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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 | import OAI.Geometry.CylinderCovering.Singular.BasicDefinitions
import OAI.Geometry.CylinderCovering.Singular.EdgeOneDefinitions
namespace OAI
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)
def height : ℝ := 1 / Real.sqrt 2
def tetrahedron : Set Space :=
{z | ∃ s : ℝ, 0 ≤ s ∧ s ≤ 1 ∧ |z 0| ≤ s / 2 ∧
|z 1| ≤ (1 - s) / 2 ∧ z 2 = height * s}
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))))}
def lowerTip (d : ℝ) : Set Space :=
{z | ∃ s : ℝ, 0 ≤ s ∧ s ≤ 2 * d ∧ |z 0| ≤ s / 2 ∧ z 2 = height * s}
def upperTip (d : ℝ) : Set Space :=
{z | ∃ s : ℝ, 0 ≤ s ∧ s ≤ 2 * d ∧ |z 1| ≤ s / 2 ∧ z 2 = height * (1 - s)}
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)
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))
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))
end CylinderCovering.SingularEdgeOneCover
end
end OAI
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