openAI-math-dataset / lean /OAI /Geometry /CylinderCovering /Rational /DictionaryStatementDefinitions.lean
| import OAI.Geometry.CylinderCovering.Rational.DictionaryDefinitions | |
| import OAI.Geometry.CylinderCovering.Rational.RealizationDefinitions | |
| namespace OAI | |
| noncomputable section | |
| open Set MeasureTheory | |
| namespace RationalDictionary | |
| def CoordinateCostStatement : Prop := | |
| ∀ d0 : ℝ, 0 < d0 → ∀ eta0 k : ℝ, | |
| RationalSimilarity.phi '' RationalSimilarity.uncenteredTetrahedron = | |
| SlopeRational.referenceTetrahedron ∧ | |
| RationalSimilarity.phi '' uncenteredFirstSweep d0 eta0 k = | |
| SlopeRational.lowerSweep (centeredRegularization d0) (centeredTilt d0 k) | |
| (centeredPadding eta0 k) ∧ | |
| RationalSimilarity.phi '' uncenteredSecondSweep d0 eta0 k = | |
| SlopeRational.upperSweep (centeredRegularization d0) (centeredTilt d0 k) | |
| (centeredPadding eta0 k) ∧ | |
| SlopeRational.weightedCost (centeredRegularization d0) (centeredTilt d0 k) | |
| (centeredPadding eta0 k) = uncenteredCost d0 eta0 k / 8 ∧ | |
| centeredPadding eta0 k = centeredMargin d0 eta0 * centeredTilt d0 k ^ 2 ∧ | |
| ∀ S : Set Space, | |
| (Measure.euclideanHausdorffMeasure 2 (RationalSimilarity.phi '' S)).toReal = | |
| (1 / 8 : ℝ) * (Measure.euclideanHausdorffMeasure 2 S).toReal | |
| end RationalDictionary | |
| end | |
| end OAI | |