awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- 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
h4₁ : b n1 = 2
⊢ Eq (α := ℝ) (∑ x ∈ Finset.range n1, ofDigitsTerm a x) (∑ x ∈ Finset.range n1, ofDigitsTerm b x)

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 :
  (∑ 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)
⊢ ∑ 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
-- 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 :
  (∑ 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

Tactic:
replace h : ofDigitsTerm a n1 + (3⁻¹ ^ n1 * ofDigits fun i ↦ a (1 + n1 + i)) * (1 / 3) =
      (3⁻¹ ^ n1 * ofDigits fun i ↦ b (1 + n1 + i)) * (1 / 3) + ofDigitsTerm b n1 := by
    ring_nf at h
    linarith
-- 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
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 :
  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
⊢ ((HPow.hPow (α := ℝ) 3 n1)⁻¹ * ofDigits (b := 3) fun (i : ℕ) ↦ a (1 + n1 + i)) * 3⁻¹ =
    ((HPow.hPow (α := ℝ) 3 n1)⁻¹ * ofDigits (b := 3) fun (i : ℕ) ↦ b (1 + n1 + i)) * 3⁻¹ +
      2 * (3⁻¹ * (HPow.hPow (α := ℝ) 3 n1)⁻¹)

Tactic:
simp only [ofDigitsTerm, h3, Fin.isValue, Fin.coe_ofNat_eq_mod, Nat.zero_mod, CharP.cast_eq_zero,
    Nat.cast_ofNat, pow_succ, mul_inv_rev, zero_mul, inv_pow, one_div, zero_add, h4,
    Nat.mod_succ] at h

Premises used: CharP.cast_eq_zero, Nat.cast_ofNat, pow_succ, MulZeroClass.zero_mul, inv_pow, zero_add, Fin.isValue, one_div, Real.ofDigitsTerm, Fin.coe_ofNat_eq_mod, mul_inv_rev, Nat.zero_mod, Nat.mod_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
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 :
  ((HPow.hPow (α := ℝ) 3 n1)⁻¹ * ofDigits (b := 3) fun (i : ℕ) ↦ a (1 + n1 + i)) * 3⁻¹ =
    ((HPow.hPow (α := ℝ) 3 n1)⁻¹ * ofDigits (b := 3) fun (i : ℕ) ↦ b (1 + n1 + i)) * 3⁻¹ +
      2 * (3⁻¹ * (HPow.hPow (α := ℝ) 3 n1)⁻¹)
⊢ (HPow.hPow (α := ℝ) 3 n1)⁻¹ ≠ 0

Tactic:
positivity
-- 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
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 :
  ((HPow.hPow (α := ℝ) 3 n1)⁻¹ * ofDigits (b := 3) fun (i : ℕ) ↦ a (1 + n1 + i)) * 3⁻¹ =
    ((HPow.hPow (α := ℝ) 3 n1)⁻¹ * ofDigits (b := 3) fun (i : ℕ) ↦ b (1 + n1 + i)) * 3⁻¹ +
      2 * (3⁻¹ * (HPow.hPow (α := ℝ) 3 n1)⁻¹)
rw :
  (HPow.hPow (α := ℝ) 3 n1)⁻¹ * ((ofDigits (b := 3) fun (i : ℕ) ↦ a (1 + n1 + i)) * 3⁻¹) =
    (HPow.hPow (α := ℝ) 3 n1)⁻¹ * ((ofDigits (b := 3) fun (i : ℕ) ↦ b (1 + n1 + i)) * 3⁻¹ + HMul.hMul (α := ℝ) 2 3⁻¹)
⊢ (ofDigits (b := 3) fun (i : ℕ) ↦ a (1 + n1 + i)) * 3⁻¹ =
    (ofDigits (b := 3) fun (i : ℕ) ↦ b (1 + n1 + i)) * 3⁻¹ + HMul.hMul (α := ℝ) 2 3⁻¹

Tactic:
rw [← mul_right_inj' (show ((3 : ℝ) ^ n1)⁻¹ ≠ 0 by positivity)]

Premises used: mul_right_inj'
-- 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
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 :
  ((HPow.hPow (α := ℝ) 3 n1)⁻¹ * ofDigits (b := 3) fun (i : ℕ) ↦ a (1 + n1 + i)) * 3⁻¹ =
    ((HPow.hPow (α := ℝ) 3 n1)⁻¹ * ofDigits (b := 3) fun (i : ℕ) ↦ b (1 + n1 + i)) * 3⁻¹ +
      2 * (3⁻¹ * (HPow.hPow (α := ℝ) 3 n1)⁻¹)
⊢ (ofDigits (b := 3) fun (i : ℕ) ↦ a (1 + n1 + i)) * 3⁻¹ =
    (ofDigits (b := 3) fun (i : ℕ) ↦ b (1 + n1 + i)) * 3⁻¹ + HMul.hMul (α := ℝ) 2 3⁻¹

Tactic:
replace h : (ofDigits fun i ↦ a (1 + n1 + i)) * 3⁻¹ =
      (ofDigits fun i ↦ b (1 + n1 + i)) * 3⁻¹ + 2 * 3⁻¹ := by
    rw [← mul_right_inj' (show ((3 : ℝ) ^ n1)⁻¹ ≠ 0 by positivity)]
    linarith
-- 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
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 :
  (ofDigits (b := 3) fun (i : ℕ) ↦ a (1 + n1 + i)) * 3⁻¹ =
    (ofDigits (b := 3) fun (i : ℕ) ↦ b (1 + n1 + i)) * 3⁻¹ + HMul.hMul (α := ℝ) 2 3⁻¹
linarith : 3 / 3 = 1
linarith₁ linarith₂ : HMul.hMul (α := ℝ) 1 3 = 3
linarith₃ : 3 / 3 = 1
linarith₄ linarith₅ : HMul.hMul (α := ℝ) 1 3 = 3
linarith₆ : 3 / 3 = 1
linarith₇ : HMul.hMul (α := ℝ) 1 3 = 3
linarith₈ : GT.gt (α := ℝ) 3 0
linarith₉ :
  -1 +
          -((1 * ofDigits (b := 3) fun (i : ℕ) ↦ a (1 + n1 + i)) * HMul.hMul (α := ℝ) 1 1 -
              ((1 * ofDigits (b := 3) fun (i : ℕ) ↦ b (1 + n1 + i)) * HMul.hMul (α := ℝ) 1 1 +
                HMul.hMul (α := ℝ) 1 2)) +
        (0 - ofDigits (b := 3) fun (i : ℕ) ↦ b (1 + n1 + i)) +
      ((ofDigits (b := 3) fun (i : ℕ) ↦ a (1 + n1 + i)) - 1) =
    0
⊢ False

Tactic:
linarith [ofDigits_nonneg (fun i ↦ b (1 + n1 + i)), ofDigits_le_one (fun i ↦ a (1 + n1 + i))]

Premises used: Real.ofDigits_nonneg, HAdd.hAdd, OfNat.ofNat, Real.ofDigits_le_one
-- Module: Mathlib.Topology.Instances.CantorSet
Goal:
x : ℝ
hx : x ∈ cantorSet
⊢ Membership.mem (γ := Set ℝ) ((fun (x : ℝ) ↦ x / 3) '' cantorSet ∪ (fun (x : ℝ) ↦ (2 + x) / 3) '' cantorSet) x

Tactic:
rw [cantorSet_eq_union_halves] at hx

Premises used: cantorSet_eq_union_halves
-- Module: Mathlib.Topology.Instances.CantorSet
Goal:
x : ℝ
hx : Membership.mem (γ := Set ℝ) ((fun (x : ℝ) ↦ x / 3) '' cantorSet ∪ (fun (x : ℝ) ↦ (2 + x) / 3) '' cantorSet) x
⊢ ∃ y ∈ cantorSet, y / 3 = x ∨ (2 + y) / 3 = x

Tactic:
grind
-- Module: Mathlib.Topology.Instances.CantorSet
Goal:
x : ℝ
hx : x ∈ cantorSet
inl :
  ∀ (y : ℝ) (hy : y ∈ cantorSet) (hx : y / 3 ∈ cantorSet),
    (if y / 3 ∈ Set.Icc 0 (1 / 3) then 3 * (y / 3) else 3 * (y / 3) - 2) ∈ cantorSet
inr :
  ∀ (y : ℝ) (hy : y ∈ cantorSet) (hx : (2 + y) / 3 ∈ cantorSet),
    (if (2 + y) / 3 ∈ Set.Icc 0 (1 / 3) then 3 * ((2 + y) / 3) else 3 * ((2 + y) / 3) - 2) ∈ cantorSet
⊢ (if x ∈ Set.Icc 0 (1 / 3) then 3 * x else 3 * x - 2) ∈ cantorSet

Tactic:
obtain ⟨y, hy, rfl | rfl⟩ : ∃ y ∈ cantorSet, y / 3 = x ∨ (2 + y) / 3 = x := by
    rw [cantorSet_eq_union_halves] at hx
    grind

Premises used: cantorSet, rfl
-- Module: Mathlib.Topology.Instances.CantorSet
Goal:
y : ℝ
hy : y ∈ cantorSet
hx : y / 3 ∈ cantorSet
⊢ (if y / 3 ∈ Set.Icc 0 (1 / 3) then 3 * (y / 3) else 3 * (y / 3) - 2) ∈ cantorSet

Tactic:
grind [cantorSet_subset_unitInterval]

Premises used: cantorSet_subset_unitInterval
-- Module: Mathlib.Topology.Instances.CantorSet
Goal:
y : ℝ
hy : y ∈ cantorSet
hx : (2 + y) / 3 ∈ cantorSet
⊢ (if (2 + y) / 3 ∈ Set.Icc 0 (1 / 3) then 3 * ((2 + y) / 3) else 3 * ((2 + y) / 3) - 2) ∈ cantorSet

Tactic:
grind [cantorSet_subset_unitInterval]

Premises used: cantorSet_subset_unitInterval
-- Module: Mathlib.Topology.Instances.CantorSet
Goal:
x : ℝ
hx : x ∈ cantorSet
⊢ (cantorSequence x).get 0 ∈ cantorSet

Tactic:
simpa [cantorSequence]

Premises used: cantorSequence
-- Module: Mathlib.Topology.Instances.CantorSet
Goal:
x : ℝ
hx : x ∈ cantorSet
n : ℕ
ih : (cantorSequence x).get n ∈ cantorSet
⊢ (cantorSequence x).get (n + 1) ∈ cantorSet

Tactic:
cantorStep_mem_cantorSet ih

Premises used: Set.Mem, cantorStep_mem_cantorSet, Stream'.iterate, Stream'.get, cantorStep, cantorSequence, ite
-- Module: Mathlib.Topology.Instances.CantorSet
Goal:
x : ℝ
hx : x ∈ cantorSet
n : ℕ
ih : (cantorSequence x).get n ∈ cantorSet
⊢ (cantorSequence x).get (n + 1) ∈ cantorSet

Tactic:
exact cantorStep_mem_cantorSet ih

Premises used: cantorStep_mem_cantorSet
-- Module: Mathlib.Topology.Instances.CantorSet
Goal:
x : ℝ
hx : x ∈ cantorSet
n : ℕ
zero : (cantorSequence x).get 0 ∈ cantorSet
succ : ∀ (n : ℕ) (ih : (cantorSequence x).get n ∈ cantorSet), (cantorSequence x).get (n + 1) ∈ cantorSet
⊢ (cantorSequence x).get n ∈ cantorSet

Tactic:
induction n with
  | zero => simpa [cantorSequence]
  | succ n ih => exact cantorStep_mem_cantorSet ih

Premises used: cantorSequence
-- Module: Mathlib.Topology.Instances.CantorSet
Goal:
f : ℕ → Bool
⊢ ∀ (n : ℕ), Ne (α := Fin 3) (bif f n then 2 else 0) 1

Tactic:
grind
-- Module: Mathlib.Topology.Instances.CantorSet
Goal:
x : ℝ
n : ℕ
⊢ (cantorToTernary x).get n ≠ 1

Tactic:
grind [cantorToTernary, Stream'.get_map]

Premises used: cantorToTernary, Stream'.get_map
-- Module: Mathlib.Topology.Instances.CantorSet
Goal:
x : ℝ
n : ℕ
simp :
  (if 0 ≤ (Stream'.iterate cantorStep x).get n ∧ (Stream'.iterate cantorStep x).get n ≤ 1 / 3 then
      3 * (Stream'.iterate cantorStep x).get n
    else 3 * (Stream'.iterate cantorStep x).get n - 2) =
    3 *
      ((Stream'.iterate cantorStep x).get n -
        HPow.hPow (α := ℝ) 3 n *
          ((↑(↑(if 0 ≤ (Stream'.iterate cantorStep x).get n ∧ (Stream'.iterate cantorStep x).get n ≤ 1 / 3 then 0
                    else 2) :
                  ℕ) :
              ℝ) *
            (HPow.hPow (α := ℝ) (↑3 : ℝ) (n + 1))⁻¹))
⊢ (cantorSequence x).get (n + 1) =
    3 * ((cantorSequence x).get n - HPow.hPow (α := ℝ) 3 n * ofDigitsTerm (cantorToTernary x).get n)

Tactic:
simp only [cantorSequence, ofDigitsTerm, cantorToTernary, cantorToBinary, Set.mem_Icc,
    Bool.if_true_right, Bool.or_false, Stream'.get_map, Bool.cond_not, Bool.cond_decide,
    Stream'.get_succ_iterate', cantorStep]

Premises used: cantorToTernary, Stream'.get_map, cantorToBinary, Bool.cond_not, cantorStep, Real.ofDigitsTerm, Set.mem_Icc, Stream'.get_succ_iterate', cantorSequence, Bool.if_true_right, Bool.or_false, Bool.cond_decide
-- Module: Mathlib.Topology.Instances.CantorSet
Goal:
x : ℝ
n : ℕ
pos :
  0 ≤ (Stream'.iterate cantorStep x).get n ∧ (Stream'.iterate cantorStep x).get n ≤ 1 / 3 →
    3 * (Stream'.iterate cantorStep x).get n =
      3 *
        ((Stream'.iterate cantorStep x).get n -
          HPow.hPow (α := ℝ) 3 n * ((↑(↑0 : ℕ) : ℝ) * (HPow.hPow (α := ℝ) (↑3 : ℝ) (n + 1))⁻¹))
neg :
  ¬(0 ≤ (Stream'.iterate cantorStep x).get n ∧ (Stream'.iterate cantorStep x).get n ≤ 1 / 3) →
    3 * (Stream'.iterate cantorStep x).get n - 2 =
      3 *
        ((Stream'.iterate cantorStep x).get n -
          HPow.hPow (α := ℝ) 3 n * ((↑(↑2 : ℕ) : ℝ) * (HPow.hPow (α := ℝ) (↑3 : ℝ) (n + 1))⁻¹))
⊢ (if 0 ≤ (Stream'.iterate cantorStep x).get n ∧ (Stream'.iterate cantorStep x).get n ≤ 1 / 3 then
      3 * (Stream'.iterate cantorStep x).get n
    else 3 * (Stream'.iterate cantorStep x).get n - 2) =
    3 *
      ((Stream'.iterate cantorStep x).get n -
        HPow.hPow (α := ℝ) 3 n *
          ((↑(↑(if 0 ≤ (Stream'.iterate cantorStep x).get n ∧ (Stream'.iterate cantorStep x).get n ≤ 1 / 3 then 0
                    else 2) :
                  ℕ) :
              ℝ) *
            (HPow.hPow (α := ℝ) (↑3 : ℝ) (n + 1))⁻¹))

Tactic:
split_ifs
-- Module: Mathlib.Topology.Instances.CantorSet
Goal:
x : ℝ
n : ℕ
h : 0 ≤ (Stream'.iterate cantorStep x).get n ∧ (Stream'.iterate cantorStep x).get n ≤ 1 / 3
⊢ 3 * (Stream'.iterate cantorStep x).get n =
    3 *
      ((Stream'.iterate cantorStep x).get n -
        HPow.hPow (α := ℝ) 3 n * ((↑(↑0 : ℕ) : ℝ) * (HPow.hPow (α := ℝ) (↑3 : ℝ) (n + 1))⁻¹))

Tactic:
simp
-- Module: Mathlib.Topology.Instances.CantorSet
Goal:
x : ℝ
n : ℕ
h : ¬(0 ≤ (Stream'.iterate cantorStep x).get n ∧ (Stream'.iterate cantorStep x).get n ≤ 1 / 3)
⊢ 3 * (Stream'.iterate cantorStep x).get n - 2 =
    3 * ((Stream'.iterate cantorStep x).get n - HPow.hPow (α := ℝ) 3 n * (2 * (HPow.hPow (α := ℝ) 3 (n + 1))⁻¹))

Tactic:
field
-- Module: Mathlib.Topology.Instances.CantorSet
Goal:
x : ℝ
⊢ (cantorSequence x).get 0 =
    (x - ∑ i ∈ Finset.range 0, ofDigitsTerm (cantorToTernary x).get i) * HPow.hPow (α := ℝ) (β := ℕ) 3 0

Tactic:
simp [cantorSequence]

Premises used: cantorSequence
-- Module: Mathlib.Topology.Instances.CantorSet
Goal:
x : ℝ
n : ℕ
ih :
  (cantorSequence x).get n = (x - ∑ i ∈ Finset.range n, ofDigitsTerm (cantorToTernary x).get i) * HPow.hPow (α := ℝ) 3 n
succ :
  3 *
      ((x - ∑ i ∈ Finset.range n, ofDigitsTerm (cantorToTernary x).get i) * HPow.hPow (α := ℝ) 3 n -
        HPow.hPow (α := ℝ) 3 n * ofDigitsTerm (cantorToTernary x).get n) =
    (x - (∑ x_1 ∈ Finset.range n, ofDigitsTerm (cantorToTernary x).get x_1 + ofDigitsTerm (cantorToTernary x).get n)) *
      HPow.hPow (α := ℝ) 3 (n + 1)
⊢ (cantorSequence x).get (n + 1) =
    (x - ∑ i ∈ Finset.range (n + 1), ofDigitsTerm (cantorToTernary x).get i) * HPow.hPow (α := ℝ) 3 (n + 1)

Tactic:
rw [cantorSequence_get_succ, ih, Finset.sum_range_succ]

Premises used: Finset.sum_range_succ, cantorSequence_get_succ
-- Module: Mathlib.Topology.Instances.CantorSet
Goal:
x : ℝ
n : ℕ
ih :
  (cantorSequence x).get n = (x - ∑ i ∈ Finset.range n, ofDigitsTerm (cantorToTernary x).get i) * HPow.hPow (α := ℝ) 3 n
⊢ 3 *
      ((x - ∑ i ∈ Finset.range n, ofDigitsTerm (cantorToTernary x).get i) * HPow.hPow (α := ℝ) 3 n -
        HPow.hPow (α := ℝ) 3 n * ofDigitsTerm (cantorToTernary x).get n) =
    (x - (∑ x_1 ∈ Finset.range n, ofDigitsTerm (cantorToTernary x).get x_1 + ofDigitsTerm (cantorToTernary x).get n)) *
      HPow.hPow (α := ℝ) 3 (n + 1)

Tactic:
ring
-- Module: Mathlib.Topology.Instances.CantorSet
Goal:
x : ℝ
n : ℕ
zero :
  (cantorSequence x).get 0 =
    (x - ∑ i ∈ Finset.range 0, ofDigitsTerm (cantorToTernary x).get i) * HPow.hPow (α := ℝ) (β := ℕ) 3 0
succ :
  ∀ (n : ℕ)
    (ih :
      (cantorSequence x).get n =
        (x - ∑ i ∈ Finset.range n, ofDigitsTerm (cantorToTernary x).get i) * HPow.hPow (α := ℝ) 3 n),
    (cantorSequence x).get (n + 1) =
      (x - ∑ i ∈ Finset.range (n + 1), ofDigitsTerm (cantorToTernary x).get i) * HPow.hPow (α := ℝ) 3 (n + 1)
⊢ (cantorSequence x).get n = (x - ∑ i ∈ Finset.range n, ofDigitsTerm (cantorToTernary x).get i) * HPow.hPow (α := ℝ) 3 n

Tactic:
induction n with
  | zero => simp [cantorSequence]
  | succ n ih => rw [cantorSequence_get_succ, ih, Finset.sum_range_succ]; ring

Premises used: cantorSequence, Finset.sum_range_succ, cantorSequence_get_succ
-- Module: Mathlib.Topology.Instances.CantorSet
Goal:
x : ℝ
hx : x ∈ cantorSet
n : ℕ
⊢ (cantorSequence x).get n ∈ cantorSet

Tactic:
have h_mem := cantorSequence_mem_cantorSet hx n
-- Module: Mathlib.Topology.Instances.CantorSet
Goal:
x : ℝ
hx : x ∈ cantorSet
n : ℕ
h_mem : (cantorSequence x).get n ∈ cantorSet
⊢ (x - ∑ i ∈ Finset.range n, ofDigitsTerm (cantorToTernary x).get i) * HPow.hPow (α := ℝ) 3 n ∈ cantorSet

Tactic:
rw [cantorSequence_eq_self_sub_sum_cantorToTernary x n] at h_mem

Premises used: cantorSequence_eq_self_sub_sum_cantorToTernary
-- Module: Mathlib.Topology.Instances.CantorSet
Goal:
x : ℝ
hx : x ∈ cantorSet
n : ℕ
h_mem : (x - ∑ i ∈ Finset.range n, ofDigitsTerm (cantorToTernary x).get i) * HPow.hPow (α := ℝ) 3 n ∈ cantorSet
⊢ Membership.mem (γ := Set ℝ) (Set.Icc 0 1)
    ((x - ∑ i ∈ Finset.range n, ofDigitsTerm (cantorToTernary x).get i) * HPow.hPow (α := ℝ) 3 n)

Tactic:
apply cantorSet_subset_unitInterval at h_mem

Premises used: cantorSet_subset_unitInterval
-- Module: Mathlib.Topology.Instances.CantorSet
Goal:
x : ℝ
hx : x ∈ cantorSet
n : ℕ
h_mem :
  Membership.mem (γ := Set ℝ) (Set.Icc 0 1)
    ((x - ∑ i ∈ Finset.range n, ofDigitsTerm (cantorToTernary x).get i) * HPow.hPow (α := ℝ) 3 n)
⊢ 0 ≤ (x - ∑ i ∈ Finset.range n, ofDigitsTerm (cantorToTernary x).get i) * HPow.hPow (α := ℝ) 3 n ∧
    (x - ∑ i ∈ Finset.range n, ofDigitsTerm (cantorToTernary x).get i) * HPow.hPow (α := ℝ) 3 n ≤ 1

Tactic:
simp only [Set.mem_Icc] at h_mem

Premises used: Set.mem_Icc
-- Module: Mathlib.Topology.Instances.CantorSet
Goal:
x : ℝ
hx : x ∈ cantorSet
n : ℕ
h_mem :
  0 ≤ (x - ∑ i ∈ Finset.range n, ofDigitsTerm (cantorToTernary x).get i) * HPow.hPow (α := ℝ) 3 n ∧
    (x - ∑ i ∈ Finset.range n, ofDigitsTerm (cantorToTernary x).get i) * HPow.hPow (α := ℝ) 3 n ≤ 1
⊢ ∑ i ∈ Finset.range n, ofDigitsTerm (cantorToTernary x) i ≤ x

Tactic:
simpa using h_mem.left

Premises used: And.left
-- Module: Mathlib.Topology.Instances.CantorSet
Goal:
x : ℝ
hx : x ∈ cantorSet
n : ℕ
h_mem :
  0 ≤ (x - ∑ i ∈ Finset.range n, ofDigitsTerm (cantorToTernary x).get i) * HPow.hPow (α := ℝ) 3 n ∧
    (x - ∑ i ∈ Finset.range n, ofDigitsTerm (cantorToTernary x).get i) * HPow.hPow (α := ℝ) 3 n ≤ 1
⊢ 0 < HPow.hPow (α := ℝ) 3 n

Tactic:
positivity
-- Module: Mathlib.Topology.Instances.CantorSet
Goal:
x : ℝ
hx : x ∈ cantorSet
n : ℕ
h_mem :
  0 ≤ (x - ∑ i ∈ Finset.range n, ofDigitsTerm (cantorToTernary x).get i) * HPow.hPow (α := ℝ) 3 n ∧
    (x - ∑ i ∈ Finset.range n, ofDigitsTerm (cantorToTernary x).get i) * HPow.hPow (α := ℝ) 3 n ≤ 1
⊢ HPow.hPow (α := ℝ) 3 n ≠ 0

Tactic:
simp
-- Module: Mathlib.Topology.Instances.CantorSet
Goal:
x : ℝ
hx : x ∈ cantorSet
n : ℕ
h_mem :
  0 ≤ (x - ∑ i ∈ Finset.range n, ofDigitsTerm (cantorToTernary x).get i) * HPow.hPow (α := ℝ) 3 n ∧
    (x - ∑ i ∈ Finset.range n, ofDigitsTerm (cantorToTernary x).get i) * HPow.hPow (α := ℝ) 3 n ≤ 1
rw :
  x * HPow.hPow (α := ℝ) 3 n - 1 ≤ (∑ i ∈ Finset.range n, ofDigitsTerm (cantorToTernary x) i) * HPow.hPow (α := ℝ) 3 n
⊢ x - HPow.hPow (α := ℝ) 3⁻¹ n ≤ ∑ i ∈ Finset.range n, ofDigitsTerm (cantorToTernary x) i

Tactic:
rw [← mul_le_mul_iff_left₀ (show 0 < (3 : ℝ) ^ n by positivity), sub_mul, inv_pow,
    inv_mul_cancel₀ (by simp)]

Premises used: inv_pow, sub_mul, mul_le_mul_iff_left₀, inv_mul_cancel₀
-- Module: Mathlib.Topology.Instances.CantorSet
Goal:
x : ℝ
hx : x ∈ cantorSet
n : ℕ
h_mem :
  0 ≤ (x - ∑ i ∈ Finset.range n, ofDigitsTerm (cantorToTernary x).get i) * HPow.hPow (α := ℝ) 3 n ∧
    (x - ∑ i ∈ Finset.range n, ofDigitsTerm (cantorToTernary x).get i) * HPow.hPow (α := ℝ) 3 n ≤ 1
linarith! :
  (∑ i ∈ Finset.range n, ofDigitsTerm (cantorToTernary x) i) * HPow.hPow (α := ℝ) 3 n < x * HPow.hPow (α := ℝ) 3 n - 1 →
    (x - ∑ i ∈ Finset.range n, ofDigitsTerm (cantorToTernary x).get i) * HPow.hPow (α := ℝ) 3 n - 1 +
        ((∑ i ∈ Finset.range n, ofDigitsTerm (cantorToTernary x) i) * HPow.hPow (α := ℝ) 3 n -
          (x * HPow.hPow (α := ℝ) 3 n - 1)) =
      0
⊢ x * HPow.hPow (α := ℝ) 3 n - 1 ≤ (∑ i ∈ Finset.range n, ofDigitsTerm (cantorToTernary x) i) * HPow.hPow (α := ℝ) 3 n

Tactic:
linarith!
-- Module: Mathlib.Topology.Instances.CantorSet
Goal:
x : ℝ
hx : x ∈ cantorSet
rw : HasSum (ofDigitsTerm (cantorToTernary x).get) x
⊢ ∑' (n : ℕ), ofDigitsTerm (cantorToTernary x).get n = x

Tactic:
rw [HasSum.tsum_eq]

Premises used: HasSum.tsum_eq
-- Module: Mathlib.Topology.Instances.CantorSet
Goal:
x : ℝ
hx : x ∈ cantorSet
rw : Filter.Tendsto (fun (n : ℕ) ↦ ∑ i ∈ Finset.range n, ofDigitsTerm (cantorToTernary x).get i) Filter.atTop (nhds x)
rw₁ : Summable (α := ℝ) fun (i : ℕ) ↦ ‖ofDigitsTerm (cantorToTernary x).get i‖
⊢ HasSum (ofDigitsTerm (cantorToTernary x).get) x

Tactic:
rw [hasSum_iff_tendsto_nat_of_summable_norm]

Premises used: hasSum_iff_tendsto_nat_of_summable_norm
-- Module: Mathlib.Topology.Instances.CantorSet
Goal:
x : ℝ
hx : x ∈ cantorSet
⊢ Summable (α := ℝ) fun (i : ℕ) ↦ ‖ofDigitsTerm (cantorToTernary x).get i‖

Tactic:
simpa only [norm_of_nonneg ofDigitsTerm_nonneg] using summable_ofDigitsTerm

Premises used: Real.summable_ofDigitsTerm, Real.ofDigitsTerm_nonneg, Real.norm_of_nonneg
-- Module: Mathlib.Topology.Instances.CantorSet
Goal:
x : ℝ
hx : x ∈ cantorSet
hg : Filter.Tendsto (fun (n : ℕ) ↦ x - HPow.hPow (α := ℝ) 3⁻¹ n) Filter.atTop (nhds x)
hh : Filter.Tendsto (fun (x_1 : ℕ) ↦ x) Filter.atTop (nhds x)
hgf :
  LE.le (α := ℕ → ℝ) (fun (n : ℕ) ↦ x - HPow.hPow (α := ℝ) 3⁻¹ n) fun (n : ℕ) ↦
    ∑ i ∈ Finset.range n, ofDigitsTerm (cantorToTernary x).get i
hfh : LE.le (α := ℕ → ℝ) (fun (n : ℕ) ↦ ∑ i ∈ Finset.range n, ofDigitsTerm (cantorToTernary x).get i) fun (x_1 : ℕ) ↦ x
⊢ Filter.Tendsto (fun (n : ℕ) ↦ ∑ i ∈ Finset.range n, ofDigitsTerm (cantorToTernary x).get i) Filter.atTop (nhds x)

Tactic:
apply tendsto_of_tendsto_of_tendsto_of_le_of_le (g := fun n ↦ x - (3⁻¹ : ℝ) ^ n) (h := fun _ ↦ x)

Premises used: tendsto_of_tendsto_of_tendsto_of_le_of_le
-- Module: Mathlib.Topology.Instances.CantorSet
Goal:
x : ℝ
hx : x ∈ cantorSet
hg : Filter.Tendsto (β := ℝ) (fun (x_1 : ℕ) ↦ x - HPow.hPow (α := ℝ) 3⁻¹ x_1 - x) Filter.atTop (nhds 0)
⊢ Filter.Tendsto (fun (n : ℕ) ↦ x - HPow.hPow (α := ℝ) 3⁻¹ n) Filter.atTop (nhds x)

Tactic:
rw [← tendsto_sub_nhds_zero_iff]

Premises used: tendsto_sub_nhds_zero_iff
-- Module: Mathlib.Topology.Instances.CantorSet
Goal:
x : ℝ
hx : x ∈ cantorSet
hg : Filter.Tendsto (β := ℝ) (fun (x : ℕ) ↦ -HPow.hPow (α := ℝ) 3⁻¹ x) Filter.atTop (nhds 0)
⊢ Filter.Tendsto (β := ℝ) (fun (x_1 : ℕ) ↦ x - HPow.hPow (α := ℝ) 3⁻¹ x_1 - x) Filter.atTop (nhds 0)

Tactic:
simp only [sub_sub_cancel_left]

Premises used: sub_sub_cancel_left
-- Module: Mathlib.Topology.Instances.CantorSet
Goal:
x : ℝ
hx : x ∈ cantorSet
⊢ Eq (α := ℝ) 0 (-0)

Tactic:
simp
-- Module: Mathlib.Topology.Instances.CantorSet
Goal:
x : ℝ
hx : x ∈ cantorSet
hg : Filter.Tendsto (β := ℝ) (fun (x : ℕ) ↦ -HPow.hPow (α := ℝ) 3⁻¹ x) Filter.atTop (nhds (-0))
⊢ Filter.Tendsto (β := ℝ) (fun (x : ℕ) ↦ -HPow.hPow (α := ℝ) 3⁻¹ x) Filter.atTop (nhds 0)

Tactic:
rw [show 0 = -(0 : ℝ) by simp]
-- Module: Mathlib.Topology.Instances.CantorSet
Goal:
x : ℝ
hx : x ∈ cantorSet
⊢ LT.lt (α := ℝ) |3⁻¹| 1

Tactic:
norm_num
-- Module: Mathlib.Topology.Instances.CantorSet
Goal:
x : ℝ
hx : x ∈ cantorSet
⊢ Filter.Tendsto (β := ℝ) (fun (x : ℕ) ↦ -HPow.hPow (α := ℝ) 3⁻¹ x) Filter.atTop (nhds (-0))

Tactic:
exact (tendsto_pow_atTop_nhds_zero_of_abs_lt_one (by norm_num)).neg

Premises used: Filter.Tendsto.neg, tendsto_pow_atTop_nhds_zero_of_abs_lt_one