Term Rewriting System R:

   [x]
   p(s(x)) -> x
   fac(0) -> s(0)
   fac(s(x)) -> times(s(x), fac(p(s(x))))

Termination of R to be shown.


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


R contains the following Dependency Pairs: 


   FAC(s(x)) -> FAC(p(s(x)))
   FAC(s(x)) -> P(s(x))

Furthermore, R contains one SCC.


SCC1:

FAC(s(x)) -> FAC(p(s(x)))

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

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


   FAC(s(x)) -> FAC(x)

The transformation is resulting in one subcycle:


SCC1.Rewr1:

FAC(s(x)) -> FAC(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(FAC(x_1)) = 1 + x_1

The following Dependency Pairs can be deleted:

   FAC(s(x)) -> FAC(x)



 This transformation is resulting in no new subcycles.




Termination of R successfully shown.


Duration: 0.493 seconds.