(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.