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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 | import OAI.Geometry.TriangularCovering.Basic
import OAI.Geometry.CylinderCovering.Angular.MarginsDefinitions
namespace OAI
noncomputable section
namespace CylinderCovering.Cubic
open Set TriangularCovering
def K4 : Set Space :=
{p | ∃ x y z : ℝ, -1 ≤ z ∧ z ≤ 1 ∧ |x| ≤ 1 - z ∧ |y| ≤ 1 + z ∧
p = point x y z}
def CoversK4 {m : ℕ} (c : Fin m → TriCylinder) : Prop :=
∀ p ∈ K4, ∃ i : Fin m, p ∈ (c i).carrier
def cubicCost (ε : ℝ) : ℝ :=
H * ∑ j : Fin (n ε),
step (n ε) * cap ε (node (n ε) j) (node (n ε) (j + 1)) ^ 2 /
Real.sqrt (1 + ε ^ 2 *
(alpha (node (n ε) j) (node (n ε) (j + 1)) ^ 2 +
2 * beta (node (n ε) j) (node (n ε) (j + 1)) ^ 2))
def lowerDirection (ε a b : ℝ) : Space :=
!₂[1, ε * alpha a b, H * ε * beta a b]
def upperDirection (ε a b : ℝ) : Space :=
!₂[-ε * alpha a b, 1, H * ε * beta a b]
def normalizedDirection (v : Space) : Space := ‖v‖⁻¹ • v
def lowerInterceptVertices (ε a b : ℝ) : Fin 3 → Space :=
![point 0 0 (-1), point 0 (a * cap ε a b) (cap ε a b - 1),
point 0 (b * cap ε a b) (cap ε a b - 1)]
def upperInterceptVertices (ε a b : ℝ) : Fin 3 → Space :=
![point 0 0 1, point (a * cap ε a b) 0 (1 - cap ε a b),
point (b * cap ε a b) 0 (1 - cap ε a b)]
def lowerIndex (ε : ℝ) (j : Fin (n ε)) : Fin (2 * n ε) :=
Fin.cast (two_mul (n ε)).symm (Fin.castAdd (n ε) j)
def upperIndex (ε : ℝ) (j : Fin (n ε)) : Fin (2 * n ε) :=
Fin.cast (two_mul (n ε)).symm (Fin.natAdd (n ε) j)
def PrescribedFamily (ε : ℝ) (c : Fin (2 * n ε) → TriCylinder) : Prop :=
∀ j : Fin (n ε),
let a := node (n ε) j
let b := node (n ε) (j + 1)
(c (lowerIndex ε j)).axis = normalizedDirection (lowerDirection ε a b) ∧
(c (lowerIndex ε j)).base = projection (normalizedDirection (lowerDirection ε a b)) ''
convexHull ℝ (Set.range (lowerInterceptVertices ε a b)) ∧
(c (upperIndex ε j)).axis = normalizedDirection (upperDirection ε a b) ∧
(c (upperIndex ε j)).base = projection (normalizedDirection (upperDirection ε a b)) ''
convexHull ℝ (Set.range (upperInterceptVertices ε a b))
def FiniteCoverStatement : Prop :=
∀ ε : ℝ, 0 < ε → ε ≤ 1 / 2 →
∃ c : Fin (2 * n ε) → TriCylinder, CoversK4 c ∧ totalArea c = cubicCost ε ∧ PrescribedFamily ε c
def minimumProjection4 : ℝ :=
sInf {a : ℝ | ∃ u : Space, ‖u‖ = 1 ∧ a = area (projection u '' K4)}
def relativeCost4 {m : ℕ} (c : Fin m → TriCylinder) : ℝ :=
∑ i : Fin m, area (c i).base / area (projection (c i).axis '' K4)
def MainStatement : Prop :=
minimumProjection4 = 4 * H ∧
∃ C δ : ℝ, 0 < C ∧ 0 < δ ∧
∀ ε : ℝ, 0 < ε → ε ≤ 1 / 2 →
∃ c : Fin (2 * n ε) → TriCylinder,
CoversK4 c ∧ PrescribedFamily ε c ∧ totalArea c = cubicCost ε ∧
|totalArea c - (2 * H - H / 15 * ε ^ 2)| ≤ C * ε ^ 3 ∧
(ε < δ → totalArea c < minimumProjection4 / 2)
def scaledLowerInterceptVertices (ε a b : ℝ) : Fin 3 → Space :=
![point 0 0 0, point 0 (a * (cap ε a b / 2)) (cap ε a b / 2),
point 0 (b * (cap ε a b / 2)) (cap ε a b / 2)]
def scaledUpperInterceptVertices (ε a b : ℝ) : Fin 3 → Space :=
![point 0 0 1, point (a * (cap ε a b / 2)) 0 (1 - cap ε a b / 2),
point (b * (cap ε a b / 2)) 0 (1 - cap ε a b / 2)]
def lowerIndexN (N : ℕ) (j : Fin N) : Fin (2 * N) :=
Fin.cast (two_mul N).symm (Fin.castAdd N j)
def upperIndexN (N : ℕ) (j : Fin N) : Fin (2 * N) :=
Fin.cast (two_mul N).symm (Fin.natAdd N j)
def ScaledPrescribedFamily (ε : ℝ) (N : ℕ) (c : Fin (2 * N) → TriCylinder) : Prop :=
∀ j : Fin N,
let a := node N j
let b := node N (j + 1)
(c (lowerIndexN N j)).axis = normalizedDirection (lowerDirection ε a b) ∧
(c (lowerIndexN N j)).base = projection (normalizedDirection (lowerDirection ε a b)) ''
convexHull ℝ (Set.range (scaledLowerInterceptVertices ε a b)) ∧
(c (upperIndexN N j)).axis = normalizedDirection (upperDirection ε a b) ∧
(c (upperIndexN N j)).base = projection (normalizedDirection (upperDirection ε a b)) ''
convexHull ℝ (Set.range (scaledUpperInterceptVertices ε a b))
def ScaledFiniteCoverStatement : Prop :=
∀ τ : ℝ, 0 < τ → τ ≤ 1 →
∃ c : Fin (2 * ⌈8 / τ ^ 2⌉₊) → TriCylinder,
Covers c ∧ totalArea c = cubicCost (τ / 2) / 4 ∧
ScaledPrescribedFamily (τ / 2) ⌈8 / τ ^ 2⌉₊ c
def ScaledMainStatement : Prop :=
∃ C δ : ℝ, 0 < C ∧ 0 < δ ∧
∀ τ : ℝ, 0 < τ → τ ≤ 1 →
∃ c : Fin (2 * ⌈8 / τ ^ 2⌉₊) → TriCylinder,
Covers c ∧ ScaledPrescribedFamily (τ / 2) ⌈8 / τ ^ 2⌉₊ c ∧
totalArea c = cubicCost (τ / 2) / 4 ∧
|totalArea c / H - (1 / 2 - τ ^ 2 / 240)| ≤ C * τ ^ 3 ∧
(τ < δ → totalArea c < A_min / 2)
end CylinderCovering.Cubic
end
end OAI
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