l3lab/ntp-mathlib huggingface dataset:
/-- The second theorem states that locally compact spaces are Baire. -/
instance (priority := 100) BaireSpace.of_t2Space_locallyCompactSpace
    [TopologicalSpace α] [T2Space α] [LocallyCompactSpace α] : BaireSpace α := by
  constructor
  intro f ho hd
  /- To prove that an intersection of open dense subsets is dense, prove that its intersection
    with any open neighbourhood `U` is dense. Define recursively a decreasing sequence `K` of
    compact neighbourhoods: start with some compact neighbourhood inside `U`, then at each step,
    take its interior, intersect with `f n`, then choose a compact neighbourhood inside the
    intersection. -/
  apply dense_iff_inter_open.2
  intro U U_open U_nonempty
  rcases exists_positiveCompacts_subset U_open U_nonempty with ⟨K₀, hK₀⟩
  have : ∀ (n) (K : PositiveCompacts α), ∃ K' : PositiveCompacts α, ↑K' ⊆ f n ∩ interior K := by
    refine' fun n K => exists_positiveCompacts_subset ((ho n).inter isOpen_interior) _
    rw [inter_comm]
    exact (hd n).inter_open_nonempty _ isOpen_interior K.interior_nonempty
  choose K_next hK_next using this
  

Proof state:
case baire_property.intro
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Type u_4
inst✝² : TopologicalSpace α
inst✝¹ : T2Space α
inst✝ : LocallyCompactSpace α
f : ℕ → Set α
ho : ∀ (n : ℕ), IsOpen (f n)
hd : ∀ (n : ℕ), Dense (f n)
U : Set α
U_open : IsOpen U
U_nonempty : Set.Nonempty U
K₀ : PositiveCompacts α
hK₀ : ↑K₀ ⊆ U
K_next : ℕ → PositiveCompacts α → PositiveCompacts α
hK_next : ∀ (n : ℕ) (K : PositiveCompacts α), ↑(K_next n K) ⊆ f n ∩ interior ↑K
⊢ Set.Nonempty (U ∩ ⋂ n, f n)

Next tactic:
let K : ℕ → PositiveCompacts α := fun n => Nat.recOn n K₀ K_next
/-- The second theorem states that locally compact spaces are Baire. -/
instance (priority := 100) BaireSpace.of_t2Space_locallyCompactSpace
    [TopologicalSpace α] [T2Space α] [LocallyCompactSpace α] : BaireSpace α := by
  constructor
  intro f ho hd
  /- To prove that an intersection of open dense subsets is dense, prove that its intersection
    with any open neighbourhood `U` is dense. Define recursively a decreasing sequence `K` of
    compact neighbourhoods: start with some compact neighbourhood inside `U`, then at each step,
    take its interior, intersect with `f n`, then choose a compact neighbourhood inside the
    intersection. -/
  apply dense_iff_inter_open.2
  intro U U_open U_nonempty
  rcases exists_positiveCompacts_subset U_open U_nonempty with ⟨K₀, hK₀⟩
  have : ∀ (n) (K : PositiveCompacts α), ∃ K' : PositiveCompacts α, ↑K' ⊆ f n ∩ interior K := by
    refine' fun n K => exists_positiveCompacts_subset ((ho n).inter isOpen_interior) _
    rw [inter_comm]
    exact (hd n).inter_open_nonempty _ isOpen_interior K.interior_nonempty
  choose K_next hK_next using this
  let K : ℕ → PositiveCompacts α := fun n => Nat.recOn n K₀ K_next
  -- This is a decreasing sequence of positive compacts contained in suitable open sets `f n`.
  

Proof state:
case baire_property.intro
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Type u_4
inst✝² : TopologicalSpace α
inst✝¹ : T2Space α
inst✝ : LocallyCompactSpace α
f : ℕ → Set α
ho : ∀ (n : ℕ), IsOpen (f n)
hd : ∀ (n : ℕ), Dense (f n)
U : Set α
U_open : IsOpen U
U_nonempty : Set.Nonempty U
K₀ : PositiveCompacts α
hK₀ : ↑K₀ ⊆ U
K_next : ℕ → PositiveCompacts α → PositiveCompacts α
hK_next : ∀ (n : ℕ) (K : PositiveCompacts α), ↑(K_next n K) ⊆ f n ∩ interior ↑K
K : ℕ → PositiveCompacts α := fun n => Nat.recOn n K₀ K_next
⊢ Set.Nonempty (U ∩ ⋂ n, f n)

Next tactic:
have hK_decreasing : ∀ n : ℕ, ((K (n + 1)).carrier) ⊆ (f n ∩ (K n).carrier) :=
    fun n => (hK_next n (K n)).trans <| inter_subset_inter_right _ interior_subset
/-- The second theorem states that locally compact spaces are Baire. -/
instance (priority := 100) BaireSpace.of_t2Space_locallyCompactSpace
    [TopologicalSpace α] [T2Space α] [LocallyCompactSpace α] : BaireSpace α := by
  constructor
  intro f ho hd
  /- To prove that an intersection of open dense subsets is dense, prove that its intersection
    with any open neighbourhood `U` is dense. Define recursively a decreasing sequence `K` of
    compact neighbourhoods: start with some compact neighbourhood inside `U`, then at each step,
    take its interior, intersect with `f n`, then choose a compact neighbourhood inside the
    intersection. -/
  apply dense_iff_inter_open.2
  intro U U_open U_nonempty
  rcases exists_positiveCompacts_subset U_open U_nonempty with ⟨K₀, hK₀⟩
  have : ∀ (n) (K : PositiveCompacts α), ∃ K' : PositiveCompacts α, ↑K' ⊆ f n ∩ interior K := by
    refine' fun n K => exists_positiveCompacts_subset ((ho n).inter isOpen_interior) _
    rw [inter_comm]
    exact (hd n).inter_open_nonempty _ isOpen_interior K.interior_nonempty
  choose K_next hK_next using this
  let K : ℕ → PositiveCompacts α := fun n => Nat.recOn n K₀ K_next
  -- This is a decreasing sequence of positive compacts contained in suitable open sets `f n`.
  have hK_decreasing : ∀ n : ℕ, ((K (n + 1)).carrier) ⊆ (f n ∩ (K n).carrier) :=
    fun n => (hK_next n (K n)).trans <| inter_subset_inter_right _ interior_subset
  -- Prove that ̀`⋂ n : ℕ, K n` is inside `U ∩ ⋂ n : ℕ, f n`.
  

Proof state:
case baire_property.intro
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Type u_4
inst✝² : TopologicalSpace α
inst✝¹ : T2Space α
inst✝ : LocallyCompactSpace α
f : ℕ → Set α
ho : ∀ (n : ℕ), IsOpen (f n)
hd : ∀ (n : ℕ), Dense (f n)
U : Set α
U_open : IsOpen U
U_nonempty : Set.Nonempty U
K₀ : PositiveCompacts α
hK₀ : ↑K₀ ⊆ U
K_next : ℕ → PositiveCompacts α → PositiveCompacts α
hK_next : ∀ (n : ℕ) (K : PositiveCompacts α), ↑(K_next n K) ⊆ f n ∩ interior ↑K
K : ℕ → PositiveCompacts α := fun n => Nat.recOn n K₀ K_next
hK_decreasing : ∀ (n : ℕ), (K (n + 1)).carrier ⊆ f n ∩ (K n).carrier
⊢ Set.Nonempty (U ∩ ⋂ n, f n)

Next tactic:
have hK_subset : (⋂ n, (K n).carrier : Set α) ⊆ U ∩ ⋂ n, f n := by
    intro x hx
    simp only [mem_iInter] at hx
    simp only [mem_inter_iff, mem_inter] at hx ⊢
    refine' ⟨hK₀ <| hx 0, _⟩
    simp only [mem_iInter]
    exact fun n => (hK_decreasing n (hx (n + 1))).1
/-- The second theorem states that locally compact spaces are Baire. -/
instance (priority := 100) BaireSpace.of_t2Space_locallyCompactSpace
    [TopologicalSpace α] [T2Space α] [LocallyCompactSpace α] : BaireSpace α := by
  constructor
  intro f ho hd
  /- To prove that an intersection of open dense subsets is dense, prove that its intersection
    with any open neighbourhood `U` is dense. Define recursively a decreasing sequence `K` of
    compact neighbourhoods: start with some compact neighbourhood inside `U`, then at each step,
    take its interior, intersect with `f n`, then choose a compact neighbourhood inside the
    intersection. -/
  apply dense_iff_inter_open.2
  intro U U_open U_nonempty
  rcases exists_positiveCompacts_subset U_open U_nonempty with ⟨K₀, hK₀⟩
  have : ∀ (n) (K : PositiveCompacts α), ∃ K' : PositiveCompacts α, ↑K' ⊆ f n ∩ interior K := by
    refine' fun n K => exists_positiveCompacts_subset ((ho n).inter isOpen_interior) _
    rw [inter_comm]
    exact (hd n).inter_open_nonempty _ isOpen_interior K.interior_nonempty
  choose K_next hK_next using this
  let K : ℕ → PositiveCompacts α := fun n => Nat.recOn n K₀ K_next
  -- This is a decreasing sequence of positive compacts contained in suitable open sets `f n`.
  have hK_decreasing : ∀ n : ℕ, ((K (n + 1)).carrier) ⊆ (f n ∩ (K n).carrier) :=
    fun n => (hK_next n (K n)).trans <| inter_subset_inter_right _ interior_subset
  -- Prove that ̀`⋂ n : ℕ, K n` is inside `U ∩ ⋂ n : ℕ, f n`.
  have hK_subset : (⋂ n, (K n).carrier : Set α) ⊆ U ∩ ⋂ n, f n := by
    

Proof state:
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Type u_4
inst✝² : TopologicalSpace α
inst✝¹ : T2Space α
inst✝ : LocallyCompactSpace α
f : ℕ → Set α
ho : ∀ (n : ℕ), IsOpen (f n)
hd : ∀ (n : ℕ), Dense (f n)
U : Set α
U_open : IsOpen U
U_nonempty : Set.Nonempty U
K₀ : PositiveCompacts α
hK₀ : ↑K₀ ⊆ U
K_next : ℕ → PositiveCompacts α → PositiveCompacts α
hK_next : ∀ (n : ℕ) (K : PositiveCompacts α), ↑(K_next n K) ⊆ f n ∩ interior ↑K
K : ℕ → PositiveCompacts α := fun n => Nat.recOn n K₀ K_next
hK_decreasing : ∀ (n : ℕ), (K (n + 1)).carrier ⊆ f n ∩ (K n).carrier
⊢ ⋂ n, (K n).carrier ⊆ U ∩ ⋂ n, f n

Next tactic:
intro x hx
/-- The second theorem states that locally compact spaces are Baire. -/
instance (priority := 100) BaireSpace.of_t2Space_locallyCompactSpace
    [TopologicalSpace α] [T2Space α] [LocallyCompactSpace α] : BaireSpace α := by
  constructor
  intro f ho hd
  /- To prove that an intersection of open dense subsets is dense, prove that its intersection
    with any open neighbourhood `U` is dense. Define recursively a decreasing sequence `K` of
    compact neighbourhoods: start with some compact neighbourhood inside `U`, then at each step,
    take its interior, intersect with `f n`, then choose a compact neighbourhood inside the
    intersection. -/
  apply dense_iff_inter_open.2
  intro U U_open U_nonempty
  rcases exists_positiveCompacts_subset U_open U_nonempty with ⟨K₀, hK₀⟩
  have : ∀ (n) (K : PositiveCompacts α), ∃ K' : PositiveCompacts α, ↑K' ⊆ f n ∩ interior K := by
    refine' fun n K => exists_positiveCompacts_subset ((ho n).inter isOpen_interior) _
    rw [inter_comm]
    exact (hd n).inter_open_nonempty _ isOpen_interior K.interior_nonempty
  choose K_next hK_next using this
  let K : ℕ → PositiveCompacts α := fun n => Nat.recOn n K₀ K_next
  -- This is a decreasing sequence of positive compacts contained in suitable open sets `f n`.
  have hK_decreasing : ∀ n : ℕ, ((K (n + 1)).carrier) ⊆ (f n ∩ (K n).carrier) :=
    fun n => (hK_next n (K n)).trans <| inter_subset_inter_right _ interior_subset
  -- Prove that ̀`⋂ n : ℕ, K n` is inside `U ∩ ⋂ n : ℕ, f n`.
  have hK_subset : (⋂ n, (K n).carrier : Set α) ⊆ U ∩ ⋂ n, f n := by
    intro x hx
    

Proof state:
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Type u_4
inst✝² : TopologicalSpace α
inst✝¹ : T2Space α
inst✝ : LocallyCompactSpace α
f : ℕ → Set α
ho : ∀ (n : ℕ), IsOpen (f n)
hd : ∀ (n : ℕ), Dense (f n)
U : Set α
U_open : IsOpen U
U_nonempty : Set.Nonempty U
K₀ : PositiveCompacts α
hK₀ : ↑K₀ ⊆ U
K_next : ℕ → PositiveCompacts α → PositiveCompacts α
hK_next : ∀ (n : ℕ) (K : PositiveCompacts α), ↑(K_next n K) ⊆ f n ∩ interior ↑K
K : ℕ → PositiveCompacts α := fun n => Nat.recOn n K₀ K_next
hK_decreasing : ∀ (n : ℕ), (K (n + 1)).carrier ⊆ f n ∩ (K n).carrier
x : α
hx : x ∈ ⋂ n, (K n).carrier
⊢ x ∈ U ∩ ⋂ n, f n

Next tactic:
simp only [mem_iInter] at hx
/-- The second theorem states that locally compact spaces are Baire. -/
instance (priority := 100) BaireSpace.of_t2Space_locallyCompactSpace
    [TopologicalSpace α] [T2Space α] [LocallyCompactSpace α] : BaireSpace α := by
  constructor
  intro f ho hd
  /- To prove that an intersection of open dense subsets is dense, prove that its intersection
    with any open neighbourhood `U` is dense. Define recursively a decreasing sequence `K` of
    compact neighbourhoods: start with some compact neighbourhood inside `U`, then at each step,
    take its interior, intersect with `f n`, then choose a compact neighbourhood inside the
    intersection. -/
  apply dense_iff_inter_open.2
  intro U U_open U_nonempty
  rcases exists_positiveCompacts_subset U_open U_nonempty with ⟨K₀, hK₀⟩
  have : ∀ (n) (K : PositiveCompacts α), ∃ K' : PositiveCompacts α, ↑K' ⊆ f n ∩ interior K := by
    refine' fun n K => exists_positiveCompacts_subset ((ho n).inter isOpen_interior) _
    rw [inter_comm]
    exact (hd n).inter_open_nonempty _ isOpen_interior K.interior_nonempty
  choose K_next hK_next using this
  let K : ℕ → PositiveCompacts α := fun n => Nat.recOn n K₀ K_next
  -- This is a decreasing sequence of positive compacts contained in suitable open sets `f n`.
  have hK_decreasing : ∀ n : ℕ, ((K (n + 1)).carrier) ⊆ (f n ∩ (K n).carrier) :=
    fun n => (hK_next n (K n)).trans <| inter_subset_inter_right _ interior_subset
  -- Prove that ̀`⋂ n : ℕ, K n` is inside `U ∩ ⋂ n : ℕ, f n`.
  have hK_subset : (⋂ n, (K n).carrier : Set α) ⊆ U ∩ ⋂ n, f n := by
    intro x hx
    simp only [mem_iInter] at hx
    

Proof state:
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Type u_4
inst✝² : TopologicalSpace α
inst✝¹ : T2Space α
inst✝ : LocallyCompactSpace α
f : ℕ → Set α
ho : ∀ (n : ℕ), IsOpen (f n)
hd : ∀ (n : ℕ), Dense (f n)
U : Set α
U_open : IsOpen U
U_nonempty : Set.Nonempty U
K₀ : PositiveCompacts α
hK₀ : ↑K₀ ⊆ U
K_next : ℕ → PositiveCompacts α → PositiveCompacts α
hK_next : ∀ (n : ℕ) (K : PositiveCompacts α), ↑(K_next n K) ⊆ f n ∩ interior ↑K
K : ℕ → PositiveCompacts α := fun n => Nat.recOn n K₀ K_next
hK_decreasing : ∀ (n : ℕ), (K (n + 1)).carrier ⊆ f n ∩ (K n).carrier
x : α
hx : ∀ (i : ℕ), x ∈ (Nat.rec K₀ K_next i).toCompacts.carrier
⊢ x ∈ U ∩ ⋂ n, f n

Next tactic:
simp only [mem_inter_iff, mem_inter] at hx ⊢
/-- The second theorem states that locally compact spaces are Baire. -/
instance (priority := 100) BaireSpace.of_t2Space_locallyCompactSpace
    [TopologicalSpace α] [T2Space α] [LocallyCompactSpace α] : BaireSpace α := by
  constructor
  intro f ho hd
  /- To prove that an intersection of open dense subsets is dense, prove that its intersection
    with any open neighbourhood `U` is dense. Define recursively a decreasing sequence `K` of
    compact neighbourhoods: start with some compact neighbourhood inside `U`, then at each step,
    take its interior, intersect with `f n`, then choose a compact neighbourhood inside the
    intersection. -/
  apply dense_iff_inter_open.2
  intro U U_open U_nonempty
  rcases exists_positiveCompacts_subset U_open U_nonempty with ⟨K₀, hK₀⟩
  have : ∀ (n) (K : PositiveCompacts α), ∃ K' : PositiveCompacts α, ↑K' ⊆ f n ∩ interior K := by
    refine' fun n K => exists_positiveCompacts_subset ((ho n).inter isOpen_interior) _
    rw [inter_comm]
    exact (hd n).inter_open_nonempty _ isOpen_interior K.interior_nonempty
  choose K_next hK_next using this
  let K : ℕ → PositiveCompacts α := fun n => Nat.recOn n K₀ K_next
  -- This is a decreasing sequence of positive compacts contained in suitable open sets `f n`.
  have hK_decreasing : ∀ n : ℕ, ((K (n + 1)).carrier) ⊆ (f n ∩ (K n).carrier) :=
    fun n => (hK_next n (K n)).trans <| inter_subset_inter_right _ interior_subset
  -- Prove that ̀`⋂ n : ℕ, K n` is inside `U ∩ ⋂ n : ℕ, f n`.
  have hK_subset : (⋂ n, (K n).carrier : Set α) ⊆ U ∩ ⋂ n, f n := by
    intro x hx
    simp only [mem_iInter] at hx
    simp only [mem_inter_iff, mem_inter] at hx ⊢
    

Proof state:
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Type u_4
inst✝² : TopologicalSpace α
inst✝¹ : T2Space α
inst✝ : LocallyCompactSpace α
f : ℕ → Set α
ho : ∀ (n : ℕ), IsOpen (f n)
hd : ∀ (n : ℕ), Dense (f n)
U : Set α
U_open : IsOpen U
U_nonempty : Set.Nonempty U
K₀ : PositiveCompacts α
hK₀ : ↑K₀ ⊆ U
K_next : ℕ → PositiveCompacts α → PositiveCompacts α
hK_next : ∀ (n : ℕ) (K : PositiveCompacts α), ↑(K_next n K) ⊆ f n ∩ interior ↑K
K : ℕ → PositiveCompacts α := fun n => Nat.recOn n K₀ K_next
hK_decreasing : ∀ (n : ℕ), (K (n + 1)).carrier ⊆ f n ∩ (K n).carrier
x : α
hx : ∀ (i : ℕ), x ∈ (Nat.rec K₀ K_next i).toCompacts.carrier
⊢ x ∈ U ∧ x ∈ ⋂ n, f n

Next tactic:
refine' ⟨hK₀ <| hx 0, _⟩
/-- The second theorem states that locally compact spaces are Baire. -/
instance (priority := 100) BaireSpace.of_t2Space_locallyCompactSpace
    [TopologicalSpace α] [T2Space α] [LocallyCompactSpace α] : BaireSpace α := by
  constructor
  intro f ho hd
  /- To prove that an intersection of open dense subsets is dense, prove that its intersection
    with any open neighbourhood `U` is dense. Define recursively a decreasing sequence `K` of
    compact neighbourhoods: start with some compact neighbourhood inside `U`, then at each step,
    take its interior, intersect with `f n`, then choose a compact neighbourhood inside the
    intersection. -/
  apply dense_iff_inter_open.2
  intro U U_open U_nonempty
  rcases exists_positiveCompacts_subset U_open U_nonempty with ⟨K₀, hK₀⟩
  have : ∀ (n) (K : PositiveCompacts α), ∃ K' : PositiveCompacts α, ↑K' ⊆ f n ∩ interior K := by
    refine' fun n K => exists_positiveCompacts_subset ((ho n).inter isOpen_interior) _
    rw [inter_comm]
    exact (hd n).inter_open_nonempty _ isOpen_interior K.interior_nonempty
  choose K_next hK_next using this
  let K : ℕ → PositiveCompacts α := fun n => Nat.recOn n K₀ K_next
  -- This is a decreasing sequence of positive compacts contained in suitable open sets `f n`.
  have hK_decreasing : ∀ n : ℕ, ((K (n + 1)).carrier) ⊆ (f n ∩ (K n).carrier) :=
    fun n => (hK_next n (K n)).trans <| inter_subset_inter_right _ interior_subset
  -- Prove that ̀`⋂ n : ℕ, K n` is inside `U ∩ ⋂ n : ℕ, f n`.
  have hK_subset : (⋂ n, (K n).carrier : Set α) ⊆ U ∩ ⋂ n, f n := by
    intro x hx
    simp only [mem_iInter] at hx
    simp only [mem_inter_iff, mem_inter] at hx ⊢
    refine' ⟨hK₀ <| hx 0, _⟩
    

Proof state:
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Type u_4
inst✝² : TopologicalSpace α
inst✝¹ : T2Space α
inst✝ : LocallyCompactSpace α
f : ℕ → Set α
ho : ∀ (n : ℕ), IsOpen (f n)
hd : ∀ (n : ℕ), Dense (f n)
U : Set α
U_open : IsOpen U
U_nonempty : Set.Nonempty U
K₀ : PositiveCompacts α
hK₀ : ↑K₀ ⊆ U
K_next : ℕ → PositiveCompacts α → PositiveCompacts α
hK_next : ∀ (n : ℕ) (K : PositiveCompacts α), ↑(K_next n K) ⊆ f n ∩ interior ↑K
K : ℕ → PositiveCompacts α := fun n => Nat.recOn n K₀ K_next
hK_decreasing : ∀ (n : ℕ), (K (n + 1)).carrier ⊆ f n ∩ (K n).carrier
x : α
hx : ∀ (i : ℕ), x ∈ (Nat.rec K₀ K_next i).toCompacts.carrier
⊢ x ∈ ⋂ n, f n

Next tactic:
simp only [mem_iInter]
/-- The second theorem states that locally compact spaces are Baire. -/
instance (priority := 100) BaireSpace.of_t2Space_locallyCompactSpace
    [TopologicalSpace α] [T2Space α] [LocallyCompactSpace α] : BaireSpace α := by
  constructor
  intro f ho hd
  /- To prove that an intersection of open dense subsets is dense, prove that its intersection
    with any open neighbourhood `U` is dense. Define recursively a decreasing sequence `K` of
    compact neighbourhoods: start with some compact neighbourhood inside `U`, then at each step,
    take its interior, intersect with `f n`, then choose a compact neighbourhood inside the
    intersection. -/
  apply dense_iff_inter_open.2
  intro U U_open U_nonempty
  rcases exists_positiveCompacts_subset U_open U_nonempty with ⟨K₀, hK₀⟩
  have : ∀ (n) (K : PositiveCompacts α), ∃ K' : PositiveCompacts α, ↑K' ⊆ f n ∩ interior K := by
    refine' fun n K => exists_positiveCompacts_subset ((ho n).inter isOpen_interior) _
    rw [inter_comm]
    exact (hd n).inter_open_nonempty _ isOpen_interior K.interior_nonempty
  choose K_next hK_next using this
  let K : ℕ → PositiveCompacts α := fun n => Nat.recOn n K₀ K_next
  -- This is a decreasing sequence of positive compacts contained in suitable open sets `f n`.
  have hK_decreasing : ∀ n : ℕ, ((K (n + 1)).carrier) ⊆ (f n ∩ (K n).carrier) :=
    fun n => (hK_next n (K n)).trans <| inter_subset_inter_right _ interior_subset
  -- Prove that ̀`⋂ n : ℕ, K n` is inside `U ∩ ⋂ n : ℕ, f n`.
  have hK_subset : (⋂ n, (K n).carrier : Set α) ⊆ U ∩ ⋂ n, f n := by
    intro x hx
    simp only [mem_iInter] at hx
    simp only [mem_inter_iff, mem_inter] at hx ⊢
    refine' ⟨hK₀ <| hx 0, _⟩
    simp only [mem_iInter]
    

Proof state:
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Type u_4
inst✝² : TopologicalSpace α
inst✝¹ : T2Space α
inst✝ : LocallyCompactSpace α
f : ℕ → Set α
ho : ∀ (n : ℕ), IsOpen (f n)
hd : ∀ (n : ℕ), Dense (f n)
U : Set α
U_open : IsOpen U
U_nonempty : Set.Nonempty U
K₀ : PositiveCompacts α
hK₀ : ↑K₀ ⊆ U
K_next : ℕ → PositiveCompacts α → PositiveCompacts α
hK_next : ∀ (n : ℕ) (K : PositiveCompacts α), ↑(K_next n K) ⊆ f n ∩ interior ↑K
K : ℕ → PositiveCompacts α := fun n => Nat.recOn n K₀ K_next
hK_decreasing : ∀ (n : ℕ), (K (n + 1)).carrier ⊆ f n ∩ (K n).carrier
x : α
hx : ∀ (i : ℕ), x ∈ (Nat.rec K₀ K_next i).toCompacts.carrier
⊢ ∀ (i : ℕ), x ∈ f i

Next tactic:
exact fun n => (hK_decreasing n (hx (n + 1))).1
/-- The second theorem states that locally compact spaces are Baire. -/
instance (priority := 100) BaireSpace.of_t2Space_locallyCompactSpace
    [TopologicalSpace α] [T2Space α] [LocallyCompactSpace α] : BaireSpace α := by
  constructor
  intro f ho hd
  /- To prove that an intersection of open dense subsets is dense, prove that its intersection
    with any open neighbourhood `U` is dense. Define recursively a decreasing sequence `K` of
    compact neighbourhoods: start with some compact neighbourhood inside `U`, then at each step,
    take its interior, intersect with `f n`, then choose a compact neighbourhood inside the
    intersection. -/
  apply dense_iff_inter_open.2
  intro U U_open U_nonempty
  rcases exists_positiveCompacts_subset U_open U_nonempty with ⟨K₀, hK₀⟩
  have : ∀ (n) (K : PositiveCompacts α), ∃ K' : PositiveCompacts α, ↑K' ⊆ f n ∩ interior K := by
    refine' fun n K => exists_positiveCompacts_subset ((ho n).inter isOpen_interior) _
    rw [inter_comm]
    exact (hd n).inter_open_nonempty _ isOpen_interior K.interior_nonempty
  choose K_next hK_next using this
  let K : ℕ → PositiveCompacts α := fun n => Nat.recOn n K₀ K_next
  -- This is a decreasing sequence of positive compacts contained in suitable open sets `f n`.
  have hK_decreasing : ∀ n : ℕ, ((K (n + 1)).carrier) ⊆ (f n ∩ (K n).carrier) :=
    fun n => (hK_next n (K n)).trans <| inter_subset_inter_right _ interior_subset
  -- Prove that ̀`⋂ n : ℕ, K n` is inside `U ∩ ⋂ n : ℕ, f n`.
  have hK_subset : (⋂ n, (K n).carrier : Set α) ⊆ U ∩ ⋂ n, f n := by
    intro x hx
    simp only [mem_iInter] at hx
    simp only [mem_inter_iff, mem_inter] at hx ⊢
    refine' ⟨hK₀ <| hx 0, _⟩
    simp only [mem_iInter]
    exact fun n => (hK_decreasing n (hx (n + 1))).1
  /- Prove that `⋂ n : ℕ, K n` is not empty, as an intersection of a decreasing sequence
    of nonempty compact subsets. -/
  

Proof state:
case baire_property.intro
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Type u_4
inst✝² : TopologicalSpace α
inst✝¹ : T2Space α
inst✝ : LocallyCompactSpace α
f : ℕ → Set α
ho : ∀ (n : ℕ), IsOpen (f n)
hd : ∀ (n : ℕ), Dense (f n)
U : Set α
U_open : IsOpen U
U_nonempty : Set.Nonempty U
K₀ : PositiveCompacts α
hK₀ : ↑K₀ ⊆ U
K_next : ℕ → PositiveCompacts α → PositiveCompacts α
hK_next : ∀ (n : ℕ) (K : PositiveCompacts α), ↑(K_next n K) ⊆ f n ∩ interior ↑K
K : ℕ → PositiveCompacts α := fun n => Nat.recOn n K₀ K_next
hK_decreasing : ∀ (n : ℕ), (K (n + 1)).carrier ⊆ f n ∩ (K n).carrier
hK_subset : ⋂ n, (K n).carrier ⊆ U ∩ ⋂ n, f n
⊢ Set.Nonempty (U ∩ ⋂ n, f n)

Next tactic:
have hK_nonempty : (⋂ n, (K n).carrier : Set α).Nonempty :=
    IsCompact.nonempty_iInter_of_sequence_nonempty_compact_closed _
      (fun n => (hK_decreasing n).trans (inter_subset_right _ _)) (fun n => (K n).nonempty)
      (K 0).isCompact fun n => (K n).isCompact.isClosed