(0) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
f(0) → s(0)
f(s(0)) → s(0)
f(s(s(x))) → f(f(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:
f(0) → s(0)
f(s(0)) → s(0)
f(s(s(x))) → f(f(s(x)))
The set Q consists of the following terms:
f(0)
f(s(0))
f(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:
F(s(s(x))) → F(f(s(x)))
F(s(s(x))) → F(s(x))
The TRS R consists of the following rules:
f(0) → s(0)
f(s(0)) → s(0)
f(s(s(x))) → f(f(s(x)))
The set Q consists of the following terms:
f(0)
f(s(0))
f(s(s(x0)))
We have to consider all minimal (P,Q,R)-chains.
(5) QDPOrderProof (EQUIVALENT transformation)
We use the reduction pair processor [LPAR04].
The following pairs can be oriented strictly and are deleted.
F(s(s(x))) → F(s(x))
The remaining pairs can at least be oriented weakly.
Used ordering: Recursive path order with status [RPO].
Quasi-Precedence:
[F1, s1, f1, 0]
Status:
f1: [1]
s1: [1]
0: multiset
F1: multiset
The following usable rules [FROCOS05] were oriented:
f(0) → s(0)
f(s(0)) → s(0)
f(s(s(x))) → f(f(s(x)))
(6) Obligation:
Q DP problem:
The TRS P consists of the following rules:
F(s(s(x))) → F(f(s(x)))
The TRS R consists of the following rules:
f(0) → s(0)
f(s(0)) → s(0)
f(s(s(x))) → f(f(s(x)))
The set Q consists of the following terms:
f(0)
f(s(0))
f(s(s(x0)))
We have to consider all minimal (P,Q,R)-chains.