| import OAI.Geometry.RuledCovering.Model |
| import OAI.Geometry.CylinderCovering.Singular.CutoffDefinitions |
|
|
| namespace OAI |
|
|
| noncomputable section |
| open Set MeasureTheory Filter Topology |
|
|
| namespace CylinderCovering.FlexibleSingular |
| open RuledApproximation |
|
|
| def labelDomain (p c d q : ℝ) : Set Plane := |
| {b | d ≤ b 1 ∧ b 1 ≤ 2 ∧ |b 0| ≤ b 1 ∧ |
| b 1 ≤ (1 / 2 : ℝ) + c ^ 2 * ceilingPolynomial p (b 0) + q} |
|
|
| def lowerFamily (p c d q : ℝ) : Set Space := |
| physicalRuled (Real.sqrt 2) (labelDomain p c d q) |
| (fun b => c • singularField p b) 1 |
|
|
| def upperFamily (p c d q : ℝ) : Set Space := |
| {z | ∃ b ∈ labelDomain p c d q, ∃ y : ℝ, |y| ≤ 1 ∧ |
| z = vec3 (b 0 - c * y * singularField p b 0) y |
| (Real.sqrt 2 * (1 - (b 1 - c * y * singularField p b 1)))} |
|
|
| def cost (p c d q : ℝ) : ℝ := |
| Real.sqrt 2 * ∫ b in labelDomain p c d q, |
| physicalWeight (Real.sqrt 2) (c • singularField p b) |
|
|
| def Statement : Prop := |
| ∀ (p : ℝ) (ρ q : ℝ → ℝ), 1 < p → p < (Real.sqrt 51 - 5) / 2 → |
| SingularCutoff.Admissible p ρ q → |
| singularCoefficient p < 0 ∧ |
| Asymptotics.IsLittleO (𝓝[>] (0 : ℝ)) |
| (fun c => 2 * cost p c (ρ c) (q c) + 8 * Real.sqrt 2 * (ρ c) ^ 2 - |
| (Real.sqrt 2 / 2 + 2 * Real.sqrt 2 * singularCoefficient p * c ^ 2)) |
| (fun c => c ^ 2) ∧ |
| ∀ᶠ c : ℝ in 𝓝[>] (0 : ℝ), |
| ∃ 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 c (ρ c) (q c) ⊆ familySet C ∧ |
| upperFamily p c (ρ c) (q c) ⊆ familySet D ∧ |
| B₁.carrier = lowerTip (ρ c) ∧ B₂.carrier = upperTip (ρ c) ∧ |
| B₁.area = 4 * Real.sqrt 2 * (ρ c) ^ 2 ∧ |
| B₂.area = 4 * Real.sqrt 2 * (ρ c) ^ 2 ∧ |
| tetrahedron ⊆ familySet C ∪ familySet D ∪ B₁.carrier ∪ B₂.carrier ∧ |
| familyArea C + familyArea D + B₁.area + B₂.area < Real.sqrt 2 / 2 |
|
|
| def OrthogonalStatement : Prop := |
| ∀ (p : ℝ) (ρ q : ℝ → ℝ), 1 < p → p < 2 → |
| SingularCutoff.Admissible p ρ q → |
| (∀ c : ℝ, volume (frontier (labelDomain p c (ρ c) (q c))) = 0) ∧ |
| (∀ e : ℝ, 0 < e → ∀ᶠ c : ℝ in 𝓝[>] 0, |
| ∀ z ∈ labelDomain p c (ρ c) (q c), |
| ‖fderiv ℝ (fun w => c • singularField p w) z‖ < e ∧ |
| ‖fderiv ℝ (fun w => -c • singularField p w) z‖ < e) ∧ |
| ∀ᶠ c : ℝ in 𝓝[>] 0, ∀ e : ℝ, 0 < e → |
| HasFiniteCover Cylinder.parallelogram (lowerFamily p c (ρ c) (q c)) |
| (cost p c (ρ c) (q c)+e) ∧ |
| HasFiniteCover Cylinder.parallelogram (upperFamily p c (ρ c) (q c)) |
| (cost p c (ρ c) (q c)+e) |
|
|
| end CylinderCovering.FlexibleSingular |
|
|
| end |
|
|
| end OAI |
|
|
