awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- 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
y x : (↑s : Type)
hxy : dist x y < 1
hne : x ≠ y
hne' : (↑x : ℕ → ℕ) ≠ (↑y : ℕ → ℕ)
dist' : dist x y = dist (↑x : ℕ → ℕ) (↑y : ℕ → ℕ)
n : ℕ := firstDiff (E := fun (n : ℕ) ↦ ℕ) (↑x : ℕ → ℕ) (↑y : ℕ → ℕ) - 1
diff_pos : 0 < firstDiff (E := fun (n : ℕ) ↦ ℕ) (↑x : ℕ → ℕ) (↑y : ℕ → ℕ)
hn : firstDiff (E := fun (n : ℕ) ↦ ℕ) (↑x : ℕ → ℕ) (↑y : ℕ → ℕ) = n + 1
B : (↑x : ℕ → ℕ) n = (↑y : ℕ → ℕ) n
⊢ dist (g x) (g y) ≤ 4 * (1 / 2) ^ (n + 1)
have dist : dist (g x) (u ((↑x : ℕ → ℕ) n)) + dist (g y) (u ((↑x : ℕ → ℕ) n)) =
  dist (g x) (u ((↑x : ℕ → ℕ) n)) + dist (g y) (u ((↑y : ℕ → ℕ) n))
Tactic: calc
      dist (g x) (g y) ≤ dist (g x) (u (x.1 n)) + dist (g y) (u (x.1 n)) :=
        dist_triangle_right _ _ _
      _ = dist (g x) (u (x.1 n)) + dist (g y) (u (y.1 n)) := by rw [← B]
      _ ≤ (1 / 2) ^ n + (1 / 2) ^ n := add_le_add (A x n) (A y n)
      _ = 4 * (1 / 2) ^ (n + 1) := by ring
-- 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.dist (g x) (u ((↑x : ℕ → ℕ) n)) ≤ (1 / 2) ^ n
y x : (↑s : Type)
hxy : Dist.dist x y < 1
hne : x ≠ y
hne' : (↑x : ℕ → ℕ) ≠ (↑y : ℕ → ℕ)
dist' : Dist.dist x y = Dist.dist (↑x : ℕ → ℕ) (↑y : ℕ → ℕ)
n : ℕ := firstDiff (E := fun (n : ℕ) ↦ ℕ) (↑x : ℕ → ℕ) (↑y : ℕ → ℕ) - 1
diff_pos : 0 < firstDiff (E := fun (n : ℕ) ↦ ℕ) (↑x : ℕ → ℕ) (↑y : ℕ → ℕ)
hn : firstDiff (E := fun (n : ℕ) ↦ ℕ) (↑x : ℕ → ℕ) (↑y : ℕ → ℕ) = n + 1
B : (↑x : ℕ → ℕ) n = (↑y : ℕ → ℕ) n
dist :
  Dist.dist (g x) (u ((↑x : ℕ → ℕ) n)) + Dist.dist (g y) (u ((↑x : ℕ → ℕ) n)) =
    Dist.dist (g x) (u ((↑x : ℕ → ℕ) n)) + Dist.dist (g y) (u ((↑y : ℕ → ℕ) n))
⊢ Dist.dist (g x) (g y) ≤ 4 * (1 / 2) ^ (n + 1)
have dist₁ : (1 / 2) ^ n + (1 / 2) ^ n = 4 * (1 / 2) ^ (n + 1)
Tactic: calc
      dist (g x) (g y) ≤ dist (g x) (u (x.1 n)) + dist (g y) (u (x.1 n)) :=
        dist_triangle_right _ _ _
      _ = dist (g x) (u (x.1 n)) + dist (g y) (u (y.1 n)) := by rw [← B]
      _ ≤ (1 / 2) ^ n + (1 / 2) ^ n := add_le_add (A x n) (A y n)
      _ = 4 * (1 / 2) ^ (n + 1) := by ring
-- 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
⊢ ∃ (f : (ℕ → ℕ) → α), Continuous[_, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] f ∧ Surjective f
have g_cont : Continuous[_, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] g
Tactic: have g_cont : Continuous g := by
    refine continuous_iff_continuousAt.2 fun y => ?_
    refine continuousAt_of_locally_lipschitz zero_lt_one 4 fun x hxy => ?_
    rcases eq_or_ne x y with (rfl | hne)
    · simp
    have hne' : x.1 ≠ y.1 := Subtype.coe_injective.ne hne
    have dist' : dist x y = dist x.1 y.1 := rfl
    let n := firstDiff x.1 y.1 - 1
    have diff_pos : 0 < firstDiff x.1 y.1 := by
      by_contra! h
      apply apply_firstDiff_ne hne'
      rw [Nat.le_zero.1 h]
      apply apply_eq_of_dist_lt _ le_rfl
      rw [pow_zero]
      exact hxy
    have hn : firstDiff x.1 y.1 = n + 1 := (Nat.succ_pred_eq_of_pos diff_pos).symm
    rw [dist', dist_eq_of_ne hne', hn]
    have B : x.1 n = y.1 n := mem_cylinder_firstDiff x.1 y.1 n (Nat.pred_lt diff_pos.ne')
    calc
      dist (g x) (g y) ≤ dist (g x) (u (x.1 n)) + dist (g y) (u (x.1 n)) :=
        dist_triangle_right _ _ _
      _ = dist (g x) (u (x.1 n)) + dist (g y) (u (y.1 n)) := by rw [← B]
      _ ≤ (1 / 2) ^ n + (1 / 2) ^ n := add_le_add (A x n) (A y n)
      _ = 4 * (1 / 2) ^ (n + 1) := by ring
-- 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
y : α
n : ℕ
⊢ ∃ (j : ℕ), y ∈ closedBall (u j) ((1 / 2) ^ n)
have rcases : ∀ (j : ℕ) (hj : dist y (u j) < (1 / 2) ^ n), ∃ (j : ℕ), y ∈ closedBall (u j) ((1 / 2) ^ n)
Tactic: rcases hu.exists_dist_lt y (by simp : (0 : ℝ) < (1 / 2) ^ n) with ⟨j, hj⟩
-- 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
y : α
⊢ ∃ (a : (↑s : Type)), g a = y
have this₁ : ∀ (n : ℕ), ∃ (j : ℕ), y ∈ closedBall (u j) ((1 / 2) ^ n)
Tactic: have : ∀ n : ℕ, ∃ j, y ∈ closedBall (u j) ((1 / 2) ^ n) := fun n ↦ by
      rcases hu.exists_dist_lt y (by simp : (0 : ℝ) < (1 / 2) ^ n) with ⟨j, hj⟩
      exact ⟨j, hj.le⟩
-- 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
y : α
this : ∀ (n : ℕ), ∃ (j : ℕ), y ∈ closedBall (u j) ((1 / 2) ^ n)
⊢ ∃ (a : (↑s : Type)), g a = y
have choose : ∀ (x : (n : ℕ) → ℕ) (hx : ∀ (n : ℕ), y ∈ closedBall (u (x n)) ((1 / 2) ^ n)), ∃ (a : (↑s : Type)), g a = y
Tactic: choose x hx using this
Cleared hypotheses: 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
y : α
x : (n : ℕ) → ℕ
hx : ∀ (n : ℕ), y ∈ closedBall (u (x n)) ((1 / 2) ^ n)
⊢ ∃ (a : (↑s : Type)), g a = y
have I : Set.Nonempty.{u_2} (α := α) (⋂ (n : ℕ), closedBall (u (x n)) ((1 / 2) ^ n))
Tactic: have I : (⋂ n : ℕ, closedBall (u (x n)) ((1 / 2) ^ n)).Nonempty := ⟨y, mem_iInter.2 hx⟩
-- 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
y : α
x : (n : ℕ) → ℕ
hx : ∀ (n : ℕ), y ∈ closedBall (u (x n)) ((1 / 2) ^ n)
I : Set.Nonempty.{u_2} (α := α) (⋂ (n : ℕ), closedBall (u (x n)) ((1 / 2) ^ n))
⊢ ∃ (a : (↑s : Type)), g a = y
have refine : g (Subtype.mk (p := fun (x : ℕ → ℕ) ↦ x ∈ s) x I) = y
Tactic: refine ⟨⟨x, I⟩, ?_⟩
-- 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
y : α
x : (n : ℕ) → ℕ
hx : ∀ (n : ℕ), y ∈ closedBall (u (x n)) ((1 / 2) ^ n)
I : Set.Nonempty.{u_2} (α := α) (⋂ (n : ℕ), closedBall (u (x n)) ((1 / 2) ^ n))
⊢ g (Subtype.mk (p := fun (x : ℕ → ℕ) ↦ x ∈ s) x I) = y
have refine : dist (g (Subtype.mk (p := fun (x : ℕ → ℕ) ↦ x ∈ s) x I)) y ≤ 0
Tactic: refine dist_le_zero.1 ?_
-- 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
y : α
x : (n : ℕ) → ℕ
hx : ∀ (n : ℕ), y ∈ closedBall (u (x n)) ((1 / 2) ^ n)
I : Set.Nonempty.{u_2} (α := α) (⋂ (n : ℕ), closedBall (u (x n)) ((1 / 2) ^ n))
⊢ dist (g (Subtype.mk (p := fun (x : ℕ → ℕ) ↦ x ∈ s) x I)) y ≤ 0
have J : ∀ (n : ℕ), dist (g (Subtype.mk (p := fun (x : ℕ → ℕ) ↦ x ∈ s) x I)) y ≤ (1 / 2) ^ n + (1 / 2) ^ n
Tactic: have J : ∀ n : ℕ, dist (g ⟨x, I⟩) y ≤ (1 / 2) ^ n + (1 / 2) ^ n := fun n =>
      calc
        dist (g ⟨x, I⟩) y ≤ dist (g ⟨x, I⟩) (u (x n)) + dist y (u (x n)) :=
          dist_triangle_right _ _ _
        _ ≤ (1 / 2) ^ n + (1 / 2) ^ n := add_le_add (A ⟨x, I⟩ n) (hx n)
-- 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
y : α
x : (n : ℕ) → ℕ
hx : ∀ (n : ℕ), y ∈ closedBall (u (x n)) ((1 / 2) ^ n)
I : Set.Nonempty.{u_2} (α := α) (⋂ (n : ℕ), closedBall (u (x n)) ((1 / 2) ^ n))
J : ∀ (n : ℕ), dist (g (Subtype.mk (p := fun (x : ℕ → ℕ) ↦ x ∈ s) x I)) y ≤ (1 / 2) ^ n + (1 / 2) ^ n
⊢ dist (g (Subtype.mk (p := fun (x : ℕ → ℕ) ↦ x ∈ s) x I)) y ≤ 0
have L : Tendsto (fun (n : ℕ) ↦ (1 / 2) ^ n + (1 / 2) ^ n) atTop (𝓝 (HAdd.hAdd (α := ℝ) 0 0))
Tactic: have L : Tendsto (fun n : ℕ => (1 / 2 : ℝ) ^ n + (1 / 2) ^ n) atTop (𝓝 (0 + 0)) :=
      (tendsto_pow_atTop_nhds_zero_of_lt_one I0.le I1).add
        (tendsto_pow_atTop_nhds_zero_of_lt_one I0.le I1)
-- 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
y : α
x : (n : ℕ) → ℕ
hx : ∀ (n : ℕ), y ∈ closedBall (u (x n)) ((1 / 2) ^ n)
I : Set.Nonempty.{u_2} (α := α) (⋂ (n : ℕ), closedBall (u (x n)) ((1 / 2) ^ n))
J : ∀ (n : ℕ), dist (g (Subtype.mk (p := fun (x : ℕ → ℕ) ↦ x ∈ s) x I)) y ≤ (1 / 2) ^ n + (1 / 2) ^ n
L : Tendsto (fun (n : ℕ) ↦ (1 / 2) ^ n + (1 / 2) ^ n) atTop (𝓝 (HAdd.hAdd (α := ℝ) 0 0))
⊢ dist (g (Subtype.mk (p := fun (x : ℕ → ℕ) ↦ x ∈ s) x I)) y ≤ 0
have L₁ : Tendsto (β := ℝ) (fun (n : ℕ) ↦ (1 / 2) ^ n + (1 / 2) ^ n) atTop (𝓝 0)
Tactic: rw [add_zero] at L
Cleared hypotheses: L
-- 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
⊢ ∃ (f : (ℕ → ℕ) → α), Continuous[_, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] f ∧ Surjective f
have g_surj : Surjective g
Tactic: have g_surj : Surjective g := fun y ↦ by
    have : ∀ n : ℕ, ∃ j, y ∈ closedBall (u j) ((1 / 2) ^ n) := fun n ↦ by
      rcases hu.exists_dist_lt y (by simp : (0 : ℝ) < (1 / 2) ^ n) with ⟨j, hj⟩
      exact ⟨j, hj.le⟩
    choose x hx using this
    have I : (⋂ n : ℕ, closedBall (u (x n)) ((1 / 2) ^ n)).Nonempty := ⟨y, mem_iInter.2 hx⟩
    refine ⟨⟨x, I⟩, ?_⟩
    refine dist_le_zero.1 ?_
    have J : ∀ n : ℕ, dist (g ⟨x, I⟩) y ≤ (1 / 2) ^ n + (1 / 2) ^ n := fun n =>
      calc
        dist (g ⟨x, I⟩) y ≤ dist (g ⟨x, I⟩) (u (x n)) + dist y (u (x n)) :=
          dist_triangle_right _ _ _
        _ ≤ (1 / 2) ^ n + (1 / 2) ^ n := add_le_add (A ⟨x, I⟩ n) (hx n)
    have L : Tendsto (fun n : ℕ => (1 / 2 : ℝ) ^ n + (1 / 2) ^ n) atTop (𝓝 (0 + 0)) :=
      (tendsto_pow_atTop_nhds_zero_of_lt_one I0.le I1).add
        (tendsto_pow_atTop_nhds_zero_of_lt_one I0.le I1)
    rw [add_zero] at L
    exact ge_of_tendsto' L J
-- 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
⊢ IsClosed s
have refine : ∀ (x : ℕ → ℕ) (hx : ClusterPt x (𝓟 s)), x ∈ s
Tactic: refine isClosed_iff_clusterPt.mpr fun x hx ↦ ?_
-- 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)
⊢ Tendsto (β := ℝ) (fun (n : ℕ) ↦ diam (closedBall (u (x n)) ((1 / 2) ^ n))) atTop (𝓝 0)
have this₁ : Tendsto (fun (n : ℕ) ↦ 2 * (1 / 2) ^ n) atTop (𝓝 (HMul.hMul (α := ℝ) 2 0))
Tactic: 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 _
-- 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_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)
this : Tendsto (fun (n : ℕ) ↦ 2 * (1 / 2) ^ n) atTop (𝓝 (HMul.hMul (α := ℝ) 2 0))
⊢ Tendsto (β := ℝ) (fun (n : ℕ) ↦ diam (closedBall (u (x n)) ((1 / 2) ^ n))) atTop (𝓝 0)
have this₁ : Tendsto (β := ℝ) (fun (n : ℕ) ↦ 2 * (1 / 2) ^ n) atTop (𝓝 0)
Tactic: rw [mul_zero] at this
Cleared hypotheses: 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)
⊢ x ∈ s
have L : Tendsto (β := ℝ) (fun (n : ℕ) ↦ diam (closedBall (u (x n)) ((1 / 2) ^ n))) atTop (𝓝 0)
Tactic: 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
-- 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)
⊢ x ∈ s
have refine : ∀ (N : ℕ), Set.Nonempty.{u_2} (α := α) (⋂ (n : ℕ), ⋂ (_ : n ≤ N), closedBall (u (x n)) ((1 / 2) ^ n))
Tactic: refine nonempty_iInter_of_nonempty_biInter (fun n => isClosed_closedBall)
      (fun n => isBounded_closedBall) (fun N ↦ ?_) L
-- 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
⊢ Set.Nonempty.{u_2} (α := α) (⋂ (n : ℕ), ⋂ (_ : n ≤ N), closedBall (u (x n)) ((1 / 2) ^ n))
have obtain : ∀ (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))
Tactic: 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))
-- 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
⊢ 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))
have refine : ∀ (n : ℕ) (hn : n ≤ N), closedBall (u (x n)) ((1 / 2) ^ n) = closedBall (u (y n)) ((1 / 2) ^ n)
Tactic: refine iInter_congr fun n ↦ iInter_congr fun hn ↦ ?_