(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 [RPO].
Precedence:
f > [F2, a]


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.