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a2ee19a | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 | 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
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