pair.

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

p1: mark#(U71(X1,X2,X3)) -> mark#(X1)

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:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^2
        order: lexicographic order
        interpretations:
          mark#_A(x1) = ((1,0),(1,1)) x1 + (1,2)
          U71_A(x1,x2,x3) = ((1,0),(1,1)) x1 + ((1,0),(1,1)) x2 + ((1,0),(1,1)) x3 + (2,1)
  
    precedence:
    
      mark# > U71
    
    partial status:
  
      pi(mark#) = [1]
      pi(U71) = [1, 3]

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__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:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^2
        order: lexicographic order
        interpretations:
          a__U41#_A(x1,x2) = ((1,0),(1,1)) x1 + ((1,0),(0,0)) x2 + (1,1)
          tt_A() = (3,4)
          a__isNatIList#_A(x1) = ((1,0),(1,0)) x1 + (2,9)
          cons_A(x1,x2) = ((1,0),(1,1)) x1 + ((1,0),(1,1)) x2 + (14,0)
          a__isNat#_A(x1) = ((1,0),(0,0)) x1
          s_A(x1) = ((1,0),(1,1)) x1 + (0,25)
          length_A(x1) = ((1,0),(1,1)) x1 + (10,10)
          a__isNatList#_A(x1) = ((1,0),(0,0)) x1 + (1,5)
          take_A(x1,x2) = ((1,0),(1,1)) x1 + ((1,0),(1,1)) x2 + (16,0)
          a__U61#_A(x1,x2) = ((1,0),(0,0)) x1 + ((1,0),(1,0)) x2 + (1,10)
          a__isNat_A(x1) = (12,24)
          a__U51#_A(x1,x2) = ((1,0),(0,0)) x1 + x2 + (1,6)
          a__U42_A(x1) = (4,5)
          U42_A(x1) = (0,0)
          a__U31_A(x1) = ((0,0),(1,0)) x1 + (4,0)
          a__U41_A(x1,x2) = ((0,0),(1,0)) x1 + ((0,0),(1,0)) x2 + (5,3)
          a__isNatIList_A(x1) = ((0,0),(1,0)) x1 + (12,8)
          U31_A(x1) = (0,0)
          U41_A(x1,x2) = (0,0)
          a__U52_A(x1) = (4,5)
          a__U62_A(x1) = (4,5)
          a__isNatList_A(x1) = ((1,0),(1,1)) x1 + (9,9)
          zeros_A() = (1,1)
          isNatIList_A(x1) = (0,0)
          U52_A(x1) = (0,0)
          U62_A(x1) = (0,0)
          a__U51_A(x1,x2) = ((1,0),(1,1)) x1 + (3,1)
          a__U61_A(x1,x2) = ((1,0),(1,1)) x1 + (2,0)
          U51_A(x1,x2) = (0,0)
          U61_A(x1,x2) = (0,0)
          a__U11_A(x1) = ((0,0),(1,0)) x1 + (11,16)
          a__U21_A(x1) = ((0,0),(1,0)) x1 + (4,11)
          nil_A() = (1,1)
          U11_A(x1) = (10,17)
          U21_A(x1) = (0,0)
          isNatList_A(x1) = (0,0)
          |0|_A() = (4,0)
          isNat_A(x1) = (0,0)
  
    precedence:
    
      a__isNat# = a__isNatList = nil > isNatIList > cons = take = a__isNat = a__U41 = a__isNatIList > s = a__U42 > U62 = a__U51 > tt = a__U62 > zeros > U42 = a__U52 > length > a__U11 > a__U41# > U52 > U51 = a__U21 > isNatList = |0| = isNat > U21 > a__isNatIList# > a__isNatList# = a__U61# = a__U51# > U31 > U11 > a__U31 = U41 = a__U61 = U61
    
    partial status:
  
      pi(a__U41#) = []
      pi(tt) = []
      pi(a__isNatIList#) = []
      pi(cons) = [1]
      pi(a__isNat#) = []
      pi(s) = []
      pi(length) = [1]
      pi(a__isNatList#) = []
      pi(take) = []
      pi(a__U61#) = []
      pi(a__isNat) = []
      pi(a__U51#) = []
      pi(a__U42) = []
      pi(U42) = []
      pi(a__U31) = []
      pi(a__U41) = []
      pi(a__isNatIList) = []
      pi(U31) = []
      pi(U41) = []
      pi(a__U52) = []
      pi(a__U62) = []
      pi(a__isNatList) = []
      pi(zeros) = []
      pi(isNatIList) = []
      pi(U52) = []
      pi(U62) = []
      pi(a__U51) = []
      pi(a__U61) = [1]
      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:

  p2

We remove them from the problem.

-- SCC decomposition.

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

p1: a__U41#(tt(),V2) -> a__isNatIList#(V2)
p2: a__isNat#(s(V1)) -> a__isNat#(V1)
p3: a__isNat#(length(V1)) -> a__isNatList#(V1)
p4: a__isNatList#(take(V1,V2)) -> a__isNat#(V1)
p5: a__isNatList#(take(V1,V2)) -> a__U61#(a__isNat(V1),V2)
p6: a__U61#(tt(),V2) -> a__isNatIList#(V2)
p7: a__isNatIList#(cons(V1,V2)) -> a__U41#(a__isNat(V1),V2)
p8: a__isNatIList#(V) -> a__isNatList#(V)
p9: a__isNatList#(cons(V1,V2)) -> a__isNat#(V1)
p10: a__isNatList#(cons(V1,V2)) -> a__U51#(a__isNat(V1),V2)
p11: 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 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: a__U41#(tt(),V2) -> a__isNatIList#(V2)
p2: a__isNatIList#(V) -> a__isNatList#(V)
p3: a__isNatList#(cons(V1,V2)) -> a__U51#(a__isNat(V1),V2)
p4: a__U51#(tt(),V2) -> a__isNatList#(V2)
p5: a__isNatList#(cons(V1,V2)) -> a__isNat#(V1)
p6: a__isNat#(length(V1)) -> a__isNatList#(V1)
p7: a__isNatList#(take(V1,V2)) -> a__U61#(a__isNat(V1),V2)
p8: a__U61#(tt(),V2) -> a__isNatIList#(V2)
p9: a__isNatIList#(cons(V1,V2)) -> a__U41#(a__isNat(V1),V2)
p10: a__isNatList#(take(V1,V2)) -> a__isNat#(V1)
p11: a__isNat#(s(V1)) -> a__isNat#(V1)

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:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^2
        order: lexicographic order
        interpretations:
          a__U41#_A(x1,x2) = ((1,0),(1,1)) x1 + ((1,0),(0,0)) x2 + (1,1)
          tt_A() = (24,22)
          a__isNatIList#_A(x1) = ((1,0),(1,1)) x1 + (12,1)
          a__isNatList#_A(x1) = ((1,0),(1,1)) x1 + (11,23)
          cons_A(x1,x2) = ((1,0),(1,1)) x1 + ((1,0),(1,1)) x2 + (26,25)
          a__U51#_A(x1,x2) = ((1,0),(1,1)) x1 + ((1,0),(1,1)) x2 + (1,1)
          a__isNat_A(x1) = ((1,0),(1,1)) x1 + (24,19)
          a__isNat#_A(x1) = x1 + (1,2)
          length_A(x1) = ((1,0),(1,1)) x1 + (27,24)
          take_A(x1,x2) = ((1,0),(1,1)) x1 + ((1,0),(1,1)) x2 + (14,3)
          a__U61#_A(x1,x2) = ((1,0),(1,1)) x2 + (13,2)
          s_A(x1) = ((1,0),(1,1)) x1 + (25,20)
          a__U42_A(x1) = (24,23)
          U42_A(x1) = (0,0)
          a__U31_A(x1) = ((0,0),(1,0)) x1 + (25,1)
          a__U41_A(x1,x2) = ((1,0),(1,0)) x1 + ((1,0),(0,0)) x2
          a__isNatIList_A(x1) = ((1,0),(1,0)) x1 + (27,28)
          U31_A(x1) = (0,0)
          U41_A(x1,x2) = (0,0)
          a__U52_A(x1) = (24,22)
          a__U62_A(x1) = (25,23)
          a__isNatList_A(x1) = ((1,0),(1,1)) x1 + (26,1)
          zeros_A() = (0,1)
          isNatIList_A(x1) = (0,0)
          U52_A(x1) = (0,0)
          U62_A(x1) = (1,24)
          a__U51_A(x1,x2) = ((1,0),(1,1)) x1 + (1,10)
          a__U61_A(x1,x2) = ((1,0),(1,1)) x1 + (4,1)
          U51_A(x1,x2) = (0,0)
          U61_A(x1,x2) = (1,2)
          a__U11_A(x1) = ((1,0),(1,0)) x1 + (2,1)
          a__U21_A(x1) = ((1,0),(1,1)) x1 + (2,1)
          nil_A() = (25,23)
          U11_A(x1) = (0,0)
          U21_A(x1) = (0,0)
          isNatList_A(x1) = (1,2)
          |0|_A() = (1,1)
          isNat_A(x1) = (24,1)
  
    precedence:
    
      isNatIList > a__U41# = cons > U31 > a__isNatIList > a__U51 > a__U52 > tt = a__isNatIList# = a__U42 = a__U31 = a__U41 = zeros = a__U61 > a__isNatList# = a__U51# = a__isNat = a__U62 > length = a__U61# = a__isNatList > s = U51 = U61 = a__U11 = a__U21 > a__isNat# = take > U42 = nil = U11 = U21 = isNatList = |0| = isNat > U41 = U52 = U62
    
    partial status:
  
      pi(a__U41#) = []
      pi(tt) = []
      pi(a__isNatIList#) = []
      pi(a__isNatList#) = [1]
      pi(cons) = [1]
      pi(a__U51#) = [2]
      pi(a__isNat) = [1]
      pi(a__isNat#) = []
      pi(length) = [1]
      pi(take) = []
      pi(a__U61#) = []
      pi(s) = [1]
      pi(a__U42) = []
      pi(U42) = []
      pi(a__U31) = []
      pi(a__U41) = []
      pi(a__isNatIList) = []
      pi(U31) = []
      pi(U41) = []
      pi(a__U52) = []
      pi(a__U62) = []
      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

We remove them from the problem.

-- SCC decomposition.

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

p1: a__isNatIList#(V) -> a__isNatList#(V)
p2: a__isNatList#(cons(V1,V2)) -> a__U51#(a__isNat(V1),V2)
p3: a__U51#(tt(),V2) -> a__isNatList#(V2)
p4: a__isNatList#(cons(V1,V2)) -> a__isNat#(V1)
p5: a__isNat#(length(V1)) -> a__isNatList#(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__isNatList#(take(V1,V2)) -> a__isNat#(V1)
p10: a__isNat#(s(V1)) -> a__isNat#(V1)

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, p9, p10}


-- Reduction pair.

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

p1: a__isNatIList#(V) -> a__isNatList#(V)
p2: a__isNatList#(take(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__U61#(a__isNat(V1),V2)
p6: a__U61#(tt(),V2) -> a__isNatIList#(V2)
p7: a__isNatList#(cons(V1,V2)) -> a__isNat#(V1)
p8: a__isNatList#(cons(V1,V2)) -> a__U51#(a__isNat(V1),V2)
p9: 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:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^2
        order: lexicographic order
        interpretations:
          a__isNatIList#_A(x1) = ((1,0),(1,0)) x1 + (9,22)
          a__isNatList#_A(x1) = ((1,0),(0,0)) x1 + (8,3)
          take_A(x1,x2) = ((1,0),(1,1)) x1 + ((1,0),(1,1)) x2 + (11,19)
          a__isNat#_A(x1) = ((1,0),(0,0)) x1 + (1,1)
          s_A(x1) = ((1,0),(1,1)) x1 + (3,14)
          length_A(x1) = ((1,0),(1,1)) x1 + (9,2)
          a__U61#_A(x1,x2) = ((0,0),(1,0)) x1 + ((1,0),(1,0)) x2 + (10,16)
          a__isNat_A(x1) = ((1,0),(1,1)) x1 + (2,13)
          tt_A() = (7,21)
          cons_A(x1,x2) = ((1,0),(1,1)) x1 + ((1,0),(1,0)) x2 + (4,2)
          a__U51#_A(x1,x2) = ((1,0),(1,1)) x2 + (9,4)
          a__U42_A(x1) = ((1,0),(1,1)) x1 + (2,2)
          U42_A(x1) = (1,1)
          a__U31_A(x1) = ((1,0),(1,1)) x1 + (1,32)
          a__U41_A(x1,x2) = ((1,0),(1,1)) x1 + ((1,0),(1,0)) x2 + (1,26)
          a__isNatIList_A(x1) = ((1,0),(1,1)) x1 + (4,34)
          U31_A(x1) = (0,0)
          U41_A(x1,x2) = (0,0)
          a__U52_A(x1) = ((1,0),(1,0)) x1 + (2,18)
          a__U62_A(x1) = (8,20)
          a__isNatList_A(x1) = ((1,0),(1,0)) x1 + (2,1)
          zeros_A() = (8,22)
          isNatIList_A(x1) = (0,0)
          U52_A(x1) = (1,19)
          U62_A(x1) = (1,21)
          a__U51_A(x1,x2) = ((1,0),(1,1)) x1 + ((1,0),(1,1)) x2 + (1,1)
          a__U61_A(x1,x2) = ((1,0),(1,1)) x1 + (2,5)
          U51_A(x1,x2) = (0,0)
          U61_A(x1,x2) = (1,6)
          a__U11_A(x1) = ((1,0),(1,1)) x1 + (8,1)
          a__U21_A(x1) = x1 + (2,18)
          nil_A() = (21,22)
          U11_A(x1) = (0,0)
          U21_A(x1) = (1,19)
          isNatList_A(x1) = (1,2)
          |0|_A() = (6,1)
          isNat_A(x1) = (0,0)
  
    precedence:
    
      a__U52 = a__U51 = nil = U11 = U21 > isNatList > U52 > a__U61# = isNat > cons > tt = a__U42 = a__U41 = a__isNatList = zeros > length > s > take > a__isNat# > a__isNatIList# = a__isNatList# > a__U51# = U42 > U31 > a__U31 = U62 = a__U61 > a__U62 > a__isNat = isNatIList > U51 > a__isNatIList = a__U11 > |0| > U41 = U61 = a__U21
    
    partial status:
  
      pi(a__isNatIList#) = []
      pi(a__isNatList#) = []
      pi(take) = []
      pi(a__isNat#) = []
      pi(s) = []
      pi(length) = [1]
      pi(a__U61#) = []
      pi(a__isNat) = []
      pi(tt) = []
      pi(cons) = []
      pi(a__U51#) = []
      pi(a__U42) = []
      pi(U42) = []
      pi(a__U31) = [1]
      pi(a__U41) = []
      pi(a__isNatIList) = []
      pi(U31) = []
      pi(U41) = []
      pi(a__U52) = []
      pi(a__U62) = []
      pi(a__isNatList) = []
      pi(zeros) = []
      pi(isNatIList) = []
      pi(U52) = []
      pi(U62) = []
      pi(a__U51) = [2]
      pi(a__U61) = []
      pi(U51) = []
      pi(U61) = []
      pi(a__U11) = []
      pi(a__U21) = [1]
      pi(nil) = []
      pi(U11) = []
      pi(U21) = []
      pi(isNatList) = []
      pi(|0|) = []
      pi(isNat) = []

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

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:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^2
        order: lexicographic order
        interpretations:
          a__isNatIList#_A(x1) = ((0,0),(1,0)) x1 + (1,25)
          a__isNatList#_A(x1) = ((0,0),(1,0)) x1 + (1,24)
          cons_A(x1,x2) = ((1,0),(1,1)) x1 + ((1,0),(1,1)) x2 + (16,16)
          a__U51#_A(x1,x2) = ((0,0),(1,0)) x2 + (1,25)
          a__isNat_A(x1) = ((0,0),(1,0)) x1 + (4,37)
          tt_A() = (2,9)
          a__isNat#_A(x1) = ((0,0),(1,0)) x1 + (1,39)
          length_A(x1) = ((1,0),(0,0)) x1 + (4,0)
          take_A(x1,x2) = ((1,0),(1,1)) x1 + ((1,0),(1,1)) x2 + (7,27)
          a__U61#_A(x1,x2) = ((0,0),(1,0)) x2 + (1,26)
          s_A(x1) = ((1,0),(1,1)) x1 + (6,38)
          a__U42_A(x1) = x1 + (2,2)
          U42_A(x1) = (1,1)
          a__U31_A(x1) = x1 + (1,1)
          a__U41_A(x1,x2) = ((1,0),(1,0)) x2 + (8,15)
          a__isNatIList_A(x1) = ((1,0),(1,0)) x1 + (5,12)
          U31_A(x1) = (0,0)
          U41_A(x1,x2) = (0,0)
          a__U52_A(x1) = ((1,0),(1,1)) x1 + (2,3)
          a__U62_A(x1) = ((0,0),(1,0)) x1 + (3,8)
          a__isNatList_A(x1) = ((1,0),(1,1)) x1 + (3,4)
          zeros_A() = (3,10)
          isNatIList_A(x1) = (0,13)
          U52_A(x1) = (1,4)
          U62_A(x1) = (0,0)
          a__U51_A(x1,x2) = ((0,0),(1,0)) x1 + ((1,0),(1,1)) x2 + (6,1)
          a__U61_A(x1,x2) = x1 + (2,2)
          U51_A(x1,x2) = (0,0)
          U61_A(x1,x2) = (0,0)
          a__U11_A(x1) = ((0,0),(1,0)) x1 + (3,8)
          a__U21_A(x1) = ((1,0),(1,1)) x1 + (0,1)
          nil_A() = (3,1)
          U11_A(x1) = (1,9)
          U21_A(x1) = (0,0)
          isNatList_A(x1) = (0,0)
          |0|_A() = (3,10)
          isNat_A(x1) = (0,0)
  
    precedence:
    
      a__isNat# = U42 > a__U31 = zeros > a__isNatIList = a__isNatList = a__U51 > U62 > tt = a__U42 = a__U41 = a__U62 = isNatIList = U61 = |0| > a__isNat > U52 > a__U52 > a__isNatIList# = a__isNatList# = a__U51# = a__U61# > U41 > take = U51 = a__U11 = nil = U11 = U21 = isNatList > cons = length > isNat > U31 = a__U61 = a__U21 > s
    
    partial status:
  
      pi(a__isNatIList#) = []
      pi(a__isNatList#) = []
      pi(cons) = [2]
      pi(a__U51#) = []
      pi(a__isNat) = []
      pi(tt) = []
      pi(a__isNat#) = []
      pi(length) = []
      pi(take) = [1]
      pi(a__U61#) = []
      pi(s) = []
      pi(a__U42) = []
      pi(U42) = []
      pi(a__U31) = []
      pi(a__U41) = []
      pi(a__isNatIList) = []
      pi(U31) = []
      pi(U41) = []
      pi(a__U52) = [1]
      pi(a__U62) = []
      pi(a__isNatList) = []
      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:

  p4

We remove them from the problem.

-- SCC decomposition.

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

p1: a__isNatIList#(V) -> a__isNatList#(V)
p2: a__isNatList#(cons(V1,V2)) -> a__U51#(a__isNat(V1),V2)
p3: a__U51#(tt(),V2) -> a__isNatList#(V2)
p4: a__isNat#(length(V1)) -> a__isNatList#(V1)
p5: a__isNatList#(take(V1,V2)) -> a__U61#(a__isNat(V1),V2)
p6: a__U61#(tt(),V2) -> a__isNatIList#(V2)
p7: a__isNat#(s(V1)) -> a__isNat#(V1)

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


-- Reduction pair.

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

p1: a__isNat#(s(V1)) -> a__isNat#(V1)

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:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^2
        order: lexicographic order
        interpretations:
          a__isNat#_A(x1) = ((1,0),(1,0)) x1 + (2,2)
          s_A(x1) = ((1,0),(0,0)) x1 + (1,1)
  
    precedence:
    
      a__isNat# = s
    
    partial status:
  
      pi(a__isNat#) = []
      pi(s) = []

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__isNatIList#(V) -> a__isNatList#(V)
p2: a__isNatList#(take(V1,V2)) -> a__U61#(a__isNat(V1),V2)
p3: a__U61#(tt(),V2) -> a__isNatIList#(V2)
p4: a__isNatList#(cons(V1,V2)) -> a__U51#(a__isNat(V1),V2)
p5: 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:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^2
        order: lexicographic order
        interpretations:
          a__isNatIList#_A(x1) = ((1,0),(0,0)) x1 + (3,6)
          a__isNatList#_A(x1) = x1 + (2,3)
          take_A(x1,x2) = ((1,0),(0,0)) x1 + ((1,0),(0,0)) x2 + (5,1)
          a__U61#_A(x1,x2) = ((1,0),(0,0)) x1 + x2 + (3,5)
          a__isNat_A(x1) = (3,15)
          tt_A() = (1,2)
          cons_A(x1,x2) = ((1,0),(1,1)) x2 + (8,5)
          a__U51#_A(x1,x2) = x2 + (3,4)
          a__U42_A(x1) = (1,2)
          U42_A(x1) = (0,0)
          a__U31_A(x1) = (2,3)
          a__U41_A(x1,x2) = ((1,0),(0,0)) x1 + ((1,0),(1,1)) x2 + (4,1)
          a__isNatIList_A(x1) = ((1,0),(1,1)) x1 + (4,1)
          U31_A(x1) = (0,0)
          U41_A(x1,x2) = (0,0)
          a__U52_A(x1) = (2,1)
          a__U62_A(x1) = (2,3)
          a__isNatList_A(x1) = (10,16)
          zeros_A() = (2,3)
          isNatIList_A(x1) = x1
          U52_A(x1) = (0,0)
          U62_A(x1) = (0,0)
          a__U51_A(x1,x2) = (9,17)
          a__U61_A(x1,x2) = (3,4)
          U51_A(x1,x2) = (0,0)
          U61_A(x1,x2) = (0,0)
          a__U11_A(x1) = (2,1)
          a__U21_A(x1) = (2,3)
          nil_A() = (2,1)
          U11_A(x1) = (0,0)
          U21_A(x1) = (1,4)
          isNatList_A(x1) = (10,16)
          |0|_A() = (2,3)
          length_A(x1) = (11,2)
          s_A(x1) = (4,16)
          isNat_A(x1) = (0,0)
  
    precedence:
    
      U52 > a__U61# > a__isNat = cons = isNatIList > a__U51# > a__U62 = a__U51 = a__U61 > a__U52 > take > a__U42 > U42 = U62 > a__isNatIList# > a__isNatList# > U31 > a__U41 = U61 = a__U21 = nil = |0| > tt = a__U31 = a__isNatIList = a__isNatList = a__U11 > zeros > U51 = U11 = U21 = isNatList > U41 > length = s = isNat
    
    partial status:
  
      pi(a__isNatIList#) = []
      pi(a__isNatList#) = [1]
      pi(take) = []
      pi(a__U61#) = []
      pi(a__isNat) = []
      pi(tt) = []
      pi(cons) = [2]
      pi(a__U51#) = [2]
      pi(a__U42) = []
      pi(U42) = []
      pi(a__U31) = []
      pi(a__U41) = []
      pi(a__isNatIList) = [1]
      pi(U31) = []
      pi(U41) = []
      pi(a__U52) = []
      pi(a__U62) = []
      pi(a__isNatList) = []
      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(length) = []
      pi(s) = []
      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: a__isNatList#(take(V1,V2)) -> a__U61#(a__isNat(V1),V2)
p2: a__U61#(tt(),V2) -> a__isNatIList#(V2)
p3: a__isNatList#(cons(V1,V2)) -> a__U51#(a__isNat(V1),V2)
p4: 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 estimated dependency graph contains the following SCCs:

  {p3, p4}


-- Reduction pair.

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

p1: a__U51#(tt(),V2) -> a__isNatList#(V2)
p2: a__isNatList#(cons(V1,V2)) -> a__U51#(a__isNat(V1),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:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^2
        order: lexicographic order
        interpretations:
          a__U51#_A(x1,x2) = x2 + (2,1)
          tt_A() = (0,0)
          a__isNatList#_A(x1) = ((1,0),(1,1)) x1 + (1,2)
          cons_A(x1,x2) = ((1,0),(1,1)) x1 + ((1,0),(1,1)) x2 + (8,5)
          a__isNat_A(x1) = (3,17)
          a__U42_A(x1) = ((1,0),(1,0)) x1 + (0,1)
          U42_A(x1) = (0,0)
          a__U31_A(x1) = (1,1)
          a__U41_A(x1,x2) = x2 + (7,4)
          a__isNatIList_A(x1) = ((1,0),(1,1)) x1 + (6,3)
          U31_A(x1) = (0,0)
          U41_A(x1,x2) = (0,0)
          a__U52_A(x1) = (2,1)
          a__U62_A(x1) = (0,0)
          a__isNatList_A(x1) = (5,20)
          zeros_A() = (1,1)
          isNatIList_A(x1) = x1 + (5,1)
          U52_A(x1) = (1,2)
          U62_A(x1) = (0,0)
          a__U51_A(x1,x2) = (3,2)
          a__U61_A(x1,x2) = x1 + (1,2)
          U51_A(x1,x2) = (0,0)
          U61_A(x1,x2) = (0,0)
          a__U11_A(x1) = (1,1)
          a__U21_A(x1) = (2,2)
          nil_A() = (1,1)
          take_A(x1,x2) = (5,18)
          U11_A(x1) = (0,0)
          U21_A(x1) = (1,1)
          isNatList_A(x1) = (1,21)
          |0|_A() = (1,1)
          length_A(x1) = (6,21)
          s_A(x1) = (4,18)
          isNat_A(x1) = (3,17)
  
    precedence:
    
      a__isNat = U21 = isNatList > a__isNatList# = a__isNatIList = a__isNatList = isNatIList = a__U51 = |0| = s > a__U31 = a__U52 = U52 > cons = a__U42 = U42 = a__U41 = U41 = zeros = U51 = U61 = take > a__U51# = U11 > a__U11 = a__U21 > nil = length = isNat > tt = U31 = a__U62 = U62 = a__U61
    
    partial status:
  
      pi(a__U51#) = []
      pi(tt) = []
      pi(a__isNatList#) = []
      pi(cons) = []
      pi(a__isNat) = []
      pi(a__U42) = []
      pi(U42) = []
      pi(a__U31) = []
      pi(a__U41) = []
      pi(a__isNatIList) = [1]
      pi(U31) = []
      pi(U41) = []
      pi(a__U52) = []
      pi(a__U62) = []
      pi(a__isNatList) = []
      pi(zeros) = []
      pi(isNatIList) = []
      pi(U52) = []
      pi(U62) = []
      pi(a__U51) = []
      pi(a__U61) = [1]
      pi(U51) = []
      pi(U61) = []
      pi(a__U11) = []
      pi(a__U21) = []
      pi(nil) = []
      pi(take) = []
      pi(U11) = []
      pi(U21) = []
      pi(isNatList) = []
      pi(|0|) = []
      pi(length) = []
      pi(s) = []
      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: a__isNatList#(cons(V1,V2)) -> a__U51#(a__isNat(V1),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 estimated dependency graph contains the following SCCs:

  (no SCCs)