awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.Closeds (goal)
Goal:
α : Type u_1
β : Type u_2
inst : EMetricSpace α
inst_1 : EMetricSpace β
s_1 : Set α
inst_2 : CompleteSpace α
B : ℕ → ℝ≥0∞ := fun (n : ℕ) ↦ HPow.hPow (α := ℝ≥0∞) 2⁻¹ n
B_pos : ∀ (n : ℕ), 0 < B n
B_ne_top : ∀ (n : ℕ), B n ≠ ∞
s : ℕ → Closeds α
hs : ∀ (N n m : ℕ), N ≤ n → N ≤ m → edist (s n) (s m) < B N
t0 : Set α :=
  ⋂ (n : ℕ), closure[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] (⋃ (m : ℕ), ⋃ (_ : m ≥ n), (↑(s m) : Set α))
t : Closeds α := { carrier := t0, isClosed' := ⋯ }
n : ℕ
x : α
hx : x ∈ s n
this :
  ∀ (l : ℕ) (z : ↥(s (n + l))), ∃ (z' : ↥(s (n + l + 1))), edist (↑z : α) (↑z' : α) ≤ B n / HPow.hPow (α := ℝ≥0∞) 2 l
⊢ ∃ (z : (l : ℕ) → ↥(s (n + l))),
    (↑(z 0) : α) = x ∧ ∀ (k : ℕ), edist (↑(z k) : α) (↑(z (k + 1)) : α) ≤ B n / HPow.hPow (α := ℝ≥0∞) 2 k
have h : (↑((fun (k : ℕ) ↦
            (Nat.recOn (motive := fun (x : ℕ) ↦ ↥(s (n + x))) k
                (Subtype.mk (p := fun (x : α) ↦ x ∈ s (n + Nat.zero)) x hx) fun (l : ℕ) (z : ↥(s (n + l))) ↦
                sorry.choose (p := fun (z' : ↥(s (n + l + 1))) ↦
                  edist (↑z : α) (↑z' : α) ≤ B n / HPow.hPow (α := ℝ≥0∞) 2 l) :
              ↥(s (n + k))))
          0) :
      α) =
    x ∧
  ∀ (k : ℕ),
    edist.{u_1} (α := α)
        (↑((fun (k : ℕ) ↦
                (Nat.recOn (motive := fun (x : ℕ) ↦ ↥(s (n + x))) k
                    (Subtype.mk (p := fun (x : α) ↦ x ∈ s (n + Nat.zero)) x hx) fun (l : ℕ) (z : ↥(s (n + l))) ↦
                    sorry.choose (p := fun (z' : ↥(s (n + l + 1))) ↦
                      edist (↑z : α) (↑z' : α) ≤ B n / HPow.hPow (α := ℝ≥0∞) 2 l) :
                  ↥(s (n + k))))
              k) :
          α)
        (↑((fun (k : ℕ) ↦
                (Nat.recOn (motive := fun (x : ℕ) ↦ ↥(s (n + x))) k
                    (Subtype.mk (p := fun (x : α) ↦ x ∈ s (n + Nat.zero)) x hx) fun (l : ℕ) (z : ↥(s (n + l))) ↦
                    sorry.choose (p := fun (z' : ↥(s (n + l + 1))) ↦
                      edist (↑z : α) (↑z' : α) ≤ B n / HPow.hPow (α := ℝ≥0∞) 2 l) :
                  ↥(s (n + k))))
              (k + 1)) :
          α) ≤
      B n / HPow.hPow (α := ℝ≥0∞) 2 k
Tactic: use fun k => Nat.recOn k ⟨x, hx⟩ fun l z => (this l z).choose
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.Closeds (goal)
Goal:
α : Type u_1
β : Type u_2
inst : EMetricSpace α
inst_1 : EMetricSpace β
s_1 : Set α
inst_2 : CompleteSpace α
B : ℕ → ℝ≥0∞ := fun (n : ℕ) ↦ HPow.hPow (α := ℝ≥0∞) 2⁻¹ n
B_pos : ∀ (n : ℕ), 0 < B n
B_ne_top : ∀ (n : ℕ), B n ≠ ∞
s : ℕ → Closeds α
hs : ∀ (N n m : ℕ), N ≤ n → N ≤ m → edist (s n) (s m) < B N
t0 : Set α :=
  ⋂ (n : ℕ), closure[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] (⋃ (m : ℕ), ⋃ (_ : m ≥ n), (↑(s m) : Set α))
t : Closeds α := { carrier := t0, isClosed' := ⋯ }
n : ℕ
x : α
hx : x ∈ s n
this :
  ∀ (l : ℕ) (z : ↥(s (n + l))), ∃ (z' : ↥(s (n + l + 1))), edist (↑z : α) (↑z' : α) ≤ B n / HPow.hPow (α := ℝ≥0∞) 2 l
⊢ (↑((fun (k : ℕ) ↦
              (Nat.recOn (motive := fun (x : ℕ) ↦ ↥(s (n + x))) k
                  (Subtype.mk (p := fun (x : α) ↦ x ∈ s (n + Nat.zero)) x hx) fun (l : ℕ) (z : ↥(s (n + l))) ↦
                  ⋯.choose (p := fun (z' : ↥(s (n + l + 1))) ↦
                    edist (↑z : α) (↑z' : α) ≤ B n / HPow.hPow (α := ℝ≥0∞) 2 l) :
                ↥(s (n + k))))
            0) :
        α) =
      x ∧
    ∀ (k : ℕ),
      edist.{u_1} (α := α)
          (↑((fun (k : ℕ) ↦
                  (Nat.recOn (motive := fun (x : ℕ) ↦ ↥(s (n + x))) k
                      (Subtype.mk (p := fun (x : α) ↦ x ∈ s (n + Nat.zero)) x hx) fun (l : ℕ) (z : ↥(s (n + l))) ↦
                      ⋯.choose (p := fun (z' : ↥(s (n + l + 1))) ↦
                        edist (↑z : α) (↑z' : α) ≤ B n / HPow.hPow (α := ℝ≥0∞) 2 l) :
                    ↥(s (n + k))))
                k) :
            α)
          (↑((fun (k : ℕ) ↦
                  (Nat.recOn (motive := fun (x : ℕ) ↦ ↥(s (n + x))) k
                      (Subtype.mk (p := fun (x : α) ↦ x ∈ s (n + Nat.zero)) x hx) fun (l : ℕ) (z : ↥(s (n + l))) ↦
                      ⋯.choose (p := fun (z' : ↥(s (n + l + 1))) ↦
                        edist (↑z : α) (↑z' : α) ≤ B n / HPow.hPow (α := ℝ≥0∞) 2 l) :
                    ↥(s (n + k))))
                (k + 1)) :
            α) ≤
        B n / HPow.hPow (α := ℝ≥0∞) 2 k
have h : ∀ (k : ℕ),
  edist
      (↑(Nat.rec (motive := fun (x : ℕ) ↦ ↥(s (n + x))) (Subtype.mk (p := fun (x : α) ↦ x ∈ s n) x hx)
            (fun (l : ℕ) (z : ↥(s (n + l))) ↦
              sorry.choose (p := fun (z' : ↥(s (n + l + 1))) ↦ edist (↑z : α) (↑z' : α) ≤ B n / HPow.hPow (α := ℝ≥0∞) 2 l))
            k) :
        α)
      (↑(Exists.choose (p := fun (z' : ↥(s (n + k + 1))) ↦
            edist
                (↑(Nat.rec (motive := fun (x : ℕ) ↦ ↥(s (n + x))) (Subtype.mk (p := fun (x : α) ↦ x ∈ s n) x hx)
                      (fun (l : ℕ) (z : ↥(s (n + l))) ↦
                        sorry.choose (p := fun (z' : ↥(s (n + l + 1))) ↦
                          edist (↑z : α) (↑z' : α) ≤ B n / HPow.hPow (α := ℝ≥0∞) 2 l))
                      k) :
                  α)
                (↑z' : α) ≤
              B n / HPow.hPow (α := ℝ≥0∞) 2 k)
            sorry) :
        α) ≤
    B n / HPow.hPow (α := ℝ≥0∞) 2 k
Tactic: simp only [Nat.add_zero, Nat.rec_zero, true_and]
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.Closeds (goal)
Goal:
α : Type u_1
β : Type u_2
inst : EMetricSpace α
inst_1 : EMetricSpace β
s_1 : Set α
inst_2 : CompleteSpace α
B : ℕ → ℝ≥0∞ := fun (n : ℕ) ↦ HPow.hPow (α := ℝ≥0∞) 2⁻¹ n
B_pos : ∀ (n : ℕ), 0 < B n
B_ne_top : ∀ (n : ℕ), B n ≠ ∞
s : ℕ → Closeds α
hs : ∀ (N n m : ℕ), N ≤ n → N ≤ m → edist (s n) (s m) < B N
t0 : Set α :=
  ⋂ (n : ℕ), closure[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] (⋃ (m : ℕ), ⋃ (_ : m ≥ n), (↑(s m) : Set α))
t : Closeds α := { carrier := t0, isClosed' := ⋯ }
n : ℕ
x : α
hx : x ∈ s n
⊢ ∃ y ∈ t0, edist x y ≤ 2 * B n
have obtain : ∀ (z : (l : ℕ) → ↥(s (n + l))) (hz₀ : (↑(z 0) : α) = x)
  (hz : ∀ (k : ℕ), edist (↑(z k) : α) (↑(z (k + 1)) : α) ≤ B n / HPow.hPow (α := ℝ≥0∞) 2 k),
  ∃ y ∈ t0, edist x y ≤ 2 * B n
Tactic: obtain ⟨z, hz₀, hz⟩ :
      ∃ z : ∀ l, s (n + l), (z 0 : α) = x ∧ ∀ k, edist (z k : α) (z (k + 1) : α) ≤ B n / 2 ^ k := by
      -- We prove existence of the sequence by induction.
      have : ∀ (l) (z : s (n + l)), ∃ z' : s (n + l + 1), edist (z : α) z' ≤ B n / 2 ^ l := by
        intro l z
        obtain ⟨z', z'_mem, hz'⟩ : ∃ z' ∈ s (n + l + 1), edist (z : α) z' < B n / 2 ^ l := by
          refine exists_edist_lt_of_hausdorffEDist_lt (s := s (n + l)) z.2 ?_
          simp only [ENNReal.inv_pow, div_eq_mul_inv]
          rw [← pow_add]
          apply hs <;> simp
        exact ⟨⟨z', z'_mem⟩, le_of_lt hz'⟩
      use fun k => Nat.recOn k ⟨x, hx⟩ fun l z => (this l z).choose
      simp only [Nat.add_zero, Nat.rec_zero, true_and]
      exact fun k => (this k _).choose_spec
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.Closeds (hypothesis)
Goal:
α : Type u_1
β : Type u_2
inst : EMetricSpace α
inst_1 : EMetricSpace β
s_1 : Set α
inst_2 : CompleteSpace α
B : ℕ → ℝ≥0∞ := fun (n : ℕ) ↦ HPow.hPow (α := ℝ≥0∞) 2⁻¹ n
B_pos : ∀ (n : ℕ), 0 < B n
B_ne_top : ∀ (n : ℕ), B n ≠ ∞
s : ℕ → Closeds α
hs : ∀ (N n m : ℕ), N ≤ n → N ≤ m → edist (s n) (s m) < B N
t0 : Set α :=
  ⋂ (n : ℕ), closure[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] (⋃ (m : ℕ), ⋃ (_ : m ≥ n), (↑(s m) : Set α))
t : Closeds α := { carrier := t0, isClosed' := ⋯ }
n : ℕ
x : α
hx : x ∈ s n
z : (l : ℕ) → ↥(s (n + l))
hz₀ : (↑(z 0) : α) = x
hz : ∀ (k : ℕ), edist (↑(z k) : α) (↑(z (k + 1)) : α) ≤ B n / HPow.hPow (α := ℝ≥0∞) 2 k
⊢ ∃ y ∈ t0, edist x y ≤ 2 * B n
have this : CauchySeq.{u_1, 0} (α := α) fun (k : ℕ) ↦ (↑(z k) : α)
Tactic: have : CauchySeq fun k => (z k : α) := cauchySeq_of_edist_le_geometric_two (B n) (B_ne_top n) hz
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.Closeds (goal)
Goal:
α : Type u_1
β : Type u_2
inst : EMetricSpace α
inst_1 : EMetricSpace β
s_1 : Set α
inst_2 : CompleteSpace α
B : ℕ → ℝ≥0∞ := fun (n : ℕ) ↦ HPow.hPow (α := ℝ≥0∞) 2⁻¹ n
B_pos : ∀ (n : ℕ), 0 < B n
B_ne_top : ∀ (n : ℕ), B n ≠ ∞
s : ℕ → Closeds α
hs : ∀ (N n m : ℕ), N ≤ n → N ≤ m → edist (s n) (s m) < B N
t0 : Set α :=
  ⋂ (n : ℕ), closure[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] (⋃ (m : ℕ), ⋃ (_ : m ≥ n), (↑(s m) : Set α))
t : Closeds α := { carrier := t0, isClosed' := ⋯ }
n : ℕ
x : α
hx : x ∈ s n
z : (l : ℕ) → ↥(s (n + l))
hz₀ : (↑(z 0) : α) = x
hz : ∀ (k : ℕ), edist (↑(z k) : α) (↑(z (k + 1)) : α) ≤ B n / HPow.hPow (α := ℝ≥0∞) 2 k
this : CauchySeq.{u_1, 0} (α := α) fun (k : ℕ) ↦ (↑(z k) : α)
⊢ ∃ y ∈ t0, edist x y ≤ 2 * B n
have rcases : ∀ (y : α) (y_lim : Tendsto (fun (k : ℕ) ↦ (↑(z k) : α)) atTop (𝓝 y)), ∃ y ∈ t0, edist x y ≤ 2 * B n
Tactic: rcases cauchySeq_tendsto_of_complete this with ⟨y, y_lim⟩
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.Closeds (goal)
Goal:
α : Type u_1
β : Type u_2
inst : EMetricSpace α
inst_1 : EMetricSpace β
s_1 : Set α
inst_2 : CompleteSpace α
B : ℕ → ℝ≥0∞ := fun (n : ℕ) ↦ HPow.hPow (α := ℝ≥0∞) 2⁻¹ n
B_pos : ∀ (n : ℕ), 0 < B n
B_ne_top : ∀ (n : ℕ), B n ≠ ∞
s : ℕ → Closeds α
hs : ∀ (N n m : ℕ), N ≤ n → N ≤ m → edist (s n) (s m) < B N
t0 : Set α :=
  ⋂ (n : ℕ), closure[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] (⋃ (m : ℕ), ⋃ (_ : m ≥ n), (↑(s m) : Set α))
t : Closeds α := { carrier := t0, isClosed' := ⋯ }
n : ℕ
x : α
hx : x ∈ s n
z : (l : ℕ) → ↥(s (n + l))
hz₀ : (↑(z 0) : α) = x
hz : ∀ (k : ℕ), edist (↑(z k) : α) (↑(z (k + 1)) : α) ≤ B n / HPow.hPow (α := ℝ≥0∞) 2 k
this : CauchySeq.{u_1, 0} (α := α) fun (k : ℕ) ↦ (↑(z k) : α)
y : α
y_lim : Tendsto (fun (k : ℕ) ↦ (↑(z k) : α)) atTop (𝓝 y)
⊢ ∃ y ∈ t0, edist x y ≤ 2 * B n
have h : y ∈ t0 ∧ edist x y ≤ 2 * B n
Tactic: use y
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.Closeds (goal)
Goal:
α : Type u_1
β : Type u_2
inst : EMetricSpace α
inst_1 : EMetricSpace β
s_1 : Set α
inst_2 : CompleteSpace α
B : ℕ → ℝ≥0∞ := fun (n : ℕ) ↦ HPow.hPow (α := ℝ≥0∞) 2⁻¹ n
B_pos : ∀ (n : ℕ), 0 < B n
B_ne_top : ∀ (n : ℕ), B n ≠ ∞
s : ℕ → Closeds α
hs : ∀ (N n m : ℕ), N ≤ n → N ≤ m → edist (s n) (s m) < B N
t0 : Set α :=
  ⋂ (n : ℕ), closure[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] (⋃ (m : ℕ), ⋃ (_ : m ≥ n), (↑(s m) : Set α))
t : Closeds α := { carrier := t0, isClosed' := ⋯ }
n : ℕ
x : α
hx : x ∈ s n
z : (l : ℕ) → ↥(s (n + l))
hz₀ : (↑(z 0) : α) = x
hz : ∀ (k : ℕ), edist (↑(z k) : α) (↑(z (k + 1)) : α) ≤ B n / HPow.hPow (α := ℝ≥0∞) 2 k
this : CauchySeq.{u_1, 0} (α := α) fun (k : ℕ) ↦ (↑(z k) : α)
y : α
y_lim : Tendsto (fun (k : ℕ) ↦ (↑(z k) : α)) atTop (𝓝 y)
k : ℕ
⊢ ∀ᶠ (x : ℕ) in atTop, Membership.mem (γ := Set α) (⋃ (m : ℕ), ⋃ (_ : m ≥ k), (↑(s m) : Set α)) (↑(z x) : α)
have simp : ∃ (a : ℕ), ∀ b ≥ a, ∃ i ≥ k, (↑(z b) : α) ∈ (↑(s i) : Set α)
Tactic: simp only [exists_prop, Set.mem_iUnion, Filter.eventually_atTop]
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.Closeds (hypothesis)
Goal:
α : Type u_1
β : Type u_2
inst : EMetricSpace α
inst_1 : EMetricSpace β
s_1 : Set α
inst_2 : CompleteSpace α
B : ℕ → ℝ≥0∞ := fun (n : ℕ) ↦ HPow.hPow (α := ℝ≥0∞) 2⁻¹ n
B_pos : ∀ (n : ℕ), 0 < B n
B_ne_top : ∀ (n : ℕ), B n ≠ ∞
s : ℕ → Closeds α
hs : ∀ (N n m : ℕ), N ≤ n → N ≤ m → edist (s n) (s m) < B N
t0 : Set α :=
  ⋂ (n : ℕ), closure[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] (⋃ (m : ℕ), ⋃ (_ : m ≥ n), (↑(s m) : Set α))
t : Closeds α := { carrier := t0, isClosed' := ⋯ }
n : ℕ
x : α
hx : x ∈ s n
z : (l : ℕ) → ↥(s (n + l))
hz₀ : (↑(z 0) : α) = x
hz : ∀ (k : ℕ), edist (↑(z k) : α) (↑(z (k + 1)) : α) ≤ B n / HPow.hPow (α := ℝ≥0∞) 2 k
this : CauchySeq.{u_1, 0} (α := α) fun (k : ℕ) ↦ (↑(z k) : α)
y : α
y_lim : Tendsto (fun (k : ℕ) ↦ (↑(z k) : α)) atTop (𝓝 y)
⊢ y ∈ t0 ∧ edist x y ≤ 2 * B n
have this₁ : y ∈ t0
Tactic: have : y ∈ t0 :=
      mem_iInter.2 fun k =>
        mem_closure_of_tendsto y_lim
          (by
            simp only [exists_prop, Set.mem_iUnion, Filter.eventually_atTop]
            exact ⟨k, fun m hm => ⟨n + m, by lia, (z m).2⟩⟩)
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.Closeds (goal)
Goal:
α : Type u_1
β : Type u_2
inst : EMetricSpace α
inst_1 : EMetricSpace β
s_1 : Set α
inst_2 : CompleteSpace α
B : ℕ → ℝ≥0∞ := fun (n : ℕ) ↦ HPow.hPow (α := ℝ≥0∞) 2⁻¹ n
B_pos : ∀ (n : ℕ), 0 < B n
B_ne_top : ∀ (n : ℕ), B n ≠ ∞
s : ℕ → Closeds α
hs : ∀ (N n m : ℕ), N ≤ n → N ≤ m → edist (s n) (s m) < B N
t0 : Set α :=
  ⋂ (n : ℕ), closure[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] (⋃ (m : ℕ), ⋃ (_ : m ≥ n), (↑(s m) : Set α))
t : Closeds α := { carrier := t0, isClosed' := ⋯ }
n : ℕ
x : α
hx : x ∈ s n
z : (l : ℕ) → ↥(s (n + l))
hz₀ : (↑(z 0) : α) = x
hz : ∀ (k : ℕ), edist (↑(z k) : α) (↑(z (k + 1)) : α) ≤ B n / HPow.hPow (α := ℝ≥0∞) 2 k
this_1 : CauchySeq.{u_1, 0} (α := α) fun (k : ℕ) ↦ (↑(z k) : α)
y : α
y_lim : Tendsto (fun (k : ℕ) ↦ (↑(z k) : α)) atTop (𝓝 y)
this : y ∈ t0
⊢ y ∈ t0 ∧ edist x y ≤ 2 * B n
have right : edist x y ≤ 2 * B n
Tactic: use this
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.Closeds (goal)
Goal:
α : Type u_1
β : Type u_2
inst : EMetricSpace α
inst_1 : EMetricSpace β
s_1 : Set α
inst_2 : CompleteSpace α
B : ℕ → ℝ≥0∞ := fun (n : ℕ) ↦ HPow.hPow (α := ℝ≥0∞) 2⁻¹ n
B_pos : ∀ (n : ℕ), 0 < B n
B_ne_top : ∀ (n : ℕ), B n ≠ ∞
s : ℕ → Closeds α
hs : ∀ (N n m : ℕ), N ≤ n → N ≤ m → edist (s n) (s m) < B N
t0 : Set α :=
  ⋂ (n : ℕ), closure[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] (⋃ (m : ℕ), ⋃ (_ : m ≥ n), (↑(s m) : Set α))
t : Closeds α := { carrier := t0, isClosed' := ⋯ }
n : ℕ
x : α
hx : x ∈ s n
z : (l : ℕ) → ↥(s (n + l))
hz₀ : (↑(z 0) : α) = x
hz : ∀ (k : ℕ), edist (↑(z k) : α) (↑(z (k + 1)) : α) ≤ B n / HPow.hPow (α := ℝ≥0∞) 2 k
this_1 : CauchySeq.{u_1, 0} (α := α) fun (k : ℕ) ↦ (↑(z k) : α)
y : α
y_lim : Tendsto (fun (k : ℕ) ↦ (↑(z k) : α)) atTop (𝓝 y)
this : y ∈ t0
⊢ edist x y ≤ 2 * B n
have right : edist (↑(z 0) : α) y ≤ 2 * B n
Tactic: rw [← hz₀]
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.Closeds (hypothesis)
Goal:
α : Type u_1
β : Type u_2
inst : EMetricSpace α
inst_1 : EMetricSpace β
s_1 : Set α
inst_2 : CompleteSpace α
B : ℕ → ℝ≥0∞ := fun (n : ℕ) ↦ HPow.hPow (α := ℝ≥0∞) 2⁻¹ n
B_pos : ∀ (n : ℕ), 0 < B n
B_ne_top : ∀ (n : ℕ), B n ≠ ∞
s : ℕ → Closeds α
hs : ∀ (N n m : ℕ), N ≤ n → N ≤ m → edist (s n) (s m) < B N
t0 : Set α :=
  ⋂ (n : ℕ), closure[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] (⋃ (m : ℕ), ⋃ (_ : m ≥ n), (↑(s m) : Set α))
t : Closeds α := { carrier := t0, isClosed' := ⋯ }
⊢ Tendsto s atTop (𝓝 t)
have I1 : ∀ (n : ℕ), ∀ x ∈ s n, ∃ y ∈ t0, edist x y ≤ 2 * B n
Tactic: have I1 : ∀ n, ∀ x ∈ s n, ∃ y ∈ t0, edist x y ≤ 2 * B n := by
    /- This is the main difficulty of the proof. Starting from `x ∈ s n`, we want
           to find a point in `t0` which is close to `x`. Define inductively a sequence of
           points `z m` with `z n = x` and `z m ∈ s m` and `edist (z m) (z (m+1)) ≤ B m`. This is
           possible since the Hausdorff distance between `s m` and `s (m+1)` is at most `B m`.
           This sequence is a Cauchy sequence, therefore converging as the space is complete, to
           a limit which satisfies the required properties. -/
    intro n x hx
    obtain ⟨z, hz₀, hz⟩ :
      ∃ z : ∀ l, s (n + l), (z 0 : α) = x ∧ ∀ k, edist (z k : α) (z (k + 1) : α) ≤ B n / 2 ^ k := by
      -- We prove existence of the sequence by induction.
      have : ∀ (l) (z : s (n + l)), ∃ z' : s (n + l + 1), edist (z : α) z' ≤ B n / 2 ^ l := by
        intro l z
        obtain ⟨z', z'_mem, hz'⟩ : ∃ z' ∈ s (n + l + 1), edist (z : α) z' < B n / 2 ^ l := by
          refine exists_edist_lt_of_hausdorffEDist_lt (s := s (n + l)) z.2 ?_
          simp only [ENNReal.inv_pow, div_eq_mul_inv]
          rw [← pow_add]
          apply hs <;> simp
        exact ⟨⟨z', z'_mem⟩, le_of_lt hz'⟩
      use fun k => Nat.recOn k ⟨x, hx⟩ fun l z => (this l z).choose
      simp only [Nat.add_zero, Nat.rec_zero, true_and]
      exact fun k => (this k _).choose_spec
    -- it follows from the previous bound that `z` is a Cauchy sequence
    have : CauchySeq fun k => (z k : α) := cauchySeq_of_edist_le_geometric_two (B n) (B_ne_top n) hz
    -- therefore, it converges
    rcases cauchySeq_tendsto_of_complete this with ⟨y, y_lim⟩
    use y
    -- the limit point `y` will be the desired point, in `t0` and close to our initial point `x`.
    -- First, we check it belongs to `t0`.
    have : y ∈ t0 :=
      mem_iInter.2 fun k =>
        mem_closure_of_tendsto y_lim
          (by
            simp only [exists_prop, Set.mem_iUnion, Filter.eventually_atTop]
            exact ⟨k, fun m hm => ⟨n + m, by lia, (z m).2⟩⟩)
    use this
    -- Then, we check that `y` is close to `x = z n`. This follows from the fact that `y`
    -- is the limit of `z k`, and the distance between `z n` and `z k` has already been estimated.
    rw [← hz₀]
    exact edist_le_of_edist_le_geometric_two_of_tendsto₀ (B n) hz y_lim
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.Closeds (hypothesis)
Goal:
α : Type u_1
β : Type u_2
inst : EMetricSpace α
inst_1 : EMetricSpace β
s_1 : Set α
inst_2 : CompleteSpace α
B : ℕ → ℝ≥0∞ := fun (n : ℕ) ↦ HPow.hPow (α := ℝ≥0∞) 2⁻¹ n
B_pos : ∀ (n : ℕ), 0 < B n
B_ne_top : ∀ (n : ℕ), B n ≠ ∞
s : ℕ → Closeds α
hs : ∀ (N n m : ℕ), N ≤ n → N ≤ m → edist (s n) (s m) < B N
t0 : Set α :=
  ⋂ (n : ℕ), closure[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] (⋃ (m : ℕ), ⋃ (_ : m ≥ n), (↑(s m) : Set α))
t : Closeds α := { carrier := t0, isClosed' := ⋯ }
I1 : ∀ (n : ℕ), ∀ x ∈ s n, ∃ y ∈ t0, edist x y ≤ 2 * B n
n : ℕ
x : α
xt0 : x ∈ t0
⊢ ∃ y ∈ s n, edist x y ≤ 2 * B n
have this : Membership.mem (γ := Set α)
  (closure[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] (⋃ (m : ℕ), ⋃ (_ : m ≥ n), (↑(s m) : Set α))) x
Tactic: have : x ∈ closure (⋃ m ≥ n, s m : Set α) := by apply mem_iInter.1 xt0 n
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.Closeds (goal)
Goal:
α : Type u_1
β : Type u_2
inst : EMetricSpace α
inst_1 : EMetricSpace β
s_1 : Set α
inst_2 : CompleteSpace α
B : ℕ → ℝ≥0∞ := fun (n : ℕ) ↦ HPow.hPow (α := ℝ≥0∞) 2⁻¹ n
B_pos : ∀ (n : ℕ), 0 < B n
B_ne_top : ∀ (n : ℕ), B n ≠ ∞
s : ℕ → Closeds α
hs : ∀ (N n m : ℕ), N ≤ n → N ≤ m → edist (s n) (s m) < B N
t0 : Set α :=
  ⋂ (n : ℕ), closure[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] (⋃ (m : ℕ), ⋃ (_ : m ≥ n), (↑(s m) : Set α))
t : Closeds α := { carrier := t0, isClosed' := ⋯ }
I1 : ∀ (n : ℕ), ∀ x ∈ s n, ∃ y ∈ t0, edist x y ≤ 2 * B n
n : ℕ
x : α
xt0 : x ∈ t0
this :
  Membership.mem (γ := Set α)
    (closure[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] (⋃ (m : ℕ), ⋃ (_ : m ≥ n), (↑(s m) : Set α))) x
⊢ ∃ y ∈ s n, edist x y ≤ 2 * B n
have obtain : ∀ (z : α) (hz : Membership.mem (γ := Set α) (⋃ (m : ℕ), ⋃ (_ : m ≥ n), (↑(s m) : Set α)) z) (Dxz : edist x z < B n),
  ∃ y ∈ s n, edist x y ≤ 2 * B n
Tactic: obtain ⟨z : α, hz, Dxz : edist x z < B n⟩ := EMetric.mem_closure_iff.1 this (B n) (B_pos n)
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.Closeds (hypothesis)
Goal:
α : Type u_1
β : Type u_2
inst : EMetricSpace α
inst_1 : EMetricSpace β
s_1 : Set α
inst_2 : CompleteSpace α
B : ℕ → ℝ≥0∞ := fun (n : ℕ) ↦ HPow.hPow (α := ℝ≥0∞) 2⁻¹ n
B_pos : ∀ (n : ℕ), 0 < B n
B_ne_top : ∀ (n : ℕ), B n ≠ ∞
s : ℕ → Closeds α
hs : ∀ (N n m : ℕ), N ≤ n → N ≤ m → edist (s n) (s m) < B N
t0 : Set α :=
  ⋂ (n : ℕ), closure[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] (⋃ (m : ℕ), ⋃ (_ : m ≥ n), (↑(s m) : Set α))
t : Closeds α := { carrier := t0, isClosed' := ⋯ }
I1 : ∀ (n : ℕ), ∀ x ∈ s n, ∃ y ∈ t0, edist x y ≤ 2 * B n
n : ℕ
x : α
xt0 : x ∈ t0
this :
  Membership.mem (γ := Set α)
    (closure[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] (⋃ (m : ℕ), ⋃ (_ : m ≥ n), (↑(s m) : Set α))) x
z : α
hz : Membership.mem (γ := Set α) (⋃ (m : ℕ), ⋃ (_ : m ≥ n), (↑(s m) : Set α)) z
Dxz : edist x z < B n
⊢ ∃ y ∈ s n, edist x y ≤ 2 * B n
have hz₁ : ∃ i ≥ n, z ∈ (↑(s i) : Set α)
Tactic: simp only [exists_prop, Set.mem_iUnion] at hz
Cleared hypotheses: hz
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.Closeds (goal)
Goal:
α : Type u_1
β : Type u_2
inst : EMetricSpace α
inst_1 : EMetricSpace β
s_1 : Set α
inst_2 : CompleteSpace α
B : ℕ → ℝ≥0∞ := fun (n : ℕ) ↦ HPow.hPow (α := ℝ≥0∞) 2⁻¹ n
B_pos : ∀ (n : ℕ), 0 < B n
B_ne_top : ∀ (n : ℕ), B n ≠ ∞
s : ℕ → Closeds α
hs : ∀ (N n m : ℕ), N ≤ n → N ≤ m → edist (s n) (s m) < B N
t0 : Set α :=
  ⋂ (n : ℕ), closure[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] (⋃ (m : ℕ), ⋃ (_ : m ≥ n), (↑(s m) : Set α))
t : Closeds α := { carrier := t0, isClosed' := ⋯ }
I1 : ∀ (n : ℕ), ∀ x ∈ s n, ∃ y ∈ t0, edist x y ≤ 2 * B n
n : ℕ
x : α
xt0 : x ∈ t0
this :
  Membership.mem (γ := Set α)
    (closure[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] (⋃ (m : ℕ), ⋃ (_ : m ≥ n), (↑(s m) : Set α))) x
z : α
Dxz : edist x z < B n
hz : ∃ i ≥ n, z ∈ (↑(s i) : Set α)
⊢ ∃ y ∈ s n, edist x y ≤ 2 * B n
have obtain : ∀ (m : ℕ) (m_ge_n : m ≥ n) (hm : z ∈ (↑(s m) : Set α)), ∃ y ∈ s n, edist x y ≤ 2 * B n
Tactic: obtain ⟨m : ℕ, m_ge_n : m ≥ n, hm : z ∈ (s m : Set α)⟩ := hz
Cleared hypotheses: hz
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.Closeds (hypothesis)
Goal:
α : Type u_1
β : Type u_2
inst : EMetricSpace α
inst_1 : EMetricSpace β
s_1 : Set α
inst_2 : CompleteSpace α
B : ℕ → ℝ≥0∞ := fun (n : ℕ) ↦ HPow.hPow (α := ℝ≥0∞) 2⁻¹ n
B_pos : ∀ (n : ℕ), 0 < B n
B_ne_top : ∀ (n : ℕ), B n ≠ ∞
s : ℕ → Closeds α
hs : ∀ (N n m : ℕ), N ≤ n → N ≤ m → edist (s n) (s m) < B N
t0 : Set α :=
  ⋂ (n : ℕ), closure[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] (⋃ (m : ℕ), ⋃ (_ : m ≥ n), (↑(s m) : Set α))
t : Closeds α := { carrier := t0, isClosed' := ⋯ }
I1 : ∀ (n : ℕ), ∀ x ∈ s n, ∃ y ∈ t0, edist x y ≤ 2 * B n
n : ℕ
x : α
xt0 : x ∈ t0
this :
  Membership.mem (γ := Set α)
    (closure[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] (⋃ (m : ℕ), ⋃ (_ : m ≥ n), (↑(s m) : Set α))) x
z : α
Dxz : edist x z < B n
m : ℕ
m_ge_n : m ≥ n
hm : z ∈ (↑(s m) : Set α)
⊢ ∃ y ∈ s n, edist x y ≤ 2 * B n
have this₁ : hausdorffEDist (↑(s m) : Set α) (↑(s n) : Set α) < B n
Tactic: have : hausdorffEDist (s m : Set α) (s n) < B n := hs n m n m_ge_n (le_refl n)
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.Closeds (goal)
Goal:
α : Type u_1
β : Type u_2
inst : EMetricSpace α
inst_1 : EMetricSpace β
s_1 : Set α
inst_2 : CompleteSpace α
B : ℕ → ℝ≥0∞ := fun (n : ℕ) ↦ HPow.hPow (α := ℝ≥0∞) 2⁻¹ n
B_pos : ∀ (n : ℕ), 0 < B n
B_ne_top : ∀ (n : ℕ), B n ≠ ∞
s : ℕ → Closeds α
hs : ∀ (N n m : ℕ), N ≤ n → N ≤ m → edist (s n) (s m) < B N
t0 : Set α :=
  ⋂ (n : ℕ), closure[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] (⋃ (m : ℕ), ⋃ (_ : m ≥ n), (↑(s m) : Set α))
t : Closeds α := { carrier := t0, isClosed' := ⋯ }
I1 : ∀ (n : ℕ), ∀ x ∈ s n, ∃ y ∈ t0, edist x y ≤ 2 * B n
n : ℕ
x : α
xt0 : x ∈ t0
this_1 :
  Membership.mem (γ := Set α)
    (closure[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] (⋃ (m : ℕ), ⋃ (_ : m ≥ n), (↑(s m) : Set α))) x
z : α
Dxz : edist x z < B n
m : ℕ
m_ge_n : m ≥ n
hm : z ∈ (↑(s m) : Set α)
this : hausdorffEDist (↑(s m) : Set α) (↑(s n) : Set α) < B n
⊢ ∃ y ∈ s n, edist x y ≤ 2 * B n
have obtain : ∀ (y : α) (hy : y ∈ (↑(s n) : Set α)) (Dzy : edist z y < B n), ∃ y ∈ s n, edist x y ≤ 2 * B n
Tactic: obtain ⟨y : α, hy : y ∈ (s n : Set α), Dzy : edist z y < B n⟩ :=
      exists_edist_lt_of_hausdorffEDist_lt hm this