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. matrix interpretations:
    
      carrier: N^3
      order: lexicographic order
      interpretations:
        a__U11#_A(x1,x2) = ((1,0,0),(0,0,0),(1,0,0)) x1 + ((0,0,0),(1,0,0),(0,0,0)) x2 + (0,115,175)
        tt_A() = (58,59,96)
        a__isNatList#_A(x1) = ((0,0,0),(1,0,0),(0,0,0)) x1 + (58,114,232)
        cons_A(x1,x2) = ((1,0,0),(1,0,0),(0,1,0)) x1 + ((1,0,0),(1,1,0),(0,0,1)) x2 + (58,56,90)
        a__isNatKind#_A(x1) = ((0,0,0),(0,0,0),(1,0,0)) x1 + (58,85,231)
        s_A(x1) = x1 + (0,3,1)
        length_A(x1) = ((1,0,0),(0,1,0),(1,0,1)) x1 + (30,30,1)
        a__isNatIListKind#_A(x1) = ((0,0,0),(0,0,0),(1,0,0)) x1 + (58,85,174)
        a__and#_A(x1,x2) = ((1,0,0),(0,0,0),(1,0,0)) x1 + ((1,0,0),(0,1,0),(1,0,1)) x2 + (0,85,142)
        a__isNatKind_A(x1) = ((0,0,0),(0,0,0),(1,0,0)) x1 + (58,59,97)
        isNatIListKind_A(x1) = ((0,0,0),(0,0,0),(1,0,0)) x1 + (0,0,30)
        mark#_A(x1) = ((1,0,0),(0,1,0),(1,0,1)) x1 + (58,85,199)
        isNatKind_A(x1) = ((0,0,0),(0,0,0),(1,0,0)) x1 + (0,1,97)
        and_A(x1,x2) = x1 + ((1,0,0),(0,1,0),(0,1,1)) x2 + (0,0,2)
        mark_A(x1) = ((1,0,0),(0,1,0),(1,1,1)) x1 + (58,59,89)
        a__length#_A(x1) = ((1,0,0),(1,1,0),(0,0,0)) x1 + (1,0,231)
        a__U51#_A(x1,x2,x3) = x1 + ((0,0,0),(1,0,0),(0,1,0)) x2 + ((0,0,0),(1,0,0),(0,0,0)) x3 + (0,71,233)
        a__and_A(x1,x2) = x1 + ((1,0,0),(0,1,0),(1,1,1)) x2 + (0,0,2)
        a__isNat#_A(x1) = ((0,0,0),(1,0,0),(0,1,0)) x1 + (58,86,234)
        a__U21#_A(x1,x2) = ((0,0,0),(1,0,0),(0,1,0)) x2 + (58,86,235)
        a__isNatIListKind_A(x1) = ((0,0,0),(0,0,0),(1,0,0)) x1 + (58,59,96)
        a__U52#_A(x1,x2) = ((1,0,0),(0,1,0),(1,0,1)) x1 + ((0,0,0),(1,0,0),(0,0,0)) x2 + (0,68,80)
        a__isNat_A(x1) = ((0,0,0),(1,0,0),(0,1,0)) x1 + (58,61,94)
        a__isNatList_A(x1) = ((0,0,0),(1,0,0),(1,1,0)) x1 + (58,48,52)
        isNat_A(x1) = ((0,0,0),(1,0,0),(1,1,0)) x1 + (0,3,1)
        a__U61#_A(x1,x2) = x1 + ((1,0,0),(1,1,0),(1,0,1)) x2 + (1,61,230)
        U61_A(x1,x2) = ((1,0,0),(0,1,0),(1,0,0)) x1 + x2 + (30,34,0)
        U53_A(x1) = x1 + (0,1,8)
        U52_A(x1,x2) = ((1,0,0),(0,1,0),(0,1,0)) x1 + ((0,0,0),(1,0,0),(1,1,0)) x2 + (0,43,0)
        U51_A(x1,x2,x3) = x1 + ((0,0,0),(1,0,0),(0,1,0)) x2 + ((0,0,0),(1,0,0),(0,0,0)) x3 + (0,46,35)
        isNatIList_A(x1) = x1 + (4,120,1)
        a__isNatIList#_A(x1) = ((1,0,0),(0,1,0),(1,0,1)) x1 + (60,146,234)
        a__U41#_A(x1,x2,x3) = ((0,0,0),(1,0,0),(1,1,0)) x1 + ((0,0,0),(1,0,0),(0,0,0)) x2 + x3 + (62,92,95)
        a__U42#_A(x1,x2) = ((1,0,0),(1,1,0),(1,1,1)) x1 + x2 + (3,30,0)
        a__U31#_A(x1,x2) = ((1,0,0),(1,0,0),(1,1,0)) x1 + (1,87,116)
        U43_A(x1) = ((1,0,0),(0,0,0),(0,1,0)) x1 + (1,2,39)
        U42_A(x1,x2) = ((1,0,0),(0,0,0),(1,0,0)) x1 + ((1,0,0),(1,0,0),(0,1,0)) x2 + (5,68,1)
        U41_A(x1,x2,x3) = ((1,0,0),(0,0,0),(1,0,0)) x1 + x2 + ((1,0,0),(1,1,0),(0,0,1)) x3 + (6,66,1)
        U32_A(x1) = x1 + (1,1,137)
        U31_A(x1,x2) = ((1,0,0),(0,1,0),(0,1,1)) x1 + ((0,0,0),(0,0,0),(1,0,0)) x2 + (2,59,33)
        U22_A(x1) = x1 + (0,0,1)
        U21_A(x1,x2) = ((1,0,0),(0,1,0),(0,1,0)) x1 + ((0,0,0),(1,0,0),(0,1,0)) x2 + (0,2,37)
        isNatList_A(x1) = ((0,0,0),(1,0,0),(1,1,0)) x1 + (1,30,53)
        U12_A(x1) = x1 + (0,1,8)
        U11_A(x1,x2) = x1 + ((0,0,0),(1,0,0),(0,0,0)) x2 + (0,31,1)
        a__zeros_A() = (59,59,92)
        |0|_A() = (0,1,16)
        zeros_A() = (1,1,1)
        a__U11_A(x1,x2) = x1 + ((0,0,0),(1,0,0),(0,0,0)) x2 + (0,31,98)
        a__U12_A(x1) = x1 + (0,1,97)
        a__U21_A(x1,x2) = ((1,0,0),(0,1,0),(0,1,0)) x1 + ((0,0,0),(1,0,0),(0,1,0)) x2 + (0,2,37)
        a__U22_A(x1) = x1 + (0,0,1)
        a__U31_A(x1,x2) = ((1,0,0),(0,1,0),(0,1,1)) x1 + ((0,0,0),(0,0,0),(1,0,0)) x2 + (2,59,34)
        a__U32_A(x1) = x1 + (1,1,138)
        a__U41_A(x1,x2,x3) = ((1,0,0),(0,0,0),(1,0,0)) x1 + ((1,0,0),(0,0,0),(1,0,0)) x2 + ((1,0,0),(1,1,0),(1,0,1)) x3 + (6,125,2)
        a__U42_A(x1,x2) = ((1,0,0),(0,0,0),(1,1,0)) x1 + ((1,0,0),(1,0,0),(0,0,0)) x2 + (5,126,47)
        a__U43_A(x1) = ((1,0,0),(0,0,0),(0,1,0)) x1 + (1,60,38)
        a__isNatIList_A(x1) = ((1,0,0),(0,0,0),(1,0,0)) x1 + (61,119,0)
        a__U51_A(x1,x2,x3) = x1 + ((0,0,0),(1,0,0),(0,1,0)) x2 + ((0,0,0),(1,0,0),(0,0,0)) x3 + (0,46,36)
        a__U52_A(x1,x2) = ((1,0,0),(0,1,0),(0,1,0)) x1 + ((0,0,0),(1,0,0),(1,1,0)) x2 + (0,43,72)
        a__U53_A(x1) = x1 + (0,1,97)
        a__U61_A(x1,x2) = ((1,0,0),(0,1,0),(1,0,0)) x1 + x2 + (30,34,1)
        a__length_A(x1) = ((1,0,0),(0,1,0),(1,0,1)) x1 + (30,30,2)
        nil_A() = (12,0,1)
    
    2. lexicographic path order with precedence:
    
      precedence:
      
        U41 > isNatIList > U32 > a__length > a__U61 > U61 > isNat > U42 > a__U61# > a__length# > a__and# > isNatList > a__isNatList > a__U11 > a__U12 > U12 > a__isNatIListKind > tt > a__isNat > mark > a__U42 > a__U43 > a__isNatIList# > a__U41# > a__U51 > a__U21 > U21 > a__U31# > a__U52 > s > U52 > U22 > a__U22 > a__U31 > |0| > nil > zeros > a__zeros > U31 > a__U41 > a__isNatIList > a__U53 > U53 > mark# > a__U51# > a__U52# > U43 > a__U11# > a__isNatList# > a__isNat# > a__U21# > a__U32 > a__isNatKind# > a__U42# > U51 > U11 > a__isNatIListKind# > a__isNatKind > a__and > and > length > isNatKind > isNatIListKind > cons
      
      argument filter:
    
        pi(a__U11#) = []
        pi(tt) = []
        pi(a__isNatList#) = []
        pi(cons) = [2]
        pi(a__isNatKind#) = []
        pi(s) = []
        pi(length) = 1
        pi(a__isNatIListKind#) = []
        pi(a__and#) = []
        pi(a__isNatKind) = []
        pi(isNatIListKind) = []
        pi(mark#) = []
        pi(isNatKind) = []
        pi(and) = 1
        pi(mark) = 1
        pi(a__length#) = []
        pi(a__U51#) = []
        pi(a__and) = 1
        pi(a__isNat#) = []
        pi(a__U21#) = []
        pi(a__isNatIListKind) = []
        pi(a__U52#) = []
        pi(a__isNat) = []
        pi(a__isNatList) = []
        pi(isNat) = []
        pi(a__U61#) = 2
        pi(U61) = 2
        pi(U53) = []
        pi(U52) = []
        pi(U51) = 1
        pi(isNatIList) = []
        pi(a__isNatIList#) = [1]
        pi(a__U41#) = [3]
        pi(a__U42#) = [2]
        pi(a__U31#) = []
        pi(U43) = []
        pi(U42) = []
        pi(U41) = 3
        pi(U32) = []
        pi(U31) = []
        pi(U22) = 1
        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) = 1
        pi(a__U31) = []
        pi(a__U32) = []
        pi(a__U41) = [3]
        pi(a__U42) = []
        pi(a__U43) = []
        pi(a__isNatIList) = []
        pi(a__U51) = 1
        pi(a__U52) = []
        pi(a__U53) = []
        pi(a__U61) = [2]
        pi(a__length) = [1]
        pi(nil) = []
    

The next rules are strictly ordered:

  p1, p2, p4, p5, p6, p7, p8, p9, p10, p11, p12, p13, p14, p15, p16, p17, p18, p19, p20, p21, p22, p23, p24, p25, p26, p27, p28, p29, p30, p31, p32, p33, p34, p35, p36, p37, p38, p39, p40, p41, p42, p43, p44, p45, p46, p47, p48, p49, p50, p51, p52, p53, p54, p55, p56, p57, p58, 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)

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}


-- 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. matrix interpretations:
    
      carrier: N^3
      order: lexicographic order
      interpretations:
        a__isNatKind#_A(x1) = ((1,0,0),(0,0,0),(1,1,0)) x1
        s_A(x1) = ((1,0,0),(1,1,0),(1,1,1)) x1 + (1,1,1)
    
    2. lexicographic path order with precedence:
    
      precedence:
      
        s > a__isNatKind#
      
      argument filter:
    
        pi(a__isNatKind#) = []
        pi(s) = []
    

The next rules are strictly ordered:

  p1

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