awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- 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
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
⊢ 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 linarith₁ : 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 →
  3 / 3 = 1
Tactic: linarith
-- 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
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
linarith₁ :
  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 →
    3 / 3 = 1
⊢ 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 linarith₂ : 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 3 = 3
Tactic: linarith
-- 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
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
linarith₁ :
  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 →
    3 / 3 = 1
linarith₂ linarith₃ :
  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 3 = 3
linarith₄ :
  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
linarith₅ :
  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 →
    3 / 3 = 1
linarith₆ linarith₇ :
  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 3 = 3
⊢ 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 linarith₈ : 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 →
  GT.gt (α := ℝ) 3 0
Tactic: linarith
-- 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)
⊢ False
have h₁ : 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
Cleared hypotheses: h
-- 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
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
⊢ False
have h₁ : ((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
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 :
  ((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
have 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)
Tactic: rw [← mul_right_inj' (show ((3 : ℝ) ^ n1)⁻¹ ≠ 0 by positivity)]
-- 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 :
  ((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 * ((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)
have linarith : (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) →
  3 / 3 = 1
Tactic: linarith
-- 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 :
  ((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)
linarith :
  (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) →
    3 / 3 = 1
⊢ (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)
have linarith₁ : (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) →
  HMul.hMul (α := ℝ) 1 3 = 3
Tactic: linarith
-- 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 :
  ((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)
linarith :
  (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) →
    3 / 3 = 1
linarith₁ linarith₂ linarith₃ :
  (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) →
    HMul.hMul (α := ℝ) 1 3 = 3
linarith₄ :
  (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) →
    3 / 3 = 1
linarith₅ linarith₆ :
  (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) →
    HMul.hMul (α := ℝ) 1 3 = 3
linarith₇ :
  (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) →
    3 / 3 = 1
linarith₈ linarith₉ :
  (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) →
    HMul.hMul (α := ℝ) 1 3 = 3
⊢ (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)
have linarith₁₀ : (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) →
  GT.gt (α := ℝ) 3 0
Tactic: linarith
-- 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 :
  ((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)
linarith :
  (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) →
    3 / 3 = 1
linarith₁ linarith₂ linarith₃ :
  (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) →
    HMul.hMul (α := ℝ) 1 3 = 3
linarith₄ :
  (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) →
    3 / 3 = 1
linarith₅ linarith₆ :
  (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) →
    HMul.hMul (α := ℝ) 1 3 = 3
linarith₇ :
  (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) →
    3 / 3 = 1
linarith₈ linarith₉ :
  (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) →
    HMul.hMul (α := ℝ) 1 3 = 3
linarith₁₀ :
  (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) →
    GT.gt (α := ℝ) 3 0
⊢ (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)
have linarith₁₁ : (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) →
  HMul.hMul (α := ℝ) 1 1 = 1
Tactic: linarith
-- 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
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)
⊢ False
have h₁ : (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
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 :
  (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
⊢ False
have linarith : 3 / 3 = 1
Tactic: linarith [ofDigits_nonneg (fun i ↦ b (1 + n1 + i)), ofDigits_le_one (fun i ↦ a (1 + n1 + i))]
-- 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 :
  (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
⊢ False
have linarith₁ : HMul.hMul (α := ℝ) 1 3 = 3
Tactic: linarith [ofDigits_nonneg (fun i ↦ b (1 + n1 + i)), ofDigits_le_one (fun i ↦ a (1 + n1 + i))]
-- 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 :
  (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
⊢ False
have linarith₈ : GT.gt (α := ℝ) 3 0
Tactic: linarith [ofDigits_nonneg (fun i ↦ b (1 + n1 + i)), ofDigits_le_one (fun i ↦ a (1 + n1 + i))]
-- 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 :
  (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
⊢ False
have 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
Tactic: linarith [ofDigits_nonneg (fun i ↦ b (1 + n1 + i)), ofDigits_le_one (fun i ↦ a (1 + n1 + i))]
-- Lean 4 proof step from Mathlib.Topology.Instances.CantorSet (hypothesis)
Goal:
x : ℝ
hx : x ∈ cantorSet
⊢ ∃ y ∈ cantorSet, y / 3 = x ∨ (2 + y) / 3 = x
have hx₁ : 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
Cleared hypotheses: hx
-- Lean 4 proof step from Mathlib.Topology.Instances.CantorSet (goal)
Goal:
x : ℝ
hx : x ∈ cantorSet
⊢ (if x ∈ Set.Icc 0 (1 / 3) then 3 * x else 3 * x - 2) ∈ cantorSet
have 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
Tactic: obtain ⟨y, hy, rfl | rfl⟩ : ∃ y ∈ cantorSet, y / 3 = x ∨ (2 + y) / 3 = x := by
    rw [cantorSet_eq_union_halves] at hx
    grind
Cleared hypotheses: x, hx
-- Lean 4 proof step from Mathlib.Topology.Instances.CantorSet (goal)
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
⊢ (if x ∈ Set.Icc 0 (1 / 3) then 3 * x else 3 * x - 2) ∈ cantorSet
have 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
Tactic: obtain ⟨y, hy, rfl | rfl⟩ : ∃ y ∈ cantorSet, y / 3 = x ∨ (2 + y) / 3 = x := by
    rw [cantorSet_eq_union_halves] at hx
    grind
Cleared hypotheses: x, hx
-- Lean 4 proof step from Mathlib.Topology.Instances.CantorSet (goal)
Goal:
x : ℝ
hx : x ∈ cantorSet
n : ℕ
⊢ (cantorSequence x).get n ∈ cantorSet
have zero : (cantorSequence x).get 0 ∈ cantorSet
Tactic: induction n with
  | zero => simpa [cantorSequence]
  | succ n ih => exact cantorStep_mem_cantorSet ih
Cleared hypotheses: n
-- Lean 4 proof step from Mathlib.Topology.Instances.CantorSet (goal)
Goal:
x : ℝ
hx : x ∈ cantorSet
n : ℕ
zero : (cantorSequence x).get 0 ∈ cantorSet
⊢ (cantorSequence x).get n ∈ cantorSet
have succ : ∀ (n : ℕ) (ih : (cantorSequence x).get n ∈ cantorSet), (cantorSequence x).get (n + 1) ∈ cantorSet
Tactic: induction n with
  | zero => simpa [cantorSequence]
  | succ n ih => exact cantorStep_mem_cantorSet ih
Cleared hypotheses: n
-- Lean 4 proof step from Mathlib.Topology.Instances.CantorSet (goal)
Goal:
x : ℝ
n : ℕ
⊢ (cantorSequence x).get (n + 1) =
    3 * ((cantorSequence x).get n - HPow.hPow (α := ℝ) 3 n * ofDigitsTerm (cantorToTernary x).get n)
have 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))
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]