From ef75329b745707e13aed536d9207d936862b5ee6 Mon Sep 17 00:00:00 2001 From: Jlh18 Date: Tue, 19 Oct 2021 16:58:41 +0100 Subject: [PATCH] groupid laws --- 0Trinitarianism/Quest4Solutions.agda | 67 +++++++++++++++++- .../0Trinitarianism/Quest4Solutions.agdai | Bin 20112 -> 53491 bytes 2 files changed, 65 insertions(+), 2 deletions(-) diff --git a/0Trinitarianism/Quest4Solutions.agda b/0Trinitarianism/Quest4Solutions.agda index 2298650..e880107 100644 --- a/0Trinitarianism/Quest4Solutions.agda +++ b/0Trinitarianism/Quest4Solutions.agda @@ -1,7 +1,70 @@ module 0Trinitarianism.Quest4Solutions where open import Cubical.Foundations.Prelude +open import Cubical.Foundations.Isomorphism renaming (Iso to _≅_) -data _≣_ {A : Type} : (x y : A) → Type where +data Id {A : Type} : (x y : A) → Type where - rfl : (x : A) → x ≣ x + rfl : {x : A} → Id x x + +idSym : (A : Type) (x y : A) → Id x y → Id y x +idSym A x .x rfl = rfl + +Sym : {A : Type} {x y : A} → Id x y → Id y x +Sym rfl = rfl + +_*_ : {A : Type} {x y z : A} → Id x y → Id y z → Id x z +rfl * q = q + +_*0_ : {A : Type} {x y z : A} → Id x y → Id y z → Id x z +p *0 rfl = p + +_*1_ : {A : Type} {x y z : A} → Id x y → Id y z → Id x z +rfl *1 rfl = rfl + +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 + + 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 + + rightInv : section fun inv + rightInv 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 + +------------Groupoid Laws---------------- + +rfl* : {x y : A} (p : Id x y) → Id (rfl * p) p +rfl* p = rfl + +*rfl : {x y : A} (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 rfl = rfl + +Sym* : {A : Type} {x y : A} (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) + → Id ((p * q) * r) (p * (q * r)) +Assoc rfl q r = rfl diff --git a/_build/2.6.2/agda/0Trinitarianism/Quest4Solutions.agdai b/_build/2.6.2/agda/0Trinitarianism/Quest4Solutions.agdai index 1f731ce9750dcd4c80c51aee76e8b2952446002c..747f402409ff245de064c5c4f7903a4e015f108a 100644 GIT binary patch literal 53491 zcmXuL3p`ZY`#-++-g9%EaVaCWU8X3NK{p9oDnv|tKII(66cwk7azYU{U8KaMPU&

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