(0) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
p(f(f(x))) → q(f(g(x)))
p(g(g(x))) → q(g(f(x)))
q(f(f(x))) → p(f(g(x)))
q(g(g(x))) → p(g(f(x)))
Q is empty.
(1) AAECC Innermost (EQUIVALENT transformation)
We have applied [NOC,AAECCNOC] to switch to innermost. The TRS R 1 is none
The TRS R 2 is
p(f(f(x))) → q(f(g(x)))
p(g(g(x))) → q(g(f(x)))
q(f(f(x))) → p(f(g(x)))
q(g(g(x))) → p(g(f(x)))
The signature Sigma is {
p,
q}
(2) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
p(f(f(x))) → q(f(g(x)))
p(g(g(x))) → q(g(f(x)))
q(f(f(x))) → p(f(g(x)))
q(g(g(x))) → p(g(f(x)))
The set Q consists of the following terms:
p(f(f(x0)))
p(g(g(x0)))
q(f(f(x0)))
q(g(g(x0)))
(3) DependencyPairsProof (EQUIVALENT transformation)
Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
(4) Obligation:
Q DP problem:
The TRS P consists of the following rules:
P(f(f(x))) → Q(f(g(x)))
P(g(g(x))) → Q(g(f(x)))
Q(f(f(x))) → P(f(g(x)))
Q(g(g(x))) → P(g(f(x)))
The TRS R consists of the following rules:
p(f(f(x))) → q(f(g(x)))
p(g(g(x))) → q(g(f(x)))
q(f(f(x))) → p(f(g(x)))
q(g(g(x))) → p(g(f(x)))
The set Q consists of the following terms:
p(f(f(x0)))
p(g(g(x0)))
q(f(f(x0)))
q(g(g(x0)))
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 4 less nodes.
(6) TRUE