-- Mathlib/Computability/AkraBazzi/AkraBazzi.lean
lemma rpow_p_mul_one_sub_smoothingFn_le :
    ∀ᶠ (n : ℕ) in atTop, ∀ i, (r i n) ^ (p a b) * (1 - ε (r i n))
      ≤ (b i) ^ (p a b) * n ^ (p a b) * (1 - ε n)

Proof state:
α : Type u_1
inst✝¹ : Fintype α
T : ℕ → ℝ
g : ℝ → ℝ
a b : α → ℝ
r : α → ℕ → ℕ
inst✝ : Nonempty α
R : AkraBazziRecurrence T g a b r
i : α
q : ℝ → ℝ := fun x => x ^ p a b * (1 - ε x)
h_diff_q : DifferentiableOn ℝ q (Set.Ioi 1)
h_deriv_q : deriv q =O[atTop] fun x => x ^ (p a b - 1)
⊢ (fun x => q ↑(r i x) - q (b i * ↑x)) =o[atTop] fun x => b i ^ p a b * ↑x ^ p a b * (ε (b i * ↑x) - ε ↑x)

Tactic:
calc
  (fun (n : ℕ) => q (r i n) - q (b i * n))
      =O[atTop] fun n => (deriv q n) * (r i n - b i * n) := by
          exact R.isBigO_apply_r_sub_b q h_diff_q
            (growsPolynomially_deriv_rpow_p_mul_one_sub_smoothingFn (p a b)) i
    _ =o[atTop] fun n => (deriv q n) * (n / log n ^ 2) := by
          exact IsBigO.mul_isLittleO (isBigO_refl _ _) (R.dist_r_b i)
    _ =O[atTop] fun n => n^((p a b) - 1) * (n / log n ^ 2) := by
          exact IsBigO.mul (IsBigO.natCast_atTop h_deriv_q) (isBigO_refl _ _)
    _ =ᶠ[atTop] fun n => n^(p a b) / (log n) ^ 2 := by
          filter_upwards [eventually_ne_atTop 0] with n hn
          have hn' : (n : ℝ) ≠ 0 := by positivity
          simp [← mul_div_assoc, ← Real.rpow_add_one hn']
    _ = fun (n : ℕ) => (n : ℝ) ^ (p a b) * (1 / (log n)^2) := by
          simp_rw [mul_div, mul_one]
    _ =Θ[atTop] fun (n : ℕ) => (b i) ^ (p a b) * n ^ (p a b) * (1 / (log n)^2) := by
          refine IsTheta.symm ?_
          simp_rw [mul_assoc]
          refine IsTheta.const_mul_left ?_ (isTheta_refl _ _)
          have := R.b_pos i; positivity
    _ =Θ[atTop] fun (n : ℕ) => (b i)^(p a b) * n^(p a b) * (ε (b i * n) - ε n) := by
          exact IsTheta.symm <| IsTheta.mul (isTheta_refl _ _)
            <| R.isTheta_smoothingFn_sub_self i

Result:
no goals

Proof state:
α : Type u_1
inst✝¹ : Fintype α
T : ℕ → ℝ
g : ℝ → ℝ
a b : α → ℝ
r : α → ℕ → ℕ
inst✝ : Nonempty α
R : AkraBazziRecurrence T g a b r
i : α
q : ℝ → ℝ := fun x => x ^ p a b * (1 - ε x)
h_diff_q : DifferentiableOn ℝ q (Set.Ioi 1)
h_deriv_q : deriv q =O[atTop] fun x => x ^ (p a b - 1)
h_main_norm :
  (fun n => ‖q ↑(r i n) - q (b i * ↑n)‖) ≤ᶠ[atTop] fun n => ‖b i ^ p a b * ↑n ^ p a b * (ε (b i * ↑n) - ε ↑n)‖
n : ℕ
hn : ⌈(b i)⁻¹⌉₊ < n
hn' : 1 < n
h₁ : 0 < b i
⊢ b i ≠ 0

Tactic:
positivity

Result:
no goals

Proof state:
α : Type u_1
inst✝¹ : Fintype α
T : ℕ → ℝ
g : ℝ → ℝ
a b : α → ℝ
r : α → ℕ → ℕ
inst✝ : Nonempty α
R : AkraBazziRecurrence T g a b r
i : α
q : ℝ → ℝ := fun x => x ^ p a b * (1 - ε x)
h_diff_q : DifferentiableOn ℝ q (Set.Ioi 1)
h_deriv_q : deriv q =O[atTop] fun x => x ^ (p a b - 1)
⊢ (fun n => ‖q ↑(r i n) - q (b i * ↑n)‖) ≤ᶠ[atTop] fun n => ‖b i ^ p a b * ↑n ^ p a b * (ε (b i * ↑n) - ε ↑n)‖

Tactic:
refine IsLittleO.eventuallyLE ?_

Result:
α : Type u_1
inst✝¹ : Fintype α
T : ℕ → ℝ
g : ℝ → ℝ
a b : α → ℝ
r : α → ℕ → ℕ
inst✝ : Nonempty α
R : AkraBazziRecurrence T g a b r
i : α
q : ℝ → ℝ := fun x => x ^ p a b * (1 - ε x)
h_diff_q : DifferentiableOn ℝ q (Set.Ioi 1)
h_deriv_q : deriv q =O[atTop] fun x => x ^ (p a b - 1)
⊢ (fun x => q ↑(r i x) - q (b i * ↑x)) =o[atTop] fun x => b i ^ p a b * ↑x ^ p a b * (ε (b i * ↑x) - ε ↑x)

Proof state:
α : Type u_1
inst✝¹ : Fintype α
T : ℕ → ℝ
g : ℝ → ℝ
a b : α → ℝ
r : α → ℕ → ℕ
inst✝ : Nonempty α
R : AkraBazziRecurrence T g a b r
i : α
q : ℝ → ℝ := fun x => x ^ p a b * (1 - ε x)
h_diff_q : DifferentiableOn ℝ q (Set.Ioi 1)
⊢ (fun x => deriv (fun z => z ^ p a b) x * (1 - ε x) + x ^ p a b * deriv (fun z => 1 - ε z) x) =O[atTop] fun x =>
    x ^ (p a b - 1)

Tactic:
refine IsBigO.add ?left ?right

Result:
case left
α : Type u_1
inst✝¹ : Fintype α
T : ℕ → ℝ
g : ℝ → ℝ
a b : α → ℝ
r : α → ℕ → ℕ
inst✝ : Nonempty α
R : AkraBazziRecurrence T g a b r
i : α
q : ℝ → ℝ := fun x => x ^ p a b * (1 - ε x)
h_diff_q : DifferentiableOn ℝ q (Set.Ioi 1)
⊢ (fun x => deriv (fun z => z ^ p a b) x * (1 - ε x)) =O[atTop] fun x => x ^ (p a b - 1)

case right
α : Type u_1
inst✝¹ : Fintype α
T : ℕ → ℝ
g : ℝ → ℝ
a b : α → ℝ
r : α → ℕ → ℕ
inst✝ : Nonempty α
R : AkraBazziRecurrence T g a b r
i : α
q : ℝ → ℝ := fun x => x ^ p a b * (1 - ε x)
h_diff_q : DifferentiableOn ℝ q (Set.Ioi 1)
⊢ (fun x => x ^ p a b * deriv (fun z => 1 - ε z) x) =O[atTop] fun x => x ^ (p a b - 1)

Proof state:
case h
α : Type u_1
inst✝¹ : Fintype α
T : ℕ → ℝ
g : ℝ → ℝ
a b : α → ℝ
r : α → ℕ → ℕ
inst✝ : Nonempty α
R : AkraBazziRecurrence T g a b r
i : α
q : ℝ → ℝ := fun x => x ^ p a b * (1 - ε x)
h_diff_q : DifferentiableOn ℝ q (Set.Ioi 1)
h_deriv_q : deriv q =O[atTop] fun x => x ^ (p a b - 1)
h_main_norm :
  (fun n => ‖q ↑(r i n) - q (b i * ↑n)‖) ≤ᶠ[atTop] fun n => ‖b i ^ p a b * ↑n ^ p a b * (ε (b i * ↑n) - ε ↑n)‖
n : ℕ
hn : ⌈(b i)⁻¹⌉₊ < n
hn' : 1 < n
h₁ : 0 < b i
⊢ (b i)⁻¹ ≤ ↑⌈(b i)⁻¹⌉₊

Tactic:
exact Nat.le_ceil _

Result:
no goals

Proof state:
α : Type u_1
inst✝¹ : Fintype α
T : ℕ → ℝ
g : ℝ → ℝ
a b : α → ℝ
r : α → ℕ → ℕ
inst✝ : Nonempty α
R : AkraBazziRecurrence T g a b r
i : α
q : ℝ → ℝ := fun x => x ^ p a b * (1 - ε x)
h_diff_q : DifferentiableOn ℝ q (Set.Ioi 1)
⊢ (fun x => x ^ (p a b - 1) * (1 - ε x)) =O[atTop] fun x => x ^ (p a b - 1) * 1

Tactic:
refine IsBigO.mul (isBigO_refl _ _)
  isEquivalent_one_sub_smoothingFn_one.isBigO

Result:
no goals

Proof state:
α : Type u_1
inst✝¹ : Fintype α
T : ℕ → ℝ
g : ℝ → ℝ
a b : α → ℝ
r : α → ℕ → ℕ
inst✝ : Nonempty α
R : AkraBazziRecurrence T g a b r
i : α
q : ℝ → ℝ := fun x => x ^ p a b * (1 - ε x)
h_diff_q : DifferentiableOn ℝ q (Set.Ioi 1)
h_deriv_q : deriv q =O[atTop] fun x => x ^ (p a b - 1)
h_main_norm :
  (fun n => ‖q ↑(r i n) - q (b i * ↑n)‖) ≤ᶠ[atTop] fun n => ‖b i ^ p a b * ↑n ^ p a b * (ε (b i * ↑n) - ε ↑n)‖
h_main : (fun n => q ↑(r i n) - q (b i * ↑n)) ≤ᶠ[atTop] fun n => b i ^ p a b * ↑n ^ p a b * (ε (b i * ↑n) - ε ↑n)
n : ℕ
hn : q ↑(r i n) - q (b i * ↑n) ≤ b i ^ p a b * ↑n ^ p a b * (ε (b i * ↑n) - ε ↑n)
this : 0 < b i
⊢ 0 ≤ b i

Tactic:
positivity

Result:
no goals

Proof state:
α : Type u_1
inst✝¹ : Fintype α
T : ℕ → ℝ
g : ℝ → ℝ
a b : α → ℝ
r : α → ℕ → ℕ
inst✝ : Nonempty α
R : AkraBazziRecurrence T g a b r
⊢ ∀ᶠ (n : ℕ) in atTop, ∀ (i : α), ↑(r i n) ^ p a b * (1 - ε ↑(r i n)) ≤ b i ^ p a b * ↑n ^ p a b * (1 - ε ↑n)

Tactic:
rw [Filter.eventually_all]

Result:
α : Type u_1
inst✝¹ : Fintype α
T : ℕ → ℝ
g : ℝ → ℝ
a b : α → ℝ
r : α → ℕ → ℕ
inst✝ : Nonempty α
R : AkraBazziRecurrence T g a b r
⊢ ∀ (i : α), ∀ᶠ (x : ℕ) in atTop, ↑(r i x) ^ p a b * (1 - ε ↑(r i x)) ≤ b i ^ p a b * ↑x ^ p a b * (1 - ε ↑x)