YES

We show the termination of the TRS R:

  a__zeros() -> cons(|0|(),zeros())
  a__U11(tt(),V1) -> a__U12(a__isNatList(V1))
  a__U12(tt()) -> tt()
  a__U21(tt(),V1) -> a__U22(a__isNat(V1))
  a__U22(tt()) -> tt()
  a__U31(tt(),V) -> a__U32(a__isNatList(V))
  a__U32(tt()) -> tt()
  a__U41(tt(),V1,V2) -> a__U42(a__isNat(V1),V2)
  a__U42(tt(),V2) -> a__U43(a__isNatIList(V2))
  a__U43(tt()) -> tt()
  a__U51(tt(),V1,V2) -> a__U52(a__isNat(V1),V2)
  a__U52(tt(),V2) -> a__U53(a__isNatList(V2))
  a__U53(tt()) -> tt()
  a__U61(tt(),L) -> s(a__length(mark(L)))
  a__and(tt(),X) -> mark(X)
  a__isNat(|0|()) -> tt()
  a__isNat(length(V1)) -> a__U11(a__isNatIListKind(V1),V1)
  a__isNat(s(V1)) -> a__U21(a__isNatKind(V1),V1)
  a__isNatIList(V) -> a__U31(a__isNatIListKind(V),V)
  a__isNatIList(zeros()) -> tt()
  a__isNatIList(cons(V1,V2)) -> a__U41(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2)
  a__isNatIListKind(nil()) -> tt()
  a__isNatIListKind(zeros()) -> tt()
  a__isNatIListKind(cons(V1,V2)) -> a__and(a__isNatKind(V1),isNatIListKind(V2))
  a__isNatKind(|0|()) -> tt()
  a__isNatKind(length(V1)) -> a__isNatIListKind(V1)
  a__isNatKind(s(V1)) -> a__isNatKind(V1)
  a__isNatList(nil()) -> tt()
  a__isNatList(cons(V1,V2)) -> a__U51(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2)
  a__length(nil()) -> |0|()
  a__length(cons(N,L)) -> a__U61(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L)
  mark(zeros()) -> a__zeros()
  mark(U11(X1,X2)) -> a__U11(mark(X1),X2)
  mark(U12(X)) -> a__U12(mark(X))
  mark(isNatList(X)) -> a__isNatList(X)
  mark(U21(X1,X2)) -> a__U21(mark(X1),X2)
  mark(U22(X)) -> a__U22(mark(X))
  mark(isNat(X)) -> a__isNat(X)
  mark(U31(X1,X2)) -> a__U31(mark(X1),X2)
  mark(U32(X)) -> a__U32(mark(X))
  mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3)
  mark(U42(X1,X2)) -> a__U42(mark(X1),X2)
  mark(U43(X)) -> a__U43(mark(X))
  mark(isNatIList(X)) -> a__isNatIList(X)
  mark(U51(X1,X2,X3)) -> a__U51(mark(X1),X2,X3)
  mark(U52(X1,X2)) -> a__U52(mark(X1),X2)
  mark(U53(X)) -> a__U53(mark(X))
  mark(U61(X1,X2)) -> a__U61(mark(X1),X2)
  mark(length(X)) -> a__length(mark(X))
  mark(and(X1,X2)) -> a__and(mark(X1),X2)
  mark(isNatIListKind(X)) -> a__isNatIListKind(X)
  mark(isNatKind(X)) -> a__isNatKind(X)
  mark(cons(X1,X2)) -> cons(mark(X1),X2)
  mark(|0|()) -> |0|()
  mark(tt()) -> tt()
  mark(s(X)) -> s(mark(X))
  mark(nil()) -> nil()
  a__zeros() -> zeros()
  a__U11(X1,X2) -> U11(X1,X2)
  a__U12(X) -> U12(X)
  a__isNatList(X) -> isNatList(X)
  a__U21(X1,X2) -> U21(X1,X2)
  a__U22(X) -> U22(X)
  a__isNat(X) -> isNat(X)
  a__U31(X1,X2) -> U31(X1,X2)
  a__U32(X) -> U32(X)
  a__U41(X1,X2,X3) -> U41(X1,X2,X3)
  a__U42(X1,X2) -> U42(X1,X2)
  a__U43(X) -> U43(X)
  a__isNatIList(X) -> isNatIList(X)
  a__U51(X1,X2,X3) -> U51(X1,X2,X3)
  a__U52(X1,X2) -> U52(X1,X2)
  a__U53(X) -> U53(X)
  a__U61(X1,X2) -> U61(X1,X2)
  a__length(X) -> length(X)
  a__and(X1,X2) -> and(X1,X2)
  a__isNatIListKind(X) -> isNatIListKind(X)
  a__isNatKind(X) -> isNatKind(X)

-- SCC decomposition.

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

p1: a__U11#(tt(),V1) -> a__U12#(a__isNatList(V1))
p2: a__U11#(tt(),V1) -> a__isNatList#(V1)
p3: a__U21#(tt(),V1) -> a__U22#(a__isNat(V1))
p4: a__U21#(tt(),V1) -> a__isNat#(V1)
p5: a__U31#(tt(),V) -> a__U32#(a__isNatList(V))
p6: a__U31#(tt(),V) -> a__isNatList#(V)
p7: a__U41#(tt(),V1,V2) -> a__U42#(a__isNat(V1),V2)
p8: a__U41#(tt(),V1,V2) -> a__isNat#(V1)
p9: a__U42#(tt(),V2) -> a__U43#(a__isNatIList(V2))
p10: a__U42#(tt(),V2) -> a__isNatIList#(V2)
p11: a__U51#(tt(),V1,V2) -> a__U52#(a__isNat(V1),V2)
p12: a__U51#(tt(),V1,V2) -> a__isNat#(V1)
p13: a__U52#(tt(),V2) -> a__U53#(a__isNatList(V2))
p14: a__U52#(tt(),V2) -> a__isNatList#(V2)
p15: a__U61#(tt(),L) -> a__length#(mark(L))
p16: a__U61#(tt(),L) -> mark#(L)
p17: a__and#(tt(),X) -> mark#(X)
p18: a__isNat#(length(V1)) -> a__U11#(a__isNatIListKind(V1),V1)
p19: a__isNat#(length(V1)) -> a__isNatIListKind#(V1)
p20: a__isNat#(s(V1)) -> a__U21#(a__isNatKind(V1),V1)
p21: a__isNat#(s(V1)) -> a__isNatKind#(V1)
p22: a__isNatIList#(V) -> a__U31#(a__isNatIListKind(V),V)
p23: a__isNatIList#(V) -> a__isNatIListKind#(V)
p24: a__isNatIList#(cons(V1,V2)) -> a__U41#(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2)
p25: a__isNatIList#(cons(V1,V2)) -> a__and#(a__isNatKind(V1),isNatIListKind(V2))
p26: a__isNatIList#(cons(V1,V2)) -> a__isNatKind#(V1)
p27: a__isNatIListKind#(cons(V1,V2)) -> a__and#(a__isNatKind(V1),isNatIListKind(V2))
p28: a__isNatIListKind#(cons(V1,V2)) -> a__isNatKind#(V1)
p29: a__isNatKind#(length(V1)) -> a__isNatIListKind#(V1)
p30: a__isNatKind#(s(V1)) -> a__isNatKind#(V1)
p31: a__isNatList#(cons(V1,V2)) -> a__U51#(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2)
p32: a__isNatList#(cons(V1,V2)) -> a__and#(a__isNatKind(V1),isNatIListKind(V2))
p33: a__isNatList#(cons(V1,V2)) -> a__isNatKind#(V1)
p34: a__length#(cons(N,L)) -> a__U61#(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L)
p35: a__length#(cons(N,L)) -> a__and#(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N)))
p36: a__length#(cons(N,L)) -> a__and#(a__isNatList(L),isNatIListKind(L))
p37: a__length#(cons(N,L)) -> a__isNatList#(L)
p38: mark#(zeros()) -> a__zeros#()
p39: mark#(U11(X1,X2)) -> a__U11#(mark(X1),X2)
p40: mark#(U11(X1,X2)) -> mark#(X1)
p41: mark#(U12(X)) -> a__U12#(mark(X))
p42: mark#(U12(X)) -> mark#(X)
p43: mark#(isNatList(X)) -> a__isNatList#(X)
p44: mark#(U21(X1,X2)) -> a__U21#(mark(X1),X2)
p45: mark#(U21(X1,X2)) -> mark#(X1)
p46: mark#(U22(X)) -> a__U22#(mark(X))
p47: mark#(U22(X)) -> mark#(X)
p48: mark#(isNat(X)) -> a__isNat#(X)
p49: mark#(U31(X1,X2)) -> a__U31#(mark(X1),X2)
p50: mark#(U31(X1,X2)) -> mark#(X1)
p51: mark#(U32(X)) -> a__U32#(mark(X))
p52: mark#(U32(X)) -> mark#(X)
p53: mark#(U41(X1,X2,X3)) -> a__U41#(mark(X1),X2,X3)
p54: mark#(U41(X1,X2,X3)) -> mark#(X1)
p55: mark#(U42(X1,X2)) -> a__U42#(mark(X1),X2)
p56: mark#(U42(X1,X2)) -> mark#(X1)
p57: mark#(U43(X)) -> a__U43#(mark(X))
p58: mark#(U43(X)) -> mark#(X)
p59: mark#(isNatIList(X)) -> a__isNatIList#(X)
p60: mark#(U51(X1,X2,X3)) -> a__U51#(mark(X1),X2,X3)
p61: mark#(U51(X1,X2,X3)) -> mark#(X1)
p62: mark#(U52(X1,X2)) -> a__U52#(mark(X1),X2)
p63: mark#(U52(X1,X2)) -> mark#(X1)
p64: mark#(U53(X)) -> a__U53#(mark(X))
p65: mark#(U53(X)) -> mark#(X)
p66: mark#(U61(X1,X2)) -> a__U61#(mark(X1),X2)
p67: mark#(U61(X1,X2)) -> mark#(X1)
p68: mark#(length(X)) -> a__length#(mark(X))
p69: mark#(length(X)) -> mark#(X)
p70: mark#(and(X1,X2)) -> a__and#(mark(X1),X2)
p71: mark#(and(X1,X2)) -> mark#(X1)
p72: mark#(isNatIListKind(X)) -> a__isNatIListKind#(X)
p73: mark#(isNatKind(X)) -> a__isNatKind#(X)
p74: mark#(cons(X1,X2)) -> mark#(X1)
p75: mark#(s(X)) -> mark#(X)

and R consists of:

r1: a__zeros() -> cons(|0|(),zeros())
r2: a__U11(tt(),V1) -> a__U12(a__isNatList(V1))
r3: a__U12(tt()) -> tt()
r4: a__U21(tt(),V1) -> a__U22(a__isNat(V1))
r5: a__U22(tt()) -> tt()
r6: a__U31(tt(),V) -> a__U32(a__isNatList(V))
r7: a__U32(tt()) -> tt()
r8: a__U41(tt(),V1,V2) -> a__U42(a__isNat(V1),V2)
r9: a__U42(tt(),V2) -> a__U43(a__isNatIList(V2))
r10: a__U43(tt()) -> tt()
r11: a__U51(tt(),V1,V2) -> a__U52(a__isNat(V1),V2)
r12: a__U52(tt(),V2) -> a__U53(a__isNatList(V2))
r13: a__U53(tt()) -> tt()
r14: a__U61(tt(),L) -> s(a__length(mark(L)))
r15: a__and(tt(),X) -> mark(X)
r16: a__isNat(|0|()) -> tt()
r17: a__isNat(length(V1)) -> a__U11(a__isNatIListKind(V1),V1)
r18: a__isNat(s(V1)) -> a__U21(a__isNatKind(V1),V1)
r19: a__isNatIList(V) -> a__U31(a__isNatIListKind(V),V)
r20: a__isNatIList(zeros()) -> tt()
r21: a__isNatIList(cons(V1,V2)) -> a__U41(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2)
r22: a__isNatIListKind(nil()) -> tt()
r23: a__isNatIListKind(zeros()) -> tt()
r24: a__isNatIListKind(cons(V1,V2)) -> a__and(a__isNatKind(V1),isNatIListKind(V2))
r25: a__isNatKind(|0|()) -> tt()
r26: a__isNatKind(length(V1)) -> a__isNatIListKind(V1)
r27: a__isNatKind(s(V1)) -> a__isNatKind(V1)
r28: a__isNatList(nil()) -> tt()
r29: a__isNatList(cons(V1,V2)) -> a__U51(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2)
r30: a__length(nil()) -> |0|()
r31: a__length(cons(N,L)) -> a__U61(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L)
r32: mark(zeros()) -> a__zeros()
r33: mark(U11(X1,X2)) -> a__U11(mark(X1),X2)
r34: mark(U12(X)) -> a__U12(mark(X))
r35: mark(isNatList(X)) -> a__isNatList(X)
r36: mark(U21(X1,X2)) -> a__U21(mark(X1),X2)
r37: mark(U22(X)) -> a__U22(mark(X))
r38: mark(isNat(X)) -> a__isNat(X)
r39: mark(U31(X1,X2)) -> a__U31(mark(X1),X2)
r40: mark(U32(X)) -> a__U32(mark(X))
r41: mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3)
r42: mark(U42(X1,X2)) -> a__U42(mark(X1),X2)
r43: mark(U43(X)) -> a__U43(mark(X))
r44: mark(isNatIList(X)) -> a__isNatIList(X)
r45: mark(U51(X1,X2,X3)) -> a__U51(mark(X1),X2,X3)
r46: mark(U52(X1,X2)) -> a__U52(mark(X1),X2)
r47: mark(U53(X)) -> a__U53(mark(X))
r48: mark(U61(X1,X2)) -> a__U61(mark(X1),X2)
r49: mark(length(X)) -> a__length(mark(X))
r50: mark(and(X1,X2)) -> a__and(mark(X1),X2)
r51: mark(isNatIListKind(X)) -> a__isNatIListKind(X)
r52: mark(isNatKind(X)) -> a__isNatKind(X)
r53: mark(cons(X1,X2)) -> cons(mark(X1),X2)
r54: mark(|0|()) -> |0|()
r55: mark(tt()) -> tt()
r56: mark(s(X)) -> s(mark(X))
r57: mark(nil()) -> nil()
r58: a__zeros() -> zeros()
r59: a__U11(X1,X2) -> U11(X1,X2)
r60: a__U12(X) -> U12(X)
r61: a__isNatList(X) -> isNatList(X)
r62: a__U21(X1,X2) -> U21(X1,X2)
r63: a__U22(X) -> U22(X)
r64: a__isNat(X) -> isNat(X)
r65: a__U31(X1,X2) -> U31(X1,X2)
r66: a__U32(X) -> U32(X)
r67: a__U41(X1,X2,X3) -> U41(X1,X2,X3)
r68: a__U42(X1,X2) -> U42(X1,X2)
r69: a__U43(X) -> U43(X)
r70: a__isNatIList(X) -> isNatIList(X)
r71: a__U51(X1,X2,X3) -> U51(X1,X2,X3)
r72: a__U52(X1,X2) -> U52(X1,X2)
r73: a__U53(X) -> U53(X)
r74: a__U61(X1,X2) -> U61(X1,X2)
r75: a__length(X) -> length(X)
r76: a__and(X1,X2) -> and(X1,X2)
r77: a__isNatIListKind(X) -> isNatIListKind(X)
r78: a__isNatKind(X) -> isNatKind(X)

The estimated dependency graph contains the following SCCs:

  {p2, p4, p6, p7, p8, p10, p11, p12, p14, p15, p16, p17, p18, p19, p20, p21, p22, p23, p24, p25, p26, p27, p28, p29, p30, p31, p32, p33, p34, p35, p36, p37, p39, p40, p42, p43, p44, p45, p47, p48, p49, p50, p52, p53, p54, p55, p56, p58, p59, p60, p61, p62, p63, p65, p66, p67, p68, p69, p70, p71, p72, p73, p74, p75}


-- Reduction pair.

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

p1: a__U11#(tt(),V1) -> a__isNatList#(V1)
p2: a__isNatList#(cons(V1,V2)) -> a__isNatKind#(V1)
p3: a__isNatKind#(s(V1)) -> a__isNatKind#(V1)
p4: a__isNatKind#(length(V1)) -> a__isNatIListKind#(V1)
p5: a__isNatIListKind#(cons(V1,V2)) -> a__isNatKind#(V1)
p6: a__isNatIListKind#(cons(V1,V2)) -> a__and#(a__isNatKind(V1),isNatIListKind(V2))
p7: a__and#(tt(),X) -> mark#(X)
p8: mark#(s(X)) -> mark#(X)
p9: mark#(cons(X1,X2)) -> mark#(X1)
p10: mark#(isNatKind(X)) -> a__isNatKind#(X)
p11: mark#(isNatIListKind(X)) -> a__isNatIListKind#(X)
p12: mark#(and(X1,X2)) -> mark#(X1)
p13: mark#(and(X1,X2)) -> a__and#(mark(X1),X2)
p14: mark#(length(X)) -> mark#(X)
p15: mark#(length(X)) -> a__length#(mark(X))
p16: a__length#(cons(N,L)) -> a__isNatList#(L)
p17: a__isNatList#(cons(V1,V2)) -> a__and#(a__isNatKind(V1),isNatIListKind(V2))
p18: a__isNatList#(cons(V1,V2)) -> a__U51#(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2)
p19: a__U51#(tt(),V1,V2) -> a__isNat#(V1)
p20: a__isNat#(s(V1)) -> a__isNatKind#(V1)
p21: a__isNat#(s(V1)) -> a__U21#(a__isNatKind(V1),V1)
p22: a__U21#(tt(),V1) -> a__isNat#(V1)
p23: a__isNat#(length(V1)) -> a__isNatIListKind#(V1)
p24: a__isNat#(length(V1)) -> a__U11#(a__isNatIListKind(V1),V1)
p25: a__U51#(tt(),V1,V2) -> a__U52#(a__isNat(V1),V2)
p26: a__U52#(tt(),V2) -> a__isNatList#(V2)
p27: a__length#(cons(N,L)) -> a__and#(a__isNatList(L),isNatIListKind(L))
p28: a__length#(cons(N,L)) -> a__and#(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N)))
p29: a__length#(cons(N,L)) -> a__U61#(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L)
p30: a__U61#(tt(),L) -> mark#(L)
p31: mark#(U61(X1,X2)) -> mark#(X1)
p32: mark#(U61(X1,X2)) -> a__U61#(mark(X1),X2)
p33: a__U61#(tt(),L) -> a__length#(mark(L))
p34: mark#(U53(X)) -> mark#(X)
p35: mark#(U52(X1,X2)) -> mark#(X1)
p36: mark#(U52(X1,X2)) -> a__U52#(mark(X1),X2)
p37: mark#(U51(X1,X2,X3)) -> mark#(X1)
p38: mark#(U51(X1,X2,X3)) -> a__U51#(mark(X1),X2,X3)
p39: mark#(isNatIList(X)) -> a__isNatIList#(X)
p40: a__isNatIList#(cons(V1,V2)) -> a__isNatKind#(V1)
p41: a__isNatIList#(cons(V1,V2)) -> a__and#(a__isNatKind(V1),isNatIListKind(V2))
p42: a__isNatIList#(cons(V1,V2)) -> a__U41#(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2)
p43: a__U41#(tt(),V1,V2) -> a__isNat#(V1)
p44: a__U41#(tt(),V1,V2) -> a__U42#(a__isNat(V1),V2)
p45: a__U42#(tt(),V2) -> a__isNatIList#(V2)
p46: a__isNatIList#(V) -> a__isNatIListKind#(V)
p47: a__isNatIList#(V) -> a__U31#(a__isNatIListKind(V),V)
p48: a__U31#(tt(),V) -> a__isNatList#(V)
p49: mark#(U43(X)) -> mark#(X)
p50: mark#(U42(X1,X2)) -> mark#(X1)
p51: mark#(U42(X1,X2)) -> a__U42#(mark(X1),X2)
p52: mark#(U41(X1,X2,X3)) -> mark#(X1)
p53: mark#(U41(X1,X2,X3)) -> a__U41#(mark(X1),X2,X3)
p54: mark#(U32(X)) -> mark#(X)
p55: mark#(U31(X1,X2)) -> mark#(X1)
p56: mark#(U31(X1,X2)) -> a__U31#(mark(X1),X2)
p57: mark#(isNat(X)) -> a__isNat#(X)
p58: mark#(U22(X)) -> mark#(X)
p59: mark#(U21(X1,X2)) -> mark#(X1)
p60: mark#(U21(X1,X2)) -> a__U21#(mark(X1),X2)
p61: mark#(isNatList(X)) -> a__isNatList#(X)
p62: mark#(U12(X)) -> mark#(X)
p63: mark#(U11(X1,X2)) -> mark#(X1)
p64: mark#(U11(X1,X2)) -> a__U11#(mark(X1),X2)

and R consists of:

r1: a__zeros() -> cons(|0|(),zeros())
r2: a__U11(tt(),V1) -> a__U12(a__isNatList(V1))
r3: a__U12(tt()) -> tt()
r4: a__U21(tt(),V1) -> a__U22(a__isNat(V1))
r5: a__U22(tt()) -> tt()
r6: a__U31(tt(),V) -> a__U32(a__isNatList(V))
r7: a__U32(tt()) -> tt()
r8: a__U41(tt(),V1,V2) -> a__U42(a__isNat(V1),V2)
r9: a__U42(tt(),V2) -> a__U43(a__isNatIList(V2))
r10: a__U43(tt()) -> tt()
r11: a__U51(tt(),V1,V2) -> a__U52(a__isNat(V1),V2)
r12: a__U52(tt(),V2) -> a__U53(a__isNatList(V2))
r13: a__U53(tt()) -> tt()
r14: a__U61(tt(),L) -> s(a__length(mark(L)))
r15: a__and(tt(),X) -> mark(X)
r16: a__isNat(|0|()) -> tt()
r17: a__isNat(length(V1)) -> a__U11(a__isNatIListKind(V1),V1)
r18: a__isNat(s(V1)) -> a__U21(a__isNatKind(V1),V1)
r19: a__isNatIList(V) -> a__U31(a__isNatIListKind(V),V)
r20: a__isNatIList(zeros()) -> tt()
r21: a__isNatIList(cons(V1,V2)) -> a__U41(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2)
r22: a__isNatIListKind(nil()) -> tt()
r23: a__isNatIListKind(zeros()) -> tt()
r24: a__isNatIListKind(cons(V1,V2)) -> a__and(a__isNatKind(V1),isNatIListKind(V2))
r25: a__isNatKind(|0|()) -> tt()
r26: a__isNatKind(length(V1)) -> a__isNatIListKind(V1)
r27: a__isNatKind(s(V1)) -> a__isNatKind(V1)
r28: a__isNatList(nil()) -> tt()
r29: a__isNatList(cons(V1,V2)) -> a__U51(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2)
r30: a__length(nil()) -> |0|()
r31: a__length(cons(N,L)) -> a__U61(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L)
r32: mark(zeros()) -> a__zeros()
r33: mark(U11(X1,X2)) -> a__U11(mark(X1),X2)
r34: mark(U12(X)) -> a__U12(mark(X))
r35: mark(isNatList(X)) -> a__isNatList(X)
r36: mark(U21(X1,X2)) -> a__U21(mark(X1),X2)
r37: mark(U22(X)) -> a__U22(mark(X))
r38: mark(isNat(X)) -> a__isNat(X)
r39: mark(U31(X1,X2)) -> a__U31(mark(X1),X2)
r40: mark(U32(X)) -> a__U32(mark(X))
r41: mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3)
r42: mark(U42(X1,X2)) -> a__U42(mark(X1),X2)
r43: mark(U43(X)) -> a__U43(mark(X))
r44: mark(isNatIList(X)) -> a__isNatIList(X)
r45: mark(U51(X1,X2,X3)) -> a__U51(mark(X1),X2,X3)
r46: mark(U52(X1,X2)) -> a__U52(mark(X1),X2)
r47: mark(U53(X)) -> a__U53(mark(X))
r48: mark(U61(X1,X2)) -> a__U61(mark(X1),X2)
r49: mark(length(X)) -> a__length(mark(X))
r50: mark(and(X1,X2)) -> a__and(mark(X1),X2)
r51: mark(isNatIListKind(X)) -> a__isNatIListKind(X)
r52: mark(isNatKind(X)) -> a__isNatKind(X)
r53: mark(cons(X1,X2)) -> cons(mark(X1),X2)
r54: mark(|0|()) -> |0|()
r55: mark(tt()) -> tt()
r56: mark(s(X)) -> s(mark(X))
r57: mark(nil()) -> nil()
r58: a__zeros() -> zeros()
r59: a__U11(X1,X2) -> U11(X1,X2)
r60: a__U12(X) -> U12(X)
r61: a__isNatList(X) -> isNatList(X)
r62: a__U21(X1,X2) -> U21(X1,X2)
r63: a__U22(X) -> U22(X)
r64: a__isNat(X) -> isNat(X)
r65: a__U31(X1,X2) -> U31(X1,X2)
r66: a__U32(X) -> U32(X)
r67: a__U41(X1,X2,X3) -> U41(X1,X2,X3)
r68: a__U42(X1,X2) -> U42(X1,X2)
r69: a__U43(X) -> U43(X)
r70: a__isNatIList(X) -> isNatIList(X)
r71: a__U51(X1,X2,X3) -> U51(X1,X2,X3)
r72: a__U52(X1,X2) -> U52(X1,X2)
r73: a__U53(X) -> U53(X)
r74: a__U61(X1,X2) -> U61(X1,X2)
r75: a__length(X) -> length(X)
r76: a__and(X1,X2) -> and(X1,X2)
r77: a__isNatIListKind(X) -> isNatIListKind(X)
r78: a__isNatKind(X) -> isNatKind(X)

The set of usable rules consists of

  r1, r2, r3, r4, r5, r6, r7, r8, r9, r10, r11, r12, r13, r14, r15, r16, r17, r18, r19, r20, r21, r22, r23, r24, r25, r26, r27, r28, r29, r30, r31, r32, r33, r34, r35, r36, r37, r38, r39, r40, r41, r42, r43, r44, r45, r46, r47, r48, r49, r50, r51, r52, r53, r54, r55, r56, r57, r58, r59, r60, r61, r62, r63, r64, r65, r66, r67, r68, r69, r70, r71, r72, r73, r74, r75, r76, r77, r78

Take the reduction pair:

  lexicographic combination of reduction pairs:
  
    1. weighted path order
    
      base order:
    
        max/plus interpretations on natural numbers:
        
          a__U11#_A(x1,x2) = max{1042, x1 - 36, x2 + 682}
          tt_A = 213
          a__isNatList#_A(x1) = x1 + 681
          cons_A(x1,x2) = max{x1, x2 + 223}
          a__isNatKind#_A(x1) = max{159, x1 - 424}
          s_A(x1) = max{247, x1}
          length_A(x1) = max{1324, x1 + 1077}
          a__isNatIListKind#_A(x1) = x1 + 566
          a__and#_A(x1,x2) = max{x1 - 4, x2 + 461}
          a__isNatKind_A(x1) = max{214, x1 - 883}
          isNatIListKind_A(x1) = x1 + 193
          mark#_A(x1) = x1 + 460
          isNatKind_A(x1) = max{214, x1 - 883}
          and_A(x1,x2) = max{157, x1, x2 + 156}
          mark_A(x1) = max{246, x1}
          a__length#_A(x1) = max{1079, x1 + 832}
          a__U51#_A(x1,x2,x3) = max{x1 - 52, x2 + 677, x3 + 903}
          a__and_A(x1,x2) = max{246, x1, x2 + 156}
          a__isNat#_A(x1) = max{160, x1 - 34}
          a__U21#_A(x1,x2) = max{0, x1 - 1, x2 - 34}
          a__isNatIListKind_A(x1) = max{195, x1 + 193}
          a__U52#_A(x1,x2) = max{x1 + 377, x2 + 901}
          a__isNat_A(x1) = x1 + 300
          a__isNatList_A(x1) = x1 + 679
          isNat_A(x1) = x1 + 300
          a__U61#_A(x1,x2) = max{1079, x1 - 89, x2 + 833}
          U61_A(x1,x2) = max{1324, x1 + 620, x2 + 1078}
          U53_A(x1) = max{589, x1 + 221}
          U52_A(x1,x2) = max{x1 + 377, x2 + 901}
          U51_A(x1,x2,x3) = max{x1 + 461, x2 + 678, x3 + 902}
          isNatIList_A(x1) = max{1095, x1 + 1061}
          a__isNatIList#_A(x1) = x1 + 1033
          a__U41#_A(x1,x2,x3) = max{x1 - 36, x2 + 762, x3 + 1256}
          a__U42#_A(x1,x2) = max{x1 - 1, x2 + 1255}
          a__U31#_A(x1,x2) = max{x1 + 16, x2 + 682}
          U43_A(x1) = max{406, x1 + 159}
          U42_A(x1,x2) = max{x1 + 461, x2 + 1255}
          U41_A(x1,x2,x3) = max{x1 + 884, x2 + 762, x3 + 1257}
          U32_A(x1) = x1
          U31_A(x1,x2) = max{x1 + 16, x2 + 682}
          U22_A(x1) = max{247, x1}
          U21_A(x1,x2) = max{404, x1 + 157, x2 + 300}
          isNatList_A(x1) = x1 + 679
          U12_A(x1) = max{264, x1 + 17}
          U11_A(x1,x2) = max{x1 + 885, x2 + 1132}
          a__zeros_A = 245
          |0|_A = 244
          zeros_A = 21
          a__U11_A(x1,x2) = max{x1 + 885, x2 + 1132}
          a__U12_A(x1) = max{264, x1 + 17}
          a__U21_A(x1,x2) = max{404, x1 + 157, x2 + 300}
          a__U22_A(x1) = max{247, x1}
          a__U31_A(x1,x2) = max{x1 + 16, x2 + 682}
          a__U32_A(x1) = max{246, x1}
          a__U41_A(x1,x2,x3) = max{x1 + 884, x2 + 762, x3 + 1257}
          a__U42_A(x1,x2) = max{x1 + 461, x2 + 1255}
          a__U43_A(x1) = max{406, x1 + 159}
          a__isNatIList_A(x1) = max{1095, x1 + 1061}
          a__U51_A(x1,x2,x3) = max{x1 + 461, x2 + 678, x3 + 902}
          a__U52_A(x1,x2) = max{x1 + 377, x2 + 901}
          a__U53_A(x1) = max{589, x1 + 221}
          a__U61_A(x1,x2) = max{1324, x1 + 620, x2 + 1078}
          a__length_A(x1) = max{1324, x1 + 1077}
          nil_A = 247
    
      precedence:
      
        a__isNatKind# > U12 = a__zeros = zeros = a__U12 > cons = isNatIListKind = and = mark = a__and = a__isNatIListKind = a__isNatList = isNatIList = U41 = a__U41 = a__isNatIList = a__U52 = a__U53 > a__U61 = a__length > a__isNatList# = a__length# = a__U61# > a__U51# > U61 > s = a__isNat# = a__U21# = U53 > U31 = a__U31 > a__isNatKind > a__U43 > U51 = isNatList = a__U51 > isNatKind > a__U11# = length = a__isNatIListKind# = a__and# = mark# = a__U52# = a__isNat = isNat = U11 = a__U11 = a__U21 > a__U31# > |0| = a__U32 > tt = U52 > a__isNatIList# = a__U41# > a__U42# = U43 = U42 = U32 = U22 = U21 = a__U22 = a__U42 = nil
      
      partial status:
    
        pi(a__U11#) = []
        pi(tt) = []
        pi(a__isNatList#) = [1]
        pi(cons) = [2]
        pi(a__isNatKind#) = []
        pi(s) = []
        pi(length) = [1]
        pi(a__isNatIListKind#) = [1]
        pi(a__and#) = []
        pi(a__isNatKind) = []
        pi(isNatIListKind) = []
        pi(mark#) = []
        pi(isNatKind) = []
        pi(and) = [2]
        pi(mark) = [1]
        pi(a__length#) = []
        pi(a__U51#) = [3]
        pi(a__and) = [2]
        pi(a__isNat#) = []
        pi(a__U21#) = []
        pi(a__isNatIListKind) = [1]
        pi(a__U52#) = [2]
        pi(a__isNat) = [1]
        pi(a__isNatList) = [1]
        pi(isNat) = [1]
        pi(a__U61#) = []
        pi(U61) = [2]
        pi(U53) = [1]
        pi(U52) = []
        pi(U51) = [1, 2, 3]
        pi(isNatIList) = []
        pi(a__isNatIList#) = [1]
        pi(a__U41#) = []
        pi(a__U42#) = []
        pi(a__U31#) = []
        pi(U43) = [1]
        pi(U42) = []
        pi(U41) = []
        pi(U32) = []
        pi(U31) = [1]
        pi(U22) = []
        pi(U21) = [1]
        pi(isNatList) = [1]
        pi(U12) = []
        pi(U11) = []
        pi(a__zeros) = []
        pi(|0|) = []
        pi(zeros) = []
        pi(a__U11) = [2]
        pi(a__U12) = []
        pi(a__U21) = []
        pi(a__U22) = []
        pi(a__U31) = [1]
        pi(a__U32) = []
        pi(a__U41) = []
        pi(a__U42) = []
        pi(a__U43) = []
        pi(a__isNatIList) = [1]
        pi(a__U51) = [1, 2, 3]
        pi(a__U52) = []
        pi(a__U53) = []
        pi(a__U61) = []
        pi(a__length) = [1]
        pi(nil) = []
    
    2. weighted path order
    
      base order:
    
        max/plus interpretations on natural numbers:
        
          a__U11#_A(x1,x2) = 112
          tt_A = 984
          a__isNatList#_A(x1) = 977
          cons_A(x1,x2) = x2 + 398
          a__isNatKind#_A(x1) = 991
          s_A(x1) = 676
          length_A(x1) = x1 + 563
          a__isNatIListKind#_A(x1) = 1
          a__and#_A(x1,x2) = 81
          a__isNatKind_A(x1) = 1707
          isNatIListKind_A(x1) = 677
          mark#_A(x1) = 113
          isNatKind_A(x1) = 1708
          and_A(x1,x2) = max{116, x2 - 118}
          mark_A(x1) = max{993, x1 + 674}
          a__length#_A(x1) = 1872
          a__U51#_A(x1,x2,x3) = x3 + 1670
          a__and_A(x1,x2) = x2 + 1195
          a__isNat#_A(x1) = 992
          a__U21#_A(x1,x2) = 992
          a__isNatIListKind_A(x1) = 1872
          a__U52#_A(x1,x2) = 1669
          a__isNat_A(x1) = x1 + 1314
          a__isNatList_A(x1) = x1 + 1591
          isNat_A(x1) = max{642, x1 + 640}
          a__U61#_A(x1,x2) = 1872
          U61_A(x1,x2) = max{424, x2 + 109}
          U53_A(x1) = 1353
          U52_A(x1,x2) = 995
          U51_A(x1,x2,x3) = max{x1 + 994, x2 + 994, x3 + 994}
          isNatIList_A(x1) = 1
          a__isNatIList#_A(x1) = 82
          a__U41#_A(x1,x2,x3) = 81
          a__U42#_A(x1,x2) = 113
          a__U31#_A(x1,x2) = 995
          U43_A(x1) = max{679, x1 + 1}
          U42_A(x1,x2) = 1665
          U41_A(x1,x2,x3) = 0
          U32_A(x1) = 112
          U31_A(x1,x2) = max{990, x1 - 1196}
          U22_A(x1) = 112
          U21_A(x1,x2) = max{987, x1 - 675}
          isNatList_A(x1) = x1 + 1591
          U12_A(x1) = 984
          U11_A(x1,x2) = 988
          a__zeros_A = 114
          |0|_A = 0
          zeros_A = 113
          a__U11_A(x1,x2) = x2 + 989
          a__U12_A(x1) = 985
          a__U21_A(x1,x2) = 986
          a__U22_A(x1) = 983
          a__U31_A(x1,x2) = max{1665, x1 - 1196}
          a__U32_A(x1) = 983
          a__U41_A(x1,x2,x3) = 681
          a__U42_A(x1,x2) = 1665
          a__U43_A(x1) = 678
          a__isNatIList_A(x1) = max{676, x1 + 283}
          a__U51_A(x1,x2,x3) = max{x1 + 995, x2 + 1670, x3 + 2344}
          a__U52_A(x1,x2) = 1669
          a__U53_A(x1) = 2344
          a__U61_A(x1,x2) = 782
          a__length_A(x1) = 1081
          nil_A = 994
    
      precedence:
      
        mark > zeros = a__U12 = a__isNatIList > a__U31 > a__isNatKind = isNatKind = and = a__and = a__isNatIListKind > a__U22 > a__U61 > s > U51 = a__zeros = a__U51 = a__U52 > cons = a__U41 > U41 > a__isNat = U52 = U12 = |0| = a__U11 = a__U21 > U32 = a__U32 > isNatIList > a__U42 > a__U43 > a__U53 = nil > U43 > isNat = U11 > length = isNatIListKind = U53 > mark# = a__length# = a__U61# > a__U42# > a__isNatIList# > a__U52# = a__U31# = U22 > a__isNatList# = a__U51# = a__isNatList = a__U41# = isNatList > a__isNat# = a__U21# > a__isNatKind# > a__isNatIListKind# > tt > a__U11# > a__and# = U61 > U42 = U31 = U21 = a__length
      
      partial status:
    
        pi(a__U11#) = []
        pi(tt) = []
        pi(a__isNatList#) = []
        pi(cons) = []
        pi(a__isNatKind#) = []
        pi(s) = []
        pi(length) = []
        pi(a__isNatIListKind#) = []
        pi(a__and#) = []
        pi(a__isNatKind) = []
        pi(isNatIListKind) = []
        pi(mark#) = []
        pi(isNatKind) = []
        pi(and) = []
        pi(mark) = []
        pi(a__length#) = []
        pi(a__U51#) = []
        pi(a__and) = []
        pi(a__isNat#) = []
        pi(a__U21#) = []
        pi(a__isNatIListKind) = []
        pi(a__U52#) = []
        pi(a__isNat) = [1]
        pi(a__isNatList) = [1]
        pi(isNat) = []
        pi(a__U61#) = []
        pi(U61) = []
        pi(U53) = []
        pi(U52) = []
        pi(U51) = []
        pi(isNatIList) = []
        pi(a__isNatIList#) = []
        pi(a__U41#) = []
        pi(a__U42#) = []
        pi(a__U31#) = []
        pi(U43) = []
        pi(U42) = []
        pi(U41) = []
        pi(U32) = []
        pi(U31) = []
        pi(U22) = []
        pi(U21) = []
        pi(isNatList) = []
        pi(U12) = []
        pi(U11) = []
        pi(a__zeros) = []
        pi(|0|) = []
        pi(zeros) = []
        pi(a__U11) = []
        pi(a__U12) = []
        pi(a__U21) = []
        pi(a__U22) = []
        pi(a__U31) = []
        pi(a__U32) = []
        pi(a__U41) = []
        pi(a__U42) = []
        pi(a__U43) = []
        pi(a__isNatIList) = []
        pi(a__U51) = [1]
        pi(a__U52) = []
        pi(a__U53) = []
        pi(a__U61) = []
        pi(a__length) = []
        pi(nil) = []
    

The next rules are strictly ordered:

  p1, p2, p5, p6, p7, p10, p11, p13, p15, p16, p17, p18, p20, p23, p24, p25, p32, p36, p38, p39, p40, p41, p42, p43, p44, p45, p46, p47, p48, p53, p56, p57, p60, p61, p64

We remove them from the problem.

-- SCC decomposition.

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

p1: a__isNatKind#(s(V1)) -> a__isNatKind#(V1)
p2: a__isNatKind#(length(V1)) -> a__isNatIListKind#(V1)
p3: mark#(s(X)) -> mark#(X)
p4: mark#(cons(X1,X2)) -> mark#(X1)
p5: mark#(and(X1,X2)) -> mark#(X1)
p6: mark#(length(X)) -> mark#(X)
p7: a__U51#(tt(),V1,V2) -> a__isNat#(V1)
p8: a__isNat#(s(V1)) -> a__U21#(a__isNatKind(V1),V1)
p9: a__U21#(tt(),V1) -> a__isNat#(V1)
p10: a__U52#(tt(),V2) -> a__isNatList#(V2)
p11: a__length#(cons(N,L)) -> a__and#(a__isNatList(L),isNatIListKind(L))
p12: a__length#(cons(N,L)) -> a__and#(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N)))
p13: a__length#(cons(N,L)) -> a__U61#(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L)
p14: a__U61#(tt(),L) -> mark#(L)
p15: mark#(U61(X1,X2)) -> mark#(X1)
p16: a__U61#(tt(),L) -> a__length#(mark(L))
p17: mark#(U53(X)) -> mark#(X)
p18: mark#(U52(X1,X2)) -> mark#(X1)
p19: mark#(U51(X1,X2,X3)) -> mark#(X1)
p20: mark#(U43(X)) -> mark#(X)
p21: mark#(U42(X1,X2)) -> mark#(X1)
p22: mark#(U42(X1,X2)) -> a__U42#(mark(X1),X2)
p23: mark#(U41(X1,X2,X3)) -> mark#(X1)
p24: mark#(U32(X)) -> mark#(X)
p25: mark#(U31(X1,X2)) -> mark#(X1)
p26: mark#(U22(X)) -> mark#(X)
p27: mark#(U21(X1,X2)) -> mark#(X1)
p28: mark#(U12(X)) -> mark#(X)
p29: mark#(U11(X1,X2)) -> mark#(X1)

and R consists of:

r1: a__zeros() -> cons(|0|(),zeros())
r2: a__U11(tt(),V1) -> a__U12(a__isNatList(V1))
r3: a__U12(tt()) -> tt()
r4: a__U21(tt(),V1) -> a__U22(a__isNat(V1))
r5: a__U22(tt()) -> tt()
r6: a__U31(tt(),V) -> a__U32(a__isNatList(V))
r7: a__U32(tt()) -> tt()
r8: a__U41(tt(),V1,V2) -> a__U42(a__isNat(V1),V2)
r9: a__U42(tt(),V2) -> a__U43(a__isNatIList(V2))
r10: a__U43(tt()) -> tt()
r11: a__U51(tt(),V1,V2) -> a__U52(a__isNat(V1),V2)
r12: a__U52(tt(),V2) -> a__U53(a__isNatList(V2))
r13: a__U53(tt()) -> tt()
r14: a__U61(tt(),L) -> s(a__length(mark(L)))
r15: a__and(tt(),X) -> mark(X)
r16: a__isNat(|0|()) -> tt()
r17: a__isNat(length(V1)) -> a__U11(a__isNatIListKind(V1),V1)
r18: a__isNat(s(V1)) -> a__U21(a__isNatKind(V1),V1)
r19: a__isNatIList(V) -> a__U31(a__isNatIListKind(V),V)
r20: a__isNatIList(zeros()) -> tt()
r21: a__isNatIList(cons(V1,V2)) -> a__U41(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2)
r22: a__isNatIListKind(nil()) -> tt()
r23: a__isNatIListKind(zeros()) -> tt()
r24: a__isNatIListKind(cons(V1,V2)) -> a__and(a__isNatKind(V1),isNatIListKind(V2))
r25: a__isNatKind(|0|()) -> tt()
r26: a__isNatKind(length(V1)) -> a__isNatIListKind(V1)
r27: a__isNatKind(s(V1)) -> a__isNatKind(V1)
r28: a__isNatList(nil()) -> tt()
r29: a__isNatList(cons(V1,V2)) -> a__U51(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2)
r30: a__length(nil()) -> |0|()
r31: a__length(cons(N,L)) -> a__U61(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L)
r32: mark(zeros()) -> a__zeros()
r33: mark(U11(X1,X2)) -> a__U11(mark(X1),X2)
r34: mark(U12(X)) -> a__U12(mark(X))
r35: mark(isNatList(X)) -> a__isNatList(X)
r36: mark(U21(X1,X2)) -> a__U21(mark(X1),X2)
r37: mark(U22(X)) -> a__U22(mark(X))
r38: mark(isNat(X)) -> a__isNat(X)
r39: mark(U31(X1,X2)) -> a__U31(mark(X1),X2)
r40: mark(U32(X)) -> a__U32(mark(X))
r41: mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3)
r42: mark(U42(X1,X2)) -> a__U42(mark(X1),X2)
r43: mark(U43(X)) -> a__U43(mark(X))
r44: mark(isNatIList(X)) -> a__isNatIList(X)
r45: mark(U51(X1,X2,X3)) -> a__U51(mark(X1),X2,X3)
r46: mark(U52(X1,X2)) -> a__U52(mark(X1),X2)
r47: mark(U53(X)) -> a__U53(mark(X))
r48: mark(U61(X1,X2)) -> a__U61(mark(X1),X2)
r49: mark(length(X)) -> a__length(mark(X))
r50: mark(and(X1,X2)) -> a__and(mark(X1),X2)
r51: mark(isNatIListKind(X)) -> a__isNatIListKind(X)
r52: mark(isNatKind(X)) -> a__isNatKind(X)
r53: mark(cons(X1,X2)) -> cons(mark(X1),X2)
r54: mark(|0|()) -> |0|()
r55: mark(tt()) -> tt()
r56: mark(s(X)) -> s(mark(X))
r57: mark(nil()) -> nil()
r58: a__zeros() -> zeros()
r59: a__U11(X1,X2) -> U11(X1,X2)
r60: a__U12(X) -> U12(X)
r61: a__isNatList(X) -> isNatList(X)
r62: a__U21(X1,X2) -> U21(X1,X2)
r63: a__U22(X) -> U22(X)
r64: a__isNat(X) -> isNat(X)
r65: a__U31(X1,X2) -> U31(X1,X2)
r66: a__U32(X) -> U32(X)
r67: a__U41(X1,X2,X3) -> U41(X1,X2,X3)
r68: a__U42(X1,X2) -> U42(X1,X2)
r69: a__U43(X) -> U43(X)
r70: a__isNatIList(X) -> isNatIList(X)
r71: a__U51(X1,X2,X3) -> U51(X1,X2,X3)
r72: a__U52(X1,X2) -> U52(X1,X2)
r73: a__U53(X) -> U53(X)
r74: a__U61(X1,X2) -> U61(X1,X2)
r75: a__length(X) -> length(X)
r76: a__and(X1,X2) -> and(X1,X2)
r77: a__isNatIListKind(X) -> isNatIListKind(X)
r78: a__isNatKind(X) -> isNatKind(X)

The estimated dependency graph contains the following SCCs:

  {p1}
  {p13, p16}
  {p3, p4, p5, p6, p15, p17, p18, p19, p20, p21, p23, p24, p25, p26, p27, p28, p29}
  {p8, p9}


-- Reduction pair.

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

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

and R consists of:

r1: a__zeros() -> cons(|0|(),zeros())
r2: a__U11(tt(),V1) -> a__U12(a__isNatList(V1))
r3: a__U12(tt()) -> tt()
r4: a__U21(tt(),V1) -> a__U22(a__isNat(V1))
r5: a__U22(tt()) -> tt()
r6: a__U31(tt(),V) -> a__U32(a__isNatList(V))
r7: a__U32(tt()) -> tt()
r8: a__U41(tt(),V1,V2) -> a__U42(a__isNat(V1),V2)
r9: a__U42(tt(),V2) -> a__U43(a__isNatIList(V2))
r10: a__U43(tt()) -> tt()
r11: a__U51(tt(),V1,V2) -> a__U52(a__isNat(V1),V2)
r12: a__U52(tt(),V2) -> a__U53(a__isNatList(V2))
r13: a__U53(tt()) -> tt()
r14: a__U61(tt(),L) -> s(a__length(mark(L)))
r15: a__and(tt(),X) -> mark(X)
r16: a__isNat(|0|()) -> tt()
r17: a__isNat(length(V1)) -> a__U11(a__isNatIListKind(V1),V1)
r18: a__isNat(s(V1)) -> a__U21(a__isNatKind(V1),V1)
r19: a__isNatIList(V) -> a__U31(a__isNatIListKind(V),V)
r20: a__isNatIList(zeros()) -> tt()
r21: a__isNatIList(cons(V1,V2)) -> a__U41(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2)
r22: a__isNatIListKind(nil()) -> tt()
r23: a__isNatIListKind(zeros()) -> tt()
r24: a__isNatIListKind(cons(V1,V2)) -> a__and(a__isNatKind(V1),isNatIListKind(V2))
r25: a__isNatKind(|0|()) -> tt()
r26: a__isNatKind(length(V1)) -> a__isNatIListKind(V1)
r27: a__isNatKind(s(V1)) -> a__isNatKind(V1)
r28: a__isNatList(nil()) -> tt()
r29: a__isNatList(cons(V1,V2)) -> a__U51(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2)
r30: a__length(nil()) -> |0|()
r31: a__length(cons(N,L)) -> a__U61(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L)
r32: mark(zeros()) -> a__zeros()
r33: mark(U11(X1,X2)) -> a__U11(mark(X1),X2)
r34: mark(U12(X)) -> a__U12(mark(X))
r35: mark(isNatList(X)) -> a__isNatList(X)
r36: mark(U21(X1,X2)) -> a__U21(mark(X1),X2)
r37: mark(U22(X)) -> a__U22(mark(X))
r38: mark(isNat(X)) -> a__isNat(X)
r39: mark(U31(X1,X2)) -> a__U31(mark(X1),X2)
r40: mark(U32(X)) -> a__U32(mark(X))
r41: mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3)
r42: mark(U42(X1,X2)) -> a__U42(mark(X1),X2)
r43: mark(U43(X)) -> a__U43(mark(X))
r44: mark(isNatIList(X)) -> a__isNatIList(X)
r45: mark(U51(X1,X2,X3)) -> a__U51(mark(X1),X2,X3)
r46: mark(U52(X1,X2)) -> a__U52(mark(X1),X2)
r47: mark(U53(X)) -> a__U53(mark(X))
r48: mark(U61(X1,X2)) -> a__U61(mark(X1),X2)
r49: mark(length(X)) -> a__length(mark(X))
r50: mark(and(X1,X2)) -> a__and(mark(X1),X2)
r51: mark(isNatIListKind(X)) -> a__isNatIListKind(X)
r52: mark(isNatKind(X)) -> a__isNatKind(X)
r53: mark(cons(X1,X2)) -> cons(mark(X1),X2)
r54: mark(|0|()) -> |0|()
r55: mark(tt()) -> tt()
r56: mark(s(X)) -> s(mark(X))
r57: mark(nil()) -> nil()
r58: a__zeros() -> zeros()
r59: a__U11(X1,X2) -> U11(X1,X2)
r60: a__U12(X) -> U12(X)
r61: a__isNatList(X) -> isNatList(X)
r62: a__U21(X1,X2) -> U21(X1,X2)
r63: a__U22(X) -> U22(X)
r64: a__isNat(X) -> isNat(X)
r65: a__U31(X1,X2) -> U31(X1,X2)
r66: a__U32(X) -> U32(X)
r67: a__U41(X1,X2,X3) -> U41(X1,X2,X3)
r68: a__U42(X1,X2) -> U42(X1,X2)
r69: a__U43(X) -> U43(X)
r70: a__isNatIList(X) -> isNatIList(X)
r71: a__U51(X1,X2,X3) -> U51(X1,X2,X3)
r72: a__U52(X1,X2) -> U52(X1,X2)
r73: a__U53(X) -> U53(X)
r74: a__U61(X1,X2) -> U61(X1,X2)
r75: a__length(X) -> length(X)
r76: a__and(X1,X2) -> and(X1,X2)
r77: a__isNatIListKind(X) -> isNatIListKind(X)
r78: a__isNatKind(X) -> isNatKind(X)

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  lexicographic combination of reduction pairs:
  
    1. weighted path order
    
      base order:
    
        max/plus interpretations on natural numbers:
        
          a__isNatKind#_A(x1) = max{4, x1 + 3}
          s_A(x1) = max{3, x1 + 2}
    
      precedence:
      
        a__isNatKind# = s
      
      partial status:
    
        pi(a__isNatKind#) = [1]
        pi(s) = [1]
    
    2. weighted path order
    
      base order:
    
        max/plus interpretations on natural numbers:
        
          a__isNatKind#_A(x1) = max{1, x1 - 1}
          s_A(x1) = x1
    
      precedence:
      
        a__isNatKind# = s
      
      partial status:
    
        pi(a__isNatKind#) = []
        pi(s) = [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__length#(cons(N,L)) -> a__U61#(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L)
p2: a__U61#(tt(),L) -> a__length#(mark(L))

and R consists of:

r1: a__zeros() -> cons(|0|(),zeros())
r2: a__U11(tt(),V1) -> a__U12(a__isNatList(V1))
r3: a__U12(tt()) -> tt()
r4: a__U21(tt(),V1) -> a__U22(a__isNat(V1))
r5: a__U22(tt()) -> tt()
r6: a__U31(tt(),V) -> a__U32(a__isNatList(V))
r7: a__U32(tt()) -> tt()
r8: a__U41(tt(),V1,V2) -> a__U42(a__isNat(V1),V2)
r9: a__U42(tt(),V2) -> a__U43(a__isNatIList(V2))
r10: a__U43(tt()) -> tt()
r11: a__U51(tt(),V1,V2) -> a__U52(a__isNat(V1),V2)
r12: a__U52(tt(),V2) -> a__U53(a__isNatList(V2))
r13: a__U53(tt()) -> tt()
r14: a__U61(tt(),L) -> s(a__length(mark(L)))
r15: a__and(tt(),X) -> mark(X)
r16: a__isNat(|0|()) -> tt()
r17: a__isNat(length(V1)) -> a__U11(a__isNatIListKind(V1),V1)
r18: a__isNat(s(V1)) -> a__U21(a__isNatKind(V1),V1)
r19: a__isNatIList(V) -> a__U31(a__isNatIListKind(V),V)
r20: a__isNatIList(zeros()) -> tt()
r21: a__isNatIList(cons(V1,V2)) -> a__U41(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2)
r22: a__isNatIListKind(nil()) -> tt()
r23: a__isNatIListKind(zeros()) -> tt()
r24: a__isNatIListKind(cons(V1,V2)) -> a__and(a__isNatKind(V1),isNatIListKind(V2))
r25: a__isNatKind(|0|()) -> tt()
r26: a__isNatKind(length(V1)) -> a__isNatIListKind(V1)
r27: a__isNatKind(s(V1)) -> a__isNatKind(V1)
r28: a__isNatList(nil()) -> tt()
r29: a__isNatList(cons(V1,V2)) -> a__U51(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2)
r30: a__length(nil()) -> |0|()
r31: a__length(cons(N,L)) -> a__U61(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L)
r32: mark(zeros()) -> a__zeros()
r33: mark(U11(X1,X2)) -> a__U11(mark(X1),X2)
r34: mark(U12(X)) -> a__U12(mark(X))
r35: mark(isNatList(X)) -> a__isNatList(X)
r36: mark(U21(X1,X2)) -> a__U21(mark(X1),X2)
r37: mark(U22(X)) -> a__U22(mark(X))
r38: mark(isNat(X)) -> a__isNat(X)
r39: mark(U31(X1,X2)) -> a__U31(mark(X1),X2)
r40: mark(U32(X)) -> a__U32(mark(X))
r41: mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3)
r42: mark(U42(X1,X2)) -> a__U42(mark(X1),X2)
r43: mark(U43(X)) -> a__U43(mark(X))
r44: mark(isNatIList(X)) -> a__isNatIList(X)
r45: mark(U51(X1,X2,X3)) -> a__U51(mark(X1),X2,X3)
r46: mark(U52(X1,X2)) -> a__U52(mark(X1),X2)
r47: mark(U53(X)) -> a__U53(mark(X))
r48: mark(U61(X1,X2)) -> a__U61(mark(X1),X2)
r49: mark(length(X)) -> a__length(mark(X))
r50: mark(and(X1,X2)) -> a__and(mark(X1),X2)
r51: mark(isNatIListKind(X)) -> a__isNatIListKind(X)
r52: mark(isNatKind(X)) -> a__isNatKind(X)
r53: mark(cons(X1,X2)) -> cons(mark(X1),X2)
r54: mark(|0|()) -> |0|()
r55: mark(tt()) -> tt()
r56: mark(s(X)) -> s(mark(X))
r57: mark(nil()) -> nil()
r58: a__zeros() -> zeros()
r59: a__U11(X1,X2) -> U11(X1,X2)
r60: a__U12(X) -> U12(X)
r61: a__isNatList(X) -> isNatList(X)
r62: a__U21(X1,X2) -> U21(X1,X2)
r63: a__U22(X) -> U22(X)
r64: a__isNat(X) -> isNat(X)
r65: a__U31(X1,X2) -> U31(X1,X2)
r66: a__U32(X) -> U32(X)
r67: a__U41(X1,X2,X3) -> U41(X1,X2,X3)
r68: a__U42(X1,X2) -> U42(X1,X2)
r69: a__U43(X) -> U43(X)
r70: a__isNatIList(X) -> isNatIList(X)
r71: a__U51(X1,X2,X3) -> U51(X1,X2,X3)
r72: a__U52(X1,X2) -> U52(X1,X2)
r73: a__U53(X) -> U53(X)
r74: a__U61(X1,X2) -> U61(X1,X2)
r75: a__length(X) -> length(X)
r76: a__and(X1,X2) -> and(X1,X2)
r77: a__isNatIListKind(X) -> isNatIListKind(X)
r78: a__isNatKind(X) -> isNatKind(X)

The set of usable rules consists of

  r1, r2, r3, r4, r5, r6, r7, r8, r9, r10, r11, r12, r13, r14, r15, r16, r17, r18, r19, r20, r21, r22, r23, r24, r25, r26, r27, r28, r29, r30, r31, r32, r33, r34, r35, r36, r37, r38, r39, r40, r41, r42, r43, r44, r45, r46, r47, r48, r49, r50, r51, r52, r53, r54, r55, r56, r57, r58, r59, r60, r61, r62, r63, r64, r65, r66, r67, r68, r69, r70, r71, r72, r73, r74, r75, r76, r77, r78

Take the reduction pair:

  lexicographic combination of reduction pairs:
  
    1. weighted path order
    
      base order:
    
        max/plus interpretations on natural numbers:
        
          a__length#_A(x1) = max{65, x1 - 41}
          cons_A(x1,x2) = max{110, x2 + 67}
          a__U61#_A(x1,x2) = max{66, x1 - 40, x2 + 26}
          a__and_A(x1,x2) = max{x1, x2 + 33}
          a__isNatList_A(x1) = max{108, x1 - 3}
          isNatIListKind_A(x1) = 1
          and_A(x1,x2) = max{x1, x2 + 25}
          isNat_A(x1) = 75
          isNatKind_A(x1) = 43
          tt_A = 115
          mark_A(x1) = max{115, x1 + 32}
          a__zeros_A = 114
          |0|_A = 0
          zeros_A = 0
          a__U11_A(x1,x2) = max{115, x1 - 7}
          a__U12_A(x1) = 115
          a__U21_A(x1,x2) = max{83, x1}
          a__U22_A(x1) = 115
          a__isNat_A(x1) = 115
          a__U31_A(x1,x2) = x2 + 321
          a__U32_A(x1) = max{119, x1 + 4}
          a__U41_A(x1,x2,x3) = max{x1 + 236, x3 + 387}
          a__U42_A(x1,x2) = max{386, x1 + 270}
          a__U43_A(x1) = 115
          a__isNatIList_A(x1) = x1 + 321
          a__U51_A(x1,x2,x3) = max{108, x3 + 63}
          a__U52_A(x1,x2) = max{92, x1 - 7, x2 + 62}
          a__U53_A(x1) = max{1, x1}
          a__U61_A(x1,x2) = 266
          s_A(x1) = max{266, x1 - 50}
          a__length_A(x1) = 287
          length_A(x1) = 286
          a__isNatIListKind_A(x1) = 115
          a__isNatKind_A(x1) = 115
          nil_A = 119
          U11_A(x1,x2) = max{109, x1 - 7}
          U12_A(x1) = 0
          U21_A(x1,x2) = max{83, x1}
          U22_A(x1) = 114
          U31_A(x1,x2) = x2 + 290
          U32_A(x1) = max{119, x1 + 4}
          U41_A(x1,x2,x3) = max{x1 + 236, x3 + 355}
          U42_A(x1,x2) = max{386, x1 + 270}
          U43_A(x1) = 0
          isNatIList_A(x1) = max{291, x1 + 290}
          U51_A(x1,x2,x3) = max{84, x3 + 63}
          U52_A(x1,x2) = max{x1 - 7, x2 + 61}
          U53_A(x1) = max{1, x1}
          U61_A(x1,x2) = 265
          isNatList_A(x1) = max{77, x1 - 4}
    
      precedence:
      
        a__and = mark = a__isNatIListKind = a__isNatKind > a__isNat > a__length > length > a__isNatIList = isNatIList > a__isNatList > a__U61 > a__U41 > a__zeros = a__U51 = U51 > nil > isNat = a__U11 > a__U32 = U32 > U11 > and > a__U21 = a__U42 > a__U52 = a__U53 > isNatList > U43 > U31 > a__U22 > U22 > isNatKind > U41 > a__U12 = a__U31 > |0| > cons > a__length# > isNatIListKind > a__U43 > a__U61# > tt > U52 = U61 > s > U42 = U53 > U21 > zeros > U12
      
      partial status:
    
        pi(a__length#) = []
        pi(cons) = []
        pi(a__U61#) = []
        pi(a__and) = []
        pi(a__isNatList) = []
        pi(isNatIListKind) = []
        pi(and) = []
        pi(isNat) = []
        pi(isNatKind) = []
        pi(tt) = []
        pi(mark) = []
        pi(a__zeros) = []
        pi(|0|) = []
        pi(zeros) = []
        pi(a__U11) = []
        pi(a__U12) = []
        pi(a__U21) = []
        pi(a__U22) = []
        pi(a__isNat) = []
        pi(a__U31) = []
        pi(a__U32) = [1]
        pi(a__U41) = [1]
        pi(a__U42) = []
        pi(a__U43) = []
        pi(a__isNatIList) = [1]
        pi(a__U51) = [3]
        pi(a__U52) = [2]
        pi(a__U53) = []
        pi(a__U61) = []
        pi(s) = []
        pi(a__length) = []
        pi(length) = []
        pi(a__isNatIListKind) = []
        pi(a__isNatKind) = []
        pi(nil) = []
        pi(U11) = []
        pi(U12) = []
        pi(U21) = []
        pi(U22) = []
        pi(U31) = []
        pi(U32) = []
        pi(U41) = []
        pi(U42) = []
        pi(U43) = []
        pi(isNatIList) = []
        pi(U51) = []
        pi(U52) = []
        pi(U53) = []
        pi(U61) = []
        pi(isNatList) = []
    
    2. weighted path order
    
      base order:
    
        max/plus interpretations on natural numbers:
        
          a__length#_A(x1) = 57
          cons_A(x1,x2) = 31
          a__U61#_A(x1,x2) = 24
          a__and_A(x1,x2) = 57
          a__isNatList_A(x1) = 60
          isNatIListKind_A(x1) = 1
          and_A(x1,x2) = 57
          isNat_A(x1) = 30
          isNatKind_A(x1) = 56
          tt_A = 45
          mark_A(x1) = 57
          a__zeros_A = 32
          |0|_A = 26
          zeros_A = 166
          a__U11_A(x1,x2) = 57
          a__U12_A(x1) = 56
          a__U21_A(x1,x2) = 63
          a__U22_A(x1) = 56
          a__isNat_A(x1) = 64
          a__U31_A(x1,x2) = 57
          a__U32_A(x1) = max{57, x1}
          a__U41_A(x1,x2,x3) = x1 + 130
          a__U42_A(x1,x2) = 189
          a__U43_A(x1) = 92
          a__isNatIList_A(x1) = 188
          a__U51_A(x1,x2,x3) = x3 + 136
          a__U52_A(x1,x2) = x2 + 95
          a__U53_A(x1) = 92
          a__U61_A(x1,x2) = 26
          s_A(x1) = 25
          a__length_A(x1) = 27
          length_A(x1) = 25
          a__isNatIListKind_A(x1) = 57
          a__isNatKind_A(x1) = 57
          nil_A = 25
          U11_A(x1,x2) = 57
          U12_A(x1) = 22
          U21_A(x1,x2) = 33
          U22_A(x1) = 25
          U31_A(x1,x2) = 57
          U32_A(x1) = 0
          U41_A(x1,x2,x3) = 59
          U42_A(x1,x2) = 24
          U43_A(x1) = 92
          isNatIList_A(x1) = 1
          U51_A(x1,x2,x3) = 24
          U52_A(x1,x2) = 94
          U53_A(x1) = 58
          U61_A(x1,x2) = 58
          isNatList_A(x1) = 97
    
      precedence:
      
        isNatIListKind > a__length# = a__and = a__isNatList = mark = a__isNatIList = length = a__isNatIListKind = a__isNatKind > a__U61# > a__isNat > a__zeros = a__U41 > a__U53 = U53 > cons = and > a__U42 = a__U43 = U41 = U43 > U22 = isNatList > a__U22 > tt = |0| = zeros = a__U12 > a__U51 > a__U32 > a__length > a__U31 = U31 > a__U61 = U42 = U61 > isNat > isNatKind = a__U11 > a__U52 = U32 = isNatIList = U52 > a__U21 = s = nil = U11 = U12 = U21 > U51
      
      partial status:
    
        pi(a__length#) = []
        pi(cons) = []
        pi(a__U61#) = []
        pi(a__and) = []
        pi(a__isNatList) = []
        pi(isNatIListKind) = []
        pi(and) = []
        pi(isNat) = []
        pi(isNatKind) = []
        pi(tt) = []
        pi(mark) = []
        pi(a__zeros) = []
        pi(|0|) = []
        pi(zeros) = []
        pi(a__U11) = []
        pi(a__U12) = []
        pi(a__U21) = []
        pi(a__U22) = []
        pi(a__isNat) = []
        pi(a__U31) = []
        pi(a__U32) = [1]
        pi(a__U41) = []
        pi(a__U42) = []
        pi(a__U43) = []
        pi(a__isNatIList) = []
        pi(a__U51) = [3]
        pi(a__U52) = []
        pi(a__U53) = []
        pi(a__U61) = []
        pi(s) = []
        pi(a__length) = []
        pi(length) = []
        pi(a__isNatIListKind) = []
        pi(a__isNatKind) = []
        pi(nil) = []
        pi(U11) = []
        pi(U12) = []
        pi(U21) = []
        pi(U22) = []
        pi(U31) = []
        pi(U32) = []
        pi(U41) = []
        pi(U42) = []
        pi(U43) = []
        pi(isNatIList) = []
        pi(U51) = []
        pi(U52) = []
        pi(U53) = []
        pi(U61) = []
        pi(isNatList) = []
    

The next rules are strictly ordered:

  p2

We remove them from the problem.

-- SCC decomposition.

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

p1: a__length#(cons(N,L)) -> a__U61#(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L)

and R consists of:

r1: a__zeros() -> cons(|0|(),zeros())
r2: a__U11(tt(),V1) -> a__U12(a__isNatList(V1))
r3: a__U12(tt()) -> tt()
r4: a__U21(tt(),V1) -> a__U22(a__isNat(V1))
r5: a__U22(tt()) -> tt()
r6: a__U31(tt(),V) -> a__U32(a__isNatList(V))
r7: a__U32(tt()) -> tt()
r8: a__U41(tt(),V1,V2) -> a__U42(a__isNat(V1),V2)
r9: a__U42(tt(),V2) -> a__U43(a__isNatIList(V2))
r10: a__U43(tt()) -> tt()
r11: a__U51(tt(),V1,V2) -> a__U52(a__isNat(V1),V2)
r12: a__U52(tt(),V2) -> a__U53(a__isNatList(V2))
r13: a__U53(tt()) -> tt()
r14: a__U61(tt(),L) -> s(a__length(mark(L)))
r15: a__and(tt(),X) -> mark(X)
r16: a__isNat(|0|()) -> tt()
r17: a__isNat(length(V1)) -> a__U11(a__isNatIListKind(V1),V1)
r18: a__isNat(s(V1)) -> a__U21(a__isNatKind(V1),V1)
r19: a__isNatIList(V) -> a__U31(a__isNatIListKind(V),V)
r20: a__isNatIList(zeros()) -> tt()
r21: a__isNatIList(cons(V1,V2)) -> a__U41(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2)
r22: a__isNatIListKind(nil()) -> tt()
r23: a__isNatIListKind(zeros()) -> tt()
r24: a__isNatIListKind(cons(V1,V2)) -> a__and(a__isNatKind(V1),isNatIListKind(V2))
r25: a__isNatKind(|0|()) -> tt()
r26: a__isNatKind(length(V1)) -> a__isNatIListKind(V1)
r27: a__isNatKind(s(V1)) -> a__isNatKind(V1)
r28: a__isNatList(nil()) -> tt()
r29: a__isNatList(cons(V1,V2)) -> a__U51(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2)
r30: a__length(nil()) -> |0|()
r31: a__length(cons(N,L)) -> a__U61(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L)
r32: mark(zeros()) -> a__zeros()
r33: mark(U11(X1,X2)) -> a__U11(mark(X1),X2)
r34: mark(U12(X)) -> a__U12(mark(X))
r35: mark(isNatList(X)) -> a__isNatList(X)
r36: mark(U21(X1,X2)) -> a__U21(mark(X1),X2)
r37: mark(U22(X)) -> a__U22(mark(X))
r38: mark(isNat(X)) -> a__isNat(X)
r39: mark(U31(X1,X2)) -> a__U31(mark(X1),X2)
r40: mark(U32(X)) -> a__U32(mark(X))
r41: mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3)
r42: mark(U42(X1,X2)) -> a__U42(mark(X1),X2)
r43: mark(U43(X)) -> a__U43(mark(X))
r44: mark(isNatIList(X)) -> a__isNatIList(X)
r45: mark(U51(X1,X2,X3)) -> a__U51(mark(X1),X2,X3)
r46: mark(U52(X1,X2)) -> a__U52(mark(X1),X2)
r47: mark(U53(X)) -> a__U53(mark(X))
r48: mark(U61(X1,X2)) -> a__U61(mark(X1),X2)
r49: mark(length(X)) -> a__length(mark(X))
r50: mark(and(X1,X2)) -> a__and(mark(X1),X2)
r51: mark(isNatIListKind(X)) -> a__isNatIListKind(X)
r52: mark(isNatKind(X)) -> a__isNatKind(X)
r53: mark(cons(X1,X2)) -> cons(mark(X1),X2)
r54: mark(|0|()) -> |0|()
r55: mark(tt()) -> tt()
r56: mark(s(X)) -> s(mark(X))
r57: mark(nil()) -> nil()
r58: a__zeros() -> zeros()
r59: a__U11(X1,X2) -> U11(X1,X2)
r60: a__U12(X) -> U12(X)
r61: a__isNatList(X) -> isNatList(X)
r62: a__U21(X1,X2) -> U21(X1,X2)
r63: a__U22(X) -> U22(X)
r64: a__isNat(X) -> isNat(X)
r65: a__U31(X1,X2) -> U31(X1,X2)
r66: a__U32(X) -> U32(X)
r67: a__U41(X1,X2,X3) -> U41(X1,X2,X3)
r68: a__U42(X1,X2) -> U42(X1,X2)
r69: a__U43(X) -> U43(X)
r70: a__isNatIList(X) -> isNatIList(X)
r71: a__U51(X1,X2,X3) -> U51(X1,X2,X3)
r72: a__U52(X1,X2) -> U52(X1,X2)
r73: a__U53(X) -> U53(X)
r74: a__U61(X1,X2) -> U61(X1,X2)
r75: a__length(X) -> length(X)
r76: a__and(X1,X2) -> and(X1,X2)
r77: a__isNatIListKind(X) -> isNatIListKind(X)
r78: a__isNatKind(X) -> isNatKind(X)

The estimated dependency graph contains the following SCCs:

  (no SCCs)

-- Reduction pair.

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

p1: mark#(s(X)) -> mark#(X)
p2: mark#(U11(X1,X2)) -> mark#(X1)
p3: mark#(U12(X)) -> mark#(X)
p4: mark#(U21(X1,X2)) -> mark#(X1)
p5: mark#(U22(X)) -> mark#(X)
p6: mark#(U31(X1,X2)) -> mark#(X1)
p7: mark#(U32(X)) -> mark#(X)
p8: mark#(U41(X1,X2,X3)) -> mark#(X1)
p9: mark#(U42(X1,X2)) -> mark#(X1)
p10: mark#(U43(X)) -> mark#(X)
p11: mark#(U51(X1,X2,X3)) -> mark#(X1)
p12: mark#(U52(X1,X2)) -> mark#(X1)
p13: mark#(U53(X)) -> mark#(X)
p14: mark#(U61(X1,X2)) -> mark#(X1)
p15: mark#(length(X)) -> mark#(X)
p16: mark#(and(X1,X2)) -> mark#(X1)
p17: mark#(cons(X1,X2)) -> mark#(X1)

and R consists of:

r1: a__zeros() -> cons(|0|(),zeros())
r2: a__U11(tt(),V1) -> a__U12(a__isNatList(V1))
r3: a__U12(tt()) -> tt()
r4: a__U21(tt(),V1) -> a__U22(a__isNat(V1))
r5: a__U22(tt()) -> tt()
r6: a__U31(tt(),V) -> a__U32(a__isNatList(V))
r7: a__U32(tt()) -> tt()
r8: a__U41(tt(),V1,V2) -> a__U42(a__isNat(V1),V2)
r9: a__U42(tt(),V2) -> a__U43(a__isNatIList(V2))
r10: a__U43(tt()) -> tt()
r11: a__U51(tt(),V1,V2) -> a__U52(a__isNat(V1),V2)
r12: a__U52(tt(),V2) -> a__U53(a__isNatList(V2))
r13: a__U53(tt()) -> tt()
r14: a__U61(tt(),L) -> s(a__length(mark(L)))
r15: a__and(tt(),X) -> mark(X)
r16: a__isNat(|0|()) -> tt()
r17: a__isNat(length(V1)) -> a__U11(a__isNatIListKind(V1),V1)
r18: a__isNat(s(V1)) -> a__U21(a__isNatKind(V1),V1)
r19: a__isNatIList(V) -> a__U31(a__isNatIListKind(V),V)
r20: a__isNatIList(zeros()) -> tt()
r21: a__isNatIList(cons(V1,V2)) -> a__U41(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2)
r22: a__isNatIListKind(nil()) -> tt()
r23: a__isNatIListKind(zeros()) -> tt()
r24: a__isNatIListKind(cons(V1,V2)) -> a__and(a__isNatKind(V1),isNatIListKind(V2))
r25: a__isNatKind(|0|()) -> tt()
r26: a__isNatKind(length(V1)) -> a__isNatIListKind(V1)
r27: a__isNatKind(s(V1)) -> a__isNatKind(V1)
r28: a__isNatList(nil()) -> tt()
r29: a__isNatList(cons(V1,V2)) -> a__U51(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2)
r30: a__length(nil()) -> |0|()
r31: a__length(cons(N,L)) -> a__U61(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L)
r32: mark(zeros()) -> a__zeros()
r33: mark(U11(X1,X2)) -> a__U11(mark(X1),X2)
r34: mark(U12(X)) -> a__U12(mark(X))
r35: mark(isNatList(X)) -> a__isNatList(X)
r36: mark(U21(X1,X2)) -> a__U21(mark(X1),X2)
r37: mark(U22(X)) -> a__U22(mark(X))
r38: mark(isNat(X)) -> a__isNat(X)
r39: mark(U31(X1,X2)) -> a__U31(mark(X1),X2)
r40: mark(U32(X)) -> a__U32(mark(X))
r41: mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3)
r42: mark(U42(X1,X2)) -> a__U42(mark(X1),X2)
r43: mark(U43(X)) -> a__U43(mark(X))
r44: mark(isNatIList(X)) -> a__isNatIList(X)
r45: mark(U51(X1,X2,X3)) -> a__U51(mark(X1),X2,X3)
r46: mark(U52(X1,X2)) -> a__U52(mark(X1),X2)
r47: mark(U53(X)) -> a__U53(mark(X))
r48: mark(U61(X1,X2)) -> a__U61(mark(X1),X2)
r49: mark(length(X)) -> a__length(mark(X))
r50: mark(and(X1,X2)) -> a__and(mark(X1),X2)
r51: mark(isNatIListKind(X)) -> a__isNatIListKind(X)
r52: mark(isNatKind(X)) -> a__isNatKind(X)
r53: mark(cons(X1,X2)) -> cons(mark(X1),X2)
r54: mark(|0|()) -> |0|()
r55: mark(tt()) -> tt()
r56: mark(s(X)) -> s(mark(X))
r57: mark(nil()) -> nil()
r58: a__zeros() -> zeros()
r59: a__U11(X1,X2) -> U11(X1,X2)
r60: a__U12(X) -> U12(X)
r61: a__isNatList(X) -> isNatList(X)
r62: a__U21(X1,X2) -> U21(X1,X2)
r63: a__U22(X) -> U22(X)
r64: a__isNat(X) -> isNat(X)
r65: a__U31(X1,X2) -> U31(X1,X2)
r66: a__U32(X) -> U32(X)
r67: a__U41(X1,X2,X3) -> U41(X1,X2,X3)
r68: a__U42(X1,X2) -> U42(X1,X2)
r69: a__U43(X) -> U43(X)
r70: a__isNatIList(X) -> isNatIList(X)
r71: a__U51(X1,X2,X3) -> U51(X1,X2,X3)
r72: a__U52(X1,X2) -> U52(X1,X2)
r73: a__U53(X) -> U53(X)
r74: a__U61(X1,X2) -> U61(X1,X2)
r75: a__length(X) -> length(X)
r76: a__and(X1,X2) -> and(X1,X2)
r77: a__isNatIListKind(X) -> isNatIListKind(X)
r78: a__isNatKind(X) -> isNatKind(X)

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  lexicographic combination of reduction pairs:
  
    1. weighted path order
    
      base order:
    
        max/plus interpretations on natural numbers:
        
          mark#_A(x1) = x1 + 2
          s_A(x1) = x1
          U11_A(x1,x2) = max{x1, x2}
          U12_A(x1) = x1
          U21_A(x1,x2) = max{x1, x2}
          U22_A(x1) = x1
          U31_A(x1,x2) = max{x1 + 1, x2 + 1}
          U32_A(x1) = x1
          U41_A(x1,x2,x3) = max{x1, x3}
          U42_A(x1,x2) = max{x1, x2}
          U43_A(x1) = x1
          U51_A(x1,x2,x3) = max{x1 + 1, x2 + 3, x3 + 3}
          U52_A(x1,x2) = max{x1, x2}
          U53_A(x1) = x1
          U61_A(x1,x2) = max{x1, x2}
          length_A(x1) = x1 + 2
          and_A(x1,x2) = max{x1, x2}
          cons_A(x1,x2) = max{x1, x2}
    
      precedence:
      
        mark# = s = U11 = U12 = U21 = U22 = U31 = U32 = U41 = U42 = U43 = U51 = U52 = U53 = U61 = length = and = cons
      
      partial status:
    
        pi(mark#) = [1]
        pi(s) = [1]
        pi(U11) = [1, 2]
        pi(U12) = [1]
        pi(U21) = [1, 2]
        pi(U22) = [1]
        pi(U31) = [1, 2]
        pi(U32) = [1]
        pi(U41) = [1, 3]
        pi(U42) = [1, 2]
        pi(U43) = [1]
        pi(U51) = [1, 3]
        pi(U52) = [1, 2]
        pi(U53) = [1]
        pi(U61) = [1, 2]
        pi(length) = [1]
        pi(and) = [1, 2]
        pi(cons) = [1, 2]
    
    2. weighted path order
    
      base order:
    
        max/plus interpretations on natural numbers:
        
          mark#_A(x1) = max{4, x1 + 3}
          s_A(x1) = x1
          U11_A(x1,x2) = max{x1 + 4, x2 + 4}
          U12_A(x1) = x1
          U21_A(x1,x2) = max{x1, x2}
          U22_A(x1) = x1
          U31_A(x1,x2) = max{x1, x2}
          U32_A(x1) = x1 + 2
          U41_A(x1,x2,x3) = max{x1, x3}
          U42_A(x1,x2) = max{x1, x2}
          U43_A(x1) = x1
          U51_A(x1,x2,x3) = max{x1, x3}
          U52_A(x1,x2) = max{x1, x2}
          U53_A(x1) = x1
          U61_A(x1,x2) = max{x1, x2}
          length_A(x1) = x1 + 2
          and_A(x1,x2) = max{x1, x2}
          cons_A(x1,x2) = max{1, x1, x2}
    
      precedence:
      
        mark# = s = U11 = U12 = U21 = U22 = U31 = U32 = U41 = U42 = U43 = U51 = U52 = U53 = U61 = length = and = cons
      
      partial status:
    
        pi(mark#) = [1]
        pi(s) = [1]
        pi(U11) = [1, 2]
        pi(U12) = [1]
        pi(U21) = [1, 2]
        pi(U22) = [1]
        pi(U31) = [1, 2]
        pi(U32) = [1]
        pi(U41) = [1, 3]
        pi(U42) = [1, 2]
        pi(U43) = [1]
        pi(U51) = [1, 3]
        pi(U52) = [1, 2]
        pi(U53) = [1]
        pi(U61) = [1, 2]
        pi(length) = [1]
        pi(and) = [1, 2]
        pi(cons) = [1, 2]
    

The next rules are strictly ordered:

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

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__isNat#(s(V1)) -> a__U21#(a__isNatKind(V1),V1)
p2: a__U21#(tt(),V1) -> a__isNat#(V1)

and R consists of:

r1: a__zeros() -> cons(|0|(),zeros())
r2: a__U11(tt(),V1) -> a__U12(a__isNatList(V1))
r3: a__U12(tt()) -> tt()
r4: a__U21(tt(),V1) -> a__U22(a__isNat(V1))
r5: a__U22(tt()) -> tt()
r6: a__U31(tt(),V) -> a__U32(a__isNatList(V))
r7: a__U32(tt()) -> tt()
r8: a__U41(tt(),V1,V2) -> a__U42(a__isNat(V1),V2)
r9: a__U42(tt(),V2) -> a__U43(a__isNatIList(V2))
r10: a__U43(tt()) -> tt()
r11: a__U51(tt(),V1,V2) -> a__U52(a__isNat(V1),V2)
r12: a__U52(tt(),V2) -> a__U53(a__isNatList(V2))
r13: a__U53(tt()) -> tt()
r14: a__U61(tt(),L) -> s(a__length(mark(L)))
r15: a__and(tt(),X) -> mark(X)
r16: a__isNat(|0|()) -> tt()
r17: a__isNat(length(V1)) -> a__U11(a__isNatIListKind(V1),V1)
r18: a__isNat(s(V1)) -> a__U21(a__isNatKind(V1),V1)
r19: a__isNatIList(V) -> a__U31(a__isNatIListKind(V),V)
r20: a__isNatIList(zeros()) -> tt()
r21: a__isNatIList(cons(V1,V2)) -> a__U41(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2)
r22: a__isNatIListKind(nil()) -> tt()
r23: a__isNatIListKind(zeros()) -> tt()
r24: a__isNatIListKind(cons(V1,V2)) -> a__and(a__isNatKind(V1),isNatIListKind(V2))
r25: a__isNatKind(|0|()) -> tt()
r26: a__isNatKind(length(V1)) -> a__isNatIListKind(V1)
r27: a__isNatKind(s(V1)) -> a__isNatKind(V1)
r28: a__isNatList(nil()) -> tt()
r29: a__isNatList(cons(V1,V2)) -> a__U51(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2)
r30: a__length(nil()) -> |0|()
r31: a__length(cons(N,L)) -> a__U61(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L)
r32: mark(zeros()) -> a__zeros()
r33: mark(U11(X1,X2)) -> a__U11(mark(X1),X2)
r34: mark(U12(X)) -> a__U12(mark(X))
r35: mark(isNatList(X)) -> a__isNatList(X)
r36: mark(U21(X1,X2)) -> a__U21(mark(X1),X2)
r37: mark(U22(X)) -> a__U22(mark(X))
r38: mark(isNat(X)) -> a__isNat(X)
r39: mark(U31(X1,X2)) -> a__U31(mark(X1),X2)
r40: mark(U32(X)) -> a__U32(mark(X))
r41: mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3)
r42: mark(U42(X1,X2)) -> a__U42(mark(X1),X2)
r43: mark(U43(X)) -> a__U43(mark(X))
r44: mark(isNatIList(X)) -> a__isNatIList(X)
r45: mark(U51(X1,X2,X3)) -> a__U51(mark(X1),X2,X3)
r46: mark(U52(X1,X2)) -> a__U52(mark(X1),X2)
r47: mark(U53(X)) -> a__U53(mark(X))
r48: mark(U61(X1,X2)) -> a__U61(mark(X1),X2)
r49: mark(length(X)) -> a__length(mark(X))
r50: mark(and(X1,X2)) -> a__and(mark(X1),X2)
r51: mark(isNatIListKind(X)) -> a__isNatIListKind(X)
r52: mark(isNatKind(X)) -> a__isNatKind(X)
r53: mark(cons(X1,X2)) -> cons(mark(X1),X2)
r54: mark(|0|()) -> |0|()
r55: mark(tt()) -> tt()
r56: mark(s(X)) -> s(mark(X))
r57: mark(nil()) -> nil()
r58: a__zeros() -> zeros()
r59: a__U11(X1,X2) -> U11(X1,X2)
r60: a__U12(X) -> U12(X)
r61: a__isNatList(X) -> isNatList(X)
r62: a__U21(X1,X2) -> U21(X1,X2)
r63: a__U22(X) -> U22(X)
r64: a__isNat(X) -> isNat(X)
r65: a__U31(X1,X2) -> U31(X1,X2)
r66: a__U32(X) -> U32(X)
r67: a__U41(X1,X2,X3) -> U41(X1,X2,X3)
r68: a__U42(X1,X2) -> U42(X1,X2)
r69: a__U43(X) -> U43(X)
r70: a__isNatIList(X) -> isNatIList(X)
r71: a__U51(X1,X2,X3) -> U51(X1,X2,X3)
r72: a__U52(X1,X2) -> U52(X1,X2)
r73: a__U53(X) -> U53(X)
r74: a__U61(X1,X2) -> U61(X1,X2)
r75: a__length(X) -> length(X)
r76: a__and(X1,X2) -> and(X1,X2)
r77: a__isNatIListKind(X) -> isNatIListKind(X)
r78: a__isNatKind(X) -> isNatKind(X)

The set of usable rules consists of

  r1, r2, r3, r4, r5, r6, r7, r8, r9, r10, r11, r12, r13, r14, r15, r16, r17, r18, r19, r20, r21, r22, r23, r24, r25, r26, r27, r28, r29, r30, r31, r32, r33, r34, r35, r36, r37, r38, r39, r40, r41, r42, r43, r44, r45, r46, r47, r48, r49, r50, r51, r52, r53, r54, r55, r56, r57, r58, r59, r60, r61, r62, r63, r64, r65, r66, r67, r68, r69, r70, r71, r72, r73, r74, r75, r76, r77, r78

Take the reduction pair:

  lexicographic combination of reduction pairs:
  
    1. weighted path order
    
      base order:
    
        max/plus interpretations on natural numbers:
        
          a__isNat#_A(x1) = max{3, x1}
          s_A(x1) = max{143, x1 + 4}
          a__U21#_A(x1,x2) = max{5, x1 - 1, x2 + 3}
          a__isNatKind_A(x1) = max{140, x1 - 16}
          tt_A = 140
          a__zeros_A = 100
          cons_A(x1,x2) = max{99, x1 - 1, x2 + 55}
          |0|_A = 99
          zeros_A = 0
          a__U11_A(x1,x2) = max{x1 + 111, x2 + 253}
          a__U12_A(x1) = max{112, x1 + 1}
          a__isNatList_A(x1) = max{29, x1 + 27}
          a__U21_A(x1,x2) = max{x1, x2 - 12}
          a__U22_A(x1) = 140
          a__isNat_A(x1) = max{140, x1 - 15}
          a__U31_A(x1,x2) = max{220, x1 + 110, x2 + 30}
          a__U32_A(x1) = max{29, x1}
          a__U41_A(x1,x2,x3) = max{x2 + 284, x3 + 341}
          a__U42_A(x1,x2) = x2 + 318
          a__U43_A(x1) = max{140, x1 + 31}
          a__isNatIList_A(x1) = x1 + 286
          a__U51_A(x1,x2,x3) = max{112, x3 + 82}
          a__U52_A(x1,x2) = max{111, x2 + 28}
          a__U53_A(x1) = max{110, x1 + 1}
          a__U61_A(x1,x2) = max{414, x1 + 280, x2 + 323}
          a__length_A(x1) = max{415, x1 + 269}
          mark_A(x1) = max{140, x1 + 31}
          length_A(x1) = max{408, x1 + 269}
          a__isNatIListKind_A(x1) = max{140, x1 + 42}
          a__and_A(x1,x2) = max{x1, x2 + 32}
          isNatIListKind_A(x1) = max{66, x1 + 12}
          nil_A = 114
          and_A(x1,x2) = max{32, x1, x2 + 2}
          isNat_A(x1) = max{99, x1 - 45}
          isNatKind_A(x1) = max{100, x1 - 46}
          U11_A(x1,x2) = max{253, x1 + 111, x2 + 252}
          U12_A(x1) = max{111, x1 + 1}
          isNatList_A(x1) = max{29, x1 - 1}
          U21_A(x1,x2) = max{x1, x2 - 43}
          U22_A(x1) = 112
          U31_A(x1,x2) = max{220, x1 + 110, x2}
          U32_A(x1) = max{29, x1}
          U41_A(x1,x2,x3) = max{x2 + 283, x3 + 340}
          U42_A(x1,x2) = max{317, x2 + 288}
          U43_A(x1) = max{140, x1 + 31}
          isNatIList_A(x1) = max{286, x1 + 256}
          U51_A(x1,x2,x3) = max{112, x3 + 82}
          U52_A(x1,x2) = max{110, x2 + 27}
          U53_A(x1) = max{110, x1 + 1}
          U61_A(x1,x2) = max{390, x1 + 280, x2 + 323}
    
      precedence:
      
        isNat > a__isNatList = mark = a__isNatIListKind = a__and = nil = U22 > a__length > length = isNatList > a__U51 > a__U43 > a__isNatKind > a__isNat > a__U11 > a__U21 > isNatIListKind > U11 > zeros > a__U31 = a__U32 > U32 > a__zeros = a__isNatIList > a__U12 = U31 > a__U52 = isNatKind > a__U53 = and > U21 > |0| > a__isNat# > a__U22 > tt > U43 > a__U21# = U51 > U42 > isNatIList > a__U61 = U53 = U61 > s > U41 > a__U41 = U52 > cons = a__U42 = U12
      
      partial status:
    
        pi(a__isNat#) = [1]
        pi(s) = []
        pi(a__U21#) = [2]
        pi(a__isNatKind) = []
        pi(tt) = []
        pi(a__zeros) = []
        pi(cons) = []
        pi(|0|) = []
        pi(zeros) = []
        pi(a__U11) = [2]
        pi(a__U12) = [1]
        pi(a__isNatList) = []
        pi(a__U21) = []
        pi(a__U22) = []
        pi(a__isNat) = []
        pi(a__U31) = [1]
        pi(a__U32) = []
        pi(a__U41) = [2, 3]
        pi(a__U42) = []
        pi(a__U43) = []
        pi(a__isNatIList) = []
        pi(a__U51) = [3]
        pi(a__U52) = []
        pi(a__U53) = []
        pi(a__U61) = [1]
        pi(a__length) = []
        pi(mark) = []
        pi(length) = []
        pi(a__isNatIListKind) = []
        pi(a__and) = []
        pi(isNatIListKind) = [1]
        pi(nil) = []
        pi(and) = []
        pi(isNat) = []
        pi(isNatKind) = []
        pi(U11) = [1]
        pi(U12) = [1]
        pi(isNatList) = []
        pi(U21) = []
        pi(U22) = []
        pi(U31) = []
        pi(U32) = []
        pi(U41) = [2]
        pi(U42) = []
        pi(U43) = []
        pi(isNatIList) = [1]
        pi(U51) = []
        pi(U52) = [2]
        pi(U53) = []
        pi(U61) = []
    
    2. weighted path order
    
      base order:
    
        max/plus interpretations on natural numbers:
        
          a__isNat#_A(x1) = 128
          s_A(x1) = 126
          a__U21#_A(x1,x2) = x2 + 129
          a__isNatKind_A(x1) = 209
          tt_A = 127
          a__zeros_A = 173
          cons_A(x1,x2) = 172
          |0|_A = 303
          zeros_A = 126
          a__U11_A(x1,x2) = 234
          a__U12_A(x1) = 174
          a__isNatList_A(x1) = 241
          a__U21_A(x1,x2) = 129
          a__U22_A(x1) = 128
          a__isNat_A(x1) = 235
          a__U31_A(x1,x2) = 220
          a__U32_A(x1) = 149
          a__U41_A(x1,x2,x3) = max{x2 + 260, x3 + 260}
          a__U42_A(x1,x2) = 241
          a__U43_A(x1) = 126
          a__isNatIList_A(x1) = 257
          a__U51_A(x1,x2,x3) = 241
          a__U52_A(x1,x2) = 237
          a__U53_A(x1) = 126
          a__U61_A(x1,x2) = 242
          a__length_A(x1) = 302
          mark_A(x1) = 241
          length_A(x1) = 230
          a__isNatIListKind_A(x1) = 241
          a__and_A(x1,x2) = 241
          isNatIListKind_A(x1) = 11
          nil_A = 242
          and_A(x1,x2) = 170
          isNat_A(x1) = 124
          isNatKind_A(x1) = 126
          U11_A(x1,x2) = max{233, x1 - 71}
          U12_A(x1) = 7
          isNatList_A(x1) = 1
          U21_A(x1,x2) = 56
          U22_A(x1) = 129
          U31_A(x1,x2) = 74
          U32_A(x1) = 76
          U41_A(x1,x2,x3) = x2 + 260
          U42_A(x1,x2) = 166
          U43_A(x1) = 1
          isNatIList_A(x1) = x1 + 185
          U51_A(x1,x2,x3) = 240
          U52_A(x1,x2) = 236
          U53_A(x1) = 1
          U61_A(x1,x2) = 242
    
      precedence:
      
        a__U32 = a__U41 > a__U12 > |0| = zeros = a__U61 > a__isNatKind > a__U43 = a__length > mark = a__isNatIListKind = a__and > a__isNat# = s = a__U21# = cons = a__isNatList = a__U22 = a__U31 = a__U52 > a__U53 > a__zeros > a__U11 > length > a__isNat > tt > U51 > U52 > a__U51 > a__U21 > a__U42 = isNatIListKind > a__isNatIList > U22 > and = isNat = isNatKind = U11 = U12 = isNatList = U21 = U31 = U32 = U41 = U42 = U43 = isNatIList = U53 = U61 > nil
      
      partial status:
    
        pi(a__isNat#) = []
        pi(s) = []
        pi(a__U21#) = []
        pi(a__isNatKind) = []
        pi(tt) = []
        pi(a__zeros) = []
        pi(cons) = []
        pi(|0|) = []
        pi(zeros) = []
        pi(a__U11) = []
        pi(a__U12) = []
        pi(a__isNatList) = []
        pi(a__U21) = []
        pi(a__U22) = []
        pi(a__isNat) = []
        pi(a__U31) = []
        pi(a__U32) = []
        pi(a__U41) = []
        pi(a__U42) = []
        pi(a__U43) = []
        pi(a__isNatIList) = []
        pi(a__U51) = []
        pi(a__U52) = []
        pi(a__U53) = []
        pi(a__U61) = []
        pi(a__length) = []
        pi(mark) = []
        pi(length) = []
        pi(a__isNatIListKind) = []
        pi(a__and) = []
        pi(isNatIListKind) = []
        pi(nil) = []
        pi(and) = []
        pi(isNat) = []
        pi(isNatKind) = []
        pi(U11) = []
        pi(U12) = []
        pi(isNatList) = []
        pi(U21) = []
        pi(U22) = []
        pi(U31) = []
        pi(U32) = []
        pi(U41) = [2]
        pi(U42) = []
        pi(U43) = []
        pi(isNatIList) = []
        pi(U51) = []
        pi(U52) = []
        pi(U53) = []
        pi(U61) = []
    

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__U21#(tt(),V1) -> a__isNat#(V1)

and R consists of:

r1: a__zeros() -> cons(|0|(),zeros())
r2: a__U11(tt(),V1) -> a__U12(a__isNatList(V1))
r3: a__U12(tt()) -> tt()
r4: a__U21(tt(),V1) -> a__U22(a__isNat(V1))
r5: a__U22(tt()) -> tt()
r6: a__U31(tt(),V) -> a__U32(a__isNatList(V))
r7: a__U32(tt()) -> tt()
r8: a__U41(tt(),V1,V2) -> a__U42(a__isNat(V1),V2)
r9: a__U42(tt(),V2) -> a__U43(a__isNatIList(V2))
r10: a__U43(tt()) -> tt()
r11: a__U51(tt(),V1,V2) -> a__U52(a__isNat(V1),V2)
r12: a__U52(tt(),V2) -> a__U53(a__isNatList(V2))
r13: a__U53(tt()) -> tt()
r14: a__U61(tt(),L) -> s(a__length(mark(L)))
r15: a__and(tt(),X) -> mark(X)
r16: a__isNat(|0|()) -> tt()
r17: a__isNat(length(V1)) -> a__U11(a__isNatIListKind(V1),V1)
r18: a__isNat(s(V1)) -> a__U21(a__isNatKind(V1),V1)
r19: a__isNatIList(V) -> a__U31(a__isNatIListKind(V),V)
r20: a__isNatIList(zeros()) -> tt()
r21: a__isNatIList(cons(V1,V2)) -> a__U41(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2)
r22: a__isNatIListKind(nil()) -> tt()
r23: a__isNatIListKind(zeros()) -> tt()
r24: a__isNatIListKind(cons(V1,V2)) -> a__and(a__isNatKind(V1),isNatIListKind(V2))
r25: a__isNatKind(|0|()) -> tt()
r26: a__isNatKind(length(V1)) -> a__isNatIListKind(V1)
r27: a__isNatKind(s(V1)) -> a__isNatKind(V1)
r28: a__isNatList(nil()) -> tt()
r29: a__isNatList(cons(V1,V2)) -> a__U51(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2)
r30: a__length(nil()) -> |0|()
r31: a__length(cons(N,L)) -> a__U61(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L)
r32: mark(zeros()) -> a__zeros()
r33: mark(U11(X1,X2)) -> a__U11(mark(X1),X2)
r34: mark(U12(X)) -> a__U12(mark(X))
r35: mark(isNatList(X)) -> a__isNatList(X)
r36: mark(U21(X1,X2)) -> a__U21(mark(X1),X2)
r37: mark(U22(X)) -> a__U22(mark(X))
r38: mark(isNat(X)) -> a__isNat(X)
r39: mark(U31(X1,X2)) -> a__U31(mark(X1),X2)
r40: mark(U32(X)) -> a__U32(mark(X))
r41: mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3)
r42: mark(U42(X1,X2)) -> a__U42(mark(X1),X2)
r43: mark(U43(X)) -> a__U43(mark(X))
r44: mark(isNatIList(X)) -> a__isNatIList(X)
r45: mark(U51(X1,X2,X3)) -> a__U51(mark(X1),X2,X3)
r46: mark(U52(X1,X2)) -> a__U52(mark(X1),X2)
r47: mark(U53(X)) -> a__U53(mark(X))
r48: mark(U61(X1,X2)) -> a__U61(mark(X1),X2)
r49: mark(length(X)) -> a__length(mark(X))
r50: mark(and(X1,X2)) -> a__and(mark(X1),X2)
r51: mark(isNatIListKind(X)) -> a__isNatIListKind(X)
r52: mark(isNatKind(X)) -> a__isNatKind(X)
r53: mark(cons(X1,X2)) -> cons(mark(X1),X2)
r54: mark(|0|()) -> |0|()
r55: mark(tt()) -> tt()
r56: mark(s(X)) -> s(mark(X))
r57: mark(nil()) -> nil()
r58: a__zeros() -> zeros()
r59: a__U11(X1,X2) -> U11(X1,X2)
r60: a__U12(X) -> U12(X)
r61: a__isNatList(X) -> isNatList(X)
r62: a__U21(X1,X2) -> U21(X1,X2)
r63: a__U22(X) -> U22(X)
r64: a__isNat(X) -> isNat(X)
r65: a__U31(X1,X2) -> U31(X1,X2)
r66: a__U32(X) -> U32(X)
r67: a__U41(X1,X2,X3) -> U41(X1,X2,X3)
r68: a__U42(X1,X2) -> U42(X1,X2)
r69: a__U43(X) -> U43(X)
r70: a__isNatIList(X) -> isNatIList(X)
r71: a__U51(X1,X2,X3) -> U51(X1,X2,X3)
r72: a__U52(X1,X2) -> U52(X1,X2)
r73: a__U53(X) -> U53(X)
r74: a__U61(X1,X2) -> U61(X1,X2)
r75: a__length(X) -> length(X)
r76: a__and(X1,X2) -> and(X1,X2)
r77: a__isNatIListKind(X) -> isNatIListKind(X)
r78: a__isNatKind(X) -> isNatKind(X)

The estimated dependency graph contains the following SCCs:

  (no SCCs)