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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 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 | import OAI.Geometry.TriangularCovering.Basic
namespace OAI
noncomputable section
open Set MeasureTheory
open scoped InnerProductSpace
namespace CylinderCovering.AngularGeneral
abbrev Space := TriangularCovering.Space
structure Data where
count : ℕ
node : ℕ → ℝ
alpha : ℕ → ℝ
beta : ℕ → ℝ
radius : ℕ → ℝ → ℝ
def interval (D : Data) (j : ℕ) : Set ℝ := Icc (D.node j) (D.node (j+1))
def profile (q : ℝ) : ℝ := (q^2+q^4)/16
def lowerDirection (τ : ℝ) (D : Data) (j : ℕ) : Space :=
TriangularCovering.point 1 (τ*D.alpha j) (τ*D.beta j)
def upperDirection (τ : ℝ) (D : Data) (j : ℕ) : Space :=
TriangularCovering.point (-τ*D.alpha j) 1 (τ*D.beta j)
def unitDirection (v : Space) : Space := ‖v‖⁻¹ • v
def lowerAxis (τ : ℝ) (D : Data) (j : ℕ) : Space :=
unitDirection (lowerDirection τ D j)
def upperAxis (τ : ℝ) (D : Data) (j : ℕ) : Space :=
unitDirection (upperDirection τ D j)
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}
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)}
def lowerBase (τ : ℝ) (D : Data) (j : ℕ) : Set Space :=
TriangularCovering.projection (lowerAxis τ D j) '' lowerIntercept D j
def upperBase (τ : ℝ) (D : Data) (j : ℕ) : Set Space :=
TriangularCovering.projection (upperAxis τ D j) '' upperIntercept D j
def carrier (u : Space) (B : Set Space) : Set Space :=
{z | ∃ b ∈ B, ∃ r : ℝ, z = b+r • u}
def lowerCylinder (τ : ℝ) (D : Data) (j : ℕ) : Set Space :=
carrier (lowerAxis τ D j) (lowerBase τ D j)
def upperCylinder (τ : ℝ) (D : Data) (j : ℕ) : Set Space :=
carrier (upperAxis τ D j) (upperBase τ D j)
def covers (τ : ℝ) (D : Data) : Prop :=
∀ z ∈ TriangularCovering.K, ∃ j : Fin D.count,
z ∈ lowerCylinder τ D j ∨ z ∈ upperCylinder τ D j
def totalArea (τ : ℝ) (D : Data) : ℝ :=
∑ j : Fin D.count,
(TriangularCovering.area (lowerBase τ D j) +
TriangularCovering.area (upperBase τ D j))
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))
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))
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)
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 ≠ ⊤
def IsNondegenerateTriangle (B : Set Space) : Prop :=
∃ v : Fin 3 → Space, AffineIndependent ℝ v ∧ B = convexHull ℝ (Set.range v)
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
end CylinderCovering.AngularGeneral
end
end OAI
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