| import Mathlib |
| import OAI.Analysis.HardSphere.Comparison.PhysicalCellReference |
| import OAI.Analysis.HardSphere.Comparison.PhysicalObservedLaw |
|
|
| namespace OAI |
|
|
| section |
|
|
| noncomputable section |
| open MeasureTheory ProbabilityTheory Set |
| open scoped ENNReal BigOperators |
| namespace HardSphere.Comparison |
| variable {n : ℕ} {E : Type*} [MeasurableSpace E] |
| attribute [local instance] permBathSpace permBathSingleton |
|
|
| def adjoinPoint (z : (Fin (n+1)→E)×E) : Option (Fin (n+1))→E := |
| fun i ↦ i.elim z.2 z.1 |
| lemma adjoinPoint_measurable : Measurable (adjoinPoint (n:=n) (E:=E)) := by |
| apply Measurable.of_eval |
| intro i |
| cases i with |
| | none => exact measurable_snd |
| | some i => exact (measurable_pi_apply i).comp measurable_fst |
|
|
| def cellObservation (b : Bool) (z : (E×(Fin n→E))×E) : E×E×(Fin n→E) := |
| if b then (z.1.1,z.2,z.1.2) else (z.2,z.1.1,z.1.2) |
| lemma cellObservation_measurable (b : Bool) : Measurable (cellObservation (n:=n) (E:=E) b) := by |
| unfold cellObservation |
| split_ifs <;> fun_prop |
|
|
| theorem primaryBath_product (P : Measure (Fin (n+1)→E)) [SFinite P] |
| (η : Measure E) [SFinite η] (b : Bool) (i : Fin (n+1)) : |
| primaryBathKernel b i ∘ₘ (P.prod η).map adjoinPoint = |
| (((P.prod (uniformBathOrder n)).map (orderedObservation i)).prod η).map (cellObservation b) := by |
| ext s hs |
| rw [Measure.bind_apply hs (primaryBathKernel b i).aemeasurable, |
| lintegral_map ((primaryBathKernel b i).measurable_coe hs) adjoinPoint_measurable, |
| lintegral_prod _ (show AEMeasurable (fun a ↦ primaryBathKernel b i (adjoinPoint a) s) (P.prod η) from |
| (((primaryBathKernel b i).measurable_coe hs).comp adjoinPoint_measurable).aemeasurable), |
| Measure.map_apply (cellObservation_measurable b) hs,Measure.prod_apply |
| ((cellObservation_measurable b) hs)] |
| have hm := measurable_measure_prodMk_left (ν:=η) ((cellObservation_measurable b) hs) |
| rw [lintegral_map hm (orderedObservation_measurable i), |
| lintegral_prod _ (show AEMeasurable (fun a ↦ η (Prod.mk (orderedObservation i a) ⁻¹' cellObservation b ⁻¹' s)) _ from |
| (hm.comp (orderedObservation_measurable i)).aemeasurable)] |
| apply lintegral_congr |
| intro X |
| simp_rw [primaryBathKernel_apply] |
| have hh : Measurable (fun z : E×Equiv.Perm (Fin n) ↦ |
| (s.indicator (fun _ ↦ (1:ℝ≥0∞))) |
| (orderedPrimaryObservation b i (adjoinPoint (X,z.1),z.2))) := |
| (measurable_const.indicator hs).comp ((orderedPrimaryObservation_measurable b i).comp |
| ((adjoinPoint_measurable.comp (measurable_const.prodMk measurable_fst)).prodMk measurable_snd)) |
| convert (lintegral_lintegral_swap (μ:=η) (ν:=uniformBathOrder n) |
| (f:=fun e τ ↦ s.indicator (fun _ ↦ (1:ℝ≥0∞)) |
| (orderedPrimaryObservation b i (adjoinPoint (X,e),τ))) hh.aemeasurable) using 1 |
| · apply lintegral_congr |
| intro e |
| rw [←lintegral_indicator_one hs] |
| exact lintegral_map (measurable_const.indicator hs) |
| (show Measurable (fun τ ↦ orderedPrimaryObservation b i (adjoinPoint (X,e),τ)) from |
| (orderedPrimaryObservation_measurable b i).comp (measurable_const.prodMk measurable_id)) |
| · apply lintegral_congr |
| intro τ |
| have he : (fun e ↦ orderedPrimaryObservation b i (adjoinPoint (X,e),τ))= |
| (fun e ↦ cellObservation b (orderedObservation i (X,τ),e)) := by |
| funext e |
| cases b <;> rfl |
| change η ((fun e ↦ cellObservation b (orderedObservation i (X,τ),e)) ⁻¹' s)=_ |
| have hm : Measurable (fun e ↦ cellObservation b (orderedObservation i (X,τ),e)) := |
| (cellObservation_measurable b).comp measurable_prodMk_left |
| rw [←Measure.map_apply hm hs,←lintegral_indicator_one hs] |
| refine (lintegral_map (measurable_const.indicator hs) hm).trans ?_ |
| exact congrArg (fun f ↦ ∫⁻e,s.indicator (fun _ ↦ (1:ℝ≥0∞)) (f e) ∂η) he.symm |
| |
| end HardSphere.Comparison |
| end |
| end |
| |
| section |
| |
| noncomputable section |
| open MeasureTheory ProbabilityTheory Set |
| open scoped ENNReal BigOperators |
| namespace HardSphere.Comparison |
| variable {n : ℕ} {E : Type*} [MeasurableSpace E] |
| attribute [local instance] permBathSpace permBathSingleton |
| |
| lemma prod_restrict_first (P : Measure E) [SFinite P] {Y : Type*} [MeasurableSpace Y] |
| (η : Measure Y) [SFinite η] (A : Set E) : |
| (P.restrict A).prod η=(P.prod η).restrict (Prod.fst ⁻¹' A) := by |
| have H := Measure.prod_restrict (μ:=P) (ν:=η) A univ |
| simpa only [Measure.restrict_univ,Set.prod_univ] using H |
|
|
| lemma orderedKernel_restrict (P : Measure (Fin (n+1)→E)) [SFinite P] |
| (i : Fin (n+1)) (A : Set E) (hA : MeasurableSet A) : |
| (orderedKernel (uniformBathOrder n) i ∘ₘ P).restrict (Prod.fst ⁻¹' A)= |
| orderedKernel (uniformBathOrder n) i ∘ₘ P.restrict ((fun X ↦ X i) ⁻¹' A) := by |
| rw [orderedKernel_comp,orderedKernel_comp,Measure.restrict_map (orderedObservation_measurable i) |
| (hA.preimage measurable_fst),prod_restrict_first] |
| rfl |
|
|
| theorem primaryBath_cell_reference (P : Measure (Fin (n+1)→E)) [SFinite P] |
| (η : Measure E) [SFinite η] (b : Bool) (i : Fin (n+1)) |
| (A : Set E) (hA : MeasurableSet A) : |
| primaryBathKernel b i ∘ₘ ((P.prod η).map adjoinPoint).restrict ((fun F ↦ F (some i)) ⁻¹' A)= |
| (((cellExtraKernel (Y:=Fin n→E) η A univ hA).map (cellObservation b)) ∘ₖ |
| orderedKernel (uniformBathOrder n) i) ∘ₘ P := by |
| rw [Measure.restrict_map adjoinPoint_measurable (hA.preimage (measurable_pi_apply (some i)))] |
| have he : adjoinPoint ⁻¹' ((fun F : Option (Fin (n+1))→E ↦ F (some i)) ⁻¹' A)= |
| Prod.fst ⁻¹' ((fun X : Fin (n+1)→E ↦ X i) ⁻¹' A) := rfl |
| rw [he,←prod_restrict_first,primaryBath_product,←orderedKernel_comp, |
| ←Measure.comp_assoc,←Measure.map_comp _ _ (cellObservation_measurable b), |
| cellExtraKernel_comp _ η A univ hA MeasurableSet.univ] |
| have he' : {p : (E×(Fin n→E))×E | p.1.1∈A ∧ p.2∈univ}= |
| Prod.fst ⁻¹' (Prod.fst ⁻¹' A) := by ext p; simp |
| rw [he',←prod_restrict_first,orderedKernel_restrict _ _ A hA] |
| |
| end HardSphere.Comparison |
| end |
| end |
| |
| end OAI |
| |
