(0) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
f(s(x)) → s(s(f(p(s(x)))))
f(0) → 0
p(s(x)) → 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:
f(s(x)) → s(s(f(p(s(x)))))
f(0) → 0
p(s(x)) → x
The set Q consists of the following terms:
f(s(x0))
f(0)
p(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:
F(s(x)) → F(p(s(x)))
F(s(x)) → P(s(x))
The TRS R consists of the following rules:
f(s(x)) → s(s(f(p(s(x)))))
f(0) → 0
p(s(x)) → x
The set Q consists of the following terms:
f(s(x0))
f(0)
p(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:
F(s(x)) → F(p(s(x)))
The TRS R consists of the following rules:
f(s(x)) → s(s(f(p(s(x)))))
f(0) → 0
p(s(x)) → x
The set Q consists of the following terms:
f(s(x0))
f(0)
p(s(x0))
We have to consider all minimal (P,Q,R)-chains.