awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.NumberTheory.ZetaValues (goal)
Goal:
k : ℕ
hk : k ≠ 0
x : ℝ
hx : Membership.mem (γ := Set ℝ) (Icc 0 1) x
this :
  HasSum
    (fun (n : ℕ) ↦
      1 / HPow.hPow (α := ℂ) (↑n : ℂ) (2 * k + 1) *
        HSub.hSub (α := ℂ) (β := ℂ) ((fourier (↑n : ℤ) : (a : AddCircle 1) → ℂ) (↑x : ℝ ⧸ AddSubgroup.zmultiples 1) : ℂ)
          ((fourier (-(↑n : ℤ)) : (a : AddCircle 1) → ℂ) (↑x : ℝ ⧸ AddSubgroup.zmultiples 1) : ℂ))
    (HPow.hPow (α := ℂ) (-1) (k + 1) * I * (2 * (↑π : ℂ)) ^ (2 * k + 1) / (↑(2 * k + 1)! : ℂ) *
      (↑(bernoulliFun (2 * k + 1) x) : ℂ))
ofReal_two : (↑2 : ℂ) = 2
⊢ HPow.hPow (α := ℂ) (-1) (k + 1) * I * (2 * (↑π : ℂ)) ^ (2 * k + 1) / (↑(2 * k + 1)! : ℂ) *
        (↑(bernoulliFun (2 * k + 1) x) : ℂ) /
      (2 * I) =
    (↑(HPow.hPow (α := ℝ) (-1) (k + 1) * (2 * π) ^ (2 * k + 1) / 2 / (↑(2 * k + 1)! : ℝ) *
          Polynomial.eval x (Polynomial.map (algebraMap ℚ ℝ) (Polynomial.bernoulli (2 * k + 1)))) :
      ℂ)
have h : -(HPow.hPow (α := ℂ) (-1) (k + 1) * I * (2 * (↑π : ℂ)) ^ (2 * k + 1) / (↑(2 * k + 1)! : ℂ) *
          (↑(bernoulliFun (2 * k + 1) x) : ℂ) *
        I /
      2) =
  HPow.hPow (α := ℂ) (-1) (k + 1) * (2 * (↑π : ℂ)) ^ (2 * k + 1) / 2 / (↑(2 * k + 1)! : ℂ) *
    (↑(Polynomial.eval x (Polynomial.map (algebraMap ℚ ℝ) (Polynomial.bernoulli (2 * k + 1)))) : ℂ)
Tactic: rw [ofReal_mul, ofReal_div, ofReal_div, ofReal_mul, ofReal_pow, ofReal_pow, ofReal_neg,
      ofReal_natCast, ofReal_mul, ofReal_two, ofReal_one, ← div_div, div_I,
      div_mul_eq_mul_div₀]
-- Lean 4 proof step from Mathlib.NumberTheory.ZetaValues (hypothesis)
Goal:
k : ℕ
hk : k ≠ 0
x : ℝ
hx : Membership.mem (γ := Set ℝ) (Icc 0 1) x
this :
  HasSum
    (fun (n : ℕ) ↦
      1 / HPow.hPow (α := ℂ) (↑n : ℂ) (2 * k + 1) *
        HSub.hSub (α := ℂ) (β := ℂ) ((fourier (↑n : ℤ) : (a : AddCircle 1) → ℂ) (↑x : ℝ ⧸ AddSubgroup.zmultiples 1) : ℂ)
          ((fourier (-(↑n : ℤ)) : (a : AddCircle 1) → ℂ) (↑x : ℝ ⧸ AddSubgroup.zmultiples 1) : ℂ))
    (HPow.hPow (α := ℂ) (-1) (k + 1) * I * (2 * (↑π : ℂ)) ^ (2 * k + 1) / (↑(2 * k + 1)! : ℂ) *
      (↑(bernoulliFun (2 * k + 1) x) : ℂ))
ofReal_two : (↑2 : ℂ) = 2
⊢ -(HPow.hPow (α := ℂ) (-1) (k + 1) * I * (2 * (↑π : ℂ)) ^ (2 * k + 1) / (↑(2 * k + 1)! : ℂ) *
            (↑(bernoulliFun (2 * k + 1) x) : ℂ) *
          I /
        2) =
    HPow.hPow (α := ℂ) (-1) (k + 1) * (2 * (↑π : ℂ)) ^ (2 * k + 1) / 2 / (↑(2 * k + 1)! : ℂ) *
      (↑(Polynomial.eval x (Polynomial.map (algebraMap ℚ ℝ) (Polynomial.bernoulli (2 * k + 1)))) : ℂ)
have this₁ : ∀ (α β γ δ : ℂ), α * I * β / γ * δ * I = HPow.hPow (β := ℕ) I 2 * α * β / γ * δ
Tactic: have : ∀ α β γ δ : ℂ, α * I * β / γ * δ * I = I ^ 2 * α * β / γ * δ := by intros; ring
-- Lean 4 proof step from Mathlib.NumberTheory.ZetaValues (goal)
Goal:
k : ℕ
hk : k ≠ 0
x : ℝ
hx : Membership.mem (γ := Set ℝ) (Icc 0 1) x
this_1 :
  HasSum
    (fun (n : ℕ) ↦
      1 / HPow.hPow (α := ℂ) (↑n : ℂ) (2 * k + 1) *
        HSub.hSub (α := ℂ) (β := ℂ) ((fourier (↑n : ℤ) : (a : AddCircle 1) → ℂ) (↑x : ℝ ⧸ AddSubgroup.zmultiples 1) : ℂ)
          ((fourier (-(↑n : ℤ)) : (a : AddCircle 1) → ℂ) (↑x : ℝ ⧸ AddSubgroup.zmultiples 1) : ℂ))
    (HPow.hPow (α := ℂ) (-1) (k + 1) * I * (2 * (↑π : ℂ)) ^ (2 * k + 1) / (↑(2 * k + 1)! : ℂ) *
      (↑(bernoulliFun (2 * k + 1) x) : ℂ))
ofReal_two : (↑2 : ℂ) = 2
this : ∀ (α β γ δ : ℂ), α * I * β / γ * δ * I = HPow.hPow (β := ℕ) I 2 * α * β / γ * δ
⊢ -(HPow.hPow (α := ℂ) (-1) (k + 1) * I * (2 * (↑π : ℂ)) ^ (2 * k + 1) / (↑(2 * k + 1)! : ℂ) *
            (↑(bernoulliFun (2 * k + 1) x) : ℂ) *
          I /
        2) =
    HPow.hPow (α := ℂ) (-1) (k + 1) * (2 * (↑π : ℂ)) ^ (2 * k + 1) / 2 / (↑(2 * k + 1)! : ℂ) *
      (↑(Polynomial.eval x (Polynomial.map (algebraMap ℚ ℝ) (Polynomial.bernoulli (2 * k + 1)))) : ℂ)
have h : -(-1 * HPow.hPow (α := ℂ) (-1) (k + 1) * (2 * (↑π : ℂ)) ^ (2 * k + 1) / (↑(2 * k + 1)! : ℂ) *
        (↑(bernoulliFun (2 * k + 1) x) : ℂ) /
      2) =
  HPow.hPow (α := ℂ) (-1) (k + 1) * (2 * (↑π : ℂ)) ^ (2 * k + 1) / 2 / (↑(2 * k + 1)! : ℂ) *
    (↑(Polynomial.eval x (Polynomial.map (algebraMap ℚ ℝ) (Polynomial.bernoulli (2 * k + 1)))) : ℂ)
Tactic: rw [this, I_sq]
-- Lean 4 proof step from Mathlib.NumberTheory.ZetaValues (goal)
Goal:
k : ℕ
hk : k ≠ 0
⊢ Eq (α := ℕ → ℝ) (fun (n : ℕ) ↦ 1 / HPow.hPow (α := ℝ) (↑n : ℝ) (2 * k)) fun (n : ℕ) ↦
    1 / HPow.hPow (α := ℝ) (↑n : ℝ) (2 * k) * Real.cos (2 * π * (↑n : ℝ) * 0)
have h : ∀ (n : ℕ),
  1 / HPow.hPow (α := ℝ) (↑n : ℝ) (2 * k) = 1 / HPow.hPow (α := ℝ) (↑n : ℝ) (2 * k) * Real.cos (2 * π * (↑n : ℝ) * 0)
Tactic: ext1 n
-- Lean 4 proof step from Mathlib.NumberTheory.ZetaValues (goal)
Goal:
k : ℕ
hk : k ≠ 0
n : ℕ
⊢ 1 / HPow.hPow (α := ℝ) (↑n : ℝ) (2 * k) = 1 / HPow.hPow (α := ℝ) (↑n : ℝ) (2 * k) * Real.cos (2 * π * (↑n : ℝ) * 0)
have h : 1 / HPow.hPow (α := ℝ) (↑n : ℝ) (2 * k) = 1 / HPow.hPow (α := ℝ) (↑n : ℝ) (2 * k) * Real.cos 0
Tactic: rw [mul_zero, Real.cos_zero, mul_one]
-- Lean 4 proof step from Mathlib.NumberTheory.ZetaValues (goal)
Goal:
k : ℕ
hk : k ≠ 0
n : ℕ
h : 1 / HPow.hPow (α := ℝ) (↑n : ℝ) (2 * k) = 1 / HPow.hPow (α := ℝ) (↑n : ℝ) (2 * k) * Real.cos 0
⊢ 1 / HPow.hPow (α := ℝ) (↑n : ℝ) (2 * k) = 1 / HPow.hPow (α := ℝ) (↑n : ℝ) (2 * k) * Real.cos (2 * π * (↑n : ℝ) * 0)
have h₁ : 1 / HPow.hPow (α := ℝ) (↑n : ℝ) (2 * k) = 1 / HPow.hPow (α := ℝ) (↑n : ℝ) (2 * k) * 1
Tactic: rw [mul_zero, Real.cos_zero, mul_one]
-- Lean 4 proof step from Mathlib.NumberTheory.ZetaValues (goal)
Goal:
k : ℕ
hk : k ≠ 0
n : ℕ
h : 1 / HPow.hPow (α := ℝ) (↑n : ℝ) (2 * k) = 1 / HPow.hPow (α := ℝ) (↑n : ℝ) (2 * k) * Real.cos 0
h₁ : 1 / HPow.hPow (α := ℝ) (↑n : ℝ) (2 * k) = 1 / HPow.hPow (α := ℝ) (↑n : ℝ) (2 * k) * 1
⊢ 1 / HPow.hPow (α := ℝ) (↑n : ℝ) (2 * k) = 1 / HPow.hPow (α := ℝ) (↑n : ℝ) (2 * k) * Real.cos (2 * π * (↑n : ℝ) * 0)
have h₂ : 1 / HPow.hPow (α := ℝ) (↑n : ℝ) (2 * k) = 1 / HPow.hPow (α := ℝ) (↑n : ℝ) (2 * k)
Tactic: rw [mul_zero, Real.cos_zero, mul_one]
-- Lean 4 proof step from Mathlib.NumberTheory.ZetaValues (goal)
Goal:
k : ℕ
hk : k ≠ 0
⊢ HPow.hPow (α := ℝ) (-1) (k + 1) * HPow.hPow (α := ℝ) 2 (2 * k - 1) * π ^ (2 * k) * (↑(bernoulli (2 * k)) : ℝ) /
      (↑(2 * k)! : ℝ) =
    HPow.hPow (α := ℝ) (-1) (k + 1) * (2 * π) ^ (2 * k) / 2 / (↑(2 * k)! : ℝ) *
      Polynomial.eval 0 (Polynomial.map (algebraMap ℚ ℝ) (Polynomial.bernoulli (2 * k)))
have h : HPow.hPow (α := ℝ) (-1) (k + 1) * HPow.hPow (α := ℝ) 2 (2 * k - 1) * π ^ (2 * k) * (↑(bernoulli (2 * k)) : ℝ) /
    (↑(2 * k)! : ℝ) =
  HPow.hPow (α := ℝ) (-1) (k + 1) * (2 * π) ^ (2 * k) / 2 / (↑(2 * k)! : ℝ) * (↑(bernoulli (2 * k)) : ℝ)
Tactic: rw [Polynomial.eval_zero_map, Polynomial.bernoulli_eval_zero, eq_ratCast]
-- Lean 4 proof step from Mathlib.NumberTheory.ZetaValues (goal)
Goal:
k : ℕ
hk : k ≠ 0
⊢ HPow.hPow (α := ℝ) 2 (2 * k - 1) = HPow.hPow (α := ℝ) 2 (2 * k) / 2
have rw : HPow.hPow (α := ℝ) 2 (2 * k - 1) * 2 = HPow.hPow (α := ℝ) 2 (2 * k)
Tactic: rw [eq_div_iff (two_ne_zero' ℝ)]
-- Lean 4 proof step from Mathlib.NumberTheory.ZetaValues (goal)
Goal:
k : ℕ
hk : k ≠ 0
| HPow.hPow (α := ℝ) 2 (2 * k - 1) * 2
have a : HPow.hPow (α := ℝ) 2 (2 * k - 1)
Tactic: congr
-- Lean 4 proof step from Mathlib.NumberTheory.ZetaValues (goal)
Goal:
k : ℕ
hk : k ≠ 0
a : HPow.hPow (α := ℝ) 2 (2 * k - 1)
| HPow.hPow (α := ℝ) 2 (2 * k - 1) * 2
have a₁ : 2
Tactic: congr
-- Lean 4 proof step from Mathlib.NumberTheory.ZetaValues (goal)
Goal:
k : ℕ
hk : k ≠ 0
| 2
have a : HPow.hPow (α := ℝ) (β := ℕ) 2 1
Tactic: rw [← pow_one (2 : ℝ)]
-- Lean 4 proof step from Mathlib.NumberTheory.ZetaValues (goal)
Goal:
k : ℕ
hk : k ≠ 0
| 2
have a : HPow.hPow (α := ℝ) (β := ℕ) 2 1
Tactic: · rw [← pow_one (2 : ℝ)]
-- Lean 4 proof step from Mathlib.NumberTheory.ZetaValues (goal)
Goal:
k : ℕ
hk : k ≠ 0
| HPow.hPow (α := ℝ) 2 (2 * k - 1) * 2
have a : HPow.hPow (α := ℝ) 2 (2 * k - 1)
Tactic: congr
      · skip
      · rw [← pow_one (2 : ℝ)]
-- Lean 4 proof step from Mathlib.NumberTheory.ZetaValues (goal)
Goal:
k : ℕ
hk : k ≠ 0
a : HPow.hPow (α := ℝ) 2 (2 * k - 1)
| HPow.hPow (α := ℝ) 2 (2 * k - 1) * 2
have a₁ : 2
Tactic: congr
      · skip
      · rw [← pow_one (2 : ℝ)]
-- Lean 4 proof step from Mathlib.NumberTheory.ZetaValues (goal)
Goal:
k : ℕ
hk : k ≠ 0
⊢ HPow.hPow (α := ℝ) 2 (2 * k - 1) * 2 = HPow.hPow (α := ℝ) 2 (2 * k)
have conv_lhs : HPow.hPow (α := ℝ) 2 (2 * k - 1) * HPow.hPow (α := ℝ) (β := ℕ) 2 1 = HPow.hPow (α := ℝ) 2 (2 * k)
Tactic: conv_lhs =>
      congr
      · skip
      · rw [← pow_one (2 : ℝ)]
-- Lean 4 proof step from Mathlib.NumberTheory.ZetaValues (goal)
Goal:
k : ℕ
hk : k ≠ 0
⊢ HPow.hPow (α := ℝ) 2 (2 * k - 1) * HPow.hPow (α := ℝ) (β := ℕ) 2 1 = HPow.hPow (α := ℝ) 2 (2 * k)
have rw : 1 ≤ 2 * k
Tactic: rw [← pow_add, Nat.sub_add_cancel]
-- Lean 4 proof step from Mathlib.NumberTheory.ZetaValues (hypothesis)
Goal:
k : ℕ
hk : k ≠ 0
⊢ HPow.hPow (α := ℝ) (-1) (k + 1) * HPow.hPow (α := ℝ) 2 (2 * k - 1) * π ^ (2 * k) * (↑(bernoulli (2 * k)) : ℝ) /
      (↑(2 * k)! : ℝ) =
    HPow.hPow (α := ℝ) (-1) (k + 1) * (2 * π) ^ (2 * k) / 2 / (↑(2 * k)! : ℝ) * (↑(bernoulli (2 * k)) : ℝ)
have this : HPow.hPow (α := ℝ) 2 (2 * k - 1) = HPow.hPow (α := ℝ) 2 (2 * k) / 2
Tactic: have : (2 : ℝ) ^ (2 * k - 1) = (2 : ℝ) ^ (2 * k) / 2 := by
    rw [eq_div_iff (two_ne_zero' ℝ)]
    conv_lhs =>
      congr
      · skip
      · rw [← pow_one (2 : ℝ)]
    rw [← pow_add, Nat.sub_add_cancel]
    lia
-- Lean 4 proof step from Mathlib.NumberTheory.ZetaValues (goal)
Goal:
k : ℕ
hk : k ≠ 0
this : HPow.hPow (α := ℝ) 2 (2 * k - 1) = HPow.hPow (α := ℝ) 2 (2 * k) / 2
⊢ HPow.hPow (α := ℝ) (-1) (k + 1) * HPow.hPow (α := ℝ) 2 (2 * k - 1) * π ^ (2 * k) * (↑(bernoulli (2 * k)) : ℝ) /
      (↑(2 * k)! : ℝ) =
    HPow.hPow (α := ℝ) (-1) (k + 1) * (2 * π) ^ (2 * k) / 2 / (↑(2 * k)! : ℝ) * (↑(bernoulli (2 * k)) : ℝ)
have h : HPow.hPow (α := ℝ) (-1) (k + 1) * (HPow.hPow (α := ℝ) 2 (2 * k) / 2) * π ^ (2 * k) * (↑(bernoulli (2 * k)) : ℝ) /
    (↑(2 * k)! : ℝ) =
  HPow.hPow (α := ℝ) (-1) (k + 1) * (HPow.hPow (α := ℝ) 2 (2 * k) * π ^ (2 * k)) / 2 / (↑(2 * k)! : ℝ) *
    (↑(bernoulli (2 * k)) : ℝ)
Tactic: rw [this, mul_pow]
-- Lean 4 proof step from Mathlib.NumberTheory.ZetaValues (goal)
Goal:
⊢ HasSum (fun (n : ℕ) ↦ 1 / HPow.hPow (α := ℝ) (β := ℕ) (↑n : ℝ) 2) (HPow.hPow (β := ℕ) π 2 / 6)
have h : HPow.hPow (β := ℕ) π 2 / 6 =
  HPow.hPow (α := ℝ) (-1) (HAdd.hAdd (α := ℕ) 1 1) * HPow.hPow (α := ℝ) 2 (HMul.hMul (α := ℕ) 2 1 - 1) *
        π ^ HMul.hMul (α := ℕ) 2 1 *
      (↑(bernoulli (HMul.hMul (α := ℕ) 2 1)) : ℝ) /
    (↑(HMul.hMul (α := ℕ) 2 1)! : ℝ)
Tactic: convert! hasSum_zeta_nat one_ne_zero using 1
-- Lean 4 proof step from Mathlib.NumberTheory.ZetaValues (goal)
Goal:
⊢ HPow.hPow (β := ℕ) π 2 / 6 =
    HPow.hPow (α := ℝ) (-1) (HAdd.hAdd (α := ℕ) 1 1) * HPow.hPow (α := ℝ) 2 (HSub.hSub (α := ℕ) 2 1) *
          HPow.hPow (β := ℕ) π 2 *
        (↑(bernoulli 2) : ℝ) /
      (↑2! : ℝ)
have h : HPow.hPow (β := ℕ) π 2 / 6 =
  HPow.hPow (α := ℝ) (-1) (HAdd.hAdd (α := ℕ) 1 1) * HPow.hPow (α := ℝ) 2 (HSub.hSub (α := ℕ) 2 1) *
        HPow.hPow (β := ℕ) π 2 *
      (↑(1 / 6) : ℝ) /
    (↑2! : ℝ)
Tactic: rw [bernoulli_eq_bernoulli'_of_ne_one (by decide : 2 ≠ 1), bernoulli'_two]
-- Lean 4 proof step from Mathlib.NumberTheory.ZetaValues (goal)
Goal:
⊢ HPow.hPow (β := ℕ) π 2 / 6 =
    HPow.hPow (α := ℝ) (-1) (HAdd.hAdd (α := ℕ) 1 1) * HPow.hPow (α := ℝ) 2 (HSub.hSub (α := ℕ) 2 1) *
          HPow.hPow (β := ℕ) π 2 *
        (↑(1 / 6) : ℝ) /
      (↑2! : ℝ)
have h : HPow.hPow (β := ℕ) π 2 / 6 = 2 * HPow.hPow (β := ℕ) π 2 * 6⁻¹ / 2
Tactic: simp [Nat.factorial]
-- Lean 4 proof step from Mathlib.NumberTheory.ZetaValues (goal)
Goal:
⊢ HasSum (fun (n : ℕ) ↦ 1 / HPow.hPow (α := ℝ) (β := ℕ) (↑n : ℝ) 4) (HPow.hPow (β := ℕ) π 4 / 90)
have h : HPow.hPow (β := ℕ) π 4 / 90 =
  HPow.hPow (α := ℝ) (-1) (HAdd.hAdd (α := ℕ) 2 1) * HPow.hPow (α := ℝ) 2 (HMul.hMul (α := ℕ) 2 2 - 1) *
        π ^ HMul.hMul (α := ℕ) 2 2 *
      (↑(bernoulli (HMul.hMul (α := ℕ) 2 2)) : ℝ) /
    (↑(HMul.hMul (α := ℕ) 2 2)! : ℝ)
Tactic: convert! hasSum_zeta_nat two_ne_zero using 1
-- Lean 4 proof step from Mathlib.NumberTheory.ZetaValues (goal)
Goal:
⊢ HPow.hPow (β := ℕ) π 4 / 90 =
    HPow.hPow (α := ℝ) (β := ℕ) (-1) 3 * HPow.hPow (α := ℝ) (β := ℕ) 2 3 * HPow.hPow (β := ℕ) π 4 *
        (↑(bernoulli 4) : ℝ) /
      (↑4! : ℝ)
have h : HPow.hPow (β := ℕ) π 4 / 90 =
  HPow.hPow (α := ℝ) (β := ℕ) (-1) 3 * HPow.hPow (α := ℝ) (β := ℕ) 2 3 * HPow.hPow (β := ℕ) π 4 * (↑(-1 / 30) : ℝ) /
    (↑4! : ℝ)
Tactic: rw [bernoulli_eq_bernoulli'_of_ne_one, bernoulli'_four]
-- Lean 4 proof step from Mathlib.NumberTheory.ZetaValues (goal)
Goal:
h :
  HPow.hPow (β := ℕ) π 4 / 90 =
    HPow.hPow (α := ℝ) (β := ℕ) (-1) 3 * HPow.hPow (α := ℝ) (β := ℕ) 2 3 * HPow.hPow (β := ℕ) π 4 * (↑(-1 / 30) : ℝ) /
      (↑4! : ℝ)
⊢ HPow.hPow (β := ℕ) π 4 / 90 =
    HPow.hPow (α := ℝ) (β := ℕ) (-1) 3 * HPow.hPow (α := ℝ) (β := ℕ) 2 3 * HPow.hPow (β := ℕ) π 4 *
        (↑(bernoulli 4) : ℝ) /
      (↑4! : ℝ)
have h₁ : Ne (α := ℕ) 4 1
Tactic: rw [bernoulli_eq_bernoulli'_of_ne_one, bernoulli'_four]
-- Lean 4 proof step from Mathlib.NumberTheory.ZetaValues (goal)
Goal:
⊢ HPow.hPow (β := ℕ) π 4 / 90 =
    HPow.hPow (α := ℝ) (β := ℕ) (-1) 3 * HPow.hPow (α := ℝ) (β := ℕ) 2 3 * HPow.hPow (β := ℕ) π 4 * (↑(-1 / 30) : ℝ) /
      (↑4! : ℝ)
have h : HPow.hPow (β := ℕ) π 4 / 90 =
  HPow.hPow (α := ℝ) (β := ℕ) (-1) 3 * HPow.hPow (α := ℝ) (β := ℕ) 2 3 * HPow.hPow (β := ℕ) π 4 *
      HDiv.hDiv (α := ℝ) (-1) 30 /
    24
Tactic: simp [Nat.factorial]
-- Lean 4 proof step from Mathlib.NumberTheory.ZetaValues (goal)
Goal:
⊢ HasSum (fun (n : ℕ) ↦ 1 / HPow.hPow (α := ℝ) (β := ℕ) (↑n : ℝ) 3 * Real.sin (π * (↑n : ℝ) / 2))
    (HPow.hPow (β := ℕ) π 3 / 32)
have apply : Eq (α := ℕ → ℝ)
  (fun (n : ℕ) ↦ 1 / HPow.hPow (α := ℝ) (↑n : ℝ) (HMul.hMul (α := ℕ) 2 1 + 1) * Real.sin (2 * π * (↑n : ℝ) * (1 / 4)))
  fun (n : ℕ) ↦ 1 / HPow.hPow (α := ℝ) (β := ℕ) (↑n : ℝ) 3 * Real.sin (π * (↑n : ℝ) / 2)
Tactic: apply (congr_arg₂ HasSum ?_ ?_).to_iff.mp <|
    hasSum_one_div_nat_pow_mul_sin one_ne_zero (?_ : 1 / 4 ∈ Icc (0 : ℝ) 1)
-- Lean 4 proof step from Mathlib.NumberTheory.ZetaValues (goal)
Goal:
apply :
  Eq (α := ℕ → ℝ)
    (fun (n : ℕ) ↦ 1 / HPow.hPow (α := ℝ) (↑n : ℝ) (HMul.hMul (α := ℕ) 2 1 + 1) * Real.sin (2 * π * (↑n : ℝ) * (1 / 4)))
    fun (n : ℕ) ↦ 1 / HPow.hPow (α := ℝ) (β := ℕ) (↑n : ℝ) 3 * Real.sin (π * (↑n : ℝ) / 2)
⊢ HasSum (fun (n : ℕ) ↦ 1 / HPow.hPow (α := ℝ) (β := ℕ) (↑n : ℝ) 3 * Real.sin (π * (↑n : ℝ) / 2))
    (HPow.hPow (β := ℕ) π 3 / 32)
have apply₁ : HPow.hPow (α := ℝ) (-1) (HAdd.hAdd (α := ℕ) 1 1) * (2 * π) ^ (HMul.hMul (α := ℕ) 2 1 + 1) / 2 /
      (↑(HMul.hMul (α := ℕ) 2 1 + 1)! : ℝ) *
    Polynomial.eval (1 / 4) (Polynomial.map (algebraMap ℚ ℝ) (Polynomial.bernoulli (HMul.hMul (α := ℕ) 2 1 + 1))) =
  HPow.hPow (β := ℕ) π 3 / 32
Tactic: apply (congr_arg₂ HasSum ?_ ?_).to_iff.mp <|
    hasSum_one_div_nat_pow_mul_sin one_ne_zero (?_ : 1 / 4 ∈ Icc (0 : ℝ) 1)
-- Lean 4 proof step from Mathlib.NumberTheory.ZetaValues (goal)
Goal:
apply :
  Eq (α := ℕ → ℝ)
    (fun (n : ℕ) ↦ 1 / HPow.hPow (α := ℝ) (↑n : ℝ) (HMul.hMul (α := ℕ) 2 1 + 1) * Real.sin (2 * π * (↑n : ℝ) * (1 / 4)))
    fun (n : ℕ) ↦ 1 / HPow.hPow (α := ℝ) (β := ℕ) (↑n : ℝ) 3 * Real.sin (π * (↑n : ℝ) / 2)
apply₁ :
  HPow.hPow (α := ℝ) (-1) (HAdd.hAdd (α := ℕ) 1 1) * (2 * π) ^ (HMul.hMul (α := ℕ) 2 1 + 1) / 2 /
        (↑(HMul.hMul (α := ℕ) 2 1 + 1)! : ℝ) *
      Polynomial.eval (1 / 4) (Polynomial.map (algebraMap ℚ ℝ) (Polynomial.bernoulli (HMul.hMul (α := ℕ) 2 1 + 1))) =
    HPow.hPow (β := ℕ) π 3 / 32
⊢ HasSum (fun (n : ℕ) ↦ 1 / HPow.hPow (α := ℝ) (β := ℕ) (↑n : ℝ) 3 * Real.sin (π * (↑n : ℝ) / 2))
    (HPow.hPow (β := ℕ) π 3 / 32)
have apply₂ : Membership.mem (γ := Set ℝ) (Icc 0 1) (1 / 4)
Tactic: apply (congr_arg₂ HasSum ?_ ?_).to_iff.mp <|
    hasSum_one_div_nat_pow_mul_sin one_ne_zero (?_ : 1 / 4 ∈ Icc (0 : ℝ) 1)
-- Lean 4 proof step from Mathlib.NumberTheory.ZetaValues (goal)
Goal:
⊢ Eq (α := ℕ → ℝ)
    (fun (n : ℕ) ↦ 1 / HPow.hPow (α := ℝ) (↑n : ℝ) (HMul.hMul (α := ℕ) 2 1 + 1) * Real.sin (2 * π * (↑n : ℝ) * (1 / 4)))
    fun (n : ℕ) ↦ 1 / HPow.hPow (α := ℝ) (β := ℕ) (↑n : ℝ) 3 * Real.sin (π * (↑n : ℝ) / 2)
have h : ∀ (n : ℕ),
  1 / HPow.hPow (α := ℝ) (↑n : ℝ) (HMul.hMul (α := ℕ) 2 1 + 1) * Real.sin (2 * π * (↑n : ℝ) * (1 / 4)) =
    1 / HPow.hPow (α := ℝ) (β := ℕ) (↑n : ℝ) 3 * Real.sin (π * (↑n : ℝ) / 2)
Tactic: ext1 n
-- Lean 4 proof step from Mathlib.NumberTheory.ZetaValues (hypothesis)
Goal:
⊢ HPow.hPow (α := ℝ) (-1) (HAdd.hAdd (α := ℕ) 1 1) * (2 * π) ^ (HMul.hMul (α := ℕ) 2 1 + 1) / 2 /
        (↑(HMul.hMul (α := ℕ) 2 1 + 1)! : ℝ) *
      Polynomial.eval (1 / 4) (Polynomial.map (algebraMap ℚ ℝ) (Polynomial.bernoulli (HMul.hMul (α := ℕ) 2 1 + 1))) =
    HPow.hPow (β := ℕ) π 3 / 32
have this : 1 / 4 = (algebraMap ℚ ℝ : (a : ℚ) → ℝ) (1 / 4)
Tactic: have : (1 / 4 : ℝ) = (algebraMap ℚ ℝ) (1 / 4 : ℚ) := by simp
-- Lean 4 proof step from Mathlib.NumberTheory.ZetaValues (goal)
Goal:
this : 1 / 4 = (algebraMap ℚ ℝ : (a : ℚ) → ℝ) (1 / 4)
⊢ HPow.hPow (α := ℝ) (-1) (HAdd.hAdd (α := ℕ) 1 1) * (2 * π) ^ (HMul.hMul (α := ℕ) 2 1 + 1) / 2 /
        (↑(HMul.hMul (α := ℕ) 2 1 + 1)! : ℝ) *
      Polynomial.eval (1 / 4) (Polynomial.map (algebraMap ℚ ℝ) (Polynomial.bernoulli (HMul.hMul (α := ℕ) 2 1 + 1))) =
    HPow.hPow (β := ℕ) π 3 / 32
have rw : HMul.hMul (β := ℝ)
    (HPow.hPow (α := ℝ) (-1) (HAdd.hAdd (α := ℕ) 1 1) * (HPow.hPow (α := ℝ) (β := ℕ) 2 3 * HPow.hPow (β := ℕ) π 3) / 2 /
      (↑3! : ℝ))
    ((algebraMap ℚ ℝ : (a : ℚ) → ℝ) (3 / 64) : ℝ) =
  HPow.hPow (β := ℕ) π 3 / 32
Tactic: rw [this, mul_pow, Polynomial.eval_map, Polynomial.eval₂_at_apply, (by decide : 2 * 1 + 1 = 3),
      Polynomial.bernoulli_three_eval_one_quarter]
-- Lean 4 proof step from Mathlib.NumberTheory.ZetaValues (goal)
Goal:
this : 1 / 4 = (algebraMap ℚ ℝ : (a : ℚ) → ℝ) (1 / 4)
⊢ HMul.hMul (β := ℝ)
      (HPow.hPow (α := ℝ) (-1) (HAdd.hAdd (α := ℕ) 1 1) * (HPow.hPow (α := ℝ) (β := ℕ) 2 3 * HPow.hPow (β := ℕ) π 3) /
          2 /
        (↑3! : ℝ))
      ((algebraMap ℚ ℝ : (a : ℚ) → ℝ) (3 / 64) : ℝ) =
    HPow.hPow (β := ℕ) π 3 / 32
have simp : HPow.hPow (α := ℝ) (β := ℕ) 2 3 * HPow.hPow (β := ℕ) π 3 / 2 / 6 * HDiv.hDiv (α := ℝ) 3 64 = HPow.hPow (β := ℕ) π 3 / 32
Tactic: simp [Nat.factorial]
-- Lean 4 proof step from Mathlib.NumberTheory.ZetaValues (goal)
Goal:
⊢ 0 ≤ 1 / 4 ∧ 1 / 4 ≤ 1
have left : 0 ≤ 1 / 4
Tactic: constructor
-- Lean 4 proof step from Mathlib.NumberTheory.ZetaValues (goal)
Goal:
left : 0 ≤ 1 / 4
⊢ 0 ≤ 1 / 4 ∧ 1 / 4 ≤ 1
have right : 1 / 4 ≤ 1
Tactic: constructor
-- Lean 4 proof step from Mathlib.NumberTheory.ZetaValues (goal)
Goal:
⊢ 0 ≤ 1 / 4
have linarith : 1 / 4 < 0 → 4 / 4 = 1
Tactic: linarith
-- Lean 4 proof step from Mathlib.NumberTheory.ZetaValues (goal)
Goal:
linarith : 1 / 4 < 0 → 4 / 4 = 1
⊢ 0 ≤ 1 / 4
have linarith₁ : 1 / 4 < 0 → HMul.hMul (α := ℝ) 1 4 = 4
Tactic: linarith
-- Lean 4 proof step from Mathlib.NumberTheory.ZetaValues (goal)
Goal:
linarith : 1 / 4 < 0 → 4 / 4 = 1
linarith₁ : 1 / 4 < 0 → HMul.hMul (α := ℝ) 1 4 = 4
⊢ 0 ≤ 1 / 4
have linarith₂ : 1 / 4 < 0 → GT.gt (α := ℝ) 4 0
Tactic: linarith
-- Lean 4 proof step from Mathlib.NumberTheory.ZetaValues (goal)
Goal:
linarith : 1 / 4 < 0 → 4 / 4 = 1
linarith₁ : 1 / 4 < 0 → HMul.hMul (α := ℝ) 1 4 = 4
linarith₂ : 1 / 4 < 0 → GT.gt (α := ℝ) 4 0
⊢ 0 ≤ 1 / 4
have linarith₃ : 1 / 4 < 0 → -1 + HMul.hMul (α := ℝ) 1 1 = 0
Tactic: linarith
-- Lean 4 proof step from Mathlib.NumberTheory.ZetaValues (goal)
Goal:
⊢ 1 / 4 ≤ 1
have linarith : 1 < 1 / 4 → 4 / 4 = 1
Tactic: linarith
-- Lean 4 proof step from Mathlib.NumberTheory.ZetaValues (goal)
Goal:
linarith : 1 < 1 / 4 → 4 / 4 = 1
⊢ 1 / 4 ≤ 1
have linarith₁ : 1 < 1 / 4 → HMul.hMul (α := ℝ) 1 4 = 4
Tactic: linarith
-- Lean 4 proof step from Mathlib.NumberTheory.ZetaValues (goal)
Goal:
linarith : 1 < 1 / 4 → 4 / 4 = 1
linarith₁ : 1 < 1 / 4 → HMul.hMul (α := ℝ) 1 4 = 4
⊢ 1 / 4 ≤ 1
have linarith₂ : 1 < 1 / 4 → GT.gt (α := ℝ) 4 0
Tactic: linarith
-- Lean 4 proof step from Mathlib.NumberTheory.ZetaValues (goal)
Goal:
linarith : 1 < 1 / 4 → 4 / 4 = 1
linarith₁ : 1 < 1 / 4 → HMul.hMul (α := ℝ) 1 4 = 4
linarith₂ : 1 < 1 / 4 → GT.gt (α := ℝ) 4 0
⊢ 1 / 4 ≤ 1
have linarith₃ : 1 < 1 / 4 → HMul.hMul (α := ℝ) 3 (-1) + HMul.hMul (α := ℝ) 1 3 = 0
Tactic: linarith
-- Lean 4 proof step from Mathlib.NumberTheory.ZetaValues (goal)
Goal:
linarith : 1 < 1 / 4 → 4 / 4 = 1
linarith₁ : 1 < 1 / 4 → HMul.hMul (α := ℝ) 1 4 = 4
linarith₂ : 1 < 1 / 4 → GT.gt (α := ℝ) 4 0
linarith₃ : 1 < 1 / 4 → HMul.hMul (α := ℝ) 3 (-1) + HMul.hMul (α := ℝ) 1 3 = 0
⊢ 1 / 4 ≤ 1
have linarith₄ : 1 < 1 / 4 → GT.gt (α := ℝ) 3 0
Tactic: linarith
-- Lean 4 proof step from Mathlib.NumberTheory.Zsqrtd.Basic (goal)
Goal:
d : ℤ
⊢ AddCommGroup (ℤ√d)
have refine_1 : ∀ (a b c : ℤ√d), a + b + c = a + (b + c)
Tactic: refine
  { sub := fun a b => a + -b
    nsmul := @nsmulRec (ℤ√d) ⟨0⟩ ⟨(· + ·)⟩
    zsmul := @zsmulRec (ℤ√d) ⟨0⟩ ⟨(· + ·)⟩ ⟨Neg.neg⟩ (@nsmulRec (ℤ√d) ⟨0⟩ ⟨(· + ·)⟩)
    add_assoc := ?_
    zero_add := ?_
    add_zero := ?_
    neg_add_cancel := ?_
    add_comm := ?_ }
-- Lean 4 proof step from Mathlib.NumberTheory.Zsqrtd.Basic (goal)
Goal:
d : ℤ
refine_1 : ∀ (a b c : ℤ√d), a + b + c = a + (b + c)
⊢ AddCommGroup (ℤ√d)
have refine_2 : ∀ (a : ℤ√d), 0 + a = a
Tactic: refine
  { sub := fun a b => a + -b
    nsmul := @nsmulRec (ℤ√d) ⟨0⟩ ⟨(· + ·)⟩
    zsmul := @zsmulRec (ℤ√d) ⟨0⟩ ⟨(· + ·)⟩ ⟨Neg.neg⟩ (@nsmulRec (ℤ√d) ⟨0⟩ ⟨(· + ·)⟩)
    add_assoc := ?_
    zero_add := ?_
    add_zero := ?_
    neg_add_cancel := ?_
    add_comm := ?_ }
-- Lean 4 proof step from Mathlib.NumberTheory.Zsqrtd.Basic (goal)
Goal:
d : ℤ
refine_1 : ∀ (a b c : ℤ√d), a + b + c = a + (b + c)
refine_2 : ∀ (a : ℤ√d), 0 + a = a
⊢ AddCommGroup (ℤ√d)
have refine_3 : ∀ (a : ℤ√d), a + 0 = a
Tactic: refine
  { sub := fun a b => a + -b
    nsmul := @nsmulRec (ℤ√d) ⟨0⟩ ⟨(· + ·)⟩
    zsmul := @zsmulRec (ℤ√d) ⟨0⟩ ⟨(· + ·)⟩ ⟨Neg.neg⟩ (@nsmulRec (ℤ√d) ⟨0⟩ ⟨(· + ·)⟩)
    add_assoc := ?_
    zero_add := ?_
    add_zero := ?_
    neg_add_cancel := ?_
    add_comm := ?_ }
-- Lean 4 proof step from Mathlib.NumberTheory.Zsqrtd.Basic (goal)
Goal:
d : ℤ
refine_1 : ∀ (a b c : ℤ√d), a + b + c = a + (b + c)
refine_2 : ∀ (a : ℤ√d), 0 + a = a
refine_3 : ∀ (a : ℤ√d), a + 0 = a
⊢ AddCommGroup (ℤ√d)
have refine_4 : ∀ (a : ℤ√d), -a + a = 0
Tactic: refine
  { sub := fun a b => a + -b
    nsmul := @nsmulRec (ℤ√d) ⟨0⟩ ⟨(· + ·)⟩
    zsmul := @zsmulRec (ℤ√d) ⟨0⟩ ⟨(· + ·)⟩ ⟨Neg.neg⟩ (@nsmulRec (ℤ√d) ⟨0⟩ ⟨(· + ·)⟩)
    add_assoc := ?_
    zero_add := ?_
    add_zero := ?_
    neg_add_cancel := ?_
    add_comm := ?_ }
-- Lean 4 proof step from Mathlib.NumberTheory.Zsqrtd.Basic (goal)
Goal:
d : ℤ
refine_1 : ∀ (a b c : ℤ√d), a + b + c = a + (b + c)
refine_2 : ∀ (a : ℤ√d), 0 + a = a
refine_3 : ∀ (a : ℤ√d), a + 0 = a
refine_4 : ∀ (a : ℤ√d), -a + a = 0
⊢ AddCommGroup (ℤ√d)
have refine_5 : ∀ (a b : ℤ√d), a + b = b + a
Tactic: refine
  { sub := fun a b => a + -b
    nsmul := @nsmulRec (ℤ√d) ⟨0⟩ ⟨(· + ·)⟩
    zsmul := @zsmulRec (ℤ√d) ⟨0⟩ ⟨(· + ·)⟩ ⟨Neg.neg⟩ (@nsmulRec (ℤ√d) ⟨0⟩ ⟨(· + ·)⟩)
    add_assoc := ?_
    zero_add := ?_
    add_zero := ?_
    neg_add_cancel := ?_
    add_comm := ?_ }
-- Lean 4 proof step from Mathlib.NumberTheory.Zsqrtd.Basic (goal)
Goal:
d : ℤ
a b c : ℤ√d
⊢ a + b + c = a + (b + c)
have refine_1 : (a + b + c).re = (a + (b + c)).re
Tactic: ext
-- Lean 4 proof step from Mathlib.NumberTheory.Zsqrtd.Basic (goal)
Goal:
d : ℤ
a b c : ℤ√d
refine_1 : (a + b + c).re = (a + (b + c)).re
⊢ a + b + c = a + (b + c)
have refine_1₁ : (a + b + c).im = (a + (b + c)).im
Tactic: ext
-- Lean 4 proof step from Mathlib.NumberTheory.Zsqrtd.Basic (goal)
Goal:
d : ℤ
a : ℤ√d
⊢ 0 + a = a
have refine_2 : (0 + a).re = a.re
Tactic: ext
-- Lean 4 proof step from Mathlib.NumberTheory.Zsqrtd.Basic (goal)
Goal:
d : ℤ
a : ℤ√d
refine_2 : (0 + a).re = a.re
⊢ 0 + a = a
have refine_2₁ : (0 + a).im = a.im
Tactic: ext
-- Lean 4 proof step from Mathlib.NumberTheory.Zsqrtd.Basic (goal)
Goal:
d : ℤ
a : ℤ√d
⊢ a + 0 = a
have refine_3 : (a + 0).re = a.re
Tactic: ext
-- Lean 4 proof step from Mathlib.NumberTheory.Zsqrtd.Basic (goal)
Goal:
d : ℤ
a : ℤ√d
refine_3 : (a + 0).re = a.re
⊢ a + 0 = a
have refine_3₁ : (a + 0).im = a.im
Tactic: ext
-- Lean 4 proof step from Mathlib.NumberTheory.Zsqrtd.Basic (goal)
Goal:
d : ℤ
a : ℤ√d
⊢ -a + a = 0
have refine_4 : (-a + a).re = re (d := d) 0
Tactic: ext
-- Lean 4 proof step from Mathlib.NumberTheory.Zsqrtd.Basic (goal)
Goal:
d : ℤ
a : ℤ√d
refine_4 : (-a + a).re = re (d := d) 0
⊢ -a + a = 0
have refine_4₁ : (-a + a).im = im (d := d) 0
Tactic: ext