(0) Obligation:

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

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

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

The TRS R consists of the following rules:

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

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(a, f(x, a)) → F(a, x)

The TRS R consists of the following rules:

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

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(a, f(x, a)) → F(a, x)
The remaining pairs can at least be oriented weakly.
Used ordering: SCNP Order with the following components:
Level mapping:
Top level AFS:
F(x1, x2)  =  F(x2)

Tags:
F has tags [1,0]

Comparison: MAX
Underlying order for the size change arcs and the rules of R:
Lexicographic path order with status [LPO].
Quasi-Precedence:
f2 > a

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


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

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