/-- If `X` is a uniform space with countably generated uniformity filter, there exists a
`PseudoMetricSpace` structure compatible with the `UniformSpace` structure. Use
`UniformSpace.pseudoMetricSpace` or `UniformSpace.metricSpace` instead. -/
protected theorem UniformSpace.metrizable_uniformity (X : Type*) [UniformSpace X]
    [IsCountablyGenerated (𝓤 X)] : ∃ I : PseudoMetricSpace X, I.toUniformSpace = ‹_› := by
  classical
  /- Choose a fast decreasing antitone basis `U : ℕ → set (X × X)` of the uniformity filter `𝓤 X`.
    Define `d x y : ℝ≥0` to be `(1 / 2) ^ n`, where `n` is the minimal index of `U n` that
    separates `x` and `y`: `(x, y) ∉ U n`, or `0` if `x` is not separated from `y`. This function
    satisfies the assumptions of `PseudoMetricSpace.ofPreNNDist` and
    `PseudoMetricSpace.le_two_mul_dist_ofPreNNDist`, hence the distance given by the former pseudo
    metric space structure is Lipschitz equivalent to the `d`. Thus the uniformities generated by
    `d` and `dist` are equal. Since the former uniformity is equal to `𝓤 X`, the latter is equal to
    `𝓤 X` as well. -/
  obtain ⟨U, hU_symm, hU_comp, hB⟩ :
    ∃ U : ℕ → Set (X × X),
      (∀ n, SymmetricRel (U n)) ∧
        (∀ ⦃m n⦄, m < n → U n ○ (U n ○ U n) ⊆ U m) ∧ (𝓤 X).HasAntitoneBasis U := by
    rcases UniformSpace.has_seq_basis X with ⟨V, hB, hV_symm⟩
    rcases hB.subbasis_with_rel fun m =>
        hB.tendsto_smallSets.eventually
          (eventually_uniformity_iterate_comp_subset (hB.mem m) 2) with
      ⟨φ, -, hφ_comp, hφB⟩
    exact ⟨V ∘ φ, fun n => hV_symm _, hφ_comp, hφB⟩
  letI := UniformSpace.separationSetoid X
  set d : X → X → ℝ≥0 := fun x y => if h : ∃ n, (x, y) ∉ U n then (1 / 2) ^ Nat.find h else 0
  have hd₀ : ∀ {x y}, d x y = 0 ↔ x ≈ y := by
    intro x y
    refine' Iff.trans _ hB.mem_separationRel.symm
    simp only [true_imp_iff]
    split_ifs with h
    · rw [← not_forall] at h
      simp [h, pow_eq_zero_iff']
    · simpa only [not_exists, Classical.not_not, eq_self_iff_true, true_iff_iff] using h
  have hd_symm : ∀ x y, d x y = d y x := by
    intro x y
    simp only [@SymmetricRel.mk_mem_comm _ _ (hU_symm _) x y]
  have hr : (1 / 2 : ℝ≥0) ∈ Ioo (0 : ℝ≥0) 1 := ⟨half_pos one_pos, NNReal.half_lt_self one_ne_zero⟩
  letI I := PseudoMetricSpace.ofPreNNDist d (fun x => hd₀.2 (Setoid.refl _)) hd_symm
  have hdist_le : ∀ x y, dist x y ≤ d x y := PseudoMetricSpace.dist_ofPreNNDist_le _ _ _
  have hle_d : ∀ {x y : X} {n : ℕ}, (1 / 2) ^ n ≤ d x y ↔ (x, y) ∉ U n := by
    intro x y n
    dsimp only []
    split_ifs with h
    · rw [(pow_right_strictAnti hr.1 hr.2).le_iff_le, Nat.find_le_iff]
      exact ⟨fun ⟨m, hmn, hm⟩ hn => hm (hB.antitone hmn hn), fun h => ⟨n, le_rfl, h⟩⟩
    · push_neg at h
      simp only [h, not_true, (pow_pos hr.1 _).not_le]
  have hd_le : ∀ x y, ↑(d x y) ≤ 2 * dist x y := by
    refine' PseudoMetricSpace.le_two_mul_dist_ofPreNNDist _ _ _ fun x₁ x₂ x₃ x₄ => _
    

Proof state:
X✝ : Type u_1
X : Type u_2
inst✝¹ : UniformSpace X
inst✝ : IsCountablyGenerated (𝓤 X)
U : ℕ → Set (X × X)
hU_symm : ∀ (n : ℕ), SymmetricRel (U n)
hU_comp : ∀ ⦃m n : ℕ⦄, m < n → U n ○ (U n ○ U n) ⊆ U m
hB : HasAntitoneBasis (𝓤 X) U
this : Setoid X := separationSetoid X
d : X → X → ℝ≥0 := fun x y => if h : ∃ n, (x, y) ∉ U n then (1 / 2) ^ Nat.find h else 0
hd₀ : ∀ {x y : X}, d x y = 0 ↔ x ≈ y
hd_symm : ∀ (x y : X), d x y = d y x
hr : 1 / 2 ∈ Ioo 0 1
I : PseudoMetricSpace X := PseudoMetricSpace.ofPreNNDist d (_ : ∀ (x : X), d x x = 0) hd_symm
hdist_le : ∀ (x y : X), dist x y ≤ ↑(d x y)
hle_d : ∀ {x y : X} {n : ℕ}, (1 / 2) ^ n ≤ d x y ↔ (x, y) ∉ U n
x₁ x₂ x₃ x₄ : X
⊢ d x₁ x₄ ≤ 2 * max (d x₁ x₂) (max (d x₂ x₃) (d x₃ x₄))

Next tactic:
by_cases H : ∃ n, (x₁, x₄) ∉ U n