Term Rewriting System R:

   [X]
   f(X, g(X)) -> f(1, g(X))
   g(1) -> g(0)

Termination of R to be shown.


R contains the following Dependency Pairs: 


   G(1) -> G(0)
   F(X, g(X)) -> F(1, g(X))

Furthermore, R contains one SCC.


SCC1:

F(X, g(X)) -> F(1, 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(1) = 0
POL(F(x_1, x_2)) = 1 + x_1 + x_2
POL(0) = 0

No Dependency Pairs can be deleted.

The following rules of R can be deleted: 

   f(X, g(X)) -> f(1, g(X))


 This transformation is resulting in one new subcycle:


SCC1.MRR1:

F(X, g(X)) -> F(1, g(X))

Found an infinite P-chain over R:
P = F(X, g(X)) -> F(1, g(X))
R = [g(1) -> g(0)]
s = F(1, g(1)) evaluates to t = F(1, g(1))
Thus, s starts an infinite reduction.


Non-Termination of R could be shown.

Duration: 0.479 seconds.