openAI-math-dataset / lean /OAI /MeasureTheory /FalconerCompletion /Shells /WeightedDistanceCollisionLower.lean
| import Mathlib | |
| import OAI.MeasureTheory.Falconer.Fourier.DistanceSincSpectralIdentity | |
| import OAI.MeasureTheory.Falconer.Model | |
| import OAI.MeasureTheory.FalconerCompletion.Fourier.MinimalMask | |
| namespace OAI | |
| /-! Dyadic shells and prefix decompositions. -/ | |
| open _root_.OAI.PlanarFalconer | |
| section | |
| section | |
| open Set MeasureTheory | |
| open scoped ComplexConjugate RealInnerProductSpace | |
| namespace GraphLiteralSupport | |
| namespace CurrentFalconer.TreeND | |
| open Set MeasureTheory PlanarFalconer _root_.OAI.GraphLiteralSupport.Attempt4FullFar | |
| open scoped ENNReal | |
| variable {n : ℕ} | |
| namespace PositiveDistanceState | |
| variable (e : PositiveDistanceState (n+1)) | |
| lemma weightedDistance_collision_lower [IsFiniteMeasure e.ρ] {K : ℝ} (hK : 32*(n+2:ℝ)≤K) | |
| (hX : e.X.length=e.a) (hY : e.Y.length=e.a) | |
| (hs : farSeparatedGridND e.a K (fun j => (pathIndex e.X j:ℤ)) (fun j => (pathIndex e.Y j:ℤ))) | |
| (S : Set (ℝ × ℝ)) (hS : MeasurableSet S) : | |
| (e.levelMass^(-2:ℤ)*(8*K)^(-(n:ℝ)/2))^2 * (e.distance.prod e.distance).real S ≤ | |
| (e.weightedDistance.prod e.weightedDistance).real S := by | |
| let := e.weightedDistance_isFinite hK hX hY hs | |
| have hKp : 0<K := by have hn : (0:ℝ)≤n := Nat.cast_nonneg n; linarith | |
| have hc : 0≤e.levelMass^(-2:ℤ)*(8*K)^(-(n:ℝ)/2) := by positivity | |
| have hp := Measure.prod_mono (e.weightedDistance_lower hK hX hY hs) | |
| (e.weightedDistance_lower hK hX hY hs) | |
| have h := ENNReal.toReal_mono (measure_ne_top _ S) (Measure.le_iff.mp hp S hS) | |
| rw [Measure.prod_smul_left,Measure.prod_smul_right,smul_smul,Measure.smul_apply, | |
| smul_eq_mul,ENNReal.toReal_mul,ENNReal.toReal_mul,ENNReal.toReal_ofReal hc] at h | |
| simpa only [pow_two,measureReal_def] using h | |
| end PositiveDistanceState | |
| theorem actual_prefix_lowpass {K : ℝ} (hK : 32*(n+2:ℝ)≤K) : | |
| ∃ H : ℝ, 0<H ∧ ∀ (e : PositiveDistanceState (n+1)) [IsFiniteMeasure e.ρ], | |
| e.X.length=e.a → e.Y.length=e.a → | |
| farSeparatedGridND e.a K (fun j => (pathIndex e.X j:ℤ)) (fun j => (pathIndex e.Y j:ℤ)) → | |
| e.coincidence.toReal ≤ H*(2:ℝ)^(-(e.a:ℝ))* | |
| ∫ r in Icc (-((2:ℝ)^e.t)) ((2:ℝ)^e.t), ‖charFun e.weightedDistance r‖^2 := by | |
| obtain ⟨C,hC,hfourier⟩ := finite_measure_distance_collision_fourier_wide | |
| have hKp : 0<K := by have hn : (0:ℝ)≤n := Nat.cast_nonneg n; linarith | |
| let c : ℝ := (8*K)^(-(n:ℝ)/2) | |
| have hcp : 0<c := by dsimp [c]; positivity | |
| refine ⟨((n+2:ℝ)*C)/c^2,by positivity,?_⟩ | |
| intro e he hX hY hs | |
| let := e.weightedDistance_isFinite hK hX hY hs | |
| have hc := e.weightedDistance_collision_lower hK hX hY hs | |
| {z : ℝ × ℝ | |z.1-z.2|≤100*(↑(n+1)+1:ℝ)*e.width} (by measurability) | |
| have hM : 0<e.levelMass := by unfold PositiveDistanceState.levelMass; positivity | |
| have hn : e.normalization.toReal = ((2:ℝ)^((e.t:ℝ)-e.a)/c^2)* | |
| (e.levelMass^(-2:ℤ)*c)^2 := by | |
| rw [PositiveDistanceState.normalization,ENNReal.toReal_ofReal (by positivity)] | |
| simp only [zpow_neg,zpow_ofNat,mul_pow] | |
| field_simp | |
| have hw : 100*(↑(n+1)+1:ℝ)*e.width=100*(n+2:ℝ)/(2:ℝ)^e.t := by | |
| simp only [PositiveDistanceState.width,Real.rpow_neg (by norm_num : (0:ℝ)≤2),Real.rpow_natCast] | |
| push_cast | |
| ring | |
| have hco : e.coincidence.toReal=e.normalization.toReal* | |
| (e.distance.prod e.distance).real {z : ℝ × ℝ | |z.1-z.2|≤100*(↑(n+1)+1:ℝ)*e.width} := by | |
| rw [PositiveDistanceState.coincidence,ENNReal.toReal_mul,measureReal_def,e.distance_collision] | |
| have hd : e.coincidence.toReal ≤ ((2:ℝ)^((e.t:ℝ)-e.a)/c^2)* | |
| (e.weightedDistance.prod e.weightedDistance).real | |
| {z : ℝ × ℝ | |z.1-z.2|≤100*(n+2:ℝ)/(2:ℝ)^e.t} := by | |
| rw [hco,hn,mul_assoc,← hw] | |
| exact mul_le_mul_of_nonneg_left hc (by positivity) | |
| have hf := hfourier e.weightedDistance ((2:ℝ)^e.t) (n+2) (by positivity) (by have hn : (0:ℝ)≤n := Nat.cast_nonneg n; linarith) | |
| have hp : (2:ℝ)^((e.t:ℝ)-e.a)=(2:ℝ)^(-(e.a:ℝ))*(2:ℝ)^e.t := by | |
| rw [← Real.rpow_natCast,← Real.rpow_add (by norm_num : (0:ℝ)<2)] | |
| congr 1 | |
| ring | |
| calc | |
| _ ≤ ((2:ℝ)^(-(e.a:ℝ))/c^2)*(((2:ℝ)^e.t)* | |
| (e.weightedDistance.prod e.weightedDistance).real | |
| {z : ℝ × ℝ | |z.1-z.2|≤100*(n+2:ℝ)/(2:ℝ)^e.t}) := by | |
| convert hd using 1; rw [hp]; ring | |
| _ ≤ ((2:ℝ)^(-(e.a:ℝ))/c^2)*((n+2:ℝ)*C* | |
| ∫ r in Icc (-((2:ℝ)^e.t)) ((2:ℝ)^e.t), ‖charFun e.weightedDistance r‖^2) := | |
| mul_le_mul_of_nonneg_left hf (by positivity) | |
| _ = _ := by ring | |
| end CurrentFalconer.TreeND | |
| end GraphLiteralSupport | |
| end | |
| end | |
| section | |
| section | |
| open Set MeasureTheory | |
| open scoped ComplexConjugate RealInnerProductSpace | |
| namespace GraphLiteralSupport | |
| namespace CurrentFalconer.TreeND | |
| open Set MeasureTheory PlanarFalconer _root_.OAI.GraphLiteralSupport.Attempt4FullFar | |
| open scoped ENNReal | |
| variable {n : ℕ} | |
| namespace PositiveDistanceState | |
| variable (e : PositiveDistanceState (n+1)) | |
| lemma weightedDistance_mass_le [IsFiniteMeasure e.ρ] {K : ℝ} (hK : 32*(n+2:ℝ)≤K) | |
| (hX : e.X.length=e.a) (hY : e.Y.length=e.a) | |
| (hs : farSeparatedGridND e.a K (fun j => (pathIndex e.X j:ℤ)) (fun j => (pathIndex e.Y j:ℤ))) : | |
| e.weightedDistance.real univ≤e.levelMass^(-2:ℤ)*e.cellPair.real univ := by | |
| have h := Measure.le_iff.mp (e.weightedPair_le hK hX hY hs) univ MeasurableSet.univ | |
| have ht : (ENNReal.ofReal (e.levelMass^(-2:ℤ)) • e.cellPair) univ≠⊤ := by | |
| rw [Measure.smul_apply,smul_eq_mul] | |
| exact ENNReal.mul_ne_top ENNReal.ofReal_ne_top (measure_ne_top _ _) | |
| have h' := ENNReal.toReal_mono ht h | |
| rw [Measure.smul_apply,smul_eq_mul,ENNReal.toReal_mul, | |
| ENNReal.toReal_ofReal (by positivity : 0≤e.levelMass^(-2:ℤ))] at h' | |
| simpa only [weightedDistance,measureReal_def,Measure.map_apply | |
| (continuous_fst.dist continuous_snd).measurable MeasurableSet.univ,preimage_univ] using h' | |
| lemma weightedDistance_profile_mass {K : ℝ} (hK : 32*(n+2:ℝ)≤K) | |
| {B : ℕ} {s lam : ℝ} (hd : RegularizedProfileData e.ρ e.N B s lam e.mass) | |
| (ha : e.a≤e.N) (hX : e.X.length=e.a) (hY : e.Y.length=e.a) | |
| (hs : farSeparatedGridND e.a K (fun j => (pathIndex e.X j:ℤ)) (fun j => (pathIndex e.Y j:ℤ))) : | |
| e.weightedDistance.real univ≤((2:ℝ)^e.N)^(2*(n+2:ℝ)*lam) := by | |
| let := hd.probability | |
| apply (e.weightedDistance_mass_le hK hX hY hs).trans | |
| simpa only [Nat.cast_add,Nat.cast_one,add_assoc,one_add_one_eq_two] using | |
| e.normalized_cellPair_profile_mass hd ha hX hY | |
| lemma prefix_low_frequency_stopping {K : ℝ} (hK : 32*(n+2:ℝ)≤K) | |
| {B : ℕ} {s lam : ℝ} (hd : RegularizedProfileData e.ρ e.N B s lam e.mass) | |
| (ha : e.a≤e.N) (hX : e.X.length=e.a) (hY : e.Y.length=e.a) | |
| (hs : farSeparatedGridND e.a K (fun j => (pathIndex e.X j:ℤ)) (fun j => (pathIndex e.Y j:ℤ))) : | |
| (2:ℝ)^(-(e.a:ℝ))*(∫ r in Icc (-((2:ℝ)^e.a)) ((2:ℝ)^e.a), ‖charFun e.weightedDistance r‖^2) ≤ | |
| 2*((2:ℝ)^e.N)^(4*(n+2:ℝ)*lam) := by | |
| let := hd.probability | |
| let := e.weightedDistance_isFinite hK hX hY hs | |
| have hm := e.weightedDistance_profile_mass hK hd ha hX hY hs | |
| have hr : (((2:ℝ)^e.N)^(2*(n+2:ℝ)*lam))^2=((2:ℝ)^e.N)^(4*(n+2:ℝ)*lam) := by | |
| rw [← Real.rpow_mul_natCast (by positivity : 0≤(2:ℝ)^e.N)] | |
| congr 1 | |
| norm_num | |
| ring | |
| have hb (r : ℝ) : ‖charFun e.weightedDistance r‖^2≤((2:ℝ)^e.N)^(4*(n+2:ℝ)*lam) := by | |
| rw [← hr] | |
| exact pow_le_pow_left₀ (norm_nonneg _) ((norm_charFun_le r).trans hm) 2 | |
| have hi : Integrable (fun r => ‖charFun e.weightedDistance r‖^2) | |
| (volume.restrict (Icc (-((2:ℝ)^e.a)) ((2:ℝ)^e.a))) := by | |
| simpa only [one_mul] using charFun_sq_integrable_finite e.weightedDistance | |
| (volume.restrict (Icc (-((2:ℝ)^e.a)) ((2:ℝ)^e.a))) 1 | |
| have h := integral_mono_ae hi (integrable_const (((2:ℝ)^e.N)^(4*(n+2:ℝ)*lam))) | |
| (Filter.Eventually.of_forall hb) | |
| apply (mul_le_mul_of_nonneg_left h (by positivity : 0≤(2:ℝ)^(-(e.a:ℝ)))).trans_eq | |
| have hwidth : 0≤(2:ℝ)^e.a - -((2:ℝ)^e.a) := by | |
| have hp : 0<(2:ℝ)^e.a := by positivity | |
| linarith | |
| rw [integral_const,Measure.real,Measure.restrict_apply_univ,Real.volume_Icc, | |
| ENNReal.toReal_ofReal hwidth,smul_eq_mul, | |
| Real.rpow_neg (by norm_num : (0:ℝ)≤2),Real.rpow_natCast] | |
| field_simp | |
| ring | |
| end PositiveDistanceState | |
| end CurrentFalconer.TreeND | |
| end GraphLiteralSupport | |
| end | |
| end | |
| end OAI | |