| import OAI.MeasureTheory.Falconer.Estimates.SpatialAnchorMasks |
| import OAI.MeasureTheory.Falconer.Recursion.PolicyDescriptors |
|
|
| namespace OAI |
|
|
| namespace PlanarFalconer |
| open Set Metric MeasureTheory |
| open scoped ContDiff Topology |
|
|
| def ValidDescriptor (N : ℕ) : PolicyDescriptor → Prop |
| | .parallel g p => g ≤ p ∧ p ≤ N |
| | .projection p u => p ≤ u ∧ u ≤ N |
|
|
| noncomputable def descriptorGoodSet (Q : DyadicSquare) (ρ : Measure Plane) |
| (m : ℝ → ℝ) (N : ℕ) (lam sig : ℝ) (J i : ℕ) (P : CellPath) : PolicyDescriptor → Set UnitCircle |
| | .parallel g p => parallelGoodSet Q ρ g p P (sourceAngularWidth N sig J i p) |
| (((2:ℝ)^N)^(height (fun n : ℕ => m n-(n:ℝ)/(N:ℝ)) g p+2000*(lam+sig))) |
| | .projection p u => projectionGoodSet Q ρ p u P (sourceAngularWidth N sig J i u) |
| (((2:ℝ)^N)^(m u-m p+height (fun n : ℕ => m n-(n:ℝ)/(N:ℝ)) p u+2000*(lam+sig))) |
|
|
| noncomputable def descriptorSpatialMask (Q : DyadicSquare) (ρ : Measure Plane) |
| (m : ℝ → ℝ) (N : ℕ) (lam sig : ℝ) (J i : ℕ) (b : PolicyDescriptor) : Plane → UnitCircle → ℝ := |
| spatialAnchorMask Q ρ (descriptorAnchor b) (fun P => descriptorGoodSet Q ρ m N lam sig J i P b) |
| (sourceAngularRadius N sig (descriptorLength b)) (sourceAngularRadius_pos _ _ _) |
|
|
| noncomputable def descriptorListMask (Q : DyadicSquare) (ρ : Measure Plane) |
| (m : ℝ → ℝ) (N : ℕ) (lam sig : ℝ) (J i : ℕ) (S : Finset PolicyDescriptor) |
| (x : Plane) (dir : UnitCircle) : ℝ := |
| ∏ b ∈ S, descriptorSpatialMask Q ρ m N lam sig J i b x dir |
|
|
| theorem measurable_descriptorGoodSet (Q : DyadicSquare) (ρ : Measure Plane) |
| (m : ℝ → ℝ) (N : ℕ) (lam sig : ℝ) (J i : ℕ) (P : CellPath) (b : PolicyDescriptor) : |
| MeasurableSet (descriptorGoodSet Q ρ m N lam sig J i P b) := by |
| cases b with |
| | parallel g p => exact measurable_parallelGoodSet _ _ _ _ _ _ _ |
| | projection p u => exact measurable_projectionGoodSet _ _ _ _ _ _ _ |
|
|
| theorem descriptorSpatialMask_bounds (Q : DyadicSquare) (hQ : 0 < Q.side) (ρ : Measure Plane) |
| (m : ℝ → ℝ) (N : ℕ) (lam sig : ℝ) (J i : ℕ) (b : PolicyDescriptor) (x : Plane) (dir : UnitCircle) : |
| 0 ≤ descriptorSpatialMask Q ρ m N lam sig J i b x dir ∧ |
| descriptorSpatialMask Q ρ m N lam sig J i b x dir ≤ 1 := spatialAnchorMask_bounds Q hQ _ _ _ _ _ _ _ |
|
|
| @[fun_prop] theorem measurable_descriptorSpatialMask (Q : DyadicSquare) (ρ : Measure Plane) |
| (m : ℝ → ℝ) (N : ℕ) (lam sig : ℝ) (J i : ℕ) (b : PolicyDescriptor) : |
| Measurable (fun z : Plane × UnitCircle => descriptorSpatialMask Q ρ m N lam sig J i b z.1 z.2) := |
| measurable_spatialAnchorMask _ _ _ _ _ _ |
|
|
| @[fun_prop] theorem measurable_descriptorListMask (Q : DyadicSquare) (ρ : Measure Plane) |
| (m : ℝ → ℝ) (N : ℕ) (lam sig : ℝ) (J i : ℕ) (S : Finset PolicyDescriptor) : |
| Measurable (fun z : Plane × UnitCircle => descriptorListMask Q ρ m N lam sig J i S z.1 z.2) := by |
| apply Finset.measurable_prod |
| intro b _ |
| exact measurable_descriptorSpatialMask _ _ _ _ _ _ _ _ _ |
|
|
| theorem descriptorListMask_bounds (Q : DyadicSquare) (hQ : 0 < Q.side) (ρ : Measure Plane) |
| (m : ℝ → ℝ) (N : ℕ) (lam sig : ℝ) (J i : ℕ) (S : Finset PolicyDescriptor) (x : Plane) (dir : UnitCircle) : |
| 0 ≤ descriptorListMask Q ρ m N lam sig J i S x dir ∧ |
| descriptorListMask Q ρ m N lam sig J i S x dir ≤ 1 := by |
| constructor |
| · exact Finset.prod_nonneg (fun b _ => (descriptorSpatialMask_bounds Q hQ ρ m N lam sig J i b x dir).1) |
| · exact Finset.prod_le_one₀ (fun b _ => (descriptorSpatialMask_bounds Q hQ ρ m N lam sig J i b x dir).1) |
| (fun b _ => (descriptorSpatialMask_bounds Q hQ ρ m N lam sig J i b x dir).2) |
|
|
| theorem descriptorGoodSet_exceptional (Q : DyadicSquare) (ρ : Measure Plane) |
| (m : ℝ → ℝ) (N : ℕ) (lam sig : ℝ) (J : ℕ) |
| (ha : SourceAngularBounds Q ρ m N lam sig) {b : PolicyDescriptor} (hb : ValidDescriptor N b) |
| {P : CellPath} (hP : P.length = descriptorAnchor b) : |
| sourceArcLength (descriptorGoodSet Q ρ m N lam sig J 0 P b)ᶜ ≤ |
| ENNReal.ofReal (((2:ℝ)^N)^(-(lam+sig))) := by |
| cases b with |
| | parallel g p => simpa only [descriptorGoodSet,sourceAngularWidth_zero] using ha.parallel g p hb.1 hb.2 P hP |
| | projection p u => simpa only [descriptorGoodSet,sourceAngularWidth_zero] using ha.projection p u hb.1 hb.2 P hP |
|
|
| theorem policyDescriptor_valid (A : ℕ → ℝ) {N q a t : ℕ} (hq : 0 < q) (ht : t ≤ N) |
| {b : PolicyDescriptor} (hb : b ∈ policyDescriptorList A N q a t) : ValidDescriptor N b := by |
| obtain ⟨s,u,_,hut,_,_,hpos,_,hbound⟩ := policyDescriptor_origin A hq hb |
| cases b <;> simp only [descriptorLength,descriptorAnchor,ValidDescriptor] at hpos hbound ⊢ <;> omega |
|
|
| end PlanarFalconer |
|
|
| end OAI |
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|
