Non-Constant Binomial Terms Vanish for Chebyshev Angles

  -- We need to show:
  -- 1. (2r - j) is odd
  -- 2. |2r - j| < 2N
  -- Then apply sum_cos_chebyshev_angles_vanishes

  -- First establish the bound: r ≤ j
  have hr_le : r ≤ j := by
    simp only [mem_range] at hr
    omega

  -- Split by parity of j
  by_cases h_parity : Odd j
  · -- Case 1: j is odd, so (2r - j) is odd
    have h_odd : Odd (2 * r - j : ℤ) := by
      obtain ⟨m, hm⟩ := h_parity
      use (r : ℤ) - m - 1
      have : (j : ℤ) = 2 * m + 1 := by exact_mod_cast hm
      calc (2 * r - j : ℤ)
        _ = 2 * (r : ℤ) - (2 * m + 1) := by rw [this]
        _ = 2 * ((r : ℤ) - m - 1) + 1 := by ring

    -- Prove |2r - j| < 2N
    have h_bound : |(2 * r - j : ℤ)| < 2 * N := by
      by_cases hr_case : 2 * r ≥ j
      · rw [abs_of_nonneg (Int.sub_nonneg_of_le (by omega : (j : ℤ) ≤ (2 * r : ℤ)))]
        calc (2 * r - j : ℤ)
          _ ≤ (2 * j - j : ℤ) := by omega
          _ = (j : ℤ) := by ring
          _ < (N : ℤ) := by omega
          _ < (2 * N : ℤ) := by omega
      · push_neg at hr_case
        rw [abs_of_neg (Int.sub_neg_of_lt (by omega : (2 * r : ℤ) < (j : ℤ)))]
        calc -(2 * r - j : ℤ)
          _ = (j - 2 * r : ℤ) := by ring
          _ ≤ (j : ℤ) := by omega
          _ < (N : ℤ) := by omega
          _ < (2 * N : ℤ) := by omega

    -- Apply odd vanishing lemma
    convert sum_cos_chebyshev_angles_vanishes N (2 * r - j) hN hr_neq h_odd h_bound using 2

  · -- Case 2: j is even, so (2r - j) is even
    have j_even : Even j := Nat.not_odd_iff_even.mp h_parity
    have h_even : Even (2 * r - j : ℤ) := by
      obtain ⟨m, hm⟩ := j_even
      use (r : ℤ) - m
      have : (j : ℤ) = (m + m : ℕ) := by simp [hm]
      calc (2 * r - j : ℤ)
        _ = 2 * (r : ℤ) - ((m : ℤ) + (m : ℤ)) := by rw [this]; simp
        _ = 2 * (r : ℤ) - 2 * (m : ℤ) := by ring
        _ = ((r : ℤ) - (m : ℤ)) + ((r : ℤ) - (m : ℤ)) := by ring

    -- Convert to natural number case
    -- Need to handle both positive and negative cases of (2r - j)
    by_cases h_sign : 2 * r ≥ j
    · -- Positive case: 2r - j ≥ 0
      have h_pos : 0 ≤ (2 * r - j : ℤ) := Int.sub_nonneg_of_le (by omega : (j : ℤ) ≤ (2 * r : ℤ))
      obtain ⟨m_nat, hm_nat⟩ := Int.eq_ofNat_of_zero_le h_pos

      -- Show m_nat is even and bounded
      have hm_even : Even m_nat := by
        have := h_even
        rwa [hm_nat, Int.even_coe_nat] at this

      have hm_pos : 0 < m_nat := by
        by_contra h
        push_neg at h
        have : m_nat = 0 := by omega
        rw [this] at hm_nat
        simp at hm_nat
        omega

      have hm_bound : m_nat < N := by
        have : (m_nat : ℤ) < N := by
          calc (m_nat : ℤ)
            _ = 2 * r - j := by rw [← hm_nat]
            _ ≤ 2 * j - j := by omega
            _ = j := by ring
            _ < N := by omega
        omega

      -- Apply even vanishing lemma
      calc ∑ k ∈ range N, Real.cos ((2 * r - j : ℤ) * (2 * k + 1 : ℝ) * π / (2 * N))
          _ = ∑ k ∈ range N, Real.cos ((m_nat : ℝ) * (2 * k + 1 : ℝ) * π / (2 * N)) := by
              congr 1; ext k; congr 1
              rw [hm_nat]
              norm_cast
          _ = 0 := sum_cos_chebyshev_angles_even_vanishes N m_nat hN hm_pos hm_even hm_bound

    · -- Negative case: 2r - j < 0, use cos(-x) = cos(x)
      have h_neg : (2 * r - j : ℤ) < 0 := Int.sub_neg_of_lt (by omega : (2 * r : ℤ) < (j : ℤ))
      have h_pos : 0 < -(2 * r - j : ℤ) := by omega

      obtain ⟨m_nat, hm_nat⟩ := Int.eq_ofNat_of_zero_le (by omega : 0 ≤ -(2 * r - j : ℤ))

      -- Show m_nat is even and bounded
      have hm_even : Even m_nat := by
        have : Even (-(2 * r - j : ℤ)) := by
          obtain ⟨k, hk⟩ := h_even
          use -k
          calc -(2 * r - j : ℤ)
            _ = -(k + k) := by rw [hk]
            _ = -k + -k := by ring
        rwa [hm_nat, Int.even_coe_nat] at this

      have hm_pos : 0 < m_nat := by
        have : (m_nat : ℤ) ≠ 0 := by
          intro h
          rw [h] at hm_nat
          simp at hm_nat
          omega
        omega

      have hm_bound : m_nat < N := by
        have : (m_nat : ℤ) < N := by
          calc (m_nat : ℤ)
            _ = -(2 * r - j) := by rw [← hm_nat]
            _ = j - 2 * r := by ring
            _ ≤ j := by omega
            _ < N := by omega
        omega

      -- Apply even vanishing lemma with cos(-x) = cos(x)
      calc ∑ k ∈ range N, Real.cos ((2 * r - j : ℤ) * (2 * k + 1 : ℝ) * π / (2 * N))
          _ = ∑ k ∈ range N, Real.cos ((-(m_nat : ℤ)) * (2 * k + 1 : ℝ) * π / (2 * N)) := by
              congr 1; ext k; congr 1
              have : (2 * r - j : ℤ) = -(m_nat : ℤ) := by rw [← hm_nat]; ring
              rw [this]
              norm_cast
          _ = ∑ k ∈ range N, Real.cos ((m_nat : ℝ) * (2 * k + 1 : ℝ) * π / (2 * N)) := by
              congr 1; ext k
              have : ((-(m_nat : ℤ)) * (2 * k + 1 : ℝ) * π / (2 * N) : ℝ) =
                  -((m_nat : ℤ) * (2 * k + 1 : ℝ) * π / (2 * N)) := by
                push_cast; ring
              rw [this, Real.cos_neg]
              norm_cast
          _ = 0 := sum_cos_chebyshev_angles_even_vanishes N m_nat hN hm_pos hm_even hm_bound