-\documentclass[]{kluwer}
+\documentclass[draft]{kluwer}
\usepackage{color}
\usepackage{graphicx}
% \usepackage{amssymb,amsmath}
\newcommand{\AUTO}{\textsc{Auto}}
\newcommand{\COQ}{Coq}
+\newcommand{\COQIDE}{CoqIde}
\newcommand{\ELIM}{\textsc{Elim}}
\newcommand{\GDOME}{Gdome}
\newcommand{\HELM}{Helm}
\newcommand{\LOCATE}{\textsc{Locate}}
\newcommand{\MATCH}{\textsc{Match}}
\newcommand{\MATITA}{Matita}
+\newcommand{\MATITAC}{\texttt{matitac}}
+\newcommand{\MATITADEP}{\texttt{matitadep}}
\newcommand{\METAHEADING}{Symbol & Position \\ \hline\hline}
\newcommand{\MOWGLI}{MoWGLI}
\newcommand{\NAT}{\ensuremath{\mathit{nat}}}
\institute{Department of Computer Science, University of Bologna\\
Mura Anteo Zamboni, 7 --- 40127 Bologna, ITALY}
-\runningtitle{The Matita proof assistant}
+\runningtitle{The \MATITA{} proof assistant}
\runningauthor{Asperti, Sacerdoti Coen, Tassi, Zacchiroli}
% \date{data}
\end{itemize}
According to our content-centric commitment, the library exported from
-Coq was conceived as being distributed and most of the tools were developed
+\COQ{} was conceived as being distributed and most of the tools were developed
as Web services. The user could interact with the library and the tools by
means of a Web interface that orchestrates the Web services.
\begin{itemize}
\item scelta del sistema fondazionale
- \item sistema indipendente (da Coq)
+ \item sistema indipendente (da \COQ)
\item compatibilit\`a con sistemi legacy
\end{itemize}
For those \components{} whose functionalities are also provided by the
aforementioned Web services, it is also possible to link stub code that
forwards the request to a remote Web service. For instance, the Getter
-is just a wrapper to the \texttt{getter} \component{} that allows the
+is just a wrapper to the \GETTER \component{} that allows the
\component{} to be used as a Web service. \MATITA{} can directly link the code
-of the \texttt{getter} \component, or it can use a stub library with the same
+of the \GETTER \component, or it can use a stub library with the same
API that forwards every request to the Getter.
To better understand the architecture of \MATITA{} and the role of each
Terms may reference other mathematical notions in the library.
One commitment of our project is that the library should be physically
- distributed. The \texttt{getter} \component{} manages the distribution,
+ distributed. The \GETTER \component{} manages the distribution,
providing a mapping from logical names (URIs) to the physical location
of a notion (an URL). The \texttt{urimanager} \component{} provides the URI
data type and several utility functions over URIs. The
- \texttt{cic\_proof\_checking} \component{} calls the \texttt{getter}
+ \texttt{cic\_proof\_checking} \component{} calls the \GETTER
\component{} every time it needs to retrieve the definition of a mathematical
notion referenced by a term that is being type-checked.
of preserving the coherence of the library and the database. For instance,
when a notion is removed, all the notions that depend on it and their
metadata are removed from the library. This aspect will be better detailed
- in Sect.~\ref{decompilazione}.
+ in Sect.~\ref{sec:libmanagement}.
\subsection{Partially specified terms}
\emph{Partially specified terms} are CIC terms where subterms can be omitted.
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{disambiguazione}.
+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
\subsection{Compilation and decompilation}
-\label{compilazione}
+\label{sec:libmanagement}
-The aim of this section is to describe the way matita
+%
+%goals: consentire sviluppo di una librearia mantenendo integrita' referenziale e usando le teconologie nostre (quindi con metadati, XML, libreria visibile)
+%\subsubsection{Composition}
+%scripts.ma, .moo, XML, metadata
+%\subsubsection{Compilation}
+%analogie con compilazione classica dso.\\
+%granularita' differenti per uso interattivo e non
+%\paragraph{Batch}
+%- granularita' .ma/buri \\
+%-- motivazioni\\
+%- come si calcolano le dipendenze\\
+%- quando la si usa\\
+%- metodi (cc e clean)\\
+%- garanzie
+%\paragraph{Interactive}
+%- granularita' fine\\
+%-- motivazioni
+%\label{sec:libmanagement}
+%consistenza: integrita' referenziale
+%Goals: mantenere consistente la rappresentazione della libreria su memoria persistente consentendo di compilare e decompilare le compilation unit (.ma).\\
+%Vincoli: dipendenze oggetti-oggetti e metadati-oggetti\\
+%Due livelli di gestione libreria, uno e' solo in fase interattiva dove la compilazione e' passo passo: \\
+%--- granularita' oggetto per matita interactive\\
+%--- granularita' baseuri (compilation unit) per la libreria\\
+%In entrmbi i casi ora:\\
+%--- matitaSync: add, remove, timetravel(facility-macro tra 2 stati)[obj]\\
+%--- matitaCleanLib: clean\_baseuri (che poi usa matitaSync a sua volta)[comp1]\\
+%Vincoli di add: typecheck ( ==$>$ tutto quello che usa sta in lib)\\
+%Vincoli di remove: \\
+%--- la remove di mSync non li controlla (ma sa cosa cancellare per ogni uri)\\
+%--- la clean\_baseuri calcola le dipendenze con i metadati (o anche i moo direi) e li rispetta\\
+%Undo di matita garantisce la consistenza a patto che l'history che tiene sia ok\\
+%Undo della lib (mClean) garantisce la consistenza (usando moo o Db).\\
+
+The aim of this section is to describe the way \MATITA{}
preserves the consistency and the availability of the library
-trough the \WHELP{} technology, in response to the user addition or
-deletion of mathematical objects.
+using the \WHELP{} technology, in response to the user addition or
+removal of mathematical objects.
As already sketched in \ref{fully-spec} the output of the
compilation of a script is split among two storage media, a
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{}. In addition the \emph{getter} component
-should be updated with the the new mapping between the logical URI
-and the physical path of objects.
+\WHELP{}.
+% Non serve piu' l'update: by --Zack
+% In addition the \GETTER{} component
+% should be updated with the the new mapping between the logical URI
+% and the physical path of objects.
While this kind of consistency has nothing to do with the nature of
-the content of the library and is thus of poor interest (but really
-tedious to implement and keep bug-free), there is a more deep
+the content of the library and is thus uninteresting (but really
+tedious to implement and keep bug-free), there is a deeper
notion of mathematical consistency we need to provide. Each object
must reference only defined object (i.e. each proof must use only
already proved theorems).
-We will focus on how matita ensures the interesting kind
+We will focus on how \MATITA{} ensures the interesting kind
of consistency during the formalization of a mathematical theory,
-giving the user the freedom of adding, deleting, modifying objects
+giving the user the freedom of adding, removing, modifying objects
without loosing the feeling of an always visible and browsable
library.
\subsubsection{Compilation}
+
The typechecker component guarantees that if an object is well typed
-it depends only on well defined objects available in the library,
+it depends only on well typed objects available in the library,
that is exactly what we need to be sure that the logic consistency of
-the library is preserved. We have only find the right order of
+the library is preserved. We have only to find the right order of
compilation of the scripts that compose the user development.
-For this purpose we developed a low level tool called \emph{matitadep}
+For this purpose we provide a tool called \MATITADEP{}
that takes in input the list of files that compose the development and
-outputs their dependencies in a format suitable for the make utility.
-The user is not asked to run \emph{matitadep} nor make by hand, but
-simply to tell matita the root directory of his development (where all
-script files can be found) and matita will handle all the compilation
-tasks.\\
-To calculate dependencies it is enough to look at the script file for
-its inclusions of other parts of the development or for explicit
+outputs their dependencies in a format suitable for the GNU \texttt{make} tool.
+The user is not asked to run \MATITADEP{} by hand, but
+simply to tell \MATITA{} the root directory of his development (where all
+script files can be found) and \MATITA{} will handle all the compilation
+related tasks, including dependencies calculation.
+To compute dependencies it is enough to look at the script files for
+inclusions of other parts of the development or for explicit
references to other objects (i.e. with explicit aliases, see
-\ref{aliases}).
+\ref{sec:disambaliases}).
The output of the compilation is immediately available to the user
trough the \WHELP{} technology, since all metadata are stored in a
user-specific area of the database where the search engine has read
access, and all the automated tactics that operates on the whole
-library, like auto, have full visibility of the newly defined objects.
+library, like \AUTO, have full visibility of the newly defined objects.
Compilation is rather simple, and the only tricky case is when we want
-to compile again the same script, maybe after the deletion of a
+to compile again the same script, maybe after the removal of a
theorem. Here the policy is simple: decompile it before recompiling.
As we will see in the next section decompilation will ensure that
there will be no theorems in the development that depends on the
-removed item.
+removed items.
\subsubsection{Decompilation}
-Decompiling an object involves,
-recursively, the decompilation of all the objects that depend on it.
+
+Decompiling an object involves the (recursive)
+decompilation of all the objects that depend on it.
The calculation of the reverse dependencies can be computed in two
ways, using the relational database or using a simpler set of metadata
-that matita saves in the filesystem as a result of compilation. The
+that \MATITA{} saves in the filesystem as a result of compilation. The
former technique is the same used by the \emph{Dependency Analyzer}
described in \cite{zack-master} and really depends on a relational
-database.\\
-The latter is a fall-back in case the database is not available. Due to
-the complex deployment of a complex peace of software like a database,
-it is a common usage for the \HELM{} team to use a single and remote
-database, that may result unavailable if the user workstation lacks
-connectivity. This facility has to be intended only as a fall-back,
-since the whole \WHELP{} technology depends on the database.
+database.
+
+The latter is a fall-back in case the database is not available.\footnote{Due to
+the complex deployment of a large piece of software like a database,
+it is a common practice for the \HELM{} team to use a shared remote
+database, that may be unavailable if the user workstation lacks
+network connectivity.} This facility has to be intended only as a fall-back,
+since the queries of the \WHELP{} technology depend require a working database.
Decompilation guarantees that if an object is removed there are no
dandling references to it, and that the part of the library still
compiled is logically consistent. Since decompilation involves the
-deletion of all the outputs of the compilation, metadata included, the
-library browsable trough the \WHELP{} technology is always up to date.
+removal of all the results of the compilation, metadata included, the
+library browsable trough the \WHELP{} technology is always kept up to date.
\subsubsection{Interactive and batch (de)compilation}
-Matita includes an interactive graphical interface and a batch
-compiler. Only the former is intended to be used directly by the
-user, the latter is automatically invoked when a not yet compiled
-part of the user development is required.
+
+\MATITA{} includes an interactive graphical interface and a batch
+compiler (\MATITAC). Only the former is intended to be used directly by the
+user, the latter is automatically invoked when a
+part of the user development is required (for example issuing an
+\texttt{include} command) but not yet compiled.
While they share the same engine for compilation and decompilation,
they provide different granularity. The batch compiler is only able to
-compile a whole script file and reciprocally it can decompile only a whole
+compile a whole script file and reciprocally to decompile only a whole
script, and consequently all the other scripts that rely on an object
defined in it. The interactive interface is able to execute single steps
of compilation, that may include the definition of an object, and
-symmetrically to undo single steps, thus removing single objects.
-
-%
-%goals: consentire sviluppo di una librearia mantenendo integrita' referenziale e usando le teconologie nostre (quindi con metadati, XML, libreria visibile)
-%\subsubsection{Composition}
-%scripts.ma, .moo, XML, metadata
-%\subsubsection{Compilation}
-%analogie con compilazione classica dso.\\
-%granularita' differenti per uso interattivo e non
-%\paragraph{Batch}
-%- granularita' .ma/buri \\
-%-- motivazioni\\
-%- come si calcolano le dipendenze\\
-%- quando la si usa\\
-%- metodi (cc e clean)\\
-%- garanzie
-%\paragraph{Interactive}
-%- granularita' fine\\
-%-- motivazioni
-%\label{sec:libmanagement}
-%consistenza: integrita' referenziale
-%Goals: mantenere consistente la rappresentazione della libreria su memoria persistente consentendo di compilare e decompilare le compilation unit (.ma).\\
-%Vincoli: dipendenze oggetti-oggetti e metadati-oggetti\\
-%Due livelli di gestione libreria, uno e' solo in fase interattiva dove la compilazione e' passo passo: \\
-%--- granularita' oggetto per matita interactive\\
-%--- granularita' baseuri (compilation unit) per la libreria\\
-%In entrmbi i casi ora:\\
-%--- matitaSync: add, remove, timetravel(facility-macro tra 2 stati)[obj]\\
-%--- matitaCleanLib: clean\_baseuri (che poi usa matitaSync a sua volta)[comp1]\\
-%Vincoli di add: typecheck ( ==$>$ tutto quello che usa sta in lib)\\
-%Vincoli di remove: \\
-%--- la remove di mSync non li controlla (ma sa cosa cancellare per ogni uri)\\
-%--- la clean\_baseuri calcola le dipendenze con i metadati (o anche i moo direi) e li rispetta\\
-%Undo di matita garantisce la consistenza a patto che l'history che tiene sia ok\\
-%Undo della lib (mClean) garantisce la consistenza (usando moo o Db).\\
+consequently to undo single steps, thus removing single objects.
\subsection{Automation}
-\subsection{Matita's naming convention}
-A minor but not entirely negligible aspect of Matita is that of
+\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
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.
+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
\section{The \MATITA{} user interface}
-
-
\subsection{Disambiguation}
+\label{sec:disambiguation}
Software applications that involve input of mathematical content should strive
to require the user as less drift from informal mathematics as possible. We
Sect.~\ref{sec:contentintro}.
The key component of the translation is the generic disambiguation algorithm
-implemented in the \texttt{disambiguation} library of Fig.~\ref{fig:libraries}
+implemented in the \texttt{disambiguation} component of Fig.~\ref{fig:libraries}
and presented in~\cite{disambiguation}. In this section we present how to use
such an algorithm in the context of the development of a library of formalized
mathematics. We will see that using multiple passes of the algorithm, varying
expressiveness.
\subsubsection{Disambiguation aliases}
-\label{aliases}
+\label{sec:disambaliases}
Let's start with the definition of the ``strictly greater then'' notion over
(Peano) natural numbers.
The \texttt{include} statement adds the requirement that the part of the library
defining the notion of natural numbers should be defined before
-processing the following definition. Note indeed that the algorithm presented
+processing the what follows. Note indeed that the algorithm presented
in~\cite{disambiguation} does not describe where interpretations for ambiguous
expressions come from, since it is application-specific. As a first
approximation, we will assume that in \MATITA{} they come from the library (i.e.
Unfortunately, none of the passes described above is able to disambiguate its
type, no matter how aliases are defined. This is because the \OP{<} operator
occurs twice in the content level term (it has two \emph{instances}) and two
-different interpretation for it have to be used in order to obtain a refinable
+different interpretations for it have to be used in order to obtain a refinable
partially specified term. To address this issue, we have the ability to consider
each instance of a single symbol as a different ambiguous expression in the
content level term, and thus we can assign a different interpretation to each of
others). For this reason we always attempt a fresh instances pass only after
attempting a non-fresh one.
+\paragraph{One-shot aliases} Disambiguation aliases as seen so far are
+instance-independent. However, aliases obtained as a result of a disambiguation
+pass which uses fresh instances ought to be instance-dependent, that is: to
+ensure a term can be disambiguated in a batch fashion we may need to state that
+an \emph{i}-th instance of a symbol should be mapped to a given partially
+specified term. Instance-depend aliases are meaningful only for the term whose
+disambiguation generated it. For this reason we call them \emph{one-shot
+aliases} and \MATITA{} doesn't use it to disambiguate further terms down in the
+script.
+
\subsubsection{Implicit coercions}
Let's now consider a (rather hypothetical) theorem about derivation:
twice that coercion on the left hand side of the implication. The obtained
partially specified term however would not probably be the expected one, being a
theorem which prove a trivial implication. For this reason we choose to always
-prefer fresh instances over implicit coercion, i.e. we always attempt
-disambiguation passes with fresh instances before attempting passes with
-implicit coercions.
+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}
-\TODO{spiegazione della tabella}
-
-\begin{center}
- \begin{tabular}{c|c|c|c}
- \multicolumn{1}{p{1.5cm}|}{\centering\raisebox{-1.5ex}{\textbf{Pass}}}
- & \multicolumn{1}{p{2.5cm}|}{\centering\textbf{Operator instances}}
- & \multicolumn{1}{p{3.1cm}|}{\centering\textbf{Disambiguation aliases}}
- & \multicolumn{1}{p{2.5cm}}{\centering\textbf{Implicit coercions}} \\
- \hline
- \PASS & Normal & Mono & Disabled \\
- \PASS & Normal & Multi & Disabled \\
- \PASS & Fresh & Mono & Disabled \\
- \PASS & Fresh & Multi & Disabled \\
- \PASS & Fresh & Mono & Enabled \\
- \PASS & Fresh & Multi & Enabled \\
- \PASS & Fresh & Library & Enabled
- \end{tabular}
-\end{center}
-
-\TODO{alias one shot}
-
-
-
-
-
+According to the criteria described above in \MATITA{} we choose to perform the
+sequence of disambiguation passes depicted in Tab.~\ref{tab:disambpasses}. In
+our experience that choice implements a good trade off among disambiguation time
+and admitted ambiguity in terms input by users.
+\begin{table}[ht]
+ \caption{Sequence of disambiguation passes used in \MATITA.\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{Disambiguation aliases}}
+ & \multicolumn{1}{p{2.5cm}|}{\centering\textbf{Operator instances}}
+ & \multicolumn{1}{p{2.5cm}}{\centering\textbf{Implicit coercions}} \\
+ \hline
+ \PASS & Mono aliases & Shared & Disabled \\
+ \PASS & Multi aliases & Shared & Disabled \\
+ \PASS & Mono aliases & Fresh instances & Disabled \\
+ \PASS & Multi aliases & Fresh instances & Disabled \\
+ \PASS & Mono aliases & Fresh instances & Enabled \\
+ \PASS & Multi aliases & Fresh instances & Enabled \\
+ \PASS & Library aliases& Fresh instances & Enabled
+ \end{tabular}
+ \end{center}
+\end{table}
\subsection{Patterns}
Patterns are the textual counterpart of the MathML widget graphical
selection.
-Matita benefits of a graphical interface and a powerful MathML rendering
+\MATITA{} benefits of a graphical interface and a powerful MathML rendering
widget that allows the user to select pieces of the sequent he is working
on. While this is an extremely intuitive way for the user to
restrict the application of tactics, for example, to some subterms of the
in un pattern phase1only come faccio nell'ultimo esempio. lo si fa
con una pattern\_of(select(pattern))}
-\subsubsection{Comparison with Coq}
-Coq has a two diffrent ways of restricting the application of tactis to
+\subsubsection{Comparison with \COQ{}}
+\COQ{} has a two diffrent ways of restricting the application of tactis to
subterms of the sequent, both relaying on the same special syntax to identify
a term occurrence.
\MATITA{} tacticals syntax is reported in table \ref{tab:tacsyn}.
While one would expect to find structured constructs like
$\verb+do+~n~\NT{tactic}$ the syntax allows pieces of tacticals to be written.
-This is essential for base idea behind matita tacticals: step-by-step execution.
+This is essential for 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
\item[Proof structuring]
is much easier. Consider for example a proof by induction, and imagine you
are using classical tacticals in one of the state of the
- art graphical interfaces for proof assistant like Proof General or Coq Ide.
+ art graphical interfaces for proof assistant like Proof General or \COQIDE.
After applying the induction principle you have to choose: structure
the proof or not. If you decide for the former you have to branch with
``\texttt{[}'' and write tactics for all the cases separated by
goal) gives you the feeling of what is going on.
\end{description}
-\section{The Matita library}
+\section{The \MATITA{} library}
-Matita is Coq compatible, in the sense that every theorem of Coq
+\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.
+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 $\phi$.
factorization.ma & chinese\_reminder.ma & fermat\_little\_th.ma \\
totient.ma& & \\
\end{array}$
-\caption{\label{scripts}Matita scripts on natural numbers}
+\caption{\label{scripts}\MATITA{} scripts on natural numbers}
\end{figure}
We do not plan to maintain the library in a centralized way,
\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
+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.
projects / helm.git / blobdiff