edits to trinitarianism/quest 5

0Trinitarianism/Quest5Solutions.agda

@@ -3,7 +3,7 @@ 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 Cubical.Foundations.Path renaming (PathPIsoPath to PathPIsoPathD)
 open import 1FundamentalGroup.Quest0Solutions
 open import Cubical.Data.Bool
 
@@ -17,33 +17,23 @@ 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)
---     ∎)
+outOfS¹D B b p x = outOfS¹P B b (_≅_.inv (PathPIsoPathD (λ i → B (loop i)) b b) p) x
 
 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 = {!!}
+  lem : (x : Bool) → pathToFun (λ i → flipPath i → flipPath i) Flip x ≡ Flip x
+  lem false = refl
+  lem true = refl
+
+  p : PathP (λ i → flipPath i → flipPath i) Flip Flip
+  p = _≅_.inv (PathPIsoPathD (λ i → flipPath i → flipPath i) Flip Flip) (funExt lem)
+
+-- repeating the example but using our API
 
 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
+example' = outOfS¹D (λ x → doubleCover x → doubleCover x) Flip (funExt lem) where
 
   lem : (x : Bool) → pathToFun (λ i → flipPath i → flipPath i) Flip x ≡ Flip x
   lem false = refl