| import Mathlib |
| import OAI.Analysis.HardSphere.Allocation.PhysicalAmbientRN |
| import OAI.Analysis.HardSphere.Allocation.PhysicalFamilyLikelihood |
|
|
| namespace OAI |
|
|
| section |
|
|
| noncomputable section |
| open MeasureTheory ProbabilityTheory Set |
| open scoped ENNReal NNReal |
| 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] |
|
|
| omit [NeZero K] in |
| lemma simpleEncounter_comm (R : ℕ) (p q : SimplePath x y) : |
| simpleEncounter R p q=simpleEncounter R q p := by |
| unfold simpleEncounter |
| apply Finset.card_bij (fun z _ ↦ z.swap) |
| · intro z hz |
| obtain ⟨hp,hq,hn⟩ := (mem_encounterSet _ _ _ _ _).mp hz |
| exact (mem_encounterSet _ _ _ _ _).mpr ⟨hq,hp,torusNear_symm hn⟩ |
| · intro z _ w _ he |
| exact Prod.swap_injective he |
| · intro z hz |
| obtain ⟨hp,hq,hn⟩ := (mem_encounterSet _ _ _ _ _).mp hz |
| exact ⟨z.swap,(mem_encounterSet _ _ _ _ _).mpr ⟨hq,hp,torusNear_symm hn⟩,rfl⟩ |
|
|
| theorem physicalCoded_moment {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) (d : PhysicalDatum N K M G L) (hxy : x≠y) |
| (p p' : SimplePath x y) (z z' : DiscreteTorus K→Option J) |
| (hz : ∀v,z v∈paddedLabels (branchAccessible hL g m t r R C S u particle v d)) |
| (hz' : ∀v,z' v∈paddedLabels (branchAccessible hL g m t r R C S u particle v d)) |
| (ha : ∀i : Fin p.val.length,(branchAccessible hL g m t r R C S u particle (pathVertexEmbedding p i.castSucc) d).Nonempty) |
| (ha' : ∀i : Fin p'.val.length,(branchAccessible hL g m t r R C S u particle (pathVertexEmbedding p' i.castSucc) d).Nonempty) |
| (hm : ∀w,(Finset.univ.filter (fun j ↦ datumHome hL d (particle j)=w)).card≤M) |
| (hp : ∀j,IsProbabilityMeasure (datumIndividualReference hL g m r R C S t u (particle j) d)) |
| (q a : ℝ≥0∞) (ha0 : 0<a) |
| (hq : ∀ 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) |
| (hsmall : a+(4*(49^3*(M+1)):ℕ)*(routeDensityBound t*q)≤1) (b : Bool) : |
| (∫⁻w,physicalRouteLikelihood hL g m t r R C S u particle hxy b (d,(p,z)) w * |
| physicalRouteLikelihood hL g m t r R C S u particle hxy b (d,(p',z')) w |
| ∂physicalPrimaryReference hL g m t r R C S u particle (siteIndex (if b then y else x)) d)≤ |
| (routeDensityBound t)^(1+(fieldLabelImage p z∩fieldLabelImage p' z').card)* |
| (a⁻¹)^(8*simpleEncounter 30 p p') := by |
| have hv := routeFieldValid_of_supported hL g m t r R C S u particle d p z hz ha |
| have hv' := routeFieldValid_of_supported hL g m t r R C S u particle d p' z' hz' ha' |
| let f := fieldRouteLabels hxy p z hv |
| let f' := fieldRouteLabels hxy p' z' hv' |
| let v := simpleRouteVertices p (p.val.length-1) |
| let v' := simpleRouteVertices p' (p'.val.length-1) |
| let steps := simpleRouteSteps hL p _ (simpleRouteLength_eq hxy p) |
| let steps' := simpleRouteSteps hL p' _ (simpleRouteLength_eq hxy p') |
| have hc := simpleRouteSteps_period hL p _ (simpleRouteLength_eq hxy p) |
| have hc' := simpleRouteSteps_period hL p' _ (simpleRouteLength_eq hxy p') |
| have hh := fieldRouteLabels_physical hL g m t r R C S u particle d hxy p z hz ha hv |
| have hh' := fieldRouteLabels_physical hL g m t r R C S u particle d hxy p' z' hz' ha' hv' |
| let A := Classical.choice (physicalRouteAux_nonempty hL g m t r R C S u particle f v steps d ht hc |
| (fun i ↦ (hh i).1) (fun i ↦ (hh i).2) hp) |
| let B := Classical.choice (physicalRouteAux_nonempty hL g m t r R C S u particle f' v' steps' d ht hc' |
| (fun i ↦ (hh' i).1) (fun i ↦ (hh' i).2) hp) |
| have hn := simpleRouteVertices_distance hL p _ (simpleRouteLength_eq hxy p) |
| have hn' := simpleRouteVertices_distance hL p' _ (simpleRouteLength_eq hxy p') |
| have ha1 : 1≤a⁻¹ := ENNReal.one_le_inv.mpr ((le_add_right le_rfl).trans hsmall) |
| have hover : (routeSlots f∩routeSlots f').card=1+(fieldLabelImage p z∩fieldLabelImage p' z').card := |
| fieldRoute_overlap hxy p p' z z' hv hv' |
| have he0 : v 0=siteIndex x := simpleRouteVertices_zero p _ |
| have he0' : v' 0=siteIndex x := simpleRouteVertices_zero p' _ |
| have he1 : v (Fin.last (p.val.length-1+1))=siteIndex y := simpleRouteVertices_last p _ (simpleRouteLength_eq hxy p) |
| have he1' : v' (Fin.last (p'.val.length-1+1))=siteIndex y := simpleRouteVertices_last p' _ (simpleRouteLength_eq hxy p') |
| have HL := lintegral_congr_ae ((physicalRouteLikelihood_measure hL g m t r R C S u particle hxy b d (p,z)).mul |
| (physicalRouteLikelihood_measure hL g m t r R C S u particle hxy b d (p',z'))) |
| simp only [Pi.mul_apply] at HL |
| rw [HL] |
| simp only [physicalCodedRoute,dite_eq_left hv,dite_eq_left hv'] |
| cases b with |
| | true => |
| simp only [ite_true] |
| have H := physical_first_likelihood_moment hL g m t r R C S ht u particle f v steps f' v' steps' d A B |
| hc hc' (fun i ↦ (hh i).1) (fun i ↦ (hh' i).1) (fun i ↦ (hh i).2) (fun i ↦ (hh' i).2) |
| hn hn' (he1'.trans he1.symm) hm hp q a ha0 hq hsmall |
| have Hrn := physical_first_RN_cross hL g m t r R C S ht hr u particle f v steps f' v' steps' d A B |
| hc hc' (fun i ↦ (hh i).1) (fun i ↦ (hh' i).1) (fun i ↦ (hh i).2) (fun i ↦ (hh' i).2) hp (he1'.trans he1.symm) |
| rw [he1] at Hrn H |
| rw [Hrn] |
| apply H.trans |
| rw [div_eq_mul_inv,ENNReal.inv_pow,hover] |
| apply mul_le_mul' le_rfl |
| apply pow_le_pow_right₀ ha1 |
| have h1 := simpleRoute_near_first p p' (simpleRouteLength_eq hxy p) (simpleRouteLength_eq hxy p') |
| (fun j ↦ datumHome hL d (particle j)) f f' (fun i ↦ (hh i).1) (fun i ↦ (hh' i).1) 30 |
| have h2 := simpleRoute_near_first p' p (simpleRouteLength_eq hxy p') (simpleRouteLength_eq hxy p) |
| (fun j ↦ datumHome hL d (particle j)) f' f (fun i ↦ (hh' i).1) (fun i ↦ (hh i).1) 30 |
| rw [simpleEncounter_comm 30 p' p] at h2 |
| omega |
| | false => |
| simp only [Bool.false_eq_true,ite_false] |
| have H := physical_second_likelihood_moment hL g m t r R C S ht u particle f v steps f' v' steps' d A B |
| hc hc' (fun i ↦ (hh i).1) (fun i ↦ (hh' i).1) (fun i ↦ (hh i).2) (fun i ↦ (hh' i).2) |
| hn hn' (he0'.trans he0.symm) hm hp q a ha0 hq hsmall |
| have Hrn := physical_second_RN_cross hL g m t r R C S ht hr u particle f v steps f' v' steps' d A B |
| hc hc' (fun i ↦ (hh i).1) (fun i ↦ (hh' i).1) (fun i ↦ (hh i).2) (fun i ↦ (hh' i).2) hp (he0'.trans he0.symm) |
| rw [he0] at Hrn H |
| rw [Hrn] |
| apply H.trans |
| rw [div_eq_mul_inv,ENNReal.inv_pow,hover] |
| apply mul_le_mul' le_rfl |
| apply pow_le_pow_right₀ ha1 |
| have h1 := simpleRoute_near_second p p' (simpleRouteLength_eq hxy p) (simpleRouteLength_eq hxy p') |
| (fun j ↦ datumHome hL d (particle j)) f f' (fun i ↦ (hh i).1) (fun i ↦ (hh' i).1) 30 |
| have h2 := simpleRoute_near_second p' p (simpleRouteLength_eq hxy p') (simpleRouteLength_eq hxy p) |
| (fun j ↦ datumHome hL d (particle j)) f' f (fun i ↦ (hh' i).1) (fun i ↦ (hh i).1) 30 |
| rw [simpleEncounter_comm 30 p' p] at h2 |
| omega |
|
|
| end HardSphere.Groups |
| end |
| end |
|
|
| section |
|
|
| noncomputable section |
| open MeasureTheory ProbabilityTheory Set |
| open scoped ENNReal NNReal |
| 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] |
|
|
| theorem physicalCoded_density {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) (d : PhysicalDatum N K M G L) (hxy : x≠y) |
| (p : SimplePath x y) (z : DiscreteTorus K→Option J) |
| (hz : ∀v,z v∈paddedLabels (branchAccessible hL g m t r R C S u particle v d)) |
| (ha : ∀i : Fin p.val.length,(branchAccessible hL g m t r R C S u particle (pathVertexEmbedding p i.castSucc) d).Nonempty) |
| (hp : ∀j,IsProbabilityMeasure (datumIndividualReference hL g m r R C S t u (particle j) d)) |
| (b : Bool) : |
| (physicalPrimaryReference hL g m t r R C S u particle (siteIndex (if b then y else x)) d).withDensity |
| (physicalRouteLikelihood hL g m t r R C S u particle hxy b (d,(p,z))) = |
| (physicalCodedRoute hL g m t r R C S u particle hxy (p,z) d).map |
| (if b then Prod.fst else Prod.snd) := by |
| have hv := routeFieldValid_of_supported hL g m t r R C S u particle d p z hz ha |
| let f := fieldRouteLabels hxy p z hv |
| let v := simpleRouteVertices p (p.val.length-1) |
| let steps := simpleRouteSteps hL p _ (simpleRouteLength_eq hxy p) |
| have hc := simpleRouteSteps_period hL p _ (simpleRouteLength_eq hxy p) |
| have hh := fieldRouteLabels_physical hL g m t r R C S u particle d hxy p z hz ha hv |
| let A := Classical.choice (physicalRouteAux_nonempty hL g m t r R C S u particle f v steps d ht hc |
| (fun i ↦ (hh i).1) (fun i ↦ (hh i).2) hp) |
| have hd := physicalPairedRoute_densities hL g m t r R C S ht hr u particle f v steps hc d |
| (fun i ↦ (hh i).1) (fun i ↦ (hh i).2) A hp |
| rw [withDensity_congr_ae (physicalRouteLikelihood_measure hL g m t r R C S u particle hxy b d (p,z))] |
| have : IsFiniteMeasure (physicalCodedRoute hL g m t r R C S u particle hxy (p,z) d) := |
| IsFiniteKernel.isFiniteMeasure (κ:=physicalCodedRoute hL g m t r R C S u particle hxy (p,z)) d |
| have : IsFiniteMeasure (physicalPrimaryReference hL g m t r R C S u particle (siteIndex (if b then y else x)) d) := |
| IsFiniteKernel.isFiniteMeasure (κ:=physicalPrimaryReference hL g m t r R C S u particle (siteIndex (if b then y else x))) d |
| have : IsFiniteMeasure ((physicalCodedRoute hL g m t r R C S u particle hxy (p,z) d).map |
| (if b then Prod.fst else Prod.snd)) := Measure.isFiniteMeasure_map _ _ |
| have := Measure.haveLebesgueDecomposition_of_sigmaFinite |
| (μ:=(physicalCodedRoute hL g m t r R C S u particle hxy (p,z) d).map (if b then Prod.fst else Prod.snd)) |
| (ν:=physicalPrimaryReference hL g m t r R C S u particle (siteIndex (if b then y else x)) d) |
| apply Measure.withDensity_rnDeriv_eq |
| simp only [physicalCodedRoute,dite_eq_left hv] |
| have hf : Measurable (routeConfigMap (fun i ↦ (Subtype.val : physicalSlotAllowed hL g m r R C S u particle d i→_))) := |
| routeConfigMap_measurable _ (fun _ ↦ measurable_subtype_coe) |
| cases b with |
| | true => |
| simp only [ite_true] |
| rw [hd.1,←simpleRouteVertices_last p _ (simpleRouteLength_eq hxy p), |
| ←physicalPrimary_on_map hL g m t r R C S hr u particle _ d] |
| exact (withDensity_absolutelyContinuous _ _).map hf |
| | false => |
| simp only [Bool.false_eq_true,ite_false] |
| rw [hd.2,←simpleRouteVertices_zero p _,←physicalPrimary_on_map hL g m t r R C S hr u particle _ d] |
| exact (withDensity_absolutelyContinuous _ _).map hf |
|
|
| end HardSphere.Groups |
| end |
| end |
|
|
| end OAI |
|
|
