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67aee1d | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 | 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
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