Lemmas about integer division needed to bootstrap omega.

dvd

theorem Int.dvd_def (a : Int) (b : Int) : (a ∣ b) = ∃ (c : Int), b = a * c

theorem Int.dvd_zero (n : Int) : n ∣ 0

theorem Int.dvd_refl (n : Int) : n ∣ n

theorem Int.one_dvd (n : Int) : 1 ∣ n

theorem Int.dvd_trans {a : Int} {b : Int} {c : Int} : a ∣ b → b ∣ c → a ∣ c

theorem Int.ofNat_dvd {m : Nat} {n : Nat} : ↑m ∣ ↑n ↔ m ∣ n

theorem Int.dvd_antisymm {a : Int} {b : Int} (H1 : 0 ≤ a) (H2 : 0 ≤ b) : a ∣ b → b ∣ a → a = b

@[simp]

theorem Int.zero_dvd {n : Int} : 0 ∣ n ↔ n = 0

theorem Int.dvd_mul_right (a : Int) (b : Int) : a ∣ a * b

theorem Int.dvd_mul_left (a : Int) (b : Int) : b ∣ a * b

theorem Int.neg_dvd {a : Int} {b : Int} : -a ∣ b ↔ a ∣ b

theorem Int.dvd_neg {a : Int} {b : Int} : a ∣ -b ↔ a ∣ b

@[simp]

theorem Int.natAbs_dvd_natAbs {a : Int} {b : Int} : a.natAbs ∣ b.natAbs ↔ a ∣ b

theorem Int.ofNat_dvd_left {n : Nat} {z : Int} : ↑n ∣ z ↔ n ∣ z.natAbs

theorem Int.dvd_add {a : Int} {b : Int} {c : Int} : a ∣ b → a ∣ c → a ∣ b + c

theorem Int.dvd_sub {a : Int} {b : Int} {c : Int} : a ∣ b → a ∣ c → a ∣ b - c

theorem Int.dvd_add_left {a : Int} {b : Int} {c : Int} (H : a ∣ c) : a ∣ b + c ↔ a ∣ b

theorem Int.dvd_add_right {a : Int} {b : Int} {c : Int} (H : a ∣ b) : a ∣ b + c ↔ a ∣ c

theorem Int.dvd_iff_dvd_of_dvd_sub {a : Int} {b : Int} {c : Int} (H : a ∣ b - c) : a ∣ b ↔ a ∣ c

theorem Int.dvd_iff_dvd_of_dvd_add {a : Int} {b : Int} {c : Int} (H : a ∣ b + c) : a ∣ b ↔ a ∣ c

theorem Int.le_of_dvd {a : Int} {b : Int} (bpos : 0 < b) (H : a ∣ b) : a ≤ b

theorem Int.natAbs_dvd {a : Int} {b : Int} : ↑a.natAbs ∣ b ↔ a ∣ b

theorem Int.dvd_natAbs {a : Int} {b : Int} : a ∣ ↑b.natAbs ↔ a ∣ b

theorem Int.natAbs_dvd_self {a : Int} : ↑a.natAbs ∣ a

theorem Int.dvd_natAbs_self {a : Int} : a ∣ ↑a.natAbs

theorem Int.ofNat_dvd_right {n : Nat} {z : Int} : z ∣ ↑n ↔ z.natAbs ∣ n

theorem Int.eq_one_of_dvd_one {a : Int} (H : 0 ≤ a) (H' : a ∣ 1) : a = 1

theorem Int.eq_one_of_mul_eq_one_right {a : Int} {b : Int} (H : 0 ≤ a) (H' : a * b = 1) : a = 1

theorem Int.eq_one_of_mul_eq_one_left {a : Int} {b : Int} (H : 0 ≤ b) (H' : a * b = 1) : b = 1

*div zero

@[simp]

theorem Int.zero_ediv (b : Int) : 0 / b = 0

@[simp]

theorem Int.ediv_zero (a : Int) : a / 0 = 0

@[simp]

theorem Int.zero_div (b : Int) : Int.div 0 b = 0

@[simp]

theorem Int.div_zero (a : Int) : a.div 0 = 0

@[simp]

theorem Int.zero_fdiv (b : Int) : Int.fdiv 0 b = 0

@[simp]

theorem Int.fdiv_zero (a : Int) : a.fdiv 0 = 0

div equivalences

theorem Int.div_eq_ediv {a : Int} {b : Int} : 0 ≤ a → 0 ≤ b → a.div b = a / b

theorem Int.fdiv_eq_ediv (a : Int) {b : Int} : 0 ≤ b → a.fdiv b = a / b

theorem Int.fdiv_eq_div {a : Int} {b : Int} (Ha : 0 ≤ a) (Hb : 0 ≤ b) : a.fdiv b = a.div b

mod zero

@[simp]

theorem Int.zero_emod (b : Int) : 0 % b = 0

@[simp]

theorem Int.emod_zero (a : Int) : a % 0 = a

@[simp]

theorem Int.zero_mod (b : Int) : Int.mod 0 b = 0

@[simp]

theorem Int.mod_zero (a : Int) : a.mod 0 = a

@[simp]

theorem Int.zero_fmod (b : Int) : Int.fmod 0 b = 0

@[simp]

theorem Int.fmod_zero (a : Int) : a.fmod 0 = a

ofNat mod

@[simp]

theorem Int.ofNat_emod (m : Nat) (n : Nat) : ↑(m % n) = ↑m % ↑n

mod definitiions

theorem Int.emod_add_ediv (a : Int) (b : Int) : a % b + b * (a / b) = a

theorem Int.emod_add_ediv.aux (m : Nat) (n : Nat) : ↑n - (↑m % ↑n + 1) - (↑n * (↑m / ↑n) + ↑n) = Int.negSucc m

theorem Int.emod_add_ediv' (a : Int) (b : Int) : a % b + a / b * b = a

theorem Int.ediv_add_emod (a : Int) (b : Int) : b * (a / b) + a % b = a

theorem Int.ediv_add_emod' (a : Int) (b : Int) : a / b * b + a % b = a

theorem Int.emod_def (a : Int) (b : Int) : a % b = a - b * (a / b)

theorem Int.mod_add_div (a : Int) (b : Int) : a.mod b + b * a.div b = a

theorem Int.div_add_mod (a : Int) (b : Int) : b * a.div b + a.mod b = a

theorem Int.mod_add_div' (m : Int) (k : Int) : m.mod k + m.div k * k = m

theorem Int.div_add_mod' (m : Int) (k : Int) : m.div k * k + m.mod k = m

theorem Int.mod_def (a : Int) (b : Int) : a.mod b = a - b * a.div b

theorem Int.fmod_add_fdiv (a : Int) (b : Int) : a.fmod b + b * a.fdiv b = a

theorem Int.fdiv_add_fmod (a : Int) (b : Int) : b * a.fdiv b + a.fmod b = a

theorem Int.fmod_def (a : Int) (b : Int) : a.fmod b = a - b * a.fdiv b

mod equivalences

theorem Int.fmod_eq_emod (a : Int) {b : Int} (hb : 0 ≤ b) : a.fmod b = a % b

theorem Int.mod_eq_emod {a : Int} {b : Int} (ha : 0 ≤ a) (hb : 0 ≤ b) : a.mod b = a % b

theorem Int.fmod_eq_mod {a : Int} {b : Int} (Ha : 0 ≤ a) (Hb : 0 ≤ b) : a.fmod b = a.mod b

/ ediv

@[simp]

theorem Int.ediv_neg (a : Int) (b : Int) : a / -b = -(a / b)

theorem Int.ediv_neg' {a : Int} {b : Int} (Ha : a < 0) (Hb : 0 < b) : a / b < 0

theorem Int.div_def (a : Int) (b : Int) : a / b = a.ediv b

theorem Int.negSucc_ediv (m : Nat) {b : Int} (H : 0 < b) : Int.negSucc m / b = -((↑m).div b + 1)

theorem Int.ediv_nonneg {a : Int} {b : Int} (Ha : 0 ≤ a) (Hb : 0 ≤ b) : 0 ≤ a / b

theorem Int.ediv_nonpos {a : Int} {b : Int} (Ha : 0 ≤ a) (Hb : b ≤ 0) : a / b ≤ 0

theorem Int.add_mul_ediv_right (a : Int) (b : Int) {c : Int} (H : c ≠ 0) : (a + b * c) / c = a / c + b

theorem Int.add_ediv_of_dvd_right {a : Int} {b : Int} {c : Int} (H : c ∣ b) : (a + b) / c = a / c + b / c

theorem Int.add_ediv_of_dvd_left {a : Int} {b : Int} {c : Int} (H : c ∣ a) : (a + b) / c = a / c + b / c

@[simp]

theorem Int.mul_ediv_cancel (a : Int) {b : Int} (H : b ≠ 0) : a * b / b = a

@[simp]

theorem Int.mul_ediv_cancel_left {a : Int} (b : Int) (H : a ≠ 0) : a * b / a = b

theorem Int.div_nonneg_iff_of_pos {a : Int} {b : Int} (h : 0 < b) : a / b ≥ 0 ↔ a ≥ 0

theorem Int.ediv_eq_zero_of_lt {a : Int} {b : Int} (H1 : 0 ≤ a) (H2 : a < b) : a / b = 0

theorem Int.add_mul_ediv_left (a : Int) {b : Int} (c : Int) (H : b ≠ 0) : (a + b * c) / b = a / b + c

@[simp]

theorem Int.mul_ediv_mul_of_pos {a : Int} (b : Int) (c : Int) (H : 0 < a) : a * b / (a * c) = b / c

@[simp]

theorem Int.mul_ediv_mul_of_pos_left (a : Int) {b : Int} (c : Int) (H : 0 < b) : a * b / (c * b) = a / c

theorem Int.ediv_eq_of_eq_mul_right {a : Int} {b : Int} {c : Int} (H1 : b ≠ 0) (H2 : a = b * c) : a / b = c

theorem Int.eq_ediv_of_mul_eq_right {a : Int} {b : Int} {c : Int} (H1 : a ≠ 0) (H2 : a * b = c) : b = c / a

theorem Int.ediv_eq_of_eq_mul_left {a : Int} {b : Int} {c : Int} (H1 : b ≠ 0) (H2 : a = c * b) : a / b = c

emod

theorem Int.mod_def' (m : Int) (n : Int) : m % n = m.emod n

theorem Int.negSucc_emod (m : Nat) {b : Int} (bpos : 0 < b) : Int.negSucc m % b = b - 1 - ↑m % b

theorem Int.emod_negSucc (m : Nat) (n : Int) : Int.negSucc m % n = Int.subNatNat n.natAbs (m % n.natAbs).succ

theorem Int.ofNat_mod_ofNat (m : Nat) (n : Nat) : ↑m % ↑n = ↑(m % n)

theorem Int.emod_nonneg (a : Int) {b : Int} : b ≠ 0 → 0 ≤ a % b

theorem Int.emod_lt_of_pos (a : Int) {b : Int} (H : 0 < b) : a % b < b

theorem Int.mul_ediv_self_le {x : Int} {k : Int} (h : k ≠ 0) : k * (x / k) ≤ x

theorem Int.lt_mul_ediv_self_add {x : Int} {k : Int} (h : 0 < k) : x < k * (x / k) + k

@[simp]

theorem Int.add_mul_emod_self {a : Int} {b : Int} {c : Int} : (a + b * c) % c = a % c

@[simp]

theorem Int.add_mul_emod_self_left (a : Int) (b : Int) (c : Int) : (a + b * c) % b = a % b

@[simp]

theorem Int.add_emod_self {a : Int} {b : Int} : (a + b) % b = a % b

@[simp]

theorem Int.add_emod_self_left {a : Int} {b : Int} : (a + b) % a = b % a

theorem Int.neg_emod {a : Int} {b : Int} : -a % b = (b - a) % b

@[simp]

theorem Int.emod_neg (a : Int) (b : Int) : a % -b = a % b

@[simp]

theorem Int.emod_add_emod (m : Int) (n : Int) (k : Int) : (m % n + k) % n = (m + k) % n

@[simp]

theorem Int.add_emod_emod (m : Int) (n : Int) (k : Int) : (m + n % k) % k = (m + n) % k

theorem Int.add_emod (a : Int) (b : Int) (n : Int) : (a + b) % n = (a % n + b % n) % n

theorem Int.add_emod_eq_add_emod_right {m : Int} {n : Int} {k : Int} (i : Int) (H : m % n = k % n) : (m + i) % n = (k + i) % n

theorem Int.emod_add_cancel_right {m : Int} {n : Int} {k : Int} (i : Int) : (m + i) % n = (k + i) % n ↔ m % n = k % n

@[simp]

theorem Int.mul_emod_left (a : Int) (b : Int) : a * b % b = 0

@[simp]

theorem Int.mul_emod_right (a : Int) (b : Int) : a * b % a = 0

theorem Int.mul_emod (a : Int) (b : Int) (n : Int) : a * b % n = a % n * (b % n) % n

theorem Int.emod_self {a : Int} : a % a = 0

@[simp]

theorem Int.emod_emod_of_dvd (n : Int) {m : Int} {k : Int} (h : m ∣ k) : n % k % m = n % m

@[simp]

theorem Int.emod_emod (a : Int) (b : Int) : a % b % b = a % b

theorem Int.sub_emod (a : Int) (b : Int) (n : Int) : (a - b) % n = (a % n - b % n) % n

theorem Int.emod_eq_of_lt {a : Int} {b : Int} (H1 : 0 ≤ a) (H2 : a < b) : a % b = a

@[simp]

theorem Int.emod_self_add_one {x : Int} (h : 0 ≤ x) : x % (x + 1) = x

properties of / and %

theorem Int.mul_ediv_cancel_of_emod_eq_zero {a : Int} {b : Int} (H : a % b = 0) : b * (a / b) = a

theorem Int.ediv_mul_cancel_of_emod_eq_zero {a : Int} {b : Int} (H : a % b = 0) : a / b * b = a

theorem Int.emod_two_eq (x : Int) : x % 2 = 0 ∨ x % 2 = 1

theorem Int.add_emod_eq_add_emod_left {m : Int} {n : Int} {k : Int} (i : Int) (H : m % n = k % n) : (i + m) % n = (i + k) % n

theorem Int.emod_add_cancel_left {m : Int} {n : Int} {k : Int} {i : Int} : (i + m) % n = (i + k) % n ↔ m % n = k % n

theorem Int.emod_sub_cancel_right {m : Int} {n : Int} {k : Int} (i : Int) : (m - i) % n = (k - i) % n ↔ m % n = k % n

theorem Int.emod_eq_emod_iff_emod_sub_eq_zero {m : Int} {n : Int} {k : Int} : m % n = k % n ↔ (m - k) % n = 0

theorem Int.ediv_emod_unique {a : Int} {b : Int} {r : Int} {q : Int} (h : 0 < b) : a / b = q ∧ a % b = r ↔ r + b * q = a ∧ 0 ≤ r ∧ r < b

@[simp]

theorem Int.mul_emod_mul_of_pos {a : Int} (b : Int) (c : Int) (H : 0 < a) : a * b % (a * c) = a * (b % c)

theorem Int.lt_ediv_add_one_mul_self (a : Int) {b : Int} (H : 0 < b) : a < (a / b + 1) * b

theorem Int.natAbs_div_le_natAbs (a : Int) (b : Int) : (a / b).natAbs ≤ a.natAbs

theorem Int.natAbs_div_le_natAbs.aux (a : Int) (n : Nat) : (a / ↑n).natAbs ≤ a.natAbs

theorem Int.ediv_le_self {a : Int} (b : Int) (Ha : 0 ≤ a) : a / b ≤ a

theorem Int.dvd_of_emod_eq_zero {a : Int} {b : Int} (H : b % a = 0) : a ∣ b

theorem Int.dvd_emod_sub_self {x : Int} {m : Nat} : ↑m ∣ x % ↑m - x

theorem Int.emod_eq_zero_of_dvd {a : Int} {b : Int} : a ∣ b → b % a = 0

theorem Int.dvd_iff_emod_eq_zero (a : Int) (b : Int) : a ∣ b ↔ b % a = 0

instance Int.decidableDvd : DecidableRel fun (x x_1 : Int) => x ∣ x_1

Equations

• x✝.decidableDvd x = decidable_of_decidable_of_iff ⋯

theorem Int.emod_pos_of_not_dvd {a : Int} {b : Int} (h : ¬a ∣ b) : a = 0 ∨ 0 < b % a

theorem Int.mul_ediv_assoc (a : Int) {b : Int} {c : Int} : c ∣ b → a * b / c = a * (b / c)

theorem Int.mul_ediv_assoc' (b : Int) {a : Int} {c : Int} (h : c ∣ a) : a * b / c = a / c * b

theorem Int.neg_ediv_of_dvd {a : Int} {b : Int} : b ∣ a → -a / b = -(a / b)

theorem Int.sub_ediv_of_dvd (a : Int) {b : Int} {c : Int} (hcb : c ∣ b) : (a - b) / c = a / c - b / c

@[simp]

theorem Int.ediv_one (a : Int) : a / 1 = a

@[simp]

theorem Int.emod_one (a : Int) : a % 1 = 0

@[simp]

theorem Int.ediv_self {a : Int} (H : a ≠ 0) : a / a = 1

@[simp]

theorem Int.Int.emod_sub_cancel (x : Int) (y : Int) : (x - y) % y = x % y

theorem Int.dvd_sub_of_emod_eq {a : Int} {b : Int} {c : Int} (h : a % b = c) : b ∣ a - c

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

theorem Int.ediv_mul_cancel {a : Int} {b : Int} (H : b ∣ a) : a / b * b = a

theorem Int.mul_ediv_cancel' {a : Int} {b : Int} (H : a ∣ b) : a * (b / a) = b

theorem Int.eq_mul_of_ediv_eq_right {a : Int} {b : Int} {c : Int} (H1 : b ∣ a) (H2 : a / b = c) : a = b * c

theorem Int.ediv_eq_iff_eq_mul_right {a : Int} {b : Int} {c : Int} (H : b ≠ 0) (H' : b ∣ a) : a / b = c ↔ a = b * c

theorem Int.ediv_eq_iff_eq_mul_left {a : Int} {b : Int} {c : Int} (H : b ≠ 0) (H' : b ∣ a) : a / b = c ↔ a = c * b

theorem Int.eq_mul_of_ediv_eq_left {a : Int} {b : Int} {c : Int} (H1 : b ∣ a) (H2 : a / b = c) : a = c * b

theorem Int.eq_zero_of_ediv_eq_zero {d : Int} {n : Int} (h : d ∣ n) (H : n / d = 0) : n = 0

theorem Int.sub_ediv_of_dvd_sub {a : Int} {b : Int} {c : Int} (hcab : c ∣ a - b) : (a - b) / c = a / c - b / c

@[simp]

theorem Int.ediv_left_inj {a : Int} {b : Int} {d : Int} (hda : d ∣ a) (hdb : d ∣ b) : a / d = b / d ↔ a = b

theorem Int.ediv_sign (a : Int) (b : Int) : a / b.sign = a * b.sign

/ and ordering

theorem Int.ediv_mul_le (a : Int) {b : Int} (H : b ≠ 0) : a / b * b ≤ a

theorem Int.le_of_mul_le_mul_left {a : Int} {b : Int} {c : Int} (w : a * b ≤ a * c) (h : 0 < a) : b ≤ c

theorem Int.le_of_mul_le_mul_right {a : Int} {b : Int} {c : Int} (w : b * a ≤ c * a) (h : 0 < a) : b ≤ c

theorem Int.ediv_le_of_le_mul {a : Int} {b : Int} {c : Int} (H : 0 < c) (H' : a ≤ b * c) : a / c ≤ b

theorem Int.mul_lt_of_lt_ediv {a : Int} {b : Int} {c : Int} (H : 0 < c) (H3 : a < b / c) : a * c < b

theorem Int.mul_le_of_le_ediv {a : Int} {b : Int} {c : Int} (H1 : 0 < c) (H2 : a ≤ b / c) : a * c ≤ b

theorem Int.ediv_lt_of_lt_mul {a : Int} {b : Int} {c : Int} (H : 0 < c) (H' : a < b * c) : a / c < b

theorem Int.lt_of_mul_lt_mul_left {a : Int} {b : Int} {c : Int} (w : a * b < a * c) (h : 0 ≤ a) : b < c

theorem Int.lt_of_mul_lt_mul_right {a : Int} {b : Int} {c : Int} (w : b * a < c * a) (h : 0 ≤ a) : b < c

theorem Int.le_ediv_of_mul_le {a : Int} {b : Int} {c : Int} (H1 : 0 < c) (H2 : a * c ≤ b) : a ≤ b / c

theorem Int.le_ediv_iff_mul_le {a : Int} {b : Int} {c : Int} (H : 0 < c) : a ≤ b / c ↔ a * c ≤ b

theorem Int.ediv_le_ediv {a : Int} {b : Int} {c : Int} (H : 0 < c) (H' : a ≤ b) : a / c ≤ b / c

theorem Int.lt_mul_of_ediv_lt {a : Int} {b : Int} {c : Int} (H1 : 0 < c) (H2 : a / c < b) : a < b * c

theorem Int.ediv_lt_iff_lt_mul {a : Int} {b : Int} {c : Int} (H : 0 < c) : a / c < b ↔ a < b * c

theorem Int.le_mul_of_ediv_le {a : Int} {b : Int} {c : Int} (H1 : 0 ≤ b) (H2 : b ∣ a) (H3 : a / b ≤ c) : a ≤ c * b

theorem Int.lt_ediv_of_mul_lt {a : Int} {b : Int} {c : Int} (H1 : 0 ≤ b) (H2 : b ∣ c) (H3 : a * b < c) : a < c / b

theorem Int.lt_ediv_iff_mul_lt {a : Int} {b : Int} (c : Int) (H : 0 < c) (H' : c ∣ b) : a < b / c ↔ a * c < b

theorem Int.ediv_pos_of_pos_of_dvd {a : Int} {b : Int} (H1 : 0 < a) (H2 : 0 ≤ b) (H3 : b ∣ a) : 0 < a / b

theorem Int.ediv_eq_ediv_of_mul_eq_mul {a : Int} {b : Int} {c : Int} {d : Int} (H2 : d ∣ c) (H3 : b ≠ 0) (H4 : d ≠ 0) (H5 : a * d = b * c) : a / b = c / d

div

@[simp]

theorem Int.div_one (a : Int) : a.div 1 = a

@[simp]

theorem Int.div_neg (a : Int) (b : Int) : a.div (-b) = -a.div b

@[simp]

theorem Int.neg_div (a : Int) (b : Int) : (-a).div b = -a.div b

theorem Int.neg_div_neg (a : Int) (b : Int) : (-a).div (-b) = a.div b

theorem Int.div_nonneg {a : Int} {b : Int} (Ha : 0 ≤ a) (Hb : 0 ≤ b) : 0 ≤ a.div b

theorem Int.div_nonpos {a : Int} {b : Int} (Ha : 0 ≤ a) (Hb : b ≤ 0) : a.div b ≤ 0

theorem Int.div_eq_zero_of_lt {a : Int} {b : Int} (H1 : 0 ≤ a) (H2 : a < b) : a.div b = 0

@[simp]

theorem Int.mul_div_cancel (a : Int) {b : Int} (H : b ≠ 0) : (a * b).div b = a

@[simp]

theorem Int.mul_div_cancel_left {a : Int} (b : Int) (H : a ≠ 0) : (a * b).div a = b

@[simp]

theorem Int.div_self {a : Int} (H : a ≠ 0) : a.div a = 1

theorem Int.mul_div_cancel_of_mod_eq_zero {a : Int} {b : Int} (H : a.mod b = 0) : b * a.div b = a

theorem Int.div_mul_cancel_of_mod_eq_zero {a : Int} {b : Int} (H : a.mod b = 0) : a.div b * b = a

theorem Int.dvd_of_mod_eq_zero {a : Int} {b : Int} (H : b.mod a = 0) : a ∣ b

theorem Int.mul_div_assoc (a : Int) {b : Int} {c : Int} : c ∣ b → (a * b).div c = a * b.div c

theorem Int.mul_div_assoc' (b : Int) {a : Int} {c : Int} (h : c ∣ a) : (a * b).div c = a.div c * b

theorem Int.div_dvd_div {a : Int} {b : Int} {c : Int} : a ∣ b → b ∣ c → b.div a ∣ c.div a

@[simp]

theorem Int.natAbs_div (a : Int) (b : Int) : (a.div b).natAbs = a.natAbs.div b.natAbs

theorem Int.div_eq_of_eq_mul_right {a : Int} {b : Int} {c : Int} (H1 : b ≠ 0) (H2 : a = b * c) : a.div b = c

theorem Int.eq_div_of_mul_eq_right {a : Int} {b : Int} {c : Int} (H1 : a ≠ 0) (H2 : a * b = c) : b = c.div a

(t-)mod

theorem Int.ofNat_mod (m : Nat) (n : Nat) : ↑(m % n) = (↑m).mod ↑n

@[simp]

theorem Int.mod_one (a : Int) : a.mod 1 = 0

theorem Int.mod_eq_of_lt {a : Int} {b : Int} (H1 : 0 ≤ a) (H2 : a < b) : a.mod b = a

theorem Int.mod_lt_of_pos (a : Int) {b : Int} (H : 0 < b) : a.mod b < b

theorem Int.mod_nonneg {a : Int} (b : Int) : 0 ≤ a → 0 ≤ a.mod b

@[simp]

theorem Int.mod_neg (a : Int) (b : Int) : a.mod (-b) = a.mod b

@[simp]

theorem Int.mul_mod_left (a : Int) (b : Int) : (a * b).mod b = 0

@[simp]

theorem Int.mul_mod_right (a : Int) (b : Int) : (a * b).mod a = 0

theorem Int.mod_eq_zero_of_dvd {a : Int} {b : Int} : a ∣ b → b.mod a = 0

theorem Int.dvd_iff_mod_eq_zero (a : Int) (b : Int) : a ∣ b ↔ b.mod a = 0

theorem Int.div_mul_cancel {a : Int} {b : Int} (H : b ∣ a) : a.div b * b = a

theorem Int.mul_div_cancel' {a : Int} {b : Int} (H : a ∣ b) : a * b.div a = b

theorem Int.eq_mul_of_div_eq_right {a : Int} {b : Int} {c : Int} (H1 : b ∣ a) (H2 : a.div b = c) : a = b * c

@[simp]

theorem Int.mod_self {a : Int} : a.mod a = 0

theorem Int.lt_div_add_one_mul_self (a : Int) {b : Int} (H : 0 < b) : a < (a.div b + 1) * b

theorem Int.div_eq_iff_eq_mul_right {a : Int} {b : Int} {c : Int} (H : b ≠ 0) (H' : b ∣ a) : a.div b = c ↔ a = b * c

theorem Int.div_eq_iff_eq_mul_left {a : Int} {b : Int} {c : Int} (H : b ≠ 0) (H' : b ∣ a) : a.div b = c ↔ a = c * b

theorem Int.eq_mul_of_div_eq_left {a : Int} {b : Int} {c : Int} (H1 : b ∣ a) (H2 : a.div b = c) : a = c * b

theorem Int.div_eq_of_eq_mul_left {a : Int} {b : Int} {c : Int} (H1 : b ≠ 0) (H2 : a = c * b) : a.div b = c

theorem Int.eq_zero_of_div_eq_zero {d : Int} {n : Int} (h : d ∣ n) (H : n.div d = 0) : n = 0

@[simp]

theorem Int.div_left_inj {a : Int} {b : Int} {d : Int} (hda : d ∣ a) (hdb : d ∣ b) : a.div d = b.div d ↔ a = b

theorem Int.div_sign (a : Int) (b : Int) : a.div b.sign = a * b.sign

theorem Int.sign_eq_div_abs (a : Int) : a.sign = a.div ↑a.natAbs

fdiv

theorem Int.fdiv_nonneg {a : Int} {b : Int} (Ha : 0 ≤ a) (Hb : 0 ≤ b) : 0 ≤ a.fdiv b

theorem Int.fdiv_nonpos {a : Int} {b : Int} : 0 ≤ a → b ≤ 0 → a.fdiv b ≤ 0

theorem Int.fdiv_neg' {a : Int} {b : Int} : a < 0 → 0 < b → a.fdiv b < 0

@[simp]

theorem Int.fdiv_one (a : Int) : a.fdiv 1 = a

@[simp]

theorem Int.mul_fdiv_cancel (a : Int) {b : Int} (H : b ≠ 0) : (a * b).fdiv b = a

@[simp]

theorem Int.mul_fdiv_cancel_left {a : Int} (b : Int) (H : a ≠ 0) : (a * b).fdiv a = b

@[simp]

theorem Int.fdiv_self {a : Int} (H : a ≠ 0) : a.fdiv a = 1

theorem Int.lt_fdiv_add_one_mul_self (a : Int) {b : Int} (H : 0 < b) : a < (a.fdiv b + 1) * b

fmod

theorem Int.ofNat_fmod (m : Nat) (n : Nat) : ↑(m % n) = (↑m).fmod ↑n

@[simp]

theorem Int.fmod_one (a : Int) : a.fmod 1 = 0

theorem Int.fmod_eq_of_lt {a : Int} {b : Int} (H1 : 0 ≤ a) (H2 : a < b) : a.fmod b = a

theorem Int.fmod_nonneg {a : Int} {b : Int} (ha : 0 ≤ a) (hb : 0 ≤ b) : 0 ≤ a.fmod b

theorem Int.fmod_nonneg' (a : Int) {b : Int} (hb : 0 < b) : 0 ≤ a.fmod b

theorem Int.fmod_lt_of_pos (a : Int) {b : Int} (H : 0 < b) : a.fmod b < b

@[simp]

theorem Int.mul_fmod_left (a : Int) (b : Int) : (a * b).fmod b = 0

@[simp]

theorem Int.mul_fmod_right (a : Int) (b : Int) : (a * b).fmod a = 0

@[simp]

theorem Int.fmod_self {a : Int} : a.fmod a = 0

Theorems crossing div/mod versions

theorem Int.div_eq_ediv_of_dvd {a : Int} {b : Int} (h : b ∣ a) : a.div b = a / b

theorem Int.fdiv_eq_ediv_of_dvd {a : Int} {b : Int} : b ∣ a → a.fdiv b = a / b

bmod

@[simp]

theorem Int.bmod_emod {x : Int} {m : Nat} : x.bmod m % ↑m = x % ↑m

@[simp]

theorem Int.emod_bmod_congr (x : Int) (n : Nat) : (x % ↑n).bmod n = x.bmod n

theorem Int.bmod_def (x : Int) (m : Nat) : x.bmod m = if x % ↑m < (↑m + 1) / 2 then x % ↑m else x % ↑m - ↑m

theorem Int.bmod_pos (x : Int) (m : Nat) (p : x % ↑m < (↑m + 1) / 2) : x.bmod m = x % ↑m

theorem Int.bmod_neg (x : Int) (m : Nat) (p : x % ↑m ≥ (↑m + 1) / 2) : x.bmod m = x % ↑m - ↑m

@[simp]

theorem Int.bmod_one_is_zero (x : Int) : x.bmod 1 = 0

@[simp]

theorem Int.bmod_add_cancel (x : Int) (n : Nat) : (x + ↑n).bmod n = x.bmod n

@[simp]

theorem Int.bmod_add_mul_cancel (x : Int) (n : Nat) (k : Int) : (x + ↑n * k).bmod n = x.bmod n

@[simp]

theorem Int.bmod_sub_cancel (x : Int) (n : Nat) : (x - ↑n).bmod n = x.bmod n

@[simp]

theorem Int.emod_add_bmod_congr {y : Int} (x : Int) (n : Nat) : (x % ↑n + y).bmod n = (x + y).bmod n

@[simp]

theorem Int.emod_mul_bmod_congr {y : Int} (x : Int) (n : Nat) : (x % ↑n * y).bmod n = (x * y).bmod n

@[simp]

theorem Int.bmod_add_bmod_congr {x : Int} {n : Nat} {y : Int} : (x.bmod n + y).bmod n = (x + y).bmod n

@[simp]

theorem Int.add_bmod_bmod {x : Int} {y : Int} {n : Nat} : (x + y.bmod n).bmod n = (x + y).bmod n

@[simp]

theorem Int.bmod_mul_bmod {x : Int} {n : Nat} {y : Int} : (x.bmod n * y).bmod n = (x * y).bmod n

@[simp]

theorem Int.mul_bmod_bmod {x : Int} {y : Int} {n : Nat} : (x * y.bmod n).bmod n = (x * y).bmod n

theorem Int.add_bmod (a : Int) (b : Int) (n : Nat) : (a + b).bmod n = (a.bmod n + b.bmod n).bmod n

theorem Int.emod_bmod {x : Int} {m : Nat} : (x % ↑m).bmod m = x.bmod m

@[simp]

theorem Int.bmod_bmod {x : Int} {m : Nat} : (x.bmod m).bmod m = x.bmod m

@[simp]

theorem Int.bmod_zero {m : Nat} : Int.bmod 0 m = 0

theorem Int.dvd_bmod_sub_self {x : Int} {m : Nat} : ↑m ∣ x.bmod m - x

theorem Int.le_bmod {x : Int} {m : Nat} (h : 0 < m) : -(↑m / 2) ≤ x.bmod m

theorem Int.bmod_lt {x : Int} {m : Nat} (h : 0 < m) : x.bmod m < (↑m + 1) / 2

theorem Int.bmod_le {x : Int} {m : Nat} (h : 0 < m) : x.bmod m ≤ (↑m - 1) / 2

theorem Int.bmod_natAbs_plus_one (x : Int) (w : 1 < x.natAbs) : x.bmod (x.natAbs + 1) = -x.sign