Term Rewriting System R:

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

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: 


   F(g(x)) -> F(a(g(g(f(x))), g(f(x))))
   F(g(x)) -> F(x)

Furthermore, R contains one SCC.


SCC1:

F(g(x)) -> 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(x_1)) = 1 + x_1
POL(F(x_1)) = 1 + x_1

The following Dependency Pairs can be deleted:

   F(g(x)) -> F(x)



 This transformation is resulting in no new subcycles.




Termination of R successfully shown.


Duration: 0.424 seconds.