| 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 |
|
|
