(0) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
a(f, a(f, a(g, a(g, x)))) → a(g, a(g, a(g, a(f, a(f, a(f, x))))))
Q is empty.
(1) Overlay + Local Confluence (EQUIVALENT transformation)
The TRS is overlay and locally confluent. By [NOC] we can switch to innermost.
(2) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
a(f, a(f, a(g, a(g, x)))) → a(g, a(g, a(g, a(f, a(f, a(f, x))))))
The set Q consists of the following terms:
a(f, a(f, a(g, a(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:
A(f, a(f, a(g, a(g, x)))) → A(g, a(g, a(g, a(f, a(f, a(f, x))))))
A(f, a(f, a(g, a(g, x)))) → A(g, a(g, a(f, a(f, a(f, x)))))
A(f, a(f, a(g, a(g, x)))) → A(g, a(f, a(f, a(f, x))))
A(f, a(f, a(g, a(g, x)))) → A(f, a(f, a(f, x)))
A(f, a(f, a(g, a(g, x)))) → A(f, a(f, x))
A(f, a(f, a(g, a(g, x)))) → A(f, x)
The TRS R consists of the following rules:
a(f, a(f, a(g, a(g, x)))) → a(g, a(g, a(g, a(f, a(f, a(f, x))))))
The set Q consists of the following terms:
a(f, a(f, a(g, a(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 1 SCC with 3 less nodes.
(6) Obligation:
Q DP problem:
The TRS P consists of the following rules:
A(f, a(f, a(g, a(g, x)))) → A(f, a(f, x))
A(f, a(f, a(g, a(g, x)))) → A(f, a(f, a(f, x)))
A(f, a(f, a(g, a(g, x)))) → A(f, x)
The TRS R consists of the following rules:
a(f, a(f, a(g, a(g, x)))) → a(g, a(g, a(g, a(f, a(f, a(f, x))))))
The set Q consists of the following terms:
a(f, a(f, a(g, a(g, x0))))
We have to consider all minimal (P,Q,R)-chains.