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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 33 34 35 36 37 38 39 | import OAI.Geometry.CylinderCovering.Cartesian.GeneralDefinitions
import OAI.Geometry.CylinderCovering.Angular.OrdinaryCapsDefinitions
namespace OAI
noncomputable section
namespace CylinderCovering.CartesianCaps
def cartesianData (N : ℕ) (a : ℕ → ℝ) (τ η : ℝ) : AngularGeneral.Data :=
withCartesianCaps τ η (AngularOrdinaryCaps.ordinaryData .radial N a τ η)
end CylinderCovering.CartesianCaps
namespace CylinderCovering
def CartesianCoverStatement : Prop :=
∀ η : ℝ, 0 < η → η < 1/480 → ∀ M : ℝ, 0 ≤ M →
∃ δ A : ℝ, 0 < δ ∧ 0 < A ∧
∀ τ : ℝ, 0 < τ → τ < δ → ∀ N : ℕ, ∀ a : ℕ → ℝ,
0 < N → a 0 = -1 → a N = 1 →
(∀ j < N, a j < a (j+1)) →
(∀ j < N, a (j+1) - a j ≤ M*τ^2) →
AngularGeneral.Conclusion τ η A (CartesianCaps.cartesianData N a τ η) ∧
∀ b ∈ Set.Icc (3/4 : ℝ) 1, ∀ j < N,
AngularGeneral.lowerIntercept (CartesianCaps.cartesianData N a τ η) j =
{z | ∃ q ∈ Set.Icc (a j) (a (j+1)),
∃ s ∈ CartesianCaps.aperture τ η b q,
z = TriangularCovering.point 0 (q*s) s} ∧
AngularGeneral.upperIntercept (CartesianCaps.cartesianData N a τ η) j =
{z | ∃ q ∈ Set.Icc (a j) (a (j+1)),
∃ s ∈ CartesianCaps.aperture τ η b q,
z = TriangularCovering.point (q*s) 0 (1-s)}
end CylinderCovering
end
end OAI
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