mathlib documentation

algebra.order.with_zero

Linearly ordered commutative groups and monoids 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.

Note that to avoid issues with import cycles, linear_ordered_comm_monoid_with_zero is defined in another file. However, the lemmas about it are stated here.

source
@[instance]
def linear_ordered_comm_group_with_zero.to_linear_ordered_comm_monoid_with_zero (α : Type u_1) [self : linear_ordered_comm_group_with_zero α] :
linear_ordered_comm_monoid_with_zero α
source
@[instance]
def linear_ordered_comm_group_with_zero.to_comm_group_with_zero (α : Type u_1) [self : linear_ordered_comm_group_with_zero α] :
comm_group_with_zero α
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@[class]
structure linear_ordered_comm_group_with_zero (α : Type u_1) :
Type u_1

A linearly ordered commutative group with a zero element.

Instances
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@[protected, instance]
def multiplicative.linear_ordered_comm_monoid_with_zero {α : Type u_1} [linear_ordered_add_comm_monoid_with_top α] :
linear_ordered_comm_monoid_with_zero (multiplicative (order_dual α))
Equations
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@[protected, instance]
def multiplicative.linear_ordered_comm_group_with_zero {α : Type u_1} [linear_ordered_add_comm_group_with_top α] :
linear_ordered_comm_group_with_zero (multiplicative (order_dual α))
Equations
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theorem left.one_le_pow_of_le {α : Type u_1} [monoid α] [preorder α] [covariant_class α α has_mul.mul has_le.le] {n : ℕ} {x : α} :
1 ≤ x → 1 ≤ x ^ n
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theorem right.one_le_pow_of_le {α : Type u_1} [monoid α] [preorder α] [covariant_class α α (function.swap has_mul.mul) has_le.le] {x : α} (H : 1 ≤ x) {n : ℕ} :
1 ≤ x ^ n
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theorem right.pow_le_one_of_le {α : Type u_1} [monoid α] [preorder α] [covariant_class α α (function.swap has_mul.mul) has_le.le] {x : α} (H : x ≤ 1) {n : ℕ} :
x ^ n ≤ 1
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theorem pow_le_pow_of_le {α : Type u_1} [monoid α] [preorder α] [covariant_class α α has_mul.mul has_le.le] [covariant_class α α (function.swap has_mul.mul) has_le.le] {x y : α} (H : x ≤ y) {n : ℕ} :
x ^ n ≤ y ^ n
source
theorem left.pow_lt_one_of_lt {α : Type u_1} [monoid α] [preorder α] [covariant_class α α has_mul.mul has_lt.lt] {n : ℕ} {x : α} (n0 : 0 < n) (H : x < 1) :
x ^ n < 1
source
theorem left.pow_lt_one_iff {α : Type u_1} [monoid α] [linear_order α] [covariant_class α α has_mul.mul has_lt.lt] {n : ℕ} {x : α} (n0 : 0 < n) :
x ^ n < 1 ↔ x < 1
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theorem right.pow_lt_one_of_lt {α : Type u_1} [monoid α] [preorder α] [covariant_class α α (function.swap has_mul.mul) has_lt.lt] {n : ℕ} {x : α} (n0 : 0 < n) (H : x < 1) :
x ^ n < 1
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theorem right.pow_lt_one_iff {α : Type u_1} [monoid α] [linear_order α] [covariant_class α α (function.swap has_mul.mul) has_lt.lt] {n : ℕ} {x : α} (n0 : 0 < n) :
x ^ n < 1 ↔ x < 1
source
@[protected, instance]
def with_zero.linear_ordered_comm_monoid_with_zero {α : Type u_1} [linear_ordered_comm_monoid α] :
linear_ordered_comm_monoid_with_zero (with_zero α)
Equations
source
@[protected, instance]
def with_zero.linear_ordered_comm_group_with_zero {α : Type u_1} [linear_ordered_comm_group α] :
linear_ordered_comm_group_with_zero (with_zero α)
Equations
source
def function.injective.linear_ordered_comm_monoid_with_zero {α : Type u_1} [linear_ordered_comm_monoid_with_zero α] {β : Type u_2} [has_zero β] [has_one β] [has_mul β] [has_pow β ℕ] (f : β → α) (hf : function.injective f) (zero : f 0 = 0) (one : f 1 = 1) (mul : ∀ (x y : β), f (x * y) = (f x) * f y) (npow : ∀ (x : β) (n : ℕ), f (x ^ n) = f x ^ n) :
linear_ordered_comm_monoid_with_zero β

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

Equations
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theorem zero_le_one' {α : Type u_1} [linear_ordered_comm_monoid_with_zero α] :
0 ≤ 1
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@[simp]
theorem zero_le' {α : Type u_1} {a : α} [linear_ordered_comm_monoid_with_zero α] :
0 ≤ a
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@[simp]
theorem not_lt_zero' {α : Type u_1} {a : α} [linear_ordered_comm_monoid_with_zero α] :
¬a < 0
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@[simp]
theorem le_zero_iff {α : Type u_1} {a : α} [linear_ordered_comm_monoid_with_zero α] :
a ≤ 0 ↔ a = 0
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theorem zero_lt_iff {α : Type u_1} {a : α} [linear_ordered_comm_monoid_with_zero α] :
0 < a ↔ a ≠ 0
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theorem ne_zero_of_lt {α : Type u_1} {a b : α} [linear_ordered_comm_monoid_with_zero α] (h : b < a) :
a ≠ 0
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theorem pow_pos_iff {α : Type u_1} {a : α} [linear_ordered_comm_monoid_with_zero α] [no_zero_divisors α] {n : ℕ} (hn : 0 < n) :
0 < a ^ n ↔ 0 < a
source
@[protected, instance]
def additive.linear_ordered_add_comm_monoid_with_top {α : Type u_1} [linear_ordered_comm_monoid_with_zero α] :
linear_ordered_add_comm_monoid_with_top (additive (order_dual α))
Equations
source
theorem zero_lt_one₀ {α : Type u_1} [linear_ordered_comm_group_with_zero α] :
0 < 1
source
theorem le_of_le_mul_right {α : Type u_1} {a b c : α} [linear_ordered_comm_group_with_zero α] (h : c ≠ 0) (hab : a * c ≤ b * c) :
a ≤ b
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theorem le_mul_inv_of_mul_le {α : Type u_1} {a b c : α} [linear_ordered_comm_group_with_zero α] (h : c ≠ 0) (hab : a * c ≤ b) :
a ≤ b * c⁻¹
source
theorem mul_inv_le_of_le_mul {α : Type u_1} {a b c : α} [linear_ordered_comm_group_with_zero α] (hab : a ≤ b * c) :
a * c⁻¹ ≤ b
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theorem le_mul_inv_iff₀ {α : Type u_1} {a b c : α} [linear_ordered_comm_group_with_zero α] (hc : c ≠ 0) :
a ≤ b * c⁻¹ ↔ a * c ≤ b
source
theorem mul_inv_le_iff₀ {α : Type u_1} {a b c : α} [linear_ordered_comm_group_with_zero α] (hc : c ≠ 0) :
a * c⁻¹ ≤ b ↔ a ≤ b * c
source
theorem div_le_div₀ {α : Type u_1} [linear_ordered_comm_group_with_zero α] (a b c d : α) (hb : b ≠ 0) (hd : d ≠ 0) :
a * b⁻¹ ≤ c * d⁻¹ ↔ a * d ≤ c * b
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@[simp]
theorem units.zero_lt {α : Type u_1} [linear_ordered_comm_group_with_zero α] (u : αˣ) :
0 < ↑u
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theorem mul_lt_mul_of_lt_of_le₀ {α : Type u_1} {a b c d : α} [linear_ordered_comm_group_with_zero α] (hab : a ≤ b) (hb : b ≠ 0) (hcd : c < d) :
a * c < b * d
source
theorem mul_lt_mul₀ {α : Type u_1} {a b c d : α} [linear_ordered_comm_group_with_zero α] (hab : a < b) (hcd : c < d) :
a * c < b * d
source
theorem mul_inv_lt_of_lt_mul₀ {α : Type u_1} {x y z : α} [linear_ordered_comm_group_with_zero α] (h : x < y * z) :
x * z⁻¹ < y
source
theorem inv_mul_lt_of_lt_mul₀ {α : Type u_1} {x y z : α} [linear_ordered_comm_group_with_zero α] (h : x < y * z) :
y⁻¹ * x < z
source
theorem mul_lt_right₀ {α : Type u_1} {a b : α} [linear_ordered_comm_group_with_zero α] (c : α) (h : a < b) (hc : c ≠ 0) :
a * c < b * c
source
theorem pow_lt_pow_succ {α : Type u_1} [linear_ordered_comm_group_with_zero α] {x : α} {n : ℕ} (hx : 1 < x) :
x ^ n < x ^ n.succ
source
theorem pow_lt_pow₀ {α : Type u_1} [linear_ordered_comm_group_with_zero α] {x : α} {m n : ℕ} (hx : 1 < x) (hmn : m < n) :
x ^ m < x ^ n
source
theorem inv_lt_inv₀ {α : Type u_1} {a b : α} [linear_ordered_comm_group_with_zero α] (ha : a ≠ 0) (hb : b ≠ 0) :
a⁻¹ < b⁻¹ ↔ b < a
source
theorem inv_le_inv₀ {α : Type u_1} {a b : α} [linear_ordered_comm_group_with_zero α] (ha : a ≠ 0) (hb : b ≠ 0) :
a⁻¹ ≤ b⁻¹ ↔ b ≤ a
source
theorem lt_of_mul_lt_mul_of_le₀ {α : Type u_1} {a b c d : α} [linear_ordered_comm_group_with_zero α] (h : a * b < c * d) (hc : 0 < c) (hh : c ≤ a) :
b < d
source
theorem mul_le_mul_right₀ {α : Type u_1} {a b c : α} [linear_ordered_comm_group_with_zero α] (hc : c ≠ 0) :
a * c ≤ b * c ↔ a ≤ b
source
theorem div_le_div_right₀ {α : Type u_1} {a b c : α} [linear_ordered_comm_group_with_zero α] (hc : c ≠ 0) :
a / c ≤ b / c ↔ a ≤ b
source
theorem le_div_iff₀ {α : Type u_1} {a b c : α} [linear_ordered_comm_group_with_zero α] (hc : c ≠ 0) :
a ≤ b / c ↔ a * c ≤ b
source
theorem div_le_iff₀ {α : Type u_1} {a b c : α} [linear_ordered_comm_group_with_zero α] (hc : c ≠ 0) :
a / c ≤ b ↔ a ≤ b * c
source
@[protected, instance]
def additive.linear_ordered_add_comm_group_with_top {α : Type u_1} [linear_ordered_comm_group_with_zero α] :
linear_ordered_add_comm_group_with_top (additive (order_dual α))
Equations
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theorem monoid_hom.map_neg_one {α : Type u_1} [linear_ordered_comm_group_with_zero α] {R : Type u_2} [ring R] (f : R →* α) :
⇑f (-1) = 1
source
@[simp]
theorem monoid_hom.map_neg {α : Type u_1} [linear_ordered_comm_group_with_zero α] {R : Type u_2} [ring R] (f : R →* α) (x : R) :
⇑f (-x) = ⇑f x
source
theorem monoid_hom.map_sub_swap {α : Type u_1} [linear_ordered_comm_group_with_zero α] {R : Type u_2} [ring R] (f : R →* α) (x y : R) :
⇑f (x - y) = ⇑f (y - x)