(0) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
f(f(x)) → g(f(x))
g(g(x)) → f(x)
Q is empty.
(1) DependencyPairsProof (EQUIVALENT transformation)
Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
(2) Obligation:
Q DP problem:
The TRS P consists of the following rules:
F(f(x)) → G(f(x))
G(g(x)) → F(x)
The TRS R consists of the following rules:
f(f(x)) → g(f(x))
g(g(x)) → f(x)
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(3) QDPOrderProof (EQUIVALENT transformation)
We use the reduction pair processor [LPAR04].
The following pairs can be oriented strictly and are deleted.
G(g(x)) → F(x)
The remaining pairs can at least be oriented weakly.
Used ordering: Recursive path order with status [RPO].
Quasi-Precedence:
[F1, f1, G1, g1]
Status:
F1: [1]
f1: multiset
G1: [1]
g1: multiset
The following usable rules [FROCOS05] were oriented:
g(g(x)) → f(x)
f(f(x)) → g(f(x))
(4) Obligation:
Q DP problem:
The TRS P consists of the following rules:
F(f(x)) → G(f(x))
The TRS R consists of the following rules:
f(f(x)) → g(f(x))
g(g(x)) → f(x)
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(5) DependencyGraphProof (EQUIVALENT transformation)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node.
(6) TRUE