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

active(dbl(0)) → mark(0)
active(dbl(s(X))) → mark(s(s(dbl(X))))
active(dbls(nil)) → mark(nil)
active(dbls(cons(X, Y))) → mark(cons(dbl(X), dbls(Y)))
active(sel(0, cons(X, Y))) → mark(X)
active(sel(s(X), cons(Y, Z))) → mark(sel(X, Z))
active(indx(nil, X)) → mark(nil)
active(indx(cons(X, Y), Z)) → mark(cons(sel(X, Z), indx(Y, Z)))
active(from(X)) → mark(cons(X, from(s(X))))
active(dbl(X)) → dbl(active(X))
active(dbls(X)) → dbls(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(indx(X1, X2)) → indx(active(X1), X2)
dbl(mark(X)) → mark(dbl(X))
dbls(mark(X)) → mark(dbls(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
indx(mark(X1), X2) → mark(indx(X1, X2))
proper(dbl(X)) → dbl(proper(X))
proper(0) → ok(0)
proper(s(X)) → s(proper(X))
proper(dbls(X)) → dbls(proper(X))
proper(nil) → ok(nil)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(indx(X1, X2)) → indx(proper(X1), proper(X2))
proper(from(X)) → from(proper(X))
dbl(ok(X)) → ok(dbl(X))
s(ok(X)) → ok(s(X))
dbls(ok(X)) → ok(dbls(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
indx(ok(X1), ok(X2)) → ok(indx(X1, X2))
from(ok(X)) → ok(from(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'(dbl'(0')) → mark'(0')
active'(dbl'(s'(X))) → mark'(s'(s'(dbl'(X))))
active'(dbls'(nil')) → mark'(nil')
active'(dbls'(cons'(X, Y))) → mark'(cons'(dbl'(X), dbls'(Y)))
active'(sel'(0', cons'(X, Y))) → mark'(X)
active'(sel'(s'(X), cons'(Y, Z))) → mark'(sel'(X, Z))
active'(indx'(nil', X)) → mark'(nil')
active'(indx'(cons'(X, Y), Z)) → mark'(cons'(sel'(X, Z), indx'(Y, Z)))
active'(from'(X)) → mark'(cons'(X, from'(s'(X))))
active'(dbl'(X)) → dbl'(active'(X))
active'(dbls'(X)) → dbls'(active'(X))
active'(sel'(X1, X2)) → sel'(active'(X1), X2)
active'(sel'(X1, X2)) → sel'(X1, active'(X2))
active'(indx'(X1, X2)) → indx'(active'(X1), X2)
dbl'(mark'(X)) → mark'(dbl'(X))
dbls'(mark'(X)) → mark'(dbls'(X))
sel'(mark'(X1), X2) → mark'(sel'(X1, X2))
sel'(X1, mark'(X2)) → mark'(sel'(X1, X2))
indx'(mark'(X1), X2) → mark'(indx'(X1, X2))
proper'(dbl'(X)) → dbl'(proper'(X))
proper'(0') → ok'(0')
proper'(s'(X)) → s'(proper'(X))
proper'(dbls'(X)) → dbls'(proper'(X))
proper'(nil') → ok'(nil')
proper'(cons'(X1, X2)) → cons'(proper'(X1), proper'(X2))
proper'(sel'(X1, X2)) → sel'(proper'(X1), proper'(X2))
proper'(indx'(X1, X2)) → indx'(proper'(X1), proper'(X2))
proper'(from'(X)) → from'(proper'(X))
dbl'(ok'(X)) → ok'(dbl'(X))
s'(ok'(X)) → ok'(s'(X))
dbls'(ok'(X)) → ok'(dbls'(X))
cons'(ok'(X1), ok'(X2)) → ok'(cons'(X1, X2))
sel'(ok'(X1), ok'(X2)) → ok'(sel'(X1, X2))
indx'(ok'(X1), ok'(X2)) → ok'(indx'(X1, X2))
from'(ok'(X)) → ok'(from'(X))
top'(mark'(X)) → top'(proper'(X))
top'(ok'(X)) → top'(active'(X))

Rewrite Strategy: INNERMOST

Infered types.

Rules:
active'(dbl'(0')) → mark'(0')
active'(dbl'(s'(X))) → mark'(s'(s'(dbl'(X))))
active'(dbls'(nil')) → mark'(nil')
active'(dbls'(cons'(X, Y))) → mark'(cons'(dbl'(X), dbls'(Y)))
active'(sel'(0', cons'(X, Y))) → mark'(X)
active'(sel'(s'(X), cons'(Y, Z))) → mark'(sel'(X, Z))
active'(indx'(nil', X)) → mark'(nil')
active'(indx'(cons'(X, Y), Z)) → mark'(cons'(sel'(X, Z), indx'(Y, Z)))
active'(from'(X)) → mark'(cons'(X, from'(s'(X))))
active'(dbl'(X)) → dbl'(active'(X))
active'(dbls'(X)) → dbls'(active'(X))
active'(sel'(X1, X2)) → sel'(active'(X1), X2)
active'(sel'(X1, X2)) → sel'(X1, active'(X2))
active'(indx'(X1, X2)) → indx'(active'(X1), X2)
dbl'(mark'(X)) → mark'(dbl'(X))
dbls'(mark'(X)) → mark'(dbls'(X))
sel'(mark'(X1), X2) → mark'(sel'(X1, X2))
sel'(X1, mark'(X2)) → mark'(sel'(X1, X2))
indx'(mark'(X1), X2) → mark'(indx'(X1, X2))
proper'(dbl'(X)) → dbl'(proper'(X))
proper'(0') → ok'(0')
proper'(s'(X)) → s'(proper'(X))
proper'(dbls'(X)) → dbls'(proper'(X))
proper'(nil') → ok'(nil')
proper'(cons'(X1, X2)) → cons'(proper'(X1), proper'(X2))
proper'(sel'(X1, X2)) → sel'(proper'(X1), proper'(X2))
proper'(indx'(X1, X2)) → indx'(proper'(X1), proper'(X2))
proper'(from'(X)) → from'(proper'(X))
dbl'(ok'(X)) → ok'(dbl'(X))
s'(ok'(X)) → ok'(s'(X))
dbls'(ok'(X)) → ok'(dbls'(X))
cons'(ok'(X1), ok'(X2)) → ok'(cons'(X1, X2))
sel'(ok'(X1), ok'(X2)) → ok'(sel'(X1, X2))
indx'(ok'(X1), ok'(X2)) → ok'(indx'(X1, X2))
from'(ok'(X)) → ok'(from'(X))
top'(mark'(X)) → top'(proper'(X))
top'(ok'(X)) → top'(active'(X))

Types:
active' :: 0':mark':nil':ok' → 0':mark':nil':ok'
dbl' :: 0':mark':nil':ok' → 0':mark':nil':ok'
0' :: 0':mark':nil':ok'
mark' :: 0':mark':nil':ok' → 0':mark':nil':ok'
s' :: 0':mark':nil':ok' → 0':mark':nil':ok'
dbls' :: 0':mark':nil':ok' → 0':mark':nil':ok'
nil' :: 0':mark':nil':ok'
cons' :: 0':mark':nil':ok' → 0':mark':nil':ok' → 0':mark':nil':ok'
sel' :: 0':mark':nil':ok' → 0':mark':nil':ok' → 0':mark':nil':ok'
indx' :: 0':mark':nil':ok' → 0':mark':nil':ok' → 0':mark':nil':ok'
from' :: 0':mark':nil':ok' → 0':mark':nil':ok'
proper' :: 0':mark':nil':ok' → 0':mark':nil':ok'
ok' :: 0':mark':nil':ok' → 0':mark':nil':ok'
top' :: 0':mark':nil':ok' → top'
_hole_0':mark':nil':ok'1 :: 0':mark':nil':ok'
_hole_top'2 :: top'
_gen_0':mark':nil':ok'3 :: Nat → 0':mark':nil':ok'

Heuristically decided to analyse the following defined symbols:
active', s', dbl', cons', dbls', sel', indx', from', proper', top'

They will be analysed ascendingly in the following order:
s' < active'
dbl' < active'
cons' < active'
dbls' < active'
sel' < active'
indx' < active'
from' < active'
active' < top'
s' < proper'
dbl' < proper'
cons' < proper'
dbls' < proper'
sel' < proper'
indx' < proper'
from' < proper'
proper' < top'

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

The following defined symbols remain to be analysed:
s', active', dbl', cons', dbls', sel', indx', from', proper', top'

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

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

The following defined symbols remain to be analysed:
dbl', active', cons', dbls', sel', indx', from', proper', top'

They will be analysed ascendingly in the following order:
dbl' < active'
cons' < active'
dbls' < active'
sel' < active'
indx' < active'
from' < active'
active' < top'
dbl' < proper'
cons' < proper'
dbls' < proper'
sel' < proper'
indx' < proper'
from' < proper'
proper' < top'

Proved the following rewrite lemma:
dbl'(_gen_0':mark':nil':ok'3(+(1, _n11))) → _*4, rt ∈ Ω(_n11)

Induction Base:
dbl'(_gen_0':mark':nil':ok'3(+(1, 0)))

Induction Step:
dbl'(_gen_0':mark':nil':ok'3(+(1, +(_$n12, 1)))) →_R^Ω(1)
mark'(dbl'(_gen_0':mark':nil':ok'3(+(1, _$n12)))) →_IH
mark'(_*4)

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

Lemmas:
dbl'(_gen_0':mark':nil':ok'3(+(1, _n11))) → _*4, rt ∈ Ω(_n11)

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

The following defined symbols remain to be analysed:
cons', active', dbls', sel', indx', from', proper', top'

They will be analysed ascendingly in the following order:
cons' < active'
dbls' < active'
sel' < active'
indx' < active'
from' < active'
active' < top'
cons' < proper'
dbls' < proper'
sel' < proper'
indx' < proper'
from' < proper'
proper' < top'

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

Lemmas:
dbl'(_gen_0':mark':nil':ok'3(+(1, _n11))) → _*4, rt ∈ Ω(_n11)

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

The following defined symbols remain to be analysed:
dbls', active', sel', indx', from', proper', top'

They will be analysed ascendingly in the following order:
dbls' < active'
sel' < active'
indx' < active'
from' < active'
active' < top'
dbls' < proper'
sel' < proper'
indx' < proper'
from' < proper'
proper' < top'

Proved the following rewrite lemma:
dbls'(_gen_0':mark':nil':ok'3(+(1, _n1328))) → _*4, rt ∈ Ω(_n1328)

Induction Base:
dbls'(_gen_0':mark':nil':ok'3(+(1, 0)))

Induction Step:
dbls'(_gen_0':mark':nil':ok'3(+(1, +(_$n1329, 1)))) →_R^Ω(1)
mark'(dbls'(_gen_0':mark':nil':ok'3(+(1, _$n1329)))) →_IH
mark'(_*4)

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

Lemmas:
dbl'(_gen_0':mark':nil':ok'3(+(1, _n11))) → _*4, rt ∈ Ω(_n11)
dbls'(_gen_0':mark':nil':ok'3(+(1, _n1328))) → _*4, rt ∈ Ω(_n1328)

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

The following defined symbols remain to be analysed:
sel', active', indx', from', proper', top'

They will be analysed ascendingly in the following order:
sel' < active'
indx' < active'
from' < active'
active' < top'
sel' < proper'
indx' < proper'
from' < proper'
proper' < top'

Proved the following rewrite lemma:
sel'(_gen_0':mark':nil':ok'3(+(1, _n2747)), _gen_0':mark':nil':ok'3(b)) → _*4, rt ∈ Ω(_n2747)

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

Induction Step:
sel'(_gen_0':mark':nil':ok'3(+(1, +(_$n2748, 1))), _gen_0':mark':nil':ok'3(_b4000)) →_R^Ω(1)
mark'(sel'(_gen_0':mark':nil':ok'3(+(1, _$n2748)), _gen_0':mark':nil':ok'3(_b4000))) →_IH
mark'(_*4)

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

Lemmas:
dbl'(_gen_0':mark':nil':ok'3(+(1, _n11))) → _*4, rt ∈ Ω(_n11)
dbls'(_gen_0':mark':nil':ok'3(+(1, _n1328))) → _*4, rt ∈ Ω(_n1328)
sel'(_gen_0':mark':nil':ok'3(+(1, _n2747)), _gen_0':mark':nil':ok'3(b)) → _*4, rt ∈ Ω(_n2747)

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

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

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

Proved the following rewrite lemma:
indx'(_gen_0':mark':nil':ok'3(+(1, _n5575)), _gen_0':mark':nil':ok'3(b)) → _*4, rt ∈ Ω(_n5575)

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

Induction Step:
indx'(_gen_0':mark':nil':ok'3(+(1, +(_$n5576, 1))), _gen_0':mark':nil':ok'3(_b6936)) →_R^Ω(1)
mark'(indx'(_gen_0':mark':nil':ok'3(+(1, _$n5576)), _gen_0':mark':nil':ok'3(_b6936))) →_IH
mark'(_*4)

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

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

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

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

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

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

No more defined symbols left to analyse.

The lowerbound Ω(n) was proven with the following lemma:
dbl'(_gen_0':mark':nil':ok'3(+(1, _n11))) → _*4, rt ∈ Ω(_n11)