Term Rewriting System R:

   [X, Y, Z]
   primes -> sieve(from(s(s(0))))
   from(X) -> cons(X, from(s(X)))
   head(cons(X, Y)) -> X
   tail(cons(X, Y)) -> Y
   if(true, X, Y) -> X
   if(false, X, Y) -> Y
   filter(s(s(X)), cons(Y, Z)) -> if(divides(s(s(X)), Y), filter(s(s(X)), Z), cons(Y, filter(X, sieve(Y))))
   sieve(cons(X, Y)) -> cons(X, filter(X, sieve(Y)))

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: 


   FROM(X) -> FROM(s(X))
   FILTER(s(s(X)), cons(Y, Z)) -> IF(divides(s(s(X)), Y), filter(s(s(X)), Z), cons(Y, filter(X, sieve(Y))))
   FILTER(s(s(X)), cons(Y, Z)) -> FILTER(s(s(X)), Z)
   FILTER(s(s(X)), cons(Y, Z)) -> FILTER(X, sieve(Y))
   FILTER(s(s(X)), cons(Y, Z)) -> SIEVE(Y)
   SIEVE(cons(X, Y)) -> FILTER(X, sieve(Y))
   SIEVE(cons(X, Y)) -> SIEVE(Y)
   PRIMES -> SIEVE(from(s(s(0))))
   PRIMES -> FROM(s(s(0)))

Furthermore, R contains two SCCs.


SCC1:

FROM(X) -> FROM(s(X))

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


Non-Termination of R could be shown.

Duration: 0.733 seconds.