   8open Classical
   9
  10/-- Closed interval with endpoints `lo ≤ hi`. -/
  11structure Interval where
  12  lo : ℝ
  13  hi : ℝ
  14  lo_le_hi : lo ≤ hi
  15
  16@[simp] def memI (I : Interval) (x : ℝ) : Prop := I.lo ≤ x ∧ x ≤ I.hi
  17
  18@[simp] def width (I : Interval) : ℝ := I.hi - I.lo
  19
  20/-- If `x,y` lie in the same interval `I`, then `|x − y| ≤ width(I)`. -/
  21lemma abs_sub_le_width_of_memI {I : Interval} {x y : ℝ}
  22  (hx : memI I x) (hy : memI I y) : |x - y| ≤ width I := by
  23  have : I.lo ≤ x ∧ x ≤ I.hi := hx
  24  have : I.lo ≤ y ∧ y ≤ I.hi := hy
  25  have : x - y ≤ I.hi - I.lo := by linarith
  26  have : y - x ≤ I.hi - I.lo := by linarith
  27  have : -(I.hi - I.lo) ≤ x - y ∧ x - y ≤ I.hi - I.lo := by
  28    constructor
  29    · linarith
  30    · linarith
  31  simpa [width, abs_le] using this
  32
  33/-! ## Certificate schema -/
  34
  35/-- Anchor certificate: per-species residue intervals plus charge-wise gap intervals. -/
  36structure AnchorCert (Species : Type) where
  37  M0 : Interval
  38  Ires : Species → Interval
  39  center : Int → ℝ
  40  eps : Int → ℝ
  41  eps_nonneg : ∀ z, 0 ≤ eps z

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