let R be the TRS under consideration

sieve(cons(_1,_2)) -> sieve(activate(_2)) is in elim_R(R)
let r0 be the right-hand side of this rule
p0 = 0 is a position in r0
we have r0|p0 = activate(_2)
activate(n__from(_3)) -> from(_3) is in R
let l'0 be the left-hand side of this rule
theta0 = {_2/n__from(_3)} is a mgu of r0|p0 and l'0

==> sieve(cons(_1,n__from(_2))) -> sieve(from(_2)) is in EU_R^1
let r1 be the right-hand side of this rule
p1 = 0 is a position in r1
we have r1|p1 = from(_2)
from(_3) -> cons(_3,n__from(s(_3))) is in R
let l'1 be the left-hand side of this rule
theta1 = {_2/_3} is a mgu of r1|p1 and l'1

==> sieve(cons(_1,n__from(_2))) -> sieve(cons(_2,n__from(s(_2)))) is in EU_R^2
let l be the left-hand side and r be the right-hand side of this rule
let p = epsilon
let theta = {}
let theta' = {_1/_2, _2/s(_2)}
we have r|p = sieve(cons(_2,n__from(s(_2)))) and
theta'(theta(l)) = theta(r|p)
so, theta(l) = sieve(cons(_1,n__from(_2))) is non-terminating w.r.t. R

Termination disproved by the forward process
proof stopped at iteration i=2, depth k=4
599 rule(s) generated