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785c727 | 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 | 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
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