(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) 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:

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)

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

(3) 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: SCNP Order with the following components:
Level mapping:
Top level AFS:
H(x0, x1, x2)  =  H(x0)
G(x0, x1, x2)  =  G(x0, x1)

Tags:
H has argument tags [4,3,6] and root tag 1
G has argument tags [3,4,7] and root tag 1

Comparison: MIN
Underlying order for the size change arcs and the rules of R:
Combined order from the following AFS and order.
H(x1, x2)  =  x1
f(x1)  =  f(x1)
G(x1, x2)  =  G(x1, x2)

Homeomorphic Embedding Order
The following usable rules [FROCOS05] were oriented: none

(4) 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)

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

(5) DependencyGraphProof (EQUIVALENT transformation)

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

(6) TRUE