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7d9acbd | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 | 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
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