mathlib documentation

algebra.linear_ordered_comm_group_with_zero

Linearly ordered commutative groups with a zero element adjoined

This file sets up a special class of linearly ordered commutative monoids that show up as the target of so-called “valuations” in algebraic number theory.

Usually, in the informal literature, these objects are constructed by taking a linearly ordered commutative group Γ and formally adjoining a zero element: Γ ∪ {0}.

The disadvantage is that a type such as nnreal is not of that form, whereas it is a very common target for valuations. The solutions is to use a typeclass, and that is exactly what we do in this file.

@[instance]

def linear_ordered_comm_group_with_zero.to_linear_order (α : Type u_1) [s : linear_ordered_comm_group_with_zero α] :

linear_order α

@[instance]

def linear_ordered_comm_group_with_zero.to_comm_group_with_zero (α : Type u_1) [s : linear_ordered_comm_group_with_zero α] :

comm_group_with_zero α

@[class]

structure linear_ordered_comm_group_with_zero :

Type u_1 → Type u_1

le : α → α → Prop
lt : α → α → Prop
le_refl : ∀ (a : α), a ≤ a
le_trans : ∀ (a b c_1 : α), a ≤ b → b ≤ c_1 → a ≤ c_1
lt_iff_le_not_le : (∀ (a b : α), a < b ↔ a ≤ b ∧ ¬b ≤ a) . "order_laws_tac"
le_antisymm : ∀ (a b : α), a ≤ b → b ≤ a → a = b
le_total : ∀ (a b : α), a ≤ b ∨ b ≤ a
decidable_le : decidable_rel has_le.le
decidable_eq : decidable_eq α
decidable_lt : decidable_rel has_lt.lt
mul : α → α → α
mul_assoc : ∀ (a b c_1 : α), (a * b) * c_1 = a * b * c_1
one : α
one_mul : ∀ (a : α), 1 * a = a
mul_one : ∀ (a : α), a * 1 = a
mul_comm : ∀ (a b : α), a * b = b * a
zero : α
zero_mul : ∀ (a : α), 0 * a = 0
mul_zero : ∀ (a : α), a * 0 = 0
inv : α → α
exists_pair_ne : ∃ (x y : α), x ≠ y
inv_zero : 0⁻¹ = 0
mul_inv_cancel : ∀ (a : α), a ≠ 0 → a * a⁻¹ = 1
mul_le_mul_left : ∀ {a b : α}, a ≤ b → ∀ (c_1 : α), c_1 * a ≤ c_1 * b
zero_le_one : 0 ≤ 1

A linearly ordered commutative group with a zero element.

Instances

nnreal.linear_ordered_comm_group_with_zero

@[instance]

def linear_ordered_comm_group_with_zero.to_ordered_comm_monoid {α : Type u_1} [linear_ordered_comm_group_with_zero α] :

ordered_comm_monoid α

Every linearly ordered commutative group with zero is an ordered commutative monoid.

Equations

linear_ordered_comm_group_with_zero.to_ordered_comm_monoid = {mul := linear_ordered_comm_group_with_zero.mul _inst_1, mul_assoc := _, one := linear_ordered_comm_group_with_zero.one _inst_1, one_mul := _, mul_one := _, mul_comm := _, le := linear_ordered_comm_group_with_zero.le _inst_1, lt := linear_ordered_comm_group_with_zero.lt _inst_1, le_refl := _, le_trans := _, lt_iff_le_not_le := _, le_antisymm := _, mul_le_mul_left := _, lt_of_mul_lt_mul_left := _}

theorem one_le_pow_of_one_le' {α : Type u_1} [linear_ordered_comm_group_with_zero α] {x : α} {n : ℕ} :

1 ≤ x → 1 ≤ x ^ n

theorem pow_le_one_of_le_one {α : Type u_1} [linear_ordered_comm_group_with_zero α] {x : α} {n : ℕ} :

x ≤ 1 → x ^ n ≤ 1

theorem eq_one_of_pow_eq_one {α : Type u_1} [linear_ordered_comm_group_with_zero α] {x : α} {n : ℕ} :

n ≠ 0 → x ^ n = 1 → x = 1

theorem pow_eq_one_iff {α : Type u_1} [linear_ordered_comm_group_with_zero α] {x : α} {n : ℕ} :

n ≠ 0 → (x ^ n = 1 ↔ x = 1)

theorem one_le_pow_iff {α : Type u_1} [linear_ordered_comm_group_with_zero α] {x : α} {n : ℕ} :

n ≠ 0 → (1 ≤ x ^ n ↔ 1 ≤ x)

theorem pow_le_one_iff {α : Type u_1} [linear_ordered_comm_group_with_zero α] {x : α} {n : ℕ} :

n ≠ 0 → (x ^ n ≤ 1 ↔ x ≤ 1)

theorem zero_le_one' {α : Type u_1} [linear_ordered_comm_group_with_zero α] :

0 ≤ 1

theorem zero_lt_one'' {α : Type u_1} [linear_ordered_comm_group_with_zero α] :

0 < 1

@[simp]

theorem zero_le' {α : Type u_1} [linear_ordered_comm_group_with_zero α] {a : α} :

0 ≤ a

@[simp]

theorem not_lt_zero' {α : Type u_1} [linear_ordered_comm_group_with_zero α] {a : α} :

¬a < 0

@[simp]

theorem le_zero_iff {α : Type u_1} [linear_ordered_comm_group_with_zero α] {a : α} :

a ≤ 0 ↔ a = 0

theorem zero_lt_iff {α : Type u_1} [linear_ordered_comm_group_with_zero α] {a : α} :

0 < a ↔ a ≠ 0

theorem le_of_le_mul_right {α : Type u_1} [linear_ordered_comm_group_with_zero α] {a b c : α} :

c ≠ 0 → a * c ≤ b * c → a ≤ b

theorem le_mul_inv_of_mul_le {α : Type u_1} [linear_ordered_comm_group_with_zero α] {a b c : α} :

c ≠ 0 → a * c ≤ b → a ≤ b * c⁻¹

theorem mul_inv_le_of_le_mul {α : Type u_1} [linear_ordered_comm_group_with_zero α] {a b c : α} :

c ≠ 0 → a ≤ b * c → a * c⁻¹ ≤ b

theorem div_le_div' {α : Type u_1} [linear_ordered_comm_group_with_zero α] (a b c d : α) :

b ≠ 0 → d ≠ 0 → (a * b⁻¹ ≤ c * d⁻¹ ↔ a * d ≤ c * b)

theorem ne_zero_of_lt {α : Type u_1} [linear_ordered_comm_group_with_zero α] {a b : α} :

b < a → a ≠ 0

@[simp]

theorem units.zero_lt {α : Type u_1} [linear_ordered_comm_group_with_zero α] (u : units α) :

0 < ↑u

theorem mul_lt_mul'''' {α : Type u_1} [linear_ordered_comm_group_with_zero α] {a b c d : α} :

a < b → c < d → a * c < b * d

theorem mul_inv_lt_of_lt_mul' {α : Type u_1} [linear_ordered_comm_group_with_zero α] {x y z : α} :

x < y * z → x * z⁻¹ < y

theorem mul_lt_right' {α : Type u_1} [linear_ordered_comm_group_with_zero α] {a b : α} (c : α) :

a < b → c ≠ 0 → a * c < b * c

theorem pow_lt_pow_succ {α : Type u_1} [linear_ordered_comm_group_with_zero α] {x : α} {n : ℕ} :

1 < x → x ^ n < x ^ n.succ

theorem pow_lt_pow' {α : Type u_1} [linear_ordered_comm_group_with_zero α] {x : α} {m n : ℕ} :

1 < x → m < n → x ^ m < x ^ n

theorem inv_lt_inv'' {α : Type u_1} [linear_ordered_comm_group_with_zero α] {a b : α} :

a ≠ 0 → b ≠ 0 → (a⁻¹ < b⁻¹ ↔ b < a)

theorem inv_le_inv'' {α : Type u_1} [linear_ordered_comm_group_with_zero α] {a b : α} :

a ≠ 0 → b ≠ 0 → (a⁻¹ ≤ b⁻¹ ↔ b ≤ a)