Runtime Complexity TRS:
The TRS R consists of the following rules:

active(filter(cons(X, Y), 0, M)) → mark(cons(0, filter(Y, M, M)))
active(filter(cons(X, Y), s(N), M)) → mark(cons(X, filter(Y, N, M)))
active(sieve(cons(0, Y))) → mark(cons(0, sieve(Y)))
active(sieve(cons(s(N), Y))) → mark(cons(s(N), sieve(filter(Y, N, N))))
active(nats(N)) → mark(cons(N, nats(s(N))))
active(zprimes) → mark(sieve(nats(s(s(0)))))
active(filter(X1, X2, X3)) → filter(active(X1), X2, X3)
active(filter(X1, X2, X3)) → filter(X1, active(X2), X3)
active(filter(X1, X2, X3)) → filter(X1, X2, active(X3))
active(cons(X1, X2)) → cons(active(X1), X2)
active(s(X)) → s(active(X))
active(sieve(X)) → sieve(active(X))
active(nats(X)) → nats(active(X))
filter(mark(X1), X2, X3) → mark(filter(X1, X2, X3))
filter(X1, mark(X2), X3) → mark(filter(X1, X2, X3))
filter(X1, X2, mark(X3)) → mark(filter(X1, X2, X3))
cons(mark(X1), X2) → mark(cons(X1, X2))
s(mark(X)) → mark(s(X))
sieve(mark(X)) → mark(sieve(X))
nats(mark(X)) → mark(nats(X))
proper(filter(X1, X2, X3)) → filter(proper(X1), proper(X2), proper(X3))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(s(X)) → s(proper(X))
proper(sieve(X)) → sieve(proper(X))
proper(nats(X)) → nats(proper(X))
proper(zprimes) → ok(zprimes)
filter(ok(X1), ok(X2), ok(X3)) → ok(filter(X1, X2, X3))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
s(ok(X)) → ok(s(X))
sieve(ok(X)) → ok(sieve(X))
nats(ok(X)) → ok(nats(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Rewrite Strategy: INNERMOST

Renamed function symbols to avoid clashes with predefined symbol.

Runtime Complexity TRS:
The TRS R consists of the following rules:


active'(filter'(cons'(X, Y), 0', M)) → mark'(cons'(0', filter'(Y, M, M)))
active'(filter'(cons'(X, Y), s'(N), M)) → mark'(cons'(X, filter'(Y, N, M)))
active'(sieve'(cons'(0', Y))) → mark'(cons'(0', sieve'(Y)))
active'(sieve'(cons'(s'(N), Y))) → mark'(cons'(s'(N), sieve'(filter'(Y, N, N))))
active'(nats'(N)) → mark'(cons'(N, nats'(s'(N))))
active'(zprimes') → mark'(sieve'(nats'(s'(s'(0')))))
active'(filter'(X1, X2, X3)) → filter'(active'(X1), X2, X3)
active'(filter'(X1, X2, X3)) → filter'(X1, active'(X2), X3)
active'(filter'(X1, X2, X3)) → filter'(X1, X2, active'(X3))
active'(cons'(X1, X2)) → cons'(active'(X1), X2)
active'(s'(X)) → s'(active'(X))
active'(sieve'(X)) → sieve'(active'(X))
active'(nats'(X)) → nats'(active'(X))
filter'(mark'(X1), X2, X3) → mark'(filter'(X1, X2, X3))
filter'(X1, mark'(X2), X3) → mark'(filter'(X1, X2, X3))
filter'(X1, X2, mark'(X3)) → mark'(filter'(X1, X2, X3))
cons'(mark'(X1), X2) → mark'(cons'(X1, X2))
s'(mark'(X)) → mark'(s'(X))
sieve'(mark'(X)) → mark'(sieve'(X))
nats'(mark'(X)) → mark'(nats'(X))
proper'(filter'(X1, X2, X3)) → filter'(proper'(X1), proper'(X2), proper'(X3))
proper'(cons'(X1, X2)) → cons'(proper'(X1), proper'(X2))
proper'(0') → ok'(0')
proper'(s'(X)) → s'(proper'(X))
proper'(sieve'(X)) → sieve'(proper'(X))
proper'(nats'(X)) → nats'(proper'(X))
proper'(zprimes') → ok'(zprimes')
filter'(ok'(X1), ok'(X2), ok'(X3)) → ok'(filter'(X1, X2, X3))
cons'(ok'(X1), ok'(X2)) → ok'(cons'(X1, X2))
s'(ok'(X)) → ok'(s'(X))
sieve'(ok'(X)) → ok'(sieve'(X))
nats'(ok'(X)) → ok'(nats'(X))
top'(mark'(X)) → top'(proper'(X))
top'(ok'(X)) → top'(active'(X))

Rewrite Strategy: INNERMOST

Infered types.

Rules:
active'(filter'(cons'(X, Y), 0', M)) → mark'(cons'(0', filter'(Y, M, M)))
active'(filter'(cons'(X, Y), s'(N), M)) → mark'(cons'(X, filter'(Y, N, M)))
active'(sieve'(cons'(0', Y))) → mark'(cons'(0', sieve'(Y)))
active'(sieve'(cons'(s'(N), Y))) → mark'(cons'(s'(N), sieve'(filter'(Y, N, N))))
active'(nats'(N)) → mark'(cons'(N, nats'(s'(N))))
active'(zprimes') → mark'(sieve'(nats'(s'(s'(0')))))
active'(filter'(X1, X2, X3)) → filter'(active'(X1), X2, X3)
active'(filter'(X1, X2, X3)) → filter'(X1, active'(X2), X3)
active'(filter'(X1, X2, X3)) → filter'(X1, X2, active'(X3))
active'(cons'(X1, X2)) → cons'(active'(X1), X2)
active'(s'(X)) → s'(active'(X))
active'(sieve'(X)) → sieve'(active'(X))
active'(nats'(X)) → nats'(active'(X))
filter'(mark'(X1), X2, X3) → mark'(filter'(X1, X2, X3))
filter'(X1, mark'(X2), X3) → mark'(filter'(X1, X2, X3))
filter'(X1, X2, mark'(X3)) → mark'(filter'(X1, X2, X3))
cons'(mark'(X1), X2) → mark'(cons'(X1, X2))
s'(mark'(X)) → mark'(s'(X))
sieve'(mark'(X)) → mark'(sieve'(X))
nats'(mark'(X)) → mark'(nats'(X))
proper'(filter'(X1, X2, X3)) → filter'(proper'(X1), proper'(X2), proper'(X3))
proper'(cons'(X1, X2)) → cons'(proper'(X1), proper'(X2))
proper'(0') → ok'(0')
proper'(s'(X)) → s'(proper'(X))
proper'(sieve'(X)) → sieve'(proper'(X))
proper'(nats'(X)) → nats'(proper'(X))
proper'(zprimes') → ok'(zprimes')
filter'(ok'(X1), ok'(X2), ok'(X3)) → ok'(filter'(X1, X2, X3))
cons'(ok'(X1), ok'(X2)) → ok'(cons'(X1, X2))
s'(ok'(X)) → ok'(s'(X))
sieve'(ok'(X)) → ok'(sieve'(X))
nats'(ok'(X)) → ok'(nats'(X))
top'(mark'(X)) → top'(proper'(X))
top'(ok'(X)) → top'(active'(X))

Types:
active' :: 0':mark':zprimes':ok' → 0':mark':zprimes':ok'
filter' :: 0':mark':zprimes':ok' → 0':mark':zprimes':ok' → 0':mark':zprimes':ok' → 0':mark':zprimes':ok'
cons' :: 0':mark':zprimes':ok' → 0':mark':zprimes':ok' → 0':mark':zprimes':ok'
0' :: 0':mark':zprimes':ok'
mark' :: 0':mark':zprimes':ok' → 0':mark':zprimes':ok'
s' :: 0':mark':zprimes':ok' → 0':mark':zprimes':ok'
sieve' :: 0':mark':zprimes':ok' → 0':mark':zprimes':ok'
nats' :: 0':mark':zprimes':ok' → 0':mark':zprimes':ok'
zprimes' :: 0':mark':zprimes':ok'
proper' :: 0':mark':zprimes':ok' → 0':mark':zprimes':ok'
ok' :: 0':mark':zprimes':ok' → 0':mark':zprimes':ok'
top' :: 0':mark':zprimes':ok' → top'
_hole_0':mark':zprimes':ok'1 :: 0':mark':zprimes':ok'
_hole_top'2 :: top'
_gen_0':mark':zprimes':ok'3 :: Nat → 0':mark':zprimes':ok'

Heuristically decided to analyse the following defined symbols:
active', cons', filter', sieve', s', nats', proper', top'

They will be analysed ascendingly in the following order:
cons' < active'
filter' < active'
sieve' < active'
s' < active'
nats' < active'
active' < top'
cons' < proper'
filter' < proper'
sieve' < proper'
s' < proper'
nats' < proper'
proper' < top'

Rules:
active'(filter'(cons'(X, Y), 0', M)) → mark'(cons'(0', filter'(Y, M, M)))
active'(filter'(cons'(X, Y), s'(N), M)) → mark'(cons'(X, filter'(Y, N, M)))
active'(sieve'(cons'(0', Y))) → mark'(cons'(0', sieve'(Y)))
active'(sieve'(cons'(s'(N), Y))) → mark'(cons'(s'(N), sieve'(filter'(Y, N, N))))
active'(nats'(N)) → mark'(cons'(N, nats'(s'(N))))
active'(zprimes') → mark'(sieve'(nats'(s'(s'(0')))))
active'(filter'(X1, X2, X3)) → filter'(active'(X1), X2, X3)
active'(filter'(X1, X2, X3)) → filter'(X1, active'(X2), X3)
active'(filter'(X1, X2, X3)) → filter'(X1, X2, active'(X3))
active'(cons'(X1, X2)) → cons'(active'(X1), X2)
active'(s'(X)) → s'(active'(X))
active'(sieve'(X)) → sieve'(active'(X))
active'(nats'(X)) → nats'(active'(X))
filter'(mark'(X1), X2, X3) → mark'(filter'(X1, X2, X3))
filter'(X1, mark'(X2), X3) → mark'(filter'(X1, X2, X3))
filter'(X1, X2, mark'(X3)) → mark'(filter'(X1, X2, X3))
cons'(mark'(X1), X2) → mark'(cons'(X1, X2))
s'(mark'(X)) → mark'(s'(X))
sieve'(mark'(X)) → mark'(sieve'(X))
nats'(mark'(X)) → mark'(nats'(X))
proper'(filter'(X1, X2, X3)) → filter'(proper'(X1), proper'(X2), proper'(X3))
proper'(cons'(X1, X2)) → cons'(proper'(X1), proper'(X2))
proper'(0') → ok'(0')
proper'(s'(X)) → s'(proper'(X))
proper'(sieve'(X)) → sieve'(proper'(X))
proper'(nats'(X)) → nats'(proper'(X))
proper'(zprimes') → ok'(zprimes')
filter'(ok'(X1), ok'(X2), ok'(X3)) → ok'(filter'(X1, X2, X3))
cons'(ok'(X1), ok'(X2)) → ok'(cons'(X1, X2))
s'(ok'(X)) → ok'(s'(X))
sieve'(ok'(X)) → ok'(sieve'(X))
nats'(ok'(X)) → ok'(nats'(X))
top'(mark'(X)) → top'(proper'(X))
top'(ok'(X)) → top'(active'(X))

Types:
active' :: 0':mark':zprimes':ok' → 0':mark':zprimes':ok'
filter' :: 0':mark':zprimes':ok' → 0':mark':zprimes':ok' → 0':mark':zprimes':ok' → 0':mark':zprimes':ok'
cons' :: 0':mark':zprimes':ok' → 0':mark':zprimes':ok' → 0':mark':zprimes':ok'
0' :: 0':mark':zprimes':ok'
mark' :: 0':mark':zprimes':ok' → 0':mark':zprimes':ok'
s' :: 0':mark':zprimes':ok' → 0':mark':zprimes':ok'
sieve' :: 0':mark':zprimes':ok' → 0':mark':zprimes':ok'
nats' :: 0':mark':zprimes':ok' → 0':mark':zprimes':ok'
zprimes' :: 0':mark':zprimes':ok'
proper' :: 0':mark':zprimes':ok' → 0':mark':zprimes':ok'
ok' :: 0':mark':zprimes':ok' → 0':mark':zprimes':ok'
top' :: 0':mark':zprimes':ok' → top'
_hole_0':mark':zprimes':ok'1 :: 0':mark':zprimes':ok'
_hole_top'2 :: top'
_gen_0':mark':zprimes':ok'3 :: Nat → 0':mark':zprimes':ok'

Generator Equations:
_gen_0':mark':zprimes':ok'3(0) ⇔ 0'
_gen_0':mark':zprimes':ok'3(+(x, 1)) ⇔ mark'(_gen_0':mark':zprimes':ok'3(x))

The following defined symbols remain to be analysed:
cons', active', filter', sieve', s', nats', proper', top'

They will be analysed ascendingly in the following order:
cons' < active'
filter' < active'
sieve' < active'
s' < active'
nats' < active'
active' < top'
cons' < proper'
filter' < proper'
sieve' < proper'
s' < proper'
nats' < proper'
proper' < top'

Proved the following rewrite lemma:
cons'(_gen_0':mark':zprimes':ok'3(+(1, _n5)), _gen_0':mark':zprimes':ok'3(b)) → _*4, rt ∈ Ω(n5)

Induction Base:
cons'(_gen_0':mark':zprimes':ok'3(+(1, 0)), _gen_0':mark':zprimes':ok'3(b))

Induction Step:
cons'(_gen_0':mark':zprimes':ok'3(+(1, +(_$n6, 1))), _gen_0':mark':zprimes':ok'3(_b610)) →RΩ(1)
mark'(cons'(_gen_0':mark':zprimes':ok'3(+(1, _$n6)), _gen_0':mark':zprimes':ok'3(_b610))) →IH
mark'(_*4)

We have rt ∈ Ω(n) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).

Rules:
active'(filter'(cons'(X, Y), 0', M)) → mark'(cons'(0', filter'(Y, M, M)))
active'(filter'(cons'(X, Y), s'(N), M)) → mark'(cons'(X, filter'(Y, N, M)))
active'(sieve'(cons'(0', Y))) → mark'(cons'(0', sieve'(Y)))
active'(sieve'(cons'(s'(N), Y))) → mark'(cons'(s'(N), sieve'(filter'(Y, N, N))))
active'(nats'(N)) → mark'(cons'(N, nats'(s'(N))))
active'(zprimes') → mark'(sieve'(nats'(s'(s'(0')))))
active'(filter'(X1, X2, X3)) → filter'(active'(X1), X2, X3)
active'(filter'(X1, X2, X3)) → filter'(X1, active'(X2), X3)
active'(filter'(X1, X2, X3)) → filter'(X1, X2, active'(X3))
active'(cons'(X1, X2)) → cons'(active'(X1), X2)
active'(s'(X)) → s'(active'(X))
active'(sieve'(X)) → sieve'(active'(X))
active'(nats'(X)) → nats'(active'(X))
filter'(mark'(X1), X2, X3) → mark'(filter'(X1, X2, X3))
filter'(X1, mark'(X2), X3) → mark'(filter'(X1, X2, X3))
filter'(X1, X2, mark'(X3)) → mark'(filter'(X1, X2, X3))
cons'(mark'(X1), X2) → mark'(cons'(X1, X2))
s'(mark'(X)) → mark'(s'(X))
sieve'(mark'(X)) → mark'(sieve'(X))
nats'(mark'(X)) → mark'(nats'(X))
proper'(filter'(X1, X2, X3)) → filter'(proper'(X1), proper'(X2), proper'(X3))
proper'(cons'(X1, X2)) → cons'(proper'(X1), proper'(X2))
proper'(0') → ok'(0')
proper'(s'(X)) → s'(proper'(X))
proper'(sieve'(X)) → sieve'(proper'(X))
proper'(nats'(X)) → nats'(proper'(X))
proper'(zprimes') → ok'(zprimes')
filter'(ok'(X1), ok'(X2), ok'(X3)) → ok'(filter'(X1, X2, X3))
cons'(ok'(X1), ok'(X2)) → ok'(cons'(X1, X2))
s'(ok'(X)) → ok'(s'(X))
sieve'(ok'(X)) → ok'(sieve'(X))
nats'(ok'(X)) → ok'(nats'(X))
top'(mark'(X)) → top'(proper'(X))
top'(ok'(X)) → top'(active'(X))

Types:
active' :: 0':mark':zprimes':ok' → 0':mark':zprimes':ok'
filter' :: 0':mark':zprimes':ok' → 0':mark':zprimes':ok' → 0':mark':zprimes':ok' → 0':mark':zprimes':ok'
cons' :: 0':mark':zprimes':ok' → 0':mark':zprimes':ok' → 0':mark':zprimes':ok'
0' :: 0':mark':zprimes':ok'
mark' :: 0':mark':zprimes':ok' → 0':mark':zprimes':ok'
s' :: 0':mark':zprimes':ok' → 0':mark':zprimes':ok'
sieve' :: 0':mark':zprimes':ok' → 0':mark':zprimes':ok'
nats' :: 0':mark':zprimes':ok' → 0':mark':zprimes':ok'
zprimes' :: 0':mark':zprimes':ok'
proper' :: 0':mark':zprimes':ok' → 0':mark':zprimes':ok'
ok' :: 0':mark':zprimes':ok' → 0':mark':zprimes':ok'
top' :: 0':mark':zprimes':ok' → top'
_hole_0':mark':zprimes':ok'1 :: 0':mark':zprimes':ok'
_hole_top'2 :: top'
_gen_0':mark':zprimes':ok'3 :: Nat → 0':mark':zprimes':ok'

Lemmas:
cons'(_gen_0':mark':zprimes':ok'3(+(1, _n5)), _gen_0':mark':zprimes':ok'3(b)) → _*4, rt ∈ Ω(n5)

Generator Equations:
_gen_0':mark':zprimes':ok'3(0) ⇔ 0'
_gen_0':mark':zprimes':ok'3(+(x, 1)) ⇔ mark'(_gen_0':mark':zprimes':ok'3(x))

The following defined symbols remain to be analysed:
filter', active', sieve', s', nats', proper', top'

They will be analysed ascendingly in the following order:
filter' < active'
sieve' < active'
s' < active'
nats' < active'
active' < top'
filter' < proper'
sieve' < proper'
s' < proper'
nats' < proper'
proper' < top'

Proved the following rewrite lemma:
filter'(_gen_0':mark':zprimes':ok'3(+(1, _n2222)), _gen_0':mark':zprimes':ok'3(b), _gen_0':mark':zprimes':ok'3(c)) → _*4, rt ∈ Ω(n2222)

Induction Base:
filter'(_gen_0':mark':zprimes':ok'3(+(1, 0)), _gen_0':mark':zprimes':ok'3(b), _gen_0':mark':zprimes':ok'3(c))

Induction Step:
filter'(_gen_0':mark':zprimes':ok'3(+(1, +(_$n2223, 1))), _gen_0':mark':zprimes':ok'3(_b4531), _gen_0':mark':zprimes':ok'3(_c4532)) →RΩ(1)
mark'(filter'(_gen_0':mark':zprimes':ok'3(+(1, _$n2223)), _gen_0':mark':zprimes':ok'3(_b4531), _gen_0':mark':zprimes':ok'3(_c4532))) →IH
mark'(_*4)

We have rt ∈ Ω(n) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).

Rules:
active'(filter'(cons'(X, Y), 0', M)) → mark'(cons'(0', filter'(Y, M, M)))
active'(filter'(cons'(X, Y), s'(N), M)) → mark'(cons'(X, filter'(Y, N, M)))
active'(sieve'(cons'(0', Y))) → mark'(cons'(0', sieve'(Y)))
active'(sieve'(cons'(s'(N), Y))) → mark'(cons'(s'(N), sieve'(filter'(Y, N, N))))
active'(nats'(N)) → mark'(cons'(N, nats'(s'(N))))
active'(zprimes') → mark'(sieve'(nats'(s'(s'(0')))))
active'(filter'(X1, X2, X3)) → filter'(active'(X1), X2, X3)
active'(filter'(X1, X2, X3)) → filter'(X1, active'(X2), X3)
active'(filter'(X1, X2, X3)) → filter'(X1, X2, active'(X3))
active'(cons'(X1, X2)) → cons'(active'(X1), X2)
active'(s'(X)) → s'(active'(X))
active'(sieve'(X)) → sieve'(active'(X))
active'(nats'(X)) → nats'(active'(X))
filter'(mark'(X1), X2, X3) → mark'(filter'(X1, X2, X3))
filter'(X1, mark'(X2), X3) → mark'(filter'(X1, X2, X3))
filter'(X1, X2, mark'(X3)) → mark'(filter'(X1, X2, X3))
cons'(mark'(X1), X2) → mark'(cons'(X1, X2))
s'(mark'(X)) → mark'(s'(X))
sieve'(mark'(X)) → mark'(sieve'(X))
nats'(mark'(X)) → mark'(nats'(X))
proper'(filter'(X1, X2, X3)) → filter'(proper'(X1), proper'(X2), proper'(X3))
proper'(cons'(X1, X2)) → cons'(proper'(X1), proper'(X2))
proper'(0') → ok'(0')
proper'(s'(X)) → s'(proper'(X))
proper'(sieve'(X)) → sieve'(proper'(X))
proper'(nats'(X)) → nats'(proper'(X))
proper'(zprimes') → ok'(zprimes')
filter'(ok'(X1), ok'(X2), ok'(X3)) → ok'(filter'(X1, X2, X3))
cons'(ok'(X1), ok'(X2)) → ok'(cons'(X1, X2))
s'(ok'(X)) → ok'(s'(X))
sieve'(ok'(X)) → ok'(sieve'(X))
nats'(ok'(X)) → ok'(nats'(X))
top'(mark'(X)) → top'(proper'(X))
top'(ok'(X)) → top'(active'(X))

Types:
active' :: 0':mark':zprimes':ok' → 0':mark':zprimes':ok'
filter' :: 0':mark':zprimes':ok' → 0':mark':zprimes':ok' → 0':mark':zprimes':ok' → 0':mark':zprimes':ok'
cons' :: 0':mark':zprimes':ok' → 0':mark':zprimes':ok' → 0':mark':zprimes':ok'
0' :: 0':mark':zprimes':ok'
mark' :: 0':mark':zprimes':ok' → 0':mark':zprimes':ok'
s' :: 0':mark':zprimes':ok' → 0':mark':zprimes':ok'
sieve' :: 0':mark':zprimes':ok' → 0':mark':zprimes':ok'
nats' :: 0':mark':zprimes':ok' → 0':mark':zprimes':ok'
zprimes' :: 0':mark':zprimes':ok'
proper' :: 0':mark':zprimes':ok' → 0':mark':zprimes':ok'
ok' :: 0':mark':zprimes':ok' → 0':mark':zprimes':ok'
top' :: 0':mark':zprimes':ok' → top'
_hole_0':mark':zprimes':ok'1 :: 0':mark':zprimes':ok'
_hole_top'2 :: top'
_gen_0':mark':zprimes':ok'3 :: Nat → 0':mark':zprimes':ok'

Lemmas:
cons'(_gen_0':mark':zprimes':ok'3(+(1, _n5)), _gen_0':mark':zprimes':ok'3(b)) → _*4, rt ∈ Ω(n5)
filter'(_gen_0':mark':zprimes':ok'3(+(1, _n2222)), _gen_0':mark':zprimes':ok'3(b), _gen_0':mark':zprimes':ok'3(c)) → _*4, rt ∈ Ω(n2222)

Generator Equations:
_gen_0':mark':zprimes':ok'3(0) ⇔ 0'
_gen_0':mark':zprimes':ok'3(+(x, 1)) ⇔ mark'(_gen_0':mark':zprimes':ok'3(x))

The following defined symbols remain to be analysed:
sieve', active', s', nats', proper', top'

They will be analysed ascendingly in the following order:
sieve' < active'
s' < active'
nats' < active'
active' < top'
sieve' < proper'
s' < proper'
nats' < proper'
proper' < top'

Proved the following rewrite lemma:
sieve'(_gen_0':mark':zprimes':ok'3(+(1, _n6764))) → _*4, rt ∈ Ω(n6764)

Induction Base:
sieve'(_gen_0':mark':zprimes':ok'3(+(1, 0)))

Induction Step:
sieve'(_gen_0':mark':zprimes':ok'3(+(1, +(_$n6765, 1)))) →RΩ(1)
mark'(sieve'(_gen_0':mark':zprimes':ok'3(+(1, _$n6765)))) →IH
mark'(_*4)

We have rt ∈ Ω(n) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).

Rules:
active'(filter'(cons'(X, Y), 0', M)) → mark'(cons'(0', filter'(Y, M, M)))
active'(filter'(cons'(X, Y), s'(N), M)) → mark'(cons'(X, filter'(Y, N, M)))
active'(sieve'(cons'(0', Y))) → mark'(cons'(0', sieve'(Y)))
active'(sieve'(cons'(s'(N), Y))) → mark'(cons'(s'(N), sieve'(filter'(Y, N, N))))
active'(nats'(N)) → mark'(cons'(N, nats'(s'(N))))
active'(zprimes') → mark'(sieve'(nats'(s'(s'(0')))))
active'(filter'(X1, X2, X3)) → filter'(active'(X1), X2, X3)
active'(filter'(X1, X2, X3)) → filter'(X1, active'(X2), X3)
active'(filter'(X1, X2, X3)) → filter'(X1, X2, active'(X3))
active'(cons'(X1, X2)) → cons'(active'(X1), X2)
active'(s'(X)) → s'(active'(X))
active'(sieve'(X)) → sieve'(active'(X))
active'(nats'(X)) → nats'(active'(X))
filter'(mark'(X1), X2, X3) → mark'(filter'(X1, X2, X3))
filter'(X1, mark'(X2), X3) → mark'(filter'(X1, X2, X3))
filter'(X1, X2, mark'(X3)) → mark'(filter'(X1, X2, X3))
cons'(mark'(X1), X2) → mark'(cons'(X1, X2))
s'(mark'(X)) → mark'(s'(X))
sieve'(mark'(X)) → mark'(sieve'(X))
nats'(mark'(X)) → mark'(nats'(X))
proper'(filter'(X1, X2, X3)) → filter'(proper'(X1), proper'(X2), proper'(X3))
proper'(cons'(X1, X2)) → cons'(proper'(X1), proper'(X2))
proper'(0') → ok'(0')
proper'(s'(X)) → s'(proper'(X))
proper'(sieve'(X)) → sieve'(proper'(X))
proper'(nats'(X)) → nats'(proper'(X))
proper'(zprimes') → ok'(zprimes')
filter'(ok'(X1), ok'(X2), ok'(X3)) → ok'(filter'(X1, X2, X3))
cons'(ok'(X1), ok'(X2)) → ok'(cons'(X1, X2))
s'(ok'(X)) → ok'(s'(X))
sieve'(ok'(X)) → ok'(sieve'(X))
nats'(ok'(X)) → ok'(nats'(X))
top'(mark'(X)) → top'(proper'(X))
top'(ok'(X)) → top'(active'(X))

Types:
active' :: 0':mark':zprimes':ok' → 0':mark':zprimes':ok'
filter' :: 0':mark':zprimes':ok' → 0':mark':zprimes':ok' → 0':mark':zprimes':ok' → 0':mark':zprimes':ok'
cons' :: 0':mark':zprimes':ok' → 0':mark':zprimes':ok' → 0':mark':zprimes':ok'
0' :: 0':mark':zprimes':ok'
mark' :: 0':mark':zprimes':ok' → 0':mark':zprimes':ok'
s' :: 0':mark':zprimes':ok' → 0':mark':zprimes':ok'
sieve' :: 0':mark':zprimes':ok' → 0':mark':zprimes':ok'
nats' :: 0':mark':zprimes':ok' → 0':mark':zprimes':ok'
zprimes' :: 0':mark':zprimes':ok'
proper' :: 0':mark':zprimes':ok' → 0':mark':zprimes':ok'
ok' :: 0':mark':zprimes':ok' → 0':mark':zprimes':ok'
top' :: 0':mark':zprimes':ok' → top'
_hole_0':mark':zprimes':ok'1 :: 0':mark':zprimes':ok'
_hole_top'2 :: top'
_gen_0':mark':zprimes':ok'3 :: Nat → 0':mark':zprimes':ok'

Lemmas:
cons'(_gen_0':mark':zprimes':ok'3(+(1, _n5)), _gen_0':mark':zprimes':ok'3(b)) → _*4, rt ∈ Ω(n5)
filter'(_gen_0':mark':zprimes':ok'3(+(1, _n2222)), _gen_0':mark':zprimes':ok'3(b), _gen_0':mark':zprimes':ok'3(c)) → _*4, rt ∈ Ω(n2222)
sieve'(_gen_0':mark':zprimes':ok'3(+(1, _n6764))) → _*4, rt ∈ Ω(n6764)

Generator Equations:
_gen_0':mark':zprimes':ok'3(0) ⇔ 0'
_gen_0':mark':zprimes':ok'3(+(x, 1)) ⇔ mark'(_gen_0':mark':zprimes':ok'3(x))

The following defined symbols remain to be analysed:
s', active', nats', proper', top'

They will be analysed ascendingly in the following order:
s' < active'
nats' < active'
active' < top'
s' < proper'
nats' < proper'
proper' < top'

Proved the following rewrite lemma:
s'(_gen_0':mark':zprimes':ok'3(+(1, _n8580))) → _*4, rt ∈ Ω(n8580)

Induction Base:
s'(_gen_0':mark':zprimes':ok'3(+(1, 0)))

Induction Step:
s'(_gen_0':mark':zprimes':ok'3(+(1, +(_$n8581, 1)))) →RΩ(1)
mark'(s'(_gen_0':mark':zprimes':ok'3(+(1, _$n8581)))) →IH
mark'(_*4)

We have rt ∈ Ω(n) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).

Rules:
active'(filter'(cons'(X, Y), 0', M)) → mark'(cons'(0', filter'(Y, M, M)))
active'(filter'(cons'(X, Y), s'(N), M)) → mark'(cons'(X, filter'(Y, N, M)))
active'(sieve'(cons'(0', Y))) → mark'(cons'(0', sieve'(Y)))
active'(sieve'(cons'(s'(N), Y))) → mark'(cons'(s'(N), sieve'(filter'(Y, N, N))))
active'(nats'(N)) → mark'(cons'(N, nats'(s'(N))))
active'(zprimes') → mark'(sieve'(nats'(s'(s'(0')))))
active'(filter'(X1, X2, X3)) → filter'(active'(X1), X2, X3)
active'(filter'(X1, X2, X3)) → filter'(X1, active'(X2), X3)
active'(filter'(X1, X2, X3)) → filter'(X1, X2, active'(X3))
active'(cons'(X1, X2)) → cons'(active'(X1), X2)
active'(s'(X)) → s'(active'(X))
active'(sieve'(X)) → sieve'(active'(X))
active'(nats'(X)) → nats'(active'(X))
filter'(mark'(X1), X2, X3) → mark'(filter'(X1, X2, X3))
filter'(X1, mark'(X2), X3) → mark'(filter'(X1, X2, X3))
filter'(X1, X2, mark'(X3)) → mark'(filter'(X1, X2, X3))
cons'(mark'(X1), X2) → mark'(cons'(X1, X2))
s'(mark'(X)) → mark'(s'(X))
sieve'(mark'(X)) → mark'(sieve'(X))
nats'(mark'(X)) → mark'(nats'(X))
proper'(filter'(X1, X2, X3)) → filter'(proper'(X1), proper'(X2), proper'(X3))
proper'(cons'(X1, X2)) → cons'(proper'(X1), proper'(X2))
proper'(0') → ok'(0')
proper'(s'(X)) → s'(proper'(X))
proper'(sieve'(X)) → sieve'(proper'(X))
proper'(nats'(X)) → nats'(proper'(X))
proper'(zprimes') → ok'(zprimes')
filter'(ok'(X1), ok'(X2), ok'(X3)) → ok'(filter'(X1, X2, X3))
cons'(ok'(X1), ok'(X2)) → ok'(cons'(X1, X2))
s'(ok'(X)) → ok'(s'(X))
sieve'(ok'(X)) → ok'(sieve'(X))
nats'(ok'(X)) → ok'(nats'(X))
top'(mark'(X)) → top'(proper'(X))
top'(ok'(X)) → top'(active'(X))

Types:
active' :: 0':mark':zprimes':ok' → 0':mark':zprimes':ok'
filter' :: 0':mark':zprimes':ok' → 0':mark':zprimes':ok' → 0':mark':zprimes':ok' → 0':mark':zprimes':ok'
cons' :: 0':mark':zprimes':ok' → 0':mark':zprimes':ok' → 0':mark':zprimes':ok'
0' :: 0':mark':zprimes':ok'
mark' :: 0':mark':zprimes':ok' → 0':mark':zprimes':ok'
s' :: 0':mark':zprimes':ok' → 0':mark':zprimes':ok'
sieve' :: 0':mark':zprimes':ok' → 0':mark':zprimes':ok'
nats' :: 0':mark':zprimes':ok' → 0':mark':zprimes':ok'
zprimes' :: 0':mark':zprimes':ok'
proper' :: 0':mark':zprimes':ok' → 0':mark':zprimes':ok'
ok' :: 0':mark':zprimes':ok' → 0':mark':zprimes':ok'
top' :: 0':mark':zprimes':ok' → top'
_hole_0':mark':zprimes':ok'1 :: 0':mark':zprimes':ok'
_hole_top'2 :: top'
_gen_0':mark':zprimes':ok'3 :: Nat → 0':mark':zprimes':ok'

Lemmas:
cons'(_gen_0':mark':zprimes':ok'3(+(1, _n5)), _gen_0':mark':zprimes':ok'3(b)) → _*4, rt ∈ Ω(n5)
filter'(_gen_0':mark':zprimes':ok'3(+(1, _n2222)), _gen_0':mark':zprimes':ok'3(b), _gen_0':mark':zprimes':ok'3(c)) → _*4, rt ∈ Ω(n2222)
sieve'(_gen_0':mark':zprimes':ok'3(+(1, _n6764))) → _*4, rt ∈ Ω(n6764)
s'(_gen_0':mark':zprimes':ok'3(+(1, _n8580))) → _*4, rt ∈ Ω(n8580)

Generator Equations:
_gen_0':mark':zprimes':ok'3(0) ⇔ 0'
_gen_0':mark':zprimes':ok'3(+(x, 1)) ⇔ mark'(_gen_0':mark':zprimes':ok'3(x))

The following defined symbols remain to be analysed:
nats', active', proper', top'

They will be analysed ascendingly in the following order:
nats' < active'
active' < top'
nats' < proper'
proper' < top'

Proved the following rewrite lemma:
nats'(_gen_0':mark':zprimes':ok'3(+(1, _n10520))) → _*4, rt ∈ Ω(n10520)

Induction Base:
nats'(_gen_0':mark':zprimes':ok'3(+(1, 0)))

Induction Step:
nats'(_gen_0':mark':zprimes':ok'3(+(1, +(_$n10521, 1)))) →RΩ(1)
mark'(nats'(_gen_0':mark':zprimes':ok'3(+(1, _$n10521)))) →IH
mark'(_*4)

We have rt ∈ Ω(n) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).

Rules:
active'(filter'(cons'(X, Y), 0', M)) → mark'(cons'(0', filter'(Y, M, M)))
active'(filter'(cons'(X, Y), s'(N), M)) → mark'(cons'(X, filter'(Y, N, M)))
active'(sieve'(cons'(0', Y))) → mark'(cons'(0', sieve'(Y)))
active'(sieve'(cons'(s'(N), Y))) → mark'(cons'(s'(N), sieve'(filter'(Y, N, N))))
active'(nats'(N)) → mark'(cons'(N, nats'(s'(N))))
active'(zprimes') → mark'(sieve'(nats'(s'(s'(0')))))
active'(filter'(X1, X2, X3)) → filter'(active'(X1), X2, X3)
active'(filter'(X1, X2, X3)) → filter'(X1, active'(X2), X3)
active'(filter'(X1, X2, X3)) → filter'(X1, X2, active'(X3))
active'(cons'(X1, X2)) → cons'(active'(X1), X2)
active'(s'(X)) → s'(active'(X))
active'(sieve'(X)) → sieve'(active'(X))
active'(nats'(X)) → nats'(active'(X))
filter'(mark'(X1), X2, X3) → mark'(filter'(X1, X2, X3))
filter'(X1, mark'(X2), X3) → mark'(filter'(X1, X2, X3))
filter'(X1, X2, mark'(X3)) → mark'(filter'(X1, X2, X3))
cons'(mark'(X1), X2) → mark'(cons'(X1, X2))
s'(mark'(X)) → mark'(s'(X))
sieve'(mark'(X)) → mark'(sieve'(X))
nats'(mark'(X)) → mark'(nats'(X))
proper'(filter'(X1, X2, X3)) → filter'(proper'(X1), proper'(X2), proper'(X3))
proper'(cons'(X1, X2)) → cons'(proper'(X1), proper'(X2))
proper'(0') → ok'(0')
proper'(s'(X)) → s'(proper'(X))
proper'(sieve'(X)) → sieve'(proper'(X))
proper'(nats'(X)) → nats'(proper'(X))
proper'(zprimes') → ok'(zprimes')
filter'(ok'(X1), ok'(X2), ok'(X3)) → ok'(filter'(X1, X2, X3))
cons'(ok'(X1), ok'(X2)) → ok'(cons'(X1, X2))
s'(ok'(X)) → ok'(s'(X))
sieve'(ok'(X)) → ok'(sieve'(X))
nats'(ok'(X)) → ok'(nats'(X))
top'(mark'(X)) → top'(proper'(X))
top'(ok'(X)) → top'(active'(X))

Types:
active' :: 0':mark':zprimes':ok' → 0':mark':zprimes':ok'
filter' :: 0':mark':zprimes':ok' → 0':mark':zprimes':ok' → 0':mark':zprimes':ok' → 0':mark':zprimes':ok'
cons' :: 0':mark':zprimes':ok' → 0':mark':zprimes':ok' → 0':mark':zprimes':ok'
0' :: 0':mark':zprimes':ok'
mark' :: 0':mark':zprimes':ok' → 0':mark':zprimes':ok'
s' :: 0':mark':zprimes':ok' → 0':mark':zprimes':ok'
sieve' :: 0':mark':zprimes':ok' → 0':mark':zprimes':ok'
nats' :: 0':mark':zprimes':ok' → 0':mark':zprimes':ok'
zprimes' :: 0':mark':zprimes':ok'
proper' :: 0':mark':zprimes':ok' → 0':mark':zprimes':ok'
ok' :: 0':mark':zprimes':ok' → 0':mark':zprimes':ok'
top' :: 0':mark':zprimes':ok' → top'
_hole_0':mark':zprimes':ok'1 :: 0':mark':zprimes':ok'
_hole_top'2 :: top'
_gen_0':mark':zprimes':ok'3 :: Nat → 0':mark':zprimes':ok'

Lemmas:
cons'(_gen_0':mark':zprimes':ok'3(+(1, _n5)), _gen_0':mark':zprimes':ok'3(b)) → _*4, rt ∈ Ω(n5)
filter'(_gen_0':mark':zprimes':ok'3(+(1, _n2222)), _gen_0':mark':zprimes':ok'3(b), _gen_0':mark':zprimes':ok'3(c)) → _*4, rt ∈ Ω(n2222)
sieve'(_gen_0':mark':zprimes':ok'3(+(1, _n6764))) → _*4, rt ∈ Ω(n6764)
s'(_gen_0':mark':zprimes':ok'3(+(1, _n8580))) → _*4, rt ∈ Ω(n8580)
nats'(_gen_0':mark':zprimes':ok'3(+(1, _n10520))) → _*4, rt ∈ Ω(n10520)

Generator Equations:
_gen_0':mark':zprimes':ok'3(0) ⇔ 0'
_gen_0':mark':zprimes':ok'3(+(x, 1)) ⇔ mark'(_gen_0':mark':zprimes':ok'3(x))

The following defined symbols remain to be analysed:
active', proper', top'

They will be analysed ascendingly in the following order:
active' < top'
proper' < top'

Could not prove a rewrite lemma for the defined symbol active'.

Rules:
active'(filter'(cons'(X, Y), 0', M)) → mark'(cons'(0', filter'(Y, M, M)))
active'(filter'(cons'(X, Y), s'(N), M)) → mark'(cons'(X, filter'(Y, N, M)))
active'(sieve'(cons'(0', Y))) → mark'(cons'(0', sieve'(Y)))
active'(sieve'(cons'(s'(N), Y))) → mark'(cons'(s'(N), sieve'(filter'(Y, N, N))))
active'(nats'(N)) → mark'(cons'(N, nats'(s'(N))))
active'(zprimes') → mark'(sieve'(nats'(s'(s'(0')))))
active'(filter'(X1, X2, X3)) → filter'(active'(X1), X2, X3)
active'(filter'(X1, X2, X3)) → filter'(X1, active'(X2), X3)
active'(filter'(X1, X2, X3)) → filter'(X1, X2, active'(X3))
active'(cons'(X1, X2)) → cons'(active'(X1), X2)
active'(s'(X)) → s'(active'(X))
active'(sieve'(X)) → sieve'(active'(X))
active'(nats'(X)) → nats'(active'(X))
filter'(mark'(X1), X2, X3) → mark'(filter'(X1, X2, X3))
filter'(X1, mark'(X2), X3) → mark'(filter'(X1, X2, X3))
filter'(X1, X2, mark'(X3)) → mark'(filter'(X1, X2, X3))
cons'(mark'(X1), X2) → mark'(cons'(X1, X2))
s'(mark'(X)) → mark'(s'(X))
sieve'(mark'(X)) → mark'(sieve'(X))
nats'(mark'(X)) → mark'(nats'(X))
proper'(filter'(X1, X2, X3)) → filter'(proper'(X1), proper'(X2), proper'(X3))
proper'(cons'(X1, X2)) → cons'(proper'(X1), proper'(X2))
proper'(0') → ok'(0')
proper'(s'(X)) → s'(proper'(X))
proper'(sieve'(X)) → sieve'(proper'(X))
proper'(nats'(X)) → nats'(proper'(X))
proper'(zprimes') → ok'(zprimes')
filter'(ok'(X1), ok'(X2), ok'(X3)) → ok'(filter'(X1, X2, X3))
cons'(ok'(X1), ok'(X2)) → ok'(cons'(X1, X2))
s'(ok'(X)) → ok'(s'(X))
sieve'(ok'(X)) → ok'(sieve'(X))
nats'(ok'(X)) → ok'(nats'(X))
top'(mark'(X)) → top'(proper'(X))
top'(ok'(X)) → top'(active'(X))

Types:
active' :: 0':mark':zprimes':ok' → 0':mark':zprimes':ok'
filter' :: 0':mark':zprimes':ok' → 0':mark':zprimes':ok' → 0':mark':zprimes':ok' → 0':mark':zprimes':ok'
cons' :: 0':mark':zprimes':ok' → 0':mark':zprimes':ok' → 0':mark':zprimes':ok'
0' :: 0':mark':zprimes':ok'
mark' :: 0':mark':zprimes':ok' → 0':mark':zprimes':ok'
s' :: 0':mark':zprimes':ok' → 0':mark':zprimes':ok'
sieve' :: 0':mark':zprimes':ok' → 0':mark':zprimes':ok'
nats' :: 0':mark':zprimes':ok' → 0':mark':zprimes':ok'
zprimes' :: 0':mark':zprimes':ok'
proper' :: 0':mark':zprimes':ok' → 0':mark':zprimes':ok'
ok' :: 0':mark':zprimes':ok' → 0':mark':zprimes':ok'
top' :: 0':mark':zprimes':ok' → top'
_hole_0':mark':zprimes':ok'1 :: 0':mark':zprimes':ok'
_hole_top'2 :: top'
_gen_0':mark':zprimes':ok'3 :: Nat → 0':mark':zprimes':ok'

Lemmas:
cons'(_gen_0':mark':zprimes':ok'3(+(1, _n5)), _gen_0':mark':zprimes':ok'3(b)) → _*4, rt ∈ Ω(n5)
filter'(_gen_0':mark':zprimes':ok'3(+(1, _n2222)), _gen_0':mark':zprimes':ok'3(b), _gen_0':mark':zprimes':ok'3(c)) → _*4, rt ∈ Ω(n2222)
sieve'(_gen_0':mark':zprimes':ok'3(+(1, _n6764))) → _*4, rt ∈ Ω(n6764)
s'(_gen_0':mark':zprimes':ok'3(+(1, _n8580))) → _*4, rt ∈ Ω(n8580)
nats'(_gen_0':mark':zprimes':ok'3(+(1, _n10520))) → _*4, rt ∈ Ω(n10520)

Generator Equations:
_gen_0':mark':zprimes':ok'3(0) ⇔ 0'
_gen_0':mark':zprimes':ok'3(+(x, 1)) ⇔ mark'(_gen_0':mark':zprimes':ok'3(x))

The following defined symbols remain to be analysed:
proper', top'

They will be analysed ascendingly in the following order:
proper' < top'

Could not prove a rewrite lemma for the defined symbol proper'.

Rules:
active'(filter'(cons'(X, Y), 0', M)) → mark'(cons'(0', filter'(Y, M, M)))
active'(filter'(cons'(X, Y), s'(N), M)) → mark'(cons'(X, filter'(Y, N, M)))
active'(sieve'(cons'(0', Y))) → mark'(cons'(0', sieve'(Y)))
active'(sieve'(cons'(s'(N), Y))) → mark'(cons'(s'(N), sieve'(filter'(Y, N, N))))
active'(nats'(N)) → mark'(cons'(N, nats'(s'(N))))
active'(zprimes') → mark'(sieve'(nats'(s'(s'(0')))))
active'(filter'(X1, X2, X3)) → filter'(active'(X1), X2, X3)
active'(filter'(X1, X2, X3)) → filter'(X1, active'(X2), X3)
active'(filter'(X1, X2, X3)) → filter'(X1, X2, active'(X3))
active'(cons'(X1, X2)) → cons'(active'(X1), X2)
active'(s'(X)) → s'(active'(X))
active'(sieve'(X)) → sieve'(active'(X))
active'(nats'(X)) → nats'(active'(X))
filter'(mark'(X1), X2, X3) → mark'(filter'(X1, X2, X3))
filter'(X1, mark'(X2), X3) → mark'(filter'(X1, X2, X3))
filter'(X1, X2, mark'(X3)) → mark'(filter'(X1, X2, X3))
cons'(mark'(X1), X2) → mark'(cons'(X1, X2))
s'(mark'(X)) → mark'(s'(X))
sieve'(mark'(X)) → mark'(sieve'(X))
nats'(mark'(X)) → mark'(nats'(X))
proper'(filter'(X1, X2, X3)) → filter'(proper'(X1), proper'(X2), proper'(X3))
proper'(cons'(X1, X2)) → cons'(proper'(X1), proper'(X2))
proper'(0') → ok'(0')
proper'(s'(X)) → s'(proper'(X))
proper'(sieve'(X)) → sieve'(proper'(X))
proper'(nats'(X)) → nats'(proper'(X))
proper'(zprimes') → ok'(zprimes')
filter'(ok'(X1), ok'(X2), ok'(X3)) → ok'(filter'(X1, X2, X3))
cons'(ok'(X1), ok'(X2)) → ok'(cons'(X1, X2))
s'(ok'(X)) → ok'(s'(X))
sieve'(ok'(X)) → ok'(sieve'(X))
nats'(ok'(X)) → ok'(nats'(X))
top'(mark'(X)) → top'(proper'(X))
top'(ok'(X)) → top'(active'(X))

Types:
active' :: 0':mark':zprimes':ok' → 0':mark':zprimes':ok'
filter' :: 0':mark':zprimes':ok' → 0':mark':zprimes':ok' → 0':mark':zprimes':ok' → 0':mark':zprimes':ok'
cons' :: 0':mark':zprimes':ok' → 0':mark':zprimes':ok' → 0':mark':zprimes':ok'
0' :: 0':mark':zprimes':ok'
mark' :: 0':mark':zprimes':ok' → 0':mark':zprimes':ok'
s' :: 0':mark':zprimes':ok' → 0':mark':zprimes':ok'
sieve' :: 0':mark':zprimes':ok' → 0':mark':zprimes':ok'
nats' :: 0':mark':zprimes':ok' → 0':mark':zprimes':ok'
zprimes' :: 0':mark':zprimes':ok'
proper' :: 0':mark':zprimes':ok' → 0':mark':zprimes':ok'
ok' :: 0':mark':zprimes':ok' → 0':mark':zprimes':ok'
top' :: 0':mark':zprimes':ok' → top'
_hole_0':mark':zprimes':ok'1 :: 0':mark':zprimes':ok'
_hole_top'2 :: top'
_gen_0':mark':zprimes':ok'3 :: Nat → 0':mark':zprimes':ok'

Lemmas:
cons'(_gen_0':mark':zprimes':ok'3(+(1, _n5)), _gen_0':mark':zprimes':ok'3(b)) → _*4, rt ∈ Ω(n5)
filter'(_gen_0':mark':zprimes':ok'3(+(1, _n2222)), _gen_0':mark':zprimes':ok'3(b), _gen_0':mark':zprimes':ok'3(c)) → _*4, rt ∈ Ω(n2222)
sieve'(_gen_0':mark':zprimes':ok'3(+(1, _n6764))) → _*4, rt ∈ Ω(n6764)
s'(_gen_0':mark':zprimes':ok'3(+(1, _n8580))) → _*4, rt ∈ Ω(n8580)
nats'(_gen_0':mark':zprimes':ok'3(+(1, _n10520))) → _*4, rt ∈ Ω(n10520)

Generator Equations:
_gen_0':mark':zprimes':ok'3(0) ⇔ 0'
_gen_0':mark':zprimes':ok'3(+(x, 1)) ⇔ mark'(_gen_0':mark':zprimes':ok'3(x))

The following defined symbols remain to be analysed:
top'

Could not prove a rewrite lemma for the defined symbol top'.

Rules:
active'(filter'(cons'(X, Y), 0', M)) → mark'(cons'(0', filter'(Y, M, M)))
active'(filter'(cons'(X, Y), s'(N), M)) → mark'(cons'(X, filter'(Y, N, M)))
active'(sieve'(cons'(0', Y))) → mark'(cons'(0', sieve'(Y)))
active'(sieve'(cons'(s'(N), Y))) → mark'(cons'(s'(N), sieve'(filter'(Y, N, N))))
active'(nats'(N)) → mark'(cons'(N, nats'(s'(N))))
active'(zprimes') → mark'(sieve'(nats'(s'(s'(0')))))
active'(filter'(X1, X2, X3)) → filter'(active'(X1), X2, X3)
active'(filter'(X1, X2, X3)) → filter'(X1, active'(X2), X3)
active'(filter'(X1, X2, X3)) → filter'(X1, X2, active'(X3))
active'(cons'(X1, X2)) → cons'(active'(X1), X2)
active'(s'(X)) → s'(active'(X))
active'(sieve'(X)) → sieve'(active'(X))
active'(nats'(X)) → nats'(active'(X))
filter'(mark'(X1), X2, X3) → mark'(filter'(X1, X2, X3))
filter'(X1, mark'(X2), X3) → mark'(filter'(X1, X2, X3))
filter'(X1, X2, mark'(X3)) → mark'(filter'(X1, X2, X3))
cons'(mark'(X1), X2) → mark'(cons'(X1, X2))
s'(mark'(X)) → mark'(s'(X))
sieve'(mark'(X)) → mark'(sieve'(X))
nats'(mark'(X)) → mark'(nats'(X))
proper'(filter'(X1, X2, X3)) → filter'(proper'(X1), proper'(X2), proper'(X3))
proper'(cons'(X1, X2)) → cons'(proper'(X1), proper'(X2))
proper'(0') → ok'(0')
proper'(s'(X)) → s'(proper'(X))
proper'(sieve'(X)) → sieve'(proper'(X))
proper'(nats'(X)) → nats'(proper'(X))
proper'(zprimes') → ok'(zprimes')
filter'(ok'(X1), ok'(X2), ok'(X3)) → ok'(filter'(X1, X2, X3))
cons'(ok'(X1), ok'(X2)) → ok'(cons'(X1, X2))
s'(ok'(X)) → ok'(s'(X))
sieve'(ok'(X)) → ok'(sieve'(X))
nats'(ok'(X)) → ok'(nats'(X))
top'(mark'(X)) → top'(proper'(X))
top'(ok'(X)) → top'(active'(X))

Types:
active' :: 0':mark':zprimes':ok' → 0':mark':zprimes':ok'
filter' :: 0':mark':zprimes':ok' → 0':mark':zprimes':ok' → 0':mark':zprimes':ok' → 0':mark':zprimes':ok'
cons' :: 0':mark':zprimes':ok' → 0':mark':zprimes':ok' → 0':mark':zprimes':ok'
0' :: 0':mark':zprimes':ok'
mark' :: 0':mark':zprimes':ok' → 0':mark':zprimes':ok'
s' :: 0':mark':zprimes':ok' → 0':mark':zprimes':ok'
sieve' :: 0':mark':zprimes':ok' → 0':mark':zprimes':ok'
nats' :: 0':mark':zprimes':ok' → 0':mark':zprimes':ok'
zprimes' :: 0':mark':zprimes':ok'
proper' :: 0':mark':zprimes':ok' → 0':mark':zprimes':ok'
ok' :: 0':mark':zprimes':ok' → 0':mark':zprimes':ok'
top' :: 0':mark':zprimes':ok' → top'
_hole_0':mark':zprimes':ok'1 :: 0':mark':zprimes':ok'
_hole_top'2 :: top'
_gen_0':mark':zprimes':ok'3 :: Nat → 0':mark':zprimes':ok'

Lemmas:
cons'(_gen_0':mark':zprimes':ok'3(+(1, _n5)), _gen_0':mark':zprimes':ok'3(b)) → _*4, rt ∈ Ω(n5)
filter'(_gen_0':mark':zprimes':ok'3(+(1, _n2222)), _gen_0':mark':zprimes':ok'3(b), _gen_0':mark':zprimes':ok'3(c)) → _*4, rt ∈ Ω(n2222)
sieve'(_gen_0':mark':zprimes':ok'3(+(1, _n6764))) → _*4, rt ∈ Ω(n6764)
s'(_gen_0':mark':zprimes':ok'3(+(1, _n8580))) → _*4, rt ∈ Ω(n8580)
nats'(_gen_0':mark':zprimes':ok'3(+(1, _n10520))) → _*4, rt ∈ Ω(n10520)

Generator Equations:
_gen_0':mark':zprimes':ok'3(0) ⇔ 0'
_gen_0':mark':zprimes':ok'3(+(x, 1)) ⇔ mark'(_gen_0':mark':zprimes':ok'3(x))

No more defined symbols left to analyse.

The lowerbound Ω(n) was proven with the following lemma:
cons'(_gen_0':mark':zprimes':ok'3(+(1, _n5)), _gen_0':mark':zprimes':ok'3(b)) → _*4, rt ∈ Ω(n5)