openAI-math-dataset / lean /OAI /Combinatorics /LowSpaceSubsetSum /Separators /SeparatorEncodedBounds.lean
| import OAI.Combinatorics.LowSpaceSubsetSum.Separators.SeparatorEncodedCheck | |
| import OAI.Combinatorics.LowSpaceSubsetSum.Separators.SeparatorWorkBound | |
| namespace OAI | |
| /-! # Fixed-pair coverage and scan bounds for original indexed entries -/ | |
| noncomputable section | |
| namespace LowSpaceSubsetSum.TopCoordinates | |
| open SeparatorExpectation RepresentationMoments | |
| variable {n g : ℕ} (C : TopCoordinates n g) | |
| theorem finds_fixed_pair (families : Fin 4 → Array (Finset (Fin g))) (target : ℤ) | |
| (xs ys : List (Entry (Fin n))) (hsize : xs.length + ys.length ≤ 2 ^ n) | |
| (x y : Entry (Fin n)) (hx : x ∈ xs) (hy : y ∈ ys) | |
| (hw : x.value + y.value = target) | |
| (hc : SeparatorCheck.Covered families (C.encodeEntry x) (C.encodeEntry y)) : | |
| (C.check families target xs ys).accepted = true := | |
| SeparatorCheck.finds_fixed_pair n families target (xs.map C.encodeEntry) (ys.map C.encodeEntry) | |
| (by simpa using hsize) (C.encodeEntry x) (C.encodeEntry y) | |
| (List.mem_map.mpr ⟨x, hx, rfl⟩) (List.mem_map.mpr ⟨y, hy, rfl⟩) hw hc | |
| theorem coverage_probability (S : RoundedParameters g) (x y : Entry (Fin n)) | |
| (hd : Disjoint x.mask y.mask) | |
| (hc : ∀ i, (SeparatorCheck.localMask i (C.encodeEntry x).mask).card = S.c i) | |
| (hhalf : ∀ i, S.c i + (SeparatorCheck.localMask i (C.encodeEntry y).mask).card = g / 2) : | |
| 1 - 4 * Real.exp (-10) ≤ finiteUniformProbability | |
| (fun draws : SeparatorCheck.Draws g S.e S.c => | |
| SeparatorCheck.Covered (SeparatorCheck.sampledFamilies draws) (C.encodeEntry x) (C.encodeEntry y)) := | |
| SeparatorCheck.coverage_probability S.e S.c (C.encodeEntry x) (C.encodeEntry y) | |
| S.even (C.encoded_disjoint hd) hc hhalf S.count_le S.size_le S.feasible | |
| theorem mean_scans_polynomial (S : RoundedParameters g) (u : ℕ) | |
| (hu : 256 ≤ u) (hnu : n ≤ u) (hgu : g + 1 ≤ u) | |
| (target : ℤ) (xs ys : List (Entry (Fin n))) | |
| (hleft : ∀ x ∈ xs, ∀ i, (SeparatorCheck.localMask i (C.encodeEntry x).mask).card = S.c i) | |
| (hright : ∀ x ∈ ys, ∀ i, | |
| (SeparatorCheck.localMask i (C.encodeEntry x).mask).card = g / 2 - S.c i) : | |
| mean (fun draws : IndependentFamilies g S.e S.counts => | |
| ((C.check (familyArrays draws) target xs ys).scans : ℝ)) ≤ | |
| ((xs.length + ys.length : ℕ) : ℝ) * (u : ℝ) ^ 22 * | |
| (2 : ℝ) ^ (((4 * g : ℕ) : ℝ) * (1 - binaryEntropy (1 / 4))) := by | |
| have h := S.mean_scans_polynomial n u hu hnu hgu target | |
| (xs.map C.encodeEntry) (ys.map C.encodeEntry) (by | |
| intro z hz | |
| obtain ⟨x, hx, rfl⟩ := List.mem_map.mp hz | |
| exact hleft x hx) (by | |
| intro z hz | |
| obtain ⟨y, hy, rfl⟩ := List.mem_map.mp hz | |
| exact hright y hy) | |
| simpa only [List.length_map, check] using h | |
| end LowSpaceSubsetSum.TopCoordinates | |
| end | |
| end OAI | |