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

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

Rewrite Strategy: INNERMOST

Infered types.

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

Types:
active' :: a':b':mark':ok' → a':b':mark':ok'
f' :: a':b':mark':ok' → a':b':mark':ok' → a':b':mark':ok'
mark' :: a':b':mark':ok' → a':b':mark':ok'
a' :: a':b':mark':ok'
b' :: a':b':mark':ok'
proper' :: a':b':mark':ok' → a':b':mark':ok'
ok' :: a':b':mark':ok' → a':b':mark':ok'
top' :: a':b':mark':ok' → top'
_hole_a':b':mark':ok'1 :: a':b':mark':ok'
_hole_top'2 :: top'
_gen_a':b':mark':ok'3 :: Nat → a':b':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'

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

Types:
active' :: a':b':mark':ok' → a':b':mark':ok'
f' :: a':b':mark':ok' → a':b':mark':ok' → a':b':mark':ok'
mark' :: a':b':mark':ok' → a':b':mark':ok'
a' :: a':b':mark':ok'
b' :: a':b':mark':ok'
proper' :: a':b':mark':ok' → a':b':mark':ok'
ok' :: a':b':mark':ok' → a':b':mark':ok'
top' :: a':b':mark':ok' → top'
_hole_a':b':mark':ok'1 :: a':b':mark':ok'
_hole_top'2 :: top'
_gen_a':b':mark':ok'3 :: Nat → a':b':mark':ok'

Generator Equations:
_gen_a':b':mark':ok'3(0) ⇔ a'
_gen_a':b':mark':ok'3(+(x, 1)) ⇔ mark'(_gen_a':b':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_a':b':mark':ok'3(+(1, _n5)), _gen_a':b':mark':ok'3(b)) → _*4, rt ∈ Ω(n5)

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

Induction Step:
f'(_gen_a':b':mark':ok'3(+(1, +(_\$n6, 1))), _gen_a':b':mark':ok'3(_b610)) →RΩ(1)
mark'(f'(_gen_a':b':mark':ok'3(+(1, _\$n6)), _gen_a':b':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'(X, X)) → mark'(f'(a', b'))
active'(b') → mark'(a')
active'(f'(X1, X2)) → f'(active'(X1), X2)
f'(mark'(X1), X2) → mark'(f'(X1, X2))
proper'(f'(X1, X2)) → f'(proper'(X1), proper'(X2))
proper'(a') → ok'(a')
proper'(b') → ok'(b')
f'(ok'(X1), ok'(X2)) → ok'(f'(X1, X2))
top'(mark'(X)) → top'(proper'(X))
top'(ok'(X)) → top'(active'(X))

Types:
active' :: a':b':mark':ok' → a':b':mark':ok'
f' :: a':b':mark':ok' → a':b':mark':ok' → a':b':mark':ok'
mark' :: a':b':mark':ok' → a':b':mark':ok'
a' :: a':b':mark':ok'
b' :: a':b':mark':ok'
proper' :: a':b':mark':ok' → a':b':mark':ok'
ok' :: a':b':mark':ok' → a':b':mark':ok'
top' :: a':b':mark':ok' → top'
_hole_a':b':mark':ok'1 :: a':b':mark':ok'
_hole_top'2 :: top'
_gen_a':b':mark':ok'3 :: Nat → a':b':mark':ok'

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

Generator Equations:
_gen_a':b':mark':ok'3(0) ⇔ a'
_gen_a':b':mark':ok'3(+(x, 1)) ⇔ mark'(_gen_a':b':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'(X, X)) → mark'(f'(a', b'))
active'(b') → mark'(a')
active'(f'(X1, X2)) → f'(active'(X1), X2)
f'(mark'(X1), X2) → mark'(f'(X1, X2))
proper'(f'(X1, X2)) → f'(proper'(X1), proper'(X2))
proper'(a') → ok'(a')
proper'(b') → ok'(b')
f'(ok'(X1), ok'(X2)) → ok'(f'(X1, X2))
top'(mark'(X)) → top'(proper'(X))
top'(ok'(X)) → top'(active'(X))

Types:
active' :: a':b':mark':ok' → a':b':mark':ok'
f' :: a':b':mark':ok' → a':b':mark':ok' → a':b':mark':ok'
mark' :: a':b':mark':ok' → a':b':mark':ok'
a' :: a':b':mark':ok'
b' :: a':b':mark':ok'
proper' :: a':b':mark':ok' → a':b':mark':ok'
ok' :: a':b':mark':ok' → a':b':mark':ok'
top' :: a':b':mark':ok' → top'
_hole_a':b':mark':ok'1 :: a':b':mark':ok'
_hole_top'2 :: top'
_gen_a':b':mark':ok'3 :: Nat → a':b':mark':ok'

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

Generator Equations:
_gen_a':b':mark':ok'3(0) ⇔ a'
_gen_a':b':mark':ok'3(+(x, 1)) ⇔ mark'(_gen_a':b':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'(X, X)) → mark'(f'(a', b'))
active'(b') → mark'(a')
active'(f'(X1, X2)) → f'(active'(X1), X2)
f'(mark'(X1), X2) → mark'(f'(X1, X2))
proper'(f'(X1, X2)) → f'(proper'(X1), proper'(X2))
proper'(a') → ok'(a')
proper'(b') → ok'(b')
f'(ok'(X1), ok'(X2)) → ok'(f'(X1, X2))
top'(mark'(X)) → top'(proper'(X))
top'(ok'(X)) → top'(active'(X))

Types:
active' :: a':b':mark':ok' → a':b':mark':ok'
f' :: a':b':mark':ok' → a':b':mark':ok' → a':b':mark':ok'
mark' :: a':b':mark':ok' → a':b':mark':ok'
a' :: a':b':mark':ok'
b' :: a':b':mark':ok'
proper' :: a':b':mark':ok' → a':b':mark':ok'
ok' :: a':b':mark':ok' → a':b':mark':ok'
top' :: a':b':mark':ok' → top'
_hole_a':b':mark':ok'1 :: a':b':mark':ok'
_hole_top'2 :: top'
_gen_a':b':mark':ok'3 :: Nat → a':b':mark':ok'

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

Generator Equations:
_gen_a':b':mark':ok'3(0) ⇔ a'
_gen_a':b':mark':ok'3(+(x, 1)) ⇔ mark'(_gen_a':b':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'(X, X)) → mark'(f'(a', b'))
active'(b') → mark'(a')
active'(f'(X1, X2)) → f'(active'(X1), X2)
f'(mark'(X1), X2) → mark'(f'(X1, X2))
proper'(f'(X1, X2)) → f'(proper'(X1), proper'(X2))
proper'(a') → ok'(a')
proper'(b') → ok'(b')
f'(ok'(X1), ok'(X2)) → ok'(f'(X1, X2))
top'(mark'(X)) → top'(proper'(X))
top'(ok'(X)) → top'(active'(X))

Types:
active' :: a':b':mark':ok' → a':b':mark':ok'
f' :: a':b':mark':ok' → a':b':mark':ok' → a':b':mark':ok'
mark' :: a':b':mark':ok' → a':b':mark':ok'
a' :: a':b':mark':ok'
b' :: a':b':mark':ok'
proper' :: a':b':mark':ok' → a':b':mark':ok'
ok' :: a':b':mark':ok' → a':b':mark':ok'
top' :: a':b':mark':ok' → top'
_hole_a':b':mark':ok'1 :: a':b':mark':ok'
_hole_top'2 :: top'
_gen_a':b':mark':ok'3 :: Nat → a':b':mark':ok'

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

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

No more defined symbols left to analyse.

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