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

(0) Obligation:

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

fac(s(x)) → *(fac(p(s(x))), s(x))
p(s(0)) → 0
p(s(s(x))) → s(p(s(x)))

Q is empty.

(1) Overlay + Local Confluence (EQUIVALENT transformation)

The TRS is overlay and locally confluent. By [NOC] we can switch to innermost.

(2) Obligation:

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

fac(s(x)) → *(fac(p(s(x))), s(x))
p(s(0)) → 0
p(s(s(x))) → s(p(s(x)))

The set Q consists of the following terms:

fac(s(x0))
p(s(0))
p(s(s(x0)))

(3) DependencyPairsProof (EQUIVALENT transformation)

Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.

(4) Obligation:

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

FAC(s(x)) → FAC(p(s(x)))
FAC(s(x)) → P(s(x))
P(s(s(x))) → P(s(x))

The TRS R consists of the following rules:

fac(s(x)) → *(fac(p(s(x))), s(x))
p(s(0)) → 0
p(s(s(x))) → s(p(s(x)))

The set Q consists of the following terms:

fac(s(x0))
p(s(0))
p(s(s(x0)))

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

(5) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs with 1 less node.

(6) Complex Obligation (AND)

(7) Obligation:

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

P(s(s(x))) → P(s(x))

The TRS R consists of the following rules:

fac(s(x)) → *(fac(p(s(x))), s(x))
p(s(0)) → 0
p(s(s(x))) → s(p(s(x)))

The set Q consists of the following terms:

fac(s(x0))
p(s(0))
p(s(s(x0)))

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.

P(s(s(x))) → P(s(x))

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
P(x1)  =  P(x1)
s(x1)  =  s(x1)
fac(x1)  =  fac(x1)
*(x1, x2)  =  *(x2)
p(x1)  =  p(x1)
0  =  0

Lexicographic Path Order [LPO].
Precedence:

[fac₁, *₁] > [s₁, p₁] > 0 > P₁

The following usable rules [FROCOS05] were oriented:

fac(s(x)) → *(fac(p(s(x))), s(x))
p(s(0)) → 0
p(s(s(x))) → s(p(s(x)))

(9) Obligation:

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

fac(s(x)) → *(fac(p(s(x))), s(x))
p(s(0)) → 0
p(s(s(x))) → s(p(s(x)))

The set Q consists of the following terms:

fac(s(x0))
p(s(0))
p(s(s(x0)))

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:

FAC(s(x)) → FAC(p(s(x)))

The TRS R consists of the following rules:

fac(s(x)) → *(fac(p(s(x))), s(x))
p(s(0)) → 0
p(s(s(x))) → s(p(s(x)))

The set Q consists of the following terms:

fac(s(x0))
p(s(0))
p(s(s(x0)))

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