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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 60 61 62 63 64 | import OAI.Probability.MatroidSecretary.Secretary.LawTransfer
import OAI.Probability.MatroidSecretary.Secretary.ReplayTransport
namespace OAI
open MeasureTheory ProbabilityTheory
namespace MatroidProphet.SecretaryReplay
lemma prefix_set_eq {n : ℕ} {K : ℕ} {σ π : ArrivalOrder n}
(h : SamePrefix K σ π) : orderPrefix σ K = orderPrefix π K := by
ext e
simp only [mem_orderPrefix]
constructor
· intro he
have hp := h (σ.symm e) he
have heq : π.symm e = σ.symm e := by
apply π.injective
simpa using hp
simpa [heq] using he
· intro he
have hp := h (π.symm e) he
have heq : σ.symm e = π.symm e := by
apply σ.injective
simpa using hp.symm
simpa [heq] using he
/-- A finite product coupling supplies the entire arbitrary-ambient-space lower
bound. In particular nothing here requires independence of the executed order. -/
theorem reconstructed_product_lower {n bits : ℕ} {Q : Type*}
[Fintype Q] [MeasurableSpace Q] [MeasurableSingletonClass Q]
(A : HiddenRule n bits) (K : Q → Fin (n + 1))
(decode : Q → Finset (Fin n) → Seed bits)
(νQ : Measure Q) (η : Measure (ArrivalOrder n)) (ν : Measure (Seed bits))
(hproductLaw : (νQ.prod η).map
(fun z => decode z.1 (orderPrefix z.2 (K z.1).val)) = ν)
(hproductMask : ∀ᵐ z ∂νQ.prod η,
A.mask (decode z.1 (orderPrefix z.2 (K z.1).val)) =
orderPrefix z.2 (K z.1).val)
(w : Weights n) {Ω : Type*} [MeasurableSpace Ω]
(μ : Measure Ω) [IsProbabilityMeasure μ]
(q : Ω → Q) (σ π : Ω → ArrivalOrder n)
(hq : Measurable q) (hσ : Measurable σ) (hπ : Measurable π)
(hqlaw : μ.map q = νQ) (hσlaw : μ.map σ = η)
(hind : IndepFun q σ μ)
(hprefix : ∀ᵐ ω ∂μ, SamePrefix (K (q ω)).val (σ ω) (π ω)) :
Integrable (fun ω => secretaryReward (reconstructedRule A K decode) (q ω) w (π ω)) μ ∧
(∫ r, hiddenWorstReward A w r ∂ν) ≤
∫ ω, secretaryReward (reconstructedRule A K decode) (q ω) w (π ω) ∂μ := by
have hlaw := secretary_reconstruction_law μ νQ η ν
(fun z => decode z.1 (orderPrefix z.2 (K z.1).val))
(measurable_of_finite _) hproductLaw q σ hq hσ hqlaw hσlaw hind
have hmask := secretary_product_ae μ νQ η q σ hq hσ hqlaw hσlaw hind
(fun z => A.mask (decode z.1 (orderPrefix z.2 (K z.1).val)) =
orderPrefix z.2 (K z.1).val) hproductMask
apply reconstructed_integral_lower A K decode w μ ν q σ π hq hσ hπ
_ hmask hlaw.map_eq
filter_upwards [hprefix] with ω hω
exact (prefix_set_eq hω).symm
end MatroidProphet.SecretaryReplay
end OAI
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