Term Rewriting System R:

   [x]
   h(f(f(x))) -> h(f(g(f(x))))
   f(g(f(x))) -> f(f(x))

Termination of R to be shown.


R contains the following Dependency Pairs: 


   F(g(f(x))) -> F(f(x))
   H(f(f(x))) -> H(f(g(f(x))))
   H(f(f(x))) -> F(g(f(x)))

Furthermore, R contains one SCC.


SCC1:

H(f(f(x))) -> H(f(g(f(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)) = x_1
POL(H(x_1)) = 1 + x_1
POL(f(x_1)) = x_1

No Dependency Pairs can be deleted.

The following rules of R can be deleted: 

   h(f(f(x))) -> h(f(g(f(x))))


 This transformation is resulting in one new subcycle:


SCC1.MRR1:

H(f(f(x))) -> H(f(g(f(x))))

Found an infinite P-chain over R:
P = H(f(f(x))) -> H(f(g(f(x))))
R = [f(g(f(x))) -> f(f(x))]
s = H(f(f(x'''))) evaluates to t = H(f(f(x''')))
Thus, s starts an infinite reduction.


Non-Termination of R could be shown.

Duration: 1.183 seconds.