openAI-math-dataset / lean /OAI /Geometry /CylinderCovering /Rational /DictionaryCostDefinitions.lean
| import OAI.Geometry.CylinderCovering.Rational.DictionaryDefinitions | |
| import OAI.Geometry.CylinderCovering.Rational.RealizationDefinitions | |
| namespace OAI | |
| noncomputable section | |
| open MeasureTheory | |
| namespace RationalDictionary | |
| def orthonormalWeight (d0 k : ℝ) (p : Plane) : ℝ := | |
| 1 / Real.sqrt (1 + k ^ 2 * (v1 d0 (p 0) (p 1) ^ 2 + v2 d0 (p 0) (p 1) ^ 2)) | |
| def orthonormalCost (d0 eta0 k : ℝ) : ℝ := | |
| ∫ p in uncenteredDomain d0 eta0 k, orthonormalWeight d0 k p | |
| def CostDictionaryStatement : Prop := | |
| ∀ d0 : ℝ, 0 < d0 → ∀ eta0 k : ℝ, | |
| SlopeRational.weightedCost (centeredRegularization d0) (centeredTilt d0 k) | |
| (centeredPadding eta0 k) = orthonormalCost d0 eta0 k / 8 | |
| end RationalDictionary | |
| end | |
| end OAI | |