Termination w.r.t. Q proof of /tmp/tmpJkpR6P/PEANO_nokinds_noand

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

0 QTRS

↳1 DependencyPairsProof (⇔)

↳2 QDP

↳3 DependencyGraphProof (⇔)

↳4 AND

  ↳5 QDP

    ↳6 QDPOrderProof (⇔)

    ↳7 QDP

    ↳8 QDPOrderProof (⇔)

    ↳9 QDP

    ↳10 PisEmptyProof (⇔)

    ↳11 TRUE

  ↳12 QDP

(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, N) → mark(N)
a__U41(tt, M, N) → a__U42(a__isNat(N), M, N)
a__U42(tt, M, N) → s(a__plus(mark(N), mark(M)))
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__plus(N, 0) → a__U31(a__isNat(N), N)
a__plus(N, s(M)) → a__U41(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(U41(X1, X2, X3)) → a__U41(mark(X1), X2, X3)
mark(U42(X1, X2, X3)) → a__U42(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) → 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__U41(X1, X2, X3) → U41(X1, X2, X3)
a__U42(X1, X2, X3) → U42(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, V2) → A__U12(a__isNat(V2))
A__U11(tt, V2) → A__ISNAT(V2)
A__U31(tt, N) → MARK(N)
A__U41(tt, M, N) → A__U42(a__isNat(N), M, N)
A__U41(tt, M, N) → A__ISNAT(N)
A__U42(tt, M, N) → A__PLUS(mark(N), mark(M))
A__U42(tt, M, N) → MARK(N)
A__U42(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__PLUS(N, 0) → A__U31(a__isNat(N), N)
A__PLUS(N, 0) → A__ISNAT(N)
A__PLUS(N, s(M)) → A__U41(a__isNat(M), M, N)
A__PLUS(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(U41(X1, X2, X3)) → A__U41(mark(X1), X2, X3)
MARK(U41(X1, X2, X3)) → MARK(X1)
MARK(U42(X1, X2, X3)) → A__U42(mark(X1), X2, X3)
MARK(U42(X1, X2, X3)) → MARK(X1)
MARK(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, V2) → a__U12(a__isNat(V2))
a__U12(tt) → tt
a__U21(tt) → tt
a__U31(tt, N) → mark(N)
a__U41(tt, M, N) → a__U42(a__isNat(N), M, N)
a__U42(tt, M, N) → s(a__plus(mark(N), mark(M)))
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__plus(N, 0) → a__U31(a__isNat(N), N)
a__plus(N, s(M)) → a__U41(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(U41(X1, X2, X3)) → a__U41(mark(X1), X2, X3)
mark(U42(X1, X2, X3)) → a__U42(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) → 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__U41(X1, X2, X3) → U41(X1, X2, X3)
a__U42(X1, X2, X3) → U42(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 2 SCCs with 9 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)

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, N) → mark(N)
a__U41(tt, M, N) → a__U42(a__isNat(N), M, N)
a__U42(tt, M, N) → s(a__plus(mark(N), mark(M)))
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__plus(N, 0) → a__U31(a__isNat(N), N)
a__plus(N, s(M)) → a__U41(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(U41(X1, X2, X3)) → a__U41(mark(X1), X2, X3)
mark(U42(X1, X2, X3)) → a__U42(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) → 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__U41(X1, X2, X3) → U41(X1, X2, X3)
a__U42(X1, X2, X3) → U42(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__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)

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(x1, x2)
tt = tt
A__ISNAT(x1) = x1
plus(x1, x2) = plus(x1, x2)
a__isNat(x1) = x1
s(x1) = x1
a__U21(x1) = x1
a__U12(x1) = a__U12
a__U11(x1, x2) = a__U11
0 = 0
U11(x1, x2) = U11
isNat(x1) = x1
U12(x1) = U12
U21(x1) = x1

Lexicographic path order with status [LPO].
Precedence:

plus₂ > A_{_U11₂}
plus₂ > a_{_U11} > a_{_U12} > tt
plus₂ > a_{_U11} > a_{_U12} > U12
plus₂ > a_{_U11} > U11
0 > tt

Status:

plus₂: [1,2]
tt: []
A_{_U11₂}: [1,2]
a_{_U11}: []
U12: []
U11: []
a_{_U12}: []
0: []

The following usable rules [FROCOS05] were oriented:

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

(7) 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, N) → mark(N)
a__U41(tt, M, N) → a__U42(a__isNat(N), M, N)
a__U42(tt, M, N) → s(a__plus(mark(N), mark(M)))
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__plus(N, 0) → a__U31(a__isNat(N), N)
a__plus(N, s(M)) → a__U41(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(U41(X1, X2, X3)) → a__U41(mark(X1), X2, X3)
mark(U42(X1, X2, X3)) → a__U42(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) → 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__U41(X1, X2, X3) → U41(X1, X2, X3)
a__U42(X1, X2, X3) → U42(X1, X2, X3)
a__plus(X1, X2) → plus(X1, X2)

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

(8) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

A__ISNAT(s(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)

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

s₁: [1]

The following usable rules [FROCOS05] were oriented: none

(9) 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, N) → mark(N)
a__U41(tt, M, N) → a__U42(a__isNat(N), M, N)
a__U42(tt, M, N) → s(a__plus(mark(N), mark(M)))
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__plus(N, 0) → a__U31(a__isNat(N), N)
a__plus(N, s(M)) → a__U41(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(U41(X1, X2, X3)) → a__U41(mark(X1), X2, X3)
mark(U42(X1, X2, X3)) → a__U42(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) → 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__U41(X1, X2, X3) → U41(X1, X2, X3)
a__U42(X1, X2, X3) → U42(X1, X2, X3)
a__plus(X1, X2) → plus(X1, X2)

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

(10) PisEmptyProof (EQUIVALENT transformation)

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

(11) TRUE

(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)) → A__U31(mark(X1), X2)
A__U31(tt, N) → MARK(N)
MARK(U31(X1, X2)) → MARK(X1)
MARK(U41(X1, X2, X3)) → A__U41(mark(X1), X2, X3)
A__U41(tt, M, N) → A__U42(a__isNat(N), M, N)
A__U42(tt, M, N) → A__PLUS(mark(N), mark(M))
A__PLUS(N, 0) → A__U31(a__isNat(N), N)
A__PLUS(N, s(M)) → A__U41(a__isNat(M), M, N)
A__U42(tt, M, N) → MARK(N)
MARK(U41(X1, X2, X3)) → MARK(X1)
MARK(U42(X1, X2, X3)) → A__U42(mark(X1), X2, X3)
A__U42(tt, M, N) → MARK(M)
MARK(U42(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, V2) → a__U12(a__isNat(V2))
a__U12(tt) → tt
a__U21(tt) → tt
a__U31(tt, N) → mark(N)
a__U41(tt, M, N) → a__U42(a__isNat(N), M, N)
a__U42(tt, M, N) → s(a__plus(mark(N), mark(M)))
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__plus(N, 0) → a__U31(a__isNat(N), N)
a__plus(N, s(M)) → a__U41(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(U41(X1, X2, X3)) → a__U41(mark(X1), X2, X3)
mark(U42(X1, X2, X3)) → a__U42(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) → 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__U41(X1, X2, X3) → U41(X1, X2, X3)
a__U42(X1, X2, X3) → U42(X1, X2, X3)
a__plus(X1, X2) → plus(X1, X2)

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