Term Rewriting System R:

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

Termination of R to be shown.


R contains the following Dependency Pairs: 


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

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


Non-Termination of R could be shown.

Duration: 0.466 seconds.