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67aee1d | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 | 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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