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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 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 | 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
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