openAI-math-dataset / lean /OAI /Geometry /CylinderCovering /Ruled /LogarithmicLocalDefinitions.lean
| import OAI.Geometry.CylinderCovering.Ruled.LocalDefinitions | |
| namespace OAI | |
| noncomputable section | |
| namespace CylinderCovering.ProductLocal | |
| open MeasureTheory RuledApproximation | |
| def LogarithmicLocalStatement : Prop := | |
| ∀ (D : Set Plane) (v : Plane → Plane) (h M : ℝ), | |
| IsCompact D → D.Nonempty → 0 < h → 0 < M → SmoothNear 1 D v → | |
| (∀ p ∈ D, HyperbolicAt M v p) → | |
| ∀ (eta m gamma : ℝ), 0 < eta → 1 < m → 0 < gamma → | |
| (1+gamma)*(1/m^2) < 1 → ProductCells.areaRatio (1/m^2) gamma ≤ 1+eta → | |
| ∃ t0 : ℝ, 0 < t0 ∧ ∀ p ∈ D, | |
| ∀ (lam : ℝ) (L : (ℝ × ℝ) ≃L[ℝ] Plane), | |
| IsUnitEigenframe (fderiv ℝ v p) lam L → 0 < M*lam → M*lam < 1 → | |
| ∀ t : ℝ, 0 < t → t < t0 → | |
| affineCell p t L (ProductCells.logarithmicCell m) ⊆ D → | |
| ∃ C : Cylinder, | |
| C.axis = ‖vec3 1 (v p 0) (h*v p 1)‖⁻¹ • vec3 1 (v p 0) (h*v p 1) ∧ | |
| C.carrier = firstPhysicalMap h '' auxiliaryCylinder | |
| (affineCell p t L (ProductCells.logarithmicIntercept m gamma)) (v p) ∧ | |
| C.openBounded ∧ | |
| physicalRuled h (affineCell p t L (ProductCells.logarithmicCell m)) v M ⊆ C.carrier ∧ | |
| C.area ≤ h*(1+eta)^2 * | |
| (∫ q in affineCell p t L (ProductCells.logarithmicCell m), physicalWeight h (v q)) | |
| def SourceLogarithmicLocalStatement : Prop := | |
| ∀ (X k : ℝ) (D : Set Plane) (v : Plane → Plane), | |
| 0 < X → 0 < k → IsCompact D → D.Nonempty → OpenRectangleC2 D v → | |
| (∀ p ∈ D, LinearMap.trace ℝ Plane (fderiv ℝ v p).toLinearMap = 0 ∧ | |
| LinearMap.det (fderiv ℝ v p).toLinearMap < 0 ∧ | |
| k*X*Real.sqrt (-LinearMap.det (fderiv ℝ v p).toLinearMap) < 1) → | |
| ∀ eta : ℝ, 0 < eta → | |
| ∃ m : ℝ, 1 < m ∧ ∃ gamma : ℝ, 0 < gamma ∧ | |
| (1+gamma)*(1/m^2) < 1 ∧ ProductCells.areaRatio (1/m^2) gamma < 1+eta ∧ | |
| ∃ t0 : ℝ, 0 < t0 ∧ ∀ p ∈ D, | |
| ∀ (lam : ℝ) (L : (ℝ × ℝ) ≃L[ℝ] Plane), | |
| 0 < lam → IsUnitEigenframe (fderiv ℝ v p) lam L → | |
| ∀ t : ℝ, 0 < t → t < t0 → | |
| affineCell p t L (ProductCells.logarithmicCell m) ⊆ D → | |
| ∃ C : Cylinder, | |
| C.axis = ‖vec3 1 (k*v p 0) (k*v p 1)‖⁻¹ • vec3 1 (k*v p 0) (k*v p 1) ∧ | |
| C.carrier = firstPhysicalMap 1 '' auxiliaryCylinder | |
| (affineCell p t L (ProductCells.logarithmicIntercept m gamma)) (k • v p) ∧ | |
| C.openBounded ∧ | |
| physicalRuled 1 (affineCell p t L (ProductCells.logarithmicCell m)) | |
| (fun q => k • v q) X ⊆ C.carrier ∧ | |
| C.area ≤ (1+eta)^2 * | |
| (∫ q in affineCell p t L (ProductCells.logarithmicCell m), | |
| 1 / Real.sqrt (1+k^2*‖v q‖^2)) | |
| def LogarithmicRemainderStatement : Prop := | |
| ∀ (X k : ℝ) (D : Set Plane) (v : Plane → Plane), | |
| 0 < X → 0 < k → IsCompact D → OpenRectangleC2 D v → | |
| (∀ p ∈ D, LinearMap.trace ℝ Plane (fderiv ℝ v p).toLinearMap = 0 ∧ | |
| LinearMap.det (fderiv ℝ v p).toLinearMap < 0) → | |
| ∀ m : ℝ, 1 < m → ∃ t0 B : ℝ, 0 < t0 ∧ 0 < B ∧ | |
| ∀ p ∈ D, ∀ (lam : ℝ) (L : (ℝ × ℝ) ≃L[ℝ] Plane), | |
| 0 < lam → IsUnitEigenframe (fderiv ℝ v p) lam L → | |
| ∀ t : ℝ, 0 < t → t < t0 → | |
| ∀ s ∈ ProductCells.logarithmicCell m, ∀ x : ℝ, |x| ≤ X → | |
| ∃ E : ℝ × ℝ, | |
| (let q := p + t • L s | |
| q + (x*k) • (v q-v p) = | |
| p + t • L (((1+x*k*lam)*s.1+E.1,(1-x*k*lam)*s.2+E.2))) ∧ | |
| ‖vec2 E.1 E.2‖ ≤ B*t | |
| def CompleteLocalStatement : Prop := | |
| ProductCells.GeometryStatement ∧ LocalStatement ∧ | |
| LogarithmicLocalStatement ∧ SourceLogarithmicLocalStatement ∧ LogarithmicRemainderStatement | |
| end CylinderCovering.ProductLocal | |
| end | |
| end OAI | |