Term Rewriting System R:

   [X, Y, M, N, X1, X2, X3]
   filter(cons(X, Y), 0, M) -> cons(0, n__filter(activate(Y), M, M))
   filter(cons(X, Y), s(N), M) -> cons(X, n__filter(activate(Y), N, M))
   filter(X1, X2, X3) -> n__filter(X1, X2, X3)
   sieve(cons(0, Y)) -> cons(0, n__sieve(activate(Y)))
   sieve(cons(s(N), Y)) -> cons(s(N), n__sieve(filter(activate(Y), N, N)))
   sieve(X) -> n__sieve(X)
   nats(N) -> cons(N, n__nats(s(N)))
   nats(X) -> n__nats(X)
   zprimes -> sieve(nats(s(s(0))))
   activate(n__filter(X1, X2, X3)) -> filter(X1, X2, X3)
   activate(n__sieve(X)) -> sieve(X)
   activate(n__nats(X)) -> nats(X)
   activate(X) -> X

Termination of R to be shown.


R contains the following Dependency Pairs: 


   ACTIVATE(n__nats(X)) -> NATS(X)
   ACTIVATE(n__filter(X1, X2, X3)) -> FILTER(X1, X2, X3)
   ACTIVATE(n__sieve(X)) -> SIEVE(X)
   ZPRIMES -> SIEVE(nats(s(s(0))))
   ZPRIMES -> NATS(s(s(0)))
   SIEVE(cons(s(N), Y)) -> FILTER(activate(Y), N, N)
   SIEVE(cons(s(N), Y)) -> ACTIVATE(Y)
   SIEVE(cons(0, Y)) -> ACTIVATE(Y)
   FILTER(cons(X, Y), 0, M) -> ACTIVATE(Y)
   FILTER(cons(X, Y), s(N), M) -> ACTIVATE(Y)

Furthermore, R contains one SCC.


SCC1:

SIEVE(cons(0, Y)) -> ACTIVATE(Y)
SIEVE(cons(s(N), Y)) -> ACTIVATE(Y)
FILTER(cons(X, Y), s(N), M) -> ACTIVATE(Y)
SIEVE(cons(s(N), Y)) -> FILTER(activate(Y), N, N)
ACTIVATE(n__sieve(X)) -> SIEVE(X)
FILTER(cons(X, Y), 0, M) -> ACTIVATE(Y)
ACTIVATE(n__filter(X1, X2, X3)) -> FILTER(X1, X2, X3)

By using a polynomial ordering, at least one Dependency Pair of this SCC can be strictly oriented.

Additionally, the following rules can be oriented: 

   activate(n__nats(X)) -> nats(X)
   activate(X) -> X
   activate(n__filter(X1, X2, X3)) -> filter(X1, X2, X3)
   activate(n__sieve(X)) -> sieve(X)
   sieve(cons(s(N), Y)) -> cons(s(N), n__sieve(filter(activate(Y), N, N)))
   sieve(X) -> n__sieve(X)
   sieve(cons(0, Y)) -> cons(0, n__sieve(activate(Y)))
   filter(cons(X, Y), 0, M) -> cons(0, n__filter(activate(Y), M, M))
   filter(cons(X, Y), s(N), M) -> cons(X, n__filter(activate(Y), N, M))
   filter(X1, X2, X3) -> n__filter(X1, X2, X3)
   nats(N) -> cons(N, n__nats(s(N)))
   nats(X) -> n__nats(X)


Used ordering: Polynomial ordering with Polynomial interpretation:
POL(activate(x_1)) = x_1
POL(s(x_1)) = 0
POL(sieve(x_1)) = 1 + x_1
POL(nats(x_1)) = 0
POL(n__nats(x_1)) = 0
POL(filter(x_1, x_2, x_3)) = x_1
POL(0) = 0
POL(cons(x_1, x_2)) = x_2
POL(n__sieve(x_1)) = 1 + x_1
POL(FILTER(x_1, x_2, x_3)) = x_1
POL(ACTIVATE(x_1)) = x_1
POL(n__filter(x_1, x_2, x_3)) = x_1
POL(SIEVE(x_1)) = 1 + x_1

 resulting in one subcycle.


SCC1.Polo1:

FILTER(cons(X, Y), 0, M) -> ACTIVATE(Y)
ACTIVATE(n__filter(X1, X2, X3)) -> FILTER(X1, X2, X3)
FILTER(cons(X, Y), s(N), M) -> ACTIVATE(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)) = x_1
POL(FILTER(x_1, x_2, x_3)) = x_1 + x_2 + x_3
POL(ACTIVATE(x_1)) = x_1
POL(n__filter(x_1, x_2, x_3)) = x_1 + x_2 + x_3
POL(0) = 0
POL(cons(x_1, x_2)) = x_1 + x_2

The following Dependency Pairs can be deleted:

   FILTER(cons(X, Y), 0, M) -> ACTIVATE(Y)
   ACTIVATE(n__filter(X1, X2, X3)) -> FILTER(X1, X2, X3)
   FILTER(cons(X, Y), s(N), M) -> ACTIVATE(Y)



 This transformation is resulting in no new subcycles.




Termination of R successfully shown.


Duration: 1.18 seconds.