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