| import OAI.Geometry.CylinderCovering.Physical.CostDefinitions |
| import OAI.Geometry.CylinderCovering.Convexity.SimilarityDefinitions |
|
|
| namespace OAI |
|
|
| noncomputable section |
| namespace CylinderCovering.UnitEdgeFootprints |
|
|
| def linearReflection (z : Space) : Space := !₂[-z 1, z 0, -z 2] |
|
|
| def reflection (z : Space) : Space := !₂[-z 1, z 0, height - z 2] |
|
|
| def secondRepresentedPoint (η s ρ y : ℝ) : Space := |
| !₂[s * ρ - η * y * (1 / 4 + s ^ 2), y, height * (1 - ρ + 2 * η * y * s)] |
|
|
| def secondSweep (η : ℝ) : Set Space := |
| {z | ∃ s ∈ Set.Icc (-1 / 2 : ℝ) (1 / 2), ∃ ρ ∈ Set.Icc (0 : ℝ) (sourceCap η s), |
| ∃ y : ℝ, |y| ≤ 1 / 2 ∧ z = secondRepresentedPoint η s ρ y} |
|
|
| def secondAxis (η m : ℝ) : Space := linearReflection (axis η m) |
|
|
| def secondBase (η d M m : ℝ) : Set Space := |
| TriangularCovering.projection (secondAxis η m) '' (reflection '' foot η d M m) |
|
|
| def PairedFamilyStatement : Prop := |
| ∀ η : ℝ, 0 < η → |
| ∃ C₁ C₂ : (N : ℕ) → Fin N → TriangularCovering.TriCylinder, |
| (∀ N (i : Fin N), |
| (C₁ N i).axis = axis η (midpoint N i) ∧ |
| (C₁ N i).base = base η (mesh N) (maximumCap (sourceCap η) N i) (midpoint N i) ∧ |
| (C₂ N i).axis = secondAxis η (midpoint N i) ∧ |
| (C₂ N i).base = secondBase η (mesh N) (maximumCap (sourceCap η) N i) (midpoint N i) ∧ |
| TriangularCovering.area (C₁ N i).base = |
| areaFormula η (mesh N) (maximumCap (sourceCap η) N i) (midpoint N i) ∧ |
| TriangularCovering.area (C₂ N i).base = |
| areaFormula η (mesh N) (maximumCap (sourceCap η) N i) (midpoint N i)) ∧ |
| (∀ N, 0 < N → ∀ z ∈ sweep η (sourceCap η) ∪ secondSweep η, |
| ∃ i : Fin N, z ∈ (C₁ N i).carrier ∨ z ∈ (C₂ N i).carrier) ∧ |
| Filter.Tendsto (fun N => TriangularCovering.totalArea (C₁ N) + |
| TriangularCovering.totalArea (C₂ N)) Filter.atTop |
| (nhds (2 * limitingArea η (sourceCap η))) |
|
|
| def tetrahedron : Set Space := |
| {z | ∃ t ∈ Set.Icc (0 : ℝ) 1, z 2 = height * t ∧ |
| |z 0| ≤ (1 - t) / 2 ∧ |z 1| ≤ t / 2} |
|
|
| def CoverageStatement : Prop := |
| ∃ δ : ℝ, 0 < δ ∧ ∀ η : ℝ, 0 < η → η < δ → |
| tetrahedron ⊆ sweep η (sourceCap η) ∪ secondSweep η |
|
|
| def FiniteStrictStatement : Prop := |
| ∃ δ : ℝ, 0 < δ ∧ ∀ η : ℝ, 0 < η → η < δ → |
| ∃ N : ℕ, 0 < N ∧ ∃ C₁ C₂ : Fin N → TriangularCovering.TriCylinder, |
| (∀ i, (C₁ i).axis = axis η (midpoint N i) ∧ |
| (C₁ i).base = base η (mesh N) (maximumCap (sourceCap η) N i) (midpoint N i) ∧ |
| (C₂ i).axis = secondAxis η (midpoint N i) ∧ |
| (C₂ i).base = secondBase η (mesh N) (maximumCap (sourceCap η) N i) (midpoint N i)) ∧ |
| (∀ z ∈ tetrahedron, ∃ i : Fin N, z ∈ (C₁ i).carrier ∨ z ∈ (C₂ i).carrier) ∧ |
| TriangularCovering.totalArea C₁ + TriangularCovering.totalArea C₂ < height / 4 |
|
|
| def FullStatement : Prop := |
| LocalRealizationStatement ∧ FamilyRealizationStatement ∧ PairedFamilyStatement ∧ |
| SourceCostStatement ∧ CoverageStatement ∧ FiniteStrictStatement ∧ |
| Similarity.minimumArea tetrahedron = height / 2 |
|
|
| end CylinderCovering.UnitEdgeFootprints |
|
|
| end |
|
|
| end OAI |
|
|
