(0) Obligation:

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

ag(c)
g(a) → b
f(g(X), b) → f(a, 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:

AG(c)
F(g(X), b) → F(a, X)
F(g(X), b) → A

The TRS R consists of the following rules:

ag(c)
g(a) → b
f(g(X), b) → f(a, 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:

F(g(X), b) → F(a, X)

The TRS R consists of the following rules:

ag(c)
g(a) → b
f(g(X), b) → f(a, 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.


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

Recursive path order with status [RPO].
Quasi-Precedence:
[F2, b] > a > c

Status:
F2: multiset
b: multiset
a: multiset
c: multiset


The following usable rules [FROCOS05] were oriented:

ag(c)

(6) Obligation:

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

ag(c)
g(a) → b
f(g(X), b) → f(a, 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