mathlib documentation

order.basic

Basic definitions about `≤` and `<` #

This file proves basic results about orders, provides extensive dot notation, defines useful order classes and allows to transfer order instances.

Type synonyms #

order_dual α : A type synonym reversing the meaning of all inequalities.
as_linear_order α: A type synonym to promote partial_order α to linear_order α using is_total α (≤).

Transfering orders #

order.preimage, preorder.lift: Transfers a (pre)order on β to an order on α using a function f : α → β.
partial_order.lift, linear_order.lift: Transfers a partial (resp., linear) order on β to a partial (resp., linear) order on α using an injective function f.

Extra classes #

no_top_order, no_bot_order: An order without a maximal/minimal element.
densely_ordered: An order with no gap, i.e. for any two elements a < b there exists c such that a < c < b.

Notes #

≤ and < are highly favored over ≥ and > in mathlib. The reason is that we can formulate all lemmas using ≤/<, and rw has trouble unifying ≤ and ≥. Hence choosing one direction spares us useless duplication. This is enforced by a linter. See Note [nolint_ge] for more infos.

Dot notation is particularly useful on ≤ (has_le.le) and < (has_lt.lt). To that end, we provide many aliases to dot notation-less lemmas. For example, le_trans is aliased with has_le.le.trans and can be used to construct hab.trans hbc : a ≤ c when hab : a ≤ b, hbc : b ≤ c, lt_of_le_of_lt is aliased as has_le.le.trans_lt and can be used to construct hab.trans hbc : a < c when hab : a ≤ b, hbc : b < c.

TODO #

expand module docs
automatic construction of dual definitions / theorems

See also #

algebra.order.basic for basic lemmas about orders, and projection notation for orders

Tags #

preorder, order, partial order, poset, linear order, chain

theorem has_le.ext {α : Type u} (x y : has_le α) (h : has_le.le = has_le.le) :

x = y

theorem has_le.ext_iff {α : Type u} (x y : has_le α) :

x = y ↔ has_le.le = has_le.le

theorem has_le.le.trans {α : Type u} [preorder α] {a b c : α} :

a ≤ b → b ≤ c → a ≤ c

Alias of le_trans.

theorem has_le.le.trans_lt {α : Type u} [preorder α] {a b c : α} :

a ≤ b → b < c → a < c

Alias of lt_of_le_of_lt.

theorem has_le.le.antisymm {α : Type u} [partial_order α] {a b : α} :

a ≤ b → b ≤ a → a = b

Alias of le_antisymm.

theorem has_le.le.lt_of_ne {α : Type u} [partial_order α] {a b : α} :

a ≤ b → a ≠ b → a < b

Alias of lt_of_le_of_ne.

theorem has_le.le.lt_of_not_le {α : Type u} [preorder α] {a b : α} :

a ≤ b → ¬b ≤ a → a < b

Alias of lt_of_le_not_le.

theorem has_le.le.lt_or_eq {α : Type u} [partial_order α] {a b : α} :

a ≤ b → a < b ∨ a = b

Alias of lt_or_eq_of_le.

@[nolint]

theorem has_le.le.lt_or_eq_dec {α : Type u} [partial_order α] [decidable_rel has_le.le] {a b : α} (hab : a ≤ b) :

a < b ∨ a = b

Alias of decidable.lt_or_eq_of_le.

theorem has_lt.lt.le {α : Type u} [preorder α] {a b : α} :

a < b → a ≤ b

Alias of le_of_lt.

theorem has_lt.lt.trans {α : Type u} [preorder α] {a b c : α} :

a < b → b < c → a < c

Alias of lt_trans.

theorem has_lt.lt.trans_le {α : Type u} [preorder α] {a b c : α} :

a < b → b ≤ c → a < c

Alias of lt_of_lt_of_le.

theorem has_lt.lt.ne {α : Type u} [preorder α] {a b : α} (h : a < b) :

a ≠ b

Alias of ne_of_lt.

theorem has_lt.lt.asymm {α : Type u} [preorder α] {a b : α} (h : a < b) :

¬b < a

Alias of lt_asymm.

theorem has_lt.lt.not_lt {α : Type u} [preorder α] {a b : α} (h : a < b) :

¬b < a

Alias of lt_asymm.

theorem eq.le {α : Type u} [preorder α] {a b : α} :

a = b → a ≤ b

Alias of le_of_eq.

theorem le_rfl {α : Type u} [preorder α] {x : α} :

x ≤ x

A version of le_refl where the argument is implicit

@[simp]

theorem lt_self_iff_false {α : Type u} [preorder α] (x : α) :

x < x ↔ false

theorem eq.ge {α : Type u} [preorder α] {x y : α} (h : x = y) :

y ≤ x

If x = y then y ≤ x. Note: this lemma uses y ≤ x instead of x ≥ y, because le is used almost exclusively in mathlib.

theorem eq.trans_le {α : Type u} [preorder α] {x y z : α} (h1 : x = y) (h2 : y ≤ z) :

x ≤ z

theorem eq.not_lt {α : Type u} [partial_order α] {x y : α} (h : x = y) :

¬x < y

theorem eq.not_gt {α : Type u} [partial_order α] {x y : α} (h : x = y) :

¬y < x

@[nolint]

theorem has_le.le.ge {α : Type u} [has_le α] {x y : α} (h : x ≤ y) :

y ≥ x

theorem has_le.le.trans_eq {α : Type u} [preorder α] {x y z : α} (h1 : x ≤ y) (h2 : y = z) :

x ≤ z

theorem has_le.le.lt_iff_ne {α : Type u} [partial_order α] {x y : α} (h : x ≤ y) :

x < y ↔ x ≠ y

theorem has_le.le.le_iff_eq {α : Type u} [partial_order α] {x y : α} (h : x ≤ y) :

y ≤ x ↔ y = x

theorem has_le.le.lt_or_le {α : Type u} [linear_order α] {a b : α} (h : a ≤ b) (c : α) :

a < c ∨ c ≤ b

theorem has_le.le.le_or_lt {α : Type u} [linear_order α] {a b : α} (h : a ≤ b) (c : α) :

a ≤ c ∨ c < b

theorem has_le.le.le_or_le {α : Type u} [linear_order α] {a b : α} (h : a ≤ b) (c : α) :

a ≤ c ∨ c ≤ b

@[nolint]

theorem has_lt.lt.gt {α : Type u} [has_lt α] {x y : α} (h : x < y) :

y > x

theorem has_lt.lt.false {α : Type u} [preorder α] {x : α} :

x < x → false

theorem has_lt.lt.ne' {α : Type u} [preorder α] {x y : α} (h : x < y) :

y ≠ x

theorem has_lt.lt.lt_or_lt {α : Type u} [linear_order α] {x y : α} (h : x < y) (z : α) :

x < z ∨ z < y

@[nolint]

theorem ge.le {α : Type u} [has_le α] {x y : α} (h : x ≥ y) :

y ≤ x

@[nolint]

theorem gt.lt {α : Type u} [has_lt α] {x y : α} (h : x > y) :

y < x

@[nolint]

theorem ge_of_eq {α : Type u} [preorder α] {a b : α} (h : a = b) :

a ≥ b

@[simp, nolint]

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

a ≥ b ↔ b ≤ a

@[simp, nolint]

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

a > b ↔ b < a

theorem not_le_of_lt {α : Type u} [preorder α] {a b : α} (h : a < b) :

theorem has_lt.lt.not_le {α : Type u} [preorder α] {a b : α} (h : a < b) :

Alias of not_le_of_lt.

theorem not_lt_of_le {α : Type u} [preorder α] {a b : α} (h : a ≤ b) :

¬b < a

theorem has_le.le.not_lt {α : Type u} [preorder α] {a b : α} (h : a ≤ b) :

¬b < a

Alias of not_lt_of_le.

theorem decidable.le_iff_eq_or_lt {α : Type u} [partial_order α] [decidable_rel has_le.le] {a b : α} :

a ≤ b ↔ a = b ∨ a < b

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

a ≤ b ↔ a = b ∨ a < b

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

a < b ↔ a ≤ b ∧ a ≠ b

theorem decidable.eq_iff_le_not_lt {α : Type u} [partial_order α] [decidable_rel has_le.le] {a b : α} :

a = b ↔ a ≤ b ∧ ¬a < b

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

a = b ↔ a ≤ b ∧ ¬a < b

theorem eq_or_lt_of_le {α : Type u} [partial_order α] {a b : α} (h : a ≤ b) :

a = b ∨ a < b

@[nolint]

theorem has_le.le.eq_or_lt_dec {α : Type u} [partial_order α] [decidable_rel has_le.le] {a b : α} (hab : a ≤ b) :

a = b ∨ a < b

Alias of decidable.eq_or_lt_of_le.

theorem has_le.le.eq_or_lt {α : Type u} [partial_order α] {a b : α} (h : a ≤ b) :

a = b ∨ a < b

Alias of eq_or_lt_of_le.

theorem ne.le_iff_lt {α : Type u} [partial_order α] {a b : α} (h : a ≠ b) :

a ≤ b ↔ a < b

theorem decidable.ne_iff_lt_iff_le {α : Type u} [partial_order α] [decidable_rel has_le.le] {a b : α} :

a ≠ b ↔ a < b ↔ a ≤ b

@[simp]

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

a ≠ b ↔ a < b ↔ a ≤ b

theorem lt_of_not_ge' {α : Type u} [linear_order α] {a b : α} (h : ¬b ≤ a) :

a < b

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

x < y ↔ ¬y ≤ x

theorem ne.lt_or_lt {α : Type u} [linear_order α] {x y : α} (h : x ≠ y) :

x < y ∨ y < x

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

¬a < b ↔ a = b ∨ b < a

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

∃ (c : α), a ≤ c ∧ b ≤ c

theorem lt_imp_lt_of_le_imp_le {α : Type u} {β : Type u_1} [linear_order α] [preorder β] {a b : α} {c d : β} (H : a ≤ b → c ≤ d) (h : d < c) :

b < a

theorem le_imp_le_iff_lt_imp_lt {α : Type u} {β : Type u_1} [linear_order α] [linear_order β] {a b : α} {c d : β} :

a ≤ b → c ≤ d ↔ d < c → b < a

theorem lt_iff_lt_of_le_iff_le' {α : Type u} {β : Type u_1} [preorder α] [preorder β] {a b : α} {c d : β} (H : a ≤ b ↔ c ≤ d) (H' : b ≤ a ↔ d ≤ c) :

b < a ↔ d < c

theorem lt_iff_lt_of_le_iff_le {α : Type u} {β : Type u_1} [linear_order α] [linear_order β] {a b : α} {c d : β} (H : a ≤ b ↔ c ≤ d) :

b < a ↔ d < c

theorem le_iff_le_iff_lt_iff_lt {α : Type u} {β : Type u_1} [linear_order α] [linear_order β] {a b : α} {c d : β} :

a ≤ b ↔ c ≤ d ↔ (b < a ↔ d < c)

theorem eq_of_forall_le_iff {α : Type u} [partial_order α] {a b : α} (H : ∀ (c : α), c ≤ a ↔ c ≤ b) :

a = b

theorem le_of_forall_le {α : Type u} [preorder α] {a b : α} (H : ∀ (c : α), c ≤ a → c ≤ b) :

a ≤ b

theorem le_of_forall_le' {α : Type u} [preorder α] {a b : α} (H : ∀ (c : α), a ≤ c → b ≤ c) :

b ≤ a

theorem le_of_forall_lt {α : Type u} [linear_order α] {a b : α} (H : ∀ (c : α), c < a → c < b) :

a ≤ b

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

(∀ ⦃c : α⦄, c < a → c < b) ↔ a ≤ b

theorem le_of_forall_lt' {α : Type u} [linear_order α] {a b : α} (H : ∀ (c : α), a < c → b < c) :

b ≤ a

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

(∀ ⦃c : α⦄, a < c → b < c) ↔ b ≤ a

theorem eq_of_forall_ge_iff {α : Type u} [partial_order α] {a b : α} (H : ∀ (c : α), a ≤ c ↔ b ≤ c) :

a = b

theorem le_implies_le_of_le_of_le {α : Type u} {a b c d : α} [preorder α] (hca : c ≤ a) (hbd : b ≤ d) :

a ≤ b → c ≤ d

monotonicity of ≤ with respect to →

@[ext]

theorem preorder.to_has_le_injective {α : Type u_1} :

function.injective (preorder.to_has_le α)

@[ext]

theorem partial_order.to_preorder_injective {α : Type u_1} :

function.injective (partial_order.to_preorder α)

@[ext]

theorem linear_order.to_partial_order_injective {α : Type u_1} :

function.injective (linear_order.to_partial_order α)

theorem preorder.ext {α : Type u_1} {A B : preorder α} (H : ∀ (x y : α), x ≤ y ↔ x ≤ y) :

A = B

theorem partial_order.ext {α : Type u_1} {A B : partial_order α} (H : ∀ (x y : α), x ≤ y ↔ x ≤ y) :

A = B

theorem linear_order.ext {α : Type u_1} {A B : linear_order α} (H : ∀ (x y : α), x ≤ y ↔ x ≤ y) :

A = B

@[simp]

def order.preimage {α : Sort u_1} {β : Sort u_2} (f : α → β) (s : β → β → Prop) (x y : α) :

Prop

Given a relation R on β and a function f : α → β, the preimage relation on α is defined by x ≤ y ↔ f x ≤ f y. It is the unique relation on α making f a rel_embedding (assuming f is injective).

Equations

(f ⁻¹'o s) x y = s (f x) (f y)

@[instance]

def order.preimage.decidable {α : Sort u_1} {β : Sort u_2} (f : α → β) (s : β → β → Prop) [H : decidable_rel s] :

decidable_rel (f ⁻¹'o s)

The preimage of a decidable order is decidable.

Equations

order.preimage.decidable f s = λ (x y : α), H (f x) (f y)

Order dual #

def order_dual (α : Type u_1) :

Type u_1

Type synonym to equip a type with the dual order: ≤ means ≥ and < means >.

Equations

order_dual α = α

@[instance]

def order_dual.nonempty (α : Type u_1) [h : nonempty α] :

nonempty (order_dual α)

@[instance]

def order_dual.subsingleton (α : Type u_1) [h : subsingleton α] :

subsingleton (order_dual α)

@[instance]

def order_dual.has_le (α : Type u_1) [has_le α] :

has_le (order_dual α)

Equations

order_dual.has_le α = {le := λ (x y : α), y ≤ x}

@[instance]

def order_dual.has_lt (α : Type u_1) [has_lt α] :

has_lt (order_dual α)

Equations

order_dual.has_lt α = {lt := λ (x y : α), y < x}

@[instance]

def order_dual.has_zero (α : Type u_1) [has_zero α] :

has_zero (order_dual α)

Equations

order_dual.has_zero α = {zero := 0}

theorem order_dual.dual_le {α : Type u} [has_le α] {a b : α} :

a ≤ b ↔ b ≤ a

theorem order_dual.dual_lt {α : Type u} [has_lt α] {a b : α} :

a < b ↔ b < a

@[instance]

def order_dual.preorder (α : Type u_1) [preorder α] :

preorder (order_dual α)

Equations

order_dual.preorder α = {le := has_le.le (order_dual.has_le α), lt := has_lt.lt (order_dual.has_lt α), le_refl := _, le_trans := _, lt_iff_le_not_le := _}

@[instance]

def order_dual.partial_order (α : Type u_1) [partial_order α] :

partial_order (order_dual α)

Equations

order_dual.partial_order α = {le := preorder.le (order_dual.preorder α), lt := preorder.lt (order_dual.preorder α), le_refl := _, le_trans := _, lt_iff_le_not_le := _, le_antisymm := _}

@[instance]

def order_dual.linear_order (α : Type u_1) [linear_order α] :

linear_order (order_dual α)

Equations

order_dual.linear_order α = {le := partial_order.le (order_dual.partial_order α), lt := partial_order.lt (order_dual.partial_order α), le_refl := _, le_trans := _, lt_iff_le_not_le := _, le_antisymm := _, le_total := _, decidable_le := infer_instance (λ (a b : α), has_le.le.decidable b a), decidable_eq := decidable_eq_of_decidable_le infer_instance, decidable_lt := infer_instance (λ (a b : α), has_lt.lt.decidable b a), max := min _inst_1, max_def := _, min := max _inst_1, min_def := _}

@[instance]

def order_dual.inhabited {α : Type u} [inhabited α] :

inhabited (order_dual α)

Equations

order_dual.inhabited = id

theorem order_dual.preorder.dual_dual (α : Type u_1) [H : preorder α] :

order_dual.preorder (order_dual α) = H

theorem order_dual.partial_order.dual_dual (α : Type u_1) [H : partial_order α] :

order_dual.partial_order (order_dual α) = H

theorem order_dual.linear_order.dual_dual (α : Type u_1) [H : linear_order α] :

order_dual.linear_order (order_dual α) = H

Order instances on the function space #

@[instance]

def pi.preorder {ι : Type u} {α : ι → Type v} [Π (i : ι), preorder (α i)] :

preorder (Π (i : ι), α i)

Equations

pi.preorder = {le := λ (x y : Π (i : ι), α i), ∀ (i : ι), x i ≤ y i, lt := λ (a b : Π (i : ι), α i), (∀ (i : ι), a i ≤ b i) ∧ ¬∀ (i : ι), b i ≤ a i, le_refl := _, le_trans := _, lt_iff_le_not_le := _}

theorem pi.le_def {ι : Type u} {α : ι → Type v} [Π (i : ι), preorder (α i)] {x y : Π (i : ι), α i} :

x ≤ y ↔ ∀ (i : ι), x i ≤ y i

theorem pi.lt_def {ι : Type u} {α : ι → Type v} [Π (i : ι), preorder (α i)] {x y : Π (i : ι), α i} :

x < y ↔ x ≤ y ∧ ∃ (i : ι), x i < y i

theorem le_update_iff {ι : Type u} {α : ι → Type v} [Π (i : ι), preorder (α i)] [decidable_eq ι] {x y : Π (i : ι), α i} {i : ι} {a : α i} :

x ≤ function.update y i a ↔ x i ≤ a ∧ ∀ (j : ι), j ≠ i → x j ≤ y j

theorem update_le_iff {ι : Type u} {α : ι → Type v} [Π (i : ι), preorder (α i)] [decidable_eq ι] {x y : Π (i : ι), α i} {i : ι} {a : α i} :

function.update x i a ≤ y ↔ a ≤ y i ∧ ∀ (j : ι), j ≠ i → x j ≤ y j

theorem update_le_update_iff {ι : Type u} {α : ι → Type v} [Π (i : ι), preorder (α i)] [decidable_eq ι] {x y : Π (i : ι), α i} {i : ι} {a b : α i} :

function.update x i a ≤ function.update y i b ↔ a ≤ b ∧ ∀ (j : ι), j ≠ i → x j ≤ y j

@[instance]

def pi.partial_order {ι : Type u} {α : ι → Type v} [Π (i : ι), partial_order (α i)] :

partial_order (Π (i : ι), α i)

Equations

pi.partial_order = {le := preorder.le pi.preorder, lt := preorder.lt pi.preorder, le_refl := _, le_trans := _, lt_iff_le_not_le := _, le_antisymm := _}

Lifts of order instances #

def preorder.lift {α : Type u_1} {β : Type u_2} [preorder β] (f : α → β) :

Transfer a preorder on β to a preorder on α using a function f : α → β. See note [reducible non-instances].

Equations

preorder.lift f = {le := λ (x y : α), f x ≤ f y, lt := λ (x y : α), f x < f y, le_refl := _, le_trans := _, lt_iff_le_not_le := _}

def partial_order.lift {α : Type u_1} {β : Type u_2} [partial_order β] (f : α → β) (inj : function.injective f) :

partial_order α

Transfer a partial_order on β to a partial_order on α using an injective function f : α → β. See note [reducible non-instances].

Equations

partial_order.lift f inj = {le := preorder.le (preorder.lift f), lt := preorder.lt (preorder.lift f), le_refl := _, le_trans := _, lt_iff_le_not_le := _, le_antisymm := _}

def linear_order.lift {α : Type u_1} {β : Type u_2} [linear_order β] (f : α → β) (inj : function.injective f) :

linear_order α

Transfer a linear_order on β to a linear_order on α using an injective function f : α → β. See note [reducible non-instances].

Equations

linear_order.lift f inj = {le := partial_order.le (partial_order.lift f inj), lt := partial_order.lt (partial_order.lift f inj), le_refl := _, le_trans := _, lt_iff_le_not_le := _, le_antisymm := _, le_total := _, decidable_le := λ (x y : α), infer_instance, decidable_eq := λ (x y : α), decidable_of_iff (f x = f y) _, decidable_lt := λ (x y : α), infer_instance, max := max_default (λ (a b : α), infer_instance), max_def := _, min := min_default (λ (a b : α), infer_instance), min_def := _}

@[instance]

def subtype.preorder {α : Type u_1} [preorder α] (p : α → Prop) :

preorder (subtype p)

Equations

subtype.preorder p = preorder.lift subtype.val

@[simp]

theorem subtype.mk_le_mk {α : Type u_1} [preorder α] {p : α → Prop} {x y : α} {hx : p x} {hy : p y} :

⟨x, hx⟩ ≤ ⟨y, hy⟩ ↔ x ≤ y

@[simp]

theorem subtype.mk_lt_mk {α : Type u_1} [preorder α] {p : α → Prop} {x y : α} {hx : p x} {hy : p y} :

⟨x, hx⟩ < ⟨y, hy⟩ ↔ x < y

@[simp]

theorem subtype.coe_le_coe {α : Type u_1} [preorder α] {p : α → Prop} {x y : subtype p} :

↑x ≤ ↑y ↔ x ≤ y

@[simp]

theorem subtype.coe_lt_coe {α : Type u_1} [preorder α] {p : α → Prop} {x y : subtype p} :

↑x < ↑y ↔ x < y

@[instance]

def subtype.partial_order {α : Type u_1} [partial_order α] (p : α → Prop) :

partial_order (subtype p)

Equations

subtype.partial_order p = partial_order.lift subtype.val subtype.val_injective

@[instance]

def subtype.linear_order {α : Type u_1} [linear_order α] (p : α → Prop) :

linear_order (subtype p)

Equations

subtype.linear_order p = linear_order.lift subtype.val subtype.val_injective

@[instance]

def prod.has_le (α : Type u) (β : Type v) [has_le α] [has_le β] :

has_le (α × β)

Equations

prod.has_le α β = {le := λ (p q : α × β), p.fst ≤ q.fst ∧ p.snd ≤ q.snd}

theorem prod.le_def {α : Type u_1} {β : Type u_2} [has_le α] [has_le β] {x y : α × β} :

x ≤ y ↔ x.fst ≤ y.fst ∧ x.snd ≤ y.snd

@[simp]

theorem prod.mk_le_mk {α : Type u_1} {β : Type u_2} [has_le α] [has_le β] {x₁ x₂ : α} {y₁ y₂ : β} :

(x₁, y₁) ≤ (x₂, y₂) ↔ x₁ ≤ x₂ ∧ y₁ ≤ y₂

@[instance]

def prod.preorder (α : Type u) (β : Type v) [preorder α] [preorder β] :

preorder (α × β)

Equations

prod.preorder α β = {le := has_le.le (prod.has_le α β), lt := λ (a b : α × β), a ≤ b ∧ ¬b ≤ a, le_refl := _, le_trans := _, lt_iff_le_not_le := _}

@[instance]

def prod.partial_order (α : Type u) (β : Type v) [partial_order α] [partial_order β] :

partial_order (α × β)

The pointwise partial order on a product. (The lexicographic ordering is defined in order/lexicographic.lean, and the instances are available via the type synonym lex α β = α × β.)

Equations

prod.partial_order α β = {le := preorder.le (prod.preorder α β), lt := preorder.lt (prod.preorder α β), le_refl := _, le_trans := _, lt_iff_le_not_le := _, le_antisymm := _}

Additional order classes #

@[class]

structure no_top_order (α : Type u) [preorder α] :

Prop

no_top : ∀ (a : α), ∃ (a' : α), a < a'

Order without a maximal element. Sometimes called cofinal.

Instances

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

∃ (a' : α), a < a'

@[instance]

def nonempty_gt {α : Type u} [preorder α] [no_top_order α] (a : α) :

nonempty {x // a < x}

def is_top {α : Type u} [has_le α] (a : α) :

Prop

a : α is a top element of α if it is greater than or equal to any other element of α. This predicate is useful, e.g., to make some statements and proofs work in both cases [order_top α] and [no_top_order α].

Equations

is_top a = ∀ (b : α), b ≤ a

@[simp]

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

theorem is_top.unique {α : Type u} [partial_order α] {a b : α} (ha : is_top a) (hb : a ≤ b) :

a = b

@[class]

structure no_bot_order (α : Type u) [preorder α] :

Prop

no_bot : ∀ (a : α), ∃ (a' : α), a' < a

Order without a minimal element. Sometimes called coinitial or dense.

Instances

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

∃ (a' : α), a' < a

def is_bot {α : Type u} [has_le α] (a : α) :

Prop

a : α is a bottom element of α if it is less than or equal to any other element of α. This predicate is useful, e.g., to make some statements and proofs work in both cases [order_bot α] and [no_bot_order α].

Equations

is_bot a = ∀ (b : α), a ≤ b

@[simp]

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

theorem is_bot.unique {α : Type u} [partial_order α] {a b : α} (ha : is_bot a) (hb : b ≤ a) :

a = b

@[instance]

def order_dual.no_top_order (α : Type u) [preorder α] [no_bot_order α] :

no_top_order (order_dual α)

@[instance]

def order_dual.no_bot_order (α : Type u) [preorder α] [no_top_order α] :

no_bot_order (order_dual α)

@[instance]

def nonempty_lt {α : Type u} [preorder α] [no_bot_order α] (a : α) :

nonempty {x // x < a}

@[class]

structure densely_ordered (α : Type u) [preorder α] :

Prop

dense : ∀ (a₁ a₂ : α), a₁ < a₂ → (∃ (a : α), a₁ < a ∧ a < a₂)

An order is dense if there is an element between any pair of distinct elements.

Instances

theorem exists_between {α : Type u} [preorder α] [densely_ordered α] {a₁ a₂ : α} :

a₁ < a₂ → (∃ (a : α), a₁ < a ∧ a < a₂)

@[instance]

def order_dual.densely_ordered (α : Type u) [preorder α] [densely_ordered α] :

densely_ordered (order_dual α)

theorem le_of_forall_le_of_dense {α : Type u} [linear_order α] [densely_ordered α] {a₁ a₂ : α} (h : ∀ (a : α), a₂ < a → a₁ ≤ a) :

a₁ ≤ a₂

theorem eq_of_le_of_forall_le_of_dense {α : Type u} [linear_order α] [densely_ordered α] {a₁ a₂ : α} (h₁ : a₂ ≤ a₁) (h₂ : ∀ (a : α), a₂ < a → a₁ ≤ a) :

a₁ = a₂

theorem le_of_forall_ge_of_dense {α : Type u} [linear_order α] [densely_ordered α] {a₁ a₂ : α} (h : ∀ (a₃ : α), a₃ < a₁ → a₃ ≤ a₂) :

a₁ ≤ a₂

theorem eq_of_le_of_forall_ge_of_dense {α : Type u} [linear_order α] [densely_ordered α] {a₁ a₂ : α} (h₁ : a₂ ≤ a₁) (h₂ : ∀ (a₃ : α), a₃ < a₁ → a₃ ≤ a₂) :

a₁ = a₂

theorem dense_or_discrete {α : Type u} [linear_order α] (a₁ a₂ : α) :

(∃ (a : α), a₁ < a ∧ a < a₂) ∨ (∀ (a : α), a₁ < a → a₂ ≤ a) ∧ ∀ (a : α), a < a₂ → a ≤ a₁

Linear order from a total partial order #

def as_linear_order (α : Type u) :

Type u

Type synonym to create an instance of linear_order from a partial_order and is_total α (≤)

Equations

as_linear_order α = α

@[instance]

def as_linear_order.inhabited {α : Type u_1} [inhabited α] :

inhabited (as_linear_order α)

Equations

as_linear_order.inhabited = {default := default α _inst_1}

@[instance]

def as_linear_order.linear_order {α : Type u_1} [partial_order α] [is_total α has_le.le] :

linear_order (as_linear_order α)

Equations

as_linear_order.linear_order = {le := partial_order.le _inst_1, lt := partial_order.lt _inst_1, le_refl := _, le_trans := _, lt_iff_le_not_le := _, le_antisymm := _, le_total := _, decidable_le := classical.dec_rel has_le.le, decidable_eq := decidable_eq_of_decidable_le (classical.dec_rel has_le.le), decidable_lt := decidable_lt_of_decidable_le (classical.dec_rel has_le.le), max := max_default (λ (a b : as_linear_order α), classical.dec_rel has_le.le a b), max_def := _, min := min_default (λ (a b : as_linear_order α), classical.dec_rel has_le.le a b), min_def := _}