-\documentclass[draft]{kluwer}
+\documentclass[]{kluwer}
\usepackage{color}
\usepackage{graphicx}
% \usepackage{amssymb,amsmath}
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
-following teps:
+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 for future system, 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};
+ 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; in particular, 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.
+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.
\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 libraries, collectively called the \HELM{} libraries.
+to a set of software libraries, collectively called the \HELM{} libraries.
At the end of the \MOWGLI{} project we already disposed of the following
-techniques and libraries:
+tools and software libraries:
\begin{itemize}
\item XML specifications for the Calculus of Inductive Constructions,
with libraries for parsing and saving mathematical objects in such a format
partially specified terms, used by the disambiguating parser;
\item complex transformation algorithms for proof rendering in natural
language \cite{remathematization};
-\item an innovative rendering widget \cite{padovani}, supporting
+\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
a different text area.
\end{itemize}
-Starting from all this, the further step of developing our own
-proof assistant was too
-small and too tempting to be neglected. Essentially, we ``just'' had to
+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.
\end{center}
\end{figure}
+\begin{figure}[t]
+ \begin{center}
+ \includegraphics[width=0.9\textwidth]{libraries.ps}
+ \caption{\MATITA{} libraries}
+ \label{fig:libraries}
+ \end{center}
+\end{figure}
+
\section{Overview of the Architecture}
Fig.~\ref{fig:libraries} shows the architecture of the \emph{libraries} (circle nodes)
and \emph{applications} (squared nodes) developed in the HELM project.
\TODO{alias one shot}
+\section{The logical library}
+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: $logic$ (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}
+
+We do not plan to maintain the library in a centralized way,
+as most of the systems do. On the contary we are currently
+developing wiki-technologies to support a collaborative
+development of the library, encouraging people to expand,
+modify and elaborate previous contributions.
+
+\subsection{Matita's naming convention}
+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.
+The convention is only applied to identifiers for theorems
+(not definitions), and relates the name of a proof to its statement.
+The basic rules are the following:
+\begin{itemize}
+\item each identifier is composed by an ordered list of (short)
+names occurring in a left to right traversal of the statement;
+\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_+'';
+\item the identifier may be followed by a numerical suffix, or a
+single or duoble apostrophe.
+
+\end{itemize}
+Take for instance the theorem
+\[\forall n:nat. n = plus \; n\; O\]
+Possible legal names are: \verb+plus_n_O+, \verb+plus_O+,
+\verb+eq_n_plus_n_O+ and so on.
+Similarly, consider the theorem
+\[\forall n,m:nat. n<m \to n \leq m\]
+In this case \verb+lt_to_le+ is a legal name,
+while \verb+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
+
+\[definition\;symmetric\;= \lambda A:Type.\lambda R.\forall x,y:A.R x y \to R y x \]
+
+Then, you may state the symmetry of equality as
+\[ \forall A:Type. symmetric \;A\;(eq \; A)\]
+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,
+forcing the user to define abstract notions 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 word
+``elim'' or ``case''.
+For instance the name \verb+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{verbatim}
+ \forall n,m:nat.
+ match (eqb n m) with
+ [ true \Rightarrow n = m
+ | false \Rightarrow n \neq m]
+\end{verbatim}
+where $eqb$ is boolean equality.
+In this cases, the name can be build starting from the matched
+expression and the suffix \verb+_to_Prop+. In the above example,
+\verb+eqb_to_Prop+ is accepted.
+
+
+\section{Conclusions}
+
\acknowledgements
We would like to thank all the students that during the past
five years collaborated in the \HELM{} project and contributed to
the development of Matita, and in particular
-A.~Griggio, F.~Guidi, P.~Di~Lena, L.~Padovani, I.~Schena, M.~Selmi,
+M.~Galat\`a, A.~Griggio, F.~Guidi, P.~Di~Lena, L.~Padovani, I.~Schena, M.~Selmi,
and V.~Tamburrelli.
\theendnotes
projects / helm.git / blobdiff