| import Mathlib |
| import OAI.Analysis.HardSphere.Kernels.TrialKernel |
| import OAI.Analysis.HardSphere.Bridges.GaussianMixture |
| import OAI.Analysis.HardSphere.Conditioning.RouteObservation |
|
|
| namespace OAI |
|
|
| section |
|
|
| section |
| noncomputable section |
| open MeasureTheory ProbabilityTheory Set |
| open scoped ENNReal |
| namespace HardSphere.Comparison |
| open Bridge Wiener GaussianPacking |
|
|
| lemma trialKernel_midpoint (t : ℝ) (y z u : Point) : |
| ∀ᵐ ω ∂trialKernel t y z u,ω midpointTime=u := by |
| rw [trialKernel_apply] |
| apply (ae_map_iff ((trialPath_continuous t y z).measurable.comp |
| (measurable_const.prodMk measurable_id)).aemeasurable (measurableSet_eq_fun ((continuous_eval_const midpointTime).measurable : Measurable (fun ω : C(Time,Point) ↦ ω midpointTime)) measurable_const)).mpr |
| exact ae_of_all _ (fun β ↦ trial_midpoint t y z u β) |
|
|
| lemma paired_trial_midpoint (μ : Measure Point) [SFinite μ] (t : ℝ) (y z y' z' : Point) |
| (S T : Set C(Time,Point)) (hS : MeasurableSet S) (hT : MeasurableSet T) : |
| ∀ᵐ p ∂pairedFactor μ (trialKernel t y z) (trialKernel t y' z') S T hS hT, |
| p.2.1 midpointTime=p.1 ∧ p.2.2 midpointTime=p.1 := by |
| apply pairedFactor_midpoint μ (trialKernel t y z) (trialKernel t y' z') |
| (fun ω ↦ ω midpointTime) (fun ω ↦ ω midpointTime) id |
| (continuous_eval_const midpointTime).measurable (continuous_eval_const midpointTime).measurable measurable_id |
| · exact ae_of_all _ (trialKernel_midpoint t y z) |
| · exact ae_of_all _ (trialKernel_midpoint t y' z') |
|
|
| lemma route_observed_trial_eq {W : Type*} [MeasurableSpace W] (d : ℕ) |
| (τ : Fin (d+2) → Measure (Point×(C(Time,Point)×C(Time,Point)))) |
| [∀k,IsProbabilityMeasure (τ k)] (ξ : Measure W) [SFinite ξ] |
| (hτ : ∀k,∀ᵐ p ∂τ k,p.2.1 midpointTime=p.1 ∧ p.2.2 midpointTime=p.1) |
| (C : Set ((Fin (d+2) → Point×(C(Time,Point)×C(Time,Point)))×W)) : |
| ((((Measure.pi τ).prod ξ) C)⁻¹ • ((Measure.pi τ).prod ξ).restrict C).map |
| (observeFirst d (fun ω ↦ ω midpointTime)) = |
| ((((Measure.pi τ).prod ξ) C)⁻¹ • ((Measure.pi τ).prod ξ).restrict C).map |
| (observeSecond d (fun ω ↦ ω midpointTime)) := by |
| apply conditioned_observed_measures_eq |
| apply (Measure.ae_prod_iff_ae_ae (by measurability)).mpr |
| have h : ∀k,∀ᵐ v ∂Measure.pi τ,(v k).2.1 midpointTime=(v k).1 ∧ (v k).2.2 midpointTime=(v k).1 := |
| fun k ↦ (measurePreserving_eval τ k).quasiMeasurePreserving.ae (hτ k) |
| filter_upwards [ae_all_iff.mpr h] with v hv |
| exact ae_of_all _ (fun _ ↦ hv) |
|
|
| end HardSphere.Comparison |
|
|
| end |
| end |
|
|
| end |
|
|
| end OAI |
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|
