lean/ComparatorChallenges/TriangularCovering.lean

import Mathlib

namespace OAI

open Set MeasureTheory
open scoped InnerProductSpace BigOperators Interval

noncomputable section
namespace TriangularCovering

abbrev Space := EuclideanSpace ℝ (Fin 3)

def H : ℝ := Real.sqrt 2

def point (x y t : ℝ) : Space := !₂[x, y, H * t]

def K : Set Space :=
  {z | ∃ x y t : ℝ, 0 ≤ t ∧ t ≤ 1 ∧ |x| ≤ 1 - t ∧ |y| ≤ t ∧ z = point x y t}

def projection (u z : Space) : Space := z - (inner ℝ u z) • u

def area (B : Set Space) : ℝ := (Measure.euclideanHausdorffMeasure 2 B).toReal

def shadowArea (u : Space) : ℝ := area (projection u '' K)

def A_min : ℝ := sInf {a : ℝ | ∃ u : Space, ‖u‖ = 1 ∧ a = shadowArea u}

structure TriCylinder where
  axis : Space
  base : Set Space
  unit_axis : ‖axis‖ = 1
  perpendicular : ∀ z ∈ base, inner ℝ axis z = 0
  triangular : ∃ v : Fin 3 → Space, AffineIndependent ℝ v ∧
    base = convexHull ℝ (Set.range v)
  compact : IsCompact base
  convex : Convex ℝ base
  measurable : MeasurableSet base
  finite_area : Measure.euclideanHausdorffMeasure 2 base ≠ ⊤

def TriCylinder.carrier (c : TriCylinder) : Set Space :=
  {z | ∃ b ∈ c.base, ∃ r : ℝ, z = b + r • c.axis}

def Covers {m : ℕ} (c : Fin m → TriCylinder) : Prop :=
  ∀ z ∈ K, ∃ i : Fin m, z ∈ (c i).carrier

def totalArea {m : ℕ} (c : Fin m → TriCylinder) : ℝ :=
  ∑ i : Fin m, area (c i).base

def relativeCost {m : ℕ} (c : Fin m → TriCylinder) : ℝ :=
  ∑ i : Fin m, area (c i).base / shadowArea (c i).axis

def n (ε : ℝ) : ℕ := ⌈2 / ε ^ 2⌉₊

def MainAsymptotic : Prop :=
  ∃ δ C : ℝ, 0 < δ ∧ 0 < C ∧
    ∀ ε : ℝ, 0 < ε → ε < δ →
      ∃ c : Fin (2 * n ε) → TriCylinder,
        Covers c ∧
        |totalArea c / H - (1 / 2 - (13 / 6000 : ℝ) * ε ^ 2)| ≤ C * ε ^ 4 ∧
        totalArea c < A_min / 2

def NormalizedCounterexample : Prop :=
  ∃ m : ℕ, ∃ c : Fin m → TriCylinder, Covers c ∧ relativeCost c < 1 / 2

theorem main : A_min = H ∧ MainAsymptotic ∧ NormalizedCounterexample := by
  sorry

end TriangularCovering

end

end OAI