l3lab/ntp-mathlib huggingface dataset:
theorem riemannZeta_neg_nat_eq_bernoulli (k : ℕ) :
    riemannZeta (-k) = (-1 : ℂ) ^ k * bernoulli (k + 1) / (k + 1) := by
  rcases Nat.even_or_odd' k with ⟨m, rfl | rfl⟩
  · cases' m with m m
    ·-- k = 0 : evaluate explicitly
      rw [Nat.zero_eq, mul_zero, Nat.cast_zero, pow_zero, one_mul, zero_add, neg_zero, zero_add,
        div_one, bernoulli_one, riemannZeta_zero]
      norm_num
    · -- k = 2 * (m + 1) : both sides "trivially" zero
      rw [Nat.cast_mul, ← neg_mul, Nat.cast_two, Nat.cast_succ, riemannZeta_neg_two_mul_nat_add_one,
        bernoulli_eq_bernoulli'_of_ne_one]
      swap; · apply ne_of_gt; norm_num
      rw [bernoulli'_odd_eq_zero ⟨m + 1, rfl⟩ (by norm_num), Rat.cast_zero, mul_zero,
        zero_div]
  · -- k = 2 * m + 1 : the interesting case
    rw [Odd.neg_one_pow ⟨m, rfl⟩]
    rw [show -(↑(2 * m + 1) : ℂ) = 1 - (2 * m + 2) by push_cast; ring]
    rw [riemannZeta_one_sub]
    

Proof state:
case intro.inr
m : ℕ
⊢ 2 ^ (1 - (2 * ↑m + 2)) * ↑π ^ (-(2 * ↑m + 2)) * Complex.Gamma (2 * ↑m + 2) *
        Complex.sin (↑π * (1 - (2 * ↑m + 2)) / 2) *
      riemannZeta (2 * ↑m + 2) =
    -1 * ↑(bernoulli (2 * m + 1 + 1)) / (↑(2 * m + 1) + 1)
case intro.inr.hs
m : ℕ
⊢ ∀ (n : ℕ), 2 * ↑m + 2 ≠ -↑n
case intro.inr.hs' m : ℕ ⊢ 2 * ↑m + 2 ≠ 1

Next tactic:
rotate_left
theorem riemannZeta_neg_nat_eq_bernoulli (k : ℕ) :
    riemannZeta (-k) = (-1 : ℂ) ^ k * bernoulli (k + 1) / (k + 1) := by
  rcases Nat.even_or_odd' k with ⟨m, rfl | rfl⟩
  · cases' m with m m
    ·-- k = 0 : evaluate explicitly
      rw [Nat.zero_eq, mul_zero, Nat.cast_zero, pow_zero, one_mul, zero_add, neg_zero, zero_add,
        div_one, bernoulli_one, riemannZeta_zero]
      norm_num
    · -- k = 2 * (m + 1) : both sides "trivially" zero
      rw [Nat.cast_mul, ← neg_mul, Nat.cast_two, Nat.cast_succ, riemannZeta_neg_two_mul_nat_add_one,
        bernoulli_eq_bernoulli'_of_ne_one]
      swap; · apply ne_of_gt; norm_num
      rw [bernoulli'_odd_eq_zero ⟨m + 1, rfl⟩ (by norm_num), Rat.cast_zero, mul_zero,
        zero_div]
  · -- k = 2 * m + 1 : the interesting case
    rw [Odd.neg_one_pow ⟨m, rfl⟩]
    rw [show -(↑(2 * m + 1) : ℂ) = 1 - (2 * m + 2) by push_cast; ring]
    rw [riemannZeta_one_sub]
    rotate_left
    · 

Proof state:
case intro.inr.hs
m : ℕ
⊢ ∀ (n : ℕ), 2 * ↑m + 2 ≠ -↑n

Next tactic:
intro n
theorem riemannZeta_neg_nat_eq_bernoulli (k : ℕ) :
    riemannZeta (-k) = (-1 : ℂ) ^ k * bernoulli (k + 1) / (k + 1) := by
  rcases Nat.even_or_odd' k with ⟨m, rfl | rfl⟩
  · cases' m with m m
    ·-- k = 0 : evaluate explicitly
      rw [Nat.zero_eq, mul_zero, Nat.cast_zero, pow_zero, one_mul, zero_add, neg_zero, zero_add,
        div_one, bernoulli_one, riemannZeta_zero]
      norm_num
    · -- k = 2 * (m + 1) : both sides "trivially" zero
      rw [Nat.cast_mul, ← neg_mul, Nat.cast_two, Nat.cast_succ, riemannZeta_neg_two_mul_nat_add_one,
        bernoulli_eq_bernoulli'_of_ne_one]
      swap; · apply ne_of_gt; norm_num
      rw [bernoulli'_odd_eq_zero ⟨m + 1, rfl⟩ (by norm_num), Rat.cast_zero, mul_zero,
        zero_div]
  · -- k = 2 * m + 1 : the interesting case
    rw [Odd.neg_one_pow ⟨m, rfl⟩]
    rw [show -(↑(2 * m + 1) : ℂ) = 1 - (2 * m + 2) by push_cast; ring]
    rw [riemannZeta_one_sub]
    rotate_left
    · intro n
      

Proof state:
case intro.inr.hs
m n : ℕ
⊢ 2 * ↑m + 2 ≠ -↑n

Next tactic:
rw [(by norm_cast : 2 * (m : ℂ) + 2 = ↑(2 * m + 2)), ← Int.cast_neg_natCast, ← Int.cast_ofNat,
        Ne.def, Int.cast_inj]
theorem riemannZeta_neg_nat_eq_bernoulli (k : ℕ) :
    riemannZeta (-k) = (-1 : ℂ) ^ k * bernoulli (k + 1) / (k + 1) := by
  rcases Nat.even_or_odd' k with ⟨m, rfl | rfl⟩
  · cases' m with m m
    ·-- k = 0 : evaluate explicitly
      rw [Nat.zero_eq, mul_zero, Nat.cast_zero, pow_zero, one_mul, zero_add, neg_zero, zero_add,
        div_one, bernoulli_one, riemannZeta_zero]
      norm_num
    · -- k = 2 * (m + 1) : both sides "trivially" zero
      rw [Nat.cast_mul, ← neg_mul, Nat.cast_two, Nat.cast_succ, riemannZeta_neg_two_mul_nat_add_one,
        bernoulli_eq_bernoulli'_of_ne_one]
      swap; · apply ne_of_gt; norm_num
      rw [bernoulli'_odd_eq_zero ⟨m + 1, rfl⟩ (by norm_num), Rat.cast_zero, mul_zero,
        zero_div]
  · -- k = 2 * m + 1 : the interesting case
    rw [Odd.neg_one_pow ⟨m, rfl⟩]
    rw [show -(↑(2 * m + 1) : ℂ) = 1 - (2 * m + 2) by push_cast; ring]
    rw [riemannZeta_one_sub]
    rotate_left
    · intro n
      rw [(by 

Proof state:
m n : ℕ
⊢ 2 * ↑m + 2 = ↑(2 * m + 2)

Next tactic:
norm_cast
theorem riemannZeta_neg_nat_eq_bernoulli (k : ℕ) :
    riemannZeta (-k) = (-1 : ℂ) ^ k * bernoulli (k + 1) / (k + 1) := by
  rcases Nat.even_or_odd' k with ⟨m, rfl | rfl⟩
  · cases' m with m m
    ·-- k = 0 : evaluate explicitly
      rw [Nat.zero_eq, mul_zero, Nat.cast_zero, pow_zero, one_mul, zero_add, neg_zero, zero_add,
        div_one, bernoulli_one, riemannZeta_zero]
      norm_num
    · -- k = 2 * (m + 1) : both sides "trivially" zero
      rw [Nat.cast_mul, ← neg_mul, Nat.cast_two, Nat.cast_succ, riemannZeta_neg_two_mul_nat_add_one,
        bernoulli_eq_bernoulli'_of_ne_one]
      swap; · apply ne_of_gt; norm_num
      rw [bernoulli'_odd_eq_zero ⟨m + 1, rfl⟩ (by norm_num), Rat.cast_zero, mul_zero,
        zero_div]
  · -- k = 2 * m + 1 : the interesting case
    rw [Odd.neg_one_pow ⟨m, rfl⟩]
    rw [show -(↑(2 * m + 1) : ℂ) = 1 - (2 * m + 2) by push_cast; ring]
    rw [riemannZeta_one_sub]
    rotate_left
    · intro n
      rw [(by norm_cast : 2 * (m : ℂ) + 2 = ↑(2 * m + 2)), ← Int.cast_neg_natCast, ← Int.cast_ofNat,
        Ne.def, Int.cast_inj]
      

Proof state:
case intro.inr.hs
m n : ℕ
⊢ ¬↑(2 * m + 2) = -↑n

Next tactic:
apply ne_of_gt
theorem riemannZeta_neg_nat_eq_bernoulli (k : ℕ) :
    riemannZeta (-k) = (-1 : ℂ) ^ k * bernoulli (k + 1) / (k + 1) := by
  rcases Nat.even_or_odd' k with ⟨m, rfl | rfl⟩
  · cases' m with m m
    ·-- k = 0 : evaluate explicitly
      rw [Nat.zero_eq, mul_zero, Nat.cast_zero, pow_zero, one_mul, zero_add, neg_zero, zero_add,
        div_one, bernoulli_one, riemannZeta_zero]
      norm_num
    · -- k = 2 * (m + 1) : both sides "trivially" zero
      rw [Nat.cast_mul, ← neg_mul, Nat.cast_two, Nat.cast_succ, riemannZeta_neg_two_mul_nat_add_one,
        bernoulli_eq_bernoulli'_of_ne_one]
      swap; · apply ne_of_gt; norm_num
      rw [bernoulli'_odd_eq_zero ⟨m + 1, rfl⟩ (by norm_num), Rat.cast_zero, mul_zero,
        zero_div]
  · -- k = 2 * m + 1 : the interesting case
    rw [Odd.neg_one_pow ⟨m, rfl⟩]
    rw [show -(↑(2 * m + 1) : ℂ) = 1 - (2 * m + 2) by push_cast; ring]
    rw [riemannZeta_one_sub]
    rotate_left
    · intro n
      rw [(by norm_cast : 2 * (m : ℂ) + 2 = ↑(2 * m + 2)), ← Int.cast_neg_natCast, ← Int.cast_ofNat,
        Ne.def, Int.cast_inj]
      apply ne_of_gt
      

Proof state:
case intro.inr.hs.h
m n : ℕ
⊢ -↑n < ↑(2 * m + 2)

Next tactic:
exact lt_of_le_of_lt (by norm_num : (-n : ℤ) ≤ 0) (by positivity)
theorem riemannZeta_neg_nat_eq_bernoulli (k : ℕ) :
    riemannZeta (-k) = (-1 : ℂ) ^ k * bernoulli (k + 1) / (k + 1) := by
  rcases Nat.even_or_odd' k with ⟨m, rfl | rfl⟩
  · cases' m with m m
    ·-- k = 0 : evaluate explicitly
      rw [Nat.zero_eq, mul_zero, Nat.cast_zero, pow_zero, one_mul, zero_add, neg_zero, zero_add,
        div_one, bernoulli_one, riemannZeta_zero]
      norm_num
    · -- k = 2 * (m + 1) : both sides "trivially" zero
      rw [Nat.cast_mul, ← neg_mul, Nat.cast_two, Nat.cast_succ, riemannZeta_neg_two_mul_nat_add_one,
        bernoulli_eq_bernoulli'_of_ne_one]
      swap; · apply ne_of_gt; norm_num
      rw [bernoulli'_odd_eq_zero ⟨m + 1, rfl⟩ (by norm_num), Rat.cast_zero, mul_zero,
        zero_div]
  · -- k = 2 * m + 1 : the interesting case
    rw [Odd.neg_one_pow ⟨m, rfl⟩]
    rw [show -(↑(2 * m + 1) : ℂ) = 1 - (2 * m + 2) by push_cast; ring]
    rw [riemannZeta_one_sub]
    rotate_left
    · intro n
      rw [(by norm_cast : 2 * (m : ℂ) + 2 = ↑(2 * m + 2)), ← Int.cast_neg_natCast, ← Int.cast_ofNat,
        Ne.def, Int.cast_inj]
      apply ne_of_gt
      exact lt_of_le_of_lt (by 

Proof state:
m n : ℕ
⊢ -↑n ≤ 0

Next tactic:
norm_num
theorem riemannZeta_neg_nat_eq_bernoulli (k : ℕ) :
    riemannZeta (-k) = (-1 : ℂ) ^ k * bernoulli (k + 1) / (k + 1) := by
  rcases Nat.even_or_odd' k with ⟨m, rfl | rfl⟩
  · cases' m with m m
    ·-- k = 0 : evaluate explicitly
      rw [Nat.zero_eq, mul_zero, Nat.cast_zero, pow_zero, one_mul, zero_add, neg_zero, zero_add,
        div_one, bernoulli_one, riemannZeta_zero]
      norm_num
    · -- k = 2 * (m + 1) : both sides "trivially" zero
      rw [Nat.cast_mul, ← neg_mul, Nat.cast_two, Nat.cast_succ, riemannZeta_neg_two_mul_nat_add_one,
        bernoulli_eq_bernoulli'_of_ne_one]
      swap; · apply ne_of_gt; norm_num
      rw [bernoulli'_odd_eq_zero ⟨m + 1, rfl⟩ (by norm_num), Rat.cast_zero, mul_zero,
        zero_div]
  · -- k = 2 * m + 1 : the interesting case
    rw [Odd.neg_one_pow ⟨m, rfl⟩]
    rw [show -(↑(2 * m + 1) : ℂ) = 1 - (2 * m + 2) by push_cast; ring]
    rw [riemannZeta_one_sub]
    rotate_left
    · intro n
      rw [(by norm_cast : 2 * (m : ℂ) + 2 = ↑(2 * m + 2)), ← Int.cast_neg_natCast, ← Int.cast_ofNat,
        Ne.def, Int.cast_inj]
      apply ne_of_gt
      exact lt_of_le_of_lt (by norm_num : (-n : ℤ) ≤ 0) (by 

Proof state:
m n : ℕ
⊢ 0 < ↑(2 * m + 2)

Next tactic:
positivity
theorem riemannZeta_neg_nat_eq_bernoulli (k : ℕ) :
    riemannZeta (-k) = (-1 : ℂ) ^ k * bernoulli (k + 1) / (k + 1) := by
  rcases Nat.even_or_odd' k with ⟨m, rfl | rfl⟩
  · cases' m with m m
    ·-- k = 0 : evaluate explicitly
      rw [Nat.zero_eq, mul_zero, Nat.cast_zero, pow_zero, one_mul, zero_add, neg_zero, zero_add,
        div_one, bernoulli_one, riemannZeta_zero]
      norm_num
    · -- k = 2 * (m + 1) : both sides "trivially" zero
      rw [Nat.cast_mul, ← neg_mul, Nat.cast_two, Nat.cast_succ, riemannZeta_neg_two_mul_nat_add_one,
        bernoulli_eq_bernoulli'_of_ne_one]
      swap; · apply ne_of_gt; norm_num
      rw [bernoulli'_odd_eq_zero ⟨m + 1, rfl⟩ (by norm_num), Rat.cast_zero, mul_zero,
        zero_div]
  · -- k = 2 * m + 1 : the interesting case
    rw [Odd.neg_one_pow ⟨m, rfl⟩]
    rw [show -(↑(2 * m + 1) : ℂ) = 1 - (2 * m + 2) by push_cast; ring]
    rw [riemannZeta_one_sub]
    rotate_left
    · intro n
      rw [(by norm_cast : 2 * (m : ℂ) + 2 = ↑(2 * m + 2)), ← Int.cast_neg_natCast, ← Int.cast_ofNat,
        Ne.def, Int.cast_inj]
      apply ne_of_gt
      exact lt_of_le_of_lt (by norm_num : (-n : ℤ) ≤ 0) (by positivity)
    · 

Proof state:
case intro.inr.hs'
m : ℕ
⊢ 2 * ↑m + 2 ≠ 1

Next tactic:
rw [(by norm_cast : 2 * (m : ℂ) + 2 = ↑(2 * m + 2)), Ne.def, Nat.cast_eq_one]
theorem riemannZeta_neg_nat_eq_bernoulli (k : ℕ) :
    riemannZeta (-k) = (-1 : ℂ) ^ k * bernoulli (k + 1) / (k + 1) := by
  rcases Nat.even_or_odd' k with ⟨m, rfl | rfl⟩
  · cases' m with m m
    ·-- k = 0 : evaluate explicitly
      rw [Nat.zero_eq, mul_zero, Nat.cast_zero, pow_zero, one_mul, zero_add, neg_zero, zero_add,
        div_one, bernoulli_one, riemannZeta_zero]
      norm_num
    · -- k = 2 * (m + 1) : both sides "trivially" zero
      rw [Nat.cast_mul, ← neg_mul, Nat.cast_two, Nat.cast_succ, riemannZeta_neg_two_mul_nat_add_one,
        bernoulli_eq_bernoulli'_of_ne_one]
      swap; · apply ne_of_gt; norm_num
      rw [bernoulli'_odd_eq_zero ⟨m + 1, rfl⟩ (by norm_num), Rat.cast_zero, mul_zero,
        zero_div]
  · -- k = 2 * m + 1 : the interesting case
    rw [Odd.neg_one_pow ⟨m, rfl⟩]
    rw [show -(↑(2 * m + 1) : ℂ) = 1 - (2 * m + 2) by push_cast; ring]
    rw [riemannZeta_one_sub]
    rotate_left
    · intro n
      rw [(by norm_cast : 2 * (m : ℂ) + 2 = ↑(2 * m + 2)), ← Int.cast_neg_natCast, ← Int.cast_ofNat,
        Ne.def, Int.cast_inj]
      apply ne_of_gt
      exact lt_of_le_of_lt (by norm_num : (-n : ℤ) ≤ 0) (by positivity)
    · rw [(by 

Proof state:
m : ℕ
⊢ 2 * ↑m + 2 = ↑(2 * m + 2)

Next tactic:
norm_cast
theorem riemannZeta_neg_nat_eq_bernoulli (k : ℕ) :
    riemannZeta (-k) = (-1 : ℂ) ^ k * bernoulli (k + 1) / (k + 1) := by
  rcases Nat.even_or_odd' k with ⟨m, rfl | rfl⟩
  · cases' m with m m
    ·-- k = 0 : evaluate explicitly
      rw [Nat.zero_eq, mul_zero, Nat.cast_zero, pow_zero, one_mul, zero_add, neg_zero, zero_add,
        div_one, bernoulli_one, riemannZeta_zero]
      norm_num
    · -- k = 2 * (m + 1) : both sides "trivially" zero
      rw [Nat.cast_mul, ← neg_mul, Nat.cast_two, Nat.cast_succ, riemannZeta_neg_two_mul_nat_add_one,
        bernoulli_eq_bernoulli'_of_ne_one]
      swap; · apply ne_of_gt; norm_num
      rw [bernoulli'_odd_eq_zero ⟨m + 1, rfl⟩ (by norm_num), Rat.cast_zero, mul_zero,
        zero_div]
  · -- k = 2 * m + 1 : the interesting case
    rw [Odd.neg_one_pow ⟨m, rfl⟩]
    rw [show -(↑(2 * m + 1) : ℂ) = 1 - (2 * m + 2) by push_cast; ring]
    rw [riemannZeta_one_sub]
    rotate_left
    · intro n
      rw [(by norm_cast : 2 * (m : ℂ) + 2 = ↑(2 * m + 2)), ← Int.cast_neg_natCast, ← Int.cast_ofNat,
        Ne.def, Int.cast_inj]
      apply ne_of_gt
      exact lt_of_le_of_lt (by norm_num : (-n : ℤ) ≤ 0) (by positivity)
    · rw [(by norm_cast : 2 * (m : ℂ) + 2 = ↑(2 * m + 2)), Ne.def, Nat.cast_eq_one]; 

Proof state:
case intro.inr.hs'
m : ℕ
⊢ ¬2 * m + 2 = 1

Next tactic:
norm_num
theorem riemannZeta_neg_nat_eq_bernoulli (k : ℕ) :
    riemannZeta (-k) = (-1 : ℂ) ^ k * bernoulli (k + 1) / (k + 1) := by
  rcases Nat.even_or_odd' k with ⟨m, rfl | rfl⟩
  · cases' m with m m
    ·-- k = 0 : evaluate explicitly
      rw [Nat.zero_eq, mul_zero, Nat.cast_zero, pow_zero, one_mul, zero_add, neg_zero, zero_add,
        div_one, bernoulli_one, riemannZeta_zero]
      norm_num
    · -- k = 2 * (m + 1) : both sides "trivially" zero
      rw [Nat.cast_mul, ← neg_mul, Nat.cast_two, Nat.cast_succ, riemannZeta_neg_two_mul_nat_add_one,
        bernoulli_eq_bernoulli'_of_ne_one]
      swap; · apply ne_of_gt; norm_num
      rw [bernoulli'_odd_eq_zero ⟨m + 1, rfl⟩ (by norm_num), Rat.cast_zero, mul_zero,
        zero_div]
  · -- k = 2 * m + 1 : the interesting case
    rw [Odd.neg_one_pow ⟨m, rfl⟩]
    rw [show -(↑(2 * m + 1) : ℂ) = 1 - (2 * m + 2) by push_cast; ring]
    rw [riemannZeta_one_sub]
    rotate_left
    · intro n
      rw [(by norm_cast : 2 * (m : ℂ) + 2 = ↑(2 * m + 2)), ← Int.cast_neg_natCast, ← Int.cast_ofNat,
        Ne.def, Int.cast_inj]
      apply ne_of_gt
      exact lt_of_le_of_lt (by norm_num : (-n : ℤ) ≤ 0) (by positivity)
    · rw [(by norm_cast : 2 * (m : ℂ) + 2 = ↑(2 * m + 2)), Ne.def, Nat.cast_eq_one]; norm_num
    -- get rid of sine term
    

Proof state:
case intro.inr
m : ℕ
⊢ 2 ^ (1 - (2 * ↑m + 2)) * ↑π ^ (-(2 * ↑m + 2)) * Complex.Gamma (2 * ↑m + 2) *
        Complex.sin (↑π * (1 - (2 * ↑m + 2)) / 2) *
      riemannZeta (2 * ↑m + 2) =
    -1 * ↑(bernoulli (2 * m + 1 + 1)) / (↑(2 * m + 1) + 1)

Next tactic:
rw [show Complex.sin (↑π * (1 - (2 * ↑m + 2)) / 2) = -(-1 : ℂ) ^ m by
        rw [(by field_simp; ring : (π : ℂ) * (1 - (2 * ↑m + 2)) / 2 = π / 2 - (π * m + π))]
        rw [Complex.sin_pi_div_two_sub, Complex.cos_add_pi, neg_inj]
        rcases Nat.even_or_odd' m with ⟨t, rfl | rfl⟩
        · rw [pow_mul, neg_one_sq, one_pow]
          convert Complex.cos_nat_mul_two_pi t using 2
          push_cast; ring_nf
        · rw [pow_add, pow_one, pow_mul, neg_one_sq, one_pow, one_mul]
          convert Complex.cos_nat_mul_two_pi_add_pi t using 2
          push_cast; ring_nf]
theorem riemannZeta_neg_nat_eq_bernoulli (k : ℕ) :
    riemannZeta (-k) = (-1 : ℂ) ^ k * bernoulli (k + 1) / (k + 1) := by
  rcases Nat.even_or_odd' k with ⟨m, rfl | rfl⟩
  · cases' m with m m
    ·-- k = 0 : evaluate explicitly
      rw [Nat.zero_eq, mul_zero, Nat.cast_zero, pow_zero, one_mul, zero_add, neg_zero, zero_add,
        div_one, bernoulli_one, riemannZeta_zero]
      norm_num
    · -- k = 2 * (m + 1) : both sides "trivially" zero
      rw [Nat.cast_mul, ← neg_mul, Nat.cast_two, Nat.cast_succ, riemannZeta_neg_two_mul_nat_add_one,
        bernoulli_eq_bernoulli'_of_ne_one]
      swap; · apply ne_of_gt; norm_num
      rw [bernoulli'_odd_eq_zero ⟨m + 1, rfl⟩ (by norm_num), Rat.cast_zero, mul_zero,
        zero_div]
  · -- k = 2 * m + 1 : the interesting case
    rw [Odd.neg_one_pow ⟨m, rfl⟩]
    rw [show -(↑(2 * m + 1) : ℂ) = 1 - (2 * m + 2) by push_cast; ring]
    rw [riemannZeta_one_sub]
    rotate_left
    · intro n
      rw [(by norm_cast : 2 * (m : ℂ) + 2 = ↑(2 * m + 2)), ← Int.cast_neg_natCast, ← Int.cast_ofNat,
        Ne.def, Int.cast_inj]
      apply ne_of_gt
      exact lt_of_le_of_lt (by norm_num : (-n : ℤ) ≤ 0) (by positivity)
    · rw [(by norm_cast : 2 * (m : ℂ) + 2 = ↑(2 * m + 2)), Ne.def, Nat.cast_eq_one]; norm_num
    -- get rid of sine term
    rw [show Complex.sin (↑π * (1 - (2 * ↑m + 2)) / 2) = -(-1 : ℂ) ^ m by
        

Proof state:
m : ℕ
⊢ Complex.sin (↑π * (1 - (2 * ↑m + 2)) / 2) = -(-1) ^ m

Next tactic:
rw [(by field_simp; ring : (π : ℂ) * (1 - (2 * ↑m + 2)) / 2 = π / 2 - (π * m + π))]
theorem riemannZeta_neg_nat_eq_bernoulli (k : ℕ) :
    riemannZeta (-k) = (-1 : ℂ) ^ k * bernoulli (k + 1) / (k + 1) := by
  rcases Nat.even_or_odd' k with ⟨m, rfl | rfl⟩
  · cases' m with m m
    ·-- k = 0 : evaluate explicitly
      rw [Nat.zero_eq, mul_zero, Nat.cast_zero, pow_zero, one_mul, zero_add, neg_zero, zero_add,
        div_one, bernoulli_one, riemannZeta_zero]
      norm_num
    · -- k = 2 * (m + 1) : both sides "trivially" zero
      rw [Nat.cast_mul, ← neg_mul, Nat.cast_two, Nat.cast_succ, riemannZeta_neg_two_mul_nat_add_one,
        bernoulli_eq_bernoulli'_of_ne_one]
      swap; · apply ne_of_gt; norm_num
      rw [bernoulli'_odd_eq_zero ⟨m + 1, rfl⟩ (by norm_num), Rat.cast_zero, mul_zero,
        zero_div]
  · -- k = 2 * m + 1 : the interesting case
    rw [Odd.neg_one_pow ⟨m, rfl⟩]
    rw [show -(↑(2 * m + 1) : ℂ) = 1 - (2 * m + 2) by push_cast; ring]
    rw [riemannZeta_one_sub]
    rotate_left
    · intro n
      rw [(by norm_cast : 2 * (m : ℂ) + 2 = ↑(2 * m + 2)), ← Int.cast_neg_natCast, ← Int.cast_ofNat,
        Ne.def, Int.cast_inj]
      apply ne_of_gt
      exact lt_of_le_of_lt (by norm_num : (-n : ℤ) ≤ 0) (by positivity)
    · rw [(by norm_cast : 2 * (m : ℂ) + 2 = ↑(2 * m + 2)), Ne.def, Nat.cast_eq_one]; norm_num
    -- get rid of sine term
    rw [show Complex.sin (↑π * (1 - (2 * ↑m + 2)) / 2) = -(-1 : ℂ) ^ m by
        rw [(by 

Proof state:
m : ℕ
⊢ ↑π * (1 - (2 * ↑m + 2)) / 2 = ↑π / 2 - (↑π * ↑m + ↑π)

Next tactic:
field_simp
theorem riemannZeta_neg_nat_eq_bernoulli (k : ℕ) :
    riemannZeta (-k) = (-1 : ℂ) ^ k * bernoulli (k + 1) / (k + 1) := by
  rcases Nat.even_or_odd' k with ⟨m, rfl | rfl⟩
  · cases' m with m m
    ·-- k = 0 : evaluate explicitly
      rw [Nat.zero_eq, mul_zero, Nat.cast_zero, pow_zero, one_mul, zero_add, neg_zero, zero_add,
        div_one, bernoulli_one, riemannZeta_zero]
      norm_num
    · -- k = 2 * (m + 1) : both sides "trivially" zero
      rw [Nat.cast_mul, ← neg_mul, Nat.cast_two, Nat.cast_succ, riemannZeta_neg_two_mul_nat_add_one,
        bernoulli_eq_bernoulli'_of_ne_one]
      swap; · apply ne_of_gt; norm_num
      rw [bernoulli'_odd_eq_zero ⟨m + 1, rfl⟩ (by norm_num), Rat.cast_zero, mul_zero,
        zero_div]
  · -- k = 2 * m + 1 : the interesting case
    rw [Odd.neg_one_pow ⟨m, rfl⟩]
    rw [show -(↑(2 * m + 1) : ℂ) = 1 - (2 * m + 2) by push_cast; ring]
    rw [riemannZeta_one_sub]
    rotate_left
    · intro n
      rw [(by norm_cast : 2 * (m : ℂ) + 2 = ↑(2 * m + 2)), ← Int.cast_neg_natCast, ← Int.cast_ofNat,
        Ne.def, Int.cast_inj]
      apply ne_of_gt
      exact lt_of_le_of_lt (by norm_num : (-n : ℤ) ≤ 0) (by positivity)
    · rw [(by norm_cast : 2 * (m : ℂ) + 2 = ↑(2 * m + 2)), Ne.def, Nat.cast_eq_one]; norm_num
    -- get rid of sine term
    rw [show Complex.sin (↑π * (1 - (2 * ↑m + 2)) / 2) = -(-1 : ℂ) ^ m by
        rw [(by field_simp; 

Proof state:
m : ℕ
⊢ ↑π * (1 - (2 * ↑m + 2)) = ↑π - 2 * (↑π * ↑m + ↑π)

Next tactic:
ring
theorem riemannZeta_neg_nat_eq_bernoulli (k : ℕ) :
    riemannZeta (-k) = (-1 : ℂ) ^ k * bernoulli (k + 1) / (k + 1) := by
  rcases Nat.even_or_odd' k with ⟨m, rfl | rfl⟩
  · cases' m with m m
    ·-- k = 0 : evaluate explicitly
      rw [Nat.zero_eq, mul_zero, Nat.cast_zero, pow_zero, one_mul, zero_add, neg_zero, zero_add,
        div_one, bernoulli_one, riemannZeta_zero]
      norm_num
    · -- k = 2 * (m + 1) : both sides "trivially" zero
      rw [Nat.cast_mul, ← neg_mul, Nat.cast_two, Nat.cast_succ, riemannZeta_neg_two_mul_nat_add_one,
        bernoulli_eq_bernoulli'_of_ne_one]
      swap; · apply ne_of_gt; norm_num
      rw [bernoulli'_odd_eq_zero ⟨m + 1, rfl⟩ (by norm_num), Rat.cast_zero, mul_zero,
        zero_div]
  · -- k = 2 * m + 1 : the interesting case
    rw [Odd.neg_one_pow ⟨m, rfl⟩]
    rw [show -(↑(2 * m + 1) : ℂ) = 1 - (2 * m + 2) by push_cast; ring]
    rw [riemannZeta_one_sub]
    rotate_left
    · intro n
      rw [(by norm_cast : 2 * (m : ℂ) + 2 = ↑(2 * m + 2)), ← Int.cast_neg_natCast, ← Int.cast_ofNat,
        Ne.def, Int.cast_inj]
      apply ne_of_gt
      exact lt_of_le_of_lt (by norm_num : (-n : ℤ) ≤ 0) (by positivity)
    · rw [(by norm_cast : 2 * (m : ℂ) + 2 = ↑(2 * m + 2)), Ne.def, Nat.cast_eq_one]; norm_num
    -- get rid of sine term
    rw [show Complex.sin (↑π * (1 - (2 * ↑m + 2)) / 2) = -(-1 : ℂ) ^ m by
        rw [(by field_simp; ring : (π : ℂ) * (1 - (2 * ↑m + 2)) / 2 = π / 2 - (π * m + π))]
        

Proof state:
m : ℕ
⊢ Complex.sin (↑π / 2 - (↑π * ↑m + ↑π)) = -(-1) ^ m

Next tactic:
rw [Complex.sin_pi_div_two_sub, Complex.cos_add_pi, neg_inj]
theorem riemannZeta_neg_nat_eq_bernoulli (k : ℕ) :
    riemannZeta (-k) = (-1 : ℂ) ^ k * bernoulli (k + 1) / (k + 1) := by
  rcases Nat.even_or_odd' k with ⟨m, rfl | rfl⟩
  · cases' m with m m
    ·-- k = 0 : evaluate explicitly
      rw [Nat.zero_eq, mul_zero, Nat.cast_zero, pow_zero, one_mul, zero_add, neg_zero, zero_add,
        div_one, bernoulli_one, riemannZeta_zero]
      norm_num
    · -- k = 2 * (m + 1) : both sides "trivially" zero
      rw [Nat.cast_mul, ← neg_mul, Nat.cast_two, Nat.cast_succ, riemannZeta_neg_two_mul_nat_add_one,
        bernoulli_eq_bernoulli'_of_ne_one]
      swap; · apply ne_of_gt; norm_num
      rw [bernoulli'_odd_eq_zero ⟨m + 1, rfl⟩ (by norm_num), Rat.cast_zero, mul_zero,
        zero_div]
  · -- k = 2 * m + 1 : the interesting case
    rw [Odd.neg_one_pow ⟨m, rfl⟩]
    rw [show -(↑(2 * m + 1) : ℂ) = 1 - (2 * m + 2) by push_cast; ring]
    rw [riemannZeta_one_sub]
    rotate_left
    · intro n
      rw [(by norm_cast : 2 * (m : ℂ) + 2 = ↑(2 * m + 2)), ← Int.cast_neg_natCast, ← Int.cast_ofNat,
        Ne.def, Int.cast_inj]
      apply ne_of_gt
      exact lt_of_le_of_lt (by norm_num : (-n : ℤ) ≤ 0) (by positivity)
    · rw [(by norm_cast : 2 * (m : ℂ) + 2 = ↑(2 * m + 2)), Ne.def, Nat.cast_eq_one]; norm_num
    -- get rid of sine term
    rw [show Complex.sin (↑π * (1 - (2 * ↑m + 2)) / 2) = -(-1 : ℂ) ^ m by
        rw [(by field_simp; ring : (π : ℂ) * (1 - (2 * ↑m + 2)) / 2 = π / 2 - (π * m + π))]
        rw [Complex.sin_pi_div_two_sub, Complex.cos_add_pi, neg_inj]
        

Proof state:
m : ℕ
⊢ Complex.cos (↑π * ↑m) = (-1) ^ m

Next tactic:
rcases Nat.even_or_odd' m with ⟨t, rfl | rfl⟩