(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) 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:Lexicographic path order with status [LPO].
Quasi-Precedence:
[g1, f1]
Status:
g1: [1]
f1: [1]
AFS:
g(x1) = g(x1)
f(x1) = f(x1)
From the DPs we obtained the following set of size-change graphs:
- G(g(x)) → F(x) (allowed arguments on rhs = {1})
The graph contains the following edges 1 > 1
- F(f(x)) → G(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].
g(g(x)) → f(x)
f(f(x)) → g(f(x))
(4) TRUE