(0) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
f(a, f(a, f(b, f(x, y)))) → f(b, f(c, f(b, f(a, f(a, f(a, f(x, y)))))))
f(a, f(c, f(x, y))) → f(b, f(x, y))
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(b, f(x, y)))) → F(b, f(c, f(b, f(a, f(a, f(a, f(x, y)))))))
F(a, f(a, f(b, f(x, y)))) → F(c, f(b, f(a, f(a, f(a, f(x, y))))))
F(a, f(a, f(b, f(x, y)))) → F(b, f(a, f(a, f(a, f(x, y)))))
F(a, f(a, f(b, f(x, y)))) → F(a, f(a, f(a, f(x, y))))
F(a, f(a, f(b, f(x, y)))) → F(a, f(a, f(x, y)))
F(a, f(a, f(b, f(x, y)))) → F(a, f(x, y))
F(a, f(c, f(x, y))) → F(b, f(x, y))
The TRS R consists of the following rules:
f(a, f(a, f(b, f(x, y)))) → f(b, f(c, f(b, f(a, f(a, f(a, f(x, y)))))))
f(a, f(c, f(x, y))) → f(b, f(x, y))
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 1 SCC with 4 less nodes.
(4) Obligation:
Q DP problem:
The TRS P consists of the following rules:
F(a, f(a, f(b, f(x, y)))) → F(a, f(a, f(x, y)))
F(a, f(a, f(b, f(x, y)))) → F(a, f(a, f(a, f(x, y))))
F(a, f(a, f(b, f(x, y)))) → F(a, f(x, y))
The TRS R consists of the following rules:
f(a, f(a, f(b, f(x, y)))) → f(b, f(c, f(b, f(a, f(a, f(a, f(x, y)))))))
f(a, f(c, f(x, y))) → f(b, f(x, y))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.