-\documentclass[draft]{kluwer}
+\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{\component}{component}
\newcommand{\components}{components}
\newcommand{\AUTO}{\textsc{Auto}}
+\newcommand{\BOXML}{BoxML}
\newcommand{\COQ}{Coq}
\newcommand{\COQIDE}{CoqIde}
\newcommand{\ELIM}{\textsc{Elim}}
\newcommand{\GDOME}{Gdome}
+\newcommand{\GTKMATHVIEW}{\textsc{GtkMathView}}
\newcommand{\HELM}{Helm}
\newcommand{\HINT}{\textsc{Hint}}
\newcommand{\IN}{\ensuremath{\dN}}
\newcommand{\LIBXSLT}{LibXSLT}
\newcommand{\LOCATE}{\textsc{Locate}}
\newcommand{\MATCH}{\textsc{Match}}
+\newcommand{\MATHML}{MathML}
\newcommand{\MATITA}{Matita}
\newcommand{\MATITAC}{\texttt{matitac}}
\newcommand{\MATITADEP}{\texttt{matitadep}}
-\newcommand{\METAHEADING}{Symbol & Position \\ \hline\hline}
\newcommand{\MOWGLI}{MoWGLI}
+\newcommand{\MOWGLIIST}{IST-2001-33562}
\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{\REWRITEHINT}{\textsc{RewriteHint}}
\newcommand{\TEXMACRO}[1]{\texttt{\char92 #1}}
\newcommand{\UWOBO}{UWOBO}
\newcommand{\GETTER}{Getter}
\newcommand{\WHELP}{Whelp}
+
\newcommand{\DOT}{\ensuremath{\mbox{\textbf{.}}}}
\newcommand{\SEMICOLON}{\ensuremath{\mbox{\textbf{;}}}}
\newcommand{\BRANCH}{\ensuremath{\mbox{\textbf{[}}}}
\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{\NT}[1]{\ensuremath{\langle\mathit{#1}\rangle}}
\newcommand{\URI}[1]{\texttt{#1}}
\newcommand{\OP}[1]{``\texttt{#1}''}
+\newcommand{\FILE}[1]{\texttt{#1}}
+\newcommand{\NOTE}[1]{\ednote{#1}{}}
+\newcommand{\TODO}[1]{\textbf{TODO: #1}}
+
+\definecolor{gray}{gray}{0.85} % 1 -> white; 0 -> black
-%{\end{SaveVerbatim}\setlength{\fboxrule}{.5mm}\setlength{\fboxsep}{2mm}%
\newenvironment{grafite}{\VerbatimEnvironment
\begin{SaveVerbatim}{boxtmp}}%
{\end{SaveVerbatim}\setlength{\fboxsep}{3mm}%
\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}}
-
\newcounter{pass}
\newcommand{\PASS}{\stepcounter{pass}\arabic{pass}}
\fcolorbox{black}{gray}{\usebox{\tmpxyz}}
\end{center}}
-\bibliographystyle{plain}
+\bibliographystyle{klunum}
\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}
+ \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}}
-% \date{data}
+ \institute{Department of Computer Science, University of Bologna\\
+ Mura Anteo Zamboni, 7 --- 40127 Bologna, ITALY}
-\begin{motto}
-``We are nearly bug-free'' -- \emph{CSC, Oct 2005}
-\end{motto}
+ \runningtitle{The \MATITA{} proof assistant}
+ \runningauthor{Asperti, Sacerdoti Coen, Tassi, Zacchiroli}
-\begin{abstract}
- abstract qui
-\end{abstract}
+ \begin{motto}
+ ``We are nearly bug-free'' -- \emph{CSC, Oct 2005}
+ \end{motto}
-\keywords{Proof Assistant, Mathematical Knowledge Management, XML, Authoring,
-Digital Libraries}
+ \begin{abstract}
+ \TODO{scrivere abstract}
+ \end{abstract}
+ \keywords{Proof Assistant, Mathematical Knowledge Management, XML, Authoring,
+ Digital Libraries}
\end{opening}
+% toc & co: to be removed in the final paper version
\tableofcontents
+\listoffigures
+\listoftables
\section{Introduction}
\label{sec:intro}
-\MATITA{} is the Proof Assistant under development by the \HELM{} team
-\cite{mkm-helm} at the University of Bologna, under the direction of
-Prof.~Asperti. \\
-The paper describes the overall architecture of
-the system, focusing on its most distintive and innovative
+
+\MATITA{} is the Proof Assistant under development by the \HELM{}
+team~\cite{mkm-helm} at the University of Bologna, under the direction of
+Prof.~Asperti. The paper describes the overall architecture of
+the system, focusing on its most distinctive and innovative
features.
-\subsection{Historical Perspective}
+\subsection{Historical perspective}
+
The origins of \MATITA{} go back to 1999. At the time we were mostly
interested to develop tools and techniques to enhance the accessibility
via Web of formal libraries of mathematics. Due to its dimension, the
library of the \COQ~\cite{CoqManual} proof assistant (of the order of 35'000 theorems)
-was choosed as a privileged test bench for our work, although experiments
+was chosen as a privileged test bench for our work, although experiments
have been also conducted with other systems, and notably
-with \NUPRL{}\cite{nuprl-book}.
+with \NUPRL~\cite{nuprl-book}.
The work, mostly performed in the framework of the recently concluded
-European project IST-33562 \MOWGLI{}~\cite{pechino}, mainly consisted in the
+European project \MOWGLIIST{} \MOWGLI~\cite{pechino}, mainly consisted in the
following steps:
\begin{itemize}
\item exporting the information from the internal representation of
\COQ{} to a system and platform independent format. Since XML was at the
time an emerging standard, we naturally adopted this technology, fostering
-a content-centric architecture\cite{content-centric} where the documents
+a content-centric architecture~\cite{content-centric} where the documents
of the library were the the main components around which everything else
has to be build;
\item developing indexing and searching techniques supporting semantic
queries to the library;
-%these efforts gave birth to our \WHELP{}
-%search engine, described in~\cite{whelp};
\item developing languages and tools for a high-quality notational
rendering of mathematical information\footnote{We have been
-active in the MathML Working group since 1999.};
-%and developed inside
-%\HELM{} a MathML-compliant widget for the GTK graphical environment
-%which can be integrated in any application.
+active in the \MATHML{} Working group since 1999.};
\end{itemize}
According to our content-centric commitment, the library exported from
means of a Web interface that orchestrates the Web services.
The Web services and the other tools have been implemented as front-ends
-to a set of software libraries, collectively called the \HELM{} libraries.
+to a set of software components, collectively called the \HELM{} components.
At the end of the \MOWGLI{} project we already disposed of the following
-tools and software libraries:
+tools and software components:
\begin{itemize}
\item XML specifications for the Calculus of Inductive Constructions,
-with libraries for parsing and saving mathematical objects in such a format
-\cite{exportation-module};
-\item metadata specifications with libraries for indexing and querying the
+with components for parsing and saving mathematical objects in such a
+format~\cite{exportation-module};
+\item metadata specifications with components for indexing and querying the
XML knowledge base;
\item a proof checker library (i.e. the {\em kernel} of a proof assistant),
-implemented to check that we exported form the \COQ{} library all the
+implemented to check that we exported from the \COQ{} library all the
logically relevant content;
\item a sophisticated parser (used by the search engine), able to deal
with potentially ambiguous and incomplete information, typical of the
-mathematical notation \cite{disambiguation};
+mathematical notation~\cite{disambiguation};
\item a {\em refiner} library, i.e. a type inference system, based on
partially specified terms, used by the disambiguating parser;
\item complex transformation algorithms for proof rendering in natural
-language \cite{remathematization};
-\item an innovative, MathML-compliant rendering widget for the GTK
-graphical environment\cite{padovani}, supporting
+language~\cite{remathematization};
+\item an innovative, \MATHML-compliant rendering widget for the GTK
+graphical environment~\cite{padovani}, supporting
high-quality bidimensional
rendering, and semantic selection, i.e. the possibility to select semantically
-meaningful rendering expressions, and to past the respective content into
+meaningful rendering expressions, and to paste the respective content into
a different text area.
\end{itemize}
Starting from all this, developing our own proof assistant was not
overall management of the library, integrating everything into a
single system. \MATITA{} is the result of this effort.
+\subsection{The system}
+
+\MATITA{} is a proof assistant (also called interactive theorem prover).
+It is based on the Calculus of (Co)Inductive Constructions
+(CIC)~\cite{Werner} that is a dependently typed lambda-calculus \`a la
+Church enriched with primitive inductive and co-inductive data types.
+Via the Curry-Howard isomorphism, the calculus can be seen as a very
+rich higher order logic and proofs can be simply represented and
+stored as lambda-terms. \COQ{} and Lego are other systems that adopt
+(variations of) CIC as their foundation.
+
+The proof language of \MATITA{} is procedural, in the tradition of the LCF
+theorem prover. \COQ, \NUPRL, PVS, Isabelle are all examples of others systems
+whose proof language is procedural. Traditionally, in a procedural system
+the user interacts only with the \emph{script}, while proof terms are internal
+records kept by the system. On the contrary, in \MATITA{} proof terms are
+praised as declarative versions of the proof. With this role, they are the
+primary mean of communication of proofs (once rendered to natural language
+for human audiences).
+
+The user interfaces now adopted by all the proof assistants based on a
+procedural proof language have been inspired by the CtCoq pioneering
+system~\cite{ctcoq1}. One successful incarnation of the ideas introduced
+by CtCoq is the Proof General generic interface~\cite{proofgeneral},
+that has set a sort of
+standard way to interact with the system. Several procedural proof assistants
+have either adopted or cloned Proof General as their main user interface.
+The authoring interface of \MATITA{} is a clone of the Proof General interface.
+
+\TODO{item che seguono:}
+\begin{itemize}
+ \item sistema indipendente (da \COQ)
+ \item compatibilit\`a con sistemi legacy
+\end{itemize}
+
\subsection{Relationship with \COQ{}}
At first sight, \MATITA{} looks as (and partly is) a \COQ{} clone. This is
more the effect of the circumstances of its creation described
above than the result of a deliberate design. In particular, we
(essentially) share the same foundational dialect of \COQ{} (the
-Calculus of (Co)Inductive Constructions), the same implementative
+Calculus of (Co)Inductive Constructions), the same implementation
language (\OCAML{}), and the same (script based) authoring philosophy.
However, the analogy essentially stops here and no code is shared by the
two systems.
-In a sense; we like to think of \MATITA{} as the way \COQ{} would
+In a sense, we like to think of \MATITA{} as the way \COQ{} would
look like if entirely rewritten from scratch: just to give an
idea, although \MATITA{} currently supports almost all functionalities of
\COQ{}, it links 60'000 lines of \OCAML{} code, against the 166'000 lines linked
by \COQ{} (and we are convinced that, starting from scratch again,
-we could furtherly reduce our code in sensible way).
+we could reduce our code even further in sensible way).
Moreover, the complexity of the code of \MATITA{} is greatly reduced with
-respect to \COQ. For instance, the API of the libraries of \MATITA{} comprise
+respect to \COQ. For instance, the API of the components of \MATITA{} comprise
989 functions, to be compared with the 4'286 functions of \COQ.
-Finally, \MATITA{} has several innovatives features over \COQ{} that derive
+Finally, \MATITA{} has several innovative features over \COQ{} that derive
from the integration of Mathematical Knowledge Management tools with proof
assistants. Among them, the advanced indexing tools over the library and
the parser for ambiguous mathematical notation.
The size and complexity improvements over \COQ{} must be understood
historically. \COQ{} is a quite old
-system whose development started 15\NOTE{Verificare} years ago. Since then
+system whose development started 20 years ago. Since then
several developers have took over the code and several new research ideas
that were not considered in the original architecture have been experimented
and integrated in the system. Moreover, there exists a lot of developments
for \COQ{} that require backward compatibility between each pair of releases;
since many central functionalities of a proof assistant are based on heuristics
or arbitrary choices to overcome undecidability (e.g. for higher order
-unification), changing these functionalities mantaining backward compatibility
+unification), changing these functionalities maintaining backward compatibility
is very difficult. Finally, the code of \COQ{} has been greatly optimized
-over the years; optimization reduces maintenability and rises the complexity
+over the years; optimization reduces maintainability and rises the complexity
of the code.
In writing \MATITA{} we have not been hindered by backward compatibility and
we have took advantage of the research results and experiences previously
developed by others, comprising the authors of \COQ. Moreover, starting from
-scratch, we have designed in advance the architecture and we have splitted
-the code in coherent minimally coupled libraries.
+scratch, we have designed in advance the architecture and we have split
+the code in coherent minimally coupled components.
In the future we plan to exploit \MATITA{} as a test bench for new ideas and
-extensions. Keeping the single libraries and the whole architecture as
+extensions. Keeping the single components and the whole architecture as
simple as possible is thus crucial to foster future experiments and to
allow other developers to quickly understand our code and contribute.
%be able to contribute to \COQ{}'s code is quite steep and requires direct
%and frequent interactions with \COQ{} developers.
-\subsection{The system}
-
-DESCRIZIONE DEL SISTEMA DAL PUNTO DI VISTA ``UTENTE''\\
-ROBA CHE MANCA:
-\begin{itemize}
- \item scelta del sistema fondazionale
- \item sistema indipendente (da \COQ)
- \item compatibilit\`a con sistemi legacy
-\end{itemize}
-
-\begin{figure}[t]
- \begin{center}
-% \includegraphics[width=0.9\textwidth]{a.eps}
- \caption{\MATITA{} screenshot}
- \label{fig:screenshot}
- \end{center}
-\end{figure}
-
-\MATITA{} has a script based user interface. As can be seen in Fig.~... it is
-split in two main windows: on the left a textual widget is used to edit the
-script, on the right the list of open goal is shown using a MathML rendering
-widget. A distinguished part of the script (shaded in the screenshot) represent
-the commands already executed and can't be edited without undoing them. The
-remaining part can be freely edited and commands from that part can be executed
-moving down the execution point. An additional window --- the ``cicBrowser'' ---
-can be used to browse the library, including the proof being developed, and
-enable content based search on it. In the cicBrowser proofs are rendered in
-natural language, automatically generated from the low-level $\lambda$-terms
-using techniques inspired by \cite{natural,YANNTHESIS}.
-
-In the \MATITA{} philosophy the script is not relevant \emph{per se}, but is
-only seen as a convenient way to create mathematical objects. The universe of
-all these objects makes up the \HELM{} library, which is always completely
-visible to the user. The mathematical library is thus conceived as a global
-hypertext, where objects may freely reference each other. It is a duty of
-the system to guide the user through the relevant parts of the library.
-
-This methodological assumption has many important consequences
-which will be discussed in the next section.
-
-%on one side
-%it requires functionalities for the overall management of the library,
-%%%%%comprising efficient indexing techniques to retrieve and filter the
-%information;
-%on the other it introduces overloading in the use of
-%identifiers and mathematical notation, requiring sophisticated disambiguation
-%techniques for interpreting the user inputs.
-%In the next two sections we shall separately discuss the two previous
-%points.
-
-%In order to maximize accessibility mathematical objects are encoded in XML. (As%discussed in the introduction,) the modular architecture of \MATITA{} is
-%organized in components which work on data in this format. For instance the
-%rendering engine, which transform $\lambda$-terms encoded as XML document to
-%MathML Presentation documents, can be used apart from \MATITA{} to print ...
-%FINIRE
-
-A final section is devoted to some innovative aspects
-of the authoring system, such as a step by step tactical execution,
-content selection and copy-paste.
-
\section{Architecture}
\label{architettura}
-\begin{figure}[ht]
+\begin{figure}[!ht]
\begin{center}
- \includegraphics[width=0.9\textwidth]{librariesCluster.ps}
- \caption{\MATITA{} libraries}
+ \includegraphics[width=0.9\textwidth,height=0.8\textheight]{pics/libraries-clusters}
+ \caption[\MATITA{} components and related applications]{\MATITA{}
+ components and related applications, with thousands of line of
+ codes (klocs)\strut}
\label{fig:libraries}
\end{center}
\end{figure}
Fig.~\ref{fig:libraries} shows the architecture of the \emph{\components}
(circle nodes) and \emph{applications} (squared nodes) developed in the HELM
-project.
+project. Each node is annotated with the number of lines of source code
+(comprising comments).
Applications and \components{} depend over other \components{} forming a
directed acyclic graph (DAG). Each \component{} can be decomposed in
Modules and \components{} provide coherent sets of functionalities
at different scales. Applications that require only a few functionalities
-depend on a restricted set of \components{}.
+depend on a restricted set of \components.
Only the proof assistant \MATITA{} and the \WHELP{} search engine are
applications 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
+ \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{zack-master}.
- \item \emph{Whelp} is a search engine to index and locate mathematical
+ \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
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
+ the query is only approximated 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
+ \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
+ 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{zack-master}.
+ The Getter is used by \UWOBO{} to retrieve the document to be rendered.
+ \UWOBO{} has been described in~\cite{zack-master}.
\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
dependencies, checking in turn every notion for correctness.
The proof checker has been described in~\cite{zack-master}.
\item The \emph{Dependency Analyzer} is a Web service that can produce
- a textual or graphical representation of the dependecies of an object.
+ a textual or graphical representation of the dependencies of an object.
The dependency analyzer has been described in~\cite{zack-master}.
\end{itemize}
For those \components{} 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 \GETTER \component{} that allows the
+is just a wrapper to the \GETTER{} \component{} that allows the
\component{} to be used as a Web service. \MATITA{} can directly link the code
-of the \GETTER \component, or it can use a stub library with the same
+of the \GETTER{} \component, 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
content level terms; presentation level terms.
\subsection{Fully specified terms}
-\label{fully-spec}
+\label{sec:fullyintro}
+
\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
Terms may reference other mathematical notions in the library.
One commitment of our project is that the library should be physically
- distributed. The \GETTER \component{} manages the distribution,
+ distributed. The \GETTER{} \component{} manages the distribution,
providing a mapping from logical names (URIs) to the physical location
of a notion (an URL). The \texttt{urimanager} \component{} provides the URI
data type and several utility functions over URIs. The
- \texttt{cic\_proof\_checking} \component{} calls the \GETTER
+ \texttt{cic\_proof\_checking} \component{} calls the \GETTER{}
\component{} every time it needs to retrieve the definition of a mathematical
notion referenced by a term that is being type-checked.
in Sect.~\ref{sec:libmanagement}.
\subsection{Partially specified terms}
+\label{sec:partiallyintro}
+
\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
The \texttt{grafite} \component{} defines the abstract syntax tree (AST) for the
commands of the \MATITA{} proof assistant. Most of the commands are tactics.
Other commands are used to give definitions and axioms or to state theorems
-and lemmas. The \texttt{grafite\_engine} \component{} is the core of \MATITA{}.
+and lemmas. The \texttt{grafite\_engine} \component{} is the core of \MATITA.
It implements the semantics of each command in the grafite AST as a function
from status to status. It implements also an undo function to go back to
previous statuses.
\subsection{Content level terms}
\label{sec:contentintro}
-The language used to communicate proofs and expecially formulae with the
-user does not only needs to be extendible and accomodate the usual mathematical
+The language used to communicate proofs and especially formulae with the
+user does not only needs to be extendible and accommodate the usual mathematical
notation. It must also reflect the comfortable degree of imprecision and
ambiguity that the mathematical language provides.
The translation from partially specified terms to content level terms must
discriminate between terms used to represent proofs and terms used to represent
formulae. The firsts are translated to a content level representation of
-proof steps that can easily be rendered in natural language. The representation
+proof steps that can in turn easily be rendered in natural language
+using techniques inspired by~\cite{natural,YANNTHESIS}. The representation
adopted has greatly influenced the OMDoc~\cite{omdoc} proof format that is now
-isomorphic to it. Terms that represent formulae are translated to MathML
-Content formulae. MathML Content~\cite{mathml} is a W3C standard
+isomorphic to it. Terms that represent formulae are translated to \MATHML{}
+Content formulae. \MATHML{} Content~\cite{mathml} is a W3C standard
for the representation of content level formulae in an XML extensible format.
The translation to content level is implemented in the
every ambiguous formula one partially specified term. The
\texttt{cic\_disambiguation} \component{} implements the
disambiguation algorithm we presented in~\cite{disambiguation} that is
-responsible of building in an efficicent way the set of all ``correct''
+responsible of building in an efficient way the set of all ``correct''
interpretations. An interpretation is correct if the partially specified term
obtained using the interpretation is refinable.
-In the last section we have described the semantics of a command as a
+In Sect.~\ref{sec:partiallyintro} the last section we described the semantics of
+a command as a
function from status to status. We also suggested that the formulae in a
command are encoded as partially specified terms. However, consider the
command ``\texttt{replace} $x$ \texttt{with} $y^2$''. Until the occurrence
context of the occurrence $x$ it must replace.
The elegant solution we have implemented consists in representing terms
-in a command as function from a context to a partially refined term. The
+in a command as functions from a context to a partially refined term. The
function is obtained by partially applying our disambiguation function to
the content term to be disambiguated. Our solution should be compared with
-the one adopted in the Coq system (where ambiguity is only relative to
-DeBrujin indexes). In Coq variables can be bound either by name or by
-position. This makes more complex every operation over terms (i.e. according
-to our architecture every module that depends on \texttt{cic}). Moreover,
-this solution cannot cope with other forms of ambiguity (as the meaning
-of the $~^2$ exponent in the previous example that depends on the context).
+the one adopted in the \COQ{} system, where ambiguity is only relative to
+De Brujin indexes.
+In \COQ{} variables can be bound either by name or by position. A term
+occurring in a command has all its variables bound by name to avoid the need of
+a context during disambiguation. Moreover, this makes more complex every
+operation over terms (i.e. according to our architecture every module that
+depends on \texttt{cic}) since the code must deal consistently with both kinds
+of binding. Also, this solution cannot cope with other forms of ambiguity (as
+the context dependent meaning of the exponent in the previous example).
\subsection{Presentation level terms}
+\label{sec:presentationintro}
Content level terms are a sort of abstract syntax trees for mathematical
formulae and proofs. The concrete syntax given to these abstract trees
single day, but they stick to a set of symbols that is more or less fixed.
The fact that the presentation language is finite allows the definition of
-standard languages. In particular, for formulae we have adopt MathML
+standard languages. In particular, for formulae we have adopt \MATHML{}
Presentation~\cite{mathml} that is an XML dialect standardized by the W3C. To
visually
represent proofs it is enough to embed formulae in plain text enriched with
formatting boxes. Since the language of formatting boxes is very simple,
many equivalent specifications exist and we have adopted our own, called
-BoxML.
+\BOXML.
The \texttt{content\_pres} \component{} contains the implementation of the
translation from content level terms to presentation level terms. The
rendering of presentation level terms is left to the application that uses
the \component. However, in the \texttt{hgdome} \component{} we provide a few
-utility functions to build a \GDOME~\cite{gdome2} MathML+BoxML tree from our
+utility functions to build a \GDOME~\cite{gdome2} \MATHML+\BOXML{} tree from our
presentation
-level terms. \GDOME{} MathML+BoxML trees can be rendered by the GtkMathView
-widget developed by Luca Padovani \cite{padovani}. The widget is
+level terms. \GDOME{} \MATHML+\BOXML{} trees can be rendered by the
+\GTKMATHVIEW{}
+widget developed by Luca Padovani~\cite{padovani}. The widget is
particularly interesting since it allows to implement \emph{semantic
selection}.
against the \texttt{grafite\_parser} \components{}
since they provide an interface to the user. In both cases the formulae
written by the user are parsed using the \texttt{content\_pres} \component{} and
-then disambiguated using the \texttt{cic\_disambiguation} \component.
-However, only \MATITA{} is linked against the \texttt{grafite\_engine} and
-\texttt{tactics} components since \WHELP{} can only execute those ASTs that
-correspond to queries (implemented in the \texttt{whelp} component).
+then disambiguated using the \texttt{cic\_disambiguation} \component. However,
+only \MATITA{} is linked against the \texttt{grafite\_engine} and
+\texttt{tactics} components (summing up to a total of 11'200 lines of code)
+since \WHELP{} can only execute those ASTs that correspond to queries
+(implemented in the \texttt{whelp} component).
The \UWOBO{} Web service wraps the \texttt{content\_pres} \component,
providing a rendering service for the documents in the distributed library.
\GETTER{} obtaining a document with fully specified terms. Then it translates
it to the presentation level passing through the content level. Finally
it returns the result document to be rendered by the user's
-browser.\footnote{\TODO{manca la passata verso HTML}}
-
+browser.\NOTE{\TODO{manca la passata verso HTML}}
The \components{} not yet described (\texttt{extlib}, \texttt{xml},
\texttt{logger}, \texttt{registry} and \texttt{utf8\_macros}) are
%In particular, the \texttt{xml} \component{} is used to easily represent,
%parse and pretty-print XML files.
-\section{Library Management}
+\section{The interface to the library}
+\label{sec:library}
+
+A proof assistant provides both an interface to interact with its library and
+an \emph{authoring} interface to develop new proofs and theories. According
+to its historical origins, \MATITA{} strives to provide innovative
+functionalities for the interaction with the library. It is more traditional
+in its script based authoring interface.
+
+In the remaining part of the paper we focus on the user view of \MATITA.
+This section is devoted to the aspects of the tool that arise from the
+document centric approach to the library. Sect.~\ref{sec:authoring} describes
+the peculiarities of the authoring interface.
+
+The library of \MATITA{} comprises mathematical concepts (theorems,
+axioms, definitions) and notation. The concepts are authored sequentially
+using scripts that are (ordered) sequences of procedural commands.
+However, once they are produced we store them independently in the library.
+The only relation implicitly kept between the notions are the logical,
+acyclic dependencies among them. This way the library forms a global (and
+distributed) hypertext.
+
+\begin{figure}[!ht]
+ \begin{center}
+ \includegraphics[width=0.40\textwidth]{pics/cicbrowser-screenshot-browsing}
+ \hspace{0.05\textwidth}
+ \includegraphics[width=0.40\textwidth]{pics/cicbrowser-screenshot-query}
+ \caption{Browsing and searching the library\strut}
+ \label{fig:cicbrowser1}
+ \end{center}
+\end{figure}
+
+\begin{figure}[!ht]
+ \begin{center}
+ \includegraphics[width=0.70\textwidth]{pics/cicbrowser-screenshot-con}
+ \caption[Natural language rendering]{Natural language rendering of a theorem
+ from the library\strut}
+ \label{fig:cicbrowser2}
+ \end{center}
+\end{figure}
+
+Several useful operations can be implemented on the library only,
+regardless of the scripts. For instance, searching and browsing is
+implemented by the ``cicBrowser'' window available from the \MATITA{}
+GUI. Using it, the hierarchical structure of the library can be
+explored (on the left of Fig.~\ref{fig:cicbrowser1}), the natural
+language rendering of proofs can be inspected
+(Fig.~\ref{fig:cicbrowser2}), and content based searches on the
+library can be performed (on the right of Fig.~\ref{fig:cicbrowser1}).
+Available content based searches are described in
+Sect.~\ref{sec:indexing}. Other examples of library operations are
+disambiguation of content level terms (see
+Sect.~\ref{sec:disambiguation}) and automatic proof searching (see
+Sect.~\ref{sec:automation}).
+
+The key requisite for the previous operations is that the library must
+be fully accessible and in a logically consistent state. To preserve
+consistency, a concept cannot be altered or removed unless the part of the
+library that depends on it is modified accordingly. To allow incremental
+changes and cooperative development, consistent revisions are necessary.
+For instance, to modify a definition, the user could fork a new version
+of the library where the definition is updated and all the concepts that
+used to rely on it are absent. The user is then responsible to restore
+the removed part in the new branch, merging the branch when the library is
+fully restored.
+
+To implement the proposed versioning system on top of a standard one
+it is necessary to implement \emph{invalidation} first. Invalidation
+is the operation that locates and removes from the library all the concepts
+that depend on a given one. As described in Sect.~\ref{sec:libmanagement} removing
+a concept from the library also involves deleting its metadata from the
+database.
+
+For non collaborative development, full versioning can be avoided, but
+invalidation is still required. Since nobody else is relying on the
+user development, the user is free to change and invalidate part of the library
+without branching. Invalidation is still necessary to avoid using a
+concept that is no longer valid.
+So far, in \MATITA{} we address only this non collaborative scenario
+(see Sect.~\ref{sec:libmanagement}). Collaborative development and versioning
+is still under design.
+
+Scripts are not seen as constituents of the library. They are not published
+and indexed, so they cannot be searched or browsed using \HELM{} tools.
+However, they play a central role for the maintenance of the library.
+Indeed, once a notion is invalidated, the only way to restore it is to
+fix the possibly broken script that used to generate it.
+Moreover, during the authoring phase, scripts are a natural way to
+group notions together. They also constitute a less fine grained clustering
+of notions for invalidation.
+
+In the rest of this section we present in more details the functionalities of
+\MATITA{} related to library management and exploitation.
+Sect.~\ref{sec:authoring} is devoted to the description of the peculiarities of
+the \MATITA{} authoring interface.
\subsection{Indexing and searching}
+\label{sec:indexing}
+
+The \MATITA{} system is first of all an interface between the user and
+the mathematical library. For this reason, it is important to be
+able to search and retrieve mathematical concepts in a quick and
+effective way, assuming as little knowledge as possible about the
+library. To this aim, \MATITA{} uses a sophisticated indexing mechanism
+for mathematical concepts, based on a rich metadata set that has been
+tuned along the European project \MOWGLIIST{} \MOWGLI. The metadata
+set, and the searching facilites built on top of them --- collected
+in the so called \WHELP{} search engine --- have been
+extensively described in~\cite{whelp}. Let us just recall here that
+the \WHELP{} metadata model is essentially based a single ternary relation
+\REF{p}{s}{t} stating that an object $s$ refers an object $t$ at a
+ given position $p$, where the position specify the place of the
+occurrence of $t$ inside $s$ (we currently work with a fixed set of
+positions, discriminating the hypothesis from the conclusion and
+outermost form innermost occurrences). This approach is extremely
+flexible, since extending the set of positions
+we may improve the granularity and the precision of our indexing technique,
+with no additional architectural impact.
+
+Every time a new mathematical concept is created and saved by the user it gets
+indexed, and becomes immediately visible in the library. Several
+interesting and innovative features of \MATITA{} described in the following
+sections rely in a direct or indirect way on its metadata system and
+the search features. Here, we shall just recall some of its most
+direct applications.
+
+A first, very simple but not negligeable feature is the check for duplicates.
+As soon as a theorem is stated, just before starting its proof,
+the library is searched
+to check that no other equivalent statement has been already proved
+(based on the pattern matching functionality of \WHELP); if this is the case,
+a warning is raised to the user. At present, the notion of equivalence
+adopted by \MATITA{} is convertibility, but we may imagine to weaken it
+in the future, covering for instance isomorphisms.
+
+Another useful \WHELP{} operation is \HINT; we may invoke this query
+at any moment during the authoring of a proof, resulting in the list
+of all theorems of the library which can be applied to the current
+goal. In practice, this is mostly used not really to discover what theorems
+can be applied to a given goal, but to actually retrieve a theorem that
+we wish to apply, but whose name we have forgotten.
+In fact, even if \MATITA{} adopts a semi-rigid naming convention for
+statements (see Sect.~\ref{sec:naming}) that greatly simplifies the effort
+of recalling names, the naming discipline remains one of the most
+annoying aspects of formal developments, and \HINT{} provides
+a very friendly solution.
+In the near feature, we expect to extend the \HINT{} operation to
+a \REWRITEHINT, resulting in all equational statements that
+can be applied to rewrite the current goal.
+
+\subsection{Disambiguation}
+\label{sec:disambiguation}
+
+Software applications that involve input of mathematical content should strive
+to require the user as less drift from informal mathematics as possible. We
+believe this to be a fundamental aspect of such applications user interfaces.
+Being that drift in general very large when inputing
+proofs~\cite{debrujinfactor}, in \MATITA{} we achieved good results for
+mathematical formulae which can be input using a \TeX-like encoding (the
+concrete syntax corresponding to presentation level terms) and are then
+translated (in multiple steps) to partially specified terms as sketched in
+Sect.~\ref{sec:contentintro}.
+
+The key component of the translation is the generic disambiguation algorithm
+implemented in the \texttt{disambiguation} component of Fig.~\ref{fig:libraries}
+and presented in~\cite{disambiguation}. In this section we present how to use
+that algorithm in the context of the development of a library of formalized
+mathematics. We will see that using multiple passes of the algorithm, varying
+some of its parameters, helps in keeping the input terse without sacrificing
+expressiveness.
+
+\subsubsection{Disambiguation aliases}
+\label{sec:disambaliases}
+
+Consider the following command to state a theorem over integer numbers:
+
+\begin{grafite}
+theorem Zlt_compat:
+ \forall x, y, z. x < y \to y < z \to x < z.
+\end{grafite}
+
+The symbol \OP{<} is likely to be overloaded in the library
+(at least over natural numbers).
+Thus, according to the disambiguation algorithm, two different
+refinable partially specified terms could be associated to it.
+\MATITA{} asks the user what interpretation he meant. However, to avoid
+posing the same question in case of a future re-execution (e.g. undo/redo),
+the choice must be recorded. Since scripts need to be re-executed after
+invalidation, the choice record must be permanently stored somewhere. The most
+natural place is in the script itself.
+
+In \MATITA{} disambiguation is governed by \emph{disambiguation aliases}.
+They are mappings, stored in the library, from ambiguity sources
+(identifiers, symbols and literal numbers at the content level) to partially
+specified terms. In case of overloaded sources there exists multiple aliases
+with the same source. It is possible to record \emph{disambiguation
+preferences} to select one of the aliases of an overloaded source.
+
+Preferences can be explicitely given in the script (using the
+misleading \texttt{alias} commands), but
+are also implicitly added when a new concept is introduced (\emph{implicit
+preferences}) or after a sucessfull disambiguation that did not require
+user interaction. Explicit preferences are added automatically by \MATITA{} to
+record the disambiguation choices of the user. For instance, after the
+disambiguation of the command above, the script is altered as follows:
+
+\begin{grafite}
+alias symbol "lt" = "integer 'less than'".
+theorem Zlt_compat:
+ \forall x, y, z. x < y \to y < z \to x < z.
+\end{grafite}
+The ``alias'' command in the example sets the preferred alias for the
+\OP{lt} symbol.
+
+Implicit preferences for new concepts are set since a concept just defined is
+likely to be the preferred one in the rest of the script. Implicit preferences
+learned from disambiguation of previous commands grant the coherence of
+the disambiguation in the rest of the script and speed up disambiguation
+reducing the search space.
+
+Disambiguation preferences are included in the lexicon status
+(see Sect.~\ref{sec:presentationintro}) that is part of the authoring interface
+status. Unlike aliases, they are not part of the library.
+
+When starting a new authoring session the set of disambiguation preferences
+is empty. Until it contains a preference for each overloaded symbol to be
+used in the script, the user can be faced with questions from the disambiguator.
+To reduce the likelyhood of user interactions, we introduced
+the \texttt{include} command. With \texttt{include} it is possible to import
+at once in the current session the set of preferences that was in effect
+at the end of the execution of a given script.
+
+Preferences can be changed. For instance, at the start of the development
+of integer numbers the preference for the symbol \OP{<} is likely
+to be the one over natural numbers; sooner or later it will be set to the one
+over integer numbers.
+
+Nothing forbids the set of preferences to become incoherent. For this reason
+the disambiguator cannot always respect the user preferences.
+Consider, for example:
+\begin{grafite}
+theorem Zlt_mono:
+ \forall x, y, k. x < y \to x < y + k.
+\end{grafite}
-\subsection{Compilation and cleaning}
+No refinable partially specified term corresponds to the preferences:
+\OP{+} over natural numbers, \OP{<} over integer numbers. To overcome this
+limitation we organized disambiguation in \emph{multiple passes}: when the
+disambiguator fails, disambiguation is tried again with a less strict set of
+preferences.
+
+Several disambiguation parameters can vary among passes. With respect to
+preference handling we implemented three passes. In the first pass, called
+\emph{mono-preferences}, we consider only the aliases corresponding to the
+current preferences. In the second pass, called \emph{multi-preferences}, we
+consider every alias corresponding to a current or past preference. For
+instance, in the example above disambiguation succeeds in the multi-preference
+pass. In the third pass, called \emph{library-preferences}, all aliases
+available in the library are considered.
+
+The rationale behind this choice is trying to respect user preferences in early
+passes that complete quickly in case of failure; later passes are slower but
+have more chances of success.
+
+\subsubsection{Operator instances}
+\label{sec:disambinstances}
+
+Consider now the following theorem:
+\begin{grafite}
+theorem lt_to_Zlt_pos_pos:
+ \forall n, m: nat. n < m \to pos n < pos m.
+\end{grafite}
+and assume that there exist in the library aliases for \OP{<} over natural
+numbers and over integer numbers. None of the passes described above is able to
+disambiguate \texttt{lt\_to\_Zlt\_pos\_pos}, no matter how preferences are set.
+This is because the \OP{<} operator occurs twice in the content level term (it
+has two \emph{instances}) and two different interpretations for it have to be
+used in order to obtain a refinable partially specified term.
+
+To address this issue, we have the ability to consider each instance of a single
+symbol as a different ambiguous expression in the content level term, and thus
+we can use a different alias for each of them. Exploiting or not this feature is
+one of the disambiguation pass parameters. A disambiguation pass which exploit
+it is said to be using \emph{fresh instances} (opposed to a \emph{shared
+instances} pass).
+
+Fresh instances lead to a non negligible performance loss (since the choice of
+an alias for one instance does not constraint the choice of the others). For
+this reason we always attempt a fresh instances pass only after attempting a
+non-fresh one.
+
+\paragraph{One-shot preferences} Disambiguation preferecens as seen so far are
+instance-independent. However, implicit preferences obtained as a result of a
+disambiguation pass which uses fresh instances ought to be instance-dependent.
+Informally, the set of preferences that can be respected by the disambiguator on
+the theorem above is: ``the first instance of the \OP{<} symbol is over natural
+numbers, while the second is on integer numbers''.
+
+Instance-dependent preferences are meaningful only for the term whose
+disambiguation generated it. For this reason we call them \emph{one-shot
+preferences} and \MATITA{} does not use them to disambiguate further terms in
+the script.
+
+\subsubsection{Implicit coercions}
+\label{sec:disambcoercions}
+
+Consider the following theorem about derivation:
+\begin{grafite}
+theorem power_deriv:
+ \forall n: nat, x: R. d x ^ n dx = n * x ^ (n - 1).
+\end{grafite}
+and assume that in the library there is an alias mapping \OP{\^} to a partially
+specified term having type: \texttt{R \TEXMACRO{to} nat \TEXMACRO{to} R}. In
+order to disambiguate \texttt{power\_deriv}, the occurrence of \texttt{n} on the
+right hand side of the equality need to be ``injected'' from \texttt{nat} to
+\texttt{R}. The refiner of \MATITA{} supports \emph{implicit coercions} for
+this reason: given as input the above presentation level term, it will return a
+partially specified term where in place of \texttt{n} the application of a
+coercion from \texttt{nat} to \texttt{R} appears (assuming such a coercion has
+been defined in advance).
+
+Coercions are not always desirable. For example, in disambiguating
+\texttt{\TEXMACRO{forall} x: nat. n < n + 1} we do not want the term which uses
+two coercions from \texttt{nat} to \texttt{R} around \OP{<} arguments to show up
+among the possible partially specified term choices. For this reason we always
+attempt a disambiguation pass which require the refiner not to use the coercions
+before attempting a coercion-enabled pass.
+
+The choice of whether implicit coercions are enabled or not interact with the
+choice about operator instances. Indeed, consider again
+\texttt{lt\_to\_Zlt\_pos\_pos}, which can be disambiguated using fresh operator
+instances. In case there exists a coercion from natural numbers to (positive)
+integers (which indeed does), the
+theorem can be disambiguated using twice that coercion on the left hand side of
+the implication. The obtained partially specified term however would not
+probably be the expected one, being a theorem which prove a trivial implication.
+Motivated by this and similar examples we choose to always prefer fresh
+instances over implicit coercions, i.e. we always attempt disambiguation
+passes with fresh instances
+and no implicit coercions before attempting passes with implicit coercions.
+
+\subsubsection{Disambiguation passes}
+\label{sec:disambpasses}
+
+According to the criteria described above, in \MATITA{} we perform the
+disambiguation passes depicted in Tab.~\ref{tab:disambpasses}. In
+our experience that choice gives reasonable performance and minimize the need of
+user interaction during the disambiguation.
+
+\begin{table}[ht]
+ \caption{Disambiguation passes sequence\strut}
+ \label{tab:disambpasses}
+ \begin{center}
+ \begin{tabular}{c|c|c|c}
+ \multicolumn{1}{p{1.5cm}|}{\centering\raisebox{-1.5ex}{\textbf{Pass}}}
+ & \multicolumn{1}{p{3.1cm}|}{\centering\textbf{Preferences}}
+ & \multicolumn{1}{p{2.5cm}|}{\centering\textbf{Operator instances}}
+ & \multicolumn{1}{p{2.5cm}}{\centering\textbf{Implicit coercions}} \\
+ \hline
+ \PASS & Mono-preferences & Shared instances & Disabled \\
+ \PASS & Multi-preferences & Shared instances & Disabled \\
+ \PASS & Mono-preferences & Fresh instances & Disabled \\
+ \PASS & Multi-preferences & Fresh instances & Disabled \\
+ \PASS & Mono-preferences & Fresh instances & Enabled \\
+ \PASS & Multi-preferences & Fresh instances & Enabled \\
+ \PASS & Library-preferences & Fresh instances & Enabled
+ \end{tabular}
+ \end{center}
+\end{table}
+
+\subsection{Generation and invalidation}
\label{sec:libmanagement}
+%The aim of this section is to describe the way \MATITA{}
+%preserves the consistency and the availability of the library
+%using the \WHELP{} technology, in response to the user alteration or
+%removal of mathematical objects.
+%
+%As already sketched in Sect.~\ref{sec:fullyintro} what we generate
+%from a script is split among two storage media, a
+%classical filesystem and a relational database. The former is used to
+%store the XML encoding of the objects defined in the script, the
+%disambiguation aliases and the interpretation and notational convention defined,
+%while the latter is used to store all the metadata needed by
+%\WHELP.
%
-%goals: consentire sviluppo di una librearia mantenendo integrita' referenziale e usando le teconologie nostre (quindi con metadati, XML, libreria visibile)
-%\subsubsection{Composition}
-%scripts.ma, .moo, XML, metadata
-%\subsubsection{Compilation}
-%analogie con compilazione classica dso.\\
-%granularita' differenti per uso interattivo e non
-%\paragraph{Batch}
-%- granularita' .ma/buri \\
-%-- motivazioni\\
-%- come si calcolano le dipendenze\\
-%- quando la si usa\\
-%- metodi (cc e clean)\\
-%- garanzie
-%\paragraph{Interactive}
-%- granularita' fine\\
-%-- motivazioni
-%\label{sec:libmanagement}
-%consistenza: integrita' referenziale
-%Goals: mantenere consistente la rappresentazione della libreria su
-%memoria persistente consentendo di compilare e pulire le compilation
-%unit (.ma).\\
-%Vincoli: dipendenze oggetti-oggetti e metadati-oggetti\\
-%Due livelli di gestione libreria, uno e' solo in fase interattiva dove la compilazione e' passo passo: \\
-%--- granularita' oggetto per matita interactive\\
-%--- granularita' baseuri (compilation unit) per la libreria\\
-%In entrmbi i casi ora:\\
-%--- matitaSync: add, remove, timetravel(facility-macro tra 2 stati)[obj]\\
-%--- matitaCleanLib: clean\_baseuri (che poi usa matitaSync a sua volta)[comp1]\\
-%Vincoli di add: typecheck ( ==$>$ tutto quello che usa sta in lib)\\
-%Vincoli di remove: \\
-%--- la remove di mSync non li controlla (ma sa cosa cancellare per ogni uri)\\
-%--- la clean\_baseuri calcola le dipendenze con i metadati (o anche i moo direi) e li rispetta\\
-%Undo di matita garantisce la consistenza a patto che l'history che tiene sia ok\\
-%Undo della lib (mClean) garantisce la consistenza (usando moo o Db).\\
-
-The aim of this section is to describe the way \MATITA{}
-preserves the consistency and the availability of the library
-using the \WHELP{} technology, in response to the user addition or
-removal of mathematical objects.
-
-As already sketched in \ref{fully-spec} the output of the
-compilation of a script is split among two storage media, a
-classical filesystem and a relational database. The former is used to
-store the XML encoding of the objects defined in the script, the
-disambiguation aliases and the interpretation and notational convention defined,
-while the latter is used to store all the metadata needed by
-\WHELP{}.
-
-While the consistency of the data store in the two media has
-nothing to do with the nature of
-the content of the library and is thus uninteresting (but really
-tedious to implement and keep bug-free), there is a deeper
-notion of mathematical consistency we need to provide. Each object
-must reference only defined object (i.e. each proof must use only
-already proved theorems).
-
-We will focus on how \MATITA{} ensures the interesting kind
-of consistency during the formalization of a mathematical theory,
+%While the consistency of the data store in the two media has
+%nothing to do with the nature of
+%the content of the library and is thus uninteresting (but really
+%tedious to implement and keep bug-free), there is a deeper
+%notion of mathematical consistency we need to provide. Each object
+%must reference only defined object (i.e. each proof must use only
+%already proved theorems).
+
+In this section we will focus on how \MATITA{} ensures the library
+consistency during the formalization of a mathematical theory,
giving the user the freedom of adding, removing, modifying objects
without loosing the feeling of an always visible and browsable
library.
-\subsubsection{Compilation}
+\subsubsection{Invalidation}
+
+Invalidation (see Sect.~\ref{sec:library}) is implemented in two phases.
+
+The first one is the calculation of all the concepts that recursively
+depend on the ones we are invalidating. The calculation of the
+reverse dependencies can be computed using the relational database
+that stores metadata.
+This technique is the same used by the \emph{Dependency Analyzer}
+and is described in~\cite{zack-master}.
+
+The second phase is the removal of all the results of the generation,
+metadata included.
+
+\subsubsection{Regeneration}
-The typechecker component guarantees that if an object is well typed
-it depends only on well typed objects available in the library,
-that is exactly what we need to be sure that the logic consistency of
-the library is preserved. We have only to find the right order of
-compilation of the scripts that compose the user development.
+%The typechecker component guarantees that if an object is well typed
+%it depends only on well typed objects available in the library,
+%that is exactly what we need to be sure that the logic consistency of
+%the library is preserved.
+
+To regenerate an invalidated part of the library \MATITA{} re-executes
+the script files that produced the invalidated concepts. The main
+problem is to find a suitable order of execution of the scripts.
For this purpose we provide a tool called \MATITADEP{}
-that takes in input the list of files that compose the development and
+that takes in input the list of scripts that compose the development and
outputs their dependencies in a format suitable for the GNU \texttt{make} tool.
The user is not asked to run \MATITADEP{} by hand, but
simply to tell \MATITA{} the root directory of his development (where all
-script files can be found) and \MATITA{} will handle all the compilation
+script files can be found) and \MATITA{} will handle all the generation
related tasks, including dependencies calculation.
-To compute dependencies it is enough to look at the script files for
-inclusions of other parts of the development or for explicit
-references to other objects (i.e. with explicit aliases, see
-\ref{sec:disambaliases}).
-
-The output of the compilation is immediately available to the user
-trough the \WHELP{} technology, since all metadata are stored in a
-user-specific area of the database where the search engine has read
-access, and all the automated tactics that operates on the whole
-library, like \AUTO, have full visibility of the newly defined objects.
-
-Compilation is rather simple, and the only tricky case is when we want
-to compile again the same script, maybe after the removal of a
-theorem. Here the policy is simple: clean the output before recompiling.
-As we will see in the next section cleaning will ensure that
-there will be no theorems in the development that depends on the
-removed items.
-
-\subsubsection{Cleaning}
-
-With the term ``cleaning'' we mean the process of removing all the
-results of an object compilation. In order to keep the consistency of
-the library, cleaning an object requires the (recursive) cleaning
-of all the objects that depend on it (\emph{reverse dependencies}).
-
-The calculation of the reverse dependencies can be computed in two
-ways, using the relational database or using a simpler set of metadata
-that \MATITA{} saves in the filesystem as a result of compilation. The
-former technique is the same used by the \emph{Dependency Analyzer}
-described in \cite{zack-master} and really depends on a relational
-database.
-The latter is a fall-back in case the database is not
-available.\footnote{Due to the complex deployment of a large piece of
-software like a database, it is a common practice for the \HELM{} team
-to use a shared remote database, that may be unavailable if the user
-workstation lacks network connectivity.} This facility has to be
-intended only as a fall-back, since the queries of the \WHELP{}
-technology depend require a working database.
+To compute dependencies it is enough to look at the script files for
+disambiguation preferences declared or imported from other scripts
+(see \ref{sec:disambaliases}).
-Cleaning guarantees that if an object is removed there are no dandling
-references to it, and that the part of the library still compiled is
-consistent. Since cleaning involves the removal of all the results of
-the compilation, metadata included, the library browsable trough the
-\WHELP{} technology is always kept up to date.
+Regenerating the content of a modified script file involves the preliminary
+invalidation of all its old content.
\subsubsection{Batch vs Interactive}
-\MATITA{} includes an interactive graphical interface and a batch
-compiler (\MATITAC). Only the former is intended to be used directly by the
-user, the latter is automatically invoked when a
-part of the user development is required (for example issuing an
-\texttt{include} command) but not yet compiled.
-
-While they share the same engine for compilation and cleaning, they
-provide different granularity. The batch compiler is only able to
-compile a whole script and similarly to clean only a whole script
-(together with all the other scripts that rely on an object defined in
-it). The interactive interface is able to execute single steps of
-compilation, that may include the definition of an object, and
-similarly to undo single steps. Note that in the latter case there is
-no risk of introducing dangling references since the \MATITA{} user
-interface inhibit undoing a step which is not the last executed.
+\MATITA{} includes an interactive authoring interface and a batch
+``compiler'' (\MATITAC).
+
+Only the former is intended to be used directly by the
+user, the latter is automatically invoked by \MATITA{}
+to try to regenerate parts of the library previously invalidated.
+
+While they share the same engine for generation and invalidation, they
+provide different granularity. \MATITAC{} is only able to reexecute a
+whole script and similarly to invalidate the whole content of a script
+(together with all the other scripts that rely on an concept defined
+in it).
\subsection{Automation}
+\label{sec:automation}
+
+\TODO{sezione sull'automazione}
\subsection{Naming convention}
+\label{sec:naming}
+
A minor but not entirely negligible aspect of \MATITA{} is that of
adopting a (semi)-rigid naming convention for identifiers, derived by
our studies about metadata for statements.
\item all identifiers should (but this is not strictly compulsory)
separated by an underscore,
\item identifiers in two different hypothesis, or in an hypothesis
-and in the conlcusion must be separated by the string ``\verb+_to_+'';
+and in the conclusion must be separated by the string ``\verb+_to_+'';
\item the identifier may be followed by a numerical suffix, or a
-single or duoble apostrophe.
+single or double apostrophe.
\end{itemize}
Take for instance the theorem
\[ \forall A:Type. symmetric \;A\;(eq \; A)\]
and \verb+symmetric_eq+ is valid \MATITA{} name for such a theorem.
So, somehow unexpectedly, the introduction of semi-rigid naming convention
-has an important benefical effect on the global organization of the library,
+has an important beneficial effect on the global organization of the library,
forcing the user to define abstract notions and properties before
using them (and formalizing such use).
expression and the suffix \verb+_to_Prop+. In the above example,
\verb+eqb_to_Prop+ is accepted.
-\section{User interface}
-
-\subsection{Disambiguation}
-\label{sec:disambiguation}
-
-Software applications that involve input of mathematical content should strive
-to require the user as less drift from informal mathematics as possible. We
-believe this to be a fundamental aspect of such applications user interfaces.
-Being that drift in general very large when inputing
-proofs~\cite{debrujinfactor}, in \MATITA{} we achieved good results for
-mathematical formulae which can be input using a \TeX-like encoding (the
-concrete syntax corresponding to presentation level terms) and are then
-translated (in multiple steps) to partially specified terms as sketched in
-Sect.~\ref{sec:contentintro}.
-
-The key component of the translation is the generic disambiguation algorithm
-implemented in the \texttt{disambiguation} component of Fig.~\ref{fig:libraries}
-and presented in~\cite{disambiguation}. In this section we present how to use
-such an algorithm in the context of the development of a library of formalized
-mathematics. We will see that using multiple passes of the algorithm, varying
-some of its parameters, helps in keeping the input terse without sacrificing
-expressiveness.
-
-\subsubsection{Disambiguation aliases}
-\label{sec:disambaliases}
-Let's start with the definition of the ``strictly greater then'' notion over
-(Peano) natural numbers.
-
-\begin{grafite}
-include "nat/nat.ma".
-..
-definition gt: nat \to nat \to Prop \def
- \lambda n, m. m < n.
-\end{grafite}
-
-The \texttt{include} statement adds the requirement that the part of the library
-defining the notion of natural numbers should be defined before
-processing the what follows. Note indeed that the algorithm presented
-in~\cite{disambiguation} does not describe where interpretations for ambiguous
-expressions come from, since it is application-specific. As a first
-approximation, we will assume that in \MATITA{} they come from the library (i.e.
-all interpretations available in the library are used) and the \texttt{include}
-statements are used to ensure the availability of required library slices (see
-Sect.~\ref{sec:libmanagement}).
-
-While processing the \texttt{gt} definition, \MATITA{} has to disambiguate two
-terms: its type and its body. Being available in the required library only one
-interpretation both for the unbound identifier \texttt{nat} and for the
-\OP{<} operator, and being the resulting partially specified term refinable,
-both type and body are easily disambiguated.
-
-Now suppose we have defined integers as signed natural numbers, and that we want
-to prove a theorem about an order relationship already defined on them (which of
-course overload the \OP{<} operator):
-
-\begin{grafite}
-include "Z/z.ma".
-..
-theorem Zlt_compat:
- \forall x, y, z. x < y \to y < z \to x < z.
-\end{grafite}
-
-Since integers are defined on top of natural numbers, the part of the library
-concerning the latters is available when disambiguating \texttt{Zlt\_compat}'s
-type. Thus, according to the disambiguation algorithm, two different partially
-specified terms could be associated to it. At first, this might not be seen as a
-problem, since the user is asked and can choose interactively which of the two
-she had in mind. However in the long run it has the drawbacks of inhibiting
-batch compilation of the library (a technique used in \MATITA{} for behind the
-scene compilation when needed, e.g. when an \texttt{include} is issued) and
-yields to poor user interaction (imagine how tedious would be to be asked for a
-choice each time you re-evaluate \texttt{Zlt\_compat}!).
-
-For this reason we added to \MATITA{} the concept of \emph{disambiguation
-aliases}. Disambiguation aliases are one-to-many mappings from ambiguous
-expressions to partially specified terms, which are part of the runtime status
-of \MATITA. They can be provided by users with the \texttt{alias} statement, but
-are usually automatically added when evaluating \texttt{include} statements
-(\emph{implicit aliases}). Aliases implicitely inferred during disambiguation
-are remembered as well. Moreover, \MATITA{} does it best to ensure that terms
-which require interactive choice are saved in batch compilable format. Thus,
-after evaluating the above theorem the script will be changed to the following
-snippet (assuming that the interpretation of \OP{<} over integers has been
-choosed):
-
-\begin{grafite}
-alias symbol "lt" = "integer 'less than'".
-theorem Zlt_compat:
- \forall x, y, z. x < y \to y < z \to x < z.
-\end{grafite}
-
-But how are disambiguation aliases used? Since they come from the parts of the
-library explicitely included we may be tempted of using them as the only
-available interpretations. This would speed up the disambiguation, but may fail.
-Consider for example:
-
-\begin{grafite}
-theorem lt_mono: \forall x, y, k. x < y \to x < y + k.
-\end{grafite}
-
-and suppose that the \OP{+} operator is defined only on natural numbers. If
-the alias for \OP{<} points to the integer version of the operator, no
-refinable partially specified term matching the term could be found.
-
-For this reason we choosed to attempt \emph{multiple disambiguation passes}. A
-first pass attempt to disambiguate using the last available disambiguation
-aliases (\emph{mono aliases} pass), in case of failure the next pass try again
-the disambiguation forgetting the aliases and using the whole library to
-retrieve interpretation for ambiguous expressions (\emph{library aliases} pass).
-Since the latter pass may lead to too many choices we intertwined an additional
-pass among the two which use as interpretations all the aliases coming for
-included parts of the library (\emph{multi aliases} phase). This is the reason
-why aliases are \emph{one-to-many} mappings instead of one-to-one. This choice
-turned out to be a well-balanced trade-off among performances (earlier passes
-fail quickly) and degree of ambiguity supported for presentation level terms.
-
-\subsubsection{Operator instances}
-
-Let's suppose now we want to define a theorem relating ordering relations on
-natural and integer numbers. The way we would like to write such a theorem (as
-we can read it in the \MATITA{} standard library) is:
-
-\begin{grafite}
-include "Z/z.ma".
-include "nat/orders.ma".
-..
-theorem lt_to_Zlt_pos_pos:
- \forall n, m: nat. n < m \to pos n < pos m.
-\end{grafite}
-
-Unfortunately, none of the passes described above is able to disambiguate its
-type, no matter how aliases are defined. This is because the \OP{<} operator
-occurs twice in the content level term (it has two \emph{instances}) and two
-different interpretations for it have to be used in order to obtain a refinable
-partially specified term. To address this issue, we have the ability to consider
-each instance of a single symbol as a different ambiguous expression in the
-content level term, and thus we can assign a different interpretation to each of
-them. A disambiguation pass which exploit this feature is said to be using
-\emph{fresh instances}.
-
-Fresh instances lead to a non negligible performance loss (since the choice of
-an interpretation for one instances does not constraint the choice for the
-others). For this reason we always attempt a fresh instances pass only after
-attempting a non-fresh one.
-
-\paragraph{One-shot aliases} Disambiguation aliases as seen so far are
-instance-independent. However, aliases obtained as a result of a disambiguation
-pass which uses fresh instances ought to be instance-dependent, that is: to
-ensure a term can be disambiguated in a batch fashion we may need to state that
-an \emph{i}-th instance of a symbol should be mapped to a given partially
-specified term. Instance-depend aliases are meaningful only for the term whose
-disambiguation generated it. For this reason we call them \emph{one-shot
-aliases} and \MATITA{} doesn't use it to disambiguate further terms down in the
-script.
-
-\subsubsection{Implicit coercions}
-
-Let's now consider a (rather hypothetical) theorem about derivation:
-
-\begin{grafite}
-theorem power_deriv:
- \forall n: nat, x: R. d x ^ n dx = n * x ^ (n - 1).
-\end{grafite}
-
-and suppose there exists a \texttt{R \TEXMACRO{to} nat \TEXMACRO{to} R}
-interpretation for \OP{\^}, and a real number interpretation for \OP{*}.
-Mathematichians would write the term that way since it is well known that the
-natural number \texttt{n} could be ``injected'' in \IR{} and considered a real
-number for the purpose of real multiplication. The refiner of \MATITA{} supports
-\emph{implicit coercions} for this reason: given as input the above content
-level term, it will return a partially specified term where in place of
-\texttt{n} the application of a coercion from \texttt{nat} to \texttt{R} appears
-(assuming it has been defined as such of course).
-
-Nonetheless coercions are not always desirable. For example, in disambiguating
-\texttt{\TEXMACRO{forall} x: nat. n < n + 1} we don't want the term which uses
-two coercions from \texttt{nat} to \texttt{R} around \OP{<} arguments to show up
-among the possible partially specified term choices. For this reason in
-\MATITA{} we always try first a disambiguation pass which require the refiner
-not to use the coercions and only in case of failure we attempt a
-coercion-enabled pass.
-
-It is interesting to observe also the relationship among operator instances and
-implicit coercions. Consider again the theorem \texttt{lt\_to\_Zlt\_pos\_pos},
-which \MATITA{} disambiguated using fresh instances. In case there exists a
-coercion from natural numbers to (positive) integers (which indeed does, it is
-the \texttt{pos} constructor itself), the theorem can be disambiguated using
-twice that coercion on the left hand side of the implication. The obtained
-partially specified term however would not probably be the expected one, being a
-theorem which prove a trivial implication. For this reason we choose to always
-prefer fresh instances over implicit coercions, i.e. we always attempt
-disambiguation passes with fresh instances and no implicit coercions before
-attempting passes with implicit coercions.
-
-\subsubsection{Disambiguation passes}
-
-According to the criteria described above in \MATITA{} we choose to perform the
-sequence of disambiguation passes depicted in Tab.~\ref{tab:disambpasses}. In
-our experience that choice implements a good trade off among disambiguation time
-and admitted ambiguity in terms input by users.
+\section{The authoring interface}
+\label{sec:authoring}
+
+The authoring interface of \MATITA{} is very similar to Proof General. We
+chose not to build the \MATITA{} UI over Proof General for two reasons. First
+of all we wanted to integrate our XML-based rendering technologies, mainly
+\GTKMATHVIEW. At the time of writing Proof General supports only text based
+rendering.\footnote{This may change with the future release of Proof General
+based on Eclipse, but is not yet the case.} The second reason is that we wanted
+to build the \MATITA{} UI on top of a state-of-the-art and widespread toolkit
+as GTK is.
+
+Fig.~\ref{fig:screenshot} is a screenshot of the \MATITA{} authoring interface,
+featuring two windows. The background one is very like to the Proof General
+interface. The main difference is that we use the \GTKMATHVIEW{} widget to
+render sequents. Since \GTKMATHVIEW{} renders \MATHML{} markup we take
+advantage of the whole bidimensional mathematical notation. The foreground
+window is an instance of the cicBrowser used to render the proof being
+developed.
+
+Note that the syntax used in the script view is \TeX-like, however Unicode is
+fully supported so that mathematical glyphs can be input as such.
+
+\begin{figure}[!ht]
+ \begin{center}
+ \includegraphics[width=0.95\textwidth]{pics/matita-screenshot}
+ \caption{Authoring interface\strut}
+ \label{fig:screenshot}
+ \end{center}
+\end{figure}
-\begin{table}[ht]
- \caption{Sequence of disambiguation passes used in \MATITA.\strut}
- \label{tab:disambpasses}
+Since the concepts of script based proof authoring are well-known, the
+remaining part of this section is dedicated to the distinguishing
+features of the \MATITA{} authoring interface.
+
+\subsection{Direct manipulation of terms}
+\label{sec:directmanip}
+
+While terms are input as \TeX-like formulae in \MATITA, they are converted to a
+mixed \MATHML+\BOXML{} markup for output purposes and then rendered by
+\GTKMATHVIEW. As described in~\cite{latexmathml} this mixed choice enables both
+high-quality bidimensional rendering of terms (including the use of fancy
+layout schemata like radicals and matrices) and the use of a
+concise and widespread textual syntax.
+
+Keeping pointers from the presentations level terms down to the
+partially specified ones \MATITA{} enable direct manipulation of
+rendered (sub)terms in the form of hyperlinks and semantic selection.
+
+\emph{Hyperlinks} have anchors on the occurrences of constant and
+inductive type constructors and point to the corresponding definitions
+in the library. Anchors are available notwithstanding the use of
+user-defined mathematical notation: as can be seen on the right of
+Fig.~\ref{fig:directmanip}, where we clicked on $\not|$, symbols
+encoding complex notations retain all the hyperlinks of constants or
+constructors used in the notation.
+
+\emph{Semantic selection} enables the selection of mixed
+\MATHML+\BOXML{} markup, constraining the selection to markup
+representing meaningful CIC (sub)terms. In the example on the left of
+Fig.~\ref{fig:directmanip} is thus possible to select the subterm
+$\mathrm{prime}~n$, whereas it would not be possible to select
+$\to n$ since the former denotes an application while the
+latter it not a subterm. Once a meaningful (sub)term has been
+selected actions can be done on it like reductions or tactic
+applications.
+
+\begin{figure}[!ht]
\begin{center}
- \begin{tabular}{c|c|c|c}
- \multicolumn{1}{p{1.5cm}|}{\centering\raisebox{-1.5ex}{\textbf{Pass}}}
- & \multicolumn{1}{p{3.1cm}|}{\centering\textbf{Disambiguation aliases}}
- & \multicolumn{1}{p{2.5cm}|}{\centering\textbf{Operator instances}}
- & \multicolumn{1}{p{2.5cm}}{\centering\textbf{Implicit coercions}} \\
- \hline
- \PASS & Mono aliases & Shared & Disabled \\
- \PASS & Multi aliases & Shared & Disabled \\
- \PASS & Mono aliases & Fresh instances & Disabled \\
- \PASS & Multi aliases & Fresh instances & Disabled \\
- \PASS & Mono aliases & Fresh instances & Enabled \\
- \PASS & Multi aliases & Fresh instances & Enabled \\
- \PASS & Library aliases& Fresh instances & Enabled
- \end{tabular}
+ \includegraphics[width=0.40\textwidth]{pics/matita-screenshot-selection}
+ \hspace{0.05\textwidth}
+ \raisebox{0.4cm}{\includegraphics[width=0.50\textwidth]{pics/matita-screenshot-href}}
+ \caption[Semantic selection and hyperlinks]{Semantic selection (on the left)
+ and hyperlinks (on the right)\strut}
+ \label{fig:directmanip}
\end{center}
-\end{table}
+\end{figure}
\subsection{Patterns}
+\label{sec:patterns}
-serve una intro che almeno cita il widget (per i patterns) e che fa
-il resoconto delle cose che abbiamo e che non descriviamo,
-sottolineando che abbiamo qualcosa da dire sui pattern e sui
-tattichini.\\
-
-Patterns are the textual counterpart of the MathML widget graphical
-selection.
+In several situations working with direct manipulation of terms is
+simpler and faster than typing the corresponding textual
+commands~\cite{proof-by-pointing}.
+Nonetheless we need to record actions and selections in scripts.
-\MATITA{} benefits of a graphical interface and a powerful MathML rendering
-widget that allows the user to select pieces of the sequent he is working
-on. While this is an extremely intuitive way for the user to
-restrict the application of tactics, for example, to some subterms of the
-conclusion or some hypothesis, the way this action is recorded to the text
-script is not obvious.\\
-In \MATITA{} this issue is addressed by patterns.
+In \MATITA{} \emph{patterns} are textual representations of selections.
+Users can select using the GUI and then ask the system to paste the
+corresponding pattern in this script, but more often this process is
+transparent: once an action is performed on a selection, the corresponding
+textual command is computed and inserted in the script.
\subsubsection{Pattern syntax}
-A pattern is composed of two terms: a $\NT{sequent\_path}$ and a
-$\NT{wanted}$.
-The former mocks-up a sequent, discharging unwanted subterms with $?$ and
-selecting the interesting parts with the placeholder $\%$.
-The latter is a term that lives in the context of the placeholders.
-
-The concrete syntax is reported in table \ref{tab:pathsyn}
-\NOTE{uso nomi diversi dalla grammatica ma che hanno + senso}
+
+Patterns are composed of two parts: \NT{sequent\_path} and
+\NT{wanted}; their concrete syntax is reported in Tab.~\ref{tab:pathsyn}.
+
+\NT{sequent\_path} mocks-up a sequent, discharging unwanted subterms
+with $?$ and selecting the interesting parts with the placeholder
+$\%$. \NT{wanted} is a term that lives in the context of the
+placeholders.
+
+Textual patterns produced from a graphical selection are made of the
+\NT{sequent\_path} only. Such patterns can represent every selection,
+but are quite verbose. The \NT{wanted} part of the syntax is meant to
+help the users in writing concise and elegant patterns by hand.
+
\begin{table}
- \caption{\label{tab:pathsyn} Concrete syntax of \MATITA{} patterns.\strut}
+ \caption{Patterns concrete syntax\strut}
+ \label{tab:pathsyn}
\hrule
\[
\begin{array}{@{}rcll@{}}
\NT{pattern} &
- ::= & [~\verb+in match+~\NT{wanted}~]~[~\verb+in+~\NT{sequent\_path}~] & \\
+ ::= & [~\verb+in+~\NT{sequent\_path}~]~[~\verb+match+~\NT{wanted}~] & \\
\NT{sequent\_path} &
::= & \{~\NT{ident}~[~\verb+:+~\NT{multipath}~]~\}~
[~\verb+\vdash+~\NT{multipath}~] & \\
- \NT{wanted} & ::= & \NT{term} & \\
\NT{multipath} & ::= & \NT{term\_with\_placeholders} & \\
+ \NT{wanted} & ::= & \NT{term} & \\
\end{array}
\]
\hrule
\end{table}
-\subsubsection{How patterns work}
-Patterns mimic the user's selection in two steps. The first one
-selects roots (subterms) of the sequent, using the
-$\NT{sequent\_path}$, while the second
-one searches the $\NT{wanted}$ term starting from these roots. Both are
-optional steps, and by convention the empty pattern selects the whole
-conclusion.
+\subsubsection{Pattern evaluation}
+
+Patterns are evaluated in two phases. The first selects roots
+(subterms) of the sequent, using the $\NT{sequent\_path}$, while the
+second searches the $\NT{wanted}$ term starting from these roots.
+% Both are optional steps, and by convention the empty pattern selects
+% the whole conclusion.
\begin{description}
\item[Phase 1]
$\NT{sequent\_path}$ starting from a visual selection.
\NOTE{Questo ancora non va in matita}
- A $\NT{multipath}$ is a CiC term in which a special constant $\%$
+ A $\NT{multipath}$ is a CIC term in which a special constant $\%$
is allowed.
The roots of discharged subterms are marked with $?$, while $\%$
is used to select roots. The default $\NT{multipath}$, the one that
the source of an arrow and the head of the application that is
found in the arrow target.
- The first phase selects not only terms (roots of subterms) but also
- their context that will be eventually used in the second phase.
+ The first phase not only selects terms (roots of subterms) but
+ determines also their context that will be eventually used in the
+ second phase.
\item[Phase 2]
- plays a role only if the $[~\verb+in match+~\NT{wanted}~]$
+ plays a role only if the $[~\verb+match+~\NT{wanted}~]$
part is specified. From the first phase we have some terms, that we
will see as subterm roots, and their context. For each of these
contexts the $\NT{wanted}$ term is disambiguated in it and the
- corresponding root is searched for a subterm $\alpha$-equivalent to
+ corresponding root is searched for a subterm that can be unified to
$\NT{wanted}$. The result of this search is the selection the
pattern represents.
\end{description}
-\noindent
-Since the first step is equipotent to the composition of the two
-steps, the system uses it to represent each visual selection.
-The second step is only meant for the
-experienced user that writes patterns by hand, since it really
-helps in writing concise patterns as we will see in the
-following examples.
-
\subsubsection{Examples}
-To explain how the first step works let's give an example. Consider
-you want to prove the uniqueness of the identity element $0$ for natural
-sum, and that you can relay on the previously demonstrated left
-injectivity of the sum, that is $inj\_plus\_l:\forall x,y,z.x+y=z+y \to x =z$.
-Typing
-\begin{grafite}
-theorem valid_name: \forall n,m. m + n = n \to m = O.
- intros (n m H).
-\end{grafite}
-\noindent
-leads you to the following sequent
+%To explain how the first phase works let us give an example. Consider
+%you want to prove the uniqueness of the identity element $0$ for natural
+%sum, and that you can rely on the previously demonstrated left
+%injectivity of the sum, that is $inj\_plus\_l:\forall x,y,z.x+y=z+y \to x =z$.
+%Typing
+%\begin{grafite}
+%theorem valid_name: \forall n,m. m + n = n \to m = O.
+% intros (n m H).
+%\end{grafite}
+%\noindent
+Consider the following sequent
\sequent{
n:nat\\
m:nat\\
m=O
}
\noindent
-where you want to change the right part of the equivalence of the $H$
-hypothesis with $O + n$ and then use $inj\_plus\_l$ to prove $m=O$.
+To change the right part of the equivalence of the $H$
+hypothesis with $O + n$ the user selects and pastes it as the pattern
+in the following statement.
\begin{grafite}
change in H:(? ? ? %) with (O + n).
\end{grafite}
\noindent
-This pattern, that is a simple instance of the $\NT{sequent\_path}$
-grammar entry, acts on $H$ that has type (without notation) $(eq~nat~(m+n)~n)$
-and discharges the head of the application and the first two arguments with a
-$?$ and selects the last argument with $\%$. The syntax may seem uncomfortable,
-but the user can simply select with the mouse the right part of the equivalence
-and left to the system the burden of writing down in the script file the
-corresponding pattern with $?$ and $\%$ in the right place (that is not
-trivial, expecially where implicit arguments are hidden by the notation, like
-the type $nat$ in this example).
-
-Changing all the occurrences of $n$ in the hypothesis $H$ with $O+n$
-works too and can be done, by the experienced user, writing directly
-a simpler pattern that uses the second phase.
+To understand the pattern (or produce it by hand) the user should be
+aware that the notation $m+n=n$ hides the term $(eq~nat~(m+n)~n)$, so
+that the pattern selects only the third argument of $eq$.
+
+The experienced user may also write by hand a concise pattern
+to change at once all the occurrences of $n$ in the hypothesis $H$:
\begin{grafite}
- change in match n in H with (O + n).
+ change in H match n with (O + n).
\end{grafite}
\noindent
In this case the $\NT{sequent\_path}$ selects the whole $H$, while
-the second phase searches the wanted $n$ inside it by
-$\alpha$-equivalence. The resulting
-equivalence will be $m+(O+n)=O+n$ since the second phase found two
-occurrences of $n$ in $H$ and the tactic changed both.
+the second phase locates $n$.
-Just for completeness the second pattern is equivalent to the
-following one, that is less readable but uses only the first phase.
+The latter pattern is equivalent to the following one, that the system
+can automatically generate from the selection.
\begin{grafite}
change in H:(? ? (? ? %) %) with (O + n).
\end{grafite}
\noindent
\subsubsection{Tactics supporting patterns}
+
+\TODO{mergiare con il successivo facendo notare che i patterns sono una
+interfaccia comune per le tattiche}
+
In \MATITA{} all the tactics that can be restricted to subterm of the working
sequent accept the pattern syntax. In particular these tactics are: simplify,
change, fold, unfold, generalize, replace and rewrite.
con una pattern\_of(select(pattern))}
\subsubsection{Comparison with \COQ{}}
-\COQ{} has a two diffrent ways of restricting the application of tactis to
+
+\COQ{} has two different ways of restricting the application of tactics to
subterms of the sequent, both relaying on the same special syntax to identify
a term occurrence.
-The first way is to use this special syntax to specify directly to the
-tactic the occurrnces of a wanted term that should be affected, while
-the second is to prepare the sequent with another tactic called
-pattern and the apply the real tactic. Note that the choice is not
+The first way is to use this special syntax to tell the
+tactic what occurrences of a wanted term should be affected.
+The second is to prepare the sequent with another tactic called
+pattern and then apply the real tactic. Note that the choice is not
left to the user, since some tactics needs the sequent to be prepared
with pattern and do not accept directly this special syntax.
The base idea is that to identify a subterm of the sequent we can
write it and say that we want, for example, the third and the fifth
-occurce of it (counting from left to right). In our previous example,
+occurrences of it (counting from left to right). In our previous example,
to change only the left part of the equivalence, the correct command
-is
+is:
+
\begin{grafite}
change n at 2 in H with (O + n)
\end{grafite}
-\noindent
+
meaning that in the hypothesis $H$ the $n$ we want to change is the
-second we encounter proceeding from left toright.
+second we encounter proceeding from left to right.
The tactic pattern computes a
$\beta$-expansion of a part of the sequent with respect to some
occurrences of the given term. In the previous example the following
-command
+command:
\begin{grafite}
pattern n at 2 in H
\end{grafite}
-\noindent
-would have resulted in this sequent
+
+would have resulted in this sequent:
+
\begin{grafite}
n : nat
m : nat
============================
m = 0
\end{grafite}
-\noindent
+
where $H$ is $\beta$-expanded over the second $n$
-occurrence. This is a trick to make the unification algorithm ignore
-the head of the application (since the unification is essentially
-first-order) but normally operate on the arguments.
-This works for some tactics, like rewrite and replace,
-but for example not for change and other tactics that do not relay on
-unification.
+occurrence.
+
+At this point, since \COQ{} unification algorithm is essentially
+first-order, the application of an elimination principle (of the
+form $\forall P.\forall x.(H~x)\to (P~x)$) will unify
+$x$ with \texttt{n} and $P$ with \texttt{(fun n0 : nat => m + n = n0)}.
+
+Since rewriting, replacing and several other tactics boils down to
+the application of the equality elimination principle, the previous
+trick deals the expected behaviour.
The idea behind this way of identifying subterms in not really far
-from the idea behind patterns, but really fails in extending to
+from the idea behind patterns, but fails in extending to
complex notation, since it relays on a mono-dimensional sequent representation.
Real math notation places arguments upside-down (like in indexed sums or
integrations) or even puts them inside a bidimensional matrix.
In these cases using the mouse to select the wanted term is probably the
-only way to tell the system exactly what you want to do.
+more effective way to tell the system what to do.
-One of the goals of \MATITA{} is to use modern publishing techiques, and
+One of the goals of \MATITA{} is to use modern publishing techniques, and
adopting a method for restricting tactics application domain that discourages
using heavy math notation, would definitively be a bad choice.
-
\subsection{Tacticals}
-There are mainly two kinds of languages used by proof assistants to recorder
-proofs: tactic based and declarative. We will not investigate the philosophy
-aroud the choice that many proof assistant made, \MATITA{} included, and we
-will not compare the two diffrent approaches. We will describe the common
-issues of the tactic-based language approach and how \MATITA{} tries to solve
-them.
-
-\subsubsection{Tacticals overview}
-
-Tacticals first appeared in LCF and can be seen as programming
-constructs, like looping, branching, error recovery or sequential composition.
-The following simple example shows three tacticals in action
+\label{sec:tinycals}
+
+%There are mainly two kinds of languages used by proof assistants to recorder
+%proofs: tactic based and declarative. We will not investigate the philosophy
+%around the choice that many proof assistant made, \MATITA{} included, and we
+%will not compare the two different approaches. We will describe the common
+%issues of the tactic-based language approach and how \MATITA{} tries to solve
+%them.
+
+The procedural proof language implemented in \MATITA{} is pretty standard,
+with a notable exception for tacticals.
+
+%\subsubsection{Tacticals overview}
+
+Tacticals first appeared in LCF as higher order tactics. They can be
+seen as control flow constructs, like looping, branching, error
+recovery or sequential composition.
+
+
+The following simple example
+shows a Coq script made of four dot-terminated commands
+
\begin{grafite}
-theorem trivial:
- \forall A,B:Prop.
- A = B \to ((A \to B) \land (B \to A)).
- intros (A B H).
+Theorem trivial:
+ forall A B:Prop,
+ A = B -> ((A -> B) /\ (B -> A)).
+ intros [A B H].
split; intro;
- [ rewrite < H. assumption.
- | rewrite > H. assumption.
- ]
-qed.
+ [ rewrite < H; assumption
+ | rewrite > H; assumption
+ ].
+Qed.
\end{grafite}
-The first is ``\texttt{;}'' that combines the tactic \texttt{split}
-with \texttt{intro}, applying the latter to each goal opened by the
-former. Then we have ``\texttt{[}'' that branches on the goals (here
-we have two goals, the two sides of the logic and).
-The first goal $B$ (with $A$ in the context)
-is proved by the first sequence of tactics
-\texttt{rewrite} and \texttt{assumption}. Then we move to the second
-goal with the separator ``\texttt{|}''. The last tactical we see here
-is ``\texttt{.}'' that is a sequential composition that selects the
-first goal opened for the following tactic (instead of applying it to
-them all like ``\texttt{;}''). Note that usually ``\texttt{.}'' is
-not considered a tactical, but a sentence terminator (i.e. the
-delimiter of commands the proof assistant executes).
-
-Giving serious examples here is rather difficult, since they are hard
-to read without the interactive tool. To help the reader in
-understanding the following considerations we just give few common
-usage examples without a proof context.
+The third command is an application of the sequencing tactical
+``$\ldots$\texttt{;}$\ldots$'', that combines the tactic
+\texttt{split} with the application of the branching tactical
+``$\ldots$\texttt{;[}$\ldots$\texttt{|}$\ldots$\texttt{|}$\ldots$\texttt{]}''
+to other tactics and tacticals.
-\begin{grafite}
- elim z; try assumption; [ ... | ... ].
- elim z; first [ assumption | reflexivity | id ].
-\end{grafite}
+The usual implementation of tacticals executes them atomically as any
+other command. In \MATITA{} thi is not true since each punctuation is
+executed as a single command.
-The first example goes by induction on a term \texttt{z} and applies
-the tactic \texttt{assumption} to each opened goal eventually recovering if
-\texttt{assumption} fails. Here we are asking the system to close all
-trivial cases and then we branch on the remaining with ``\texttt{[}''.
-The second example goes again by induction on \texttt{z} and tries to
-close each opened goal first with \texttt{assumption}, if it fails it
-tries \texttt{reflexivity} and finally \texttt{id}
-that is the tactic that leaves the goal untouched without failing.
-
-Note that in the common implementation of tacticals both lines are
-compositions of tacticals and in particular they are a single
-statement (i.e. derived from the same non terminal entry of the
-grammar) ended with ``\texttt{.}''. As we will see later in \MATITA{}
-this is not true, since each atomic tactic or punctuation is considered
-a single statement.
+%The latter is applied to all the goals opened by \texttt{split}
+%
+%(here we have two goals, the two sides of the logic and). The first
+%goal $B$ (with $A$ in the context) is proved by the first sequence of
+%tactics \texttt{rewrite} and \texttt{assumption}. Then we move to the
+%second goal with the separator ``\texttt{|}''.
+%
+%Giving serious examples here is rather difficult, since they are hard
+%to read without the interactive tool. To help the reader in
+%understanding the following considerations we just give few common
+%usage examples without a proof context.
+%
+%\begin{grafite}
+% elim z; try assumption; [ ... | ... ].
+% elim z; first [ assumption | reflexivity | id ].
+%\end{grafite}
+%
+%The first example goes by induction on a term \texttt{z} and applies
+%the tactic \texttt{assumption} to each opened goal eventually recovering if
+%\texttt{assumption} fails. Here we are asking the system to close all
+%trivial cases and then we branch on the remaining with ``\texttt{[}''.
+%The second example goes again by induction on \texttt{z} and tries to
+%close each opened goal first with \texttt{assumption}, if it fails it
+%tries \texttt{reflexivity} and finally \texttt{id}
+%that is the tactic that leaves the goal untouched without failing.
+%
+%Note that in the common implementation of tacticals both lines are
+%compositions of tacticals and in particular they are a single
+%statement (i.e. derived from the same non terminal entry of the
+%grammar) ended with ``\texttt{.}''. As we will see later in \MATITA{}
+%this is not true, since each atomic tactic or punctuation is considered
+%a single statement.
\subsubsection{Common issues of tactic(als)-based proof languages}
We will examine the two main problems of tactic(als)-based proof script:
maintainability and readability.
-Huge libraries of formal mathematics have been developed, and backward
-compatibility is a really time consuming task. \\
-A real-life example in the history of \MATITA{} was the reordering of
-goals opened by a tactic application. We noticed that some tactics
-were not opening goals in the expected order. In particular the
-\texttt{elim} tactic on a term of an inductive type with constructors
-$c_1, \ldots, c_n$ used to open goals in order $g_1, g_n, g_{n-1}
-\ldots, g_2$. The library of \MATITA{} was still in an embryonic state
-but some theorems about integers were there. The inductive type of
-$\mathcal{Z}$ has three constructors: $zero$, $pos$ and $neg$. All the
-induction proofs on this type where written without tacticals and,
-obviously, considering the three induction cases in the wrong order.
-Fixing the behavior of the tactic broke the library and two days of
-work were needed to make it compile again. The whole time was spent in
-finding the list of tactics used to prove the third induction case and
-swap it with the list of tactics used to prove the second case. If
-the proofs was structured with the branch tactical this task could
-have been done automatically.
-
-From this experience we learned that the use of tacticals for
-structuring proofs gives some help but may have some drawbacks in
-proof script readability. We must highlight that proof scripts
-readability is poor by itself, but in conjunction with tacticals it
-can be nearly impossible. The main cause is the fact that in proof
-scripts there is no trace of what you are working on. It is not rare
-for two different theorems to have the same proof script (while the
-proof is completely different).\\
-Bad readability is not a big deal for the user while he is
-constructing the proof, but is considerably a problem when he tries to
-reread what he did or when he shows his work to someone else. The
-workaround commonly used to read a script is to execute it again
-step-by-step, so that you can see the proof goal changing and you can
-follow the proof steps. This works fine until you reach a tactical. A
-compound statement, made by some basic tactics glued with tacticals,
-is executed in a single step, while it obviously performs lot of proof
-steps. In the fist example of the previous section the whole branch
-over the two goals (respectively the left and right part of the logic
-and) result in a single step of execution. The workaround doesn't work
-anymore unless you de-structure on the fly the proof, putting some
-``\texttt{.}'' where you want the system to stop.\\
-
-Now we can understand the tradeoff between script readability and
-proof structuring with tacticals. Using tacticals helps in maintaining
-scripts, but makes it really hard to read them again, cause of the way
-they are executed.
-
-\MATITA{} uses a language of tactics and tacticals, but tries to avoid
-this tradeoff, alluring the user to write structured proof without
-making it impossible to read them again.
+%Huge libraries of formal mathematics have been developed, and backward
+%compatibility is a really time consuming task. \\
+%A real-life example in the history of \MATITA{} was the reordering of
+%goals opened by a tactic application. We noticed that some tactics
+%were not opening goals in the expected order. In particular the
+%\texttt{elim} tactic on a term of an inductive type with constructors
+%$c_1, \ldots, c_n$ used to open goals in order $g_1, g_n, g_{n-1}
+%\ldots, g_2$. The library of \MATITA{} was still in an embryonic state
+%but some theorems about integers were there. The inductive type of
+%$\mathcal{Z}$ has three constructors: $zero$, $pos$ and $neg$. All the
+%induction proofs on this type where written without tacticals and,
+%obviously, considering the three induction cases in the wrong order.
+%Fixing the behavior of the tactic broke the library and two days of
+%work were needed to make it compile again. The whole time was spent in
+%finding the list of tactics used to prove the third induction case and
+%swap it with the list of tactics used to prove the second case. If
+%the proofs was structured with the branch tactical this task could
+%have been done automatically.
+%
+%From this experience we learned that the use of tacticals for
+%structuring proofs gives some help but may have some drawbacks in
+%proof script readability.
+
+Tacticals are not only used to make scripts shorter by factoring out
+common cases and repeating commands. They are a primary way of making
+scripts more mainteable. Moreover, they also have the well-known
+role of structuring the proof.
+
+However, authoring a proof structured with tacticals is annoying.
+Consider for example a proof by induction, and imagine you
+are using one of the state of the art graphical interfaces for proof assistant
+like Proof General. After applying the induction principle you have to choose:
+immediately structure the proof or postpone the structuring.
+If you decide for the former you have to apply the branching tactical and write
+at once tactics for all the cases. Since the user does not even know the
+generated goals yet, she can only replace all the cases with the identity
+tactic and execute the command, just to receive feedback on the first
+goal. Then she has to go one step back to replace the first identity
+tactic with the wanted one and repeat the process until all the
+branches are closed.
+
+One could imagine that a structured script is simpler to understand.
+This is not the case.
+A proof script, being not declarative, is not meant to be read.
+However, the user has the need of explaining it to others.
+This is achieved by interactively re-playing the script to show each
+intermediate proof status. Tacticals make this operation uncomfortable.
+Indeed, a tactical is executed atomically, while it is obvious that it
+performs lot of smaller steps we are interested in.
+To show the intermediate steps, the proof must be de-structured on the
+fly, for example replacing ``\texttt{;}'' with ``\texttt{.}'' where
+possible.\\
+
+%Proof scripts
+%readability is poor by itself, but in conjunction with tacticals it
+%can be nearly impossible. The main cause is the fact that in proof
+%scripts there is no trace of what you are working on. It is not rare
+%for two different theorems to have the same proof script.\\
+%Bad readability is not a big deal for the user while he is
+%constructing the proof, but is considerably a problem when he tries to
+%reread what he did or when he shows his work to someone else. The
+%workaround commonly used to read a script is to execute it again
+%step-by-step, so that you can see the proof goal changing and you can
+%follow the proof steps. This works fine until you reach a tactical. A
+%compound statement, made by some basic tactics glued with tacticals,
+%is executed in a single step, while it obviously performs lot of proof
+%steps. In the fist example of the previous section the whole branch
+%over the two goals (respectively the left and right part of the logic
+%and) result in a single step of execution. The workaround does not work
+%anymore unless you de-structure on the fly the proof, putting some
+%``\texttt{.}'' where you want the system to stop.\\
+
+%Now we can understand the tradeoff between script readability and
+%proof structuring with tacticals. Using tacticals helps in maintaining
+%scripts, but makes it really hard to read them again, cause of the way
+%they are executed.
+
+\MATITA{} has a peculiar tacticals implementation that provides the
+same benefits as classical tacticals, while not burdening the user
+during proof authoring and re-playing.
+
+%\MATITA{} uses a language of tactics and tacticals, but tries to avoid
+%this tradeoff, alluring the user to write structured proof without
+%making it impossible to read them again.
\subsubsection{The \MATITA{} approach: Tinycals}
\begin{table}
- \caption{\label{tab:tacsyn} Concrete syntax of \MATITA{} tacticals.\strut}
+ \caption{Concrete syntax of tacticals\strut}
+ \label{tab:tacsyn}
\hrule
\[
\begin{array}{@{}rcll@{}}
\hrule
\end{table}
-\MATITA{} tacticals syntax is reported in table \ref{tab:tacsyn}.
+\MATITA{} tacticals syntax is reported in Tab.~\ref{tab:tacsyn}.
While one would expect to find structured constructs like
$\verb+do+~n~\NT{tactic}$ the syntax allows pieces of tacticals to be written.
-This is essential for base idea behind \MATITA{} tacticals: step-by-step
+This is essential for the base idea behind \MATITA{} tacticals: step-by-step
execution.
The low-level tacticals implementation of \MATITA{} allows a step-by-step
execution of a tactical, that substantially means that a $\NT{block\_kind}$ is
-not executed as an atomic operation. This has two major benefits for the user,
-even being a so simple idea:
-\begin{description}
-\item[Proof structuring]
- is much easier. Consider for example a proof by induction, and imagine you
- are using classical tacticals in one of the state of the
- art graphical interfaces for proof assistant like Proof General or \COQIDE.
- After applying the induction principle you have to choose: structure
- the proof or not. If you decide for the former you have to branch with
- ``\texttt{[}'' and write tactics for all the cases separated by
- ``\texttt{|}'' and then close the tactical with ``\texttt{]}''.
- You can replace most of the cases by the identity tactic just to
- concentrate only on the first goal, but you will have to go one step back and
- one further every time you add something inside the tactical. Again this is
- caused by the one step execution of tacticals and by the fact that to modify
- the already executed script you have to undo one step.
- And if you are board of doing so, you will finish in giving up structuring
- the proof and write a plain list of tactics.\\
- With step-by-step tacticals you can apply the induction principle, and just
- open the branching tactical ``\texttt{[}''. Then you can interact with the
- system reaching a proof of the first case, without having to specify any
- tactic for the other goals. When you have proved all the induction cases, you
- close the branching tactical with ``\texttt{]}'' and you are done with a
- structured proof. \\
- While \MATITA{} tacticals help in structuring proofs they allow you to
- choose the amount of structure you want. There are no constraints imposed by
- the system, and if the user wants he can even write completely plain proofs.
+not executed as an atomic operation. This has major benefits for the
+user during proof structuring and re-playing.
+
+For instance, reconsider the previous example of a proof by induction.
+With step-by-step tacticals the user can apply the induction principle, and just
+open the branching tactical ``\texttt{[}''. Then she can interact with the
+system until the proof of the first case is terminated. After that
+``\texttt{|}'' is used to move to the next goal, until all goals are
+closed. After the last goal, the user closes the branching tactical with
+``\texttt{]}'' and is done with a structured proof. \\
+While \MATITA{} tacticals help in structuring proofs they allow you to
+choose the amount of structure you want. There are no constraints imposed by
+the system, and if the user wants he can even write completely plain proofs.
-\item[Rereading]
- is possible. Going on step by step shows exactly what is going on. Consider
- again a proof by induction, that starts applying the induction principle and
- suddenly branches with a ``\texttt{[}''. This clearly separates all the
- induction cases, but if the square brackets content is executed in one single
- step you completely loose the possibility of rereading it and you have to
- temporary remove the branching tactical to execute in a satisfying way the
- branches. Again, executing step-by-step is the way you would like to review
- the demonstration. Remember that understanding the proof from the script is
- not easy, and only the execution of tactics (and the resulting transformed
- goal) gives you the feeling of what is going on.
-\end{description}
+Re-playing a proof is also made simpler. There is no longer any need
+to destructure the proof on the fly since \MATITA{} executes each
+tactical not atomically.
+
+%\item[Rereading]
+% is possible. Going on step by step shows exactly what is going on. Consider
+% again a proof by induction, that starts applying the induction principle and
+% suddenly branches with a ``\texttt{[}''. This clearly separates all the
+% induction cases, but if the square brackets content is executed in one single
+% step you completely loose the possibility of rereading it and you have to
+% temporary remove the branching tactical to execute in a satisfying way the
+% branches. Again, executing step-by-step is the way you would like to review
+% the demonstration. Remember that understanding the proof from the script is
+% not easy, and only the execution of tactics (and the resulting transformed
+% goal) gives you the feeling of what is going on.
+%\end{description}
\section{Standard library}
+\label{sec:stdlib}
\MATITA{} is \COQ{} compatible, in the sense that every theorem of \COQ{}
can be read, checked and referenced in further developments.
The current library just comprises about one thousand theorems in
elementary aspects of arithmetics up to the multiplicative property for
Eulers' totient function $\phi$.
-The library is organized in five main directories: $logic$ (connectives,
-quantifiers, equality, $\dots$), $datatypes$ (basic datatypes and type
-constructors), $nat$ (natural numbers), $Z$ (integers), $Q$ (rationals).
-The most complex development is $nat$, organized in 25 scripts, listed
-in Figure\ref{scripts}
-\begin{figure}[htb]
-$\begin{array}{lll}
-nat.ma & plus.ma & times.ma \\
-minus.ma & exp.ma & compare.ma \\
-orders.ma & le\_arith.ma & lt\_arith.ma \\
-factorial.ma & sigma\_and\_pi.ma & minimization.ma \\
-div\_and\_mod.ma & gcd.ma & congruence.ma \\
-primes.ma & nth\_prime.ma & ord.ma\\
-count.ma & relevant\_equations.ma & permutation.ma \\
-factorization.ma & chinese\_reminder.ma & fermat\_little\_th.ma \\
-totient.ma& & \\
-\end{array}$
-\caption{\label{scripts}\MATITA{} scripts on natural numbers}
-\end{figure}
+The library is organized in five main directories: \texttt{logic} (connectives,
+quantifiers, equality, \ldots), \texttt{datatypes} (basic datatypes and type
+constructors), \texttt{nat} (natural numbers), \texttt{Z} (integers), \texttt{Q}
+(rationals). The most complex development is \texttt{nat}, organized in 25
+scripts, listed in Tab.~\ref{tab:scripts}.
+
+\begin{table}[ht]
+ \begin{tabular}{lll}
+ \FILE{nat.ma} & \FILE{plus.ma} & \FILE{times.ma} \\
+ \FILE{minus.ma} & \FILE{exp.ma} & \FILE{compare.ma} \\
+ \FILE{orders.ma} & \FILE{le\_arith.ma} & \FILE{lt\_arith.ma} \\
+ \FILE{factorial.ma} & \FILE{sigma\_and\_pi.ma} & \FILE{minimization.ma} \\
+ \FILE{div\_and\_mod.ma} & \FILE{gcd.ma} & \FILE{congruence.ma} \\
+ \FILE{primes.ma} & \FILE{nth\_prime.ma} & \FILE{ord.ma} \\
+ \FILE{count.ma} & \FILE{relevant\_equations.ma} & \FILE{permutation.ma} \\
+ \FILE{factorization.ma} & \FILE{chinese\_reminder.ma} &
+ \FILE{fermat\_little\_th.ma} \\
+ \FILE{totient.ma} & & \\
+ \end{tabular}
+ \caption{Scripts on natural numbers in the standard library\strut}
+ \label{tab:scripts}
+\end{table}
We do not plan to maintain the library in a centralized way,
-as most of the systems do. On the contary we are currently
+as most of the systems do. On the contrary we are currently
developing wiki-technologies to support a collaborative
development of the library, encouraging people to expand,
modify and elaborate previous contributions.
\section{Conclusions}
+\label{sec:conclusion}
+
+\TODO{conclusioni}
\acknowledgements
-We would like to thank all the students that during the past
-five years collaborated in the \HELM{} project and contributed to
+We would like to thank all the people that during the past
+7 years collaborated in the \HELM{} project and contributed to
the development of \MATITA{}, and in particular
M.~Galat\`a, A.~Griggio, F.~Guidi, P.~Di~Lena, L.~Padovani, I.~Schena, M.~Selmi,
and V.~Tamburrelli.
\bibliography{matita}
\end{document}
-
projects / helm.git / blobdiff