From 27d2849922bf7d6f3a1b2b5e2ddbcb3a059fdc4c Mon Sep 17 00:00:00 2001 From: Michael Zhang Date: Fri, 18 Oct 2024 08:21:49 -0500 Subject: [PATCH] changes --- src/CubicalHott/Theorem7-1-11.agda | 16 +- talks/2024-grads/.gitignore | 8 + talks/2024-grads/Demo.agda | 20 ++ talks/2024-grads/Makefile | 10 +- talks/2024-grads/images/box.png | Bin 0 -> 87826 bytes talks/2024-grads/images/homotopy.png | Bin 0 -> 132609 bytes talks/2024-grads/images/id.png | Bin 0 -> 79889 bytes talks/2024-grads/images/id2.png | Bin 0 -> 92051 bytes talks/2024-grads/images/interval.png | Bin 0 -> 72923 bytes talks/2024-grads/images/maps.png | Bin 0 -> 127164 bytes talks/2024-grads/images/mapsid.png | Bin 0 -> 87477 bytes talks/2024-grads/images/nat.png | Bin 0 -> 130273 bytes talks/2024-grads/images/paths.png | Bin 0 -> 143665 bytes talks/2024-grads/images/s1.png | Bin 0 -> 66809 bytes talks/2024-grads/images/sigma.png | Bin 0 -> 155955 bytes talks/2024-grads/images/square.png | Bin 0 -> 87826 bytes talks/2024-grads/images/type.png | Bin 0 -> 69545 bytes talks/2024-grads/images/ua.png | Bin 0 -> 91042 bytes talks/2024-grads/images/zquot.png | Bin 0 -> 133035 bytes talks/2024-grads/main.tex | 374 +++++++++++++++++++++------ 20 files changed, 346 insertions(+), 82 deletions(-) create mode 100644 talks/2024-grads/images/box.png create mode 100644 talks/2024-grads/images/homotopy.png create mode 100644 talks/2024-grads/images/id.png create mode 100644 talks/2024-grads/images/id2.png create mode 100644 talks/2024-grads/images/interval.png create mode 100644 talks/2024-grads/images/maps.png create mode 100644 talks/2024-grads/images/mapsid.png create mode 100644 talks/2024-grads/images/nat.png create mode 100644 talks/2024-grads/images/paths.png create mode 100644 talks/2024-grads/images/s1.png create mode 100644 talks/2024-grads/images/sigma.png create mode 100644 talks/2024-grads/images/square.png create mode 100644 talks/2024-grads/images/type.png create mode 100644 talks/2024-grads/images/ua.png create mode 100644 talks/2024-grads/images/zquot.png diff --git a/src/CubicalHott/Theorem7-1-11.agda b/src/CubicalHott/Theorem7-1-11.agda index e97f8bb..ea14b3b 100644 --- a/src/CubicalHott/Theorem7-1-11.agda +++ b/src/CubicalHott/Theorem7-1-11.agda @@ -22,15 +22,17 @@ lemma zero (A , a , Acontr) (B , b , Bcontr) i = A≡B i , a≡b i , λ y j → a≡b : PathP (λ i → A≡B i) a b a≡b j = glue (λ { (j = i0) → a ; (j = i1) → b }) b - eqv2 : ((y : A) → a ≡ y) ≡ ((y : B) → b ≡ y) - eqv2 = λ i → (y : A≡B i) → a≡b i ≡ y + -- eqv2 : ((y : A) → a ≡ y) ≡ ((y : B) → b ≡ y) + -- eqv2 = λ i → (y : A≡B i) → a≡b i ≡ y - T1 = λ { (i = i0) → ((y : A) → a ≡ y) ; (i = i1) → ((y : B) → b ≡ y) } - e1 = λ { (i = i0) → pathToEquiv eqv2 ; (i = i1) → idEquiv ((y : B) → b ≡ y) } - Uhh = primGlue ((y : B) → b ≡ y) T1 e1 + -- T1 = λ { (i = i0) → ((y : A) → a ≡ y) ; (i = i1) → ((y : B) → b ≡ y) } + -- e1 = λ { (i = i0) → pathToEquiv eqv2 ; (i = i1) → idEquiv ((y : B) → b ≡ y) } + -- Uhh = primGlue ((y : B) → b ≡ y) T1 e1 - uhh : Uhh - uhh = glue {T = T1} {e = e1} (λ { (i = i0) → Acontr ; (i = i1) → Bcontr }) λ y j → {! Bcontr y j !} + -- uhh : Uhh + -- uhh = glue {T = T1} {e = e1} (λ { (i = i0) → Acontr ; (i = i1) → Bcontr }) λ y j → {! Bcontr y j !} + + Acontr-is-Prop : isProp ((y : A) → a ≡ y) wtf : PathP (λ i → (y : A≡B i) → a≡b i ≡ y) Acontr Bcontr wtf = λ i y j → {! !} diff --git a/talks/2024-grads/.gitignore b/talks/2024-grads/.gitignore index f0de8a3..21bd4dd 100644 --- a/talks/2024-grads/.gitignore +++ b/talks/2024-grads/.gitignore @@ -1 +1,9 @@ main.pdf +*-blx.bib +*.aux +*.log +*.nav +*.out +*.run.xml +*.snm +*.toc \ No newline at end of file diff --git a/talks/2024-grads/Demo.agda b/talks/2024-grads/Demo.agda index b344d9e..c3c6ab0 100644 --- a/talks/2024-grads/Demo.agda +++ b/talks/2024-grads/Demo.agda @@ -6,12 +6,28 @@ open import Relation.Binary.PropositionalEquality open import Relation.Binary.PropositionalEquality using (module ≡-Reasoning) open import Tactic.RingSolver.Core.AlmostCommutativeRing using (AlmostCommutativeRing) open import Data.Nat.Tactic.RingSolver +open import Data.Bool +open import Data.Bool.Properties open ≡-Reasoning sumTo : ℕ → ℕ sumTo zero = zero sumTo (suc n) = (suc n) + (sumTo n) +isEven : ℕ → Bool +isEven zero = true +isEven (suc x) = not (isEven x) + +example1 : isEven 5 ≡ false +example1 = refl + +eee : ∀ (m n : ℕ) → isEven m ≡ true → isEven n ≡ true → isEven (m + n) ≡ true +eee zero n m-even n-even = n-even +eee (2+ m) n m-even n-even = + let IH = eee m n (trans (sym (not-involutive (isEven m))) m-even) n-even in + trans (not-involutive (isEven (m + n))) IH + + lemma : ∀ (n : ℕ) → 2 * suc n + n * (n + 1) ≡ suc n * (suc n + 1) lemma = solve-∀ @@ -30,3 +46,7 @@ theorem (suc n) = ≡⟨ lemma n ⟩ (suc n) * ((suc n) + 1) ∎ + +data Vec (A : Set) (n : ℕ) : Set where + +z = let z1 = Vec in {! !} \ No newline at end of file diff --git a/talks/2024-grads/Makefile b/talks/2024-grads/Makefile index ba84f79..e84ef0b 100644 --- a/talks/2024-grads/Makefile +++ b/talks/2024-grads/Makefile @@ -1,7 +1,11 @@ -.PHONY: watch +.PHONY: watch clean main.pdf: main.tex - tectonic $< + make clean + pdflatex main.tex watch: - watchexec -e tex,bib make main.pdf \ No newline at end of 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-\usepackage[backend=bibtex,style=numeric-comp,sorting=none]{biblatex} +% \usepackage{hyperref} \usepackage{bussproofs} +\usepackage{colortbl} \usepackage{geometry} -\addbibresource{references.bib} -\nocite{*} +% \usepackage[backend=bibtex,style=numeric-comp,sorting=none]{biblatex} +% \addbibresource{references.bib} +% \nocite{*} + +\setbeamercovered{transparent} % \setbeameroption{show notes on second screen=right} \title{Formalizing mathematics with cubical type theory} @@ -42,19 +45,20 @@ \begin{itemize} \item Why formalize proofs? \note[item]{Why: Would be nice to get computers to check our work for us} + \pause \item What can we formalize? \note[item]{What: Program properties, compiler optimizations, correctness} - \note[item]{Notably, can NOT prove some things, Rice's theorem} + \pause - \note[item]{Talk about proofs vs. testing} - \item How to formalize proofs? - \note[item]{How: lot of different techniques involving model checking / SMT approaches, etc} - \note[item]{I'm going to focus here on proofs of mathematical statements, like $$ x + y = y + x $$} + % \item How to formalize proofs? + % \note[item]{How: lot of different techniques involving model checking / SMT approaches, etc} + % \note[item]{I'm going to focus here on proofs of mathematical statements, like $$ x + y = y + x $$} + % \pause \item What are some existing theorem provers? \begin{itemize} - \item Rocq (Coq), Lean, Agda, F*, Idris, Metamath, HOL, ... + \item Rocq (Coq), Lean, Agda \end{itemize} \end{itemize} \end{frame} @@ -66,16 +70,19 @@ \section{Type theory} \begin{frame} - \frametitle{Proofs are programs} + \frametitle{Set theory} \begin{itemize} - \item Curry-Howard isomorphism - \item Hypotheses are free variables - \item Functions are implication - \item Pairs are AND - \item Unions are OR - \item Universal quantification ($\forall$) is dependent types - \item Existential quantification ($\exists$) is dependent sums + \item Math is built upon sets + \pause + + \item We often operate at a much higher level + \pause + + \item Skip the details + \pause + + \item We can't do this for mechanized systems! \end{itemize} \end{frame} @@ -87,19 +94,85 @@ Talk about set theory \end{frame} +\begin{frame} + \frametitle{Introducing the type} + + \begin{figure} + \centering + \includegraphics[width=0.75\textwidth]{images/type.png} + \end{figure} +\end{frame} + +\begin{frame} + \frametitle{Proofs are programs} + + \begin{table}[] + \begin{tabular}{c|c} + \hline + Logic side & Programming side \\ \hline \hline + formula & type \\ \hline + proof & term \\ \hline + formula is true & type has an element \\ \hline + formula is false & type does not have an element \\ \hline + logical constant $\top$ (truth) & unit type \\ \hline + logical constant $\bot$ (falsehood) & empty type \\ \hline + implication & function type \\ \hline + conjunction & product type \\ \hline + disjunction & sum type \\ \hline \onslide<+(1)-> + universal quantification & dependent product type \\ \hline + existential quantification & dependent sum type \\ \hline + \end{tabular} + \end{table} +\end{frame} + \begin{frame} \frametitle{Important features of Martin-L{\"o}f type theory} \begin{itemize} - \item Types instead of sets - \item Predicative universes: $\mathsf{Type}_0, \mathsf{Type}_1, \mathsf{Type}_2, \dots$ - \item Inductive types: for example, $\mathbb{N}$ is defined with 2 possible constructors: $\mathsf{zero}$ and $\mathsf{suc}$. + \pause + \item Stratified universes: $\mathsf{Type}_0, \mathsf{Type}_1, \mathsf{Type}_2, \dots$ + \pause + \item Inductive types, defined by type former, constructors, eliminators, and computation rules + \end{itemize} +\end{frame} + +\begin{frame} + \frametitle{Important features of Martin-L{\"o}f type theory} + + \begin{figure} + \centering + \includegraphics[width=0.5\textwidth]{images/nat.png} + \end{figure} + \pause + + \begin{itemize} + \item + For example, $\mathbb{N}$ is defined with 2 possible constructors: $\mathsf{zero}$ and $\mathsf{suc}$. + + \begin{columns} + \begin{column}{.3\textwidth} + \begin{prooftree} + \AxiomC{} + \UnaryInfC{$\Gamma \vdash \mathsf{zero} : \mathbb{N}$} + \end{prooftree} + \end{column} + \begin{column}{.3\textwidth} + \begin{prooftree} + \AxiomC{$\Gamma \vdash n : \mathbb{N}$} + \UnaryInfC{$\Gamma \vdash \mathsf{suc}(n) : \mathbb{N}$} + \end{prooftree} + \end{column} + \end{columns} + + \pause + \item + Elimination rule for $\mathbb{N}$: \begin{prooftree} - \AxiomC{$ \Gamma \vdash c_0 : C $} - \AxiomC{$ \Gamma , x : \mathbb{N} , y : C \vdash c_s : C $} + \AxiomC{$ \Gamma \vdash c_z : C $} + \AxiomC{$ \Gamma , (x : \mathbb{N}) , (y : C) \vdash (c_s : C) $} \AxiomC{$ \Gamma \vdash n : \mathbb{N} $} - \TrinaryInfC{$ \Gamma \vdash \mathsf{ind}_{\mathbb{N}} (c_0 , x . y . c_s , n) : C $} + \TrinaryInfC{$ \Gamma \vdash \mathsf{ind}_{\mathbb{N}} (c_z , x . y . c_s , n) : C $} \end{prooftree} \end{itemize} \end{frame} @@ -108,61 +181,162 @@ \frametitle{Important features of Martin-L{\"o}f type theory} \begin{itemize} - \item \emph{Dependent types}, or $\Pi$-types: - - $$ \mathsf{Vec} : \mathsf{Type} \rightarrow \mathbb{N} \rightarrow \mathsf{Type} $$ - $$ \mathsf{append} : \prod_{A : \mathsf{Type}} (\prod_{m, n : \mathbb{N}} (\mathsf{Vec}(A, m) \rightarrow \mathsf{Vec}(A, n) \rightarrow \mathsf{Vec}(A, m + n))) $$ - \item \emph{Dependent sums}, or $\Sigma$-types - - $$ \mathsf{isEven} : \mathbb{N} \rightarrow \mathsf{Type} $$ - $$ \mathsf{Even} : \sum_{n : \mathbb{N}} \mathsf{isEven}(n) $$ \end{itemize} - % \footnotetext[1]{This is the non-dependent version of the induction principle} + \begin{figure} + \centering + \includegraphics[width=0.7\textwidth]{images/sigma.png} + \end{figure} + + \pause + \begin{columns} + \begin{column}{.3\textwidth} + $$ \mathsf{isEven} : \mathbb{N} \rightarrow \mathsf{Type} $$ + \end{column} + \begin{column}{.3\textwidth} + $$ \mathsf{Even} : \sum_{n : \mathbb{N}} \mathsf{isEven}(n) $$ + \end{column} + \end{columns} \end{frame} \begin{frame} \frametitle{Important features of Martin-L{\"o}f type theory} + \begin{itemize} + \item \emph{Dependent functions}, or $\Pi$-types + \pause + + \begin{equation*} + \begin{aligned} + \mathsf{List} &: \mathsf{Type} \rightarrow \mathsf{Type} \\ + \onslide<+(1)-> + \mathsf{Vec} &: \mathsf{Type} \rightarrow \mathbb{N} \rightarrow \mathsf{Type} + \end{aligned} + \end{equation*} + + \end{itemize} + + \pause + $$ \mathsf{append} : \prod_{A : \mathsf{Type}} \left( \prod_{m, n : \mathbb{N}} \left( \mathsf{Vec}(A, m) \rightarrow \mathsf{Vec}(A, n) \rightarrow \mathsf{Vec}(A, m + n) \right) \right) $$ + +\end{frame} + +\begin{frame} + \frametitle{Important features of Martin-L{\"o}f type theory} + + \begin{figure} + \centering + \includegraphics[width=0.5\textwidth]{images/id.png} + \end{figure} + \begin{itemize} \item The identity type, $\mathsf{Id}_A(x, y)$, for some $x, y : A$ + \pause \item Also written as $x \equiv_A y$, or just $x \equiv y$ if $A$ is obvious + \pause \item Single constructor: $\mathsf{refl}_x : \mathsf{Id}_A(x, x)$ \end{itemize} \end{frame} +\begin{frame} + \frametitle{Important features of Martin-L{\"o}f type theory} + + \begin{figure} + \centering + \includegraphics[width=0.5\textwidth]{images/id2.png} + \end{figure} + + \begin{itemize} + \item There can be multiple paths between points! + \end{itemize} +\end{frame} + \begin{frame} \frametitle{Homotopy type theory} - \newcommand{\htcolor}[1]{\textcolor{Melon}{####1}} - \newcommand{\ttcolor}[1]{\textcolor{Periwinkle}{####1}} + \newcommand{\htcolor}[1]{\textcolor{red}{####1}} + \newcommand{\ttcolor}[1]{\textcolor{blue}{####1}} \begin{itemize} - \item A \htcolor{"homotopy"}, from algebraic topology, is a way to continuously deform one path into another ($A \times [0, 1] \rightarrow B$) - \item In type theory, we can imagine two functions \ttcolor{$f, g : A \rightarrow B$} as being homotopic if we can inhabit $h : (x : A) \rightarrow \mathsf{Id}_B (f(x), g(x))$ - \item Note: assume all functions are continuous. - \item Interpret \htcolor{paths between points in a space} as the \ttcolor{identity type $\mathsf{Id}$} + \item A homotopy, from algebraic topology, is a way to continuously deform one path into another ($A \times [0, 1] \rightarrow B$) + + \item In type theory, we consider two functions $f, g : A \rightarrow B$ as being homotopic if we can inhabit $h : (x : A) \rightarrow f(x) \equiv g(x)$ + % \item Note: assume all functions are continuous. + + \begin{figure} + \centering + \includegraphics[width=0.5\textwidth]{images/homotopy.png} + \end{figure} + + \note[item]{Interpret \htcolor{paths between points in a space} as the \ttcolor{identity type $\mathsf{Id}$}} \end{itemize} \end{frame} \begin{frame} - \frametitle{Equivalences} + \frametitle{Homotopy type theory} - \begin{itemize} - \item In math, we consider a relation to be an equivalence if it is reflexive, symmetric, and transitive. - \item In HoTT, a homotopy equivalence is a specific notion between two types $A$ and $B$ + \begin{figure} + \centering + \includegraphics[width=0.8\textwidth]{images/paths.png} + \end{figure} + + \note[item]{We consider these "paths" to be our identities} + \note[item]{The continuousness is important because it means that we can't just connect any two elements, they still have to be constructed uniformly} +\end{frame} + +\begin{frame} + \frametitle{Isomorphism} + + \begin{columns} + \begin{column}{0.5\textwidth} \begin{itemize} \item A function $f : A \rightarrow B$ \item A function $g : B \rightarrow A$ - \item A proof that $g \circ f$ is homotopic to the identity function $\mathsf{id}_A$ - \item A proof that $f \circ g$ is homotopic to the identity function $\mathsf{id}_B$ \end{itemize} + \end{column} + \begin{column}{0.5\textwidth} + \begin{figure} + \centering + \includegraphics[width=0.8\textwidth]{images/maps.png} + \end{figure} + \end{column} + \end{columns} + + + \begin{columns} + \begin{column}{0.5\textwidth} + \begin{itemize} + \item $\prod_{(a: A)} g(f(a)) \equiv a$ + \item $\prod_{(b: B)} f(g(b)) \equiv b$ + \end{itemize} + \end{column} + \begin{column}{0.5\textwidth} + \begin{figure} + \centering + \includegraphics[width=0.8\textwidth]{images/mapsid.png} + \end{figure} + \end{column} + \end{columns} + + \note[item]{Equivalence is sort of like an upgraded isomorphism} +\end{frame} + +\begin{frame} + \frametitle{Univalence} + + \begin{figure} + \centering + \includegraphics[width=0.7\textwidth]{images/ua.png} + \end{figure} + + \begin{itemize} + \item Equivalences \emph{are} identities \end{itemize} \end{frame} \begin{frame} - \frametitle{Univalence axiom} + \frametitle{Univalence} \begin{itemize} \item Equivalences \emph{are} identities @@ -172,20 +346,63 @@ \item Intuitively, if you wanted to use a $B$ where you wanted to use an $A$, just convert it and then convert it back later + \pause \item Importantly, there can be multiple inhabitants of $A \equiv B$ - \begin{itemize} \item Booleans! \end{itemize} + \begin{itemize} + \item Booleans! \\ + $\mathsf{ua}(\mathsf{id})$ and $\mathsf{ua}(\mathsf{not})$ are different elements of $\mathsf{Bool} \equiv \mathsf{Bool}$ + \end{itemize} + \pause \item Transport allows you to use $A$ and $B$ interchangeably \end{itemize} + + $$ \mathsf{transport} : (P : (X : \mathsf{Type}) \rightarrow \mathsf{Type}) \rightarrow (p : x \equiv_X y) \rightarrow P(x) \rightarrow P(y) $$ \end{frame} \begin{frame} \frametitle{Univalence axiom?} \begin{itemize} - \item Unfortunately, univalence does not compute in the "base" version of homotopy type theory, also known as Book HoTT - \item Since $\mathsf{Id}_A$ is an inductive type, the only way you can define it is using the constructor $\mathsf{refl}$. + \item Unfortunately, univalence does not compute in Book HoTT + \pause + \item Assume the functions we want as axioms + \end{itemize} + + $$ + \mathsf{ua} : (A \simeq B) \rightarrow (A \equiv B) + $$ +\end{frame} + +\begin{frame} + \frametitle{Higher inductive types} + + \note[item]{Another way to inject paths into a type} + + \begin{itemize} + \item Normally inductive types give us ways to construct \emph{points} + \pause + \item Higher inductive types give us ways to construct \emph{paths} + \end{itemize} + + \begin{figure} + \centering + \includegraphics[width=0.5\textwidth]{images/zquot.png} + \end{figure} +\end{frame} + +\begin{frame} + \frametitle{Circle ($S^1$)} + + \begin{figure} + \centering + \includegraphics[width=0.4\textwidth]{images/s1.png} + \end{figure} + + \begin{itemize} + \item $\mathsf{base} : S^1$, some arbitrary base point + \item $\mathsf{loop} : \mathsf{base} \equiv \mathsf{base}$, a path representing the rest of the circle \end{itemize} \end{frame} @@ -194,13 +411,32 @@ \begin{itemize} \item Make paths based on the interval type $I$ which represents $[0, 1]$ - \item The interval is a primitive construct + \item The interval is a primitive construct! + \end{itemize} + + \begin{figure} + \centering + \includegraphics[width=0.4\textwidth]{images/interval.png} + \end{figure} +\end{frame} + +\begin{frame} + \frametitle{Cubical type theory} + + \begin{figure} + \centering + \includegraphics[width=0.3\textwidth]{images/box.png} + \end{figure} + + \begin{itemize} + \note[item]{I'll get into \emph{how} to construct squares and paths later} \item Construct squares and cubes to create paths by composition and filling + \pause \item Consequences \begin{itemize} - \item Transport must become a primitive notion, and path induction defined in terms of transport instead - \item Gives us a way to make the univalence axiom a provable theorem - \item Function extensionality is easy + \item Transport is a primitive notion + % \item Path induction defined in terms of transport instead + \item Makes the univalence "axiom" definable \end{itemize} \end{itemize} \end{frame} @@ -211,29 +447,23 @@ \frametitle{The fundamental group $\pi_1$} \begin{itemize} - \item Main idea of algebraic topology: identify which spaces are different from each other - \item The fundamental group is one metric of identifying spaces (i.e donuts and coffee mugs would have the same group) + \item One central idea of algebraic topology: identify which spaces are different from each other + (i.e donuts and coffee mugs are different from a solid sphere) + \pause + \item The fundamental group is one metric of identifying spaces + \note[item]{fundamental group is a special case of a homotopy group} \end{itemize} \end{frame} \begin{frame} - \frametitle{The circle} + \frametitle{Revisiting the circle} \begin{itemize} - \item The circle ($S^1$) is an example of a simple but non-trivial space - \note[item]{it's called $S^1$ because it generalizes into higher dimensions} - - \item Finding fundamental groups of $n$-spheres is one of the interesting problems in algebraic topology - \end{itemize} -\end{frame} - -\begin{frame} - \frametitle{The circle} - - \begin{itemize} - \item Constructed as a higher inductive type - \item $\mathsf{base} : S^1$, some arbitrary base point - \item $\mathsf{loop} : \mathsf{base} \equiv \mathsf{base}$, a path representing the rest of the circle + \item The circle ($S^1$) is an example of a simple but non-trivial space because of the loop + \pause + \item Determining fundamental groups of $n$-spheres is a difficult problem in algebraic topology + \pause + \item Fortunately, $\pi_1(S^1)$ is easy \end{itemize} \end{frame} @@ -373,7 +603,7 @@ \bigskip - \printbibliography + % \printbibliography \end{frame} \end{document} \ No newline at end of file