. cases Nat.lt_or_ge (a + b) (2 ^ Nat.size a) with
  | inl no_carry =>
    exact a₁_hi_of_no_uflow_of_no_carry_of_lt_round
      npos hfaithful₁ hcorrect₁ hfa hfb hba no_uflow no_carry lt_round
  | inr carry =>
    have ulp_le : ulp n (a + b) ≤ b := by
      rewrite [← Nat.not_lt]
      intro lt_ulp
      apply Nat.lt_le_asymm lt_round
      exact round_le_of_no_uflow_of_carry_of_lt_ulp
        npos hfaithful₀ hfaithful₁ hcorrect₁ hfa hba no_uflow carry lt_ulp
    exact a₁_hi_of_ulp_le_of_lt_round npos hfaithful₁ hfa hfb hba ulp_le lt_round
case inr.inl.right
round : ℕ → ℕ
n a b : ℕ
npos : 0 < n
hfaithful₀ : faithful₀ n round
hfaithful₁ : faithful₁ n round
hcorrect₁ : correct₁ n round
hfa : trunc n a = a
hfb : trunc n b = b
hba : b ≤ a
no_uflow : a + b ≥ 2 ^ n
lt_round : a + b < round (a + b)
⊢ trunc n (round (a + b) - (a + b)) = round (a + b) - (a + b)
no goals