Theorem: a₁' (Fl.lean)

State before:
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
⊢ trunc n (a + b - round (a + b)) = a + b - round (a + b) ∧ trunc n (round (a + b) - (a + b)) = round (a + b) - (a + b)
Tactic:
cases Nat.lt_or_ge (a + b) (2 ^ n) with
| inl uflow => exact a₁_of_uflow npos hfaithful₀ uflow
| inr no_uflow =>
  cases Nat.lt_or_ge (a + b) (round (a + b)) with
  | inl lt_round =>
    constructor
    . exact a₁_lo_of_lt_round npos hfaithful₁ lt_round
    . 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
  | inr round_le =>
    constructor
    . cases Nat.lt_or_ge (a + b) (2 ^ Nat.size a) with
      | inl no_carry =>
        exact a₁_lo_of_no_carry_of_round_le
          npos hfaithful₁ hfa hfb hba no_carry round_le
      | inr carry =>
        exact a₁_lo_of_no_uflow_of_carry_of_round_le
          npos hfaithful₁ hfa hfb hba no_uflow carry round_le
    . exact a₁_hi_of_round_le npos hfaithful₁ round_le
State after:
no goals

State before:
case inl
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
uflow : a + b < 2 ^ n
⊢ trunc n (a + b - round (a + b)) = a + b - round (a + b) ∧ trunc n (round (a + b) - (a + b)) = round (a + b) - (a + b)
Tactic:
exact a₁_of_uflow npos hfaithful₀ uflow
State after:
no goals

State before:
case inr
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
⊢ trunc n (a + b - round (a + b)) = a + b - round (a + b) ∧ trunc n (round (a + b) - (a + b)) = round (a + b) - (a + b)
Tactic:
cases Nat.lt_or_ge (a + b) (round (a + b)) with
| inl lt_round =>
  constructor
  . exact a₁_lo_of_lt_round npos hfaithful₁ lt_round
  . 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
| inr round_le =>
  constructor
  . cases Nat.lt_or_ge (a + b) (2 ^ Nat.size a) with
    | inl no_carry =>
      exact a₁_lo_of_no_carry_of_round_le
        npos hfaithful₁ hfa hfb hba no_carry round_le
    | inr carry =>
      exact a₁_lo_of_no_uflow_of_carry_of_round_le
        npos hfaithful₁ hfa hfb hba no_uflow carry round_le
  . exact a₁_hi_of_round_le npos hfaithful₁ round_le
State after:
no goals

State before:
case inr.inl
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 (a + b - round (a + b)) = a + b - round (a + b) ∧ trunc n (round (a + b) - (a + b)) = round (a + b) - (a + b)
Tactic:
constructor
State after:
case inr.inl.left
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 (a + b - round (a + b)) = a + b - round (a + b)

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)

State before:
case inr.inl.left
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 (a + b - round (a + b)) = a + b - round (a + b)

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)
Tactic:
. exact a₁_lo_of_lt_round npos hfaithful₁ lt_round
State after:
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)

State before:
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)
Tactic:
. 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
State after:
no goals

State before:
case inr.inl.left
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 (a + b - round (a + b)) = a + b - round (a + b)
Tactic:
exact a₁_lo_of_lt_round npos hfaithful₁ lt_round
State after:
no goals

State before:
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)
Tactic:
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
State after:
no goals

State before:
case inr.inl.right.inl
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)
no_carry : a + b < 2 ^ Nat.size a
⊢ trunc n (round (a + b) - (a + b)) = round (a + b) - (a + b)
Tactic:
exact a₁_hi_of_no_uflow_of_no_carry_of_lt_round
  npos hfaithful₁ hcorrect₁ hfa hfb hba no_uflow no_carry lt_round
State after:
no goals

State before:
case inr.inl.right.inr
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)
carry : a + b ≥ 2 ^ Nat.size a
⊢ trunc n (round (a + b) - (a + b)) = round (a + b) - (a + b)
Tactic:
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
State after:
case inr.inl.right.inr
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)
carry : a + b ≥ 2 ^ Nat.size a
ulp_le : ulp n (a + b) ≤ b
⊢ trunc n (round (a + b) - (a + b)) = round (a + b) - (a + b)

State before:
case inr.inl.right.inr
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)
carry : a + b ≥ 2 ^ Nat.size a
ulp_le : ulp n (a + b) ≤ b
⊢ trunc n (round (a + b) - (a + b)) = round (a + b) - (a + b)
Tactic:
exact a₁_hi_of_ulp_le_of_lt_round npos hfaithful₁ hfa hfb hba ulp_le lt_round
State after:
no goals

State before:
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)
carry : a + b ≥ 2 ^ Nat.size a
⊢ ulp n (a + b) ≤ b
Tactic:
rewrite [← Nat.not_lt]
State after:
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)
carry : a + b ≥ 2 ^ Nat.size a
⊢ ¬b < ulp n (a + b)

State before:
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)
carry : a + b ≥ 2 ^ Nat.size a
⊢ ¬b < ulp n (a + b)
Tactic:
intro lt_ulp
State after:
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)
carry : a + b ≥ 2 ^ Nat.size a
lt_ulp : b < ulp n (a + b)
⊢ False

State before:
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)
carry : a + b ≥ 2 ^ Nat.size a
lt_ulp : b < ulp n (a + b)
⊢ False
Tactic:
apply Nat.lt_le_asymm lt_round
State after:
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)
carry : a + b ≥ 2 ^ Nat.size a
lt_ulp : b < ulp n (a + b)
⊢ round (a + b) ≤ a + b

State before:
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)
carry : a + b ≥ 2 ^ Nat.size a
lt_ulp : b < ulp n (a + b)
⊢ round (a + b) ≤ a + b
Tactic:
exact round_le_of_no_uflow_of_carry_of_lt_ulp
  npos hfaithful₀ hfaithful₁ hcorrect₁ hfa hba no_uflow carry lt_ulp
State after:
no goals

State before:
case inr.inr
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
round_le : a + b ≥ round (a + b)
⊢ trunc n (a + b - round (a + b)) = a + b - round (a + b) ∧ trunc n (round (a + b) - (a + b)) = round (a + b) - (a + b)
Tactic:
constructor
State after:
case inr.inr.left
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
round_le : a + b ≥ round (a + b)
⊢ trunc n (a + b - round (a + b)) = a + b - round (a + b)

case inr.inr.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
round_le : a + b ≥ round (a + b)
⊢ trunc n (round (a + b) - (a + b)) = round (a + b) - (a + b)

State before:
case inr.inr.left
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
round_le : a + b ≥ round (a + b)
⊢ trunc n (a + b - round (a + b)) = a + b - round (a + b)

case inr.inr.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
round_le : a + b ≥ round (a + b)
⊢ trunc n (round (a + b) - (a + b)) = round (a + b) - (a + b)
Tactic:
. cases Nat.lt_or_ge (a + b) (2 ^ Nat.size a) with
  | inl no_carry =>
    exact a₁_lo_of_no_carry_of_round_le
      npos hfaithful₁ hfa hfb hba no_carry round_le
  | inr carry =>
    exact a₁_lo_of_no_uflow_of_carry_of_round_le
      npos hfaithful₁ hfa hfb hba no_uflow carry round_le
State after:
case inr.inr.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
round_le : a + b ≥ round (a + b)
⊢ trunc n (round (a + b) - (a + b)) = round (a + b) - (a + b)

State before:
case inr.inr.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
round_le : a + b ≥ round (a + b)
⊢ trunc n (round (a + b) - (a + b)) = round (a + b) - (a + b)
Tactic:
. exact a₁_hi_of_round_le npos hfaithful₁ round_le
State after:
no goals

State before:
case inr.inr.left
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
round_le : a + b ≥ round (a + b)
⊢ trunc n (a + b - round (a + b)) = a + b - round (a + b)
Tactic:
cases Nat.lt_or_ge (a + b) (2 ^ Nat.size a) with
| inl no_carry =>
  exact a₁_lo_of_no_carry_of_round_le
    npos hfaithful₁ hfa hfb hba no_carry round_le
| inr carry =>
  exact a₁_lo_of_no_uflow_of_carry_of_round_le
    npos hfaithful₁ hfa hfb hba no_uflow carry round_le
State after:
no goals

State before:
case inr.inr.left.inl
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
round_le : a + b ≥ round (a + b)
no_carry : a + b < 2 ^ Nat.size a
⊢ trunc n (a + b - round (a + b)) = a + b - round (a + b)
Tactic:
exact a₁_lo_of_no_carry_of_round_le
  npos hfaithful₁ hfa hfb hba no_carry round_le
State after:
no goals

State before:
case inr.inr.left.inr
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
round_le : a + b ≥ round (a + b)
carry : a + b ≥ 2 ^ Nat.size a
⊢ trunc n (a + b - round (a + b)) = a + b - round (a + b)
Tactic:
exact a₁_lo_of_no_uflow_of_carry_of_round_le
  npos hfaithful₁ hfa hfb hba no_uflow carry round_le
State after:
no goals

State before:
case inr.inr.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
round_le : a + b ≥ round (a + b)
⊢ trunc n (round (a + b) - (a + b)) = round (a + b) - (a + b)
Tactic:
exact a₁_hi_of_round_le npos hfaithful₁ round_le
State after:
no goals