(0) Obligation:

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

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

The set Q consists of the following terms:

f(g(x0))

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

The TRS R consists of the following rules:

f(g(x)) → f(a(g(g(f(x))), g(f(x))))

The set Q consists of the following terms:

f(g(x0))

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 1 less node.

(6) Obligation:

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

F(g(x)) → F(x)

The TRS R consists of the following rules:

f(g(x)) → f(a(g(g(f(x))), g(f(x))))

The set Q consists of the following terms:

f(g(x0))

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(g(x)) → F(x)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
F(x1)  =  F(x1)
g(x1)  =  g(x1)
f(x1)  =  x1
a(x1, x2)  =  a

Recursive Path Order [RPO].
Precedence:
[F1, g1] > a


The following usable rules [FROCOS05] were oriented:

f(g(x)) → f(a(g(g(f(x))), g(f(x))))

(8) Obligation:

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

f(g(x)) → f(a(g(g(f(x))), g(f(x))))

The set Q consists of the following terms:

f(g(x0))

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