(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) 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:

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))

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

(3) QDPSizeChangeProof (EQUIVALENT transformation)

We used the following order and afs together with the size-change analysis [AAECC05] to show that there are no infinite chains for this DP problem.

Order:Homeomorphic Embedding Order

AFS:
h(x1)  =  h(x1)

From the DPs we obtained the following set of size-change graphs:

  • G(h(x), y) → F(x, y) (allowed arguments on rhs = {1, 2})
    The graph contains the following edges 1 > 1, 2 >= 2

  • G(h(x), y) → G(x, y) (allowed arguments on rhs = {1, 2})
    The graph contains the following edges 1 > 1, 2 >= 2

  • F(x, y) → G(x, y) (allowed arguments on rhs = {1, 2})
    The graph contains the following edges 1 >= 1, 2 >= 2

We oriented the following set of usable rules [AAECC05,FROCOS05]. none

(4) TRUE