| import Mathlib |
| import OAI.Analysis.HardSphere.Allocation.PhysicalRouteAuxiliary |
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| namespace OAI |
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| section |
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| noncomputable section |
| open MeasureTheory ProbabilityTheory Set |
| open scoped ENNReal NNReal |
| namespace HardSphere.Comparison |
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| lemma piSubtype_measurableEmbedding {I : Type*} [Countable I] |
| {X : I→Type*} [∀i,MeasurableSpace (X i)] (A : ∀i,Set (X i)) |
| (hA : ∀i,MeasurableSet (A i)) : |
| MeasurableEmbedding (fun z : ∀i,A i ↦ fun i ↦ (z i).val) := by |
| let f : (∀i,A i)→∀i,X i := fun z i ↦ (z i).val |
| have hr : range f={z | ∀i,z i∈A i} := by |
| ext z |
| constructor |
| · rintro ⟨w,rfl⟩ i |
| exact (w i).property |
| · intro hz |
| exact ⟨fun i ↦ ⟨z i,hz i⟩,rfl⟩ |
| let g : range f→∀i,A i := fun z i ↦ ⟨z.val i,by |
| exact ((Set.ext_iff.mp hr z.val).mp z.property) i⟩ |
| apply MeasurableEmbedding.of_measurable_inverse_on_range |
| (g:=g) (Measurable.of_eval fun i ↦ measurable_subtype_coe.comp (measurable_pi_apply i)) |
| · rw [hr] |
| rw [Set.ofPred_forall] |
| exact MeasurableSet.iInter fun i ↦ (hA i).preimage (measurable_pi_apply i) |
| · exact Measurable.of_eval fun i ↦ ((measurable_pi_apply i).comp measurable_subtype_coe).subtype_mk |
| · intro z |
| rfl |
|
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| lemma mapped_likelihood_cross {X Y : Type*} [MeasurableSpace X] [MeasurableSpace Y] |
| (P μ ν : Measure X) [IsFiniteMeasure P] [IsFiniteMeasure μ] [IsFiniteMeasure ν] |
| (f g : X→ℝ≥0∞) (hf : Measurable f) (hg : Measurable g) |
| (hμ : μ=P.withDensity f) (hν : ν=P.withDensity g) |
| (e : X→Y) (he : MeasurableEmbedding e) : |
| (∫⁻y,(μ.map e).rnDeriv (P.map e) y*(ν.map e).rnDeriv (P.map e) y ∂P.map e)= |
| ∫⁻x,f x*g x ∂P := by |
| rw [he.lintegral_map] |
| apply lintegral_congr_ae |
| have hf' : (fun x ↦ (μ.map e).rnDeriv (P.map e) (e x))=ᵐ[P] f := by |
| refine (he.rnDeriv_map μ P).trans ?_ |
| rw [hμ] |
| exact Measure.rnDeriv_withDensity P hf |
| have hg' : (fun x ↦ (ν.map e).rnDeriv (P.map e) (e x))=ᵐ[P] g := by |
| refine (he.rnDeriv_map ν P).trans ?_ |
| rw [hν] |
| exact Measure.rnDeriv_withDensity P hg |
| exact hf'.mul hg' |
|
|
| end HardSphere.Comparison |
| namespace HardSphere.Groups |
| open Wiener Bridge Sampling Comparison GaussianPacking PathCutoff CellOrbital |
| variable {J : Type*} [Fintype J] [DecidableEq J] |
| omit [DecidableEq J] in |
| lemma physicalConfig_embedding {N K M G : ℕ} {L : Length} (hL : L.val=(K:ℝ)) |
| (g : Fin G) (m : ℕ) (r R C S : ℝ) (u : Time) |
| (particle : J→Fin N) (d : PhysicalDatum N K M G L) : |
| MeasurableEmbedding (routeConfigMap (fun i ↦ (Subtype.val : physicalSlotAllowed hL g m r R C S u particle d i→_))) := |
| piSubtype_measurableEmbedding _ (physicalSlotAllowed_measurable hL g m r R C S u particle d) |
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| variable {N K M G n : 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} |
| {labels : Fin (n+1)↪J} {v : Fin (n+2)→Fin 3→Fin K} {steps : Fin (n+1)→Fin 7} |
| {d : PhysicalDatum N K M G L} |
| lemma PhysicalRouteAux.firstDensity_measurable |
| (A : PhysicalRouteAux hL g m t r R C S u particle labels v steps d) |
| (hp : ∀j,IsProbabilityMeasure (datumIndividualReference hL g m r R C S t u (particle j) d)) : |
| Measurable A.firstDensity := by |
| have := physicalOn_probability hL g m t r R C S u particle (v (Fin.last (n+1))) d hp |
| exact measurable_conditionalLikelihood _ _ _ _ _ _ _ |
| lemma PhysicalRouteAux.secondDensity_measurable |
| (A : PhysicalRouteAux hL g m t r R C S u particle labels v steps d) |
| (hp : ∀j,IsProbabilityMeasure (datumIndividualReference hL g m r R C S t u (particle j) d)) : |
| Measurable A.secondDensity := by |
| have := physicalOn_probability hL g m t r R C S u particle (v 0) d hp |
| have : ∀i,IsFiniteKernel (A.backward i) := A.backward_finite |
| exact measurable_conditionalLikelihood _ _ _ _ _ _ _ |
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|
| end HardSphere.Groups |
| end |
| end |
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| end OAI |
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