(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