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

active(f(b, X, c)) → mark(f(X, c, X))
active(c) → mark(b)
active(f(X1, X2, X3)) → f(X1, active(X2), X3)
f(X1, mark(X2), X3) → mark(f(X1, X2, X3))
proper(f(X1, X2, X3)) → f(proper(X1), proper(X2), proper(X3))
proper(b) → ok(b)
proper(c) → ok(c)
f(ok(X1), ok(X2), ok(X3)) → ok(f(X1, X2, X3))
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'(f'(b', X, c')) → mark'(f'(X, c', X))
active'(c') → mark'(b')
active'(f'(X1, X2, X3)) → f'(X1, active'(X2), X3)
f'(X1, mark'(X2), X3) → mark'(f'(X1, X2, X3))
proper'(f'(X1, X2, X3)) → f'(proper'(X1), proper'(X2), proper'(X3))
proper'(b') → ok'(b')
proper'(c') → ok'(c')
f'(ok'(X1), ok'(X2), ok'(X3)) → ok'(f'(X1, X2, X3))
top'(mark'(X)) → top'(proper'(X))
top'(ok'(X)) → top'(active'(X))

Rewrite Strategy: INNERMOST

Infered types.

Rules:
active'(f'(b', X, c')) → mark'(f'(X, c', X))
active'(c') → mark'(b')
active'(f'(X1, X2, X3)) → f'(X1, active'(X2), X3)
f'(X1, mark'(X2), X3) → mark'(f'(X1, X2, X3))
proper'(f'(X1, X2, X3)) → f'(proper'(X1), proper'(X2), proper'(X3))
proper'(b') → ok'(b')
proper'(c') → ok'(c')
f'(ok'(X1), ok'(X2), ok'(X3)) → ok'(f'(X1, X2, X3))
top'(mark'(X)) → top'(proper'(X))
top'(ok'(X)) → top'(active'(X))

Types:
active' :: b':c':mark':ok' → b':c':mark':ok'
f' :: b':c':mark':ok' → b':c':mark':ok' → b':c':mark':ok' → b':c':mark':ok'
b' :: b':c':mark':ok'
c' :: b':c':mark':ok'
mark' :: b':c':mark':ok' → b':c':mark':ok'
proper' :: b':c':mark':ok' → b':c':mark':ok'
ok' :: b':c':mark':ok' → b':c':mark':ok'
top' :: b':c':mark':ok' → top'
_hole_b':c':mark':ok'1 :: b':c':mark':ok'
_hole_top'2 :: top'
_gen_b':c':mark':ok'3 :: Nat → b':c':mark':ok'

Heuristically decided to analyse the following defined symbols:
active', f', proper', top'

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

Generator Equations:
_gen_b':c':mark':ok'3(0) ⇔ b'
_gen_b':c':mark':ok'3(+(x, 1)) ⇔ mark'(_gen_b':c':mark':ok'3(x))

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

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

Proved the following rewrite lemma:
f'(_gen_b':c':mark':ok'3(a), _gen_b':c':mark':ok'3(+(1, _n5)), _gen_b':c':mark':ok'3(c)) → _*4, rt ∈ Ω(_n5)

Induction Base:
f'(_gen_b':c':mark':ok'3(a), _gen_b':c':mark':ok'3(+(1, 0)), _gen_b':c':mark':ok'3(c))

Induction Step:
f'(_gen_b':c':mark':ok'3(_a1018), _gen_b':c':mark':ok'3(+(1, +(_$n6, 1))), _gen_b':c':mark':ok'3(_c1019)) →_R^Ω(1)
mark'(f'(_gen_b':c':mark':ok'3(_a1018), _gen_b':c':mark':ok'3(+(1, _$n6)), _gen_b':c':mark':ok'3(_c1019))) →_IH
mark'(_*4)

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

Lemmas:
f'(_gen_b':c':mark':ok'3(a), _gen_b':c':mark':ok'3(+(1, _n5)), _gen_b':c':mark':ok'3(c)) → _*4, rt ∈ Ω(_n5)

Generator Equations:
_gen_b':c':mark':ok'3(0) ⇔ b'
_gen_b':c':mark':ok'3(+(x, 1)) ⇔ mark'(_gen_b':c':mark':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'.

Lemmas:
f'(_gen_b':c':mark':ok'3(a), _gen_b':c':mark':ok'3(+(1, _n5)), _gen_b':c':mark':ok'3(c)) → _*4, rt ∈ Ω(_n5)

Generator Equations:
_gen_b':c':mark':ok'3(0) ⇔ b'
_gen_b':c':mark':ok'3(+(x, 1)) ⇔ mark'(_gen_b':c':mark':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'.

Lemmas:
f'(_gen_b':c':mark':ok'3(a), _gen_b':c':mark':ok'3(+(1, _n5)), _gen_b':c':mark':ok'3(c)) → _*4, rt ∈ Ω(_n5)

Generator Equations:
_gen_b':c':mark':ok'3(0) ⇔ b'
_gen_b':c':mark':ok'3(+(x, 1)) ⇔ mark'(_gen_b':c':mark':ok'3(x))

The following defined symbols remain to be analysed:
top'

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

Lemmas:
f'(_gen_b':c':mark':ok'3(a), _gen_b':c':mark':ok'3(+(1, _n5)), _gen_b':c':mark':ok'3(c)) → _*4, rt ∈ Ω(_n5)

Generator Equations:
_gen_b':c':mark':ok'3(0) ⇔ b'
_gen_b':c':mark':ok'3(+(x, 1)) ⇔ mark'(_gen_b':c':mark':ok'3(x))

No more defined symbols left to analyse.

The lowerbound Ω(n) was proven with the following lemma:
f'(_gen_b':c':mark':ok'3(a), _gen_b':c':mark':ok'3(+(1, _n5)), _gen_b':c':mark':ok'3(c)) → _*4, rt ∈ Ω(_n5)