Term Rewriting System R:
[X, Y, M, N, X1, X2, X3]
filter(cons(X, Y), 0, M) -> cons(0, nfilter(activate(Y), M, M))
filter(cons(X, Y), s(N), M) -> cons(X, nfilter(activate(Y), N, M))
filter(X1, X2, X3) -> nfilter(X1, X2, X3)
sieve(cons(0, Y)) -> cons(0, nsieve(activate(Y)))
sieve(cons(s(N), Y)) -> cons(s(N), nsieve(nfilter(activate(Y), N, N)))
sieve(X) -> nsieve(X)
nats(N) -> cons(N, nnats(ns(N)))
nats(X) -> nnats(X)
zprimes -> sieve(nats(s(s(0))))
s(X) -> ns(X)
activate(nfilter(X1, X2, X3)) -> filter(activate(X1), activate(X2), activate(X3))
activate(nsieve(X)) -> sieve(activate(X))
activate(nnats(X)) -> nats(activate(X))
activate(ns(X)) -> s(activate(X))
activate(X) -> X

Innermost Termination of R to be shown. 



   R
     ↳Dependency Pair Analysis



R contains the following Dependency Pairs:

FILTER(cons(X, Y), 0, M) -> ACTIVATE(Y)
FILTER(cons(X, Y), s(N), M) -> ACTIVATE(Y)
SIEVE(cons(0, Y)) -> ACTIVATE(Y)
SIEVE(cons(s(N), Y)) -> ACTIVATE(Y)
ZPRIMES -> SIEVE(nats(s(s(0))))
ZPRIMES -> NATS(s(s(0)))
ZPRIMES -> S(s(0))
ZPRIMES -> S(0)
ACTIVATE(nfilter(X1, X2, X3)) -> FILTER(activate(X1), activate(X2), activate(X3))
ACTIVATE(nfilter(X1, X2, X3)) -> ACTIVATE(X1)
ACTIVATE(nfilter(X1, X2, X3)) -> ACTIVATE(X2)
ACTIVATE(nfilter(X1, X2, X3)) -> ACTIVATE(X3)
ACTIVATE(nsieve(X)) -> SIEVE(activate(X))
ACTIVATE(nsieve(X)) -> ACTIVATE(X)
ACTIVATE(nnats(X)) -> NATS(activate(X))
ACTIVATE(nnats(X)) -> ACTIVATE(X)
ACTIVATE(ns(X)) -> S(activate(X))
ACTIVATE(ns(X)) -> ACTIVATE(X)

Furthermore, R contains one SCC. 


   R
     ↳DPs
       →DP Problem 1
         ↳Remaining Obligation(s)




The following remains to be proven:
Dependency Pairs:

ACTIVATE(ns(X)) -> ACTIVATE(X)
ACTIVATE(nnats(X)) -> ACTIVATE(X)
ACTIVATE(nsieve(X)) -> ACTIVATE(X)
SIEVE(cons(0, Y)) -> ACTIVATE(Y)
ACTIVATE(nsieve(X)) -> SIEVE(activate(X))
ACTIVATE(nfilter(X1, X2, X3)) -> ACTIVATE(X3)
ACTIVATE(nfilter(X1, X2, X3)) -> ACTIVATE(X2)
ACTIVATE(nfilter(X1, X2, X3)) -> ACTIVATE(X1)
ACTIVATE(nfilter(X1, X2, X3)) -> FILTER(activate(X1), activate(X2), activate(X3))
FILTER(cons(X, Y), 0, M) -> ACTIVATE(Y)


Rules:


filter(cons(X, Y), 0, M) -> cons(0, nfilter(activate(Y), M, M))
filter(cons(X, Y), s(N), M) -> cons(X, nfilter(activate(Y), N, M))
filter(X1, X2, X3) -> nfilter(X1, X2, X3)
sieve(cons(0, Y)) -> cons(0, nsieve(activate(Y)))
sieve(cons(s(N), Y)) -> cons(s(N), nsieve(nfilter(activate(Y), N, N)))
sieve(X) -> nsieve(X)
nats(N) -> cons(N, nnats(ns(N)))
nats(X) -> nnats(X)
zprimes -> sieve(nats(s(s(0))))
s(X) -> ns(X)
activate(nfilter(X1, X2, X3)) -> filter(activate(X1), activate(X2), activate(X3))
activate(nsieve(X)) -> sieve(activate(X))
activate(nnats(X)) -> nats(activate(X))
activate(ns(X)) -> s(activate(X))
activate(X) -> X


Strategy:

innermost



Innermost Termination of R could not be shown.
Duration:
0:01 minutes