File size: 4,544 Bytes
b224933 | 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 | import Mathlib
import OAI.MeasureTheory.FalconerCompletion.Graphs.ProfileChildCard
namespace OAI
/-! Recursive policies and quantitative depth estimates. -/
section
section
open Set MeasureTheory
open scoped ComplexConjugate RealInnerProductSpace
namespace GraphLiteralSupport
namespace Attempt4FullFar
open Set MeasureTheory Metric
lemma far_common_depth_upperND {b h k : ℕ} {D : ℝ}
(hlo : 1/(4*(2:ℝ)^h)≤D) (hhi : D≤(2:ℝ)^k/(2:ℝ)^b) :
b≤h+k+2 := by
have hh := hlo.trans hhi
have hp : 0<(2:ℝ)^b := by positivity
have hq : 0<(4*(2:ℝ)^h) := by positivity
have he := (div_le_div_iff₀ hq hp).mp hh
have hpow : (2:ℝ)^b≤(2:ℝ)^(h+k+2) := by
calc
_ ≤ (2:ℝ)^k*(4*(2:ℝ)^h) := by simpa using he
_ = _ := by rw [pow_add,pow_add]; ring
exact (pow_le_pow_iff_right₀ (by norm_num : (1:ℝ)<2)).mp hpow
lemma mem_ambientGridCell_floorND {d : ℕ} (b : ℕ) (x : EuclideanSpace ℝ (Fin d)) :
x∈ambientGridCell b (fun i => ⌊(2:ℝ)^b*x i⌋) := by
apply mem_ambientGridCell_iff.mpr
intro i
rfl
lemma ambientGridCell_subset_of_common_pointND {d a b : ℕ} (hab : a≤b)
{P Q : Fin d → ℤ} {x : EuclideanSpace ℝ (Fin d)}
(hx : x∈ambientGridCell a P) (hy : x∈ambientGridCell b Q) :
ambientGridCell b Q⊆ambientGridCell a P := by
have hs := ambientGridCell_subset_ancestor hab Q
have hz := hs hy
have he : (fun i => Q i / (2:ℤ)^(b-a))=P := by
funext i
exact ((mem_ambientGridCell_iff.mp hz) i).symm.trans ((mem_ambientGridCell_iff.mp hx) i)
simpa only [he] using hs
theorem exists_far_grid_coverND {d a h k : ℕ}
(hK : 16*(d+1:ℝ)≤(2:ℝ)^k) {X Y : Fin d → ℤ}
{x x' y y' : EuclideanSpace ℝ (Fin d)}
(hx : x∈ambientGridCell a X) (hx' : x'∈ambientGridCell a X)
(hy : y∈ambientGridCell a Y) (hy' : y'∈ambientGridCell a Y)
(hfar : 1/(4*(2:ℝ)^h)≤ max (dist x x') (dist y y'))
(hcomp : max (dist x x') (dist y y')≤2*min (dist x x') (dist y y')) :
∃ b∈Finset.Icc (a+1) (h+k+2),
∃ P P' Q Q' : Fin d → ℤ,
x∈ambientGridCell b P ∧ x'∈ambientGridCell b P' ∧
y∈ambientGridCell b Q ∧ y'∈ambientGridCell b Q' ∧
ambientGridCell b P⊆ambientGridCell a X ∧
ambientGridCell b P'⊆ambientGridCell a X ∧
ambientGridCell b Q⊆ambientGridCell a Y ∧
ambientGridCell b Q'⊆ambientGridCell a Y ∧
farSeparatedGridND b ((2:ℝ)^k) P P' ∧ farSeparatedGridND b ((2:ℝ)^k) Q Q' := by
have hD : 0< max (dist x x') (dist y y') :=
(show 0<1/(4*(2:ℝ)^h) by positivity).trans_le hfar
have hm : 0< min (dist x x') (dist y y') := by linarith
have hD₁ := hm.trans_le (min_le_left _ _)
have hD₂ := hm.trans_le (min_le_right _ _)
have hupper : max (dist x x') (dist y y')≤(d+1:ℝ)/(2:ℝ)^a := by
apply max_le
· simpa only [dist_comm] using ambientGridCell_dist_le hx hx'
· simpa only [dist_comm] using ambientGridCell_dist_le hy hy'
have hsize : max (dist x x') (dist y y')≤(2:ℝ)^k := by
have hs : 1≤(2:ℝ)^a := one_le_pow₀ (by norm_num)
have hdu : (d+1:ℝ)/(2:ℝ)^a≤d+1 := div_le_self (by positivity) hs
have hd : (0:ℝ)≤d := Nat.cast_nonneg d
exact hupper.trans (hdu.trans (by linarith))
obtain ⟨b,hbl₁,hbu₁,hbl₂,hbu₂,hbmax⟩ := exists_far_common_depthND hD₁ hD₂ hcomp hsize
have hab := far_common_depth_strictND hK hbmax hupper
have hbh := far_common_depth_upperND hfar (max_le hbu₁ hbu₂)
let P : Fin d → ℤ := fun i => ⌊(2:ℝ)^b*x i⌋
let P' : Fin d → ℤ := fun i => ⌊(2:ℝ)^b*x' i⌋
let Q : Fin d → ℤ := fun i => ⌊(2:ℝ)^b*y i⌋
let Q' : Fin d → ℤ := fun i => ⌊(2:ℝ)^b*y' i⌋
have hxP : x∈ambientGridCell b P := mem_ambientGridCell_floorND b x
have hxP' : x'∈ambientGridCell b P' := mem_ambientGridCell_floorND b x'
have hyQ : y∈ambientGridCell b Q := mem_ambientGridCell_floorND b y
have hyQ' : y'∈ambientGridCell b Q' := mem_ambientGridCell_floorND b y'
exact ⟨b,Finset.mem_Icc.mpr ⟨hab,hbh⟩,P,P',Q,Q',hxP,hxP',hyQ,hyQ',
ambientGridCell_subset_of_common_pointND hab.le hx hxP,
ambientGridCell_subset_of_common_pointND hab.le hx' hxP',
ambientGridCell_subset_of_common_pointND hab.le hy hyQ,
ambientGridCell_subset_of_common_pointND hab.le hy' hyQ',
farSeparatedGridND_of_lengths hK hxP hxP' hbl₁ hbu₁,
farSeparatedGridND_of_lengths hK hyQ hyQ' hbl₂ hbu₂⟩
end Attempt4FullFar
end GraphLiteralSupport
end
end
end OAI
|