Module: Mathlib.Combinatorics.HalesJewett

Goal state before:
pp: case neg.refine_3
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'))) (↑l' x) = C'
mono_of_mono : (∃ l, IsMono C' l) → ∃ l, IsMono C l
s : ColorFocused C'
sr : s.lines.card = r
h : ¬∃ p ∈ s.lines, p.color = C' s.focus
⊢ ∀
    p ∈
      { line := vertical s.focus (map some l'), color := C' s.focus, has_color := ⋯ } ::ₘ
        Multiset.map (fun p ↦ { line := p.line.prod (map some l'), color := p.color, has_color := ⋯ }) s.lines,
    ↑p.line none = Sum.elim s.focus (↑(map some l') none)

pp: case neg.refine_4
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'))) (↑l' x) = C'
mono_of_mono : (∃ l, IsMono C' l) → ∃ l, IsMono C l
s : ColorFocused C'
sr : s.lines.card = r
h : ¬∃ p ∈ s.lines, p.color = C' s.focus
⊢ (Multiset.map AlmostMono.color
      ({ line := vertical s.focus (map some l'), color := C' s.focus, has_color := ⋯ } ::ₘ
        Multiset.map (fun p ↦ { line := p.line.prod (map some l'), color := p.color, has_color := ⋯ }) s.lines)).Nodup

pp: 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'))) (↑l' x) = C'
mono_of_mono : (∃ l, IsMono C' l) → ∃ l, IsMono C l
s : ColorFocused C'
sr : s.lines.card = r
h : ¬∃ p ∈ s.lines, p.color = C' s.focus
⊢ {
          lines :=
            { line := vertical s.focus (map some l'), color := C' s.focus, has_color := ⋯ } ::ₘ
              Multiset.map (fun p ↦ { line := p.line.prod (map some l'), color := p.color, has_color := ⋯ }) s.lines,
          focus := Sum.elim s.focus (↑(map some l') none), is_focused := ?neg.refine_3✝,
          distinct_colors := ?neg.refine_4✝ }.lines.card =
    r + 1

Tactic:
· 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]

Goal state after:
case neg.refine_4
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'))) (↑l' x) = C'
mono_of_mono : (∃ l, IsMono C' l) → ∃ l, IsMono C l
s : ColorFocused C'
sr : s.lines.card = r
h : ¬∃ p ∈ s.lines, p.color = C' s.focus
⊢ (Multiset.map AlmostMono.color
      ({ line := vertical s.focus (map some l'), color := C' s.focus, has_color := ⋯ } ::ₘ
        Multiset.map (fun p ↦ { line := p.line.prod (map some l'), color := p.color, has_color := ⋯ }) s.lines)).Nodup

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'))) (↑l' x) = C'
mono_of_mono : (∃ l, IsMono C' l) → ∃ l, IsMono C l
s : ColorFocused C'
sr : s.lines.card = r
h : ¬∃ p ∈ s.lines, p.color = C' s.focus
⊢ {
          lines :=
            { line := vertical s.focus (map some l'), color := C' s.focus, has_color := ⋯ } ::ₘ
              Multiset.map (fun p ↦ { line := p.line.prod (map some l'), color := p.color, has_color := ⋯ }) s.lines,
          focus := Sum.elim s.focus (↑(map some l') none), is_focused := ⋯,
          distinct_colors := ?neg.refine_4✝ }.lines.card =
    r + 1