internlm/Lean-Github huggingface dataset:
Theorem: b₁_lo_of_round_lt_of_no_uflow_of_ulp_lt_ulp_of_two_mul_lt_of_pos_of_le_ulp
Proof state before:
round : ℕ → ℕ
n a b : ℕ
npos : 0 < n
hfaithful₀ : faithful₀ n round
hfaithful₁ : faithful₁ n round
hcorrect₀ : correct₀ n round
hfa : trunc n a = a
hfb : trunc n b = b
hba : b ≤ a
round_lt : round (a - b) < a - b
no_uflow : a - b ≥ 2 ^ n
ulp_lt_ulp : ulp n (a - b) < ulp n a
two_mul_lt : 2 * b < a
bpos : b > 0
le_ulp : b ≤ ulp n (a - b)
ulp_even : 2 ∣ ulp n (a - b)
lt_ulp : b < ulp n a
lt_size : n < Nat.size a
size_sub_one_pos : 0 < Nat.size a - 1
size_pos : 0 < Nat.size a
apos : 0 < a
eq_pow : a = 2 ^ (Nat.size a - 1)
even : 2 ∣ 2 ^ (Nat.size a - 1)
ulp_eq_half_ulp : ulp n (a - b) = ulp n a / 2
one_le_size_sub : 1 ≤ Nat.size a - n
le_size_sub_one : n ≤ Nat.size a - 1
k₁ : a = 2 ^ n * ulp n (a - b)
⊢ a - b - trunc n (a - b) = ulp n (a - b) - b
Tactic:
have k₂ : a - b = (2 ^ n - 1) * ulp n (a - b) + (ulp n (a - b) - b) := by
  rewrite [Nat.sub_one, Nat.mul_pred_left]
  rewrite [← add_tsub_assoc_of_le le_ulp]
  rewrite [tsub_add_cancel_of_le (Nat.le_mul_of_pos_left two_pow_pos)]
  rw [← k₁]
Proof state after:
round : ℕ → ℕ
n a b : ℕ
npos : 0 < n
hfaithful₀ : faithful₀ n round
hfaithful₁ : faithful₁ n round
hcorrect₀ : correct₀ n round
hfa : trunc n a = a
hfb : trunc n b = b
hba : b ≤ a
round_lt : round (a - b) < a - b
no_uflow : a - b ≥ 2 ^ n
ulp_lt_ulp : ulp n (a - b) < ulp n a
two_mul_lt : 2 * b < a
bpos : b > 0
le_ulp : b ≤ ulp n (a - b)
ulp_even : 2 ∣ ulp n (a - b)
lt_ulp : b < ulp n a
lt_size : n < Nat.size a
size_sub_one_pos : 0 < Nat.size a - 1
size_pos : 0 < Nat.size a
apos : 0 < a
eq_pow : a = 2 ^ (Nat.size a - 1)
even : 2 ∣ 2 ^ (Nat.size a - 1)
ulp_eq_half_ulp : ulp n (a - b) = ulp n a / 2
one_le_size_sub : 1 ≤ Nat.size a - n
le_size_sub_one : n ≤ Nat.size a - 1
k₁ : a = 2 ^ n * ulp n (a - b)
k₂ : a - b = (2 ^ n - 1) * ulp n (a - b) + (ulp n (a - b) - b)
⊢ a - b - trunc n (a - b) = ulp n (a - b) - b
Theorem: b₁_lo_of_round_lt_of_no_uflow_of_ulp_lt_ulp_of_two_mul_lt_of_pos_of_le_ulp
Proof state before:
round : ℕ → ℕ
n a b : ℕ
npos : 0 < n
hfaithful₀ : faithful₀ n round
hfaithful₁ : faithful₁ n round
hcorrect₀ : correct₀ n round
hfa : trunc n a = a
hfb : trunc n b = b
hba : b ≤ a
round_lt : round (a - b) < a - b
no_uflow : a - b ≥ 2 ^ n
ulp_lt_ulp : ulp n (a - b) < ulp n a
two_mul_lt : 2 * b < a
bpos : b > 0
le_ulp : b ≤ ulp n (a - b)
ulp_even : 2 ∣ ulp n (a - b)
lt_ulp : b < ulp n a
lt_size : n < Nat.size a
size_sub_one_pos : 0 < Nat.size a - 1
size_pos : 0 < Nat.size a
apos : 0 < a
eq_pow : a = 2 ^ (Nat.size a - 1)
even : 2 ∣ 2 ^ (Nat.size a - 1)
ulp_eq_half_ulp : ulp n (a - b) = ulp n a / 2
one_le_size_sub : 1 ≤ Nat.size a - n
le_size_sub_one : n ≤ Nat.size a - 1
k₁ : a = 2 ^ n * ulp n (a - b)
k₂ : a - b = (2 ^ n - 1) * ulp n (a - b) + (ulp n (a - b) - b)
⊢ a - b - trunc n (a - b) = ulp n (a - b) - b
Tactic:
have k₃ : trunc n (a - b) = (2 ^ n - 1) * ulp n (a - b) := by
  unfold trunc
  conv => lhs ; left ; left ; rewrite [k₂]
  rewrite [Nat.add_comm]
  rewrite [Nat.add_mul_div_right _ _ (ulp_pos _ _)]
  rewrite [Nat.div_eq_of_lt (tsub_lt_self (ulp_pos _ _) bpos)]
  rw [Nat.zero_add]
Proof state after:
round : ℕ → ℕ
n a b : ℕ
npos : 0 < n
hfaithful₀ : faithful₀ n round
hfaithful₁ : faithful₁ n round
hcorrect₀ : correct₀ n round
hfa : trunc n a = a
hfb : trunc n b = b
hba : b ≤ a
round_lt : round (a - b) < a - b
no_uflow : a - b ≥ 2 ^ n
ulp_lt_ulp : ulp n (a - b) < ulp n a
two_mul_lt : 2 * b < a
bpos : b > 0
le_ulp : b ≤ ulp n (a - b)
ulp_even : 2 ∣ ulp n (a - b)
lt_ulp : b < ulp n a
lt_size : n < Nat.size a
size_sub_one_pos : 0 < Nat.size a - 1
size_pos : 0 < Nat.size a
apos : 0 < a
eq_pow : a = 2 ^ (Nat.size a - 1)
even : 2 ∣ 2 ^ (Nat.size a - 1)
ulp_eq_half_ulp : ulp n (a - b) = ulp n a / 2
one_le_size_sub : 1 ≤ Nat.size a - n
le_size_sub_one : n ≤ Nat.size a - 1
k₁ : a = 2 ^ n * ulp n (a - b)
k₂ : a - b = (2 ^ n - 1) * ulp n (a - b) + (ulp n (a - b) - b)
k₃ : trunc n (a - b) = (2 ^ n - 1) * ulp n (a - b)
⊢ a - b - trunc n (a - b) = ulp n (a - b) - b
Theorem: b₁_lo_of_round_lt_of_no_uflow_of_ulp_lt_ulp_of_two_mul_lt_of_pos_of_le_ulp
Proof state before:
round : ℕ → ℕ
n a b : ℕ
npos : 0 < n
hfaithful₀ : faithful₀ n round
hfaithful₁ : faithful₁ n round
hcorrect₀ : correct₀ n round
hfa : trunc n a = a
hfb : trunc n b = b
hba : b ≤ a
round_lt : round (a - b) < a - b
no_uflow : a - b ≥ 2 ^ n
ulp_lt_ulp : ulp n (a - b) < ulp n a
two_mul_lt : 2 * b < a
bpos : b > 0
le_ulp : b ≤ ulp n (a - b)
ulp_even : 2 ∣ ulp n (a - b)
lt_ulp : b < ulp n a
lt_size : n < Nat.size a
size_sub_one_pos : 0 < Nat.size a - 1
size_pos : 0 < Nat.size a
apos : 0 < a
eq_pow : a = 2 ^ (Nat.size a - 1)
even : 2 ∣ 2 ^ (Nat.size a - 1)
ulp_eq_half_ulp : ulp n (a - b) = ulp n a / 2
one_le_size_sub : 1 ≤ Nat.size a - n
le_size_sub_one : n ≤ Nat.size a - 1
k₁ : a = 2 ^ n * ulp n (a - b)
k₂ : a - b = (2 ^ n - 1) * ulp n (a - b) + (ulp n (a - b) - b)
k₃ : trunc n (a - b) = (2 ^ n - 1) * ulp n (a - b)
⊢ a - b - trunc n (a - b) = ulp n (a - b) - b
Tactic:
conv => lhs ; left ; rewrite [k₂]
Proof state after:
round : ℕ → ℕ
n a b : ℕ
npos : 0 < n
hfaithful₀ : faithful₀ n round
hfaithful₁ : faithful₁ n round
hcorrect₀ : correct₀ n round
hfa : trunc n a = a
hfb : trunc n b = b
hba : b ≤ a
round_lt : round (a - b) < a - b
no_uflow : a - b ≥ 2 ^ n
ulp_lt_ulp : ulp n (a - b) < ulp n a
two_mul_lt : 2 * b < a
bpos : b > 0
le_ulp : b ≤ ulp n (a - b)
ulp_even : 2 ∣ ulp n (a - b)
lt_ulp : b < ulp n a
lt_size : n < Nat.size a
size_sub_one_pos : 0 < Nat.size a - 1
size_pos : 0 < Nat.size a
apos : 0 < a
eq_pow : a = 2 ^ (Nat.size a - 1)
even : 2 ∣ 2 ^ (Nat.size a - 1)
ulp_eq_half_ulp : ulp n (a - b) = ulp n a / 2
one_le_size_sub : 1 ≤ Nat.size a - n
le_size_sub_one : n ≤ Nat.size a - 1
k₁ : a = 2 ^ n * ulp n (a - b)
k₂ : a - b = (2 ^ n - 1) * ulp n (a - b) + (ulp n (a - b) - b)
k₃ : trunc n (a - b) = (2 ^ n - 1) * ulp n (a - b)
⊢ (2 ^ n - 1) * ulp n (a - b) + (ulp n (a - b) - b) - trunc n (a - b) = ulp n (a - b) - b
Theorem: b₁_lo_of_round_lt_of_no_uflow_of_ulp_lt_ulp_of_two_mul_lt_of_pos_of_le_ulp
Proof state before:
round : ℕ → ℕ
n a b : ℕ
npos : 0 < n
hfaithful₀ : faithful₀ n round
hfaithful₁ : faithful₁ n round
hcorrect₀ : correct₀ n round
hfa : trunc n a = a
hfb : trunc n b = b
hba : b ≤ a
round_lt : round (a - b) < a - b
no_uflow : a - b ≥ 2 ^ n
ulp_lt_ulp : ulp n (a - b) < ulp n a
two_mul_lt : 2 * b < a
bpos : b > 0
le_ulp : b ≤ ulp n (a - b)
ulp_even : 2 ∣ ulp n (a - b)
lt_ulp : b < ulp n a
lt_size : n < Nat.size a
size_sub_one_pos : 0 < Nat.size a - 1
size_pos : 0 < Nat.size a
apos : 0 < a
eq_pow : a = 2 ^ (Nat.size a - 1)
even : 2 ∣ 2 ^ (Nat.size a - 1)
ulp_eq_half_ulp : ulp n (a - b) = ulp n a / 2
one_le_size_sub : 1 ≤ Nat.size a - n
le_size_sub_one : n ≤ Nat.size a - 1
k₁ : a = 2 ^ n * ulp n (a - b)
k₂ : a - b = (2 ^ n - 1) * ulp n (a - b) + (ulp n (a - b) - b)
k₃ : trunc n (a - b) = (2 ^ n - 1) * ulp n (a - b)
⊢ (2 ^ n - 1) * ulp n (a - b) + (ulp n (a - b) - b) - trunc n (a - b) = ulp n (a - b) - b
Tactic:
rewrite [k₃]
Proof state after:
round : ℕ → ℕ
n a b : ℕ
npos : 0 < n
hfaithful₀ : faithful₀ n round
hfaithful₁ : faithful₁ n round
hcorrect₀ : correct₀ n round
hfa : trunc n a = a
hfb : trunc n b = b
hba : b ≤ a
round_lt : round (a - b) < a - b
no_uflow : a - b ≥ 2 ^ n
ulp_lt_ulp : ulp n (a - b) < ulp n a
two_mul_lt : 2 * b < a
bpos : b > 0
le_ulp : b ≤ ulp n (a - b)
ulp_even : 2 ∣ ulp n (a - b)
lt_ulp : b < ulp n a
lt_size : n < Nat.size a
size_sub_one_pos : 0 < Nat.size a - 1
size_pos : 0 < Nat.size a
apos : 0 < a
eq_pow : a = 2 ^ (Nat.size a - 1)
even : 2 ∣ 2 ^ (Nat.size a - 1)
ulp_eq_half_ulp : ulp n (a - b) = ulp n a / 2
one_le_size_sub : 1 ≤ Nat.size a - n
le_size_sub_one : n ≤ Nat.size a - 1
k₁ : a = 2 ^ n * ulp n (a - b)
k₂ : a - b = (2 ^ n - 1) * ulp n (a - b) + (ulp n (a - b) - b)
k₃ : trunc n (a - b) = (2 ^ n - 1) * ulp n (a - b)
⊢ (2 ^ n - 1) * ulp n (a - b) + (ulp n (a - b) - b) - (2 ^ n - 1) * ulp n (a - b) = ulp n (a - b) - b
Theorem: b₁_lo_of_round_lt_of_no_uflow_of_ulp_lt_ulp_of_two_mul_lt_of_pos_of_le_ulp
Proof state before:
round : ℕ → ℕ
n a b : ℕ
npos : 0 < n
hfaithful₀ : faithful₀ n round
hfaithful₁ : faithful₁ n round
hcorrect₀ : correct₀ n round
hfa : trunc n a = a
hfb : trunc n b = b
hba : b ≤ a
round_lt : round (a - b) < a - b
no_uflow : a - b ≥ 2 ^ n
ulp_lt_ulp : ulp n (a - b) < ulp n a
two_mul_lt : 2 * b < a
bpos : b > 0
le_ulp : b ≤ ulp n (a - b)
ulp_even : 2 ∣ ulp n (a - b)
lt_ulp : b < ulp n a
lt_size : n < Nat.size a
size_sub_one_pos : 0 < Nat.size a - 1
size_pos : 0 < Nat.size a
apos : 0 < a
eq_pow : a = 2 ^ (Nat.size a - 1)
even : 2 ∣ 2 ^ (Nat.size a - 1)
ulp_eq_half_ulp : ulp n (a - b) = ulp n a / 2
one_le_size_sub : 1 ≤ Nat.size a - n
le_size_sub_one : n ≤ Nat.size a - 1
k₁ : a = 2 ^ n * ulp n (a - b)
k₂ : a - b = (2 ^ n - 1) * ulp n (a - b) + (ulp n (a - b) - b)
k₃ : trunc n (a - b) = (2 ^ n - 1) * ulp n (a - b)
⊢ (2 ^ n - 1) * ulp n (a - b) + (ulp n (a - b) - b) - (2 ^ n - 1) * ulp n (a - b) = ulp n (a - b) - b
Tactic:
exact add_tsub_cancel_left _ _
Proof state after:
no goals
Theorem: b₁_lo_of_round_lt_of_no_uflow_of_ulp_lt_ulp_of_two_mul_lt_of_pos_of_le_ulp
Proof state before:
round : ℕ → ℕ
n a b : ℕ
npos : 0 < n
hfaithful₀ : faithful₀ n round
hfaithful₁ : faithful₁ n round
hcorrect₀ : correct₀ n round
hfa : trunc n a = a
hfb : trunc n b = b
hba : b ≤ a
round_lt : round (a - b) < a - b
no_uflow : a - b ≥ 2 ^ n
ulp_lt_ulp : ulp n (a - b) < ulp n a
two_mul_lt : 2 * b < a
bpos : b > 0
le_ulp : b ≤ ulp n (a - b)
ulp_even : 2 ∣ ulp n (a - b)
lt_ulp : b < ulp n a
lt_size : n < Nat.size a
size_sub_one_pos : 0 < Nat.size a - 1
size_pos : 0 < Nat.size a
apos : 0 < a
eq_pow : a = 2 ^ (Nat.size a - 1)
even : 2 ∣ 2 ^ (Nat.size a - 1)
ulp_eq_half_ulp : ulp n (a - b) = ulp n a / 2
one_le_size_sub : 1 ≤ Nat.size a - n
le_size_sub_one : n ≤ Nat.size a - 1
⊢ a = 2 ^ n * ulp n (a - b)
Tactic:
rewrite [ulp_eq_half_ulp]
Proof state after:
round : ℕ → ℕ
n a b : ℕ
npos : 0 < n
hfaithful₀ : faithful₀ n round
hfaithful₁ : faithful₁ n round
hcorrect₀ : correct₀ n round
hfa : trunc n a = a
hfb : trunc n b = b
hba : b ≤ a
round_lt : round (a - b) < a - b
no_uflow : a - b ≥ 2 ^ n
ulp_lt_ulp : ulp n (a - b) < ulp n a
two_mul_lt : 2 * b < a
bpos : b > 0
le_ulp : b ≤ ulp n (a - b)
ulp_even : 2 ∣ ulp n (a - b)
lt_ulp : b < ulp n a
lt_size : n < Nat.size a
size_sub_one_pos : 0 < Nat.size a - 1
size_pos : 0 < Nat.size a
apos : 0 < a
eq_pow : a = 2 ^ (Nat.size a - 1)
even : 2 ∣ 2 ^ (Nat.size a - 1)
ulp_eq_half_ulp : ulp n (a - b) = ulp n a / 2
one_le_size_sub : 1 ≤ Nat.size a - n
le_size_sub_one : n ≤ Nat.size a - 1
⊢ a = 2 ^ n * (ulp n a / 2)
Theorem: b₁_lo_of_round_lt_of_no_uflow_of_ulp_lt_ulp_of_two_mul_lt_of_pos_of_le_ulp
Proof state before:
round : ℕ → ℕ
n a b : ℕ
npos : 0 < n
hfaithful₀ : faithful₀ n round
hfaithful₁ : faithful₁ n round
hcorrect₀ : correct₀ n round
hfa : trunc n a = a
hfb : trunc n b = b
hba : b ≤ a
round_lt : round (a - b) < a - b
no_uflow : a - b ≥ 2 ^ n
ulp_lt_ulp : ulp n (a - b) < ulp n a
two_mul_lt : 2 * b < a
bpos : b > 0
le_ulp : b ≤ ulp n (a - b)
ulp_even : 2 ∣ ulp n (a - b)
lt_ulp : b < ulp n a
lt_size : n < Nat.size a
size_sub_one_pos : 0 < Nat.size a - 1
size_pos : 0 < Nat.size a
apos : 0 < a
eq_pow : a = 2 ^ (Nat.size a - 1)
even : 2 ∣ 2 ^ (Nat.size a - 1)
ulp_eq_half_ulp : ulp n (a - b) = ulp n a / 2
one_le_size_sub : 1 ≤ Nat.size a - n
le_size_sub_one : n ≤ Nat.size a - 1
⊢ a = 2 ^ n * (ulp n a / 2)
Tactic:
unfold ulp expt
Proof state after:
round : ℕ → ℕ
n a b : ℕ
npos : 0 < n
hfaithful₀ : faithful₀ n round
hfaithful₁ : faithful₁ n round
hcorrect₀ : correct₀ n round
hfa : trunc n a = a
hfb : trunc n b = b
hba : b ≤ a
round_lt : round (a - b) < a - b
no_uflow : a - b ≥ 2 ^ n
ulp_lt_ulp : ulp n (a - b) < ulp n a
two_mul_lt : 2 * b < a
bpos : b > 0
le_ulp : b ≤ ulp n (a - b)
ulp_even : 2 ∣ ulp n (a - b)
lt_ulp : b < ulp n a
lt_size : n < Nat.size a
size_sub_one_pos : 0 < Nat.size a - 1
size_pos : 0 < Nat.size a
apos : 0 < a
eq_pow : a = 2 ^ (Nat.size a - 1)
even : 2 ∣ 2 ^ (Nat.size a - 1)
ulp_eq_half_ulp : ulp n (a - b) = ulp n a / 2
one_le_size_sub : 1 ≤ Nat.size a - n
le_size_sub_one : n ≤ Nat.size a - 1
⊢ a = 2 ^ n * (2 ^ (Nat.size a - n) / 2)
Theorem: b₁_lo_of_round_lt_of_no_uflow_of_ulp_lt_ulp_of_two_mul_lt_of_pos_of_le_ulp
Proof state before:
round : ℕ → ℕ
n a b : ℕ
npos : 0 < n
hfaithful₀ : faithful₀ n round
hfaithful₁ : faithful₁ n round
hcorrect₀ : correct₀ n round
hfa : trunc n a = a
hfb : trunc n b = b
hba : b ≤ a
round_lt : round (a - b) < a - b
no_uflow : a - b ≥ 2 ^ n
ulp_lt_ulp : ulp n (a - b) < ulp n a
two_mul_lt : 2 * b < a
bpos : b > 0
le_ulp : b ≤ ulp n (a - b)
ulp_even : 2 ∣ ulp n (a - b)
lt_ulp : b < ulp n a
lt_size : n < Nat.size a
size_sub_one_pos : 0 < Nat.size a - 1
size_pos : 0 < Nat.size a
apos : 0 < a
eq_pow : a = 2 ^ (Nat.size a - 1)
even : 2 ∣ 2 ^ (Nat.size a - 1)
ulp_eq_half_ulp : ulp n (a - b) = ulp n a / 2
one_le_size_sub : 1 ≤ Nat.size a - n
le_size_sub_one : n ≤ Nat.size a - 1
⊢ a = 2 ^ n * (2 ^ (Nat.size a - n) / 2)
Tactic:
conv => rhs ; right ; right ; rewrite [← pow_one 2]
Proof state after:
round : ℕ → ℕ
n a b : ℕ
npos : 0 < n
hfaithful₀ : faithful₀ n round
hfaithful₁ : faithful₁ n round
hcorrect₀ : correct₀ n round
hfa : trunc n a = a
hfb : trunc n b = b
hba : b ≤ a
round_lt : round (a - b) < a - b
no_uflow : a - b ≥ 2 ^ n
ulp_lt_ulp : ulp n (a - b) < ulp n a
two_mul_lt : 2 * b < a
bpos : b > 0
le_ulp : b ≤ ulp n (a - b)
ulp_even : 2 ∣ ulp n (a - b)
lt_ulp : b < ulp n a
lt_size : n < Nat.size a
size_sub_one_pos : 0 < Nat.size a - 1
size_pos : 0 < Nat.size a
apos : 0 < a
eq_pow : a = 2 ^ (Nat.size a - 1)
even : 2 ∣ 2 ^ (Nat.size a - 1)
ulp_eq_half_ulp : ulp n (a - b) = ulp n a / 2
one_le_size_sub : 1 ≤ Nat.size a - n
le_size_sub_one : n ≤ Nat.size a - 1
⊢ a = 2 ^ n * (2 ^ (Nat.size a - n) / 2 ^ 1)
Theorem: b₁_lo_of_round_lt_of_no_uflow_of_ulp_lt_ulp_of_two_mul_lt_of_pos_of_le_ulp
Proof state before:
round : ℕ → ℕ
n a b : ℕ
npos : 0 < n
hfaithful₀ : faithful₀ n round
hfaithful₁ : faithful₁ n round
hcorrect₀ : correct₀ n round
hfa : trunc n a = a
hfb : trunc n b = b
hba : b ≤ a
round_lt : round (a - b) < a - b
no_uflow : a - b ≥ 2 ^ n
ulp_lt_ulp : ulp n (a - b) < ulp n a
two_mul_lt : 2 * b < a
bpos : b > 0
le_ulp : b ≤ ulp n (a - b)
ulp_even : 2 ∣ ulp n (a - b)
lt_ulp : b < ulp n a
lt_size : n < Nat.size a
size_sub_one_pos : 0 < Nat.size a - 1
size_pos : 0 < Nat.size a
apos : 0 < a
eq_pow : a = 2 ^ (Nat.size a - 1)
even : 2 ∣ 2 ^ (Nat.size a - 1)
ulp_eq_half_ulp : ulp n (a - b) = ulp n a / 2
one_le_size_sub : 1 ≤ Nat.size a - n
le_size_sub_one : n ≤ Nat.size a - 1
⊢ a = 2 ^ n * (2 ^ (Nat.size a - n) / 2 ^ 1)
Tactic:
rewrite [Nat.pow_div one_le_size_sub two_pos, ← pow_add]
Proof state after:
round : ℕ → ℕ
n a b : ℕ
npos : 0 < n
hfaithful₀ : faithful₀ n round
hfaithful₁ : faithful₁ n round
hcorrect₀ : correct₀ n round
hfa : trunc n a = a
hfb : trunc n b = b
hba : b ≤ a
round_lt : round (a - b) < a - b
no_uflow : a - b ≥ 2 ^ n
ulp_lt_ulp : ulp n (a - b) < ulp n a
two_mul_lt : 2 * b < a
bpos : b > 0
le_ulp : b ≤ ulp n (a - b)
ulp_even : 2 ∣ ulp n (a - b)
lt_ulp : b < ulp n a
lt_size : n < Nat.size a
size_sub_one_pos : 0 < Nat.size a - 1
size_pos : 0 < Nat.size a
apos : 0 < a
eq_pow : a = 2 ^ (Nat.size a - 1)
even : 2 ∣ 2 ^ (Nat.size a - 1)
ulp_eq_half_ulp : ulp n (a - b) = ulp n a / 2
one_le_size_sub : 1 ≤ Nat.size a - n
le_size_sub_one : n ≤ Nat.size a - 1
⊢ a = 2 ^ (n + (Nat.size a - n - 1))
Theorem: b₁_lo_of_round_lt_of_no_uflow_of_ulp_lt_ulp_of_two_mul_lt_of_pos_of_le_ulp
Proof state before:
round : ℕ → ℕ
n a b : ℕ
npos : 0 < n
hfaithful₀ : faithful₀ n round
hfaithful₁ : faithful₁ n round
hcorrect₀ : correct₀ n round
hfa : trunc n a = a
hfb : trunc n b = b
hba : b ≤ a
round_lt : round (a - b) < a - b
no_uflow : a - b ≥ 2 ^ n
ulp_lt_ulp : ulp n (a - b) < ulp n a
two_mul_lt : 2 * b < a
bpos : b > 0
le_ulp : b ≤ ulp n (a - b)
ulp_even : 2 ∣ ulp n (a - b)
lt_ulp : b < ulp n a
lt_size : n < Nat.size a
size_sub_one_pos : 0 < Nat.size a - 1
size_pos : 0 < Nat.size a
apos : 0 < a
eq_pow : a = 2 ^ (Nat.size a - 1)
even : 2 ∣ 2 ^ (Nat.size a - 1)
ulp_eq_half_ulp : ulp n (a - b) = ulp n a / 2
one_le_size_sub : 1 ≤ Nat.size a - n
le_size_sub_one : n ≤ Nat.size a - 1
⊢ a = 2 ^ (n + (Nat.size a - n - 1))
Tactic:
rewrite [tsub_right_comm]
Proof state after:
round : ℕ → ℕ
n a b : ℕ
npos : 0 < n
hfaithful₀ : faithful₀ n round
hfaithful₁ : faithful₁ n round
hcorrect₀ : correct₀ n round
hfa : trunc n a = a
hfb : trunc n b = b
hba : b ≤ a
round_lt : round (a - b) < a - b
no_uflow : a - b ≥ 2 ^ n
ulp_lt_ulp : ulp n (a - b) < ulp n a
two_mul_lt : 2 * b < a
bpos : b > 0
le_ulp : b ≤ ulp n (a - b)
ulp_even : 2 ∣ ulp n (a - b)
lt_ulp : b < ulp n a
lt_size : n < Nat.size a
size_sub_one_pos : 0 < Nat.size a - 1
size_pos : 0 < Nat.size a
apos : 0 < a
eq_pow : a = 2 ^ (Nat.size a - 1)
even : 2 ∣ 2 ^ (Nat.size a - 1)
ulp_eq_half_ulp : ulp n (a - b) = ulp n a / 2
one_le_size_sub : 1 ≤ Nat.size a - n
le_size_sub_one : n ≤ Nat.size a - 1
⊢ a = 2 ^ (n + (Nat.size a - 1 - n))
Theorem: b₁_lo_of_round_lt_of_no_uflow_of_ulp_lt_ulp_of_two_mul_lt_of_pos_of_le_ulp
Proof state before:
round : ℕ → ℕ
n a b : ℕ
npos : 0 < n
hfaithful₀ : faithful₀ n round
hfaithful₁ : faithful₁ n round
hcorrect₀ : correct₀ n round
hfa : trunc n a = a
hfb : trunc n b = b
hba : b ≤ a
round_lt : round (a - b) < a - b
no_uflow : a - b ≥ 2 ^ n
ulp_lt_ulp : ulp n (a - b) < ulp n a
two_mul_lt : 2 * b < a
bpos : b > 0
le_ulp : b ≤ ulp n (a - b)
ulp_even : 2 ∣ ulp n (a - b)
lt_ulp : b < ulp n a
lt_size : n < Nat.size a
size_sub_one_pos : 0 < Nat.size a - 1
size_pos : 0 < Nat.size a
apos : 0 < a
eq_pow : a = 2 ^ (Nat.size a - 1)
even : 2 ∣ 2 ^ (Nat.size a - 1)
ulp_eq_half_ulp : ulp n (a - b) = ulp n a / 2
one_le_size_sub : 1 ≤ Nat.size a - n
le_size_sub_one : n ≤ Nat.size a - 1
⊢ a = 2 ^ (n + (Nat.size a - 1 - n))
Tactic:
rewrite [add_tsub_cancel_of_le le_size_sub_one]
Proof state after:
round : ℕ → ℕ
n a b : ℕ
npos : 0 < n
hfaithful₀ : faithful₀ n round
hfaithful₁ : faithful₁ n round
hcorrect₀ : correct₀ n round
hfa : trunc n a = a
hfb : trunc n b = b
hba : b ≤ a
round_lt : round (a - b) < a - b
no_uflow : a - b ≥ 2 ^ n
ulp_lt_ulp : ulp n (a - b) < ulp n a
two_mul_lt : 2 * b < a
bpos : b > 0
le_ulp : b ≤ ulp n (a - b)
ulp_even : 2 ∣ ulp n (a - b)
lt_ulp : b < ulp n a
lt_size : n < Nat.size a
size_sub_one_pos : 0 < Nat.size a - 1
size_pos : 0 < Nat.size a
apos : 0 < a
eq_pow : a = 2 ^ (Nat.size a - 1)
even : 2 ∣ 2 ^ (Nat.size a - 1)
ulp_eq_half_ulp : ulp n (a - b) = ulp n a / 2
one_le_size_sub : 1 ≤ Nat.size a - n
le_size_sub_one : n ≤ Nat.size a - 1
⊢ a = 2 ^ (Nat.size a - 1)
Theorem: b₁_lo_of_round_lt_of_no_uflow_of_ulp_lt_ulp_of_two_mul_lt_of_pos_of_le_ulp
Proof state before:
round : ℕ → ℕ
n a b : ℕ
npos : 0 < n
hfaithful₀ : faithful₀ n round
hfaithful₁ : faithful₁ n round
hcorrect₀ : correct₀ n round
hfa : trunc n a = a
hfb : trunc n b = b
hba : b ≤ a
round_lt : round (a - b) < a - b
no_uflow : a - b ≥ 2 ^ n
ulp_lt_ulp : ulp n (a - b) < ulp n a
two_mul_lt : 2 * b < a
bpos : b > 0
le_ulp : b ≤ ulp n (a - b)
ulp_even : 2 ∣ ulp n (a - b)
lt_ulp : b < ulp n a
lt_size : n < Nat.size a
size_sub_one_pos : 0 < Nat.size a - 1
size_pos : 0 < Nat.size a
apos : 0 < a
eq_pow : a = 2 ^ (Nat.size a - 1)
even : 2 ∣ 2 ^ (Nat.size a - 1)
ulp_eq_half_ulp : ulp n (a - b) = ulp n a / 2
one_le_size_sub : 1 ≤ Nat.size a - n
le_size_sub_one : n ≤ Nat.size a - 1
⊢ a = 2 ^ (Nat.size a - 1)
Tactic:
exact eq_pow
Proof state after:
no goals
Theorem: b₁_lo_of_round_lt_of_no_uflow_of_ulp_lt_ulp_of_two_mul_lt_of_pos_of_le_ulp
Proof state before:
round : ℕ → ℕ
n a b : ℕ
npos : 0 < n
hfaithful₀ : faithful₀ n round
hfaithful₁ : faithful₁ n round
hcorrect₀ : correct₀ n round
hfa : trunc n a = a
hfb : trunc n b = b
hba : b ≤ a
round_lt : round (a - b) < a - b
no_uflow : a - b ≥ 2 ^ n
ulp_lt_ulp : ulp n (a - b) < ulp n a
two_mul_lt : 2 * b < a
bpos : b > 0
le_ulp : b ≤ ulp n (a - b)
ulp_even : 2 ∣ ulp n (a - b)
lt_ulp : b < ulp n a
lt_size : n < Nat.size a
size_sub_one_pos : 0 < Nat.size a - 1
size_pos : 0 < Nat.size a
apos : 0 < a
eq_pow : a = 2 ^ (Nat.size a - 1)
even : 2 ∣ 2 ^ (Nat.size a - 1)
ulp_eq_half_ulp : ulp n (a - b) = ulp n a / 2
one_le_size_sub : 1 ≤ Nat.size a - n
le_size_sub_one : n ≤ Nat.size a - 1
k₁ : a = 2 ^ n * ulp n (a - b)
⊢ a - b = (2 ^ n - 1) * ulp n (a - b) + (ulp n (a - b) - b)
Tactic:
rewrite [Nat.sub_one, Nat.mul_pred_left]
Proof state after:
round : ℕ → ℕ
n a b : ℕ
npos : 0 < n
hfaithful₀ : faithful₀ n round
hfaithful₁ : faithful₁ n round
hcorrect₀ : correct₀ n round
hfa : trunc n a = a
hfb : trunc n b = b
hba : b ≤ a
round_lt : round (a - b) < a - b
no_uflow : a - b ≥ 2 ^ n
ulp_lt_ulp : ulp n (a - b) < ulp n a
two_mul_lt : 2 * b < a
bpos : b > 0
le_ulp : b ≤ ulp n (a - b)
ulp_even : 2 ∣ ulp n (a - b)
lt_ulp : b < ulp n a
lt_size : n < Nat.size a
size_sub_one_pos : 0 < Nat.size a - 1
size_pos : 0 < Nat.size a
apos : 0 < a
eq_pow : a = 2 ^ (Nat.size a - 1)
even : 2 ∣ 2 ^ (Nat.size a - 1)
ulp_eq_half_ulp : ulp n (a - b) = ulp n a / 2
one_le_size_sub : 1 ≤ Nat.size a - n
le_size_sub_one : n ≤ Nat.size a - 1
k₁ : a = 2 ^ n * ulp n (a - b)
⊢ a - b = 2 ^ n * ulp n (a - b) - ulp n (a - b) + (ulp n (a - b) - b)
Theorem: b₁_lo_of_round_lt_of_no_uflow_of_ulp_lt_ulp_of_two_mul_lt_of_pos_of_le_ulp
Proof state before:
round : ℕ → ℕ
n a b : ℕ
npos : 0 < n
hfaithful₀ : faithful₀ n round
hfaithful₁ : faithful₁ n round
hcorrect₀ : correct₀ n round
hfa : trunc n a = a
hfb : trunc n b = b
hba : b ≤ a
round_lt : round (a - b) < a - b
no_uflow : a - b ≥ 2 ^ n
ulp_lt_ulp : ulp n (a - b) < ulp n a
two_mul_lt : 2 * b < a
bpos : b > 0
le_ulp : b ≤ ulp n (a - b)
ulp_even : 2 ∣ ulp n (a - b)
lt_ulp : b < ulp n a
lt_size : n < Nat.size a
size_sub_one_pos : 0 < Nat.size a - 1
size_pos : 0 < Nat.size a
apos : 0 < a
eq_pow : a = 2 ^ (Nat.size a - 1)
even : 2 ∣ 2 ^ (Nat.size a - 1)
ulp_eq_half_ulp : ulp n (a - b) = ulp n a / 2
one_le_size_sub : 1 ≤ Nat.size a - n
le_size_sub_one : n ≤ Nat.size a - 1
k₁ : a = 2 ^ n * ulp n (a - b)
⊢ a - b = 2 ^ n * ulp n (a - b) - ulp n (a - b) + (ulp n (a - b) - b)
Tactic:
rewrite [← add_tsub_assoc_of_le le_ulp]
Proof state after:
round : ℕ → ℕ
n a b : ℕ
npos : 0 < n
hfaithful₀ : faithful₀ n round
hfaithful₁ : faithful₁ n round
hcorrect₀ : correct₀ n round
hfa : trunc n a = a
hfb : trunc n b = b
hba : b ≤ a
round_lt : round (a - b) < a - b
no_uflow : a - b ≥ 2 ^ n
ulp_lt_ulp : ulp n (a - b) < ulp n a
two_mul_lt : 2 * b < a
bpos : b > 0
le_ulp : b ≤ ulp n (a - b)
ulp_even : 2 ∣ ulp n (a - b)
lt_ulp : b < ulp n a
lt_size : n < Nat.size a
size_sub_one_pos : 0 < Nat.size a - 1
size_pos : 0 < Nat.size a
apos : 0 < a
eq_pow : a = 2 ^ (Nat.size a - 1)
even : 2 ∣ 2 ^ (Nat.size a - 1)
ulp_eq_half_ulp : ulp n (a - b) = ulp n a / 2
one_le_size_sub : 1 ≤ Nat.size a - n
le_size_sub_one : n ≤ Nat.size a - 1
k₁ : a = 2 ^ n * ulp n (a - b)
⊢ a - b = 2 ^ n * ulp n (a - b) - ulp n (a - b) + ulp n (a - b) - b