Termination w.r.t. Q proof of /tmp/tmpUpgf2d/LengthOfFiniteLists_complete_noand

Termination w.r.t. Q of the given QTRS could not be shown:

0 QTRS

↳1 DependencyPairsProof (⇔)

↳2 QDP

↳3 DependencyGraphProof (⇔)

↳4 AND

  ↳5 QDP

  ↳6 QDP

  ↳7 QDP

    ↳8 QDPOrderProof (⇔)

    ↳9 QDP

    ↳10 DependencyGraphProof (⇔)

    ↳11 TRUE

  ↳12 QDP

(0) Obligation:

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

a__zeros → cons(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__zeros → zeros
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__zeros → cons(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__zeros → zeros
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__zeros → cons(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__zeros → zeros
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__zeros → cons(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__zeros → zeros
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__zeros → cons(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__zeros → zeros
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) = a__U13
a__isNatList(x1) = a__isNatList
a__U21(x1, x2) = a__U21
a__U22(x1, x2) = a__U22
a__U23(x1) = a__U23
a__U31(x1, x2) = a__U31(x2)
a__U32(x1, x2) = a__U32(x1)
a__U33(x1) = x1
a__U41(x1, x2, x3) = a__U41(x3)
a__U42(x1, x2, x3) = a__U42(x3)
a__U43(x1, x2, x3) = a__U43(x1)
a__U44(x1, x2, x3) = a__U44
a__U45(x1, x2) = a__U45
a__U46(x1) = a__U46
a__isNatIList(x1) = a__isNatIList(x1)
a__U51(x1, x2) = a__U51
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
a__U85(x1, x2) = a__U85
a__U86(x1) = a__U86
a__U91(x1, x2, x3) = x1
a__U92(x1, x2, x3) = a__U92
a__U93(x1, x2, x3) = a__U93
a__U94(x1, x2) = a__U94
s(x1) = s
a__length(x1) = a__length
mark(x1) = mark(x1)
length(x1) = length
nil = nil
U11(x1, x2) = U11(x1)
U12(x1, x2) = U12(x1)
isNatIListKind(x1) = isNatIListKind
U13(x1) = U13
isNatList(x1) = isNatList
U21(x1, x2) = U21
U22(x1, x2) = U22
isNatKind(x1) = isNatKind
U23(x1) = U23
isNat(x1) = isNat
U31(x1, x2) = U31(x2)
U32(x1, x2) = U32(x1)
U33(x1) = x1
U41(x1, x2, x3) = U41(x3)
U42(x1, x2, x3) = U42(x3)
U43(x1, x2, x3) = U43(x1)
U44(x1, x2, x3) = U44
U45(x1, x2) = U45
U46(x1) = U46
isNatIList(x1) = isNatIList(x1)
U51(x1, x2) = U51
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
U85(x1, x2) = U85
U86(x1) = U86
U91(x1, x2, x3) = x1
U92(x1, x2, x3) = U92
U93(x1, x2, x3) = U93
U94(x1, x2) = U94

Recursive Path Order [RPO].
Precedence:

mark₁ > a_{_isNat} > a_{_U11₁} > a_{_{isNatIListKind}} > a_{_U51} > a_{_U52} > tt > a_{_U92} > a_{_U93} > a_{_U94} > s
mark₁ > a_{_isNat} > a_{_U11₁} > a_{_{isNatIListKind}} > a_{_U51} > a_{_U52} > tt > a_{_U92} > a_{_U93} > a_{_U94} > U94
mark₁ > a_{_isNat} > a_{_U11₁} > a_{_{isNatIListKind}} > a_{_U51} > a_{_U52} > tt > a_{_U92} > a_{_U93} > U93
mark₁ > a_{_isNat} > a_{_U11₁} > a_{_{isNatIListKind}} > a_{_U51} > a_{_U52} > tt > a_{_U92} > U92
mark₁ > a_{_isNat} > a_{_U11₁} > a_{_{isNatIListKind}} > a_{_U51} > a_{_U52} > U52
mark₁ > a_{_isNat} > a_{_U11₁} > a_{_{isNatIListKind}} > a_{_U51} > U51
mark₁ > a_{_isNat} > a_{_U11₁} > a_{_{isNatIListKind}} > isNatIListKind
mark₁ > a_{_isNat} > a_{_U11₁} > a_{_U12₁} > a_{_U13} > tt > a_{_U92} > a_{_U93} > a_{_U94} > s
mark₁ > a_{_isNat} > a_{_U11₁} > a_{_U12₁} > a_{_U13} > tt > a_{_U92} > a_{_U93} > a_{_U94} > U94
mark₁ > a_{_isNat} > a_{_U11₁} > a_{_U12₁} > a_{_U13} > tt > a_{_U92} > a_{_U93} > U93
mark₁ > a_{_isNat} > a_{_U11₁} > a_{_U12₁} > a_{_U13} > tt > a_{_U92} > U92
mark₁ > a_{_isNat} > a_{_U11₁} > a_{_U12₁} > a_{_U13} > U13
mark₁ > a_{_isNat} > a_{_U11₁} > a_{_U12₁} > U12₁
mark₁ > a_{_isNat} > a_{_U11₁} > U11₁
mark₁ > a_{_isNat} > a_{_U21} > a_{_isNatKind} > a_{_{isNatIListKind}} > a_{_U51} > a_{_U52} > tt > a_{_U92} > a_{_U93} > a_{_U94} > s
mark₁ > a_{_isNat} > a_{_U21} > a_{_isNatKind} > a_{_{isNatIListKind}} > a_{_U51} > a_{_U52} > tt > a_{_U92} > a_{_U93} > a_{_U94} > U94
mark₁ > a_{_isNat} > a_{_U21} > a_{_isNatKind} > a_{_{isNatIListKind}} > a_{_U51} > a_{_U52} > tt > a_{_U92} > a_{_U93} > U93
mark₁ > a_{_isNat} > a_{_U21} > a_{_isNatKind} > a_{_{isNatIListKind}} > a_{_U51} > a_{_U52} > tt > a_{_U92} > U92
mark₁ > a_{_isNat} > a_{_U21} > a_{_isNatKind} > a_{_{isNatIListKind}} > a_{_U51} > a_{_U52} > U52
mark₁ > a_{_isNat} > a_{_U21} > a_{_isNatKind} > a_{_{isNatIListKind}} > a_{_U51} > U51
mark₁ > a_{_isNat} > a_{_U21} > a_{_isNatKind} > a_{_{isNatIListKind}} > isNatIListKind
mark₁ > a_{_isNat} > a_{_U21} > a_{_isNatKind} > a_{_U61} > tt > a_{_U92} > a_{_U93} > a_{_U94} > s
mark₁ > a_{_isNat} > a_{_U21} > a_{_isNatKind} > a_{_U61} > tt > a_{_U92} > a_{_U93} > a_{_U94} > U94
mark₁ > a_{_isNat} > a_{_U21} > a_{_isNatKind} > a_{_U61} > tt > a_{_U92} > a_{_U93} > U93
mark₁ > a_{_isNat} > a_{_U21} > a_{_isNatKind} > a_{_U61} > tt > a_{_U92} > U92
mark₁ > a_{_isNat} > a_{_U21} > a_{_isNatKind} > a_{_U61} > U61
mark₁ > a_{_isNat} > a_{_U21} > a_{_isNatKind} > isNatKind
mark₁ > a_{_isNat} > a_{_U21} > a_{_U22} > a_{_U23} > tt > a_{_U92} > a_{_U93} > a_{_U94} > s
mark₁ > a_{_isNat} > a_{_U21} > a_{_U22} > a_{_U23} > tt > a_{_U92} > a_{_U93} > a_{_U94} > U94
mark₁ > a_{_isNat} > a_{_U21} > a_{_U22} > a_{_U23} > tt > a_{_U92} > a_{_U93} > U93
mark₁ > a_{_isNat} > a_{_U21} > a_{_U22} > a_{_U23} > tt > a_{_U92} > U92
mark₁ > a_{_isNat} > a_{_U21} > a_{_U22} > a_{_U23} > U23
mark₁ > a_{_isNat} > a_{_U21} > a_{_U22} > U22
mark₁ > a_{_isNat} > a_{_U21} > U21
mark₁ > a_{_isNat} > isNat
mark₁ > a_{_zeros} > cons₁ > A_{_U41₁} > a_{_isNatKind} > a_{_{isNatIListKind}} > a_{_U51} > a_{_U52} > tt > a_{_U92} > a_{_U93} > a_{_U94} > s
mark₁ > a_{_zeros} > cons₁ > A_{_U41₁} > a_{_isNatKind} > a_{_{isNatIListKind}} > a_{_U51} > a_{_U52} > tt > a_{_U92} > a_{_U93} > a_{_U94} > U94
mark₁ > a_{_zeros} > cons₁ > A_{_U41₁} > a_{_isNatKind} > a_{_{isNatIListKind}} > a_{_U51} > a_{_U52} > tt > a_{_U92} > a_{_U93} > U93
mark₁ > a_{_zeros} > cons₁ > A_{_U41₁} > a_{_isNatKind} > a_{_{isNatIListKind}} > a_{_U51} > a_{_U52} > tt > a_{_U92} > U92
mark₁ > a_{_zeros} > cons₁ > A_{_U41₁} > a_{_isNatKind} > a_{_{isNatIListKind}} > a_{_U51} > a_{_U52} > U52
mark₁ > a_{_zeros} > cons₁ > A_{_U41₁} > a_{_isNatKind} > a_{_{isNatIListKind}} > a_{_U51} > U51
mark₁ > a_{_zeros} > cons₁ > A_{_U41₁} > a_{_isNatKind} > a_{_{isNatIListKind}} > isNatIListKind
mark₁ > a_{_zeros} > cons₁ > A_{_U41₁} > a_{_isNatKind} > a_{_U61} > tt > a_{_U92} > a_{_U93} > a_{_U94} > s
mark₁ > a_{_zeros} > cons₁ > A_{_U41₁} > a_{_isNatKind} > a_{_U61} > tt > a_{_U92} > a_{_U93} > a_{_U94} > U94
mark₁ > a_{_zeros} > cons₁ > A_{_U41₁} > a_{_isNatKind} > a_{_U61} > tt > a_{_U92} > a_{_U93} > U93
mark₁ > a_{_zeros} > cons₁ > A_{_U41₁} > a_{_isNatKind} > a_{_U61} > tt > a_{_U92} > U92
mark₁ > a_{_zeros} > cons₁ > A_{_U41₁} > a_{_isNatKind} > a_{_U61} > U61
mark₁ > a_{_zeros} > cons₁ > A_{_U41₁} > a_{_isNatKind} > isNatKind
mark₁ > a_{_zeros} > cons₁ > A_{_U41₁} > A_{_U42₁} > a_{_{isNatIListKind}} > a_{_U51} > a_{_U52} > tt > a_{_U92} > a_{_U93} > a_{_U94} > s
mark₁ > a_{_zeros} > cons₁ > A_{_U41₁} > A_{_U42₁} > a_{_{isNatIListKind}} > a_{_U51} > a_{_U52} > tt > a_{_U92} > a_{_U93} > a_{_U94} > U94
mark₁ > a_{_zeros} > cons₁ > A_{_U41₁} > A_{_U42₁} > a_{_{isNatIListKind}} > a_{_U51} > a_{_U52} > tt > a_{_U92} > a_{_U93} > U93
mark₁ > a_{_zeros} > cons₁ > A_{_U41₁} > A_{_U42₁} > a_{_{isNatIListKind}} > a_{_U51} > a_{_U52} > tt > a_{_U92} > U92
mark₁ > a_{_zeros} > cons₁ > A_{_U41₁} > A_{_U42₁} > a_{_{isNatIListKind}} > a_{_U51} > a_{_U52} > U52
mark₁ > a_{_zeros} > cons₁ > A_{_U41₁} > A_{_U42₁} > a_{_{isNatIListKind}} > a_{_U51} > U51
mark₁ > a_{_zeros} > cons₁ > A_{_U41₁} > A_{_U42₁} > a_{_{isNatIListKind}} > isNatIListKind
mark₁ > a_{_zeros} > cons₁ > a_{_isNatList} > a_{_isNatKind} > a_{_{isNatIListKind}} > a_{_U51} > a_{_U52} > tt > a_{_U92} > a_{_U93} > a_{_U94} > s
mark₁ > a_{_zeros} > cons₁ > a_{_isNatList} > a_{_isNatKind} > a_{_{isNatIListKind}} > a_{_U51} > a_{_U52} > tt > a_{_U92} > a_{_U93} > a_{_U94} > U94
mark₁ > a_{_zeros} > cons₁ > a_{_isNatList} > a_{_isNatKind} > a_{_{isNatIListKind}} > a_{_U51} > a_{_U52} > tt > a_{_U92} > a_{_U93} > U93
mark₁ > a_{_zeros} > cons₁ > a_{_isNatList} > a_{_isNatKind} > a_{_{isNatIListKind}} > a_{_U51} > a_{_U52} > tt > a_{_U92} > U92
mark₁ > a_{_zeros} > cons₁ > a_{_isNatList} > a_{_isNatKind} > a_{_{isNatIListKind}} > a_{_U51} > a_{_U52} > U52
mark₁ > a_{_zeros} > cons₁ > a_{_isNatList} > a_{_isNatKind} > a_{_{isNatIListKind}} > a_{_U51} > U51
mark₁ > a_{_zeros} > cons₁ > a_{_isNatList} > a_{_isNatKind} > a_{_{isNatIListKind}} > isNatIListKind
mark₁ > a_{_zeros} > cons₁ > a_{_isNatList} > a_{_isNatKind} > a_{_U61} > tt > a_{_U92} > a_{_U93} > a_{_U94} > s
mark₁ > a_{_zeros} > cons₁ > a_{_isNatList} > a_{_isNatKind} > a_{_U61} > tt > a_{_U92} > a_{_U93} > a_{_U94} > U94
mark₁ > a_{_zeros} > cons₁ > a_{_isNatList} > a_{_isNatKind} > a_{_U61} > tt > a_{_U92} > a_{_U93} > U93
mark₁ > a_{_zeros} > cons₁ > a_{_isNatList} > a_{_isNatKind} > a_{_U61} > tt > a_{_U92} > U92
mark₁ > a_{_zeros} > cons₁ > a_{_isNatList} > a_{_isNatKind} > a_{_U61} > U61
mark₁ > a_{_zeros} > cons₁ > a_{_isNatList} > a_{_isNatKind} > isNatKind
mark₁ > a_{_zeros} > cons₁ > a_{_isNatList} > a_{_U81} > a_{_U82} > a_{_U83} > a_{_U84} > a_{_U85} > a_{_U86} > tt > a_{_U92} > a_{_U93} > a_{_U94} > s
mark₁ > a_{_zeros} > cons₁ > a_{_isNatList} > a_{_U81} > a_{_U82} > a_{_U83} > a_{_U84} > a_{_U85} > a_{_U86} > tt > a_{_U92} > a_{_U93} > a_{_U94} > U94
mark₁ > a_{_zeros} > cons₁ > a_{_isNatList} > a_{_U81} > a_{_U82} > a_{_U83} > a_{_U84} > a_{_U85} > a_{_U86} > tt > a_{_U92} > a_{_U93} > U93
mark₁ > a_{_zeros} > cons₁ > a_{_isNatList} > a_{_U81} > a_{_U82} > a_{_U83} > a_{_U84} > a_{_U85} > a_{_U86} > tt > a_{_U92} > U92
mark₁ > a_{_zeros} > cons₁ > a_{_isNatList} > a_{_U81} > a_{_U82} > a_{_U83} > a_{_U84} > a_{_U85} > a_{_U86} > U86
mark₁ > a_{_zeros} > cons₁ > a_{_isNatList} > a_{_U81} > a_{_U82} > a_{_U83} > a_{_U84} > a_{_U85} > U85
mark₁ > a_{_zeros} > cons₁ > a_{_isNatList} > a_{_U81} > a_{_U82} > a_{_U83} > a_{_U84} > U84
mark₁ > a_{_zeros} > cons₁ > a_{_isNatList} > a_{_U81} > a_{_U82} > a_{_U83} > U83
mark₁ > a_{_zeros} > cons₁ > a_{_isNatList} > a_{_U81} > a_{_U82} > U82
mark₁ > a_{_zeros} > cons₁ > a_{_isNatList} > a_{_U81} > U81
mark₁ > a_{_zeros} > cons₁ > a_{_isNatList} > isNatList
mark₁ > a_{_zeros} > 0
mark₁ > a_{_zeros} > zeros > tt > a_{_U92} > a_{_U93} > a_{_U94} > s
mark₁ > a_{_zeros} > zeros > tt > a_{_U92} > a_{_U93} > a_{_U94} > U94
mark₁ > a_{_zeros} > zeros > tt > a_{_U92} > a_{_U93} > U93
mark₁ > a_{_zeros} > zeros > tt > a_{_U92} > U92
mark₁ > a_{_{isNatIList₁}} > a_{_U31₁} > a_{_U32₁} > a_{_isNatList} > a_{_isNatKind} > a_{_{isNatIListKind}} > a_{_U51} > a_{_U52} > tt > a_{_U92} > a_{_U93} > a_{_U94} > s
mark₁ > a_{_{isNatIList₁}} > a_{_U31₁} > a_{_U32₁} > a_{_isNatList} > a_{_isNatKind} > a_{_{isNatIListKind}} > a_{_U51} > a_{_U52} > tt > a_{_U92} > a_{_U93} > a_{_U94} > U94
mark₁ > a_{_{isNatIList₁}} > a_{_U31₁} > a_{_U32₁} > a_{_isNatList} > a_{_isNatKind} > a_{_{isNatIListKind}} > a_{_U51} > a_{_U52} > tt > a_{_U92} > a_{_U93} > U93
mark₁ > a_{_{isNatIList₁}} > a_{_U31₁} > a_{_U32₁} > a_{_isNatList} > a_{_isNatKind} > a_{_{isNatIListKind}} > a_{_U51} > a_{_U52} > tt > a_{_U92} > U92
mark₁ > a_{_{isNatIList₁}} > a_{_U31₁} > a_{_U32₁} > a_{_isNatList} > a_{_isNatKind} > a_{_{isNatIListKind}} > a_{_U51} > a_{_U52} > U52
mark₁ > a_{_{isNatIList₁}} > a_{_U31₁} > a_{_U32₁} > a_{_isNatList} > a_{_isNatKind} > a_{_{isNatIListKind}} > a_{_U51} > U51
mark₁ > a_{_{isNatIList₁}} > a_{_U31₁} > a_{_U32₁} > a_{_isNatList} > a_{_isNatKind} > a_{_{isNatIListKind}} > isNatIListKind
mark₁ > a_{_{isNatIList₁}} > a_{_U31₁} > a_{_U32₁} > a_{_isNatList} > a_{_isNatKind} > a_{_U61} > tt > a_{_U92} > a_{_U93} > a_{_U94} > s
mark₁ > a_{_{isNatIList₁}} > a_{_U31₁} > a_{_U32₁} > a_{_isNatList} > a_{_isNatKind} > a_{_U61} > tt > a_{_U92} > a_{_U93} > a_{_U94} > U94
mark₁ > a_{_{isNatIList₁}} > a_{_U31₁} > a_{_U32₁} > a_{_isNatList} > a_{_isNatKind} > a_{_U61} > tt > a_{_U92} > a_{_U93} > U93
mark₁ > a_{_{isNatIList₁}} > a_{_U31₁} > a_{_U32₁} > a_{_isNatList} > a_{_isNatKind} > a_{_U61} > tt > a_{_U92} > U92
mark₁ > a_{_{isNatIList₁}} > a_{_U31₁} > a_{_U32₁} > a_{_isNatList} > a_{_isNatKind} > a_{_U61} > U61
mark₁ > a_{_{isNatIList₁}} > a_{_U31₁} > a_{_U32₁} > a_{_isNatList} > a_{_isNatKind} > isNatKind
mark₁ > a_{_{isNatIList₁}} > a_{_U31₁} > a_{_U32₁} > a_{_isNatList} > a_{_U81} > a_{_U82} > a_{_U83} > a_{_U84} > a_{_U85} > a_{_U86} > tt > a_{_U92} > a_{_U93} > a_{_U94} > s
mark₁ > a_{_{isNatIList₁}} > a_{_U31₁} > a_{_U32₁} > a_{_isNatList} > a_{_U81} > a_{_U82} > a_{_U83} > a_{_U84} > a_{_U85} > a_{_U86} > tt > a_{_U92} > a_{_U93} > a_{_U94} > U94
mark₁ > a_{_{isNatIList₁}} > a_{_U31₁} > a_{_U32₁} > a_{_isNatList} > a_{_U81} > a_{_U82} > a_{_U83} > a_{_U84} > a_{_U85} > a_{_U86} > tt > a_{_U92} > a_{_U93} > U93
mark₁ > a_{_{isNatIList₁}} > a_{_U31₁} > a_{_U32₁} > a_{_isNatList} > a_{_U81} > a_{_U82} > a_{_U83} > a_{_U84} > a_{_U85} > a_{_U86} > tt > a_{_U92} > U92
mark₁ > a_{_{isNatIList₁}} > a_{_U31₁} > a_{_U32₁} > a_{_isNatList} > a_{_U81} > a_{_U82} > a_{_U83} > a_{_U84} > a_{_U85} > a_{_U86} > U86
mark₁ > a_{_{isNatIList₁}} > a_{_U31₁} > a_{_U32₁} > a_{_isNatList} > a_{_U81} > a_{_U82} > a_{_U83} > a_{_U84} > a_{_U85} > U85
mark₁ > a_{_{isNatIList₁}} > a_{_U31₁} > a_{_U32₁} > a_{_isNatList} > a_{_U81} > a_{_U82} > a_{_U83} > a_{_U84} > U84
mark₁ > a_{_{isNatIList₁}} > a_{_U31₁} > a_{_U32₁} > a_{_isNatList} > a_{_U81} > a_{_U82} > a_{_U83} > U83
mark₁ > a_{_{isNatIList₁}} > a_{_U31₁} > a_{_U32₁} > a_{_isNatList} > a_{_U81} > a_{_U82} > U82
mark₁ > a_{_{isNatIList₁}} > a_{_U31₁} > a_{_U32₁} > a_{_isNatList} > a_{_U81} > U81
mark₁ > a_{_{isNatIList₁}} > a_{_U31₁} > a_{_U32₁} > a_{_isNatList} > isNatList
mark₁ > a_{_{isNatIList₁}} > a_{_U31₁} > a_{_U32₁} > U32₁
mark₁ > a_{_{isNatIList₁}} > a_{_U31₁} > U31₁
mark₁ > a_{_{isNatIList₁}} > a_{_U41₁} > a_{_isNatKind} > a_{_{isNatIListKind}} > a_{_U51} > a_{_U52} > tt > a_{_U92} > a_{_U93} > a_{_U94} > s
mark₁ > a_{_{isNatIList₁}} > a_{_U41₁} > a_{_isNatKind} > a_{_{isNatIListKind}} > a_{_U51} > a_{_U52} > tt > a_{_U92} > a_{_U93} > a_{_U94} > U94
mark₁ > a_{_{isNatIList₁}} > a_{_U41₁} > a_{_isNatKind} > a_{_{isNatIListKind}} > a_{_U51} > a_{_U52} > tt > a_{_U92} > a_{_U93} > U93
mark₁ > a_{_{isNatIList₁}} > a_{_U41₁} > a_{_isNatKind} > a_{_{isNatIListKind}} > a_{_U51} > a_{_U52} > tt > a_{_U92} > U92
mark₁ > a_{_{isNatIList₁}} > a_{_U41₁} > a_{_isNatKind} > a_{_{isNatIListKind}} > a_{_U51} > a_{_U52} > U52
mark₁ > a_{_{isNatIList₁}} > a_{_U41₁} > a_{_isNatKind} > a_{_{isNatIListKind}} > a_{_U51} > U51
mark₁ > a_{_{isNatIList₁}} > a_{_U41₁} > a_{_isNatKind} > a_{_{isNatIListKind}} > isNatIListKind
mark₁ > a_{_{isNatIList₁}} > a_{_U41₁} > a_{_isNatKind} > a_{_U61} > tt > a_{_U92} > a_{_U93} > a_{_U94} > s
mark₁ > a_{_{isNatIList₁}} > a_{_U41₁} > a_{_isNatKind} > a_{_U61} > tt > a_{_U92} > a_{_U93} > a_{_U94} > U94
mark₁ > a_{_{isNatIList₁}} > a_{_U41₁} > a_{_isNatKind} > a_{_U61} > tt > a_{_U92} > a_{_U93} > U93
mark₁ > a_{_{isNatIList₁}} > a_{_U41₁} > a_{_isNatKind} > a_{_U61} > tt > a_{_U92} > U92
mark₁ > a_{_{isNatIList₁}} > a_{_U41₁} > a_{_isNatKind} > a_{_U61} > U61
mark₁ > a_{_{isNatIList₁}} > a_{_U41₁} > a_{_isNatKind} > isNatKind
mark₁ > a_{_{isNatIList₁}} > a_{_U41₁} > a_{_U42₁} > a_{_U43₁} > a_{_{isNatIListKind}} > a_{_U51} > a_{_U52} > tt > a_{_U92} > a_{_U93} > a_{_U94} > s
mark₁ > a_{_{isNatIList₁}} > a_{_U41₁} > a_{_U42₁} > a_{_U43₁} > a_{_{isNatIListKind}} > a_{_U51} > a_{_U52} > tt > a_{_U92} > a_{_U93} > a_{_U94} > U94
mark₁ > a_{_{isNatIList₁}} > a_{_U41₁} > a_{_U42₁} > a_{_U43₁} > a_{_{isNatIListKind}} > a_{_U51} > a_{_U52} > tt > a_{_U92} > a_{_U93} > U93
mark₁ > a_{_{isNatIList₁}} > a_{_U41₁} > a_{_U42₁} > a_{_U43₁} > a_{_{isNatIListKind}} > a_{_U51} > a_{_U52} > tt > a_{_U92} > U92
mark₁ > a_{_{isNatIList₁}} > a_{_U41₁} > a_{_U42₁} > a_{_U43₁} > a_{_{isNatIListKind}} > a_{_U51} > a_{_U52} > U52
mark₁ > a_{_{isNatIList₁}} > a_{_U41₁} > a_{_U42₁} > a_{_U43₁} > a_{_{isNatIListKind}} > a_{_U51} > U51
mark₁ > a_{_{isNatIList₁}} > a_{_U41₁} > a_{_U42₁} > a_{_U43₁} > a_{_{isNatIListKind}} > isNatIListKind
mark₁ > a_{_{isNatIList₁}} > a_{_U41₁} > a_{_U42₁} > a_{_U43₁} > a_{_U44} > a_{_U45} > a_{_U46} > tt > a_{_U92} > a_{_U93} > a_{_U94} > s
mark₁ > a_{_{isNatIList₁}} > a_{_U41₁} > a_{_U42₁} > a_{_U43₁} > a_{_U44} > a_{_U45} > a_{_U46} > tt > a_{_U92} > a_{_U93} > a_{_U94} > U94
mark₁ > a_{_{isNatIList₁}} > a_{_U41₁} > a_{_U42₁} > a_{_U43₁} > a_{_U44} > a_{_U45} > a_{_U46} > tt > a_{_U92} > a_{_U93} > U93
mark₁ > a_{_{isNatIList₁}} > a_{_U41₁} > a_{_U42₁} > a_{_U43₁} > a_{_U44} > a_{_U45} > a_{_U46} > tt > a_{_U92} > U92
mark₁ > a_{_{isNatIList₁}} > a_{_U41₁} > a_{_U42₁} > a_{_U43₁} > a_{_U44} > a_{_U45} > a_{_U46} > U46
mark₁ > a_{_{isNatIList₁}} > a_{_U41₁} > a_{_U42₁} > a_{_U43₁} > a_{_U44} > a_{_U45} > U45
mark₁ > a_{_{isNatIList₁}} > a_{_U41₁} > a_{_U42₁} > a_{_U43₁} > a_{_U44} > U44
mark₁ > a_{_{isNatIList₁}} > a_{_U41₁} > a_{_U42₁} > a_{_U43₁} > U43₁
mark₁ > a_{_{isNatIList₁}} > a_{_U41₁} > a_{_U42₁} > U42₁
mark₁ > a_{_{isNatIList₁}} > a_{_U41₁} > U41₁
mark₁ > a_{_{isNatIList₁}} > isNatIList₁
mark₁ > a_{_length} > 0
mark₁ > a_{_length} > a_{_isNatList} > a_{_isNatKind} > a_{_{isNatIListKind}} > a_{_U51} > a_{_U52} > tt > a_{_U92} > a_{_U93} > a_{_U94} > s
mark₁ > a_{_length} > a_{_isNatList} > a_{_isNatKind} > a_{_{isNatIListKind}} > a_{_U51} > a_{_U52} > tt > a_{_U92} > a_{_U93} > a_{_U94} > U94
mark₁ > a_{_length} > a_{_isNatList} > a_{_isNatKind} > a_{_{isNatIListKind}} > a_{_U51} > a_{_U52} > tt > a_{_U92} > a_{_U93} > U93
mark₁ > a_{_length} > a_{_isNatList} > a_{_isNatKind} > a_{_{isNatIListKind}} > a_{_U51} > a_{_U52} > tt > a_{_U92} > U92
mark₁ > a_{_length} > a_{_isNatList} > a_{_isNatKind} > a_{_{isNatIListKind}} > a_{_U51} > a_{_U52} > U52
mark₁ > a_{_length} > a_{_isNatList} > a_{_isNatKind} > a_{_{isNatIListKind}} > a_{_U51} > U51
mark₁ > a_{_length} > a_{_isNatList} > a_{_isNatKind} > a_{_{isNatIListKind}} > isNatIListKind
mark₁ > a_{_length} > a_{_isNatList} > a_{_isNatKind} > a_{_U61} > tt > a_{_U92} > a_{_U93} > a_{_U94} > s
mark₁ > a_{_length} > a_{_isNatList} > a_{_isNatKind} > a_{_U61} > tt > a_{_U92} > a_{_U93} > a_{_U94} > U94
mark₁ > a_{_length} > a_{_isNatList} > a_{_isNatKind} > a_{_U61} > tt > a_{_U92} > a_{_U93} > U93
mark₁ > a_{_length} > a_{_isNatList} > a_{_isNatKind} > a_{_U61} > tt > a_{_U92} > U92
mark₁ > a_{_length} > a_{_isNatList} > a_{_isNatKind} > a_{_U61} > U61
mark₁ > a_{_length} > a_{_isNatList} > a_{_isNatKind} > isNatKind
mark₁ > a_{_length} > a_{_isNatList} > a_{_U81} > a_{_U82} > a_{_U83} > a_{_U84} > a_{_U85} > a_{_U86} > tt > a_{_U92} > a_{_U93} > a_{_U94} > s
mark₁ > a_{_length} > a_{_isNatList} > a_{_U81} > a_{_U82} > a_{_U83} > a_{_U84} > a_{_U85} > a_{_U86} > tt > a_{_U92} > a_{_U93} > a_{_U94} > U94
mark₁ > a_{_length} > a_{_isNatList} > a_{_U81} > a_{_U82} > a_{_U83} > a_{_U84} > a_{_U85} > a_{_U86} > tt > a_{_U92} > a_{_U93} > U93
mark₁ > a_{_length} > a_{_isNatList} > a_{_U81} > a_{_U82} > a_{_U83} > a_{_U84} > a_{_U85} > a_{_U86} > tt > a_{_U92} > U92
mark₁ > a_{_length} > a_{_isNatList} > a_{_U81} > a_{_U82} > a_{_U83} > a_{_U84} > a_{_U85} > a_{_U86} > U86
mark₁ > a_{_length} > a_{_isNatList} > a_{_U81} > a_{_U82} > a_{_U83} > a_{_U84} > a_{_U85} > U85
mark₁ > a_{_length} > a_{_isNatList} > a_{_U81} > a_{_U82} > a_{_U83} > a_{_U84} > U84
mark₁ > a_{_length} > a_{_isNatList} > a_{_U81} > a_{_U82} > a_{_U83} > U83
mark₁ > a_{_length} > a_{_isNatList} > a_{_U81} > a_{_U82} > U82
mark₁ > a_{_length} > a_{_isNatList} > a_{_U81} > U81
mark₁ > a_{_length} > a_{_isNatList} > isNatList
mark₁ > a_{_length} > length > a_{_U11₁} > a_{_{isNatIListKind}} > a_{_U51} > a_{_U52} > tt > a_{_U92} > a_{_U93} > a_{_U94} > s
mark₁ > a_{_length} > length > a_{_U11₁} > a_{_{isNatIListKind}} > a_{_U51} > a_{_U52} > tt > a_{_U92} > a_{_U93} > a_{_U94} > U94
mark₁ > a_{_length} > length > a_{_U11₁} > a_{_{isNatIListKind}} > a_{_U51} > a_{_U52} > tt > a_{_U92} > a_{_U93} > U93
mark₁ > a_{_length} > length > a_{_U11₁} > a_{_{isNatIListKind}} > a_{_U51} > a_{_U52} > tt > a_{_U92} > U92
mark₁ > a_{_length} > length > a_{_U11₁} > a_{_{isNatIListKind}} > a_{_U51} > a_{_U52} > U52
mark₁ > a_{_length} > length > a_{_U11₁} > a_{_{isNatIListKind}} > a_{_U51} > U51
mark₁ > a_{_length} > length > a_{_U11₁} > a_{_{isNatIListKind}} > isNatIListKind
mark₁ > a_{_length} > length > a_{_U11₁} > a_{_U12₁} > a_{_U13} > tt > a_{_U92} > a_{_U93} > a_{_U94} > s
mark₁ > a_{_length} > length > a_{_U11₁} > a_{_U12₁} > a_{_U13} > tt > a_{_U92} > a_{_U93} > a_{_U94} > U94
mark₁ > a_{_length} > length > a_{_U11₁} > a_{_U12₁} > a_{_U13} > tt > a_{_U92} > a_{_U93} > U93
mark₁ > a_{_length} > length > a_{_U11₁} > a_{_U12₁} > a_{_U13} > tt > a_{_U92} > U92
mark₁ > a_{_length} > length > a_{_U11₁} > a_{_U12₁} > a_{_U13} > U13
mark₁ > a_{_length} > length > a_{_U11₁} > a_{_U12₁} > U12₁
mark₁ > a_{_length} > length > a_{_U11₁} > U11₁
mark₁ > a_{_length} > length > a_{_U61} > tt > a_{_U92} > a_{_U93} > a_{_U94} > s
mark₁ > a_{_length} > length > a_{_U61} > tt > a_{_U92} > a_{_U93} > a_{_U94} > U94
mark₁ > a_{_length} > length > a_{_U61} > tt > a_{_U92} > a_{_U93} > U93
mark₁ > a_{_length} > length > a_{_U61} > tt > a_{_U92} > U92
mark₁ > a_{_length} > length > a_{_U61} > U61
mark₁ > nil > tt > a_{_U92} > a_{_U93} > a_{_U94} > s
mark₁ > nil > tt > a_{_U92} > a_{_U93} > a_{_U94} > U94
mark₁ > nil > tt > a_{_U92} > a_{_U93} > U93
mark₁ > nil > tt > a_{_U92} > U92
mark₁ > nil > 0

The following usable rules [FROCOS05] were oriented:

a__zeros → cons(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__zeros → zeros
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__zeros → cons(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__zeros → zeros
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__zeros → cons(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__zeros → zeros
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.