lemma eventually_atTop_sumTransform_ge :
    ∃ c > 0, ∀ᶠ (n:ℕ) in atTop, ∀ i, c * g n ≤ sumTransform (p a b) g (r i n) n := by
  obtain ⟨c₁, hc₁_mem, hc₁⟩ := R.exists_eventually_const_mul_le_r
  obtain ⟨c₂, hc₂_mem, hc₂⟩ := R.g_grows_poly.eventually_atTop_ge_nat hc₁_mem
  obtain ⟨c₃, hc₃_mem, hc₃⟩ := R.exists_eventually_r_le_const_mul
  have hc₁_pos : 0 < c₁ := hc₁_mem.1
  have hc₃' : 0 < (1 - c₃) := by have := hc₃_mem.2; linarith
  refine ⟨min (c₂ * (1 - c₃)) ((1 - c₃) * c₂ / c₁^((p a b) + 1)), by positivity, ?_⟩
  filter_upwards [hc₁, hc₂, hc₃, R.eventually_r_pos, R.eventually_r_lt_n, eventually_gt_atTop 0]
    with n hn₁ hn₂ hn₃ hrpos hr_lt_n hn_pos
  intro i
  have hrpos_i := hrpos i
  have g_nonneg : 0 ≤ g n := R.g_nonneg n (by positivity)
  rcases le_or_gt 0 (p a b + 1) with hp|hp
  case h.inl =>   -- 0 ≤ (p a b) + 1
    calc sumTransform (p a b) g (r i n) n
           = n ^ (p a b) * (∑ u in Finset.Ico (r i n) n, g u / u ^ ((p a b) + 1))     := by rfl
         _ ≥ n ^ (p a b) * (∑ u in Finset.Ico (r i n) n, c₂ * g n / u^((p a b) + 1)) := by
                gcongr with u hu
                rw [Finset.mem_Ico] at hu
                have hu' : u ∈ Set.Icc (r i n) n := ⟨hu.1, by linarith⟩
                refine hn₂ u ?_
                rw [Set.mem_Icc]
                refine ⟨?_, by norm_cast; linarith⟩
                calc c₁ * n ≤ r i n      := by exact hn₁ i
                          _ ≤ u            := by exact_mod_cast hu'.1
         _ ≥ n ^ (p a b) * (∑ _u in Finset.Ico (r i n) n, c₂ * g n / n ^ ((p a b) + 1)) := by
                gcongr with u hu
                · rw [Finset.mem_Ico] at hu
                  have := calc 0 < r i n := hrpos_i
                              _ ≤ u := hu.1
                  positivity
                · rw [Finset.mem_Ico] at hu
                  exact le_of_lt hu.2
         _ ≥ n ^ (p a b) * (Finset.Ico (r i n) n).card • (c₂ * g n / n ^ ((p a b) + 1)) := by
                gcongr; exact Finset.card_nsmul_le_sum _ _ _ (fun x _ => by rfl)
         _ = n ^ (p a b) * (Finset.Ico (r i n) n).card * (c₂ * g n / n ^ ((p a b) + 1)) := by
                rw [nsmul_eq_mul, mul_assoc]
         _ = n ^ (p a b) * (n - r i n) * (c₂ * g n / n ^ ((p a b) + 1)) := by
                congr; rw [Nat.card_Ico, Nat.cast_sub (le_of_lt <| hr_lt_n i)]
         _ ≥ n ^ (p a b) * (n - c₃ * n) * (c₂ * g n / n ^ ((p a b) + 1)) := by
                gcongr; exact hn₃ i
         _ = n ^ (p a b) * n * (1 - c₃) * (c₂ * g n / n ^ ((p a b) + 1)) := by ring
         _ = c₂ * (1 - c₃) * g n * (n ^ ((p a b) + 1) / n ^ ((p a b) + 1)) := by
                rw [← Real.rpow_add_one (by positivity) (p a b)]; ring
         _ = c₂ * (1 - c₃) * g n := by rw [div_self (by positivity), mul_one]
         _ ≥ min (c₂ * (1 - c₃)) ((1 - c₃) * c₂ / c₁ ^ ((p a b) + 1)) * g n := by
                gcongr; exact min_le_left _ _
  case h.inr =>  -- (p a b) + 1 < 0
    calc sumTransform (p a b) g (r i n) n
        = n ^ (p a b) * (∑ u in Finset.Ico (r i n) n, g u / u^((p a b) + 1))     := by rfl
      _ ≥ n ^ (p a b) * (∑ u in Finset.Ico (r i n) n, c₂ * g n / u ^ ((p a b) + 1)) := by
             gcongr with u hu
             rw [Finset.mem_Ico] at hu
             have hu' : u ∈ Set.Icc (r i n) n := ⟨hu.1, by linarith⟩
             refine hn₂ u ?_
             rw [Set.mem_Icc]
             refine ⟨?_, by norm_cast; linarith⟩
             calc c₁ * n ≤ r i n := by exact hn₁ i
                       _ ≤ u := by exact_mod_cast hu'.1
      _ ≥ n ^ (p a b) * (∑ _u in Finset.Ico (r i n) n, c₂ * g n / (r i n) ^ ((p a b) + 1)) := by
             gcongr n^(p a b) * (Finset.Ico (r i n) n).sum (fun _ => c₂ * g n / ?_) with u hu
             · rw [Finset.mem_Ico] at hu
               have := calc 0 < r i n := hrpos_i
                           _ ≤ u := hu.1
               

Proof state:
case h.h.hc
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
c₁ : ℝ
hc₁_mem : c₁ ∈ Set.Ioo 0 1
hc₁ : ∀ᶠ (n : ℕ) in atTop, ∀ (i : α), c₁ * ↑n ≤ ↑(r i n)
c₂ : ℝ
hc₂_mem : c₂ > 0
hc₂ : ∀ᶠ (n : ℕ) in atTop, ∀ u ∈ Set.Icc (c₁ * ↑n) ↑n, c₂ * g ↑n ≤ g u
c₃ : ℝ
hc₃_mem : c₃ ∈ Set.Ioo 0 1
hc₃ : ∀ᶠ (n : ℕ) in atTop, ∀ (i : α), ↑(r i n) ≤ c₃ * ↑n
hc₁_pos : 0 < c₁
hc₃' : 0 < 1 - c₃
n : ℕ
hn₁ : ∀ (i : α), c₁ * ↑n ≤ ↑(r i n)
hn₂ : ∀ u ∈ Set.Icc (c₁ * ↑n) ↑n, c₂ * g ↑n ≤ g u
hn₃ : ∀ (i : α), ↑(r i n) ≤ c₃ * ↑n
hrpos : ∀ (i : α), 0 < r i n
hr_lt_n : ∀ (i : α), r i n < n
hn_pos : 0 < n
i : α
hrpos_i : 0 < r i n
g_nonneg : 0 ≤ g ↑n
hp : 0 > p a b + 1
u : ℕ
hu : r i n ≤ u ∧ u < n
this : 0 < u
⊢ 0 < ↑u ^ (p a b + 1)

Next tactic:
positivity