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:

  max/plus interpretations on natural numbers:
  
    a__U93#_A(x1,x2,x3,x4) = max{x1 + 26, x2 + 36, x4 + 36}
    tt_A = 10
    mark#_A(x1) = max{36, x1 + 15}
    s_A(x1) = max{21, x1}
    cons_A(x1,x2) = max{21, x1, x2}
    take_A(x1,x2) = max{x1 + 13, x2 + 21}
    a__take#_A(x1,x2) = x2 + 36
    mark_A(x1) = x1
    a__U91#_A(x1,x2,x3,x4) = max{x1 + 18, x2 + 36, x4 + 36}
    a__isNatIList_A(x1) = 10
    a__U92#_A(x1,x2,x3,x4) = max{x1 + 17, x2 + 36, x4 + 36}
    a__isNat_A(x1) = 10
    U93_A(x1,x2,x3,x4) = max{x1 + 11, x2 + 21, x3 + 13, x4 + 21}
    U92_A(x1,x2,x3,x4) = max{x1 + 3, x2 + 21, x3 + 13, x4 + 21}
    U91_A(x1,x2,x3,x4) = max{x1 + 3, x2 + 21, x3 + 13, x4 + 21}
    U81_A(x1) = max{21, x1}
    length_A(x1) = max{21, x1 + 2}
    a__length#_A(x1) = x1 + 16
    a__U71#_A(x1,x2,x3) = max{x1 + 26, x2 + 16, x3 + 3}
    a__isNatList_A(x1) = 10
    a__U72#_A(x1,x2) = max{x1 + 26, x2 + 16}
    U72_A(x1,x2) = max{x1 + 11, x2 + 2}
    U71_A(x1,x2,x3) = max{x1 + 11, x2 + 2, x3 - 12}
    U62_A(x1) = max{9, x1}
    U61_A(x1,x2) = max{8, x1}
    U52_A(x1) = max{9, x1}
    U51_A(x1,x2) = x1
    U42_A(x1) = max{10, x1}
    U41_A(x1,x2) = max{1, x1}
    U31_A(x1) = max{10, x1}
    U21_A(x1) = x1
    U11_A(x1) = max{10, x1}
    a__zeros_A = 21
    |0|_A = 21
    zeros_A = 21
    a__U11_A(x1) = max{10, x1}
    a__U21_A(x1) = x1
    a__U31_A(x1) = max{10, x1}
    a__U41_A(x1,x2) = max{1, x1}
    a__U42_A(x1) = max{10, x1}
    a__U51_A(x1,x2) = x1
    a__U52_A(x1) = max{9, x1}
    a__U61_A(x1,x2) = max{8, x1}
    a__U62_A(x1) = max{9, x1}
    a__U71_A(x1,x2,x3) = max{x1 + 11, x2 + 2, x3 - 12}
    a__U72_A(x1,x2) = max{x1 + 11, x2 + 2}
    a__length_A(x1) = max{21, x1 + 2}
    a__U81_A(x1) = max{21, x1}
    nil_A = 11
    a__U91_A(x1,x2,x3,x4) = max{x1 + 3, x2 + 21, x3 + 13, x4 + 21}
    a__U92_A(x1,x2,x3,x4) = max{x1 + 3, x2 + 21, x3 + 13, x4 + 21}
    a__U93_A(x1,x2,x3,x4) = max{x1 + 11, x2 + 21, x3 + 13, x4 + 21}
    a__take_A(x1,x2) = max{x1 + 13, x2 + 21}
    isNatIList_A(x1) = 10
    isNatList_A(x1) = 10
    isNat_A(x1) = 10

The next rules are strictly ordered:

  p18

We remove them from the problem.

-- SCC decomposition.

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

p1: a__U93#(tt(),IL,M,N) -> mark#(N)
p2: mark#(s(X)) -> mark#(X)
p3: mark#(cons(X1,X2)) -> mark#(X1)
p4: mark#(take(X1,X2)) -> mark#(X2)
p5: mark#(take(X1,X2)) -> mark#(X1)
p6: mark#(take(X1,X2)) -> a__take#(mark(X1),mark(X2))
p7: a__take#(s(M),cons(N,IL)) -> a__U91#(a__isNatIList(IL),IL,M,N)
p8: a__U91#(tt(),IL,M,N) -> a__U92#(a__isNat(M),IL,M,N)
p9: 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: 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, p6, p7, p8, p9, p10, p11, p12, p13, p14, p15, p16, p17, p18, p19, p20, p21, p22, p23, p24, p25, p26, p27, p28, p29, p30, p31, p32, p33, p34}


-- 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)) -> 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)) -> a__take#(mark(X1),mark(X2))
p30: a__take#(s(M),cons(N,IL)) -> a__U91#(a__isNatIList(IL),IL,M,N)
p31: mark#(take(X1,X2)) -> mark#(X1)
p32: mark#(take(X1,X2)) -> mark#(X2)
p33: mark#(cons(X1,X2)) -> mark#(X1)
p34: 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:

  max/plus interpretations on natural numbers:
  
    a__U93#_A(x1,x2,x3,x4) = max{x1 - 6, x2 + 34, x3 + 18, x4 + 3}
    tt_A = 26
    mark#_A(x1) = max{19, x1 + 3}
    U11_A(x1) = max{21, x1 + 1}
    U21_A(x1) = max{13, x1}
    U31_A(x1) = max{27, x1}
    U41_A(x1,x2) = max{27, x1, x2 + 13}
    U42_A(x1) = max{17, x1}
    U51_A(x1,x2) = max{x1, x2 + 13}
    U52_A(x1) = max{26, x1}
    U61_A(x1,x2) = max{x1, x2 + 27}
    U62_A(x1) = max{12, x1}
    U71_A(x1,x2,x3) = max{x1 + 26, x2 + 39, x3 + 25}
    a__U71#_A(x1,x2,x3) = max{x1 + 5, x2 + 32}
    mark_A(x1) = max{13, x1}
    a__U72#_A(x1,x2) = x2 + 32
    a__isNat_A(x1) = max{21, x1 + 13}
    a__length#_A(x1) = max{21, x1 + 19}
    cons_A(x1,x2) = max{x1, x2 + 13}
    a__isNatList_A(x1) = x1 + 13
    U72_A(x1,x2) = max{52, x1 + 1, x2 + 39}
    length_A(x1) = max{39, x1 + 26}
    U81_A(x1) = max{17, x1 + 1}
    U91_A(x1,x2,x3,x4) = max{x1 + 18, x2 + 31, x3 + 20, x4 + 18}
    a__U91#_A(x1,x2,x3,x4) = max{x1 - 7, x2 + 34, x3 + 22, x4 + 21}
    a__U92#_A(x1,x2,x3,x4) = max{x1 - 18, x2 + 34, x3 + 22, x4 + 21}
    U92_A(x1,x2,x3,x4) = max{44, x1, x2 + 31, x3 + 20, x4 + 18}
    U93_A(x1,x2,x3,x4) = max{44, x1 + 5, x2 + 31, x3 + 20, x4 + 18}
    take_A(x1,x2) = max{31, x1 + 7, x2 + 18}
    a__take#_A(x1,x2) = max{x1 + 9, x2 + 21}
    s_A(x1) = max{38, x1 + 13}
    a__isNatIList_A(x1) = max{27, x1 + 13}
    a__zeros_A = 13
    |0|_A = 13
    zeros_A = 0
    a__U11_A(x1) = max{21, x1 + 1}
    a__U21_A(x1) = max{13, x1}
    a__U31_A(x1) = max{27, x1}
    a__U41_A(x1,x2) = max{27, x1, x2 + 13}
    a__U42_A(x1) = max{17, x1}
    a__U51_A(x1,x2) = max{x1, x2 + 13}
    a__U52_A(x1) = max{26, x1}
    a__U61_A(x1,x2) = max{x1, x2 + 27}
    a__U62_A(x1) = max{13, x1}
    a__U71_A(x1,x2,x3) = max{x1 + 26, x2 + 39, x3 + 25}
    a__U72_A(x1,x2) = max{52, x1 + 1, x2 + 39}
    a__length_A(x1) = max{39, x1 + 26}
    a__U81_A(x1) = max{17, x1 + 1}
    nil_A = 18
    a__U91_A(x1,x2,x3,x4) = max{x1 + 18, x2 + 31, x3 + 20, x4 + 18}
    a__U92_A(x1,x2,x3,x4) = max{44, x1, x2 + 31, x3 + 20, x4 + 18}
    a__U93_A(x1,x2,x3,x4) = max{44, x1 + 5, x2 + 31, x3 + 20, x4 + 18}
    a__take_A(x1,x2) = max{31, x1 + 7, x2 + 18}
    isNatIList_A(x1) = max{27, x1 + 13}
    isNatList_A(x1) = x1 + 13
    isNat_A(x1) = max{21, x1 + 13}

The next rules are strictly ordered:

  p2, p11, p15, p16, p17, p18, p19, p20, p24, p28, p31, p32, p34

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#(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: a__U71#(tt(),L,N) -> a__U72#(a__isNat(N),L)
p11: a__U72#(tt(),L) -> a__length#(mark(L))
p12: a__length#(cons(N,L)) -> a__U71#(a__isNatList(L),L,N)
p13: mark#(U91(X1,X2,X3,X4)) -> a__U91#(mark(X1),X2,X3,X4)
p14: a__U91#(tt(),IL,M,N) -> a__U92#(a__isNat(M),IL,M,N)
p15: a__U92#(tt(),IL,M,N) -> a__U93#(a__isNat(N),IL,M,N)
p16: mark#(U92(X1,X2,X3,X4)) -> a__U92#(mark(X1),X2,X3,X4)
p17: mark#(U92(X1,X2,X3,X4)) -> mark#(X1)
p18: mark#(U93(X1,X2,X3,X4)) -> a__U93#(mark(X1),X2,X3,X4)
p19: mark#(take(X1,X2)) -> a__take#(mark(X1),mark(X2))
p20: a__take#(s(M),cons(N,IL)) -> a__U91#(a__isNatIList(IL),IL,M,N)
p21: mark#(cons(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 estimated dependency graph contains the following SCCs:

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


-- Reduction pair.

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

p1: a__U93#(tt(),IL,M,N) -> mark#(N)
p2: mark#(cons(X1,X2)) -> mark#(X1)
p3: mark#(take(X1,X2)) -> a__take#(mark(X1),mark(X2))
p4: a__take#(s(M),cons(N,IL)) -> a__U91#(a__isNatIList(IL),IL,M,N)
p5: a__U91#(tt(),IL,M,N) -> a__U92#(a__isNat(M),IL,M,N)
p6: a__U92#(tt(),IL,M,N) -> a__U93#(a__isNat(N),IL,M,N)
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)) -> a__U91#(mark(X1),X2,X3,X4)
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

  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:

  max/plus interpretations on natural numbers:
  
    a__U93#_A(x1,x2,x3,x4) = max{x1 + 9, x2 + 3, x3 + 36, x4 + 8}
    tt_A = 11
    mark#_A(x1) = x1 + 8
    cons_A(x1,x2) = max{9, x1, x2}
    take_A(x1,x2) = max{x1 + 38, x2 + 38}
    a__take#_A(x1,x2) = max{x1 + 37, x2 + 23}
    mark_A(x1) = x1 + 9
    s_A(x1) = x1
    a__U91#_A(x1,x2,x3,x4) = max{x1 - 4, x2 + 22, x3 + 37, x4 + 23}
    a__isNatIList_A(x1) = max{41, x1 + 27}
    a__U92#_A(x1,x2,x3,x4) = max{x1 - 8, x2 + 21, x3 + 36, x4 + 8}
    a__isNat_A(x1) = 11
    U93_A(x1,x2,x3,x4) = max{x1 + 10, x2 + 29, x3 + 29, x4 + 12}
    U92_A(x1,x2,x3,x4) = max{x1, x2 + 29, x3 + 29, x4 + 12}
    U91_A(x1,x2,x3,x4) = max{x1 - 3, x2 + 29, x3 + 29, x4 + 15}
    U62_A(x1) = max{2, x1 + 1}
    U61_A(x1,x2) = max{x1 + 1, x2 + 33}
    U52_A(x1) = max{1, x1}
    U51_A(x1,x2) = max{4, x1, x2 - 5}
    U42_A(x1) = max{1, x1}
    U41_A(x1,x2) = max{x1 + 30, x2 + 18}
    U31_A(x1) = max{19, x1 + 15}
    U21_A(x1) = max{1, x1}
    a__zeros_A = 9
    |0|_A = 6
    zeros_A = 0
    a__U11_A(x1) = 11
    a__U21_A(x1) = max{10, x1}
    a__U31_A(x1) = max{28, x1 + 15}
    a__U41_A(x1,x2) = max{x1 + 30, x2 + 27}
    a__U42_A(x1) = max{9, x1}
    a__U51_A(x1,x2) = max{13, x1, x2 + 4}
    a__U52_A(x1) = max{9, x1}
    a__isNatList_A(x1) = max{12, x1 + 4}
    a__U61_A(x1,x2) = max{x1 + 1, x2 + 42}
    a__U62_A(x1) = max{10, x1 + 1}
    a__U71_A(x1,x2,x3) = 12
    a__U72_A(x1,x2) = x1 + 1
    a__length_A(x1) = 12
    a__U81_A(x1) = max{6, x1 - 6}
    nil_A = 6
    a__U91_A(x1,x2,x3,x4) = max{x1 - 3, x2 + 38, x3 + 38, x4 + 21}
    a__U92_A(x1,x2,x3,x4) = max{x1, x2 + 38, x3 + 38, x4 + 21}
    a__U93_A(x1,x2,x3,x4) = max{x1 + 10, x2 + 38, x3 + 38, x4 + 21}
    a__take_A(x1,x2) = max{x1 + 38, x2 + 38}
    U11_A(x1) = 2
    isNatList_A(x1) = max{5, x1 - 5}
    U71_A(x1,x2,x3) = 4
    U72_A(x1,x2) = x1 + 1
    length_A(x1) = 3
    U81_A(x1) = max{0, x1 - 6}
    isNatIList_A(x1) = max{32, x1 + 18}
    isNat_A(x1) = 2

The next rules are strictly ordered:

  p5, p9, p11, p12, p16, p17

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#(cons(X1,X2)) -> mark#(X1)
p3: mark#(take(X1,X2)) -> a__take#(mark(X1),mark(X2))
p4: a__take#(s(M),cons(N,IL)) -> a__U91#(a__isNatIList(IL),IL,M,N)
p5: a__U92#(tt(),IL,M,N) -> a__U93#(a__isNat(N),IL,M,N)
p6: mark#(U93(X1,X2,X3,X4)) -> a__U93#(mark(X1),X2,X3,X4)
p7: mark#(U92(X1,X2,X3,X4)) -> mark#(X1)
p8: mark#(U91(X1,X2,X3,X4)) -> a__U91#(mark(X1),X2,X3,X4)
p9: mark#(U52(X)) -> mark#(X)
p10: mark#(U51(X1,X2)) -> mark#(X1)
p11: mark#(U42(X)) -> mark#(X)
p12: 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, p6, p7, p9, p10, p11, p12}


-- Reduction pair.

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

p1: a__U93#(tt(),IL,M,N) -> mark#(N)
p2: mark#(U21(X)) -> mark#(X)
p3: mark#(U42(X)) -> mark#(X)
p4: mark#(U51(X1,X2)) -> mark#(X1)
p5: mark#(U52(X)) -> mark#(X)
p6: mark#(U92(X1,X2,X3,X4)) -> mark#(X1)
p7: mark#(U93(X1,X2,X3,X4)) -> a__U93#(mark(X1),X2,X3,X4)
p8: mark#(cons(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

  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:

  max/plus interpretations on natural numbers:
  
    a__U93#_A(x1,x2,x3,x4) = max{x1 + 2, x2 + 15, x4 + 9}
    tt_A = 4
    mark#_A(x1) = x1 + 9
    U21_A(x1) = max{1, x1}
    U42_A(x1) = max{1, x1}
    U51_A(x1,x2) = max{18, x1 + 15, x2 - 4}
    U52_A(x1) = max{1, x1}
    U92_A(x1,x2,x3,x4) = max{x1, x2 + 7, x4 + 7}
    U93_A(x1,x2,x3,x4) = max{x1 + 18, x2 + 7, x4 + 7}
    mark_A(x1) = x1 + 24
    cons_A(x1,x2) = max{x1 + 7, x2 + 7}
    a__zeros_A = 24
    |0|_A = 17
    zeros_A = 0
    a__U11_A(x1) = 4
    a__U21_A(x1) = max{3, x1}
    a__U31_A(x1) = max{22, x1 + 1}
    a__U41_A(x1,x2) = max{x1 + 24, x2 + 24}
    a__U42_A(x1) = max{24, x1}
    a__isNatIList_A(x1) = max{22, x1 + 21}
    a__U51_A(x1,x2) = max{x1 + 15, x2 + 20}
    a__U52_A(x1) = max{3, x1}
    a__isNatList_A(x1) = x1 + 20
    a__U61_A(x1,x2) = max{x1 - 2, x2 + 26}
    a__U62_A(x1) = x1 + 4
    a__U71_A(x1,x2,x3) = 28
    a__U72_A(x1,x2) = x1 + 24
    a__isNat_A(x1) = 4
    s_A(x1) = 16
    a__length_A(x1) = 28
    a__U81_A(x1) = max{6, x1 + 1}
    nil_A = 6
    a__U91_A(x1,x2,x3,x4) = max{x1 + 3, x2 + 31, x4 + 31}
    a__U92_A(x1,x2,x3,x4) = max{x1, x2 + 31, x4 + 31}
    a__U93_A(x1,x2,x3,x4) = max{x1 + 18, x2 + 31, x4 + 31}
    take_A(x1,x2) = x2 + 24
    length_A(x1) = 4
    a__take_A(x1,x2) = x2 + 24
    U11_A(x1) = 0
    U31_A(x1) = max{2, x1 + 1}
    U41_A(x1,x2) = max{x1 + 24, x2}
    isNatIList_A(x1) = max{22, x1 + 21}
    isNatList_A(x1) = max{20, x1 - 4}
    U61_A(x1,x2) = max{23, x1 - 2, x2 + 2}
    U62_A(x1) = x1 + 4
    U71_A(x1,x2,x3) = 10
    U72_A(x1,x2) = x1 + 24
    isNat_A(x1) = 4
    U81_A(x1) = max{2, x1 + 1}
    U91_A(x1,x2,x3,x4) = max{x1 + 3, x2 + 7, x4 + 7}

The next rules are strictly ordered:

  p4, p7, p8

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#(U21(X)) -> mark#(X)
p3: mark#(U42(X)) -> mark#(X)
p4: mark#(U52(X)) -> mark#(X)
p5: mark#(U92(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:

  {p2, p3, p4, p5}


-- Reduction pair.

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

p1: mark#(U21(X)) -> mark#(X)
p2: mark#(U92(X1,X2,X3,X4)) -> mark#(X1)
p3: mark#(U52(X)) -> mark#(X)
p4: 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:

  max/plus interpretations on natural numbers:
  
    mark#_A(x1) = x1 + 1
    U21_A(x1) = x1
    U92_A(x1,x2,x3,x4) = max{x1, x2, x3, x4}
    U52_A(x1) = x1 + 1
    U42_A(x1) = x1

The next rules are strictly ordered:

  p3

We remove them from the problem.

-- SCC decomposition.

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

p1: mark#(U21(X)) -> mark#(X)
p2: mark#(U92(X1,X2,X3,X4)) -> mark#(X1)
p3: 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}


-- 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#(U92(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 set of usable rules consists of

  (no rules)

Take the reduction pair:

  max/plus interpretations on natural numbers:
  
    mark#_A(x1) = x1 + 1
    U21_A(x1) = x1
    U42_A(x1) = x1 + 1
    U92_A(x1,x2,x3,x4) = max{x1, x2, x3, x4}

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#(U92(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}


-- Reduction pair.

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

p1: mark#(U21(X)) -> mark#(X)
p2: mark#(U92(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 set of usable rules consists of

  (no rules)

Take the reduction pair:

  max/plus interpretations on natural numbers:
  
    mark#_A(x1) = x1
    U21_A(x1) = x1 + 1
    U92_A(x1,x2,x3,x4) = max{x1, x3, x4}

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#(U92(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}


-- Reduction pair.

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

p1: mark#(U92(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 set of usable rules consists of

  (no rules)

Take the reduction pair:

  max/plus interpretations on natural numbers:
  
    mark#_A(x1) = x1
    U92_A(x1,x2,x3,x4) = max{x1 + 1, x2 + 1, x3 + 1, x4 + 1}

The next rules are strictly ordered:

  p1

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

-- Reduction pair.

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

p1: a__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:

  max/plus interpretations on natural numbers:
  
    a__U71#_A(x1,x2,x3) = max{x1, x2 + 8}
    tt_A = 26
    a__U72#_A(x1,x2) = max{8, x1 - 24, x2 + 7}
    a__isNat_A(x1) = 26
    a__length#_A(x1) = x1 + 1
    mark_A(x1) = max{7, x1}
    cons_A(x1,x2) = x2 + 7
    a__isNatList_A(x1) = x1 + 6
    a__zeros_A = 7
    |0|_A = 18
    zeros_A = 0
    a__U11_A(x1) = 26
    a__U21_A(x1) = max{25, x1}
    a__U31_A(x1) = 26
    a__U41_A(x1,x2) = max{16, x1}
    a__U42_A(x1) = max{26, x1 - 1}
    a__isNatIList_A(x1) = 26
    a__U51_A(x1,x2) = max{x1 - 30, x2 + 10}
    a__U52_A(x1) = max{10, x1}
    a__U61_A(x1,x2) = max{8, x1}
    a__U62_A(x1) = max{10, x1}
    a__U71_A(x1,x2,x3) = max{x1, x2 + 6}
    a__U72_A(x1,x2) = max{x1, x2 + 6}
    s_A(x1) = max{26, x1 + 7}
    a__length_A(x1) = max{0, x1 - 1}
    a__U81_A(x1) = 39
    nil_A = 39
    a__U91_A(x1,x2,x3,x4) = max{46, x1 - 26, x3 + 27}
    a__U92_A(x1,x2,x3,x4) = max{x1 + 20, x3 + 27}
    a__U93_A(x1,x2,x3,x4) = max{x1 + 20, x3 + 27}
    take_A(x1,x2) = max{39, x1 + 20}
    a__take_A(x1,x2) = max{39, x1 + 20}
    U11_A(x1) = 26
    U21_A(x1) = max{25, x1}
    U31_A(x1) = 26
    U41_A(x1,x2) = max{16, x1}
    U42_A(x1) = max{26, x1 - 1}
    isNatIList_A(x1) = 26
    U51_A(x1,x2) = max{x1 - 30, x2 + 10}
    U52_A(x1) = max{10, x1}
    U61_A(x1,x2) = max{8, x1}
    U62_A(x1) = max{10, x1}
    U71_A(x1,x2,x3) = max{x1, x2 + 6}
    U72_A(x1,x2) = max{x1, x2 + 6}
    length_A(x1) = max{0, x1 - 1}
    U81_A(x1) = 39
    U91_A(x1,x2,x3,x4) = max{46, x1 - 26, x3 + 27}
    U92_A(x1,x2,x3,x4) = max{x1 + 20, x3 + 27}
    U93_A(x1,x2,x3,x4) = max{x1 + 20, x3 + 27}
    isNatList_A(x1) = x1 + 6
    isNat_A(x1) = 26

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__U72#(tt(),L) -> a__length#(mark(L))
p2: 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 estimated dependency graph contains the following SCCs:

  (no SCCs)

-- Reduction pair.

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

p1: a__U41#(tt(),V2) -> a__isNatIList#(V2)
p2: a__isNatIList#(cons(V1,V2)) -> a__isNat#(V1)
p3: a__isNat#(s(V1)) -> a__isNat#(V1)
p4: a__isNat#(length(V1)) -> a__isNatList#(V1)
p5: a__isNatList#(take(V1,V2)) -> a__isNat#(V1)
p6: a__isNatList#(take(V1,V2)) -> a__U61#(a__isNat(V1),V2)
p7: a__U61#(tt(),V2) -> a__isNatIList#(V2)
p8: a__isNatIList#(cons(V1,V2)) -> a__U41#(a__isNat(V1),V2)
p9: a__isNatIList#(V) -> a__isNatList#(V)
p10: a__isNatList#(cons(V1,V2)) -> a__isNat#(V1)
p11: a__isNatList#(cons(V1,V2)) -> a__U51#(a__isNat(V1),V2)
p12: a__U51#(tt(),V2) -> a__isNatList#(V2)

and R consists of:

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

The set of usable rules consists of

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

Take the reduction pair:

  max/plus interpretations on natural numbers:
  
    a__U41#_A(x1,x2) = x2 + 11
    tt_A = 6
    a__isNatIList#_A(x1) = max{9, x1}
    cons_A(x1,x2) = max{x1 + 11, x2 + 11}
    a__isNat#_A(x1) = x1 + 11
    s_A(x1) = x1 + 2
    length_A(x1) = x1 + 2
    a__isNatList#_A(x1) = max{9, x1}
    take_A(x1,x2) = max{x1 + 12, x2 + 14}
    a__U61#_A(x1,x2) = max{x1 + 7, x2 + 10}
    a__isNat_A(x1) = max{6, x1 + 5}
    a__U51#_A(x1,x2) = x2 + 9
    a__U42_A(x1) = 16
    U42_A(x1) = 14
    a__U31_A(x1) = max{6, x1 - 3}
    a__U41_A(x1,x2) = max{16, x1 + 9}
    a__isNatIList_A(x1) = max{6, x1 + 5}
    U31_A(x1) = max{0, x1 - 3}
    U41_A(x1,x2) = max{16, x1 + 9}
    a__U52_A(x1) = 14
    a__U62_A(x1) = 17
    a__isNatList_A(x1) = max{8, x1 + 5}
    zeros_A = 0
    isNatIList_A(x1) = max{6, x1 + 5}
    U52_A(x1) = 14
    U62_A(x1) = 17
    a__U51_A(x1,x2) = x1 + 10
    a__U61_A(x1,x2) = max{16, x1 + 11}
    U51_A(x1,x2) = x1 + 10
    U61_A(x1,x2) = max{16, x1 + 11}
    a__U11_A(x1) = max{7, x1 - 1}
    a__U21_A(x1) = max{7, x1 + 1}
    nil_A = 0
    U11_A(x1) = max{7, x1 - 1}
    U21_A(x1) = max{7, x1 + 1}
    isNatList_A(x1) = max{8, x1 + 5}
    |0|_A = 0
    isNat_A(x1) = max{6, x1 + 5}

The next rules are strictly ordered:

  p1, p3, p4, p5, p7, p11

We remove them from the problem.

-- SCC decomposition.

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

p1: a__isNatIList#(cons(V1,V2)) -> a__isNat#(V1)
p2: a__isNatList#(take(V1,V2)) -> a__U61#(a__isNat(V1),V2)
p3: a__isNatIList#(cons(V1,V2)) -> a__U41#(a__isNat(V1),V2)
p4: a__isNatIList#(V) -> a__isNatList#(V)
p5: a__isNatList#(cons(V1,V2)) -> a__isNat#(V1)
p6: 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:

  (no SCCs)