| import Mathlib |
| import OAI.Analysis.HardSphere.Conditioning.PhysicalFullReference |
|
|
| 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 |
|
|
| theorem physicalBranch_primary_midpoint {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) (v : Fin 3→Fin K) : |
| ((physicalBranchLaw hL μ g b 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) v)).map |
| (physicalFullMidpoint (Function.Embedding.subtype (fun i ↦ i∈groupUnion g a))) = |
| (((rawPhysicalBranch hL μ g b a m r R C S s₀ (2*t) u).map |
| (fun z ↦ labelBridgePath N L (2*t) z midpointTime)).prod |
| ((cellPathReference v).map (fun e ↦ projectPoint L (e midpointTime)))).map |
| (fun z i ↦ i.elim z.2 z.1) := by |
| have H := congrArg (fun ν ↦ ν.map Prod.snd) |
| (physicalBranch_full_midpoint_raw hL hK h50 μ g b a m r R C S s₀ t u v) |
| rw [Measure.map_map measurable_snd |
| (measurable_fst.prodMk (physicalFullMidpoint_measurable _)), |
| Measure.map_map measurable_snd (by |
| apply Measurable.prodMk |
| · exact (physicalDatum_measurable g m r R C S s₀ u).comp |
| (((labelBridgePath_measurable N L (2*t)).comp measurable_fst).prodMk measurable_const) |
| · apply Measurable.of_eval |
| intro i |
| cases i with |
| | none => |
| exact (projectPoint_continuous L).measurable.comp |
| ((continuous_eval_const midpointTime).measurable.comp measurable_snd) |
| | some i => |
| exact (measurable_pi_apply i).comp |
| ((continuous_eval_const midpointTime).measurable.comp |
| ((labelBridgePath_measurable N L (2*t)).comp measurable_fst)))] at H |
| have hf : Measurable (fun z ↦ labelBridgePath N L (2*t) z midpointTime) := |
| (continuous_eval_const midpointTime).measurable.comp (labelBridgePath_measurable N L (2*t)) |
| have he : Measurable (fun e : C(Time,Point) ↦ projectPoint L (e midpointTime)) := |
| (projectPoint_continuous L).measurable.comp (continuous_eval_const midpointTime).measurable |
| rw [Measure.map_prod_map _ _ hf he,Measure.map_map (by |
| apply Measurable.of_eval |
| intro i |
| cases i with |
| | none => exact measurable_snd |
| | some i => exact (measurable_pi_apply i).comp measurable_fst) (hf.prodMap he)] |
| exact H |
|
|
| end HardSphere.Groups |
| end |
| end |
|
|
| end OAI |
|
|
