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

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

Rewrite Strategy: INNERMOST

Infered types.

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

Types:
a__f' :: g':f':h' → g':f':h'
g' :: g':f':h'
mark' :: g':f':h' → g':f':h'
f' :: g':f':h' → g':f':h'
h' :: g':f':h' → g':f':h'
_hole_g':f':h'1 :: g':f':h'
_gen_g':f':h'2 :: Nat → g':f':h'

Heuristically decided to analyse the following defined symbols:
mark'

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

Types:
a__f' :: g':f':h' → g':f':h'
g' :: g':f':h'
mark' :: g':f':h' → g':f':h'
f' :: g':f':h' → g':f':h'
h' :: g':f':h' → g':f':h'
_hole_g':f':h'1 :: g':f':h'
_gen_g':f':h'2 :: Nat → g':f':h'

Generator Equations:
_gen_g':f':h'2(0) ⇔ g'
_gen_g':f':h'2(+(x, 1)) ⇔ f'(_gen_g':f':h'2(x))

The following defined symbols remain to be analysed:
mark'

Proved the following rewrite lemma:
mark'(_gen_g':f':h'2(_n4)) → _gen_g':f':h'2(0), rt ∈ Ω(1 + n4)

Induction Base:
mark'(_gen_g':f':h'2(0)) →RΩ(1)
g'

Induction Step:
mark'(_gen_g':f':h'2(+(_$n5, 1))) →RΩ(1)
a__f'(mark'(_gen_g':f':h'2(_$n5))) →IH
a__f'(_gen_g':f':h'2(0)) →RΩ(1)
g'

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

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

Types:
a__f' :: g':f':h' → g':f':h'
g' :: g':f':h'
mark' :: g':f':h' → g':f':h'
f' :: g':f':h' → g':f':h'
h' :: g':f':h' → g':f':h'
_hole_g':f':h'1 :: g':f':h'
_gen_g':f':h'2 :: Nat → g':f':h'

Lemmas:
mark'(_gen_g':f':h'2(_n4)) → _gen_g':f':h'2(0), rt ∈ Ω(1 + n4)

Generator Equations:
_gen_g':f':h'2(0) ⇔ g'
_gen_g':f':h'2(+(x, 1)) ⇔ f'(_gen_g':f':h'2(x))

No more defined symbols left to analyse.

The lowerbound Ω(n) was proven with the following lemma:
mark'(_gen_g':f':h'2(_n4)) → _gen_g':f':h'2(0), rt ∈ Ω(1 + n4)