| import OAI.Geometry.TriangularCovering.Main | |
| namespace OAI | |
| open Set MeasureTheory | |
| open scoped InnerProductSpace | |
| noncomputable section | |
| namespace TriangularCovering | |
| def lowerAngularIndex {N : ℕ} (j : Fin N) : Fin (2*N) := | |
| Fin.cast (two_mul N).symm (Fin.castAdd N j) | |
| def upperAngularIndex {N : ℕ} (j : Fin N) : Fin (2*N) := | |
| Fin.cast (two_mul N).symm (Fin.natAdd N j) | |
| def lowerAngularAxis (ε : ℝ) (j : ℕ) : Space := | |
| let a := mesh (n ε) j | |
| let b := mesh (n ε) (j+1) | |
| unit (tiltV ε (alpha a b) (beta a b)) | |
| def upperAngularAxis (ε : ℝ) (j : ℕ) : Space := | |
| let a := mesh (n ε) j | |
| let b := mesh (n ε) (j+1) | |
| unit (tiltW ε (alpha a b) (beta a b)) | |
| def lowerAngularBase (ε : ℝ) (j : ℕ) : Set Space := | |
| let a := mesh (n ε) j | |
| let b := mesh (n ε) (j+1) | |
| let v := lowerVertices a b (cutoff ε a b) | |
| projection (lowerAngularAxis ε j) '' triangle (v 0) (v 1) (v 2) | |
| def upperAngularBase (ε : ℝ) (j : ℕ) : Set Space := | |
| let a := mesh (n ε) j | |
| let b := mesh (n ε) (j+1) | |
| let v := upperVertices a b (cutoff ε a b) | |
| projection (upperAngularAxis ε j) '' triangle (v 0) (v 1) (v 2) | |
| def ExplicitAngularFamily (ε : ℝ) (c : Fin (2*n ε) → TriCylinder) : Prop := | |
| ∀ j : Fin (n ε), | |
| (c (lowerAngularIndex j)).axis = lowerAngularAxis ε j ∧ | |
| (c (lowerAngularIndex j)).base = lowerAngularBase ε j ∧ | |
| (c (upperAngularIndex j)).axis = upperAngularAxis ε j ∧ | |
| (c (upperAngularIndex j)).base = upperAngularBase ε j | |
| def AngularAreaStatement : Prop := | |
| ∀ ε : ℝ, 0 < ε → ε ≤ 1 → | |
| ∃ c : Fin (2*n ε) → TriCylinder, | |
| ExplicitAngularFamily ε c ∧ | |
| totalArea c/H = ∑ j : Fin (n ε), | |
| normalizedTerm ε (mesh (n ε) j) (mesh (n ε) (j+1)) ∧ | |
| |totalArea c/H - (1/2-(13/6000:ℝ)*ε^2)| ≤ 2*ε^4 | |
| def ExplicitAngularStatement : Prop := | |
| A_min = H ∧ | |
| ∀ ε : ℝ, 0 < ε → ε ≤ 1/2000 → | |
| ∃ c : Fin (2*n ε) → TriCylinder, | |
| ExplicitAngularFamily ε c ∧ Covers c ∧ | |
| |totalArea c/H - (1/2-(13/6000:ℝ)*ε^2)| ≤ 2*ε^4 ∧ | |
| totalArea c < A_min/2 ∧ relativeCost c < 1/2 | |
| end TriangularCovering | |
| end | |
| end OAI | |