awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.PiNat (hypothesis)
Goal:
α : Type u_2
inst : MetricSpace α
inst_1 : CompleteSpace α
inst_2 : SecondCountableTopology α
inst_3 : Nonempty α
this : MetricSpace (ℕ → ℕ) := metricSpaceNatNat
I0 : 0 < 1 / 2
I1 : 1 / 2 < 1
u : ℕ → α
hu : DenseRange u
s : Set (ℕ → ℕ) := {x : ℕ → ℕ | Set.Nonempty.{u_2} (α := α) (⋂ (n : ℕ), closedBall (u (x n)) ((1 / 2) ^ n))}
g : (↑s : Type) → α :=
  fun (x : (↑s : Type)) ↦ Set.Nonempty.some (s := ⋂ (n : ℕ), closedBall (u ((↑x : ℕ → ℕ) n)) ((1 / 2) ^ n)) ⋯
A : ∀ (x : (↑s : Type)) (n : ℕ), dist (g x) (u ((↑x : ℕ → ℕ) n)) ≤ (1 / 2) ^ n
g_cont : Continuous[_, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] g
g_surj : Surjective g
x : ℕ → ℕ
hx : ClusterPt x (𝓟 s)
L : Tendsto (β := ℝ) (fun (n : ℕ) ↦ diam (closedBall (u (x n)) ((1 / 2) ^ n))) atTop (𝓝 0)
N : ℕ
y : ℕ → ℕ
hxy : y ∈ ball x ((1 / 2) ^ N)
ys : y ∈ s
n : ℕ
hn : n ≤ N
⊢ closedBall (u (x n)) ((1 / 2) ^ n) = closedBall (u (y n)) ((1 / 2) ^ n)
have this₁ : x n = y n
Tactic: have : x n = y n := apply_eq_of_dist_lt (mem_ball'.1 hxy) hn
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.PiNat (goal)
Goal:
α : Type u_2
inst : MetricSpace α
inst_1 : CompleteSpace α
inst_2 : SecondCountableTopology α
inst_3 : Nonempty α
this_1 : MetricSpace (ℕ → ℕ) := metricSpaceNatNat
I0 : 0 < 1 / 2
I1 : 1 / 2 < 1
u : ℕ → α
hu : DenseRange u
s : Set (ℕ → ℕ) := {x : ℕ → ℕ | Set.Nonempty.{u_2} (α := α) (⋂ (n : ℕ), closedBall (u (x n)) ((1 / 2) ^ n))}
g : (↑s : Type) → α :=
  fun (x : (↑s : Type)) ↦ Set.Nonempty.some (s := ⋂ (n : ℕ), closedBall (u ((↑x : ℕ → ℕ) n)) ((1 / 2) ^ n)) ⋯
A : ∀ (x : (↑s : Type)) (n : ℕ), dist (g x) (u ((↑x : ℕ → ℕ) n)) ≤ (1 / 2) ^ n
g_cont : Continuous[_, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] g
g_surj : Surjective g
x : ℕ → ℕ
hx : ClusterPt x (𝓟 s)
L : Tendsto (β := ℝ) (fun (n : ℕ) ↦ diam (closedBall (u (x n)) ((1 / 2) ^ n))) atTop (𝓝 0)
N : ℕ
y : ℕ → ℕ
hxy : y ∈ ball x ((1 / 2) ^ N)
ys : y ∈ s
n : ℕ
hn : n ≤ N
this : x n = y n
⊢ closedBall (u (x n)) ((1 / 2) ^ n) = closedBall (u (y n)) ((1 / 2) ^ n)
have rw : closedBall (u (y n)) ((1 / 2) ^ n) = closedBall (u (y n)) ((1 / 2) ^ n)
Tactic: rw [this]
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.PiNat (hypothesis)
Goal:
α : Type u_2
inst : MetricSpace α
inst_1 : CompleteSpace α
inst_2 : SecondCountableTopology α
inst_3 : Nonempty α
this : MetricSpace (ℕ → ℕ) := metricSpaceNatNat
I0 : 0 < 1 / 2
I1 : 1 / 2 < 1
u : ℕ → α
hu : DenseRange u
s : Set (ℕ → ℕ) := {x : ℕ → ℕ | Set.Nonempty.{u_2} (α := α) (⋂ (n : ℕ), closedBall (u (x n)) ((1 / 2) ^ n))}
g : (↑s : Type) → α :=
  fun (x : (↑s : Type)) ↦ Set.Nonempty.some (s := ⋂ (n : ℕ), closedBall (u ((↑x : ℕ → ℕ) n)) ((1 / 2) ^ n)) ⋯
A : ∀ (x : (↑s : Type)) (n : ℕ), dist (g x) (u ((↑x : ℕ → ℕ) n)) ≤ (1 / 2) ^ n
g_cont : Continuous[_, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] g
g_surj : Surjective g
x : ℕ → ℕ
hx : ClusterPt x (𝓟 s)
L : Tendsto (β := ℝ) (fun (n : ℕ) ↦ diam (closedBall (u (x n)) ((1 / 2) ^ n))) atTop (𝓝 0)
N : ℕ
y : ℕ → ℕ
hxy : y ∈ ball x ((1 / 2) ^ N)
ys : y ∈ s
⊢ Set.Nonempty.{u_2} (α := α) (⋂ (n : ℕ), ⋂ (_ : n ≤ N), closedBall (u (x n)) ((1 / 2) ^ n))
have E : Eq.{u_2 + 1} (α := Set α) (⋂ (n : ℕ), ⋂ (_ : n ≤ N), closedBall (u (x n)) ((1 / 2) ^ n))
  (⋂ (n : ℕ), ⋂ (_ : n ≤ N), closedBall (u (y n)) ((1 / 2) ^ n))
Tactic: have E :
      ⋂ (n : ℕ) (H : n ≤ N), closedBall (u (x n)) ((1 / 2) ^ n) =
        ⋂ (n : ℕ) (H : n ≤ N), closedBall (u (y n)) ((1 / 2) ^ n) := by
      refine iInter_congr fun n ↦ iInter_congr fun hn ↦ ?_
      have : x n = y n := apply_eq_of_dist_lt (mem_ball'.1 hxy) hn
      rw [this]
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.PiNat (goal)
Goal:
α : Type u_2
inst : MetricSpace α
inst_1 : CompleteSpace α
inst_2 : SecondCountableTopology α
inst_3 : Nonempty α
this : MetricSpace (ℕ → ℕ) := metricSpaceNatNat
I0 : 0 < 1 / 2
I1 : 1 / 2 < 1
u : ℕ → α
hu : DenseRange u
s : Set (ℕ → ℕ) := {x : ℕ → ℕ | Set.Nonempty.{u_2} (α := α) (⋂ (n : ℕ), closedBall (u (x n)) ((1 / 2) ^ n))}
g : (↑s : Type) → α :=
  fun (x : (↑s : Type)) ↦ Set.Nonempty.some (s := ⋂ (n : ℕ), closedBall (u ((↑x : ℕ → ℕ) n)) ((1 / 2) ^ n)) ⋯
A : ∀ (x : (↑s : Type)) (n : ℕ), dist (g x) (u ((↑x : ℕ → ℕ) n)) ≤ (1 / 2) ^ n
g_cont : Continuous[_, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] g
g_surj : Surjective g
x : ℕ → ℕ
hx : ClusterPt x (𝓟 s)
L : Tendsto (β := ℝ) (fun (n : ℕ) ↦ diam (closedBall (u (x n)) ((1 / 2) ^ n))) atTop (𝓝 0)
N : ℕ
y : ℕ → ℕ
hxy : y ∈ ball x ((1 / 2) ^ N)
ys : y ∈ s
E :
  Eq.{u_2 + 1} (α := Set α) (⋂ (n : ℕ), ⋂ (_ : n ≤ N), closedBall (u (x n)) ((1 / 2) ^ n))
    (⋂ (n : ℕ), ⋂ (_ : n ≤ N), closedBall (u (y n)) ((1 / 2) ^ n))
⊢ Set.Nonempty.{u_2} (α := α) (⋂ (n : ℕ), ⋂ (_ : n ≤ N), closedBall (u (x n)) ((1 / 2) ^ n))
have rw : Set.Nonempty.{u_2} (α := α) (⋂ (n : ℕ), ⋂ (_ : n ≤ N), closedBall (u (y n)) ((1 / 2) ^ n))
Tactic: rw [E]
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.PiNat (goal)
Goal:
α : Type u_2
inst : MetricSpace α
inst_1 : CompleteSpace α
inst_2 : SecondCountableTopology α
inst_3 : Nonempty α
this : MetricSpace (ℕ → ℕ) := metricSpaceNatNat
I0 : 0 < 1 / 2
I1 : 1 / 2 < 1
u : ℕ → α
hu : DenseRange u
s : Set (ℕ → ℕ) := {x : ℕ → ℕ | Set.Nonempty.{u_2} (α := α) (⋂ (n : ℕ), closedBall (u (x n)) ((1 / 2) ^ n))}
g : (↑s : Type) → α :=
  fun (x : (↑s : Type)) ↦ Set.Nonempty.some (s := ⋂ (n : ℕ), closedBall (u ((↑x : ℕ → ℕ) n)) ((1 / 2) ^ n)) ⋯
A : ∀ (x : (↑s : Type)) (n : ℕ), dist (g x) (u ((↑x : ℕ → ℕ) n)) ≤ (1 / 2) ^ n
g_cont : Continuous[_, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] g
g_surj : Surjective g
x : ℕ → ℕ
hx : ClusterPt x (𝓟 s)
L : Tendsto (β := ℝ) (fun (n : ℕ) ↦ diam (closedBall (u (x n)) ((1 / 2) ^ n))) atTop (𝓝 0)
N : ℕ
y : ℕ → ℕ
hxy : y ∈ ball x ((1 / 2) ^ N)
ys : y ∈ s
E :
  Eq.{u_2 + 1} (α := Set α) (⋂ (n : ℕ), ⋂ (_ : n ≤ N), closedBall (u (x n)) ((1 / 2) ^ n))
    (⋂ (n : ℕ), ⋂ (_ : n ≤ N), closedBall (u (y n)) ((1 / 2) ^ n))
⊢ Set.Nonempty.{u_2} (α := α) (⋂ (n : ℕ), ⋂ (_ : n ≤ N), closedBall (u (y n)) ((1 / 2) ^ n))
have apply : Subset.{u_2} (α := Set α) (⋂ (n : ℕ), closedBall (u (y n)) ((1 / 2) ^ n))
  (⋂ (n : ℕ), ⋂ (_ : n ≤ N), closedBall (u (y n)) ((1 / 2) ^ n))
Tactic: apply Nonempty.mono _ ys
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.PiNat (hypothesis)
Goal:
α : Type u_2
inst : MetricSpace α
inst_1 : CompleteSpace α
inst_2 : SecondCountableTopology α
inst_3 : Nonempty α
this : MetricSpace (ℕ → ℕ) := metricSpaceNatNat
I0 : 0 < 1 / 2
I1 : 1 / 2 < 1
u : ℕ → α
hu : DenseRange u
s : Set (ℕ → ℕ) := {x : ℕ → ℕ | Set.Nonempty.{u_2} (α := α) (⋂ (n : ℕ), closedBall (u (x n)) ((1 / 2) ^ n))}
g : (↑s : Type) → α :=
  fun (x : (↑s : Type)) ↦ Set.Nonempty.some (s := ⋂ (n : ℕ), closedBall (u ((↑x : ℕ → ℕ) n)) ((1 / 2) ^ n)) ⋯
A : ∀ (x : (↑s : Type)) (n : ℕ), dist (g x) (u ((↑x : ℕ → ℕ) n)) ≤ (1 / 2) ^ n
g_cont : Continuous[_, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] g
g_surj : Surjective g
⊢ ∃ (f : (ℕ → ℕ) → α), Continuous[_, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] f ∧ Surjective f
have s_closed : IsClosed s
Tactic: have s_closed : IsClosed s := by
    refine isClosed_iff_clusterPt.mpr fun x hx ↦ ?_
    have L : Tendsto (fun n : ℕ => diam (closedBall (u (x n)) ((1 / 2) ^ n))) atTop (𝓝 0) := by
      have : Tendsto (fun n : ℕ => (2 : ℝ) * (1 / 2) ^ n) atTop (𝓝 (2 * 0)) :=
        (tendsto_pow_atTop_nhds_zero_of_lt_one I0.le I1).const_mul _
      rw [mul_zero] at this
      exact
        squeeze_zero (fun n => diam_nonneg) (fun n => diam_closedBall (pow_nonneg I0.le _)) this
    refine nonempty_iInter_of_nonempty_biInter (fun n => isClosed_closedBall)
      (fun n => isBounded_closedBall) (fun N ↦ ?_) L
    obtain ⟨y, hxy, ys⟩ : ∃ y, y ∈ ball x ((1 / 2) ^ N) ∩ s :=
      clusterPt_principal_iff.1 hx _ (ball_mem_nhds x (pow_pos I0 N))
    have E :
      ⋂ (n : ℕ) (H : n ≤ N), closedBall (u (x n)) ((1 / 2) ^ n) =
        ⋂ (n : ℕ) (H : n ≤ N), closedBall (u (y n)) ((1 / 2) ^ n) := by
      refine iInter_congr fun n ↦ iInter_congr fun hn ↦ ?_
      have : x n = y n := apply_eq_of_dist_lt (mem_ball'.1 hxy) hn
      rw [this]
    rw [E]
    apply Nonempty.mono _ ys
    apply iInter_subset_iInter₂
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.PiNat (goal)
Goal:
α : Type u_2
inst : MetricSpace α
inst_1 : CompleteSpace α
inst_2 : SecondCountableTopology α
inst_3 : Nonempty α
this : MetricSpace (ℕ → ℕ) := metricSpaceNatNat
I0 : 0 < 1 / 2
I1 : 1 / 2 < 1
u : ℕ → α
hu : DenseRange u
s : Set (ℕ → ℕ) := {x : ℕ → ℕ | Set.Nonempty.{u_2} (α := α) (⋂ (n : ℕ), closedBall (u (x n)) ((1 / 2) ^ n))}
g : (↑s : Type) → α :=
  fun (x : (↑s : Type)) ↦ Set.Nonempty.some (s := ⋂ (n : ℕ), closedBall (u ((↑x : ℕ → ℕ) n)) ((1 / 2) ^ n)) ⋯
A : ∀ (x : (↑s : Type)) (n : ℕ), dist (g x) (u ((↑x : ℕ → ℕ) n)) ≤ (1 / 2) ^ n
g_cont : Continuous[_, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] g
g_surj : Surjective g
s_closed : IsClosed s
⊢ ∃ (f : (ℕ → ℕ) → (↑s : Type)), (∀ (x : (↑s : Type)), f (↑x : ℕ → ℕ) = x) ∧ Surjective f ∧ Continuous f
have apply : s.Nonempty
Tactic: apply exists_retraction_subtype_of_isClosed s_closed
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.PiNat (goal)
Goal:
α : Type u_2
inst : MetricSpace α
inst_1 : CompleteSpace α
inst_2 : SecondCountableTopology α
inst_3 : Nonempty α
this : MetricSpace (ℕ → ℕ) := metricSpaceNatNat
I0 : 0 < 1 / 2
I1 : 1 / 2 < 1
u : ℕ → α
hu : DenseRange u
s : Set (ℕ → ℕ) := {x : ℕ → ℕ | Set.Nonempty.{u_2} (α := α) (⋂ (n : ℕ), closedBall (u (x n)) ((1 / 2) ^ n))}
g : (↑s : Type) → α :=
  fun (x : (↑s : Type)) ↦ Set.Nonempty.some (s := ⋂ (n : ℕ), closedBall (u ((↑x : ℕ → ℕ) n)) ((1 / 2) ^ n)) ⋯
A : ∀ (x : (↑s : Type)) (n : ℕ), dist (g x) (u ((↑x : ℕ → ℕ) n)) ≤ (1 / 2) ^ n
g_cont : Continuous[_, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] g
g_surj : Surjective g
s_closed : IsClosed s
⊢ ∃ (f : (ℕ → ℕ) → α), Continuous[_, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] f ∧ Surjective f
have obtain : ∀ (f : (ℕ → ℕ) → (↑s : Type)) (f_surj : Surjective f) (f_cont : Continuous f),
  ∃ (f : (ℕ → ℕ) → α), Continuous[_, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] f ∧ Surjective f
Tactic: obtain ⟨f, -, f_surj, f_cont⟩ :
    ∃ f : (ℕ → ℕ) → s, (∀ x : s, f x = x) ∧ Surjective f ∧ Continuous f := by
    apply exists_retraction_subtype_of_isClosed s_closed
    simpa only [nonempty_coe_sort] using g_surj.nonempty
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.PiNat (goal)
Goal:
ι : Type u_2
inst : Encodable ι
F : ι → Type u_3
inst_1 : (i : ι) → EDist (F i)
x y : (i : ι) → F i
⊢ LE.le (α := ℝ≥0∞) (∑' (i : ι), min (HPow.hPow (α := ℝ≥0∞) 2⁻¹ (encode i)) (edist (x i) (y i)))
    (∑' (i : ι), HPow.hPow (α := ℝ≥0∞) 2⁻¹ (encode i))
have h : ∀ (i : ι), min (HPow.hPow (α := ℝ≥0∞) 2⁻¹ (encode i)) (edist (x i) (y i)) ≤ HPow.hPow (α := ℝ≥0∞) 2⁻¹ (encode i)
Tactic: gcongr
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.PiNat (goal)
Goal:
E : ℕ → Type u_1
ι : Type u_2
inst : Encodable ι
F : ι → Type u_3
inst_1 : (i : ι) → PseudoEMetricSpace (F i)
x y z : (i : ι) → F i
⊢ LE.le (α := ℝ≥0∞) (∑' (i : ι), min (HPow.hPow (α := ℝ≥0∞) 2⁻¹ (encode i)) (edist (x i) (z i)))
    (∑' (i : ι),
      (min (HPow.hPow (α := ℝ≥0∞) 2⁻¹ (encode i)) (edist (x i) (y i)) +
        min (HPow.hPow (α := ℝ≥0∞) 2⁻¹ (encode i)) (edist (y i) (z i))))
have h : ∀ (n : ι),
  min (HPow.hPow (α := ℝ≥0∞) 2⁻¹ (encode n)) (edist (x n) (z n)) ≤
    min (HPow.hPow (α := ℝ≥0∞) 2⁻¹ (encode n)) (edist (x n) (y n)) +
      min (HPow.hPow (α := ℝ≥0∞) 2⁻¹ (encode n)) (edist (y n) (z n))
Tactic: gcongr with n
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.PiNat (goal)
Goal:
E : ℕ → Type u_1
ι : Type u_2
inst : Encodable ι
F : ι → Type u_3
inst_1 : (i : ι) → PseudoEMetricSpace (F i)
x y z : (i : ι) → F i
n : ι
⊢ min (HPow.hPow (α := ℝ≥0∞) 2⁻¹ (encode n)) (edist (x n) (z n)) ≤
    min (HPow.hPow (α := ℝ≥0∞) 2⁻¹ (encode n)) (edist (x n) (y n)) +
      min (HPow.hPow (α := ℝ≥0∞) 2⁻¹ (encode n)) (edist (y n) (z n))
have h : min (HPow.hPow (α := ℝ≥0∞) 2⁻¹ (encode n)) (edist (x n) (y n) + edist (y n) (z n)) ≤
  min (HPow.hPow (α := ℝ≥0∞) 2⁻¹ (encode n)) (edist (x n) (y n)) +
    min (HPow.hPow (α := ℝ≥0∞) 2⁻¹ (encode n)) (edist (y n) (z n))
Tactic: grw [edist_triangle _ (y n), min_add_distrib, min_le_right]
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.PiNat (goal)
Goal:
E : ℕ → Type u_1
ι : Type u_2
inst : Encodable ι
F : ι → Type u_3
inst_1 : (i : ι) → PseudoEMetricSpace (F i)
x y z : (i : ι) → F i
n : ι
h :
  min (HPow.hPow (α := ℝ≥0∞) 2⁻¹ (encode n)) (edist (x n) (y n) + edist (y n) (z n)) ≤
    min (HPow.hPow (α := ℝ≥0∞) 2⁻¹ (encode n)) (edist (x n) (y n)) +
      min (HPow.hPow (α := ℝ≥0∞) 2⁻¹ (encode n)) (edist (y n) (z n))
⊢ min (HPow.hPow (α := ℝ≥0∞) 2⁻¹ (encode n)) (edist (x n) (z n)) ≤
    min (HPow.hPow (α := ℝ≥0∞) 2⁻¹ (encode n)) (edist (x n) (y n)) +
      min (HPow.hPow (α := ℝ≥0∞) 2⁻¹ (encode n)) (edist (y n) (z n))
have h₁ : min (HPow.hPow (α := ℝ≥0∞) 2⁻¹ (encode n))
    (min (HPow.hPow (α := ℝ≥0∞) 2⁻¹ (encode n)) (edist (x n) (y n)) +
      min (HPow.hPow (α := ℝ≥0∞) 2⁻¹ (encode n)) (edist (y n) (z n))) ≤
  min (HPow.hPow (α := ℝ≥0∞) 2⁻¹ (encode n)) (edist (x n) (y n)) +
    min (HPow.hPow (α := ℝ≥0∞) 2⁻¹ (encode n)) (edist (y n) (z n))
Tactic: grw [edist_triangle _ (y n), min_add_distrib, min_le_right]
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.PiNat (goal)
Goal:
E : ℕ → Type u_1
ι : Type u_2
inst : Encodable ι
F : ι → Type u_3
inst_1 : (i : ι) → PseudoEMetricSpace (F i)
x y z : (i : ι) → F i
n : ι
h :
  min (HPow.hPow (α := ℝ≥0∞) 2⁻¹ (encode n)) (edist (x n) (y n) + edist (y n) (z n)) ≤
    min (HPow.hPow (α := ℝ≥0∞) 2⁻¹ (encode n)) (edist (x n) (y n)) +
      min (HPow.hPow (α := ℝ≥0∞) 2⁻¹ (encode n)) (edist (y n) (z n))
h₁ :
  min (HPow.hPow (α := ℝ≥0∞) 2⁻¹ (encode n))
      (min (HPow.hPow (α := ℝ≥0∞) 2⁻¹ (encode n)) (edist (x n) (y n)) +
        min (HPow.hPow (α := ℝ≥0∞) 2⁻¹ (encode n)) (edist (y n) (z n))) ≤
    min (HPow.hPow (α := ℝ≥0∞) 2⁻¹ (encode n)) (edist (x n) (y n)) +
      min (HPow.hPow (α := ℝ≥0∞) 2⁻¹ (encode n)) (edist (y n) (z n))
⊢ min (HPow.hPow (α := ℝ≥0∞) 2⁻¹ (encode n)) (edist (x n) (z n)) ≤
    min (HPow.hPow (α := ℝ≥0∞) 2⁻¹ (encode n)) (edist (x n) (y n)) +
      min (HPow.hPow (α := ℝ≥0∞) 2⁻¹ (encode n)) (edist (y n) (z n))
have h₂ : min (HPow.hPow (α := ℝ≥0∞) 2⁻¹ (encode n)) (edist (x n) (y n)) +
    min (HPow.hPow (α := ℝ≥0∞) 2⁻¹ (encode n)) (edist (y n) (z n)) ≤
  min (HPow.hPow (α := ℝ≥0∞) 2⁻¹ (encode n)) (edist (x n) (y n)) +
    min (HPow.hPow (α := ℝ≥0∞) 2⁻¹ (encode n)) (edist (y n) (z n))
Tactic: grw [edist_triangle _ (y n), min_add_distrib, min_le_right]
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.PiNat (goal)
Goal:
E : ℕ → Type u_1
ι : Type u_2
inst : Encodable ι
F : ι → Type u_3
inst_1 : (i : ι) → PseudoEMetricSpace (F i)
⊢ Eq (α := Filter (((i : ι) → F i) × ((i : ι) → F i))) (uniformity ((i : ι) → F i))
    (⨅ (ε : ℝ≥0∞), ⨅ (_ : ε > 0), 𝓟 {p : ((i : ι) → F i) × ((i : ι) → F i) | edist p.1 p.2 < ε})
have simp : (∀ (i : ℝ≥0∞) (i_1 : 0 < i),
    Membership.mem (γ := Filter (((i : ι) → F i) × ((i : ι) → F i)))
      (⨅ (i : ι),
        ⨅ (i_2 : ℝ≥0∞), ⨅ (_ : 0 < i_2), 𝓟 {a : ((i : ι) → F i) × ((i : ι) → F i) | edist (a.1 i) (a.2 i) < i_2})
      {p : ((i : ι) → F i) × ((i : ι) → F i) | edist p.1 p.2 < i}) ∧
  ∀ (i : ι) (i_1 : ℝ≥0∞) (i_2 : 0 < i_1),
    Membership.mem (γ := Filter (((i : ι) → F i) × ((i : ι) → F i)))
      (⨅ (ε : ℝ≥0∞), ⨅ (_ : 0 < ε), 𝓟 {p : ((i : ι) → F i) × ((i : ι) → F i) | edist p.1 p.2 < ε})
      {a : ((i : ι) → F i) × ((i : ι) → F i) | edist (a.1 i) (a.2 i) < i_1}
Tactic: simp only [Pi.uniformity, comap_iInf, gt_iff_lt, preimage_setOf_eq, comap_principal,
      PseudoEMetricSpace.uniformity_edist, le_antisymm_iff, le_iInf_iff, le_principal_iff]
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.PiNat (goal)
Goal:
E : ℕ → Type u_1
ι : Type u_2
inst : Encodable ι
F : ι → Type u_3
inst_1 : (i : ι) → PseudoEMetricSpace (F i)
⊢ (∀ (i : ℝ≥0∞) (i_1 : 0 < i),
      Membership.mem (γ := Filter (((i : ι) → F i) × ((i : ι) → F i)))
        (⨅ (i : ι),
          ⨅ (i_2 : ℝ≥0∞), ⨅ (_ : 0 < i_2), 𝓟 {a : ((i : ι) → F i) × ((i : ι) → F i) | edist (a.1 i) (a.2 i) < i_2})
        {p : ((i : ι) → F i) × ((i : ι) → F i) | edist p.1 p.2 < i}) ∧
    ∀ (i : ι) (i_1 : ℝ≥0∞) (i_2 : 0 < i_1),
      Membership.mem (γ := Filter (((i : ι) → F i) × ((i : ι) → F i)))
        (⨅ (ε : ℝ≥0∞), ⨅ (_ : 0 < ε), 𝓟 {p : ((i : ι) → F i) × ((i : ι) → F i) | edist p.1 p.2 < ε})
        {a : ((i : ι) → F i) × ((i : ι) → F i) | edist (a.1 i) (a.2 i) < i_1}
have left : ∀ (i : ℝ≥0∞) (i_1 : 0 < i),
  Membership.mem (γ := Filter (((i : ι) → F i) × ((i : ι) → F i)))
    (⨅ (i : ι),
      ⨅ (i_2 : ℝ≥0∞), ⨅ (_ : 0 < i_2), 𝓟 {a : ((i : ι) → F i) × ((i : ι) → F i) | edist (a.1 i) (a.2 i) < i_2})
    {p : ((i : ι) → F i) × ((i : ι) → F i) | edist p.1 p.2 < i}
Tactic: constructor
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.PiNat (goal)
Goal:
E : ℕ → Type u_1
ι : Type u_2
inst : Encodable ι
F : ι → Type u_3
inst_1 : (i : ι) → PseudoEMetricSpace (F i)
left :
  ∀ (i : ℝ≥0∞) (i_1 : 0 < i),
    Membership.mem (γ := Filter (((i : ι) → F i) × ((i : ι) → F i)))
      (⨅ (i : ι),
        ⨅ (i_2 : ℝ≥0∞), ⨅ (_ : 0 < i_2), 𝓟 {a : ((i : ι) → F i) × ((i : ι) → F i) | edist (a.1 i) (a.2 i) < i_2})
      {p : ((i : ι) → F i) × ((i : ι) → F i) | edist p.1 p.2 < i}
⊢ (∀ (i : ℝ≥0∞) (i_1 : 0 < i),
      Membership.mem (γ := Filter (((i : ι) → F i) × ((i : ι) → F i)))
        (⨅ (i : ι),
          ⨅ (i_2 : ℝ≥0∞), ⨅ (_ : 0 < i_2), 𝓟 {a : ((i : ι) → F i) × ((i : ι) → F i) | edist (a.1 i) (a.2 i) < i_2})
        {p : ((i : ι) → F i) × ((i : ι) → F i) | edist p.1 p.2 < i}) ∧
    ∀ (i : ι) (i_1 : ℝ≥0∞) (i_2 : 0 < i_1),
      Membership.mem (γ := Filter (((i : ι) → F i) × ((i : ι) → F i)))
        (⨅ (ε : ℝ≥0∞), ⨅ (_ : 0 < ε), 𝓟 {p : ((i : ι) → F i) × ((i : ι) → F i) | edist p.1 p.2 < ε})
        {a : ((i : ι) → F i) × ((i : ι) → F i) | edist (a.1 i) (a.2 i) < i_1}
have right : ∀ (i : ι) (i_1 : ℝ≥0∞) (i_2 : 0 < i_1),
  Membership.mem (γ := Filter (((i : ι) → F i) × ((i : ι) → F i)))
    (⨅ (ε : ℝ≥0∞), ⨅ (_ : 0 < ε), 𝓟 {p : ((i : ι) → F i) × ((i : ι) → F i) | edist p.1 p.2 < ε})
    {a : ((i : ι) → F i) × ((i : ι) → F i) | edist (a.1 i) (a.2 i) < i_1}
Tactic: constructor
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.PiNat (goal)
Goal:
E : ℕ → Type u_1
ι : Type u_2
inst : Encodable ι
F : ι → Type u_3
inst_1 : (i : ι) → PseudoEMetricSpace (F i)
ε : ℝ≥0∞
hε : 0 < ε
⊢ Membership.mem (γ := Filter (((i : ι) → F i) × ((i : ι) → F i)))
    (⨅ (i : ι),
      ⨅ (i_1 : ℝ≥0∞), ⨅ (_ : 0 < i_1), 𝓟 {a : ((i : ι) → F i) × ((i : ι) → F i) | edist (a.1 i) (a.2 i) < i_1})
    {p : ((i : ι) → F i) × ((i : ι) → F i) | edist p.1 p.2 < ε}
have left : ∀ (K : Finset ι) (hK : ∑' (i : { j : ι // j ∉ K }), HPow.hPow (α := ℝ≥0∞) 2⁻¹ (encode (↑i : ι)) < ε / 2),
  Membership.mem (γ := Filter (((i : ι) → F i) × ((i : ι) → F i)))
    (⨅ (i : ι),
      ⨅ (i_1 : ℝ≥0∞), ⨅ (_ : 0 < i_1), 𝓟 {a : ((i : ι) → F i) × ((i : ι) → F i) | edist (a.1 i) (a.2 i) < i_1})
    {p : ((i : ι) → F i) × ((i : ι) → F i) | edist p.1 p.2 < ε}
Tactic: obtain ⟨K, hK⟩ : ∃ K : Finset ι, ∑' i : {j // j ∉ K}, 2⁻¹ ^ encode (i : ι) < ε / 2 :=
        ((tendsto_order.1 <| ENNReal.tendsto_tsum_compl_atTop_zero
          (tsum_geometric_encode_lt_top ENNReal.one_half_lt_one).ne).2 _
            <| by simpa using hε.ne').exists
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.PiNat (goal)
Goal:
E : ℕ → Type u_1
ι : Type u_2
inst : Encodable ι
F : ι → Type u_3
inst_1 : (i : ι) → PseudoEMetricSpace (F i)
ε : ℝ≥0∞
hε : 0 < ε
K : Finset ι
hK : ∑' (i : { j : ι // j ∉ K }), HPow.hPow (α := ℝ≥0∞) 2⁻¹ (encode (↑i : ι)) < ε / 2
⊢ Membership.mem (γ := Filter (((i : ι) → F i) × ((i : ι) → F i)))
    (⨅ (i : ι),
      ⨅ (i_1 : ℝ≥0∞), ⨅ (_ : 0 < i_1), 𝓟 {a : ((i : ι) → F i) × ((i : ι) → F i) | edist (a.1 i) (a.2 i) < i_1})
    {p : ((i : ι) → F i) × ((i : ι) → F i) | edist p.1 p.2 < ε}
have left : ∀ (δ : ℝ≥0∞) (δpos : 0 < δ) (hδ : δ * (↑K.card : ℝ≥0∞) < ε / 2),
  Membership.mem (γ := Filter (((i : ι) → F i) × ((i : ι) → F i)))
    (⨅ (i : ι),
      ⨅ (i_1 : ℝ≥0∞), ⨅ (_ : 0 < i_1), 𝓟 {a : ((i : ι) → F i) × ((i : ι) → F i) | edist (a.1 i) (a.2 i) < i_1})
    {p : ((i : ι) → F i) × ((i : ι) → F i) | edist p.1 p.2 < ε}
Tactic: obtain ⟨δ, δpos, hδ⟩ : ∃ δ, 0 < δ ∧ δ * K.card < ε / 2 :=
        ENNReal.exists_pos_mul_lt (by simp) (by simpa using hε.ne')
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.PiNat (goal)
Goal:
E : ℕ → Type u_1
ι : Type u_2
inst : Encodable ι
F : ι → Type u_3
inst_1 : (i : ι) → PseudoEMetricSpace (F i)
ε : ℝ≥0∞
hε : 0 < ε
K : Finset ι
hK : ∑' (i : { j : ι // j ∉ K }), HPow.hPow (α := ℝ≥0∞) 2⁻¹ (encode (↑i : ι)) < ε / 2
δ : ℝ≥0∞
δpos : 0 < δ
hδ : δ * (↑K.card : ℝ≥0∞) < ε / 2
⊢ Membership.mem (γ := Filter (((i : ι) → F i) × ((i : ι) → F i)))
    (⨅ (i : ι),
      ⨅ (i_1 : ℝ≥0∞), ⨅ (_ : 0 < i_1), 𝓟 {a : ((i : ι) → F i) × ((i : ι) → F i) | edist (a.1 i) (a.2 i) < i_1})
    {p : ((i : ι) → F i) × ((i : ι) → F i) | edist p.1 p.2 < ε}
have left : ∀ (i : (↑(↑K : Set ι) : Type u_2)),
  Membership.mem (γ := Filter (((i : ι) → F i) × ((i : ι) → F i)))
    (⨅ (i_1 : ℝ≥0∞),
      ⨅ (_ : 0 < i_1), 𝓟 {a : ((i : ι) → F i) × ((i : ι) → F i) | edist (a.1 (↑i : ι)) (a.2 (↑i : ι)) < i_1})
    {p : ((i : ι) → F i) × ((i : ι) → F i) | edist (p.1 (↑i : ι)) (p.2 (↑i : ι)) < δ}
Tactic: apply @mem_iInf_of_iInter _ _ _ _ _ K.finite_toSet fun i =>
          {p : (∀ i : ι, F i) × ∀ i : ι, F i | edist (p.fst i) (p.snd i) < δ}
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.PiNat (goal)
Goal:
E : ℕ → Type u_1
ι : Type u_2
inst : Encodable ι
F : ι → Type u_3
inst_1 : (i : ι) → PseudoEMetricSpace (F i)
ε : ℝ≥0∞
hε : 0 < ε
K : Finset ι
hK : ∑' (i : { j : ι // j ∉ K }), HPow.hPow (α := ℝ≥0∞) 2⁻¹ (encode (↑i : ι)) < ε / 2
δ : ℝ≥0∞
δpos : 0 < δ
hδ : δ * (↑K.card : ℝ≥0∞) < ε / 2
left :
  ∀ (i : (↑(↑K : Set ι) : Type u_2)),
    Membership.mem (γ := Filter (((i : ι) → F i) × ((i : ι) → F i)))
      (⨅ (i_1 : ℝ≥0∞),
        ⨅ (_ : 0 < i_1), 𝓟 {a : ((i : ι) → F i) × ((i : ι) → F i) | edist (a.1 (↑i : ι)) (a.2 (↑i : ι)) < i_1})
      {p : ((i : ι) → F i) × ((i : ι) → F i) | edist (p.1 (↑i : ι)) (p.2 (↑i : ι)) < δ}
⊢ Membership.mem (γ := Filter (((i : ι) → F i) × ((i : ι) → F i)))
    (⨅ (i : ι),
      ⨅ (i_1 : ℝ≥0∞), ⨅ (_ : 0 < i_1), 𝓟 {a : ((i : ι) → F i) × ((i : ι) → F i) | edist (a.1 i) (a.2 i) < i_1})
    {p : ((i : ι) → F i) × ((i : ι) → F i) | edist p.1 p.2 < ε}
have left₁ : Subset (α := Set (((i : ι) → F i) × ((i : ι) → F i)))
  (⋂ (i : (↑(↑K : Set ι) : Type u_2)),
    {p : ((i : ι) → F i) × ((i : ι) → F i) | edist (p.1 (↑i : ι)) (p.2 (↑i : ι)) < δ})
  {p : ((i : ι) → F i) × ((i : ι) → F i) | edist p.1 p.2 < ε}
Tactic: apply @mem_iInf_of_iInter _ _ _ _ _ K.finite_toSet fun i =>
          {p : (∀ i : ι, F i) × ∀ i : ι, F i | edist (p.fst i) (p.snd i) < δ}