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

f(f(X)) → c(n__f(n__g(n__f(X))))
c(X) → d(activate(X))
h(X) → c(n__d(X))
f(X) → n__f(X)
g(X) → n__g(X)
d(X) → n__d(X)
activate(n__f(X)) → f(activate(X))
activate(n__g(X)) → g(X)
activate(n__d(X)) → d(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'(f'(X)) → c'(n__f'(n__g'(n__f'(X))))
c'(X) → d'(activate'(X))
h'(X) → c'(n__d'(X))
f'(X) → n__f'(X)
g'(X) → n__g'(X)
d'(X) → n__d'(X)
activate'(n__f'(X)) → f'(activate'(X))
activate'(n__g'(X)) → g'(X)
activate'(n__d'(X)) → d'(X)
activate'(X) → X

Rewrite Strategy: INNERMOST

Sliced the following arguments:
n__g'/0
g'/0

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

f'(f'(X)) → c'(n__f'(n__g'))
c'(X) → d'(activate'(X))
h'(X) → c'(n__d'(X))
f'(X) → n__f'(X)
g' → n__g'
d'(X) → n__d'(X)
activate'(n__f'(X)) → f'(activate'(X))
activate'(n__g') → g'
activate'(n__d'(X)) → d'(X)
activate'(X) → X

Rewrite Strategy: INNERMOST

Infered types.

Rules:
f'(f'(X)) → c'(n__f'(n__g'))
c'(X) → d'(activate'(X))
h'(X) → c'(n__d'(X))
f'(X) → n__f'(X)
g' → n__g'
d'(X) → n__d'(X)
activate'(n__f'(X)) → f'(activate'(X))
activate'(n__g') → g'
activate'(n__d'(X)) → d'(X)
activate'(X) → X

Types:
f' :: n__g':n__f':n__d' → n__g':n__f':n__d'
c' :: n__g':n__f':n__d' → n__g':n__f':n__d'
n__f' :: n__g':n__f':n__d' → n__g':n__f':n__d'
n__g' :: n__g':n__f':n__d'
d' :: n__g':n__f':n__d' → n__g':n__f':n__d'
activate' :: n__g':n__f':n__d' → n__g':n__f':n__d'
h' :: n__g':n__f':n__d' → n__g':n__f':n__d'
n__d' :: n__g':n__f':n__d' → n__g':n__f':n__d'
g' :: n__g':n__f':n__d'
_hole_n__g':n__f':n__d'1 :: n__g':n__f':n__d'
_gen_n__g':n__f':n__d'2 :: Nat → n__g':n__f':n__d'

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

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

Generator Equations:
_gen_n__g':n__f':n__d'2(0) ⇔ n__g'
_gen_n__g':n__f':n__d'2(+(x, 1)) ⇔ n__f'(_gen_n__g':n__f':n__d'2(x))

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

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

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

Generator Equations:
_gen_n__g':n__f':n__d'2(0) ⇔ n__g'
_gen_n__g':n__f':n__d'2(+(x, 1)) ⇔ n__f'(_gen_n__g':n__f':n__d'2(x))

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

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

Proved the following rewrite lemma:
activate'(_gen_n__g':n__f':n__d'2(_n1972)) → _gen_n__g':n__f':n__d'2(_n1972), rt ∈ Ω(1 + _n1972)

Induction Base:
activate'(_gen_n__g':n__f':n__d'2(0)) →_R^Ω(1)
_gen_n__g':n__f':n__d'2(0)

Induction Step:
activate'(_gen_n__g':n__f':n__d'2(+(_$n1973, 1))) →_R^Ω(1)
f'(activate'(_gen_n__g':n__f':n__d'2(_$n1973))) →_IH
f'(_gen_n__g':n__f':n__d'2(_$n1973)) →_R^Ω(1)
n__f'(_gen_n__g':n__f':n__d'2(_$n1973))

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

Lemmas:
activate'(_gen_n__g':n__f':n__d'2(_n1972)) → _gen_n__g':n__f':n__d'2(_n1972), rt ∈ Ω(1 + _n1972)

Generator Equations:
_gen_n__g':n__f':n__d'2(0) ⇔ n__g'
_gen_n__g':n__f':n__d'2(+(x, 1)) ⇔ n__f'(_gen_n__g':n__f':n__d'2(x))

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

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

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

Lemmas:
activate'(_gen_n__g':n__f':n__d'2(_n1972)) → _gen_n__g':n__f':n__d'2(_n1972), rt ∈ Ω(1 + _n1972)

Generator Equations:
_gen_n__g':n__f':n__d'2(0) ⇔ n__g'
_gen_n__g':n__f':n__d'2(+(x, 1)) ⇔ n__f'(_gen_n__g':n__f':n__d'2(x))

The following defined symbols remain to be analysed:
c'

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

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

Lemmas:
activate'(_gen_n__g':n__f':n__d'2(_n1972)) → _gen_n__g':n__f':n__d'2(_n1972), rt ∈ Ω(1 + _n1972)

Generator Equations:
_gen_n__g':n__f':n__d'2(0) ⇔ n__g'
_gen_n__g':n__f':n__d'2(+(x, 1)) ⇔ n__f'(_gen_n__g':n__f':n__d'2(x))

No more defined symbols left to analyse.

The lowerbound Ω(n) was proven with the following lemma:
activate'(_gen_n__g':n__f':n__d'2(_n1972)) → _gen_n__g':n__f':n__d'2(_n1972), rt ∈ Ω(1 + _n1972)