| 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 |
|
|
