/-- 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]
      

Proof state:
case intro.intro.intro
α : Type u
inst✝¹ : Fintype α
ihα : ∀ (κ : Type (max v u)) [inst : Finite κ], ∃ ι x, ∀ (C : (ι → α) → κ), ∃ l, IsMono C l
κ : Type (max v u)
inst✝ : Finite κ
val✝ : Fintype κ
h : Nonempty α
key : ∀ (r : ℕ), ∃ ι x, ∀ (C : (ι → Option α) → κ), (∃ s, Multiset.card s.lines = r) ∨ ∃ l, IsMono C l
ι : Type
_inst : Fintype ι
hι : ∀ (C : (ι → Option α) → κ), (∃ s, Multiset.card s.lines = Fintype.card κ + 1) ∨ ∃ l, IsMono C l
C : (ι → Option α) → κ
s : ColorFocused C
sr : Multiset.card s.lines = Fintype.card κ + 1
⊢ Finset.card { val := Multiset.map ?intro.intro.intro.f s.lines, nodup := ?m.16677 } ≤ Fintype.card κ
α : Type u
inst✝¹ : Fintype α
ihα : ∀ (κ : Type (max v u)) [inst : Finite κ], ∃ ι x, ∀ (C : (ι → α) → κ), ∃ l, IsMono C l
κ : Type (max v u)
inst✝ : Finite κ
val✝ : Fintype κ
h : Nonempty α
key : ∀ (r : ℕ), ∃ ι x, ∀ (C : (ι → Option α) → κ), (∃ s, Multiset.card s.lines = r) ∨ ∃ l, IsMono C l
ι : Type
_inst : Fintype ι
hι : ∀ (C : (ι → Option α) → κ), (∃ s, Multiset.card s.lines = Fintype.card κ + 1) ∨ ∃ l, IsMono C l
C : (ι → Option α) → κ
s : ColorFocused C
sr : Multiset.card s.lines = Fintype.card κ + 1
⊢ Type ?u.16674
case intro.intro.intro.f
α : Type u
inst✝¹ : Fintype α
ihα : ∀ (κ : Type (max v u)) [inst : Finite κ], ∃ ι x, ∀ (C : (ι → α) → κ), ∃ l, IsMono C l
κ : Type (max v u)
inst✝ : Finite κ
val✝ : Fintype κ
h : Nonempty α
key : ∀ (r : ℕ), ∃ ι x, ∀ (C : (ι → Option α) → κ), (∃ s, Multiset.card s.lines = r) ∨ ∃ l, IsMono C l
ι : Type
_inst : Fintype ι
hι : ∀ (C : (ι → Option α) → κ), (∃ s, Multiset.card s.lines = Fintype.card κ + 1) ∨ ∃ l, IsMono C l
C : (ι → Option α) → κ
s : ColorFocused C
sr : Multiset.card s.lines = Fintype.card κ + 1
⊢ AlmostMono C → ?m.16675
α : Type u
inst✝¹ : Fintype α
ihα : ∀ (κ : Type (max v u)) [inst : Finite κ], ∃ ι x, ∀ (C : (ι → α) → κ), ∃ l, IsMono C l
κ : Type (max v u)
inst✝ : Finite κ
val✝ : Fintype κ
h : Nonempty α
key : ∀ (r : ℕ), ∃ ι x, ∀ (C : (ι → Option α) → κ), (∃ s, Multiset.card s.lines = r) ∨ ∃ l, IsMono C l
ι : Type
_inst : Fintype ι
hι : ∀ (C : (ι → Option α) → κ), (∃ s, Multiset.card s.lines = Fintype.card κ + 1) ∨ ∃ l, IsMono C l
C : (ι → Option α) → κ
s : ColorFocused C
sr : Multiset.card s.lines = Fintype.card κ + 1
⊢ Multiset.Nodup (Multiset.map ?intro.intro.intro.f s.lines)
case intro.intro.intro.f
α : Type u
inst✝¹ : Fintype α
ihα : ∀ (κ : Type (max v u)) [inst : Finite κ], ∃ ι x, ∀ (C : (ι → α) → κ), ∃ l, IsMono C l
κ : Type (max v u)
inst✝ : Finite κ
val✝ : Fintype κ
h : Nonempty α
key : ∀ (r : ℕ), ∃ ι x, ∀ (C : (ι → Option α) → κ), (∃ s, Multiset.card s.lines = r) ∨ ∃ l, IsMono C l
ι : Type
_inst : Fintype ι
hι : ∀ (C : (ι → Option α) → κ), (∃ s, Multiset.card s.lines = Fintype.card κ + 1) ∨ ∃ l, IsMono C l
C : (ι → Option α) → κ
s : ColorFocused C
sr : Multiset.card s.lines = Fintype.card κ + 1
⊢ AlmostMono C → ?m.16675

Next tactic:
exact Finset.card_le_univ ⟨_, s.distinct_colors⟩