Term Rewriting System R:

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

Termination of R to be shown.


R contains the following Dependency Pairs: 


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

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

Found an infinite P-chain over R:
P = F(g(x), s(0)) -> F(g(x), g(x))
R = [g(0) -> 0, g(s(x)) -> s(g(x)), f(g(x), s(0)) -> f(g(x), g(x))]
s = F(g(s(0)), s(0)) evaluates to t = F(g(s(0)), s(0))
Thus, s starts an infinite reduction.


Non-Termination of R could be shown.

Duration: 0.579 seconds.