mathlib documentation

order.order_dual

Initial lemmas to work with the order_dual #

Definitions #

to_dual and of_dual the order reversing identity maps, bundled as equivalences.

Basic Lemmas to convert between an order and its dual #

This file is similar to algebra/group/type_tags.lean

@[instance]
def order_dual.nontrivial {α : Type u} [nontrivial α] :
def order_dual.to_dual {α : Type u} :

to_dual is the identity function to the order_dual of a linear order.

Equations
def order_dual.of_dual {α : Type u} :

of_dual is the identity function from the order_dual of a linear order.

Equations
@[simp]
@[simp]
theorem order_dual.to_dual_inj {α : Type u} {a b : α} :
@[simp]
@[simp]
theorem order_dual.to_dual_lt_to_dual {α : Type u} [has_lt α] {a b : α} :
@[simp]
@[simp]
theorem order_dual.le_to_dual {α : Type u} [has_le α] {a : order_dual α} {b : α} :
theorem order_dual.lt_to_dual {α : Type u} [has_lt α] {a : order_dual α} {b : α} :
theorem order_dual.to_dual_le {α : Type u} [has_le α] {a : α} {b : order_dual α} :
theorem order_dual.to_dual_lt {α : Type u} [has_lt α] {a : α} {b : order_dual α} :
def order_dual.rec {α : Type u} {C : order_dual αSort u_1} (h₂ : Π (a : α), C (order_dual.to_dual a)) (a : order_dual α) :
C a

Recursor for order_dual α.

Equations
@[simp]
theorem order_dual.forall {α : Type u} {p : order_dual α → Prop} :
(∀ (a : order_dual α), p a) ∀ (a : α), p (order_dual.to_dual a)
@[simp]
theorem order_dual.exists {α : Type u} {p : order_dual α → Prop} :
(∃ (a : order_dual α), p a) ∃ (a : α), p (order_dual.to_dual a)
theorem has_lt.lt.dual {α : Type u} [has_lt α] {a b : α} :

Alias of order_dual.to_dual_lt_to_dual.

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

Alias of order_dual.to_dual_le_to_dual.