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

active(nats) → mark(cons(0, incr(nats)))
active(pairs) → mark(cons(0, incr(odds)))
active(odds) → mark(incr(pairs))
active(incr(cons(X, XS))) → mark(cons(s(X), incr(XS)))
active(head(cons(X, XS))) → mark(X)
active(tail(cons(X, XS))) → mark(XS)
active(cons(X1, X2)) → cons(active(X1), X2)
active(incr(X)) → incr(active(X))
active(s(X)) → s(active(X))
active(head(X)) → head(active(X))
active(tail(X)) → tail(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
incr(mark(X)) → mark(incr(X))
s(mark(X)) → mark(s(X))
head(mark(X)) → mark(head(X))
tail(mark(X)) → mark(tail(X))
proper(nats) → ok(nats)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(incr(X)) → incr(proper(X))
proper(pairs) → ok(pairs)
proper(odds) → ok(odds)
proper(s(X)) → s(proper(X))
proper(head(X)) → head(proper(X))
proper(tail(X)) → tail(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
incr(ok(X)) → ok(incr(X))
s(ok(X)) → ok(s(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'(nats') → mark'(cons'(0', incr'(nats')))
active'(pairs') → mark'(cons'(0', incr'(odds')))
active'(odds') → mark'(incr'(pairs'))
active'(incr'(cons'(X, XS))) → mark'(cons'(s'(X), incr'(XS)))
active'(head'(cons'(X, XS))) → mark'(X)
active'(tail'(cons'(X, XS))) → mark'(XS)
active'(cons'(X1, X2)) → cons'(active'(X1), X2)
active'(incr'(X)) → incr'(active'(X))
active'(s'(X)) → s'(active'(X))
active'(head'(X)) → head'(active'(X))
active'(tail'(X)) → tail'(active'(X))
cons'(mark'(X1), X2) → mark'(cons'(X1, X2))
incr'(mark'(X)) → mark'(incr'(X))
s'(mark'(X)) → mark'(s'(X))
head'(mark'(X)) → mark'(head'(X))
tail'(mark'(X)) → mark'(tail'(X))
proper'(nats') → ok'(nats')
proper'(cons'(X1, X2)) → cons'(proper'(X1), proper'(X2))
proper'(0') → ok'(0')
proper'(incr'(X)) → incr'(proper'(X))
proper'(pairs') → ok'(pairs')
proper'(odds') → ok'(odds')
proper'(s'(X)) → s'(proper'(X))
proper'(head'(X)) → head'(proper'(X))
proper'(tail'(X)) → tail'(proper'(X))
cons'(ok'(X1), ok'(X2)) → ok'(cons'(X1, X2))
incr'(ok'(X)) → ok'(incr'(X))
s'(ok'(X)) → ok'(s'(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'(nats') → mark'(cons'(0', incr'(nats')))
active'(pairs') → mark'(cons'(0', incr'(odds')))
active'(odds') → mark'(incr'(pairs'))
active'(incr'(cons'(X, XS))) → mark'(cons'(s'(X), incr'(XS)))
active'(head'(cons'(X, XS))) → mark'(X)
active'(tail'(cons'(X, XS))) → mark'(XS)
active'(cons'(X1, X2)) → cons'(active'(X1), X2)
active'(incr'(X)) → incr'(active'(X))
active'(s'(X)) → s'(active'(X))
active'(head'(X)) → head'(active'(X))
active'(tail'(X)) → tail'(active'(X))
cons'(mark'(X1), X2) → mark'(cons'(X1, X2))
incr'(mark'(X)) → mark'(incr'(X))
s'(mark'(X)) → mark'(s'(X))
head'(mark'(X)) → mark'(head'(X))
tail'(mark'(X)) → mark'(tail'(X))
proper'(nats') → ok'(nats')
proper'(cons'(X1, X2)) → cons'(proper'(X1), proper'(X2))
proper'(0') → ok'(0')
proper'(incr'(X)) → incr'(proper'(X))
proper'(pairs') → ok'(pairs')
proper'(odds') → ok'(odds')
proper'(s'(X)) → s'(proper'(X))
proper'(head'(X)) → head'(proper'(X))
proper'(tail'(X)) → tail'(proper'(X))
cons'(ok'(X1), ok'(X2)) → ok'(cons'(X1, X2))
incr'(ok'(X)) → ok'(incr'(X))
s'(ok'(X)) → ok'(s'(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' :: nats':0':mark':pairs':odds':ok' → nats':0':mark':pairs':odds':ok'
nats' :: nats':0':mark':pairs':odds':ok'
mark' :: nats':0':mark':pairs':odds':ok' → nats':0':mark':pairs':odds':ok'
cons' :: nats':0':mark':pairs':odds':ok' → nats':0':mark':pairs':odds':ok' → nats':0':mark':pairs':odds':ok'
0' :: nats':0':mark':pairs':odds':ok'
incr' :: nats':0':mark':pairs':odds':ok' → nats':0':mark':pairs':odds':ok'
pairs' :: nats':0':mark':pairs':odds':ok'
odds' :: nats':0':mark':pairs':odds':ok'
s' :: nats':0':mark':pairs':odds':ok' → nats':0':mark':pairs':odds':ok'
head' :: nats':0':mark':pairs':odds':ok' → nats':0':mark':pairs':odds':ok'
tail' :: nats':0':mark':pairs':odds':ok' → nats':0':mark':pairs':odds':ok'
proper' :: nats':0':mark':pairs':odds':ok' → nats':0':mark':pairs':odds':ok'
ok' :: nats':0':mark':pairs':odds':ok' → nats':0':mark':pairs':odds':ok'
top' :: nats':0':mark':pairs':odds':ok' → top'
_hole_nats':0':mark':pairs':odds':ok'1 :: nats':0':mark':pairs':odds':ok'
_hole_top'2 :: top'
_gen_nats':0':mark':pairs':odds':ok'3 :: Nat → nats':0':mark':pairs':odds':ok'

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

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

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

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

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

Induction Base:
cons'(_gen_nats':0':mark':pairs':odds':ok'3(+(1, 0)), _gen_nats':0':mark':pairs':odds':ok'3(b))

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

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

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

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

Proved the following rewrite lemma:
incr'(_gen_nats':0':mark':pairs':odds':ok'3(+(1, _n1673))) → _*4, rt ∈ Ω(_n1673)

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

Induction Step:
incr'(_gen_nats':0':mark':pairs':odds':ok'3(+(1, +(_$n1674, 1)))) →_R^Ω(1)
mark'(incr'(_gen_nats':0':mark':pairs':odds':ok'3(+(1, _$n1674)))) →_IH
mark'(_*4)

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

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

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

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

Proved the following rewrite lemma:
s'(_gen_nats':0':mark':pairs':odds':ok'3(+(1, _n2844))) → _*4, rt ∈ Ω(_n2844)

Induction Base:
s'(_gen_nats':0':mark':pairs':odds':ok'3(+(1, 0)))

Induction Step:
s'(_gen_nats':0':mark':pairs':odds':ok'3(+(1, +(_$n2845, 1)))) →_R^Ω(1)
mark'(s'(_gen_nats':0':mark':pairs':odds':ok'3(+(1, _$n2845)))) →_IH
mark'(_*4)

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

Generator Equations:
_gen_nats':0':mark':pairs':odds':ok'3(0) ⇔ nats'
_gen_nats':0':mark':pairs':odds':ok'3(+(x, 1)) ⇔ mark'(_gen_nats':0':mark':pairs':odds':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_nats':0':mark':pairs':odds':ok'3(+(1, _n4139))) → _*4, rt ∈ Ω(_n4139)

Induction Base:
head'(_gen_nats':0':mark':pairs':odds':ok'3(+(1, 0)))

Induction Step:
head'(_gen_nats':0':mark':pairs':odds':ok'3(+(1, +(_$n4140, 1)))) →_R^Ω(1)
mark'(head'(_gen_nats':0':mark':pairs':odds':ok'3(+(1, _$n4140)))) →_IH
mark'(_*4)

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

Lemmas:
cons'(_gen_nats':0':mark':pairs':odds':ok'3(+(1, _n5)), _gen_nats':0':mark':pairs':odds':ok'3(b)) → _*4, rt ∈ Ω(_n5)
incr'(_gen_nats':0':mark':pairs':odds':ok'3(+(1, _n1673))) → _*4, rt ∈ Ω(_n1673)
s'(_gen_nats':0':mark':pairs':odds':ok'3(+(1, _n2844))) → _*4, rt ∈ Ω(_n2844)
head'(_gen_nats':0':mark':pairs':odds':ok'3(+(1, _n4139))) → _*4, rt ∈ Ω(_n4139)

Generator Equations:
_gen_nats':0':mark':pairs':odds':ok'3(0) ⇔ nats'
_gen_nats':0':mark':pairs':odds':ok'3(+(x, 1)) ⇔ mark'(_gen_nats':0':mark':pairs':odds':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_nats':0':mark':pairs':odds':ok'3(+(1, _n5558))) → _*4, rt ∈ Ω(_n5558)

Induction Base:
tail'(_gen_nats':0':mark':pairs':odds':ok'3(+(1, 0)))

Induction Step:
tail'(_gen_nats':0':mark':pairs':odds':ok'3(+(1, +(_$n5559, 1)))) →_R^Ω(1)
mark'(tail'(_gen_nats':0':mark':pairs':odds':ok'3(+(1, _$n5559)))) →_IH
mark'(_*4)

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

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

No more defined symbols left to analyse.

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