| import Mathlib |
| import OAI.Analysis.HardSphere.Allocation.PhysicalMixedObservation |
|
|
| namespace OAI |
|
|
| section |
|
|
| noncomputable section |
| open MeasureTheory ProbabilityTheory Set |
| open scoped ENNReal |
| namespace HardSphere.Groups |
| open Wiener Bridge Sampling Comparison GaussianPacking PathCutoff CellOrbital HardSphere.Lattice |
| attribute [local instance] Classical.propDecidable |
| variable {K : ℕ} [NeZero K] {x y : DiscreteTorus K} |
| variable {J : Type*} [Fintype J] [DecidableEq J] |
|
|
| def physicalObservedMixture {N M G : ℕ} {L : Length} (hL : L.val=(K:ℝ)) |
| (g : Fin G) (m : ℕ) (t r R C S : ℝ) (u : Time) (particle : J↪Fin (N+1)) |
| (hxy : x≠y) (ν : Kernel (PhysicalDatum (N+1) K M G L) (SimplePath x y)) [IsMarkovKernel ν] |
| (b : Bool) : Kernel (PhysicalDatum (N+1) K M G L) (Torus L×Torus L×(Fin N→Torus L)) := |
| ((physicalMixedRoutes hL g m t r R C S u particle hxy ν) ⊗ₖ |
| physicalCodeObserver particle hxy b).snd |
| instance physicalObservedMixture_finite {N M G : ℕ} {L : Length} (hL : L.val=(K:ℝ)) |
| (g : Fin G) (m : ℕ) (t r R C S : ℝ) (u : Time) (particle : J↪Fin (N+1)) |
| (hxy : x≠y) (ν : Kernel (PhysicalDatum (N+1) K M G L) (SimplePath x y)) [IsMarkovKernel ν] |
| (b : Bool) : IsFiniteKernel (physicalObservedMixture hL g m t r R C S u particle hxy ν b) := by |
| unfold physicalObservedMixture; infer_instance |
| |
| theorem physicalObservedMixture_same {N M G : ℕ} {L : Length} (hL : L.val=(K:ℝ)) |
| (g : Fin G) (m : ℕ) (t r R C S : ℝ) (ht : 0<t) (u : Time) |
| (particle : J↪Fin (N+1)) (d : PhysicalDatum (N+1) K M G L) (hxy : x≠y) |
| (ν : Kernel (PhysicalDatum (N+1) K M G L) (SimplePath x y)) [IsMarkovKernel ν] |
| (ha : ∀ᵐp ∂ν d,∀i : Fin p.val.length, |
| (branchAccessible hL g m t r R C S u particle (pathVertexEmbedding p i.castSucc) d).Nonempty) : |
| physicalObservedMixture hL g m t r R C S u particle hxy ν true d = |
| physicalObservedMixture hL g m t r R C S u particle hxy ν false d := by |
| ext A hA |
| simp only [physicalObservedMixture,Kernel.snd_apply' _ _ hA,Kernel.compProd_apply (measurable_snd hA)] |
| have hex (b : Bool) : |
| (∫⁻w,(physicalCodeObserver particle hxy b) (d,w) A ∂physicalMixedRoutes hL g m t r R C S u particle hxy ν d)= |
| ∫⁻c,∫⁻w,(physicalCodeObserver particle hxy b) (d,(c,w)) A |
| ∂physicalCodedRoute hL g m t r R C S u particle hxy c d |
| ∂(ν.prod (physicalLabels hL g m t r R C S u particle)) d := by |
| rw [physicalMixedRoutes,Kernel.lintegral_compProd] |
| rfl |
| exact (physicalCodeObserver particle hxy b).measurable_coe hA |>.comp measurable_prodMk_left |
| change (∫⁻w,(physicalCodeObserver particle hxy true) (d,w) A ∂physicalMixedRoutes hL g m t r R C S u particle hxy ν d)= |
| (∫⁻w,(physicalCodeObserver particle hxy false) (d,w) A ∂physicalMixedRoutes hL g m t r R C S u particle hxy ν d) |
| rw [hex true,hex false] |
| apply lintegral_congr_ae |
| rw [Kernel.prod_apply] |
| apply (Measure.ae_prod_iff_ae_ae (Set.to_countable _).measurableSet).mpr |
| filter_upwards [ha] with p hp |
| filter_upwards [physicalLabels_supported hL g m t r R C S u particle d] with z hz |
| have H := congrArg (fun μ ↦ μ A) (physicalCodedRoute_same_observed hL g m t r R C S ht u particle d hxy p z hz hp) |
| simpa only [Measure.bind_apply hA (Kernel.aemeasurable _),Kernel.comap_apply] using H |
| |
| theorem physicalObservedMixture_same_joint {N M G : ℕ} {L : Length} (hL : L.val=(K:ℝ)) |
| (g : Fin G) (m : ℕ) (t r R C S : ℝ) (ht : 0<t) (u : Time) |
| (particle : J↪Fin (N+1)) (hxy : x≠y) |
| (ν : Kernel (PhysicalDatum (N+1) K M G L) (SimplePath x y)) [IsMarkovKernel ν] |
| (μ : Measure (PhysicalDatum (N+1) K M G L)) [SFinite μ] |
| (E : Set (PhysicalDatum (N+1) K M G L)) |
| (ha : ∀ᵐd ∂μ.restrict E,∀ᵐp ∂ν d,∀i : Fin p.val.length, |
| (branchAccessible hL g m t r R C S u particle (pathVertexEmbedding p i.castSucc) d).Nonempty) : |
| (μ.restrict E).compProd (physicalObservedMixture hL g m t r R C S u particle hxy ν true)= |
| (μ.restrict E).compProd (physicalObservedMixture hL g m t r R C S u particle hxy ν false) := by |
| apply Measure.compProd_congr |
| filter_upwards [ha] with d hd |
| exact physicalObservedMixture_same hL g m t r R C S ht u particle d hxy ν hd |
| |
| end HardSphere.Groups |
| end |
| end |
| |
| end OAI |
| |
