(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) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


F(f(x)) → G(f(x))
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
F(x1)  =  F(x1)
f(x1)  =  f(x1)
G(x1)  =  x1
g(x1)  =  g(x1)

Recursive path order with status [RPO].
Quasi-Precedence:
[F1, f1, g1]

Status:
f1: multiset
g1: multiset
F1: multiset


The following usable rules [FROCOS05] were oriented:

g(g(x)) → f(x)
f(f(x)) → g(f(x))

(4) Obligation:

Q DP problem:
The TRS P consists of the following rules:

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.

(5) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node.

(6) TRUE