awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.CantorScheme (goal)
Goal:
β : Type u_1
α : Type u_2
A : List β → Set α
x y : (↑((inducedMap A).fst (β := fun (s : Set (ℕ → β)) ↦ (↑s : Type u_1) → α)) : Type u_1)
hxy :
  (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) y
n : ℕ
ih : res (↑x : ℕ → β) n = res (↑y : ℕ → β) n
hA : ¬(↑x : ℕ → β) n = (↑y : ℕ → β) n
⊢ ¬_root_.Disjoint (A ((↑x : ℕ → β) n :: res (↑x : ℕ → β) n)) (A ((↑y : ℕ → β) n :: res (↑x : ℕ → β) n))
have a : ∃ x_1 ∈ A ((↑x : ℕ → β) n :: res (↑x : ℕ → β) n), x_1 ∈ A ((↑y : ℕ → β) n :: res (↑x : ℕ → β) n)
Tactic: rw [not_disjoint_iff]
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.CantorScheme (goal)
Goal:
β : Type u_1
α : Type u_2
A : List β → Set α
x y : (↑((inducedMap A).fst (β := fun (s : Set (ℕ → β)) ↦ (↑s : Type u_1) → α)) : Type u_1)
hxy :
  (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) y
n : ℕ
ih : res (↑x : ℕ → β) n = res (↑y : ℕ → β) n
hA : ¬(↑x : ℕ → β) n = (↑y : ℕ → β) n
⊢ ∃ x_1 ∈ A ((↑x : ℕ → β) n :: res (↑x : ℕ → β) n), x_1 ∈ A ((↑y : ℕ → β) n :: res (↑x : ℕ → β) n)
have a : (inducedMap A).snd (β := fun (s : Set (ℕ → β)) ↦ (↑s : Type u_1) → α) x ∈ A ((↑x : ℕ → β) n :: res (↑x : ℕ → β) n)
Tactic: refine ⟨(inducedMap A).2 x, ?_, ?_⟩
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.CantorScheme (goal)
Goal:
β : Type u_1
α : Type u_2
A : List β → Set α
x y : (↑((inducedMap A).fst (β := fun (s : Set (ℕ → β)) ↦ (↑s : Type u_1) → α)) : Type u_1)
hxy :
  (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) y
n : ℕ
ih : res (↑x : ℕ → β) n = res (↑y : ℕ → β) n
hA : ¬(↑x : ℕ → β) n = (↑y : ℕ → β) n
a : (inducedMap A).snd (β := fun (s : Set (ℕ → β)) ↦ (↑s : Type u_1) → α) x ∈ A ((↑x : ℕ → β) n :: res (↑x : ℕ → β) n)
⊢ ∃ x_1 ∈ A ((↑x : ℕ → β) n :: res (↑x : ℕ → β) n), x_1 ∈ A ((↑y : ℕ → β) n :: res (↑x : ℕ → β) n)
have a₁ : (inducedMap A).snd (β := fun (s : Set (ℕ → β)) ↦ (↑s : Type u_1) → α) x ∈ A ((↑y : ℕ → β) n :: res (↑x : ℕ → β) n)
Tactic: refine ⟨(inducedMap A).2 x, ?_, ?_⟩
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.CantorScheme (goal)
Goal:
β : Type u_1
α : Type u_2
A : List β → Set α
x y : (↑((inducedMap A).fst (β := fun (s : Set (ℕ → β)) ↦ (↑s : Type u_1) → α)) : Type u_1)
hxy :
  (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) y
n : ℕ
ih : res (↑x : ℕ → β) n = res (↑y : ℕ → β) n
hA : ¬(↑x : ℕ → β) n = (↑y : ℕ → β) n
⊢ (inducedMap A).snd (β := fun (s : Set (ℕ → β)) ↦ (↑s : Type u_1) → α) x ∈ A ((↑x : ℕ → β) n :: res (↑x : ℕ → β) n)
have a : (inducedMap A).snd (β := fun (s : Set (ℕ → β)) ↦ (↑s : Type u_1) → α) x ∈ A (res (↑x : ℕ → β) n.succ)
Tactic: rw [← res_succ]
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.CantorScheme (goal)
Goal:
β : Type u_1
α : Type u_2
A : List β → Set α
x y : (↑((inducedMap A).fst (β := fun (s : Set (ℕ → β)) ↦ (↑s : Type u_1) → α)) : Type u_1)
hxy :
  (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) y
n : ℕ
ih : res (↑x : ℕ → β) n = res (↑y : ℕ → β) n
hA : ¬(↑x : ℕ → β) n = (↑y : ℕ → β) n
⊢ (inducedMap A).snd (β := fun (s : Set (ℕ → β)) ↦ (↑s : Type u_1) → α) x ∈ A ((↑y : ℕ → β) n :: res (↑x : ℕ → β) n)
have a : (inducedMap A).snd (β := fun (s : Set (ℕ → β)) ↦ (↑s : Type u_1) → α) y ∈ A (res (↑y : ℕ → β) n.succ)
Tactic: rw [hxy, ih, ← res_succ]
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.CantorScheme (goal)
Goal:
β : Type u_1
α : Type u_2
A : List β → Set α
hA : CantorScheme.Disjoint A
x y : (↑((inducedMap A).fst (β := fun (s : Set (ℕ → β)) ↦ (↑s : Type u_1) → α)) : Type u_1)
hxy :
  (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) y
n : ℕ
⊢ res (↑x : ℕ → β) n = res (↑y : ℕ → β) n
have a : res (↑x : ℕ → β) 0 = res (↑y : ℕ → β) 0
Tactic: induction n with
  | zero => simp
  | succ n ih =>
    simp only [res_succ, cons.injEq]
    refine ⟨?_, ih⟩
    contrapose hA
    simp only [CantorScheme.Disjoint, _root_.Pairwise, Ne, not_forall, exists_prop]
    refine ⟨res x n, _, _, hA, ?_⟩
    rw [not_disjoint_iff]
    refine ⟨(inducedMap A).2 x, ?_, ?_⟩
    · rw [← res_succ]
      apply map_mem
    rw [hxy, ih, ← res_succ]
    apply map_mem
Cleared hypotheses: n
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.CantorScheme (goal)
Goal:
β : Type u_1
α : Type u_2
A : List β → Set α
hA : CantorScheme.Disjoint A
x y : (↑((inducedMap A).fst (β := fun (s : Set (ℕ → β)) ↦ (↑s : Type u_1) → α)) : Type u_1)
hxy :
  (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) y
n : ℕ
a : res (↑x : ℕ → β) 0 = res (↑y : ℕ → β) 0
⊢ res (↑x : ℕ → β) n = res (↑y : ℕ → β) n
have a₁ : ∀ (n : ℕ) (ih : res (↑x : ℕ → β) n = res (↑y : ℕ → β) n), res (↑x : ℕ → β) (n + 1) = res (↑y : ℕ → β) (n + 1)
Tactic: induction n with
  | zero => simp
  | succ n ih =>
    simp only [res_succ, cons.injEq]
    refine ⟨?_, ih⟩
    contrapose hA
    simp only [CantorScheme.Disjoint, _root_.Pairwise, Ne, not_forall, exists_prop]
    refine ⟨res x n, _, _, hA, ?_⟩
    rw [not_disjoint_iff]
    refine ⟨(inducedMap A).2 x, ?_, ?_⟩
    · rw [← res_succ]
      apply map_mem
    rw [hxy, ih, ← res_succ]
    apply map_mem
Cleared hypotheses: n
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.CantorScheme (hypothesis)
Goal:
β : Type u_1
α : Type u_2
A : List β → Set α
inst : PseudoMetricSpace α
hA : VanishingDiam A
ε : ℝ
ε_pos : 0 < ε
x : ℕ → β
⊢ ∃ (n : ℕ), ∀ y ∈ A (res x n), ∀ z ∈ A (res x n), dist y z < ε
have hA₁ : Tendsto (β := ENNReal) (fun (n : ℕ) ↦ Metric.ediam (A (res x n))) atTop (𝓝 0)
Tactic: specialize hA x
Cleared hypotheses: hA
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.CantorScheme (hypothesis)
Goal:
β : Type u_1
α : Type u_2
A : List β → Set α
inst : PseudoMetricSpace α
ε : ℝ
ε_pos : 0 < ε
x : ℕ → β
hA : Tendsto (β := ENNReal) (fun (n : ℕ) ↦ Metric.ediam (A (res x n))) atTop (𝓝 0)
⊢ ∃ (n : ℕ), ∀ y ∈ A (res x n), ∀ z ∈ A (res x n), dist y z < ε
have hA₁ : ∀ ε > 0, ∃ (N : ℕ), ∀ n ≥ N, Metric.ediam (A (res x n)) ≤ ε
Tactic: rw [ENNReal.tendsto_atTop_zero] at hA
Cleared hypotheses: hA
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.CantorScheme (goal)
Goal:
β : Type u_1
α : Type u_2
A : List β → Set α
inst : PseudoMetricSpace α
ε : ℝ
ε_pos : 0 < ε
x : ℕ → β
hA : ∀ ε > 0, ∃ (N : ℕ), ∀ n ≥ N, Metric.ediam (A (res x n)) ≤ ε
⊢ ENNReal.ofReal (ε / 2) > 0
have simp : 0 < ε / 2
Tactic: simp only [gt_iff_lt, ENNReal.ofReal_pos]
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.CantorScheme (goal)
Goal:
β : Type u_1
α : Type u_2
A : List β → Set α
inst : PseudoMetricSpace α
ε : ℝ
ε_pos : 0 < ε
x : ℕ → β
hA : ∀ ε > 0, ∃ (N : ℕ), ∀ n ≥ N, Metric.ediam (A (res x n)) ≤ ε
⊢ 0 < ε / 2
have linarith : ε / 2 ≤ 0 → 2 / 2 = 1
Tactic: linarith
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.CantorScheme (goal)
Goal:
β : Type u_1
α : Type u_2
A : List β → Set α
inst : PseudoMetricSpace α
ε : ℝ
ε_pos : 0 < ε
x : ℕ → β
hA : ∀ ε > 0, ∃ (N : ℕ), ∀ n ≥ N, Metric.ediam (A (res x n)) ≤ ε
linarith : ε / 2 ≤ 0 → 2 / 2 = 1
⊢ 0 < ε / 2
have linarith₁ : ε / 2 ≤ 0 → HMul.hMul (α := ℝ) 1 2 = 2
Tactic: linarith
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.CantorScheme (goal)
Goal:
β : Type u_1
α : Type u_2
A : List β → Set α
inst : PseudoMetricSpace α
ε : ℝ
ε_pos : 0 < ε
x : ℕ → β
hA : ∀ ε > 0, ∃ (N : ℕ), ∀ n ≥ N, Metric.ediam (A (res x n)) ≤ ε
linarith : ε / 2 ≤ 0 → 2 / 2 = 1
linarith₁ : ε / 2 ≤ 0 → HMul.hMul (α := ℝ) 1 2 = 2
⊢ 0 < ε / 2
have linarith₂ : ε / 2 ≤ 0 → GT.gt (α := ℝ) 2 0
Tactic: linarith
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.CantorScheme (goal)
Goal:
β : Type u_1
α : Type u_2
A : List β → Set α
inst : PseudoMetricSpace α
ε : ℝ
ε_pos : 0 < ε
x : ℕ → β
hA : ∀ ε > 0, ∃ (N : ℕ), ∀ n ≥ N, Metric.ediam (A (res x n)) ≤ ε
linarith : ε / 2 ≤ 0 → 2 / 2 = 1
linarith₁ : ε / 2 ≤ 0 → HMul.hMul (α := ℝ) 1 2 = 2
linarith₂ : ε / 2 ≤ 0 → GT.gt (α := ℝ) 2 0
⊢ 0 < ε / 2
have linarith₃ : ε / 2 ≤ 0 → 0 - ε + (1 * ε - HMul.hMul (α := ℝ) 2 0) = 0
Tactic: linarith
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.CantorScheme (goal)
Goal:
β : Type u_1
α : Type u_2
A : List β → Set α
inst : PseudoMetricSpace α
ε : ℝ
ε_pos : 0 < ε
x : ℕ → β
hA : ∀ ε > 0, ∃ (N : ℕ), ∀ n ≥ N, Metric.ediam (A (res x n)) ≤ ε
⊢ ∃ (n : ℕ), ∀ y ∈ A (res x n), ∀ z ∈ A (res x n), dist y z < ε
have obtain : ∀ (n : ℕ) (hn : ∀ n_1 ≥ n, Metric.ediam (A (res x n_1)) ≤ ENNReal.ofReal (ε / 2)),
  ∃ (n : ℕ), ∀ y ∈ A (res x n), ∀ z ∈ A (res x n), dist y z < ε
Tactic: obtain ⟨n, hn⟩ := hA (ENNReal.ofReal (ε / 2)) (by
    simp only [gt_iff_lt, ENNReal.ofReal_pos]; linarith)
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.CantorScheme (goal)
Goal:
β : Type u_1
α : Type u_2
A : List β → Set α
inst : PseudoMetricSpace α
ε : ℝ
ε_pos : 0 < ε
x : ℕ → β
hA : ∀ ε > 0, ∃ (N : ℕ), ∀ n ≥ N, Metric.ediam (A (res x n)) ≤ ε
n : ℕ
hn : ∀ n_1 ≥ n, Metric.ediam (A (res x n_1)) ≤ ENNReal.ofReal (ε / 2)
⊢ ∃ (n : ℕ), ∀ y ∈ A (res x n), ∀ z ∈ A (res x n), dist y z < ε
have h : ∀ y ∈ A (res x n), ∀ z ∈ A (res x n), dist y z < ε
Tactic: use n
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.CantorScheme (goal)
Goal:
β : Type u_1
α : Type u_2
A : List β → Set α
inst : PseudoMetricSpace α
ε : ℝ
ε_pos : 0 < ε
x : ℕ → β
hA : ∀ ε > 0, ∃ (N : ℕ), ∀ n ≥ N, Metric.ediam (A (res x n)) ≤ ε
n : ℕ
hn : ∀ n_1 ≥ n, Metric.ediam (A (res x n_1)) ≤ ENNReal.ofReal (ε / 2)
y : α
hy : y ∈ A (res x n)
z : α
hz : z ∈ A (res x n)
⊢ dist y z < ε
have h : edist y z < ENNReal.ofReal ε
Tactic: rw [← ENNReal.ofReal_lt_ofReal_iff ε_pos, ← edist_dist]
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.CantorScheme (goal)
Goal:
β : Type u_1
α : Type u_2
A : List β → Set α
inst : PseudoMetricSpace α
ε : ℝ
ε_pos : 0 < ε
x : ℕ → β
hA : ∀ ε > 0, ∃ (N : ℕ), ∀ n ≥ N, Metric.ediam (A (res x n)) ≤ ε
n : ℕ
hn : ∀ n_1 ≥ n, Metric.ediam (A (res x n_1)) ≤ ENNReal.ofReal (ε / 2)
y : α
hy : y ∈ A (res x n)
z : α
hz : z ∈ A (res x n)
⊢ edist y z < ENNReal.ofReal ε
have h : Metric.ediam (A (res x n)) < ENNReal.ofReal ε
Tactic: apply lt_of_le_of_lt (Metric.edist_le_ediam_of_mem hy hz)
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.CantorScheme (goal)
Goal:
β : Type u_1
α : Type u_2
A : List β → Set α
inst : PseudoMetricSpace α
ε : ℝ
ε_pos : 0 < ε
x : ℕ → β
hA : ∀ ε > 0, ∃ (N : ℕ), ∀ n ≥ N, Metric.ediam (A (res x n)) ≤ ε
n : ℕ
hn : ∀ n_1 ≥ n, Metric.ediam (A (res x n_1)) ≤ ENNReal.ofReal (ε / 2)
y : α
hy : y ∈ A (res x n)
z : α
hz : z ∈ A (res x n)
⊢ Metric.ediam (A (res x n)) < ENNReal.ofReal ε
have h : ENNReal.ofReal (ε / 2) < ENNReal.ofReal ε
Tactic: apply lt_of_le_of_lt (hn _ (le_refl _))
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.CantorScheme (goal)
Goal:
β : Type u_1
α : Type u_2
A : List β → Set α
inst : PseudoMetricSpace α
ε : ℝ
ε_pos : 0 < ε
x : ℕ → β
hA : ∀ ε > 0, ∃ (N : ℕ), ∀ n ≥ N, Metric.ediam (A (res x n)) ≤ ε
n : ℕ
hn : ∀ n_1 ≥ n, Metric.ediam (A (res x n_1)) ≤ ENNReal.ofReal (ε / 2)
y : α
hy : y ∈ A (res x n)
z : α
hz : z ∈ A (res x n)
⊢ ENNReal.ofReal (ε / 2) < ENNReal.ofReal ε
have h : ε / 2 < ε
Tactic: rw [ENNReal.ofReal_lt_ofReal_iff ε_pos]
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.CantorScheme (goal)
Goal:
β : Type u_1
α : Type u_2
A : List β → Set α
inst : PseudoMetricSpace α
ε : ℝ
ε_pos : 0 < ε
x : ℕ → β
hA : ∀ ε > 0, ∃ (N : ℕ), ∀ n ≥ N, Metric.ediam (A (res x n)) ≤ ε
n : ℕ
hn : ∀ n_1 ≥ n, Metric.ediam (A (res x n_1)) ≤ ENNReal.ofReal (ε / 2)
y : α
hy : y ∈ A (res x n)
z : α
hz : z ∈ A (res x n)
⊢ ε / 2 < ε
have linarith : ε ≤ ε / 2 → 2 / 2 = 1
Tactic: linarith
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.CantorScheme (goal)
Goal:
β : Type u_1
α : Type u_2
A : List β → Set α
inst : PseudoMetricSpace α
ε : ℝ
ε_pos : 0 < ε
x : ℕ → β
hA : ∀ ε > 0, ∃ (N : ℕ), ∀ n ≥ N, Metric.ediam (A (res x n)) ≤ ε
n : ℕ
hn : ∀ n_1 ≥ n, Metric.ediam (A (res x n_1)) ≤ ENNReal.ofReal (ε / 2)
y : α
hy : y ∈ A (res x n)
z : α
hz : z ∈ A (res x n)
linarith : ε ≤ ε / 2 → 2 / 2 = 1
⊢ ε / 2 < ε
have linarith₁ : ε ≤ ε / 2 → HMul.hMul (α := ℝ) 1 2 = 2
Tactic: linarith
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.CantorScheme (goal)
Goal:
β : Type u_1
α : Type u_2
A : List β → Set α
inst : PseudoMetricSpace α
ε : ℝ
ε_pos : 0 < ε
x : ℕ → β
hA : ∀ ε > 0, ∃ (N : ℕ), ∀ n ≥ N, Metric.ediam (A (res x n)) ≤ ε
n : ℕ
hn : ∀ n_1 ≥ n, Metric.ediam (A (res x n_1)) ≤ ENNReal.ofReal (ε / 2)
y : α
hy : y ∈ A (res x n)
z : α
hz : z ∈ A (res x n)
linarith : ε ≤ ε / 2 → 2 / 2 = 1
linarith₁ : ε ≤ ε / 2 → HMul.hMul (α := ℝ) 1 2 = 2
⊢ ε / 2 < ε
have linarith₂ : ε ≤ ε / 2 → GT.gt (α := ℝ) 2 0
Tactic: linarith
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.CantorScheme (goal)
Goal:
β : Type u_1
α : Type u_2
A : List β → Set α
inst : PseudoMetricSpace α
ε : ℝ
ε_pos : 0 < ε
x : ℕ → β
hA : ∀ ε > 0, ∃ (N : ℕ), ∀ n ≥ N, Metric.ediam (A (res x n)) ≤ ε
n : ℕ
hn : ∀ n_1 ≥ n, Metric.ediam (A (res x n_1)) ≤ ENNReal.ofReal (ε / 2)
y : α
hy : y ∈ A (res x n)
z : α
hz : z ∈ A (res x n)
linarith : ε ≤ ε / 2 → 2 / 2 = 1
linarith₁ : ε ≤ ε / 2 → HMul.hMul (α := ℝ) 1 2 = 2
linarith₂ : ε ≤ ε / 2 → GT.gt (α := ℝ) 2 0
⊢ ε / 2 < ε
have linarith₃ : ε ≤ ε / 2 → 0 - ε + (2 * ε - 1 * ε) = 0
Tactic: linarith
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.CantorScheme (goal)
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
⊢ Continuous[instTopologicalSpaceSubtype, PseudoMetricSpace.toUniformSpace.toTopologicalSpace]
    ((inducedMap A).snd (β := fun (s : Set (ℕ → β)) ↦ (↑s : Type u_1) → α))
have 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
Tactic: rw [Metric.continuous_iff']
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.CantorScheme (goal)
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
⊢ ∀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
have 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
Tactic: obtain ⟨n, hn⟩ := hA.dist_lt _ ε_pos x
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.CantorScheme (goal)
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
⊢ ∀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
have 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
Tactic: rw [_root_.eventually_nhds_iff]
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.CantorScheme (goal)
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
⊢ ∃ (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
have 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
Tactic: refine ⟨(↑)⁻¹' cylinder x.1 n, ?_, ?_, by simp⟩
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.CantorScheme (goal)
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
⊢ ∃ (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
have 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)
Tactic: refine ⟨(↑)⁻¹' cylinder x.1 n, ?_, ?_, by simp⟩
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.CantorScheme (hypothesis)
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
⊢ 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
have hyx₁ : res (↑y : ℕ → β) n = res (↑x : ℕ → β) n
Tactic: rw [mem_preimage, Subtype.coe_mk, cylinder_eq_res, mem_setOf] at hyx
Cleared hypotheses: hyx
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.CantorScheme (goal)
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
⊢ 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
have refine_1 : (inducedMap A).snd (β := fun (s : Set (ℕ → β)) ↦ (↑s : Type u_1) → α) y ∈ A (res (↑x : ℕ → β) n)
Tactic: apply hn
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.CantorScheme (goal)
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)
⊢ 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
have refine_1₁ : (inducedMap A).snd (β := fun (s : Set (ℕ → β)) ↦ (↑s : Type u_1) → α) x ∈ A (res (↑x : ℕ → β) n)
Tactic: apply hn
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.CantorScheme (goal)
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 (↑x : ℕ → β) n)
have refine_1 : (inducedMap A).snd (β := fun (s : Set (ℕ → β)) ↦ (↑s : Type u_1) → α) y ∈ A (res (↑y : ℕ → β) n)
Tactic: rw [← hyx]
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.CantorScheme (goal)
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[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)
have refine_2 : IsOpen[Pi.topologicalSpace] (cylinder (E := fun (n : ℕ) ↦ β) (↑x : ℕ → β) n)
Tactic: apply continuous_subtype_val.isOpen_preimage