| import OAI.Geometry.CylinderCovering.Cartesian.DomainDefinitions |
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| namespace OAI |
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| noncomputable section |
| namespace CylinderCovering |
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| def CartesianApertureStatement : Prop := |
| ∀ τ η : ℝ, τ ∈ Set.Icc (0 : ℝ) (1/4) → η ∈ Set.Ioc (0 : ℝ) 1 → |
| ContinuousOn (CartesianCaps.cap τ η) (Set.Icc (-1 : ℝ) 1) ∧ |
| (∀ q ∈ Set.Icc (-1 : ℝ) 1, |
| CartesianCaps.cap τ η q ∈ Set.Icc (1/2 : ℝ) (3/4) ∧ |
| CartesianCaps.residual τ η q (CartesianCaps.cap τ η q) = 0 ∧ |
| (∀ s ∈ Set.Icc (0 : ℝ) 1, CartesianCaps.residual τ η q s = 0 → |
| s = CartesianCaps.cap τ η q) ∧ |
| |CartesianCaps.cap τ η q - (1/2 + τ^2 * (CartesianCaps.d q + η))| ≤ 12 * τ^4 ∧ |
| ∀ b ∈ Set.Icc (3/4 : ℝ) 1, |
| CartesianCaps.aperture τ η b q = Set.Icc 0 (CartesianCaps.cap τ η q)) ∧ |
| ∀ b ∈ Set.Icc (3/4 : ℝ) 1, |
| IsCompact (CartesianCaps.domain τ η b) ∧ |
| CartesianCaps.domain τ η b = {p : ℝ×ℝ | ∃ q ∈ Set.Icc (-1 : ℝ) 1, |
| ∃ s ∈ Set.Icc (0 : ℝ) (CartesianCaps.cap τ η q), p = (q*s,s)} ∧ |
| 1/4 + τ^2 * (1/30 + η) ≤ MeasureTheory.volume.real (CartesianCaps.domain τ η b) ∧ |
| MeasureTheory.volume.real (CartesianCaps.domain τ η b) ≤ |
| 1/4 + τ^2 * (1/30 + η) + (81/8)*τ^4 |
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| end CylinderCovering |
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| end |
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| end OAI |
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