openAI-math-dataset / lean /OAI /Geometry /CylinderCovering /Singular /EdgeOneDifferentialDefinitions.lean
| import OAI.Geometry.CylinderCovering.Singular.EdgeOneGeometryDefinitions | |
| namespace OAI | |
| noncomputable section | |
| namespace SingularEdgeOne | |
| open RuledApproximation | |
| def labelLinear : Plane ≃ₗ[ℝ] Plane where | |
| toFun := labelMap | |
| invFun z := vec2 (2 * z 0) (z 1) | |
| left_inv := by | |
| intro z | |
| ext i | |
| fin_cases i <;> simp [labelMap, vec2] | |
| ring | |
| right_inv := by | |
| intro z | |
| ext i | |
| fin_cases i <;> simp [labelMap, vec2] | |
| map_add' := by | |
| intro x y | |
| ext i | |
| fin_cases i <;> simp [labelMap, vec2] | |
| ring | |
| map_smul' := by | |
| intro c z | |
| ext i | |
| fin_cases i <;> simp [labelMap, vec2] | |
| ring | |
| def labelContinuous : Plane ≃L[ℝ] Plane := labelLinear.toContinuousLinearEquiv | |
| def edgeHamiltonian (p w l : ℝ) : ℝ := | |
| (2 : ℝ) ^ (p - 3) * (w ^ 2 * l ^ (p - 2) - l ^ p / 4) | |
| end SingularEdgeOne | |
| end | |
| end OAI | |