openAI-math-dataset / lean /OAI /Geometry /CylinderCovering /Rational /DictionaryExpansionDefinitions.lean
| import OAI.Geometry.CylinderCovering.Rational.DictionaryCoefficientsDefinitions | |
| namespace OAI | |
| noncomputable section | |
| open Filter | |
| open scoped Topology | |
| namespace RationalDictionary | |
| def uncenteredQuadraticCoefficient (d0 eta0 : ℝ) : ℝ := | |
| (∫ u in (-c0)..c0, a0 d0 u + eta0) - uncenteredEnergy d0 | |
| def UncenteredExpansionStatement : Prop := | |
| (∀ d0 : ℝ, 0 < d0 → ∀ eta0 : ℝ, 0 ≤ eta0 → | |
| Asymptotics.IsBigO (𝓝[>] (0 : ℝ)) | |
| (fun k => orthonormalCost d0 eta0 k - | |
| (c0 + k ^ 2 * uncenteredQuadraticCoefficient d0 eta0)) (fun k => k ^ 4)) ∧ | |
| (∀ d0 : ℝ, 0 < d0 → 0 < coefficientGap d0 → | |
| ∀ eta0 : ℝ, 0 < eta0 → eta0 < coefficientGap d0 / (2 * c0) → | |
| ∀ᶠ k : ℝ in 𝓝[>] (0 : ℝ), orthonormalCost d0 eta0 k < c0) ∧ | |
| ∃ d0 : ℝ, 0 < d0 ∧ ∃ eta0 : ℝ, 0 < eta0 ∧ | |
| eta0 < coefficientGap d0 / (2 * c0) ∧ | |
| ∀ᶠ k : ℝ in 𝓝[>] (0 : ℝ), orthonormalCost d0 eta0 k < c0 | |
| end RationalDictionary | |
| end | |
| end OAI | |