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src/HoTTEST/Agda/Lecture-Notes/files/vector-and-list-isomorphisms.lagda.md create mode 100644 src/HoTTEST/Agda/Q-and-A/README.md create mode 100644 src/HoTTEST/Agda/Q-and-A/qa1.pdf create mode 100644 src/HoTTEST/Agda/Q-and-A/qa3.pdf create mode 100644 src/HoTTEST/Agda/Q-and-A/qa4.pdf create mode 100644 src/HoTTEST/Agda/Q-and-A/qa5.pdf create mode 100644 src/HoTTEST/Agda/Q-and-A/qa6.pdf create mode 100644 src/HoTTEST/Agda/Q-and-A/qa7.pdf create mode 100644 src/HoTTEST/Agda/Q-and-A/qa9.pdf create mode 100644 src/HoTTEST/Agda/README.md create mode 100644 src/HoTTEST/Agda/agda-lectures.agda-lib create mode 100644 src/HoTTEST/Agda/support/web/template.html create mode 100644 src/MayConcise/Chapter1.pdf create mode 100644 src/MayConcise/Chapter1.typ diff --git a/src/CubicalHott/Chapter2.lagda.md b/src/CubicalHott/Chapter2.lagda.md index 1ecb81b..1fca2d5 100644 --- a/src/CubicalHott/Chapter2.lagda.md +++ b/src/CubicalHott/Chapter2.lagda.md @@ -186,6 +186,34 @@ module lemma2∙4∙12 where -- in g ∘ f , qinv-to-isequiv (mkQinv (f-inv ∘ g-inv) forward backward) ``` +### Lemma 2.9.2 + +``` +happly : {A B : Type} + → {f g : A → B} + → (p : f ≡ g) + → (x : A) + → f x ≡ g x +happly {A} {B} {f} {g} p x = ap (λ h → h x) p + +happlyd : {A : Type} {B : A → Type} + → {f g : (a : A) → B a} + → (p : f ≡ g) + → (x : A) + → f x ≡ g x +happlyd {A} {B} {f} {g} p x = ap (λ h → h x) p +``` + +### Theorem 2.9.3 (Function extensionality) + +``` +funext : {l : Level} {A B : Type l} + → {f g : A → B} + → ((x : A) → f x ≡ g x) + → f ≡ g +funext h i x = h x i +``` + ## 2.10 Universes and the univalence axiom ### Lemma 2.10.1 @@ -222,6 +250,31 @@ module axiom2∙10∙3 where open axiom2∙10∙3 ``` +### Theorem 2.11.2 + +``` +module lemma2∙11∙2 where + open ≡-Reasoning + + i : {l : Level} {A : Type l} {a x1 x2 : A} + → (p : x1 ≡ x2) + → (q : a ≡ x1) + → transport (λ y → a ≡ y) p q ≡ q ∙ p + i {l} {A} {a} {x1} {x2} p q j = {! !} + + ii : {l : Level} {A : Type l} {a x1 x2 : A} + → (p : x1 ≡ x2) + → (q : x1 ≡ a) + → transport (λ y → y ≡ a) p q ≡ sym p ∙ q + ii {l} {A} {a} {x1} {x2} p q = {! !} + + iii : {l : Level} {A : Type l} {a x1 x2 : A} + → (p : x1 ≡ x2) + → (q : x1 ≡ x1) + → transport (λ y → y ≡ y) p q ≡ sym p ∙ q ∙ p + iii {l} {A} {a} {x1} {x2} p q = {! !} +``` + ### Remark 2.12.6 ``` diff --git a/src/CubicalHott/Chapter6.lagda.md b/src/CubicalHott/Chapter6.lagda.md index c62650a..3025b3b 100644 --- a/src/CubicalHott/Chapter6.lagda.md +++ b/src/CubicalHott/Chapter6.lagda.md @@ -32,4 +32,22 @@ lemma6∙4∙1 p = p ≡⟨ {! !} ⟩ p ≡⟨ {! !} ⟩ refl ∎ +``` + +### Lemma 6.4.2 + +``` +lemma6∙4∙2 : Σ ((x : S¹) → x ≡ x) (λ y → y ≢ (λ x → refl)) +lemma6∙4∙2 = f , g + where + f : (x : S¹) → x ≡ x + f base = loop + f (loop i) i₁ = loop {! !} + + g : f ≢ (λ x → refl) + g p = let + z = happlyd p + z2 = z base + z3 = lemma6∙4∙1 z2 + in z3 ``` \ No newline at end of file diff --git a/src/HoTTEST/Agda/Agda-installation.md b/src/HoTTEST/Agda/Agda-installation.md new file mode 100644 index 0000000..4ed3b9e --- /dev/null +++ b/src/HoTTEST/Agda/Agda-installation.md @@ -0,0 +1,78 @@ +# Instructions on how to get Agda 2.6.2.2 + +## Online + +Thanks to Amélia Liao, there's an [online version](https://hcomp.io/) available, +which means you won't have to install anything. But if you would like to +install Agda locally, please see below. + +## Mac + +If you're a Mac user who already uses homebrew, the process is painless: just +run `brew install` agda and get ready to wait a while. + +If you're a Mac user who doesn't already use homebrew, follow the [instructions +here](https://brew.sh/) to get it working. + +## Linux + +If you're an Arch or Fedora user, then `pacman -S agda` or `yum install agda` +should work. See also [theses Arch instructions](https://pastebin.com/jj2c2dqR). + +If you're Debian/Ubuntu, then you should install Agda through `cabal`, because +the Agda in the repositories of the operating system is outdated. See [these +instructions](https://agda.readthedocs.io/en/v2.6.2.2/getting-started/installation.html). + +## Windows + +For Windows, see [these instructions](https://pastebin.com/Zguv4743). (Thanks +Todd Waugh Ambridge!) + +## Editing Agda code + +Most people use emacs with agda-mode to edit, but you can also use neovim with +cornelis or vscode with the agda-mode port. + +To get emacs with agda-mode set up, there's a [tutorial +here](https://agda.readthedocs.io/en/v2.6.2.2/getting-started/installation.html#running-the-agda-mode-program) + +To get neovim with corenlis setup, there's installation instructions on [their +GitHub](https://github.com/isovector/cornelis) + +To get the vscode plugin setup, see +[here](https://marketplace.visualstudio.com/items?itemName=banacorn.agda-mode) + +## Agda in Markdown files + +If you're using emacs, then add the following at the very end of your `.emacs` +file (in your home directory) + +``` +(add-to-list 'auto-mode-alist '("\\.lagda.md\\'" . agda2-mode)) +``` + +to enable agda-mode for Markdown files with Agda code, like the lecture notes +and exercises. + +## Checking that everything works + +After installing Agda, you should check that it's installed correctly by +type-checking the [Hello, Agda! +program](https://agda.readthedocs.io/en/v2.6.2.2/getting-started/hello-world.html#hello-agda) + +## Working outside the repository + +If you would like to work on the exercises and have the lecture notes available +outside (for `open import`) the repository, then you should register the +repository as a library: + +In `~/.agda/libraries` add the line +``` +~/HoTTEST-Summer-School/Agda/agda-lectures.agda-lib +``` +and then add the line `agda-lectures` to `~/.agda/defaults`. + +## Getting help + +If you're having trouble installing Agda, please ask for help in the +`#agda-installation` channel on the summer school's Discord server. diff --git a/src/HoTTEST/Agda/Auxiliary/Hurkens.lagda.md b/src/HoTTEST/Agda/Auxiliary/Hurkens.lagda.md new file mode 100644 index 0000000..830f436 --- /dev/null +++ b/src/HoTTEST/Agda/Auxiliary/Hurkens.lagda.md @@ -0,0 +1,69 @@ +# Hurkens' paradox +(File provided by Ulrik Buchholtz) + +Hurkens' simplification of Girard's paradox. +This derives a contradiction using only Π-types +and “type in type”, a universe that contains itself. +For a description of how it works, and the +similarity to Burali-Forti's paradox, see: + +https://www.cs.cmu.edu/~kw/scans/hurkens95tlca.pdf + +```agda +{-# OPTIONS --without-K --type-in-type #-} +module Hurkens where + +Type = Set + +P : Type → Type +P X = X → Type + +PP : Type → Type +PP X = P (P X) + +⊥ : Type +⊥ = (X : Type) → X + +¬ : Type → Type +¬ X = X → ⊥ + +U : Type +U = (X : Type) → (PP X → X) → PP X + +τ : PP U → U +τ t X f p = t λ x → p (f (x X f)) + +σ : U → PP U +σ s = s U λ t → τ t + +Δ : P U +Δ y = ¬ ((p : P U) → σ y p → p (τ (σ y))) + +Ω : U +Ω = τ λ p → (x : U) → σ x p → p x + +D : Type +D = (p : P U) → σ Ω p → p (τ (σ Ω)) + +lemma : (p : P U) → ((x : U) → σ x p → p x) → p Ω +lemma p H = H Ω λ x → H (τ (σ x)) + +nd : ¬ D +nd = lemma Δ λ x H K → K Δ H λ p → K λ y → p (τ (σ y)) + +d : D +d p = lemma λ y → p (τ (σ y)) + +boom : ⊥ +boom = nd d +``` + +With this encoding of false we can of course inhabit all types, +inlucing the inductively defined empty type. + +```agda +data ∅ : Type where + +boom' : ∅ +boom' = boom ∅ +``` diff --git a/src/HoTTEST/Agda/Auxiliary/README.md b/src/HoTTEST/Agda/Auxiliary/README.md new file mode 100644 index 0000000..89358b5 --- /dev/null +++ b/src/HoTTEST/Agda/Auxiliary/README.md @@ -0,0 +1,10 @@ +# Auxiliary Agda files + +This folder contains some auxiliary files +for the Agda track of the HoTTEST Summer School +2022. +Everything here is optional material. + +1. Paradoxes derived from “type in type”: + 1. [Hurkens' paradox](Hurkens.lagda.md) + 1. [Russell's paradox](Russell.lagda.md) diff --git a/src/HoTTEST/Agda/Auxiliary/Russell.lagda.md b/src/HoTTEST/Agda/Auxiliary/Russell.lagda.md new file mode 100644 index 0000000..d1b4db8 --- /dev/null +++ b/src/HoTTEST/Agda/Auxiliary/Russell.lagda.md @@ -0,0 +1,83 @@ +# Russell's paradox +(File provided by Ulrik Buchholtz) + +Here we derive a contradiction from “type in type”, +a universe that contains itself, assuming that +this is additionally closed under generalized +inductive types. +Then we can directly encode Russell's paradox. + +This development is inspired by Aczel's +[“sets as trees” interpretation of constructive set +theory](https://doi.org/10.1016/S0049-237X(08)71989-X), +from which this very simple paradox was +[deduced by Thierry Coquand](https://doi.org/10.1007/BF01995104). +N.G. de Bruijn also noticed a similar paradox. + +```agda +{-# OPTIONS --without-K --type-in-type #-} +module Russell where + +Type = Set +``` + +We need some standard inductive types: + Σ, ∅, and identity types. + +```agda +data Σ (A : Type) (B : A → Type) : Type where + _,_ : (a : A) → B a → Σ A B + +pr₁ : {A : Type} {B : A → Type} → Σ A B → A +pr₁ (a , b) = a + +data ∅ : Type where + +data _≡_ {A : Type} (a : A) : A → Type where + refl : a ≡ a +``` + +Then we use the following generalized inductive type. +(It's called generalized because it has a recursive +argument that is a function type.) +It represents trees that can branch according to +any type in the universe. +Without “type in type”, these would be well-founded +trees, like the membership trees in the cumulative +hierarchy model of set theory. + +```agda +data V : Type where + sup : (X : Type) → (X → V) → V +``` + +We can similarly define membership and non-membership. + +```agda +_∈_ : V → V → Type +x ∈ sup X f = Σ X λ x' → x ≡ f x' + +_∉_ : V → V → Type +x ∉ y = x ∈ y → ∅ +``` + +Now it's straight-forward to define Russell's +paradoxical “set” of all sets that don't contain +themselves, and derive a contradiction. + +```agda +R : V +R = sup (Σ V λ x → x ∉ x) pr₁ + +x∈R→x∉x : (x : V) → x ∈ R → x ∉ x +x∈R→x∉x x ((.x , x∉x) , refl) = x∉x + +x∉x→x∈R : (x : V) → x ∉ x → x ∈ R +x∉x→x∈R x x∉x = (x , x∉x) , refl + +R∉R : R ∉ R +R∉R R∈R = x∈R→x∉x R R∈R R∈R + +boom : ∅ +boom = R∉R (x∉x→x∈R R R∉R) +``` diff --git a/src/HoTTEST/Agda/Cubical/Exercises7.lagda.md b/src/HoTTEST/Agda/Cubical/Exercises7.lagda.md new file mode 100644 index 0000000..6dbe8a4 --- /dev/null +++ b/src/HoTTEST/Agda/Cubical/Exercises7.lagda.md @@ -0,0 +1,224 @@ +# Week 7 - Cubical Agda Exercises + +## Standard disclaimer: + +**The exercises are designed to increase in difficulty so that we can cater to +our large and diverse audience. This also means that it is *perfectly fine* if +you don't manage to do all exercises: some of them are definitely a bit hard for +beginners and there are likely too many exercises! You *may* wish to come back +to them later when you have learned more.** + +Having said that, here we go! + +In case you haven't done the other Agda exercises: +This is a markdown file with Agda code, which means that it displays nicely on +GitHub, but at the same time you can load this file in Agda and fill the holes +to solve exercises. + +**When solving the problems, +please make a copy of this file to work in, so that it doesn't get overwritten +(in case we update the exercises through `git`)!** + + +```agda +{-# OPTIONS --cubical --allow-unsolved-metas #-} + +module Exercises7 where + +open import cubical-prelude +open import Lecture7-notes +``` + +```agda +private + variable + A : Type ℓ + B : A → Type ℓ +``` + +## Part I: Generalizing to dependent functions + +### Exercise 1 (★) + +State and prove funExt for dependent functions `f g : (x : A) → B x` + +### Exercise 2 (★) + +Generalize the type of ap to dependent function `f : (x : A) → B x` +(hint: the result should be a `PathP`) + + +## Part II: Some facts about (homotopy) propositions and sets + +The first three homotopy levels `isContr`, `isProp` and `isSet` +are defined in `cubical-prelude` in the usual way +(only using path types of course). + +### Exercise 3 (★) + +State and prove that inhabited propositions are contractible + + +### Exercise 4 (★) + +Prove + +```agda +isPropΠ : (h : (x : A) → isProp (B x)) → isProp ((x : A) → B x) +isPropΠ = {!!} +``` + +### Exercise 5 (★) + +Prove the inverse of `funExt` (sometimes called `happly`): + +```agda +funExt⁻ : {f g : (x : A) → B x} → f ≡ g → ((x : A) → f x ≡ g x) +funExt⁻ = {!!} +``` + +### Exercise 6 (★★) + +Use funExt⁻ to prove isSetΠ: + +```agda +isSetΠ : (h : (x : A) → isSet (B x)) → isSet ((x : A) → B x) +isSetΠ = {!!} +``` + +### Exercise 7 (★★★): alternative contractibility of singletons + +We could have defined the type of singletons as follows + +```agda +singl' : {A : Type ℓ} (a : A) → Type ℓ +singl' {A = A} a = Σ x ꞉ A , x ≡ a +``` + +Prove the corresponding version of contractibility of singetons for +`singl'` (hint: use a suitable combinations of connections and `~_`) + +```agda +isContrSingl' : (x : A) → isContr (singl' x) +isContrSingl' x = {!!} +``` + +## Part III: Equality in Σ-types +### Exercise 8 (★★) + +Having the primitive notion of equality be heterogeneous is an +easily overlooked, but very important, strength of cubical type +theory. This allows us to work directly with equality in Σ-types +without using transport. + +Fill the following holes (some of them are easier than you might think): + +```agda +module _ {A : Type ℓ} {B : A → Type ℓ'} {x y : Σ A B} where + + ΣPathP : Σ p ꞉ pr₁ x ≡ pr₁ y , PathP (λ i → B (p i)) (pr₂ x) (pr₂ y) + → x ≡ y + ΣPathP = {!!} + + PathPΣ : x ≡ y + → Σ p ꞉ pr₁ x ≡ pr₁ y , PathP (λ i → B (p i)) (pr₂ x) (pr₂ y) + PathPΣ = {!!} + + ΣPathP-PathPΣ : ∀ p → PathPΣ (ΣPathP p) ≡ p + ΣPathP-PathPΣ = {!!} + + PathPΣ-ΣPathP : ∀ p → ΣPathP (PathPΣ p) ≡ p + PathPΣ-ΣPathP = {!!} +``` + +If one looks carefully the proof of prf in Lecture 7 uses ΣPathP +inlined, in fact this is used all over the place when working +cubically and simplifies many proofs which in Book HoTT requires long +complicated reasoning about transport. + + +## Part IV: Some HITs + +Now we want prove some identities of HITs analogous to `Torus ≡ S¹ × S¹` +Hint: you can just use `isoToPath` in all of them. + + +### Exercise 9 (★★) + +We saw two definitions of the torus: +`Torus` having a dependent path constructor `square` +and `Torus'` with a path constructor `square` that involves composition. + +Using these two ideas, define the *Klein bottle* in two different ways. + +### Exercise 10 (★★) + +Prove + +```agda +suspUnitChar : Susp Unit ≡ Interval +suspUnitChar = {!!} +``` + + +### Exercise 11 (★★★) + +Define suspension using the Pushout HIT and prove that it's equal to Susp. + + +### Exercise 12 (🌶) + +The goal of this exercise is to prove + +```agda +suspBoolChar : Susp Bool ≡ S¹ +suspBoolChar = {!!} +``` + +For the map `Susp Bool → S¹`, we have to specify the behavior of two +path constructors. The idea is to map one to `loop` and one to `refl`. + +For the other direction, we have to fix one base point in `Susp Bool` +and give a non-trivial loop on it, i.e. one that shouldn't cancel out +to `refl`. + +Proving that the two maps cancel each other requires some primitives +of `Cubical Agda` that we won't really discuss in the lectures, +namely `hcomp` and `hfill`. These are used (among other things) +to define path composition and ensure that it behaves correctly. + +Path composition corresponds to the top of the following square: + +```text + p∙q + x --------> z + ^ ^ + ¦ ¦ + refl ¦ sq ¦ q + ¦ ¦ + ¦ ¦ + x --------> y + p +``` + +The type of `sq` is a `PathP` sending `p` to `p ∙ q` +over `q` and the following lemma lets us construct such a *square*: + +```agda +comp-filler : {x y z : A} (p : x ≡ y) (q : y ≡ z) + → PathP (λ j → refl {x = x} j ≡ q j) p (p ∙ q) +comp-filler {x = x} p q j i = hfill (λ k → λ { (i = i0) → x + ; (i = i1) → q k }) + (inS (p i)) j +``` + +You can use this `comp-filler` as a black-box for proving the round-trips +in this exercise. + +**Hint:** For one of the round-trips you only need the following familiar +result, that is a direct consequence of `comp-filler` in `Cubical Agda` + +```agda +rUnit : {x y : A} (p : x ≡ y) → p ∙ refl ≡ p +rUnit p = sym (comp-filler p refl) +``` diff --git a/src/HoTTEST/Agda/Cubical/Exercises8.lagda.md b/src/HoTTEST/Agda/Cubical/Exercises8.lagda.md new file mode 100644 index 0000000..dcce9e2 --- /dev/null +++ b/src/HoTTEST/Agda/Cubical/Exercises8.lagda.md @@ -0,0 +1,126 @@ +# Week 8 - Cubical Agda Exercises + +## Standard disclaimer: + +**The exercises are designed to increase in difficulty so that we can cater to +our large and diverse audience. This also means that it is *perfectly fine* if +you don't manage to do all exercises: some of them are definitely a bit hard for +beginners and there are likely too many exercises! You *may* wish to come back +to them later when you have learned more.** + +Having said that, here we go! + +In case you haven't done the other Agda exercises: +This is a markdown file with Agda code, which means that it displays nicely on +GitHub, but at the same time you can load this file in Agda and fill the holes +to solve exercises. + +**When solving the problems, +please make a copy of this file to work in, so that it doesn't get overwritten +(in case we update the exercises through `git`)!** + + +```agda +{-# OPTIONS --cubical --allow-unsolved-metas #-} + +module Exercises8 where + +open import cubical-prelude +open import Lecture7-notes +open import Lecture8-notes +open import Solutions7 +``` + +## Part I: `transp` and `hcomp` + +### Exercise 1 (★) + +Prove the propositional computation law for `J`: + +```agda +JRefl : {A : Type ℓ} {x : A} (P : (z : A) → x ≡ z → Type ℓ'') + (d : P x refl) → J P d refl ≡ d +JRefl P d = {!!} +``` + +### Exercise 2 (★★) + +Using `transp`, construct a method for turning dependent paths into paths. + +**Hint**: +Transport the point `p i` to the fibre `A i1`, and set `φ = i` such that the +transport computes away at `i = i1`. +```text + x ----(p i)----> y + A i0 A i A i1 +``` + +```agda +fromPathP : {A : I → Type ℓ} {x : A i0} {y : A i1} → + PathP A x y → transport (λ i → A i) x ≡ y +fromPathP {A = A} p i = {!!} +``` + +### Exercise 3 (★★★) + +Using `hcomp`, cunstruct a method for turning paths into dependent paths. + +**Hint**: +At each point `i`, the verical fibre of the following diagram should lie in +`A i`. The key is to parametrise the bottom line with a dependent path from `x` +to `transport A x`. This requires writing a `transp` that computes away at +`i = i0`. + +```text + x - - - -> y + ^ ^ + ¦ ¦ + refl ¦ ¦ p i + ¦ ¦ + ¦ ¦ + x ---> transport A x +``` + + +```agda +toPathP : {A : I → Type ℓ} {x : A i0} {y : A i1} → + transport (λ i → A i) x ≡ y → PathP A x y +toPathP {A = A} {x = x} p i = + hcomp + (λ {j (i = i0) → {!!} ; + j (i = i1) → {!!} }) + {!!} +``` + +### Exercise 4 (★) + +Using `toPathP`, prove the following lemma, which lets you fill in dependent +lines in hProps, provided their boundary. + +```agda +isProp→PathP : {A : I → Type ℓ} (p : (i : I) → isProp (A i)) + (a₀ : A i0) (a₁ : A i1) → PathP A a₀ a₁ +isProp→PathP p a₀ a₁ = {!!} +``` + +### Exercise 5 (★★) + +Prove the following lemma charictarising equality in subtypes: + +```agda +Σ≡Prop : {A : Type ℓ} {B : A → Type ℓ'} {u v : Σ A B} (h : (x : A) → isProp (B x)) + → (p : pr₁ u ≡ pr₁ v) → u ≡ v +Σ≡Prop {B = B} {u = u} {v = v} h p = {!!} +``` + +### Exercise 6 (★★★) + +Prove that `isContr` is a proposition: + +**Hint**: +This requires drawing a cube (yes, an actual 3D one)! + +```agda +isPropIsContr : {A : Type ℓ} → isProp (isContr A) +isPropIsContr (c0 , h0) (c1 , h1) j = {!!} +``` diff --git a/src/HoTTEST/Agda/Cubical/Exercises9.lagda.md b/src/HoTTEST/Agda/Cubical/Exercises9.lagda.md new file mode 100644 index 0000000..17696c7 --- /dev/null +++ b/src/HoTTEST/Agda/Cubical/Exercises9.lagda.md @@ -0,0 +1,205 @@ +# Week 9 - Cubical Agda Exercises + +## Standard disclaimer: + +**The exercises are designed to increase in difficulty so that we can cater to +our large and diverse audience. This also means that it is *perfectly fine* if +you don't manage to do all exercises: some of them are definitely a bit hard for +beginners and there are likely too many exercises! You *may* wish to come back +to them later when you have learned more.** + +Having said that, here we go! + +In case you haven't done the other Agda exercises: +This is a markdown file with Agda code, which means that it displays nicely on +GitHub, but at the same time you can load this file in Agda and fill the holes +to solve exercises. + +**When solving the problems, +please make a copy of this file to work in, so that it doesn't get overwritten +(in case we update the exercises through `git`)!** + + +```agda +{-# OPTIONS --cubical --allow-unsolved-metas #-} + +module Exercises9 where + +open import cubical-prelude +open import Lecture7-notes +open import Lecture8-notes +open import Lecture9-notes +open import Solutions7 hiding (rUnit) +open import Solutions8 +open import Lecture9-live using (SemiGroupℕ) +``` + +## Part I: More hcomps + +### Exercise 1 (★★) +### (a) +Show the left cancellation law for path composition using an hcomp. +Hint: one hcomp should suffice. Use `comp-filler` and connections + +```agda +lUnit : {A : Type ℓ} {x y : A} (p : x ≡ y) → refl ∙ p ≡ p +lUnit = {!!} + +``` +### (b) +Try to mimic the construction of lUnit for rUnit (i.e. redefine it) +in such a way that `rUnit refl ≡ lUnit refl` holds by `refl`. +Hint: use (almost) the exact same hcomp. + +```agda +rUnit : {A : Type ℓ} {x y : A} (p : x ≡ y) → p ∙ refl ≡ p +rUnit = {!!} + +-- uncomment to see if it type-checks + +{- +rUnit≡lUnit : ∀ {ℓ} {A : Type ℓ} {x : A} → rUnit (refl {x = x}) ≡ lUnit refl +rUnit≡lUnit = refl +-} + +``` + + +### Exercise 2 (★★) +Show the associativity law for path composition +Hint: one hcomp should suffice. This one can be done without connections + (but you might need comp-filler in more than one place) + +```agda +assoc : {A : Type ℓ} {x y z w : A} (p : x ≡ y) (q : y ≡ z) (r : z ≡ w) + → p ∙ (q ∙ r) ≡ (p ∙ q) ∙ r +assoc = {!!} + +``` + +### Exercise 3 (Master class in connections) (🌶) +The goal of this exercise is to give a cubical proof of the Eckmann-Hilton argument, +which says that path composition for higher loops is commutative + +(a) While we cannot get `p ∙ q ≡ q ∙ p` as a one-liner, we can get a +one-liner showing that the identiy holds up to some annoying +coherences. Try to understand the following statement (and why it's +well-typed). After that, fill the holes + +Hint: each hole will need a `∨` or a `∧` + +```agda +pre-EH : {A : Type ℓ} {x : A} (p q : refl {x = x} ≡ refl) + → ap (λ x → x ∙ refl) p ∙ ap (λ x → refl ∙ x) q + ≡ ap (λ x → refl ∙ x) q ∙ ap (λ x → x ∙ refl) p +pre-EH {x = x} p q i = (λ j → p {!!} ∙ q {!!}) + ∙ (λ j → p {!!} ∙ q {!!}) + +``` +(b) If we manage to cancel out all of the annoying aps, we get Eckmann-Hilton: +For paths (p q : refl ≡ refl), we have p ∙ q ≡ q ∙ p. Try to prove this, using the above lemma. + +Hint: Use the pre-EH as the bottom of an hcomp (one should be enough). +For the sides, use lUnit and rUnit wherever they're needed. Note that this will only work out smoothly if +you've solved Exercise 1 (b). + +```agda +Eckmann-Hilton : {A : Type ℓ} {x : A} (p q : refl {x = x} ≡ refl) → p ∙ q ≡ q ∙ p +Eckmann-Hilton = {!!} + +``` +# Part 2: Binary numbers as a HIT +Here is another HIT describing binary numbers. The idea is that a binary number is a list of booleans, modulo trailing zeros. + +For instance, `true ∷ true ∷ true ∷ []` is the binary number 110 ... +... and so is `true ∷ true ∷ false ∷ false ∷ false ∷ []` + +(!) Note that we're interpreting 110 as 1·2⁰ + 1·2¹ + 0·2² here. + +```agda +0B = false +1B = true + +data ListBin : Type where + [] : ListBin + _∷_ : (x : Bool) (xs : ListBin) → ListBin + drop0 : 0B ∷ [] ≡ [] + +-- 1 as a binary number +1LB : ListBin +1LB = 1B ∷ [] +``` +### Exercise 4 (★) +Define the successor function on ListBin +```agda + +sucListBin : ListBin → ListBin +sucListBin = {!!} + +``` +### Exercise 5 (★★) +Define an addition `+LB` on ListBin and prove that `x +LB [] ≡ x` +Do this by mutual induction! Make sure the three cases for the right unit law hold by refl. +```agda + +_+LB_ : ListBin → ListBin → ListBin +rUnit+LB : (x : ListBin) → x +LB [] ≡ x +x +LB y = {!!} +rUnit+LB = {!!} + +``` +(c) Prove that sucListBin is left distributive over `+LB` +Hint: If you pattern match deep enough, there should be a lot of refls... +```agda + +sucListBinDistrL : (x y : ListBin) → sucListBin (x +LB y) ≡ (sucListBin x +LB y) +sucListBinDistrL = {!!} +``` + +### Exercise 6 (★) +Define a map `LB→ℕ : ListBin → ℕ` and show that it preserves addition + +```agda +ℕ→ListBin : ℕ → ListBin +ℕ→ListBin = {!!} + +ℕ→ListBin-pres+ : (x y : ℕ) → ℕ→ListBin (x + y) ≡ (ℕ→ListBin x +LB ℕ→ListBin y) +ℕ→ListBin-pres+ = {!!} + +``` + +### Exercise 7 (★★★) +Show that `ℕ ≃ ListBin`. + +```agda + +ListBin→ℕ : ListBin → ℕ +ListBin→ℕ = {!!} + +ListBin→ℕ→ListBin : (x : ListBin) → ℕ→ListBin (ListBin→ℕ x) ≡ x +ListBin→ℕ→ListBin = {!!} + +ℕ→ListBin→ℕ : (x : ℕ) → ListBin→ℕ (ℕ→ListBin x) ≡ x +ℕ→ListBin→ℕ = {!!} + +ℕ≃ListBin : ℕ ≃ ListBin +ℕ≃ListBin = {!!} + +``` +# Part 3: The SIP +### Exericise 8 (★★) +Show that, using an SIP inspired argument, if `(A , _+A_)` is a semigroup and `(B , _+B_)` is some other type with a composition satisfying: + +(i) `e : A ≃ B` + +(ii) `((x y : A) → e (x +A y) ≡ e x +B e y` + +then `(B , _+B_)` defines a semigroup. + +Conclude that `(ListBin , _+LB_)` is a semigroup + +For inspiration, see Lecture9-notes +```agda + +SemiGroupListBin : SemiGroup ListBin +SemiGroupListBin = {!!} diff --git a/src/HoTTEST/Agda/Cubical/Lecture7-live.agda b/src/HoTTEST/Agda/Cubical/Lecture7-live.agda new file mode 100644 index 0000000..2b9eca5 --- /dev/null +++ b/src/HoTTEST/Agda/Cubical/Lecture7-live.agda @@ -0,0 +1,196 @@ +-- Lecture 7: Cubical Agda + +{-# OPTIONS --cubical #-} + +module Lecture7-live where + +-- Today: +-- - Path types and the cubical interval +-- - Cubical higher inductive types + +open import cubical-prelude + +-- Libraries: +-- - agda/cubical library +-- - 1lab + +-- The interval in Cubical Agda is written I +-- Endpoints are called i0 and i1 + +apply0 : (A : Type) → (p : I → A) → A +apply0 A p = p i0 + +-- Path types: +-- p : x ≡ y consists of p : I → A s.t. p i0 ≐ x and p i1 ≐ y + +refl : {A : Type} {x : A} → x ≡ x +refl {x = x} = λ i → x + +-- Cannot pattern-match on r: +-- oops : {A : Type} → I → A +-- oops r = {!r!} + +variable + A B : Type ℓ -- ℓ is \ell + +-- ap is called cong in the agda/cubical library +ap : (f : A → B) {x y : A} → x ≡ y → f x ≡ f y +ap f p i = f (p i) + +-- Satisfies definitional/judgmental/strict computation rules (see exercises) + +-- Funext can be seen as turning +-- A → (I → B) +-- into +-- I → (A → B) + +funExt : {f g : A → B} → (p : (x : A) → f x ≡ g x) → f ≡ g +funExt p = λ i → λ x → p x i +-- To convince oneself that this makes sense: plug in i0 and i1 for i + +-- In Cubical Agda we also have three operations on I + +-- Minimum: _∧_ : I → I → I +-- Maximum: _∨_ : I → I → I +-- Symmetry: ~_ : I → I + +-- ! or ⁻¹ +sym : {x y : A} → x ≡ y → y ≡ x +sym p i = p (~ i) + +-- For examples of _∧_ and _∨_ see the notes for now + +-- Path overs in Book HoTT are called PathP in Cubical Agda +-- In fact, x ≡ y is short for PathP (λ i → A) x y + +reflP : {x : A} → PathP (λ i → A) x x +reflP {x = x} = λ i → x + + + +-- Cubical higher inductive types + +-- HITs: S1, Circle2, Torus, ... + +-- Dan introduced these using postulates + +-- In Cubical Agda you can just write them as data types, +-- in particular you don't have postulate the recursor/eliminator +-- and can instead use pattern matching + +data S¹ : Type where + base : S¹ + loop : base ≡ base -- I → S¹ with endpoints base and base + +double : S¹ → S¹ +double base = base +double (loop i) = (loop ∙ loop) i + +helix : S¹ → Type +helix base = ℤ +helix (loop i) = sucPath i + +winding : base ≡ base → ℤ +winding p = transp (λ i → helix (p i)) i0 (pos 0) + +_ : winding (loop ∙ loop) ≡ pos 2 +_ = refl + +-- Following Dan's lecture one can prove that winding is an equivalence +-- using the encode-decode, see Cubical.HITs.S1.Base + + +-- Torus + + -- line1 + -- p ----------> p + -- ^ ^ + -- ¦ ¦ + -- line2 ¦ square ¦ line2 + -- ¦ ¦ + -- p ----------> p + -- line1 + +data Torus : Type where + point : Torus + line1 : point ≡ point + line2 : point ≡ point + square : PathP (λ i → line1 i ≡ line1 i) line2 line2 + +-- Exercise: define the Klein bottle + +t2c : Torus → S¹ × S¹ +t2c point = base , base +t2c (line1 i) = (loop i) , base +t2c (line2 i) = base , (loop i) +t2c (square i j) = (loop i) , (loop j) + +c2t : S¹ × S¹ → Torus +c2t (base , base) = point +c2t (base , loop i) = line2 i +c2t (loop i , base) = line1 i +c2t (loop i , loop j) = square i j + +c2t-t2c : (t : Torus) → c2t (t2c t) ≡ t +c2t-t2c point = refl +c2t-t2c (line1 i) = refl +c2t-t2c (line2 i) = refl +c2t-t2c (square i j) = refl + +t2c-c2t : (t : S¹ × S¹) → t2c (c2t t) ≡ t +t2c-c2t (base , base) = refl +t2c-c2t (base , loop i) = refl +t2c-c2t (loop i , base) = refl +t2c-c2t (loop i , loop j) = refl + +Torus≡S¹×S¹ : Torus ≡ S¹ × S¹ +Torus≡S¹×S¹ = isoToPath (iso t2c c2t t2c-c2t c2t-t2c) + + +-- Alternative definition: +data Torus' : Type where + point : Torus' + line1 : point ≡ point + line2 : point ≡ point + square : line1 ∙ line2 ≡ line2 ∙ line1 + +-- (Probably very hard) Exercise: prove that Torus ≃ Torus' + + +-- More HITs: + +-- Suspension + +data Susp (A : Type) : Type where + north : Susp A + south : Susp A + merid : (a : A) → north ≡ south + +flip : Susp S¹ → Susp S¹ +flip north = south +flip south = north +flip (merid a i) = merid a (~ i) + +-- Pushout + +-- f +-- A ---> B +-- | | +-- g| | +-- V V +-- C ---> Pushout + + +data Pushout {A B C : Type} (f : A → B) (g : A → C) : Type where + inl : B → Pushout f g + inr : C → Pushout f g + push : (a : A) → inl (f a) ≡ inr (g a) + + +-- Today: +-- - Interval and path types +-- - Cubical HITs + +-- Next time (Friday) +-- - Cubical transport and path induction +-- - Set truncated HITs and how to work with them diff --git a/src/HoTTEST/Agda/Cubical/Lecture7-notes.lagda.md b/src/HoTTEST/Agda/Cubical/Lecture7-notes.lagda.md new file mode 100644 index 0000000..64b299b --- /dev/null +++ b/src/HoTTEST/Agda/Cubical/Lecture7-notes.lagda.md @@ -0,0 +1,529 @@ +# Lecture 7: Cubical Agda + +Contents: + +- The interval and Path/PathP types +- Cubical higher inductive types + +## Some introductory pointers for further reading + +Documentation of the Cubical Agda mode can be found +[here](https://agda.readthedocs.io/en/v2.6.2.2/language/cubical.html). + +These lectures about Cubical Agda will be inspired by my material from +the [2020 EPIT school on HoTT](https://github.com/HoTT/EPIT-2020/tree/main/04-cubical-type-theory). + +For students interested in a more in depth introduction to cubical +type theory see [my lecture notes for the 2019 HoTT school](https://www.cambridge.org/core/journals/mathematical-structures-in-computer-science/article/cubical-methods-in-homotopy-type-theory-and-univalent-foundations/ECB3FE6B4A0B19AED2D3A2D785C38AF9). +Those notes contain a lot more background and motivation to cubical +methods in HoTT and an extensive list of references for those that +want to read more background material. + +If one wants a library to work with when doing Cubical Agda there is +the [agda/cubical](https://github.com/agda/cubical/) library that I +started developing with Andrea Vezzosi (the implementor of Cubical +Agda) in 2018 and which has now over 70 contributors. It contains a +variety of things, including data structures, algebra, synthetic +homotopy and cohomology theory, etc.. + +For a slower-paced introduction to mathematics in Cubical Agda, there +is the more recent [1lab](https://1lab.dev/). Although meant to be +read on the web, it also doubles as a community-maintained library of +formalized mathematics, most notably category theory. + +# Cubical Agda + +To make Agda cubical simply add the following option: + +```agda +{-# OPTIONS --cubical #-} + +module Lecture7-notes where +``` + +We also have a small cubical prelude which sets things up to work +nicely and which provides whatever we might need for the lectures. + +```agda +open import cubical-prelude +``` + +The key idea in cubical type theories like Cubical Agda is to not have +equality be inductively defined as in Book HoTT, but rather we assume +that there is a primitive interval and define equality literally as +paths, i.e. as functions out of the interval. By iterating these paths +we get squares, cubes, hypercubes, ..., making the type theory +inherently cubical. + +# The interval and path types + +The interval is a primitive concept in Cubical Agda. It's written `I`. +It has two endpoints: + +```text + i0 : I + i1 : I +``` + +These stand for "interval 0" and "interval 1". + +We can apply a function out of the interval to an endpoint just +like we would with any Agda function: + +```agda +apply0 : (A : Type ℓ) (p : I → A) → A +apply0 A p = p i0 +``` + +The equality type `_≡_` is not inductively defined in Cubical Agda, +instead it's a builtin primitive notion defined using the interval. An +element of `x ≡ y` consists of a function `p : I → A` such that `p i0` +is definitionally `x` and `p i1` is definitionally `y`. The check that +the endpoints are correct when we provide a `p : I → A` is +automatically performed by Agda during typechecking, so introducing an +element of `x ≡ y` is done just like how we introduce elements of +`I → A` but Agda will check the side conditions. + +We can hence write paths using λ-abstraction: + +```agda +mypath : {A : Type ℓ} (x : A) → x ≡ x +mypath x = λ i → x +``` + +As explained above Agda checks that whatever we written as definition +matches the path that we have provided (so the endpoints have to be +correct). In this case everything is fine and mypath can be thought of +as a proof reflexivity. Let's give it a nicer name and more implicit +arguments: + +```agda +refl : {A : Type ℓ} {x : A} → x ≡ x +refl {x = x} = λ i → x +``` + +The notation `{x = x}` lets us access the implicit argument `x` (the +`x` in the LHS of `x = x`) and rename it to `x` (the `x` in the RHS +`x = x`) in the body of `refl`. We could just as well have written: + +```text +refl : {A : Type ℓ} {x : A} → x ≡ x +refl {x = y} = λ i → y +``` + +Note that we cannot pattern-match on interval variables as `I` is not +inductively defined. Try uncommenting and typing `C-c C-c` in the hole: + +```text +oops : {A : Type} → I → A +oops r = {!r!} +``` + +It quickly gets tiring to write `{A : Type ℓ}` everywhere, so let's +assume that we have some types (in fact, we've already assumed that +`ℓ` is a `Level` in the cubical-prelude): + +```agda +private + variable + A B : Type ℓ +``` + +This will make `A` and `B` elements of different universes (all +arguments is maximally generalized) and all definitions that use them +will have them as implicit arguments. + +We can now implement some basic operations on `_≡_`. Let's start with +`ap`: + +```agda +ap : (f : A → B) {x y : A} → x ≡ y → f x ≡ f y +ap f p i = f (p i) +``` + +Note that the definition differs from the Book HoTT definition in that +it is not defined by path induction or pattern-matching on `p`, but +rather it's just a direct definition as a composition of functions. +Agda treats `p : x ≡ y` like any function, so we can apply it to `i` +to get an element of `A` which at `i0` is `x` and at `i1` is `y`. By +applying `f` to this element we hence get an element of `B` which at +`i0` is `f x` and at `i1` is `f y`. + +As this is just function composition it satisfies lots of nice +definitional equalities, see the exercises. Some of these are not +satisfied by the Book HoTT definition of `ap`. + +In Book HoTT function extensionality is proved as a consequence of +univalence using a rather ingenious proof due to Voevodsky, but in +cubical systems it has a much more direct proof. As paths are just +functions we can get it by swapping the arguments to `p`: + +```agda +funExt : {f g : A → B} (p : (x : A) → f x ≡ g x) → f ≡ g +funExt p i x = p x i +``` + +To see that this has the correct type, note that when `i` is `i0` we +have `p x i0 = f x` and when `i` is `i1` we have `p x i1 = g x`, so by +η for function types we have a path `f ≡ g` as desired. + +The interval has additional operations: + +```text +Minimum: _∧_ : I → I → I (corresponds to min(i,j)) +Maximum: _∨_ : I → I → I (corresponds to max(i,j)) +Symmetry: ~_ : I → I (corresponds to 1 - i) +``` + +These satisfy the equations of a De Morgan algebra (i.e. a +distributive lattice (_∧_ , _∨_ , i0 , i1) with an "De Morgan" +involution ~). This just means that we have the following kinds of +equations definitionally: + +```text +i0 ∨ i ≐ i +i ∨ i1 ≐ i1 +i ∨ j ≐ j ∨ i +i0 ∧ i ≐ i0 +i1 ∧ i ≐ i +i ∧ j ≐ j ∧ i +~ (~ i) ≐ i +i0 ≐ ~ i1 +~ (i ∨ j) ≐ ~ i ∧ ~ j +~ (i ∧ j) ≐ ~ i ∨ ~ j +``` + +However, we do not have `i ∨ ~ i = i1` and `i ∧ ~ i = i0`. The reason +is that I represents an abstract interval, so we if we think of it as +the real interval [0,1] ⊂ ℝ we clearly don't always have +"max(i,1-i) = 1" or "min(i,1-i) = 0)" for all i ∈ [0,1]. + +These operations on `I` are very useful as they let us define even +more things directly. For example symmetry of paths is easily defined +using `~_`. + +```agda +sym : {x y : A} → x ≡ y → y ≡ x +sym p i = p (~ i) +``` + +Remark: this has been called `⁻¹` and `!` in the previous +lectures. Here we stick to `sym` for the cubical version following the +agda/cubical notation. + +The operations `_∧_` and `_∨_` are called *connections* and let us +build higher dimensional cubes from lower dimensional ones, for +example if we have a path `p : x ≡ y` then + +```text + sq i j = p (i ∧ j) +``` + +is a square (as we've parametrized by `i` and `j`) with the following +boundary: + +```text + sq i0 j = p (i0 ∧ j) = p i0 = x + sq i1 j = p (i1 ∧ j) = p j + sq i i0 = p (i ∧ i0) = p i0 = x + sq i i1 = p (i ∧ i1) = p i +``` + +If we draw this we get: + +```text + p + x --------> y + ^ ^ + ¦ ¦ + refl ¦ sq ¦ p + ¦ ¦ + ¦ ¦ + x --------> x + refl +``` + +Being able to make this square directly is very useful. It for +example let's prove that singletons are contractible (a.k.a. based +path induction). + +We define the type of singletons as follows + +```agda +singl : {A : Type ℓ} (a : A) → Type ℓ +singl {A = A} a = Σ x ꞉ A , a ≡ x +``` + +To show that this type is contractible we need to provide a center +of contraction and the fact that any element of it is path-equal to +the center + +```agda +isContrSingl : (x : A) → isContr (singl x) +isContrSingl x = ctr , prf + where + -- The center is just a pair with x and refl + ctr : singl x + ctr = x , refl + + -- We then need to prove that ctr is equal to any element s : singl x. + -- This is an equality in a Σ-type, so the first component is a path + -- and the second is a path over the path we pick as first argument, + -- i.e. the second component is a square. In fact, we need a square + -- relating refl and pax over a path between refl and pax, so we can + -- use an _∧_ connection. + prf : (s : singl x) → ctr ≡ s + prf (y , pax) i = (pax i) , λ j → pax (i ∧ j) +``` + +As we saw in the second component of prf we often need squares when +proving things. In fact, `pax (i ∧ j)` is a path relating `refl` to +`pax` *over* another path `λ j → x ≡ pax j`. This notion of path over +a path is very useful when working in Book HoTT as we've seen in the +previous lectures, this is also the case when working cubically. In +Cubical Agda path-overs are a primitive notion called `PathP` ("Path +over a Path"). In general `PathP A x y` has + +```text + A : I → Type ℓ + x : A i0 + y : A i1 +``` + +So PathP lets us natively define heteorgeneous paths, i.e. paths +where the endpoints are in different types. This allows us to +specify the type of the second component of `prf`: + +```agda +prf' : (x : A) (s : singl x) → (x , refl) ≡ s +prf' x (y , pax) i = (pax i) , λ j → goal i j + where + goal : PathP (λ j → x ≡ pax j) refl pax + goal i j = pax (i ∧ j) +``` + +Just like `_×_` is a special case of Σ-types we have that `_≡_` is a +special case of PathP. In fact, `x ≡ y` is just short for +`PathP (λ _ → A) x y`: + +```agda +reflP : {x : A} → PathP (λ _ → A) x x +reflP = refl +``` + +Working directly with paths and equalities makes many proofs from Book +HoTT very short: + +```agda +isContrΠ : {B : A → Type ℓ} (h : (x : A) → isContr (B x)) + → isContr ((x : A) → B x) +isContrΠ h = (λ x → pr₁ (h x)) , (λ f i x → pr₂ (h x) (f x) i) +``` + +# Cubical higher inductive types + +We have seen various HITs earlier in the course. These were added +axiomatically to Agda by postulating their existence together with +suitable elimination/induction principles. In Cubical Agda they are +instead added just like any inductive data type, but with path +constructors. This is made possible by the fact that paths in Cubical +Agda are just fancy functions. + +## The circle + +We can define the circle as the following simple data declaration: + +```agda +data S¹ : Type₀ where + base : S¹ + loop : base ≡ base +``` + +We can write functions on `S¹` using pattern-matching: + +```agda +double : S¹ → S¹ +double base = base +double (loop i) = (loop ∙ loop) i +``` + +Note that loop takes an `i : I` argument. This is not very surprising +as it's a path of type `base ≡ base`, but it's an important difference +to Book HoTT where we instead would have to state the equation using +`ap`. Having the native notion of equality be heterogeneous makes it +possible to quite directly define a general schema for a large class +of HITs and use it in the implementation of a system like Cubical Agda. + +Let's use univalence to compute some winding numbers on the +circle. We first define a family of types over the circle with +fibers being the integers. + +```agda +helix : S¹ → Type₀ +helix base = ℤ +helix (loop i) = sucPath i +``` + +Here univalence is baked into `sucPath : ℤ ≡ ℤ`. The loopspace of the +circle is then defined as + +```agda +ΩS¹ : Type₀ +ΩS¹ = base ≡ base +``` + +and we can then define a function computing how many times we've +looped around the circle by: + +```agda +winding : ΩS¹ → ℤ +winding p = transp (λ i → helix (p i)) i0 (pos 0) +``` + +Here `transp` is a cubical transport function. We'll talk about it in +more detail in the next lecture, but for now we can observe that it +reduces as expected: + +```agda +_ : winding (λ i → double ((loop ∙ loop) i)) ≡ pos 4 +_ = refl +``` + +This would not reduce definitionally in Book HoTT as univalence is an +axiom. Having things compute definitionally makes it possible to +substantially simplify many proofs from Book HoTT in Cubical Agda. + +We can in fact prove that `winding` is an equivalence, this is very +similar to the Book HoTT proof and uses the encode-decode method. For +details about how this proof looks in Cubical Agda see the +Cubical.HITs.S1.Base file in the agda/cubical library. + +## The torus + +We can define the torus as: + +```agda +data Torus : Type₀ where + point : Torus + line1 : point ≡ point + line2 : point ≡ point + square : PathP (λ i → line1 i ≡ line1 i) line2 line2 +``` + +The square corresponds to the usual folding diagram from topology +(where `p` is short for `point`): + +```text + line1 + p ----------> p + ^ ^ + ¦ ¦ + line2 ¦ ¦ line2 + ¦ ¦ + p ----------> p + line1 +``` + +Proving that it is equivalent to two circles is pretty much trivial as +we have definitional computation rules for all constructors, including +higher ones: + +```agda +t2c : Torus → S¹ × S¹ +t2c point = (base , base) +t2c (line1 i) = (loop i , base) +t2c (line2 j) = (base , loop j) +t2c (square i j) = (loop i , loop j) + +c2t : S¹ × S¹ → Torus +c2t (base , base) = point +c2t (loop i , base) = line1 i +c2t (base , loop j) = line2 j +c2t (loop i , loop j) = square i j + +c2t-t2c : (t : Torus) → c2t (t2c t) ≡ t +c2t-t2c point = refl +c2t-t2c (line1 _) = refl +c2t-t2c (line2 _) = refl +c2t-t2c (square _ _) = refl + +t2c-c2t : (p : S¹ × S¹) → t2c (c2t p) ≡ p +t2c-c2t (base , base) = refl +t2c-c2t (base , loop _) = refl +t2c-c2t (loop _ , base) = refl +t2c-c2t (loop _ , loop _) = refl +``` + +Using univalence we get the following equality: + +```agda +Torus≡S¹×S¹ : Torus ≡ S¹ × S¹ +Torus≡S¹×S¹ = isoToPath (iso t2c c2t t2c-c2t c2t-t2c) +``` + +We can also directly compute winding numbers on the torus + +```agda +windingTorus : point ≡ point → ℤ × ℤ +windingTorus l = ( winding (λ i → pr₁ (t2c (l i))) + , winding (λ i → pr₂ (t2c (l i)))) + +_ : windingTorus (line1 ∙ sym line2) ≡ (pos 1 , negsuc 0) +_ = refl +``` + +# Bonus content if there is time + +## Interval + +```agda +data Interval : Type₀ where + zero : Interval + one : Interval + seg : zero ≡ one +``` + +## Suspension + +```agda +data Susp (A : Type ℓ) : Type ℓ where + north : Susp A + south : Susp A + merid : (a : A) → north ≡ south +``` + +We can define Dan's Circle2 as the suspension of Bool, or we can do it directly as: + +```agda +data Circle2 : Type₀ where + north : Circle2 + south : Circle2 + west : north ≡ south + east : north ≡ south +``` + +## Pushouts + +```agda +data Pushout {ℓ ℓ' ℓ''} {A : Type ℓ} {B : Type ℓ'} {C : Type ℓ''} + (f : A → B) (g : A → C) : Type (ℓ ⊔ ℓ' ⊔ ℓ'') where + inl : B → Pushout f g + inr : C → Pushout f g + push : (a : A) → inl (f a) ≡ inr (g a) +``` + +## Relation quotient + +```agda +data _/ₜ_ {ℓ ℓ'} (A : Type ℓ) (R : A → A → Type ℓ') : Type (ℓ ⊔ ℓ') where + [_] : (a : A) → A /ₜ R + eq/ : (a b : A) → (r : R a b) → [ a ] ≡ [ b ] +``` + +## Propositional truncation + +```agda +data ∥_∥₋₁ {ℓ} (A : Type ℓ) : Type ℓ where + ∣_∣₋₁ : A → ∥ A ∥₋₁ + squash₁ : (x y : ∥ A ∥₋₁) → x ≡ y +``` diff --git a/src/HoTTEST/Agda/Cubical/Lecture8-live.agda b/src/HoTTEST/Agda/Cubical/Lecture8-live.agda new file mode 100644 index 0000000..afc2822 --- /dev/null +++ b/src/HoTTEST/Agda/Cubical/Lecture8-live.agda @@ -0,0 +1,232 @@ +-- Lecture 8: Cubical Agda + +{-# OPTIONS --cubical #-} + +module Lecture8-live where + +{- + +Last time: +- Path types and the cubical interval (I with endpoints i0, i1. _≡_ is now a PathP) +- Cubical higher inductive types + +Today: +- Set quotients +- Cubical transport (`transp`) and path induction +- Homogeneous composition (`hcomp`) + +-} + +open import cubical-prelude +open import Lecture7-notes + +private + variable + A B : Type ℓ + + +-- Set quotients + +-- A Type +-- R : A → A → Type (hProp) +-- Want a quotient type A / R + +data _/_ (A : Type) (R : A → A → Type) : Type where + [_] : A → A / R + eq/ : (a b : A) → R a b → [ a ] ≡ [ b ] + trunc : isSet (A / R) -- (a b : A / R) (p q : a ≡ b) → p ≡ q + + + +-- (a , b) ~ (c , d) iff a + d ≡ c + b +ℤ' : Type +ℤ' = (ℕ × ℕ) / rel + where + rel : ℕ × ℕ → ℕ × ℕ → Type -- Press C-c C-s and get the type automagically + rel (a , b) (c , d) = a + d ≡ b + c + +-- Exercise: prove ℤ ≃ ℤ' +-- Exercise: define 0, 1, +, -, * and prove standard properties + +-- Can also do ℤ / nℤ + +-- Similarly we can define ℚ as a quotient pairs of a number with a nonzero number + +-- And so on... + +-- All of these can be defined without quotients in type theory, but +-- the quotient definition is sometimes more efficient. + +-- ℚ can be defined not as a quotient by taking pairs of coprime numbers (i.e. as normalized fractions) + +-- Problem: need to normalize constantly + +-- With the quotient definition we don't have to normalize unless we want to + +-- Practical benefit with quotients: can get more efficient because you don't have to normalize + + + +-- Q: What happens if A is a set and R is prop valued, do you need trunc? + +-- A = Unit +-- R is the total + +-- Unit / total ≃ S¹ (if we don't have trunc constructor) + + +-- Q: What happens if A is not a set, but R is not prop valued? + +-- It will be a set by def. + +-- Nothing too bad... But A / R won't have all the properties you want. For example: + +-- effectivity : (a b : A) → [ a ] ≡ [ b ] → R a b + +-- (Also need R equivalence relation, and proof of effectivity uses univalence for hProp (prop. ext.)) + + + + + + + +-- Another example: finite multisets + +infixr 5 _∷_ + +data FMSet (A : Type) : Type where + [] : FMSet A + _∷_ : (x : A) → (xs : FMSet A) → FMSet A + comm : (x y : A) (xs : FMSet A) → x ∷ y ∷ xs ≡ y ∷ x ∷ xs + trunc : isSet (FMSet A) + + +-- Didn't use _/_ because it gets a bit longer and annoying +-- Also nice to have a simple path constructor comm to pattern-match on + +infixr 30 _++_ + +_++_ : (xs ys : FMSet A) → FMSet A +[] ++ ys = ys +(x ∷ xs) ++ ys = x ∷ (xs ++ ys) +comm x y xs i ++ ys = comm x y (xs ++ ys) i +trunc xs zs p q i j ++ ys = + trunc (xs ++ ys) (zs ++ ys) (λ k → p k ++ ys) (λ k → q k ++ ys) i j + +-- unitr-++ : (xs : FMSet A) → xs ++ [] ≡ xs +-- unitr-++ [] = refl +-- unitr-++ (x ∷ xs) = ap (x ∷_) (unitr-++ xs) +-- unitr-++ (comm x y xs i) j = comm x y (unitr-++ xs j) i +-- unitr-++ (trunc xs xs₁ x y i i₁) = {!!} -- Ugh... + +-- unitr-++ : (xs : FMSet A) → xs ++ [] ≡ xs +-- unitr-++ = ElimProp.f (trunc _ _) refl (λ x p → ap (x ∷_) p) + +-- Can do set quotients this way, but need good lemmas and combinators! + + +-- In the notes there's a more efficient representation of FMSet, but don't have time to cover it now... + + +-- Set quotients: +-- Either use _/_ -- get good lemmas, but maybe not the constructors you want +-- Or write your own -- need to prove good lemmas (but they are boilerplate), but get good constructors to pattern-match on + + + + +-- Cubical transport and path induction + +-- Last time we saw that we can prove many things for path types using +-- the operations on the interval I (like ~_ , _∧_ , _∨_) + +-- But we cannot yet compose paths or transport along paths or prove things by path induction... + + +-- Basic/primitive operation in Cubical Agda for transport is called `transp` + +-- Special case is cubical transport: + +transport : A ≡ B → A → B +transport p a = transp (λ i → p i) i0 a + +-- In general transp has this (slightly simplified type): + +-- transp : (L : I → Type) → (r : I) → L i0 → L i1 + +-- The r lets us specify where the transp is the identity function +-- +-- transp L i1 a ≐ a +-- +-- For this to make sense L has to be "constant" whenever r = i1 + + +-- What we cubists call subst, Book HoTT calls transport + +subst : (B : A → Type) {x y : A} → x ≡ y → B x → B y +subst B p bx = transport (ap B p) bx + + +-- Because we are *not* defining things by path-induction transport along refl +-- isn't the identity function + +transportRefl : (x : A) → transport refl x ≡ x +transportRefl {A = A} x i = transp (λ _ → A) i x + + +-- Path-induction is not a primitive concept in Cubical Agda, rather +-- we have to prove it + +-- General fact: subst + contractibility of singletons ⇒ path induction + +-- Book HoTT version: transport + singleton contractibility ⇒ based path induction ⇒ path induction + +-- Path-induction has traditionally been called J in the type theory community + +J : {x : A} + (P : (y : A) → x ≡ y → Type) + (d : P x refl) + {y : A} + (p : x ≡ y) + → + P y p +J {A = A} {x = x} P d p = subst (λ X → P (pr₁ X) (pr₂ X)) (lem .pr₂ (_ , p)) d + where + lem : isContr (Σ y ꞉ A , x ≡ y) + lem = isContrSingl x + + +-- A little bit annoying to use because it doesn't satisfy the computation rule at refl + +-- Also can lead to very big terms when Agda unfolds things (bad for readability and efficiency) + +-- Often better to avoid J and stick to cubical primitives directly + + + + +-- Homogenous composition (`hcomp`) + +-- Special case is path composition + +compPath : {x y z : A} → x ≡ y → y ≡ z → x ≡ z +compPath {x = x} p q i = hcomp (λ j → λ { (i = i0) → x + ; (i = i1) → q j }) + (p i) + +{- + + hcomp + x - - - - -> z + ^ ^ + ¦ ¦ + x ¦ ¦ q j + ¦ ¦ + x ----------> y + p i + + +-} + + diff --git a/src/HoTTEST/Agda/Cubical/Lecture8-notes.lagda.md b/src/HoTTEST/Agda/Cubical/Lecture8-notes.lagda.md new file mode 100644 index 0000000..91f99fe --- /dev/null +++ b/src/HoTTEST/Agda/Cubical/Lecture8-notes.lagda.md @@ -0,0 +1,381 @@ +# Lecture 8 + +Contents: + +- Set quotients +- Cubical transport and path induction +- Homogeneous composition (`hcomp`) + +```agda +{-# OPTIONS --cubical #-} + +module Lecture8-notes where + +open import cubical-prelude +open import Lecture7-notes + +private + variable + A B : Type ℓ +``` + +## Set quotients + +Last time we saw some HITs that are useful for topology and homotopy +theory. Now we'll look at some HITs that are not very interesting from +a homotopical perspective, but still very useful for other purposes. +In particular we will look at how we can construct quotient types and +in order for the result to be a set we will also set truncate the +type. This kind of types has many applications in both computer +science and mathematics. + +In general these are written as: + +```text +data _/_ (A : Type ℓ) (R : A → A → Type ℓ') : Type (ℓ ⊔ ℓ') where + [_] : A → A / R + eq/ : (a b : A) → R a b → [ a ] ≡ [ b ] + trunc : (a b : A / R) (p q : a ≡ b) → p ≡ q +``` + +The type of the `trunc` constructor can simply be written as: + +```agda +data _/_ (A : Type ℓ) (R : A → A → Type ℓ') : Type (ℓ ⊔ ℓ') where + [_] : A → A / R + eq/ : (a b : A) → R a b → [ a ] ≡ [ b ] + trunc : isSet (A / R) +``` + +The idea is that `[_]` injects elements of `A` into the quotient while +`eq/` ensures that elements that are related by `R` are sent to equal +elements. Finally we have to truncate the type to be a set in order +for this to be a set quotient. If we leave out the `trunc` constructor +quite weird things can happen, for example if one quotients the unit +type by the total relation then one obtains a type which is equivalent +to the circle! The `trunc` constructor is hence crucial to ensure that +the result is a set **even** if `A` is one already (so quotienting can +raise the truncation level). + +Various sources require that `R` is proposition-valued, i.e. `R : A → +A → hProp`. However this is not necessary to define `_/_` as seen +above. There are however various important properties that are only +satisfied when one quotients by a prop-valued relation. The key +example being effectivity, i.e. that + +```text +(a b : A) → [ a ] ≡ [ b ] → R a b +``` + +To prove this one first of all needs that `R` is prop-valued and the +proof then uses univalence for propositions (i.e., logically +equivalent propositions are equal, a.k.a. "propositional +extensionality"). + +Having the ability to define set quotients using `_/_` lets us do many +examples from mathematics and computer science. For example we can +define the integers as: + +```agda +ℤ' : Type +ℤ' = (ℕ × ℕ) / rel + where + rel : ℕ × ℕ → ℕ × ℕ → Type + rel (x₀ , y₀) (x₁ , y₁) = x₀ + y₁ ≡ x₁ + y₀ +``` + +If you haven't seen this construction before see +https://en.wikipedia.org/wiki/Integer#Construction + +Exercises: define `0`, `1`, `+`, `-`, `*` + +Similarly we can define the rational numbers as a quotient of pairs of +a number and a nonzero number (and more generally we can define the +field of fractions of a commutative ring R using `_/_`). Both the +integers and rationals can of course be defined without using +quotients in type theory, but in the case of the rationals this has +some efficiency problems. Indeed, when defining the rationals not as a +quotient we need to do it as normalized fractions, that is, pairs of +coprime numbers. All operations, like addition and multiplication, +then have to ensure that the resulting fractions are normalized which +can become quite inefficient because of repeated GCD computations. A +more efficient way is to work with the equivalence classes of +not-necessarily-normalized fractions and then normalize whenever one +wants to. This can be done when defining the rationals using `_/_`. + +Let us now look at a more computer science inspired example: finite +multisets. These can be represented as lists modulo permutations. +Defining these using `_/_` is a bit annoying, luckily there is an +easier way: + +```agda +infixr 5 _∷_ + +data FMSet (A : Type ℓ) : Type ℓ where + [] : FMSet A + _∷_ : (x : A) → (xs : FMSet A) → FMSet A + comm : (x y : A) (xs : FMSet A) → x ∷ y ∷ xs ≡ y ∷ x ∷ xs + trunc : (xs ys : FMSet A) (p q : xs ≡ ys) → p ≡ q +``` + +Programming and proving with this is quite straightforward: + +```agda +infixr 30 _++_ + +_++_ : (xs ys : FMSet A) → FMSet A +[] ++ ys = ys +(x ∷ xs) ++ ys = x ∷ xs ++ ys +comm x y xs i ++ ys = comm x y (xs ++ ys) i +trunc xs zs p q i j ++ ys = + trunc (xs ++ ys) (zs ++ ys) (λ k → p k ++ ys) (λ k → q k ++ ys) i j + +unitl-++ : (xs : FMSet A) → [] ++ xs ≡ xs +unitl-++ xs = refl + +unitr-++ : (xs : FMSet A) → xs ++ [] ≡ xs +unitr-++ [] = refl +unitr-++ (x ∷ xs) = ap (x ∷_) (unitr-++ xs) +unitr-++ (comm x y xs i) j = comm x y (unitr-++ xs j) i +unitr-++ (trunc xs ys p q i j) k = + trunc (unitr-++ xs k) (unitr-++ ys k) + (λ l → unitr-++ (p l) k) (λ l → unitr-++ (q l) k) i j +``` + +Filling the goals for `comm` and `trunc` quickly gets tiresome and +when working with HITs like this it's very strongly recommended to +prove special lemmas for eliminating into propositions (which is the +case above as `FMSet` is a set, so its `≡`-type is a proposition). If +we do this the proof of `unitr-++` can be simplified to a one-liner: + +```text +unitr-++ : (xs : FMSet A) → xs ++ [] ≡ xs +unitr-++ = ElimProp.f (trunc _ _) refl (λ x p → cong (_∷_ x) p) +``` + +We recommend the interested reader to look at the code in +Cubical.HITs.FiniteMultiset.Base and +Cubical.HITs.FiniteMultiset.Properties in agda/cubical to see how +these lemmas are stated and proved. This is a very common pattern when +working with set truncated HITs: first define the HIT, then prove +special purpose recursors and eliminators for eliminating into types +of different truncation levels. All definitions are then written using +these recursors and eliminators and one get very short proofs. + +A more efficient version of finite multisets based on association +lists can be found in Cubical.HITs.AssocList.Base. It looks like this: + +```agda +data AssocList (A : Type ℓ) : Type ℓ where + ⟨⟩ : AssocList A + ⟨_,_⟩∷_ : (a : A) (n : ℕ) (xs : AssocList A) → AssocList A + per : (a b : A) (m n : ℕ) (xs : AssocList A) + → ⟨ a , m ⟩∷ ⟨ b , n ⟩∷ xs ≡ ⟨ b , n ⟩∷ ⟨ a , m ⟩∷ xs + agg : (a : A) (m n : ℕ) (xs : AssocList A) + → ⟨ a , m ⟩∷ ⟨ a , n ⟩∷ xs ≡ ⟨ a , m + n ⟩∷ xs + del : (a : A) (xs : AssocList A) → ⟨ a , 0 ⟩∷ xs ≡ xs + trunc : (xs ys : AssocList A) (p q : xs ≡ ys) → p ≡ q +``` + +Programming and proving is more complicated with `AssocList` compared +to `FMSet`. This kind of example occurs a lot in programming and +mathematics: one representation is easier to work with, but not +efficient, while another is efficient but difficult to work with. +Next time we will see how we can use univalence and the structure +identity principle (SIP) to get the best of both worlds (if someone +can't wait they can look at https://dl.acm.org/doi/10.1145/3434293 for +more details). + +It's sometimes easier to work directly with `_/_` instead of defining +special HITs as one can reuse lemmas for `_/_` instead of reproving +things. For example, general lemmas about eliminating into +propositions have already been proved for `_/_` so we do not have to +reprove them for our special purpose HIT. However on the other hand it +can sometimes be much easier to write the HIT directly (imagine +implementing `AssocList` using `_/_`...) and one also gets the benefit +that one can pattern-match directly with nice names for the +constructors (it might still be better to use the handcrafted +recursor/eliminator for the HIT to avoid having to give cases for path +construtors though). + +## Cubical transport and path induction + +While path types are great for reasoning about equality they don't let +us transport along paths between types or even compose paths. +Furthermore, as paths are not inductively defined we don't +automatically get an induction principle for them. In order to remedy +this Cubical Agda also has a built-in cubical transport operation and +homogeneous composition operation from which the induction principle +is derivable (and much more!). + +The basic operation is called `transp` and we will soon explain it, +but let's first focus on the special case of cubical transport which +we used to define `winding` last time: + +```agda +transport : A ≡ B → A → B +transport p a = transp (λ i → p i) i0 a +``` + +This is a more primitive operation than "transport" in Book HoTT as it +only lets us turn a path into a function. However, the transport of +HoTT can easily be proved from cubical transport and in order to avoid +a name clash we call it "subst": + +```agda +subst : (B : A → Type ℓ') {x y : A} (p : x ≡ y) → B x → B y +subst B p pa = transport (λ i → B (p i)) pa +``` + +The `transp` operation is a generalized transport in the sense that it +lets us specify where the transport is the identity function (this is +why there is an `i0` in the definition of `transport` above). The +general type of `transp` is: + +```text +transp : (A : I → Type ℓ) (r : I) (a : A i0) → A i1 +``` + +There is an additional side condition which has to be satisfied for +Cubical Agda to typecheck `transp A r a`. This is that `A` has to be +"constant" on `r`. This means that `A` should be a constant function +whenever `r = i1` is satisfied. This side condition is vacuously true +when `r = i0`, so there is nothing to check when writing transport as +above. However, when `r` is equal to `i1` the `transp` function will +compute as the identity function. + +```text +transp A i1 a ≐ a +``` + +(Here `≐` is definitional/judgmental equality) + +Having this extra generality is useful for quite technical reasons, +for instance we can easily relate a term `a` with its transport over a +path `p`: + +```agda +transportFill : (p : A ≡ B) (a : A) → PathP (λ i → p i) a (transport p a) +transportFill p a i = transp (λ j → p (i ∧ j)) (~ i) a +``` + +Another result that follows easily from `transp` is that transporting +in a constant family is the identity function (up to a path). Note +that this is *not* proved by `refl`. This is maybe not so surprising +as `transport` is not defined by pattern-matching on `p`. + +```agda +transportRefl : (x : A) → transport refl x ≡ x +transportRefl {A = A} x i = transp (λ _ → A) i x +``` + +Having `transp` lets us prove many more useful lemmas like this. For +details see Cubical.Foundations.Transport in the agda/cubical library. + +Using contractibility of singletons we can also define the `J` +eliminator for paths (a.k.a. path induction): + +```agda +J : {x : A} (P : (y : A) → x ≡ y → Type ℓ'') + (d : P x refl) {y : A} (p : x ≡ y) → P y p +J {x = x} P d p = subst (λ X → P (pr₁ X) (pr₂ X)) (isContrSingl x .pr₂ (_ , p)) d +``` + +Unfolded version: + +```text +transport (λ i → P (p i) (λ j → p (i ∧ j))) d +``` + +So `J` is provable, but it doesn't satisfy the computation rule of +`refl` definitionally as `_≡_` is not inductively defined. See +exercises for how to prove it. Not having this hold definitionally is +almost never a problem in practice as the cubical primitives satisfy +many new definitional equalities (c.f. `ap`). + +## Homogeneous composition (`hcomp`) + +As we now have `J` we can define path concatenation and many more +things, however this is not the way to do things in Cubical Agda. One +of the key features of cubical type theory is that the `transp` +primitive reduces differently for different types formers (see CCHM +[1] or the Cubical Agda paper [2] for details). For paths it reduces +to another primitive operation called `hcomp`. This primitive is much +better suited for concatenating paths than `J` as it is much more +general. In particular, it lets us compose multiple higher dimensional +cubes directly. We will explain it by example. + +In order to compose two paths we write: + +```agda +compPath : {x y z : A} → x ≡ y → y ≡ z → x ≡ z +compPath {x = x} p q i = hcomp (λ j → λ { (i = i0) → x + ; (i = i1) → q j }) + (p i) +``` + +This is best understood with the following drawing: + +```text + x z + ^ ^ + ¦ ¦ + x ¦ ¦ q j + ¦ ¦ + x ----------> y + p i +``` + +In the drawing the direction `i` goes left-to-right and `j` goes +bottom-to-top. As we are constructing a path from `x` to `z` along `i` +we have `i : I` in the context already and we put `p i` as bottom. The +direction `j` that we are doing the composition in is abstracted in +the first argument to `hcomp`. + +Having `hcomp` as a primitive operation lets us prove many things very +directly. For instance, we can prove that any proposition is also a +set using a higher dimensional `hcomp`. + +```agda +isProp→isSet : isProp A → isSet A +isProp→isSet h a b p q j i = + hcomp (λ k → λ { (i = i0) → h a a k + ; (i = i1) → h a b k + ; (j = i0) → h a (p i) k + ; (j = i1) → h a (q i) k }) a +``` + +Geometric picture: start with a square with `a` everywhere as base, +then change its sides so that they connect `p` with `q` over `refl a` +and `refl b`. + +This has some useful consequences: + +```agda +isPropIsProp : isProp (isProp A) +isPropIsProp f g i a b = isProp→isSet f a b (f a b) (g a b) i + +isPropIsSet : isProp (isSet A) +isPropIsSet h1 h2 i x y = isPropIsProp (h1 x y) (h2 x y) i +``` + +In order to really understand what the second argument to `hcomp` is +and how to use `hfill` we will need to explain partial elements and +cubical subtypes. This is documented in the Cubical Agda documentation +(https://agda.readthedocs.io/en/v2.6.2.2/language/cubical.html#partial-elements) +and in lecture 9 these will be discussed in more detail. + + +However, beginners often doesn't have to write `hcomp` to prove things +as the library provides many basic lemmas. This is especially true +when reasoning about sets. But when reasoning about types that have +higher truncation level it's very convenient to be able to construct +squares and cubes directly and being able to use `hcomp` is quite +necessary (see exercise 12 on +[Exercises7](https://github.com/martinescardo/HoTTEST-Summer-School/blob/main/Agda/Cubical/Exercises7.lagda.md)). + +References +========== + +[1] https://arxiv.org/abs/1611.02108 +[2] https://staff.math.su.se/anders.mortberg/papers/cubicalagda2.pdf diff --git a/src/HoTTEST/Agda/Cubical/Lecture9-live.agda b/src/HoTTEST/Agda/Cubical/Lecture9-live.agda new file mode 100644 index 0000000..13fe671 --- /dev/null +++ b/src/HoTTEST/Agda/Cubical/Lecture9-live.agda @@ -0,0 +1,309 @@ +-- Lecture 9: Cubical Agda + +{-# OPTIONS --cubical #-} + +module Lecture9-live where + +open import cubical-prelude hiding (_∙_) +open import Lecture7-notes +open import Lecture8-notes hiding (compPath) + +private + variable + A B : Type ℓ + +-- Last time: +-- - Set quotients +-- - Cubical transport and path induction +-- - Homogeneous composition (`hcomp`) + + +-- Today: +-- - More about homogeneous composition (`hcomp`) +-- - Cubical univalence (Glue types) +-- - The structure identity principle (SIP) + + + + +-- More about hcomp + + +compPath : {x y z : A} → x ≡ y → y ≡ z → x ≡ z +compPath {x = x} p q i = + hcomp (λ j → λ { (i = i0) → x + ; (i = i1) → q j }) -- System of sides + (p i) -- The base of the hcomp + +{- + x z + ^ ^ + ¦ ¦ + x ¦ ¦ q j + ¦ ¦ + x ----------> y + p i +-} + +_∙∙_∙∙_ : {x y z w : A} → x ≡ y → y ≡ z → z ≡ w → x ≡ w +_∙∙_∙∙_ p q r i = + hcomp (λ j → λ { (i = i0) → p (~ j) + ; (i = i1) → r j }) + (q i) + +{- + x w + ^ ^ + ¦ ¦ + p (~ j) ¦ ¦ r j + ¦ ¦ + y ----------> z + q i +-} + +_∙_ : {x y z : A} → x ≡ y → y ≡ z → x ≡ z +_∙_ p q = refl ∙∙ p ∙∙ q + + +-- The hcomp operation is a cubical version of lifting conditions for Kan complexes + + +-- What is the type of the weird first argument to hcomp? + +-- Answer: I → .(IsOne φ) → A + +-- IsOne φ represents (φ = i1) + +-- Partial φ A the type of partial elements of A, i.e. the type of cubes in A that are defined when IsOne φ holds + +-- Question: is Partial φ A the same as (IsOne φ) → A ? +-- Answer: not exactly, Partial φ A is "more extensional" (elements equal up to permutation, duplication...) + +-- Partial (i ∨ ~ i) Bool ≐ Partial (~ i ∨ i) Bool + +partialBool : (i : I) → Partial (i ∨ ~ i) Bool +partialBool i = λ { (i = i0) → true + ; (i = i1) → false } + +partialBool' : (i j : I) → Partial (~ i ∨ i ∨ (i ∧ j)) Bool +partialBool' i j (i = i0) = true +partialBool' i j (i = i1) = true +partialBool' i j (i = i1) (j = i1) = true + + +-- partialBoolBad : (i : I) → Bool +-- partialBoolBad i0 = true +-- partialBoolBad i1 = false + + + +-- Cubical subtypes + +{- + +_[_↦_] : (A : Type ℓ) (φ : I) (u : Partial φ A) → SSet ℓ +A [ φ ↦ u ] = Sub A φ u + +inS : {A : Type ℓ} {φ : I} (u : A) → A [ φ ↦ (λ _ → u) ] + +outS : {A : Type ℓ} {φ : I} {u : Partial φ A} → A [ φ ↦ u ] → A + +They satisfy the following equalities: + +outS (inS a) ≐ a + +inS {φ = φ} (outS {φ = φ} a) ≐ a + +outS {φ = i1} {u} _ ≐ u 1=1 + +-} + + +hcomp' : {A : Type} {φ : I} (u : I → Partial φ A) (u0 : A [ φ ↦ u i0 ]) → A [ φ ↦ u i1 ] +hcomp' u u0 = inS (hcomp u (outS u0)) + + + + + +-- Cubical univalence (Glue types) + + +ua : {A B : Type} → A ≃ B → A ≡ B +ua {A = A} {B = B} e i = + Glue B λ { (i = i0) → A , e + ; (i = i1) → B , idEquiv B } + + +uaβ : (e : A ≃ B) → (a : A) → transport (ua e) a ≡ pr₁ e a +uaβ e a = transportRefl (pr₁ e a) + + +uaβℕ : (e : ℕ ≃ ℕ) → (a : ℕ) → transport (ua e) a ≡ pr₁ e a +uaβℕ e a = refl + + +-- Fact: ua + uaβ ⇒ univalence axiom + + +-- Examples + +not : Bool → Bool +not true = false +not false = true + +notPath : Bool ≡ Bool +notPath = ua (isoToEquiv (iso not not rem rem)) + where + rem : (b : Bool) → not (not b) ≡ b + rem true = refl + rem false = refl + + + +-- The structure identity principle (SIP) + + +substEquiv : (S : Type → Type) (e : A ≃ B) → S A → S B +substEquiv S e = subst S (ua e) + + +-- Easy exercise (in the notes): substEquiv induces and equivalence + +-- Example: S could be IsMonoid + + +-- This is useful for many things! + +-- One application: Can program with one type and prove using another equivalent type + +{- + +data Pos : Type where + pos1 : Pos + x0 : Pos → Pos + x1 : Pos → Pos + +-- x1 (x0 (x1 pos1)) = 1101 = 13 + +data Bin : Type where + bin0 : Bin + binPos : Pos → Bin + +-} + + +ℕ≃Bin : ℕ ≃ Bin +ℕ≃Bin = isoToEquiv (iso ℕ→Bin Bin→ℕ Bin→ℕ→Bin ℕ→Bin→ℕ) + + +SemiGroup : Type → Type +SemiGroup A = Σ _·_ ꞉ (A → A → A) , ((x y z : A) → x · (y · z) ≡ (x · y) · z) + +SemiGroupℕ : SemiGroup ℕ +SemiGroupℕ = _+_ , +-assoc + +SemiGroupBin : SemiGroup Bin +SemiGroupBin = substEquiv SemiGroup ℕ≃Bin SemiGroupℕ + +_+Bin_ : Bin → Bin → Bin +_+Bin_ = pr₁ SemiGroupBin + ++Bin-assoc : (x y z : Bin) → x +Bin (y +Bin z) ≡ (x +Bin y) +Bin z ++Bin-assoc = pr₂ SemiGroupBin + + + +mutual + _+P_ : Pos → Pos → Pos + pos1 +P y = sucPos y + x0 x +P pos1 = x1 x + x0 x +P x0 y = x0 (x +P y) + x0 x +P x1 y = x1 (x +P y) + x1 x +P pos1 = x0 (sucPos x) + x1 x +P x0 y = x1 (x +P y) + x1 x +P x1 y = x0 (x +PC y) + + _+B_ : Bin → Bin → Bin + bin0 +B y = y + x +B bin0 = x + binPos x +B binPos y = binPos (x +P y) + + -- Add with carry + _+PC_ : Pos → Pos → Pos + pos1 +PC pos1 = x1 pos1 + pos1 +PC x0 y = x0 (sucPos y) + pos1 +PC x1 y = x1 (sucPos y) + x0 x +PC pos1 = x0 (sucPos x) + x0 x +PC x0 y = x1 (x +P y) + x0 x +PC x1 y = x0 (x +PC y) + x1 x +PC pos1 = x1 (sucPos x) + x1 x +PC x0 y = x0 (x +PC y) + x1 x +PC x1 y = x1 (x +PC y) + + + + + +-- How to prove that +B is associative? (By hand = total pain) + + + + + + +-- Correctness: ++PC-suc : (x y : Pos) → x +PC y ≡ sucPos (x +P y) ++PC-suc pos1 pos1 = refl ++PC-suc pos1 (x0 y) = refl ++PC-suc pos1 (x1 y) = refl ++PC-suc (x0 x) pos1 = refl ++PC-suc (x0 x) (x0 y) = refl ++PC-suc (x0 x) (x1 y) = ap x0 (+PC-suc x y) ++PC-suc (x1 x) pos1 = refl ++PC-suc (x1 x) (x0 y) = ap x0 (+PC-suc x y) ++PC-suc (x1 x) (x1 y) = refl + +sucPos-+P : (x y : Pos) → sucPos (x +P y) ≡ sucPos x +P y +sucPos-+P pos1 pos1 = refl +sucPos-+P pos1 (x0 y) = refl +sucPos-+P pos1 (x1 y) = refl +sucPos-+P (x0 x) pos1 = refl +sucPos-+P (x0 x) (x0 y) = refl +sucPos-+P (x0 x) (x1 y) = ap x0 (sym (+PC-suc x y)) +sucPos-+P (x1 x) pos1 = refl +sucPos-+P (x1 x) (x0 y) = ap x0 (sucPos-+P x y) +sucPos-+P (x1 x) (x1 y) = ap x1 (+PC-suc x y ∙ sucPos-+P x y) + +ℕ→Pos-+P : (x y : ℕ) → ℕ→Pos (suc x + suc y) ≡ ℕ→Pos (suc x) +P ℕ→Pos (suc y) +ℕ→Pos-+P zero _ = refl +ℕ→Pos-+P (suc x) y = ap sucPos (ℕ→Pos-+P x y) ∙ sucPos-+P (ℕ→Pos (suc x)) (ℕ→Pos (suc y)) + +ℕ→Bin-+B : (x y : ℕ) → ℕ→Bin (x + y) ≡ ℕ→Bin x +B ℕ→Bin y +ℕ→Bin-+B zero y = refl +ℕ→Bin-+B (suc x) zero = ap (λ x → binPos (ℕ→Pos (suc x))) (+-zero x) +ℕ→Bin-+B (suc x) (suc y) = ap binPos (ℕ→Pos-+P x y) + + + + ++B≡+Bin : _+B_ ≡ _+Bin_ ++B≡+Bin i x y = goal x y i + where + goal : (x y : Bin) → x +B y ≡ ℕ→Bin (Bin→ℕ x + Bin→ℕ y) + goal x y = (λ i → Bin→ℕ→Bin x (~ i) +B Bin→ℕ→Bin y (~ i)) + ∙ sym (ℕ→Bin-+B (Bin→ℕ x) (Bin→ℕ y)) + + + + ++B-assoc : (m n o : Bin) → m +B (n +B o) ≡ (m +B n) +B o ++B-assoc m n o = + (λ i → +B≡+Bin i m (+B≡+Bin i n o)) + ∙∙ +Bin-assoc m n o + ∙∙ (λ i → +B≡+Bin (~ i) (+B≡+Bin (~ i) m n) o) + + + +-- The agda/cubical library has convenient automation for making a lot +-- of this easier diff --git a/src/HoTTEST/Agda/Cubical/Lecture9-notes.lagda.md b/src/HoTTEST/Agda/Cubical/Lecture9-notes.lagda.md new file mode 100644 index 0000000..46748be --- /dev/null +++ b/src/HoTTEST/Agda/Cubical/Lecture9-notes.lagda.md @@ -0,0 +1,608 @@ +# Lecture 9 + +Contents: + +- More about homogeneous composition (`hcomp`) +- Cubical univalence (Glue types) +- The structure identity principle (SIP) + +```agda +{-# OPTIONS --cubical #-} + +module Lecture9-notes where + +open import cubical-prelude +open import Lecture7-notes +open import Lecture8-notes hiding (compPath) + +private + variable + A B : Type ℓ +``` + +## More about homogeneous composition (`hcomp`) + +Recall from Lecture 8: in order to compose two paths we write: + +```agda +compPath : {x y z : A} → x ≡ y → y ≡ z → x ≡ z +compPath {x = x} p q i = hcomp (λ j → λ { (i = i0) → x + ; (i = i1) → q j }) + (p i) +``` + +This is best understood with the following drawing: + +```text + x z + ^ ^ + ¦ ¦ + x ¦ ¦ q j + ¦ ¦ + x ----------> y + p i +``` + +In the drawing the direction `i` goes left-to-right and `j` goes +bottom-to-top. As we are constructing a path from `x` to `z` along `i` +we have `i : I` in the context already and we put `p i` as bottom. The +direction `j` that we are doing the composition in is abstracted in +the first argument to `hcomp`. + +As we can see here the `hcomp` operation has a very natural geometric +motivation. The following YouTube video might be very helpful to +clarify this: https://www.youtube.com/watch?v=MVtlD22Y8SQ + +Side remark: this operation is related to lifting conditions for Kan +cubical sets, i.e. it's a form of open box filling analogous to horn +filling in Kan complexes. + +A more natural form of composition of paths in Cubical Agda is +ternary composition: + +```text + x w + ^ ^ + ¦ ¦ + p (~ j) ¦ ¦ r j + ¦ ¦ + y ----------> z + q i +``` + +This is written `p ∙∙ q ∙∙ r` in the agda/cubical library and +implemented by: + +```agda +_∙∙_∙∙_ : {x y z w : A} → x ≡ y → y ≡ z → z ≡ w → x ≡ w +(p ∙∙ q ∙∙ r) i = hcomp (λ j → λ { (i = i0) → p (~ j) + ; (i = i1) → r j }) + (q i) +``` + +Using this we can define `compPath` much slicker: + +```text +_∙_ : {x y z : A} → x ≡ y → y ≡ z → x ≡ z +p ∙ q = refl ∙∙ p ∙∙ q +``` + +In order to understand the second argument to `hcomp` we need to talk +about partial elements and cubical subtypes. These allow us to write +partially specified n-dimensional cubes (i.e. cubes where some faces +are missing as in the input to `hcomp`). + +### Partial elements + +Given an element of the interval `r : I` there is a predicate `IsOne` +which represents the constraint `r = i1`. This comes with a proof that +`i1` is in fact equal to `i1` called `1=1 : IsOne i1`. We use Greek +letters like `φ` or `ψ` when such an `r` should be thought of as being +in the domain of `IsOne`. + +Using this we introduce a type of partial elements called `Partial φ +A`, this is a special version of `IsOne φ → A` with a more extensional +judgmental equality (two elements of `Partial φ A` are considered +equal if they represent the same subcube, so the faces of the cubes +can for example be given in different order and the two elements will +still be considered the same). The idea is that `Partial φ A` is the +type of cubes in `A` that are only defined when `IsOne φ`. There is also a +dependent version of this called `PartialP φ A` which allows `A` to be +defined only when `IsOne φ`. + +```text +Partial : ∀ {ℓ} → I → Set ℓ → SSet ℓ + +PartialP : ∀ {ℓ} → (φ : I) → Partial φ (Set ℓ) → SSet ℓ +``` + +Here `SSet` is the universe of *strict* sets. + +Cubical Agda has a new form of pattern matching that can be used to +introduce partial elements: + +```agda +partialBool : (i : I) → Partial (i ∨ ~ i) Bool +partialBool i (i = i0) = true +partialBool i (i = i1) = false +``` + +The term `partialBool i` should be thought of a boolean with different +values when `(i = i0)` and `(i = i1)`. Note that this is different +from pattern matching on the interval which is not allowed, so we +couldn't have written: + +```text +partialBool : (i : I) → Bool +partialBool i0 = true +partialBool i1 = false +``` + +Terms of type `Partial φ A` can also be introduced using a pattern +matching lambda and this is what we have been using when we wrote +`hcomp`'s above. + +```agda +partialBool' : (i : I) → Partial (i ∨ ~ i) Bool +partialBool' i = λ { (i = i0) → true + ; (i = i1) → false } +``` + +When the cases overlap they must agree (note that the order of the +cases doesn’t have to match the interval formula exactly): + +```agda +partialBool'' : (i j : I) → Partial (~ i ∨ i ∨ (i ∧ j)) Bool +partialBool'' i j = λ { (i = i1) → true + ; (i = i1) (j = i1) → true + ; (i = i0) → false } +``` + +Furthermore `IsOne i0` is actually absurd. + +```agda +_ : {A : Type} → Partial i0 A +_ = λ { () } +``` + +With this cubical infrastructure we can now describe the type of the +`hcomp` operation. + +```text +hcomp : {A : Type ℓ} {φ : I} (u : I → Partial φ A) (u0 : A) → A +``` + +When calling `hcomp {φ = φ} u u0` Agda makes sure that `u0` agrees +with `u i0` on `φ`. The result is then an element of `A` which agrees +with `u i1` on `φ`. The idea being that `u0` is the base and `u` +specifies the sides of an open box while `hcomp u u0` is the lid of +the box. In fact, we can use yet another cubical construction to +specify these side conditions in the type of `hcomp`. For this we need +to talk about cubical subtypes. + +### Cubical subtypes and filling + +Cubical Agda also has cubical subtypes as in the CCHM type theory: + +```text +_[_↦_] : (A : Type ℓ) (φ : I) (u : Partial φ A) → SSet ℓ +A [ φ ↦ u ] = Sub A φ u +``` + +A term `v : A [ φ ↦ u ]` should be thought of as a term of type `A` +which is definitionally equal to `u : A` when `IsOne φ` is +satisfied. Any term `u : A` can be seen as a term of `A [ φ ↦ u ]` +which agrees with itself on `φ`: + +```text +inS : {A : Type ℓ} {φ : I} (u : A) → A [ φ ↦ (λ _ → u) ] +``` + +One can also forget that a partial element agrees with `u` on `φ`: + +```text +outS : {A : Type ℓ} {φ : I} {u : Partial φ A} → A [ φ ↦ u ] → A +``` + +They satisfy the following equalities: + +```text +outS (inS a) ≐ a + +inS {φ = φ} (outS {φ = φ} a) ≐ a + +outS {φ = i1} {u} _ ≐ u 1=1 +``` + +Note that given `a : A [ φ ↦ u ]` and `α : IsOne φ`, it is not the case +that `outS a ≐ u α`; however, underneath the pattern binding `(φ = i1)`, +one has `outS a ≐ u 1=1`. + +With this we can now give `hcomp` the following more specific type: + +```agda +hcomp' : {A : Type ℓ} {φ : I} (u : I → Partial φ A) (u0 : A [ φ ↦ u i0 ]) → A [ φ ↦ u i1 ] +hcomp' u u0 = inS (hcomp u (outS u0)) +``` + +This more specific type is of course more informative, but it quickly +gets quite annoying to write `inS`/`outS` everywhere. So the builtin +`hcomp` operation comes with the slightly less informative type and +the side conditions are then implicit and checked internally. + +Another very useful operation is open box *filling*. This produces an +element corresponding to the interior of an open box: + +```agda +hfill' : {A : Type ℓ} + {φ : I} + (u : ∀ i → Partial φ A) + (u0 : A [ φ ↦ u i0 ]) + (i : I) → + A [ φ ∨ ~ i ∨ i ↦ + (λ { (φ = i1) → u i 1=1 + ; (i = i0) → outS u0 + ; (i = i1) → hcomp u (outS u0) }) ] +hfill' {φ = φ} u u0 i = inS (hcomp (λ j → λ { (φ = i1) → u (i ∧ j) 1=1 + ; (i = i0) → outS u0 }) + (outS u0)) +``` + +This has a slightly less informative type in the cubical-prelude: + +```text +hfill : {A : Type ℓ} + {φ : I} + (u : ∀ i → Partial φ A) + (u0 : A [ φ ↦ u i0 ]) + (i : I) → + A +hfill {φ = φ} u u0 i = + hcomp (λ j → λ { (φ = i1) → u (i ∧ j) 1=1 + ; (i = i0) → outS u0 }) + (outS u0) +``` + +Having defined this operation we can prove various groupoid laws for +`_≡_` very cubically as `_∙_` is defined using `hcomp`. + +```agda +compPathRefl : {A : Type ℓ} {x y : A} (p : x ≡ y) → p ∙ refl ≡ p +compPathRefl {x = x} {y = y} p j i = hfill (λ _ → λ { (i = i0) → x + ; (i = i1) → y }) + (inS (p i)) + (~ j) +``` + +For more examples see Cubical.Foundations.GroupoidLaws in agda/cubical. + + +## Cubical univalence (Glue types) + SIP + +A key concept in HoTT/UF is univalence. As we have seen earlier in the +course this is assumed as an axiom in Book HoTT. In Cubical Agda it is +instead provable and has computational content. This means that +transporting with paths constructed using univalence reduces as +opposed to Book HoTT where they would be stuck. This simplifies many +proofs and make it possible to actually do concrete computations using +univalence. + +The part of univalence which is most useful for our applications is to +be able to turn equivalences (written `_≃_` and defined as a Σ-type of +a function and a proof that it has contractible fibers) into paths: + +```agda +ua : {A B : Type ℓ} → A ≃ B → A ≡ B +ua {A = A} {B = B} e i = Glue B (λ { (i = i0) → (A , e) + ; (i = i1) → (B , idEquiv B) }) +``` + +This term is defined using "Glue types" which were introduced in the +CCHM paper. We won't have time to go into too much details about them +today, but for practical applications one can usually forget about +them and use `ua` as a black box. The key idea though is that they are +similar to `hcomp`'s, but instead of attaching (higher dimensional) +paths to a base we attach equivalences to a type. One of the original +applications of Glue types was to give computational meaning to +transporting in a line of types constructed by `hcomp` in the universe +`Type`. + +For us today the important point is that transporting along the path +constructed using `ua` applies the function underlying the +equivalence. This is easily proved using `transportRefl`: + +```agda +uaβ : (e : A ≃ B) (x : A) → transport (ua e) x ≡ pr₁ e x +uaβ e x = transportRefl (equivFun e x) +``` + +Note that for concrete types this typically holds definitionally, but +for an arbitrary equivalence `e` between abstract types `A` and `B` we +have to prove it. + +```agda +uaβℕ : (e : ℕ ≃ ℕ) (x : ℕ) → transport (ua e) x ≡ pr₁ e x +uaβℕ e x = refl +``` + +The fact that we have both `ua` and `uaβ` suffices to be able to prove +the standard formulation of univalence. For details see +Cubical.Foundations.Univalence in the agda/cubical library. + +The standard way of constructing equivalences is to start with an +isomorphism and then improve it into an equivalence. The lemma in the +agda/cubical library that does this is + +```text +isoToEquiv : {A B : Type ℓ} → Iso A B → A ≃ B +``` + +Composing this with `ua` lets us directly turn isomorphisms into paths: + +```text +isoToPath : {A B : Type ℓ} → Iso A B → A ≡ B +isoToPath e = ua (isoToEquiv e) +``` + +Time for an example! + +```agda +not : Bool → Bool +not false = true +not true = false + +notPath : Bool ≡ Bool +notPath = isoToPath (iso not not rem rem) + where + rem : (b : Bool) → not (not b) ≡ b + rem false = refl + rem true = refl + +_ : transport notPath true ≡ false +_ = refl +``` + +Another example, integers: + +```agda +_ : transport sucPath (pos 0) ≡ pos 1 +_ = refl + +_ : transport (sucPath ∙ sucPath) (pos 0) ≡ pos 2 +_ = refl + +_ : transport (sym sucPath) (pos 0) ≡ negsuc 0 +_ = refl +``` + +Note that we have already used this in the `winding` function in +Lecture 7. + +One can do more fun things with `sucPath`. For example by composing it +with itself `n` times and then transporting we get a new addition +function `_+ℤ'_`. It is direct to prove `isEquiv (λ n → n +ℤ' m)` as +`_+ℤ'_` is defined by `transport`. + +```agda +addPath : ℕ → ℤ ≡ ℤ +addPath zero = refl +addPath (suc n) = addPath n ∙ sucPath + +predPath : ℤ ≡ ℤ +predPath = isoToPath (iso predℤ sucℤ predSuc sucPred) + +subPath : ℕ → ℤ ≡ ℤ +subPath zero = refl +subPath (suc n) = subPath n ∙ predPath + +_+ℤ'_ : ℤ → ℤ → ℤ +m +ℤ' pos n = transport (addPath n) m +m +ℤ' negsuc n = transport (subPath (suc n)) m + +-- This agrees with regular addition defined by pattern-matching ++ℤ'≡+ℤ : _+ℤ'_ ≡ _+ℤ_ ++ℤ'≡+ℤ i m (pos zero) = m ++ℤ'≡+ℤ i m (pos (suc n)) = sucℤ (+ℤ'≡+ℤ i m (pos n)) ++ℤ'≡+ℤ i m (negsuc zero) = predℤ m ++ℤ'≡+ℤ i m (negsuc (suc n)) = predℤ (+ℤ'≡+ℤ i m (negsuc n)) + +-- We can prove that transport is an equivalence easily +isEquivTransport : {A B : Type ℓ} (p : A ≡ B) → isEquiv (transport p) +isEquivTransport p = + transport (λ i → isEquiv (transp (λ j → p (i ∧ j)) (~ i))) (idEquiv _ .pr₂) + +-- So +ℤ' with a fixed element is an equivalence +isEquivAddℤ' : (m : ℤ) → isEquiv (λ n → n +ℤ' m) +isEquivAddℤ' (pos n) = isEquivTransport (addPath n) +isEquivAddℤ' (negsuc n) = isEquivTransport (subPath (suc n)) + +isEquivAddℤ : (m : ℤ) → isEquiv (λ n → n +ℤ m) +isEquivAddℤ = subst (λ add → (m : ℤ) → isEquiv (λ n → add n m)) +ℤ'≡+ℤ isEquivAddℤ' +``` + +### The structure identity principle + +Combining `subst` and `ua` lets us transport any structure on `A` to +get a structure on an equivalent type `B`: + +```agda +substEquiv : (S : Type → Type) (e : A ≃ B) → S A → S B +substEquiv S e = subst S (ua e) +``` + +In fact this induces an equivalence: + +```agda +substEquiv≃ : (S : Type → Type) (e : A ≃ B) → S A ≃ S B +substEquiv≃ S e = (substEquiv S e) , (isEquivTransport (ap S (ua e))) +``` + +What this says is that any structure on types must be invariant under +equivalence. We can for example take `IsMonoid` for `S` and get that +any two monoid structures on equivalent types `A` and `B` are +themselves equivalent. This is a simple version of the *structure +identity principle* (SIP). There are various more high powered and +generalized versions which also work for *structured* types +(e.g. monoids, groups, etc.) together with their corresponding notions +of *structured* equivalences (e.g. monoid and group isomorphism, +etc.). We will look more at this later, but for now let's look at a +nice example where we can use `substEquiv` to transport functions and +their properties between equivalent types. + +When programming and proving properties of programs there are usually +various tradeoffs between different data representations. Very often +one version is suitable for proofs while the other is more suitable +for efficient programming. The standard example of them is unary and +binary numbers. One way to define binary numbers in Agda is as: + +```text +data Pos : Type where + pos1 : Pos + x0 : Pos → Pos + x1 : Pos → Pos + +data Bin : Type where + bin0 : Bin + binPos : Pos → Bin +``` + +With some work one can prove that this is equivalent to unary numbers +(see the cubical-prelude for details): + +```agda +ℕ≃Bin : ℕ ≃ Bin +ℕ≃Bin = isoToEquiv (iso ℕ→Bin Bin→ℕ Bin→ℕ→Bin ℕ→Bin→ℕ) +``` + +We can now use `substEquiv` to transport addition on `ℕ` together with +the fact that it's associative over to `Bin`: + +```agda +SemiGroup : Type → Type +SemiGroup X = Σ _+_ ꞉ (X → X → X) , ((x y z : X) → x + (y + z) ≡ (x + y) + z) + +SemiGroupBin : SemiGroup Bin +SemiGroupBin = substEquiv SemiGroup ℕ≃Bin (_+_ , +-assoc) + +_+Bin_ : Bin → Bin → Bin +_+Bin_ = pr₁ SemiGroupBin + ++Bin-assoc : (m n o : Bin) → m +Bin (n +Bin o) ≡ (m +Bin n) +Bin o ++Bin-assoc = pr₂ SemiGroupBin +``` + +This is nice as it helps us avoid having to repeat work on `Bin` that +we have already done on `ℕ`. This is however not always what we want +as `_+Bin_` is not very efficient as an addition function on binary +numbers. In fact, it will translate its input to unary numbers, add +using unary addition, and then translate back. This is of course very +naive and what we instead want to do is to use efficient addition on +binary numbers, but get the associativity proof for free. + +This can be achieved by first defining our fast addition `_+B_ : Bin → +Bin → Bin` and then prove that the map `ℕ→Bin : ℕ → Bin` maps `x + y : +ℕ` to `ℕ→Bin x +B ℕ→Bin y : Bin`. + +``` +mutual + _+P_ : Pos → Pos → Pos + pos1 +P y = sucPos y + x0 x +P pos1 = x1 x + x0 x +P x0 y = x0 (x +P y) + x0 x +P x1 y = x1 (x +P y) + x1 x +P pos1 = x0 (sucPos x) + x1 x +P x0 y = x1 (x +P y) + x1 x +P x1 y = x0 (x +PC y) + + _+B_ : Bin → Bin → Bin + bin0 +B y = y + x +B bin0 = x + binPos x +B binPos y = binPos (x +P y) + + -- Add with carry + _+PC_ : Pos → Pos → Pos + pos1 +PC pos1 = x1 pos1 + pos1 +PC x0 y = x0 (sucPos y) + pos1 +PC x1 y = x1 (sucPos y) + x0 x +PC pos1 = x0 (sucPos x) + x0 x +PC x0 y = x1 (x +P y) + x0 x +PC x1 y = x0 (x +PC y) + x1 x +PC pos1 = x1 (sucPos x) + x1 x +PC x0 y = x0 (x +PC y) + x1 x +PC x1 y = x1 (x +PC y) + +-- Correctness: ++PC-suc : (x y : Pos) → x +PC y ≡ sucPos (x +P y) ++PC-suc pos1 pos1 = refl ++PC-suc pos1 (x0 y) = refl ++PC-suc pos1 (x1 y) = refl ++PC-suc (x0 x) pos1 = refl ++PC-suc (x0 x) (x0 y) = refl ++PC-suc (x0 x) (x1 y) = ap x0 (+PC-suc x y) ++PC-suc (x1 x) pos1 = refl ++PC-suc (x1 x) (x0 y) = ap x0 (+PC-suc x y) ++PC-suc (x1 x) (x1 y) = refl + +sucPos-+P : (x y : Pos) → sucPos (x +P y) ≡ sucPos x +P y +sucPos-+P pos1 pos1 = refl +sucPos-+P pos1 (x0 y) = refl +sucPos-+P pos1 (x1 y) = refl +sucPos-+P (x0 x) pos1 = refl +sucPos-+P (x0 x) (x0 y) = refl +sucPos-+P (x0 x) (x1 y) = ap x0 (sym (+PC-suc x y)) +sucPos-+P (x1 x) pos1 = refl +sucPos-+P (x1 x) (x0 y) = ap x0 (sucPos-+P x y) +sucPos-+P (x1 x) (x1 y) = ap x1 (+PC-suc x y ∙ sucPos-+P x y) + +ℕ→Pos-+P : (x y : ℕ) → ℕ→Pos (suc x + suc y) ≡ ℕ→Pos (suc x) +P ℕ→Pos (suc y) +ℕ→Pos-+P zero _ = refl +ℕ→Pos-+P (suc x) y = ap sucPos (ℕ→Pos-+P x y) ∙ sucPos-+P (ℕ→Pos (suc x)) (ℕ→Pos (suc y)) + +ℕ→Bin-+B : (x y : ℕ) → ℕ→Bin (x + y) ≡ ℕ→Bin x +B ℕ→Bin y +ℕ→Bin-+B zero y = refl +ℕ→Bin-+B (suc x) zero = ap (λ x → binPos (ℕ→Pos (suc x))) (+-zero x) +ℕ→Bin-+B (suc x) (suc y) = ap binPos (ℕ→Pos-+P x y) +``` + +Having done this it's now straightforward to prove that `_+B_` is +associative using the fact that `_+Bin_` is: + +```agda ++B≡+Bin : _+B_ ≡ _+Bin_ ++B≡+Bin i x y = goal x y i + where + goal : (x y : Bin) → x +B y ≡ ℕ→Bin (Bin→ℕ x + Bin→ℕ y) + goal x y = (λ i → Bin→ℕ→Bin x (~ i) +B Bin→ℕ→Bin y (~ i)) + ∙ sym (ℕ→Bin-+B (Bin→ℕ x) (Bin→ℕ y)) + ++B-assoc : (m n o : Bin) → m +B (n +B o) ≡ (m +B n) +B o ++B-assoc m n o = + (λ i → +B≡+Bin i m (+B≡+Bin i n o)) + ∙∙ +Bin-assoc m n o + ∙∙ (λ i → +B≡+Bin (~ i) (+B≡+Bin (~ i) m n) o) +``` + +This kind of proofs quickly get tedious and it turns out we can +streamline them using a more high prowered version of the SIP. +Indeed, what we really want to do is to characterize the equality of +T-structured types, i.e. we want a proof that two types equipped with +T-structures are equal if there is a T-structure preserving +equivalence between them. In the semigroup example above `T` would +just be the structure of a magma (i.e. a type with a binary operation) +together with magma preserving equivalence so that we can transport +for example the fact that the magma is a semigroup over to the other +magma. + +We formalize this and make it more convenient to use using a lot of +automation in the agda/cubical library. This is documented with +various more examples in: + +Internalizing Representation Independence with Univalence +Carlo Angiuli, Evan Cavallo, Anders Mörtberg, Max Zeuner +https://dl.acm.org/doi/10.1145/3434293 + +The binary numbers example with efficient addition is spelled out in +detail in Section 2.1.1 of: + +https://www.doi.org/10.1017/S0956796821000034 +(Can be downloaded from: https://staff.math.su.se/anders.mortberg/papers/cubicalagda2.pdf) diff --git a/src/HoTTEST/Agda/Cubical/Solutions7.lagda.md b/src/HoTTEST/Agda/Cubical/Solutions7.lagda.md new file mode 100644 index 0000000..3cd7517 --- /dev/null +++ b/src/HoTTEST/Agda/Cubical/Solutions7.lagda.md @@ -0,0 +1,249 @@ +# Week 7 - Cubical Agda Exercises - Solutions + +```agda +{-# OPTIONS --cubical #-} + +module Solutions7 where + +open import cubical-prelude +open import Lecture7-notes +``` + +```agda +private + variable + A : Type ℓ + B : A → Type ℓ +``` + +## Part I: + +### Exercise 1 +```agda +depFunExt : {f g : (x : A) → B x} + → (∀ x → f x ≡ g x) + → f ≡ g +depFunExt h i x = h x i +``` + +### Exercise 2 +```agda +apd : (f : (x : A) → B x) {x y : A} (p : x ≡ y) → PathP (λ i → B (p i)) (f x) (f y) +apd f p i = f (p i) + +``` + + +## Part II + +### Exercise 3 +```agda +inhPropIsContr : isProp A → A → isContr A +inhPropIsContr h a = a , h a +``` + +### Exercise 4 +```agda +isPropΠ : (h : (x : A) → isProp (B x)) → isProp ((x : A) → B x) +isPropΠ h f g i x = h x (f x) (g x) i +``` + +### Exercise 5 +```agda +funExt⁻ : {f g : (x : A) → B x} → f ≡ g → ((x : A) → f x ≡ g x) +funExt⁻ p x i = p i x +``` + +### Exercise 6 +```agda +isSetΠ : (h : (x : A) → isSet (B x)) → isSet ((x : A) → B x) +isSetΠ h f g p q i j x = h x (f x) (g x) (funExt⁻ p x) (funExt⁻ q x) i j +``` + +### Exercise 7 +```agda +singl' : {A : Type ℓ} (a : A) → Type ℓ +singl' {A = A} a = Σ x ꞉ A , x ≡ a + +isContrSingl' : (x : A) → isContr (singl' x) +isContrSingl' x = ctr , prf + where + ctr : singl' x + ctr = (x , refl) + + prf : (s : singl' x) → ctr ≡ s + prf (y , p) i = p (~ i) , λ j → p ((~ i) ∨ j) +``` + + +## Part III: Equality in Σ-types +### Exercise 8 +```agda +module _ {A : Type ℓ} {B : A → Type ℓ'} {x y : Σ A B} where + + ΣPathP : Σ p ꞉ pr₁ x ≡ pr₁ y , PathP (λ i → B (p i)) (pr₂ x) (pr₂ y) + → x ≡ y + ΣPathP (p , q) i = p i , q i + + PathPΣ : x ≡ y + → Σ p ꞉ pr₁ x ≡ pr₁ y , PathP (λ i → B (p i)) (pr₂ x) (pr₂ y) + PathPΣ p = (ap pr₁ p) , (apd pr₂ p) + + ΣPathP-PathPΣ : ∀ p → PathPΣ (ΣPathP p) ≡ p + ΣPathP-PathPΣ _ = refl + + PathPΣ-ΣPathP : ∀ p → ΣPathP (PathPΣ p) ≡ p + PathPΣ-ΣPathP _ = refl +``` + + +## Part III: Some HITs +### Exercise 9 +```agda +{- + + +The torus: + + ∙ ->->->-> ∙ + ∧ ∧ + | | + ∧ ∧ + | | + ∙ ->->->-> ∙ + + + + +The Klein bottle: + + ∙ <-<-<-<- ∙ + ∧ ∧ + | | + ∧ ∧ + | | + ∙ ->->->-> ∙ + + +-} +data KleinBottle : Type where + point : KleinBottle + line1 : point ≡ point + line2 : point ≡ point + square : PathP (λ i → line1 (~ i) ≡ line1 i) line2 line2 + +data KleinBottle' : Type where + point : KleinBottle' + line1 : point ≡ point + line2 : point ≡ point + square : (sym line1) ∙ line2 ∙ line1 ≡ line2 +``` + + + +### Exercise 10 +```agda +suspUnitChar : Susp Unit ≡ Interval +suspUnitChar = isoToPath (iso to fro toFro froTo) + where + to : Susp Unit → Interval + to north = zero + to south = one + to (merid ⋆ i) = seg i + + fro : Interval → Susp Unit + fro zero = north + fro one = south + fro (seg i) = merid ⋆ i + + toFro : (b : Interval) → to (fro b) ≡ b + toFro zero = refl + toFro one = refl + toFro (seg i) = refl + + froTo : (s : Susp Unit) → fro (to s) ≡ s + froTo north = refl + froTo south = refl + froTo (merid ⋆ i) = refl +``` + + + +### Exercise 11 +```agda +SuspPushout : Type ℓ → Type ℓ +SuspPushout A = Pushout {A = A} {B = Unit} {C = Unit} (λ a → ⋆) (λ a → ⋆) + +Susp≡SuspPushout : (A : Type ℓ) → Susp A ≡ SuspPushout A +Susp≡SuspPushout A = isoToPath (iso to fro toFro froTo) + where + to : Susp A → SuspPushout A + to north = inl ⋆ + to south = inr ⋆ + to (merid a i) = push a i + + fro : SuspPushout A → Susp A + fro (inl ⋆) = north + fro (inr ⋆) = south + fro (push a i) = merid a i + + toFro : (b : SuspPushout A) → to (fro b) ≡ b + toFro (inl ⋆) = refl + toFro (inr ⋆) = refl + toFro (push a i) = refl + + froTo : (s : Susp A) → fro (to s) ≡ s + froTo north = refl + froTo south = refl + froTo (merid a i) = refl +``` + +### Exercise 12 + +```agda +comp-filler : {x y z : A} (p : x ≡ y) (q : y ≡ z) + → PathP (λ j → refl {x = x} j ≡ q j) p (p ∙ q) +comp-filler {x = x} p q j i = hfill (λ k → λ { (i = i0) → x + ; (i = i1) → q k }) + (inS (p i)) j + +rUnit : {x y : A} (p : x ≡ y) → p ∙ refl ≡ p +rUnit p = sym (comp-filler p refl) + + + +suspBoolChar : Susp Bool ≡ S¹ +suspBoolChar = isoToPath (iso to fro toFro froTo) + where + to : Susp Bool → S¹ + to north = base + to south = base + to (merid true i) = loop i + to (merid false i) = base + + fro : S¹ → Susp Bool + fro base = north + fro (loop i) = (merid true ∙ sym (merid false)) i + + toFro : (b : S¹) → to (fro b) ≡ b + toFro base = refl + toFro (loop i) j = mySquare j i + where + mySquare : ap to (merid true ∙ sym (merid false)) ≡ loop + mySquare = ap to (merid true ∙ sym (merid false)) ≡⟨ refl ⟩ + ap to (merid true) ∙ ap to (sym (merid false)) ≡⟨ refl ⟩ + ap to (merid true) ∙ refl ≡⟨ refl ⟩ + loop ∙ refl ≡⟨ rUnit loop ⟩ + loop ∎ + + froTo : (s : Susp Bool) → fro (to s) ≡ s + froTo north = refl + froTo south = merid false + froTo (merid true i) j = mySquare (~ j) i + where + mySquare : PathP (λ j → north ≡ sym (merid false) j) + (merid true) + (merid true ∙ sym (merid false)) + mySquare = comp-filler (merid true) (sym (merid false)) + froTo (merid false i) j = merid false (i ∧ j) +``` diff --git a/src/HoTTEST/Agda/Cubical/Solutions8.lagda.md b/src/HoTTEST/Agda/Cubical/Solutions8.lagda.md new file mode 100644 index 0000000..589d0ce --- /dev/null +++ b/src/HoTTEST/Agda/Cubical/Solutions8.lagda.md @@ -0,0 +1,114 @@ +# Week 8 - Cubical Agda Exercises - Solutions + +```agda +{-# OPTIONS --cubical #-} + +module Solutions8 where + +open import cubical-prelude +open import Lecture7-notes +open import Lecture8-notes +``` + +## Part I: `transp` and `hcomp` + +### Exercise 1 (★) + +Prove the propositional computation law for `J`: + +```agda +-- J = λ {x} P d p → transp (λ i → P (p i) (λ j → p (i ∧ j))) i0 d + +JRefl : {A : Type ℓ} {x : A} (P : (z : A) → x ≡ z → Type ℓ'') + (d : P x refl) → J P d refl ≡ d +JRefl {x = x} P d i = transp (λ _ → P x (λ _ → x)) i d +``` + +### Exercise 2 (★★) + +Using `transp`, construct a method for turning dependent paths into paths. + +**Hint**: +Transport the point `p i` to the fibre `A i1`, and set `φ = i` such that the +transport computes away at `i = i1`. +```text + x ----(p i)----> y + A i0 A i A i1 +``` + +```agda +fromPathP : {A : I → Type ℓ} {x : A i0} {y : A i1} → + PathP A x y → transport (λ i → A i) x ≡ y +fromPathP {A = A} p i = transp (λ j → A (i ∨ j)) i (p i) +``` + +### Exercise 3 (★★★) + +Using `hcomp`, construct a method for turning paths into dependent paths. + +**Hint**: +At each point `i`, the verical fibre of the following diagram should lie in +`A i`. The key is to parametrise the bottom line with a dependent path from `x` +to `transport A x`. This requires writing a `transp` that computes away at +`i = i0`. + +```text + x - - - -> y + ^ ^ + ¦ ¦ + refl ¦ ¦ p i + ¦ ¦ + ¦ ¦ + x ---> transport A x +``` + + +```agda +toPathP : {A : I → Type ℓ} {x : A i0} {y : A i1} → + transport (λ i → A i) x ≡ y → PathP A x y +toPathP {A = A} {x = x} p i = + hcomp + (λ {j (i = i0) → x ; + j (i = i1) → p j }) + (transpFill i) + where + transpFill : PathP A x (transport (λ i → A i) x) + transpFill i = transp (λ j → A (i ∧ j)) (~ i) x +``` + +### Exercise 4 (★) + +Using `toPathP`, prove the following lemma, which lets you fill in dependent +lines in hProps, provided their boundary. + +```agda +isProp→PathP : {A : I → Type ℓ} (p : (i : I) → isProp (A i)) + (a₀ : A i0) (a₁ : A i1) → PathP A a₀ a₁ +isProp→PathP p a₀ a₁ = toPathP (p i1 _ a₁) +``` + +### Exercise 5 (★★) + +Prove the following lemma characterizing equality in subtypes: + +```agda +Σ≡Prop : {A : Type ℓ} {B : A → Type ℓ'} {u v : Σ A B} (h : (x : A) → isProp (B x)) + → (p : pr₁ u ≡ pr₁ v) → u ≡ v +Σ≡Prop {B = B} {u = u} {v = v} h p i = p i , isProp→PathP (λ i → h (p i)) (pr₂ u) (pr₂ v) i +``` + +### Exercise 6 (★★★) + +Prove that `isContr` is a proposition: + +**Hint**: +This requires drawing a cube (yes, an actual 3D one)! + +```agda +isPropIsContr : {A : Type ℓ} → isProp (isContr A) +isPropIsContr (c0 , h0) (c1 , h1) j = h0 c1 j , λ y i → + hcomp (λ {k (i = i0) → h0 c1 j ; + k (i = i1) → h1 y k ; + k (j = i0) → h0 (h1 y k) i ; + k (j = i1) → h1 y (i ∧ k) }) (h0 c1 (i ∨ j)) +``` diff --git a/src/HoTTEST/Agda/Cubical/Solutions9.lagda.md b/src/HoTTEST/Agda/Cubical/Solutions9.lagda.md new file mode 100644 index 0000000..d69809c --- /dev/null +++ b/src/HoTTEST/Agda/Cubical/Solutions9.lagda.md @@ -0,0 +1,359 @@ +# Week 9 - Cubical Agda Exercises + +## Standard disclaimer: + +**The exercises are designed to increase in difficulty so that we can cater to +our large and diverse audience. This also means that it is *perfectly fine* if +you don't manage to do all exercises: some of them are definitely a bit hard for +beginners and there are likely too many exercises! You *may* wish to come back +to them later when you have learned more.** + +Having said that, here we go! + +In case you haven't done the other Agda exercises: +This is a markdown file with Agda code, which means that it displays nicely on +GitHub, but at the same time you can load this file in Agda and fill the holes +to solve exercises. + +**When solving the problems, +please make a copy of this file to work in, so that it doesn't get overwritten +(in case we update the exercises through `git`)!** + + +```agda +{-# OPTIONS --cubical --allow-unsolved-metas #-} + +module Solutions9 where + +open import cubical-prelude +open import Lecture7-notes +open import Lecture8-notes +open import Lecture9-notes +open import Solutions7 hiding (rUnit) +open import Solutions8 +open import Lecture9-live using (SemiGroupℕ) +``` + +## Part I: More hcomps + +### Exercise 1 (★★) +### (a) +Show the left cancellation law for path composition using an hcomp. +Hint: one hcomp should suffice. Use `comp-filler` and connections + +```agda +lUnit : {A : Type ℓ} {x y : A} (p : x ≡ y) → refl ∙ p ≡ p +lUnit {x = x} {y = y} p i j = + hcomp (λ k → λ {(i = i0) → comp-filler refl p k j + ; (i = i1) → p (k ∧ j) + ; (j = i0) → x + ; (j = i1) → p k}) + x + + +``` +### (b) +Try to mimic the construction of lUnit for rUnit (i.e. redefine it) +in such a way that `rUnit refl ≡ lUnit refl` holds by `refl`. +Hint: use (almost) the exact same hcomp. + +```agda +rUnit : {A : Type ℓ} {x y : A} (p : x ≡ y) → p ∙ refl ≡ p +rUnit {x = x} {y = y} p i j = + hcomp (λ k → λ {(i = i0) → comp-filler p refl k j + ; (i = i1) → p j + ; (j = i0) → x + ; (j = i1) → y}) + (p j) + +-- uncomment to see if it type-checks + +{- +rUnit≡lUnit : ∀ {ℓ} {A : Type ℓ} {x : A} → rUnit (refl {x = x}) ≡ lUnit refl +rUnit≡lUnit = refl +-} + +``` + + +### Exercise 2 (★★) +Show the associativity law for path composition +Hint: one hcomp should suffice. This one can be done without connections + (but you might need comp-filler in more than one place) + +```agda +assoc : {A : Type ℓ} {x y z w : A} (p : x ≡ y) (q : y ≡ z) (r : z ≡ w) + → p ∙ (q ∙ r) ≡ (p ∙ q) ∙ r +assoc {x = x} {y = y} p q r i j = + hcomp (λ k → λ {(i = i0) → (p ∙ comp-filler q r k) j + ; (i = i1) → comp-filler (p ∙ q) r k j + ; (j = i0) → x + ; (j = i1) → r k}) + ((p ∙ q) j) + +``` + +### Exercise 3 (Master class in connections) (🌶) +The goal of this exercise is to give a cubical proof of the Eckmann-Hilton argument, +which says that path composition for higher loops is commutative + +(a) While we cannot get `p ∙ q ≡ q ∙ p` as a one-liner, we can get a +one-liner showing that the identiy holds up to some annoying +coherences. Try to understand the following statement (and why it's +well-typed). After that, fill the holes + +Hint: each hole will need a `∨` or a `∧` + +```agda +pre-EH : {A : Type ℓ} {x : A} (p q : refl {x = x} ≡ refl) + → ap (λ x → x ∙ refl) p ∙ ap (λ x → refl ∙ x) q + ≡ ap (λ x → refl ∙ x) q ∙ ap (λ x → x ∙ refl) p +pre-EH {x = x} p q i = (λ j → p (~ i ∧ j) ∙ q (i ∧ j)) + ∙ (λ j → p (~ i ∨ j) ∙ q (i ∨ j)) + +``` +(b) If we manage to cancel out all of the annoying aps, we get Eckmann-Hilton: +For paths (p q : refl ≡ refl), we have p ∙ q ≡ q ∙ p. Try to prove this, using the above lemma. + +Hint: Use the pre-EH as the bottom of an hcomp (one should be enough). +For the sides, use lUnit and rUnit wherever they're needed. Note that this will only work out smoothly if +you've solved Exercise 1 (b). + +```agda +Eckmann-Hilton : {A : Type ℓ} {x : A} (p q : refl {x = x} ≡ refl) → p ∙ q ≡ q ∙ p +Eckmann-Hilton {x = x} p q k i = + hcomp (λ r → λ {(i = i0) → rUnit (refl {x = x}) r + ; (i = i1) → rUnit (refl {x = x}) r + ; (k = i0) → ((λ j → rUnit (p j) r) ∙ (λ j → lUnit (q j) r)) i + ; (k = i1) → ((λ j → lUnit (q j) r) ∙ (λ j → rUnit (p j) r)) i}) + (pre-EH p q k i) + +``` +# # Part 2: Binary numbers as a HIT +Here is another HIT describing binary numbers. The idea is that a binary number is a list of booleans, modulo trailing zeros. + +For instance, `true ∷ true ∷ true ∷ []` is the binary number 110 ... +... and so is `true ∷ true ∷ false ∷ false ∷ false ∷ []` + +(!) Note that we're interpreting 110 as 1·2⁰ + 1·2¹ + 0·2² here. + +```agda +0B = false +1B = true + +data ListBin : Type where + [] : ListBin + _∷_ : (x : Bool) (xs : ListBin) → ListBin + drop0 : 0B ∷ [] ≡ [] + +-- 1 as a binary number +1LB : ListBin +1LB = 1B ∷ [] +``` +### Exercise 4 (★) +Define the successor function on ListBin +```agda + + + +sucListBin : ListBin → ListBin +sucListBin [] = 1LB +sucListBin (true ∷ xs) = false ∷ sucListBin xs +sucListBin (false ∷ xs) = 1B ∷ xs +sucListBin (drop0 i) = 1LB + +``` +### Exercise 5 (★★) +Define an addition `+LB` on ListBin and prove that `x +LB [] ≡ x` +Do this by mutual induction! Make sure the three cases for the right unit law hold by refl. +```agda + +_+LB_ : ListBin → ListBin → ListBin +rUnit+LB : (x : ListBin) → x +LB [] ≡ x +[] +LB y = y +(x ∷ xs) +LB [] = x ∷ xs +(true ∷ xs) +LB (true ∷ ys) = false ∷ sucListBin (xs +LB ys) +(true ∷ xs) +LB (false ∷ ys) = true ∷ (xs +LB ys) +(false ∷ xs) +LB (y ∷ ys) = y ∷ (xs +LB ys) +(true ∷ xs) +LB drop0 i = true ∷ (rUnit+LB xs i) +(false ∷ xs) +LB drop0 i = false ∷ (rUnit+LB xs i) +drop0 i +LB [] = drop0 i +drop0 i +LB (y ∷ ys) = y ∷ ys +drop0 i +LB drop0 j = drop0 (j ∧ i) +rUnit+LB [] = refl +rUnit+LB (x ∷ x₁) = refl +rUnit+LB (drop0 i) = refl + +``` +(c) Prove that sucListBin is left distributive over `+LB` +Hint: If you pattern match deep enough, there should be a lot of refls... +```agda + +sucListBinDistrL : (x y : ListBin) → sucListBin (x +LB y) ≡ (sucListBin x +LB y) +sucListBinDistrL [] [] = refl +sucListBinDistrL [] (true ∷ y) = refl +sucListBinDistrL [] (false ∷ y) = refl +sucListBinDistrL [] (drop0 i) = refl +sucListBinDistrL (true ∷ xs) [] = refl +sucListBinDistrL (false ∷ xs) [] = refl +sucListBinDistrL (true ∷ xs) (true ∷ y) = ap (1B ∷_) (sucListBinDistrL xs y) +sucListBinDistrL (true ∷ xs) (false ∷ y) = ap (0B ∷_) (sucListBinDistrL xs y) +sucListBinDistrL (false ∷ xs) (true ∷ y) = refl +sucListBinDistrL (false ∷ xs) (false ∷ y) = refl +sucListBinDistrL (true ∷ []) (drop0 i) = refl +sucListBinDistrL (true ∷ (true ∷ xs)) (drop0 i) = refl +sucListBinDistrL (true ∷ (false ∷ xs)) (drop0 i) = refl +sucListBinDistrL (true ∷ drop0 i) (drop0 j) = refl +sucListBinDistrL (false ∷ xs) (drop0 i) = refl +sucListBinDistrL (drop0 i) [] = refl +sucListBinDistrL (drop0 i) (true ∷ y) = refl +sucListBinDistrL (drop0 i) (false ∷ y) = refl +sucListBinDistrL (drop0 i) (drop0 j) = refl +``` + +### Exercise 6 (★) +Define a map `LB→ℕ : ListBin → ℕ` and show that it preserves addition + +```agda +ℕ→ListBin : ℕ → ListBin +ℕ→ListBin zero = [] +ℕ→ListBin (suc x) = sucListBin (ℕ→ListBin x) + +ℕ→ListBin-pres+ : (x y : ℕ) → ℕ→ListBin (x + y) ≡ (ℕ→ListBin x +LB ℕ→ListBin y) +ℕ→ListBin-pres+ zero y = refl +ℕ→ListBin-pres+ (suc x) zero = + ap sucListBin (ap ℕ→ListBin (+-comm x zero)) + ∙ sym (rUnit+LB (sucListBin (ℕ→ListBin x))) +ℕ→ListBin-pres+ (suc x) (suc y) = + ap sucListBin (ℕ→ListBin-pres+ x (suc y)) + ∙ sucListBinDistrL (ℕ→ListBin x) (sucListBin (ℕ→ListBin y)) + +``` + +### Exercise 7 (★★★) +Show that `ℕ ≃ ListBin`. + +```agda + +-- useful lemma +move4 : (x y z w : ℕ) → (x + y) + (z + w) ≡ (x + z) + (y + w) +move4 x y z w = + (x + y) + (z + w) ≡⟨ +-assoc (x + y) z w ⟩ + ((x + y) + z) + w ≡⟨ ap (_+ w) (sym (+-assoc x y z) + ∙ ap (x +_) (+-comm y z) + ∙ (+-assoc x z y)) ⟩ + ((x + z) + y) + w ≡⟨ sym (+-assoc (x + z) y w) ⟩ + ((x + z) + (y + w)) ∎ + +ListBin→ℕ : ListBin → ℕ +ListBin→ℕ [] = 0 +ListBin→ℕ (true ∷ xs) = suc (2 · ListBin→ℕ xs) +ListBin→ℕ (false ∷ xs) = 2 · ListBin→ℕ xs +ListBin→ℕ (drop0 i) = 0 + +ListBin→ℕ-suc : (x : ListBin) → ListBin→ℕ (sucListBin x) ≡ suc (ListBin→ℕ x) +ListBin→ℕ-suc [] = refl +ListBin→ℕ-suc (true ∷ xs) = + ap (λ x → x + x) (ListBin→ℕ-suc xs) + ∙ ap suc (sym (+-suc (ListBin→ℕ xs) (ListBin→ℕ xs))) +ListBin→ℕ-suc (false ∷ xs) = refl +ListBin→ℕ-suc (drop0 i) = refl + +x+x-charac : (xs : ListBin) → (xs +LB xs) ≡ (false ∷ xs) +x+x-charac [] = sym drop0 +x+x-charac (true ∷ xs) = ap (false ∷_) (ap sucListBin (x+x-charac xs)) +x+x-charac (false ∷ xs) = ap (false ∷_) (x+x-charac xs) +x+x-charac (drop0 i) j = + hcomp (λ k → λ {(i = i0) → false ∷ drop0 (~ j ∨ ~ k) + ; (i = i1) → drop0 (~ j) + ; (j = i0) → drop0 i + ; (j = i1) → false ∷ drop0 (i ∨ ~ k)}) + (drop0 (~ j ∧ i)) + +suc-x+x-charac : (xs : ListBin) → sucListBin (xs +LB xs) ≡ (true ∷ xs) +suc-x+x-charac xs = ap sucListBin (x+x-charac xs) + +ListBin→ℕ→ListBin : (x : ListBin) → ℕ→ListBin (ListBin→ℕ x) ≡ x +ListBin→ℕ→ListBin [] = refl +ListBin→ℕ→ListBin (true ∷ xs) = + ap sucListBin (ℕ→ListBin-pres+ (ListBin→ℕ xs) (ListBin→ℕ xs) + ∙ ap (λ x → x +LB x) (ListBin→ℕ→ListBin xs)) + ∙ suc-x+x-charac xs +ListBin→ℕ→ListBin (false ∷ xs) = + (ℕ→ListBin-pres+ (ListBin→ℕ xs) (ListBin→ℕ xs) + ∙ (ap (λ x → x +LB x) (ListBin→ℕ→ListBin xs))) + ∙ x+x-charac xs +ListBin→ℕ→ListBin (drop0 i) = help i + where + lem : (refl ∙ refl) ∙ sym drop0 ≡ sym drop0 + lem = ap (_∙ sym drop0) (rUnit refl) ∙ lUnit (sym drop0) + + help : PathP (λ i → [] ≡ drop0 i) ((refl ∙ refl) ∙ sym drop0) refl + help i j = hcomp (λ k → λ {(i = i0) → lem (~ k) j + ; (i = i1) → [] + ; (j = i0) → [] + ; (j = i1) → drop0 i}) + (drop0 (i ∨ ~ j)) + +ℕ→ListBin→ℕ : (x : ℕ) → ListBin→ℕ (ℕ→ListBin x) ≡ x +ℕ→ListBin→ℕ zero = refl +ℕ→ListBin→ℕ (suc x) = + ListBin→ℕ-suc (ℕ→ListBin x) + ∙ ap suc (ℕ→ListBin→ℕ x) + +ℕ≃ListBin : ℕ ≃ ListBin +ℕ≃ListBin = isoToEquiv (iso ℕ→ListBin ListBin→ℕ ListBin→ℕ→ListBin ℕ→ListBin→ℕ) + +``` +# Part 3: The SIP +### Exericise 8 (★★) +Show that, using an SIP inspired argument, if `(A , _+A_)` is a semigroup and `(B , _+B_)` is some other type with a composition satisfying: + +(i) `e : A ≃ B` + +(ii) `((x y : A) → e (x +A y) ≡ e x +B e y` + +then `(B , _+B_)` defines a semigroup. + +Conclude that `(ListBin , _+LB_)` is a semigroup + +For inspiration, see Lecture9-notes +```agda + + +module _ {A B : Type} (SA : SemiGroup A) (e : A ≃ B) + (_+B_ : B → B → B) + (hom : (x y : A) → pr₁ e (pr₁ SA x y) ≡ ((pr₁ e x) +B pr₁ e y)) + where + f = pr₁ e + f⁻ = invEq e + + _+A_ = pr₁ SA + + SemiGroupSIPAux : SemiGroup B + SemiGroupSIPAux = substEquiv SemiGroup e SA + + _+B'_ = pr₁ SemiGroupSIPAux + + +B'≡+B : (x y : B) → x +B' y ≡ (x +B y) + +B'≡+B x y = + (x +B' y) ≡⟨ transportRefl (f (f⁻ (transport refl x) + +A (f⁻ (transport refl y)))) ⟩ + f (f⁻ (transport refl x) +A f⁻ (transport refl y)) ≡⟨ (λ i → f (f⁻ (transportRefl x i) + +A f⁻ (transportRefl y i))) ⟩ + f (f⁻ x +A f⁻ y) ≡⟨ hom (f⁻ x) (f⁻ y) ⟩ + (f (f⁻ x) +B f (f⁻ y)) ≡⟨ (λ i → secEq e x i +B secEq e y i) ⟩ + x +B y ∎ + + assoc+B : ∀ m n o → m +B (n +B o) ≡ (m +B n) +B o + assoc+B m n o = + (m +B (n +B o)) ≡⟨ (λ i → +B'≡+B m (+B'≡+B n o (~ i)) (~ i)) ⟩ + (m +B' (n +B' o)) ≡⟨ pr₂ SemiGroupSIPAux m n o ⟩ + ((m +B' n) +B' o) ≡⟨ (λ i → +B'≡+B (+B'≡+B m n i) o i) ⟩ + ((m +B n) +B o) ∎ + + inducedSemiGroup : SemiGroup B + inducedSemiGroup = _+B_ , assoc+B + +SemiGroupListBin : SemiGroup ListBin +SemiGroupListBin = inducedSemiGroup SemiGroupℕ ℕ≃ListBin _+LB_ ℕ→ListBin-pres+ diff --git a/src/HoTTEST/Agda/Cubical/cubical-prelude.lagda.md b/src/HoTTEST/Agda/Cubical/cubical-prelude.lagda.md new file mode 100644 index 0000000..deb0c6e --- /dev/null +++ b/src/HoTTEST/Agda/Cubical/cubical-prelude.lagda.md @@ -0,0 +1,502 @@ +This file contains whatever is needed from the agda/cubical library to +get the lectures to typecheck. Warning: it is not very organized or +documented. + +```agda +{-# OPTIONS --cubical #-} + +module cubical-prelude where + +open import Agda.Builtin.Cubical.Path public +open import Agda.Builtin.Cubical.Sub public + renaming ( primSubOut to outS + ) +open import Agda.Primitive.Cubical public + renaming ( primIMin to _∧_ -- I → I → I + ; primIMax to _∨_ -- I → I → I + ; primINeg to ~_ -- I → I + ; isOneEmpty to empty + ; primComp to comp + ; primHComp to hcomp + ; primTransp to transp + ; itIsOne to 1=1 ) + + +open import Agda.Builtin.Cubical.Glue public + using ( isEquiv -- ∀ {ℓ ℓ'} {A : Type ℓ} {B : Type ℓ'} (f : A → B) → Type (ℓ ⊔ ℓ') + + ; equiv-proof -- ∀ (y : B) → isContr (fiber f y) + + ; _≃_ -- ∀ {ℓ ℓ'} (A : Type ℓ) (B : Type ℓ') → Type (ℓ ⊔ ℓ') + + ; equivFun -- ∀ {ℓ ℓ'} {A : Type ℓ} {B : Type ℓ'} → A ≃ B → A → B + + ; equivProof -- ∀ {ℓ ℓ'} (T : Type ℓ) (A : Type ℓ') (w : T ≃ A) (a : A) φ → + -- Partial φ (fiber (equivFun w) a) → fiber (equivFun w) a + + ; primGlue -- ∀ {ℓ ℓ'} (A : Type ℓ) {φ : I} (T : Partial φ (Type ℓ')) + -- → (e : PartialP φ (λ o → T o ≃ A)) → Type ℓ' + + ; prim^unglue -- ∀ {ℓ ℓ'} {A : Type ℓ} {φ : I} {T : Partial φ (Type ℓ')} + -- → {e : PartialP φ (λ o → T o ≃ A)} → primGlue A T e → A + + -- The ∀ operation on I. This is commented out as it is not currently used for anything + -- ; primFaceForall -- (I → I) → I + ) + renaming ( prim^glue to glue -- ∀ {ℓ ℓ'} {A : Type ℓ} {φ : I} {T : Partial φ (Type ℓ')} + -- → {e : PartialP φ (λ o → T o ≃ A)} + -- → PartialP φ T → A → primGlue A T e + ) + + +open import Agda.Primitive public + using ( Level + ; SSet + ; lzero + ; lsuc + ; _⊔_ ) + renaming ( Set to Type ) + +-- We parametrize everything by some universe levels +variable + ℓ ℓ' ℓ'' : Level + +-- Non dependent path types + +Path : ∀ {ℓ} (A : Type ℓ) → A → A → Type ℓ +Path A a b = PathP (λ _ → A) a b + +-- PathP (λ i → A) x y gets printed as x ≡ y when A does not mention i. +-- _≡_ : ∀ {ℓ} {A : Type ℓ} → A → A → Type ℓ +-- _≡_ {A = A} = PathP (λ _ → A) + +-- Path composition +_∙_ : {A : Type ℓ} {x y z : A} → x ≡ y → y ≡ z → x ≡ z +_∙_ {x = x} p q i = hcomp (λ j → λ { (i = i0) → x + ; (i = i1) → q j }) + (p i) + +infixr 30 _∙_ + + +-- Equality reasoning +infix 3 _∎ +infixr 2 _≡⟨_⟩_ _≡⟨⟩_ + +_≡⟨_⟩_ : {A : Type ℓ} (x : A) {y z : A} → x ≡ y → y ≡ z → x ≡ z +_ ≡⟨ x≡y ⟩ y≡z = x≡y ∙ y≡z + +≡⟨⟩-syntax : {A : Type ℓ} (x : A) {y z : A} → x ≡ y → y ≡ z → x ≡ z +≡⟨⟩-syntax = _≡⟨_⟩_ +infixr 2 ≡⟨⟩-syntax +syntax ≡⟨⟩-syntax x (λ i → B) y = x ≡[ i ]⟨ B ⟩ y + +_≡⟨⟩_ : {A : Type ℓ} (x : A) {y : A} → x ≡ y → x ≡ y +_ ≡⟨⟩ x≡y = x≡y + +_∎ : {A : Type ℓ} (x : A) → x ≡ x +x ∎ = λ _ → x + + +_[_↦_] : ∀ {ℓ} (A : Type ℓ) (φ : I) (u : Partial φ A) → SSet ℓ +A [ φ ↦ u ] = Sub A φ u + +infix 4 _[_↦_] + +--- Homogeneous filling +hfill : {A : Type ℓ} {φ : I} (u : ∀ i → Partial φ A) (u0 : A [ φ ↦ u i0 ]) (i : I) → A +hfill {φ = φ} u u0 i = + hcomp (λ j → λ { (φ = i1) → u (i ∧ j) 1=1 + ; (i = i0) → outS u0 }) + (outS u0) + +-- Use built in Σ types to avoid problems with the imported Cubical +-- Agda operations that use Σ types +open import Agda.Builtin.Sigma public renaming (fst to pr₁ ; snd to pr₂) + +Sigma : {l1 l2 : Level} (A : Type l1) (B : A → Type l2) → Type (l1 ⊔ l2) +Sigma {l1} {l2} A B = Σ {l1} {l2} A B + +syntax Sigma A (λ x → b) = Σ x ꞉ A , b + +infix -1 Sigma + +_×_ : ∀ {l1 l2} → Type l1 → Type l2 → Type (l1 ⊔ l2) +A × B = Sigma A (\ _ → B) + +infixr 5 _×_ + +-- Contractible types, propositions and sets: + +isContr : Type ℓ → Type ℓ +isContr A = Σ x ꞉ A , ((y : A) → x ≡ y) + +isProp : Type ℓ → Type ℓ +isProp A = (x y : A) → x ≡ y + +isSet : Type ℓ → Type ℓ +isSet A = (x y : A) → isProp (x ≡ y) + +-- Fibers +fiber : {A : Type ℓ} {B : Type ℓ'} (f : A → B) (y : B) → Set (ℓ ⊔ ℓ') +fiber {A = A} f y = Σ x ꞉ A , f x ≡ y + +-- In the agda/cubical library we call these h-levels following +-- Voevodsky instead of n-types and index by natural numbers instead +-- of ℕ₋₂. So isContr is isOfHLevel 0, isProp is isOfHLevel 1, isSet +-- is isOfHLevel 2, etc. For details see Cubical/Foundations/HLevels.agda + + +-- Sections and retractions +module _ {ℓ ℓ'} {A : Type ℓ} {B : Type ℓ'} where + section : (f : A → B) → (g : B → A) → Type ℓ' + section f g = ∀ b → f (g b) ≡ b + + -- NB: `g` is the retraction! + retract : (f : A → B) → (g : B → A) → Type ℓ + retract f g = ∀ a → g (f a) ≡ a + +module _ {A B : Type} {f : A → B} (equivF : isEquiv f) where + funIsEq : A → B + funIsEq = f + + invIsEq : B → A + invIsEq y = equivF .equiv-proof y .pr₁ .pr₁ + + secIsEq : section f invIsEq + secIsEq y = equivF .equiv-proof y .pr₁ .pr₂ + + retIsEq : retract f invIsEq + retIsEq a i = equivF .equiv-proof (f a) .pr₂ (a , λ _ → f a) i .pr₁ + +module _ {A B : Type} (w : A ≃ B) where + invEq : B → A + invEq = invIsEq (pr₂ w) + + retEq : retract (w .pr₁) invEq + retEq = retIsEq (pr₂ w) + + secEq : section (w .pr₁) invEq + secEq = secIsEq (pr₂ w) + +-- Isomorphisms +record Iso {ℓ ℓ'} (A : Type ℓ) (B : Type ℓ') : Type (ℓ ⊔ ℓ') where + no-eta-equality + constructor iso + field + fun : A → B + inv : B → A + rightInv : section fun inv + leftInv : retract fun inv + +-- Any iso is an equivalence (called "improve" by Dan, here we use the +-- contractible fibers version of equivalences) +module _ {ℓ ℓ'} {A : Type ℓ} {B : Type ℓ'} (i : Iso A B) where + open Iso i renaming ( fun to f + ; inv to g + ; rightInv to s + ; leftInv to t) + + private + module _ (y : B) (x0 x1 : A) (p0 : f x0 ≡ y) (p1 : f x1 ≡ y) where + fill0 : I → I → A + fill0 i = hfill (λ k → λ { (i = i1) → t x0 k + ; (i = i0) → g y }) + (inS (g (p0 (~ i)))) + + fill1 : I → I → A + fill1 i = hfill (λ k → λ { (i = i1) → t x1 k + ; (i = i0) → g y }) + (inS (g (p1 (~ i)))) + + fill2 : I → I → A + fill2 i = hfill (λ k → λ { (i = i1) → fill1 k i1 + ; (i = i0) → fill0 k i1 }) + (inS (g y)) + + p : x0 ≡ x1 + p i = fill2 i i1 + + sq : I → I → A + sq i j = hcomp (λ k → λ { (i = i1) → fill1 j (~ k) + ; (i = i0) → fill0 j (~ k) + ; (j = i1) → t (fill2 i i1) (~ k) + ; (j = i0) → g y }) + (fill2 i j) + + sq1 : I → I → B + sq1 i j = hcomp (λ k → λ { (i = i1) → s (p1 (~ j)) k + ; (i = i0) → s (p0 (~ j)) k + ; (j = i1) → s (f (p i)) k + ; (j = i0) → s y k }) + (f (sq i j)) + + lemIso : (x0 , p0) ≡ (x1 , p1) + lemIso i .pr₁ = p i + lemIso i .pr₂ = λ j → sq1 i (~ j) + + isoToIsEquiv : isEquiv f + isoToIsEquiv .equiv-proof y .pr₁ .pr₁ = g y + isoToIsEquiv .equiv-proof y .pr₁ .pr₂ = s y + isoToIsEquiv .equiv-proof y .pr₂ z = lemIso y (g y) (pr₁ z) (s y) (pr₂ z) + + +isoToEquiv : {ℓ ℓ' : Level} {A : Type ℓ} {B : Type ℓ'} → Iso A B → A ≃ B +isoToEquiv i .pr₁ = i .Iso.fun +isoToEquiv i .pr₂ = isoToIsEquiv i + +Glue : (A : Type ℓ) {φ : I} + → (Te : Partial φ (Σ T ꞉ Type ℓ' , T ≃ A)) + → Type ℓ' +Glue A Te = primGlue A (λ x → Te x .pr₁) (λ x → Te x .pr₂) + +idEquiv : ∀ {ℓ} (A : Type ℓ) → A ≃ A +idEquiv A .pr₁ = λ x → x +equiv-proof (idEquiv A .pr₂) = λ y → (y , (λ i → y)) , (λ h i → (h .pr₂ (~ i)) , λ j → h .pr₂ ((j ∨ ~ i))) + +isoToPath : {A B : Type ℓ} → Iso A B → A ≡ B +isoToPath {A = A} {B = B} f i = + Glue B (λ { (i = i0) → (A , isoToEquiv f) + ; (i = i1) → (B , idEquiv B) }) + +-- Natural numbers. We use the built in ones to be able to use +-- numerals. +open import introduction public + using (ℕ; zero; suc; _+_) + renaming (_*_ to _·_) + +_-_ : ℕ → ℕ → ℕ +n - zero = n +zero - suc m = zero +suc n - suc m = n - m + +{-# BUILTIN NATMINUS _-_ #-} + + +-- Integers (slightly different from how Dan did them in order to have +-- one less constructor to pattern match on) +data ℤ : Type₀ where + pos : (n : ℕ) → ℤ + negsuc : (n : ℕ) → ℤ + +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) + +sucPred : ∀ i → sucℤ (predℤ i) ≡ i +sucPred (pos zero) = λ i → pos zero +sucPred (pos (suc n)) = λ i → pos (suc n) +sucPred (negsuc n) = λ i → negsuc n + +predSuc : ∀ i → predℤ (sucℤ i) ≡ i +predSuc (pos n) = λ i → pos n +predSuc (negsuc zero) = λ i → negsuc zero +predSuc (negsuc (suc n)) = λ i → negsuc (suc n) + +sucPath : ℤ ≡ ℤ +sucPath = isoToPath (iso sucℤ predℤ sucPred predSuc) + +_+ℤ_ : ℤ → ℤ → ℤ +m +ℤ pos n = m +pos n + where + _+pos_ : ℤ → ℕ → ℤ + z +pos 0 = z + z +pos (suc n) = sucℤ (z +pos n) +m +ℤ negsuc n = m +negsuc n + where + _+negsuc_ : ℤ → ℕ → ℤ + z +negsuc 0 = predℤ z + z +negsuc (suc n) = predℤ (z +negsuc n) + + + +-- 'Data' types from Martín's prelude +record Unit : Type where + constructor + ⋆ + +open Unit public + +data Bool : Type where + true false : Bool + +if_then_else_ : {A : Type ℓ} → Bool → A → A → A +if true then x else y = x +if false then x else y = y +``` + + +```agda + + +funExt₂ : {A : Type ℓ} {B : A → Type} {C : (x : A) → B x → I → Type} + {f : (x : A) → (y : B x) → C x y i0} + {g : (x : A) → (y : B x) → C x y i1} + → ((x : A) (y : B x) → PathP (C x y) (f x y) (g x y)) + → PathP (λ i → ∀ x y → C x y i) f g +funExt₂ p i x y = p x y i + +doubleℕ : ℕ → ℕ +doubleℕ zero = zero +doubleℕ (suc x) = suc (suc (doubleℕ x)) + ++-suc : ∀ m n → suc (m + n) ≡ m + suc n ++-suc zero n i = suc n ++-suc (suc m) n i = suc (+-suc m n i) + ++-comm : ∀ m n → m + n ≡ n + m ++-comm zero zero i = 0 ++-comm zero (suc n) i = suc (+-comm zero n i) ++-comm (suc m) zero i = suc (+-comm m zero i) ++-comm (suc m) (suc n) i = + suc (((λ i → (+-suc m n) (~ i)) + ∙ (λ j → suc (+-comm m n j)) + ∙ +-suc n m) i) + ++-zero : ∀ m → m + 0 ≡ m ++-zero zero i = zero ++-zero (suc m) i = suc (+-zero m i) + ++-assoc : ∀ m n o → m + (n + o) ≡ (m + n) + o ++-assoc zero n o i = n + o ++-assoc (suc m) n o i = suc (+-assoc m n o i) + + +data ⊥ : Type₀ where + + +⊥-elim : {A : ⊥ → Type ℓ} → (x : ⊥) → A x +⊥-elim () + +⊥-rec : {A : Type ℓ} → ⊥ → A +⊥-rec () + +¬_ : Type ℓ → Type ℓ +¬ A = A → ⊥ + +``` + + +Binary numbers code: + +```agda + +localrefl : {A : Type} {x : A} → x ≡ x +localrefl {x = x} = λ i → x + +localap : {A B : Type} (f : A → B) {x y : A} → x ≡ y → f x ≡ f y +localap f p i = f (p i) + +localfunExt : {A B : Type} {f g : A → B} (p : (x : A) → f x ≡ g x) → f ≡ g +localfunExt p i x = p x i + + +data Pos : Type where + pos1 : Pos + x0 : Pos → Pos + x1 : Pos → Pos + +sucPos : Pos → Pos +sucPos pos1 = x0 pos1 +sucPos (x0 ps) = x1 ps +sucPos (x1 ps) = x0 (sucPos ps) + +Pos→ℕ : Pos → ℕ +Pos→ℕ pos1 = suc zero +Pos→ℕ (x0 ps) = doubleℕ (Pos→ℕ ps) +Pos→ℕ (x1 ps) = suc (doubleℕ (Pos→ℕ ps)) + +posInd : {P : Pos → Type} → P pos1 → ((p : Pos) → P p → P (sucPos p)) → (p : Pos) → P p +posInd {P} h1 hs = f + where + H : (p : Pos) → P (x0 p) → P (x0 (sucPos p)) + H p hx0p = hs (x1 p) (hs (x0 p) hx0p) + + f : (ps : Pos) → P ps + f pos1 = h1 + f (x0 ps) = posInd (hs pos1 h1) H ps + f (x1 ps) = hs (x0 ps) (posInd (hs pos1 h1) H ps) + +Pos→ℕsucPos : (p : Pos) → Pos→ℕ (sucPos p) ≡ suc (Pos→ℕ p) +Pos→ℕsucPos pos1 = localrefl +Pos→ℕsucPos (x0 p) = localrefl +Pos→ℕsucPos (x1 p) = λ i → doubleℕ (Pos→ℕsucPos p i) + +caseNat : ∀ {ℓ} → {A : Type ℓ} → (a0 aS : A) → ℕ → A +caseNat a0 aS zero = a0 +caseNat a0 aS (suc n) = aS + +-- zero is not in the image of Pos→ℕ. +znots : {n : ℕ} → ¬ (0 ≡ suc n) +znots eq = transp (λ i → caseNat ℕ ⊥ (eq i)) i0 0 + +zero≠Pos→ℕ : (p : Pos) → ¬ (zero ≡ Pos→ℕ p) +zero≠Pos→ℕ p = posInd (λ prf → znots prf) hs p + where + hs : (p : Pos) → ¬ (zero ≡ Pos→ℕ p) → zero ≡ Pos→ℕ (sucPos p) → ⊥ + hs p neq ieq = ⊥-rec (znots (ieq ∙ Pos→ℕsucPos p)) + +ℕ→Pos : ℕ → Pos +ℕ→Pos zero = pos1 +ℕ→Pos (suc zero) = pos1 +ℕ→Pos (suc (suc n)) = sucPos (ℕ→Pos (suc n)) + +ℕ→PosSuc : ∀ n → ¬ (zero ≡ n) → ℕ→Pos (suc n) ≡ sucPos (ℕ→Pos n) +ℕ→PosSuc zero neq = ⊥-elim (neq localrefl) +ℕ→PosSuc (suc n) neq = localrefl + +Pos→ℕ→Pos : (p : Pos) → ℕ→Pos (Pos→ℕ p) ≡ p +Pos→ℕ→Pos p = posInd localrefl hs p + where + hs : (p : Pos) → ℕ→Pos (Pos→ℕ p) ≡ p → ℕ→Pos (Pos→ℕ (sucPos p)) ≡ sucPos p + hs p hp = + ℕ→Pos (Pos→ℕ (sucPos p)) ≡⟨ localap ℕ→Pos (Pos→ℕsucPos p) ⟩ + ℕ→Pos (suc (Pos→ℕ p)) ≡⟨ ℕ→PosSuc (Pos→ℕ p) (zero≠Pos→ℕ p) ⟩ + sucPos (ℕ→Pos (Pos→ℕ p)) ≡⟨ localap sucPos hp ⟩ + sucPos p ∎ + +ℕ→Pos→ℕ : (n : ℕ) → Pos→ℕ (ℕ→Pos (suc n)) ≡ suc n +ℕ→Pos→ℕ zero = localrefl +ℕ→Pos→ℕ (suc n) = + Pos→ℕ (sucPos (ℕ→Pos (suc n))) ≡⟨ Pos→ℕsucPos (ℕ→Pos (suc n)) ⟩ + suc (Pos→ℕ (ℕ→Pos (suc n))) ≡⟨ localap suc (ℕ→Pos→ℕ n) ⟩ + suc (suc n) ∎ + +-- Binary numbers +data Bin : Type where + bin0 : Bin + binPos : Pos → Bin + +ℕ→Bin : ℕ → Bin +ℕ→Bin zero = bin0 +ℕ→Bin (suc n) = binPos (ℕ→Pos (suc n)) + +Bin→ℕ : Bin → ℕ +Bin→ℕ bin0 = zero +Bin→ℕ (binPos x) = Pos→ℕ x + +ℕ→Bin→ℕ : (n : ℕ) → Bin→ℕ (ℕ→Bin n) ≡ n +ℕ→Bin→ℕ zero = localrefl +ℕ→Bin→ℕ (suc zero) = localrefl +ℕ→Bin→ℕ (suc (suc n)) = + Pos→ℕ (sucPos (ℕ→Pos (suc n))) ≡⟨ Pos→ℕsucPos (ℕ→Pos (suc n)) ⟩ + suc (Pos→ℕ (ℕ→Pos (suc n))) ≡⟨ localap suc (ℕ→Bin→ℕ (suc n)) ⟩ + suc (suc n) ∎ + +Bin→ℕ→Bin : (n : Bin) → ℕ→Bin (Bin→ℕ n) ≡ n +Bin→ℕ→Bin bin0 = localrefl +Bin→ℕ→Bin (binPos p) = posInd localrefl (λ p _ → rem p) p + where + rem : (p : Pos) → ℕ→Bin (Pos→ℕ (sucPos p)) ≡ binPos (sucPos p) + rem p = + ℕ→Bin (Pos→ℕ (sucPos p)) ≡⟨ localap ℕ→Bin (Pos→ℕsucPos p) ⟩ + binPos (ℕ→Pos (suc (Pos→ℕ p))) ≡⟨ localap binPos (ℕ→PosSuc (Pos→ℕ p) (zero≠Pos→ℕ p) ∙ + (localap sucPos (Pos→ℕ→Pos p))) ⟩ + binPos (sucPos p) ∎ +``` diff --git a/src/HoTTEST/Agda/Exercises/01-Exercises.lagda.md b/src/HoTTEST/Agda/Exercises/01-Exercises.lagda.md new file mode 100644 index 0000000..c5fc2a7 --- /dev/null +++ b/src/HoTTEST/Agda/Exercises/01-Exercises.lagda.md @@ -0,0 +1,279 @@ +# Week 01 - Agda Exercises + +## Please read before starting the exercises + +**The exercises are designed to increase in difficulty so that we can cater to +our large and diverse audience. This also means that it is *perfectly fine* if +you don't manage to do all exercises: some of them are definitely a bit hard for +beginners and there are likely too many exercises! You *may* wish to come back +to them later when you have learned more.** + +Having said that, here we go! + +This is a markdown file with Agda code, which means that it displays nicely on +GitHub, but at the same time you can load this file in Agda and fill the holes +to solve exercises. + +**Please make a copy of this file to work in, so that it doesn't get overwritten + (in case we update the exercises through `git`)!** + +```agda +{-# OPTIONS --without-K --allow-unsolved-metas #-} + +module 01-Exercises where + +open import prelude hiding (not-is-involution) +``` + +## Part I: Writing functions on Booleans, natural numbers and lists (★/★★) + +### Exercise 1 (★) + +In the lectures we defined `&&` (logical and) on `Bool` by pattern matching on +the leftmost argument only. + +*NB*: We didn't get round to doing this in the lecture, but see `_&&_` in + [introduction.lagda.md](../Lecture-Notes/files/introduction.lagda.md). + +**Define** the same operation but this time by pattern matching (case splitting) + on both arguments. + +```agda +_&&'_ : Bool → Bool → Bool +a &&' b = {!!} +``` + +One advantage of this definition is that it reads just like a Boolean truth +table. Later on in this exercise sheet, we will see a disadvantange of this more +verbose definition. + +### Exercise 2 (★) + +**Define** `xor` (excluse or) on `Bool`. Exclusive or is true if and only if +*exactly one* of its arguments is true. + +```agda +_xor_ : Bool → Bool → Bool +a xor b = {!!} +``` + +### Exercise 3 (★) + +**Define** the exponential and factorial functions on natural numbers. + +If you do things correctly, then the examples should compute correctly, i.e. the +proof that 3 ^ 4 ≡ 81 should simply be given by `refl _` which says that the +left hand side and the right hand side compute to the same value. + +```agda +_^_ : ℕ → ℕ → ℕ +n ^ m = {!!} + +^-example : 3 ^ 4 ≡ 81 +^-example = {!!} -- refl 81 should fill the hole here + +_! : ℕ → ℕ +n ! = {!!} + +!-example : 4 ! ≡ 24 +!-example = {!!} -- refl 24 should fill the hole here +``` + +### Exercise 4 (★) + +We can recursively compute the maximum of two natural numbers as follows. +```agda +max : ℕ → ℕ → ℕ +max zero m = m +max (suc n) zero = suc n +max (suc n) (suc m) = suc (max n m) +``` + +**Define** the minimum of two natural numbers analogously. + +```agda +min : ℕ → ℕ → ℕ +min = {!!} + +min-example : min 5 3 ≡ 3 +min-example = {!!} -- refl 3 should fill the hole here +``` + +### Exercise 5 (★) + +Use pattern matching on lists to **define** `map`. + +This function should behave as follows: +`map f [x₁ , x₂ , ... , xₙ] = [f x₁ , f x₂ , ... , f xₙ]`. +That is, `map f xs` applies the given function `f` to every +element of the list `xs` and returns the resulting list. + +```agda +map : {X Y : Type} → (X → Y) → List X → List Y +map f xs = {!!} + +map-example : map (_+ 3) (1 :: 2 :: 3 :: []) ≡ 4 :: 5 :: 6 :: [] +map-example = {!!} -- refl _ should fill the hole here + + -- We write the underscore, because we don't wish to repeat + -- the relatively long "4 :: 5 :: 6 :: []" and Agda can + -- figure out what is supposed to go there. +``` + +### Exercise 6 (★★) + +**Define** a function `filter` that takes predicate `p : X → Bool` and a list + `xs` that returns the list of elements of `xs` for which `p` is true. + +For example, filtering the non-zero elements of the list [4 , 3 , 0 , 1 , 0] +should return [4 , 3 , 1], see the code below. + +```agda +filter : {X : Type} (p : X → Bool) → List X → List X +filter = {!!} + +is-non-zero : ℕ → Bool +is-non-zero zero = false +is-non-zero (suc _) = true + +filter-example : filter is-non-zero (4 :: 3 :: 0 :: 1 :: 0 :: []) ≡ 4 :: 3 :: 1 :: [] +filter-example = {!!} -- refl _ should fill the hole here +``` + +## Part II: The identity type of the Booleans (★/★★) + +In the lectures we saw a definition of `≣` on natural numbers where the idea was +that `x ≣ y` is a type which either has precisely one element, if `x` and `y` +are the same natural number, or else is empty, if `x` and `y` are different. + +### Exercise 1 (★) + +**Define** `≣` for Booleans this time. + +```agda +_≣_ : Bool → Bool → Type +a ≣ b = {!!} +``` + +### Exercise 2 (★) + +**Show** that for every Boolean `b` we can find an element of the type `b ≣ b`. + +```agda +Bool-refl : (b : Bool) → b ≣ b +Bool-refl = {!!} +``` + +### Exercise 3 (★★) + +Just like we did in the lectures for natural numbers, **show** that we can go +back and forth between `a ≣ b` and `a ≡ b`. + +*NB*: Again, we didn't have time to do this in the lectures, but see + [introduction.lagda.md](../Lecture-Notes/files/introduction.lagda.md), + specifically the functions `back` and `forth` there. + +```agda +≡-to-≣ : (a b : Bool) → a ≡ b → a ≣ b +≡-to-≣ = {!!} + +≣-to-≡ : (a b : Bool) → a ≣ b → a ≡ b +≣-to-≡ = {!!} +``` + +## Part III: Proving in Agda (★★/★★★) + +We now turn to *proving* things in Agda: one of its key features. + +For example, here is a proof that `not (not b) ≡ b` for every Boolean `b`. + +```agda +not-is-involution : (b : Bool) → not (not b) ≡ b +not-is-involution true = refl true +not-is-involution false = refl false +``` + +### Exercise 1 (★★) + +Use pattern matching to **prove** that `||` is commutative. + +```agda +||-is-commutative : (a b : Bool) → a || b ≡ b || a +||-is-commutative a b = {!!} +``` + +### Exercise 2 (★★) + +Taking inspiration from the above, try to **state** and **prove** that `&&` is +commutative. + +```agda +&&-is-commutative : {!!} +&&-is-commutative = {!!} +``` + +### Exercise 3 (★★) + +**Prove** that `&&` and `&&'` are both associative. + +```agda +&&-is-associative : (a b c : Bool) → a && (b && c) ≡ (a && b) && c +&&-is-associative = {!!} + +&&'-is-associative : (a b c : Bool) → a &&' (b &&' c) ≡ (a &&' b) &&' c +&&'-is-associative = {!!} +``` + +**Question**: Can you spot a downside of the more verbose definition of `&&'` + now? + +### Exercise 4 (★★★) + +Another key feature of Agda is its ability to carry out inductive proofs. For +example, here is a commented inductive proof that `max` is commutative. + +```agda +max-is-commutative : (n m : ℕ) → max n m ≡ max m n +max-is-commutative zero zero = refl zero +max-is-commutative zero (suc m) = refl (suc m) +max-is-commutative (suc n) zero = refl (suc n) +max-is-commutative (suc n) (suc m) = to-show + where + IH : max n m ≡ max m n -- induction hypothesis + IH = max-is-commutative n m -- recursive call on smaller arguments + to-show : suc (max n m) ≡ suc (max m n) + to-show = ap suc IH -- ap(ply) suc on both sides of the equation +``` + +**Prove** analogously that `min` is commutative. + +```agda +min-is-commutative : (n m : ℕ) → min n m ≡ min m n +min-is-commutative = {!!} +``` + +### Exercise 5 (★★★) + +Using another inductive proof, **show** that `n ≡ n + 0` for every natural +number `n`. + +```agda +0-right-neutral : (n : ℕ) → n ≡ n + 0 +0-right-neutral = {!!} +``` + +### Exercise 6 (★★★) + +The function `map` on lists is a so-called *functor*, which means that it +respects the identity and composition, as formally expressed below. + +Try to **prove** these equations using pattern matching and inductive proofs. + +```agda +map-id : {X : Type} (xs : List X) → map id xs ≡ xs +map-id xs = {!!} + +map-comp : {X Y Z : Type} (f : X → Y) (g : Y → Z) + (xs : List X) → map (g ∘ f) xs ≡ map g (map f xs) +map-comp f g xs = {!!} +``` diff --git a/src/HoTTEST/Agda/Exercises/01-Solutions.lagda.md b/src/HoTTEST/Agda/Exercises/01-Solutions.lagda.md new file mode 100644 index 0000000..fac047d --- /dev/null +++ b/src/HoTTEST/Agda/Exercises/01-Solutions.lagda.md @@ -0,0 +1,182 @@ +# Week 01 - Agda Exercises - Solutions + +```agda +{-# OPTIONS --without-K --safe #-} + +module 01-Solutions where + +open import prelude hiding (not-is-involution) +``` + +## Part I + +```agda +_&&'_ : Bool → Bool → Bool +true &&' true = true +true &&' false = false +false &&' true = false +false &&' false = false + +_xor_ : Bool → Bool → Bool +true xor true = false +true xor false = true +false xor true = true +false xor false = false + +_^_ : ℕ → ℕ → ℕ +n ^ zero = 1 +n ^ suc m = n * (n ^ m) + +^-example : 3 ^ 4 ≡ 81 +^-example = refl 81 + +_! : ℕ → ℕ +zero ! = 1 +suc n ! = suc n * (n !) + +!-example : 4 ! ≡ 24 +!-example = refl 24 + +max : ℕ → ℕ → ℕ +max zero m = m +max (suc n) zero = suc n +max (suc n) (suc m) = suc (max n m) + +min : ℕ → ℕ → ℕ +min zero m = zero +min (suc n) zero = zero +min (suc n) (suc m) = suc (min n m) + +min-example : min 5 3 ≡ 3 +min-example = refl 3 + +map : {X Y : Type} → (X → Y) → List X → List Y +map f [] = [] +map f (x :: xs) = f x :: map f xs + +map-example : map (_+ 3) (1 :: 2 :: 3 :: []) ≡ 4 :: 5 :: 6 :: [] +map-example = refl _ + +filter : {X : Type} (p : X → Bool) → List X → List X +filter {X} p [] = [] +filter {X} p (x :: xs) = if (p x) then (x :: ys) else ys + where + ys : List X + ys = filter p xs + +is-non-zero : ℕ → Bool +is-non-zero zero = false +is-non-zero (suc _) = true + +filter-example : filter is-non-zero (4 :: 3 :: 0 :: 1 :: 0 :: []) ≡ 4 :: 3 :: 1 :: [] +filter-example = refl _ +``` + +## Part II + +```agda +_≣_ : Bool → Bool → Type +true ≣ true = 𝟙 +true ≣ false = 𝟘 +false ≣ true = 𝟘 +false ≣ false = 𝟙 + +Bool-refl : (b : Bool) → b ≣ b +Bool-refl true = ⋆ +Bool-refl false = ⋆ + +≡-to-≣ : (a b : Bool) → a ≡ b → a ≣ b +≡-to-≣ b b (refl b) = Bool-refl b + +≣-to-≡ : (a b : Bool) → a ≣ b → a ≡ b +≣-to-≡ true true ⋆ = refl true +≣-to-≡ false false ⋆ = refl false +``` + +## Part III + +```agda +not-is-involution : (b : Bool) → not (not b) ≡ b +not-is-involution true = refl true +not-is-involution false = refl false + +||-is-commutative : (a b : Bool) → a || b ≡ b || a +||-is-commutative true true = refl true +||-is-commutative true false = refl true +||-is-commutative false true = refl true +||-is-commutative false false = refl false + +&&-is-commutative : (a b : Bool) → a && b ≡ b && a +&&-is-commutative true true = refl true +&&-is-commutative true false = refl false +&&-is-commutative false true = refl false +&&-is-commutative false false = refl false + +-- Notice how we choose to pattern match on the leftmost argument, because +-- that's also how we defined &&. +&&-is-associative : (a b c : Bool) → a && (b && c) ≡ (a && b) && c +&&-is-associative true b c = refl (b && c) +&&-is-associative false b c = refl false + +-- Because &&' is defined by pattern matching on all its arguments, we now need +-- to consider 2^3 = 8 cases. +&&'-is-associative : (a b c : Bool) → a &&' (b &&' c) ≡ (a &&' b) &&' c +&&'-is-associative true true true = refl true +&&'-is-associative true true false = refl false +&&'-is-associative true false true = refl false +&&'-is-associative true false false = refl false +&&'-is-associative false true true = refl false +&&'-is-associative false true false = refl false +&&'-is-associative false false true = refl false +&&'-is-associative false false false = refl false + +max-is-commutative : (n m : ℕ) → max n m ≡ max m n +max-is-commutative zero zero = refl zero +max-is-commutative zero (suc m) = refl (suc m) +max-is-commutative (suc n) zero = refl (suc n) +max-is-commutative (suc n) (suc m) = to-show + where + IH : max n m ≡ max m n -- induction hypothesis + IH = max-is-commutative n m -- recursive call on smaller arguments + to-show : suc (max n m) ≡ suc (max m n) + to-show = ap suc IH -- ap(ply) suc on both sides of the equation + +min-is-commutative : (n m : ℕ) → min n m ≡ min m n +min-is-commutative zero zero = refl zero +min-is-commutative zero (suc m) = refl zero +min-is-commutative (suc n) zero = refl zero +min-is-commutative (suc n) (suc m) = to-show + where + IH : min n m ≡ min m n -- induction hypothesis + IH = min-is-commutative n m -- recursive call on smaller arguments + to-show : suc (min n m) ≡ suc (min m n) + to-show = ap suc IH -- ap(ply) suc on both sides of the equation + +0-right-neutral : (n : ℕ) → n ≡ n + 0 +0-right-neutral zero = refl zero +0-right-neutral (suc n) = to-show + where + IH : n ≡ n + 0 -- induction hypothesis + IH = 0-right-neutral n -- recursive call on smaller argument + to-show : suc n ≡ suc (n + 0) + to-show = ap suc IH -- ap(ply) suc on both sides of the equation + +map-id : {X : Type} (xs : List X) → map id xs ≡ xs +map-id [] = refl [] +map-id (x :: xs) = to-show + where + IH : map id xs ≡ xs + IH = map-id xs + to-show : x :: map id xs ≡ x :: xs + to-show = ap (x ::_) IH + +map-comp : {X Y Z : Type} (f : X → Y) (g : Y → Z) + (xs : List X) → map (g ∘ f) xs ≡ map g (map f xs) +map-comp f g [] = refl [] +map-comp f g (x :: xs) = to-show + where + IH : map (g ∘ f) xs ≡ map g (map f xs) + IH = map-comp f g xs + to-show : g (f x) :: map (g ∘ f) xs ≡ g (f x) :: map g (map f xs) + to-show = ap (g (f x) ::_) IH +``` diff --git a/src/HoTTEST/Agda/Exercises/02-Exercises.lagda.md b/src/HoTTEST/Agda/Exercises/02-Exercises.lagda.md new file mode 100644 index 0000000..1303126 --- /dev/null +++ b/src/HoTTEST/Agda/Exercises/02-Exercises.lagda.md @@ -0,0 +1,182 @@ +# Week 02 - Agda Exercises + +## Please read before starting the exercises + +**The exercises are designed to increase in difficulty so that we can cater to +our large and diverse audience. This also means that it is *perfectly fine* if +you don't manage to do all exercises: some of them are definitely a bit hard for +beginners and there are likely too many exercises! You *may* wish to come back +to them later when you have learned more.** + +Having said that, here we go! + +This is a markdown file with Agda code, which means that it displays nicely on +GitHub, but at the same time you can load this file in Agda and fill the holes +to solve exercises. + +**Please make a copy of this file to work in, so that it doesn't get overwritten + (in case we update the exercises through `git`)!** + +```agda +{-# OPTIONS --without-K --allow-unsolved-metas #-} + +module 02-Exercises where + +open import prelude +open import decidability +open import sums +``` + +## Part I: Propositions as types + + +### Exercise 1 (★) + +Prove +```agda +uncurry : {A B X : Type} → (A → B → X) → (A × B → X) +uncurry = {!!} + +curry : {A B X : Type} → (A × B → X) → (A → B → X) +curry = {!!} +``` +You might know these functions from programming e.g. in Haskell. +But what do they say under the propositions-as-types interpretation? + + +### Exercise 2 (★) + +Consider the following goals: +```agda +[i] : {A B C : Type} → (A × B) ∔ C → (A ∔ C) × (B ∔ C) +[i] = {!!} + +[ii] : {A B C : Type} → (A ∔ B) × C → (A × C) ∔ (B × C) +[ii] = {!!} + +[iii] : {A B : Type} → ¬ (A ∔ B) → ¬ A × ¬ B +[iii] = {!!} + +[iv] : {A B : Type} → ¬ (A × B) → ¬ A ∔ ¬ B +[iv] = {!!} + +[v] : {A B : Type} → (A → B) → ¬ B → ¬ A +[v] = {!!} + +[vi] : {A B : Type} → (¬ A → ¬ B) → B → A +[vi] = {!!} + +[vii] : {A B : Type} → ((A → B) → A) → A +[vii] = {!!} + +[viii] : {A : Type} {B : A → Type} + → ¬ (Σ a ꞉ A , B a) → (a : A) → ¬ B a +[viii] = {!!} + +[ix] : {A : Type} {B : A → Type} + → ¬ ((a : A) → B a) → (Σ a ꞉ A , ¬ B a) +[ix] = {!!} + +[x] : {A B : Type} {C : A → B → Type} + → ((a : A) → (Σ b ꞉ B , C a b)) + → Σ f ꞉ (A → B) , ((a : A) → C a (f a)) +[x] = {!!} +``` +For each goal determine whether it is provable or not. +If it is, fill it. If not, explain why it shouldn't be possible. +Propositions-as-types might help. + + +### Exercise 3 (★★) + +```agda +¬¬_ : Type → Type +¬¬ A = ¬ (¬ A) + +¬¬¬ : Type → Type +¬¬¬ A = ¬ (¬¬ A) +``` +In the lecture we have discussed that we can't prove `∀ {A : Type} → ¬¬ A → A`. +What you can prove however, is +```agda +tne : ∀ {A : Type} → ¬¬¬ A → ¬ A +tne = {!!} +``` + + +### Exercise 4 (★★★) +Prove +```agda +¬¬-functor : {A B : Type} → (A → B) → ¬¬ A → ¬¬ B +¬¬-functor = {!!} + +¬¬-kleisli : {A B : Type} → (A → ¬¬ B) → ¬¬ A → ¬¬ B +¬¬-kleisli = {!!} +``` +Hint: For the second goal use `tne` from the previous exercise + + + + + +## Part II: `_≡_` for `Bool` + +**In this exercise we want to investigate what equality of booleans looks like. +In particular we want to show that for `true false : Bool` we have `true ≢ false`.** + +### Exercise 1 (★) + +Under the propositions-as-types paradigm, an inhabited type corresponds +to a true proposition while an uninhabited type corresponds to a false proposition. +With this in mind construct a family +```agda +bool-as-type : Bool → Type +bool-as-type = {!!} +``` +such that `bool-as-type true` corresponds to "true" and +`bool-as-type false` corresponds to "false". (Hint: +we have seen canonical types corresponding true and false in the lectures) + + +### Exercise 2 (★★) + +Prove +```agda +bool-≡-char₁ : ∀ (b b' : Bool) → b ≡ b' → (bool-as-type b ⇔ bool-as-type b') +bool-≡-char₁ = {!!} +``` + + +### Exercise 3 (★★) + +Using ex. 2, conclude that +```agda +true≢false : ¬ (true ≡ false) +true≢false () +``` +You can actually prove this much easier! How? + + +### Exercise 4 (★★★) + +Finish our characterisation of `_≡_` by proving +```agda +bool-≡-char₂ : ∀ (b b' : Bool) → (bool-as-type b ⇔ bool-as-type b') → b ≡ b' +bool-≡-char₂ = {!!} +``` + + +## Part III (🌶) +A type `A` is called *discrete* if it has decidable equality. +Consider the following predicate on types: +```agda +has-bool-dec-fct : Type → Type +has-bool-dec-fct A = Σ f ꞉ (A → A → Bool) , (∀ x y → x ≡ y ⇔ (f x y) ≡ true) +``` + +Prove that + +```agda +decidable-equality-char : (A : Type) → has-decidable-equality A ⇔ has-bool-dec-fct A +decidable-equality-char = ? +``` diff --git a/src/HoTTEST/Agda/Exercises/02-Solutions.lagda.md b/src/HoTTEST/Agda/Exercises/02-Solutions.lagda.md new file mode 100644 index 0000000..694b74d --- /dev/null +++ b/src/HoTTEST/Agda/Exercises/02-Solutions.lagda.md @@ -0,0 +1,166 @@ +# Week 02 - Agda Exercises - Solutions + +```agda +{-# OPTIONS --without-K --safe #-} + +module 02-Solutions where + +open import prelude +open import decidability +open import sums + +``` + +## Part I + +### Exercise 1 +```agda +uncurry : {A B X : Type} → (A → B → X) → (A × B → X) +uncurry f (a , b) = f a b + +curry : {A B X : Type} → (A × B → X) → (A → B → X) +curry f a b = f (a , b) +``` +Under the propositions-as-types interpretation curry and uncurry +tell us that "if A then if B then X" is (logically) equivalent +to "if A and B then X" + +### Exercise 2 +```agda +[i] : {A B C : Type} → (A × B) ∔ C → (A ∔ C) × (B ∔ C) +[i] (inl (a , b)) = inl a , inl b +[i] (inr c) = inr c , inr c + +[ii] : {A B C : Type} → (A ∔ B) × C → (A × C) ∔ (B × C) +[ii] (inl a , c) = inl (a , c) +[ii] (inr b , c) = inr (b , c) + +[iii] : {A B : Type} → ¬ (A ∔ B) → ¬ A × ¬ B +pr₁ ([iii] f) a = f (inl a) +pr₂ ([iii] f) b = f (inr b) +``` + +The next goal `[iv] : {A B : Type} → ¬ (A × B) → ¬ A ∔ ¬ B` +is not provable. Under propositions-as-types it says that +"not (A and B) implies not A or not B", which is not true +in constructive logic. At the end we have to give a term of +the form `inl ...` or `inr ...` but for abstract `A B` we +can not say of which form it should be. +```agda +[v] : {A B : Type} → (A → B) → ¬ B → ¬ A -- also known as contraposition +[v] f g a = g (f a) +``` + +Neither of `[vi] : {A B : Type} → (¬ A → ¬ B) → B → A` +and `[vii] : {A B : Type} → ((A → B) → A) → A` are provable +Under propositions-as-types `[vi]` is known as *inverse contraposition* +and `[vii]` is known as *Peirce's law*. At the end we have to construct +something of type `A` but this is not possible with all the assumptions +being functions. +```agda +[viii] : {A : Type} {B : A → Type} + → ¬ (Σ a ꞉ A , B a) → (a : A) → ¬ B a +[viii] f a bₐ = f (a , bₐ) +``` +The next goal +`[ix] : {A : Type} {B : A → Type} → ¬ ((a : A) → B a) → (Σ a ꞉ A , ¬ B a)` +reads as: "If not for all a, B(a), then there is an a such that not B(a)" +This is not true in constructive logic. Again, we have to construct +an `a : A` as the first projection of the Sigma-type in the conclusion, +which is not possible from our assumptions. + +```agda +[x] : {A B : Type} {C : A → B → Type} + → ((a : A) → (Σ b ꞉ B , C a b)) + → Σ f ꞉ (A → B) , ((a : A) → C a (f a)) +pr₁ ([x] g) a = g a .pr₁ +pr₂ ([x] g) a = g a .pr₂ +``` +Note that under propositions-as-types `[x]` reads somewhat like the +*axiom of choice*. Yet it is still provable. This result is often +referred to as the *distributivity of Π over Σ* and shows that +propositions-as-types should be taken with a grain of salt sometimes. + +### Exercise 3 +```agda +¬¬_ : Type → Type +¬¬ A = ¬ (¬ A) + +¬¬¬ : Type → Type +¬¬¬ A = ¬ (¬¬ A) + +tne : ∀ {A : Type} → ¬¬¬ A → ¬ A +tne f a = f (λ g → g a) +``` + +### Exercise 4 +```agda +¬¬-functor : {A B : Type} → (A → B) → ¬¬ A → ¬¬ B +¬¬-functor f = [v] ([v] f) + +¬¬-kleisli : {A B : Type} → (A → ¬¬ B) → ¬¬ A → ¬¬ B +¬¬-kleisli f g = tne (¬¬-functor f g) +``` + + + + +## Part II + +```agda +bool-as-type : Bool → Type +bool-as-type true = 𝟙 +bool-as-type false = 𝟘 + +bool-≡-char₁ : ∀ (b b' : Bool) → b ≡ b' → (bool-as-type b ⇔ bool-as-type b') +bool-≡-char₁ b .b (refl .b) = ⇔-refl + where + ⇔-refl : {X : Type} → X ⇔ X + pr₁ ⇔-refl x = x + pr₂ ⇔-refl x = x + +true≢false : ¬ (true ≡ false) +true≢false p = bool-≡-char₁ true false p .pr₁ ⋆ +-- also true≢false () + +bool-≡-char₂ : ∀ (b b' : Bool) → (bool-as-type b ⇔ bool-as-type b') → b ≡ b' +bool-≡-char₂ true true (b→b' , b'→b) = refl true +bool-≡-char₂ true false (b→b' , b'→b) = 𝟘-elim (b→b' ⋆) +bool-≡-char₂ false true (b→b' , b'→b) = 𝟘-elim (b'→b ⋆) +bool-≡-char₂ false false (b→b' , b'→b) = refl false +``` + + + + +## Part III + +```agda +has-bool-dec-fct : Type → Type +has-bool-dec-fct A = Σ f ꞉ (A → A → Bool) , (∀ x y → x ≡ y ⇔ (f x y) ≡ true) + +decidable-equality-char : (A : Type) → has-decidable-equality A ⇔ has-bool-dec-fct A +pr₁ (decidable-equality-char A) discA = f , f-dec -- left to right implication + where + f' : (a b : A) → is-decidable (a ≡ b) → Bool + f' a b (inl _) = true + f' a b (inr _) = false + + f'-refl : (x : A) (d : is-decidable (x ≡ x)) → f' x x d ≡ true + f'-refl x (inl _) = refl true + f'-refl x (inr x≢x) = 𝟘-nondep-elim (x≢x (refl x)) + + f : A → A → Bool + f a b = f' a b (discA a b) + + f-dec : ∀ x y → x ≡ y ⇔ (f x y) ≡ true + pr₁ (f-dec x .x) (refl .x) = f'-refl x (discA x x) + pr₂ (f-dec x y) with discA x y + ... | (inl p) = λ _ → p + ... | (inr _) = λ q → 𝟘-nondep-elim (true≢false (q ⁻¹)) + +pr₂ (decidable-equality-char A) (f , f-dec) x y -- right to left implication + with Bool-has-decidable-equality (f x y) true +... | (inl p) = inl (f-dec x y .pr₂ p) +... | (inr g) = inr λ p → g (f-dec x y .pr₁ p) +``` diff --git a/src/HoTTEST/Agda/Exercises/03-Exercises.lagda.md b/src/HoTTEST/Agda/Exercises/03-Exercises.lagda.md new file mode 100644 index 0000000..61f9055 --- /dev/null +++ b/src/HoTTEST/Agda/Exercises/03-Exercises.lagda.md @@ -0,0 +1,349 @@ +# Week 03 - Agda Exercises + +## Please read before starting the exercises + +**The exercises are designed to increase in difficulty so that we can cater to +our large and diverse audience. This also means that it is *perfectly fine* if +you don't manage to do all exercises: some of them are definitely a bit hard for +beginners and there are likely too many exercises! You *may* wish to come back +to them later when you have learned more.** + +Having said that, here we go! + +This is a markdown file with Agda code, which means that it displays nicely on +GitHub, but at the same time you can load this file in Agda and fill the holes +to solve exercises. + +**Please make a copy of this file to work in, so that it doesn't get overwritten + (in case we update the exercises through `git`)!** + +```agda +{-# OPTIONS --without-K --allow-unsolved-metas #-} + +module 03-Exercises where + +open import prelude hiding (_∼_) +``` + +## Part I -- Homotopies + +It is often convenient to work with *pointwise equality* of functions, defined as follows. +```agda +module _ {A : Type} {B : A → Type} where + _∼_ : ((x : A) → B x) → ((x : A) → B x) → Type + f ∼ g = ∀ x → f x ≡ g x +``` +An element of `f ∼ g` is usually called a homotopy. + +### Exercise 1 (⋆⋆) + +Unsurprisingly, many properties of this type of pointwise equalities +can be inferred directly from the same operations on paths. + +Try to prove reflexivity, symmetry and transitivity of `_∼_` by filling these holes. +```agda + ∼-refl : (f : (x : A) → B x) → f ∼ f + ∼-refl f = {!!} + + ∼-inv : (f g : (x : A) → B x) → (f ∼ g) → (g ∼ f) + ∼-inv f g H x = {!!} + + ∼-concat : (f g h : (x : A) → B x) → f ∼ g → g ∼ h → f ∼ h + ∼-concat f g h H K x = {!!} + + infix 0 _∼_ +``` + +## Part II -- Isomorphisms + +A function `f : A → B` is called a *bijection* if there is a function `g : B → A` in the opposite direction such that `g ∘ f ∼ id` and `f ∘ g ∼ id`. Recall that `_∼_` is [pointwise equality](../Lecture-Notes/files/identity-type.lagda.md) and that `id` is the [identity function](../Lecture-Notes/files/products.lagda.md). This means that we can convert back and forth between the types `A` and `B` landing at the same element we started with, either from `A` or from `B`. In this case, we say that the types `A` and `B` are *isomorphic*, and we write `A ≅ B`. Bijections are also called type *isomorphisms*. We can define these concepts in Agda using [sum types](../Lecture-Notes/files/sums.lagda.md) or [records](https://agda.readthedocs.io/en/latest/language/record-types.html). We will adopt the latter, but we include both definitions for the sake of illustration. +Recall that we [defined](../Lecture-Notes/files/general-notation.lagda.md) the domain of a function `f : A → B` to be `A` and its codomain to be `B`. + +We adopt this definition of isomorphisms using records. +```agda +record is-bijection {A B : Type} (f : A → B) : Type where + constructor + Inverse + field + inverse : B → A + η : inverse ∘ f ∼ id + ε : f ∘ inverse ∼ id + +record _≅_ (A B : Type) : Type where + constructor + Isomorphism + field + bijection : A → B + bijectivity : is-bijection bijection + +infix 0 _≅_ +``` + +### Exercise 2 (⋆) + +Reformulate the same definition using Sigma-types. +```agda +is-bijection' : {A B : Type} → (A → B) → Type +is-bijection' f = {!!} + +_≅'_ : Type → Type → Type +A ≅' B = {!!} +``` +The definition with `Σ` is probably more intuitive, but, as discussed above, +the definition with a record is often easier to work with, +because we can easily extract the components of the definitions using the names of the fields. +It also often allows Agda to infer more types, and to give us more sensible goals in the +interactive development of Agda programs and proofs. + +Notice that `inverse` plays the role of `g`. +The reason we use `inverse` is that then we can use the word +`inverse` to extract the inverse of a bijection. +Similarly we use `bijection` for `f`, as we can use the word +`bijection` to extract the bijection from a record. + +This type can be defined to be `𝟙 ∔ 𝟙` using the coproduct, +but we give a direct definition which will allow us to discuss some relationships between the various type formers of Basic MLTT. + +```agda +data 𝟚 : Type where + 𝟎 𝟏 : 𝟚 +``` + +### Exercise 3 (⋆⋆) + +Prove that 𝟚 and Bool are isomorphic + +```agda +Bool-𝟚-isomorphism : Bool ≅ 𝟚 +Bool-𝟚-isomorphism = record { bijection = {!!} ; bijectivity = {!!} } + where + f : Bool → 𝟚 + f false = {!!} + f true = {!!} + + g : 𝟚 → Bool + g 𝟎 = {!!} + g 𝟏 = {!!} + + gf : g ∘ f ∼ id + gf true = {!!} + gf false = {!!} + + fg : f ∘ g ∼ id + fg 𝟎 = {!!} + fg 𝟏 = {!!} + + f-is-bijection : is-bijection f + f-is-bijection = record { inverse = {!!} ; η = {!!} ; ε = {!!} } +``` + + +## Part III - Finite Types + +In the last HoTT Exercises we encountered two definitions of the finite types. +Here we prove that they are isomorphic. +Note that `zero` was called `pt` and suc `i` on the HoTT exercise sheet. + +```agda +data Fin : ℕ → Type where + zero : {n : ℕ} → Fin (suc n) + suc : {n : ℕ} → Fin n → Fin (suc n) +``` + +### Exercise 4 (⋆) + +Prove the elimination principle of `Fin`. +```agda +Fin-elim : (A : {n : ℕ} → Fin n → Type) + → ({n : ℕ} → A {suc n} zero) + → ({n : ℕ} (k : Fin n) → A k → A (suc k)) + → {n : ℕ} (k : Fin n) → A k +Fin-elim A a f = h + where + h : {n : ℕ} (k : Fin n) → A k + h zero = {!!} + h (suc k) = {!!} +``` + +We give the other definition of the finite types and introduce some notation. + +```agda +Fin' : ℕ → Type +Fin' 0 = 𝟘 +Fin' (suc n) = 𝟙 ∔ Fin' n + +zero' : {n : ℕ} → Fin' (suc n) +zero' = inl ⋆ + +suc' : {n : ℕ} → Fin' n → Fin' (suc n) +suc' = inr +``` + +### Exercise 5 (⋆⋆⋆) + +Prove that `Fin n` and `Fin' n` are isomorphic for every `n`. + +```agda +Fin-isomorphism : (n : ℕ) → Fin n ≅ Fin' n +Fin-isomorphism n = record { bijection = f n ; bijectivity = f-is-bijection n } + where + f : (n : ℕ) → Fin n → Fin' n + f (suc n) zero = {!!} + f (suc n) (suc k) = {!!} + + g : (n : ℕ) → Fin' n → Fin n + g (suc n) (inl ⋆) = {!!} + g (suc n) (inr k) = {!!} + + gf : (n : ℕ) → g n ∘ f n ∼ id + gf (suc n) zero = {!!} + gf (suc n) (suc k) = γ + where + IH : g n (f n k) ≡ k + IH = gf n k + + γ = g (suc n) (f (suc n) (suc k)) ≡⟨ {!!} ⟩ + g (suc n) (suc' (f n k)) ≡⟨ {!!} ⟩ + suc (g n (f n k)) ≡⟨ {!!} ⟩ + suc k ∎ + + fg : (n : ℕ) → f n ∘ g n ∼ id + fg (suc n) (inl ⋆) = {!!} + fg (suc n) (inr k) = γ + where + IH : f n (g n k) ≡ k + IH = fg n k + + γ = f (suc n) (g (suc n) (suc' k)) ≡⟨ {!!} ⟩ + f (suc n) (suc (g n k)) ≡⟨ {!!} ⟩ + suc' (f n (g n k)) ≡⟨ {!!} ⟩ + suc' k ∎ + + f-is-bijection : (n : ℕ) → is-bijection (f n) + f-is-bijection n = record { inverse = g n ; η = gf n ; ε = fg n} +``` + +## Part IV -- minimal elements in the natural numbers + +In this section we establish some definitions which are needed to state and prove the well-ordering +principle of the natural numbers. + +### Exercise 6 (⋆) + +Give the recursive definition of the less than or equals relation on the natural numbers. + +```agda +_≤₁_ : ℕ → ℕ → Type +0 ≤₁ y = {!!} +suc x ≤₁ 0 = {!!} +suc x ≤₁ suc y = {!!} +``` + +### Exercise 7 (⋆) + +Given a type family `P` over the naturals, a lower bound `n` is a natural number such that +for all other naturals `m`, we have that if `P(m)` holds, then`n ≤₁ m`. +Translate this definition into HoTT. + +```agda +is-lower-bound : (P : ℕ → Type) (n : ℕ) → Type +is-lower-bound P n = {!!} +``` + +We define the type of minimal elements of a type family over the naturals. +```agda +minimal-element : (P : ℕ → Type) → Type +minimal-element P = Σ n ꞉ ℕ , (P n) × (is-lower-bound P n) +``` + +### Exercise 8 (⋆) + +Prove that all numbers are at least as large as zero. +```agda +leq-zero : (n : ℕ) → 0 ≤₁ n +leq-zero n = {!!} +``` + + +## Part V -- The well-ordering principle of ℕ + +Classically, the well-ordering principle states that every non-empty set +of natural numbers has a least element. + +In HoTT, such subsets can be translated as decidable type family. +Recall the definition of decidability: +```agda +open import decidability + using (is-decidable; is-decidable-predicate) +``` + +The well-ordering principle reads +```agda +Well-ordering-principle = (P : ℕ → Type) → (d : is-decidable-predicate P) → (n : ℕ) → P n → minimal-element P +``` + +We shall prove this statement via induction on `n`. +In order to make the proof more readable, we first prove two lemmas. + +### Exercise 9 (🌶) + +What is the statement of `is-minimal-element-suc` +under the Curry-Howard interpretation? +Prove this lemma. + +```agda +is-minimal-element-suc : + (P : ℕ → Type) (d : is-decidable-predicate P) + (m : ℕ) (pm : P (suc m)) + (is-lower-bound-m : is-lower-bound (λ x → P (suc x)) m) → + ¬ (P 0) → is-lower-bound P (suc m) +is-minimal-element-suc P d m pm is-lower-bound neg-p0 = {! !} +``` + +### Exercise 10 (🌶) + +What is the statement of `well-ordering-principle-suc` +under the Curry-Howard interpretation? +Prove this lemma. + +```agda +well-ordering-principle-suc : + (P : ℕ → Type) (d : is-decidable-predicate P) + (n : ℕ) (p : P (suc n)) → + is-decidable (P 0) → + minimal-element (λ m → P (suc m)) → minimal-element P +well-ordering-principle-suc P d n p (inl p0) _ = {!!} +well-ordering-principle-suc P d n p (inr neg-p0) (m , (pm , is-min-m)) = {!!} +``` + +### Exercise 11 (🌶) + +Use the previous two lemmas to prove the well-ordering principle +```agda +well-ordering-principle : (P : ℕ → Type) → (d : is-decidable-predicate P) → (n : ℕ) → P n → minimal-element P +well-ordering-principle P d 0 p = {!!} +well-ordering-principle P d (suc n) p = well-ordering-principle-suc P d n p (d 0) {!!} +``` + +### Exercise 12 (🌶) + +Prove that the well-ordering principle returns 0 if `P 0` holds. + +```agda +is-zero-well-ordering-principle-suc : + (P : ℕ → Type) (d : is-decidable-predicate P) + (n : ℕ) (p : P (suc n)) (d0 : is-decidable (P 0)) → + (x : minimal-element (λ m → P (suc m))) (p0 : P 0) → + (pr₁ (well-ordering-principle-suc P d n p d0 x)) ≡ 0 +is-zero-well-ordering-principle-suc P d n p (inl p0) x q0 = {!!} +is-zero-well-ordering-principle-suc P d n p (inr np0) x q0 = {!!} + +is-zero-well-ordering-principle : + (P : ℕ → Type) (d : is-decidable-predicate P) → + (n : ℕ) → (pn : P n) → + P 0 → + pr₁ (well-ordering-principle P d n pn) ≡ 0 +is-zero-well-ordering-principle P d zero p p0 = {! !} +is-zero-well-ordering-principle P d (suc m) pm = + is-zero-well-ordering-principle-suc P d m pm (d 0) {!!} +``` diff --git a/src/HoTTEST/Agda/Exercises/03-Solutions.lagda.md b/src/HoTTEST/Agda/Exercises/03-Solutions.lagda.md new file mode 100644 index 0000000..12da7fc --- /dev/null +++ b/src/HoTTEST/Agda/Exercises/03-Solutions.lagda.md @@ -0,0 +1,359 @@ +# Week 03 - Agda Exercises - Solutions + +```agda +{-# OPTIONS --without-K --safe #-} + +module 03-Solutions where + +open import prelude hiding (_∼_) +``` + +## Part I -- Homotopies + +It is often convenient to work with *pointwise equality* of functions, defined as follows. +```agda +module _ {A : Type} {B : A → Type} where + _∼_ : ((x : A) → B x) → ((x : A) → B x) → Type + f ∼ g = ∀ x → f x ≡ g x +``` +An element of `f ∼ g` is usually called a homotopy. + +### Exercise 1 (⋆⋆) + +Unsurprisingly, many properties of this type of pointwise equalities +can be inferred directly from the same operations on paths. + +Try to prove reflexivity, symmetry and transitivity of `_∼_` by filling these holes. +```agda + ∼-refl : (f : (x : A) → B x) → f ∼ f + ∼-refl f = λ x → refl (f x) + + ∼-inv : (f g : (x : A) → B x) → (f ∼ g) → (g ∼ f) + ∼-inv f g H x = sym (H x) + + ∼-concat : (f g h : (x : A) → B x) → f ∼ g → g ∼ h → f ∼ h + ∼-concat f g h H K x = trans (H x) (K x) + + infix 0 _∼_ +``` + +## Part II -- Isomorphisms + +A function `f : A → B` is called a *bijection* if there is a function `g : B → A` in the opposite direction such that `g ∘ f ∼ id` and `f ∘ g ∼ id`. Recall that `_∼_` is [pointwise equality](../Lecture-Notes/files/identity-type.lagda.md) and that `id` is the [identity function](../Lecture-Notes/files/products.lagda.md). This means that we can convert back and forth between the types `A` and `B` landing at the same element we started with, either from `A` or from `B`. In this case, we say that the types `A` and `B` are *isomorphic*, and we write `A ≅ B`. Bijections are also called type *isomorphisms*. We can define these concepts in Agda using [sum types](../Lecture-Notes/files/sums.lagda.md) or [records](https://agda.readthedocs.io/en/latest/language/record-types.html). We will adopt the latter, but we include both definitions for the sake of illustration. +Recall that we [defined](../Lecture-Notes/files/general-notation.lagda.md) the domain of a function `f : A → B` to be `A` and its codomain to be `B`. + +We adopt this definition of isomorphisms using records. +```agda +record is-bijection {A B : Type} (f : A → B) : Type where + constructor + Inverse + field + inverse : B → A + η : inverse ∘ f ∼ id + ε : f ∘ inverse ∼ id + +record _≅_ (A B : Type) : Type where + constructor + Isomorphism + field + bijection : A → B + bijectivity : is-bijection bijection + +infix 0 _≅_ +``` + +### Exercise 2 (⋆) +Reformulate the same definition using Sigma-types. +```agda +is-bijection' : {A B : Type} → (A → B) → Type +is-bijection' f = Σ g ꞉ (codomain f → domain f) , ((g ∘ f ∼ id) × (f ∘ g ∼ id)) + +_≅'_ : Type → Type → Type +A ≅' B = Σ f ꞉ (A → B) , is-bijection' f +``` +The definition with `Σ` is probably more intuitive, but, as discussed above, +the definition with a record is often easier to work with, +because we can easily extract the components of the definitions using the names of the fields. +It also often allows Agda to infer more types, and to give us more sensible goals in the +interactive development of Agda programs and proofs. + +Notice that `inverse` plays the role of `g`. +The reason we use `inverse` is that then we can use the word +`inverse` to extract the inverse of a bijection. +Similarly we use `bijection` for `f`, as we can use the word +`bijection` to extract the bijection from a record. + +This type can be defined to be `𝟙 ∔ 𝟙` using the coproduct, +but we give a direct definition which will allow us to discuss some relationships between the various type formers of Basic MLTT. +```agda +data 𝟚 : Type where + 𝟎 𝟏 : 𝟚 +``` + +### Exercise 3 (⋆⋆) +Prove that 𝟚 and Bool are isomorphic + +```agda +Bool-𝟚-isomorphism : Bool ≅ 𝟚 +Bool-𝟚-isomorphism = record { bijection = f ; bijectivity = f-is-bijection } + where + f : Bool → 𝟚 + f false = 𝟎 + f true = 𝟏 + + g : 𝟚 → Bool + g 𝟎 = false + g 𝟏 = true + + gf : g ∘ f ∼ id + gf true = refl true + gf false = refl false + + fg : f ∘ g ∼ id + fg 𝟎 = refl 𝟎 + fg 𝟏 = refl 𝟏 + + f-is-bijection : is-bijection f + f-is-bijection = record { inverse = g ; η = gf ; ε = fg } +``` + + +## Part III - Finite Types + +In the last HoTT Exercises we encountered two definitions of the finite types. +Here we prove that they are isomorphic. +Note that `zero` was called `pt` and suc `i` on the HoTT exercise sheet. + +```agda +data Fin : ℕ → Type where + zero : {n : ℕ} → Fin (suc n) + suc : {n : ℕ} → Fin n → Fin (suc n) +``` + +### Exercise 4 (⋆) + +Prove the elimination principle of `Fin`. +```agda +Fin-elim : (A : {n : ℕ} → Fin n → Type) + → ({n : ℕ} → A {suc n} zero) + → ({n : ℕ} (k : Fin n) → A k → A (suc k)) + → {n : ℕ} (k : Fin n) → A k +Fin-elim A a f = h + where + h : {n : ℕ} (k : Fin n) → A k + h zero = a + h (suc k) = f k (h k) +``` + +We give the other definition of the finite types and introduce some notation. + +```agda +Fin' : ℕ → Type +Fin' 0 = 𝟘 +Fin' (suc n) = 𝟙 ∔ Fin' n + +zero' : {n : ℕ} → Fin' (suc n) +zero' = inl ⋆ + +suc' : {n : ℕ} → Fin' n → Fin' (suc n) +suc' = inr +``` + +### Exercise 5 (⋆⋆⋆) + +Prove that `Fin n` and `Fin' n` are isomorphic for every `n`. + +```agda +Fin-isomorphism : (n : ℕ) → Fin n ≅ Fin' n +Fin-isomorphism n = record { bijection = f n ; bijectivity = f-is-bijection n } + where + f : (n : ℕ) → Fin n → Fin' n + f (suc n) zero = zero' + f (suc n) (suc k) = suc' (f n k) + + g : (n : ℕ) → Fin' n → Fin n + g (suc n) (inl ⋆) = zero + g (suc n) (inr k) = suc (g n k) + + gf : (n : ℕ) → g n ∘ f n ∼ id + gf (suc n) zero = refl zero + gf (suc n) (suc k) = γ + where + IH : g n (f n k) ≡ k + IH = gf n k + + γ = g (suc n) (f (suc n) (suc k)) ≡⟨ refl _ ⟩ + g (suc n) (suc' (f n k)) ≡⟨ refl _ ⟩ + suc (g n (f n k)) ≡⟨ ap suc IH ⟩ + suc k ∎ + + fg : (n : ℕ) → f n ∘ g n ∼ id + fg (suc n) (inl ⋆) = refl (inl ⋆) + fg (suc n) (inr k) = γ + where + IH : f n (g n k) ≡ k + IH = fg n k + + γ = f (suc n) (g (suc n) (suc' k)) ≡⟨ refl _ ⟩ + f (suc n) (suc (g n k)) ≡⟨ refl _ ⟩ + suc' (f n (g n k)) ≡⟨ ap suc' IH ⟩ + suc' k ∎ + + f-is-bijection : (n : ℕ) → is-bijection (f n) + f-is-bijection n = record { inverse = g n ; η = gf n ; ε = fg n} +``` + +## Part IV -- minimal elements in the natural numbers + +In this section we establish some definitions which are needed to state and prove the well-ordering +principle of the natural numbers. + +### Exercise 6 (⋆) + +Give the recursive definition of the less than or equals relation on the natural numbers. + +```agda +_≤₁_ : ℕ → ℕ → Type +0 ≤₁ y = 𝟙 +suc x ≤₁ 0 = 𝟘 +suc x ≤₁ suc y = x ≤₁ y +``` + +### Exercise 7 (⋆) + +Given a type family `P` over the naturals, a lower bound `n` is a natural number such that +for all other naturals `m`, we have that if `P(m)` holds, then`n ≤₁ m`. +Translate this definition into HoTT. + +```agda +is-lower-bound : (P : ℕ → Type) (n : ℕ) → Type +is-lower-bound P n = (m : ℕ) → P m → n ≤₁ m +``` + +We define the type of minimal elements of a type family over the naturals. +```agda +minimal-element : (P : ℕ → Type) → Type +minimal-element P = Σ n ꞉ ℕ , (P n) × (is-lower-bound P n) +``` + +### Exercise 8 (⋆) + +Prove that all numbers are at least as large as zero. +```agda +leq-zero : (n : ℕ) → 0 ≤₁ n +leq-zero n = ⋆ +``` + + +## Part V -- The well-ordering principle of ℕ + +Classically, the well-ordering principle states that every non-empty set +of natural numbers has a least element. + +In HoTT, such subsets can be translated as decidable type family. +Recall the definition of decidability: +```agda +open import decidability + using (is-decidable; is-decidable-predicate) +``` + +The well-ordering principle reads +```agda +Well-ordering-principle = (P : ℕ → Type) → (d : is-decidable-predicate P) → (n : ℕ) → P n → minimal-element P +``` + +We shall prove this statement via induction on `n`. +In order to make the proof more readable, we first prove two lemmas. + +### Exercise 9 (🌶) + +What is the statement of `is-minimal-element-suc` +under the Curry-Howard interpretation? +Prove this lemma. + +```agda +is-minimal-element-suc : + (P : ℕ → Type) (d : is-decidable-predicate P) + (m : ℕ) (pm : P (suc m)) + (is-lower-bound-m : is-lower-bound (λ x → P (suc x)) m) → + ¬ (P 0) → is-lower-bound P (suc m) +is-minimal-element-suc P d m pm is-lower-bound-m neg-p0 0 p0 = 𝟘-nondep-elim (neg-p0 p0) +-- In the previous clause, 𝟘-nondep-elim is superfluous, because neg-p0 p0 : ∅ already. +is-minimal-element-suc P d 0 pm is-lower-bound-m neg-p0 (suc n) psuccn = ⋆ +is-minimal-element-suc P d (suc m) pm is-lower-bound-m neg-p0 (suc n) psuccn = h + where + h : suc m ≤₁ n + h = is-minimal-element-suc (λ x → P (suc x)) + (λ x → d (suc x)) m pm + (λ m → is-lower-bound-m (suc m)) + (is-lower-bound-m 0) + (n) + (psuccn) + -- alternative solution + h' : suc m ≤₁ n + h' = is-lower-bound-m n psuccn +``` +The lemma states that for a decidable type family `P`, if `m` is a lower bound +for `P ∘ suc`, and `P 0` is false, then `m + 1` is a lower bound for `P`. +Note that the assumptions `d` and `pm` are not used. + +### Exercise 10 (🌶) + +What is the statement of `well-ordering-principle-suc` +under the Curry-Howard interpretation? +Prove this lemma. + +```agda +well-ordering-principle-suc : + (P : ℕ → Type) (d : is-decidable-predicate P) + (n : ℕ) (p : P (suc n)) → + is-decidable (P 0) → + minimal-element (λ m → P (suc m)) → minimal-element P +well-ordering-principle-suc P d n p (inl p0) _ = 0 , (p0 , (λ m q → leq-zero m)) +well-ordering-principle-suc P d n p (inr neg-p0) (m , (pm , is-min-m)) = (suc m) , (pm , h) + where + h : is-lower-bound P (suc m) + h = is-minimal-element-suc P d m pm is-min-m neg-p0 + + -- alternative solution + h' : is-lower-bound P (suc m) + h' zero q = 𝟘-nondep-elim (neg-p0 q) + h' (suc k) q = is-min-m k q +``` +This lemma states that for a decidable type family `P`, if `P ∘ suc` is true for some `n`, +and `P 0` is decidable, then minimal elements of `P ∘ suc` yield minimal elements of `P`. +Note that `d` and `p` are not used. + +### Exercise 11 (🌶) + +Use the previous two lemmas to prove the well-ordering principle +```agda +well-ordering-principle : (P : ℕ → Type) → (d : is-decidable-predicate P) → (n : ℕ) → P n → minimal-element P +well-ordering-principle P d 0 p = 0 , (p , (λ m q → leq-zero m)) +well-ordering-principle P d (suc n) p = well-ordering-principle-suc P d n p (d 0) (well-ordering-principle (λ m → P (suc m)) (λ m → d (suc m)) n p) +``` + +### Exercise 12 (⋆⋆⋆) + +Prove that the well-ordering principle returns 0 if `P 0` holds. + +```agda +is-zero-well-ordering-principle-suc : + (P : ℕ → Type) (d : is-decidable-predicate P) + (n : ℕ) (p : P (suc n)) (d0 : is-decidable (P 0)) → + (x : minimal-element (λ m → P (suc m))) (p0 : P 0) → + (pr₁ (well-ordering-principle-suc P d n p d0 x)) ≡ 0 +is-zero-well-ordering-principle-suc P d n p (inl p0) x q0 = refl 0 +is-zero-well-ordering-principle-suc P d n p (inr np0) x q0 = 𝟘-nondep-elim (np0 q0) + +is-zero-well-ordering-principle : + (P : ℕ → Type) (d : is-decidable-predicate P) → + (n : ℕ) → (pn : P n) → P 0 → pr₁ (well-ordering-principle P d n pn) ≡ 0 +is-zero-well-ordering-principle P d 0 p p0 = refl 0 +is-zero-well-ordering-principle P d (suc m) pm = + is-zero-well-ordering-principle-suc P d m pm (d 0) + (well-ordering-principle + (λ z → P (suc z)) + (λ x → d (suc x)) + m pm) +``` diff --git a/src/HoTTEST/Agda/Exercises/Pool/Homework/homework2-solutions.lagda.md b/src/HoTTEST/Agda/Exercises/Pool/Homework/homework2-solutions.lagda.md new file mode 100644 index 0000000..17c9d1a --- /dev/null +++ b/src/HoTTEST/Agda/Exercises/Pool/Homework/homework2-solutions.lagda.md @@ -0,0 +1,69 @@ +Week 2 - Homework Sheet - Solutions + +```agda +{-# OPTIONS --without-K --safe #-} + +module Pool.Homework.homework2-solutions where + +open import prelude hiding (id ; _∘_) + +ap² : {A : Type} (f : A → A) {x y : A} → x ≡ y → f (f x) ≡ f (f y) +ap² f e = ap f (ap f e) + +double-not-eq : {a b : Bool} → a ≡ b → not (not a) ≡ not (not b) +double-not-eq e = ap² not e + +transport-vec-A : {A : Type} {n m : ℕ} → n ≡ m + → Vector A n → Vector A m +transport-vec-A (refl n) v = v + + +map : {A B : Type} → (A → B) → List A → List B +map f [] = [] +map f (x :: xs) = f x :: map f xs + +id : {A : Type} → A → A +id = λ x → x + +_∘_ : {A B C : Type} → (g : B → C) → (f : A → B)→ A → C +g ∘ f = λ x → g (f x) + +map-law1 : {A : Type} (xs : List A) → map id xs ≡ xs +map-law1 [] = refl [] +map-law1 (x :: xs) = ap (x ::_) IH + where + IH : map id xs ≡ xs -- IH is short for induction hypothesis + IH = map-law1 xs + +map-law2 : {A B C : Type} (g : B → C) (f : A → B) (xs : List A) + → map (g ∘ f) xs ≡ map g (map f xs) +map-law2 g f [] = refl [] +map-law2 g f (x :: xs) = ap (g (f x) ::_) IH + where + IH : map (g ∘ f) xs ≡ map g (map f xs) + IH = map-law2 g f xs + + +and : List Bool → Bool +and [] = true +and (true :: bs) = and bs +and (false :: bs) = false + +and-example1 : and (true :: false :: []) ≡ false +and-example1 = refl false + +and-example2 : and [] ≡ true +and-example2 = refl true + +for-all : {A : Type} → (A → Bool) → List A → Bool +for-all f xs = and (map f xs) + +filter : {A : Type} → (A → Bool) → List A → List A +filter p [] = [] +filter p (x :: xs) = if (p x) then x :: ys else ys + where + ys = filter p xs + +filter-soundness : {A : Type} → Type +filter-soundness {A} = (p : A → Bool) (xs : List A) + → for-all p (filter p xs) ≡ true diff --git a/src/HoTTEST/Agda/Exercises/Pool/Homework/homework2.lagda.md b/src/HoTTEST/Agda/Exercises/Pool/Homework/homework2.lagda.md new file mode 100644 index 0000000..a0beb10 --- /dev/null +++ b/src/HoTTEST/Agda/Exercises/Pool/Homework/homework2.lagda.md @@ -0,0 +1,182 @@ +# Week 2 - Homework Sheet + +Please complete Part II of this week's Lab Sheet before moving on to these exercises. + +## Section 1: Identity Type + +```agda +{-# OPTIONS --without-K --safe #-} + +module Pool.Homework.homework2 where + +open import prelude hiding (id ; _∘_) +``` + +### Exercise 1.1 + +Recall the definitions in `trans`, `sym` and especially `ap` from the +[introduction](../introduction.lagda.md). + +We could define "double `ap`" like so: + +```agda +ap² : {A : Type} (f : A → A) {x y : A} → x ≡ y → f (f x) ≡ f (f y) +ap² f e = {!!} +``` + +For example, this would help us prove the following without pattern matching: + +```agda +double-not-eq : {a b : Bool} → a ≡ b → not (not a) ≡ not (not b) +double-not-eq e = ap² not e +``` + +**Define** "double `ap`" *without* using pattern matching. + +### Exercise 1.2 + +We have seen that `ap` allows us to "transport" an equality along a function. + +We will now see how we can "transport" equalities along types; in our example, +we use the type for Vectors. + +```agda +transport-vec-A : {A : Type} {n m : ℕ} → n ≡ m + → Vector A n → Vector A m +transport-vec-A e v = {!!} +``` + +**Complete** the above function. + +## Section 2: Laws for `map` + +Recall that we defined `map` as follows: +```agda +map : {A B : Type} → (A → B) → List A → List B +map f [] = [] +map f (x :: xs) = f x :: map f xs +``` + +In Haskell, `map` is supposed to satisfy, for every list `xs`, two laws: + + 1. `map id xs = xs` (where `id` the identity function); + 1. `map (g . f) xs = map g (map f xs)` (where `.` is function composition). + +But checking these laws is left to a pen-and-paper check by the programmer. + +In Agda, we can both *state* and *prove* these laws. + +We first define function composition and the identity function. + +```agda +id : {A : Type} → A → A +id = λ x → x + +_∘_ : {A B C : Type} → (g : B → C) → (f : A → B)→ A → C +g ∘ f = λ x → g (f x) +``` + +Tip: Use `\comp` to type `∘`. + +### Exercise 2.1 + +The first `map` law can now be written in Agda as +```agda +map-law1 : {A : Type} (xs : List A) → map id xs ≡ xs +map-law1 xs = {!!} +``` + +**Define** this function. (Hint: An induction hypothesis comes in helpful). + +### Exercise 2.2 + +A partial statement of the second `map` law is the following: +```agda +map-law2 : {A B C : Type} (g : {!!}) (f : {!!}) (xs : List A) + → {!!} ≡ {!!} +map-law2 g f xs = {!!} +``` + +**Complete** the holes in `map-law2 : ...`, i.e. write down the types of `g` and +`f`, and translate the second `map` law `map (g ∘ f) xs = map g (map f xs)` to +Agda. + + +### Exercise 2.3 + +Now **complete** the proof of `map-law2`. + +## Section 3: Filtering lists + +**Define** a function `and` that takes a list of Booleans and computes the +conjunction of all the Booleans in it: + +### Exercise 3.1 + +```agda +and : List Bool → Bool +and bs = {!!} +``` + +For example, `and true false` should equal to `false` + +```agda +and-example1 : and (true :: false :: []) ≡ false +and-example1 = {! refl false !} + -- make sure that your code is correct by checking + -- that the equality in this type is definitional. + -- The proof should be just `refl false`. +``` + +```agda +and-example2 : and [] ≡ true +and-example2 = {! refl true!} + -- make sure that your code is correct by checking + -- that the equality in this type is definitional. + -- The proof should be just `refl true`. +``` + +### Exercise 3.2 + +**Define** a function `for-all` that takes a list of element of some type `A` +together with a predicate `p : A → Bool`, and computes the Boolean truth value +denoting if *everything* in the list satisfy the predicate `p`. You should be +able to this just by composing `map` and your implementation of `and`. + +```agda +for-all : {A : Type} → (A → Bool) → List A → Bool +for-all = {!!} +``` + +Next, we will define a function `filter` that filters out a subset of a list of +elements that satisfy a certain predicate. You have probably worked with a +function like this in Haskell before. + +### Exercise 3.3 + +```agda +filter : {A : Type} → (A → Bool) → List A → List A +filter = {!!} +``` + +`filter p xs` should give you the sublist of `xs` consisting of elements that +satisfy `p`. + +### Exercise 3.4 + +You have probably worked with similar functions in Haskell before. The nice +thing about working in Agda is that, we can actually write down *specifications* +of such functions. + +Using your `for-all`, **define** a function that expresses: *for every predicate +`p` and every list `xs`, everything in `filter p xs` satisfies the predicate +`p`*. Call this type `filter-soundness`. + +```agda +filter-soundness : {A : Type} → Type +filter-soundness {A} = {!!} +``` + +Note that the type of the function here is `Type`. This means that you are not +writing an inhabitant of a type but you are *writing a type*, i.e. you are not +asked to *prove* filter soundness, but rather to *state* it. diff --git a/src/HoTTEST/Agda/Exercises/Pool/Homework/homework3-solutions.lagda.md b/src/HoTTEST/Agda/Exercises/Pool/Homework/homework3-solutions.lagda.md new file mode 100644 index 0000000..5e9603c --- /dev/null +++ b/src/HoTTEST/Agda/Exercises/Pool/Homework/homework3-solutions.lagda.md @@ -0,0 +1,79 @@ +# Week 3 - Homework Sheet - Solutions + +```agda +{-# OPTIONS --without-K --safe #-} + +module Pool.Homework.homework3-solutions where + +open import prelude hiding (𝟘-nondep-elim) +open import Pool.Lab.lab3-solutions + +∔-assoc : {A B C : Type} → A ∔ (B ∔ C) → (A ∔ B) ∔ C +∔-assoc (inl a) = inl (inl a) +∔-assoc (inr (inl b)) = inl (inr b) +∔-assoc (inr (inr c)) = inr c + +×-assoc : {A B C : Type} → A × (B × C) → (A × B) × C +×-assoc (a , (b , c)) = (a , b) , c + +∔-comm : {A B : Type} → A ∔ B → B ∔ A +∔-comm (inl a) = inr a +∔-comm (inr b) = inl b + +×-comm : {A B : Type} → A × B → B × A +×-comm (a , b) = b , a + + +not-A-and-not-A : {A : Type} → ¬ (A × ¬ A) +not-A-and-not-A (a , p) = p a + +A-and-not-A-implies-B : {A B : Type} → A × ¬ A → B +A-and-not-A-implies-B p = 𝟘-nondep-elim (not-A-and-not-A p) + +LEM : {A : Type} → A ∔ ¬ A +LEM = {!!} + +not-not-LEM : {A : Type} → ¬¬ (A ∔ ¬ A) +not-not-LEM p = p (inr (λ a → p (inl a))) + +DNE : {A : Type} → ¬¬ A → A +DNE = {!!} + +LEM' : {A : Type} → A ∔ ¬ A +LEM' = DNE not-not-LEM + +DNE' : {A : Type} → ¬¬ A → A +DNE' {A} p = γ LEM + where + γ : A ∔ ¬ A → A + γ (inl a) = a + γ (inr q) = 𝟘-nondep-elim (p q) + +not-not-DNE : {A : Type} → ¬¬ (¬¬ A → A) +not-not-DNE {A} p = p dne + where + r : ¬ A + r a = p (λ _ → a) + dne : ¬¬ A → A + dne q = 𝟘-nondep-elim (q r) + + +Σ-∔-distributivity : {A : Type} {B C : A → Type} + → (Σ a ꞉ A , (B a ∔ C a)) + → (Σ a ꞉ A , B a) ∔ (Σ a ꞉ A , C a) +Σ-∔-distributivity (a , inl b) = inl (a , b) +Σ-∔-distributivity (a , inr c) = inr (a , c) + +¬Σ-if-forall-not : {A : Type} {B : A → Type} + → ((a : A) → ¬ B a) → ¬ (Σ a ꞉ A , B a) +¬Σ-if-forall-not p (a , b) = p a b + +forall-not-if-¬Σ : {A : Type} {B : A → Type} + → ¬ (Σ a ꞉ A , B a) → (a : A) → ¬ B a +forall-not-if-¬Σ p a b = p (a , b) + +Π-Σ-distributivity : {A B : Type} {C : A → B → Type} + → ((a : A) → (Σ b ꞉ B , C a b)) + → Σ f ꞉ (A → B) , ((a : A) → C a (f a)) +Π-Σ-distributivity p = (λ a → fst (p a)) , (λ a → snd (p a)) +``` diff --git a/src/HoTTEST/Agda/Exercises/Pool/Homework/homework3.lagda.md b/src/HoTTEST/Agda/Exercises/Pool/Homework/homework3.lagda.md new file mode 100644 index 0000000..7cb1b35 --- /dev/null +++ b/src/HoTTEST/Agda/Exercises/Pool/Homework/homework3.lagda.md @@ -0,0 +1,199 @@ +# Week 3 - Homework Sheet + +**Please finish the lab sheet before moving on to these exercises.** + +```agda +{-# OPTIONS --without-K --safe #-} + +module Pool.Homework.homework3 where + +open import prelude +open import Pool.Lab.lab3-solutions +``` + +## Part I: Associativity and Commutativity of ∔ and × + +We have already seen that the Boolean operations `_||_` and `_&&_` are +associative and commutative. + +The type formers that represent these logical connectives are also associative +and commutative. + +### Exercise 1.1 + +**Prove** that `_∔_` is associative. + +```agda +∔-assoc : {A B C : Type} → A ∔ (B ∔ C) → (A ∔ B) ∔ C +∔-assoc = {!!} +``` + +### Exercise 1.2 + +**Prove** that `_×_` is associative. + +```agda +×-assoc : {A B C : Type} → A × (B × C) → (A × B) × C +×-assoc = {!!} +``` + +### Exercise 1.3 + +**Prove** that `_∔_` is commutative. + +```agda +∔-comm : {A B : Type} → A ∔ B → B ∔ A +∔-comm = {!!} +``` +### Exercise 1.4 + +**Prove** that `_×_` is commutative. + +```agda +×-comm : {A B : Type} → A × B → B × A +×-comm = {!!} +``` + +## Part II: Law of excluded middle and double-negation elimination + +In classical logic, we have the law of excluded middle (LEM): `A + ¬ A`, for any +logical formula `A`. + +### Exercise 2.1 + +This seems intuitive to us, as having both `A` and `¬ A` gives us a +contradiction. + +```agda +not-A-and-not-A : {A : Type} → ¬ (A × ¬ A) +not-A-and-not-A = {!!} +``` + +**Complete** the proof that `¬ (A x ¬ A)`. + +### Exercise 2.2 + +Furthermore, if we had both `A` and `¬ A`, we could prove anything. + +```agda +A-and-not-A-implies-B : {A B : Type} → A × ¬ A → B +A-and-not-A-implies-B p = {!!} +``` + +**Complete** the above statement *without* using pattern matching. + +Hint: Use `𝟘-nondep-elim`. + +### Exercise 2.3 + +However, it turns out that LEM is not provable (or disprovable) in Agda. + +You can try this out yourself (but you won't succeed): + +```agda +LEM : {A : Type} → A ∔ ¬ A +LEM = {!!} -- You are not expected to complete this hole. + -- In fact, it's impossible. +``` + +However, we *can* prove the *double-negation* of `LEM`. + +```agda +not-not-LEM : {A : Type} → ¬¬ (A ∔ ¬ A) +not-not-LEM = {!!} +``` + +**Prove** the double-negation of the law of excluded middle. + +### Exercise 2.4 + +If we had access to double-negation elimination (DNE), as in classical logic, we +could give `LEM`. + +Note: Do not try to prove DNE (see Exercise 2.5). + +**Complete** `LEM'` using `DNE`. + +```agda +DNE : {A : Type} → ¬¬ A → A +DNE = {!!} -- You are not expected to complete this hole. + -- In fact, it's impossible. + +LEM' : {A : Type} → A ∔ ¬ A +LEM' = {!!} +``` + +### Exercise 2.5 + +However, `DNE` is *also* not provable or disprovable in Agda. + +It is the case, however, that if we had access to `LEM`, we could prove `DNE`. + +```agda +DNE' : {A : Type} → ¬¬ A → A +DNE' {A} = {!!} +``` + +**Complete** `DNE'` using `LEM`. + +### Exercise 2.6 + +So we have seen that `LEM` and `DNE` are both not provable in Agda, yet are +equivalent in the sense that having one gives us the other. + +Finally, we can also show that the *double-negation* of DNE *is* provable in +Agda. + +```agda +not-not-DNE : {A : Type} → ¬¬ (¬¬ A → A) +not-not-DNE {A} = {!!} +``` + +**Prove** the double-negation of the law of excluded middle. + +## Part III: Sums and products + +### Exercise 3.1 + +**Complete** the following proof of distributivity of `Σ` over `_∔_`. + +```agda +Σ-∔-distributivity : {A : Type} {B C : A → Type} + → (Σ a ꞉ A , (B a ∔ C a)) + → (Σ a ꞉ A , B a) ∔ (Σ a ꞉ A , C a) +Σ-∔-distributivity = {!!} +``` + +### Exercise 3.2 + +If, for every `a : A` we have `¬ B a` (the type `B a` is empty), then there +does not exist any `a : A` satisfying `B a` (the type `Σ B` is empty). + +```agda +¬Σ-if-forall-not : {A : Type} {B : A → Type} + → ((a : A) → ¬ B a) → ¬ (Σ a ꞉ A , B a) +¬Σ-if-forall-not = {!!} +``` + +**Complete** the proof of the above statement. + +### Exercise 3.3 + +**Prove** that the converse of the above also holds. + +```agda +forall-not-if-¬Σ : {A : Type} {B : A → Type} + → ¬ (Σ a ꞉ A , B a) → (a : A) → ¬ B a +forall-not-if-¬Σ = {!!} +``` + +### Exercise 3.4 + +Finally, **prove** that `Σ` distributes over "for all". + +```agda +Π-Σ-distributivity : {A B : Type} {C : A → B → Type} + → ((a : A) → (Σ b ꞉ B , C a b)) + → Σ f ꞉ (A → B) , ((a : A) → C a (f a)) +Π-Σ-distributivity = {!!} +``` diff --git a/src/HoTTEST/Agda/Exercises/Pool/Homework/homework4-solutions.lagda.md b/src/HoTTEST/Agda/Exercises/Pool/Homework/homework4-solutions.lagda.md new file mode 100644 index 0000000..556b119 --- /dev/null +++ b/src/HoTTEST/Agda/Exercises/Pool/Homework/homework4-solutions.lagda.md @@ -0,0 +1,79 @@ +```agda +{-# OPTIONS --without-K --safe #-} + +module Pool.Homework.homework4-solutions where + +open import prelude +open import List-functions hiding (++-assoc) + ++-comm : (x y : ℕ) → x + y ≡ y + x ++-comm 0 0 = refl zero ++-comm 0 (suc y) = ap suc (+-comm zero y) ++-comm (suc x) 0 = ap suc (+-comm x zero) ++-comm (suc x) (suc y) + = suc (x + suc y) ≡⟨ ap suc (+-comm x (suc y)) ⟩ + suc (suc (y + x)) ≡⟨ ap (suc ∘ suc) (+-comm y x) ⟩ + suc (suc x + y) ≡⟨ ap suc (+-comm (suc x) y) ⟩ + suc (y + suc x) ∎ + +data _≼_ {X : Type} : List X → List X → Type where + []-is-prefix : (xs : List X) → [] ≼ xs + ::-is-prefix : (x : X) (xs ys : List X) + → xs ≼ ys → (x :: xs) ≼ (x :: ys) + +_≼'_ : {X : Type} → List X → List X → Type +_≼'_ {X} xs ys = Σ zs ꞉ List X , xs ++ zs ≡ ys + + +++-assoc : {X : Type} (xs ys zs : List X) + → (xs ++ ys) ++ zs ≡ xs ++ (ys ++ zs) +++-assoc [] ys zs = refl (ys ++ zs) +++-assoc (x :: xs) ys zs = ap (λ - → x :: -) IH + where + IH : (xs ++ ys) ++ zs ≡ xs ++ (ys ++ zs) + IH = ++-assoc xs ys zs + +≼'-is-transitive : {X : Type} (xs ys zs : List X) + → xs ≼' ys → ys ≼' zs → xs ≼' zs +≼'-is-transitive xs ys zs (l , e) (l' , e') = ((l ++ l') , e'') + where + e'' : xs ++ l ++ l' ≡ zs + e'' = xs ++ l ++ l' ≡⟨ sym (++-assoc xs l l') ⟩ + (xs ++ l) ++ l' ≡⟨ ap (λ - → - ++ l') e ⟩ + ys ++ l' ≡⟨ e' ⟩ + zs ∎ + + +≼-++ : {X : Type} (xs ys : List X) → xs ≼ (xs ++ ys) +≼-++ [] ys = []-is-prefix ys +≼-++ (x :: xs) ys = ::-is-prefix x xs (xs ++ ys) IH + where + IH : xs ≼ (xs ++ ys) + IH = ≼-++ xs ys + +≼-unprime : {X : Type} (xs ys : List X) → xs ≼' ys → xs ≼ ys +≼-unprime xs _ (zs , refl _) = ≼-++ xs zs + + +≼'-[] : {X : Type} (xs : List X) → [] ≼' xs +≼'-[] xs = (xs , refl xs) + +≼'-cons : {X : Type} (x : X) (xs ys : List X) + → xs ≼' ys + → (x :: xs) ≼' (x :: ys) +≼'-cons x xs ys (zs , e) = (zs , ap (λ - → x :: -) e) + +≼-prime : {X : Type} (xs ys : List X) → xs ≼ ys → xs ≼' ys +≼-prime [] ys ([]-is-prefix ys) = ≼'-[] ys +≼-prime (x :: xs) (x :: ys) (::-is-prefix x xs ys h) = ≼'-cons x xs ys IH + where + IH : xs ≼' ys + IH = ≼-prime xs ys h + + +≼-is-transitive : {X : Type} (xs ys zs : List X) + → xs ≼ ys → ys ≼ zs → xs ≼ zs +≼-is-transitive xs ys zs p q = ≼-unprime xs zs + (≼'-is-transitive xs ys zs (≼-prime xs ys p) + (≼-prime ys zs q)) +``` diff --git a/src/HoTTEST/Agda/Exercises/Pool/Homework/homework4.lagda.md b/src/HoTTEST/Agda/Exercises/Pool/Homework/homework4.lagda.md new file mode 100644 index 0000000..665e4ad --- /dev/null +++ b/src/HoTTEST/Agda/Exercises/Pool/Homework/homework4.lagda.md @@ -0,0 +1,191 @@ +# Week 4 - Homework Sheet + +**Please finish the lab sheet before moving on to these exercises.** + +```agda +{-# OPTIONS --without-K --safe #-} +module Pool.Homework.homework4 where + +open import prelude +open import List-functions hiding (++-assoc) +``` + +## Part I: Commutativity of addition of natural numbers + +We wish to prove the commutativity of `_+_` on the natural numbers. + +The following proof skeleton has been provided for you, using +[notation for equational reasoning](https://git.cs.bham.ac.uk/mhe/afp-learning/-/blob/main/files/LectureNotes/files/identity-type.lagda.md#notation-for-equality-reasoning). + +```agda ++-comm : (x y : ℕ) → x + y ≡ y + x ++-comm 0 0 = {!!} ++-comm 0 (suc y) = {!!} ++-comm (suc x) 0 = {!!} ++-comm (suc x) (suc y) + = suc (x + suc y) ≡⟨ {!!} ⟩ + suc (suc (y + x)) ≡⟨ {!!} ⟩ + suc (suc x + y) ≡⟨ {!!} ⟩ + suc (y + suc x) ∎ +``` + +**Fill** the holes of our proof that `_+_` is commutative. + +## Part II: Prefixes of lists + +In this part we will study two ways of expressing that a list is prefix of +another list. + +This will be similar to how we had two ways of expressing less-than-or-equal +`≤` on natural numbers, as seen in the Lab Sheet for Week 4. In particular, +you will notice how to the structure of the proofs below mirror the structure +of the proofs in this week's Lab Sheet. + +The first definition is an inductive one. + +```agda +data _≼_ {X : Type} : List X → List X → Type where + []-is-prefix : (xs : List X) → [] ≼ xs + ::-is-prefix : (x : X) (xs ys : List X) + → xs ≼ ys → (x :: xs) ≼ (x :: ys) +``` + +It says: +1. that the empty list is a prefix of any list; +1. if `xs` is a prefix of `ys`, then `x :: xs` is a prefix of `x :: ys`. + +The second item says that you can extend prefixes by adding the same element at +the start. + +The second definition uses a `Σ`-type. + +```agda +_≼'_ : {X : Type} → List X → List X → Type +_≼'_ {X} xs ys = Σ zs ꞉ List X , xs ++ zs ≡ ys +``` + +It says that `xs` is a prefix of `ys` if we have another list `zs` such that +`xs ++ zs ≡ ys`. In other words, `xs` can be extended to `ys.` + +### Examples + +Here are some examples of prefixes of lists. + +```agda +≼'-example₀ : [] ≼' (1 :: 2 :: [ 3 ]) +≼'-example₀ = ((1 :: 2 :: [ 3 ]) , refl (1 :: 2 :: [ 3 ])) + +≼'-example₁ : [ 1 ] ≼' (1 :: [ 2 ]) +≼'-example₁ = ([ 2 ] , refl (1 :: [ 2 ])) + +≼'-example₂ : (7 :: [ 3 ]) ≼' (7 :: 3 :: 4 :: [ 9 ]) +≼'-example₂ = ((4 :: [ 9 ]) , refl (7 :: 3 :: 4 :: [ 9 ])) +``` + +For comparison, here are some examples using `≼` instead of `≼'`. + +```agda +≼-example₀ : [] ≼ (1 :: 2 :: [ 3 ]) +≼-example₀ = []-is-prefix (1 :: 2 :: [ 3 ]) + +≼-example₁ : [ 1 ] ≼ (1 :: [ 2 ]) +≼-example₁ = ::-is-prefix 1 [] [ 2 ] + ([]-is-prefix [ 2 ]) + +≼-example₂ : (7 :: [ 3 ]) ≼ (7 :: 3 :: 4 :: [ 9 ]) +≼-example₂ = ::-is-prefix 7 [ 3 ] (3 :: 4 :: [ 9 ]) + (::-is-prefix 3 [] (4 :: [ 9 ]) + ([]-is-prefix (4 :: [ 9 ]))) +``` + +Notice how we build up the proofs by consecutive uses of `::-is-prefix`, +finishing with `[]-is-prefix`. This reflects the inductive definition of `≼`. + +We will prove that `x ≼ y` and `x ≼' y` are logically equivalent, but first we +include a useful exercise on associativity. + +### Exercise 2.1 + +**Prove** that concatenation of lists, `++`, is associative. + +```agda +++-assoc : {X : Type} (xs ys zs : List X) + → (xs ++ ys) ++ zs ≡ xs ++ (ys ++ zs) +++-assoc = {!!} +``` + +### Exercise 2.2 + +**Complete** the holes in the equational reasoning below to prove that `≼'` is +transitive. + +```agda +≼'-is-transitive : {X : Type} (xs ys zs : List X) + → xs ≼' ys → ys ≼' zs → xs ≼' zs +≼'-is-transitive xs ys zs (l , e) (l' , e') = ((l ++ l') , e'') + where + e'' : xs ++ l ++ l' ≡ zs + e'' = xs ++ (l ++ l') ≡⟨ {!!} ⟩ + (xs ++ l) ++ l' ≡⟨ {!!} ⟩ + ys ++ l' ≡⟨ {!!} ⟩ + zs ∎ +``` + +### Exercise 2.3 + +**Prove** the following about `≼`. + +```agda +≼-++ : {X : Type} (xs ys : List X) → xs ≼ (xs ++ ys) +≼-++ [] ys = {!!} +≼-++ (x :: xs) ys = {!!} +``` + +### Exercise 2.4 + +**Complete** the function below, showing that we can go from `≼'` to `≼`. + +*Hint*: Use `≼-++`. + +```agda +≼-unprime : {X : Type} (xs ys : List X) → xs ≼' ys → xs ≼ ys +≼-unprime = {!!} +``` + +### Exercise 2.5 + +**Prove** the following facts about `≼'`. + +```agda +≼'-[] : {X : Type} (xs : List X) → [] ≼' xs +≼'-[] = {!!} + +≼'-cons : {X : Type} (x : X) (xs ys : List X) + → xs ≼' ys + → (x :: xs) ≼' (x :: ys) +≼'-cons x xs ys (zs , e) = {!!} +``` + +Note that these amount to the constructors of `≼`. + +### Exercise 2.6 + +**Complete** the function below, showing that we can go from `≼` to `≼'`. + +*Hint*: Use `≼'-[]` and `≼'-cons`. + +```agda +≼-prime : {X : Type} (xs ys : List X) → xs ≼ ys → xs ≼' ys +≼-prime = {!!} +``` + +### Exercise 2.7 + +Using the functions `≼-unprime` and `≼-prime`, and the fact that `≼'` is +transitive, **prove** that `≼` is transitive too. + +```agda +≼-is-transitive : {X : Type} (xs ys zs : List X) + → xs ≼ ys → ys ≼ zs → xs ≼ zs +≼-is-transitive = {!!} +``` diff --git a/src/HoTTEST/Agda/Exercises/Pool/Homework/homework5-solutions.lagda.md b/src/HoTTEST/Agda/Exercises/Pool/Homework/homework5-solutions.lagda.md new file mode 100644 index 0000000..2fa3810 --- /dev/null +++ b/src/HoTTEST/Agda/Exercises/Pool/Homework/homework5-solutions.lagda.md @@ -0,0 +1,240 @@ +```agda +{-# OPTIONS --without-K --safe #-} + +module Pool.Homework.homework5-solutions where + +open import prelude hiding (Bool-elim) +open import natural-numbers-functions hiding (_≤_ ; is-even ; +-assoc ; +-comm) +open import List-functions +open import isomorphisms +open import negation + +open import Pool.Homework.homework4-solutions +open import Pool.Lab.lab4-solutions +open import Pool.Lab.lab5-solutions + +length-of-reverse : {A : Type} (xs : List A) + → length (reverse xs) ≡ length xs +length-of-reverse [] = refl 0 +length-of-reverse (x :: xs) = + length (reverse xs ++ [ x ]) ≡⟨ length-of-++ (reverse xs) [ x ] ⟩ + length (reverse xs) + length [ x ] ≡⟨ refl _ ⟩ + length (reverse xs) + 1 ≡⟨ ap (_+ 1) IH ⟩ + length xs + 1 ≡⟨ +-comm (length xs) 1 ⟩ + 1 + length xs ∎ + where + IH : length (reverse xs) ≡ length xs + IH = length-of-reverse xs + +ℕ-[⋆]-iso : ℕ ≅ List 𝟙 +ℕ-[⋆]-iso = record { bijection = f ; bijectivity = f-is-bijection } + where + f : ℕ → List 𝟙 + f 0 = [] + f (suc n) = ⋆ :: f n + + g : List 𝟙 → ℕ + g [] = 0 + g (⋆ :: ⋆s) = suc (g ⋆s) + + gf : g ∘ f ∼ id + gf 0 = refl 0 + gf (suc n) = ap suc (gf n) + + fg : f ∘ g ∼ id + fg [] = refl [] + fg (⋆ :: ⋆s) = ap (⋆ ::_) (fg ⋆s) + + f-is-bijection : is-bijection f + f-is-bijection = record { inverse = g ; η = gf ; ε = fg } + +open _≅_ + +ℕ→[⋆]-preserves-length : (n : ℕ) → length (bijection ℕ-[⋆]-iso n) ≡ n +ℕ→[⋆]-preserves-length zero = refl 0 +ℕ→[⋆]-preserves-length (suc n) = ap suc (ℕ→[⋆]-preserves-length n) + +principle-of-bivalence : (b : Bool) → (b ≡ true) ∔ (b ≡ false) +principle-of-bivalence true = inl (refl true) +principle-of-bivalence false = inr (refl false) + +is-even-is-decidable : (n : ℕ) → is-decidable (is-even n) +is-even-is-decidable n = + ∔-nondep-elim goal₁ goal₂ (principle-of-bivalence (check-even n)) + where + goal₁ : check-even n ≡ true → is-decidable (is-even n) + goal₁ p = inl (check-even⇒even n p) + + goal₂ : check-even n ≡ false → is-decidable (is-even n) + goal₂ p = inr subgoal + where + subgoal : ¬ is-even n + subgoal q = true-is-not-false (true ≡⟨ sym (even⇒check-even n q) ⟩ + check-even n ≡⟨ p ⟩ + false ∎) + +filter : {A : Type} → (A → Bool) → List A → List A +filter P [] = [] +filter P (x :: xs) = if P x then (x :: ys) else ys + where + ys = filter P xs + +≤-suc-lemma : (n : ℕ) → n ≤ (1 + n) +≤-suc-lemma 0 = ≤-zero (1 + 0) +≤-suc-lemma (suc n) = goal + where + IH : n ≤ (1 + n) + IH = ≤-suc-lemma n + goal : suc n ≤ suc (suc n) + goal = ≤-suc n (suc n) IH + +Bool-elim : (A : Bool → Type) + → A false + → A true + → (x : Bool) → A x +Bool-elim A x₀ x₁ false = x₀ +Bool-elim A x₀ x₁ true = x₁ + +length-of-filter : {A : Type} (P : A → Bool) (xs : List A) + → length (filter P xs) ≤ length xs +length-of-filter P [] = ≤-zero 0 +length-of-filter P (x :: xs) = Bool-elim goal-statement false-case + true-case + (P x) + where + ys = filter P xs + + {- Note that by definition + filter P (x :: xs) = if P x then (x :: ys) else ys + and so goal-statement is almost + length (filter P (x :: xs)) ≤ length (x :: xs) + except that we replace "P x" by the Boolean "b". -} + goal-statement : Bool → Type + goal-statement b = length (if b then (x :: ys) else ys) ≤ length (x :: xs) + + IH : length ys ≤ length xs + IH = length-of-filter P xs + + {- The type of "false-case" is equal to "goal-statement false". -} + false-case : length ys ≤ length (x :: xs) + false-case = ≤-trans (length ys) (length xs) (length (x :: xs)) + IH (≤-suc-lemma (length xs)) + + {- The type of "true-case" is equal to "goal-statement true". -} + true-case : length (x :: ys) ≤ length (x :: xs) + true-case = ≤-suc (length ys) (length xs) IH + +{- Here is a version that uses Agda's built-in with-keyword (as shown by Eric in + the lab of 28 Feb) instead of Bool-elim. (Under the hood, they amount to the + same thing.) -} +length-of-filter' : {A : Type} (P : A → Bool) (xs : List A) + → length (filter P xs) ≤ length xs +length-of-filter' P [] = ≤-zero 0 + +length-of-filter' P (x :: xs) with P x +length-of-filter' P (x :: xs) | true = true-case + where + ys = filter P xs + + IH : length ys ≤ length xs + IH = length-of-filter' P xs + + true-case : length (x :: ys) ≤ length (x :: xs) + true-case = ≤-suc (length ys) (length xs) IH + +length-of-filter' P (x :: xs) | false = false-case + where + ys = filter P xs + + IH : length ys ≤ length xs + IH = length-of-filter' P xs + + false-case : length ys ≤ length (x :: xs) + false-case = ≤-trans (length ys) (length xs) (length (x :: xs)) + IH (≤-suc-lemma (length xs)) + +length-of-filters : {A : Type} (P : A → Bool) (xs : List A) + → length (filter P xs) + length (filter (not ∘ P) xs) + ≡ length xs +length-of-filters P [] = refl _ +length-of-filters P (x :: xs) = Bool-elim goal-statement false-case + true-case + (P x) + where + ys = filter P xs + ys' = filter (not ∘ P) xs + + IH : length ys + length ys' ≡ length xs + IH = length-of-filters P xs + + {- Note that by definition + filter P (x :: xs) = if P x then (x :: ys) else ys + and so goal-statement is almost + length (filter P (x :: xs)) + + length (filter (not ∘ P) (x :: xs)) + ≡ length (x :: xs) + except that we replace "P x" by the Boolean "b". -} + goal-statement : Bool → Type + goal-statement b = length (if b then (x :: ys ) else ys ) + + length (if not b then (x :: ys') else ys') + ≡ 1 + length xs + + {- The type of "false-case" is equal to "goal-statement false. -} + false-case : length ys + length (x :: ys') ≡ length (x :: xs) + false-case = + length ys + length (x :: ys') ≡⟨ refl _ ⟩ + length ys + (1 + length ys') ≡⟨ +-assoc (length ys) 1 (length ys') ⟩ + (length ys + 1) + length ys' ≡⟨ ap (_+ length ys') (+-comm (length ys) 1) ⟩ + (1 + length ys) + length ys' ≡⟨ sym (+-assoc 1 (length ys) (length ys')) ⟩ + 1 + (length ys + length ys') ≡⟨ ap (1 +_) IH ⟩ + 1 + length xs ∎ + + {- The type of "true-case" is equal to "goal-statement true". -} + true-case : length (x :: ys) + length ys' ≡ length (x :: xs) + true-case = + length (x :: ys) + length ys' ≡⟨ refl _ ⟩ + (1 + length ys) + length ys' ≡⟨ +-assoc 1 (length ys) (length ys') ⟩ + 1 + (length ys + length ys') ≡⟨ ap (1 +_) IH ⟩ + 1 + length xs ∎ + +{- Here is a version that uses Agda's built-in with-keyword (as shown by Eric in + the lab of 28 Feb) instead of Bool-elim. (Under the hood, they amount to the + same thing.) -} +length-of-filters' : {A : Type} (P : A → Bool) (xs : List A) + → length (filter P xs) + length (filter (not ∘ P) xs) + ≡ length xs +length-of-filters' P [] = refl _ + +length-of-filters' P (x :: xs) with P x +length-of-filters' P (x :: xs) | true = true-case + where + ys = filter P xs + ys' = filter (not ∘ P) xs + + IH : length ys + length ys' ≡ length xs + IH = length-of-filters P xs + + true-case : length (x :: ys) + length ys' ≡ length (x :: xs) + true-case = + length (x :: ys) + length ys' ≡⟨ refl _ ⟩ + (1 + length ys) + length ys' ≡⟨ +-assoc 1 (length ys) (length ys') ⟩ + 1 + (length ys + length ys') ≡⟨ ap (1 +_) IH ⟩ + 1 + length xs ∎ + +length-of-filters' P (x :: xs) | false = false-case + where + ys = filter P xs + ys' = filter (not ∘ P) xs + + IH : length ys + length ys' ≡ length xs + IH = length-of-filters P xs + + false-case : length ys + length (x :: ys') ≡ length (x :: xs) + false-case = + length ys + length (x :: ys') ≡⟨ refl _ ⟩ + length ys + (1 + length ys') ≡⟨ +-assoc (length ys) 1 (length ys') ⟩ + (length ys + 1) + length ys' ≡⟨ ap (_+ length ys') (+-comm (length ys) 1) ⟩ + (1 + length ys) + length ys' ≡⟨ sym (+-assoc 1 (length ys) (length ys')) ⟩ + 1 + (length ys + length ys') ≡⟨ ap (1 +_) IH ⟩ + 1 + length xs ∎ +``` diff --git a/src/HoTTEST/Agda/Exercises/Pool/Homework/homework5.lagda.md b/src/HoTTEST/Agda/Exercises/Pool/Homework/homework5.lagda.md new file mode 100644 index 0000000..53b6786 --- /dev/null +++ b/src/HoTTEST/Agda/Exercises/Pool/Homework/homework5.lagda.md @@ -0,0 +1,196 @@ +# Week 5 - Homework Sheet + +**Please finish the lab sheet before moving on to these exercises.** + +```agda +{-# OPTIONS --without-K --safe #-} + +module Pool.Homework.homework5 where + +open import prelude hiding (Bool-elim) +open import natural-numbers-functions hiding (_≤_ ; is-even ; +-assoc ; +-comm) +open import List-functions +open import isomorphisms +``` +We will also want to use some things from the Lab and Homework sheet of Week 4: + +```agda +open import Pool.Homework.homework4-solutions +open import Pool.Lab.lab4-solutions +``` + +## Part I: More on length + +Besides `map`, the function `reverse` is another example of a length-preserving +operation. + +```agda +length-of-reverse : {A : Type} (xs : List A) + → length (reverse xs) ≡ length xs +length-of-reverse = {!!} +``` + +**Prove** the above. + +## Part II: More on isomorphisms + +### Exercise 2a + +```agda +ℕ-[⋆]-iso : ℕ ≅ List 𝟙 +ℕ-[⋆]-iso = record { bijection = f ; bijectivity = f-is-bijection } + where + f : ℕ → List 𝟙 + f = {!!} + + g : List 𝟙 → ℕ + g = {!!} + + gf : g ∘ f ∼ id + gf = {!!} + + fg : f ∘ g ∼ id + fg = {!!} + + f-is-bijection : is-bijection f + f-is-bijection = record { inverse = g ; η = gf ; ε = fg } +``` + +**Show** that the type of natural numbers `ℕ` is isomorphic to the type of lists +over the unit type `𝟙`. + +Hint: The statement of Exercise 2b may help you. + +### Exercise 2b + +```agda +open _≅_ + +ℕ→[⋆]-preserves-length : (n : ℕ) → length (bijection ℕ-[⋆]-iso n) ≡ n +ℕ→[⋆]-preserves-length = {!!} +``` + +Notice how `bijection` extracts the function `f` you defined in `ℕ-[⋆]-iso`. + +**Prove** that for any `n : ℕ`, the length of the list `f n : List 𝟙` +(where `f : ℕ → List 𝟙` is as you defined in Exercise 3a) is `n`. + +## Part III: More on evenness + +In this exercise, we will continue where we left off in the lab exercises on +evenness. Recall the predicates `is-even` and `check-even`: + +```agda +is-even : ℕ → Type +is-even x = Σ y ꞉ ℕ , x ≡ 2 * y +``` + +```agda +check-even : ℕ → Bool +check-even zero = true +check-even (suc zero) = false +check-even (suc (suc n)) = check-even n +``` + +Now recall the discussion about decidable predicates that we had in the previous +weeks. When you proved that `check-even` and `is-even` are logically equivalent +in the lab, you have in fact implicitly proved that `is-even` is a decidable +predicate! In this exercise, we will make this implicit proof _explicit_. + +**Complete** the remaining holes in the following proof outline; starting with +proving a lemma stating that a Boolean is either `true` or `false`. + +```agda +principle-of-bivalence : (b : Bool) → (b ≡ true) ∔ (b ≡ false) +principle-of-bivalence = {!!} + +is-even-is-decidable : (n : ℕ) → is-decidable (is-even n) +is-even-is-decidable n = + ∔-nondep-elim goal₁ goal₂ (principle-of-bivalence (check-even n)) + where + goal₁ : check-even n ≡ true → is-decidable (is-even n) + goal₁ p = {!!} + + goal₂ : check-even n ≡ false → is-decidable (is-even n) + goal₂ p = inr subgoal + where + subgoal : ¬ is-even n + subgoal q = {!!} +``` + +## Part IV: Stretcher exercises on length + +*The following two exercises are rather hard and are should be interesting to +students that like a challenge.* + +Recall that we can define `filter` as +```agda +filter : {A : Type} → (A → Bool) → List A → List A +filter P [] = [] +filter P (x :: xs) = if P x then (x :: ys) else ys + where + ys = filter P xs +``` + +We also remind you of the inductively defined less-than-or-equal relation `≤` +on the natural numbers. + +```agdacode +data _≤_ : ℕ → ℕ → Type where + ≤-zero : ( y : ℕ) → 0 ≤ y + ≤-suc : (x y : ℕ) → x ≤ y → suc x ≤ suc y +``` + +Finally, the following lemmas are provided to you for your convenience. + +```agda +≤-suc-lemma : (n : ℕ) → n ≤ (1 + n) +≤-suc-lemma 0 = ≤-zero (1 + 0) +≤-suc-lemma (suc n) = goal + where + IH : n ≤ (1 + n) + IH = ≤-suc-lemma n + goal : suc n ≤ suc (suc n) + goal = ≤-suc n (suc n) IH + +Bool-elim : (A : Bool → Type) + → A false + → A true + → (x : Bool) → A x +Bool-elim A x₀ x₁ false = x₀ +Bool-elim A x₀ x₁ true = x₁ +``` + +### Exercise 4a (stretcher 🌶) + +**Prove** that filtering a list decreases its length. + +```agda +length-of-filter : {A : Type} (P : A → Bool) (xs : List A) + → length (filter P xs) ≤ length xs +length-of-filter = {!!} +``` + +*Hints*: +- You can use `≤-trans` from the [sample solutions to + Lab 4](lab4-solutions.lagda.md) if you need that `≤` is transitive. + (The sample solutions are already imported for you.) +- Think about how to use `Bool-elim`. + +### Exercise 4b (stretcher 🌶🌶) + +Given a predicate `P : A → Bool` and a list `xs : List A`, we could filter `xs` +by `P` and by `not ∘ P`. If we do this and compute the lengths of the resulting +lists, then we expect their sum to be equal to the length of the unfiltered list +`xs`. **Prove** this fact. + +```agda +length-of-filters : {A : Type} (P : A → Bool) (xs : List A) + → length (filter P xs) + length (filter (not ∘ P) xs) + ≡ length xs +length-of-filters = {!!} +``` + +*Hint*: You can use associativity (`+-assoc`) and commutativity (`+-comm`) from +the sample solutions to last week's exercises. (The necessary files are already +imported for you.) diff --git a/src/HoTTEST/Agda/Exercises/Pool/Homework/homework6-solutions.lagda.md b/src/HoTTEST/Agda/Exercises/Pool/Homework/homework6-solutions.lagda.md new file mode 100644 index 0000000..992e63a --- /dev/null +++ b/src/HoTTEST/Agda/Exercises/Pool/Homework/homework6-solutions.lagda.md @@ -0,0 +1,96 @@ +```agda +{-# OPTIONS --without-K --safe #-} + +module Pool.Homework.homework6-solutions where + +open import prelude +open import isomorphisms +open import natural-numbers-functions +open import Fin +open import Vector + + +×-distributivity-+ : (X Y Z : Type) → (X ∔ Y) × Z ≅ (X × Z) ∔ (Y × Z) +×-distributivity-+ X Y Z = + record { bijection = f + ; bijectivity = record { inverse = g + ; η = section + ; ε = retraction } } + where + f : (X ∔ Y) × Z → (X × Z) ∔ (Y × Z) + f (inl x , z) = inl (x , z) + f (inr y , z) = inr (y , z) + + g : (X × Z) ∔ (Y × Z) → (X ∔ Y) × Z + g (inl (x , z)) = inl x , z + g (inr (y , z)) = inr y , z + + section : (g ∘ f) ∼ id + section (inl x , z) = refl (inl x , z) + section (inr y , z) = refl (inr y , z) + + retraction : (f ∘ g) ∼ id + retraction (inl (x , z)) = refl (inl (x , z)) + retraction (inr (y , z)) = refl (inr (y , z)) + +alternate : Type → Type → Bool → Type +alternate X _ true = X +alternate _ Y false = Y + +∔-isomorphic-to-Σ-bool : (X Y : Type) → (X ∔ Y) ≅ (Σ b ꞉ Bool , alternate X Y b) +∔-isomorphic-to-Σ-bool X Y = + record { bijection = f ; bijectivity = record { inverse = g + ; η = section + ; ε = retraction } } + where + f : X ∔ Y → Σ b ꞉ Bool , alternate X Y b + f (inl x) = true , x + f (inr y) = false , y + + g : (Σ b ꞉ Bool , alternate X Y b) → X ∔ Y + g (true , X) = inl X + g (false , Y) = inr Y + + section : (g ∘ f) ∼ id + section (inl x) = refl (inl x) + section (inr y) = refl (inr y) + + retraction : (f ∘ g) ∼ id + retraction (true , X) = refl (true , X) + retraction (false , Y) = refl (false , Y) + + +fib : ℕ → ℕ +fib 0 = 0 +fib 1 = 1 +fib (suc (suc n)) = fib n + fib (suc n) + +fib-aux : ℕ → ℕ → ℕ → ℕ +fib-aux a b 0 = b +fib-aux a b (suc n) = fib-aux (b + a) a n + +fib-fast : ℕ → ℕ +fib-fast = fib-aux 1 0 + +fib-aux-fib-lemma : (k n : ℕ) → fib-aux (fib (suc n)) (fib n) k ≡ fib (k + n) +fib-aux-fib-lemma zero n = refl (fib n) +fib-aux-fib-lemma (suc k) n = + fib-aux (fib n + fib (1 + n)) (fib (1 + n)) k ≡⟨ refl _ ⟩ + fib-aux (fib (2 + n)) (fib (1 + n)) k ≡⟨ IH ⟩ + fib (k + suc n) ≡⟨ ap fib (+-step k n) ⟩ + fib (suc (k + n)) ∎ + where + IH : fib-aux (fib (2 + n)) (fib (1 + n)) k ≡ fib (k + suc n) + IH = fib-aux-fib-lemma k (suc n) + +fib-fast-is-correct : (n : ℕ) → fib-fast n ≡ fib n +fib-fast-is-correct n = fib-fast n ≡⟨ refl _ ⟩ + fib-aux 1 0 n ≡⟨ claim ⟩ + fib (n + 0) ≡⟨ ap fib (+-base n) ⟩ + fib n ∎ + where + lemma : fib-aux (fib 1) (fib 0) n ≡ fib (n + 0) + lemma = fib-aux-fib-lemma n 0 + claim : fib-aux 1 0 n ≡ fib (n + 0) + claim = lemma +``` diff --git a/src/HoTTEST/Agda/Exercises/Pool/Homework/homework6.lagda.md b/src/HoTTEST/Agda/Exercises/Pool/Homework/homework6.lagda.md new file mode 100644 index 0000000..5caf9f6 --- /dev/null +++ b/src/HoTTEST/Agda/Exercises/Pool/Homework/homework6.lagda.md @@ -0,0 +1,109 @@ +# Week 6 - Homework Sheet + +```agda +{-# OPTIONS --without-K --safe #-} + +module Pool.Homework.homework6 where + +open import prelude +open import isomorphisms +open import natural-numbers-functions +open import Fin +open import Vector +``` + +## Part I: Isomorphisms + +### Exercise 1.1 + +**Prove** the following isomorphism: + +```agda +×-distributivity-+ : (X Y Z : Type) → (X ∔ Y) × Z ≅ (X × Z) ∔ (Y × Z) +×-distributivity-+ = {!!} +``` + +### Exercise 1.2 + +We now define a function `alternate` that takes two types `X` and `Y` and +evaluates to `X` on `true` and evaluates to `Y` on `false`. + +```agda +alternate : Type → Type → Bool → Type +alternate X _ true = X +alternate _ Y false = Y +``` + +It can be proved that `Σ b ꞉ Bool , alternate X Y b` is the same thing as `X ∔ +Y`. **Prove** this by constructing the following isomorphism: + +```agda +∔-isomorphic-to-Σ-bool : (X Y : Type) → (X ∔ Y) ≅ (Σ b ꞉ Bool , alternate X Y b) +∔-isomorphic-to-Σ-bool = {!!} +``` + +## Part II: Proving correctness of efficient Fibonacci + +In Functional Programming you saw two ways of defining the Fibonacci function. +The first one, `fib` was fairly simple, but inefficient + +```agda +fib : ℕ → ℕ +fib 0 = 0 +fib 1 = 1 +fib (suc (suc n)) = fib n + fib (suc n) +``` + +Therefore, we introduce a second version, `fib-fast`, which uses an accumulating +parameter to make it more efficient. + +```agda +fib-aux : ℕ → ℕ → ℕ → ℕ +fib-aux a b 0 = b +fib-aux a b (suc n) = fib-aux (b + a) a n + +fib-fast : ℕ → ℕ +fib-fast = fib-aux 1 0 +``` + +It is not obvious, however, that `fib-fast` implements the same behaviour as +`fib`. In Agda, we can *prove* this, showing that `fib-fast` is correct. + +The following lemma relates `fib-aux` and `fib` and is fundamental in proving +the correctness of `fib-fast`. You will be asked to prove it later. + +```agda +fib-aux-fib-lemma : (k n : ℕ) → fib-aux (fib (suc n)) (fib n) k ≡ fib (k + n) +fib-aux-fib-lemma = {!!} -- Come back to this later +``` + +### Exercise 2.1 + +Using `fib-aux-fib-lemma`, **prove** the correctness of `fib-fast`. + +```agda +fib-fast-is-correct : (n : ℕ) → fib-fast n ≡ fib n +fib-fast-is-correct n = {!!} +``` + +*Hints*: +1. The lemma allows you to prove this fairly directly. There is no need to do +induction on `n`. +1. You may also find the `+-base` function from the +[natural-numbers-functions](../natural-numbers-functions.lagda.md) module useful. + +### Exercise 2.2 + +Now **complete** the proof of fundamental lemma `fib-aux-fib-lemma` above. + +*Hint*: You will probably want to use `+-step` from +[natural-numbers-functions](../natural-numbers-functions.lagda.md) at some +point. + +### References + +The exercises and solutions of Part 2 were based on [Neil Mitchell's +proof][mitchell] in the programming language [Idris][idris]. + +[mitchell]: http://neilmitchell.blogspot.com/2017/05/proving-fib-equivalence.html +[idris]: https://en.wikipedia.org/wiki/Idris_(programming_language) diff --git a/src/HoTTEST/Agda/Exercises/Pool/Lab/lab2-solutions.lagda.md b/src/HoTTEST/Agda/Exercises/Pool/Lab/lab2-solutions.lagda.md new file mode 100644 index 0000000..e8d6436 --- /dev/null +++ b/src/HoTTEST/Agda/Exercises/Pool/Lab/lab2-solutions.lagda.md @@ -0,0 +1,183 @@ +# Week 2 - Lab Sheet - Solutions + +```agda +{-# OPTIONS --without-K --safe #-} + +module Pool.Lab.lab2-solutions where + +open import prelude hiding (_||_) + +_≣_ : Bool → Bool → Type +true ≣ true = 𝟙 +true ≣ false = 𝟘 +false ≣ true = 𝟘 +false ≣ false = 𝟙 + +_==_ : Bool → Bool → Bool +true == true = true +true == false = false +false == true = false +false == false = true + +≡-to-≣ : (x y : Bool) → x ≡ y → x ≣ y +≡-to-≣ true true e = ⋆ +≡-to-≣ false false e = ⋆ + +is-true : Bool → Type +is-true b = b ≡ true + +≣-to-== : (x y : Bool) → x ≣ y → is-true (x == y) +≣-to-== true true e = refl true +≣-to-== false false e = refl true + +==-to-≡ : (x y : Bool) → is-true (x == y) → x ≡ y +==-to-≡ true true e = refl true +==-to-≡ false false e = refl false + +≡-to-== : (x y : Bool) → x ≡ y → is-true (x == y) +≡-to-== x y e = ≣-to-== x y (≡-to-≣ x y e) + +≣-to-≡ : (x y : Bool) → x ≣ y → x ≡ y +≣-to-≡ x y e = ==-to-≡ x y (≣-to-== x y e) + +==-to-≣ : (x y : Bool) → is-true (x == y) → x ≣ y +==-to-≣ x y e = ≡-to-≣ x y (==-to-≡ x y e) + + +data Either (A B : Type) : Type where + left : A → Either A B + right : B → Either A B + +if'_then_else_ : {A B : Type} → Bool → A → B → Either A B +if' true then x else y = left x +if' false then x else y = right y + + +_||_ : Bool → Bool → Bool +true || y = true +false || y = y + +_||'_ : Bool → Bool → Bool +true ||' true = true +true ||' false = true +false ||' true = true +false ||' false = false + + +||-assoc : (a b c : Bool) → a || (b || c) ≡ (a || b) || c +||-assoc true b c = refl true +||-assoc false b c = refl (b || c) + +||'-assoc : (a b c : Bool) → a ||' (b ||' c) ≡ (a ||' b) ||' c +||'-assoc true true true = refl true +||'-assoc true true false = refl true +||'-assoc true false true = refl true +||'-assoc true false false = refl true +||'-assoc false true true = refl true +||'-assoc false true false = refl true +||'-assoc false false true = refl true +||'-assoc false false false = refl false + +||-is-commutative : (a b : Bool) → a || b ≡ b || a +||-is-commutative true true = refl true +||-is-commutative true false = refl true +||-is-commutative false true = refl true +||-is-commutative false false = refl false + +false-left-unit-for-|| : (b : Bool) → false || b ≡ b +false-left-unit-for-|| = refl + +false-right-unit-for-|| : (b : Bool) → b || false ≡ b +false-right-unit-for-|| true = refl true +false-right-unit-for-|| false = refl false + +&&-is-associative : (a b c : Bool) → a && (b && c) ≡ (a && b) && c +&&-is-associative true b c = refl (b && c) +&&-is-associative false b c = refl false + +&&-is-commutative : (a b : Bool) → a && b ≡ b && a +&&-is-commutative true true = refl true +&&-is-commutative true false = refl false +&&-is-commutative false true = refl false +&&-is-commutative false false = refl false + +true-left-unit-for-&& : (b : Bool) → true && b ≡ b +true-left-unit-for-&& = refl + +true-right-unit-for-&& : (b : Bool) → b && true ≡ b +true-right-unit-for-&& true = refl true +true-right-unit-for-&& false = refl false + +&&-distributes-over-|| : (a b c : Bool) → a && (b || c) ≡ (a && b) || (a && c) +&&-distributes-over-|| true true c = refl true +&&-distributes-over-|| true false true = refl true +&&-distributes-over-|| true false false = refl false +&&-distributes-over-|| false b c = refl false + +||-distributes-over-&& : (a b c : Bool) → a || (b && c) ≡ (a || b) && (a || c) +||-distributes-over-&& true b c = refl true +||-distributes-over-&& false true c = refl c +||-distributes-over-&& false false c = refl false + + ++-suc-on-left : (x y : ℕ) → (suc x) + y ≡ suc (x + y) ++-suc-on-left zero y = refl (suc y) ++-suc-on-left (suc x) y = refl (suc (suc x) + y) + +max : ℕ → ℕ → ℕ +max zero n = n +max (suc m) zero = suc m +max (suc m) (suc n) = suc (max m n) + +min : ℕ → ℕ → ℕ +min zero n = zero +min (suc m) zero = zero +min (suc m) (suc n) = suc (min m n) + + ++-0-on-right : (x : ℕ) → x + 0 ≡ x ++-0-on-right zero = refl zero ++-0-on-right (suc x) = ap suc IH + where + IH : x + 0 ≡ x -- IH is short for induction hypothesis + IH = +-0-on-right x + ++-suc-on-right : (x y : ℕ) → x + suc y ≡ suc (x + y) ++-suc-on-right zero y = refl (suc y) ++-suc-on-right (suc x) y = ap suc IH + where + IH : x + suc y ≡ suc (x + y) + IH = +-suc-on-right x y + +max-idempotent : (x : ℕ) → max x x ≡ x +max-idempotent zero = refl zero +max-idempotent (suc x) = ap suc IH + where + IH : max x x ≡ x + IH = max-idempotent x + +max-commutative : (x y : ℕ) → max x y ≡ max y x +max-commutative zero zero = refl zero +max-commutative zero (suc y) = refl (suc y) +max-commutative (suc x) zero = refl (suc x) +max-commutative (suc x) (suc y) = ap suc IH + where + IH : max x y ≡ max y x + IH = max-commutative x y + +min-idempotent : (x : ℕ) → min x x ≡ x +min-idempotent zero = refl zero +min-idempotent (suc x) = ap suc IH + where + IH : min x x ≡ x + IH = min-idempotent x + +min-commutative : (x y : ℕ) → min x y ≡ min y x +min-commutative zero zero = refl zero +min-commutative zero (suc y) = refl zero +min-commutative (suc x) zero = refl zero +min-commutative (suc x) (suc y) = ap suc IH + where + IH : min x y ≡ min y x + IH = min-commutative x y +``` diff --git a/src/HoTTEST/Agda/Exercises/Pool/Lab/lab2.lagda.md b/src/HoTTEST/Agda/Exercises/Pool/Lab/lab2.lagda.md new file mode 100644 index 0000000..efa9e74 --- /dev/null +++ b/src/HoTTEST/Agda/Exercises/Pool/Lab/lab2.lagda.md @@ -0,0 +1,345 @@ +# Week 2 - Lab Sheet + +```agda +{-# OPTIONS --without-K --safe #-} +module Pool.Lab.lab2 where + +open import prelude hiding (if_then_else_;_||_) +``` + +## Part I: Booleans + +### Section 1: Operations on Booleans + +#### Exercise 1.1 + +We have already defined `if-then-else` like so: + +```agda +if_then_else_ : {A : Type} → Bool → A → A → A +if true then x else y = x +if false then x else y = y +``` + +But this requires our two branches to be of the same type A. Instead, we could +define `if-then-else` to have branches of different types, using the `Either` +datatype + +```agda +data Either (A B : Type) : Type where + left : A → Either A B + right : B → Either A B + +if'_then_else_ : {A B : Type} → Bool → A → B → Either A B +if' b then x else y = {!!} +``` + +**Define** this function. + +#### Exercise 1.2 + +We can define the `_||_` function in (at least) two different ways, depending on +how much pattern-matching we wish to perform: + +```agda +_||_ : Bool → Bool → Bool +true || y = {!!} +false || y = {!!} + +_||'_ : Bool → Bool → Bool +true ||' true = {!!} +true ||' false = {!!} +false ||' true = {!!} +false ||' false = {!!} +``` + +**Complete** the two holes for `_||_` and the four holes for `_||'_`. + +The `_||_` operator is *associative*, meaning `a || (b || c) ≡ (a || b) || c`. + +We can prove this for both of our definitions: + +```agda +||-assoc : (a b c : Bool) → a || (b || c) ≡ (a || b) || c +||-assoc a b c = {!!} + +||'-assoc : (a b c : Bool) → a ||' (b ||' c) ≡ (a ||' b) ||' c +||'-assoc a b c = {!!} +``` + +**Complete** both of these proofs. +(Hint: Because `||` was defined by pattern matching on the first argument, it +may be helpful to use the same strategy.) + +Which of these did you prefer proving, and why? + +#### Exercise 1.3 + +**Complete** the proof that Boolean disjunction is commutative: + +```agda +||-is-commutative : (a b : Bool) → a || b ≡ b || a +||-is-commutative a b = {!!} +``` + +#### Exercise 1.4 + +**Complete** the proof that `false` is the left unit element of `||`: + +```agda +false-left-unit-for-|| : (b : Bool) → false || b ≡ b +false-left-unit-for-|| b = {!!} +``` + +**Complete** the proof that `false` is the right unit element of `||`: + +```agda +false-right-unit-for-|| : (b : Bool) → b || false ≡ b +false-right-unit-for-|| b = {!!} +``` + +#### Exercise 1.5 + +Now prove the analogous properties for `&&`. + +**Complete** the proof that Boolean conjunction is associative: + +```agda +&&-is-associative : (a b c : Bool) → a && (b && c) ≡ (a && b) && c +&&-is-associative a b c = {!!} +``` + +**Complete** the proof that Boolean conjunction is commutative: + +```agda +&&-is-commutative : (a b : Bool) → a && b ≡ b && a +&&-is-commutative a b = {!!} +``` + +**Complete** the proof that that `true` is the left unit element of `&&`: + +```agda +true-left-unit-for-&& : (b : Bool) → true && b ≡ b +true-left-unit-for-&& b = {!!} +``` + +**Complete** the proof that that `true` is the right unit element of `&&`: + +```agda +true-right-unit-for-&& : (b : Bool) → b && true ≡ b +true-right-unit-for-&& b = {!!} +``` + +#### Exercise 1.6 + +An important algebraic property between two operators is *distributivity*. For +example, multiplication distributes over addition since `x * (y + z) = (x * y) + +(x * z)` for every `x, y, z ∈ ℕ`. The same is the case for the Boolean +conjunction and disjunction operators we have defined. + +**Complete** the proof that Boolean conjunction distributes over Boolean +disjunction: + +```agda +&&-distributes-over-|| : (a b c : Bool) → a && (b || c) ≡ (a && b) || (a && c) +&&-distributes-over-|| a b c = {!!} +``` + +```agda +||-distributes-over-&& : (a b c : Bool) → a || (b && c) ≡ (a || b) && (a || c) +||-distributes-over-&& a b c = {!!} +``` + + +### Section 2: Identity type and `Bool` + +With these exercises, you will practise with the identity type and `Bool`. + +#### Exercise 2.1 + +Recall how we defined `_≣_` for [natural +numbers](../introduction.lagda.md#the-empty-type-and-the-unit-type). We could do +the same for the Booleans. + +```agda +_≣_ : Bool → Bool → Type +x ≣ y = {!!} +``` + +**Define** this function. + +Tip: you can type `≣` as `\===`. + +#### Exercise 2.2 + +We can also define a Boolean-valued equality function. + +```agda +_==_ : Bool → Bool → Bool +x == y = {!!} +``` + +**Define** this function. + +#### Exercise 2.3 + +Now we write a conversion function from `x ≡ y` to `x ≣ y`, just like the +function `back` in +[introduction](../introduction.lagda.md#the-identity-type-former-__). + +```agda +≡-to-≣ : (x y : Bool) → x ≡ y → x ≣ y +≡-to-≣ x y e = {!!} +``` + +**Complete** this function. + +#### Exercise 2.4 + +The following helper function makes some functions nicer to read. + +```agda +is-true : Bool → Type +is-true b = b ≡ true +``` + +Now we can consider another conversion function. + +```agda +≣-to-== : (x y : Bool) → x ≣ y → is-true (x == y) +≣-to-== x y e = {!!} +``` + +**Define** this function. + +#### Exercise 2.5 + +And similarly, we have + +```agda +==-to-≡ : (x y : Bool) → is-true (x == y) → x ≡ y +==-to-≡ x y e = {!!} +``` + +**Define** this function. + +#### Exercise 2.6 + +Finally, we can use the above functions to define the three remaining +implications + +```agda +≡-to-== : (x y : Bool) → x ≡ y → is-true (x == y) +≡-to-== x y e = {!!} + +≣-to-≡ : (x y : Bool) → x ≣ y → x ≡ y +≣-to-≡ x y e = {!!} + +==-to-≣ : (x y : Bool) → is-true (x == y) → x ≣ y +==-to-≣ x y e = {!!} +``` + +**Complete** these functions. + + +## Part II: Natural numbers + +### Section 1: Some functions on natural numbers + +#### Exercise 1.1 + +**Complete** the proof of the fact that `(suc x) + y ≡ suc (x + y)` + +```agda ++-suc-on-left : (x y : ℕ) → (suc x) + y ≡ suc (x + y) ++-suc-on-left x y = {!!} +``` + +#### Exercise 1.2 + +We can define `max` operation on natural numbers as follows: + +```agda +max : ℕ → ℕ → ℕ +max zero n = n +max (suc m) zero = suc m +max (suc m) (suc n) = suc (max m n) +``` + +**Define** the function `min` analogously: + +```agda +min : ℕ → ℕ → ℕ +min m n = {!!} +``` + +### Section 2: Natural numbers and proofs using induction + +The function `+-0-on-right` expresses the fact that `0` is a right unit of the +operation `_+_`. Notice how we use an inductive hypothesis (the recursive call) +in the proof. (Spelling this out, `IH` tells us that `x + 0 ≡ x`, so that `ap +suc IH` witnesses that `suc (x + 0) ≡ suc x`.) + +```agda ++-0-on-right : (x : ℕ) → x + 0 ≡ x ++-0-on-right zero = refl zero ++-0-on-right (suc x) = ap suc IH + where + IH : x + 0 ≡ x -- IH is short for induction hypothesis + IH = +-0-on-right x +``` + +Using similar induction hypotheses, try to complete the following exercises. + + +#### Exercise 2.1 + +**Complete** the proof: + +```agda ++-suc-on-right : (x y : ℕ) → x + suc y ≡ suc (x + y) ++-suc-on-right x y = {!!} +``` + +#### Exercise 2.2 + +In algebra, an operator `_*_` is called idempotent iff `x * x = x` for every +`x`. `max` is an example of an idempotent operator: + +**Complete** the hole to prove that `max` is idempotent: + +```agda +max-idempotent : (x : ℕ) → max x x ≡ x +max-idempotent x = {!!} +``` + +#### Exercise 2.3 + +**Complete** the hole by constructing a proof of the fact that `max` is a +commutative operator: + +```agda +max-commutative : (x y : ℕ) → max x y ≡ max y x +max-commutative x y = {!!} +``` + +#### Exercise 2.4 + +Now recall that we defined `min` in Exercise 1.2. Similarly, we can show that +`min` is idempotent and commutative too. + +**Complete** the proof that `min` is idempotent: + +```agda +min-idempotent : (x : ℕ) → min x x ≡ x +min-idempotent x = {!!} +``` + +#### Exercise 2.5 + +**Complete** the proof that `min` is commutative: + +```agda +min-commutative : (x y : ℕ) → min x y ≡ min y x +min-commutative x y = {!!} +``` diff --git a/src/HoTTEST/Agda/Exercises/Pool/Lab/lab3-solutions.lagda.md b/src/HoTTEST/Agda/Exercises/Pool/Lab/lab3-solutions.lagda.md new file mode 100644 index 0000000..075327e --- /dev/null +++ b/src/HoTTEST/Agda/Exercises/Pool/Lab/lab3-solutions.lagda.md @@ -0,0 +1,122 @@ +# Week 3 - Lab Sheet - Solutions + +```agda +{-# OPTIONS --without-K --safe #-} +module Pool.Lab.lab3-solutions where + +open import prelude hiding (𝟘-nondep-elim) + +¬¬_ : Type → Type +¬¬ A = ¬ (¬ A) + +¬¬¬ : Type → Type +¬¬¬ A = ¬ (¬¬ A) + +∔-introduction-left : {A B : Type} → A → A ∔ B +∔-introduction-left a = inl a + +∔-introduction-right : {A B : Type} → B → A ∔ B +∔-introduction-right b = inr b + +∔-elimination : {A B X : Type} → (A → X) → (B → X) → (A ∔ B → X) +∔-elimination p q (inl a) = p a +∔-elimination p q (inr b) = q b + + +×-elimination-left : {A B : Type} → A × B → A +×-elimination-left (a , b) = a + +×-elimination-right : {A B : Type} → A × B → B +×-elimination-right (a , b) = b + +×-introduction : {A B : Type} → A → B → A × B +×-introduction a b = (a , b) + +×-introduction' : {A B X : Type} → (X → A) → (X → B) → (X → A × B) +×-introduction' p q x = (p x , q x) + + +uncurry : {A B X : Type} → (A → B → X) → (A × B → X) +uncurry p (a , b) = p a b + +curry : {A B X : Type} → (A × B → X) → (A → B → X) +curry p a b = p (a , b) + +→-trans : {A B C : Type} → (A → B) → (B → C) → (A → C) +→-trans p q = λ a → q (p a) + + +𝟘-nondep-elim : {A : Type} → 𝟘 → A +𝟘-nondep-elim = λ () + +¬¬-introduction : {A : Type} → A → ¬¬ A +¬¬-introduction a = λ p → p a + +not-implies-not³ : {A : Type} → ¬ A → ¬¬¬ A +not-implies-not³ p = λ q → q p + +not³-implies-not : {A : Type} → ¬¬¬ A → ¬ A +not³-implies-not p = λ a → p (λ q → q a) + +contraposition : {A B : Type} → (A → B) → ¬ B → ¬ A +contraposition p q = λ a → q (p a) + +¬¬-functor : {A B : Type} → (A → B) → ¬¬ A → ¬¬ B +¬¬-functor p = contraposition (contraposition p) + +¬¬-kleisli : {A B : Type} → (A → ¬¬ B) → ¬¬ A → ¬¬ B +¬¬-kleisli p = contraposition (λ q → λ a → p a q) + + +de-morgan₁ : {A B : Type} → ¬ (A ∔ B) → ¬ A × ¬ B +de-morgan₁ {A} {B} p = ×-introduction p₁ p₂ + where + p₁ : ¬ A + p₁ = p ∘ ∔-introduction-left + p₂ : ¬ B + p₂ = p ∘ ∔-introduction-right + +de-morgan₂ : {A B : Type} → ¬ A ∔ ¬ B → ¬ (A × B) +de-morgan₂ = ∔-elimination (→-trans ×-elimination-left) + (→-trans ×-elimination-right) + +¬-and-+-exercise₁ : {A B : Type} → ¬ A ∔ B → A → B +¬-and-+-exercise₁ (inl p) a = 𝟘-nondep-elim (p a) +¬-and-+-exercise₁ (inr q) _ = q + +¬-and-+-exercise₂ : {A B : Type} → ¬ A ∔ B → ¬ (A × ¬ B) +¬-and-+-exercise₂ (inl p) = λ { (a , _) → p a } +¬-and-+-exercise₂ (inr q) = λ { (a , r) → r q } + +distributivity₁ : {A B C : Type} → (A × B) ∔ C → (A ∔ C) × (B ∔ C) +distributivity₁ {A} {B} {C} = ∔-elimination q₁ q₂ + where + q₁ : A × B → (A ∔ C) × (B ∔ C) + q₁ (a , b) = (inl a , inl b) + q₂ : C → A ∔ C × B ∔ C + q₂ c = (inr c , inr c) + +distributivity₂ : {A B C : Type} → (A ∔ B) × C → (A × C) ∔ (B × C) +distributivity₂ (p , c) = + ∔-elimination (λ a → inl (a , c)) (λ b → inr (b , c)) p + + +Σ-introduction : {A : Type} {B : (A → Type)} → (a : A) → B a → (Σ a ꞉ A , B a) +Σ-introduction a b = (a , b) + +Σ-to-× : {A : Type} {B : (A → Type)} → ((a , _) : (Σ a ꞉ A , B a)) → A × B a +Σ-to-× (a , b) = (a , b) + +Σ-on-Bool : {B : Bool → Type} → (Σ x ꞉ Bool , B x) → B true ∔ B false +Σ-on-Bool (true , b) = inl b +Σ-on-Bool (false , b) = inr b + + +Π-apply : {A : Type} {B : (A → Type)} → ((a : A) → B a) → (a : A) → B a +Π-apply p a = p a + +Π-→ : {A : Type} {B C : A → Type} + → ((a : A) → B a → C a) + → ((a : A) → B a) → ((a : A) → C a) +Π-→ p q = λ a → p a (q a) +``` diff --git a/src/HoTTEST/Agda/Exercises/Pool/Lab/lab3.lagda.md b/src/HoTTEST/Agda/Exercises/Pool/Lab/lab3.lagda.md new file mode 100644 index 0000000..b70849c --- /dev/null +++ b/src/HoTTEST/Agda/Exercises/Pool/Lab/lab3.lagda.md @@ -0,0 +1,261 @@ +# Week 3 - Lab Sheet + +```agda +{-# OPTIONS --without-K --safe #-} + +module Pool.Lab.lab3 where + +open import prelude hiding (𝟘-nondep-elim) +``` + +Before we proceed with the exercises, we introduce a some convenient notation +for multiple negations. + +```agda +¬¬_ : Type → Type +¬¬ A = ¬ (¬ A) + +¬¬¬ : Type → Type +¬¬¬ A = ¬ (¬¬ A) +``` + +## Part I: Propositional logic + +### Section 1: Disjunction + +#### Exercise 1.1 + +**Complete** the following proofs involving disjunctions. + +```agda +∔-introduction-left : {A B : Type} → A → A ∔ B +∔-introduction-left = {!!} + +∔-introduction-right : {A B : Type} → B → A ∔ B +∔-introduction-right = {!!} +``` + +#### Exercise 1.2 + +**Complete** the proof about disjunctions. + +```agda +∔-elimination : {A B X : Type} → (A → X) → (B → X) → (A ∔ B → X) +∔-elimination = {!!} +``` + +### Section 2: Conjunction + +#### Exercise 2.1 + +**Complete** the following proofs involving conjunctions. + +```agda +×-elimination-left : {A B : Type} → A × B → A +×-elimination-left = {!!} + +×-elimination-right : {A B : Type} → A × B → B +×-elimination-right = {!!} +``` + +#### Exercise 2.2 + +**Prove** the following: + +```agda +×-introduction : {A B : Type} → A → B → A × B +×-introduction = {!!} + +×-introduction' : {A B X : Type} → (X → A) → (X → B) → (X → A × B) +×-introduction' = {!!} +``` + +### Section 3: Implication + +#### Exercise 3.1 + +**Complete** the following proofs involving implications. + +```agda +uncurry : {A B X : Type} → (A → B → X) → (A × B → X) +uncurry = {!!} + +curry : {A B X : Type} → (A × B → X) → (A → B → X) +curry = {!!} +``` + +You probably already know `curry` and `uncurry` from Haskell, but notice how we +can read them from a logical perspective: `uncurry` says that if `A` implies +that `B` implies `X`, then the conjunction of `A` and `B` implies `X`. + +#### Exercise 3.2 + +**Prove** that implication is transitive. + +``` +→-trans : {A B C : Type} → (A → B) → (B → C) → (A → C) +→-trans = {!!} +``` + +Notice that the proof that implication is transitive is just function +composition. + + +### Section 4: Negation + +The fact that falsity implies everything is known as the [_principle of +explosion_](https://en.wikipedia.org/wiki/Principle_of_explosion) or _ex falso +quodlibet_. + +**Complete** the proof of the principle of explosion in Agda. + +#### Exercise 4.1 + +```agda +𝟘-nondep-elim : {A : Type} → 𝟘 → A +𝟘-nondep-elim = {!!} +``` + +#### Exercise 4.2 + +**Complete** the proof a proposition implies its own double negation. + +```agda +¬¬-introduction : {A : Type} → A → ¬¬ A +¬¬-introduction = {!!} +``` + +#### Exercise 4.3 + +**Prove** that having three negations is the logically equivalent to having a +single negation. + +```agda +not-implies-not³ : {A : Type} → ¬ A → ¬¬¬ A +not-implies-not³ = {!!} + +not³-implies-not : {A : Type} → ¬¬¬ A → ¬ A +not³-implies-not = {!!} +``` + +#### Exercise 4.4 + +**Complete** the proof of contraposition. + +```agda +contraposition : {A B : Type} → (A → B) → ¬ B → ¬ A +contraposition = {!!} +``` + +#### Exercise 4.5 + +Use `contraposition` to **complete** the following two proofs. + +```agda +¬¬-functor : {A B : Type} → (A → B) → ¬¬ A → ¬¬ B +¬¬-functor = {!!} + +¬¬-kleisli : {A B : Type} → (A → ¬¬ B) → ¬¬ A → ¬¬ B +¬¬-kleisli = {!!} +``` +{- + Better HINT: + start with f : A → ¬¬ B + apply ¬¬-functor to get ¬¬ A → ¬¬¬¬ B + Now use ¬¬¬¬ B → ¬¬ B which is a particular case of not-implies-not³ with A = ¬ B + + HINT: + start with f : A → ¬¬ B + apply contraposition to get ¬¬¬ B → ¬A + do this again to get ¬¬ A → ¬¬¬¬ B + Now use ¬¬¬¬ B → ¬¬ B which is a particular case of not-implies-not³ with A = ¬ B +-} + +### Section 5: De Morgan Laws and logical laws + +The De Morgan laws cannot be proved in Agda, though some of the implications +involved in the De Morgan laws _can_ be. The following exercises will involve +proving these (and some other similar laws) for Agda types. + +#### Exercise 5.1 + +**Complete** the proofs. + +```agda +de-morgan₁ : {A B : Type} → ¬ (A ∔ B) → ¬ A × ¬ B +de-morgan₁ = {!!} + +de-morgan₂ : {A B : Type} → ¬ A ∔ ¬ B → ¬ (A × B) +de-morgan₂ = {!!} +``` + +#### Exercise 5.2 + +**Complete** the proofs. + +```agda +¬-and-+-exercise₁ : {A B : Type} → ¬ A ∔ B → A → B +¬-and-+-exercise₁ = {!!} + +¬-and-+-exercise₂ : {A B : Type} → ¬ A ∔ B → ¬ (A × ¬ B) +¬-and-+-exercise₂ = {!!} +``` + +#### Exercise 5.3 + +**Prove** the distributivity laws for `×` and `∔`. + +```agda +distributivity₁ : {A B C : Type} → (A × B) ∔ C → (A ∔ C) × (B ∔ C) +distributivity₁ = {!!} + +distributivity₂ : {A B C : Type} → (A ∔ B) × C → (A × C) ∔ (B × C) +distributivity₂ = {!!} +``` + +## Part II: Logic with quantifiers + +### Section 1: Sums + +#### Exercise 1.1 + +**Complete** the following constructions. + +```agda +Σ-introduction : {A : Type} {B : (A → Type)} → (a : A) → B a → (Σ a ꞉ A , B a) +Σ-introduction = {!!} + +Σ-to-× : {A : Type} {B : (A → Type)} → ((a , _) : (Σ a ꞉ A , B a)) → A × B a +Σ-to-× = {!!} +``` + +#### Exercise 1.2 + +**Complete** the following proof about sums over Booleans. + +```agda +Σ-on-Bool : {B : Bool → Type} → (Σ x ꞉ Bool , B x) → B true ∔ B false +Σ-on-Bool = {!!} +``` + +### Section 2: Products + +#### Exercise 2.1 + +Complete the proof. + +```agda +Π-apply : {A : Type} {B : (A → Type)} → ((a : A) → B a) → (a : A) → B a +Π-apply = {!!} +``` + +#### Exercise 2.2 + +**Prove** the following. + +```agda +Π-→ : {A : Type} {B C : A → Type} + → ((a : A) → B a → C a) + → ((a : A) → B a) → ((a : A) → C a) +Π-→ = {!!} +``` diff --git a/src/HoTTEST/Agda/Exercises/Pool/Lab/lab4-solutions.lagda.md b/src/HoTTEST/Agda/Exercises/Pool/Lab/lab4-solutions.lagda.md new file mode 100644 index 0000000..fc8cb1c --- /dev/null +++ b/src/HoTTEST/Agda/Exercises/Pool/Lab/lab4-solutions.lagda.md @@ -0,0 +1,89 @@ +```agda +{-# OPTIONS --without-K --safe #-} + +module Pool.Lab.lab4-solutions where + +open import prelude +open import List-functions + +reverse-lemma : {X : Type} (x : X) (xs : List X) + → x :: reverse xs ≡ reverse (xs ++ [ x ]) +reverse-lemma x [] = refl (x :: []) +reverse-lemma x (y :: xs) = ap (_++ [ y ]) (reverse-lemma x xs) + +reverse-is-involution : {X : Type} → (xs : List X) → xs ≡ reverse (reverse xs) +reverse-is-involution {X} [] = refl [] +reverse-is-involution {X} (x :: xs) + = x :: xs ≡⟨ ap (x ::_) (reverse-is-involution xs) ⟩ + x :: reverse (reverse xs) ≡⟨ reverse-lemma x (reverse xs) ⟩ + reverse (reverse xs ++ [ x ]) ≡⟨ refl (reverse (reverse xs ++ [ x ])) ⟩ + reverse (reverse (x :: xs)) ∎ + ++-assoc : (x y z : ℕ) → x + (y + z) ≡ (x + y) + z ++-assoc zero y z = refl (y + z) ++-assoc (suc x) y z = ap suc (+-assoc x y z) + + +data _≤_ : ℕ → ℕ → Type where + ≤-zero : ( y : ℕ) → 0 ≤ y + ≤-suc : (x y : ℕ) → x ≤ y → suc x ≤ suc y + +_≤'_ : ℕ → ℕ → Type +x ≤' y = Σ k ꞉ ℕ , x + k ≡ y + + +≤'-zero : (y : ℕ) → 0 ≤' y +≤'-zero y = y , refl y + +≤'-suc : (x y : ℕ) → x ≤' y → suc x ≤' suc y +≤'-suc x y (n , p) = n , ap suc p + +≤-prime : (x y : ℕ) → x ≤ y → x ≤' y +≤-prime 0 y (≤-zero y) = ≤'-zero y +≤-prime (suc x) (suc y) (≤-suc x y p) = ≤'-suc x y (≤-prime x y p) + +≤-unprime : (x y : ℕ) → x ≤' y → x ≤ y +≤-unprime zero y (n , p) = ≤-zero y +≤-unprime (suc x) (suc .(x + n)) (n , refl .(suc (x + n))) + = ≤-suc x (x + n) (≤-unprime x (x + n) (n , refl (x + n))) + +≤-trans : (x y z : ℕ) → x ≤ y → y ≤ z → x ≤ z +≤-trans zero y z p q = ≤-zero z +≤-trans (suc x) .(suc y) .(suc z) (≤-suc .x y p) (≤-suc .y z q) + = ≤-suc x z (≤-trans x y z p q) + +≤'-trans : (x y z : ℕ) → x ≤' y → y ≤' z → x ≤' z +≤'-trans x .(x + n) .((x + n) + m) (n , refl .(x + n)) (m , refl .((x + n) + m)) + = (n + m) , +-assoc x n m + +is-decidable : Type → Type +is-decidable A = A ∔ ¬ A + +has-decidable-equality : Type → Type +has-decidable-equality A = (x y : A) → is-decidable (x ≡ y) + +bool-has-decidable-equality : has-decidable-equality Bool +bool-has-decidable-equality true true = inl (refl true) +bool-has-decidable-equality true false = inr (λ ()) +bool-has-decidable-equality false true = inr (λ ()) +bool-has-decidable-equality false false = inl (refl false) + +≤-lemma : (m n : ℕ) → suc m ≤ suc n → m ≤ n +≤-lemma m n (≤-suc m n p) = p + +not-suc-≤-zero : (n : ℕ) → ¬ (suc n ≤ 0) +not-suc-≤-zero n () + +≤-is-decidable : (m n : ℕ) → is-decidable (m ≤ n) +≤-is-decidable zero zero = inl (≤-zero zero) +≤-is-decidable zero (suc n) = inl (≤-zero (suc n)) +≤-is-decidable (suc m) zero = inr (not-suc-≤-zero m) +≤-is-decidable (suc m) (suc n) = ∔-nondep-elim f g IH + where + IH : (m ≤ n) ∔ ¬ (m ≤ n) + IH = ≤-is-decidable m n + f : m ≤ n → is-decidable (suc m ≤ suc n) + f p = inl (≤-suc m n p) + g : ¬ (m ≤ n) → is-decidable (suc m ≤ suc n) + g p = inr (λ q → p (≤-lemma m n q)) +``` diff --git a/src/HoTTEST/Agda/Exercises/Pool/Lab/lab4.lagda.md b/src/HoTTEST/Agda/Exercises/Pool/Lab/lab4.lagda.md new file mode 100644 index 0000000..9a3848f --- /dev/null +++ b/src/HoTTEST/Agda/Exercises/Pool/Lab/lab4.lagda.md @@ -0,0 +1,210 @@ +# Week 4 - Lab Sheet + +```agda +{-# OPTIONS --without-K --safe #-} + +module Pool.Lab.lab4 where + +open import prelude +open import List-functions +``` + +## Part I: Reverse is an involution + +We wish to prove that the `reverse` function on lists is an involution; +that is to say that reversing a list twice is the same as doing nothing. + +### Exercise 1.1 + +First, we will prove the following lemma. + +```agda +reverse-lemma : {X : Type} (x : X) (xs : List X) + → x :: reverse xs ≡ reverse (xs ++ [ x ]) +reverse-lemma = {!!} +``` + +**Prove** the lemma about `reverse`. + +### Exercise 1.2 + +The following proof skeleton has been provided for you, using notation for +equational reasoning. + +```agda +reverse-is-involution : {X : Type} → (xs : List X) → xs ≡ reverse (reverse xs) +reverse-is-involution [] = {!!} +reverse-is-involution (x :: xs) + = x :: xs ≡⟨ {!!} ⟩ + x :: reverse (reverse xs) ≡⟨ {!!} ⟩ + reverse (reverse xs ++ [ x ]) ≡⟨ {!!} ⟩ + reverse (reverse (x :: xs)) ∎ +``` + +**Fill** the holes of our proof that `reverse` is an involution. + +## Part II: Associativity of addition of natural numbers + +We wish to prove the associativity of `_+_` on the natural numbers. + +```agda ++-assoc : (x y z : ℕ) → x + (y + z) ≡ (x + y) + z ++-assoc = {!!} +``` + +**Complete** the proof that `_+_` is associative. + +## Part III: Order on the natural numbers + +In this part we will study two ways of expressing that a natural number is less +than or equal to another natural number. + +The first definition is an inductive one. + +```agda +data _≤_ : ℕ → ℕ → Type where + ≤-zero : ( y : ℕ) → 0 ≤ y + ≤-suc : (x y : ℕ) → x ≤ y → suc x ≤ suc y +``` + +It says: +1. that zero is less than or equal to any natural number; +1. if `x` is less than or equal to `y`, then `suc x` is less than or equal to `suc y`. + +The second definition uses a `Σ`-type. + +```agda +_≤'_ : ℕ → ℕ → Type +x ≤' y = Σ k ꞉ ℕ , x + k ≡ y +``` + +It says that `x` is less than or equal to `y` if we have some `k : ℕ` +such that `x + k ≡ y`. + +We will prove that the two definitions are logically equivalent. + +### Exercise 3.1 + +In order to prove that the first definition implies the second, we first +prove two little lemmas about `_≤'_`. + +Note that they amount to the constructors of `_≤_`. + +```agda +≤'-zero : ( y : ℕ) → 0 ≤' y +≤'-zero = {!!} + +≤'-suc : (x y : ℕ) → x ≤' y → suc x ≤' suc y +≤'-suc = {!!} +``` + +**Prove** the two little lemmas about `_≤'_`. + +### Exercise 3.2 + +We now prove that the first definition implies the second. + +```agda +≤-prime : (x y : ℕ) → x ≤ y → x ≤' y +≤-prime = {!!} +``` + +**Prove** that `x ≤ y` implies `x ≤' y` using the little lemmas from 3.1. + +### Exercise 3.3 + +The other direction is slightly trickier. + +```agda +≤-unprime : (x y : ℕ) → x ≤' y → x ≤ y +≤-unprime = {!!} +``` + +**Prove** that `x ≤' y` implies `x ≤ y`. + +*Hint:* The base case only requires pattern matching on `x`, whereas +the inductive case requires further pattern matching. + +### Exercise 3.4 + +The order on the natural numbers is transitive, meaning that if +`x ≤ y` and `y ≤ z` then also `x ≤ z`. We can prove this for +both our definitions of the order. + +```agda +≤-trans : (x y z : ℕ) → x ≤ y → y ≤ z → x ≤ z +≤-trans = {!!} + +≤'-trans : (x y z : ℕ) → x ≤' y → y ≤' z → x ≤' z +≤'-trans = {!!} +``` + +**Complete** the proofs that `_≤_` and `_≤'_` are transitive. + +## Part IV: Decidability and decidable equality + +In last week's exercises, you constructed proofs of logical formulae by writing +Agda programs through the interpretation of Agda types as propositions. We +mentioned, however, that this interpretation does not make sense _a priori_ in +the setting of classical logic and we said that we have to work _constructively_ +to make logical sense of Agda's types. More specifically, we mentioned in +Exercise 2.3 of Homework 3 that the logical interpretation of the law of +excluded middle `(A : Type) → A ∔ ¬ A`, is not provable in Agda. + +Notice, however, that the statement + +```txt + (A : Type) → A ∔ ¬ A +``` + +is to be viewed as asserting that _the law of excluded middle holds for all +types_. Even though this fails to hold in the context of Agda's type system, it +doesn't mean that the law of excluded middle does not hold for _some_ types. In +this exercise, this is precisely the question that we will be interested in; we +will study “types that admit the law of excluded middle”. Such types are called +**decidable types**. We will express this notion through the Agda predicate +`is-decidable`: + +```agda +is-decidable : Type → Type +is-decidable A = A ∔ ¬ A +``` + +To assert `is-decidable A` for some type `A` is to say that type `A` satisfies +the law of excluded middle: we can either construct an inhabitant of `A` or +prove that the existence of an inhabitant for `A` is impossible. + +We can consider the decidability of any type but we will often be interested in +the _decidability of equality types_. A type `A` is said to have _decidable +equality_ iff for any two `x y : A`, the identity type `x ≡ y` is decidable. We +can write this notion down in Agda as follows: + +```agda +has-decidable-equality : Type → Type +has-decidable-equality A = (x y : A) → is-decidable (x ≡ y) +``` + +### Exercise 4.1 + +To familiarise yourself with the notion of decidable equality, **prove** that +the `Bool` type has decidable equality: + +```agda +bool-has-decidable-equality : has-decidable-equality Bool +bool-has-decidable-equality = {!!} +``` + +### Exercise 4.2 + +**Prove** that the `_≤_` relation on `ℕ` is decidable. You will have to use +the following lemma: + +```agda +≤-lemma : (m n : ℕ) → suc m ≤ suc n → m ≤ n +≤-lemma m n (≤-suc m n p) = p +``` + +```agda +≤-is-decidable : (m n : ℕ) → is-decidable (m ≤ n) +≤-is-decidable = {!!} +``` diff --git a/src/HoTTEST/Agda/Exercises/Pool/Lab/lab5-solutions.lagda.md b/src/HoTTEST/Agda/Exercises/Pool/Lab/lab5-solutions.lagda.md new file mode 100644 index 0000000..fd2acc9 --- /dev/null +++ b/src/HoTTEST/Agda/Exercises/Pool/Lab/lab5-solutions.lagda.md @@ -0,0 +1,219 @@ +```agda +{-# OPTIONS --without-K --safe #-} + +module Pool.Lab.lab5-solutions where + +open import prelude +open import natural-numbers-functions hiding (_≤_ ; is-even ; +-assoc ; +-comm) +open import List-functions +open import isomorphisms + +open import decidability +open import negation + +open import Pool.Homework.homework4-solutions +open import Pool.Lab.lab4-solutions + + +{- Verbose solution, manually performing and showing many equalities that hold + by definition. -} +map-preserves-length : {A B : Type} (f : A → B) (xs : List A) + → length (map f xs) ≡ length xs +map-preserves-length f [] = refl _ +map-preserves-length f (x :: xs) = + length (map f (x :: xs)) ≡⟨ refl _ ⟩ + length (f x :: (map f xs)) ≡⟨ refl _ ⟩ + 1 + length (map f xs) ≡⟨ ap (1 +_) IH ⟩ + 1 + length xs ∎ + where + IH : length (map f xs) ≡ length xs + IH = map-preserves-length f xs + +{- Alternative, shorter solution that relies on Agda computing the necessary + equalities for us. + We come up with such proofs by looking at the goal, using `C-c C-,`, which + we sometimes normalize (by prefixing the command by `C-u C-u`) to let Agda + do even more work for us. -} +map-preserves-length' : {A B : Type} (f : A → B) (xs : List A) + → length (map f xs) ≡ length xs +map-preserves-length' f [] = refl _ +map-preserves-length' f (x :: xs) = ap suc (map-preserves-length' f xs) + +{- Verbose solution, manually performing and showing many equalities that hold + by definition. -} +length-of-++ : {A : Type} (xs ys : List A) + → length (xs ++ ys) ≡ length xs + length ys +length-of-++ [] ys = refl (length ys) +length-of-++ (x :: xs) ys = + length ((x :: xs) ++ ys) ≡⟨ refl _ ⟩ + length (x :: (xs ++ ys)) ≡⟨ refl _ ⟩ + (1 + length (xs ++ ys)) ≡⟨ ap (1 +_) IH ⟩ + (1 + (length xs + length ys)) ≡⟨ +-assoc 1 (length xs) (length ys) ⟩ + (1 + length xs) + length ys ≡⟨ refl _ ⟩ + length (x :: xs) + length ys ∎ + where + IH : length (xs ++ ys) ≡ length xs + length ys + IH = length-of-++ xs ys + + {- Alternative, shorter solution that relies on Agda computing the necessary + equalities for us. + We come up with such proofs by looking at the goal, using `C-c C-,`, which + we sometimes normalize (by prefixing the command by `C-u C-u`) to let Agda + do even more work for us. -} +length-of-++' : {A : Type} (xs ys : List A) + → length (xs ++ ys) ≡ length xs + length ys +length-of-++' [] ys = refl (length ys) +length-of-++' (x :: xs) ys = ap suc IH + where + IH : length (xs ++ ys) ≡ length xs + length ys + IH = length-of-++ xs ys + +length-of-prefix : {A : Type} (xs ys : List A) + → xs ≼' ys + → length xs ≤' length ys +length-of-prefix xs ys (zs , e) = (length zs , goal) + where + goal = length xs + length zs ≡⟨ sym (length-of-++ xs zs) ⟩ + length (xs ++ zs) ≡⟨ ap length e ⟩ + length ys ∎ + +is-nonempty : {A : Type} → List A → Type +is-nonempty xs = 1 ≤' length xs + +tail : {A : Type} (xs : List A) → is-nonempty xs → List A +tail (x :: xs) p = xs + +head : {A : Type} (xs : List A) → is-nonempty xs → A +head (x :: xs) p = x + +length-of-tail : {A : Type} (xs : List A) (p : is-nonempty xs) + → 1 + length (tail xs p) ≡ length xs +length-of-tail (x :: xs) p = + 1 + length (tail (x :: xs) p) ≡⟨ refl _ ⟩ + 1 + length xs ≡⟨ refl _ ⟩ + length (x :: xs) ∎ + +≤'-suc-lemma : (n : ℕ) → n ≤' (1 + n) +≤'-suc-lemma zero = (1 , refl 1) +≤'-suc-lemma (suc n) = (1 , goal) + where + goal : suc n + 1 ≡ 1 + suc n + goal = +-comm (suc n) 1 + +length-of-tail-decreases : {A : Type} (xs : List A) (p : is-nonempty xs) + → length (tail xs p) ≤' length xs +length-of-tail-decreases (x :: xs) p = goal + where + goal : length xs ≤' (1 + length xs) + goal = ≤'-suc-lemma (length xs) + +×-iso : (X Y : Type) → X × Y ≅ Y × X +×-iso X Y = record { bijection = f ; bijectivity = f-is-bijection } + where + f : X × Y → Y × X + f (x , y) = y , x + + g : Y × X → X × Y + g (y , x) = x , y + + gf : g ∘ f ∼ id + gf (x , y) = refl (x , y) + + fg : f ∘ g ∼ id + fg (y , x) = refl (y , x) + + f-is-bijection : is-bijection f + f-is-bijection = record { inverse = g ; η = gf ; ε = fg } + ++-iso : (X Y : Type) → X ∔ Y ≅ Y ∔ X ++-iso X Y = record { bijection = f ; bijectivity = f-is-bijection } + where + f : X ∔ Y → Y ∔ X + f (inl x) = inr x + f (inr y) = inl y + + g : Y ∔ X → X ∔ Y + g (inl y) = inr y + g (inr x) = inl x + + gf : g ∘ f ∼ id + gf (inl x) = refl (inl x) + gf (inr y) = refl (inr y) + + fg : f ∘ g ∼ id + fg (inl y) = refl (inl y) + fg (inr x) = refl (inr x) + + f-is-bijection : is-bijection f + f-is-bijection = record { inverse = g ; η = gf ; ε = fg } + +lists-from-vectors : {A : Type} → List A ≅ (Σ n ꞉ ℕ , Vector A n) +lists-from-vectors {A} + = record { bijection = f ; bijectivity = f-is-bijection } + where + f : List A → Σ n ꞉ ℕ , Vector A n + f [] = 0 , [] + f (x :: xs) = suc (fst (f xs)) , (x :: snd (f xs)) + + g : Σ n ꞉ ℕ , Vector A n → List A + g (0 , [] ) = [] + g (suc n , (x :: xs)) = x :: g (n , xs) + + gf : g ∘ f ∼ id + gf [] = refl [] + gf (x :: xs) = ap (x ::_) (gf xs) + + fg : f ∘ g ∼ id + fg (0 , [] ) = refl (0 , []) + fg (suc n , (x :: xs)) + = ap (λ - → suc (fst -) , (x :: snd -)) (fg (n , xs)) + + f-is-bijection : is-bijection f + f-is-bijection = record { inverse = g ; η = gf ; ε = fg } + +open _≅_ + +lfv-preserves-length : {A : Type} (xs : List A) + → length xs + ≡ fst (bijection lists-from-vectors xs) +lfv-preserves-length [] = refl 0 +lfv-preserves-length (x :: xs) = ap suc (lfv-preserves-length xs) + +is-even : ℕ → Type +is-even x = Σ y ꞉ ℕ , x ≡ 2 * y + +check-even : ℕ → Bool +check-even zero = true +check-even (suc zero) = false +check-even (suc (suc n)) = check-even n + +evenness-lemma₁ : (n : ℕ) → is-even (2 + n) → is-even n +evenness-lemma₁ n (suc k , p) = k , goal + where + subgoal : suc (suc n) ≡ suc (suc (2 * k)) + subgoal = suc (suc n) ≡⟨ p ⟩ + suc k + suc k ≡⟨ ap suc (+-comm k (suc k)) ⟩ + suc ((suc k) + k) ∎ + + goal : n ≡ 2 * k + goal = suc-is-injective (suc-is-injective subgoal) + +evenness-lemma₂ : (n : ℕ) → is-even n → is-even (2 + n) +evenness-lemma₂ n (k , p) = suc k , goal + where + goal : 2 + n ≡ 2 * suc k + goal = 2 + n ≡⟨ ap (λ - → 2 + -) p ⟩ + 2 + (k + k) ≡⟨ ap suc (+-comm (suc k) k) ⟩ + suc k + suc k ∎ + +even⇒check-even : (n : ℕ) → is-even n → check-even n ≡ true +even⇒check-even zero _ = refl true +even⇒check-even (suc zero) (suc zero , ()) +even⇒check-even (suc zero) (suc (suc _) , ()) +even⇒check-even (suc (suc n)) p = even⇒check-even n (evenness-lemma₁ n p) + +check-even⇒even : (n : ℕ) → check-even n ≡ true → is-even n +check-even⇒even zero (refl true) = zero , refl zero +check-even⇒even (suc zero) () +check-even⇒even (suc (suc n)) p = evenness-lemma₂ n (check-even⇒even n p) +``` diff --git a/src/HoTTEST/Agda/Exercises/Pool/Lab/lab5.lagda.md b/src/HoTTEST/Agda/Exercises/Pool/Lab/lab5.lagda.md new file mode 100644 index 0000000..808e403 --- /dev/null +++ b/src/HoTTEST/Agda/Exercises/Pool/Lab/lab5.lagda.md @@ -0,0 +1,312 @@ +# Week 5 - Lab Sheet + +```agda +{-# OPTIONS --without-K --safe #-} + +module Pool.Lab.lab5 where + +open import prelude +open import natural-numbers-functions hiding (_≤_ ; is-even ; is-odd ; +-comm) +open import List-functions +open import isomorphisms +``` + +We will also want to use some things from the Lab and Homework sheet of Week 4: + +```agda +open import Pool.Homework.homework4-solutions +open import Pool.Lab.lab4-solutions +``` + +## Part I: Length + +In the file [List-functions.lagda.md](../List-functions.lagda.md), the +function `length` is recursively defined as follows. + +```agdacode +length : {A : Type} → List A → ℕ +length [] = 0 +length (x :: xs) = 1 + length xs +``` + +In the following exercises we will prove some facts involving the length of +lists. In doing so, you will practise with inductive proofs. + +### Exercise 1.1 + +Recall that we defined `map` as follows (see +[List-functions.lagda.md](../List-functions.lagda.md)). + +```agdacode +map : {A B : Type} → (A → B) → List A → List B +map f [] = [] +map f (x :: xs) = f x :: map f xs +``` + +**Prove** that `map` preserves the length of a list. + +```agda +map-preserves-length : {A B : Type} (f : A → B) (xs : List A) + → length (map f xs) ≡ length xs +map-preserves-length = {!!} +``` + +### Exercise 1.2 + +Another useful fact is that the length of two concatenated lists is the sum of +their respective lengths. **Complete** the proof of this fact. + +```agda +length-of-++ : {A : Type} (xs ys : List A) + → length (xs ++ ys) ≡ length xs + length ys +length-of-++ = {!!} +``` + +### Exercise 1.3 + +Recall `≤'` from Lab Sheet 4 and `≼'` from Homework Sheet 4 + +```agdacode +_≤'_ : ℕ → ℕ → Type +x ≤' y = Σ k ꞉ ℕ , x + k ≡ y +``` + +```agdacode +_≼'_ : {X : Type} → List X → List X → Type +_≼'_ {X} xs ys = Σ zs ꞉ List X , xs ++ zs ≡ ys +``` + +and that `x ≤' y` expresses that the natural number `x` is less than or equal to +the natural number `y`, while `xs ≼' ys` expresses that the list `xs` is a +prefix of the list `ys`. + +**Prove** that the length of a prefix of a list `ys` is less than the length of +`ys`, relating the two notions above. + +```agda +length-of-prefix : {A : Type} (xs ys : List A) + → xs ≼' ys + → length xs ≤' length ys +length-of-prefix = {!!} +``` + +### Exercise 1.4 + +A nice use case of the length function is that we are now able to define safe +`head` and `tail` operations on nonempty lists. + +We say that a list is *nonempty* if its length is at least 1. +```agda +is-nonempty : {A : Type} → List A → Type +is-nonempty xs = 1 ≤' length xs +``` + +We can then define `tail` as follows. +```agda +tail : {A : Type} (xs : List A) → is-nonempty xs → List A +tail (x :: xs) p = xs +``` + +Agda accepts this definition and does not complain about missing the `[]`-case, +because it realizes that `[]` does not satisfy `is-nonempty`. + +#### Exercise 1.4a + +```agda +head : {A : Type} (xs : List A) → is-nonempty xs → A +head = {!!} +``` + +**Complete** the definition of `head` yourself. + +#### Exercise 1.4b + +```agda +length-of-tail : {A : Type} (xs : List A) (p : 1 ≤' length xs) + → 1 + length (tail xs p) ≡ length xs +length-of-tail = {!!} +``` + +**Prove** that the length of a list is obtained by adding 1 to the length of the +tail. + +#### Exercise 1.4c + +```agda +≤'-suc-lemma : (n : ℕ) → n ≤' (1 + n) +≤'-suc-lemma = {!!} + +length-of-tail-decreases : {A : Type} (xs : List A) (p : 1 ≤' length xs) + → length (tail xs p) ≤' length xs +length-of-tail-decreases = {!!} +``` + +**Complete** the proof of the following lemma and use it to prove that the +length of the tail of a list is less than the length of the full list. + +## Part II: Isomorphisms + +Make sure you have read the [file on isomorphisms](../isomorphisms.lagda.md), +where we define ismorphisms and show that `Bool ≅ 𝟚`. + +You will now give three more isomorphisms. In each case, you should think +about *why* and *how* each pair of types are isomorphic before attemping to +formalise the isomorphism. + +### Exercise 2.1 + +```agda +×-iso : (X Y : Type) → X × Y ≅ Y × X +×-iso X Y = record { bijection = f ; bijectivity = f-is-bijection } + where + f : X × Y → Y × X + f = {!!} + + g : Y × X → X × Y + g = {!!} + + gf : g ∘ f ∼ id + gf = {!!} + + fg : f ∘ g ∼ id + fg = {!!} + + f-is-bijection : is-bijection f + f-is-bijection = record { inverse = g ; η = gf ; ε = fg } +``` + +**Show** that X × Y is isomorphic to Y × X using the above template. + +### Exercise 2.2 + +```agda ++-iso : (X Y : Type) → X ∔ Y ≅ Y ∔ X ++-iso X Y = record { bijection = f ; bijectivity = f-is-bijection } + where + f : X ∔ Y → Y ∔ X + f = {!!} + + g : Y ∔ X → X ∔ Y + g = {!!} + + gf : g ∘ f ∼ id + gf = {!!} + + fg : f ∘ g ∼ id + fg = {!!} + + f-is-bijection : is-bijection f + f-is-bijection = record { inverse = g ; η = gf ; ε = fg } +``` + +### Exercise 2.3 + +```agda +lists-from-vectors : {A : Type} → List A ≅ (Σ n ꞉ ℕ , Vector A n) +lists-from-vectors {A} + = record { bijection = f ; bijectivity = f-is-bijection } + where + f : List A → Σ n ꞉ ℕ , Vector A n + f = {!!} + + g : Σ n ꞉ ℕ , Vector A n → List A + g = {!!} + + gf : g ∘ f ∼ id + gf = {!!} + + fg : f ∘ g ∼ id + fg = {!!} + + f-is-bijection : is-bijection f + f-is-bijection = record { inverse = g ; η = gf ; ε = fg } +``` + +**Show** that the the type `List A` is isomorphic to the type `Σ (Vector A)`. + +(Note that this is the first question in [this question +sheet](../vector-and-list-isomorphisms.lagda.md)). + +Hint: The statement of Exercise 2.3b may help you. + +### Exercise 2.3b + +```agda +open _≅_ + +lfv-preserves-length : {A : Type} (xs : List A) + → fst (bijection lists-from-vectors xs) + ≡ length xs +lfv-preserves-length = {!!} +``` + +Notice how `bijection` extracts the function `f` you defined in +`lists-from-vectors`. + +**Prove** that for any `xs : List A`, the length of `xs` is the same as the +first projection of `f xs : Σ (Vector A)` (where `f : ℕ → List 𝟙` is as you +defined in Exercise 4a). + +## Part III: Evenness + +In the lecture notes, you have seen the predicates `is-even` and `is-odd`: + +```agda +is-even is-odd : ℕ → Type +is-even x = Σ y ꞉ ℕ , x ≡ 2 * y +is-odd x = Σ y ꞉ ℕ , x ≡ 1 + 2 * y +``` + +In these exercises, we will define a Boolean-valued version of the `is-even` +predicate and prove that the two versions are _logically_ equivalent: + +```agda +check-even : ℕ → Bool +check-even zero = true +check-even (suc zero) = false +check-even (suc (suc n)) = check-even n +``` + +### Exercise 3.1 + +First, we will have to prove two lemmas that we will use in Exercise 3.2, where +we will prove our main result. + +```agda +evenness-lemma₁ : (n : ℕ) → is-even (2 + n) → is-even n +evenness-lemma₁ n (suc k , p) = k , goal + where + subgoal : suc (suc n) ≡ suc (suc (2 * k)) + subgoal = suc (suc n) ≡⟨ {!!} ⟩ + suc k + suc k ≡⟨ {!!} ⟩ + suc ((suc k) + k) ∎ + + goal : n ≡ 2 * k + goal = suc-is-injective (suc-is-injective subgoal) + +evenness-lemma₂ : (n : ℕ) → is-even n → is-even (2 + n) +evenness-lemma₂ n (k , p) = suc k , goal + where + goal : 2 + n ≡ 2 * suc k + goal = 2 + n ≡⟨ {!!} ⟩ + 2 + (k + k) ≡⟨ {!!} ⟩ + suc k + suc k ∎ +``` + +**Complete** the above proofs. + +### Exercise 3.2 + +**Prove** that if `is-even n` is inhabited, then `check-even n ≡ true`. + +```agda +even⇒check-even : (n : ℕ) → is-even n → check-even n ≡ true +even⇒check-even = {!!} +``` + +**Prove** that if `check-even n ≡ true` then `is-even n` is inhabited: + +```agda +check-even⇒even : (n : ℕ) → check-even n ≡ true → is-even n +check-even⇒even = {!!} +``` diff --git a/src/HoTTEST/Agda/Exercises/Pool/README.md b/src/HoTTEST/Agda/Exercises/Pool/README.md new file mode 100644 index 0000000..b917f90 --- /dev/null +++ b/src/HoTTEST/Agda/Exercises/Pool/README.md @@ -0,0 +1,19 @@ +Tom de Jong. + +A pool of Agda exercises originally intended for lab sessions and homework in +the module "Advanced Functional Programming" at the [School of Computer Science](https://www.birmingham.ac.uk/schools/computer-science/index.aspx) +of the [University of Birmingham](https://www.birmingham.ac.uk/index.aspx), UK. + +These exercises were created together with +[Ayberk Tosun](https://www.cs.bham.ac.uk/~axt978/) and +[Todd Waugh Ambridge](https://www.cs.bham.ac.uk/~txw467/). + + +In general (but not always) the [Homework](Homework) exercises were designed +to be a little more challenging than the exercises selected for the [Lab](Lab). +The numbers in the file names, e.g. `Lab4.lagda.md`, mark the week numbers of +the original module. Thus, a higher number typically indicates more advanced +exercises. + +All exercises are solved in a corresponding file marked with the suffix +`-solutions.lagda.md`. diff --git a/src/HoTTEST/Agda/Exercises/README.md b/src/HoTTEST/Agda/Exercises/README.md new file mode 100644 index 0000000..d614f56 --- /dev/null +++ b/src/HoTTEST/Agda/Exercises/README.md @@ -0,0 +1,21 @@ +# Links to the Agda Exercise Sheets + +1. Week 01: [Exercises](01-Exercises.lagda.md); + [Solutions](01-Solutions.lagda.md). +1. Week 02: [Exercises](02-Exercises.lagda.md); + [Solutions](02-Solutions.lagda.md). +1. Week 03: [Exercises](03-Exercises.lagda.md); + [Solutions](03-Solutions.lagda.md). +1. Week 04: [Exercises](../HITs/Exercises4.lagda.md); + [Solutions](../HITs/Solutions4.lagda.md). +1. Week 05: [Exercises](../HITs/Exercises5.lagda.md); + [Solutions (by Dan Licata)](../HITs/Solutions5-dan.lagda.md); + [Solutions (by Tom de Jong)](../HITs/Solutions5-tom.lagda.md). +1. Week 06: [Exercises](../HITs/Exercises6.lagda.md); + [Solutions](../HITs/Solutions6.lagda.md). +1. Week 07: [Exercises](../Cubical/Exercises7.lagda.md); + [Solutions](../Cubical/Solutions7.lagda.md). +1. Week 08: [Exercises](../Cubical/Exercises8.lagda.md); + [Solutions](../Cubical/Solutions8.lagda.md). +1. Week 09: [Exercises](../Cubical/Exercises9.lagda.md); + [Solutions](../Cubical/Solutions9.lagda.md). diff --git a/src/HoTTEST/Agda/HITs/Exercises4.lagda.md b/src/HoTTEST/Agda/HITs/Exercises4.lagda.md new file mode 100644 index 0000000..65dbdd2 --- /dev/null +++ b/src/HoTTEST/Agda/HITs/Exercises4.lagda.md @@ -0,0 +1,152 @@ +```agda +{-# OPTIONS --without-K --rewriting --allow-unsolved-metas #-} + +open import new-prelude +open import Lecture4-notes + +module Exercises4 where +``` + +# Constructing homotopies between paths + +(⋆) Give two "different" path-between-paths/homotopies between (loop ∙ ! +loop) ∙ loop and loop. They should be different in the sense that one +should cancel the !loop with the first loop, and one with the second +loop. But these aren't really *really* different, in that there will be +a path-between-paths-between-paths between the two! + +```agda +homotopy1 : (loop ∙ ! loop) ∙ loop ≡ loop +homotopy1 = {!!} + +homotopy2 : (loop ∙ ! loop) ∙ loop ≡ loop +homotopy2 = {!!} +``` + +(Harder exercise (🌶️): give a path between homotopy1 and +homotopy2! I'd recommend saving this until later though, because there +is a trick to it that we haven't covered in lecture yet.) + +```agda +path-between-paths-between-paths : homotopy1 ≡ homotopy2 +path-between-paths-between-paths = {!!} +``` + +# Functions are group homomorphism + +(⋆⋆) State and prove a general lemma about what ap of a function on the +inverse of a path (! p) does (see ap-∙ for inspiration). + +State and prove a general lemma about what ! (p ∙ q) is. + +Use them to prove that the double function takes loop-inverse to +loop-inverse concatenated with itself. + +```agda +double-!loop : ap double (! loop) ≡ ! loop ∙ ! loop +double-!loop = {!!} +``` + +(⋆) Define a function invert : S1 → S1 such that (ap invert) inverts a path +on the circle, i.e. sends the n-fold loop to the -n-fold loop. + +```agda +invert : S1 → S1 +invert = {!!} +``` + +# Circles equivalence + +See the maps between the 1 point circle and the 2 point circle in the +lecture code. Check that the composite map S1 → S1 +is homotopic to the identity on base and loop: + +(⋆) + +```agda +to-from-base : from (to base) ≡ base +to-from-base = {!!} +``` + +(⋆⋆⋆) + +``` +to-from-loop : ap from (ap to loop) ≡ loop +to-from-loop = {!!} +``` + +Note: the problems below here are progressively more optional, so if you +run out of time/energy at some point that is totally fine. + +# Torus to circles + +The torus is equivalent to the product of two circles S1 × S1. The idea +for the map from the torus to S1 × S1 that is part of this equivalence +is that one loop on on the torus is sent to to the circle loop in one +copy of S1, and the other loop on the torus to the loop in the other +copy of the circle. Define this map. + +Hint: for the image of the square, you might want a lemma saying how +paths in product types compose (⋆⋆⋆): + +```agda +compose-pair≡ : {A B : Type} {x1 x2 x3 : A} {y1 y2 y3 : B} + (p12 : x1 ≡ x2) (p23 : x2 ≡ x3) + (q12 : y1 ≡ y2) (q23 : y2 ≡ y3) + → ((pair≡ p12 q12) ∙ (pair≡ p23 q23)) ≡ {!!} [ (x1 , y1) ≡ (x3 , y3) [ A × B ] ] +compose-pair≡ = {!!} +``` + +(🌶️) +``` +torus-to-circles : Torus → S1 × S1 +torus-to-circles = {!!} +``` + +# Suspensions (🌶️) + +Find a type X such that the two-point circle Circle2 is equivalent to +the suspension Susp X. Check your answer by defining a logical +equivalence (functions back and forth), since we haven't seen how to +prove that such functions are inverse yet. + +```agda +c2s : Circle2 → Susp {!!} +c2s = {!!} + +s2c : Susp {!!} → Circle2 +s2c = {!!} +``` + +Suspension is a functor from types, which means that it acts on +functions as well as types. Define the action of Susp on functions: + +```agda +susp-func : {X Y : Type} → (f : X → Y) → Susp X → Susp Y +susp-func f = {!!} +``` + +To really prove that Susp is a functor, we should check that this action +on functions preserves the identity function and composition of +functions. But to do that we would need the dependent elimination rule +for suspensions, which we haven't talked about yet. + +# Pushouts (🌶️) + +Fix a type X. Find types A,B,C with functions f : C → A and g : C → B +such that the suspension Susp X is equivalent to the Pushout C A B f g. +Check your answer by defining a logical equivalence (functions back and +forth), since we haven't seen how to prove that such functions are +inverse yet. + +```agda +SuspFromPush : Type → Type +SuspFromPush A = {!!} + +s2p : (A : Type) → Susp A → SuspFromPush A +s2p A = {!!} + +p2s : (A : Type) → SuspFromPush A → Susp A +p2s A = {!!} +``` + diff --git a/src/HoTTEST/Agda/HITs/Exercises5.lagda.md b/src/HoTTEST/Agda/HITs/Exercises5.lagda.md new file mode 100644 index 0000000..fd7b108 --- /dev/null +++ b/src/HoTTEST/Agda/HITs/Exercises5.lagda.md @@ -0,0 +1,207 @@ + +```agda +{-# OPTIONS --rewriting --without-K --allow-unsolved-metas #-} + +open import new-prelude +open import Lecture5-notes +open import Solutions4 using (ap-!; to-from-base; to-from-loop; s2c; c2s; susp-func) + +module Exercises5 where +``` + +# 1 point and 2 point circles are equivalent (⋆) + +In lecture, we defined maps between the one point circle (S1) and the +two point circle (Circle2) and checked that the round-trip on Circle2 is +the identity. + +Prove that the round-trip on S1 is the identity (use to-from-base +and to-from-loop from the Lecture 4 exercises), and package all of +these items up as an equivalence S1 ≃ Circle2. + +```agda +to-from : (x : S1) → from (to x) ≡ x +to-from = {!!} + +circles-equivalent : S1 ≃ Circle2 +circles-equivalent = {!!} +``` + +# Reversing the circle (⋆⋆) + +Define a map S1 → S1 that "reverses the orientation of the circle", +i.e. sends loop to ! loop. + +```agda +rev : S1 → S1 +rev = {!!} +``` + +Prove that rev is an equivalence. Hint: you will need to state and prove +one new generalized "path algebra" lemma and to use one of the lemmas from +the "Functions are group homomorphism" section of Lecture 4's exercises. +```agda +rev-equiv : is-equiv rev +rev-equiv = {!!} +``` + + +# Circles to torus (⋆⋆) + +In the Lecture 4 exercises, you defined a map from the Torus to S1 × S1. +In this problem, you will define a converse map. The goal is for these +two maps to be part of an equivalence, but we won't prove that in these +exercises. + +You will need to state and prove a lemma characterizing a path over a +path in a path fibration. Then, to define the map S1 × S1 → Torus, you +will want to curry it and use S1-rec and/or S1-elim on each circle. + +```agda +PathOver-path≡ : ∀ {A B : Type} {g : A → B} {f : A → B} + {a a' : A} {p : a ≡ a'} + {q : (f a) ≡ (g a)} + {r : (f a') ≡ (g a')} + → {!!} + → q ≡ r [ (\ x → (f x) ≡ (g x)) ↓ p ] +PathOver-path≡ {A}{B}{g}{f}{a}{a'}{p}{q}{r} h = {!!} + +circles-to-torus : S1 → (S1 → Torus) +circles-to-torus = {!!} + +circles-to-torus' : S1 × S1 → Torus +circles-to-torus' (x , y) = circles-to-torus x y +``` + +**Below are some "extra credit" exercise if you want more to do. These +are (even more) optional: nothing in the next lecture will depend on you +understanding them. The next section (H space) is harder code but uses +only the circle, whereas the following sections are a bit easier code +but require understanding the suspension type, which we haven't talked +about too much yet.** + +# H space + +The multiplication operation on the circle discussed in lecture is part +of what is called an "H space" structure on the circle. One part of +this structure is that the circle's basepoint is a unit element for +multiplication. + +(⋆) Show that base is a left unit. +```agda +mult-unit-l : (y : S1) → mult base y ≡ y +mult-unit-l y = {!!} +``` + +(⋆) Because we'll need it in a second, show that ap distributes over +function composition: +```agda +ap-∘ : ∀ {l1 l2 l3 : Level} {A : Type l1} {B : Type l2} {C : Type l3} + (f : A → B) (g : B → C) + {a a' : A} + (p : a ≡ a') + → ap (g ∘ f) p ≡ ap g (ap f p) +ap-∘ = {!!} +``` + +(⋆⋆) Suppose we have a curried function f : S1 → A → B. Under the +equivalence between paths in S1 × A and pairs of paths discussed in +Lecture 3, we can then "apply" (the uncurried version of) f to a pair of +paths (p : x ≡ y [ S1 ] , q : z ≡ w [ A ]) to get a path (f x z ≡ f y w +[ B ]). In the special case where q is reflexivity on a, this +application to p and q can be simplified to ap (\ x → f x a) p : f x a ≡ +f y a [ B ]. + +Now, suppose that f is defined by circle recursion. We would expect +some kind of reduction for applying f to the pair of paths (loop , q) --- +i.e. we should have reductions for *nested* pattern matching on HITs. +In the case where q is reflexivity, applying f to the pair (loop , refl) +can reduce like this: +```agda +S1-rec-loop-1 : ∀ {A B : Type} {f : A → B} {h : f ≡ f} {a : A} + → ap (\ x → S1-rec f h x a) loop ≡ app≡ h a +S1-rec-loop-1 {A}{B}{f}{h}{a} = {!!} +``` +Prove this reduction using ap-∘ and the reduction rule for S1-rec on the loop. + +(⋆⋆⋆) Show that base is a right unit for multiplication. You will need +a slightly different path-over lemma than before. + +```agda +PathOver-endo≡ : ∀ {A : Type} {f : A → A} + {a a' : A} {p : a ≡ a'} + {q : (f a) ≡ a} + {r : (f a') ≡ a'} + → the Type {! !} + → q ≡ r [ (\ x → f x ≡ x) ↓ p ] +PathOver-endo≡ {p = (refl _)} {q = q} {r} h = {!!} + +mult-unit-r : (x : S1) → mult x base ≡ x +mult-unit-r = {!!} +``` + +# Suspensions and the 2-point circle + +(⋆) Postulate the computation rules for the non-dependent susp-rec and +declare rewrites for the reduction rules on the point constructors. +```agda +postulate + Susp-rec-north : {l : Level} {A : Type} {X : Type l} + (n : X) (s : X) (m : A → n ≡ s) + → Susp-rec n s m northS ≡ {!!} + Susp-rec-south : {l : Level} {A : Type} {X : Type l} + (n : X) (s : X) (m : A → n ≡ s) + → Susp-rec n s m southS ≡ {!!} +-- {-# REWRITE Susp-rec-north #-} +-- {-# REWRITE Susp-rec-south #-} +postulate + Susp-rec-merid : {l : Level} {A : Type} {X : Type l} + (n : X) (s : X) (m : A → n ≡ s) + → (x : A) → ap (Susp-rec n s m) (merid x) ≡ {!!} +``` + +(⋆) Postulate the dependent elimination rule for suspensions: + +```agda +postulate + Susp-elim : {l : Level} {A : Type} (P : Susp A → Type l) + → (n : {!!}) + → (s : {!!}) + → (m : {!!}) + → (x : Susp A) → P x +``` + +(⋆⋆) Show that the maps s2c and c2s from the Lecture 4 exercises are mutually inverse: + +```agda +c2s2c : (x : Circle2) → s2c (c2s x) ≡ x +c2s2c = {!!} + +s2c2s : (x : Susp Bool) → c2s (s2c x) ≡ x +s2c2s = {!!} +``` + +(⋆) Conclude that Circle2 is equivalent to Susp Bool: + +```agda +Circle2-Susp-Bool : Circle2 ≃ Susp Bool +Circle2-Susp-Bool = {!!} +``` + +# Functoriality of suspension (⋆⋆) + +In the Lecture 4 exercises, we defined functoriality for the suspension +type, which given a function X → Y gives a function Σ X → Σ Y. Show +that this operation is functorial, meaning that it preserves identity +and composition of functions: +```agda +susp-func-id : ∀ {X : Type} → susp-func {X} id ∼ id +susp-func-id = {!!} + +susp-func-∘ : ∀ {X Y Z : Type} (f : X → Y) (g : Y → Z) + → susp-func {X} (g ∘ f) ∼ susp-func g ∘ susp-func f +susp-func-∘ f g = {!!} +``` + + + diff --git a/src/HoTTEST/Agda/HITs/Exercises6.lagda.md b/src/HoTTEST/Agda/HITs/Exercises6.lagda.md new file mode 100644 index 0000000..c4e590f --- /dev/null +++ b/src/HoTTEST/Agda/HITs/Exercises6.lagda.md @@ -0,0 +1,292 @@ + +```agda + +{-# OPTIONS --rewriting --without-K --allow-unsolved-metas #-} + +open import new-prelude + +open import Lecture6-notes +open import Lecture5-notes + +module Exercises6 where +``` + +In this problem set, you will look at a variation on the circle, a +higher inductive type for a "bowtie", i.e. two loops at a point. +(Unscaffolded harder exercise: do these problems for a "wedge of k +circles" for any natural number k.) + +# HIT recursion from induction + +In general, the dependent elimination rule for a higher inductive type +implies the simple/non-dependent elimination rule. In this problem, you +will show this for the bowtie. We could have done this for the circles +in the past lectures, but I wanted to introduce the non-dependent +elimination rule first, and then left both as postulates. + +Note that this problem has a bit of a "metamathematical" flavor (showing +that a set of axioms is implied by a shorter set). If you prefer to +jump right to the more "mathematical" problem of characterizing the loop +space of the bowtie below, I recommend turning Bowtie-rec and its +associated reductions into postulates like we have done for previous +higher inductive types, and adding a rewrite for the reduction on the +base point. This will make Agda display things in a more easy to read +way (otherwise, it will display Bowtie-rec as a meta-variable). + +Here is the definition of the bowtie and its dependent elimination rule: + +```agda +postulate + Bowtie : Set + baseB : Bowtie + loop1 : baseB ≡ baseB + loop2 : baseB ≡ baseB + Bowtie-elim : {l : Level} (X : Bowtie → Type l) + (x : X baseB) + (p : x ≡ x [ X ↓ loop1 ]) + (q : x ≡ x [ X ↓ loop2 ]) + → (x : Bowtie) → X x + Bowtie-elim-base : {l : Level} (X : Bowtie → Type l) + (x : X baseB) + (p : x ≡ x [ X ↓ loop1 ]) + (q : x ≡ x [ X ↓ loop2 ]) + → Bowtie-elim X x p q baseB ≡ x +{-# REWRITE Bowtie-elim-base #-} + +postulate + Bowtie-elim-loop1 : {l : Level} (X : Bowtie → Type l) + (x : X baseB) + (p : x ≡ x [ X ↓ loop1 ]) + (q : x ≡ x [ X ↓ loop2 ]) + → apd (Bowtie-elim X x p q) loop1 ≡ p + Bowtie-elim-loop2 : {l : Level} (X : Bowtie → Type l) + (x : X baseB) + (p : x ≡ x [ X ↓ loop1 ]) + (q : x ≡ x [ X ↓ loop2 ]) + → apd (Bowtie-elim X x p q) loop2 ≡ q +``` + +Next, we will prove the non-dependent elim/"recursion principle" from +these. First, we need some lemmas. + +(⋆) Paths over a path in a constant fibration are equivalent to paths. +It is simple to prove this by path induction. + +```agda +PathOver-constant : {l1 l2 : Level} {A : Type l1} {B : Type l2} + → {a1 a2 : A} + → (p : a1 ≡ a2) + → {b1 b2 : B} + → b1 ≡ b2 + → b1 ≡ b2 [ (\ _ → B) ↓ p ] +PathOver-constant = {!!} + +PathOver-constant-inverse : {l1 l2 : Level} {A : Type l1} {B : Type l2} + → {a1 a2 : A} + → (p : a1 ≡ a2) + → {b1 b2 : B} + → b1 ≡ b2 [ (\ _ → B) ↓ p ] + → b1 ≡ b2 +PathOver-constant-inverse = {!!} + +PathOver-constant-inverse-cancel1 : {l1 l2 : Level} {A : Type l1} {B : Type l2} + → {a1 a2 : A} + → (p : a1 ≡ a2) + → {b1 b2 : B} + → (q : b1 ≡ b2) + → PathOver-constant-inverse p (PathOver-constant p q) ≡ q +PathOver-constant-inverse-cancel1 = {!!} + +PathOver-constant-inverse-cancel2 : {l1 l2 : Level} {A : Type l1} {B : Type l2} + → {a1 a2 : A} + → (p : a1 ≡ a2) + → {b1 b2 : B} + → (q : b1 ≡ b2 [ _ ↓ p ]) + → PathOver-constant p (PathOver-constant-inverse p q) ≡ q +PathOver-constant-inverse-cancel2 = {!!} + +PathOver-constant-equiv : {l1 l2 : Level} {A : Type l1} {B : Type l2} + → {a1 a2 : A} + → (p : a1 ≡ a2) + → {b1 b2 : B} + → (b1 ≡ b2) ≃ (b1 ≡ b2 [ (\ _ → B) ↓ p ]) +PathOver-constant-equiv p = improve (Isomorphism (PathOver-constant p) + (Inverse (PathOver-constant-inverse p) + (PathOver-constant-inverse-cancel1 p) + (PathOver-constant-inverse-cancel2 p))) + +``` + +(⋆) Next, for a non-dependent function f, there is an annoying mismatch between +ap f and apd f, which we can reconcile as follows: + +```agda +ap-apd-constant : {l1 l2 : Level} {A : Type l1} {B : Type l2} + → {a1 a2 : A} + → (p : a1 ≡ a2) + → (f : A → B) + → ap f p ≡ PathOver-constant-inverse _ (apd f p) +ap-apd-constant = {!!} +``` + +(⋆) Define Bowtie-rec and prove the reduction for base: + +```agda +Bowtie-rec : {l : Level} {X : Type l} + (x : X) + (p : x ≡ x [ X ]) + (q : x ≡ x [ X ]) + → (Bowtie) → X +Bowtie-rec {_} {X} x p q = {!!} + +Bowtie-rec-base : {l : Level} {X : Type l} + (x : X) + (p : x ≡ x [ X ]) + (q : x ≡ x [ X ]) + → Bowtie-rec x p q baseB ≡ x +Bowtie-rec-base _ _ _ = {!!} +``` + +(⋆⋆) Prove the reductions for loop: + +```agda +Bowtie-rec-loop1 : {l : Level} {X : Type l} + (x : X) + (p : x ≡ x [ X ]) + (q : x ≡ x [ X ]) + → ap (Bowtie-rec x p q) loop1 ≡ p [ x ≡ x ] +Bowtie-rec-loop1 x p q = {!!} + +Bowtie-rec-loop2 : {l : Level} {X : Type l} + (x : X) + (p : x ≡ x [ X ]) + (q : x ≡ x [ X ]) + → ap (Bowtie-rec x p q) loop2 ≡ q [ x ≡ x ] +Bowtie-rec-loop2 x p q = {!!} +``` + +# Loop space of the bowtie + +In this problem, you will show that the loop space of the bowtie is the +"free group on two generators", which we will write in Agda as F2. The +point of this problem is mainly for you to read and really understand +the proof that the loop space of the circle is ℤ. All of the code is +essentially a rearrangement of code from that proof. I'd suggest +trying the proof yourself, and looking at the analogous bits of the +Circle proof if you get stuck. + +## Some lemmas (⋆⋆) + +In the Circle proof in lecture, I inlined a couple of things that +can be proved more generally. You might want to prove these general +versions in advance and use them in your proof, or, if that seems +confusing, you might first do the proof without these lemmas +to motivate them. + +```agda +concat-equiv : ∀ {A : Type} (a : A) {a' a'' : A} + → (p : a' ≡ a'') + → (a ≡ a') ≃ (a ≡ a'') +concat-equiv = {!!} + +concat-equiv-map : ∀ {A : Type} {a a' a'' : A} + → (p : a' ≡ a'') + → fwd (concat-equiv a p) ≡ \ q → q ∙ p +concat-equiv-map = {!!} +``` + +(Note: you could also write out all of the components, but this was easier.) + +```agda +transport-∙ : {l1 l2 : Level} {A : Type l1} {B : A → Type l2} + {a1 a2 a3 : A} (p : a1 ≡ a2) (q : a2 ≡ a3) + → transport B (p ∙ q) ∼ transport B q ∘ transport B p +transport-∙ = {!!} +``` +## Calculating the loop space + +First, we will assume a type F2 representing the free group on 2 +generators. + +ℤ is the free group on one generator, with 0 as the neutral element and +succℤ corresponding to "addition" with the one generator. succℤ is an +equivalence, with the inverse representing "addition" with -1. + +For other groups, it is somewhat more common to think of the group +operation as "multiplication" rather than "addition", so we will name +the neutral element as "1" and the action of the elements as +"multiplication". Thus, we assume a type with an element 1F, and two +equivalences, which we think of as "multiplication with generator 1" and +"multiplication with generator 2". + +Unscaffolded hard exercise: You can implement F2 as lists whose +elements are a four-element type g1, g2, g1⁻¹, g2⁻¹ with no adjacent +pairs of inverse elements. Then the forward directions of mult1/mult2 +will be implement by cons'ing g1/g2 on and "reducing" if that creates +two adjacent inverses, the backwards directions by consing g1⁻¹ and g2⁻¹ +on and reducing, and the inverse laws will hold because the reduction +cancels the inverses. + +For this problem, we will simply assume the nice universal property for +this type: that it maps uniquely into any other type with a point and +two equivalences, and that it is a set. + +```agda +module AssumeF2 + (F2 : Type) + (1F : F2) + (mult1 : F2 ≃ F2) + (mult2 : F2 ≃ F2) + (F2-rec : {X : Type} + (o : X) + (m1 : X ≃ X) + (m2 : X ≃ X) + → F2 → X) + (F2-rec-1 : {X : Type} + (z : X) + (m1 : X ≃ X) + (m2 : X ≃ X) + → F2-rec z m1 m2 1F ≡ z) + (F2-rec-mult1 : {X : Type} + (z : X) + (m1 : X ≃ X) + (m2 : X ≃ X) + (a : F2) → F2-rec z m1 m2 (fwd mult1 a) ≡ fwd m1 (F2-rec z m1 m2 a)) + (F2-rec-mult2 : {X : Type} + (z : X) + (m1 : X ≃ X) + (m2 : X ≃ X) + (a : F2) → F2-rec z m1 m2 (fwd mult2 a) ≡ fwd m2 (F2-rec z m1 m2 a)) + (F2-rec-unique : {X : Type} + (f : F2 → X) + (z : X) + (m1 : X ≃ X) + (m2 : X ≃ X) + → f 1F ≡ z + → ((f ∘ fwd mult1) ∼ (fwd m1 ∘ f)) + → ((f ∘ fwd mult2) ∼ (fwd m2 ∘ f)) + → (x : F2) → f x ≡ F2-rec z m1 m2 x) + (hSetF : is-set F2) where +``` + +(⋆⋆⋆) Prove that the loop space of the Bowtie is F2. Each bit of the +proof will be analogous to the corresponding part of the Circle proof. + +```agda + Cover : Bowtie → Type + Cover = {!!} + + encode : (x : Bowtie) → baseB ≡ x → Cover x + encode = {!!} + + decode : (x : Bowtie) → Cover x → baseB ≡ x + decode = {!!} + + encode-decode : (x : Bowtie) (p : baseB ≡ x) → decode x (encode x p) ≡ p + encode-decode = {!!} + + decode-encode : (x : Bowtie) (p : Cover x) → encode x (decode x p) ≡ p + decode-encode = {!!} + +``` + diff --git a/src/HoTTEST/Agda/HITs/Lecture4-live.lagda.md b/src/HoTTEST/Agda/HITs/Lecture4-live.lagda.md new file mode 100644 index 0000000..84054b7 --- /dev/null +++ b/src/HoTTEST/Agda/HITs/Lecture4-live.lagda.md @@ -0,0 +1,137 @@ +```agda + +{-# OPTIONS --without-K --rewriting #-} + +open import new-prelude + +module Lecture4-live where + + {- + postulate + Bool : Type + true : Bool + false : Bool + -} + + postulate + S1 : Type + base : S1 + loop : base ≡ base [ S1 ] + + example-path : base ≡ base + example-path = loop ∙ ! loop ∙ loop ∙ ! loop ∙ loop + + -- groupoid laws + + ∙assoc : {A : Type} {x y z w : A} + (p : x ≡ y) + (q : y ≡ z) + (r : z ≡ w) + → (p ∙ (q ∙ r)) ≡ ((p ∙ q) ∙ r) [ x ≡ w [ A ] ] + ∙assoc p q (refl _) = refl _ + + ∙unit-l : {A : Type} {x y : A} + → (p : x ≡ y) + → (refl _ ∙ p) ≡ p + ∙unit-l (refl _) = refl _ + + ∙unit-r : {A : Type} {x y : A} + → (p : x ≡ y) → (p ∙ refl _) ≡ p + ∙unit-r p = refl p + + !-inv-l : {A : Type} {x y : A} + → (p : x ≡ y) + → (! p ∙ p) ≡ refl _ + !-inv-l (refl _) = refl _ + + !-inv-r : {A : Type} {x y : A} → (p : x ≡ y) + → (p ∙ ! p) ≡ refl _ + !-inv-r (refl _) = refl _ + + example : loop ∙ ! loop ∙ loop ∙ ! loop ∙ loop + ≡ loop + example = loop ∙ ! loop ∙ loop ∙ ! loop ∙ loop ≡⟨ refl _ ⟩ + (((loop ∙ ! loop) ∙ loop) ∙ ! loop) ∙ loop ≡⟨ ap (\ H → H ∙ loop ∙ ! loop ∙ loop) (!-inv-r loop) ⟩ + ((refl _ ∙ loop) ∙ ! loop) ∙ loop ≡⟨ ap (λ H → H ∙ ! loop ∙ loop) (∙unit-l loop) ⟩ + ((loop ∙ ! loop) ∙ loop) ≡⟨ ! (∙assoc _ _ loop) ⟩ + (loop ∙ (! loop ∙ loop)) ≡⟨ ap (\ H → loop ∙ H) (!-inv-l loop) ⟩ + (loop ∙ refl _) ≡⟨ ∙unit-r loop ⟩ + loop ∎ + + -- Circle recursion + postulate + S1-rec : {X : Type} (x : X) (l : x ≡ x [ X ]) → (S1 → X) + S1-rec-base : {X : Type} (x : X) (l : x ≡ x [ X ]) + → (S1-rec x l) base ≡ x + + {-# BUILTIN REWRITE _≡_ #-} + {-# REWRITE S1-rec-base #-} + + postulate + S1-rec-loop : {X : Type} (x : X) (l : x ≡ x [ X ]) + → ap (S1-rec x l) loop ≡ l + + double : S1 → S1 + double = S1-rec base (loop ∙ loop) + + double-loop : ap double loop ≡ loop ∙ loop + double-loop = S1-rec-loop _ _ + + ap-∙ : {A B : Type} {f : A → B} {x y z : A} + (p : x ≡ y) + (q : y ≡ z) + → ap f (p ∙ q) ≡ ap f p ∙ ap f q + ap-∙ (refl _) (refl _) = refl _ + + double-2-loops : ap double (loop ∙ loop) ≡ (loop ∙ loop) ∙ (loop ∙ loop) + double-2-loops = + ap double (loop ∙ loop) ≡⟨ ap-∙ loop loop ⟩ + ap double loop ∙ ap double loop ≡⟨ ap₂ (\ p q → p ∙ q) + (S1-rec-loop _ _) + (S1-rec-loop _ _) ⟩ + (loop ∙ loop) ∙ (loop ∙ loop) ∎ + + postulate + Circle2 : Type + north : Circle2 + south : Circle2 + west : north ≡ south + east : north ≡ south + Circle2-rec : {X : Type} + (n : X) (s : X) (w : n ≡ s) (e : n ≡ s) + → Circle2 → X + Circle2-rec-north : {X : Type} (n : X) (s : X) (w : n ≡ s) (e : n ≡ s) + → Circle2-rec n s w e north ≡ n + Circle2-rec-south : {X : Type} (n : X) (s : X) (w : n ≡ s) (e : n ≡ s) + → Circle2-rec n s w e south ≡ s + {-# REWRITE Circle2-rec-north #-} + {-# REWRITE Circle2-rec-south #-} + + postulate + Circle2-rec-west : {X : Type} (n : X) (s : X) (w : n ≡ s) (e : n ≡ s) + → ap (Circle2-rec n s w e) west ≡ w + Circle2-rec-east : {X : Type} (n : X) (s : X) (w : n ≡ s) (e : n ≡ s) + → ap (Circle2-rec n s w e) east ≡ e + + to : S1 → Circle2 + to = S1-rec north (east ∙ ! west) + + from : Circle2 → S1 + from = Circle2-rec base base (refl _) loop + + from-to-north : to (from north) ≡ north + from-to-north = refl _ + + from-to-south : to (from south) ≡ south + from-to-south = west + + from-to-west : ap to (ap from west) ∙ from-to-south ≡ west + from-to-west = ap to (ap from west) ∙ west ≡⟨ ap (\ H → ap to H ∙ west) (Circle2-rec-west _ _ _ _) ⟩ + ap to (refl base) ∙ west ≡⟨ ∙unit-l west ⟩ + west ∎ + + + -- Q: is loop distinct from refl? + -- A: so far, maybe + +``` diff --git a/src/HoTTEST/Agda/HITs/Lecture4-notes.lagda.md b/src/HoTTEST/Agda/HITs/Lecture4-notes.lagda.md new file mode 100644 index 0000000..706b365 --- /dev/null +++ b/src/HoTTEST/Agda/HITs/Lecture4-notes.lagda.md @@ -0,0 +1,349 @@ +```agda + +{-# OPTIONS --without-K --rewriting #-} + +open import new-prelude + +module Lecture4-notes where +``` + +# The circle + +In general, a type should be thought of as an infinity-groupoid: it has +points, paths (or identifications, equalities, depending on what you +want to call them), and then paths between paths, and so on. The +*regular* inductive types that we've, like booleans or natural numbers, +have constructors only for points. *Higher* inductive types (HITs) also have +constructors for paths, paths between paths, etc. + +A good first example is the circle, which we will write as S1. As as a +higher inductive type, it is presented as "the free infinity groupoid +generated by a point and a loop at that point". + +```agda +postulate + S1 : Type + base : S1 + loop : base ≡ base +``` + +base is a point constructor, like zero for the natural numbers or true +and false for the booleans. loop is a *path* constructor, which means +that instead of constructing an element of S1, it constructs a path in S1 +from base to base. We represent this using the identity type ≡. Since +Agda doesn't have HITs built-in (until we get to Cubical Agda in Lecture +7), we will "postulate" these, which means to assume constants of these +types for the rest of the development. A postulate is roughly the same +as making the whole rest of the codebase depend on a variable of that +type. + +Recall from previous lectures that for any type we can define operations +of path concatenation and path inverses. We will write concatenation as +p ∙ q and inverse as ! p. (This was written (-)⁻¹ in previous lectures, +but I find a prefix notation easier to read when Agda displays terms.) +You can review the definitions in the [prelude](new-prelude.lagda.md). + +Using these we can construct other paths on the circle from loop: + +```agda +example-path : base ≡ base +example-path = loop ∙ ! loop ∙ loop ∙ ! loop ∙ loop +``` + +There should be a homotopy between example (forwards, backwards, +forwards, backwards, forwards) and loop (forwards). To define it, +we can generalize and prove the groupoid laws for paths. + +# Groupoid laws for paths + +If you've seen some abstract algebra, you know that the *group laws* say +that the "multiplication" of the group is associative and has the +neutral element as a unit on both sides, and that the inverse of an +element multiplied with that element is the neutral element. The +*groupoid laws* for paths are a "typed" version of these. They say that +the identity path (refl) is a unit for path concatenation, path +concatenation is associative, and the inverse of a path composed with +that path is the identity. Let's state and prove them. + +```agda +∙unit-r : {A : Type} {x y : A} → (p : x ≡ y) → (p ∙ refl _) ≡ p +∙unit-r p = refl _ + +∙unit-l : {A : Type} {x y : A} → (p : x ≡ y) → (refl _ ∙ p) ≡ p +∙unit-l (refl _) = refl _ + +∙assoc : {A : Type} {x y z w : A} (p : x ≡ y) (q : y ≡ z) (r : z ≡ w) + → p ∙ (q ∙ r) ≡ (p ∙ q) ∙ r +∙assoc p (refl _) (refl _) = (refl _) + +!-inv-l : {A : Type} {x y : A} → (p : x ≡ y) → (! p ∙ p) ≡ refl _ +!-inv-l (refl _) = refl _ + +!-inv-r : {A : Type} {x y : A} → (p : x ≡ y) → (p ∙ ! p) ≡ refl _ +!-inv-r (refl _) = refl _ +``` + +It's kind of amazing that all of this structure emerges from a very +simple axiom (path induction!). + +OK, now we can use these to construct a path/homotopy between the path +above and loop: + +```agda +example : (loop ∙ ! loop ∙ loop ∙ ! loop ∙ loop) ≡ loop [ base ≡ base ] +example = loop ∙ ! loop ∙ loop ∙ ! loop ∙ loop ≡⟨ refl _ ⟩ + (((loop ∙ ! loop) ∙ loop) ∙ ! loop) ∙ loop ≡⟨ ap (\ H → H ∙ loop ∙ ! loop ∙ loop) (!-inv-r loop) ⟩ + refl _ ∙ loop ∙ ! loop ∙ loop ≡⟨ ap (\ H → H ∙ ! loop ∙ loop) (∙unit-l (loop)) ⟩ + loop ∙ ! loop ∙ loop ≡⟨ ! (∙assoc _ (! loop) loop) ⟩ + loop ∙ (! loop ∙ loop) ≡⟨ ap (\ H → loop ∙ H) (!-inv-l loop) ⟩ + loop ∙ (refl _) ≡⟨ refl _ ⟩ + loop ∎ +``` + +We'll use the notation x ≡ y [ A ] if we want to make the type a path is +in explicit. We'll sometimes write (\ x → e) for (λ x → e) because it +easier to type and de-emphasizes the function a little (since ap almost +always needs a lambda abstraction as input, it would IMO be nicer to have +a syntax with a bound variable like ap (H → ...) p). + +Up to homotopy, there should be ℤ many paths on the circle: +... , ! loop ∙ ! loop, ! loop, refl, loop, loop ∙ loop, ... + +# Circle "recursion" + +Terminology note: for an inductive type like booleans, natural numbers, +lists, etc., it is common to call the dependent elimination principle +the "elimination" or "induction" principle, and the non-dependent +elimination principle the "recursion" principle. This makes perfect +sense for natural numbers, but the analogy gets a little strained for +types like booleans. Since bool isn't properly inductive (there are no +constructors for bool that take a boolean as input, like successor for +natural numbers), the "recursion" principle is just if-then-else, and +the "induction" principle is just proof by case analysis. Nonetheless, +for uniformity, we will refer to non-dependent elimination rules as +"recursion principles" even when the type is not properly inductive. + +With this in mind, the "recursion" principle for the circle expresses +that it is the *initial* type with a point and a loop at that point: +given any other type with a point and a loop, we can map the circle into +it (in a unique way, but we'll get to that later). + +```agda +postulate + S1-rec : {l : Level} {X : Type l} (x : X) (p : x ≡ x) → S1 → X + S1-rec-base : {l : Level} {X : Type l} (x : X) (p : x ≡ x) + → S1-rec {l} {X} x p base ≡ x + S1-rec-loop' : {l : Level} {X : Type l} (x : X) (p : x ≡ x) + → ap (S1-rec x p) loop ≡ (S1-rec-base x p) ∙ p ∙ ! (S1-rec-base x p) +``` +The base and loop postulates are the reduction rules for S1-rec. +Note that we use ap to apply S1-rec to the loop. + +It's common to assume that at least the point reductions like +S1-rec-base are definitional equalities. We can do this in Agda like +this: +```agda +{-# BUILTIN REWRITE _≡_ #-} +{-# REWRITE S1-rec-base #-} +``` +Then, for example, we don't need to explicitly mention S1-rec-base to state the loop reduction. +```agda +postulate + S1-rec-loop : {l : Level} {X : Type l} (x : X) (p : x ≡ x) + → ap (S1-rec x p) loop ≡ p +``` + +This is mainly for convenience, but it's a big convenience! When we get +to Cubical Agda, both of these reductions will be definitional, but for +axiomatic HoTT we only put in the point one. + +## Doubling function + +As a first example of circle recursion, we can define the function from +the circle to the circle that sends the loop to the concatenation of two +loops. Thinking of the paths as integers, this sends x to 2*x. + +```agda +double : S1 → S1 +double = S1-rec base (loop ∙ loop) + +calculate-double-loop : ap double loop ≡ (loop ∙ loop) +calculate-double-loop = S1-rec-loop _ _ +``` + +How do we "reduce" this function on two loops? We need a general fact +that functions are functors, which in particular means that they are +group homomorphisms on paths, seen as groups with composition as the +group operation. + +```agda +ap-∙ : {A B : Type} {f : A → B} {x y z : A} (p : x ≡ y) (q : y ≡ z) + → ap f (p ∙ q) ≡ ap f p ∙ ap f q +ap-∙ (refl _) (refl _) = refl _ + +calculate-double-2-loops : ap double (loop ∙ loop) ≡ loop ∙ loop ∙ loop ∙ loop +calculate-double-2-loops = + ap double (loop ∙ loop) ≡⟨ ap-∙ loop loop ⟩ + ap double loop ∙ ap double loop ≡⟨ ap₂ (\ p q → p ∙ q) (S1-rec-loop _ _) (S1-rec-loop _ _) ⟩ + (loop ∙ loop) ∙ (loop ∙ loop) ≡⟨ (∙assoc (loop ∙ loop) loop loop ) ⟩ + ((loop ∙ loop) ∙ loop) ∙ loop ∎ +``` + +# 2 point version of the circle + +We can also "subdivide" the circle into two points with two paths +between them: + +```agda +postulate + Circle2 : Type + north : Circle2 + south : Circle2 + west : north ≡ south + east : north ≡ south + Circle2-rec : {X : Type} (n : X) (s : X) (w : n ≡ s) (e : n ≡ s) + → Circle2 → X + Circle2-rec-north : {X : Type} (n : X) (s : X) (w : n ≡ s) (e : n ≡ s) + → Circle2-rec n s w e north ≡ n + Circle2-rec-south : {X : Type} (n : X) (s : X) (w : n ≡ s) (e : n ≡ s) + → Circle2-rec n s w e south ≡ s +{-# REWRITE Circle2-rec-north #-} +{-# REWRITE Circle2-rec-south #-} + +postulate + Circle2-rec-west : {X : Type} (n : X) (s : X) (w : n ≡ s) (e : n ≡ s) + → ap (Circle2-rec n s w e) west ≡ w + Circle2-rec-east : {X : Type} (n : X) (s : X) (w : n ≡ s) (e : n ≡ s) + → ap (Circle2-rec n s w e) east ≡ e +``` + +Let's work towards showing that these two definitions of the circle give +equivalent types. This means defining functions back and forth whose +composites are *homotopic* to the identity function. But inside type +theory, this looks just like giving a "isomorphism" in the sense defined +in the Lecture 3 exercises. + +First, to map the 1-point circle to the 2-point circle, we need a point +and a loop. Of course, there are lots of choices, but let's pick +```agda +to : S1 → Circle2 +to = S1-rec north (east ∙ ! west) +``` + +In the other direction, we need to map north and south each to some +point of the circle, but we've only got one point, base. For the paths, +we can go "twice as fast" on one and "stand still" on the other. +```agda +from : Circle2 → S1 +from = Circle2-rec base base (refl base) loop +``` + +We can check that on the constructors, the composite from-then-to is the identity: + +```agda +from-to-north : to (from north) ≡ north +from-to-north = refl _ + +from-to-south : to (from south) ≡ south +from-to-south = west + +from-to-west : ap to (ap from west) ∙ from-to-south ≡ west +from-to-west = ap to (ap from west) ∙ west ≡⟨ ap (\ H → ap to H ∙ west) (Circle2-rec-west _ _ _ _) ⟩ + ap to (refl base) ∙ west ≡⟨ ∙unit-l west ⟩ + west ∎ + +from-to-east : ap to (ap from east) ∙ from-to-south ≡ east +from-to-east = ap to (ap from east) ∙ west ≡⟨ ap (\ H → ap to H ∙ west) (Circle2-rec-east _ _ _ _) ⟩ + ap to loop ∙ west ≡⟨ ap (\ H → H ∙ west) (S1-rec-loop _ _) ⟩ + east ∙ ! west ∙ west ≡⟨ ! (∙assoc _ (! west) west) ⟩ + east ∙ (! west ∙ west) ≡⟨ ap (\ H → east ∙ H) (!-inv-l west) ⟩ + east ∎ +``` + +Note that it doesn't even type check to ask that ap to (ap from west) ≡ +west, because to (from south) ≡ north. So we need to compare them "up +to" a path. We'll talk about why (- ∙ from-to-south) is the right path +to pick next time. + +# The torus + +The torus is a nice example of a type with a path-between-paths constructor: + +```agda +postulate + Torus : Type + baseT : Torus + pT : baseT ≡ baseT + qT : baseT ≡ baseT + sT : pT ∙ qT ≡ qT ∙ pT + T-rec : {l : Level} {X : Type l} (x : X) (p : x ≡ x) (q : x ≡ x) (s : p ∙ q ≡ q ∙ p) → Torus → X + T-rec-base : {l : Level} {X : Type l} (x : X) (p : x ≡ x) (q : x ≡ x) (s : p ∙ q ≡ q ∙ p) + → T-rec {l} {X} x p q s baseT ≡ x +{-# REWRITE T-rec-base #-} +postulate + T-rec-pT : {l : Level} {X : Type l} (x : X) (p : x ≡ x) (q : x ≡ x) (s : p ∙ q ≡ q ∙ p) + → ap (T-rec x p q s) pT ≡ p + T-rec-qT : {l : Level} {X : Type l} (x : X) (p : x ≡ x) (q : x ≡ x) (s : p ∙ q ≡ q ∙ p) + → ap (T-rec x p q s) qT ≡ q + -- there needs to be an analogous reduction for sT, but it's a little tricky to state it, + -- so we will defer it for now +``` + +Lots of other important concepts can be defined with HITs, including +quotients, suspensions, pushouts, truncations. + +# Suspensions + +```agda +postulate + Susp : Type → Type + northS : {A : Type} → Susp A + southS : {A : Type} → Susp A + merid : {A : Type} → A → northS ≡ southS [ Susp A ] + Susp-rec : {l : Level} {A : Type} {X : Type l} + (n : X) (s : X) (m : A → n ≡ s) + → Susp A → X +``` + +# Pushouts + +```agda +postulate + Pushout : (C : Type) (A : Type) (B : Type) (f : C → A) (g : C → B) → Type + +module _ {C : Type} {A : Type} {B : Type} {f : C → A} {g : C → B} where + + postulate + inl : A → Pushout C A B f g + inr : B → Pushout C A B f g + glue : (c : C) → inl (f c) ≡ inr (g c) + Push-rec : {X : Type} (l : A → X) (r : B → X) (gl : (c : C) → l (f c) ≡ r (g c)) + → Pushout C A B f g → X +``` + +# Relation quotient + +Suppressing some details, think of this as the quotient of A by the +equivalence relation closure of R (technically, this description is +precisely right when R(x,y) is a proposition and A is a set; otherwise, +you need to think about the higher homotopies induced by the +constructors). + +```agda +postulate + _/_ : (A : Type) → (A → A → Type) → Type + i : {A : Type} {R : A → A → Type} → A → A / R + q : {A : Type} {R : A → A → Type} (x y : A) → R x y → i x ≡ i y [ A / R ] + +``` + +# Propositional truncation + +The universal way of making a type into a -1-type. This is a nice +example of a (properly) inductive HIT. +```agda +postulate + ∥_∥₋₁ : Type → Type + ∣_∣ : {A : Type} → A → ∥ A ∥₋₁ + trunc : {A : Type} → is-prop ∥ A ∥₋₁ +``` diff --git a/src/HoTTEST/Agda/HITs/Lecture4-pictures.pdf b/src/HoTTEST/Agda/HITs/Lecture4-pictures.pdf new file mode 100644 index 0000000000000000000000000000000000000000..6fc738df1b11fdbd766cdcd203f8af00adabc450 GIT binary patch literal 3697958 zcmV(+K;6G3P((&8F)lL-CB)_O4?5av(28Y+-a|L}g=dWMv>eJ_>Vma%Ev{3U~pG{Yj2xyOynK*0mMdfl!w` zDSIKxQSt^{xcMtw3&6wr55)HmF43d1w;J5+;o%;cN^+4i&&d6M|NQ^_{J(!b{QQ4C z{CeaajUCwnwFRzcEL7rfuzaQJGzaL&7u0`J- zo?e^v?4Zw&A6Bxag#x_2eYA`(KfFE<^8WOGAdj!mcxJsm8T0b%_3ioTjCtO<%zu3U z_+g#*r^l@W8}BdIv>TX{KL2`setm59r(d7%5063KA0MAP`%}k!`eEa+_WU`>^Mf<) zxUUbdPw#_0KEY%NczYj)o_;-jJPu9YiS(}@|7%-)_}F4^zdl|KLpqy&`h4gE@cHrb zKFI6iAg{Lbeh~8~8h#u!{0aKG<9vR;S#+p*Mtj?OeSCU2$a;YG@y^rB=NV%wZ!Lv+ z-d^4g>@gyNe7@qIt@Qj3+xLuOp8Y<5JjY_MzaBnc$70WK&!5kOyga-@!`rXdhsTFd 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