(0) Obligation:

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

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

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(g(x), a) → F(x, g(a))
F(g(x), g(y)) → H(g(y), x, g(y))
H(g(x), y, z) → F(y, h(x, y, z))
H(g(x), y, z) → H(x, y, z)

The TRS R consists of the following rules:

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

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:
a  =  a
g(x1)  =  g(x1)

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

  • F(g(x), g(y)) → H(g(y), x, g(y)) (allowed arguments on rhs = {1, 2, 3})
    The graph contains the following edges 2 >= 1, 1 > 2, 2 >= 3

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

  • H(g(x), y, z) → H(x, y, z) (allowed arguments on rhs = {1, 2, 3})
    The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3

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

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

(4) TRUE