(0) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
f(x, f(a, f(f(a, a), a))) → f(f(a, x), x)
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(x, f(a, f(f(a, a), a))) → f(f(a, x), x)
The set Q consists of the following terms:
f(x0, f(a, f(f(a, a), 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(x, f(a, f(f(a, a), a))) → F(f(a, x), x)
F(x, f(a, f(f(a, a), a))) → F(a, x)
The TRS R consists of the following rules:
f(x, f(a, f(f(a, a), a))) → f(f(a, x), x)
The set Q consists of the following terms:
f(x0, f(a, f(f(a, a), a)))
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(x, f(a, f(f(a, a), a))) → F(a, x)
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 with status [RPO].
Quasi-Precedence:
[F2, f] > a
Status:
F2: multiset
f: multiset
a: multiset
The following usable rules [FROCOS05] were oriented:
f(x, f(a, f(f(a, a), a))) → f(f(a, x), x)
(6) Obligation:
Q DP problem:
The TRS P consists of the following rules:
F(x, f(a, f(f(a, a), a))) → F(f(a, x), x)
The TRS R consists of the following rules:
f(x, f(a, f(f(a, a), a))) → f(f(a, x), x)
The set Q consists of the following terms:
f(x0, f(a, f(f(a, a), a)))
We have to consider all minimal (P,Q,R)-chains.