Martin Escardo, 29 June 2018

The type Ordinals of ordinals in a universe 𝓤, and the subtype Ordinalsᵀ of
ordinals with a top element.

\begin{code}

{-# OPTIONS --safe --without-K #-}

open import MLTT.Spartan
open import Ordinals.Notions
open import Ordinals.Underlying
open import UF.FunExt
open import UF.Sets
open import UF.Subsingletons
open import UF.Subsingletons-FunExt

module Ordinals.Type where

\end{code}

An ordinal is a type equipped with ordinal structure. Such a type is
automatically a set.

\begin{code}

OrdinalStructure : 𝓤 ̇ → 𝓤 ⁺ ̇
OrdinalStructure {𝓤} X = Σ _<_ ꞉ (X → X → 𝓤 ̇ ) , (is-well-order _<_)

Ordinal : ∀ 𝓤 → 𝓤 ⁺ ̇
Ordinal 𝓤 = Σ X ꞉ 𝓤 ̇ , OrdinalStructure X

Ord = Ordinal 𝓤₀

\end{code}

NB. Perhaps we will eventually need to have two parameters 𝓤 (the
universe where the underlying type X lives) and 𝓥 (the universe where
_<_ takes values in) for Ordinal.

Ordinals are ranged over by α,β,γ.

The underlying type of an ordinal (which happens to be necessarily a
set):

\begin{code}

instance
underlying-type-of-ordinal : Underlying (Ordinal 𝓤)
⟨_⟩ {{underlying-type-of-ordinal}} (X , _) = X
underlying-order {{underlying-type-of-ordinal}} (X , _<_ , o) = _<_

structure : (α : Ordinal 𝓤) → OrdinalStructure ⟨ α ⟩
structure (X , s) = s

is-trichotomous : Ordinal 𝓤 → 𝓤 ̇
is-trichotomous α = is-trichotomous-order (underlying-order α)

is-well-ordered : (α : Ordinal 𝓤) → is-well-order (underlying-order α)
is-well-ordered (X , _<_ , o) = o

Prop-valuedness : (α : Ordinal 𝓤) → is-prop-valued (underlying-order α)
Prop-valuedness α = prop-valuedness (underlying-order α) (is-well-ordered α)

Reflexivity : (α : Ordinal 𝓤) {x : ⟨ α ⟩} → x ≼⟨ α ⟩ x
Reflexivity α = ≼-refl (underlying-order α)

Transitivity : (α : Ordinal 𝓤) → is-transitive (underlying-order α)
Transitivity α = transitivity (underlying-order α) (is-well-ordered α)

Well-foundedness : (α : Ordinal 𝓤) (x : ⟨ α ⟩) → is-accessible (underlying-order α) x
Well-foundedness α = well-foundedness (underlying-order α) (is-well-ordered α)

Transfinite-induction : (α : Ordinal 𝓤)
→ (P : ⟨ α ⟩ → 𝓦 ̇ )
→ ((x : ⟨ α ⟩)
→ ((y : ⟨ α ⟩) → y ≺⟨ α ⟩ x → P y) → P x)
→ (x : ⟨ α ⟩) → P x
Transfinite-induction α = transfinite-induction
(underlying-order α)
(Well-foundedness α)

Transfinite-recursion : (α : Ordinal 𝓤) {Y : 𝓥 ̇ }
→ ((x : ⟨ α ⟩)
→ ((y : ⟨ α ⟩) → y ≺⟨ α ⟩ x → Y) → Y)
→ ⟨ α ⟩ → Y
Transfinite-recursion α = transfinite-recursion
(underlying-order α)
(Well-foundedness α)
\end{code}

Added 31 October 2022 by Tom de Jong.
We record the (computational) behaviour of transfinite induction for use in
other constructions.

\begin{code}

Transfinite-induction-behaviour : FunExt
→ (α : Ordinal 𝓤)
→ (P : ⟨ α ⟩ → 𝓦 ̇ )
→ (f : (x : ⟨ α ⟩) → ((y : ⟨ α ⟩) → y ≺⟨ α ⟩ x → P y) → P x)
→ (x : ⟨ α ⟩)
→ Transfinite-induction α P f x
＝ f x (λ y l → Transfinite-induction α P f y)
Transfinite-induction-behaviour fe α = transfinite-induction-behaviour
(underlying-order α) fe
(Well-foundedness α)
\end{code}

\begin{code}

Transfinite-recursion-behaviour : FunExt
→ (α : Ordinal 𝓤)
→ {Y : 𝓥 ̇ }
→ (f : (x : ⟨ α ⟩) → ((y : ⟨ α ⟩) → y ≺⟨ α ⟩ x → Y) → Y)
→ (x : ⟨ α ⟩)
→ Transfinite-recursion α f x
＝ f x (λ y l → Transfinite-recursion α f y)
Transfinite-recursion-behaviour fe α =
transfinite-recursion-behaviour (underlying-order α) fe (Well-foundedness α)

Extensionality : (α : Ordinal 𝓤) → is-extensional (underlying-order α)
Extensionality α = extensionality (underlying-order α) (is-well-ordered α)

Antisymmetry : (α : Ordinal 𝓤) (x y : ⟨ α ⟩) → x ≼⟨ α ⟩ y → y ≼⟨ α ⟩ x → x ＝ y
Antisymmetry = Extensionality

underlying-type-is-set : FunExt
→ (α : Ordinal 𝓤)
→ is-set ⟨ α ⟩
underlying-type-is-set fe α =
extensionally-ordered-types-are-sets
(underlying-order α)
fe
(Prop-valuedness α)
(Extensionality α)

is-least : (α : Ordinal 𝓤) → ⟨ α ⟩ → 𝓤 ̇
is-least α x = (y : ⟨ α ⟩) → x ≼⟨ α ⟩ y

being-least-is-prop : Fun-Ext → (α : Ordinal 𝓤) (x : ⟨ α ⟩) → is-prop (is-least α x)
being-least-is-prop fe α x = Π₃-is-prop fe (λ y u _ → Prop-valuedness α u y)

at-most-one-least : (α : Ordinal 𝓤) (x y : ⟨ α ⟩)
→ is-least α x
→ is-least α y
→ x ＝ y
at-most-one-least α x y l l' = Antisymmetry α x y (l y) (l' x)

has-least : Ordinal 𝓤 → 𝓤 ̇
has-least α = Σ ⊥ ꞉ ⟨ α ⟩ , is-least α ⊥

having-least-is-prop : Fun-Ext → (α : Ordinal 𝓤) → is-prop (has-least α)
having-least-is-prop fe α (⊥ , l) (⊥' , l') =
to-subtype-＝
(being-least-is-prop fe α)
(at-most-one-least α ⊥ ⊥' l l')

is-largest : (α : Ordinal 𝓤) → ⟨ α ⟩ → 𝓤 ̇
is-largest α x = (y : ⟨ α ⟩) → y ≼⟨ α ⟩ x

being-largest-is-prop : Fun-Ext → (α : Ordinal 𝓤) (x : ⟨ α ⟩) → is-prop (is-largest α x)
being-largest-is-prop fe α x = Π₃-is-prop fe (λ y u _ → Prop-valuedness α u x)

at-most-one-largest : (α : Ordinal 𝓤) (x y : ⟨ α ⟩)
→ is-largest α x
→ is-largest α y
→ x ＝ y
at-most-one-largest α x y l l' = Antisymmetry α x y (l' x) (l y)

has-largest : Ordinal 𝓤 → 𝓤 ̇
has-largest α = Σ ⊤ ꞉ ⟨ α ⟩ , is-largest α ⊤

having-largest-is-prop : Fun-Ext → (α : Ordinal 𝓤) → is-prop (has-largest α)
having-largest-is-prop fe α (⊥ , l) (⊥' , l') =
to-subtype-＝
(being-largest-is-prop fe α)
(at-most-one-largest α ⊥ ⊥' l l')

\end{code}

TODO. We should add further properties of the order from the module
Notions. For the moment, we add this:

\begin{code}

irrefl : (α : Ordinal 𝓤) (x : ⟨ α ⟩) → ¬(x ≺⟨ α ⟩ x)
irrefl α x = irreflexive (underlying-order α) x (Well-foundedness α x)

≼-gives-≾ : (α : Ordinal 𝓤) (x y : ⟨ α ⟩) → x ≼⟨ α ⟩ y → x ≾⟨ α ⟩ y
≼-gives-≾ {𝓤} α x y = ≼-coarser-than-≾ (underlying-order α)
y (Well-foundedness α y) x
\end{code}