| import OAI.MeasureTheory.Falconer.Recursion.DescriptorMasks |
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| namespace OAI |
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| namespace PlanarFalconer |
| open Set Metric MeasureTheory |
| open scoped ENNReal |
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| theorem descriptor_mask_failure_bound (Q : DyadicSquare) (hQ : 0 < Q.side) (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) {x : Plane} |
| (hx : ∃ P ∈ occupiedAnchorPaths Q ρ (descriptorAnchor b), x ∈ (Q.descendant P).carrier) : |
| sourceArcLength {dir | descriptorSpatialMask Q ρ m N lam sig J 0 b x dir ≠ 1} ≤ |
| ENNReal.ofReal (((2:ℝ)^N)^(-(lam+sig))) := by |
| obtain ⟨P,hP,hx⟩ := hx |
| simp only [descriptorSpatialMask,spatialAnchorMask_eq Q hQ ρ _ _ _ _ hP hx] |
| apply (circleTestMask_failure_le _ (measurable_descriptorGoodSet _ _ _ _ _ _ _ _ _ _) _ _).trans |
| exact descriptorGoodSet_exceptional Q ρ m N lam sig J ha hb |
| ((mem_dyadicPaths _ P).mp (Finset.mem_filter.mp hP).1) |
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| theorem listMask_failure_subset (Q : DyadicSquare) (ρ : Measure Plane) |
| (m : ℝ → ℝ) (N : ℕ) (lam sig : ℝ) (J i : ℕ) (S : Finset PolicyDescriptor) (x : Plane) : |
| {dir | descriptorListMask Q ρ m N lam sig J i S x dir ≠ 1} ⊆ |
| ⋃ b ∈ S, {dir | descriptorSpatialMask Q ρ m N lam sig J i b x dir ≠ 1} := by |
| intro dir hdir |
| by_contra hn |
| have hh : ∀ b ∈ S, descriptorSpatialMask Q ρ m N lam sig J i b x dir = 1 := by |
| intro b hb |
| by_contra h |
| exact hn (mem_iUnion.mpr ⟨b,mem_iUnion.mpr ⟨hb,h⟩⟩) |
| exact hdir (by simp only [descriptorListMask,Finset.prod_eq_one hh]) |
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| theorem ae_listMask_failure_bound (Q : DyadicSquare) (hQ : 0 < Q.side) (ρ : Measure Plane) |
| (hcarry : ρ Q.carrierᶜ = 0) (m : ℝ → ℝ) (N : ℕ) (lam sig : ℝ) (J : ℕ) |
| (ha : SourceAngularBounds Q ρ m N lam sig) (S : Finset PolicyDescriptor) |
| (hS : ∀ b ∈ S, ValidDescriptor N b) : |
| ∀ᵐ x ∂ρ, sourceArcLength {dir | descriptorListMask Q ρ m N lam sig J 0 S x dir ≠ 1} ≤ |
| (S.card:ENNReal)*ENNReal.ofReal (((2:ℝ)^N)^(-(lam+sig))) := by |
| have hx : ∀ᵐ x ∂ρ, ∀ n : ℕ, ∃ P ∈ occupiedAnchorPaths Q ρ n, x ∈ (Q.descendant P).carrier := |
| ae_all_iff.mpr (fun n => ae_occupied_anchor Q ρ hcarry n) |
| filter_upwards [hx] with x hx |
| apply (measure_mono (listMask_failure_subset Q ρ m N lam sig J 0 S x)).trans |
| apply (measure_biUnion_finset_le _ _).trans |
| calc |
| _ ≤ ∑ _b ∈ S, ENNReal.ofReal (((2:ℝ)^N)^(-(lam+sig))) := by |
| apply Finset.sum_le_sum |
| intro b hb |
| exact descriptor_mask_failure_bound Q hQ ρ m N lam sig J ha (hS b hb) (hx _) |
| _ = _ := by simp |
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| end PlanarFalconer |
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| end OAI |
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