(0) Obligation:

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

f(x, a) → x
f(x, g(y)) → f(g(x), y)

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(x, a) → x
f(x, g(y)) → f(g(x), y)

The set Q consists of the following terms:

f(x0, a)
f(x0, g(x1))

(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(x, g(y)) → F(g(x), y)

The TRS R consists of the following rules:

f(x, a) → x
f(x, g(y)) → f(g(x), y)

The set Q consists of the following terms:

f(x0, a)
f(x0, g(x1))

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.


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

Recursive path order with status [RPO].
Precedence:
f2 > g1
a > g1

Status:
g1: multiset
f2: [2,1]
a: multiset

The following usable rules [FROCOS05] were oriented:

f(x, a) → x
f(x, g(y)) → f(g(x), y)

(6) Obligation:

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

f(x, a) → x
f(x, g(y)) → f(g(x), y)

The set Q consists of the following terms:

f(x0, a)
f(x0, g(x1))

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