openAI-math-dataset / lean /OAI /MeasureTheory /FalconerCompletion /Recursion /PrefixShellLeSourceShellSuccessor.lean
| import Mathlib | |
| import OAI.MeasureTheory.Falconer.Fourier.DistanceSincSpectralIdentity | |
| import OAI.MeasureTheory.Falconer.Model | |
| import OAI.MeasureTheory.FalconerCompletion.Recursion.PrefixDescriptors | |
| import OAI.MeasureTheory.FalconerCompletion.Shells.PrefixShell | |
| import OAI.MeasureTheory.FalconerCompletion.Shells.WeightedMaskTransform | |
| namespace OAI | |
| /-! Recursive policies and quantitative depth estimates. -/ | |
| open _root_.OAI.PlanarFalconer | |
| section | |
| section | |
| open Set MeasureTheory | |
| open scoped ComplexConjugate RealInnerProductSpace | |
| namespace GraphLiteralSupport.CurrentFalconer.TreeND | |
| open Set MeasureTheory PlanarFalconer GraphLiteralSupport.Attempt4FullFar | |
| open scoped ContDiff | |
| variable {n : ℕ} | |
| namespace PositiveDistanceState | |
| variable (e : PositiveDistanceState (n+1)) | |
| lemma prefixShell_le_sourceShell_successor [IsFiniteMeasure e.weightedDistance] (j : ℕ) : | |
| e.prefixShell j ≤ (2:ℝ)^(-(e.a:ℝ))* | |
| (∫ r in Icc ((2:ℝ)^(j+1:ℕ)/2) (2*(2:ℝ)^(j+1:ℕ)), | |
| ‖Attempt4PrefixShell.weightedMaskTransformND e.prefixDescriptors e.prefixDescriptors | |
| e.prefixEndpointSmooth e.prefixEndpointSmooth (fun _ => 1) (fun _ => 1) | |
| e.ρ e.ρ (treeCell e.X) (treeCell e.Y) e.levelMass e.levelMass e.a r‖^2) := by | |
| simp_rw [e.prefix_shell_transform_exact] | |
| have hsub : Icc ((2:ℝ)^j) ((2:ℝ)^(j+1))⊆Icc ((2:ℝ)^(j+1)/2) (2*(2:ℝ)^(j+1)) := by | |
| intro r hr | |
| constructor | |
| · simpa only [pow_succ,mul_div_cancel_right₀ _ (by norm_num : (2:ℝ) ≠ 0)] using hr.1 | |
| · have hp : 0 ≤ (2:ℝ)^(j+1) := by positivity | |
| linarith [hr.2] | |
| have hi : Integrable (fun r => ‖charFun e.weightedDistance r‖^2) | |
| (volume.restrict (Icc ((2:ℝ)^(j+1)/2) (2*(2:ℝ)^(j+1)))) := by | |
| simpa only [one_mul] using charFun_sq_integrable_finite e.weightedDistance | |
| (volume.restrict (Icc ((2:ℝ)^(j+1)/2) (2*(2:ℝ)^(j+1)))) 1 | |
| exact mul_le_mul_of_nonneg_left | |
| (integral_mono_measure (Measure.restrict_mono hsub le_rfl) | |
| (Filter.Eventually.of_forall (fun r => sq_nonneg ‖charFun e.weightedDistance r‖)) hi) (by positivity) | |
| lemma prefixHigh_shell_regime {sig : ℝ} (hsig : 0 < sig) | |
| (htN : e.t ≤ e.N) (hwidth : e.a+100*e.q < e.t) (hq : sig*e.N ≤ e.q) | |
| {v : ℕ} (hv : v+1 ≤ e.t) (hhigh : (e.t:ℝ)-(v+1:ℕ) < sig*e.N) : | |
| 20*sig*e.N ≤ (v+1:ℕ)-(e.a:ℝ) ∧ | |
| (((e.t-e.a)/2:ℕ):ℝ) ≤ ((v+1:ℕ)-(e.a:ℝ))/2+sig*e.N ∧ | |
| (v+1:ℕ)-(e.a:ℝ) ≤ 2*e.N := by | |
| have hat : e.a ≤ e.t := by omega | |
| have hw : (e.a:ℝ)+100*e.q < e.t := by exact_mod_cast hwidth | |
| have hv' : ((v+1:ℕ):ℝ) ≤ e.N := by exact_mod_cast hv.trans htN | |
| have hfloor : 2*(((e.t-e.a)/2:ℕ):ℝ) ≤ (e.t:ℝ)-e.a := by | |
| have hh := Nat.mul_div_le (e.t-e.a) 2 | |
| have hhc : (2:ℝ)*(((e.t-e.a)/2:ℕ):ℝ) ≤ ((e.t-e.a:ℕ):ℝ) := by exact_mod_cast hh | |
| simpa only [Nat.cast_sub hat] using hhc | |
| have hna : 0 ≤ (e.a:ℝ) := Nat.cast_nonneg _ | |
| have hnN : 0 ≤ (e.N:ℝ) := Nat.cast_nonneg _ | |
| have hsigN : 0 ≤ sig*e.N := mul_nonneg hsig.le hnN | |
| constructor | |
| · linarith | |
| constructor <;> linarith | |
| lemma prefixShell_normalized_mass {B : ℕ} {s sig : ℝ} | |
| (hd : RegularizedProfileData e.ρ e.N B s sig e.mass) | |
| (ha : e.a ≤ e.N) (hX : e.X.length=e.a) (hY : e.Y.length=e.a) : | |
| |(e.levelMass*e.levelMass)⁻¹| * | |
| ((e.ρ.restrict (treeCell e.X)).prod (e.ρ.restrict (treeCell e.Y))).real univ ≤ | |
| ((2:ℝ)^e.N)^(2*(n+2:ℝ)*sig) := by | |
| have h := e.normalized_cellPair_profile_mass hd ha hX hY | |
| have hm : |(e.levelMass*e.levelMass)⁻¹| = e.levelMass^(-2:ℤ) := by | |
| rw [abs_of_nonneg (inv_nonneg.mpr (mul_self_nonneg e.levelMass))] | |
| simp [zpow_neg,pow_two] | |
| rw [hm] | |
| convert h using 1 <;> push_cast <;> congr 1 | |
| ring | |
| lemma prefixDescriptors_card_max (hN : 1 ≤ e.N) (hq : 0 < e.q) : | |
| max 1 ((e.prefixDescriptors.card+e.prefixDescriptors.card:ℕ):ℝ) ≤ 16*(e.N:ℝ)^3 := by | |
| have hN' : (1:ℝ) ≤ e.N := by exact_mod_cast hN | |
| exact max_le (by nlinarith [one_le_pow₀ hN' (n:=3)]) (e.prefixDescriptors_card hN hq) | |
| end PositiveDistanceState | |
| end GraphLiteralSupport.CurrentFalconer.TreeND | |
| end | |
| end | |
| end OAI | |