| import Mathlib |
| import OAI.Computability.VertexCover.PCP.SpectralCut |
| import OAI.Computability.VertexCover.Analysis.IntegralUpdate |
| import OAI.Computability.VertexCover.Analysis.VariancePi |
|
|
| namespace OAI |
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| namespace VertexCover.Product |
| open MeasureTheory |
|
|
| theorem update_right_measurePreserving {ι E : Type*} [Fintype ι] [DecidableEq ι] |
| [MeasurableSpace E] (μ : Measure E) [IsProbabilityMeasure μ] (j : ι) : |
| MeasurePreserving (fun p : (ι → E) × E => Function.update p.1 j p.2) |
| ((Measure.pi (fun _ : ι => μ)).prod μ) (Measure.pi (fun _ : ι => μ)) := by |
| exact (update_measurePreserving μ j).comp Measure.measurePreserving_swap |
|
|
| theorem update_right_ae {ι E : Type*} [Fintype ι] [DecidableEq ι] |
| [MeasurableSpace E] (μ : Measure E) [IsProbabilityMeasure μ] (j : ι) |
| {P : (ι → E) → Prop} (hP : ∀ᵐ s ∂Measure.pi (fun _ : ι => μ), P s) : |
| ∀ᵐ s ∂Measure.pi (fun _ : ι => μ), ∀ᵐ x ∂μ, P (Function.update s j x) := by |
| exact Measure.ae_ae_of_ae_prod ((update_right_measurePreserving μ j).quasiMeasurePreserving.ae hP) |
|
|
| theorem integral_update_right {ι E : Type*} [Fintype ι] [DecidableEq ι] |
| [MeasurableSpace E] (μ : Measure E) [IsProbabilityMeasure μ] (j : ι) |
| {f : (ι → E) → ℝ} (hf : Integrable f (Measure.pi (fun _ : ι => μ))) : |
| (∫ s, ∫ x, f (Function.update s j x) ∂μ ∂Measure.pi (fun _ : ι => μ)) = |
| ∫ s, f s ∂Measure.pi (fun _ : ι => μ) := by |
| have hp := update_right_measurePreserving μ j |
| calc |
| _ = ∫ p, f (Function.update p.1 j p.2) |
| ∂((Measure.pi (fun _ : ι => μ)).prod μ) := by |
| simpa only [Function.comp_apply] using |
| (integral_prod _ (hp.integrable_comp_of_integrable hf)).symm |
| _ = ∫ s, f s ∂Measure.map (fun p : (ι → E) × E => Function.update p.1 j p.2) |
| ((Measure.pi (fun _ : ι => μ)).prod μ) := by |
| exact (integral_map hp.measurable.aemeasurable |
| (by simpa only [hp.map_eq] using hf.aestronglyMeasurable)).symm |
| _ = _ := by rw [hp.map_eq] |
|
|
| end VertexCover.Product |
|
|
| namespace VertexCover.Cube |
| open MeasureTheory ProbabilityTheory |
| open scoped ENNReal |
|
|
| abbrev carrier (n : ℕ) : Set (Fin n → ℝ) := Set.univ.pi (fun _ => Set.Icc (-1 : ℝ) 1) |
|
|
| theorem measurableSet_carrier (n : ℕ) : MeasurableSet (carrier n) := |
| MeasurableSet.univ_pi (fun _ => measurableSet_Icc) |
|
|
| theorem ae_carrier (n : ℕ) : ∀ᵐ s ∂law (Fin n), s ∈ carrier n := by |
| rw [law_eq_restrict_volume] |
| exact (Measure.ae_ennreal_smul_measure_iff (pow_ne_zero _ (by norm_num) : |
| (1/2 : ℝ≥0∞) ^ Fintype.card (Fin n) ≠ 0)).mpr |
| (ae_restrict_mem (measurableSet_carrier n)) |
|
|
| theorem norm_le_one_of_mem_carrier {n : ℕ} {s : Fin n → ℝ} (hs : s ∈ carrier n) : |
| ‖s‖ ≤ 1 := by |
| apply (pi_norm_le_iff_of_nonneg (by norm_num : (0:ℝ) ≤ 1)).mpr |
| intro k |
| exact (abs_le.mpr (Set.mem_univ_pi.mp hs k)) |
|
|
| noncomputable def boundedVersion {n : ℕ} (f : (Fin n → ℝ) → ℝ) : (Fin n → ℝ) → ℝ := |
| (carrier n).indicator f |
|
|
| theorem boundedVersion_measurable {n : ℕ} {f : (Fin n → ℝ) → ℝ} |
| (hf : Measurable f) : Measurable (boundedVersion f) := |
| hf.indicator (measurableSet_carrier n) |
|
|
| theorem boundedVersion_ae {n : ℕ} (f : (Fin n → ℝ) → ℝ) : |
| boundedVersion f =ᵐ[law (Fin n)] f := by |
| filter_upwards [ae_carrier n] with s hs |
| exact Set.indicator_of_mem hs f |
|
|
| theorem boundedVersion_bound {n : ℕ} {f : (Fin n → ℝ) → ℝ} |
| (hf : LipschitzWith 1 f) (s : Fin n → ℝ) : |
| ‖boundedVersion f s‖ ≤ ‖f 0‖ + 1 := by |
| by_cases hs : s ∈ carrier n |
| · rw [boundedVersion, Set.indicator_of_mem hs] |
| have h := hf.dist_le_mul s 0 |
| simp only [NNReal.coe_one, one_mul, dist_zero_right, Real.dist_eq] at h |
| calc |
| ‖f s‖ ≤ ‖f s - f 0‖ + ‖f 0‖ := by simpa using norm_add_le (f s - f 0) (f 0) |
| _ ≤ 1 + ‖f 0‖ := by simpa only [Real.norm_eq_abs, add_comm] using add_le_add_right (h.trans (norm_le_one_of_mem_carrier hs)) ‖f 0‖ |
| _ = _ := add_comm _ _ |
| · rw [boundedVersion, Set.indicator_of_notMem hs] |
| simp only [norm_zero] |
| positivity |
|
|
| theorem variance_le_coordinate_sum {n : ℕ} {f : (Fin n → ℝ) → ℝ} |
| (hf : LipschitzWith 1 f) : |
| variance f (law (Fin n)) ≤ |
| ∑ j, ∫ s, VertexCover.Product.coordinateVariance intervalLaw f j s ∂law (Fin n) := by |
| have he := boundedVersion_ae f |
| have hb := VertexCover.Product.variance_pi_le intervalLaw n (boundedVersion f) |
| (boundedVersion_measurable hf.continuous.measurable) (boundedVersion_bound hf) |
| have hv : variance (boundedVersion f) (law (Fin n)) = variance f (law (Fin n)) := |
| variance_congr he |
| change variance (boundedVersion f) (law (Fin n)) ≤ _ at hb |
| rw [hv] at hb |
| convert hb using 1 |
| apply Finset.sum_congr rfl |
| intro j hj |
| apply integral_congr_ae |
| filter_upwards [VertexCover.Product.update_right_ae intervalLaw j he] with s hs |
| exact (variance_congr hs).symm |
|
|
| end VertexCover.Cube |
|
|
| end |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
|
|
| end OAI |
|
|
