(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