mathlib3 documentation

algebra.order.lattice_group

Lattice ordered groups #

THIS FILE IS SYNCHRONIZED WITH MATHLIB4. Any changes to this file require a corresponding PR to mathlib4.

Lattice ordered groups were introduced by Birkhoff. They form the algebraic underpinnings of vector lattices, Banach lattices, AL-space, AM-space etc.

This file develops the basic theory, concentrating on the commutative case.

Main statements #

pos_div_neg: Every element a of a lattice ordered commutative group has a decomposition a⁺-a⁻ into the difference of the positive and negative component.
pos_inf_neg_eq_one: The positive and negative components are coprime.
abs_triangle: The absolute value operation satisfies the triangle inequality.

It is shown that the inf and sup operations are related to the absolute value operation by a number of equations and inequalities.

Notations #

a⁺ = a ⊔ 0: The positive component of an element a of a lattice ordered commutative group
a⁻ = (-a) ⊔ 0: The negative component of an element a of a lattice ordered commutative group
|a| = a⊔(-a): The absolute value of an element a of a lattice ordered commutative group

Implementation notes #

A lattice ordered commutative group is a type α satisfying:

[lattice α]
[comm_group α]
[covariant_class α α (*) (≤)]

The remainder of the file establishes basic properties of lattice ordered commutative groups. A number of these results also hold in the non-commutative case (Birkhoff, Fuchs) but we have not developed that here, since we are primarily interested in vector lattices.

References #

Tags #

lattice, ordered, group

theorem add_sup {α : Type u} [lattice α] [add_comm_group α] [covariant_class α α has_add.add has_le.le] (a b c : α) :

c + a ⊔ b = (c + a) ⊔ (c + b)

theorem mul_sup {α : Type u} [lattice α] [comm_group α] [covariant_class α α has_mul.mul has_le.le] (a b c : α) :

c * (a ⊔ b) = c * a ⊔ c * b

theorem sup_add {α : Type u} [lattice α] [add_comm_group α] [covariant_class α α has_add.add has_le.le] (a b c : α) :

a ⊔ b + c = (a + c) ⊔ (b + c)

theorem sup_mul {α : Type u} [lattice α] [comm_group α] [covariant_class α α has_mul.mul has_le.le] (a b c : α) :

(a ⊔ b) * c = a * c ⊔ b * c

theorem add_inf {α : Type u} [lattice α] [add_comm_group α] [covariant_class α α has_add.add has_le.le] (a b c : α) :

c + a ⊓ b = (c + a) ⊓ (c + b)

theorem mul_inf {α : Type u} [lattice α] [comm_group α] [covariant_class α α has_mul.mul has_le.le] (a b c : α) :

c * (a ⊓ b) = c * a ⊓ c * b

theorem inf_mul {α : Type u} [lattice α] [comm_group α] [covariant_class α α has_mul.mul has_le.le] (a b c : α) :

(a ⊓ b) * c = a * c ⊓ b * c

theorem inf_add {α : Type u} [lattice α] [add_comm_group α] [covariant_class α α has_add.add has_le.le] (a b c : α) :

a ⊓ b + c = (a + c) ⊓ (b + c)

theorem neg_sup_eq_neg_inf_neg {α : Type u} [lattice α] [add_comm_group α] [covariant_class α α has_add.add has_le.le] (a b : α) :

-(a ⊔ b) = -a ⊓ -b

theorem inv_sup_eq_inv_inf_inv {α : Type u} [lattice α] [comm_group α] [covariant_class α α has_mul.mul has_le.le] (a b : α) :

(a ⊔ b)⁻¹ = a⁻¹ ⊓ b⁻¹

theorem inv_inf_eq_sup_inv {α : Type u} [lattice α] [comm_group α] [covariant_class α α has_mul.mul has_le.le] (a b : α) :

(a ⊓ b)⁻¹ = a⁻¹ ⊔ b⁻¹

theorem neg_inf_eq_sup_neg {α : Type u} [lattice α] [add_comm_group α] [covariant_class α α has_add.add has_le.le] (a b : α) :

-(a ⊓ b) = -a ⊔ -b

theorem inf_add_sup {α : Type u} [lattice α] [add_comm_group α] [covariant_class α α has_add.add has_le.le] (a b : α) :

a ⊓ b + a ⊔ b = a + b

theorem inf_mul_sup {α : Type u} [lattice α] [comm_group α] [covariant_class α α has_mul.mul has_le.le] (a b : α) :

(a ⊓ b) * (a ⊔ b) = a * b

@[protected, instance]

def lattice_ordered_comm_group.has_one_lattice_has_pos_part {α : Type u} [lattice α] [comm_group α] :

has_pos_part α

Let α be a lattice ordered commutative group with identity 1. For an element a of type α, the element a ⊔ 1 is said to be the positive component of a, denoted a⁺.

Equations

lattice_ordered_comm_group.has_one_lattice_has_pos_part = {pos := λ (a : α), a ⊔ 1}

@[protected, instance]

def lattice_ordered_comm_group.has_zero_lattice_has_pos_part {α : Type u} [lattice α] [add_comm_group α] :

has_pos_part α

Let α be a lattice ordered commutative group with identity 0. For an element a of type α, the element a ⊔ 0 is said to be the positive component of a, denoted a⁺.

Equations

lattice_ordered_comm_group.has_zero_lattice_has_pos_part = {pos := λ (a : α), a ⊔ 0}

theorem lattice_ordered_comm_group.pos_part_def {α : Type u} [lattice α] [add_comm_group α] (a : α) :

a⁺ = a ⊔ 0

theorem lattice_ordered_comm_group.m_pos_part_def {α : Type u} [lattice α] [comm_group α] (a : α) :

a⁺ = a ⊔ 1

@[protected, instance]

def lattice_ordered_comm_group.has_zero_lattice_has_neg_part {α : Type u} [lattice α] [add_comm_group α] :

has_neg_part α

Let α be a lattice ordered commutative group with identity 0. For an element a of type α, the element (-a) ⊔ 0 is said to be the negative component of a, denoted a⁻.

Equations

lattice_ordered_comm_group.has_zero_lattice_has_neg_part = {neg := λ (a : α), -a ⊔ 0}

@[protected, instance]

def lattice_ordered_comm_group.has_one_lattice_has_neg_part {α : Type u} [lattice α] [comm_group α] :

has_neg_part α

Let α be a lattice ordered commutative group with identity 1. For an element a of type α, the element (-a) ⊔ 1 is said to be the negative component of a, denoted a⁻.

Equations

lattice_ordered_comm_group.has_one_lattice_has_neg_part = {neg := λ (a : α), a⁻¹ ⊔ 1}

theorem lattice_ordered_comm_group.m_neg_part_def {α : Type u} [lattice α] [comm_group α] (a : α) :

a⁻ = a⁻¹ ⊔ 1

theorem lattice_ordered_comm_group.neg_part_def {α : Type u} [lattice α] [add_comm_group α] (a : α) :

a⁻ = -a ⊔ 0

@[simp]

theorem lattice_ordered_comm_group.pos_one {α : Type u} [lattice α] [comm_group α] :

1⁺ = 1

@[simp]

theorem lattice_ordered_comm_group.pos_zero {α : Type u} [lattice α] [add_comm_group α] :

0⁺ = 0

@[simp]

theorem lattice_ordered_comm_group.neg_zero {α : Type u} [lattice α] [add_comm_group α] :

0⁻ = 0

@[simp]

theorem lattice_ordered_comm_group.neg_one {α : Type u} [lattice α] [comm_group α] :

1⁻ = 1

theorem lattice_ordered_comm_group.neg_eq_inv_inf_one {α : Type u} [lattice α] [comm_group α] [covariant_class α α has_mul.mul has_le.le] (a : α) :

a⁻ = (a ⊓ 1)⁻¹

theorem lattice_ordered_comm_group.neg_eq_neg_inf_zero {α : Type u} [lattice α] [add_comm_group α] [covariant_class α α has_add.add has_le.le] (a : α) :

a⁻ = -(a ⊓ 0)

theorem lattice_ordered_comm_group.le_mabs {α : Type u} [lattice α] [comm_group α] (a : α) :

theorem lattice_ordered_comm_group.le_abs {α : Type u} [lattice α] [add_comm_group α] (a : α) :

theorem lattice_ordered_comm_group.neg_le_abs {α : Type u} [lattice α] [add_comm_group α] (a : α) :

theorem lattice_ordered_comm_group.inv_le_abs {α : Type u} [lattice α] [comm_group α] (a : α) :

theorem lattice_ordered_comm_group.one_le_pos {α : Type u} [lattice α] [comm_group α] (a : α) :

theorem lattice_ordered_comm_group.pos_nonneg {α : Type u} [lattice α] [add_comm_group α] (a : α) :

theorem lattice_ordered_comm_group.neg_nonneg {α : Type u} [lattice α] [add_comm_group α] (a : α) :

theorem lattice_ordered_comm_group.one_le_neg {α : Type u} [lattice α] [comm_group α] (a : α) :

theorem lattice_ordered_comm_group.pos_le_one_iff {α : Type u} [lattice α] [comm_group α] {a : α} :

a⁺ ≤ 1 ↔ a ≤ 1

theorem lattice_ordered_comm_group.pos_nonpos_iff {α : Type u} [lattice α] [add_comm_group α] {a : α} :

a⁺ ≤ 0 ↔ a ≤ 0

theorem lattice_ordered_comm_group.neg_nonpos_iff {α : Type u} [lattice α] [add_comm_group α] {a : α} :

a⁻ ≤ 0 ↔ -a ≤ 0

theorem lattice_ordered_comm_group.neg_le_one_iff {α : Type u} [lattice α] [comm_group α] {a : α} :

a⁻ ≤ 1 ↔ a⁻¹ ≤ 1

theorem lattice_ordered_comm_group.pos_eq_zero_iff {α : Type u} [lattice α] [add_comm_group α] {a : α} :

a⁺ = 0 ↔ a ≤ 0

theorem lattice_ordered_comm_group.pos_eq_one_iff {α : Type u} [lattice α] [comm_group α] {a : α} :

a⁺ = 1 ↔ a ≤ 1

theorem lattice_ordered_comm_group.neg_eq_one_iff' {α : Type u} [lattice α] [comm_group α] {a : α} :

a⁻ = 1 ↔ a⁻¹ ≤ 1

theorem lattice_ordered_comm_group.neg_eq_zero_iff' {α : Type u} [lattice α] [add_comm_group α] {a : α} :

a⁻ = 0 ↔ -a ≤ 0

theorem lattice_ordered_comm_group.neg_eq_one_iff {α : Type u} [lattice α] [comm_group α] [covariant_class α α has_mul.mul has_le.le] {a : α} :

a⁻ = 1 ↔ 1 ≤ a

theorem lattice_ordered_comm_group.neg_eq_zero_iff {α : Type u} [lattice α] [add_comm_group α] [covariant_class α α has_add.add has_le.le] {a : α} :

a⁻ = 0 ↔ 0 ≤ a

theorem lattice_ordered_comm_group.le_pos {α : Type u} [lattice α] [add_comm_group α] (a : α) :

theorem lattice_ordered_comm_group.m_le_pos {α : Type u} [lattice α] [comm_group α] (a : α) :

theorem lattice_ordered_comm_group.neg_le_neg {α : Type u} [lattice α] [add_comm_group α] (a : α) :

theorem lattice_ordered_comm_group.inv_le_neg {α : Type u} [lattice α] [comm_group α] (a : α) :

a⁻¹ ≤ a⁻

theorem lattice_ordered_comm_group.neg_eq_pos_inv {α : Type u} [lattice α] [comm_group α] (a : α) :

a⁻ = a⁻¹ ⁺

theorem lattice_ordered_comm_group.neg_eq_pos_neg {α : Type u} [lattice α] [add_comm_group α] (a : α) :

theorem lattice_ordered_comm_group.pos_eq_neg_neg {α : Type u} [lattice α] [add_comm_group α] (a : α) :

theorem lattice_ordered_comm_group.pos_eq_neg_inv {α : Type u} [lattice α] [comm_group α] (a : α) :

a⁺ = a⁻¹ ⁻

@[simp]

theorem lattice_ordered_comm_group.pos_div_neg {α : Type u} [lattice α] [comm_group α] [covariant_class α α has_mul.mul has_le.le] (a : α) :

a⁺ / a⁻ = a

@[simp]

theorem lattice_ordered_comm_group.pos_sub_neg {α : Type u} [lattice α] [add_comm_group α] [covariant_class α α has_add.add has_le.le] (a : α) :

a⁺ - a⁻ = a

theorem lattice_ordered_comm_group.pos_inf_neg_eq_one {α : Type u} [lattice α] [comm_group α] [covariant_class α α has_mul.mul has_le.le] (a : α) :

a⁺ ⊓ a⁻ = 1

theorem lattice_ordered_comm_group.pos_inf_neg_eq_zero {α : Type u} [lattice α] [add_comm_group α] [covariant_class α α has_add.add has_le.le] (a : α) :

a⁺ ⊓ a⁻ = 0

theorem lattice_ordered_comm_group.sup_eq_add_pos_sub {α : Type u} [lattice α] [add_comm_group α] [covariant_class α α has_add.add has_le.le] (a b : α) :

a ⊔ b = b + (a - b)⁺

theorem lattice_ordered_comm_group.sup_eq_mul_pos_div {α : Type u} [lattice α] [comm_group α] [covariant_class α α has_mul.mul has_le.le] (a b : α) :

a ⊔ b = b * (a / b)⁺

theorem lattice_ordered_comm_group.inf_eq_sub_pos_sub {α : Type u} [lattice α] [add_comm_group α] [covariant_class α α has_add.add has_le.le] (a b : α) :

a ⊓ b = a - (a - b)⁺

theorem lattice_ordered_comm_group.inf_eq_div_pos_div {α : Type u} [lattice α] [comm_group α] [covariant_class α α has_mul.mul has_le.le] (a b : α) :

a ⊓ b = a / (a / b)⁺

theorem lattice_ordered_comm_group.m_le_iff_pos_le_neg_ge {α : Type u} [lattice α] [comm_group α] [covariant_class α α has_mul.mul has_le.le] (a b : α) :

a ≤ b ↔ a⁺ ≤ b⁺ ∧ b⁻ ≤ a⁻

theorem lattice_ordered_comm_group.le_iff_pos_le_neg_ge {α : Type u} [lattice α] [add_comm_group α] [covariant_class α α has_add.add has_le.le] (a b : α) :

a ≤ b ↔ a⁺ ≤ b⁺ ∧ b⁻ ≤ a⁻

theorem lattice_ordered_comm_group.neg_abs {α : Type u} [lattice α] [add_comm_group α] [covariant_class α α has_add.add has_le.le] (a : α) :

theorem lattice_ordered_comm_group.m_neg_abs {α : Type u} [lattice α] [comm_group α] [covariant_class α α has_mul.mul has_le.le] (a : α) :

theorem lattice_ordered_comm_group.pos_abs {α : Type u} [lattice α] [add_comm_group α] [covariant_class α α has_add.add has_le.le] (a : α) :

theorem lattice_ordered_comm_group.m_pos_abs {α : Type u} [lattice α] [comm_group α] [covariant_class α α has_mul.mul has_le.le] (a : α) :

theorem lattice_ordered_comm_group.abs_nonneg {α : Type u} [lattice α] [add_comm_group α] [covariant_class α α has_add.add has_le.le] (a : α) :

theorem lattice_ordered_comm_group.one_le_abs {α : Type u} [lattice α] [comm_group α] [covariant_class α α has_mul.mul has_le.le] (a : α) :

theorem lattice_ordered_comm_group.pos_mul_neg {α : Type u} [lattice α] [comm_group α] [covariant_class α α has_mul.mul has_le.le] (a : α) :

|a| = a⁺ * a⁻

theorem lattice_ordered_comm_group.pos_add_neg {α : Type u} [lattice α] [add_comm_group α] [covariant_class α α has_add.add has_le.le] (a : α) :

|a| = a⁺ + a⁻

theorem lattice_ordered_comm_group.sup_div_inf_eq_abs_div {α : Type u} [lattice α] [comm_group α] [covariant_class α α has_mul.mul has_le.le] (a b : α) :

(a ⊔ b) / (a ⊓ b) = |b / a|

theorem lattice_ordered_comm_group.sup_sub_inf_eq_abs_sub {α : Type u} [lattice α] [add_comm_group α] [covariant_class α α has_add.add has_le.le] (a b : α) :

a ⊔ b - a ⊓ b = |b - a|

theorem lattice_ordered_comm_group.two_sup_eq_add_add_abs_sub {α : Type u} [lattice α] [add_comm_group α] [covariant_class α α has_add.add has_le.le] (a b : α) :

2 • (a ⊔ b) = a + b + |b - a|

theorem lattice_ordered_comm_group.sup_sq_eq_mul_mul_abs_div {α : Type u} [lattice α] [comm_group α] [covariant_class α α has_mul.mul has_le.le] (a b : α) :

(a ⊔ b) ^ 2 = a * b * |b / a|

theorem lattice_ordered_comm_group.inf_sq_eq_mul_div_abs_div {α : Type u} [lattice α] [comm_group α] [covariant_class α α has_mul.mul has_le.le] (a b : α) :

(a ⊓ b) ^ 2 = a * b / |b / a|

theorem lattice_ordered_comm_group.two_inf_eq_add_sub_abs_sub {α : Type u} [lattice α] [add_comm_group α] [covariant_class α α has_add.add has_le.le] (a b : α) :

2 • (a ⊓ b) = a + b - |b - a|

def lattice_ordered_comm_group.lattice_ordered_add_comm_group_to_distrib_lattice (α : Type u) [s : lattice α] [add_comm_group α] [covariant_class α α has_add.add has_le.le] :

distrib_lattice α

Every lattice ordered commutative additive group is a distributive lattice

Equations

lattice_ordered_comm_group.lattice_ordered_add_comm_group_to_distrib_lattice α = {sup := lattice.sup s, le := lattice.le s, lt := lattice.lt s, le_refl := _, le_trans := _, lt_iff_le_not_le := _, le_antisymm := _, le_sup_left := _, le_sup_right := _, sup_le := _, inf := lattice.inf s, inf_le_left := _, inf_le_right := _, le_inf := _, le_sup_inf := _}

def lattice_ordered_comm_group.lattice_ordered_comm_group_to_distrib_lattice (α : Type u) [s : lattice α] [comm_group α] [covariant_class α α has_mul.mul has_le.le] :

distrib_lattice α

Every lattice ordered commutative group is a distributive lattice

Equations

lattice_ordered_comm_group.lattice_ordered_comm_group_to_distrib_lattice α = {sup := lattice.sup s, le := lattice.le s, lt := lattice.lt s, le_refl := _, le_trans := _, lt_iff_le_not_le := _, le_antisymm := _, le_sup_left := _, le_sup_right := _, sup_le := _, inf := lattice.inf s, inf_le_left := _, inf_le_right := _, le_inf := _, le_sup_inf := _}

theorem lattice_ordered_comm_group.abs_div_sup_mul_abs_div_inf {α : Type u} [lattice α] [comm_group α] [covariant_class α α has_mul.mul has_le.le] (a b c : α) :

|(a ⊔ c) / (b ⊔ c)| * |(a ⊓ c) / (b ⊓ c)| = |a / b|

theorem lattice_ordered_comm_group.abs_sub_sup_add_abs_sub_inf {α : Type u} [lattice α] [add_comm_group α] [covariant_class α α has_add.add has_le.le] (a b c : α) :

|a ⊔ c - b ⊔ c| + |a ⊓ c - b ⊓ c| = |a - b|

theorem lattice_ordered_comm_group.pos_of_nonneg {α : Type u} [lattice α] [add_comm_group α] (a : α) (h : 0 ≤ a) :

a⁺ = a

If a is positive, then it is equal to its positive component a⁺.

theorem lattice_ordered_comm_group.pos_of_one_le {α : Type u} [lattice α] [comm_group α] (a : α) (h : 1 ≤ a) :

a⁺ = a

If a is positive, then it is equal to its positive component a⁺.

theorem lattice_ordered_comm_group.pos_eq_self_of_one_lt_pos {α : Type u_1} [linear_order α] [comm_group α] {x : α} (hx : 1 < x⁺) :

x⁺ = x

theorem lattice_ordered_comm_group.pos_eq_self_of_pos_pos {α : Type u_1} [linear_order α] [add_comm_group α] {x : α} (hx : 0 < x⁺) :

x⁺ = x

theorem lattice_ordered_comm_group.pos_of_nonpos {α : Type u} [lattice α] [add_comm_group α] (a : α) (h : a ≤ 0) :

a⁺ = 0

theorem lattice_ordered_comm_group.pos_of_le_one {α : Type u} [lattice α] [comm_group α] (a : α) (h : a ≤ 1) :

a⁺ = 1

theorem lattice_ordered_comm_group.neg_of_one_le_inv {α : Type u} [lattice α] [comm_group α] (a : α) (h : 1 ≤ a⁻¹) :

theorem lattice_ordered_comm_group.neg_of_inv_nonneg {α : Type u} [lattice α] [add_comm_group α] (a : α) (h : 0 ≤ -a) :

theorem lattice_ordered_comm_group.neg_of_inv_le_one {α : Type u} [lattice α] [comm_group α] (a : α) (h : a⁻¹ ≤ 1) :

a⁻ = 1

theorem lattice_ordered_comm_group.neg_of_neg_nonpos {α : Type u} [lattice α] [add_comm_group α] (a : α) (h : -a ≤ 0) :

a⁻ = 0

theorem lattice_ordered_comm_group.neg_of_nonpos {α : Type u} [lattice α] [add_comm_group α] [covariant_class α α has_add.add has_le.le] (a : α) (h : a ≤ 0) :

theorem lattice_ordered_comm_group.neg_of_le_one {α : Type u} [lattice α] [comm_group α] [covariant_class α α has_mul.mul has_le.le] (a : α) (h : a ≤ 1) :

theorem lattice_ordered_comm_group.neg_of_one_le {α : Type u} [lattice α] [comm_group α] [covariant_class α α has_mul.mul has_le.le] (a : α) (h : 1 ≤ a) :

a⁻ = 1

theorem lattice_ordered_comm_group.neg_of_nonneg {α : Type u} [lattice α] [add_comm_group α] [covariant_class α α has_add.add has_le.le] (a : α) (h : 0 ≤ a) :

a⁻ = 0

theorem lattice_ordered_comm_group.mabs_of_one_le {α : Type u} [lattice α] [comm_group α] [covariant_class α α has_mul.mul has_le.le] (a : α) (h : 1 ≤ a) :

|a| = a

theorem lattice_ordered_comm_group.abs_of_nonneg {α : Type u} [lattice α] [add_comm_group α] [covariant_class α α has_add.add has_le.le] (a : α) (h : 0 ≤ a) :

|a| = a

@[simp]

theorem lattice_ordered_comm_group.mabs_mabs {α : Type u} [lattice α] [comm_group α] [covariant_class α α has_mul.mul has_le.le] (a : α) :

The unary operation of taking the absolute value is idempotent.

@[simp]

theorem lattice_ordered_comm_group.abs_abs {α : Type u} [lattice α] [add_comm_group α] [covariant_class α α has_add.add has_le.le] (a : α) :

The unary operation of taking the absolute value is idempotent.

theorem lattice_ordered_comm_group.mabs_sup_div_sup_le_mabs {α : Type u} [lattice α] [comm_group α] [covariant_class α α has_mul.mul has_le.le] (a b c : α) :

|(a ⊔ c) / (b ⊔ c)| ≤ |a / b|

theorem lattice_ordered_comm_group.abs_sup_sub_sup_le_abs {α : Type u} [lattice α] [add_comm_group α] [covariant_class α α has_add.add has_le.le] (a b c : α) :

|a ⊔ c - b ⊔ c| ≤ |a - b|

theorem lattice_ordered_comm_group.abs_inf_sub_inf_le_abs {α : Type u} [lattice α] [add_comm_group α] [covariant_class α α has_add.add has_le.le] (a b c : α) :

|a ⊓ c - b ⊓ c| ≤ |a - b|

theorem lattice_ordered_comm_group.mabs_inf_div_inf_le_mabs {α : Type u} [lattice α] [comm_group α] [covariant_class α α has_mul.mul has_le.le] (a b c : α) :

|(a ⊓ c) / (b ⊓ c)| ≤ |a / b|

theorem lattice_ordered_comm_group.m_Birkhoff_inequalities {α : Type u} [lattice α] [comm_group α] [covariant_class α α has_mul.mul has_le.le] (a b c : α) :

|(a ⊔ c) / (b ⊔ c)| ⊔ |(a ⊓ c) / (b ⊓ c)| ≤ |a / b|

theorem lattice_ordered_comm_group.Birkhoff_inequalities {α : Type u} [lattice α] [add_comm_group α] [covariant_class α α has_add.add has_le.le] (a b c : α) :

|a ⊔ c - b ⊔ c| ⊔ |a ⊓ c - b ⊓ c| ≤ |a - b|

theorem lattice_ordered_comm_group.mabs_mul_le {α : Type u} [lattice α] [comm_group α] [covariant_class α α has_mul.mul has_le.le] (a b : α) :

|a * b| ≤ |a| * |b|

The absolute value satisfies the triangle inequality.

theorem lattice_ordered_comm_group.abs_add_le {α : Type u} [lattice α] [add_comm_group α] [covariant_class α α has_add.add has_le.le] (a b : α) :

|a + b| ≤ |a| + |b|

The absolute value satisfies the triangle inequality.

theorem lattice_ordered_comm_group.abs_neg_comm {α : Type u} [lattice α] [add_comm_group α] (a b : α) :

|a - b| = |b - a|

theorem lattice_ordered_comm_group.abs_inv_comm {α : Type u} [lattice α] [comm_group α] (a b : α) :

|a / b| = |b / a|

theorem lattice_ordered_comm_group.abs_abs_div_abs_le {α : Type u} [lattice α] [comm_group α] [covariant_class α α has_mul.mul has_le.le] (a b : α) :

||a| / |b|| ≤ |a / b|

theorem lattice_ordered_comm_group.abs_abs_sub_abs_le {α : Type u} [lattice α] [add_comm_group α] [covariant_class α α has_add.add has_le.le] (a b : α) :

||a| - |b|| ≤ |a - b|

def lattice_ordered_add_comm_group.is_solid {β : Type u} [lattice β] [add_comm_group β] (s : set β) :

Prop

A subset s ⊆ β, with β an add_comm_group with a lattice structure, is solid if for all x ∈ s and all y ∈ β such that |y| ≤ |x|, then y ∈ s.

Equations

lattice_ordered_add_comm_group.is_solid s = ∀ ⦃x : β⦄, x ∈ s → ∀ ⦃y : β⦄, |y| ≤ |x| → y ∈ s

def lattice_ordered_add_comm_group.solid_closure {β : Type u} [lattice β] [add_comm_group β] (s : set β) :

set β

The solid closure of a subset s is the smallest superset of s that is solid.

Equations

lattice_ordered_add_comm_group.solid_closure s = {y : β | ∃ (x : β) (H : x ∈ s), |y| ≤ |x|}

theorem lattice_ordered_add_comm_group.is_solid_solid_closure {β : Type u} [lattice β] [add_comm_group β] (s : set β) :

lattice_ordered_add_comm_group.is_solid (lattice_ordered_add_comm_group.solid_closure s)

theorem lattice_ordered_add_comm_group.solid_closure_min {β : Type u} [lattice β] [add_comm_group β] (s t : set β) (h1 : s ⊆ t) (h2 : lattice_ordered_add_comm_group.is_solid t) :

lattice_ordered_add_comm_group.solid_closure s ⊆ t