l3lab/ntp-mathlib huggingface dataset:
/-- 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⟩
  

Proof state:
case intro.intro.intro
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
⊢ ∃ I, PseudoMetricSpace.toUniformSpace = inst✝¹

Next tactic:
letI := UniformSpace.separationSetoid X
/-- 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
  

Proof state:
case intro.intro.intro
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
⊢ ∃ I, PseudoMetricSpace.toUniformSpace = inst✝¹

Next tactic:
set d : X → X → ℝ≥0 := fun x y => if h : ∃ n, (x, y) ∉ U n then (1 / 2) ^ Nat.find h else 0
/-- 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
  

Proof state:
case intro.intro.intro
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
⊢ ∃ I, PseudoMetricSpace.toUniformSpace = inst✝¹

Next tactic:
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
/-- 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
    

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
⊢ ∀ {x y : X}, d x y = 0 ↔ x ≈ y

Next tactic:
intro x y
/-- 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
    

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
x y : X
⊢ d x y = 0 ↔ x ≈ y

Next tactic:
refine' Iff.trans _ hB.mem_separationRel.symm
/-- 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
    

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
x y : X
⊢ d x y = 0 ↔ ∀ (i : ℕ), True → (x, y) ∈ U i

Next tactic:
simp only [true_imp_iff]
/-- 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]
    

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
x y : X
⊢ (if h : ∃ n, (x, y) ∉ U n then (1 / 2) ^ Nat.find h else 0) = 0 ↔ ∀ (i : ℕ), (x, y) ∈ U i

Next tactic:
split_ifs with h
/-- 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
    · 

Proof state:
case pos
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
x y : X
h : ∃ n, (x, y) ∉ U n
⊢ (1 / 2) ^ Nat.find h = 0 ↔ ∀ (i : ℕ), (x, y) ∈ U i

Next tactic:
rw [← not_forall] at h
/-- 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
      

Proof state:
case pos
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
x y : X
h✝ : ∃ n, (x, y) ∉ U n
h : ¬∀ (x_1 : ℕ), (x, y) ∈ U x_1
⊢ (1 / 2) ^ Nat.find h✝ = 0 ↔ ∀ (i : ℕ), (x, y) ∈ U i

Next tactic:
simp [h, pow_eq_zero_iff']
/-- 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']
    · 

Proof state:
case neg
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
x y : X
h : ¬∃ n, (x, y) ∉ U n
⊢ 0 = 0 ↔ ∀ (i : ℕ), (x, y) ∈ U i

Next tactic:
simpa only [not_exists, Classical.not_not, eq_self_iff_true, true_iff_iff] using h