Even Chebyshev Angle Sum Vanishes

  -- Extract s from m = 2s
  obtain ⟨s, hs⟩ := hm_even
  rw [hs]

  -- Bound on s
  have hs_pos : 0 < s := by omega
  have hs_bound : s < N := by omega

  -- First convert s + s to 2 * s
  conv_lhs => arg 2; ext k; arg 1; rw [show (s + s : ℕ) = 2 * s by omega]

  -- Push down the cast and simplify
  conv_lhs => arg 2; ext k; arg 1; rw [show ((2 * s : ℕ) : ℝ) = 2 * (s : ℝ) by norm_cast]

  -- Simplify 2s * (2k+1) * π / (2N) = s * (2k+1) * π / N
  conv_lhs => arg 2; ext k; arg 1; rw [show (2 * (s : ℝ)) * (2 * k + 1 : ℝ) * π / (2 * N) =
      (s : ℝ) * (2 * k + 1 : ℝ) * π / N by field_simp]

  -- Now prove: ∑ k, cos(s*(2k+1)*π/N) = 0 using complex exponentials
  -- First prove the complex exponential sum is zero
  have geom_sum_zero : ∑ k ∈ range N, Complex.exp ((s * k : ℕ) * (2 * π * Complex.I / N)) = 0 := by
    -- Let ω = exp(2πi/N), primitive N-th root
    let ω := Complex.exp (2 * π * Complex.I / N)
    have ω_prim : IsPrimitiveRoot ω N := Complex.isPrimitiveRoot_exp N (by omega)

    -- Rewrite as sum of powers of ω^s
    have h_rewrite : ∑ k ∈ range N, Complex.exp ((s * k : ℕ) * (2 * π * Complex.I / N)) =
        ∑ k ∈ range N, ω ^ (s * k) := by
      refine Finset.sum_congr rfl fun k _ => ?_
      rw [← Complex.exp_nat_mul]
    rw [h_rewrite]

    conv_lhs =>
      arg 2
      ext k
      rw [pow_mul]

    -- Show ω^s ≠ 1
    have ωs_ne_one : ω ^ s ≠ 1 := by
      intro h_eq
      have h_div : N ∣ s := (ω_prim.pow_eq_one_iff_dvd s).mp h_eq
      have : N ≤ s := Nat.le_of_dvd hs_pos h_div
      omega

    -- Show (ω^s)^N = 1
    have ωs_pow_N : (ω ^ s) ^ N = 1 := by
      rw [← pow_mul, mul_comm, pow_mul, ω_prim.pow_eq_one, one_pow]

    -- Geometric sum
    have h_geom : (ω ^ s - 1) * ∑ k ∈ range N, (ω ^ s) ^ k = (ω ^ s) ^ N - 1 :=
      mul_geom_sum (ω ^ s) N
    rw [ωs_pow_N] at h_geom
    have : (ω ^ s - 1) * ∑ k ∈ range N, (ω ^ s) ^ k = 0 := by
      rw [h_geom]; ring
    exact eq_zero_of_ne_zero_of_mul_left_eq_zero (sub_ne_zero_of_ne ωs_ne_one) this

  have complex_sum :
      ∑ k ∈ range N, Complex.exp (Complex.I * ((s : ℝ) * (2 * k + 1) * π / N)) = 0 := by
    -- Factor: s*(2k+1)*π/N = s*π/N + s*2k*π/N
    calc ∑ k ∈ range N, Complex.exp (Complex.I * ((s : ℝ) * (2 * k + 1) * π / N))
        = ∑ k ∈ range N, Complex.exp (Complex.I * (s : ℝ) * π / N) *
            Complex.exp (Complex.I * ((s : ℝ) * (2 * k) * π / N)) := by
          refine Finset.sum_congr rfl fun k _ => ?_
          rw [← Complex.exp_add]
          congr 1
          push_cast
          ring
      _ = Complex.exp (Complex.I * (s : ℝ) * π / N) *
            ∑ k ∈ range N, Complex.exp (Complex.I * ((s : ℝ) * (2 * k) * π / N)) := by
          rw [Finset.mul_sum]
      _ = Complex.exp (Complex.I * (s : ℝ) * π / N) *
            ∑ k ∈ range N, Complex.exp ((s * k : ℕ) * (2 * π * Complex.I / N)) := by
          congr 2
          funext k
          congr 1
          push_cast
          ring
      _ = Complex.exp (Complex.I * (s : ℝ) * π / N) * 0 := by rw [geom_sum_zero]
      _ = 0 := by simp

  -- Convert cosine to exponential and use complex_sum
  have cos_eq : ∀ k : ℕ, Real.cos ((s : ℝ) * (2 * k + 1) * π / N) =
      (Complex.exp (Complex.I * ((s : ℝ) * (2 * k + 1) * π / N))).re := by
    intro k
    have h1 : (Complex.I * (((s : ℝ) * (2 * k + 1) * π / N) : ℂ)) =
        ((((s : ℝ) * (2 * k + 1) * π / N) : ℝ) : ℂ) * Complex.I := by
      push_cast
      ring
    rw [show (Complex.I * ((s : ℝ) * (2 * k + 1) * π / N) : ℂ) =
        Complex.I * (((s : ℝ) * (2 * k + 1) * π / N) : ℂ) by push_cast; rfl]
    rw [h1]
    exact (Complex.exp_ofReal_mul_I_re _).symm

  simp_rw [cos_eq]
  rw [← Complex.re_sum, complex_sum]
  simp