Term Rewriting System R:

   [x]
   f(x) -> s(x)
   f(s(s(x))) -> s(f(f(x)))

Termination of R to be shown.


R contains the following Dependency Pairs: 


   F(s(s(x))) -> F(f(x))
   F(s(s(x))) -> F(x)

Furthermore, R contains one SCC.


SCC1:

F(s(s(x))) -> F(x)
F(s(s(x))) -> F(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(s(x_1)) = 1 + x_1
POL(F(x_1)) = 1 + x_1
POL(f(x_1)) = 1 + x_1

The following Dependency Pairs can be deleted:

   F(s(s(x))) -> F(x)
   F(s(s(x))) -> F(f(x))



 This transformation is resulting in no new subcycles.




Termination of R successfully shown.


Duration: 2.129 seconds.