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785c727 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 | 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
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