(0) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
f(a, f(a, f(a, f(x, b)))) → f(f(a, f(a, f(a, x))), b)
f(f(f(a, x), b), b) → f(f(a, f(f(x, b), b)), b)
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(a, f(a, f(a, f(x, b)))) → F(f(a, f(a, f(a, x))), b)
F(a, f(a, f(a, f(x, b)))) → F(a, f(a, f(a, x)))
F(a, f(a, f(a, f(x, b)))) → F(a, f(a, x))
F(a, f(a, f(a, f(x, b)))) → F(a, x)
F(f(f(a, x), b), b) → F(f(a, f(f(x, b), b)), b)
F(f(f(a, x), b), b) → F(a, f(f(x, b), b))
F(f(f(a, x), b), b) → F(f(x, b), b)
F(f(f(a, x), b), b) → F(x, b)
The TRS R consists of the following rules:
f(a, f(a, f(a, f(x, b)))) → f(f(a, f(a, f(a, x))), b)
f(f(f(a, x), b), b) → f(f(a, f(f(x, b), b)), b)
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(3) DependencyGraphProof (EQUIVALENT transformation)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs with 2 less nodes.
(4) Complex Obligation (AND)
(5) Obligation:
Q DP problem:
The TRS P consists of the following rules:
F(f(f(a, x), b), b) → F(f(x, b), b)
F(f(f(a, x), b), b) → F(f(a, f(f(x, b), b)), b)
F(f(f(a, x), b), b) → F(x, b)
The TRS R consists of the following rules:
f(a, f(a, f(a, f(x, b)))) → f(f(a, f(a, f(a, x))), b)
f(f(f(a, x), b), b) → f(f(a, f(f(x, b), b)), b)
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(6) Obligation:
Q DP problem:
The TRS P consists of the following rules:
F(a, f(a, f(a, f(x, b)))) → F(a, f(a, x))
F(a, f(a, f(a, f(x, b)))) → F(a, f(a, f(a, x)))
F(a, f(a, f(a, f(x, b)))) → F(a, x)
The TRS R consists of the following rules:
f(a, f(a, f(a, f(x, b)))) → f(f(a, f(a, f(a, x))), b)
f(f(f(a, x), b), b) → f(f(a, f(f(x, b), b)), b)
Q is empty.
We have to consider all minimal (P,Q,R)-chains.