Extend a partial order to a linear order

This file constructs a linear order which is an extension of the given partial order, using Zorn's lemma.

theorem extend_partialOrder {α : Type u} (r : α → α → Prop) [IsPartialOrder α r] : ∃ (s : α → α → Prop), IsLinearOrder α s ∧ r ≤ s

Szpilrajn extension theorem: any partial order can be extended to a linear order.

def LinearExtension (α : Type u) : Type u

A type alias for α, intended to extend a partial order on α to a linear order.

noncomputable instance instLinearOrderLinearExtensionOfPartialOrder {α : Type u} [PartialOrder α] : LinearOrder (LinearExtension α)

noncomputable def toLinearExtension {α : Type u} [PartialOrder α] : α →o LinearExtension α

The embedding of α into LinearExtension α as an order homomorphism.

instance instInhabitedLinearExtension {α : Type u} [Inhabited α] : Inhabited (LinearExtension α)