File size: 3,082 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 | import OAI.MeasureTheory.Falconer.Recursion.DescriptorMasks
namespace OAI
namespace PlanarFalconer
open Set Metric MeasureTheory
open scoped ContDiff Topology
theorem descriptorGoodSet_antipodal (Q : DyadicSquare) (ρ : Measure Plane)
(m : ℝ → ℝ) (N : ℕ) (lam sig : ℝ) (J i : ℕ) (P : CellPath) (b : PolicyDescriptor) (dir : UnitCircle) :
circleAntipode dir ∈ descriptorGoodSet Q ρ m N lam sig J i P b ↔
dir ∈ descriptorGoodSet Q ρ m N lam sig J i P b := by
cases b with
| parallel g p =>
change (parallelTestCount Q ρ g p P _ (-(dir:Plane)):ℝ) ≤ _ ↔ _
rw [parallelTestCount_neg]
rfl
| projection p u =>
change (projectionTestCount Q ρ p u P _ (-(dir:Plane)):ℝ) ≤ _ ↔ _
rw [projectionTestCount_neg]
rfl
theorem spatialAnchorMask_antipodal (Q : DyadicSquare) (ρ : Measure Plane) (n : ℕ)
(G : CellPath → Set UnitCircle) (d : ℝ) (hd : 0 < d)
(hG : ∀ P : CellPath, ∀ dir : UnitCircle, circleAntipode dir ∈ G P ↔ dir ∈ G P)
(x : Plane) (dir : UnitCircle) :
spatialAnchorMask Q ρ n G d hd x (circleAntipode dir) = spatialAnchorMask Q ρ n G d hd x dir := by
classical
unfold spatialAnchorMask
apply Finset.sum_congr rfl
intro P _
rw [circleTestMask_antipodal (hG P)]
theorem descriptorSpatialMask_antipodal (Q : DyadicSquare) (ρ : Measure Plane)
(m : ℝ → ℝ) (N : ℕ) (lam sig : ℝ) (J i : ℕ) (b : PolicyDescriptor) (x : Plane) (dir : UnitCircle) :
descriptorSpatialMask Q ρ m N lam sig J i b x (circleAntipode dir) =
descriptorSpatialMask Q ρ m N lam sig J i b x dir :=
spatialAnchorMask_antipodal Q ρ _ _ _ _ (fun P dir => descriptorGoodSet_antipodal Q ρ m N lam sig J i P b dir) x dir
theorem descriptorListMask_antipodal (Q : DyadicSquare) (ρ : Measure Plane)
(m : ℝ → ℝ) (N : ℕ) (lam sig : ℝ) (J i : ℕ) (S : Finset PolicyDescriptor) (x : Plane) (dir : UnitCircle) :
descriptorListMask Q ρ m N lam sig J i S x (circleAntipode dir) =
descriptorListMask Q ρ m N lam sig J i S x dir := by
unfold descriptorListMask
apply Finset.prod_congr rfl
intro b _
exact descriptorSpatialMask_antipodal Q ρ m N lam sig J i b x dir
theorem spatialAnchorMask_derivative_bound (Q : DyadicSquare) (hQ : 0 < Q.side) (ρ : Measure Plane) (n : ℕ)
(G : CellPath → Set UnitCircle) (d : ℝ) (hd : 0 < d) (x : Plane) (j : ℕ) (θ : ℝ) :
‖iteratedDeriv j (fun φ => spatialAnchorMask Q ρ n G d hd x (sourceUnitDirection φ)) θ‖ ≤
((100:ℝ)^j*angleDerivativeConstant j)*(d⁻¹)^j := by
by_cases hx : ∃ P ∈ occupiedAnchorPaths Q ρ n, x ∈ (Q.descendant P).carrier
· obtain ⟨P,hP,hx⟩ := hx
simpa only [spatialAnchorMask_eq Q hQ ρ n G d hd hP hx,circleTestMask_comp] using
sourceTestMask_derivatives (G P) d hd j θ
· push Not at hx
simp only [spatialAnchorMask_zero Q ρ n G d hd hx,iteratedDeriv_fun_const_zero,norm_zero]
exact mul_nonneg (mul_nonneg (by positivity) (angleDerivativeConstant_nonneg j)) (by positivity)
end PlanarFalconer
end OAI
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