| import Mathlib | |
| namespace OAI | |
| open Set Metric | |
| noncomputable section | |
| namespace CompactBanach | |
| theorem normalized_embedding_of_finite | |
| {X E : Type*} [MetricSpace X] [NormedAddCommGroup E] [ProperSpace E] | |
| (o : X) {A : ℝ} (hA : 0 ≤ A) | |
| (hfinite : ∀ s : Finset X, ∃ f : s → E, ∀ x y : s, | |
| dist x y ≤ dist (f x) (f y) ∧ dist (f x) (f y) ≤ A * dist x y) : | |
| ∃ f : X → E, ∀ x y : X, | |
| dist x y ≤ dist (f x) (f y) ∧ dist (f x) (f y) ≤ A * dist x y := by | |
| classical | |
| let Q : Set (X → E) := {f | ∀ x, f x ∈ closedBall 0 (A * dist o x)} | |
| have hQ : IsCompact Q := isCompact_pi_infinite fun x => isCompact_closedBall _ _ | |
| let C : X × X → Set (X → E) := fun p => {f | | |
| dist p.1 p.2 ≤ dist (f p.1) (f p.2) ∧ | |
| dist (f p.1) (f p.2) ≤ A * dist p.1 p.2} | |
| have hC (p : X × X) : IsClosed (C p) := | |
| (isClosed_le continuous_const ((continuous_apply p.1).dist (continuous_apply p.2))).inter | |
| (isClosed_le ((continuous_apply p.1).dist (continuous_apply p.2)) continuous_const) | |
| have hFIP (u : Finset (X × X)) : (Q ∩ ⋂ p ∈ u, C p).Nonempty := by | |
| let s : Finset X := insert o (u.image Prod.fst ∪ u.image Prod.snd) | |
| have ho : o ∈ s := by simp [s] | |
| obtain ⟨f, hf⟩ := hfinite s | |
| let g : X → E := fun x => if hx : x ∈ s then f ⟨x, hx⟩ - f ⟨o, ho⟩ else 0 | |
| refine ⟨g, ?_, ?_⟩ | |
| · intro x | |
| by_cases hx : x ∈ s | |
| · simp only [g, dite_eq_left hx, mem_closedBall, dist_zero_right] | |
| simpa only [dist_eq_norm, Subtype.dist_eq, dist_comm o x] using (hf ⟨x, hx⟩ ⟨o, ho⟩).2 | |
| · simp only [g, dite_eq_right hx, mem_closedBall, dist_self] | |
| exact mul_nonneg hA dist_nonneg | |
| · simp only [mem_iInter] | |
| intro p hp | |
| have hx : p.1 ∈ s := by simp [s, Finset.mem_image_of_mem _ hp] | |
| have hy : p.2 ∈ s := by simp [s, Finset.mem_image_of_mem _ hp] | |
| change dist p.1 p.2 ≤ dist (g p.1) (g p.2) ∧ | |
| dist (g p.1) (g p.2) ≤ A * dist p.1 p.2 | |
| simpa only [g, dite_eq_left hx, dite_eq_left hy, dist_sub_right, Subtype.dist_eq] using | |
| hf ⟨p.1, hx⟩ ⟨p.2, hy⟩ | |
| obtain ⟨f, _, hf⟩ := hQ.inter_iInter_nonempty C hC hFIP | |
| refine ⟨f, fun x y => ?_⟩ | |
| exact mem_iInter.mp hf (x, y) | |
| theorem normalized_embedding_of_scaled | |
| {X E : Type*} [MetricSpace X] [NormedAddCommGroup E] [NormedSpace ℝ E] | |
| {f : X → E} {a A : ℝ} (ha : 0 < a) | |
| (hf : ∀ x y, a * dist x y ≤ dist (f x) (f y) ∧ | |
| dist (f x) (f y) ≤ A * a * dist x y) : | |
| ∃ g : X → E, ∀ x y, | |
| dist x y ≤ dist (g x) (g y) ∧ dist (g x) (g y) ≤ A * dist x y := by | |
| let g : X → E := fun x => a⁻¹ • f x | |
| have hd (x y : X) : dist (g x) (g y) = a⁻¹ * dist (f x) (f y) := by | |
| simp only [g, dist_smul₀, Real.norm_eq_abs, abs_inv, abs_of_pos ha] | |
| refine ⟨g, fun x y => ?_⟩ | |
| rw [hd] | |
| constructor | |
| · calc | |
| dist x y = a⁻¹ * (a * dist x y) := by field_simp | |
| _ ≤ _ := mul_le_mul_of_nonneg_left (hf x y).1 (inv_nonneg.mpr ha.le) | |
| · calc | |
| a⁻¹ * dist (f x) (f y) ≤ a⁻¹ * (A * a * dist x y) := | |
| mul_le_mul_of_nonneg_left (hf x y).2 (inv_nonneg.mpr ha.le) | |
| _ = A * dist x y := by field_simp | |
| theorem finite_witness_of_no_embedding | |
| {X E : Type*} [MetricSpace X] [NormedAddCommGroup E] [NormedSpace ℝ E] | |
| [ProperSpace E] (o : X) {A : ℝ} (hA : 0 ≤ A) | |
| (hno : ¬ ∃ f : X → E, ∀ x y, | |
| dist x y ≤ dist (f x) (f y) ∧ dist (f x) (f y) ≤ A * dist x y) : | |
| ∃ s : Finset X, ¬ ∃ f : s → E, ∃ a : ℝ, 0 < a ∧ | |
| ∀ x y : s, a * dist x y ≤ dist (f x) (f y) ∧ | |
| dist (f x) (f y) ≤ A * a * dist x y := by | |
| classical | |
| by_contra! h | |
| apply hno | |
| apply normalized_embedding_of_finite o hA | |
| intro s | |
| obtain ⟨f, a, ha, hf⟩ := h s | |
| exact normalized_embedding_of_scaled ha hf | |
| end CompactBanach | |
| end | |
| end OAI | |