openAI-math-dataset / lean /OAI /MeasureTheory /FalconerCompletion /Angular /InnerNormalizedOfClose.lean
| import Mathlib | |
| import OAI.MeasureTheory.Falconer.Model | |
| import OAI.MeasureTheory.FalconerCompletion.Graphs.NearGrid | |
| import OAI.MeasureTheory.FalconerCompletion.Recursion.ExistsFarSeparationPower2 | |
| namespace OAI | |
| /-! Angular cutoffs and derivative estimates. -/ | |
| open _root_.OAI.PlanarFalconer | |
| section | |
| section | |
| open Set MeasureTheory | |
| open scoped ComplexConjugate RealInnerProductSpace | |
| namespace GraphLiteralSupport | |
| namespace CurrentFalconer.TreeND | |
| open Set Metric MeasureTheory PlanarFalconer _root_.OAI.GraphLiteralSupport.Attempt4FullFar | |
| open scoped ContDiff Topology | |
| lemma inner_normalized_of_close {V : Type*} [NormedAddCommGroup V] [InnerProductSpace ℝ V] | |
| {c u : V} (hc : 0<‖c‖) (hu : ‖u-c‖≤‖c‖/2) : | |
| (1/3:ℝ)≤ inner ℝ (‖u‖⁻¹ • u) (‖c‖⁻¹ • c) := by | |
| have ht : ‖u‖≤‖u-c‖+‖c‖ := by simpa only [sub_add_cancel] using norm_add_le (u-c) c | |
| have ht' : ‖c‖≤‖u-c‖+‖u‖ := by | |
| simpa only [sub_add_cancel,norm_sub_rev] using norm_add_le (c-u) u | |
| have hp : 0<‖u‖ := by linarith | |
| have hi := (abs_le.mp (abs_real_inner_le_norm (u-c) c)).1 | |
| rw [inner_sub_left,real_inner_self_eq_norm_sq] at hi | |
| have hi' : ‖c‖^2/2≤ inner ℝ u c := by nlinarith | |
| rw [real_inner_smul_left,real_inner_smul_right] | |
| have h : (1/3:ℝ)*(‖u‖*‖c‖)≤ inner ℝ u c := by nlinarith | |
| apply (le_div_iff₀ (mul_pos hp hc)).mpr at h | |
| simpa only [div_eq_mul_inv,mul_inv_rev,mul_assoc,mul_comm,mul_left_comm] using h | |
| variable {d : ℕ} | |
| noncomputable def separatedAxis (a : ℕ) (X Y : Fin d → ℤ) : EuclideanSpace ℝ (Fin d) := | |
| radialVector (Attempt4FullFar.ambientGridCenter a X) (Attempt4FullFar.ambientGridCenter a Y) | |
| noncomputable def separatedCutoff (a : ℕ) (X Y : Fin d → ℤ) (w : EuclideanSpace ℝ (Fin d)) : ℂ := | |
| Real.smoothTransition (4*inner ℝ w (separatedAxis a X Y)+1/2) | |
| lemma separatedCutoff_contDiff (a : ℕ) (X Y : Fin d → ℤ) : | |
| ContDiff ℝ ∞ (separatedCutoff a X Y) := | |
| Complex.ofRealCLM.contDiff.comp (Real.smoothTransition.contDiff.comp | |
| ((contDiff_const.mul (contDiff_id.inner ℝ contDiff_const)).add contDiff_const)) | |
| lemma separatedCutoff_norm_le (a : ℕ) (X Y : Fin d → ℤ) (w : EuclideanSpace ℝ (Fin d)) : | |
| ‖separatedCutoff a X Y w‖≤1 := by | |
| rw [separatedCutoff,Complex.norm_real,Real.norm_eq_abs,abs_of_nonneg (Real.smoothTransition.nonneg _)] | |
| exact Real.smoothTransition.le_one _ | |
| lemma separated_direction_inner {a : ℕ} {K : ℝ} (hK : 32*(d+1:ℝ)≤K) | |
| {X Y : Fin d → ℤ} (hs : farSeparatedGridND a K X Y) | |
| {x y : EuclideanSpace ℝ (Fin d)} (hx : x∈Attempt4FullFar.ambientGridCell a X) (hy : y∈Attempt4FullFar.ambientGridCell a Y) : | |
| (1/3:ℝ)≤ inner ℝ (radialVector x y) (separatedAxis a X Y) := by | |
| have hp : 0<(2:ℝ)^a := by positivity | |
| have hKp : 0<K := by have hd : (0:ℝ)≤d := Nat.cast_nonneg d; linarith | |
| have hc : 0<‖Attempt4FullFar.ambientGridCenter a X-Attempt4FullFar.ambientGridCenter a Y‖ := | |
| (div_pos hKp (mul_pos (by norm_num) hp)).trans_le hs.1 | |
| have hcx := Attempt4FullFar.ambientGridCell_dist_le hx (Attempt4FullFar.ambientGridCenter_mem a X) | |
| have hcy := Attempt4FullFar.ambientGridCell_dist_le hy (Attempt4FullFar.ambientGridCenter_mem a Y) | |
| have he : (x-y)-(Attempt4FullFar.ambientGridCenter a X-Attempt4FullFar.ambientGridCenter a Y)= | |
| (x-Attempt4FullFar.ambientGridCenter a X)-(y-Attempt4FullFar.ambientGridCenter a Y) := by abel | |
| have hb := norm_sub_le (x-Attempt4FullFar.ambientGridCenter a X) (y-Attempt4FullFar.ambientGridCenter a Y) | |
| have hk := div_le_div_of_nonneg_right hK hp.le | |
| have hlo := hs.1 | |
| have heq : K/(8*(2:ℝ)^a)=(K/(2:ℝ)^a)/8 := by ring | |
| rw [heq,dist_eq_norm] at hlo | |
| have hk' : 32*((d+1:ℝ)/(2:ℝ)^a)≤K/(2:ℝ)^a := by simpa only [mul_div_assoc] using hk | |
| rw [dist_eq_norm,norm_sub_rev] at hcx hcy | |
| apply inner_normalized_of_close hc | |
| rw [he] | |
| linarith | |
| lemma separatedCutoff_local {a : ℕ} {K : ℝ} (hK : 32*(d+1:ℝ)≤K) | |
| {X Y : Fin d → ℤ} (hs : farSeparatedGridND a K X Y) | |
| {x y : EuclideanSpace ℝ (Fin d)} (hx : x∈Attempt4FullFar.ambientGridCell a X) (hy : y∈Attempt4FullFar.ambientGridCell a Y) : | |
| separatedCutoff a X Y =ᶠ[𝓝 (radialVector x y)] (fun _ => 1) ∧ | |
| separatedCutoff a X Y =ᶠ[𝓝 (-(radialVector x y))] (fun _ => 0) := by | |
| have hi := separated_direction_inner hK hs hx hy | |
| have hf : Continuous (fun w : EuclideanSpace ℝ (Fin d) => inner ℝ w (separatedAxis a X Y)) := | |
| continuous_id.inner continuous_const | |
| constructor | |
| · have hh : ∀ᶠ w in 𝓝 (radialVector x y), (1/4:ℝ)< inner ℝ w (separatedAxis a X Y) := | |
| hf.continuousAt.eventually (Ioi_mem_nhds (by linarith)) | |
| filter_upwards [hh] with w hw | |
| rw [separatedCutoff,Real.smoothTransition.one_of_one_le (by linarith),Complex.ofReal_one] | |
| · have hh : ∀ᶠ w in 𝓝 (-(radialVector x y)), inner ℝ w (separatedAxis a X Y)< -(1/4:ℝ) := | |
| hf.continuousAt.eventually (Iio_mem_nhds (by simp only [inner_neg_left]; linarith)) | |
| filter_upwards [hh] with w hw | |
| rw [separatedCutoff,Real.smoothTransition.zero_of_nonpos (by linarith),Complex.ofReal_zero] | |
| end CurrentFalconer.TreeND | |
| end GraphLiteralSupport | |
| end | |
| end | |
| section | |
| section | |
| open Set MeasureTheory | |
| open scoped ComplexConjugate RealInnerProductSpace | |
| namespace GraphLiteralSupport | |
| namespace CurrentFalconer.TreeND | |
| open Set Metric MeasureTheory | |
| open scoped ContDiff Topology | |
| variable {d : ℕ} | |
| noncomputable def gridOffsetVector (Z : Fin d → ℤ) : EuclideanSpace ℝ (Fin d) := | |
| WithLp.toLp 2 (fun j => (Z j:ℝ)) | |
| lemma grid_centers_sub (a : ℕ) (X Y : Fin d → ℤ) : | |
| Attempt4FullFar.ambientGridCenter a X-Attempt4FullFar.ambientGridCenter a Y= | |
| ((2:ℝ)^a)⁻¹ • gridOffsetVector (X-Y) := by | |
| ext j | |
| change ((X j:ℝ)+1/2)/(2:ℝ)^a-((Y j:ℝ)+1/2)/(2:ℝ)^a= | |
| ((2:ℝ)^a)⁻¹*((X j-Y j:ℤ):ℝ) | |
| rw [Int.cast_sub] | |
| ring | |
| lemma normalized_pos_smul {V : Type*} [NormedAddCommGroup V] [NormedSpace ℝ V] | |
| {r : ℝ} (hr : 0<r) (v : V) : | |
| ‖r • v‖⁻¹ • (r • v)=‖v‖⁻¹ • v := by | |
| simp [norm_smul,Real.norm_eq_abs,abs_of_pos hr,smul_smul,mul_inv_rev,hr.ne'] | |
| lemma separatedAxis_offset (a : ℕ) (X Y : Fin d → ℤ) : | |
| separatedAxis a X Y=‖gridOffsetVector (X-Y)‖⁻¹ • gridOffsetVector (X-Y) := by | |
| unfold separatedAxis radialVector | |
| rw [grid_centers_sub,normalized_pos_smul (by positivity : 0<((2:ℝ)^a)⁻¹)] | |
| lemma separatedCutoff_offset (a : ℕ) (X Y : Fin d → ℤ) : | |
| separatedCutoff a X Y=separatedCutoff 0 (X-Y) 0 := by | |
| funext w | |
| simp only [separatedCutoff,separatedAxis_offset,sub_zero] | |
| noncomputable def separatedOffsets (K : ℝ) : Set (Fin d → ℤ) := | |
| {Z | ∀ j, Z j∈Icc (-⌈4*K⌉) ⌈4*K⌉} | |
| lemma separatedOffsets_finite (K : ℝ) : (separatedOffsets (d:=d) K).Finite := | |
| Set.Finite.pi' (fun _ => Set.finite_Icc _ _) | |
| lemma gridOffset_mem {a : ℕ} {K : ℝ} {X Y : Fin d → ℤ} | |
| (hs : Attempt4FullFar.farSeparatedGridND a K X Y) : X-Y∈separatedOffsets K := by | |
| intro j | |
| have h := Attempt4FullFar.ambientGridCenter_coordinate_bound hs.2 j | |
| have he : (X j:ℝ)+1/2-(2:ℝ)^a*Attempt4FullFar.ambientGridCenter a Y j=(X j-Y j:ℝ) := by | |
| rw [Attempt4FullFar.ambientGridCenter_apply] | |
| field_simp | |
| ring | |
| have he' : (2:ℝ)^a*(4*K/(2:ℝ)^a)=4*K := by field_simp | |
| rw [he,he'] at h | |
| have hc := Int.le_ceil (4*K) | |
| have hlo : -(⌈4*K⌉:ℤ)≤X j-Y j := by | |
| have hh : -((⌈4*K⌉:ℤ):ℝ)≤(X j-Y j:ℝ) := by linarith [abs_le.mp h] | |
| exact_mod_cast hh | |
| have hhi : X j-Y j≤(⌈4*K⌉:ℤ) := by | |
| have hh : (X j-Y j:ℝ)≤((⌈4*K⌉:ℤ):ℝ) := by linarith [abs_le.mp h] | |
| exact_mod_cast hh | |
| exact ⟨hlo,hhi⟩ | |
| lemma eventually_all_separated_cutoffs (K : ℝ) | |
| (P : (EuclideanSpace ℝ (Fin d) → ℂ) → ℕ → Prop) | |
| (hP : ∀ Z∈separatedOffsets K, ∀ᶠ N : ℕ in Filter.atTop, P (separatedCutoff 0 Z 0) N) : | |
| ∀ᶠ N : ℕ in Filter.atTop, ∀ (a : ℕ) (X Y : Fin d → ℤ), | |
| Attempt4FullFar.farSeparatedGridND a K X Y → P (separatedCutoff a X Y) N := by | |
| have h := (Filter.eventually_all_finite (separatedOffsets_finite K)).mpr hP | |
| filter_upwards [h] with N hN | |
| intro a X Y hs | |
| rw [separatedCutoff_offset] | |
| exact hN (X-Y) (gridOffset_mem hs) | |
| lemma uniform_separatedCutoff_derivatives (K : ℝ) : | |
| ∃ C : ℕ → ℝ, (∀ m, 0≤C m) ∧ ∀ (a : ℕ) (X Y : Fin d → ℤ), | |
| Attempt4FullFar.farSeparatedGridND a K X Y → ∀ (m : ℕ) (w : EuclideanSpace ℝ (Fin d)), | |
| ‖w‖=1 → ‖iteratedFDeriv ℝ m (separatedCutoff a X Y) w‖≤C m := by | |
| classical | |
| have hm (m : ℕ) : ∃ C : ℝ, 0≤C ∧ ∀ (a : ℕ) (X Y : Fin d → ℤ), | |
| Attempt4FullFar.farSeparatedGridND a K X Y → ∀ (w : EuclideanSpace ℝ (Fin d)), | |
| ‖w‖=1 → ‖iteratedFDeriv ℝ m (separatedCutoff a X Y) w‖≤C := by | |
| have hz (Z : Fin d → ℤ) : ∃ B : ℝ, 0≤B ∧ ∀ (w : EuclideanSpace ℝ (Fin d)), | |
| ‖w‖=1 → ‖iteratedFDeriv ℝ m (separatedCutoff 0 Z 0) w‖≤B := by | |
| have hf := (ContDiff.continuous_iteratedFDeriv (m:=m) (n:=∞) (by exact_mod_cast (le_top : (m:ℕ∞)≤⊤)) | |
| (separatedCutoff_contDiff 0 Z 0)).norm | |
| obtain ⟨B,hB,hb⟩ := (bddAbove_iff_exists_ge (0:ℝ)).mp | |
| ((isCompact_sphere (0:EuclideanSpace ℝ (Fin d)) 1).bddAbove_image hf.continuousOn) | |
| refine ⟨B,hB,?_⟩ | |
| intro w hw | |
| exact hb _ ⟨w,by simpa [Metric.mem_sphere] using hw,rfl⟩ | |
| choose B hB hb using hz | |
| obtain ⟨C,hC,hc⟩ := (bddAbove_iff_exists_ge (0:ℝ)).mp | |
| (((separatedOffsets_finite (d:=d) K).image B).bddAbove) | |
| refine ⟨C,hC,?_⟩ | |
| intro a X Y hs w hw | |
| rw [separatedCutoff_offset] | |
| exact (hb (X-Y) w hw).trans (hc _ ⟨X-Y,gridOffset_mem hs,rfl⟩) | |
| choose C hC hbound using hm | |
| exact ⟨C,hC,fun a X Y hs m w hw => hbound m a X Y hs w hw⟩ | |
| end CurrentFalconer.TreeND | |
| end GraphLiteralSupport | |
| end | |
| end | |
| end OAI | |