(0) Obligation:

Q restricted rewrite system:
The TRS R consists of the following rules:

f(f(a, x), a) → f(f(f(x, a), f(a, a)), a)

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(a, x), a) → F(f(f(x, a), f(a, a)), a)
F(f(a, x), a) → F(f(x, a), f(a, a))
F(f(a, x), a) → F(x, a)
F(f(a, x), a) → F(a, a)

The TRS R consists of the following rules:

f(f(a, x), a) → f(f(f(x, a), f(a, a)), a)

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 3 less nodes.

(4) Obligation:

Q DP problem:
The TRS P consists of the following rules:

F(f(a, x), a) → F(x, a)

The TRS R consists of the following rules:

f(f(a, x), a) → f(f(f(x, a), f(a, a)), a)

Q is empty.
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(a, x), a) → F(x, a)
The remaining pairs can at least be oriented weakly.
Used ordering: Lexicographic path order with status [LPO].
Quasi-Precedence:
a > [F2, f2]

Status:
F2: [1,2]
f2: [1,2]
a: []


The following usable rules [FROCOS05] were oriented: none

(6) Obligation:

Q DP problem:
P is empty.
The TRS R consists of the following rules:

f(f(a, x), a) → f(f(f(x, a), f(a, a)), a)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(7) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(8) TRUE