case succ x IH =>
  intro hx
  rcases (IH (lt_of_le_of_lt (by simp) hx) : ∃ y ≤ a, Exponential x y) with ⟨y, hy, H⟩
  have : 0 < a := by by_contra A; rcases (show a = 0 from by simpa using A); simp_all
  have : 2 * y ≤ a := (le_iff_le_log_of_exp H.succ this).mpr (le_log_of_lt_length hx)
  exact ⟨2 * y, this, H.succ⟩
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 x : M
IH : x < ‖a‖ → ∃ y ≤ a, Exponential x y
⊢ x + 1 < ‖a‖ → ∃ y ≤ a, Exponential (x + 1) y
no goals