awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.Topology.Instances.CantorSet (goal)
Goal:
x : ℝ
n : ℕ
⊢ (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))
have 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))
Tactic: split_ifs
-- Lean 4 proof step from Mathlib.Topology.Instances.CantorSet (goal)
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))
⊢ (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))
have 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))
Tactic: split_ifs
-- Lean 4 proof step from Mathlib.Topology.Instances.CantorSet (goal)
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
⊢ (cantorSequence x).get (n + 1) =
    (x - ∑ i ∈ Finset.range (n + 1), ofDigitsTerm (cantorToTernary x).get i) * HPow.hPow (α := ℝ) 3 (n + 1)
have 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)
Tactic: rw [cantorSequence_get_succ, ih, Finset.sum_range_succ]
-- Lean 4 proof step from Mathlib.Topology.Instances.CantorSet (goal)
Goal:
x : ℝ
n : ℕ
⊢ (cantorSequence x).get n = (x - ∑ i ∈ Finset.range n, ofDigitsTerm (cantorToTernary x).get i) * HPow.hPow (α := ℝ) 3 n
have zero : (cantorSequence x).get 0 =
  (x - ∑ i ∈ Finset.range 0, ofDigitsTerm (cantorToTernary x).get i) * HPow.hPow (α := ℝ) (β := ℕ) 3 0
Tactic: induction n with
  | zero => simp [cantorSequence]
  | succ n ih => rw [cantorSequence_get_succ, ih, Finset.sum_range_succ]; ring
Cleared hypotheses: n
-- Lean 4 proof step from Mathlib.Topology.Instances.CantorSet (goal)
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
⊢ (cantorSequence x).get n = (x - ∑ i ∈ Finset.range n, ofDigitsTerm (cantorToTernary x).get i) * HPow.hPow (α := ℝ) 3 n
have 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)
Tactic: induction n with
  | zero => simp [cantorSequence]
  | succ n ih => rw [cantorSequence_get_succ, ih, Finset.sum_range_succ]; ring
Cleared hypotheses: n
-- Lean 4 proof step from Mathlib.Topology.Instances.CantorSet (hypothesis)
Goal:
x : ℝ
hx : x ∈ cantorSet
n : ℕ
⊢ ∑ i ∈ Finset.range n, ofDigitsTerm (cantorToTernary x) i ≤ x
have h_mem : (cantorSequence x).get n ∈ cantorSet
Tactic: have h_mem := cantorSequence_mem_cantorSet hx n
-- Lean 4 proof step from Mathlib.Topology.Instances.CantorSet (hypothesis)
Goal:
x : ℝ
hx : x ∈ cantorSet
n : ℕ
h_mem : (cantorSequence x).get n ∈ cantorSet
⊢ ∑ i ∈ Finset.range n, ofDigitsTerm (cantorToTernary x) i ≤ x
have h_mem₁ : (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
Cleared hypotheses: h_mem
-- Lean 4 proof step from Mathlib.Topology.Instances.CantorSet (hypothesis)
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
⊢ ∑ i ∈ Finset.range n, ofDigitsTerm (cantorToTernary x) i ≤ x
have h_mem₁ : 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
Cleared hypotheses: h_mem
-- Lean 4 proof step from Mathlib.Topology.Instances.CantorSet (hypothesis)
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)
⊢ ∑ i ∈ Finset.range n, ofDigitsTerm (cantorToTernary x) i ≤ x
have h_mem₁ : 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
Cleared hypotheses: h_mem
-- Lean 4 proof step from Mathlib.Topology.Instances.CantorSet (hypothesis)
Goal:
x : ℝ
hx : x ∈ cantorSet
n : ℕ
⊢ x - HPow.hPow (α := ℝ) 3⁻¹ n ≤ ∑ i ∈ Finset.range n, ofDigitsTerm (cantorToTernary x) i
have h_mem : (cantorSequence x).get n ∈ cantorSet
Tactic: have h_mem := cantorSequence_mem_cantorSet hx n
-- Lean 4 proof step from Mathlib.Topology.Instances.CantorSet (hypothesis)
Goal:
x : ℝ
hx : x ∈ cantorSet
n : ℕ
h_mem : (cantorSequence x).get n ∈ cantorSet
⊢ x - HPow.hPow (α := ℝ) 3⁻¹ n ≤ ∑ i ∈ Finset.range n, ofDigitsTerm (cantorToTernary x) i
have h_mem₁ : (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
Cleared hypotheses: h_mem
-- Lean 4 proof step from Mathlib.Topology.Instances.CantorSet (hypothesis)
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
⊢ x - HPow.hPow (α := ℝ) 3⁻¹ n ≤ ∑ i ∈ Finset.range n, ofDigitsTerm (cantorToTernary x) i
have h_mem₁ : 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
Cleared hypotheses: h_mem
-- Lean 4 proof step from Mathlib.Topology.Instances.CantorSet (hypothesis)
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)
⊢ x - HPow.hPow (α := ℝ) 3⁻¹ n ≤ ∑ i ∈ Finset.range n, ofDigitsTerm (cantorToTernary x) i
have h_mem₁ : 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
Cleared hypotheses: h_mem
-- Lean 4 proof step from Mathlib.Topology.Instances.CantorSet (goal)
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
⊢ x - HPow.hPow (α := ℝ) 3⁻¹ n ≤ ∑ i ∈ Finset.range n, ofDigitsTerm (cantorToTernary x) i
have rw : x * HPow.hPow (α := ℝ) 3 n - 1 ≤ (∑ i ∈ Finset.range n, ofDigitsTerm (cantorToTernary x) i) * HPow.hPow (α := ℝ) 3 n
Tactic: rw [← mul_le_mul_iff_left₀ (show 0 < (3 : ℝ) ^ n by positivity), sub_mul, inv_pow,
    inv_mul_cancel₀ (by simp)]
-- Lean 4 proof step from Mathlib.Topology.Instances.CantorSet (goal)
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
⊢ x * HPow.hPow (α := ℝ) 3 n - 1 ≤ (∑ i ∈ Finset.range n, ofDigitsTerm (cantorToTernary x) i) * HPow.hPow (α := ℝ) 3 n
have 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
Tactic: linarith!
-- Lean 4 proof step from Mathlib.Topology.Instances.CantorSet (goal)
Goal:
x : ℝ
hx : x ∈ cantorSet
⊢ ∑' (n : ℕ), ofDigitsTerm (cantorToTernary x).get n = x
have rw : HasSum (ofDigitsTerm (cantorToTernary x).get) x
Tactic: rw [HasSum.tsum_eq]
-- Lean 4 proof step from Mathlib.Topology.Instances.CantorSet (goal)
Goal:
x : ℝ
hx : x ∈ cantorSet
⊢ HasSum (ofDigitsTerm (cantorToTernary x).get) x
have rw : Filter.Tendsto (fun (n : ℕ) ↦ ∑ i ∈ Finset.range n, ofDigitsTerm (cantorToTernary x).get i) Filter.atTop (nhds x)
Tactic: rw [hasSum_iff_tendsto_nat_of_summable_norm]
-- Lean 4 proof step from Mathlib.Topology.Instances.CantorSet (goal)
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)
⊢ HasSum (ofDigitsTerm (cantorToTernary x).get) x
have rw₁ : Summable (α := ℝ) fun (i : ℕ) ↦ ‖ofDigitsTerm (cantorToTernary x).get i‖
Tactic: rw [hasSum_iff_tendsto_nat_of_summable_norm]
-- Lean 4 proof step from Mathlib.Topology.Instances.CantorSet (goal)
Goal:
x : ℝ
hx : x ∈ cantorSet
⊢ Filter.Tendsto (fun (n : ℕ) ↦ ∑ i ∈ Finset.range n, ofDigitsTerm (cantorToTernary x).get i) Filter.atTop (nhds x)
have hg : Filter.Tendsto (fun (n : ℕ) ↦ x - HPow.hPow (α := ℝ) 3⁻¹ n) 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)
-- Lean 4 proof step from Mathlib.Topology.Instances.CantorSet (goal)
Goal:
x : ℝ
hx : x ∈ cantorSet
hg : Filter.Tendsto (fun (n : ℕ) ↦ x - HPow.hPow (α := ℝ) 3⁻¹ n) Filter.atTop (nhds x)
⊢ Filter.Tendsto (fun (n : ℕ) ↦ ∑ i ∈ Finset.range n, ofDigitsTerm (cantorToTernary x).get i) Filter.atTop (nhds x)
have hh : Filter.Tendsto (fun (x_1 : ℕ) ↦ x) 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)
-- Lean 4 proof step from Mathlib.Topology.Instances.CantorSet (goal)
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)
⊢ Filter.Tendsto (fun (n : ℕ) ↦ ∑ i ∈ Finset.range n, ofDigitsTerm (cantorToTernary x).get i) Filter.atTop (nhds x)
have 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
Tactic: apply tendsto_of_tendsto_of_tendsto_of_le_of_le (g := fun n ↦ x - (3⁻¹ : ℝ) ^ n) (h := fun _ ↦ x)
-- Lean 4 proof step from Mathlib.Topology.Instances.CantorSet (goal)
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
⊢ Filter.Tendsto (fun (n : ℕ) ↦ ∑ i ∈ Finset.range n, ofDigitsTerm (cantorToTernary x).get i) Filter.atTop (nhds x)
have hfh : LE.le (α := ℕ → ℝ) (fun (n : ℕ) ↦ ∑ i ∈ Finset.range n, ofDigitsTerm (cantorToTernary x).get i) fun (x_1 : ℕ) ↦ x
Tactic: apply tendsto_of_tendsto_of_tendsto_of_le_of_le (g := fun n ↦ x - (3⁻¹ : ℝ) ^ n) (h := fun _ ↦ x)
-- Lean 4 proof step from Mathlib.Topology.Instances.CantorSet (goal)
Goal:
x : ℝ
hx : x ∈ cantorSet
⊢ Filter.Tendsto (fun (n : ℕ) ↦ x - HPow.hPow (α := ℝ) 3⁻¹ n) Filter.atTop (nhds x)
have hg : Filter.Tendsto (β := ℝ) (fun (x_1 : ℕ) ↦ x - HPow.hPow (α := ℝ) 3⁻¹ x_1 - x) Filter.atTop (nhds 0)
Tactic: rw [← tendsto_sub_nhds_zero_iff]
-- Lean 4 proof step from Mathlib.Topology.Instances.CantorSet (goal)
Goal:
x : ℝ
hx : x ∈ cantorSet
⊢ Filter.Tendsto (β := ℝ) (fun (x_1 : ℕ) ↦ x - HPow.hPow (α := ℝ) 3⁻¹ x_1 - x) Filter.atTop (nhds 0)
have hg : Filter.Tendsto (β := ℝ) (fun (x : ℕ) ↦ -HPow.hPow (α := ℝ) 3⁻¹ x) Filter.atTop (nhds 0)
Tactic: simp only [sub_sub_cancel_left]
-- Lean 4 proof step from Mathlib.Topology.Instances.CantorSet (goal)
Goal:
x : ℝ
hx : x ∈ cantorSet
⊢ Filter.Tendsto (β := ℝ) (fun (x : ℕ) ↦ -HPow.hPow (α := ℝ) 3⁻¹ x) Filter.atTop (nhds 0)
have hg : 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]
-- Lean 4 proof step from Mathlib.Topology.Instances.CantorSet (goal)
Goal:
⊢ cantorSet = {x : ℝ | ∃ (a : ℕ → Fin 3), (∀ (i : ℕ), a i ≠ 1) ∧ ofDigits a = x}
have h : ∀ (x : ℝ),
  x ∈ cantorSet ↔ Membership.mem (γ := Set ℝ) {x : ℝ | ∃ (a : ℕ → Fin 3), (∀ (i : ℕ), a i ≠ 1) ∧ ofDigits a = x} x
Tactic: ext x
-- Lean 4 proof step from Mathlib.Topology.Instances.CantorSet (goal)
Goal:
x : ℝ
⊢ x ∈ cantorSet ↔ Membership.mem (γ := Set ℝ) {x : ℝ | ∃ (a : ℕ → Fin 3), (∀ (i : ℕ), a i ≠ 1) ∧ ofDigits a = x} x
have h : ∀ (h : x ∈ cantorSet), Membership.mem (γ := Set ℝ) {x : ℝ | ∃ (a : ℕ → Fin 3), (∀ (i : ℕ), a i ≠ 1) ∧ ofDigits a = x} x
Tactic: refine ⟨fun h ↦ ?_, fun ⟨a, ha⟩ ↦ ?_⟩
-- Lean 4 proof step from Mathlib.Topology.Instances.CantorSet (goal)
Goal:
x : ℝ
h :
  ∀ (h : x ∈ cantorSet),
    Membership.mem (γ := Set ℝ) {x : ℝ | ∃ (a : ℕ → Fin 3), (∀ (i : ℕ), a i ≠ 1) ∧ ofDigits a = x} x
⊢ x ∈ cantorSet ↔ Membership.mem (γ := Set ℝ) {x : ℝ | ∃ (a : ℕ → Fin 3), (∀ (i : ℕ), a i ≠ 1) ∧ ofDigits a = x} x
have h₁ : Membership.mem (γ := Set ℝ) {x : ℝ | ∃ (a : ℕ → Fin 3), (∀ (i : ℕ), a i ≠ 1) ∧ ofDigits a = x} x →
  ∀ (a : ℕ → Fin 3) (ha : (∀ (i : ℕ), a i ≠ 1) ∧ ofDigits a = x), x ∈ cantorSet
Tactic: refine ⟨fun h ↦ ?_, fun ⟨a, ha⟩ ↦ ?_⟩
-- Lean 4 proof step from Mathlib.Topology.Instances.CantorSet (goal)
Goal:
x : ℝ
h : x ∈ cantorSet
⊢ Membership.mem (γ := Set ℝ) {x : ℝ | ∃ (a : ℕ → Fin 3), (∀ (i : ℕ), a i ≠ 1) ∧ ofDigits a = x} x
have h₁ : (∀ (i : ℕ), cantorToTernary x i ≠ 1) ∧ ofDigits (cantorToTernary x) = x
Tactic: use cantorToTernary x
-- Lean 4 proof step from Mathlib.Topology.Instances.CantorSet (goal)
Goal:
x : ℝ
h : Membership.mem (γ := Set ℝ) {x : ℝ | ∃ (a : ℕ → Fin 3), (∀ (i : ℕ), a i ≠ 1) ∧ ofDigits a = x} x
a : ℕ → Fin 3
ha : (∀ (i : ℕ), a i ≠ 1) ∧ ofDigits a = x
⊢ x ∈ cantorSet
have h : ofDigits a ∈ cantorSet
Tactic: rw [← ha.right]
-- Lean 4 proof step from Mathlib.Topology.Instances.CantorSet (goal)
Goal:
x : ℝ
hx : x ∈ cantorSet
⊢ (fun (y : ℕ → Bool) ↦
        Subtype.mk (p := fun (x : ℝ) ↦ x ∈ cantorSet) (ofDigits (b := 3) fun (i : ℕ) ↦ bif y i then 2 else 0) ⋯)
      ((fun (x : (↑cantorSet : Type)) ↦
          (match (motive := (↑cantorSet : Type) → ℕ → Bool) x with
            | ⟨x, h⟩ => (cantorToBinary x).get :
            ℕ → Bool))
        (Subtype.mk (p := fun (x : ℝ) ↦ x ∈ cantorSet) x hx)) =
    Subtype.mk (p := fun (x : ℝ) ↦ x ∈ cantorSet) x hx
have simp : (ofDigits (b := 3) fun (i : ℕ) ↦ bif (cantorToBinary x).get i then 2 else 0) = x
Tactic: simp only [Fin.isValue, Subtype.mk.injEq]
-- Lean 4 proof step from Mathlib.Topology.Instances.CantorSet (hypothesis)
Goal:
y : ℕ → Bool
x : ℝ := ofDigits (b := 3) fun (i : ℕ) ↦ bif y i then 2 else 0
⊢ (cantorToBinary x).get = y
have this : ofDigits (cantorToTernary (ofDigits (b := 3) fun (i : ℕ) ↦ bif y i then 2 else 0)).get =
  ofDigits (b := 3) fun (i : ℕ) ↦ bif y i then 2 else 0
Tactic: have := ofDigits_cantorToTernary (ofDigits_bool_to_fin_three_mem_cantorSet y)
-- Lean 4 proof step from Mathlib.Topology.Instances.CantorSet (goal)
Goal:
y : ℕ → Bool
x : ℝ := ofDigits (b := 3) fun (i : ℕ) ↦ bif y i then 2 else 0
this :
  ofDigits (cantorToTernary (ofDigits (b := 3) fun (i : ℕ) ↦ bif y i then 2 else 0)).get =
    ofDigits (b := 3) fun (i : ℕ) ↦ bif y i then 2 else 0
⊢ (cantorToBinary x).get = y
have apply : ∀
  (this :
    (cantorToTernary (ofDigits (b := 3) fun (i : ℕ) ↦ bif y i then 2 else 0)).get = fun (i : ℕ) ↦
      bif y i then 2 else 0),
  (cantorToBinary x).get = y
Tactic: apply ofDigits_zero_two_sequence_unique at this
Cleared hypotheses: this
-- Lean 4 proof step from Mathlib.Topology.Instances.CantorSet (goal)
Goal:
y : ℕ → Bool
x : ℝ := ofDigits (b := 3) fun (i : ℕ) ↦ bif y i then 2 else 0
this :
  ofDigits (cantorToTernary (ofDigits (b := 3) fun (i : ℕ) ↦ bif y i then 2 else 0)).get =
    ofDigits (b := 3) fun (i : ℕ) ↦ bif y i then 2 else 0
apply :
  ∀
    (this :
      (cantorToTernary (ofDigits (b := 3) fun (i : ℕ) ↦ bif y i then 2 else 0)).get = fun (i : ℕ) ↦
        bif y i then 2 else 0),
    (cantorToBinary x).get = y
⊢ (cantorToBinary x).get = y
have ha : ∀ (n : ℕ), (cantorToTernary (ofDigits (b := 3) fun (i : ℕ) ↦ bif y i then 2 else 0)).get n ≠ 1
Tactic: apply ofDigits_zero_two_sequence_unique at this
-- Lean 4 proof step from Mathlib.Topology.Instances.CantorSet (goal)
Goal:
y : ℕ → Bool
x : ℝ := ofDigits (b := 3) fun (i : ℕ) ↦ bif y i then 2 else 0
this :
  ofDigits (cantorToTernary (ofDigits (b := 3) fun (i : ℕ) ↦ bif y i then 2 else 0)).get =
    ofDigits (b := 3) fun (i : ℕ) ↦ bif y i then 2 else 0
apply :
  ∀
    (this :
      (cantorToTernary (ofDigits (b := 3) fun (i : ℕ) ↦ bif y i then 2 else 0)).get = fun (i : ℕ) ↦
        bif y i then 2 else 0),
    (cantorToBinary x).get = y
ha : ∀ (n : ℕ), (cantorToTernary (ofDigits (b := 3) fun (i : ℕ) ↦ bif y i then 2 else 0)).get n ≠ 1
⊢ (cantorToBinary x).get = y
have hb : ∀ (n : ℕ), Ne (α := Fin 3) (bif y n then 2 else 0) 1
Tactic: apply ofDigits_zero_two_sequence_unique at this
-- Lean 4 proof step from Mathlib.Topology.Instances.CantorSet (goal)
Goal:
y : ℕ → Bool
x : ℝ := ofDigits (b := 3) fun (i : ℕ) ↦ bif y i then 2 else 0
this :
  (cantorToTernary (ofDigits (b := 3) fun (i : ℕ) ↦ bif y i then 2 else 0)).get = fun (i : ℕ) ↦ bif y i then 2 else 0
⊢ (cantorToBinary x).get = y
have h : ∀ (n : ℕ), (cantorToBinary x).get n = y n
Tactic: ext n
-- Lean 4 proof step from Mathlib.Topology.Instances.CantorSet (hypothesis)
Goal:
y : ℕ → Bool
x : ℝ := ofDigits (b := 3) fun (i : ℕ) ↦ bif y i then 2 else 0
this :
  (cantorToTernary (ofDigits (b := 3) fun (i : ℕ) ↦ bif y i then 2 else 0)).get = fun (i : ℕ) ↦ bif y i then 2 else 0
n : ℕ
⊢ (cantorToBinary x).get n = y n
have this₁ : (cantorToTernary (ofDigits (b := 3) fun (i : ℕ) ↦ bif y i then 2 else 0)).get n = bif y n then 2 else 0
Tactic: apply congrFun (a := n) at this
Cleared hypotheses: this
-- Lean 4 proof step from Mathlib.Topology.Instances.Complex (hypothesis)
Goal:
K : Subfield ℂ
hc : IsClosed (↑K : Set ℂ)
this : range ofReal ⊆ (↑K : Set ℂ)
⊢ K = ofRealHom.fieldRange ∨ K = ⊤
have this₁ : ∀ (y : ℝ), Membership.mem (α := ℂ) (↑K : Set ℂ) ((algebraMap ℝ ℂ : (a : ℝ) → ℂ) y : ℂ)
Tactic: rw [range_subset_iff, ← coe_algebraMap] at this
Cleared hypotheses: this
-- Lean 4 proof step from Mathlib.Topology.Instances.Complex (hypothesis)
Goal:
K : Subfield ℂ
hc : IsClosed (↑K : Set ℂ)
this : ∀ (y : ℝ), Membership.mem (α := ℂ) (↑K : Set ℂ) ((algebraMap ℝ ℂ : (a : ℝ) → ℂ) y : ℂ)
⊢ K = ofRealHom.fieldRange ∨ K = ⊤
have this₁ : (K.toIntermediateField this).toSubalgebra = ⊥ ∨ (K.toIntermediateField this).toSubalgebra = ⊤
Tactic: have :=
      (Subalgebra.isSimpleOrder_of_finrank finrank_real_complex).eq_bot_or_eq_top
        (Subfield.toIntermediateField K this).toSubalgebra
-- Lean 4 proof step from Mathlib.Topology.Instances.Complex (goal)
Goal:
K : Subfield ℂ
hc : IsClosed (↑K : Set ℂ)
this_1 : ∀ (y : ℝ), Membership.mem (α := ℂ) (↑K : Set ℂ) ((algebraMap ℝ ℂ : (a : ℝ) → ℂ) y : ℂ)
this : (K.toIntermediateField this_1).toSubalgebra = ⊥ ∨ (K.toIntermediateField this_1).toSubalgebra = ⊤
⊢ K = ofRealHom.fieldRange ∨ K = ⊤
have simp_rw : ∀
  (this :
    (↑(K.toIntermediateField this_1) : Set ℂ) = (↑⊥ : Set ℂ) ∨
      (↑(K.toIntermediateField this_1) : Set ℂ) = (↑⊤ : Set ℂ)),
  (↑K : Set ℂ) = (↑ofRealHom.fieldRange : Set ℂ) ∨ (↑K : Set ℂ) = (↑⊤ : Set ℂ)
Tactic: simp_rw [← SetLike.coe_set_eq, IntermediateField.coe_toSubalgebra] at this ⊢
Cleared hypotheses: this
-- Lean 4 proof step from Mathlib.Topology.Instances.Complex (goal)
Goal:
K : Subfield ℂ
hc : IsClosed (↑K : Set ℂ)
⊢ K = ofRealHom.fieldRange ∨ K = ⊤
have suffices : range ofReal ⊆ (↑K : Set ℂ)
Tactic: suffices range (ofReal : ℝ → ℂ) ⊆ K by
    rw [range_subset_iff, ← coe_algebraMap] at this
    have :=
      (Subalgebra.isSimpleOrder_of_finrank finrank_real_complex).eq_bot_or_eq_top
        (Subfield.toIntermediateField K this).toSubalgebra
    simp_rw [← SetLike.coe_set_eq, IntermediateField.coe_toSubalgebra] at this ⊢
    exact this
-- Lean 4 proof step from Mathlib.Topology.Instances.Complex (goal)
Goal:
K : Subfield ℂ
hc : IsClosed (↑K : Set ℂ)
this : range ofReal ⊆ closure (range (ofReal ∘ Rat.cast))
⊢ range ofReal ⊆ (↑K : Set ℂ)
have refine : closure (range (ofReal ∘ Rat.cast)) ⊆ (↑K : Set ℂ)
Tactic: refine subset_trans this ?_
-- Lean 4 proof step from Mathlib.Topology.Instances.Complex (goal)
Goal:
K : Subfield ℂ
hc : IsClosed (↑K : Set ℂ)
this : range ofReal ⊆ closure (range (ofReal ∘ Rat.cast))
⊢ closure (range (ofReal ∘ Rat.cast)) ⊆ (↑K : Set ℂ)
have rw : closure (range (ofReal ∘ Rat.cast)) ⊆ closure (↑K : Set ℂ)
Tactic: rw [← IsClosed.closure_eq hc]
-- Lean 4 proof step from Mathlib.Topology.Instances.Complex (goal)
Goal:
K : Subfield ℂ
hc : IsClosed (↑K : Set ℂ)
this : range ofReal ⊆ closure (range (ofReal ∘ Rat.cast))
⊢ closure (range (ofReal ∘ Rat.cast)) ⊆ closure (↑K : Set ℂ)
have h : range (ofReal ∘ Rat.cast) ⊆ (↑K : Set ℂ)
Tactic: apply closure_mono
-- Lean 4 proof step from Mathlib.Topology.Instances.Complex (goal)
Goal:
K : Subfield ℂ
hc : IsClosed (↑K : Set ℂ)
⊢ range ofReal ⊆ (↑K : Set ℂ)
have suffices : range ofReal ⊆ closure (range (ofReal ∘ Rat.cast))
Tactic: suffices range (ofReal : ℝ → ℂ) ⊆ closure (Set.range ((ofReal : ℝ → ℂ) ∘ ((↑) : ℚ → ℝ))) by
    refine subset_trans this ?_
    rw [← IsClosed.closure_eq hc]
    apply closure_mono
    rintro _ ⟨_, rfl⟩
    simp only [Function.comp_apply, ofReal_ratCast, SetLike.mem_coe, SubfieldClass.ratCast_mem]
-- Lean 4 proof step from Mathlib.Topology.Instances.Complex (goal)
Goal:
K : Subfield ℂ
hc : IsClosed (↑K : Set ℂ)
⊢ range ofReal ⊆ closure (range (ofReal ∘ Rat.cast))
have nth_rw : range ofReal ⊆ closure (ofReal '' range Rat.cast)
Tactic: nth_rw 1 [range_comp]
-- Lean 4 proof step from Mathlib.Topology.Instances.Complex (goal)
Goal:
K : Subfield ℂ
hc : IsClosed (↑K : Set ℂ)
⊢ range ofReal ⊆ closure (ofReal '' range Rat.cast)
have refine : range ofReal ⊆ ofReal '' closure (range Rat.cast)
Tactic: refine subset_trans ?_ (image_closure_subset_closure_image continuous_ofReal)
-- Lean 4 proof step from Mathlib.Topology.Instances.Complex (goal)
Goal:
K : Subfield ℂ
hc : IsClosed (↑K : Set ℂ)
⊢ range ofReal ⊆ ofReal '' closure (range Rat.cast)
have rw : range ofReal ⊆ ofReal '' univ
Tactic: rw [DenseRange.closure_range Rat.isDenseEmbedding_coe_real.dense]
-- Lean 4 proof step from Mathlib.Topology.Instances.Complex (goal)
Goal:
K : Subfield ℂ
hc : IsClosed (↑K : Set ℂ)
⊢ range ofReal ⊆ ofReal '' univ
have simp : range ofReal ⊆ range ofReal
Tactic: simp only [image_univ]