Term Rewriting System R:

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

Termination of R to be shown.


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

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

where the Polynomial interpretation:
POL(g) = 0
POL(f) = 1
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: 


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

Furthermore, R contains one SCC.


SCC1:

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

Removing rules from R by ordering and analyzing Dependency Pairs, Usable Rules, and Usable Equations.

This is possible by using the following (C_E-compatible) Polynomial ordering.

Polynomial interpretation:
POL(g) = 1
POL(APP(x_1, x_2)) = 1 + x_1 + x_2
POL(f) = 1
POL(app(x_1, x_2)) = 1 + x_1 + x_2

The following Dependency Pairs can be deleted:

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



 This transformation is resulting in no new subcycles.




Termination of R successfully shown.


Duration: 0.462 seconds.