openAI-math-dataset / lean /OAI /Probability /FreeIsing /Stochastic /FiniteObservationFiltrationLimit.lean
| import OAI.Probability.FreeIsing.Posterior.FiniteIsingFilteringRepresentation | |
| namespace OAI | |
| noncomputable section | |
| open MeasureTheory ProbabilityTheory | |
| open scoped NNReal | |
| namespace Problem367.FiniteIsingFiltering | |
| variable {d : ℕ} {Ω : Type*} [MeasurableSpace Ω] | |
| variable (B : TreeVertex d → ℝ≥0 → Ω → ℝ) | |
| variable (hmB : ∀ i t, Measurable (B i t)) | |
| theorem observationFiltration_mono_of_subset | |
| {D E : Finset (TreeVertex d)} (hDE : D ⊆ E) : | |
| observationFiltration B hmB D ≤ observationFiltration B hmB E := by | |
| intro t | |
| change (⨆ i : D, Filtration.natural (plantedArrayObservation B signal i) | |
| (fun s => (plantedArrayObservation_measurable hmB measurable_signal i s).stronglyMeasurable) t) ≤ | |
| ⨆ i : E, Filtration.natural (plantedArrayObservation B signal i) | |
| (fun s => (plantedArrayObservation_measurable hmB measurable_signal i s).stronglyMeasurable) t | |
| exact iSup_le fun i => le_iSup_of_le ⟨i.1, hDE i.2⟩ le_rfl | |
| theorem observationFiltration_le_arrayObservationFiltration | |
| (D : Finset (TreeVertex d)) : | |
| observationFiltration B hmB D ≤ | |
| arrayObservationFiltration B hmB signal measurable_signal := by | |
| intro t | |
| change (⨆ i : D, Filtration.natural (plantedArrayObservation B signal i) | |
| (fun s => (plantedArrayObservation_measurable hmB measurable_signal i s).stronglyMeasurable) t) ≤ | |
| ⨆ i : TreeVertex d, Filtration.natural (plantedArrayObservation B signal i) | |
| (fun s => (plantedArrayObservation_measurable hmB measurable_signal i s).stronglyMeasurable) t | |
| exact iSup_le fun i => le_iSup_of_le i.1 le_rfl | |
| theorem observationFiltration_monotone | |
| {D : ℕ → Finset (TreeVertex d)} (hD : Monotone D) : | |
| Monotone (fun n => observationFiltration B hmB (D n)) := by | |
| intro n m hnm | |
| exact observationFiltration_mono_of_subset B hmB (hD hnm) | |
| theorem observationFiltration_time_monotone | |
| {D : ℕ → Finset (TreeVertex d)} (hD : Monotone D) (t : ℝ≥0) : | |
| Monotone (fun n => observationFiltration B hmB (D n) t) := by | |
| intro n m hnm | |
| exact observationFiltration_mono_of_subset B hmB (hD hnm) t | |
| theorem iSup_observationFiltration_eq_arrayObservationFiltration | |
| {κ : Type*} (D : κ → Finset (TreeVertex d)) | |
| (hcover : ∀ v, ∃ n, v ∈ D n) (t : ℝ≥0) : | |
| (⨆ n, observationFiltration B hmB (D n) t) = | |
| arrayObservationFiltration B hmB signal measurable_signal t := by | |
| apply le_antisymm | |
| · exact iSup_le fun n => observationFiltration_le_arrayObservationFiltration B hmB (D n) t | |
| · change (⨆ i : TreeVertex d, Filtration.natural (plantedArrayObservation B signal i) | |
| (fun s => (plantedArrayObservation_measurable hmB measurable_signal i s).stronglyMeasurable) t) ≤ _ | |
| refine iSup_le fun i => ?_ | |
| obtain ⟨n, hn⟩ := hcover i | |
| apply le_iSup_of_le n | |
| change _ ≤ ⨆ j : D n, Filtration.natural (plantedArrayObservation B signal j) | |
| (fun s => (plantedArrayObservation_measurable hmB measurable_signal j s).stronglyMeasurable) t | |
| exact le_iSup_of_le ⟨i, hn⟩ le_rfl | |
| end Problem367.FiniteIsingFiltering | |
| end | |
| end OAI | |