openAI-math-dataset / lean /OAI /Geometry /CylinderCovering /Caps /TrapezoidGeometryDefinitions.lean
| import OAI.Geometry.CylinderCovering.Caps.FiniteDefinitions | |
| import OAI.Geometry.CylinderCovering.Convexity.SimilarityDefinitions | |
| namespace OAI | |
| open Set Filter | |
| open scoped Topology | |
| noncomputable section | |
| namespace CylinderCovering.HalfheightFinite | |
| def NondegenerateTrapezoid (B : Set Space) : Prop := | |
| ∃ v : Fin 4 → Space, | |
| B = convexHull ℝ (range v) ∧ Function.Injective v ∧ | |
| AffineIndependent ℝ ![v 0, v 1, v 2] ∧ | |
| ∃ a b : ℝ, 0 < a ∧ 0 < b ∧ ∃ w : Space, w ≠ 0 ∧ | |
| v 1 - v 0 = a • w ∧ v 3 - v 2 = b • w | |
| structure GeometryStatement : Prop where | |
| unitEdge : ∃ v : Fin 4 → Space, AffineIndependent ℝ v ∧ | |
| (∀ i j : Fin 4, i ≠ j → dist (v i) (v j) = 1) ∧ | |
| tetrahedron = convexHull ℝ (range v) | |
| minimumShadow : Similarity.minimumArea tetrahedron = height / 2 | |
| sourceTrapezoids : ∀ δ : ℝ, 0 < δ → ∀ N : ℕ, ∀ i : Fin N, | |
| NondegenerateTrapezoid (firstBase δ (cubicCap δ) N i) ∧ | |
| NondegenerateTrapezoid (secondBase δ (cubicCap δ) N i) ∧ | |
| NondegenerateTrapezoid (firstBase δ (fractionalCap δ) N i) ∧ | |
| NondegenerateTrapezoid (secondBase δ (fractionalCap δ) N i) | |
| eventualHalfMinimum : ∀ᶠ δ : ℝ in 𝓝[>] 0, | |
| (∀ᶠ N in atTop, 0 < N ∧ | |
| totalArea δ (cubicCap δ) N < Similarity.minimumArea tetrahedron / 2) ∧ | |
| (∀ᶠ N in atTop, 0 < N ∧ | |
| totalArea δ (fractionalCap δ) N < Similarity.minimumArea tetrahedron / 2) | |
| end CylinderCovering.HalfheightFinite | |
| end | |
| end OAI | |