openAI-math-dataset / lean /OAI /Geometry /CylinderCovering /Physical /UnitEdgeFootprintsDefinitions.lean
| import OAI.Geometry.TriangularCovering.Basic | |
| import OAI.Geometry.CylinderCovering.Radial.RiemannDefinitions | |
| namespace OAI | |
| noncomputable section | |
| namespace CylinderCovering.UnitEdgeFootprints | |
| abbrev Space := TriangularCovering.Space | |
| def height : ℝ := 1 / Real.sqrt 2 | |
| def intercept (u v : ℝ) : Space := !₂[0, v, height * u] | |
| def rawDirection (η m : ℝ) : Space := | |
| !₂[1, η * (1 / 4 + m ^ 2), 2 * height * η * m] | |
| def axis (η m : ℝ) : Space := ‖rawDirection η m‖⁻¹ • rawDirection η m | |
| def vertices (η d M m : ℝ) : Fin 3 → Space := | |
| ![intercept (-η * d) (m * (-η * d)), | |
| intercept (M + η * d) (m * (M + η * d) - d / 2 * (M + 2 * η * d)), | |
| intercept (M + η * d) (m * (M + η * d) + d / 2 * (M + 2 * η * d))] | |
| def foot (η d M m : ℝ) : Set Space := convexHull ℝ (Set.range (vertices η d M m)) | |
| def base (η d M m : ℝ) : Set Space := | |
| TriangularCovering.projection (axis η m) '' foot η d M m | |
| def representedPoint (η s ρ x : ℝ) : Space := | |
| !₂[x, s * ρ + η * x * (1 / 4 + s ^ 2), height * (ρ + 2 * η * x * s)] | |
| def areaFormula (η d M m : ℝ) : ℝ := | |
| height * d * (M + 2 * η * d) ^ 2 / | |
| (2 * Real.sqrt (1 + η ^ 2 * ((1 / 4 + m ^ 2) ^ 2 + 2 * m ^ 2))) | |
| def LocalRealizationStatement : Prop := | |
| ∀ η d M m : ℝ, 0 < η → 0 < d → 0 ≤ M → | |
| ∃ C : TriangularCovering.TriCylinder, | |
| C.axis = axis η m ∧ C.base = base η d M m ∧ | |
| TriangularCovering.area C.base = areaFormula η d M m ∧ | |
| ∀ s ρ x : ℝ, |s - m| ≤ d / 2 → 0 ≤ ρ → ρ ≤ M → |x| ≤ 1 / 2 → | |
| representedPoint η s ρ x ∈ C.carrier | |
| def sourceCap (η s : ℝ) : ℝ := | |
| Real.sqrt (1 / 4 + η ^ 2 * (s ^ 2 / 4 + s ^ 4 + 1 / 960)) | |
| def sweep (η : ℝ) (R : ℝ → ℝ) : Set Space := | |
| {z | ∃ s ∈ Set.Icc (-1 / 2 : ℝ) (1 / 2), ∃ ρ ∈ Set.Icc (0 : ℝ) (R s), | |
| ∃ x : ℝ, |x| ≤ 1 / 2 ∧ z = representedPoint η s ρ x} | |
| def mesh (N : ℕ) : ℝ := RadialRiemann.step (-1 / 2) (1 / 2) N | |
| def midpoint (N : ℕ) (i : Fin N) : ℝ := | |
| (RadialRiemann.point (-1 / 2) (1 / 2) N i + | |
| RadialRiemann.point (-1 / 2) (1 / 2) N (i + 1)) / 2 | |
| def maximumCap (R : ℝ → ℝ) (N : ℕ) (i : Fin N) : ℝ := | |
| RadialRiemann.capMax R (-1 / 2) (1 / 2) N i | |
| def limitingArea (η : ℝ) (R : ℝ → ℝ) : ℝ := | |
| height * ∫ s in (-1 / 2 : ℝ)..(1 / 2), | |
| R s ^ 2 / (2 * Real.sqrt (1 + η ^ 2 * ((1 / 4 + s ^ 2) ^ 2 + 2 * s ^ 2))) | |
| def FamilyRealizationStatement : Prop := | |
| ∀ η : ℝ, 0 < η → ∀ R : ℝ → ℝ, | |
| ContinuousOn R (Set.Icc (-1 / 2 : ℝ) (1 / 2)) → | |
| (∀ s ∈ Set.Icc (-1 / 2 : ℝ) (1 / 2), 0 ≤ R s) → | |
| ∃ C : (N : ℕ) → Fin N → TriangularCovering.TriCylinder, | |
| (∀ N (i : Fin N), | |
| (C N i).axis = axis η (midpoint N i) ∧ | |
| (C N i).base = base η (mesh N) (maximumCap R N i) (midpoint N i) ∧ | |
| TriangularCovering.area (C N i).base = | |
| areaFormula η (mesh N) (maximumCap R N i) (midpoint N i)) ∧ | |
| (∀ N, 0 < N → ∀ z ∈ sweep η R, ∃ i : Fin N, z ∈ (C N i).carrier) ∧ | |
| Filter.Tendsto (fun N => TriangularCovering.totalArea (C N)) Filter.atTop | |
| (nhds (limitingArea η R)) | |
| end CylinderCovering.UnitEdgeFootprints | |
| end | |
| end OAI | |