The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).

(0) Obligation:

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: FULL

(1) DecreasingLoopProof (EQUIVALENT transformation)

The following loop(s) give(s) rise to the lower bound Ω(n¹):
The rewrite sequence
filter(mark(X1), X2, X3) →⁺ mark(filter(X1, X2, X3))
gives rise to a decreasing loop by considering the right hand sides subterm at position [0].
The pumping substitution is [X1 / mark(X1)].
The result substitution is [ ].

(2) BOUNDS(n^1, INF)

(3) RenamingProof (EQUIVALENT transformation)

Renamed function symbols to avoid clashes with predefined symbol.

(4) Obligation:

Runtime Complexity Relative 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))

S is empty.
Rewrite Strategy: FULL

(5) TypeInferenceProof (BOTH BOUNDS(ID, ID) transformation)

Infered types.

(6) Obligation:

TRS:
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:ok1_0 :: 0':mark:zprimes:ok
hole_top2_0 :: top
gen_0':mark:zprimes:ok3_0 :: Nat → 0':mark:zprimes:ok

(7) OrderProof (LOWER BOUND(ID) transformation)

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

(8) Obligation:

TRS:
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:ok1_0 :: 0':mark:zprimes:ok
hole_top2_0 :: top
gen_0':mark:zprimes:ok3_0 :: Nat → 0':mark:zprimes:ok

Generator Equations:
gen_0':mark:zprimes:ok3_0(0) ⇔ 0'
gen_0':mark:zprimes:ok3_0(+(x, 1)) ⇔ mark(gen_0':mark:zprimes:ok3_0(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

(9) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
cons(gen_0':mark:zprimes:ok3_0(+(1, n5_0)), gen_0':mark:zprimes:ok3_0(b)) → *4_0, rt ∈ Ω(n5₀)

Induction Base:
cons(gen_0':mark:zprimes:ok3_0(+(1, 0)), gen_0':mark:zprimes:ok3_0(b))

Induction Step:
cons(gen_0':mark:zprimes:ok3_0(+(1, +(n5_0, 1))), gen_0':mark:zprimes:ok3_0(b)) →_R^Ω(1)
mark(cons(gen_0':mark:zprimes:ok3_0(+(1, n5_0)), gen_0':mark:zprimes:ok3_0(b))) →_IH
mark(*4_0)

We have rt ∈ Ω(n¹) and sz ∈ O(n). Thus, we have irc_R ∈ Ω(n).

(10) Complex Obligation (BEST)

(11) Obligation:

TRS:
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:ok1_0 :: 0':mark:zprimes:ok
hole_top2_0 :: top
gen_0':mark:zprimes:ok3_0 :: Nat → 0':mark:zprimes:ok

Lemmas:
cons(gen_0':mark:zprimes:ok3_0(+(1, n5_0)), gen_0':mark:zprimes:ok3_0(b)) → *4_0, rt ∈ Ω(n5₀)

Generator Equations:
gen_0':mark:zprimes:ok3_0(0) ⇔ 0'
gen_0':mark:zprimes:ok3_0(+(x, 1)) ⇔ mark(gen_0':mark:zprimes:ok3_0(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

(12) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
filter(gen_0':mark:zprimes:ok3_0(+(1, n950_0)), gen_0':mark:zprimes:ok3_0(b), gen_0':mark:zprimes:ok3_0(c)) → *4_0, rt ∈ Ω(n950₀)

Induction Base:
filter(gen_0':mark:zprimes:ok3_0(+(1, 0)), gen_0':mark:zprimes:ok3_0(b), gen_0':mark:zprimes:ok3_0(c))

Induction Step:
filter(gen_0':mark:zprimes:ok3_0(+(1, +(n950_0, 1))), gen_0':mark:zprimes:ok3_0(b), gen_0':mark:zprimes:ok3_0(c)) →_R^Ω(1)
mark(filter(gen_0':mark:zprimes:ok3_0(+(1, n950_0)), gen_0':mark:zprimes:ok3_0(b), gen_0':mark:zprimes:ok3_0(c))) →_IH
mark(*4_0)

We have rt ∈ Ω(n¹) and sz ∈ O(n). Thus, we have irc_R ∈ Ω(n).

(13) Complex Obligation (BEST)

(14) Obligation:

TRS:
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:ok1_0 :: 0':mark:zprimes:ok
hole_top2_0 :: top
gen_0':mark:zprimes:ok3_0 :: Nat → 0':mark:zprimes:ok

Lemmas:
cons(gen_0':mark:zprimes:ok3_0(+(1, n5_0)), gen_0':mark:zprimes:ok3_0(b)) → *4_0, rt ∈ Ω(n5₀)
filter(gen_0':mark:zprimes:ok3_0(+(1, n950_0)), gen_0':mark:zprimes:ok3_0(b), gen_0':mark:zprimes:ok3_0(c)) → *4_0, rt ∈ Ω(n950₀)

Generator Equations:
gen_0':mark:zprimes:ok3_0(0) ⇔ 0'
gen_0':mark:zprimes:ok3_0(+(x, 1)) ⇔ mark(gen_0':mark:zprimes:ok3_0(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

(15) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
sieve(gen_0':mark:zprimes:ok3_0(+(1, n3774_0))) → *4_0, rt ∈ Ω(n3774₀)

Induction Base:
sieve(gen_0':mark:zprimes:ok3_0(+(1, 0)))

Induction Step:
sieve(gen_0':mark:zprimes:ok3_0(+(1, +(n3774_0, 1)))) →_R^Ω(1)
mark(sieve(gen_0':mark:zprimes:ok3_0(+(1, n3774_0)))) →_IH
mark(*4_0)

We have rt ∈ Ω(n¹) and sz ∈ O(n). Thus, we have irc_R ∈ Ω(n).

(16) Complex Obligation (BEST)

(17) Obligation:

TRS:
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:ok1_0 :: 0':mark:zprimes:ok
hole_top2_0 :: top
gen_0':mark:zprimes:ok3_0 :: Nat → 0':mark:zprimes:ok

Lemmas:
cons(gen_0':mark:zprimes:ok3_0(+(1, n5_0)), gen_0':mark:zprimes:ok3_0(b)) → *4_0, rt ∈ Ω(n5₀)
filter(gen_0':mark:zprimes:ok3_0(+(1, n950_0)), gen_0':mark:zprimes:ok3_0(b), gen_0':mark:zprimes:ok3_0(c)) → *4_0, rt ∈ Ω(n950₀)
sieve(gen_0':mark:zprimes:ok3_0(+(1, n3774_0))) → *4_0, rt ∈ Ω(n3774₀)

Generator Equations:
gen_0':mark:zprimes:ok3_0(0) ⇔ 0'
gen_0':mark:zprimes:ok3_0(+(x, 1)) ⇔ mark(gen_0':mark:zprimes:ok3_0(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

(18) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
s(gen_0':mark:zprimes:ok3_0(+(1, n4492_0))) → *4_0, rt ∈ Ω(n4492₀)

Induction Base:
s(gen_0':mark:zprimes:ok3_0(+(1, 0)))

Induction Step:
s(gen_0':mark:zprimes:ok3_0(+(1, +(n4492_0, 1)))) →_R^Ω(1)
mark(s(gen_0':mark:zprimes:ok3_0(+(1, n4492_0)))) →_IH
mark(*4_0)

We have rt ∈ Ω(n¹) and sz ∈ O(n). Thus, we have irc_R ∈ Ω(n).

(19) Complex Obligation (BEST)

(20) Obligation:

TRS:
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:ok1_0 :: 0':mark:zprimes:ok
hole_top2_0 :: top
gen_0':mark:zprimes:ok3_0 :: Nat → 0':mark:zprimes:ok

Lemmas:
cons(gen_0':mark:zprimes:ok3_0(+(1, n5_0)), gen_0':mark:zprimes:ok3_0(b)) → *4_0, rt ∈ Ω(n5₀)
filter(gen_0':mark:zprimes:ok3_0(+(1, n950_0)), gen_0':mark:zprimes:ok3_0(b), gen_0':mark:zprimes:ok3_0(c)) → *4_0, rt ∈ Ω(n950₀)
sieve(gen_0':mark:zprimes:ok3_0(+(1, n3774_0))) → *4_0, rt ∈ Ω(n3774₀)
s(gen_0':mark:zprimes:ok3_0(+(1, n4492_0))) → *4_0, rt ∈ Ω(n4492₀)

Generator Equations:
gen_0':mark:zprimes:ok3_0(0) ⇔ 0'
gen_0':mark:zprimes:ok3_0(+(x, 1)) ⇔ mark(gen_0':mark:zprimes:ok3_0(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

(21) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
nats(gen_0':mark:zprimes:ok3_0(+(1, n5311_0))) → *4_0, rt ∈ Ω(n5311₀)

Induction Base:
nats(gen_0':mark:zprimes:ok3_0(+(1, 0)))

Induction Step:
nats(gen_0':mark:zprimes:ok3_0(+(1, +(n5311_0, 1)))) →_R^Ω(1)
mark(nats(gen_0':mark:zprimes:ok3_0(+(1, n5311_0)))) →_IH
mark(*4_0)

We have rt ∈ Ω(n¹) and sz ∈ O(n). Thus, we have irc_R ∈ Ω(n).

(22) Complex Obligation (BEST)

(23) Obligation:

TRS:
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:ok1_0 :: 0':mark:zprimes:ok
hole_top2_0 :: top
gen_0':mark:zprimes:ok3_0 :: Nat → 0':mark:zprimes:ok

Lemmas:
cons(gen_0':mark:zprimes:ok3_0(+(1, n5_0)), gen_0':mark:zprimes:ok3_0(b)) → *4_0, rt ∈ Ω(n5₀)
filter(gen_0':mark:zprimes:ok3_0(+(1, n950_0)), gen_0':mark:zprimes:ok3_0(b), gen_0':mark:zprimes:ok3_0(c)) → *4_0, rt ∈ Ω(n950₀)
sieve(gen_0':mark:zprimes:ok3_0(+(1, n3774_0))) → *4_0, rt ∈ Ω(n3774₀)
s(gen_0':mark:zprimes:ok3_0(+(1, n4492_0))) → *4_0, rt ∈ Ω(n4492₀)
nats(gen_0':mark:zprimes:ok3_0(+(1, n5311_0))) → *4_0, rt ∈ Ω(n5311₀)

Generator Equations:
gen_0':mark:zprimes:ok3_0(0) ⇔ 0'
gen_0':mark:zprimes:ok3_0(+(x, 1)) ⇔ mark(gen_0':mark:zprimes:ok3_0(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

(24) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

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

(25) Obligation:

TRS:
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:ok1_0 :: 0':mark:zprimes:ok
hole_top2_0 :: top
gen_0':mark:zprimes:ok3_0 :: Nat → 0':mark:zprimes:ok

Lemmas:
cons(gen_0':mark:zprimes:ok3_0(+(1, n5_0)), gen_0':mark:zprimes:ok3_0(b)) → *4_0, rt ∈ Ω(n5₀)
filter(gen_0':mark:zprimes:ok3_0(+(1, n950_0)), gen_0':mark:zprimes:ok3_0(b), gen_0':mark:zprimes:ok3_0(c)) → *4_0, rt ∈ Ω(n950₀)
sieve(gen_0':mark:zprimes:ok3_0(+(1, n3774_0))) → *4_0, rt ∈ Ω(n3774₀)
s(gen_0':mark:zprimes:ok3_0(+(1, n4492_0))) → *4_0, rt ∈ Ω(n4492₀)
nats(gen_0':mark:zprimes:ok3_0(+(1, n5311_0))) → *4_0, rt ∈ Ω(n5311₀)

Generator Equations:
gen_0':mark:zprimes:ok3_0(0) ⇔ 0'
gen_0':mark:zprimes:ok3_0(+(x, 1)) ⇔ mark(gen_0':mark:zprimes:ok3_0(x))

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

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

(26) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

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

(27) Obligation:

TRS:
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:ok1_0 :: 0':mark:zprimes:ok
hole_top2_0 :: top
gen_0':mark:zprimes:ok3_0 :: Nat → 0':mark:zprimes:ok

Lemmas:
cons(gen_0':mark:zprimes:ok3_0(+(1, n5_0)), gen_0':mark:zprimes:ok3_0(b)) → *4_0, rt ∈ Ω(n5₀)
filter(gen_0':mark:zprimes:ok3_0(+(1, n950_0)), gen_0':mark:zprimes:ok3_0(b), gen_0':mark:zprimes:ok3_0(c)) → *4_0, rt ∈ Ω(n950₀)
sieve(gen_0':mark:zprimes:ok3_0(+(1, n3774_0))) → *4_0, rt ∈ Ω(n3774₀)
s(gen_0':mark:zprimes:ok3_0(+(1, n4492_0))) → *4_0, rt ∈ Ω(n4492₀)
nats(gen_0':mark:zprimes:ok3_0(+(1, n5311_0))) → *4_0, rt ∈ Ω(n5311₀)

Generator Equations:
gen_0':mark:zprimes:ok3_0(0) ⇔ 0'
gen_0':mark:zprimes:ok3_0(+(x, 1)) ⇔ mark(gen_0':mark:zprimes:ok3_0(x))

The following defined symbols remain to be analysed:
top

(28) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

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

(29) Obligation:

TRS:
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:ok1_0 :: 0':mark:zprimes:ok
hole_top2_0 :: top
gen_0':mark:zprimes:ok3_0 :: Nat → 0':mark:zprimes:ok

Lemmas:
cons(gen_0':mark:zprimes:ok3_0(+(1, n5_0)), gen_0':mark:zprimes:ok3_0(b)) → *4_0, rt ∈ Ω(n5₀)
filter(gen_0':mark:zprimes:ok3_0(+(1, n950_0)), gen_0':mark:zprimes:ok3_0(b), gen_0':mark:zprimes:ok3_0(c)) → *4_0, rt ∈ Ω(n950₀)
sieve(gen_0':mark:zprimes:ok3_0(+(1, n3774_0))) → *4_0, rt ∈ Ω(n3774₀)
s(gen_0':mark:zprimes:ok3_0(+(1, n4492_0))) → *4_0, rt ∈ Ω(n4492₀)
nats(gen_0':mark:zprimes:ok3_0(+(1, n5311_0))) → *4_0, rt ∈ Ω(n5311₀)

Generator Equations:
gen_0':mark:zprimes:ok3_0(0) ⇔ 0'
gen_0':mark:zprimes:ok3_0(+(x, 1)) ⇔ mark(gen_0':mark:zprimes:ok3_0(x))

No more defined symbols left to analyse.

(30) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
cons(gen_0':mark:zprimes:ok3_0(+(1, n5_0)), gen_0':mark:zprimes:ok3_0(b)) → *4_0, rt ∈ Ω(n5₀)

(31) BOUNDS(n^1, INF)

(32) Obligation:

TRS:
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:ok1_0 :: 0':mark:zprimes:ok
hole_top2_0 :: top
gen_0':mark:zprimes:ok3_0 :: Nat → 0':mark:zprimes:ok

Lemmas:
cons(gen_0':mark:zprimes:ok3_0(+(1, n5_0)), gen_0':mark:zprimes:ok3_0(b)) → *4_0, rt ∈ Ω(n5₀)
filter(gen_0':mark:zprimes:ok3_0(+(1, n950_0)), gen_0':mark:zprimes:ok3_0(b), gen_0':mark:zprimes:ok3_0(c)) → *4_0, rt ∈ Ω(n950₀)
sieve(gen_0':mark:zprimes:ok3_0(+(1, n3774_0))) → *4_0, rt ∈ Ω(n3774₀)
s(gen_0':mark:zprimes:ok3_0(+(1, n4492_0))) → *4_0, rt ∈ Ω(n4492₀)
nats(gen_0':mark:zprimes:ok3_0(+(1, n5311_0))) → *4_0, rt ∈ Ω(n5311₀)

Generator Equations:
gen_0':mark:zprimes:ok3_0(0) ⇔ 0'
gen_0':mark:zprimes:ok3_0(+(x, 1)) ⇔ mark(gen_0':mark:zprimes:ok3_0(x))

No more defined symbols left to analyse.

(33) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
cons(gen_0':mark:zprimes:ok3_0(+(1, n5_0)), gen_0':mark:zprimes:ok3_0(b)) → *4_0, rt ∈ Ω(n5₀)

(34) BOUNDS(n^1, INF)

(35) Obligation:

TRS:
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:ok1_0 :: 0':mark:zprimes:ok
hole_top2_0 :: top
gen_0':mark:zprimes:ok3_0 :: Nat → 0':mark:zprimes:ok

Lemmas:
cons(gen_0':mark:zprimes:ok3_0(+(1, n5_0)), gen_0':mark:zprimes:ok3_0(b)) → *4_0, rt ∈ Ω(n5₀)
filter(gen_0':mark:zprimes:ok3_0(+(1, n950_0)), gen_0':mark:zprimes:ok3_0(b), gen_0':mark:zprimes:ok3_0(c)) → *4_0, rt ∈ Ω(n950₀)
sieve(gen_0':mark:zprimes:ok3_0(+(1, n3774_0))) → *4_0, rt ∈ Ω(n3774₀)
s(gen_0':mark:zprimes:ok3_0(+(1, n4492_0))) → *4_0, rt ∈ Ω(n4492₀)

Generator Equations:
gen_0':mark:zprimes:ok3_0(0) ⇔ 0'
gen_0':mark:zprimes:ok3_0(+(x, 1)) ⇔ mark(gen_0':mark:zprimes:ok3_0(x))

No more defined symbols left to analyse.

(36) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
cons(gen_0':mark:zprimes:ok3_0(+(1, n5_0)), gen_0':mark:zprimes:ok3_0(b)) → *4_0, rt ∈ Ω(n5₀)

(37) BOUNDS(n^1, INF)

(38) Obligation:

TRS:
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:ok1_0 :: 0':mark:zprimes:ok
hole_top2_0 :: top
gen_0':mark:zprimes:ok3_0 :: Nat → 0':mark:zprimes:ok

Lemmas:
cons(gen_0':mark:zprimes:ok3_0(+(1, n5_0)), gen_0':mark:zprimes:ok3_0(b)) → *4_0, rt ∈ Ω(n5₀)
filter(gen_0':mark:zprimes:ok3_0(+(1, n950_0)), gen_0':mark:zprimes:ok3_0(b), gen_0':mark:zprimes:ok3_0(c)) → *4_0, rt ∈ Ω(n950₀)
sieve(gen_0':mark:zprimes:ok3_0(+(1, n3774_0))) → *4_0, rt ∈ Ω(n3774₀)

Generator Equations:
gen_0':mark:zprimes:ok3_0(0) ⇔ 0'
gen_0':mark:zprimes:ok3_0(+(x, 1)) ⇔ mark(gen_0':mark:zprimes:ok3_0(x))

No more defined symbols left to analyse.

(39) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
cons(gen_0':mark:zprimes:ok3_0(+(1, n5_0)), gen_0':mark:zprimes:ok3_0(b)) → *4_0, rt ∈ Ω(n5₀)

(40) BOUNDS(n^1, INF)

(41) Obligation:

TRS:
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:ok1_0 :: 0':mark:zprimes:ok
hole_top2_0 :: top
gen_0':mark:zprimes:ok3_0 :: Nat → 0':mark:zprimes:ok

Lemmas:
cons(gen_0':mark:zprimes:ok3_0(+(1, n5_0)), gen_0':mark:zprimes:ok3_0(b)) → *4_0, rt ∈ Ω(n5₀)
filter(gen_0':mark:zprimes:ok3_0(+(1, n950_0)), gen_0':mark:zprimes:ok3_0(b), gen_0':mark:zprimes:ok3_0(c)) → *4_0, rt ∈ Ω(n950₀)

Generator Equations:
gen_0':mark:zprimes:ok3_0(0) ⇔ 0'
gen_0':mark:zprimes:ok3_0(+(x, 1)) ⇔ mark(gen_0':mark:zprimes:ok3_0(x))

No more defined symbols left to analyse.

(42) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
cons(gen_0':mark:zprimes:ok3_0(+(1, n5_0)), gen_0':mark:zprimes:ok3_0(b)) → *4_0, rt ∈ Ω(n5₀)

(43) BOUNDS(n^1, INF)

(44) Obligation:

TRS:
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:ok1_0 :: 0':mark:zprimes:ok
hole_top2_0 :: top
gen_0':mark:zprimes:ok3_0 :: Nat → 0':mark:zprimes:ok

Lemmas:
cons(gen_0':mark:zprimes:ok3_0(+(1, n5_0)), gen_0':mark:zprimes:ok3_0(b)) → *4_0, rt ∈ Ω(n5₀)

Generator Equations:
gen_0':mark:zprimes:ok3_0(0) ⇔ 0'
gen_0':mark:zprimes:ok3_0(+(x, 1)) ⇔ mark(gen_0':mark:zprimes:ok3_0(x))

No more defined symbols left to analyse.

(45) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
cons(gen_0':mark:zprimes:ok3_0(+(1, n5_0)), gen_0':mark:zprimes:ok3_0(b)) → *4_0, rt ∈ Ω(n5₀)

(46) BOUNDS(n^1, INF)