(0) Obligation:

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

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

The set Q consists of the following terms:

f(x0, x1)
g(h(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:

F(x, y) → G(x, y)
G(h(x), y) → F(x, y)
G(h(x), y) → G(x, y)

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

f(x0, x1)
g(h(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.


F(x, y) → G(x, y)
G(h(x), y) → F(x, y)
G(h(x), y) → 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)  =  F(x1)
G(x1, x2)  =  G(x1)
h(x1)  =  h(x1)
f(x1, x2)  =  x1
g(x1, x2)  =  x1

Lexicographic path order with status [LPO].
Precedence:
h1 > F1 > G1

Status:
h1: [1]
G1: [1]
F1: [1]

The following usable rules [FROCOS05] were oriented:

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

(6) Obligation:

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

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

The set Q consists of the following terms:

f(x0, x1)
g(h(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