mathlib documentation

data.int.basic

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@[instance]
def int.inhabited  :
inhabited ℤ
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@[instance]
def int.nontrivial  :
nontrivial ℤ
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@[instance]
def int.comm_ring  :
comm_ring ℤ
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Extra instances to short-circuit type class resolution #

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@[instance]
def int.add_comm_monoid  :
add_comm_monoid ℤ
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@[instance]
def int.add_monoid  :
add_monoid ℤ
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@[instance]
def int.monoid  :
monoid ℤ
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@[instance]
def int.comm_monoid  :
comm_monoid ℤ
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@[instance]
def int.comm_semigroup  :
comm_semigroup ℤ
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@[instance]
def int.semigroup  :
semigroup ℤ
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@[instance]
def int.add_comm_semigroup  :
add_comm_semigroup ℤ
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@[instance]
def int.add_semigroup  :
add_semigroup ℤ
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@[instance]
def int.comm_semiring  :
comm_semiring ℤ
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@[instance]
def int.semiring  :
semiring ℤ
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@[instance]
def int.ring  :
ring ℤ
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@[instance]
def int.distrib  :
distrib ℤ
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@[instance]
def int.linear_ordered_comm_ring  :
linear_ordered_comm_ring ℤ
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@[instance]
def int.linear_ordered_add_comm_group  :
linear_ordered_add_comm_group ℤ
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@[simp]
theorem int.add_neg_one (i : ℤ) :
i + -1 = i - 1
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theorem int.abs_eq_nat_abs (a : ℤ) :
abs a = ↑(a.nat_abs)
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theorem int.nat_abs_abs (a : ℤ) :
(abs a).nat_abs = a.nat_abs
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theorem int.sign_mul_abs (a : ℤ) :
(a.sign) * abs a = a
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@[simp]
theorem int.default_eq_zero  :
default ℤ = 0
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@[instance]
meta def int.has_to_format  :
has_to_format ℤ
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@[instance]
meta def int.has_reflect  :
has_reflect ℤ
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@[simp]
theorem int.add_def {a b : ℤ} :
a.add b = a + b
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@[simp]
theorem int.mul_def {a b : ℤ} :
a.mul b = a * b
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@[simp]
theorem int.neg_succ_not_nonneg (n : ℕ) :
0 ≤ -[1+ n] ↔ false
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@[simp]
theorem int.neg_succ_not_pos (n : ℕ) :
0 < -[1+ n] ↔ false
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@[simp]
theorem int.neg_succ_sub_one (n : ℕ) :
-[1+ n] - 1 = -[1+ n + 1]
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@[simp]
theorem int.coe_nat_mul_neg_succ (m n : ℕ) :
(↑m) * -[1+ n] = -(↑m) * ↑(n.succ)
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@[simp]
theorem int.neg_succ_mul_coe_nat (m n : ℕ) :
-[1+ m] * ↑n = -(↑(m.succ)) * ↑n
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@[simp]
theorem int.neg_succ_mul_neg_succ (m n : ℕ) :
-[1+ m] * -[1+ n] = (↑(m.succ)) * ↑(n.succ)
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@[simp]
theorem int.coe_nat_le {m n : ℕ} :
↑m ≤ ↑n ↔ m ≤ n
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@[simp]
theorem int.coe_nat_lt {m n : ℕ} :
↑m < ↑n ↔ m < n
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@[simp]
theorem int.coe_nat_inj' {m n : ℕ} :
↑m = ↑n ↔ m = n
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@[simp]
theorem int.coe_nat_pos {n : ℕ} :
0 < ↑n ↔ 0 < n
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@[simp]
theorem int.coe_nat_eq_zero {n : ℕ} :
↑n = 0 ↔ n = 0
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theorem int.coe_nat_ne_zero {n : ℕ} :
↑n ≠ 0 ↔ n ≠ 0
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@[simp]
theorem int.coe_nat_nonneg (n : ℕ) :
0 ≤ ↑n
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theorem int.coe_nat_ne_zero_iff_pos {n : ℕ} :
↑n ≠ 0 ↔ 0 < n
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theorem int.coe_nat_succ_pos (n : ℕ) :
0 < ↑(n.succ)
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@[simp]
theorem int.coe_nat_abs (n : ℕ) :
abs ↑n = ↑n

succ and pred #

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def int.succ (a : ℤ) :
ℤ

Immediate successor of an integer: succ n = n + 1

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def int.pred (a : ℤ) :
ℤ

Immediate predecessor of an integer: pred n = n - 1

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theorem int.nat_succ_eq_int_succ (n : ℕ) :
↑(n.succ) = ↑n.succ
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theorem int.pred_succ (a : ℤ) :
a.succ.pred = a
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theorem int.succ_pred (a : ℤ) :
a.pred.succ = a
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theorem int.neg_succ (a : ℤ) :
-a.succ = (-a).pred
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theorem int.succ_neg_succ (a : ℤ) :
(-a.succ).succ = -a
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theorem int.neg_pred (a : ℤ) :
-a.pred = (-a).succ
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theorem int.pred_neg_pred (a : ℤ) :
(-a.pred).pred = -a
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theorem int.pred_nat_succ (n : ℕ) :
↑(n.succ).pred = ↑n
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theorem int.neg_nat_succ (n : ℕ) :
-↑(n.succ) = (-↑n).pred
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theorem int.succ_neg_nat_succ (n : ℕ) :
(-↑(n.succ)).succ = -↑n
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theorem int.lt_succ_self (a : ℤ) :
a < a.succ
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theorem int.pred_self_lt (a : ℤ) :
a.pred < a
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theorem int.add_one_le_iff {a b : ℤ} :
a + 1 ≤ b ↔ a < b
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theorem int.lt_add_one_iff {a b : ℤ} :
a < b + 1 ↔ a ≤ b
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@[simp]
theorem int.succ_coe_nat_pos (n : ℕ) :
0 < ↑n + 1
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theorem int.coe_pred_of_pos (n : ℕ) (h : 0 < n) :
↑(n - 1) = ↑n - 1
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theorem int.le_add_one {a b : ℤ} (h : a ≤ b) :
a ≤ b + 1
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theorem int.sub_one_lt_iff {a b : ℤ} :
a - 1 < b ↔ a ≤ b
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theorem int.le_sub_one_iff {a b : ℤ} :
a ≤ b - 1 ↔ a < b
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@[simp]
theorem int.eq_zero_iff_abs_lt_one {a : ℤ} :
abs a < 1 ↔ a = 0
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theorem int.induction_on {p : ℤ → Prop} (i : ℤ) (hz : p 0) (hp : ∀ (i : ℕ), p ↑i → p (↑i + 1)) (hn : ∀ (i : ℕ), p (-↑i) → p (-↑i - 1)) :
p i
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def int.induction_on' {C : ℤ → Sort u_1} (z b : ℤ) :
C b → (Π (k : ℤ), b ≤ k → C k → C (k + 1)) → (Π (k : ℤ), k ≤ b → C k → C (k - 1)) → C z

Inductively define a function on ℤ by defining it at b, for the succ of a number greater than b, and the pred of a number less than b.

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nat abs #

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theorem int.nat_abs_add_le (a b : ℤ) :
(a + b).nat_abs ≤ a.nat_abs + b.nat_abs
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theorem int.nat_abs_neg_of_nat (n : ℕ) :
(int.neg_of_nat n).nat_abs = n
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theorem int.nat_abs_mul (a b : ℤ) :
(a * b).nat_abs = (a.nat_abs) * b.nat_abs
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theorem int.nat_abs_mul_nat_abs_eq {a b : ℤ} {c : ℕ} (h : a * b = ↑c) :
(a.nat_abs) * b.nat_abs = c
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@[simp]
theorem int.nat_abs_mul_self' (a : ℤ) :
(↑(a.nat_abs)) * ↑(a.nat_abs) = a * a
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theorem int.neg_succ_of_nat_eq' (m : ℕ) :
-[1+ m] = -↑m - 1
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theorem int.nat_abs_ne_zero_of_ne_zero {z : ℤ} (hz : z ≠ 0) :
z.nat_abs ≠ 0
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@[simp]
theorem int.nat_abs_eq_zero {a : ℤ} :
a.nat_abs = 0 ↔ a = 0
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theorem int.nat_abs_ne_zero {a : ℤ} :
a.nat_abs ≠ 0 ↔ a ≠ 0
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theorem int.nat_abs_lt_nat_abs_of_nonneg_of_lt {a b : ℤ} (w₁ : 0 ≤ a) (w₂ : a < b) :
a.nat_abs < b.nat_abs
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theorem int.nat_abs_eq_nat_abs_iff {a b : ℤ} :
a.nat_abs = b.nat_abs ↔ a = b ∨ a = -b
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theorem int.nat_abs_eq_iff {a : ℤ} {n : ℕ} :
a.nat_abs = n ↔ a = ↑n ∨ a = -↑n
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theorem int.nat_abs_eq_iff_mul_self_eq {a b : ℤ} :
a.nat_abs = b.nat_abs ↔ a * a = b * b
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theorem int.nat_abs_lt_iff_mul_self_lt {a b : ℤ} :
a.nat_abs < b.nat_abs ↔ a * a < b * b
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theorem int.nat_abs_le_iff_mul_self_le {a b : ℤ} :
a.nat_abs ≤ b.nat_abs ↔ a * a ≤ b * b
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theorem int.nat_abs_eq_iff_sq_eq {a b : ℤ} :
a.nat_abs = b.nat_abs ↔ a ^ 2 = b ^ 2
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theorem int.nat_abs_lt_iff_sq_lt {a b : ℤ} :
a.nat_abs < b.nat_abs ↔ a ^ 2 < b ^ 2
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theorem int.nat_abs_le_iff_sq_le {a b : ℤ} :
a.nat_abs ≤ b.nat_abs ↔ a ^ 2 ≤ b ^ 2

/ #

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@[simp]
theorem int.of_nat_div (m n : ℕ) :
int.of_nat (m / n) = int.of_nat m / int.of_nat n
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@[simp]
theorem int.coe_nat_div (m n : ℕ) :
↑(m / n) = ↑m / ↑n
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theorem int.neg_succ_of_nat_div (m : ℕ) {b : ℤ} (H : 0 < b) :
-[1+ m] / b = -(↑m / b + 1)
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theorem int.zero_div (b : ℤ) :
0 / b = 0
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theorem int.div_zero (a : ℤ) :
a / 0 = 0
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@[simp]
theorem int.div_neg (a b : ℤ) :
a / -b = -(a / b)
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theorem int.div_of_neg_of_pos {a b : ℤ} (Ha : a < 0) (Hb : 0 < b) :
a / b = -((-a - 1) / b + 1)
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theorem int.div_nonneg {a b : ℤ} (Ha : 0 ≤ a) (Hb : 0 ≤ b) :
0 ≤ a / b
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theorem int.div_nonpos {a b : ℤ} (Ha : 0 ≤ a) (Hb : b ≤ 0) :
a / b ≤ 0
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theorem int.div_neg' {a b : ℤ} (Ha : a < 0) (Hb : 0 < b) :
a / b < 0
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@[simp]
theorem int.div_one (a : ℤ) :
a / 1 = a
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theorem int.div_eq_zero_of_lt {a b : ℤ} (H1 : 0 ≤ a) (H2 : a < b) :
a / b = 0
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theorem int.div_eq_zero_of_lt_abs {a b : ℤ} (H1 : 0 ≤ a) (H2 : a < abs b) :
a / b = 0
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theorem int.add_mul_div_right (a b : ℤ) {c : ℤ} (H : c ≠ 0) :
(a + b * c) / c = a / c + b
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theorem int.add_mul_div_left (a : ℤ) {b : ℤ} (c : ℤ) (H : b ≠ 0) :
(a + b * c) / b = a / b + c
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theorem int.add_div_of_dvd_right {a b c : ℤ} (H : c ∣ b) :
(a + b) / c = a / c + b / c
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theorem int.add_div_of_dvd_left {a b c : ℤ} (H : c ∣ a) :
(a + b) / c = a / c + b / c
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@[simp]
theorem int.mul_div_cancel (a : ℤ) {b : ℤ} (H : b ≠ 0) :
a * b / b = a
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@[simp]
theorem int.mul_div_cancel_left {a : ℤ} (b : ℤ) (H : a ≠ 0) :
a * b / a = b
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@[simp]
theorem int.div_self {a : ℤ} (H : a ≠ 0) :
a / a = 1

mod #

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theorem int.of_nat_mod (m n : ℕ) :
↑m % ↑n = int.of_nat (m % n)
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@[simp]
theorem int.coe_nat_mod (m n : ℕ) :
↑(m % n) = ↑m % ↑n
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theorem int.neg_succ_of_nat_mod (m : ℕ) {b : ℤ} (bpos : 0 < b) :
-[1+ m] % b = b - 1 - ↑m % b
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@[simp]
theorem int.mod_neg (a b : ℤ) :
a % -b = a % b
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@[simp]
theorem int.mod_abs (a b : ℤ) :
a % abs b = a % b
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theorem int.zero_mod (b : ℤ) :
0 % b = 0
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theorem int.mod_zero (a : ℤ) :
a % 0 = a
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theorem int.mod_one (a : ℤ) :
a % 1 = 0
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theorem int.mod_eq_of_lt {a b : ℤ} (H1 : 0 ≤ a) (H2 : a < b) :
a % b = a
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theorem int.mod_nonneg (a : ℤ) {b : ℤ} :
b ≠ 0 → 0 ≤ a % b
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theorem int.mod_lt_of_pos (a : ℤ) {b : ℤ} (H : 0 < b) :
a % b < b
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theorem int.mod_lt (a : ℤ) {b : ℤ} (H : b ≠ 0) :
a % b < abs b
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theorem int.mod_add_div_aux (m n : ℕ) :
↑n - (↑m % ↑n + 1) - ((↑n) * (↑m / ↑n) + ↑n) = -[1+ m]
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theorem int.mod_add_div (a b : ℤ) :
a % b + b * (a / b) = a
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theorem int.div_add_mod (a b : ℤ) :
b * (a / b) + a % b = a
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theorem int.mod_add_div' (m k : ℤ) :
m % k + (m / k) * k = m
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theorem int.div_add_mod' (m k : ℤ) :
(m / k) * k + m % k = m
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theorem int.mod_def (a b : ℤ) :
a % b = a - b * (a / b)
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@[simp]
theorem int.add_mul_mod_self {a b c : ℤ} :
(a + b * c) % c = a % c
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@[simp]
theorem int.add_mul_mod_self_left (a b c : ℤ) :
(a + b * c) % b = a % b
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@[simp]
theorem int.add_mod_self {a b : ℤ} :
(a + b) % b = a % b
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@[simp]
theorem int.add_mod_self_left {a b : ℤ} :
(a + b) % a = b % a
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@[simp]
theorem int.mod_add_mod (m n k : ℤ) :
(m % n + k) % n = (m + k) % n
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@[simp]
theorem int.add_mod_mod (m n k : ℤ) :
(m + n % k) % k = (m + n) % k
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theorem int.add_mod (a b n : ℤ) :
(a + b) % n = (a % n + b % n) % n
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theorem int.add_mod_eq_add_mod_right {m n k : ℤ} (i : ℤ) (H : m % n = k % n) :
(m + i) % n = (k + i) % n
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theorem int.add_mod_eq_add_mod_left {m n k : ℤ} (i : ℤ) (H : m % n = k % n) :
(i + m) % n = (i + k) % n
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theorem int.mod_add_cancel_right {m n k : ℤ} (i : ℤ) :
(m + i) % n = (k + i) % n ↔ m % n = k % n
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theorem int.mod_add_cancel_left {m n k i : ℤ} :
(i + m) % n = (i + k) % n ↔ m % n = k % n
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theorem int.mod_sub_cancel_right {m n k : ℤ} (i : ℤ) :
(m - i) % n = (k - i) % n ↔ m % n = k % n
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theorem int.mod_eq_mod_iff_mod_sub_eq_zero {m n k : ℤ} :
m % n = k % n ↔ (m - k) % n = 0
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@[simp]
theorem int.mul_mod_left (a b : ℤ) :
a * b % b = 0
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@[simp]
theorem int.mul_mod_right (a b : ℤ) :
a * b % a = 0
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theorem int.mul_mod (a b n : ℤ) :
a * b % n = (a % n) * (b % n) % n
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@[simp]
theorem int.neg_mod_two (i : ℤ) :
-i % 2 = i % 2
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theorem int.mod_self {a : ℤ} :
a % a = 0
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@[simp]
theorem int.mod_mod_of_dvd (n : ℤ) {m k : ℤ} (h : m ∣ k) :
n % k % m = n % m
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@[simp]
theorem int.mod_mod (a b : ℤ) :
a % b % b = a % b
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theorem int.sub_mod (a b n : ℤ) :
(a - b) % n = (a % n - b % n) % n

properties of / and % #

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@[simp]
theorem int.mul_div_mul_of_pos {a : ℤ} (b c : ℤ) (H : 0 < a) :
a * b / a * c = b / c
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@[simp]
theorem int.mul_div_mul_of_pos_left (a : ℤ) {b : ℤ} (c : ℤ) (H : 0 < b) :
a * b / c * b = a / c
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@[simp]
theorem int.mul_mod_mul_of_pos {a : ℤ} (b c : ℤ) (H : 0 < a) :
a * b % a * c = a * (b % c)
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theorem int.lt_div_add_one_mul_self (a : ℤ) {b : ℤ} (H : 0 < b) :
a < (a / b + 1) * b
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theorem int.abs_div_le_abs (a b : ℤ) :
abs (a / b) ≤ abs a
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theorem int.div_le_self {a : ℤ} (b : ℤ) (Ha : 0 ≤ a) :
a / b ≤ a
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theorem int.mul_div_cancel_of_mod_eq_zero {a b : ℤ} (H : a % b = 0) :
b * (a / b) = a
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theorem int.div_mul_cancel_of_mod_eq_zero {a b : ℤ} (H : a % b = 0) :
(a / b) * b = a
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theorem int.mod_two_eq_zero_or_one (n : ℤ) :
n % 2 = 0 ∨ n % 2 = 1

dvd #

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theorem int.coe_nat_dvd {m n : ℕ} :
↑m ∣ ↑n ↔ m ∣ n
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theorem int.coe_nat_dvd_left {n : ℕ} {z : ℤ} :
↑n ∣ z ↔ n ∣ z.nat_abs
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theorem int.coe_nat_dvd_right {n : ℕ} {z : ℤ} :
z ∣ ↑n ↔ z.nat_abs ∣ n
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theorem int.dvd_antisymm {a b : ℤ} (H1 : 0 ≤ a) (H2 : 0 ≤ b) :
a ∣ b → b ∣ a → a = b
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theorem int.dvd_of_mod_eq_zero {a b : ℤ} (H : b % a = 0) :
a ∣ b
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theorem int.mod_eq_zero_of_dvd {a b : ℤ} :
a ∣ b → b % a = 0
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theorem int.dvd_iff_mod_eq_zero (a b : ℤ) :
a ∣ b ↔ b % a = 0
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theorem int.dvd_sub_of_mod_eq {a b c : ℤ} (h : a % b = c) :
b ∣ a - c

If a % b = c then b divides a - c.

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theorem int.nat_abs_dvd {a b : ℤ} :
↑(a.nat_abs) ∣ b ↔ a ∣ b
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theorem int.dvd_nat_abs {a b : ℤ} :
a ∣ ↑(b.nat_abs) ↔ a ∣ b
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@[instance]
def int.decidable_dvd  :
decidable_rel has_dvd.dvd
Equations
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theorem int.div_mul_cancel {a b : ℤ} (H : b ∣ a) :
(a / b) * b = a
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theorem int.mul_div_cancel' {a b : ℤ} (H : a ∣ b) :
a * (b / a) = b
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theorem int.mul_div_assoc (a : ℤ) {b c : ℤ} :
c ∣ b → a * b / c = a * (b / c)
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theorem int.mul_div_assoc' (b : ℤ) {a c : ℤ} (h : c ∣ a) :
a * b / c = (a / c) * b
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theorem int.div_dvd_div {a b c : ℤ} (H1 : a ∣ b) (H2 : b ∣ c) :
b / a ∣ c / a
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theorem int.eq_mul_of_div_eq_right {a b c : ℤ} (H1 : b ∣ a) (H2 : a / b = c) :
a = b * c
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theorem int.div_eq_of_eq_mul_right {a b c : ℤ} (H1 : b ≠ 0) (H2 : a = b * c) :
a / b = c
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theorem int.eq_div_of_mul_eq_right {a b c : ℤ} (H1 : a ≠ 0) (H2 : a * b = c) :
b = c / a
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theorem int.div_eq_iff_eq_mul_right {a b c : ℤ} (H : b ≠ 0) (H' : b ∣ a) :
a / b = c ↔ a = b * c
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theorem int.div_eq_iff_eq_mul_left {a b c : ℤ} (H : b ≠ 0) (H' : b ∣ a) :
a / b = c ↔ a = c * b
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theorem int.eq_mul_of_div_eq_left {a b c : ℤ} (H1 : b ∣ a) (H2 : a / b = c) :
a = c * b
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theorem int.div_eq_of_eq_mul_left {a b c : ℤ} (H1 : b ≠ 0) (H2 : a = c * b) :
a / b = c
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theorem int.neg_div_of_dvd {a b : ℤ} (H : b ∣ a) :
-a / b = -(a / b)
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theorem int.sub_div_of_dvd {a b c : ℤ} (hcb : c ∣ b) :
(a - b) / c = a / c - b / c
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theorem int.sub_div_of_dvd_sub {a b c : ℤ} (hcab : c ∣ a - b) :
(a - b) / c = a / c - b / c
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theorem int.div_sign (a b : ℤ) :
a / b.sign = a * b.sign
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@[simp]
theorem int.sign_mul (a b : ℤ) :
(a * b).sign = (a.sign) * b.sign
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theorem int.sign_eq_div_abs (a : ℤ) :
a.sign = a / abs a
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theorem int.mul_sign (i : ℤ) :
i * i.sign = ↑(i.nat_abs)
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@[simp]
theorem int.sign_pow_bit1 (k : ℕ) (n : ℤ) :
n.sign ^ bit1 k = n.sign
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theorem int.le_of_dvd {a b : ℤ} (bpos : 0 < b) (H : a ∣ b) :
a ≤ b
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theorem int.eq_one_of_dvd_one {a : ℤ} (H : 0 ≤ a) (H' : a ∣ 1) :
a = 1
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theorem int.eq_one_of_mul_eq_one_right {a b : ℤ} (H : 0 ≤ a) (H' : a * b = 1) :
a = 1
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theorem int.eq_one_of_mul_eq_one_left {a b : ℤ} (H : 0 ≤ b) (H' : a * b = 1) :
b = 1
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theorem int.of_nat_dvd_of_dvd_nat_abs {a : ℕ} {z : ℤ} (haz : a ∣ z.nat_abs) :
↑a ∣ z
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theorem int.dvd_nat_abs_of_of_nat_dvd {a : ℕ} {z : ℤ} (haz : ↑a ∣ z) :
a ∣ z.nat_abs
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theorem int.pow_dvd_of_le_of_pow_dvd {p m n : ℕ} {k : ℤ} (hmn : m ≤ n) (hdiv : ↑(p ^ n) ∣ k) :
↑(p ^ m) ∣ k
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theorem int.dvd_of_pow_dvd {p k : ℕ} {m : ℤ} (hk : 1 ≤ k) (hpk : ↑(p ^ k) ∣ m) :
↑p ∣ m
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theorem int.exists_lt_and_lt_iff_not_dvd (m : ℤ) {n : ℤ} (hn : 0 < n) :
(∃ (k : ℤ), n * k < m ∧ m < n * (k + 1)) ↔ ¬n ∣ m

If n > 0 then m is not divisible by n iff it is between n * k and n * (k + 1) for some k.

/ and ordering #

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theorem int.div_mul_le (a : ℤ) {b : ℤ} (H : b ≠ 0) :
(a / b) * b ≤ a
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theorem int.div_le_of_le_mul {a b c : ℤ} (H : 0 < c) (H' : a ≤ b * c) :
a / c ≤ b
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theorem int.mul_lt_of_lt_div {a b c : ℤ} (H : 0 < c) (H3 : a < b / c) :
a * c < b
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theorem int.mul_le_of_le_div {a b c : ℤ} (H1 : 0 < c) (H2 : a ≤ b / c) :
a * c ≤ b
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theorem int.le_div_of_mul_le {a b c : ℤ} (H1 : 0 < c) (H2 : a * c ≤ b) :
a ≤ b / c
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theorem int.le_div_iff_mul_le {a b c : ℤ} (H : 0 < c) :
a ≤ b / c ↔ a * c ≤ b
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theorem int.div_le_div {a b c : ℤ} (H : 0 < c) (H' : a ≤ b) :
a / c ≤ b / c
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theorem int.div_lt_of_lt_mul {a b c : ℤ} (H : 0 < c) (H' : a < b * c) :
a / c < b
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theorem int.lt_mul_of_div_lt {a b c : ℤ} (H1 : 0 < c) (H2 : a / c < b) :
a < b * c
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theorem int.div_lt_iff_lt_mul {a b c : ℤ} (H : 0 < c) :
a / c < b ↔ a < b * c
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theorem int.le_mul_of_div_le {a b c : ℤ} (H1 : 0 ≤ b) (H2 : b ∣ a) (H3 : a / b ≤ c) :
a ≤ c * b
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theorem int.lt_div_of_mul_lt {a b c : ℤ} (H1 : 0 ≤ b) (H2 : b ∣ c) (H3 : a * b < c) :
a < c / b
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theorem int.lt_div_iff_mul_lt {a b : ℤ} (c : ℤ) (H : 0 < c) (H' : c ∣ b) :
a < b / c ↔ a * c < b
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theorem int.div_pos_of_pos_of_dvd {a b : ℤ} (H1 : 0 < a) (H2 : 0 ≤ b) (H3 : b ∣ a) :
0 < a / b
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theorem int.div_eq_div_of_mul_eq_mul {a b c d : ℤ} (H2 : d ∣ c) (H3 : b ≠ 0) (H4 : d ≠ 0) (H5 : a * d = b * c) :
a / b = c / d
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theorem int.eq_mul_div_of_mul_eq_mul_of_dvd_left {a b c d : ℤ} (hb : b ≠ 0) (hbc : b ∣ c) (h : b * a = c * d) :
a = (c / b) * d
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theorem int.eq_zero_of_dvd_of_nat_abs_lt_nat_abs {a b : ℤ} (w : a ∣ b) (h : b.nat_abs < a.nat_abs) :
b = 0

If an integer with larger absolute value divides an integer, it is zero.

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theorem int.eq_zero_of_dvd_of_nonneg_of_lt {a b : ℤ} (w₁ : 0 ≤ a) (w₂ : a < b) (h : b ∣ a) :
a = 0
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theorem int.eq_of_mod_eq_of_nat_abs_sub_lt_nat_abs {a b c : ℤ} (h1 : a % b = c) (h2 : (a - c).nat_abs < b.nat_abs) :
a = c

If two integers are congruent to a sufficiently large modulus, they are equal.

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theorem int.of_nat_add_neg_succ_of_nat_of_lt {m n : ℕ} (h : m < n.succ) :
int.of_nat m + -[1+ n] = -[1+ n - m]
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theorem int.of_nat_add_neg_succ_of_nat_of_ge {m n : ℕ} (h : n.succ ≤ m) :
int.of_nat m + -[1+ n] = int.of_nat (m - n.succ)
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@[simp]
theorem int.neg_add_neg (m n : ℕ) :
-[1+ m] + -[1+ n] = -[1+ (m + n).succ]

to_nat #

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theorem int.to_nat_eq_max (a : ℤ) :
↑(a.to_nat) = max a 0
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@[simp]
theorem int.to_nat_zero  :
0.to_nat = 0
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@[simp]
theorem int.to_nat_one  :
1.to_nat = 1
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@[simp]
theorem int.to_nat_of_nonneg {a : ℤ} (h : 0 ≤ a) :
↑(a.to_nat) = a
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@[simp]
theorem int.to_nat_sub_of_le (a b : ℤ) (h : b ≤ a) :
↑((a + -b).to_nat) = a + -b
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@[simp]
theorem int.to_nat_coe_nat (n : ℕ) :
↑n.to_nat = n
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@[simp]
theorem int.to_nat_coe_nat_add_one {n : ℕ} :
(↑n + 1).to_nat = n + 1
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theorem int.le_to_nat (a : ℤ) :
a ≤ ↑(a.to_nat)
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@[simp]
theorem int.to_nat_le {a : ℤ} {n : ℕ} :
a.to_nat ≤ n ↔ a ≤ ↑n
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@[simp]
theorem int.lt_to_nat {n : ℕ} {a : ℤ} :
n < a.to_nat ↔ ↑n < a
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theorem int.to_nat_le_to_nat {a b : ℤ} (h : a ≤ b) :
a.to_nat ≤ b.to_nat
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theorem int.to_nat_lt_to_nat {a b : ℤ} (hb : 0 < b) :
a.to_nat < b.to_nat ↔ a < b
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theorem int.lt_of_to_nat_lt {a b : ℤ} (h : a.to_nat < b.to_nat) :
a < b
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theorem int.to_nat_add {a b : ℤ} (ha : 0 ≤ a) (hb : 0 ≤ b) :
(a + b).to_nat = a.to_nat + b.to_nat
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theorem int.to_nat_add_nat {a : ℤ} (ha : 0 ≤ a) (n : ℕ) :
(a + ↑n).to_nat = a.to_nat + n
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@[simp]
theorem int.pred_to_nat (i : ℤ) :
(i - 1).to_nat = i.to_nat - 1
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@[simp]
theorem int.to_nat_pred_coe_of_pos {i : ℤ} (h : 0 < i) :
↑(i.to_nat - 1) = i - 1
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def int.to_nat'  :
ℤ → option ℕ

If n : ℕ, then int.to_nat' n = some n, if n : ℤ is negative, then int.to_nat' n = none.

Equations
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theorem int.mem_to_nat' (a : ℤ) (n : ℕ) :
n ∈ a.to_nat' ↔ a = ↑n
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theorem int.to_nat_of_nonpos {z : ℤ} :
z ≤ 0 → z.to_nat = 0

units #

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@[simp]
theorem int.units_nat_abs (u : units ℤ) :
↑u.nat_abs = 1
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theorem int.units_eq_one_or (u : units ℤ) :
u = 1 ∨ u = -1
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theorem int.is_unit_eq_one_or {a : ℤ} :
is_unit a → a = 1 ∨ a = -1
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theorem int.is_unit_iff {a : ℤ} :
is_unit a ↔ a = 1 ∨ a = -1
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theorem int.is_unit_iff_nat_abs_eq {n : ℤ} :
is_unit n ↔ n.nat_abs = 1
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theorem int.units_inv_eq_self (u : units ℤ) :
u⁻¹ = u
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@[simp]
theorem int.units_mul_self (u : units ℤ) :
u * u = 1
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@[simp]
theorem int.units_coe_mul_self (u : units ℤ) :
(↑u) * ↑u = 1
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@[simp]
theorem int.neg_one_pow_ne_zero {n : ℕ} :
(-1) ^ n ≠ 0

bitwise ops #

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@[simp]
theorem int.bodd_zero  :
0.bodd = ff
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@[simp]
theorem int.bodd_one  :
1.bodd = tt
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theorem int.bodd_two  :
2.bodd = ff
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@[simp]
theorem int.bodd_coe (n : ℕ) :
↑n.bodd = n.bodd
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@[simp]
theorem int.bodd_sub_nat_nat (m n : ℕ) :
(int.sub_nat_nat m n).bodd = bxor m.bodd n.bodd
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@[simp]
theorem int.bodd_neg_of_nat (n : ℕ) :
(int.neg_of_nat n).bodd = n.bodd
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@[simp]
theorem int.bodd_neg (n : ℤ) :
(-n).bodd = n.bodd
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@[simp]
theorem int.bodd_add (m n : ℤ) :
(m + n).bodd = bxor m.bodd n.bodd
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@[simp]
theorem int.bodd_mul (m n : ℤ) :
(m * n).bodd = m.bodd && n.bodd
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theorem int.bodd_add_div2 (n : ℤ) :
cond n.bodd 1 0 + 2 * n.div2 = n
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theorem int.div2_val (n : ℤ) :
n.div2 = n / 2
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theorem int.bit0_val (n : ℤ) :
bit0 n = 2 * n
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theorem int.bit1_val (n : ℤ) :
bit1 n = 2 * n + 1
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theorem int.bit_val (b : bool) (n : ℤ) :
int.bit b n = 2 * n + cond b 1 0
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theorem int.bit_decomp (n : ℤ) :
int.bit n.bodd n.div2 = n
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def int.bit_cases_on {C : ℤ → Sort u} (n : ℤ) (h : Π (b : bool) (n : ℤ), C (int.bit b n)) :
C n

Defines a function from ℤ conditionally, if it is defined for odd and even integers separately using bit.

Equations
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@[simp]
theorem int.bit_zero  :
int.bit ff 0 = 0
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@[simp]
theorem int.bit_coe_nat (b : bool) (n : ℕ) :
int.bit b ↑n = ↑(nat.bit b n)
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@[simp]
theorem int.bit_neg_succ (b : bool) (n : ℕ) :
int.bit b -[1+ n] = -[1+ nat.bit (!b) n]
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@[simp]
theorem int.bodd_bit (b : bool) (n : ℤ) :
(int.bit b n).bodd = b
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@[simp]
theorem int.bodd_bit0 (n : ℤ) :
(bit0 n).bodd = ff
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@[simp]
theorem int.bodd_bit1 (n : ℤ) :
(bit1 n).bodd = tt
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@[simp]
theorem int.div2_bit (b : bool) (n : ℤ) :
(int.bit b n).div2 = n
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theorem int.bit0_ne_bit1 (m n : ℤ) :
bit0 m ≠ bit1 n
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theorem int.bit1_ne_bit0 (m n : ℤ) :
bit1 m ≠ bit0 n
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theorem int.bit1_ne_zero (m : ℤ) :
bit1 m ≠ 0
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@[simp]
theorem int.test_bit_zero (b : bool) (n : ℤ) :
(int.bit b n).test_bit 0 = b
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@[simp]
theorem int.test_bit_succ (m : ℕ) (b : bool) (n : ℤ) :
(int.bit b n).test_bit m.succ = n.test_bit m
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theorem int.bitwise_or  :
int.bitwise bor = int.lor
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theorem int.bitwise_and  :
int.bitwise band = int.land
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theorem int.bitwise_diff  :
int.bitwise (λ (a b : bool), a && !b) = int.ldiff
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theorem int.bitwise_xor  :
int.bitwise bxor = int.lxor
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@[simp]
theorem int.bitwise_bit (f : bool → bool → bool) (a : bool) (m : ℤ) (b : bool) (n : ℤ) :
int.bitwise f (int.bit a m) (int.bit b n) = int.bit (f a b) (int.bitwise f m n)
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@[simp]
theorem int.lor_bit (a : bool) (m : ℤ) (b : bool) (n : ℤ) :
(int.bit a m).lor (int.bit b n) = int.bit (a || b) (m.lor n)
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@[simp]
theorem int.land_bit (a : bool) (m : ℤ) (b : bool) (n : ℤ) :
(int.bit a m).land (int.bit b n) = int.bit (a && b) (m.land n)
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@[simp]
theorem int.ldiff_bit (a : bool) (m : ℤ) (b : bool) (n : ℤ) :
(int.bit a m).ldiff (int.bit b n) = int.bit (a && !b) (m.ldiff n)
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@[simp]
theorem int.lxor_bit (a : bool) (m : ℤ) (b : bool) (n : ℤ) :
(int.bit a m).lxor (int.bit b n) = int.bit (bxor a b) (m.lxor n)
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@[simp]
theorem int.lnot_bit (b : bool) (n : ℤ) :
(int.bit b n).lnot = int.bit (!b) n.lnot
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@[simp]
theorem int.test_bit_bitwise (f : bool → bool → bool) (m n : ℤ) (k : ℕ) :
(int.bitwise f m n).test_bit k = f (m.test_bit k) (n.test_bit k)
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@[simp]
theorem int.test_bit_lor (m n : ℤ) (k : ℕ) :
(m.lor n).test_bit k = m.test_bit k || n.test_bit k
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@[simp]
theorem int.test_bit_land (m n : ℤ) (k : ℕ) :
(m.land n).test_bit k = m.test_bit k && n.test_bit k
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@[simp]
theorem int.test_bit_ldiff (m n : ℤ) (k : ℕ) :
(m.ldiff n).test_bit k = m.test_bit k && !n.test_bit k
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@[simp]
theorem int.test_bit_lxor (m n : ℤ) (k : ℕ) :
(m.lxor n).test_bit k = bxor (m.test_bit k) (n.test_bit k)
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@[simp]
theorem int.test_bit_lnot (n : ℤ) (k : ℕ) :
n.lnot.test_bit k = !n.test_bit k
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theorem int.shiftl_add (m : ℤ) (n : ℕ) (k : ℤ) :
m.shiftl (↑n + k) = (m.shiftl ↑n).shiftl k
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theorem int.shiftl_sub (m : ℤ) (n : ℕ) (k : ℤ) :
m.shiftl (↑n - k) = (m.shiftl ↑n).shiftr k
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@[simp]
theorem int.shiftl_neg (m n : ℤ) :
m.shiftl (-n) = m.shiftr n
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@[simp]
theorem int.shiftr_neg (m n : ℤ) :
m.shiftr (-n) = m.shiftl n
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@[simp]
theorem int.shiftl_coe_nat (m n : ℕ) :
↑m.shiftl ↑n = ↑(m.shiftl n)
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@[simp]
theorem int.shiftr_coe_nat (m n : ℕ) :
↑m.shiftr ↑n = ↑(m.shiftr n)
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@[simp]
theorem int.shiftl_neg_succ (m n : ℕ) :
-[1+ m].shiftl ↑n = -[1+ nat.shiftl' tt m n]
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@[simp]
theorem int.shiftr_neg_succ (m n : ℕ) :
-[1+ m].shiftr ↑n = -[1+ m.shiftr n]
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theorem int.shiftr_add (m : ℤ) (n k : ℕ) :
m.shiftr (↑n + ↑k) = (m.shiftr ↑n).shiftr ↑k
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theorem int.shiftl_eq_mul_pow (m : ℤ) (n : ℕ) :
m.shiftl ↑n = m * ↑(2 ^ n)
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theorem int.shiftr_eq_div_pow (m : ℤ) (n : ℕ) :
m.shiftr ↑n = m / ↑(2 ^ n)
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theorem int.one_shiftl (n : ℕ) :
1.shiftl ↑n = ↑(2 ^ n)
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@[simp]
theorem int.zero_shiftl (n : ℤ) :
0.shiftl n = 0
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@[simp]
theorem int.zero_shiftr (n : ℤ) :
0.shiftr n = 0

Least upper bound property for integers #

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def int.least_of_bdd {P : ℤ → Prop} [decidable_pred P] (b : ℤ) (Hb : ∀ (z : ℤ), P z → b ≤ z) (Hinh : ∃ (z : ℤ), P z) :
{lb // P lb ∧ ∀ (z : ℤ), P z → lb ≤ z}

A computable version of exists_least_of_bdd: given a decidable predicate on the integers, with an explicit lower bound and a proof that it is somewhere true, return the least value for which the predicate is true.

Equations
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theorem int.exists_least_of_bdd {P : ℤ → Prop} (Hbdd : ∃ (b : ℤ), ∀ (z : ℤ), P z → b ≤ z) (Hinh : ∃ (z : ℤ), P z) :
∃ (lb : ℤ), P lb ∧ ∀ (z : ℤ), P z → lb ≤ z
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def int.greatest_of_bdd {P : ℤ → Prop} [decidable_pred P] (b : ℤ) (Hb : ∀ (z : ℤ), P z → z ≤ b) (Hinh : ∃ (z : ℤ), P z) :
{ub // P ub ∧ ∀ (z : ℤ), P z → z ≤ ub}

A computable version of exists_greatest_of_bdd: given a decidable predicate on the integers, with an explicit upper bound and a proof that it is somewhere true, return the greatest value for which the predicate is true.

Equations
  • b.greatest_of_bdd Hb Hinh = have Hbdd' : ∀ (z : ℤ), P (-z) → -b ≤ z, from _, have Hinh' : ∃ (z : ℤ), P (-z), from _, int.greatest_of_bdd._match_2 ((-b).least_of_bdd Hbdd' Hinh')
  • int.greatest_of_bdd._match_2 ⟨lb, _⟩ = ⟨-lb, _⟩
  • _ = _
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theorem int.exists_greatest_of_bdd {P : ℤ → Prop} (Hbdd : ∃ (b : ℤ), ∀ (z : ℤ), P z → z ≤ b) (Hinh : ∃ (z : ℤ), P z) :
∃ (ub : ℤ), P ub ∧ ∀ (z : ℤ), P z → z ≤ ub