(0) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
f(0, 1, x) → f(s(x), x, x)
f(x, y, s(z)) → s(f(0, 1, 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(0, 1, x) → F(s(x), x, x)
F(x, y, s(z)) → F(0, 1, z)
The TRS R consists of the following rules:
f(0, 1, x) → f(s(x), x, x)
f(x, y, s(z)) → s(f(0, 1, 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:
0 = 0
1 = 1
s(x1) = s(x1)
From the DPs we obtained the following set of size-change graphs:
- F(x, y, s(z)) → F(0, 1, z) (allowed arguments on rhs = {3})
The graph contains the following edges 3 > 3
- F(0, 1, x) → F(s(x), x, x) (allowed arguments on rhs = {1, 2, 3})
The graph contains the following edges 3 >= 2, 3 >= 3
We oriented the following set of usable rules [AAECC05,FROCOS05].
none
(4) TRUE