openAI-math-dataset / lean /OAI /Geometry /CylinderCovering /Footprints /ConvergenceDefinitions.lean
| import OAI.Geometry.CylinderCovering.Footprints.FootprintsDefinitions | |
| import OAI.Geometry.CylinderCovering.Radial.RiemannDefinitions | |
| namespace OAI | |
| noncomputable section | |
| namespace RadialFootprints | |
| open Set MeasureTheory Filter | |
| open scoped Topology | |
| open CylinderCovering.RadialRiemann | |
| def AbsoluteConvergenceStatement : Prop := | |
| ∀ {a b : ℝ}, a < b → | |
| ∀ {R W : ℝ → ℝ}, ContinuousOn R (Icc a b) → ContinuousOn W (Icc a b) → | |
| (∀ t ∈ Icc a b, 0 ≤ R t) → ∀ {A B : ℝ}, 0 ≤ A → 0 ≤ B → | |
| ∀ (tag shear : (N : ℕ) → Fin N → ℝ), (∀ N, IsTagged a b N (tag N)) → | |
| Tendsto (fun N => ∑ i : Fin N, | |
| volume.real (absoluteFootprint A B (capMax R a b N i) (step a b N) (shear N i)) * | |
| W (tag N i)) atTop (𝓝 ((1/2 : ℝ) * ∫ t in a..b, R t^2 * W t)) | |
| def AffineConvergenceStatement : Prop := | |
| ∀ {a b : ℝ}, a < b → | |
| ∀ {R W : ℝ → ℝ}, ContinuousOn R (Icc a b) → ContinuousOn W (Icc a b) → | |
| (∀ t ∈ Icc a b, 0 ≤ R t) → ∀ {A B : ℝ}, 0 ≤ A → A/2 ≤ B → | |
| ∀ (tag shear : (N : ℕ) → Fin N → ℝ), (∀ N, IsTagged a b N (tag N)) → | |
| Tendsto (fun N => ∑ i : Fin N, | |
| volume.real (affineFootprint A B (capMax R a b N i) (step a b N) (shear N i)) * | |
| W (tag N i)) atTop (𝓝 ((1/2 : ℝ) * ∫ t in a..b, R t^2 * W t)) | |
| def HalfheightConvergenceStatement : Prop := | |
| ∀ {a b : ℝ}, a < b → | |
| ∀ {R W : ℝ → ℝ}, ContinuousOn R (Icc a b) → ContinuousOn W (Icc a b) → | |
| (∀ t ∈ Icc a b, 0 ≤ R t) → ∀ {E : ℝ}, 0 ≤ E → | |
| ∀ (tag shear : (N : ℕ) → Fin N → ℝ), (∀ N, IsTagged a b N (tag N)) → | |
| Tendsto (fun N => ∑ i : Fin N, | |
| volume.real (halfheightFootprint (step a b N / 2) (E * step a b N) | |
| (capMax R a b N i) (shear N i)) * W (tag N i)) | |
| atTop (𝓝 ((1/2 : ℝ) * ∫ t in a..b, R t^2 * W t)) | |
| def FootprintConvergenceStatement : Prop := | |
| AbsoluteConvergenceStatement ∧ AffineConvergenceStatement ∧ HalfheightConvergenceStatement | |
| end RadialFootprints | |
| end | |
| end OAI | |