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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 40 41 42 43 44 45 46 47 48 | import OAI.Geometry.CylinderCovering.Beams.StatementsDefinitions
import OAI.Geometry.CylinderCovering.Beams.StripDefinitions
import OAI.Geometry.CylinderCovering.Convexity.GlobalDefinitions
namespace OAI
noncomputable section
open Set MeasureTheory
namespace ShortenedBeam
def HasTriangularBase (C : RuledApproximation.Cylinder) : Prop :=
∃ v : Fin 3 → Space, AffineIndependent ℝ v ∧
((fun z : ((Submodule.span ℝ ({C.axis} : Set Space))ᗮ) =>
(z : Space)) '' C.base) = convexHull ℝ (range v)
def HasMixedBase (C : RuledApproximation.Cylinder) : Prop :=
HasTriangularBase C ∨ CylinderCovering.BeamStrip.HasRectangularBase C
def FiniteStripStatement : Prop :=
∀ a ε : ℝ, 0 < ε →
RuledApproximation.HasFiniteCover CylinderCovering.BeamStrip.IsHorizontalRectangle
(strip a) ((Real.sqrt 2/4)*CylinderCovering.BeamCost.stripCost a + ε)
def CoreFiniteMixedStatement : Prop :=
∀ a ε : ℝ, 0 < a → a < 1/2 → 0 < ε →
RuledApproximation.HasFiniteCover HasMixedBase tetrahedron
((Real.sqrt 2/4)*(CylinderCovering.BeamCost.beamCost a+
CylinderCovering.BeamCost.stripCost a)+ε)
def CanonicalCounterexampleStatement : Prop :=
∃ n : ℕ, ∃ C : Fin n → RuledApproximation.Cylinder,
(∀ i, HasMixedBase (C i)) ∧ tetrahedron ⊆ RuledApproximation.familySet C ∧
RuledApproximation.familyArea C < Real.sqrt 2
def RegularMixedCounterexampleStatement : Prop :=
∀ K : Set Space, CylinderCovering.IsRegularTetrahedron K →
∃ n : ℕ, ∃ C : Fin n → RuledApproximation.Cylinder,
(∀ i, HasMixedBase (C i)) ∧ K ⊆ RuledApproximation.familySet C ∧
RuledApproximation.familyArea C < CylinderCovering.minimumShadowArea K / 2 ∧
CylinderCovering.relativeCost K C < 1/2
end ShortenedBeam
end
end OAI
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