Further lemmas about integer division, now that omega is available.

theorem Int.exists_add_of_le {a b : Int} (h : a ≤ b) : ∃ (c : Nat), b = a + ↑c

theorem Int.toNat_emod {x y : Int} (hx : 0 ≤ x) (hy : 0 ≤ y) : (x % y).toNat = x.toNat % y.toNat

dvd

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

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

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

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

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

@[simp]

theorem Int.dvd_add_mul_self {a b c : Int} : a ∣ b + c * a ↔ a ∣ b

@[simp]

theorem Int.dvd_add_self_mul {a b c : Int} : a ∣ b + a * c ↔ a ∣ b

@[simp]

theorem Int.dvd_mul_self_add {a b c : Int} : a ∣ b * a + c ↔ a ∣ c

@[simp]

theorem Int.dvd_self_mul_add {a b c : Int} : a ∣ a * b + c ↔ a ∣ c

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

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

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

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

theorem Int.dvd_natAbs {a 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.natAbs_eq_of_dvd_dvd {m n : Int} (hmn : m ∣ n) (hnm : n ∣ m) : m.natAbs = n.natAbs

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

@[simp]

theorem Int.negSucc_dvd {a : Nat} {b : Int} : negSucc a ∣ b ↔ ↑(a + 1) ∣ b

@[simp]

theorem Int.dvd_negSucc {a : Int} {b : Nat} : a ∣ negSucc b ↔ a ∣ ↑(b + 1)

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 b : Int} (H : 0 ≤ a) (H' : a * b = 1) : a = 1

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

instance Int.decidableDvd : DecidableRel fun (x1 x2 : Int) => x1 ∣ x2

Equations

• x✝¹.decidableDvd x✝ = decidable_of_decidable_of_iff ⋯

theorem Int.mul_dvd_mul_iff_left {a b c : Int} (h : a ≠ 0) : a * b ∣ a * c ↔ b ∣ c

theorem Int.mul_dvd_mul_iff_right {a b c : Int} (h : a ≠ 0) : b * a ∣ c * a ↔ b ∣ c

theorem Int.mul_dvd_mul {a b c d : Int} : a ∣ b → c ∣ d → a * c ∣ b * d

theorem Int.mul_dvd_mul_left {b c : Int} (a : Int) (h : b ∣ c) : a * b ∣ a * c

theorem Int.mul_dvd_mul_right {b c : Int} (a : Int) (h : b ∣ c) : b * a ∣ c * a

theorem Int.dvd_of_mul_dvd_mul_left {a m n : Int} (ha : a ≠ 0) (h : a * m ∣ a * n) : m ∣ n

theorem Int.dvd_of_mul_dvd_mul_right {a m n : Int} (ha : a ≠ 0) (h : m * a ∣ n * a) : m ∣ n

*div zero

@[simp]

theorem Int.zero_tdiv (b : Int) : tdiv 0 b = 0

@[simp]

theorem Int.tdiv_zero (a : Int) : a.tdiv 0 = 0

@[simp]

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

@[simp]

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

preliminaries for div equivalences

theorem Int.negSucc_emod_ofNat_succ_eq_zero_iff {a b : Nat} : negSucc a % (↑b + 1) = 0 ↔ (a + 1) % (b + 1) = 0

theorem Int.negSucc_emod_negSucc_eq_zero_iff {a b : Nat} : negSucc a % negSucc b = 0 ↔ (a + 1) % (b + 1) = 0

div equivalences

theorem Int.tdiv_eq_ediv_of_nonneg {a b : Int} : 0 ≤ a → a.tdiv b = a / b

theorem Int.tdiv_eq_ediv {a b : Int} : a.tdiv b = a / b + if 0 ≤ a ∨ b ∣ a then 0 else b.sign

theorem Int.ediv_eq_tdiv {a b : Int} : a / b = a.tdiv b - if 0 ≤ a ∨ b ∣ a then 0 else b.sign

theorem Int.divExact_eq_tdiv {a b : Int} (h : b ∣ a) : a.divExact b h = a.tdiv b

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

theorem Int.fdiv_eq_ediv {a b : Int} : a.fdiv b = a / b - if 0 ≤ b ∨ b ∣ a then 0 else 1

theorem Int.ediv_eq_fdiv {a b : Int} : a / b = a.fdiv b + if 0 ≤ b ∨ b ∣ a then 0 else 1

theorem Int.divExact_eq_fdiv {a b : Int} (h : b ∣ a) : a.divExact b h = a.fdiv b

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

theorem Int.fdiv_eq_tdiv {a b : Int} : a.fdiv b = a.tdiv b - if b ∣ a then 0 else if 0 ≤ a then if 0 ≤ b then 0 else 1 else if 0 ≤ b then b.sign else 1 + b.sign

theorem Int.tdiv_eq_fdiv {a b : Int} : a.tdiv b = a.fdiv b + if b ∣ a then 0 else if 0 ≤ a then if 0 ≤ b then 0 else 1 else if 0 ≤ b then b.sign else 1 + b.sign

theorem Int.tdiv_eq_ediv_of_dvd {a b : Int} (h : b ∣ a) : a.tdiv b = a / b

theorem Int.fdiv_eq_ediv_of_dvd {a b : Int} (h : b ∣ a) : a.fdiv b = a / b

mod zero

@[simp]

theorem Int.zero_tmod (b : Int) : tmod 0 b = 0

@[simp]

theorem Int.tmod_zero (a : Int) : a.tmod 0 = a

@[simp]

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

@[simp]

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

mod definitions

theorem Int.tmod_add_tdiv (a b : Int) : a.tmod b + b * a.tdiv b = a

theorem Int.tdiv_add_tmod (a b : Int) : b * a.tdiv b + a.tmod b = a

theorem Int.tmod_add_tdiv' (m k : Int) : m.tmod k + m.tdiv k * k = m

Variant of tmod_add_tdiv with the multiplication written the other way around.

theorem Int.tdiv_add_tmod' (m k : Int) : m.tdiv k * k + m.tmod k = m

Variant of tdiv_add_tmod with the multiplication written the other way around.

theorem Int.tmod_def (a b : Int) : a.tmod b = a - b * a.tdiv b

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

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

Variant of fmod_add_fdiv with the multiplication written the other way around.

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

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

Variant of fdiv_add_fmod with the multiplication written the other way around.

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

mod equivalences

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

theorem Int.fmod_eq_emod {a b : Int} : a.fmod b = a % b + if 0 ≤ b ∨ b ∣ a then 0 else b

theorem Int.emod_eq_fmod {a b : Int} : a % b = a.fmod b - if 0 ≤ b ∨ b ∣ a then 0 else b

theorem Int.tmod_eq_emod_of_nonneg {a b : Int} (ha : 0 ≤ a) : a.tmod b = a % b

theorem Int.tmod_eq_emod {a b : Int} : a.tmod b = a % b - ↑(if 0 ≤ a ∨ b ∣ a then 0 else b.natAbs)

theorem Int.emod_eq_tmod {a b : Int} : a % b = a.tmod b + ↑(if 0 ≤ a ∨ b ∣ a then 0 else b.natAbs)

theorem Int.fmod_eq_tmod_of_nonneg {a b : Int} (ha : 0 ≤ a) (hb : 0 ≤ b) : a.fmod b = a.tmod b

theorem Int.fmod_eq_tmod {a b : Int} : a.fmod b = a.tmod b + if b ∣ a then 0 else if 0 ≤ a then if 0 ≤ b then 0 else b else if 0 ≤ b then ↑b.natAbs else 2 * ↑b.toNat

theorem Int.tmod_eq_fmod {a b : Int} : a.tmod b = a.fmod b - if b ∣ a then 0 else if 0 ≤ a then if 0 ≤ b then 0 else b else if 0 ≤ b then ↑b.natAbs else 2 * ↑b.toNat

/ ediv

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

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

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

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

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

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

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

@[reducible, inline, deprecated Int.ediv_neg_of_neg_of_pos (since := "2025-03-04")]

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

Equations

• @Int.ediv_neg' = @Int.ediv_neg_of_neg_of_pos

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

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

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

theorem Int.ediv_pos_of_neg_of_neg {a b : Int} (ha : a < 0) (hb : b < 0) : 0 < a / b

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

@[reducible, inline, deprecated Int.ediv_nonpos_of_nonneg_of_nonpos (since := "2025-03-04")]

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

Equations

• @Int.ediv_nonpos = @Int.ediv_nonpos_of_nonneg_of_nonpos

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

theorem Int.ediv_eq_neg_one_of_neg_of_le {a b : Int} (H1 : a < 0) (H2 : -a ≤ b) : a / b = -1

theorem Int.ediv_eq_one_of_neg_of_le {a b : Int} (H1 : a < 0) (H2 : b ≤ a) : a / b = 1

theorem Int.one_ediv (b : Int) : 1 / b = if b.natAbs = 1 then b else 0

theorem Int.neg_one_ediv (b : Int) : -1 / b = -b.sign

@[simp]

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

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

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

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

@[simp]

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

@[simp]

theorem Int.neg_ediv_self (a : Int) (h : a ≠ 0) : -a / a = -1

theorem Int.sign_ediv (a b : Int) : (a / b).sign = if 0 ≤ a ∧ a < ↑b.natAbs then 0 else a.sign * b.sign

emod

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

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

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

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

@[simp]

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

@[deprecated Int.add_neg_mul_emod_self_right (since := "2025-04-11")]

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

@[simp]

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

@[simp]

theorem Int.add_emod_right (a b : Int) : (a + b) % b = a % b

@[deprecated Int.add_emod_right (since := "2025-04-11")]

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

@[simp]

theorem Int.add_emod_left (a b : Int) : (a + b) % a = b % a

@[deprecated Int.add_emod_left (since := "2025-04-11")]

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

@[simp]

theorem Int.sub_mul_emod_self_right (a b c : Int) : (a - b * c) % c = a % c

@[simp]

theorem Int.sub_mul_emod_self_left (a b c : Int) : (a - b * c) % b = a % b

@[simp]

theorem Int.mul_sub_emod_self_right (a b c : Int) : (a * b - c) % b = -c % b

@[simp]

theorem Int.mul_sub_emod_self_left (a b c : Int) : (a * b - c) % a = -c % a

@[simp]

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

@[simp]

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

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

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

@[simp]

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

@[simp]

theorem Int.neg_emod_self (a : Int) : -a % a = 0

@[simp]

theorem Int.emod_sub_emod (m n k : Int) : (m % n - k) % n = (m - k) % n

@[simp]

theorem Int.sub_emod_emod (m n k : Int) : (m - n % k) % k = (m - n) % k

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

theorem Int.emod_lt_of_neg (a : Int) {b : Int} (h : b < 0) : a % b < -b

theorem Int.emod_lt (a : Int) {b : Int} (h : b ≠ 0) : a % b < ↑b.natAbs

@[simp]

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

theorem Int.sign_emod (a : Int) {b : Int} (h : b ≠ 0) : (a % b).sign = if b ∣ a then 0 else 1

theorem Int.one_emod {b : Int} : 1 % b = if b.natAbs = 1 then 0 else 1

@[simp]

theorem Int.neg_emod_two (i : Int) : -i % 2 = i % 2

properties of / and %

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

theorem Int.ediv_mul_cancel_of_dvd {a b : Int} (H : b ∣ a) : 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 n k : Int} (i : Int) (H : m % n = k % n) : (i + m) % n = (i + k) % n

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

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

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

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

theorem Int.ediv_emod_unique' {a b r q : Int} (h : b < 0) : 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 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.neg_ediv {a b : Int} : -a / b = -(a / b) - if b ∣ a then 0 else b.sign

theorem Int.tdiv_cases (n m : Int) : n.tdiv m = if 0 ≤ n then if 0 ≤ m then n / m else -(n / -m) else if 0 ≤ m then -(-n / m) else -n / -m

theorem Int.neg_emod {a b : Int} : -a % b = if b ∣ a then 0 else ↑b.natAbs - a % b

theorem Int.natAbs_ediv (a b : Int) : (a / b).natAbs = a.natAbs / b.natAbs + if 0 ≤ a ∨ b = 0 ∨ b ∣ a then 0 else 1

theorem Int.natAbs_ediv_of_nonneg {a b : Int} (ha : 0 ≤ a) : (a / b).natAbs = a.natAbs / b.natAbs

theorem Int.natAbs_ediv_of_dvd {a b : Int} (hab : b ∣ a) : (a / b).natAbs = a.natAbs / b.natAbs

theorem Int.natAbs_emod_of_nonneg {a : Int} (h : 0 ≤ a) (b : Int) : (a % b).natAbs = a.natAbs % b.natAbs

theorem Int.natAbs_emod (a : Int) {b : Int} (hb : b ≠ 0) : (a % b).natAbs = if 0 ≤ a ∨ b ∣ a then a.natAbs % b.natAbs else b.natAbs - a.natAbs % b.natAbs

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

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

@[reducible, inline, deprecated Int.natAbs_ediv_le_natAbs (since := "2025-03-05")]

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

Equations

• Int.natAbs_div_le_natAbs = Int.natAbs_ediv_le_natAbs

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

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

theorem Int.dvd_self_sub_emod {x m : Int} : m ∣ x - x % m

theorem Int.emod_eq_iff {a b c : Int} (hb : b ≠ 0) : a % b = c ↔ 0 ≤ c ∧ c < ↑b.natAbs ∧ b ∣ c - a

@[simp]

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

@[simp]

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

@[deprecated Int.mul_ediv_cancel (since := "2025-03-05")]

theorem Int.neg_mul_ediv_cancel (a b : Int) (h : b ≠ 0) : -(a * b) / b = -a

@[deprecated Int.mul_ediv_cancel (since := "2025-03-05")]

theorem Int.neg_mul_ediv_cancel_left (a b : Int) (h : a ≠ 0) : -(a * b) / a = -b

@[simp]

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

@[simp]

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

@[deprecated Int.sub_emod_right (since := "2025-04-11")]

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

@[simp]

theorem Int.add_neg_emod_self (a b : Int) : (a + -b) % b = a % b

@[simp]

theorem Int.neg_add_emod_self (a b : Int) : (-a + b) % a = b % a

theorem Int.dvd_self_sub_of_emod_eq {a b c : Int} : a % b = c → b ∣ a - c

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

@[deprecated Int.dvd_self_sub_of_emod_eq (since := "2025-04-12")]

theorem Int.dvd_sub_of_emod_eq {a b c : Int} : a % b = c → b ∣ a - c

theorem Int.dvd_sub_self_of_emod_eq {a b c : Int} : a % b = c → b ∣ c - a

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

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

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

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

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

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

@[simp]

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

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

theorem Int.sign_eq_ediv_natAbs (a : Int) : a.sign = a / ↑a.natAbs

theorem Int.dvd_mul_of_ediv_dvd {a b c : Int} (h : b ∣ a) (hdiv : a / b ∣ c) : a ∣ b * c

@[simp]

theorem Int.ediv_dvd_iff_dvd_mul {a b c : Int} (h : b ∣ a) (hb : b ≠ 0) : a / b ∣ c ↔ a ∣ b * c

theorem Int.mul_dvd_of_dvd_ediv {a b c : Int} (hcb : c ∣ b) (h : a ∣ b / c) : c * a ∣ b

theorem Int.dvd_ediv_of_mul_dvd {a b c : Int} (h : a * b ∣ c) : b ∣ c / a

@[simp]

theorem Int.dvd_ediv_iff_mul_dvd {a b c : Int} (hbc : c ∣ b) : a ∣ b / c ↔ c * a ∣ b

theorem Int.ediv_dvd_ediv {a b c : Int} : a ∣ b → b ∣ c → b / a ∣ c / a

theorem Int.ediv_dvd_of_dvd {m n : Int} (hmn : m ∣ n) : n / m ∣ n

/ 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 b c : Int} (w : a * b ≤ a * c) (h : 0 < a) : b ≤ c

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

theorem Int.mul_le_mul_iff_of_pos_right {a b c : Int} (ha : 0 < a) : b * a ≤ c * a ↔ b ≤ c

theorem Int.mul_nonneg_iff_of_pos_right {a b : Int} (hb : 0 < b) : 0 ≤ a * b ↔ 0 ≤ a

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

theorem Int.ediv_nonpos_of_nonpos_of_neg {n s : Int} (h : n ≤ 0) (h2 : 0 < s) : n / s ≤ 0

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

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

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

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

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

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

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

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

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

theorem Int.ediv_le_iff_le_mul {k x y : Int} (h : 0 < k) : x / k ≤ y ↔ x < y * k + k

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

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

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

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

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

theorem Int.ediv_eq_iff_of_pos {k x y : Int} (h : 0 < k) : x / k = y ↔ y * k ≤ x ∧ x < y * k + k

theorem Int.add_ediv_of_pos {a b c : Int} (h : 0 < c) : (a + b) / c = a / c + b / c + if c ≤ a % c + b % c then 1 else 0

theorem Int.add_ediv {a b c : Int} (h : c ≠ 0) : (a + b) / c = a / c + b / c + if ↑c.natAbs ≤ a % c + b % c then c.sign else 0

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

theorem Int.not_dvd_iff_lt_mul_succ {n : Int} (m : Int) (hn : 0 < n) : ¬n ∣ m ↔ ∃ (k : Int), n * k < m ∧ m < n * (k + 1)

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

tdiv

@[simp]

theorem Int.tdiv_neg (a b : Int) : a.tdiv (-b) = -a.tdiv b

There are no lemmas

• add_mul_tdiv_right : c ≠ 0 → (a + b * c).tdiv c = a.tdiv c + b
• add_mul_tdiv_left : b ≠ 0 → (a + b * c).tdiv b = a.tdiv b + c
• (similarly mul_add_tdiv_right, mul_add_tdiv_left)
• add_tdiv_of_dvd_right : c ∣ b → (a + b).tdiv c = a.tdiv c + b.tdiv c
• add_tdiv_of_dvd_left : c ∣ a → (a + b).tdiv c = a.tdiv c + b.tdiv c because these statements are all incorrect, and require awkward conditional off-by-one corrections.

@[simp]

theorem Int.mul_tdiv_cancel (a : Int) {b : Int} (H : b ≠ 0) : (a * b).tdiv b = a

@[simp]

theorem Int.mul_tdiv_cancel_left {a : Int} (b : Int) (H : a ≠ 0) : (a * b).tdiv a = b

theorem Int.tdiv_nonneg {a b : Int} (Ha : 0 ≤ a) (Hb : 0 ≤ b) : 0 ≤ a.tdiv b

theorem Int.tdiv_nonneg_of_nonpos_of_nonpos {a b : Int} (Ha : a ≤ 0) (Hb : b ≤ 0) : 0 ≤ a.tdiv b

theorem Int.tdiv_nonpos_of_nonneg_of_nonpos {a b : Int} (Ha : 0 ≤ a) (Hb : b ≤ 0) : a.tdiv b ≤ 0

@[reducible, inline, deprecated Int.tdiv_nonpos_of_nonneg_of_nonpos (since := "2025-03-04")]

abbrev Int.tdiv_nonpos {a b : Int} (Ha : 0 ≤ a) (Hb : b ≤ 0) : a.tdiv b ≤ 0

Equations

• @Int.tdiv_nonpos = @Int.tdiv_nonpos_of_nonneg_of_nonpos

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

@[simp]

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

@[simp]

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

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

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

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

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

@[simp]

theorem Int.tdiv_self {a : Int} (H : a ≠ 0) : a.tdiv a = 1

@[simp]

theorem Int.neg_tdiv (a b : Int) : (-a).tdiv b = -a.tdiv b

theorem Int.neg_tdiv_neg (a b : Int) : (-a).tdiv (-b) = a.tdiv b

theorem Int.sign_tdiv (a b : Int) : (a.tdiv b).sign = if a.natAbs < b.natAbs then 0 else a.sign * b.sign

@[simp]

theorem Int.natAbs_tdiv (a b : Int) : (a.tdiv b).natAbs = a.natAbs.div b.natAbs

tmod

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

theorem Int.tmod_nonneg {a : Int} (b : Int) : 0 ≤ a → 0 ≤ a.tmod b

@[simp]

theorem Int.tmod_neg (a b : Int) : a.tmod (-b) = a.tmod b

@[simp]

theorem Int.neg_tmod (a b : Int) : (-a).tmod b = -a.tmod b

theorem Int.tmod_lt_of_pos (a : Int) {b : Int} (H : 0 < b) : a.tmod b < b

theorem Int.lt_tmod_of_pos (a : Int) {b : Int} (H : 0 < b) : -b < a.tmod b

@[simp]

theorem Int.mul_tmod_left (a b : Int) : (a * b).tmod b = 0

@[simp]

theorem Int.mul_tmod_right (a b : Int) : (a * b).tmod a = 0

theorem Int.mul_tmod (a b n : Int) : (a * b).tmod n = (a.tmod n * b.tmod n).tmod n

@[simp]

theorem Int.tmod_self {a : Int} : a.tmod a = 0

@[simp]

theorem Int.tmod_tmod_of_dvd (n : Int) {m k : Int} (h : m ∣ k) : (n.tmod k).tmod m = n.tmod m

@[simp]

theorem Int.tmod_tmod (a b : Int) : (a.tmod b).tmod b = a.tmod b

theorem Int.tmod_eq_zero_of_dvd {a b : Int} : a ∣ b → b.tmod a = 0

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

theorem Int.sign_tmod (a b : Int) : (a.tmod b).sign = if b ∣ a then 0 else a.sign

@[simp]

theorem Int.natAbs_tmod (a b : Int) : (a.tmod b).natAbs = a.natAbs % b.natAbs

properties of tdiv and tmod

theorem Int.mul_tdiv_cancel_of_tmod_eq_zero {a b : Int} (H : a.tmod b = 0) : b * a.tdiv b = a

theorem Int.tdiv_mul_cancel_of_tmod_eq_zero {a b : Int} (H : a.tmod b = 0) : a.tdiv b * b = a

theorem Int.dvd_of_tmod_eq_zero {a b : Int} (H : b.tmod a = 0) : a ∣ b

theorem Int.dvd_iff_tmod_eq_zero {a b : Int} : a ∣ b ↔ b.tmod a = 0

theorem Int.tdiv_mul_cancel {a b : Int} (H : b ∣ a) : a.tdiv b * b = a

theorem Int.mul_tdiv_cancel' {a b : Int} (H : a ∣ b) : a * b.tdiv a = b

@[simp]

theorem Int.neg_tmod_self (a : Int) : (-a).tmod a = 0

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

theorem Int.mul_tdiv_assoc (a : Int) {b c : Int} : c ∣ b → (a * b).tdiv c = a * b.tdiv c

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

theorem Int.neg_tdiv_of_dvd {a b : Int} : b ∣ a → (-a).tdiv b = -a.tdiv b

theorem Int.tdiv_dvd_tdiv {a b c : Int} : a ∣ b → b ∣ c → b.tdiv a ∣ c.tdiv a

theorem Int.mul_tdiv_cancel_of_dvd {a b : Int} (H : b ∣ a) : b * a.tdiv b = a

theorem Int.tdiv_mul_cancel_of_dvd {a b : Int} (H : b ∣ a) : a.tdiv b * b = a

theorem Int.tmod_two_eq (x : Int) : x.tmod 2 = -1 ∨ x.tmod 2 = 0 ∨ x.tmod 2 = 1

theorem Int.tdiv_tmod_unique {a b r q : Int} (ha : 0 ≤ a) (hb : b ≠ 0) : a.tdiv b = q ∧ a.tmod b = r ↔ r + b * q = a ∧ 0 ≤ r ∧ r < ↑b.natAbs

theorem Int.tdiv_tmod_unique' {a b r q : Int} (ha : a ≤ 0) (hb : b ≠ 0) : a.tdiv b = q ∧ a.tmod b = r ↔ r + b * q = a ∧ -↑b.natAbs < r ∧ r ≤ 0

@[simp]

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

theorem Int.natAbs_tdiv_le_natAbs (a b : Int) : (a.tdiv b).natAbs ≤ a.natAbs

theorem Int.tdiv_le_self {a : Int} (b : Int) (Ha : 0 ≤ a) : a.tdiv b ≤ a

theorem Int.dvd_tmod_sub_self {x m : Int} : m ∣ x.tmod m - x

theorem Int.dvd_self_sub_tmod {x m : Int} : m ∣ x - x.tmod m

@[simp]

theorem Int.neg_mul_tmod_right (a b : Int) : (-(a * b)).tmod a = 0

@[simp]

theorem Int.neg_mul_tmod_left (a b : Int) : (-(a * b)).tmod b = 0

@[simp]

theorem Int.tdiv_one (a : Int) : a.tdiv 1 = a

@[simp]

theorem Int.tmod_one (a : Int) : a.tmod 1 = 0

theorem Int.dvd_self_sub_of_tmod_eq {a b c : Int} (h : a.tmod b = c) : b ∣ a - c

theorem Int.dvd_sub_self_of_tmod_eq {a b c : Int} (h : a.tmod b = c) : b ∣ c - a

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

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

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

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

theorem Int.eq_zero_of_tdiv_eq_zero {d n : Int} (h : d ∣ n) (H : n.tdiv d = 0) : n = 0

@[simp]

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

theorem Int.tdiv_sign (a b : Int) : a.tdiv b.sign = a * b.sign

theorem Int.sign_eq_tdiv_abs (a : Int) : a.sign = a.tdiv ↑a.natAbs

@[simp]

theorem Int.mul_le_mul_left {a b c : Int} (ha : 0 < a) : a * b ≤ a * c ↔ b ≤ c

@[simp]

theorem Int.mul_le_mul_right {a b c : Int} (ha : 0 < a) : b * a ≤ c * a ↔ b ≤ c

@[simp]

theorem Int.mul_lt_mul_left {a b c : Int} (ha : 0 < a) : a * b < a * c ↔ b < c

@[simp]

theorem Int.mul_lt_mul_right {a b c : Int} (ha : 0 < a) : b * a < c * a ↔ b < c

@[simp]

theorem Int.mul_le_mul_left_of_neg {a b c : Int} (ha : a < 0) : a * b ≤ a * c ↔ c ≤ b

@[simp]

theorem Int.mul_le_mul_right_of_neg {a b c : Int} (ha : a < 0) : b * a ≤ c * a ↔ c ≤ b

@[simp]

theorem Int.mul_lt_mul_left_of_neg {a b c : Int} (ha : a < 0) : a * b < a * c ↔ c < b

@[simp]

theorem Int.mul_lt_mul_right_of_neg {a b c : Int} (ha : a < 0) : b * a < c * a ↔ c < b

theorem Int.ediv_le_iff_of_dvd_of_pos {a b c : Int} (hb : 0 < b) (hba : b ∣ a) : a / b ≤ c ↔ a ≤ b * c

theorem Int.ediv_le_iff_of_dvd_of_neg {a b c : Int} (hb : b < 0) (hba : b ∣ a) : a / b ≤ c ↔ b * c ≤ a

theorem Int.ediv_lt_iff_of_dvd_of_pos {a b c : Int} (hb : 0 < b) (hba : b ∣ a) : a / b < c ↔ a < b * c

theorem Int.ediv_lt_iff_of_dvd_of_neg {a b c : Int} (hb : b < 0) (hba : b ∣ a) : a / b < c ↔ b * c < a

theorem Int.le_ediv_iff_of_dvd_of_pos {a b c : Int} (hc : 0 < c) (hcb : c ∣ b) : a ≤ b / c ↔ c * a ≤ b

theorem Int.le_ediv_iff_of_dvd_of_neg {a b c : Int} (hc : c < 0) (hcb : c ∣ b) : a ≤ b / c ↔ b ≤ c * a

theorem Int.lt_ediv_iff_of_dvd_of_pos {a b c : Int} (hc : 0 < c) (hcb : c ∣ b) : a < b / c ↔ c * a < b

theorem Int.lt_ediv_iff_of_dvd_of_neg {a b c : Int} (hc : c < 0) (hcb : c ∣ b) : a < b / c ↔ b < c * a

theorem Int.ediv_le_ediv_iff_of_dvd_of_pos_of_pos {a b c d : Int} (hb : 0 < b) (hd : 0 < d) (hba : b ∣ a) (hdc : d ∣ c) : a / b ≤ c / d ↔ d * a ≤ c * b

theorem Int.ediv_le_ediv_iff_of_dvd_of_pos_of_neg {a b c d : Int} (hb : 0 < b) (hd : d < 0) (hba : b ∣ a) (hdc : d ∣ c) : a / b ≤ c / d ↔ c * b ≤ d * a

theorem Int.ediv_le_ediv_iff_of_dvd_of_neg_of_pos {a b c d : Int} (hb : b < 0) (hd : 0 < d) (hba : b ∣ a) (hdc : d ∣ c) : a / b ≤ c / d ↔ c * b ≤ d * a

theorem Int.ediv_le_ediv_iff_of_dvd_of_neg_of_neg {a b c d : Int} (hb : b < 0) (hd : d < 0) (hba : b ∣ a) (hdc : d ∣ c) : a / b ≤ c / d ↔ d * a ≤ c * b

theorem Int.ediv_lt_ediv_iff_of_dvd_of_pos {a b c d : Int} (hb : 0 < b) (hd : 0 < d) (hba : b ∣ a) (hdc : d ∣ c) : a / b < c / d ↔ d * a < c * b

theorem Int.ediv_lt_ediv_iff_of_dvd_of_pos_of_neg {a b c d : Int} (hb : 0 < b) (hd : d < 0) (hba : b ∣ a) (hdc : d ∣ c) : a / b < c / d ↔ c * b < d * a

theorem Int.ediv_lt_ediv_iff_of_dvd_of_neg_of_pos {a b c d : Int} (hb : b < 0) (hd : 0 < d) (hba : b ∣ a) (hdc : d ∣ c) : a / b < c / d ↔ c * b < d * a

theorem Int.ediv_lt_ediv_iff_of_dvd_of_neg_of_neg {a b c d : Int} (hb : b < 0) (hd : d < 0) (hba : b ∣ a) (hdc : d ∣ c) : a / b < c / d ↔ d * a < c * b

tdiv and ordering

theorem Int.mul_tdiv_self_le {x k : Int} (h : 0 ≤ x) : k * x.tdiv k ≤ x

theorem Int.lt_mul_tdiv_self_add {x k : Int} (h : 0 < k) : x < k * x.tdiv k + k

theorem Int.tdiv_mul_le (a : Int) {b : Int} (hb : b ≠ 0) : a.tdiv b * b ≤ a + ↑(if 0 ≤ a then 0 else b.natAbs - 1)

theorem Int.tdiv_le_of_le_mul {a b c : Int} (Hc : 0 < c) (H' : a ≤ b * c) : a.tdiv c ≤ b + if 0 ≤ a then 0 else 1

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

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

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

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

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

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

theorem Int.le_emod_self_add_one_iff {a b : Int} (h : 0 < b) : b ≤ a % b + 1 ↔ b ∣ a + 1

@[deprecated Int.le_emod_self_add_one_iff (since := "2025-04-12")]

theorem Int.le_mod_self_add_one_iff {a b : Int} (h : 0 < b) : b ≤ a % b + 1 ↔ b ∣ a + 1

theorem Int.add_one_tdiv_of_pos {a b : Int} (h : 0 < b) : (a + 1).tdiv b = a.tdiv b + if 0 < a + 1 ∧ b ∣ a + 1 ∨ a < 0 ∧ b ∣ a then 1 else 0

theorem Int.add_one_tdiv {a b : Int} : (a + 1).tdiv b = a.tdiv b + if 0 < a + 1 ∧ b ∣ a + 1 ∨ a < 0 ∧ b ∣ a then b.sign else 0

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

fdiv

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

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

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

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

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

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

theorem Int.mul_sub_fdiv_right (a c : Int) {b : Int} (H : b ≠ 0) : (a * b - c).fdiv b = a + (-c).fdiv b

theorem Int.mul_sub_fdiv_left (b : Int) {a : Int} (c : Int) (H : a ≠ 0) : (a * b - c).fdiv a = b + (-c).fdiv 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

theorem Int.add_fdiv_of_dvd_right {a b c : Int} (H : c ∣ b) : (a + b).fdiv c = a.fdiv c + b.fdiv c

theorem Int.add_fdiv_of_dvd_left {a b c : Int} (H : c ∣ a) : (a + b).fdiv c = a.fdiv c + b.fdiv c

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

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

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

@[reducible, inline, deprecated Int.fdiv_nonpos_of_nonneg_of_nonpos (since := "2025-03-04")]

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

Equations

• @Int.fdiv_nonpos = @Int.fdiv_nonpos_of_nonneg_of_nonpos

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

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

@[simp]

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

@[simp]

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

@[simp]

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

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

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

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

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

@[simp]

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

theorem Int.neg_fdiv {a b : Int} : (-a).fdiv b = -a.fdiv b - if b = 0 ∨ b ∣ a then 0 else 1

@[simp]

theorem Int.neg_fdiv_neg (a b : Int) : (-a).fdiv (-b) = a.fdiv b

theorem Int.fdiv_neg {a b : Int} (h : b ≠ 0) : a.fdiv (-b) = if b ∣ a then -a.fdiv b else -a.fdiv b - 1

One could prove the following, but as the statements are quite awkward, so far it doesn't seem worthwhile.

theorem natAbs_fdiv {a b : Int} (h : b ≠ 0) :
    natAbs (a.fdiv b) = a.natAbs / b.natAbs + if a.sign = b.sign ∨ b ∣ a then 0 else 1 := ...

theorem sign_fdiv (a b : Int) :
    sign (a.fdiv b) = if a.sign = b.sign ∧ natAbs a < natAbs b then 0 else sign a * sign b := ...

fmod

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

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

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

@[reducible, inline, deprecated Int.fmod_nonneg_of_pos (since := "2025-03-04")]

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

Equations

• Int.fmod_nonneg' = Int.fmod_nonneg_of_pos

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

@[simp]

theorem Int.add_mul_fmod_self_right (a b c : Int) : (a + b * c).fmod c = a.fmod c

@[deprecated Int.add_mul_fmod_self_right (since := "2025-04-11")]

theorem Int.add_mul_fmod_self {a b c : Int} : (a + b * c).fmod c = a.fmod c

@[simp]

theorem Int.add_mul_fmod_self_left (a b c : Int) : (a + b * c).fmod b = a.fmod b

@[simp]

theorem Int.mul_add_fmod_self_right (a b c : Int) : (a * b + c).fmod b = c.fmod b

@[simp]

theorem Int.mul_add_fmod_self_left (a b c : Int) : (a * b + c).fmod a = c.fmod a

@[simp]

theorem Int.add_neg_mul_fmod_self_right (a b c : Int) : (a + -(b * c)).fmod c = a.fmod c

@[simp]

theorem Int.add_neg_mul_fmod_self_left (a b c : Int) : (a + -(b * c)).fmod b = a.fmod b

@[simp]

theorem Int.add_fmod_right (a b : Int) : (a + b).fmod b = a.fmod b

@[simp]

theorem Int.add_fmod_left (a b : Int) : (a + b).fmod a = b.fmod a

@[simp]

theorem Int.sub_mul_fmod_self_right (a b c : Int) : (a - b * c).fmod c = a.fmod c

@[simp]

theorem Int.sub_mul_fmod_self_left (a b c : Int) : (a - b * c).fmod b = a.fmod b

@[simp]

theorem Int.mul_sub_fmod_self_right (a b c : Int) : (a * b - c).fmod b = (-c).fmod b

@[simp]

theorem Int.mul_sub_fmod_self_left (a b c : Int) : (a * b - c).fmod a = (-c).fmod a

@[simp]

theorem Int.sub_fmod_right (a b : Int) : (a - b).fmod b = a.fmod b

@[simp]

theorem Int.sub_fmod_left (a b : Int) : (a - b).fmod a = (-b).fmod a

@[simp]

theorem Int.fmod_add_fmod (m n k : Int) : (m.fmod n + k).fmod n = (m + k).fmod n

@[simp]

theorem Int.add_fmod_fmod (m n k : Int) : (m + n.fmod k).fmod k = (m + n).fmod k

theorem Int.add_fmod (a b n : Int) : (a + b).fmod n = (a.fmod n + b.fmod n).fmod n

theorem Int.add_fmod_eq_add_fmod_right {m n k : Int} (i : Int) (H : m.fmod n = k.fmod n) : (m + i).fmod n = (k + i).fmod n

theorem Int.fmod_add_cancel_right {m n k : Int} (i : Int) : (m + i).fmod n = (k + i).fmod n ↔ m.fmod n = k.fmod n

@[simp]

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

@[simp]

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

theorem Int.mul_fmod (a b n : Int) : (a * b).fmod n = (a.fmod n * b.fmod n).fmod n

@[simp]

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

@[simp]

theorem Int.fmod_fmod_of_dvd (n : Int) {m k : Int} (h : m ∣ k) : (n.fmod k).fmod m = n.fmod m

@[simp]

theorem Int.fmod_fmod (a b : Int) : (a.fmod b).fmod b = a.fmod b

theorem Int.sub_fmod (a b n : Int) : (a - b).fmod n = (a.fmod n - b.fmod n).fmod n

theorem Int.fmod_eq_zero_of_dvd {a b : Int} : a ∣ b → b.fmod a = 0

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

@[simp]

theorem Int.neg_fmod_neg (a b : Int) : (-a).fmod (-b) = -a.fmod b

properties of fdiv and fmod

theorem Int.mul_fdiv_cancel_of_fmod_eq_zero {a b : Int} (H : a.fmod b = 0) : b * a.fdiv b = a

theorem Int.fdiv_mul_cancel_of_fmod_eq_zero {a b : Int} (H : a.fmod b = 0) : a.fdiv b * b = a

theorem Int.dvd_of_fmod_eq_zero {a b : Int} (H : b.fmod a = 0) : a ∣ b

theorem Int.dvd_iff_fmod_eq_zero {a b : Int} : a ∣ b ↔ b.fmod a = 0

theorem Int.fdiv_mul_cancel {a b : Int} (H : b ∣ a) : a.fdiv b * b = a

theorem Int.mul_fdiv_cancel' {a b : Int} (H : a ∣ b) : a * b.fdiv a = b

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

@[simp]

theorem Int.neg_fmod_self (a : Int) : (-a).fmod a = 0

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

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

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

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

theorem Int.eq_zero_of_fdiv_eq_zero {d n : Int} (h : d ∣ n) (H : n.fdiv d = 0) : n = 0

@[simp]

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

theorem Int.mul_fdiv_assoc (a : Int) {b c : Int} : c ∣ b → (a * b).fdiv c = a * b.fdiv c

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

theorem Int.neg_fdiv_of_dvd {a b : Int} : b ∣ a → (-a).fdiv b = -a.fdiv b

theorem Int.sub_fdiv_of_dvd (a : Int) {b c : Int} (hcb : c ∣ b) : (a - b).fdiv c = a.fdiv c - b.fdiv c

theorem Int.fdiv_dvd_fdiv {a b c : Int} : a ∣ b → b ∣ c → b.fdiv a ∣ c.fdiv a

theorem Int.mul_fdiv_cancel_of_dvd {a b : Int} (H : b ∣ a) : b * a.fdiv b = a

theorem Int.fdiv_mul_cancel_of_dvd {a b : Int} (H : b ∣ a) : a.fdiv b * b = a

theorem Int.fmod_two_eq (x : Int) : x.fmod 2 = 0 ∨ x.fmod 2 = 1

theorem Int.add_fmod_eq_add_fmod_left {m n k : Int} (i : Int) (H : m.fmod n = k.fmod n) : (i + m).fmod n = (i + k).fmod n

theorem Int.fmod_add_cancel_left {m n k i : Int} : (i + m).fmod n = (i + k).fmod n ↔ m.fmod n = k.fmod n

theorem Int.fmod_sub_cancel_right {m n k : Int} (i : Int) : (m - i).fmod n = (k - i).fmod n ↔ m.fmod n = k.fmod n

theorem Int.fmod_eq_fmod_iff_fmod_sub_eq_zero {m n k : Int} : m.fmod n = k.fmod n ↔ (m - k).fmod n = 0

theorem Int.fmod_sub_cancel_left {m n k : Int} (i : Int) : (i - m).fmod n = (i - k).fmod n ↔ m.fmod n = k.fmod n

theorem Int.fdiv_fmod_unique {a b r q : Int} (h : 0 < b) : a.fdiv b = q ∧ a.fmod b = r ↔ r + b * q = a ∧ 0 ≤ r ∧ r < b

theorem Int.fdiv_fmod_unique' {a b r q : Int} (h : b < 0) : a.fdiv b = q ∧ a.fmod b = r ↔ r + b * q = a ∧ b < r ∧ r ≤ 0

@[simp]

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

theorem Int.natAbs_fdiv_le_natAbs (a b : Int) : (a.fdiv b).natAbs ≤ a.natAbs

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

theorem Int.dvd_fmod_sub_self {x m : Int} : m ∣ x.fmod m - x

theorem Int.dvd_self_sub_fmod {x m : Int} : m ∣ x - x.fmod m

theorem Int.dvd_self_sub_of_fmod_eq {a b c : Int} (h : a.fmod b = c) : b ∣ a - c

theorem Int.dvd_sub_self_of_fmod_eq {a b c : Int} (h : a.fmod b = c) : b ∣ c - a

@[simp]

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

@[simp]

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

@[simp]

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

@[deprecated Int.sub_fmod_right (since := "2025-04-12")]

theorem Int.fmod_sub_cancel (x y : Int) : (x - y).fmod y = x.fmod y

@[simp]

theorem Int.add_neg_fmod_self (a b : Int) : (a + -b).fmod b = a.fmod b

@[simp]

theorem Int.neg_add_fmod_self (a b : Int) : (-a + b).fmod a = b.fmod a

@[deprecated Int.dvd_self_sub_of_fmod_eq (since := "2025-04-12")]

theorem Int.dvd_sub_of_fmod_eq {a b c : Int} (h : a.fmod b = c) : b ∣ a - c

theorem Int.fdiv_sign {a b : Int} : a.fdiv b.sign = a * b.sign

theorem Int.sign_eq_fdiv_abs (a : Int) : a.sign = a.fdiv ↑a.natAbs

fdiv and ordering

theorem Int.mul_fdiv_self_le {x k : Int} (h : 0 < k) : k * x.fdiv k ≤ x

theorem Int.lt_mul_fdiv_self_add {x k : Int} (h : 0 < k) : x < k * x.fdiv k + k

bmod

@[simp]

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

@[deprecated Int.emod_bmod (since := "2025-04-11")]

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

theorem Int.bdiv_add_bmod (x : Int) (m : Nat) : ↑m * x.bdiv m + x.bmod m = x

theorem Int.bmod_add_bdiv (x : Int) (m : Nat) : x.bmod m + ↑m * x.bdiv m = x

theorem Int.bdiv_add_bmod' (x : Int) (m : Nat) : x.bdiv m * ↑m + x.bmod m = x

theorem Int.bmod_add_bdiv' (x : Int) (m : Nat) : x.bmod m + x.bdiv m * ↑m = x

theorem Int.bmod_eq_self_sub_mul_bdiv (x : Int) (m : Nat) : x.bmod m = x - ↑m * x.bdiv m

theorem Int.bmod_eq_self_sub_bdiv_mul (x : Int) (m : Nat) : x.bmod m = x - x.bdiv 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

theorem Int.bmod_eq_emod (x : Int) (m : Nat) : x.bmod m = x % ↑m - ↑(if x % ↑m ≥ (↑m + 1) / 2 then m else 0)

@[simp]

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

@[reducible, inline, deprecated Int.bmod_one (since := "2025-04-10")]

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

Equations

• Int.bmod_one_is_zero = Int.bmod_one

@[simp]

theorem Int.add_bmod_right (a : Int) (b : Nat) : (a + ↑b).bmod b = a.bmod b

@[simp]

theorem Int.add_bmod_left (a : Nat) (b : Int) : (↑a + b).bmod a = b.bmod a

@[simp]

theorem Int.add_mul_bmod_self_right (a b : Int) (c : Nat) : (a + b * ↑c).bmod c = a.bmod c

@[simp]

theorem Int.add_mul_bmod_self_left (a : Int) (b : Nat) (c : Int) : (a + ↑b * c).bmod b = a.bmod b

@[simp]

theorem Int.mul_add_bmod_self_right (a : Int) (b : Nat) (c : Int) : (a * ↑b + c).bmod b = c.bmod b

@[simp]

theorem Int.mul_add_bmod_self_left (a : Nat) (b c : Int) : (↑a * b + c).bmod a = c.bmod a

@[simp]

theorem Int.sub_bmod_right (a : Int) (b : Nat) : (a - ↑b).bmod b = a.bmod b

@[simp]

theorem Int.sub_bmod_left (a : Nat) (b : Int) : (↑a - b).bmod a = (-b).bmod a

@[simp]

theorem Int.sub_mul_bmod_self_right (a b : Int) (c : Nat) : (a - b * ↑c).bmod c = a.bmod c

@[simp]

theorem Int.sub_mul_bmod_self_left (a : Int) (b : Nat) (c : Int) : (a - ↑b * c).bmod b = a.bmod b

@[simp]

theorem Int.mul_sub_bmod_self_right (a : Int) (b : Nat) (c : Int) : (a * ↑b - c).bmod b = (-c).bmod b

@[simp]

theorem Int.mul_sub_bmod_self_left (a : Nat) (b c : Int) : (↑a * b - c).bmod a = (-c).bmod a

@[simp]

theorem Int.add_neg_mul_bmod_self_right (a b : Int) (c : Nat) : (a + -(b * ↑c)).bmod c = a.bmod c

@[simp]

theorem Int.add_neg_mul_bmod_self_left (a : Int) (b : Nat) (c : Int) : (a + -(↑b * c)).bmod b = a.bmod b

@[deprecated Int.add_bmod_right (since := "2025-04-10")]

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

@[deprecated Int.add_mul_bmod_self_left (since := "2025-04-10")]

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

@[deprecated Int.sub_bmod_right (since := "2025-04-10")]

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

@[deprecated Int.sub_mul_bmod_self_left (since := "2025-04-10")]

theorem Int.Int.bmod_sub_mul_cancel (x : Int) (n : Nat) (k : Int) : (x - ↑n * k).bmod n = x.bmod n

@[simp]

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

@[deprecated Int.emod_add_bmod (since := "2025-04-11")]

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_sub_bmod {y : Int} (x : Int) (n : Nat) : (x % ↑n - y).bmod n = (x - y).bmod n

@[deprecated Int.emod_sub_bmod (since := "2025-04-11")]

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

@[simp]

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

@[deprecated Int.sub_emod_bmod (since := "2025-04-11")]

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

@[simp]

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

@[deprecated Int.emod_mul_bmod (since := "2025-04-11")]

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

@[deprecated Int.bmod_add_bmod (since := "2025-04-11")]

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.bmod_sub_bmod {x : Int} {n : Nat} {y : Int} : (x.bmod n - y).bmod n = (x - y).bmod n

@[deprecated Int.bmod_sub_bmod (since := "2025-04-11")]

theorem Int.bmod_sub_bmod_congr {x : Int} {n : Nat} {y : Int} : (x.bmod n - y).bmod n = (x - y).bmod n

theorem Int.add_bmod_eq_add_bmod_right {x : Int} {n : Nat} {y : Int} (i : Int) (H : x.bmod n = y.bmod n) : (x + i).bmod n = (y + i).bmod n

theorem Int.bmod_add_cancel_right {x : Int} {n : Nat} {y : Int} (i : Int) : (x + i).bmod n = (y + i).bmod n ↔ x.bmod n = y.bmod n

theorem Int.bmod_add_cancel_left {x : Int} {n : Nat} {y : Int} (i : Int) : (i + x).bmod n = (i + y).bmod n ↔ x.bmod n = y.bmod n

theorem Int.add_bmod_eq_add_bmod_left {x : Int} {n : Nat} {y : Int} (i : Int) (h : x.bmod n = y.bmod n) : (i + x).bmod n = (i + y).bmod n

@[simp]

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

@[simp]

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

@[simp]

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

@[simp]

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

@[simp]

theorem Int.mul_bmod_left (a : Int) (b : Nat) : (a * ↑b).bmod b = 0

@[simp]

theorem Int.mul_bmod_right (a : Nat) (b : Int) : (↑a * b).bmod a = 0

theorem Int.mul_bmod (a b : Int) (n : Nat) : (a * b).bmod n = (a.bmod n * b.bmod n).bmod n

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

theorem Int.sub_bmod (a b : Int) (n : Nat) : (a - b).bmod n = (a.bmod n - b.bmod n).bmod n

@[simp]

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

@[simp]

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

@[simp]

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

@[simp]

theorem Int.bmod_self {a : Nat} : (↑a).bmod a = 0

@[simp]

theorem Int.neg_bmod_self {a : Nat} : (-↑a).bmod a = 0

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

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

theorem Int.dvd_of_bmod_eq_zero {a : Nat} {b : Int} (h : b.bmod a = 0) : ↑a ∣ b

theorem Int.bmod_eq_zero_of_dvd {a : Nat} {b : Int} (h : ↑a ∣ b) : b.bmod a = 0

theorem Int.dvd_iff_bmod_eq_zero {a : Nat} {b : Int} : ↑a ∣ b ↔ b.bmod a = 0

theorem Int.divExact_eq_bdiv {a : Int} {b : Nat} (h : ↑b ∣ a) : a.divExact (↑b) h = a.bdiv b

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_eq_iff {a : Int} {b : Nat} {c : Int} (hb : 0 < b) : a.bmod b = c ↔ -(↑b / 2) ≤ c ∧ c < (↑b + 1) / 2 ∧ ↑b ∣ c - a

theorem Int.bmod_eq_emod_of_lt {x : Int} {m : Nat} (hx : x % ↑m < (↑m + 1) / 2) : x.bmod m = x % ↑m

theorem Int.bmod_eq_neg {n : Nat} {m : Int} (hm : 0 ≤ m) (hn : ↑n = 2 * m) : m.bmod n = -m

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

theorem Int.bmod_natAbs_add_one (x : Int) (w : x ≠ -1) : x.bmod (x.natAbs + 1) = -x.sign

@[reducible, inline, deprecated Int.bmod_natAbs_add_one (since := "2025-04-04")]

abbrev Int.bmod_natAbs_plus_one (x : Int) (w : x ≠ -1) : x.bmod (x.natAbs + 1) = -x.sign

Equations

• Int.bmod_natAbs_plus_one = Int.bmod_natAbs_add_one

theorem Int.bmod_self_add_one {x : Nat} : (↑x).bmod (x + 1) = if x = 0 then 0 else -1

theorem Int.one_bmod_two : bmod 1 2 = -1

theorem Int.one_bmod {b : Nat} (h : 3 ≤ b) : bmod 1 b = 1

theorem Int.bmod_two_eq (x : Int) : x.bmod 2 = -1 ∨ x.bmod 2 = 0

theorem Int.bmod_sub_cancel_right {x : Int} {n : Nat} {y : Int} (i : Int) : (x - i).bmod n = (y - i).bmod n ↔ x.bmod n = y.bmod n

theorem Int.bmod_eq_bmod_iff_bmod_sub_eq_zero {m : Int} {n : Nat} {k : Int} : m.bmod n = k.bmod n ↔ (m - k).bmod n = 0

theorem Int.bmod_sub_cancel_left {x : Int} {n : Nat} {y : Int} (i : Int) : (i - x).bmod n = (i - y).bmod n ↔ x.bmod n = y.bmod n

theorem Int.mod_bmod_mul_of_pos {a : Nat} (b : Int) (c : Nat) (h : 0 < a) (hce : 2 ∣ c) : (↑a * b).bmod (a * c) = ↑a * b.bmod c

@[simp]

theorem Int.bmod_neg_bmod {x : Int} {n : Nat} : (-x.bmod n).bmod n = (-x).bmod n

theorem Int.neg_bmod {a : Int} {b : Nat} : (-a).bmod b = if ↑b ∣ 2 * a then a.bmod b else -a.bmod b

@[simp]

theorem Int.neg_mul_bmod_left (a : Int) (b : Nat) : (-(a * ↑b)).bmod b = 0

@[simp]

theorem Int.neg_mul_bmod_right (a : Nat) (b : Int) : (-(↑a * b)).bmod a = 0

@[simp]

theorem Int.add_neg_bmod_self (a : Int) (b : Nat) : (a + -↑b).bmod b = a.bmod b

@[simp]

theorem Int.neg_add_bmod_self (a : Nat) (b : Int) : (-↑a + b).bmod a = b.bmod a

theorem Int.dvd_self_sub_of_bmod_eq {a : Int} {b : Nat} {c : Int} (h : a.bmod b = c) : ↑b ∣ a - c

theorem Int.dvd_sub_self_of_bmod_eq {a : Int} {b : Nat} {c : Int} (h : a.bmod b = c) : ↑b ∣ c - a

theorem Int.bmod_neg_iff {m : Nat} {x : Int} (h2 : -↑m ≤ x) (h1 : x < ↑m) : x.bmod m < 0 ↔ -(↑m / 2) ≤ x ∧ x < 0 ∨ (↑m + 1) / 2 ≤ x

theorem Int.bmod_eq_of_le {n : Int} {m : Nat} (hn' : -(↑m / 2) ≤ n) (hn : n < (↑m + 1) / 2) : n.bmod m = n

@[deprecated Int.bmod_eq_of_le (since := "2025-04-11")]

theorem Int.bmod_eq_self_of_le {n : Int} {m : Nat} (hn' : -(↑m / 2) ≤ n) (hn : n < (↑m + 1) / 2) : n.bmod m = n

theorem Int.bmod_bmod_of_dvd {a : Int} {n m : Nat} (hnm : n ∣ m) : (a.bmod m).bmod n = a.bmod n

theorem Int.bmod_eq_of_le_mul_two {x : Int} {y : Nat} (hle : -↑y ≤ x * 2) (hlt : x * 2 < ↑y) : x.bmod y = x

@[deprecated Int.bmod_eq_of_le_mul_two (since := "2025-04-11")]

theorem Int.bmod_eq_self_of_le_mul_two {x : Int} {y : Nat} (hle : -↑y ≤ x * 2) (hlt : x * 2 < ↑y) : x.bmod y = x

theorem Int.ediv_lt_self_of_pos_of_ne_one {x y : Int} (hx : 0 < x) (hy : y ≠ 1) : x / y < x

theorem Int.ediv_nonneg_of_nonneg_of_nonneg {x y : Int} (hx : 0 ≤ x) (hy : 0 ≤ y) : 0 ≤ x / y

theorem Int.ediv_lt_natAbs_self_of_lt_neg_one_of_lt_neg_one {x y : Int} (hx : x < -1) (hy : y < -1) : x / y < ↑x.natAbs

When both x and y are negative we need stricter bounds on x and y to establish the upper bound of x/y, i.e., x / y < x.natAbs. In particular, consider the following counter examples: · (-1) / (-2) = 1 and ¬ 1 < (-1).natAbs (note that Int.neg_one_ediv already handles ediv where the numerator is -1) · (-2) / (-1) = 2 and ¬ 1 < (-2).natAbs To exclude these cases, we enforce stricter bounds on the values of x and y.

theorem Int.self_le_ediv_of_nonpos_of_nonneg {x y : Int} (hx : x ≤ 0) (hy : 0 ≤ y) : x ≤ x / y

theorem Int.neg_self_le_ediv_of_nonneg_of_nonpos (x y : Int) (hx : 0 ≤ x) (hy : y ≤ 0) : -x ≤ x / y

Helper theorems for dvd simproc

theorem Int.dvd_eq_true_of_mod_eq_zero {a b : Int} (h : (b % a == 0) = true) : (a ∣ b) = True

theorem Int.dvd_eq_false_of_mod_ne_zero {a b : Int} (h : (b % a != 0) = true) : (a ∣ b) = False