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) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

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

(10) 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:
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

(11) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

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

(12) 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:
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

(13) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

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

(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

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

(15) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

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

(16) 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:
nats, active, proper, top

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

(17) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

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

(18) 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:
active, proper, top

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

(19) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

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

(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

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

(21) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

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

(22) 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:
top

(23) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

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

(24) 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))

No more defined symbols left to analyse.