/-- The **Hales-Jewett theorem**. This version has a restriction on universe levels which is
necessary for the proof. See `exists_mono_in_high_dimension` for a fully universe-polymorphic
version. -/
private theorem exists_mono_in_high_dimension' :
    ∀ (α : Type u) [Finite α] (κ : Type max v u) [Finite κ],
      ∃ (ι : Type) (_ : Fintype ι), ∀ C : (ι → α) → κ, ∃ l : Line α ι, l.IsMono C :=
-- The proof proceeds by induction on `α`.
  Finite.induction_empty_option
  (-- We have to show that the theorem is invariant under `α ≃ α'` for the induction to work.
  fun {α α'} e =>
    forall_imp fun κ =>
      forall_imp fun _ =>
        Exists.imp fun ι =>
          Exists.imp fun _ h C =>
            let ⟨l, c, lc⟩ := h fun v => C (e ∘ v)
            ⟨l.map e, c, e.forall_congr_left.mp fun x => by rw [← lc x, Line.map_apply]⟩)
  (by
    -- This deals with the degenerate case where `α` is empty.
    intro κ _
    by_cases h : Nonempty κ
    · refine' ⟨Unit, inferInstance, fun C => ⟨default, Classical.arbitrary _, PEmpty.rec⟩⟩
    · exact ⟨Empty, inferInstance, fun C => (h ⟨C (Empty.rec)⟩).elim⟩)
  (by
    -- Now we have to show that the theorem holds for `Option α` if it holds for `α`.
    intro α _ ihα κ _
    cases nonempty_fintype κ
    -- Later we'll need `α` to be nonempty. So we first deal with the trivial case where `α` is
    -- empty.
    -- Then `Option α` has only one element, so any line is monochromatic.
    by_cases h : Nonempty α
    case neg =>
      refine' ⟨Unit, inferInstance, fun C => ⟨diagonal _ Unit, C fun _ => none, ?_⟩⟩
      rintro (_ | ⟨a⟩)
      · rfl
      · exact (h ⟨a⟩).elim
    -- The key idea is to show that for every `r`, in high dimension we can either find
    -- `r` color focused lines or a monochromatic line.
    suffices key :
      ∀ r : ℕ,
        ∃ (ι : Type) (_ : Fintype ι),
          ∀ C : (ι → Option α) → κ,
            (∃ s : ColorFocused C, Multiset.card s.lines = r) ∨ ∃ l, IsMono C l by
      -- Given the key claim, we simply take `r = |κ| + 1`. We cannot have this many distinct colors
      -- so we must be in the second case, where there is a monochromatic line.
      obtain ⟨ι, _inst, hι⟩ := key (Fintype.card κ + 1)
      refine' ⟨ι, _inst, fun C => (hι C).resolve_left _⟩
      rintro ⟨s, sr⟩
      apply Nat.not_succ_le_self (Fintype.card κ)
      rw [← Nat.add_one, ← sr, ← Multiset.card_map, ← Finset.card_mk]
      exact Finset.card_le_univ ⟨_, s.distinct_colors⟩
    -- We now prove the key claim, by induction on `r`.
    intro r
    induction' r with r ihr
    -- The base case `r = 0` is trivial as the empty collection is color-focused.
    · exact ⟨Empty, inferInstance, fun C => Or.inl ⟨default, Multiset.card_zero⟩⟩
    -- Supposing the key claim holds for `r`, we need to show it for `r+1`. First pick a high
    -- enough dimension `ι` for `r`.
    obtain ⟨ι, _inst, hι⟩ := ihr
    -- Then since the theorem holds for `α` with any number of colors, pick a dimension `ι'` such
    -- that `ι' → α` always has a monochromatic line whenever it is `(ι → Option α) → κ`-colored.
    specialize ihα ((ι → Option α) → κ)
    obtain ⟨ι', _inst, hι'⟩ := ihα
    -- We claim that `ι ⊕ ι'` works for `Option α` and `κ`-coloring.
    refine' ⟨Sum ι ι', inferInstance, _⟩
    intro C
    -- A `κ`-coloring of `ι ⊕ ι' → Option α` induces an `(ι → Option α) → κ`-coloring of `ι' → α`.
    specialize hι' fun v' v => C (Sum.elim v (some ∘ v'))
    -- By choice of `ι'` this coloring has a monochromatic line `l'` with color class `C'`, where
    -- `C'` is a `κ`-coloring of `ι → α`.
    obtain ⟨l', C', hl'⟩ := hι'
    -- If `C'` has a monochromatic line, then so does `C`. We use this in two places below.
    have mono_of_mono : (∃ l, IsMono C' l) → ∃ l, IsMono C l := by
      rintro ⟨l, c, hl⟩
      refine' ⟨l.horizontal (some ∘ l' (Classical.arbitrary α)), c, fun x => _⟩
      

Proof state:
case intro.intro
α : Type u
inst✝¹ : Fintype α
κ : Type (max v u)
inst✝ : Finite κ
val✝ : Fintype κ
h : Nonempty α
r : ℕ
ι : Type
_inst✝ : Fintype ι
hι : ∀ (C : (ι → Option α) → κ), (∃ s, Multiset.card s.lines = r) ∨ ∃ l, IsMono C l
ι' : Type
_inst : Fintype ι'
C : (ι ⊕ ι' → Option α) → κ
l' : Line α ι'
C' : (ι → Option α) → κ
hl' : ∀ (x : α), (fun v' v => C (Sum.elim v (some ∘ v'))) ((fun x i => Option.getD (idxFun l' i) x) x) = C'
l : Line (Option α) ι
c : κ
hl : ∀ (x : Option α), C' ((fun x i => Option.getD (idxFun l i) x) x) = c
x : Option α
⊢ C
      ((fun x i =>
          Option.getD
            (idxFun (horizontal l (some ∘ (fun x i => Option.getD (idxFun l' i) x) (Classical.arbitrary α))) i) x)
        x) =
    c

Next tactic:
rw [Line.horizontal_apply, ← hl, ← hl']