From a2f10662ff31ee3f54f2fc012f61cb21f8f02513 Mon Sep 17 00:00:00 2001 From: Viktor Kuncak Date: Thu, 28 Dec 2023 21:27:18 +0100 Subject: [PATCH] Formatted kernel proof system to fit pages --- refman/kernel.tex | 73 ++++++++++++++++++++++++------------------ refman/lisa.pdf | Bin 237952 -> 240499 bytes refman/lisa.tex | 68 +++++++++++++-------------------------- refman/macro.tex | 17 ++-------- refman/quickguide.tex | 48 +++++++++++++-------------- refman/r | 1 + refman/titlepage.tex | 35 ++++++++++++++++++++ 7 files changed, 124 insertions(+), 118 deletions(-) create mode 100755 refman/r create mode 100644 refman/titlepage.tex diff --git a/refman/kernel.tex b/refman/kernel.tex index 41115024..a12768c7 100644 --- a/refman/kernel.tex +++ b/refman/kernel.tex @@ -1,5 +1,5 @@ %\part{Reference Manual} -\chapter{Lisa's trusted code: The Kernel} +\chapter{Lisa's Trusted Kernel} \label{chapt:kernel} Lisa's kernel is the starting point of Lisa, formalising the foundations of the whole theorem prover. It is the only trusted code base, meaning that if it is bug-free then no further erroneous code can violate the soundness property and prove invalid statements. Hence, the two main goals of the kernel are to be efficient and trustworthy. @@ -378,7 +378,7 @@ \subsection{Sequent Calculus} \end{definition} A sequent $\phi \vdash \psi$ is logically (but not conceptually) equivalent to a sequent $\vdash \phi \rightarrow \psi$. The distinction is similar to the distinction between meta-implication and inner implication in Isabelle \cite{paulsonIsabelleNext7001993}, for example. Typically, a theorem or a lemma should have its various assumptions on the left-hand side of the sequent and a single conclusion on the right. During proofs however, there may be multiple elements on the right side. \footnote{In a strict description of Sequent Calculus, this is in particular needed to have double negation elimination.} -Sequents are manipulated in a proof using \emph{deduction rules}. A deduction rule, also called a proof step, has zero or more prerequisite sequents (which we call \emph{premises} of the rule) and one conclusion sequent. All the basic deduction rules used in Lisa's kernel are shown in \autoref{fig:deduct_rules_1}. +Sequents are manipulated in a proof using \emph{deduction rules}. A deduction rule, also called a proof step, has zero or more prerequisite sequents (which we call \emph{premises} of the rule) and one conclusion sequent. All the basic deduction rules used in Lisa's kernel are shown in \autoref{fig:deduct_rules_1} and \autoref{fig:deduct_rules_2}. This includes first rules of propositional logic, then rules for quantifiers, then equality rules. Moreover, we include equal-for-equal and equivalent-for-equivalent substitutions. While those substitution rules are deduced steps, and hence could technically be omitted, simulating them can sometimes take a high number of steps, so they are included as base steps for efficiency. Finally, the two rules Restate and Weakening leverage the $\FOLalg$ algorithm. @@ -387,24 +387,20 @@ \subsection{Sequent Calculus} \begin{figure} - \scalebox{.8}{ + \scalebox{.88}{ \begin{minipage}{\textwidth} \begin{center} \begin{tabular}{l l} - \multicolumn{2}{c}{ - \AxiomC{} - \RightLabel{\text { Hypothesis}} - \UnaryInfC{$\Gamma, \phi \vdash \phi, \Delta$} - \DisplayProof - } \\[5ex] - - \multicolumn{2}{c}{ - \AxiomC{$\Gamma \vdash \phi, \Delta$} - \AxiomC{$\Sigma, \phi \vdash \Pi$} - \RightLabel{\text{ Cut}} - \BinaryInfC{$\Gamma, \Sigma \vdash \Delta, \Pi$} - \DisplayProof - } \\[5ex] + \AxiomC{} + \RightLabel{\text { Hypothesis}} + \UnaryInfC{$\Gamma, \phi \vdash \phi, \Delta$} + \DisplayProof & + \AxiomC{$\Gamma \vdash \phi, \Delta$} + \AxiomC{$\Sigma, \phi \vdash \Pi$} + \RightLabel{\text{ Cut}} + \BinaryInfC{$\Gamma, \Sigma \vdash \Delta, \Pi$} + \DisplayProof + \\[5ex] \AxiomC{$\Gamma, \phi, \psi \vdash \Delta$} \RightLabel{\text { LeftAnd}} @@ -478,16 +474,32 @@ \subsection{Sequent Calculus} \RightLabel{\text { RightExists}} \UnaryInfC{$\Gamma \vdash \exists \schem{x}. \phi, \Delta$} \DisplayProof - \\[5ex] + \end{tabular} + \end{center} + \end{minipage}} + \caption{Deduction rules allowed by Lisa's kernel (Part 1). These are typical rules for a sequent calculus for FOL. Different occurrences of the same symbols need not represent equal elements, but only elements with the same \FOLalg{} normal form.} + \label{fig:deduct_rules_1} +\end{figure} - \AxiomC{$\Gamma, \exists y \forall x. (x=y) \leftrightarrow \phi \vdash \Delta$} - \RightLabel{\text { LeftExistsOne}} - \UnaryInfC{$\Gamma, \exists ! x. \phi \vdash \Delta$} - \DisplayProof & - \AxiomC{$\Gamma \vdash \exists y \forall x. (x=y) \leftrightarrow \phi , \Delta$} - \RightLabel{\text { RightExistsOne}} - \UnaryInfC{$\Gamma \vdash \exists ! x. \phi, \Delta$} - \DisplayProof +\begin{figure} + \scalebox{.9}{ + \begin{minipage}{\textwidth} + \begin{center} + \begin{tabular}{l l} + \multicolumn{2}{c}{ + \AxiomC{$\Gamma, \exists y \forall x. (x=y) \leftrightarrow \phi \vdash \Delta$} + \RightLabel{\text { LeftExistsOne}} + \UnaryInfC{$\Gamma, \exists ! x. \phi \vdash \Delta$} + \DisplayProof + } + \\[5ex] + + \multicolumn{2}{c}{ + \AxiomC{$\Gamma \vdash \exists y \forall x. (x=y) \leftrightarrow \phi , \Delta$} + \RightLabel{\text { RightExistsOne}} + \UnaryInfC{$\Gamma \vdash \exists ! x. \phi, \Delta$} + \DisplayProof + } \\[5ex] \multicolumn{2}{c}{ @@ -550,7 +562,7 @@ \subsection{Sequent Calculus} \multicolumn{2}{c}{ \AxiomC{} - \RightLabel{\text { Sorry, admit any statement as true. Usage transitively tracked with a warning.}} + \RightLabel{\text { Sorry: admit a statement as axiom. Usage transitively tracked.}} \UnaryInfC{$\Gamma \vdash \Delta$} \DisplayProof } @@ -558,12 +570,9 @@ \subsection{Sequent Calculus} \end{center} \end{minipage} } - \caption{Deduction rules allowed by Lisa's kernel. Different occurrences of the same symbols need not represent equal elements, but only elements with the same \FOLalg{} normal form.} - \label{fig:deduct_rules_1} - % \label{fig:deduct_rules_2} + \caption{Deduction rules allowed by Lisa's kernel (Part 2). Different occurrences of the same symbols need not represent equal elements, but only elements with the same \FOLalg{} normal form.} + \label{fig:deduct_rules_2} \end{figure} -\newpage - \subsection{Proofs} A sequent calculus proof is a tree whose nodes are proof steps. diff --git a/refman/lisa.pdf b/refman/lisa.pdf index 8989a6184e2cab29c27c9a41ae1ea7dd6a0a7eb5..6b4f4297efecc47a2b57e554c27b8633612d8919 100644 GIT binary patch delta 188123 zcmZ6yQ*_4o3qbOtP?Qq$6#=8LllIlQGV;pY9}4*CH5`RqByrCn;ibQnJMPVr{w#zsd2ny#cLRuT|BRtLSBysKa~7NG9t;z#`ZTFK7$C9S^7!$yq!!|ZheGq z^g3a<#^U*|rV3Y6;n>XOG|ECd#p~tzdjmM0<%+j+yV7f$G0lG6q;P`s%{3h;%euMo*F(sR@ER3bG>+_htct1d~w4+G3q*j1ApE~b);G1@@Fyeos9QmwJ|5V^|uA`u{|3; zg~GOyQs^~>Mx!$xp#T=pS;6NVX@EOBmovKH_2Zy;>vVx9&LRc)cI)&2i>|h*n*`d0 z4$Ac(c$%QKNgEKPNoHUu*tF!h^>Ai(rvC@X|6caS)^Hq2M$D8zRyIzS|B3OH_O#t0 z7fR2K+8N|jBKs#K7%^NEs23n%(->Km;AKht<2&qL<4Ac$JF>BliWQ8s!6sqw%{+8> z%Hh==1^fkQ%)%ngA@3k4@N<|a@P2XQBgmDv_+yY4YT)M%E^S_oI=JY`V>#0lx}T2` 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clone https://github.com/epfl-lara/lisa \end{lstlisting} @@ -20,37 +20,35 @@ \section{Installation} > cd lisa > sbt \end{lstlisting} -SBT is a tool to run scala project and manage versions and dependencies. When it finished loading, do +\lstinline|sbt| is a tool to run a Scala project and manage versions and dependencies. Once inside sbt, run the following commands: \begin{lstlisting}[language=console] > project lisa-examples > run \end{lstlisting} -Wait for the Lisa codebase to be compiled and then press the number corresponding to "Example". You should obtain the following result: +Wait for the Lisa codebase to be compiled and then press the number corresponding to "Example". You should obtain the result demonstrating some example theorems proven, such as the following: \noindent\begin{minipage}{\linewidth}\vspace{1em} \begin{lstlisting}[language=console] @*Theorem fixedPointDoubleApplication := - ∀'x. 'P('x) ==> 'P('f('x)) ⊢ 'P('x) ==> 'P('f('f('x))) + ∀'x. 'P('x) ==> 'P('f('x)) |- 'P('x) ==> 'P('f('f('x))) - Theorem emptySetIsASubset := ⊢ subsetOf(emptySet, 'x) + Theorem emptySetIsASubset := |- subsetOf(emptySet, 'x) Theorem setWithElementNonEmpty := - elem('y, 'x) ⊢ ¬('x = emptySet) + elem('y, 'x) |- ¬('x = emptySet) - Theorem powerSetNonEmpty := ⊢ ¬(powerSet('x) = emptySet) + Theorem powerSetNonEmpty := |- ¬(powerSet('x) = emptySet) *@ \end{lstlisting} \end{minipage} - - \section{Development Environment} -To write Lisa proofs, you can use any text editor or IDE. We recommend using \emph{Visual Studio Code} with the \emph{Metals} plugin. +To write Lisa proofs, you can use any text editor or IDE. We recommend using \emph{Visual Studio Code} (henceforth VSCode) with the \emph{Metals} plugin. \subsection*{A Note on Special Characters} -Math is full of special character. Lisa usually admits both an ascii name and a unicode name for such symbols. By enabling ligatures, common ascii characters such as \lstinline|=|\lstinline|=|\lstinline|>| are rendered as \lstinline|==>|. -The present document uses the font \href{https://github.com/tonsky/FiraCode}{Fira Code}. Once installed on your system, you can activate it and ligatures on VSCode the following way: +Math often uses symbols beyond the Latin alphabet. Lisa usually admits both an English alphabet name and a unicode name for such symbols. By enabling \emph{font ligatures}, common character sequences, such as \lstinline|=|\lstinline|=|\lstinline|>| are rendered as \lstinline|==>|. +The present document also uses the font \href{https://github.com/tonsky/FiraCode}{Fira Code}. Once installed on your system, you can activate it and ligatures on VSCode the following way: \begin{enumerate} - \item Press ctrl-shift-P + \item Press Ctrl-Shift-P \item Search for ``Open User Settings (JSON)'' \item in the \lstinline|settings.json| file, add: \begin{lstlisting} @@ -58,7 +56,7 @@ \subsection*{A Note on Special Characters} "editor.fontLigatures": true, \end{lstlisting} \end{enumerate} -Other symbols such as \lstinline|∀| are unicode symbols, which can be entered via their unicode code, depending on your OS\footnote{alt+numpad on windows, ctrl-shift-U+code on Linux}, or by using an extension for VS Code such as \emph{Fast Unicode Math Characters}, \emph{Insert Unicode} or \emph{Unicode Latex}. +Other symbols such as \lstinline|∀| are Unicode symbols, which can be entered via their unicode code, depending on your OS\footnote{alt+numpad on windows, Ctrl-Shift-U+code on Linux}, or by using an extension for VS Code such as \emph{Fast Unicode Math Characters}, \emph{Insert Unicode} or \emph{Unicode Latex}. A cheat sheet of the most common symbols and how to input them is in \autoref{tab:Unicode}. \begin{table} \center @@ -75,13 +73,13 @@ \subsection*{A Note on Special Characters} \lstinline| ⊆ | & U+2286 & subseteq \\ \hline \lstinline| ∅ | & U+2205 & emptyset \\ \end{tabular} - \caption{Most frequently used Unicode symbols and ligatures.} + \caption{Frequently used Unicode symbols and ligatures.} \label{tab:Unicode} \end{table} -Note that by default, unicode characters will not be printed correctly on a Windows console. You will need to activate the corresponding charset and pick a font with support for unicode in your console's options, such as Consolas. +Note that by default, Unicode characters may not be printed correctly on a Windows console. You will need to activate the corresponding charset and pick a font with support for unicode in your console's options, such as Consolas. \section{Writing theory files} -Lisa provides a canonical way of writing and organizing kernel proofs by mean of a set of utilities and a DSL made possible by some of Scala 3's features. +Lisa provides a canonical way of writing and organizing kernel proofs by mean of a set of utilities and a domain-specific language (DSL) made possible by some of Scala's features. To prove some theorems by yourself, start by creating a file named \lstinline|MyTheoryName.scala| right next to the Example.scala file\footnote{The relative path is lisa/lisa-examples/src/main/scala}. Then simply write: diff --git a/refman/r b/refman/r new file mode 100755 index 00000000..35a76bb8 --- /dev/null +++ b/refman/r @@ -0,0 +1 @@ +latexmk lisa.tex diff --git a/refman/titlepage.tex b/refman/titlepage.tex new file mode 100644 index 00000000..bf9b8656 --- /dev/null +++ b/refman/titlepage.tex @@ -0,0 +1,35 @@ +%\newfontfamily\titlefont{Arial} +\begin{titlepage} +\sf +%\fontfamily{\sfdefault}\selectfont + + \begin{center} + \vspace*{1cm} + + \textbf{\Huge \ourtitle} + + \vspace{1.5cm} + + %\textbf{\large Version 0.2.1} + + \textbf{\large \today} + + \vspace{1.5cm} + + {\Large Laboratory for Automated Reasoning and Analysis \\ \ \\ EPFL, Switzerland} + + \vspace{1.5cm} + + \date{} + + \end{center} + + \vfill + + Contributors: + \begin{flushright} + \large + \ournames + \end{flushright} + + \end{titlepage}