From ee0f734da2622f4f49a34f9c04fe6a06cb4d68c6 Mon Sep 17 00:00:00 2001 From: Jlh18 Date: Sun, 3 Oct 2021 15:55:33 +0100 Subject: [PATCH] added Z=NuN and Groupoid laws exercises in quest2 --- 1FundamentalGroup/Preambles/P2.agda | 6 ++ 1FundamentalGroup/Quest0.agda | 1 + 1FundamentalGroup/Quest2Solutions.agda | 81 +++++++++++------- .../agda/1FundamentalGroup/Preambles/P2.agdai | Bin 52059 -> 55213 bytes .../1FundamentalGroup/Quest2Solutions.agdai | Bin 38075 -> 66348 bytes 5 files changed, 59 insertions(+), 29 deletions(-) diff --git a/1FundamentalGroup/Preambles/P2.agda b/1FundamentalGroup/Preambles/P2.agda index 83a8993..7722895 100644 --- a/1FundamentalGroup/Preambles/P2.agda +++ b/1FundamentalGroup/Preambles/P2.agda @@ -9,3 +9,9 @@ open import Cubical.Foundations.Prelude ) public open import Cubical.HITs.S1 using (S¹ ; base ; loop) public open import 1FundamentalGroup.Quest1Solutions public + +refl∙refl : {A : Type} {a : A} → refl ∙ refl ≡ refl {x = a} +refl∙refl {a = a} = sym (λ i j → compPath-filler (refl {x = a}) refl i j) + +symRefl : {A : Type} {a : A} → sym refl ≡ refl {x = a} +symRefl = refl diff --git a/1FundamentalGroup/Quest0.agda b/1FundamentalGroup/Quest0.agda index 7256b77..430de35 100644 --- a/1FundamentalGroup/Quest0.agda +++ b/1FundamentalGroup/Quest0.agda @@ -62,3 +62,4 @@ true≢false' : true ≡ false → ⊥ true≢false' = {!!} -} + diff --git a/1FundamentalGroup/Quest2Solutions.agda b/1FundamentalGroup/Quest2Solutions.agda index 45df76b..a458de9 100644 --- a/1FundamentalGroup/Quest2Solutions.agda +++ b/1FundamentalGroup/Quest2Solutions.agda @@ -2,40 +2,63 @@ module 1FundamentalGroup.Quest2Solutions where open import 1FundamentalGroup.Preambles.P2 --- sucℤ : ℤ → ℤ --- sucℤ (pos n) = pos (suc n) --- sucℤ (negsuc zero) = pos zero --- sucℤ (negsuc (suc n)) = negsuc n --- predℤ : ℤ → ℤ --- predℤ (pos zero) = negsuc zero --- predℤ (pos (suc n)) = pos n --- predℤ (negsuc n) = negsuc (suc n) +data _⊔_ (A B : Type) : Type where --- sucℤIso : ℤ ≅ ℤ --- sucℤIso = iso sucℤ predℤ s r where + inl : A → A ⊔ B + inr : B → A ⊔ B --- s : section sucℤ predℤ --- s (pos zero) = refl --- s (pos (suc n)) = refl --- s (negsuc zero) = refl --- s (negsuc (suc n)) = refl +ℤ≡ℕ⊔ℕ : ℤ ≡ ℕ ⊔ ℕ +ℤ≡ℕ⊔ℕ = isoToPath (iso fun inv rightInv leftInv) where --- r : retract sucℤ predℤ --- r (pos zero) = refl --- r (pos (suc n)) = refl --- r (negsuc zero) = refl --- r (negsuc (suc n)) = refl + fun : ℤ → ℕ ⊔ ℕ + fun (pos n) = inl n + fun (negsuc n) = inr n --- sucℤPath : ℤ ≡ ℤ --- sucℤPath = isoToPath sucℤIso + inv : ℕ ⊔ ℕ → ℤ + inv (inl n) = pos n + inv (inr n) = negsuc n --- helix : S¹ → Type --- helix base = ℤ --- helix (loop i) = sucℤPath i + rightInv : section fun inv + rightInv (inl n) = refl + rightInv (inr n) = refl --- windingNumberBase : base ≡ base → ℤ --- windingNumberBase p = endPt helix p (pos zero) + leftInv : retract fun inv + leftInv (pos n) = refl + leftInv (negsuc n) = refl --- windingNumber : (x : S¹) → base ≡ x → helix x --- windingNumber x p = endPt helix p (pos zero) +∙refl : {A : Type} {x y : A} (p : x ≡ y) → p ∙ refl ≡ p +∙refl = J (λ y p → p ∙ refl ≡ p) refl∙refl + +refl∙ : {A : Type} {x y : A} (p : x ≡ y) → refl ∙ p ≡ p +refl∙ = J (λ y p → refl ∙ p ≡ p) refl∙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∙ : {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 ∎) + +assoc : {A : Type} {w x : A} (p : w ≡ x) {y : A} (q : x ≡ y) {z : A} (r : y ≡ z) + → (p ∙ q) ∙ r ≡ p ∙ (q ∙ r) +assoc {A} {w} = J + -- casing on p + (λ x p → {y : A} (q : x ≡ y) {z : A} (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 ∙ 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