Term Rewriting System R:

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

Innermost Termination of R to be shown.


R contains the following Dependency Pairs: 


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

Furthermore, R contains two SCCs.


SCC1:

G(s(x)) -> G(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(s(x_1)) = 1 + x_1

The following Dependency Pairs can be deleted:

   G(s(x)) -> G(x)



 This transformation is resulting in no new subcycles.


SCC2:

F(g(x), s(0), y) -> F(g(s(0)), y, g(x))

On this Scc, a Rewriting SCC transformation can be performed.
 
As a result of transforming the rule 

   F(g(x), s(0), y) -> F(g(s(0)), y, g(x))
one new Dependency Pair
is created:


   F(g(x), s(0), y) -> F(s(g(0)), y, g(x))

The transformation is resulting in no subcycles.




Innermost Termination of R successfully shown.


Duration: 0.441 seconds.