openAI-math-dataset / lean /OAI /Geometry /CylinderCovering /Footprints /PhysicalConvergenceDefinitions.lean
| import OAI.Geometry.CylinderCovering.Footprints.ProjectionDefinitions | |
| import OAI.Geometry.CylinderCovering.Radial.RiemannDefinitions | |
| namespace OAI | |
| noncomputable section | |
| namespace RadialFootprints | |
| open Set MeasureTheory Filter TriangularCovering | |
| open scoped Topology InnerProductSpace | |
| open CylinderCovering.RadialRiemann | |
| def AbsolutePhysicalConvergenceStatement : Prop := | |
| ∀ {α β : ℝ}, α < β → | |
| ∀ {e : Space}, ‖e‖ = 1 → ∀ (p₀ a b : Space), | |
| inner ℝ e a = 0 → inner ℝ e b = 0 → | |
| ∀ {R : ℝ → ℝ} {w : ℝ → Space}, ContinuousOn R (Icc α β) → | |
| (∀ t ∈ Icc α β, 0 ≤ R t) → ContinuousOn w (Icc α β) → | |
| (∀ t ∈ Icc α β, inner ℝ e (w t) = 0) → | |
| ∀ {A B : ℝ}, 0 ≤ A → 0 ≤ B → | |
| ∀ (tag shear : (N : ℕ) → Fin N → ℝ), (∀ N, IsTagged α β N (tag N)) → | |
| Tendsto (fun N => ∑ i : Fin N, | |
| area (projectedCoordinateFootprint e (w (tag N i)) p₀ a b | |
| (absoluteFootprint A B (capMax R α β N i) (step α β N) (shear N i)))) atTop | |
| (𝓝 (Real.sqrt (inner ℝ a a * inner ℝ b b - (inner ℝ a b)^2) * | |
| ∫ t in α..β, R t^2 / (2 * Real.sqrt (1 + ‖w t‖^2)))) | |
| def AffinePhysicalConvergenceStatement : Prop := | |
| ∀ {α β : ℝ}, α < β → | |
| ∀ {e : Space}, ‖e‖ = 1 → ∀ (p₀ a b : Space), | |
| inner ℝ e a = 0 → inner ℝ e b = 0 → | |
| ∀ {R : ℝ → ℝ} {w : ℝ → Space}, ContinuousOn R (Icc α β) → | |
| (∀ t ∈ Icc α β, 0 ≤ R t) → ContinuousOn w (Icc α β) → | |
| (∀ t ∈ Icc α β, inner ℝ e (w t) = 0) → | |
| ∀ {A B : ℝ}, 0 ≤ A → A/2 ≤ B → | |
| ∀ (tag shear : (N : ℕ) → Fin N → ℝ), (∀ N, IsTagged α β N (tag N)) → | |
| Tendsto (fun N => ∑ i : Fin N, | |
| area (projectedCoordinateFootprint e (w (tag N i)) p₀ a b | |
| (affineFootprint A B (capMax R α β N i) (step α β N) (shear N i)))) atTop | |
| (𝓝 (Real.sqrt (inner ℝ a a * inner ℝ b b - (inner ℝ a b)^2) * | |
| ∫ t in α..β, R t^2 / (2 * Real.sqrt (1 + ‖w t‖^2)))) | |
| def HalfheightPhysicalConvergenceStatement : Prop := | |
| ∀ {α β : ℝ}, α < β → | |
| ∀ {e : Space}, ‖e‖ = 1 → ∀ (p₀ a b : Space), | |
| inner ℝ e a = 0 → inner ℝ e b = 0 → | |
| ∀ {R : ℝ → ℝ} {w : ℝ → Space}, ContinuousOn R (Icc α β) → | |
| (∀ t ∈ Icc α β, 0 ≤ R t) → ContinuousOn w (Icc α β) → | |
| (∀ t ∈ Icc α β, inner ℝ e (w t) = 0) → | |
| ∀ {E : ℝ}, 0 ≤ E → | |
| ∀ (tag shear : (N : ℕ) → Fin N → ℝ), (∀ N, IsTagged α β N (tag N)) → | |
| Tendsto (fun N => ∑ i : Fin N, | |
| area (projectedCoordinateFootprint e (w (tag N i)) p₀ a b | |
| (halfheightFootprint (step α β N / 2) (E * step α β N) | |
| (capMax R α β N i) (shear N i)))) atTop | |
| (𝓝 (Real.sqrt (inner ℝ a a * inner ℝ b b - (inner ℝ a b)^2) * | |
| ∫ t in α..β, R t^2 / (2 * Real.sqrt (1 + ‖w t‖^2)))) | |
| def PhysicalFootprintConvergenceStatement : Prop := | |
| AbsolutePhysicalConvergenceStatement ∧ AffinePhysicalConvergenceStatement ∧ | |
| HalfheightPhysicalConvergenceStatement | |
| end RadialFootprints | |
| end | |
| end OAI | |