Term Rewriting System R:

   [y, x]
   f(0, y) -> 0
   f(s(x), y) -> f(f(x, y), y)

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(s(x), y) -> F(f(x, y), y)
   F(s(x), y) -> F(x, y)

Furthermore, R contains one SCC.


SCC1:

F(s(x), y) -> F(x, y)

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(s(x_1)) = 1 + x_1
POL(F(x_1, x_2)) = 1 + x_1 + x_2

The following Dependency Pairs can be deleted:

   F(s(x), y) -> F(x, y)



 This transformation is resulting in no new subcycles.




Termination of R successfully shown.


Duration: 0.423 seconds.