test

html/src/SUMMARY.md

@@ -2,6 +2,14 @@
 
 - [Front](./front.md)
 
+# HoTT Book
+
+- [Chapter 1](./generated/HottBook.Chapter1.md)
+- [Chapter 2](./generated/HottBook.Chapter2.md)
+- [Chapter 3](./generated/HottBook.Chapter3.md)
+- [Chapter 4](./generated/HottBook.Chapter4.md)
+- [Chapter 5](./generated/HottBook.Chapter5.md)
+
 # HoTT Book (Cubical formulation)
 
 - [Chapter 1](./generated/CubicalHott.Chapter1.md)

src/CEKCoinductive.agda

@@ -0,0 +1,22 @@
+{-# OPTIONS --guardedness #-}
+
+module CEKCoinductive where
+
+open import Agda.Primitive
+
+private
+  variable
+    l l1 l2 l3 : Level
+
+    E : Set → Set
+    R : Set
+
+data ITreeF (ITree : Set) : Set where
+  Ret : (r : R) → ITreeF ITree
+  Tau : (t : ITree) → ITreeF ITree
+  Vis : {A : Set l1} → (e : E A) → (k : A → ITreeF E R) → ITreeF E R
+
+record ITree : Set where
+  coinductive
+  field
+    _observe : ITreeF ITree

src/CubicalHott/Chapter6.lagda.md

@@ -6,6 +6,21 @@ open import CubicalHott.Chapter1
 open import CubicalHott.Chapter2
 ```
 
+## 6.3 The interval
+
+```
+data Iv : Type where
+  0I : Iv
+  1I : Iv
+  seg : 0I ≡ 1I
+```
+
+### Lemma 6.3.1
+
+```
+lemma6∙3∙1 : isContr Iv
+```
+
 ## 6.4 Circles and spheres
 
 ```
@@ -18,36 +33,36 @@ data S¹ : Type where
 
 ```
 lemma6∙4∙1 : loop ≢ refl
-lemma6∙4∙1 p =
-  {!   !}
-  where
-    open ≡-Reasoning
+-- lemma6∙4∙1 p =
+--   {!   !}
+--   where
+--     open ≡-Reasoning
 
-    f : {A : Type} (x : A) → (p : x ≡ x) → S¹ → A
-    f x p base = x
-    f x p (loop i) = p i
+--     f : {A : Type} (x : A) → (p : x ≡ x) → S¹ → A
+--     f x p base = x
+--     f x p (loop i) = p i
 
-    bad : {A : Type} (x : A) → (p : x ≡ x) → p ≡ refl
-    bad x p = begin
-      p ≡⟨ {!   !} ⟩
-      p ≡⟨ {!   !} ⟩
-      refl ∎
+--     bad : {A : Type} (x : A) → (p : x ≡ x) → p ≡ refl
+--     bad x p = begin
+--       p ≡⟨ {!   !} ⟩
+--       p ≡⟨ {!   !} ⟩
+--       refl ∎
 ```
 
 ### Lemma 6.4.2
 
 ```
 lemma6∙4∙2 : Σ ((x : S¹) → x ≡ x) (λ y → y ≢ (λ x → refl))
-lemma6∙4∙2 = f , g
-  where
-    f : (x : S¹) → x ≡ x
-    f base = loop
-    f (loop i) i₁ = loop {!   !}
+-- lemma6∙4∙2 = f , g
+--   where
+--     f : (x : S¹) → x ≡ x
+--     f base = loop
+--     f (loop i) i₁ = loop {!   !}
 
-    g : f ≢ (λ x → refl)
-    g p = let 
-        z = happlyd p
-        z2 = z base
-        z3 = lemma6∙4∙1 z2
-      in z3
+--     g : f ≢ (λ x → refl)
+--     g p = let 
+--         z = happlyd p
+--         z2 = z base
+--         z3 = lemma6∙4∙1 z2
+--       in z3
 ```

src/HottBook/Chapter2.lagda.md

@@ -724,11 +724,8 @@ module axiom2∙10∙3 where
   postulate
     ua : {l : Level} {A B : Set l} → (A ≃ B) → (A ≡ B)
 
-    forward : {l : Level} {A B : Set l} → (eqv : A ≃ B) → (idtoeqv ∘ ua) eqv ≡ eqv
-    -- forward eqv = {!   !}
-
     backward : {l : Level} {A B : Set l} → (p : A ≡ B) → (ua ∘ idtoeqv) p ≡ p
-    -- backward p = {!   !}
+    forward : {l : Level} {A B : Set l} → (eqv : A ≃ B) → (idtoeqv ∘ ua) eqv ≡ eqv
 
   ua-eqv : {A B : Set l} → (A ≃ B) ≃ (A ≡ B)
   ua-eqv = ua , qinv-to-isequiv (mkQinv idtoeqv backward forward)

src/HottBook/Chapter3.lagda.md

@@ -184,18 +184,8 @@ theorem3∙2∙2 double-neg = conclusion
       let y = what ∙ x in
       sym y ∙ foranyu
 
-    -- postulate
-    huhh : (Σ.fst e) (fbool u) ≡ fbool u
-    huhh =
-      let
-        equiv1 = ap ua (axiom2∙10∙3.forward e)
-
-        x : {A B : Set l} → (e : A ≃ B) → (a : A) → transport id (ua e) a ≡ Σ.fst e a
-        x e a = 
-          {!  axiom2∙10∙3.forward ? !}
-      in
-      {!   !}
-    --   sym (x e (fbool u)) ∙ nextStep
+    postulate
+      huhh : (Σ.fst e) (fbool u) ≡ fbool u
 
     finalStep : (x : bool) → ¬ ((Σ.fst e) x ≡ x)
     finalStep (lift true) p = 
@@ -310,5 +300,9 @@ lemma3∙9∙1 {P} prop = lemma3∙3∙3 prop prop2 ∣_∣ g
         concat : g (∣ g x ∣) ≡ g (∣ g y ∣)
         concat = a ∙ eqProp ∙ (sym b)
       in
-      {!  prop ?  !}
+      admit
+      where
+        postulate
+          -- TODO: Finish this 
+          admit : x ≡ y
 ```

src/HottBook/Chapter6.lagda.md

@@ -0,0 +1,53 @@
+```
+module HottBook.Chapter6 where
+
+open import HottBook.Chapter1
+open import HottBook.Chapter2
+open import HottBook.Chapter3
+
+private
+  variable
+    l : Level
+```
+
+# Chapter 6 Higher Inductive Types
+
+```
+postulate
+  S¹ : Set
+  base : S¹
+  loop : base ≡ base
+```
+
+## 6.2 Induction principles and dependent paths
+
+```
+dep-path : {A : Set} {x y : A} → (P : A → Set) → (p : x ≡ y) → (u : P x) → (v : P y) → Set
+dep-path P p u v = transport P p u ≡ v
+
+syntax dep-path P p u v = u ≡[ P , p ] v
+```
+
+## 6.3 The interval
+
+```
+postulate
+  I : Set
+  0I : I
+  1I : I
+  seg : 0I ≡ 1I
+
+record I-rec (B : Set) (b0 b1 : B) (s : b0 ≡ b1) : Set where
+  field
+    f : I → B
+    prop1 : f 0I ≡ b0
+    prop2 : f 1I ≡ b1
+    prop3 : apd f seg ≡ {! s  !}
+
+record I-ind (P : I → Set) (b0 : P 0I) (b1 : P 1I) (s : b0 ≡[ P , seg ] b1) : Set where
+  field
+    f : (x : I) → P x
+    prop1 : f 0I ≡ b0
+    prop2 : f 1I ≡ b1
+    prop3 : apd f seg ≡ {! s  !}
+```

src/Hurewicz.agda

@@ -0,0 +1,2 @@
+module Hurewicz where
+