Term Rewriting System R:

   [x]
   f(s(x)) -> s(s(f(p(s(x)))))
   f(0) -> 0
   p(s(x)) -> x

Termination of R to be shown.


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

   f(0) -> 0

where the Polynomial interpretation:
POL(s(x_1)) = x_1
POL(f(x_1)) = 2*x_1
POL(0) = 1
POL(p(x_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(s(x)) -> F(p(s(x)))
   F(s(x)) -> P(s(x))

Furthermore, R contains one SCC.


SCC1:

F(s(x)) -> F(p(s(x)))

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

   F(s(x)) -> F(p(s(x)))
one new Dependency Pair
is created:


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

The transformation is resulting in one subcycle:


SCC1.Rewr1:

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

The following Dependency Pairs can be deleted:

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



 This transformation is resulting in no new subcycles.




Termination of R successfully shown.


Duration: 0.495 seconds.