(0) Obligation:

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

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

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

The TRS R consists of the following rules:

a(f, a(f, x)) → a(x, x)
a(h, x) → a(f, a(g, a(f, 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 2 less nodes.

(4) Obligation:

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

A(h, x) → A(f, x)
A(f, a(f, x)) → A(x, x)

The TRS R consists of the following rules:

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


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

Recursive path order with status [RPO].
Quasi-Precedence:
h > f > A2
a1 > A2

Status:
f: multiset
a1: multiset
A2: multiset
h: multiset


The following usable rules [FROCOS05] were oriented: none

(6) Obligation:

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

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