| import OAI.MeasureTheory.Falconer.Recursion.DescriptorMasks |
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| namespace OAI |
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| namespace PlanarFalconer |
| open Set Metric MeasureTheory |
| open scoped ContDiff Topology |
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| 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 |
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| 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)] |
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| 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 |
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| 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 |
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| 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) |
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| end PlanarFalconer |
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| end OAI |
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