From 79db381ac197f8a3e6073e3d0c31d8b037a8aa22 Mon Sep 17 00:00:00 2001 From: Jlh18 Date: Wed, 15 Sep 2021 11:54:35 +0100 Subject: [PATCH] Edited 1FundamentalGroup/Quest0.agda --- 1FundamentalGroup/Quest0.agda | 202 +++++++++++++++++- .../2.6.2/agda/1FundamentalGroup/Quest0.agdai | Bin 34177 -> 59559 bytes 2 files changed, 195 insertions(+), 7 deletions(-) diff --git a/1FundamentalGroup/Quest0.agda b/1FundamentalGroup/Quest0.agda index 6c9b8db..2fdf1a9 100644 --- a/1FundamentalGroup/Quest0.agda +++ b/1FundamentalGroup/Quest0.agda @@ -1,21 +1,209 @@ module 1FundamentalGroup.Quest0 where -open import Cubical.Core.Everything open import Cubical.Data.Empty +open import Cubical.Data.Unit renaming ( Unit to ⊤ ) +open import Cubical.Data.Bool open import Cubical.Foundations.Prelude +open import Cubical.Foundations.Isomorphism open import Cubical.Foundations.Path -open import Cubical.Foundations.GroupoidLaws + +private + variable + u : Level data S¹ : Type where base : S¹ loop : base ≡ base -¬ : Type → Type +-- if you don't know how to input a character +-- go to evil-mode, put your cursor on the character +-- and do `SPC h '` + +¬ : Type u → Type u ¬ A = A → ⊥ -doubleCover : type -doubleCover = ? +_≢_ : {A : Type u} → (x y : A) → Type u +x ≢ y = ¬ (x ≡ y) + +_≅_ = Iso + +{- Bool + +data Bool : Type where + true : Bool + false : Bool + +The above definition for the Booleans +can be interpreted as + +- a construction with only two recipes + `true` and `false` +- a space with two points `true` and `false`. + This space is discrete in the sense that + we haven't specified any paths. + +Our goal is to show + + refl ≢ loop (input \nequiv) + +that there is path (aka homotopy) from `refl` to `loop`. +To do so we must assume there is such a path and derive +a contradiction. +The contradiction we will try to reach is that `true ≡ false`. +Indeed it does not hold: + +-} -refl≢loop : ¬ ( refl ≡ loop ) -refl≢loop h = {!!} +{- transport + +To follow a point in `a : A` along a path `p : A ≡ B` +we use + + transport : {A B : Type u} → A ≡ B → A → B + +Why do we propify? Discuss. + +-} + +true≢false' : true ≢ false +true≢false' h = transport ⊤≡⊥ tt where + + propify : Bool → Type + propify false = ⊥ + propify true = ⊤ + + ⊤≡⊥ : ⊤ ≡ ⊥ + ⊤≡⊥ = cong propify h + + + +Flip : Bool → Bool +Flip false = true +Flip true = false + +{- Iso + +We show that Flip is an isomorphism from Bool → Bool +with inverse Flip. + +A proof of `A ≅ B` (input \cong or write Iso A B) is given by + + iso f i s r + +where + + f : A → B and i : B → A + +are the map and its inverse, +here both `f` and `i` are Flip + +`s` is a proof that `f` is a section with +right inverse `i` and +`r` is a proof that `f` is a retraction +with left inverse `i` + +-} + +flipIso : Bool ≅ Bool +flipIso = iso Flip Flip s r where + s : section Flip Flip + s false = refl + s true = refl + + r : retract Flip Flip + r false = refl + r true = refl + +{- Path ≡ + +A corollary of univalence is +`isoToPath` which takes an isomorphism +`f : A ≅ B` and gives a path +`fPath : A ≡ B`. +The resulting path has the important property +that when you follow (transport/subst) +a point in `A` along the path +you will get the point `f(a)` in `B` + +-} + +flipPath : Bool ≡ Bool +flipPath = isoToPath flipIso + +{- + +Try out `transport` on `true : Bool` and +`flipPath` by doing `C-c C-n` +and typing in `transport flipPath true` + +-} + +{- bundle over S¹ + +We want to construct a bundle over S¹ +that looks like this: +-- insert image of double cover + +to do so we use flipPath +to specify the fibers of the bundle +at each point on the `loop`. +These fibers must coincide at the end-points +with the fiber we set for `base`, which is `Bool`. + +-} + +-- the bundle +doubleCover : S¹ → Type +doubleCover base = Bool +doubleCover (loop i) = flipPath i + +{- subst + +Given a bundle `B : A → Type u` +over a space `A` and a path `p : x ≡ y` +between points in `x y : A`, + + subst : (B : A → Type u) (p : x ≡ y) → B x → B y + +follows the path _over_ `p`, taking one +end point of the path in fiber `B x` to +the other end point in fiber `B y`. + +We use `subst` to get a function that takes a path `p : base ≡ base` +and follows the point `true` in the fiber `doubleCover base` +along the path over `p` to some point in `doubleCover base`. + +Note that `doubleCover base` is just `Bool` (externally). + +-} + +SubstTrue : (p : base ≡ base) → doubleCover base +SubstTrue p = subst doubleCover p true + +{- + +You can check that `SubstTrue refl` and `SubstTrue loop` +are using `C-c C-n` + +-} + +{- cong + +Given a function `f : A → B` +and a path `p : x ≡ y` between points `x y : A` + + cong : (f : A → B) → (p : x ≡ y) → f x ≡ f y + +gives us a path in `B` from `f x` to `f y` + +We can use the above to get the contradiction we want +by + +- assuming `p : refl ≡ loop` +- deducing `SubstTrue refl ≡ SubstTrue loop` using `cong` + +-} + +refl≢loop : refl ≢ loop +refl≢loop p = true≢false (cong SubstTrue p) diff --git a/_build/2.6.2/agda/1FundamentalGroup/Quest0.agdai 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