awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.NumberTheory.ZetaValues (goal)
Goal:
m : ℕ
m0 : m ≠ 0
x : ℝ
m0' : Ne (α := ℝ) (↑m : ℝ) 0
f : (k : ℕ) → (x : ℝ) → ℝ :=
  fun (k : ℕ) (x : ℝ) ↦
    bernoulliFun k ((↑m : ℝ) * x) -
      HPow.hPow (α := ℝ) (↑m : ℝ) k / (↑m : ℝ) *
        ∑ i ∈ Finset.range m, bernoulliFun k (x + HDiv.hDiv (α := ℝ) (↑i : ℝ) (↑m : ℝ))
k : ℕ
h : ∀ (x : ℝ), f k x = 0
d : ∀ (x : ℝ), HasDerivAt (f (k + 1)) 0 x
c : ℝ
fc : ∀ (x : ℝ), f (k + 1) x = c
rw :
  HPow.hPow (α := ℝ) (↑m : ℝ) (k + 1) / (↑m : ℝ) = 0 ∨
    Eq (α := ℝ)
      (∫ (x : ℝ) in 0..(↑m : ℝ)⁻¹,
        ∑ i ∈ Finset.range m, bernoulliFun (k + 1) (x + HDiv.hDiv (α := ℝ) (↑i : ℝ) (↑m : ℝ)))
      0
hf : IntervalIntegrable (ε := ℝ) (fun (x : ℝ) ↦ bernoulliFun (k + 1) ((↑m : ℝ) * x)) volume 0 (↑m : ℝ)⁻¹
⊢ Eq (α := ℝ)
    (∫ (x : ℝ) in 0..(↑m : ℝ)⁻¹,
      bernoulliFun (k + 1) ((↑m : ℝ) * x) -
        HPow.hPow (α := ℝ) (↑m : ℝ) (k + 1) / (↑m : ℝ) *
          ∑ i ∈ Finset.range m, bernoulliFun (k + 1) (x + HDiv.hDiv (α := ℝ) (↑i : ℝ) (↑m : ℝ)))
    0
have hg : IntervalIntegrable (ε := ℝ)
  (fun (x : ℝ) ↦
    HPow.hPow (α := ℝ) (↑m : ℝ) (k + 1) / (↑m : ℝ) *
      ∑ i ∈ Finset.range m, bernoulliFun (k + 1) (x + HDiv.hDiv (α := ℝ) (↑i : ℝ) (↑m : ℝ)))
  volume 0 (↑m : ℝ)⁻¹
Tactic: rw [intervalIntegral.integral_sub, intervalIntegral.integral_comp_mul_left _ m0', mul_zero,
        mul_inv_cancel₀ m0', integral_bernoulliFun_eq_zero (by lia), smul_zero, sub_eq_zero,
        intervalIntegral.integral_const_mul, eq_comm (a := 0), mul_eq_zero]
-- Lean 4 proof step from Mathlib.NumberTheory.ZetaValues (goal)
Goal:
m : ℕ
m0 : m ≠ 0
x : ℝ
m0' : Ne (α := ℝ) (↑m : ℝ) 0
f : (k : ℕ) → (x : ℝ) → ℝ :=
  fun (k : ℕ) (x : ℝ) ↦
    bernoulliFun k ((↑m : ℝ) * x) -
      HPow.hPow (α := ℝ) (↑m : ℝ) k / (↑m : ℝ) *
        ∑ i ∈ Finset.range m, bernoulliFun k (x + HDiv.hDiv (α := ℝ) (↑i : ℝ) (↑m : ℝ))
k : ℕ
h : ∀ (x : ℝ), f k x = 0
d : ∀ (x : ℝ), HasDerivAt (f (k + 1)) 0 x
c : ℝ
fc : ∀ (x : ℝ), f (k + 1) x = c
⊢ HPow.hPow (α := ℝ) (↑m : ℝ) (k + 1) / (↑m : ℝ) = 0 ∨
    Eq (α := ℝ)
      (∫ (x : ℝ) in 0..(↑m : ℝ)⁻¹,
        ∑ i ∈ Finset.range m, bernoulliFun (k + 1) (x + HDiv.hDiv (α := ℝ) (↑i : ℝ) (↑m : ℝ)))
      0
have h₁ : Eq (α := ℝ)
  (∫ (x : ℝ) in 0..(↑m : ℝ)⁻¹, ∑ i ∈ Finset.range m, bernoulliFun (k + 1) (x + HDiv.hDiv (α := ℝ) (↑i : ℝ) (↑m : ℝ))) 0
Tactic: right
-- Lean 4 proof step from Mathlib.NumberTheory.ZetaValues (goal)
Goal:
m : ℕ
m0 : m ≠ 0
x : ℝ
m0' : Ne (α := ℝ) (↑m : ℝ) 0
f : (k : ℕ) → (x : ℝ) → ℝ :=
  fun (k : ℕ) (x : ℝ) ↦
    bernoulliFun k ((↑m : ℝ) * x) -
      HPow.hPow (α := ℝ) (↑m : ℝ) k / (↑m : ℝ) *
        ∑ i ∈ Finset.range m, bernoulliFun k (x + HDiv.hDiv (α := ℝ) (↑i : ℝ) (↑m : ℝ))
k : ℕ
h : ∀ (x : ℝ), f k x = 0
d : ∀ (x : ℝ), HasDerivAt (f (k + 1)) 0 x
c : ℝ
fc : ∀ (x : ℝ), f (k + 1) x = c
⊢ Eq (α := ℝ)
    (∫ (x : ℝ) in 0..(↑m : ℝ)⁻¹, ∑ i ∈ Finset.range m, bernoulliFun (k + 1) (x + HDiv.hDiv (α := ℝ) (↑i : ℝ) (↑m : ℝ)))
    0
have h₁ : Eq (α := ℝ)
  (∑ i ∈ Finset.range m, ∫ (x : ℝ) in 0..(↑m : ℝ)⁻¹, bernoulliFun (k + 1) (x + HDiv.hDiv (α := ℝ) (↑i : ℝ) (↑m : ℝ))) 0
Tactic: rw [intervalIntegral.integral_finsetSum]
-- Lean 4 proof step from Mathlib.NumberTheory.ZetaValues (goal)
Goal:
m : ℕ
m0 : m ≠ 0
x : ℝ
m0' : Ne (α := ℝ) (↑m : ℝ) 0
f : (k : ℕ) → (x : ℝ) → ℝ :=
  fun (k : ℕ) (x : ℝ) ↦
    bernoulliFun k ((↑m : ℝ) * x) -
      HPow.hPow (α := ℝ) (↑m : ℝ) k / (↑m : ℝ) *
        ∑ i ∈ Finset.range m, bernoulliFun k (x + HDiv.hDiv (α := ℝ) (↑i : ℝ) (↑m : ℝ))
k : ℕ
h : ∀ (x : ℝ), f k x = 0
d : ∀ (x : ℝ), HasDerivAt (f (k + 1)) 0 x
c : ℝ
fc : ∀ (x : ℝ), f (k + 1) x = c
h₁ :
  Eq (α := ℝ)
    (∑ i ∈ Finset.range m, ∫ (x : ℝ) in 0..(↑m : ℝ)⁻¹, bernoulliFun (k + 1) (x + HDiv.hDiv (α := ℝ) (↑i : ℝ) (↑m : ℝ)))
    0
⊢ Eq (α := ℝ)
    (∫ (x : ℝ) in 0..(↑m : ℝ)⁻¹, ∑ i ∈ Finset.range m, bernoulliFun (k + 1) (x + HDiv.hDiv (α := ℝ) (↑i : ℝ) (↑m : ℝ)))
    0
have h₂ : ∀ i ∈ Finset.range m,
  IntervalIntegrable (ε := ℝ) (fun (x : ℝ) ↦ bernoulliFun (k + 1) (x + HDiv.hDiv (α := ℝ) (↑i : ℝ) (↑m : ℝ))) volume 0
    (↑m : ℝ)⁻¹
Tactic: rw [intervalIntegral.integral_finsetSum]
-- Lean 4 proof step from Mathlib.NumberTheory.ZetaValues (goal)
Goal:
m : ℕ
m0 : m ≠ 0
x : ℝ
m0' : Ne (α := ℝ) (↑m : ℝ) 0
f : (k : ℕ) → (x : ℝ) → ℝ :=
  fun (k : ℕ) (x : ℝ) ↦
    bernoulliFun k ((↑m : ℝ) * x) -
      HPow.hPow (α := ℝ) (↑m : ℝ) k / (↑m : ℝ) *
        ∑ i ∈ Finset.range m, bernoulliFun k (x + HDiv.hDiv (α := ℝ) (↑i : ℝ) (↑m : ℝ))
k : ℕ
h : ∀ (x : ℝ), f k x = 0
d : ∀ (x : ℝ), HasDerivAt (f (k + 1)) 0 x
c : ℝ
fc : ∀ (x : ℝ), f (k + 1) x = c
⊢ Eq (α := ℝ)
    (∑ i ∈ Finset.range m, ∫ (x : ℝ) in 0..(↑m : ℝ)⁻¹, bernoulliFun (k + 1) (x + HDiv.hDiv (α := ℝ) (↑i : ℝ) (↑m : ℝ)))
    0
have h₁ : Eq (α := ℝ)
  (∑ x ∈ Finset.range m,
    ∫ (x : ℝ) in HDiv.hDiv (α := ℝ) (↑x : ℝ) (↑m : ℝ)..HDiv.hDiv (α := ℝ) (↑(x + 1) : ℝ) (↑m : ℝ),
      bernoulliFun (k + 1) x)
  0
Tactic: simp only [intervalIntegral.integral_comp_add_right, zero_add, ← one_div, ← add_div,
            add_comm (1 : ℝ), ← Nat.cast_add_one]
-- Lean 4 proof step from Mathlib.NumberTheory.ZetaValues (hypothesis)
Goal:
m : ℕ
m0 : m ≠ 0
x : ℝ
m0' : Ne (α := ℝ) (↑m : ℝ) 0
f : (k : ℕ) → (x : ℝ) → ℝ :=
  fun (k : ℕ) (x : ℝ) ↦
    bernoulliFun k ((↑m : ℝ) * x) -
      HPow.hPow (α := ℝ) (↑m : ℝ) k / (↑m : ℝ) *
        ∑ i ∈ Finset.range m, bernoulliFun k (x + HDiv.hDiv (α := ℝ) (↑i : ℝ) (↑m : ℝ))
k : ℕ
h : ∀ (x : ℝ), f k x = 0
d : ∀ (x : ℝ), HasDerivAt (f (k + 1)) 0 x
c : ℝ
fc : ∀ (x : ℝ), f (k + 1) x = c
⊢ ∀ (x : ℝ), f (k + 1) x = 0
have i : Eq (α := ℝ) (∫ (x : ℝ) in 0..(↑m : ℝ)⁻¹, f (k + 1) x) 0
Tactic: have i : ∫ x in (0 : ℝ)..m⁻¹, f (k + 1) x = 0 := by
      simp only [f]
      rw [intervalIntegral.integral_sub, intervalIntegral.integral_comp_mul_left _ m0', mul_zero,
        mul_inv_cancel₀ m0', integral_bernoulliFun_eq_zero (by lia), smul_zero, sub_eq_zero,
        intervalIntegral.integral_const_mul, eq_comm (a := 0), mul_eq_zero]
      · right
        rw [intervalIntegral.integral_finsetSum]
        · simp only [intervalIntegral.integral_comp_add_right, zero_add, ← one_div, ← add_div,
            add_comm (1 : ℝ), ← Nat.cast_add_one]
          rw [intervalIntegral.sum_integral_adjacent_intervals]
          · simp [div_self m0', integral_bernoulliFun_eq_zero]
          · intros; exact Continuous.intervalIntegrable (by fun_prop) _ _
        · intros; exact Continuous.intervalIntegrable (by fun_prop) _ _
      · exact Continuous.intervalIntegrable (by fun_prop) _ _
      · exact Continuous.intervalIntegrable (by fun_prop) _ _
-- Lean 4 proof step from Mathlib.NumberTheory.ZetaValues (hypothesis)
Goal:
m : ℕ
m0 : m ≠ 0
x : ℝ
m0' : Ne (α := ℝ) (↑m : ℝ) 0
f : (k : ℕ) → (x : ℝ) → ℝ :=
  fun (k : ℕ) (x : ℝ) ↦
    bernoulliFun k ((↑m : ℝ) * x) -
      HPow.hPow (α := ℝ) (↑m : ℝ) k / (↑m : ℝ) *
        ∑ i ∈ Finset.range m, bernoulliFun k (x + HDiv.hDiv (α := ℝ) (↑i : ℝ) (↑m : ℝ))
k : ℕ
h : ∀ (x : ℝ), f k x = 0
d : ∀ (x : ℝ), HasDerivAt (f (k + 1)) 0 x
c : ℝ
fc : ∀ (x : ℝ), f (k + 1) x = c
i : Eq (α := ℝ) (∫ (x : ℝ) in 0..(↑m : ℝ)⁻¹, f (k + 1) x) 0
⊢ ∀ (x : ℝ), f (k + 1) x = 0
have i₁ : c = 0
Tactic: simp only [fc, intervalIntegral.integral_const, sub_zero, smul_eq_mul, mul_eq_zero, inv_eq_zero,
      Nat.cast_eq_zero, m0, false_or] at i
Cleared hypotheses: i
-- Lean 4 proof step from Mathlib.NumberTheory.ZetaValues (goal)
Goal:
k m : ℕ
m0 : m ≠ 0
x : ℝ
m0' : Ne (α := ℝ) (↑m : ℝ) 0
f : (k : ℕ) → (x : ℝ) → ℝ :=
  fun (k : ℕ) (x : ℝ) ↦
    bernoulliFun k ((↑m : ℝ) * x) -
      HPow.hPow (α := ℝ) (↑m : ℝ) k / (↑m : ℝ) *
        ∑ i ∈ Finset.range m, bernoulliFun k (x + HDiv.hDiv (α := ℝ) (↑i : ℝ) (↑m : ℝ))
⊢ ∀ (x : ℝ), f k x = 0
have zero : ∀ (x : ℝ), f 0 x = 0
Tactic: induction k with
  | zero =>
    intro x
    simp only [f, bernoulliFun_zero, pow_zero, one_div, Finset.sum_const, Finset.card_range,
      nsmul_eq_mul, mul_one, sub_eq_zero]
    rw [inv_mul_cancel₀ (Nat.cast_ne_zero.mpr m0)]
  | succ k h =>
    have d (x) : HasDerivAt (f (k + 1)) (m * (k + 1) * f k x) x := by
      simp only [f, mul_sub, Finset.mul_sum, pow_succ, mul_div_cancel_right₀ _ m0',
        ← mul_assoc, mul_comm _ (_ / _), div_mul_cancel₀ _ m0']
      apply HasDerivAt.sub
      · rw [mul_assoc, mul_comm (m : ℝ) _, ← Nat.cast_add_one]
        exact (hasDerivAt_bernoulliFun _ _).comp _ (hasDerivAt_const_mul ..)
      · refine HasDerivAt.fun_sum fun i _ ↦ ?_
        simp only [mul_assoc, ← Nat.cast_add_one]
        apply HasDerivAt.const_mul
        rw [← mul_one (_ * _)]
        exact (hasDerivAt_bernoulliFun _ _).comp _ ((hasDerivAt_id' _).add_const _)
    simp only [h, mul_zero] at d
    have fc (x) : f (k + 1) x = f (k + 1) 0 :=
      is_const_of_deriv_eq_zero (fun _ ↦ (d _).differentiableAt) (fun _ ↦ (d _).deriv) x 0
    generalize f (k + 1) 0 = c at fc
    have i : ∫ x in (0 : ℝ)..m⁻¹, f (k + 1) x = 0 := by
      simp only [f]
      rw [intervalIntegral.integral_sub, intervalIntegral.integral_comp_mul_left _ m0', mul_zero,
        mul_inv_cancel₀ m0', integral_bernoulliFun_eq_zero (by lia), smul_zero, sub_eq_zero,
        intervalIntegral.integral_const_mul, eq_comm (a := 0), mul_eq_zero]
      · right
        rw [intervalIntegral.integral_finsetSum]
        · simp only [intervalIntegral.integral_comp_add_right, zero_add, ← one_div, ← add_div,
            add_comm (1 : ℝ), ← Nat.cast_add_one]
          rw [intervalIntegral.sum_integral_adjacent_intervals]
          · simp [div_self m0', integral_bernoulliFun_eq_zero]
          · intros; exact Continuous.intervalIntegrable (by fun_prop) _ _
        · intros; exact Continuous.intervalIntegrable (by fun_prop) _ _
      · exact Continuous.intervalIntegrable (by fun_prop) _ _
      · exact Continuous.intervalIntegrable (by fun_prop) _ _
    simp only [fc, intervalIntegral.integral_const, sub_zero, smul_eq_mul, mul_eq_zero, inv_eq_zero,
      Nat.cast_eq_zero, m0, false_or] at i
    simpa only [i] using fc
Cleared hypotheses: k
-- Lean 4 proof step from Mathlib.NumberTheory.ZetaValues (goal)
Goal:
k m : ℕ
m0 : m ≠ 0
x : ℝ
m0' : Ne (α := ℝ) (↑m : ℝ) 0
f : (k : ℕ) → (x : ℝ) → ℝ :=
  fun (k : ℕ) (x : ℝ) ↦
    bernoulliFun k ((↑m : ℝ) * x) -
      HPow.hPow (α := ℝ) (↑m : ℝ) k / (↑m : ℝ) *
        ∑ i ∈ Finset.range m, bernoulliFun k (x + HDiv.hDiv (α := ℝ) (↑i : ℝ) (↑m : ℝ))
zero : ∀ (x : ℝ), f 0 x = 0
⊢ ∀ (x : ℝ), f k x = 0
have succ : ∀ (k : ℕ) (h : ∀ (x : ℝ), f k x = 0) (x : ℝ), f (k + 1) x = 0
Tactic: induction k with
  | zero =>
    intro x
    simp only [f, bernoulliFun_zero, pow_zero, one_div, Finset.sum_const, Finset.card_range,
      nsmul_eq_mul, mul_one, sub_eq_zero]
    rw [inv_mul_cancel₀ (Nat.cast_ne_zero.mpr m0)]
  | succ k h =>
    have d (x) : HasDerivAt (f (k + 1)) (m * (k + 1) * f k x) x := by
      simp only [f, mul_sub, Finset.mul_sum, pow_succ, mul_div_cancel_right₀ _ m0',
        ← mul_assoc, mul_comm _ (_ / _), div_mul_cancel₀ _ m0']
      apply HasDerivAt.sub
      · rw [mul_assoc, mul_comm (m : ℝ) _, ← Nat.cast_add_one]
        exact (hasDerivAt_bernoulliFun _ _).comp _ (hasDerivAt_const_mul ..)
      · refine HasDerivAt.fun_sum fun i _ ↦ ?_
        simp only [mul_assoc, ← Nat.cast_add_one]
        apply HasDerivAt.const_mul
        rw [← mul_one (_ * _)]
        exact (hasDerivAt_bernoulliFun _ _).comp _ ((hasDerivAt_id' _).add_const _)
    simp only [h, mul_zero] at d
    have fc (x) : f (k + 1) x = f (k + 1) 0 :=
      is_const_of_deriv_eq_zero (fun _ ↦ (d _).differentiableAt) (fun _ ↦ (d _).deriv) x 0
    generalize f (k + 1) 0 = c at fc
    have i : ∫ x in (0 : ℝ)..m⁻¹, f (k + 1) x = 0 := by
      simp only [f]
      rw [intervalIntegral.integral_sub, intervalIntegral.integral_comp_mul_left _ m0', mul_zero,
        mul_inv_cancel₀ m0', integral_bernoulliFun_eq_zero (by lia), smul_zero, sub_eq_zero,
        intervalIntegral.integral_const_mul, eq_comm (a := 0), mul_eq_zero]
      · right
        rw [intervalIntegral.integral_finsetSum]
        · simp only [intervalIntegral.integral_comp_add_right, zero_add, ← one_div, ← add_div,
            add_comm (1 : ℝ), ← Nat.cast_add_one]
          rw [intervalIntegral.sum_integral_adjacent_intervals]
          · simp [div_self m0', integral_bernoulliFun_eq_zero]
          · intros; exact Continuous.intervalIntegrable (by fun_prop) _ _
        · intros; exact Continuous.intervalIntegrable (by fun_prop) _ _
      · exact Continuous.intervalIntegrable (by fun_prop) _ _
      · exact Continuous.intervalIntegrable (by fun_prop) _ _
    simp only [fc, intervalIntegral.integral_const, sub_zero, smul_eq_mul, mul_eq_zero, inv_eq_zero,
      Nat.cast_eq_zero, m0, false_or] at i
    simpa only [i] using fc
Cleared hypotheses: k
-- Lean 4 proof step from Mathlib.NumberTheory.ZetaValues (hypothesis)
Goal:
k : ℕ
⊢ bernoulliFun (2 * k + 1) 2⁻¹ = 0
have h : bernoulliFun (2 * k + 1) (HSub.hSub (α := ℝ) 1 2⁻¹) = HPow.hPow (α := ℝ) (-1) (2 * k + 1) * bernoulliFun (2 * k + 1) 2⁻¹
Tactic: have h := bernoulliFun_eval_one_sub (k := 2 * k + 1) (x := 2⁻¹)
-- Lean 4 proof step from Mathlib.NumberTheory.ZetaValues (hypothesis)
Goal:
k : ℕ
h :
  bernoulliFun (2 * k + 1) (HSub.hSub (α := ℝ) 1 2⁻¹) =
    HPow.hPow (α := ℝ) (-1) (2 * k + 1) * bernoulliFun (2 * k + 1) 2⁻¹
⊢ bernoulliFun (2 * k + 1) 2⁻¹ = 0
have h₁ : bernoulliFun (2 * k + 1) (1 / 2) = -bernoulliFun (2 * k + 1) (1 / 2)
Tactic: simp only [pow_succ, even_two, Even.mul_right, Even.neg_pow, one_pow, mul_neg, mul_one, neg_mul,
    one_mul, ← one_div, (sub_eq_of_eq_add (add_halves (1 : ℝ)).symm)] at h
Cleared hypotheses: h
-- Lean 4 proof step from Mathlib.NumberTheory.ZetaValues (goal)
Goal:
k : ℕ
h : bernoulliFun (2 * k + 1) (1 / 2) = -bernoulliFun (2 * k + 1) (1 / 2)
⊢ bernoulliFun (2 * k + 1) 2⁻¹ = 0
have linarith : bernoulliFun (2 * k + 1) 2⁻¹ < 0 →
  -(1 * bernoulliFun (2 * k + 1) (1 / 2) - -(1 * bernoulliFun (2 * k + 1) (1 / 2))) +
      2 * (1 * bernoulliFun (2 * k + 1) (1 / 2) - HMul.hMul (α := ℝ) 1 0) =
    0
Tactic: linarith
-- Lean 4 proof step from Mathlib.NumberTheory.ZetaValues (goal)
Goal:
k : ℕ
h : bernoulliFun (2 * k + 1) (1 / 2) = -bernoulliFun (2 * k + 1) (1 / 2)
linarith :
  bernoulliFun (2 * k + 1) 2⁻¹ < 0 →
    -(1 * bernoulliFun (2 * k + 1) (1 / 2) - -(1 * bernoulliFun (2 * k + 1) (1 / 2))) +
        2 * (1 * bernoulliFun (2 * k + 1) (1 / 2) - HMul.hMul (α := ℝ) 1 0) =
      0
⊢ bernoulliFun (2 * k + 1) 2⁻¹ = 0
have linarith₁ : bernoulliFun (2 * k + 1) 2⁻¹ < 0 → GT.gt (α := ℝ) 2 0
Tactic: linarith
-- Lean 4 proof step from Mathlib.NumberTheory.ZetaValues (goal)
Goal:
k : ℕ
h : bernoulliFun (2 * k + 1) (1 / 2) = -bernoulliFun (2 * k + 1) (1 / 2)
linarith :
  bernoulliFun (2 * k + 1) 2⁻¹ < 0 →
    -(1 * bernoulliFun (2 * k + 1) (1 / 2) - -(1 * bernoulliFun (2 * k + 1) (1 / 2))) +
        2 * (1 * bernoulliFun (2 * k + 1) (1 / 2) - HMul.hMul (α := ℝ) 1 0) =
      0
linarith₁ : bernoulliFun (2 * k + 1) 2⁻¹ < 0 → GT.gt (α := ℝ) 2 0
⊢ bernoulliFun (2 * k + 1) 2⁻¹ = 0
have linarith₂ : 0 < bernoulliFun (2 * k + 1) 2⁻¹ →
  1 * bernoulliFun (2 * k + 1) (1 / 2) - -(1 * bernoulliFun (2 * k + 1) (1 / 2)) +
      2 * (HMul.hMul (α := ℝ) 1 0 - 1 * bernoulliFun (2 * k + 1) (1 / 2)) =
    0
Tactic: linarith
-- Lean 4 proof step from Mathlib.NumberTheory.ZetaValues (goal)
Goal:
k : ℕ
h : bernoulliFun (2 * k + 1) (1 / 2) = -bernoulliFun (2 * k + 1) (1 / 2)
linarith :
  bernoulliFun (2 * k + 1) 2⁻¹ < 0 →
    -(1 * bernoulliFun (2 * k + 1) (1 / 2) - -(1 * bernoulliFun (2 * k + 1) (1 / 2))) +
        2 * (1 * bernoulliFun (2 * k + 1) (1 / 2) - HMul.hMul (α := ℝ) 1 0) =
      0
linarith₁ : bernoulliFun (2 * k + 1) 2⁻¹ < 0 → GT.gt (α := ℝ) 2 0
linarith₂ :
  0 < bernoulliFun (2 * k + 1) 2⁻¹ →
    1 * bernoulliFun (2 * k + 1) (1 / 2) - -(1 * bernoulliFun (2 * k + 1) (1 / 2)) +
        2 * (HMul.hMul (α := ℝ) 1 0 - 1 * bernoulliFun (2 * k + 1) (1 / 2)) =
      0
⊢ bernoulliFun (2 * k + 1) 2⁻¹ = 0
have linarith₃ : 0 < bernoulliFun (2 * k + 1) 2⁻¹ → GT.gt (α := ℝ) 2 0
Tactic: linarith
-- Lean 4 proof step from Mathlib.NumberTheory.ZetaValues (goal)
Goal:
k : ℕ
⊢ bernoulliFun k 2⁻¹ = (2 / HPow.hPow (α := ℝ) 2 k - 1) * (↑(bernoulli k) : ℝ)
have pos : ∀ (k1 : k = 1), bernoulliFun k 2⁻¹ = (2 / HPow.hPow (α := ℝ) 2 k - 1) * (↑(bernoulli k) : ℝ)
Tactic: by_cases k1 : k = 1
-- Lean 4 proof step from Mathlib.NumberTheory.ZetaValues (goal)
Goal:
k : ℕ
pos : ∀ (k1 : k = 1), bernoulliFun k 2⁻¹ = (2 / HPow.hPow (α := ℝ) 2 k - 1) * (↑(bernoulli k) : ℝ)
⊢ bernoulliFun k 2⁻¹ = (2 / HPow.hPow (α := ℝ) 2 k - 1) * (↑(bernoulli k) : ℝ)
have neg : ∀ (k1 : ¬k = 1), bernoulliFun k 2⁻¹ = (2 / HPow.hPow (α := ℝ) 2 k - 1) * (↑(bernoulli k) : ℝ)
Tactic: by_cases k1 : k = 1
-- Lean 4 proof step from Mathlib.NumberTheory.ZetaValues (hypothesis)
Goal:
k : ℕ
k1 : ¬k = 1
⊢ bernoulliFun k 2⁻¹ = (2 / HPow.hPow (α := ℝ) 2 k - 1) * (↑(bernoulli k) : ℝ)
have m : bernoulliFun k (HMul.hMul (α := ℝ) (↑2 : ℝ) 2⁻¹) =
  HPow.hPow (α := ℝ) (↑2 : ℝ) k / (↑2 : ℝ) *
    ∑ i ∈ Finset.range 2, bernoulliFun k (2⁻¹ + HDiv.hDiv (α := ℝ) (↑i : ℝ) (↑2 : ℝ))
Tactic: have m := bernoulliFun_mul k two_ne_zero 2⁻¹
-- Lean 4 proof step from Mathlib.NumberTheory.ZetaValues (hypothesis)
Goal:
k : ℕ
k1 : ¬k = 1
m :
  bernoulliFun k (HMul.hMul (α := ℝ) (↑2 : ℝ) 2⁻¹) =
    HPow.hPow (α := ℝ) (↑2 : ℝ) k / (↑2 : ℝ) *
      ∑ i ∈ Finset.range 2, bernoulliFun k (2⁻¹ + HDiv.hDiv (α := ℝ) (↑i : ℝ) (↑2 : ℝ))
⊢ bernoulliFun k 2⁻¹ = (2 / HPow.hPow (α := ℝ) 2 k - 1) * (↑(bernoulli k) : ℝ)
have m₁ : (↑(bernoulli k) : ℝ) = HPow.hPow (α := ℝ) 2 k / 2 * (bernoulliFun k (1 / 2) + (↑(bernoulli k) : ℝ))
Tactic: simp_rw [Nat.cast_ofNat, mul_inv_cancel₀ (two_ne_zero' ℝ), Finset.sum_range_succ,
      Finset.sum_range_zero, Nat.cast_zero, Nat.cast_one, ← one_div, add_halves,
      bernoulliFun_eval_one, if_neg k1, bernoulliFun_eval_zero, zero_div, add_zero, zero_add] at m
Cleared hypotheses: m
-- Lean 4 proof step from Mathlib.NumberTheory.ZetaValues (hypothesis)
Goal:
k : ℕ
k1 : ¬k = 1
m : (↑(bernoulli k) : ℝ) = HPow.hPow (α := ℝ) 2 k / 2 * (bernoulliFun k (1 / 2) + (↑(bernoulli k) : ℝ))
⊢ bernoulliFun k 2⁻¹ = (2 / HPow.hPow (α := ℝ) 2 k - 1) * (↑(bernoulli k) : ℝ)
have m₁ : (2 / HPow.hPow (α := ℝ) 2 k - 1) * (↑(bernoulli k) : ℝ) = bernoulliFun k (1 / 2)
Tactic: rw [← inv_mul_eq_iff_eq_mul₀ (by positivity), ← sub_eq_iff_eq_add, ← sub_one_mul, inv_div] at m
Cleared hypotheses: m
-- Lean 4 proof step from Mathlib.NumberTheory.ZetaValues (goal)
Goal:
k : ℕ
k1 : ¬k = 1
m : (2 / HPow.hPow (α := ℝ) 2 k - 1) * (↑(bernoulli k) : ℝ) = bernoulliFun k (1 / 2)
⊢ bernoulliFun k 2⁻¹ = (2 / HPow.hPow (α := ℝ) 2 k - 1) * (↑(bernoulli k) : ℝ)
have neg : bernoulliFun k 2⁻¹ = bernoulliFun k (1 / 2)
Tactic: rw [m, one_div]
-- Lean 4 proof step from Mathlib.NumberTheory.ZetaValues (goal)
Goal:
k : ℕ
k1 : ¬k = 1
m : (2 / HPow.hPow (α := ℝ) 2 k - 1) * (↑(bernoulli k) : ℝ) = bernoulliFun k (1 / 2)
neg : bernoulliFun k 2⁻¹ = bernoulliFun k (1 / 2)
⊢ bernoulliFun k 2⁻¹ = (2 / HPow.hPow (α := ℝ) 2 k - 1) * (↑(bernoulli k) : ℝ)
have neg₁ : bernoulliFun k 2⁻¹ = bernoulliFun k 2⁻¹
Tactic: rw [m, one_div]
-- Lean 4 proof step from Mathlib.NumberTheory.ZetaValues (goal)
Goal:
k : ℕ
n : ℤ
hn : n ≠ 0
⊢ fourierCoeffOn bernoulliFourierCoeff._proof_1 (fun (x : ℝ) ↦ (↑(bernoulliFun k x) : ℂ)) n =
    1 / (-2 * (↑π : ℂ) * I * (↑n : ℂ)) *
      ((if k = 1 then 1 else 0) -
        HMul.hMul (α := ℂ) (↑k : ℂ)
          (fourierCoeffOn bernoulliFourierCoeff._proof_1 (fun (x : ℝ) ↦ (↑(bernoulliFun (k - 1) x) : ℂ)) n))
have rw : 1 / (-2 * (↑π : ℂ) * I * (↑n : ℂ)) *
    (HMul.hMul (α := ℂ) ((fourier (-n) : (a : AddCircle (1 - 0)) → ℂ) (↑0 : ℝ ⧸ AddSubgroup.zmultiples (1 - 0)) : ℂ)
        ((↑(bernoulliFun k 1) : ℂ) - (↑(bernoulliFun k 0) : ℂ)) -
      ((↑1 : ℂ) - (↑0 : ℂ)) *
        fourierCoeffOn (a := 0) (b := 1) sorry (fun (x : ℝ) ↦ (↑((↑k : ℝ) * bernoulliFun (k - 1) x) : ℂ)) n) =
  1 / (-2 * (↑π : ℂ) * I * (↑n : ℂ)) *
    ((if k = 1 then 1 else 0) -
      HMul.hMul (α := ℂ) (↑k : ℂ)
        (fourierCoeffOn bernoulliFourierCoeff._proof_1 (fun (x : ℝ) ↦ (↑(bernoulliFun (k - 1) x) : ℂ)) n))
Tactic: rw [fourierCoeffOn_of_hasDerivAt zero_lt_one hn
      (fun x _ => (hasDerivAt_bernoulliFun k x).ofReal_comp)
      ((continuous_ofReal.comp <|
            continuous_const.mul <| Polynomial.continuous _).intervalIntegrable
        _ _)]
-- Lean 4 proof step from Mathlib.NumberTheory.ZetaValues (goal)
Goal:
k : ℕ
n : ℤ
hn : n ≠ 0
⊢ 1 / (-2 * (↑π : ℂ) * I * (↑n : ℂ)) *
      (HMul.hMul (α := ℂ) ((fourier (-n) : (a : AddCircle (1 - 0)) → ℂ) (↑0 : ℝ ⧸ AddSubgroup.zmultiples (1 - 0)) : ℂ)
          ((↑(bernoulliFun k 1) : ℂ) - (↑(bernoulliFun k 0) : ℂ)) -
        ((↑1 : ℂ) - (↑0 : ℂ)) *
          fourierCoeffOn (a := 0) (b := 1) ⋯ (fun (x : ℝ) ↦ (↑((↑k : ℝ) * bernoulliFun (k - 1) x) : ℂ)) n) =
    1 / (-2 * (↑π : ℂ) * I * (↑n : ℂ)) *
      ((if k = 1 then 1 else 0) -
        HMul.hMul (α := ℂ) (↑k : ℂ)
          (fourierCoeffOn bernoulliFourierCoeff._proof_1 (fun (x : ℝ) ↦ (↑(bernoulliFun (k - 1) x) : ℂ)) n))
have simp_rw : 1 / (-2 * (↑π : ℂ) * I * (↑n : ℂ)) *
    (HMul.hMul (α := ℂ) ((fourier (-n) : (a : AddCircle (1 - 0)) → ℂ) (↑0 : ℝ ⧸ AddSubgroup.zmultiples (1 - 0)) : ℂ)
        ((↑(bernoulliFun k 1) : ℂ) - (↑(bernoulliFun k 0) : ℂ)) -
      fourierCoeffOn (a := 0) (b := 1) sorry (fun (x : ℝ) ↦ (↑((↑k : ℝ) * bernoulliFun (k - 1) x) : ℂ)) n) =
  1 / (-2 * (↑π : ℂ) * I * (↑n : ℂ)) *
    ((if k = 1 then 1 else 0) -
      HMul.hMul (α := ℂ) (↑k : ℂ)
        (fourierCoeffOn bernoulliFourierCoeff._proof_1 (fun (x : ℝ) ↦ (↑(bernoulliFun (k - 1) x) : ℂ)) n))
Tactic: simp_rw [ofReal_one, ofReal_zero, sub_zero, one_mul]
-- Lean 4 proof step from Mathlib.NumberTheory.ZetaValues (goal)
Goal:
k : ℕ
n : ℤ
hn : n ≠ 0
⊢ 1 / (-2 * (↑π : ℂ) * I * (↑n : ℂ)) *
      (HMul.hMul (α := ℂ) ((fourier (-n) : (a : AddCircle (1 - 0)) → ℂ) (↑0 : ℝ ⧸ AddSubgroup.zmultiples (1 - 0)) : ℂ)
          ((↑(bernoulliFun k 1) : ℂ) - (↑(bernoulliFun k 0) : ℂ)) -
        fourierCoeffOn (a := 0) (b := 1) ⋯ (fun (x : ℝ) ↦ (↑((↑k : ℝ) * bernoulliFun (k - 1) x) : ℂ)) n) =
    1 / (-2 * (↑π : ℂ) * I * (↑n : ℂ)) *
      ((if k = 1 then 1 else 0) -
        HMul.hMul (α := ℂ) (↑k : ℂ)
          (fourierCoeffOn bernoulliFourierCoeff._proof_1 (fun (x : ℝ) ↦ (↑(bernoulliFun (k - 1) x) : ℂ)) n))
have rw : 1 / (-2 * (↑π : ℂ) * I * (↑n : ℂ)) *
    ((↑(if k = 1 then 1 else 0) : ℂ) -
      fourierCoeffOn (a := 0) (b := 1) sorry (fun (x : ℝ) ↦ (↑((↑k : ℝ) * bernoulliFun (k - 1) x) : ℂ)) n) =
  1 / (-2 * (↑π : ℂ) * I * (↑n : ℂ)) *
    ((if k = 1 then 1 else 0) -
      HMul.hMul (α := ℂ) (↑k : ℂ)
        (fourierCoeffOn bernoulliFourierCoeff._proof_1 (fun (x : ℝ) ↦ (↑(bernoulliFun (k - 1) x) : ℂ)) n))
Tactic: rw [QuotientAddGroup.mk_zero, fourier_eval_zero, one_mul, ← ofReal_sub, bernoulliFun_eval_one,
    add_sub_cancel_left]
-- Lean 4 proof step from Mathlib.NumberTheory.ZetaValues (goal)
Goal:
k : ℕ
n : ℤ
hn : n ≠ 0
⊢ 1 / (-2 * (↑π : ℂ) * I * (↑n : ℂ)) *
      ((↑(if k = 1 then 1 else 0) : ℂ) -
        fourierCoeffOn (a := 0) (b := 1) ⋯ (fun (x : ℝ) ↦ (↑((↑k : ℝ) * bernoulliFun (k - 1) x) : ℂ)) n) =
    1 / (-2 * (↑π : ℂ) * I * (↑n : ℂ)) *
      ((if k = 1 then 1 else 0) -
        HMul.hMul (α := ℂ) (↑k : ℂ)
          (fourierCoeffOn bernoulliFourierCoeff._proof_1 (fun (x : ℝ) ↦ (↑(bernoulliFun (k - 1) x) : ℂ)) n))
have e_a : (↑(if k = 1 then 1 else 0) : ℂ) = if k = 1 then 1 else 0
Tactic: congr 2
-- Lean 4 proof step from Mathlib.NumberTheory.ZetaValues (goal)
Goal:
k : ℕ
n : ℤ
hn : n ≠ 0
e_a : (↑(if k = 1 then 1 else 0) : ℂ) = if k = 1 then 1 else 0
⊢ 1 / (-2 * (↑π : ℂ) * I * (↑n : ℂ)) *
      ((↑(if k = 1 then 1 else 0) : ℂ) -
        fourierCoeffOn (a := 0) (b := 1) ⋯ (fun (x : ℝ) ↦ (↑((↑k : ℝ) * bernoulliFun (k - 1) x) : ℂ)) n) =
    1 / (-2 * (↑π : ℂ) * I * (↑n : ℂ)) *
      ((if k = 1 then 1 else 0) -
        HMul.hMul (α := ℂ) (↑k : ℂ)
          (fourierCoeffOn bernoulliFourierCoeff._proof_1 (fun (x : ℝ) ↦ (↑(bernoulliFun (k - 1) x) : ℂ)) n))
have e_a₁ : fourierCoeffOn (a := 0) (b := 1) sorry (fun (x : ℝ) ↦ (↑((↑k : ℝ) * bernoulliFun (k - 1) x) : ℂ)) n =
  HMul.hMul (α := ℂ) (↑k : ℂ)
    (fourierCoeffOn bernoulliFourierCoeff._proof_1 (fun (x : ℝ) ↦ (↑(bernoulliFun (k - 1) x) : ℂ)) n)
Tactic: congr 2
-- Lean 4 proof step from Mathlib.NumberTheory.ZetaValues (goal)
Goal:
k : ℕ
n : ℤ
hn : n ≠ 0
⊢ (↑(if k = 1 then 1 else 0) : ℂ) = if k = 1 then 1 else 0
have pos : k = 1 → (↑1 : ℂ) = 1
Tactic: split_ifs