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dd14c26 | 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 | import OAI.Probability.MatroidSecretary.Secretary.Assembly
import OAI.Probability.MatroidSecretary.Secretary.Seed
import OAI.Probability.MatroidProphet.Main
namespace OAI
/-! The actual hidden-vector performance theorem supplies the secretary bound.
The finite coupling laws below are internal inputs; the final endpoint discharges
them with the concrete source-prefix/table construction. -/
open MeasureTheory ProbabilityTheory
namespace MatroidProphet.SecretaryReplay
theorem reconstructed_source_guarantee.{u} {n : ℕ} {Q : Type*}
[Fintype Q] [MeasurableSpace Q] [MeasurableSingletonClass Q]
(M : Matroid (Fin n)) (hE : M.E = Set.univ)
(K : Q → Fin (n + 1))
(decode : Q → Finset (Fin n) → Seed (mainSeedBits n))
(νQ : Measure Q)
(hproductLaw : (νQ.prod (Secretary.uniformArrivalLaw n)).map
(fun z => decode z.1 (orderPrefix z.2 (K z.1).val)) = sourceSeedLaw n)
(hproductMask : ∀ᵐ z ∂νQ.prod (Secretary.uniformArrivalLaw n),
(completeHiddenRule M hE).mask (decode z.1 (orderPrefix z.2 (K z.1).val)) =
orderPrefix z.2 (K z.1).val) :
SecretaryGuarantee.{u} M (reconstructedRule (completeHiddenRule M hE) K decode)
νQ (((2 : ℝ) ^ 293)⁻¹) := by
intro w hw Ω mΩ μ hμ q σ π hq hσ hπ hqlaw hσlaw hind hprefix
obtain ⟨hi, hl⟩ := reconstructed_product_lower (completeHiddenRule M hE) K decode
νQ (Secretary.uniformArrivalLaw n) (sourceSeedLaw n) hproductLaw hproductMask
w μ q σ π hq hσ hπ hqlaw hσlaw hind hprefix
exact ⟨hi, (complete_hidden_guarantee M hE w hw).trans hl⟩
/-- A concrete finite coupling completes the entire existential for one matroid,
including pointwise feasibility even at null seeds and the full adversary clause. -/
theorem reconstructed_source_witness.{u} {n : ℕ} {Q : Type*}
[Fintype Q] [MeasurableSpace Q] [MeasurableSingletonClass Q]
(M : Matroid (Fin n)) (hE : M.E = Set.univ)
(K : Q → Fin (n + 1))
(decode : Q → Finset (Fin n) → Seed (mainSeedBits n))
(νQ : Measure Q) [IsProbabilityMeasure νQ]
(hproductLaw : (νQ.prod (Secretary.uniformArrivalLaw n)).map
(fun z => decode z.1 (orderPrefix z.2 (K z.1).val)) = sourceSeedLaw n)
(hproductMask : ∀ᵐ z ∂νQ.prod (Secretary.uniformArrivalLaw n),
(completeHiddenRule M hE).mask (decode z.1 (orderPrefix z.2 (K z.1).val)) =
orderPrefix z.2 (K z.1).val) :
∃ (q : ℕ) (ν : Measure (Fin q)), IsProbabilityMeasure ν ∧
∃ (A : SecretaryRule n (Fin q)), SecretaryFeasible M A ∧
SecretaryGuarantee.{u} M A ν (((2 : ℝ) ^ 293)⁻¹) := by
exact secretaryFinSeed_witness (reconstructedRule (completeHiddenRule M hE) K decode)
M νQ (((2 : ℝ) ^ 293)⁻¹)
(reconstructed_feasible M (completeHiddenRule M hE) K decode
(completeHiddenRule_feasible M hE))
(reconstructed_source_guarantee M hE K decode νQ hproductLaw hproductMask)
end MatroidProphet.SecretaryReplay
end OAI
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