| import OAI.Geometry.CylinderCovering.Physical.FiniteDefinitions | |
| import OAI.Geometry.CylinderCovering.Convexity.GlobalDefinitions | |
| namespace OAI | |
| open Set | |
| noncomputable section | |
| namespace RadialPhysical.PhysicalMinimum | |
| open CylinderCovering | |
| def actualRelativeCost (centered : Bool) (b eta : ℝ) (T : ℝ → ℝ) | |
| (N : ℕ) (tagU tagL : Fin N → ℝ) : ℝ := | |
| (∑ i, TriangularCovering.area (upperPhysicalBase centered b eta T N tagU i) / | |
| CylinderCovering.shadowArea (modelBody centered b) (upperPhysicalAxis b eta (tagU i))) + | |
| (∑ i, TriangularCovering.area (lowerPhysicalBase centered b eta T N tagL i) / | |
| CylinderCovering.shadowArea (modelBody centered b) (lowerPhysicalAxis b eta (tagL i))) | |
| def Statement : Prop := | |
| (∀ (centered : Bool) (b : ℝ), 0 < b → | |
| minimumShadowArea (modelBody centered b) = (if centered then 4 else 1)*b*height b ∧ | |
| (∃ u : Space, ‖u‖ = 1 ∧ | |
| CylinderCovering.shadowArea (modelBody centered b) u = | |
| minimumShadowArea (modelBody centered b)) ∧ | |
| (∀ u : Space, ‖u‖ = 1 → | |
| (if centered then 4 else 1)*b*height b ≤ | |
| CylinderCovering.shadowArea (modelBody centered b) u) ∧ | |
| (∀ (n : ℕ) (C : Fin n → TriangularCovering.TriCylinder), | |
| TriangularCovering.totalArea C < ((if centered then 4 else 1)*b*height b)/2 → | |
| TriangularCovering.totalArea C < minimumShadowArea (modelBody centered b)/2 ∧ | |
| triangularRelativeCost (modelBody centered b) C < 1/2)) ∧ | |
| minimumShadowArea {z : Space | ∃ p q t : ℝ, | |
| |p| ≤ 1+t ∧ |q| ≤ 1-t ∧ z = nonorthPoint p q t} = 2*Real.sqrt 2 ∧ | |
| (∀ (centered : Bool) (b : ℝ), 0 < b → ∀ (eta : ℝ) (T : ℝ → ℝ) | |
| (N : ℕ) (tagU tagL : Fin N → ℝ), | |
| physicalActualCost centered b eta T N tagU tagL < | |
| ((if centered then 4 else 1)*b*height b)/2 → | |
| physicalActualCost centered b eta T N tagU tagL < | |
| minimumShadowArea (modelBody centered b)/2 ∧ | |
| actualRelativeCost centered b eta T N tagU tagL < 1/2) | |
| end RadialPhysical.PhysicalMinimum | |
| end | |
| end OAI | |