| import OAI.Geometry.CylinderCovering.Singular.BasicDefinitions |
| import OAI.Geometry.CylinderCovering.Singular.EdgeOneDefinitions |
|
|
| namespace OAI |
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| noncomputable section |
| open Set MeasureTheory Filter Topology |
|
|
| namespace CylinderCovering.SingularEdgeOneCover |
| open RuledApproximation |
|
|
| def map (z : Space) : Space := vec3 (z 1 / 2) (z 0 / 2) (z 2 / 2) |
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| def height : ℝ := 1 / Real.sqrt 2 |
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| def tetrahedron : Set Space := |
| {z | ∃ s : ℝ, 0 ≤ s ∧ s ≤ 1 ∧ |z 0| ≤ s / 2 ∧ |
| |z 1| ≤ (1 - s) / 2 ∧ z 2 = height * s} |
|
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| def labelDomain (p k d q : ℝ) : Set Plane := |
| {b | d ≤ b 1 ∧ b 1 ≤ 1 ∧ |b 0| ≤ b 1 / 2 ∧ |
| b 1 ≤ (1 / 2 : ℝ) + k ^ 2 * SingularEdgeOne.edgeCeiling p (b 0) + q} |
|
|
| def lowerFamily (p k d q : ℝ) : Set Space := |
| {z | ∃ b ∈ labelDomain p k d q, ∃ y : ℝ, |y| ≤ 1 / 2 ∧ |
| z = vec3 (b 0 + k * y * SingularEdgeOne.edgeU p (b 0) (b 1)) y |
| (height * (b 1 + k * y * SingularEdgeOne.edgeV p (b 0) (b 1)))} |
|
|
| def upperFamily (p k d q : ℝ) : Set Space := |
| {z | ∃ b ∈ labelDomain p k d q, ∃ x : ℝ, |x| ≤ 1 / 2 ∧ |
| z = vec3 x (b 0 - k * x * SingularEdgeOne.edgeU p (b 0) (b 1)) |
| (height * (1 - (b 1 - k * x * SingularEdgeOne.edgeV p (b 0) (b 1))))} |
|
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| def lowerTip (d : ℝ) : Set Space := |
| {z | ∃ s : ℝ, 0 ≤ s ∧ s ≤ 2 * d ∧ |z 0| ≤ s / 2 ∧ z 2 = height * s} |
|
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| def upperTip (d : ℝ) : Set Space := |
| {z | ∃ s : ℝ, 0 ≤ s ∧ s ≤ 2 * d ∧ |z 1| ≤ s / 2 ∧ z 2 = height * (1 - s)} |
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|
| def cost (p k d q : ℝ) : ℝ := |
| height * ∫ b in labelDomain p k d q, |
| physicalWeight height |
| (k • vec2 (SingularEdgeOne.edgeU p (b 0) (b 1)) |
| (SingularEdgeOne.edgeV p (b 0) (b 1))) |
|
|
| def FiniteCoverAt (p k d q : ℝ) : Prop := |
| ∃ n m : ℕ, ∃ C : Fin n → Cylinder, ∃ D : Fin m → Cylinder, |
| ∃ B₁ B₂ : Cylinder, |
| (∀ i, (C i).parallelogram) ∧ (∀ j, (D j).parallelogram) ∧ |
| B₁.triangle ∧ B₂.triangle ∧ |
| lowerFamily p k d q ⊆ familySet C ∧ upperFamily p k d q ⊆ familySet D ∧ |
| B₁.carrier = lowerTip d ∧ B₂.carrier = upperTip d ∧ |
| B₁.area = 2 * height * d ^ 2 ∧ B₂.area = 2 * height * d ^ 2 ∧ |
| tetrahedron ⊆ familySet C ∪ familySet D ∪ B₁.carrier ∪ B₂.carrier ∧ |
| familyArea C + familyArea D + B₁.area + B₂.area < height / 4 |
|
|
| def AlternativeStatement : Prop := |
| ∀ p β : ℝ, 1 < p → p < (Real.sqrt 51 - 5) / 2 → |
| 1 < β → β < 1 / (2 - p) → |
| ∀ᶠ k : ℝ in 𝓝[>] 0, FiniteCoverAt p k (k ^ β) (k ^ (5 / 2 : ℝ)) |
|
|
| def FlexibleStatement : Prop := |
| ∀ (p : ℝ) (ρ q : ℝ → ℝ), 1 < p → p < (Real.sqrt 51-5)/2 → |
| SingularCutoff.Admissible p ρ q → |
| ∀ᶠ k : ℝ in 𝓝[>] 0, FiniteCoverAt p k (ρ (k/4)) (q (k/4)) |
|
|
| def ExactStatement : Prop := |
| ∀ p γ : ℝ, 1 < p → p < (Real.sqrt 51-5)/2 → |
| 1 < γ → γ < 1/(2-p) → |
| ∀ᶠ k : ℝ in 𝓝[>] 0, |
| FiniteCoverAt p k ((4:ℝ)^(-γ)*k^γ) (k^(5/2:ℝ)/32) |
|
|
| def CostAt (p k d q : ℝ) : Prop := |
| 2*cost p k d q+4*height*d^2 < height/4 ∧ |
| ∀ e : ℝ, 0 < e → |
| HasFiniteCover Cylinder.parallelogram (lowerFamily p k d q) (cost p k d q+e) ∧ |
| HasFiniteCover Cylinder.parallelogram (upperFamily p k d q) (cost p k d q+e) |
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| def CostStatement : Prop := |
| ∀ (p : ℝ) (ρ q : ℝ → ℝ), 1 < p → p < (Real.sqrt 51-5)/2 → |
| SingularCutoff.Admissible p ρ q → |
| Asymptotics.IsLittleO (𝓝[>] (0:ℝ)) |
| (fun k => cost p k (ρ (k/4)) (q (k/4))/height- |
| (1/8+singularCoefficient p/32*k^2)) (fun k => k^2) ∧ |
| ∀ᶠ k : ℝ in 𝓝[>] 0, CostAt p k (ρ (k/4)) (q (k/4)) |
|
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| def AlternativeCostStatement : Prop := |
| ∀ p β : ℝ, 1 < p → p < (Real.sqrt 51-5)/2 → |
| 1 < β → β < 1/(2-p) → |
| Asymptotics.IsLittleO (𝓝[>] (0:ℝ)) |
| (fun k => cost p k (k^β) (k^(5/2:ℝ))/height- |
| (1/8+singularCoefficient p/32*k^2)) (fun k => k^2) ∧ |
| ∀ᶠ k : ℝ in 𝓝[>] 0, CostAt p k (k^β) (k^(5/2:ℝ)) |
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| end CylinderCovering.SingularEdgeOneCover |
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| end |
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| end OAI |
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