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