by_cases hi : i ∈ a <;> simp only [hi, true_and, not_true_eq_false, false_and,
  or_false, insert, bitInsert, ↓reduceIte, imp_self,
  not_false_eq_true, true_and, false_or, forall_exists_index, and_imp]
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* 𝐈𝚺₁
b i a : M
⊢ (i ∈ a ∧ b = a ∨ i ∉ a ∧ ∃ e ≤ b, e = exp i ∧ b = a + e) → b = insert i a
case neg
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* 𝐈𝚺₁
b i a : M
hi : i ∉ a
⊢ ∀ x ≤ b, x = exp i → b = a + x → b = a + exp i