| import OAI.Geometry.CylinderCovering.Barycentric.SweepsDefinitions | |
| namespace OAI | |
| namespace CylinderCovering | |
| def BarycentricCoreStatement : Prop := | |
| (∀ k : ℝ, IsCompact (RadialCoordinates.firstSweep k) ∧ | |
| IsCompact (RadialCoordinates.secondSweep k)) ∧ | |
| (∀ k : ℝ, 0 < k → k ≤ 1 / 5000 → | |
| RadialCoordinates.barycentricSimplex ⊆ | |
| RadialCoordinates.firstSweep k ∪ RadialCoordinates.secondSweep k) ∧ | |
| (∀ k : ℝ, |k| ≤ 1 / 2 → | |
| |RadialCoordinates.J k / Real.sqrt 2 - (1 / 8 - (11 / 3000) * k ^ 2)| ≤ | |
| (47 / 16) * k ^ 4) ∧ | |
| (∀ k : ℝ, 0 < k → k ≤ 1 / 5000 → | |
| 2 * RadialCoordinates.J k < Real.sqrt 2 / 4) ∧ | |
| (∀ a b z : ℝ, 0 < a → 0 < b → | |
| ∃! t : ℝ, t ∈ Set.Ioo 0 1 ∧ a / t - b / (1 - t) = z) ∧ | |
| (∀ a b z : ℝ, 0 ≤ a → 0 ≤ b → | |
| ∃ t S : ℝ, t ∈ Set.Icc 0 1 ∧ 0 ≤ S ∧ | |
| a = S * t + z * t ^ 2 ∧ b = S * (1 - t) - z * (1 - t) ^ 2) | |
| end CylinderCovering | |
| end OAI | |