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

f(X, n__g(X), Y) → f(activate(Y), activate(Y), activate(Y))
g(b) → c
bc
g(X) → n__g(X)
activate(n__g(X)) → g(activate(X))
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'(X, n__g'(X), Y) → f'(activate'(Y), activate'(Y), activate'(Y))
g'(b') → c'
b'c'
g'(X) → n__g'(X)
activate'(n__g'(X)) → g'(activate'(X))
activate'(X) → X

Rewrite Strategy: INNERMOST

Infered types.

Rules:
f'(X, n__g'(X), Y) → f'(activate'(Y), activate'(Y), activate'(Y))
g'(b') → c'
b'c'
g'(X) → n__g'(X)
activate'(n__g'(X)) → g'(activate'(X))
activate'(X) → X

Types:
f' :: n__g':c' → n__g':c' → n__g':c' → f'
n__g' :: n__g':c' → n__g':c'
activate' :: n__g':c' → n__g':c'
g' :: n__g':c' → n__g':c'
b' :: n__g':c'
c' :: n__g':c'
_hole_f'1 :: f'
_hole_n__g':c'2 :: n__g':c'
_gen_n__g':c'3 :: Nat → n__g':c'

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

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

Rules:
f'(X, n__g'(X), Y) → f'(activate'(Y), activate'(Y), activate'(Y))
g'(b') → c'
b'c'
g'(X) → n__g'(X)
activate'(n__g'(X)) → g'(activate'(X))
activate'(X) → X

Types:
f' :: n__g':c' → n__g':c' → n__g':c' → f'
n__g' :: n__g':c' → n__g':c'
activate' :: n__g':c' → n__g':c'
g' :: n__g':c' → n__g':c'
b' :: n__g':c'
c' :: n__g':c'
_hole_f'1 :: f'
_hole_n__g':c'2 :: n__g':c'
_gen_n__g':c'3 :: Nat → n__g':c'

Generator Equations:
_gen_n__g':c'3(0) ⇔ c'
_gen_n__g':c'3(+(x, 1)) ⇔ n__g'(_gen_n__g':c'3(x))

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

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

Proved the following rewrite lemma:
activate'(_gen_n__g':c'3(+(1, _n5))) → _*4, rt ∈ Ω(n5)

Induction Base:
activate'(_gen_n__g':c'3(+(1, 0)))

Induction Step:
activate'(_gen_n__g':c'3(+(1, +(_\$n6, 1)))) →RΩ(1)
g'(activate'(_gen_n__g':c'3(+(1, _\$n6)))) →IH
g'(_*4)

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

Rules:
f'(X, n__g'(X), Y) → f'(activate'(Y), activate'(Y), activate'(Y))
g'(b') → c'
b'c'
g'(X) → n__g'(X)
activate'(n__g'(X)) → g'(activate'(X))
activate'(X) → X

Types:
f' :: n__g':c' → n__g':c' → n__g':c' → f'
n__g' :: n__g':c' → n__g':c'
activate' :: n__g':c' → n__g':c'
g' :: n__g':c' → n__g':c'
b' :: n__g':c'
c' :: n__g':c'
_hole_f'1 :: f'
_hole_n__g':c'2 :: n__g':c'
_gen_n__g':c'3 :: Nat → n__g':c'

Lemmas:
activate'(_gen_n__g':c'3(+(1, _n5))) → _*4, rt ∈ Ω(n5)

Generator Equations:
_gen_n__g':c'3(0) ⇔ c'
_gen_n__g':c'3(+(x, 1)) ⇔ n__g'(_gen_n__g':c'3(x))

The following defined symbols remain to be analysed:
f'

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

Rules:
f'(X, n__g'(X), Y) → f'(activate'(Y), activate'(Y), activate'(Y))
g'(b') → c'
b'c'
g'(X) → n__g'(X)
activate'(n__g'(X)) → g'(activate'(X))
activate'(X) → X

Types:
f' :: n__g':c' → n__g':c' → n__g':c' → f'
n__g' :: n__g':c' → n__g':c'
activate' :: n__g':c' → n__g':c'
g' :: n__g':c' → n__g':c'
b' :: n__g':c'
c' :: n__g':c'
_hole_f'1 :: f'
_hole_n__g':c'2 :: n__g':c'
_gen_n__g':c'3 :: Nat → n__g':c'

Lemmas:
activate'(_gen_n__g':c'3(+(1, _n5))) → _*4, rt ∈ Ω(n5)

Generator Equations:
_gen_n__g':c'3(0) ⇔ c'
_gen_n__g':c'3(+(x, 1)) ⇔ n__g'(_gen_n__g':c'3(x))

No more defined symbols left to analyse.

The lowerbound Ω(n) was proven with the following lemma:
activate'(_gen_n__g':c'3(+(1, _n5))) → _*4, rt ∈ Ω(n5)