mathlib documentation

algebra.ordered_ring

source
@[instance]
def ordered_semiring.to_semiring (α : Type u) [s : ordered_semiring α] :
semiring α

source
@[class]
structure ordered_semiring (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
  • add_comm : ∀ (a b : α), a + b = b + a
  • 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
  • zero_mul : ∀ (a : α), 0 * a = 0
  • mul_zero : ∀ (a : α), a * 0 = 0
  • left_distrib : ∀ (a b c_1 : α), a * (b + c_1) = a * b + a * c_1
  • right_distrib : ∀ (a b c_1 : α), (a + b) * c_1 = a * c_1 + b * c_1
  • add_left_cancel : ∀ (a b c_1 : α), a + b = a + c_1 → b = c_1
  • add_right_cancel : ∀ (a b c_1 : α), a + b = c_1 + b → a = c_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
  • add_le_add_left : ∀ (a b : α), a ≤ b → ∀ (c_1 : α), c_1 + a ≤ c_1 + b
  • le_of_add_le_add_left : ∀ (a b c_1 : α), a + b ≤ a + c_1 → b ≤ c_1
  • mul_lt_mul_of_pos_left : ∀ (a b c_1 : α), a < b → 0 < c_1 → c_1 * a < c_1 * b
  • mul_lt_mul_of_pos_right : ∀ (a b c_1 : α), a < b → 0 < c_1 → a * c_1 < b * c_1

An ordered_semiring α is a semiring α with a partial order such that multiplication with a positive number and addition are monotone.

Instances
source
@[instance]
def ordered_semiring.to_ordered_cancel_add_comm_monoid (α : Type u) [s : ordered_semiring α] :
ordered_cancel_add_comm_monoid α

source
theorem mul_lt_mul_of_pos_left {α : Type u} [ordered_semiring α] {a b c : α} (h₁ : a < b) (h₂ : 0 < c) :
c * a < c * b

source
theorem mul_lt_mul_of_pos_right {α : Type u} [ordered_semiring α] {a b c : α} (h₁ : a < b) (h₂ : 0 < c) :
a * c < b * c

source
theorem mul_le_mul_of_nonneg_left {α : Type u} [ordered_semiring α] {a b c : α} (h₁ : a ≤ b) (h₂ : 0 ≤ c) :
c * a ≤ c * b

source
theorem mul_le_mul_of_nonneg_right {α : Type u} [ordered_semiring α] {a b c : α} (h₁ : a ≤ b) (h₂ : 0 ≤ c) :
a * c ≤ b * c

source
theorem mul_le_mul {α : Type u} [ordered_semiring α] {a b c d : α} (hac : a ≤ c) (hbd : b ≤ d) (nn_b : 0 ≤ b) (nn_c : 0 ≤ c) :
a * b ≤ c * d

source
theorem mul_nonneg {α : Type u} [ordered_semiring α] {a b : α} (ha : 0 ≤ a) (hb : 0 ≤ b) :
0 ≤ a * b

source
theorem mul_nonpos_of_nonneg_of_nonpos {α : Type u} [ordered_semiring α] {a b : α} (ha : 0 ≤ a) (hb : b ≤ 0) :
a * b ≤ 0

source
theorem mul_nonpos_of_nonpos_of_nonneg {α : Type u} [ordered_semiring α] {a b : α} (ha : a ≤ 0) (hb : 0 ≤ b) :
a * b ≤ 0

source
theorem mul_lt_mul {α : Type u} [ordered_semiring α] {a b c d : α} (hac : a < c) (hbd : b ≤ d) (pos_b : 0 < b) (nn_c : 0 ≤ c) :
a * b < c * d

source
theorem mul_lt_mul' {α : Type u} [ordered_semiring α] {a b c d : α} (h1 : a ≤ c) (h2 : b < d) (h3 : 0 ≤ b) (h4 : 0 < c) :
a * b < c * d

source
theorem mul_pos {α : Type u} [ordered_semiring α] {a b : α} (ha : 0 < a) (hb : 0 < b) :
0 < a * b

source
theorem mul_neg_of_pos_of_neg {α : Type u} [ordered_semiring α] {a b : α} (ha : 0 < a) (hb : b < 0) :
a * b < 0

source
theorem mul_neg_of_neg_of_pos {α : Type u} [ordered_semiring α] {a b : α} (ha : a < 0) (hb : 0 < b) :
a * b < 0

source
theorem mul_self_le_mul_self {α : Type u} [ordered_semiring α] {a b : α} (h1 : 0 ≤ a) (h2 : a ≤ b) :
a * a ≤ b * b

source
theorem mul_self_lt_mul_self {α : Type u} [ordered_semiring α] {a b : α} (h1 : 0 ≤ a) (h2 : a < b) :
a * a < b * b

source
@[instance]
def linear_ordered_semiring.to_ordered_semiring (α : Type u) [s : linear_ordered_semiring α] :
ordered_semiring α

source
@[class]
structure linear_ordered_semiring (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
  • add_comm : ∀ (a b : α), a + b = b + a
  • 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
  • zero_mul : ∀ (a : α), 0 * a = 0
  • mul_zero : ∀ (a : α), a * 0 = 0
  • left_distrib : ∀ (a b c_1 : α), a * (b + c_1) = a * b + a * c_1
  • right_distrib : ∀ (a b c_1 : α), (a + b) * c_1 = a * c_1 + b * c_1
  • add_left_cancel : ∀ (a b c_1 : α), a + b = a + c_1 → b = c_1
  • add_right_cancel : ∀ (a b c_1 : α), a + b = c_1 + b → a = c_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
  • add_le_add_left : ∀ (a b : α), a ≤ b → ∀ (c_1 : α), c_1 + a ≤ c_1 + b
  • le_of_add_le_add_left : ∀ (a b c_1 : α), a + b ≤ a + c_1 → b ≤ c_1
  • mul_lt_mul_of_pos_left : ∀ (a b c_1 : α), a < b → 0 < c_1 → c_1 * a < c_1 * b
  • mul_lt_mul_of_pos_right : ∀ (a b c_1 : α), a < b → 0 < c_1 → a * c_1 < b * c_1
  • le_total : ∀ (a b : α), a ≤ b ∨ b ≤ a
  • zero_lt_one : linear_ordered_semiring.zero < linear_ordered_semiring.one

A linear_ordered_semiring α is a semiring α with a linear order such that multiplication with a positive number and addition are monotone.

Instances
source
@[instance]
def linear_ordered_semiring.to_linear_order (α : Type u) [s : linear_ordered_semiring α] :
linear_order α

source
theorem zero_lt_one {α : Type u} [linear_ordered_semiring α] :
0 < 1

source
theorem zero_le_one {α : Type u} [linear_ordered_semiring α] :
0 ≤ 1

source
theorem zero_lt_two {α : Type u} [linear_ordered_semiring α] :
0 < 2

source
theorem two_ne_zero {α : Type u} [linear_ordered_semiring α] :
2 ≠ 0

source
theorem one_lt_two {α : Type u} [linear_ordered_semiring α] :
1 < 2

source
theorem one_le_two {α : Type u} [linear_ordered_semiring α] :
1 ≤ 2

source
theorem zero_lt_four {α : Type u} [linear_ordered_semiring α] :
0 < 4

source
theorem lt_of_mul_lt_mul_left {α : Type u} [linear_ordered_semiring α] {a b c : α} (h : c * a < c * b) (hc : 0 ≤ c) :
a < b

source
theorem lt_of_mul_lt_mul_right {α : Type u} [linear_ordered_semiring α] {a b c : α} (h : a * c < b * c) (hc : 0 ≤ c) :
a < b

source
theorem le_of_mul_le_mul_left {α : Type u} [linear_ordered_semiring α] {a b c : α} (h : c * a ≤ c * b) (hc : 0 < c) :
a ≤ b

source
theorem le_of_mul_le_mul_right {α : Type u} [linear_ordered_semiring α] {a b c : α} (h : a * c ≤ b * c) (hc : 0 < c) :
a ≤ b

source
theorem pos_of_mul_pos_left {α : Type u} [linear_ordered_semiring α] {a b : α} (h : 0 < a * b) (h1 : 0 ≤ a) :
0 < b

source
theorem pos_of_mul_pos_right {α : Type u} [linear_ordered_semiring α] {a b : α} (h : 0 < a * b) (h1 : 0 ≤ b) :
0 < a

source
theorem nonneg_of_mul_nonneg_left {α : Type u} [linear_ordered_semiring α] {a b : α} (h : 0 ≤ a * b) (h1 : 0 < a) :
0 ≤ b

source
theorem nonneg_of_mul_nonneg_right {α : Type u} [linear_ordered_semiring α] {a b : α} (h : 0 ≤ a * b) (h1 : 0 < b) :
0 ≤ a

source
theorem neg_of_mul_neg_left {α : Type u} [linear_ordered_semiring α] {a b : α} (h : a * b < 0) (h1 : 0 ≤ a) :
b < 0

source
theorem neg_of_mul_neg_right {α : Type u} [linear_ordered_semiring α] {a b : α} (h : a * b < 0) (h1 : 0 ≤ b) :
a < 0

source
theorem nonpos_of_mul_nonpos_left {α : Type u} [linear_ordered_semiring α] {a b : α} (h : a * b ≤ 0) (h1 : 0 < a) :
b ≤ 0

source
theorem nonpos_of_mul_nonpos_right {α : Type u} [linear_ordered_semiring α] {a b : α} (h : a * b ≤ 0) (h1 : 0 < b) :
a ≤ 0

source
@[simp]
theorem mul_le_mul_left {α : Type u} [linear_ordered_semiring α] {a b c : α} (h : 0 < c) :
c * a ≤ c * b ↔ a ≤ b

source
@[simp]
theorem mul_le_mul_right {α : Type u} [linear_ordered_semiring α] {a b c : α} (h : 0 < c) :
a * c ≤ b * c ↔ a ≤ b

source
@[simp]
theorem mul_lt_mul_left {α : Type u} [linear_ordered_semiring α] {a b c : α} (h : 0 < c) :
c * a < c * b ↔ a < b

source
@[simp]
theorem mul_lt_mul_right {α : Type u} [linear_ordered_semiring α] {a b c : α} (h : 0 < c) :
a * c < b * c ↔ a < b

source
@[simp]
theorem zero_le_mul_left {α : Type u} [linear_ordered_semiring α] {b c : α} (h : 0 < c) :
0 ≤ c * b ↔ 0 ≤ b

source
@[simp]
theorem zero_le_mul_right {α : Type u} [linear_ordered_semiring α] {b c : α} (h : 0 < c) :
0 ≤ b * c ↔ 0 ≤ b

source
@[simp]
theorem zero_lt_mul_left {α : Type u} [linear_ordered_semiring α] {b c : α} (h : 0 < c) :
0 < c * b ↔ 0 < b

source
@[simp]
theorem zero_lt_mul_right {α : Type u} [linear_ordered_semiring α] {b c : α} (h : 0 < c) :
0 < b * c ↔ 0 < b

source
@[simp]
theorem bit0_le_bit0 {α : Type u} [linear_ordered_semiring α] {a b : α} :
bit0 a ≤ bit0 b ↔ a ≤ b

source
@[simp]
theorem bit0_lt_bit0 {α : Type u} [linear_ordered_semiring α] {a b : α} :
bit0 a < bit0 b ↔ a < b

source
@[simp]
theorem bit1_le_bit1 {α : Type u} [linear_ordered_semiring α] {a b : α} :
bit1 a ≤ bit1 b ↔ a ≤ b

source
@[simp]
theorem bit1_lt_bit1 {α : Type u} [linear_ordered_semiring α] {a b : α} :
bit1 a < bit1 b ↔ a < b

source
@[simp]
theorem one_le_bit1 {α : Type u} [linear_ordered_semiring α] {a : α} :
1 ≤ bit1 a ↔ 0 ≤ a

source
@[simp]
theorem one_lt_bit1 {α : Type u} [linear_ordered_semiring α] {a : α} :
1 < bit1 a ↔ 0 < a

source
@[simp]
theorem zero_le_bit0 {α : Type u} [linear_ordered_semiring α] {a : α} :
0 ≤ bit0 a ↔ 0 ≤ a

source
@[simp]
theorem zero_lt_bit0 {α : Type u} [linear_ordered_semiring α] {a : α} :
0 < bit0 a ↔ 0 < a

source
theorem mul_lt_mul'' {α : Type u} [linear_ordered_semiring α] {a b c d : α} (h1 : a < c) (h2 : b < d) (h3 : 0 ≤ a) (h4 : 0 ≤ b) :
a * b < c * d

source
theorem le_mul_iff_one_le_left {α : Type u} [linear_ordered_semiring α] {a b : α} (hb : 0 < b) :
b ≤ a * b ↔ 1 ≤ a

source
theorem lt_mul_iff_one_lt_left {α : Type u} [linear_ordered_semiring α] {a b : α} (hb : 0 < b) :
b < a * b ↔ 1 < a

source
theorem le_mul_iff_one_le_right {α : Type u} [linear_ordered_semiring α] {a b : α} (hb : 0 < b) :
b ≤ b * a ↔ 1 ≤ a

source
theorem lt_mul_iff_one_lt_right {α : Type u} [linear_ordered_semiring α] {a b : α} (hb : 0 < b) :
b < b * a ↔ 1 < a

source
theorem lt_mul_of_one_lt_right {α : Type u} [linear_ordered_semiring α] {a b : α} (hb : 0 < b) (a_1 : 1 < a) :
b < b * a

source
theorem le_mul_of_one_le_right {α : Type u} [linear_ordered_semiring α] {a b : α} (hb : 0 ≤ b) (h : 1 ≤ a) :
b ≤ b * a

source
theorem le_mul_of_one_le_left {α : Type u} [linear_ordered_semiring α] {a b : α} (hb : 0 ≤ b) (h : 1 ≤ a) :
b ≤ a * b

source
theorem mul_nonneg_iff_right_nonneg_of_pos {α : Type u} [linear_ordered_semiring α] {a b : α} (h : 0 < a) :
0 ≤ b * a ↔ 0 ≤ b

source
theorem bit1_pos {α : Type u} [linear_ordered_semiring α] {a : α} (h : 0 ≤ a) :
0 < bit1 a

source
theorem bit1_pos' {α : Type u} [linear_ordered_semiring α] {a : α} (h : 0 < a) :
0 < bit1 a

source
theorem lt_add_one {α : Type u} [linear_ordered_semiring α] (a : α) :
a < a + 1

source
theorem lt_one_add {α : Type u} [linear_ordered_semiring α] (a : α) :
a < 1 + a

source
theorem one_lt_mul {α : Type u} [linear_ordered_semiring α] {a b : α} (ha : 1 ≤ a) (hb : 1 < b) :
1 < a * b

source
theorem mul_le_one {α : Type u} [linear_ordered_semiring α] {a b : α} (ha : a ≤ 1) (hb' : 0 ≤ b) (hb : b ≤ 1) :
a * b ≤ 1

source
theorem one_lt_mul_of_le_of_lt {α : Type u} [linear_ordered_semiring α] {a b : α} (ha : 1 ≤ a) (hb : 1 < b) :
1 < a * b

source
theorem one_lt_mul_of_lt_of_le {α : Type u} [linear_ordered_semiring α] {a b : α} (ha : 1 < a) (hb : 1 ≤ b) :
1 < a * b

source
theorem mul_le_of_le_one_right {α : Type u} [linear_ordered_semiring α] {a b : α} (ha : 0 ≤ a) (hb1 : b ≤ 1) :
a * b ≤ a

source
theorem mul_le_of_le_one_left {α : Type u} [linear_ordered_semiring α] {a b : α} (hb : 0 ≤ b) (ha1 : a ≤ 1) :
a * b ≤ b

source
theorem mul_lt_one_of_nonneg_of_lt_one_left {α : Type u} [linear_ordered_semiring α] {a b : α} (ha0 : 0 ≤ a) (ha : a < 1) (hb : b ≤ 1) :
a * b < 1

source
theorem mul_lt_one_of_nonneg_of_lt_one_right {α : Type u} [linear_ordered_semiring α] {a b : α} (ha : a ≤ 1) (hb0 : 0 ≤ b) (hb : b < 1) :
a * b < 1

source
theorem mul_le_iff_le_one_left {α : Type u} [linear_ordered_semiring α] {a b : α} (hb : 0 < b) :
a * b ≤ b ↔ a ≤ 1

source
theorem mul_lt_iff_lt_one_left {α : Type u} [linear_ordered_semiring α] {a b : α} (hb : 0 < b) :
a * b < b ↔ a < 1

source
theorem mul_le_iff_le_one_right {α : Type u} [linear_ordered_semiring α] {a b : α} (hb : 0 < b) :
b * a ≤ b ↔ a ≤ 1

source
theorem mul_lt_iff_lt_one_right {α : Type u} [linear_ordered_semiring α] {a b : α} (hb : 0 < b) :
b * a < b ↔ a < 1

source
theorem nonpos_of_mul_nonneg_left {α : Type u} [linear_ordered_semiring α] {a b : α} (h : 0 ≤ a * b) (hb : b < 0) :
a ≤ 0

source
theorem nonpos_of_mul_nonneg_right {α : Type u} [linear_ordered_semiring α] {a b : α} (h : 0 ≤ a * b) (ha : a < 0) :
b ≤ 0

source
theorem neg_of_mul_pos_left {α : Type u} [linear_ordered_semiring α] {a b : α} (h : 0 < a * b) (hb : b ≤ 0) :
a < 0

source
theorem neg_of_mul_pos_right {α : Type u} [linear_ordered_semiring α] {a b : α} (h : 0 < a * b) (ha : a ≤ 0) :
b < 0

source
@[instance]
def linear_ordered_semiring.to_nontrivial {α : Type u_1} [linear_ordered_semiring α] :
nontrivial α

source
@[instance]
def linear_ordered_semiring.to_no_top_order {α : Type u_1} [linear_ordered_semiring α] :
no_top_order α

source
theorem monotone_mul_left_of_nonneg {α : Type u} [linear_ordered_semiring α] {a : α} (ha : 0 ≤ a) :
monotone (λ (x : α), a * x)

source
theorem monotone_mul_right_of_nonneg {α : Type u} [linear_ordered_semiring α] {a : α} (ha : 0 ≤ a) :
monotone (λ (x : α), x * a)

source
theorem monotone.mul_const {α : Type u} {β : Type u_1} [linear_ordered_semiring α] [preorder β] {f : β → α} {a : α} (hf : monotone f) (ha : 0 ≤ a) :
monotone (λ (x : β), (f x) * a)

source
theorem monotone.const_mul {α : Type u} {β : Type u_1} [linear_ordered_semiring α] [preorder β] {f : β → α} {a : α} (hf : monotone f) (ha : 0 ≤ a) :
monotone (λ (x : β), a * f x)

source
theorem monotone.mul {α : Type u} {β : Type u_1} [linear_ordered_semiring α] [preorder β] {f g : β → α} (hf : monotone f) (hg : monotone g) (hf0 : ∀ (x : β), 0 ≤ f x) (hg0 : ∀ (x : β), 0 ≤ g x) :
monotone (λ (x : β), (f x) * g x)

source
theorem strict_mono_mul_left_of_pos {α : Type u} [linear_ordered_semiring α] {a : α} (ha : 0 < a) :
strict_mono (λ (x : α), a * x)

source
theorem strict_mono_mul_right_of_pos {α : Type u} [linear_ordered_semiring α] {a : α} (ha : 0 < a) :
strict_mono (λ (x : α), x * a)

source
theorem strict_mono.mul_const {α : Type u} {β : Type u_1} [linear_ordered_semiring α] [preorder β] {f : β → α} {a : α} (hf : strict_mono f) (ha : 0 < a) :
strict_mono (λ (x : β), (f x) * a)

source
theorem strict_mono.const_mul {α : Type u} {β : Type u_1} [linear_ordered_semiring α] [preorder β] {f : β → α} {a : α} (hf : strict_mono f) (ha : 0 < a) :
strict_mono (λ (x : β), a * f x)

source
theorem strict_mono.mul_monotone {α : Type u} {β : Type u_1} [linear_ordered_semiring α] [preorder β] {f g : β → α} (hf : strict_mono f) (hg : monotone g) (hf0 : ∀ (x : β), 0 ≤ f x) (hg0 : ∀ (x : β), 0 < g x) :
strict_mono (λ (x : β), (f x) * g x)

source
theorem monotone.mul_strict_mono {α : Type u} {β : Type u_1} [linear_ordered_semiring α] [preorder β] {f g : β → α} (hf : monotone f) (hg : strict_mono g) (hf0 : ∀ (x : β), 0 < f x) (hg0 : ∀ (x : β), 0 ≤ g x) :
strict_mono (λ (x : β), (f x) * g x)

source
theorem strict_mono.mul {α : Type u} {β : Type u_1} [linear_ordered_semiring α] [preorder β] {f g : β → α} (hf : strict_mono f) (hg : strict_mono g) (hf0 : ∀ (x : β), 0 ≤ f x) (hg0 : ∀ (x : β), 0 ≤ g x) :
strict_mono (λ (x : β), (f x) * g x)

source
@[class]
structure decidable_linear_ordered_semiring (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
  • add_comm : ∀ (a b : α), a + b = b + a
  • 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
  • zero_mul : ∀ (a : α), 0 * a = 0
  • mul_zero : ∀ (a : α), a * 0 = 0
  • left_distrib : ∀ (a b c_1 : α), a * (b + c_1) = a * b + a * c_1
  • right_distrib : ∀ (a b c_1 : α), (a + b) * c_1 = a * c_1 + b * c_1
  • add_left_cancel : ∀ (a b c_1 : α), a + b = a + c_1 → b = c_1
  • add_right_cancel : ∀ (a b c_1 : α), a + b = c_1 + b → a = c_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
  • add_le_add_left : ∀ (a b : α), a ≤ b → ∀ (c_1 : α), c_1 + a ≤ c_1 + b
  • le_of_add_le_add_left : ∀ (a b c_1 : α), a + b ≤ a + c_1 → b ≤ c_1
  • mul_lt_mul_of_pos_left : ∀ (a b c_1 : α), a < b → 0 < c_1 → c_1 * a < c_1 * b
  • mul_lt_mul_of_pos_right : ∀ (a b c_1 : α), a < b → 0 < c_1 → a * c_1 < b * c_1
  • le_total : ∀ (a b : α), a ≤ b ∨ b ≤ a
  • zero_lt_one : decidable_linear_ordered_semiring.zero < decidable_linear_ordered_semiring.one
  • decidable_le : decidable_rel has_le.le
  • decidable_eq : decidable_eq α
  • decidable_lt : decidable_rel has_lt.lt

A decidable_linear_ordered_semiring α is a semiring α with a decidable linear order such that multiplication with a positive number and addition are monotone.

Instances
source
@[instance]
def decidable_linear_ordered_semiring.to_linear_ordered_semiring (α : Type u) [s : decidable_linear_ordered_semiring α] :
linear_ordered_semiring α

source
@[instance]
def decidable_linear_ordered_semiring.to_decidable_linear_order (α : Type u) [s : decidable_linear_ordered_semiring α] :
decidable_linear_order α

source
@[simp]
theorem decidable.mul_le_mul_left {α : Type u} [decidable_linear_ordered_semiring α] {a b c : α} (h : 0 < c) :
c * a ≤ c * b ↔ a ≤ b

source
@[simp]
theorem decidable.mul_le_mul_right {α : Type u} [decidable_linear_ordered_semiring α] {a b c : α} (h : 0 < c) :
a * c ≤ b * c ↔ a ≤ b

source
@[instance]
def ordered_ring.to_nontrivial (α : Type u) [s : ordered_ring α] :
nontrivial α

source
@[instance]
def ordered_ring.to_ordered_add_comm_group (α : Type u) [s : ordered_ring α] :
ordered_add_comm_group α

source
@[class]
structure ordered_ring (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
  • neg : α → α
  • add_left_neg : ∀ (a : α), -a + a = 0
  • add_comm : ∀ (a b : α), a + b = b + a
  • 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
  • left_distrib : ∀ (a b c_1 : α), a * (b + c_1) = a * b + a * c_1
  • right_distrib : ∀ (a b c_1 : α), (a + b) * c_1 = a * c_1 + b * c_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
  • add_le_add_left : ∀ (a b : α), a ≤ b → ∀ (c_1 : α), c_1 + a ≤ c_1 + b
  • exists_pair_ne : ∃ (x y : α), x ≠ y
  • mul_pos : ∀ (a b : α), 0 < a → 0 < b → 0 < a * b

An ordered_ring α is a ring α with a partial order such that multiplication with a positive number and addition are monotone.

Instances
source
@[instance]
def ordered_ring.to_ring (α : Type u) [s : ordered_ring α] :
ring α

source
theorem ordered_ring.mul_nonneg {α : Type u} [ordered_ring α] (a b : α) (h₁ : 0 ≤ a) (h₂ : 0 ≤ b) :
0 ≤ a * b

source
theorem ordered_ring.mul_le_mul_of_nonneg_left {α : Type u} [ordered_ring α] {a b c : α} (h₁ : a ≤ b) (h₂ : 0 ≤ c) :
c * a ≤ c * b

source
theorem ordered_ring.mul_le_mul_of_nonneg_right {α : Type u} [ordered_ring α] {a b c : α} (h₁ : a ≤ b) (h₂ : 0 ≤ c) :
a * c ≤ b * c

source
theorem ordered_ring.mul_lt_mul_of_pos_left {α : Type u} [ordered_ring α] {a b c : α} (h₁ : a < b) (h₂ : 0 < c) :
c * a < c * b

source
theorem ordered_ring.mul_lt_mul_of_pos_right {α : Type u} [ordered_ring α] {a b c : α} (h₁ : a < b) (h₂ : 0 < c) :
a * c < b * c

source
@[instance]
def ordered_ring.to_ordered_semiring {α : Type u} [ordered_ring α] :
ordered_semiring α

Equations
source
theorem mul_le_mul_of_nonpos_left {α : Type u} [ordered_ring α] {a b c : α} (h : b ≤ a) (hc : c ≤ 0) :
c * a ≤ c * b

source
theorem mul_le_mul_of_nonpos_right {α : Type u} [ordered_ring α] {a b c : α} (h : b ≤ a) (hc : c ≤ 0) :
a * c ≤ b * c

source
theorem mul_nonneg_of_nonpos_of_nonpos {α : Type u} [ordered_ring α] {a b : α} (ha : a ≤ 0) (hb : b ≤ 0) :
0 ≤ a * b

source
theorem mul_lt_mul_of_neg_left {α : Type u} [ordered_ring α] {a b c : α} (h : b < a) (hc : c < 0) :
c * a < c * b

source
theorem mul_lt_mul_of_neg_right {α : Type u} [ordered_ring α] {a b c : α} (h : b < a) (hc : c < 0) :
a * c < b * c

source
theorem mul_pos_of_neg_of_neg {α : Type u} [ordered_ring α] {a b : α} (ha : a < 0) (hb : b < 0) :
0 < a * b

source
@[instance]
def linear_ordered_ring.to_linear_order (α : Type u) [s : linear_ordered_ring α] :
linear_order α

source
@[class]
structure linear_ordered_ring (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
  • neg : α → α
  • add_left_neg : ∀ (a : α), -a + a = 0
  • add_comm : ∀ (a b : α), a + b = b + a
  • 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
  • left_distrib : ∀ (a b c_1 : α), a * (b + c_1) = a * b + a * c_1
  • right_distrib : ∀ (a b c_1 : α), (a + b) * c_1 = a * c_1 + b * c_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
  • add_le_add_left : ∀ (a b : α), a ≤ b → ∀ (c_1 : α), c_1 + a ≤ c_1 + b
  • exists_pair_ne : ∃ (x y : α), x ≠ y
  • mul_pos : ∀ (a b : α), 0 < a → 0 < b → 0 < a * b
  • le_total : ∀ (a b : α), a ≤ b ∨ b ≤ a
  • zero_lt_one : linear_ordered_ring.zero < linear_ordered_ring.one

A linear_ordered_ring α is a ring α with a linear order such that multiplication with a positive number and addition are monotone.

Instances
source
@[instance]
def linear_ordered_ring.to_ordered_ring (α : Type u) [s : linear_ordered_ring α] :
ordered_ring α

source
@[instance]
def linear_ordered_ring.to_linear_ordered_semiring {α : Type u} [linear_ordered_ring α] :
linear_ordered_semiring α

Equations
source
theorem mul_self_nonneg {α : Type u} [linear_ordered_ring α] (a : α) :
0 ≤ a * a

source
theorem pos_and_pos_or_neg_and_neg_of_mul_pos {α : Type u} [linear_ordered_ring α] {a b : α} (hab : 0 < a * b) :
0 < a ∧ 0 < b ∨ a < 0 ∧ b < 0

source
theorem gt_of_mul_lt_mul_neg_left {α : Type u} [linear_ordered_ring α] {a b c : α} (h : c * a < c * b) (hc : c ≤ 0) :
b < a

source
theorem neg_one_lt_zero {α : Type u} [linear_ordered_ring α] :
-1 < 0

source
theorem le_of_mul_le_of_one_le {α : Type u} [linear_ordered_ring α] {a b c : α} (h : a * c ≤ b) (hb : 0 ≤ b) (hc : 1 ≤ c) :
a ≤ b

source
theorem nonneg_le_nonneg_of_squares_le {α : Type u} [linear_ordered_ring α] {a b : α} (hb : 0 ≤ b) (h : a * a ≤ b * b) :
a ≤ b

source
theorem mul_self_le_mul_self_iff {α : Type u} [linear_ordered_ring α] {a b : α} (h1 : 0 ≤ a) (h2 : 0 ≤ b) :
a ≤ b ↔ a * a ≤ b * b

source
theorem mul_self_lt_mul_self_iff {α : Type u} [linear_ordered_ring α] {a b : α} (h1 : 0 ≤ a) (h2 : 0 ≤ b) :
a < b ↔ a * a < b * b

source
theorem linear_ordered_ring.eq_zero_or_eq_zero_of_mul_eq_zero {α : Type u} [linear_ordered_ring α] {a b : α} (h : a * b = 0) :
a = 0 ∨ b = 0

source
@[instance]
def linear_ordered_ring.to_no_bot_order {α : Type u} [linear_ordered_ring α] :
no_bot_order α

source
@[instance]
def linear_ordered_ring.to_domain {α : Type u} [linear_ordered_ring α] :
domain α

Equations
source
@[simp]
theorem mul_le_mul_left_of_neg {α : Type u} [linear_ordered_ring α] {a b c : α} (h : c < 0) :
c * a ≤ c * b ↔ b ≤ a

source
@[simp]
theorem mul_le_mul_right_of_neg {α : Type u} [linear_ordered_ring α] {a b c : α} (h : c < 0) :
a * c ≤ b * c ↔ b ≤ a

source
@[simp]
theorem mul_lt_mul_left_of_neg {α : Type u} [linear_ordered_ring α] {a b c : α} (h : c < 0) :
c * a < c * b ↔ b < a

source
@[simp]
theorem mul_lt_mul_right_of_neg {α : Type u} [linear_ordered_ring α] {a b c : α} (h : c < 0) :
a * c < b * c ↔ b < a

source
theorem sub_one_lt {α : Type u} [linear_ordered_ring α] (a : α) :
a - 1 < a

source
theorem mul_self_pos {α : Type u} [linear_ordered_ring α] {a : α} (ha : a ≠ 0) :
0 < a * a

source
theorem mul_self_le_mul_self_of_le_of_neg_le {α : Type u} [linear_ordered_ring α] {x y : α} (h₁ : x ≤ y) (h₂ : -x ≤ y) :
x * x ≤ y * y

source
theorem nonneg_of_mul_nonpos_left {α : Type u} [linear_ordered_ring α] {a b : α} (h : a * b ≤ 0) (hb : b < 0) :
0 ≤ a

source
theorem nonneg_of_mul_nonpos_right {α : Type u} [linear_ordered_ring α] {a b : α} (h : a * b ≤ 0) (ha : a < 0) :
0 ≤ b

source
theorem pos_of_mul_neg_left {α : Type u} [linear_ordered_ring α] {a b : α} (h : a * b < 0) (hb : b ≤ 0) :
0 < a

source
theorem pos_of_mul_neg_right {α : Type u} [linear_ordered_ring α] {a b : α} (h : a * b < 0) (ha : a ≤ 0) :
0 < b

source
theorem mul_self_add_mul_self_eq_zero {α : Type u} [linear_ordered_ring α] {x y : α} :
x * x + y * y = 0 ↔ x = 0 ∧ y = 0

source
@[class]
structure linear_ordered_comm_ring (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
  • neg : α → α
  • add_left_neg : ∀ (a : α), -a + a = 0
  • add_comm : ∀ (a b : α), a + b = b + a
  • 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
  • left_distrib : ∀ (a b c_1 : α), a * (b + c_1) = a * b + a * c_1
  • right_distrib : ∀ (a b c_1 : α), (a + b) * c_1 = a * c_1 + b * c_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
  • add_le_add_left : ∀ (a b : α), a ≤ b → ∀ (c_1 : α), c_1 + a ≤ c_1 + b
  • exists_pair_ne : ∃ (x y : α), x ≠ y
  • mul_pos : ∀ (a b : α), 0 < a → 0 < b → 0 < a * b
  • le_total : ∀ (a b : α), a ≤ b ∨ b ≤ a
  • zero_lt_one : linear_ordered_comm_ring.zero < linear_ordered_comm_ring.one
  • mul_comm : ∀ (a b : α), a * b = b * a

A linear_ordered_comm_ring α is a commutative ring α with a linear order such that multiplication with a positive number and addition are monotone.

Instances
source
@[instance]
def linear_ordered_comm_ring.to_linear_ordered_ring (α : Type u) [s : linear_ordered_comm_ring α] :
linear_ordered_ring α

source
@[instance]
def linear_ordered_comm_ring.to_comm_monoid (α : Type u) [s : linear_ordered_comm_ring α] :
comm_monoid α

source
@[instance]
def linear_ordered_comm_ring.to_integral_domain {α : Type u} [s : linear_ordered_comm_ring α] :
integral_domain α

Equations
source
@[instance]
def decidable_linear_ordered_comm_ring.to_decidable_linear_ordered_add_comm_group (α : Type u) [s : decidable_linear_ordered_comm_ring α] :
decidable_linear_ordered_add_comm_group α

source
@[class]
structure decidable_linear_ordered_comm_ring (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
  • neg : α → α
  • add_left_neg : ∀ (a : α), -a + a = 0
  • add_comm : ∀ (a b : α), a + b = b + a
  • 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
  • left_distrib : ∀ (a b c_1 : α), a * (b + c_1) = a * b + a * c_1
  • right_distrib : ∀ (a b c_1 : α), (a + b) * c_1 = a * c_1 + b * c_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
  • add_le_add_left : ∀ (a b : α), a ≤ b → ∀ (c_1 : α), c_1 + a ≤ c_1 + b
  • exists_pair_ne : ∃ (x y : α), x ≠ y
  • mul_pos : ∀ (a b : α), 0 < a → 0 < b → 0 < a * b
  • le_total : ∀ (a b : α), a ≤ b ∨ b ≤ a
  • zero_lt_one : decidable_linear_ordered_comm_ring.zero < decidable_linear_ordered_comm_ring.one
  • mul_comm : ∀ (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

A decidable_linear_ordered_comm_ring α is a commutative ring α with a decidable linear order such that multiplication with a positive number and addition are monotone.

Instances
source
@[instance]
def decidable_linear_ordered_comm_ring.to_linear_ordered_comm_ring (α : Type u) [s : decidable_linear_ordered_comm_ring α] :
linear_ordered_comm_ring α

source
@[instance]
def decidable_linear_ordered_comm_ring.to_decidable_linear_ordered_semiring {α : Type u} [d : decidable_linear_ordered_comm_ring α] :
decidable_linear_ordered_semiring α

Equations
source
theorem abs_mul {α : Type u} [decidable_linear_ordered_comm_ring α] (a b : α) :
abs (a * b) = (abs a) * abs b

source
theorem abs_mul_abs_self {α : Type u} [decidable_linear_ordered_comm_ring α] (a : α) :
(abs a) * abs a = a * a

source
theorem abs_mul_self {α : Type u} [decidable_linear_ordered_comm_ring α] (a : α) :
abs (a * a) = a * a

source
theorem sub_le_of_abs_sub_le_left {α : Type u} [decidable_linear_ordered_comm_ring α] {a b c : α} (h : abs (a - b) ≤ c) :
b - c ≤ a

source
theorem sub_le_of_abs_sub_le_right {α : Type u} [decidable_linear_ordered_comm_ring α] {a b c : α} (h : abs (a - b) ≤ c) :
a - c ≤ b

source
theorem sub_lt_of_abs_sub_lt_left {α : Type u} [decidable_linear_ordered_comm_ring α] {a b c : α} (h : abs (a - b) < c) :
b - c < a

source
theorem sub_lt_of_abs_sub_lt_right {α : Type u} [decidable_linear_ordered_comm_ring α] {a b c : α} (h : abs (a - b) < c) :
a - c < b

source
theorem abs_sub_square {α : Type u} [decidable_linear_ordered_comm_ring α] (a b : α) :
(abs (a - b)) * abs (a - b) = a * a + b * b - ((1 + 1) * a) * b

source
theorem eq_zero_of_mul_self_add_mul_self_eq_zero {α : Type u} [decidable_linear_ordered_comm_ring α] {a b : α} (h : a * a + b * b = 0) :
a = 0

source
theorem abs_abs_sub_abs_le_abs_sub {α : Type u} [decidable_linear_ordered_comm_ring α] (a b : α) :
abs (abs a - abs b) ≤ abs (a - b)

source
@[simp]
theorem abs_two {α : Type u} [decidable_linear_ordered_comm_ring α] :
abs 2 = 2

source
@[instance]
def nonneg_ring.to_ring (α : Type u_1) [s : nonneg_ring α] :
ring α

source
@[class]
structure nonneg_ring (α : Type u_1) :
Type u_1

Extend nonneg_add_comm_group to support ordered rings specified by their nonnegative elements

Instances
source
@[instance]
def nonneg_ring.to_nonneg_add_comm_group (α : Type u_1) [s : nonneg_ring α] :
nonneg_add_comm_group α

source
@[instance]
def nonneg_ring.to_nontrivial (α : Type u_1) [s : nonneg_ring α] :
nontrivial α

source
@[class]
structure linear_nonneg_ring (α : Type u_1) :
Type u_1

Extend nonneg_add_comm_group to support linearly ordered rings specified by their nonnegative elements

source
@[instance]
def linear_nonneg_ring.to_domain (α : Type u_1) [s : linear_nonneg_ring α] :
domain α

source
@[instance]
def linear_nonneg_ring.to_nonneg_add_comm_group (α : Type u_1) [s : linear_nonneg_ring α] :
nonneg_add_comm_group α

source
@[instance]
def nonneg_ring.to_ordered_ring {α : Type u} [nonneg_ring α] :
ordered_ring α

Equations
source
def nonneg_ring.to_linear_nonneg_ring {α : Type u} [nonneg_ring α] (nonneg_total : ∀ (a : α), nonneg_ring.nonneg a ∨ nonneg_ring.nonneg (-a)) :
linear_nonneg_ring α

to_linear_nonneg_ring shows that a nonneg_ring with a total order is a domain, hence a linear_nonneg_ring.

Equations
source
@[instance]
def linear_nonneg_ring.to_nonneg_ring {α : Type u} [linear_nonneg_ring α] :
nonneg_ring α

Equations
source
@[instance]
def linear_nonneg_ring.to_linear_order {α : Type u} [linear_nonneg_ring α] :
linear_order α

Equations
source
@[instance]
def linear_nonneg_ring.to_linear_ordered_ring {α : Type u} [linear_nonneg_ring α] :
linear_ordered_ring α

Equations
source
def linear_nonneg_ring.to_decidable_linear_ordered_comm_ring {α : Type u} [linear_nonneg_ring α] [decidable_pred linear_nonneg_ring.nonneg] [comm : is_commutative α has_mul.mul] :
decidable_linear_ordered_comm_ring α

Convert a linear_nonneg_ring with a commutative multiplication and decidable non-negativity into a decidable_linear_ordered_comm_ring

Equations
source
@[class]
structure canonically_ordered_comm_semiring (α : Type u_1) :
Type u_1
  • 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
  • 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 ≤ 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
  • 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_1 → b < c_1
  • bot : α
  • bot_le : ∀ (a : α), ⊥ ≤ a
  • le_iff_exists_add : ∀ (a b : α), a ≤ b ↔ ∃ (c_1 : α), b = a + c_1
  • 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
  • zero_mul : ∀ (a : α), 0 * a = 0
  • mul_zero : ∀ (a : α), a * 0 = 0
  • left_distrib : ∀ (a b c_1 : α), a * (b + c_1) = a * b + a * c_1
  • right_distrib : ∀ (a b c_1 : α), (a + b) * c_1 = a * c_1 + b * c_1
  • mul_comm : ∀ (a b : α), a * b = b * a
  • exists_pair_ne : ∃ (x y : α), x ≠ y
  • eq_zero_or_eq_zero_of_mul_eq_zero : ∀ (a b : α), a * b = 0 → a = 0 ∨ b = 0

A canonically ordered commutative semiring is an ordered, commutative semiring in which a ≤ b iff there exists c with b = a + c. This is satisfied by the natural numbers, for example, but not the integers or other ordered groups.

Instances
source
@[instance]
def canonically_ordered_comm_semiring.to_nontrivial (α : Type u_1) [s : canonically_ordered_comm_semiring α] :
nontrivial α

source
@[instance]
def canonically_ordered_comm_semiring.to_comm_semiring (α : Type u_1) [s : canonically_ordered_comm_semiring α] :
comm_semiring α

source
@[instance]
def canonically_ordered_comm_semiring.to_canonically_ordered_add_monoid (α : Type u_1) [s : canonically_ordered_comm_semiring α] :
canonically_ordered_add_monoid α

source
@[instance]
def canonically_ordered_semiring.canonically_ordered_comm_semiring.to_no_zero_divisors {α : Type u} [canonically_ordered_comm_semiring α] :
no_zero_divisors α

source
theorem canonically_ordered_semiring.mul_le_mul {α : Type u} [canonically_ordered_comm_semiring α] {a b c d : α} (hab : a ≤ b) (hcd : c ≤ d) :
a * c ≤ b * d

source
theorem canonically_ordered_semiring.mul_le_mul_left' {α : Type u} [canonically_ordered_comm_semiring α] {b c : α} (h : b ≤ c) (a : α) :
a * b ≤ a * c

source
theorem canonically_ordered_semiring.mul_le_mul_right' {α : Type u} [canonically_ordered_comm_semiring α] {b c : α} (h : b ≤ c) (a : α) :
b * a ≤ c * a

source
theorem canonically_ordered_semiring.zero_lt_one {α : Type u} [canonically_ordered_comm_semiring α] :
0 < 1

A version of zero_lt_one : 0 < 1 for a canonically_ordered_comm_semiring.

source
theorem canonically_ordered_semiring.mul_pos {α : Type u} [canonically_ordered_comm_semiring α] {a b : α} :
0 < a * b ↔ 0 < a ∧ 0 < b

source
@[instance]
def with_top.nontrivial {α : Type u} [nonempty α] :
nontrivial (with_top α)

source
@[instance]
def with_top.mul_zero_class {α : Type u} [decidable_eq α] [has_zero α] [has_mul α] :
mul_zero_class (with_top α)

Equations
source
theorem with_top.mul_def {α : Type u} [decidable_eq α] [has_zero α] [has_mul α] {a b : with_top α} :
a * b = ite (a = 0 ∨ b = 0) 0 (option.bind a (λ (a : α), option.bind b (λ (b : α), ↑a * b)))

source
@[simp]
theorem with_top.mul_top {α : Type u} [decidable_eq α] [has_zero α] [has_mul α] {a : with_top α} (h : a ≠ 0) :
a * ⊤ = ⊤

source
@[simp]
theorem with_top.top_mul {α : Type u} [decidable_eq α] [has_zero α] [has_mul α] {a : with_top α} (h : a ≠ 0) :
⊤ * a = ⊤

source
@[simp]
theorem with_top.top_mul_top {α : Type u} [decidable_eq α] [has_zero α] [has_mul α] :
⊤ * ⊤ = ⊤

source
theorem with_top.coe_mul {α : Type u} [decidable_eq α] [mul_zero_class α] {a b : α} :
↑a * b = (↑a) * ↑b

source
theorem with_top.mul_coe {α : Type u} [decidable_eq α] [mul_zero_class α] {b : α} (hb : b ≠ 0) {a : with_top α} :
a * ↑b = option.bind a (λ (a : α), ↑a * b)

source
@[simp]
theorem with_top.mul_eq_top_iff {α : Type u} [decidable_eq α] [mul_zero_class α] {a b : with_top α} :
a * b = ⊤ ↔ a ≠ 0 ∧ b = ⊤ ∨ a = ⊤ ∧ b ≠ 0

source
@[instance]
def with_top.no_zero_divisors {α : Type u} [decidable_eq α] [mul_zero_class α] [no_zero_divisors α] :
no_zero_divisors (with_top α)

source
@[instance]
def with_top.canonically_ordered_comm_semiring {α : Type u} [decidable_eq α] [canonically_ordered_comm_semiring α] :
canonically_ordered_comm_semiring (with_top α)

Equations