Termination w.r.t. Q proof of Transformed_CSR_04_MYNAT_nokinds_noand

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

0 QTRS

↳1 DependencyPairsProof (⇔, 0 ms)

↳2 QDP

↳3 DependencyGraphProof (⇔, 0 ms)

↳4 AND

    ↳5 QDP

        ↳6 UsableRulesProof (⇔, 0 ms)

        ↳7 QDP

        ↳8 QDPSizeChangeProof (⇔, 0 ms)

        ↳9 YES

    ↳10 QDP

        ↳11 QDPOrderProof (⇔, 446 ms)

        ↳12 QDP

        ↳13 DependencyGraphProof (⇔, 0 ms)

        ↳14 QDP

        ↳15 UsableRulesProof (⇔, 0 ms)

        ↳16 QDP

        ↳17 QDPSizeChangeProof (⇔, 0 ms)

        ↳18 YES

(0) Obligation:

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

a__U11(tt, V2) → a__U12(a__isNat(V2))
a__U12(tt) → tt
a__U21(tt) → tt
a__U31(tt, V2) → a__U32(a__isNat(V2))
a__U32(tt) → tt
a__U41(tt, N) → mark(N)
a__U51(tt, M, N) → a__U52(a__isNat(N), M, N)
a__U52(tt, M, N) → s(a__plus(mark(N), mark(M)))
a__U61(tt) → 0
a__U71(tt, M, N) → a__U72(a__isNat(N), M, N)
a__U72(tt, M, N) → a__plus(a__x(mark(N), mark(M)), mark(N))
a__isNat(0) → tt
a__isNat(plus(V1, V2)) → a__U11(a__isNat(V1), V2)
a__isNat(s(V1)) → a__U21(a__isNat(V1))
a__isNat(x(V1, V2)) → a__U31(a__isNat(V1), V2)
a__plus(N, 0) → a__U41(a__isNat(N), N)
a__plus(N, s(M)) → a__U51(a__isNat(M), M, N)
a__x(N, 0) → a__U61(a__isNat(N))
a__x(N, s(M)) → a__U71(a__isNat(M), M, N)
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(U12(X)) → a__U12(mark(X))
mark(isNat(X)) → a__isNat(X)
mark(U21(X)) → a__U21(mark(X))
mark(U31(X1, X2)) → a__U31(mark(X1), X2)
mark(U32(X)) → a__U32(mark(X))
mark(U41(X1, X2)) → a__U41(mark(X1), X2)
mark(U51(X1, X2, X3)) → a__U51(mark(X1), X2, X3)
mark(U52(X1, X2, X3)) → a__U52(mark(X1), X2, X3)
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(U61(X)) → a__U61(mark(X))
mark(U71(X1, X2, X3)) → a__U71(mark(X1), X2, X3)
mark(U72(X1, X2, X3)) → a__U72(mark(X1), X2, X3)
mark(x(X1, X2)) → a__x(mark(X1), mark(X2))
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(0) → 0
a__U11(X1, X2) → U11(X1, X2)
a__U12(X) → U12(X)
a__isNat(X) → isNat(X)
a__U21(X) → U21(X)
a__U31(X1, X2) → U31(X1, X2)
a__U32(X) → U32(X)
a__U41(X1, X2) → U41(X1, X2)
a__U51(X1, X2, X3) → U51(X1, X2, X3)
a__U52(X1, X2, X3) → U52(X1, X2, X3)
a__plus(X1, X2) → plus(X1, X2)
a__U61(X) → U61(X)
a__U71(X1, X2, X3) → U71(X1, X2, X3)
a__U72(X1, X2, X3) → U72(X1, X2, X3)
a__x(X1, X2) → x(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, V2) → A__U12(a__isNat(V2))
A__U11(tt, V2) → A__ISNAT(V2)
A__U31(tt, V2) → A__U32(a__isNat(V2))
A__U31(tt, V2) → A__ISNAT(V2)
A__U41(tt, N) → MARK(N)
A__U51(tt, M, N) → A__U52(a__isNat(N), M, N)
A__U51(tt, M, N) → A__ISNAT(N)
A__U52(tt, M, N) → A__PLUS(mark(N), mark(M))
A__U52(tt, M, N) → MARK(N)
A__U52(tt, M, N) → MARK(M)
A__U71(tt, M, N) → A__U72(a__isNat(N), M, N)
A__U71(tt, M, N) → A__ISNAT(N)
A__U72(tt, M, N) → A__PLUS(a__x(mark(N), mark(M)), mark(N))
A__U72(tt, M, N) → A__X(mark(N), mark(M))
A__U72(tt, M, N) → MARK(N)
A__U72(tt, M, N) → MARK(M)
A__ISNAT(plus(V1, V2)) → A__U11(a__isNat(V1), V2)
A__ISNAT(plus(V1, V2)) → A__ISNAT(V1)
A__ISNAT(s(V1)) → A__U21(a__isNat(V1))
A__ISNAT(s(V1)) → A__ISNAT(V1)
A__ISNAT(x(V1, V2)) → A__U31(a__isNat(V1), V2)
A__ISNAT(x(V1, V2)) → A__ISNAT(V1)
A__PLUS(N, 0) → A__U41(a__isNat(N), N)
A__PLUS(N, 0) → A__ISNAT(N)
A__PLUS(N, s(M)) → A__U51(a__isNat(M), M, N)
A__PLUS(N, s(M)) → A__ISNAT(M)
A__X(N, 0) → A__U61(a__isNat(N))
A__X(N, 0) → A__ISNAT(N)
A__X(N, s(M)) → A__U71(a__isNat(M), M, N)
A__X(N, s(M)) → A__ISNAT(M)
MARK(U11(X1, X2)) → A__U11(mark(X1), X2)
MARK(U11(X1, X2)) → MARK(X1)
MARK(U12(X)) → A__U12(mark(X))
MARK(U12(X)) → MARK(X)
MARK(isNat(X)) → A__ISNAT(X)
MARK(U21(X)) → A__U21(mark(X))
MARK(U21(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(X1, X2)) → A__U41(mark(X1), X2)
MARK(U41(X1, X2)) → MARK(X1)
MARK(U51(X1, X2, X3)) → A__U51(mark(X1), X2, X3)
MARK(U51(X1, X2, X3)) → MARK(X1)
MARK(U52(X1, X2, X3)) → A__U52(mark(X1), X2, X3)
MARK(U52(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(U61(X)) → A__U61(mark(X))
MARK(U61(X)) → MARK(X)
MARK(U71(X1, X2, X3)) → A__U71(mark(X1), X2, X3)
MARK(U71(X1, X2, X3)) → MARK(X1)
MARK(U72(X1, X2, X3)) → A__U72(mark(X1), X2, X3)
MARK(U72(X1, X2, X3)) → MARK(X1)
MARK(x(X1, X2)) → A__X(mark(X1), mark(X2))
MARK(x(X1, X2)) → MARK(X1)
MARK(x(X1, X2)) → MARK(X2)
MARK(s(X)) → MARK(X)

The TRS R consists of the following rules:

a__U11(tt, V2) → a__U12(a__isNat(V2))
a__U12(tt) → tt
a__U21(tt) → tt
a__U31(tt, V2) → a__U32(a__isNat(V2))
a__U32(tt) → tt
a__U41(tt, N) → mark(N)
a__U51(tt, M, N) → a__U52(a__isNat(N), M, N)
a__U52(tt, M, N) → s(a__plus(mark(N), mark(M)))
a__U61(tt) → 0
a__U71(tt, M, N) → a__U72(a__isNat(N), M, N)
a__U72(tt, M, N) → a__plus(a__x(mark(N), mark(M)), mark(N))
a__isNat(0) → tt
a__isNat(plus(V1, V2)) → a__U11(a__isNat(V1), V2)
a__isNat(s(V1)) → a__U21(a__isNat(V1))
a__isNat(x(V1, V2)) → a__U31(a__isNat(V1), V2)
a__plus(N, 0) → a__U41(a__isNat(N), N)
a__plus(N, s(M)) → a__U51(a__isNat(M), M, N)
a__x(N, 0) → a__U61(a__isNat(N))
a__x(N, s(M)) → a__U71(a__isNat(M), M, N)
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(U12(X)) → a__U12(mark(X))
mark(isNat(X)) → a__isNat(X)
mark(U21(X)) → a__U21(mark(X))
mark(U31(X1, X2)) → a__U31(mark(X1), X2)
mark(U32(X)) → a__U32(mark(X))
mark(U41(X1, X2)) → a__U41(mark(X1), X2)
mark(U51(X1, X2, X3)) → a__U51(mark(X1), X2, X3)
mark(U52(X1, X2, X3)) → a__U52(mark(X1), X2, X3)
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(U61(X)) → a__U61(mark(X))
mark(U71(X1, X2, X3)) → a__U71(mark(X1), X2, X3)
mark(U72(X1, X2, X3)) → a__U72(mark(X1), X2, X3)
mark(x(X1, X2)) → a__x(mark(X1), mark(X2))
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(0) → 0
a__U11(X1, X2) → U11(X1, X2)
a__U12(X) → U12(X)
a__isNat(X) → isNat(X)
a__U21(X) → U21(X)
a__U31(X1, X2) → U31(X1, X2)
a__U32(X) → U32(X)
a__U41(X1, X2) → U41(X1, X2)
a__U51(X1, X2, X3) → U51(X1, X2, X3)
a__U52(X1, X2, X3) → U52(X1, X2, X3)
a__plus(X1, X2) → plus(X1, X2)
a__U61(X) → U61(X)
a__U71(X1, X2, X3) → U71(X1, X2, X3)
a__U72(X1, X2, X3) → U72(X1, X2, X3)
a__x(X1, X2) → x(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 2 SCCs with 17 less nodes.

(4) Complex Obligation (AND)

(5) Obligation:

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

A__U11(tt, V2) → A__ISNAT(V2)
A__ISNAT(plus(V1, V2)) → A__U11(a__isNat(V1), V2)
A__ISNAT(plus(V1, V2)) → A__ISNAT(V1)
A__ISNAT(s(V1)) → A__ISNAT(V1)
A__ISNAT(x(V1, V2)) → A__U31(a__isNat(V1), V2)
A__U31(tt, V2) → A__ISNAT(V2)
A__ISNAT(x(V1, V2)) → A__ISNAT(V1)

The TRS R consists of the following rules:

a__U11(tt, V2) → a__U12(a__isNat(V2))
a__U12(tt) → tt
a__U21(tt) → tt
a__U31(tt, V2) → a__U32(a__isNat(V2))
a__U32(tt) → tt
a__U41(tt, N) → mark(N)
a__U51(tt, M, N) → a__U52(a__isNat(N), M, N)
a__U52(tt, M, N) → s(a__plus(mark(N), mark(M)))
a__U61(tt) → 0
a__U71(tt, M, N) → a__U72(a__isNat(N), M, N)
a__U72(tt, M, N) → a__plus(a__x(mark(N), mark(M)), mark(N))
a__isNat(0) → tt
a__isNat(plus(V1, V2)) → a__U11(a__isNat(V1), V2)
a__isNat(s(V1)) → a__U21(a__isNat(V1))
a__isNat(x(V1, V2)) → a__U31(a__isNat(V1), V2)
a__plus(N, 0) → a__U41(a__isNat(N), N)
a__plus(N, s(M)) → a__U51(a__isNat(M), M, N)
a__x(N, 0) → a__U61(a__isNat(N))
a__x(N, s(M)) → a__U71(a__isNat(M), M, N)
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(U12(X)) → a__U12(mark(X))
mark(isNat(X)) → a__isNat(X)
mark(U21(X)) → a__U21(mark(X))
mark(U31(X1, X2)) → a__U31(mark(X1), X2)
mark(U32(X)) → a__U32(mark(X))
mark(U41(X1, X2)) → a__U41(mark(X1), X2)
mark(U51(X1, X2, X3)) → a__U51(mark(X1), X2, X3)
mark(U52(X1, X2, X3)) → a__U52(mark(X1), X2, X3)
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(U61(X)) → a__U61(mark(X))
mark(U71(X1, X2, X3)) → a__U71(mark(X1), X2, X3)
mark(U72(X1, X2, X3)) → a__U72(mark(X1), X2, X3)
mark(x(X1, X2)) → a__x(mark(X1), mark(X2))
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(0) → 0
a__U11(X1, X2) → U11(X1, X2)
a__U12(X) → U12(X)
a__isNat(X) → isNat(X)
a__U21(X) → U21(X)
a__U31(X1, X2) → U31(X1, X2)
a__U32(X) → U32(X)
a__U41(X1, X2) → U41(X1, X2)
a__U51(X1, X2, X3) → U51(X1, X2, X3)
a__U52(X1, X2, X3) → U52(X1, X2, X3)
a__plus(X1, X2) → plus(X1, X2)
a__U61(X) → U61(X)
a__U71(X1, X2, X3) → U71(X1, X2, X3)
a__U72(X1, X2, X3) → U72(X1, X2, X3)
a__x(X1, X2) → x(X1, X2)

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

(6) UsableRulesProof (EQUIVALENT transformation)

We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R.

(7) Obligation:

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

A__U11(tt, V2) → A__ISNAT(V2)
A__ISNAT(plus(V1, V2)) → A__U11(a__isNat(V1), V2)
A__ISNAT(plus(V1, V2)) → A__ISNAT(V1)
A__ISNAT(s(V1)) → A__ISNAT(V1)
A__ISNAT(x(V1, V2)) → A__U31(a__isNat(V1), V2)
A__U31(tt, V2) → A__ISNAT(V2)
A__ISNAT(x(V1, V2)) → A__ISNAT(V1)

The TRS R consists of the following rules:

a__isNat(0) → tt
a__isNat(plus(V1, V2)) → a__U11(a__isNat(V1), V2)
a__isNat(s(V1)) → a__U21(a__isNat(V1))
a__isNat(x(V1, V2)) → a__U31(a__isNat(V1), V2)
a__isNat(X) → isNat(X)
a__U31(tt, V2) → a__U32(a__isNat(V2))
a__U31(X1, X2) → U31(X1, X2)
a__U32(tt) → tt
a__U32(X) → U32(X)
a__U21(tt) → tt
a__U21(X) → U21(X)
a__U11(tt, V2) → a__U12(a__isNat(V2))
a__U11(X1, X2) → U11(X1, X2)
a__U12(tt) → tt
a__U12(X) → U12(X)

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

(8) QDPSizeChangeProof (EQUIVALENT transformation)

By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem.

From the DPs we obtained the following set of size-change graphs:

A__ISNAT(plus(V1, V2)) → A__U11(a__isNat(V1), V2)
The graph contains the following edges 1 > 2
A__ISNAT(x(V1, V2)) → A__U31(a__isNat(V1), V2)
The graph contains the following edges 1 > 2
A__U11(tt, V2) → A__ISNAT(V2)
The graph contains the following edges 2 >= 1
A__U31(tt, V2) → A__ISNAT(V2)
The graph contains the following edges 2 >= 1
A__ISNAT(plus(V1, V2)) → A__ISNAT(V1)
The graph contains the following edges 1 > 1
A__ISNAT(s(V1)) → A__ISNAT(V1)
The graph contains the following edges 1 > 1
A__ISNAT(x(V1, V2)) → A__ISNAT(V1)
The graph contains the following edges 1 > 1

(9) YES

(10) Obligation:

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

MARK(U11(X1, X2)) → MARK(X1)
MARK(U12(X)) → MARK(X)
MARK(U21(X)) → MARK(X)
MARK(U31(X1, X2)) → MARK(X1)
MARK(U32(X)) → MARK(X)
MARK(U41(X1, X2)) → A__U41(mark(X1), X2)
A__U41(tt, N) → MARK(N)
MARK(U41(X1, X2)) → MARK(X1)
MARK(U51(X1, X2, X3)) → A__U51(mark(X1), X2, X3)
A__U51(tt, M, N) → A__U52(a__isNat(N), M, N)
A__U52(tt, M, N) → A__PLUS(mark(N), mark(M))
A__PLUS(N, 0) → A__U41(a__isNat(N), N)
A__PLUS(N, s(M)) → A__U51(a__isNat(M), M, N)
A__U52(tt, M, N) → MARK(N)
MARK(U51(X1, X2, X3)) → MARK(X1)
MARK(U52(X1, X2, X3)) → A__U52(mark(X1), X2, X3)
A__U52(tt, M, N) → MARK(M)
MARK(U52(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(U61(X)) → MARK(X)
MARK(U71(X1, X2, X3)) → A__U71(mark(X1), X2, X3)
A__U71(tt, M, N) → A__U72(a__isNat(N), M, N)
A__U72(tt, M, N) → A__PLUS(a__x(mark(N), mark(M)), mark(N))
A__U72(tt, M, N) → A__X(mark(N), mark(M))
A__X(N, s(M)) → A__U71(a__isNat(M), M, N)
A__U72(tt, M, N) → MARK(N)
MARK(U71(X1, X2, X3)) → MARK(X1)
MARK(U72(X1, X2, X3)) → A__U72(mark(X1), X2, X3)
A__U72(tt, M, N) → MARK(M)
MARK(U72(X1, X2, X3)) → MARK(X1)
MARK(x(X1, X2)) → A__X(mark(X1), mark(X2))
MARK(x(X1, X2)) → MARK(X1)
MARK(x(X1, X2)) → MARK(X2)
MARK(s(X)) → MARK(X)

The TRS R consists of the following rules:

a__U11(tt, V2) → a__U12(a__isNat(V2))
a__U12(tt) → tt
a__U21(tt) → tt
a__U31(tt, V2) → a__U32(a__isNat(V2))
a__U32(tt) → tt
a__U41(tt, N) → mark(N)
a__U51(tt, M, N) → a__U52(a__isNat(N), M, N)
a__U52(tt, M, N) → s(a__plus(mark(N), mark(M)))
a__U61(tt) → 0
a__U71(tt, M, N) → a__U72(a__isNat(N), M, N)
a__U72(tt, M, N) → a__plus(a__x(mark(N), mark(M)), mark(N))
a__isNat(0) → tt
a__isNat(plus(V1, V2)) → a__U11(a__isNat(V1), V2)
a__isNat(s(V1)) → a__U21(a__isNat(V1))
a__isNat(x(V1, V2)) → a__U31(a__isNat(V1), V2)
a__plus(N, 0) → a__U41(a__isNat(N), N)
a__plus(N, s(M)) → a__U51(a__isNat(M), M, N)
a__x(N, 0) → a__U61(a__isNat(N))
a__x(N, s(M)) → a__U71(a__isNat(M), M, N)
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(U12(X)) → a__U12(mark(X))
mark(isNat(X)) → a__isNat(X)
mark(U21(X)) → a__U21(mark(X))
mark(U31(X1, X2)) → a__U31(mark(X1), X2)
mark(U32(X)) → a__U32(mark(X))
mark(U41(X1, X2)) → a__U41(mark(X1), X2)
mark(U51(X1, X2, X3)) → a__U51(mark(X1), X2, X3)
mark(U52(X1, X2, X3)) → a__U52(mark(X1), X2, X3)
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(U61(X)) → a__U61(mark(X))
mark(U71(X1, X2, X3)) → a__U71(mark(X1), X2, X3)
mark(U72(X1, X2, X3)) → a__U72(mark(X1), X2, X3)
mark(x(X1, X2)) → a__x(mark(X1), mark(X2))
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(0) → 0
a__U11(X1, X2) → U11(X1, X2)
a__U12(X) → U12(X)
a__isNat(X) → isNat(X)
a__U21(X) → U21(X)
a__U31(X1, X2) → U31(X1, X2)
a__U32(X) → U32(X)
a__U41(X1, X2) → U41(X1, X2)
a__U51(X1, X2, X3) → U51(X1, X2, X3)
a__U52(X1, X2, X3) → U52(X1, X2, X3)
a__plus(X1, X2) → plus(X1, X2)
a__U61(X) → U61(X)
a__U71(X1, X2, X3) → U71(X1, X2, X3)
a__U72(X1, X2, X3) → U72(X1, X2, X3)
a__x(X1, X2) → x(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,JAR06].

The following pairs can be oriented strictly and are deleted.

MARK(U41(X1, X2)) → A__U41(mark(X1), X2)
MARK(U41(X1, X2)) → MARK(X1)
MARK(U51(X1, X2, X3)) → A__U51(mark(X1), X2, X3)
A__U52(tt, M, N) → A__PLUS(mark(N), mark(M))
A__PLUS(N, 0) → A__U41(a__isNat(N), N)
A__PLUS(N, s(M)) → A__U51(a__isNat(M), M, N)
A__U52(tt, M, N) → MARK(N)
MARK(U51(X1, X2, X3)) → MARK(X1)
MARK(U52(X1, X2, X3)) → A__U52(mark(X1), X2, X3)
A__U52(tt, M, N) → MARK(M)
MARK(U52(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(U71(X1, X2, X3)) → A__U71(mark(X1), X2, X3)
A__U72(tt, M, N) → A__PLUS(a__x(mark(N), mark(M)), mark(N))
A__U72(tt, M, N) → A__X(mark(N), mark(M))
A__X(N, s(M)) → A__U71(a__isNat(M), M, N)
A__U72(tt, M, N) → MARK(N)
MARK(U71(X1, X2, X3)) → MARK(X1)
MARK(U72(X1, X2, X3)) → A__U72(mark(X1), X2, X3)
A__U72(tt, M, N) → MARK(M)
MARK(U72(X1, X2, X3)) → MARK(X1)
MARK(x(X1, X2)) → A__X(mark(X1), mark(X2))
MARK(x(X1, X2)) → MARK(X1)
MARK(x(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) = x1
U12(x1) = x1
U21(x1) = x1
U31(x1, x2) = x1
U32(x1) = x1
U41(x1, x2) = U41(x1, x2)
A__U41(x1, x2) = A__U41(x2)
mark(x1) = x1
tt = tt
U51(x1, x2, x3) = U51(x1, x2, x3)
A__U51(x1, x2, x3) = A__U51(x1, x2, x3)
A__U52(x1, x2, x3) = A__U52(x1, x2, x3)
a__isNat(x1) = a__isNat
A__PLUS(x1, x2) = A__PLUS(x1, x2)
0 = 0
s(x1) = s(x1)
U52(x1, x2, x3) = U52(x1, x2, x3)
plus(x1, x2) = plus(x1, x2)
U61(x1) = x1
U71(x1, x2, x3) = U71(x1, x2, x3)
A__U71(x1, x2, x3) = A__U71(x1, x2, x3)
A__U72(x1, x2, x3) = A__U72(x1, x2, x3)
a__x(x1, x2) = a__x(x1, x2)
A__X(x1, x2) = A__X(x1, x2)
U72(x1, x2, x3) = U72(x1, x2, x3)
x(x1, x2) = x(x1, x2)
a__U11(x1, x2) = x1
a__U12(x1) = x1
isNat(x1) = isNat
a__U21(x1) = x1
a__U31(x1, x2) = x1
a__U32(x1) = x1
a__U41(x1, x2) = a__U41(x1, x2)
a__plus(x1, x2) = a__plus(x1, x2)
a__U71(x1, x2, x3) = a__U71(x1, x2, x3)
a__U72(x1, x2, x3) = a__U72(x1, x2, x3)
a__U51(x1, x2, x3) = a__U51(x1, x2, x3)
a__U52(x1, x2, x3) = a__U52(x1, x2, x3)
a__U61(x1) = x1

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

[U71₃, A_{_U71₃}, A_{_U72₃}, a_{_x₂}, A_{_X₂}, U72₃, x₂, a_{_U71₃}, a_{_U72₃}] > [U51₃, U52₃, plus₂, a_{_plus₂}, a_{_U51₃}, a_{_U52₃}] > [MARK₁, U41₂, A_{_U41₁}, tt, A_{_U51₃}, A_{_U52₃}, a_{_isNat}, A_{_PLUS₂}, s₁, isNat, a_{_U41₂}] > 0

Status:

MARK₁: multiset
U41₂: multiset
A_{_U41₁}: multiset
tt: multiset
U51₃: [3,2,1]
A_{_U51₃}: multiset
A_{_U52₃}: multiset
a_{_isNat}: []
A_{_PLUS₂}: multiset
0: multiset
s₁: [1]
U52₃: [3,2,1]
plus₂: [1,2]
U71₃: multiset
A_{_U71₃}: multiset
A_{_U72₃}: multiset
a_{_x₂}: multiset
A_{_X₂}: multiset
U72₃: multiset
x₂: multiset
isNat: []
a_{_U41₂}: multiset
a_{_plus₂}: [1,2]
a_{_U71₃}: multiset
a_{_U72₃}: multiset
a_{_U51₃}: [3,2,1]
a_{_U52₃}: [3,2,1]

The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:

mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(U12(X)) → a__U12(mark(X))
mark(isNat(X)) → a__isNat(X)
mark(U21(X)) → a__U21(mark(X))
mark(U31(X1, X2)) → a__U31(mark(X1), X2)
mark(U32(X)) → a__U32(mark(X))
mark(U41(X1, X2)) → a__U41(mark(X1), X2)
a__U41(tt, N) → mark(N)
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
a__plus(N, 0) → a__U41(a__isNat(N), N)
mark(U71(X1, X2, X3)) → a__U71(mark(X1), X2, X3)
a__U71(tt, M, N) → a__U72(a__isNat(N), M, N)
a__U72(tt, M, N) → a__plus(a__x(mark(N), mark(M)), mark(N))
mark(U72(X1, X2, X3)) → a__U72(mark(X1), X2, X3)
mark(x(X1, X2)) → a__x(mark(X1), mark(X2))
a__x(N, s(M)) → a__U71(a__isNat(M), M, N)
mark(U51(X1, X2, X3)) → a__U51(mark(X1), X2, X3)
mark(U52(X1, X2, X3)) → a__U52(mark(X1), X2, X3)
mark(U61(X)) → a__U61(mark(X))
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(0) → 0
a__isNat(0) → tt
a__isNat(plus(V1, V2)) → a__U11(a__isNat(V1), V2)
a__isNat(s(V1)) → a__U21(a__isNat(V1))
a__isNat(x(V1, V2)) → a__U31(a__isNat(V1), V2)
a__isNat(X) → isNat(X)
a__x(N, 0) → a__U61(a__isNat(N))
a__x(X1, X2) → x(X1, X2)
a__U52(tt, M, N) → s(a__plus(mark(N), mark(M)))
a__plus(X1, X2) → plus(X1, X2)
a__U11(tt, V2) → a__U12(a__isNat(V2))
a__U11(X1, X2) → U11(X1, X2)
a__U12(tt) → tt
a__U12(X) → U12(X)
a__U21(tt) → tt
a__U21(X) → U21(X)
a__U31(tt, V2) → a__U32(a__isNat(V2))
a__U31(X1, X2) → U31(X1, X2)
a__U32(tt) → tt
a__U32(X) → U32(X)
a__U41(X1, X2) → U41(X1, X2)
a__U71(X1, X2, X3) → U71(X1, X2, X3)
a__U72(X1, X2, X3) → U72(X1, X2, X3)
a__U51(X1, X2, X3) → U51(X1, X2, X3)
a__U52(X1, X2, X3) → U52(X1, X2, X3)
a__U61(tt) → 0
a__U61(X) → U61(X)
a__U51(tt, M, N) → a__U52(a__isNat(N), M, N)
a__plus(N, s(M)) → a__U51(a__isNat(M), M, N)

(12) Obligation:

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

MARK(U11(X1, X2)) → MARK(X1)
MARK(U12(X)) → MARK(X)
MARK(U21(X)) → MARK(X)
MARK(U31(X1, X2)) → MARK(X1)
MARK(U32(X)) → MARK(X)
A__U41(tt, N) → MARK(N)
A__U51(tt, M, N) → A__U52(a__isNat(N), M, N)
MARK(U61(X)) → MARK(X)
A__U71(tt, M, N) → A__U72(a__isNat(N), M, N)

The TRS R consists of the following rules:

a__U11(tt, V2) → a__U12(a__isNat(V2))
a__U12(tt) → tt
a__U21(tt) → tt
a__U31(tt, V2) → a__U32(a__isNat(V2))
a__U32(tt) → tt
a__U41(tt, N) → mark(N)
a__U51(tt, M, N) → a__U52(a__isNat(N), M, N)
a__U52(tt, M, N) → s(a__plus(mark(N), mark(M)))
a__U61(tt) → 0
a__U71(tt, M, N) → a__U72(a__isNat(N), M, N)
a__U72(tt, M, N) → a__plus(a__x(mark(N), mark(M)), mark(N))
a__isNat(0) → tt
a__isNat(plus(V1, V2)) → a__U11(a__isNat(V1), V2)
a__isNat(s(V1)) → a__U21(a__isNat(V1))
a__isNat(x(V1, V2)) → a__U31(a__isNat(V1), V2)
a__plus(N, 0) → a__U41(a__isNat(N), N)
a__plus(N, s(M)) → a__U51(a__isNat(M), M, N)
a__x(N, 0) → a__U61(a__isNat(N))
a__x(N, s(M)) → a__U71(a__isNat(M), M, N)
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(U12(X)) → a__U12(mark(X))
mark(isNat(X)) → a__isNat(X)
mark(U21(X)) → a__U21(mark(X))
mark(U31(X1, X2)) → a__U31(mark(X1), X2)
mark(U32(X)) → a__U32(mark(X))
mark(U41(X1, X2)) → a__U41(mark(X1), X2)
mark(U51(X1, X2, X3)) → a__U51(mark(X1), X2, X3)
mark(U52(X1, X2, X3)) → a__U52(mark(X1), X2, X3)
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(U61(X)) → a__U61(mark(X))
mark(U71(X1, X2, X3)) → a__U71(mark(X1), X2, X3)
mark(U72(X1, X2, X3)) → a__U72(mark(X1), X2, X3)
mark(x(X1, X2)) → a__x(mark(X1), mark(X2))
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(0) → 0
a__U11(X1, X2) → U11(X1, X2)
a__U12(X) → U12(X)
a__isNat(X) → isNat(X)
a__U21(X) → U21(X)
a__U31(X1, X2) → U31(X1, X2)
a__U32(X) → U32(X)
a__U41(X1, X2) → U41(X1, X2)
a__U51(X1, X2, X3) → U51(X1, X2, X3)
a__U52(X1, X2, X3) → U52(X1, X2, X3)
a__plus(X1, X2) → plus(X1, X2)
a__U61(X) → U61(X)
a__U71(X1, X2, X3) → U71(X1, X2, X3)
a__U72(X1, X2, X3) → U72(X1, X2, X3)
a__x(X1, X2) → x(X1, X2)

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

(13) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 3 less nodes.

(14) Obligation:

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

MARK(U12(X)) → MARK(X)
MARK(U11(X1, X2)) → MARK(X1)
MARK(U21(X)) → MARK(X)
MARK(U31(X1, X2)) → MARK(X1)
MARK(U32(X)) → MARK(X)
MARK(U61(X)) → MARK(X)

The TRS R consists of the following rules:

a__U11(tt, V2) → a__U12(a__isNat(V2))
a__U12(tt) → tt
a__U21(tt) → tt
a__U31(tt, V2) → a__U32(a__isNat(V2))
a__U32(tt) → tt
a__U41(tt, N) → mark(N)
a__U51(tt, M, N) → a__U52(a__isNat(N), M, N)
a__U52(tt, M, N) → s(a__plus(mark(N), mark(M)))
a__U61(tt) → 0
a__U71(tt, M, N) → a__U72(a__isNat(N), M, N)
a__U72(tt, M, N) → a__plus(a__x(mark(N), mark(M)), mark(N))
a__isNat(0) → tt
a__isNat(plus(V1, V2)) → a__U11(a__isNat(V1), V2)
a__isNat(s(V1)) → a__U21(a__isNat(V1))
a__isNat(x(V1, V2)) → a__U31(a__isNat(V1), V2)
a__plus(N, 0) → a__U41(a__isNat(N), N)
a__plus(N, s(M)) → a__U51(a__isNat(M), M, N)
a__x(N, 0) → a__U61(a__isNat(N))
a__x(N, s(M)) → a__U71(a__isNat(M), M, N)
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(U12(X)) → a__U12(mark(X))
mark(isNat(X)) → a__isNat(X)
mark(U21(X)) → a__U21(mark(X))
mark(U31(X1, X2)) → a__U31(mark(X1), X2)
mark(U32(X)) → a__U32(mark(X))
mark(U41(X1, X2)) → a__U41(mark(X1), X2)
mark(U51(X1, X2, X3)) → a__U51(mark(X1), X2, X3)
mark(U52(X1, X2, X3)) → a__U52(mark(X1), X2, X3)
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(U61(X)) → a__U61(mark(X))
mark(U71(X1, X2, X3)) → a__U71(mark(X1), X2, X3)
mark(U72(X1, X2, X3)) → a__U72(mark(X1), X2, X3)
mark(x(X1, X2)) → a__x(mark(X1), mark(X2))
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(0) → 0
a__U11(X1, X2) → U11(X1, X2)
a__U12(X) → U12(X)
a__isNat(X) → isNat(X)
a__U21(X) → U21(X)
a__U31(X1, X2) → U31(X1, X2)
a__U32(X) → U32(X)
a__U41(X1, X2) → U41(X1, X2)
a__U51(X1, X2, X3) → U51(X1, X2, X3)
a__U52(X1, X2, X3) → U52(X1, X2, X3)
a__plus(X1, X2) → plus(X1, X2)
a__U61(X) → U61(X)
a__U71(X1, X2, X3) → U71(X1, X2, X3)
a__U72(X1, X2, X3) → U72(X1, X2, X3)
a__x(X1, X2) → x(X1, X2)

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

(15) UsableRulesProof (EQUIVALENT transformation)

(16) Obligation:

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

MARK(U12(X)) → MARK(X)
MARK(U11(X1, X2)) → MARK(X1)
MARK(U21(X)) → MARK(X)
MARK(U31(X1, X2)) → MARK(X1)
MARK(U32(X)) → MARK(X)
MARK(U61(X)) → MARK(X)

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

(17) QDPSizeChangeProof (EQUIVALENT transformation)

By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem.

From the DPs we obtained the following set of size-change graphs:

MARK(U12(X)) → MARK(X)
The graph contains the following edges 1 > 1
MARK(U11(X1, X2)) → MARK(X1)
The graph contains the following edges 1 > 1
MARK(U21(X)) → MARK(X)
The graph contains the following edges 1 > 1
MARK(U31(X1, X2)) → MARK(X1)
The graph contains the following edges 1 > 1
MARK(U32(X)) → MARK(X)
The graph contains the following edges 1 > 1
MARK(U61(X)) → MARK(X)
The graph contains the following edges 1 > 1

(0) Obligation:

(1) DependencyPairsProof (EQUIVALENT transformation)

(2) Obligation:

(3) DependencyGraphProof (EQUIVALENT transformation)

(4) Complex Obligation (AND)

(5) Obligation:

(6) UsableRulesProof (EQUIVALENT transformation)

(7) Obligation:

(8) QDPSizeChangeProof (EQUIVALENT transformation)

(9) YES

(10) Obligation:

(11) QDPOrderProof (EQUIVALENT transformation)

(12) Obligation:

(13) DependencyGraphProof (EQUIVALENT transformation)

(14) Obligation:

(15) UsableRulesProof (EQUIVALENT transformation)

(16) Obligation:

(17) QDPSizeChangeProof (EQUIVALENT transformation)

(18) YES