import Mathlib

namespace OAI

namespace DoublingHilbert

abbrev RealL2 := lp (fun _ : ℕ => ℝ) 2

/-- Intrinsic open-ball doubling with a specified integer bound and centers in the subset. -/
def DoublingAtMost (S : Set RealL2) (bound : ℕ) : Prop :=
  ∀ x : S, ∀ r : ℝ, 0 < r →
    ∃ centers : Finset S, centers.card ≤ bound ∧
      ∀ y : S, dist y x < r → ∃ c ∈ centers, dist y c < r / 2

/-- Finite distortion includes an arbitrary positive change of scale. -/
def AdmitsBiLipschitzEmbedding (S : Set RealL2) (k : ℕ) : Prop :=
  ∃ f : S → EuclideanSpace ℝ (Fin k),
    ∃ a : ℝ, 0 < a ∧ ∃ D : ℝ, 1 ≤ D ∧
      ∀ x y : S, a * dist x y ≤ dist (f x) (f y) ∧
        dist (f x) (f y) ≤ D * a * dist x y

end DoublingHilbert

end OAI