openAI-math-dataset / lean /OAI /Geometry /CylinderCovering /Physical /CapNormalizedDefinitions.lean
| import OAI.Geometry.CylinderCovering.Physical.CapFiniteDefinitions | |
| import OAI.Geometry.CylinderCovering.Convexity.GlobalDefinitions | |
| namespace OAI | |
| open Set Filter | |
| open scoped Topology | |
| noncomputable section | |
| namespace RadialPhysical | |
| open CylinderCovering | |
| def physicalActualRelativeCost (centered : Bool) (b eta : ℝ) (T : ℝ → ℝ) | |
| (N : ℕ) (tagU tagL : Fin N → ℝ) : ℝ := | |
| (∑ i : Fin N, TriangularCovering.area (upperPhysicalBase centered b eta T N tagU i) / | |
| shadowArea (modelBody centered b) (upperPhysicalAxis b eta (tagU i))) + | |
| (∑ i : Fin N, TriangularCovering.area (lowerPhysicalBase centered b eta T N tagL i) / | |
| shadowArea (modelBody centered b) (lowerPhysicalAxis b eta (tagL i))) | |
| def PhysicalBothHalfBounds (centered : Bool) (b eta : ℝ) (T : ℝ → ℝ) | |
| (N : ℕ) (tagU tagL : Fin N → ℝ) | |
| (C : Fin (2*N) → TriangularCovering.TriCylinder) : Prop := | |
| TriangularCovering.totalArea C < minimumShadowArea (modelBody centered b)/2 ∧ | |
| triangularRelativeCost (modelBody centered b) C < 1/2 ∧ | |
| physicalActualCost centered b eta T N tagU tagL < minimumShadowArea (modelBody centered b)/2 ∧ | |
| physicalActualRelativeCost centered b eta T N tagU tagL < 1/2 | |
| def PhysicalCapNormalizedStatement : Prop := | |
| ∀ (centered square : Bool) (b beta : ℝ), 0 < b → 0 < beta → beta < 1/60 → | |
| ∀ᶠ eta : ℝ in 𝓝[>] (0 : ℝ), | |
| ∀ tagU tagL : (N : ℕ) → Fin N → ℝ, | |
| (∀ N, RadialRiemann.IsTagged (-b) b N (tagU N)) → | |
| (∀ N, RadialRiemann.IsTagged (-b) b N (tagL N)) → | |
| ∃ C : (N : ℕ) → Fin (2*N) → TriangularCovering.TriCylinder, | |
| PhysicalFamilyConclusion centered b eta (physicalModelCap centered square b eta beta) | |
| tagU tagL C ∧ | |
| ∀ᶠ N in atTop, 0 < N ∧ | |
| PhysicalBothHalfBounds centered b eta (physicalModelCap centered square b eta beta) | |
| N (tagU N) (tagL N) (C N) | |
| end RadialPhysical | |
| end | |
| end OAI | |