/me reviewed automation subsection

[helm.git] / helm / papers / matita / matita2.tex

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\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}

 \begin{motto}
  ``We are nearly bug-free'' -- \emph{CSC, Oct 2005}
 \end{motto}

 \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. This paper describes the overall architecture of
the system, focusing on its most distinctive and innovative 
features.

\subsection{Historical perspective}

The origins of \MATITA{} go back to 1999. At the time we were mostly 
interested in developing tools and techniques to enhance the accessibility
via Web of libraries of formalized mathematics. Due to its dimension, the
library of the \COQ~\cite{CoqManual} proof assistant (of the order of 35'000 theorems) 
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}.\TODO{citare la tesi di vincenzo(?)}
The work, mostly performed in the framework of the recently concluded 
European project \MOWGLIIST{} \MOWGLI~\cite{pechino}, mainly consisted in the 
following steps:
\begin{enumerate}

 \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 that technology,
  fostering 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 built;

 \item developing indexing and searching techniques supporting semantic
  queries to the library; 

 \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.} 

\end{enumerate}

According to our content-centric commitment, the library exported from
\COQ{} was conceived as being distributed and most of the tools were developed
as Web services. The user can interact with the library and the tools by
means of a Web interface that orchestrates the Web services.

Web services and other tools have been implemented as front-ends
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 components:
\begin{itemize}

 \item XML specifications for the Calculus of Inductive Constructions,
  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 (i.e. the \emph{kernel} of a proof assistant), 
  implemented to check that we exported from the \COQ{} library all the 
  logically relevant content;

 \item a sophisticated term parser (used by the search engine), able to deal 
  with potentially ambiguous and incomplete information, typical of the 
  mathematical notation~\cite{disambiguation};

 \item a \emph{refiner} component, 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~\cite{padovani}
  for the \GTK{} graphical environment,\footnote{\url{http://www.gtk.org/}}
  supporting high-quality bidimensional
  rendering, and semantic selection, i.e. the possibility to select semantically
  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
too far away: essentially, we ``just'' had to
add an authoring interface, and a set of functionalities for the
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~\cite{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~\cite{lcf}. \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. Playing that 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.
On the contrary, the interface to interact with the library is rather
innovative and directly inspired by the Web interfaces to our Web servers.

\MATITA{} is backward compatible with the XML library of proof objects exported
from \COQ{}, but, in order to test the actual usability of the system, we are
also developing a new library of basic results from scratch.

\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 implementation
language (\OCAML\footnote{\url{http://caml.inria.fr/}}),
and the same (procedural, script based) authoring philosophy.
However, the analogy essentially stops here and no code is shared
between the two systems.

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 reduce our code even further in a sensible way).

Moreover, the complexity of the code of \MATITA{} is greatly reduced with
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 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{}\cite{CoqArt} is a quite old
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 maintaining backward compatibility
is very difficult. Finally, the code of \COQ{} has been greatly optimized
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 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 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.

%For direct experience of the authors, the learning curve to understand and
%be able to contribute to \COQ{}'s code is quite steep and requires direct
%and frequent interactions with \COQ{} developers.

\section{Architecture}
\label{architettura}

\begin{figure}[!ht]
 \begin{center}
  \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. Each node is annotated with the number of lines of
source code (comprising comments).

Applications and \components{} depend on other \components{} forming a
directed acyclic graph (DAG). Each \component{} can be decomposed in
a set of \emph{modules} also forming a DAG.

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.

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}~\cite{zack-master} 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.

 \item \emph{\WHELP}~\cite{whelp} is a search engine to index and
  locate mathematical concepts (axioms, theorems, definitions) in the
  logical library managed by the Getter. Typical examples of
  \WHELP{} queries are those that search for a theorem that generalize or
  instantiate a given formula, or that can be immediately applied to
  prove a given goal. The output of Whelp is an XML document that lists
  the URIs of a complete set of candidates that are likely to satisfy
  the given query. The set is complete in the sense that no concept that
  actually satisfies the query is thrown away. However, the query is
  only approximated in the sense that false matches can be returned.

 \item \emph{\UWOBO}~\cite{zack-master} is a Web service that, given the
  URI of a mathematical concept in the distributed library, renders it
  according to the user provided two dimensional mathematical notation.
  \UWOBO{} may also inline the rendering of mathematical concepts into
  arbitrary documents before returning them.  The Getter is used by
  \UWOBO{} to retrieve the document to be rendered.

 \item The \emph{Proof Checker}~\cite{zack-master} is a Web service
  that, given the URI of a concept in the distributed library, checks its
  correctness. Since the concept is likely to depend in an acyclic way
  on other concepts, the proof checker is also responsible of building
  in a top-down way the DAG of all dependencies, checking in turn every
  concept for correctness.

 \item The \emph{Dependency Analyzer}~\cite{zack-master} is a Web
  service that can produce a textual or graphical representation of the
  dependencies of a concept.

\end{itemize}

The dependency of a \component{} or application over another \component{} can
be satisfied by linking the \component{} in the same executable.
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{} application is just a wrapper to the \GETTER{} \component{}
that allows it 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 API that forwards every request to the Web service.

To better understand the architecture of \MATITA{} and the role of each
\component, we can focus on the representation of the mathematical
information.  In CIC terms are used to represent mathematical formulae,
types and proofs. \MATITA{} is able to handle terms at four different
levels of specification. On each level it is possible to provide a
different set of functionalities. The four different levels are: fully
specified terms; partially specified terms; content level terms;
presentation level terms.

\subsection{Fully specified terms}
\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 concepts (axioms, definitions, theorems) that are stored
   in our mathematical library are fully specified and well-typed terms.
   Fully specified terms are extremely verbose (to make type-checking
   decidable). Their syntax is fixed and does not resemble the usual
   extendible mathematical notation. They are not meant for direct user
   consumption.

   The \texttt{cic} \component{} defines the data type that represents CIC terms
   and provides a parser for terms stored in XML format.

   The most important \component{} that deals with fully specified terms is
   \texttt{cic\_proof\_checking}. It implements the procedure that verifies
   if a fully specified term is well-typed. It also implements the
   \emph{conversion} judgement that verifies if two given terms are
   computationally equivalent (i.e. they share the same normal form).

   Terms may reference other mathematical concepts in the library.
   One commitment of our project is that the library should be physically
   distributed. The \GETTER{} \component{} manages the distribution,
   providing a mapping from logical names (URIs) to the physical location
   of a concept (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{}
   \component{} every time it needs to retrieve the definition of a mathematical
   concept referenced by a term that is being type-checked. 

   The Proof Checker application is the Web service that provides an interface
   to the \texttt{cic\_proof\_checking} \component.

   We use metadata and a sort of crawler to index the mathematical concepts
   in the distributed library. We are interested in retrieving a concept
   by matching, instantiation or generalization of a user or system provided
   mathematical formula. Thus we need to collect metadata over the fully
   specified terms and to store the metadata in some kind of (relational)
   database for later usage. The \texttt{hmysql} \component{} provides
   a simplified
   interface to a (possibly remote) MySQL\footnote{\url{http://www.mysql.com/}}
   database system used to store the metadata.
   The \texttt{metadata} \component{} defines the data type of the metadata
   we are collecting and the functions that extracts the metadata from the
   mathematical concepts (the main functionality of the crawler).
   The \texttt{whelp} \component{} implements a search engine that performs
   approximated queries by matching/instantiation/generalization. The queries
   operate only on the metadata and do not involve any actual matching
   (see the \texttt{cic\_unification} \component in
   Sect.~\ref{sec:partiallyintro}). Not performing any actual matching
   a query only returns a complete and hopefully small set of matching
   candidates. The process that has issued the query is responsible of
   actually retrieving from the distributed library the candidates to prune
   out false matches if interested in doing so.

   The \WHELP{} application is the Web service that provides an interface to
   the \texttt{whelp} \component.

   According to our vision, the library is developed collaboratively so that
   changing or removing a concept can invalidate other concepts in the library.
   Moreover, changing or removing a concept requires a corresponding change
   in the metadata database. The \texttt{library} \component{} is responsible
   of preserving the coherence of the library and the database. For instance,
   when a concept is removed, all the concepts that depend on it and their
   metadata are removed from the library. This aspect will be better detailed
   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
linearly. The latters may occur multiple times and are called
\emph{metavariables}. An \emph{explicit substitution} is applied to each
occurrence of a metavariable. A metavariable stands for a term whose type is
given by the conclusion of the sequent. The term must be closed in the
context that is given by the ordered list of hypotheses of the sequent.
The explicit substitution instantiates every hypothesis with an actual
value for the variable bound by the hypothesis.

Partially specified terms are not required to be well-typed. However a
partially specified term should be \emph{refinable}. A \emph{refiner} is
a type-inference procedure that can instantiate implicit terms and
metavariables and that can introduce
\emph{implicit coercions}~\cite{barthe95implicit} to make a
partially specified term well-typed. The refiner of \MATITA{} is implemented
in the \texttt{cic\_unification} \component. As the type checker is based on
the conversion check, the refiner is based on \emph{unification} that is
a procedure that makes two partially specified term convertible by instantiating
as few as possible metavariables that occur in them.

Since terms are used in CIC to represent proofs, correct incomplete
proofs are represented by refinable partially specified terms. The metavariables
that occur in the proof correspond to the conjectures still to be proved.
The sequent associated to the metavariable is the conjecture the user needs to
prove.

\emph{Tactics} are the procedures that the user can apply to progress in the
proof. A tactic proves a conjecture possibly creating new (and hopefully
simpler) conjectures. The implementation of tactics is given in the
\texttt{tactics} \component. It is heavily based on the refinement and
unification procedures of the \texttt{cic\_unification} \component.

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.
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.

As fully specified terms, partially specified terms are not well suited
for user consumption since their syntax is not extendible and it is not
possible to adopt the usual mathematical notation. However they are already
an improvement over fully specified terms since they allow to omit redundant
information that can be inferred by the refiner.

\subsection{Content level terms}
\label{sec:contentintro}

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.

For instance, it is common practice in mathematics to speak of a generic
equality that can be used to compare any two terms. However, it is well known
that several equalities can be distinguished as soon as we care for decidability
or for their computational properties. For instance equality over real
numbers is well known to be undecidable, whereas it is decidable over
rational numbers.

Similarly, we usually speak of natural numbers and their operations and
properties without caring about their representation. However the computational
properties of addition over the binary representation are very different from
those of addition over the unary representation. And addition over two natural
numbers is definitely different from addition over two real numbers.

Formalized mathematics cannot hide these differences and obliges the user to be
very precise on the types he is using and their representation. However,
to communicate formulae with the user and with external tools, it seems good
practice to stick to the usual imprecise mathematical ontology. In the
Mathematical Knowledge Management community this imprecise language is called
the \emph{content level}~\cite{adams} representation of formulae.

In \MATITA{} we provide translations from partially specified terms
to content level terms and the other way around. The first translation can also
be applied to fully specified terms since a fully specified term is a special
case of partially specified term where no metavariable or implicit term occurs.

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 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
for the representation of content level formulae in an extensible XML format.

The translation to content level is implemented in the
\texttt{acic\_content} \component. Its input are \emph{annotated partially
specified terms}, that are maximally unshared
partially specified terms enriched with additional typing information for each
subterm. This information is used to discriminate between terms that represent
proofs and terms that represent formulae. Part of it is also stored at the
content level since it is required to generate the natural language rendering
of proofs. The terms need to be maximally unshared (i.e. they must be a tree
and not a DAG). The reason is that to different occurrences of a subterm
we need to associate different typing information.
This association is made easier when the term is represented as a tree since
it is possible to label each node with an unique identifier and associate
the typing information using a map on the identifiers.
The \texttt{cic\_acic} \component{} unshares and annotates terms. It is used
by the \texttt{library} \component{} since fully specified terms are stored
in the library in their annotated form.

We do not provide yet a reverse translation from content level proofs to
partially specified terms. But in \texttt{cic\_disambiguation} we do provide
the reverse translation for formulae. The mapping from
content level formulae to partially specified terms is not unique due to
the ambiguity of the content level. As a consequence the translation
is guided by an \emph{interpretation}, that is a function that chooses for
every ambiguous formula one partially specified term. The
\texttt{cic\_disambiguation} \component{} implements the
disambiguation algorithm presented in~\cite{disambiguation} that is
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 Sect.~\ref{sec:partiallyintro} we described the semantics of
a command as a
function from status to status. We also hinted 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
of $x$ to be replaced is located, its context is unknown. Since $y^2$ must
replace $x$ in that context, its encoding as a term cannot be computed
until $x$ is located. In other words, $y^2$ must be disambiguated in the
context of the occurrence $x$ it must replace.

The elegant solution we have implemented consists in representing terms
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 level term to be disambiguated. Our solution should be compared with
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.  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. Moreover, 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
is called \emph{presentation level}.

The main important difference between the content level language and the
presentation level language is that only the former is extendible. Indeed,
the presentation level language is a finite language that comprises all
the usual mathematical symbols. Mathematicians invent new notions every
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{}
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.

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
presentation
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 the implementation of \emph{semantic
selection}.

Semantic selection is a technique that consists in enriching the presentation
level terms with pointers to the content level terms and to the partially
specified terms they correspond to. Highlight of formulae in the widget is
constrained to selection of meaningful expressions, i.e. expressions that
correspond to a lower level term, that is a content term or a partially or
fully specified term.
Once the rendering of a lower level term is
selected it is possible for the application to retrieve the pointer to the
lower level term. An example of applications of semantic selection is
\emph{semantic copy \& paste}: the user can select an expression and paste it
elsewhere preserving its semantics (i.e. the partially specified term),
possibly performing some semantic transformation over it (e.g. renaming
variables that would be captured or lambda-lifting free variables).

The reverse translation from presentation level terms to content level terms
is implemented by a parser that is also found in \texttt{content\_pres}.
Differently from the translation from content level terms to partially
refined terms, this translation is not ambiguous. The reason is that the
parsing tool we have adopted (CamlP4) is not able to parse ambiguous
grammars. Thus we require the mapping from presentation level terms
(concrete syntax) to content level terms (abstract syntax) to be unique.
This means that the user must fix once and for all the associativity and
precedence level of every operator he is using. In practice this limitation
does not seem too strong. The reason is that the target of the
translation is an ambiguous language and the user is free to associate
to every content level term several different interpretations (as a
partially specified term).

Both the direct and reverse translation from presentation to content level
terms are parameterized over the user provided mathematical notation. 
The \texttt{lexicon} \component{} is responsible of managing the lexicon,
that is the set of active notations. It defines an abstract syntax tree
of commands to declare and activate new notations and it implements the
semantics of these commands. It also implements undoing of the semantic
actions. Among the commands there are hints to the
disambiguation algorithm that are used to control and speed up disambiguation.
These mechanisms will be further discussed in Sect.~\ref{sec:disambiguation}.

Finally, the \texttt{grafite\_parser} \component{} implements a parser for
the concrete syntax of the commands of \MATITA. The parser process a stream
of characters and returns a stream of abstract syntax trees (the ones
defined by the \texttt{grafite} component and whose semantics is given
by \texttt{grafite\_engine}). When the parser meets a command that changes
the lexicon, it invokes the \texttt{lexicon} \component{} to immediately
process the command. When the parser needs to parse a term at the presentation
level, it invokes the already described parser for terms contained in
\texttt{content\_pres}.

The \MATITA{} proof assistant and the \WHELP{} search engine are both linked
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 (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.
To render a document given its URI, \UWOBO{} retrieves it using the
\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.

The \components{} not yet described (\texttt{extlib}, \texttt{xml},
\texttt{logger}, \texttt{registry} and \texttt{utf8\_macros}) are 
minor \components{} that provide a core of useful functions and basic
services missing from the standard library of the programming language.
%In particular, the \texttt{xml} \component{} is used to easily represent,
%parse and pretty-print XML files.

\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.

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.
Once they are produced we store them independently in the library.
The only relation implicitly kept between the concepts are the logical,
acyclic dependencies among them. This way the library forms a global (and
distributed) hypertext.

\begin{figure}[!ht]
 \begin{center}
  \includegraphics[width=0.45\textwidth]{pics/cicbrowser-screenshot-browsing}
  \hspace{0.05\textwidth}
  \includegraphics[width=0.45\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}).
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 concept 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 concepts together. They also constitute a less fine grained clustering
of concepts 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 a concept $s$ refers a concept $t$ 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 \emph{duplicate check}.
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 future, we expect to extend the \HINT{} query 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 ingredient 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 detail 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 that states 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 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}

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 3 passes.  In the first pass, called
\emph{mono-preferences}, we consider only the aliases corresponding to the
current set of 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,
enabling the use of 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
shared instances pass.

\paragraph{One-shot preferences} Disambiguation preferences 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}~\cite{barthe95implicit} 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).

Implicitc 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
2 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 proves 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 minimizes 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.
%
%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{Invalidation}

Invalidation (see Sect.~\ref{sec:library}) is implemented in 2 phases.

The first one is the calculation of all the concepts that recursively
depend on the ones we are invalidating. It can be performed
using the relational database that stores the 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.

To regenerate an invalidated part of the library \MATITA{} re-executes
the scripts 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 scripts that compose the development and
outputs their dependencies in a format suitable for the GNU \texttt{make}
tool.\footnote{\url{http://www.gnu.org/software/make/}}
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 generation
related tasks, including dependencies calculation.

To compute dependencies it is enough to look at the script files for
literal of included explicit disambiguation preferences
(see Sect.~\ref{sec:disambaliases}). 

\TODO{da rivedere: da dove salta fuori ``regenerating content''?}
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 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 regenerate parts of the library previously invalidated.

\TODO{come sopra: ``content of a script''?}
While they share the same engine for generation and invalidation, they
provide different granularity. \MATITAC{} is only able to re-execute a
whole script and similarly to invalidate the whole content of a script
(together with all the other scripts that rely on a concept defined
in it). 

\subsection{Automation}
\label{sec:automation}

In the long run, one would expect to work with a proof assistant 
like \MATITA, using only 3 basic tactics: \TAC{intro}, \TAC{elim},
and \TAC{auto}
(possibly integrated by a moderate use of \TAC{cut}). The state of the art
in automated deduction is still far away from this goal, but 
this is one of the main development direction of \MATITA. 

Even in this field, the underlying philosophy of \MATITA{} is to 
free the user from any burden relative to the overall management
of the library. For instance, in \COQ, the user is responsible to 
define small collections of theorems to be used as a parameter 
by the \TAC{auto} tactic;
in \MATITA, it is the system itself that automatically retrieves, from
the whole library, a subset of theorems worth to be considered 
according to the signature of the current goal and context. 

The basic tactic merely iterates the use of the \TAC{apply} tactic
(with no \TAC{intro}). The search tree may be pruned according to 2 
main parameters: the \emph{depth} (whit the obvious meaning), and the 
\emph{width} that is the maximum number of (new) open goals allowed at
any instant. \MATITA{} has only one notion of metavariable, corresponding
to the so called existential variables of Coq; so, \MATITA's \TAC{auto}
tactic should be compared with \COQ's \TAC{EAuto} tactic.

Recently we have extended automation with paramodulation based 
techniques. At present, the system works reasonably well with
equational rewriting, where the notion of equality is parametric
and can be specified by the user: the system only requires 
a proof of {\em reflexivity} and {\em paramodulation} (or rewriting, 
as it is usually called in the proof assistant community).

Given an equational goal, \MATITA{} recovers all known equational facts
from the library (and the local context), applying a variant of
the so called {\em given-clause algorithm}~\cite{paramodulation}, 
that is the the procedure currently used by the majority of modern
automatic theorem provers. 

The given-clause algorithm is essentially composed by an alternation
of a \emph{saturation} phase and a \emph{demodulation} phase.
The former derives new facts by a set of active
facts and a new \emph{given} clause suitably selected from a set of passive
equations. The latter tries to simplify the equations
orienting them according to a suitable weight associated to terms.
\MATITA{} currently supports several different weigthing functions
comprising Knuth-Bendix ordering (kbo) and recursive path ordering (rpo), 
that integrates particularly well with normalization.

Demodulation alone is already a quite powerful technique, and 
it has been turned into a tactic by itself: the \TAC{demodulate}
tactic, which can be seen as a kind of generalization of \TAC{simplify}. 
The following portion of script describes two
interesting cases of application of this tactic (both of them relying 
on elementary arithmetic equations):

\begin{grafite}
theorem example1: 
  \forall x: nat. (x+1)*(x-1) = x*x - 1.
intro.
apply (nat_case x)
[ simplify; reflexivity;
| intro; demodulate; reflexivity; ]
qed.
\end{grafite}

\begin{grafite}
theorem example2: 
  \forall x, y: nat. (x+y)*(x+y) = x*x + 2*x*y + y*y.
intros; demodulate; reflexivity;
qed.
\end{grafite}

In the future we expect to integrate applicative and equational 
rewriting. In particular, the overall idea would be to integrate
applicative rewriting with demodulation, treating saturation as an
operation to be performed in batch mode, e.g. during the night. 

\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 concept names, derived by 
our studies about metadata for statements. 
The convention is only applied to theorems 
(not definitions), and relates theorem names to their statements.
The basic rules are the following:
\begin{itemize}

 \item each name is composed by an ordered list of (short)
  identifiers occurring in a left to right traversal of the statement; 

 \item all names should (but this is not strictly compulsory) 
  separated by an underscore;

 \item names occurring in 2 different hypotheses, or in an hypothesis
  and in the conclusion must be separated by the string \texttt{\_to\_};

 \item the identifier may be followed by a numerical suffix, or a
  single or double apostrophe.

\end{itemize}

Take for instance the statement:
\begin{grafite}
 \forall n: nat. n = plus n O
\end{grafite}
Possible legal names are: \texttt{plus\_n\_O}, \texttt{plus\_O}, 
\texttt{eq\_n\_plus\_n\_O} and so on.

Similarly, consider the theorem:
\begin{grafite}
 \forall n, m: nat. n < m to n \leq m
\end{grafite}
In this case \texttt{lt\_to\_le} is a legal name, 
while \texttt{lt\_le} is not.

But what about, say, the symmetric law of equality? Probably you would like 
to name such a theorem with something explicitly recalling symmetry.
The correct approach, 
in this case, is the following. You should start with defining the 
symmetric property for relations:
\begin{grafite}
definition symmetric =
  \lambda A: Type. \lambda R. \forall x, y: A.
    R x y \to R y x
\end{grafite}
Then, you may state the symmetry of equality as:
\begin{grafite}
\forall A: Type. symmetric A (eq A)
\end{grafite}
and \texttt{symmetric\_eq} is a legal name for such a theorem. 

So, somehow unexpectedly, the introduction of semi-rigid naming convention
has an important beneficial effect on the global organization of the library, 
forcing the user to define abstract concepts and properties before 
using them (and formalizing such use).

Two cases have a special treatment. The first one concerns theorems whose
conclusion is a (universally quantified) predicate variable, i.e. 
theorems of the shape
$\forall P,\dots,.P(t)$.
In this case you may replace the conclusion with the string
\texttt{elim} or \texttt{case}.
For instance the name \texttt{nat\_elim2} is a legal name for the double
induction principle.

The other special case is that of statements whose conclusion is a
match expression. 
A typical example is the following:
\begin{grafite}
\forall n,m:nat. 
  match (eqb n m) with
  [ true  \Rightarrow n = m 
  | false \Rightarrow n \neq m]
\end{grafite}
where \texttt{eqb} is boolean equality.
In this cases, the name can be build starting from the matched
expression and the suffix \texttt{\_to\_Prop}. In the above example, 
\texttt{eqb\_to\_Prop} is accepted. 

\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}

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}
  \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{figure}

\subsection{Patterns}
\label{sec:patterns}

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.

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}

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{Patterns concrete syntax\strut}
 \label{tab:pathsyn}
\hrule
\[
\begin{array}{@{}rcll@{}}
  \NT{pattern} & 
    ::= & [~\verb+in+~\NT{sequent\_path}~]~[~\verb+match+~\NT{wanted}~] & \\
  \NT{sequent\_path} & 
    ::= & \{~\NT{ident}~[~\verb+:+~\NT{multipath}~]~\}~
      [~\verb+\vdash+~\NT{multipath}~] & \\
  \NT{multipath} & ::= & \NT{term\_with\_placeholders} & \\
  \NT{wanted} & ::= & \NT{term} & \\
\end{array}
\]
\hrule
\end{table}

\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]
  concerns only the $[~\verb+in+~\NT{sequent\_path}~]$
  part of the syntax. $\NT{ident}$ is an hypothesis name and
  selects the assumption where the following optional $\NT{multipath}$
  will operate. \verb+\vdash+ can be considered the name for the goal.
  If the whole pattern is omitted, the whole goal will be selected.
  If one or more hypotheses names are given the selection is restricted to 
  these assumptions. If a $\NT{multipath}$ is omitted the whole
  assumption is selected. Remember that the user can be mostly
  unaware of this syntax, since the system is able to write down a 
  $\NT{sequent\_path}$ starting from a visual selection.
  \NOTE{Questo ancora non va in matita}

  A $\NT{multipath}$ is a CIC term in which a special constant $\%$
  is allowed.
  The roots of discharged subterms are marked with $?$, while $\%$
  is used to select roots. The default $\NT{multipath}$, the one that
  selects the whole term, is simply $\%$.
  Valid $\NT{multipath}$ are, for example, $(?~\%~?)$ or $\%~\verb+\to+~(\%~?)$
  that respectively select the first argument of an application or
  the source of an arrow and the head of the application that is
  found in the arrow target.

  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 $[~\verb+match+~\NT{wanted}~]$
  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 that can be unified to
  $\NT{wanted}$. The result of this search is the selection the
  pattern represents.

\end{description}

\subsubsection{Examples}
%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}

Consider the following sequent 
\sequent{
n:nat\\
m:nat\\
H: m + n = n}{
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}

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 H match n with (O + n).
\end{grafite}

In this case the $\NT{sequent\_path}$ selects the whole $H$, while
the second phase locates $n$.

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}

\subsubsection{Tactics supporting patterns}

\TODO{Grazie ai pattern, rispetto a Coq noi abbiamo per esempio la possibilita' di fare riduzioni profonde!!!}

\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.

\NOTE{attualmente rewrite e fold non supportano phase 2. per
supportarlo bisogna far loro trasformare il pattern phase1+phase2 
in un pattern phase1only come faccio nell'ultimo esempio. lo si fa
con una pattern\_of(select(pattern))}

\subsubsection{Comparison with \COQ{}}

\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 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
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:

\begin{grafite}
  change n at 2 in H with (O + n)
\end{grafite} 

meaning that in the hypothesis $H$ the $n$ we want to change is the
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:
\begin{grafite}
  pattern n at 2 in H
\end{grafite}

would have resulted in this sequent:

\begin{grafite}
  n : nat
  m : nat
  H : (fun n0 : nat => m + n = n0) n
  ============================
   m = 0
\end{grafite}

where $H$ is $\beta$-expanded over the second $n$
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 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 
more effective way to tell the system what to do. 

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}
\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 -> ((A -> B) /\ (B -> A)).
  intros [A B H].
  split; intro; 
    [ rewrite < H; assumption
    | rewrite > H; assumption
    ].
Qed.
\end{grafite}

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.

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 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. 

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, he can only replace all the cases with the identity
tactic and execute the command, just to receive feedback on the first
goal. Then he 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{Concrete syntax of tacticals\strut}
 \label{tab:tacsyn}
\hrule
\[
\begin{array}{@{}rcll@{}}
  \NT{punctuation} & 
    ::= & \SEMICOLON \quad|\quad \DOT \quad|\quad \SHIFT \quad|\quad \BRANCH \quad|\quad \MERGE \quad|\quad \POS{\mathrm{NUMBER}~} & \\
  \NT{block\_kind} & 
    ::= & \verb+focus+ ~|~ \verb+try+ ~|~ \verb+solve+ ~|~ \verb+first+ ~|~ \verb+repeat+ ~|~ \verb+do+~\mathrm{NUMBER} & \\
  \NT{block\_delimiter} & 
    ::= & \verb+begin+ ~|~ \verb+end+ & \\
  \NT{tactical} & 
    ::= & \verb+skip+ ~|~ \NT{tactic} ~|~ \NT{block\_delimiter} ~|~ \NT{block\_kind} ~|~ \NT{punctuation} ~|~& \\
\end{array}
\]
\hrule
\end{table}

\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 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 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 he 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.
  
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. 
However, in order to test the actual usability of the system, a
new library of results has been started from scratch. In this case, 
of course, we wrote (and offer) the source script files, 
while, in the case of \COQ, \MATITA{} may only rely on XML files of
\COQ{} objects. 
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: \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 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 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.

\theendnotes

\TODO{rivedere bibliografia, \'e un po' povera}

\TODO{aggiungere entry per le coercion implicite}

\bibliography{matita}

\end{document}