openAI-math-dataset / lean /OAI /Probability /PosteriorReplicas /Geometry /SphereTubeQuantitative.lean
| import OAI.Probability.PosteriorReplicas.Gaussian.GaussianGramMoment | |
| namespace OAI | |
| open MeasureTheory ProbabilityTheory Set Matrix InnerProductSpace Metric EuclideanGeometry Filter | |
| open scoped ENNReal NNReal RealInnerProductSpace Topology | |
| noncomputable section | |
| namespace PosteriorReplicas | |
| lemma sphereLaw_closedBall {d : ℕ} [NeZero d] (z : Vec d) {r : ℝ} (hr : 0 < r) : | |
| sphereLaw d (closedBall z r) ≤ ENNReal.ofReal ((3:ℝ)*4^d) * ENNReal.ofReal (r^(d-1)) := by | |
| simpa only [sphereLaw, PosteriorRegression.ambientSphereLaw, PosteriorRegression.sphereLaw, | |
| finrank_euclideanSpace_fin] using | |
| PosteriorRegression.ambientSphereLaw_closedBall z hr (d-1) (by simp) | |
| lemma sphereLaw_support_unit {d : ℕ} : (sphereLaw d).support ⊆ closedBall 0 1 := by | |
| exact (PosteriorRegression.ambientSphereLaw_support (E := Vec d)).trans sphere_subset_closedBall | |
| lemma sphere_affine_tube_quantitative {d k : ℕ} [NeZero d] | |
| (L : AffineSubspace ℝ (Vec d)) [Nonempty L] | |
| (hd : Module.finrank ℝ L.direction + k ≤ d-1) {r : ℝ} (hr : 0 < r) (hr1 : r ≤ 1) : | |
| sphereLaw d {x | Metric.infDist x L ≤ r} ≤ ENNReal.ofReal ((72:ℝ)^d) * ENNReal.ofReal (r^k) := by | |
| classical | |
| obtain ⟨s,hcard,hcover⟩ := PosteriorRegression.exists_affine_tube_cover L hr hr1 | |
| let c : L.direction → Vec d := fun y => (y : Vec d) + (orthogonalProjection L 0 : Vec d) | |
| have hsub : {x | Metric.infDist x L ≤ r} ≤ᵐ[sphereLaw d] | |
| ⋃ y ∈ s, closedBall (c y) (2*r) := by | |
| filter_upwards [(sphereLaw d).support_mem_ae] with x hx | |
| intro hdist | |
| have hxunit : ‖x‖ ≤ 1 := by simpa using sphereLaw_support_unit hx | |
| obtain ⟨y,hy,hyx⟩ := hcover x hxunit hdist | |
| exact mem_iUnion₂.mpr ⟨y,hy,hyx⟩ | |
| let a := Module.finrank ℝ L.direction | |
| have hd0 : 1 ≤ d := NeZero.one_le | |
| have ha : a ≤ d := by dsimp [a]; omega | |
| have hpow : (3/r)^a * ((3:ℝ)*4^d) * (2*r)^(d-1) ≤ (72:ℝ)^d*r^k := by | |
| have hcan : (3/r)^a * ((3:ℝ)*4^d) * (2*r)^(d-1) = | |
| (3*4^d*3^a*2^(d-1)) * r^(d-1-a) := by | |
| have hdim : d-1 = a+(d-1-a) := by dsimp [a]; omega | |
| nth_rw 1 [hdim] | |
| rw [mul_pow, div_pow, pow_add] | |
| field_simp | |
| rw [hdim, pow_add] | |
| simp only [Nat.add_sub_cancel_left] | |
| ring | |
| rw [hcan] | |
| have hc : (3:ℝ)*4^d*3^a*2^(d-1) ≤ (72:ℝ)^d := by | |
| have h3 : (3:ℝ) ≤ 3^d := by simpa using pow_le_pow_right₀ (by norm_num : (1:ℝ) ≤ 3) hd0 | |
| have h3a : (3:ℝ)^a ≤ 3^d := pow_le_pow_right₀ (by norm_num) ha | |
| have h2 : (2:ℝ)^(d-1) ≤ 2^d := pow_le_pow_right₀ (by norm_num) (Nat.sub_le _ _) | |
| calc | |
| _ ≤ 3^d*4^d*3^d*2^d := by gcongr | |
| _ = (72:ℝ)^d := by simp only [← mul_pow]; congr 1; norm_num | |
| apply mul_le_mul hc ?_ (by positivity) (by positivity) | |
| exact pow_le_pow_of_le_one hr.le hr1 (by dsimp [a]; omega) | |
| calc | |
| _ ≤ sphereLaw d (⋃ y ∈ s, closedBall (c y) (2*r)) := measure_mono_ae hsub | |
| _ ≤ ∑ y ∈ s, sphereLaw d (closedBall (c y) (2*r)) := measure_biUnion_finset_le _ _ | |
| _ ≤ ∑ _y ∈ s, ENNReal.ofReal ((3:ℝ)*4^d)*ENNReal.ofReal ((2*r)^(d-1)) := by | |
| exact Finset.sum_le_sum (fun y _ => sphereLaw_closedBall (c y) (by positivity)) | |
| _ = (s.card : ℝ≥0∞) * (ENNReal.ofReal ((3:ℝ)*4^d)*ENNReal.ofReal ((2*r)^(d-1))) := by simp | |
| _ ≤ ENNReal.ofReal ((3/r)^a) * (ENNReal.ofReal ((3:ℝ)*4^d)*ENNReal.ofReal ((2*r)^(d-1))) := by | |
| gcongr | |
| exact_mod_cast ENNReal.ofReal_le_ofReal hcard | |
| _ ≤ _ := by | |
| rw [← ENNReal.ofReal_mul (by positivity), ← ENNReal.ofReal_mul (by positivity), | |
| ← ENNReal.ofReal_mul (by positivity)] | |
| apply ENNReal.ofReal_le_ofReal | |
| nlinarith [hpow] | |
| lemma spherical_tilt_affine_tube {d k : ℕ} [NeZero d] | |
| {f : Vec d → ℝ≥0∞} {M : ℝ≥0∞} (hfM : ∀ x, f x ≤ M) | |
| (L : AffineSubspace ℝ (Vec d)) [Nonempty L] | |
| (hd : Module.finrank ℝ L.direction+k ≤ d-1) {r : ℝ} (hr : 0 < r) (hr1 : r ≤ 1) : | |
| (sphereLaw d).withDensity f {x | Metric.infDist x L ≤ r} ≤ | |
| (M*ENNReal.ofReal ((72:ℝ)^d)) * ENNReal.ofReal (r^k) := by | |
| rw [withDensity_apply _ (measurableSet_le (Metric.continuous_infDist_pt _).measurable measurable_const)] | |
| calc | |
| _ ≤ ∫⁻ _x in {x | Metric.infDist x L ≤ r}, M ∂sphereLaw d := lintegral_mono hfM | |
| _ = M * sphereLaw d {x | Metric.infDist x L ≤ r} := by simp | |
| _ ≤ _ := by | |
| rw [mul_assoc] | |
| exact mul_le_mul' le_rfl (sphere_affine_tube_quantitative L hd hr hr1) | |
| end PosteriorReplicas | |
| end | |
| end OAI | |