From 61a70598370b63cf2a6da36401f7d11807252a04 Mon Sep 17 00:00:00 2001
From: Martin Escardo <m.escardo@cs.bham.ac.uk>
Date: Mon, 21 Oct 2024 15:30:12 +0100
Subject: [PATCH] remove question - I tried and it doesn't improve the
 universes

---
 source/InjectiveTypes/Blackboard.lagda | 31 +++++++++++++++++++++++++-
 source/InjectiveTypes/Subtypes.lagda   |  3 ---
 2 files changed, 30 insertions(+), 4 deletions(-)

diff --git a/source/InjectiveTypes/Blackboard.lagda b/source/InjectiveTypes/Blackboard.lagda
index 6f00822ac..28918ab26 100644
--- a/source/InjectiveTypes/Blackboard.lagda
+++ b/source/InjectiveTypes/Blackboard.lagda
@@ -556,7 +556,7 @@ retract-of-ainjective : (D' : 𝓦' ̇ ) (D : 𝓦 ̇ )
                       → ainjective-type D 𝓤 𝓥
                       → retract D' of D
                       → ainjective-type D' 𝓤 𝓥
-retract-of-ainjective D' D i (r , (s , rs)) {X} {Y} j e f = φ a
+retract-of-ainjective D' D i (r , s , rs) {X} {Y} j e f = φ a
   where
    a : Σ f' ꞉ (Y → D) , f' ∘ j ∼ s ∘ f
    a = i j e (s ∘ f)
@@ -1779,6 +1779,35 @@ Here are some corollaries:
 
 \end{code}
 
+Added 21st October 2024.
+
+\begin{code}
+
+aflabby-embedding-retract : (D : 𝓤 ̇ ) (Y : 𝓥 ̇ ) (j : D → Y)
+                          → is-embedding j
+                          → aflabby D (𝓤 ⊔ 𝓥)
+                          → retract D of Y
+aflabby-embedding-retract D Y j e a =
+ embedding-retract D Y j e (aflabby-types-are-ainjective D a)
+
+retract-of-aflabby : (D : 𝓤 ̇ ) (E : 𝓥 ̇ )
+                   → aflabby D 𝓣
+                   → retract E of D
+                   → aflabby E 𝓣
+retract-of-aflabby D E D-aflabby (r , s , rs) P P-is-prop f
+ = r d ,
+   (λ (p : P) → r d         ＝⟨ ap r (I p) ⟩
+                r (s (f p)) ＝⟨ rs (f p) ⟩
+                f p         ∎)
+  where
+   d : D
+   d = aflabby-extension D-aflabby (P , P-is-prop) (s ∘ f)
+
+   I : (p : P) → d ＝ s (f p)
+   I = aflabby-extension-property D-aflabby (P , P-is-prop) (s ∘ f)
+
+\end{code}
+
 Fixities:
 
 \begin{code}
diff --git a/source/InjectiveTypes/Subtypes.lagda b/source/InjectiveTypes/Subtypes.lagda
index 53c3b6d82..fdbd997ce 100644
--- a/source/InjectiveTypes/Subtypes.lagda
+++ b/source/InjectiveTypes/Subtypes.lagda
@@ -106,6 +106,3 @@ module _ (D : 𝓤 ̇ )
 
 TODO. Can the above logically equivalence be made into a type
 equivalence?
-
-TODO. Perhaps using aflabbiness we would get more general universe
-levels.