exact ⟨x + 1, by simp; exact lt_of_lt_of_le hxy (by simp),
  2 * y', le_trans (by simpa using gey) le_self_add, Exponential.exponential_succ_mul_two.mpr H, two_mul_add_one_lt_two_mul_of_lt lty⟩
case inr.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
IH : 0 < y → ∃ x < y, ∃ y' ≤ y, Exponential x y' ∧ y < 2 * y'
hy : 0 < 2 * y + 1
pos : 0 < y
x : M
hxy : x < y
y' : M
gey : y' ≤ y
H : Exponential x y'
lty : y < 2 * y'
⊢ ∃ x < 2 * y + 1, ∃ y' ≤ 2 * y + 1, Exponential x y' ∧ 2 * y + 1 < 2 * y'
no goals