∀ {m : ℤ} {C : FixedDetMatrix (Fin 2) ℤ m → Sort u_3} (base : (A : FixedDetMatrix (Fin 2) ℤ m) → ↑A 1 0 = 0 → C A)
  (step : (A : FixedDetMatrix (Fin 2) ℤ m) → ↑A 1 0 ≠ 0 → C (FixedDetMatrices.reduceStep A) → C A)
  (A : FixedDetMatrix (Fin 2) ℤ m),
  FixedDetMatrices.reduce_rec base step A =
    if h : ↑A 1 0 = 0 then base A h
    else step A h (FixedDetMatrices.reduce_rec base step (FixedDetMatrices.reduceStep A))