/-- The main proof of the lower bound part of the Akra-Bazzi theorem. The factor
`1 + ε n` does not change the asymptotic order, but is needed for the induction step to go
through. -/
lemma smoothingFn_mul_asympBound_isBigO_T :
    (fun (n : ℕ) => (1 + ε n) * asympBound g a b n) =O[atTop] T := by
  let b' := b (min_bi b) / 2
  have hb_pos : 0 < b' := R.bi_min_div_two_pos
  rw [isBigO_atTop_iff_eventually_exists_pos]
  obtain ⟨c₁, hc₁, h_sumTransform_aux⟩ := R.eventually_atTop_sumTransform_le
  filter_upwards [eventually_ge_atTop R.n₀,                                 -- n₀_ge_Rn₀
      (tendsto_nat_floor_mul_atTop b' hb_pos).eventually_gt_atTop 0,        -- h_b_floor
      eventually_forall_ge_atTop.mpr eventually_one_add_smoothingFn_pos,    -- h_smoothing_pos
      (tendsto_nat_floor_mul_atTop b' hb_pos).eventually_forall_ge_atTop
        eventually_one_add_smoothingFn_pos,                                 -- h_smoothing_pos'
      eventually_forall_ge_atTop.mpr R.eventually_asympBound_pos,            -- h_asympBound_pos
      eventually_forall_ge_atTop.mpr R.eventually_asympBound_r_pos,          -- h_asympBound_r_pos
      (tendsto_nat_floor_mul_atTop b' hb_pos).eventually_forall_ge_atTop
        R.eventually_asympBound_pos,                                         -- h_asympBound_floor
      eventually_gt_atTop 0,                                                -- n₀_pos
      eventually_forall_ge_atTop.mpr R.eventually_one_add_smoothingFn_r_pos,  -- h_smoothing_r_pos
      eventually_forall_ge_atTop.mpr R.rpow_p_mul_one_add_smoothingFn_ge,   -- bound2
      (tendsto_nat_floor_mul_atTop b' hb_pos).eventually_forall_ge_atTop
        eventually_one_add_smoothingFn_pos,                                 -- h_smoothingFn_floor
      eventually_forall_ge_atTop.mpr h_sumTransform_aux,                    -- h_sumTransform
      eventually_forall_ge_atTop.mpr R.eventually_bi_mul_le_r,              -- h_bi_le_r
      eventually_forall_ge_atTop.mpr (eventually_ge_atTop ⌈exp 1⌉₊)]        -- h_exp
    with n₀ n₀_ge_Rn₀ h_b_floor h_smoothing_pos h_smoothing_pos' h_asympBound_pos h_asympBound_r_pos
      h_asympBound_floor n₀_pos h_smoothing_r_pos bound2 h_smoothingFn_floor h_sumTransform
      h_bi_le_r h_exp
  

Proof state:
case h
α : Type u_1
inst✝¹ : Fintype α
inst✝ : Nonempty α
T : ℕ → ℝ
g : ℝ → ℝ
a b : α → ℝ
r : α → ℕ → ℕ
R : AkraBazziRecurrence T g a b r
b' : ℝ := b (min_bi b) / 2
hb_pos : 0 < b'
c₁ : ℝ
hc₁ : c₁ > 0
h_sumTransform_aux : ∀ᶠ (n : ℕ) in atTop, ∀ (i : α), sumTransform (p a b) g (r i n) n ≤ c₁ * g ↑n
n₀ : ℕ
n₀_ge_Rn₀ : R.n₀ ≤ n₀
h_b_floor : 0 < ⌊b' * ↑n₀⌋₊
h_smoothing_pos : ∀ (y : ℕ), n₀ ≤ y → 0 < 1 + ε ↑y
h_smoothing_pos' : ∀ (y : ℕ), ⌊b' * ↑n₀⌋₊ ≤ y → 0 < 1 + ε ↑y
h_asympBound_pos : ∀ (y : ℕ), n₀ ≤ y → 0 < asympBound g a b y
h_asympBound_r_pos : ∀ (y : ℕ), n₀ ≤ y → ∀ (i : α), 0 < asympBound g a b (r i y)
h_asympBound_floor : ∀ (y : ℕ), ⌊b' * ↑n₀⌋₊ ≤ y → 0 < asympBound g a b y
n₀_pos : 0 < n₀
h_smoothing_r_pos : ∀ (y : ℕ), n₀ ≤ y → ∀ (i : α), 0 < 1 + ε ↑(r i y)
bound2 : ∀ (y : ℕ), n₀ ≤ y → ∀ (i : α), b i ^ p a b * ↑y ^ p a b * (1 + ε ↑y) ≤ ↑(r i y) ^ p a b * (1 + ε ↑(r i y))
h_smoothingFn_floor : ∀ (y : ℕ), ⌊b' * ↑n₀⌋₊ ≤ y → 0 < 1 + ε ↑y
h_sumTransform : ∀ (y : ℕ), n₀ ≤ y → ∀ (i : α), sumTransform (p a b) g (r i y) y ≤ c₁ * g ↑y
h_bi_le_r : ∀ (y : ℕ), n₀ ≤ y → ∀ (i : α), b (min_bi b) / 2 * ↑y ≤ ↑(r i y)
h_exp : ∀ (y : ℕ), n₀ ≤ y → ⌈rexp 1⌉₊ ≤ y
⊢ ∃ c > 0, ∀ n ≥ n₀, c * ‖(1 + ε ↑n) * asympBound g a b n‖ ≤ ‖T n‖

Next tactic:
have h_base_nonempty := R.base_nonempty n₀_pos