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

active(f(g(X), Y)) → mark(f(X, f(g(X), Y)))
active(f(X1, X2)) → f(active(X1), X2)
active(g(X)) → g(active(X))
f(mark(X1), X2) → mark(f(X1, X2))
g(mark(X)) → mark(g(X))
proper(f(X1, X2)) → f(proper(X1), proper(X2))
proper(g(X)) → g(proper(X))
f(ok(X1), ok(X2)) → ok(f(X1, X2))
g(ok(X)) → ok(g(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'(f'(g'(X), Y)) → mark'(f'(X, f'(g'(X), Y)))
active'(f'(X1, X2)) → f'(active'(X1), X2)
active'(g'(X)) → g'(active'(X))
f'(mark'(X1), X2) → mark'(f'(X1, X2))
g'(mark'(X)) → mark'(g'(X))
proper'(f'(X1, X2)) → f'(proper'(X1), proper'(X2))
proper'(g'(X)) → g'(proper'(X))
f'(ok'(X1), ok'(X2)) → ok'(f'(X1, X2))
g'(ok'(X)) → ok'(g'(X))
top'(mark'(X)) → top'(proper'(X))
top'(ok'(X)) → top'(active'(X))

Rewrite Strategy: INNERMOST

Infered types.

Rules:
active'(f'(g'(X), Y)) → mark'(f'(X, f'(g'(X), Y)))
active'(f'(X1, X2)) → f'(active'(X1), X2)
active'(g'(X)) → g'(active'(X))
f'(mark'(X1), X2) → mark'(f'(X1, X2))
g'(mark'(X)) → mark'(g'(X))
proper'(f'(X1, X2)) → f'(proper'(X1), proper'(X2))
proper'(g'(X)) → g'(proper'(X))
f'(ok'(X1), ok'(X2)) → ok'(f'(X1, X2))
g'(ok'(X)) → ok'(g'(X))
top'(mark'(X)) → top'(proper'(X))
top'(ok'(X)) → top'(active'(X))

Types:
active' :: mark':ok' → mark':ok'
f' :: mark':ok' → mark':ok' → mark':ok'
g' :: mark':ok' → mark':ok'
mark' :: mark':ok' → mark':ok'
proper' :: mark':ok' → mark':ok'
ok' :: mark':ok' → mark':ok'
top' :: mark':ok' → top'
_hole_mark':ok'1 :: mark':ok'
_hole_top'2 :: top'
_gen_mark':ok'3 :: Nat → mark':ok'

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

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

Rules:
active'(f'(g'(X), Y)) → mark'(f'(X, f'(g'(X), Y)))
active'(f'(X1, X2)) → f'(active'(X1), X2)
active'(g'(X)) → g'(active'(X))
f'(mark'(X1), X2) → mark'(f'(X1, X2))
g'(mark'(X)) → mark'(g'(X))
proper'(f'(X1, X2)) → f'(proper'(X1), proper'(X2))
proper'(g'(X)) → g'(proper'(X))
f'(ok'(X1), ok'(X2)) → ok'(f'(X1, X2))
g'(ok'(X)) → ok'(g'(X))
top'(mark'(X)) → top'(proper'(X))
top'(ok'(X)) → top'(active'(X))

Types:
active' :: mark':ok' → mark':ok'
f' :: mark':ok' → mark':ok' → mark':ok'
g' :: mark':ok' → mark':ok'
mark' :: mark':ok' → mark':ok'
proper' :: mark':ok' → mark':ok'
ok' :: mark':ok' → mark':ok'
top' :: mark':ok' → top'
_hole_mark':ok'1 :: mark':ok'
_hole_top'2 :: top'
_gen_mark':ok'3 :: Nat → mark':ok'

Generator Equations:
_gen_mark':ok'3(0) ⇔ _hole_mark':ok'1
_gen_mark':ok'3(+(x, 1)) ⇔ mark'(_gen_mark':ok'3(x))

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

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

Proved the following rewrite lemma:
f'(_gen_mark':ok'3(+(1, _n5)), _gen_mark':ok'3(b)) → _*4, rt ∈ Ω(n5)

Induction Base:
f'(_gen_mark':ok'3(+(1, 0)), _gen_mark':ok'3(b))

Induction Step:
f'(_gen_mark':ok'3(+(1, +(_\$n6, 1))), _gen_mark':ok'3(_b610)) →RΩ(1)
mark'(f'(_gen_mark':ok'3(+(1, _\$n6)), _gen_mark':ok'3(_b610))) →IH
mark'(_*4)

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

Rules:
active'(f'(g'(X), Y)) → mark'(f'(X, f'(g'(X), Y)))
active'(f'(X1, X2)) → f'(active'(X1), X2)
active'(g'(X)) → g'(active'(X))
f'(mark'(X1), X2) → mark'(f'(X1, X2))
g'(mark'(X)) → mark'(g'(X))
proper'(f'(X1, X2)) → f'(proper'(X1), proper'(X2))
proper'(g'(X)) → g'(proper'(X))
f'(ok'(X1), ok'(X2)) → ok'(f'(X1, X2))
g'(ok'(X)) → ok'(g'(X))
top'(mark'(X)) → top'(proper'(X))
top'(ok'(X)) → top'(active'(X))

Types:
active' :: mark':ok' → mark':ok'
f' :: mark':ok' → mark':ok' → mark':ok'
g' :: mark':ok' → mark':ok'
mark' :: mark':ok' → mark':ok'
proper' :: mark':ok' → mark':ok'
ok' :: mark':ok' → mark':ok'
top' :: mark':ok' → top'
_hole_mark':ok'1 :: mark':ok'
_hole_top'2 :: top'
_gen_mark':ok'3 :: Nat → mark':ok'

Lemmas:
f'(_gen_mark':ok'3(+(1, _n5)), _gen_mark':ok'3(b)) → _*4, rt ∈ Ω(n5)

Generator Equations:
_gen_mark':ok'3(0) ⇔ _hole_mark':ok'1
_gen_mark':ok'3(+(x, 1)) ⇔ mark'(_gen_mark':ok'3(x))

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

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

Proved the following rewrite lemma:
g'(_gen_mark':ok'3(+(1, _n1085))) → _*4, rt ∈ Ω(n1085)

Induction Base:
g'(_gen_mark':ok'3(+(1, 0)))

Induction Step:
g'(_gen_mark':ok'3(+(1, +(_\$n1086, 1)))) →RΩ(1)
mark'(g'(_gen_mark':ok'3(+(1, _\$n1086)))) →IH
mark'(_*4)

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

Rules:
active'(f'(g'(X), Y)) → mark'(f'(X, f'(g'(X), Y)))
active'(f'(X1, X2)) → f'(active'(X1), X2)
active'(g'(X)) → g'(active'(X))
f'(mark'(X1), X2) → mark'(f'(X1, X2))
g'(mark'(X)) → mark'(g'(X))
proper'(f'(X1, X2)) → f'(proper'(X1), proper'(X2))
proper'(g'(X)) → g'(proper'(X))
f'(ok'(X1), ok'(X2)) → ok'(f'(X1, X2))
g'(ok'(X)) → ok'(g'(X))
top'(mark'(X)) → top'(proper'(X))
top'(ok'(X)) → top'(active'(X))

Types:
active' :: mark':ok' → mark':ok'
f' :: mark':ok' → mark':ok' → mark':ok'
g' :: mark':ok' → mark':ok'
mark' :: mark':ok' → mark':ok'
proper' :: mark':ok' → mark':ok'
ok' :: mark':ok' → mark':ok'
top' :: mark':ok' → top'
_hole_mark':ok'1 :: mark':ok'
_hole_top'2 :: top'
_gen_mark':ok'3 :: Nat → mark':ok'

Lemmas:
f'(_gen_mark':ok'3(+(1, _n5)), _gen_mark':ok'3(b)) → _*4, rt ∈ Ω(n5)
g'(_gen_mark':ok'3(+(1, _n1085))) → _*4, rt ∈ Ω(n1085)

Generator Equations:
_gen_mark':ok'3(0) ⇔ _hole_mark':ok'1
_gen_mark':ok'3(+(x, 1)) ⇔ mark'(_gen_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'.

Rules:
active'(f'(g'(X), Y)) → mark'(f'(X, f'(g'(X), Y)))
active'(f'(X1, X2)) → f'(active'(X1), X2)
active'(g'(X)) → g'(active'(X))
f'(mark'(X1), X2) → mark'(f'(X1, X2))
g'(mark'(X)) → mark'(g'(X))
proper'(f'(X1, X2)) → f'(proper'(X1), proper'(X2))
proper'(g'(X)) → g'(proper'(X))
f'(ok'(X1), ok'(X2)) → ok'(f'(X1, X2))
g'(ok'(X)) → ok'(g'(X))
top'(mark'(X)) → top'(proper'(X))
top'(ok'(X)) → top'(active'(X))

Types:
active' :: mark':ok' → mark':ok'
f' :: mark':ok' → mark':ok' → mark':ok'
g' :: mark':ok' → mark':ok'
mark' :: mark':ok' → mark':ok'
proper' :: mark':ok' → mark':ok'
ok' :: mark':ok' → mark':ok'
top' :: mark':ok' → top'
_hole_mark':ok'1 :: mark':ok'
_hole_top'2 :: top'
_gen_mark':ok'3 :: Nat → mark':ok'

Lemmas:
f'(_gen_mark':ok'3(+(1, _n5)), _gen_mark':ok'3(b)) → _*4, rt ∈ Ω(n5)
g'(_gen_mark':ok'3(+(1, _n1085))) → _*4, rt ∈ Ω(n1085)

Generator Equations:
_gen_mark':ok'3(0) ⇔ _hole_mark':ok'1
_gen_mark':ok'3(+(x, 1)) ⇔ mark'(_gen_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'.

Rules:
active'(f'(g'(X), Y)) → mark'(f'(X, f'(g'(X), Y)))
active'(f'(X1, X2)) → f'(active'(X1), X2)
active'(g'(X)) → g'(active'(X))
f'(mark'(X1), X2) → mark'(f'(X1, X2))
g'(mark'(X)) → mark'(g'(X))
proper'(f'(X1, X2)) → f'(proper'(X1), proper'(X2))
proper'(g'(X)) → g'(proper'(X))
f'(ok'(X1), ok'(X2)) → ok'(f'(X1, X2))
g'(ok'(X)) → ok'(g'(X))
top'(mark'(X)) → top'(proper'(X))
top'(ok'(X)) → top'(active'(X))

Types:
active' :: mark':ok' → mark':ok'
f' :: mark':ok' → mark':ok' → mark':ok'
g' :: mark':ok' → mark':ok'
mark' :: mark':ok' → mark':ok'
proper' :: mark':ok' → mark':ok'
ok' :: mark':ok' → mark':ok'
top' :: mark':ok' → top'
_hole_mark':ok'1 :: mark':ok'
_hole_top'2 :: top'
_gen_mark':ok'3 :: Nat → mark':ok'

Lemmas:
f'(_gen_mark':ok'3(+(1, _n5)), _gen_mark':ok'3(b)) → _*4, rt ∈ Ω(n5)
g'(_gen_mark':ok'3(+(1, _n1085))) → _*4, rt ∈ Ω(n1085)

Generator Equations:
_gen_mark':ok'3(0) ⇔ _hole_mark':ok'1
_gen_mark':ok'3(+(x, 1)) ⇔ mark'(_gen_mark':ok'3(x))

The following defined symbols remain to be analysed:
top'

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

Rules:
active'(f'(g'(X), Y)) → mark'(f'(X, f'(g'(X), Y)))
active'(f'(X1, X2)) → f'(active'(X1), X2)
active'(g'(X)) → g'(active'(X))
f'(mark'(X1), X2) → mark'(f'(X1, X2))
g'(mark'(X)) → mark'(g'(X))
proper'(f'(X1, X2)) → f'(proper'(X1), proper'(X2))
proper'(g'(X)) → g'(proper'(X))
f'(ok'(X1), ok'(X2)) → ok'(f'(X1, X2))
g'(ok'(X)) → ok'(g'(X))
top'(mark'(X)) → top'(proper'(X))
top'(ok'(X)) → top'(active'(X))

Types:
active' :: mark':ok' → mark':ok'
f' :: mark':ok' → mark':ok' → mark':ok'
g' :: mark':ok' → mark':ok'
mark' :: mark':ok' → mark':ok'
proper' :: mark':ok' → mark':ok'
ok' :: mark':ok' → mark':ok'
top' :: mark':ok' → top'
_hole_mark':ok'1 :: mark':ok'
_hole_top'2 :: top'
_gen_mark':ok'3 :: Nat → mark':ok'

Lemmas:
f'(_gen_mark':ok'3(+(1, _n5)), _gen_mark':ok'3(b)) → _*4, rt ∈ Ω(n5)
g'(_gen_mark':ok'3(+(1, _n1085))) → _*4, rt ∈ Ω(n1085)

Generator Equations:
_gen_mark':ok'3(0) ⇔ _hole_mark':ok'1
_gen_mark':ok'3(+(x, 1)) ⇔ mark'(_gen_mark':ok'3(x))

No more defined symbols left to analyse.

The lowerbound Ω(n) was proven with the following lemma:
f'(_gen_mark':ok'3(+(1, _n5)), _gen_mark':ok'3(b)) → _*4, rt ∈ Ω(n5)