openAI-math-dataset / lean /OAI /Geometry /CylinderCovering /Singular /EdgeOneNormalizationDefinitions.lean
| import OAI.Geometry.CylinderCovering.Singular.EdgeOneCoverDefinitions | |
| import OAI.Geometry.CylinderCovering.Rational.RealizationDefinitions | |
| import OAI.Geometry.CylinderCovering.Convexity.GlobalDefinitions | |
| namespace OAI | |
| noncomputable section | |
| open Set | |
| namespace CylinderCovering.SingularEdgeOneCover | |
| open RuledApproximation | |
| def NormalizationStatement : Prop := | |
| tetrahedron = | |
| (fun z : Space => z + vec3 0 0 (height / 2)) '' SlopeRational.referenceTetrahedron ∧ | |
| minimumShadowArea tetrahedron = height / 2 ∧ | |
| ∀ (n m : ℕ) (C : Fin n → Cylinder) (D : Fin m → Cylinder) (B₁ B₂ : Cylinder), | |
| familyArea C + familyArea D + B₁.area + B₂.area < height / 4 → | |
| familyArea C + familyArea D + B₁.area + B₂.area < | |
| minimumShadowArea tetrahedron / 2 ∧ | |
| relativeCost tetrahedron C + relativeCost tetrahedron D + | |
| B₁.area / shadowArea tetrahedron B₁.axis + | |
| B₂.area / shadowArea tetrahedron B₂.axis < 1 / 2 | |
| end CylinderCovering.SingularEdgeOneCover | |
| end | |
| end OAI | |