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9b6e014 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 | import Mathlib
namespace OAI
noncomputable section
open Filter Asymptotics
namespace BinaryCoordinateSweeps.Sparse
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
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
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
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