Termination w.r.t. Q proof of /tmp/tmp2RNkLs/PEANO_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 QDPOrderProof (⇔)

    ↳7 QDP

    ↳8 DependencyGraphProof (⇔)

    ↳9 QDP

    ↳10 QDPOrderProof (⇔)

    ↳11 QDP

    ↳12 PisEmptyProof (⇔)

    ↳13 TRUE

  ↳14 QDP

    ↳15 QDPOrderProof (⇔)

    ↳16 QDP

    ↳17 DependencyGraphProof (⇔)

    ↳18 TRUE

  ↳19 QDP

    ↳20 QDPOrderProof (⇔)

    ↳21 QDP

    ↳22 DependencyGraphProof (⇔)

    ↳23 QDP

(0) Obligation:

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

a__U11(tt, V1, V2) → a__U12(a__isNatKind(V1), V1, V2)
a__U12(tt, V1, V2) → a__U13(a__isNatKind(V2), V1, V2)
a__U13(tt, V1, V2) → a__U14(a__isNatKind(V2), V1, V2)
a__U14(tt, V1, V2) → a__U15(a__isNat(V1), V2)
a__U15(tt, V2) → a__U16(a__isNat(V2))
a__U16(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, V2) → a__U32(a__isNatKind(V2))
a__U32(tt) → tt
a__U41(tt) → tt
a__U51(tt, N) → a__U52(a__isNatKind(N), N)
a__U52(tt, N) → mark(N)
a__U61(tt, M, N) → a__U62(a__isNatKind(M), M, N)
a__U62(tt, M, N) → a__U63(a__isNat(N), M, N)
a__U63(tt, M, N) → a__U64(a__isNatKind(N), M, N)
a__U64(tt, M, N) → s(a__plus(mark(N), mark(M)))
a__isNat(0) → tt
a__isNat(plus(V1, V2)) → a__U11(a__isNatKind(V1), V1, V2)
a__isNat(s(V1)) → a__U21(a__isNatKind(V1), V1)
a__isNatKind(0) → tt
a__isNatKind(plus(V1, V2)) → a__U31(a__isNatKind(V1), V2)
a__isNatKind(s(V1)) → a__U41(a__isNatKind(V1))
a__plus(N, 0) → a__U51(a__isNat(N), N)
a__plus(N, s(M)) → a__U61(a__isNat(M), M, N)
mark(U11(X1, X2, X3)) → a__U11(mark(X1), X2, X3)
mark(U12(X1, X2, X3)) → a__U12(mark(X1), X2, X3)
mark(isNatKind(X)) → a__isNatKind(X)
mark(U13(X1, X2, X3)) → a__U13(mark(X1), X2, X3)
mark(U14(X1, X2, X3)) → a__U14(mark(X1), X2, X3)
mark(U15(X1, X2)) → a__U15(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
mark(U16(X)) → a__U16(mark(X))
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(U22(X1, X2)) → a__U22(mark(X1), X2)
mark(U23(X)) → a__U23(mark(X))
mark(U31(X1, X2)) → a__U31(mark(X1), X2)
mark(U32(X)) → a__U32(mark(X))
mark(U41(X)) → a__U41(mark(X))
mark(U51(X1, X2)) → a__U51(mark(X1), X2)
mark(U52(X1, X2)) → a__U52(mark(X1), X2)
mark(U61(X1, X2, X3)) → a__U61(mark(X1), X2, X3)
mark(U62(X1, X2, X3)) → a__U62(mark(X1), X2, X3)
mark(U63(X1, X2, X3)) → a__U63(mark(X1), X2, X3)
mark(U64(X1, X2, X3)) → a__U64(mark(X1), X2, X3)
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(0) → 0
a__U11(X1, X2, X3) → U11(X1, X2, X3)
a__U12(X1, X2, X3) → U12(X1, X2, X3)
a__isNatKind(X) → isNatKind(X)
a__U13(X1, X2, X3) → U13(X1, X2, X3)
a__U14(X1, X2, X3) → U14(X1, X2, X3)
a__U15(X1, X2) → U15(X1, X2)
a__isNat(X) → isNat(X)
a__U16(X) → U16(X)
a__U21(X1, X2) → U21(X1, X2)
a__U22(X1, X2) → U22(X1, X2)
a__U23(X) → U23(X)
a__U31(X1, X2) → U31(X1, X2)
a__U32(X) → U32(X)
a__U41(X) → U41(X)
a__U51(X1, X2) → U51(X1, X2)
a__U52(X1, X2) → U52(X1, X2)
a__U61(X1, X2, X3) → U61(X1, X2, X3)
a__U62(X1, X2, X3) → U62(X1, X2, X3)
a__U63(X1, X2, X3) → U63(X1, X2, X3)
a__U64(X1, X2, X3) → U64(X1, X2, X3)
a__plus(X1, X2) → plus(X1, X2)

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, V2) → A__U12(a__isNatKind(V1), V1, V2)
A__U11(tt, V1, V2) → A__ISNATKIND(V1)
A__U12(tt, V1, V2) → A__U13(a__isNatKind(V2), V1, V2)
A__U12(tt, V1, V2) → A__ISNATKIND(V2)
A__U13(tt, V1, V2) → A__U14(a__isNatKind(V2), V1, V2)
A__U13(tt, V1, V2) → A__ISNATKIND(V2)
A__U14(tt, V1, V2) → A__U15(a__isNat(V1), V2)
A__U14(tt, V1, V2) → A__ISNAT(V1)
A__U15(tt, V2) → A__U16(a__isNat(V2))
A__U15(tt, V2) → A__ISNAT(V2)
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, V2) → A__U32(a__isNatKind(V2))
A__U31(tt, V2) → A__ISNATKIND(V2)
A__U51(tt, N) → A__U52(a__isNatKind(N), N)
A__U51(tt, N) → A__ISNATKIND(N)
A__U52(tt, N) → MARK(N)
A__U61(tt, M, N) → A__U62(a__isNatKind(M), M, N)
A__U61(tt, M, N) → A__ISNATKIND(M)
A__U62(tt, M, N) → A__U63(a__isNat(N), M, N)
A__U62(tt, M, N) → A__ISNAT(N)
A__U63(tt, M, N) → A__U64(a__isNatKind(N), M, N)
A__U63(tt, M, N) → A__ISNATKIND(N)
A__U64(tt, M, N) → A__PLUS(mark(N), mark(M))
A__U64(tt, M, N) → MARK(N)
A__U64(tt, M, N) → MARK(M)
A__ISNAT(plus(V1, V2)) → A__U11(a__isNatKind(V1), V1, V2)
A__ISNAT(plus(V1, V2)) → A__ISNATKIND(V1)
A__ISNAT(s(V1)) → A__U21(a__isNatKind(V1), V1)
A__ISNAT(s(V1)) → A__ISNATKIND(V1)
A__ISNATKIND(plus(V1, V2)) → A__U31(a__isNatKind(V1), V2)
A__ISNATKIND(plus(V1, V2)) → A__ISNATKIND(V1)
A__ISNATKIND(s(V1)) → A__U41(a__isNatKind(V1))
A__ISNATKIND(s(V1)) → A__ISNATKIND(V1)
A__PLUS(N, 0) → A__U51(a__isNat(N), N)
A__PLUS(N, 0) → A__ISNAT(N)
A__PLUS(N, s(M)) → A__U61(a__isNat(M), M, N)
A__PLUS(N, s(M)) → A__ISNAT(M)
MARK(U11(X1, X2, X3)) → A__U11(mark(X1), X2, X3)
MARK(U11(X1, X2, X3)) → MARK(X1)
MARK(U12(X1, X2, X3)) → A__U12(mark(X1), X2, X3)
MARK(U12(X1, X2, X3)) → MARK(X1)
MARK(isNatKind(X)) → A__ISNATKIND(X)
MARK(U13(X1, X2, X3)) → A__U13(mark(X1), X2, X3)
MARK(U13(X1, X2, X3)) → MARK(X1)
MARK(U14(X1, X2, X3)) → A__U14(mark(X1), X2, X3)
MARK(U14(X1, X2, X3)) → MARK(X1)
MARK(U15(X1, X2)) → A__U15(mark(X1), X2)
MARK(U15(X1, X2)) → MARK(X1)
MARK(isNat(X)) → A__ISNAT(X)
MARK(U16(X)) → A__U16(mark(X))
MARK(U16(X)) → MARK(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(U23(X)) → A__U23(mark(X))
MARK(U23(X)) → MARK(X)
MARK(U31(X1, X2)) → A__U31(mark(X1), X2)
MARK(U31(X1, X2)) → MARK(X1)
MARK(U32(X)) → A__U32(mark(X))
MARK(U32(X)) → MARK(X)
MARK(U41(X)) → A__U41(mark(X))
MARK(U41(X)) → MARK(X)
MARK(U51(X1, X2)) → A__U51(mark(X1), X2)
MARK(U51(X1, X2)) → MARK(X1)
MARK(U52(X1, X2)) → A__U52(mark(X1), X2)
MARK(U52(X1, X2)) → MARK(X1)
MARK(U61(X1, X2, X3)) → A__U61(mark(X1), X2, X3)
MARK(U61(X1, X2, X3)) → MARK(X1)
MARK(U62(X1, X2, X3)) → A__U62(mark(X1), X2, X3)
MARK(U62(X1, X2, X3)) → MARK(X1)
MARK(U63(X1, X2, X3)) → A__U63(mark(X1), X2, X3)
MARK(U63(X1, X2, X3)) → MARK(X1)
MARK(U64(X1, X2, X3)) → A__U64(mark(X1), X2, X3)
MARK(U64(X1, X2, X3)) → MARK(X1)
MARK(plus(X1, X2)) → A__PLUS(mark(X1), mark(X2))
MARK(plus(X1, X2)) → MARK(X1)
MARK(plus(X1, X2)) → MARK(X2)
MARK(s(X)) → MARK(X)

The TRS R consists of the following rules:

a__U11(tt, V1, V2) → a__U12(a__isNatKind(V1), V1, V2)
a__U12(tt, V1, V2) → a__U13(a__isNatKind(V2), V1, V2)
a__U13(tt, V1, V2) → a__U14(a__isNatKind(V2), V1, V2)
a__U14(tt, V1, V2) → a__U15(a__isNat(V1), V2)
a__U15(tt, V2) → a__U16(a__isNat(V2))
a__U16(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, V2) → a__U32(a__isNatKind(V2))
a__U32(tt) → tt
a__U41(tt) → tt
a__U51(tt, N) → a__U52(a__isNatKind(N), N)
a__U52(tt, N) → mark(N)
a__U61(tt, M, N) → a__U62(a__isNatKind(M), M, N)
a__U62(tt, M, N) → a__U63(a__isNat(N), M, N)
a__U63(tt, M, N) → a__U64(a__isNatKind(N), M, N)
a__U64(tt, M, N) → s(a__plus(mark(N), mark(M)))
a__isNat(0) → tt
a__isNat(plus(V1, V2)) → a__U11(a__isNatKind(V1), V1, V2)
a__isNat(s(V1)) → a__U21(a__isNatKind(V1), V1)
a__isNatKind(0) → tt
a__isNatKind(plus(V1, V2)) → a__U31(a__isNatKind(V1), V2)
a__isNatKind(s(V1)) → a__U41(a__isNatKind(V1))
a__plus(N, 0) → a__U51(a__isNat(N), N)
a__plus(N, s(M)) → a__U61(a__isNat(M), M, N)
mark(U11(X1, X2, X3)) → a__U11(mark(X1), X2, X3)
mark(U12(X1, X2, X3)) → a__U12(mark(X1), X2, X3)
mark(isNatKind(X)) → a__isNatKind(X)
mark(U13(X1, X2, X3)) → a__U13(mark(X1), X2, X3)
mark(U14(X1, X2, X3)) → a__U14(mark(X1), X2, X3)
mark(U15(X1, X2)) → a__U15(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
mark(U16(X)) → a__U16(mark(X))
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(U22(X1, X2)) → a__U22(mark(X1), X2)
mark(U23(X)) → a__U23(mark(X))
mark(U31(X1, X2)) → a__U31(mark(X1), X2)
mark(U32(X)) → a__U32(mark(X))
mark(U41(X)) → a__U41(mark(X))
mark(U51(X1, X2)) → a__U51(mark(X1), X2)
mark(U52(X1, X2)) → a__U52(mark(X1), X2)
mark(U61(X1, X2, X3)) → a__U61(mark(X1), X2, X3)
mark(U62(X1, X2, X3)) → a__U62(mark(X1), X2, X3)
mark(U63(X1, X2, X3)) → a__U63(mark(X1), X2, X3)
mark(U64(X1, X2, X3)) → a__U64(mark(X1), X2, X3)
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(0) → 0
a__U11(X1, X2, X3) → U11(X1, X2, X3)
a__U12(X1, X2, X3) → U12(X1, X2, X3)
a__isNatKind(X) → isNatKind(X)
a__U13(X1, X2, X3) → U13(X1, X2, X3)
a__U14(X1, X2, X3) → U14(X1, X2, X3)
a__U15(X1, X2) → U15(X1, X2)
a__isNat(X) → isNat(X)
a__U16(X) → U16(X)
a__U21(X1, X2) → U21(X1, X2)
a__U22(X1, X2) → U22(X1, X2)
a__U23(X) → U23(X)
a__U31(X1, X2) → U31(X1, X2)
a__U32(X) → U32(X)
a__U41(X) → U41(X)
a__U51(X1, X2) → U51(X1, X2)
a__U52(X1, X2) → U52(X1, X2)
a__U61(X1, X2, X3) → U61(X1, X2, X3)
a__U62(X1, X2, X3) → U62(X1, X2, X3)
a__U63(X1, X2, X3) → U63(X1, X2, X3)
a__U64(X1, X2, X3) → U64(X1, X2, X3)
a__plus(X1, X2) → plus(X1, X2)

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 3 SCCs with 30 less nodes.

(4) Complex Obligation (AND)

(5) Obligation:

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

A__U31(tt, V2) → A__ISNATKIND(V2)
A__ISNATKIND(plus(V1, V2)) → A__U31(a__isNatKind(V1), V2)
A__ISNATKIND(plus(V1, V2)) → A__ISNATKIND(V1)
A__ISNATKIND(s(V1)) → A__ISNATKIND(V1)

The TRS R consists of the following rules:

a__U11(tt, V1, V2) → a__U12(a__isNatKind(V1), V1, V2)
a__U12(tt, V1, V2) → a__U13(a__isNatKind(V2), V1, V2)
a__U13(tt, V1, V2) → a__U14(a__isNatKind(V2), V1, V2)
a__U14(tt, V1, V2) → a__U15(a__isNat(V1), V2)
a__U15(tt, V2) → a__U16(a__isNat(V2))
a__U16(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, V2) → a__U32(a__isNatKind(V2))
a__U32(tt) → tt
a__U41(tt) → tt
a__U51(tt, N) → a__U52(a__isNatKind(N), N)
a__U52(tt, N) → mark(N)
a__U61(tt, M, N) → a__U62(a__isNatKind(M), M, N)
a__U62(tt, M, N) → a__U63(a__isNat(N), M, N)
a__U63(tt, M, N) → a__U64(a__isNatKind(N), M, N)
a__U64(tt, M, N) → s(a__plus(mark(N), mark(M)))
a__isNat(0) → tt
a__isNat(plus(V1, V2)) → a__U11(a__isNatKind(V1), V1, V2)
a__isNat(s(V1)) → a__U21(a__isNatKind(V1), V1)
a__isNatKind(0) → tt
a__isNatKind(plus(V1, V2)) → a__U31(a__isNatKind(V1), V2)
a__isNatKind(s(V1)) → a__U41(a__isNatKind(V1))
a__plus(N, 0) → a__U51(a__isNat(N), N)
a__plus(N, s(M)) → a__U61(a__isNat(M), M, N)
mark(U11(X1, X2, X3)) → a__U11(mark(X1), X2, X3)
mark(U12(X1, X2, X3)) → a__U12(mark(X1), X2, X3)
mark(isNatKind(X)) → a__isNatKind(X)
mark(U13(X1, X2, X3)) → a__U13(mark(X1), X2, X3)
mark(U14(X1, X2, X3)) → a__U14(mark(X1), X2, X3)
mark(U15(X1, X2)) → a__U15(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
mark(U16(X)) → a__U16(mark(X))
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(U22(X1, X2)) → a__U22(mark(X1), X2)
mark(U23(X)) → a__U23(mark(X))
mark(U31(X1, X2)) → a__U31(mark(X1), X2)
mark(U32(X)) → a__U32(mark(X))
mark(U41(X)) → a__U41(mark(X))
mark(U51(X1, X2)) → a__U51(mark(X1), X2)
mark(U52(X1, X2)) → a__U52(mark(X1), X2)
mark(U61(X1, X2, X3)) → a__U61(mark(X1), X2, X3)
mark(U62(X1, X2, X3)) → a__U62(mark(X1), X2, X3)
mark(U63(X1, X2, X3)) → a__U63(mark(X1), X2, X3)
mark(U64(X1, X2, X3)) → a__U64(mark(X1), X2, X3)
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(0) → 0
a__U11(X1, X2, X3) → U11(X1, X2, X3)
a__U12(X1, X2, X3) → U12(X1, X2, X3)
a__isNatKind(X) → isNatKind(X)
a__U13(X1, X2, X3) → U13(X1, X2, X3)
a__U14(X1, X2, X3) → U14(X1, X2, X3)
a__U15(X1, X2) → U15(X1, X2)
a__isNat(X) → isNat(X)
a__U16(X) → U16(X)
a__U21(X1, X2) → U21(X1, X2)
a__U22(X1, X2) → U22(X1, X2)
a__U23(X) → U23(X)
a__U31(X1, X2) → U31(X1, X2)
a__U32(X) → U32(X)
a__U41(X) → U41(X)
a__U51(X1, X2) → U51(X1, X2)
a__U52(X1, X2) → U52(X1, X2)
a__U61(X1, X2, X3) → U61(X1, X2, X3)
a__U62(X1, X2, X3) → U62(X1, X2, X3)
a__U63(X1, X2, X3) → U63(X1, X2, X3)
a__U64(X1, X2, X3) → U64(X1, X2, X3)
a__plus(X1, X2) → plus(X1, X2)

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

(6) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

A__ISNATKIND(plus(V1, V2)) → A__U31(a__isNatKind(V1), V2)
A__ISNATKIND(plus(V1, V2)) → A__ISNATKIND(V1)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
A__U31(x1, x2) = A__U31(x2)
tt = tt
A__ISNATKIND(x1) = A__ISNATKIND(x1)
plus(x1, x2) = plus(x1, x2)
a__isNatKind(x1) = a__isNatKind(x1)
s(x1) = x1
a__U11(x1, x2, x3) = a__U11(x2, x3)
a__U12(x1, x2, x3) = a__U12(x2, x3)
a__U13(x1, x2, x3) = a__U13(x2, x3)
a__U14(x1, x2, x3) = a__U14(x2)
a__U15(x1, x2) = x1
a__isNat(x1) = a__isNat(x1)
a__U16(x1) = a__U16
a__U21(x1, x2) = a__U21(x2)
a__U22(x1, x2) = a__U22(x2)
a__U23(x1) = x1
a__U31(x1, x2) = a__U31(x2)
a__U32(x1) = x1
a__U41(x1) = x1
a__U51(x1, x2) = x2
a__U52(x1, x2) = x2
mark(x1) = x1
a__U61(x1, x2, x3) = a__U61(x2, x3)
a__U62(x1, x2, x3) = a__U62(x2, x3)
a__U63(x1, x2, x3) = a__U63(x2, x3)
a__U64(x1, x2, x3) = a__U64(x2, x3)
a__plus(x1, x2) = a__plus(x1, x2)
0 = 0
U11(x1, x2, x3) = U11(x2, x3)
U12(x1, x2, x3) = U12(x2, x3)
isNatKind(x1) = isNatKind(x1)
U13(x1, x2, x3) = U13(x2, x3)
U14(x1, x2, x3) = U14(x2)
U15(x1, x2) = x1
isNat(x1) = isNat(x1)
U16(x1) = U16
U21(x1, x2) = U21(x2)
U22(x1, x2) = U22(x2)
U23(x1) = x1
U31(x1, x2) = U31(x2)
U32(x1) = x1
U41(x1) = x1
U51(x1, x2) = x2
U52(x1, x2) = x2
U61(x1, x2, x3) = U61(x2, x3)
U62(x1, x2, x3) = U62(x2, x3)
U63(x1, x2, x3) = U63(x2, x3)
U64(x1, x2, x3) = U64(x2, x3)

Recursive Path Order [RPO].
Precedence:

[A_{_U31₁}, A_{_ISNATKIND₁}] > [a_{_isNatKind₁}, a_{_U31₁}, isNatKind₁, U31₁] > [a_{_U11₂}, a_{_U12₂}, a_{_U13₂}, a_{_U14₁}, a_{_isNat₁}, a_{_U21₁}, a_{_U22₁}, U11₂, U12₂, U13₂, U14₁, isNat₁, U21₁, U22₁]
0 > [tt, a_{_U16}, U16] > [plus₂, a_{_U61₂}, a_{_U62₂}, a_{_U63₂}, a_{_U64₂}, a_{_plus₂}, U61₂, U62₂, U63₂, U64₂] > [a_{_U11₂}, a_{_U12₂}, a_{_U13₂}, a_{_U14₁}, a_{_isNat₁}, a_{_U21₁}, a_{_U22₁}, U11₂, U12₂, U13₂, U14₁, isNat₁, U21₁, U22₁]
0 > [tt, a_{_U16}, U16] > [a_{_isNatKind₁}, a_{_U31₁}, isNatKind₁, U31₁] > [a_{_U11₂}, a_{_U12₂}, a_{_U13₂}, a_{_U14₁}, a_{_isNat₁}, a_{_U21₁}, a_{_U22₁}, U11₂, U12₂, U13₂, U14₁, isNat₁, U21₁, U22₁]

The following usable rules [FROCOS05] were oriented:

a__U11(tt, V1, V2) → a__U12(a__isNatKind(V1), V1, V2)
a__U12(tt, V1, V2) → a__U13(a__isNatKind(V2), V1, V2)
a__U13(tt, V1, V2) → a__U14(a__isNatKind(V2), V1, V2)
a__U14(tt, V1, V2) → a__U15(a__isNat(V1), V2)
a__U15(tt, V2) → a__U16(a__isNat(V2))
a__U16(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, V2) → a__U32(a__isNatKind(V2))
a__U32(tt) → tt
a__U41(tt) → tt
a__U51(tt, N) → a__U52(a__isNatKind(N), N)
a__U52(tt, N) → mark(N)
a__U61(tt, M, N) → a__U62(a__isNatKind(M), M, N)
a__U62(tt, M, N) → a__U63(a__isNat(N), M, N)
a__U63(tt, M, N) → a__U64(a__isNatKind(N), M, N)
a__U64(tt, M, N) → s(a__plus(mark(N), mark(M)))
a__isNat(0) → tt
a__isNat(plus(V1, V2)) → a__U11(a__isNatKind(V1), V1, V2)
a__isNat(s(V1)) → a__U21(a__isNatKind(V1), V1)
a__isNatKind(0) → tt
a__isNatKind(plus(V1, V2)) → a__U31(a__isNatKind(V1), V2)
a__isNatKind(s(V1)) → a__U41(a__isNatKind(V1))
a__plus(N, 0) → a__U51(a__isNat(N), N)
a__plus(N, s(M)) → a__U61(a__isNat(M), M, N)
mark(U11(X1, X2, X3)) → a__U11(mark(X1), X2, X3)
mark(U12(X1, X2, X3)) → a__U12(mark(X1), X2, X3)
mark(isNatKind(X)) → a__isNatKind(X)
mark(U13(X1, X2, X3)) → a__U13(mark(X1), X2, X3)
mark(U14(X1, X2, X3)) → a__U14(mark(X1), X2, X3)
mark(U15(X1, X2)) → a__U15(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
mark(U16(X)) → a__U16(mark(X))
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(U22(X1, X2)) → a__U22(mark(X1), X2)
mark(U23(X)) → a__U23(mark(X))
mark(U31(X1, X2)) → a__U31(mark(X1), X2)
mark(U32(X)) → a__U32(mark(X))
mark(U41(X)) → a__U41(mark(X))
mark(U51(X1, X2)) → a__U51(mark(X1), X2)
mark(U52(X1, X2)) → a__U52(mark(X1), X2)
mark(U61(X1, X2, X3)) → a__U61(mark(X1), X2, X3)
mark(U62(X1, X2, X3)) → a__U62(mark(X1), X2, X3)
mark(U63(X1, X2, X3)) → a__U63(mark(X1), X2, X3)
mark(U64(X1, X2, X3)) → a__U64(mark(X1), X2, X3)
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(0) → 0
a__U11(X1, X2, X3) → U11(X1, X2, X3)
a__U12(X1, X2, X3) → U12(X1, X2, X3)
a__isNatKind(X) → isNatKind(X)
a__U13(X1, X2, X3) → U13(X1, X2, X3)
a__U14(X1, X2, X3) → U14(X1, X2, X3)
a__U15(X1, X2) → U15(X1, X2)
a__isNat(X) → isNat(X)
a__U16(X) → U16(X)
a__U21(X1, X2) → U21(X1, X2)
a__U22(X1, X2) → U22(X1, X2)
a__U23(X) → U23(X)
a__U31(X1, X2) → U31(X1, X2)
a__U32(X) → U32(X)
a__U41(X) → U41(X)
a__U51(X1, X2) → U51(X1, X2)
a__U52(X1, X2) → U52(X1, X2)
a__U61(X1, X2, X3) → U61(X1, X2, X3)
a__U62(X1, X2, X3) → U62(X1, X2, X3)
a__U63(X1, X2, X3) → U63(X1, X2, X3)
a__U64(X1, X2, X3) → U64(X1, X2, X3)
a__plus(X1, X2) → plus(X1, X2)

(7) Obligation:

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

A__U31(tt, V2) → A__ISNATKIND(V2)
A__ISNATKIND(s(V1)) → A__ISNATKIND(V1)

The TRS R consists of the following rules:

a__U11(tt, V1, V2) → a__U12(a__isNatKind(V1), V1, V2)
a__U12(tt, V1, V2) → a__U13(a__isNatKind(V2), V1, V2)
a__U13(tt, V1, V2) → a__U14(a__isNatKind(V2), V1, V2)
a__U14(tt, V1, V2) → a__U15(a__isNat(V1), V2)
a__U15(tt, V2) → a__U16(a__isNat(V2))
a__U16(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, V2) → a__U32(a__isNatKind(V2))
a__U32(tt) → tt
a__U41(tt) → tt
a__U51(tt, N) → a__U52(a__isNatKind(N), N)
a__U52(tt, N) → mark(N)
a__U61(tt, M, N) → a__U62(a__isNatKind(M), M, N)
a__U62(tt, M, N) → a__U63(a__isNat(N), M, N)
a__U63(tt, M, N) → a__U64(a__isNatKind(N), M, N)
a__U64(tt, M, N) → s(a__plus(mark(N), mark(M)))
a__isNat(0) → tt
a__isNat(plus(V1, V2)) → a__U11(a__isNatKind(V1), V1, V2)
a__isNat(s(V1)) → a__U21(a__isNatKind(V1), V1)
a__isNatKind(0) → tt
a__isNatKind(plus(V1, V2)) → a__U31(a__isNatKind(V1), V2)
a__isNatKind(s(V1)) → a__U41(a__isNatKind(V1))
a__plus(N, 0) → a__U51(a__isNat(N), N)
a__plus(N, s(M)) → a__U61(a__isNat(M), M, N)
mark(U11(X1, X2, X3)) → a__U11(mark(X1), X2, X3)
mark(U12(X1, X2, X3)) → a__U12(mark(X1), X2, X3)
mark(isNatKind(X)) → a__isNatKind(X)
mark(U13(X1, X2, X3)) → a__U13(mark(X1), X2, X3)
mark(U14(X1, X2, X3)) → a__U14(mark(X1), X2, X3)
mark(U15(X1, X2)) → a__U15(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
mark(U16(X)) → a__U16(mark(X))
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(U22(X1, X2)) → a__U22(mark(X1), X2)
mark(U23(X)) → a__U23(mark(X))
mark(U31(X1, X2)) → a__U31(mark(X1), X2)
mark(U32(X)) → a__U32(mark(X))
mark(U41(X)) → a__U41(mark(X))
mark(U51(X1, X2)) → a__U51(mark(X1), X2)
mark(U52(X1, X2)) → a__U52(mark(X1), X2)
mark(U61(X1, X2, X3)) → a__U61(mark(X1), X2, X3)
mark(U62(X1, X2, X3)) → a__U62(mark(X1), X2, X3)
mark(U63(X1, X2, X3)) → a__U63(mark(X1), X2, X3)
mark(U64(X1, X2, X3)) → a__U64(mark(X1), X2, X3)
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(0) → 0
a__U11(X1, X2, X3) → U11(X1, X2, X3)
a__U12(X1, X2, X3) → U12(X1, X2, X3)
a__isNatKind(X) → isNatKind(X)
a__U13(X1, X2, X3) → U13(X1, X2, X3)
a__U14(X1, X2, X3) → U14(X1, X2, X3)
a__U15(X1, X2) → U15(X1, X2)
a__isNat(X) → isNat(X)
a__U16(X) → U16(X)
a__U21(X1, X2) → U21(X1, X2)
a__U22(X1, X2) → U22(X1, X2)
a__U23(X) → U23(X)
a__U31(X1, X2) → U31(X1, X2)
a__U32(X) → U32(X)
a__U41(X) → U41(X)
a__U51(X1, X2) → U51(X1, X2)
a__U52(X1, X2) → U52(X1, X2)
a__U61(X1, X2, X3) → U61(X1, X2, X3)
a__U62(X1, X2, X3) → U62(X1, X2, X3)
a__U63(X1, X2, X3) → U63(X1, X2, X3)
a__U64(X1, X2, X3) → U64(X1, X2, X3)
a__plus(X1, X2) → plus(X1, X2)

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

(8) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.

(9) Obligation:

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

A__ISNATKIND(s(V1)) → A__ISNATKIND(V1)

The TRS R consists of the following rules:

a__U11(tt, V1, V2) → a__U12(a__isNatKind(V1), V1, V2)
a__U12(tt, V1, V2) → a__U13(a__isNatKind(V2), V1, V2)
a__U13(tt, V1, V2) → a__U14(a__isNatKind(V2), V1, V2)
a__U14(tt, V1, V2) → a__U15(a__isNat(V1), V2)
a__U15(tt, V2) → a__U16(a__isNat(V2))
a__U16(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, V2) → a__U32(a__isNatKind(V2))
a__U32(tt) → tt
a__U41(tt) → tt
a__U51(tt, N) → a__U52(a__isNatKind(N), N)
a__U52(tt, N) → mark(N)
a__U61(tt, M, N) → a__U62(a__isNatKind(M), M, N)
a__U62(tt, M, N) → a__U63(a__isNat(N), M, N)
a__U63(tt, M, N) → a__U64(a__isNatKind(N), M, N)
a__U64(tt, M, N) → s(a__plus(mark(N), mark(M)))
a__isNat(0) → tt
a__isNat(plus(V1, V2)) → a__U11(a__isNatKind(V1), V1, V2)
a__isNat(s(V1)) → a__U21(a__isNatKind(V1), V1)
a__isNatKind(0) → tt
a__isNatKind(plus(V1, V2)) → a__U31(a__isNatKind(V1), V2)
a__isNatKind(s(V1)) → a__U41(a__isNatKind(V1))
a__plus(N, 0) → a__U51(a__isNat(N), N)
a__plus(N, s(M)) → a__U61(a__isNat(M), M, N)
mark(U11(X1, X2, X3)) → a__U11(mark(X1), X2, X3)
mark(U12(X1, X2, X3)) → a__U12(mark(X1), X2, X3)
mark(isNatKind(X)) → a__isNatKind(X)
mark(U13(X1, X2, X3)) → a__U13(mark(X1), X2, X3)
mark(U14(X1, X2, X3)) → a__U14(mark(X1), X2, X3)
mark(U15(X1, X2)) → a__U15(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
mark(U16(X)) → a__U16(mark(X))
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(U22(X1, X2)) → a__U22(mark(X1), X2)
mark(U23(X)) → a__U23(mark(X))
mark(U31(X1, X2)) → a__U31(mark(X1), X2)
mark(U32(X)) → a__U32(mark(X))
mark(U41(X)) → a__U41(mark(X))
mark(U51(X1, X2)) → a__U51(mark(X1), X2)
mark(U52(X1, X2)) → a__U52(mark(X1), X2)
mark(U61(X1, X2, X3)) → a__U61(mark(X1), X2, X3)
mark(U62(X1, X2, X3)) → a__U62(mark(X1), X2, X3)
mark(U63(X1, X2, X3)) → a__U63(mark(X1), X2, X3)
mark(U64(X1, X2, X3)) → a__U64(mark(X1), X2, X3)
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(0) → 0
a__U11(X1, X2, X3) → U11(X1, X2, X3)
a__U12(X1, X2, X3) → U12(X1, X2, X3)
a__isNatKind(X) → isNatKind(X)
a__U13(X1, X2, X3) → U13(X1, X2, X3)
a__U14(X1, X2, X3) → U14(X1, X2, X3)
a__U15(X1, X2) → U15(X1, X2)
a__isNat(X) → isNat(X)
a__U16(X) → U16(X)
a__U21(X1, X2) → U21(X1, X2)
a__U22(X1, X2) → U22(X1, X2)
a__U23(X) → U23(X)
a__U31(X1, X2) → U31(X1, X2)
a__U32(X) → U32(X)
a__U41(X) → U41(X)
a__U51(X1, X2) → U51(X1, X2)
a__U52(X1, X2) → U52(X1, X2)
a__U61(X1, X2, X3) → U61(X1, X2, X3)
a__U62(X1, X2, X3) → U62(X1, X2, X3)
a__U63(X1, X2, X3) → U63(X1, X2, X3)
a__U64(X1, X2, X3) → U64(X1, X2, X3)
a__plus(X1, X2) → plus(X1, X2)

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

(10) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

A__ISNATKIND(s(V1)) → A__ISNATKIND(V1)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
A__ISNATKIND(x1) = x1
s(x1) = s(x1)
a__U11(x1, x2, x3) = x1
tt = tt
a__U12(x1, x2, x3) = x1
a__isNatKind(x1) = a__isNatKind
a__U13(x1, x2, x3) = x1
a__U14(x1, x2, x3) = x1
a__U15(x1, x2) = x1
a__isNat(x1) = a__isNat
a__U16(x1) = a__U16
a__U21(x1, x2) = x1
a__U22(x1, x2) = x1
a__U23(x1) = a__U23
a__U31(x1, x2) = a__U31
a__U32(x1) = a__U32
a__U41(x1) = a__U41
a__U51(x1, x2) = x2
a__U52(x1, x2) = x2
mark(x1) = x1
a__U61(x1, x2, x3) = a__U61(x1, x2, x3)
a__U62(x1, x2, x3) = a__U62(x1, x2, x3)
a__U63(x1, x2, x3) = a__U63(x1, x2, x3)
a__U64(x1, x2, x3) = a__U64(x1, x2, x3)
a__plus(x1, x2) = a__plus(x1, x2)
0 = 0
plus(x1, x2) = plus(x1, x2)
U11(x1, x2, x3) = x1
U12(x1, x2, x3) = x1
isNatKind(x1) = isNatKind
U13(x1, x2, x3) = x1
U14(x1, x2, x3) = x1
U15(x1, x2) = x1
isNat(x1) = isNat
U16(x1) = U16
U21(x1, x2) = x1
U22(x1, x2) = x1
U23(x1) = U23
U31(x1, x2) = U31
U32(x1) = U32
U41(x1) = U41
U51(x1, x2) = x2
U52(x1, x2) = x2
U61(x1, x2, x3) = U61(x1, x2, x3)
U62(x1, x2, x3) = U62(x1, x2, x3)
U63(x1, x2, x3) = U63(x1, x2, x3)
U64(x1, x2, x3) = U64(x1, x2, x3)

Recursive Path Order [RPO].
Precedence:

[a_{_U61₃}, a_{_U62₃}, a_{_U63₃}, a_{_U64₃}, a_{_plus₂}, plus₂, U61₃, U62₃, U63₃, U64₃] > s₁ > [tt, a_{_isNatKind}, a_{_isNat}, a_{_U16}, a_{_U23}, a_{_U31}, a_{_U32}, a_{_U41}, isNatKind, isNat, U16, U23, U31, U32, U41]
0 > [tt, a_{_isNatKind}, a_{_isNat}, a_{_U16}, a_{_U23}, a_{_U31}, a_{_U32}, a_{_U41}, isNatKind, isNat, U16, U23, U31, U32, U41]

The following usable rules [FROCOS05] were oriented:

a__U11(tt, V1, V2) → a__U12(a__isNatKind(V1), V1, V2)
a__U12(tt, V1, V2) → a__U13(a__isNatKind(V2), V1, V2)
a__U13(tt, V1, V2) → a__U14(a__isNatKind(V2), V1, V2)
a__U14(tt, V1, V2) → a__U15(a__isNat(V1), V2)
a__U15(tt, V2) → a__U16(a__isNat(V2))
a__U16(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, V2) → a__U32(a__isNatKind(V2))
a__U32(tt) → tt
a__U41(tt) → tt
a__U51(tt, N) → a__U52(a__isNatKind(N), N)
a__U52(tt, N) → mark(N)
a__U61(tt, M, N) → a__U62(a__isNatKind(M), M, N)
a__U62(tt, M, N) → a__U63(a__isNat(N), M, N)
a__U63(tt, M, N) → a__U64(a__isNatKind(N), M, N)
a__U64(tt, M, N) → s(a__plus(mark(N), mark(M)))
a__isNat(0) → tt
a__isNat(plus(V1, V2)) → a__U11(a__isNatKind(V1), V1, V2)
a__isNat(s(V1)) → a__U21(a__isNatKind(V1), V1)
a__isNatKind(0) → tt
a__isNatKind(plus(V1, V2)) → a__U31(a__isNatKind(V1), V2)
a__isNatKind(s(V1)) → a__U41(a__isNatKind(V1))
a__plus(N, 0) → a__U51(a__isNat(N), N)
a__plus(N, s(M)) → a__U61(a__isNat(M), M, N)
mark(U11(X1, X2, X3)) → a__U11(mark(X1), X2, X3)
mark(U12(X1, X2, X3)) → a__U12(mark(X1), X2, X3)
mark(isNatKind(X)) → a__isNatKind(X)
mark(U13(X1, X2, X3)) → a__U13(mark(X1), X2, X3)
mark(U14(X1, X2, X3)) → a__U14(mark(X1), X2, X3)
mark(U15(X1, X2)) → a__U15(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
mark(U16(X)) → a__U16(mark(X))
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(U22(X1, X2)) → a__U22(mark(X1), X2)
mark(U23(X)) → a__U23(mark(X))
mark(U31(X1, X2)) → a__U31(mark(X1), X2)
mark(U32(X)) → a__U32(mark(X))
mark(U41(X)) → a__U41(mark(X))
mark(U51(X1, X2)) → a__U51(mark(X1), X2)
mark(U52(X1, X2)) → a__U52(mark(X1), X2)
mark(U61(X1, X2, X3)) → a__U61(mark(X1), X2, X3)
mark(U62(X1, X2, X3)) → a__U62(mark(X1), X2, X3)
mark(U63(X1, X2, X3)) → a__U63(mark(X1), X2, X3)
mark(U64(X1, X2, X3)) → a__U64(mark(X1), X2, X3)
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(0) → 0
a__U11(X1, X2, X3) → U11(X1, X2, X3)
a__U12(X1, X2, X3) → U12(X1, X2, X3)
a__isNatKind(X) → isNatKind(X)
a__U13(X1, X2, X3) → U13(X1, X2, X3)
a__U14(X1, X2, X3) → U14(X1, X2, X3)
a__U15(X1, X2) → U15(X1, X2)
a__isNat(X) → isNat(X)
a__U16(X) → U16(X)
a__U21(X1, X2) → U21(X1, X2)
a__U22(X1, X2) → U22(X1, X2)
a__U23(X) → U23(X)
a__U31(X1, X2) → U31(X1, X2)
a__U32(X) → U32(X)
a__U41(X) → U41(X)
a__U51(X1, X2) → U51(X1, X2)
a__U52(X1, X2) → U52(X1, X2)
a__U61(X1, X2, X3) → U61(X1, X2, X3)
a__U62(X1, X2, X3) → U62(X1, X2, X3)
a__U63(X1, X2, X3) → U63(X1, X2, X3)
a__U64(X1, X2, X3) → U64(X1, X2, X3)
a__plus(X1, X2) → plus(X1, X2)

(11) Obligation:

Q DP problem:
P is empty.
The TRS R consists of the following rules:

a__U11(tt, V1, V2) → a__U12(a__isNatKind(V1), V1, V2)
a__U12(tt, V1, V2) → a__U13(a__isNatKind(V2), V1, V2)
a__U13(tt, V1, V2) → a__U14(a__isNatKind(V2), V1, V2)
a__U14(tt, V1, V2) → a__U15(a__isNat(V1), V2)
a__U15(tt, V2) → a__U16(a__isNat(V2))
a__U16(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, V2) → a__U32(a__isNatKind(V2))
a__U32(tt) → tt
a__U41(tt) → tt
a__U51(tt, N) → a__U52(a__isNatKind(N), N)
a__U52(tt, N) → mark(N)
a__U61(tt, M, N) → a__U62(a__isNatKind(M), M, N)
a__U62(tt, M, N) → a__U63(a__isNat(N), M, N)
a__U63(tt, M, N) → a__U64(a__isNatKind(N), M, N)
a__U64(tt, M, N) → s(a__plus(mark(N), mark(M)))
a__isNat(0) → tt
a__isNat(plus(V1, V2)) → a__U11(a__isNatKind(V1), V1, V2)
a__isNat(s(V1)) → a__U21(a__isNatKind(V1), V1)
a__isNatKind(0) → tt
a__isNatKind(plus(V1, V2)) → a__U31(a__isNatKind(V1), V2)
a__isNatKind(s(V1)) → a__U41(a__isNatKind(V1))
a__plus(N, 0) → a__U51(a__isNat(N), N)
a__plus(N, s(M)) → a__U61(a__isNat(M), M, N)
mark(U11(X1, X2, X3)) → a__U11(mark(X1), X2, X3)
mark(U12(X1, X2, X3)) → a__U12(mark(X1), X2, X3)
mark(isNatKind(X)) → a__isNatKind(X)
mark(U13(X1, X2, X3)) → a__U13(mark(X1), X2, X3)
mark(U14(X1, X2, X3)) → a__U14(mark(X1), X2, X3)
mark(U15(X1, X2)) → a__U15(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
mark(U16(X)) → a__U16(mark(X))
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(U22(X1, X2)) → a__U22(mark(X1), X2)
mark(U23(X)) → a__U23(mark(X))
mark(U31(X1, X2)) → a__U31(mark(X1), X2)
mark(U32(X)) → a__U32(mark(X))
mark(U41(X)) → a__U41(mark(X))
mark(U51(X1, X2)) → a__U51(mark(X1), X2)
mark(U52(X1, X2)) → a__U52(mark(X1), X2)
mark(U61(X1, X2, X3)) → a__U61(mark(X1), X2, X3)
mark(U62(X1, X2, X3)) → a__U62(mark(X1), X2, X3)
mark(U63(X1, X2, X3)) → a__U63(mark(X1), X2, X3)
mark(U64(X1, X2, X3)) → a__U64(mark(X1), X2, X3)
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(0) → 0
a__U11(X1, X2, X3) → U11(X1, X2, X3)
a__U12(X1, X2, X3) → U12(X1, X2, X3)
a__isNatKind(X) → isNatKind(X)
a__U13(X1, X2, X3) → U13(X1, X2, X3)
a__U14(X1, X2, X3) → U14(X1, X2, X3)
a__U15(X1, X2) → U15(X1, X2)
a__isNat(X) → isNat(X)
a__U16(X) → U16(X)
a__U21(X1, X2) → U21(X1, X2)
a__U22(X1, X2) → U22(X1, X2)
a__U23(X) → U23(X)
a__U31(X1, X2) → U31(X1, X2)
a__U32(X) → U32(X)
a__U41(X) → U41(X)
a__U51(X1, X2) → U51(X1, X2)
a__U52(X1, X2) → U52(X1, X2)
a__U61(X1, X2, X3) → U61(X1, X2, X3)
a__U62(X1, X2, X3) → U62(X1, X2, X3)
a__U63(X1, X2, X3) → U63(X1, X2, X3)
a__U64(X1, X2, X3) → U64(X1, X2, X3)
a__plus(X1, X2) → plus(X1, X2)

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

(12) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(13) TRUE

(14) Obligation:

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

A__U12(tt, V1, V2) → A__U13(a__isNatKind(V2), V1, V2)
A__U13(tt, V1, V2) → A__U14(a__isNatKind(V2), V1, V2)
A__U14(tt, V1, V2) → A__U15(a__isNat(V1), V2)
A__U15(tt, V2) → A__ISNAT(V2)
A__ISNAT(plus(V1, V2)) → A__U11(a__isNatKind(V1), V1, V2)
A__U11(tt, V1, V2) → A__U12(a__isNatKind(V1), V1, V2)
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)
A__U14(tt, V1, V2) → A__ISNAT(V1)

The TRS R consists of the following rules:

a__U11(tt, V1, V2) → a__U12(a__isNatKind(V1), V1, V2)
a__U12(tt, V1, V2) → a__U13(a__isNatKind(V2), V1, V2)
a__U13(tt, V1, V2) → a__U14(a__isNatKind(V2), V1, V2)
a__U14(tt, V1, V2) → a__U15(a__isNat(V1), V2)
a__U15(tt, V2) → a__U16(a__isNat(V2))
a__U16(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, V2) → a__U32(a__isNatKind(V2))
a__U32(tt) → tt
a__U41(tt) → tt
a__U51(tt, N) → a__U52(a__isNatKind(N), N)
a__U52(tt, N) → mark(N)
a__U61(tt, M, N) → a__U62(a__isNatKind(M), M, N)
a__U62(tt, M, N) → a__U63(a__isNat(N), M, N)
a__U63(tt, M, N) → a__U64(a__isNatKind(N), M, N)
a__U64(tt, M, N) → s(a__plus(mark(N), mark(M)))
a__isNat(0) → tt
a__isNat(plus(V1, V2)) → a__U11(a__isNatKind(V1), V1, V2)
a__isNat(s(V1)) → a__U21(a__isNatKind(V1), V1)
a__isNatKind(0) → tt
a__isNatKind(plus(V1, V2)) → a__U31(a__isNatKind(V1), V2)
a__isNatKind(s(V1)) → a__U41(a__isNatKind(V1))
a__plus(N, 0) → a__U51(a__isNat(N), N)
a__plus(N, s(M)) → a__U61(a__isNat(M), M, N)
mark(U11(X1, X2, X3)) → a__U11(mark(X1), X2, X3)
mark(U12(X1, X2, X3)) → a__U12(mark(X1), X2, X3)
mark(isNatKind(X)) → a__isNatKind(X)
mark(U13(X1, X2, X3)) → a__U13(mark(X1), X2, X3)
mark(U14(X1, X2, X3)) → a__U14(mark(X1), X2, X3)
mark(U15(X1, X2)) → a__U15(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
mark(U16(X)) → a__U16(mark(X))
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(U22(X1, X2)) → a__U22(mark(X1), X2)
mark(U23(X)) → a__U23(mark(X))
mark(U31(X1, X2)) → a__U31(mark(X1), X2)
mark(U32(X)) → a__U32(mark(X))
mark(U41(X)) → a__U41(mark(X))
mark(U51(X1, X2)) → a__U51(mark(X1), X2)
mark(U52(X1, X2)) → a__U52(mark(X1), X2)
mark(U61(X1, X2, X3)) → a__U61(mark(X1), X2, X3)
mark(U62(X1, X2, X3)) → a__U62(mark(X1), X2, X3)
mark(U63(X1, X2, X3)) → a__U63(mark(X1), X2, X3)
mark(U64(X1, X2, X3)) → a__U64(mark(X1), X2, X3)
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(0) → 0
a__U11(X1, X2, X3) → U11(X1, X2, X3)
a__U12(X1, X2, X3) → U12(X1, X2, X3)
a__isNatKind(X) → isNatKind(X)
a__U13(X1, X2, X3) → U13(X1, X2, X3)
a__U14(X1, X2, X3) → U14(X1, X2, X3)
a__U15(X1, X2) → U15(X1, X2)
a__isNat(X) → isNat(X)
a__U16(X) → U16(X)
a__U21(X1, X2) → U21(X1, X2)
a__U22(X1, X2) → U22(X1, X2)
a__U23(X) → U23(X)
a__U31(X1, X2) → U31(X1, X2)
a__U32(X) → U32(X)
a__U41(X) → U41(X)
a__U51(X1, X2) → U51(X1, X2)
a__U52(X1, X2) → U52(X1, X2)
a__U61(X1, X2, X3) → U61(X1, X2, X3)
a__U62(X1, X2, X3) → U62(X1, X2, X3)
a__U63(X1, X2, X3) → U63(X1, X2, X3)
a__U64(X1, X2, X3) → U64(X1, X2, X3)
a__plus(X1, X2) → plus(X1, X2)

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

(15) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

A__U13(tt, V1, V2) → A__U14(a__isNatKind(V2), V1, V2)
A__U14(tt, V1, V2) → A__U15(a__isNat(V1), V2)
A__U15(tt, V2) → A__ISNAT(V2)
A__ISNAT(plus(V1, V2)) → A__U11(a__isNatKind(V1), V1, V2)
A__U11(tt, V1, V2) → A__U12(a__isNatKind(V1), V1, V2)
A__ISNAT(s(V1)) → A__U21(a__isNatKind(V1), V1)
A__U14(tt, V1, V2) → A__ISNAT(V1)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
A__U12(x1, x2, x3) = A__U12(x2, x3)
tt = tt
A__U13(x1, x2, x3) = A__U13(x2, x3)
a__isNatKind(x1) = a__isNatKind
A__U14(x1, x2, x3) = A__U14(x1, x2, x3)
A__U15(x1, x2) = A__U15(x1, x2)
a__isNat(x1) = a__isNat
A__ISNAT(x1) = x1
plus(x1, x2) = plus(x1, x2)
A__U11(x1, x2, x3) = A__U11(x1, x2, x3)
s(x1) = s(x1)
A__U21(x1, x2) = x2
A__U22(x1, x2) = x2
a__U11(x1, x2, x3) = x1
a__U12(x1, x2, x3) = x1
a__U13(x1, x2, x3) = x1
a__U14(x1, x2, x3) = x1
a__U15(x1, x2) = x1
a__U16(x1) = x1
a__U21(x1, x2) = x1
a__U22(x1, x2) = x1
a__U23(x1) = x1
a__U31(x1, x2) = x1
a__U32(x1) = x1
a__U41(x1) = x1
a__U51(x1, x2) = a__U51(x2)
a__U52(x1, x2) = x2
mark(x1) = x1
a__U61(x1, x2, x3) = a__U61(x1, x2, x3)
a__U62(x1, x2, x3) = a__U62(x1, x2, x3)
a__U63(x1, x2, x3) = a__U63(x1, x2, x3)
a__U64(x1, x2, x3) = a__U64(x1, x2, x3)
a__plus(x1, x2) = a__plus(x1, x2)
0 = 0
U11(x1, x2, x3) = x1
U12(x1, x2, x3) = x1
isNatKind(x1) = isNatKind
U13(x1, x2, x3) = x1
U14(x1, x2, x3) = x1
U15(x1, x2) = x1
isNat(x1) = isNat
U16(x1) = x1
U21(x1, x2) = x1
U22(x1, x2) = x1
U23(x1) = x1
U31(x1, x2) = x1
U32(x1) = x1
U41(x1) = x1
U51(x1, x2) = U51(x2)
U52(x1, x2) = x2
U61(x1, x2, x3) = U61(x1, x2, x3)
U62(x1, x2, x3) = U62(x1, x2, x3)
U63(x1, x2, x3) = U63(x1, x2, x3)
U64(x1, x2, x3) = U64(x1, x2, x3)

Recursive Path Order [RPO].
Precedence:

[plus₂, a_{_U61₃}, a_{_U62₃}, a_{_U63₃}, a_{_U64₃}, a_{_plus₂}, U61₃, U62₃, U63₃, U64₃] > A_{_U11₃} > [A_{_U12₂}, A_{_U13₂}] > [tt, a_{_isNatKind}, A_{_U14₃}, a_{_isNat}, isNatKind, isNat] > A_{_U15₂} > [a_{_U51₁}, 0, U51₁]
[plus₂, a_{_U61₃}, a_{_U62₃}, a_{_U63₃}, a_{_U64₃}, a_{_plus₂}, U61₃, U62₃, U63₃, U64₃] > s₁ > [tt, a_{_isNatKind}, A_{_U14₃}, a_{_isNat}, isNatKind, isNat] > A_{_U15₂} > [a_{_U51₁}, 0, U51₁]

The following usable rules [FROCOS05] were oriented:

a__U11(tt, V1, V2) → a__U12(a__isNatKind(V1), V1, V2)
a__U12(tt, V1, V2) → a__U13(a__isNatKind(V2), V1, V2)
a__U13(tt, V1, V2) → a__U14(a__isNatKind(V2), V1, V2)
a__U14(tt, V1, V2) → a__U15(a__isNat(V1), V2)
a__U15(tt, V2) → a__U16(a__isNat(V2))
a__U16(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, V2) → a__U32(a__isNatKind(V2))
a__U32(tt) → tt
a__U41(tt) → tt
a__U51(tt, N) → a__U52(a__isNatKind(N), N)
a__U52(tt, N) → mark(N)
a__U61(tt, M, N) → a__U62(a__isNatKind(M), M, N)
a__U62(tt, M, N) → a__U63(a__isNat(N), M, N)
a__U63(tt, M, N) → a__U64(a__isNatKind(N), M, N)
a__U64(tt, M, N) → s(a__plus(mark(N), mark(M)))
a__isNat(0) → tt
a__isNat(plus(V1, V2)) → a__U11(a__isNatKind(V1), V1, V2)
a__isNat(s(V1)) → a__U21(a__isNatKind(V1), V1)
a__isNatKind(0) → tt
a__isNatKind(plus(V1, V2)) → a__U31(a__isNatKind(V1), V2)
a__isNatKind(s(V1)) → a__U41(a__isNatKind(V1))
a__plus(N, 0) → a__U51(a__isNat(N), N)
a__plus(N, s(M)) → a__U61(a__isNat(M), M, N)
mark(U11(X1, X2, X3)) → a__U11(mark(X1), X2, X3)
mark(U12(X1, X2, X3)) → a__U12(mark(X1), X2, X3)
mark(isNatKind(X)) → a__isNatKind(X)
mark(U13(X1, X2, X3)) → a__U13(mark(X1), X2, X3)
mark(U14(X1, X2, X3)) → a__U14(mark(X1), X2, X3)
mark(U15(X1, X2)) → a__U15(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
mark(U16(X)) → a__U16(mark(X))
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(U22(X1, X2)) → a__U22(mark(X1), X2)
mark(U23(X)) → a__U23(mark(X))
mark(U31(X1, X2)) → a__U31(mark(X1), X2)
mark(U32(X)) → a__U32(mark(X))
mark(U41(X)) → a__U41(mark(X))
mark(U51(X1, X2)) → a__U51(mark(X1), X2)
mark(U52(X1, X2)) → a__U52(mark(X1), X2)
mark(U61(X1, X2, X3)) → a__U61(mark(X1), X2, X3)
mark(U62(X1, X2, X3)) → a__U62(mark(X1), X2, X3)
mark(U63(X1, X2, X3)) → a__U63(mark(X1), X2, X3)
mark(U64(X1, X2, X3)) → a__U64(mark(X1), X2, X3)
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(0) → 0
a__U11(X1, X2, X3) → U11(X1, X2, X3)
a__U12(X1, X2, X3) → U12(X1, X2, X3)
a__isNatKind(X) → isNatKind(X)
a__U13(X1, X2, X3) → U13(X1, X2, X3)
a__U14(X1, X2, X3) → U14(X1, X2, X3)
a__U15(X1, X2) → U15(X1, X2)
a__isNat(X) → isNat(X)
a__U16(X) → U16(X)
a__U21(X1, X2) → U21(X1, X2)
a__U22(X1, X2) → U22(X1, X2)
a__U23(X) → U23(X)
a__U31(X1, X2) → U31(X1, X2)
a__U32(X) → U32(X)
a__U41(X) → U41(X)
a__U51(X1, X2) → U51(X1, X2)
a__U52(X1, X2) → U52(X1, X2)
a__U61(X1, X2, X3) → U61(X1, X2, X3)
a__U62(X1, X2, X3) → U62(X1, X2, X3)
a__U63(X1, X2, X3) → U63(X1, X2, X3)
a__U64(X1, X2, X3) → U64(X1, X2, X3)
a__plus(X1, X2) → plus(X1, X2)

(16) Obligation:

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

A__U12(tt, V1, V2) → A__U13(a__isNatKind(V2), V1, V2)
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__U11(tt, V1, V2) → a__U12(a__isNatKind(V1), V1, V2)
a__U12(tt, V1, V2) → a__U13(a__isNatKind(V2), V1, V2)
a__U13(tt, V1, V2) → a__U14(a__isNatKind(V2), V1, V2)
a__U14(tt, V1, V2) → a__U15(a__isNat(V1), V2)
a__U15(tt, V2) → a__U16(a__isNat(V2))
a__U16(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, V2) → a__U32(a__isNatKind(V2))
a__U32(tt) → tt
a__U41(tt) → tt
a__U51(tt, N) → a__U52(a__isNatKind(N), N)
a__U52(tt, N) → mark(N)
a__U61(tt, M, N) → a__U62(a__isNatKind(M), M, N)
a__U62(tt, M, N) → a__U63(a__isNat(N), M, N)
a__U63(tt, M, N) → a__U64(a__isNatKind(N), M, N)
a__U64(tt, M, N) → s(a__plus(mark(N), mark(M)))
a__isNat(0) → tt
a__isNat(plus(V1, V2)) → a__U11(a__isNatKind(V1), V1, V2)
a__isNat(s(V1)) → a__U21(a__isNatKind(V1), V1)
a__isNatKind(0) → tt
a__isNatKind(plus(V1, V2)) → a__U31(a__isNatKind(V1), V2)
a__isNatKind(s(V1)) → a__U41(a__isNatKind(V1))
a__plus(N, 0) → a__U51(a__isNat(N), N)
a__plus(N, s(M)) → a__U61(a__isNat(M), M, N)
mark(U11(X1, X2, X3)) → a__U11(mark(X1), X2, X3)
mark(U12(X1, X2, X3)) → a__U12(mark(X1), X2, X3)
mark(isNatKind(X)) → a__isNatKind(X)
mark(U13(X1, X2, X3)) → a__U13(mark(X1), X2, X3)
mark(U14(X1, X2, X3)) → a__U14(mark(X1), X2, X3)
mark(U15(X1, X2)) → a__U15(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
mark(U16(X)) → a__U16(mark(X))
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(U22(X1, X2)) → a__U22(mark(X1), X2)
mark(U23(X)) → a__U23(mark(X))
mark(U31(X1, X2)) → a__U31(mark(X1), X2)
mark(U32(X)) → a__U32(mark(X))
mark(U41(X)) → a__U41(mark(X))
mark(U51(X1, X2)) → a__U51(mark(X1), X2)
mark(U52(X1, X2)) → a__U52(mark(X1), X2)
mark(U61(X1, X2, X3)) → a__U61(mark(X1), X2, X3)
mark(U62(X1, X2, X3)) → a__U62(mark(X1), X2, X3)
mark(U63(X1, X2, X3)) → a__U63(mark(X1), X2, X3)
mark(U64(X1, X2, X3)) → a__U64(mark(X1), X2, X3)
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(0) → 0
a__U11(X1, X2, X3) → U11(X1, X2, X3)
a__U12(X1, X2, X3) → U12(X1, X2, X3)
a__isNatKind(X) → isNatKind(X)
a__U13(X1, X2, X3) → U13(X1, X2, X3)
a__U14(X1, X2, X3) → U14(X1, X2, X3)
a__U15(X1, X2) → U15(X1, X2)
a__isNat(X) → isNat(X)
a__U16(X) → U16(X)
a__U21(X1, X2) → U21(X1, X2)
a__U22(X1, X2) → U22(X1, X2)
a__U23(X) → U23(X)
a__U31(X1, X2) → U31(X1, X2)
a__U32(X) → U32(X)
a__U41(X) → U41(X)
a__U51(X1, X2) → U51(X1, X2)
a__U52(X1, X2) → U52(X1, X2)
a__U61(X1, X2, X3) → U61(X1, X2, X3)
a__U62(X1, X2, X3) → U62(X1, X2, X3)
a__U63(X1, X2, X3) → U63(X1, X2, X3)
a__U64(X1, X2, X3) → U64(X1, X2, X3)
a__plus(X1, X2) → plus(X1, X2)

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

(17) DependencyGraphProof (EQUIVALENT transformation)

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

(18) TRUE

(19) Obligation:

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

MARK(U11(X1, X2, X3)) → MARK(X1)
MARK(U12(X1, X2, X3)) → MARK(X1)
MARK(U13(X1, X2, X3)) → MARK(X1)
MARK(U14(X1, X2, X3)) → MARK(X1)
MARK(U15(X1, X2)) → MARK(X1)
MARK(U16(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(X)) → MARK(X)
MARK(U41(X)) → MARK(X)
MARK(U51(X1, X2)) → A__U51(mark(X1), X2)
A__U51(tt, N) → A__U52(a__isNatKind(N), N)
A__U52(tt, N) → MARK(N)
MARK(U51(X1, X2)) → MARK(X1)
MARK(U52(X1, X2)) → A__U52(mark(X1), X2)
MARK(U52(X1, X2)) → MARK(X1)
MARK(U61(X1, X2, X3)) → A__U61(mark(X1), X2, X3)
A__U61(tt, M, N) → A__U62(a__isNatKind(M), M, N)
A__U62(tt, M, N) → A__U63(a__isNat(N), M, N)
A__U63(tt, M, N) → A__U64(a__isNatKind(N), M, N)
A__U64(tt, M, N) → A__PLUS(mark(N), mark(M))
A__PLUS(N, 0) → A__U51(a__isNat(N), N)
A__PLUS(N, s(M)) → A__U61(a__isNat(M), M, N)
A__U64(tt, M, N) → MARK(N)
MARK(U61(X1, X2, X3)) → MARK(X1)
MARK(U62(X1, X2, X3)) → A__U62(mark(X1), X2, X3)
MARK(U62(X1, X2, X3)) → MARK(X1)
MARK(U63(X1, X2, X3)) → A__U63(mark(X1), X2, X3)
MARK(U63(X1, X2, X3)) → MARK(X1)
MARK(U64(X1, X2, X3)) → A__U64(mark(X1), X2, X3)
A__U64(tt, M, N) → MARK(M)
MARK(U64(X1, X2, X3)) → MARK(X1)
MARK(plus(X1, X2)) → A__PLUS(mark(X1), mark(X2))
MARK(plus(X1, X2)) → MARK(X1)
MARK(plus(X1, X2)) → MARK(X2)
MARK(s(X)) → MARK(X)

The TRS R consists of the following rules:

a__U11(tt, V1, V2) → a__U12(a__isNatKind(V1), V1, V2)
a__U12(tt, V1, V2) → a__U13(a__isNatKind(V2), V1, V2)
a__U13(tt, V1, V2) → a__U14(a__isNatKind(V2), V1, V2)
a__U14(tt, V1, V2) → a__U15(a__isNat(V1), V2)
a__U15(tt, V2) → a__U16(a__isNat(V2))
a__U16(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, V2) → a__U32(a__isNatKind(V2))
a__U32(tt) → tt
a__U41(tt) → tt
a__U51(tt, N) → a__U52(a__isNatKind(N), N)
a__U52(tt, N) → mark(N)
a__U61(tt, M, N) → a__U62(a__isNatKind(M), M, N)
a__U62(tt, M, N) → a__U63(a__isNat(N), M, N)
a__U63(tt, M, N) → a__U64(a__isNatKind(N), M, N)
a__U64(tt, M, N) → s(a__plus(mark(N), mark(M)))
a__isNat(0) → tt
a__isNat(plus(V1, V2)) → a__U11(a__isNatKind(V1), V1, V2)
a__isNat(s(V1)) → a__U21(a__isNatKind(V1), V1)
a__isNatKind(0) → tt
a__isNatKind(plus(V1, V2)) → a__U31(a__isNatKind(V1), V2)
a__isNatKind(s(V1)) → a__U41(a__isNatKind(V1))
a__plus(N, 0) → a__U51(a__isNat(N), N)
a__plus(N, s(M)) → a__U61(a__isNat(M), M, N)
mark(U11(X1, X2, X3)) → a__U11(mark(X1), X2, X3)
mark(U12(X1, X2, X3)) → a__U12(mark(X1), X2, X3)
mark(isNatKind(X)) → a__isNatKind(X)
mark(U13(X1, X2, X3)) → a__U13(mark(X1), X2, X3)
mark(U14(X1, X2, X3)) → a__U14(mark(X1), X2, X3)
mark(U15(X1, X2)) → a__U15(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
mark(U16(X)) → a__U16(mark(X))
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(U22(X1, X2)) → a__U22(mark(X1), X2)
mark(U23(X)) → a__U23(mark(X))
mark(U31(X1, X2)) → a__U31(mark(X1), X2)
mark(U32(X)) → a__U32(mark(X))
mark(U41(X)) → a__U41(mark(X))
mark(U51(X1, X2)) → a__U51(mark(X1), X2)
mark(U52(X1, X2)) → a__U52(mark(X1), X2)
mark(U61(X1, X2, X3)) → a__U61(mark(X1), X2, X3)
mark(U62(X1, X2, X3)) → a__U62(mark(X1), X2, X3)
mark(U63(X1, X2, X3)) → a__U63(mark(X1), X2, X3)
mark(U64(X1, X2, X3)) → a__U64(mark(X1), X2, X3)
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(0) → 0
a__U11(X1, X2, X3) → U11(X1, X2, X3)
a__U12(X1, X2, X3) → U12(X1, X2, X3)
a__isNatKind(X) → isNatKind(X)
a__U13(X1, X2, X3) → U13(X1, X2, X3)
a__U14(X1, X2, X3) → U14(X1, X2, X3)
a__U15(X1, X2) → U15(X1, X2)
a__isNat(X) → isNat(X)
a__U16(X) → U16(X)
a__U21(X1, X2) → U21(X1, X2)
a__U22(X1, X2) → U22(X1, X2)
a__U23(X) → U23(X)
a__U31(X1, X2) → U31(X1, X2)
a__U32(X) → U32(X)
a__U41(X) → U41(X)
a__U51(X1, X2) → U51(X1, X2)
a__U52(X1, X2) → U52(X1, X2)
a__U61(X1, X2, X3) → U61(X1, X2, X3)
a__U62(X1, X2, X3) → U62(X1, X2, X3)
a__U63(X1, X2, X3) → U63(X1, X2, X3)
a__U64(X1, X2, X3) → U64(X1, X2, X3)
a__plus(X1, X2) → plus(X1, X2)

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

(20) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

MARK(U51(X1, X2)) → A__U51(mark(X1), X2)
MARK(U51(X1, X2)) → MARK(X1)
MARK(U52(X1, X2)) → A__U52(mark(X1), X2)
MARK(U52(X1, X2)) → MARK(X1)
MARK(U61(X1, X2, X3)) → A__U61(mark(X1), X2, X3)
A__PLUS(N, 0) → A__U51(a__isNat(N), N)
A__PLUS(N, s(M)) → A__U61(a__isNat(M), M, N)
A__U64(tt, M, N) → MARK(N)
MARK(U61(X1, X2, X3)) → MARK(X1)
MARK(U62(X1, X2, X3)) → A__U62(mark(X1), X2, X3)
MARK(U62(X1, X2, X3)) → MARK(X1)
MARK(U63(X1, X2, X3)) → A__U63(mark(X1), X2, X3)
MARK(U63(X1, X2, X3)) → MARK(X1)
MARK(U64(X1, X2, X3)) → A__U64(mark(X1), X2, X3)
A__U64(tt, M, N) → MARK(M)
MARK(U64(X1, X2, X3)) → MARK(X1)
MARK(plus(X1, X2)) → A__PLUS(mark(X1), mark(X2))
MARK(plus(X1, X2)) → MARK(X1)
MARK(plus(X1, X2)) → MARK(X2)
MARK(s(X)) → MARK(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
MARK(x1) = MARK(x1)
U11(x1, x2, x3) = x1
U12(x1, x2, x3) = x1
U13(x1, x2, x3) = x1
U14(x1, x2, x3) = x1
U15(x1, x2) = x1
U16(x1) = x1
U21(x1, x2) = x1
U22(x1, x2) = x1
U23(x1) = x1
U31(x1, x2) = x1
U32(x1) = x1
U41(x1) = x1
U51(x1, x2) = U51(x1, x2)
A__U51(x1, x2) = A__U51(x2)
mark(x1) = x1
tt = tt
A__U52(x1, x2) = A__U52(x2)
a__isNatKind(x1) = a__isNatKind
U52(x1, x2) = U52(x1, x2)
U61(x1, x2, x3) = U61(x1, x2, x3)
A__U61(x1, x2, x3) = A__U61(x2, x3)
A__U62(x1, x2, x3) = A__U62(x2, x3)
A__U63(x1, x2, x3) = A__U63(x2, x3)
a__isNat(x1) = a__isNat
A__U64(x1, x2, x3) = A__U64(x2, x3)
A__PLUS(x1, x2) = A__PLUS(x1, x2)
0 = 0
s(x1) = s(x1)
U62(x1, x2, x3) = U62(x1, x2, x3)
U63(x1, x2, x3) = U63(x1, x2, x3)
U64(x1, x2, x3) = U64(x1, x2, x3)
plus(x1, x2) = plus(x1, x2)
a__U11(x1, x2, x3) = x1
a__U12(x1, x2, x3) = x1
a__U13(x1, x2, x3) = x1
a__U14(x1, x2, x3) = x1
a__U15(x1, x2) = x1
a__U16(x1) = x1
a__U21(x1, x2) = x1
a__U22(x1, x2) = x1
a__U23(x1) = x1
a__U31(x1, x2) = x1
a__U32(x1) = x1
a__U41(x1) = x1
a__U51(x1, x2) = a__U51(x1, x2)
a__U52(x1, x2) = a__U52(x1, x2)
a__U61(x1, x2, x3) = a__U61(x1, x2, x3)
a__U62(x1, x2, x3) = a__U62(x1, x2, x3)
a__U63(x1, x2, x3) = a__U63(x1, x2, x3)
a__U64(x1, x2, x3) = a__U64(x1, x2, x3)
a__plus(x1, x2) = a__plus(x1, x2)
isNatKind(x1) = isNatKind
isNat(x1) = isNat

Recursive Path Order [RPO].
Precedence:

[U61₃, A_{_U61₂}, A_{_U62₂}, A_{_U63₂}, A_{_U64₂}, A_{_PLUS₂}, U62₃, U63₃, U64₃, plus₂, a_{_U61₃}, a_{_U62₃}, a_{_U63₃}, a_{_U64₃}, a_{_plus₂}] > [MARK₁, U51₂, A_{_U51₁}, A_{_U52₁}, a_{_U51₂}] > [tt, a_{_isNatKind}, a_{_isNat}, s₁, isNatKind, isNat] > [U52₂, a_{_U52₂}]

The following usable rules [FROCOS05] were oriented:

a__U11(tt, V1, V2) → a__U12(a__isNatKind(V1), V1, V2)
a__U12(tt, V1, V2) → a__U13(a__isNatKind(V2), V1, V2)
a__U13(tt, V1, V2) → a__U14(a__isNatKind(V2), V1, V2)
a__U14(tt, V1, V2) → a__U15(a__isNat(V1), V2)
a__U15(tt, V2) → a__U16(a__isNat(V2))
a__U16(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, V2) → a__U32(a__isNatKind(V2))
a__U32(tt) → tt
a__U41(tt) → tt
a__U51(tt, N) → a__U52(a__isNatKind(N), N)
a__U52(tt, N) → mark(N)
a__U61(tt, M, N) → a__U62(a__isNatKind(M), M, N)
a__U62(tt, M, N) → a__U63(a__isNat(N), M, N)
a__U63(tt, M, N) → a__U64(a__isNatKind(N), M, N)
a__U64(tt, M, N) → s(a__plus(mark(N), mark(M)))
a__isNat(0) → tt
a__isNat(plus(V1, V2)) → a__U11(a__isNatKind(V1), V1, V2)
a__isNat(s(V1)) → a__U21(a__isNatKind(V1), V1)
a__isNatKind(0) → tt
a__isNatKind(plus(V1, V2)) → a__U31(a__isNatKind(V1), V2)
a__isNatKind(s(V1)) → a__U41(a__isNatKind(V1))
a__plus(N, 0) → a__U51(a__isNat(N), N)
a__plus(N, s(M)) → a__U61(a__isNat(M), M, N)
mark(U11(X1, X2, X3)) → a__U11(mark(X1), X2, X3)
mark(U12(X1, X2, X3)) → a__U12(mark(X1), X2, X3)
mark(isNatKind(X)) → a__isNatKind(X)
mark(U13(X1, X2, X3)) → a__U13(mark(X1), X2, X3)
mark(U14(X1, X2, X3)) → a__U14(mark(X1), X2, X3)
mark(U15(X1, X2)) → a__U15(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
mark(U16(X)) → a__U16(mark(X))
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(U22(X1, X2)) → a__U22(mark(X1), X2)
mark(U23(X)) → a__U23(mark(X))
mark(U31(X1, X2)) → a__U31(mark(X1), X2)
mark(U32(X)) → a__U32(mark(X))
mark(U41(X)) → a__U41(mark(X))
mark(U51(X1, X2)) → a__U51(mark(X1), X2)
mark(U52(X1, X2)) → a__U52(mark(X1), X2)
mark(U61(X1, X2, X3)) → a__U61(mark(X1), X2, X3)
mark(U62(X1, X2, X3)) → a__U62(mark(X1), X2, X3)
mark(U63(X1, X2, X3)) → a__U63(mark(X1), X2, X3)
mark(U64(X1, X2, X3)) → a__U64(mark(X1), X2, X3)
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(0) → 0
a__U11(X1, X2, X3) → U11(X1, X2, X3)
a__U12(X1, X2, X3) → U12(X1, X2, X3)
a__isNatKind(X) → isNatKind(X)
a__U13(X1, X2, X3) → U13(X1, X2, X3)
a__U14(X1, X2, X3) → U14(X1, X2, X3)
a__U15(X1, X2) → U15(X1, X2)
a__isNat(X) → isNat(X)
a__U16(X) → U16(X)
a__U21(X1, X2) → U21(X1, X2)
a__U22(X1, X2) → U22(X1, X2)
a__U23(X) → U23(X)
a__U31(X1, X2) → U31(X1, X2)
a__U32(X) → U32(X)
a__U41(X) → U41(X)
a__U51(X1, X2) → U51(X1, X2)
a__U52(X1, X2) → U52(X1, X2)
a__U61(X1, X2, X3) → U61(X1, X2, X3)
a__U62(X1, X2, X3) → U62(X1, X2, X3)
a__U63(X1, X2, X3) → U63(X1, X2, X3)
a__U64(X1, X2, X3) → U64(X1, X2, X3)
a__plus(X1, X2) → plus(X1, X2)

(21) Obligation:

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

MARK(U11(X1, X2, X3)) → MARK(X1)
MARK(U12(X1, X2, X3)) → MARK(X1)
MARK(U13(X1, X2, X3)) → MARK(X1)
MARK(U14(X1, X2, X3)) → MARK(X1)
MARK(U15(X1, X2)) → MARK(X1)
MARK(U16(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(X)) → MARK(X)
MARK(U41(X)) → MARK(X)
A__U51(tt, N) → A__U52(a__isNatKind(N), N)
A__U52(tt, N) → MARK(N)
A__U61(tt, M, N) → A__U62(a__isNatKind(M), M, N)
A__U62(tt, M, N) → A__U63(a__isNat(N), M, N)
A__U63(tt, M, N) → A__U64(a__isNatKind(N), M, N)
A__U64(tt, M, N) → A__PLUS(mark(N), mark(M))

The TRS R consists of the following rules:

a__U11(tt, V1, V2) → a__U12(a__isNatKind(V1), V1, V2)
a__U12(tt, V1, V2) → a__U13(a__isNatKind(V2), V1, V2)
a__U13(tt, V1, V2) → a__U14(a__isNatKind(V2), V1, V2)
a__U14(tt, V1, V2) → a__U15(a__isNat(V1), V2)
a__U15(tt, V2) → a__U16(a__isNat(V2))
a__U16(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, V2) → a__U32(a__isNatKind(V2))
a__U32(tt) → tt
a__U41(tt) → tt
a__U51(tt, N) → a__U52(a__isNatKind(N), N)
a__U52(tt, N) → mark(N)
a__U61(tt, M, N) → a__U62(a__isNatKind(M), M, N)
a__U62(tt, M, N) → a__U63(a__isNat(N), M, N)
a__U63(tt, M, N) → a__U64(a__isNatKind(N), M, N)
a__U64(tt, M, N) → s(a__plus(mark(N), mark(M)))
a__isNat(0) → tt
a__isNat(plus(V1, V2)) → a__U11(a__isNatKind(V1), V1, V2)
a__isNat(s(V1)) → a__U21(a__isNatKind(V1), V1)
a__isNatKind(0) → tt
a__isNatKind(plus(V1, V2)) → a__U31(a__isNatKind(V1), V2)
a__isNatKind(s(V1)) → a__U41(a__isNatKind(V1))
a__plus(N, 0) → a__U51(a__isNat(N), N)
a__plus(N, s(M)) → a__U61(a__isNat(M), M, N)
mark(U11(X1, X2, X3)) → a__U11(mark(X1), X2, X3)
mark(U12(X1, X2, X3)) → a__U12(mark(X1), X2, X3)
mark(isNatKind(X)) → a__isNatKind(X)
mark(U13(X1, X2, X3)) → a__U13(mark(X1), X2, X3)
mark(U14(X1, X2, X3)) → a__U14(mark(X1), X2, X3)
mark(U15(X1, X2)) → a__U15(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
mark(U16(X)) → a__U16(mark(X))
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(U22(X1, X2)) → a__U22(mark(X1), X2)
mark(U23(X)) → a__U23(mark(X))
mark(U31(X1, X2)) → a__U31(mark(X1), X2)
mark(U32(X)) → a__U32(mark(X))
mark(U41(X)) → a__U41(mark(X))
mark(U51(X1, X2)) → a__U51(mark(X1), X2)
mark(U52(X1, X2)) → a__U52(mark(X1), X2)
mark(U61(X1, X2, X3)) → a__U61(mark(X1), X2, X3)
mark(U62(X1, X2, X3)) → a__U62(mark(X1), X2, X3)
mark(U63(X1, X2, X3)) → a__U63(mark(X1), X2, X3)
mark(U64(X1, X2, X3)) → a__U64(mark(X1), X2, X3)
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(0) → 0
a__U11(X1, X2, X3) → U11(X1, X2, X3)
a__U12(X1, X2, X3) → U12(X1, X2, X3)
a__isNatKind(X) → isNatKind(X)
a__U13(X1, X2, X3) → U13(X1, X2, X3)
a__U14(X1, X2, X3) → U14(X1, X2, X3)
a__U15(X1, X2) → U15(X1, X2)
a__isNat(X) → isNat(X)
a__U16(X) → U16(X)
a__U21(X1, X2) → U21(X1, X2)
a__U22(X1, X2) → U22(X1, X2)
a__U23(X) → U23(X)
a__U31(X1, X2) → U31(X1, X2)
a__U32(X) → U32(X)
a__U41(X) → U41(X)
a__U51(X1, X2) → U51(X1, X2)
a__U52(X1, X2) → U52(X1, X2)
a__U61(X1, X2, X3) → U61(X1, X2, X3)
a__U62(X1, X2, X3) → U62(X1, X2, X3)
a__U63(X1, X2, X3) → U63(X1, X2, X3)
a__U64(X1, X2, X3) → U64(X1, X2, X3)
a__plus(X1, X2) → plus(X1, X2)

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

(22) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 6 less nodes.

(23) Obligation:

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

MARK(U12(X1, X2, X3)) → MARK(X1)
MARK(U11(X1, X2, X3)) → MARK(X1)
MARK(U13(X1, X2, X3)) → MARK(X1)
MARK(U14(X1, X2, X3)) → MARK(X1)
MARK(U15(X1, X2)) → MARK(X1)
MARK(U16(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(X)) → MARK(X)
MARK(U41(X)) → MARK(X)

The TRS R consists of the following rules:

a__U11(tt, V1, V2) → a__U12(a__isNatKind(V1), V1, V2)
a__U12(tt, V1, V2) → a__U13(a__isNatKind(V2), V1, V2)
a__U13(tt, V1, V2) → a__U14(a__isNatKind(V2), V1, V2)
a__U14(tt, V1, V2) → a__U15(a__isNat(V1), V2)
a__U15(tt, V2) → a__U16(a__isNat(V2))
a__U16(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, V2) → a__U32(a__isNatKind(V2))
a__U32(tt) → tt
a__U41(tt) → tt
a__U51(tt, N) → a__U52(a__isNatKind(N), N)
a__U52(tt, N) → mark(N)
a__U61(tt, M, N) → a__U62(a__isNatKind(M), M, N)
a__U62(tt, M, N) → a__U63(a__isNat(N), M, N)
a__U63(tt, M, N) → a__U64(a__isNatKind(N), M, N)
a__U64(tt, M, N) → s(a__plus(mark(N), mark(M)))
a__isNat(0) → tt
a__isNat(plus(V1, V2)) → a__U11(a__isNatKind(V1), V1, V2)
a__isNat(s(V1)) → a__U21(a__isNatKind(V1), V1)
a__isNatKind(0) → tt
a__isNatKind(plus(V1, V2)) → a__U31(a__isNatKind(V1), V2)
a__isNatKind(s(V1)) → a__U41(a__isNatKind(V1))
a__plus(N, 0) → a__U51(a__isNat(N), N)
a__plus(N, s(M)) → a__U61(a__isNat(M), M, N)
mark(U11(X1, X2, X3)) → a__U11(mark(X1), X2, X3)
mark(U12(X1, X2, X3)) → a__U12(mark(X1), X2, X3)
mark(isNatKind(X)) → a__isNatKind(X)
mark(U13(X1, X2, X3)) → a__U13(mark(X1), X2, X3)
mark(U14(X1, X2, X3)) → a__U14(mark(X1), X2, X3)
mark(U15(X1, X2)) → a__U15(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
mark(U16(X)) → a__U16(mark(X))
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(U22(X1, X2)) → a__U22(mark(X1), X2)
mark(U23(X)) → a__U23(mark(X))
mark(U31(X1, X2)) → a__U31(mark(X1), X2)
mark(U32(X)) → a__U32(mark(X))
mark(U41(X)) → a__U41(mark(X))
mark(U51(X1, X2)) → a__U51(mark(X1), X2)
mark(U52(X1, X2)) → a__U52(mark(X1), X2)
mark(U61(X1, X2, X3)) → a__U61(mark(X1), X2, X3)
mark(U62(X1, X2, X3)) → a__U62(mark(X1), X2, X3)
mark(U63(X1, X2, X3)) → a__U63(mark(X1), X2, X3)
mark(U64(X1, X2, X3)) → a__U64(mark(X1), X2, X3)
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(0) → 0
a__U11(X1, X2, X3) → U11(X1, X2, X3)
a__U12(X1, X2, X3) → U12(X1, X2, X3)
a__isNatKind(X) → isNatKind(X)
a__U13(X1, X2, X3) → U13(X1, X2, X3)
a__U14(X1, X2, X3) → U14(X1, X2, X3)
a__U15(X1, X2) → U15(X1, X2)
a__isNat(X) → isNat(X)
a__U16(X) → U16(X)
a__U21(X1, X2) → U21(X1, X2)
a__U22(X1, X2) → U22(X1, X2)
a__U23(X) → U23(X)
a__U31(X1, X2) → U31(X1, X2)
a__U32(X) → U32(X)
a__U41(X) → U41(X)
a__U51(X1, X2) → U51(X1, X2)
a__U52(X1, X2) → U52(X1, X2)
a__U61(X1, X2, X3) → U61(X1, X2, X3)
a__U62(X1, X2, X3) → U62(X1, X2, X3)
a__U63(X1, X2, X3) → U63(X1, X2, X3)
a__U64(X1, X2, X3) → U64(X1, X2, X3)
a__plus(X1, X2) → plus(X1, X2)

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