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. 



   R
     ↳Overlay and local confluence Check



The TRS is overlay and locally confluent (all critical pairs are trivially joinable).Hence, we can switch to innermost. 


   R
     ↳OC
       →TRS2
         ↳Dependency Pair Analysis



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. 


   R
     ↳OC
       →TRS2
         ↳DPs
           →DP Problem 1
             ↳Argument Filtering and Ordering


Dependency Pairs:

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))


Rules:


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


Strategy:

innermost




The following dependency pair can be strictly oriented:

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


There are no usable rules w.r.t. the AFS that need to be oriented.
Used ordering: Lexicographic Path Order with Precedence:
trivial

resulting in one new DP problem.
Used Argument Filtering System:
AP(x1, x2) -> x1
ap(x1, x2) -> ap(x1, x2)


   R
     ↳OC
       →TRS2
         ↳DPs
           →DP Problem 1
             ↳AFS
             ...
               →DP Problem 2
                 ↳Non Termination


Dependency Pair:

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


Rules:


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


Strategy:

innermost




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(ap(f, s), 0))
evaluates to t =AP(ap(ap(g, f), s), ap(ap(f, s), 0))

Thus, s starts an infinite chain.

Non-Termination of R could be shown.
Duration:
0:00 minutes