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

active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(U12(tt, L))
active(U12(tt, L)) → mark(s(length(L)))
active(U21(tt, IL, M, N)) → mark(U22(tt, IL, M, N))
active(U22(tt, IL, M, N)) → mark(U23(tt, IL, M, N))
active(U23(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U11(tt, L))
active(take(0, IL)) → mark(nil)
active(take(s(M), cons(N, IL))) → mark(U21(tt, IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X1, X2)) → U12(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U21(X1, X2, X3, X4)) → U21(active(X1), X2, X3, X4)
active(U22(X1, X2, X3, X4)) → U22(active(X1), X2, X3, X4)
active(U23(X1, X2, X3, X4)) → U23(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X1), X2) → mark(U12(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U21(mark(X1), X2, X3, X4) → mark(U21(X1, X2, X3, X4))
U22(mark(X1), X2, X3, X4) → mark(U22(X1, X2, X3, X4))
U23(mark(X1), X2, X3, X4) → mark(U23(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U21(X1, X2, X3, X4)) → U21(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U22(X1, X2, X3, X4)) → U22(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U23(X1, X2, X3, X4)) → U23(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U21(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U21(X1, X2, X3, X4))
U22(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U22(X1, X2, X3, X4))
U23(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U23(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
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'(zeros') → mark'(cons'(0', zeros'))
active'(U11'(tt', L)) → mark'(U12'(tt', L))
active'(U12'(tt', L)) → mark'(s'(length'(L)))
active'(U21'(tt', IL, M, N)) → mark'(U22'(tt', IL, M, N))
active'(U22'(tt', IL, M, N)) → mark'(U23'(tt', IL, M, N))
active'(U23'(tt', IL, M, N)) → mark'(cons'(N, take'(M, IL)))
active'(length'(nil')) → mark'(0')
active'(length'(cons'(N, L))) → mark'(U11'(tt', L))
active'(take'(0', IL)) → mark'(nil')
active'(take'(s'(M), cons'(N, IL))) → mark'(U21'(tt', IL, M, N))
active'(cons'(X1, X2)) → cons'(active'(X1), X2)
active'(U11'(X1, X2)) → U11'(active'(X1), X2)
active'(U12'(X1, X2)) → U12'(active'(X1), X2)
active'(s'(X)) → s'(active'(X))
active'(length'(X)) → length'(active'(X))
active'(U21'(X1, X2, X3, X4)) → U21'(active'(X1), X2, X3, X4)
active'(U22'(X1, X2, X3, X4)) → U22'(active'(X1), X2, X3, X4)
active'(U23'(X1, X2, X3, X4)) → U23'(active'(X1), X2, X3, X4)
active'(take'(X1, X2)) → take'(active'(X1), X2)
active'(take'(X1, X2)) → take'(X1, active'(X2))
cons'(mark'(X1), X2) → mark'(cons'(X1, X2))
U11'(mark'(X1), X2) → mark'(U11'(X1, X2))
U12'(mark'(X1), X2) → mark'(U12'(X1, X2))
s'(mark'(X)) → mark'(s'(X))
length'(mark'(X)) → mark'(length'(X))
U21'(mark'(X1), X2, X3, X4) → mark'(U21'(X1, X2, X3, X4))
U22'(mark'(X1), X2, X3, X4) → mark'(U22'(X1, X2, X3, X4))
U23'(mark'(X1), X2, X3, X4) → mark'(U23'(X1, X2, X3, X4))
take'(mark'(X1), X2) → mark'(take'(X1, X2))
take'(X1, mark'(X2)) → mark'(take'(X1, X2))
proper'(zeros') → ok'(zeros')
proper'(cons'(X1, X2)) → cons'(proper'(X1), proper'(X2))
proper'(0') → ok'(0')
proper'(U11'(X1, X2)) → U11'(proper'(X1), proper'(X2))
proper'(tt') → ok'(tt')
proper'(U12'(X1, X2)) → U12'(proper'(X1), proper'(X2))
proper'(s'(X)) → s'(proper'(X))
proper'(length'(X)) → length'(proper'(X))
proper'(U21'(X1, X2, X3, X4)) → U21'(proper'(X1), proper'(X2), proper'(X3), proper'(X4))
proper'(U22'(X1, X2, X3, X4)) → U22'(proper'(X1), proper'(X2), proper'(X3), proper'(X4))
proper'(U23'(X1, X2, X3, X4)) → U23'(proper'(X1), proper'(X2), proper'(X3), proper'(X4))
proper'(take'(X1, X2)) → take'(proper'(X1), proper'(X2))
proper'(nil') → ok'(nil')
cons'(ok'(X1), ok'(X2)) → ok'(cons'(X1, X2))
U11'(ok'(X1), ok'(X2)) → ok'(U11'(X1, X2))
U12'(ok'(X1), ok'(X2)) → ok'(U12'(X1, X2))
s'(ok'(X)) → ok'(s'(X))
length'(ok'(X)) → ok'(length'(X))
U21'(ok'(X1), ok'(X2), ok'(X3), ok'(X4)) → ok'(U21'(X1, X2, X3, X4))
U22'(ok'(X1), ok'(X2), ok'(X3), ok'(X4)) → ok'(U22'(X1, X2, X3, X4))
U23'(ok'(X1), ok'(X2), ok'(X3), ok'(X4)) → ok'(U23'(X1, X2, X3, X4))
take'(ok'(X1), ok'(X2)) → ok'(take'(X1, X2))
top'(mark'(X)) → top'(proper'(X))
top'(ok'(X)) → top'(active'(X))

Rewrite Strategy: INNERMOST

Infered types.

Rules:
active'(zeros') → mark'(cons'(0', zeros'))
active'(U11'(tt', L)) → mark'(U12'(tt', L))
active'(U12'(tt', L)) → mark'(s'(length'(L)))
active'(U21'(tt', IL, M, N)) → mark'(U22'(tt', IL, M, N))
active'(U22'(tt', IL, M, N)) → mark'(U23'(tt', IL, M, N))
active'(U23'(tt', IL, M, N)) → mark'(cons'(N, take'(M, IL)))
active'(length'(nil')) → mark'(0')
active'(length'(cons'(N, L))) → mark'(U11'(tt', L))
active'(take'(0', IL)) → mark'(nil')
active'(take'(s'(M), cons'(N, IL))) → mark'(U21'(tt', IL, M, N))
active'(cons'(X1, X2)) → cons'(active'(X1), X2)
active'(U11'(X1, X2)) → U11'(active'(X1), X2)
active'(U12'(X1, X2)) → U12'(active'(X1), X2)
active'(s'(X)) → s'(active'(X))
active'(length'(X)) → length'(active'(X))
active'(U21'(X1, X2, X3, X4)) → U21'(active'(X1), X2, X3, X4)
active'(U22'(X1, X2, X3, X4)) → U22'(active'(X1), X2, X3, X4)
active'(U23'(X1, X2, X3, X4)) → U23'(active'(X1), X2, X3, X4)
active'(take'(X1, X2)) → take'(active'(X1), X2)
active'(take'(X1, X2)) → take'(X1, active'(X2))
cons'(mark'(X1), X2) → mark'(cons'(X1, X2))
U11'(mark'(X1), X2) → mark'(U11'(X1, X2))
U12'(mark'(X1), X2) → mark'(U12'(X1, X2))
s'(mark'(X)) → mark'(s'(X))
length'(mark'(X)) → mark'(length'(X))
U21'(mark'(X1), X2, X3, X4) → mark'(U21'(X1, X2, X3, X4))
U22'(mark'(X1), X2, X3, X4) → mark'(U22'(X1, X2, X3, X4))
U23'(mark'(X1), X2, X3, X4) → mark'(U23'(X1, X2, X3, X4))
take'(mark'(X1), X2) → mark'(take'(X1, X2))
take'(X1, mark'(X2)) → mark'(take'(X1, X2))
proper'(zeros') → ok'(zeros')
proper'(cons'(X1, X2)) → cons'(proper'(X1), proper'(X2))
proper'(0') → ok'(0')
proper'(U11'(X1, X2)) → U11'(proper'(X1), proper'(X2))
proper'(tt') → ok'(tt')
proper'(U12'(X1, X2)) → U12'(proper'(X1), proper'(X2))
proper'(s'(X)) → s'(proper'(X))
proper'(length'(X)) → length'(proper'(X))
proper'(U21'(X1, X2, X3, X4)) → U21'(proper'(X1), proper'(X2), proper'(X3), proper'(X4))
proper'(U22'(X1, X2, X3, X4)) → U22'(proper'(X1), proper'(X2), proper'(X3), proper'(X4))
proper'(U23'(X1, X2, X3, X4)) → U23'(proper'(X1), proper'(X2), proper'(X3), proper'(X4))
proper'(take'(X1, X2)) → take'(proper'(X1), proper'(X2))
proper'(nil') → ok'(nil')
cons'(ok'(X1), ok'(X2)) → ok'(cons'(X1, X2))
U11'(ok'(X1), ok'(X2)) → ok'(U11'(X1, X2))
U12'(ok'(X1), ok'(X2)) → ok'(U12'(X1, X2))
s'(ok'(X)) → ok'(s'(X))
length'(ok'(X)) → ok'(length'(X))
U21'(ok'(X1), ok'(X2), ok'(X3), ok'(X4)) → ok'(U21'(X1, X2, X3, X4))
U22'(ok'(X1), ok'(X2), ok'(X3), ok'(X4)) → ok'(U22'(X1, X2, X3, X4))
U23'(ok'(X1), ok'(X2), ok'(X3), ok'(X4)) → ok'(U23'(X1, X2, X3, X4))
take'(ok'(X1), ok'(X2)) → ok'(take'(X1, X2))
top'(mark'(X)) → top'(proper'(X))
top'(ok'(X)) → top'(active'(X))

Types:
active' :: zeros':0':mark':tt':nil':ok' → zeros':0':mark':tt':nil':ok'
zeros' :: zeros':0':mark':tt':nil':ok'
mark' :: zeros':0':mark':tt':nil':ok' → zeros':0':mark':tt':nil':ok'
cons' :: zeros':0':mark':tt':nil':ok' → zeros':0':mark':tt':nil':ok' → zeros':0':mark':tt':nil':ok'
0' :: zeros':0':mark':tt':nil':ok'
U11' :: zeros':0':mark':tt':nil':ok' → zeros':0':mark':tt':nil':ok' → zeros':0':mark':tt':nil':ok'
tt' :: zeros':0':mark':tt':nil':ok'
U12' :: zeros':0':mark':tt':nil':ok' → zeros':0':mark':tt':nil':ok' → zeros':0':mark':tt':nil':ok'
s' :: zeros':0':mark':tt':nil':ok' → zeros':0':mark':tt':nil':ok'
length' :: zeros':0':mark':tt':nil':ok' → zeros':0':mark':tt':nil':ok'
U21' :: zeros':0':mark':tt':nil':ok' → zeros':0':mark':tt':nil':ok' → zeros':0':mark':tt':nil':ok' → zeros':0':mark':tt':nil':ok' → zeros':0':mark':tt':nil':ok'
U22' :: zeros':0':mark':tt':nil':ok' → zeros':0':mark':tt':nil':ok' → zeros':0':mark':tt':nil':ok' → zeros':0':mark':tt':nil':ok' → zeros':0':mark':tt':nil':ok'
U23' :: zeros':0':mark':tt':nil':ok' → zeros':0':mark':tt':nil':ok' → zeros':0':mark':tt':nil':ok' → zeros':0':mark':tt':nil':ok' → zeros':0':mark':tt':nil':ok'
take' :: zeros':0':mark':tt':nil':ok' → zeros':0':mark':tt':nil':ok' → zeros':0':mark':tt':nil':ok'
nil' :: zeros':0':mark':tt':nil':ok'
proper' :: zeros':0':mark':tt':nil':ok' → zeros':0':mark':tt':nil':ok'
ok' :: zeros':0':mark':tt':nil':ok' → zeros':0':mark':tt':nil':ok'
top' :: zeros':0':mark':tt':nil':ok' → top'
_hole_zeros':0':mark':tt':nil':ok'1 :: zeros':0':mark':tt':nil':ok'
_hole_top'2 :: top'
_gen_zeros':0':mark':tt':nil':ok'3 :: Nat → zeros':0':mark':tt':nil':ok'

Heuristically decided to analyse the following defined symbols:
active', cons', U12', s', length', U22', U23', take', U11', U21', proper', top'

They will be analysed ascendingly in the following order:
cons' < active'
U12' < active'
s' < active'
length' < active'
U22' < active'
U23' < active'
take' < active'
U11' < active'
U21' < active'
active' < top'
cons' < proper'
U12' < proper'
s' < proper'
length' < proper'
U22' < proper'
U23' < proper'
take' < proper'
U11' < proper'
U21' < proper'
proper' < top'

Generator Equations:
_gen_zeros':0':mark':tt':nil':ok'3(0) ⇔ zeros'
_gen_zeros':0':mark':tt':nil':ok'3(+(x, 1)) ⇔ mark'(_gen_zeros':0':mark':tt':nil':ok'3(x))

The following defined symbols remain to be analysed:
cons', active', U12', s', length', U22', U23', take', U11', U21', proper', top'

Proved the following rewrite lemma:
cons'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _n5)), _gen_zeros':0':mark':tt':nil':ok'3(b)) → _*4, rt ∈ Ω(_n5)

Induction Base:
cons'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, 0)), _gen_zeros':0':mark':tt':nil':ok'3(b))

Induction Step:
cons'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, +(_$n6, 1))), _gen_zeros':0':mark':tt':nil':ok'3(_b610)) →_R^Ω(1)
mark'(cons'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _$n6)), _gen_zeros':0':mark':tt':nil':ok'3(_b610))) →_IH
mark'(_*4)

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

Lemmas:
cons'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _n5)), _gen_zeros':0':mark':tt':nil':ok'3(b)) → _*4, rt ∈ Ω(_n5)

Generator Equations:
_gen_zeros':0':mark':tt':nil':ok'3(0) ⇔ zeros'
_gen_zeros':0':mark':tt':nil':ok'3(+(x, 1)) ⇔ mark'(_gen_zeros':0':mark':tt':nil':ok'3(x))

The following defined symbols remain to be analysed:
U12', active', s', length', U22', U23', take', U11', U21', proper', top'

They will be analysed ascendingly in the following order:
U12' < active'
s' < active'
length' < active'
U22' < active'
U23' < active'
take' < active'
U11' < active'
U21' < active'
active' < top'
U12' < proper'
s' < proper'
length' < proper'
U22' < proper'
U23' < proper'
take' < proper'
U11' < proper'
U21' < proper'
proper' < top'

Proved the following rewrite lemma:
U12'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _n3596)), _gen_zeros':0':mark':tt':nil':ok'3(b)) → _*4, rt ∈ Ω(_n3596)

Induction Base:
U12'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, 0)), _gen_zeros':0':mark':tt':nil':ok'3(b))

Induction Step:
U12'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, +(_$n3597, 1))), _gen_zeros':0':mark':tt':nil':ok'3(_b4525)) →_R^Ω(1)
mark'(U12'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _$n3597)), _gen_zeros':0':mark':tt':nil':ok'3(_b4525))) →_IH
mark'(_*4)

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

Generator Equations:
_gen_zeros':0':mark':tt':nil':ok'3(0) ⇔ zeros'
_gen_zeros':0':mark':tt':nil':ok'3(+(x, 1)) ⇔ mark'(_gen_zeros':0':mark':tt':nil':ok'3(x))

The following defined symbols remain to be analysed:
s', active', length', U22', U23', take', U11', U21', proper', top'

They will be analysed ascendingly in the following order:
s' < active'
length' < active'
U22' < active'
U23' < active'
take' < active'
U11' < active'
U21' < active'
active' < top'
s' < proper'
length' < proper'
U22' < proper'
U23' < proper'
take' < proper'
U11' < proper'
U21' < proper'
proper' < top'

Proved the following rewrite lemma:
s'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _n7555))) → _*4, rt ∈ Ω(_n7555)

Induction Base:
s'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, 0)))

Induction Step:
s'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, +(_$n7556, 1)))) →_R^Ω(1)
mark'(s'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _$n7556)))) →_IH
mark'(_*4)

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

Generator Equations:
_gen_zeros':0':mark':tt':nil':ok'3(0) ⇔ zeros'
_gen_zeros':0':mark':tt':nil':ok'3(+(x, 1)) ⇔ mark'(_gen_zeros':0':mark':tt':nil':ok'3(x))

The following defined symbols remain to be analysed:
length', active', U22', U23', take', U11', U21', proper', top'

They will be analysed ascendingly in the following order:
length' < active'
U22' < active'
U23' < active'
take' < active'
U11' < active'
U21' < active'
active' < top'
length' < proper'
U22' < proper'
U23' < proper'
take' < proper'
U11' < proper'
U21' < proper'
proper' < top'

Proved the following rewrite lemma:
length'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _n10234))) → _*4, rt ∈ Ω(_n10234)

Induction Base:
length'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, 0)))

Induction Step:
length'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, +(_$n10235, 1)))) →_R^Ω(1)
mark'(length'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _$n10235)))) →_IH
mark'(_*4)

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

Generator Equations:
_gen_zeros':0':mark':tt':nil':ok'3(0) ⇔ zeros'
_gen_zeros':0':mark':tt':nil':ok'3(+(x, 1)) ⇔ mark'(_gen_zeros':0':mark':tt':nil':ok'3(x))

The following defined symbols remain to be analysed:
U22', active', U23', take', U11', U21', proper', top'

They will be analysed ascendingly in the following order:
U22' < active'
U23' < active'
take' < active'
U11' < active'
U21' < active'
active' < top'
U22' < proper'
U23' < proper'
take' < proper'
U11' < proper'
U21' < proper'
proper' < top'

Proved the following rewrite lemma:
U22'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _n13037)), _gen_zeros':0':mark':tt':nil':ok'3(b), _gen_zeros':0':mark':tt':nil':ok'3(c), _gen_zeros':0':mark':tt':nil':ok'3(d)) → _*4, rt ∈ Ω(_n13037)

Induction Base:
U22'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, 0)), _gen_zeros':0':mark':tt':nil':ok'3(b), _gen_zeros':0':mark':tt':nil':ok'3(c), _gen_zeros':0':mark':tt':nil':ok'3(d))

Induction Step:
U22'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, +(_$n13038, 1))), _gen_zeros':0':mark':tt':nil':ok'3(_b16834), _gen_zeros':0':mark':tt':nil':ok'3(_c16835), _gen_zeros':0':mark':tt':nil':ok'3(_d16836)) →_R^Ω(1)
mark'(U22'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _$n13038)), _gen_zeros':0':mark':tt':nil':ok'3(_b16834), _gen_zeros':0':mark':tt':nil':ok'3(_c16835), _gen_zeros':0':mark':tt':nil':ok'3(_d16836))) →_IH
mark'(_*4)

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

Generator Equations:
_gen_zeros':0':mark':tt':nil':ok'3(0) ⇔ zeros'
_gen_zeros':0':mark':tt':nil':ok'3(+(x, 1)) ⇔ mark'(_gen_zeros':0':mark':tt':nil':ok'3(x))

The following defined symbols remain to be analysed:
U23', active', take', U11', U21', proper', top'

They will be analysed ascendingly in the following order:
U23' < active'
take' < active'
U11' < active'
U21' < active'
active' < top'
U23' < proper'
take' < proper'
U11' < proper'
U21' < proper'
proper' < top'

Proved the following rewrite lemma:
U23'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _n22066)), _gen_zeros':0':mark':tt':nil':ok'3(b), _gen_zeros':0':mark':tt':nil':ok'3(c), _gen_zeros':0':mark':tt':nil':ok'3(d)) → _*4, rt ∈ Ω(_n22066)

Induction Base:
U23'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, 0)), _gen_zeros':0':mark':tt':nil':ok'3(b), _gen_zeros':0':mark':tt':nil':ok'3(c), _gen_zeros':0':mark':tt':nil':ok'3(d))

Induction Step:
U23'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, +(_$n22067, 1))), _gen_zeros':0':mark':tt':nil':ok'3(_b26943), _gen_zeros':0':mark':tt':nil':ok'3(_c26944), _gen_zeros':0':mark':tt':nil':ok'3(_d26945)) →_R^Ω(1)
mark'(U23'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _$n22067)), _gen_zeros':0':mark':tt':nil':ok'3(_b26943), _gen_zeros':0':mark':tt':nil':ok'3(_c26944), _gen_zeros':0':mark':tt':nil':ok'3(_d26945))) →_IH
mark'(_*4)

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

Generator Equations:
_gen_zeros':0':mark':tt':nil':ok'3(0) ⇔ zeros'
_gen_zeros':0':mark':tt':nil':ok'3(+(x, 1)) ⇔ mark'(_gen_zeros':0':mark':tt':nil':ok'3(x))

The following defined symbols remain to be analysed:
take', active', U11', U21', proper', top'

They will be analysed ascendingly in the following order:
take' < active'
U11' < active'
U21' < active'
active' < top'
take' < proper'
U11' < proper'
U21' < proper'
proper' < top'

Proved the following rewrite lemma:
take'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _n32311)), _gen_zeros':0':mark':tt':nil':ok'3(b)) → _*4, rt ∈ Ω(_n32311)

Induction Base:
take'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, 0)), _gen_zeros':0':mark':tt':nil':ok'3(b))

Induction Step:
take'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, +(_$n32312, 1))), _gen_zeros':0':mark':tt':nil':ok'3(_b35292)) →_R^Ω(1)
mark'(take'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _$n32312)), _gen_zeros':0':mark':tt':nil':ok'3(_b35292))) →_IH
mark'(_*4)

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

Lemmas:
cons'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _n5)), _gen_zeros':0':mark':tt':nil':ok'3(b)) → _*4, rt ∈ Ω(_n5)
U12'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _n3596)), _gen_zeros':0':mark':tt':nil':ok'3(b)) → _*4, rt ∈ Ω(_n3596)
s'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _n7555))) → _*4, rt ∈ Ω(_n7555)
length'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _n10234))) → _*4, rt ∈ Ω(_n10234)
U22'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _n13037)), _gen_zeros':0':mark':tt':nil':ok'3(b), _gen_zeros':0':mark':tt':nil':ok'3(c), _gen_zeros':0':mark':tt':nil':ok'3(d)) → _*4, rt ∈ Ω(_n13037)
U23'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _n22066)), _gen_zeros':0':mark':tt':nil':ok'3(b), _gen_zeros':0':mark':tt':nil':ok'3(c), _gen_zeros':0':mark':tt':nil':ok'3(d)) → _*4, rt ∈ Ω(_n22066)
take'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _n32311)), _gen_zeros':0':mark':tt':nil':ok'3(b)) → _*4, rt ∈ Ω(_n32311)

Generator Equations:
_gen_zeros':0':mark':tt':nil':ok'3(0) ⇔ zeros'
_gen_zeros':0':mark':tt':nil':ok'3(+(x, 1)) ⇔ mark'(_gen_zeros':0':mark':tt':nil':ok'3(x))

The following defined symbols remain to be analysed:
U11', active', U21', proper', top'

They will be analysed ascendingly in the following order:
U11' < active'
U21' < active'
active' < top'
U11' < proper'
U21' < proper'
proper' < top'

Proved the following rewrite lemma:
U11'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _n38583)), _gen_zeros':0':mark':tt':nil':ok'3(b)) → _*4, rt ∈ Ω(_n38583)

Induction Base:
U11'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, 0)), _gen_zeros':0':mark':tt':nil':ok'3(b))

Induction Step:
U11'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, +(_$n38584, 1))), _gen_zeros':0':mark':tt':nil':ok'3(_b41672)) →_R^Ω(1)
mark'(U11'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _$n38584)), _gen_zeros':0':mark':tt':nil':ok'3(_b41672))) →_IH
mark'(_*4)

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

Generator Equations:
_gen_zeros':0':mark':tt':nil':ok'3(0) ⇔ zeros'
_gen_zeros':0':mark':tt':nil':ok'3(+(x, 1)) ⇔ mark'(_gen_zeros':0':mark':tt':nil':ok'3(x))

The following defined symbols remain to be analysed:
U21', active', proper', top'

They will be analysed ascendingly in the following order:
U21' < active'
active' < top'
U21' < proper'
proper' < top'

Proved the following rewrite lemma:
U21'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _n45004)), _gen_zeros':0':mark':tt':nil':ok'3(b), _gen_zeros':0':mark':tt':nil':ok'3(c), _gen_zeros':0':mark':tt':nil':ok'3(d)) → _*4, rt ∈ Ω(_n45004)

Induction Base:
U21'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, 0)), _gen_zeros':0':mark':tt':nil':ok'3(b), _gen_zeros':0':mark':tt':nil':ok'3(c), _gen_zeros':0':mark':tt':nil':ok'3(d))

Induction Step:
U21'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, +(_$n45005, 1))), _gen_zeros':0':mark':tt':nil':ok'3(_b52257), _gen_zeros':0':mark':tt':nil':ok'3(_c52258), _gen_zeros':0':mark':tt':nil':ok'3(_d52259)) →_R^Ω(1)
mark'(U21'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _$n45005)), _gen_zeros':0':mark':tt':nil':ok'3(_b52257), _gen_zeros':0':mark':tt':nil':ok'3(_c52258), _gen_zeros':0':mark':tt':nil':ok'3(_d52259))) →_IH
mark'(_*4)

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

Generator Equations:
_gen_zeros':0':mark':tt':nil':ok'3(0) ⇔ zeros'
_gen_zeros':0':mark':tt':nil':ok'3(+(x, 1)) ⇔ mark'(_gen_zeros':0':mark':tt':nil':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'.

Generator Equations:
_gen_zeros':0':mark':tt':nil':ok'3(0) ⇔ zeros'
_gen_zeros':0':mark':tt':nil':ok'3(+(x, 1)) ⇔ mark'(_gen_zeros':0':mark':tt':nil':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'.

Generator Equations:
_gen_zeros':0':mark':tt':nil':ok'3(0) ⇔ zeros'
_gen_zeros':0':mark':tt':nil':ok'3(+(x, 1)) ⇔ mark'(_gen_zeros':0':mark':tt':nil':ok'3(x))

The following defined symbols remain to be analysed:
top'

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

Generator Equations:
_gen_zeros':0':mark':tt':nil':ok'3(0) ⇔ zeros'
_gen_zeros':0':mark':tt':nil':ok'3(+(x, 1)) ⇔ mark'(_gen_zeros':0':mark':tt':nil':ok'3(x))

No more defined symbols left to analyse.

The lowerbound Ω(n) was proven with the following lemma:
cons'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _n5)), _gen_zeros':0':mark':tt':nil':ok'3(b)) → _*4, rt ∈ Ω(_n5)