openAI-math-dataset / lean /OAI /Combinatorics /BalancedRyser /Selection /RobustOccupancyAssembly.lean
| import OAI.Combinatorics.BalancedRyser.Deletion.UniformDeletionTolerance | |
| import OAI.Combinatorics.BalancedRyser.Probability.RobustMultiplicity | |
| import OAI.Combinatorics.BalancedRyser.Selection.OccupancyFailureClass | |
| import OAI.Combinatorics.BalancedRyser.Selection.OccupancyWitness | |
| namespace OAI | |
| namespace Balanced | |
| open scoped _root_.Balanced | |
| open Filter | |
| open scoped Topology | |
| universe u | |
| theorem robustOccupancy_of_profileCounting_of_visible | |
| (hcounting : ProfileCountingResult.{u}) | |
| (k : ℕ) (hk : 1 ≤ k) (K lam η : ℝ) | |
| (hη : 0 < η) (hηone : η ≤ 1) | |
| (hvisible : ∀ ε : ℕ → ℝ, Tendsto ε atTop (𝓝 0) → ∀ high : Bool, | |
| ProfileVisibleExcess (occupancyFailureClass.{u} k (η / 4) lam ε high)) : | |
| UniformDeletionVanishing.{u} k K | |
| (fun F _ _ _ D => η * (Fintype.card F : ℝ) ≤ | |
| ((exceptionalOccupancyLines k lam D.val).card : ℝ)) := by | |
| obtain ⟨ε, _hpos, _hanti, hlim, hbadTV⟩ := exists_uniform_poisson_tolerance k K | |
| (fun τ hτ => robustMultiplicity_of_profileCounting hcounting k hk K τ hτ) | |
| have hclass (high : Bool) : UniformDeletionVanishing.{u} k K | |
| (fun F _ _ _ D => ∃ C : Finset (Line F), | |
| occupancyFailureClass k (η / 4) lam ε high F D.val C) := | |
| uniformDeletionVanishing_of_profileCounting hcounting k hk K _ | |
| (occupancyFailureClass_valid k (η / 4) lam ε high) | |
| ⟨η / 4, by positivity, occupancyFailureClass_sizeGap k (η / 4) lam ε high⟩ | |
| (hvisible ε hlim high) | |
| have hsigns := uniformDeletionVanishing_or k K _ _ (hclass false) (hclass true) | |
| have hall := uniformDeletionVanishing_or k K _ _ hbadTV hsigns | |
| apply uniformDeletionVanishing_eventually_mono k K _ _ _ hall | |
| obtain ⟨N, hN⟩ := exists_order_for_occupancy_gap η hη | |
| refine ⟨N, ?_⟩ | |
| intro F _ _ _ hn D hbad | |
| by_cases hTV : ε (Fintype.card F) ≤ deletionMultiplicityTV k D.val | |
| · exact Or.inl hTV | |
| · obtain ⟨high, C, hC⟩ := occupancyFailureClass_witness_of_exceptional | |
| k hη.le hηone ε F D (hN _ hn) hbad (le_of_not_ge hTV) | |
| cases high | |
| · exact Or.inr (Or.inl ⟨C, hC⟩) | |
| · exact Or.inr (Or.inr ⟨C, hC⟩) | |
| theorem robustOccupancy_extend_fraction | |
| (k : ℕ) (K lam : ℝ) | |
| (hsmall : ∀ η : ℝ, 0 < η → η ≤ 1 → UniformDeletionVanishing.{u} k K | |
| (fun F _ _ _ D => η * (Fintype.card F : ℝ) ≤ | |
| ((exceptionalOccupancyLines k lam D.val).card : ℝ))) : | |
| ∀ η : ℝ, 0 < η → UniformDeletionVanishing.{u} k K | |
| (fun F _ _ _ D => η * (Fintype.card F : ℝ) ≤ | |
| ((exceptionalOccupancyLines k lam D.val).card : ℝ)) := by | |
| intro η hη | |
| apply uniformDeletionVanishing_mono k K _ _ _ | |
| (hsmall (min η 1) (lt_min hη (by norm_num)) (min_le_right _ _)) | |
| intro F _ _ _ D hD | |
| exact (mul_le_mul_of_nonneg_right (min_le_left η 1) | |
| (Nat.cast_nonneg (Fintype.card F))).trans hD | |
| end Balanced | |
| end OAI | |