openAI-math-dataset / lean /OAI /Combinatorics /BalancedRyser /Selection /DistinctInitialOccupancy.lean
| import OAI.Combinatorics.BalancedRyser.Probability.DistinctRoleLaw | |
| import OAI.Combinatorics.BalancedRyser.Selection.LineOccupancyConcentration | |
| namespace OAI | |
| noncomputable section | |
| open Filter Topology | |
| open scoped Classical | |
| namespace Balanced.DistinctRole | |
| variable {F : Type*} [Field F] [Fintype F] [DecidableEq F] | |
| theorem map_initial_data : | |
| ((offsetLaw F).filter {M | DistinctConstituents M} distinct_support).map | |
| (fun M => (initialDeletedLines M, initiallyRetained M)) = | |
| (offsetLaw F).map (fun M => (initialDeletedLines M, initiallyRetained M)) := by | |
| let f : (Direction F → F) → Finset (Line F) × Finset (Point F) := | |
| fun r => (initialDeletedLines (fun d _ => r d), initiallyRetained (fun d _ => r d)) | |
| exact map_role_one_observable f | |
| theorem initial_event_probability (E : Set (Finset (Line F) × Finset (Point F))) : | |
| (((offsetLaw F).filter {M | DistinctConstituents M} distinct_support).toOuterMeasure | |
| {M | (initialDeletedLines M, initiallyRetained M) ∈ E}).toReal = | |
| offsetEventProbability (F := F) | |
| {M | (initialDeletedLines M, initiallyRetained M) ∈ E} := by | |
| have h := congrArg (fun p : PMF (Finset (Line F) × Finset (Point F)) => | |
| (p.toOuterMeasure E).toReal) (map_initial_data (F := F)) | |
| simp only [PMF.toOuterMeasure_map_apply] at h | |
| convert h using 1 <;> congr 2 | |
| theorem initial_failure_probability_eq : | |
| (((offsetLaw F).filter {M | DistinctConstituents M} distinct_support).toOuterMeasure | |
| {M | ∃ l, l ∉ initialDeletedLines M ∧ | |
| ((linePoints l ∩ initiallyRetained M).card : ℝ) < (Fintype.card F : ℝ) / 5}).toReal = | |
| offsetEventProbability (F := F) | |
| {M | ∃ l, l ∉ initialDeletedLines M ∧ | |
| ((linePoints l ∩ initiallyRetained M).card : ℝ) < (Fintype.card F : ℝ) / 5} := by | |
| exact initial_event_probability | |
| {z | ∃ l, l ∉ z.1 ∧ ((linePoints l ∩ z.2).card : ℝ) < (Fintype.card F : ℝ) / 5} | |
| theorem initial_failure_probability_le | |
| (hgap : (Fintype.card F : ℝ) / 20 ≤ | |
| (Fintype.card F : ℝ) * (1 - 1 / (Fintype.card F : ℝ)) ^ Fintype.card F - | |
| (Fintype.card F : ℝ) / 5) : | |
| (((offsetLaw F).filter {M | DistinctConstituents M} distinct_support).toOuterMeasure | |
| {M | ∃ l, l ∉ initialDeletedLines M ∧ | |
| ((linePoints l ∩ initiallyRetained M).card : ℝ) < (Fintype.card F : ℝ) / 5}).toReal ≤ | |
| (Fintype.card F : ℝ) * ((Fintype.card F : ℝ) + 1) * | |
| (2 * Real.exp (-(Fintype.card F : ℝ) / 3200)) := by | |
| rw [initial_failure_probability_eq] | |
| exact LineOccupancy.initial_failure_probability_le hgap | |
| theorem tendsto_initial_failure (F : ℕ → Type*) | |
| [∀ n, Field (F n)] [∀ n, Fintype (F n)] [∀ n, DecidableEq (F n)] | |
| (hq : Tendsto (fun n => Fintype.card (F n)) atTop atTop) : | |
| Tendsto (fun n => | |
| (((offsetLaw (F n)).filter {M | DistinctConstituents M} distinct_support).toOuterMeasure | |
| {M | ∃ l, l ∉ initialDeletedLines M ∧ | |
| ((linePoints l ∩ initiallyRetained M).card : ℝ) < (Fintype.card (F n) : ℝ) / 5}).toReal) | |
| atTop (𝓝 0) := by | |
| simpa only [initial_failure_probability_eq] using LineOccupancy.tendsto_initial_failure F hq | |
| end Balanced.DistinctRole | |
| end | |
| end OAI | |