awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Module: Mathlib.Topology.Instances.CantorSet
Goal:
⊢ Function.Bijective (β := ℝ) fun (x : ℝ) ↦ (2 + x) / 3

Tactic:
exact (mulRight_bijective₀ 3⁻¹ (by simp)).comp (AddGroup.addLeft_bijective 2)

Premises used: Inv.inv, mulRight_bijective₀, AddGroup.addLeft_bijective, Function.Bijective.comp, OfNat.ofNat
-- Module: Mathlib.Topology.Instances.CantorSet
Goal:
⊢ Function.Bijective (β := ℝ) fun (x : ℝ) ↦ x / 3

Tactic:
exact mulRight_bijective₀ 3⁻¹ (by simp)

Premises used: Inv.inv, mulRight_bijective₀, OfNat.ofNat
-- Module: Mathlib.Topology.Instances.CantorSet
Goal:
simp_rw : Eq (α := Set ℝ) (⋂ (n : ℕ), preCantorSet n) (⋂ (i : ℕ), preCantorSet (i + 1))
⊢ 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:
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]

Premises used: preCantorSet_antitone, Function.comp_def, preCantorSet_succ, Set.monotone_image, Monotone.comp_antitone, Set.iInter_union_of_antitone
-- Module: Mathlib.Topology.Instances.CantorSet
Goal:
⊢ Eq (α := Set ℝ) (⋂ (n : ℕ), preCantorSet n) (⋂ (i : ℕ), preCantorSet (i + 1))

Tactic:
(preCantorSet_antitone.iInter_nat_add _).symm

Premises used: preCantorSet_antitone, Antitone.iInter_nat_add, Eq.symm
-- Module: Mathlib.Topology.Instances.CantorSet
Goal:
⊢ Eq (α := Set ℝ) (⋂ (n : ℕ), preCantorSet n) (⋂ (i : ℕ), preCantorSet (i + 1))

Tactic:
exact (preCantorSet_antitone.iInter_nat_add _).symm

Premises used: preCantorSet_antitone, Antitone.iInter_nat_add, Eq.symm
-- Module: Mathlib.Topology.Instances.CantorSet
Goal:
n : ℕ
⊢ 1 / 3 ≠ 0

Tactic:
simp
-- Module: Mathlib.Topology.Instances.CantorSet
Goal:
f : ℝ ≃ₜ ℝ := Homeomorph.mulLeft₀ (1 / 3) ⋯
g : ℝ ≃ₜ ℝ := Homeomorph.trans (Homeomorph.addLeft 2) f
⊢ IsClosed (preCantorSet 0)

Tactic:
isClosed_Icc

Premises used: isClosed_Icc, preCantorSet
-- Module: Mathlib.Topology.Instances.CantorSet
Goal:
f : ℝ ≃ₜ ℝ := Homeomorph.mulLeft₀ (1 / 3) ⋯
g : ℝ ≃ₜ ℝ := Homeomorph.trans (Homeomorph.addLeft 2) f
⊢ IsClosed (preCantorSet 0)

Tactic:
exact isClosed_Icc

Premises used: isClosed_Icc
-- Module: Mathlib.Topology.Instances.CantorSet
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))

Tactic:
IsClosed.union ?_ ?_

Premises used: IsClosed.union, preCantorSet
-- Module: Mathlib.Topology.Instances.CantorSet
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)
succ₁ : IsClosed (X := ℝ) ((fun (x : ℝ) ↦ (2 + x) / 3) '' preCantorSet n)
⊢ IsClosed (preCantorSet (n + 1))

Tactic:
refine IsClosed.union ?_ ?_

Premises used: IsClosed.union
-- Module: Mathlib.Topology.Instances.CantorSet
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 (X := ℝ) ((fun (x : ℝ) ↦ x / 3) '' preCantorSet n)

Tactic:
simpa [f, div_eq_inv_mul] using f.isClosedEmbedding.isClosed_iff_image_isClosed.mp ih

Premises used: Topology.IsClosedEmbedding.isClosed_iff_image_isClosed, Iff.mp, div_eq_inv_mul, Homeomorph.isClosedEmbedding
-- Module: Mathlib.Topology.Instances.CantorSet
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 (X := ℝ) ((fun (x : ℝ) ↦ (2 + x) / 3) '' preCantorSet n)

Tactic:
simpa [g, f, div_eq_inv_mul] using g.isClosedEmbedding.isClosed_iff_image_isClosed.mp ih

Premises used: Topology.IsClosedEmbedding.isClosed_iff_image_isClosed, Iff.mp, div_eq_inv_mul, Homeomorph.isClosedEmbedding
-- Module: Mathlib.Topology.Instances.CantorSet
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)
succ : ∀ (n : ℕ) (ih : IsClosed (preCantorSet n)), IsClosed (preCantorSet (n + 1))
⊢ IsClosed (preCantorSet n)

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

Premises used: IsClosed.union, isClosed_Icc, div_eq_inv_mul
-- Module: Mathlib.Topology.Instances.CantorSet
Goal:
⊢ IsClosed cantorSet

Tactic:
isClosed_iInter isClosed_preCantorSet

Premises used: isClosed_preCantorSet, cantorSet, isClosed_iInter
-- Module: Mathlib.Topology.Instances.CantorSet
Goal:
⊢ IsCompact cantorSet

Tactic:
isCompact_Icc.of_isClosed_subset isClosed_cantorSet cantorSet_subset_unitInterval

Premises used: CompactIccSpace.isCompact_Icc, isClosed_cantorSet, cantorSet_subset_unitInterval, IsCompact.of_isClosed_subset
-- Module: Mathlib.Topology.Instances.CantorSet
Goal:
a : ℕ → Fin 3
h : ∀ (n : ℕ), a n ≠ 1
simp : ∀ (i : ℕ), ofDigits a ∈ preCantorSet i
⊢ ofDigits a ∈ cantorSet

Tactic:
simp only [cantorSet, Set.mem_iInter]

Premises used: Set.mem_iInter, cantorSet
-- Module: Mathlib.Topology.Instances.CantorSet
Goal:
a : ℕ → Fin 3
h : ∀ (n : ℕ), a n ≠ 1
⊢ 0 ≤ ofDigits a ∧ ofDigits a ≤ 1

Tactic:
⟨ofDigits_nonneg a, ofDigits_le_one a⟩

Premises used: Real.ofDigits_nonneg, And.intro, Real.ofDigits_le_one
-- Module: Mathlib.Topology.Instances.CantorSet
Goal:
a : ℕ → Fin 3
h : ∀ (n : ℕ), a n ≠ 1
⊢ 0 ≤ ofDigits a ∧ ofDigits a ≤ 1

Tactic:
exact ⟨ofDigits_nonneg a, ofDigits_le_one a⟩
-- Module: Mathlib.Topology.Instances.CantorSet
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
succ : ∃ (x : ℝ), x ∈ preCantorSet i ∧ x / 3 = ofDigits a ∨ x ∈ preCantorSet i ∧ (2 + x) / 3 = ofDigits a
⊢ ofDigits a ∈ preCantorSet (i + 1)

Tactic:
simp only [preCantorSet, Set.mem_union, Set.mem_image, ← exists_or]

Premises used: Set.mem_union, exists_or, Set.mem_image, preCantorSet
-- Module: Mathlib.Topology.Instances.CantorSet
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
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
⊢ ∃ (x : ℝ), x ∈ preCantorSet i ∧ x / 3 = ofDigits a ∨ x ∈ preCantorSet i ∧ (2 + x) / 3 = ofDigits a

Tactic:
use ofDigits (fun i ↦ a (i + 1))

Premises used: HAdd.hAdd, Real.ofDigits, OfNat.ofNat
-- Module: Mathlib.Topology.Instances.CantorSet
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
⊢ ∀ (n : ℕ), a (n + 1) ≠ 1

Tactic:
solve_by_elim
-- Module: Mathlib.Topology.Instances.CantorSet
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

Tactic:
have : (ofDigits fun i ↦ a (i + 1)) ∈ preCantorSet i := ih (by solve_by_elim)
-- Module: Mathlib.Topology.Instances.CantorSet
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
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)
⊢ (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:
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]

Premises used: Nat.cast_ofNat, Real.ofDigits_eq_sum_add_ofDigits, zero_add, Finset.sum_singleton, pow_one, Real.ofDigitsTerm, Finset.range_one, OfNat.ofNat, true_and
-- Module: Mathlib.Topology.Instances.CantorSet
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
⊢ a 0 ≠ 1

Tactic:
specialize h 0

Premises used: OfNat.ofNat
-- Module: Mathlib.Topology.Instances.CantorSet
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
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)
⊢ ((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:
generalize a 0 = x at h

Premises used: OfNat.ofNat
-- Module: Mathlib.Topology.Instances.CantorSet
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)
h₃ :
  ∀ (h : Ne (α := Fin 3) ((fun (i : Fin 3) ↦ i) ⟨2, ⋯⟩) 1),
    ((ofDigits (b := 3) fun (i : ℕ) ↦ a (i + 1)) =
        (↑(↑((fun (i : Fin 3) ↦ i) ⟨2, ⋯⟩) : 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, ⋯⟩) : 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)

Tactic:
fin_cases x
-- Module: Mathlib.Topology.Instances.CantorSet
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 : 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)

Tactic:
simp at ⊢ h
-- Module: Mathlib.Topology.Instances.CantorSet
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 : 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)

Tactic:
simp at ⊢ h
-- Module: Mathlib.Topology.Instances.CantorSet
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 : Ne (α := Fin 3) ((fun (i : Fin 3) ↦ i) ⟨2, ⋯⟩) 1
⊢ ((ofDigits (b := 3) fun (i : ℕ) ↦ a (i + 1)) =
      (↑(↑((fun (i : Fin 3) ↦ i) ⟨2, ⋯⟩) : 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, ⋯⟩) : ℕ) : ℝ) + ofDigits (b := 3) fun (i : ℕ) ↦ a (i + 1)

Tactic:
simp at ⊢ h
-- Module: Mathlib.Topology.Instances.CantorSet
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
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)
⊢ ofDigits a ∈ preCantorSet i

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

Premises used: Set.mem_union, Nat.cast_ofNat, exists_or, Set.mem_image, Real.ofDigits_eq_sum_add_ofDigits, zero_add, Finset.sum_singleton, HAdd.hAdd, pow_one, preCantorSet_zero, Real.ofDigitsTerm, preCantorSet, Real.ofDigits, Set.mem_Icc, Finset.range_one, OfNat.ofNat, true_and
-- Module: Mathlib.Topology.Instances.CantorSet
Goal:
a b : ℕ → Fin 3
ha : ∀ (n : ℕ), a n ≠ 1
hb : ∀ (n : ℕ), b n ≠ 1
h : ofDigits a = ofDigits b
by_contra! : ∀ (h : a ≠ b), False
⊢ a = b

Tactic:
by_contra! h
-- Module: Mathlib.Topology.Instances.CantorSet
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
⊢ ∃ (a_1 : ℕ), a a_1 ≠ b a_1

Tactic:
rw [Function.ne_iff] at h

Premises used: Function.ne_iff
-- Module: Mathlib.Topology.Instances.CantorSet
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
n : ℕ
hn : n < n0
⊢ a n = b n

Tactic:
simpa using Nat.find_min h hn

Premises used: Nat.find_min
-- Module: Mathlib.Topology.Instances.CantorSet
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
⊢ ∀ (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
-- Module: Mathlib.Topology.Instances.CantorSet
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
⊢ a n0 ≠ b n0

Tactic:
simpa using Nat.find_spec h

Premises used: Nat.find_spec
-- Module: Mathlib.Topology.Instances.CantorSet
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
⊢ a n0 ≠ b n0

Tactic:
have h2 : a n0 ≠ b n0 := by simpa using Nat.find_spec h
-- Module: Mathlib.Topology.Instances.CantorSet
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
generalize : ∀ (n1 : ℕ) (h1 : ∀ (n : ℕ) (hn : n < n1), a n = b n) (h2 : a n1 ≠ b n1), False
⊢ False

Tactic:
generalize n0 = n1 at h1 h2
-- Module: Mathlib.Topology.Instances.CantorSet
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
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
⊢ False

Tactic:
wlog h3 : a n1 = 0 ∧ b n1 = 2 generalizing a b
-- Module: Mathlib.Topology.Instances.CantorSet
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
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)
⊢ b n1 = 0 ∧ a n1 = 2

Tactic:
grind
-- Module: Mathlib.Topology.Instances.CantorSet
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
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:
exact this hb ha h.symm (fun n hn ↦ (h1 n hn).symm) h2.symm (by grind)

Premises used: Ne.symm, Eq.symm
-- Module: Mathlib.Topology.Instances.CantorSet
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
⊢ a n1 = 0

Tactic:
obtain ⟨h3, h4⟩ := h3
-- Module: Mathlib.Topology.Instances.CantorSet
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
⊢ b n1 = 2

Tactic:
obtain ⟨h3, h4⟩ := h3
-- Module: Mathlib.Topology.Instances.CantorSet
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
apply : ∀ x ∈ Finset.range n1, ofDigitsTerm a x = ofDigitsTerm b x
⊢ Eq (α := ℝ) (∑ x ∈ Finset.range n1, ofDigitsTerm a x) (∑ x ∈ Finset.range n1, ofDigitsTerm b x)

Tactic:
apply Finset.sum_congr rfl

Premises used: Finset.sum_congr, rfl
-- Module: Mathlib.Topology.Instances.CantorSet
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
⊢ ∀ x ∈ Finset.range n1, ofDigitsTerm a x = ofDigitsTerm b x

Tactic:
grind [ofDigitsTerm]

Premises used: Real.ofDigitsTerm
-- Module: Mathlib.Topology.Instances.CantorSet
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)

Tactic:
have : ∑ x ∈ Finset.range n1, ofDigitsTerm a x = ∑ x ∈ Finset.range n1, ofDigitsTerm b x := by
    apply Finset.sum_congr rfl
    grind [ofDigitsTerm]
-- Module: Mathlib.Topology.Instances.CantorSet
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)
⊢ (∑ 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

Premises used: Real.ofDigits_eq_sum_add_ofDigits, HAdd.hAdd, OfNat.ofNat, Finset.sum_range_succ
-- Module: Mathlib.Topology.Instances.CantorSet
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)
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))
⊢ ∀ (n : ℕ) (hn : n < n1), a n = b n

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

Premises used: Real.ofDigits_eq_sum_add_ofDigits, HAdd.hAdd, OfNat.ofNat, Finset.sum_range_succ
-- Module: Mathlib.Topology.Instances.CantorSet
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)
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))
h1₁ : ∀ (n : ℕ) (hn : n < n1), a n = b n
⊢ a n1 = 0

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

Premises used: Real.ofDigits_eq_sum_add_ofDigits, HAdd.hAdd, OfNat.ofNat, Finset.sum_range_succ
-- Module: Mathlib.Topology.Instances.CantorSet
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)
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))
h1₁ : ∀ (n : ℕ) (hn : n < n1), a n = b n
h3₁ : a n1 = 0
⊢ b n1 = 2

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

Premises used: Real.ofDigits_eq_sum_add_ofDigits, HAdd.hAdd, OfNat.ofNat, Finset.sum_range_succ