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8e4d533 | 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 | import Mathlib
import OAI.Geometry.Cannon.Boundary.PolynomialPacking
namespace OAI
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noncomputable section
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open Set Metric
open scoped NNReal ENNReal
namespace CannonNode
open Set Metric
theorem annular_net {X : Type*} [MetricSpace X] [CompactSpace X]
(x : X) (A B R r : ℝ) (hr : 0 < r) :
∃ T : Finset X,
(∀ y ∈ T, (A-1)*R ≤ dist y x ∧ dist y x ≤ (B+1)*R) ∧
(∀ y ∈ T, ∀ z ∈ T, y ≠ z → r ≤ dist y z) ∧
(∀ v, (A-1)*R ≤ dist v x → dist v x ≤ (B+1)*R →
∃ y ∈ T, dist v y ≤ r) := by
let W := {v : X | (A-1)*R ≤ dist v x ∧ dist v x ≤ (B+1)*R}
obtain ⟨T,hT,hsep,hcover⟩ := CannonNet.exists_finite_separated_net W
(isCompact_univ.totallyBounded.subset (subset_univ W)) r hr
exact ⟨T,fun y hy => hT hy,hsep,fun v hl hu => hcover v ⟨hl,hu⟩⟩
end CannonNode
end
end
end
end
end
end
end
end
end
end
end
end
end OAI
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