added arc0/quest5 and arc1/quest3

0Trinitarianism/Quest4Solutions.agda

@@ -104,10 +104,10 @@ Trans {x = x} {z = z} = J (λ y1 p → y1 ≡ z → x ≡ z) λ q → q
 
 _∙_ = Trans -- input \.
 
-_≡⟨_⟩_ : (x : A) → x ≡ y → y ≡ z → x ≡ z
+_≡⟨_⟩_ : (x : A) → x ≡ y → y ≡ z → x ≡ z -- input \< and \>
 _ ≡⟨ x≡y ⟩ y≡z = x≡y ∙ y≡z
 
-_∎ : (x : A) → x ≡ x
+_∎ : (x : A) → x ≡ x -- input \qed
 _ ∎ = refl
 
 infixr 30 _∙_

0Trinitarianism/Quest5Solutions.agda

@@ -0,0 +1,50 @@
+module 0Trinitarianism.Quest5Solutions where
+
+open import Cubical.Foundations.Prelude renaming (subst to endPt ; transport to pathToFun)
+open import Cubical.HITs.S1 using ( S¹ ; base ; loop )
+open import Cubical.Foundations.Isomorphism renaming (Iso to _≅_)
+open import Cubical.Foundations.Path
+open import 1FundamentalGroup.Quest0Solutions
+open import Cubical.Data.Bool
+
+PathD : {A0 A1 : Type} (A : A0 ≡ A1) (x : A0) (y : A1) → Type
+PathD A x y = pathToFun A x ≡ y
+
+syntax PathD A x y = x ≡ y along A
+
+outOfS¹P : (B : S¹ → Type) → (b : B base) → PathP (λ i → B (loop i)) b b → (x : S¹) → B x
+outOfS¹P B b p base = b
+outOfS¹P B b p (loop i) = p i
+
+outOfS¹D : (B : S¹ → Type) → (b : B base) → b ≡ b along (λ i → B (loop i)) → (x : S¹) → B x
+outOfS¹D B b p x = outOfS¹P B b (_≅_.inv (PathPIsoPath (λ i → B (loop i)) b b) p) x
+
+-- example : (x : S¹) → doubleCover x → Bool
+-- example = outOfS¹D (λ x → doubleCover x → Bool) (λ x → true)
+--     (pathToFun (λ i → (doubleCover (loop i)) → Bool) (λ x → true)
+--   ≡⟨ refl ⟩
+--     pathToFun (λ i → flipPath i → Bool) (λ x → true)
+--   ≡⟨ refl ⟩
+--     (λ x → true)
+--     ∎)
+
+example : (x : S¹) → doubleCover x → doubleCover x
+example base = Flip
+example (loop i) = p i where
+
+  p : PathP (λ i → doubleCover (loop i) → doubleCover (loop i)) Flip Flip
+  p = {!!}
+
+example' : (x : S¹) → doubleCover x → doubleCover x
+example' = outOfS¹D (λ x → doubleCover x → doubleCover x) Flip
+  (
+    pathToFun (λ i → doubleCover (loop i) → doubleCover (loop i)) Flip
+  ≡⟨ refl ⟩
+   pathToFun (λ i → flipPath i → flipPath i) Flip
+  ≡⟨ funExt lem ⟩
+    Flip
+  ∎) where
+
+  lem : (x : Bool) → pathToFun (λ i → flipPath i → flipPath i) Flip x ≡ Flip x
+  lem false = refl
+  lem true = refl

1FundamentalGroup/Quest3.agda

@@ -0,0 +1 @@
+module 1FundamentalGroup.Quest3 where

1FundamentalGroup/Quest3Solutions.agda

@@ -1,9 +1,21 @@
 module 1FundamentalGroup.Quest3Solutions where
 
 open import Cubical.Foundations.Prelude
-open import Cubical.HITs.S1
+open import Cubical.HITs.S1 using ( S¹ ; base ; loop )
 open import 1FundamentalGroup.Quest1Solutions
+open import Cubical.Foundations.Path
 
 outOfS¹ : (B : S¹ → Type) → (b : B base) → PathP (λ i → B (loop i)) b b → (x : S¹) → B x
 outOfS¹ B b p base = b
 outOfS¹ B b p (loop i) = p i
+
+
+
+rewind : (x : S¹) → helix x → base ≡ x
+rewind base = loop_times
+rewind (loop i) = path i where
+
+   path : PathP (λ i → helix (loop i) → base ≡ loop i) loop_times loop_times
+   path  = {!!}
+
+thing = PathPIsoPath