# mathlibdocumentation

algebra.ordered_group

# Ordered groups #

This file develops the basics of ordered groups.

## Implementation details #

Unfortunately, the number of `'` appended to lemmas in this file may differ between the multiplicative and the additive version of a lemma. The reason is that we did not want to change existing names in the library.

@[instance]
@[instance]
@[instance]
@[instance]
@[instance]
@[class]
structure ordered_add_comm_group (α : Type u) :
Type u
• add : α → α → α
• add_assoc : ∀ (a b c_1 : α), a + b + c_1 = a + (b + c_1)
• zero : α
• zero_add : ∀ (a : α), 0 + a = a
• add_zero : ∀ (a : α), a + 0 = a
• nsmul : α → α
• nsmul_zero' : (∀ (x : α), . "try_refl_tac"
• nsmul_succ' : (∀ (n : ) (x : α), . "try_refl_tac"
• neg : α → α
• sub : α → α → α
• sub_eq_add_neg : (∀ (a b : α), a - b = a + -b) . "try_refl_tac"
• gsmul : α → α
• gsmul_zero' : (∀ (a : α), . "try_refl_tac"
• gsmul_succ' : (∀ (n : ) (a : α), . "try_refl_tac"
• gsmul_neg' : (∀ (n : ) (a : α), . "try_refl_tac"
• add_left_neg : ∀ (a : α), -a + a = 0
• add_comm : ∀ (a b : α), a + b = b + a
• 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
• add_le_add_left : ∀ (a b : α), a b∀ (c_1 : α), c_1 + a c_1 + b

An ordered additive commutative group is an additive commutative group with a partial order in which addition is strictly monotone.

Instances
@[instance]
@[instance]
def ordered_comm_group.to_comm_group (α : Type u) [self : ordered_comm_group α] :
@[class]
structure ordered_comm_group (α : Type u) :
Type u
• 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
• npow : α → α
• npow_zero' : (∀ (x : α), = 1) . "try_refl_tac"
• npow_succ' : (∀ (n : ) (x : α), = x * . "try_refl_tac"
• inv : α → α
• div : α → α → α
• div_eq_mul_inv : (∀ (a b : α), a / b = a * b⁻¹) . "try_refl_tac"
• gpow : α → α
• gpow_zero' : (∀ (a : α), = 1) . "try_refl_tac"
• gpow_succ' : (∀ (n : ) (a : α), = a * . "try_refl_tac"
• gpow_neg' : (∀ (n : ) (a : α), = a)⁻¹) . "try_refl_tac"
• mul_left_inv : ∀ (a : α), a⁻¹ * a = 1
• mul_comm : ∀ (a b : α), a * b = b * a
• 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
• mul_le_mul_left : ∀ (a b : α), a b∀ (c_1 : α), c_1 * a c_1 * b

An ordered commutative group is an commutative group with a partial order in which multiplication is strictly monotone.

Instances
@[instance]
def ordered_comm_group.to_partial_order (α : Type u) [self : ordered_comm_group α] :
@[instance]
@[instance]
@[instance]
@[instance]
def units.ordered_comm_group {α : Type u}  :

The units of an ordered commutative monoid form an ordered commutative group.

Equations
@[instance]
@[instance]
Equations
@[instance]
@[instance]
@[simp]
theorem left.neg_nonpos_iff {α : Type u} [add_group α] [has_le α] {a : α} :
-a 0 0 a
@[simp]
theorem left.inv_le_one_iff {α : Type u} [group α] [has_le α] {a : α} :
a⁻¹ 1 1 a

Uses `left` co(ntra)variant.

@[simp]
theorem left.nonneg_neg_iff {α : Type u} [add_group α] [has_le α] {a : α} :
0 -a a 0
@[simp]
theorem left.one_le_inv_iff {α : Type u} [group α] [has_le α] {a : α} :
1 a⁻¹ a 1

Uses `left` co(ntra)variant.

@[simp]
b -a + c a + b c
@[simp]
theorem le_inv_mul_iff_mul_le {α : Type u} [group α] [has_le α] {a b c : α} :
b a⁻¹ * c a * b c
@[simp]
theorem inv_mul_le_iff_le_mul {α : Type u} [group α] [has_le α] {a b c : α} :
b⁻¹ * a c a b * c
@[simp]
-b + a c a b + c
theorem inv_le_iff_one_le_mul' {α : Type u} [group α] [has_le α] {a b : α} :
a⁻¹ b 1 a * b
theorem neg_le_iff_add_nonneg' {α : Type u} [add_group α] [has_le α] {a b : α} :
-a b 0 a + b
theorem le_inv_iff_mul_le_one_left {α : Type u} [group α] [has_le α] {a b : α} :
a b⁻¹ b * a 1
theorem le_neg_iff_add_nonpos_left {α : Type u} [add_group α] [has_le α] {a b : α} :
a -b b + a 0
theorem le_neg_add_iff_le {α : Type u} [add_group α] [has_le α] {a b : α} :
0 -b + a b a
theorem le_inv_mul_iff_le {α : Type u} [group α] [has_le α] {a b : α} :
1 b⁻¹ * a b a
theorem neg_add_nonpos_iff {α : Type u} [add_group α] [has_le α] {a b : α} :
-a + b 0 b a
theorem inv_mul_le_one_iff {α : Type u} [group α] [has_le α] {a b : α} :
a⁻¹ * b 1 b a
@[simp]
theorem left.one_lt_inv_iff {α : Type u} [group α] [has_lt α] {a : α} :
1 < a⁻¹ a < 1

Uses `left` co(ntra)variant.

@[simp]
theorem left.neg_pos_iff {α : Type u} [add_group α] [has_lt α] {a : α} :
0 < -a a < 0
@[simp]
theorem left.neg_neg_iff {α : Type u} [add_group α] [has_lt α] {a : α} :
-a < 0 0 < a
@[simp]
theorem left.inv_lt_one_iff {α : Type u} [group α] [has_lt α] {a : α} :
a⁻¹ < 1 1 < a

Uses `left` co(ntra)variant.

@[simp]
theorem lt_inv_mul_iff_mul_lt {α : Type u} [group α] [has_lt α] {a b c : α} :
b < a⁻¹ * c a * b < c
@[simp]
b < -a + c a + b < c
@[simp]
theorem inv_mul_lt_iff_lt_mul {α : Type u} [group α] [has_lt α] {a b c : α} :
b⁻¹ * a < c a < b * c
@[simp]
-b + a < c a < b + c
theorem inv_lt_iff_one_lt_mul' {α : Type u} [group α] [has_lt α] {a b : α} :
a⁻¹ < b 1 < a * b
theorem neg_lt_iff_pos_add' {α : Type u} [add_group α] [has_lt α] {a b : α} :
-a < b 0 < a + b
theorem lt_inv_iff_mul_lt_one' {α : Type u} [group α] [has_lt α] {a b : α} :
a < b⁻¹ b * a < 1
theorem lt_neg_iff_add_neg' {α : Type u} [add_group α] [has_lt α] {a b : α} :
a < -b b + a < 0
theorem lt_inv_mul_iff_lt {α : Type u} [group α] [has_lt α] {a b : α} :
1 < b⁻¹ * a b < a
theorem lt_neg_add_iff_lt {α : Type u} [add_group α] [has_lt α] {a b : α} :
0 < -b + a b < a
theorem inv_mul_lt_one_iff {α : Type u} [group α] [has_lt α] {a b : α} :
a⁻¹ * b < 1 b < a
theorem neg_add_neg_iff {α : Type u} [add_group α] [has_lt α] {a b : α} :
-a + b < 0 b < a
@[simp]
theorem right.neg_nonpos_iff {α : Type u} [add_group α] [has_le α] {a : α} :
-a 0 0 a
@[simp]
theorem right.inv_le_one_iff {α : Type u} [group α] [has_le α] {a : α} :
a⁻¹ 1 1 a

Uses `right` co(ntra)variant.

@[simp]
theorem right.nonneg_neg_iff {α : Type u} [add_group α] [has_le α] {a : α} :
0 -a a 0
@[simp]
theorem right.one_le_inv_iff {α : Type u} [group α] [has_le α] {a : α} :
1 a⁻¹ a 1

Uses `right` co(ntra)variant.

theorem inv_le_iff_one_le_mul {α : Type u} [group α] [has_le α] {a b : α} :
a⁻¹ b 1 b * a
theorem neg_le_iff_add_nonneg {α : Type u} [add_group α] [has_le α] {a b : α} :
-a b 0 b + a
theorem le_neg_iff_add_nonpos_right {α : Type u} [add_group α] [has_le α] {a b : α} :
a -b a + b 0
theorem le_inv_iff_mul_le_one_right {α : Type u} [group α] [has_le α] {a b : α} :
a b⁻¹ a * b 1
@[simp]
theorem mul_inv_le_iff_le_mul {α : Type u} [group α] [has_le α] {a b c : α} :
a * b⁻¹ c a c * b
@[simp]
a + -b c a c + b
@[simp]
theorem le_mul_inv_iff_mul_le {α : Type u} [group α] [has_le α] {a b c : α} :
c a * b⁻¹ c * b a
@[simp]
c a + -b c + b a
@[simp]
theorem mul_inv_le_one_iff_le {α : Type u} [group α] [has_le α] {a b : α} :
a * b⁻¹ 1 a b
@[simp]
theorem add_neg_nonpos_iff_le {α : Type u} [add_group α] [has_le α] {a b : α} :
a + -b 0 a b
theorem le_add_neg_iff_le {α : Type u} [add_group α] [has_le α] {a b : α} :
0 a + -b b a
theorem le_mul_inv_iff_le {α : Type u} [group α] [has_le α] {a b : α} :
1 a * b⁻¹ b a
theorem mul_inv_le_one_iff {α : Type u} [group α] [has_le α] {a b : α} :
b * a⁻¹ 1 b a
theorem add_neg_nonpos_iff {α : Type u} [add_group α] [has_le α] {a b : α} :
b + -a 0 b a
@[simp]
theorem right.neg_neg_iff {α : Type u} [add_group α] [has_lt α] {a : α} :
-a < 0 0 < a
@[simp]
theorem right.inv_lt_one_iff {α : Type u} [group α] [has_lt α] {a : α} :
a⁻¹ < 1 1 < a

Uses `right` co(ntra)variant.

@[simp]
theorem right.one_lt_inv_iff {α : Type u} [group α] [has_lt α] {a : α} :
1 < a⁻¹ a < 1

Uses `right` co(ntra)variant.

@[simp]
theorem right.neg_pos_iff {α : Type u} [add_group α] [has_lt α] {a : α} :
0 < -a a < 0
theorem neg_lt_iff_pos_add {α : Type u} [add_group α] [has_lt α] {a b : α} :
-a < b 0 < b + a
theorem inv_lt_iff_one_lt_mul {α : Type u} [group α] [has_lt α] {a b : α} :
a⁻¹ < b 1 < b * a
theorem lt_inv_iff_mul_lt_one {α : Type u} [group α] [has_lt α] {a b : α} :
a < b⁻¹ a * b < 1
theorem lt_neg_iff_add_neg {α : Type u} [add_group α] [has_lt α] {a b : α} :
a < -b a + b < 0
@[simp]
theorem mul_inv_lt_iff_lt_mul {α : Type u} [group α] [has_lt α] {a b c : α} :
a * b⁻¹ < c a < c * b
@[simp]
a + -b < c a < c + b
@[simp]
c < a + -b c + b < a
@[simp]
theorem lt_mul_inv_iff_mul_lt {α : Type u} [group α] [has_lt α] {a b c : α} :
c < a * b⁻¹ c * b < a
@[simp]
theorem neg_add_neg_iff_lt {α : Type u} [add_group α] [has_lt α] {a b : α} :
a + -b < 0 a < b
@[simp]
theorem inv_mul_lt_one_iff_lt {α : Type u} [group α] [has_lt α] {a b : α} :
a * b⁻¹ < 1 a < b
theorem lt_mul_inv_iff_lt {α : Type u} [group α] [has_lt α] {a b : α} :
1 < a * b⁻¹ b < a
theorem lt_add_neg_iff_lt {α : Type u} [add_group α] [has_lt α] {a b : α} :
0 < a + -b b < a
theorem add_neg_neg_iff {α : Type u} [add_group α] [has_lt α] {a b : α} :
b + -a < 0 b < a
theorem mul_inv_lt_one_iff {α : Type u} [group α] [has_lt α] {a b : α} :
b * a⁻¹ < 1 b < a
@[simp]
theorem neg_le_neg_iff {α : Type u} [add_group α] [has_le α] {a b : α} :
-a -b b a
@[simp]
theorem inv_le_inv_iff {α : Type u} [group α] [has_le α] {a b : α} :
theorem le_of_neg_le_neg {α : Type u} [add_group α] [has_le α] {a b : α} :
-a -bb a

Alias of `neg_le_neg_iff`.

theorem inv_le_of_inv_le {α : Type u} [group α] [has_le α] {a b : α} (h : a⁻¹ b) :
theorem neg_le_of_neg_le {α : Type u} [add_group α] [has_le α] {a b : α} (h : -a b) :
-b a
theorem neg_le {α : Type u} [add_group α] [has_le α] {a b : α} :
-a b -b a
theorem inv_le' {α : Type u} [group α] [has_le α] {a b : α} :
theorem le_neg {α : Type u} [add_group α] [has_le α] {a b : α} :
a -b b -a
theorem le_inv' {α : Type u} [group α] [has_le α] {a b : α} :
theorem mul_inv_le_inv_mul_iff {α : Type u} [group α] [has_le α] {a b c d : α} :
a * b⁻¹ d⁻¹ * c d * a c * b
theorem add_neg_le_neg_add_iff {α : Type u} [add_group α] [has_le α] {a b c d : α} :
a + -b -d + c d + a c + b
@[simp]
theorem sub_le_self_iff {α : Type u} [add_group α] [has_le α] (a : α) {b : α} :
a - b a 0 b
@[simp]
theorem div_le_self_iff {α : Type u} [group α] [has_le α] (a : α) {b : α} :
a / b a 1 b
theorem sub_le_self {α : Type u} [add_group α] [has_le α] (a : α) {b : α} :
0 ba - b a

Alias of `sub_le_self_iff`.

@[simp]
theorem inv_lt_inv_iff {α : Type u} [group α] [has_lt α] {a b : α} :
a⁻¹ < b⁻¹ b < a
@[simp]
theorem neg_lt_neg_iff {α : Type u} [add_group α] [has_lt α] {a b : α} :
-a < -b b < a
theorem lt_neg_of_lt_neg {α : Type u} [add_group α] [has_lt α] {a b : α} (h : a < -b) :
b < -a
theorem lt_inv_of_lt_inv {α : Type u} [group α] [has_lt α] {a b : α} (h : a < b⁻¹) :
b < a⁻¹
theorem inv_lt_of_inv_lt {α : Type u} [group α] [has_lt α] {a b : α} (h : a⁻¹ < b) :
b⁻¹ < a
theorem neg_lt_of_neg_lt {α : Type u} [add_group α] [has_lt α] {a b : α} (h : -a < b) :
-b < a
theorem neg_lt {α : Type u} [add_group α] [has_lt α] {a b : α} :
-a < b -b < a
theorem inv_lt' {α : Type u} [group α] [has_lt α] {a b : α} :
a⁻¹ < b b⁻¹ < a
theorem lt_inv' {α : Type u} [group α] [has_lt α] {a b : α} :
a < b⁻¹ b < a⁻¹
theorem lt_neg {α : Type u} [add_group α] [has_lt α] {a b : α} :
a < -b b < -a
theorem mul_inv_lt_inv_mul_iff {α : Type u} [group α] [has_lt α] {a b c d : α} :
a * b⁻¹ < d⁻¹ * c d * a < c * b
theorem add_neg_lt_neg_add_iff {α : Type u} [add_group α] [has_lt α] {a b c d : α} :
a + -b < -d + c d + a < c + b
@[simp]
theorem div_lt_self_iff {α : Type u} [group α] [has_lt α] (a : α) {b : α} :
a / b < a 1 < b
@[simp]
theorem sub_lt_self_iff {α : Type u} [add_group α] [has_lt α] (a : α) {b : α} :
a - b < a 0 < b
theorem sub_lt_self {α : Type u} [add_group α] [has_lt α] (a : α) {b : α} :
0 < ba - b < a

Alias of `sub_lt_self_iff`.

theorem left.neg_le_self {α : Type u} [add_group α] [preorder α] {a : α} (h : 0 a) :
-a a
theorem left.inv_le_self {α : Type u} [group α] [preorder α] {a : α} (h : 1 a) :
theorem neg_le_self {α : Type u} [add_group α] [preorder α] {a : α} (h : 0 a) :
-a a

Alias of `left.neg_le_self`.

theorem left.self_le_neg {α : Type u} [add_group α] [preorder α] {a : α} (h : a 0) :
a -a
theorem left.self_le_inv {α : Type u} [group α] [preorder α] {a : α} (h : a 1) :
theorem left.inv_lt_self {α : Type u} [group α] [preorder α] {a : α} (h : 1 < a) :
a⁻¹ < a
theorem left.neg_lt_self {α : Type u} [add_group α] [preorder α] {a : α} (h : 0 < a) :
-a < a
theorem neg_lt_self {α : Type u} [add_group α] [preorder α] {a : α} (h : 0 < a) :
-a < a

Alias of `left.neg_lt_self`.

theorem left.self_lt_neg {α : Type u} [add_group α] [preorder α] {a : α} (h : a < 0) :
a < -a
theorem left.self_lt_inv {α : Type u} [group α] [preorder α] {a : α} (h : a < 1) :
a < a⁻¹
theorem right.neg_le_self {α : Type u} [add_group α] [preorder α] {a : α} (h : 0 a) :
-a a
theorem right.inv_le_self {α : Type u} [group α] [preorder α] {a : α} (h : 1 a) :
theorem right.self_le_neg {α : Type u} [add_group α] [preorder α] {a : α} (h : a 0) :
a -a
theorem right.self_le_inv {α : Type u} [group α] [preorder α] {a : α} (h : a 1) :
theorem right.neg_lt_self {α : Type u} [add_group α] [preorder α] {a : α} (h : 0 < a) :
-a < a
theorem right.inv_lt_self {α : Type u} [group α] [preorder α] {a : α} (h : 1 < a) :
a⁻¹ < a
theorem right.self_lt_inv {α : Type u} [group α] [preorder α] {a : α} (h : a < 1) :
a < a⁻¹
theorem right.self_lt_neg {α : Type u} [add_group α] [preorder α] {a : α} (h : a < 0) :
a < -a
theorem inv_mul_le_iff_le_mul' {α : Type u} [comm_group α] [has_le α] {a b c : α} :
c⁻¹ * a b a b * c
theorem neg_add_le_iff_le_add' {α : Type u} [has_le α] {a b c : α} :
-c + a b a b + c
@[simp]
theorem mul_inv_le_iff_le_mul' {α : Type u} [comm_group α] [has_le α] {a b c : α} :
a * b⁻¹ c a b * c
@[simp]
theorem add_neg_le_iff_le_add' {α : Type u} [has_le α] {a b c : α} :
a + -b c a b + c
theorem add_neg_le_add_neg_iff {α : Type u} [has_le α] {a b c d : α} :
a + -b c + -d a + d c + b
theorem mul_inv_le_mul_inv_iff' {α : Type u} [comm_group α] [has_le α] {a b c d : α} :
a * b⁻¹ c * d⁻¹ a * d c * b
theorem neg_add_lt_iff_lt_add' {α : Type u} [has_lt α] {a b c : α} :
-c + a < b a < b + c
theorem inv_mul_lt_iff_lt_mul' {α : Type u} [comm_group α] [has_lt α] {a b c : α} :
c⁻¹ * a < b a < b * c
@[simp]
theorem mul_inv_lt_iff_le_mul' {α : Type u} [comm_group α] [has_lt α] {a b c : α} :
a * b⁻¹ < c a < b * c
@[simp]
theorem add_neg_lt_iff_le_add' {α : Type u} [has_lt α] {a b c : α} :
a + -b < c a < b + c
theorem add_neg_lt_add_neg_iff {α : Type u} [has_lt α] {a b c d : α} :
a + -b < c + -d a + d < c + b
theorem mul_inv_lt_mul_inv_iff' {α : Type u} [comm_group α] [has_lt α] {a b c d : α} :
a * b⁻¹ < c * d⁻¹ a * d < c * b
theorem le_inv_of_le_inv {α : Type u} [group α] [has_le α] {a b : α} :
a b⁻¹b a⁻¹

Alias of `le_inv'`.

theorem le_neg_of_le_neg {α : Type u} [add_group α] [has_le α] {a b : α} :
a -bb -a
theorem one_le_of_inv_le_one {α : Type u} [group α] [has_le α] {a : α} :
a⁻¹ 11 a

Alias of `left.inv_le_one_iff`.

theorem nonneg_of_neg_nonpos {α : Type u} [add_group α] [has_le α] {a : α} :
-a 00 a
theorem le_one_of_one_le_inv {α : Type u} [group α] [has_le α] {a : α} :
1 a⁻¹a 1

Alias of `left.one_le_inv_iff`.

theorem nonpos_of_neg_nonneg {α : Type u} [add_group α] [has_le α] {a : α} :
0 -aa 0
theorem lt_of_inv_lt_inv {α : Type u} [group α] [has_lt α] {a b : α} :
a⁻¹ < b⁻¹b < a

Alias of `inv_lt_inv_iff`.

theorem lt_of_neg_lt_neg {α : Type u} [add_group α] [has_lt α] {a b : α} :
-a < -bb < a
theorem one_lt_of_inv_lt_one {α : Type u} [group α] [has_lt α] {a : α} :
a⁻¹ < 11 < a

Alias of `left.inv_lt_one_iff`.

theorem pos_of_neg_neg {α : Type u} [add_group α] [has_lt α] {a : α} :
-a < 00 < a
theorem inv_lt_one_iff_one_lt {α : Type u} [group α] [has_lt α] {a : α} :
a⁻¹ < 1 1 < a

Alias of `left.inv_lt_one_iff`.

theorem neg_neg_iff_pos {α : Type u} [add_group α] [has_lt α] {a : α} :
-a < 0 0 < a
theorem inv_lt_one' {α : Type u} [group α] [has_lt α] {a : α} :
a⁻¹ < 1 1 < a

Alias of `left.inv_lt_one_iff`.

theorem neg_lt_zero {α : Type u} [add_group α] [has_lt α] {a : α} :
-a < 0 0 < a
theorem inv_of_one_lt_inv {α : Type u} [group α] [has_lt α] {a : α} :
1 < a⁻¹a < 1

Alias of `left.one_lt_inv_iff`.

theorem neg_of_neg_pos {α : Type u} [add_group α] [has_lt α] {a : α} :
0 < -aa < 0
theorem one_lt_inv_of_inv {α : Type u} [group α] [has_lt α] {a : α} :
a < 11 < a⁻¹

Alias of `left.one_lt_inv_iff`.

theorem neg_pos_of_neg {α : Type u} [add_group α] [has_lt α] {a : α} :
a < 00 < -a
theorem mul_le_of_le_inv_mul {α : Type u} [group α] [has_le α] {a b c : α} :
b a⁻¹ * ca * b c

Alias of `le_inv_mul_iff_mul_le`.

b -a + ca + b c
theorem le_inv_mul_of_mul_le {α : Type u} [group α] [has_le α] {a b c : α} :
a * b cb a⁻¹ * c

Alias of `le_inv_mul_iff_mul_le`.

a + b cb -a + c
theorem inv_mul_le_of_le_mul {α : Type u} [group α] [has_le α] {a b c : α} :
a b * cb⁻¹ * a c

Alias of `inv_mul_le_iff_le_mul`.

theorem mul_lt_of_lt_inv_mul {α : Type u} [group α] [has_lt α] {a b c : α} :
b < a⁻¹ * ca * b < c

Alias of `lt_inv_mul_iff_mul_lt`.

b < -a + ca + b < c
theorem lt_inv_mul_of_mul_lt {α : Type u} [group α] [has_lt α] {a b c : α} :
a * b < cb < a⁻¹ * c

Alias of `lt_inv_mul_iff_mul_lt`.

a + b < cb < -a + c
theorem inv_mul_lt_of_lt_mul {α : Type u} [group α] [has_lt α] {a b c : α} :
a < b * cb⁻¹ * a < c

Alias of `inv_mul_lt_iff_lt_mul`.

theorem lt_mul_of_inv_mul_lt {α : Type u} [group α] [has_lt α] {a b c : α} :
b⁻¹ * a < ca < b * c

Alias of `inv_mul_lt_iff_lt_mul`.

-b + a < ca < b + c
a < b + c-b + a < c
theorem lt_mul_of_inv_mul_lt_left {α : Type u} [group α] [has_lt α] {a b c : α} :
b⁻¹ * a < ca < b * c

Alias of `lt_mul_of_inv_mul_lt`.

-b + a < ca < b + c
theorem inv_le_one' {α : Type u} [group α] [has_le α] {a : α} :
a⁻¹ 1 1 a

Alias of `left.inv_le_one_iff`.

theorem neg_nonpos {α : Type u} [add_group α] [has_le α] {a : α} :
-a 0 0 a
theorem one_le_inv' {α : Type u} [group α] [has_le α] {a : α} :
1 a⁻¹ a 1

Alias of `left.one_le_inv_iff`.

theorem neg_nonneg {α : Type u} [add_group α] [has_le α] {a : α} :
0 -a a 0
theorem one_lt_inv' {α : Type u} [group α] [has_lt α] {a : α} :
1 < a⁻¹ a < 1

Alias of `left.one_lt_inv_iff`.

theorem neg_pos {α : Type u} [add_group α] [has_lt α] {a : α} :
0 < -a a < 0
theorem ordered_comm_group.mul_lt_mul_left' {α : Type u_1} [has_mul α] [has_lt α] {b c : α} (bc : b < c) (a : α) :
a * b < a * c

Alias of `mul_lt_mul_left'`.

theorem ordered_add_comm_group.add_lt_add_left {α : Type u_1} [has_add α] [has_lt α] {b c : α} (bc : b < c) (a : α) :
a + b < a + c
theorem ordered_comm_group.le_of_mul_le_mul_left {α : Type u_1} [has_mul α] [has_le α] {a b c : α} (bc : a * b a * c) :
b c

Alias of `le_of_mul_le_mul_left'`.

theorem ordered_add_comm_group.le_of_add_le_add_left {α : Type u_1} [has_add α] [has_le α] {a b c : α} (bc : a + b a + c) :
b c
theorem ordered_comm_group.lt_of_mul_lt_mul_left {α : Type u_1} [has_mul α] [has_lt α] {a b c : α} (bc : a * b < a * c) :
b < c

Alias of `lt_of_mul_lt_mul_left'`.

theorem ordered_add_comm_group.lt_of_add_lt_add_left {α : Type u_1} [has_add α] [has_lt α] {a b c : α} (bc : a + b < a + c) :
b < c
def function.injective.ordered_add_comm_group {α : Type u} {β : Type u_1} [has_zero β] [has_add β] [has_neg β] [has_sub β] (f : β → α) (hf : function.injective f) (one : f 0 = 0) (mul : ∀ (x y : β), f (x + y) = f x + f y) (inv : ∀ (x : β), f (-x) = -f x) (div : ∀ (x y : β), f (x - y) = f x - f y) :

Pullback an `ordered_add_comm_group` under an injective map.

def function.injective.ordered_comm_group {α : Type u} {β : Type u_1} [has_one β] [has_mul β] [has_inv β] [has_div β] (f : β → α) (hf : function.injective f) (one : f 1 = 1) (mul : ∀ (x y : β), f (x * y) = (f x) * f y) (inv : ∀ (x : β), f x⁻¹ = (f x)⁻¹) (div : ∀ (x y : β), f (x / y) = f x / f y) :

Pullback an `ordered_comm_group` under an injective map. See note [reducible non-instances].

Equations
@[simp]
theorem div_le_div_iff_right {α : Type u} [group α] [has_le α] {a b : α} (c : α) :
a / c b / c a b
@[simp]
theorem sub_le_sub_iff_right {α : Type u} [add_group α] [has_le α] {a b : α} (c : α) :
a - c b - c a b
theorem sub_le_sub_right {α : Type u} [add_group α] [has_le α] {a b : α} (h : a b) (c : α) :
a - c b - c
theorem div_le_div_right' {α : Type u} [group α] [has_le α] {a b : α} (h : a b) (c : α) :
a / c b / c
@[simp]
theorem sub_nonneg {α : Type u} [add_group α] [has_le α] {a b : α} :
0 a - b b a
@[simp]
theorem one_le_div' {α : Type u} [group α] [has_le α] {a b : α} :
1 a / b b a
theorem le_of_sub_nonneg {α : Type u} [add_group α] [has_le α] {a b : α} :
0 a - bb a

Alias of `sub_nonneg`.

theorem sub_nonneg_of_le {α : Type u} [add_group α] [has_le α] {a b : α} :
b a0 a - b

Alias of `sub_nonneg`.

@[simp]
theorem div_le_one' {α : Type u} [group α] [has_le α] {a b : α} :
a / b 1 a b
@[simp]
theorem sub_nonpos {α : Type u} [add_group α] [has_le α] {a b : α} :
a - b 0 a b
theorem sub_nonpos_of_le {α : Type u} [add_group α] [has_le α] {a b : α} :
a ba - b 0

Alias of `sub_nonpos`.

theorem le_of_sub_nonpos {α : Type u} [add_group α] [has_le α] {a b : α} :
a - b 0a b

Alias of `sub_nonpos`.

theorem le_div_iff_mul_le {α : Type u} [group α] [has_le α] {a b c : α} :
a c / b a * b c
theorem le_sub_iff_add_le {α : Type u} [add_group α] [has_le α] {a b c : α} :
a c - b a + b c
theorem add_le_of_le_sub_right {α : Type u} [add_group α] [has_le α] {a b c : α} :
a c - ba + b c

Alias of `le_sub_iff_add_le`.

theorem le_sub_right_of_add_le {α : Type u} [add_group α] [has_le α] {a b c : α} :
a + b ca c - b

Alias of `le_sub_iff_add_le`.

theorem sub_le_iff_le_add {α : Type u} [add_group α] [has_le α] {a b c : α} :
a - c b a b + c
theorem div_le_iff_le_mul {α : Type u} [group α] [has_le α] {a b c : α} :
a / c b a b * c
@[simp]
theorem order_iso.mul_right_apply {α : Type u} [group α] [has_le α] (a ᾰ : α) :
= * a
@[simp]
theorem order_iso.mul_right_symm_apply {α : Type u} [group α] [has_le α] (a ᾰ : α) :
= * a⁻¹
def order_iso.mul_right {α : Type u} [group α] [has_le α] (a : α) :
α ≃o α

`equiv.mul_right` as an order_iso.

Equations
@[simp]
theorem order_iso.mul_left_apply {α : Type u} [group α] [has_le α] (a ᾰ : α) :
= a *
@[simp]
theorem order_iso.mul_left_symm_apply {α : Type u} [group α] [has_le α] (a ᾰ : α) :
= a⁻¹ *
def order_iso.mul_left {α : Type u} [group α] [has_le α] (a : α) :
α ≃o α

`equiv.mul_left` as an order_iso.

Equations
@[simp]
theorem div_le_div_iff_left {α : Type u} [group α] [has_le α] {b c : α} (a : α) :
a / b a / c c b
@[simp]
theorem sub_le_sub_iff_left {α : Type u} [add_group α] [has_le α] {b c : α} (a : α) :
a - b a - c c b
theorem div_le_div_left' {α : Type u} [group α] [has_le α] {a b : α} (h : a b) (c : α) :
c / b c / a
theorem sub_le_sub_left {α : Type u} [add_group α] [has_le α] {a b : α} (h : a b) (c : α) :
c - b c - a
theorem sub_le_sub_iff {α : Type u} [has_le α] {a b c d : α} :
a - b c - d a + d c + b
theorem div_le_div_iff' {α : Type u} [comm_group α] [has_le α] {a b c d : α} :
a / b c / d a * d c * b
theorem le_sub_iff_add_le' {α : Type u} [has_le α] {a b c : α} :
b c - a a + b c
theorem le_div_iff_mul_le' {α : Type u} [comm_group α] [has_le α] {a b c : α} :
b c / a a * b c
theorem add_le_of_le_sub_left {α : Type u} [has_le α] {a b c : α} :
b c - aa + b c

Alias of `le_sub_iff_add_le'`.

theorem le_sub_left_of_add_le {α : Type u} [has_le α] {a b c : α} :
a + b cb c - a

Alias of `le_sub_iff_add_le'`.

theorem sub_le_iff_le_add' {α : Type u} [has_le α] {a b c : α} :
a - b c a b + c
theorem div_le_iff_le_mul' {α : Type u} [comm_group α] [has_le α] {a b c : α} :
a / b c a b * c
theorem le_add_of_sub_left_le {α : Type u} [has_le α] {a b c : α} :
a - b ca b + c

Alias of `sub_le_iff_le_add'`.

theorem sub_left_le_of_le_add {α : Type u} [has_le α] {a b c : α} :
a b + ca - b c

Alias of `sub_le_iff_le_add'`.

@[simp]
theorem neg_le_sub_iff_le_add {α : Type u} [has_le α] {a b c : α} :
-b a - c c a + b
@[simp]
theorem inv_le_div_iff_le_mul {α : Type u} [comm_group α] [has_le α] {a b c : α} :
b⁻¹ a / c c a * b
theorem inv_le_div_iff_le_mul' {α : Type u} [comm_group α] [has_le α] {a b c : α} :
a⁻¹ b / c c a * b
theorem neg_le_sub_iff_le_add' {α : Type u} [has_le α] {a b c : α} :
-a b - c c a + b
theorem sub_le {α : Type u} [has_le α] {a b c : α} :
a - b c a - c b
theorem div_le'' {α : Type u} [comm_group α] [has_le α] {a b c : α} :
a / b c a / c b
theorem le_sub {α : Type u} [has_le α] {a b c : α} :
a b - c c b - a
theorem le_div'' {α : Type u} [comm_group α] [has_le α] {a b c : α} :
a b / c c b / a
theorem div_le_div'' {α : Type u} [comm_group α] [preorder α] {a b c d : α} (hab : a b) (hcd : c d) :
a / d b / c
theorem sub_le_sub {α : Type u} [preorder α] {a b c d : α} (hab : a b) (hcd : c d) :
a - d b - c
@[simp]
theorem sub_lt_sub_iff_right {α : Type u} [add_group α] [has_lt α] {a b : α} (c : α) :
a - c < b - c a < b
@[simp]
theorem div_lt_div_iff_right {α : Type u} [group α] [has_lt α] {a b : α} (c : α) :
a / c < b / c a < b
theorem div_lt_div_right' {α : Type u} [group α] [has_lt α] {a b : α} (h : a < b) (c : α) :
a / c < b / c
theorem sub_lt_sub_right {α : Type u} [add_group α] [has_lt α] {a b : α} (h : a < b) (c : α) :
a - c < b - c
@[simp]
theorem sub_pos {α : Type u} [add_group α] [has_lt α] {a b : α} :
0 < a - b b < a
@[simp]
theorem one_lt_div' {α : Type u} [group α] [has_lt α] {a b : α} :
1 < a / b b < a
theorem lt_of_sub_pos {α : Type u} [add_group α] [has_lt α] {a b : α} :
0 < a - bb < a

Alias of `sub_pos`.

theorem sub_pos_of_lt {α : Type u} [add_group α] [has_lt α] {a b : α} :
b < a0 < a - b

Alias of `sub_pos`.

@[simp]
theorem div_lt_one' {α : Type u} [group α] [has_lt α] {a b : α} :
a / b < 1 a < b
@[simp]
theorem sub_neg {α : Type u} [add_group α] [has_lt α] {a b : α} :
a - b < 0 a < b
theorem lt_of_sub_neg {α : Type u} [add_group α] [has_lt α] {a b : α} :
a - b < 0a < b

Alias of `sub_neg`.

theorem sub_neg_of_lt {α : Type u} [add_group α] [has_lt α] {a b : α} :
a < ba - b < 0

Alias of `sub_neg`.

theorem sub_lt_zero {α : Type u} [add_group α] [has_lt α] {a b : α} :
a - b < 0 a < b

Alias of `sub_neg`.

theorem lt_sub_iff_add_lt {α : Type u} [add_group α] [has_lt α] {a b c : α} :
a < c - b a + b < c
theorem lt_div_iff_mul_lt {α : Type u} [group α] [has_lt α] {a b c : α} :
a < c / b a * b < c
theorem add_lt_of_lt_sub_right {α : Type u} [add_group α] [has_lt α] {a b c : α} :
a < c - ba + b < c

Alias of `lt_sub_iff_add_lt`.

theorem lt_sub_right_of_add_lt {α : Type u} [add_group α] [has_lt α] {a b c : α} :
a + b < ca < c - b

Alias of `lt_sub_iff_add_lt`.

theorem sub_lt_iff_lt_add {α : Type u} [add_group α] [has_lt α] {a b c : α} :
a - c < b a < b + c
theorem div_lt_iff_lt_mul {α : Type u} [group α] [has_lt α] {a b c : α} :
a / c < b a < b * c
theorem lt_add_of_sub_right_lt {α : Type u} [add_group α] [has_lt α] {a b c : α} :
a - c < ba < b + c

Alias of `sub_lt_iff_lt_add`.

theorem sub_right_lt_of_lt_add {α : Type u} [add_group α] [has_lt α] {a b c : α} :
a < b + ca - c < b

Alias of `sub_lt_iff_lt_add`.

@[simp]
theorem div_lt_div_iff_left {α : Type u} [group α] [has_lt α] {b c : α} (a : α) :
a / b < a / c c < b
@[simp]
theorem sub_lt_sub_iff_left {α : Type u} [add_group α] [has_lt α] {b c : α} (a : α) :
a - b < a - c c < b
@[simp]
theorem neg_lt_sub_iff_lt_add {α : Type u} [add_group α] [has_lt α] {a b c : α} :
-a < b - c c < a + b
@[simp]
theorem inv_lt_div_iff_lt_mul {α : Type u} [group α] [has_lt α] {a b c : α} :
a⁻¹ < b / c c < a * b
theorem sub_lt_sub_left {α : Type u} [add_group α] [has_lt α] {a b : α} (h : a < b) (c : α) :
c - b < c - a
theorem div_lt_div_left' {α : Type u} [group α] [has_lt α] {a b : α} (h : a < b) (c : α) :
c / b < c / a
theorem div_lt_div_iff' {α : Type u} [comm_group α] [has_lt α] {a b c d : α} :
a / b < c / d a * d < c * b
theorem sub_lt_sub_iff {α : Type u} [has_lt α] {a b c d : α} :
a - b < c - d a + d < c + b
theorem lt_div_iff_mul_lt' {α : Type u} [comm_group α] [has_lt α] {a b c : α} :
b < c / a a * b < c
theorem lt_sub_iff_add_lt' {α : Type u} [has_lt α] {a b c : α} :
b < c - a a + b < c
theorem add_lt_of_lt_sub_left {α : Type u} [has_lt α] {a b c : α} :
b < c - aa + b < c

Alias of `lt_sub_iff_add_lt'`.

theorem lt_sub_left_of_add_lt {α : Type u} [has_lt α] {a b c : α} :
a + b < cb < c - a

Alias of `lt_sub_iff_add_lt'`.

theorem div_lt_iff_lt_mul' {α : Type u} [comm_group α] [has_lt α] {a b c : α} :
a / b < c a < b * c
theorem sub_lt_iff_lt_add' {α : Type u} [has_lt α] {a b c : α} :
a - b < c a < b + c
theorem sub_left_lt_of_lt_add {α : Type u} [has_lt α] {a b c : α} :
a < b + ca - b < c

Alias of `sub_lt_iff_lt_add'`.

theorem lt_add_of_sub_left_lt {α : Type u} [has_lt α] {a b c : α} :
a - b < ca < b + c

Alias of `sub_lt_iff_lt_add'`.

theorem inv_lt_div_iff_lt_mul' {α : Type u} [comm_group α] [has_lt α] {a b c : α} :
b⁻¹ < a / c c < a * b
theorem neg_lt_sub_iff_lt_add' {α : Type u} [has_lt α] {a b c : α} :
-b < a - c c < a + b
theorem sub_lt {α : Type u} [has_lt α] {a b c : α} :
a - b < c a - c < b
theorem div_lt'' {α : Type u} [comm_group α] [has_lt α] {a b c : α} :
a / b < c a / c < b
theorem lt_div'' {α : Type u} [comm_group α] [has_lt α] {a b c : α} :
a < b / c c < b / a
theorem lt_sub {α : Type u} [has_lt α] {a b c : α} :
a < b - c c < b - a
theorem div_lt_div'' {α : Type u} [comm_group α] [preorder α] {a b c d : α} (hab : a < b) (hcd : c < d) :
a / d < b / c
theorem sub_lt_sub {α : Type u} [preorder α] {a b c d : α} (hab : a < b) (hcd : c < d) :
a - d < b - c
theorem le_of_forall_one_lt_lt_mul {α : Type u} [group α] [linear_order α] {a b : α} (h : ∀ (ε : α), 1 < εa < b * ε) :
a b
theorem le_of_forall_pos_lt_add {α : Type u} [add_group α] [linear_order α] {a b : α} (h : ∀ (ε : α), 0 < εa < b + ε) :
a b
theorem le_iff_forall_pos_lt_add {α : Type u} [add_group α] [linear_order α] {a b : α} :
a b ∀ (ε : α), 0 < εa < b + ε
theorem le_iff_forall_one_lt_lt_mul {α : Type u} [group α] [linear_order α] {a b : α} :
a b ∀ (ε : α), 1 < εa < b * ε
theorem sub_le_neg_add_iff {α : Type u} [add_group α] [linear_order α] {a b : α}  :
a - b -a + b a b
theorem div_le_inv_mul_iff {α : Type u} [group α] [linear_order α] {a b : α}  :
a / b a⁻¹ * b a b
@[simp]
theorem sub_le_sub_flip {α : Type u_1} [linear_order α] {a b : α} :
a - b b - a a b
@[simp]
theorem div_le_div_flip {α : Type u_1} [comm_group α] [linear_order α] {a b : α} :
a / b b / a a b
@[simp]
theorem max_zero_sub_max_neg_zero_eq_self {α : Type u} [add_group α] [linear_order α] (a : α) :
max a 0 - max (-a) 0 = a
@[simp]
theorem max_one_div_max_inv_one_eq_self {α : Type u} [group α] [linear_order α] (a : α) :
max a 1 / 1 = a
theorem max_zero_sub_eq_self {α : Type u} [add_group α] [linear_order α] (a : α) :
max a 0 - max (-a) 0 = a

Alias of `max_zero_sub_max_neg_zero_eq_self`.

theorem le_of_forall_one_lt_le_mul {α : Type u} [group α] [linear_order α] {a b : α} (h : ∀ (ε : α), 1 < εa b * ε) :
a b
theorem le_of_forall_pos_le_add {α : Type u} [add_group α] [linear_order α] {a b : α} (h : ∀ (ε : α), 0 < εa b + ε) :
a b
theorem le_iff_forall_pos_le_add {α : Type u} [add_group α] [linear_order α] {a b : α} :
a b ∀ (ε : α), 0 < εa b + ε
theorem le_iff_forall_one_lt_le_mul {α : Type u} [group α] [linear_order α] {a b : α} :
a b ∀ (ε : α), 1 < εa b * ε

### Linearly ordered commutative groups #

@[class]
structure linear_ordered_add_comm_group (α : Type u) :
Type u
• add : α → α → α
• add_assoc : ∀ (a b c_1 : α), a + b + c_1 = a + (b + c_1)
• zero : α
• zero_add : ∀ (a : α), 0 + a = a
• add_zero : ∀ (a : α), a + 0 = a
• nsmul : α → α
• nsmul_zero' : (∀ (x : α), . "try_refl_tac"
• nsmul_succ' : (∀ (n : ) (x : α), . "try_refl_tac"
• neg : α → α
• sub : α → α → α
• sub_eq_add_neg : (∀ (a b : α), a - b = a + -b) . "try_refl_tac"
• gsmul : α → α
• gsmul_zero' : (∀ (a : α), . "try_refl_tac"
• gsmul_succ' : (∀ (n : ) (a : α), . "try_refl_tac"
• gsmul_neg' : (∀ (n : ) (a : α), . "try_refl_tac"
• add_left_neg : ∀ (a : α), -a + a = 0
• add_comm : ∀ (a b : α), a + b = b + a
• 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
• add_le_add_left : ∀ (a b : α), a b∀ (c_1 : α), c_1 + a c_1 + b
• le_total : ∀ (a b : α), a b b a
• decidable_le :
• decidable_eq :
• decidable_lt :

A linearly ordered additive commutative group is an additive commutative group with a linear order in which addition is monotone.

Instances
@[class]
structure linear_ordered_add_comm_group_with_top (α : 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 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_eq :
• decidable_lt :
• add : α → α → α
• add_assoc : ∀ (a b c_1 : α), a + b + c_1 = a + (b + c_1)
• zero : α
• zero_add : ∀ (a : α), 0 + a = a
• add_zero : ∀ (a : α), a + 0 = a
• nsmul : α → α
• nsmul_zero' : (∀ (x : α), . "try_refl_tac"
• nsmul_succ' : (∀ (n : ) (x : α), . "try_refl_tac"
• add_comm : ∀ (a b : α), a + b = b + a
• add_le_add_left : ∀ (a b : α), a b∀ (c_1 : α), c_1 + a c_1 + b
• lt_of_add_lt_add_left : ∀ (a b c_1 : α), a + b < a + c_1b < c_1
• top : α
• le_top : ∀ (a : α), a
• top_add' : ∀ (x : α), + x =
• neg : α → α
• sub : α → α → α
• sub_eq_add_neg : (∀ (a b : α), a - b = a + -b) . "try_refl_tac"
• gsmul : α → α
• gsmul_zero' : (∀ (a : α), . "try_refl_tac"
• gsmul_succ' : (∀ (n : ) (a : α), . "try_refl_tac"
• gsmul_neg' : (∀ (n : ) (a : α), . "try_refl_tac"
• exists_pair_ne : ∃ (x y : α), x y
• neg_top :
• add_neg_cancel : ∀ (a : α), a a + -a = 0

A linearly ordered commutative monoid with an additively absorbing `⊤` element. Instances should include number systems with an infinite element adjoined.`

Instances
@[instance]
@[instance]
@[class]
structure linear_ordered_comm_group (α : Type u) :
Type u
• 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
• npow : α → α
• npow_zero' : (∀ (x : α), . "try_refl_tac"
• npow_succ' : (∀ (n : ) (x : α), . "try_refl_tac"
• inv : α → α
• div : α → α → α
• div_eq_mul_inv : (∀ (a b : α), a / b = a * b⁻¹) . "try_refl_tac"
• gpow : α → α
• gpow_zero' : (∀ (a : α), . "try_refl_tac"
• gpow_succ' : (∀ (n : ) (a : α), . "try_refl_tac"
• gpow_neg' : (∀ (n : ) (a : α), . "try_refl_tac"
• mul_left_inv : ∀ (a : α), a⁻¹ * a = 1
• mul_comm : ∀ (a b : α), a * b = b * a
• 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
• mul_le_mul_left : ∀ (a b : α), a b∀ (c_1 : α), c_1 * a c_1 * b
• le_total : ∀ (a b : α), a b b a
• decidable_le :
• decidable_eq :
• decidable_lt :

A linearly ordered commutative group is a commutative group with a linear order in which multiplication is monotone.

Instances
@[instance]
@[instance]
def function.injective.linear_ordered_comm_group {α : Type u} {β : Type u_1} [has_one β] [has_mul β] [has_inv β] [has_div β] (f : β → α) (hf : function.injective f) (one : f 1 = 1) (mul : ∀ (x y : β), f (x * y) = (f x) * f y) (inv : ∀ (x : β), f x⁻¹ = (f x)⁻¹) (div : ∀ (x y : β), f (x / y) = f x / f y) :

Pullback a `linear_ordered_comm_group` under an injective map. See note [reducible non-instances].

Equations
def function.injective.linear_ordered_add_comm_group {α : Type u} {β : Type u_1} [has_zero β] [has_add β] [has_neg β] [has_sub β] (f : β → α) (hf : function.injective f) (one : f 0 = 0) (mul : ∀ (x y : β), f (x + y) = f x + f y) (inv : ∀ (x : β), f (-x) = -f x) (div : ∀ (x y : β), f (x - y) = f x - f y) :

Pullback a `linear_ordered_add_comm_group` under an injective map.

theorem linear_ordered_comm_group.mul_lt_mul_left' {α : Type u} (a b : α) (h : a < b) (c : α) :
c * a < c * b
theorem linear_ordered_add_comm_group.add_lt_add_left {α : Type u} (a b : α) (h : a < b) (c : α) :
c + a < c + b
theorem min_neg_neg {α : Type u} (a b : α) :
min (-a) (-b) = -max a b
theorem min_inv_inv' {α : Type u} (a b : α) :
b⁻¹ = (max a b)⁻¹
theorem max_inv_inv' {α : Type u} (a b : α) :
b⁻¹ = (min a b)⁻¹
theorem max_neg_neg {α : Type u} (a b : α) :
max (-a) (-b) = -min a b
theorem min_div_div_right' {α : Type u} (a b c : α) :
min (a / c) (b / c) = min a b / c
theorem min_sub_sub_right {α : Type u} (a b c : α) :
min (a - c) (b - c) = min a b - c
theorem max_div_div_right' {α : Type u} (a b c : α) :
max (a / c) (b / c) = max a b / c
theorem max_sub_sub_right {α : Type u} (a b c : α) :
max (a - c) (b - c) = max a b - c
theorem min_div_div_left' {α : Type u} (a b c : α) :
min (a / b) (a / c) = a / max b c
theorem min_sub_sub_left {α : Type u} (a b c : α) :
min (a - b) (a - c) = a - max b c
theorem max_div_div_left' {α : Type u} (a b c : α) :
max (a / b) (a / c) = a / min b c
theorem max_sub_sub_left {α : Type u} (a b c : α) :
max (a - b) (a - c) = a - min b c
theorem eq_zero_of_neg_eq {α : Type u} {a : α} (h : -a = a) :
a = 0
theorem eq_one_of_inv_eq' {α : Type u} {a : α} (h : a⁻¹ = a) :
a = 1
theorem exists_one_lt' {α : Type u} [nontrivial α] :
∃ (a : α), 1 < a
theorem exists_zero_lt {α : Type u} [nontrivial α] :
∃ (a : α), 0 < a
@[instance]
@[instance]
@[instance]
@[instance]
def abs {α : Type u_1} [has_neg α] [linear_order α] (a : α) :
α

`abs a` is the absolute value of `a`.

Equations
theorem abs_choice {α : Type u} [has_neg α] [linear_order α] (x : α) :
abs x = x abs x = -x
theorem abs_le' {α : Type u} [has_neg α] [linear_order α] {a b : α} :
abs a b a b -a b
theorem le_abs {α : Type u} [has_neg α] [linear_order α] {a b : α} :
a abs b a b a -b
theorem le_abs_self {α : Type u} [has_neg α] [linear_order α] (a : α) :
a abs a
theorem neg_le_abs_self {α : Type u} [has_neg α] [linear_order α] (a : α) :
-a abs a
theorem lt_abs {α : Type u} [has_neg α] [linear_order α] {a b : α} :
a < abs b a < b a < -b
theorem abs_le_abs {α : Type u} [has_neg α] [linear_order α] {a b : α} (h₀ : a b) (h₁ : -a b) :
abs a abs b
theorem abs_by_cases {α : Type u} [has_neg α] [linear_order α] (P : α → Prop) {a : α} (h1 : P a) (h2 : P (-a)) :
P (abs a)
@[simp]
theorem abs_neg {α : Type u} [add_group α] [linear_order α] (a : α) :
abs (-a) = abs a
theorem eq_or_eq_neg_of_abs_eq {α : Type u} [add_group α] [linear_order α] {a b : α} (h : abs a = b) :
a = b a = -b