import Mathlib
import OAI.Geometry.Cannon.Boundary.PolynomialPacking

namespace OAI

section
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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