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use Quotient in Ordinals.v #1811

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Jan 13, 2024
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use Quotient in Ordinals.v #1811

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21 changes: 10 additions & 11 deletions theories/Sets/Ordinals.v
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Original file line number Diff line number Diff line change
@@ -1,6 +1,6 @@
From HoTT Require Import TruncType ExcludedMiddle Modalities.ReflectiveSubuniverse abstract_algebra.
From HoTT Require Import PropResizing.PropResizing.
From HoTT Require Import HIT.quotient.
From HoTT Require Import Colimits.Quotient.

(** This file contains a definition of ordinals and some fundamental results,
roughly following the presentation in the HoTT book. *)
Expand Down Expand Up @@ -806,7 +806,7 @@ Qed.
(** * Ordinal limit *)

Definition image@{i j} {A : Type@{i}} {B : HSet@{j}} (f : A -> B) : Type@{i}
:= quotient (fun a a' : A => f a = f a').
:= Quotient (fun a a' : A => f a = f a').

Definition factor1 {A} {B : HSet} (f : A -> B)
: A -> image f
Expand All @@ -818,29 +818,28 @@ Lemma image_ind_prop {A} {B : HSet} (f : A -> B)
-> forall x : image f, P x.
Proof.
intros step.
srefine (quotient_ind_prop _ _ _); intros a; cbn.
srefine (Quotient_ind_hprop _ _ _); intros a; cbn.
apply step.
Qed.

Definition image_rec {A} {B : HSet} (f : A -> B)
{C : HSet} (step : A -> C)
: (forall a a', f a = f a' -> step a = step a')
-> image f -> C
:= quotient_rec _ step.

:= Quotient_rec _ _ step.


Definition factor2 {A} {B : HSet} (f : A -> B)
: image f -> B
:= quotient_rec _ f (fun a a' fa_fa' => fa_fa').
:= Quotient_rec _ _ f (fun a a' fa_fa' => fa_fa').

Global Instance isinjective_factor2 `{Funext} {A} {B : HSet} (f : A -> B)
: IsInjective (factor2 f).
Proof.
unfold IsInjective, image.
refine (quotient_ind_prop _ _ _); intros x; cbn.
refine (quotient_ind_prop _ _ _); intros y; cbn.
rapply related_classes_eq.
refine (Quotient_ind_hprop _ _ _); intros x; cbn.
refine (Quotient_ind_hprop _ _ _); intros y; cbn.
rapply qglue.
Qed.


Expand All @@ -862,7 +861,7 @@ Proof.
+ rapply image_ind_prop; intros a. cbn.
intros B B_fa. apply tr.
exists (factor1 f (a.1; out (bound B_fa))). cbn.
unfold lt, relation, f; cbn.
unfold lt, relation, f; simpl.
assert (↓(out (bound B_fa)) = B) as ->. {
rewrite (path_initial_segment_simulation out).
symmetry. apply bound_property.
Expand Down Expand Up @@ -893,7 +892,7 @@ Proof.
intros a_u. apply equiv_resize_hprop in a_u. cbn in a_u.
apply tr. exists (out (bound a_u)). split.
+ apply initial_segment_property.
+ apply (injective (factor2 _)). cbn.
+ apply (injective (factor2 _)); simpl.
rewrite (path_initial_segment_simulation out).
symmetry. apply bound_property.
Qed.
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