| import Mathlib |
| import OAI.Analysis.HardSphere.Allocation.PhysicalBranchLaw |
|
|
| namespace OAI |
|
|
| section |
|
|
| noncomputable section |
| open MeasureTheory ProbabilityTheory Set |
| open scoped ENNReal |
| namespace HardSphere.Groups |
| open Wiener Bridge Sampling Comparison GaussianPacking PathCutoff CellOrbital |
| attribute [local instance] Classical.propDecidable |
|
|
| def rawPhysicalBranch {N K M G : ℕ} {L : Length} (hL : L.val=(K:ℝ)) |
| (μ : Measure (BridgeEndpoints N L)) [SFinite μ] |
| (_group : Fin G) (b : Fin N→Bool) (a : GridAllocation N K (G*M)) |
| (m : ℕ) (r R C S s₀ T : ℝ) (u : Time) := |
| ((allocationBase μ (gridAllocationKernel hL (G*M)) b a).prod |
| (Measure.pi fun _ : Fin N ↦ Bridge.law)).restrict |
| (labelInteraction N L r T∩flagEvent (eligibilityFlag (K:=K) m r R C S s₀ T u) b) |
| instance rawPhysicalBranch_sfinite {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) : |
| SFinite (rawPhysicalBranch hL μ g b a m r R C S s₀ T u) := by |
| unfold rawPhysicalBranch; infer_instance |
| |
| theorem physicalBranch_reference_raw {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] |
| (g : Fin G) (b : Fin N→Bool) (a : GridAllocation N K (G*M)) |
| (m : ℕ) (r R C S s₀ t : ℝ) (u : Time) : |
| (physicalBranchLaw hL μ g b a m r R C S s₀ (2*t) u).compProd |
| (datumPhysicalReference hL g a m r R C S t u)= |
| (rawPhysicalBranch hL μ g b a m r R C S s₀ (2*t) u).map |
| (fun z ↦ (physicalDatum g m r R C S s₀ u (labelBridgePath N L (2*t) z,a), |
| fun i : {i // i∈groupUnion g a} ↦ onePathAt i (labelBridgePath N L (2*t) z))) := by |
| 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 hm : Measurable (fun z : BridgeEndpoints N L×(Fin N→Bridge.Noise) ↦ |
| (physicalDatum g m r R C S s₀ u (labelBridgePath N L (2*t) z,a), |
| fun i : {i // p i} ↦ onePathAt i (labelBridgePath N L (2*t) z))) := |
| ((physicalDatum_measurable g m r R C S s₀ u).comp |
| ((labelBridgePath_measurable N L (2*t)).prodMk measurable_const)).prodMk |
| (Measurable.of_eval fun i ↦ (onePathAt_continuous (i:Fin N)).measurable.comp |
| (labelBridgePath_measurable N L (2*t))) |
| rw [physicalBranchLaw,←actual_physical_branch_joint hL hK h50 μ m r R C S s₀ t u g b a] |
| have : SFinite (exposedLaw (allocationBase μ (gridAllocationKernel hL (G*M)) b a) |
| (fun _ : Fin N ↦ Bridge.law) p B) := by unfold exposedLaw; infer_instance |
| have H := congrArg (fun ν ↦ ν.map (fun z ↦ |
| (physicalDatum g m r R C S s₀ u (labelBridgePath N L (2*t) z,a), |
| fun i : {i // p i} ↦ onePathAt i (labelBridgePath N L (2*t) z)))) |
| (exposed_block_reconstruct (allocationBase μ (gridAllocationKernel hL (G*M)) b a) |
| (fun _ : Fin N ↦ Bridge.law) p hB) |
| rw [Measure.map_map hm (blockGlue_measurable p)] at H |
| dsimp only [Function.comp_def,rawPhysicalBranch,selectedPhysicalPath,p,B] at H ⊢ |
| convert H using 1 |
| congr 20 |
| all_goals try exact Subsingleton.elim _ _ |
| funext z |
| unfold selectedPhysicalPath |
| congr 20 |
| all_goals first | exact Subsingleton.elim _ _ | (funext index; congr 20) |
| |
| def physicalRecoveredPath {N K M G : ℕ} {L : Length} (g : Fin G) |
| (a : GridAllocation N K (G*M)) |
| (z : PhysicalDatum N K M G L×({i // i∈groupUnion g a}→ ConfigurationPath 1 L)) : |
| Fin N→ ConfigurationPath 1 L := |
| fun i ↦ if hi : i∈groupUnion g a then z.2 ⟨i,hi⟩ else z.1.2.2.2.2 i |
| lemma physicalRecoveredPath_measurable {N K M G : ℕ} {L : Length} (g : Fin G) |
| (a : GridAllocation N K (G*M)) : |
| Measurable (physicalRecoveredPath (L:=L) g a) := by |
| apply Measurable.of_eval |
| intro i |
| unfold physicalRecoveredPath |
| by_cases hi : i∈groupUnion g a |
| · simp only [dite_eq_left hi] |
| exact (measurable_pi_apply (⟨i,hi⟩ : {i // i∈groupUnion g a})).comp measurable_snd |
| · simp only [dite_eq_right hi] |
| exact (datumFrozen_measurable i).comp measurable_fst |
| lemma physicalRecoveredPath_actual {N K M G : ℕ} {L : Length} (g : Fin G) |
| (a : GridAllocation N K (G*M)) (m : ℕ) (r R C S s₀ : ℝ) (u : Time) |
| (F : ConfigurationPath N L) : |
| physicalRecoveredPath g a (physicalDatum g m r R C S s₀ u (F,a),fun i ↦ onePathAt i F)= |
| fun i ↦ onePathAt i F := by |
| funext i |
| by_cases hi : i∈groupUnion g a <;> simp [physicalRecoveredPath,physicalDatum,hi] |
| |
| def physicalRecoveredMidpoint {N K M G : ℕ} {L : Length} (g : Fin G) |
| (a : GridAllocation N K (G*M)) |
| (z : PhysicalDatum N K M G L×({i // i∈groupUnion g a}→ ConfigurationPath 1 L)) : Configuration N L := |
| fun i ↦ physicalRecoveredPath g a z i midpointTime 0 |
| lemma physicalRecoveredMidpoint_measurable {N K M G : ℕ} {L : Length} (g : Fin G) |
| (a : GridAllocation N K (G*M)) : Measurable (physicalRecoveredMidpoint (L:=L) g a) := |
| Measurable.of_eval fun i ↦ (measurable_pi_apply 0).comp |
| ((continuous_eval_const midpointTime).measurable.comp |
| ((measurable_pi_apply i).comp (physicalRecoveredPath_measurable g a))) |
| lemma physicalRecoveredMidpoint_actual {N K M G : ℕ} {L : Length} (g : Fin G) |
| (a : GridAllocation N K (G*M)) (m : ℕ) (r R C S s₀ : ℝ) (u : Time) |
| (F : ConfigurationPath N L) : |
| physicalRecoveredMidpoint g a (physicalDatum g m r R C S s₀ u (F,a),fun i ↦ onePathAt i F)= |
| F midpointTime := by |
| unfold physicalRecoveredMidpoint |
| rw [physicalRecoveredPath_actual] |
| rfl |
| |
| theorem physicalBranch_midpoint_raw {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] |
| (g : Fin G) (b : Fin N→Bool) (a : GridAllocation N K (G*M)) |
| (m : ℕ) (r R C S s₀ t : ℝ) (u : Time) : |
| ((physicalBranchLaw hL μ g b a m r R C S s₀ (2*t) u).compProd |
| (datumGroupReference hL g a m r R C S t u)).map |
| (fun z ↦ (z.1,physicalRecoveredMidpoint g a (z.1,projectGroup z.2)))= |
| (rawPhysicalBranch hL μ g b a m r R C S s₀ (2*t) u).map |
| (fun z ↦ (physicalDatum g m r R C S s₀ u (labelBridgePath N L (2*t) z,a), |
| labelBridgePath N L (2*t) z midpointTime)) := by |
| have hm := measurable_fst.prodMk (physicalRecoveredMidpoint_measurable (L:=L) g a) |
| have H := congrArg (fun ν ↦ ν.map (fun z ↦ (z.1,physicalRecoveredMidpoint g a z))) |
| (physicalBranch_reference_raw hL hK h50 μ g b a m r R C S s₀ t u) |
| rw [datumPhysicalReference,Measure.compProd_map projectGroup_measurable, |
| Measure.map_map hm (measurable_id.prodMap projectGroup_measurable)] at H |
| have hr : Measurable (fun z : BridgeEndpoints N L×(Fin N→Bridge.Noise) ↦ |
| (physicalDatum g m r R C S s₀ u (labelBridgePath N L (2*t) z,a), |
| fun i : {i // i∈groupUnion g a} ↦ onePathAt i (labelBridgePath N L (2*t) z))) := |
| ((physicalDatum_measurable g m r R C S s₀ u).comp |
| ((labelBridgePath_measurable N L (2*t)).prodMk measurable_const)).prodMk |
| (Measurable.of_eval fun i ↦ (onePathAt_continuous (i:Fin N)).measurable.comp |
| (labelBridgePath_measurable N L (2*t))) |
| rw [Measure.map_map hm hr] at H |
| simpa only [Function.comp_def,Prod.map,Function.id_def,physicalRecoveredMidpoint_actual] using H |
| |
| end HardSphere.Groups |
| end |
| end |
| |
| end OAI |
| |
