/-- 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
  have h_base_nonempty := R.base_nonempty n₀_pos
  -- Min of the ratio T(n) / asympBound(n) over the base case n ∈ [b * n₀, n₀)
  set base_min : ℝ :=
    (Finset.Ico (⌊b' * n₀⌋₊) n₀).inf' h_base_nonempty
      (fun n => T n / ((1 + ε n) * asympBound g a b n)) with base_min_def
  -- The big-O constant we are aiming for: min of the base case ratio and what we need to cancel
  -- out the g(n) term in the calculation below
  let C := min (2 * c₁)⁻¹ base_min
  have hC_pos : 0 < C := by
    refine lt_min (by positivity) ?_
    obtain ⟨m, hm_mem, hm⟩ :=
      Finset.exists_mem_eq_inf' h_base_nonempty (fun n => T n / ((1 + ε n) * asympBound g a b n))
    calc 0 < T m / ((1 + ε m) * asympBound g a b m) := by
              have H₁ : 0 < T m := by exact R.T_pos _
              have H₂ : 0 < 1 + ε m := by rw [Finset.mem_Ico] at hm_mem
                                          exact h_smoothing_pos' m hm_mem.1
              have H₃ : 0 < asympBound g a b m := by
                refine R.asympBound_pos m ?_
                calc 0 < ⌊b' * n₀⌋₊ := by exact h_b_floor
                     _ ≤ m := by rw [Finset.mem_Ico] at hm_mem; exact hm_mem.1
              positivity
         _ = base_min := by rw [base_min_def, hm]
  refine ⟨C, hC_pos, fun n hn => ?_⟩
  -- Base case: statement is true for `b' * n₀ ≤ n < n₀`
  have h_base : ∀ n ∈ Finset.Ico (⌊b' * n₀⌋₊) n₀, C * ((1 + ε n) * asympBound g a b n) ≤ T 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_.le_div_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).inf' h_base_nonempty
                  fun z => T z / ((1 + ε z) * asympBound g a b z) :=
                    Finset.inf'_le_of_le _ (b := n) hn <| le_refl _
         _ ≥ C := min_le_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
  

Proof state:
case h.ind
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
h_base_nonempty : Finset.Nonempty (Ico ⌊b (min_bi b) / 2 * ↑n₀⌋₊ n₀)
base_min : ℝ := inf' (Ico ⌊b' * ↑n₀⌋₊ n₀) h_base_nonempty fun n => T n / ((1 + ε ↑n) * asympBound g a b n)
base_min_def : base_min = inf' (Ico ⌊b' * ↑n₀⌋₊ n₀) h_base_nonempty fun n => T n / ((1 + ε ↑n) * asympBound g a b n)
C : ℝ := min (2 * c₁)⁻¹ base_min
hC_pos : 0 < C
h_base : ∀ n ∈ Ico ⌊b' * ↑n₀⌋₊ n₀, C * ((1 + ε ↑n) * asympBound g a b n) ≤ T n
n✝ : ℕ
a✝ : ∀ m < n✝, m ≥ n₀ → 0 < asympBound g a b m → 0 < 1 + ε ↑m → C * ((1 + ε ↑m) * asympBound g a b m) ≤ T m
hn : n✝ ≥ n₀
h_asympBound_pos' : 0 < asympBound g a b n✝
h_one_sub_smoothingFn_pos' : 0 < 1 + ε ↑n✝
⊢ C * ((1 + ε ↑n✝) * asympBound g a b n✝) ≤ T n✝

Next tactic:
case ind n h_ind =>
    have b_mul_n₀_le_ri i : ⌊b' * ↑n₀⌋₊ ≤ r i n := by
      exact_mod_cast calc ⌊b' * ↑n₀⌋₊ ≤ 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 _
              · rcases lt_or_le (r i n) n₀ with ri_lt_n₀ | n₀_le_ri
                case inl => exact h_base _ <| Finset.mem_Ico.mpr ⟨b_mul_n₀_le_ri i, ri_lt_n₀⟩
                case inr =>
                  exact h_ind (r i n) (R.r_lt_n _ _ (n₀_ge_Rn₀.trans hn)) n₀_le_ri
                    (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 bound2 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
              gcongr
              refine mul_nonneg ?_ g_pos
              rw [sub_nonneg]
              calc C * c₁ * (1 + ε n) ≤ C * c₁ * 2 := by
                        gcongr
                        refine one_add_smoothingFn_le_two ?_
                        calc exp 1 ≤ ⌈exp 1⌉₊ := by exact Nat.le_ceil _
                                 _ ≤ n := by exact_mod_cast h_exp n hn
                    _ = C * (2 * c₁) := by ring
                    _ ≤ (2 * c₁)⁻¹ * (2 * c₁) := by gcongr; exact min_le_left _ _
                    _ = c₁⁻¹ * c₁ := by ring
                    _ = 1 := inv_mul_cancel (by positivity)
        _ = C * ((1 + ε n) * asympBound g a b n) := by ring