| import OAI.Geometry.CylinderCovering.Cubic.RescalingDefinitions | |
| import OAI.Geometry.CylinderCovering.Angular.ConstructionDefinitions | |
| namespace OAI | |
| noncomputable section | |
| open Set Filter Asymptotics | |
| open scoped Topology | |
| namespace CylinderCovering.Cubic | |
| open TriangularCovering | |
| def FullCubicRescaledStatement : Prop := | |
| contraction '' K4 = K ∧ | |
| (∀ a b : ℝ, a ≤ b → | |
| IsGreatest (TriangularCovering.d '' Icc a b) | |
| (max (TriangularCovering.d a) (TriangularCovering.d b))) ∧ | |
| (∀ τ a b : ℝ, cap (τ / 2) a b / 2 = | |
| 1 / 2 + τ ^ 2 * | |
| (max (TriangularCovering.d a) (TriangularCovering.d b) + 25 / 4 * τ)) ∧ | |
| Tendsto (fun τ : ℝ => 25 / 4 * τ) (𝓝[>] (0 : ℝ)) (𝓝 0) ∧ | |
| RescaledFiniteCoverStatement ∧ | |
| (∀ τ : ℝ, 0 < τ → τ ≤ 1 → | |
| |cubicCost (τ / 2) / (4 * H) - (1 / 2 - τ ^ 2 / 240)| ≤ | |
| (401 / 4 : ℝ) * τ ^ 3) ∧ | |
| ((fun τ : ℝ => cubicCost (τ / 2) / (4 * H) - (1 / 2 - τ ^ 2 / 240)) | |
| =O[𝓝[>] (0 : ℝ)] (fun τ => τ ^ 3)) ∧ | |
| ∃ delta : ℝ, 0 < delta ∧ | |
| ∀ τ : ℝ, 0 < τ → τ < delta → | |
| ∃ C : Fin (2 * n (τ / 2)) → TriCylinder, | |
| Covers C ∧ ContractedFamily (τ / 2) C ∧ | |
| totalArea C = cubicCost (τ / 2) / 4 ∧ totalArea C < A_min / 2 | |
| end CylinderCovering.Cubic | |
| end | |
| end OAI | |