import Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic
import Mathlib.Tactic.FieldSimp
import QFD.Lepton.KoideRelation

open Real
open QFD.Lepton.KoideRelation

namespace QFD.Lepton.KoideRelation

/--
Sum of squared cosines at symmetric angles equals 3/2.
This is the key algebraic step for proving the Koide relation.

Uses double-angle formula cos²x = (1 + cos(2x))/2 and sum_cos_symm.
-/
lemma sum_cos_sq_symm (δ : ℝ) :
  cos δ ^ 2 + cos (δ + 2*π/3) ^ 2 + cos (δ + 4*π/3) ^ 2 = 3/2 := by
  -- Apply double-angle formula to each term
  rw [cos_sq δ, cos_sq (δ + 2*π/3), cos_sq (δ + 4*π/3)]
  -- Normalize angles: 2*(δ + 2π/3) = 2*δ + 4*π/3 and 2*(δ + 4*π/3) = 2*δ + 8*π/3
  have h1 : (2:ℝ) * (δ + 2*π/3) = 2*δ + 4*π/3 := by ring
  have h2 : (2:ℝ) * (δ + 4*π/3) = 2*δ + 8*π/3 := by ring
  rw [h1, h2]
  -- Use periodicity: cos(2δ + 8π/3) = cos(2δ + 2π/3) since 8π/3 - 2π = 2π/3
  have sum_cos_zero : cos (2*δ) + cos (2*δ + 4*π/3) + cos (2*δ + 8*π/3) = 0 := by
    have h_period : cos (2*δ + 8*π/3) = cos (2*δ + 2*π/3) := by
      have : (2:ℝ)*δ + 8*π/3 = 2*δ + 2*π/3 + 2*π := by ring
      rw [this, cos_add_two_pi]
    calc cos (2*δ) + cos (2*δ + 4*π/3) + cos (2*δ + 8*π/3)
        = cos (2*δ) + cos (2*δ + 4*π/3) + cos (2*δ + 2*π/3) := by rw [h_period]
      _ = cos (2*δ) + cos (2*δ + 2*π/3) + cos (2*δ + 4*π/3) := by ring
      _ = 0 := sum_cos_symm (2*δ)
  -- Final algebraic simplification: combine fractions and apply sum_cos_zero
  calc 1 / 2 + cos (2 * δ) / 2 + (1 / 2 + cos (2 * δ + 4 * π / 3) / 2)
          + (1 / 2 + cos (2 * δ + 8 * π / 3) / 2)
      = (1 + 1 + 1 + cos (2*δ) + cos (2*δ + 4*π/3) + cos (2*δ + 8*π/3)) / 2 := by ring
    _ = (3 + (cos (2*δ) + cos (2*δ + 4*π/3) + cos (2*δ + 8*π/3))) / 2 := by ring
    _ = (3 + 0) / 2 := by rw [sum_cos_zero]
    _ = 3 / 2 := by norm_num

end QFD.Lepton.KoideRelation