| import Mathlib |
| import OAI.Analysis.HardSphere.Thermodynamics.PhysicalScaleData |
| import OAI.Analysis.HardSphere.Allocation.PhysicalTaggedObservation |
|
|
| namespace OAI |
|
|
| section |
|
|
| noncomputable section |
| open MeasureTheory ProbabilityTheory Set |
| open scoped ENNReal |
| namespace HardSphere.Groups |
| open Wiener Bridge Sampling Comparison GaussianPacking PathCutoff CellOrbital _root_.OAI.HardSphere.Lattice |
| attribute [local instance] Classical.propDecidable |
| variable {K : ℕ} [NeZero K] {x y : DiscreteTorus K} |
| variable {J : Type*} [Fintype J] [DecidableEq J] [MeasurableSpace J] [MeasurableSingletonClass J] |
|
|
| omit [MeasurableSpace J] [MeasurableSingletonClass J] in |
| lemma physicalObservedMixture_mass_eq {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) (d : PhysicalDatum (N+1) K M G L) : |
| physicalObservedMixture hL g m t r R C S u particle hxy ν b d univ= |
| physicalMixedRoutes hL g m t r R C S u particle hxy ν d univ := by |
| simp only [physicalObservedMixture,Kernel.snd_apply' _ _ MeasurableSet.univ,preimage_univ, |
| Kernel.compProd_apply_univ] |
|
|
| omit [MeasurableSpace J] [MeasurableSingletonClass J] in |
| theorem physicalObservedMixture_mass {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) (hr : r≤1) (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)) |
| (q a : ℝ≥0∞) (ha0 : 0<a) |
| (hsmall : a+(4*(49^3*(M+1)):ℕ)*(routeDensityBound t*q)≤1) |
| (hdata : ∀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) ∧ |
| (∀w,(Finset.univ.filter (fun j ↦ datumHome hL d (particle j)=w)).card≤M) ∧ |
| (∀j,IsProbabilityMeasure (datumIndividualReference hL g m r R C S t u (particle j) d)) ∧ |
| (∀ j k, j ≠ k → ∀f : physicalSlotAllowed hL g m r R C S u particle d (some k), |
| physicalCollisionReference hL g m t r R C S u particle d j |
| {p | TrialCollision r (projectCurve L f.val) p.val}≤q) ∧ |
| IsProbabilityMeasure (physicalPrimaryReference hL g m t r R C S u particle (siteIndex y) d)) |
| (b : Bool) : |
| ((μ.restrict E).compProd (physicalObservedMixture hL g m t r R C S u particle hxy ν b)) univ=μ E := by |
| rw [Measure.compProd_apply MeasurableSet.univ] |
| simp only [preimage_univ] |
| have hh : ∀ᵐd ∂μ.restrict E,physicalObservedMixture hL g m t r R C S u particle hxy ν b d univ=1 := by |
| filter_upwards [hdata] with d hd |
| rw [physicalObservedMixture_mass_eq,physicalMixedRoutes_mass hL g m t r R C S ht hr u |
| particle d hxy ν hd.1 hd.2.1 hd.2.2.1 q a ha0 hd.2.2.2.1 hsmall] |
| let := hd.2.2.2.2 |
| exact measure_univ |
| rw [lintegral_congr_ae hh] |
| simp |
|
|
| end HardSphere.Groups |
| end |
| end |
|
|
| 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] [MeasurableSpace J] [MeasurableSingletonClass J] |
|
|
| instance physicalTaggedReference_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)) |
| (b : Bool) (μ : Measure (PhysicalDatum (N+1) K M G L)) [IsFiniteMeasure μ] : |
| IsFiniteMeasure (physicalTaggedReference (x:=x) (y:=y) hL g m t r R C S u particle b μ) := by |
| unfold physicalTaggedReference taggedReference |
| infer_instance |
|
|
| omit [MeasurableSpace J] [MeasurableSingletonClass J] in |
| lemma physicalTaggedMixture_unretained {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) (b : Bool) |
| (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)) : |
| taggedPhysicalBathKernel particle b ∘ₘ physicalTaggedMixture hL g m t r R C S u particle hxy b ν μ E = |
| physicalObservedMixture hL g m t r R C S u particle hxy ν b ∘ₘ μ.restrict E := by |
| have H := congrArg (fun ρ ↦ ρ.map Prod.snd) |
| (physicalTaggedMixture_observation hL g m t r R C S u particle hxy b ν μ E) |
| rw [Measure.map_comp _ _ measurable_snd,←Measure.snd,Measure.snd_compProd] at H |
| have he : (retainedTaggedPhysicalBath (K:=K) (M:=M) (G:=G) (L:=L) particle b).map Prod.snd= |
| taggedPhysicalBathKernel particle b := by |
| rw [←Kernel.snd_eq] |
| unfold retainedTaggedPhysicalBath |
| rw [Kernel.snd_prod] |
| rw [he] at H |
| exact H |
|
|
| end HardSphere.Groups |
| end |
| end |
|
|
| section |
|
|
| noncomputable section |
| open MeasureTheory ProbabilityTheory Set |
| open scoped ENNReal BigOperators |
| 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] [MeasurableSpace J] [MeasurableSingletonClass J] |
|
|
| def physicalCommonObserved {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] |
| (μ : Measure (PhysicalDatum (N+1) K M G L)) (E : Set (PhysicalDatum (N+1) K M G L)) := |
| taggedPhysicalBathKernel particle false ∘ₘ physicalTaggedMixture hL g m t r R C S u particle hxy false ν μ E |
| instance physicalCommonObserved_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] |
| (μ : Measure (PhysicalDatum (N+1) K M G L)) [IsFiniteMeasure μ] (E : Set (PhysicalDatum (N+1) K M G L)) : |
| IsFiniteMeasure (physicalCommonObserved hL g m t r R C S u particle hxy ν μ E) := by |
| unfold physicalCommonObserved; infer_instance |
|
|
| def physicalObservedReference {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)) |
| (b : Bool) (μ : Measure (PhysicalDatum (N+1) K M G L)) := |
| taggedPhysicalBathKernel particle b ∘ₘ physicalTaggedReference (x:=x) (y:=y) hL g m t r R C S u particle b μ |
| instance physicalObservedReference_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)) |
| (b : Bool) (μ : Measure (PhysicalDatum (N+1) K M G L)) [IsFiniteMeasure μ] : |
| IsFiniteMeasure (physicalObservedReference (x:=x) (y:=y) hL g m t r R C S u particle b μ) := by |
| unfold physicalObservedReference; infer_instance |
|
|
| omit [MeasurableSpace J] [MeasurableSingletonClass J] in |
| lemma physicalCommonObserved_eq {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) |
| (b : Bool) : |
| physicalCommonObserved hL g m t r R C S u particle hxy ν μ E= |
| taggedPhysicalBathKernel particle b ∘ₘ physicalTaggedMixture hL g m t r R C S u particle hxy b ν μ E := by |
| cases b |
| · rfl |
| · have H := congrArg (fun ρ ↦ ρ.map Prod.snd) |
| (physicalTaggedMixture_same hL g m t r R C S ht u particle hxy ν μ E ha) |
| have he (b : Bool) : (retainedTaggedPhysicalBath (K:=K) (M:=M) (G:=G) (L:=L) particle b).map Prod.snd= |
| taggedPhysicalBathKernel particle b := by |
| rw [←Kernel.snd_eq]; unfold retainedTaggedPhysicalBath; rw [Kernel.snd_prod] |
| rw [Measure.map_comp _ _ measurable_snd,Measure.map_comp _ _ measurable_snd,he,he] at H |
| exact H.symm |
|
|
| theorem physical_common_observed_estimates {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) (hr : r≤1) (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)) [IsFiniteMeasure μ] |
| (E : Set (PhysicalDatum (N+1) K M G L)) (hE : MeasurableSet E) |
| (hM : 0<M) (q : ℝ≥0∞) (δ B κ Cpath : ℝ) (hδ : 0<δ) (hB : 1≤B) (hCpath : 0≤Cpath) |
| (hBt : routeDensityBound t≤ENNReal.ofReal B) |
| (hκ : 2*(B-1)/(M:ℝ)-8*Real.log δ≤κ) |
| (hsmall : ENNReal.ofReal δ+(4*(49^3*(M+1)):ℕ)*(routeDensityBound t*q)≤1) |
| (hdata : ∀ᵐd ∂μ.restrict E,PhysicalMomentData hL g m t r R C S u particle ν q d) |
| (hprimary : ∀ᵐd ∂μ.restrict E,IsProbabilityMeasure (physicalPrimaryReference hL g m t r R C S u particle (siteIndex y) d)) |
| (henc : (∫⁻d in E,∫⁻p,∫⁻p',ENNReal.ofReal (Real.exp (κ*(simpleEncounter 30 p p':ℝ))) ∂ν d ∂ν d ∂μ)≤ENNReal.ofReal Cpath) : |
| let σ := physicalCommonObserved hL g m t r R C S u particle hxy ν μ E |
| σ univ=μ E ∧ ∀b, |
| let P := physicalObservedReference (x:=x) (y:=y) hL g m t r R C S u particle b μ |
| σ≪P ∧ Integrable (fun z ↦ (σ.rnDeriv P z).toReal^2) P ∧ |
| (∫z,(σ.rnDeriv P z).toReal^2 ∂P)≤B*Cpath := by |
| dsimp only |
| have 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 := |
| hdata.mono (fun d hd ↦ hd.accessible hM) |
| constructor |
| · have H := physicalObservedMixture_mass hL g m t r R C S ht hr u particle hxy ν μ E q |
| (ENNReal.ofReal δ) (ENNReal.ofReal_pos.mpr hδ) hsmall |
| (by filter_upwards [hdata,hprimary] with d hd hp; exact ⟨hd.accessible hM,hd.2.1,hd.2.2.1,hd.2.2.2,hp⟩) false |
| rw [←physicalTaggedMixture_observation,Measure.comp_apply_univ] at H |
| simpa only [physicalCommonObserved,Measure.comp_apply_univ] using H |
| · intro b |
| rw [physicalCommonObserved_eq hL g m t r R C S ht u particle hxy ν μ E ha b] |
| refine ⟨physicalTagged_observed_ac hL g m t r R C S ht hr u particle hxy ν μ E hE |
| (by filter_upwards [hdata] with d hd; exact ⟨hd.accessible hM,hd.2.2.1⟩) b (taggedPhysicalBathKernel particle b),?_⟩ |
| have H := physicalTagged_observed_moment hL g m t r R C S ht hr u particle hxy ν μ E hE hM |
| q δ B κ Cpath hδ hB hCpath hBt hκ hsmall hdata henc b (taggedPhysicalBathKernel particle b) |
| rw [←Measure.snd,Measure.snd_compProd,←Measure.snd,Measure.snd_compProd] at H |
| exact H |
|
|
| end HardSphere.Groups |
| end |
| end |
|
|
| end OAI |
|
|
