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algebra.gcd_monoid.basic

Monoids with normalization functions, gcd, and lcm #

This file defines extra structures on comm_cancel_monoid_with_zeros, including is_domains.

Main Definitions #

For the normalized_gcd_monoid instances on and , see ring_theory.int.basic.

Implementation Notes #

TODO #

Tags #

divisibility, gcd, lcm, normalize

@[class]
structure normalization_monoid (α : Type u_2) [comm_cancel_monoid_with_zero α] :
Type u_2

Normalization monoid: multiplying with norm_unit gives a normal form for associated elements.

Instances
@[simp]

Chooses an element of each associate class, by multiplying by norm_unit

Equations
@[simp]
theorem normalize_apply {α : Type u_1} [comm_cancel_monoid_with_zero α] [normalization_monoid α] (x : α) :
@[simp]
@[simp]
theorem normalize_eq_zero {α : Type u_1} [comm_cancel_monoid_with_zero α] [normalization_monoid α] {x : α} :
@[simp]
theorem norm_unit_mul_norm_unit {α : Type u_1} [comm_cancel_monoid_with_zero α] [normalization_monoid α] (a : α) :
theorem normalize_eq_normalize {α : Type u_1} [comm_cancel_monoid_with_zero α] [normalization_monoid α] {a b : α} (hab : a b) (hba : b a) :
theorem dvd_antisymm_of_normalize_eq {α : Type u_1} [comm_cancel_monoid_with_zero α] [normalization_monoid α] {a b : α} (ha : normalize a = a) (hb : normalize b = b) (hab : a b) (hba : b a) :
a = b
theorem dvd_normalize_iff {α : Type u_1} [comm_cancel_monoid_with_zero α] [normalization_monoid α] {a b : α} :
theorem normalize_dvd_iff {α : Type u_1} [comm_cancel_monoid_with_zero α] [normalization_monoid α] {a b : α} :
def associates.out {α : Type u_1} [comm_cancel_monoid_with_zero α] [normalization_monoid α] :
associates α → α

Maps an element of associates back to the normalized element of its associate class

Equations
@[simp]
theorem associates.out_one {α : Type u_1} [comm_cancel_monoid_with_zero α] [normalization_monoid α] :
1.out = 1
theorem associates.out_mul {α : Type u_1} [comm_cancel_monoid_with_zero α] [normalization_monoid α] (a b : associates α) :
(a * b).out = (a.out) * b.out
theorem associates.dvd_out_iff {α : Type u_1} [comm_cancel_monoid_with_zero α] [normalization_monoid α] (a : α) (b : associates α) :
theorem associates.out_dvd_iff {α : Type u_1} [comm_cancel_monoid_with_zero α] [normalization_monoid α] (a : α) (b : associates α) :
@[simp]
@[class]
structure gcd_monoid (α : Type u_2) [comm_cancel_monoid_with_zero α] :
Type u_2
  • gcd : α → α → α
  • lcm : α → α → α
  • gcd_dvd_left : ∀ (a b : α), gcd a b a
  • gcd_dvd_right : ∀ (a b : α), gcd a b b
  • dvd_gcd : ∀ {a b c : α}, a ca ba gcd c b
  • gcd_mul_lcm : ∀ (a b : α), associated ((gcd a b) * lcm a b) (a * b)
  • lcm_zero_left : ∀ (a : α), lcm 0 a = 0
  • lcm_zero_right : ∀ (a : α), lcm a 0 = 0

GCD monoid: a comm_cancel_monoid_with_zero with gcd (greatest common divisor) and lcm (least common multiple) operations, determined up to a unit. The type class focuses on gcd and we derive the corresponding lcm facts from gcd.

Instances
@[class]
structure normalized_gcd_monoid (α : Type u_2) [comm_cancel_monoid_with_zero α] :
Type u_2

Normalized GCD monoid: a comm_cancel_monoid_with_zero with normalization and gcd (greatest common divisor) and lcm (least common multiple) operations. In this setting gcd and lcm form a bounded lattice on the associated elements where gcd is the infimum, lcm is the supremum, 1 is bottom, and 0 is top. The type class focuses on gcd and we derive the corresponding lcm facts from gcd.

Instances
@[simp]
theorem normalize_gcd {α : Type u_1} [comm_cancel_monoid_with_zero α] [normalized_gcd_monoid α] (a b : α) :
normalize (gcd a b) = gcd a b
theorem gcd_mul_lcm {α : Type u_1} [comm_cancel_monoid_with_zero α] [gcd_monoid α] (a b : α) :
associated ((gcd a b) * lcm a b) (a * b)
theorem dvd_gcd_iff {α : Type u_1} [comm_cancel_monoid_with_zero α] [gcd_monoid α] (a b c : α) :
a gcd b c a b a c
theorem gcd_comm {α : Type u_1} [comm_cancel_monoid_with_zero α] [normalized_gcd_monoid α] (a b : α) :
gcd a b = gcd b a
theorem gcd_comm' {α : Type u_1} [comm_cancel_monoid_with_zero α] [gcd_monoid α] (a b : α) :
associated (gcd a b) (gcd b a)
theorem gcd_assoc {α : Type u_1} [comm_cancel_monoid_with_zero α] [normalized_gcd_monoid α] (m n k : α) :
gcd (gcd m n) k = gcd m (gcd n k)
theorem gcd_assoc' {α : Type u_1} [comm_cancel_monoid_with_zero α] [gcd_monoid α] (m n k : α) :
associated (gcd (gcd m n) k) (gcd m (gcd n k))
theorem gcd_eq_normalize {α : Type u_1} [comm_cancel_monoid_with_zero α] [normalized_gcd_monoid α] {a b c : α} (habc : gcd a b c) (hcab : c gcd a b) :
@[simp]
theorem gcd_zero_left {α : Type u_1} [comm_cancel_monoid_with_zero α] [normalized_gcd_monoid α] (a : α) :
theorem gcd_zero_left' {α : Type u_1} [comm_cancel_monoid_with_zero α] [gcd_monoid α] (a : α) :
associated (gcd 0 a) a
@[simp]
theorem gcd_zero_right {α : Type u_1} [comm_cancel_monoid_with_zero α] [normalized_gcd_monoid α] (a : α) :
theorem gcd_zero_right' {α : Type u_1} [comm_cancel_monoid_with_zero α] [gcd_monoid α] (a : α) :
associated (gcd a 0) a
@[simp]
theorem gcd_eq_zero_iff {α : Type u_1} [comm_cancel_monoid_with_zero α] [gcd_monoid α] (a b : α) :
gcd a b = 0 a = 0 b = 0
@[simp]
theorem gcd_one_left {α : Type u_1} [comm_cancel_monoid_with_zero α] [normalized_gcd_monoid α] (a : α) :
gcd 1 a = 1
@[simp]
theorem gcd_one_left' {α : Type u_1} [comm_cancel_monoid_with_zero α] [gcd_monoid α] (a : α) :
associated (gcd 1 a) 1
@[simp]
theorem gcd_one_right {α : Type u_1} [comm_cancel_monoid_with_zero α] [normalized_gcd_monoid α] (a : α) :
gcd a 1 = 1
@[simp]
theorem gcd_one_right' {α : Type u_1} [comm_cancel_monoid_with_zero α] [gcd_monoid α] (a : α) :
associated (gcd a 1) 1
theorem gcd_dvd_gcd {α : Type u_1} [comm_cancel_monoid_with_zero α] [gcd_monoid α] {a b c d : α} (hab : a b) (hcd : c d) :
gcd a c gcd b d
@[simp]
theorem gcd_same {α : Type u_1} [comm_cancel_monoid_with_zero α] [normalized_gcd_monoid α] (a : α) :
@[simp]
theorem gcd_mul_left {α : Type u_1} [comm_cancel_monoid_with_zero α] [normalized_gcd_monoid α] (a b c : α) :
gcd (a * b) (a * c) = (normalize a) * gcd b c
theorem gcd_mul_left' {α : Type u_1} [comm_cancel_monoid_with_zero α] [gcd_monoid α] (a b c : α) :
associated (gcd (a * b) (a * c)) (a * gcd b c)
@[simp]
theorem gcd_mul_right {α : Type u_1} [comm_cancel_monoid_with_zero α] [normalized_gcd_monoid α] (a b c : α) :
gcd (b * a) (c * a) = (gcd b c) * normalize a
@[simp]
theorem gcd_mul_right' {α : Type u_1} [comm_cancel_monoid_with_zero α] [gcd_monoid α] (a b c : α) :
associated (gcd (b * a) (c * a)) ((gcd b c) * a)
theorem gcd_eq_left_iff {α : Type u_1} [comm_cancel_monoid_with_zero α] [normalized_gcd_monoid α] (a b : α) (h : normalize a = a) :
gcd a b = a a b
theorem gcd_eq_right_iff {α : Type u_1} [comm_cancel_monoid_with_zero α] [normalized_gcd_monoid α] (a b : α) (h : normalize b = b) :
gcd a b = b b a
theorem gcd_dvd_gcd_mul_left {α : Type u_1} [comm_cancel_monoid_with_zero α] [gcd_monoid α] (m n k : α) :
gcd m n gcd (k * m) n
theorem gcd_dvd_gcd_mul_right {α : Type u_1} [comm_cancel_monoid_with_zero α] [gcd_monoid α] (m n k : α) :
gcd m n gcd (m * k) n
theorem gcd_dvd_gcd_mul_left_right {α : Type u_1} [comm_cancel_monoid_with_zero α] [gcd_monoid α] (m n k : α) :
gcd m n gcd m (k * n)
theorem gcd_dvd_gcd_mul_right_right {α : Type u_1} [comm_cancel_monoid_with_zero α] [gcd_monoid α] (m n k : α) :
gcd m n gcd m (n * k)
theorem associated.gcd_eq_left {α : Type u_1} [comm_cancel_monoid_with_zero α] [normalized_gcd_monoid α] {m n : α} (h : associated m n) (k : α) :
gcd m k = gcd n k
theorem associated.gcd_eq_right {α : Type u_1} [comm_cancel_monoid_with_zero α] [normalized_gcd_monoid α] {m n : α} (h : associated m n) (k : α) :
gcd k m = gcd k n
theorem dvd_gcd_mul_of_dvd_mul {α : Type u_1} [comm_cancel_monoid_with_zero α] [gcd_monoid α] {m n k : α} (H : k m * n) :
k (gcd k m) * n
theorem dvd_mul_gcd_of_dvd_mul {α : Type u_1} [comm_cancel_monoid_with_zero α] [gcd_monoid α] {m n k : α} (H : k m * n) :
k m * gcd k n
theorem exists_dvd_and_dvd_of_dvd_mul {α : Type u_1} [comm_cancel_monoid_with_zero α] [gcd_monoid α] {m n k : α} (H : k m * n) :
∃ (d₁ : α) (hd₁ : d₁ m) (d₂ : α) (hd₂ : d₂ n), k = d₁ * d₂

Represent a divisor of m * n as a product of a divisor of m and a divisor of n.

Note: In general, this representation is highly non-unique.

theorem gcd_mul_dvd_mul_gcd {α : Type u_1} [comm_cancel_monoid_with_zero α] [gcd_monoid α] (k m n : α) :
gcd k (m * n) (gcd k m) * gcd k n
theorem gcd_pow_right_dvd_pow_gcd {α : Type u_1} [comm_cancel_monoid_with_zero α] [gcd_monoid α] {a b : α} {k : } :
gcd a (b ^ k) gcd a b ^ k
theorem gcd_pow_left_dvd_pow_gcd {α : Type u_1} [comm_cancel_monoid_with_zero α] [gcd_monoid α] {a b : α} {k : } :
gcd (a ^ k) b gcd a b ^ k
theorem pow_dvd_of_mul_eq_pow {α : Type u_1} [comm_cancel_monoid_with_zero α] [gcd_monoid α] {a b c d₁ d₂ : α} (ha : a 0) (hab : is_unit (gcd a b)) {k : } (h : a * b = c ^ k) (hc : c = d₁ * d₂) (hd₁ : d₁ a) :
d₁ ^ k 0 d₁ ^ k a
theorem exists_associated_pow_of_mul_eq_pow {α : Type u_1} [comm_cancel_monoid_with_zero α] [gcd_monoid α] {a b c : α} (hab : is_unit (gcd a b)) {k : } (h : a * b = c ^ k) :
∃ (d : α), associated (d ^ k) a
theorem lcm_dvd_iff {α : Type u_1} [comm_cancel_monoid_with_zero α] [gcd_monoid α] {a b c : α} :
lcm a b c a c b c
theorem dvd_lcm_left {α : Type u_1} [comm_cancel_monoid_with_zero α] [gcd_monoid α] (a b : α) :
a lcm a b
theorem dvd_lcm_right {α : Type u_1} [comm_cancel_monoid_with_zero α] [gcd_monoid α] (a b : α) :
b lcm a b
theorem lcm_dvd {α : Type u_1} [comm_cancel_monoid_with_zero α] [gcd_monoid α] {a b c : α} (hab : a b) (hcb : c b) :
lcm a c b
@[simp]
theorem lcm_eq_zero_iff {α : Type u_1} [comm_cancel_monoid_with_zero α] [gcd_monoid α] (a b : α) :
lcm a b = 0 a = 0 b = 0
@[simp]
theorem normalize_lcm {α : Type u_1} [comm_cancel_monoid_with_zero α] [normalized_gcd_monoid α] (a b : α) :
normalize (lcm a b) = lcm a b
theorem lcm_comm {α : Type u_1} [comm_cancel_monoid_with_zero α] [normalized_gcd_monoid α] (a b : α) :
lcm a b = lcm b a
theorem lcm_comm' {α : Type u_1} [comm_cancel_monoid_with_zero α] [gcd_monoid α] (a b : α) :
associated (lcm a b) (lcm b a)
theorem lcm_assoc {α : Type u_1} [comm_cancel_monoid_with_zero α] [normalized_gcd_monoid α] (m n k : α) :
lcm (lcm m n) k = lcm m (lcm n k)
theorem lcm_assoc' {α : Type u_1} [comm_cancel_monoid_with_zero α] [gcd_monoid α] (m n k : α) :
associated (lcm (lcm m n) k) (lcm m (lcm n k))
theorem lcm_eq_normalize {α : Type u_1} [comm_cancel_monoid_with_zero α] [normalized_gcd_monoid α] {a b c : α} (habc : lcm a b c) (hcab : c lcm a b) :
theorem lcm_dvd_lcm {α : Type u_1} [comm_cancel_monoid_with_zero α] [gcd_monoid α] {a b c d : α} (hab : a b) (hcd : c d) :
lcm a c lcm b d
@[simp]
theorem lcm_units_coe_left {α : Type u_1} [comm_cancel_monoid_with_zero α] [normalized_gcd_monoid α] (u : units α) (a : α) :
@[simp]
theorem lcm_units_coe_right {α : Type u_1} [comm_cancel_monoid_with_zero α] [normalized_gcd_monoid α] (a : α) (u : units α) :
@[simp]
theorem lcm_one_left {α : Type u_1} [comm_cancel_monoid_with_zero α] [normalized_gcd_monoid α] (a : α) :
@[simp]
theorem lcm_one_right {α : Type u_1} [comm_cancel_monoid_with_zero α] [normalized_gcd_monoid α] (a : α) :
@[simp]
theorem lcm_same {α : Type u_1} [comm_cancel_monoid_with_zero α] [normalized_gcd_monoid α] (a : α) :
@[simp]
theorem lcm_eq_one_iff {α : Type u_1} [comm_cancel_monoid_with_zero α] [normalized_gcd_monoid α] (a b : α) :
lcm a b = 1 a 1 b 1
@[simp]
theorem lcm_mul_left {α : Type u_1} [comm_cancel_monoid_with_zero α] [normalized_gcd_monoid α] (a b c : α) :
lcm (a * b) (a * c) = (normalize a) * lcm b c
@[simp]
theorem lcm_mul_right {α : Type u_1} [comm_cancel_monoid_with_zero α] [normalized_gcd_monoid α] (a b c : α) :
lcm (b * a) (c * a) = (lcm b c) * normalize a
theorem lcm_eq_left_iff {α : Type u_1} [comm_cancel_monoid_with_zero α] [normalized_gcd_monoid α] (a b : α) (h : normalize a = a) :
lcm a b = a b a
theorem lcm_eq_right_iff {α : Type u_1} [comm_cancel_monoid_with_zero α] [normalized_gcd_monoid α] (a b : α) (h : normalize b = b) :
lcm a b = b a b
theorem lcm_dvd_lcm_mul_left {α : Type u_1} [comm_cancel_monoid_with_zero α] [gcd_monoid α] (m n k : α) :
lcm m n lcm (k * m) n
theorem lcm_dvd_lcm_mul_right {α : Type u_1} [comm_cancel_monoid_with_zero α] [gcd_monoid α] (m n k : α) :
lcm m n lcm (m * k) n
theorem lcm_dvd_lcm_mul_left_right {α : Type u_1} [comm_cancel_monoid_with_zero α] [gcd_monoid α] (m n k : α) :
lcm m n lcm m (k * n)
theorem lcm_dvd_lcm_mul_right_right {α : Type u_1} [comm_cancel_monoid_with_zero α] [gcd_monoid α] (m n k : α) :
lcm m n lcm m (n * k)
theorem lcm_eq_of_associated_left {α : Type u_1} [comm_cancel_monoid_with_zero α] [normalized_gcd_monoid α] {m n : α} (h : associated m n) (k : α) :
lcm m k = lcm n k
theorem lcm_eq_of_associated_right {α : Type u_1} [comm_cancel_monoid_with_zero α] [normalized_gcd_monoid α] {m n : α} (h : associated m n) (k : α) :
lcm k m = lcm k n
theorem gcd_monoid.prime_of_irreducible {α : Type u_1} [comm_cancel_monoid_with_zero α] [gcd_monoid α] {x : α} (hi : irreducible x) :
@[simp]
theorem norm_unit_eq_one {α : Type u_1} [comm_cancel_monoid_with_zero α] [unique (units α)] (x : α) :
@[simp]
theorem normalize_eq {α : Type u_1} [comm_cancel_monoid_with_zero α] [unique (units α)] (x : α) :
theorem gcd_eq_of_dvd_sub_right {α : Type u_1} [comm_ring α] [is_domain α] [normalized_gcd_monoid α] {a b c : α} (h : a b - c) :
gcd a b = gcd a c
theorem gcd_eq_of_dvd_sub_left {α : Type u_1} [comm_ring α] [is_domain α] [normalized_gcd_monoid α] {a b c : α} (h : a b - c) :
gcd b a = gcd c a
def gcd_monoid_of_gcd {α : Type u_1} [comm_cancel_monoid_with_zero α] [decidable_eq α] (gcd : α → α → α) (gcd_dvd_left : ∀ (a b : α), gcd a b a) (gcd_dvd_right : ∀ (a b : α), gcd a b b) (dvd_gcd : ∀ {a b c : α}, a ca ba gcd c b) :

Define gcd_monoid on a structure just from the gcd and its properties.

Equations
def normalized_gcd_monoid_of_gcd {α : Type u_1} [comm_cancel_monoid_with_zero α] [normalization_monoid α] [decidable_eq α] (gcd : α → α → α) (gcd_dvd_left : ∀ (a b : α), gcd a b a) (gcd_dvd_right : ∀ (a b : α), gcd a b b) (dvd_gcd : ∀ {a b c : α}, a ca ba gcd c b) (normalize_gcd : ∀ (a b : α), normalize (gcd a b) = gcd a b) :

Define normalized_gcd_monoid on a structure just from the gcd and its properties.

Equations
def gcd_monoid_of_lcm {α : Type u_1} [comm_cancel_monoid_with_zero α] [decidable_eq α] (lcm : α → α → α) (dvd_lcm_left : ∀ (a b : α), a lcm a b) (dvd_lcm_right : ∀ (a b : α), b lcm a b) (lcm_dvd : ∀ {a b c : α}, c ab alcm c b a) :

Define gcd_monoid on a structure just from the lcm and its properties.

Equations
def normalized_gcd_monoid_of_lcm {α : Type u_1} [comm_cancel_monoid_with_zero α] [normalization_monoid α] [decidable_eq α] (lcm : α → α → α) (dvd_lcm_left : ∀ (a b : α), a lcm a b) (dvd_lcm_right : ∀ (a b : α), b lcm a b) (lcm_dvd : ∀ {a b c : α}, c ab alcm c b a) (normalize_lcm : ∀ (a b : α), normalize (lcm a b) = lcm a b) :

Define normalized_gcd_monoid on a structure just from the lcm and its properties.

Equations
def gcd_monoid_of_exists_gcd {α : Type u_1} [comm_cancel_monoid_with_zero α] [decidable_eq α] (h : ∀ (a b : α), ∃ (c : α), ∀ (d : α), d a d b d c) :

Define a gcd_monoid structure on a monoid just from the existence of a gcd.

Equations
def normalized_gcd_monoid_of_exists_gcd {α : Type u_1} [comm_cancel_monoid_with_zero α] [normalization_monoid α] [decidable_eq α] (h : ∀ (a b : α), ∃ (c : α), ∀ (d : α), d a d b d c) :

Define a normalized_gcd_monoid structure on a monoid just from the existence of a gcd.

Equations
def gcd_monoid_of_exists_lcm {α : Type u_1} [comm_cancel_monoid_with_zero α] [decidable_eq α] (h : ∀ (a b : α), ∃ (c : α), ∀ (d : α), a d b d c d) :

Define a gcd_monoid structure on a monoid just from the existence of an lcm.

Equations
def normalized_gcd_monoid_of_exists_lcm {α : Type u_1} [comm_cancel_monoid_with_zero α] [normalization_monoid α] [decidable_eq α] (h : ∀ (a b : α), ∃ (c : α), ∀ (d : α), a d b d c d) :

Define a normalized_gcd_monoid structure on a monoid just from the existence of an lcm.

Equations
@[instance]
Equations
@[simp]
theorem comm_group_with_zero.coe_norm_unit (G₀ : Type u_2) [comm_group_with_zero G₀] [decidable_eq G₀] {a : G₀} (h0 : a 0) :
theorem comm_group_with_zero.normalize_eq_one (G₀ : Type u_2) [comm_group_with_zero G₀] [decidable_eq G₀] {a : G₀} (h0 : a 0) :