| 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 |
|
|
