Termination w.r.t. Q proof of /tmp/tmpwWLdzJ/PEANO_complete_noand

Termination w.r.t. Q of the given QTRS could be proven:

0 QTRS

↳1 DependencyPairsProof (⇔)

↳2 QDP

↳3 DependencyGraphProof (⇔)

↳4 AND

    ↳5 QDP

        ↳6 QDPOrderProof (⇔)

        ↳7 QDP

        ↳8 DependencyGraphProof (⇔)

        ↳9 TRUE

    ↳10 QDP

        ↳11 QDPOrderProof (⇔)

        ↳12 QDP

        ↳13 QDPOrderProof (⇔)

        ↳14 QDP

        ↳15 DependencyGraphProof (⇔)

        ↳16 TRUE

    ↳17 QDP

        ↳18 QDPOrderProof (⇔)

        ↳19 QDP

        ↳20 DependencyGraphProof (⇔)

        ↳21 QDP

        ↳22 QDPOrderProof (⇔)

        ↳23 QDP

        ↳24 QDPOrderProof (⇔)

        ↳25 QDP

        ↳26 QDPOrderProof (⇔)

        ↳27 QDP

        ↳28 QDPOrderProof (⇔)

        ↳29 QDP

        ↳30 QDPOrderProof (⇔)

        ↳31 QDP

        ↳32 QDPOrderProof (⇔)

        ↳33 QDP

        ↳34 QDPOrderProof (⇔)

        ↳35 QDP

        ↳36 PisEmptyProof (⇔)

        ↳37 TRUE

(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__U31(tt, V2) → A__ISNATKIND(V2)
A__ISNATKIND(plus(V1, V2)) → A__ISNATKIND(V1)
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__U31(x1, x2) = A__U31(x1, x2)
tt = tt
A__ISNATKIND(x1) = x1
plus(x1, x2) = plus(x1, x2)
a__isNatKind(x1) = x1
s(x1) = s(x1)
0 = 0
a__U31(x1, x2) = a__U31(x1, x2)
a__U41(x1) = a__U41(x1)
isNatKind(x1) = x1
a__U32(x1) = a__U32(x1)
U31(x1, x2) = U31(x1, x2)
U41(x1) = U41(x1)
U32(x1) = U32(x1)

Recursive path order with status [RPO].
Quasi-Precedence:

[A_{_U31₂}, plus₂] > [a_{_U31₂}, U31₂] > [a_{_U32₁}, U32₁]
s₁ > [a_{_U41₁}, U41₁]
0 > tt > [a_{_U32₁}, U32₁]

Status:

A_{_U31₂}: [2,1]
tt: multiset
plus₂: [2,1]
s₁: multiset
0: multiset
a_{_U31₂}: multiset
a_{_U41₁}: [1]
a_{_U32₁}: multiset
U31₂: multiset
U41₁: [1]
U32₁: multiset

The following usable rules [FROCOS05] were oriented:

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__isNatKind(X) → isNatKind(X)
a__U31(tt, V2) → a__U32(a__isNatKind(V2))
a__U31(X1, X2) → U31(X1, X2)
a__U41(tt) → tt
a__U41(X) → U41(X)
a__U32(tt) → tt
a__U32(X) → U32(X)

(7) Obligation:

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

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

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 0 SCCs with 1 less node.

(9) TRUE

(10) 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.

(11) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

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__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(x1, x2, x3)
tt = tt
A__U13(x1, x2, x3) = A__U13(x2, x3)
a__isNatKind(x1) = x1
A__U14(x1, x2, x3) = A__U14(x2, x3)
A__U15(x1, x2) = A__U15(x2)
a__isNat(x1) = a__isNat(x1)
A__ISNAT(x1) = x1
plus(x1, x2) = plus(x1, x2)
A__U11(x1, x2, x3) = A__U11(x1, x2, x3)
s(x1) = x1
A__U21(x1, x2) = x2
A__U22(x1, x2) = x2
0 = 0
a__U31(x1, x2) = a__U31(x1, x2)
a__U41(x1) = x1
isNatKind(x1) = isNatKind
a__U11(x1, x2, x3) = a__U11(x1, x2, x3)
a__U21(x1, x2) = a__U21
isNat(x1) = isNat(x1)
a__U12(x1, x2, x3) = a__U12(x1)
a__U22(x1, x2) = a__U22
a__U13(x1, x2, x3) = a__U13
a__U23(x1) = a__U23(x1)
a__U14(x1, x2, x3) = a__U14(x1, x2, x3)
a__U15(x1, x2) = a__U15
a__U16(x1) = a__U16
U13(x1, x2, x3) = U13(x1, x2, x3)
U14(x1, x2, x3) = U14(x1, x2, x3)
U15(x1, x2) = U15
U11(x1, x2, x3) = U11
U12(x1, x2, x3) = U12(x1, x2, x3)
U21(x1, x2) = U21(x1, x2)
U22(x1, x2) = U22(x1, x2)
U23(x1) = U23
U16(x1) = U16(x1)
a__U32(x1) = a__U32
U31(x1, x2) = U31(x1, x2)
U41(x1) = U41
U32(x1) = U32

Recursive path order with status [RPO].
Quasi-Precedence:

a_{_isNat₁} > [A_{_U12₃}, tt, 0, a_{_U32}, U32] > A_{_U13₂} > A_{_U14₂} > [A_{_U15₁}, isNatKind, U21₂, U41]
a_{_isNat₁} > [A_{_U12₃}, tt, 0, a_{_U32}, U32] > [a_{_U12₁}, a_{_U13}] > U13₃ > [A_{_U15₁}, isNatKind, U21₂, U41]
a_{_isNat₁} > [A_{_U12₃}, tt, 0, a_{_U32}, U32] > a_{_U14₃} > U14₃ > [A_{_U15₁}, isNatKind, U21₂, U41]
a_{_isNat₁} > [A_{_U12₃}, tt, 0, a_{_U32}, U32] > a_{_U16} > U16₁ > [A_{_U15₁}, isNatKind, U21₂, U41]
a_{_isNat₁} > a_{_U11₃} > [A_{_U15₁}, isNatKind, U21₂, U41]
a_{_isNat₁} > a_{_U21} > [A_{_U15₁}, isNatKind, U21₂, U41]
plus₂ > A_{_U11₃} > [A_{_U12₃}, tt, 0, a_{_U32}, U32] > A_{_U13₂} > A_{_U14₂} > [A_{_U15₁}, isNatKind, U21₂, U41]
plus₂ > A_{_U11₃} > [A_{_U12₃}, tt, 0, a_{_U32}, U32] > [a_{_U12₁}, a_{_U13}] > U13₃ > [A_{_U15₁}, isNatKind, U21₂, U41]
plus₂ > A_{_U11₃} > [A_{_U12₃}, tt, 0, a_{_U32}, U32] > a_{_U14₃} > U14₃ > [A_{_U15₁}, isNatKind, U21₂, U41]
plus₂ > A_{_U11₃} > [A_{_U12₃}, tt, 0, a_{_U32}, U32] > a_{_U16} > U16₁ > [A_{_U15₁}, isNatKind, U21₂, U41]
plus₂ > [a_{_U31₂}, U31₂] > [A_{_U15₁}, isNatKind, U21₂, U41]
plus₂ > a_{_U11₃} > [A_{_U15₁}, isNatKind, U21₂, U41]
isNat₁ > [A_{_U15₁}, isNatKind, U21₂, U41]
a_{_U22} > a_{_U23₁} > [A_{_U12₃}, tt, 0, a_{_U32}, U32] > A_{_U13₂} > A_{_U14₂} > [A_{_U15₁}, isNatKind, U21₂, U41]
a_{_U22} > a_{_U23₁} > [A_{_U12₃}, tt, 0, a_{_U32}, U32] > [a_{_U12₁}, a_{_U13}] > U13₃ > [A_{_U15₁}, isNatKind, U21₂, U41]
a_{_U22} > a_{_U23₁} > [A_{_U12₃}, tt, 0, a_{_U32}, U32] > a_{_U14₃} > U14₃ > [A_{_U15₁}, isNatKind, U21₂, U41]
a_{_U22} > a_{_U23₁} > [A_{_U12₃}, tt, 0, a_{_U32}, U32] > a_{_U16} > U16₁ > [A_{_U15₁}, isNatKind, U21₂, U41]
a_{_U22} > a_{_U23₁} > U23 > [A_{_U15₁}, isNatKind, U21₂, U41]
a_{_U22} > U22₂ > [A_{_U15₁}, isNatKind, U21₂, U41]
[a_{_U15}, U15] > a_{_U16} > U16₁ > [A_{_U15₁}, isNatKind, U21₂, U41]
U11 > [A_{_U15₁}, isNatKind, U21₂, U41]
U12₃ > [A_{_U15₁}, isNatKind, U21₂, U41]

Status:

A_{_U12₃}: multiset
tt: multiset
A_{_U13₂}: multiset
A_{_U14₂}: multiset
A_{_U15₁}: multiset
a_{_isNat₁}: multiset
plus₂: [2,1]
A_{_U11₃}: multiset
0: multiset
a_{_U31₂}: multiset
isNatKind: multiset
a_{_U11₃}: multiset
a_{_U21}: multiset
isNat₁: multiset
a_{_U12₁}: multiset
a_{_U22}: multiset
a_{_U13}: multiset
a_{_U23₁}: multiset
a_{_U14₃}: multiset
a_{_U15}: []
a_{_U16}: multiset
U13₃: [1,2,3]
U14₃: [3,1,2]
U15: multiset
U11: []
U12₃: [1,2,3]
U21₂: [1,2]
U22₂: [2,1]
U23: []
U16₁: [1]
a_{_U32}: []
U31₂: multiset
U41: []
U32: []

The following usable rules [FROCOS05] were oriented:

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__isNatKind(X) → isNatKind(X)
a__U31(tt, V2) → a__U32(a__isNatKind(V2))
a__U31(X1, X2) → U31(X1, X2)
a__U41(tt) → tt
a__U41(X) → U41(X)
a__U32(tt) → tt
a__U32(X) → U32(X)

(12) Obligation:

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

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__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.

(13) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

A__U22(tt, V1) → A__ISNAT(V1)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
A__ISNAT(x1) = x1
s(x1) = s(x1)
A__U21(x1, x2) = A__U21(x2)
a__isNatKind(x1) = a__isNatKind(x1)
tt = tt
A__U22(x1, x2) = A__U22(x2)
0 = 0
plus(x1, x2) = plus
a__U31(x1, x2) = a__U31
a__U41(x1) = a__U41
isNatKind(x1) = isNatKind(x1)
a__U32(x1) = a__U32(x1)
U31(x1, x2) = U31(x1, x2)
U41(x1) = U41(x1)
U32(x1) = U32(x1)

Recursive path order with status [RPO].
Quasi-Precedence:

[a_{_isNatKind₁}, tt, a_{_U41}, isNatKind₁, U41₁] > [s₁, A_{_U21₁}, A_{_U22₁}]
[a_{_isNatKind₁}, tt, a_{_U41}, isNatKind₁, U41₁] > [plus, a_{_U31}, a_{_U32₁}, U31₂]

Status:

s₁: [1]
A_{_U21₁}: [1]
a_{_isNatKind₁}: multiset
tt: multiset
A_{_U22₁}: [1]
0: multiset
plus: multiset
a_{_U31}: multiset
a_{_U41}: multiset
isNatKind₁: multiset
a_{_U32₁}: multiset
U31₂: multiset
U41₁: multiset
U32₁: multiset

The following usable rules [FROCOS05] were oriented: none

(14) Obligation:

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

A__ISNAT(s(V1)) → A__U21(a__isNatKind(V1), V1)
A__U21(tt, V1) → A__U22(a__isNatKind(V1), 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) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 2 less nodes.

(16) TRUE

(17) 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.

(18) 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)
A__U51(tt, N) → A__U52(a__isNatKind(N), 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__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 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(x1, 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(x1, x2, x3)
A__U62(x1, x2, x3) = A__U62(x1, x2, x3)
A__U63(x1, x2, x3) = A__U63(x1, x2, x3)
a__isNat(x1) = a__isNat
A__U64(x1, x2, x3) = A__U64(x1, 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
isNatKind(x1) = isNatKind
a__U13(x1, x2, x3) = x1
a__U14(x1, x2, x3) = x1
a__U15(x1, x2) = x1
isNat(x1) = isNat
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__plus(x1, x2) = a__plus(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)

Recursive path order with status [RPO].
Quasi-Precedence:

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

Status:

MARK₁: multiset
U51₂: [2,1]
A_{_U51₂}: multiset
tt: multiset
A_{_U52₁}: multiset
a_{_isNatKind}: []
U52₂: multiset
U61₃: multiset
A_{_U61₃}: multiset
A_{_U62₃}: multiset
A_{_U63₃}: multiset
a_{_isNat}: []
A_{_U64₃}: multiset
A_{_PLUS₂}: multiset
0: multiset
s₁: [1]
U62₃: multiset
U63₃: multiset
U64₃: multiset
plus₂: multiset
isNatKind: []
isNat: []
a_{_U51₂}: [2,1]
a_{_U52₂}: multiset
a_{_plus₂}: multiset
a_{_U61₃}: multiset
a_{_U62₃}: multiset
a_{_U63₃}: multiset
a_{_U64₃}: multiset

The following usable rules [FROCOS05] were oriented:

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

(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)
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)

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) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 4 less nodes.

(21) 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.

(22) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

MARK(U32(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)
U12(x1, x2, x3) = x1
U11(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) = U32(x1)
U41(x1) = x1

Recursive path order with status [RPO].
Quasi-Precedence:

[MARK₁, U32₁]

Status:

MARK₁: multiset
U32₁: multiset

The following usable rules [FROCOS05] were oriented: none

(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(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.

(24) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

MARK(U11(X1, X2, X3)) → MARK(X1)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
MARK(x1) = x1
U12(x1, x2, x3) = x1
U11(x1, x2, x3) = U11(x1, x2, x3)
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
U41(x1) = x1

Recursive path order with status [RPO].
Quasi-Precedence:

trivial

Status:

U11₃: multiset

The following usable rules [FROCOS05] were oriented: none

(25) Obligation:

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

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(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.

(26) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

MARK(U15(X1, X2)) → MARK(X1)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
MARK(x1) = MARK(x1)
U12(x1, x2, x3) = x1
U13(x1, x2, x3) = x1
U14(x1, x2, x3) = x1
U15(x1, x2) = U15(x1, x2)
U16(x1) = x1
U21(x1, x2) = x1
U22(x1, x2) = x1
U23(x1) = x1
U31(x1, x2) = x1
U41(x1) = x1

Recursive path order with status [RPO].
Quasi-Precedence:

trivial

Status:

MARK₁: [1]
U15₂: [2,1]

The following usable rules [FROCOS05] were oriented: none

(27) Obligation:

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

MARK(U12(X1, X2, X3)) → MARK(X1)
MARK(U13(X1, X2, X3)) → MARK(X1)
MARK(U14(X1, X2, X3)) → 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(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.

(28) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

MARK(U12(X1, X2, X3)) → MARK(X1)
MARK(U13(X1, X2, X3)) → MARK(X1)
MARK(U14(X1, X2, X3)) → MARK(X1)
MARK(U21(X1, X2)) → MARK(X1)
MARK(U22(X1, X2)) → MARK(X1)
MARK(U31(X1, X2)) → MARK(X1)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
MARK(x1) = MARK(x1)
U12(x1, x2, x3) = U12(x1, x2, x3)
U13(x1, x2, x3) = U13(x1, x2, x3)
U14(x1, x2, x3) = U14(x1, x2, x3)
U16(x1) = x1
U21(x1, x2) = U21(x1, x2)
U22(x1, x2) = U22(x1, x2)
U23(x1) = x1
U31(x1, x2) = U31(x1, x2)
U41(x1) = x1

Recursive path order with status [RPO].
Quasi-Precedence:

U14₃ > MARK₁

Status:

MARK₁: [1]
U12₃: multiset
U13₃: multiset
U14₃: multiset
U21₂: multiset
U22₂: multiset
U31₂: [2,1]

The following usable rules [FROCOS05] were oriented: none

(29) Obligation:

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

MARK(U16(X)) → MARK(X)
MARK(U23(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.

(30) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

MARK(U41(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)
U16(x1) = x1
U23(x1) = x1
U41(x1) = U41(x1)

Recursive path order with status [RPO].
Quasi-Precedence:

[MARK₁, U41₁]

Status:

MARK₁: multiset
U41₁: multiset

The following usable rules [FROCOS05] were oriented: none

(31) Obligation:

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

MARK(U16(X)) → MARK(X)
MARK(U23(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.

(32) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

MARK(U16(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)
U16(x1) = U16(x1)
U23(x1) = x1

Recursive path order with status [RPO].
Quasi-Precedence:

[MARK₁, U16₁]

Status:

MARK₁: [1]
U16₁: multiset

The following usable rules [FROCOS05] were oriented: none

(33) Obligation:

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

MARK(U23(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.

(34) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

MARK(U23(X)) → MARK(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Recursive path order with status [RPO].
Quasi-Precedence:

U23₁ > MARK₁

Status:

MARK₁: multiset
U23₁: multiset

The following usable rules [FROCOS05] were oriented: none

(35) 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.

(36) PisEmptyProof (EQUIVALENT transformation)

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

(0) Obligation:

(1) DependencyPairsProof (EQUIVALENT transformation)

(2) Obligation:

(3) DependencyGraphProof (EQUIVALENT transformation)

(4) Complex Obligation (AND)

(5) Obligation:

(6) QDPOrderProof (EQUIVALENT transformation)

(7) Obligation:

(8) DependencyGraphProof (EQUIVALENT transformation)

(9) TRUE

(10) Obligation:

(11) QDPOrderProof (EQUIVALENT transformation)

(12) Obligation:

(13) QDPOrderProof (EQUIVALENT transformation)

(14) Obligation:

(15) DependencyGraphProof (EQUIVALENT transformation)

(16) TRUE

(17) Obligation:

(18) QDPOrderProof (EQUIVALENT transformation)

(19) Obligation:

(20) DependencyGraphProof (EQUIVALENT transformation)

(21) Obligation:

(22) QDPOrderProof (EQUIVALENT transformation)

(23) Obligation:

(24) QDPOrderProof (EQUIVALENT transformation)

(25) Obligation:

(26) QDPOrderProof (EQUIVALENT transformation)

(27) Obligation:

(28) QDPOrderProof (EQUIVALENT transformation)

(29) Obligation:

(30) QDPOrderProof (EQUIVALENT transformation)

(31) Obligation:

(32) QDPOrderProof (EQUIVALENT transformation)

(33) Obligation:

(34) QDPOrderProof (EQUIVALENT transformation)

(35) Obligation:

(36) PisEmptyProof (EQUIVALENT transformation)

(37) TRUE