apply WellFounded.recursion (InvImage.wf familyMeasure wellFounded_lt) 𝒜
α : Type u_1
inst✝ : LinearOrder α
s U V : Finset α
n r : ℕ
𝒜 : Finset (Finset (Fin n))
⊢ Set.Sized r ↑𝒜 →
    ∃ ℬ,
      (∂ ℬ).card ≤ (∂ 𝒜).card ∧
        𝒜.card = ℬ.card ∧
          Set.Sized r ↑ℬ ∧ ∀ (U V : Finset (Fin n)), Finset.UV.UsefulCompression U V → IsCompressed U V ℬ
α : Type u_1
inst✝ : LinearOrder α
s U V : Finset α
n r : ℕ
𝒜 : Finset (Finset (Fin n))
⊢ ∀ (x : Finset (Finset (Fin n))),
    (∀ (y : Finset (Finset (Fin n))),
        InvImage (fun x x_1 => x < x_1) Finset.UV.familyMeasure y x →
          Set.Sized r ↑y →
            ∃ ℬ,
              (∂ ℬ).card ≤ (∂ y).card ∧
                y.card = ℬ.card ∧
                  Set.Sized r ↑ℬ ∧ ∀ (U V : Finset (Fin n)), Finset.UV.UsefulCompression U V → IsCompressed U V ℬ) →
      Set.Sized r ↑x →
        ∃ ℬ,
          (∂ ℬ).card ≤ (∂ x).card ∧
            x.card = ℬ.card ∧
              Set.Sized r ↑ℬ ∧ ∀ (U V : Finset (Fin n)), Finset.UV.UsefulCompression U V → IsCompressed U V ℬ