File size: 2,146 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 | 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
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