(0) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
f(f(f(a, f(a, a)), a), x) → f(x, f(x, 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(f(a, f(a, a)), a), x) → f(x, f(x, a))
The set Q consists of the following terms:
f(f(f(a, f(a, a)), a), 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:
F(f(f(a, f(a, a)), a), x) → F(x, f(x, a))
F(f(f(a, f(a, a)), a), x) → F(x, a)
The TRS R consists of the following rules:
f(f(f(a, f(a, a)), a), x) → f(x, f(x, a))
The set Q consists of the following terms:
f(f(f(a, f(a, a)), a), x0)
We have to consider all minimal (P,Q,R)-chains.
(5) QDPOrderProof (EQUIVALENT transformation)
We use the reduction pair processor [LPAR04].
The following pairs can be oriented strictly and are deleted.
F(f(f(a, f(a, a)), a), x) → F(x, a)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
F(
x1,
x2) =
F(
x1,
x2)
f(
x1,
x2) =
f
a =
a
Recursive Path Order [RPO].
Precedence:
F2 > f > a
The following usable rules [FROCOS05] were oriented:
f(f(f(a, f(a, a)), a), x) → f(x, f(x, a))
(6) Obligation:
Q DP problem:
The TRS P consists of the following rules:
F(f(f(a, f(a, a)), a), x) → F(x, f(x, a))
The TRS R consists of the following rules:
f(f(f(a, f(a, a)), a), x) → f(x, f(x, a))
The set Q consists of the following terms:
f(f(f(a, f(a, a)), a), x0)
We have to consider all minimal (P,Q,R)-chains.