openAI-math-dataset / lean /OAI /Geometry /CylinderCovering /Singular /ParameterRangeDefinitions.lean
| import OAI.Geometry.RuledCovering.Model | |
| namespace OAI | |
| open Filter Topology | |
| namespace CylinderCovering.SingularParameterRange | |
| open RuledApproximation | |
| structure Statement : Prop where | |
| negative_range : ∀ p : ℝ, 1 < p → | |
| (singularCoefficient p < 0 ↔ p < (Real.sqrt 51 - 5) / 2) | |
| critical_zero : singularCoefficient ((Real.sqrt 51 - 5) / 2) = 0 | |
| endpoint_value : singularCoefficient 1 = -(1 / 480 : ℝ) | |
| endpoint_limit : Tendsto singularCoefficient (𝓝 1) (𝓝 (-(1 / 480 : ℝ))) | |
| cutoff_window : ∀ p : ℝ, | |
| (∃ γ : ℝ, 1 < γ ∧ γ < 1 / (2 - p)) ↔ 1 < p ∧ p < 2 | |
| endpoint_degenerate : ∀ z : Plane, 0 < z 1 → | |
| LinearMap.det (fderiv ℝ (singularField 1) z).toLinearMap = 0 ∧ | |
| ∀ μ : ℝ, Module.End.HasEigenvalue (fderiv ℝ (singularField 1) z).toLinearMap μ → μ = 0 | |
| end CylinderCovering.SingularParameterRange | |
| end OAI | |