exact (le_iff_lt_length_of_exp (exponential_hash a 1)).mpr (by
  simp [mul_comm 2 a]
  have : ‖a * 2 + 1‖ = ‖a‖ + 1 := by
    simpa using length_mul_pow2_add_of_lt pos (show Pow2 2 from by simp) one_lt_two
  simp [this])
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 + 𝛀₁
a : M
pos : 0 < a
⊢ a # 1 ≤ 2 * a + 1
no goals