(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))
G(g(x)) → F(x)
The remaining pairs can at least be oriented weakly.
Used ordering: SCNP Order with the following components:
Level mapping:
Top level AFS:
F(x0, x1)  =  F(x0)
G(x0, x1)  =  G(x0, x1)

Tags:
F has argument tags [0,1] and root tag 1
G has argument tags [0,3] and root tag 0

Comparison: MAX
Underlying order for the size change arcs and the rules of R:
Recursive path order with status [RPO].
Quasi-Precedence:
[F1, G1] > [f1, g1]

Status:
F1: [1]
f1: multiset
G1: [1]
g1: multiset


The following usable rules [FROCOS05] were oriented:

g(g(x)) → f(x)
f(f(x)) → g(f(x))

(4) Obligation:

Q DP problem:
P is empty.
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) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(6) TRUE