Term Rewriting System R:

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


   G(0, x) -> G(f(x, x), x)
   G(0, x) -> F(x, x)
   F(s(x), s(y)) -> F(x, y)

Furthermore, R contains one SCC.


SCC1:

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



 This transformation is resulting in no new subcycles.




Termination of R successfully shown.


Duration: 0.453 seconds.