(0) Obligation:

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

a(b(x)) → b(a(x))
a(c(x)) → 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:

a(b(x)) → b(a(x))
a(c(x)) → x

The set Q consists of the following terms:

a(b(x0))
a(c(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:

A(b(x)) → A(x)

The TRS R consists of the following rules:

a(b(x)) → b(a(x))
a(c(x)) → x

The set Q consists of the following terms:

a(b(x0))
a(c(x0))

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

Lexicographic Path Order [LPO].
Precedence:
[A1, b1]


The following usable rules [FROCOS05] were oriented:

a(b(x)) → b(a(x))
a(c(x)) → x

(6) Obligation:

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

a(b(x)) → b(a(x))
a(c(x)) → x

The set Q consists of the following terms:

a(b(x0))
a(c(x0))

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