(0) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
a(f, a(g, a(f, x))) → a(f, a(g, a(g, a(f, x))))
a(g, a(f, a(g, x))) → a(g, a(f, a(f, a(g, 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:
A(f, a(g, a(f, x))) → A(f, a(g, a(g, a(f, x))))
A(f, a(g, a(f, x))) → A(g, a(g, a(f, x)))
A(g, a(f, a(g, x))) → A(g, a(f, a(f, a(g, x))))
A(g, a(f, a(g, x))) → A(f, a(f, a(g, x)))
The TRS R consists of the following rules:
a(f, a(g, a(f, x))) → a(f, a(g, a(g, a(f, x))))
a(g, a(f, a(g, x))) → a(g, a(f, a(f, a(g, 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]. Here, we combined the reduction pair processor with the A-transformation [FROCOS05] which results in the following intermediate Q-DP Problem.
The a-transformed P is
f1(g(f(x))) → f1(g(g(f(x))))
f1(g(f(x))) → g1(g(f(x)))
g1(f(g(x))) → g1(f(f(g(x))))
g1(f(g(x))) → f1(f(g(x)))
The a-transformed usable rules are
f(g(f(x))) → f(g(g(f(x))))
g(f(g(x))) → g(f(f(g(x))))
The following pairs can be oriented strictly and are deleted.
A(f, a(g, a(f, x))) → A(g, a(g, a(f, x)))
A(g, a(f, a(g, x))) → A(f, a(f, a(g, x)))
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
f1(
x1) =
x1
g(
x1) =
g
f(
x1) =
f
g1(
x1) =
g1
Recursive Path Order [RPO].
Precedence:
g > g1 > f
The following usable rules [FROCOS05] were oriented:
a(f, a(g, a(f, x))) → a(f, a(g, a(g, a(f, x))))
a(g, a(f, a(g, x))) → a(g, a(f, a(f, a(g, x))))
(4) Obligation:
Q DP problem:
The TRS P consists of the following rules:
A(f, a(g, a(f, x))) → A(f, a(g, a(g, a(f, x))))
A(g, a(f, a(g, x))) → A(g, a(f, a(f, a(g, x))))
The TRS R consists of the following rules:
a(f, a(g, a(f, x))) → a(f, a(g, a(g, a(f, x))))
a(g, a(f, a(g, x))) → a(g, a(f, a(f, a(g, 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 2 SCCs.
(6) Complex Obligation (AND)
(7) Obligation:
Q DP problem:
The TRS P consists of the following rules:
A(g, a(f, a(g, x))) → A(g, a(f, a(f, a(g, x))))
The TRS R consists of the following rules:
a(f, a(g, a(f, x))) → a(f, a(g, a(g, a(f, x))))
a(g, a(f, a(g, x))) → a(g, a(f, a(f, a(g, x))))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(8) Obligation:
Q DP problem:
The TRS P consists of the following rules:
A(f, a(g, a(f, x))) → A(f, a(g, a(g, a(f, x))))
The TRS R consists of the following rules:
a(f, a(g, a(f, x))) → a(f, a(g, a(g, a(f, x))))
a(g, a(f, a(g, x))) → a(g, a(f, a(f, a(g, x))))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.