Theorem: AkraBazziRecurrence.eventually_bi_mul_le_r
File: Mathlib/Computability/AkraBazzi/AkraBazzi.lean

Proof steps:
state_before: 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
⊢ ∀ᶠ (n : ℕ) in atTop, ∀ (i : α), b (min_bi b) / 2 * ↑n ≤ ↑(r i n)
state_after: 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
gt_zero : 0 < b (min_bi b)
⊢ ∀ᶠ (n : ℕ) in atTop, ∀ (i : α), b (min_bi b) / 2 * ↑n ≤ ↑(r i n)
tactic: have gt_zero : 0 < b (min_bi b) := R.b_pos (min_bi b)
state_before: 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
gt_zero : 0 < b (min_bi b)
⊢ ∀ᶠ (n : ℕ) in atTop, ∀ (i : α), b (min_bi b) / 2 * ↑n ≤ ↑(r i n)
state_after: 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
gt_zero : 0 < b (min_bi b)
hlo : (fun n => ↑n / log ↑n ^ 2) =o[atTop] fun n => ↑n
⊢ ∀ᶠ (n : ℕ) in atTop, ∀ (i : α), b (min_bi b) / 2 * ↑n ≤ ↑(r i n)
tactic: have hlo := isLittleO_self_div_log_id
state_before: 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
gt_zero : 0 < b (min_bi b)
hlo : (fun n => ↑n / log ↑n ^ 2) =o[atTop] fun n => ↑n
⊢ ∀ᶠ (n : ℕ) in atTop, ∀ (i : α), b (min_bi b) / 2 * ↑n ≤ ↑(r i n)
state_after: 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
gt_zero : 0 < b (min_bi b)
hlo : ∀ ⦃c : ℝ⦄, 0 < c → ∀ᶠ (x : ℕ) in atTop, ‖↑x / log ↑x ^ 2‖ ≤ c * ‖↑x‖
⊢ ∀ᶠ (n : ℕ) in atTop, ∀ (i : α), b (min_bi b) / 2 * ↑n ≤ ↑(r i n)
tactic: rw [Asymptotics.isLittleO_iff] at hlo
state_before: 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
gt_zero : 0 < b (min_bi b)
hlo : ∀ ⦃c : ℝ⦄, 0 < c → ∀ᶠ (x : ℕ) in atTop, ‖↑x / log ↑x ^ 2‖ ≤ c * ‖↑x‖
⊢ ∀ᶠ (n : ℕ) in atTop, ∀ (i : α), b (min_bi b) / 2 * ↑n ≤ ↑(r i n)
state_after: 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
gt_zero : 0 < b (min_bi b)
hlo : ∀ ⦃c : ℝ⦄, 0 < c → ∀ᶠ (x : ℕ) in atTop, ‖↑x / log ↑x ^ 2‖ ≤ c * ‖↑x‖
hlo' : ∀ᶠ (x : ℕ) in atTop, ‖↑x / log ↑x ^ 2‖ ≤ b (min_bi b) / 2 * ‖↑x‖
⊢ ∀ᶠ (n : ℕ) in atTop, ∀ (i : α), b (min_bi b) / 2 * ↑n ≤ ↑(r i n)
tactic: have hlo' := hlo (by positivity : 0 < b (min_bi b) / 2)
state_before: 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
gt_zero : 0 < b (min_bi b)
hlo : ∀ ⦃c : ℝ⦄, 0 < c → ∀ᶠ (x : ℕ) in atTop, ‖↑x / log ↑x ^ 2‖ ≤ c * ‖↑x‖
hlo' : ∀ᶠ (x : ℕ) in atTop, ‖↑x / log ↑x ^ 2‖ ≤ b (min_bi b) / 2 * ‖↑x‖
⊢ ∀ᶠ (n : ℕ) in atTop, ∀ (i : α), b (min_bi b) / 2 * ↑n ≤ ↑(r i n)
state_after: case h
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
gt_zero : 0 < b (min_bi b)
hlo : ∀ ⦃c : ℝ⦄, 0 < c → ∀ᶠ (x : ℕ) in atTop, ‖↑x / log ↑x ^ 2‖ ≤ c * ‖↑x‖
hlo' : ∀ᶠ (x : ℕ) in atTop, ‖↑x / log ↑x ^ 2‖ ≤ b (min_bi b) / 2 * ‖↑x‖
n : ℕ
hn : ‖↑n / log ↑n ^ 2‖ ≤ b (min_bi b) / 2 * ‖↑n‖
hn' : ∀ (i : α), b i * ↑n - ↑n / log ↑n ^ 2 ≤ ↑(r i n)
⊢ ∀ (i : α), b (min_bi b) / 2 * ↑n ≤ ↑(r i n)
tactic: filter_upwards [hlo', R.eventually_b_le_r] with n hn hn'
state_before: case h
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
gt_zero : 0 < b (min_bi b)
hlo : ∀ ⦃c : ℝ⦄, 0 < c → ∀ᶠ (x : ℕ) in atTop, ‖↑x / log ↑x ^ 2‖ ≤ c * ‖↑x‖
hlo' : ∀ᶠ (x : ℕ) in atTop, ‖↑x / log ↑x ^ 2‖ ≤ b (min_bi b) / 2 * ‖↑x‖
n : ℕ
hn : ‖↑n / log ↑n ^ 2‖ ≤ b (min_bi b) / 2 * ‖↑n‖
hn' : ∀ (i : α), b i * ↑n - ↑n / log ↑n ^ 2 ≤ ↑(r i n)
⊢ ∀ (i : α), b (min_bi b) / 2 * ↑n ≤ ↑(r i n)
state_after: case h
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
gt_zero : 0 < b (min_bi b)
hlo : ∀ ⦃c : ℝ⦄, 0 < c → ∀ᶠ (x : ℕ) in atTop, ‖↑x / log ↑x ^ 2‖ ≤ c * ‖↑x‖
hlo' : ∀ᶠ (x : ℕ) in atTop, ‖↑x / log ↑x ^ 2‖ ≤ b (min_bi b) / 2 * ‖↑x‖
n : ℕ
hn : ‖↑n / log ↑n ^ 2‖ ≤ b (min_bi b) / 2 * ‖↑n‖
hn' : ∀ (i : α), b i * ↑n - ↑n / log ↑n ^ 2 ≤ ↑(r i n)
i : α
⊢ b (min_bi b) / 2 * ↑n ≤ ↑(r i n)
tactic: intro i
state_before: case h
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
gt_zero : 0 < b (min_bi b)
hlo : ∀ ⦃c : ℝ⦄, 0 < c → ∀ᶠ (x : ℕ) in atTop, ‖↑x / log ↑x ^ 2‖ ≤ c * ‖↑x‖
hlo' : ∀ᶠ (x : ℕ) in atTop, ‖↑x / log ↑x ^ 2‖ ≤ b (min_bi b) / 2 * ‖↑x‖
n : ℕ
hn : ‖↑n / log ↑n ^ 2‖ ≤ b (min_bi b) / 2 * ‖↑n‖
hn' : ∀ (i : α), b i * ↑n - ↑n / log ↑n ^ 2 ≤ ↑(r i n)
i : α
⊢ b (min_bi b) / 2 * ↑n ≤ ↑(r i n)
state_after: case h
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
gt_zero : 0 < b (min_bi b)
hlo : ∀ ⦃c : ℝ⦄, 0 < c → ∀ᶠ (x : ℕ) in atTop, ‖↑x / log ↑x ^ 2‖ ≤ c * ‖↑x‖
hlo' : ∀ᶠ (x : ℕ) in atTop, ‖↑x / log ↑x ^ 2‖ ≤ b (min_bi b) / 2 * ‖↑x‖
n : ℕ
hn' : ∀ (i : α), b i * ↑n - ↑n / log ↑n ^ 2 ≤ ↑(r i n)
i : α
hn : ‖↑n / log ↑n ^ 2‖ ≤ b (min_bi b) / 2 * ↑n
⊢ b (min_bi b) / 2 * ↑n ≤ ↑(r i n)
tactic: simp only [Real.norm_of_nonneg (by positivity : 0 ≤ (n : ℝ))] at hn
state_before: case h
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
gt_zero : 0 < b (min_bi b)
hlo : ∀ ⦃c : ℝ⦄, 0 < c → ∀ᶠ (x : ℕ) in atTop, ‖↑x / log ↑x ^ 2‖ ≤ c * ‖↑x‖
hlo' : ∀ᶠ (x : ℕ) in atTop, ‖↑x / log ↑x ^ 2‖ ≤ b (min_bi b) / 2 * ‖↑x‖
n : ℕ
hn' : ∀ (i : α), b i * ↑n - ↑n / log ↑n ^ 2 ≤ ↑(r i n)
i : α
hn : ‖↑n / log ↑n ^ 2‖ ≤ b (min_bi b) / 2 * ↑n
⊢ b (min_bi b) / 2 * ↑n ≤ ↑(r i n)
state_after: no goals
tactic: calc b (min_bi b) / 2 * n = b (min_bi b) * n - b (min_bi b) / 2 * n := by ring
                          _ ≤ b (min_bi b) * n - ‖n / log n ^ 2‖ := by gcongr
                          _ ≤ b i * n - ‖n / log n ^ 2‖ := by gcongr; aesop
                          _ = b i * n - n / log n ^ 2 := by
                                congr
                                exact Real.norm_of_nonneg <| by positivity
                          _ ≤ r i n := hn' i