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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 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 | 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
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