openAI-math-dataset / lean /OAI /Geometry /CylinderCovering /Rational /LogarithmicStatementDefinitions.lean
| import OAI.Geometry.CylinderCovering.Rational.DictionaryCoefficientsDefinitions | |
| import OAI.Geometry.CylinderCovering.Convexity.GlobalDefinitions | |
| namespace OAI | |
| noncomputable section | |
| open Set Filter | |
| open scoped Topology | |
| namespace RationalDictionary | |
| def LogarithmicApproximationStatement : Prop := | |
| ∀ d0 : ℝ, 0 < d0 → ∀ k : ℝ, 0 < k → k < d0 / (1 + d0) → | |
| ∀ eta0 : ℝ, ∀ e : ℝ, 0 < e → | |
| RuledApproximation.HasFiniteCover RuledApproximation.Cylinder.openBounded | |
| (uncenteredFirstSweep d0 eta0 k) (orthonormalCost d0 eta0 k + e) ∧ | |
| RuledApproximation.HasFiniteCover RuledApproximation.Cylinder.openBounded | |
| (uncenteredSecondSweep d0 eta0 k) (orthonormalCost d0 eta0 k + e) | |
| def HasStrictLogarithmicCover (d0 eta0 k : ℝ) : Prop := | |
| ∃ n m : ℕ, ∃ C : Fin n → RuledApproximation.Cylinder, | |
| ∃ D : Fin m → RuledApproximation.Cylinder, | |
| (∀ i, (C i).openBounded) ∧ (∀ j, (D j).openBounded) ∧ | |
| uncenteredFirstSweep d0 eta0 k ⊆ RuledApproximation.familySet C ∧ | |
| uncenteredSecondSweep d0 eta0 k ⊆ RuledApproximation.familySet D ∧ | |
| RationalSimilarity.uncenteredTetrahedron ⊆ | |
| RuledApproximation.familySet C ∪ RuledApproximation.familySet D ∧ | |
| RuledApproximation.familyArea C + RuledApproximation.familyArea D < | |
| CylinderCovering.minimumShadowArea RationalSimilarity.uncenteredTetrahedron / 2 ∧ | |
| CylinderCovering.relativeCost RationalSimilarity.uncenteredTetrahedron C + | |
| CylinderCovering.relativeCost RationalSimilarity.uncenteredTetrahedron D < 1 / 2 | |
| def LogarithmicStatement : Prop := | |
| LogarithmicApproximationStatement ∧ | |
| CylinderCovering.minimumShadowArea RationalSimilarity.uncenteredTetrahedron = 4 * c0 ∧ | |
| (∀ d0 : ℝ, 0 < d0 → 0 < coefficientGap d0 → | |
| ∀ eta0 : ℝ, 0 < eta0 → eta0 < coefficientGap d0 / (2 * c0) → | |
| ∀ᶠ k : ℝ in 𝓝[>] (0 : ℝ), | |
| orthonormalCost d0 eta0 k < c0 ∧ HasStrictLogarithmicCover d0 eta0 k) ∧ | |
| ∃ d0 : ℝ, 0 < d0 ∧ ∃ eta0 : ℝ, 0 < eta0 ∧ | |
| eta0 < coefficientGap d0 / (2 * c0) ∧ | |
| ∀ᶠ k : ℝ in 𝓝[>] (0 : ℝ), | |
| orthonormalCost d0 eta0 k < c0 ∧ HasStrictLogarithmicCover d0 eta0 k | |
| end RationalDictionary | |
| end | |
| end OAI | |