| import OAI.Analysis.FreeEntropy.BlockEnsembleLower |
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| namespace OAI |
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|
| noncomputable section |
| open MeasureTheory ProbabilityTheory Finset |
| open scoped ENNReal |
| namespace FiniteEntropySeparation |
| variable {p d : ℕ} |
| theorem two_pow_le_exp (k : ℕ) : (2:ℝ)^k≤Real.exp k := by |
| have hh := pow_le_pow_left₀ (by norm_num : (0:ℝ)≤2) (show (2:ℝ)≤Real.exp 1 from by linarith [Real.add_one_le_exp (1:ℝ)]) k |
| simpa only [← Real.exp_nat_mul,mul_one] using hh |
| theorem blockEvent_gaussian_four (hd : 64≤d) {v : ℝ} (hv : 0<v) |
| (hvs : Real.sqrt v≤(2:ℝ)^25) (hpv : 32768*v≤p) (hp : (p:ℝ)≤Real.exp 64) : |
| (normalLaw (v/d)) (blockEvent p d (Real.sqrt ((p:ℝ)*d)/4) (4*Real.sqrt (v*p*d)))≤ |
| ENNReal.ofReal (Real.exp (-4*(d:ℝ)^2)) := by |
| have hd0 : 0<d := by omega |
| have hd' : (64:ℝ)≤d := by exact_mod_cast hd |
| have hdpos : (0:ℝ)<d := by linarith |
| have hp0 : 0<(p:ℝ) := by nlinarith |
| have hpd : 0<(p:ℝ)*d := mul_pos hp0 hdpos |
| have hsq : 0<Real.sqrt v := Real.sqrt_pos.mpr hv |
| let ε : ℝ := 1/(64*Real.sqrt v) |
| have hε : 0<ε := by dsimp [ε]; positivity |
| have hroot : Real.sqrt (v*p*d)=Real.sqrt v*Real.sqrt ((p:ℝ)*d) := by |
| rw [mul_assoc,Real.sqrt_mul hv.le] |
| have hgap : Real.sqrt ((p:ℝ)*d)/4-2*ε*(4*Real.sqrt (v*p*d))=Real.sqrt ((p:ℝ)*d)/8 := by |
| rw [hroot] |
| dsimp [ε] |
| field_simp |
| ring |
| have hbase : 1+2/ε≤Real.exp 34 := by |
| have hb : 1+2/ε≤(2:ℝ)^34 := by |
| have he : 1+2/ε=1+128*Real.sqrt v := by dsimp [ε]; field_simp; ring |
| rw [he] |
| norm_num at hvs ⊢ |
| linarith |
| exact hb.trans (two_pow_le_exp 34) |
| have htail : -((Real.sqrt ((p:ℝ)*d)/8)^2)/(4*(v/d))≤-128*(d:ℝ)^2 := by |
| rw [div_pow,Real.sq_sqrt hpd.le] |
| have he : -((p:ℝ)*d/(8:ℝ)^2)/(4*(v/d))=-(p:ℝ)*(d:ℝ)^2/(256*v) := by field_simp; ring |
| rw [he] |
| apply (div_le_iff₀ (by positivity : 0<256*v)).mpr |
| have hh := mul_le_mul_of_nonneg_right hpv (sq_nonneg (d:ℝ)) |
| nlinarith |
| have h1 : (p:ℝ)^d≤Real.exp ((d:ℝ)^2) := by |
| calc |
| _ ≤ (Real.exp 64)^d := pow_le_pow_left₀ hp0.le hp d |
| _ = Real.exp ((d:ℝ)*64) := (Real.exp_nat_mul _ _).symm |
| _ ≤ _ := Real.exp_le_exp.mpr (by nlinarith) |
| have h2 : (1+2/ε)^(2*d^2)≤Real.exp (68*(d:ℝ)^2) := by |
| calc |
| _ ≤ (Real.exp 34)^(2*d^2) := pow_le_pow_left₀ (by positivity) hbase _ |
| _ = _ := by rw [← Real.exp_nat_mul]; congr 1; push_cast; ring |
| have h3 : Real.exp (-((Real.sqrt ((p:ℝ)*d)/8)^2)/(4*(v/d)))≤Real.exp (-128*(d:ℝ)^2) := Real.exp_le_exp.mpr htail |
| have h4 : Real.sqrt 2^(2*d^2)≤Real.exp (2*(d:ℝ)^2) := by |
| have hs : Real.sqrt 2≤Real.exp 1 := by |
| apply le_trans (show Real.sqrt 2≤2 from by nlinarith [Real.sq_sqrt (show (0:ℝ)≤2 by norm_num),Real.sqrt_nonneg (2:ℝ)]) |
| linarith [Real.add_one_le_exp (1:ℝ)] |
| calc |
| _ ≤ (Real.exp 1)^(2*d^2) := pow_le_pow_left₀ (Real.sqrt_nonneg _) hs _ |
| _ = _ := by rw [← Real.exp_nat_mul]; congr 1; push_cast; ring |
| have hb := blockEvent_gaussian_bound (p := p) hd0 (div_pos hv hdpos) hε |
| (show 0≤Real.sqrt ((p:ℝ)*d)/4-2*ε*(4*Real.sqrt (v*p*d)) by rw [hgap]; positivity) |
| apply hb.trans |
| apply ENNReal.ofReal_le_ofReal |
| rw [hgap] |
| calc |
| _ ≤ Real.exp ((d:ℝ)^2)*Real.exp (68*(d:ℝ)^2)*Real.exp (-128*(d:ℝ)^2)*Real.exp (2*(d:ℝ)^2) := |
| mul_le_mul (mul_le_mul (mul_le_mul h1 h2 (by positivity) (Real.exp_pos _).le) h3 (by positivity) (by positivity)) h4 (by positivity) (by positivity) |
| _ = Real.exp (-57*(d:ℝ)^2) := by simp only [← Real.exp_add]; congr 1; ring |
| _ ≤ _ := Real.exp_le_exp.mpr (by nlinarith [sq_nonneg (d:ℝ)]) |
| def chosenP : ℕ := 2^64 |
| def chosenTime : ℝ := 2^48 |
| instance chosenP_neZero : NeZero chosenP := ⟨by norm_num [chosenP]⟩ |
| theorem chosenP_ge_two : 2≤chosenP := by norm_num [chosenP] |
| theorem chosenTime_pos : 0<chosenTime := by norm_num [chosenTime] |
| theorem chosen_time_noise_budget : 256*chosenTime≤(chosenP:ℝ) := by norm_num [chosenTime,chosenP] |
| theorem chosen_gaussian_block (hd : 64≤d) {h : ℝ} (hh : 0<h) (hh1 : h<1) : |
| (normalLaw ((1+h+chosenTime)/d)) |
| (blockEvent chosenP d (Real.sqrt ((chosenP:ℝ)*d)/4) (4*Real.sqrt ((1+h+chosenTime)*chosenP*d)))≤ |
| ENNReal.ofReal (Real.exp (-4*(d:ℝ)^2)) := by |
| apply blockEvent_gaussian_four hd (by have := chosenTime_pos; linarith) |
| · apply (Real.sqrt_le_left (by positivity : (0:ℝ)≤(2:ℝ)^25)).mpr |
| norm_num [chosenTime] at * |
| linarith |
| · norm_num [chosenP,chosenTime] at * |
| linarith |
| · simpa only [chosenP,Nat.cast_pow,Nat.cast_ofNat] using two_pow_le_exp 64 |
| end FiniteEntropySeparation |
| |
| end |
| |
| end OAI |
| |
