| import OAI.Geometry.TriangularCovering.Basic |
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| namespace OAI |
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| noncomputable section |
| open Set MeasureTheory |
| open scoped InnerProductSpace |
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| namespace CylinderCovering.AngularGeneral |
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| abbrev Space := TriangularCovering.Space |
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|
| structure Data where |
| count : ℕ |
| node : ℕ → ℝ |
| alpha : ℕ → ℝ |
| beta : ℕ → ℝ |
| radius : ℕ → ℝ → ℝ |
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| def interval (D : Data) (j : ℕ) : Set ℝ := Icc (D.node j) (D.node (j+1)) |
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| def profile (q : ℝ) : ℝ := (q^2+q^4)/16 |
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| def lowerDirection (τ : ℝ) (D : Data) (j : ℕ) : Space := |
| TriangularCovering.point 1 (τ*D.alpha j) (τ*D.beta j) |
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| def upperDirection (τ : ℝ) (D : Data) (j : ℕ) : Space := |
| TriangularCovering.point (-τ*D.alpha j) 1 (τ*D.beta j) |
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| def unitDirection (v : Space) : Space := ‖v‖⁻¹ • v |
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| def lowerAxis (τ : ℝ) (D : Data) (j : ℕ) : Space := |
| unitDirection (lowerDirection τ D j) |
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| def upperAxis (τ : ℝ) (D : Data) (j : ℕ) : Space := |
| unitDirection (upperDirection τ D j) |
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| def lowerIntercept (D : Data) (j : ℕ) : Set Space := |
| {z | ∃ q ∈ interval D j, ∃ S ∈ Icc (0:ℝ) (D.radius j q), |
| z = TriangularCovering.point 0 (q*S) S} |
|
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| def upperIntercept (D : Data) (j : ℕ) : Set Space := |
| {z | ∃ q ∈ interval D j, ∃ T ∈ Icc (0:ℝ) (D.radius j q), |
| z = TriangularCovering.point (q*T) 0 (1-T)} |
|
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| def lowerBase (τ : ℝ) (D : Data) (j : ℕ) : Set Space := |
| TriangularCovering.projection (lowerAxis τ D j) '' lowerIntercept D j |
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| def upperBase (τ : ℝ) (D : Data) (j : ℕ) : Set Space := |
| TriangularCovering.projection (upperAxis τ D j) '' upperIntercept D j |
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| def carrier (u : Space) (B : Set Space) : Set Space := |
| {z | ∃ b ∈ B, ∃ r : ℝ, z = b+r • u} |
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| def lowerCylinder (τ : ℝ) (D : Data) (j : ℕ) : Set Space := |
| carrier (lowerAxis τ D j) (lowerBase τ D j) |
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| def upperCylinder (τ : ℝ) (D : Data) (j : ℕ) : Set Space := |
| carrier (upperAxis τ D j) (upperBase τ D j) |
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| def covers (τ : ℝ) (D : Data) : Prop := |
| ∀ z ∈ TriangularCovering.K, ∃ j : Fin D.count, |
| z ∈ lowerCylinder τ D j ∨ z ∈ upperCylinder τ D j |
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| def totalArea (τ : ℝ) (D : Data) : ℝ := |
| ∑ j : Fin D.count, |
| (TriangularCovering.area (lowerBase τ D j) + |
| TriangularCovering.area (upperBase τ D j)) |
|
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| def relativeCost (τ : ℝ) (D : Data) : ℝ := |
| ∑ j : Fin D.count, |
| (TriangularCovering.area (lowerBase τ D j) / |
| TriangularCovering.shadowArea (lowerAxis τ D j) + |
| TriangularCovering.area (upperBase τ D j) / |
| TriangularCovering.shadowArea (upperAxis τ D j)) |
|
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| def integralCost (τ : ℝ) (D : Data) : ℝ := |
| ∑ j : Fin D.count, ∫ q in D.node j..D.node (j+1), |
| (D.radius j q)^2 / Real.sqrt (1+τ^2*((D.alpha j)^2+2*(D.beta j)^2)) |
|
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| def Admissible (τ η E : ℝ) (D : Data) : Prop := |
| 0 < D.count ∧ D.node 0 = -1 ∧ D.node D.count = 1 ∧ |
| (∀ j < D.count, D.node j < D.node (j+1)) ∧ |
| (∀ j ≤ D.count, |D.node j| ≤ 1) ∧ |
| (∃ L : ℝ → ℝ, ContinuousOn L (Icc (-1:ℝ) 1) ∧ L (-1) = 0 ∧ L 1 = 0 ∧ |
| ∀ j < D.count, ∀ q ∈ interval D j, D.alpha j-q*D.beta j = L q) ∧ |
| (∀ j < D.count, ContinuousOn (D.radius j) (interval D j)) ∧ |
| (∀ j < D.count, ∀ q ∈ interval D j, 0 < D.radius j q) ∧ |
| (∀ j < D.count, ∀ q ∈ interval D j, |
| |D.alpha j-(1+q^2)/4| ≤ E*τ^2 ∧ |
| |D.beta j-q/2| ≤ E*τ^2 ∧ |
| |D.radius j q-(1/2+τ^2*(profile q+η))| ≤ E*τ^4) |
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| def BaseProperties (u : Space) (B : Set Space) : Prop := |
| ‖u‖ = 1 ∧ (∀ z ∈ B, inner ℝ u z = 0) ∧ |
| IsCompact B ∧ MeasurableSet B ∧ Measure.euclideanHausdorffMeasure 2 B ≠ ⊤ |
|
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| def IsNondegenerateTriangle (B : Set Space) : Prop := |
| ∃ v : Fin 3 → Space, AffineIndependent ℝ v ∧ B = convexHull ℝ (Set.range v) |
|
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| def Conclusion (τ η A : ℝ) (D : Data) : Prop := |
| covers τ D ∧ |
| (∀ j < D.count, |
| BaseProperties (lowerAxis τ D j) (lowerBase τ D j) ∧ |
| BaseProperties (upperAxis τ D j) (upperBase τ D j)) ∧ |
| totalArea τ D / TriangularCovering.H = integralCost τ D ∧ |
| |totalArea τ D / TriangularCovering.H - (1/2+(2*η-1/240)*τ^2)| ≤ A*τ^4 ∧ |
| totalArea τ D < TriangularCovering.A_min/2 ∧ relativeCost τ D < 1/2 ∧ |
| (∀ j < D.count, (∃ r : ℝ, ∀ q ∈ interval D j, D.radius j q = r) → |
| IsNondegenerateTriangle (lowerBase τ D j) ∧ |
| IsNondegenerateTriangle (upperBase τ D j)) |
|
|
| def Statement : Prop := |
| ∀ η : ℝ, 0 < η → η < 1/480 → ∀ E : ℝ, 0 ≤ E → |
| ∃ δ A : ℝ, 0 < δ ∧ 0 < A ∧ |
| ∀ τ : ℝ, 0 < τ → τ < δ → ∀ D : Data, |
| Admissible τ η E D → Conclusion τ η A D |
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| end CylinderCovering.AngularGeneral |
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| end |
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| end OAI |
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