Lemmas about integer division

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theorem Int.ofNat_div (m : Nat) (n : Nat) : ↑(m / n) = Int.div ↑m ↑n

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theorem Int.ofNat_fdiv (m : Nat) (n : Nat) : ↑(m / n) = Int.fdiv ↑m ↑n

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@[simp]

theorem Int.ofNat_ediv (m : Nat) (n : Nat) : ↑(m / n) = ↑m / ↑n

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

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@[simp]

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

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

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

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

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

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

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

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

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

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@[simp]

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

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

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

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@[simp]

theorem Int.div_def' (a : Int) (b : Int) : a / b = Int.div a b

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@[simp]

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

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

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

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

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

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

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

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

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

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

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@[simp]

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

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

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@[simp]

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

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

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

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

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

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

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

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@[simp]

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

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

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

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

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

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

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

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

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

mod

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

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

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

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

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@[simp]

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

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

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@[simp]

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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theorem Int.emod_add_ediv' (m : Int) (k : Int) : m % k + m / k * k = m

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theorem Int.ediv_add_emod' (m : Int) (k : Int) : m / k * k + m % k = m

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

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

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

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

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

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

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

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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

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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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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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

properties of / and %

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

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

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

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

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

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

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

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

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

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

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

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

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

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

dvd

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Equations

• Int.decidableDvd x x = decidable_of_decidable_of_iff (_ : Int.mod x x = 0 ↔ x ∣ x)

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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.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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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

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

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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

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

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

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

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

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

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

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

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

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

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

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

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@[simp]

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

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@[simp]

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

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

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

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

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theorem Int.mul_sign (i : Int) : i * Int.sign i = ↑(Int.natAbs i)

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

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

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

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