| import Mathlib | |
| namespace OAI | |
| universe u v | |
| namespace CompactBanach | |
| def DoublingAtMost {B : Type*} [MetricSpace B] (K : Set B) (bound : ℕ) : Prop := | |
| ∀ x : K, ∀ r : ℝ, 0 < r → | |
| ∃ centers : Finset K, centers.card ≤ bound ∧ | |
| ∀ y : K, dist y x < r → ∃ c ∈ centers, dist y c < r / ((2 : ℕ) : ℝ) | |
| def AdmitsBiLipschitzEmbedding {B : Type*} [MetricSpace B] (K : Set B) | |
| (E : Type*) [MetricSpace E] : Prop := | |
| ∃ f : K → E, ∃ a : ℝ, 0 < a ∧ ∃ D : ℝ, 1 ≤ D ∧ | |
| ∀ x y : K, a * dist x y ≤ dist (f x) (f y) ∧ | |
| dist (f x) (f y) ≤ D * a * dist x y | |
| theorem main : ∃ Λ : ℕ, ∀ (B : Type u) [NormedAddCommGroup B] [NormedSpace ℝ B] | |
| [CompleteSpace B], ¬ FiniteDimensional ℝ B → | |
| ∃ K : Set B, IsCompact K ∧ DoublingAtMost K Λ ∧ | |
| ∀ (E : Type v) [NormedAddCommGroup E] [NormedSpace ℝ E] | |
| [FiniteDimensional ℝ E], ¬ AdmitsBiLipschitzEmbedding K E := by | |
| sorry | |
| end CompactBanach | |
| end OAI | |