Term Rewriting System R:

   [x, y]
   ap(ap(g, x), y) -> y
   ap(f, x) -> ap(f, app(g, x))

Termination of R to be shown.


Removing the following rules from R which fullfill a polynomial ordering: 

   ap(ap(g, x), y) -> y

where the Polynomial interpretation:
POL(g) = 0
POL(ap(x_1, x_2)) = 1 + x_1 + x_2
POL(f) = 2
POL(app(x_1, x_2)) = x_1 + x_2
was used. 



Not all Rules of R can be deleted, so we still have to regard a part of R.


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


R contains the following Dependency Pairs: 


   AP(f, x) -> AP(f, app(g, x))

Furthermore, R contains one SCC.


SCC1:

AP(f, x) -> AP(f, app(g, x))

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


Non-Termination of R could be shown.

Duration: 0.452 seconds.