From 27524b92506b5401a550dc0e01a45feeccfa8cd1 Mon Sep 17 00:00:00 2001
From: Michael Zhang <mail@mzhang.io>
Date: Wed, 10 Jul 2024 23:03:58 -0500
Subject: [PATCH] update

---
 src/CubicalHott/Chapter1.lagda.md             |  5 +++
 src/HottBook/Chapter2Exercises.lagda.md       | 22 +++++++++---
 src/HottBook/Chapter4.lagda.md                | 19 +++++++---
 .../Preliminaries.lagda.md                    | 35 ++++++++++++++++++-
 4 files changed, 70 insertions(+), 11 deletions(-)

diff --git a/src/CubicalHott/Chapter1.lagda.md b/src/CubicalHott/Chapter1.lagda.md
index afec5ac..b8d802d 100644
--- a/src/CubicalHott/Chapter1.lagda.md
+++ b/src/CubicalHott/Chapter1.lagda.md
@@ -32,6 +32,11 @@ open Σ public
 
 {-# BUILTIN SIGMA Σ #-}
 infixr 4 _,_
+
+syntax Σ A (λ x → B) = Σ[ x ∈ A ] B
+
+_×_ : {l : Level} (A B : Type l) → Type l
+_×_ A B = Σ A (λ _ → B)
 ```
 
 Unit type:
diff --git a/src/HottBook/Chapter2Exercises.lagda.md b/src/HottBook/Chapter2Exercises.lagda.md
index 2166a7a..5b6bae5 100644
--- a/src/HottBook/Chapter2Exercises.lagda.md
+++ b/src/HottBook/Chapter2Exercises.lagda.md
@@ -167,9 +167,21 @@ TODO: Generalizing to dependent functions.
 ## Exercise 2.10
 
 ```
-postulate
-  exercise2∙10 : {A : Set} {B : A → Set} {C : Σ A B → Set}
-    → Σ A (λ x → Σ (B x) (λ y → C (x , y))) ≃ Σ (Σ A B) (λ p → C p)
+exercise2∙10 : {A : Set} {B : A → Set} {C : Σ A B → Set}
+  → (Σ A (λ x → Σ (B x) (λ y → C (x , y)))) ≃ Σ (Σ A B) C
+exercise2∙10 {A} {B} {C} = f , qinv-to-isequiv (mkQinv g f-g g-f)
+  where
+    f : (Σ A (λ x → Σ (B x) (λ y → C (x , y)))) → Σ (Σ A B) C
+    f (x , (y , cxy)) = (x , y) , cxy
+
+    g : Σ (Σ A B) C → (Σ A (λ x → Σ (B x) (λ y → C (x , y))))
+    g ((x , y) , cxy) = x , (y , cxy)
+
+    f-g : (f ∘ g) ∼ id
+    f-g ((x , y) , cxy) = refl
+
+    g-f : (g ∘ f) ∼ id
+    g-f (x , (y , cxy)) = refl
 ```
 
 ## Exercise 2.13
@@ -311,5 +323,5 @@ module exercise2∙17 where
       B-id = ua B-eqv
 
       combined-id = pair-≡ A-id B-id
-    in idtoeqv combined-id
-```
\ No newline at end of file
+   in idtoeqv combined-id
+``` 
\ No newline at end of file
diff --git a/src/HottBook/Chapter4.lagda.md b/src/HottBook/Chapter4.lagda.md
index 1bcb079..4ffd44f 100644
--- a/src/HottBook/Chapter4.lagda.md
+++ b/src/HottBook/Chapter4.lagda.md
@@ -82,12 +82,18 @@ lemma4∙1∙2 {A} {a} q prop1 g prop3 = (λ x → {!   !}) , {!   !}
 There exist types A and B and a function f : A → B such that qinv( f ) is not a mere proposition.
 
 ```
-theorem4∙1∙3 : ∀ {l} {A B : Set l}
-  → Σ (A → B) (λ f → isProp (qinv f) → ⊥)
-theorem4∙1∙3 = {! !} , {!   !}
-  where
-    goal : Σ (Set (lsuc l)) (λ X → isProp ((x : X) → x ≡ x) → ⊥)
+theorem4∙1∙3 : ∀ {l} → Σ (Set l) (λ A → Σ (Set l) (λ B → Σ (A → B) (λ f → isProp (qinv f) → ⊥)))
+-- theorem4∙1∙3 {l} = goal
+--   where
+--     goal : ∀ {l} → Σ (Set l) (λ A → Σ (Set l) (λ B → Σ (A → B) (λ f → isProp (qinv f) → ⊥)))
 
+--     goal2 : ∀ {l} → Σ (Set l) (λ X → isProp ((x : X) → x ≡ x) → ⊥)
+--     goal {l} = Σ.fst (goal2 {l}) , {!   !} , {!   !} , {!   !}
+
+--     X : ∀ {l} → Set (lsuc l)
+--     X {l} = Σ (Set l) (λ A → ∥ Lift 𝟚 ≡ A ∥)
+
+--     goal2 {l} = X {l} , ?
 ```
 
 ## 4.2 Half adjoint equivalences
@@ -106,6 +112,9 @@ record ishae {A B : Set} (f : A → B) : Set where
 
 ### Lemma 4.2.2
 
+```
+```
+
 ### Theorem 4.2.3
 
 ```
diff --git a/src/VanDoornDissertation/Preliminaries.lagda.md b/src/VanDoornDissertation/Preliminaries.lagda.md
index 51014f8..f3be99e 100644
--- a/src/VanDoornDissertation/Preliminaries.lagda.md
+++ b/src/VanDoornDissertation/Preliminaries.lagda.md
@@ -1,5 +1,5 @@
 ```
-{-# OPTIONS --cubical #-}
+{-# OPTIONS --cubical --no-load-primitives #-}
 module VanDoornDissertation.Preliminaries where
 
 open import CubicalHott.Chapter1
@@ -57,4 +57,37 @@ Square p q s r = PathP (λ i → p i ≡ r i) q s
 
 ```
 
+```
+
+### 2.2.5 Pointed Types
+
+#### Definition 2.2.6
+
+```
+data pointed (A : Type) : Type where
+  mkPointed : (a₀ : A) → pointed A
+
+1-is-pointed : pointed 𝟙
+1-is-pointed = mkPointed tt
+
+2-is-pointed : pointed 𝟚
+2-is-pointed = mkPointed false
+
+S⁰ = 2-is-pointed
+
+_×_-is-pointed : {A B : Type} → pointed A → pointed B → pointed (A × B)
+_×_-is-pointed {A} {B} (mkPointed a₀) (mkPointed b₀) = mkPointed (a₀ , b₀)
+```
+
+### 2.2.6 Higher inductive types
+
+```
+data Interval : Type where
+  0I : Interval
+  1I : Interval
+  seg : 0I ≡ 1I
+
+data Quotient (A : Type) (R : A → A → Type) : Type where
+  x : A → Quotient A R
+  glue : (a a' : A) → R a a' → x a ≡ x a'
 ```
\ No newline at end of file