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theorem proved term proof

tick_phases_equally_spaced

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formal statement (Lean)

  67theorem tick_phases_equally_spaced (j k : Fin 8) (hjk : j < k) :
  68    -- The quotient tickPhase k / tickPhase j has argument (k - j) * π/4 modulo 2π
  69    tickPhase k / tickPhase j = Complex.exp ((k.val - j.val : ℝ) * π / 4 * I) := by

proof body

Term-mode proof.

  70  unfold tickPhase
  71  -- Use exp_sub: exp(a) / exp(b) = exp(a - b)
  72  rw [← Complex.exp_sub]
  73  congr 1
  74  -- Show: I * π * k / 4 - I * π * j / 4 = (k - j) * π / 4 * I
  75  push_cast
  76  ring
  77
  78/-! ## Why Real Numbers Are Insufficient -/
  79
  80/-- The problem with real numbers: they can't represent rotation.
  81    In ℝ, multiplication is just scaling. No rotation. -/

depends on (6)

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