(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)
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)

Recursive path order with status [RPO].
Quasi-Precedence:
f1 > [H1, G1]

Status:
H1: multiset
f1: multiset
G1: multiset


The following usable rules [FROCOS05] were oriented: none

(6) Obligation:

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

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.

(7) DependencyGraphProof (EQUIVALENT transformation)

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

(8) TRUE