42 lines
1.0 KiB
Agda
42 lines
1.0 KiB
Agda
module 1FundamentalGroup.Quest0Solutions where
|
||
|
||
open import Cubical.Data.Empty
|
||
open import Cubical.Data.Unit renaming ( Unit to ⊤ )
|
||
open import Cubical.Data.Bool
|
||
open import Cubical.Foundations.Prelude renaming ( subst to endPt )
|
||
open import Cubical.Foundations.Isomorphism renaming ( Iso to _≅_ )
|
||
open import Cubical.Foundations.Path
|
||
open import Cubical.HITs.S1
|
||
|
||
Refl : base ≡ base
|
||
Refl = λ i → base
|
||
|
||
Flip : Bool → Bool
|
||
Flip false = true
|
||
Flip true = false
|
||
|
||
-- notice we used `refl` instead of `λ i → false`,
|
||
-- more on `refl` in Quest1
|
||
flipIso : Bool ≅ Bool
|
||
flipIso = iso Flip Flip s r where
|
||
s : section Flip Flip
|
||
s false = refl
|
||
s true = refl
|
||
|
||
r : retract Flip Flip
|
||
r false = refl
|
||
r true = refl
|
||
|
||
flipPath : Bool ≡ Bool
|
||
flipPath = isoToPath flipIso
|
||
|
||
doubleCover : S¹ → Type
|
||
doubleCover base = Bool
|
||
doubleCover (loop i) = flipPath i
|
||
|
||
endPtOfTrue : (p : base ≡ base) → doubleCover base
|
||
endPtOfTrue p = endPt doubleCover p true
|
||
|
||
Refl≢loop : Refl ≡ loop → ⊥
|
||
Refl≢loop p = true≢false (cong endPtOfTrue p)
|