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

active(incr(nil)) → mark(nil)
active(incr(cons(X, L))) → mark(cons(s(X), incr(L)))
active(adx(nil)) → mark(nil)
active(adx(cons(X, L))) → mark(incr(cons(X, adx(L))))
active(nats) → mark(adx(zeros))
active(zeros) → mark(cons(0, zeros))
active(head(cons(X, L))) → mark(X)
active(tail(cons(X, L))) → mark(L)
active(incr(X)) → incr(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(s(X)) → s(active(X))
active(adx(X)) → adx(active(X))
active(head(X)) → head(active(X))
active(tail(X)) → tail(active(X))
incr(mark(X)) → mark(incr(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
s(mark(X)) → mark(s(X))
adx(mark(X)) → mark(adx(X))
head(mark(X)) → mark(head(X))
tail(mark(X)) → mark(tail(X))
proper(incr(X)) → incr(proper(X))
proper(nil) → ok(nil)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(adx(X)) → adx(proper(X))
proper(nats) → ok(nats)
proper(zeros) → ok(zeros)
proper(0) → ok(0)
proper(head(X)) → head(proper(X))
proper(tail(X)) → tail(proper(X))
incr(ok(X)) → ok(incr(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
s(ok(X)) → ok(s(X))
adx(ok(X)) → ok(adx(X))
head(ok(X)) → ok(head(X))
tail(ok(X)) → ok(tail(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'(incr'(nil')) → mark'(nil')
active'(incr'(cons'(X, L))) → mark'(cons'(s'(X), incr'(L)))
active'(adx'(nil')) → mark'(nil')
active'(adx'(cons'(X, L))) → mark'(incr'(cons'(X, adx'(L))))
active'(nats') → mark'(adx'(zeros'))
active'(zeros') → mark'(cons'(0', zeros'))
active'(head'(cons'(X, L))) → mark'(X)
active'(tail'(cons'(X, L))) → mark'(L)
active'(incr'(X)) → incr'(active'(X))
active'(cons'(X1, X2)) → cons'(active'(X1), X2)
active'(s'(X)) → s'(active'(X))
active'(adx'(X)) → adx'(active'(X))
active'(head'(X)) → head'(active'(X))
active'(tail'(X)) → tail'(active'(X))
incr'(mark'(X)) → mark'(incr'(X))
cons'(mark'(X1), X2) → mark'(cons'(X1, X2))
s'(mark'(X)) → mark'(s'(X))
adx'(mark'(X)) → mark'(adx'(X))
head'(mark'(X)) → mark'(head'(X))
tail'(mark'(X)) → mark'(tail'(X))
proper'(incr'(X)) → incr'(proper'(X))
proper'(nil') → ok'(nil')
proper'(cons'(X1, X2)) → cons'(proper'(X1), proper'(X2))
proper'(s'(X)) → s'(proper'(X))
proper'(adx'(X)) → adx'(proper'(X))
proper'(nats') → ok'(nats')
proper'(zeros') → ok'(zeros')
proper'(0') → ok'(0')
proper'(head'(X)) → head'(proper'(X))
proper'(tail'(X)) → tail'(proper'(X))
incr'(ok'(X)) → ok'(incr'(X))
cons'(ok'(X1), ok'(X2)) → ok'(cons'(X1, X2))
s'(ok'(X)) → ok'(s'(X))
adx'(ok'(X)) → ok'(adx'(X))
head'(ok'(X)) → ok'(head'(X))
tail'(ok'(X)) → ok'(tail'(X))
top'(mark'(X)) → top'(proper'(X))
top'(ok'(X)) → top'(active'(X))

Rewrite Strategy: INNERMOST

Infered types.

Rules:
active'(incr'(nil')) → mark'(nil')
active'(incr'(cons'(X, L))) → mark'(cons'(s'(X), incr'(L)))
active'(adx'(nil')) → mark'(nil')
active'(adx'(cons'(X, L))) → mark'(incr'(cons'(X, adx'(L))))
active'(nats') → mark'(adx'(zeros'))
active'(zeros') → mark'(cons'(0', zeros'))
active'(head'(cons'(X, L))) → mark'(X)
active'(tail'(cons'(X, L))) → mark'(L)
active'(incr'(X)) → incr'(active'(X))
active'(cons'(X1, X2)) → cons'(active'(X1), X2)
active'(s'(X)) → s'(active'(X))
active'(adx'(X)) → adx'(active'(X))
active'(head'(X)) → head'(active'(X))
active'(tail'(X)) → tail'(active'(X))
incr'(mark'(X)) → mark'(incr'(X))
cons'(mark'(X1), X2) → mark'(cons'(X1, X2))
s'(mark'(X)) → mark'(s'(X))
adx'(mark'(X)) → mark'(adx'(X))
head'(mark'(X)) → mark'(head'(X))
tail'(mark'(X)) → mark'(tail'(X))
proper'(incr'(X)) → incr'(proper'(X))
proper'(nil') → ok'(nil')
proper'(cons'(X1, X2)) → cons'(proper'(X1), proper'(X2))
proper'(s'(X)) → s'(proper'(X))
proper'(adx'(X)) → adx'(proper'(X))
proper'(nats') → ok'(nats')
proper'(zeros') → ok'(zeros')
proper'(0') → ok'(0')
proper'(head'(X)) → head'(proper'(X))
proper'(tail'(X)) → tail'(proper'(X))
incr'(ok'(X)) → ok'(incr'(X))
cons'(ok'(X1), ok'(X2)) → ok'(cons'(X1, X2))
s'(ok'(X)) → ok'(s'(X))
adx'(ok'(X)) → ok'(adx'(X))
head'(ok'(X)) → ok'(head'(X))
tail'(ok'(X)) → ok'(tail'(X))
top'(mark'(X)) → top'(proper'(X))
top'(ok'(X)) → top'(active'(X))

Types:
active' :: nil':mark':nats':zeros':0':ok' → nil':mark':nats':zeros':0':ok'
incr' :: nil':mark':nats':zeros':0':ok' → nil':mark':nats':zeros':0':ok'
nil' :: nil':mark':nats':zeros':0':ok'
mark' :: nil':mark':nats':zeros':0':ok' → nil':mark':nats':zeros':0':ok'
cons' :: nil':mark':nats':zeros':0':ok' → nil':mark':nats':zeros':0':ok' → nil':mark':nats':zeros':0':ok'
s' :: nil':mark':nats':zeros':0':ok' → nil':mark':nats':zeros':0':ok'
adx' :: nil':mark':nats':zeros':0':ok' → nil':mark':nats':zeros':0':ok'
nats' :: nil':mark':nats':zeros':0':ok'
zeros' :: nil':mark':nats':zeros':0':ok'
0' :: nil':mark':nats':zeros':0':ok'
head' :: nil':mark':nats':zeros':0':ok' → nil':mark':nats':zeros':0':ok'
tail' :: nil':mark':nats':zeros':0':ok' → nil':mark':nats':zeros':0':ok'
proper' :: nil':mark':nats':zeros':0':ok' → nil':mark':nats':zeros':0':ok'
ok' :: nil':mark':nats':zeros':0':ok' → nil':mark':nats':zeros':0':ok'
top' :: nil':mark':nats':zeros':0':ok' → top'
_hole_nil':mark':nats':zeros':0':ok'1 :: nil':mark':nats':zeros':0':ok'
_hole_top'2 :: top'
_gen_nil':mark':nats':zeros':0':ok'3 :: Nat → nil':mark':nats':zeros':0':ok'

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

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

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

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

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

Induction Base:
cons'(_gen_nil':mark':nats':zeros':0':ok'3(+(1, 0)), _gen_nil':mark':nats':zeros':0':ok'3(b))

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

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

The following defined symbols remain to be analysed:
s', active', incr', adx', head', tail', proper', top'

They will be analysed ascendingly in the following order:
s' < active'
incr' < active'
adx' < active'
head' < active'
tail' < active'
active' < top'
s' < proper'
incr' < proper'
adx' < proper'
head' < proper'
tail' < proper'
proper' < top'

Proved the following rewrite lemma:
s'(_gen_nil':mark':nats':zeros':0':ok'3(+(1, _n1871))) → _*4, rt ∈ Ω(_n1871)

Induction Base:
s'(_gen_nil':mark':nats':zeros':0':ok'3(+(1, 0)))

Induction Step:
s'(_gen_nil':mark':nats':zeros':0':ok'3(+(1, +(_$n1872, 1)))) →_R^Ω(1)
mark'(s'(_gen_nil':mark':nats':zeros':0':ok'3(+(1, _$n1872)))) →_IH
mark'(_*4)

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

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

The following defined symbols remain to be analysed:
incr', active', adx', head', tail', proper', top'

They will be analysed ascendingly in the following order:
incr' < active'
adx' < active'
head' < active'
tail' < active'
active' < top'
incr' < proper'
adx' < proper'
head' < proper'
tail' < proper'
proper' < top'

Proved the following rewrite lemma:
incr'(_gen_nil':mark':nats':zeros':0':ok'3(+(1, _n3168))) → _*4, rt ∈ Ω(_n3168)

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

Induction Step:
incr'(_gen_nil':mark':nats':zeros':0':ok'3(+(1, +(_$n3169, 1)))) →_R^Ω(1)
mark'(incr'(_gen_nil':mark':nats':zeros':0':ok'3(+(1, _$n3169)))) →_IH
mark'(_*4)

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

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

The following defined symbols remain to be analysed:
adx', active', head', tail', proper', top'

They will be analysed ascendingly in the following order:
adx' < active'
head' < active'
tail' < active'
active' < top'
adx' < proper'
head' < proper'
tail' < proper'
proper' < top'

Proved the following rewrite lemma:
adx'(_gen_nil':mark':nats':zeros':0':ok'3(+(1, _n4589))) → _*4, rt ∈ Ω(_n4589)

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

Induction Step:
adx'(_gen_nil':mark':nats':zeros':0':ok'3(+(1, +(_$n4590, 1)))) →_R^Ω(1)
mark'(adx'(_gen_nil':mark':nats':zeros':0':ok'3(+(1, _$n4590)))) →_IH
mark'(_*4)

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

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

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

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

Proved the following rewrite lemma:
head'(_gen_nil':mark':nats':zeros':0':ok'3(+(1, _n6134))) → _*4, rt ∈ Ω(_n6134)

Induction Base:
head'(_gen_nil':mark':nats':zeros':0':ok'3(+(1, 0)))

Induction Step:
head'(_gen_nil':mark':nats':zeros':0':ok'3(+(1, +(_$n6135, 1)))) →_R^Ω(1)
mark'(head'(_gen_nil':mark':nats':zeros':0':ok'3(+(1, _$n6135)))) →_IH
mark'(_*4)

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

Lemmas:
cons'(_gen_nil':mark':nats':zeros':0':ok'3(+(1, _n5)), _gen_nil':mark':nats':zeros':0':ok'3(b)) → _*4, rt ∈ Ω(_n5)
s'(_gen_nil':mark':nats':zeros':0':ok'3(+(1, _n1871))) → _*4, rt ∈ Ω(_n1871)
incr'(_gen_nil':mark':nats':zeros':0':ok'3(+(1, _n3168))) → _*4, rt ∈ Ω(_n3168)
adx'(_gen_nil':mark':nats':zeros':0':ok'3(+(1, _n4589))) → _*4, rt ∈ Ω(_n4589)
head'(_gen_nil':mark':nats':zeros':0':ok'3(+(1, _n6134))) → _*4, rt ∈ Ω(_n6134)

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

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

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

Proved the following rewrite lemma:
tail'(_gen_nil':mark':nats':zeros':0':ok'3(+(1, _n7803))) → _*4, rt ∈ Ω(_n7803)

Induction Base:
tail'(_gen_nil':mark':nats':zeros':0':ok'3(+(1, 0)))

Induction Step:
tail'(_gen_nil':mark':nats':zeros':0':ok'3(+(1, +(_$n7804, 1)))) →_R^Ω(1)
mark'(tail'(_gen_nil':mark':nats':zeros':0':ok'3(+(1, _$n7804)))) →_IH
mark'(_*4)

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

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

No more defined symbols left to analyse.

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