awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.Topology.Instances.CantorSet (goal)
Goal:
n : ℕ
zero : 0 ∈ preCantorSet 0
⊢ 0 ∈ preCantorSet n
have succ : ∀ (n : ℕ) (ih : 0 ∈ preCantorSet n), 0 ∈ preCantorSet (n + 1)
Tactic: induction n with
  | zero =>
    simp [preCantorSet]
  | succ n ih =>
    exact Or.inl ⟨0, ih, by simp only [zero_div]⟩
Cleared hypotheses: n
-- Lean 4 proof step from Mathlib.Topology.Instances.CantorSet (goal)
Goal:
⊢ Antitone preCantorSet
have refine : ∀ (m : ℕ), preCantorSet (m + 1) ≤ preCantorSet m
Tactic: refine antitone_nat_of_succ_le fun m ↦ ?_
-- Lean 4 proof step from Mathlib.Topology.Instances.CantorSet (goal)
Goal:
m : ℕ
⊢ preCantorSet (m + 1) ⊆ preCantorSet m
have zero : preCantorSet (HAdd.hAdd (α := ℕ) 0 1) ⊆ preCantorSet 0
Tactic: induction m with grind [preCantorSet_zero, preCantorSet_succ]
Cleared hypotheses: m
-- Lean 4 proof step from Mathlib.Topology.Instances.CantorSet (goal)
Goal:
m : ℕ
zero : preCantorSet (HAdd.hAdd (α := ℕ) 0 1) ⊆ preCantorSet 0
⊢ preCantorSet (m + 1) ⊆ preCantorSet m
have succ : ∀ (n : ℕ), preCantorSet (n + 1) ⊆ preCantorSet n → preCantorSet (n + 1 + 1) ⊆ preCantorSet (n + 1)
Tactic: induction m with grind [preCantorSet_zero, preCantorSet_succ]
Cleared hypotheses: m
-- Lean 4 proof step from Mathlib.Topology.Instances.CantorSet (goal)
Goal:
⊢ Eq (α := Set ℝ) (⋂ (n : ℕ), preCantorSet n)
    (((fun (x : ℝ) ↦ x / 3) '' ⋂ (n : ℕ), preCantorSet n) ∪ (fun (x : ℝ) ↦ (2 + x) / 3) '' ⋂ (n : ℕ), preCantorSet n)
have rw : Eq (α := Set ℝ) (⋂ (n : ℕ), preCantorSet n)
  ((⋂ (i : ℕ), (fun (x : ℝ) ↦ x / 3) '' preCantorSet i) ∪ ⋂ (i : ℕ), (fun (x : ℝ) ↦ (2 + x) / 3) '' preCantorSet i)
Tactic: rw [Set.image_iInter, Set.image_iInter]
-- Lean 4 proof step from Mathlib.Topology.Instances.CantorSet (goal)
Goal:
rw :
  Eq (α := Set ℝ) (⋂ (n : ℕ), preCantorSet n)
    ((⋂ (i : ℕ), (fun (x : ℝ) ↦ x / 3) '' preCantorSet i) ∪ ⋂ (i : ℕ), (fun (x : ℝ) ↦ (2 + x) / 3) '' preCantorSet i)
⊢ Eq (α := Set ℝ) (⋂ (n : ℕ), preCantorSet n)
    (((fun (x : ℝ) ↦ x / 3) '' ⋂ (n : ℕ), preCantorSet n) ∪ (fun (x : ℝ) ↦ (2 + x) / 3) '' ⋂ (n : ℕ), preCantorSet n)
have hf : Function.Bijective (β := ℝ) fun (x : ℝ) ↦ (2 + x) / 3
Tactic: rw [Set.image_iInter, Set.image_iInter]
-- Lean 4 proof step from Mathlib.Topology.Instances.CantorSet (goal)
Goal:
rw :
  Eq (α := Set ℝ) (⋂ (n : ℕ), preCantorSet n)
    ((⋂ (i : ℕ), (fun (x : ℝ) ↦ x / 3) '' preCantorSet i) ∪ ⋂ (i : ℕ), (fun (x : ℝ) ↦ (2 + x) / 3) '' preCantorSet i)
hf : Function.Bijective (β := ℝ) fun (x : ℝ) ↦ (2 + x) / 3
⊢ Eq (α := Set ℝ) (⋂ (n : ℕ), preCantorSet n)
    (((fun (x : ℝ) ↦ x / 3) '' ⋂ (n : ℕ), preCantorSet n) ∪ (fun (x : ℝ) ↦ (2 + x) / 3) '' ⋂ (n : ℕ), preCantorSet n)
have hf₁ : Function.Bijective (β := ℝ) fun (x : ℝ) ↦ x / 3
Tactic: rw [Set.image_iInter, Set.image_iInter]
-- Lean 4 proof step from Mathlib.Topology.Instances.CantorSet (goal)
Goal:
⊢ Eq (α := Set ℝ) (⋂ (n : ℕ), preCantorSet n)
    ((⋂ (i : ℕ), (fun (x : ℝ) ↦ x / 3) '' preCantorSet i) ∪ ⋂ (i : ℕ), (fun (x : ℝ) ↦ (2 + x) / 3) '' preCantorSet i)
have simp_rw : Eq (α := Set ℝ) (⋂ (n : ℕ), preCantorSet n) (⋂ (i : ℕ), preCantorSet (i + 1))
Tactic: simp_rw [← Function.comp_def,
    ← Set.iInter_union_of_antitone
      (Set.monotone_image.comp_antitone preCantorSet_antitone)
      (Set.monotone_image.comp_antitone preCantorSet_antitone),
    Function.comp_def, ← preCantorSet_succ]
-- Lean 4 proof step from Mathlib.Topology.Instances.CantorSet (goal)
Goal:
f : ℝ ≃ₜ ℝ := Homeomorph.mulLeft₀ (1 / 3) ⋯
g : ℝ ≃ₜ ℝ := Homeomorph.trans (Homeomorph.addLeft 2) f
n : ℕ
ih : IsClosed (preCantorSet n)
⊢ IsClosed (preCantorSet (n + 1))
have succ : IsClosed (X := ℝ) ((fun (x : ℝ) ↦ x / 3) '' preCantorSet n)
Tactic: refine IsClosed.union ?_ ?_
-- Lean 4 proof step from Mathlib.Topology.Instances.CantorSet (goal)
Goal:
f : ℝ ≃ₜ ℝ := Homeomorph.mulLeft₀ (1 / 3) ⋯
g : ℝ ≃ₜ ℝ := Homeomorph.trans (Homeomorph.addLeft 2) f
n : ℕ
ih : IsClosed (preCantorSet n)
succ : IsClosed (X := ℝ) ((fun (x : ℝ) ↦ x / 3) '' preCantorSet n)
⊢ IsClosed (preCantorSet (n + 1))
have succ₁ : IsClosed (X := ℝ) ((fun (x : ℝ) ↦ (2 + x) / 3) '' preCantorSet n)
Tactic: refine IsClosed.union ?_ ?_
-- Lean 4 proof step from Mathlib.Topology.Instances.CantorSet (goal)
Goal:
n : ℕ
f : ℝ ≃ₜ ℝ := Homeomorph.mulLeft₀ (1 / 3) ⋯
g : ℝ ≃ₜ ℝ := Homeomorph.trans (Homeomorph.addLeft 2) f
⊢ IsClosed (preCantorSet n)
have zero : IsClosed (preCantorSet 0)
Tactic: induction n with
  | zero => exact isClosed_Icc
  | succ n ih =>
    refine IsClosed.union ?_ ?_
    · simpa [f, div_eq_inv_mul] using f.isClosedEmbedding.isClosed_iff_image_isClosed.mp ih
    · simpa [g, f, div_eq_inv_mul] using g.isClosedEmbedding.isClosed_iff_image_isClosed.mp ih
Cleared hypotheses: n
-- Lean 4 proof step from Mathlib.Topology.Instances.CantorSet (goal)
Goal:
n : ℕ
f : ℝ ≃ₜ ℝ := Homeomorph.mulLeft₀ (1 / 3) ⋯
g : ℝ ≃ₜ ℝ := Homeomorph.trans (Homeomorph.addLeft 2) f
zero : IsClosed (preCantorSet 0)
⊢ IsClosed (preCantorSet n)
have succ : ∀ (n : ℕ) (ih : IsClosed (preCantorSet n)), IsClosed (preCantorSet (n + 1))
Tactic: induction n with
  | zero => exact isClosed_Icc
  | succ n ih =>
    refine IsClosed.union ?_ ?_
    · simpa [f, div_eq_inv_mul] using f.isClosedEmbedding.isClosed_iff_image_isClosed.mp ih
    · simpa [g, f, div_eq_inv_mul] using g.isClosedEmbedding.isClosed_iff_image_isClosed.mp ih
Cleared hypotheses: n
-- Lean 4 proof step from Mathlib.Topology.Instances.CantorSet (goal)
Goal:
a : ℕ → Fin 3
h : ∀ (n : ℕ), a n ≠ 1
⊢ ofDigits a ∈ cantorSet
have simp : ∀ (i : ℕ), ofDigits a ∈ preCantorSet i
Tactic: simp only [cantorSet, Set.mem_iInter]
-- Lean 4 proof step from Mathlib.Topology.Instances.CantorSet (goal)
Goal:
i : ℕ
ih : ∀ {a : ℕ → Fin 3} (h : ∀ (n : ℕ), a n ≠ 1), ofDigits a ∈ preCantorSet i
a : ℕ → Fin 3
h : ∀ (n : ℕ), a n ≠ 1
⊢ ofDigits a ∈ preCantorSet (i + 1)
have succ : ∃ (x : ℝ), x ∈ preCantorSet i ∧ x / 3 = ofDigits a ∨ x ∈ preCantorSet i ∧ (2 + x) / 3 = ofDigits a
Tactic: simp only [preCantorSet, Set.mem_union, Set.mem_image, ← exists_or]
-- Lean 4 proof step from Mathlib.Topology.Instances.CantorSet (goal)
Goal:
i : ℕ
ih : ∀ {a : ℕ → Fin 3} (h : ∀ (n : ℕ), a n ≠ 1), ofDigits a ∈ preCantorSet i
a : ℕ → Fin 3
h : ∀ (n : ℕ), a n ≠ 1
⊢ ∃ (x : ℝ), x ∈ preCantorSet i ∧ x / 3 = ofDigits a ∨ x ∈ preCantorSet i ∧ (2 + x) / 3 = ofDigits a
have h₁ : (ofDigits (b := 3) fun (i : ℕ) ↦ a (i + 1)) ∈ preCantorSet i ∧
    (ofDigits (b := 3) fun (i : ℕ) ↦ a (i + 1)) / 3 = ofDigits a ∨
  (ofDigits (b := 3) fun (i : ℕ) ↦ a (i + 1)) ∈ preCantorSet i ∧
    (2 + ofDigits (b := 3) fun (i : ℕ) ↦ a (i + 1)) / 3 = ofDigits a
Tactic: use ofDigits (fun i ↦ a (i + 1))
-- Lean 4 proof step from Mathlib.Topology.Instances.CantorSet (hypothesis)
Goal:
i : ℕ
ih : ∀ {a : ℕ → Fin 3} (h : ∀ (n : ℕ), a n ≠ 1), ofDigits a ∈ preCantorSet i
a : ℕ → Fin 3
h : ∀ (n : ℕ), a n ≠ 1
⊢ (ofDigits (b := 3) fun (i : ℕ) ↦ a (i + 1)) ∈ preCantorSet i ∧
      (ofDigits (b := 3) fun (i : ℕ) ↦ a (i + 1)) / 3 = ofDigits a ∨
    (ofDigits (b := 3) fun (i : ℕ) ↦ a (i + 1)) ∈ preCantorSet i ∧
      (2 + ofDigits (b := 3) fun (i : ℕ) ↦ a (i + 1)) / 3 = ofDigits a
have this : (ofDigits (b := 3) fun (i : ℕ) ↦ a (i + 1)) ∈ preCantorSet i
Tactic: have : (ofDigits fun i ↦ a (i + 1)) ∈ preCantorSet i := ih (by solve_by_elim)
-- Lean 4 proof step from Mathlib.Topology.Instances.CantorSet (goal)
Goal:
i : ℕ
ih : ∀ {a : ℕ → Fin 3} (h : ∀ (n : ℕ), a n ≠ 1), ofDigits a ∈ preCantorSet i
a : ℕ → Fin 3
h : ∀ (n : ℕ), a n ≠ 1
this : (ofDigits (b := 3) fun (i : ℕ) ↦ a (i + 1)) ∈ preCantorSet i
⊢ (ofDigits (b := 3) fun (i : ℕ) ↦ a (i + 1)) ∈ preCantorSet i ∧
      (ofDigits (b := 3) fun (i : ℕ) ↦ a (i + 1)) / 3 = ofDigits a ∨
    (ofDigits (b := 3) fun (i : ℕ) ↦ a (i + 1)) ∈ preCantorSet i ∧
      (2 + ofDigits (b := 3) fun (i : ℕ) ↦ a (i + 1)) / 3 = ofDigits a
have h₁ : ((ofDigits (b := 3) fun (i : ℕ) ↦ a (i + 1)) = (↑(↑(a 0) : ℕ) : ℝ) + ofDigits (b := 3) fun (i : ℕ) ↦ a (i + 1)) ∨
  (2 + ofDigits (b := 3) fun (i : ℕ) ↦ a (i + 1)) = (↑(↑(a 0) : ℕ) : ℝ) + ofDigits (b := 3) fun (i : ℕ) ↦ a (i + 1)
Tactic: simp only [this, ofDigits_eq_sum_add_ofDigits a 1, Finset.range_one, ofDigitsTerm,
      Nat.cast_ofNat, Finset.sum_singleton, zero_add, pow_one, true_and, field]
-- Lean 4 proof step from Mathlib.Topology.Instances.CantorSet (hypothesis)
Goal:
i : ℕ
ih : ∀ {a : ℕ → Fin 3} (h : ∀ (n : ℕ), a n ≠ 1), ofDigits a ∈ preCantorSet i
a : ℕ → Fin 3
h : ∀ (n : ℕ), a n ≠ 1
this : (ofDigits (b := 3) fun (i : ℕ) ↦ a (i + 1)) ∈ preCantorSet i
⊢ ((ofDigits (b := 3) fun (i : ℕ) ↦ a (i + 1)) = (↑(↑(a 0) : ℕ) : ℝ) + ofDigits (b := 3) fun (i : ℕ) ↦ a (i + 1)) ∨
    (2 + ofDigits (b := 3) fun (i : ℕ) ↦ a (i + 1)) = (↑(↑(a 0) : ℕ) : ℝ) + ofDigits (b := 3) fun (i : ℕ) ↦ a (i + 1)
have h₁ : a 0 ≠ 1
Tactic: specialize h 0
Cleared hypotheses: h
-- Lean 4 proof step from Mathlib.Topology.Instances.CantorSet (goal)
Goal:
i : ℕ
ih : ∀ {a : ℕ → Fin 3} (h : ∀ (n : ℕ), a n ≠ 1), ofDigits a ∈ preCantorSet i
a : ℕ → Fin 3
this : (ofDigits (b := 3) fun (i : ℕ) ↦ a (i + 1)) ∈ preCantorSet i
h : a 0 ≠ 1
⊢ ((ofDigits (b := 3) fun (i : ℕ) ↦ a (i + 1)) = (↑(↑(a 0) : ℕ) : ℝ) + ofDigits (b := 3) fun (i : ℕ) ↦ a (i + 1)) ∨
    (2 + ofDigits (b := 3) fun (i : ℕ) ↦ a (i + 1)) = (↑(↑(a 0) : ℕ) : ℝ) + ofDigits (b := 3) fun (i : ℕ) ↦ a (i + 1)
have h₁ : ∀ (x : Fin 3) (h : x ≠ 1),
  ((ofDigits (b := 3) fun (i : ℕ) ↦ a (i + 1)) = (↑(↑x : ℕ) : ℝ) + ofDigits (b := 3) fun (i : ℕ) ↦ a (i + 1)) ∨
    (2 + ofDigits (b := 3) fun (i : ℕ) ↦ a (i + 1)) = (↑(↑x : ℕ) : ℝ) + ofDigits (b := 3) fun (i : ℕ) ↦ a (i + 1)
Tactic: generalize a 0 = x at h
Cleared hypotheses: h
-- Lean 4 proof step from Mathlib.Topology.Instances.CantorSet (goal)
Goal:
i : ℕ
ih : ∀ {a : ℕ → Fin 3} (h : ∀ (n : ℕ), a n ≠ 1), ofDigits a ∈ preCantorSet i
a : ℕ → Fin 3
this : (ofDigits (b := 3) fun (i : ℕ) ↦ a (i + 1)) ∈ preCantorSet i
x : Fin 3
h : x ≠ 1
⊢ ((ofDigits (b := 3) fun (i : ℕ) ↦ a (i + 1)) = (↑(↑x : ℕ) : ℝ) + ofDigits (b := 3) fun (i : ℕ) ↦ a (i + 1)) ∨
    (2 + ofDigits (b := 3) fun (i : ℕ) ↦ a (i + 1)) = (↑(↑x : ℕ) : ℝ) + ofDigits (b := 3) fun (i : ℕ) ↦ a (i + 1)
have h₁ : ∀ (h : Ne (α := Fin 3) ((fun (i : Fin 3) ↦ i) ⟨0, sorry⟩) 1),
  ((ofDigits (b := 3) fun (i : ℕ) ↦ a (i + 1)) =
      (↑(↑((fun (i : Fin 3) ↦ i) ⟨0, sorry⟩) : ℕ) : ℝ) + ofDigits (b := 3) fun (i : ℕ) ↦ a (i + 1)) ∨
    (2 + ofDigits (b := 3) fun (i : ℕ) ↦ a (i + 1)) =
      (↑(↑((fun (i : Fin 3) ↦ i) ⟨0, sorry⟩) : ℕ) : ℝ) + ofDigits (b := 3) fun (i : ℕ) ↦ a (i + 1)
Tactic: fin_cases x
Cleared hypotheses: x, h
-- Lean 4 proof step from Mathlib.Topology.Instances.CantorSet (goal)
Goal:
i : ℕ
ih : ∀ {a : ℕ → Fin 3} (h : ∀ (n : ℕ), a n ≠ 1), ofDigits a ∈ preCantorSet i
a : ℕ → Fin 3
this : (ofDigits (b := 3) fun (i : ℕ) ↦ a (i + 1)) ∈ preCantorSet i
x : Fin 3
h : x ≠ 1
h₁ :
  ∀ (h : Ne (α := Fin 3) ((fun (i : Fin 3) ↦ i) ⟨0, ⋯⟩) 1),
    ((ofDigits (b := 3) fun (i : ℕ) ↦ a (i + 1)) =
        (↑(↑((fun (i : Fin 3) ↦ i) ⟨0, ⋯⟩) : ℕ) : ℝ) + ofDigits (b := 3) fun (i : ℕ) ↦ a (i + 1)) ∨
      (2 + ofDigits (b := 3) fun (i : ℕ) ↦ a (i + 1)) =
        (↑(↑((fun (i : Fin 3) ↦ i) ⟨0, ⋯⟩) : ℕ) : ℝ) + ofDigits (b := 3) fun (i : ℕ) ↦ a (i + 1)
⊢ ((ofDigits (b := 3) fun (i : ℕ) ↦ a (i + 1)) = (↑(↑x : ℕ) : ℝ) + ofDigits (b := 3) fun (i : ℕ) ↦ a (i + 1)) ∨
    (2 + ofDigits (b := 3) fun (i : ℕ) ↦ a (i + 1)) = (↑(↑x : ℕ) : ℝ) + ofDigits (b := 3) fun (i : ℕ) ↦ a (i + 1)
have h₂ : ∀ (h : Ne (α := Fin 3) ((fun (i : Fin 3) ↦ i) ⟨1, sorry⟩) 1),
  ((ofDigits (b := 3) fun (i : ℕ) ↦ a (i + 1)) =
      (↑(↑((fun (i : Fin 3) ↦ i) ⟨1, sorry⟩) : ℕ) : ℝ) + ofDigits (b := 3) fun (i : ℕ) ↦ a (i + 1)) ∨
    (2 + ofDigits (b := 3) fun (i : ℕ) ↦ a (i + 1)) =
      (↑(↑((fun (i : Fin 3) ↦ i) ⟨1, sorry⟩) : ℕ) : ℝ) + ofDigits (b := 3) fun (i : ℕ) ↦ a (i + 1)
Tactic: fin_cases x
Cleared hypotheses: x, h
-- Lean 4 proof step from Mathlib.Topology.Instances.CantorSet (goal)
Goal:
i : ℕ
ih : ∀ {a : ℕ → Fin 3} (h : ∀ (n : ℕ), a n ≠ 1), ofDigits a ∈ preCantorSet i
a : ℕ → Fin 3
this : (ofDigits (b := 3) fun (i : ℕ) ↦ a (i + 1)) ∈ preCantorSet i
x : Fin 3
h : x ≠ 1
h₁ :
  ∀ (h : Ne (α := Fin 3) ((fun (i : Fin 3) ↦ i) ⟨0, ⋯⟩) 1),
    ((ofDigits (b := 3) fun (i : ℕ) ↦ a (i + 1)) =
        (↑(↑((fun (i : Fin 3) ↦ i) ⟨0, ⋯⟩) : ℕ) : ℝ) + ofDigits (b := 3) fun (i : ℕ) ↦ a (i + 1)) ∨
      (2 + ofDigits (b := 3) fun (i : ℕ) ↦ a (i + 1)) =
        (↑(↑((fun (i : Fin 3) ↦ i) ⟨0, ⋯⟩) : ℕ) : ℝ) + ofDigits (b := 3) fun (i : ℕ) ↦ a (i + 1)
h₂ :
  ∀ (h : Ne (α := Fin 3) ((fun (i : Fin 3) ↦ i) ⟨1, ⋯⟩) 1),
    ((ofDigits (b := 3) fun (i : ℕ) ↦ a (i + 1)) =
        (↑(↑((fun (i : Fin 3) ↦ i) ⟨1, ⋯⟩) : ℕ) : ℝ) + ofDigits (b := 3) fun (i : ℕ) ↦ a (i + 1)) ∨
      (2 + ofDigits (b := 3) fun (i : ℕ) ↦ a (i + 1)) =
        (↑(↑((fun (i : Fin 3) ↦ i) ⟨1, ⋯⟩) : ℕ) : ℝ) + ofDigits (b := 3) fun (i : ℕ) ↦ a (i + 1)
⊢ ((ofDigits (b := 3) fun (i : ℕ) ↦ a (i + 1)) = (↑(↑x : ℕ) : ℝ) + ofDigits (b := 3) fun (i : ℕ) ↦ a (i + 1)) ∨
    (2 + ofDigits (b := 3) fun (i : ℕ) ↦ a (i + 1)) = (↑(↑x : ℕ) : ℝ) + ofDigits (b := 3) fun (i : ℕ) ↦ a (i + 1)
have h₃ : ∀ (h : Ne (α := Fin 3) ((fun (i : Fin 3) ↦ i) ⟨2, sorry⟩) 1),
  ((ofDigits (b := 3) fun (i : ℕ) ↦ a (i + 1)) =
      (↑(↑((fun (i : Fin 3) ↦ i) ⟨2, sorry⟩) : ℕ) : ℝ) + ofDigits (b := 3) fun (i : ℕ) ↦ a (i + 1)) ∨
    (2 + ofDigits (b := 3) fun (i : ℕ) ↦ a (i + 1)) =
      (↑(↑((fun (i : Fin 3) ↦ i) ⟨2, sorry⟩) : ℕ) : ℝ) + ofDigits (b := 3) fun (i : ℕ) ↦ a (i + 1)
Tactic: fin_cases x
Cleared hypotheses: x, h
-- Lean 4 proof step from Mathlib.Topology.Instances.CantorSet (goal)
Goal:
a : ℕ → Fin 3
h : ∀ (n : ℕ), a n ≠ 1
i : ℕ
⊢ ofDigits a ∈ preCantorSet i
have zero : ∀ {a : ℕ → Fin 3} (h : ∀ (n : ℕ), a n ≠ 1), ofDigits a ∈ preCantorSet 0
Tactic: induction i generalizing a with
  | zero =>
    simp only [preCantorSet_zero, Set.mem_Icc]
    exact ⟨ofDigits_nonneg a, ofDigits_le_one a⟩
  | succ i ih =>
    simp only [preCantorSet, Set.mem_union, Set.mem_image, ← exists_or]
    use ofDigits (fun i ↦ a (i + 1))
    have : (ofDigits fun i ↦ a (i + 1)) ∈ preCantorSet i := ih (by solve_by_elim)
    simp only [this, ofDigits_eq_sum_add_ofDigits a 1, Finset.range_one, ofDigitsTerm,
      Nat.cast_ofNat, Finset.sum_singleton, zero_add, pow_one, true_and, field]
    specialize h 0
    generalize a 0 = x at h
    fin_cases x <;> simp at ⊢ h
Cleared hypotheses: a, h, i
-- Lean 4 proof step from Mathlib.Topology.Instances.CantorSet (goal)
Goal:
a : ℕ → Fin 3
h : ∀ (n : ℕ), a n ≠ 1
i : ℕ
zero : ∀ {a : ℕ → Fin 3} (h : ∀ (n : ℕ), a n ≠ 1), ofDigits a ∈ preCantorSet 0
⊢ ofDigits a ∈ preCantorSet i
have succ : ∀ (i : ℕ) (ih : ∀ {a : ℕ → Fin 3} (h : ∀ (n : ℕ), a n ≠ 1), ofDigits a ∈ preCantorSet i) {a : ℕ → Fin 3}
  (h : ∀ (n : ℕ), a n ≠ 1), ofDigits a ∈ preCantorSet (i + 1)
Tactic: induction i generalizing a with
  | zero =>
    simp only [preCantorSet_zero, Set.mem_Icc]
    exact ⟨ofDigits_nonneg a, ofDigits_le_one a⟩
  | succ i ih =>
    simp only [preCantorSet, Set.mem_union, Set.mem_image, ← exists_or]
    use ofDigits (fun i ↦ a (i + 1))
    have : (ofDigits fun i ↦ a (i + 1)) ∈ preCantorSet i := ih (by solve_by_elim)
    simp only [this, ofDigits_eq_sum_add_ofDigits a 1, Finset.range_one, ofDigitsTerm,
      Nat.cast_ofNat, Finset.sum_singleton, zero_add, pow_one, true_and, field]
    specialize h 0
    generalize a 0 = x at h
    fin_cases x <;> simp at ⊢ h
Cleared hypotheses: a, h, i
-- Lean 4 proof step from Mathlib.Topology.Instances.CantorSet (goal)
Goal:
a b : ℕ → Fin 3
ha : ∀ (n : ℕ), a n ≠ 1
hb : ∀ (n : ℕ), b n ≠ 1
h : ofDigits a = ofDigits b
⊢ a = b
have by_contra! : ∀ (h : a ≠ b), False
Tactic: by_contra! h
-- Lean 4 proof step from Mathlib.Topology.Instances.CantorSet (hypothesis)
Goal:
a b : ℕ → Fin 3
ha : ∀ (n : ℕ), a n ≠ 1
hb : ∀ (n : ℕ), b n ≠ 1
h_1 : ofDigits a = ofDigits b
h : a ≠ b
⊢ False
have h₁ : ∃ (a_1 : ℕ), a a_1 ≠ b a_1
Tactic: rw [Function.ne_iff] at h
Cleared hypotheses: h
-- Lean 4 proof step from Mathlib.Topology.Instances.CantorSet (hypothesis)
Goal:
a b : ℕ → Fin 3
ha : ∀ (n : ℕ), a n ≠ 1
hb : ∀ (n : ℕ), b n ≠ 1
h_1 : ofDigits a = ofDigits b
h : ∃ (a_1 : ℕ), a a_1 ≠ b a_1
n0 : ℕ := Nat.find (p := fun (n : ℕ) ↦ a n ≠ b n) h
⊢ False
have h1 : ∀ (n : ℕ) (hn : n < n0), a n = b n
Tactic: have h1 (n) (hn : n < n0) : a n = b n := by simpa using Nat.find_min h hn
-- Lean 4 proof step from Mathlib.Topology.Instances.CantorSet (hypothesis)
Goal:
a b : ℕ → Fin 3
ha : ∀ (n : ℕ), a n ≠ 1
hb : ∀ (n : ℕ), b n ≠ 1
h_1 : ofDigits a = ofDigits b
h : ∃ (a_1 : ℕ), a a_1 ≠ b a_1
n0 : ℕ := Nat.find (p := fun (n : ℕ) ↦ a n ≠ b n) h
h1 : ∀ (n : ℕ) (hn : n < n0), a n = b n
⊢ False
have h2 : a n0 ≠ b n0
Tactic: have h2 : a n0 ≠ b n0 := by simpa using Nat.find_spec h
-- Lean 4 proof step from Mathlib.Topology.Instances.CantorSet (goal)
Goal:
a b : ℕ → Fin 3
ha : ∀ (n : ℕ), a n ≠ 1
hb : ∀ (n : ℕ), b n ≠ 1
h_1 : ofDigits a = ofDigits b
h : ∃ (a_1 : ℕ), a a_1 ≠ b a_1
n0 : ℕ := Nat.find (p := fun (n : ℕ) ↦ a n ≠ b n) h
h1 : ∀ (n : ℕ) (hn : n < n0), a n = b n
h2 : a n0 ≠ b n0
⊢ False
have generalize : ∀ (n1 : ℕ) (h1 : ∀ (n : ℕ) (hn : n < n1), a n = b n) (h2 : a n1 ≠ b n1), False
Tactic: generalize n0 = n1 at h1 h2
Cleared hypotheses: h1, h2
-- Lean 4 proof step from Mathlib.Topology.Instances.CantorSet (goal)
Goal:
a b : ℕ → Fin 3
ha : ∀ (n : ℕ), a n ≠ 1
hb : ∀ (n : ℕ), b n ≠ 1
h : ofDigits a = ofDigits b
n1 : ℕ
h1 : ∀ (n : ℕ) (hn : n < n1), a n = b n
h2 : a n1 ≠ b n1
⊢ False
have inr : ∀
  (this :
    ∀ {a b : ℕ → Fin 3} (ha : ∀ (n : ℕ), a n ≠ 1) (hb : ∀ (n : ℕ), b n ≠ 1) (h : ofDigits a = ofDigits b)
      (h1 : ∀ (n : ℕ) (hn : n < n1), a n = b n) (h2 : a n1 ≠ b n1) (h3 : a n1 = 0 ∧ b n1 = 2), False)
  (h3 : ¬(a n1 = 0 ∧ b n1 = 2)), False
Tactic: wlog h3 : a n1 = 0 ∧ b n1 = 2 generalizing a b
-- Lean 4 proof step from Mathlib.Topology.Instances.CantorSet (goal)
Goal:
a b : ℕ → Fin 3
ha : ∀ (n : ℕ), a n ≠ 1
hb : ∀ (n : ℕ), b n ≠ 1
h : ofDigits a = ofDigits b
n1 : ℕ
h1 : ∀ (n : ℕ) (hn : n < n1), a n = b n
h2 : a n1 ≠ b n1
inr :
  ∀
    (this :
      ∀ {a b : ℕ → Fin 3} (ha : ∀ (n : ℕ), a n ≠ 1) (hb : ∀ (n : ℕ), b n ≠ 1) (h : ofDigits a = ofDigits b)
        (h1 : ∀ (n : ℕ) (hn : n < n1), a n = b n) (h2 : a n1 ≠ b n1) (h3 : a n1 = 0 ∧ b n1 = 2), False)
    (h3 : ¬(a n1 = 0 ∧ b n1 = 2)), False
⊢ False
have wlog : ∀ {a b : ℕ → Fin 3} (ha : ∀ (n : ℕ), a n ≠ 1) (hb : ∀ (n : ℕ), b n ≠ 1) (h : ofDigits a = ofDigits b)
  (h1 : ∀ (n : ℕ) (hn : n < n1), a n = b n) (h2 : a n1 ≠ b n1) (h3 : a n1 = 0 ∧ b n1 = 2), False
Tactic: wlog h3 : a n1 = 0 ∧ b n1 = 2 generalizing a b
Cleared hypotheses: a, b, ha, hb, h, h1, h2
-- Lean 4 proof step from Mathlib.Topology.Instances.CantorSet (hypothesis)
Goal:
n1 : ℕ
a b : ℕ → Fin 3
ha : ∀ (n : ℕ), a n ≠ 1
hb : ∀ (n : ℕ), b n ≠ 1
h : ofDigits a = ofDigits b
h1 : ∀ (n : ℕ) (hn : n < n1), a n = b n
h2 : a n1 ≠ b n1
h3 : a n1 = 0 ∧ b n1 = 2
⊢ False
have h3₁ : a n1 = 0
Tactic: obtain ⟨h3, h4⟩ := h3
Cleared hypotheses: h3
-- Lean 4 proof step from Mathlib.Topology.Instances.CantorSet (hypothesis)
Goal:
n1 : ℕ
a b : ℕ → Fin 3
ha : ∀ (n : ℕ), a n ≠ 1
hb : ∀ (n : ℕ), b n ≠ 1
h : ofDigits a = ofDigits b
h1 : ∀ (n : ℕ) (hn : n < n1), a n = b n
h2 : a n1 ≠ b n1
h3 : a n1 = 0 ∧ b n1 = 2
h3₁ : a n1 = 0
⊢ False
have h4 : b n1 = 2
Tactic: obtain ⟨h3, h4⟩ := h3
Cleared hypotheses: h3
-- Lean 4 proof step from Mathlib.Topology.Instances.CantorSet (goal)
Goal:
n1 : ℕ
a b : ℕ → Fin 3
ha : ∀ (n : ℕ), a n ≠ 1
hb : ∀ (n : ℕ), b n ≠ 1
h : ofDigits a = ofDigits b
h1 : ∀ (n : ℕ) (hn : n < n1), a n = b n
h3 : a n1 = 0
h4 : b n1 = 2
⊢ Eq (α := ℝ) (∑ x ∈ Finset.range n1, ofDigitsTerm a x) (∑ x ∈ Finset.range n1, ofDigitsTerm b x)
have apply : ∀ x ∈ Finset.range n1, ofDigitsTerm a x = ofDigitsTerm b x
Tactic: apply Finset.sum_congr rfl
-- Lean 4 proof step from Mathlib.Topology.Instances.CantorSet (hypothesis)
Goal:
n1 : ℕ
a b : ℕ → Fin 3
ha : ∀ (n : ℕ), a n ≠ 1
hb : ∀ (n : ℕ), b n ≠ 1
h : ofDigits a = ofDigits b
h1 : ∀ (n : ℕ) (hn : n < n1), a n = b n
h3 : a n1 = 0
h4 : b n1 = 2
⊢ False
have this : Eq (α := ℝ) (∑ x ∈ Finset.range n1, ofDigitsTerm a x) (∑ x ∈ Finset.range n1, ofDigitsTerm b x)
Tactic: have : ∑ x ∈ Finset.range n1, ofDigitsTerm a x = ∑ x ∈ Finset.range n1, ofDigitsTerm b x := by
    apply Finset.sum_congr rfl
    grind [ofDigitsTerm]
-- Lean 4 proof step from Mathlib.Topology.Instances.CantorSet (hypothesis)
Goal:
n1 : ℕ
a b : ℕ → Fin 3
ha : ∀ (n : ℕ), a n ≠ 1
hb : ∀ (n : ℕ), b n ≠ 1
h : ofDigits a = ofDigits b
h1 : ∀ (n : ℕ) (hn : n < n1), a n = b n
h3 : a n1 = 0
h4 : b n1 = 2
this : Eq (α := ℝ) (∑ x ∈ Finset.range n1, ofDigitsTerm a x) (∑ x ∈ Finset.range n1, ofDigitsTerm b x)
⊢ False
have h₁ : (∑ x ∈ Finset.range n1, ofDigitsTerm b x + ofDigitsTerm a n1 +
    (HPow.hPow (α := ℝ) (↑3 : ℝ) (n1 + 1))⁻¹ * ofDigits (b := 3) fun (i : ℕ) ↦ a (i + (n1 + 1))) =
  ∑ x ∈ Finset.range n1, ofDigitsTerm b x + ofDigitsTerm b n1 +
    (HPow.hPow (α := ℝ) (↑3 : ℝ) (n1 + 1))⁻¹ * ofDigits (b := 3) fun (i : ℕ) ↦ b (i + (n1 + 1))
Tactic: rw [ofDigits_eq_sum_add_ofDigits a (n1 + 1),
    ofDigits_eq_sum_add_ofDigits b (n1 + 1), Finset.sum_range_succ,
    Finset.sum_range_succ, this] at h
Cleared hypotheses: h, h1, h3, h4, this
-- Lean 4 proof step from Mathlib.Topology.Instances.CantorSet (hypothesis)
Goal:
n1 : ℕ
a b : ℕ → Fin 3
ha : ∀ (n : ℕ), a n ≠ 1
hb : ∀ (n : ℕ), b n ≠ 1
h :
  (∑ x ∈ Finset.range n1, ofDigitsTerm b x + ofDigitsTerm a n1 +
      (HPow.hPow (α := ℝ) (↑3 : ℝ) (n1 + 1))⁻¹ * ofDigits (b := 3) fun (i : ℕ) ↦ a (i + (n1 + 1))) =
    ∑ x ∈ Finset.range n1, ofDigitsTerm b x + ofDigitsTerm b n1 +
      (HPow.hPow (α := ℝ) (↑3 : ℝ) (n1 + 1))⁻¹ * ofDigits (b := 3) fun (i : ℕ) ↦ b (i + (n1 + 1))
h1 : ∀ (n : ℕ) (hn : n < n1), a n = b n
h3 : a n1 = 0
h4 : b n1 = 2
this : Eq (α := ℝ) (∑ x ∈ Finset.range n1, ofDigitsTerm a x) (∑ x ∈ Finset.range n1, ofDigitsTerm b x)
⊢ ofDigitsTerm a n1 + (HPow.hPow (α := ℝ) 3⁻¹ n1 * ofDigits (b := 3) fun (i : ℕ) ↦ a (1 + n1 + i)) * (1 / 3) =
    (HPow.hPow (α := ℝ) 3⁻¹ n1 * ofDigits (b := 3) fun (i : ℕ) ↦ b (1 + n1 + i)) * (1 / 3) + ofDigitsTerm b n1
have h₁ : ∑ x ∈ Finset.range n1, ofDigitsTerm b x + ofDigitsTerm a n1 +
    (HPow.hPow (α := ℝ) 3⁻¹ n1 * ofDigits (b := 3) fun (i : ℕ) ↦ a (1 + n1 + i)) * (1 / 3) =
  ∑ x ∈ Finset.range n1, ofDigitsTerm b x +
      (HPow.hPow (α := ℝ) 3⁻¹ n1 * ofDigits (b := 3) fun (i : ℕ) ↦ b (1 + n1 + i)) * (1 / 3) +
    ofDigitsTerm b n1
Tactic: ring_nf at h
Cleared hypotheses: h
-- Lean 4 proof step from Mathlib.Topology.Instances.CantorSet (goal)
Goal:
n1 : ℕ
a b : ℕ → Fin 3
ha : ∀ (n : ℕ), a n ≠ 1
hb : ∀ (n : ℕ), b n ≠ 1
h1 : ∀ (n : ℕ) (hn : n < n1), a n = b n
h3 : a n1 = 0
h4 : b n1 = 2
this : Eq (α := ℝ) (∑ x ∈ Finset.range n1, ofDigitsTerm a x) (∑ x ∈ Finset.range n1, ofDigitsTerm b x)
h :
  ∑ x ∈ Finset.range n1, ofDigitsTerm b x + ofDigitsTerm a n1 +
      (HPow.hPow (α := ℝ) 3⁻¹ n1 * ofDigits (b := 3) fun (i : ℕ) ↦ a (1 + n1 + i)) * (1 / 3) =
    ∑ x ∈ Finset.range n1, ofDigitsTerm b x +
        (HPow.hPow (α := ℝ) 3⁻¹ n1 * ofDigits (b := 3) fun (i : ℕ) ↦ b (1 + n1 + i)) * (1 / 3) +
      ofDigitsTerm b n1
⊢ ofDigitsTerm a n1 + (HPow.hPow (α := ℝ) 3⁻¹ n1 * ofDigits (b := 3) fun (i : ℕ) ↦ a (1 + n1 + i)) * (1 / 3) =
    (HPow.hPow (α := ℝ) 3⁻¹ n1 * ofDigits (b := 3) fun (i : ℕ) ↦ b (1 + n1 + i)) * (1 / 3) + ofDigitsTerm b n1
have linarith : ofDigitsTerm a n1 + (HPow.hPow (α := ℝ) 3⁻¹ n1 * ofDigits (b := 3) fun (i : ℕ) ↦ a (1 + n1 + i)) * (1 / 3) <
    (HPow.hPow (α := ℝ) 3⁻¹ n1 * ofDigits (b := 3) fun (i : ℕ) ↦ b (1 + n1 + i)) * (1 / 3) + ofDigitsTerm b n1 →
  HMul.hMul (α := ℝ) 1 1 = 1
Tactic: linarith