File size: 4,794 Bytes
d4317d7 | 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 | 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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