/-- UV-compression reduces the size of the shadow of `𝒜` if, for all `x ∈ u` there is `y ∈ v` such
that `𝒜` is `(u.erase x, v.erase y)`-compressed. This is the key fact about compression for
Kruskal-Katona. -/
theorem shadow_compression_subset_compression_shadow (u v : Finset α)
    (huv : ∀ x ∈ u, ∃ y ∈ v, IsCompressed (u.erase x) (v.erase y) 𝒜) :
    ∂ (𝓒 u v 𝒜) ⊆ 𝓒 u v (∂ 𝒜) := by
  set 𝒜' := 𝓒 u v 𝒜
  suffices H : ∀ s ∈ ∂ 𝒜',
      s ∉ ∂ 𝒜 → u ⊆ s ∧ Disjoint v s ∧ (s ∪ v) \ u ∈ ∂ 𝒜 ∧ (s ∪ v) \ u ∉ ∂ 𝒜'
  · rintro s hs'
    rw [mem_compression]
    by_cases hs : s ∈ 𝒜.shadow
    swap
    · obtain ⟨hus, hvs, h, _⟩ := H _ hs' hs
      exact Or.inr ⟨hs, _, h, compress_of_disjoint_of_le' hvs hus⟩
    refine' Or.inl ⟨hs, _⟩
    rw [compress]
    split_ifs with huvs
    swap
    · exact hs
    rw [mem_shadow_iff] at hs'
    obtain ⟨t, Ht, a, hat, rfl⟩ := hs'
    have hav : a ∉ v := not_mem_mono huvs.2 (not_mem_erase a t)
    have hvt : v ≤ t := huvs.2.trans (erase_subset _ t)
    have ht : t ∈ 𝒜 := mem_of_mem_compression Ht hvt (aux huv)
    by_cases hau : a ∈ u
    · obtain ⟨b, hbv, Hcomp⟩ := huv a hau
      refine' mem_shadow_iff_insert_mem.2 ⟨b, not_mem_sdiff_of_mem_right hbv, _⟩
      rw [← Hcomp.eq] at ht
      have hsb :=
        sup_sdiff_mem_of_mem_compression ht ((erase_subset _ _).trans hvt)
          (disjoint_erase_comm.2 huvs.1)
      rwa [sup_eq_union, sdiff_erase (mem_union_left _ <| hvt hbv), union_erase_of_mem hat, ←
        erase_union_of_mem hau] at hsb
    · refine'
        mem_shadow_iff.2
          ⟨(t ⊔ u) \ v,
            sup_sdiff_mem_of_mem_compression Ht hvt <| disjoint_of_erase_right hau huvs.1, a, _, _⟩
      · rw [sup_eq_union, mem_sdiff, mem_union]
        exact ⟨Or.inl hat, hav⟩
      · rw [← erase_sdiff_comm, sup_eq_union, erase_union_distrib, erase_eq_of_not_mem hau]
  intro s hs𝒜' hs𝒜
  -- This is gonna be useful a couple of times so let's name it.
  have m : ∀ y, y ∉ s → insert y s ∉ 𝒜 := fun y h a => hs𝒜 (mem_shadow_iff_insert_mem.2 ⟨y, h, a⟩)
  obtain ⟨x, _, _⟩ := mem_shadow_iff_insert_mem.1 hs𝒜'
  have hus : u ⊆ insert x s := le_of_mem_compression_of_not_mem ‹_ ∈ 𝒜'› (m _ ‹x ∉ s›)
  have hvs : Disjoint v (insert x s) := disjoint_of_mem_compression_of_not_mem ‹_› (m _ ‹x ∉ s›)
  have : (insert x s ∪ v) \ u ∈ 𝒜 := sup_sdiff_mem_of_mem_compression_of_not_mem ‹_› (m _ ‹x ∉ s›)
  have hsv : Disjoint s v := hvs.symm.mono_left (subset_insert _ _)
  have hvu : Disjoint v u := disjoint_of_subset_right hus hvs
  have hxv : x ∉ v := disjoint_right.1 hvs (mem_insert_self _ _)
  have : v \ u = v := ‹Disjoint v u›.sdiff_eq_left
  -- The first key part is that `x ∉ u`
  have : x ∉ u := by
    intro hxu
    obtain ⟨y, hyv, hxy⟩ := huv x hxu
    -- If `x ∈ u`, we can get `y ∈ v` so that `𝒜` is `(u.erase x, v.erase y)`-compressed
    apply m y (disjoint_right.1 hsv hyv)
    -- and we will use this `y` to contradict `m`, so we would like to show `insert y s ∈ 𝒜`.
    -- We do this by showing the below
    have : ((insert x s ∪ v) \ u ∪ erase u x) \ erase v y ∈ 𝒜 := by
      refine'
        sup_sdiff_mem_of_mem_compression (by rwa [hxy.eq]) _
          (disjoint_of_subset_left (erase_subset _ _) disjoint_sdiff)
      rw [union_sdiff_distrib, ‹v \ u = v›]
      exact (erase_subset _ _).trans (subset_union_right _ _)
    -- and then arguing that it's the same
    convert this using 1
    rw [sdiff_union_erase_cancel (hus.trans <| subset_union_left _ _) ‹x ∈ u›, erase_union_distrib,
      erase_insert ‹x ∉ s›, erase_eq_of_not_mem ‹x ∉ v›, sdiff_erase (mem_union_right _ hyv),
      union_sdiff_cancel_right hsv]
  -- Now that this is done, it's immediate that `u ⊆ s`
  have hus : u ⊆ s := by rwa [← erase_eq_of_not_mem ‹x ∉ u›, ← subset_insert_iff]
  -- and we already had that `v` and `s` are disjoint,
  -- so it only remains to get `(s ∪ v) \ u ∈ ∂ 𝒜 \ ∂ 𝒜'`
  

Proof state:
case H.intro.intro
α : Type u_1
inst✝ : DecidableEq α
𝒜 : Finset (Finset α)
u✝ v✝ a : Finset α
r : ℕ
u v : Finset α
huv : ∀ x ∈ u, ∃ y ∈ v, IsCompressed (erase u x) (erase v y) 𝒜
𝒜' : Finset (Finset α) := 𝓒 u v 𝒜
s : Finset α
hs𝒜' : s ∈ ∂ 𝒜'
hs𝒜 : s ∉ ∂ 𝒜
m : ∀ y ∉ s, insert y s ∉ 𝒜
x : α
left✝ : x ∉ s
right✝ : insert x s ∈ 𝒜'
hus✝ : u ⊆ insert x s
hvs : Disjoint v (insert x s)
this✝¹ : (insert x s ∪ v) \ u ∈ 𝒜
hsv : Disjoint s v
hvu : Disjoint v u
hxv : x ∉ v
this✝ : v \ u = v
this : x ∉ u
hus : u ⊆ s
⊢ u ⊆ s ∧ Disjoint v s ∧ (s ∪ v) \ u ∈ ∂ 𝒜 ∧ (s ∪ v) \ u ∉ ∂ 𝒜'

Next tactic:
simp_rw [mem_shadow_iff_insert_mem]