Term Rewriting System R:

   [f, g, x]
   app(app(app(comp, f), g), x) -> app(f, app(g, x))
   app(twice, f) -> app(app(comp, f), f)

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: 


   APP(app(app(comp, f), g), x) -> APP(f, app(g, x))
   APP(app(app(comp, f), g), x) -> APP(g, x)
   APP(twice, f) -> APP(app(comp, f), f)
   APP(twice, f) -> APP(comp, f)

Furthermore, R contains one SCC.


SCC1:

APP(twice, f) -> APP(app(comp, f), f)
APP(app(app(comp, f), g), x) -> APP(g, x)
APP(app(app(comp, f), g), x) -> APP(f, app(g, x))

On this Scc, a Narrowing SCC transformation can be performed.
 
As a result of transforming the rule 

   APP(twice, f) -> APP(app(comp, f), f)
no new Dependency Pairs
are created.

none
The transformation is resulting in one subcycle:


SCC1.Nar1:

APP(app(app(comp, f), g), x) -> APP(f, app(g, x))
APP(app(app(comp, f), g), x) -> APP(g, x)

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(APP(x_1, x_2)) = x_1
POL(comp) = 1
POL(twice) = 0
POL(app(x_1, x_2)) = x_1 + x_2

 resulting in no subcycles.




Termination of R successfully shown.


Duration: 0.551 seconds.