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--- /dev/null
+\documentclass{kluwer}
+\usepackage{color}
+\usepackage{graphicx}
+% \usepackage{amssymb,amsmath}
+\usepackage{hyperref}
+% \usepackage{picins}
+\usepackage{color}
+\usepackage{fancyvrb}
+\usepackage[show]{ed}
+
+\definecolor{gray}{gray}{0.85}
+%\newcommand{\logo}[3]{
+%\parpic(0cm,0cm)(#2,#3)[l]{\includegraphics[width=#1]{whelp-bw}}
+%}
+
+\newcommand{\AUTO}{\textsc{Auto}}
+\newcommand{\COQ}{Coq}
+\newcommand{\ELIM}{\textsc{Elim}}
+\newcommand{\HELM}{Helm}
+\newcommand{\HINT}{\textsc{Hint}}
+\newcommand{\IN}{\ensuremath{\dN}}
+\newcommand{\INSTANCE}{\textsc{Instance}}
+\newcommand{\IR}{\ensuremath{\dR}}
+\newcommand{\IZ}{\ensuremath{\dZ}}
+\newcommand{\LIBXSLT}{LibXSLT}
+\newcommand{\LOCATE}{\textsc{Locate}}
+\newcommand{\MATCH}{\textsc{Match}}
+\newcommand{\MATITA}{Matita}
+\newcommand{\METAHEADING}{Symbol & Position \\ \hline\hline}
+\newcommand{\MOWGLI}{MoWGLI}
+\newcommand{\NAT}{\ensuremath{\mathit{nat}}}
+\newcommand{\NATIND}{\mathit{nat\_ind}}
+\newcommand{\NUPRL}{NuPRL}
+\newcommand{\OCAML}{OCaml}
+\newcommand{\PROP}{\mathit{Prop}}
+\newcommand{\REF}[3]{\ensuremath{\mathit{Ref}_{#1}(#2,#3)}}
+\newcommand{\TEXMACRO}[1]{\texttt{\char92 #1}}
+\newcommand{\UWOBO}{UWOBO}
+\newcommand{\WHELP}{Whelp}
+\newcommand{\DOT}{\ensuremath{\mbox{\textbf{.}}}}
+\newcommand{\SEMICOLON}{\ensuremath{\mbox{\textbf{;}}}}
+\newcommand{\BRANCH}{\ensuremath{\mbox{\textbf{[}}}}
+\newcommand{\SHIFT}{\ensuremath{\mbox{\textbf{\textbar}}}}
+\newcommand{\POS}[1]{\ensuremath{#1\mbox{\textbf{:}}}}
+\newcommand{\MERGE}{\ensuremath{\mbox{\textbf{]}}}}
+\newcommand{\FOCUS}[1]{\ensuremath{\mathtt{focus}~#1}}
+\newcommand{\UNFOCUS}{\ensuremath{\mathtt{unfocus}}}
+\newcommand{\SKIP}{\MATHTT{skip}}
+\newcommand{\TACTIC}[1]{\ensuremath{\mathtt{tactic}~#1}}
+
+\definecolor{gray}{gray}{0.85} % 1 -> white; 0 -> black
+\newcommand{\NT}[1]{\langle\mathit{#1}\rangle}
+\newcommand{\URI}[1]{\texttt{#1}}
+
+%{\end{SaveVerbatim}\setlength{\fboxrule}{.5mm}\setlength{\fboxsep}{2mm}%
+\newenvironment{grafite}{\VerbatimEnvironment
+ \begin{SaveVerbatim}{boxtmp}}%
+ {\end{SaveVerbatim}\setlength{\fboxsep}{3mm}%
+ \begin{center}
+ \fcolorbox{black}{gray}{\BUseVerbatim[boxwidth=0.9\linewidth]{boxtmp}}
+ \end{center}}
+
+\newcounter{example}
+\newenvironment{example}{\stepcounter{example}\vspace{0.5em}\noindent\emph{Example} \arabic{example}.}
+ {}
+\newcommand{\ASSIGNEDTO}[1]{\textbf{Assigned to:} #1}
+\newcommand{\FILE}[1]{\texttt{#1}}
+% \newcommand{\NOTE}[1]{\ifodd \arabic{page} \else \hspace{-2cm}\fi\ednote{#1}}
+\newcommand{\NOTE}[1]{\ednote{#1}{foo}}
+\newcommand{\TODO}[1]{\textbf{TODO: #1}}
+
+\newsavebox{\tmpxyz}
+\newcommand{\sequent}[2]{
+ \savebox{\tmpxyz}[0.9\linewidth]{
+ \begin{minipage}{0.9\linewidth}
+ \ensuremath{#1} \\
+ \rule{3cm}{0.03cm}\\
+ \ensuremath{#2}
+ \end{minipage}}\setlength{\fboxsep}{3mm}%
+ \begin{center}
+ \fcolorbox{black}{gray}{\usebox{\tmpxyz}}
+ \end{center}}
+
+\bibliographystyle{plain}
+
+\begin{document}
+
+\begin{opening}
+
+ \title{The \MATITA{} Proof Assistant}
+
+\author{Andrea \surname{Asperti} \email{asperti@cs.unibo.it}}
+\author{Claudio \surname{Sacerdoti Coen} \email{sacerdot@cs.unibo.it}}
+\author{Enrico \surname{Tassi} \email{tassi@cs.unibo.it}}
+\author{Stefano \surname{Zacchiroli} \email{zacchiro@cs.unibo.it}}
+\institute{Department of Computer Science, University of Bologna\\
+ Mura Anteo Zamboni, 7 --- 40127 Bologna, ITALY}
+
+\runningtitle{The Matita proof assistant}
+\runningauthor{Asperti, Sacerdoti Coen, Tassi, Zacchiroli}
+
+% \date{data}
+
+\begin{motto}
+``We are nearly bug-free'' -- \emph{CSC, Oct 2005}
+\end{motto}
+
+\begin{abstract}
+ abstract qui
+\end{abstract}
+
+\keywords{Proof Assistant, Mathematical Knowledge Management, XML, Authoring,
+Digital Libraries}
+
+\end{opening}
+
+\begin{figure}[t]
+ \begin{center}
+ \includegraphics[width=0.9\textwidth]{libraries}
+ \caption{\MATITA{} libraries}
+ \end{center}
+ \label{fig:libraries}
+\end{figure}
+
+\section{Overview of the Architecture}
+Fig.~\ref{fig:libraries} shows the architecture of the \emph{libraries} (circle nodes)
+and \emph{applications} (squared nodes) developed in the HELM project.
+
+Applications and libraries depend over other libraries forming a
+directed acyclic graph (DAG). Each library can be decomposed in
+a a set of \emph{modules} also forming a DAG.
+
+Modules and libraries provide coherent sets of functionalities
+at different scales. Applications that require only a few functionalities
+depend on a restricted set of libraries. \MATITA, our most complex
+application, depends on every library.
+
+Only the proof assistant \MATITA{} is an application meant to be used directly
+by the user. All the other applications are Web services developed in the
+HELM and MoWGLI projects and already described elsewhere. In particular:
+\begin{itemize}
+ \item The \emph{Getter} is a Web service to retrieve an (XML) document
+ from a physical location (URL) given its logical name (URI). The Getter is
+ responsible of updating a table that maps URIs to URLs. Thanks to the Getter
+ it is possible to work on a logically monolithic library that is physically
+ distributed on the network. More information on the Getter can be found
+ in~\cite{getter}.
+ \item \emph{Whelp} is a search engine to index and locate mathematical
+ notions (axioms, theorems, definitions) in the logical library managed
+ by the Getter. Typical examples of a query to Whelp are queries that search
+ for a theorem that generalize or instantiate a given formula, or that
+ can be immediately applied to prove a given goal. The output of Whelp is
+ an XML document that lists the URIs of a complete set of candidates that
+ are likely to satisfy the given query. The set is complete in the sense
+ that no notion that actually satisfies the query is thrown away. However,
+ the query is only approssimated in the sense that false matches can be
+ returned. Whelp has been described in~\cite{whelp}.
+ \item \emph{Uwobo} is a Web service that, given the URI of a mathematical
+ notion in the distributed library, renders it according to the user provided
+ two dimensional mathematical notation. Uwobo may also embed the rendering
+ of mathematical notions into arbitrary documents before returning them.
+ The Getter is used by Uwobo to retrieve the document to be rendered.
+ Uwobo has been described in~\cite{uwobo}.
+ \item The \emph{Proof Checker} is a Web service that, given the URI of
+ notion in the distributed library, checks its correctness. Since the notion
+ is likely to depend in an acyclic way over other notions, the proof checker
+ is also responsible of building in a top-down way the DAG of all
+ dependencies, checking in turn every notion for correctness.
+ The proof checker has been described in~\cite{proofchecker}.
+ \item The \emph{Dependency Analyzer} is a Web service that can produce
+ a textual or graphical representation of the dependecies of an object.
+ The dependency analyzer has been described in~\cite{dependencyanalyzer}.
+\end{itemize}
+
+The dependency of a library or application over another library can
+be satisfied by linking the library in the same executable.
+For those libraries whose functionalities are also provided by the
+aforementioned Web services, it is also possible to link stub code that
+forwards the request to a remote Web service. For instance, the Getter
+is just a wrapper to the \texttt{getter} library that allows the library
+to be used as a Web service. \MATITA{} can directly link the code of the
+\texttt{getter} library, or it can use a stub library with the same API
+that forwards every request to the Getter.
+
+To better understand the architecture of \MATITA{} and the role of each
+library, we can focus on the rappresentation of the mathematical information.
+\MATITA{} is based on (a variant of) the Calculus of (Co)Inductive
+Constructions (CIC). In CIC terms are used to represent mathematical
+expressions, types and proofs. \MATITA{} is able to handle terms at
+four different levels of refinement. On each level it is possible to provide a
+different set of functionalities. The four different levels are:
+fully specified terms; partially specified terms; terms
+at the content level; terms at the presentation level.
+
+\subsection{Fully specified terms}
+ \emph{Fully specified terms} are CIC terms where no information is
+ missing or left implicit. A fully specified term should be well-typed.
+ The mathematical notions (axioms, definitions, theorems) that are stored
+ in our mathematical library are fully specified and well-typed terms.
+ Fully specified terms are extremely verbose (to make type-checking
+ decidable). Their syntax is fixed and does not resemble the usual
+ extendible mathematical notation. They are not meant for direct user
+ consumption.
+
+ The \texttt{cic} library defines the data type that represents CIC terms
+ and provides a parser for terms stored in an XML format.
+
+ The most important library that deals with fully specified terms is
+ \texttt{cic\_proof\_checking}. It implements the procedure that verifies
+ if a fully specified term is well-typed. It also implements the
+ \emph{conversion} judgement that verifies if two given terms are
+ computationally equivalent (i.e. they share the same normal form).
+
+ Terms may reference other mathematical notions in the library.
+ One commitment of our project is that the library should be physically
+ distributed. The \texttt{getter} library manages the distribution,
+ providing a mapping from logical names (URIs) to the physical location
+ of a notion (an URL). The \texttt{urimanager} library provides the URI
+ data type and several utility functions over URIs. The
+ \texttt{cic\_proof\_checking} library calls the \texttt{getter} library
+ every time it needs to retrieve the definition of a mathematical notion
+ referenced by a term that is being type-checked.
+
+ The Proof Checker is the Web service that provides an HTTP interface
+ to the \texttt{cic\_proof\_checking} library.
+
+ We use metadata and a sort of crawler to index the mathematical notions
+ in the distributed library. We are interested in retrieving a notion
+ by matching, instantiation or generalization of a user or system provided
+ mathematical expression. Thus we need to collect metadata over the fully
+ specified terms and to store the metadata in some kind of (relational)
+ database for later usage. The \texttt{hmysql} library provides a simplified
+ interface to a (possibly remote) MySql database system used to store the
+ metadata. The \texttt{metadata} library defines the data type of the metadata
+ we are collecting and the functions that extracts the metadata from the
+ mathematical notions (the main functionality of the crawler).
+ The \texttt{whelp} library implements a search engine that performs
+ approximated queries by matching/instantiation/generalization. The queries
+ operate only on the metadata and do not involve any actual matching
+ (that will be described later on and that is implemented in the
+ \texttt{cic\_unification} library). Not performing any actual matching
+ the query only returns a complete and hopefully small set of matching
+ candidates. The process that has issued the query is responsible of
+ actually retrieving from the distributed library the candidates to prune
+ out false matches if interested in doing so.
+
+ The Whelp search engine is the Web service that provides an interface to
+ the \texttt{whelp} library.
+
+\subsection{Partially specified terms}
+\emph{Partially specified terms} are CIC terms where subterms can be omitted.
+Omitted subterms can bear no information at all or they may be associated to
+a sequent. The formers are called \emph{implicit terms} and they occur only
+linearly. The latters may occur multiple times and are called
+\emph{metavariables}. An \emph{explicit substitution} is applied to each
+occurrence of a metavariable. A metavariable stand for a term whose type is
+given by the conclusion of the sequent. The term must be closed in the
+context that is given by the ordered list of hypotheses of the sequent.
+The explicit substitution instantiates every hypothesis with an actual
+value for the term bound by the hypothesis.
+
+Partially specified terms are not required to be well-typed. However a
+partially specified term should be \emph{refinable}. A \emph{refiner} is
+a type-inference procedure that can instantiate implicit terms and
+metavariables and that can introduce \emph{implicit coercions} to make a
+partially specified term be well-typed. The refiner of \MATITA{} is implemented
+in the \texttt{cic\_unification} library. As the type checker is based on
+the conversion check, the refiner is based on \emph{unification} that is
+a procedure that makes two partially specified term convertible by instantiating
+as few as possible metavariables that occur in them.
+
+Since terms are use in CIC to represent proofs, so far correct incomplete
+proofs are represented by refinable partially specified terms. The metavariables
+that occur in the proof correspond to the conjectures still to be proved.
+The sequent associated to the metavariable is the conjecture the user needs to
+prove.
+
+\emph{Tactics} are the procedures that the user can apply to progress in the
+proof. A tactic proves a conjecture possibly creating new (and hopefully
+simpler) conjectures. The implementation of tactics is given in the
+\texttt{tactics} library. It is heavily based on the refinement and unification
+procedures of the \texttt{cic\_unification} library.
+
+As fully specified terms, partially specified terms are not well suited
+for user consumption since their syntax is not extendible and it is not
+possible to adopt the usual mathematical notation. However they are already
+an improvement over fully specified terms since they allow to omit redundant
+information that can be inferred by the refiner.
+
+\subsection{Terms at the content level}
+
+\subsection{Terms at the presentation level}
+
+\hrule
+
+At the bottom of the DAG we have a few libraries (\texttt{extlib},
+\texttt{xml} and the \texttt{registry}) that provide a core of
+useful functions used everywhere else. In particular, the \texttt{xml} library
+to easily represent, parse and pretty-print XML files is a central component
+since in HELM every piece of information is stored in \ldots. [FINIRE]
+The other basic libraries provide often needed operations over generic
+data structures (\texttt{extlib}) and central storage for configuration options
+(the \texttt{registry}).
+
+\texttt{urimanager}
+
+\texttt{getter}
+
+\texttt{cic}
+
+\acknowledgements
+We would like to thank all the students that during the past
+five years collaborated in the \HELM{} project and contributed to
+the development of Matita, and in particular
+A.Griggio, F.Guidi, P. Di Lena, L.Padovani, I.Schena, M.Selmi,
+V.Tamburrelli.
+
+\theendnotes
+
+\bibliography{matita}
+
+
+\end{document}
+
projects / helm.git / commitdiff