openAI-math-dataset / lean /OAI /Combinatorics /LowSpaceSubsetSum /Separators /SeparatorSamplerCoupling.lean
| import OAI.Combinatorics.LowSpaceSubsetSum.Representation.RepresentationSamplerCoupling | |
| namespace OAI | |
| /-! # Bounded sampling of the four actual separator families -/ | |
| noncomputable section | |
| namespace LowSpaceSubsetSum | |
| open scoped BigOperators | |
| attribute [local instance] Classical.propDecidable | |
| namespace SeparatorExpectation.RoundedParameters | |
| variable {g : ℕ} (S : RoundedParameters g) | |
| private instance sampleNonempty (i : Fin 4) : | |
| Nonempty (SeparatorSample (Finset.univ : Finset (Fin g)) (S.e i)) := | |
| ⟨⟨initialSubset g (S.e i) (S.size_le i), | |
| Finset.mem_powersetCard.mpr ⟨Finset.subset_univ _, initialSubset_card _ _ _⟩⟩⟩ | |
| def samplerCoupling (n : ℕ) : SamplerCoupling (SeparatorCheck.Draws g S.e S.c) := | |
| SamplerCoupling.pi 4 (fun i => Fin (S.counts i) → | |
| SeparatorSample (Finset.univ : Finset (Fin g)) (S.e i)) | |
| (fun i => SamplerCoupling.pi (S.counts i) (fun _ => | |
| SeparatorSample (Finset.univ : Finset (Fin g)) (S.e i)) | |
| (fun _ => fixedSizeCoupling n g (S.e i) (S.size_le i))) | |
| theorem samplerCoupling_uniform (n : ℕ) : | |
| (S.samplerCoupling n).idealLaw = FiniteLaw.uniform := by | |
| apply SamplerCoupling.pi_uniform | |
| intro i | |
| apply SamplerCoupling.pi_uniform | |
| intro j | |
| exact fixedSizeCoupling_uniform n g (S.e i) (S.size_le i) | |
| theorem counts_le (i : Fin 4) : S.counts i ≤ 11 * 2 ^ g := | |
| rounded_family_size g (S.e i) (S.c i) (S.size_le i) (S.feasible i) | |
| (separator_denominator_le g (S.e i) (S.c i) (S.count_half i) (S.count_le i)) | |
| /-- Each subset uses at most `g` range calls. Repeated subset occurrences | |
| remain independent calls, exactly as in the stored sampled family. -/ | |
| theorem samplerCoupling_failure (n : ℕ) : | |
| (S.samplerCoupling n).failure ≤ (44 * g * 2 ^ g : ℕ) / (2 : ℝ) ^ (n ^ 3) := by | |
| have hi (i : Fin 4) : | |
| (SamplerCoupling.pi (S.counts i) (fun _ => | |
| SeparatorSample (Finset.univ : Finset (Fin g)) (S.e i)) | |
| (fun _ => fixedSizeCoupling n g (S.e i) (S.size_le i))).failure ≤ | |
| (S.counts i : ℝ) * (g / (2 : ℝ) ^ (n ^ 3)) := by | |
| simpa using SamplerCoupling.pi_failure_le (S.counts i) (fun _ => | |
| SeparatorSample (Finset.univ : Finset (Fin g)) (S.e i)) | |
| (fun _ => fixedSizeCoupling n g (S.e i) (S.size_le i)) | |
| (fun _ => g / (2 : ℝ) ^ (n ^ 3)) | |
| (fun _ => fixedSizeCoupling_failure n g (S.e i) (S.size_le i)) | |
| have h := SamplerCoupling.pi_failure_le 4 (fun i => Fin (S.counts i) → | |
| SeparatorSample (Finset.univ : Finset (Fin g)) (S.e i)) | |
| (fun i => SamplerCoupling.pi (S.counts i) (fun _ => | |
| SeparatorSample (Finset.univ : Finset (Fin g)) (S.e i)) | |
| (fun _ => fixedSizeCoupling n g (S.e i) (S.size_le i))) | |
| (fun i => (S.counts i : ℝ) * (g / (2 : ℝ) ^ (n ^ 3))) hi | |
| apply h.trans | |
| calc | |
| _ ≤ ∑ _i : Fin 4, ((11 * 2 ^ g : ℕ) : ℝ) * (g / (2 : ℝ) ^ (n ^ 3)) := by | |
| apply Finset.sum_le_sum | |
| intro i _ | |
| exact mul_le_mul_of_nonneg_right (by exact_mod_cast S.counts_le i) (by positivity) | |
| _ = _ := by simp; ring | |
| end SeparatorExpectation.RoundedParameters | |
| namespace RepresentationSetup | |
| variable {n : ℕ} (R : RepresentationSetup n) | |
| def separatorCoupling (h : R.PrimeHistory) (hn : 2200 ≤ n) : | |
| SamplerCoupling (R.SeparatorDraws h hn) := | |
| (separatorParameters h.1.1.2 hn).samplerCoupling n | |
| theorem separatorCoupling_uniform (h : R.PrimeHistory) (hn : 2200 ≤ n) : | |
| (R.separatorCoupling h hn).idealLaw = R.separatorLaw h hn := | |
| (separatorParameters h.1.1.2 hn).samplerCoupling_uniform n | |
| theorem separatorCoupling_failure (h : R.PrimeHistory) (hn : 2200 ≤ n) : | |
| (R.separatorCoupling h hn).failure ≤ | |
| (44 * (2 * blockHalf n 0) * 2 ^ (2 * blockHalf n 0) : ℕ) / (2 : ℝ) ^ (n ^ 3) := | |
| (separatorParameters h.1.1.2 hn).samplerCoupling_failure n | |
| end RepresentationSetup | |
| end LowSpaceSubsetSum | |
| end | |
| end OAI | |