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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 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 | import OAI.MeasureTheory.Falconer.Estimates.SuffixScales
import OAI.MeasureTheory.Falconer.Graphs.EligibleDyadicGraph
namespace OAI
/-! # Actual descriptor counts supply the local graph witnesses -/
namespace Falconer
open Set MeasureTheory Metric
open scoped ENNReal
attribute [local instance] Classical.propDecidable
variable {d : ℕ}
theorem parallelGoodSet_local_test (Q : DyadicCube d) (hQ : Q.side = 1)
(ρ : Measure (Space d)) {a g p : ℕ} (hag : a ≤ g) (hgp : g ≤ p)
(X P : List (Fin d → Bool)) (hX : X.length = a)
(w : Sphere d) (W T : ℝ) (hT : 0 ≤ T)
(hw : w ∈ Regularization.parallelGoodSet Q ρ g p (X++P) W T) :
DyadicParallelTest (Q.descendant X) ((Q.descendant X).occupiedPaths ρ (p-a))
(g-a) P w W ⌈T⌉₊ := by
have hscale : (Q.descendant X).side/(2:ℝ)^(g-a) = 1/(2:ℝ)^g := by
rw [Q.descendant_side,hX,hQ,div_div,← pow_add,Nat.add_sub_of_le hag]
let S := ((Q.descendant X).occupiedPaths ρ (p-a)).filter fun z =>
z.take (g-a) = P.take (g-a) ∧
‖((Q.descendant X).descendant z).center-((Q.descendant X).descendant P).center‖ ≤
Real.sqrt d*((Q.descendant X).side/(2:ℝ)^(g-a)) ∧
‖perpendicularPart (w:Space d)
(((Q.descendant X).descendant z).center-((Q.descendant X).descendant P).center)‖ ≤ W
have hinj : Function.Injective (fun z : List (Fin d → Bool) => X++z) := by
intro u v huv
exact List.append_cancel_left huv
have hsub : S.image (fun z => X++z) ⊆
(Regularization.parallelTestCells Q ρ g p (X++P)).filter (fun z =>
‖perpendicularPart (w:Space d) ((Q.descendant z).center-(Q.descendant (X++P)).center)‖ ≤ W) := by
intro z hz
obtain ⟨u,hu,rfl⟩ := Finset.mem_image.mp hz
obtain ⟨hu,_,hdist,hperp⟩ := Finset.mem_filter.mp hu
obtain ⟨hu,hocc⟩ := Finset.mem_filter.mp hu
have hulen := (mem_cubePaths _ _ _).mp hu
apply Finset.mem_filter.mpr
constructor
· simp only [Regularization.parallelTestCells,Regularization.occupiedCenterCells,
Finset.mem_filter]
refine ⟨?_,?_,?_⟩
· apply (mem_cubePaths _ _ _).mpr
simp only [List.length_append,hX,hulen]
omega
· simpa only [Q.descendant_append] using hocc
· rw [mem_ball,dist_eq_norm,Q.descendant_append,Q.descendant_append]
apply hdist.trans_lt
rw [hscale]
have hs := sqrt_dimension_le d
have hr : Real.sqrt d < 10*((d:ℝ)+1) := by
nlinarith [Nat.cast_nonneg (α := ℝ) d]
simpa only [div_eq_mul_inv,one_mul] using
mul_lt_mul_of_pos_right hr (by positivity : 0 < 1/(2:ℝ)^g)
· simpa only [Q.descendant_append] using hperp
have hcard : S.card ≤ Regularization.parallelTestCount Q ρ g p (X++P) W w := by
calc
S.card = (S.image (fun z => X++z)).card :=
(Finset.card_image_of_injective S hinj).symm
_ ≤ _ := Finset.card_le_card hsub
have hcount : (Regularization.parallelTestCount Q ρ g p (X++P) W w : ℝ) ≤ T := by
unfold Regularization.parallelGoodSet at hw
split_ifs at hw with hocc
· simpa only [ENNReal.toReal_natCast,ENNReal.toReal_ofReal hT] using
ENNReal.toReal_mono ENNReal.ofReal_ne_top hw
· exact False.elim hw
exact hcard.trans ((Nat.cast_le (α := ℝ)).mp (hcount.trans (Nat.le_ceil T)))
theorem descriptorEndpointGood_graph_test (Q : DyadicCube d) (hQ : Q.side = 1)
(ρ : Measure (Space d)) (m : ℝ → ℝ) (N : ℕ) (σ : ℝ) (J i : ℕ)
{a g p : ℕ} (hag : a ≤ g) (hgp : g ≤ p)
(X P : List (Fin d → Bool)) (hX : X.length = a) (hP : P.length = p-a)
{x : Space d} (hx : x ∈ ((Q.descendant X).descendant P).carrier)
(w : Sphere d)
(hw : (x,w) ∈ descriptorEndpointGood Q ρ m N σ J i (.parallel g p)) :
DyadicParallelTest (Q.descendant X) ((Q.descendant X).occupiedPaths ρ (p-a))
(g-a) P w (angularTestWidth N σ J i p)
⌈((2:ℝ)^N)^(PlanarFalconer.height
(fun n : ℕ => m n-((d:ℝ)/2)*(n:ℝ)/N) g p+1000*((d:ℝ)+1)*σ)⌉₊ := by
apply parallelGoodSet_local_test Q hQ ρ hag hgp X P hX w _ _ (by positivity)
apply (descriptorEndpointGood_on_cell Q ρ m N σ J i (.parallel g p) (X++P) ?_
(by simpa only [Q.descendant_append] using hx) w).mp hw
simp only [PlanarFalconer.descriptorAnchor,List.length_append,hX,hP]
omega
theorem descriptorMask_graph_test (Q : DyadicCube d) (hQ : Q.side = 1)
(ρ : Measure (Space d)) (m : ℝ → ℝ) (N : ℕ) (σ : ℝ) (J i : ℕ)
{a g p : ℕ} (hag : a ≤ g) (hgp : g ≤ p)
(X P : List (Fin d → Bool)) (hX : X.length = a) (hP : P.length = p-a)
(S : Finset (Sphere d))
(hcover : ∀ w : Sphere d, ∃ c ∈ S,
dist w c ≤ descriptorMaskRadius N σ (.parallel g p)/2)
{x : Space d} (hx : x ∈ ((Q.descendant X).descendant P).carrier) (w : Sphere d)
(hw : descriptorMask Q ρ m N σ J i (.parallel g p) S x w ≠ 0) :
DyadicParallelTest (Q.descendant X) ((Q.descendant X).occupiedPaths ρ (p-a))
(g-a) P w (angularTestWidth N σ J (i+1) p)
⌈((2:ℝ)^N)^(PlanarFalconer.height
(fun n : ℕ => m n-((d:ℝ)/2)*(n:ℝ)/N) g p+1000*((d:ℝ)+1)*σ)⌉₊ :=
descriptorEndpointGood_graph_test Q hQ ρ m N σ J (i+1) hag hgp X P hX hP hx w
(descriptorMask_support Q ρ m N σ J i (.parallel g p) S hcover w hw)
end Falconer
end OAI
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