(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(f, a(f, x)) → A(x, x)
The remaining pairs can at least be oriented weakly.
Used ordering: SCNP Order with the following components:
Level mapping:
Top level AFS:
A(x0, x1, x2)  =  A(x2)

Tags:
A has argument tags [3,1,0] and root tag 0

Comparison: MAX
Underlying order for the size change arcs and the rules of R:
Combined order from the following AFS and order.
A(x1, x2)  =  x1
h  =  h
f  =  f
a(x1, x2)  =  a(x1, x2)

Lexicographic path order with status [LPO].
Quasi-Precedence:
h > f

Status:
h: []
f: []
a2: [1,2]


The following usable rules [FROCOS05] were oriented: none

(6) Obligation:

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

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.

(7) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node.

(8) TRUE