awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.NumberTheory.ZetaValues (goal)
Goal:
k : ℕ
hk : 2 ≤ k
x : ℝ
hx : Membership.mem (γ := Set ℝ) (Icc 0 1) x
y : ℝ
hy : Membership.mem (γ := Set ℝ) (Ico 0 1) y
B : C(𝕌, ℂ) := { toFun := ofReal ∘ periodizedBernoulli k, continuous_toFun := ⋯ }
step1 : ∀ (n : ℤ), fourierCoeff (T := 1) (⇑B : (a : 𝕌) → ℂ) n = -(↑k ! : ℂ) / (2 * (↑π : ℂ) * I * (↑n : ℂ)) ^ k
step2 :
  HasSum (α := ℂ)
    (fun (i : ℤ) ↦
      HSMul.hSMul (β := ℂ) (-(↑k ! : ℂ) / (2 * (↑π : ℂ) * I * (↑i : ℂ)) ^ k)
        ((fourier i : (a : AddCircle 1) → ℂ) (↑y : ℝ ⧸ AddSubgroup.zmultiples 1) : ℂ))
    ((B : (a : AddCircle 1) → ℂ) (↑y : ℝ ⧸ AddSubgroup.zmultiples 1) : ℂ)
⊢ (↑(bernoulliFun k y) : ℂ) = (B : (a : AddCircle 1) → ℂ) (↑y : ℝ ⧸ AddSubgroup.zmultiples 1)
have h : (↑(bernoulliFun k y) : ℂ) = (ofReal ∘ periodizedBernoulli k) (↑y : ℝ ⧸ AddSubgroup.zmultiples 1)
Tactic: rw [ContinuousMap.coe_mk, Function.comp_apply, ofReal_inj, periodizedBernoulli,
      AddCircle.liftIco_coe_apply (show y ∈ Ico 0 (0 + 1) by rwa [zero_add])]
-- Lean 4 proof step from Mathlib.NumberTheory.ZetaValues (goal)
Goal:
k : ℕ
hk : 2 ≤ k
x : ℝ
hx : Membership.mem (γ := Set ℝ) (Icc 0 1) x
y : ℝ
hy : Membership.mem (γ := Set ℝ) (Ico 0 1) y
B : C(𝕌, ℂ) := { toFun := ofReal ∘ periodizedBernoulli k, continuous_toFun := ⋯ }
step1 : ∀ (n : ℤ), fourierCoeff (T := 1) (⇑B : (a : 𝕌) → ℂ) n = -(↑k ! : ℂ) / (2 * (↑π : ℂ) * I * (↑n : ℂ)) ^ k
step2 :
  HasSum (α := ℂ)
    (fun (i : ℤ) ↦
      HSMul.hSMul (β := ℂ) (-(↑k ! : ℂ) / (2 * (↑π : ℂ) * I * (↑i : ℂ)) ^ k)
        ((fourier i : (a : AddCircle 1) → ℂ) (↑y : ℝ ⧸ AddSubgroup.zmultiples 1) : ℂ))
    ((B : (a : AddCircle 1) → ℂ) (↑y : ℝ ⧸ AddSubgroup.zmultiples 1) : ℂ)
h : (↑(bernoulliFun k y) : ℂ) = (ofReal ∘ periodizedBernoulli k) (↑y : ℝ ⧸ AddSubgroup.zmultiples 1)
⊢ (↑(bernoulliFun k y) : ℂ) = (B : (a : AddCircle 1) → ℂ) (↑y : ℝ ⧸ AddSubgroup.zmultiples 1)
have h₁ : (↑(bernoulliFun k y) : ℂ) = (↑(periodizedBernoulli k (↑y : ℝ ⧸ AddSubgroup.zmultiples 1)) : ℂ)
Tactic: rw [ContinuousMap.coe_mk, Function.comp_apply, ofReal_inj, periodizedBernoulli,
      AddCircle.liftIco_coe_apply (show y ∈ Ico 0 (0 + 1) by rwa [zero_add])]
-- Lean 4 proof step from Mathlib.NumberTheory.ZetaValues (goal)
Goal:
k : ℕ
hk : 2 ≤ k
x : ℝ
hx : Membership.mem (γ := Set ℝ) (Icc 0 1) x
y : ℝ
hy : Membership.mem (γ := Set ℝ) (Ico 0 1) y
B : C(𝕌, ℂ) := { toFun := ofReal ∘ periodizedBernoulli k, continuous_toFun := ⋯ }
step1 : ∀ (n : ℤ), fourierCoeff (T := 1) (⇑B : (a : 𝕌) → ℂ) n = -(↑k ! : ℂ) / (2 * (↑π : ℂ) * I * (↑n : ℂ)) ^ k
step2 :
  HasSum (α := ℂ)
    (fun (i : ℤ) ↦
      HSMul.hSMul (β := ℂ) (-(↑k ! : ℂ) / (2 * (↑π : ℂ) * I * (↑i : ℂ)) ^ k)
        ((fourier i : (a : AddCircle 1) → ℂ) (↑y : ℝ ⧸ AddSubgroup.zmultiples 1) : ℂ))
    ((B : (a : AddCircle 1) → ℂ) (↑y : ℝ ⧸ AddSubgroup.zmultiples 1) : ℂ)
h : (↑(bernoulliFun k y) : ℂ) = (ofReal ∘ periodizedBernoulli k) (↑y : ℝ ⧸ AddSubgroup.zmultiples 1)
h₁ : (↑(bernoulliFun k y) : ℂ) = (↑(periodizedBernoulli k (↑y : ℝ ⧸ AddSubgroup.zmultiples 1)) : ℂ)
⊢ (↑(bernoulliFun k y) : ℂ) = (B : (a : AddCircle 1) → ℂ) (↑y : ℝ ⧸ AddSubgroup.zmultiples 1)
have h₂ : bernoulliFun k y = periodizedBernoulli k (↑y : ℝ ⧸ AddSubgroup.zmultiples 1)
Tactic: rw [ContinuousMap.coe_mk, Function.comp_apply, ofReal_inj, periodizedBernoulli,
      AddCircle.liftIco_coe_apply (show y ∈ Ico 0 (0 + 1) by rwa [zero_add])]
-- Lean 4 proof step from Mathlib.NumberTheory.ZetaValues (goal)
Goal:
k : ℕ
hk : 2 ≤ k
x : ℝ
hx : Membership.mem (γ := Set ℝ) (Icc 0 1) x
y : ℝ
hy : Membership.mem (γ := Set ℝ) (Ico 0 1) y
B : C(𝕌, ℂ) := { toFun := ofReal ∘ periodizedBernoulli k, continuous_toFun := ⋯ }
step1 : ∀ (n : ℤ), fourierCoeff (T := 1) (⇑B : (a : 𝕌) → ℂ) n = -(↑k ! : ℂ) / (2 * (↑π : ℂ) * I * (↑n : ℂ)) ^ k
step2 :
  HasSum (α := ℂ)
    (fun (i : ℤ) ↦
      HSMul.hSMul (β := ℂ) (-(↑k ! : ℂ) / (2 * (↑π : ℂ) * I * (↑i : ℂ)) ^ k)
        ((fourier i : (a : AddCircle 1) → ℂ) (↑y : ℝ ⧸ AddSubgroup.zmultiples 1) : ℂ))
    ((B : (a : AddCircle 1) → ℂ) (↑y : ℝ ⧸ AddSubgroup.zmultiples 1) : ℂ)
h : (↑(bernoulliFun k y) : ℂ) = (ofReal ∘ periodizedBernoulli k) (↑y : ℝ ⧸ AddSubgroup.zmultiples 1)
h₁ : (↑(bernoulliFun k y) : ℂ) = (↑(periodizedBernoulli k (↑y : ℝ ⧸ AddSubgroup.zmultiples 1)) : ℂ)
h₂ : bernoulliFun k y = periodizedBernoulli k (↑y : ℝ ⧸ AddSubgroup.zmultiples 1)
⊢ (↑(bernoulliFun k y) : ℂ) = (B : (a : AddCircle 1) → ℂ) (↑y : ℝ ⧸ AddSubgroup.zmultiples 1)
have h₃ : bernoulliFun k y = AddCircle.liftIco 1 0 (bernoulliFun k) (↑y : ℝ ⧸ AddSubgroup.zmultiples 1)
Tactic: rw [ContinuousMap.coe_mk, Function.comp_apply, ofReal_inj, periodizedBernoulli,
      AddCircle.liftIco_coe_apply (show y ∈ Ico 0 (0 + 1) by rwa [zero_add])]
-- Lean 4 proof step from Mathlib.NumberTheory.ZetaValues (goal)
Goal:
k : ℕ
hk : 2 ≤ k
x : ℝ
hx : Membership.mem (γ := Set ℝ) (Icc 0 1) x
y : ℝ
hy : Membership.mem (γ := Set ℝ) (Ico 0 1) y
B : C(𝕌, ℂ) := { toFun := ofReal ∘ periodizedBernoulli k, continuous_toFun := ⋯ }
step1 : ∀ (n : ℤ), fourierCoeff (T := 1) (⇑B : (a : 𝕌) → ℂ) n = -(↑k ! : ℂ) / (2 * (↑π : ℂ) * I * (↑n : ℂ)) ^ k
step2 :
  HasSum (α := ℂ)
    (fun (i : ℤ) ↦
      HSMul.hSMul (β := ℂ) (-(↑k ! : ℂ) / (2 * (↑π : ℂ) * I * (↑i : ℂ)) ^ k)
        ((fourier i : (a : AddCircle 1) → ℂ) (↑y : ℝ ⧸ AddSubgroup.zmultiples 1) : ℂ))
    ((B : (a : AddCircle 1) → ℂ) (↑y : ℝ ⧸ AddSubgroup.zmultiples 1) : ℂ)
h : (↑(bernoulliFun k y) : ℂ) = (ofReal ∘ periodizedBernoulli k) (↑y : ℝ ⧸ AddSubgroup.zmultiples 1)
h₁ : (↑(bernoulliFun k y) : ℂ) = (↑(periodizedBernoulli k (↑y : ℝ ⧸ AddSubgroup.zmultiples 1)) : ℂ)
h₂ : bernoulliFun k y = periodizedBernoulli k (↑y : ℝ ⧸ AddSubgroup.zmultiples 1)
h₃ : bernoulliFun k y = AddCircle.liftIco 1 0 (bernoulliFun k) (↑y : ℝ ⧸ AddSubgroup.zmultiples 1)
⊢ (↑(bernoulliFun k y) : ℂ) = (B : (a : AddCircle 1) → ℂ) (↑y : ℝ ⧸ AddSubgroup.zmultiples 1)
have h₄ : bernoulliFun k y = bernoulliFun k y
Tactic: rw [ContinuousMap.coe_mk, Function.comp_apply, ofReal_inj, periodizedBernoulli,
      AddCircle.liftIco_coe_apply (show y ∈ Ico 0 (0 + 1) by rwa [zero_add])]
-- Lean 4 proof step from Mathlib.NumberTheory.ZetaValues (goal)
Goal:
k : ℕ
hk : 2 ≤ k
x : ℝ
hx : Membership.mem (γ := Set ℝ) (Icc 0 1) x
⊢ HasSum
    (fun (n : ℕ) ↦
      1 / HPow.hPow (α := ℂ) (↑n : ℂ) k *
        HAdd.hAdd (α := ℂ) ((fourier (↑n : ℤ) : (a : AddCircle 1) → ℂ) (↑x : ℝ ⧸ AddSubgroup.zmultiples 1) : ℂ)
          (HMul.hMul (β := ℂ) (HPow.hPow (α := ℂ) (-1) k)
            ((fourier (-(↑n : ℤ)) : (a : AddCircle 1) → ℂ) (↑x : ℝ ⧸ AddSubgroup.zmultiples 1) : ℂ)))
    (-(2 * (↑π : ℂ) * I) ^ k / (↑k ! : ℂ) * (↑(bernoulliFun k x) : ℂ))
have h : Eq (α := ℕ → ℂ)
  (fun (n : ℕ) ↦
    1 / HPow.hPow (α := ℂ) (↑n : ℂ) k *
      HAdd.hAdd (α := ℂ) ((fourier (↑n : ℤ) : (a : AddCircle 1) → ℂ) (↑x : ℝ ⧸ AddSubgroup.zmultiples 1) : ℂ)
        (HMul.hMul (β := ℂ) (HPow.hPow (α := ℂ) (-1) k)
          ((fourier (-(↑n : ℤ)) : (a : AddCircle 1) → ℂ) (↑x : ℝ ⧸ AddSubgroup.zmultiples 1) : ℂ)))
  fun (n : ℕ) ↦
  HMul.hMul (β := ℂ) (1 / HPow.hPow (α := ℂ) (↑(↑n : ℤ) : ℂ) k)
      ((fourier (↑n : ℤ) : (a : AddCircle 1) → ℂ) (↑x : ℝ ⧸ AddSubgroup.zmultiples 1) : ℂ) +
    HMul.hMul (β := ℂ) (1 / HPow.hPow (α := ℂ) (↑(-(↑n : ℤ)) : ℂ) k)
      ((fourier (-(↑n : ℤ)) : (a : AddCircle 1) → ℂ) (↑x : ℝ ⧸ AddSubgroup.zmultiples 1) : ℂ)
Tactic: convert! (hasSum_one_div_pow_mul_fourier_mul_bernoulliFun hk hx).nat_add_neg using 1
-- Lean 4 proof step from Mathlib.NumberTheory.ZetaValues (goal)
Goal:
k : ℕ
hk : 2 ≤ k
x : ℝ
hx : Membership.mem (γ := Set ℝ) (Icc 0 1) x
h :
  Eq (α := ℕ → ℂ)
    (fun (n : ℕ) ↦
      1 / HPow.hPow (α := ℂ) (↑n : ℂ) k *
        HAdd.hAdd (α := ℂ) ((fourier (↑n : ℤ) : (a : AddCircle 1) → ℂ) (↑x : ℝ ⧸ AddSubgroup.zmultiples 1) : ℂ)
          (HMul.hMul (β := ℂ) (HPow.hPow (α := ℂ) (-1) k)
            ((fourier (-(↑n : ℤ)) : (a : AddCircle 1) → ℂ) (↑x : ℝ ⧸ AddSubgroup.zmultiples 1) : ℂ)))
    fun (n : ℕ) ↦
    HMul.hMul (β := ℂ) (1 / HPow.hPow (α := ℂ) (↑(↑n : ℤ) : ℂ) k)
        ((fourier (↑n : ℤ) : (a : AddCircle 1) → ℂ) (↑x : ℝ ⧸ AddSubgroup.zmultiples 1) : ℂ) +
      HMul.hMul (β := ℂ) (1 / HPow.hPow (α := ℂ) (↑(-(↑n : ℤ)) : ℂ) k)
        ((fourier (-(↑n : ℤ)) : (a : AddCircle 1) → ℂ) (↑x : ℝ ⧸ AddSubgroup.zmultiples 1) : ℂ)
⊢ HasSum
    (fun (n : ℕ) ↦
      1 / HPow.hPow (α := ℂ) (↑n : ℂ) k *
        HAdd.hAdd (α := ℂ) ((fourier (↑n : ℤ) : (a : AddCircle 1) → ℂ) (↑x : ℝ ⧸ AddSubgroup.zmultiples 1) : ℂ)
          (HMul.hMul (β := ℂ) (HPow.hPow (α := ℂ) (-1) k)
            ((fourier (-(↑n : ℤ)) : (a : AddCircle 1) → ℂ) (↑x : ℝ ⧸ AddSubgroup.zmultiples 1) : ℂ)))
    (-(2 * (↑π : ℂ) * I) ^ k / (↑k ! : ℂ) * (↑(bernoulliFun k x) : ℂ))
have h₁ : -(2 * (↑π : ℂ) * I) ^ k / (↑k ! : ℂ) * (↑(bernoulliFun k x) : ℂ) =
  -(2 * (↑π : ℂ) * I) ^ k / (↑k ! : ℂ) * (↑(bernoulliFun k x) : ℂ) +
    HMul.hMul (β := ℂ) (1 / HPow.hPow (α := ℂ) (↑0 : ℂ) k)
      ((fourier 0 : (a : AddCircle 1) → ℂ) (↑x : ℝ ⧸ AddSubgroup.zmultiples 1) : ℂ)
Tactic: convert! (hasSum_one_div_pow_mul_fourier_mul_bernoulliFun hk hx).nat_add_neg using 1
-- Lean 4 proof step from Mathlib.NumberTheory.ZetaValues (goal)
Goal:
k : ℕ
hk : 2 ≤ k
x : ℝ
hx : Membership.mem (γ := Set ℝ) (Icc 0 1) x
⊢ Eq (α := ℕ → ℂ)
    (fun (n : ℕ) ↦
      1 / HPow.hPow (α := ℂ) (↑n : ℂ) k *
        HAdd.hAdd (α := ℂ) ((fourier (↑n : ℤ) : (a : AddCircle 1) → ℂ) (↑x : ℝ ⧸ AddSubgroup.zmultiples 1) : ℂ)
          (HMul.hMul (β := ℂ) (HPow.hPow (α := ℂ) (-1) k)
            ((fourier (-(↑n : ℤ)) : (a : AddCircle 1) → ℂ) (↑x : ℝ ⧸ AddSubgroup.zmultiples 1) : ℂ)))
    fun (n : ℕ) ↦
    HMul.hMul (β := ℂ) (1 / HPow.hPow (α := ℂ) (↑(↑n : ℤ) : ℂ) k)
        ((fourier (↑n : ℤ) : (a : AddCircle 1) → ℂ) (↑x : ℝ ⧸ AddSubgroup.zmultiples 1) : ℂ) +
      HMul.hMul (β := ℂ) (1 / HPow.hPow (α := ℂ) (↑(-(↑n : ℤ)) : ℂ) k)
        ((fourier (-(↑n : ℤ)) : (a : AddCircle 1) → ℂ) (↑x : ℝ ⧸ AddSubgroup.zmultiples 1) : ℂ)
have h : ∀ (n : ℕ),
  1 / HPow.hPow (α := ℂ) (↑n : ℂ) k *
      HAdd.hAdd (α := ℂ) ((fourier (↑n : ℤ) : (a : AddCircle 1) → ℂ) (↑x : ℝ ⧸ AddSubgroup.zmultiples 1) : ℂ)
        (HMul.hMul (β := ℂ) (HPow.hPow (α := ℂ) (-1) k)
          ((fourier (-(↑n : ℤ)) : (a : AddCircle 1) → ℂ) (↑x : ℝ ⧸ AddSubgroup.zmultiples 1) : ℂ)) =
    HMul.hMul (β := ℂ) (1 / HPow.hPow (α := ℂ) (↑(↑n : ℤ) : ℂ) k)
        ((fourier (↑n : ℤ) : (a : AddCircle 1) → ℂ) (↑x : ℝ ⧸ AddSubgroup.zmultiples 1) : ℂ) +
      HMul.hMul (β := ℂ) (1 / HPow.hPow (α := ℂ) (↑(-(↑n : ℤ)) : ℂ) k)
        ((fourier (-(↑n : ℤ)) : (a : AddCircle 1) → ℂ) (↑x : ℝ ⧸ AddSubgroup.zmultiples 1) : ℂ)
Tactic: ext1 n
-- Lean 4 proof step from Mathlib.NumberTheory.ZetaValues (goal)
Goal:
k : ℕ
hk : 2 ≤ k
x : ℝ
hx : Membership.mem (γ := Set ℝ) (Icc 0 1) x
n : ℕ
⊢ 1 / HPow.hPow (α := ℂ) (↑n : ℂ) k *
      HAdd.hAdd (α := ℂ) ((fourier (↑n : ℤ) : (a : AddCircle 1) → ℂ) (↑x : ℝ ⧸ AddSubgroup.zmultiples 1) : ℂ)
        (HMul.hMul (β := ℂ) (HPow.hPow (α := ℂ) (-1) k)
          ((fourier (-(↑n : ℤ)) : (a : AddCircle 1) → ℂ) (↑x : ℝ ⧸ AddSubgroup.zmultiples 1) : ℂ)) =
    HMul.hMul (β := ℂ) (1 / HPow.hPow (α := ℂ) (↑(↑n : ℤ) : ℂ) k)
        ((fourier (↑n : ℤ) : (a : AddCircle 1) → ℂ) (↑x : ℝ ⧸ AddSubgroup.zmultiples 1) : ℂ) +
      HMul.hMul (β := ℂ) (1 / HPow.hPow (α := ℂ) (↑(-(↑n : ℤ)) : ℂ) k)
        ((fourier (-(↑n : ℤ)) : (a : AddCircle 1) → ℂ) (↑x : ℝ ⧸ AddSubgroup.zmultiples 1) : ℂ)
have h : HMul.hMul (β := ℂ) (1 / HPow.hPow (α := ℂ) (↑n : ℂ) k)
      ((fourier (↑n : ℤ) : (a : AddCircle 1) → ℂ) (↑x : ℝ ⧸ AddSubgroup.zmultiples 1) : ℂ) +
    HMul.hMul (β := ℂ) (1 / HPow.hPow (α := ℂ) (↑n : ℂ) k * HPow.hPow (α := ℂ) (-1) k)
      ((fourier (-(↑n : ℤ)) : (a : AddCircle 1) → ℂ) (↑x : ℝ ⧸ AddSubgroup.zmultiples 1) : ℂ) =
  HMul.hMul (β := ℂ) (1 / HPow.hPow (α := ℂ) (↑(↑n : ℤ) : ℂ) k)
      ((fourier (↑n : ℤ) : (a : AddCircle 1) → ℂ) (↑x : ℝ ⧸ AddSubgroup.zmultiples 1) : ℂ) +
    HMul.hMul (β := ℂ) (1 / HPow.hPow (α := ℂ) (-(↑(↑n : ℤ) : ℂ)) k)
      ((fourier (-(↑n : ℤ)) : (a : AddCircle 1) → ℂ) (↑x : ℝ ⧸ AddSubgroup.zmultiples 1) : ℂ)
Tactic: rw [Int.cast_neg, mul_add, ← mul_assoc]
-- Lean 4 proof step from Mathlib.NumberTheory.ZetaValues (goal)
Goal:
k : ℕ
hk : 2 ≤ k
x : ℝ
hx : Membership.mem (γ := Set ℝ) (Icc 0 1) x
n : ℕ
| HMul.hMul (β := ℂ) (1 / HPow.hPow (α := ℂ) (↑(↑n : ℤ) : ℂ) k)
      ((fourier (↑n : ℤ) : (a : AddCircle 1) → ℂ) (↑x : ℝ ⧸ AddSubgroup.zmultiples 1) : ℂ) +
    HMul.hMul (β := ℂ) (1 / HPow.hPow (α := ℂ) (-(↑(↑n : ℤ) : ℂ)) k)
      ((fourier (-(↑n : ℤ)) : (a : AddCircle 1) → ℂ) (↑x : ℝ ⧸ AddSubgroup.zmultiples 1) : ℂ)
have rw : HMul.hMul (β := ℂ) (1 / HPow.hPow (α := ℂ) (↑(↑n : ℤ) : ℂ) k)
    ((fourier (↑n : ℤ) : (a : AddCircle 1) → ℂ) (↑x : ℝ ⧸ AddSubgroup.zmultiples 1) : ℂ) +
  HMul.hMul (β := ℂ) (1 / HPow.hPow (α := ℂ) (-1) k / HPow.hPow (α := ℂ) (↑(↑n : ℤ) : ℂ) k)
    ((fourier (-(↑n : ℤ)) : (a : AddCircle 1) → ℂ) (↑x : ℝ ⧸ AddSubgroup.zmultiples 1) : ℂ)
Tactic: rw [neg_eq_neg_one_mul, mul_pow, ← div_div]
-- Lean 4 proof step from Mathlib.NumberTheory.ZetaValues (goal)
Goal:
k : ℕ
hk : 2 ≤ k
x : ℝ
hx : Membership.mem (γ := Set ℝ) (Icc 0 1) x
n : ℕ
⊢ HMul.hMul (β := ℂ) (1 / HPow.hPow (α := ℂ) (↑n : ℂ) k)
        ((fourier (↑n : ℤ) : (a : AddCircle 1) → ℂ) (↑x : ℝ ⧸ AddSubgroup.zmultiples 1) : ℂ) +
      HMul.hMul (β := ℂ) (1 / HPow.hPow (α := ℂ) (↑n : ℂ) k * HPow.hPow (α := ℂ) (-1) k)
        ((fourier (-(↑n : ℤ)) : (a : AddCircle 1) → ℂ) (↑x : ℝ ⧸ AddSubgroup.zmultiples 1) : ℂ) =
    HMul.hMul (β := ℂ) (1 / HPow.hPow (α := ℂ) (↑(↑n : ℤ) : ℂ) k)
        ((fourier (↑n : ℤ) : (a : AddCircle 1) → ℂ) (↑x : ℝ ⧸ AddSubgroup.zmultiples 1) : ℂ) +
      HMul.hMul (β := ℂ) (1 / HPow.hPow (α := ℂ) (-(↑(↑n : ℤ) : ℂ)) k)
        ((fourier (-(↑n : ℤ)) : (a : AddCircle 1) → ℂ) (↑x : ℝ ⧸ AddSubgroup.zmultiples 1) : ℂ)
have h : HMul.hMul (β := ℂ) (1 / HPow.hPow (α := ℂ) (↑n : ℂ) k)
      ((fourier (↑n : ℤ) : (a : AddCircle 1) → ℂ) (↑x : ℝ ⧸ AddSubgroup.zmultiples 1) : ℂ) +
    HMul.hMul (β := ℂ) (1 / HPow.hPow (α := ℂ) (↑n : ℂ) k * HPow.hPow (α := ℂ) (-1) k)
      ((fourier (-(↑n : ℤ)) : (a : AddCircle 1) → ℂ) (↑x : ℝ ⧸ AddSubgroup.zmultiples 1) : ℂ) =
  HMul.hMul (β := ℂ) (1 / HPow.hPow (α := ℂ) (↑(↑n : ℤ) : ℂ) k)
      ((fourier (↑n : ℤ) : (a : AddCircle 1) → ℂ) (↑x : ℝ ⧸ AddSubgroup.zmultiples 1) : ℂ) +
    HMul.hMul (β := ℂ) (1 / HPow.hPow (α := ℂ) (-1) k / HPow.hPow (α := ℂ) (↑(↑n : ℤ) : ℂ) k)
      ((fourier (-(↑n : ℤ)) : (a : AddCircle 1) → ℂ) (↑x : ℝ ⧸ AddSubgroup.zmultiples 1) : ℂ)
Tactic: conv_rhs => rw [neg_eq_neg_one_mul, mul_pow, ← div_div]
-- Lean 4 proof step from Mathlib.NumberTheory.ZetaValues (goal)
Goal:
k : ℕ
hk : 2 ≤ k
x : ℝ
hx : Membership.mem (γ := Set ℝ) (Icc 0 1) x
n : ℕ
⊢ HMul.hMul (β := ℂ) (1 / HPow.hPow (α := ℂ) (↑n : ℂ) k)
        ((fourier (↑n : ℤ) : (a : AddCircle 1) → ℂ) (↑x : ℝ ⧸ AddSubgroup.zmultiples 1) : ℂ) +
      HMul.hMul (β := ℂ) (1 / HPow.hPow (α := ℂ) (↑n : ℂ) k * HPow.hPow (α := ℂ) (-1) k)
        ((fourier (-(↑n : ℤ)) : (a : AddCircle 1) → ℂ) (↑x : ℝ ⧸ AddSubgroup.zmultiples 1) : ℂ) =
    HMul.hMul (β := ℂ) (1 / HPow.hPow (α := ℂ) (↑(↑n : ℤ) : ℂ) k)
        ((fourier (↑n : ℤ) : (a : AddCircle 1) → ℂ) (↑x : ℝ ⧸ AddSubgroup.zmultiples 1) : ℂ) +
      HMul.hMul (β := ℂ) (1 / HPow.hPow (α := ℂ) (-1) k / HPow.hPow (α := ℂ) (↑(↑n : ℤ) : ℂ) k)
        ((fourier (-(↑n : ℤ)) : (a : AddCircle 1) → ℂ) (↑x : ℝ ⧸ AddSubgroup.zmultiples 1) : ℂ)
have h : 1 / HPow.hPow (α := ℂ) (↑n : ℂ) k * HPow.hPow (α := ℂ) (-1) k =
  1 / HPow.hPow (α := ℂ) (-1) k / HPow.hPow (α := ℂ) (↑(↑n : ℤ) : ℂ) k
Tactic: congr 2
-- Lean 4 proof step from Mathlib.NumberTheory.ZetaValues (goal)
Goal:
k : ℕ
hk : 2 ≤ k
x : ℝ
hx : Membership.mem (γ := Set ℝ) (Icc 0 1) x
n : ℕ
⊢ 1 / HPow.hPow (α := ℂ) (↑n : ℂ) k * HPow.hPow (α := ℂ) (-1) k =
    1 / HPow.hPow (α := ℂ) (-1) k / HPow.hPow (α := ℂ) (↑(↑n : ℤ) : ℂ) k
have h : HPow.hPow (α := ℂ) (-1) k / HPow.hPow (α := ℂ) (↑n : ℂ) k =
  1 / HPow.hPow (α := ℂ) (-1) k / HPow.hPow (α := ℂ) (↑(↑n : ℤ) : ℂ) k
Tactic: rw [div_mul_eq_mul_div₀, one_mul]
-- Lean 4 proof step from Mathlib.NumberTheory.ZetaValues (goal)
Goal:
k : ℕ
hk : 2 ≤ k
x : ℝ
hx : Membership.mem (γ := Set ℝ) (Icc 0 1) x
n : ℕ
⊢ HPow.hPow (α := ℂ) (-1) k / HPow.hPow (α := ℂ) (↑n : ℂ) k =
    1 / HPow.hPow (α := ℂ) (-1) k / HPow.hPow (α := ℂ) (↑(↑n : ℤ) : ℂ) k
have h : HPow.hPow (α := ℂ) (-1) k = 1 / HPow.hPow (α := ℂ) (-1) k
Tactic: congr 1
-- Lean 4 proof step from Mathlib.NumberTheory.ZetaValues (goal)
Goal:
k : ℕ
hk : 2 ≤ k
x : ℝ
hx : Membership.mem (γ := Set ℝ) (Icc 0 1) x
n : ℕ
⊢ HPow.hPow (α := ℂ) (-1) k = 1 / HPow.hPow (α := ℂ) (-1) k
have h : HPow.hPow (α := ℂ) (-1) k ≠ 0
Tactic: rw [eq_div_iff, ← mul_pow, ← neg_eq_neg_one_mul, neg_neg, one_pow]
-- Lean 4 proof step from Mathlib.NumberTheory.ZetaValues (goal)
Goal:
k : ℕ
hk : 2 ≤ k
x : ℝ
hx : Membership.mem (γ := Set ℝ) (Icc 0 1) x
n : ℕ
⊢ HPow.hPow (α := ℂ) (-1) k ≠ 0
have h : Ne (α := ℂ) (-1) 0
Tactic: apply pow_ne_zero
-- Lean 4 proof step from Mathlib.NumberTheory.ZetaValues (goal)
Goal:
k : ℕ
hk : 2 ≤ k
x : ℝ
hx : Membership.mem (γ := Set ℝ) (Icc 0 1) x
n : ℕ
⊢ Ne (α := ℂ) (-1) 0
have h : Ne (α := ℂ) 1 0
Tactic: rw [neg_ne_zero]
-- Lean 4 proof step from Mathlib.NumberTheory.ZetaValues (goal)
Goal:
k : ℕ
hk : 2 ≤ k
x : ℝ
hx : Membership.mem (γ := Set ℝ) (Icc 0 1) x
⊢ -(2 * (↑π : ℂ) * I) ^ k / (↑k ! : ℂ) * (↑(bernoulliFun k x) : ℂ) =
    -(2 * (↑π : ℂ) * I) ^ k / (↑k ! : ℂ) * (↑(bernoulliFun k x) : ℂ) +
      HMul.hMul (β := ℂ) (1 / HPow.hPow (α := ℂ) (↑0 : ℂ) k)
        ((fourier 0 : (a : AddCircle 1) → ℂ) (↑x : ℝ ⧸ AddSubgroup.zmultiples 1) : ℂ)
have h : -(2 * (↑π : ℂ) * I) ^ k / (↑k ! : ℂ) * (↑(bernoulliFun k x) : ℂ) =
  -(2 * (↑π : ℂ) * I) ^ k / (↑k ! : ℂ) * (↑(bernoulliFun k x) : ℂ) +
    HMul.hMul (β := ℂ) (1 / HPow.hPow (α := ℂ) 0 k)
      ((fourier 0 : (a : AddCircle 1) → ℂ) (↑x : ℝ ⧸ AddSubgroup.zmultiples 1) : ℂ)
Tactic: rw [Int.cast_zero, zero_pow (by positivity : k ≠ 0), div_zero, zero_mul, add_zero]
-- Lean 4 proof step from Mathlib.NumberTheory.ZetaValues (goal)
Goal:
k : ℕ
hk : 2 ≤ k
x : ℝ
hx : Membership.mem (γ := Set ℝ) (Icc 0 1) x
h :
  -(2 * (↑π : ℂ) * I) ^ k / (↑k ! : ℂ) * (↑(bernoulliFun k x) : ℂ) =
    -(2 * (↑π : ℂ) * I) ^ k / (↑k ! : ℂ) * (↑(bernoulliFun k x) : ℂ) +
      HMul.hMul (β := ℂ) (1 / HPow.hPow (α := ℂ) 0 k)
        ((fourier 0 : (a : AddCircle 1) → ℂ) (↑x : ℝ ⧸ AddSubgroup.zmultiples 1) : ℂ)
⊢ -(2 * (↑π : ℂ) * I) ^ k / (↑k ! : ℂ) * (↑(bernoulliFun k x) : ℂ) =
    -(2 * (↑π : ℂ) * I) ^ k / (↑k ! : ℂ) * (↑(bernoulliFun k x) : ℂ) +
      HMul.hMul (β := ℂ) (1 / HPow.hPow (α := ℂ) (↑0 : ℂ) k)
        ((fourier 0 : (a : AddCircle 1) → ℂ) (↑x : ℝ ⧸ AddSubgroup.zmultiples 1) : ℂ)
have h₁ : -(2 * (↑π : ℂ) * I) ^ k / (↑k ! : ℂ) * (↑(bernoulliFun k x) : ℂ) =
  -(2 * (↑π : ℂ) * I) ^ k / (↑k ! : ℂ) * (↑(bernoulliFun k x) : ℂ) +
    HMul.hMul (β := ℂ) (1 / 0) ((fourier 0 : (a : AddCircle 1) → ℂ) (↑x : ℝ ⧸ AddSubgroup.zmultiples 1) : ℂ)
Tactic: rw [Int.cast_zero, zero_pow (by positivity : k ≠ 0), div_zero, zero_mul, add_zero]
-- Lean 4 proof step from Mathlib.NumberTheory.ZetaValues (goal)
Goal:
k : ℕ
hk : 2 ≤ k
x : ℝ
hx : Membership.mem (γ := Set ℝ) (Icc 0 1) x
h :
  -(2 * (↑π : ℂ) * I) ^ k / (↑k ! : ℂ) * (↑(bernoulliFun k x) : ℂ) =
    -(2 * (↑π : ℂ) * I) ^ k / (↑k ! : ℂ) * (↑(bernoulliFun k x) : ℂ) +
      HMul.hMul (β := ℂ) (1 / HPow.hPow (α := ℂ) 0 k)
        ((fourier 0 : (a : AddCircle 1) → ℂ) (↑x : ℝ ⧸ AddSubgroup.zmultiples 1) : ℂ)
h₁ :
  -(2 * (↑π : ℂ) * I) ^ k / (↑k ! : ℂ) * (↑(bernoulliFun k x) : ℂ) =
    -(2 * (↑π : ℂ) * I) ^ k / (↑k ! : ℂ) * (↑(bernoulliFun k x) : ℂ) +
      HMul.hMul (β := ℂ) (1 / 0) ((fourier 0 : (a : AddCircle 1) → ℂ) (↑x : ℝ ⧸ AddSubgroup.zmultiples 1) : ℂ)
⊢ -(2 * (↑π : ℂ) * I) ^ k / (↑k ! : ℂ) * (↑(bernoulliFun k x) : ℂ) =
    -(2 * (↑π : ℂ) * I) ^ k / (↑k ! : ℂ) * (↑(bernoulliFun k x) : ℂ) +
      HMul.hMul (β := ℂ) (1 / HPow.hPow (α := ℂ) (↑0 : ℂ) k)
        ((fourier 0 : (a : AddCircle 1) → ℂ) (↑x : ℝ ⧸ AddSubgroup.zmultiples 1) : ℂ)
have h₂ : -(2 * (↑π : ℂ) * I) ^ k / (↑k ! : ℂ) * (↑(bernoulliFun k x) : ℂ) =
  -(2 * (↑π : ℂ) * I) ^ k / (↑k ! : ℂ) * (↑(bernoulliFun k x) : ℂ) +
    HMul.hMul (α := ℂ) (β := ℂ) (0 : ℂ) ((fourier 0 : (a : AddCircle 1) → ℂ) (↑x : ℝ ⧸ AddSubgroup.zmultiples 1) : ℂ)
Tactic: rw [Int.cast_zero, zero_pow (by positivity : k ≠ 0), div_zero, zero_mul, add_zero]
-- Lean 4 proof step from Mathlib.NumberTheory.ZetaValues (goal)
Goal:
k : ℕ
hk : 2 ≤ k
x : ℝ
hx : Membership.mem (γ := Set ℝ) (Icc 0 1) x
h :
  -(2 * (↑π : ℂ) * I) ^ k / (↑k ! : ℂ) * (↑(bernoulliFun k x) : ℂ) =
    -(2 * (↑π : ℂ) * I) ^ k / (↑k ! : ℂ) * (↑(bernoulliFun k x) : ℂ) +
      HMul.hMul (β := ℂ) (1 / HPow.hPow (α := ℂ) 0 k)
        ((fourier 0 : (a : AddCircle 1) → ℂ) (↑x : ℝ ⧸ AddSubgroup.zmultiples 1) : ℂ)
h₁ :
  -(2 * (↑π : ℂ) * I) ^ k / (↑k ! : ℂ) * (↑(bernoulliFun k x) : ℂ) =
    -(2 * (↑π : ℂ) * I) ^ k / (↑k ! : ℂ) * (↑(bernoulliFun k x) : ℂ) +
      HMul.hMul (β := ℂ) (1 / 0) ((fourier 0 : (a : AddCircle 1) → ℂ) (↑x : ℝ ⧸ AddSubgroup.zmultiples 1) : ℂ)
h₂ :
  -(2 * (↑π : ℂ) * I) ^ k / (↑k ! : ℂ) * (↑(bernoulliFun k x) : ℂ) =
    -(2 * (↑π : ℂ) * I) ^ k / (↑k ! : ℂ) * (↑(bernoulliFun k x) : ℂ) +
      HMul.hMul (α := ℂ) (β := ℂ) (0 : ℂ) ((fourier 0 : (a : AddCircle 1) → ℂ) (↑x : ℝ ⧸ AddSubgroup.zmultiples 1) : ℂ)
⊢ -(2 * (↑π : ℂ) * I) ^ k / (↑k ! : ℂ) * (↑(bernoulliFun k x) : ℂ) =
    -(2 * (↑π : ℂ) * I) ^ k / (↑k ! : ℂ) * (↑(bernoulliFun k x) : ℂ) +
      HMul.hMul (β := ℂ) (1 / HPow.hPow (α := ℂ) (↑0 : ℂ) k)
        ((fourier 0 : (a : AddCircle 1) → ℂ) (↑x : ℝ ⧸ AddSubgroup.zmultiples 1) : ℂ)
have h₃ : -(2 * (↑π : ℂ) * I) ^ k / (↑k ! : ℂ) * (↑(bernoulliFun k x) : ℂ) =
  -(2 * (↑π : ℂ) * I) ^ k / (↑k ! : ℂ) * (↑(bernoulliFun k x) : ℂ) + 0
Tactic: rw [Int.cast_zero, zero_pow (by positivity : k ≠ 0), div_zero, zero_mul, add_zero]
-- Lean 4 proof step from Mathlib.NumberTheory.ZetaValues (goal)
Goal:
k : ℕ
hk : 2 ≤ k
x : ℝ
hx : Membership.mem (γ := Set ℝ) (Icc 0 1) x
h :
  -(2 * (↑π : ℂ) * I) ^ k / (↑k ! : ℂ) * (↑(bernoulliFun k x) : ℂ) =
    -(2 * (↑π : ℂ) * I) ^ k / (↑k ! : ℂ) * (↑(bernoulliFun k x) : ℂ) +
      HMul.hMul (β := ℂ) (1 / HPow.hPow (α := ℂ) 0 k)
        ((fourier 0 : (a : AddCircle 1) → ℂ) (↑x : ℝ ⧸ AddSubgroup.zmultiples 1) : ℂ)
h₁ :
  -(2 * (↑π : ℂ) * I) ^ k / (↑k ! : ℂ) * (↑(bernoulliFun k x) : ℂ) =
    -(2 * (↑π : ℂ) * I) ^ k / (↑k ! : ℂ) * (↑(bernoulliFun k x) : ℂ) +
      HMul.hMul (β := ℂ) (1 / 0) ((fourier 0 : (a : AddCircle 1) → ℂ) (↑x : ℝ ⧸ AddSubgroup.zmultiples 1) : ℂ)
h₂ :
  -(2 * (↑π : ℂ) * I) ^ k / (↑k ! : ℂ) * (↑(bernoulliFun k x) : ℂ) =
    -(2 * (↑π : ℂ) * I) ^ k / (↑k ! : ℂ) * (↑(bernoulliFun k x) : ℂ) +
      HMul.hMul (α := ℂ) (β := ℂ) (0 : ℂ) ((fourier 0 : (a : AddCircle 1) → ℂ) (↑x : ℝ ⧸ AddSubgroup.zmultiples 1) : ℂ)
h₃ :
  -(2 * (↑π : ℂ) * I) ^ k / (↑k ! : ℂ) * (↑(bernoulliFun k x) : ℂ) =
    -(2 * (↑π : ℂ) * I) ^ k / (↑k ! : ℂ) * (↑(bernoulliFun k x) : ℂ) + 0
⊢ -(2 * (↑π : ℂ) * I) ^ k / (↑k ! : ℂ) * (↑(bernoulliFun k x) : ℂ) =
    -(2 * (↑π : ℂ) * I) ^ k / (↑k ! : ℂ) * (↑(bernoulliFun k x) : ℂ) +
      HMul.hMul (β := ℂ) (1 / HPow.hPow (α := ℂ) (↑0 : ℂ) k)
        ((fourier 0 : (a : AddCircle 1) → ℂ) (↑x : ℝ ⧸ AddSubgroup.zmultiples 1) : ℂ)
have h₄ : -(2 * (↑π : ℂ) * I) ^ k / (↑k ! : ℂ) * (↑(bernoulliFun k x) : ℂ) =
  -(2 * (↑π : ℂ) * I) ^ k / (↑k ! : ℂ) * (↑(bernoulliFun k x) : ℂ)
Tactic: rw [Int.cast_zero, zero_pow (by positivity : k ≠ 0), div_zero, zero_mul, add_zero]
-- 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
⊢ HasSum
    (fun (n : ℕ) ↦
      1 / HPow.hPow (α := ℂ) (↑n : ℂ) (2 * k) *
        HAdd.hAdd (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) * (2 * (↑π : ℂ)) ^ (2 * k) / (↑(2 * k)! : ℂ) * (↑(bernoulliFun (2 * k) x) : ℂ))
have h : ∀ (x_1 : ℕ),
  HAdd.hAdd (α := ℂ) (β := ℂ) ((fourier (↑x_1 : ℤ) : (a : AddCircle 1) → ℂ) (↑x : ℝ ⧸ AddSubgroup.zmultiples 1) : ℂ)
      ((fourier (-(↑x_1 : ℤ)) : (a : AddCircle 1) → ℂ) (↑x : ℝ ⧸ AddSubgroup.zmultiples 1) : ℂ) =
    HAdd.hAdd (α := ℂ) ((fourier (↑x_1 : ℤ) : (a : AddCircle 1) → ℂ) (↑x : ℝ ⧸ AddSubgroup.zmultiples 1) : ℂ)
      (HMul.hMul (β := ℂ) (HPow.hPow (α := ℂ) (-1) (2 * k))
        ((fourier (-(↑x_1 : ℤ)) : (a : AddCircle 1) → ℂ) (↑x : ℝ ⧸ AddSubgroup.zmultiples 1) : ℂ))
Tactic: convert! hasSum_one_div_nat_pow_mul_fourier (by lia : 2 ≤ 2 * k) hx using 3