(0) Obligation:

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

h(f(x), y) → f(g(x, y))
g(x, y) → h(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:

h(f(x), y) → f(g(x, y))
g(x, y) → h(x, y)

The set Q consists of the following terms:

h(f(x0), x1)
g(x0, 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:

H(f(x), y) → G(x, y)
G(x, y) → H(x, y)

The TRS R consists of the following rules:

h(f(x), y) → f(g(x, y))
g(x, y) → h(x, y)

The set Q consists of the following terms:

h(f(x0), x1)
g(x0, 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.


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

Recursive Path Order [RPO].
Precedence:
f1 > G1 > H1

The following usable rules [FROCOS05] were oriented:

h(f(x), y) → f(g(x, y))
g(x, y) → h(x, y)

(6) Obligation:

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

h(f(x), y) → f(g(x, y))
g(x, y) → h(x, y)

The set Q consists of the following terms:

h(f(x0), x1)
g(x0, 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