| import OAI.Geometry.RuledCovering.Model | |
| namespace OAI | |
| noncomputable section | |
| open Set MeasureTheory Filter | |
| open scoped Topology ContDiff | |
| namespace RuledApproximation | |
| def NilpotentC1Statement : Prop := | |
| ∀ (D : Set Plane) (G : Plane → Plane), IsCompact D → SmoothNear 1 D G → | |
| (∀ b ∈ D, (fderiv ℝ G b).comp (fderiv ℝ G b) = 0) → | |
| ∀ h eps : ℝ, 0 < h → 0 < eps → | |
| ∃ n : ℕ, ∃ T : Fin n → Set Plane, ∃ g : Fin n → Plane, | |
| (∀ i, IsSquareTile (T i)) ∧ | |
| physicalRuled 1 D G 1 ⊆ ⋃ i, auxiliaryCylinder (T i) (g i) ∧ | |
| (∑ i, volume.real (T i) * physicalWeight h (g i)) ≤ | |
| (∫ b in D, physicalWeight h (G b)) + eps ∧ | |
| ∀ a : ℝ, ∃ C₁ C₂ : Fin n → Cylinder, | |
| (∀ i, (C₁ i).parallelogram ∧ (C₂ i).parallelogram ∧ | |
| (C₁ i).carrier = firstPhysicalMap h '' auxiliaryCylinder (T i) (g i) ∧ | |
| (C₂ i).carrier = secondPhysicalMap h a '' auxiliaryCylinder (T i) (g i) ∧ | |
| (C₁ i).area = h * volume.real (T i) * physicalWeight h (g i) ∧ | |
| (C₂ i).area = h * volume.real (T i) * physicalWeight h (g i)) | |
| end RuledApproximation | |
| end | |
| end OAI | |