| import OAI.Geometry.CylinderCovering.Convexity.GlobalDefinitions |
| import OAI.Geometry.CylinderCovering.Rational.RealizationDefinitions |
| import OAI.Geometry.CylinderCovering.Rational.CeilingIntegralDefinitions |
| import OAI.Geometry.CylinderCovering.Rational.EnergyIntegralDefinitions |
|
|
| namespace OAI |
|
|
| noncomputable section |
| open Set MeasureTheory Filter |
| open scoped Topology ContDiff |
|
|
| namespace SlopeRational |
|
|
| def coefficientGap (d : ℝ) : ℝ := energyIntegral d - ceilingIntegral d |
|
|
| def AdmissiblePadding (q : ℝ → ℝ) : Prop := |
| (∀ᶠ ε : ℝ in 𝓝[>] (0 : ℝ), 0 < q ε) ∧ |
| Asymptotics.IsBigO (𝓝[>] (0 : ℝ)) q (fun ε => ε ^ 2) ∧ |
| Asymptotics.IsLittleO (𝓝[>] (0 : ℝ)) (fun ε => ε ^ 3) q |
|
|
| def HasStrictFiniteCover (d ε q : ℝ) : Prop := |
| ∃ n m : ℕ, ∃ C : Fin n → RuledApproximation.Cylinder, |
| ∃ D : Fin m → RuledApproximation.Cylinder, |
| (∀ i, (C i).parallelogram) ∧ (∀ j, (D j).parallelogram) ∧ |
| lowerSweep d ε q ⊆ RuledApproximation.familySet C ∧ |
| upperSweep d ε q ⊆ RuledApproximation.familySet D ∧ |
| referenceTetrahedron ⊆ RuledApproximation.familySet C ∪ RuledApproximation.familySet D ∧ |
| RuledApproximation.familyArea C + RuledApproximation.familyArea D < physicalScale / 4 ∧ |
| CylinderCovering.relativeCost referenceTetrahedron C + |
| CylinderCovering.relativeCost referenceTetrahedron D < 1 / 2 |
|
|
| def AsymptoticStatement : Prop := |
| (∀ d : ℝ, 0 < d → ∀ q : ℝ → ℝ, AdmissiblePadding q → |
| (∀ᶠ ε : ℝ in 𝓝[>] (0 : ℝ), |
| referenceTetrahedron ⊆ lowerSweep d ε (q ε) ∪ upperSweep d ε (q ε)) ∧ |
| Asymptotics.IsBigO (𝓝[>] (0 : ℝ)) |
| (fun ε => weightedCost d ε (q ε) / physicalScale - |
| (1 / 8 + ε ^ 2 * (ceilingIntegral d - energyIntegral d) + q ε / 2)) |
| (fun ε => ε ^ 4)) ∧ |
| Tendsto ceilingIntegral (𝓝[>] (0 : ℝ)) (𝓝 (16 / 15360 : ℝ)) ∧ |
| Tendsto energyIntegral (𝓝[>] (0 : ℝ)) (𝓝 (17 / 15360 : ℝ)) |
|
|
| def ApproximationEligibilityStatement : Prop := |
| ∀ d : ℝ, 0 < d → ∀ q : ℝ → ℝ, AdmissiblePadding q → |
| ∀ᶠ ε : ℝ in 𝓝[>] (0 : ℝ), |
| IsCompact (labelDomain d ε (q ε)) ∧ |
| volume (frontier (labelDomain d ε (q ε))) = 0 ∧ |
| RuledApproximation.SmoothNear ∞ (labelDomain d ε (q ε)) (field d) ∧ |
| (∀ p ∈ labelDomain d ε (q ε), |
| RuledApproximation.HyperbolicAt (1 / 2) (fun z => ε • field d z) p) ∧ |
| (∀ p ∈ labelDomain d ε (q ε), |
| RuledApproximation.HyperbolicAt (1 / 2) (fun z => -ε • field d z) p) |
|
|
| def PaddingSchedulesStatement : Prop := |
| (∃ d : ℝ, 0 < d ∧ 0 < coefficientGap d) ∧ |
| (∀ d : ℝ, 0 < d → 0 < coefficientGap d → |
| (∀ᶠ ε : ℝ in 𝓝[>] (0 : ℝ), |
| weightedCost d ε (ε ^ (5 / 2 : ℝ)) < physicalScale / 8 ∧ |
| HasStrictFiniteCover d ε (ε ^ (5 / 2 : ℝ))) ∧ |
| ∀ η : ℝ, 0 < η → η / 2 < coefficientGap d → |
| ∀ᶠ ε : ℝ in 𝓝[>] (0 : ℝ), |
| weightedCost d ε (η * ε ^ 2) < physicalScale / 8 ∧ |
| HasStrictFiniteCover d ε (η * ε ^ 2)) |
|
|
| def ConstructionStatement : Prop := |
| CylinderCovering.minimumShadowArea referenceTetrahedron = physicalScale / 2 ∧ |
| AsymptoticStatement ∧ ApproximationEligibilityStatement ∧ PaddingSchedulesStatement |
|
|
| end SlopeRational |
|
|
| end |
|
|
| end OAI |
|
|
