Martin Escardo, 2018, April 2022.

The type of ordinals is (algebraically) injective.

\begin{code}

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

open import UF.FunExt

module Ordinals.Injectivity where

open import MLTT.Spartan
open import Notation.CanonicalMap
open import Ordinals.Equivalence
open import Ordinals.Maps
open import Ordinals.Type
open import Ordinals.Underlying
open import Ordinals.WellOrderArithmetic
open import UF.Base
open import UF.Embeddings
open import UF.Equiv
open import UF.Subsingletons-Properties
open import UF.Univalence

module ordinals-injectivity (fe : FunExt) where

 open import InjectiveTypes.Blackboard fe

 _↗_ : {I : 𝓤  ̇ } {J : 𝓥 ̇ }
     → (I → Ordinal 𝓦)
     → (I ↪ J)
     → (J → Ordinal (𝓤 ⊔ 𝓥 ⊔ 𝓦))
 α ↗ (e , e-is-embedding) =
  λ j → ((a / e) j  ,
         Extension.order j ,
         Extension.well-order j (λ i → is-well-ordered (α i)))
  where
   a = λ i → ⟨ α i ⟩

   module Extension = extension fe a e
                       e-is-embedding
                       (λ {i} → underlying-order (α i))

 ↗-propertyₒ : {I : 𝓤  ̇ } {J : 𝓥 ̇ }
              (α : I → Ordinal 𝓦)
              (𝓮@(e , e-is-embedding) : I ↪ J)
              (i : I)
            → (α ↗ 𝓮) (e i) ≃ₒ α i
 ↗-propertyₒ {𝓤} {𝓥} {I} {J} α 𝓮@(e , e-is-embedding) i = γ
  where
   ϕ : ⟨ (α ↗ 𝓮) (e i) ⟩ ≃ ⟨ α i ⟩
   ϕ = Π-extension-property (λ i → ⟨ α i ⟩) e e-is-embedding i

   g : ⟨ (α ↗ 𝓮) (e i) ⟩ → ⟨ α i ⟩
   g = ⌜ ϕ ⌝

   g-is-equiv : is-equiv g
   g-is-equiv = ⌜⌝-is-equiv ϕ

   g-is-order-preserving : is-order-preserving ((α ↗ 𝓮) (e i)) (α i) g
   g-is-order-preserving u v ((i' , p) , l) = m
    where
     q : (i' , p) ＝ (i , refl)
     q = e-is-embedding (e i) (i' , p) (i , refl)

     m : u (i , refl) ≺⟨ α i ⟩ v (i , refl)
     m = transport (λ (i' , p) → u (i' , p) ≺⟨ α i' ⟩ v (i' , p)) q l

   g⁻¹ : ⟨ α i ⟩ → ⟨ (α ↗ 𝓮) (e i) ⟩
   g⁻¹ = ⌜ ϕ ⌝⁻¹

   g⁻¹-is-order-preserving : is-order-preserving (α i) ((α ↗ 𝓮) (e i)) g⁻¹
   g⁻¹-is-order-preserving x y l = (i , refl) , r
     where
      p : g⁻¹ x (i , refl) ＝ x
      p = inverses-are-sections g g-is-equiv x

      q : g⁻¹ y (i , refl) ＝ y
      q = inverses-are-sections g g-is-equiv y

      r : g⁻¹ x (i , refl) ≺⟨ α i ⟩ g⁻¹ y (i , refl)
      r = transport₂ (λ x y → x ≺⟨ α  i ⟩ y) (p ⁻¹) (q ⁻¹) l

   γ : (α ↗ 𝓮) (e i) ≃ₒ α i
   γ = g , g-is-order-preserving , g-is-equiv , g⁻¹-is-order-preserving

 ↗-property : is-univalent (𝓤 ⊔ 𝓥)
            → {I : 𝓤  ̇ } {J : 𝓥 ̇ }
              (α : I → Ordinal (𝓤 ⊔ 𝓥))
              (𝓮@(e , e-is-embedding) : I ↪ J)
              (i : I)
            → (α ↗ 𝓮) (e i) ＝ α i
 ↗-property ua α 𝓮@(e , e-is-embedding) i =
  eqtoidₒ ua (fe _ _) ((α ↗ 𝓮) (e i)) (α i) (↗-propertyₒ α 𝓮 i)

 Ordinal-is-ainjective : is-univalent (𝓤 ⊔ 𝓥)
                       → ainjective-type (Ordinal (𝓤 ⊔ 𝓥)) 𝓤 𝓥
 Ordinal-is-ainjective ua e e-is-embedding α =
  (α ↗ (e , e-is-embedding)) ,
  ↗-property ua α (e , e-is-embedding)

module topped-ordinals-injectivity (fe : FunExt) where

 open import InjectiveTypes.Blackboard fe
 open import Ordinals.ToppedType fe

 _↗_ : {I : 𝓤  ̇ } {J : 𝓥 ̇ }
     → (I → Ordinalᵀ 𝓦)
     → (I ↪ J)
     → (J → Ordinalᵀ (𝓤 ⊔ 𝓥 ⊔ 𝓦))
 τ ↗ (e , e-is-embedding) =
  λ j → ((t / e) j ,
         Extension.order j ,
         Extension.well-order j (λ i → tis-well-ordered (τ i))) ,
         Extension.top-preservation j (λ i → topped (τ i))
  where
   t = λ x → ⟨ τ x ⟩
   module Extension = extension fe t e
                       e-is-embedding
                       (λ {i} → underlying-order (τ i))

 ↗-propertyₒ : {I : 𝓤  ̇ } {J : 𝓥 ̇ }
               (α : I → Ordinalᵀ 𝓦)
               (𝓮@(e , e-is-embedding) : I ↪ J)
               (i : I)
             → [ (α ↗ 𝓮) (e i) ] ≃ₒ [ α i ]
 ↗-propertyₒ α = ordinals-injectivity.↗-propertyₒ fe (λ i → [ α i ])

\end{code}

TODO. The type of compact∙ ordinals is injective. The type of ordinals
that have infs of complemented subsets is injective. These two results
are already proved in other modules, but these results are not
explicitly stated. We should refactor that code.

Added 11th May 2022.

\begin{code}

module ordinals-injectivity-order (ua : Univalence) where

 open import Ordinals.OrdinalOfOrdinals ua
 open import UF.UA-FunExt

 fe : FunExt
 fe = Univalence-gives-FunExt ua

 open ordinals-injectivity fe

 ↗-preserves-⊴ : {I J : 𝓤  ̇ } (𝓮 : I ↪ J)
                 (α β : I → Ordinal 𝓤)
               → ((i : I) → α i ⊴ β i)
               → (j : J) → (α ↗ 𝓮) j ⊴ (β ↗ 𝓮) j
 ↗-preserves-⊴ {𝓤} {I} {J} 𝓮@(e , e-is-embedding) α β ℓ j = f , fi , fop
  where
   h : (i : I) → ⟨ α i ⟩ → ⟨ β i ⟩
   h i = pr₁ (ℓ i)

   hi : (i : I) → is-initial-segment (α i) (β i) (h i)
   hi i = pr₁ (pr₂ (ℓ i))

   hop : (i : I) → is-order-preserving (α i) (β i) (h i)
   hop i = pr₂ (pr₂ (ℓ i))

   f : ⟨ (α ↗ 𝓮) j ⟩ → ⟨ (β ↗ 𝓮) j ⟩
   f ϕ (i , refl) = h i (ϕ (i , refl))

   fi : is-initial-segment ((α ↗ 𝓮) j) ((β ↗ 𝓮) j) f
   fi ϕ γ ((i , refl) , m) = ⦅b⦆ ⦅a⦆
    where
     g⁻¹ : ⟨ α i ⟩ → ⟨ (α ↗ 𝓮) (e i) ⟩
     g⁻¹ = case (↗-propertyₒ α 𝓮 i) of
            (λ (g , gop , geq , g⁻¹op) → inverse g geq)

     w : fiber e (e i)
     w = (i , refl)

     u : w ＝ w
     u = e-is-embedding (e i) w w

     v : u ＝ 𝓻𝓮𝒻𝓵 w
     v = props-are-sets (e-is-embedding (e i)) _ _

     ⦅a⦆ : Σ x ꞉ ⟨ α i ⟩ , (x ≺⟨ α i ⟩ ϕ (i , refl)) × (h i x ＝ γ (i , refl))
     ⦅a⦆ = hi i (ϕ (i , refl)) (γ (i , refl)) m

     ⦅b⦆ : type-of ⦅a⦆
         → Σ ϕ' ꞉ ⟨ (α ↗ 𝓮) (e i) ⟩ , (ϕ' ≺⟨ (α ↗ 𝓮) (e i) ⟩ ϕ) × (f ϕ' ＝ γ)
     ⦅b⦆ (x , n , t) = g⁻¹ x , (w , l) , dfunext (fe 𝓤 𝓤) H
      where
       p : g⁻¹ x w ＝ x
       p = g⁻¹ x w                                     ＝⟨ refl ⟩
           transport (λ - → ⟨ α (pr₁ -) ⟩) u x         ＝⟨ ⦅0⦆ ⟩
           transport (λ - → ⟨ α (pr₁ -) ⟩) (𝓻𝓮𝒻𝓵 w) x ＝⟨ refl ⟩
           x                                           ∎
        where
         ⦅0⦆ = ap (λ - → transport (λ - → ⟨ α (pr₁ -) ⟩) - x) v

       l : g⁻¹ x w ≺⟨ α i ⟩ ϕ w
       l = transport (λ - → - ≺⟨ α i ⟩ ϕ w) (p ⁻¹) n

       H : f (g⁻¹ x) ∼ γ
       H (i' , r) =
         f (g⁻¹ x) (i' , r)                              ＝⟨ ⦅1⦆ ⟩
         transport (λ - → ⟨ β (pr₁ -) ⟩) q (f (g⁻¹ x) w) ＝⟨ ⦅3⦆ ⟩
         transport (λ - → ⟨ β (pr₁ -) ⟩) q (γ w)         ＝⟨ ⦅4⦆ ⟩
         γ (i' , r)                                      ∎
         where
          q : w ＝ (i' , r)
          q = e-is-embedding (e i) w (i' , r)

          ⦅1⦆ = (apd ( f (g⁻¹ x)) q)⁻¹

          ⦅2⦆ = f (g⁻¹ x) w   ＝⟨ refl ⟩
                h i (g⁻¹ x w) ＝⟨ ap (h i) p ⟩
                h i x         ＝⟨ t ⟩
                γ w           ∎

          ⦅3⦆ = ap (transport (λ - → ⟨ β (pr₁ -) ⟩) q) ⦅2⦆

          ⦅4⦆ = apd γ q

   fop : is-order-preserving ((α ↗ 𝓮) j) ((β ↗ 𝓮) j) f
   fop ϕ γ ((i , refl) , m) = (i , refl) , hop i (ϕ (i , refl)) (γ (i , refl)) m

module topped-ordinals-injectivity-order (ua : Univalence) where

 open import UF.UA-FunExt

 fe : FunExt
 fe = Univalence-gives-FunExt ua

 open import Ordinals.ToppedType fe
 open import Ordinals.OrdinalOfOrdinals ua

 open topped-ordinals-injectivity fe

 ↗-preserves-⊴ : {I J : 𝓤  ̇ } (𝓮 : I ↪ J)
                 (α β : I → Ordinalᵀ 𝓤)
               → ((i : I) → [ α i ] ⊴ [ β i ])
               → (j : J) → [ (α ↗ 𝓮) j ] ⊴ [ (β ↗ 𝓮) j ]
 ↗-preserves-⊴ 𝓮 α β =
   ordinals-injectivity-order.↗-preserves-⊴ ua 𝓮 (λ i → [ α i ]) (λ i → [ β i ])

\end{code}