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b17ead8 | 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 | import OAI.MeasureTheory.Falconer.Angular.AngularFarDepth
namespace OAI
namespace PlanarFalconer
open Set Metric
theorem comparable_cells_strong_separation (P Q : DyadicSquare)
(hP : 0 < P.side) (hQ : 0 < Q.side) {l : ℝ} (hl : 0 < l)
(hsP : P.side ≤ l) (hsQ : Q.side ≤ l) {x y : Plane}
(hx : x ∈ P.carrier) (hy : y ∈ Q.carrier)
(hlo : 256*l < dist x y) (hhi : dist x y ≤ 1024*l) :
100*l ≤ dist P.center Q.center ∧ dist P.center Q.center ≤ 2000*l := by
have hpx := (P.dist_le hP.le (P.center_mem hP) hx).trans (mul_le_mul_of_nonneg_left hsP (by norm_num))
have hqy := (Q.dist_le hQ.le hy (Q.center_mem hQ)).trans (mul_le_mul_of_nonneg_left hsQ (by norm_num))
have ht₁ := (dist_triangle P.center x Q.center).trans
(add_le_add (le_refl _) (dist_triangle x y Q.center))
have ht₂ := (dist_triangle x P.center y).trans
(add_le_add (le_refl _) (dist_triangle P.center Q.center y))
rw [dist_comm x P.center,dist_comm Q.center y] at ht₂
constructor <;> linarith
theorem comparable_dyadic_cells_strong_separation (Q₁ Q₂ : DyadicSquare)
(hQ₁ : Q₁.side = 1) (hQ₂ : Q₂.side = 1) (b : ℕ) (P Q : CellPath)
(hP : P.length = b) (hQ : Q.length = b) {x y : Plane}
(hx : x ∈ (Q₁.descendant P).carrier) (hy : y ∈ (Q₂.descendant Q).carrier)
(hlo : 256*(2:ℝ)^(-(b:ℝ)) < dist x y) (hhi : dist x y ≤ 1024*(2:ℝ)^(-(b:ℝ))) :
100*(2:ℝ)^(-(b:ℝ)) ≤ dist (Q₁.descendant P).center (Q₂.descendant Q).center ∧
dist (Q₁.descendant P).center (Q₂.descendant Q).center ≤ 2000*(2:ℝ)^(-(b:ℝ)) := by
have hp : 0 < Q₁.side := by rw [hQ₁]; norm_num
have hq : 0 < Q₂.side := by rw [hQ₂]; norm_num
apply comparable_cells_strong_separation (Q₁.descendant P) (Q₂.descendant Q)
(Q₁.descendant_side_pos hp P) (Q₂.descendant_side_pos hq Q) (by positivity) _ _ hx hy hlo hhi
· rw [Q₁.descendant_side,hP,hQ₁,Real.rpow_neg (by norm_num : (0:ℝ) ≤ 2),Real.rpow_natCast]
simp
· rw [Q₂.descendant_side,hQ,hQ₂,Real.rpow_neg (by norm_num : (0:ℝ) ≤ 2),Real.rpow_natCast]
simp
end PlanarFalconer
end OAI
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