| import Mathlib |
| import OAI.Analysis.HardSphere.Allocation.PhysicalRouteGeometry |
| import OAI.Analysis.HardSphere.Allocation.PhysicalBranchLaw |
| import OAI.Analysis.HardSphere.Geometry.PhysicalGeometry |
|
|
| 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 |
| instance labelOptionSpace (J : Type*) : MeasurableSpace (Option J) := ⊤ |
| instance labelOptionSingletons (J : Type*) : MeasurableSingletonClass (Option J) := ⟨fun _ ↦ trivial⟩ |
| |
| variable {J : Type*} [Fintype J] [DecidableEq J] [MeasurableSpace J] [MeasurableSingletonClass J] |
| |
| def paddedLabels (s : Finset J) : Finset (Option J) := |
| if s.Nonempty then s.image some else {none} |
| omit [Fintype J] [MeasurableSpace J] [MeasurableSingletonClass J] in |
| lemma paddedLabels_nonempty (s : Finset J) : (paddedLabels s).Nonempty := by |
| by_cases h : s.Nonempty |
| · simpa only [paddedLabels,ite_eq_left h] using h.image some |
| · simp [paddedLabels,h] |
| |
| def uniformLabels (s : Finset J) : Measure (Option J) := |
| (PMF.uniformOfFinset (paddedLabels s) (paddedLabels_nonempty s)).toMeasure |
| instance uniformLabels_probability (s : Finset J) : IsProbabilityMeasure (uniformLabels s) := by |
| unfold uniformLabels; infer_instance |
| |
| omit [MeasurableSpace J] [MeasurableSingletonClass J] in |
| lemma uniformLabels_supported (s : Finset J) : ∀ᵐ i ∂uniformLabels s,i∈paddedLabels s := by |
| rw [ae_iff] |
| change uniformLabels s ((↑(paddedLabels s) : Set (Option J))ᶜ)=0 |
| unfold uniformLabels |
| rw [PMF.toMeasure_uniformOfFinset_apply _ _ (Set.to_countable _).measurableSet] |
| simp |
| |
| def branchAccessible {N K M G : ℕ} [NeZero K] {L : Length} (hL : L.val=(K:ℝ)) |
| (g : Fin G) (m : ℕ) (t r R C S : ℝ) (u : Time) (particle : J→Fin N) |
| (v : DiscreteTorus K) (d : PhysicalDatum N K M G L) : Finset J := |
| Finset.univ.filter fun j ↦ particle j∈datumAccessibleLabels hL g m t r R C S u (siteIndex v) d |
| |
| omit [DecidableEq J] [MeasurableSpace J] [MeasurableSingletonClass J] in |
| lemma branchAccessible_measurable {N K M G : ℕ} [NeZero K] {L : Length} (hL : L.val=(K:ℝ)) |
| (g : Fin G) (m : ℕ) (t r R C S : ℝ) (u : Time) (particle : J→Fin N) |
| (v : DiscreteTorus K) : Measurable (branchAccessible (M:=M) hL g m t r R C S u particle v) := by |
| exact (measurable_of_countable (f:=fun s : Finset (Fin N) ↦ |
| Finset.univ.filter fun j ↦ particle j∈s)).comp (datumAccessibleLabels_measurable hL g m t r R C S u _) |
| |
| def physicalLabelAt {N K M G : ℕ} [NeZero K] {L : Length} (hL : L.val=(K:ℝ)) |
| (g : Fin G) (m : ℕ) (t r R C S : ℝ) (u : Time) (particle : J→Fin N) |
| (v : DiscreteTorus K) : Kernel (PhysicalDatum N K M G L) (Option J) where |
| toFun d := uniformLabels (branchAccessible hL g m t r R C S u particle v d) |
| measurable' := (measurable_of_countable (f:=uniformLabels (J:=J))).comp |
| (branchAccessible_measurable hL g m t r R C S u particle v) |
| |
| instance physicalLabelAt_markov {N K M G : ℕ} [NeZero K] {L : Length} (hL : L.val=(K:ℝ)) |
| (g : Fin G) (m : ℕ) (t r R C S : ℝ) (u : Time) (particle : J→Fin N) |
| (v : DiscreteTorus K) : IsMarkovKernel (physicalLabelAt (M:=M) hL g m t r R C S u particle v) := |
| ⟨fun _ ↦ inferInstanceAs (IsProbabilityMeasure (uniformLabels _))⟩ |
| |
| def physicalLabels {N K M G : ℕ} [NeZero K] {L : Length} (hL : L.val=(K:ℝ)) |
| (g : Fin G) (m : ℕ) (t r R C S : ℝ) (u : Time) (particle : J→Fin N) : |
| Kernel (PhysicalDatum N K M G L) (DiscreteTorus K→Option J) := |
| finiteKernelPi (physicalLabelAt hL g m t r R C S u particle) |
| instance physicalLabels_markov {N K M G : ℕ} [NeZero K] {L : Length} (hL : L.val=(K:ℝ)) |
| (g : Fin G) (m : ℕ) (t r R C S : ℝ) (u : Time) (particle : J→Fin N) : |
| IsMarkovKernel (physicalLabels (M:=M) hL g m t r R C S u particle) := by |
| constructor |
| intro d |
| rw [physicalLabels,finiteKernelPi_apply] |
| infer_instance |
| |
| omit [MeasurableSpace J] [MeasurableSingletonClass J] in |
| lemma physicalLabels_supported {N K M G : ℕ} [NeZero K] {L : Length} (hL : L.val=(K:ℝ)) |
| (g : Fin G) (m : ℕ) (t r R C S : ℝ) (u : Time) (particle : J→Fin N) |
| (d : PhysicalDatum N K M G L) : |
| ∀ᵐ z ∂physicalLabels hL g m t r R C S u particle d, |
| ∀v,z v∈paddedLabels (branchAccessible hL g m t r R C S u particle v d) := by |
| rw [physicalLabels,finiteKernelPi_apply] |
| apply ae_all_iff.mpr |
| intro v |
| exact (measurePreserving_eval (fun v ↦ physicalLabelAt hL g m t r R C S u particle v d) v).quasiMeasurePreserving.ae |
| (uniformLabels_supported _) |
| |
| 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 |
| attribute [local irreducible] datumHome datumSelected groupUnion datumStart datumEnd anchorPoint |
| |
| lemma physicalBranchLaw_ae {N K M G : ℕ} {L : Length} (hL : L.val=(K:ℝ)) (hK : 1≤K) |
| (μ : Measure (BridgeEndpoints N L)) [SFinite μ] |
| (g : Fin G) (b : Fin N→Bool) (a : GridAllocation N K (G*M)) |
| (m : ℕ) (r R C S s₀ T : ℝ) (u : Time) |
| (P : PhysicalDatum N K M G L→Prop) (hPm : MeasurableSet {d | P d}) |
| (hP : ∀ω : ConfigurationPath N L, |
| (∀v,a v∈(allocationPMF (eligibleList m r R C S s₀ u v ω) (G*M)).support) → |
| P (physicalDatum g m r R C S s₀ u (ω,a))) : |
| ∀ᵐd ∂physicalBranchLaw hL μ g b a m r R C S s₀ T u,P d := by |
| rw [physicalBranchLaw_raw] |
| have hm : Measurable (fun z : BridgeEndpoints N L×(Fin N→Bridge.Noise) ↦ |
| physicalDatum g m r R C S s₀ u (labelBridgePath N L T z,a)) := |
| (physicalDatum_measurable g m r R C S s₀ u).comp |
| ((labelBridgePath_measurable N L T).prodMk measurable_const) |
| apply (ae_map_iff hm.aemeasurable hPm).mpr |
| have hB := (labelInteraction_measurable N L r T).inter |
| (flagEvent_measurable _ (fun i ↦ eligibilityFlag_measurable (K:=K) m r R C S s₀ T u i) b) |
| have hs := allocationBase_ae_support μ (gridAllocationKernel hL (G*M)) b a |
| (fun e a' ↦ ∀v,a' v∈(allocationPMF (recordedList hL v (e,b)) (G*M)).support) |
| (fun e ↦ gridAllocation_support_ae hL (G*M) (e,b)) |
| filter_upwards [ae_restrict_of_ae (Measure.quasiMeasurePreserving_fst.ae hs), |
| ae_restrict_of_ae (label_endpoints_ae T (allocationBase μ (gridAllocationKernel hL (G*M)) b a)), |
| ae_restrict_mem hB] with z hz hz01 hzB |
| apply hP _ |
| intro v |
| have hb : (fun i ↦ eligibilityFlag (K:=K) m r R C S s₀ T u i (z.1,z.2 i))=b := funext hzB.2 |
| have he := recordedList_eq hL hK m r R C S s₀ T u v z.1 z.2 hz01.1 hz01.2 |
| rw [hb] at he |
| simpa only [←he] using hz v |
| |
| lemma physicalBranchLaw_allocation {N K M G : ℕ} {L : Length} (hL : L.val=(K:ℝ)) |
| (μ : Measure (BridgeEndpoints N L)) [SFinite μ] |
| (g : Fin G) (b : Fin N→Bool) (a : GridAllocation N K (G*M)) |
| (m : ℕ) (r R C S s₀ T : ℝ) (u : Time) : |
| ∀ᵐd ∂physicalBranchLaw hL μ g b a m r R C S s₀ T u,d.2.2.2.1=a := by |
| apply (ae_map_iff (physicalBranchDatum_measurable g b a T).aemeasurable |
| (measurableSet_eq_fun measurable_snd.snd.snd.fst measurable_const)).mpr |
| exact ae_of_all _ fun _ ↦ rfl |
| |
| lemma physicalBranchLaw_multiplicity {N K M G : ℕ} {L : Length} (hL : L.val=(K:ℝ)) (hK : 1≤K) |
| (μ : Measure (BridgeEndpoints N L)) [SFinite μ] |
| (g : Fin G) (b : Fin N→Bool) (a : GridAllocation N K (G*M)) |
| (m : ℕ) (r R C S s₀ T : ℝ) (u : Time) : |
| ∀ᵐd ∂physicalBranchLaw hL μ g b a m r R C S s₀ T u, ∀v, |
| (Finset.univ.filter (fun i : {i // i∈groupUnion g a} ↦ datumHome hL d i=v)).card≤M := by |
| apply physicalBranchLaw_ae hL hK μ g b a m r R C S s₀ T u _ ?_ |
| (fun ω hω v ↦ physicalDatum_group_multiplicity hL hK g m r R C S s₀ u ω a hω v) |
| have hf : Measurable (fun d : PhysicalDatum N K M G L ↦ fun i : {i // i∈groupUnion g a} ↦ datumHome hL d i.val) := |
| Measurable.of_eval (fun i ↦ datumHome_measurable hL i.val) |
| exact hf ((Set.to_countable {f : {i // i∈groupUnion g a}→(Fin 3→Fin K) | |
| ∀v,(Finset.univ.filter (fun i ↦ f i=v)).card≤M}).measurableSet) |
| |
| lemma datum_norm_geometry_measurableSet {N K M G : ℕ} {L : Length} (hL : L.val=(K:ℝ)) (i : Fin N) : |
| MeasurableSet {d : PhysicalDatum N K M G L | |
| ‖datumStart hL d i-anchorPoint (datumHome hL d i)‖≤3 ∧ |
| ‖datumEnd hL d i-anchorPoint (datumHome hL d i)‖≤3} := by |
| have hs := datumStart_measurable (N:=N) (K:=K) (M:=M) (G:=G) hL i |
| have he := datumEnd_measurable (N:=N) (K:=K) (M:=M) (G:=G) hL i |
| have hc := datumAnchor_measurable (N:=N) (K:=K) (M:=M) (G:=G) hL i |
| exact (measurableSet_le (hs.sub hc).norm measurable_const).inter |
| (measurableSet_le (he.sub hc).norm measurable_const) |
| |
| lemma datum_sep_geometry_measurableSet {N K M G : ℕ} {L : Length} (s₀ : ℝ) (i j : Fin N) : |
| MeasurableSet {d : PhysicalDatum N K M G L | |
| s₀<torusDistance L (d.1 i) (d.1 j) ∧ s₀<torusDistance L (d.2.1 i) (d.2.1 j)} := by |
| have h₀ : Measurable (fun d : PhysicalDatum N K M G L ↦ (d.1 i,d.1 j)) := |
| ((measurable_pi_apply i).comp measurable_fst).prodMk ((measurable_pi_apply j).comp measurable_fst) |
| have h₁ : Measurable (fun d : PhysicalDatum N K M G L ↦ (d.2.1 i,d.2.1 j)) := |
| ((measurable_pi_apply i).comp measurable_snd.fst).prodMk ((measurable_pi_apply j).comp measurable_snd.fst) |
| exact (measurableSet_lt measurable_const ((torusDistance_continuous_pair L).measurable.comp h₀)).inter |
| (measurableSet_lt measurable_const ((torusDistance_continuous_pair L).measurable.comp h₁)) |
| |
| lemma datum_geometry_measurableSet {N K M G : ℕ} {L : Length} (hL : L.val=(K:ℝ)) |
| (s₀ : ℝ) (s : Finset (Fin N)) : |
| MeasurableSet {d : PhysicalDatum N K M G L | ∀i : {i // i∈s}, |
| ‖datumStart hL d i-anchorPoint (datumHome hL d i)‖≤3 ∧ |
| ‖datumEnd hL d i-anchorPoint (datumHome hL d i)‖≤3 ∧ |
| ∀j : Fin N,j≠i→s₀<torusDistance L (projectPoint L (datumStart hL d i)) (projectPoint L (datumStart hL d j)) ∧ |
| s₀<torusDistance L (projectPoint L (datumEnd hL d i)) (projectPoint L (datumEnd hL d j))} := by |
| have hsep (i : Fin N) : MeasurableSet {d : PhysicalDatum N K M G L | ∀j : Fin N,j≠i→ |
| s₀<torusDistance L (projectPoint L (datumStart hL d i)) (projectPoint L (datumStart hL d j)) ∧ |
| s₀<torusDistance L (projectPoint L (datumEnd hL d i)) (projectPoint L (datumEnd hL d j))} := by |
| simp only [datumStart,datumEnd,nearestLift_project,Set.ofPred_forall] |
| exact MeasurableSet.iInter fun j ↦ MeasurableSet.iInter fun _ ↦ datum_sep_geometry_measurableSet s₀ i j |
| simp only [Set.ofPred_forall] |
| apply MeasurableSet.iInter |
| intro i |
| have h := (datum_norm_geometry_measurableSet (M:=M) (G:=G) hL i.val).inter (hsep i.val) |
| convert h using 1 |
| ext d |
| exact and_assoc.symm |
| |
| lemma physicalBranchLaw_geometry {N K M G : ℕ} {L : Length} (hL : L.val=(K:ℝ)) (hK : 1≤K) |
| (μ : Measure (BridgeEndpoints N L)) [SFinite μ] |
| (g : Fin G) (b : Fin N→Bool) (a : GridAllocation N K (G*M)) |
| (m : ℕ) (r R C S s₀ T : ℝ) (u : Time) : |
| ∀ᵐd ∂physicalBranchLaw hL μ g b a m r R C S s₀ T u, ∀i : {i // i∈groupUnion g a}, |
| ‖datumStart hL d i-anchorPoint (datumHome hL d i)‖≤3 ∧ |
| ‖datumEnd hL d i-anchorPoint (datumHome hL d i)‖≤3 ∧ |
| ∀j : Fin N,j≠i→s₀<torusDistance L (projectPoint L (datumStart hL d i)) (projectPoint L (datumStart hL d j)) ∧ |
| s₀<torusDistance L (projectPoint L (datumEnd hL d i)) (projectPoint L (datumEnd hL d j)) := by |
| exact physicalBranchLaw_ae hL hK μ g b a m r R C S s₀ T u _ |
| (datum_geometry_measurableSet hL s₀ (groupUnion g a)) |
| (fun ω hω (i : {i // i∈groupUnion g a}) ↦ physicalDatum_selected_geometry hL hK g m r R C S s₀ u ω a hω i.val i.property) |
| |
| lemma branchAccessible_card {N K M G : ℕ} [NeZero K] {L : Length} (hL : L.val=(K:ℝ)) |
| (g : Fin G) (a : GridAllocation N K (G*M)) (m : ℕ) (t r R C S : ℝ) (u : Time) |
| (v : HardSphere.Lattice.DiscreteTorus K) (d : PhysicalDatum N K M G L) (hd : d.2.2.2.1=a) : |
| (branchAccessible hL g m t r R C S u (fun i : {i // i∈groupUnion g a} ↦ i.val) v d).card= |
| (datumAccessibleLabels hL g m t r R C S u (siteIndex v) d).card := by |
| let s := branchAccessible hL g m t r R C S u (fun i : {i // i∈groupUnion g a} ↦ i.val) v d |
| have he : s.image Subtype.val=datumAccessibleLabels hL g m t r R C S u (siteIndex v) d := by |
| ext i |
| constructor |
| · intro hi |
| obtain ⟨j,hj,rfl⟩ := Finset.mem_image.mp hi |
| exact (Finset.mem_filter.mp hj).2 |
| · intro hi |
| have hsel : i∈groupUnion g a := by |
| have h := (Finset.mem_filter.mp hi).1 |
| simpa only [datumSelected,hd] using h |
| exact Finset.mem_image.mpr ⟨⟨i,hsel⟩,Finset.mem_filter.mpr ⟨Finset.mem_univ _,hi⟩,rfl⟩ |
| rw [←he,Finset.card_image_of_injective _ Subtype.val_injective] |
| |
| lemma branchAccessible_supply {N K M G : ℕ} [NeZero K] {L : Length} (hL : L.val=(K:ℝ)) |
| (g : Fin G) (a : GridAllocation N K (G*M)) (m : ℕ) (t r R C S : ℝ) (u : Time) |
| (v : HardSphere.Lattice.DiscreteTorus K) (d : PhysicalDatum N K M G L) (hd : d.2.2.2.1=a) |
| (hv : siteIndex v∉datumBad hL g m t r R C S u d) : |
| (M:ℝ)≤2*((branchAccessible hL g m t r R C S u (fun i : {i // i∈groupUnion g a} ↦ i.val) v d).card:ℝ) := by |
| rw [branchAccessible_card hL g a m t r R C S u v d hd] |
| have hh : M≤2*(datumAccessibleLabels hL g m t r R C S u (siteIndex v) d).card := by |
| simpa only [datumBad,Finset.mem_filter,Finset.mem_univ,true_and,not_lt] using hv |
| exact_mod_cast hh |
| |
| end HardSphere.Groups |
| end |
| end |
| |
| end OAI |
| |
