Term Rewriting System R:

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

Termination of R to be shown.


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

   f(g(a)) -> a

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



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


R contains the following Dependency Pairs: 


   G(x) -> F(g(x))
   G(x) -> G(x)

Furthermore, R contains one SCC.


SCC1:

G(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)) = 2 + x_1

No Dependency Pairs can be deleted.

The following rules of R can be deleted: 

   g(x) -> f(g(x))
   f(f(x)) -> b


 This transformation is resulting in one new subcycle:


SCC1.MRR1:

G(x) -> G(x)

Applying the non-overlappingness check to the DPs.

The transformation is resulting in one subcycle:

SCC1.MRR1.NOC1:

G(x) -> G(x)

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


Non-Termination of R could be shown.

Duration: 0.798 seconds.