/-- 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 => _⟩
      rw [Line.horizontal_apply, ← hl, ← hl']
    -- By choice of `ι`, `C'` either has `r` color-focused lines or a monochromatic line.
    specialize hι C'
    rcases hι with (⟨s, sr⟩ | h)
    on_goal 2 => exact Or.inr (mono_of_mono h)
    -- Here we assume `C'` has `r` color focused lines. We split into cases depending on whether
    -- one of these `r` lines has the same color as the focus point.
    by_cases h : ∃ p ∈ s.lines, (p : AlmostMono _).color = C' s.focus
    -- If so then this is a `C'`-monochromatic line and we are done.
    · obtain ⟨p, p_mem, hp⟩ := h
      refine' Or.inr (mono_of_mono ⟨p.line, p.color, _⟩)
      rintro (_ | _)
      rw [hp, s.is_focused p p_mem]
      apply p.has_color
    -- If not, we get `r+1` color focused lines by taking the product of the `r` lines with `l'`
    -- and adding to this the vertical line obtained by the focus point and `l`.
    refine' Or.inl ⟨⟨(s.lines.map _).cons ⟨(l'.map some).vertical s.focus, C' s.focus, fun x => _⟩,
            Sum.elim s.focus (l'.map some none), _, _⟩, _⟩
    -- Porting note: Needed to reorder the following two goals
    -- The product lines are almost monochromatic.
    · refine' fun p => ⟨p.line.prod (l'.map some), p.color, fun x => _⟩
      rw [Line.prod_apply, Line.map_apply, ← p.has_color, ← congr_fun (hl' x)]
    -- The vertical line is almost monochromatic.
    · rw [vertical_apply, ← congr_fun (hl' x), Line.map_apply]
    -- Our `r+1` lines have the same endpoint.
    · simp_rw [Multiset.mem_cons, Multiset.mem_map]
      rintro _ (rfl | ⟨q, hq, rfl⟩)
      · simp only [vertical_apply]
      · simp only [prod_apply, s.is_focused q hq]
    -- Our `r+1` lines have distinct colors (this is why we needed to split into cases above).
    · rw [Multiset.map_cons, Multiset.map_map, Multiset.nodup_cons, Multiset.mem_map]
      exact ⟨fun ⟨q, hq, he⟩ => h ⟨q, hq, he⟩, s.distinct_colors⟩
    -- Finally, we really do have `r+1` lines!
    · 

Proof state:
case neg.refine'_5
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
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'
mono_of_mono : (∃ l, IsMono C' l) → ∃ l, IsMono C l
s : ColorFocused C'
sr : Multiset.card s.lines = r
h : ¬∃ p ∈ s.lines, p.color = C' s.focus
⊢ Multiset.card
      {
          lines :=
            { line := vertical s.focus (map some l'), color := C' s.focus,
                has_color :=
                  (_ :
                    ∀ (x : α),
                      C ((fun x i => Option.getD (idxFun (vertical s.focus (map some l')) i) x) (some x)) =
                        C' s.focus) } ::ₘ
              Multiset.map
                (fun p =>
                  { line := prod p.line (map some l'), color := p.color,
                    has_color :=
                      (_ :
                        ∀ (x : α),
                          C ((fun x i => Option.getD (idxFun (prod p.line (map some l')) i) x) (some x)) = p.color) })
                s.lines,
          focus := Sum.elim s.focus ((fun x i => Option.getD (idxFun (map some l') i) x) none),
          is_focused :=
            (_ :
              ∀
                p ∈
                  { line := vertical s.focus (map some l'), color := C' s.focus,
                      has_color :=
                        (_ :
                          ∀ (x : α),
                            C ((fun x i => Option.getD (idxFun (vertical s.focus (map some l')) i) x) (some x)) =
                              C' s.focus) } ::ₘ
                    Multiset.map
                      (fun p =>
                        { line := prod p.line (map some l'), color := p.color,
                          has_color :=
                            (_ :
                              ∀ (x : α),
                                C ((fun x i => Option.getD (idxFun (prod p.line (map some l')) i) x) (some x)) =
                                  p.color) })
                      s.lines,
                (fun x i => Option.getD (idxFun p.line i) x) none =
                  Sum.elim s.focus ((fun x i => Option.getD (idxFun (map some l') i) x) none)),
          distinct_colors :=
            (_ :
              Multiset.Nodup
                (Multiset.map AlmostMono.color
                  ({ line := vertical s.focus (map some l'), color := C' s.focus,
                      has_color :=
                        (_ :
                          ∀ (x : α),
                            C ((fun x i => Option.getD (idxFun (vertical s.focus (map some l')) i) x) (some x)) =
                              C' s.focus) } ::ₘ
                    Multiset.map
                      (fun p =>
                        { line := prod p.line (map some l'), color := p.color,
                          has_color :=
                            (_ :
                              ∀ (x : α),
                                C ((fun x i => Option.getD (idxFun (prod p.line (map some l')) i) x) (some x)) =
                                  p.color) })
                      s.lines))) }.lines =
    Nat.succ r

Next tactic:
rw [Multiset.card_cons, Multiset.card_map, sr]