(0) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
f(f(a, x), a) → f(f(x, f(a, f(a, a))), a)
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:
f(f(a, x), a) → f(f(x, f(a, f(a, a))), a)
The set Q consists of the following terms:
f(f(a, x0), a)
(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:
F(f(a, x), a) → F(f(x, f(a, f(a, a))), a)
F(f(a, x), a) → F(x, f(a, f(a, a)))
F(f(a, x), a) → F(a, f(a, a))
F(f(a, x), a) → F(a, a)
The TRS R consists of the following rules:
f(f(a, x), a) → f(f(x, f(a, f(a, a))), a)
The set Q consists of the following terms:
f(f(a, x0), a)
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:
F(f(a, x), a) → F(f(x, f(a, f(a, a))), a)
The TRS R consists of the following rules:
f(f(a, x), a) → f(f(x, f(a, f(a, a))), a)
The set Q consists of the following terms:
f(f(a, x0), a)
We have to consider all minimal (P,Q,R)-chains.