openAI-math-dataset / lean /OAI /MeasureTheory /FalconerCompletion /Estimates /NormLinearErrorIdentity.lean
| import Mathlib | |
| import OAI.MeasureTheory.Falconer.Geometry.BinBands | |
| import OAI.MeasureTheory.Falconer.Model | |
| import OAI.MeasureTheory.FalconerCompletion.Measures.LocalDistanceCollisionBound | |
| namespace OAI | |
| /-! Quantitative estimates for the distance-set construction. -/ | |
| open _root_.OAI.PlanarFalconer | |
| section | |
| section | |
| open Set MeasureTheory | |
| open scoped ComplexConjugate RealInnerProductSpace | |
| namespace GraphLiteralSupport | |
| namespace CurrentFalconer | |
| open Set Metric MeasureTheory | |
| open scoped RealInnerProductSpace | |
| variable {V : Type*} [NormedAddCommGroup V] [InnerProductSpace ℝ V] | |
| lemma norm_linear_error_identityND (a v : V) (ha : 0<‖a‖) : | |
| ‖v‖-inner ℝ (‖a‖⁻¹ • a) v = (‖v-a‖^2-(‖v‖-‖a‖)^2)/(2*‖a‖) := by | |
| rw [real_inner_smul_left,norm_sub_sq_real,real_inner_comm v a] | |
| field_simp [ne_of_gt ha] | |
| ring | |
| lemma norm_linear_errorND (a v : V) (ha : 0<‖a‖) : | |
| 0 ≤ ‖v‖-inner ℝ (‖a‖⁻¹ • a) v ∧ | |
| ‖v‖-inner ℝ (‖a‖⁻¹ • a) v ≤ ‖v-a‖^2/(2*‖a‖) := by | |
| have hn : ‖‖a‖⁻¹ • a‖=1 := by | |
| rw [norm_smul,Real.norm_eq_abs,abs_of_pos (inv_pos.mpr ha),inv_mul_cancel₀ (ne_of_gt ha)] | |
| constructor | |
| · have hh := (le_abs_self (inner ℝ (‖a‖⁻¹ • a) v)).trans (abs_real_inner_le_norm _ _) | |
| rw [hn,one_mul] at hh | |
| linarith | |
| · rw [norm_linear_error_identityND a v ha] | |
| gcongr | |
| exact sub_le_self _ (sq_nonneg _) | |
| lemma distance_linearizationND (x y c e : V) {δ ρ : ℝ} | |
| (hρ : 0<ρ) (hsep : ρ ≤ ‖c-e‖) (hx : dist x c ≤ δ) (hy : dist y e ≤ δ) : | |
| |dist x y-inner ℝ (‖c-e‖⁻¹ • (c-e)) (x-y)| ≤ 2*δ^2/ρ := by | |
| have hδ : 0 ≤ δ := dist_nonneg.trans hx | |
| have hce : 0<‖c-e‖ := hρ.trans_le hsep | |
| have hh := norm_linear_errorND (c-e) (x-y) hce | |
| have hnear : ‖(x-y)-(c-e)‖ ≤ 2*δ := by | |
| have he : (x-y)-(c-e)=(x-c)-(y-e) := by abel | |
| rw [he] | |
| exact (norm_sub_le _ _).trans (by simpa only [dist_eq_norm, two_mul] using add_le_add hx hy) | |
| rw [dist_eq_norm,abs_of_nonneg hh.1] | |
| calc | |
| _ ≤ ‖(x-y)-(c-e)‖^2/(2*‖c-e‖) := hh.2 | |
| _ ≤ (2*δ)^2/(2*ρ) := by gcongr | |
| _ = _ := by ring | |
| lemma normalized_direction_distND (a v : V) (ha : 0<‖a‖) (hv : 0<‖v‖) : | |
| dist (‖a‖⁻¹ • a) (‖v‖⁻¹ • v) ≤ 2*‖a-v‖/‖a‖ := by | |
| have he : ‖a‖⁻¹ • a-‖v‖⁻¹ • v = | |
| ‖a‖⁻¹ • (a-v)+(‖a‖⁻¹-‖v‖⁻¹) • v := by | |
| simp only [smul_sub,sub_smul]; abel | |
| have hn : ‖(‖a‖⁻¹-‖v‖⁻¹) • v‖=|‖v‖-‖a‖|/‖a‖ := by | |
| rw [norm_smul,Real.norm_eq_abs] | |
| have he : ‖a‖⁻¹-‖v‖⁻¹=(‖v‖-‖a‖)/(‖a‖*‖v‖) := by field_simp | |
| rw [he,abs_div,abs_mul,abs_of_pos ha,abs_of_pos hv] | |
| field_simp | |
| rw [dist_eq_norm,he] | |
| calc | |
| _ ≤ ‖‖a‖⁻¹ • (a-v)‖+‖(‖a‖⁻¹-‖v‖⁻¹) • v‖ := norm_add_le _ _ | |
| _ = ‖a-v‖/‖a‖+|‖v‖-‖a‖|/‖a‖ := by | |
| rw [hn,norm_smul,Real.norm_eq_abs,abs_of_pos (inv_pos.mpr ha)]; ring | |
| _ ≤ ‖a-v‖/‖a‖+‖a-v‖/‖a‖ := by | |
| gcongr | |
| simpa only [norm_sub_rev] using abs_norm_sub_norm_le v a | |
| _ = _ := by ring | |
| end CurrentFalconer | |
| end GraphLiteralSupport | |
| end | |
| end | |
| section | |
| section | |
| open Set MeasureTheory | |
| open scoped ComplexConjugate RealInnerProductSpace | |
| namespace GraphLiteralSupport | |
| namespace CurrentFalconer | |
| open Set Metric MeasureTheory | |
| open scoped ENNReal RealInnerProductSpace | |
| variable {V : Type*} [NormedAddCommGroup V] [InnerProductSpace ℝ V] | |
| [MeasurableSpace V] [BorelSpace V] [SecondCountableTopology V] | |
| open PlanarFalconer | |
| omit [MeasurableSpace V] [BorelSpace V] [SecondCountableTopology V] in | |
| lemma projection_direction_changeND (y y' c e v : V) {δ ε w : ℝ} | |
| (hy : dist y c ≤ δ) (hy' : dist y' c ≤ δ) (he : dist e v ≤ ε) | |
| (hproj : |inner ℝ (y-y') e| ≤ w) : |inner ℝ (y-y') v| ≤ w+2*δ*ε := by | |
| have hδ : 0 ≤ δ := dist_nonneg.trans hy | |
| have hd : ‖y-y'‖ ≤ 2*δ := by | |
| rw [← dist_eq_norm] | |
| exact (dist_triangle y c y').trans (by rw [dist_comm c y']; linarith) | |
| have hh := abs_real_inner_le_norm (y-y') (v-e) | |
| rw [inner_sub_right] at hh | |
| have hev : ‖v-e‖ ≤ ε := by simpa only [← dist_eq_norm,dist_comm v e] using he | |
| have hh' : |inner ℝ (y-y') v-inner ℝ (y-y') e| ≤ 2*δ*ε := | |
| hh.trans (mul_le_mul hd hev (norm_nonneg _) (by positivity)) | |
| have htri := abs_add_le (inner ℝ (y-y') v-inner ℝ (y-y') e) (inner ℝ (y-y') e) | |
| simp only [sub_add_cancel] at htri | |
| linarith | |
| theorem suffix_witness_local_collisionND (μ : Measure V) [IsFiniteMeasure μ] | |
| {P Q : Set V} (hP : MeasurableSet P) (hQ : MeasurableSet Q) | |
| (c c' x₀ y₀ : V) {δ ρ w W : ℝ} | |
| (hρ : 0<ρ) (hw : 0<w) (hsep : ρ ≤ ‖c-c'‖) | |
| (hPcell : ∀ x ∈ P, dist x c ≤ δ) (hQcell : ∀ y ∈ Q, dist y c' ≤ δ) | |
| (hx₀ : x₀ ∈ P) (hy₀ : y₀ ∈ Q) (hxy₀ : x₀ ≠ y₀) | |
| (htaylor : 2*δ^2/ρ ≤ w) (hwidth : w+8*δ^2/ρ ≤ W) {B : ℝ≥0∞} | |
| (hproj : ((ProbabilityTheory.cond μ Q).prod (ProbabilityTheory.cond μ Q)) | |
| {z : V × V | |inner ℝ (z.1-z.2) (‖x₀-y₀‖⁻¹ • (x₀-y₀))| ≤ W} ≤ B) : | |
| (((μ.restrict P).prod (μ.restrict Q)).prod ((μ.restrict P).prod (μ.restrict Q))) | |
| {z | |dist z.1.1 z.1.2-dist z.2.1 z.2.2| ≤ w} ≤ | |
| ((bandOffsets 3).card:ℝ≥0∞)*B*(μ P)^2*(μ Q)^2 := by | |
| have hδ : 0 ≤ δ := dist_nonneg.trans (hPcell x₀ hx₀) | |
| have hdiff : ‖(c-c')-(x₀-y₀)‖ ≤ 2*δ := by | |
| have he : (c-c')-(x₀-y₀)=(c-x₀)-(c'-y₀) := by abel | |
| rw [he] | |
| apply (norm_sub_le _ _).trans | |
| have hp := hPcell x₀ hx₀ | |
| have hq := hQcell y₀ hy₀ | |
| simp only [dist_eq_norm,norm_sub_rev x₀ c,norm_sub_rev y₀ c'] at hp hq | |
| linarith | |
| have hdir : dist (‖c-c'‖⁻¹ • (c-c')) (‖x₀-y₀‖⁻¹ • (x₀-y₀)) ≤ 4*δ/ρ := by | |
| apply (normalized_direction_distND (c-c') (x₀-y₀) (hρ.trans_le hsep) | |
| (norm_pos_iff.mpr (sub_ne_zero.mpr hxy₀))).trans | |
| apply (div_le_div₀ (by positivity) (by nlinarith : 2*‖(c-c')-(x₀-y₀)‖ ≤ 4*δ) | |
| hρ hsep) | |
| have haff : ∀ x ∈ P, ∀ y ∈ Q, | |
| |dist x y-inner ℝ (x-y) (‖c-c'‖⁻¹ • (c-c'))| ≤ w := by | |
| intro x hx y hy | |
| have hh := distance_linearizationND x y c c' hρ hsep (hPcell x hx) (hQcell y hy) | |
| rw [real_inner_comm] at hh | |
| exact hh.trans htaylor | |
| apply local_distance_conditional_collisionND μ hP hQ (‖c-c'‖⁻¹ • (c-c')) hw haff | |
| apply le_trans _ hproj | |
| apply measure_mono_ae | |
| have hcond : ∀ᵐ y ∂ProbabilityTheory.cond μ Q, y ∈ Q := by | |
| exact Measure.ae_smul_measure (ae_restrict_mem hQ) _ | |
| filter_upwards [Measure.quasiMeasurePreserving_fst.ae hcond, | |
| Measure.quasiMeasurePreserving_snd.ae hcond] with yy hy hy' | |
| intro hh | |
| have hi := projection_direction_changeND yy.1 yy.2 c' | |
| (‖c-c'‖⁻¹ • (c-c')) (‖x₀-y₀‖⁻¹ • (x₀-y₀)) | |
| (hQcell yy.1 hy) (hQcell yy.2 hy') hdir hh | |
| apply hi.trans | |
| calc | |
| w+2*δ*(4*δ/ρ) = w+8*δ^2/ρ := by ring | |
| _ ≤ W := hwidth | |
| end CurrentFalconer | |
| end GraphLiteralSupport | |
| end | |
| end | |
| end OAI | |