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b37e936 | 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 | import OAI.Geometry.RCD.Coordinates.SplittingApproximation
namespace OAI
/-! # Restricting a pointed approximation to a smaller ball -/
open Metric
namespace RegularPointCoordinates
/-- Restriction costs a factor three in the approximation error because
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
end RegularPointCoordinates
end OAI
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