mathlib documentation

algebra.ordered_field

Linear ordered fields

A linear ordered field is a field equipped with a linear order such that

Main Definitions

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@[instance]
def linear_ordered_field.to_field (α : Type u_2) [s : linear_ordered_field α] :
field α

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@[class]
structure linear_ordered_field (α : Type u_2) :
Type u_2
  • 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_field.zero < linear_ordered_field.one
  • mul_comm : ∀ (a b : α), a * b = b * a
  • inv : α → α
  • mul_inv_cancel : ∀ {a : α}, a ≠ 0 → a * a⁻¹ = 1
  • inv_zero : 0⁻¹ = 0

A linear ordered field is a field with a linear order respecting the operations.

Instances
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@[instance]
def linear_ordered_field.to_linear_ordered_comm_ring (α : Type u_2) [s : linear_ordered_field α] :
linear_ordered_comm_ring α

Lemmas about pos, nonneg, nonpos, neg

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@[simp]
theorem inv_pos {α : Type u_1} [linear_ordered_field α] {a : α} :
0 < a⁻¹ ↔ 0 < a

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@[simp]
theorem inv_nonneg {α : Type u_1} [linear_ordered_field α] {a : α} :
0 ≤ a⁻¹ ↔ 0 ≤ a

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@[simp]
theorem inv_lt_zero {α : Type u_1} [linear_ordered_field α] {a : α} :
a⁻¹ < 0 ↔ a < 0

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@[simp]
theorem inv_nonpos {α : Type u_1} [linear_ordered_field α] {a : α} :
a⁻¹ ≤ 0 ↔ a ≤ 0

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theorem one_div_pos {α : Type u_1} [linear_ordered_field α] {a : α} :
0 < 1 / a ↔ 0 < a

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theorem one_div_neg {α : Type u_1} [linear_ordered_field α] {a : α} :
1 / a < 0 ↔ a < 0

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theorem one_div_nonneg {α : Type u_1} [linear_ordered_field α] {a : α} :
0 ≤ 1 / a ↔ 0 ≤ a

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theorem one_div_nonpos {α : Type u_1} [linear_ordered_field α] {a : α} :
1 / a ≤ 0 ↔ a ≤ 0

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theorem div_pos {α : Type u_1} [linear_ordered_field α] {a b : α} (ha : 0 < a) (hb : 0 < b) :
0 < a / b

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theorem div_pos_of_neg_of_neg {α : Type u_1} [linear_ordered_field α] {a b : α} (ha : a < 0) (hb : b < 0) :
0 < a / b

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theorem div_neg_of_neg_of_pos {α : Type u_1} [linear_ordered_field α] {a b : α} (ha : a < 0) (hb : 0 < b) :
a / b < 0

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theorem div_neg_of_pos_of_neg {α : Type u_1} [linear_ordered_field α] {a b : α} (ha : 0 < a) (hb : b < 0) :
a / b < 0

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theorem div_nonneg {α : Type u_1} [linear_ordered_field α] {a b : α} (ha : 0 ≤ a) (hb : 0 ≤ b) :
0 ≤ a / b

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theorem div_nonneg_of_nonpos {α : Type u_1} [linear_ordered_field α] {a b : α} (ha : a ≤ 0) (hb : b ≤ 0) :
0 ≤ a / b

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theorem div_nonpos_of_nonpos_of_nonneg {α : Type u_1} [linear_ordered_field α] {a b : α} (ha : a ≤ 0) (hb : 0 ≤ b) :
a / b ≤ 0

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theorem div_nonpos_of_nonneg_of_nonpos {α : Type u_1} [linear_ordered_field α] {a b : α} (ha : 0 ≤ a) (hb : b ≤ 0) :
a / b ≤ 0

Relating one division with another term.

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theorem le_div_iff {α : Type u_1} [linear_ordered_field α] {a b c : α} (hc : 0 < c) :
a ≤ b / c ↔ a * c ≤ b

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theorem le_div_iff' {α : Type u_1} [linear_ordered_field α] {a b c : α} (hc : 0 < c) :
a ≤ b / c ↔ c * a ≤ b

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theorem div_le_iff {α : Type u_1} [linear_ordered_field α] {a b c : α} (hb : 0 < b) :
a / b ≤ c ↔ a ≤ c * b

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theorem div_le_iff' {α : Type u_1} [linear_ordered_field α] {a b c : α} (hb : 0 < b) :
a / b ≤ c ↔ a ≤ b * c

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theorem lt_div_iff {α : Type u_1} [linear_ordered_field α] {a b c : α} (hc : 0 < c) :
a < b / c ↔ a * c < b

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theorem lt_div_iff' {α : Type u_1} [linear_ordered_field α] {a b c : α} (hc : 0 < c) :
a < b / c ↔ c * a < b

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theorem div_lt_iff {α : Type u_1} [linear_ordered_field α] {a b c : α} (hc : 0 < c) :
b / c < a ↔ b < a * c

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theorem div_lt_iff' {α : Type u_1} [linear_ordered_field α] {a b c : α} (hc : 0 < c) :
b / c < a ↔ b < c * a

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theorem inv_mul_le_iff {α : Type u_1} [linear_ordered_field α] {a b c : α} (h : 0 < b) :
b⁻¹ * a ≤ c ↔ a ≤ b * c

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theorem inv_mul_le_iff' {α : Type u_1} [linear_ordered_field α] {a b c : α} (h : 0 < b) :
b⁻¹ * a ≤ c ↔ a ≤ c * b

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theorem mul_inv_le_iff {α : Type u_1} [linear_ordered_field α] {a b c : α} (h : 0 < b) :
a * b⁻¹ ≤ c ↔ a ≤ b * c

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theorem mul_inv_le_iff' {α : Type u_1} [linear_ordered_field α] {a b c : α} (h : 0 < b) :
a * b⁻¹ ≤ c ↔ a ≤ c * b

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theorem inv_mul_lt_iff {α : Type u_1} [linear_ordered_field α] {a b c : α} (h : 0 < b) :
b⁻¹ * a < c ↔ a < b * c

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theorem inv_mul_lt_iff' {α : Type u_1} [linear_ordered_field α] {a b c : α} (h : 0 < b) :
b⁻¹ * a < c ↔ a < c * b

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theorem mul_inv_lt_iff {α : Type u_1} [linear_ordered_field α] {a b c : α} (h : 0 < b) :
a * b⁻¹ < c ↔ a < b * c

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theorem mul_inv_lt_iff' {α : Type u_1} [linear_ordered_field α] {a b c : α} (h : 0 < b) :
a * b⁻¹ < c ↔ a < c * b

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theorem div_le_iff_of_neg {α : Type u_1} [linear_ordered_field α] {a b c : α} (hc : c < 0) :
b / c ≤ a ↔ a * c ≤ b

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theorem div_le_iff_of_neg' {α : Type u_1} [linear_ordered_field α] {a b c : α} (hc : c < 0) :
b / c ≤ a ↔ c * a ≤ b

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theorem le_div_iff_of_neg {α : Type u_1} [linear_ordered_field α] {a b c : α} (hc : c < 0) :
a ≤ b / c ↔ b ≤ a * c

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theorem le_div_iff_of_neg' {α : Type u_1} [linear_ordered_field α] {a b c : α} (hc : c < 0) :
a ≤ b / c ↔ b ≤ c * a

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theorem div_lt_iff_of_neg {α : Type u_1} [linear_ordered_field α] {a b c : α} (hc : c < 0) :
b / c < a ↔ a * c < b

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theorem div_lt_iff_of_neg' {α : Type u_1} [linear_ordered_field α] {a b c : α} (hc : c < 0) :
b / c < a ↔ c * a < b

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theorem lt_div_iff_of_neg {α : Type u_1} [linear_ordered_field α] {a b c : α} (hc : c < 0) :
a < b / c ↔ b < a * c

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theorem lt_div_iff_of_neg' {α : Type u_1} [linear_ordered_field α] {a b c : α} (hc : c < 0) :
a < b / c ↔ b < c * a

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theorem div_le_iff_of_nonneg_of_le {α : Type u_1} [linear_ordered_field α] {a b c : α} (hb : 0 ≤ b) (hc : 0 ≤ c) (h : a ≤ c * b) :
a / b ≤ c

One direction of div_le_iff where b is allowed to be 0 (but c must be nonnegative)

Bi-implications of inequalities using inversions

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theorem inv_le_inv_of_le {α : Type u_1} [linear_ordered_field α] {a b : α} (ha : 0 < a) (h : a ≤ b) :
b⁻¹ ≤ a⁻¹

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theorem inv_le_inv {α : Type u_1} [linear_ordered_field α] {a b : α} (ha : 0 < a) (hb : 0 < b) :
a⁻¹ ≤ b⁻¹ ↔ b ≤ a

See inv_le_inv_of_le for the implication from right-to-left with one fewer assumption.

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theorem inv_le {α : Type u_1} [linear_ordered_field α] {a b : α} (ha : 0 < a) (hb : 0 < b) :
a⁻¹ ≤ b ↔ b⁻¹ ≤ a

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theorem le_inv {α : Type u_1} [linear_ordered_field α] {a b : α} (ha : 0 < a) (hb : 0 < b) :
a ≤ b⁻¹ ↔ b ≤ a⁻¹

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theorem inv_lt_inv {α : Type u_1} [linear_ordered_field α] {a b : α} (ha : 0 < a) (hb : 0 < b) :
a⁻¹ < b⁻¹ ↔ b < a

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theorem inv_lt {α : Type u_1} [linear_ordered_field α] {a b : α} (ha : 0 < a) (hb : 0 < b) :
a⁻¹ < b ↔ b⁻¹ < a

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theorem lt_inv {α : Type u_1} [linear_ordered_field α] {a b : α} (ha : 0 < a) (hb : 0 < b) :
a < b⁻¹ ↔ b < a⁻¹

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theorem inv_le_inv_of_neg {α : Type u_1} [linear_ordered_field α] {a b : α} (ha : a < 0) (hb : b < 0) :
a⁻¹ ≤ b⁻¹ ↔ b ≤ a

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theorem inv_le_of_neg {α : Type u_1} [linear_ordered_field α] {a b : α} (ha : a < 0) (hb : b < 0) :
a⁻¹ ≤ b ↔ b⁻¹ ≤ a

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theorem le_inv_of_neg {α : Type u_1} [linear_ordered_field α] {a b : α} (ha : a < 0) (hb : b < 0) :
a ≤ b⁻¹ ↔ b ≤ a⁻¹

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theorem inv_lt_inv_of_neg {α : Type u_1} [linear_ordered_field α] {a b : α} (ha : a < 0) (hb : b < 0) :
a⁻¹ < b⁻¹ ↔ b < a

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theorem inv_lt_of_neg {α : Type u_1} [linear_ordered_field α] {a b : α} (ha : a < 0) (hb : b < 0) :
a⁻¹ < b ↔ b⁻¹ < a

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theorem lt_inv_of_neg {α : Type u_1} [linear_ordered_field α] {a b : α} (ha : a < 0) (hb : b < 0) :
a < b⁻¹ ↔ b < a⁻¹

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theorem inv_lt_one {α : Type u_1} [linear_ordered_field α] {a : α} (ha : 1 < a) :
a⁻¹ < 1

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theorem one_lt_inv {α : Type u_1} [linear_ordered_field α] {a : α} (h₁ : 0 < a) (h₂ : a < 1) :
1 < a⁻¹

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theorem inv_le_one {α : Type u_1} [linear_ordered_field α] {a : α} (ha : 1 ≤ a) :
a⁻¹ ≤ 1

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theorem one_le_inv {α : Type u_1} [linear_ordered_field α] {a : α} (h₁ : 0 < a) (h₂ : a ≤ 1) :
1 ≤ a⁻¹

Relating two divisions.

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theorem div_le_div_of_le {α : Type u_1} [linear_ordered_field α] {a b c : α} (hc : 0 ≤ c) (h : a ≤ b) :
a / c ≤ b / c

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theorem div_le_div_of_le_left {α : Type u_1} [linear_ordered_field α] {a b c : α} (ha : 0 ≤ a) (hc : 0 < c) (h : c ≤ b) :
a / b ≤ a / c

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theorem div_le_div_of_le_of_nonneg {α : Type u_1} [linear_ordered_field α] {a b c : α} (hab : a ≤ b) (hc : 0 ≤ c) :
a / c ≤ b / c

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theorem div_le_div_of_nonpos_of_le {α : Type u_1} [linear_ordered_field α] {a b c : α} (hc : c ≤ 0) (h : b ≤ a) :
a / c ≤ b / c

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theorem div_lt_div_of_lt {α : Type u_1} [linear_ordered_field α] {a b c : α} (hc : 0 < c) (h : a < b) :
a / c < b / c

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theorem div_lt_div_of_neg_of_lt {α : Type u_1} [linear_ordered_field α] {a b c : α} (hc : c < 0) (h : b < a) :
a / c < b / c

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theorem div_le_div_right {α : Type u_1} [linear_ordered_field α] {a b c : α} (hc : 0 < c) :
a / c ≤ b / c ↔ a ≤ b

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theorem div_le_div_right_of_neg {α : Type u_1} [linear_ordered_field α] {a b c : α} (hc : c < 0) :
a / c ≤ b / c ↔ b ≤ a

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theorem div_lt_div_right {α : Type u_1} [linear_ordered_field α] {a b c : α} (hc : 0 < c) :
a / c < b / c ↔ a < b

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theorem div_lt_div_right_of_neg {α : Type u_1} [linear_ordered_field α] {a b c : α} (hc : c < 0) :
a / c < b / c ↔ b < a

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theorem div_lt_div_left {α : Type u_1} [linear_ordered_field α] {a b c : α} (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) :
a / b < a / c ↔ c < b

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theorem div_le_div_left {α : Type u_1} [linear_ordered_field α] {a b c : α} (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) :
a / b ≤ a / c ↔ c ≤ b

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theorem div_lt_div_iff {α : Type u_1} [linear_ordered_field α] {a b c d : α} (b0 : 0 < b) (d0 : 0 < d) :
a / b < c / d ↔ a * d < c * b

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theorem div_le_div_iff {α : Type u_1} [linear_ordered_field α] {a b c d : α} (b0 : 0 < b) (d0 : 0 < d) :
a / b ≤ c / d ↔ a * d ≤ c * b

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theorem div_le_div {α : Type u_1} [linear_ordered_field α] {a b c d : α} (hc : 0 ≤ c) (hac : a ≤ c) (hd : 0 < d) (hbd : d ≤ b) :
a / b ≤ c / d

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theorem div_lt_div {α : Type u_1} [linear_ordered_field α] {a b c d : α} (hac : a < c) (hbd : d ≤ b) (c0 : 0 ≤ c) (d0 : 0 < d) :
a / b < c / d

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theorem div_lt_div' {α : Type u_1} [linear_ordered_field α] {a b c d : α} (hac : a ≤ c) (hbd : d < b) (c0 : 0 < c) (d0 : 0 < d) :
a / b < c / d

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theorem div_lt_div_of_lt_left {α : Type u_1} [linear_ordered_field α] {a b c : α} (hb : 0 < b) (h : b < a) (hc : 0 < c) :
c / a < c / b

Relating one division and involving 1

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theorem one_le_div {α : Type u_1} [linear_ordered_field α] {a b : α} (hb : 0 < b) :
1 ≤ a / b ↔ b ≤ a

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theorem div_le_one {α : Type u_1} [linear_ordered_field α] {a b : α} (hb : 0 < b) :
a / b ≤ 1 ↔ a ≤ b

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theorem one_lt_div {α : Type u_1} [linear_ordered_field α] {a b : α} (hb : 0 < b) :
1 < a / b ↔ b < a

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theorem div_lt_one {α : Type u_1} [linear_ordered_field α] {a b : α} (hb : 0 < b) :
a / b < 1 ↔ a < b

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theorem one_le_div_of_neg {α : Type u_1} [linear_ordered_field α] {a b : α} (hb : b < 0) :
1 ≤ a / b ↔ a ≤ b

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theorem div_le_one_of_neg {α : Type u_1} [linear_ordered_field α] {a b : α} (hb : b < 0) :
a / b ≤ 1 ↔ b ≤ a

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theorem one_lt_div_of_neg {α : Type u_1} [linear_ordered_field α] {a b : α} (hb : b < 0) :
1 < a / b ↔ a < b

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theorem div_lt_one_of_neg {α : Type u_1} [linear_ordered_field α] {a b : α} (hb : b < 0) :
a / b < 1 ↔ b < a

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theorem one_div_le {α : Type u_1} [linear_ordered_field α] {a b : α} (ha : 0 < a) (hb : 0 < b) :
1 / a ≤ b ↔ 1 / b ≤ a

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theorem one_div_lt {α : Type u_1} [linear_ordered_field α] {a b : α} (ha : 0 < a) (hb : 0 < b) :
1 / a < b ↔ 1 / b < a

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theorem le_one_div {α : Type u_1} [linear_ordered_field α] {a b : α} (ha : 0 < a) (hb : 0 < b) :
a ≤ 1 / b ↔ b ≤ 1 / a

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theorem lt_one_div {α : Type u_1} [linear_ordered_field α] {a b : α} (ha : 0 < a) (hb : 0 < b) :
a < 1 / b ↔ b < 1 / a

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theorem one_div_le_of_neg {α : Type u_1} [linear_ordered_field α] {a b : α} (ha : a < 0) (hb : b < 0) :
1 / a ≤ b ↔ 1 / b ≤ a

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theorem one_div_lt_of_neg {α : Type u_1} [linear_ordered_field α] {a b : α} (ha : a < 0) (hb : b < 0) :
1 / a < b ↔ 1 / b < a

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theorem le_one_div_of_neg {α : Type u_1} [linear_ordered_field α] {a b : α} (ha : a < 0) (hb : b < 0) :
a ≤ 1 / b ↔ b ≤ 1 / a

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theorem lt_one_div_of_neg {α : Type u_1} [linear_ordered_field α] {a b : α} (ha : a < 0) (hb : b < 0) :
a < 1 / b ↔ b < 1 / a

Relating two divisions, involving 1

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theorem one_div_le_one_div_of_le {α : Type u_1} [linear_ordered_field α] {a b : α} (ha : 0 < a) (h : a ≤ b) :
1 / b ≤ 1 / a

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theorem one_div_lt_one_div_of_lt {α : Type u_1} [linear_ordered_field α] {a b : α} (ha : 0 < a) (h : a < b) :
1 / b < 1 / a

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theorem one_div_le_one_div_of_neg_of_le {α : Type u_1} [linear_ordered_field α] {a b : α} (hb : b < 0) (h : a ≤ b) :
1 / b ≤ 1 / a

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theorem one_div_lt_one_div_of_neg_of_lt {α : Type u_1} [linear_ordered_field α] {a b : α} (hb : b < 0) (h : a < b) :
1 / b < 1 / a

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theorem le_of_one_div_le_one_div {α : Type u_1} [linear_ordered_field α] {a b : α} (ha : 0 < a) (h : 1 / a ≤ 1 / b) :
b ≤ a

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theorem lt_of_one_div_lt_one_div {α : Type u_1} [linear_ordered_field α] {a b : α} (ha : 0 < a) (h : 1 / a < 1 / b) :
b < a

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theorem le_of_neg_of_one_div_le_one_div {α : Type u_1} [linear_ordered_field α] {a b : α} (hb : b < 0) (h : 1 / a ≤ 1 / b) :
b ≤ a

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theorem lt_of_neg_of_one_div_lt_one_div {α : Type u_1} [linear_ordered_field α] {a b : α} (hb : b < 0) (h : 1 / a < 1 / b) :
b < a

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theorem one_div_le_one_div {α : Type u_1} [linear_ordered_field α] {a b : α} (ha : 0 < a) (hb : 0 < b) :
1 / a ≤ 1 / b ↔ b ≤ a

For the single implications with fewer assumptions, see one_div_le_one_div_of_le and le_of_one_div_le_one_div

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theorem one_div_lt_one_div {α : Type u_1} [linear_ordered_field α] {a b : α} (ha : 0 < a) (hb : 0 < b) :
1 / a < 1 / b ↔ b < a

For the single implications with fewer assumptions, see one_div_lt_one_div_of_lt and lt_of_one_div_lt_one_div

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theorem one_div_le_one_div_of_neg {α : Type u_1} [linear_ordered_field α] {a b : α} (ha : a < 0) (hb : b < 0) :
1 / a ≤ 1 / b ↔ b ≤ a

For the single implications with fewer assumptions, see one_div_lt_one_div_of_neg_of_lt and lt_of_one_div_lt_one_div

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theorem one_div_lt_one_div_of_neg {α : Type u_1} [linear_ordered_field α] {a b : α} (ha : a < 0) (hb : b < 0) :
1 / a < 1 / b ↔ b < a

For the single implications with fewer assumptions, see one_div_lt_one_div_of_lt and lt_of_one_div_lt_one_div

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theorem one_lt_one_div {α : Type u_1} [linear_ordered_field α] {a : α} (h1 : 0 < a) (h2 : a < 1) :
1 < 1 / a

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theorem one_le_one_div {α : Type u_1} [linear_ordered_field α] {a : α} (h1 : 0 < a) (h2 : a ≤ 1) :
1 ≤ 1 / a

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theorem one_div_lt_neg_one {α : Type u_1} [linear_ordered_field α] {a : α} (h1 : a < 0) (h2 : -1 < a) :
1 / a < -1

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theorem one_div_le_neg_one {α : Type u_1} [linear_ordered_field α] {a : α} (h1 : a < 0) (h2 : -1 ≤ a) :
1 / a ≤ -1

Results about halving.

The equalities also hold in fields of characteristic 0.

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theorem add_halves {α : Type u_1} [linear_ordered_field α] (a : α) :
a / 2 + a / 2 = a

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theorem sub_self_div_two {α : Type u_1} [linear_ordered_field α] (a : α) :
a - a / 2 = a / 2

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theorem div_two_sub_self {α : Type u_1} [linear_ordered_field α] (a : α) :
a / 2 - a = -(a / 2)

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theorem add_self_div_two {α : Type u_1} [linear_ordered_field α] (a : α) :
(a + a) / 2 = a

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theorem half_pos {α : Type u_1} [linear_ordered_field α] {a : α} (h : 0 < a) :
0 < a / 2

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theorem one_half_pos {α : Type u_1} [linear_ordered_field α] :
0 < 1 / 2

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theorem div_two_lt_of_pos {α : Type u_1} [linear_ordered_field α] {a : α} (h : 0 < a) :
a / 2 < a

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theorem half_lt_self {α : Type u_1} [linear_ordered_field α] {a : α} (a_1 : 0 < a) :
a / 2 < a

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theorem one_half_lt_one {α : Type u_1} [linear_ordered_field α] :
1 / 2 < 1

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theorem add_sub_div_two_lt {α : Type u_1} [linear_ordered_field α] {a b : α} (h : a < b) :
a + (b - a) / 2 < b

Miscellaneous lemmas

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theorem mul_sub_mul_div_mul_neg_iff {α : Type u_1} [linear_ordered_field α] {a b c d : α} (hc : c ≠ 0) (hd : d ≠ 0) :
(a * d - b * c) / c * d < 0 ↔ a / c < b / d

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theorem mul_sub_mul_div_mul_nonpos_iff {α : Type u_1} [linear_ordered_field α] {a b c d : α} (hc : c ≠ 0) (hd : d ≠ 0) :
(a * d - b * c) / c * d ≤ 0 ↔ a / c ≤ b / d

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theorem mul_le_mul_of_mul_div_le {α : Type u_1} [linear_ordered_field α] {a b c d : α} (h : a * (b / c) ≤ d) (hc : 0 < c) :
b * a ≤ d * c

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theorem div_mul_le_div_mul_of_div_le_div {α : Type u_1} [linear_ordered_field α] {a b c d e : α} (h : a / b ≤ c / d) (he : 0 ≤ e) :
a / b * e ≤ c / d * e

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theorem exists_add_lt_and_pos_of_lt {α : Type u_1} [linear_ordered_field α] {a b : α} (h : b < a) :
∃ (c : α), b + c < a ∧ 0 < c

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theorem le_of_forall_sub_le {α : Type u_1} [linear_ordered_field α] {a b : α} (h : ∀ (ε : α), ε > 0 → b - ε ≤ a) :
b ≤ a

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theorem monotone.div_const {α : Type u_1} [linear_ordered_field α] {β : Type u_2} [preorder β] {f : β → α} (hf : monotone f) {c : α} (hc : 0 ≤ c) :
monotone (λ (x : β), f x / c)

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theorem strict_mono.div_const {α : Type u_1} [linear_ordered_field α] {β : Type u_2} [preorder β] {f : β → α} (hf : strict_mono f) {c : α} (hc : 0 < c) :
strict_mono (λ (x : β), f x / c)

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@[instance]
def linear_ordered_field.to_densely_ordered {α : Type u_1} [linear_ordered_field α] :
densely_ordered α

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theorem mul_self_inj_of_nonneg {α : Type u_1} [linear_ordered_field α] {a b : α} (a0 : 0 ≤ a) (b0 : 0 ≤ b) :
a * a = b * b ↔ a = b

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@[class]
structure discrete_linear_ordered_field (α : Type u_2) :
Type u_2
  • 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 : discrete_linear_ordered_field.zero < discrete_linear_ordered_field.one
  • mul_comm : ∀ (a b : α), a * b = b * a
  • inv : α → α
  • mul_inv_cancel : ∀ {a : α}, a ≠ 0 → a * a⁻¹ = 1
  • inv_zero : 0⁻¹ = 0
  • decidable_le : decidable_rel has_le.le
  • decidable_eq : decidable_eq α
  • decidable_lt : decidable_rel has_lt.lt

A discrete linear ordered field is a field with a decidable linear order respecting the operations.

Instances
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@[instance]
def discrete_linear_ordered_field.to_linear_ordered_field (α : Type u_2) [s : discrete_linear_ordered_field α] :
linear_ordered_field α

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@[instance]
def discrete_linear_ordered_field.to_decidable_linear_ordered_comm_ring (α : Type u_2) [s : discrete_linear_ordered_field α] :
decidable_linear_ordered_comm_ring α

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theorem abs_div {α : Type u_1} [discrete_linear_ordered_field α] (a b : α) :
abs (a / b) = abs a / abs b

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theorem abs_one_div {α : Type u_1} [discrete_linear_ordered_field α] (a : α) :
abs (1 / a) = 1 / abs a

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theorem abs_inv {α : Type u_1} [discrete_linear_ordered_field α] (a : α) :
abs a⁻¹ = (abs a)⁻¹