(0) Obligation:

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

p(s(x)) → x
fac(0) → s(0)
fac(s(x)) → times(s(x), fac(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:

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

The set Q consists of the following terms:

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

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

p(s(x0))
fac(0)
fac(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 1 SCC with 1 less node.

(6) 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:

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

The set Q consists of the following terms:

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

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