| import OAI.Geometry.CylinderCovering.Beams.sDefinitions |
| import OAI.Geometry.CylinderCovering.Beams.CostDefinitions |
| import OAI.Geometry.CylinderCovering.Radial.RiemannDefinitions |
| import OAI.Geometry.TriangularCovering.Basic |
|
|
| namespace OAI |
|
|
| open Filter |
| open scoped Topology |
| noncomputable section |
| namespace ShortenedBeam |
|
|
| def CoverageStatement : Prop := |
| ∀ a : ℝ, 0 < a → a < 1/2 → |
| IsCompact (beamMinus a) ∧ IsCompact (beamPlus a) ∧ |
| tetrahedron ⊆ beamMinus a ∪ beamPlus a ∪ strip a |
|
|
| def FiniteBeamsStatement : Prop := |
| ∀ a : ℝ, 0 < a → a < 1/2 → |
| ∀ tag : (N : ℕ) → Fin N → ℝ, |
| (∀ N, CylinderCovering.RadialRiemann.IsTagged 0 1 N (tag N)) → |
| ∃ Cminus Cplus : (N : ℕ) → Fin N → TriangularCovering.TriCylinder, |
| (∀ N, 0 < N → |
| (∀ i, (Cminus N i).axis = axisMinus a (tag N i) ∧ |
| (Cplus N i).axis = axisPlus a (tag N i)) ∧ |
| (∀ z ∈ beamMinus a, ∃ i : Fin N, z ∈ (Cminus N i).carrier) ∧ |
| (∀ z ∈ beamPlus a, ∃ i : Fin N, z ∈ (Cplus N i).carrier)) ∧ |
| Tendsto (fun N => TriangularCovering.totalArea (Cminus N) + |
| TriangularCovering.totalArea (Cplus N)) atTop |
| (𝓝 ((Real.sqrt 2/4)*CylinderCovering.BeamCost.beamCost a)) |
|
|
| end ShortenedBeam |
|
|
| end |
|
|
| end OAI |
|
|
