(0) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
f(f(X)) → f(g(f(g(f(X)))))
f(g(f(X))) → f(g(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)) → F(g(f(g(f(X)))))
F(f(X)) → F(g(f(X)))
F(g(f(X))) → F(g(X))
The TRS R consists of the following rules:
f(f(X)) → f(g(f(g(f(X)))))
f(g(f(X))) → f(g(X))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(3) DependencyGraphProof (EQUIVALENT transformation)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 2 less nodes.
(4) Obligation:
Q DP problem:
The TRS P consists of the following rules:
F(g(f(X))) → F(g(X))
The TRS R consists of the following rules:
f(f(X)) → f(g(f(g(f(X)))))
f(g(f(X))) → f(g(X))
Q is empty.
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(g(f(X))) → F(g(X))
The remaining pairs can at least be oriented weakly.
Used ordering: Lexicographic Path Order [LPO].
Precedence:
[F1, f1] > g1
The following usable rules [FROCOS05] were oriented:
none
(6) Obligation:
Q DP problem:
P is empty.
The TRS R consists of the following rules:
f(f(X)) → f(g(f(g(f(X)))))
f(g(f(X))) → f(g(X))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(7) PisEmptyProof (EQUIVALENT transformation)
The TRS P is empty. Hence, there is no (P,Q,R) chain.
(8) TRUE