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5d5048e | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 | 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
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