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. max/plus interpretations on natural numbers:
    
      a__U11#_A(x1,x2) = max{x1 + 107, x2 + 68}
      tt_A = 61
      a__isNatList#_A(x1) = x1 + 67
      cons_A(x1,x2) = max{x1 + 80, x2 + 89}
      a__isNatKind#_A(x1) = x1 + 126
      s_A(x1) = max{90, x1}
      length_A(x1) = max{77, x1 + 73}
      a__isNatIListKind#_A(x1) = x1 + 67
      a__and#_A(x1,x2) = max{x1 + 68, x2 + 149}
      a__isNatKind_A(x1) = x1 + 63
      isNatIListKind_A(x1) = max{6, x1 - 36}
      mark#_A(x1) = x1 + 148
      isNatKind_A(x1) = max{0, x1 - 21}
      and_A(x1,x2) = max{12, x1 + 6, x2 + 11}
      mark_A(x1) = max{89, x1 + 85}
      a__length#_A(x1) = max{190, x1 + 60}
      a__U51#_A(x1,x2,x3) = max{x1 - 59, x2 + 128, x3 + 69}
      a__and_A(x1,x2) = max{x1 + 6, x2 + 90}
      a__isNat#_A(x1) = x1 + 127
      a__U21#_A(x1,x2) = max{x1 - 62, x2 + 127}
      a__isNatIListKind_A(x1) = max{90, x1 + 10}
      a__U52#_A(x1,x2) = max{x1 - 61, x2 + 68}
      a__isNat_A(x1) = x1 + 65
      a__isNatList_A(x1) = max{82, x1 + 47}
      isNat_A(x1) = max{8, x1 - 16}
      a__U61#_A(x1,x2) = max{190, x2 + 148}
      U61_A(x1,x2) = max{158, x1 + 50, x2 + 78}
      U53_A(x1) = max{119, x1 + 39}
      U52_A(x1,x2) = max{x1 + 61, x2 + 120}
      U51_A(x1,x2,x3) = max{127, x1 + 30, x2 + 43, x3 + 35}
      isNatIList_A(x1) = max{97, x1 + 13}
      a__isNatIList#_A(x1) = max{131, x1 + 69}
      a__U41#_A(x1,x2,x3) = max{x1 + 36, x2 + 148, x3 + 133}
      a__U42#_A(x1,x2) = max{x1 + 71, x2 + 132}
      a__U31#_A(x1,x2) = max{x1 - 22, x2 + 68}
      U43_A(x1) = max{87, x1 + 1}
      U42_A(x1,x2) = max{99, x1 + 38, x2 + 42}
      U41_A(x1,x2,x3) = max{104, x1 + 28, x2 + 103, x3 + 100}
      U32_A(x1) = max{8, x1 + 4}
      U31_A(x1,x2) = max{95, x1 + 7, x2 + 10}
      U22_A(x1) = max{63, x1}
      U21_A(x1,x2) = max{x1 + 1, x2 + 64}
      isNatList_A(x1) = x1
      U12_A(x1) = max{62, x1 + 58}
      U11_A(x1,x2) = max{x1 + 49, x2 + 56}
      a__zeros_A = 89
      |0|_A = 0
      zeros_A = 0
      a__U11_A(x1,x2) = max{141, x1 + 49, x2 + 106}
      a__U12_A(x1) = max{62, x1 + 58}
      a__U21_A(x1,x2) = max{x1 + 1, x2 + 65}
      a__U22_A(x1) = max{63, x1}
      a__U31_A(x1,x2) = max{x1 + 7, x2 + 95}
      a__U32_A(x1) = max{88, x1 + 4}
      a__U41_A(x1,x2,x3) = max{x1 + 28, x2 + 104, x3 + 100}
      a__U42_A(x1,x2) = max{x1 + 38, x2 + 99}
      a__U43_A(x1) = max{87, x1 + 1}
      a__isNatIList_A(x1) = max{97, x1 + 96}
      a__U51_A(x1,x2,x3) = max{x1 + 30, x2 + 127, x3 + 120}
      a__U52_A(x1,x2) = max{x1 + 61, x2 + 120}
      a__U53_A(x1) = max{119, x1 + 39}
      a__U61_A(x1,x2) = max{x1 + 50, x2 + 162}
      a__length_A(x1) = max{156, x1 + 73}
      nil_A = 52
    
    2. max/plus interpretations on natural numbers:
    
      a__U11#_A(x1,x2) = 5
      tt_A = 29
      a__isNatList#_A(x1) = 6
      cons_A(x1,x2) = 0
      a__isNatKind#_A(x1) = 2
      s_A(x1) = 18
      length_A(x1) = 31
      a__isNatIListKind#_A(x1) = 1
      a__and#_A(x1,x2) = 0
      a__isNatKind_A(x1) = 51
      isNatIListKind_A(x1) = 0
      mark#_A(x1) = 40
      isNatKind_A(x1) = 0
      and_A(x1,x2) = 30
      mark_A(x1) = 32
      a__length#_A(x1) = 41
      a__U51#_A(x1,x2,x3) = 5
      a__and_A(x1,x2) = 31
      a__isNat#_A(x1) = 4
      a__U21#_A(x1,x2) = 4
      a__isNatIListKind_A(x1) = 50
      a__U52#_A(x1,x2) = 7
      a__isNat_A(x1) = max{28, x1 + 14}
      a__isNatList_A(x1) = 33
      isNat_A(x1) = 14
      a__U61#_A(x1,x2) = 41
      U61_A(x1,x2) = 29
      U53_A(x1) = 21
      U52_A(x1,x2) = 10
      U51_A(x1,x2,x3) = 11
      isNatIList_A(x1) = 32
      a__isNatIList#_A(x1) = 8
      a__U41#_A(x1,x2,x3) = 5
      a__U42#_A(x1,x2) = 9
      a__U31#_A(x1,x2) = 7
      U43_A(x1) = 28
      U42_A(x1,x2) = 20
      U41_A(x1,x2,x3) = 19
      U32_A(x1) = 21
      U31_A(x1,x2) = 0
      U22_A(x1) = 1
      U21_A(x1,x2) = 20
      isNatList_A(x1) = 24
      U12_A(x1) = 29
      U11_A(x1,x2) = 20
      a__zeros_A = 32
      |0|_A = 33
      zeros_A = 33
      a__U11_A(x1,x2) = 31
      a__U12_A(x1) = 30
      a__U21_A(x1,x2) = 31
      a__U22_A(x1) = 30
      a__U31_A(x1,x2) = 13
      a__U32_A(x1) = 32
      a__U41_A(x1,x2,x3) = 30
      a__U42_A(x1,x2) = 31
      a__U43_A(x1) = 28
      a__isNatIList_A(x1) = 45
      a__U51_A(x1,x2,x3) = 22
      a__U52_A(x1,x2) = 21
      a__U53_A(x1) = 32
      a__U61_A(x1,x2) = 30
      a__length_A(x1) = 31
      nil_A = 5
    

The next rules are strictly ordered:

  p1, p2, p4, p5, p6, p7, p9, p10, p11, p12, p13, p14, p15, p16, p17, p18, p19, p20, p23, p24, p25, p26, p27, p28, p30, p31, p32, p34, p35, p36, p37, p38, p39, p40, p41, p42, p43, p44, p45, p46, p47, p48, p49, p50, p51, p52, p53, p54, p55, p56, p57, p59, p60, p61, p62, p63, 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: mark#(s(X)) -> mark#(X)
p3: a__isNat#(s(V1)) -> a__U21#(a__isNatKind(V1),V1)
p4: a__U21#(tt(),V1) -> a__isNat#(V1)
p5: a__length#(cons(N,L)) -> a__U61#(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L)
p6: a__U61#(tt(),L) -> a__length#(mark(L))
p7: mark#(U22(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:

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


-- 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. max/plus interpretations on natural numbers:
    
      a__isNatKind#_A(x1) = x1
      s_A(x1) = max{2, x1 + 1}
    
    2. max/plus interpretations on natural numbers:
    
      a__isNatKind#_A(x1) = x1
      s_A(x1) = 0
    

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: mark#(s(X)) -> mark#(X)
p2: mark#(U22(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 set of usable rules consists of

  (no rules)

Take the monotone reduction pair:

  lexicographic combination of reduction pairs:
  
    1. max/plus interpretations on natural numbers:
    
      mark#_A(x1) = x1
      s_A(x1) = x1 + 1
      U22_A(x1) = x1
    
    2. max/plus interpretations on natural numbers:
    
      mark#_A(x1) = x1
      s_A(x1) = x1 + 1
      U22_A(x1) = x1 + 1
    

The next rules are strictly ordered:

  p1, p2

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. max/plus interpretations on natural numbers:
    
      a__isNat#_A(x1) = x1
      s_A(x1) = max{54, x1 + 2}
      a__U21#_A(x1,x2) = max{x1 - 95, x2 + 1}
      a__isNatKind_A(x1) = 68
      tt_A = 68
      a__zeros_A = 60
      cons_A(x1,x2) = max{59, x2 + 57}
      |0|_A = 65
      zeros_A = 0
      a__U11_A(x1,x2) = 68
      a__U12_A(x1) = 68
      a__isNatList_A(x1) = max{47, x1 + 15}
      a__U21_A(x1,x2) = max{3, x1}
      a__U22_A(x1) = max{17, x1}
      a__isNat_A(x1) = 68
      a__U31_A(x1,x2) = max{69, x1 + 18}
      a__U32_A(x1) = 86
      a__U41_A(x1,x2,x3) = max{126, x1 + 35, x3 + 123}
      a__U42_A(x1,x2) = x2 + 120
      a__U43_A(x1) = max{67, x1 + 33}
      a__isNatIList_A(x1) = max{87, x1 + 67}
      a__U51_A(x1,x2,x3) = x3 + 71
      a__U52_A(x1,x2) = x2 + 70
      a__U53_A(x1) = 70
      a__U61_A(x1,x2) = max{67, x1 + 3, x2 + 56}
      a__length_A(x1) = max{68, x1}
      mark_A(x1) = max{68, x1 + 51}
      length_A(x1) = x1
      a__isNatIListKind_A(x1) = 68
      a__and_A(x1,x2) = max{57, x1, x2 + 52}
      isNatIListKind_A(x1) = 1
      nil_A = 69
      and_A(x1,x2) = max{x1, x2 + 7}
      isNat_A(x1) = 3
      isNatKind_A(x1) = 0
      U11_A(x1,x2) = 68
      U12_A(x1) = 37
      isNatList_A(x1) = max{16, x1 + 15}
      U21_A(x1,x2) = max{3, x1}
      U22_A(x1) = max{17, x1}
      U31_A(x1,x2) = max{68, x1 + 18}
      U32_A(x1) = 86
      U41_A(x1,x2,x3) = max{x1 + 35, x3 + 123}
      U42_A(x1,x2) = max{72, x2 + 70}
      U43_A(x1) = max{67, x1 + 33}
      isNatIList_A(x1) = max{87, x1 + 66}
      U51_A(x1,x2,x3) = max{46, x3 + 20}
      U52_A(x1,x2) = max{68, x2 + 19}
      U53_A(x1) = 70
      U61_A(x1,x2) = max{55, x1 + 3, x2 + 19}
    
    2. max/plus interpretations on natural numbers:
    
      a__isNat#_A(x1) = max{1, x1}
      s_A(x1) = 19
      a__U21#_A(x1,x2) = 0
      a__isNatKind_A(x1) = 20
      tt_A = 8
      a__zeros_A = 0
      cons_A(x1,x2) = 21
      |0|_A = 0
      zeros_A = 6
      a__U11_A(x1,x2) = 10
      a__U12_A(x1) = 9
      a__isNatList_A(x1) = 12
      a__U21_A(x1,x2) = 17
      a__U22_A(x1) = 8
      a__isNat_A(x1) = 19
      a__U31_A(x1,x2) = 14
      a__U32_A(x1) = 13
      a__U41_A(x1,x2,x3) = 12
      a__U42_A(x1,x2) = 11
      a__U43_A(x1) = 9
      a__isNatIList_A(x1) = 13
      a__U51_A(x1,x2,x3) = 20
      a__U52_A(x1,x2) = 10
      a__U53_A(x1) = 9
      a__U61_A(x1,x2) = 17
      a__length_A(x1) = 19
      mark_A(x1) = 20
      length_A(x1) = 0
      a__isNatIListKind_A(x1) = 20
      a__and_A(x1,x2) = 20
      isNatIListKind_A(x1) = 4
      nil_A = 6
      and_A(x1,x2) = 20
      isNat_A(x1) = 20
      isNatKind_A(x1) = 21
      U11_A(x1,x2) = 0
      U12_A(x1) = 10
      isNatList_A(x1) = 0
      U21_A(x1,x2) = 16
      U22_A(x1) = 7
      U31_A(x1,x2) = 14
      U32_A(x1) = 13
      U41_A(x1,x2,x3) = 11
      U42_A(x1,x2) = 10
      U43_A(x1) = 0
      isNatIList_A(x1) = 10
      U51_A(x1,x2,x3) = 19
      U52_A(x1,x2) = 9
      U53_A(x1) = 6
      U61_A(x1,x2) = 16
    

The next rules are strictly ordered:

  p1, p2

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. max/plus interpretations on natural numbers:
    
      a__length#_A(x1) = max{0, x1 - 42}
      cons_A(x1,x2) = max{182, x2 + 92}
      a__U61#_A(x1,x2) = max{x1 - 41, x2 + 2}
      a__and_A(x1,x2) = max{180, x1, x2 + 45}
      a__isNatList_A(x1) = max{42, x1 + 6}
      isNatIListKind_A(x1) = 77
      and_A(x1,x2) = max{x1, x2 + 45}
      isNat_A(x1) = 0
      isNatKind_A(x1) = 0
      tt_A = 184
      mark_A(x1) = max{184, x1 + 44}
      a__zeros_A = 183
      |0|_A = 159
      zeros_A = 90
      a__U11_A(x1,x2) = max{184, x1 - 1}
      a__U12_A(x1) = 184
      a__U21_A(x1,x2) = 184
      a__U22_A(x1) = x1
      a__isNat_A(x1) = 184
      a__U31_A(x1,x2) = max{43, x2 + 6}
      a__U32_A(x1) = max{6, x1}
      a__U41_A(x1,x2,x3) = x3 + 187
      a__U42_A(x1,x2) = max{187, x2 + 185}
      a__U43_A(x1) = max{186, x1 - 1}
      a__isNatIList_A(x1) = x1 + 95
      a__U51_A(x1,x2,x3) = max{187, x1 - 81, x3 + 97}
      a__U52_A(x1,x2) = max{186, x2 + 97}
      a__U53_A(x1) = max{185, x1 + 44}
      a__U61_A(x1,x2) = max{162, x1 - 20, x2 + 21}
      s_A(x1) = max{161, x1 - 22}
      a__length_A(x1) = max{163, x1 - 1}
      length_A(x1) = max{163, x1 - 1}
      a__isNatIListKind_A(x1) = 184
      a__isNatKind_A(x1) = 184
      nil_A = 185
      U11_A(x1,x2) = max{141, x1 - 1}
      U12_A(x1) = 140
      U21_A(x1,x2) = 184
      U22_A(x1) = x1
      U31_A(x1,x2) = max{5, x2 - 38}
      U32_A(x1) = max{5, x1}
      U41_A(x1,x2,x3) = max{187, x3 + 144}
      U42_A(x1,x2) = max{143, x2 + 141}
      U43_A(x1) = max{142, x1 - 1}
      isNatIList_A(x1) = x1 + 51
      U51_A(x1,x2,x3) = max{143, x1 - 81, x3 + 97}
      U52_A(x1,x2) = max{143, x2 + 53}
      U53_A(x1) = max{185, x1 + 44}
      U61_A(x1,x2) = max{119, x1 - 20, x2 + 20}
      isNatList_A(x1) = x1 + 5
    
    2. max/plus interpretations on natural numbers:
    
      a__length#_A(x1) = 0
      cons_A(x1,x2) = 8
      a__U61#_A(x1,x2) = 1
      a__and_A(x1,x2) = 7
      a__isNatList_A(x1) = 12
      isNatIListKind_A(x1) = 8
      and_A(x1,x2) = 6
      isNat_A(x1) = 0
      isNatKind_A(x1) = 8
      tt_A = 2
      mark_A(x1) = 7
      a__zeros_A = 4
      |0|_A = 3
      zeros_A = 5
      a__U11_A(x1,x2) = 4
      a__U12_A(x1) = 3
      a__U21_A(x1,x2) = 5
      a__U22_A(x1) = max{1, x1}
      a__isNat_A(x1) = 5
      a__U31_A(x1,x2) = 4
      a__U32_A(x1) = 3
      a__U41_A(x1,x2,x3) = 2
      a__U42_A(x1,x2) = 1
      a__U43_A(x1) = 3
      a__isNatIList_A(x1) = 3
      a__U51_A(x1,x2,x3) = 6
      a__U52_A(x1,x2) = 1
      a__U53_A(x1) = 7
      a__U61_A(x1,x2) = 1
      s_A(x1) = 0
      a__length_A(x1) = 2
      length_A(x1) = 1
      a__isNatIListKind_A(x1) = 7
      a__isNatKind_A(x1) = 7
      nil_A = 6
      U11_A(x1,x2) = 3
      U12_A(x1) = 4
      U21_A(x1,x2) = 4
      U22_A(x1) = max{0, x1 - 1}
      U31_A(x1,x2) = 14
      U32_A(x1) = 2
      U41_A(x1,x2,x3) = 1
      U42_A(x1,x2) = 0
      U43_A(x1) = 0
      isNatIList_A(x1) = x1 + 9
      U51_A(x1,x2,x3) = 5
      U52_A(x1,x2) = 0
      U53_A(x1) = 7
      U61_A(x1,x2) = 0
      isNatList_A(x1) = max{13, x1 - 2}
    

The next rules are strictly ordered:

  p1, p2

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