(0) Obligation:

Q restricted rewrite system:
The TRS R consists of the following rules:

a__zeroscons(0, zeros)
a__U11(tt, V1) → a__U12(a__isNatIListKind(V1), V1)
a__U12(tt, V1) → a__U13(a__isNatList(V1))
a__U13(tt) → tt
a__U21(tt, V1) → a__U22(a__isNatKind(V1), V1)
a__U22(tt, V1) → a__U23(a__isNat(V1))
a__U23(tt) → tt
a__U31(tt, V) → a__U32(a__isNatIListKind(V), V)
a__U32(tt, V) → a__U33(a__isNatList(V))
a__U33(tt) → tt
a__U41(tt, V1, V2) → a__U42(a__isNatKind(V1), V1, V2)
a__U42(tt, V1, V2) → a__U43(a__isNatIListKind(V2), V1, V2)
a__U43(tt, V1, V2) → a__U44(a__isNatIListKind(V2), V1, V2)
a__U44(tt, V1, V2) → a__U45(a__isNat(V1), V2)
a__U45(tt, V2) → a__U46(a__isNatIList(V2))
a__U46(tt) → tt
a__U51(tt, V2) → a__U52(a__isNatIListKind(V2))
a__U52(tt) → tt
a__U61(tt) → tt
a__U71(tt) → tt
a__U81(tt, V1, V2) → a__U82(a__isNatKind(V1), V1, V2)
a__U82(tt, V1, V2) → a__U83(a__isNatIListKind(V2), V1, V2)
a__U83(tt, V1, V2) → a__U84(a__isNatIListKind(V2), V1, V2)
a__U84(tt, V1, V2) → a__U85(a__isNat(V1), V2)
a__U85(tt, V2) → a__U86(a__isNatList(V2))
a__U86(tt) → tt
a__U91(tt, L, N) → a__U92(a__isNatIListKind(L), L, N)
a__U92(tt, L, N) → a__U93(a__isNat(N), L, N)
a__U93(tt, L, N) → a__U94(a__isNatKind(N), L)
a__U94(tt, L) → s(a__length(mark(L)))
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__isNatKind(V1), V1, V2)
a__isNatIListKind(nil) → tt
a__isNatIListKind(zeros) → tt
a__isNatIListKind(cons(V1, V2)) → a__U51(a__isNatKind(V1), V2)
a__isNatKind(0) → tt
a__isNatKind(length(V1)) → a__U61(a__isNatIListKind(V1))
a__isNatKind(s(V1)) → a__U71(a__isNatKind(V1))
a__isNatList(nil) → tt
a__isNatList(cons(V1, V2)) → a__U81(a__isNatKind(V1), V1, V2)
a__length(nil) → 0
a__length(cons(N, L)) → a__U91(a__isNatList(L), L, N)
mark(zeros) → a__zeros
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(U12(X1, X2)) → a__U12(mark(X1), X2)
mark(isNatIListKind(X)) → a__isNatIListKind(X)
mark(U13(X)) → a__U13(mark(X))
mark(isNatList(X)) → a__isNatList(X)
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(U22(X1, X2)) → a__U22(mark(X1), X2)
mark(isNatKind(X)) → a__isNatKind(X)
mark(U23(X)) → a__U23(mark(X))
mark(isNat(X)) → a__isNat(X)
mark(U31(X1, X2)) → a__U31(mark(X1), X2)
mark(U32(X1, X2)) → a__U32(mark(X1), X2)
mark(U33(X)) → a__U33(mark(X))
mark(U41(X1, X2, X3)) → a__U41(mark(X1), X2, X3)
mark(U42(X1, X2, X3)) → a__U42(mark(X1), X2, X3)
mark(U43(X1, X2, X3)) → a__U43(mark(X1), X2, X3)
mark(U44(X1, X2, X3)) → a__U44(mark(X1), X2, X3)
mark(U45(X1, X2)) → a__U45(mark(X1), X2)
mark(U46(X)) → a__U46(mark(X))
mark(isNatIList(X)) → a__isNatIList(X)
mark(U51(X1, X2)) → a__U51(mark(X1), X2)
mark(U52(X)) → a__U52(mark(X))
mark(U61(X)) → a__U61(mark(X))
mark(U71(X)) → a__U71(mark(X))
mark(U81(X1, X2, X3)) → a__U81(mark(X1), X2, X3)
mark(U82(X1, X2, X3)) → a__U82(mark(X1), X2, X3)
mark(U83(X1, X2, X3)) → a__U83(mark(X1), X2, X3)
mark(U84(X1, X2, X3)) → a__U84(mark(X1), X2, X3)
mark(U85(X1, X2)) → a__U85(mark(X1), X2)
mark(U86(X)) → a__U86(mark(X))
mark(U91(X1, X2, X3)) → a__U91(mark(X1), X2, X3)
mark(U92(X1, X2, X3)) → a__U92(mark(X1), X2, X3)
mark(U93(X1, X2, X3)) → a__U93(mark(X1), X2, X3)
mark(U94(X1, X2)) → a__U94(mark(X1), X2)
mark(length(X)) → a__length(mark(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__zeroszeros
a__U11(X1, X2) → U11(X1, X2)
a__U12(X1, X2) → U12(X1, X2)
a__isNatIListKind(X) → isNatIListKind(X)
a__U13(X) → U13(X)
a__isNatList(X) → isNatList(X)
a__U21(X1, X2) → U21(X1, X2)
a__U22(X1, X2) → U22(X1, X2)
a__isNatKind(X) → isNatKind(X)
a__U23(X) → U23(X)
a__isNat(X) → isNat(X)
a__U31(X1, X2) → U31(X1, X2)
a__U32(X1, X2) → U32(X1, X2)
a__U33(X) → U33(X)
a__U41(X1, X2, X3) → U41(X1, X2, X3)
a__U42(X1, X2, X3) → U42(X1, X2, X3)
a__U43(X1, X2, X3) → U43(X1, X2, X3)
a__U44(X1, X2, X3) → U44(X1, X2, X3)
a__U45(X1, X2) → U45(X1, X2)
a__U46(X) → U46(X)
a__isNatIList(X) → isNatIList(X)
a__U51(X1, X2) → U51(X1, X2)
a__U52(X) → U52(X)
a__U61(X) → U61(X)
a__U71(X) → U71(X)
a__U81(X1, X2, X3) → U81(X1, X2, X3)
a__U82(X1, X2, X3) → U82(X1, X2, X3)
a__U83(X1, X2, X3) → U83(X1, X2, X3)
a__U84(X1, X2, X3) → U84(X1, X2, X3)
a__U85(X1, X2) → U85(X1, X2)
a__U86(X) → U86(X)
a__U91(X1, X2, X3) → U91(X1, X2, X3)
a__U92(X1, X2, X3) → U92(X1, X2, X3)
a__U93(X1, X2, X3) → U93(X1, X2, X3)
a__U94(X1, X2) → U94(X1, X2)
a__length(X) → length(X)

Q is empty.

(1) DependencyPairsProof (EQUIVALENT transformation)

Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.

(2) Obligation:

Q DP problem:
The TRS P consists of the following rules:

A__U11(tt, V1) → A__U12(a__isNatIListKind(V1), V1)
A__U11(tt, V1) → A__ISNATILISTKIND(V1)
A__U12(tt, V1) → A__U13(a__isNatList(V1))
A__U12(tt, V1) → A__ISNATLIST(V1)
A__U21(tt, V1) → A__U22(a__isNatKind(V1), V1)
A__U21(tt, V1) → A__ISNATKIND(V1)
A__U22(tt, V1) → A__U23(a__isNat(V1))
A__U22(tt, V1) → A__ISNAT(V1)
A__U31(tt, V) → A__U32(a__isNatIListKind(V), V)
A__U31(tt, V) → A__ISNATILISTKIND(V)
A__U32(tt, V) → A__U33(a__isNatList(V))
A__U32(tt, V) → A__ISNATLIST(V)
A__U41(tt, V1, V2) → A__U42(a__isNatKind(V1), V1, V2)
A__U41(tt, V1, V2) → A__ISNATKIND(V1)
A__U42(tt, V1, V2) → A__U43(a__isNatIListKind(V2), V1, V2)
A__U42(tt, V1, V2) → A__ISNATILISTKIND(V2)
A__U43(tt, V1, V2) → A__U44(a__isNatIListKind(V2), V1, V2)
A__U43(tt, V1, V2) → A__ISNATILISTKIND(V2)
A__U44(tt, V1, V2) → A__U45(a__isNat(V1), V2)
A__U44(tt, V1, V2) → A__ISNAT(V1)
A__U45(tt, V2) → A__U46(a__isNatIList(V2))
A__U45(tt, V2) → A__ISNATILIST(V2)
A__U51(tt, V2) → A__U52(a__isNatIListKind(V2))
A__U51(tt, V2) → A__ISNATILISTKIND(V2)
A__U81(tt, V1, V2) → A__U82(a__isNatKind(V1), V1, V2)
A__U81(tt, V1, V2) → A__ISNATKIND(V1)
A__U82(tt, V1, V2) → A__U83(a__isNatIListKind(V2), V1, V2)
A__U82(tt, V1, V2) → A__ISNATILISTKIND(V2)
A__U83(tt, V1, V2) → A__U84(a__isNatIListKind(V2), V1, V2)
A__U83(tt, V1, V2) → A__ISNATILISTKIND(V2)
A__U84(tt, V1, V2) → A__U85(a__isNat(V1), V2)
A__U84(tt, V1, V2) → A__ISNAT(V1)
A__U85(tt, V2) → A__U86(a__isNatList(V2))
A__U85(tt, V2) → A__ISNATLIST(V2)
A__U91(tt, L, N) → A__U92(a__isNatIListKind(L), L, N)
A__U91(tt, L, N) → A__ISNATILISTKIND(L)
A__U92(tt, L, N) → A__U93(a__isNat(N), L, N)
A__U92(tt, L, N) → A__ISNAT(N)
A__U93(tt, L, N) → A__U94(a__isNatKind(N), L)
A__U93(tt, L, N) → A__ISNATKIND(N)
A__U94(tt, L) → A__LENGTH(mark(L))
A__U94(tt, L) → MARK(L)
A__ISNAT(length(V1)) → A__U11(a__isNatIListKind(V1), V1)
A__ISNAT(length(V1)) → A__ISNATILISTKIND(V1)
A__ISNAT(s(V1)) → A__U21(a__isNatKind(V1), V1)
A__ISNAT(s(V1)) → A__ISNATKIND(V1)
A__ISNATILIST(V) → A__U31(a__isNatIListKind(V), V)
A__ISNATILIST(V) → A__ISNATILISTKIND(V)
A__ISNATILIST(cons(V1, V2)) → A__U41(a__isNatKind(V1), V1, V2)
A__ISNATILIST(cons(V1, V2)) → A__ISNATKIND(V1)
A__ISNATILISTKIND(cons(V1, V2)) → A__U51(a__isNatKind(V1), V2)
A__ISNATILISTKIND(cons(V1, V2)) → A__ISNATKIND(V1)
A__ISNATKIND(length(V1)) → A__U61(a__isNatIListKind(V1))
A__ISNATKIND(length(V1)) → A__ISNATILISTKIND(V1)
A__ISNATKIND(s(V1)) → A__U71(a__isNatKind(V1))
A__ISNATKIND(s(V1)) → A__ISNATKIND(V1)
A__ISNATLIST(cons(V1, V2)) → A__U81(a__isNatKind(V1), V1, V2)
A__ISNATLIST(cons(V1, V2)) → A__ISNATKIND(V1)
A__LENGTH(cons(N, L)) → A__U91(a__isNatList(L), L, N)
A__LENGTH(cons(N, L)) → A__ISNATLIST(L)
MARK(zeros) → A__ZEROS
MARK(U11(X1, X2)) → A__U11(mark(X1), X2)
MARK(U11(X1, X2)) → MARK(X1)
MARK(U12(X1, X2)) → A__U12(mark(X1), X2)
MARK(U12(X1, X2)) → MARK(X1)
MARK(isNatIListKind(X)) → A__ISNATILISTKIND(X)
MARK(U13(X)) → A__U13(mark(X))
MARK(U13(X)) → MARK(X)
MARK(isNatList(X)) → A__ISNATLIST(X)
MARK(U21(X1, X2)) → A__U21(mark(X1), X2)
MARK(U21(X1, X2)) → MARK(X1)
MARK(U22(X1, X2)) → A__U22(mark(X1), X2)
MARK(U22(X1, X2)) → MARK(X1)
MARK(isNatKind(X)) → A__ISNATKIND(X)
MARK(U23(X)) → A__U23(mark(X))
MARK(U23(X)) → MARK(X)
MARK(isNat(X)) → A__ISNAT(X)
MARK(U31(X1, X2)) → A__U31(mark(X1), X2)
MARK(U31(X1, X2)) → MARK(X1)
MARK(U32(X1, X2)) → A__U32(mark(X1), X2)
MARK(U32(X1, X2)) → MARK(X1)
MARK(U33(X)) → A__U33(mark(X))
MARK(U33(X)) → MARK(X)
MARK(U41(X1, X2, X3)) → A__U41(mark(X1), X2, X3)
MARK(U41(X1, X2, X3)) → MARK(X1)
MARK(U42(X1, X2, X3)) → A__U42(mark(X1), X2, X3)
MARK(U42(X1, X2, X3)) → MARK(X1)
MARK(U43(X1, X2, X3)) → A__U43(mark(X1), X2, X3)
MARK(U43(X1, X2, X3)) → MARK(X1)
MARK(U44(X1, X2, X3)) → A__U44(mark(X1), X2, X3)
MARK(U44(X1, X2, X3)) → MARK(X1)
MARK(U45(X1, X2)) → A__U45(mark(X1), X2)
MARK(U45(X1, X2)) → MARK(X1)
MARK(U46(X)) → A__U46(mark(X))
MARK(U46(X)) → MARK(X)
MARK(isNatIList(X)) → A__ISNATILIST(X)
MARK(U51(X1, X2)) → A__U51(mark(X1), X2)
MARK(U51(X1, X2)) → MARK(X1)
MARK(U52(X)) → A__U52(mark(X))
MARK(U52(X)) → MARK(X)
MARK(U61(X)) → A__U61(mark(X))
MARK(U61(X)) → MARK(X)
MARK(U71(X)) → A__U71(mark(X))
MARK(U71(X)) → MARK(X)
MARK(U81(X1, X2, X3)) → A__U81(mark(X1), X2, X3)
MARK(U81(X1, X2, X3)) → MARK(X1)
MARK(U82(X1, X2, X3)) → A__U82(mark(X1), X2, X3)
MARK(U82(X1, X2, X3)) → MARK(X1)
MARK(U83(X1, X2, X3)) → A__U83(mark(X1), X2, X3)
MARK(U83(X1, X2, X3)) → MARK(X1)
MARK(U84(X1, X2, X3)) → A__U84(mark(X1), X2, X3)
MARK(U84(X1, X2, X3)) → MARK(X1)
MARK(U85(X1, X2)) → A__U85(mark(X1), X2)
MARK(U85(X1, X2)) → MARK(X1)
MARK(U86(X)) → A__U86(mark(X))
MARK(U86(X)) → MARK(X)
MARK(U91(X1, X2, X3)) → A__U91(mark(X1), X2, X3)
MARK(U91(X1, X2, X3)) → MARK(X1)
MARK(U92(X1, X2, X3)) → A__U92(mark(X1), X2, X3)
MARK(U92(X1, X2, X3)) → MARK(X1)
MARK(U93(X1, X2, X3)) → A__U93(mark(X1), X2, X3)
MARK(U93(X1, X2, X3)) → MARK(X1)
MARK(U94(X1, X2)) → A__U94(mark(X1), X2)
MARK(U94(X1, X2)) → MARK(X1)
MARK(length(X)) → A__LENGTH(mark(X))
MARK(length(X)) → MARK(X)
MARK(cons(X1, X2)) → MARK(X1)
MARK(s(X)) → MARK(X)

The TRS R consists of the following rules:

a__zeroscons(0, zeros)
a__U11(tt, V1) → a__U12(a__isNatIListKind(V1), V1)
a__U12(tt, V1) → a__U13(a__isNatList(V1))
a__U13(tt) → tt
a__U21(tt, V1) → a__U22(a__isNatKind(V1), V1)
a__U22(tt, V1) → a__U23(a__isNat(V1))
a__U23(tt) → tt
a__U31(tt, V) → a__U32(a__isNatIListKind(V), V)
a__U32(tt, V) → a__U33(a__isNatList(V))
a__U33(tt) → tt
a__U41(tt, V1, V2) → a__U42(a__isNatKind(V1), V1, V2)
a__U42(tt, V1, V2) → a__U43(a__isNatIListKind(V2), V1, V2)
a__U43(tt, V1, V2) → a__U44(a__isNatIListKind(V2), V1, V2)
a__U44(tt, V1, V2) → a__U45(a__isNat(V1), V2)
a__U45(tt, V2) → a__U46(a__isNatIList(V2))
a__U46(tt) → tt
a__U51(tt, V2) → a__U52(a__isNatIListKind(V2))
a__U52(tt) → tt
a__U61(tt) → tt
a__U71(tt) → tt
a__U81(tt, V1, V2) → a__U82(a__isNatKind(V1), V1, V2)
a__U82(tt, V1, V2) → a__U83(a__isNatIListKind(V2), V1, V2)
a__U83(tt, V1, V2) → a__U84(a__isNatIListKind(V2), V1, V2)
a__U84(tt, V1, V2) → a__U85(a__isNat(V1), V2)
a__U85(tt, V2) → a__U86(a__isNatList(V2))
a__U86(tt) → tt
a__U91(tt, L, N) → a__U92(a__isNatIListKind(L), L, N)
a__U92(tt, L, N) → a__U93(a__isNat(N), L, N)
a__U93(tt, L, N) → a__U94(a__isNatKind(N), L)
a__U94(tt, L) → s(a__length(mark(L)))
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__isNatKind(V1), V1, V2)
a__isNatIListKind(nil) → tt
a__isNatIListKind(zeros) → tt
a__isNatIListKind(cons(V1, V2)) → a__U51(a__isNatKind(V1), V2)
a__isNatKind(0) → tt
a__isNatKind(length(V1)) → a__U61(a__isNatIListKind(V1))
a__isNatKind(s(V1)) → a__U71(a__isNatKind(V1))
a__isNatList(nil) → tt
a__isNatList(cons(V1, V2)) → a__U81(a__isNatKind(V1), V1, V2)
a__length(nil) → 0
a__length(cons(N, L)) → a__U91(a__isNatList(L), L, N)
mark(zeros) → a__zeros
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(U12(X1, X2)) → a__U12(mark(X1), X2)
mark(isNatIListKind(X)) → a__isNatIListKind(X)
mark(U13(X)) → a__U13(mark(X))
mark(isNatList(X)) → a__isNatList(X)
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(U22(X1, X2)) → a__U22(mark(X1), X2)
mark(isNatKind(X)) → a__isNatKind(X)
mark(U23(X)) → a__U23(mark(X))
mark(isNat(X)) → a__isNat(X)
mark(U31(X1, X2)) → a__U31(mark(X1), X2)
mark(U32(X1, X2)) → a__U32(mark(X1), X2)
mark(U33(X)) → a__U33(mark(X))
mark(U41(X1, X2, X3)) → a__U41(mark(X1), X2, X3)
mark(U42(X1, X2, X3)) → a__U42(mark(X1), X2, X3)
mark(U43(X1, X2, X3)) → a__U43(mark(X1), X2, X3)
mark(U44(X1, X2, X3)) → a__U44(mark(X1), X2, X3)
mark(U45(X1, X2)) → a__U45(mark(X1), X2)
mark(U46(X)) → a__U46(mark(X))
mark(isNatIList(X)) → a__isNatIList(X)
mark(U51(X1, X2)) → a__U51(mark(X1), X2)
mark(U52(X)) → a__U52(mark(X))
mark(U61(X)) → a__U61(mark(X))
mark(U71(X)) → a__U71(mark(X))
mark(U81(X1, X2, X3)) → a__U81(mark(X1), X2, X3)
mark(U82(X1, X2, X3)) → a__U82(mark(X1), X2, X3)
mark(U83(X1, X2, X3)) → a__U83(mark(X1), X2, X3)
mark(U84(X1, X2, X3)) → a__U84(mark(X1), X2, X3)
mark(U85(X1, X2)) → a__U85(mark(X1), X2)
mark(U86(X)) → a__U86(mark(X))
mark(U91(X1, X2, X3)) → a__U91(mark(X1), X2, X3)
mark(U92(X1, X2, X3)) → a__U92(mark(X1), X2, X3)
mark(U93(X1, X2, X3)) → a__U93(mark(X1), X2, X3)
mark(U94(X1, X2)) → a__U94(mark(X1), X2)
mark(length(X)) → a__length(mark(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__zeroszeros
a__U11(X1, X2) → U11(X1, X2)
a__U12(X1, X2) → U12(X1, X2)
a__isNatIListKind(X) → isNatIListKind(X)
a__U13(X) → U13(X)
a__isNatList(X) → isNatList(X)
a__U21(X1, X2) → U21(X1, X2)
a__U22(X1, X2) → U22(X1, X2)
a__isNatKind(X) → isNatKind(X)
a__U23(X) → U23(X)
a__isNat(X) → isNat(X)
a__U31(X1, X2) → U31(X1, X2)
a__U32(X1, X2) → U32(X1, X2)
a__U33(X) → U33(X)
a__U41(X1, X2, X3) → U41(X1, X2, X3)
a__U42(X1, X2, X3) → U42(X1, X2, X3)
a__U43(X1, X2, X3) → U43(X1, X2, X3)
a__U44(X1, X2, X3) → U44(X1, X2, X3)
a__U45(X1, X2) → U45(X1, X2)
a__U46(X) → U46(X)
a__isNatIList(X) → isNatIList(X)
a__U51(X1, X2) → U51(X1, X2)
a__U52(X) → U52(X)
a__U61(X) → U61(X)
a__U71(X) → U71(X)
a__U81(X1, X2, X3) → U81(X1, X2, X3)
a__U82(X1, X2, X3) → U82(X1, X2, X3)
a__U83(X1, X2, X3) → U83(X1, X2, X3)
a__U84(X1, X2, X3) → U84(X1, X2, X3)
a__U85(X1, X2) → U85(X1, X2)
a__U86(X) → U86(X)
a__U91(X1, X2, X3) → U91(X1, X2, X3)
a__U92(X1, X2, X3) → U92(X1, X2, X3)
a__U93(X1, X2, X3) → U93(X1, X2, X3)
a__U94(X1, X2) → U94(X1, X2)
a__length(X) → length(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(3) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 4 SCCs with 61 less nodes.

(4) Complex Obligation (AND)

(5) Obligation:

Q DP problem:
The TRS P consists of the following rules:

A__U51(tt, V2) → A__ISNATILISTKIND(V2)
A__ISNATILISTKIND(cons(V1, V2)) → A__U51(a__isNatKind(V1), V2)
A__ISNATILISTKIND(cons(V1, V2)) → A__ISNATKIND(V1)
A__ISNATKIND(length(V1)) → A__ISNATILISTKIND(V1)
A__ISNATKIND(s(V1)) → A__ISNATKIND(V1)

The TRS R consists of the following rules:

a__zeroscons(0, zeros)
a__U11(tt, V1) → a__U12(a__isNatIListKind(V1), V1)
a__U12(tt, V1) → a__U13(a__isNatList(V1))
a__U13(tt) → tt
a__U21(tt, V1) → a__U22(a__isNatKind(V1), V1)
a__U22(tt, V1) → a__U23(a__isNat(V1))
a__U23(tt) → tt
a__U31(tt, V) → a__U32(a__isNatIListKind(V), V)
a__U32(tt, V) → a__U33(a__isNatList(V))
a__U33(tt) → tt
a__U41(tt, V1, V2) → a__U42(a__isNatKind(V1), V1, V2)
a__U42(tt, V1, V2) → a__U43(a__isNatIListKind(V2), V1, V2)
a__U43(tt, V1, V2) → a__U44(a__isNatIListKind(V2), V1, V2)
a__U44(tt, V1, V2) → a__U45(a__isNat(V1), V2)
a__U45(tt, V2) → a__U46(a__isNatIList(V2))
a__U46(tt) → tt
a__U51(tt, V2) → a__U52(a__isNatIListKind(V2))
a__U52(tt) → tt
a__U61(tt) → tt
a__U71(tt) → tt
a__U81(tt, V1, V2) → a__U82(a__isNatKind(V1), V1, V2)
a__U82(tt, V1, V2) → a__U83(a__isNatIListKind(V2), V1, V2)
a__U83(tt, V1, V2) → a__U84(a__isNatIListKind(V2), V1, V2)
a__U84(tt, V1, V2) → a__U85(a__isNat(V1), V2)
a__U85(tt, V2) → a__U86(a__isNatList(V2))
a__U86(tt) → tt
a__U91(tt, L, N) → a__U92(a__isNatIListKind(L), L, N)
a__U92(tt, L, N) → a__U93(a__isNat(N), L, N)
a__U93(tt, L, N) → a__U94(a__isNatKind(N), L)
a__U94(tt, L) → s(a__length(mark(L)))
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__isNatKind(V1), V1, V2)
a__isNatIListKind(nil) → tt
a__isNatIListKind(zeros) → tt
a__isNatIListKind(cons(V1, V2)) → a__U51(a__isNatKind(V1), V2)
a__isNatKind(0) → tt
a__isNatKind(length(V1)) → a__U61(a__isNatIListKind(V1))
a__isNatKind(s(V1)) → a__U71(a__isNatKind(V1))
a__isNatList(nil) → tt
a__isNatList(cons(V1, V2)) → a__U81(a__isNatKind(V1), V1, V2)
a__length(nil) → 0
a__length(cons(N, L)) → a__U91(a__isNatList(L), L, N)
mark(zeros) → a__zeros
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(U12(X1, X2)) → a__U12(mark(X1), X2)
mark(isNatIListKind(X)) → a__isNatIListKind(X)
mark(U13(X)) → a__U13(mark(X))
mark(isNatList(X)) → a__isNatList(X)
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(U22(X1, X2)) → a__U22(mark(X1), X2)
mark(isNatKind(X)) → a__isNatKind(X)
mark(U23(X)) → a__U23(mark(X))
mark(isNat(X)) → a__isNat(X)
mark(U31(X1, X2)) → a__U31(mark(X1), X2)
mark(U32(X1, X2)) → a__U32(mark(X1), X2)
mark(U33(X)) → a__U33(mark(X))
mark(U41(X1, X2, X3)) → a__U41(mark(X1), X2, X3)
mark(U42(X1, X2, X3)) → a__U42(mark(X1), X2, X3)
mark(U43(X1, X2, X3)) → a__U43(mark(X1), X2, X3)
mark(U44(X1, X2, X3)) → a__U44(mark(X1), X2, X3)
mark(U45(X1, X2)) → a__U45(mark(X1), X2)
mark(U46(X)) → a__U46(mark(X))
mark(isNatIList(X)) → a__isNatIList(X)
mark(U51(X1, X2)) → a__U51(mark(X1), X2)
mark(U52(X)) → a__U52(mark(X))
mark(U61(X)) → a__U61(mark(X))
mark(U71(X)) → a__U71(mark(X))
mark(U81(X1, X2, X3)) → a__U81(mark(X1), X2, X3)
mark(U82(X1, X2, X3)) → a__U82(mark(X1), X2, X3)
mark(U83(X1, X2, X3)) → a__U83(mark(X1), X2, X3)
mark(U84(X1, X2, X3)) → a__U84(mark(X1), X2, X3)
mark(U85(X1, X2)) → a__U85(mark(X1), X2)
mark(U86(X)) → a__U86(mark(X))
mark(U91(X1, X2, X3)) → a__U91(mark(X1), X2, X3)
mark(U92(X1, X2, X3)) → a__U92(mark(X1), X2, X3)
mark(U93(X1, X2, X3)) → a__U93(mark(X1), X2, X3)
mark(U94(X1, X2)) → a__U94(mark(X1), X2)
mark(length(X)) → a__length(mark(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__zeroszeros
a__U11(X1, X2) → U11(X1, X2)
a__U12(X1, X2) → U12(X1, X2)
a__isNatIListKind(X) → isNatIListKind(X)
a__U13(X) → U13(X)
a__isNatList(X) → isNatList(X)
a__U21(X1, X2) → U21(X1, X2)
a__U22(X1, X2) → U22(X1, X2)
a__isNatKind(X) → isNatKind(X)
a__U23(X) → U23(X)
a__isNat(X) → isNat(X)
a__U31(X1, X2) → U31(X1, X2)
a__U32(X1, X2) → U32(X1, X2)
a__U33(X) → U33(X)
a__U41(X1, X2, X3) → U41(X1, X2, X3)
a__U42(X1, X2, X3) → U42(X1, X2, X3)
a__U43(X1, X2, X3) → U43(X1, X2, X3)
a__U44(X1, X2, X3) → U44(X1, X2, X3)
a__U45(X1, X2) → U45(X1, X2)
a__U46(X) → U46(X)
a__isNatIList(X) → isNatIList(X)
a__U51(X1, X2) → U51(X1, X2)
a__U52(X) → U52(X)
a__U61(X) → U61(X)
a__U71(X) → U71(X)
a__U81(X1, X2, X3) → U81(X1, X2, X3)
a__U82(X1, X2, X3) → U82(X1, X2, X3)
a__U83(X1, X2, X3) → U83(X1, X2, X3)
a__U84(X1, X2, X3) → U84(X1, X2, X3)
a__U85(X1, X2) → U85(X1, X2)
a__U86(X) → U86(X)
a__U91(X1, X2, X3) → U91(X1, X2, X3)
a__U92(X1, X2, X3) → U92(X1, X2, X3)
a__U93(X1, X2, X3) → U93(X1, X2, X3)
a__U94(X1, X2) → U94(X1, X2)
a__length(X) → length(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(6) Obligation:

Q DP problem:
The TRS P consists of the following rules:

A__U12(tt, V1) → A__ISNATLIST(V1)
A__ISNATLIST(cons(V1, V2)) → A__U81(a__isNatKind(V1), V1, V2)
A__U81(tt, V1, V2) → A__U82(a__isNatKind(V1), V1, V2)
A__U82(tt, V1, V2) → A__U83(a__isNatIListKind(V2), V1, V2)
A__U83(tt, V1, V2) → A__U84(a__isNatIListKind(V2), V1, V2)
A__U84(tt, V1, V2) → A__U85(a__isNat(V1), V2)
A__U85(tt, V2) → A__ISNATLIST(V2)
A__U84(tt, V1, V2) → A__ISNAT(V1)
A__ISNAT(length(V1)) → A__U11(a__isNatIListKind(V1), V1)
A__U11(tt, V1) → A__U12(a__isNatIListKind(V1), V1)
A__ISNAT(s(V1)) → A__U21(a__isNatKind(V1), V1)
A__U21(tt, V1) → A__U22(a__isNatKind(V1), V1)
A__U22(tt, V1) → A__ISNAT(V1)

The TRS R consists of the following rules:

a__zeroscons(0, zeros)
a__U11(tt, V1) → a__U12(a__isNatIListKind(V1), V1)
a__U12(tt, V1) → a__U13(a__isNatList(V1))
a__U13(tt) → tt
a__U21(tt, V1) → a__U22(a__isNatKind(V1), V1)
a__U22(tt, V1) → a__U23(a__isNat(V1))
a__U23(tt) → tt
a__U31(tt, V) → a__U32(a__isNatIListKind(V), V)
a__U32(tt, V) → a__U33(a__isNatList(V))
a__U33(tt) → tt
a__U41(tt, V1, V2) → a__U42(a__isNatKind(V1), V1, V2)
a__U42(tt, V1, V2) → a__U43(a__isNatIListKind(V2), V1, V2)
a__U43(tt, V1, V2) → a__U44(a__isNatIListKind(V2), V1, V2)
a__U44(tt, V1, V2) → a__U45(a__isNat(V1), V2)
a__U45(tt, V2) → a__U46(a__isNatIList(V2))
a__U46(tt) → tt
a__U51(tt, V2) → a__U52(a__isNatIListKind(V2))
a__U52(tt) → tt
a__U61(tt) → tt
a__U71(tt) → tt
a__U81(tt, V1, V2) → a__U82(a__isNatKind(V1), V1, V2)
a__U82(tt, V1, V2) → a__U83(a__isNatIListKind(V2), V1, V2)
a__U83(tt, V1, V2) → a__U84(a__isNatIListKind(V2), V1, V2)
a__U84(tt, V1, V2) → a__U85(a__isNat(V1), V2)
a__U85(tt, V2) → a__U86(a__isNatList(V2))
a__U86(tt) → tt
a__U91(tt, L, N) → a__U92(a__isNatIListKind(L), L, N)
a__U92(tt, L, N) → a__U93(a__isNat(N), L, N)
a__U93(tt, L, N) → a__U94(a__isNatKind(N), L)
a__U94(tt, L) → s(a__length(mark(L)))
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__isNatKind(V1), V1, V2)
a__isNatIListKind(nil) → tt
a__isNatIListKind(zeros) → tt
a__isNatIListKind(cons(V1, V2)) → a__U51(a__isNatKind(V1), V2)
a__isNatKind(0) → tt
a__isNatKind(length(V1)) → a__U61(a__isNatIListKind(V1))
a__isNatKind(s(V1)) → a__U71(a__isNatKind(V1))
a__isNatList(nil) → tt
a__isNatList(cons(V1, V2)) → a__U81(a__isNatKind(V1), V1, V2)
a__length(nil) → 0
a__length(cons(N, L)) → a__U91(a__isNatList(L), L, N)
mark(zeros) → a__zeros
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(U12(X1, X2)) → a__U12(mark(X1), X2)
mark(isNatIListKind(X)) → a__isNatIListKind(X)
mark(U13(X)) → a__U13(mark(X))
mark(isNatList(X)) → a__isNatList(X)
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(U22(X1, X2)) → a__U22(mark(X1), X2)
mark(isNatKind(X)) → a__isNatKind(X)
mark(U23(X)) → a__U23(mark(X))
mark(isNat(X)) → a__isNat(X)
mark(U31(X1, X2)) → a__U31(mark(X1), X2)
mark(U32(X1, X2)) → a__U32(mark(X1), X2)
mark(U33(X)) → a__U33(mark(X))
mark(U41(X1, X2, X3)) → a__U41(mark(X1), X2, X3)
mark(U42(X1, X2, X3)) → a__U42(mark(X1), X2, X3)
mark(U43(X1, X2, X3)) → a__U43(mark(X1), X2, X3)
mark(U44(X1, X2, X3)) → a__U44(mark(X1), X2, X3)
mark(U45(X1, X2)) → a__U45(mark(X1), X2)
mark(U46(X)) → a__U46(mark(X))
mark(isNatIList(X)) → a__isNatIList(X)
mark(U51(X1, X2)) → a__U51(mark(X1), X2)
mark(U52(X)) → a__U52(mark(X))
mark(U61(X)) → a__U61(mark(X))
mark(U71(X)) → a__U71(mark(X))
mark(U81(X1, X2, X3)) → a__U81(mark(X1), X2, X3)
mark(U82(X1, X2, X3)) → a__U82(mark(X1), X2, X3)
mark(U83(X1, X2, X3)) → a__U83(mark(X1), X2, X3)
mark(U84(X1, X2, X3)) → a__U84(mark(X1), X2, X3)
mark(U85(X1, X2)) → a__U85(mark(X1), X2)
mark(U86(X)) → a__U86(mark(X))
mark(U91(X1, X2, X3)) → a__U91(mark(X1), X2, X3)
mark(U92(X1, X2, X3)) → a__U92(mark(X1), X2, X3)
mark(U93(X1, X2, X3)) → a__U93(mark(X1), X2, X3)
mark(U94(X1, X2)) → a__U94(mark(X1), X2)
mark(length(X)) → a__length(mark(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__zeroszeros
a__U11(X1, X2) → U11(X1, X2)
a__U12(X1, X2) → U12(X1, X2)
a__isNatIListKind(X) → isNatIListKind(X)
a__U13(X) → U13(X)
a__isNatList(X) → isNatList(X)
a__U21(X1, X2) → U21(X1, X2)
a__U22(X1, X2) → U22(X1, X2)
a__isNatKind(X) → isNatKind(X)
a__U23(X) → U23(X)
a__isNat(X) → isNat(X)
a__U31(X1, X2) → U31(X1, X2)
a__U32(X1, X2) → U32(X1, X2)
a__U33(X) → U33(X)
a__U41(X1, X2, X3) → U41(X1, X2, X3)
a__U42(X1, X2, X3) → U42(X1, X2, X3)
a__U43(X1, X2, X3) → U43(X1, X2, X3)
a__U44(X1, X2, X3) → U44(X1, X2, X3)
a__U45(X1, X2) → U45(X1, X2)
a__U46(X) → U46(X)
a__isNatIList(X) → isNatIList(X)
a__U51(X1, X2) → U51(X1, X2)
a__U52(X) → U52(X)
a__U61(X) → U61(X)
a__U71(X) → U71(X)
a__U81(X1, X2, X3) → U81(X1, X2, X3)
a__U82(X1, X2, X3) → U82(X1, X2, X3)
a__U83(X1, X2, X3) → U83(X1, X2, X3)
a__U84(X1, X2, X3) → U84(X1, X2, X3)
a__U85(X1, X2) → U85(X1, X2)
a__U86(X) → U86(X)
a__U91(X1, X2, X3) → U91(X1, X2, X3)
a__U92(X1, X2, X3) → U92(X1, X2, X3)
a__U93(X1, X2, X3) → U93(X1, X2, X3)
a__U94(X1, X2) → U94(X1, X2)
a__length(X) → length(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(7) Obligation:

Q DP problem:
The TRS P consists of the following rules:

A__U44(tt, V1, V2) → A__U45(a__isNat(V1), V2)
A__U45(tt, V2) → A__ISNATILIST(V2)
A__ISNATILIST(cons(V1, V2)) → A__U41(a__isNatKind(V1), V1, V2)
A__U41(tt, V1, V2) → A__U42(a__isNatKind(V1), V1, V2)
A__U42(tt, V1, V2) → A__U43(a__isNatIListKind(V2), V1, V2)
A__U43(tt, V1, V2) → A__U44(a__isNatIListKind(V2), V1, V2)

The TRS R consists of the following rules:

a__zeroscons(0, zeros)
a__U11(tt, V1) → a__U12(a__isNatIListKind(V1), V1)
a__U12(tt, V1) → a__U13(a__isNatList(V1))
a__U13(tt) → tt
a__U21(tt, V1) → a__U22(a__isNatKind(V1), V1)
a__U22(tt, V1) → a__U23(a__isNat(V1))
a__U23(tt) → tt
a__U31(tt, V) → a__U32(a__isNatIListKind(V), V)
a__U32(tt, V) → a__U33(a__isNatList(V))
a__U33(tt) → tt
a__U41(tt, V1, V2) → a__U42(a__isNatKind(V1), V1, V2)
a__U42(tt, V1, V2) → a__U43(a__isNatIListKind(V2), V1, V2)
a__U43(tt, V1, V2) → a__U44(a__isNatIListKind(V2), V1, V2)
a__U44(tt, V1, V2) → a__U45(a__isNat(V1), V2)
a__U45(tt, V2) → a__U46(a__isNatIList(V2))
a__U46(tt) → tt
a__U51(tt, V2) → a__U52(a__isNatIListKind(V2))
a__U52(tt) → tt
a__U61(tt) → tt
a__U71(tt) → tt
a__U81(tt, V1, V2) → a__U82(a__isNatKind(V1), V1, V2)
a__U82(tt, V1, V2) → a__U83(a__isNatIListKind(V2), V1, V2)
a__U83(tt, V1, V2) → a__U84(a__isNatIListKind(V2), V1, V2)
a__U84(tt, V1, V2) → a__U85(a__isNat(V1), V2)
a__U85(tt, V2) → a__U86(a__isNatList(V2))
a__U86(tt) → tt
a__U91(tt, L, N) → a__U92(a__isNatIListKind(L), L, N)
a__U92(tt, L, N) → a__U93(a__isNat(N), L, N)
a__U93(tt, L, N) → a__U94(a__isNatKind(N), L)
a__U94(tt, L) → s(a__length(mark(L)))
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__isNatKind(V1), V1, V2)
a__isNatIListKind(nil) → tt
a__isNatIListKind(zeros) → tt
a__isNatIListKind(cons(V1, V2)) → a__U51(a__isNatKind(V1), V2)
a__isNatKind(0) → tt
a__isNatKind(length(V1)) → a__U61(a__isNatIListKind(V1))
a__isNatKind(s(V1)) → a__U71(a__isNatKind(V1))
a__isNatList(nil) → tt
a__isNatList(cons(V1, V2)) → a__U81(a__isNatKind(V1), V1, V2)
a__length(nil) → 0
a__length(cons(N, L)) → a__U91(a__isNatList(L), L, N)
mark(zeros) → a__zeros
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(U12(X1, X2)) → a__U12(mark(X1), X2)
mark(isNatIListKind(X)) → a__isNatIListKind(X)
mark(U13(X)) → a__U13(mark(X))
mark(isNatList(X)) → a__isNatList(X)
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(U22(X1, X2)) → a__U22(mark(X1), X2)
mark(isNatKind(X)) → a__isNatKind(X)
mark(U23(X)) → a__U23(mark(X))
mark(isNat(X)) → a__isNat(X)
mark(U31(X1, X2)) → a__U31(mark(X1), X2)
mark(U32(X1, X2)) → a__U32(mark(X1), X2)
mark(U33(X)) → a__U33(mark(X))
mark(U41(X1, X2, X3)) → a__U41(mark(X1), X2, X3)
mark(U42(X1, X2, X3)) → a__U42(mark(X1), X2, X3)
mark(U43(X1, X2, X3)) → a__U43(mark(X1), X2, X3)
mark(U44(X1, X2, X3)) → a__U44(mark(X1), X2, X3)
mark(U45(X1, X2)) → a__U45(mark(X1), X2)
mark(U46(X)) → a__U46(mark(X))
mark(isNatIList(X)) → a__isNatIList(X)
mark(U51(X1, X2)) → a__U51(mark(X1), X2)
mark(U52(X)) → a__U52(mark(X))
mark(U61(X)) → a__U61(mark(X))
mark(U71(X)) → a__U71(mark(X))
mark(U81(X1, X2, X3)) → a__U81(mark(X1), X2, X3)
mark(U82(X1, X2, X3)) → a__U82(mark(X1), X2, X3)
mark(U83(X1, X2, X3)) → a__U83(mark(X1), X2, X3)
mark(U84(X1, X2, X3)) → a__U84(mark(X1), X2, X3)
mark(U85(X1, X2)) → a__U85(mark(X1), X2)
mark(U86(X)) → a__U86(mark(X))
mark(U91(X1, X2, X3)) → a__U91(mark(X1), X2, X3)
mark(U92(X1, X2, X3)) → a__U92(mark(X1), X2, X3)
mark(U93(X1, X2, X3)) → a__U93(mark(X1), X2, X3)
mark(U94(X1, X2)) → a__U94(mark(X1), X2)
mark(length(X)) → a__length(mark(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__zeroszeros
a__U11(X1, X2) → U11(X1, X2)
a__U12(X1, X2) → U12(X1, X2)
a__isNatIListKind(X) → isNatIListKind(X)
a__U13(X) → U13(X)
a__isNatList(X) → isNatList(X)
a__U21(X1, X2) → U21(X1, X2)
a__U22(X1, X2) → U22(X1, X2)
a__isNatKind(X) → isNatKind(X)
a__U23(X) → U23(X)
a__isNat(X) → isNat(X)
a__U31(X1, X2) → U31(X1, X2)
a__U32(X1, X2) → U32(X1, X2)
a__U33(X) → U33(X)
a__U41(X1, X2, X3) → U41(X1, X2, X3)
a__U42(X1, X2, X3) → U42(X1, X2, X3)
a__U43(X1, X2, X3) → U43(X1, X2, X3)
a__U44(X1, X2, X3) → U44(X1, X2, X3)
a__U45(X1, X2) → U45(X1, X2)
a__U46(X) → U46(X)
a__isNatIList(X) → isNatIList(X)
a__U51(X1, X2) → U51(X1, X2)
a__U52(X) → U52(X)
a__U61(X) → U61(X)
a__U71(X) → U71(X)
a__U81(X1, X2, X3) → U81(X1, X2, X3)
a__U82(X1, X2, X3) → U82(X1, X2, X3)
a__U83(X1, X2, X3) → U83(X1, X2, X3)
a__U84(X1, X2, X3) → U84(X1, X2, X3)
a__U85(X1, X2) → U85(X1, X2)
a__U86(X) → U86(X)
a__U91(X1, X2, X3) → U91(X1, X2, X3)
a__U92(X1, X2, X3) → U92(X1, X2, X3)
a__U93(X1, X2, X3) → U93(X1, X2, X3)
a__U94(X1, X2) → U94(X1, X2)
a__length(X) → length(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(8) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


A__ISNATILIST(cons(V1, V2)) → A__U41(a__isNatKind(V1), V1, V2)
A__U41(tt, V1, V2) → A__U42(a__isNatKind(V1), V1, V2)
A__U42(tt, V1, V2) → A__U43(a__isNatIListKind(V2), V1, V2)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
A__U44(x1, x2, x3)  =  x3
tt  =  tt
A__U45(x1, x2)  =  x2
a__isNat(x1)  =  a__isNat
A__ISNATILIST(x1)  =  x1
cons(x1, x2)  =  cons(x2)
A__U41(x1, x2, x3)  =  A__U41(x3)
a__isNatKind(x1)  =  a__isNatKind
A__U42(x1, x2, x3)  =  A__U42(x3)
A__U43(x1, x2, x3)  =  x3
a__isNatIListKind(x1)  =  a__isNatIListKind
a__zeros  =  a__zeros
0  =  0
zeros  =  zeros
a__U11(x1, x2)  =  a__U11(x1)
a__U12(x1, x2)  =  a__U12(x1)
a__U13(x1)  =  x1
a__isNatList(x1)  =  a__isNatList
a__U21(x1, x2)  =  a__U21(x1)
a__U22(x1, x2)  =  a__U22(x1)
a__U23(x1)  =  a__U23
a__U31(x1, x2)  =  a__U31
a__U32(x1, x2)  =  a__U32
a__U33(x1)  =  a__U33
a__U41(x1, x2, x3)  =  a__U41
a__U42(x1, x2, x3)  =  a__U42
a__U43(x1, x2, x3)  =  a__U43
a__U44(x1, x2, x3)  =  a__U44
a__U45(x1, x2)  =  a__U45
a__U46(x1)  =  a__U46
a__isNatIList(x1)  =  a__isNatIList
a__U51(x1, x2)  =  a__U51(x1)
a__U52(x1)  =  a__U52
a__U61(x1)  =  a__U61
a__U71(x1)  =  x1
a__U81(x1, x2, x3)  =  a__U81
a__U82(x1, x2, x3)  =  a__U82
a__U83(x1, x2, x3)  =  a__U83
a__U84(x1, x2, x3)  =  a__U84(x1)
a__U85(x1, x2)  =  a__U85
a__U86(x1)  =  a__U86
a__U91(x1, x2, x3)  =  a__U91
a__U92(x1, x2, x3)  =  a__U92(x1)
a__U93(x1, x2, x3)  =  a__U93
a__U94(x1, x2)  =  a__U94
s(x1)  =  s
a__length(x1)  =  a__length(x1)
mark(x1)  =  mark(x1)
length(x1)  =  length(x1)
nil  =  nil
U11(x1, x2)  =  U11(x1)
U12(x1, x2)  =  U12(x1)
isNatIListKind(x1)  =  isNatIListKind
U13(x1)  =  x1
isNatList(x1)  =  isNatList
U21(x1, x2)  =  U21(x1)
U22(x1, x2)  =  U22(x1)
isNatKind(x1)  =  isNatKind
U23(x1)  =  U23
isNat(x1)  =  isNat
U31(x1, x2)  =  U31
U32(x1, x2)  =  U32
U33(x1)  =  U33
U41(x1, x2, x3)  =  U41
U42(x1, x2, x3)  =  U42
U43(x1, x2, x3)  =  U43
U44(x1, x2, x3)  =  U44
U45(x1, x2)  =  U45
U46(x1)  =  U46
isNatIList(x1)  =  isNatIList
U51(x1, x2)  =  U51(x1)
U52(x1)  =  U52
U61(x1)  =  U61
U71(x1)  =  x1
U81(x1, x2, x3)  =  U81
U82(x1, x2, x3)  =  U82
U83(x1, x2, x3)  =  U83
U84(x1, x2, x3)  =  U84(x1)
U85(x1, x2)  =  U85
U86(x1)  =  U86
U91(x1, x2, x3)  =  U91
U92(x1, x2, x3)  =  U92(x1)
U93(x1, x2, x3)  =  U93
U94(x1, x2)  =  U94

Recursive path order with status [RPO].
Precedence:
mark1 > azeros > cons1 > AU411 > AU421 > U111
mark1 > azeros > 0 > U111
mark1 > azeros > zeros > tt > AU421 > U111
mark1 > aisNatIList > aU31 > aU32 > aisNatList > aU81 > aU82 > aU83 > aisNatIListKind > aisNatKind > aU61 > tt > AU421 > U111
mark1 > aisNatIList > aU31 > aU32 > aisNatList > aU81 > aU82 > aU83 > aisNatIListKind > aisNatKind > aU61 > U61 > U111
mark1 > aisNatIList > aU31 > aU32 > aisNatList > aU81 > aU82 > aU83 > aisNatIListKind > aisNatKind > isNatKind > U111
mark1 > aisNatIList > aU31 > aU32 > aisNatList > aU81 > aU82 > aU83 > aisNatIListKind > aU511 > aU52 > tt > AU421 > U111
mark1 > aisNatIList > aU31 > aU32 > aisNatList > aU81 > aU82 > aU83 > aisNatIListKind > aU511 > aU52 > U52 > U111
mark1 > aisNatIList > aU31 > aU32 > aisNatList > aU81 > aU82 > aU83 > aisNatIListKind > aU511 > U511 > U111
mark1 > aisNatIList > aU31 > aU32 > aisNatList > aU81 > aU82 > aU83 > aisNatIListKind > isNatIListKind > U111
mark1 > aisNatIList > aU31 > aU32 > aisNatList > aU81 > aU82 > aU83 > aU841 > aU85 > aU86 > tt > AU421 > U111
mark1 > aisNatIList > aU31 > aU32 > aisNatList > aU81 > aU82 > aU83 > aU841 > aU85 > aU86 > U86 > U111
mark1 > aisNatIList > aU31 > aU32 > aisNatList > aU81 > aU82 > aU83 > aU841 > aU85 > U85 > U111
mark1 > aisNatIList > aU31 > aU32 > aisNatList > aU81 > aU82 > aU83 > aU841 > U841 > U111
mark1 > aisNatIList > aU31 > aU32 > aisNatList > aU81 > aU82 > aU83 > U83 > U111
mark1 > aisNatIList > aU31 > aU32 > aisNatList > aU81 > aU82 > U82 > U111
mark1 > aisNatIList > aU31 > aU32 > aisNatList > aU81 > U81 > U111
mark1 > aisNatIList > aU31 > aU32 > aisNatList > isNatList > U111
mark1 > aisNatIList > aU31 > aU32 > aU33 > tt > AU421 > U111
mark1 > aisNatIList > aU31 > aU32 > aU33 > U33 > U111
mark1 > aisNatIList > aU31 > aU32 > U32 > U111
mark1 > aisNatIList > aU31 > U31 > U111
mark1 > aisNatIList > aU41 > aU42 > aU43 > aU44 > aisNat > aU111 > aU121 > aisNatList > aU81 > aU82 > aU83 > aisNatIListKind > aisNatKind > aU61 > tt > AU421 > U111
mark1 > aisNatIList > aU41 > aU42 > aU43 > aU44 > aisNat > aU111 > aU121 > aisNatList > aU81 > aU82 > aU83 > aisNatIListKind > aisNatKind > aU61 > U61 > U111
mark1 > aisNatIList > aU41 > aU42 > aU43 > aU44 > aisNat > aU111 > aU121 > aisNatList > aU81 > aU82 > aU83 > aisNatIListKind > aisNatKind > isNatKind > U111
mark1 > aisNatIList > aU41 > aU42 > aU43 > aU44 > aisNat > aU111 > aU121 > aisNatList > aU81 > aU82 > aU83 > aisNatIListKind > aU511 > aU52 > tt > AU421 > U111
mark1 > aisNatIList > aU41 > aU42 > aU43 > aU44 > aisNat > aU111 > aU121 > aisNatList > aU81 > aU82 > aU83 > aisNatIListKind > aU511 > aU52 > U52 > U111
mark1 > aisNatIList > aU41 > aU42 > aU43 > aU44 > aisNat > aU111 > aU121 > aisNatList > aU81 > aU82 > aU83 > aisNatIListKind > aU511 > U511 > U111
mark1 > aisNatIList > aU41 > aU42 > aU43 > aU44 > aisNat > aU111 > aU121 > aisNatList > aU81 > aU82 > aU83 > aisNatIListKind > isNatIListKind > U111
mark1 > aisNatIList > aU41 > aU42 > aU43 > aU44 > aisNat > aU111 > aU121 > aisNatList > aU81 > aU82 > aU83 > aU841 > aU85 > aU86 > tt > AU421 > U111
mark1 > aisNatIList > aU41 > aU42 > aU43 > aU44 > aisNat > aU111 > aU121 > aisNatList > aU81 > aU82 > aU83 > aU841 > aU85 > aU86 > U86 > U111
mark1 > aisNatIList > aU41 > aU42 > aU43 > aU44 > aisNat > aU111 > aU121 > aisNatList > aU81 > aU82 > aU83 > aU841 > aU85 > U85 > U111
mark1 > aisNatIList > aU41 > aU42 > aU43 > aU44 > aisNat > aU111 > aU121 > aisNatList > aU81 > aU82 > aU83 > aU841 > U841 > U111
mark1 > aisNatIList > aU41 > aU42 > aU43 > aU44 > aisNat > aU111 > aU121 > aisNatList > aU81 > aU82 > aU83 > U83 > U111
mark1 > aisNatIList > aU41 > aU42 > aU43 > aU44 > aisNat > aU111 > aU121 > aisNatList > aU81 > aU82 > U82 > U111
mark1 > aisNatIList > aU41 > aU42 > aU43 > aU44 > aisNat > aU111 > aU121 > aisNatList > aU81 > U81 > U111
mark1 > aisNatIList > aU41 > aU42 > aU43 > aU44 > aisNat > aU111 > aU121 > aisNatList > isNatList > U111
mark1 > aisNatIList > aU41 > aU42 > aU43 > aU44 > aisNat > aU111 > aU121 > U121 > U111
mark1 > aisNatIList > aU41 > aU42 > aU43 > aU44 > aisNat > aU211 > aisNatKind > aU61 > tt > AU421 > U111
mark1 > aisNatIList > aU41 > aU42 > aU43 > aU44 > aisNat > aU211 > aisNatKind > aU61 > U61 > U111
mark1 > aisNatIList > aU41 > aU42 > aU43 > aU44 > aisNat > aU211 > aisNatKind > isNatKind > U111
mark1 > aisNatIList > aU41 > aU42 > aU43 > aU44 > aisNat > aU211 > aU221 > aU23 > tt > AU421 > U111
mark1 > aisNatIList > aU41 > aU42 > aU43 > aU44 > aisNat > aU211 > aU221 > aU23 > U23 > U111
mark1 > aisNatIList > aU41 > aU42 > aU43 > aU44 > aisNat > aU211 > aU221 > U221 > U111
mark1 > aisNatIList > aU41 > aU42 > aU43 > aU44 > aisNat > aU211 > U211 > U111
mark1 > aisNatIList > aU41 > aU42 > aU43 > aU44 > aisNat > isNat > U111
mark1 > aisNatIList > aU41 > aU42 > aU43 > aU44 > aU45 > aU46 > tt > AU421 > U111
mark1 > aisNatIList > aU41 > aU42 > aU43 > aU44 > aU45 > aU46 > U46 > U111
mark1 > aisNatIList > aU41 > aU42 > aU43 > aU44 > aU45 > U45 > U111
mark1 > aisNatIList > aU41 > aU42 > aU43 > aU44 > U44 > U111
mark1 > aisNatIList > aU41 > aU42 > aU43 > U43 > U111
mark1 > aisNatIList > aU41 > aU42 > U42 > U111
mark1 > aisNatIList > aU41 > U41 > U111
mark1 > aisNatIList > isNatIList > U111
mark1 > alength1 > 0 > U111
mark1 > alength1 > aU91 > aU921 > aisNat > aU111 > aU121 > aisNatList > aU81 > aU82 > aU83 > aisNatIListKind > aisNatKind > aU61 > tt > AU421 > U111
mark1 > alength1 > aU91 > aU921 > aisNat > aU111 > aU121 > aisNatList > aU81 > aU82 > aU83 > aisNatIListKind > aisNatKind > aU61 > U61 > U111
mark1 > alength1 > aU91 > aU921 > aisNat > aU111 > aU121 > aisNatList > aU81 > aU82 > aU83 > aisNatIListKind > aisNatKind > isNatKind > U111
mark1 > alength1 > aU91 > aU921 > aisNat > aU111 > aU121 > aisNatList > aU81 > aU82 > aU83 > aisNatIListKind > aU511 > aU52 > tt > AU421 > U111
mark1 > alength1 > aU91 > aU921 > aisNat > aU111 > aU121 > aisNatList > aU81 > aU82 > aU83 > aisNatIListKind > aU511 > aU52 > U52 > U111
mark1 > alength1 > aU91 > aU921 > aisNat > aU111 > aU121 > aisNatList > aU81 > aU82 > aU83 > aisNatIListKind > aU511 > U511 > U111
mark1 > alength1 > aU91 > aU921 > aisNat > aU111 > aU121 > aisNatList > aU81 > aU82 > aU83 > aisNatIListKind > isNatIListKind > U111
mark1 > alength1 > aU91 > aU921 > aisNat > aU111 > aU121 > aisNatList > aU81 > aU82 > aU83 > aU841 > aU85 > aU86 > tt > AU421 > U111
mark1 > alength1 > aU91 > aU921 > aisNat > aU111 > aU121 > aisNatList > aU81 > aU82 > aU83 > aU841 > aU85 > aU86 > U86 > U111
mark1 > alength1 > aU91 > aU921 > aisNat > aU111 > aU121 > aisNatList > aU81 > aU82 > aU83 > aU841 > aU85 > U85 > U111
mark1 > alength1 > aU91 > aU921 > aisNat > aU111 > aU121 > aisNatList > aU81 > aU82 > aU83 > aU841 > U841 > U111
mark1 > alength1 > aU91 > aU921 > aisNat > aU111 > aU121 > aisNatList > aU81 > aU82 > aU83 > U83 > U111
mark1 > alength1 > aU91 > aU921 > aisNat > aU111 > aU121 > aisNatList > aU81 > aU82 > U82 > U111
mark1 > alength1 > aU91 > aU921 > aisNat > aU111 > aU121 > aisNatList > aU81 > U81 > U111
mark1 > alength1 > aU91 > aU921 > aisNat > aU111 > aU121 > aisNatList > isNatList > U111
mark1 > alength1 > aU91 > aU921 > aisNat > aU111 > aU121 > U121 > U111
mark1 > alength1 > aU91 > aU921 > aisNat > aU211 > aisNatKind > aU61 > tt > AU421 > U111
mark1 > alength1 > aU91 > aU921 > aisNat > aU211 > aisNatKind > aU61 > U61 > U111
mark1 > alength1 > aU91 > aU921 > aisNat > aU211 > aisNatKind > isNatKind > U111
mark1 > alength1 > aU91 > aU921 > aisNat > aU211 > aU221 > aU23 > tt > AU421 > U111
mark1 > alength1 > aU91 > aU921 > aisNat > aU211 > aU221 > aU23 > U23 > U111
mark1 > alength1 > aU91 > aU921 > aisNat > aU211 > aU221 > U221 > U111
mark1 > alength1 > aU91 > aU921 > aisNat > aU211 > U211 > U111
mark1 > alength1 > aU91 > aU921 > aisNat > isNat > U111
mark1 > alength1 > aU91 > aU921 > aU93 > aU94 > s > aU211 > aisNatKind > aU61 > tt > AU421 > U111
mark1 > alength1 > aU91 > aU921 > aU93 > aU94 > s > aU211 > aisNatKind > aU61 > U61 > U111
mark1 > alength1 > aU91 > aU921 > aU93 > aU94 > s > aU211 > aisNatKind > isNatKind > U111
mark1 > alength1 > aU91 > aU921 > aU93 > aU94 > s > aU211 > aU221 > aU23 > tt > AU421 > U111
mark1 > alength1 > aU91 > aU921 > aU93 > aU94 > s > aU211 > aU221 > aU23 > U23 > U111
mark1 > alength1 > aU91 > aU921 > aU93 > aU94 > s > aU211 > aU221 > U221 > U111
mark1 > alength1 > aU91 > aU921 > aU93 > aU94 > s > aU211 > U211 > U111
mark1 > alength1 > aU91 > aU921 > aU93 > aU94 > U94 > U111
mark1 > alength1 > aU91 > aU921 > aU93 > U93 > U111
mark1 > alength1 > aU91 > aU921 > U921 > U111
mark1 > alength1 > aU91 > U91 > U111
mark1 > alength1 > length1 > aU111 > aU121 > aisNatList > aU81 > aU82 > aU83 > aisNatIListKind > aisNatKind > aU61 > tt > AU421 > U111
mark1 > alength1 > length1 > aU111 > aU121 > aisNatList > aU81 > aU82 > aU83 > aisNatIListKind > aisNatKind > aU61 > U61 > U111
mark1 > alength1 > length1 > aU111 > aU121 > aisNatList > aU81 > aU82 > aU83 > aisNatIListKind > aisNatKind > isNatKind > U111
mark1 > alength1 > length1 > aU111 > aU121 > aisNatList > aU81 > aU82 > aU83 > aisNatIListKind > aU511 > aU52 > tt > AU421 > U111
mark1 > alength1 > length1 > aU111 > aU121 > aisNatList > aU81 > aU82 > aU83 > aisNatIListKind > aU511 > aU52 > U52 > U111
mark1 > alength1 > length1 > aU111 > aU121 > aisNatList > aU81 > aU82 > aU83 > aisNatIListKind > aU511 > U511 > U111
mark1 > alength1 > length1 > aU111 > aU121 > aisNatList > aU81 > aU82 > aU83 > aisNatIListKind > isNatIListKind > U111
mark1 > alength1 > length1 > aU111 > aU121 > aisNatList > aU81 > aU82 > aU83 > aU841 > aU85 > aU86 > tt > AU421 > U111
mark1 > alength1 > length1 > aU111 > aU121 > aisNatList > aU81 > aU82 > aU83 > aU841 > aU85 > aU86 > U86 > U111
mark1 > alength1 > length1 > aU111 > aU121 > aisNatList > aU81 > aU82 > aU83 > aU841 > aU85 > U85 > U111
mark1 > alength1 > length1 > aU111 > aU121 > aisNatList > aU81 > aU82 > aU83 > aU841 > U841 > U111
mark1 > alength1 > length1 > aU111 > aU121 > aisNatList > aU81 > aU82 > aU83 > U83 > U111
mark1 > alength1 > length1 > aU111 > aU121 > aisNatList > aU81 > aU82 > U82 > U111
mark1 > alength1 > length1 > aU111 > aU121 > aisNatList > aU81 > U81 > U111
mark1 > alength1 > length1 > aU111 > aU121 > aisNatList > isNatList > U111
mark1 > alength1 > length1 > aU111 > aU121 > U121 > U111
mark1 > nil > tt > AU421 > U111
mark1 > nil > 0 > U111

Status:
tt: multiset
aisNat: multiset
cons1: multiset
AU411: multiset
aisNatKind: []
AU421: multiset
aisNatIListKind: multiset
azeros: multiset
0: multiset
zeros: multiset
aU111: multiset
aU121: multiset
aisNatList: multiset
aU211: multiset
aU221: multiset
aU23: multiset
aU31: multiset
aU32: multiset
aU33: multiset
aU41: multiset
aU42: multiset
aU43: multiset
aU44: multiset
aU45: multiset
aU46: multiset
aisNatIList: multiset
aU511: multiset
aU52: multiset
aU61: multiset
aU81: multiset
aU82: multiset
aU83: multiset
aU841: multiset
aU85: multiset
aU86: multiset
aU91: multiset
aU921: multiset
aU93: multiset
aU94: multiset
s: multiset
alength1: multiset
mark1: [1]
length1: multiset
nil: multiset
U111: multiset
U121: multiset
isNatIListKind: multiset
isNatList: multiset
U211: multiset
U221: multiset
isNatKind: multiset
U23: multiset
isNat: multiset
U31: multiset
U32: multiset
U33: multiset
U41: multiset
U42: multiset
U43: multiset
U44: multiset
U45: multiset
U46: multiset
isNatIList: multiset
U511: multiset
U52: multiset
U61: multiset
U81: multiset
U82: multiset
U83: multiset
U841: multiset
U85: multiset
U86: multiset
U91: multiset
U921: multiset
U93: multiset
U94: multiset

The following usable rules [FROCOS05] were oriented:

a__zeroscons(0, zeros)
a__U11(tt, V1) → a__U12(a__isNatIListKind(V1), V1)
a__U12(tt, V1) → a__U13(a__isNatList(V1))
a__U13(tt) → tt
a__U21(tt, V1) → a__U22(a__isNatKind(V1), V1)
a__U22(tt, V1) → a__U23(a__isNat(V1))
a__U23(tt) → tt
a__U31(tt, V) → a__U32(a__isNatIListKind(V), V)
a__U32(tt, V) → a__U33(a__isNatList(V))
a__U33(tt) → tt
a__U41(tt, V1, V2) → a__U42(a__isNatKind(V1), V1, V2)
a__U42(tt, V1, V2) → a__U43(a__isNatIListKind(V2), V1, V2)
a__U43(tt, V1, V2) → a__U44(a__isNatIListKind(V2), V1, V2)
a__U44(tt, V1, V2) → a__U45(a__isNat(V1), V2)
a__U45(tt, V2) → a__U46(a__isNatIList(V2))
a__U46(tt) → tt
a__U51(tt, V2) → a__U52(a__isNatIListKind(V2))
a__U52(tt) → tt
a__U61(tt) → tt
a__U71(tt) → tt
a__U81(tt, V1, V2) → a__U82(a__isNatKind(V1), V1, V2)
a__U82(tt, V1, V2) → a__U83(a__isNatIListKind(V2), V1, V2)
a__U83(tt, V1, V2) → a__U84(a__isNatIListKind(V2), V1, V2)
a__U84(tt, V1, V2) → a__U85(a__isNat(V1), V2)
a__U85(tt, V2) → a__U86(a__isNatList(V2))
a__U86(tt) → tt
a__U91(tt, L, N) → a__U92(a__isNatIListKind(L), L, N)
a__U92(tt, L, N) → a__U93(a__isNat(N), L, N)
a__U93(tt, L, N) → a__U94(a__isNatKind(N), L)
a__U94(tt, L) → s(a__length(mark(L)))
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__isNatKind(V1), V1, V2)
a__isNatIListKind(nil) → tt
a__isNatIListKind(zeros) → tt
a__isNatIListKind(cons(V1, V2)) → a__U51(a__isNatKind(V1), V2)
a__isNatKind(0) → tt
a__isNatKind(length(V1)) → a__U61(a__isNatIListKind(V1))
a__isNatKind(s(V1)) → a__U71(a__isNatKind(V1))
a__isNatList(nil) → tt
a__isNatList(cons(V1, V2)) → a__U81(a__isNatKind(V1), V1, V2)
a__length(nil) → 0
a__length(cons(N, L)) → a__U91(a__isNatList(L), L, N)
mark(zeros) → a__zeros
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(U12(X1, X2)) → a__U12(mark(X1), X2)
mark(isNatIListKind(X)) → a__isNatIListKind(X)
mark(U13(X)) → a__U13(mark(X))
mark(isNatList(X)) → a__isNatList(X)
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(U22(X1, X2)) → a__U22(mark(X1), X2)
mark(isNatKind(X)) → a__isNatKind(X)
mark(U23(X)) → a__U23(mark(X))
mark(isNat(X)) → a__isNat(X)
mark(U31(X1, X2)) → a__U31(mark(X1), X2)
mark(U32(X1, X2)) → a__U32(mark(X1), X2)
mark(U33(X)) → a__U33(mark(X))
mark(U41(X1, X2, X3)) → a__U41(mark(X1), X2, X3)
mark(U42(X1, X2, X3)) → a__U42(mark(X1), X2, X3)
mark(U43(X1, X2, X3)) → a__U43(mark(X1), X2, X3)
mark(U44(X1, X2, X3)) → a__U44(mark(X1), X2, X3)
mark(U45(X1, X2)) → a__U45(mark(X1), X2)
mark(U46(X)) → a__U46(mark(X))
mark(isNatIList(X)) → a__isNatIList(X)
mark(U51(X1, X2)) → a__U51(mark(X1), X2)
mark(U52(X)) → a__U52(mark(X))
mark(U61(X)) → a__U61(mark(X))
mark(U71(X)) → a__U71(mark(X))
mark(U81(X1, X2, X3)) → a__U81(mark(X1), X2, X3)
mark(U82(X1, X2, X3)) → a__U82(mark(X1), X2, X3)
mark(U83(X1, X2, X3)) → a__U83(mark(X1), X2, X3)
mark(U84(X1, X2, X3)) → a__U84(mark(X1), X2, X3)
mark(U85(X1, X2)) → a__U85(mark(X1), X2)
mark(U86(X)) → a__U86(mark(X))
mark(U91(X1, X2, X3)) → a__U91(mark(X1), X2, X3)
mark(U92(X1, X2, X3)) → a__U92(mark(X1), X2, X3)
mark(U93(X1, X2, X3)) → a__U93(mark(X1), X2, X3)
mark(U94(X1, X2)) → a__U94(mark(X1), X2)
mark(length(X)) → a__length(mark(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__zeroszeros
a__U11(X1, X2) → U11(X1, X2)
a__U12(X1, X2) → U12(X1, X2)
a__isNatIListKind(X) → isNatIListKind(X)
a__U13(X) → U13(X)
a__isNatList(X) → isNatList(X)
a__U21(X1, X2) → U21(X1, X2)
a__U22(X1, X2) → U22(X1, X2)
a__isNatKind(X) → isNatKind(X)
a__U23(X) → U23(X)
a__isNat(X) → isNat(X)
a__U31(X1, X2) → U31(X1, X2)
a__U32(X1, X2) → U32(X1, X2)
a__U33(X) → U33(X)
a__U41(X1, X2, X3) → U41(X1, X2, X3)
a__U42(X1, X2, X3) → U42(X1, X2, X3)
a__U43(X1, X2, X3) → U43(X1, X2, X3)
a__U44(X1, X2, X3) → U44(X1, X2, X3)
a__U45(X1, X2) → U45(X1, X2)
a__U46(X) → U46(X)
a__isNatIList(X) → isNatIList(X)
a__U51(X1, X2) → U51(X1, X2)
a__U52(X) → U52(X)
a__U61(X) → U61(X)
a__U71(X) → U71(X)
a__U81(X1, X2, X3) → U81(X1, X2, X3)
a__U82(X1, X2, X3) → U82(X1, X2, X3)
a__U83(X1, X2, X3) → U83(X1, X2, X3)
a__U84(X1, X2, X3) → U84(X1, X2, X3)
a__U85(X1, X2) → U85(X1, X2)
a__U86(X) → U86(X)
a__U91(X1, X2, X3) → U91(X1, X2, X3)
a__U92(X1, X2, X3) → U92(X1, X2, X3)
a__U93(X1, X2, X3) → U93(X1, X2, X3)
a__U94(X1, X2) → U94(X1, X2)
a__length(X) → length(X)

(9) Obligation:

Q DP problem:
The TRS P consists of the following rules:

A__U44(tt, V1, V2) → A__U45(a__isNat(V1), V2)
A__U45(tt, V2) → A__ISNATILIST(V2)
A__U43(tt, V1, V2) → A__U44(a__isNatIListKind(V2), V1, V2)

The TRS R consists of the following rules:

a__zeroscons(0, zeros)
a__U11(tt, V1) → a__U12(a__isNatIListKind(V1), V1)
a__U12(tt, V1) → a__U13(a__isNatList(V1))
a__U13(tt) → tt
a__U21(tt, V1) → a__U22(a__isNatKind(V1), V1)
a__U22(tt, V1) → a__U23(a__isNat(V1))
a__U23(tt) → tt
a__U31(tt, V) → a__U32(a__isNatIListKind(V), V)
a__U32(tt, V) → a__U33(a__isNatList(V))
a__U33(tt) → tt
a__U41(tt, V1, V2) → a__U42(a__isNatKind(V1), V1, V2)
a__U42(tt, V1, V2) → a__U43(a__isNatIListKind(V2), V1, V2)
a__U43(tt, V1, V2) → a__U44(a__isNatIListKind(V2), V1, V2)
a__U44(tt, V1, V2) → a__U45(a__isNat(V1), V2)
a__U45(tt, V2) → a__U46(a__isNatIList(V2))
a__U46(tt) → tt
a__U51(tt, V2) → a__U52(a__isNatIListKind(V2))
a__U52(tt) → tt
a__U61(tt) → tt
a__U71(tt) → tt
a__U81(tt, V1, V2) → a__U82(a__isNatKind(V1), V1, V2)
a__U82(tt, V1, V2) → a__U83(a__isNatIListKind(V2), V1, V2)
a__U83(tt, V1, V2) → a__U84(a__isNatIListKind(V2), V1, V2)
a__U84(tt, V1, V2) → a__U85(a__isNat(V1), V2)
a__U85(tt, V2) → a__U86(a__isNatList(V2))
a__U86(tt) → tt
a__U91(tt, L, N) → a__U92(a__isNatIListKind(L), L, N)
a__U92(tt, L, N) → a__U93(a__isNat(N), L, N)
a__U93(tt, L, N) → a__U94(a__isNatKind(N), L)
a__U94(tt, L) → s(a__length(mark(L)))
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__isNatKind(V1), V1, V2)
a__isNatIListKind(nil) → tt
a__isNatIListKind(zeros) → tt
a__isNatIListKind(cons(V1, V2)) → a__U51(a__isNatKind(V1), V2)
a__isNatKind(0) → tt
a__isNatKind(length(V1)) → a__U61(a__isNatIListKind(V1))
a__isNatKind(s(V1)) → a__U71(a__isNatKind(V1))
a__isNatList(nil) → tt
a__isNatList(cons(V1, V2)) → a__U81(a__isNatKind(V1), V1, V2)
a__length(nil) → 0
a__length(cons(N, L)) → a__U91(a__isNatList(L), L, N)
mark(zeros) → a__zeros
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(U12(X1, X2)) → a__U12(mark(X1), X2)
mark(isNatIListKind(X)) → a__isNatIListKind(X)
mark(U13(X)) → a__U13(mark(X))
mark(isNatList(X)) → a__isNatList(X)
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(U22(X1, X2)) → a__U22(mark(X1), X2)
mark(isNatKind(X)) → a__isNatKind(X)
mark(U23(X)) → a__U23(mark(X))
mark(isNat(X)) → a__isNat(X)
mark(U31(X1, X2)) → a__U31(mark(X1), X2)
mark(U32(X1, X2)) → a__U32(mark(X1), X2)
mark(U33(X)) → a__U33(mark(X))
mark(U41(X1, X2, X3)) → a__U41(mark(X1), X2, X3)
mark(U42(X1, X2, X3)) → a__U42(mark(X1), X2, X3)
mark(U43(X1, X2, X3)) → a__U43(mark(X1), X2, X3)
mark(U44(X1, X2, X3)) → a__U44(mark(X1), X2, X3)
mark(U45(X1, X2)) → a__U45(mark(X1), X2)
mark(U46(X)) → a__U46(mark(X))
mark(isNatIList(X)) → a__isNatIList(X)
mark(U51(X1, X2)) → a__U51(mark(X1), X2)
mark(U52(X)) → a__U52(mark(X))
mark(U61(X)) → a__U61(mark(X))
mark(U71(X)) → a__U71(mark(X))
mark(U81(X1, X2, X3)) → a__U81(mark(X1), X2, X3)
mark(U82(X1, X2, X3)) → a__U82(mark(X1), X2, X3)
mark(U83(X1, X2, X3)) → a__U83(mark(X1), X2, X3)
mark(U84(X1, X2, X3)) → a__U84(mark(X1), X2, X3)
mark(U85(X1, X2)) → a__U85(mark(X1), X2)
mark(U86(X)) → a__U86(mark(X))
mark(U91(X1, X2, X3)) → a__U91(mark(X1), X2, X3)
mark(U92(X1, X2, X3)) → a__U92(mark(X1), X2, X3)
mark(U93(X1, X2, X3)) → a__U93(mark(X1), X2, X3)
mark(U94(X1, X2)) → a__U94(mark(X1), X2)
mark(length(X)) → a__length(mark(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__zeroszeros
a__U11(X1, X2) → U11(X1, X2)
a__U12(X1, X2) → U12(X1, X2)
a__isNatIListKind(X) → isNatIListKind(X)
a__U13(X) → U13(X)
a__isNatList(X) → isNatList(X)
a__U21(X1, X2) → U21(X1, X2)
a__U22(X1, X2) → U22(X1, X2)
a__isNatKind(X) → isNatKind(X)
a__U23(X) → U23(X)
a__isNat(X) → isNat(X)
a__U31(X1, X2) → U31(X1, X2)
a__U32(X1, X2) → U32(X1, X2)
a__U33(X) → U33(X)
a__U41(X1, X2, X3) → U41(X1, X2, X3)
a__U42(X1, X2, X3) → U42(X1, X2, X3)
a__U43(X1, X2, X3) → U43(X1, X2, X3)
a__U44(X1, X2, X3) → U44(X1, X2, X3)
a__U45(X1, X2) → U45(X1, X2)
a__U46(X) → U46(X)
a__isNatIList(X) → isNatIList(X)
a__U51(X1, X2) → U51(X1, X2)
a__U52(X) → U52(X)
a__U61(X) → U61(X)
a__U71(X) → U71(X)
a__U81(X1, X2, X3) → U81(X1, X2, X3)
a__U82(X1, X2, X3) → U82(X1, X2, X3)
a__U83(X1, X2, X3) → U83(X1, X2, X3)
a__U84(X1, X2, X3) → U84(X1, X2, X3)
a__U85(X1, X2) → U85(X1, X2)
a__U86(X) → U86(X)
a__U91(X1, X2, X3) → U91(X1, X2, X3)
a__U92(X1, X2, X3) → U92(X1, X2, X3)
a__U93(X1, X2, X3) → U93(X1, X2, X3)
a__U94(X1, X2) → U94(X1, X2)
a__length(X) → length(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(10) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 3 less nodes.

(11) TRUE

(12) Obligation:

Q DP problem:
The TRS P consists of the following rules:

MARK(U11(X1, X2)) → MARK(X1)
MARK(U12(X1, X2)) → MARK(X1)
MARK(U13(X)) → MARK(X)
MARK(U21(X1, X2)) → MARK(X1)
MARK(U22(X1, X2)) → MARK(X1)
MARK(U23(X)) → MARK(X)
MARK(U31(X1, X2)) → MARK(X1)
MARK(U32(X1, X2)) → MARK(X1)
MARK(U33(X)) → MARK(X)
MARK(U41(X1, X2, X3)) → MARK(X1)
MARK(U42(X1, X2, X3)) → MARK(X1)
MARK(U43(X1, X2, X3)) → MARK(X1)
MARK(U44(X1, X2, X3)) → MARK(X1)
MARK(U45(X1, X2)) → MARK(X1)
MARK(U46(X)) → MARK(X)
MARK(U51(X1, X2)) → MARK(X1)
MARK(U52(X)) → MARK(X)
MARK(U61(X)) → MARK(X)
MARK(U71(X)) → MARK(X)
MARK(U81(X1, X2, X3)) → MARK(X1)
MARK(U82(X1, X2, X3)) → MARK(X1)
MARK(U83(X1, X2, X3)) → MARK(X1)
MARK(U84(X1, X2, X3)) → MARK(X1)
MARK(U85(X1, X2)) → MARK(X1)
MARK(U86(X)) → MARK(X)
MARK(U91(X1, X2, X3)) → A__U91(mark(X1), X2, X3)
A__U91(tt, L, N) → A__U92(a__isNatIListKind(L), L, N)
A__U92(tt, L, N) → A__U93(a__isNat(N), L, N)
A__U93(tt, L, N) → A__U94(a__isNatKind(N), L)
A__U94(tt, L) → A__LENGTH(mark(L))
A__LENGTH(cons(N, L)) → A__U91(a__isNatList(L), L, N)
A__U94(tt, L) → MARK(L)
MARK(U91(X1, X2, X3)) → MARK(X1)
MARK(U92(X1, X2, X3)) → A__U92(mark(X1), X2, X3)
MARK(U92(X1, X2, X3)) → MARK(X1)
MARK(U93(X1, X2, X3)) → A__U93(mark(X1), X2, X3)
MARK(U93(X1, X2, X3)) → MARK(X1)
MARK(U94(X1, X2)) → A__U94(mark(X1), X2)
MARK(U94(X1, X2)) → MARK(X1)
MARK(length(X)) → A__LENGTH(mark(X))
MARK(length(X)) → MARK(X)
MARK(cons(X1, X2)) → MARK(X1)
MARK(s(X)) → MARK(X)

The TRS R consists of the following rules:

a__zeroscons(0, zeros)
a__U11(tt, V1) → a__U12(a__isNatIListKind(V1), V1)
a__U12(tt, V1) → a__U13(a__isNatList(V1))
a__U13(tt) → tt
a__U21(tt, V1) → a__U22(a__isNatKind(V1), V1)
a__U22(tt, V1) → a__U23(a__isNat(V1))
a__U23(tt) → tt
a__U31(tt, V) → a__U32(a__isNatIListKind(V), V)
a__U32(tt, V) → a__U33(a__isNatList(V))
a__U33(tt) → tt
a__U41(tt, V1, V2) → a__U42(a__isNatKind(V1), V1, V2)
a__U42(tt, V1, V2) → a__U43(a__isNatIListKind(V2), V1, V2)
a__U43(tt, V1, V2) → a__U44(a__isNatIListKind(V2), V1, V2)
a__U44(tt, V1, V2) → a__U45(a__isNat(V1), V2)
a__U45(tt, V2) → a__U46(a__isNatIList(V2))
a__U46(tt) → tt
a__U51(tt, V2) → a__U52(a__isNatIListKind(V2))
a__U52(tt) → tt
a__U61(tt) → tt
a__U71(tt) → tt
a__U81(tt, V1, V2) → a__U82(a__isNatKind(V1), V1, V2)
a__U82(tt, V1, V2) → a__U83(a__isNatIListKind(V2), V1, V2)
a__U83(tt, V1, V2) → a__U84(a__isNatIListKind(V2), V1, V2)
a__U84(tt, V1, V2) → a__U85(a__isNat(V1), V2)
a__U85(tt, V2) → a__U86(a__isNatList(V2))
a__U86(tt) → tt
a__U91(tt, L, N) → a__U92(a__isNatIListKind(L), L, N)
a__U92(tt, L, N) → a__U93(a__isNat(N), L, N)
a__U93(tt, L, N) → a__U94(a__isNatKind(N), L)
a__U94(tt, L) → s(a__length(mark(L)))
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__isNatKind(V1), V1, V2)
a__isNatIListKind(nil) → tt
a__isNatIListKind(zeros) → tt
a__isNatIListKind(cons(V1, V2)) → a__U51(a__isNatKind(V1), V2)
a__isNatKind(0) → tt
a__isNatKind(length(V1)) → a__U61(a__isNatIListKind(V1))
a__isNatKind(s(V1)) → a__U71(a__isNatKind(V1))
a__isNatList(nil) → tt
a__isNatList(cons(V1, V2)) → a__U81(a__isNatKind(V1), V1, V2)
a__length(nil) → 0
a__length(cons(N, L)) → a__U91(a__isNatList(L), L, N)
mark(zeros) → a__zeros
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(U12(X1, X2)) → a__U12(mark(X1), X2)
mark(isNatIListKind(X)) → a__isNatIListKind(X)
mark(U13(X)) → a__U13(mark(X))
mark(isNatList(X)) → a__isNatList(X)
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(U22(X1, X2)) → a__U22(mark(X1), X2)
mark(isNatKind(X)) → a__isNatKind(X)
mark(U23(X)) → a__U23(mark(X))
mark(isNat(X)) → a__isNat(X)
mark(U31(X1, X2)) → a__U31(mark(X1), X2)
mark(U32(X1, X2)) → a__U32(mark(X1), X2)
mark(U33(X)) → a__U33(mark(X))
mark(U41(X1, X2, X3)) → a__U41(mark(X1), X2, X3)
mark(U42(X1, X2, X3)) → a__U42(mark(X1), X2, X3)
mark(U43(X1, X2, X3)) → a__U43(mark(X1), X2, X3)
mark(U44(X1, X2, X3)) → a__U44(mark(X1), X2, X3)
mark(U45(X1, X2)) → a__U45(mark(X1), X2)
mark(U46(X)) → a__U46(mark(X))
mark(isNatIList(X)) → a__isNatIList(X)
mark(U51(X1, X2)) → a__U51(mark(X1), X2)
mark(U52(X)) → a__U52(mark(X))
mark(U61(X)) → a__U61(mark(X))
mark(U71(X)) → a__U71(mark(X))
mark(U81(X1, X2, X3)) → a__U81(mark(X1), X2, X3)
mark(U82(X1, X2, X3)) → a__U82(mark(X1), X2, X3)
mark(U83(X1, X2, X3)) → a__U83(mark(X1), X2, X3)
mark(U84(X1, X2, X3)) → a__U84(mark(X1), X2, X3)
mark(U85(X1, X2)) → a__U85(mark(X1), X2)
mark(U86(X)) → a__U86(mark(X))
mark(U91(X1, X2, X3)) → a__U91(mark(X1), X2, X3)
mark(U92(X1, X2, X3)) → a__U92(mark(X1), X2, X3)
mark(U93(X1, X2, X3)) → a__U93(mark(X1), X2, X3)
mark(U94(X1, X2)) → a__U94(mark(X1), X2)
mark(length(X)) → a__length(mark(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__zeroszeros
a__U11(X1, X2) → U11(X1, X2)
a__U12(X1, X2) → U12(X1, X2)
a__isNatIListKind(X) → isNatIListKind(X)
a__U13(X) → U13(X)
a__isNatList(X) → isNatList(X)
a__U21(X1, X2) → U21(X1, X2)
a__U22(X1, X2) → U22(X1, X2)
a__isNatKind(X) → isNatKind(X)
a__U23(X) → U23(X)
a__isNat(X) → isNat(X)
a__U31(X1, X2) → U31(X1, X2)
a__U32(X1, X2) → U32(X1, X2)
a__U33(X) → U33(X)
a__U41(X1, X2, X3) → U41(X1, X2, X3)
a__U42(X1, X2, X3) → U42(X1, X2, X3)
a__U43(X1, X2, X3) → U43(X1, X2, X3)
a__U44(X1, X2, X3) → U44(X1, X2, X3)
a__U45(X1, X2) → U45(X1, X2)
a__U46(X) → U46(X)
a__isNatIList(X) → isNatIList(X)
a__U51(X1, X2) → U51(X1, X2)
a__U52(X) → U52(X)
a__U61(X) → U61(X)
a__U71(X) → U71(X)
a__U81(X1, X2, X3) → U81(X1, X2, X3)
a__U82(X1, X2, X3) → U82(X1, X2, X3)
a__U83(X1, X2, X3) → U83(X1, X2, X3)
a__U84(X1, X2, X3) → U84(X1, X2, X3)
a__U85(X1, X2) → U85(X1, X2)
a__U86(X) → U86(X)
a__U91(X1, X2, X3) → U91(X1, X2, X3)
a__U92(X1, X2, X3) → U92(X1, X2, X3)
a__U93(X1, X2, X3) → U93(X1, X2, X3)
a__U94(X1, X2) → U94(X1, X2)
a__length(X) → length(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.