openAI-math-dataset / lean /OAI /Probability /MatroidSecretary /Labels /LabeledTransportOneSample.lean
| import OAI.Probability.MatroidSecretary.Labels.LabeledTransport | |
| import OAI.Probability.MatroidProphet.Main | |
| namespace OAI | |
| /-! The indexed endpoint transported to any finite labeled ground type. The | |
| encoding is fixed from the labels alone, before distributions and randomness. | |
| No representability, positive-rank, atomlessness, or per-label moment premise | |
| is introduced. This is optional source-correspondence support. -/ | |
| namespace MatroidProphet.LabeledTransport | |
| open MeasureTheory ProbabilityTheory | |
| variable {α : Type*} [Fintype α] [MeasurableSpace α] [MeasurableSingletonClass α] | |
| def pairedCoordinates {Ω : Type*} (S V : Ω → α → ℝ) (i : α × Bool) : Ω → ℝ := | |
| if i.2 then fun ω => S ω i.1 else fun ω => V ω i.1 | |
| /-- The main guarantee on arbitrary finite physical labels, with a measurable | |
| history-only rule and the same all-prefix feasibility and arbitrary-law bound. -/ | |
| theorem one_sample.{u} (M : Matroid α) (hE : M.E = Set.univ) : | |
| ∃ (bits : ℕ) (ν : Measure (Seed bits)), IsProbabilityMeasure ν ∧ | |
| ∃ (A : OnlineRule α (Fintype.card α) bits), | |
| (∀ (r : Seed bits) (s v : α → ℝ) (π : ArrivalOrder α (Fintype.card α)) (t : ℕ), | |
| (∀ a, 0 ≤ s a) → (∀ a, 0 ≤ v a) → | |
| M.Indep (acceptedThrough A r s v π t : Set α)) ∧ | |
| ∀ {Ω : Type u} [MeasurableSpace Ω] (μ : Measure Ω) [IsProbabilityMeasure μ] | |
| (S V : Ω → α → ℝ) (R : Ω → Seed bits), | |
| Measurable S → Measurable V → Measurable R → | |
| (∀ᵐ ω ∂μ, ∀ a, 0 ≤ S ω a) → | |
| (∀ᵐ ω ∂μ, ∀ a, 0 ≤ V ω a) → | |
| iIndepFun (pairedCoordinates S V) μ → | |
| (∀ a, Measure.map (fun ω => S ω a) μ = Measure.map (fun ω => V ω a) μ) → | |
| IndepFun (fun ω => (S ω, V ω)) R μ → | |
| Measure.map R μ = ν → | |
| Integrable (fun ω => optimum M (V ω)) μ → | |
| ∀ (π : Ω → ArrivalOrder α (Fintype.card α)), Measurable π → | |
| Integrable (fun ω => reward A (R ω) (S ω) (V ω) (π ω)) μ ∧ | |
| ((2 : ℝ) ^ 310)⁻¹ * (∫ ω, optimum M (V ω) ∂μ) ≤ | |
| ∫ ω, reward A (R ω) (S ω) (V ω) (π ω) ∂μ := by | |
| classical | |
| let e : α ≃ Fin (Fintype.card α) := Fintype.equivFin α | |
| obtain ⟨bits, ν, hν, A, hA, hperf⟩ := | |
| MatroidProphet.one_sample.{u} (Fintype.card α) (M.mapEquiv e) (mapEquiv_ground M hE e) | |
| refine ⟨bits, ν, hν, decodeRule e A, ?_, ?_⟩ | |
| · intro r s v π t hs hv | |
| exact decode_feasible M e A hA r s v π t hs hv | |
| · intro Ω _ μ _ S V R hS hV hR hS0 hV0 hind hmatch hseed hlaw hint π hπ | |
| let S' : Ω → Weights (Fintype.card α) := fun ω => S ω ∘ e.symm | |
| let V' : Ω → Weights (Fintype.card α) := fun ω => V ω ∘ e.symm | |
| have hS' : Measurable S' := by | |
| apply Measurable.of_eval | |
| intro i | |
| exact (measurable_pi_apply (e.symm i)).comp hS | |
| have hV' : Measurable V' := by | |
| apply Measurable.of_eval | |
| intro i | |
| exact (measurable_pi_apply (e.symm i)).comp hV | |
| have hS0' : ∀ᵐ ω ∂μ, ∀ i, 0 ≤ S' ω i := | |
| hS0.mono (fun _ h i => h (e.symm i)) | |
| have hV0' : ∀ᵐ ω ∂μ, ∀ i, 0 ≤ V' ω i := | |
| hV0.mono (fun _ h i => h (e.symm i)) | |
| have hind' : iIndepFun (MatroidProphet.pairedCoordinates S' V') μ := by | |
| have h := hind.precomp (g := fun i : Fin (Fintype.card α) × Bool => (e.symm i.1, i.2)) | |
| (by | |
| intro i j hij | |
| apply Prod.ext | |
| · exact e.symm.injective (congrArg Prod.fst hij) | |
| · exact congrArg (fun p : α × Bool => p.2) hij) | |
| exact h | |
| have hmap : Measurable (fun x : (α → ℝ) × (α → ℝ) => | |
| (x.1 ∘ e.symm, x.2 ∘ e.symm)) := by | |
| apply Measurable.prodMk | |
| · apply Measurable.of_eval | |
| intro i | |
| exact (measurable_pi_apply (e.symm i)).comp measurable_fst | |
| · apply Measurable.of_eval | |
| intro i | |
| exact (measurable_pi_apply (e.symm i)).comp measurable_snd | |
| have hseed' : IndepFun (fun ω => (S' ω, V' ω)) R μ := | |
| hseed.comp hmap measurable_id | |
| have hint' : Integrable (fun ω => MatroidProphet.optimum (M.mapEquiv e) (V' ω)) μ := by | |
| simpa only [optimum_eq_indexed M e, V'] using hint | |
| have hπ' : Measurable (fun ω => (π ω).trans e) := | |
| (measurable_of_finite (fun p : ArrivalOrder α (Fintype.card α) => p.trans e)).comp hπ | |
| have h := hperf μ S' V' R hS' hV' hR hS0' hV0' hind' | |
| (fun i => hmatch (e.symm i)) hseed' hlaw hint' (fun ω => (π ω).trans e) hπ' | |
| simpa only [decode_reward, optimum_eq_indexed M e, S', V'] using h | |
| end MatroidProphet.LabeledTransport | |
| end OAI | |