Term Rewriting System R:

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

Termination of R to be shown.


Removing the following rules from R which fullfill a polynomial ordering: 

   g(f(x)) -> x

where the Polynomial interpretation:
POL(g(x_1)) = x_1
POL(a) = 0
POL(f(x_1)) = 1 + x_1
was used. 



Not all Rules of R can be deleted, so we still have to regard a part of R.


This program has no overlaps, so it is sufficient to show innermost termination. 


R contains the following Dependency Pairs: 


   F(x) -> G(f(x))
   F(x) -> F(x)

Furthermore, R contains one SCC.


SCC1:

F(x) -> F(x)

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


Non-Termination of R could be shown.

Duration: 0.450 seconds.