File size: 3,944 Bytes
a2ee19a | 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 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 | import OAI.Probability.RadialTransition.MultisetEnergy
namespace OAI
noncomputable section
open MeasureTheory Set Filter Function
open scoped ContDiff NNReal BigOperators ENNReal
namespace RadialTransition
lemma measurable_cellOf : Measurable cellOf := by
apply Measurable.of_eval
intro k
exact Int.measurable_floor.comp (measurable_const.mul (PiLp.continuous_apply 2 (fun _ : Fin 3 => ℝ) k).measurable)
def gridRegion (k : Cell) : Set Position := {x | cellOf x = k}
lemma measurableSet_gridRegion (k : Cell) : MeasurableSet (gridRegion k) :=
measurable_cellOf (measurableSet_singleton k)
abbrev CroppedCell (A : Set Position) (k : Cell) := {x : Position // x ∈ A ∩ gridRegion k}
def croppedCellMeasure (A : Set Position) (k : Cell) : Measure (CroppedCell A k) :=
volume.comap Subtype.val
def gridOrigin (k : Cell) : Position := WithLp.toLp 2 (fun i => (k i : ℝ)/4)
lemma gridRegion_subset_box (k : Cell) : gridRegion k ⊆ translateSet (gridOrigin k) (cube (1/4)) := by
intro x hx i
have hh : ⌊4*x i⌋ = k i := congrFun hx i
have hl := Int.floor_le (4*x i)
have hu := Int.lt_floor_add_one (4*x i)
rw [hh] at hl hu
change 0 ≤ x i - (k i : ℝ)/4 ∧ x i - (k i : ℝ)/4 ≤ 1/4
constructor <;> linarith
lemma volume_gridRegion_le (k : Cell) : volume (gridRegion k) ≤ ENNReal.ofReal (1/64:ℝ) := by
have hh := measure_mono (gridRegion_subset_box k) (μ := volume)
rw [volume_translateSet _ _ (measurableSet_cube _), volume_cube] at hh
rw [← ENNReal.ofReal_pow (by norm_num : (0:ℝ) ≤ 1/4)] at hh
norm_num at hh ⊢
exact hh
lemma croppedCellMeasure_univ {A : Set Position} (hA : MeasurableSet A) (k : Cell) :
croppedCellMeasure A k univ = volume (A ∩ gridRegion k) := by
rw [croppedCellMeasure, comap_subtype_coe_apply (hA.inter (measurableSet_gridRegion k))]
simp only [Set.image_univ, Subtype.range_coe]
lemma croppedCellMeasure_finite {A : Set Position} (hA : MeasurableSet A) (k : Cell) :
IsFiniteMeasure (croppedCellMeasure A k) := by
constructor
rw [croppedCellMeasure_univ hA k]
exact ((measure_mono inter_subset_right).trans (volume_gridRegion_le k)).trans_lt ENNReal.ofReal_lt_top
lemma croppedCellMeasure_volume {A : Set Position} (hA : MeasurableSet A) (k : Cell) :
((croppedCellMeasure A k) univ).toReal ≤ 1/64 := by
rw [croppedCellMeasure_univ hA k]
have hh := ENNReal.toReal_mono ENNReal.ofReal_ne_top
((measure_mono (inter_subset_right (s := A))).trans (volume_gridRegion_le k))
simpa only [ENNReal.toReal_ofReal (by norm_num : (0:ℝ) ≤ 1/64)] using hh
lemma croppedCellMeasure_map {A : Set Position} (hA : MeasurableSet A) (k : Cell) :
Measure.map Subtype.val (croppedCellMeasure A k) = volume.restrict (A ∩ gridRegion k) :=
map_comap_subtype_coe (hA.inter (measurableSet_gridRegion k)) volume
lemma croppedCell_diameter {A : Set Position} {k : Cell} (x y : CroppedCell A k) :
dist (x : Position) y ≤ 1 :=
same_cell_diameter (x.property.2.trans y.property.2.symm)
lemma croppedCell_partition {I : Type} [Fintype I] (e : I → Cell) (he : Function.Injective e)
{A : Set Position} (hA : MeasurableSet A) (hcover : ∀ x ∈ A, ∃ i, cellOf x = e i) :
volume.restrict A = ∑ i, Measure.map Subtype.val (croppedCellMeasure A (e i)) := by
have hd : Pairwise (Disjoint on (fun i => A ∩ gridRegion (e i))) := by
intro i j hij
apply Set.disjoint_left.mpr
intro x hx hy
exact hij (he (hx.2.symm.trans hy.2))
have hu : (⋃ i, A ∩ gridRegion (e i)) = A := by
ext x
simp only [mem_iUnion, mem_inter_iff, gridRegion, mem_ofPred_eq]
exact ⟨fun ⟨_,hx,_⟩ => hx, fun hx => by obtain ⟨i,hi⟩ := hcover x hx; exact ⟨i,hx,hi⟩⟩
rw [← hu, Measure.restrict_iUnion hd (fun i => hA.inter (measurableSet_gridRegion (e i))), Measure.sum_fintype]
apply Finset.sum_congr rfl
intro i _
rw [hu, croppedCellMeasure_map hA]
end RadialTransition
end
end OAI
|