case even y _ IH =>
  rcases (IH (by simpa using hy) : ∃ x < y, ∃ y' ≤ y, Exponential x y' ∧ y < 2 * y') with ⟨x, hxy, y', gey, H, lty⟩
  exact ⟨x + 1, lt_of_lt_of_le (by simp [hxy]) (succ_le_double_of_pos (pos_of_gt hxy)),
    2 * y', by simpa using gey, Exponential.exponential_succ_mul_two.mpr H, by simpa using lty⟩
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
y : M
a✝ : y > 0
IH : 0 < y → ∃ x < y, ∃ y' ≤ y, Exponential x y' ∧ y < 2 * y'
hy : 0 < 2 * y
⊢ ∃ x < 2 * y, ∃ y' ≤ 2 * y, Exponential x y' ∧ 2 * y < 2 * y'
no goals