| import OAI.Geometry.RCD.Coordinates.SplittingApproximation |
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| namespace OAI |
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| /-! # Restricting a pointed approximation to a smaller ball -/ |
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| open Metric |
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| namespace RegularPointCoordinates |
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| / |
| the preimage supplied by density must be shown to lie in the smaller ball. -/ |
| theorem IsPointedApproximation.restrict_radius {X Y : Type*} |
| [MetricSpace X] [MetricSpace Y] {p : X} {q : Y} {r R S ε : ℝ} |
| (hr : 0 < r) (hS : 0 ≤ S) (hSR : S ≤ R) (hε : 0 ≤ ε) |
| {f : X → Y} (hf : IsPointedApproximation p q r R ε f) : |
| IsPointedApproximation p q r S (3 * ε) f := by |
| have hlarge {x : X} (hx : dist x p ≤ S * r) : dist x p ≤ R * r := |
| hx.trans (mul_le_mul_of_nonneg_right hSR hr.le) |
| have hpp : dist p p ≤ R * r := by |
| simp only [dist_self] |
| exact mul_nonneg (hS.trans hSR) hr.le |
| refine ⟨hf.1.trans (by linarith), fun x y hx hy => |
| (hf.2.1 x y (hlarge hx) (hlarge hy)).trans (by linarith), ?_⟩ |
| intro y hy |
| obtain ⟨x, hx, hxy⟩ := hf.2.2 y (by linarith) |
| refine ⟨x, ?_, hxy.trans (by linarith)⟩ |
| have hd := (abs_le.mp (hf.2.1 x p hx hpp)).1 |
| have ht := dist_triangle (f x) y (f p) |
| have ht' := dist_triangle y q (f p) |
| have hbase := hf.1 |
| rw [dist_comm (f p)] at hbase |
| apply (div_le_iff₀ hr).mp |
| linarith |
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| end RegularPointCoordinates |
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| end OAI |
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