| import OAI.Combinatorics.FiniteCircle.HistogramEmbedding |
|
|
| namespace OAI |
|
|
| namespace FiniteCircle |
|
|
| open scoped BigOperators List |
| open scoped ComplexConjugate |
| open RCLike Module |
| open Classical |
|
|
| theorem log_add_one_le_dyadic_count (n : ℕ) : |
| Real.log ((n:ℝ)+1) ≤ Nat.log 2 n + 1 := by |
| have hpow : n+1 ≤ 2^(Nat.log 2 n + 1) := |
| Nat.succ_le_of_lt (Nat.lt_pow_succ_log_self (by norm_num) n) |
| have hlog := Real.log_le_log (by positivity : (0:ℝ) < (n:ℝ)+1) |
| (show (n:ℝ)+1 ≤ (2:ℝ)^(Nat.log 2 n+1) by exact_mod_cast hpow) |
| rw [Real.log_pow] at hlog |
| have hl2 : Real.log 2 ≤ 1 := by linarith [Real.log_two_lt_d9] |
| have hc : (0:ℝ) ≤ Nat.log 2 n + 1 := by positivity |
| push_cast at hlog |
| nlinarith |
|
|
| theorem log_large_aux {t : ℝ} (ht : 16 ≤ t) : |
| 1 ≤ Real.log t ∧ 4 * Real.log t ≤ t := by |
| have ht0 : 0 < t := by linarith |
| have hl16 : Real.log 16 = 4 * Real.log 2 := by |
| rw [show (16:ℝ) = 2^4 by norm_num, Real.log_pow] |
| norm_num |
| have hl2lo : (1:ℝ)/2 ≤ Real.log 2 := by linarith [Real.log_two_gt_d9] |
| have hl2hi : Real.log 2 ≤ 1 := by linarith [Real.log_two_lt_d9] |
| have hlo := Real.log_le_log (by norm_num : (0:ℝ) < 16) ht |
| rw [hl16] at hlo |
| refine ⟨by linarith, ?_⟩ |
| have hr := Real.log_le_sub_one_of_pos (div_pos ht0 (by norm_num) : 0 < t/16) |
| rw [Real.log_div ht0.ne' (by norm_num), hl16] at hr |
| linarith |
| |
| theorem natLog_le_twice_log (n : ℕ) : (Nat.log 2 n : ℝ) ≤ 2 * Real.log n := by |
| have h := Real.natLog_le_logb n 2 |
| simp only [Real.logb, Nat.cast_ofNat] at h |
| have hl : (1:ℝ)/2 ≤ Real.log 2 := by linarith [Real.log_two_gt_d9] |
| have hl0 : 0 < Real.log 2 := by linarith |
| have hn := Real.log_natCast_nonneg n |
| refine h.trans ((div_le_iff₀ hl0).mpr ?_) |
| nlinarith |
| |
| theorem exists_power_parameters {d : ℕ} (hd : 16 ≤ Real.log d) : |
| ∃ b H : ℕ, 2 ≤ b ∧ 1 ≤ H ∧ d ≤ b^H ∧ |
| (4 * (b:ℝ) * (1024 * ((Nat.log 2 (b^H) : ℝ)+1)^2)^H) ≤ |
| Real.exp (32 * Real.sqrt (Real.log d * Real.log (Real.log d))) := by |
| let t := Real.log d |
| let l := Real.log t |
| have ht : 16 ≤ t := hd |
| obtain ⟨hl, hlt⟩ := log_large_aux ht |
| change 1 ≤ l at hl |
| change 4*l ≤ t at hlt |
| have ht0 : 0 < t := by linarith |
| have hl0 : 0 < l := by linarith |
| let r := Real.sqrt (t/l) |
| let s := Real.sqrt (t*l) |
| have hr0 : 0 ≤ r := Real.sqrt_nonneg _ |
| have hs0 : 0 ≤ s := Real.sqrt_nonneg _ |
| have hr2 : r^2 = t/l := Real.sq_sqrt (div_nonneg ht0.le hl0.le) |
| have hs2 : s^2 = t*l := Real.sq_sqrt (mul_nonneg ht0.le hl0.le) |
| have hrl : r^2*l = t := by rw [hr2]; field_simp |
| have hrlo : 2 ≤ r := by |
| have h : (4:ℝ) ≤ t/l := (le_div_iff₀ hl0).mpr (by linarith) |
| nlinarith only [h, hr2, hr0] |
| have hrhi : r ≤ t := by |
| have h : t/l ≤ t := div_le_self ht0.le hl |
| nlinarith only [h, hr2, hr0, ht] |
| have hrs : r*l = s := by |
| have hsq : (r*l)^2 = s^2 := by |
| calc |
| _ = (r^2*l)*l := by ring |
| _ = t*l := by rw [hrl] |
| _ = _ := hs2.symm |
| have hp : 0 ≤ r*l := mul_nonneg hr0 hl0.le |
| nlinarith only [hsq, hp, hs0] |
| have htr : t = r*s := by rw [← hrs, ← hrl]; ring |
| have hslo : 1 ≤ s := by nlinarith only [hs2, hs0, ht, hl] |
| let H := ⌊r⌋₊ |
| have hHhi : (H:ℝ) ≤ r := Nat.floor_le hr0 |
| have hHlow : r < (H:ℝ)+1 := Nat.lt_floor_add_one r |
| have hHhalf : r/2 ≤ (H:ℝ) := by linarith |
| have hHpos : 0 < (H:ℝ) := by linarith |
| have hHnat : 1 ≤ H := by exact_mod_cast (show (1:ℝ) ≤ H by linarith only [hrlo, hHhalf]) |
| let b := ⌈Real.exp (t/H)⌉₊ |
| have hbexp : Real.exp (t/H) ≤ (b:ℝ) := Nat.le_ceil _ |
| have hbupper : (b:ℝ) ≤ 2 * Real.exp (t/H) := by |
| have hc : (b:ℝ) < Real.exp (t/H)+1 := Nat.ceil_lt_add_one (Real.exp_pos _).le |
| have he : 1 ≤ Real.exp (t/H) := Real.one_le_exp_iff.mpr (div_nonneg ht0.le hHpos.le) |
| linarith |
| have hb2 : 2 ≤ b := by |
| have he : 1 < Real.exp (t/H) := Real.one_lt_exp_iff.mpr (div_pos ht0 hHpos) |
| have hb1 : (1:ℝ) < b := he.trans_le hbexp |
| exact_mod_cast hb1 |
| have hb0 : (0:ℝ) < b := by exact_mod_cast (by omega : 0 < b) |
| have hlblo : t/H ≤ Real.log b := by |
| have h := Real.log_le_log (Real.exp_pos _) hbexp |
| simpa only [Real.log_exp] using h |
| have hlbhi : Real.log b ≤ 1 + t/H := by |
| have h := Real.log_le_log hb0 hbupper |
| rw [Real.log_mul (by norm_num) (Real.exp_pos _).ne', Real.log_exp] at h |
| linarith [Real.log_two_lt_d9] |
| let n := b^H |
| have hn0 : (0:ℝ) < n := by dsimp [n]; positivity |
| have hlogn : Real.log n = (H:ℝ)*Real.log b := by simp [n, Real.log_pow] |
| have htn : t ≤ Real.log n := by |
| rw [hlogn] |
| calc |
| _ = (H:ℝ)*(t/H) := by field_simp |
| _ ≤ _ := mul_le_mul_of_nonneg_left hlblo hHpos.le |
| have hnt : Real.log n ≤ 2*t := by |
| rw [hlogn] |
| have h := mul_le_mul_of_nonneg_left hlbhi hHpos.le |
| have he : (H:ℝ)*(1+t/H) = H+t := by field_simp |
| rw [he] at h |
| linarith |
| have hd0 : (0:ℝ) < d := by |
| by_contra h |
| have hd0 : d = 0 := by exact_mod_cast (le_antisymm (not_lt.mp h) (Nat.cast_nonneg d)) |
| norm_num [hd0] at hd |
| have hdn : d ≤ n := by |
| exact_mod_cast (Real.log_le_log_iff hd0 hn0).mp htn |
| let K : ℝ := (Nat.log 2 n : ℝ)+1 |
| have hKpos : 0 < K := by dsimp [K]; positivity |
| have hKbound : K ≤ 5*t := by |
| have h := natLog_le_twice_log n |
| dsimp [K] |
| linarith |
| have hlogK : Real.log K ≤ 4+l := by |
| have h := Real.log_le_log hKpos hKbound |
| rw [Real.log_mul (by norm_num) ht0.ne'] at h |
| have h5 := Real.log_le_sub_one_of_pos (by norm_num : (0:ℝ) < 5) |
| dsimp [l] |
| linarith |
| let B : ℝ := 1024*K^2 |
| have hBpos : 0 < B := by dsimp [B]; positivity |
| have hlogB : Real.log B ≤ 20*l := by |
| have h1024 : Real.log 1024 = 10 * Real.log 2 := by |
| rw [show (1024:ℝ)=2^10 by norm_num, Real.log_pow] |
| norm_num |
| dsimp [B] |
| rw [Real.log_mul (by norm_num) (pow_pos hKpos 2).ne', Real.log_pow, h1024] |
| norm_num |
| linarith only [hlogK, hl, Real.log_two_lt_d9] |
| have hdiv : t/(H:ℝ) ≤ 2*s := by |
| apply (div_le_iff₀ hHpos).mpr |
| rw [htr] |
| nlinarith only [hHhalf, hs0] |
| have hHl : (H:ℝ)*l ≤ s := by rw [← hrs]; exact mul_le_mul_of_nonneg_right hHhi hl0.le |
| have hprodpos : 0 < 4*(b:ℝ)*B^H := by positivity |
| have hlogprod : Real.log (4*(b:ℝ)*B^H) ≤ 32*s := by |
| rw [Real.log_mul (by positivity) (pow_pos hBpos H).ne', |
| Real.log_mul (by norm_num) hb0.ne', Real.log_pow] |
| have h4 : Real.log 4 ≤ 2 := by |
| rw [show (4:ℝ)=2^2 by norm_num, Real.log_pow] |
| norm_num only [Nat.cast_ofNat] |
| nlinarith only [Real.log_two_lt_d9] |
| have h := mul_le_mul_of_nonneg_left hlogB hHpos.le |
| nlinarith only [h4, hlbhi, hdiv, h, hHl, hslo] |
| refine ⟨b, H, hb2, hHnat, hdn, ?_⟩ |
| change 4*(b:ℝ)*B^H ≤ Real.exp (32*s) |
| exact (Real.log_le_iff_le_exp hprodpos).mp hlogprod |
|
|
| end FiniteCircle |
|
|
| end OAI |
|
|
