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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 | import OAI.MeasureTheory.Falconer.DistanceSet
import Mathlib.Tactic.Linarith
import Mathlib.Tactic.Positivity
import Mathlib.Tactic.FinCases
import Mathlib.Analysis.Normed.Lp.MeasurableSpace
namespace OAI
namespace PlanarFalconer
open Set Metric
structure DyadicSquare where
x : ℝ
y : ℝ
side : ℝ
def DyadicSquare.carrier (Q : DyadicSquare) : Set Plane :=
{z | Q.x ≤ z 0 ∧ z 0 < Q.x + Q.side ∧
Q.y ≤ z 1 ∧ z 1 < Q.y + Q.side}
noncomputable def DyadicSquare.child (Q : DyadicSquare) (i j : Bool) : DyadicSquare :=
⟨Q.x + (if i then Q.side / 2 else 0),
Q.y + (if j then Q.side / 2 else 0), Q.side / 2⟩
theorem DyadicSquare.child_side_pos (Q : DyadicSquare) (hQ : 0 < Q.side)
(i j : Bool) : 0 < (Q.child i j).side := by
dsimp [child]
positivity
theorem DyadicSquare.measurable (Q : DyadicSquare) : MeasurableSet Q.carrier := by
have h0 : Continuous (fun z : Plane => z 0) := (EuclideanSpace.proj 0).continuous
have h1 : Continuous (fun z : Plane => z 1) := (EuclideanSpace.proj 1).continuous
exact (isClosed_le continuous_const h0).measurableSet.inter
((isOpen_lt h0 continuous_const).measurableSet.inter
((isClosed_le continuous_const h1).measurableSet.inter
(isOpen_lt h1 continuous_const).measurableSet))
theorem DyadicSquare.child_subset (Q : DyadicSquare) (hQ : 0 ≤ Q.side)
(i j : Bool) : (Q.child i j).carrier ⊆ Q.carrier := by
have halfSide_nonneg : 0 ≤ Q.side / 2 := div_nonneg hQ (by norm_num)
intro z hz
cases i <;> cases j <;>
simp only [carrier, child, Bool.false_eq_true, ↓reduceIte,
add_zero, mem_ofPred_eq] at * <;>
refine ⟨by linarith [halfSide_nonneg], by linarith [halfSide_nonneg],
by linarith [halfSide_nonneg], by linarith [halfSide_nonneg]⟩
theorem DyadicSquare.cover_children (Q : DyadicSquare) :
Q.carrier = ((Q.child false false).carrier ∪ (Q.child false true).carrier) ∪
((Q.child true false).carrier ∪ (Q.child true true).carrier) := by
ext z
simp only [carrier, child, Bool.false_eq_true, ↓reduceIte,
add_zero, mem_ofPred_eq, mem_union]
constructor
· intro hz
by_cases hx : z 0 < Q.x + Q.side / 2 <;>
by_cases hy : z 1 < Q.y + Q.side / 2
· exact Or.inl (Or.inl ⟨hz.1,hx,hz.2.2.1,hy⟩)
· exact Or.inl (Or.inr ⟨hz.1,hx,by linarith,by linarith [hz.2.2.2]⟩)
· exact Or.inr (Or.inl ⟨by linarith,by linarith [hz.2.1],hz.2.2.1,hy⟩)
· exact Or.inr (Or.inr ⟨by linarith,by linarith [hz.2.1],
by linarith,by linarith [hz.2.2.2]⟩)
· intro hz
rcases hz with (hz | hz) | (hz | hz) <;>
refine ⟨by linarith [hz.1,hz.2.1], by linarith [hz.1,hz.2.1],
by linarith [hz.2.2.1,hz.2.2.2], by linarith [hz.2.2.1,hz.2.2.2]⟩
theorem DyadicSquare.children_disjoint (Q : DyadicSquare)
{i j i' j' : Bool} (h : (i,j) ≠ (i',j')) :
Disjoint (Q.child i j).carrier (Q.child i' j').carrier := by
apply Set.disjoint_left.mpr
intro z hz hz'
cases i <;> cases j <;> cases i' <;> cases j' <;>
simp_all only [carrier, child, Bool.false_eq_true, ↓reduceIte,
add_zero, mem_ofPred_eq, ne_eq, not_true_eq_false] <;> linarith
theorem DyadicSquare.dist_le (Q : DyadicSquare) (hQ : 0 ≤ Q.side)
{z w : Plane} (hz : z ∈ Q.carrier) (hw : w ∈ Q.carrier) :
dist z w ≤ 2 * Q.side := by
have h0 : |z 0 - w 0| ≤ Q.side := by
apply abs_le.mpr; constructor <;> linarith [hz.1,hz.2.1,hw.1,hw.2.1]
have h1 : |z 1 - w 1| ≤ Q.side := by
apply abs_le.mpr; constructor <;>
linarith [hz.2.2.1,hz.2.2.2,hw.2.2.1,hw.2.2.2]
have hs := EuclideanSpace.dist_sq_eq z w
simp only [Fin.sum_univ_two, Real.dist_eq, sq_abs] at hs
have hs0 := sq_le_sq₀ (abs_nonneg _) hQ |>.mpr h0
have hs1 := sq_le_sq₀ (abs_nonneg _) hQ |>.mpr h1
rw [sq_abs] at hs0 hs1
nlinarith [dist_nonneg (x := z) (y := w)]
theorem DyadicSquare.ediam_le (Q : DyadicSquare) (hQ : 0 ≤ Q.side) :
ediam Q.carrier ≤ ENNReal.ofReal (2 * Q.side) := by
apply Metric.ediam_le
intro z hz w hw
rw [edist_dist]
exact ENNReal.ofReal_le_ofReal (Q.dist_le hQ hz hw)
end PlanarFalconer
end OAI
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