(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) 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(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))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(3) QDPSizeChangeProof (EQUIVALENT transformation)
We used the following order and afs together with the size-change analysis [AAECC05] to show that there are no infinite chains for this DP problem.
Order:Combined order from the following AFS and order.
f(x1) = x1
g(x1) = g(x1)
h(x1) = h
Lexicographic path order with status [LPO].
Quasi-Precedence:
[g1, h]
Status:
g1: [1]
h: []
AFS:
f(x1) = x1
g(x1) = g(x1)
h(x1) = h
From the DPs we obtained the following set of size-change graphs:
- F(g(X)) → F(f(X)) (allowed arguments on rhs = {1})
The graph contains the following edges 1 > 1
- F(g(X)) → F(X) (allowed arguments on rhs = {1})
The graph contains the following edges 1 > 1
We oriented the following set of usable rules [AAECC05,FROCOS05].
f(g(X)) → g(f(f(X)))
f(h(X)) → h(g(X))
(4) TRUE