(0) Obligation:

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

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

The set Q consists of the following terms:

f(a, f(x0, 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(a, f(x, a)) → F(a, f(f(a, x), f(a, a)))
F(a, f(x, a)) → F(f(a, x), f(a, a))
F(a, f(x, a)) → F(a, x)
F(a, f(x, a)) → F(a, a)

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

f(a, f(x0, a))

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

(5) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 3 less nodes.

(6) 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, x), f(a, a)))

The set Q consists of the following terms:

f(a, f(x0, a))

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

(7) 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: Combined order from the following AFS and order.
F(x1, x2)  =  x2
a  =  a
f(x1, x2)  =  f(x1)

Lexicographic path order with status [LPO].
Quasi-Precedence:
f1 > a

Status:
a: []
f1: [1]


The following usable rules [FROCOS05] were oriented:

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

(8) 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, x), f(a, a)))

The set Q consists of the following terms:

f(a, f(x0, a))

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

(9) PisEmptyProof (EQUIVALENT transformation)

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

(10) TRUE