Term Rewriting System R:

   [x]
   f(h(x)) -> f(i(x))
   f(i(x)) -> a
   i(x) -> h(x)

Termination of R to be shown.


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

   f(i(x)) -> a

where the Polynomial interpretation:
POL(i(x_1)) = x_1
POL(a) = 0
POL(h(x_1)) = x_1
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(h(x)) -> F(i(x))
   F(h(x)) -> I(x)

Furthermore, R contains one SCC.


SCC1:

F(h(x)) -> F(i(x))

On this Scc, a Rewriting SCC transformation can be performed.
 
As a result of transforming the rule 

   F(h(x)) -> F(i(x))
one new Dependency Pair
is created:


   F(h(x)) -> F(h(x))

The transformation is resulting in one subcycle:


SCC1.Rewr1:

F(h(x)) -> F(h(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(F(x_1)) = 2 + x_1
POL(h(x_1)) = 2 + x_1

No Dependency Pairs can be deleted.

The following rules of R can be deleted: 

   i(x) -> h(x)


 This transformation is resulting in one new subcycle:


SCC1.Rewr1.MRR1:

F(h(x)) -> F(h(x))

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


Non-Termination of R could be shown.

Duration: 0.841 seconds.