∀ (s : Set ℤ),
  ¬BddAbove s →
    (if h : s.Nonempty ∧ BddAbove s then ↑((Classical.choose ⋯).greatestOfBdd ⋯ ⋯) else 0) =
      if h : ∅.Nonempty ∧ BddAbove ∅ then ↑((Classical.choose ⋯).greatestOfBdd ⋯ ⋯) else 0