| 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 | |