awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Module: Mathlib.Topology.MetricSpace.CantorScheme
Goal:
β : Type u_1
α : Type u_2
A : List β → Set α
inst : PseudoMetricSpace α
inst_1 : TopologicalSpace β
inst_2 : DiscreteTopology β
hA : VanishingDiam A
rw :
  ∀ (a : (↑((inducedMap A).fst (β := fun (s : Set (ℕ → β)) ↦ (↑s : Type u_1) → α)) : Type u_1)),
    ∀ ε > 0,
      ∀ᶠ (x : (↑((inducedMap A).fst (β := fun (s : Set (ℕ → β)) ↦ (↑s : Type u_1) → α)) : Type u_1)) in 𝓝 a,
        dist ((inducedMap A).snd (β := fun (s : Set (ℕ → β)) ↦ (↑s : Type u_1) → α) x)
            ((inducedMap A).snd (β := fun (s : Set (ℕ → β)) ↦ (↑s : Type u_1) → α) a) <
          ε
⊢ Continuous[instTopologicalSpaceSubtype, PseudoMetricSpace.toUniformSpace.toTopologicalSpace]
    ((inducedMap A).snd (β := fun (s : Set (ℕ → β)) ↦ (↑s : Type u_1) → α))

Tactic:
rw [Metric.continuous_iff']

Premises used: Metric.continuous_iff'
-- Module: Mathlib.Topology.MetricSpace.CantorScheme
Goal:
β : Type u_1
α : Type u_2
A : List β → Set α
inst : PseudoMetricSpace α
inst_1 : TopologicalSpace β
inst_2 : DiscreteTopology β
hA : VanishingDiam A
x : (↑((inducedMap A).fst (β := fun (s : Set (ℕ → β)) ↦ (↑s : Type u_1) → α)) : Type u_1)
ε : ℝ
ε_pos : ε > 0
obtain :
  ∀ (n : ℕ) (hn : ∀ y ∈ A (res (↑x : ℕ → β) n), ∀ z ∈ A (res (↑x : ℕ → β) n), dist y z < ε),
    ∀ᶠ (x_1 : (↑((inducedMap A).fst (β := fun (s : Set (ℕ → β)) ↦ (↑s : Type u_1) → α)) : Type u_1)) in 𝓝 x,
      dist ((inducedMap A).snd (β := fun (s : Set (ℕ → β)) ↦ (↑s : Type u_1) → α) x_1)
          ((inducedMap A).snd (β := fun (s : Set (ℕ → β)) ↦ (↑s : Type u_1) → α) x) <
        ε
⊢ ∀ᶠ (x_1 : (↑((inducedMap A).fst (β := fun (s : Set (ℕ → β)) ↦ (↑s : Type u_1) → α)) : Type u_1)) in 𝓝 x,
    dist ((inducedMap A).snd (β := fun (s : Set (ℕ → β)) ↦ (↑s : Type u_1) → α) x_1)
        ((inducedMap A).snd (β := fun (s : Set (ℕ → β)) ↦ (↑s : Type u_1) → α) x) <
      ε

Tactic:
obtain ⟨n, hn⟩ := hA.dist_lt _ ε_pos x

Premises used: CantorScheme.VanishingDiam.dist_lt, Subtype.val
-- Module: Mathlib.Topology.MetricSpace.CantorScheme
Goal:
β : Type u_1
α : Type u_2
A : List β → Set α
inst : PseudoMetricSpace α
inst_1 : TopologicalSpace β
inst_2 : DiscreteTopology β
hA : VanishingDiam A
x : (↑((inducedMap A).fst (β := fun (s : Set (ℕ → β)) ↦ (↑s : Type u_1) → α)) : Type u_1)
ε : ℝ
ε_pos : ε > 0
n : ℕ
hn : ∀ y ∈ A (res (↑x : ℕ → β) n), ∀ z ∈ A (res (↑x : ℕ → β) n), dist y z < ε
rw :
  ∃ (t : Set (↑((inducedMap A).fst (β := fun (s : Set (ℕ → β)) ↦ (↑s : Type u_1) → α)) : Type u_1)),
    (∀ y ∈ t,
        dist ((inducedMap A).snd (β := fun (s : Set (ℕ → β)) ↦ (↑s : Type u_1) → α) y)
            ((inducedMap A).snd (β := fun (s : Set (ℕ → β)) ↦ (↑s : Type u_1) → α) x) <
          ε) ∧
      IsOpen[instTopologicalSpaceSubtype] t ∧ x ∈ t
⊢ ∀ᶠ (x_1 : (↑((inducedMap A).fst (β := fun (s : Set (ℕ → β)) ↦ (↑s : Type u_1) → α)) : Type u_1)) in 𝓝 x,
    dist ((inducedMap A).snd (β := fun (s : Set (ℕ → β)) ↦ (↑s : Type u_1) → α) x_1)
        ((inducedMap A).snd (β := fun (s : Set (ℕ → β)) ↦ (↑s : Type u_1) → α) x) <
      ε

Tactic:
rw [_root_.eventually_nhds_iff]

Premises used: eventually_nhds_iff
-- Module: Mathlib.Topology.MetricSpace.CantorScheme
Goal:
β : Type u_1
α : Type u_2
A : List β → Set α
inst : PseudoMetricSpace α
inst_1 : TopologicalSpace β
inst_2 : DiscreteTopology β
hA : VanishingDiam A
x : (↑((inducedMap A).fst (β := fun (s : Set (ℕ → β)) ↦ (↑s : Type u_1) → α)) : Type u_1)
ε : ℝ
ε_pos : ε > 0
n : ℕ
hn : ∀ y ∈ A (res (↑x : ℕ → β) n), ∀ z ∈ A (res (↑x : ℕ → β) n), dist y z < ε
⊢ Membership.mem (γ := Set (↑((inducedMap A).fst (β := fun (s : Set (ℕ → β)) ↦ (↑s : Type u_1) → α)) : Type u_1))
    (Subtype.val (p := fun (x : ℕ → β) ↦ x ∈ (inducedMap A).fst (β := fun (s : Set (ℕ → β)) ↦ (↑s : Type u_1) → α)) ⁻¹'
      cylinder (E := fun (n : ℕ) ↦ β) (↑x : ℕ → β) n)
    x

Tactic:
simp
-- Module: Mathlib.Topology.MetricSpace.CantorScheme
Goal:
β : Type u_1
α : Type u_2
A : List β → Set α
inst : PseudoMetricSpace α
inst_1 : TopologicalSpace β
inst_2 : DiscreteTopology β
hA : VanishingDiam A
x : (↑((inducedMap A).fst (β := fun (s : Set (ℕ → β)) ↦ (↑s : Type u_1) → α)) : Type u_1)
ε : ℝ
ε_pos : ε > 0
n : ℕ
hn : ∀ y ∈ A (res (↑x : ℕ → β) n), ∀ z ∈ A (res (↑x : ℕ → β) n), dist y z < ε
refine_1 :
  ∀ (y : (↑((inducedMap A).fst (β := fun (s : Set (ℕ → β)) ↦ (↑s : Type u_1) → α)) : Type u_1)),
    Membership.mem (γ := Set (↑((inducedMap A).fst (β := fun (s : Set (ℕ → β)) ↦ (↑s : Type u_1) → α)) : Type u_1))
        (Subtype.val (p := fun (x : ℕ → β) ↦
            x ∈ (inducedMap A).fst (β := fun (s : Set (ℕ → β)) ↦ (↑s : Type u_1) → α)) ⁻¹'
          cylinder (E := fun (n : ℕ) ↦ β) (↑x : ℕ → β) n)
        y →
      dist ((inducedMap A).snd (β := fun (s : Set (ℕ → β)) ↦ (↑s : Type u_1) → α) y)
          ((inducedMap A).snd (β := fun (s : Set (ℕ → β)) ↦ (↑s : Type u_1) → α) x) <
        ε
refine_2 :
  IsOpen[instTopologicalSpaceSubtype]
    (Subtype.val (p := fun (x : ℕ → β) ↦ x ∈ (inducedMap A).fst (β := fun (s : Set (ℕ → β)) ↦ (↑s : Type u_1) → α)) ⁻¹'
      cylinder (E := fun (n : ℕ) ↦ β) (↑x : ℕ → β) n)
⊢ ∃ (t : Set (↑((inducedMap A).fst (β := fun (s : Set (ℕ → β)) ↦ (↑s : Type u_1) → α)) : Type u_1)),
    (∀ y ∈ t,
        dist ((inducedMap A).snd (β := fun (s : Set (ℕ → β)) ↦ (↑s : Type u_1) → α) y)
            ((inducedMap A).snd (β := fun (s : Set (ℕ → β)) ↦ (↑s : Type u_1) → α) x) <
          ε) ∧
      IsOpen[instTopologicalSpaceSubtype] t ∧ x ∈ t

Tactic:
refine ⟨(↑)⁻¹' cylinder x.1 n, ?_, ?_, by simp⟩
-- Module: Mathlib.Topology.MetricSpace.CantorScheme
Goal:
β : Type u_1
α : Type u_2
A : List β → Set α
inst : PseudoMetricSpace α
inst_1 : TopologicalSpace β
inst_2 : DiscreteTopology β
hA : VanishingDiam A
x : (↑((inducedMap A).fst (β := fun (s : Set (ℕ → β)) ↦ (↑s : Type u_1) → α)) : Type u_1)
ε : ℝ
ε_pos : ε > 0
n : ℕ
hn : ∀ y ∈ A (res (↑x : ℕ → β) n), ∀ z ∈ A (res (↑x : ℕ → β) n), dist y z < ε
y : (↑((inducedMap A).fst (β := fun (s : Set (ℕ → β)) ↦ (↑s : Type u_1) → α)) : Type u_1)
hyx :
  Membership.mem (γ := Set (↑((inducedMap A).fst (β := fun (s : Set (ℕ → β)) ↦ (↑s : Type u_1) → α)) : Type u_1))
    (Subtype.val (p := fun (x : ℕ → β) ↦ x ∈ (inducedMap A).fst (β := fun (s : Set (ℕ → β)) ↦ (↑s : Type u_1) → α)) ⁻¹'
      cylinder (E := fun (n : ℕ) ↦ β) (↑x : ℕ → β) n)
    y
⊢ res (↑y : ℕ → β) n = res (↑x : ℕ → β) n

Tactic:
rw [mem_preimage, Subtype.coe_mk, cylinder_eq_res, mem_setOf] at hyx

Premises used: Subtype.coe_mk, PiNat.cylinder_eq_res, Set.mem_setOf, Set.mem_preimage
-- Module: Mathlib.Topology.MetricSpace.CantorScheme
Goal:
β : Type u_1
α : Type u_2
A : List β → Set α
inst : PseudoMetricSpace α
inst_1 : TopologicalSpace β
inst_2 : DiscreteTopology β
hA : VanishingDiam A
x : (↑((inducedMap A).fst (β := fun (s : Set (ℕ → β)) ↦ (↑s : Type u_1) → α)) : Type u_1)
ε : ℝ
ε_pos : ε > 0
n : ℕ
hn : ∀ y ∈ A (res (↑x : ℕ → β) n), ∀ z ∈ A (res (↑x : ℕ → β) n), dist y z < ε
y : (↑((inducedMap A).fst (β := fun (s : Set (ℕ → β)) ↦ (↑s : Type u_1) → α)) : Type u_1)
hyx : res (↑y : ℕ → β) n = res (↑x : ℕ → β) n
refine_1 : (inducedMap A).snd (β := fun (s : Set (ℕ → β)) ↦ (↑s : Type u_1) → α) y ∈ A (res (↑x : ℕ → β) n)
refine_1₁ : (inducedMap A).snd (β := fun (s : Set (ℕ → β)) ↦ (↑s : Type u_1) → α) x ∈ A (res (↑x : ℕ → β) n)
⊢ dist ((inducedMap A).snd (β := fun (s : Set (ℕ → β)) ↦ (↑s : Type u_1) → α) y)
      ((inducedMap A).snd (β := fun (s : Set (ℕ → β)) ↦ (↑s : Type u_1) → α) x) <
    ε

Tactic:
apply hn
-- Module: Mathlib.Topology.MetricSpace.CantorScheme
Goal:
β : Type u_1
α : Type u_2
A : List β → Set α
inst : PseudoMetricSpace α
inst_1 : TopologicalSpace β
inst_2 : DiscreteTopology β
hA : VanishingDiam A
x : (↑((inducedMap A).fst (β := fun (s : Set (ℕ → β)) ↦ (↑s : Type u_1) → α)) : Type u_1)
ε : ℝ
ε_pos : ε > 0
n : ℕ
hn : ∀ y ∈ A (res (↑x : ℕ → β) n), ∀ z ∈ A (res (↑x : ℕ → β) n), dist y z < ε
y : (↑((inducedMap A).fst (β := fun (s : Set (ℕ → β)) ↦ (↑s : Type u_1) → α)) : Type u_1)
hyx : res (↑y : ℕ → β) n = res (↑x : ℕ → β) n
refine_1 : (inducedMap A).snd (β := fun (s : Set (ℕ → β)) ↦ (↑s : Type u_1) → α) y ∈ A (res (↑y : ℕ → β) n)
⊢ (inducedMap A).snd (β := fun (s : Set (ℕ → β)) ↦ (↑s : Type u_1) → α) y ∈ A (res (↑x : ℕ → β) n)

Tactic:
rw [← hyx]
-- Module: Mathlib.Topology.MetricSpace.CantorScheme
Goal:
β : Type u_1
α : Type u_2
A : List β → Set α
inst : PseudoMetricSpace α
inst_1 : TopologicalSpace β
inst_2 : DiscreteTopology β
hA : VanishingDiam A
x : (↑((inducedMap A).fst (β := fun (s : Set (ℕ → β)) ↦ (↑s : Type u_1) → α)) : Type u_1)
ε : ℝ
ε_pos : ε > 0
n : ℕ
hn : ∀ y ∈ A (res (↑x : ℕ → β) n), ∀ z ∈ A (res (↑x : ℕ → β) n), dist y z < ε
y : (↑((inducedMap A).fst (β := fun (s : Set (ℕ → β)) ↦ (↑s : Type u_1) → α)) : Type u_1)
hyx : res (↑y : ℕ → β) n = res (↑x : ℕ → β) n
⊢ (inducedMap A).snd (β := fun (s : Set (ℕ → β)) ↦ (↑s : Type u_1) → α) y ∈ A (res (↑y : ℕ → β) n)

Tactic:
apply map_mem

Premises used: CantorScheme.map_mem
-- Module: Mathlib.Topology.MetricSpace.CantorScheme
Goal:
β : Type u_1
α : Type u_2
A : List β → Set α
inst : PseudoMetricSpace α
inst_1 : TopologicalSpace β
inst_2 : DiscreteTopology β
hA : VanishingDiam A
x : (↑((inducedMap A).fst (β := fun (s : Set (ℕ → β)) ↦ (↑s : Type u_1) → α)) : Type u_1)
ε : ℝ
ε_pos : ε > 0
n : ℕ
hn : ∀ y ∈ A (res (↑x : ℕ → β) n), ∀ z ∈ A (res (↑x : ℕ → β) n), dist y z < ε
y : (↑((inducedMap A).fst (β := fun (s : Set (ℕ → β)) ↦ (↑s : Type u_1) → α)) : Type u_1)
hyx : res (↑y : ℕ → β) n = res (↑x : ℕ → β) n
⊢ (inducedMap A).snd (β := fun (s : Set (ℕ → β)) ↦ (↑s : Type u_1) → α) x ∈ A (res (↑x : ℕ → β) n)

Tactic:
apply map_mem

Premises used: CantorScheme.map_mem
-- Module: Mathlib.Topology.MetricSpace.CantorScheme
Goal:
β : Type u_1
α : Type u_2
A : List β → Set α
inst : PseudoMetricSpace α
inst_1 : TopologicalSpace β
inst_2 : DiscreteTopology β
hA : VanishingDiam A
x : (↑((inducedMap A).fst (β := fun (s : Set (ℕ → β)) ↦ (↑s : Type u_1) → α)) : Type u_1)
ε : ℝ
ε_pos : ε > 0
n : ℕ
hn : ∀ y ∈ A (res (↑x : ℕ → β) n), ∀ z ∈ A (res (↑x : ℕ → β) n), dist y z < ε
refine_2 : IsOpen[Pi.topologicalSpace] (cylinder (E := fun (n : ℕ) ↦ β) (↑x : ℕ → β) n)
⊢ IsOpen[instTopologicalSpaceSubtype]
    (Subtype.val (p := fun (x : ℕ → β) ↦ x ∈ (inducedMap A).fst (β := fun (s : Set (ℕ → β)) ↦ (↑s : Type u_1) → α)) ⁻¹'
      cylinder (E := fun (n : ℕ) ↦ β) (↑x : ℕ → β) n)

Tactic:
apply continuous_subtype_val.isOpen_preimage

Premises used: continuous_subtype_val, Continuous.isOpen_preimage
-- Module: Mathlib.Topology.MetricSpace.CantorScheme
Goal:
β : Type u_1
α : Type u_2
A : List β → Set α
inst : PseudoMetricSpace α
inst_1 : TopologicalSpace β
inst_2 : DiscreteTopology β
hA : VanishingDiam A
x : (↑((inducedMap A).fst (β := fun (s : Set (ℕ → β)) ↦ (↑s : Type u_1) → α)) : Type u_1)
ε : ℝ
ε_pos : ε > 0
n : ℕ
hn : ∀ y ∈ A (res (↑x : ℕ → β) n), ∀ z ∈ A (res (↑x : ℕ → β) n), dist y z < ε
⊢ IsOpen[Pi.topologicalSpace] (cylinder (E := fun (n : ℕ) ↦ β) (↑x : ℕ → β) n)

Tactic:
apply isOpen_cylinder

Premises used: PiNat.isOpen_cylinder
-- Module: Mathlib.Topology.MetricSpace.CantorScheme
Goal:
β : Type u_1
α : Type u_2
A : List β → Set α
inst : PseudoMetricSpace α
inst_1 : CompleteSpace α
hdiam : VanishingDiam A
hanti : ClosureAntitone A
hnonempty : ∀ (l : List β), (A l).Nonempty
rw : ∀ (x : ℕ → β), x ∈ (inducedMap A).fst (β := fun (s : Set (ℕ → β)) ↦ (↑s : Type u_1) → α)
⊢ (inducedMap A).fst (β := fun (s : Set (ℕ → β)) ↦ (↑s : Type u_1) → α) = univ

Tactic:
rw [eq_univ_iff_forall]

Premises used: Set.eq_univ_iff_forall
-- Module: Mathlib.Topology.MetricSpace.CantorScheme
Goal:
β : Type u_1
α : Type u_2
A : List β → Set α
inst : PseudoMetricSpace α
inst_1 : CompleteSpace α
hdiam : VanishingDiam A
hanti : ClosureAntitone A
hnonempty : ∀ (l : List β), (A l).Nonempty
x : ℕ → β
choose :
  ∀ (u : (n : ℕ) → α) (hu : ∀ (n : ℕ), u n ∈ A (res x n)),
    x ∈ (inducedMap A).fst (β := fun (s : Set (ℕ → β)) ↦ (↑s : Type u_1) → α)
⊢ x ∈ (inducedMap A).fst (β := fun (s : Set (ℕ → β)) ↦ (↑s : Type u_1) → α)

Tactic:
choose u hu using fun n => hnonempty (res x n)

Premises used: PiNat.res
-- Module: Mathlib.Topology.MetricSpace.CantorScheme
Goal:
β : Type u_1
α : Type u_2
A : List β → Set α
inst : PseudoMetricSpace α
inst_1 : CompleteSpace α
hdiam : VanishingDiam A
hanti : ClosureAntitone A
hnonempty : ∀ (l : List β), (A l).Nonempty
x : ℕ → β
u : (n : ℕ) → α
hu : ∀ (n : ℕ), u n ∈ A (res x n)
⊢ Antitone.{0, u_2} (β := Set α) fun (n : ℕ) ↦ A (res x n)

Tactic:
antitone_nat_of_succ_le ?_

Premises used: antitone_nat_of_succ_le
-- Module: Mathlib.Topology.MetricSpace.CantorScheme
Goal:
β : Type u_1
α : Type u_2
A : List β → Set α
inst : PseudoMetricSpace α
inst_1 : CompleteSpace α
hdiam : VanishingDiam A
hanti : ClosureAntitone A
hnonempty : ∀ (l : List β), (A l).Nonempty
x : ℕ → β
u : (n : ℕ) → α
hu : ∀ (n : ℕ), u n ∈ A (res x n)
refine : ∀ (n : ℕ), A (res x (n + 1)) ≤ A (res x n)
⊢ Antitone.{0, u_2} (β := Set α) fun (n : ℕ) ↦ A (res x n)

Tactic:
refine antitone_nat_of_succ_le ?_

Premises used: antitone_nat_of_succ_le
-- Module: Mathlib.Topology.MetricSpace.CantorScheme
Goal:
β : Type u_1
α : Type u_2
A : List β → Set α
inst : PseudoMetricSpace α
inst_1 : CompleteSpace α
hdiam : VanishingDiam A
hanti : ClosureAntitone A
hnonempty : ∀ (l : List β), (A l).Nonempty
x : ℕ → β
u : (n : ℕ) → α
hu : ∀ (n : ℕ), u n ∈ A (res x n)
n : ℕ
⊢ A (res x (n + 1)) ≤ A (res x n)

Tactic:
apply hanti.antitone

Premises used: CantorScheme.ClosureAntitone.antitone
-- Module: Mathlib.Topology.MetricSpace.CantorScheme
Goal:
β : Type u_1
α : Type u_2
A : List β → Set α
inst : PseudoMetricSpace α
inst_1 : CompleteSpace α
hdiam : VanishingDiam A
hanti : ClosureAntitone A
hnonempty : ∀ (l : List β), (A l).Nonempty
x : ℕ → β
u : (n : ℕ) → α
hu : ∀ (n : ℕ), u n ∈ A (res x n)
⊢ Antitone.{0, u_2} (β := Set α) fun (n : ℕ) ↦ A (res x n)

Tactic:
have : Antitone fun n : ℕ => A (res x n) := by
      refine antitone_nat_of_succ_le ?_
      intro n
      apply hanti.antitone
-- Module: Mathlib.Topology.MetricSpace.CantorScheme
Goal:
β : Type u_1
α : Type u_2
A : List β → Set α
inst : PseudoMetricSpace α
inst_1 : CompleteSpace α
hdiam : VanishingDiam A
hanti : ClosureAntitone A
hnonempty : ∀ (l : List β), (A l).Nonempty
x : ℕ → β
u : (n : ℕ) → α
hu : ∀ (n : ℕ), u n ∈ A (res x n)
this : Antitone.{0, u_2} (β := Set α) fun (n : ℕ) ↦ A (res x n)
n m : ℕ
hnm : n ≤ m
⊢ u m ∈ A (res x n)

Tactic:
this hnm (hu _)

Premises used: Set.Mem
-- Module: Mathlib.Topology.MetricSpace.CantorScheme
Goal:
β : Type u_1
α : Type u_2
A : List β → Set α
inst : PseudoMetricSpace α
inst_1 : CompleteSpace α
hdiam : VanishingDiam A
hanti : ClosureAntitone A
hnonempty : ∀ (l : List β), (A l).Nonempty
x : ℕ → β
u : (n : ℕ) → α
hu : ∀ (n : ℕ), u n ∈ A (res x n)
this : Antitone.{0, u_2} (β := Set α) fun (n : ℕ) ↦ A (res x n)
n m : ℕ
hnm : n ≤ m
⊢ u m ∈ A (res x n)

Tactic:
exact this hnm (hu _)
-- Module: Mathlib.Topology.MetricSpace.CantorScheme
Goal:
β : Type u_1
α : Type u_2
A : List β → Set α
inst : PseudoMetricSpace α
inst_1 : CompleteSpace α
hdiam : VanishingDiam A
hanti : ClosureAntitone A
hnonempty : ∀ (l : List β), (A l).Nonempty
x : ℕ → β
u : (n : ℕ) → α
hu : ∀ (n : ℕ), u n ∈ A (res x n)
⊢ ∀ (n m : ℕ), n ≤ m → u m ∈ A (res x n)

Tactic:
have umem : ∀ n m : ℕ, n ≤ m → u m ∈ A (res x n) := by
    have : Antitone fun n : ℕ => A (res x n) := by
      refine antitone_nat_of_succ_le ?_
      intro n
      apply hanti.antitone
    intro n m hnm
    exact this hnm (hu _)
-- Module: Mathlib.Topology.MetricSpace.CantorScheme
Goal:
β : Type u_1
α : Type u_2
A : List β → Set α
inst : PseudoMetricSpace α
inst_1 : CompleteSpace α
hdiam : VanishingDiam A
hanti : ClosureAntitone A
hnonempty : ∀ (l : List β), (A l).Nonempty
x : ℕ → β
u : (n : ℕ) → α
hu : ∀ (n : ℕ), u n ∈ A (res x n)
umem : ∀ (n m : ℕ), n ≤ m → u m ∈ A (res x n)
rw : ∀ ε > 0, ∃ (N : ℕ), ∀ m ≥ N, ∀ n ≥ N, dist (u m) (u n) < ε
⊢ CauchySeq u

Tactic:
rw [Metric.cauchySeq_iff]

Premises used: Metric.cauchySeq_iff
-- Module: Mathlib.Topology.MetricSpace.CantorScheme
Goal:
β : Type u_1
α : Type u_2
A : List β → Set α
inst : PseudoMetricSpace α
inst_1 : CompleteSpace α
hdiam : VanishingDiam A
hanti : ClosureAntitone A
hnonempty : ∀ (l : List β), (A l).Nonempty
x : ℕ → β
u : (n : ℕ) → α
hu : ∀ (n : ℕ), u n ∈ A (res x n)
umem : ∀ (n m : ℕ), n ≤ m → u m ∈ A (res x n)
ε : ℝ
ε_pos : ε > 0
obtain :
  ∀ (n : ℕ) (hn : ∀ y ∈ A (res x n), ∀ z ∈ A (res x n), dist y z < ε), ∃ (N : ℕ), ∀ m ≥ N, ∀ n ≥ N, dist (u m) (u n) < ε
⊢ ∃ (N : ℕ), ∀ m ≥ N, ∀ n ≥ N, dist (u m) (u n) < ε

Tactic:
obtain ⟨n, hn⟩ := hdiam.dist_lt _ ε_pos x

Premises used: CantorScheme.VanishingDiam.dist_lt
-- Module: Mathlib.Topology.MetricSpace.CantorScheme
Goal:
β : Type u_1
α : Type u_2
A : List β → Set α
inst : PseudoMetricSpace α
inst_1 : CompleteSpace α
hdiam : VanishingDiam A
hanti : ClosureAntitone A
hnonempty : ∀ (l : List β), (A l).Nonempty
x : ℕ → β
u : (n : ℕ) → α
hu : ∀ (n : ℕ), u n ∈ A (res x n)
umem : ∀ (n m : ℕ), n ≤ m → u m ∈ A (res x n)
ε : ℝ
ε_pos : ε > 0
n : ℕ
hn : ∀ y ∈ A (res x n), ∀ z ∈ A (res x n), dist y z < ε
h : ∀ m ≥ n, ∀ n_1 ≥ n, dist (u m) (u n_1) < ε
⊢ ∃ (N : ℕ), ∀ m ≥ N, ∀ n ≥ N, dist (u m) (u n) < ε

Tactic:
use n
-- Module: Mathlib.Topology.MetricSpace.CantorScheme
Goal:
β : Type u_1
α : Type u_2
A : List β → Set α
inst : PseudoMetricSpace α
inst_1 : CompleteSpace α
hdiam : VanishingDiam A
hanti : ClosureAntitone A
hnonempty : ∀ (l : List β), (A l).Nonempty
x : ℕ → β
u : (n : ℕ) → α
hu : ∀ (n : ℕ), u n ∈ A (res x n)
umem : ∀ (n m : ℕ), n ≤ m → u m ∈ A (res x n)
ε : ℝ
ε_pos : ε > 0
n : ℕ
hn : ∀ y ∈ A (res x n), ∀ z ∈ A (res x n), dist y z < ε
m₀ : ℕ
hm₀ : m₀ ≥ n
m₁ : ℕ
hm₁ : m₁ ≥ n
h : u m₀ ∈ A (res x n)
h₁ : u m₁ ∈ A (res x n)
⊢ dist (u m₀) (u m₁) < ε

Tactic:
apply hn
-- Module: Mathlib.Topology.MetricSpace.CantorScheme
Goal:
β : Type u_1
α : Type u_2
A : List β → Set α
inst : PseudoMetricSpace α
inst_1 : CompleteSpace α
hdiam : VanishingDiam A
hanti : ClosureAntitone A
hnonempty : ∀ (l : List β), (A l).Nonempty
x : ℕ → β
u : (n : ℕ) → α
hu : ∀ (n : ℕ), u n ∈ A (res x n)
umem : ∀ (n m : ℕ), n ≤ m → u m ∈ A (res x n)
ε : ℝ
ε_pos : ε > 0
n : ℕ
hn : ∀ y ∈ A (res x n), ∀ z ∈ A (res x n), dist y z < ε
m₀ : ℕ
hm₀ : m₀ ≥ n
m₁ : ℕ
hm₁ : m₁ ≥ n
h : n ≤ m₀
⊢ u m₀ ∈ A (res x n)

Tactic:
apply umem
-- Module: Mathlib.Topology.MetricSpace.CantorScheme
Goal:
β : Type u_1
α : Type u_2
A : List β → Set α
inst : PseudoMetricSpace α
inst_1 : CompleteSpace α
hdiam : VanishingDiam A
hanti : ClosureAntitone A
hnonempty : ∀ (l : List β), (A l).Nonempty
x : ℕ → β
u : (n : ℕ) → α
hu : ∀ (n : ℕ), u n ∈ A (res x n)
umem : ∀ (n m : ℕ), n ≤ m → u m ∈ A (res x n)
ε : ℝ
ε_pos : ε > 0
n : ℕ
hn : ∀ y ∈ A (res x n), ∀ z ∈ A (res x n), dist y z < ε
m₀ : ℕ
hm₀ : m₀ ≥ n
m₁ : ℕ
hm₁ : m₁ ≥ n
h : n ≤ m₁
⊢ u m₁ ∈ A (res x n)

Tactic:
apply umem
-- Module: Mathlib.Topology.MetricSpace.CantorScheme
Goal:
β : Type u_1
α : Type u_2
A : List β → Set α
inst : PseudoMetricSpace α
inst_1 : CompleteSpace α
hdiam : VanishingDiam A
hanti : ClosureAntitone A
hnonempty : ∀ (l : List β), (A l).Nonempty
x : ℕ → β
u : (n : ℕ) → α
hu : ∀ (n : ℕ), u n ∈ A (res x n)
umem : ∀ (n m : ℕ), n ≤ m → u m ∈ A (res x n)
ε : ℝ
ε_pos : ε > 0
n : ℕ
hn : ∀ y ∈ A (res x n), ∀ z ∈ A (res x n), dist y z < ε
m₀ : ℕ
hm₀ : m₀ ≥ n
m₁ : ℕ
hm₁ : m₁ ≥ n
⊢ n ≤ m₀

Tactic:
assumption
-- Module: Mathlib.Topology.MetricSpace.CantorScheme
Goal:
β : Type u_1
α : Type u_2
A : List β → Set α
inst : PseudoMetricSpace α
inst_1 : CompleteSpace α
hdiam : VanishingDiam A
hanti : ClosureAntitone A
hnonempty : ∀ (l : List β), (A l).Nonempty
x : ℕ → β
u : (n : ℕ) → α
hu : ∀ (n : ℕ), u n ∈ A (res x n)
umem : ∀ (n m : ℕ), n ≤ m → u m ∈ A (res x n)
ε : ℝ
ε_pos : ε > 0
n : ℕ
hn : ∀ y ∈ A (res x n), ∀ z ∈ A (res x n), dist y z < ε
m₀ : ℕ
hm₀ : m₀ ≥ n
m₁ : ℕ
hm₁ : m₁ ≥ n
⊢ n ≤ m₁

Tactic:
assumption
-- Module: Mathlib.Topology.MetricSpace.CantorScheme
Goal:
β : Type u_1
α : Type u_2
A : List β → Set α
inst : PseudoMetricSpace α
inst_1 : CompleteSpace α
hdiam : VanishingDiam A
hanti : ClosureAntitone A
hnonempty : ∀ (l : List β), (A l).Nonempty
x : ℕ → β
u : (n : ℕ) → α
hu : ∀ (n : ℕ), u n ∈ A (res x n)
umem : ∀ (n m : ℕ), n ≤ m → u m ∈ A (res x n)
⊢ CauchySeq u

Tactic:
have : CauchySeq u := by
    rw [Metric.cauchySeq_iff]
    intro ε ε_pos
    obtain ⟨n, hn⟩ := hdiam.dist_lt _ ε_pos x
    use n
    intro m₀ hm₀ m₁ hm₁
    apply hn <;> apply umem <;> assumption
-- Module: Mathlib.Topology.MetricSpace.CantorScheme
Goal:
β : Type u_1
α : Type u_2
A : List β → Set α
inst : PseudoMetricSpace α
inst_1 : CompleteSpace α
hdiam : VanishingDiam A
hanti : ClosureAntitone A
hnonempty : ∀ (l : List β), (A l).Nonempty
x : ℕ → β
u : (n : ℕ) → α
hu : ∀ (n : ℕ), u n ∈ A (res x n)
umem : ∀ (n m : ℕ), n ≤ m → u m ∈ A (res x n)
this : CauchySeq u
obtain :
  ∀ (y : α) (hy : Tendsto u atTop (𝓝 y)), x ∈ (inducedMap A).fst (β := fun (s : Set (ℕ → β)) ↦ (↑s : Type u_1) → α)
⊢ x ∈ (inducedMap A).fst (β := fun (s : Set (ℕ → β)) ↦ (↑s : Type u_1) → α)

Tactic:
obtain ⟨y, hy⟩ := cauchySeq_tendsto_of_complete this

Premises used: cauchySeq_tendsto_of_complete
-- Module: Mathlib.Topology.MetricSpace.CantorScheme
Goal:
β : Type u_1
α : Type u_2
A : List β → Set α
inst : PseudoMetricSpace α
inst_1 : CompleteSpace α
hdiam : VanishingDiam A
hanti : ClosureAntitone A
hnonempty : ∀ (l : List β), (A l).Nonempty
x : ℕ → β
u : (n : ℕ) → α
hu : ∀ (n : ℕ), u n ∈ A (res x n)
umem : ∀ (n m : ℕ), n ≤ m → u m ∈ A (res x n)
this : CauchySeq u
y : α
hy : Tendsto u atTop (𝓝 y)
h : Membership.mem (γ := Set α) (⋂ (n : ℕ), A (res x n)) y
⊢ x ∈ (inducedMap A).fst (β := fun (s : Set (ℕ → β)) ↦ (↑s : Type u_1) → α)

Tactic:
use y
-- Module: Mathlib.Topology.MetricSpace.CantorScheme
Goal:
β : Type u_1
α : Type u_2
A : List β → Set α
inst : PseudoMetricSpace α
inst_1 : CompleteSpace α
hdiam : VanishingDiam A
hanti : ClosureAntitone A
hnonempty : ∀ (l : List β), (A l).Nonempty
x : ℕ → β
u : (n : ℕ) → α
hu : ∀ (n : ℕ), u n ∈ A (res x n)
umem : ∀ (n m : ℕ), n ≤ m → u m ∈ A (res x n)
this : CauchySeq u
y : α
hy : Tendsto u atTop (𝓝 y)
h : ∀ (i : ℕ), y ∈ A (res x i)
⊢ Membership.mem (γ := Set α) (⋂ (n : ℕ), A (res x n)) y

Tactic:
rw [mem_iInter]

Premises used: Set.mem_iInter
-- Module: Mathlib.Topology.MetricSpace.CantorScheme
Goal:
β : Type u_1
α : Type u_2
A : List β → Set α
inst : PseudoMetricSpace α
inst_1 : CompleteSpace α
hdiam : VanishingDiam A
hanti : ClosureAntitone A
hnonempty : ∀ (l : List β), (A l).Nonempty
x : ℕ → β
u : (n : ℕ) → α
hu : ∀ (n : ℕ), u n ∈ A (res x n)
umem : ∀ (n m : ℕ), n ≤ m → u m ∈ A (res x n)
this : CauchySeq u
y : α
hy : Tendsto u atTop (𝓝 y)
n : ℕ
h : y ∈ closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (A (x n :: res x n))
⊢ y ∈ A (res x n)

Tactic:
apply hanti _ (x n)
-- Module: Mathlib.Topology.MetricSpace.CantorScheme
Goal:
β : Type u_1
α : Type u_2
A : List β → Set α
inst : PseudoMetricSpace α
inst_1 : CompleteSpace α
hdiam : VanishingDiam A
hanti : ClosureAntitone A
hnonempty : ∀ (l : List β), (A l).Nonempty
x : ℕ → β
u : (n : ℕ) → α
hu : ∀ (n : ℕ), u n ∈ A (res x n)
umem : ∀ (n m : ℕ), n ≤ m → u m ∈ A (res x n)
this : CauchySeq u
y : α
hy : Tendsto u atTop (𝓝 y)
n : ℕ
h : ∀ᶠ (x_1 : ℕ) in atTop, u x_1 ∈ A (x n :: res x n)
⊢ y ∈ closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (A (x n :: res x n))

Tactic:
apply mem_closure_of_tendsto hy

Premises used: mem_closure_of_tendsto
-- Module: Mathlib.Topology.MetricSpace.CantorScheme
Goal:
β : Type u_1
α : Type u_2
A : List β → Set α
inst : PseudoMetricSpace α
inst_1 : CompleteSpace α
hdiam : VanishingDiam A
hanti : ClosureAntitone A
hnonempty : ∀ (l : List β), (A l).Nonempty
x : ℕ → β
u : (n : ℕ) → α
hu : ∀ (n : ℕ), u n ∈ A (res x n)
umem : ∀ (n m : ℕ), n ≤ m → u m ∈ A (res x n)
this : CauchySeq u
y : α
hy : Tendsto u atTop (𝓝 y)
n : ℕ
h : ∃ (a : ℕ), ∀ b ≥ a, u b ∈ A (x n :: res x n)
⊢ ∀ᶠ (x_1 : ℕ) in atTop, u x_1 ∈ A (x n :: res x n)

Tactic:
rw [eventually_atTop]

Premises used: Filter.eventually_atTop
-- Module: Mathlib.Topology.MetricSpace.CantorScheme
Goal:
β : Type u_1
α : Type u_2
A : List β → Set α
inst : PseudoMetricSpace α
inst_1 : CompleteSpace α
hdiam : VanishingDiam A
hanti : ClosureAntitone A
hnonempty : ∀ (l : List β), (A l).Nonempty
x : ℕ → β
u : (n : ℕ) → α
hu : ∀ (n : ℕ), u n ∈ A (res x n)
umem : ∀ (n m : ℕ), n ≤ m → u m ∈ A (res x n)
this : CauchySeq u
y : α
hy : Tendsto u atTop (𝓝 y)
n : ℕ
⊢ ∃ (a : ℕ), ∀ b ≥ a, u b ∈ A (x n :: res x n)

Tactic:
⟨n.succ, umem _⟩

Premises used: Exists.intro, Nat.succ