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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 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 | 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
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