refine' ih h₁.shadow (strengthened_kk h₁ h₂ h₃) _
case succ
α : Type u_1
inst✝ : LinearOrder α
s U V : Finset α
n i _k : ℕ
ih :
  ∀ {r : ℕ} {𝒜 𝒞 : Finset (Finset (Fin n))},
    Set.Sized r ↑𝒜 → 𝒞.card ≤ 𝒜.card → IsInitSeg 𝒞 r → (∂ ^[_k] 𝒞).card ≤ (∂ ^[_k] 𝒜).card
r : ℕ
𝒜 𝒞 : Finset (Finset (Fin n))
h₁ : Set.Sized r ↑𝒜
h₂ : 𝒞.card ≤ 𝒜.card
h₃ : IsInitSeg 𝒞 r
⊢ (∂ ^[_k + 1] 𝒞).card ≤ (∂ ^[_k + 1] 𝒜).card
case succ
α : Type u_1
inst✝ : LinearOrder α
s U V : Finset α
n i _k : ℕ
ih :
  ∀ {r : ℕ} {𝒜 𝒞 : Finset (Finset (Fin n))},
    Set.Sized r ↑𝒜 → 𝒞.card ≤ 𝒜.card → IsInitSeg 𝒞 r → (∂ ^[_k] 𝒞).card ≤ (∂ ^[_k] 𝒜).card
r : ℕ
𝒜 𝒞 : Finset (Finset (Fin n))
h₁ : Set.Sized r ↑𝒜
h₂ : 𝒞.card ≤ 𝒜.card
h₃ : IsInitSeg 𝒞 r
⊢ IsInitSeg (∂ 𝒞) (r - 1)