Theorem: log_sq_isbigo_mul (PrimeNumberTheoremAnd/Wiener.lean)

State before:
n : ℕ
A a✝ b✝ c d u x y t σ' : ℝ
ψ Ψ : ℝ → ℂ
F G : ℂ → ℂ
f : ℕ → ℂ
𝕜 : Type
inst✝ : _root_.RCLike 𝕜
a b : ℝ
hb : 0 < b
⊢ (fun x => x.log ^ 2) =O[atTop] fun x => a + (x / b).log ^ 2
Tactic:
apply (log_isbigo_log_div hb).sq.trans
State after:
n : ℕ
A a✝ b✝ c d u x y t σ' : ℝ
ψ Ψ : ℝ → ℂ
F G : ℂ → ℂ
f : ℕ → ℂ
𝕜 : Type
inst✝ : _root_.RCLike 𝕜
a b : ℝ
hb : 0 < b
⊢ (fun n => (n / b).log ^ 2) =O[atTop] fun x => a + (x / b).log ^ 2

State before:
n : ℕ
A a✝ b✝ c d u x y t σ' : ℝ
ψ Ψ : ℝ → ℂ
F G : ℂ → ℂ
f : ℕ → ℂ
𝕜 : Type
inst✝ : _root_.RCLike 𝕜
a b : ℝ
hb : 0 < b
⊢ (fun n => (n / b).log ^ 2) =O[atTop] fun x => a + (x / b).log ^ 2
Tactic:
simp_rw [add_comm a]
State after:
n : ℕ
A a✝ b✝ c d u x y t σ' : ℝ
ψ Ψ : ℝ → ℂ
F G : ℂ → ℂ
f : ℕ → ℂ
𝕜 : Type
inst✝ : _root_.RCLike 𝕜
a b : ℝ
hb : 0 < b
⊢ (fun n => (n / b).log ^ 2) =O[atTop] fun x => (x / b).log ^ 2 + a

State before:
n : ℕ
A a✝ b✝ c d u x y t σ' : ℝ
ψ Ψ : ℝ → ℂ
F G : ℂ → ℂ
f : ℕ → ℂ
𝕜 : Type
inst✝ : _root_.RCLike 𝕜
a b : ℝ
hb : 0 < b
⊢ (fun n => (n / b).log ^ 2) =O[atTop] fun x => (x / b).log ^ 2 + a
Tactic:
refine IsBigO.add_isLittleO_right <| isLittleO_const_of_tendsto_atTop _ ?_
State after:
n : ℕ
A a✝ b✝ c d u x y t σ' : ℝ
ψ Ψ : ℝ → ℂ
F G : ℂ → ℂ
f : ℕ → ℂ
𝕜 : Type
inst✝ : _root_.RCLike 𝕜
a b : ℝ
hb : 0 < b
⊢ Tendsto (fun n => (n / b).log ^ 2) atTop atTop

State before:
n : ℕ
A a✝ b✝ c d u x y t σ' : ℝ
ψ Ψ : ℝ → ℂ
F G : ℂ → ℂ
f : ℕ → ℂ
𝕜 : Type
inst✝ : _root_.RCLike 𝕜
a b : ℝ
hb : 0 < b
⊢ Tendsto (fun n => (n / b).log ^ 2) atTop atTop
Tactic:
exact (tendsto_pow_atTop (two_ne_zero)).comp <| tendsto_log_atTop.comp <| tendsto_id.atTop_div_const hb
State after:
no goals