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

f(f(a)) → f(g(n__f(n__a)))
f(X) → n__f(X)
an__a
activate(n__f(X)) → f(activate(X))
activate(n__a) → a
activate(X) → X

Rewrite Strategy: INNERMOST

Renamed function symbols to avoid clashes with predefined symbol.

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

f'(f'(a')) → f'(g'(n__f'(n__a')))
f'(X) → n__f'(X)
a'n__a'
activate'(n__f'(X)) → f'(activate'(X))
activate'(n__a') → a'
activate'(X) → X

Rewrite Strategy: INNERMOST

Sliced the following arguments:
g'/0

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

f'(f'(a')) → f'(g')
f'(X) → n__f'(X)
a'n__a'
activate'(n__f'(X)) → f'(activate'(X))
activate'(n__a') → a'
activate'(X) → X

Rewrite Strategy: INNERMOST

Infered types.

Rules:
f'(f'(a')) → f'(g')
f'(X) → n__f'(X)
a'n__a'
activate'(n__f'(X)) → f'(activate'(X))
activate'(n__a') → a'
activate'(X) → X

Types:
f' :: g':n__f':n__a' → g':n__f':n__a'
a' :: g':n__f':n__a'
g' :: g':n__f':n__a'
n__f' :: g':n__f':n__a' → g':n__f':n__a'
n__a' :: g':n__f':n__a'
activate' :: g':n__f':n__a' → g':n__f':n__a'
_hole_g':n__f':n__a'1 :: g':n__f':n__a'
_gen_g':n__f':n__a'2 :: Nat → g':n__f':n__a'

Heuristically decided to analyse the following defined symbols:
f', activate'

They will be analysed ascendingly in the following order:
f' < activate'

Rules:
f'(f'(a')) → f'(g')
f'(X) → n__f'(X)
a'n__a'
activate'(n__f'(X)) → f'(activate'(X))
activate'(n__a') → a'
activate'(X) → X

Types:
f' :: g':n__f':n__a' → g':n__f':n__a'
a' :: g':n__f':n__a'
g' :: g':n__f':n__a'
n__f' :: g':n__f':n__a' → g':n__f':n__a'
n__a' :: g':n__f':n__a'
activate' :: g':n__f':n__a' → g':n__f':n__a'
_hole_g':n__f':n__a'1 :: g':n__f':n__a'
_gen_g':n__f':n__a'2 :: Nat → g':n__f':n__a'

Generator Equations:
_gen_g':n__f':n__a'2(0) ⇔ n__a'
_gen_g':n__f':n__a'2(+(x, 1)) ⇔ n__f'(_gen_g':n__f':n__a'2(x))

The following defined symbols remain to be analysed:
f', activate'

They will be analysed ascendingly in the following order:
f' < activate'

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

Rules:
f'(f'(a')) → f'(g')
f'(X) → n__f'(X)
a'n__a'
activate'(n__f'(X)) → f'(activate'(X))
activate'(n__a') → a'
activate'(X) → X

Types:
f' :: g':n__f':n__a' → g':n__f':n__a'
a' :: g':n__f':n__a'
g' :: g':n__f':n__a'
n__f' :: g':n__f':n__a' → g':n__f':n__a'
n__a' :: g':n__f':n__a'
activate' :: g':n__f':n__a' → g':n__f':n__a'
_hole_g':n__f':n__a'1 :: g':n__f':n__a'
_gen_g':n__f':n__a'2 :: Nat → g':n__f':n__a'

Generator Equations:
_gen_g':n__f':n__a'2(0) ⇔ n__a'
_gen_g':n__f':n__a'2(+(x, 1)) ⇔ n__f'(_gen_g':n__f':n__a'2(x))

The following defined symbols remain to be analysed:
activate'

Proved the following rewrite lemma:
activate'(_gen_g':n__f':n__a'2(_n12)) → _gen_g':n__f':n__a'2(_n12), rt ∈ Ω(1 + n12)

Induction Base:
activate'(_gen_g':n__f':n__a'2(0)) →RΩ(1)
_gen_g':n__f':n__a'2(0)

Induction Step:
activate'(_gen_g':n__f':n__a'2(+(_\$n13, 1))) →RΩ(1)
f'(activate'(_gen_g':n__f':n__a'2(_\$n13))) →IH
f'(_gen_g':n__f':n__a'2(_\$n13)) →RΩ(1)
n__f'(_gen_g':n__f':n__a'2(_\$n13))

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

Rules:
f'(f'(a')) → f'(g')
f'(X) → n__f'(X)
a'n__a'
activate'(n__f'(X)) → f'(activate'(X))
activate'(n__a') → a'
activate'(X) → X

Types:
f' :: g':n__f':n__a' → g':n__f':n__a'
a' :: g':n__f':n__a'
g' :: g':n__f':n__a'
n__f' :: g':n__f':n__a' → g':n__f':n__a'
n__a' :: g':n__f':n__a'
activate' :: g':n__f':n__a' → g':n__f':n__a'
_hole_g':n__f':n__a'1 :: g':n__f':n__a'
_gen_g':n__f':n__a'2 :: Nat → g':n__f':n__a'

Lemmas:
activate'(_gen_g':n__f':n__a'2(_n12)) → _gen_g':n__f':n__a'2(_n12), rt ∈ Ω(1 + n12)

Generator Equations:
_gen_g':n__f':n__a'2(0) ⇔ n__a'
_gen_g':n__f':n__a'2(+(x, 1)) ⇔ n__f'(_gen_g':n__f':n__a'2(x))

No more defined symbols left to analyse.

The lowerbound Ω(n) was proven with the following lemma:
activate'(_gen_g':n__f':n__a'2(_n12)) → _gen_g':n__f':n__a'2(_n12), rt ∈ Ω(1 + n12)