| import Mathlib |
| import OAI.Geometry.DoublingHilbert.IntrinsicCovers |
|
|
| namespace OAI |
|
|
| open Set Metric |
| open scoped BigOperators |
| noncomputable section |
| namespace CompactBanach |
|
|
| theorem intrinsic_doubling_of_distortion36 |
| {X Y : Type*} [MetricSpace X] [MetricSpace Y] {S : Set X} {L : ℕ} |
| (hS : AmbientDoubling S L) (p : Y → X) (hp : ∀ y, p y ∈ S) |
| {a : ℝ} (ha : 0 < a) |
| (hd : ∀ y z, a * dist (p y) (p z) ≤ dist y z ∧ |
| dist y z ≤ 36 * a * dist (p y) (p z)) : |
| ∀ x : Y, ∀ r : ℝ, 0 < r → |
| ∃ centers : Finset Y, centers.card ≤ L ^ 8 ∧ |
| ∀ y : Y, dist y x < r → ∃ c ∈ centers, dist y c < r / 2 := by |
| classical |
| intro x r hr |
| obtain ⟨c, _, hcard, hcov⟩ := ambient_iterated_cover hS 8 (hp x) (div_pos hr ha) |
| let meets (z : c) : Prop := ∃ y : Y, dist y x < r ∧ |
| dist (p y) z.val < r / a / (2 : ℝ) ^ 8 |
| let chooseCenter (z : c) : Y := if hz : meets z then hz.choose else x |
| let cs := c.attach.image chooseCenter |
| refine ⟨cs, Finset.card_image_le.trans (by simpa using hcard), ?_⟩ |
| intro y hy |
| have hy' : dist (p y) (p x) < r / a := (lt_div_iff₀ ha).mpr (by |
| have h := (hd y x).1 |
| nlinarith) |
| obtain ⟨z, hz, hyz⟩ := hcov (p y) (hp y) hy' |
| have hm : meets ⟨z, hz⟩ := ⟨y, hy, hyz⟩ |
| have hc : dist (p (chooseCenter ⟨z, hz⟩)) z < r / a / (2 : ℝ) ^ 8 := by |
| simp only [chooseCenter, dite_eq_left hm] |
| exact hm.choose_spec.2 |
| refine ⟨chooseCenter ⟨z, hz⟩, |
| Finset.mem_image.mpr ⟨⟨z, hz⟩, Finset.mem_attach _ _, rfl⟩, ?_⟩ |
| have htri := dist_triangle (p y) z (p (chooseCenter ⟨z, hz⟩)) |
| rw [dist_comm z] at htri |
| have hbound : a * dist (p y) (p (chooseCenter ⟨z, hz⟩)) < r / 128 := by |
| calc |
| _ ≤ a * (dist (p y) z + dist (p (chooseCenter ⟨z, hz⟩)) z) := |
| mul_le_mul_of_nonneg_left htri ha.le |
| _ < a * (r / a / (2 : ℝ) ^ 8 + r / a / (2 : ℝ) ^ 8) := |
| mul_lt_mul_of_pos_left (add_lt_add hyz hc) ha |
| _ = r / 128 := by field_simp; ring |
| have hu := (hd y (chooseCenter ⟨z, hz⟩)).2 |
| linarith |
|
|
| theorem range_doubling_of_distortion36 |
| {X Y Z : Type*} [MetricSpace X] [MetricSpace Y] {S : Set X} {L : ℕ} |
| (hS : AmbientDoubling S L) (p : Z → X) (hp : ∀ z, p z ∈ S) |
| (f : Z → Y) {a : ℝ} (ha : 0 < a) |
| (hd : ∀ z w, a * dist (p z) (p w) ≤ dist (f z) (f w) ∧ |
| dist (f z) (f w) ≤ 36 * a * dist (p z) (p w)) : |
| DoublingAtMost (Set.range f) (L ^ 8) := by |
| classical |
| let q : Set.range f → Z := fun y => y.property.choose |
| have hq (y : Set.range f) : f (q y) = y.val := y.property.choose_spec |
| apply intrinsic_doubling_of_distortion36 hS (p ∘ q) (fun y => hp (q y)) ha |
| intro y z |
| simpa only [Function.comp_apply, hq, Subtype.dist_eq] using hd (q y) (q z) |
|
|
| end CompactBanach |
| end |
|
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| end OAI |
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