calc
  ‖2 * a + 1‖ ^ 2 = (‖a‖ + 1) ^ 2         := by rw [length_two_mul_add_one a]
  _               = ‖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]
  _               ≤ 3 * (2 * a + 1)       := by simp
case inr
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
IH : ‖a‖ ^ 2 ≤ 3 * a
pos : 0 < a
⊢ ‖2 * a + 1‖ ^ 2 ≤ 3 * (2 * a + 1)
no goals