put quest0 sie quests into quest0.

1FundamentalGroup/Preambles/P0.agda

@@ -1,6 +1,6 @@
 module 1FundamentalGroup.Preambles.P0 where
 
-open import Cubical.Data.Empty public
+open import Cubical.Data.Empty using (⊥) public
 open import Cubical.Data.Unit renaming ( Unit to ⊤ ) public
 open import Cubical.Data.Bool public
 open import Cubical.Foundations.Prelude renaming ( subst to endPt ) public

1FundamentalGroup/Preambles/PEmpty.agda

@@ -1,6 +0,0 @@
-module 1FundamentalGroup.Preambles.PEmpty where
-
-open import Cubical.Foundations.Prelude public
-open import Cubical.Data.Empty renaming (rec to ⊥Rec) public
-open import Cubical.Foundations.Isomorphism renaming (Iso to _≅_) public
-

1FundamentalGroup/Preambles/PTrueNotFalse.agda

@@ -1,6 +0,0 @@
-module 1FundamentalGroup.Preambles.PTrueNotFalse where
-
-open import Cubical.Data.Empty public
-open import Cubical.Data.Unit renaming ( Unit to ⊤ ) public
-open import Cubical.Data.Bool using (Bool ; true ; false) public
-open import Cubical.Foundations.Prelude public

1FundamentalGroup/Quest0.agda

@@ -23,3 +23,42 @@ endPtOfTrue p = {!!}
 
 Refl≢loop : Refl ≡ loop → ⊥
 Refl≢loop p = {!!}
+
+------------------- Side Quest - Empty -------------------------
+
+{-
+-- This is a comment box,
+-- remove the {- and -} to do the side quests
+
+toEmpty : (A : Type) → Type
+toEmpty A = {!!}
+
+pathEmpty : (A : Type) → Type₁
+pathEmpty A = {!!}
+
+isoEmpty : (A : Type) → Type
+isoEmpty A = {!!}
+
+outOf⊥ : (A : Type) → ⊥ → A
+outOf⊥ A ()
+
+toEmpty→isoEmpty : (A : Type) → toEmpty A → isoEmpty A
+toEmpty→isoEmpty A = {!!}
+
+isoEmpty→pathEmpty : (A : Type) → isoEmpty A → pathEmpty A
+isoEmpty→pathEmpty A = {!!}
+
+pathEmpty→toEmpty : (A : Type) → pathEmpty A → toEmpty A
+pathEmpty→toEmpty A = {!!}
+-}
+
+------------------- Side Quests - true≢false --------------------
+
+{-
+-- This is a comment box,
+-- remove the {- and -} to do the side quests
+
+true≢false' : true ≡ false → ⊥
+true≢false' = {!!}
+
+-}

1FundamentalGroup/Quest0SideQuests/Empty.agda

@@ -1,21 +0,0 @@
-module 1FundamentalGroup.Quest0SideQuests.Empty where
-
-open import 1FundamentalGroup.Preambles.PEmpty
-
-toEmpty : (A : Type) → Type
-toEmpty A = {!!}
-
-pathEmpty : (A : Type) → Type₁
-pathEmpty A = {!!}
-
-isoEmpty : (A : Type) → Type
-isoEmpty A = {!!}
-
-toEmpty→isoEmpty : (A : Type) → toEmpty A → isoEmpty A
-toEmpty→isoEmpty A = {!!}
-
-isoEmpty→pathEmpty : (A : Type) → isoEmpty A → pathEmpty A
-isoEmpty→pathEmpty A = {!!}
-
-pathEmpty→toEmpty : (A : Type) → pathEmpty A → toEmpty A
-pathEmpty→toEmpty A = {!!}

1FundamentalGroup/Quest0SideQuests/EmptySolutions.agda

@@ -1,27 +0,0 @@
-module 1FundamentalGroup.Quest0SideQuests.EmptySolutions where
-
-open import 1FundamentalGroup.Preambles.PEmpty
-
-toEmpty : (A : Type) → Type
-toEmpty A = A → ⊥
-
-pathEmpty : (A : Type) → Type₁
-pathEmpty A = A ≡ ⊥
-
-isoEmpty : (A : Type) → Type
-isoEmpty A = A ≅ ⊥
-
-toEmpty→isoEmpty : (A : Type) → toEmpty A → isoEmpty A
-toEmpty→isoEmpty A f = iso f ⊥Rec rightInv leftInv where
-
-  rightInv : section f ⊥Rec
-  rightInv ()
-
-  leftInv : retract f ⊥Rec
-  leftInv x = ⊥Rec (f x)
-
-isoEmpty→pathEmpty : (A : Type) → isoEmpty A → pathEmpty A
-isoEmpty→pathEmpty A = isoToPath
-
-pathEmpty→toEmpty : (A : Type) → pathEmpty A → toEmpty A
-pathEmpty→toEmpty A p x = transport p x

1FundamentalGroup/Quest0SideQuests/TrueNotFalse.agda

@@ -1,6 +0,0 @@
-module 1FundamentalGroup.Quest0SideQuests.TrueNotFalse where
-
-open import 1FundamentalGroup.Preambles.PTrueNotFalse
-
-true≢false : true ≡ false → ⊥
-true≢false = {!!}

1FundamentalGroup/Quest0SideQuests/TrueNotFalseSolutions.agda

@@ -1,23 +0,0 @@
-module 1FundamentalGroup.Quest0SideQuests.TrueNotFalseSolutions where
-
-open import 1FundamentalGroup.Preambles.PTrueNotFalse
-
-true≢false : true ≡ false → ⊥
-true≢false h = transport ⊤≡⊥ tt where
-
-  propify : Bool → Type
-  propify false = ⊥
-  propify true = ⊤
-
-  ⊤≡⊥ : ⊤ ≡ ⊥
-  ⊤≡⊥ = cong propify h
-{- transport
-
-To follow a point in `a : A` along a path `p : A ≡ B`
-we use
-
-  transport : {A B : Type u} → A ≡ B → A → B
-
-Why do we propify? Discuss.
-
--}

1FundamentalGroup/Quest0Solutions.agda

@@ -33,3 +33,44 @@ endPtOfTrue p = endPt doubleCover p true
 
 Refl≢loop : Refl ≡ loop → ⊥
 Refl≢loop p = true≢false (cong endPtOfTrue p)
+
+------------------- Side Quest - Empty -------------------------
+
+toEmpty : (A : Type) → Type
+toEmpty A = A → ⊥
+
+pathEmpty : (A : Type) → Type₁
+pathEmpty A = A ≡ ⊥
+
+isoEmpty : (A : Type) → Type
+isoEmpty A = A ≅ ⊥
+
+outOf⊥ : (A : Type) → ⊥ → A
+outOf⊥ A ()
+
+toEmpty→isoEmpty : (A : Type) → toEmpty A → isoEmpty A
+toEmpty→isoEmpty A f = iso f (outOf⊥ A) rightInv leftInv where
+
+  rightInv : section f (outOf⊥ A)
+  rightInv ()
+
+  leftInv : retract f (outOf⊥ A)
+  leftInv x = outOf⊥ (outOf⊥ A (f x) ≡ x) (f x)
+
+isoEmpty→pathEmpty : (A : Type) → isoEmpty A → pathEmpty A
+isoEmpty→pathEmpty A = isoToPath
+
+pathEmpty→toEmpty : (A : Type) → pathEmpty A → toEmpty A
+pathEmpty→toEmpty A p x = transport p x
+
+------------------- Side Quests - true≢false --------------------
+
+true≢false' : true ≡ false → ⊥
+true≢false' h = transport ⊤≡⊥ tt where
+
+  propify : Bool → Type
+  propify false = ⊥
+  propify true = ⊤
+
+  ⊤≡⊥ : ⊤ ≡ ⊥
+  ⊤≡⊥ = cong propify h