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

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)‖
h_main : (fun n => q (b i * ↑n) - q ↑(r 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 (b i * ↑n) - q ↑(r i n) ≤ b i ^ p a b * ↑n ^ p a b * (ε (b i * ↑n) - ε ↑n)
⊢ b i ^ p a b * ↑n ^ p a b * (1 + ε ↑n) ≤ ↑(r i n) ^ p a b * (1 + ε ↑(r i n))

Tactic:
have h₁ : q (b i * n) - (b i) ^ (p a b) * n ^ (p a b) * (ε (b i * n) - ε n)
    = (b i) ^ (p a b) * n ^ (p a b) * (1 + ε n) := by
  have := R.b_pos i
  simp only [q, mul_rpow (by positivity : (0 : ℝ) ≤ b i) (by positivity : (0 : ℝ) ≤ n)]
  ring

Result:
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)‖
h_main : (fun n => q (b i * ↑n) - q ↑(r 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 (b i * ↑n) - q ↑(r i n) ≤ b i ^ p a b * ↑n ^ p a b * (ε (b i * ↑n) - ε ↑n)
h₁ : q (b i * ↑n) - b i ^ p a b * ↑n ^ p a b * (ε (b i * ↑n) - ε ↑n) = b i ^ p a b * ↑n ^ p a b * (1 + ε ↑n)
⊢ b i ^ p a b * ↑n ^ p a b * (1 + ε ↑n) ≤ ↑(r i n) ^ p a b * (1 + ε ↑(r i 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)
this : 0 < b i
⊢ b i ^ p a b ≠ 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)
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
⊢ 1 < n

Tactic:
exact hn'

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)‖
⊢ (fun n => ‖b i ^ p a b * ↑n ^ p a b * (ε (b i * ↑n) - ε ↑n)‖) =ᶠ[atTop] fun n =>
    b i ^ p a b * ↑n ^ p a b * (ε (b i * ↑n) - ε ↑n)

Tactic:
filter_upwards [eventually_gt_atTop ⌈(b i)⁻¹⌉₊, eventually_gt_atTop 1] with n hn hn'

Result:
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
⊢ ‖b i ^ p a b * ↑n ^ p a b * (ε (b i * ↑n) - ε ↑n)‖ = b i ^ p a b * ↑n ^ p a b * (ε (b i * ↑n) - 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)
⊢ (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:
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)
z : ℝ
hz : z ∈ Set.Ioi 1
⊢ z ∈ {0}ᶜ

Tactic:
rw [Set.mem_compl_singleton_iff]

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)
z : ℝ
hz : z ∈ Set.Ioi 1
⊢ z ≠ 0

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
⊢ ↑n ∈ Set.Ioi 1

Tactic:
show 1 < (n : ℝ)

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)
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
⊢ 1 < ↑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 n => b i ^ p a b * ↑n ^ p a b * (1 / log ↑n ^ 2)) =Θ[atTop] fun n => ↑n ^ p a b * (1 / log ↑n ^ 2)

Tactic:
simp_rw [mul_assoc]

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 n => b i ^ p a b * (↑n ^ p a b * (1 / log ↑n ^ 2))) =Θ[atTop] fun n => ↑n ^ p a b * (1 / log ↑n ^ 2)