(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