Term Rewriting System R:

   [x, y, z]
   ap(f, x) -> x
   ap(ap(ap(g, x), y), ap(s, z)) -> ap(ap(ap(g, x), y), ap(ap(x, y), 0))

Termination of R to be shown.


This program has no overlaps, so it is sufficient to show innermost termination. 


R contains the following Dependency Pairs: 


   AP(ap(ap(g, x), y), ap(s, z)) -> AP(ap(ap(g, x), y), ap(ap(x, y), 0))
   AP(ap(ap(g, x), y), ap(s, z)) -> AP(ap(x, y), 0)
   AP(ap(ap(g, x), y), ap(s, z)) -> AP(x, y)

Furthermore, R contains one SCC.


SCC1:

AP(ap(ap(g, x), y), ap(s, z)) -> AP(x, y)
AP(ap(ap(g, x), y), ap(s, z)) -> AP(ap(ap(g, x), y), ap(ap(x, y), 0))

By using a polynomial ordering, at least one Dependency Pair of this SCC can be strictly oriented.

No rules need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
POL(g) = 1
POL(s) = 0
POL(ap(x_1, x_2)) = x_1 + x_2
POL(AP(x_1, x_2)) = x_1
POL(f) = 0
POL(0) = 0

 resulting in one subcycle.


SCC1.Polo1:

AP(ap(ap(g, x), y), ap(s, z)) -> AP(ap(ap(g, x), y), ap(ap(x, y), 0))

Found an infinite P-chain over R:
P = AP(ap(ap(g, x), y), ap(s, z)) -> AP(ap(ap(g, x), y), ap(ap(x, y), 0))
R = [ap(f, x) -> x, ap(ap(ap(g, x), y), ap(s, z)) -> ap(ap(ap(g, x), y), ap(ap(x, y), 0))]
s = AP(ap(ap(g, f), s), ap(s, 0)) evaluates to t = AP(ap(ap(g, f), s), ap(s, 0))
Thus, s starts an infinite reduction.


Non-Termination of R could be shown.

Duration: 0.670 seconds.