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} :