45 lines
1.7 KiB
Agda
45 lines
1.7 KiB
Agda
module 0Trinitarianism.Quest5Solutions where
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open import Cubical.Foundations.Prelude renaming (subst to endPt ; transport to pathToFun)
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open import Cubical.HITs.S1 using ( S¹ ; base ; loop )
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open import Cubical.Foundations.Isomorphism renaming (Iso to _≅_)
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open import Cubical.Foundations.Path renaming (PathPIsoPath to PathPIsoPathD)
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open import 1FundamentalGroup.Quest0Solutions
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open import Cubical.Data.Bool
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PathD : {A0 A1 : Type} (A : A0 ≡ A1) (x : A0) (y : A1) → Type
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PathD A x y = pathToFun A x ≡ y
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syntax PathD A x y = x ≡ y along A
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outOfS¹P : (B : S¹ → Type) (b : B base) → PathP (λ i → B (loop i)) b b → (x : S¹) → B x
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outOfS¹P B b p base = b
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outOfS¹P B b p (loop i) = p i
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outOfS¹D : (B : S¹ → Type) (b : B base) → b ≡ b along (λ i → B (loop i)) → (x : S¹) → B x
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outOfS¹D B b p x = outOfS¹P B b (_≅_.inv (PathPIsoPathD (λ i → B (loop i)) b b) p) x
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example : (x : S¹) → doubleCover x → doubleCover x
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example base = Flip
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example (loop i) = p i where
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lem : (x : Bool) → pathToFun (λ i → flipPath i → flipPath i) Flip x ≡ Flip x
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lem false = refl
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lem true = refl
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p : PathP (λ i → flipPath i → flipPath i) Flip Flip
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p = _≅_.inv (PathPIsoPathD (λ i → flipPath i → flipPath i) Flip Flip) (funExt lem)
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-- repeating the example but using our API
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example' : (x : S¹) → doubleCover x → doubleCover x
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example' = outOfS¹D (λ x → doubleCover x → doubleCover x) Flip (funExt lem) where
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lem : (x : Bool) → pathToFun (λ i → flipPath i → flipPath i) Flip x ≡ Flip x
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lem false = refl
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lem true = refl
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outOfS¹DBase : (B : S¹ → Type) (b : B base)
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(p : b ≡ b along (λ i → B (loop i)))→ outOfS¹D B b p base ≡ b
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outOfS¹DBase B b p = refl
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