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⟩
case intro.intro.intro.intro.intro
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 : M
hxy : x < y
y' : M
gey : y' ≤ y
H : Exponential x y'
lty : y < 2 * y'
⊢ ∃ x < 2 * y, ∃ y' ≤ 2 * y, Exponential x y' ∧ 2 * y < 2 * y'
no goals