| import Mathlib | |
| namespace OAI | |
| noncomputable section | |
| open MeasureTheory ProbabilityTheory Filter | |
| open scoped Topology ENNReal | |
| namespace HoneycombLocal | |
| def sumLaw (p : PMF ℕ) : ℕ → PMF ℕ | |
| | 0 => PMF.pure 0 | |
| | k + 1 => p.bind (fun j => (sumLaw p k).map (fun r => j + r)) | |
| def renewalMass (p : PMF ℕ) (m : ℕ) : ℝ≥0∞ := | |
| ∑' k : ℕ, sumLaw p (k + 1) m | |
| def SpanOne (p : PMF ℕ) : Prop := | |
| AddSubgroup.closure {z : ℤ | ∃ j ∈ p.support, ∃ k ∈ p.support, z = (j : ℤ) - k} = ⊤ | |
| def laplace (p : PMF ℕ) (u : ℝ) : ℝ := | |
| ∫ j, Real.exp (-u * (j : ℝ)) ∂p.toMeasure | |
| def BoundedPowerInput (p : PMF ℕ) (a : ℝ) : Prop := | |
| ∃ c > 0, ∃ C > 0, ∃ x₀ > 0, | |
| (∀ x ≥ x₀, p.toMeasure.real {j | x < (j : ℝ)} ≤ C * x ^ (-a)) ∧ | |
| (∀ u > 0, u ≤ x₀⁻¹ → | |
| c * u ^ a ≤ 1 - laplace p u ∧ 1 - laplace p u ≤ C * u ^ a) | |
| end HoneycombLocal | |
| end | |
| end OAI | |