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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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 | import OAI.MeasureTheory.Falconer.Incidence.RoundedRowPencils
namespace OAI
/-! Normalize the rounded product rows into the bounded line chart without
changing their first coordinate. All size losses are explicit. -/
namespace Falconer
attribute [local instance] Classical.propDecidable
noncomputable def verticalQuarter (p : Space 2) : Space 2 := rowPoint (p 0) (p 1/4)
noncomputable def verticalFour (p : Space 2) : Space 2 := rowPoint (p 0) (4*p 1)
@[simp] theorem verticalQuarter_zero (p : Space 2) : verticalQuarter p 0 = p 0 := rfl
@[simp] theorem verticalQuarter_one (p : Space 2) : verticalQuarter p 1 = p 1/4 := rfl
theorem verticalFour_verticalQuarter (p : Space 2) : verticalFour (verticalQuarter p) = p := by
ext i
fin_cases i
· rfl
· change 4*(p 1/4) = p 1
ring
theorem verticalQuarter_verticalFour (p : Space 2) : verticalQuarter (verticalFour p) = p := by
ext i
fin_cases i
· rfl
· change (4*p 1)/4 = p 1
ring
theorem verticalQuarter_injective : Function.Injective verticalQuarter :=
Function.LeftInverse.injective verticalFour_verticalQuarter
theorem verticalQuarter_row_distance {p q : Space 2} (h : p 0 = q 0) :
dist (verticalQuarter p) (verticalQuarter q) = dist p q/4 := by
have hp : p = rowPoint (p 0) (p 1) := by ext i; fin_cases i <;> rfl
have hq : q = rowPoint (p 0) (q 1) := by
ext i
fin_cases i
· exact h.symm
· rfl
conv_rhs => rw [hp,hq,rowPoint_dist]
change dist (rowPoint (p 0) (p 1/4)) (rowPoint (q 0) (q 1/4)) = _
rw [← h,rowPoint_dist,← sub_div,abs_div]
norm_num
theorem verticalQuarter_row_separated {δ y : ℝ} (P : Finset (Space 2))
(hrow : ∀ p ∈ P, p 0 = y) (hsep : FiniteSeparatedAt δ P) :
FiniteSeparatedAt (δ/4) (P.image verticalQuarter) := by
intro p hp q hq hpq
obtain ⟨x,hx,rfl⟩ := Finset.mem_image.mp hp
obtain ⟨z,hz,rfl⟩ := Finset.mem_image.mp hq
rw [verticalQuarter_row_distance ((hrow x hx).trans (hrow z hz).symm)]
exact div_le_div_of_nonneg_right
(hsep x hx z hz (fun h => hpq (congrArg _ h))) (by norm_num)
theorem verticalQuarter_row_regular {δ s K y : ℝ} (hδ : 0 < δ)
(hs : 0 ≤ s) (hK : 1 ≤ K) (P : Finset (Space 2))
(hrow : ∀ p ∈ P, p 0 = y) (hreg : NormalizedFiniteBallBound δ s K P) :
NormalizedFiniteBallBound (δ/4) s (K*(4:ℝ)^s) (P.image verticalQuarter) := by
intro z r hr _
have hr0 : 0 < r := (by linarith : 0 < δ/4).trans_le hr
have hsub : (P.image verticalQuarter).filter (fun p => dist p z ≤ r) ⊆
(P.filter (fun p => dist p (rowPoint y (4*z 1)) ≤ 4*r)).image verticalQuarter := by
intro p hp
obtain ⟨hp,hpr⟩ := Finset.mem_filter.mp hp
obtain ⟨x,hx,rfl⟩ := Finset.mem_image.mp hp
refine Finset.mem_image.mpr ⟨x,Finset.mem_filter.mpr ⟨hx,?_⟩,rfl⟩
rw [row_dist_projection (hrow x hx)]
have hh : |x 1/4-z 1| ≤ r :=
(show |x 1/4-z 1| ≤ dist (verticalQuarter x) z by
simpa only [Real.dist_eq,verticalQuarter_one] using PiLp.dist_apply_le (verticalQuarter x) z 1).trans hpr
have he : |x 1-4*z 1| = 4*|x 1/4-z 1| := by
rw [← abs_of_pos (by norm_num : (0:ℝ) < 4),← abs_mul]
congr 1
ring
rw [he]
linarith only [hh]
have hc : (((P.image verticalQuarter).filter (fun p => dist p z ≤ r)).card:ℝ) ≤
((P.filter (fun p => dist p (rowPoint y (4*z 1)) ≤ 4*r)).card:ℝ) := by
exact_mod_cast (Finset.card_le_card hsub).trans Finset.card_image_le
rw [Finset.card_image_of_injective _ verticalQuarter_injective]
calc
_ ≤ ((P.filter (fun p => dist p (rowPoint y (4*z 1)) ≤ 4*r)).card:ℝ) := hc
_ ≤ K*(4*r)^s*(P.card:ℝ) :=
normalized_ball_bound_all_radii hs hK hreg _ _ (by linarith only [hr])
_ = _ := by rw [Real.mul_rpow (by norm_num : (0:ℝ) ≤ 4) hr0.le]; ring
theorem quarter_adjusted_incident {δ : ℝ} (hδ : 0 < δ) (p l : Space 2) :
TubeIncident (δ/4) (verticalQuarter p) (planarQuarter (rowAdjustedLine δ p l)) := by
have hd : 1 ≤ Real.sqrt (1+(planarQuarter (rowAdjustedLine δ p l) 0)^2) := by
apply (Real.le_sqrt (by norm_num) (by positivity)).2
nlinarith only [sq_nonneg (planarQuarter (rowAdjustedLine δ p l) 0)]
have he : verticalQuarter p 1-planarQuarter (rowAdjustedLine δ p l) 0*verticalQuarter p 0-
planarQuarter (rowAdjustedLine δ p l) 1 =
(p 1-rowAdjustedLine δ p l 0*p 0-rowAdjustedLine δ p l 1)/4 := by
simp only [verticalQuarter_zero,verticalQuarter_one,planarQuarter_apply]
ring
unfold TubeIncident
rw [he,abs_div]
norm_num only [abs_of_pos (by norm_num : (0:ℝ) < 4)]
apply (div_le_iff₀ (zero_lt_one.trans_le hd)).2
exact (div_le_div_of_nonneg_right (rowAdjustedLine_residual hδ p l) (by norm_num)).trans
(le_mul_of_one_le_right (by positivity) hd)
theorem quarter_adjusted_chart {δ : ℝ} (hδ : 0 < δ) (hδsmall : δ ≤ 1/8)
(p l : Space 2) (hp : |p 0| ≤ 1) (hl : |l 0| ≤ 1 ∧ |l 1| ≤ 1)
(hi : TubeIncident (3*δ) p l) :
|planarQuarter (rowAdjustedLine δ p l) 0| ≤ 1 ∧
|planarQuarter (rowAdjustedLine δ p l) 1| ≤ 1 := by
obtain ⟨h0,h1⟩ := rowAdjustedLine_coordinate_error hδ p l hp hl.1 hi
have h0' : |rowAdjustedLine δ p l 0| ≤ |l 0|+δ := by
have hh := abs_sub_le (rowAdjustedLine δ p l 0) (l 0) 0
simp only [sub_zero] at hh
rw [abs_sub_comm (rowAdjustedLine δ p l 0) (l 0)] at hh
linarith only [hh,h0]
have h1' : |rowAdjustedLine δ p l 1| ≤ |l 1|+8*δ := by
have hh := abs_sub_le (rowAdjustedLine δ p l 1) (l 1) 0
simp only [sub_zero] at hh
rw [abs_sub_comm (rowAdjustedLine δ p l 1) (l 1)] at hh
linarith only [hh,h1]
simp only [planarQuarter_apply,abs_div,abs_of_pos (by norm_num : (0:ℝ) < 4)]
constructor <;> linarith only [h0',h1',hl.1,hl.2,hδsmall]
end Falconer
end OAI
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