| import OAI.Geometry.CylinderCovering.Angular.CubicCoverDefinitions | |
| namespace OAI | |
| noncomputable section | |
| open Set Filter Asymptotics | |
| open scoped Topology | |
| namespace CylinderCovering.Cubic | |
| open TriangularCovering | |
| def minimumShadow4 : ℝ := | |
| sInf {a : ℝ | ∃ u : Space, ‖u‖ = 1 ∧ a = area (projection u '' K4)} | |
| def FullCubicStatement : Prop := | |
| minimumShadow4 = 4 * H ∧ | |
| FiniteCoverStatement ∧ | |
| (∀ eps : ℝ, 0 < eps → eps ≤ 1 / 2 → | |
| |cubicCost eps / H - (2 - eps ^ 2 / 15)| ≤ 3208 * eps ^ 3) ∧ | |
| ((fun eps : ℝ => cubicCost eps - (2 * H - H / 15 * eps ^ 2)) | |
| =O[𝓝[>] (0 : ℝ)] (fun eps => eps ^ 3)) ∧ | |
| ∃ delta : ℝ, 0 < delta ∧ | |
| ∀ eps : ℝ, 0 < eps → eps < delta → | |
| ∃ C : Fin (2 * n eps) → TriCylinder, | |
| CoversK4 C ∧ PrescribedFamily eps C ∧ | |
| totalArea C = cubicCost eps ∧ totalArea C < minimumShadow4 / 2 | |
| end CylinderCovering.Cubic | |
| end | |
| end OAI | |