awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.Topology.Metrizable.CompletelyMetrizable (goal)
Goal:
X : Type u_3
ht : TopologicalSpace X
h : IsCompletelyPseudoMetrizableSpace X
⊢ CompleteSpace X
have h₁ : completelyPseudoMetrizableMetric X =
  Exists.choose (p := fun (m : PseudoMetricSpace X) ↦ UniformSpace.toTopologicalSpace = ht ∧ CompleteSpace X) sorry
Tactic: convert! h.complete.choose_spec.2
-- Lean 4 proof step from Mathlib.Topology.Metrizable.CompletelyMetrizable (goal)
Goal:
X : Type u_1
Y : Type u_2
inst : TopologicalSpace X
inst_1 : TopologicalSpace Y
inst_2 : TopologicalSpace.IsCompletelyPseudoMetrizableSpace Y
f : X → Y
hf : IsClosedEmbedding f
this_1 : UpgradedIsCompletelyPseudoMetrizableSpace Y := upgradeIsCompletelyPseudoMetrizable Y
this : PseudoMetricSpace X := IsInducing.comapPseudoMetricSpace.{u_1, u_2} (β := Y) (f := f) ⋯
⊢ CompleteSpace X
have rw : IsComplete (range f)
Tactic: rw [completeSpace_iff_isComplete_range hf.isEmbedding.to_isometry.isUniformInducing]
-- Lean 4 proof step from Mathlib.Topology.Metrizable.CompletelyMetrizable (hypothesis)
Goal:
X : Type u_1
Y : Type u_2
inst : TopologicalSpace X
inst_1 : TopologicalSpace Y
inst_2 : TopologicalSpace.IsCompletelyPseudoMetrizableSpace Y
f : X → Y
hf : IsClosedEmbedding f
this_1 : UpgradedIsCompletelyPseudoMetrizableSpace Y := upgradeIsCompletelyPseudoMetrizable Y
this : PseudoMetricSpace X := IsInducing.comapPseudoMetricSpace.{u_1, u_2} (β := Y) (f := f) ⋯
⊢ TopologicalSpace.IsCompletelyPseudoMetrizableSpace X
have this₁ : CompleteSpace X
Tactic: have : CompleteSpace X := by
    rw [completeSpace_iff_isComplete_range hf.isEmbedding.to_isometry.isUniformInducing]
    exact hf.isClosed_range.isComplete
-- Lean 4 proof step from Mathlib.Topology.Metrizable.CompletelyMetrizable (goal)
Goal:
X : Type u_1
Y : Type u_2
inst : TopologicalSpace X
inst_1 : IsCompletelyMetrizableSpace X
⊢ IsCompletelyPseudoMetrizableSpace X
have obtain : ∀ (m : MetricSpace X), UniformSpace.toTopologicalSpace = inst ∧ CompleteSpace X → IsCompletelyPseudoMetrizableSpace X
Tactic: obtain ⟨m, _⟩ := ‹_›
-- Lean 4 proof step from Mathlib.Topology.Metrizable.CompletelyMetrizable (goal)
Goal:
X : Type u_1
Y : Type u_2
inst : TopologicalSpace X
inst_1 : IsCompletelyMetrizableSpace X
m : MetricSpace X
h : UniformSpace.toTopologicalSpace = inst ∧ CompleteSpace X
⊢ IsCompletelyPseudoMetrizableSpace X
have w : PseudoMetricSpace X
Tactic: use m.toPseudoMetricSpace
-- Lean 4 proof step from Mathlib.Topology.Metrizable.CompletelyMetrizable (goal)
Goal:
X : Type u_1
inst : TopologicalSpace X
inst_1 : IsCompletelyPseudoMetrizableSpace X
inst_2 : T0Space X
this : UpgradedIsCompletelyPseudoMetrizableSpace X := upgradeIsCompletelyPseudoMetrizable X
⊢ IsCompletelyMetrizableSpace X
have h : UniformSpace.toTopologicalSpace = inst ∧ CompleteSpace X
Tactic: use MetricSpace.ofT0PseudoMetricSpace X
-- Lean 4 proof step from Mathlib.Topology.Metrizable.CompletelyMetrizable (goal)
Goal:
X : Type u_3
ht : TopologicalSpace X
h : IsCompletelyMetrizableSpace X
⊢ CompleteSpace X
have h₁ : completelyMetrizableMetric X =
  Exists.choose (p := fun (m : MetricSpace X) ↦ UniformSpace.toTopologicalSpace = ht ∧ CompleteSpace X) sorry
Tactic: convert! h.complete.choose_spec.2
-- Lean 4 proof step from Mathlib.Topology.Metrizable.CompletelyMetrizable (goal)
Goal:
X : Type u_1
Y : Type u_2
inst : TopologicalSpace X
inst_1 : TopologicalSpace Y
inst_2 : TopologicalSpace.IsCompletelyMetrizableSpace Y
f : X → Y
hf : IsClosedEmbedding f
this_1 : UpgradedIsCompletelyMetrizableSpace Y := upgradeIsCompletelyMetrizable Y
this : MetricSpace X := IsEmbedding.comapMetricSpace f ⋯
⊢ CompleteSpace X
have rw : IsComplete (range f)
Tactic: rw [completeSpace_iff_isComplete_range hf.isEmbedding.to_isometry.isUniformInducing]
-- Lean 4 proof step from Mathlib.Topology.Metrizable.CompletelyMetrizable (hypothesis)
Goal:
X : Type u_1
Y : Type u_2
inst : TopologicalSpace X
inst_1 : TopologicalSpace Y
inst_2 : TopologicalSpace.IsCompletelyMetrizableSpace Y
f : X → Y
hf : IsClosedEmbedding f
this_1 : UpgradedIsCompletelyMetrizableSpace Y := upgradeIsCompletelyMetrizable Y
this : MetricSpace X := IsEmbedding.comapMetricSpace f ⋯
⊢ TopologicalSpace.IsCompletelyMetrizableSpace X
have this₁ : CompleteSpace X
Tactic: have : CompleteSpace X := by
    rw [completeSpace_iff_isComplete_range hf.isEmbedding.to_isometry.isUniformInducing]
    exact hf.isClosed_range.isComplete
-- Lean 4 proof step from Mathlib.Topology.Metrizable.CompletelyMetrizable (goal)
Goal:
X : Type u_1
Y : Type u_2
inst : TopologicalSpace X
inst_1 : DiscreteTopology X
x y : X
⊢ Eq (α := ℝ) (if x = y then 0 else 1) (if y = x then 0 else 1)
have inl : ∀ (h : x = y), Eq (α := ℝ) (if x = y then 0 else 1) (if y = x then 0 else 1)
Tactic: obtain h | h := eq_or_ne x y
-- Lean 4 proof step from Mathlib.Topology.Metrizable.CompletelyMetrizable (goal)
Goal:
X : Type u_1
Y : Type u_2
inst : TopologicalSpace X
inst_1 : DiscreteTopology X
x y : X
inl : ∀ (h : x = y), Eq (α := ℝ) (if x = y then 0 else 1) (if y = x then 0 else 1)
⊢ Eq (α := ℝ) (if x = y then 0 else 1) (if y = x then 0 else 1)
have inr : ∀ (h : x ≠ y), Eq (α := ℝ) (if x = y then 0 else 1) (if y = x then 0 else 1)
Tactic: obtain h | h := eq_or_ne x y
-- Lean 4 proof step from Mathlib.Topology.Metrizable.CompletelyMetrizable (goal)
Goal:
X : Type u_1
Y : Type u_2
inst : TopologicalSpace X
inst_1 : DiscreteTopology X
x y z : X
⊢ (if x = z then 0 else 1) ≤ HAdd.hAdd (α := ℝ) (if x = y then 0 else 1) (if y = z then 0 else 1)
have pos : x = y → (if x = z then 0 else 1) ≤ HAdd.hAdd (α := ℝ) (if x = y then 0 else 1) (if y = z then 0 else 1)
Tactic: by_cases x = y
-- Lean 4 proof step from Mathlib.Topology.Metrizable.CompletelyMetrizable (goal)
Goal:
X : Type u_1
Y : Type u_2
inst : TopologicalSpace X
inst_1 : DiscreteTopology X
x y z : X
pos : x = y → (if x = z then 0 else 1) ≤ HAdd.hAdd (α := ℝ) (if x = y then 0 else 1) (if y = z then 0 else 1)
⊢ (if x = z then 0 else 1) ≤ HAdd.hAdd (α := ℝ) (if x = y then 0 else 1) (if y = z then 0 else 1)
have neg : ¬x = y → (if x = z then 0 else 1) ≤ HAdd.hAdd (α := ℝ) (if x = y then 0 else 1) (if y = z then 0 else 1)
Tactic: by_cases x = y
-- Lean 4 proof step from Mathlib.Topology.Metrizable.CompletelyMetrizable (goal)
Goal:
X : Type u_1
Y : Type u_2
inst : TopologicalSpace X
inst_1 : DiscreteTopology X
x y z : X
h : x = y
⊢ (if x = z then 0 else 1) ≤ HAdd.hAdd (α := ℝ) (if x = y then 0 else 1) (if y = z then 0 else 1)
have pos : x = z → (if x = z then 0 else 1) ≤ HAdd.hAdd (α := ℝ) (if x = y then 0 else 1) (if y = z then 0 else 1)
Tactic: by_cases x = z
-- Lean 4 proof step from Mathlib.Topology.Metrizable.CompletelyMetrizable (goal)
Goal:
X : Type u_1
Y : Type u_2
inst : TopologicalSpace X
inst_1 : DiscreteTopology X
x y z : X
h : x = y
pos : x = z → (if x = z then 0 else 1) ≤ HAdd.hAdd (α := ℝ) (if x = y then 0 else 1) (if y = z then 0 else 1)
⊢ (if x = z then 0 else 1) ≤ HAdd.hAdd (α := ℝ) (if x = y then 0 else 1) (if y = z then 0 else 1)
have neg : ¬x = z → (if x = z then 0 else 1) ≤ HAdd.hAdd (α := ℝ) (if x = y then 0 else 1) (if y = z then 0 else 1)
Tactic: by_cases x = z
-- Lean 4 proof step from Mathlib.Topology.Metrizable.CompletelyMetrizable (goal)
Goal:
X : Type u_1
Y : Type u_2
inst : TopologicalSpace X
inst_1 : DiscreteTopology X
x y z : X
h : ¬x = y
⊢ (if x = z then 0 else 1) ≤ HAdd.hAdd (α := ℝ) (if x = y then 0 else 1) (if y = z then 0 else 1)
have pos : x = z → (if x = z then 0 else 1) ≤ HAdd.hAdd (α := ℝ) (if x = y then 0 else 1) (if y = z then 0 else 1)
Tactic: by_cases x = z
-- Lean 4 proof step from Mathlib.Topology.Metrizable.CompletelyMetrizable (goal)
Goal:
X : Type u_1
Y : Type u_2
inst : TopologicalSpace X
inst_1 : DiscreteTopology X
x y z : X
h : ¬x = y
pos : x = z → (if x = z then 0 else 1) ≤ HAdd.hAdd (α := ℝ) (if x = y then 0 else 1) (if y = z then 0 else 1)
⊢ (if x = z then 0 else 1) ≤ HAdd.hAdd (α := ℝ) (if x = y then 0 else 1) (if y = z then 0 else 1)
have neg : ¬x = z → (if x = z then 0 else 1) ≤ HAdd.hAdd (α := ℝ) (if x = y then 0 else 1) (if y = z then 0 else 1)
Tactic: by_cases x = z
-- Lean 4 proof step from Mathlib.Topology.Metrizable.CompletelyMetrizable (goal)
Goal:
X : Type u_1
Y : Type u_2
inst : TopologicalSpace X
inst_1 : DiscreteTopology X
x y z : X
h : x = y
h_1 : x = z
⊢ (if x = z then 0 else 1) ≤ HAdd.hAdd (α := ℝ) (if x = y then 0 else 1) (if y = z then 0 else 1)
have pos : y = z → (if x = z then 0 else 1) ≤ HAdd.hAdd (α := ℝ) (if x = y then 0 else 1) (if y = z then 0 else 1)
Tactic: by_cases y = z
-- Lean 4 proof step from Mathlib.Topology.Metrizable.CompletelyMetrizable (goal)
Goal:
X : Type u_1
Y : Type u_2
inst : TopologicalSpace X
inst_1 : DiscreteTopology X
x y z : X
h : x = y
h_1 : x = z
pos : y = z → (if x = z then 0 else 1) ≤ HAdd.hAdd (α := ℝ) (if x = y then 0 else 1) (if y = z then 0 else 1)
⊢ (if x = z then 0 else 1) ≤ HAdd.hAdd (α := ℝ) (if x = y then 0 else 1) (if y = z then 0 else 1)
have neg : ¬y = z → (if x = z then 0 else 1) ≤ HAdd.hAdd (α := ℝ) (if x = y then 0 else 1) (if y = z then 0 else 1)
Tactic: by_cases y = z
-- Lean 4 proof step from Mathlib.Topology.Metrizable.CompletelyMetrizable (goal)
Goal:
X : Type u_1
Y : Type u_2
inst : TopologicalSpace X
inst_1 : DiscreteTopology X
x y z : X
h : x = y
h_1 : ¬x = z
⊢ (if x = z then 0 else 1) ≤ HAdd.hAdd (α := ℝ) (if x = y then 0 else 1) (if y = z then 0 else 1)
have pos : y = z → (if x = z then 0 else 1) ≤ HAdd.hAdd (α := ℝ) (if x = y then 0 else 1) (if y = z then 0 else 1)
Tactic: by_cases y = z
-- Lean 4 proof step from Mathlib.Topology.Metrizable.CompletelyMetrizable (goal)
Goal:
X : Type u_1
Y : Type u_2
inst : TopologicalSpace X
inst_1 : DiscreteTopology X
x y z : X
h : x = y
h_1 : ¬x = z
pos : y = z → (if x = z then 0 else 1) ≤ HAdd.hAdd (α := ℝ) (if x = y then 0 else 1) (if y = z then 0 else 1)
⊢ (if x = z then 0 else 1) ≤ HAdd.hAdd (α := ℝ) (if x = y then 0 else 1) (if y = z then 0 else 1)
have neg : ¬y = z → (if x = z then 0 else 1) ≤ HAdd.hAdd (α := ℝ) (if x = y then 0 else 1) (if y = z then 0 else 1)
Tactic: by_cases y = z
-- Lean 4 proof step from Mathlib.Topology.Metrizable.CompletelyMetrizable (goal)
Goal:
X : Type u_1
Y : Type u_2
inst : TopologicalSpace X
inst_1 : DiscreteTopology X
x y z : X
h : ¬x = y
h_1 : x = z
⊢ (if x = z then 0 else 1) ≤ HAdd.hAdd (α := ℝ) (if x = y then 0 else 1) (if y = z then 0 else 1)
have pos : y = z → (if x = z then 0 else 1) ≤ HAdd.hAdd (α := ℝ) (if x = y then 0 else 1) (if y = z then 0 else 1)
Tactic: by_cases y = z
-- Lean 4 proof step from Mathlib.Topology.Metrizable.CompletelyMetrizable (goal)
Goal:
X : Type u_1
Y : Type u_2
inst : TopologicalSpace X
inst_1 : DiscreteTopology X
x y z : X
h : ¬x = y
h_1 : x = z
pos : y = z → (if x = z then 0 else 1) ≤ HAdd.hAdd (α := ℝ) (if x = y then 0 else 1) (if y = z then 0 else 1)
⊢ (if x = z then 0 else 1) ≤ HAdd.hAdd (α := ℝ) (if x = y then 0 else 1) (if y = z then 0 else 1)
have neg : ¬y = z → (if x = z then 0 else 1) ≤ HAdd.hAdd (α := ℝ) (if x = y then 0 else 1) (if y = z then 0 else 1)
Tactic: by_cases y = z
-- Lean 4 proof step from Mathlib.Topology.Metrizable.CompletelyMetrizable (goal)
Goal:
X : Type u_1
Y : Type u_2
inst : TopologicalSpace X
inst_1 : DiscreteTopology X
x y z : X
h : ¬x = y
h_1 : ¬x = z
⊢ (if x = z then 0 else 1) ≤ HAdd.hAdd (α := ℝ) (if x = y then 0 else 1) (if y = z then 0 else 1)
have pos : y = z → (if x = z then 0 else 1) ≤ HAdd.hAdd (α := ℝ) (if x = y then 0 else 1) (if y = z then 0 else 1)
Tactic: by_cases y = z
-- Lean 4 proof step from Mathlib.Topology.Metrizable.CompletelyMetrizable (goal)
Goal:
X : Type u_1
Y : Type u_2
inst : TopologicalSpace X
inst_1 : DiscreteTopology X
x y z : X
h : ¬x = y
h_1 : ¬x = z
pos : y = z → (if x = z then 0 else 1) ≤ HAdd.hAdd (α := ℝ) (if x = y then 0 else 1) (if y = z then 0 else 1)
⊢ (if x = z then 0 else 1) ≤ HAdd.hAdd (α := ℝ) (if x = y then 0 else 1) (if y = z then 0 else 1)
have neg : ¬y = z → (if x = z then 0 else 1) ≤ HAdd.hAdd (α := ℝ) (if x = y then 0 else 1) (if y = z then 0 else 1)
Tactic: by_cases y = z
-- Lean 4 proof step from Mathlib.Topology.Metrizable.CompletelyMetrizable (goal)
Goal:
X : Type u_1
Y : Type u_2
inst : TopologicalSpace X
inst_1 : DiscreteTopology X
m : MetricSpace X :=
  { dist := fun (x y : X) ↦ if x = y then 0 else 1, dist_self := ⋯, dist_comm := ⋯, dist_triangle := ⋯, edist_dist := ⋯,
    uniformity_dist := ⋯, cobounded_sets := ⋯, eq_of_dist_eq_zero := ⋯ }
⊢ IsCompletelyMetrizableSpace X
have refine_1 : UniformSpace.toTopologicalSpace = inst
Tactic: refine ⟨m, ?_, ?_⟩
-- Lean 4 proof step from Mathlib.Topology.Metrizable.CompletelyMetrizable (goal)
Goal:
X : Type u_1
Y : Type u_2
inst : TopologicalSpace X
inst_1 : DiscreteTopology X
m : MetricSpace X :=
  { dist := fun (x y : X) ↦ if x = y then 0 else 1, dist_self := ⋯, dist_comm := ⋯, dist_triangle := ⋯, edist_dist := ⋯,
    uniformity_dist := ⋯, cobounded_sets := ⋯, eq_of_dist_eq_zero := ⋯ }
refine_1 : UniformSpace.toTopologicalSpace = inst
⊢ IsCompletelyMetrizableSpace X
have refine_2 : CompleteSpace X
Tactic: refine ⟨m, ?_, ?_⟩
-- Lean 4 proof step from Mathlib.Topology.Metrizable.CompletelyMetrizable (goal)
Goal:
X : Type u_1
Y : Type u_2
inst : TopologicalSpace X
inst_1 : DiscreteTopology X
m : MetricSpace X :=
  { dist := fun (x y : X) ↦ if x = y then 0 else 1, dist_self := ⋯, dist_comm := ⋯, dist_triangle := ⋯, edist_dist := ⋯,
    uniformity_dist := ⋯, cobounded_sets := ⋯, eq_of_dist_eq_zero := ⋯ }
⊢ UniformSpace.toTopologicalSpace = inst
have refine_1 : Eq.{u_1 + 1} (α := TopologicalSpace X) UniformSpace.toTopologicalSpace ⊥
Tactic: rw [DiscreteTopology.eq_bot (α := X)]
-- Lean 4 proof step from Mathlib.Topology.Metrizable.CompletelyMetrizable (goal)
Goal:
X : Type u_1
Y : Type u_2
inst : TopologicalSpace X
inst_1 : DiscreteTopology X
m : MetricSpace X :=
  { dist := fun (x y : X) ↦ if x = y then 0 else 1, dist_self := ⋯, dist_comm := ⋯, dist_triangle := ⋯, edist_dist := ⋯,
    uniformity_dist := ⋯, cobounded_sets := ⋯, eq_of_dist_eq_zero := ⋯ }
⊢ Eq.{u_1 + 1} (α := TopologicalSpace X) UniformSpace.toTopologicalSpace ⊥
have refine_1 : ∀ (x : X), IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] {x}
Tactic: refine eq_bot_of_singletons_open fun x ↦ ?_
-- Lean 4 proof step from Mathlib.Topology.Metrizable.CompletelyMetrizable (goal)
Goal:
X : Type u_1
Y : Type u_2
inst : TopologicalSpace X
inst_1 : DiscreteTopology X
m : MetricSpace X :=
  { dist := fun (x y : X) ↦ if x = y then 0 else 1, dist_self := ⋯, dist_comm := ⋯, dist_triangle := ⋯, edist_dist := ⋯,
    uniformity_dist := ⋯, cobounded_sets := ⋯, eq_of_dist_eq_zero := ⋯ }
x : X
⊢ IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] {x}
have h : {x} = Metric.ball x 1
Tactic: convert! @Metric.isOpen_ball _ _ x 1
-- Lean 4 proof step from Mathlib.Topology.Metrizable.CompletelyMetrizable (goal)
Goal:
X : Type u_1
Y : Type u_2
inst : TopologicalSpace X
inst_1 : DiscreteTopology X
m : MetricSpace X :=
  { dist := fun (x y : X) ↦ if x = y then 0 else 1, dist_self := ⋯, dist_comm := ⋯, dist_triangle := ⋯, edist_dist := ⋯,
    uniformity_dist := ⋯, cobounded_sets := ⋯, eq_of_dist_eq_zero := ⋯ }
x : X
⊢ {x} = Metric.ball x 1
have h : ∀ (y : X) (hy : y ∈ Metric.ball x 1), Membership.mem (γ := Set X) {x} y
Tactic: refine subset_antisymm (singleton_subset_iff.2 (Metric.mem_ball_self (by simp)))
      fun y hy ↦ ?_
-- Lean 4 proof step from Mathlib.Topology.Metrizable.CompletelyMetrizable (hypothesis)
Goal:
X : Type u_1
Y : Type u_2
inst : TopologicalSpace X
inst_1 : DiscreteTopology X
m : MetricSpace X :=
  { dist := fun (x y : X) ↦ if x = y then 0 else 1, dist_self := ⋯, dist_comm := ⋯, dist_triangle := ⋯, edist_dist := ⋯,
    uniformity_dist := ⋯, cobounded_sets := ⋯, eq_of_dist_eq_zero := ⋯ }
x y : X
hy : y ∈ Metric.ball x 1
⊢ Membership.mem (γ := Set X) {x} y
have hy₁ : dist y x < 1
Tactic: simp only [Metric.mem_ball, mem_singleton_iff] at *
Cleared hypotheses: hy
-- Lean 4 proof step from Mathlib.Topology.Metrizable.CompletelyMetrizable (goal)
Goal:
X : Type u_1
Y : Type u_2
inst : TopologicalSpace X
inst_1 : DiscreteTopology X
m : MetricSpace X :=
  { dist := fun (x y : X) ↦ if x = y then 0 else 1, dist_self := ⋯, dist_comm := ⋯, dist_triangle := ⋯, edist_dist := ⋯,
    uniformity_dist := ⋯, cobounded_sets := ⋯, eq_of_dist_eq_zero := ⋯ }
x y : X
hy : dist y x < 1
⊢ y = x
have h : ∀ (this : ¬y = x), False
Tactic: by_contra
-- Lean 4 proof step from Mathlib.Topology.Metrizable.CompletelyMetrizable (hypothesis)
Goal:
X : Type u_1
Y : Type u_2
inst : TopologicalSpace X
inst_1 : DiscreteTopology X
m : MetricSpace X :=
  { dist := fun (x y : X) ↦ if x = y then 0 else 1, dist_self := ⋯, dist_comm := ⋯, dist_triangle := ⋯, edist_dist := ⋯,
    uniformity_dist := ⋯, cobounded_sets := ⋯, eq_of_dist_eq_zero := ⋯ }
x y : X
hy : dist y x < 1
this : ¬y = x
⊢ False
have hy₁ : LT.lt (α := ℝ) (if y = x then 0 else 1) 1
Tactic: change (if y = x then 0 else 1) < 1 at hy
Cleared hypotheses: hy
-- Lean 4 proof step from Mathlib.Topology.Metrizable.CompletelyMetrizable (goal)
Goal:
X : Type u_1
Y : Type u_2
inst : TopologicalSpace X
inst_1 : DiscreteTopology X
m : MetricSpace X :=
  { dist := fun (x y : X) ↦ if x = y then 0 else 1, dist_self := ⋯, dist_comm := ⋯, dist_triangle := ⋯, edist_dist := ⋯,
    uniformity_dist := ⋯, cobounded_sets := ⋯, eq_of_dist_eq_zero := ⋯ }
⊢ CompleteSpace X
have refine_2 : ∀ (u : ℕ → X) (hu : CauchySeq u), ∃ (a : X), Tendsto u atTop (𝓝 a)
Tactic: refine Metric.complete_of_cauchySeq_tendsto fun u hu ↦ ?_
-- Lean 4 proof step from Mathlib.Topology.Metrizable.CompletelyMetrizable (hypothesis)
Goal:
X : Type u_1
Y : Type u_2
inst : TopologicalSpace X
inst_1 : DiscreteTopology X
m : MetricSpace X :=
  { dist := fun (x y : X) ↦ if x = y then 0 else 1, dist_self := ⋯, dist_comm := ⋯, dist_triangle := ⋯, edist_dist := ⋯,
    uniformity_dist := ⋯, cobounded_sets := ⋯, eq_of_dist_eq_zero := ⋯ }
u : ℕ → X
hu : CauchySeq u
⊢ ∃ (a : X), Tendsto u atTop (𝓝 a)
have hu₁ : ∀ ε > 0, ∃ (N : ℕ), ∀ n ≥ N, dist (u n) (u N) < ε
Tactic: rw [Metric.cauchySeq_iff'] at hu
Cleared hypotheses: hu
-- Lean 4 proof step from Mathlib.Topology.Metrizable.CompletelyMetrizable (goal)
Goal:
X : Type u_1
Y : Type u_2
inst : TopologicalSpace X
inst_1 : DiscreteTopology X
m : MetricSpace X :=
  { dist := fun (x y : X) ↦ if x = y then 0 else 1, dist_self := ⋯, dist_comm := ⋯, dist_triangle := ⋯, edist_dist := ⋯,
    uniformity_dist := ⋯, cobounded_sets := ⋯, eq_of_dist_eq_zero := ⋯ }
u : ℕ → X
hu : ∀ ε > 0, ∃ (N : ℕ), ∀ n ≥ N, dist (u n) (u N) < ε
⊢ ∃ (a : X), Tendsto u atTop (𝓝 a)
have refine_2 : ∀ (N : ℕ) (hN : ∀ n ≥ N, dist (u n) (u N) < 1), ∃ (a : X), Tendsto u atTop (𝓝 a)
Tactic: obtain ⟨N, hN⟩ := hu 1 (by simp)
-- Lean 4 proof step from Mathlib.Topology.Metrizable.CompletelyMetrizable (goal)
Goal:
X : Type u_1
Y : Type u_2
inst : TopologicalSpace X
inst_1 : DiscreteTopology X
m : MetricSpace X :=
  { dist := fun (x y : X) ↦ if x = y then 0 else 1, dist_self := ⋯, dist_comm := ⋯, dist_triangle := ⋯, edist_dist := ⋯,
    uniformity_dist := ⋯, cobounded_sets := ⋯, eq_of_dist_eq_zero := ⋯ }
u : ℕ → X
hu : ∀ ε > 0, ∃ (N : ℕ), ∀ n ≥ N, dist (u n) (u N) < ε
N : ℕ
hN : ∀ n ≥ N, dist (u n) (u N) < 1
⊢ ∃ (a : X), Tendsto u atTop (𝓝 a)
have refine_2 : ∀ (n : ℕ) (hn : n ≥ N), u n = u N
Tactic: refine ⟨u N, @tendsto_atTop_of_eventually_const X UniformSpace.toTopologicalSpace (u N) _ _ _ N
      fun n hn ↦ ?_⟩
-- Lean 4 proof step from Mathlib.Topology.Metrizable.CompletelyMetrizable (hypothesis)
Goal:
X : Type u_1
Y : Type u_2
inst : TopologicalSpace X
inst_1 : DiscreteTopology X
m : MetricSpace X :=
  { dist := fun (x y : X) ↦ if x = y then 0 else 1, dist_self := ⋯, dist_comm := ⋯, dist_triangle := ⋯, edist_dist := ⋯,
    uniformity_dist := ⋯, cobounded_sets := ⋯, eq_of_dist_eq_zero := ⋯ }
u : ℕ → X
hu : ∀ ε > 0, ∃ (N : ℕ), ∀ n ≥ N, dist (u n) (u N) < ε
N : ℕ
hN : ∀ n ≥ N, dist (u n) (u N) < 1
n : ℕ
hn : n ≥ N
⊢ u n = u N
have hN₁ : dist (u n) (u N) < 1
Tactic: specialize hN n hn
Cleared hypotheses: hN
-- Lean 4 proof step from Mathlib.Topology.Metrizable.CompletelyMetrizable (goal)
Goal:
X : Type u_1
Y : Type u_2
inst : TopologicalSpace X
inst_1 : DiscreteTopology X
m : MetricSpace X :=
  { dist := fun (x y : X) ↦ if x = y then 0 else 1, dist_self := ⋯, dist_comm := ⋯, dist_triangle := ⋯, edist_dist := ⋯,
    uniformity_dist := ⋯, cobounded_sets := ⋯, eq_of_dist_eq_zero := ⋯ }
u : ℕ → X
hu : ∀ ε > 0, ∃ (N : ℕ), ∀ n ≥ N, dist (u n) (u N) < ε
N n : ℕ
hn : n ≥ N
hN : dist (u n) (u N) < 1
⊢ u n = u N
have refine_2 : ∀ (this : ¬u n = u N), False
Tactic: by_contra
-- Lean 4 proof step from Mathlib.Topology.Metrizable.CompletelyMetrizable (hypothesis)
Goal:
X : Type u_1
Y : Type u_2
inst : TopologicalSpace X
inst_1 : DiscreteTopology X
m : MetricSpace X :=
  { dist := fun (x y : X) ↦ if x = y then 0 else 1, dist_self := ⋯, dist_comm := ⋯, dist_triangle := ⋯, edist_dist := ⋯,
    uniformity_dist := ⋯, cobounded_sets := ⋯, eq_of_dist_eq_zero := ⋯ }
u : ℕ → X
hu : ∀ ε > 0, ∃ (N : ℕ), ∀ n ≥ N, dist (u n) (u N) < ε
N n : ℕ
hn : n ≥ N
hN : dist (u n) (u N) < 1
this : ¬u n = u N
⊢ False
have hN₁ : LT.lt (α := ℝ) (if u n = u N then 0 else 1) 1
Tactic: change (if u n = u N then 0 else 1) < 1 at hN
Cleared hypotheses: hN
-- Lean 4 proof step from Mathlib.Topology.Metrizable.CompletelyMetrizable (goal)
Goal:
X : Type u_1
Y : Type u_2
inst : TopologicalSpace X
inst_1 : DiscreteTopology X
let let₁ let₂ : IsCompletelyMetrizableSpace X
⊢ IsCompletelyMetrizableSpace X
have refine_1 : let m :=
  { dist := fun (x y : X) ↦ if x = y then 0 else 1, dist_self := sorry, dist_comm := sorry, dist_triangle := sorry, edist_dist := sorry,
    uniformity_dist := sorry, cobounded_sets := sorry, eq_of_dist_eq_zero := sorry : MetricSpace X };
UniformSpace.toTopologicalSpace = inst
Tactic: classical
  let m : MetricSpace X :=
    { dist x y := if x = y then 0 else 1
      dist_self x := by simp
      dist_comm x y := by
        obtain h | h := eq_or_ne x y
        · simp [h]
        · simp [h, h.symm]
      dist_triangle x y z := by
        by_cases x = y <;> by_cases x = z <;> by_cases y = z <;> simp_all
      eq_of_dist_eq_zero := by simp }
  refine ⟨m, ?_, ?_⟩
  · rw [DiscreteTopology.eq_bot (α := X)]
    refine eq_bot_of_singletons_open fun x ↦ ?_
    convert! @Metric.isOpen_ball _ _ x 1
    refine subset_antisymm (singleton_subset_iff.2 (Metric.mem_ball_self (by simp)))
      fun y hy ↦ ?_
    simp only [Metric.mem_ball, mem_singleton_iff] at *
    by_contra
    change (if y = x then 0 else 1) < 1 at hy
    simp_all
  · refine Metric.complete_of_cauchySeq_tendsto fun u hu ↦ ?_
    rw [Metric.cauchySeq_iff'] at hu
    obtain ⟨N, hN⟩ := hu 1 (by simp)
    refine ⟨u N, @tendsto_atTop_of_eventually_const X UniformSpace.toTopologicalSpace (u N) _ _ _ N
      fun n hn ↦ ?_⟩
    specialize hN n hn
    by_contra
    change (if u n = u N then 0 else 1) < 1 at hN
    simp_all