Termination w.r.t. Q proof of /tmp/tmptRpllt/MYNAT_nokinds_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 QDP

    ↳10 QDPOrderProof (⇔)

    ↳11 QDP

    ↳12 PisEmptyProof (⇔)

    ↳13 TRUE

  ↳14 QDP

    ↳15 QDPOrderProof (⇔)

    ↳16 QDP

    ↳17 DependencyGraphProof (⇔)

    ↳18 QDP

    ↳19 QDPOrderProof (⇔)

    ↳20 QDP

    ↳21 QDPOrderProof (⇔)

    ↳22 QDP

    ↳23 QDPOrderProof (⇔)

    ↳24 QDP

    ↳25 QDPOrderProof (⇔)

    ↳26 QDP

    ↳27 PisEmptyProof (⇔)

    ↳28 TRUE

(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) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

A__ISNAT(plus(V1, V2)) → A__U11(a__isNat(V1), V2)
A__ISNAT(plus(V1, V2)) → 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 remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
A__U11(x1, x2) = A__U11(x2)
tt = tt
A__ISNAT(x1) = A__ISNAT(x1)
plus(x1, x2) = plus(x1, x2)
a__isNat(x1) = a__isNat(x1)
s(x1) = x1
x(x1, x2) = x(x1, x2)
A__U31(x1, x2) = A__U31(x1, x2)
a__U12(x1) = a__U12
a__U21(x1) = x1
a__U11(x1, x2) = a__U11(x1, x2)
U11(x1, x2) = U11
U12(x1) = U12
a__U31(x1, x2) = x1
a__U32(x1) = a__U32
isNat(x1) = isNat
U21(x1) = x1
U31(x1, x2) = x1
U32(x1) = U32
0 = 0

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

[A_{_U11₁}, A_{_ISNAT₁}, a_{_isNat₁}, x₂, A_{_U31₂}, a_{_U11₂}, U11] > [a_{_U12}, U12] > [tt, a_{_U32}, U32, 0]
[A_{_U11₁}, A_{_ISNAT₁}, a_{_isNat₁}, x₂, A_{_U31₂}, a_{_U11₂}, U11] > isNat > [tt, a_{_U32}, U32, 0]
plus₂ > [tt, a_{_U32}, U32, 0]

Status:

A_{_U31₂}: [2,1]
plus₂: multiset
A_{_ISNAT₁}: [1]
U32: multiset
a_{_U11₂}: [2,1]
U12: []
U11: []
x₂: multiset
0: multiset
isNat: multiset
A_{_U11₁}: [1]
tt: multiset
a_{_isNat₁}: [1]
a_{_U12}: []
a_{_U32}: multiset

The following usable rules [FROCOS05] were oriented:

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

(7) Obligation:

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

A__U11(tt, V2) → A__ISNAT(V2)
A__ISNAT(s(V1)) → 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.

(8) DependencyGraphProof (EQUIVALENT transformation)

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

(9) Obligation:

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

A__ISNAT(s(V1)) → 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.

(10) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

A__ISNAT(s(V1)) → A__ISNAT(V1)

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

s₁ > A_{_ISNAT₁}

Status:

A_{_ISNAT₁}: multiset
s₁: multiset

The following usable rules [FROCOS05] were oriented: none

(11) Obligation:

Q DP problem:
P is empty.
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.

(12) PisEmptyProof (EQUIVALENT transformation)

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

(13) TRUE

(14) Obligation:

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

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.

(15) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

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__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__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(x1, x2)
mark(x1) = x1
tt = tt
U51(x1, x2, x3) = U51(x1, x2, x3)
A__U51(x1, x2, x3) = A__U51(x2, x3)
A__U52(x1, x2, x3) = A__U52(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) = U61(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__U52(x1, x2, x3) = a__U52(x1, x2, x3)
a__U61(x1) = a__U61(x1)
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__U51(x1, x2, x3) = a__U51(x1, x2, x3)
a__plus(x1, x2) = a__plus(x1, x2)
a__U72(x1, x2, x3) = a__U72(x1, x2, x3)
a__U71(x1, x2, x3) = a__U71(x1, x2, x3)

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

[U61₁, U71₃, A_{_U71₃}, A_{_U72₃}, a_{_x₂}, A_{_X₂}, U72₃, x₂, a_{_U61₁}, a_{_U72₃}, a_{_U71₃}] > [MARK₁, A_{_U41₂}, A_{_U51₂}, A_{_U52₂}, A_{_PLUS₂}] > [tt, a_{_isNat}, isNat] > s₁
[U61₁, U71₃, A_{_U71₃}, A_{_U72₃}, a_{_x₂}, A_{_X₂}, U72₃, x₂, a_{_U61₁}, a_{_U72₃}, a_{_U71₃}] > [U51₃, U52₃, plus₂, a_{_U52₃}, a_{_U51₃}, a_{_plus₂}] > [U41₂, a_{_U41₂}] > s₁
[U61₁, U71₃, A_{_U71₃}, A_{_U72₃}, a_{_x₂}, A_{_X₂}, U72₃, x₂, a_{_U61₁}, a_{_U72₃}, a_{_U71₃}] > [U51₃, U52₃, plus₂, a_{_U52₃}, a_{_U51₃}, a_{_plus₂}] > [tt, a_{_isNat}, isNat] > s₁
[U61₁, U71₃, A_{_U71₃}, A_{_U72₃}, a_{_x₂}, A_{_X₂}, U72₃, x₂, a_{_U61₁}, a_{_U72₃}, a_{_U71₃}] > 0 > [U41₂, a_{_U41₂}] > s₁
[U61₁, U71₃, A_{_U71₃}, A_{_U72₃}, a_{_x₂}, A_{_X₂}, U72₃, x₂, a_{_U61₁}, a_{_U72₃}, a_{_U71₃}] > 0 > [tt, a_{_isNat}, isNat] > s₁

Status:

A_{_PLUS₂}: multiset
x₂: [2,1]
A_{_U41₂}: multiset
isNat: []
a_{_plus₂}: [2,1]
a_{_U61₁}: [1]
tt: multiset
s₁: multiset
A_{_U52₂}: multiset
U51₃: [2,3,1]
plus₂: [2,1]
a_{_U41₂}: multiset
U61₁: [1]
a_{_isNat}: []
a_{_U52₃}: [2,3,1]
0: multiset
a_{_U51₃}: [2,3,1]
a_{_x₂}: [2,1]
U52₃: [2,3,1]
A_{_U71₃}: [2,3,1]
MARK₁: multiset
A_{_U51₂}: multiset
U41₂: multiset
A_{_X₂}: [2,1]
U72₃: [2,3,1]
A_{_U72₃}: [2,3,1]
a_{_U71₃}: [2,3,1]
a_{_U72₃}: [2,3,1]
U71₃: [2,3,1]

The following usable rules [FROCOS05] were oriented:

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__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__U52(X1, X2, X3) → U52(X1, X2, X3)
a__U51(X1, X2, X3) → U51(X1, X2, X3)
a__U61(X) → U61(X)
a__plus(X1, X2) → plus(X1, X2)
a__U72(X1, X2, X3) → U72(X1, X2, X3)
a__U71(X1, X2, X3) → U71(X1, X2, X3)
a__x(X1, X2) → x(X1, X2)
a__U12(tt) → tt
a__U21(tt) → tt
a__U11(tt, V2) → a__U12(a__isNat(V2))
a__U51(tt, M, N) → a__U52(a__isNat(N), M, N)
a__U31(tt, V2) → a__U32(a__isNat(V2))
a__U32(tt) → tt
a__U52(tt, M, N) → s(a__plus(mark(N), mark(M)))
a__U61(tt) → 0
a__isNat(x(V1, V2)) → a__U31(a__isNat(V1), V2)
a__isNat(s(V1)) → a__U21(a__isNat(V1))
a__isNat(plus(V1, V2)) → a__U11(a__isNat(V1), V2)
a__isNat(0) → tt
a__x(N, 0) → a__U61(a__isNat(N))
a__plus(N, s(M)) → a__U51(a__isNat(M), M, N)
mark(U21(X)) → a__U21(mark(X))
mark(isNat(X)) → a__isNat(X)
mark(U12(X)) → a__U12(mark(X))
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(U51(X1, X2, X3)) → a__U51(mark(X1), X2, X3)
mark(x(X1, X2)) → a__x(mark(X1), mark(X2))
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(U71(X1, X2, X3)) → a__U71(mark(X1), X2, X3)
a__U71(tt, M, N) → a__U72(a__isNat(N), M, N)
a__x(N, s(M)) → a__U71(a__isNat(M), M, N)
mark(U72(X1, X2, X3)) → a__U72(mark(X1), X2, X3)
a__plus(N, 0) → a__U41(a__isNat(N), N)
a__U41(tt, N) → mark(N)
a__U72(tt, M, N) → a__plus(a__x(mark(N), mark(M)), mark(N))
mark(U41(X1, X2)) → a__U41(mark(X1), X2)
mark(U32(X)) → a__U32(mark(X))
mark(U31(X1, X2)) → a__U31(mark(X1), X2)

(16) 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__U51(tt, M, N) → A__U52(a__isNat(N), M, N)
A__U52(tt, M, N) → A__PLUS(mark(N), mark(M))
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.

(17) DependencyGraphProof (EQUIVALENT transformation)

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

(18) 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)

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.

(19) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

MARK(U11(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) = x1
U11(x1, x2) = U11(x1, x2)
U21(x1) = x1
U31(x1, x2) = U31(x1, x2)
U32(x1) = x1

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

trivial

Status:

MARK₁: [1]
U31₂: multiset
U11₂: multiset

The following usable rules [FROCOS05] were oriented: none

(20) Obligation:

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

MARK(U12(X)) → MARK(X)
MARK(U21(X)) → MARK(X)
MARK(U32(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.

(21) 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) = x1
U21(x1) = x1
U32(x1) = U32(x1)

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

[MARK₁, U32₁]

Status:

MARK₁: multiset
U32₁: multiset

The following usable rules [FROCOS05] were oriented: none

(22) Obligation:

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

MARK(U12(X)) → MARK(X)
MARK(U21(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.

(23) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

MARK(U12(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) = U12(x1)
U21(x1) = x1

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

[MARK₁, U12₁]

Status:

MARK₁: [1]
U12₁: multiset

The following usable rules [FROCOS05] were oriented: none

(24) Obligation:

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

MARK(U21(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.

(25) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

MARK(U21(X)) → MARK(X)

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

U21₁ > MARK₁

Status:

MARK₁: multiset
U21₁: multiset

The following usable rules [FROCOS05] were oriented: none

(26) Obligation:

Q DP problem:
P is empty.
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.

(27) 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) Obligation:

(10) QDPOrderProof (EQUIVALENT transformation)

(11) Obligation:

(12) PisEmptyProof (EQUIVALENT transformation)

(13) TRUE

(14) Obligation:

(15) QDPOrderProof (EQUIVALENT transformation)

(16) Obligation:

(17) DependencyGraphProof (EQUIVALENT transformation)

(18) Obligation:

(19) QDPOrderProof (EQUIVALENT transformation)

(20) Obligation:

(21) QDPOrderProof (EQUIVALENT transformation)

(22) Obligation:

(23) QDPOrderProof (EQUIVALENT transformation)

(24) Obligation:

(25) QDPOrderProof (EQUIVALENT transformation)

(26) Obligation:

(27) PisEmptyProof (EQUIVALENT transformation)

(28) TRUE