| import Mathlib |
| import OAI.Analysis.HardSphere.Conditioning.SharedTransport |
| import OAI.Analysis.HardSphere.Conditioning.PhysicalRoute |
|
|
| namespace OAI |
|
|
| section |
|
|
| noncomputable section |
| open MeasureTheory ProbabilityTheory Set |
| open scoped ENNReal NNReal |
| namespace HardSphere.Comparison |
| variable {X Y : Type*} [MeasurableSpace X] [MeasurableSpace Y] |
|
|
| lemma conditioned_prod_univ (μ : Measure X) [SFinite μ] (ν : Measure Y) [IsProbabilityMeasure ν] |
| (A : Set X) : |
| conditionedMeasure (μ.prod ν) (Prod.fst ⁻¹' A)=(conditionedMeasure μ A).prod ν := by |
| have he : Prod.fst ⁻¹' A=A ×ˢ (univ : Set Y) := by ext p; simp |
| rw [he,conditionedMeasure,conditionedMeasure,Measure.prod_smul_left, |
| Measure.prod_prod,measure_univ,mul_one,←Measure.prod_restrict,Measure.restrict_univ] |
|
|
| lemma conditioned_option {J : Type*} [Fintype J] |
| (μ : Option J→Measure X) [∀i,IsFiniteMeasure (μ i)] [IsProbabilityMeasure (μ none)] |
| (A : Set (J→X)) (hA : MeasurableSet A) : |
| conditionedMeasure (Measure.pi μ) ((fun x : Option J→X ↦ fun i ↦ x (some i)) ⁻¹' A)= |
| ((conditionedMeasure (Measure.pi fun i ↦ μ (some i)) A).prod (μ none)).map |
| (MeasurableEquiv.piOptionEquivProd (fun _ : Option J ↦ X)).symm := by |
| let e := MeasurableEquiv.piOptionEquivProd (fun _ : Option J ↦ X) |
| have hB : MeasurableSet ((fun x : Option J→X ↦ fun i ↦ x (some i)) ⁻¹' A) := |
| hA.preimage (Measurable.of_eval fun i ↦ measurable_pi_apply (some i)) |
| rw [←Measure.pi_map_piOptionEquivProd μ, |
| ←conditioned_map_preimage _ e.symm e.symm.measurable hB] |
| change (conditionedMeasure _ (Prod.fst ⁻¹' A)).map e.symm = _ |
| rw [conditioned_prod_univ] |
|
|
| end HardSphere.Comparison |
| namespace HardSphere.Groups |
| open Wiener Bridge Sampling Comparison GaussianPacking PathCutoff CellOrbital |
| variable {J : Type*} [Fintype J] [DecidableEq J] |
|
|
| def physicalSlotMutual {L : Length} (r : ℝ) : Set (Option J→C(Time,Point)) := |
| (fun x ↦ fun i ↦ x (some i)) ⁻¹' physicalMutual (L:=L) r |
| omit [DecidableEq J] in |
| lemma physicalSlotMutual_measurable {L : Length} (r : ℝ) : |
| MeasurableSet (physicalSlotMutual (J:=J) (L:=L) r) := |
| (physicalMutual_measurable r).preimage |
| (Measurable.of_eval fun i ↦ measurable_pi_apply (some i)) |
|
|
| def physicalPrimaryReference {N K 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) (v : Fin 3→Fin K) : |
| Kernel (PhysicalDatum N K M G L) (Option J→C(Time,Point)) := |
| fiberNormalize (finiteKernelPi (physicalSlotReference hL g m t r R C S u particle v)) |
| (Prod.snd ⁻¹' physicalSlotMutual (L:=L) r) |
|
|
| instance physicalPrimaryReference_finite {N K 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) (v : Fin 3→Fin K) : |
| IsFiniteKernel (physicalPrimaryReference (M:=M) hL g m t r R C S u particle v) := by |
| unfold physicalPrimaryReference; infer_instance |
|
|
| omit [DecidableEq J] in |
| lemma physicalPrimaryReference_apply {N K 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) (v : Fin 3→Fin K) (d : PhysicalDatum N K M G L) : |
| physicalPrimaryReference hL g m t r R C S u particle v d= |
| conditionedMeasure (Measure.pi fun i ↦ physicalSlotReference hL g m t r R C S u particle v i d) |
| (physicalSlotMutual (L:=L) r) := by |
| unfold physicalPrimaryReference |
| change (fiberRestrict _ _ d univ)⁻¹ • fiberRestrict _ _ d=_ |
| rw [fiberRestrict_apply _ ((physicalSlotMutual_measurable r).preimage measurable_snd)] |
| simp only [Measure.restrict_apply_univ,finiteKernelPi_apply] |
| rfl |
|
|
| theorem physicalPrimaryReference_actual {N K M G : ℕ} {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 : Fin 3→Fin K) (d : PhysicalDatum N K M G L) : |
| physicalPrimaryReference hL g m t r R C S u (fun i : {i // i∈groupUnion g a} ↦ i.val) v d= |
| ((datumGroupReference hL g a m r R C S t u d).prod (cellPathReference v)).map |
| (MeasurableEquiv.piOptionEquivProd (fun _ : Option {i // i∈groupUnion g a} ↦ C(Time,Point))).symm := by |
| have : IsProbabilityMeasure (physicalSlotReference hL g m t r R C S u |
| (fun i : {i // i∈groupUnion g a} ↦ i.val) v none d) := |
| inferInstanceAs (IsProbabilityMeasure (cellPathReference v)) |
| rw [physicalPrimaryReference_apply,physicalSlotMutual,conditioned_option _ _ (physicalMutual_measurable r), |
| datumGroupReference_apply] |
| simp only [physicalSlotReference,Kernel.const_apply,datumIndividualReference_apply] |
|
|
| def physicalSharedRoute {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) : |
| Kernel (PhysicalDatum N K M G L) |
| ((Option J→C(Time,Point))×(routeSlots labels→C(Time,Point)×C(Time,Point))) := |
| ambientSharedKernel (routeSlots labels) (routePermutation labels) (routePermutation_fixed labels) |
| (physicalPrimaryReference hL g m t r R C S u particle (v (Fin.last (n+1)))) |
| (physicalSelectedFactor hL g m t r R C S u particle labels v steps) |
| (Prod.snd ⁻¹' physicalSlotMutual (L:=L) r) |
|
|
| instance physicalSharedRoute_finite {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) : |
| IsFiniteKernel (physicalSharedRoute (M:=M) hL g m t r R C S u particle labels v steps) := by |
| unfold physicalSharedRoute; infer_instance |
|
|
| 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 |
| variable {J : Type*} [Fintype J] [DecidableEq J] |
|
|
| def physicalSlotAllowed {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) : Option J→Set C(Time,Point) |
| | none => univ |
| | some i => Prod.mk d ⁻¹' datumPermitted hL g m r R C S u (particle i) |
|
|
| omit [Fintype J] [DecidableEq J] in |
| lemma physicalSlotAllowed_measurable {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) (i : Option J) : |
| MeasurableSet (physicalSlotAllowed hL g m r R C S u particle d i) := by |
| cases i with |
| | none => exact MeasurableSet.univ |
| | some i => exact (datumPermitted_measurable hL g m r R C S u (particle i)).preimage measurable_prodMk_left |
|
|
| def physicalSlotOn {N K 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) (v : Fin 3→Fin K) (d : PhysicalDatum N K M G L) : |
| (i : Option J)→Measure (physicalSlotAllowed hL g m r R C S u particle d i) |
| | none => subtypeConditional (cellPathReference v) univ |
| | some i => referenceOn t (datumStart hL d (particle i)) (datumEnd hL d (particle i)) |
| (Prod.mk d ⁻¹' datumPermitted hL g m r R C S u (particle i)) |
|
|
| omit [Fintype J] [DecidableEq J] in |
| lemma physicalSlotOn_map {N K 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) (v : Fin 3→Fin K) (d : PhysicalDatum N K M G L) (i : Option J) : |
| (physicalSlotOn hL g m t r R C S u particle v d i).map Subtype.val= |
| physicalSlotReference hL g m t r R C S u particle v i d := by |
| cases i with |
| | none => |
| change (subtypeConditional (cellPathReference v) (univ : Set C(Time,Point))).map |
| Subtype.val=cellPathReference v |
| rw [subtypeConditional_map _ MeasurableSet.univ,measure_univ,inv_one, |
| one_smul,Measure.restrict_univ] |
| | some i => |
| change (referenceOn t (datumStart hL d (particle i)) (datumEnd hL d (particle i)) |
| (Prod.mk d ⁻¹' datumPermitted hL g m r R C S u (particle i))).map Subtype.val= |
| datumIndividualReference hL g m r R C S t u (particle i) d |
| rw [referenceOn_map _ _ _ ((datumPermitted_measurable hL g m r R C S u (particle i)).preimage |
| measurable_prodMk_left),datumIndividualReference_apply] |
|
|
| instance physicalSlotOn_finite {N K 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) (v : Fin 3→Fin K) (d : PhysicalDatum N K M G L) (i : Option J) : |
| IsFiniteMeasure (physicalSlotOn hL g m t r R C S u particle v d i) := by |
| have hm := physicalSlotOn_map hL g m t r R C S u particle v d i |
| have : IsFiniteMeasure ((physicalSlotOn hL g m t r R C S u particle v d i).map Subtype.val) := by |
| rw [hm]; infer_instance |
| exact Measure.isFiniteMeasure_of_map measurable_subtype_coe.aemeasurable |
|
|
| omit [Fintype J] [DecidableEq J] in |
| lemma physicalSlotOn_probability {N K 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) (v : Fin 3→Fin K) (d : PhysicalDatum N K M G L) (i : Option J) |
| [IsProbabilityMeasure (physicalSlotReference hL g m t r R C S u particle v i d)] : |
| IsProbabilityMeasure (physicalSlotOn hL g m t r R C S u particle v d i) := by |
| have : IsProbabilityMeasure ((physicalSlotOn hL g m t r R C S u particle v d i).map Subtype.val) := by |
| rw [physicalSlotOn_map]; infer_instance |
| exact Measure.isProbabilityMeasure_of_map measurable_subtype_coe.aemeasurable |
|
|
| omit [Fintype J] [DecidableEq J] in |
| lemma physicalAllowed_confined {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) (i : J) |
| (f : physicalSlotAllowed hL g m r R C S u particle d (some i)) : |
| Confined (projectPoint L (anchorPoint (datumHome hL d (particle i)))) (projectCurve L f.val) := |
| fun s ↦ (f.property.2.1 s).2 |
|
|
| lemma physicalAllowed_checks {N K M G : ℕ} {L : Length} (hL : L.val=(K:ℝ)) |
| (g : Fin G) (m : ℕ) (r R C S : ℝ) (hr : r≤1) (u : Time) |
| (particle : J→Fin N) (d : PhysicalDatum N K M G L) : |
| (routeConfigMap (fun i ↦ (Subtype.val : physicalSlotAllowed hL g m r R C S u particle d i→_))) ⁻¹' |
| physicalSlotMutual (L:=L) r= |
| retainedEvent (primaryChecks r (fun j ↦ projectPoint L (anchorPoint (datumHome hL d (particle j)))) |
| (fun j ↦ physicalSlotAllowed hL g m r R C S u particle d (some j)) |
| some (fun _ f ↦ f) (fun _ ↦ measurable_id)) Finset.univ := by |
| ext x |
| constructor |
| · intro hx e he |
| exact hx e.2.val.1 e.2.val.2 e.2.property.1 |
| · intro hx |
| let allowed := physicalSlotAllowed hL g m r R C S u particle d |
| let anchor := fun j ↦ projectPoint L (anchorPoint (datumHome hL d (particle j))) |
| have hconf : ∀j,∀f : allowed (some j),Confined (anchor j) (projectCurve L f.val) := |
| physicalAllowed_confined hL g m r R C S u particle d |
| have hh := physicalSparseChecks_complete (I:=Option J) (V:=fun i ↦ allowed i) |
| r hr anchor (fun j ↦ allowed (some j)) hconf |
| (fun _ ↦ some) (fun _ _ f ↦ f) (fun _ _ ↦ measurable_id) x hx |
| exact hh false |
|
|
| lemma physicalPrimary_on_map {N K M G : ℕ} {L : Length} (hL : L.val=(K:ℝ)) |
| (g : Fin G) (m : ℕ) (t r R C S : ℝ) (hr : r≤1) (u : Time) |
| (particle : J→Fin N) (v : Fin 3→Fin K) (d : PhysicalDatum N K M G L) : |
| (conditionedMeasure (Measure.pi (physicalSlotOn hL g m t r R C S u particle v d)) |
| (retainedEvent (primaryChecks r (fun j ↦ projectPoint L (anchorPoint (datumHome hL d (particle j)))) |
| (fun j ↦ physicalSlotAllowed hL g m r R C S u particle d (some j)) |
| some (fun _ f ↦ f) (fun _ ↦ measurable_id)) Finset.univ)).map |
| (routeConfigMap (fun i ↦ (Subtype.val : physicalSlotAllowed hL g m r R C S u particle d i→_)))= |
| physicalPrimaryReference hL g m t r R C S u particle v d := by |
| rw [←physicalAllowed_checks hL g m r R C S hr u particle d, |
| physicalPrimaryReference_apply] |
| exact (conditioned_pi_map (physicalSlotOn hL g m t r R C S u particle v d) |
| (fun i ↦ (Subtype.val : physicalSlotAllowed hL g m r R C S u particle d i→_)) |
| (fun _ ↦ measurable_subtype_coe) (physicalSlotMutual_measurable r)).trans |
| (by simp only [physicalSlotOn_map]) |
|
|
| end HardSphere.Groups |
| end |
| end |
|
|
| end OAI |
|
|
