/-- The main proof of the upper 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 T_isBigO_smoothingFn_mul_asympBound :
    T =O[atTop] (fun n => (1 - ε n) * asympBound g a b n) := 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]
  obtain ⟨c₁, hc₁, h_sumTransform_aux⟩ := R.eventually_atTop_sumTransform_ge
  filter_upwards [eventually_ge_atTop R.n₀,       -- n₀_ge_Rn₀
      eventually_forall_ge_atTop.mpr eventually_one_sub_smoothingFn_pos,    -- h_smoothing_pos
      eventually_forall_ge_atTop.mpr
        <| eventually_one_sub_smoothingFn_gt_const (1/2) (by norm_num),    -- h_smoothing_gt_half
      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_sub_smoothingFn_r_pos,  -- h_smoothing_r_pos
      eventually_forall_ge_atTop.mpr R.rpow_p_mul_one_sub_smoothingFn_le,    -- bound1
      (tendsto_nat_floor_mul_atTop b' hb_pos).eventually_forall_ge_atTop
        eventually_one_sub_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
    with n₀ n₀_ge_Rn₀ h_smoothing_pos h_smoothing_gt_half
      h_asympBound_pos h_asympBound_r_pos h_asympBound_floor n₀_pos h_smoothing_r_pos
      bound1 h_smoothingFn_floor h_sumTransform h_bi_le_r
  -- Max of the ratio `T(n) / asympBound(n)` over the base case `n ∈ [b * n₀, n₀)`
  have h_base_nonempty := R.base_nonempty n₀_pos
  let base_max : ℝ :=
    (Finset.Ico (⌊b' * n₀⌋₊) n₀).sup' h_base_nonempty
      fun n => T n / ((1 - ε n) * asympBound g a b n)
  -- The big-O constant we are aiming for: max of the base case ratio and what we need to
  -- cancel out the `g(n)` term in the calculation below
  set C := max (2 * c₁⁻¹) base_max with hC
  refine ⟨C, fun n hn => ?_⟩
  -- Base case: statement is true for `b' * n₀ ≤ n < n₀`
  have h_base : ∀ n ∈ Finset.Ico (⌊b' * n₀⌋₊) n₀, T n ≤ C * ((1 - ε n) * asympBound g a b n) := by
    intro n hn
    rw [Finset.mem_Ico] at hn
    have htmp1 : 0 < 1 - ε n := h_smoothingFn_floor n hn.1
    have htmp2 : 0 < asympBound g a b n := h_asympBound_floor n hn.1
    rw [← _root_.div_le_iff (by positivity)]
    rw [← Finset.mem_Ico] at hn
    calc T n / ((1 - ε ↑n) * asympBound g a b n)
           ≤ (Finset.Ico (⌊b' * n₀⌋₊) n₀).sup' h_base_nonempty
                (fun z => T z / ((1 - ε z) * asympBound g a b z)) :=
                  Finset.le_sup'_of_le _ (b := n) hn <| le_refl _
         _ ≤ C := le_max_right _ _
  have h_asympBound_pos' : 0 < asympBound g a b n := h_asympBound_pos n hn
  have h_one_sub_smoothingFn_pos' : 0 < 1 - ε n := h_smoothing_pos n hn
  rw [Real.norm_of_nonneg (R.T_nonneg n), Real.norm_of_nonneg (by positivity)]
  -- We now prove all other cases by induction
  induction n using Nat.strongInductionOn
  case ind n h_ind =>
    have b_mul_n₀_le_ri i : ⌊b' * ↑n₀⌋₊ ≤ r i n := by
      exact_mod_cast calc ⌊b' * (n₀ : ℝ)⌋₊ ≤ b' * n₀      := Nat.floor_le <| by positivity
                                  _ ≤ b' * n        := by gcongr
                                  _ ≤ r i n      := h_bi_le_r n hn i
    have g_pos : 0 ≤ g n := R.g_nonneg n (by positivity)
    calc
      T n = (∑ i, a i * T (r i n)) + g n := by exact R.h_rec n <| n₀_ge_Rn₀.trans hn
        _ ≤ (∑ i, a i * (C * ((1 - ε (r i n)) * asympBound g a b (r i n)))) + g n := by
            -- Apply the induction hypothesis, or use the base case depending on how large n is
            gcongr (∑ i, a i * ?_) + g n with i _
            · exact le_of_lt <| R.a_pos _
            · by_cases ri_lt_n₀ : r i n < n₀
              case pos => exact h_base _ <| by aesop
              case neg =>
                push_neg at ri_lt_n₀
                exact h_ind (r i n) (R.r_lt_n _ _ (n₀_ge_Rn₀.trans hn)) ri_lt_n₀
                  (h_asympBound_r_pos _ hn _) (h_smoothing_r_pos n hn i)
        _ = (∑ i, a i * (C * ((1 - ε (r i n)) * ((r i n) ^ (p a b)
                * (1 + (∑ u in range (r i n), g u / u ^ ((p a b) + 1))))))) + g n := by
            simp_rw [asympBound_def']
        _ = (∑ i, C * a i * ((r i n) ^ (p a b) * (1 - ε (r i n))
                * ((1 + (∑ u in range (r i n), g u / u ^ ((p a b) + 1)))))) + g n := by
            congr; ext; ring
        _ ≤ (∑ i, C * a i * ((b i) ^ (p a b) * n ^ (p a b) * (1 - ε n)
                * ((1 + (∑ u in range (r i n), g u / u ^ ((p a b) + 1)))))) + g n := by
            gcongr (∑ i, C * a i * (?_
                * ((1 + (∑ u in range (r i n), g u / u ^ ((p a b) + 1)))))) + g n with i
            · have := R.a_pos i
              positivity
            · refine add_nonneg zero_le_one <| Finset.sum_nonneg fun j _ => ?_
              rw [div_nonneg_iff]
              exact Or.inl ⟨R.g_nonneg j (by positivity), by positivity⟩
            · exact bound1 n hn i
        _ = (∑ i, C * a i * ((b i) ^ (p a b) * n ^ (p a b) * (1 - ε n)
                * ((1 + ((∑ u in range n, g u / u ^ ((p a b) + 1))
                - (∑ u in Finset.Ico (r i n) n, g u / u ^ ((p a b) + 1))))))) + g n := by
            congr; ext i; congr
            refine eq_sub_of_add_eq ?_
            rw [add_comm]
            exact add_eq_of_eq_sub <| Finset.sum_Ico_eq_sub _
              <| le_of_lt <| R.r_lt_n i n <| n₀_ge_Rn₀.trans hn
        _ = (∑ i, C * a i * ((b i) ^ (p a b) * (1 - ε n) * ((n ^ (p a b)
                * (1 + (∑ u in range n, g u / u ^ ((p a b) + 1)))
                - n ^ (p a b) * (∑ u in Finset.Ico (r i n) n, g u / u ^ ((p a b) + 1))))))
                + g n := by
            congr; ext; ring
        _ = (∑ i, C * a i * ((b i) ^ (p a b) * (1 - ε n)
                * ((asympBound g a b n - sumTransform (p a b) g (r i n) n)))) + g n := by
            simp_rw [asympBound_def', sumTransform_def]
        _ ≤ (∑ i, C * a i * ((b i) ^ (p a b) * (1 - ε n)
                * ((asympBound g a b n - c₁ * g n)))) + g n := by
            gcongr with i
            · have := R.a_pos i
              positivity
            · have := R.b_pos i
              positivity
            · exact h_sumTransform n hn i
        _ = (∑ i, C * (1 - ε n) * ((asympBound g a b n - c₁ * g n))
                * (a i * (b i) ^ (p a b))) + g n := by
            congr; ext; ring
        _ = C * (1 - ε n) * (asympBound g a b n - c₁ * g n) + g n := by
            rw [← Finset.mul_sum, R.sumCoeffsExp_p_eq_one, mul_one]
        _ = C * (1 - ε n) * asympBound g a b n + (1 - C * c₁ * (1 - ε n)) * g n := by ring
        _ ≤ C * (1 - ε n) * asympBound g a b n + 0 := by
            

Proof state:
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), c₁ * g ↑n ≤ sumTransform (p a b) g (r i n) n
n₀ : ℕ
n₀_ge_Rn₀ : R.n₀ ≤ n₀
h_smoothing_pos : ∀ (y : ℕ), n₀ ≤ y → 0 < 1 - ε ↑y
h_smoothing_gt_half : ∀ (y : ℕ), n₀ ≤ y → 1 / 2 < 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)
bound1 : ∀ (y : ℕ), n₀ ≤ y → ∀ (i : α), ↑(r i y) ^ p a b * (1 - ε ↑(r i y)) ≤ b i ^ p a b * ↑y ^ p a b * (1 - ε ↑y)
h_smoothingFn_floor : ∀ (y : ℕ), ⌊b' * ↑n₀⌋₊ ≤ y → 0 < 1 - ε ↑y
h_sumTransform : ∀ (y : ℕ), n₀ ≤ y → ∀ (i : α), c₁ * g ↑y ≤ sumTransform (p a b) g (r i y) y
h_bi_le_r : ∀ (y : ℕ), n₀ ≤ y → ∀ (i : α), b (min_bi b) / 2 * ↑y ≤ ↑(r i y)
h_base_nonempty : Finset.Nonempty (Ico ⌊b (min_bi b) / 2 * ↑n₀⌋₊ n₀)
base_max : ℝ := sup' (Ico ⌊b' * ↑n₀⌋₊ n₀) h_base_nonempty fun n => T n / ((1 - ε ↑n) * asympBound g a b n)
C : ℝ := max (2 * c₁⁻¹) base_max
hC : C = max (2 * c₁⁻¹) base_max
h_base : ∀ n ∈ Ico ⌊b' * ↑n₀⌋₊ n₀, T n ≤ C * ((1 - ε ↑n) * asympBound g a b n)
n : ℕ
h_ind : ∀ m < n, m ≥ n₀ → 0 < asympBound g a b m → 0 < 1 - ε ↑m → T m ≤ C * ((1 - ε ↑m) * asympBound g a b m)
hn : n ≥ n₀
h_asympBound_pos' : 0 < asympBound g a b n
h_one_sub_smoothingFn_pos' : 0 < 1 - ε ↑n
b_mul_n₀_le_ri : ∀ (i : α), ⌊b' * ↑n₀⌋₊ ≤ r i n
g_pos : 0 ≤ g ↑n
⊢ C * (1 - ε ↑n) * asympBound g a b n + (1 - C * c₁ * (1 - ε ↑n)) * g ↑n ≤ C * (1 - ε ↑n) * asympBound g a b n + 0

Next tactic:
gcongr