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

active(eq(0, 0)) → mark(true)
active(eq(s(X), s(Y))) → mark(eq(X, Y))
active(eq(X, Y)) → mark(false)
active(inf(X)) → mark(cons(X, inf(s(X))))
active(take(0, X)) → mark(nil)
active(take(s(X), cons(Y, L))) → mark(cons(Y, take(X, L)))
active(length(nil)) → mark(0)
active(length(cons(X, L))) → mark(s(length(L)))
active(inf(X)) → inf(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(length(X)) → length(active(X))
inf(mark(X)) → mark(inf(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
length(mark(X)) → mark(length(X))
proper(eq(X1, X2)) → eq(proper(X1), proper(X2))
proper(0) → ok(0)
proper(true) → ok(true)
proper(s(X)) → s(proper(X))
proper(false) → ok(false)
proper(inf(X)) → inf(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(length(X)) → length(proper(X))
eq(ok(X1), ok(X2)) → ok(eq(X1, X2))
s(ok(X)) → ok(s(X))
inf(ok(X)) → ok(inf(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
length(ok(X)) → ok(length(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Rewrite Strategy: INNERMOST

Renamed function symbols to avoid clashes with predefined symbol.

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

active'(eq'(0', 0')) → mark'(true')
active'(eq'(s'(X), s'(Y))) → mark'(eq'(X, Y))
active'(eq'(X, Y)) → mark'(false')
active'(inf'(X)) → mark'(cons'(X, inf'(s'(X))))
active'(take'(0', X)) → mark'(nil')
active'(take'(s'(X), cons'(Y, L))) → mark'(cons'(Y, take'(X, L)))
active'(length'(nil')) → mark'(0')
active'(length'(cons'(X, L))) → mark'(s'(length'(L)))
active'(inf'(X)) → inf'(active'(X))
active'(take'(X1, X2)) → take'(active'(X1), X2)
active'(take'(X1, X2)) → take'(X1, active'(X2))
active'(length'(X)) → length'(active'(X))
inf'(mark'(X)) → mark'(inf'(X))
take'(mark'(X1), X2) → mark'(take'(X1, X2))
take'(X1, mark'(X2)) → mark'(take'(X1, X2))
length'(mark'(X)) → mark'(length'(X))
proper'(eq'(X1, X2)) → eq'(proper'(X1), proper'(X2))
proper'(0') → ok'(0')
proper'(true') → ok'(true')
proper'(s'(X)) → s'(proper'(X))
proper'(false') → ok'(false')
proper'(inf'(X)) → inf'(proper'(X))
proper'(cons'(X1, X2)) → cons'(proper'(X1), proper'(X2))
proper'(take'(X1, X2)) → take'(proper'(X1), proper'(X2))
proper'(nil') → ok'(nil')
proper'(length'(X)) → length'(proper'(X))
eq'(ok'(X1), ok'(X2)) → ok'(eq'(X1, X2))
s'(ok'(X)) → ok'(s'(X))
inf'(ok'(X)) → ok'(inf'(X))
cons'(ok'(X1), ok'(X2)) → ok'(cons'(X1, X2))
take'(ok'(X1), ok'(X2)) → ok'(take'(X1, X2))
length'(ok'(X)) → ok'(length'(X))
top'(mark'(X)) → top'(proper'(X))
top'(ok'(X)) → top'(active'(X))

Rewrite Strategy: INNERMOST

Infered types.

Rules:
active'(eq'(0', 0')) → mark'(true')
active'(eq'(s'(X), s'(Y))) → mark'(eq'(X, Y))
active'(eq'(X, Y)) → mark'(false')
active'(inf'(X)) → mark'(cons'(X, inf'(s'(X))))
active'(take'(0', X)) → mark'(nil')
active'(take'(s'(X), cons'(Y, L))) → mark'(cons'(Y, take'(X, L)))
active'(length'(nil')) → mark'(0')
active'(length'(cons'(X, L))) → mark'(s'(length'(L)))
active'(inf'(X)) → inf'(active'(X))
active'(take'(X1, X2)) → take'(active'(X1), X2)
active'(take'(X1, X2)) → take'(X1, active'(X2))
active'(length'(X)) → length'(active'(X))
inf'(mark'(X)) → mark'(inf'(X))
take'(mark'(X1), X2) → mark'(take'(X1, X2))
take'(X1, mark'(X2)) → mark'(take'(X1, X2))
length'(mark'(X)) → mark'(length'(X))
proper'(eq'(X1, X2)) → eq'(proper'(X1), proper'(X2))
proper'(0') → ok'(0')
proper'(true') → ok'(true')
proper'(s'(X)) → s'(proper'(X))
proper'(false') → ok'(false')
proper'(inf'(X)) → inf'(proper'(X))
proper'(cons'(X1, X2)) → cons'(proper'(X1), proper'(X2))
proper'(take'(X1, X2)) → take'(proper'(X1), proper'(X2))
proper'(nil') → ok'(nil')
proper'(length'(X)) → length'(proper'(X))
eq'(ok'(X1), ok'(X2)) → ok'(eq'(X1, X2))
s'(ok'(X)) → ok'(s'(X))
inf'(ok'(X)) → ok'(inf'(X))
cons'(ok'(X1), ok'(X2)) → ok'(cons'(X1, X2))
take'(ok'(X1), ok'(X2)) → ok'(take'(X1, X2))
length'(ok'(X)) → ok'(length'(X))
top'(mark'(X)) → top'(proper'(X))
top'(ok'(X)) → top'(active'(X))

Types:
active' :: 0':true':mark':false':nil':ok' → 0':true':mark':false':nil':ok'
eq' :: 0':true':mark':false':nil':ok' → 0':true':mark':false':nil':ok' → 0':true':mark':false':nil':ok'
0' :: 0':true':mark':false':nil':ok'
mark' :: 0':true':mark':false':nil':ok' → 0':true':mark':false':nil':ok'
true' :: 0':true':mark':false':nil':ok'
s' :: 0':true':mark':false':nil':ok' → 0':true':mark':false':nil':ok'
false' :: 0':true':mark':false':nil':ok'
inf' :: 0':true':mark':false':nil':ok' → 0':true':mark':false':nil':ok'
cons' :: 0':true':mark':false':nil':ok' → 0':true':mark':false':nil':ok' → 0':true':mark':false':nil':ok'
take' :: 0':true':mark':false':nil':ok' → 0':true':mark':false':nil':ok' → 0':true':mark':false':nil':ok'
nil' :: 0':true':mark':false':nil':ok'
length' :: 0':true':mark':false':nil':ok' → 0':true':mark':false':nil':ok'
proper' :: 0':true':mark':false':nil':ok' → 0':true':mark':false':nil':ok'
ok' :: 0':true':mark':false':nil':ok' → 0':true':mark':false':nil':ok'
top' :: 0':true':mark':false':nil':ok' → top'
_hole_0':true':mark':false':nil':ok'1 :: 0':true':mark':false':nil':ok'
_hole_top'2 :: top'
_gen_0':true':mark':false':nil':ok'3 :: Nat → 0':true':mark':false':nil':ok'

Heuristically decided to analyse the following defined symbols:
active', eq', cons', inf', s', take', length', proper', top'

They will be analysed ascendingly in the following order:
eq' < active'
cons' < active'
inf' < active'
s' < active'
take' < active'
length' < active'
active' < top'
eq' < proper'
cons' < proper'
inf' < proper'
s' < proper'
take' < proper'
length' < proper'
proper' < top'

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

The following defined symbols remain to be analysed:
eq', active', cons', inf', s', take', length', proper', top'

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

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

The following defined symbols remain to be analysed:
cons', active', inf', s', take', length', proper', top'

They will be analysed ascendingly in the following order:
cons' < active'
inf' < active'
s' < active'
take' < active'
length' < active'
active' < top'
cons' < proper'
inf' < proper'
s' < proper'
take' < proper'
length' < proper'
proper' < top'

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

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

The following defined symbols remain to be analysed:
inf', active', s', take', length', proper', top'

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

Proved the following rewrite lemma:
inf'(_gen_0':true':mark':false':nil':ok'3(+(1, _n29))) → _*4, rt ∈ Ω(_n29)

Induction Base:
inf'(_gen_0':true':mark':false':nil':ok'3(+(1, 0)))

Induction Step:
inf'(_gen_0':true':mark':false':nil':ok'3(+(1, +(_$n30, 1)))) →_R^Ω(1)
mark'(inf'(_gen_0':true':mark':false':nil':ok'3(+(1, _$n30)))) →_IH
mark'(_*4)

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

Lemmas:
inf'(_gen_0':true':mark':false':nil':ok'3(+(1, _n29))) → _*4, rt ∈ Ω(_n29)

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

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

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

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

Lemmas:
inf'(_gen_0':true':mark':false':nil':ok'3(+(1, _n29))) → _*4, rt ∈ Ω(_n29)

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

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

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

Proved the following rewrite lemma:
take'(_gen_0':true':mark':false':nil':ok'3(+(1, _n1198)), _gen_0':true':mark':false':nil':ok'3(b)) → _*4, rt ∈ Ω(_n1198)

Induction Base:
take'(_gen_0':true':mark':false':nil':ok'3(+(1, 0)), _gen_0':true':mark':false':nil':ok'3(b))

Induction Step:
take'(_gen_0':true':mark':false':nil':ok'3(+(1, +(_$n1199, 1))), _gen_0':true':mark':false':nil':ok'3(_b2235)) →_R^Ω(1)
mark'(take'(_gen_0':true':mark':false':nil':ok'3(+(1, _$n1199)), _gen_0':true':mark':false':nil':ok'3(_b2235))) →_IH
mark'(_*4)

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

Lemmas:
inf'(_gen_0':true':mark':false':nil':ok'3(+(1, _n29))) → _*4, rt ∈ Ω(_n29)
take'(_gen_0':true':mark':false':nil':ok'3(+(1, _n1198)), _gen_0':true':mark':false':nil':ok'3(b)) → _*4, rt ∈ Ω(_n1198)

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

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

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

Proved the following rewrite lemma:
length'(_gen_0':true':mark':false':nil':ok'3(+(1, _n3581))) → _*4, rt ∈ Ω(_n3581)

Induction Base:
length'(_gen_0':true':mark':false':nil':ok'3(+(1, 0)))

Induction Step:
length'(_gen_0':true':mark':false':nil':ok'3(+(1, +(_$n3582, 1)))) →_R^Ω(1)
mark'(length'(_gen_0':true':mark':false':nil':ok'3(+(1, _$n3582)))) →_IH
mark'(_*4)

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

Generator Equations:
_gen_0':true':mark':false':nil':ok'3(0) ⇔ 0'
_gen_0':true':mark':false':nil':ok'3(+(x, 1)) ⇔ mark'(_gen_0':true':mark':false':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_0':true':mark':false':nil':ok'3(0) ⇔ 0'
_gen_0':true':mark':false':nil':ok'3(+(x, 1)) ⇔ mark'(_gen_0':true':mark':false':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_0':true':mark':false':nil':ok'3(0) ⇔ 0'
_gen_0':true':mark':false':nil':ok'3(+(x, 1)) ⇔ mark'(_gen_0':true':mark':false':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_0':true':mark':false':nil':ok'3(0) ⇔ 0'
_gen_0':true':mark':false':nil':ok'3(+(x, 1)) ⇔ mark'(_gen_0':true':mark':false':nil':ok'3(x))

No more defined symbols left to analyse.

The lowerbound Ω(n) was proven with the following lemma:
inf'(_gen_0':true':mark':false':nil':ok'3(+(1, _n29))) → _*4, rt ∈ Ω(_n29)