∀ {n r : ℕ} (𝒜 : Finset (Finset (Fin n))),
  Set.Sized r ↑𝒜 →
    ∃ ℬ,
      ℬ.shadow.card ≤ 𝒜.shadow.card ∧
        𝒜.card = ℬ.card ∧
          Set.Sized r ↑ℬ ∧ ∀ (U V : Finset (Fin n)), Finset.UV.UsefulCompression✝ U V → UV.IsCompressed U V ℬ