| import OAI.Geometry.CylinderCovering.Beams.MainDefinitions |
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| namespace OAI |
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| noncomputable section |
| open Set MeasureTheory |
| open scoped InnerProductSpace |
| namespace ShortenedBeam |
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| def vertices : Fin 4 → Space := |
| ![point 2 0 (-1), point (-2) 0 (-1), point 0 2 1, point 0 (-2) 1] |
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| def GeometryStatement : Prop := |
| tetrahedron = convexHull ℝ (range vertices) ∧ AffineIndependent ℝ vertices ∧ |
| (∀ i j : Fin 4, i ≠ j → dist (vertices i) (vertices j) = 2*Real.sqrt 2) ∧ |
| volume.real tetrahedron = 8/3 ∧ |
| CylinderCovering.minimumShadowArea tetrahedron = 2*Real.sqrt 2 |
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|
| abbrev MixedShape := HasMixedBase |
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| def FiniteMixedStatement : Prop := |
| ∀ a : ℝ, 0 < a → a < 1/2 → ∀ ε : ℝ, 0 < ε → |
| ∃ n : ℕ, ∃ C : Fin n → RuledApproximation.Cylinder, |
| (∀ i, MixedShape (C i) ∧ IsCompact (C i).base) ∧ |
| tetrahedron ⊆ RuledApproximation.familySet C ∧ |
| RuledApproximation.familyArea C ≤ |
| (Real.sqrt 2/4)*(CylinderCovering.BeamCost.beamCost a+ |
| CylinderCovering.BeamCost.stripCost a)+ε |
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|
| abbrev RegularCounterexampleStatement := RegularMixedCounterexampleStatement |
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| abbrev ParameterPlane := EuclideanSpace ℝ (Fin 2) |
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| def doubleSector (a : ℝ) (N : ℕ) (i : Fin N) : Set ParameterPlane := |
| {z | ∃ p ∈ CylinderCovering.RadialRiemann.cell 0 1 N i, |
| ∃ r ∈ Icc (-(1/(N:ℝ))) ((2-a)+1/(N:ℝ)), |
| z = !₂[r*p,r*(1-p)]} |
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| def midpointTag (N : ℕ) (i : Fin N) : ℝ := |
| (CylinderCovering.RadialRiemann.point 0 1 N i+ |
| CylinderCovering.RadialRiemann.point 0 1 N (i+1))/2 |
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| def neighborhoodSector (a : ℝ) (N : ℕ) (i : Fin N) : Set ParameterPlane := |
| Metric.cthickening ((1/(N:ℝ))^2) (doubleSector a N i) |
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| def neighborhoodBaseMinus (a : ℝ) (N : ℕ) (i : Fin N) : Set Space := |
| TriangularCovering.projection (axisMinus a (midpointTag N i)) '' |
| ((fun z : ParameterPlane => point (-1+z 0) (-1+z 1) 0) '' neighborhoodSector a N i) |
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| def neighborhoodBasePlus (a : ℝ) (N : ℕ) (i : Fin N) : Set Space := |
| TriangularCovering.projection (axisPlus a (midpointTag N i)) '' |
| ((fun z : ParameterPlane => point (1-z 0) (1-z 1) 0) '' neighborhoodSector a N i) |
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| def NeighborhoodAlternativeStatement : Prop := |
| ∀ a : ℝ, 0 < a → a < 1/2 → |
| ∃ errorConstant : ℝ, 0 ≤ errorConstant ∧ |
| ∃ Cminus Cplus : (N : ℕ) → Fin N → RuledApproximation.Cylinder, |
| ∀ N : ℕ, 0 < N → |
| (∀ i : Fin N, |
| (Cminus N i).axis = axisMinus a (midpointTag N i) ∧ |
| (Cplus N i).axis = axisPlus a (midpointTag N i) ∧ |
| Subtype.val '' (Cminus N i).base = neighborhoodBaseMinus a N i ∧ |
| Subtype.val '' (Cplus N i).base = neighborhoodBasePlus a N i ∧ |
| IsCompact (Cminus N i).base ∧ IsCompact (Cplus N i).base) ∧ |
| beamMinus a ⊆ RuledApproximation.familySet (Cminus N) ∧ |
| beamPlus a ⊆ RuledApproximation.familySet (Cplus N) ∧ |
| (RuledApproximation.familyArea (Cminus N)+RuledApproximation.familyArea (Cplus N)) / |
| (Real.sqrt 2/4) ≤ |
| CylinderCovering.BeamCost.beamCost a+errorConstant/(N:ℝ) |
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| def FullStatement : Prop := |
| GeometryStatement ∧ CoverageStatement ∧ FiniteBeamsStatement ∧ FiniteStripStatement ∧ |
| CylinderCovering.BeamCost.CostStatement ∧ FiniteMixedStatement ∧ |
| CanonicalCounterexampleStatement ∧ RegularCounterexampleStatement ∧ |
| NeighborhoodAlternativeStatement |
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| end ShortenedBeam |
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| end |
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| end OAI |
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