(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:Recursive path order with status [RPO].
Quasi-Precedence:

[g1, f1]

Status:
g1: multiset
f1: multiset

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