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

a__f(f(a)) → a__f(g(f(a)))
mark(f(X)) → a__f(mark(X))
mark(a) → a
mark(g(X)) → g(X)
a__f(X) → f(X)

Rewrite Strategy: INNERMOST

Renamed function symbols to avoid clashes with predefined symbol.

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

a__f'(f'(a')) → a__f'(g'(f'(a')))
mark'(f'(X)) → a__f'(mark'(X))
mark'(a') → a'
mark'(g'(X)) → g'(X)
a__f'(X) → f'(X)

Rewrite Strategy: INNERMOST

Sliced the following arguments:
g'/0

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

a__f'(f'(a')) → a__f'(g')
mark'(f'(X)) → a__f'(mark'(X))
mark'(a') → a'
mark'(g') → g'
a__f'(X) → f'(X)

Rewrite Strategy: INNERMOST

Infered types.

Rules:
a__f'(f'(a')) → a__f'(g')
mark'(f'(X)) → a__f'(mark'(X))
mark'(a') → a'
mark'(g') → g'
a__f'(X) → f'(X)

Types:
a__f' :: a':f':g' → a':f':g'
f' :: a':f':g' → a':f':g'
a' :: a':f':g'
g' :: a':f':g'
mark' :: a':f':g' → a':f':g'
_hole_a':f':g'1 :: a':f':g'
_gen_a':f':g'2 :: Nat → a':f':g'

Heuristically decided to analyse the following defined symbols:
a__f', mark'

They will be analysed ascendingly in the following order:
a__f' < mark'

Rules:
a__f'(f'(a')) → a__f'(g')
mark'(f'(X)) → a__f'(mark'(X))
mark'(a') → a'
mark'(g') → g'
a__f'(X) → f'(X)

Types:
a__f' :: a':f':g' → a':f':g'
f' :: a':f':g' → a':f':g'
a' :: a':f':g'
g' :: a':f':g'
mark' :: a':f':g' → a':f':g'
_hole_a':f':g'1 :: a':f':g'
_gen_a':f':g'2 :: Nat → a':f':g'

Generator Equations:
_gen_a':f':g'2(0) ⇔ a'
_gen_a':f':g'2(+(x, 1)) ⇔ f'(_gen_a':f':g'2(x))

The following defined symbols remain to be analysed:
a__f', mark'

They will be analysed ascendingly in the following order:
a__f' < mark'

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

Rules:
a__f'(f'(a')) → a__f'(g')
mark'(f'(X)) → a__f'(mark'(X))
mark'(a') → a'
mark'(g') → g'
a__f'(X) → f'(X)

Types:
a__f' :: a':f':g' → a':f':g'
f' :: a':f':g' → a':f':g'
a' :: a':f':g'
g' :: a':f':g'
mark' :: a':f':g' → a':f':g'
_hole_a':f':g'1 :: a':f':g'
_gen_a':f':g'2 :: Nat → a':f':g'

Generator Equations:
_gen_a':f':g'2(0) ⇔ a'
_gen_a':f':g'2(+(x, 1)) ⇔ f'(_gen_a':f':g'2(x))

The following defined symbols remain to be analysed:
mark'

Proved the following rewrite lemma:
mark'(_gen_a':f':g'2(+(1, _n18))) → _*3, rt ∈ Ω(n18)

Induction Base:
mark'(_gen_a':f':g'2(+(1, 0)))

Induction Step:
mark'(_gen_a':f':g'2(+(1, +(_\$n19, 1)))) →RΩ(1)
a__f'(mark'(_gen_a':f':g'2(+(1, _\$n19)))) →IH
a__f'(_*3)

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

Rules:
a__f'(f'(a')) → a__f'(g')
mark'(f'(X)) → a__f'(mark'(X))
mark'(a') → a'
mark'(g') → g'
a__f'(X) → f'(X)

Types:
a__f' :: a':f':g' → a':f':g'
f' :: a':f':g' → a':f':g'
a' :: a':f':g'
g' :: a':f':g'
mark' :: a':f':g' → a':f':g'
_hole_a':f':g'1 :: a':f':g'
_gen_a':f':g'2 :: Nat → a':f':g'

Lemmas:
mark'(_gen_a':f':g'2(+(1, _n18))) → _*3, rt ∈ Ω(n18)

Generator Equations:
_gen_a':f':g'2(0) ⇔ a'
_gen_a':f':g'2(+(x, 1)) ⇔ f'(_gen_a':f':g'2(x))

No more defined symbols left to analyse.

The lowerbound Ω(n) was proven with the following lemma:
mark'(_gen_a':f':g'2(+(1, _n18))) → _*3, rt ∈ Ω(n18)