| import Mathlib | |
| import OAI.Analysis.HardSphere.Allocation.CoarsePhysical | |
| namespace OAI | |
| section | |
| noncomputable section | |
| open MeasureTheory ProbabilityTheory Set | |
| open scoped ENNReal NNReal Topology | |
| namespace HardSphere.Groups | |
| open Wiener Bridge Sampling Comparison GaussianPacking PathCutoff CellOrbital | |
| attribute [local instance] Classical.propDecidable | |
| lemma label_endpoints_ae {N : ℕ} {L : Length} (T : ℝ) | |
| (μ : Measure (BridgeEndpoints N L)) : | |
| ∀ᵐ z ∂μ.prod (Measure.pi fun _ : Fin N ↦ Bridge.law), | |
| labelBridgePath N L T z 0=z.1.2 ∧ | |
| labelBridgePath N L T z 1=bridgeEnd N L T z.1 := by | |
| apply (Measure.ae_prod_iff_ae_ae | |
| ((measurableSet_eq_fun ((continuous_eval_const (0:Time)).measurable.comp | |
| (labelBridgePath_measurable N L T)) measurable_fst.snd).inter | |
| (measurableSet_eq_fun ((continuous_eval_const (1:Time)).measurable.comp | |
| (labelBridgePath_measurable N L T)) | |
| ((bridgeEnd_continuous N L T).measurable.comp measurable_fst)))).mpr | |
| exact ae_of_all _ (exposed_endpoints T) | |
| lemma physicalDatum_branch_of {N K M G : ℕ} {L : Length} | |
| (g : Fin G) (b : Fin N→Bool) (a : GridAllocation N K (G*M)) | |
| (m : ℕ) (r R C S s₀ T : ℝ) (u : Time) | |
| (z : BridgeEndpoints N L × (Fin N→Bridge.Noise)) | |
| (h0 : labelBridgePath N L T z 0=z.1.2) | |
| (h1 : labelBridgePath N L T z 1=bridgeEnd N L T z.1) | |
| (hb : z∈flagEvent (eligibilityFlag (K:=K) m r R C S s₀ T u) b) : | |
| physicalDatum g m r R C S s₀ u (labelBridgePath N L T z,a)= | |
| physicalBranchDatum g b a T (blockExpose (fun i ↦ i∈groupUnion g a) z) := by | |
| have hf : (fun i ↦ decide (∃v : Fin 3→Fin K, | |
| Eligible m r R C S s₀ u v i (labelBridgePath N L T z)))=b := by | |
| funext i | |
| have hi := hb i | |
| simpa only [eligibilityFlag,ownEligible,Eligible,h0,h1,ownPath_eq] using hi | |
| simp only [physicalDatum,physicalBranchDatum,h0,h1,hf,blockExpose] | |
| refine Prod.ext rfl (Prod.ext rfl (Prod.ext rfl (Prod.ext rfl ?_))) | |
| funext i | |
| by_cases hi : i∈groupUnion g a | |
| · simp only [ite_eq_left hi,dite_eq_left hi] | |
| · simp only [ite_eq_right hi,dite_eq_right hi] | |
| rw [labelBridgePath_lift] | |
| rfl | |
| theorem actual_physical_branch_joint {N K M G : ℕ} {L : Length} | |
| (hL : L.val=(K:ℝ)) (hK : 1≤K) (h50 : 50<L.val) | |
| (μ : Measure (BridgeEndpoints N L)) [SFinite μ] | |
| (m : ℕ) (r R C S s₀ t : ℝ) (u : Time) | |
| (g : Fin G) (b : Fin N→Bool) (a : GridAllocation N K (G*M)) : | |
| let p := fun i ↦ i∈groupUnion g a | |
| let B := labelInteraction N L r (2*t)∩ | |
| flagEvent (eligibilityFlag (K:=K) m r R C S s₀ (2*t) u) b | |
| let ν := exposedLaw (allocationBase μ (gridAllocationKernel hL (G*M)) b a) | |
| (fun _ : Fin N ↦ Bridge.law) p B | |
| (ν.compProd (fiberNormalize (Kernel.const _ (Measure.pi fun _ : {i // p i} ↦ Bridge.law)) | |
| (blockEvent p B))).map (fun z ↦ | |
| (physicalDatum g m r R C S s₀ u (labelBridgePath N L (2*t) (blockGlue p z),a), | |
| selectedPhysicalPath (L:=L) (2*t) p z)) = | |
| (ν.map (physicalBranchDatum g b a (2*t))).compProd | |
| (datumPhysicalReference hL g a m r R C S t u) := by | |
| dsimp only | |
| let p := fun i ↦ i∈groupUnion g a | |
| let B := labelInteraction N L r (2*t)∩ | |
| flagEvent (eligibilityFlag (K:=K) m r R C S s₀ (2*t) u) b | |
| let μ₀ := allocationBase μ (gridAllocationKernel hL (G*M)) b a | |
| let ν := exposedLaw μ₀ (fun _ : Fin N ↦ Bridge.law) p B | |
| let κ := fiberNormalize (Kernel.const _ (Measure.pi fun _ : {i // p i} ↦ Bridge.law)) (blockEvent p B) | |
| have hB : MeasurableSet B := (labelInteraction_measurable N L r (2*t)).inter | |
| (flagEvent_measurable _ (fun i ↦ eligibilityFlag_measurable m r R C S s₀ (2*t) u i) b) | |
| have : SFinite ν := by dsimp [ν,exposedLaw]; infer_instance | |
| have hr := exposed_block_reconstruct μ₀ (fun _ : Fin N ↦ Bridge.law) p hB | |
| have he : ∀ᵐ z ∂((μ₀.prod (Measure.pi fun _ : Fin N ↦ Bridge.law)).restrict B), | |
| physicalDatum g m r R C S s₀ u (labelBridgePath N L (2*t) z,a)= | |
| physicalBranchDatum g b a (2*t) (blockExpose p z) := by | |
| filter_upwards [ae_restrict_of_ae (label_endpoints_ae (2*t) μ₀),ae_restrict_mem hB] with z hz hzB | |
| exact physicalDatum_branch_of g b a m r R C S s₀ (2*t) u z hz.1 hz.2 hzB.2 | |
| have he' : ∀ᵐ z ∂ν.compProd κ, | |
| physicalDatum g m r R C S s₀ u (labelBridgePath N L (2*t) (blockGlue p z),a)= | |
| physicalBranchDatum g b a (2*t) z.1 := by | |
| rw [←hr] at he | |
| have ht := ae_of_ae_map (blockGlue_measurable p).aemeasurable he | |
| have ht' := ht.mono (fun z hz ↦ show | |
| physicalDatum g m r R C S s₀ u (labelBridgePath N L (2*t) (blockGlue p z),a)= | |
| physicalBranchDatum g b a (2*t) z.1 from by | |
| have hzg := congrFun (blockExpose_glue (E:=BridgeEndpoints N L) | |
| (X:=fun _ : Fin N ↦ Bridge.Noise) p) z | |
| simpa only [Function.comp_apply] using hz.trans (congrArg (physicalBranchDatum g b a (2*t)) hzg)) | |
| dsimp only [ν,κ] | |
| convert ht' using 1 | |
| congr 20 | |
| all_goals try exact Subsingleton.elim _ _ | |
| trans (ν.compProd κ).map (fun z ↦ | |
| (physicalBranchDatum g b a (2*t) z.1,selectedPhysicalPath (L:=L) (2*t) p z)) | |
| · apply Measure.map_congr | |
| filter_upwards [he'] with z hz | |
| exact Prod.ext hz rfl | |
| · exact coarse_joint_reference ν κ _ _ (physicalBranchDatum_measurable g b a (2*t)) | |
| _ (selectedPhysicalPath_measurable (2*t) p) | |
| (actual_allocated_physical_coarse_fiber hL hK h50 μ m r R C S s₀ t u g b a) | |
| 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 | |
| attribute [local instance] Classical.propDecidable | |
| lemma individualReference_probability_of_nonzero (t : ℝ) (y z : Point) | |
| (E : Set C(Time,Point)) (h : individualReference t y z E≠0) : | |
| IsProbabilityMeasure (individualReference t y z E) := by | |
| have : IsProbabilityMeasure (law.map (leg (2*t) y z)) := | |
| (Measure.isProbabilityMeasure_map_iff | |
| (leg_continuous (2*t) y z).measurable.aemeasurable).mpr inferInstance | |
| apply conditioned_probability | |
| intro hz | |
| apply h | |
| simp only [individualReference,Measure.restrict_eq_zero.mpr hz,smul_zero] | |
| lemma datum_group_probability_factors {N K M G : ℕ} {L : Length} (hL : L.val=(K:ℝ)) | |
| (g : Fin G) (a : GridAllocation N K (G*M)) | |
| (m : ℕ) (r R C S t : ℝ) (u : Time) (d : PhysicalDatum N K M G L) | |
| (hmass : datumGroupReference hL g a m r R C S t u d univ=1) : | |
| ∀i : {i // i∈groupUnion g a}, | |
| IsProbabilityMeasure (datumIndividualReference hL g m r R C S t u i d) := by | |
| let μ := fun i : {i // i∈groupUnion g a} ↦ datumIndividualReference hL g m r R C S t u i d | |
| have : ∀i,IsFiniteMeasure (μ i) := fun _ ↦ inferInstance | |
| have hcond : conditionedMeasure (Measure.pi μ) (physicalMutual (L:=L) r) univ=1 := by | |
| simpa only [μ,datumGroupReference_apply,datumIndividualReference_apply] using hmass | |
| have hprod : (Measure.pi μ) univ≠0 := by | |
| intro hz | |
| have he : (Measure.pi μ) (physicalMutual (L:=L) r)=0 := measure_mono_null (subset_univ _) hz | |
| simp only [conditionedMeasure,Measure.restrict_eq_zero.mpr he,smul_zero,Measure.coe_zero,Pi.zero_apply] at hcond | |
| exact zero_ne_one hcond | |
| intro i | |
| rw [datumIndividualReference_apply] | |
| apply individualReference_probability_of_nonzero | |
| intro hz | |
| have hi : μ i=0 := by simpa only [μ,datumIndividualReference_apply] using hz | |
| apply hprod | |
| rw [Measure.pi_univ] | |
| exact Finset.prod_eq_zero (Finset.mem_univ i) (by rw [hi]; rfl) | |
| lemma exposed_fiber_mass {E I : Type*} [MeasurableSpace E] [Fintype I] | |
| {X : I→Type*} [∀i,MeasurableSpace (X i)] | |
| (μ : Measure E) [SFinite μ] (ν : ∀i,Measure (X i)) [∀i,IsProbabilityMeasure (ν i)] | |
| (p : I→Prop) [DecidablePred p] {B : Set (E×(∀i,X i))} (hB : MeasurableSet B) : | |
| ∀ᵐw ∂exposedLaw μ ν p B, | |
| fiberNormalize (Kernel.const _ (Measure.pi fun i : {i // p i} ↦ ν i)) | |
| (blockGlue p ⁻¹' B) w univ=1 := by | |
| filter_upwards [exposed_block_positive μ ν p hB] with w hw | |
| exact fiberNormalize_mass (Kernel.const _ (Measure.pi fun i : {i // p i} ↦ ν i)) | |
| (hB.preimage (blockGlue_measurable p)) hw | |
| theorem actual_branch_normalized {N K M G : ℕ} {L : Length} | |
| (hL : L.val=(K:ℝ)) (hK : 1≤K) (h50 : 50<L.val) | |
| (μ : Measure (BridgeEndpoints N L)) [SFinite μ] | |
| (m : ℕ) (r R C S s₀ t : ℝ) (u : Time) | |
| (g : Fin G) (b : Fin N→Bool) (a : GridAllocation N K (G*M)) : | |
| let p := fun i ↦ i∈groupUnion g a | |
| ∀ᵐw ∂exposedLaw (allocationBase μ (gridAllocationKernel hL (G*M)) b a) | |
| (fun _ : Fin N ↦ Bridge.law) p | |
| (labelInteraction N L r (2*t)∩flagEvent (eligibilityFlag (K:=K) m r R C S s₀ (2*t) u) b), | |
| IsProbabilityMeasure (datumGroupReference hL g a m r R C S t u | |
| (physicalBranchDatum g b a (2*t) w)) ∧ | |
| ∀i : {i // i∈groupUnion g a},IsProbabilityMeasure | |
| (datumIndividualReference hL g m r R C S t u i (physicalBranchDatum g b a (2*t) w)) := by | |
| dsimp only | |
| let p := fun i ↦ i∈groupUnion g a | |
| let B := labelInteraction N L r (2*t)∩ | |
| flagEvent (eligibilityFlag (K:=K) m r R C S s₀ (2*t) u) b | |
| have hB : MeasurableSet B := (labelInteraction_measurable N L r (2*t)).inter | |
| (flagEvent_measurable _ (fun i ↦ eligibilityFlag_measurable m r R C S s₀ (2*t) u i) b) | |
| have hpos := exposed_fiber_mass (allocationBase μ (gridAllocationKernel hL (G*M)) b a) | |
| (fun _ : Fin N ↦ Bridge.law) p hB | |
| filter_upwards [actual_allocated_physical_coarse_fiber hL hK h50 μ m r R C S s₀ t u g b a, | |
| hpos] with w hw hp | |
| change (_ : ℝ≥0∞)=1 at hp | |
| have hh := congrArg (fun ν ↦ ν univ) hw | |
| rw [Measure.map_apply (by exact (selectedPhysicalPath_measurable (L:=L) (2*t) (fun i ↦ i∈groupUnion g a)).comp (measurable_const.prodMk measurable_id)) MeasurableSet.univ,preimage_univ] at hh | |
| change (fiberNormalize _ (blockGlue p ⁻¹' B) w) univ = _ at hh | |
| have hh' : (datumPhysicalReference hL g a m r R C S t u | |
| (physicalBranchDatum g b a (2*t) w)) univ=1 := | |
| hh.symm.trans (by | |
| convert hp using 1 | |
| congr | |
| exact Subsingleton.elim _ _) | |
| have hmass : datumGroupReference hL g a m r R C S t u (physicalBranchDatum g b a (2*t) w) univ=1 := by | |
| rw [datumPhysicalReference,Kernel.map_apply _ projectGroup_measurable, | |
| Measure.map_apply projectGroup_measurable MeasurableSet.univ,preimage_univ] at hh' | |
| exact hh' | |
| exact ⟨⟨hmass⟩,datum_group_probability_factors hL g a m r R C S t u _ hmass⟩ | |
| end HardSphere.Groups | |
| end | |
| end | |
| end OAI | |