calc
  ‖2 * a‖ ^ 2 = (‖a‖ + 1) ^ 2         := by rw [length_two_mul_of_pos pos]
  _           = ‖a‖ ^ 2 + 2 * ‖a‖ + 1 := by simp [sq, add_mul_self_eq]
  _           ≤ 3 * a + 2 * ‖a‖ + 1   := by simpa using IH
  _           ≤ 3 * a + 2 * a + 1     := by simp
  _           ≤ 3 * a + 2 * a + a     := by simp [←pos_iff_one_le, pos]
  _           = 3 * (2 * a)           := by simp_all only [←two_add_one_eq_three, two_mul, add_mul, add_assoc, one_mul]
M : Type u_1
inst✝⁵ : Zero M
inst✝⁴ : One M
inst✝³ : Add M
inst✝² : Mul M
inst✝¹ : LT M
inst✝ : M ⊧ₘ* 𝐈𝚫₀
a : M
pos : a > 0
IH : ‖a‖ ^ 2 ≤ 3 * a
⊢ ‖2 * a‖ ^ 2 ≤ 3 * (2 * a)
no goals