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.

@[class]
structure linear_ordered_comm_group_with_zero  :
Type u_1Type u_1
  • le : α → α → Prop
  • lt : α → α → Prop
  • le_refl : ∀ (a : α), a a
  • le_trans : ∀ (a b c_1 : α), a bb c_1a 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 bb aa = 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 0a * 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
theorem one_le_pow_of_one_le' {α : Type u_1} [linear_ordered_comm_group_with_zero α] {x : α} {n : } :
1 x1 x ^ n

theorem pow_le_one_of_le_one {α : Type u_1} [linear_ordered_comm_group_with_zero α] {x : α} {n : } :
x 1x ^ n 1

theorem eq_one_of_pow_eq_one {α : Type u_1} [linear_ordered_comm_group_with_zero α] {x : α} {n : } :
n 0x ^ n = 1x = 1

theorem pow_eq_one_iff {α : Type u_1} [linear_ordered_comm_group_with_zero α] {x : α} {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 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 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 0a * c b * ca b

theorem le_mul_inv_of_mul_le {α : Type u_1} [linear_ordered_comm_group_with_zero α] {a b c : α} :
c 0a * c ba 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 0a b * ca * c⁻¹ b

theorem div_le_div' {α : Type u_1} [linear_ordered_comm_group_with_zero α] (a b c d : α) :
b 0d 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 < aa 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 < bc < da * 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 * zx * z⁻¹ < y

theorem mul_lt_right' {α : Type u_1} [linear_ordered_comm_group_with_zero α] {a b : α} (c : α) :
a < bc 0a * c < b * c

theorem pow_lt_pow_succ {α : Type u_1} [linear_ordered_comm_group_with_zero α] {x : α} {n : } :
1 < xx ^ n < x ^ n.succ

theorem pow_lt_pow' {α : Type u_1} [linear_ordered_comm_group_with_zero α] {x : α} {m n : } :
1 < xm < nx ^ m < x ^ n

theorem inv_lt_inv'' {α : Type u_1} [linear_ordered_comm_group_with_zero α] {a b : α} :
a 0b 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 0b 0(a⁻¹ b⁻¹ b a)