openAI-math-dataset / lean /OAI /Combinatorics /BalancedRyser /Profiles /ConditionalOrderDetection.lean
| import OAI.Combinatorics.BalancedRyser.Probability.ConditionalOrderVariance | |
| import OAI.Combinatorics.BalancedRyser.Profiles.ProfileDetectionConcentration | |
| namespace OAI | |
| open MeasureTheory ProbabilityTheory | |
| open scoped BigOperators | |
| namespace Balanced.ConditionalOrderDetection | |
| open ConditionalOrderVariance UncoveredSampling | |
| variable {F : Type*} [Field F] [Fintype F] [DecidableEq F] | |
| noncomputable def planeObservable (U V : Finset (Line F)) | |
| (I : Finset U) (J : Finset V) (p : OrderPair U V) : ℝ := | |
| planeFraction (populationLines U (I.map p.1.toEmbedding)) | |
| (populationLines V (J.map p.2.toEmbedding)) | |
| noncomputable def targetObservable (U V : Finset (Line F)) | |
| (I : Finset U) (J : Finset V) (x : U) (p : OrderPair U V) : ℝ := | |
| targetFraction x.val (populationLines U (I.map p.1.toEmbedding)) | |
| (populationLines V (J.map p.2.toEmbedding)) | |
| private theorem finite_memLp {Ω : Type*} [Fintype Ω] | |
| [MeasurableSpace Ω] [MeasurableSingletonClass Ω] | |
| (μ : Measure Ω) [IsFiniteMeasure μ] (f : Ω → ℝ) : MemLp f 2 μ := by | |
| apply MemLp.of_bound (measurable_of_countable f).aestronglyMeasurable | |
| (∑ ω, |f ω|) | |
| apply Filter.Eventually.of_forall | |
| intro ω | |
| simp only [Real.norm_eq_abs] | |
| exact Finset.single_le_sum (fun z _ => abs_nonneg (f z)) (Finset.mem_univ ω) | |
| section Conditional | |
| variable (U V : Finset (Line F)) | |
| variable [MeasurableSpace (OrderPair U V)] [MeasurableSingletonClass (OrderPair U V)] | |
| variable (I : Finset U) (J : Finset V) (i x : U) | |
| theorem unmarked_probability_le_of_mean_gap | |
| (hUV : Disjoint U V) (hi : i ∉ I) {ε : ℝ} (hε : 0 < ε) | |
| (hgap : 2 * ε < | |
| |(∫ p, targetObservable U V I J x p ∂(conditionedOrderLaw U V i x).toMeasure) - | |
| ∫ p, planeObservable U V I J p ∂(conditionedOrderLaw U V i x).toMeasure|) : | |
| (conditionedOrderLaw U V i x).toMeasure.real | |
| {p | |targetObservable U V I J x p - planeObservable U V I J p| ≤ ε} ≤ | |
| 32 * ((I.card : ℝ) + J.card) / ((Fintype.card F : ℝ)^2 * ε^2) := by | |
| have h := ProfileDetectionConcentration.unmarked_probability_le_of_mean_gap | |
| (finite_memLp _ (targetObservable U V I J x)) | |
| (finite_memLp _ (planeObservable U V I J)) hε hgap | |
| have htarget := target_conditioned_order_measure_variance_le U V hUV I J i x hi | |
| have hplane := plane_conditioned_order_measure_variance_le U V I J i x hi | |
| calc | |
| _ ≤ 4 * (variance (targetObservable U V I J x) | |
| (conditionedOrderLaw U V i x).toMeasure + | |
| variance (planeObservable U V I J) | |
| (conditionedOrderLaw U V i x).toMeasure) / ε^2 := h | |
| _ ≤ 4 * (4 * ((I.card : ℝ) + J.card) / (Fintype.card F : ℝ)^2 + | |
| 4 * ((I.card : ℝ) + J.card) / (Fintype.card F : ℝ)^2) / ε^2 := by | |
| gcongr | |
| · exact htarget | |
| · exact hplane | |
| _ = _ := by ring | |
| theorem unmarked_probability_le_of_profile_gap | |
| (hUV : Disjoint U V) (hi : i ∉ I) {a b ε : ℝ} | |
| (hε : 0 < ε) (hgap : 3 * ε < |a - b|) | |
| (hTa : |(∫ p, targetObservable U V I J x p | |
| ∂(conditionedOrderLaw U V i x).toMeasure) - a| ≤ ε / 2) | |
| (hPb : |(∫ p, planeObservable U V I J p | |
| ∂(conditionedOrderLaw U V i x).toMeasure) - b| ≤ ε / 2) : | |
| (conditionedOrderLaw U V i x).toMeasure.real | |
| {p | |targetObservable U V I J x p - planeObservable U V I J p| ≤ ε} ≤ | |
| 32 * ((I.card : ℝ) + J.card) / ((Fintype.card F : ℝ)^2 * ε^2) := by | |
| apply unmarked_probability_le_of_mean_gap U V I J i x hUV hi hε | |
| have h₁ := abs_sub_le a | |
| (∫ p, targetObservable U V I J x p ∂(conditionedOrderLaw U V i x).toMeasure) b | |
| have h₂ := abs_sub_le | |
| (∫ p, targetObservable U V I J x p ∂(conditionedOrderLaw U V i x).toMeasure) | |
| (∫ p, planeObservable U V I J p ∂(conditionedOrderLaw U V i x).toMeasure) b | |
| rw [abs_sub_comm a | |
| (∫ p, targetObservable U V I J x p ∂(conditionedOrderLaw U V i x).toMeasure)] at h₁ | |
| linarith | |
| theorem marked_probability_ge_of_profile_gap | |
| (hUV : Disjoint U V) (hi : i ∉ I) {a b ε : ℝ} | |
| (hε : 0 < ε) (hgap : 3 * ε < |a - b|) | |
| (hTa : |(∫ p, targetObservable U V I J x p | |
| ∂(conditionedOrderLaw U V i x).toMeasure) - a| ≤ ε / 2) | |
| (hPb : |(∫ p, planeObservable U V I J p | |
| ∂(conditionedOrderLaw U V i x).toMeasure) - b| ≤ ε / 2) : | |
| 1 - 32 * ((I.card : ℝ) + J.card) / ((Fintype.card F : ℝ)^2 * ε^2) ≤ | |
| (conditionedOrderLaw U V i x).toMeasure.real | |
| {p | ε < |targetObservable U V I J x p - planeObservable U V I J p|} := by | |
| rw [ProfileDetectionConcentration.marked_probability_eq | |
| (finite_memLp _ (targetObservable U V I J x)) | |
| (finite_memLp _ (planeObservable U V I J))] | |
| exact sub_le_sub_left | |
| (unmarked_probability_le_of_profile_gap U V I J i x hUV hi hε hgap hTa hPb) 1 | |
| theorem unmarked_probability_le_scaled | |
| (hUV : Disjoint U V) (hi : i ∉ I) {a b ε C : ℝ} | |
| (hε : 0 < ε) (hgap : 3 * ε < |a - b|) | |
| (hTa : |(∫ p, targetObservable U V I J x p | |
| ∂(conditionedOrderLaw U V i x).toMeasure) - a| ≤ ε / 2) | |
| (hPb : |(∫ p, planeObservable U V I J p | |
| ∂(conditionedOrderLaw U V i x).toMeasure) - b| ≤ ε / 2) | |
| (hsize : (I.card : ℝ) + J.card ≤ C * Fintype.card F) : | |
| (conditionedOrderLaw U V i x).toMeasure.real | |
| {p | |targetObservable U V I J x p - planeObservable U V I J p| ≤ ε} ≤ | |
| 32 * C / ((Fintype.card F : ℝ) * ε^2) := by | |
| have hq : (Fintype.card F : ℝ) ≠ 0 := by positivity | |
| calc | |
| _ ≤ 32 * ((I.card : ℝ) + J.card) / ((Fintype.card F : ℝ)^2 * ε^2) := | |
| unmarked_probability_le_of_profile_gap U V I J i x hUV hi hε hgap hTa hPb | |
| _ ≤ 32 * (C * Fintype.card F) / ((Fintype.card F : ℝ)^2 * ε^2) := by | |
| gcongr | |
| _ = _ := by field_simp | |
| end Conditional | |
| end Balanced.ConditionalOrderDetection | |
| end OAI | |