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785c727 | 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 | 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
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