/-- 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))
    

Proof state:
case intro.intro.intro.intro.refine'_1
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
⊢ 0 < min (min (δ / 2) r) (B (n + 1))
case intro.intro.intro.intro.refine'_2
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
⊢ min (min (δ / 2) r) (B (n + 1)) ≤ B (n + 1)
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 lt_min (lt_min (ENNReal.half_pos δpos) rpos) (Bpos (n + 1))