l3lab/ntp-mathlib huggingface dataset:
/-- Baire theorems asserts that various topological spaces have the Baire property.
Two versions of these theorems are given.
The first states that complete `PseudoEMetricSpace`s are Baire. -/
instance (priority := 100) BaireSpace.of_pseudoEMetricSpace_completeSpace [PseudoEMetricSpace α]
    [CompleteSpace α] : BaireSpace α := by
  refine' ⟨fun f ho hd => _⟩
  let B : ℕ → ℝ≥0∞ := fun n => 1 / 2 ^ n
  have Bpos : ∀ n, 0 < B n := by
    intro n
    simp only [one_div, one_mul, ENNReal.inv_pos]
    exact pow_ne_top two_ne_top
  /- Translate the density assumption into two functions `center` and `radius` associating
    to any n, x, δ, δpos a center and a positive radius such that
    `closedBall center radius` is included both in `f n` and in `closedBall x δ`.
    We can also require `radius ≤ (1/2)^(n+1)`, to ensure we get a Cauchy sequence later. -/
  have : ∀ n x δ, δ ≠ 0 → ∃ y r, 0 < r ∧ r ≤ B (n + 1) ∧ closedBall y r ⊆ closedBall x δ ∩ f n := by
    intro n x δ δpos
    have : x ∈ closure (f n) := hd n x
    rcases EMetric.mem_closure_iff.1 this (δ / 2) (ENNReal.half_pos δpos) with ⟨y, ys, xy⟩
    rw [edist_comm] at xy
    obtain ⟨r, rpos, hr⟩ : ∃ r > 0, closedBall y r ⊆ f n :=
      nhds_basis_closed_eball.mem_iff.1 (isOpen_iff_mem_nhds.1 (ho n) y ys)
    refine' ⟨y, min (min (δ / 2) r) (B (n + 1)), _, _, fun z hz => ⟨_, _⟩⟩
    show 0 < min (min (δ / 2) r) (B (n + 1))
    exact lt_min (lt_min (ENNReal.half_pos δpos) rpos) (Bpos (n + 1))
    show min (min (δ / 2) r) (B (n + 1)) ≤ B (n + 1)
    exact min_le_right _ _
    show z ∈ closedBall x δ
    

Proof state:
case intro.intro.intro.intro.refine'_3
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Type u_4
inst✝¹ : PseudoEMetricSpace α
inst✝ : CompleteSpace α
f : ℕ → Set α
ho : ∀ (n : ℕ), IsOpen (f n)
hd : ∀ (n : ℕ), Dense (f n)
B : ℕ → ℝ≥0∞ := fun n => 1 / 2 ^ n
Bpos : ∀ (n : ℕ), 0 < B n
n : ℕ
x : α
δ : ℝ≥0∞
δpos : δ ≠ 0
this : x ∈ closure (f n)
y : α
ys : y ∈ f n
xy : edist y x < δ / 2
r : ℝ≥0∞
rpos : r > 0
hr : closedBall y r ⊆ f n
z : α
hz : z ∈ closedBall y (min (min (δ / 2) r) (B (n + 1)))
⊢ z ∈ closedBall x δ
case intro.intro.intro.intro.refine'_4
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Type u_4
inst✝¹ : PseudoEMetricSpace α
inst✝ : CompleteSpace α
f : ℕ → Set α
ho : ∀ (n : ℕ), IsOpen (f n)
hd : ∀ (n : ℕ), Dense (f n)
B : ℕ → ℝ≥0∞ := fun n => 1 / 2 ^ n
Bpos : ∀ (n : ℕ), 0 < B n
n : ℕ
x : α
δ : ℝ≥0∞
δpos : δ ≠ 0
this : x ∈ closure (f n)
y : α
ys : y ∈ f n
xy : edist y x < δ / 2
r : ℝ≥0∞
rpos : r > 0
hr : closedBall y r ⊆ f n
z : α
hz : z ∈ closedBall y (min (min (δ / 2) r) (B (n + 1)))
⊢ z ∈ f n

Next tactic:
exact
      calc
        edist z x ≤ edist z y + edist y x := edist_triangle _ _ _
        _ ≤ min (min (δ / 2) r) (B (n + 1)) + δ / 2 := (add_le_add hz (le_of_lt xy))
        _ ≤ δ / 2 + δ / 2 := (add_le_add (le_trans (min_le_left _ _) (min_le_left _ _)) le_rfl)
        _ = δ := ENNReal.add_halves δ
/-- Baire theorems asserts that various topological spaces have the Baire property.
Two versions of these theorems are given.
The first states that complete `PseudoEMetricSpace`s are Baire. -/
instance (priority := 100) BaireSpace.of_pseudoEMetricSpace_completeSpace [PseudoEMetricSpace α]
    [CompleteSpace α] : BaireSpace α := by
  refine' ⟨fun f ho hd => _⟩
  let B : ℕ → ℝ≥0∞ := fun n => 1 / 2 ^ n
  have Bpos : ∀ n, 0 < B n := by
    intro n
    simp only [one_div, one_mul, ENNReal.inv_pos]
    exact pow_ne_top two_ne_top
  /- Translate the density assumption into two functions `center` and `radius` associating
    to any n, x, δ, δpos a center and a positive radius such that
    `closedBall center radius` is included both in `f n` and in `closedBall x δ`.
    We can also require `radius ≤ (1/2)^(n+1)`, to ensure we get a Cauchy sequence later. -/
  have : ∀ n x δ, δ ≠ 0 → ∃ y r, 0 < r ∧ r ≤ B (n + 1) ∧ closedBall y r ⊆ closedBall x δ ∩ f n := by
    intro n x δ δpos
    have : x ∈ closure (f n) := hd n x
    rcases EMetric.mem_closure_iff.1 this (δ / 2) (ENNReal.half_pos δpos) with ⟨y, ys, xy⟩
    rw [edist_comm] at xy
    obtain ⟨r, rpos, hr⟩ : ∃ r > 0, closedBall y r ⊆ f n :=
      nhds_basis_closed_eball.mem_iff.1 (isOpen_iff_mem_nhds.1 (ho n) y ys)
    refine' ⟨y, min (min (δ / 2) r) (B (n + 1)), _, _, fun z hz => ⟨_, _⟩⟩
    show 0 < min (min (δ / 2) r) (B (n + 1))
    exact lt_min (lt_min (ENNReal.half_pos δpos) rpos) (Bpos (n + 1))
    show min (min (δ / 2) r) (B (n + 1)) ≤ B (n + 1)
    exact min_le_right _ _
    show z ∈ closedBall x δ
    exact
      calc
        edist z x ≤ edist z y + edist y x := edist_triangle _ _ _
        _ ≤ min (min (δ / 2) r) (B (n + 1)) + δ / 2 := (add_le_add hz (le_of_lt xy))
        _ ≤ δ / 2 + δ / 2 := (add_le_add (le_trans (min_le_left _ _) (min_le_left _ _)) le_rfl)
        _ = δ := ENNReal.add_halves δ
    

Proof state:
case intro.intro.intro.intro.refine'_4
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Type u_4
inst✝¹ : PseudoEMetricSpace α
inst✝ : CompleteSpace α
f : ℕ → Set α
ho : ∀ (n : ℕ), IsOpen (f n)
hd : ∀ (n : ℕ), Dense (f n)
B : ℕ → ℝ≥0∞ := fun n => 1 / 2 ^ n
Bpos : ∀ (n : ℕ), 0 < B n
n : ℕ
x : α
δ : ℝ≥0∞
δpos : δ ≠ 0
this : x ∈ closure (f n)
y : α
ys : y ∈ f n
xy : edist y x < δ / 2
r : ℝ≥0∞
rpos : r > 0
hr : closedBall y r ⊆ f n
z : α
hz : z ∈ closedBall y (min (min (δ / 2) r) (B (n + 1)))
⊢ z ∈ f n

Next tactic:
show z ∈ f n
/-- Baire theorems asserts that various topological spaces have the Baire property.
Two versions of these theorems are given.
The first states that complete `PseudoEMetricSpace`s are Baire. -/
instance (priority := 100) BaireSpace.of_pseudoEMetricSpace_completeSpace [PseudoEMetricSpace α]
    [CompleteSpace α] : BaireSpace α := by
  refine' ⟨fun f ho hd => _⟩
  let B : ℕ → ℝ≥0∞ := fun n => 1 / 2 ^ n
  have Bpos : ∀ n, 0 < B n := by
    intro n
    simp only [one_div, one_mul, ENNReal.inv_pos]
    exact pow_ne_top two_ne_top
  /- Translate the density assumption into two functions `center` and `radius` associating
    to any n, x, δ, δpos a center and a positive radius such that
    `closedBall center radius` is included both in `f n` and in `closedBall x δ`.
    We can also require `radius ≤ (1/2)^(n+1)`, to ensure we get a Cauchy sequence later. -/
  have : ∀ n x δ, δ ≠ 0 → ∃ y r, 0 < r ∧ r ≤ B (n + 1) ∧ closedBall y r ⊆ closedBall x δ ∩ f n := by
    intro n x δ δpos
    have : x ∈ closure (f n) := hd n x
    rcases EMetric.mem_closure_iff.1 this (δ / 2) (ENNReal.half_pos δpos) with ⟨y, ys, xy⟩
    rw [edist_comm] at xy
    obtain ⟨r, rpos, hr⟩ : ∃ r > 0, closedBall y r ⊆ f n :=
      nhds_basis_closed_eball.mem_iff.1 (isOpen_iff_mem_nhds.1 (ho n) y ys)
    refine' ⟨y, min (min (δ / 2) r) (B (n + 1)), _, _, fun z hz => ⟨_, _⟩⟩
    show 0 < min (min (δ / 2) r) (B (n + 1))
    exact lt_min (lt_min (ENNReal.half_pos δpos) rpos) (Bpos (n + 1))
    show min (min (δ / 2) r) (B (n + 1)) ≤ B (n + 1)
    exact min_le_right _ _
    show z ∈ closedBall x δ
    exact
      calc
        edist z x ≤ edist z y + edist y x := edist_triangle _ _ _
        _ ≤ min (min (δ / 2) r) (B (n + 1)) + δ / 2 := (add_le_add hz (le_of_lt xy))
        _ ≤ δ / 2 + δ / 2 := (add_le_add (le_trans (min_le_left _ _) (min_le_left _ _)) le_rfl)
        _ = δ := ENNReal.add_halves δ
    show z ∈ f n
    

Proof state:
case intro.intro.intro.intro.refine'_4
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Type u_4
inst✝¹ : PseudoEMetricSpace α
inst✝ : CompleteSpace α
f : ℕ → Set α
ho : ∀ (n : ℕ), IsOpen (f n)
hd : ∀ (n : ℕ), Dense (f n)
B : ℕ → ℝ≥0∞ := fun n => 1 / 2 ^ n
Bpos : ∀ (n : ℕ), 0 < B n
n : ℕ
x : α
δ : ℝ≥0∞
δpos : δ ≠ 0
this : x ∈ closure (f n)
y : α
ys : y ∈ f n
xy : edist y x < δ / 2
r : ℝ≥0∞
rpos : r > 0
hr : closedBall y r ⊆ f n
z : α
hz : z ∈ closedBall y (min (min (δ / 2) r) (B (n + 1)))
⊢ z ∈ f n

Next tactic:
exact hr (calc
      edist z y ≤ min (min (δ / 2) r) (B (n + 1)) := hz
      _ ≤ r := le_trans (min_le_left _ _) (min_le_right _ _))
/-- Baire theorems asserts that various topological spaces have the Baire property.
Two versions of these theorems are given.
The first states that complete `PseudoEMetricSpace`s are Baire. -/
instance (priority := 100) BaireSpace.of_pseudoEMetricSpace_completeSpace [PseudoEMetricSpace α]
    [CompleteSpace α] : BaireSpace α := by
  refine' ⟨fun f ho hd => _⟩
  let B : ℕ → ℝ≥0∞ := fun n => 1 / 2 ^ n
  have Bpos : ∀ n, 0 < B n := by
    intro n
    simp only [one_div, one_mul, ENNReal.inv_pos]
    exact pow_ne_top two_ne_top
  /- Translate the density assumption into two functions `center` and `radius` associating
    to any n, x, δ, δpos a center and a positive radius such that
    `closedBall center radius` is included both in `f n` and in `closedBall x δ`.
    We can also require `radius ≤ (1/2)^(n+1)`, to ensure we get a Cauchy sequence later. -/
  have : ∀ n x δ, δ ≠ 0 → ∃ y r, 0 < r ∧ r ≤ B (n + 1) ∧ closedBall y r ⊆ closedBall x δ ∩ f n := by
    intro n x δ δpos
    have : x ∈ closure (f n) := hd n x
    rcases EMetric.mem_closure_iff.1 this (δ / 2) (ENNReal.half_pos δpos) with ⟨y, ys, xy⟩
    rw [edist_comm] at xy
    obtain ⟨r, rpos, hr⟩ : ∃ r > 0, closedBall y r ⊆ f n :=
      nhds_basis_closed_eball.mem_iff.1 (isOpen_iff_mem_nhds.1 (ho n) y ys)
    refine' ⟨y, min (min (δ / 2) r) (B (n + 1)), _, _, fun z hz => ⟨_, _⟩⟩
    show 0 < min (min (δ / 2) r) (B (n + 1))
    exact lt_min (lt_min (ENNReal.half_pos δpos) rpos) (Bpos (n + 1))
    show min (min (δ / 2) r) (B (n + 1)) ≤ B (n + 1)
    exact min_le_right _ _
    show z ∈ closedBall x δ
    exact
      calc
        edist z x ≤ edist z y + edist y x := edist_triangle _ _ _
        _ ≤ min (min (δ / 2) r) (B (n + 1)) + δ / 2 := (add_le_add hz (le_of_lt xy))
        _ ≤ δ / 2 + δ / 2 := (add_le_add (le_trans (min_le_left _ _) (min_le_left _ _)) le_rfl)
        _ = δ := ENNReal.add_halves δ
    show z ∈ f n
    exact hr (calc
      edist z y ≤ min (min (δ / 2) r) (B (n + 1)) := hz
      _ ≤ r := le_trans (min_le_left _ _) (min_le_right _ _))
  

Proof state:
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Type u_4
inst✝¹ : PseudoEMetricSpace α
inst✝ : CompleteSpace α
f : ℕ → Set α
ho : ∀ (n : ℕ), IsOpen (f n)
hd : ∀ (n : ℕ), Dense (f n)
B : ℕ → ℝ≥0∞ := fun n => 1 / 2 ^ n
Bpos : ∀ (n : ℕ), 0 < B n
this : ∀ (n : ℕ) (x : α) (δ : ℝ≥0∞), δ ≠ 0 → ∃ y r, 0 < r ∧ r ≤ B (n + 1) ∧ closedBall y r ⊆ closedBall x δ ∩ f n
⊢ Dense (⋂ n, f n)

Next tactic:
choose! center radius Hpos HB Hball using this
/-- Baire theorems asserts that various topological spaces have the Baire property.
Two versions of these theorems are given.
The first states that complete `PseudoEMetricSpace`s are Baire. -/
instance (priority := 100) BaireSpace.of_pseudoEMetricSpace_completeSpace [PseudoEMetricSpace α]
    [CompleteSpace α] : BaireSpace α := by
  refine' ⟨fun f ho hd => _⟩
  let B : ℕ → ℝ≥0∞ := fun n => 1 / 2 ^ n
  have Bpos : ∀ n, 0 < B n := by
    intro n
    simp only [one_div, one_mul, ENNReal.inv_pos]
    exact pow_ne_top two_ne_top
  /- Translate the density assumption into two functions `center` and `radius` associating
    to any n, x, δ, δpos a center and a positive radius such that
    `closedBall center radius` is included both in `f n` and in `closedBall x δ`.
    We can also require `radius ≤ (1/2)^(n+1)`, to ensure we get a Cauchy sequence later. -/
  have : ∀ n x δ, δ ≠ 0 → ∃ y r, 0 < r ∧ r ≤ B (n + 1) ∧ closedBall y r ⊆ closedBall x δ ∩ f n := by
    intro n x δ δpos
    have : x ∈ closure (f n) := hd n x
    rcases EMetric.mem_closure_iff.1 this (δ / 2) (ENNReal.half_pos δpos) with ⟨y, ys, xy⟩
    rw [edist_comm] at xy
    obtain ⟨r, rpos, hr⟩ : ∃ r > 0, closedBall y r ⊆ f n :=
      nhds_basis_closed_eball.mem_iff.1 (isOpen_iff_mem_nhds.1 (ho n) y ys)
    refine' ⟨y, min (min (δ / 2) r) (B (n + 1)), _, _, fun z hz => ⟨_, _⟩⟩
    show 0 < min (min (δ / 2) r) (B (n + 1))
    exact lt_min (lt_min (ENNReal.half_pos δpos) rpos) (Bpos (n + 1))
    show min (min (δ / 2) r) (B (n + 1)) ≤ B (n + 1)
    exact min_le_right _ _
    show z ∈ closedBall x δ
    exact
      calc
        edist z x ≤ edist z y + edist y x := edist_triangle _ _ _
        _ ≤ min (min (δ / 2) r) (B (n + 1)) + δ / 2 := (add_le_add hz (le_of_lt xy))
        _ ≤ δ / 2 + δ / 2 := (add_le_add (le_trans (min_le_left _ _) (min_le_left _ _)) le_rfl)
        _ = δ := ENNReal.add_halves δ
    show z ∈ f n
    exact hr (calc
      edist z y ≤ min (min (δ / 2) r) (B (n + 1)) := hz
      _ ≤ r := le_trans (min_le_left _ _) (min_le_right _ _))
  choose! center radius Hpos HB Hball using this
  

Proof state:
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Type u_4
inst✝¹ : PseudoEMetricSpace α
inst✝ : CompleteSpace α
f : ℕ → Set α
ho : ∀ (n : ℕ), IsOpen (f n)
hd : ∀ (n : ℕ), Dense (f n)
B : ℕ → ℝ≥0∞ := fun n => 1 / 2 ^ n
Bpos : ∀ (n : ℕ), 0 < B n
center : ℕ → α → ℝ≥0∞ → α
radius : ℕ → α → ℝ≥0∞ → ℝ≥0∞
Hpos : ∀ (n : ℕ) (x : α) (δ : ℝ≥0∞), δ ≠ 0 → 0 < radius n x δ
HB : ∀ (n : ℕ) (x : α) (δ : ℝ≥0∞), δ ≠ 0 → radius n x δ ≤ B (n + 1)
Hball : ∀ (n : ℕ) (x : α) (δ : ℝ≥0∞), δ ≠ 0 → closedBall (center n x δ) (radius n x δ) ⊆ closedBall x δ ∩ f n
⊢ Dense (⋂ n, f n)

Next tactic:
refine' fun x => (mem_closure_iff_nhds_basis nhds_basis_closed_eball).2 fun ε εpos => _
/-- 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
  

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
⊢ BaireSpace α

Next tactic:
constructor
/-- 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
  

Proof state:
case baire_property
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), (∀ (n : ℕ), IsOpen (f n)) → (∀ (n : ℕ), Dense (f n)) → Dense (⋂ n, f n)

Next tactic:
intro f ho hd
/-- 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. -/
  

Proof state:
case baire_property
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)
⊢ Dense (⋂ n, f n)

Next tactic:
apply dense_iff_inter_open.2
/-- 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
  

Proof state:
case baire_property
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), IsOpen U → Set.Nonempty U → Set.Nonempty (U ∩ ⋂ n, f n)

Next tactic:
intro U U_open U_nonempty
/-- 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
  

Proof state:
case baire_property
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
⊢ Set.Nonempty (U ∩ ⋂ n, f n)

Next tactic:
rcases exists_positiveCompacts_subset U_open U_nonempty with ⟨K₀, hK₀⟩
/-- 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⟩
  

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
⊢ Set.Nonempty (U ∩ ⋂ n, f n)

Next tactic:
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
/-- 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
    

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
⊢ ∀ (n : ℕ) (K : PositiveCompacts α), ∃ K', ↑K' ⊆ f n ∩ interior ↑K

Next tactic:
refine' fun n K => exists_positiveCompacts_subset ((ho n).inter isOpen_interior) _
/-- 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) _
    

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
n : ℕ
K : PositiveCompacts α
⊢ Set.Nonempty (f n ∩ interior ↑K)

Next tactic:
rw [inter_comm]
/-- 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]
    

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
n : ℕ
K : PositiveCompacts α
⊢ Set.Nonempty (interior ↑K ∩ f n)

Next tactic:
exact (hd n).inter_open_nonempty _ isOpen_interior K.interior_nonempty
/-- 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
  

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
this : ∀ (n : ℕ) (K : PositiveCompacts α), ∃ K', ↑K' ⊆ f n ∩ interior ↑K
⊢ Set.Nonempty (U ∩ ⋂ n, f n)

Next tactic:
choose K_next hK_next using this