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

active(and(tt, X)) → mark(X)
active(plus(N, 0)) → mark(N)
active(plus(N, s(M))) → mark(s(plus(N, M)))
active(and(X1, X2)) → and(active(X1), X2)
active(plus(X1, X2)) → plus(active(X1), X2)
active(plus(X1, X2)) → plus(X1, active(X2))
active(s(X)) → s(active(X))
and(mark(X1), X2) → mark(and(X1, X2))
plus(mark(X1), X2) → mark(plus(X1, X2))
plus(X1, mark(X2)) → mark(plus(X1, X2))
s(mark(X)) → mark(s(X))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(plus(X1, X2)) → plus(proper(X1), proper(X2))
proper(0) → ok(0)
proper(s(X)) → s(proper(X))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
plus(ok(X1), ok(X2)) → ok(plus(X1, X2))
s(ok(X)) → ok(s(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'(and'(tt', X)) → mark'(X)
active'(plus'(N, 0')) → mark'(N)
active'(plus'(N, s'(M))) → mark'(s'(plus'(N, M)))
active'(and'(X1, X2)) → and'(active'(X1), X2)
active'(plus'(X1, X2)) → plus'(active'(X1), X2)
active'(plus'(X1, X2)) → plus'(X1, active'(X2))
active'(s'(X)) → s'(active'(X))
and'(mark'(X1), X2) → mark'(and'(X1, X2))
plus'(mark'(X1), X2) → mark'(plus'(X1, X2))
plus'(X1, mark'(X2)) → mark'(plus'(X1, X2))
s'(mark'(X)) → mark'(s'(X))
proper'(and'(X1, X2)) → and'(proper'(X1), proper'(X2))
proper'(tt') → ok'(tt')
proper'(plus'(X1, X2)) → plus'(proper'(X1), proper'(X2))
proper'(0') → ok'(0')
proper'(s'(X)) → s'(proper'(X))
and'(ok'(X1), ok'(X2)) → ok'(and'(X1, X2))
plus'(ok'(X1), ok'(X2)) → ok'(plus'(X1, X2))
s'(ok'(X)) → ok'(s'(X))
top'(mark'(X)) → top'(proper'(X))
top'(ok'(X)) → top'(active'(X))

Rewrite Strategy: INNERMOST

Infered types.

Rules:
active'(and'(tt', X)) → mark'(X)
active'(plus'(N, 0')) → mark'(N)
active'(plus'(N, s'(M))) → mark'(s'(plus'(N, M)))
active'(and'(X1, X2)) → and'(active'(X1), X2)
active'(plus'(X1, X2)) → plus'(active'(X1), X2)
active'(plus'(X1, X2)) → plus'(X1, active'(X2))
active'(s'(X)) → s'(active'(X))
and'(mark'(X1), X2) → mark'(and'(X1, X2))
plus'(mark'(X1), X2) → mark'(plus'(X1, X2))
plus'(X1, mark'(X2)) → mark'(plus'(X1, X2))
s'(mark'(X)) → mark'(s'(X))
proper'(and'(X1, X2)) → and'(proper'(X1), proper'(X2))
proper'(tt') → ok'(tt')
proper'(plus'(X1, X2)) → plus'(proper'(X1), proper'(X2))
proper'(0') → ok'(0')
proper'(s'(X)) → s'(proper'(X))
and'(ok'(X1), ok'(X2)) → ok'(and'(X1, X2))
plus'(ok'(X1), ok'(X2)) → ok'(plus'(X1, X2))
s'(ok'(X)) → ok'(s'(X))
top'(mark'(X)) → top'(proper'(X))
top'(ok'(X)) → top'(active'(X))

Types:
active' :: tt':mark':0':ok' → tt':mark':0':ok'
and' :: tt':mark':0':ok' → tt':mark':0':ok' → tt':mark':0':ok'
tt' :: tt':mark':0':ok'
mark' :: tt':mark':0':ok' → tt':mark':0':ok'
plus' :: tt':mark':0':ok' → tt':mark':0':ok' → tt':mark':0':ok'
0' :: tt':mark':0':ok'
s' :: tt':mark':0':ok' → tt':mark':0':ok'
proper' :: tt':mark':0':ok' → tt':mark':0':ok'
ok' :: tt':mark':0':ok' → tt':mark':0':ok'
top' :: tt':mark':0':ok' → top'
_hole_tt':mark':0':ok'1 :: tt':mark':0':ok'
_hole_top'2 :: top'
_gen_tt':mark':0':ok'3 :: Nat → tt':mark':0':ok'

Heuristically decided to analyse the following defined symbols:
active', s', plus', and', proper', top'

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

Generator Equations:
_gen_tt':mark':0':ok'3(0) ⇔ tt'
_gen_tt':mark':0':ok'3(+(x, 1)) ⇔ mark'(_gen_tt':mark':0':ok'3(x))

The following defined symbols remain to be analysed:
s', active', plus', and', proper', top'

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

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

Induction Base:
s'(_gen_tt':mark':0':ok'3(+(1, 0)))

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

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

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

Generator Equations:
_gen_tt':mark':0':ok'3(0) ⇔ tt'
_gen_tt':mark':0':ok'3(+(x, 1)) ⇔ mark'(_gen_tt':mark':0':ok'3(x))

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

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

Proved the following rewrite lemma:
plus'(_gen_tt':mark':0':ok'3(+(1, _n889)), _gen_tt':mark':0':ok'3(b)) → _*4, rt ∈ Ω(_n889)

Induction Base:
plus'(_gen_tt':mark':0':ok'3(+(1, 0)), _gen_tt':mark':0':ok'3(b))

Induction Step:
plus'(_gen_tt':mark':0':ok'3(+(1, +(_$n890, 1))), _gen_tt':mark':0':ok'3(_b1926)) →_R^Ω(1)
mark'(plus'(_gen_tt':mark':0':ok'3(+(1, _$n890)), _gen_tt':mark':0':ok'3(_b1926))) →_IH
mark'(_*4)

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

Lemmas:
s'(_gen_tt':mark':0':ok'3(+(1, _n5))) → _*4, rt ∈ Ω(_n5)
plus'(_gen_tt':mark':0':ok'3(+(1, _n889)), _gen_tt':mark':0':ok'3(b)) → _*4, rt ∈ Ω(_n889)

Generator Equations:
_gen_tt':mark':0':ok'3(0) ⇔ tt'
_gen_tt':mark':0':ok'3(+(x, 1)) ⇔ mark'(_gen_tt':mark':0':ok'3(x))

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

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

Proved the following rewrite lemma:
and'(_gen_tt':mark':0':ok'3(+(1, _n2843)), _gen_tt':mark':0':ok'3(b)) → _*4, rt ∈ Ω(_n2843)

Induction Base:
and'(_gen_tt':mark':0':ok'3(+(1, 0)), _gen_tt':mark':0':ok'3(b))

Induction Step:
and'(_gen_tt':mark':0':ok'3(+(1, +(_$n2844, 1))), _gen_tt':mark':0':ok'3(_b3988)) →_R^Ω(1)
mark'(and'(_gen_tt':mark':0':ok'3(+(1, _$n2844)), _gen_tt':mark':0':ok'3(_b3988))) →_IH
mark'(_*4)

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

Lemmas:
s'(_gen_tt':mark':0':ok'3(+(1, _n5))) → _*4, rt ∈ Ω(_n5)
plus'(_gen_tt':mark':0':ok'3(+(1, _n889)), _gen_tt':mark':0':ok'3(b)) → _*4, rt ∈ Ω(_n889)
and'(_gen_tt':mark':0':ok'3(+(1, _n2843)), _gen_tt':mark':0':ok'3(b)) → _*4, rt ∈ Ω(_n2843)

Generator Equations:
_gen_tt':mark':0':ok'3(0) ⇔ tt'
_gen_tt':mark':0':ok'3(+(x, 1)) ⇔ mark'(_gen_tt':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_tt':mark':0':ok'3(0) ⇔ tt'
_gen_tt':mark':0':ok'3(+(x, 1)) ⇔ mark'(_gen_tt':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_tt':mark':0':ok'3(0) ⇔ tt'
_gen_tt':mark':0':ok'3(+(x, 1)) ⇔ mark'(_gen_tt':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_tt':mark':0':ok'3(0) ⇔ tt'
_gen_tt':mark':0':ok'3(+(x, 1)) ⇔ mark'(_gen_tt':mark':0':ok'3(x))

No more defined symbols left to analyse.

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