| import Mathlib | |
| namespace OAI | |
| noncomputable section | |
| namespace Problem355 | |
| abbrev Point := ℝ × ℝ | |
| def triangleArea (p q r : Point) : ℝ := | |
| abs ((q.1 - p.1) * (r.2 - p.2) - (q.2 - p.2) * (r.1 - p.1)) / 2 | |
| def pointInUnitSquare (p : Point) : Prop := | |
| 0 ≤ p.1 ∧ p.1 ≤ 1 ∧ 0 ≤ p.2 ∧ p.2 ≤ 1 | |
| def pointsInUnitSquare (P : Finset Point) : Prop := by | |
| classical | |
| exact ∀ p, p ∈ P → pointInUnitSquare p | |
| def triangleAreasAtLeast (P : Finset Point) (a : ℝ) : Prop := by | |
| classical | |
| exact ∀ p, p ∈ P → ∀ q, q ∈ P → ∀ r, r ∈ P → | |
| p ≠ q → p ≠ r → q ≠ r → a ≤ triangleArea p q r | |
| def hasTriangleAreaAtMost (P : Finset Point) (a : ℝ) : Prop := by | |
| classical | |
| exact ∃ p, p ∈ P ∧ ∃ q, q ∈ P ∧ ∃ r, r ∈ P ∧ | |
| p ≠ q ∧ p ≠ r ∧ q ≠ r ∧ triangleArea p q r ≤ a | |
| def heilbronnD : ℕ := 41 | |
| def heilbronnM : ℕ := Nat.choose (4 * heilbronnD - 1) heilbronnD | |
| def heilbronnT : ℕ := Nat.choose heilbronnM 3 | |
| @[irreducible] def heilbronnK : ℕ := heilbronnT ^ 2 + 1 | |
| def heilbronnExponent : ℝ := | |
| 1 / (100000 * (heilbronnK : ℝ)) | |
| def eventualAlmostUpperBound (epsilon : ℝ) : Prop := | |
| ∃ C : ℝ, 0 < C ∧ ∃ n0 : ℕ, ∀ n : ℕ, n0 ≤ n → 3 ≤ n → | |
| ∀ P : Finset Point, P.card = n → pointsInUnitSquare P → | |
| hasTriangleAreaAtMost P | |
| (C * Real.rpow (n : ℝ) (-2 + epsilon)) | |
| end Problem355 | |
| end | |
| end OAI | |