From 324568407af9822cf0e68a003a9b0099dd698622 Mon Sep 17 00:00:00 2001 From: Jlh18 Date: Thu, 7 Oct 2021 22:57:21 +0100 Subject: [PATCH] quest 2 complete --- 1FundamentalGroup/Quest2.agda | 2 +- 1FundamentalGroup/Quest2Solutions.agda | 45 +++++++++++------- .../1FundamentalGroup/Quest2Solutions.agdai | Bin 118923 -> 119027 bytes 3 files changed, 29 insertions(+), 18 deletions(-) diff --git a/1FundamentalGroup/Quest2.agda b/1FundamentalGroup/Quest2.agda index 0562894..76777aa 100644 --- a/1FundamentalGroup/Quest2.agda +++ b/1FundamentalGroup/Quest2.agda @@ -33,7 +33,7 @@ Your definition of ⊔NoConfusion goes here. Your definition of Path≡⊔NoConfusion goes here. -Your definition of isProp⊔NoConfusion goes here. +Your definition of isSet⊔NoConfusion goes here. Your definition of isSet⊔ goes here. diff --git a/1FundamentalGroup/Quest2Solutions.agda b/1FundamentalGroup/Quest2Solutions.agda index 30d4cfd..2e7ba12 100644 --- a/1FundamentalGroup/Quest2Solutions.agda +++ b/1FundamentalGroup/Quest2Solutions.agda @@ -79,17 +79,34 @@ isProp⊥ () ⊔NoConfusion (inr x) (inl y) = ⊥ ⊔NoConfusion (inr x) (inr y) = x ≡ y -- Path B x y -isProp⊔NoConfusion : isSet A → isSet B +isSet⊔NoConfusion : isSet A → isSet B → (x y : A ⊔ B) → isProp (⊔NoConfusion x y) -isProp⊔NoConfusion hA hB (inl x) (inl y) = hA x y -isProp⊔NoConfusion hA hB (inl x) (inr y) = isProp⊥ -isProp⊔NoConfusion hA hB (inr x) (inl y) = isProp⊥ -isProp⊔NoConfusion hA hB (inr x) (inr y) = hB x y +isSet⊔NoConfusion hA hB (inl x) (inl y) = hA x y +isSet⊔NoConfusion hA hB (inl x) (inr y) = isProp⊥ +isSet⊔NoConfusion hA hB (inr x) (inl y) = isProp⊥ +isSet⊔NoConfusion hA hB (inr x) (inr y) = hB x y ⊔NoConfusionSelf : (x : A ⊔ B) → ⊔NoConfusion x x ⊔NoConfusionSelf (inl x) = refl ⊔NoConfusionSelf (inr x) = refl +Path≅⊔NoConfusion' : (x y : A ⊔ B) → (x ≡ y) ≅ ⊔NoConfusion x y +Path≅⊔NoConfusion' x y = iso (fun _ _) ({!!}) ({!!}) ({!!}) where + + -- if you case on x and y you would have to show that inl and inr are injective + -- J avoids this, but leads to needing J and JRefl for showing section and retract + fun : (x y : A ⊔ B) → (x ≡ y) → ⊔NoConfusion x y + fun x y = J (λ y₂ x₂ → ⊔NoConfusion x y₂) (⊔NoConfusionSelf x) + + inv : (x y : A ⊔ B) → ⊔NoConfusion x y → x ≡ y + inv (inl x) (inl y) p = cong inl p + inv (inr x) (inr y) p = cong inr p + + rightInv : (x y : A ⊔ B) → section (fun x y) (inv x y) + rightInv {A} {B} (inl x) (inl x₁) = J ((λ y' p → fun {A} {B} (inl x) (inl y') (inv (inl x) (inl y') p) ≡ p)) {!!} + rightInv {A} {B} (inl x) (inr x₁) = {!!} + rightInv {A} {B} (inr x) y = {!!} + Path≅⊔NoConfusion : (x y : A ⊔ B) → (x ≡ y) ≅ ⊔NoConfusion x y Path≅⊔NoConfusion x y = iso (fun _ _) (inv _ _) (rightInv _ _) (leftInv _ _) where @@ -103,27 +120,21 @@ Path≅⊔NoConfusion x y = iso (fun _ _) (inv _ _) (rightInv _ _) (leftInv _ _) inv (inr x) (inr y) p = cong inr p rightInv : (x y : A ⊔ B) → section (fun x y) (inv x y) - rightInv {B = B} (inl x) (inl y) p = leml x y p where - - leml : (x y : A) (p : x ≡ y) → fun {A} {B} (inl x) (inl y) (inv (inl x) (inl y) p) ≡ p - leml {A} x y = J (λ y' p → fun {A} {B} (inl x) (inl y') (inv (inl x) (inl y') p) ≡ p) + rightInv {A} {B} (inl x) (inl y) p = J (λ y' p → fun {A} {B} (inl x) (inl y') (inv (inl x) (inl y') p) ≡ p) ( fun {A} {B} (inl x) (inl x) refl ≡⟨ JRefl {x = inl x} ((λ y' p → ⊔NoConfusion {A} {B} (inl x) y')) _ ⟩ -- uses how J computes on refl refl ∎ - ) + ) p - rightInv {A = A} (inr x) (inr y) p = lemr x y p where - - lemr : (x y : B) (p : x ≡ y) → fun {A} {B} (inr x) (inr y) (inv (inr x) (inr y) p) ≡ p - lemr {B} x y = J (λ y' p → fun {A} {B} (inr x) (inr y') (inv (inr x) (inr y') p) ≡ p) + rightInv {A} {B} (inr x) (inr y) p = J (λ y' p → fun {A} {B} (inr x) (inr y') (inv (inr x) (inr y') p) ≡ p) ( fun {A} {B} (inr x) (inr x) refl ≡⟨ JRefl {x = inr x} ((λ y' p → ⊔NoConfusion {A} {B} (inr x) y')) _ ⟩ -- uses how J computes on refl refl ∎ - ) + ) p leftInv : (x y : A ⊔ B) → retract (fun x y) (inv x y) leftInv x y = J (λ y' p → inv x y' (fun x y' p) ≡ p) @@ -144,11 +155,11 @@ Path≡⊔NoConfusion x y = isoToPath (Path≅⊔NoConfusion x y) isSet⊔ : {A B : Type} → isSet A → isSet B → isSet (A ⊔ B) isSet⊔ hA hB x y = pathToFun (cong isProp (sym (Path≡⊔NoConfusion x y))) - (isProp⊔NoConfusion hA hB x y) + (isSet⊔NoConfusion hA hB x y) isSet⊔' : {A B : Type} → isSet A → isSet B → isSet (A ⊔ B) isSet⊔' hA hB x y = endPt (λ A → isProp A) (sym (Path≡⊔NoConfusion x y)) - (isProp⊔NoConfusion hA hB x y) + (isSet⊔NoConfusion hA hB x y) isSetℤ : isSet ℤ isSetℤ = pathToFun (cong isSet (sym ℤ≡ℕ⊔ℕ)) (isSet⊔ isSetℕ isSetℕ) diff --git a/_build/2.6.2/agda/1FundamentalGroup/Quest2Solutions.agdai b/_build/2.6.2/agda/1FundamentalGroup/Quest2Solutions.agdai index 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