openAI-math-dataset / lean /OAI /Geometry /CylinderCovering /Angular /EndpointScaledDefinitions.lean
| import OAI.Geometry.CylinderCovering.Angular.EndpointCoverDefinitions | |
| import OAI.Geometry.CylinderCovering.Convexity.SimilarityDefinitions | |
| namespace OAI | |
| noncomputable section | |
| namespace CylinderCovering | |
| def endpointScaledBody : Set TriangularCovering.Space := | |
| (fun z => (1/TriangularCovering.H) • z) '' TriangularCovering.K | |
| def AngularEndpointScaledStatement : Prop := | |
| Similarity.minimumArea endpointScaledBody = TriangularCovering.H/2 ∧ | |
| ∀ k : ℕ, 14000 ≤ k → | |
| ∃ C D : Fin (2*(k^2+1)) → TriangularCovering.TriCylinder, | |
| TriangularCovering.AngularFamily.PrescribedFamily C (1/(k:ℝ)) | |
| (fun j => AngularEndpoint.boundary (k^2) j) | |
| (fun j => AngularEndpoint.boundary (k^2) (j+1)) | |
| (fun j => AngularEndpoint.alpha (k^2) j) | |
| (fun j => AngularEndpoint.beta (k^2) j) | |
| (fun j => AngularEndpoint.radius (1/(k:ℝ)) (1/1000) (k^2) j) ∧ | |
| (∀ i, (D i).axis = (C i).axis ∧ | |
| (D i).base = (fun z => (1/TriangularCovering.H) • z) '' (C i).base) ∧ | |
| (∀ z ∈ endpointScaledBody, ∃ i, z ∈ (D i).carrier) ∧ | |
| |TriangularCovering.totalArea D - (2/TriangularCovering.H)* | |
| (1/4-(13/12000:ℝ)*(1/(k:ℝ))^2)| ≤ | |
| 7*TriangularCovering.H*(1/(k:ℝ))^4 ∧ | |
| TriangularCovering.totalArea D < Similarity.minimumArea endpointScaledBody/2 ∧ | |
| (∑ i, TriangularCovering.area (D i).base / | |
| Similarity.shadowArea endpointScaledBody (D i).axis) < 1/2 | |
| end CylinderCovering | |
| end | |
| end OAI | |