| import Mathlib |
| import OAI.Analysis.HardSphere.Allocation.PhysicalMixedObservation |
|
|
| namespace OAI |
|
|
| section |
|
|
| noncomputable section |
| open MeasureTheory ProbabilityTheory Set |
| open scoped ENNReal |
| namespace HardSphere.Groups |
| open Wiener Bridge Sampling Comparison GaussianPacking PathCutoff CellOrbital |
|
|
| lemma point_chart_preserving (L : Length) (c : Point) : |
| MeasurePreserving (projectPoint L) |
| (volume.restrict (univ.pi (fun j ↦ Icc (c j-L.val/2) (c j+L.val/2)))) volume := by |
| have hp (j : Fin 3) : MeasurePreserving (fun value : ℝ ↦ (value : AddCircle L.val)) |
| (volume.restrict (Icc (c j-L.val/2) (c j+L.val/2))) volume := by |
| have hj := AddCircle.measurePreserving_mk L.val (c j-L.val/2) |
| rw [restrict_Ioc_eq_restrict_Icc,show c j-L.val/2+L.val=c j+L.val/2 by ring] at hj |
| exact hj |
| change MeasurePreserving _ ((Measure.pi (fun _ : Fin 3 ↦ (volume : Measure ℝ))).restrict _) |
| (Measure.pi (fun _ : Fin 3 ↦ (volume : Measure (AddCircle L.val)))) |
| simpa only [projectPoint,Measure.restrict_pi_pi] using measurePreserving_pi |
| (fun j : Fin 3 ↦ volume.restrict (Icc (c j-L.val/2) (c j+L.val/2))) |
| (fun _ : Fin 3 ↦ (volume : Measure (AddCircle L.val))) hp |
|
|
| lemma cellPathReference_midpoint_le {K : ℕ} [NeZero K] {L : Length} (hL : 1≤L.val) |
| (v : Fin 3→Fin K) : |
| (cellPathReference v).map (fun p ↦ projectPoint L (p midpointTime)) ≤ |
| volume.restrict (centeredCell L (gridCenter L v)) := by |
| let f := fun p : C(Time,Point) ↦ projectPoint L (p midpointTime) |
| have hf : Measurable f := (projectPoint_continuous L).measurable.comp |
| (continuous_eval_const midpointTime).measurable |
| have hs : ∀ᵐx ∂(cellPathReference v).map f,x∈centeredCell L (gridCenter L v) := by |
| exact (ae_map_iff hf.aemeasurable (centeredCell_measurable L (gridCenter L v))).mpr |
| (cellPathReference_support hL v) |
| rw [←Measure.restrict_eq_self_of_ae_mem hs] |
| apply Measure.restrict_mono_measure |
| rw [cellPathReference,Measure.map_map hf pointAsPath_measurable] |
| change (targetUniform (unitTarget (anchorPoint v))).map (projectPoint L)≤volume |
| rw [targetUniform,unitTarget_volume,inv_one,one_smul] |
| have hsub : unitTarget (anchorPoint v) ⊆ |
| univ.pi (fun j ↦ Icc (anchorPoint v j-L.val/2) (anchorPoint v j+L.val/2)) := by |
| intro x hx j hj |
| have h := hx j hj |
| constructor <;> linarith [h.1,h.2] |
| exact (Measure.map_mono (Measure.restrict_mono_set volume hsub) (projectPoint_continuous L).measurable).trans_eq |
| (point_chart_preserving L (anchorPoint v)).map_eq |
|
|
| end HardSphere.Groups |
| end |
| end |
|
|
| end OAI |
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|
