| import Mathlib |
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| namespace OAI |
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| noncomputable section |
| open Filter Asymptotics |
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| namespace BinaryCoordinateSweeps.Sparse |
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| lemma nat_pow_le_exp_of_le {x A : ℝ} (hx : 0≤x) (h : x≤Real.exp A) (n : ℕ) : |
| x^n≤Real.exp ((n:ℝ)*A) := by |
| simpa only [←Real.exp_nat_mul] using pow_le_pow_left₀ hx h n |
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| lemma log_le_small_rpow_eventually {ε : ℝ} (hε : 0<ε) : ∀ᶠ s : ℕ in atTop, |
| Real.log (s:ℝ)≤(s:ℝ)^ε := by |
| have hh := (isLittleO_log_rpow_rpow_atTop (1:ℝ) hε).bound (by norm_num : (0:ℝ)<1) |
| have hc : Tendsto (fun s : ℕ => (s:ℝ)) atTop atTop := tendsto_natCast_atTop_atTop |
| filter_upwards [hc.eventually hh,eventually_ge_atTop (1:ℕ)] with s hs hs1 |
| have hl : 0≤Real.log (s:ℝ) := Real.log_nonneg (by exact_mod_cast hs1) |
| simpa only [Real.rpow_one,Real.norm_of_nonneg hl, |
| Real.norm_of_nonneg (Real.rpow_nonneg (Nat.cast_nonneg s) ε),one_mul] using hs |
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| lemma many_hs_scalar (b h k : ℕ) {L x : ℝ} (hb : 1≤b) |
| (hh : (h:ℝ)≤100000*k) (hL : 4000000000*(b:ℝ)≤L) |
| (hx0 : 0≤x) (hx : x≤Real.exp (-L/4000)) : |
| ((2:ℝ)^k*Real.exp ((b:ℝ)*(h+k)+Real.log 4*k*b))^2 * |
| ((2:ℝ)^k*x^((k+1)/2)) ≤ Real.exp (-(k:ℝ)*L/16000) := by |
| have hL0 : 0≤L := (by positivity : (0:ℝ)≤4000000000*b).trans hL |
| have hbk : (1:ℝ)≤b := by exact_mod_cast hb |
| have hkt : (k:ℝ)≤2*(((k+1)/2:ℕ):ℝ) := by exact_mod_cast (show k≤2*((k+1)/2) from by omega) |
| have hal : x^((k+1)/2)≤Real.exp (-(k:ℝ)*L/8000) := by |
| apply (nat_pow_le_exp_of_le hx0 hx _).trans |
| apply Real.exp_le_exp.mpr |
| nlinarith [mul_le_mul_of_nonneg_right hkt hL0] |
| have hlog2 : Real.log 2≤2 := (Real.log_le_sub_one_of_pos (by norm_num : (0:ℝ)<2)).trans (by norm_num) |
| have hlog4 : Real.log 4≤4 := (Real.log_le_sub_one_of_pos (by norm_num : (0:ℝ)<4)).trans (by norm_num) |
| have hp (n : ℕ) : (2:ℝ)^n=Real.exp ((n:ℝ)*Real.log 2) := by |
| rw [Real.exp_nat_mul,Real.exp_log (by norm_num)] |
| calc |
| _ ≤ ((2:ℝ)^k*Real.exp ((b:ℝ)*(h+k)+Real.log 4*k*b))^2 * |
| ((2:ℝ)^k*Real.exp (-(k:ℝ)*L/8000)) := by gcongr |
| _ = Real.exp (3*(k:ℝ)*Real.log 2+2*((b:ℝ)*(h+k)+Real.log 4*k*b)-(k:ℝ)*L/8000) := by |
| simp only [hp,←Real.exp_add,←Real.exp_nat_mul] |
| congr 1 |
| ring |
| _ ≤ _ := by |
| apply Real.exp_le_exp.mpr |
| have hh' := mul_le_mul_of_nonneg_left hh (show (0:ℝ)≤b by positivity) |
| have h2 := mul_le_mul_of_nonneg_left hlog2 (show (0:ℝ)≤k by positivity) |
| have h4 := mul_le_mul_of_nonneg_right hlog4 (show (0:ℝ)≤k*b by positivity) |
| have hkb := mul_le_mul_of_nonneg_left hbk (show (0:ℝ)≤k by positivity) |
| have hLc := mul_le_mul_of_nonneg_right hL (show (0:ℝ)≤k by positivity) |
| nlinarith |
| |
| end BinaryCoordinateSweeps.Sparse |
| |
| end |
| |
| end OAI |
| |
