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