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 x, r ≤ s

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 instLinearOrderLinearExtension {α : Type u} [PartialOrder α] : LinearOrder (LinearExtension α)

def toLinearExtension {α : Type u} [PartialOrder α] : (fun x x_1 => x ≤ x_1) →r fun x x_1 => x ≤ x_1

The embedding of α into LinearExtension α as a relation homomorphism.

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