awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.NumberTheory.LSeries.HurwitzZetaValues (goal)
Goal:
k : ℕ
x : ℝ
hk : k ≠ 0
hx : Membership.mem (γ := Set ℝ) (Icc 0 1) x
h1 : ∀ (n : ℕ), HMul.hMul (α := ℂ) 2 (↑k : ℂ) ≠ -(↑n : ℂ)
h2 : HMul.hMul (α := ℂ) 2 (↑k : ℂ) ≠ 1
h3 : (HMul.hMul (α := ℂ) 2 (↑k : ℂ)).Gammaℂ ≠ 0
e_a : Complex.cos ((↑π : ℂ) * (HMul.hMul (α := ℂ) 2 (↑k : ℂ) / 2)) * HPow.hPow (α := ℂ) (-1) (k + 1) = -1
e_a₁ : Complex.cos ((↑π : ℂ) * (↑k : ℂ)) * HPow.hPow (α := ℂ) (-1) (k + 1) = -1
e_a₂ : Complex.cos ((↑π : ℂ) * (↑(↑k : ℝ) : ℂ)) * HPow.hPow (α := ℂ) (-1) (k + 1) = -1
⊢ Complex.cos ((↑π : ℂ) * HMul.hMul (α := ℂ) 2 (↑k : ℂ) / 2) * HPow.hPow (α := ℂ) (-1) (k + 1) = -1
have e_a₃ : Complex.cos (↑(π * (↑k : ℝ)) : ℂ) * HPow.hPow (α := ℂ) (-1) (k + 1) = -1
Tactic: rw [mul_div_assoc, mul_div_cancel_left₀ _ two_ne_zero, ← ofReal_natCast, ← ofReal_mul,
    ← ofReal_cos, mul_comm π, ← sub_zero (k * π), cos_nat_mul_pi_sub, Real.cos_zero, mul_one,
    ofReal_pow, ofReal_neg, ofReal_one, pow_succ, mul_neg_one, mul_neg, ← mul_pow, neg_one_mul,
    neg_neg, one_pow]
-- Lean 4 proof step from Mathlib.NumberTheory.LSeries.HurwitzZetaValues (goal)
Goal:
k : ℕ
x : ℝ
hk : k ≠ 0
hx : Membership.mem (γ := Set ℝ) (Icc 0 1) x
h1 : ∀ (n : ℕ), HMul.hMul (α := ℂ) 2 (↑k : ℂ) ≠ -(↑n : ℂ)
h2 : HMul.hMul (α := ℂ) 2 (↑k : ℂ) ≠ 1
h3 : (HMul.hMul (α := ℂ) 2 (↑k : ℂ)).Gammaℂ ≠ 0
e_a : Complex.cos ((↑π : ℂ) * (HMul.hMul (α := ℂ) 2 (↑k : ℂ) / 2)) * HPow.hPow (α := ℂ) (-1) (k + 1) = -1
e_a₁ : Complex.cos ((↑π : ℂ) * (↑k : ℂ)) * HPow.hPow (α := ℂ) (-1) (k + 1) = -1
e_a₂ : Complex.cos ((↑π : ℂ) * (↑(↑k : ℝ) : ℂ)) * HPow.hPow (α := ℂ) (-1) (k + 1) = -1
e_a₃ : Complex.cos (↑(π * (↑k : ℝ)) : ℂ) * HPow.hPow (α := ℂ) (-1) (k + 1) = -1
⊢ Complex.cos ((↑π : ℂ) * HMul.hMul (α := ℂ) 2 (↑k : ℂ) / 2) * HPow.hPow (α := ℂ) (-1) (k + 1) = -1
have e_a₄ : (↑(Real.cos (π * (↑k : ℝ))) : ℂ) * HPow.hPow (α := ℂ) (-1) (k + 1) = -1
Tactic: rw [mul_div_assoc, mul_div_cancel_left₀ _ two_ne_zero, ← ofReal_natCast, ← ofReal_mul,
    ← ofReal_cos, mul_comm π, ← sub_zero (k * π), cos_nat_mul_pi_sub, Real.cos_zero, mul_one,
    ofReal_pow, ofReal_neg, ofReal_one, pow_succ, mul_neg_one, mul_neg, ← mul_pow, neg_one_mul,
    neg_neg, one_pow]
-- Lean 4 proof step from Mathlib.NumberTheory.LSeries.HurwitzZetaValues (goal)
Goal:
k : ℕ
x : ℝ
hk : k ≠ 0
hx : Membership.mem (γ := Set ℝ) (Icc 0 1) x
h1 : ∀ (n : ℕ), HMul.hMul (α := ℂ) 2 (↑k : ℂ) ≠ -(↑n : ℂ)
h2 : HMul.hMul (α := ℂ) 2 (↑k : ℂ) ≠ 1
h3 : (HMul.hMul (α := ℂ) 2 (↑k : ℂ)).Gammaℂ ≠ 0
e_a : Complex.cos ((↑π : ℂ) * (HMul.hMul (α := ℂ) 2 (↑k : ℂ) / 2)) * HPow.hPow (α := ℂ) (-1) (k + 1) = -1
e_a₁ : Complex.cos ((↑π : ℂ) * (↑k : ℂ)) * HPow.hPow (α := ℂ) (-1) (k + 1) = -1
e_a₂ : Complex.cos ((↑π : ℂ) * (↑(↑k : ℝ) : ℂ)) * HPow.hPow (α := ℂ) (-1) (k + 1) = -1
e_a₃ : Complex.cos (↑(π * (↑k : ℝ)) : ℂ) * HPow.hPow (α := ℂ) (-1) (k + 1) = -1
e_a₄ : (↑(Real.cos (π * (↑k : ℝ))) : ℂ) * HPow.hPow (α := ℂ) (-1) (k + 1) = -1
⊢ Complex.cos ((↑π : ℂ) * HMul.hMul (α := ℂ) 2 (↑k : ℂ) / 2) * HPow.hPow (α := ℂ) (-1) (k + 1) = -1
have e_a₅ : (↑(Real.cos ((↑k : ℝ) * π)) : ℂ) * HPow.hPow (α := ℂ) (-1) (k + 1) = -1
Tactic: rw [mul_div_assoc, mul_div_cancel_left₀ _ two_ne_zero, ← ofReal_natCast, ← ofReal_mul,
    ← ofReal_cos, mul_comm π, ← sub_zero (k * π), cos_nat_mul_pi_sub, Real.cos_zero, mul_one,
    ofReal_pow, ofReal_neg, ofReal_one, pow_succ, mul_neg_one, mul_neg, ← mul_pow, neg_one_mul,
    neg_neg, one_pow]
-- Lean 4 proof step from Mathlib.NumberTheory.LSeries.HurwitzZetaValues (goal)
Goal:
k : ℕ
x : ℝ
hk : k ≠ 0
hx : Membership.mem (γ := Set ℝ) (Icc 0 1) x
h1 : ∀ (n : ℕ), HMul.hMul (α := ℂ) 2 (↑k : ℂ) ≠ -(↑n : ℂ)
h2 : HMul.hMul (α := ℂ) 2 (↑k : ℂ) ≠ 1
h3 : (HMul.hMul (α := ℂ) 2 (↑k : ℂ)).Gammaℂ ≠ 0
e_a : Complex.cos ((↑π : ℂ) * (HMul.hMul (α := ℂ) 2 (↑k : ℂ) / 2)) * HPow.hPow (α := ℂ) (-1) (k + 1) = -1
e_a₁ : Complex.cos ((↑π : ℂ) * (↑k : ℂ)) * HPow.hPow (α := ℂ) (-1) (k + 1) = -1
e_a₂ : Complex.cos ((↑π : ℂ) * (↑(↑k : ℝ) : ℂ)) * HPow.hPow (α := ℂ) (-1) (k + 1) = -1
e_a₃ : Complex.cos (↑(π * (↑k : ℝ)) : ℂ) * HPow.hPow (α := ℂ) (-1) (k + 1) = -1
e_a₄ : (↑(Real.cos (π * (↑k : ℝ))) : ℂ) * HPow.hPow (α := ℂ) (-1) (k + 1) = -1
e_a₅ : (↑(Real.cos ((↑k : ℝ) * π)) : ℂ) * HPow.hPow (α := ℂ) (-1) (k + 1) = -1
⊢ Complex.cos ((↑π : ℂ) * HMul.hMul (α := ℂ) 2 (↑k : ℂ) / 2) * HPow.hPow (α := ℂ) (-1) (k + 1) = -1
have e_a₆ : (↑(Real.cos ((↑k : ℝ) * π - 0)) : ℂ) * HPow.hPow (α := ℂ) (-1) (k + 1) = -1
Tactic: rw [mul_div_assoc, mul_div_cancel_left₀ _ two_ne_zero, ← ofReal_natCast, ← ofReal_mul,
    ← ofReal_cos, mul_comm π, ← sub_zero (k * π), cos_nat_mul_pi_sub, Real.cos_zero, mul_one,
    ofReal_pow, ofReal_neg, ofReal_one, pow_succ, mul_neg_one, mul_neg, ← mul_pow, neg_one_mul,
    neg_neg, one_pow]
-- Lean 4 proof step from Mathlib.NumberTheory.LSeries.HurwitzZetaValues (goal)
Goal:
k : ℕ
x : ℝ
hk : k ≠ 0
hx : Membership.mem (γ := Set ℝ) (Icc 0 1) x
h1 : ∀ (n : ℕ), HMul.hMul (α := ℂ) 2 (↑k : ℂ) ≠ -(↑n : ℂ)
h2 : HMul.hMul (α := ℂ) 2 (↑k : ℂ) ≠ 1
h3 : (HMul.hMul (α := ℂ) 2 (↑k : ℂ)).Gammaℂ ≠ 0
e_a : Complex.cos ((↑π : ℂ) * (HMul.hMul (α := ℂ) 2 (↑k : ℂ) / 2)) * HPow.hPow (α := ℂ) (-1) (k + 1) = -1
e_a₁ : Complex.cos ((↑π : ℂ) * (↑k : ℂ)) * HPow.hPow (α := ℂ) (-1) (k + 1) = -1
e_a₂ : Complex.cos ((↑π : ℂ) * (↑(↑k : ℝ) : ℂ)) * HPow.hPow (α := ℂ) (-1) (k + 1) = -1
e_a₃ : Complex.cos (↑(π * (↑k : ℝ)) : ℂ) * HPow.hPow (α := ℂ) (-1) (k + 1) = -1
e_a₄ : (↑(Real.cos (π * (↑k : ℝ))) : ℂ) * HPow.hPow (α := ℂ) (-1) (k + 1) = -1
e_a₅ : (↑(Real.cos ((↑k : ℝ) * π)) : ℂ) * HPow.hPow (α := ℂ) (-1) (k + 1) = -1
e_a₆ : (↑(Real.cos ((↑k : ℝ) * π - 0)) : ℂ) * HPow.hPow (α := ℂ) (-1) (k + 1) = -1
⊢ Complex.cos ((↑π : ℂ) * HMul.hMul (α := ℂ) 2 (↑k : ℂ) / 2) * HPow.hPow (α := ℂ) (-1) (k + 1) = -1
have e_a₇ : (↑(HPow.hPow (α := ℝ) (-1) k * Real.cos 0) : ℂ) * HPow.hPow (α := ℂ) (-1) (k + 1) = -1
Tactic: rw [mul_div_assoc, mul_div_cancel_left₀ _ two_ne_zero, ← ofReal_natCast, ← ofReal_mul,
    ← ofReal_cos, mul_comm π, ← sub_zero (k * π), cos_nat_mul_pi_sub, Real.cos_zero, mul_one,
    ofReal_pow, ofReal_neg, ofReal_one, pow_succ, mul_neg_one, mul_neg, ← mul_pow, neg_one_mul,
    neg_neg, one_pow]
-- Lean 4 proof step from Mathlib.NumberTheory.LSeries.HurwitzZetaValues (goal)
Goal:
k : ℕ
x : ℝ
hk : k ≠ 0
hx : Membership.mem (γ := Set ℝ) (Icc 0 1) x
h1 : ∀ (n : ℕ), HMul.hMul (α := ℂ) 2 (↑k : ℂ) ≠ -(↑n : ℂ)
h2 : HMul.hMul (α := ℂ) 2 (↑k : ℂ) ≠ 1
h3 : (HMul.hMul (α := ℂ) 2 (↑k : ℂ)).Gammaℂ ≠ 0
e_a : Complex.cos ((↑π : ℂ) * (HMul.hMul (α := ℂ) 2 (↑k : ℂ) / 2)) * HPow.hPow (α := ℂ) (-1) (k + 1) = -1
e_a₁ : Complex.cos ((↑π : ℂ) * (↑k : ℂ)) * HPow.hPow (α := ℂ) (-1) (k + 1) = -1
e_a₂ : Complex.cos ((↑π : ℂ) * (↑(↑k : ℝ) : ℂ)) * HPow.hPow (α := ℂ) (-1) (k + 1) = -1
e_a₃ : Complex.cos (↑(π * (↑k : ℝ)) : ℂ) * HPow.hPow (α := ℂ) (-1) (k + 1) = -1
e_a₄ : (↑(Real.cos (π * (↑k : ℝ))) : ℂ) * HPow.hPow (α := ℂ) (-1) (k + 1) = -1
e_a₅ : (↑(Real.cos ((↑k : ℝ) * π)) : ℂ) * HPow.hPow (α := ℂ) (-1) (k + 1) = -1
e_a₆ : (↑(Real.cos ((↑k : ℝ) * π - 0)) : ℂ) * HPow.hPow (α := ℂ) (-1) (k + 1) = -1
e_a₇ : (↑(HPow.hPow (α := ℝ) (-1) k * Real.cos 0) : ℂ) * HPow.hPow (α := ℂ) (-1) (k + 1) = -1
⊢ Complex.cos ((↑π : ℂ) * HMul.hMul (α := ℂ) 2 (↑k : ℂ) / 2) * HPow.hPow (α := ℂ) (-1) (k + 1) = -1
have e_a₈ : (↑(HPow.hPow (α := ℝ) (-1) k * 1) : ℂ) * HPow.hPow (α := ℂ) (-1) (k + 1) = -1
Tactic: rw [mul_div_assoc, mul_div_cancel_left₀ _ two_ne_zero, ← ofReal_natCast, ← ofReal_mul,
    ← ofReal_cos, mul_comm π, ← sub_zero (k * π), cos_nat_mul_pi_sub, Real.cos_zero, mul_one,
    ofReal_pow, ofReal_neg, ofReal_one, pow_succ, mul_neg_one, mul_neg, ← mul_pow, neg_one_mul,
    neg_neg, one_pow]
-- Lean 4 proof step from Mathlib.NumberTheory.LSeries.HurwitzZetaValues (goal)
Goal:
k : ℕ
x : ℝ
hk : k ≠ 0
hx : Membership.mem (γ := Set ℝ) (Icc 0 1) x
h1 : ∀ (n : ℕ), HMul.hMul (α := ℂ) 2 (↑k : ℂ) ≠ -(↑n : ℂ)
h2 : HMul.hMul (α := ℂ) 2 (↑k : ℂ) ≠ 1
h3 : (HMul.hMul (α := ℂ) 2 (↑k : ℂ)).Gammaℂ ≠ 0
e_a : Complex.cos ((↑π : ℂ) * (HMul.hMul (α := ℂ) 2 (↑k : ℂ) / 2)) * HPow.hPow (α := ℂ) (-1) (k + 1) = -1
e_a₁ : Complex.cos ((↑π : ℂ) * (↑k : ℂ)) * HPow.hPow (α := ℂ) (-1) (k + 1) = -1
e_a₂ : Complex.cos ((↑π : ℂ) * (↑(↑k : ℝ) : ℂ)) * HPow.hPow (α := ℂ) (-1) (k + 1) = -1
e_a₃ : Complex.cos (↑(π * (↑k : ℝ)) : ℂ) * HPow.hPow (α := ℂ) (-1) (k + 1) = -1
e_a₄ : (↑(Real.cos (π * (↑k : ℝ))) : ℂ) * HPow.hPow (α := ℂ) (-1) (k + 1) = -1
e_a₅ : (↑(Real.cos ((↑k : ℝ) * π)) : ℂ) * HPow.hPow (α := ℂ) (-1) (k + 1) = -1
e_a₆ : (↑(Real.cos ((↑k : ℝ) * π - 0)) : ℂ) * HPow.hPow (α := ℂ) (-1) (k + 1) = -1
e_a₇ : (↑(HPow.hPow (α := ℝ) (-1) k * Real.cos 0) : ℂ) * HPow.hPow (α := ℂ) (-1) (k + 1) = -1
e_a₈ : (↑(HPow.hPow (α := ℝ) (-1) k * 1) : ℂ) * HPow.hPow (α := ℂ) (-1) (k + 1) = -1
⊢ Complex.cos ((↑π : ℂ) * HMul.hMul (α := ℂ) 2 (↑k : ℂ) / 2) * HPow.hPow (α := ℂ) (-1) (k + 1) = -1
have e_a₉ : (↑(HPow.hPow (α := ℝ) (-1) k) : ℂ) * HPow.hPow (α := ℂ) (-1) (k + 1) = -1
Tactic: rw [mul_div_assoc, mul_div_cancel_left₀ _ two_ne_zero, ← ofReal_natCast, ← ofReal_mul,
    ← ofReal_cos, mul_comm π, ← sub_zero (k * π), cos_nat_mul_pi_sub, Real.cos_zero, mul_one,
    ofReal_pow, ofReal_neg, ofReal_one, pow_succ, mul_neg_one, mul_neg, ← mul_pow, neg_one_mul,
    neg_neg, one_pow]
-- Lean 4 proof step from Mathlib.NumberTheory.LSeries.HurwitzZetaValues (goal)
Goal:
k : ℕ
x : ℝ
hk : k ≠ 0
hx : Membership.mem (γ := Set ℝ) (Icc 0 1) x
h1 : ∀ (n : ℕ), HMul.hMul (α := ℂ) 2 (↑k : ℂ) ≠ -(↑n : ℂ)
h2 : HMul.hMul (α := ℂ) 2 (↑k : ℂ) ≠ 1
h3 : (HMul.hMul (α := ℂ) 2 (↑k : ℂ)).Gammaℂ ≠ 0
e_a : Complex.cos ((↑π : ℂ) * (HMul.hMul (α := ℂ) 2 (↑k : ℂ) / 2)) * HPow.hPow (α := ℂ) (-1) (k + 1) = -1
e_a₁ : Complex.cos ((↑π : ℂ) * (↑k : ℂ)) * HPow.hPow (α := ℂ) (-1) (k + 1) = -1
e_a₂ : Complex.cos ((↑π : ℂ) * (↑(↑k : ℝ) : ℂ)) * HPow.hPow (α := ℂ) (-1) (k + 1) = -1
e_a₃ : Complex.cos (↑(π * (↑k : ℝ)) : ℂ) * HPow.hPow (α := ℂ) (-1) (k + 1) = -1
e_a₄ : (↑(Real.cos (π * (↑k : ℝ))) : ℂ) * HPow.hPow (α := ℂ) (-1) (k + 1) = -1
e_a₅ : (↑(Real.cos ((↑k : ℝ) * π)) : ℂ) * HPow.hPow (α := ℂ) (-1) (k + 1) = -1
e_a₆ : (↑(Real.cos ((↑k : ℝ) * π - 0)) : ℂ) * HPow.hPow (α := ℂ) (-1) (k + 1) = -1
e_a₇ : (↑(HPow.hPow (α := ℝ) (-1) k * Real.cos 0) : ℂ) * HPow.hPow (α := ℂ) (-1) (k + 1) = -1
e_a₈ : (↑(HPow.hPow (α := ℝ) (-1) k * 1) : ℂ) * HPow.hPow (α := ℂ) (-1) (k + 1) = -1
e_a₉ : (↑(HPow.hPow (α := ℝ) (-1) k) : ℂ) * HPow.hPow (α := ℂ) (-1) (k + 1) = -1
⊢ Complex.cos ((↑π : ℂ) * HMul.hMul (α := ℂ) 2 (↑k : ℂ) / 2) * HPow.hPow (α := ℂ) (-1) (k + 1) = -1
have e_a₁₀ : (↑(-1) : ℂ) ^ k * HPow.hPow (α := ℂ) (-1) (k + 1) = -1
Tactic: rw [mul_div_assoc, mul_div_cancel_left₀ _ two_ne_zero, ← ofReal_natCast, ← ofReal_mul,
    ← ofReal_cos, mul_comm π, ← sub_zero (k * π), cos_nat_mul_pi_sub, Real.cos_zero, mul_one,
    ofReal_pow, ofReal_neg, ofReal_one, pow_succ, mul_neg_one, mul_neg, ← mul_pow, neg_one_mul,
    neg_neg, one_pow]
-- Lean 4 proof step from Mathlib.NumberTheory.LSeries.HurwitzZetaValues (goal)
Goal:
k : ℕ
x : ℝ
hk : k ≠ 0
hx : Membership.mem (γ := Set ℝ) (Icc 0 1) x
h1 : ∀ (n : ℕ), HMul.hMul (α := ℂ) 2 (↑k : ℂ) ≠ -(↑n : ℂ)
h2 : HMul.hMul (α := ℂ) 2 (↑k : ℂ) ≠ 1
h3 : (HMul.hMul (α := ℂ) 2 (↑k : ℂ)).Gammaℂ ≠ 0
e_a : Complex.cos ((↑π : ℂ) * (HMul.hMul (α := ℂ) 2 (↑k : ℂ) / 2)) * HPow.hPow (α := ℂ) (-1) (k + 1) = -1
e_a₁ : Complex.cos ((↑π : ℂ) * (↑k : ℂ)) * HPow.hPow (α := ℂ) (-1) (k + 1) = -1
e_a₂ : Complex.cos ((↑π : ℂ) * (↑(↑k : ℝ) : ℂ)) * HPow.hPow (α := ℂ) (-1) (k + 1) = -1
e_a₃ : Complex.cos (↑(π * (↑k : ℝ)) : ℂ) * HPow.hPow (α := ℂ) (-1) (k + 1) = -1
e_a₄ : (↑(Real.cos (π * (↑k : ℝ))) : ℂ) * HPow.hPow (α := ℂ) (-1) (k + 1) = -1
e_a₅ : (↑(Real.cos ((↑k : ℝ) * π)) : ℂ) * HPow.hPow (α := ℂ) (-1) (k + 1) = -1
e_a₆ : (↑(Real.cos ((↑k : ℝ) * π - 0)) : ℂ) * HPow.hPow (α := ℂ) (-1) (k + 1) = -1
e_a₇ : (↑(HPow.hPow (α := ℝ) (-1) k * Real.cos 0) : ℂ) * HPow.hPow (α := ℂ) (-1) (k + 1) = -1
e_a₈ : (↑(HPow.hPow (α := ℝ) (-1) k * 1) : ℂ) * HPow.hPow (α := ℂ) (-1) (k + 1) = -1
e_a₉ : (↑(HPow.hPow (α := ℝ) (-1) k) : ℂ) * HPow.hPow (α := ℂ) (-1) (k + 1) = -1
e_a₁₀ : (↑(-1) : ℂ) ^ k * HPow.hPow (α := ℂ) (-1) (k + 1) = -1
⊢ Complex.cos ((↑π : ℂ) * HMul.hMul (α := ℂ) 2 (↑k : ℂ) / 2) * HPow.hPow (α := ℂ) (-1) (k + 1) = -1
have e_a₁₁ : (-(↑1 : ℂ)) ^ k * HPow.hPow (α := ℂ) (-1) (k + 1) = -1
Tactic: rw [mul_div_assoc, mul_div_cancel_left₀ _ two_ne_zero, ← ofReal_natCast, ← ofReal_mul,
    ← ofReal_cos, mul_comm π, ← sub_zero (k * π), cos_nat_mul_pi_sub, Real.cos_zero, mul_one,
    ofReal_pow, ofReal_neg, ofReal_one, pow_succ, mul_neg_one, mul_neg, ← mul_pow, neg_one_mul,
    neg_neg, one_pow]
-- Lean 4 proof step from Mathlib.NumberTheory.LSeries.HurwitzZetaValues (goal)
Goal:
k : ℕ
x : ℝ
hk : k ≠ 0
hx : Membership.mem (γ := Set ℝ) (Icc 0 1) x
h1 : ∀ (n : ℕ), HMul.hMul (α := ℂ) 2 (↑k : ℂ) ≠ -(↑n : ℂ)
h2 : HMul.hMul (α := ℂ) 2 (↑k : ℂ) ≠ 1
h3 : (HMul.hMul (α := ℂ) 2 (↑k : ℂ)).Gammaℂ ≠ 0
e_a : Complex.cos ((↑π : ℂ) * (HMul.hMul (α := ℂ) 2 (↑k : ℂ) / 2)) * HPow.hPow (α := ℂ) (-1) (k + 1) = -1
e_a₁ : Complex.cos ((↑π : ℂ) * (↑k : ℂ)) * HPow.hPow (α := ℂ) (-1) (k + 1) = -1
e_a₂ : Complex.cos ((↑π : ℂ) * (↑(↑k : ℝ) : ℂ)) * HPow.hPow (α := ℂ) (-1) (k + 1) = -1
e_a₃ : Complex.cos (↑(π * (↑k : ℝ)) : ℂ) * HPow.hPow (α := ℂ) (-1) (k + 1) = -1
e_a₄ : (↑(Real.cos (π * (↑k : ℝ))) : ℂ) * HPow.hPow (α := ℂ) (-1) (k + 1) = -1
e_a₅ : (↑(Real.cos ((↑k : ℝ) * π)) : ℂ) * HPow.hPow (α := ℂ) (-1) (k + 1) = -1
e_a₆ : (↑(Real.cos ((↑k : ℝ) * π - 0)) : ℂ) * HPow.hPow (α := ℂ) (-1) (k + 1) = -1
e_a₇ : (↑(HPow.hPow (α := ℝ) (-1) k * Real.cos 0) : ℂ) * HPow.hPow (α := ℂ) (-1) (k + 1) = -1
e_a₈ : (↑(HPow.hPow (α := ℝ) (-1) k * 1) : ℂ) * HPow.hPow (α := ℂ) (-1) (k + 1) = -1
e_a₉ : (↑(HPow.hPow (α := ℝ) (-1) k) : ℂ) * HPow.hPow (α := ℂ) (-1) (k + 1) = -1
e_a₁₀ : (↑(-1) : ℂ) ^ k * HPow.hPow (α := ℂ) (-1) (k + 1) = -1
e_a₁₁ : (-(↑1 : ℂ)) ^ k * HPow.hPow (α := ℂ) (-1) (k + 1) = -1
⊢ Complex.cos ((↑π : ℂ) * HMul.hMul (α := ℂ) 2 (↑k : ℂ) / 2) * HPow.hPow (α := ℂ) (-1) (k + 1) = -1
have e_a₁₂ : HPow.hPow (α := ℂ) (-1) k * HPow.hPow (α := ℂ) (-1) (k + 1) = -1
Tactic: rw [mul_div_assoc, mul_div_cancel_left₀ _ two_ne_zero, ← ofReal_natCast, ← ofReal_mul,
    ← ofReal_cos, mul_comm π, ← sub_zero (k * π), cos_nat_mul_pi_sub, Real.cos_zero, mul_one,
    ofReal_pow, ofReal_neg, ofReal_one, pow_succ, mul_neg_one, mul_neg, ← mul_pow, neg_one_mul,
    neg_neg, one_pow]
-- Lean 4 proof step from Mathlib.NumberTheory.LSeries.HurwitzZetaValues (goal)
Goal:
k : ℕ
x : ℝ
hk : k ≠ 0
hx : Membership.mem (γ := Set ℝ) (Icc 0 1) x
h1 : ∀ (n : ℕ), HMul.hMul (α := ℂ) 2 (↑k : ℂ) ≠ -(↑n : ℂ)
h2 : HMul.hMul (α := ℂ) 2 (↑k : ℂ) ≠ 1
h3 : (HMul.hMul (α := ℂ) 2 (↑k : ℂ)).Gammaℂ ≠ 0
e_a : Complex.cos ((↑π : ℂ) * (HMul.hMul (α := ℂ) 2 (↑k : ℂ) / 2)) * HPow.hPow (α := ℂ) (-1) (k + 1) = -1
e_a₁ : Complex.cos ((↑π : ℂ) * (↑k : ℂ)) * HPow.hPow (α := ℂ) (-1) (k + 1) = -1
e_a₂ : Complex.cos ((↑π : ℂ) * (↑(↑k : ℝ) : ℂ)) * HPow.hPow (α := ℂ) (-1) (k + 1) = -1
e_a₃ : Complex.cos (↑(π * (↑k : ℝ)) : ℂ) * HPow.hPow (α := ℂ) (-1) (k + 1) = -1
e_a₄ : (↑(Real.cos (π * (↑k : ℝ))) : ℂ) * HPow.hPow (α := ℂ) (-1) (k + 1) = -1
e_a₅ : (↑(Real.cos ((↑k : ℝ) * π)) : ℂ) * HPow.hPow (α := ℂ) (-1) (k + 1) = -1
e_a₆ : (↑(Real.cos ((↑k : ℝ) * π - 0)) : ℂ) * HPow.hPow (α := ℂ) (-1) (k + 1) = -1
e_a₇ : (↑(HPow.hPow (α := ℝ) (-1) k * Real.cos 0) : ℂ) * HPow.hPow (α := ℂ) (-1) (k + 1) = -1
e_a₈ : (↑(HPow.hPow (α := ℝ) (-1) k * 1) : ℂ) * HPow.hPow (α := ℂ) (-1) (k + 1) = -1
e_a₉ : (↑(HPow.hPow (α := ℝ) (-1) k) : ℂ) * HPow.hPow (α := ℂ) (-1) (k + 1) = -1
e_a₁₀ : (↑(-1) : ℂ) ^ k * HPow.hPow (α := ℂ) (-1) (k + 1) = -1
e_a₁₁ : (-(↑1 : ℂ)) ^ k * HPow.hPow (α := ℂ) (-1) (k + 1) = -1
e_a₁₂ : HPow.hPow (α := ℂ) (-1) k * HPow.hPow (α := ℂ) (-1) (k + 1) = -1
⊢ Complex.cos ((↑π : ℂ) * HMul.hMul (α := ℂ) 2 (↑k : ℂ) / 2) * HPow.hPow (α := ℂ) (-1) (k + 1) = -1
have e_a₁₃ : HPow.hPow (α := ℂ) (-1) k * (HPow.hPow (α := ℂ) (-1) k * -1) = -1
Tactic: rw [mul_div_assoc, mul_div_cancel_left₀ _ two_ne_zero, ← ofReal_natCast, ← ofReal_mul,
    ← ofReal_cos, mul_comm π, ← sub_zero (k * π), cos_nat_mul_pi_sub, Real.cos_zero, mul_one,
    ofReal_pow, ofReal_neg, ofReal_one, pow_succ, mul_neg_one, mul_neg, ← mul_pow, neg_one_mul,
    neg_neg, one_pow]
-- Lean 4 proof step from Mathlib.NumberTheory.LSeries.HurwitzZetaValues (goal)
Goal:
k : ℕ
x : ℝ
hk : k ≠ 0
hx : Membership.mem (γ := Set ℝ) (Icc 0 1) x
h1 : ∀ (n : ℕ), HMul.hMul (α := ℂ) 2 (↑k : ℂ) ≠ -(↑n : ℂ)
h2 : HMul.hMul (α := ℂ) 2 (↑k : ℂ) ≠ 1
h3 : (HMul.hMul (α := ℂ) 2 (↑k : ℂ)).Gammaℂ ≠ 0
e_a : Complex.cos ((↑π : ℂ) * (HMul.hMul (α := ℂ) 2 (↑k : ℂ) / 2)) * HPow.hPow (α := ℂ) (-1) (k + 1) = -1
e_a₁ : Complex.cos ((↑π : ℂ) * (↑k : ℂ)) * HPow.hPow (α := ℂ) (-1) (k + 1) = -1
e_a₂ : Complex.cos ((↑π : ℂ) * (↑(↑k : ℝ) : ℂ)) * HPow.hPow (α := ℂ) (-1) (k + 1) = -1
e_a₃ : Complex.cos (↑(π * (↑k : ℝ)) : ℂ) * HPow.hPow (α := ℂ) (-1) (k + 1) = -1
e_a₄ : (↑(Real.cos (π * (↑k : ℝ))) : ℂ) * HPow.hPow (α := ℂ) (-1) (k + 1) = -1
e_a₅ : (↑(Real.cos ((↑k : ℝ) * π)) : ℂ) * HPow.hPow (α := ℂ) (-1) (k + 1) = -1
e_a₆ : (↑(Real.cos ((↑k : ℝ) * π - 0)) : ℂ) * HPow.hPow (α := ℂ) (-1) (k + 1) = -1
e_a₇ : (↑(HPow.hPow (α := ℝ) (-1) k * Real.cos 0) : ℂ) * HPow.hPow (α := ℂ) (-1) (k + 1) = -1
e_a₈ : (↑(HPow.hPow (α := ℝ) (-1) k * 1) : ℂ) * HPow.hPow (α := ℂ) (-1) (k + 1) = -1
e_a₉ : (↑(HPow.hPow (α := ℝ) (-1) k) : ℂ) * HPow.hPow (α := ℂ) (-1) (k + 1) = -1
e_a₁₀ : (↑(-1) : ℂ) ^ k * HPow.hPow (α := ℂ) (-1) (k + 1) = -1
e_a₁₁ : (-(↑1 : ℂ)) ^ k * HPow.hPow (α := ℂ) (-1) (k + 1) = -1
e_a₁₂ : HPow.hPow (α := ℂ) (-1) k * HPow.hPow (α := ℂ) (-1) (k + 1) = -1
e_a₁₃ : HPow.hPow (α := ℂ) (-1) k * (HPow.hPow (α := ℂ) (-1) k * -1) = -1
⊢ Complex.cos ((↑π : ℂ) * HMul.hMul (α := ℂ) 2 (↑k : ℂ) / 2) * HPow.hPow (α := ℂ) (-1) (k + 1) = -1
have e_a₁₄ : HPow.hPow (α := ℂ) (-1) k * -HPow.hPow (α := ℂ) (-1) k = -1
Tactic: rw [mul_div_assoc, mul_div_cancel_left₀ _ two_ne_zero, ← ofReal_natCast, ← ofReal_mul,
    ← ofReal_cos, mul_comm π, ← sub_zero (k * π), cos_nat_mul_pi_sub, Real.cos_zero, mul_one,
    ofReal_pow, ofReal_neg, ofReal_one, pow_succ, mul_neg_one, mul_neg, ← mul_pow, neg_one_mul,
    neg_neg, one_pow]
-- Lean 4 proof step from Mathlib.NumberTheory.LSeries.HurwitzZetaValues (goal)
Goal:
k : ℕ
x : ℝ
hk : k ≠ 0
hx : Membership.mem (γ := Set ℝ) (Icc 0 1) x
h1 : ∀ (n : ℕ), HMul.hMul (α := ℂ) 2 (↑k : ℂ) ≠ -(↑n : ℂ)
h2 : HMul.hMul (α := ℂ) 2 (↑k : ℂ) ≠ 1
h3 : (HMul.hMul (α := ℂ) 2 (↑k : ℂ)).Gammaℂ ≠ 0
e_a : Complex.cos ((↑π : ℂ) * (HMul.hMul (α := ℂ) 2 (↑k : ℂ) / 2)) * HPow.hPow (α := ℂ) (-1) (k + 1) = -1
e_a₁ : Complex.cos ((↑π : ℂ) * (↑k : ℂ)) * HPow.hPow (α := ℂ) (-1) (k + 1) = -1
e_a₂ : Complex.cos ((↑π : ℂ) * (↑(↑k : ℝ) : ℂ)) * HPow.hPow (α := ℂ) (-1) (k + 1) = -1
e_a₃ : Complex.cos (↑(π * (↑k : ℝ)) : ℂ) * HPow.hPow (α := ℂ) (-1) (k + 1) = -1
e_a₄ : (↑(Real.cos (π * (↑k : ℝ))) : ℂ) * HPow.hPow (α := ℂ) (-1) (k + 1) = -1
e_a₅ : (↑(Real.cos ((↑k : ℝ) * π)) : ℂ) * HPow.hPow (α := ℂ) (-1) (k + 1) = -1
e_a₆ : (↑(Real.cos ((↑k : ℝ) * π - 0)) : ℂ) * HPow.hPow (α := ℂ) (-1) (k + 1) = -1
e_a₇ : (↑(HPow.hPow (α := ℝ) (-1) k * Real.cos 0) : ℂ) * HPow.hPow (α := ℂ) (-1) (k + 1) = -1
e_a₈ : (↑(HPow.hPow (α := ℝ) (-1) k * 1) : ℂ) * HPow.hPow (α := ℂ) (-1) (k + 1) = -1
e_a₉ : (↑(HPow.hPow (α := ℝ) (-1) k) : ℂ) * HPow.hPow (α := ℂ) (-1) (k + 1) = -1
e_a₁₀ : (↑(-1) : ℂ) ^ k * HPow.hPow (α := ℂ) (-1) (k + 1) = -1
e_a₁₁ : (-(↑1 : ℂ)) ^ k * HPow.hPow (α := ℂ) (-1) (k + 1) = -1
e_a₁₂ : HPow.hPow (α := ℂ) (-1) k * HPow.hPow (α := ℂ) (-1) (k + 1) = -1
e_a₁₃ : HPow.hPow (α := ℂ) (-1) k * (HPow.hPow (α := ℂ) (-1) k * -1) = -1
e_a₁₄ : HPow.hPow (α := ℂ) (-1) k * -HPow.hPow (α := ℂ) (-1) k = -1
⊢ Complex.cos ((↑π : ℂ) * HMul.hMul (α := ℂ) 2 (↑k : ℂ) / 2) * HPow.hPow (α := ℂ) (-1) (k + 1) = -1
have e_a₁₅ : -(HPow.hPow (α := ℂ) (-1) k * HPow.hPow (α := ℂ) (-1) k) = -1
Tactic: rw [mul_div_assoc, mul_div_cancel_left₀ _ two_ne_zero, ← ofReal_natCast, ← ofReal_mul,
    ← ofReal_cos, mul_comm π, ← sub_zero (k * π), cos_nat_mul_pi_sub, Real.cos_zero, mul_one,
    ofReal_pow, ofReal_neg, ofReal_one, pow_succ, mul_neg_one, mul_neg, ← mul_pow, neg_one_mul,
    neg_neg, one_pow]
-- Lean 4 proof step from Mathlib.NumberTheory.LSeries.HurwitzZetaValues (goal)
Goal:
k : ℕ
x : ℝ
hk : k ≠ 0
hx : Membership.mem (γ := Set ℝ) (Icc 0 1) x
h1 : ∀ (n : ℕ), HMul.hMul (α := ℂ) 2 (↑k : ℂ) ≠ -(↑n : ℂ)
h2 : HMul.hMul (α := ℂ) 2 (↑k : ℂ) ≠ 1
h3 : (HMul.hMul (α := ℂ) 2 (↑k : ℂ)).Gammaℂ ≠ 0
e_a : Complex.cos ((↑π : ℂ) * (HMul.hMul (α := ℂ) 2 (↑k : ℂ) / 2)) * HPow.hPow (α := ℂ) (-1) (k + 1) = -1
e_a₁ : Complex.cos ((↑π : ℂ) * (↑k : ℂ)) * HPow.hPow (α := ℂ) (-1) (k + 1) = -1
e_a₂ : Complex.cos ((↑π : ℂ) * (↑(↑k : ℝ) : ℂ)) * HPow.hPow (α := ℂ) (-1) (k + 1) = -1
e_a₃ : Complex.cos (↑(π * (↑k : ℝ)) : ℂ) * HPow.hPow (α := ℂ) (-1) (k + 1) = -1
e_a₄ : (↑(Real.cos (π * (↑k : ℝ))) : ℂ) * HPow.hPow (α := ℂ) (-1) (k + 1) = -1
e_a₅ : (↑(Real.cos ((↑k : ℝ) * π)) : ℂ) * HPow.hPow (α := ℂ) (-1) (k + 1) = -1
e_a₆ : (↑(Real.cos ((↑k : ℝ) * π - 0)) : ℂ) * HPow.hPow (α := ℂ) (-1) (k + 1) = -1
e_a₇ : (↑(HPow.hPow (α := ℝ) (-1) k * Real.cos 0) : ℂ) * HPow.hPow (α := ℂ) (-1) (k + 1) = -1
e_a₈ : (↑(HPow.hPow (α := ℝ) (-1) k * 1) : ℂ) * HPow.hPow (α := ℂ) (-1) (k + 1) = -1
e_a₉ : (↑(HPow.hPow (α := ℝ) (-1) k) : ℂ) * HPow.hPow (α := ℂ) (-1) (k + 1) = -1
e_a₁₀ : (↑(-1) : ℂ) ^ k * HPow.hPow (α := ℂ) (-1) (k + 1) = -1
e_a₁₁ : (-(↑1 : ℂ)) ^ k * HPow.hPow (α := ℂ) (-1) (k + 1) = -1
e_a₁₂ : HPow.hPow (α := ℂ) (-1) k * HPow.hPow (α := ℂ) (-1) (k + 1) = -1
e_a₁₃ : HPow.hPow (α := ℂ) (-1) k * (HPow.hPow (α := ℂ) (-1) k * -1) = -1
e_a₁₄ : HPow.hPow (α := ℂ) (-1) k * -HPow.hPow (α := ℂ) (-1) k = -1
e_a₁₅ : -(HPow.hPow (α := ℂ) (-1) k * HPow.hPow (α := ℂ) (-1) k) = -1
⊢ Complex.cos ((↑π : ℂ) * HMul.hMul (α := ℂ) 2 (↑k : ℂ) / 2) * HPow.hPow (α := ℂ) (-1) (k + 1) = -1
have e_a₁₆ : -HMul.hMul (α := ℂ) (-1) (-1) ^ k = -1
Tactic: rw [mul_div_assoc, mul_div_cancel_left₀ _ two_ne_zero, ← ofReal_natCast, ← ofReal_mul,
    ← ofReal_cos, mul_comm π, ← sub_zero (k * π), cos_nat_mul_pi_sub, Real.cos_zero, mul_one,
    ofReal_pow, ofReal_neg, ofReal_one, pow_succ, mul_neg_one, mul_neg, ← mul_pow, neg_one_mul,
    neg_neg, one_pow]
-- Lean 4 proof step from Mathlib.NumberTheory.LSeries.HurwitzZetaValues (goal)
Goal:
k : ℕ
x : ℝ
hk : k ≠ 0
hx : Membership.mem (γ := Set ℝ) (Icc 0 1) x
h1 : ∀ (n : ℕ), HMul.hMul (α := ℂ) 2 (↑k : ℂ) ≠ -(↑n : ℂ)
h2 : HMul.hMul (α := ℂ) 2 (↑k : ℂ) ≠ 1
h3 : (HMul.hMul (α := ℂ) 2 (↑k : ℂ)).Gammaℂ ≠ 0
e_a : Complex.cos ((↑π : ℂ) * (HMul.hMul (α := ℂ) 2 (↑k : ℂ) / 2)) * HPow.hPow (α := ℂ) (-1) (k + 1) = -1
e_a₁ : Complex.cos ((↑π : ℂ) * (↑k : ℂ)) * HPow.hPow (α := ℂ) (-1) (k + 1) = -1
e_a₂ : Complex.cos ((↑π : ℂ) * (↑(↑k : ℝ) : ℂ)) * HPow.hPow (α := ℂ) (-1) (k + 1) = -1
e_a₃ : Complex.cos (↑(π * (↑k : ℝ)) : ℂ) * HPow.hPow (α := ℂ) (-1) (k + 1) = -1
e_a₄ : (↑(Real.cos (π * (↑k : ℝ))) : ℂ) * HPow.hPow (α := ℂ) (-1) (k + 1) = -1
e_a₅ : (↑(Real.cos ((↑k : ℝ) * π)) : ℂ) * HPow.hPow (α := ℂ) (-1) (k + 1) = -1
e_a₆ : (↑(Real.cos ((↑k : ℝ) * π - 0)) : ℂ) * HPow.hPow (α := ℂ) (-1) (k + 1) = -1
e_a₇ : (↑(HPow.hPow (α := ℝ) (-1) k * Real.cos 0) : ℂ) * HPow.hPow (α := ℂ) (-1) (k + 1) = -1
e_a₈ : (↑(HPow.hPow (α := ℝ) (-1) k * 1) : ℂ) * HPow.hPow (α := ℂ) (-1) (k + 1) = -1
e_a₉ : (↑(HPow.hPow (α := ℝ) (-1) k) : ℂ) * HPow.hPow (α := ℂ) (-1) (k + 1) = -1
e_a₁₀ : (↑(-1) : ℂ) ^ k * HPow.hPow (α := ℂ) (-1) (k + 1) = -1
e_a₁₁ : (-(↑1 : ℂ)) ^ k * HPow.hPow (α := ℂ) (-1) (k + 1) = -1
e_a₁₂ : HPow.hPow (α := ℂ) (-1) k * HPow.hPow (α := ℂ) (-1) (k + 1) = -1
e_a₁₃ : HPow.hPow (α := ℂ) (-1) k * (HPow.hPow (α := ℂ) (-1) k * -1) = -1
e_a₁₄ : HPow.hPow (α := ℂ) (-1) k * -HPow.hPow (α := ℂ) (-1) k = -1
e_a₁₅ : -(HPow.hPow (α := ℂ) (-1) k * HPow.hPow (α := ℂ) (-1) k) = -1
e_a₁₆ : -HMul.hMul (α := ℂ) (-1) (-1) ^ k = -1
⊢ Complex.cos ((↑π : ℂ) * HMul.hMul (α := ℂ) 2 (↑k : ℂ) / 2) * HPow.hPow (α := ℂ) (-1) (k + 1) = -1
have e_a₁₇ : -HPow.hPow (α := ℂ) (- -1) k = -1
Tactic: rw [mul_div_assoc, mul_div_cancel_left₀ _ two_ne_zero, ← ofReal_natCast, ← ofReal_mul,
    ← ofReal_cos, mul_comm π, ← sub_zero (k * π), cos_nat_mul_pi_sub, Real.cos_zero, mul_one,
    ofReal_pow, ofReal_neg, ofReal_one, pow_succ, mul_neg_one, mul_neg, ← mul_pow, neg_one_mul,
    neg_neg, one_pow]