Term Rewriting System R:

   [x, y]
   prime(0) -> false
   prime(s(0)) -> false
   prime(s(s(x))) -> prime1(s(s(x)), s(x))
   prime1(x, 0) -> false
   prime1(x, s(0)) -> true
   prime1(x, s(s(y))) -> and(not(divp(s(s(y)), x)), prime1(x, s(y)))
   divp(x, y) -> =(rem(x, y), 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: 


   PRIME(s(s(x))) -> PRIME1(s(s(x)), s(x))
   PRIME1(x, s(s(y))) -> DIVP(s(s(y)), x)
   PRIME1(x, s(s(y))) -> PRIME1(x, s(y))

Furthermore, R contains one SCC.


SCC1:

PRIME1(x, s(s(y))) -> PRIME1(x, s(y))

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(PRIME1(x_1, x_2)) = 1 + x_1 + x_2

The following Dependency Pairs can be deleted:

   PRIME1(x, s(s(y))) -> PRIME1(x, s(y))



 This transformation is resulting in no new subcycles.




Termination of R successfully shown.


Duration: 0.599 seconds.