From c1c23df3d091402959869a1daa02b76b511f298e Mon Sep 17 00:00:00 2001 From: Jlh18 Date: Sat, 23 Oct 2021 22:03:39 +0100 Subject: [PATCH] =?UTF-8?q?Path=20=E2=89=A1=20Id=20done?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- 0Trinitarianism/Quest4Solutions.agda | 167 +++++++++++++++--- .../0Trinitarianism/Quest4Solutions.agdai | Bin 53491 -> 86737 bytes .../agda/1FundamentalGroup/Preambles/P1.agdai | Bin 53398 -> 53403 bytes .../1FundamentalGroup/Quest0Solutions.agdai | Bin 65772 -> 65678 bytes .../1FundamentalGroup/Quest1Solutions.agdai | Bin 45641 -> 45631 bytes 5 files changed, 146 insertions(+), 21 deletions(-) diff --git a/0Trinitarianism/Quest4Solutions.agda b/0Trinitarianism/Quest4Solutions.agda index e880107..470809f 100644 --- a/0Trinitarianism/Quest4Solutions.agda +++ b/0Trinitarianism/Quest4Solutions.agda @@ -1,9 +1,18 @@ module 0Trinitarianism.Quest4Solutions where -open import Cubical.Foundations.Prelude +open import Cubical.Foundations.Prelude using ( Level ; Type ; _≡_ ; J ; JRefl ; refl ) open import Cubical.Foundations.Isomorphism renaming (Iso to _≅_) -data Id {A : Type} : (x y : A) → Type where +infixr 30 _∙_ +infix 3 _∎ +infixr 2 _≡⟨_⟩_ + +private + variable + u : Level + + +data Id {A : Type u} : (x y : A) → Type u where rfl : {x : A} → Id x x @@ -26,45 +35,161 @@ data _×_ (A B : Type) : Type where _,_ : A → B → A × B -id× : {A B : Type} (a0 a1 : A) (b0 b1 : B) → - (Id a0 a1 × Id b0 b1) ≅ Id {A × B} ( a0 , b0 ) ( a1 , b1 ) -id× {A} {B} a0 a1 b0 b1 = iso fun inv rightInv leftInv where +-- id× : {A B : Type} (a0 a1 : A) (b0 b1 : B) → +-- (Id a0 a1 × Id b0 b1) ≅ Id {A × B} ( a0 , b0 ) ( a1 , b1 ) +-- id× {A} {B} a0 a1 b0 b1 = iso fun inv rightInv leftInv where - fun : Id a0 a1 × Id b0 b1 → Id {A × B} ( a0 , b0 ) ( a1 , b1 ) - fun (rfl , rfl) = rfl +-- fun : Id a0 a1 × Id b0 b1 → Id {A × B} ( a0 , b0 ) ( a1 , b1 ) +-- fun (rfl , rfl) = rfl - inv : Id {A × B} ( a0 , b0 ) ( a1 , b1 ) → Id a0 a1 × Id b0 b1 - inv rfl = rfl , rfl +-- inv : Id {A × B} ( a0 , b0 ) ( a1 , b1 ) → Id a0 a1 × Id b0 b1 +-- inv rfl = rfl , rfl - rightInv : section fun inv - rightInv rfl = refl +-- rightInv : section fun inv +-- rightInv rfl = refl - leftInv : retract fun inv - leftInv (rfl , rfl) = refl +-- leftInv : retract fun inv +-- leftInv (rfl , rfl) = refl ------------Cong------------------------- private variable A B : Type - -Cong : {x y : A} (f : A → B) → Id x y → Id (f x) (f y) -Cong f rfl = rfl + w x y z : A ------------Groupoid Laws---------------- -rfl* : {x y : A} (p : Id x y) → Id (rfl * p) p +rfl* : (p : Id x y) → Id (rfl * p) p rfl* p = rfl -*rfl : {x y : A} (p : Id x y) → Id (p * rfl) p +*rfl : (p : Id x y) → Id (p * rfl) p *rfl rfl = rfl -*Sym : {A : Type} {x y : A} (p : Id x y) → Id (p * Sym p) rfl +*Sym : (p : Id x y) → Id (p * Sym p) rfl *Sym rfl = rfl -Sym* : {A : Type} {x y : A} (p : Id x y) → Id rfl (p * Sym p) +Sym* : (p : Id x y) → Id rfl (p * Sym p) Sym* rfl = rfl -Assoc : {A : Type} {w x y z : A} (p : Id w x) (q : Id x y) (r : Id y z) +Assoc : (p : Id w x) (q : Id x y) (r : Id y z) → Id ((p * q) * r) (p * (q * r)) Assoc rfl q r = rfl + + +-------------Mapping Out---------------- + +thing = JRefl + +outOfId : (M : (y : A) → Id x y → Type) → M x rfl + → {y : A} (p : Id x y) → M y p +outOfId M h rfl = h + +------------Path vs Id-------------------- + +Path→Id : x ≡ y → Id x y +Path→Id {A} {x} = J (λ y p → Id x y) rfl + +Id→Path : Id x y → x ≡ y +Id→Path rfl = refl + +-----------Basics about paths-------------- + +Cong : (f : A → B) → Id x y → Id (f x) (f y) +Cong f rfl = rfl + +cong : (f : A → B) (p : x ≡ y) → f x ≡ f y +cong {x = x} f = J (λ y p → f x ≡ f y) refl + +cong' : (f : A → B) (p : x ≡ y) → f x ≡ f y +cong' f p = Id→Path (Cong f (Path→Id p)) + +------------Path vs Id--------------------- + +Path→IdRefl : Path→Id (refl {x = x}) ≡ rfl +Path→IdRefl {x = x} = JRefl (λ y p → Id x y) rfl + +Path≡Id : (x ≡ y) ≡ (Id x y) +Path≡Id {A} {x} {y} = isoToPath (iso Path→Id Id→Path rightInv leftInv) where + + rightInv : section (Path→Id {A} {x} {y}) Id→Path + rightInv rfl = Path→IdRefl + + leftInv : retract (Path→Id {A} {x} {y}) Id→Path + leftInv = J (λ y p → Id→Path (Path→Id p) ≡ p) (cong (λ p → Id→Path p) Path→IdRefl) + + + -- leftInv : retract (Path→Id {A} {x} {y}) Id→Path + -- leftInv = J (λ y p → Id→Path (Path→Id p) ≡ p) + -- ( + -- Id→Path (Path→Id refl) + -- ≡⟨ cong (λ p → Id→Path p) Path→IdRefl ⟩ + -- Id→Path rfl + -- ≡⟨ refl ⟩ + -- refl ∎ + -- ) + +-----------Basics about Path Space----------------- + +sym : {x y : A} → x ≡ y → y ≡ x +sym {A} {x} = J (λ y1 p → y1 ≡ x) refl + +symRefl : {x : A} → sym {A} {x} {x} (refl) ≡ refl +symRefl {A} {x} = JRefl (λ y1 p → y1 ≡ x) refl + +Trans : {x y z : A} → x ≡ y → y ≡ z → x ≡ z +Trans {x = x} {z = z} = J (λ y1 p → y1 ≡ z → x ≡ z) λ q → q + +_∙_ = Trans -- input \. + +_≡⟨_⟩_ : (x : A) → x ≡ y → y ≡ z → x ≡ z +_ ≡⟨ x≡y ⟩ y≡z = x≡y ∙ y≡z + +_∎ : (x : A) → x ≡ x +_ ∎ = refl + +TransRefl : {x y : A} → Trans {A} {x} {x} {y} refl ≡ λ q → q +TransRefl {x = x} {y = y} = JRefl ((λ y1 p → y1 ≡ y → x ≡ y)) λ q → q + +refl∙refl : {x : A} → refl {_} {A} {x} ∙ refl ≡ refl +refl∙refl = cong (λ f → f refl) TransRefl + +∙refl : {x y : A} (p : x ≡ y) → Trans p refl ≡ p +∙refl {A} {x} {y} = J (λ y p → Trans 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 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)) ∎ diff --git a/_build/2.6.2/agda/0Trinitarianism/Quest4Solutions.agdai b/_build/2.6.2/agda/0Trinitarianism/Quest4Solutions.agdai index 747f402409ff245de064c5c4f7903a4e015f108a..1f93c938d2c4f3e0e8c3545863aa2ce43d4488ac 100644 GIT binary patch literal 86737 zcmXuL3tUY3|3Cgd=gh6!%yiNHau7mSk(BDhQYjksS=x1}v?wIEic+0Ya!HM1wVP^E zRR48}9l54*NntgZUAB-*R)$0>OsfCuJ-)x4$73gF&b-d+@_apCuh;v$&;0M&`Xyb* z{!09`vr=z5pE-%(KV6;Yx;A?S_)o;4IE`AeSDT;kpS>RZI@Id2d6msrp38svW{IYU zHs1}}Z?E_2*NJ+gj^@vsRrcfm{xLps(zyDEkUd5*WwVaX8Rsx*iOh+ zcAE29U8EqieJPHF6-opi4AR5@6p0xTezy@f%*DsuNRG3G0%iTgqW=(Sip(S2NI`@* zZ5kn@*(`QvgyVk%8KxU8`0oh3llYx4c(zdU0YyI2Rg4Z}v%)xnh|SFG8nl-n)@elx zj3c5Lw1hCv;D&)PGuq-x!n?B1X??#wMB00i)w7vf3i(}Varw1LOpQY=Wn@hW#@Be! zh&?{iZ+I%@+UHgueaRz2eEhntOw2_3G1+9p0#^7Tb;~1;c-`J&B5NY!oW2a{Ax|=6 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