openAI-math-dataset / lean /OAI /Combinatorics /BalancedRyser /Profiles /ProfileCountingAssembly.lean
| import OAI.Combinatorics.BalancedRyser.Profiles.ProfileSourceMarkLimit | |
| import OAI.Combinatorics.BalancedRyser.Profiles.ProfileWitnessPayment | |
| namespace OAI | |
| noncomputable section | |
| open scoped BigOperators | |
| namespace Balanced.ProfileCountingAssembly | |
| universe u | |
| open ProfileScheduledMargin ProfileWitnessPayment | |
| def SourceRareTailBound (k : ℕ) : Prop := | |
| ∀ η ε ξ : ℝ, 0 < η → η ≤ 1 → 0 < ε → 0 < ξ → | |
| ∃ L : ℝ, ∃ N : ℕ, | |
| ∀ (F : Type u) [Field F] [Fintype F] [DecidableEq F] | |
| (m : ℕ), N ≤ Fintype.card F → k ≤ Fintype.card F → | |
| m ≤ Fintype.card F → | |
| pairMarkAverage (F := F) k m (fun D C => tailProbability D.val C η ε ξ) ≤ | |
| Real.exp (L * (Fintype.card F : ℝ) - | |
| (ξ / 2) * (Fintype.card F : ℝ) * Real.log (Fintype.card F : ℝ)) / | |
| (Fintype.card F : ℝ) ^ m | |
| theorem profileCounting_of_sourceRareTailBound | |
| (hrare : ∀ k : ℕ, 1 ≤ k → SourceRareTailBound.{u} k) : | |
| ProfileCountingResult.{u} := by | |
| intro k hk A hvalid hgap hvisible | |
| obtain ⟨n, ξ, hξ, N₀, hpositive⟩ := | |
| ProfileSourceMarkLimit.exists_uniform_positive_tail k hk A hvalid hgap hvisible | |
| have hthreshold : threshold n ≤ 1 := by | |
| unfold threshold | |
| apply (div_le_one (by positivity : 0 < (n : ℝ) + 1)).mpr | |
| linarith [show (0 : ℝ) ≤ (n : ℝ) from Nat.cast_nonneg n] | |
| obtain ⟨L, N₁, hnegative⟩ := hrare k hk (threshold n) (threshold n) ξ | |
| (threshold_pos n) hthreshold (threshold_pos n) hξ | |
| have hkR : (0 : ℝ) < k := by exact_mod_cast (lt_of_lt_of_le Nat.zero_lt_one hk) | |
| apply counting_of_mark_bounds k A hvalid (ξ / (4 * (k : ℝ))) L (ξ / 2) | |
| (by positivity) (half_pos hξ) | |
| refine ⟨max N₀ N₁, ?_⟩ | |
| intro F _ _ _ hN hkq | |
| refine ⟨fun _ D C => tailProbability D.val C (threshold n) (threshold n) ξ, | |
| ?_, ?_, ?_⟩ | |
| · intro m D C | |
| unfold tailProbability | |
| apply Finset.expect_nonneg | |
| intro p _ | |
| split <;> norm_num | |
| · intro m D C _ hA | |
| exact hpositive F D.val C hA ((le_max_left N₀ N₁).trans hN) | |
| · intro m hm | |
| exact hnegative F m ((le_max_right N₀ N₁).trans hN) hkq hm | |
| end Balanced.ProfileCountingAssembly | |
| end | |
| end OAI | |