Term Rewriting System R:

   [f, x]
   app(app(F, app(app(F, f), x)), x) -> app(app(F, app(G, app(app(F, f), x))), app(f, x))

Termination of R to be shown.


R contains the following Dependency Pairs: 


   APP(app(F, app(app(F, f), x)), x) -> APP(app(F, app(G, app(app(F, f), x))), app(f, x))
   APP(app(F, app(app(F, f), x)), x) -> APP(F, app(G, app(app(F, f), x)))
   APP(app(F, app(app(F, f), x)), x) -> APP(G, app(app(F, f), x))
   APP(app(F, app(app(F, f), x)), x) -> APP(f, x)

Furthermore, R contains one SCC.


SCC1:

APP(app(F, app(app(F, f), x)), x) -> APP(f, x)
APP(app(F, app(app(F, f), x)), x) -> APP(app(F, app(G, app(app(F, f), x))), app(f, x))

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

Additionally, the following rule can be oriented: 

   app(app(F, app(app(F, f), x)), x) -> app(app(F, app(G, app(app(F, f), x))), app(f, x))


Used ordering: Polynomial ordering with Polynomial interpretation:
POL(G) = 0
POL(APP(x_1, x_2)) = x_1
POL(F) = 1
POL(app(x_1, x_2)) = x_1 + x_1*x_2

 resulting in no subcycles.




Termination of R successfully shown.


Duration: 13.982 seconds.