(0) Obligation:

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

a(b(a(x))) → b(a(b(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(b(a(x))) → A(b(x))

The TRS R consists of the following rules:

a(b(a(x))) → b(a(b(x)))

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

(3) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


A(b(a(x))) → A(b(x))
The remaining pairs can at least be oriented weakly.
Used ordering: Lexicographic path order with status [LPO].
Precedence:
A1 > b1
a1 > b1

Status:
a1: [1]
A1: [1]
b1: [1]

The following usable rules [FROCOS05] were oriented:

a(b(a(x))) → b(a(b(x)))

(4) Obligation:

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

a(b(a(x))) → b(a(b(x)))

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

(5) PisEmptyProof (EQUIVALENT transformation)

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

(6) TRUE