openAI-math-dataset / lean /OAI /MeasureTheory /FalconerCompletion /Recursion /FarCommonDepthUpper.lean
| 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 | |