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

f(X) → g(n__h(n__f(X)))
h(X) → n__h(X)
f(X) → n__f(X)
activate(n__h(X)) → h(activate(X))
activate(n__f(X)) → f(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) → g'(n__h'(n__f'(X)))
h'(X) → n__h'(X)
f'(X) → n__f'(X)
activate'(n__h'(X)) → h'(activate'(X))
activate'(n__f'(X)) → f'(activate'(X))
activate'(X) → X

Rewrite Strategy: INNERMOST

Sliced the following arguments:
g'/0

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


f'(X) → g'
h'(X) → n__h'(X)
f'(X) → n__f'(X)
activate'(n__h'(X)) → h'(activate'(X))
activate'(n__f'(X)) → f'(activate'(X))
activate'(X) → X

Rewrite Strategy: INNERMOST

Infered types.

Rules:
f'(X) → g'
h'(X) → n__h'(X)
f'(X) → n__f'(X)
activate'(n__h'(X)) → h'(activate'(X))
activate'(n__f'(X)) → f'(activate'(X))
activate'(X) → X

Types:
f' :: g':n__h':n__f' → g':n__h':n__f'
g' :: g':n__h':n__f'
h' :: g':n__h':n__f' → g':n__h':n__f'
n__h' :: g':n__h':n__f' → g':n__h':n__f'
n__f' :: g':n__h':n__f' → g':n__h':n__f'
activate' :: g':n__h':n__f' → g':n__h':n__f'
_hole_g':n__h':n__f'1 :: g':n__h':n__f'
_gen_g':n__h':n__f'2 :: Nat → g':n__h':n__f'

Heuristically decided to analyse the following defined symbols:
activate'

Rules:
f'(X) → g'
h'(X) → n__h'(X)
f'(X) → n__f'(X)
activate'(n__h'(X)) → h'(activate'(X))
activate'(n__f'(X)) → f'(activate'(X))
activate'(X) → X

Types:
f' :: g':n__h':n__f' → g':n__h':n__f'
g' :: g':n__h':n__f'
h' :: g':n__h':n__f' → g':n__h':n__f'
n__h' :: g':n__h':n__f' → g':n__h':n__f'
n__f' :: g':n__h':n__f' → g':n__h':n__f'
activate' :: g':n__h':n__f' → g':n__h':n__f'
_hole_g':n__h':n__f'1 :: g':n__h':n__f'
_gen_g':n__h':n__f'2 :: Nat → g':n__h':n__f'

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

The following defined symbols remain to be analysed:
activate'

Proved the following rewrite lemma:
activate'(_gen_g':n__h':n__f'2(+(1, _n4))) → _*3, rt ∈ Ω(n4)

Induction Base:
activate'(_gen_g':n__h':n__f'2(+(1, 0)))

Induction Step:
activate'(_gen_g':n__h':n__f'2(+(1, +(_$n5, 1)))) →RΩ(1)
h'(activate'(_gen_g':n__h':n__f'2(+(1, _$n5)))) →IH
h'(_*3)

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

Rules:
f'(X) → g'
h'(X) → n__h'(X)
f'(X) → n__f'(X)
activate'(n__h'(X)) → h'(activate'(X))
activate'(n__f'(X)) → f'(activate'(X))
activate'(X) → X

Types:
f' :: g':n__h':n__f' → g':n__h':n__f'
g' :: g':n__h':n__f'
h' :: g':n__h':n__f' → g':n__h':n__f'
n__h' :: g':n__h':n__f' → g':n__h':n__f'
n__f' :: g':n__h':n__f' → g':n__h':n__f'
activate' :: g':n__h':n__f' → g':n__h':n__f'
_hole_g':n__h':n__f'1 :: g':n__h':n__f'
_gen_g':n__h':n__f'2 :: Nat → g':n__h':n__f'

Lemmas:
activate'(_gen_g':n__h':n__f'2(+(1, _n4))) → _*3, rt ∈ Ω(n4)

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

No more defined symbols left to analyse.

The lowerbound Ω(n) was proven with the following lemma:
activate'(_gen_g':n__h':n__f'2(+(1, _n4))) → _*3, rt ∈ Ω(n4)