From be2f38fc0679c277732f3fdacea26f53ab6b5acb Mon Sep 17 00:00:00 2001 From: Jlh18 Date: Tue, 5 Oct 2021 23:50:35 +0100 Subject: [PATCH] edits to quest2 --- 1FundamentalGroup/Preambles/P2.agda | 2 + 1FundamentalGroup/Quest2.agda | 43 ++++---- 1FundamentalGroup/Quest2Solutions.agda | 94 ++++++++++++++++++ .../agda/1FundamentalGroup/Preambles/P2.agdai | Bin 55213 -> 60114 bytes .../2.6.2/agda/1FundamentalGroup/Quest2.agdai | Bin 66408 -> 45546 bytes .../1FundamentalGroup/Quest2Solutions.agdai | Bin 65958 -> 118923 bytes 6 files changed, 115 insertions(+), 24 deletions(-) diff --git a/1FundamentalGroup/Preambles/P2.agda b/1FundamentalGroup/Preambles/P2.agda index 7722895..78e8513 100644 --- a/1FundamentalGroup/Preambles/P2.agda +++ b/1FundamentalGroup/Preambles/P2.agda @@ -3,6 +3,8 @@ module 1FundamentalGroup.Preambles.P2 where open import Cubical.Data.Nat public open import Cubical.Data.Int using (ℤ ; pos ; negsuc ; -_) public open import Cubical.Foundations.Isomorphism renaming (Iso to _≅_) public +open import Cubical.Data.Empty using (⊥) public +open import Cubical.Data.Unit renaming (Unit to ⊤) public open import Cubical.Foundations.Prelude renaming ( subst to endPt ; transport to pathToFun diff --git a/1FundamentalGroup/Quest2.agda b/1FundamentalGroup/Quest2.agda index a58c810..0562894 100644 --- a/1FundamentalGroup/Quest2.agda +++ b/1FundamentalGroup/Quest2.agda @@ -6,7 +6,7 @@ data _⊔_ (A B : Type) : Type where inl : A → A ⊔ B inr : B → A ⊔ B - +{- ℤ≡ℕ⊔ℕ : ℤ ≡ ℕ ⊔ ℕ ℤ≡ℕ⊔ℕ = {!!} @@ -17,31 +17,26 @@ refl∙ : {A : Type} {x y : A} (p : x ≡ y) → refl ∙ p ≡ p refl∙ = {!!} ∙sym : {A : Type} {x y : A} (p : x ≡ y) → p ∙ sym p ≡ refl -∙sym = J (λ y p → p ∙ sym p ≡ refl) - ( - refl ∙ sym refl - ≡⟨ cong (λ p → refl ∙ p) symRefl ⟩ - refl ∙ refl - ≡⟨ refl∙refl ⟩ - refl ∎) +∙sym = {!!} sym∙ : {A : Type} {x y : A} (p : x ≡ y) → (sym p) ∙ p ≡ refl -sym∙ = J (λ y p → (sym p) ∙ p ≡ refl) - ( - (sym refl) ∙ refl - ≡⟨ cong (λ p → p ∙ refl) symRefl ⟩ - refl ∙ refl - ≡⟨ refl∙refl ⟩ - refl ∎) +sym∙ = {!!} assoc : {A : Type} {w x : A} (p : w ≡ x) {y z : A} (q : x ≡ y) (r : y ≡ z) → (p ∙ q) ∙ r ≡ p ∙ (q ∙ r) -assoc {A} = J - -- casing on p - (λ x p → {y z : A} (q : x ≡ y) (r : y ≡ z) → (p ∙ q) ∙ r ≡ p ∙ (q ∙ r)) - -- reduce to showing when p = refl - λ q r → (refl ∙ q) ∙ r - ≡⟨ cong (λ p' → p' ∙ r) (refl∙ q) ⟩ - q ∙ r - ≡⟨ sym (refl∙ (q ∙ r)) ⟩ - refl ∙ q ∙ r ∎ +assoc {A} = {!!} +-} + +{- + +Your definition of ⊔NoConfusion goes here. + +Your definition of Path≡⊔NoConfusion goes here. + +Your definition of isProp⊔NoConfusion goes here. + +Your definition of isSet⊔ goes here. + +Your definition of isSetℤ goes here. + +-} diff --git a/1FundamentalGroup/Quest2Solutions.agda b/1FundamentalGroup/Quest2Solutions.agda index febc567..30d4cfd 100644 --- a/1FundamentalGroup/Quest2Solutions.agda +++ b/1FundamentalGroup/Quest2Solutions.agda @@ -62,3 +62,97 @@ assoc {A} = J q ∙ r ≡⟨ sym (refl∙ (q ∙ r)) ⟩ refl ∙ q ∙ r ∎ + +private + variable + A B : Type + +isProp⊤ : isProp ⊤ +isProp⊤ tt tt = refl + +isProp⊥ : isProp ⊥ +isProp⊥ () + +⊔NoConfusion : A ⊔ B → A ⊔ B → Type +⊔NoConfusion (inl x) (inl y) = x ≡ y -- Path A x y +⊔NoConfusion (inl x) (inr y) = ⊥ +⊔NoConfusion (inr x) (inl y) = ⊥ +⊔NoConfusion (inr x) (inr y) = x ≡ y -- Path B x y + +isProp⊔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 + +⊔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 _ _) (inv _ _) (rightInv _ _) (leftInv _ _) 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 y' p → ⊔NoConfusion x y') (⊔NoConfusionSelf _) + + 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 {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) + ( + 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 ∎ + ) + + 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) + ( + 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 ∎ + ) + + 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) + ( + (inv x x (fun x x refl)) + ≡⟨ cong (inv x x) (JRefl ((λ y' p → ⊔NoConfusion x y')) _) ⟩ + inv x x (⊔NoConfusionSelf x) + ≡⟨ lem x ⟩ + refl ∎ + ) where + + lem : (x : A ⊔ B) → inv x x (⊔NoConfusionSelf x) ≡ refl + lem (inl x) = refl + lem (inr x) = refl + +Path≡⊔NoConfusion : (x y : A ⊔ B) → (x ≡ y) ≡ ⊔NoConfusion x y +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⊔' : {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) + +isSetℤ : isSet ℤ +isSetℤ = pathToFun (cong isSet (sym ℤ≡ℕ⊔ℕ)) (isSet⊔ isSetℕ 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