mathlib docs: core.init.algebra.order

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@[class]

structure preorder :

Type u → Type u

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"

A preorder is a reflexive, transitive relation ≤ with a < b defined in the obvious way.

Instances

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@[instance]

def preorder.to_has_lt (α : Type u) [s : preorder α] :

has_lt α

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@[instance]

def preorder.to_has_le (α : Type u) [s : preorder α] :

has_le α

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@[instance]

def partial_order.to_preorder (α : Type u) [s : partial_order α] :

preorder α

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@[class]

structure partial_order :

Type u → Type u

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

A partial order is a reflexive, transitive, antisymmetric relation ≤.

Instances

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@[class]

structure linear_order :

Type u → Type u

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
le_total : ∀ (a b : α), a ≤ b ∨ b ≤ a

A linear order is reflexive, transitive, antisymmetric and total relation ≤.

Instances

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@[instance]

def linear_order.to_partial_order (α : Type u) [s : linear_order α] :

partial_order α

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theorem le_refl {α : Type u} [preorder α] (a : α) :

a ≤ a

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theorem le_trans {α : Type u} [preorder α] {a b c : α} :

a ≤ b → b ≤ c → a ≤ c

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theorem lt_iff_le_not_le {α : Type u} [preorder α] {a b : α} :

a < b ↔ a ≤ b ∧ ¬b ≤ a

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theorem lt_of_le_not_le {α : Type u} [preorder α] {a b : α} :

a ≤ b → ¬b ≤ a → a < b

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theorem le_not_le_of_lt {α : Type u} [preorder α] {a b : α} :

a < b → a ≤ b ∧ ¬b ≤ a

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theorem le_antisymm {α : Type u} [partial_order α] {a b : α} :

a ≤ b → b ≤ a → a = b

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theorem le_of_eq {α : Type u} [preorder α] {a b : α} :

a = b → a ≤ b

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theorem le_antisymm_iff {α : Type u} [partial_order α] {a b : α} :

a = b ↔ a ≤ b ∧ b ≤ a

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theorem ge_trans {α : Type u} [preorder α] {a b c : α} :

a ≥ b → b ≥ c → a ≥ c

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theorem le_total {α : Type u} [linear_order α] (a b : α) :

a ≤ b ∨ b ≤ a

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theorem le_of_not_ge {α : Type u} [linear_order α] {a b : α} :

¬a ≥ b → a ≤ b

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theorem le_of_not_le {α : Type u} [linear_order α] {a b : α} :

¬a ≤ b → b ≤ a

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theorem lt_irrefl {α : Type u} [preorder α] (a : α) :

¬a < a

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theorem gt_irrefl {α : Type u} [preorder α] (a : α) :

¬a > a

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theorem lt_trans {α : Type u} [preorder α] {a b c : α} :

a < b → b < c → a < c

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def lt.trans {α : Type u_1} [preorder α] {a b c : α} :

a < b → b < c → a < c

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lt.trans = lt_trans

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theorem gt_trans {α : Type u} [preorder α] {a b c : α} :

a > b → b > c → a > c

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def gt.trans {α : Type u_1} [preorder α] {a b c : α} :

a > b → b > c → a > c

Equations

gt.trans = gt_trans

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theorem ne_of_lt {α : Type u} [preorder α] {a b : α} :

a < b → a ≠ b

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theorem ne_of_gt {α : Type u} [preorder α] {a b : α} :

a > b → a ≠ b

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theorem lt_asymm {α : Type u} [preorder α] {a b : α} :

a < b → ¬b < a

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theorem not_lt_of_gt {α : Type u} [linear_order α] {a b : α} :

a > b → ¬a < b

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theorem le_of_lt {α : Type u} [preorder α] {a b : α} :

a < b → a ≤ b

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theorem lt_of_lt_of_le {α : Type u} [preorder α] {a b c : α} :

a < b → b ≤ c → a < c

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theorem lt_of_le_of_lt {α : Type u} [preorder α] {a b c : α} :

a ≤ b → b < c → a < c

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theorem gt_of_gt_of_ge {α : Type u} [preorder α] {a b c : α} :

a > b → b ≥ c → a > c

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theorem gt_of_ge_of_gt {α : Type u} [preorder α] {a b c : α} :

a ≥ b → b > c → a > c

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theorem not_le_of_gt {α : Type u} [preorder α] {a b : α} :

a > b → ¬a ≤ b

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theorem not_lt_of_ge {α : Type u} [preorder α] {a b : α} :

a ≥ b → ¬a < b

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theorem lt_or_eq_of_le {α : Type u} [partial_order α] {a b : α} :

a ≤ b → a < b ∨ a = b

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theorem le_of_lt_or_eq {α : Type u} [preorder α] {a b : α} :

a < b ∨ a = b → a ≤ b

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theorem le_iff_lt_or_eq {α : Type u} [partial_order α] {a b : α} :

a ≤ b ↔ a < b ∨ a = b

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theorem lt_of_le_of_ne {α : Type u} [partial_order α] {a b : α} :

a ≤ b → a ≠ b → a < b

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theorem lt_trichotomy {α : Type u} [linear_order α] (a b : α) :

a < b ∨ a = b ∨ b < a

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theorem le_of_not_gt {α : Type u} [linear_order α] {a b : α} :

¬a > b → a ≤ b

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theorem lt_of_not_ge {α : Type u} [linear_order α] {a b : α} :

¬a ≥ b → a < b

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theorem lt_or_ge {α : Type u} [linear_order α] (a b : α) :

a < b ∨ a ≥ b

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theorem le_or_gt {α : Type u} [linear_order α] (a b : α) :

a ≤ b ∨ a > b

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theorem lt_or_gt_of_ne {α : Type u} [linear_order α] {a b : α} :

a ≠ b → a < b ∨ a > b

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theorem le_of_eq_or_lt {α : Type u} [preorder α] {a b : α} :

a = b ∨ a < b → a ≤ b

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theorem ne_iff_lt_or_gt {α : Type u} [linear_order α] {a b : α} :

a ≠ b ↔ a < b ∨ a > b

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theorem lt_iff_not_ge {α : Type u} [linear_order α] (x y : α) :

x < y ↔ ¬x ≥ y

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@[instance]

def decidable_lt_of_decidable_le {α : Type u} [preorder α] [decidable_rel has_le.le] :

decidable_rel has_lt.lt

Equations

decidable_lt_of_decidable_le a b = dite (a ≤ b) (λ (hab : a ≤ b), dite (b ≤ a) (λ (hba : b ≤ a), decidable.is_false _) (λ (hba : ¬b ≤ a), decidable.is_true _)) (λ (hab : ¬a ≤ b), decidable.is_false _)

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@[instance]

def decidable_eq_of_decidable_le {α : Type u} [partial_order α] [decidable_rel has_le.le] :

decidable_eq α

Equations

decidable_eq_of_decidable_le a b = dite (a ≤ b) (λ (hab : a ≤ b), dite (b ≤ a) (λ (hba : b ≤ a), decidable.is_true _) (λ (hba : ¬b ≤ a), decidable.is_false _)) (λ (hab : ¬a ≤ b), decidable.is_false _)

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@[instance]

def decidable_linear_order.to_linear_order (α : Type u) [s : decidable_linear_order α] :

linear_order α

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@[class]

structure decidable_linear_order :

Type u → Type u

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
le_total : ∀ (a b : α), a ≤ b ∨ b ≤ a
decidable_le : decidable_rel has_le.le
decidable_eq : decidable_eq α
decidable_lt : decidable_rel has_lt.lt

Instances

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@[instance]

def has_lt.lt.decidable {α : Type u} [decidable_linear_order α] (a b : α) :

decidable (a < b)

Equations

has_lt.lt.decidable a b = decidable_linear_order.decidable_lt a b

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@[instance]

def has_le.le.decidable {α : Type u} [decidable_linear_order α] (a b : α) :

decidable (a ≤ b)

Equations

has_le.le.decidable a b = decidable_linear_order.decidable_le a b

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@[instance]

def eq.decidable {α : Type u} [decidable_linear_order α] (a b : α) :

decidable (a = b)

Equations

eq.decidable a b = decidable_linear_order.decidable_eq a b

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theorem eq_or_lt_of_not_lt {α : Type u} [decidable_linear_order α] {a b : α} :

¬a < b → a = b ∨ b < a

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@[instance]

def has_le.le.is_total_preorder {α : Type u} [decidable_linear_order α] :

is_total_preorder α has_le.le

Equations

has_le.le.is_total_preorder = is_total_preorder.mk

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@[instance]

def is_strict_weak_order_of_decidable_linear_order {α : Type u} [decidable_linear_order α] :

is_strict_weak_order α has_lt.lt

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is_strict_weak_order_of_decidable_linear_order = _

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@[instance]

def is_strict_total_order_of_decidable_linear_order {α : Type u} [decidable_linear_order α] :

is_strict_total_order α has_lt.lt

Equations

is_strict_total_order_of_decidable_linear_order = is_strict_total_order.mk

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core.init.algebra.order

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mathlib documentation

core.​init.​algebra.​order

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core.init.algebra.order