Congruences modulo an integer

This file defines the equivalence relation a ≡ b [ZMOD n] on the integers, similarly to how Data.Nat.ModEq defines them for the natural numbers. The notation is short for n.ModEq a b, which is defined to be a % n = b % n for integers a b n.

Tags

modeq, congruence, mod, MOD, modulo, integers

def Int.ModEq (n : ℤ) (a : ℤ) (b : ℤ) : Prop

a ≡ b [ZMOD n] when a % n = b % n.

Equations

• (a ≡ b [ZMOD n]) = (a % n = b % n)

def Int.«term_≡_[ZMOD_]» : Lean.TrailingParserDescr

a ≡ b [ZMOD n] when a % n = b % n.

Equations

• One or more equations did not get rendered due to their size.

instance Int.instDecidableModEq {n : ℤ} {a : ℤ} {b : ℤ} : Decidable (a ≡ b [ZMOD n])

Equations

• Int.instDecidableModEq = decEq (a % n) (b % n)

theorem Int.ModEq.refl {n : ℤ} (a : ℤ) : a ≡ a [ZMOD n]

theorem Int.ModEq.rfl {n : ℤ} {a : ℤ} : a ≡ a [ZMOD n]

instance Int.ModEq.instIsReflIntModEq {n : ℤ} : IsRefl ℤ (Int.ModEq n)

Equations

• @Int.ModEq.instIsReflIntModEq = @Int.ModEq.instIsReflIntModEq.proof_1

theorem Int.ModEq.symm {n : ℤ} {a : ℤ} {b : ℤ} : a ≡ b [ZMOD n] → b ≡ a [ZMOD n]

theorem Int.ModEq.trans {n : ℤ} {a : ℤ} {b : ℤ} {c : ℤ} : a ≡ b [ZMOD n] → b ≡ c [ZMOD n] → a ≡ c [ZMOD n]

instance Int.ModEq.instIsTransIntModEq {n : ℤ} : IsTrans ℤ (Int.ModEq n)

Equations

• @Int.ModEq.instIsTransIntModEq = @Int.ModEq.instIsTransIntModEq.proof_1

theorem Int.ModEq.eq {n : ℤ} {a : ℤ} {b : ℤ} : a ≡ b [ZMOD n] → a % n = b % n

theorem Int.modEq_comm {n : ℤ} {a : ℤ} {b : ℤ} : a ≡ b [ZMOD n] ↔ b ≡ a [ZMOD n]

theorem Int.coe_nat_modEq_iff {a : ℕ} {b : ℕ} {n : ℕ} : ↑a ≡ ↑b [ZMOD ↑n] ↔ a ≡ b [MOD n]

theorem Int.modEq_zero_iff_dvd {n : ℤ} {a : ℤ} : a ≡ 0 [ZMOD n] ↔ n ∣ a

theorem Dvd.dvd.modEq_zero_int {n : ℤ} {a : ℤ} (h : n ∣ a) : a ≡ 0 [ZMOD n]

theorem Dvd.dvd.zero_modEq_int {n : ℤ} {a : ℤ} (h : n ∣ a) : 0 ≡ a [ZMOD n]

theorem Int.modEq_iff_dvd {n : ℤ} {a : ℤ} {b : ℤ} : a ≡ b [ZMOD n] ↔ n ∣ b - a

theorem Int.modEq_iff_add_fac {a : ℤ} {b : ℤ} {n : ℤ} : a ≡ b [ZMOD n] ↔ ∃ t, b = a + n * t

theorem Int.modEq_of_dvd {n : ℤ} {a : ℤ} {b : ℤ} : n ∣ b - a → a ≡ b [ZMOD n]

Alias of the reverse direction of Int.modEq_iff_dvd.

theorem Int.ModEq.dvd {n : ℤ} {a : ℤ} {b : ℤ} : a ≡ b [ZMOD n] → n ∣ b - a

Alias of the forward direction of Int.modEq_iff_dvd.

theorem Int.mod_modEq (a : ℤ) (n : ℤ) : a % n ≡ a [ZMOD n]

@[simp]

theorem Int.neg_modEq_neg {n : ℤ} {a : ℤ} {b : ℤ} : -a ≡ -b [ZMOD n] ↔ a ≡ b [ZMOD n]

@[simp]

theorem Int.modEq_neg {n : ℤ} {a : ℤ} {b : ℤ} : a ≡ b [ZMOD -n] ↔ a ≡ b [ZMOD n]

theorem Int.ModEq.of_dvd {m : ℤ} {n : ℤ} {a : ℤ} {b : ℤ} (d : m ∣ n) (h : a ≡ b [ZMOD n]) : a ≡ b [ZMOD m]

theorem Int.ModEq.mul_left' {n : ℤ} {a : ℤ} {b : ℤ} {c : ℤ} (h : a ≡ b [ZMOD n]) : c * a ≡ c * b [ZMOD c * n]

theorem Int.ModEq.mul_right' {n : ℤ} {a : ℤ} {b : ℤ} {c : ℤ} (h : a ≡ b [ZMOD n]) : a * c ≡ b * c [ZMOD n * c]

theorem Int.ModEq.add {n : ℤ} {a : ℤ} {b : ℤ} {c : ℤ} {d : ℤ} (h₁ : a ≡ b [ZMOD n]) (h₂ : c ≡ d [ZMOD n]) : a + c ≡ b + d [ZMOD n]

theorem Int.ModEq.add_left {n : ℤ} {a : ℤ} {b : ℤ} (c : ℤ) (h : a ≡ b [ZMOD n]) : c + a ≡ c + b [ZMOD n]

theorem Int.ModEq.add_right {n : ℤ} {a : ℤ} {b : ℤ} (c : ℤ) (h : a ≡ b [ZMOD n]) : a + c ≡ b + c [ZMOD n]

theorem Int.ModEq.add_left_cancel {n : ℤ} {a : ℤ} {b : ℤ} {c : ℤ} {d : ℤ} (h₁ : a ≡ b [ZMOD n]) (h₂ : a + c ≡ b + d [ZMOD n]) : c ≡ d [ZMOD n]

theorem Int.ModEq.add_left_cancel' {n : ℤ} {a : ℤ} {b : ℤ} (c : ℤ) (h : c + a ≡ c + b [ZMOD n]) : a ≡ b [ZMOD n]

theorem Int.ModEq.add_right_cancel {n : ℤ} {a : ℤ} {b : ℤ} {c : ℤ} {d : ℤ} (h₁ : c ≡ d [ZMOD n]) (h₂ : a + c ≡ b + d [ZMOD n]) : a ≡ b [ZMOD n]

theorem Int.ModEq.add_right_cancel' {n : ℤ} {a : ℤ} {b : ℤ} (c : ℤ) (h : a + c ≡ b + c [ZMOD n]) : a ≡ b [ZMOD n]

theorem Int.ModEq.neg {n : ℤ} {a : ℤ} {b : ℤ} (h : a ≡ b [ZMOD n]) : -a ≡ -b [ZMOD n]

theorem Int.ModEq.sub {n : ℤ} {a : ℤ} {b : ℤ} {c : ℤ} {d : ℤ} (h₁ : a ≡ b [ZMOD n]) (h₂ : c ≡ d [ZMOD n]) : a - c ≡ b - d [ZMOD n]

theorem Int.ModEq.sub_left {n : ℤ} {a : ℤ} {b : ℤ} (c : ℤ) (h : a ≡ b [ZMOD n]) : c - a ≡ c - b [ZMOD n]

theorem Int.ModEq.sub_right {n : ℤ} {a : ℤ} {b : ℤ} (c : ℤ) (h : a ≡ b [ZMOD n]) : a - c ≡ b - c [ZMOD n]

theorem Int.ModEq.mul_left {n : ℤ} {a : ℤ} {b : ℤ} (c : ℤ) (h : a ≡ b [ZMOD n]) : c * a ≡ c * b [ZMOD n]

theorem Int.ModEq.mul_right {n : ℤ} {a : ℤ} {b : ℤ} (c : ℤ) (h : a ≡ b [ZMOD n]) : a * c ≡ b * c [ZMOD n]

theorem Int.ModEq.mul {n : ℤ} {a : ℤ} {b : ℤ} {c : ℤ} {d : ℤ} (h₁ : a ≡ b [ZMOD n]) (h₂ : c ≡ d [ZMOD n]) : a * c ≡ b * d [ZMOD n]

theorem Int.ModEq.pow {n : ℤ} {a : ℤ} {b : ℤ} (m : ℕ) (h : a ≡ b [ZMOD n]) : a ^ m ≡ b ^ m [ZMOD n]

theorem Int.ModEq.of_mul_left {n : ℤ} {a : ℤ} {b : ℤ} (m : ℤ) (h : a ≡ b [ZMOD m * n]) : a ≡ b [ZMOD n]

theorem Int.ModEq.of_mul_right {n : ℤ} {a : ℤ} {b : ℤ} (m : ℤ) : a ≡ b [ZMOD n * m] → a ≡ b [ZMOD n]

theorem Int.ModEq.cancel_right_div_gcd {m : ℤ} {a : ℤ} {b : ℤ} {c : ℤ} (hm : 0 < m) (h : a * c ≡ b * c [ZMOD m]) : a ≡ b [ZMOD m / ↑(Int.gcd m c)]

To cancel a common factor c from a ModEq we must divide the modulus m by gcd m c.

theorem Int.ModEq.cancel_left_div_gcd {m : ℤ} {a : ℤ} {b : ℤ} {c : ℤ} (hm : 0 < m) (h : c * a ≡ c * b [ZMOD m]) : a ≡ b [ZMOD m / ↑(Int.gcd m c)]

To cancel a common factor c from a ModEq we must divide the modulus m by gcd m c.

theorem Int.ModEq.of_div {m : ℤ} {a : ℤ} {b : ℤ} {c : ℤ} (h : a / c ≡ b / c [ZMOD m / c]) (ha : c ∣ a) (ha : c ∣ b) (ha : c ∣ m) : a ≡ b [ZMOD m]

theorem Int.modEq_one {a : ℤ} {b : ℤ} : a ≡ b [ZMOD 1]

theorem Int.modEq_sub (a : ℤ) (b : ℤ) : a ≡ b [ZMOD a - b]

@[simp]

theorem Int.modEq_zero_iff {a : ℤ} {b : ℤ} : a ≡ b [ZMOD 0] ↔ a = b

@[simp]

theorem Int.add_modEq_left {n : ℤ} {a : ℤ} : n + a ≡ a [ZMOD n]

@[simp]

theorem Int.add_modEq_right {n : ℤ} {a : ℤ} : a + n ≡ a [ZMOD n]

theorem Int.modEq_and_modEq_iff_modEq_mul {a : ℤ} {b : ℤ} {m : ℤ} {n : ℤ} (hmn : Nat.Coprime (Int.natAbs m) (Int.natAbs n)) : a ≡ b [ZMOD m] ∧ a ≡ b [ZMOD n] ↔ a ≡ b [ZMOD m * n]

theorem Int.gcd_a_modEq (a : ℕ) (b : ℕ) : ↑a * Nat.gcdA a b ≡ ↑(Nat.gcd a b) [ZMOD ↑b]

theorem Int.modEq_add_fac {a : ℤ} {b : ℤ} {n : ℤ} (c : ℤ) (ha : a ≡ b [ZMOD n]) : a + n * c ≡ b [ZMOD n]

theorem Int.modEq_sub_fac {a : ℤ} {b : ℤ} {n : ℤ} (c : ℤ) (ha : a ≡ b [ZMOD n]) : a - n * c ≡ b [ZMOD n]

theorem Int.modEq_add_fac_self {a : ℤ} {t : ℤ} {n : ℤ} : a + n * t ≡ a [ZMOD n]

theorem Int.mod_coprime {a : ℕ} {b : ℕ} (hab : Nat.Coprime a b) : ∃ y, ↑a * y ≡ 1 [ZMOD ↑b]

theorem Int.exists_unique_equiv (a : ℤ) {b : ℤ} (hb : 0 < b) : ∃ z, 0 ≤ z ∧ z < b ∧ z ≡ a [ZMOD b]

theorem Int.exists_unique_equiv_nat (a : ℤ) {b : ℤ} (hb : 0 < b) : ∃ z, ↑z < b ∧ ↑z ≡ a [ZMOD b]

theorem Int.mod_mul_right_mod (a : ℤ) (b : ℤ) (c : ℤ) : a % (b * c) % b = a % b

theorem Int.mod_mul_left_mod (a : ℤ) (b : ℤ) (c : ℤ) : a % (b * c) % c = a % c