- Commit
- e7007e250e990bfd81563dbae22db94ae7e5b9e2
- Author
- Pablo <pablo-escobar@riseup.net>
- Date
Commit inicial
Slides of a (very) informal lecture of mine on the Curry-Howard correspondence
Commit inicial
33 files changed, 24637 insertions, 0 deletions
Status | File Name | N° Changes | Insertions | Deletions |
Added | .gitignore | 14 | 14 | 0 |
Added | Makefile | 16 | 16 | 0 |
Added | README.md | 52 | 52 | 0 |
Added | examples/example.lean | 9 | 9 | 0 |
Added | examples/exhaustion.py | 2 | 2 | 0 |
Added | examples/factorial.py | 5 | 5 | 0 |
Added | examples/gen-hacker.py | 14 | 14 | 0 |
Added | examples/hacker.txt | 15 | 15 | 0 |
Added | examples/sqrt.py | 1 | 1 | 0 |
Added | examples/types.py | 5 | 5 | 0 |
Added | images/4-colors.eps | 18142 | 18142 | 0 |
Added | images/calculator.jpg | 0 | 0 | 0 |
Added | images/coq.png | 0 | 0 | 0 |
Added | images/gap.png | 0 | 0 | 0 |
Added | images/geogebra.eps | 101 | 101 | 0 |
Added | images/hacker.jpg | 0 | 0 | 0 |
Added | images/idris.eps | 95 | 95 | 0 |
Added | images/isabelle.png | 0 | 0 | 0 |
Added | images/kepler.eps | 580 | 580 | 0 |
Added | images/lean.eps | 172 | 172 | 0 |
Added | images/mathics.eps | 94 | 94 | 0 |
Added | images/ode.eps | 2722 | 2722 | 0 |
Added | images/pentium.png | 0 | 0 | 0 |
Added | images/pet.jpg | 0 | 0 | 0 |
Added | images/sage.eps | 1834 | 1834 | 0 |
Added | images/watchmen.eps | 113 | 113 | 0 |
Added | images/wolfram.eps | 124 | 124 | 0 |
Added | main.pdf | 0 | 0 | 0 |
Added | main.tex | 43 | 43 | 0 |
Added | sections/curry.tex | 278 | 278 | 0 |
Added | sections/exploratory.tex | 115 | 115 | 0 |
Added | sections/numeric.tex | 89 | 89 | 0 |
Added | test | 2 | 2 | 0 |
diff --git a/.gitignore b/.gitignore @@ -0,0 +1,14 @@ +_minted-main +*.aux +*.bak +*.bbl +*.blg +*.log +*.nav +*.out +*.xml +*.snm +*.toc +*.fdb_latexmk +*.fls +*.vrb
diff --git a/Makefile b/Makefile @@ -0,0 +1,16 @@ +MAIN=main.tex + +.PHONY: all auto-compile clean + +# Compile the presentation +all: + latexmk $(MAIN) + +# Compile the presentation every time a file is changes +auto-compile: + find . -name '*.tex' -or -name '*.bib' | entr make || exit 0 + +# Remove os artefatos do xetex e do aspell +clean: + texclear $(MAIN) +
diff --git a/README.md b/README.md @@ -0,0 +1,52 @@ +# Plano + +* Disclaimers +* Provas numéricas + * Ideia + * Provar coisas as vezes são continhas + * Compudores são bons em vazer continhas + * Podemos usar eles + * Exemplo daquele paper de combinatória + * Exemplo mais básico? +* Provadores de teoremas + * Ideia + * Provar coisas as vezes são continhas, não só continhas aritmeticas, mas + continhas "lógicas" + * Indução por exemplo é continha + * Computadores são bons com continhas + * Podemos usar eles? + * Primeira ideia + * Usar boolans e checar todos os casos? + * Não rola se eu tenho infinitos casos + * Como eu faço então? Tento representar as afirmações lógicas no computador + * Problema: temos termos mas não temos formulas + * Solução: curry howard + * Mas como eu passo isso pra um PC? + * Preciso de uma linguagem estaticamente tipada: provar uma coisa significa + contruir um elemento do tipo correspondente, então precisamos garantir + que esse elemento é de fato do tipo correspondente + * Fixo coisas que eu quero impor que são verdade + * Exemplo: fixo que x = x + * Pra isso eu preciso de tipos dependentes + * Exemplos de provadores de teoremas: Lean, Idris, Coq?, Agda? + * Algum exemplo básico em Lean + * Isso é meio que inútil + * Não tamos ainda na posição de provar novos resultados em Lean, mas só de + "verificar formalmente" que resultados já provados são verdade + * Exemplo do paper de combinatória + * Do que adianta ter uma prova formal se não temos um real entendimento de + por que aquilo é verdade? +* Uso exploratório + * Ideia + * As vezes entender casos particulares ou desenhar as coisas ajuda muito a + gente a entender o caso geral + * Mas provar casos particulares/desenhar as coisas pode envolver muitas + continhas chatas + * Computadores são bons em continhas + * Podemos usar eles + * Manipulação simbólica + * Exemplo de integração simbólica + * Exemplo de "x + 2" em Python vs. Mathics? + * CAS + * Exemplos de CAS: SageMath, Gap, Mathics? +
diff --git a/examples/example.lean b/examples/example.lean @@ -0,0 +1,9 @@ +theorem ex : Or p q → Or q p := by + intro h + match h with + | Or.inl _ => + apply Or.inr + assumption + | Or.inr h2 => + apply Or.inl + exact h2
diff --git a/examples/exhaustion.py b/examples/exhaustion.py @@ -0,0 +1,2 @@ +for x in X: + assert p(x)
diff --git a/examples/factorial.py b/examples/factorial.py @@ -0,0 +1,5 @@ +def f(n): + if n == 0: + return 1 + else: + return n * f(n - 1)
diff --git a/examples/gen-hacker.py b/examples/gen-hacker.py @@ -0,0 +1,14 @@ +import random + +# Katakana characters +CHARS = "01" + +WIDTH = 50 +HEIGHT = 15 + +print( + "\n".join( + "".join(random.choice(CHARS) for _ in range(WIDTH)) + for _ in range(HEIGHT) + ) +)
diff --git a/examples/hacker.txt b/examples/hacker.txt @@ -0,0 +1,15 @@ +00110000100011101000011100011010111110101010011000 +00010011011001101011110011011101011010000101010110 +10111101100000110001110010100000101100111110100000 +01001110001000101000110010000001100010001100110110 +00100110000010011010010100000110101110110000101101 +11011111111010010100110100001110100110001111001110 +10101000000101000000000100101000000110001011001010 +11001000101110010001010000101101001110110011100101 +01010100011110010001010000111010101010000011001111 +01100010111011110001111101011011100010000101111100 +11011110010011011110001100010101110110100100100000 +10110010101101110111101110100101111011001111001111 +10100001010100010010111100001010011000010110111110 +10011100111000010001011101010110101110001100000010 +11011001001001110001000110110000110000111101110111
diff --git a/examples/sqrt.py b/examples/sqrt.py @@ -0,0 +1 @@ +math.sqrt(2) ** 2 # 2.0000000000000004
diff --git a/examples/types.py b/examples/types.py @@ -0,0 +1,5 @@ +class AndAB: + """A e B <-> AndAB""" + def __init__(self, a: A, b: B): + self.left = a + self.right = b
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diff --git a/main.tex b/main.tex @@ -0,0 +1,43 @@ +\input{preamble-beamer} +\usepackage{minted} +\usemintedstyle{friendly} + +\definecolor{hacker}{RGB}{4,165,90} + +\title{Como fazer matemática com um computador?} +\author{Pablo} +\date{Agosto de 2021} + +\begin{document} +\begin{frame} + \maketitle +\end{frame} + +\begin{frame}{Disclaimer} + \begin{itemize} + \item Baseado em + \href{https://www.youtube.com/watch?v=T9ZqbQh-t9E}{Machine-Assisted + Proof} + + \pause + + \item Hacker\dots + \begin{figure} + \centering + \BVerbatimInput[formatcom=\color{hacker}]{examples/hacker.txt} + \end{figure} + + \pause + + \item Me interrompa! + \end{itemize} +\end{frame} + +\input{sections/numeric} + +\input{sections/curry} + +\input{sections/exploratory} + +\end{document} +
diff --git a/sections/curry.tex b/sections/curry.tex @@ -0,0 +1,278 @@ +\section{Provadores de Teoremas} + +\begin{frame}[fragile]{Métodos Não-Numéricos?} + \begin{itemize} + \item E se as contas que eu preciso fazer não são \emph{numéricas}? + \begin{itemize} + \item Indução é conta + + \pause + + \item Continhas em algebra abstrata + \end{itemize} \pause + \[ + n! \cdot m! \mid (n + m)! + \] + + \pause + + \item Antes de tentar provar meu teorema, eu preciso enunciar ele\dots + \end{itemize} + + \pause + + \begin{columns} + \begin{column}{.5\linewidth} + \[ + n! = + \begin{cases} + 1 & \text{se}\ n = 0 \\ + n \cdot (n - 1)! & \text{se}\ n > 0 + \end{cases} + \] + \end{column} + \begin{column}{.5\linewidth} + \inputminted{python}{examples/factorial.py} + \end{column} + \end{columns} + + \pause + + \begin{columns} + \begin{column}{.5\linewidth} + \[ + n! \cdot m! \mid (n + m)! + \] + \end{column} + \begin{column}{.5\linewidth} + \begin{minted}{python} + # ??? + \end{minted} + \end{column} + \end{columns} + + \pause + + \begin{itemize} + \item Temos termos, mas não temos formulas! + \end{itemize} + + \pause + + \begin{figure} + \centering + \begin{tabular}{|l|l|} + \hline + Termos + & Fórmulas \\ + \hline + Um número \(n\) + & ``\(n > m\)'' \\ + \hline + Um conjunto \(X\) + & ``\(x \in X\)'' \\ + \hline + Pontos, retas, planos, \dots + & O quinto postulado de Euclides \\ + \hline + \end{tabular} + \end{figure} +\end{frame} + +\subsection{A Correspondência de Curry-Howard} + +\begin{frame}{A Correspondência de Curry-Howard} + \begin{itemize} + \item Provar que vale \(A\) é o mesmo que mostrar que \(\{\text{provas de}\ + A\} \ne \emptyset\) + + \pause + + \item Proposições como tipos + \begin{center} + \begin{tabular}{rcl} + \(A\) + & \(\leftrightsquigarrow\) + & \(\{\text{provas de}\ A \}\) \\ + \(A\ \text{e}\ B\) + & \(\leftrightsquigarrow\) + & \(\{\text{provas de}\ A \} \times \{\text{provas de}\ B \}\) \\ + \(A\ \text{ou}\ B\) + & \(\leftrightsquigarrow\) + & \(\{\text{provas de}\ A \} \cup \{\text{provas de}\ B \}\) \\ + \(A \Rightarrow B\) + & \(\leftrightsquigarrow\) + & \(\{\text{provas de}\ A \} \to \{\text{provas de}\ B \}\) \\ + \end{tabular} + \end{center} + + \pause + + \item Mas como eu represento isso num computador? + + \pause + + \item Tipos + \begin{figure} + \centering + \inputminted{python}{examples/types.py} + \end{figure} + + \pause + + \item Se eu conseguir contruir um valor to tipo \texttt{AndAB} eu provei + que vale \(A \ \text{e} \ B\) + + \pause + + \item Mas o Python não sabe checar se o valor que eu construi é de fato do + tipo \texttt{AndAB} + \end{itemize} +\end{frame} + +\begin{frame}{Provadores de Teoremas} + \begin{itemize} + \item Linguagens estaticamente tipadas + + \pause + + \item Fixo certas coisas que quero impor que sejam verdade + \begin{itemize} + \item Dados \(x\) e \(y\), existe o tipo \texttt{x = y} das provas de + que \(x = y\) + + \item O tipo \texttt{x = x} tem um único elemento \texttt{rfl} + \end{itemize} + + \pause + + \item O tipo \texttt{x = y} varia com \(x\) e \(y\) + + \pause + + \item Provadores de Teoremas + \begin{itemize} + \item Coq (\url{https://coq.inria.fr/}) + + \item Idris (\url{https://www.idris-lang.org/}) + + \item Lean (\url{https://leanprover.github.io/}) + + \item Isabelle (\url{https://isabelle.in.tum.de/}) + \end{itemize} + \end{itemize} + + \begin{columns} + \begin{column}{.2\linewidth} + \begin{figure} + \centering + \includegraphics[scale=.6]{images/coq.png} + \end{figure} + \end{column} + \begin{column}{.2\linewidth} + \begin{figure} + \centering + \includegraphics[scale=.095]{images/idris.eps} + \end{figure} + \end{column} + \begin{column}{.2\linewidth} + \begin{figure} + \centering + \includegraphics[scale=.15]{images/lean.eps} + \end{figure} + \end{column} + \begin{column}{.2\linewidth} + \begin{figure} + \centering + \includegraphics[scale=.4]{images/isabelle.png} + \end{figure} + \end{column} + \end{columns} +\end{frame} + +\begin{frame}[fragile]{Lean} + \begin{columns} + \begin{column}{.45\linewidth} + \begin{figure}[t] + \centering + \inputminted{lean}{examples/example.lean} + \end{figure} + \end{column} \pause + \begin{column}{.40\linewidth} + \begin{itemize} + \item + \href{https://leanprover-community.github.io/mathlib-overview.html}{mathlib} + \begin{itemize} + \item Algebra linear + + \item Algebra abstrata + + \item Topologia + + \item Análise + + \item Geometria diferencial + + \item Teoria dos números + + \item Combinatória + + \item \dots + \end{itemize} + \end{itemize} + \end{column} + \end{columns} + + \pause + + \begin{itemize} + \item A \emph{mathlib} ainda não tem suporte para tudo na matemática\dots + + \pause + + \item A \emph{mathlib} nunca foi usada para provar algo novo\dots + + \pause + + \item Pra que tudo isso então? + \end{itemize} +\end{frame} + +\subsection{Aplicações} + +\begin{frame}{Pra que tudo isso?} + \begin{columns} + \begin{column}[t]{.65\linewidth} + \begin{itemize} + \item<1-> Verificação formal de software e hardware \pause + \begin{itemize} + \item<2-> O bug do Pentium de 1994 + + \item<3-> A queda do voo 302 da Ethiopian Airlines em 2019 + \end{itemize} + + \item<4-> Provas de alta complexidade + \begin{itemize} + \item<4-> Classificação dos grupos simples finitos + + \item<5-> A prova de Mochizuki da conjetura abc + + \item<6-> A prova de Hales da conjectura de Kepler de 2015 + \end{itemize} + + \item<7-> Limitações + \begin{itemize} + \item<7-> Cadê a intuição? + + \item<8-> \emph{Who watches the Watchmen?} + \includegraphics[height=1em]{images/watchmen.eps} + \end{itemize} + \end{itemize} + \end{column} + \begin{column}[t]{.25\linewidth} + \begin{figure}[t] + \includegraphics[width=\linewidth]{images/pentium.png} + \end{figure} + \end{column} + \end{columns} +\end{frame}
diff --git a/sections/exploratory.tex b/sections/exploratory.tex @@ -0,0 +1,115 @@ +\section{Uso Exploratório} + +\begin{frame}{Uso Exploratório} + \begin{itemize} + \item Plotar coisas \pause + \begin{align*} + \frac{\dx}{\dt} & = x + y & \frac{\dy}{\dt} & = -x + y + \end{align*} + + \pause + + \begin{figure} + \centering + \includegraphics[width=.8\linewidth]{images/ode.eps} + \end{figure} + \end{itemize} +\end{frame} + +\begin{frame}{Uso Exploratório} + \begin{itemize} + \item<1-> Continhas bobas + \begin{columns} + \begin{column}{.5\linewidth} + \begin{align*} + \onslide<1->{ + \int_0^\pi \sin t \cdot \cos^2 t \; \dt & = \frac{2}{3} + } \\ + \onslide<2->{ + \int_0^\pi \sin t \cdot \cos^2 t \; \dt & \approx 0.66667 + } + \end{align*} + \end{column} + \begin{column}{.5\linewidth} + \begin{figure} + \centering + \includegraphics[scale=2]{images/wolfram.eps} + \end{figure} + \end{column} + \end{columns} + + \item<3-> Como o Wolfram resolve essa integral? + \onslide<4->{ + \begin{align*} + \int_0^\pi \sin t \cdot \cos^2 t \; \dt + & = \left. \left( - \cos^3 t \right) \right|_{t = 0}^\pi + - 2 \int_0^\pi \sin t \cdot \cos^2 t \; \dt \\ + \int_0^\pi \sin t \cdot \cos^2 t \; \dt + & = \frac{\left. \left( - \cos^3 t \right) \right|_{t = 0}^\pi}{3} \\ + \int_0^\pi \sin t \cdot \cos^2 t \; \dt + & = \frac{2}{3} + \end{align*} + } + \end{itemize} +\end{frame} + +\begin{frame}{Computação Simbólica} + \begin{columns} + \begin{column}{.7\linewidth} + \begin{itemize} + \item<1-> Calculadora \emph{simbólica} + \begin{itemize} + \item<1-> Abre as mesmas contas que abrimos na mão + \item<2-> Usa todos os truques do baralho + \item<3-> Deixa \emph{coisas indeterminadas} expressas + \begin{figure} + \centering + \inputminted{python}{examples/sqrt.py} + \end{figure} + \end{itemize} + + \item<4-> Computer Algebra System (CAS) + \begin{itemize} + \item Sage (\url{https://www.sagemath.org/}) + \item GAP (\url{https://www.gap-system.org/}) + \item Mathics (\url{https://mathics.org/}) + \item GeoGebra (\url{https://geogebra.org/}) + \end{itemize} + \end{itemize} + \end{column} + \begin{column}{.22\linewidth} + \begin{figure} + \includegraphics[width=\linewidth]{images/calculator.jpg} + \end{figure} + \end{column} + \end{columns} + + \onslide<4->{ + \begin{columns} + \begin{column}{.2\linewidth} + \begin{figure} + \centering + \includegraphics[scale=.12]{images/sage.eps} + \end{figure} + \end{column} + \begin{column}{.2\linewidth} + \begin{figure} + \centering + \includegraphics[scale=.2]{images/gap.png} + \end{figure} + \end{column} + \begin{column}{.2\linewidth} + \begin{figure} + \centering + \includegraphics[scale=.2]{images/mathics.eps} + \end{figure} + \end{column} + \begin{column}{.2\linewidth} + \begin{figure} + \centering + \includegraphics[scale=2]{images/geogebra.eps} + \end{figure} + \end{column} + \end{columns} + } +\end{frame}
diff --git a/sections/numeric.tex b/sections/numeric.tex @@ -0,0 +1,89 @@ +\section{Métodos Numéricos} + +\begin{frame}{Como fazer matemática com um computador?} + \begin{itemize} + \item Será que um computador pode nos ajudar a provar teoremas? + + \pause + + \item As vezes provar teoremas acaba caindo em continhas + + \item As vezes são \emph{muitas} continhas + \begin{figure} + \centering + \includegraphics[scale=.3]{images/pet.jpg} + \end{figure} + + \pause + + \item Será que um computador não consegue fazer as continhas pra mim? + \end{itemize} +\end{frame} + +\begin{frame}{Métodos Numéricos} + \begin{theorem}[Teorema das Quatro Cores] + Todo mapa pode ser pintado de quatro cores distintas de modo dois países + adjacentes não compartilham a mesma cor. + \end{theorem} + + \begin{figure} + \centering + \includegraphics[width=.8\linewidth]{images/4-colors.eps} + \end{figure} + + \pause + + \begin{itemize} + \item Provado em 1976 por Kenneth Appel e Wolfgang Haken usando o método + \emph{discharging} + \end{itemize} +\end{frame} + +\subsection{Provas Por Exaustão} + +\begin{frame}{Provar Por Exaustão} + \begin{columns} + \begin{column}{.35\linewidth} + \begin{figure} + \centering + \includegraphics[width=\linewidth]{images/kepler.eps} + \end{figure} + \end{column} + \begin{column}{.65\linewidth} + \begin{itemize} + \item Se eu tenho um número finito de casos\dots posso testar todos os + casos! + \begin{figure} + \centering + \inputminted{python}{examples/exhaustion.py} + \end{figure} + \end{itemize} + + \pause + + \begin{theorem}[Conjectura de Kepler] + Nenhum arranjo de esferas em três dimensões tem uma densidade média maior + do que o arranjo cúbico. + \end{theorem} + + \begin{itemize} + \item Provado em 1998 por Thomas Hales via exaustão + + \pause + + \item Limitações + \begin{itemize} + \item Pouco acessível + + \pause + + \item E se tiver um bug no programa que eu escrevi? + + \pause + + \item E se o número de casos não for finito? + \end{itemize} + \end{itemize} + \end{column} + \end{columns} +\end{frame}
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