Associated, prime, and irreducible elements.

In this file we define the predicate Prime p saying that an element of a commutative monoid with zero is prime. Namely, Prime p means that p isn't zero, it isn't a unit, and p ∣ a * b → p ∣ a ∨ p ∣ b for all a, b;

In decomposition monoids (e.g., ℕ, ℤ), this predicate is equivalent to Irreducible, however this is not true in general.

We also define an equivalence relation Associated saying that two elements of a monoid differ by a multiplication by a unit. Then we show that the quotient type Associates is a monoid and prove basic properties of this quotient.

def Prime {M : Type u_1} [CommMonoidWithZero M] (p : M) : Prop

An element p of a commutative monoid with zero (e.g., a ring) is called prime, if it's not zero, not a unit, and p ∣ a * b → p ∣ a ∨ p ∣ b for all a, b.

Equations

• Prime p = (p ≠ 0 ∧ ¬IsUnit p ∧ ∀ (a b : M), p ∣ a * b → p ∣ a ∨ p ∣ b)

theorem Prime.ne_zero {M : Type u_1} [CommMonoidWithZero M] {p : M} (hp : Prime p) : p ≠ 0

theorem Prime.not_unit {M : Type u_1} [CommMonoidWithZero M] {p : M} (hp : Prime p) : ¬IsUnit p

theorem Prime.not_dvd_one {M : Type u_1} [CommMonoidWithZero M] {p : M} (hp : Prime p) : ¬p ∣ 1

theorem Prime.ne_one {M : Type u_1} [CommMonoidWithZero M] {p : M} (hp : Prime p) : p ≠ 1

theorem Prime.dvd_or_dvd {M : Type u_1} [CommMonoidWithZero M] {p : M} (hp : Prime p) {a : M} {b : M} (h : p ∣ a * b) : p ∣ a ∨ p ∣ b

theorem Prime.dvd_mul {M : Type u_1} [CommMonoidWithZero M] {p : M} (hp : Prime p) {a : M} {b : M} : p ∣ a * b ↔ p ∣ a ∨ p ∣ b

theorem Prime.isPrimal {M : Type u_1} [CommMonoidWithZero M] {p : M} (hp : Prime p) : IsPrimal p

theorem Prime.not_dvd_mul {M : Type u_1} [CommMonoidWithZero M] {p : M} (hp : Prime p) {a : M} {b : M} (ha : ¬p ∣ a) (hb : ¬p ∣ b) : ¬p ∣ a * b

theorem Prime.dvd_of_dvd_pow {M : Type u_1} [CommMonoidWithZero M] {p : M} (hp : Prime p) {a : M} {n : ℕ} (h : p ∣ a ^ n) : p ∣ a

theorem Prime.dvd_pow_iff_dvd {M : Type u_1} [CommMonoidWithZero M] {p : M} (hp : Prime p) {a : M} {n : ℕ} (hn : n ≠ 0) : p ∣ a ^ n ↔ p ∣ a

@[simp]

theorem not_prime_zero {M : Type u_1} [CommMonoidWithZero M] : ¬Prime 0

@[simp]

theorem not_prime_one {M : Type u_1} [CommMonoidWithZero M] : ¬Prime 1

theorem comap_prime {M : Type u_1} {N : Type u_2} [CommMonoidWithZero M] [CommMonoidWithZero N] {F : Type u_3} {G : Type u_4} [FunLike F M N] [MonoidWithZeroHomClass F M N] [FunLike G N M] [MulHomClass G N M] (f : F) (g : G) {p : M} (hinv : ∀ (a : M), g (f a) = a) (hp : Prime (f p)) : Prime p

theorem MulEquiv.prime_iff {M : Type u_1} {N : Type u_2} [CommMonoidWithZero M] [CommMonoidWithZero N] {p : M} (e : M ≃* N) : Prime p ↔ Prime (e p)

theorem Prime.left_dvd_or_dvd_right_of_dvd_mul {M : Type u_1} [CancelCommMonoidWithZero M] {p : M} (hp : Prime p) {a : M} {b : M} : a ∣ p * b → p ∣ a ∨ a ∣ b

theorem Prime.pow_dvd_of_dvd_mul_left {M : Type u_1} [CancelCommMonoidWithZero M] {p : M} {a : M} {b : M} (hp : Prime p) (n : ℕ) (h : ¬p ∣ a) (h' : p ^ n ∣ a * b) : p ^ n ∣ b

theorem Prime.pow_dvd_of_dvd_mul_right {M : Type u_1} [CancelCommMonoidWithZero M] {p : M} {a : M} {b : M} (hp : Prime p) (n : ℕ) (h : ¬p ∣ b) (h' : p ^ n ∣ a * b) : p ^ n ∣ a

theorem Prime.dvd_of_pow_dvd_pow_mul_pow_of_square_not_dvd {M : Type u_1} [CancelCommMonoidWithZero M] {p : M} {a : M} {b : M} {n : ℕ} (hp : Prime p) (hpow : p ^ n.succ ∣ a ^ n.succ * b ^ n) (hb : ¬p ^ 2 ∣ b) : p ∣ a

theorem prime_pow_succ_dvd_mul {M : Type u_3} [CancelCommMonoidWithZero M] {p : M} {x : M} {y : M} (h : Prime p) {i : ℕ} (hxy : p ^ (i + 1) ∣ x * y) : p ^ (i + 1) ∣ x ∨ p ∣ y

structure Irreducible {M : Type u_1} [Monoid M] (p : M) : Prop

Irreducible p states that p is non-unit and only factors into units.

We explicitly avoid stating that p is non-zero, this would require a semiring. Assuming only a monoid allows us to reuse irreducible for associated elements.

• not_unit : ¬IsUnit p

  p is not a unit

• isUnit_or_isUnit' : ∀ (a b : M), p = a * b → IsUnit a ∨ IsUnit b

  if p factors then one factor is a unit

theorem Irreducible.not_unit {M : Type u_1} [Monoid M] {p : M} (self : Irreducible p) : ¬IsUnit p

p is not a unit

theorem Irreducible.isUnit_or_isUnit' {M : Type u_1} [Monoid M] {p : M} (self : Irreducible p) (a : M) (b : M) : p = a * b → IsUnit a ∨ IsUnit b

if p factors then one factor is a unit

theorem Irreducible.not_dvd_one {M : Type u_1} [CommMonoid M] {p : M} (hp : Irreducible p) : ¬p ∣ 1

theorem Irreducible.isUnit_or_isUnit {M : Type u_1} [Monoid M] {p : M} (hp : Irreducible p) {a : M} {b : M} (h : p = a * b) : IsUnit a ∨ IsUnit b

theorem irreducible_iff {M : Type u_1} [Monoid M] {p : M} : Irreducible p ↔ ¬IsUnit p ∧ ∀ (a b : M), p = a * b → IsUnit a ∨ IsUnit b

@[simp]

theorem not_irreducible_one {M : Type u_1} [Monoid M] : ¬Irreducible 1

theorem Irreducible.ne_one {M : Type u_1} [Monoid M] {p : M} : Irreducible p → p ≠ 1

@[simp]

theorem not_irreducible_zero {M : Type u_1} [MonoidWithZero M] : ¬Irreducible 0

theorem Irreducible.ne_zero {M : Type u_1} [MonoidWithZero M] {p : M} : Irreducible p → p ≠ 0

theorem of_irreducible_mul {M : Type u_3} [Monoid M] {x : M} {y : M} : Irreducible (x * y) → IsUnit x ∨ IsUnit y

theorem not_irreducible_pow {M : Type u_3} [Monoid M] {x : M} {n : ℕ} (hn : n ≠ 1) : ¬Irreducible (x ^ n)

theorem irreducible_or_factor {M : Type u_3} [Monoid M] (x : M) (h : ¬IsUnit x) : Irreducible x ∨ ∃ (a : M), ∃ (b : M), ¬IsUnit a ∧ ¬IsUnit b ∧ a * b = x

theorem Irreducible.dvd_symm {M : Type u_1} [Monoid M] {p : M} {q : M} (hp : Irreducible p) (hq : Irreducible q) : p ∣ q → q ∣ p

If p and q are irreducible, then p ∣ q implies q ∣ p.

theorem Irreducible.dvd_comm {M : Type u_1} [Monoid M] {p : M} {q : M} (hp : Irreducible p) (hq : Irreducible q) : p ∣ q ↔ q ∣ p

theorem Irreducible.of_map {M : Type u_1} {N : Type u_2} {F : Type u_3} [Monoid M] [Monoid N] [FunLike F M N] [MonoidHomClass F M N] {f : F} [IsLocalHom f] {x : M} (hfx : Irreducible (f x)) : Irreducible x

theorem irreducible_units_mul {M : Type u_1} [Monoid M] (a : Mˣ) (b : M) : Irreducible (↑a * b) ↔ Irreducible b

theorem irreducible_isUnit_mul {M : Type u_1} [Monoid M] {a : M} {b : M} (h : IsUnit a) : Irreducible (a * b) ↔ Irreducible b

theorem irreducible_mul_units {M : Type u_1} [Monoid M] (a : Mˣ) (b : M) : Irreducible (b * ↑a) ↔ Irreducible b

theorem irreducible_mul_isUnit {M : Type u_1} [Monoid M] {a : M} {b : M} (h : IsUnit a) : Irreducible (b * a) ↔ Irreducible b

theorem irreducible_mul_iff {M : Type u_1} [Monoid M] {a : M} {b : M} : Irreducible (a * b) ↔ Irreducible a ∧ IsUnit b ∨ Irreducible b ∧ IsUnit a

theorem Irreducible.map {M : Type u_1} {N : Type u_2} [Monoid M] [Monoid N] {F : Type u_3} [EquivLike F M N] [MulEquivClass F M N] (f : F) {x : M} (h : Irreducible x) : Irreducible (f x)

Irreducibility is preserved by multiplicative equivalences. Note that surjective + local hom is not enough. Consider the additive monoids M = ℕ ⊕ ℕ, N = ℕ, with a surjective local (additive) hom f : M →+ N sending (m, n) to 2m + n. It is local because the only add unit in N is 0, with preimage {(0, 0)} also an add unit. Then x = (1, 0) is irreducible in M, but f x = 2 = 1 + 1 is not irreducible in N.

theorem MulEquiv.irreducible_iff {M : Type u_1} {N : Type u_2} [Monoid M] [Monoid N] {F : Type u_3} [EquivLike F M N] [MulEquivClass F M N] (f : F) {a : M} : Irreducible (f a) ↔ Irreducible a

theorem Irreducible.not_square {M : Type u_1} [CommMonoid M] {a : M} (ha : Irreducible a) : ¬IsSquare a

theorem IsSquare.not_irreducible {M : Type u_1} [CommMonoid M] {a : M} (ha : IsSquare a) : ¬Irreducible a

theorem Irreducible.prime_of_isPrimal {M : Type u_1} [CommMonoidWithZero M] {a : M} (irr : Irreducible a) (primal : IsPrimal a) : Prime a

theorem Irreducible.prime {M : Type u_1} [CommMonoidWithZero M] [DecompositionMonoid M] {a : M} (irr : Irreducible a) : Prime a

theorem Prime.irreducible {M : Type u_1} [CancelCommMonoidWithZero M] {p : M} (hp : Prime p) : Irreducible p

theorem irreducible_iff_prime {M : Type u_1} [CancelCommMonoidWithZero M] [DecompositionMonoid M] {a : M} : Irreducible a ↔ Prime a

theorem succ_dvd_or_succ_dvd_of_succ_sum_dvd_mul {M : Type u_1} [CancelCommMonoidWithZero M] {p : M} (hp : Prime p) {a : M} {b : M} {k : ℕ} {l : ℕ} : p ^ k ∣ a → p ^ l ∣ b → p ^ (k + l + 1) ∣ a * b → p ^ (k + 1) ∣ a ∨ p ^ (l + 1) ∣ b

theorem Prime.not_square {M : Type u_1} [CancelCommMonoidWithZero M] {p : M} (hp : Prime p) : ¬IsSquare p

theorem IsSquare.not_prime {M : Type u_1} [CancelCommMonoidWithZero M] {a : M} (ha : IsSquare a) : ¬Prime a

theorem not_prime_pow {M : Type u_1} [CancelCommMonoidWithZero M] {a : M} {n : ℕ} (hn : n ≠ 1) : ¬Prime (a ^ n)

def Associated {M : Type u_1} [Monoid M] (x : M) (y : M) : Prop

Two elements of a Monoid are Associated if one of them is another one multiplied by a unit on the right.

Equations

• Associated x y = ∃ (u : Mˣ), x * ↑u = y

theorem Associated.refl {M : Type u_1} [Monoid M] (x : M) : Associated x x

theorem Associated.rfl {M : Type u_1} [Monoid M] {x : M} : Associated x x

instance Associated.instIsRefl {M : Type u_1} [Monoid M] : IsRefl M Associated

Equations

• ⋯ = ⋯

theorem Associated.symm {M : Type u_1} [Monoid M] {x : M} {y : M} : Associated x y → Associated y x

instance Associated.instIsSymm {M : Type u_1} [Monoid M] : IsSymm M Associated

Equations

• ⋯ = ⋯

theorem Associated.comm {M : Type u_1} [Monoid M] {x : M} {y : M} : Associated x y ↔ Associated y x

theorem Associated.trans {M : Type u_1} [Monoid M] {x : M} {y : M} {z : M} : Associated x y → Associated y z → Associated x z

instance Associated.instIsTrans {M : Type u_1} [Monoid M] : IsTrans M Associated

Equations

• ⋯ = ⋯

def Associated.setoid (M : Type u_3) [Monoid M] : Setoid M

The setoid of the relation x ~ᵤ y iff there is a unit u such that x * u = y

Equations

• Associated.setoid M = { r := Associated, iseqv := ⋯ }

theorem Associated.map {M : Type u_3} {N : Type u_4} [Monoid M] [Monoid N] {F : Type u_5} [FunLike F M N] [MonoidHomClass F M N] (f : F) {x : M} {y : M} (ha : Associated x y) : Associated (f x) (f y)

theorem unit_associated_one {M : Type u_1} [Monoid M] {u : Mˣ} : Associated (↑u) 1

@[simp]

theorem associated_one_iff_isUnit {M : Type u_1} [Monoid M] {a : M} : Associated a 1 ↔ IsUnit a

@[simp]

theorem associated_zero_iff_eq_zero {M : Type u_1} [MonoidWithZero M] (a : M) : Associated a 0 ↔ a = 0

theorem associated_one_of_mul_eq_one {M : Type u_1} [CommMonoid M] {a : M} (b : M) (hab : a * b = 1) : Associated a 1

theorem associated_one_of_associated_mul_one {M : Type u_1} [CommMonoid M] {a : M} {b : M} : Associated (a * b) 1 → Associated a 1

theorem associated_mul_unit_left {N : Type u_3} [Monoid N] (a : N) (u : N) (hu : IsUnit u) : Associated (a * u) a

theorem associated_unit_mul_left {N : Type u_3} [CommMonoid N] (a : N) (u : N) (hu : IsUnit u) : Associated (u * a) a

theorem associated_mul_unit_right {N : Type u_3} [Monoid N] (a : N) (u : N) (hu : IsUnit u) : Associated a (a * u)

theorem associated_unit_mul_right {N : Type u_3} [CommMonoid N] (a : N) (u : N) (hu : IsUnit u) : Associated a (u * a)

theorem associated_mul_isUnit_left_iff {N : Type u_3} [Monoid N] {a : N} {u : N} {b : N} (hu : IsUnit u) : Associated (a * u) b ↔ Associated a b

theorem associated_isUnit_mul_left_iff {N : Type u_3} [CommMonoid N] {u : N} {a : N} {b : N} (hu : IsUnit u) : Associated (u * a) b ↔ Associated a b

theorem associated_mul_isUnit_right_iff {N : Type u_3} [Monoid N] {a : N} {b : N} {u : N} (hu : IsUnit u) : Associated a (b * u) ↔ Associated a b

theorem associated_isUnit_mul_right_iff {N : Type u_3} [CommMonoid N] {a : N} {u : N} {b : N} (hu : IsUnit u) : Associated a (u * b) ↔ Associated a b

@[simp]

theorem associated_mul_unit_left_iff {N : Type u_3} [Monoid N] {a : N} {b : N} {u : Nˣ} : Associated (a * ↑u) b ↔ Associated a b

@[simp]

theorem associated_unit_mul_left_iff {N : Type u_3} [CommMonoid N] {a : N} {b : N} {u : Nˣ} : Associated (↑u * a) b ↔ Associated a b

@[simp]

theorem associated_mul_unit_right_iff {N : Type u_3} [Monoid N] {a : N} {b : N} {u : Nˣ} : Associated a (b * ↑u) ↔ Associated a b

@[simp]

theorem associated_unit_mul_right_iff {N : Type u_3} [CommMonoid N] {a : N} {b : N} {u : Nˣ} : Associated a (↑u * b) ↔ Associated a b

theorem Associated.mul_left {M : Type u_1} [Monoid M] (a : M) {b : M} {c : M} (h : Associated b c) : Associated (a * b) (a * c)

theorem Associated.mul_right {M : Type u_1} [CommMonoid M] {a : M} {b : M} (h : Associated a b) (c : M) : Associated (a * c) (b * c)

theorem Associated.mul_mul {M : Type u_1} [CommMonoid M] {a₁ : M} {a₂ : M} {b₁ : M} {b₂ : M} (h₁ : Associated a₁ b₁) (h₂ : Associated a₂ b₂) : Associated (a₁ * a₂) (b₁ * b₂)

theorem Associated.pow_pow {M : Type u_1} [CommMonoid M] {a : M} {b : M} {n : ℕ} (h : Associated a b) : Associated (a ^ n) (b ^ n)

theorem Associated.dvd {M : Type u_1} [Monoid M] {a : M} {b : M} : Associated a b → a ∣ b

theorem Associated.dvd' {M : Type u_1} [Monoid M] {a : M} {b : M} (h : Associated a b) : b ∣ a

theorem Associated.dvd_dvd {M : Type u_1} [Monoid M] {a : M} {b : M} (h : Associated a b) : a ∣ b ∧ b ∣ a

theorem associated_of_dvd_dvd {M : Type u_1} [CancelMonoidWithZero M] {a : M} {b : M} (hab : a ∣ b) (hba : b ∣ a) : Associated a b

theorem dvd_dvd_iff_associated {M : Type u_1} [CancelMonoidWithZero M] {a : M} {b : M} : a ∣ b ∧ b ∣ a ↔ Associated a b

instance instDecidableRelAssociatedOfDvd {M : Type u_1} [CancelMonoidWithZero M] [DecidableRel fun (x1 x2 : M) => x1 ∣ x2] : DecidableRel fun (x1 x2 : M) => Associated x1 x2

Equations

• instDecidableRelAssociatedOfDvd x✝ x = decidable_of_iff (x✝ ∣ x ∧ x ∣ x✝) ⋯

theorem Associated.dvd_iff_dvd_left {M : Type u_1} [Monoid M] {a : M} {b : M} {c : M} (h : Associated a b) : a ∣ c ↔ b ∣ c

theorem Associated.dvd_iff_dvd_right {M : Type u_1} [Monoid M] {a : M} {b : M} {c : M} (h : Associated b c) : a ∣ b ↔ a ∣ c

theorem Associated.eq_zero_iff {M : Type u_1} [MonoidWithZero M] {a : M} {b : M} (h : Associated a b) : a = 0 ↔ b = 0

theorem Associated.ne_zero_iff {M : Type u_1} [MonoidWithZero M] {a : M} {b : M} (h : Associated a b) : a ≠ 0 ↔ b ≠ 0

theorem Associated.neg_left {M : Type u_1} [Monoid M] [HasDistribNeg M] {a : M} {b : M} (h : Associated a b) : Associated (-a) b

theorem Associated.neg_right {M : Type u_1} [Monoid M] [HasDistribNeg M] {a : M} {b : M} (h : Associated a b) : Associated a (-b)

theorem Associated.neg_neg {M : Type u_1} [Monoid M] [HasDistribNeg M] {a : M} {b : M} (h : Associated a b) : Associated (-a) (-b)

theorem Associated.prime {M : Type u_1} [CommMonoidWithZero M] {p : M} {q : M} (h : Associated p q) (hp : Prime p) : Prime q

theorem prime_mul_iff {M : Type u_1} [CancelCommMonoidWithZero M] {x : M} {y : M} : Prime (x * y) ↔ Prime x ∧ IsUnit y ∨ IsUnit x ∧ Prime y

@[simp]

theorem prime_pow_iff {M : Type u_1} [CancelCommMonoidWithZero M] {p : M} {n : ℕ} : Prime (p ^ n) ↔ Prime p ∧ n = 1

theorem Irreducible.dvd_iff {M : Type u_1} [Monoid M] {x : M} {y : M} (hx : Irreducible x) : y ∣ x ↔ IsUnit y ∨ Associated x y

theorem Irreducible.associated_of_dvd {M : Type u_1} [Monoid M] {p : M} {q : M} (p_irr : Irreducible p) (q_irr : Irreducible q) (dvd : p ∣ q) : Associated p q

theorem Irreducible.dvd_irreducible_iff_associated {M : Type u_1} [Monoid M] {p : M} {q : M} (pp : Irreducible p) (qp : Irreducible q) : p ∣ q ↔ Associated p q

theorem Prime.associated_of_dvd {M : Type u_1} [CancelCommMonoidWithZero M] {p : M} {q : M} (p_prime : Prime p) (q_prime : Prime q) (dvd : p ∣ q) : Associated p q

theorem Prime.dvd_prime_iff_associated {M : Type u_1} [CancelCommMonoidWithZero M] {p : M} {q : M} (pp : Prime p) (qp : Prime q) : p ∣ q ↔ Associated p q

theorem Associated.prime_iff {M : Type u_1} [CommMonoidWithZero M] {p : M} {q : M} (h : Associated p q) : Prime p ↔ Prime q

theorem Associated.isUnit {M : Type u_1} [Monoid M] {a : M} {b : M} (h : Associated a b) : IsUnit a → IsUnit b

theorem Associated.isUnit_iff {M : Type u_1} [Monoid M] {a : M} {b : M} (h : Associated a b) : IsUnit a ↔ IsUnit b

theorem Irreducible.isUnit_iff_not_associated_of_dvd {M : Type u_1} [Monoid M] {x : M} {y : M} (hx : Irreducible x) (hy : y ∣ x) : IsUnit y ↔ ¬Associated x y

theorem Associated.irreducible {M : Type u_1} [Monoid M] {p : M} {q : M} (h : Associated p q) (hp : Irreducible p) : Irreducible q

theorem Associated.irreducible_iff {M : Type u_1} [Monoid M] {p : M} {q : M} (h : Associated p q) : Irreducible p ↔ Irreducible q

theorem Associated.of_mul_left {M : Type u_1} [CancelCommMonoidWithZero M] {a : M} {b : M} {c : M} {d : M} (h : Associated (a * b) (c * d)) (h₁ : Associated a c) (ha : a ≠ 0) : Associated b d

theorem Associated.of_mul_right {M : Type u_1} [CancelCommMonoidWithZero M] {a : M} {b : M} {c : M} {d : M} : Associated (a * b) (c * d) → Associated b d → b ≠ 0 → Associated a c

theorem Associated.of_pow_associated_of_prime {M : Type u_1} [CancelCommMonoidWithZero M] {p₁ : M} {p₂ : M} {k₁ : ℕ} {k₂ : ℕ} (hp₁ : Prime p₁) (hp₂ : Prime p₂) (hk₁ : 0 < k₁) (h : Associated (p₁ ^ k₁) (p₂ ^ k₂)) : Associated p₁ p₂

theorem Associated.of_pow_associated_of_prime' {M : Type u_1} [CancelCommMonoidWithZero M] {p₁ : M} {p₂ : M} {k₁ : ℕ} {k₂ : ℕ} (hp₁ : Prime p₁) (hp₂ : Prime p₂) (hk₂ : 0 < k₂) (h : Associated (p₁ ^ k₁) (p₂ ^ k₂)) : Associated p₁ p₂

theorem Irreducible.isRelPrime_iff_not_dvd {M : Type u_1} [Monoid M] {p : M} {n : M} (hp : Irreducible p) : IsRelPrime p n ↔ ¬p ∣ n

See also Irreducible.coprime_iff_not_dvd.

theorem Irreducible.dvd_or_isRelPrime {M : Type u_1} [Monoid M] {p : M} {n : M} (hp : Irreducible p) : p ∣ n ∨ IsRelPrime p n

theorem associated_iff_eq {M : Type u_1} [Monoid M] [Subsingleton Mˣ] {x : M} {y : M} : Associated x y ↔ x = y

theorem associated_eq_eq {M : Type u_1} [Monoid M] [Subsingleton Mˣ] : Associated = Eq

theorem prime_dvd_prime_iff_eq {M : Type u_3} [CancelCommMonoidWithZero M] [Subsingleton Mˣ] {p : M} {q : M} (pp : Prime p) (qp : Prime q) : p ∣ q ↔ p = q

theorem eq_of_prime_pow_eq {R : Type u_3} [CancelCommMonoidWithZero R] [Subsingleton Rˣ] {p₁ : R} {p₂ : R} {k₁ : ℕ} {k₂ : ℕ} (hp₁ : Prime p₁) (hp₂ : Prime p₂) (hk₁ : 0 < k₁) (h : p₁ ^ k₁ = p₂ ^ k₂) : p₁ = p₂

theorem eq_of_prime_pow_eq' {R : Type u_3} [CancelCommMonoidWithZero R] [Subsingleton Rˣ] {p₁ : R} {p₂ : R} {k₁ : ℕ} {k₂ : ℕ} (hp₁ : Prime p₁) (hp₂ : Prime p₂) (hk₁ : 0 < k₂) (h : p₁ ^ k₁ = p₂ ^ k₂) : p₁ = p₂

@[reducible, inline]

abbrev Associates (M : Type u_3) [Monoid M] : Type u_3

The quotient of a monoid by the Associated relation. Two elements x and y are associated iff there is a unit u such that x * u = y. There is a natural monoid structure on Associates M.

Equations

• Associates M = Quotient (Associated.setoid M)

@[reducible, inline]

abbrev Associates.mk {M : Type u_3} [Monoid M] (a : M) : Associates M

The canonical quotient map from a monoid M into the Associates of M

Equations

• Associates.mk a = ⟦a⟧

instance Associates.instInhabited {M : Type u_1} [Monoid M] : Inhabited (Associates M)

Equations

• Associates.instInhabited = { default := ⟦1⟧ }

theorem Associates.mk_eq_mk_iff_associated {M : Type u_1} [Monoid M] {a : M} {b : M} : Associates.mk a = Associates.mk b ↔ Associated a b

theorem Associates.quotient_mk_eq_mk {M : Type u_1} [Monoid M] (a : M) : ⟦a⟧ = Associates.mk a

theorem Associates.quot_mk_eq_mk {M : Type u_1} [Monoid M] (a : M) : Quot.mk (⇑(Associated.setoid M)) a = Associates.mk a

@[simp]

theorem Associates.quot_out {M : Type u_1} [Monoid M] (a : Associates M) : Associates.mk (Quot.out a) = a

theorem Associates.mk_quot_out {M : Type u_1} [Monoid M] (a : M) : Associated (Quot.out (Associates.mk a)) a

theorem Associates.forall_associated {M : Type u_1} [Monoid M] {p : Associates M → Prop} : (∀ (a : Associates M), p a) ↔ ∀ (a : M), p (Associates.mk a)

theorem Associates.mk_surjective {M : Type u_1} [Monoid M] : Function.Surjective Associates.mk

instance Associates.instOne {M : Type u_1} [Monoid M] : One (Associates M)

Equations

• Associates.instOne = { one := ⟦1⟧ }

@[simp]

theorem Associates.mk_one {M : Type u_1} [Monoid M] : Associates.mk 1 = 1

theorem Associates.one_eq_mk_one {M : Type u_1} [Monoid M] : 1 = Associates.mk 1

@[simp]

theorem Associates.mk_eq_one {M : Type u_1} [Monoid M] {a : M} : Associates.mk a = 1 ↔ IsUnit a

instance Associates.instBot {M : Type u_1} [Monoid M] : Bot (Associates M)

Equations

• Associates.instBot = { bot := 1 }

theorem Associates.bot_eq_one {M : Type u_1} [Monoid M] : ⊥ = 1

theorem Associates.exists_rep {M : Type u_1} [Monoid M] (a : Associates M) : ∃ (a0 : M), Associates.mk a0 = a

instance Associates.instUniqueOfSubsingleton {M : Type u_1} [Monoid M] [Subsingleton M] : Unique (Associates M)

Equations

• Associates.instUniqueOfSubsingleton = { default := 1, uniq := ⋯ }

theorem Associates.mk_injective {M : Type u_1} [Monoid M] [Subsingleton Mˣ] : Function.Injective Associates.mk

instance Associates.instMul {M : Type u_1} [CommMonoid M] : Mul (Associates M)

Equations

• Associates.instMul = { mul := Quotient.map₂ (fun (x1 x2 : M) => x1 * x2) ⋯ }

theorem Associates.mk_mul_mk {M : Type u_1} [CommMonoid M] {x : M} {y : M} : Associates.mk x * Associates.mk y = Associates.mk (x * y)

instance Associates.instCommMonoid {M : Type u_1} [CommMonoid M] : CommMonoid (Associates M)

Equations

• Associates.instCommMonoid = CommMonoid.mk ⋯

instance Associates.instPreorder {M : Type u_1} [CommMonoid M] : Preorder (Associates M)

Equations

• Associates.instPreorder = Preorder.mk ⋯ ⋯ ⋯

def Associates.mkMonoidHom {M : Type u_1} [CommMonoid M] : M →* Associates M

Associates.mk as a MonoidHom.

Equations

• Associates.mkMonoidHom = { toFun := Associates.mk, map_one' := ⋯, map_mul' := ⋯ }

@[simp]

theorem Associates.mkMonoidHom_apply {M : Type u_1} [CommMonoid M] (a : M) : Associates.mkMonoidHom a = Associates.mk a

theorem Associates.associated_map_mk {M : Type u_1} [CommMonoid M] {f : Associates M →* M} (hinv : Function.RightInverse (⇑f) Associates.mk) (a : M) : Associated a (f (Associates.mk a))

theorem Associates.mk_pow {M : Type u_1} [CommMonoid M] (a : M) (n : ℕ) : Associates.mk (a ^ n) = Associates.mk a ^ n

theorem Associates.dvd_eq_le {M : Type u_1} [CommMonoid M] : (fun (x1 x2 : Associates M) => x1 ∣ x2) = fun (x1 x2 : Associates M) => x1 ≤ x2

instance Associates.uniqueUnits {M : Type u_1} [CommMonoid M] : Unique (Associates M)ˣ

Equations

• Associates.uniqueUnits = { toInhabited := Units.instInhabited, uniq := ⋯ }

@[deprecated mul_eq_one]

theorem Associates.mul_eq_one_iff {α : Type u} [CommMonoid α] [Subsingleton αˣ] {a : α} {b : α} : a * b = 1 ↔ a = 1 ∧ b = 1

Alias of mul_eq_one.

@[deprecated Subsingleton.elim]

theorem Associates.units_eq_one {α : Sort u} [h : Subsingleton α] (a : α) (b : α) : a = b

Alias of Subsingleton.elim.

@[simp]

theorem Associates.coe_unit_eq_one {M : Type u_1} [CommMonoid M] (u : (Associates M)ˣ) : ↑u = 1

theorem Associates.isUnit_iff_eq_one {M : Type u_1} [CommMonoid M] (a : Associates M) : IsUnit a ↔ a = 1

theorem Associates.isUnit_iff_eq_bot {M : Type u_1} [CommMonoid M] {a : Associates M} : IsUnit a ↔ a = ⊥

theorem Associates.isUnit_mk {M : Type u_1} [CommMonoid M] {a : M} : IsUnit (Associates.mk a) ↔ IsUnit a

theorem Associates.mul_mono {M : Type u_1} [CommMonoid M] {a : Associates M} {b : Associates M} {c : Associates M} {d : Associates M} (h₁ : a ≤ b) (h₂ : c ≤ d) : a * c ≤ b * d

theorem Associates.one_le {M : Type u_1} [CommMonoid M] {a : Associates M} : 1 ≤ a

theorem Associates.le_mul_right {M : Type u_1} [CommMonoid M] {a : Associates M} {b : Associates M} : a ≤ a * b

theorem Associates.le_mul_left {M : Type u_1} [CommMonoid M] {a : Associates M} {b : Associates M} : a ≤ b * a

instance Associates.instOrderBot {M : Type u_1} [CommMonoid M] : OrderBot (Associates M)

Equations

• Associates.instOrderBot = OrderBot.mk ⋯

@[simp]

theorem Associates.mk_dvd_mk {M : Type u_1} [CommMonoid M] {a : M} {b : M} : Associates.mk a ∣ Associates.mk b ↔ a ∣ b

theorem Associates.dvd_of_mk_le_mk {M : Type u_1} [CommMonoid M] {a : M} {b : M} : Associates.mk a ≤ Associates.mk b → a ∣ b

theorem Associates.mk_le_mk_of_dvd {M : Type u_1} [CommMonoid M] {a : M} {b : M} : a ∣ b → Associates.mk a ≤ Associates.mk b

theorem Associates.mk_le_mk_iff_dvd {M : Type u_1} [CommMonoid M] {a : M} {b : M} : Associates.mk a ≤ Associates.mk b ↔ a ∣ b

@[deprecated Associates.mk_le_mk_iff_dvd]

theorem Associates.mk_le_mk_iff_dvd_iff {M : Type u_1} [CommMonoid M] {a : M} {b : M} : Associates.mk a ≤ Associates.mk b ↔ a ∣ b

Alias of Associates.mk_le_mk_iff_dvd.

@[simp]

theorem Associates.isPrimal_mk {M : Type u_1} [CommMonoid M] {a : M} : IsPrimal (Associates.mk a) ↔ IsPrimal a

@[deprecated Associates.isPrimal_mk]

theorem Associates.isPrimal_iff {M : Type u_1} [CommMonoid M] {a : M} : IsPrimal (Associates.mk a) ↔ IsPrimal a

Alias of Associates.isPrimal_mk.

@[simp]

theorem Associates.decompositionMonoid_iff {M : Type u_1} [CommMonoid M] : DecompositionMonoid (Associates M) ↔ DecompositionMonoid M

instance Associates.instDecompositionMonoid {M : Type u_1} [CommMonoid M] [DecompositionMonoid M] : DecompositionMonoid (Associates M)

Equations

• ⋯ = ⋯

@[simp]

theorem Associates.mk_isRelPrime_iff {M : Type u_1} [CommMonoid M] {a : M} {b : M} : IsRelPrime (Associates.mk a) (Associates.mk b) ↔ IsRelPrime a b

instance Associates.instZero {M : Type u_1} [Zero M] [Monoid M] : Zero (Associates M)

Equations

• Associates.instZero = { zero := ⟦0⟧ }

instance Associates.instTopOfZero {M : Type u_1} [Zero M] [Monoid M] : Top (Associates M)

Equations

• Associates.instTopOfZero = { top := 0 }

@[simp]

theorem Associates.mk_zero {M : Type u_1} [Zero M] [Monoid M] : Associates.mk 0 = 0

@[simp]

theorem Associates.mk_eq_zero {M : Type u_1} [MonoidWithZero M] {a : M} : Associates.mk a = 0 ↔ a = 0

@[simp]

theorem Associates.quot_out_zero {M : Type u_1} [MonoidWithZero M] : Quot.out 0 = 0

theorem Associates.mk_ne_zero {M : Type u_1} [MonoidWithZero M] {a : M} : Associates.mk a ≠ 0 ↔ a ≠ 0

instance Associates.instNontrivial {M : Type u_1} [MonoidWithZero M] [Nontrivial M] : Nontrivial (Associates M)

Equations

• ⋯ = ⋯

theorem Associates.exists_non_zero_rep {M : Type u_1} [MonoidWithZero M] {a : Associates M} : a ≠ 0 → ∃ (a0 : M), a0 ≠ 0 ∧ Associates.mk a0 = a

instance Associates.instCommMonoidWithZero {M : Type u_1} [CommMonoidWithZero M] : CommMonoidWithZero (Associates M)

Equations

• Associates.instCommMonoidWithZero = CommMonoidWithZero.mk ⋯ ⋯

instance Associates.instOrderTop {M : Type u_1} [CommMonoidWithZero M] : OrderTop (Associates M)

Equations

• Associates.instOrderTop = OrderTop.mk ⋯

@[simp]

theorem Associates.le_zero {M : Type u_1} [CommMonoidWithZero M] (a : Associates M) : a ≤ 0

instance Associates.instBoundedOrder {M : Type u_1} [CommMonoidWithZero M] : BoundedOrder (Associates M)

Equations

• Associates.instBoundedOrder = BoundedOrder.mk

instance Associates.instDecidableRelDvd {M : Type u_1} [CommMonoidWithZero M] [DecidableRel fun (x1 x2 : M) => x1 ∣ x2] : DecidableRel fun (x1 x2 : Associates M) => x1 ∣ x2

Equations

• a.instDecidableRelDvd b = Quotient.recOnSubsingleton₂ a b fun (x x_1 : M) => decidable_of_iff' (x ∣ x_1) ⋯

theorem Associates.Prime.le_or_le {M : Type u_1} [CommMonoidWithZero M] {p : Associates M} (hp : Prime p) {a : Associates M} {b : Associates M} (h : p ≤ a * b) : p ≤ a ∨ p ≤ b

@[simp]

theorem Associates.prime_mk {M : Type u_1} [CommMonoidWithZero M] {p : M} : Prime (Associates.mk p) ↔ Prime p

@[simp]

theorem Associates.irreducible_mk {M : Type u_1} [CommMonoidWithZero M] {a : M} : Irreducible (Associates.mk a) ↔ Irreducible a

@[simp]

theorem Associates.mk_dvdNotUnit_mk_iff {M : Type u_1} [CommMonoidWithZero M] {a : M} {b : M} : DvdNotUnit (Associates.mk a) (Associates.mk b) ↔ DvdNotUnit a b

theorem Associates.dvdNotUnit_of_lt {M : Type u_1} [CommMonoidWithZero M] {a : Associates M} {b : Associates M} (hlt : a < b) : DvdNotUnit a b

theorem Associates.irreducible_iff_prime_iff {M : Type u_1} [CommMonoidWithZero M] : (∀ (a : M), Irreducible a ↔ Prime a) ↔ ∀ (a : Associates M), Irreducible a ↔ Prime a

instance Associates.instPartialOrder {M : Type u_1} [CancelCommMonoidWithZero M] : PartialOrder (Associates M)

Equations

• Associates.instPartialOrder = PartialOrder.mk ⋯

instance Associates.instCancelCommMonoidWithZero {M : Type u_1} [CancelCommMonoidWithZero M] : CancelCommMonoidWithZero (Associates M)

Equations

• Associates.instCancelCommMonoidWithZero = CancelCommMonoidWithZero.mk

theorem associates_irreducible_iff_prime {M : Type u_1} [CancelCommMonoidWithZero M] [DecompositionMonoid M] {p : Associates M} : Irreducible p ↔ Prime p

instance Associates.instNoZeroDivisors {M : Type u_1} [CancelCommMonoidWithZero M] : NoZeroDivisors (Associates M)

Equations

• ⋯ = ⋯

theorem Associates.le_of_mul_le_mul_left {M : Type u_1} [CancelCommMonoidWithZero M] (a : Associates M) (b : Associates M) (c : Associates M) (ha : a ≠ 0) : a * b ≤ a * c → b ≤ c

theorem Associates.one_or_eq_of_le_of_prime {M : Type u_1} [CancelCommMonoidWithZero M] {p : Associates M} {m : Associates M} (hp : Prime p) (hle : m ≤ p) : m = 1 ∨ m = p

theorem Associates.dvdNotUnit_iff_lt {M : Type u_1} [CancelCommMonoidWithZero M] {a : Associates M} {b : Associates M} : DvdNotUnit a b ↔ a < b

theorem Associates.le_one_iff {M : Type u_1} [CancelCommMonoidWithZero M] {p : Associates M} : p ≤ 1 ↔ p = 1

theorem DvdNotUnit.isUnit_of_irreducible_right {M : Type u_1} [CommMonoidWithZero M] {p : M} {q : M} (h : DvdNotUnit p q) (hq : Irreducible q) : IsUnit p

theorem not_irreducible_of_not_unit_dvdNotUnit {M : Type u_1} [CommMonoidWithZero M] {p : M} {q : M} (hp : ¬IsUnit p) (h : DvdNotUnit p q) : ¬Irreducible q

theorem DvdNotUnit.not_unit {M : Type u_1} [CommMonoidWithZero M] {p : M} {q : M} (hp : DvdNotUnit p q) : ¬IsUnit q

theorem dvdNotUnit_of_dvdNotUnit_associated {M : Type u_1} [CommMonoidWithZero M] [Nontrivial M] {p : M} {q : M} {r : M} (h : DvdNotUnit p q) (h' : Associated q r) : DvdNotUnit p r

theorem isUnit_of_associated_mul {M : Type u_1} [CancelCommMonoidWithZero M] {p : M} {b : M} (h : Associated (p * b) p) (hp : p ≠ 0) : IsUnit b

theorem DvdNotUnit.not_associated {M : Type u_1} [CancelCommMonoidWithZero M] {p : M} {q : M} (h : DvdNotUnit p q) : ¬Associated p q

theorem DvdNotUnit.ne {M : Type u_1} [CancelCommMonoidWithZero M] {p : M} {q : M} (h : DvdNotUnit p q) : p ≠ q

theorem pow_injective_of_not_isUnit {M : Type u_1} [CancelCommMonoidWithZero M] {q : M} (hq : ¬IsUnit q) (hq' : q ≠ 0) : Function.Injective fun (n : ℕ) => q ^ n

@[deprecated pow_injective_of_not_isUnit]

theorem pow_injective_of_not_unit {M : Type u_1} [CancelCommMonoidWithZero M] {q : M} (hq : ¬IsUnit q) (hq' : q ≠ 0) : Function.Injective fun (n : ℕ) => q ^ n

Alias of pow_injective_of_not_isUnit.

theorem pow_inj_of_not_isUnit {M : Type u_1} [CancelCommMonoidWithZero M] {q : M} (hq : ¬IsUnit q) (hq' : q ≠ 0) {m : ℕ} {n : ℕ} : q ^ m = q ^ n ↔ m = n

theorem dvd_prime_pow {M : Type u_1} [CancelCommMonoidWithZero M] {p : M} {q : M} (hp : Prime p) (n : ℕ) : q ∣ p ^ n ↔ ∃ (i : ℕ), i ≤ n ∧ Associated q (p ^ i)