| import Mathlib |
| import OAI.Analysis.HardSphere.Allocation.PhysicalMixedObservation |
| import OAI.Analysis.HardSphere.Conditioning.TagPushforward |
| import OAI.Analysis.HardSphere.Allocation.PhysicalProjectedPrimary |
| import OAI.Analysis.HardSphere.Conditioning.PrimaryProduct |
|
|
| namespace OAI |
|
|
| section |
|
|
| noncomputable section |
| open MeasureTheory ProbabilityTheory Set |
| open scoped ENNReal BigOperators |
| namespace HardSphere.Groups |
| open Wiener Bridge Sampling Comparison GaussianPacking PathCutoff CellOrbital |
| attribute [local instance] Classical.propDecidable |
| attribute [local instance] Comparison.permBathSpace Comparison.permBathSingleton |
|
|
| def cellPointReference {K : ℕ} (L : Length) (v : Fin 3→Fin K) : Measure (Torus L) := |
| (cellPathReference v).map (fun e ↦ projectPoint L (e midpointTime)) |
| instance cellPointReference_probability {K : ℕ} (L : Length) (v : Fin 3→Fin K) : |
| IsProbabilityMeasure (cellPointReference L v) := by |
| unfold cellPointReference |
| exact (Measure.isProbabilityMeasure_map_iff (((projectPoint_continuous L).measurable.comp |
| (continuous_eval_const midpointTime).measurable).aemeasurable)).mpr inferInstance |
| |
| def physicalCellPost {N K : ℕ} (L : Length) (extra tag : Fin 3→Fin K) (b : Bool) : |
| Kernel (Torus L×Configuration N L) (Torus L×Torus L×Configuration N L) := |
| (cellExtraKernel (cellPointReference L extra) (centeredCell L (gridCenter L tag)) univ |
| (centeredCell_measurable L _)).map (cellObservation b) |
| |
| theorem physical_branch_observed_reference {N K M G : ℕ} [NeZero K] {L : Length} |
| (hL : L.val=(K:ℝ)) (hK : 1≤K) (h50 : 50<L.val) |
| (μ : Measure (BridgeEndpoints (N+1) L)) [SFinite μ] |
| (g : Fin G) (flags : Fin (N+1)→Bool) (a : GridAllocation (N+1) K (G*M)) |
| (m : ℕ) (r R C S s₀ t : ℝ) (u : Time) (extra tag : Fin 3→Fin K) (b : Bool) : |
| taggedPhysicalBathKernel (Function.Embedding.subtype (fun i ↦ i∈groupUnion g a)) b ∘ₘ |
| taggedReference |
| ((physicalBranchLaw hL μ g flags a m r R C S s₀ (2*t) u).compProd |
| (physicalPrimaryReference hL g m t r R C S u |
| (fun i : {i // i∈groupUnion g a} ↦ i.val) extra)) |
| (physicalTagEvent (Function.Embedding.subtype (fun i ↦ i∈groupUnion g a)) tag) = |
| ∑i : Fin (N+1), if i∈groupUnion g a then |
| ((physicalCellPost (N:=N) L extra tag b) ∘ₖ orderedKernel (uniformBathOrder N) i) ∘ₘ |
| (rawPhysicalBranch hL μ g flags a m r R C S s₀ (2*t) u).map |
| (fun z ↦ labelBridgePath (N+1) L (2*t) z midpointTime) |
| else 0 := by |
| classical |
| let particle := Function.Embedding.subtype (fun i ↦ i∈groupUnion g a) |
| let P := (physicalBranchLaw hL μ g flags a m r R C S s₀ (2*t) u).compProd |
| (physicalPrimaryReference hL g m t r R C S u |
| (fun i : {i // i∈groupUnion g a} ↦ i.val) extra) |
| change taggedPhysicalBathKernel particle b ∘ₘ taggedReference P (physicalTagEvent particle tag)=_ |
| rw [taggedReference_comp P _ (physicalTagEvent_measurable particle tag)] |
| apply Finset.sum_congr rfl |
| intro i _ |
| have hk : (taggedPhysicalBathKernel (K:=K) (M:=M) (G:=G) (L:=L) particle b).comap (Prod.mk i) measurable_prodMk_left= |
| (primaryBathKernel b i).comap (physicalFullMidpoint particle) (physicalFullMidpoint_measurable particle) := rfl |
| rw [hk] |
| have hrange : i∈Set.range particle ↔ i∈groupUnion g a := by |
| constructor |
| · rintro ⟨j,rfl⟩; exact j.property |
| · intro hi; exact ⟨⟨i,hi⟩,rfl⟩ |
| by_cases hi : i∈groupUnion g a |
| · rw [ite_eq_left hi] |
| have he : physicalTagEvent (M:=M) (G:=G) (L:=L) particle tag i=(physicalFullMidpoint particle) ⁻¹' |
| ((fun F : Option (Fin (N+1))→Torus L ↦ F (some i)) ⁻¹' centeredCell L (gridCenter L tag)) := by |
| ext z |
| simp only [physicalTagEvent,Set.mem_ofPred_eq,hrange,hi,true_and,Set.mem_preimage] |
| rw [he,comp_comap_map,←Measure.restrict_map (physicalFullMidpoint_measurable particle) |
| ((centeredCell_measurable L _).preimage (measurable_pi_apply (some i)))] |
| change primaryBathKernel b i ∘ₘ (P.map (physicalFullMidpoint particle)).restrict _=_ |
| dsimp only [P,particle] |
| rw [physicalBranch_primary_midpoint hL hK h50 μ g flags a m r R C S s₀ t u extra] |
| exact primaryBath_cell_reference _ (cellPointReference L extra) b i _ (centeredCell_measurable L _) |
| · rw [ite_eq_right hi] |
| have he : physicalTagEvent (M:=M) (G:=G) (L:=L) particle tag i=∅ := by |
| ext z; simp [physicalTagEvent,hrange,hi] |
| simp [he] |
| |
| end HardSphere.Groups |
| end |
| end |
| |
| section |
| |
| noncomputable section |
| open MeasureTheory ProbabilityTheory Set |
| open scoped ENNReal BigOperators |
| namespace HardSphere.Comparison |
| variable {Ω A G Y : Type*} [MeasurableSpace Ω] [MeasurableSpace Y] |
| [Fintype A] [DecidableEq A] [Fintype G] [DecidableEq G] |
| lemma sum_allocatedTagLaw_le_ae (μ : Measure Ω) (I : G→Ω→Finset A) |
| (hI : ∀g i,MeasurableSet {ω | i∈I g ω}) (κ : A→Kernel Ω Y) |
| (hd : ∀ᵐω ∂μ, ∀g g',g≠g' → Disjoint (I g ω) (I g' ω)) : |
| ∑g,allocatedTagLaw μ I κ g ≤ ∑i,κ i ∘ₘ μ := by |
| classical |
| apply Measure.le_iff.2 |
| intro s hs |
| have hbind (i : A) (ν : Measure Ω) : (κ i ∘ₘ ν) s=∫⁻ω,κ i ω s ∂ν := |
| Measure.bind_apply hs (κ i).aemeasurable |
| simp only [allocatedTagLaw,Measure.finsetSum_apply,hbind] |
| rw [Finset.sum_comm] |
| apply Finset.sum_le_sum |
| intro i _ |
| simp_rw [←lintegral_indicator (hI _ i)] |
| rw [←lintegral_finsetSum _ (fun g _ ↦ ((κ i).measurable_coe hs).indicator (hI g i))] |
| apply lintegral_mono_ae |
| filter_upwards [hd] with ω hdω |
| let T : Finset G := Finset.univ.filter (fun g ↦ i∈I g ω) |
| have hT : T.card≤1 := by |
| rw [Finset.card_le_one] |
| intro g hg g' hg' |
| by_contra h |
| exact Finset.disjoint_left.mp (hdω g g' h) (Finset.mem_filter.mp hg).2 (Finset.mem_filter.mp hg').2 |
| calc |
| (∑g,({ω | i∈I g ω}.indicator (fun ω ↦ κ i ω s)) ω)=∑g∈T,κ i ω s := by |
| change _=∑g∈Finset.univ.filter (fun g ↦ i∈I g ω),κ i ω s |
| rw [Finset.sum_filter] |
| apply Finset.sum_congr rfl |
| intro g _ |
| simp only [Set.indicator_apply, Set.mem_ofPred_eq] |
| _ ≤ κ i ω s := by |
| rw [Finset.sum_const] |
| simpa only [one_nsmul] using nsmul_le_nsmul_left (show 0≤κ i ω s from bot_le) hT |
| end HardSphere.Comparison |
| namespace HardSphere.Groups |
| open Sampling Wiener PathCutoff Comparison |
| attribute [local instance] Classical.propDecidable |
| |
| lemma groupAt_mem_eligible {N K M G : ℕ} {L : Length} |
| (m : ℕ) (r R C S s₀ : ℝ) (u : Time) (ω : ConfigurationPath N L) |
| (a : GridAllocation N K (G*M)) |
| (ha : ∀v,a v∈(allocationPMF (eligibleList m r R C S s₀ u v ω) (G*M)).support) |
| (g : Fin G) (v : Fin 3→Fin K) : groupAt g a v⊆eligibleList m r R C S s₀ u v ω := by |
| intro i hi |
| cases he : a v with |
| | none => simp [groupAt,he] at hi |
| | some f => |
| simp only [groupAt,he,Finset.mem_image,Finset.mem_univ,true_and] at hi |
| obtain ⟨k,rfl⟩ := hi |
| exact allocationPMF_support _ _ _ (ha v) f he _ |
| |
| lemma physical_groups_disjoint {N K M G : ℕ} {L : Length} |
| (hL : L.val=(K:ℝ)) (hK : 1≤K) (m : ℕ) (r R C S s₀ : ℝ) (u : Time) |
| (ω : ConfigurationPath N L) (a : GridAllocation N K (G*M)) |
| (ha : ∀v,a v∈(allocationPMF (eligibleList m r R C S s₀ u v ω) (G*M)).support) |
| (g h : Fin G) (hgh : g≠h) : Disjoint (groupUnion g a) (groupUnion h a) := by |
| apply Finset.disjoint_left.mpr |
| intro i hig hih |
| obtain ⟨v,_,hiv⟩ := Finset.mem_biUnion.mp hig |
| obtain ⟨w,_,hiw⟩ := Finset.mem_biUnion.mp hih |
| have hv := groupAt_mem_eligible m r R C S s₀ u ω a ha g v hiv |
| have hw := groupAt_mem_eligible m r R C S s₀ u ω a ha h w hiw |
| have hvw : v=w := by |
| by_contra hn |
| exact Finset.disjoint_left.mp (eligibleList_disjoint hL hK m r R C S s₀ u ω hn) hv hw |
| rw [←hvw] at hiw |
| cases he : a v with |
| | none => simp [groupAt,he] at hiv |
| | some f => |
| simp only [groupAt,he,Finset.mem_image,Finset.mem_univ,true_and] at hiv hiw |
| obtain ⟨k,hk⟩ := hiv |
| obtain ⟨l,hl⟩ := hiw |
| exact groupPosition_disjoint M G g h hgh k l (f.injective (hk.trans hl.symm)) |
| |
| lemma physicalAllocation_supported_ae {N K : ℕ} {L : Length} |
| (m total : ℕ) (r R C S s₀ : ℝ) (u : Time) (ω : ConfigurationPath N L) : |
| ∀ᵐa ∂physicalAllocationKernel (K:=K) m total r R C S s₀ u ω, |
| ∀v,a v∈(allocationPMF (eligibleList m r R C S s₀ u v ω) total).support := by |
| change ∀ᵐa ∂Measure.pi (fun v ↦ (allocationPMF (eligibleList m r R C S s₀ u v ω) total).toMeasure), _ |
| apply ae_all_iff.mpr |
| intro v |
| exact (MeasureTheory.measurePreserving_eval |
| (fun w ↦ (allocationPMF (eligibleList m r R C S s₀ u w ω) total).toMeasure) v).quasiMeasurePreserving.tendsto_ae |
| (allocation_support_ae _ _) |
| |
| theorem sum_physical_group_tags_le {N K M G : ℕ} {L : Length} |
| {Y : Type*} [MeasurableSpace Y] |
| (hL : L.val=(K:ℝ)) (hK : 1≤K) (m : ℕ) (r R C S s₀ : ℝ) (u : Time) |
| (μ : Measure (ConfigurationPath N L)) [SFinite μ] |
| (κ : Fin N→Kernel (ConfigurationPath N L×GridAllocation N K (G*M)) Y) : |
| (∑g,allocatedTagLaw (μ.compProd (physicalAllocationKernel m (G*M) r R C S s₀ u)) |
| (fun g p ↦ groupUnion g p.2) κ g) ≤ |
| ∑i,κ i ∘ₘ μ.compProd (physicalAllocationKernel m (G*M) r R C S s₀ u) := by |
| apply sum_allocatedTagLaw_le_ae |
| · intro g i |
| exact ((measurable_of_countable (fun a : GridAllocation N K (G*M) ↦ i∈groupUnion g a)).comp measurable_snd).setOf |
| · apply Measure.ae_compProd_of_ae_ae |
| (((measurable_of_countable (fun a : GridAllocation N K (G*M) ↦ |
| ∀g h : Fin G,g≠h→Disjoint (groupUnion g a) (groupUnion h a))).comp measurable_snd).setOf) |
| filter_upwards [] with ω |
| filter_upwards [physicalAllocation_supported_ae m (G*M) r R C S s₀ u ω] with a ha |
| exact fun g h hgh ↦ physical_groups_disjoint hL hK m r R C S s₀ u ω a ha g h hgh |
| |
| end HardSphere.Groups |
| end |
| end |
| |
| end OAI |
| |
