openAI-math-dataset / lean /OAI /Combinatorics /LowSpaceSubsetSum /Compression /CompressionSamplerCoupling.lean
| import OAI.Combinatorics.LowSpaceSubsetSum.Storage.PermutationSamplerCoupling | |
| import OAI.Combinatorics.LowSpaceSubsetSum.Compression.CompressionSoundness | |
| namespace OAI | |
| /-! # Input-only bounded permutation and prime draws for compression -/ | |
| noncomputable section | |
| namespace LowSpaceSubsetSum.CompressionData | |
| variable {n : ℕ} (D : CompressionData n) | |
| /-- Compression draws one permutation and one prime. Its residue counter is | |
| enumerated deterministically by `compressionTrial`. -/ | |
| def sampler (_D : CompressionData n) (hn : 168000000 ≤ n) : | |
| SamplerCoupling (CompressionSetup.TrialHistory n) := | |
| (permutationCoupling n n).bind (fun _ => CompressionSetup.PrimeSpace n) | |
| (fun _ => primeCoupling (by unfold compressionPrimeExponent; omega) n) | |
| theorem sampler_uniform (hn : 168000000 ≤ n) : | |
| (D.sampler hn).idealLaw = D.trialLaw hn := by | |
| rw [sampler, SamplerCoupling.bind_idealLaw, permutationCoupling_uniform] | |
| simp only [primeCoupling_uniform] | |
| rfl | |
| theorem sampler_failure (hn : 168000000 ≤ n) : | |
| (D.sampler hn).failure ≤ n / (2 : ℝ) ^ (n ^ 3) + Real.exp (-((n : ℝ) ^ 2 / 4)) := by | |
| have hprime : compressionPrimeExponent n ≤ n := by | |
| unfold compressionPrimeExponent | |
| omega | |
| have h := SamplerCoupling.bind_failure_le (permutationCoupling n n) | |
| (fun _ => CompressionSetup.PrimeSpace n) | |
| (fun _ => primeCoupling (by unfold compressionPrimeExponent; omega) n) | |
| (Real.exp (-((n : ℝ) ^ 2 / 4))) (fun _ => primeCoupling_failure _ hprime) | |
| exact h.trans (add_le_add (permutationCoupling_failure n n) le_rfl) | |
| private theorem prime_tail (hn : 168000000 ≤ n) : | |
| Real.exp (-((n : ℝ) ^ 2 / 4)) ≤ 1 / (2 : ℝ) ^ (2 * n) := by | |
| have hnR : (16 : ℝ) ≤ n := by exact_mod_cast (show 16 ≤ n by omega) | |
| have htwo : (2 : ℝ) ≤ Real.exp 1 := by linarith [Real.add_one_le_exp (1 : ℝ)] | |
| have hp : (2 : ℝ) ^ (2 * n) ≤ Real.exp ((n : ℝ) ^ 2 / 4) := by | |
| calc | |
| _ ≤ (Real.exp 1) ^ (2 * n) := pow_le_pow_left₀ (by norm_num) htwo _ | |
| _ = Real.exp ((2 * n : ℕ) : ℝ) := by rw [← Real.exp_nat_mul]; simp | |
| _ ≤ _ := Real.exp_le_exp.mpr (by push_cast; nlinarith) | |
| rw [Real.exp_neg, ← one_div] | |
| exact one_div_le_one_div_of_le (by positivity) hp | |
| theorem sampler_failure_pow (hn : 168000000 ≤ n) : | |
| (D.sampler hn).failure ≤ 1 / (2 : ℝ) ^ n := by | |
| have hcube : 3 * n ≤ n ^ 3 := by | |
| have hs : 3 ≤ n ^ 2 := by nlinarith [show 16 ≤ n by omega] | |
| calc | |
| 3 * n ≤ n ^ 2 * n := Nat.mul_le_mul_right n hs | |
| _ = n ^ 3 := by ring | |
| have hrange : (n : ℝ) / (2 : ℝ) ^ (n ^ 3) ≤ 1 / (2 : ℝ) ^ (2 * n) := by | |
| apply (div_le_div_iff₀ (by positivity) (by positivity)).mpr | |
| rw [one_mul] | |
| calc | |
| _ ≤ (2 : ℝ) ^ n * 2 ^ (2 * n) := | |
| mul_le_mul_of_nonneg_right (by exact_mod_cast (Nat.lt_two_pow_self (n := n)).le) | |
| (by positivity) | |
| _ = (2 : ℝ) ^ (3 * n) := by rw [← pow_add]; congr 1; omega | |
| _ ≤ _ := pow_le_pow_right₀ (by norm_num) hcube | |
| have hp := prime_tail hn | |
| have hb := D.sampler_failure hn | |
| have htwo : (2 : ℝ) ≤ 2 ^ n := by | |
| simpa using pow_le_pow_right₀ (by norm_num : (1 : ℝ) ≤ 2) (by omega : 1 ≤ n) | |
| calc | |
| _ ≤ 2 / (2 : ℝ) ^ (2 * n) := by | |
| simp only [div_eq_mul_inv, one_mul] at hrange hp hb ⊢ | |
| linarith | |
| _ ≤ 1 / (2 : ℝ) ^ n := by | |
| apply (div_le_div_iff₀ (by positivity) (by positivity)).mpr | |
| rw [one_mul, two_mul n, pow_add] | |
| exact mul_le_mul_of_nonneg_right htwo (by positivity) | |
| /-- Every ideal compression event loses at most the unconditional exhaustion | |
| probability. No event is conditioned on successful sampling. -/ | |
| theorem sampler_event (hn : 168000000 ≤ n) | |
| (P : CompressionSetup.TrialHistory n → Prop) : | |
| (D.trialLaw hn).probability P - 1 / (2 : ℝ) ^ n ≤ | |
| (D.sampler hn).boundedLaw.probability (fun o => ∃ h, o = some h ∧ P h) := by | |
| have h := (D.sampler hn).event_lower P | |
| rw [D.sampler_uniform hn] at h | |
| exact (sub_le_sub_left (D.sampler_failure_pow hn) _).trans h | |
| end LowSpaceSubsetSum.CompressionData | |
| end | |
| end OAI | |