(0) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
f(g(X)) → g(f(f(X)))
f(h(X)) → h(g(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(g(X)) → g(f(f(X)))
f(h(X)) → h(g(X))
The set Q consists of the following terms:
f(g(x0))
f(h(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(g(X)) → F(f(X))
F(g(X)) → F(X)
The TRS R consists of the following rules:
f(g(X)) → g(f(f(X)))
f(h(X)) → h(g(X))
The set Q consists of the following terms:
f(g(x0))
f(h(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(g(X)) → F(f(X))
F(g(X)) → F(X)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
F(
x1) =
x1
g(
x1) =
g(
x1)
f(
x1) =
x1
h(
x1) =
h
Lexicographic Path Order [LPO].
Precedence:
g1 > h
The following usable rules [FROCOS05] were oriented:
f(g(X)) → g(f(f(X)))
f(h(X)) → h(g(X))
(6) Obligation:
Q DP problem:
P is empty.
The TRS R consists of the following rules:
f(g(X)) → g(f(f(X)))
f(h(X)) → h(g(X))
The set Q consists of the following terms:
f(g(x0))
f(h(x0))
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