(0) Obligation:

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

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

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

The set Q consists of the following terms:

g(f(x0), x1)
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:

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

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

g(f(x0), x1)
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.


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

Lexicographic path order with status [LPO].
Precedence:
trivial

Status:
f1: [1]

The following usable rules [FROCOS05] were oriented:

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

(6) Obligation:

Q DP problem:
The TRS P consists of the following rules:

H(x, y) → G(x, f(y))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

We have to consider all minimal (P,Q,R)-chains.

(7) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node.

(8) TRUE