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

active(from(X)) → mark(cons(X, from(s(X))))
active(sel(0, cons(X, XS))) → mark(X)
active(sel(s(N), cons(X, XS))) → mark(sel(N, XS))
active(minus(X, 0)) → mark(0)
active(minus(s(X), s(Y))) → mark(minus(X, Y))
active(quot(0, s(Y))) → mark(0)
active(quot(s(X), s(Y))) → mark(s(quot(minus(X, Y), s(Y))))
active(zWquot(XS, nil)) → mark(nil)
active(zWquot(nil, XS)) → mark(nil)
active(zWquot(cons(X, XS), cons(Y, YS))) → mark(cons(quot(X, Y), zWquot(XS, YS)))
active(from(X)) → from(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(s(X)) → s(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(minus(X1, X2)) → minus(active(X1), X2)
active(minus(X1, X2)) → minus(X1, active(X2))
active(quot(X1, X2)) → quot(active(X1), X2)
active(quot(X1, X2)) → quot(X1, active(X2))
active(zWquot(X1, X2)) → zWquot(active(X1), X2)
active(zWquot(X1, X2)) → zWquot(X1, active(X2))
from(mark(X)) → mark(from(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
s(mark(X)) → mark(s(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
minus(mark(X1), X2) → mark(minus(X1, X2))
minus(X1, mark(X2)) → mark(minus(X1, X2))
quot(mark(X1), X2) → mark(quot(X1, X2))
quot(X1, mark(X2)) → mark(quot(X1, X2))
zWquot(mark(X1), X2) → mark(zWquot(X1, X2))
zWquot(X1, mark(X2)) → mark(zWquot(X1, X2))
proper(from(X)) → from(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(0) → ok(0)
proper(minus(X1, X2)) → minus(proper(X1), proper(X2))
proper(quot(X1, X2)) → quot(proper(X1), proper(X2))
proper(zWquot(X1, X2)) → zWquot(proper(X1), proper(X2))
proper(nil) → ok(nil)
from(ok(X)) → ok(from(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
s(ok(X)) → ok(s(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
minus(ok(X1), ok(X2)) → ok(minus(X1, X2))
quot(ok(X1), ok(X2)) → ok(quot(X1, X2))
zWquot(ok(X1), ok(X2)) → ok(zWquot(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'(from'(X)) → mark'(cons'(X, from'(s'(X))))
active'(sel'(0', cons'(X, XS))) → mark'(X)
active'(sel'(s'(N), cons'(X, XS))) → mark'(sel'(N, XS))
active'(minus'(X, 0')) → mark'(0')
active'(minus'(s'(X), s'(Y))) → mark'(minus'(X, Y))
active'(quot'(0', s'(Y))) → mark'(0')
active'(quot'(s'(X), s'(Y))) → mark'(s'(quot'(minus'(X, Y), s'(Y))))
active'(zWquot'(XS, nil')) → mark'(nil')
active'(zWquot'(nil', XS)) → mark'(nil')
active'(zWquot'(cons'(X, XS), cons'(Y, YS))) → mark'(cons'(quot'(X, Y), zWquot'(XS, YS)))
active'(from'(X)) → from'(active'(X))
active'(cons'(X1, X2)) → cons'(active'(X1), X2)
active'(s'(X)) → s'(active'(X))
active'(sel'(X1, X2)) → sel'(active'(X1), X2)
active'(sel'(X1, X2)) → sel'(X1, active'(X2))
active'(minus'(X1, X2)) → minus'(active'(X1), X2)
active'(minus'(X1, X2)) → minus'(X1, active'(X2))
active'(quot'(X1, X2)) → quot'(active'(X1), X2)
active'(quot'(X1, X2)) → quot'(X1, active'(X2))
active'(zWquot'(X1, X2)) → zWquot'(active'(X1), X2)
active'(zWquot'(X1, X2)) → zWquot'(X1, active'(X2))
from'(mark'(X)) → mark'(from'(X))
cons'(mark'(X1), X2) → mark'(cons'(X1, X2))
s'(mark'(X)) → mark'(s'(X))
sel'(mark'(X1), X2) → mark'(sel'(X1, X2))
sel'(X1, mark'(X2)) → mark'(sel'(X1, X2))
minus'(mark'(X1), X2) → mark'(minus'(X1, X2))
minus'(X1, mark'(X2)) → mark'(minus'(X1, X2))
quot'(mark'(X1), X2) → mark'(quot'(X1, X2))
quot'(X1, mark'(X2)) → mark'(quot'(X1, X2))
zWquot'(mark'(X1), X2) → mark'(zWquot'(X1, X2))
zWquot'(X1, mark'(X2)) → mark'(zWquot'(X1, X2))
proper'(from'(X)) → from'(proper'(X))
proper'(cons'(X1, X2)) → cons'(proper'(X1), proper'(X2))
proper'(s'(X)) → s'(proper'(X))
proper'(sel'(X1, X2)) → sel'(proper'(X1), proper'(X2))
proper'(0') → ok'(0')
proper'(minus'(X1, X2)) → minus'(proper'(X1), proper'(X2))
proper'(quot'(X1, X2)) → quot'(proper'(X1), proper'(X2))
proper'(zWquot'(X1, X2)) → zWquot'(proper'(X1), proper'(X2))
proper'(nil') → ok'(nil')
from'(ok'(X)) → ok'(from'(X))
cons'(ok'(X1), ok'(X2)) → ok'(cons'(X1, X2))
s'(ok'(X)) → ok'(s'(X))
sel'(ok'(X1), ok'(X2)) → ok'(sel'(X1, X2))
minus'(ok'(X1), ok'(X2)) → ok'(minus'(X1, X2))
quot'(ok'(X1), ok'(X2)) → ok'(quot'(X1, X2))
zWquot'(ok'(X1), ok'(X2)) → ok'(zWquot'(X1, X2))
top'(mark'(X)) → top'(proper'(X))
top'(ok'(X)) → top'(active'(X))

Rewrite Strategy: INNERMOST

Infered types.

Rules:
active'(from'(X)) → mark'(cons'(X, from'(s'(X))))
active'(sel'(0', cons'(X, XS))) → mark'(X)
active'(sel'(s'(N), cons'(X, XS))) → mark'(sel'(N, XS))
active'(minus'(X, 0')) → mark'(0')
active'(minus'(s'(X), s'(Y))) → mark'(minus'(X, Y))
active'(quot'(0', s'(Y))) → mark'(0')
active'(quot'(s'(X), s'(Y))) → mark'(s'(quot'(minus'(X, Y), s'(Y))))
active'(zWquot'(XS, nil')) → mark'(nil')
active'(zWquot'(nil', XS)) → mark'(nil')
active'(zWquot'(cons'(X, XS), cons'(Y, YS))) → mark'(cons'(quot'(X, Y), zWquot'(XS, YS)))
active'(from'(X)) → from'(active'(X))
active'(cons'(X1, X2)) → cons'(active'(X1), X2)
active'(s'(X)) → s'(active'(X))
active'(sel'(X1, X2)) → sel'(active'(X1), X2)
active'(sel'(X1, X2)) → sel'(X1, active'(X2))
active'(minus'(X1, X2)) → minus'(active'(X1), X2)
active'(minus'(X1, X2)) → minus'(X1, active'(X2))
active'(quot'(X1, X2)) → quot'(active'(X1), X2)
active'(quot'(X1, X2)) → quot'(X1, active'(X2))
active'(zWquot'(X1, X2)) → zWquot'(active'(X1), X2)
active'(zWquot'(X1, X2)) → zWquot'(X1, active'(X2))
from'(mark'(X)) → mark'(from'(X))
cons'(mark'(X1), X2) → mark'(cons'(X1, X2))
s'(mark'(X)) → mark'(s'(X))
sel'(mark'(X1), X2) → mark'(sel'(X1, X2))
sel'(X1, mark'(X2)) → mark'(sel'(X1, X2))
minus'(mark'(X1), X2) → mark'(minus'(X1, X2))
minus'(X1, mark'(X2)) → mark'(minus'(X1, X2))
quot'(mark'(X1), X2) → mark'(quot'(X1, X2))
quot'(X1, mark'(X2)) → mark'(quot'(X1, X2))
zWquot'(mark'(X1), X2) → mark'(zWquot'(X1, X2))
zWquot'(X1, mark'(X2)) → mark'(zWquot'(X1, X2))
proper'(from'(X)) → from'(proper'(X))
proper'(cons'(X1, X2)) → cons'(proper'(X1), proper'(X2))
proper'(s'(X)) → s'(proper'(X))
proper'(sel'(X1, X2)) → sel'(proper'(X1), proper'(X2))
proper'(0') → ok'(0')
proper'(minus'(X1, X2)) → minus'(proper'(X1), proper'(X2))
proper'(quot'(X1, X2)) → quot'(proper'(X1), proper'(X2))
proper'(zWquot'(X1, X2)) → zWquot'(proper'(X1), proper'(X2))
proper'(nil') → ok'(nil')
from'(ok'(X)) → ok'(from'(X))
cons'(ok'(X1), ok'(X2)) → ok'(cons'(X1, X2))
s'(ok'(X)) → ok'(s'(X))
sel'(ok'(X1), ok'(X2)) → ok'(sel'(X1, X2))
minus'(ok'(X1), ok'(X2)) → ok'(minus'(X1, X2))
quot'(ok'(X1), ok'(X2)) → ok'(quot'(X1, X2))
zWquot'(ok'(X1), ok'(X2)) → ok'(zWquot'(X1, X2))
top'(mark'(X)) → top'(proper'(X))
top'(ok'(X)) → top'(active'(X))

Types:
active' :: mark':0':nil':ok' → mark':0':nil':ok'
from' :: mark':0':nil':ok' → mark':0':nil':ok'
mark' :: mark':0':nil':ok' → mark':0':nil':ok'
cons' :: mark':0':nil':ok' → mark':0':nil':ok' → mark':0':nil':ok'
s' :: mark':0':nil':ok' → mark':0':nil':ok'
sel' :: mark':0':nil':ok' → mark':0':nil':ok' → mark':0':nil':ok'
0' :: mark':0':nil':ok'
minus' :: mark':0':nil':ok' → mark':0':nil':ok' → mark':0':nil':ok'
quot' :: mark':0':nil':ok' → mark':0':nil':ok' → mark':0':nil':ok'
zWquot' :: mark':0':nil':ok' → mark':0':nil':ok' → mark':0':nil':ok'
nil' :: mark':0':nil':ok'
proper' :: mark':0':nil':ok' → mark':0':nil':ok'
ok' :: mark':0':nil':ok' → mark':0':nil':ok'
top' :: mark':0':nil':ok' → top'
_hole_mark':0':nil':ok'1 :: mark':0':nil':ok'
_hole_top'2 :: top'
_gen_mark':0':nil':ok'3 :: Nat → mark':0':nil':ok'

Heuristically decided to analyse the following defined symbols:
active', cons', from', s', sel', minus', quot', zWquot', proper', top'

They will be analysed ascendingly in the following order:
cons' < active'
from' < active'
s' < active'
sel' < active'
minus' < active'
quot' < active'
zWquot' < active'
active' < top'
cons' < proper'
from' < proper'
s' < proper'
sel' < proper'
minus' < proper'
quot' < proper'
zWquot' < proper'
proper' < top'

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

The following defined symbols remain to be analysed:
cons', active', from', s', sel', minus', quot', zWquot', proper', top'

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

Induction Base:
cons'(_gen_mark':0':nil':ok'3(+(1, 0)), _gen_mark':0':nil':ok'3(b))

Induction Step:
cons'(_gen_mark':0':nil':ok'3(+(1, +(_$n6, 1))), _gen_mark':0':nil':ok'3(_b610)) →_R^Ω(1)
mark'(cons'(_gen_mark':0':nil':ok'3(+(1, _$n6)), _gen_mark':0':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_mark':0':nil':ok'3(+(1, _n5)), _gen_mark':0':nil':ok'3(b)) → _*4, rt ∈ Ω(_n5)

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

The following defined symbols remain to be analysed:
from', active', s', sel', minus', quot', zWquot', proper', top'

They will be analysed ascendingly in the following order:
from' < active'
s' < active'
sel' < active'
minus' < active'
quot' < active'
zWquot' < active'
active' < top'
from' < proper'
s' < proper'
sel' < proper'
minus' < proper'
quot' < proper'
zWquot' < proper'
proper' < top'

Proved the following rewrite lemma:
from'(_gen_mark':0':nil':ok'3(+(1, _n2981))) → _*4, rt ∈ Ω(_n2981)

Induction Base:
from'(_gen_mark':0':nil':ok'3(+(1, 0)))

Induction Step:
from'(_gen_mark':0':nil':ok'3(+(1, +(_$n2982, 1)))) →_R^Ω(1)
mark'(from'(_gen_mark':0':nil':ok'3(+(1, _$n2982)))) →_IH
mark'(_*4)

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

Lemmas:
cons'(_gen_mark':0':nil':ok'3(+(1, _n5)), _gen_mark':0':nil':ok'3(b)) → _*4, rt ∈ Ω(_n5)
from'(_gen_mark':0':nil':ok'3(+(1, _n2981))) → _*4, rt ∈ Ω(_n2981)

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

The following defined symbols remain to be analysed:
s', active', sel', minus', quot', zWquot', proper', top'

They will be analysed ascendingly in the following order:
s' < active'
sel' < active'
minus' < active'
quot' < active'
zWquot' < active'
active' < top'
s' < proper'
sel' < proper'
minus' < proper'
quot' < proper'
zWquot' < proper'
proper' < top'

Proved the following rewrite lemma:
s'(_gen_mark':0':nil':ok'3(+(1, _n5040))) → _*4, rt ∈ Ω(_n5040)

Induction Base:
s'(_gen_mark':0':nil':ok'3(+(1, 0)))

Induction Step:
s'(_gen_mark':0':nil':ok'3(+(1, +(_$n5041, 1)))) →_R^Ω(1)
mark'(s'(_gen_mark':0':nil':ok'3(+(1, _$n5041)))) →_IH
mark'(_*4)

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

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

The following defined symbols remain to be analysed:
sel', active', minus', quot', zWquot', proper', top'

They will be analysed ascendingly in the following order:
sel' < active'
minus' < active'
quot' < active'
zWquot' < active'
active' < top'
sel' < proper'
minus' < proper'
quot' < proper'
zWquot' < proper'
proper' < top'

Proved the following rewrite lemma:
sel'(_gen_mark':0':nil':ok'3(+(1, _n7223)), _gen_mark':0':nil':ok'3(b)) → _*4, rt ∈ Ω(_n7223)

Induction Base:
sel'(_gen_mark':0':nil':ok'3(+(1, 0)), _gen_mark':0':nil':ok'3(b))

Induction Step:
sel'(_gen_mark':0':nil':ok'3(+(1, +(_$n7224, 1))), _gen_mark':0':nil':ok'3(_b8800)) →_R^Ω(1)
mark'(sel'(_gen_mark':0':nil':ok'3(+(1, _$n7224)), _gen_mark':0':nil':ok'3(_b8800))) →_IH
mark'(_*4)

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

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

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

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

Proved the following rewrite lemma:
minus'(_gen_mark':0':nil':ok'3(+(1, _n11268)), _gen_mark':0':nil':ok'3(b)) → _*4, rt ∈ Ω(_n11268)

Induction Base:
minus'(_gen_mark':0':nil':ok'3(+(1, 0)), _gen_mark':0':nil':ok'3(b))

Induction Step:
minus'(_gen_mark':0':nil':ok'3(+(1, +(_$n11269, 1))), _gen_mark':0':nil':ok'3(_b13169)) →_R^Ω(1)
mark'(minus'(_gen_mark':0':nil':ok'3(+(1, _$n11269)), _gen_mark':0':nil':ok'3(_b13169))) →_IH
mark'(_*4)

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

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

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

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

Proved the following rewrite lemma:
quot'(_gen_mark':0':nil':ok'3(+(1, _n15681)), _gen_mark':0':nil':ok'3(b)) → _*4, rt ∈ Ω(_n15681)

Induction Base:
quot'(_gen_mark':0':nil':ok'3(+(1, 0)), _gen_mark':0':nil':ok'3(b))

Induction Step:
quot'(_gen_mark':0':nil':ok'3(+(1, +(_$n15682, 1))), _gen_mark':0':nil':ok'3(_b17906)) →_R^Ω(1)
mark'(quot'(_gen_mark':0':nil':ok'3(+(1, _$n15682)), _gen_mark':0':nil':ok'3(_b17906))) →_IH
mark'(_*4)

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

Lemmas:
cons'(_gen_mark':0':nil':ok'3(+(1, _n5)), _gen_mark':0':nil':ok'3(b)) → _*4, rt ∈ Ω(_n5)
from'(_gen_mark':0':nil':ok'3(+(1, _n2981))) → _*4, rt ∈ Ω(_n2981)
s'(_gen_mark':0':nil':ok'3(+(1, _n5040))) → _*4, rt ∈ Ω(_n5040)
sel'(_gen_mark':0':nil':ok'3(+(1, _n7223)), _gen_mark':0':nil':ok'3(b)) → _*4, rt ∈ Ω(_n7223)
minus'(_gen_mark':0':nil':ok'3(+(1, _n11268)), _gen_mark':0':nil':ok'3(b)) → _*4, rt ∈ Ω(_n11268)
quot'(_gen_mark':0':nil':ok'3(+(1, _n15681)), _gen_mark':0':nil':ok'3(b)) → _*4, rt ∈ Ω(_n15681)

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

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

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

Proved the following rewrite lemma:
zWquot'(_gen_mark':0':nil':ok'3(+(1, _n20462)), _gen_mark':0':nil':ok'3(b)) → _*4, rt ∈ Ω(_n20462)

Induction Base:
zWquot'(_gen_mark':0':nil':ok'3(+(1, 0)), _gen_mark':0':nil':ok'3(b))

Induction Step:
zWquot'(_gen_mark':0':nil':ok'3(+(1, +(_$n20463, 1))), _gen_mark':0':nil':ok'3(_b23011)) →_R^Ω(1)
mark'(zWquot'(_gen_mark':0':nil':ok'3(+(1, _$n20463)), _gen_mark':0':nil':ok'3(_b23011))) →_IH
mark'(_*4)

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

Generator Equations:
_gen_mark':0':nil':ok'3(0) ⇔ 0'
_gen_mark':0':nil':ok'3(+(x, 1)) ⇔ mark'(_gen_mark':0':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_mark':0':nil':ok'3(0) ⇔ 0'
_gen_mark':0':nil':ok'3(+(x, 1)) ⇔ mark'(_gen_mark':0':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_mark':0':nil':ok'3(0) ⇔ 0'
_gen_mark':0':nil':ok'3(+(x, 1)) ⇔ mark'(_gen_mark':0':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_mark':0':nil':ok'3(0) ⇔ 0'
_gen_mark':0':nil':ok'3(+(x, 1)) ⇔ mark'(_gen_mark':0':nil':ok'3(x))

No more defined symbols left to analyse.

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