l3lab/ntp-mathlib huggingface dataset:
/-- This bootstraps `nonempty_sections_of_finite_inverse_system`. In this version,
the `F` functor is between categories of the same universe, and it is an easy
corollary to `TopCat.nonempty_limitCone_of_compact_t2_cofiltered_system`. -/
theorem nonempty_sections_of_finite_cofiltered_system.init {J : Type u} [SmallCategory J]
    [IsCofilteredOrEmpty J] (F : J ⥤ Type u) [hf : ∀ j, Finite (F.obj j)]
    [hne : ∀ j, Nonempty (F.obj j)] : F.sections.Nonempty := by
  let F' : J ⥤ TopCat := F ⋙ TopCat.discrete
  haveI : ∀ j, DiscreteTopology (F'.obj j) := fun _ => ⟨rfl⟩
  haveI : ∀ j, Finite (F'.obj j) := hf
  haveI : ∀ j, Nonempty (F'.obj j) := hne
  obtain ⟨⟨u, hu⟩⟩ := TopCat.nonempty_limitCone_of_compact_t2_cofiltered_system.{u} F'
  

Proof state:
case intro.mk
J : Type u
inst✝¹ : SmallCategory J
inst✝ : IsCofilteredOrEmpty J
F : J ⥤ Type u
hf : ∀ (j : J), Finite (F.obj j)
hne : ∀ (j : J), Nonempty (F.obj j)
F' : J ⥤ TopCat := F ⋙ TopCat.discrete
this✝¹ : ∀ (j : J), DiscreteTopology ↑(F'.obj j)
this✝ : ∀ (j : J), Finite ↑(F'.obj j)
this : ∀ (j : J), Nonempty ↑(F'.obj j)
u : (j : J) → ↑(F'.obj j)
hu : u ∈ {u | ∀ {i j : J} (f : i ⟶ j), (F'.map f) (u i) = u j}
⊢ Set.Nonempty (Functor.sections F)

Next tactic:
exact ⟨u, hu⟩
/-- The cofiltered limit of nonempty finite types is nonempty.

See `nonempty_sections_of_finite_inverse_system` for a specialization to inverse limits. -/
theorem nonempty_sections_of_finite_cofiltered_system {J : Type u} [Category.{w} J]
    [IsCofilteredOrEmpty J] (F : J ⥤ Type v) [∀ j : J, Finite (F.obj j)]
    [∀ j : J, Nonempty (F.obj j)] : F.sections.Nonempty := by
  -- Step 1: lift everything to the `max u v w` universe.
  

Proof state:
J : Type u
inst✝³ : Category.{w, u} J
inst✝² : IsCofilteredOrEmpty J
F : J ⥤ Type v
inst✝¹ : ∀ (j : J), Finite (F.obj j)
inst✝ : ∀ (j : J), Nonempty (F.obj j)
⊢ Set.Nonempty (Functor.sections F)

Next tactic:
let J' : Type max w v u := AsSmall.{max w v} J
/-- The cofiltered limit of nonempty finite types is nonempty.

See `nonempty_sections_of_finite_inverse_system` for a specialization to inverse limits. -/
theorem nonempty_sections_of_finite_cofiltered_system {J : Type u} [Category.{w} J]
    [IsCofilteredOrEmpty J] (F : J ⥤ Type v) [∀ j : J, Finite (F.obj j)]
    [∀ j : J, Nonempty (F.obj j)] : F.sections.Nonempty := by
  -- Step 1: lift everything to the `max u v w` universe.
  let J' : Type max w v u := AsSmall.{max w v} J
  

Proof state:
J : Type u
inst✝³ : Category.{w, u} J
inst✝² : IsCofilteredOrEmpty J
F : J ⥤ Type v
inst✝¹ : ∀ (j : J), Finite (F.obj j)
inst✝ : ∀ (j : J), Nonempty (F.obj j)
J' : Type (max w v u) := AsSmall J
⊢ Set.Nonempty (Functor.sections F)

Next tactic:
let down : J' ⥤ J := AsSmall.down
/-- The cofiltered limit of nonempty finite types is nonempty.

See `nonempty_sections_of_finite_inverse_system` for a specialization to inverse limits. -/
theorem nonempty_sections_of_finite_cofiltered_system {J : Type u} [Category.{w} J]
    [IsCofilteredOrEmpty J] (F : J ⥤ Type v) [∀ j : J, Finite (F.obj j)]
    [∀ j : J, Nonempty (F.obj j)] : F.sections.Nonempty := by
  -- Step 1: lift everything to the `max u v w` universe.
  let J' : Type max w v u := AsSmall.{max w v} J
  let down : J' ⥤ J := AsSmall.down
  

Proof state:
J : Type u
inst✝³ : Category.{w, u} J
inst✝² : IsCofilteredOrEmpty J
F : J ⥤ Type v
inst✝¹ : ∀ (j : J), Finite (F.obj j)
inst✝ : ∀ (j : J), Nonempty (F.obj j)
J' : Type (max w v u) := AsSmall J
down : J' ⥤ J := AsSmall.down
⊢ Set.Nonempty (Functor.sections F)

Next tactic:
let F' : J' ⥤ Type max u v w := down ⋙ F ⋙ uliftFunctor.{max u w, v}
/-- The cofiltered limit of nonempty finite types is nonempty.

See `nonempty_sections_of_finite_inverse_system` for a specialization to inverse limits. -/
theorem nonempty_sections_of_finite_cofiltered_system {J : Type u} [Category.{w} J]
    [IsCofilteredOrEmpty J] (F : J ⥤ Type v) [∀ j : J, Finite (F.obj j)]
    [∀ j : J, Nonempty (F.obj j)] : F.sections.Nonempty := by
  -- Step 1: lift everything to the `max u v w` universe.
  let J' : Type max w v u := AsSmall.{max w v} J
  let down : J' ⥤ J := AsSmall.down
  let F' : J' ⥤ Type max u v w := down ⋙ F ⋙ uliftFunctor.{max u w, v}
  

Proof state:
J : Type u
inst✝³ : Category.{w, u} J
inst✝² : IsCofilteredOrEmpty J
F : J ⥤ Type v
inst✝¹ : ∀ (j : J), Finite (F.obj j)
inst✝ : ∀ (j : J), Nonempty (F.obj j)
J' : Type (max w v u) := AsSmall J
down : J' ⥤ J := AsSmall.down
F' : J' ⥤ Type (max u v w) := down ⋙ F ⋙ uliftFunctor.{max u w, v}
⊢ Set.Nonempty (Functor.sections F)

Next tactic:
haveI : ∀ i, Nonempty (F'.obj i) := fun i => ⟨⟨Classical.arbitrary (F.obj (down.obj i))⟩⟩
/-- The cofiltered limit of nonempty finite types is nonempty.

See `nonempty_sections_of_finite_inverse_system` for a specialization to inverse limits. -/
theorem nonempty_sections_of_finite_cofiltered_system {J : Type u} [Category.{w} J]
    [IsCofilteredOrEmpty J] (F : J ⥤ Type v) [∀ j : J, Finite (F.obj j)]
    [∀ j : J, Nonempty (F.obj j)] : F.sections.Nonempty := by
  -- Step 1: lift everything to the `max u v w` universe.
  let J' : Type max w v u := AsSmall.{max w v} J
  let down : J' ⥤ J := AsSmall.down
  let F' : J' ⥤ Type max u v w := down ⋙ F ⋙ uliftFunctor.{max u w, v}
  haveI : ∀ i, Nonempty (F'.obj i) := fun i => ⟨⟨Classical.arbitrary (F.obj (down.obj i))⟩⟩
  

Proof state:
J : Type u
inst✝³ : Category.{w, u} J
inst✝² : IsCofilteredOrEmpty J
F : J ⥤ Type v
inst✝¹ : ∀ (j : J), Finite (F.obj j)
inst✝ : ∀ (j : J), Nonempty (F.obj j)
J' : Type (max w v u) := AsSmall J
down : J' ⥤ J := AsSmall.down
F' : J' ⥤ Type (max u v w) := down ⋙ F ⋙ uliftFunctor.{max u w, v}
this : ∀ (i : J'), Nonempty (F'.obj i)
⊢ Set.Nonempty (Functor.sections F)

Next tactic:
haveI : ∀ i, Finite (F'.obj i) := fun i => Finite.of_equiv (F.obj (down.obj i)) Equiv.ulift.symm
/-- The cofiltered limit of nonempty finite types is nonempty.

See `nonempty_sections_of_finite_inverse_system` for a specialization to inverse limits. -/
theorem nonempty_sections_of_finite_cofiltered_system {J : Type u} [Category.{w} J]
    [IsCofilteredOrEmpty J] (F : J ⥤ Type v) [∀ j : J, Finite (F.obj j)]
    [∀ j : J, Nonempty (F.obj j)] : F.sections.Nonempty := by
  -- Step 1: lift everything to the `max u v w` universe.
  let J' : Type max w v u := AsSmall.{max w v} J
  let down : J' ⥤ J := AsSmall.down
  let F' : J' ⥤ Type max u v w := down ⋙ F ⋙ uliftFunctor.{max u w, v}
  haveI : ∀ i, Nonempty (F'.obj i) := fun i => ⟨⟨Classical.arbitrary (F.obj (down.obj i))⟩⟩
  haveI : ∀ i, Finite (F'.obj i) := fun i => Finite.of_equiv (F.obj (down.obj i)) Equiv.ulift.symm
  -- Step 2: apply the bootstrap theorem
  

Proof state:
J : Type u
inst✝³ : Category.{w, u} J
inst✝² : IsCofilteredOrEmpty J
F : J ⥤ Type v
inst✝¹ : ∀ (j : J), Finite (F.obj j)
inst✝ : ∀ (j : J), Nonempty (F.obj j)
J' : Type (max w v u) := AsSmall J
down : J' ⥤ J := AsSmall.down
F' : J' ⥤ Type (max u v w) := down ⋙ F ⋙ uliftFunctor.{max u w, v}
this✝ : ∀ (i : J'), Nonempty (F'.obj i)
this : ∀ (i : J'), Finite (F'.obj i)
⊢ Set.Nonempty (Functor.sections F)

Next tactic:
cases isEmpty_or_nonempty J
/-- The cofiltered limit of nonempty finite types is nonempty.

See `nonempty_sections_of_finite_inverse_system` for a specialization to inverse limits. -/
theorem nonempty_sections_of_finite_cofiltered_system {J : Type u} [Category.{w} J]
    [IsCofilteredOrEmpty J] (F : J ⥤ Type v) [∀ j : J, Finite (F.obj j)]
    [∀ j : J, Nonempty (F.obj j)] : F.sections.Nonempty := by
  -- Step 1: lift everything to the `max u v w` universe.
  let J' : Type max w v u := AsSmall.{max w v} J
  let down : J' ⥤ J := AsSmall.down
  let F' : J' ⥤ Type max u v w := down ⋙ F ⋙ uliftFunctor.{max u w, v}
  haveI : ∀ i, Nonempty (F'.obj i) := fun i => ⟨⟨Classical.arbitrary (F.obj (down.obj i))⟩⟩
  haveI : ∀ i, Finite (F'.obj i) := fun i => Finite.of_equiv (F.obj (down.obj i)) Equiv.ulift.symm
  -- Step 2: apply the bootstrap theorem
  cases isEmpty_or_nonempty J
  · 

Proof state:
case inl
J : Type u
inst✝³ : Category.{w, u} J
inst✝² : IsCofilteredOrEmpty J
F : J ⥤ Type v
inst✝¹ : ∀ (j : J), Finite (F.obj j)
inst✝ : ∀ (j : J), Nonempty (F.obj j)
J' : Type (max w v u) := AsSmall J
down : J' ⥤ J := AsSmall.down
F' : J' ⥤ Type (max u v w) := down ⋙ F ⋙ uliftFunctor.{max u w, v}
this✝ : ∀ (i : J'), Nonempty (F'.obj i)
this : ∀ (i : J'), Finite (F'.obj i)
h✝ : IsEmpty J
⊢ Set.Nonempty (Functor.sections F)

Next tactic:
fconstructor
/-- The cofiltered limit of nonempty finite types is nonempty.

See `nonempty_sections_of_finite_inverse_system` for a specialization to inverse limits. -/
theorem nonempty_sections_of_finite_cofiltered_system {J : Type u} [Category.{w} J]
    [IsCofilteredOrEmpty J] (F : J ⥤ Type v) [∀ j : J, Finite (F.obj j)]
    [∀ j : J, Nonempty (F.obj j)] : F.sections.Nonempty := by
  -- Step 1: lift everything to the `max u v w` universe.
  let J' : Type max w v u := AsSmall.{max w v} J
  let down : J' ⥤ J := AsSmall.down
  let F' : J' ⥤ Type max u v w := down ⋙ F ⋙ uliftFunctor.{max u w, v}
  haveI : ∀ i, Nonempty (F'.obj i) := fun i => ⟨⟨Classical.arbitrary (F.obj (down.obj i))⟩⟩
  haveI : ∀ i, Finite (F'.obj i) := fun i => Finite.of_equiv (F.obj (down.obj i)) Equiv.ulift.symm
  -- Step 2: apply the bootstrap theorem
  cases isEmpty_or_nonempty J
  · fconstructor <;> 

Proof state:
case inl.w
J : Type u
inst✝³ : Category.{w, u} J
inst✝² : IsCofilteredOrEmpty J
F : J ⥤ Type v
inst✝¹ : ∀ (j : J), Finite (F.obj j)
inst✝ : ∀ (j : J), Nonempty (F.obj j)
J' : Type (max w v u) := AsSmall J
down : J' ⥤ J := AsSmall.down
F' : J' ⥤ Type (max u v w) := down ⋙ F ⋙ uliftFunctor.{max u w, v}
this✝ : ∀ (i : J'), Nonempty (F'.obj i)
this : ∀ (i : J'), Finite (F'.obj i)
h✝ : IsEmpty J
⊢ (j : J) → F.obj j

Next tactic:
apply isEmptyElim
/-- The cofiltered limit of nonempty finite types is nonempty.

See `nonempty_sections_of_finite_inverse_system` for a specialization to inverse limits. -/
theorem nonempty_sections_of_finite_cofiltered_system {J : Type u} [Category.{w} J]
    [IsCofilteredOrEmpty J] (F : J ⥤ Type v) [∀ j : J, Finite (F.obj j)]
    [∀ j : J, Nonempty (F.obj j)] : F.sections.Nonempty := by
  -- Step 1: lift everything to the `max u v w` universe.
  let J' : Type max w v u := AsSmall.{max w v} J
  let down : J' ⥤ J := AsSmall.down
  let F' : J' ⥤ Type max u v w := down ⋙ F ⋙ uliftFunctor.{max u w, v}
  haveI : ∀ i, Nonempty (F'.obj i) := fun i => ⟨⟨Classical.arbitrary (F.obj (down.obj i))⟩⟩
  haveI : ∀ i, Finite (F'.obj i) := fun i => Finite.of_equiv (F.obj (down.obj i)) Equiv.ulift.symm
  -- Step 2: apply the bootstrap theorem
  cases isEmpty_or_nonempty J
  · fconstructor <;> 

Proof state:
case inl.h
J : Type u
inst✝³ : Category.{w, u} J
inst✝² : IsCofilteredOrEmpty J
F : J ⥤ Type v
inst✝¹ : ∀ (j : J), Finite (F.obj j)
inst✝ : ∀ (j : J), Nonempty (F.obj j)
J' : Type (max w v u) := AsSmall J
down : J' ⥤ J := AsSmall.down
F' : J' ⥤ Type (max u v w) := down ⋙ F ⋙ uliftFunctor.{max u w, v}
this✝ : ∀ (i : J'), Nonempty (F'.obj i)
this : ∀ (i : J'), Finite (F'.obj i)
h✝ : IsEmpty J
⊢ isEmptyElim ∈ Functor.sections F

Next tactic:
apply isEmptyElim
/-- The cofiltered limit of nonempty finite types is nonempty.

See `nonempty_sections_of_finite_inverse_system` for a specialization to inverse limits. -/
theorem nonempty_sections_of_finite_cofiltered_system {J : Type u} [Category.{w} J]
    [IsCofilteredOrEmpty J] (F : J ⥤ Type v) [∀ j : J, Finite (F.obj j)]
    [∀ j : J, Nonempty (F.obj j)] : F.sections.Nonempty := by
  -- Step 1: lift everything to the `max u v w` universe.
  let J' : Type max w v u := AsSmall.{max w v} J
  let down : J' ⥤ J := AsSmall.down
  let F' : J' ⥤ Type max u v w := down ⋙ F ⋙ uliftFunctor.{max u w, v}
  haveI : ∀ i, Nonempty (F'.obj i) := fun i => ⟨⟨Classical.arbitrary (F.obj (down.obj i))⟩⟩
  haveI : ∀ i, Finite (F'.obj i) := fun i => Finite.of_equiv (F.obj (down.obj i)) Equiv.ulift.symm
  -- Step 2: apply the bootstrap theorem
  cases isEmpty_or_nonempty J
  · fconstructor <;> apply isEmptyElim
  

Proof state:
case inr
J : Type u
inst✝³ : Category.{w, u} J
inst✝² : IsCofilteredOrEmpty J
F : J ⥤ Type v
inst✝¹ : ∀ (j : J), Finite (F.obj j)
inst✝ : ∀ (j : J), Nonempty (F.obj j)
J' : Type (max w v u) := AsSmall J
down : J' ⥤ J := AsSmall.down
F' : J' ⥤ Type (max u v w) := down ⋙ F ⋙ uliftFunctor.{max u w, v}
this✝ : ∀ (i : J'), Nonempty (F'.obj i)
this : ∀ (i : J'), Finite (F'.obj i)
h✝ : Nonempty J
⊢ Set.Nonempty (Functor.sections F)

Next tactic:
haveI : IsCofiltered J := ⟨⟩
/-- The cofiltered limit of nonempty finite types is nonempty.

See `nonempty_sections_of_finite_inverse_system` for a specialization to inverse limits. -/
theorem nonempty_sections_of_finite_cofiltered_system {J : Type u} [Category.{w} J]
    [IsCofilteredOrEmpty J] (F : J ⥤ Type v) [∀ j : J, Finite (F.obj j)]
    [∀ j : J, Nonempty (F.obj j)] : F.sections.Nonempty := by
  -- Step 1: lift everything to the `max u v w` universe.
  let J' : Type max w v u := AsSmall.{max w v} J
  let down : J' ⥤ J := AsSmall.down
  let F' : J' ⥤ Type max u v w := down ⋙ F ⋙ uliftFunctor.{max u w, v}
  haveI : ∀ i, Nonempty (F'.obj i) := fun i => ⟨⟨Classical.arbitrary (F.obj (down.obj i))⟩⟩
  haveI : ∀ i, Finite (F'.obj i) := fun i => Finite.of_equiv (F.obj (down.obj i)) Equiv.ulift.symm
  -- Step 2: apply the bootstrap theorem
  cases isEmpty_or_nonempty J
  · fconstructor <;> apply isEmptyElim
  haveI : IsCofiltered J := ⟨⟩
  

Proof state:
case inr
J : Type u
inst✝³ : Category.{w, u} J
inst✝² : IsCofilteredOrEmpty J
F : J ⥤ Type v
inst✝¹ : ∀ (j : J), Finite (F.obj j)
inst✝ : ∀ (j : J), Nonempty (F.obj j)
J' : Type (max w v u) := AsSmall J
down : J' ⥤ J := AsSmall.down
F' : J' ⥤ Type (max u v w) := down ⋙ F ⋙ uliftFunctor.{max u w, v}
this✝¹ : ∀ (i : J'), Nonempty (F'.obj i)
this✝ : ∀ (i : J'), Finite (F'.obj i)
h✝ : Nonempty J
this : IsCofiltered J
⊢ Set.Nonempty (Functor.sections F)

Next tactic:
obtain ⟨u, hu⟩ := nonempty_sections_of_finite_cofiltered_system.init F'
/-- The cofiltered limit of nonempty finite types is nonempty.

See `nonempty_sections_of_finite_inverse_system` for a specialization to inverse limits. -/
theorem nonempty_sections_of_finite_cofiltered_system {J : Type u} [Category.{w} J]
    [IsCofilteredOrEmpty J] (F : J ⥤ Type v) [∀ j : J, Finite (F.obj j)]
    [∀ j : J, Nonempty (F.obj j)] : F.sections.Nonempty := by
  -- Step 1: lift everything to the `max u v w` universe.
  let J' : Type max w v u := AsSmall.{max w v} J
  let down : J' ⥤ J := AsSmall.down
  let F' : J' ⥤ Type max u v w := down ⋙ F ⋙ uliftFunctor.{max u w, v}
  haveI : ∀ i, Nonempty (F'.obj i) := fun i => ⟨⟨Classical.arbitrary (F.obj (down.obj i))⟩⟩
  haveI : ∀ i, Finite (F'.obj i) := fun i => Finite.of_equiv (F.obj (down.obj i)) Equiv.ulift.symm
  -- Step 2: apply the bootstrap theorem
  cases isEmpty_or_nonempty J
  · fconstructor <;> apply isEmptyElim
  haveI : IsCofiltered J := ⟨⟩
  obtain ⟨u, hu⟩ := nonempty_sections_of_finite_cofiltered_system.init F'
  -- Step 3: interpret the results
  

Proof state:
case inr.intro
J : Type u
inst✝³ : Category.{w, u} J
inst✝² : IsCofilteredOrEmpty J
F : J ⥤ Type v
inst✝¹ : ∀ (j : J), Finite (F.obj j)
inst✝ : ∀ (j : J), Nonempty (F.obj j)
J' : Type (max w v u) := AsSmall J
down : J' ⥤ J := AsSmall.down
F' : J' ⥤ Type (max u v w) := down ⋙ F ⋙ uliftFunctor.{max u w, v}
this✝¹ : ∀ (i : J'), Nonempty (F'.obj i)
this✝ : ∀ (i : J'), Finite (F'.obj i)
h✝ : Nonempty J
this : IsCofiltered J
u : (j : J') → F'.obj j
hu : u ∈ Functor.sections F'
⊢ Set.Nonempty (Functor.sections F)

Next tactic:
use fun j => (u ⟨j⟩).down
/-- The cofiltered limit of nonempty finite types is nonempty.

See `nonempty_sections_of_finite_inverse_system` for a specialization to inverse limits. -/
theorem nonempty_sections_of_finite_cofiltered_system {J : Type u} [Category.{w} J]
    [IsCofilteredOrEmpty J] (F : J ⥤ Type v) [∀ j : J, Finite (F.obj j)]
    [∀ j : J, Nonempty (F.obj j)] : F.sections.Nonempty := by
  -- Step 1: lift everything to the `max u v w` universe.
  let J' : Type max w v u := AsSmall.{max w v} J
  let down : J' ⥤ J := AsSmall.down
  let F' : J' ⥤ Type max u v w := down ⋙ F ⋙ uliftFunctor.{max u w, v}
  haveI : ∀ i, Nonempty (F'.obj i) := fun i => ⟨⟨Classical.arbitrary (F.obj (down.obj i))⟩⟩
  haveI : ∀ i, Finite (F'.obj i) := fun i => Finite.of_equiv (F.obj (down.obj i)) Equiv.ulift.symm
  -- Step 2: apply the bootstrap theorem
  cases isEmpty_or_nonempty J
  · fconstructor <;> apply isEmptyElim
  haveI : IsCofiltered J := ⟨⟩
  obtain ⟨u, hu⟩ := nonempty_sections_of_finite_cofiltered_system.init F'
  -- Step 3: interpret the results
  use fun j => (u ⟨j⟩).down
  

Proof state:
case h
J : Type u
inst✝³ : Category.{w, u} J
inst✝² : IsCofilteredOrEmpty J
F : J ⥤ Type v
inst✝¹ : ∀ (j : J), Finite (F.obj j)
inst✝ : ∀ (j : J), Nonempty (F.obj j)
J' : Type (max w v u) := AsSmall J
down : J' ⥤ J := AsSmall.down
F' : J' ⥤ Type (max u v w) := down ⋙ F ⋙ uliftFunctor.{max u w, v}
this✝¹ : ∀ (i : J'), Nonempty (F'.obj i)
this✝ : ∀ (i : J'), Finite (F'.obj i)
h✝ : Nonempty J
this : IsCofiltered J
u : (j : J') → F'.obj j
hu : u ∈ Functor.sections F'
⊢ (fun j => (u { down := j }).down) ∈ Functor.sections F

Next tactic:
intro j j' f
/-- The cofiltered limit of nonempty finite types is nonempty.

See `nonempty_sections_of_finite_inverse_system` for a specialization to inverse limits. -/
theorem nonempty_sections_of_finite_cofiltered_system {J : Type u} [Category.{w} J]
    [IsCofilteredOrEmpty J] (F : J ⥤ Type v) [∀ j : J, Finite (F.obj j)]
    [∀ j : J, Nonempty (F.obj j)] : F.sections.Nonempty := by
  -- Step 1: lift everything to the `max u v w` universe.
  let J' : Type max w v u := AsSmall.{max w v} J
  let down : J' ⥤ J := AsSmall.down
  let F' : J' ⥤ Type max u v w := down ⋙ F ⋙ uliftFunctor.{max u w, v}
  haveI : ∀ i, Nonempty (F'.obj i) := fun i => ⟨⟨Classical.arbitrary (F.obj (down.obj i))⟩⟩
  haveI : ∀ i, Finite (F'.obj i) := fun i => Finite.of_equiv (F.obj (down.obj i)) Equiv.ulift.symm
  -- Step 2: apply the bootstrap theorem
  cases isEmpty_or_nonempty J
  · fconstructor <;> apply isEmptyElim
  haveI : IsCofiltered J := ⟨⟩
  obtain ⟨u, hu⟩ := nonempty_sections_of_finite_cofiltered_system.init F'
  -- Step 3: interpret the results
  use fun j => (u ⟨j⟩).down
  intro j j' f
  

Proof state:
case h
J : Type u
inst✝³ : Category.{w, u} J
inst✝² : IsCofilteredOrEmpty J
F : J ⥤ Type v
inst✝¹ : ∀ (j : J), Finite (F.obj j)
inst✝ : ∀ (j : J), Nonempty (F.obj j)
J' : Type (max w v u) := AsSmall J
down : J' ⥤ J := AsSmall.down
F' : J' ⥤ Type (max u v w) := down ⋙ F ⋙ uliftFunctor.{max u w, v}
this✝¹ : ∀ (i : J'), Nonempty (F'.obj i)
this✝ : ∀ (i : J'), Finite (F'.obj i)
h✝ : Nonempty J
this : IsCofiltered J
u : (j : J') → F'.obj j
hu : u ∈ Functor.sections F'
j j' : J
f : j ⟶ j'
⊢ F.map f ((fun j => (u { down := j }).down) j) = (fun j => (u { down := j }).down) j'

Next tactic:
have h := @hu (⟨j⟩ : J') (⟨j'⟩ : J') (ULift.up f)
/-- The cofiltered limit of nonempty finite types is nonempty.

See `nonempty_sections_of_finite_inverse_system` for a specialization to inverse limits. -/
theorem nonempty_sections_of_finite_cofiltered_system {J : Type u} [Category.{w} J]
    [IsCofilteredOrEmpty J] (F : J ⥤ Type v) [∀ j : J, Finite (F.obj j)]
    [∀ j : J, Nonempty (F.obj j)] : F.sections.Nonempty := by
  -- Step 1: lift everything to the `max u v w` universe.
  let J' : Type max w v u := AsSmall.{max w v} J
  let down : J' ⥤ J := AsSmall.down
  let F' : J' ⥤ Type max u v w := down ⋙ F ⋙ uliftFunctor.{max u w, v}
  haveI : ∀ i, Nonempty (F'.obj i) := fun i => ⟨⟨Classical.arbitrary (F.obj (down.obj i))⟩⟩
  haveI : ∀ i, Finite (F'.obj i) := fun i => Finite.of_equiv (F.obj (down.obj i)) Equiv.ulift.symm
  -- Step 2: apply the bootstrap theorem
  cases isEmpty_or_nonempty J
  · fconstructor <;> apply isEmptyElim
  haveI : IsCofiltered J := ⟨⟩
  obtain ⟨u, hu⟩ := nonempty_sections_of_finite_cofiltered_system.init F'
  -- Step 3: interpret the results
  use fun j => (u ⟨j⟩).down
  intro j j' f
  have h := @hu (⟨j⟩ : J') (⟨j'⟩ : J') (ULift.up f)
  

Proof state:
case h
J : Type u
inst✝³ : Category.{w, u} J
inst✝² : IsCofilteredOrEmpty J
F : J ⥤ Type v
inst✝¹ : ∀ (j : J), Finite (F.obj j)
inst✝ : ∀ (j : J), Nonempty (F.obj j)
J' : Type (max w v u) := AsSmall J
down : J' ⥤ J := AsSmall.down
F' : J' ⥤ Type (max u v w) := down ⋙ F ⋙ uliftFunctor.{max u w, v}
this✝¹ : ∀ (i : J'), Nonempty (F'.obj i)
this✝ : ∀ (i : J'), Finite (F'.obj i)
h✝ : Nonempty J
this : IsCofiltered J
u : (j : J') → F'.obj j
hu : u ∈ Functor.sections F'
j j' : J
f : j ⟶ j'
h : F'.map { down := f } (u { down := j }) = u { down := j' }
⊢ F.map f ((fun j => (u { down := j }).down) j) = (fun j => (u { down := j }).down) j'

Next tactic:
simp only [AsSmall.down, Functor.comp_map, uliftFunctor_map, Functor.op_map] at h
/-- The cofiltered limit of nonempty finite types is nonempty.

See `nonempty_sections_of_finite_inverse_system` for a specialization to inverse limits. -/
theorem nonempty_sections_of_finite_cofiltered_system {J : Type u} [Category.{w} J]
    [IsCofilteredOrEmpty J] (F : J ⥤ Type v) [∀ j : J, Finite (F.obj j)]
    [∀ j : J, Nonempty (F.obj j)] : F.sections.Nonempty := by
  -- Step 1: lift everything to the `max u v w` universe.
  let J' : Type max w v u := AsSmall.{max w v} J
  let down : J' ⥤ J := AsSmall.down
  let F' : J' ⥤ Type max u v w := down ⋙ F ⋙ uliftFunctor.{max u w, v}
  haveI : ∀ i, Nonempty (F'.obj i) := fun i => ⟨⟨Classical.arbitrary (F.obj (down.obj i))⟩⟩
  haveI : ∀ i, Finite (F'.obj i) := fun i => Finite.of_equiv (F.obj (down.obj i)) Equiv.ulift.symm
  -- Step 2: apply the bootstrap theorem
  cases isEmpty_or_nonempty J
  · fconstructor <;> apply isEmptyElim
  haveI : IsCofiltered J := ⟨⟩
  obtain ⟨u, hu⟩ := nonempty_sections_of_finite_cofiltered_system.init F'
  -- Step 3: interpret the results
  use fun j => (u ⟨j⟩).down
  intro j j' f
  have h := @hu (⟨j⟩ : J') (⟨j'⟩ : J') (ULift.up f)
  simp only [AsSmall.down, Functor.comp_map, uliftFunctor_map, Functor.op_map] at h
  

Proof state:
case h
J : Type u
inst✝³ : Category.{w, u} J
inst✝² : IsCofilteredOrEmpty J
F : J ⥤ Type v
inst✝¹ : ∀ (j : J), Finite (F.obj j)
inst✝ : ∀ (j : J), Nonempty (F.obj j)
J' : Type (max w v u) := AsSmall J
down : J' ⥤ J := AsSmall.down
F' : J' ⥤ Type (max u v w) := down ⋙ F ⋙ uliftFunctor.{max u w, v}
this✝¹ : ∀ (i : J'), Nonempty (F'.obj i)
this✝ : ∀ (i : J'), Finite (F'.obj i)
h✝ : Nonempty J
this : IsCofiltered J
u : (j : J') → F'.obj j
hu : u ∈ Functor.sections F'
j j' : J
f : j ⟶ j'
h : { down := F.map f (u { down := j }).down } = u { down := j' }
⊢ F.map f ((fun j => (u { down := j }).down) j) = (fun j => (u { down := j }).down) j'

Next tactic:
simp_rw [← h]
/-- The inverse limit of nonempty finite types is nonempty.

See `nonempty_sections_of_finite_cofiltered_system` for a generalization to cofiltered limits.
That version applies in almost all cases, and the only difference is that this version
allows `J` to be empty.

This may be regarded as a generalization of Kőnig's lemma.
To specialize: given a locally finite connected graph, take `Jᵒᵖ` to be `ℕ` and
`F j` to be length-`j` paths that start from an arbitrary fixed vertex.
Elements of `F.sections` can be read off as infinite rays in the graph. -/
theorem nonempty_sections_of_finite_inverse_system {J : Type u} [Preorder J] [IsDirected J (· ≤ ·)]
    (F : Jᵒᵖ ⥤ Type v) [∀ j : Jᵒᵖ, Finite (F.obj j)] [∀ j : Jᵒᵖ, Nonempty (F.obj j)] :
    F.sections.Nonempty := by
  

Proof state:
J : Type u
inst✝³ : Preorder J
inst✝² : IsDirected J fun x x_1 => x ≤ x_1
F : Jᵒᵖ ⥤ Type v
inst✝¹ : ∀ (j : Jᵒᵖ), Finite (F.obj j)
inst✝ : ∀ (j : Jᵒᵖ), Nonempty (F.obj j)
⊢ Set.Nonempty (Functor.sections F)

Next tactic:
cases isEmpty_or_nonempty J