Term Rewriting System R:

   []
   f(0) -> cons(0, f(s(0)))
   f(s(0)) -> f(p(s(0)))
   p(s(0)) -> 0

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: 


   F(s(0)) -> F(p(s(0)))
   F(s(0)) -> P(s(0))
   F(0) -> F(s(0))

Furthermore, R contains one SCC.


SCC1:

F(0) -> F(s(0))
F(s(0)) -> F(p(s(0)))

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

   F(s(0)) -> F(p(s(0)))
one new Dependency Pair
is created:


   F(s(0)) -> F(0)

The transformation is resulting in one subcycle:


SCC1.Rewr1:

F(s(0)) -> F(0)
F(0) -> F(s(0))

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)) = x_1
POL(F(x_1)) = x_1
POL(0) = 0

No Dependency Pairs can be deleted.

The following rules of R can be deleted: 

   p(s(0)) -> 0


 This transformation is resulting in one new subcycle:


SCC1.Rewr1.MRR1:

F(0) -> F(s(0))
F(s(0)) -> F(0)

Found an infinite P-chain over R:
P = F(0) -> F(s(0))
F(s(0)) -> F(0)
R = []
s = F(0) evaluates to t = F(0)
Thus, s starts an infinite reduction.


Non-Termination of R could be shown.

Duration: 0.538 seconds.