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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 28 29 30 31 32 | import OAI.Geometry.CylinderCovering.Cartesian.DomainDefinitions
namespace OAI
noncomputable section
namespace CylinderCovering
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
end CylinderCovering
end
end OAI
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