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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 | 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
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