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

active(nats) → mark(adx(zeros))
active(zeros) → mark(cons(0, zeros))
active(incr(cons(X, Y))) → mark(cons(s(X), incr(Y)))
active(adx(cons(X, Y))) → mark(incr(cons(X, adx(Y))))
active(hd(cons(X, Y))) → mark(X)
active(tl(cons(X, Y))) → mark(Y)
active(adx(X)) → adx(active(X))
active(incr(X)) → incr(active(X))
active(hd(X)) → hd(active(X))
active(tl(X)) → tl(active(X))
adx(mark(X)) → mark(adx(X))
incr(mark(X)) → mark(incr(X))
hd(mark(X)) → mark(hd(X))
tl(mark(X)) → mark(tl(X))
proper(nats) → ok(nats)
proper(adx(X)) → adx(proper(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(incr(X)) → incr(proper(X))
proper(s(X)) → s(proper(X))
proper(hd(X)) → hd(proper(X))
proper(tl(X)) → tl(proper(X))
adx(ok(X)) → ok(adx(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
incr(ok(X)) → ok(incr(X))
s(ok(X)) → ok(s(X))
hd(ok(X)) → ok(hd(X))
tl(ok(X)) → ok(tl(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'(nats') → mark'(adx'(zeros'))
active'(zeros') → mark'(cons'(0', zeros'))
active'(incr'(cons'(X, Y))) → mark'(cons'(s'(X), incr'(Y)))
active'(adx'(cons'(X, Y))) → mark'(incr'(cons'(X, adx'(Y))))
active'(hd'(cons'(X, Y))) → mark'(X)
active'(tl'(cons'(X, Y))) → mark'(Y)
active'(adx'(X)) → adx'(active'(X))
active'(incr'(X)) → incr'(active'(X))
active'(hd'(X)) → hd'(active'(X))
active'(tl'(X)) → tl'(active'(X))
adx'(mark'(X)) → mark'(adx'(X))
incr'(mark'(X)) → mark'(incr'(X))
hd'(mark'(X)) → mark'(hd'(X))
tl'(mark'(X)) → mark'(tl'(X))
proper'(nats') → ok'(nats')
proper'(adx'(X)) → adx'(proper'(X))
proper'(zeros') → ok'(zeros')
proper'(cons'(X1, X2)) → cons'(proper'(X1), proper'(X2))
proper'(0') → ok'(0')
proper'(incr'(X)) → incr'(proper'(X))
proper'(s'(X)) → s'(proper'(X))
proper'(hd'(X)) → hd'(proper'(X))
proper'(tl'(X)) → tl'(proper'(X))
adx'(ok'(X)) → ok'(adx'(X))
cons'(ok'(X1), ok'(X2)) → ok'(cons'(X1, X2))
incr'(ok'(X)) → ok'(incr'(X))
s'(ok'(X)) → ok'(s'(X))
hd'(ok'(X)) → ok'(hd'(X))
tl'(ok'(X)) → ok'(tl'(X))
top'(mark'(X)) → top'(proper'(X))
top'(ok'(X)) → top'(active'(X))

Rewrite Strategy: INNERMOST

Infered types.

Rules:
active'(nats') → mark'(adx'(zeros'))
active'(zeros') → mark'(cons'(0', zeros'))
active'(incr'(cons'(X, Y))) → mark'(cons'(s'(X), incr'(Y)))
active'(adx'(cons'(X, Y))) → mark'(incr'(cons'(X, adx'(Y))))
active'(hd'(cons'(X, Y))) → mark'(X)
active'(tl'(cons'(X, Y))) → mark'(Y)
active'(adx'(X)) → adx'(active'(X))
active'(incr'(X)) → incr'(active'(X))
active'(hd'(X)) → hd'(active'(X))
active'(tl'(X)) → tl'(active'(X))
adx'(mark'(X)) → mark'(adx'(X))
incr'(mark'(X)) → mark'(incr'(X))
hd'(mark'(X)) → mark'(hd'(X))
tl'(mark'(X)) → mark'(tl'(X))
proper'(nats') → ok'(nats')
proper'(adx'(X)) → adx'(proper'(X))
proper'(zeros') → ok'(zeros')
proper'(cons'(X1, X2)) → cons'(proper'(X1), proper'(X2))
proper'(0') → ok'(0')
proper'(incr'(X)) → incr'(proper'(X))
proper'(s'(X)) → s'(proper'(X))
proper'(hd'(X)) → hd'(proper'(X))
proper'(tl'(X)) → tl'(proper'(X))
adx'(ok'(X)) → ok'(adx'(X))
cons'(ok'(X1), ok'(X2)) → ok'(cons'(X1, X2))
incr'(ok'(X)) → ok'(incr'(X))
s'(ok'(X)) → ok'(s'(X))
hd'(ok'(X)) → ok'(hd'(X))
tl'(ok'(X)) → ok'(tl'(X))
top'(mark'(X)) → top'(proper'(X))
top'(ok'(X)) → top'(active'(X))

Types:
active' :: nats':zeros':mark':0':ok' → nats':zeros':mark':0':ok'
nats' :: nats':zeros':mark':0':ok'
mark' :: nats':zeros':mark':0':ok' → nats':zeros':mark':0':ok'
adx' :: nats':zeros':mark':0':ok' → nats':zeros':mark':0':ok'
zeros' :: nats':zeros':mark':0':ok'
cons' :: nats':zeros':mark':0':ok' → nats':zeros':mark':0':ok' → nats':zeros':mark':0':ok'
0' :: nats':zeros':mark':0':ok'
incr' :: nats':zeros':mark':0':ok' → nats':zeros':mark':0':ok'
s' :: nats':zeros':mark':0':ok' → nats':zeros':mark':0':ok'
hd' :: nats':zeros':mark':0':ok' → nats':zeros':mark':0':ok'
tl' :: nats':zeros':mark':0':ok' → nats':zeros':mark':0':ok'
proper' :: nats':zeros':mark':0':ok' → nats':zeros':mark':0':ok'
ok' :: nats':zeros':mark':0':ok' → nats':zeros':mark':0':ok'
top' :: nats':zeros':mark':0':ok' → top'
_hole_nats':zeros':mark':0':ok'1 :: nats':zeros':mark':0':ok'
_hole_top'2 :: top'
_gen_nats':zeros':mark':0':ok'3 :: Nat → nats':zeros':mark':0':ok'

Heuristically decided to analyse the following defined symbols:
active', adx', cons', s', incr', hd', tl', proper', top'

They will be analysed ascendingly in the following order:
adx' < active'
cons' < active'
s' < active'
incr' < active'
hd' < active'
tl' < active'
active' < top'
adx' < proper'
cons' < proper'
s' < proper'
incr' < proper'
hd' < proper'
tl' < proper'
proper' < top'

Generator Equations:
_gen_nats':zeros':mark':0':ok'3(0) ⇔ nats'
_gen_nats':zeros':mark':0':ok'3(+(x, 1)) ⇔ mark'(_gen_nats':zeros':mark':0':ok'3(x))

The following defined symbols remain to be analysed:
adx', active', cons', s', incr', hd', tl', proper', top'

Proved the following rewrite lemma:
adx'(_gen_nats':zeros':mark':0':ok'3(+(1, _n5))) → _*4, rt ∈ Ω(_n5)

Induction Base:
adx'(_gen_nats':zeros':mark':0':ok'3(+(1, 0)))

Induction Step:
adx'(_gen_nats':zeros':mark':0':ok'3(+(1, +(_$n6, 1)))) →_R^Ω(1)
mark'(adx'(_gen_nats':zeros':mark':0':ok'3(+(1, _$n6)))) →_IH
mark'(_*4)

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

Lemmas:
adx'(_gen_nats':zeros':mark':0':ok'3(+(1, _n5))) → _*4, rt ∈ Ω(_n5)

Generator Equations:
_gen_nats':zeros':mark':0':ok'3(0) ⇔ nats'
_gen_nats':zeros':mark':0':ok'3(+(x, 1)) ⇔ mark'(_gen_nats':zeros':mark':0':ok'3(x))

The following defined symbols remain to be analysed:
cons', active', s', incr', hd', tl', proper', top'

They will be analysed ascendingly in the following order:
cons' < active'
s' < active'
incr' < active'
hd' < active'
tl' < active'
active' < top'
cons' < proper'
s' < proper'
incr' < proper'
hd' < proper'
tl' < proper'
proper' < top'

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

Lemmas:
adx'(_gen_nats':zeros':mark':0':ok'3(+(1, _n5))) → _*4, rt ∈ Ω(_n5)

Generator Equations:
_gen_nats':zeros':mark':0':ok'3(0) ⇔ nats'
_gen_nats':zeros':mark':0':ok'3(+(x, 1)) ⇔ mark'(_gen_nats':zeros':mark':0':ok'3(x))

The following defined symbols remain to be analysed:
s', active', incr', hd', tl', proper', top'

They will be analysed ascendingly in the following order:
s' < active'
incr' < active'
hd' < active'
tl' < active'
active' < top'
s' < proper'
incr' < proper'
hd' < proper'
tl' < proper'
proper' < top'

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

Lemmas:
adx'(_gen_nats':zeros':mark':0':ok'3(+(1, _n5))) → _*4, rt ∈ Ω(_n5)

Generator Equations:
_gen_nats':zeros':mark':0':ok'3(0) ⇔ nats'
_gen_nats':zeros':mark':0':ok'3(+(x, 1)) ⇔ mark'(_gen_nats':zeros':mark':0':ok'3(x))

The following defined symbols remain to be analysed:
incr', active', hd', tl', proper', top'

They will be analysed ascendingly in the following order:
incr' < active'
hd' < active'
tl' < active'
active' < top'
incr' < proper'
hd' < proper'
tl' < proper'
proper' < top'

Proved the following rewrite lemma:
incr'(_gen_nats':zeros':mark':0':ok'3(+(1, _n1013))) → _*4, rt ∈ Ω(_n1013)

Induction Base:
incr'(_gen_nats':zeros':mark':0':ok'3(+(1, 0)))

Induction Step:
incr'(_gen_nats':zeros':mark':0':ok'3(+(1, +(_$n1014, 1)))) →_R^Ω(1)
mark'(incr'(_gen_nats':zeros':mark':0':ok'3(+(1, _$n1014)))) →_IH
mark'(_*4)

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

Lemmas:
adx'(_gen_nats':zeros':mark':0':ok'3(+(1, _n5))) → _*4, rt ∈ Ω(_n5)
incr'(_gen_nats':zeros':mark':0':ok'3(+(1, _n1013))) → _*4, rt ∈ Ω(_n1013)

Generator Equations:
_gen_nats':zeros':mark':0':ok'3(0) ⇔ nats'
_gen_nats':zeros':mark':0':ok'3(+(x, 1)) ⇔ mark'(_gen_nats':zeros':mark':0':ok'3(x))

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

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

Proved the following rewrite lemma:
hd'(_gen_nats':zeros':mark':0':ok'3(+(1, _n2111))) → _*4, rt ∈ Ω(_n2111)

Induction Base:
hd'(_gen_nats':zeros':mark':0':ok'3(+(1, 0)))

Induction Step:
hd'(_gen_nats':zeros':mark':0':ok'3(+(1, +(_$n2112, 1)))) →_R^Ω(1)
mark'(hd'(_gen_nats':zeros':mark':0':ok'3(+(1, _$n2112)))) →_IH
mark'(_*4)

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

Lemmas:
adx'(_gen_nats':zeros':mark':0':ok'3(+(1, _n5))) → _*4, rt ∈ Ω(_n5)
incr'(_gen_nats':zeros':mark':0':ok'3(+(1, _n1013))) → _*4, rt ∈ Ω(_n1013)
hd'(_gen_nats':zeros':mark':0':ok'3(+(1, _n2111))) → _*4, rt ∈ Ω(_n2111)

Generator Equations:
_gen_nats':zeros':mark':0':ok'3(0) ⇔ nats'
_gen_nats':zeros':mark':0':ok'3(+(x, 1)) ⇔ mark'(_gen_nats':zeros':mark':0':ok'3(x))

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

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

Proved the following rewrite lemma:
tl'(_gen_nats':zeros':mark':0':ok'3(+(1, _n3333))) → _*4, rt ∈ Ω(_n3333)

Induction Base:
tl'(_gen_nats':zeros':mark':0':ok'3(+(1, 0)))

Induction Step:
tl'(_gen_nats':zeros':mark':0':ok'3(+(1, +(_$n3334, 1)))) →_R^Ω(1)
mark'(tl'(_gen_nats':zeros':mark':0':ok'3(+(1, _$n3334)))) →_IH
mark'(_*4)

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

Generator Equations:
_gen_nats':zeros':mark':0':ok'3(0) ⇔ nats'
_gen_nats':zeros':mark':0':ok'3(+(x, 1)) ⇔ mark'(_gen_nats':zeros':mark':0':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_nats':zeros':mark':0':ok'3(0) ⇔ nats'
_gen_nats':zeros':mark':0':ok'3(+(x, 1)) ⇔ mark'(_gen_nats':zeros':mark':0':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_nats':zeros':mark':0':ok'3(0) ⇔ nats'
_gen_nats':zeros':mark':0':ok'3(+(x, 1)) ⇔ mark'(_gen_nats':zeros':mark':0':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_nats':zeros':mark':0':ok'3(0) ⇔ nats'
_gen_nats':zeros':mark':0':ok'3(+(x, 1)) ⇔ mark'(_gen_nats':zeros':mark':0':ok'3(x))

No more defined symbols left to analyse.

The lowerbound Ω(n) was proven with the following lemma:
adx'(_gen_nats':zeros':mark':0':ok'3(+(1, _n5))) → _*4, rt ∈ Ω(_n5)