File size: 2,852 Bytes
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 | 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
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