means of a Web interface that orchestrates the Web services.
The Web services and the other tools have been implemented as front-ends
-to a set of software libraries, collectively called the \HELM{} libraries.
+to a set of software components, collectively called the \HELM{} components.
At the end of the \MOWGLI{} project we already disposed of the following
-tools and software libraries:
+tools and software components:
\begin{itemize}
\item XML specifications for the Calculus of Inductive Constructions,
-with libraries for parsing and saving mathematical objects in such a format
+with components for parsing and saving mathematical objects in such a format
\cite{exportation-module};
-\item metadata specifications with libraries for indexing and querying the
+\item metadata specifications with components for indexing and querying the
XML knowledge base;
\item a proof checker library (i.e. the {\em kernel} of a proof assistant),
-implemented to check that we exported form the \COQ{} library all the
+implemented to check that we exported from the \COQ{} library all the
logically relevant content;
\item a sophisticated parser (used by the search engine), able to deal
with potentially ambiguous and incomplete information, typical of the
graphical environment\cite{padovani}, supporting
high-quality bidimensional
rendering, and semantic selection, i.e. the possibility to select semantically
-meaningful rendering expressions, and to past the respective content into
+meaningful rendering expressions, and to paste the respective content into
a different text area.
\end{itemize}
Starting from all this, developing our own proof assistant was not
overall management of the library, integrating everything into a
single system. \MATITA{} is the result of this effort.
+\subsection{The system}
+
+\MATITA{} is a proof assistant (also called interactive theorem prover).
+It is based on the Calculus of (Co)Inductive Constructions (CIC) that
+is a dependently typed lambda-calculus \`a la Church enriched with primitive
+inductive and co-indutive data types. Via the Curry-Howard isomorphism, the
+calculus can be seen as a very rich higher order logic and proofs can be
+simply represented and stored as lambda-terms. Coq and Lego are other systems
+that adopt (variations of) CIC as their foundation.
+
+The proof language of \MATITA{} is procedural, in the tradition of the LCF
+theorem prover. Coq, NuPRL, PVS, Isabelle are all examples of others systems
+whose proof language is procedural. Traditionally, in a procedural system
+the user interacts only with the \emph{script}, while proof terms are internal
+records kept by the system. On the contrary, in \MATITA{} proof terms are
+praised as declarative versions of the proof. With this role, they are the
+primary mean of communication of proofs (once rendered to natural language
+for human audiences).
+
+The user interfaces now adopted by all the proof assistants that adopt a
+procedural proof language have been inspired by the CtCoq pioneering
+system~\cite{ctcoq}. One succesfull incarnation of the ideas introduced
+by CtCoq is the Proof General generic interface, 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.
+\MATITA{} has also cloned the Proof General interface,
+
+\begin{itemize}
+ \item scelta del sistema fondazionale
+ \item sistema indipendente (da \COQ)
+ \item compatibilit\`a con sistemi legacy
+\end{itemize}
+
\subsection{Relationship with \COQ{}}
At first sight, \MATITA{} looks as (and partly is) a \COQ{} clone. This is
we could furtherly reduce our code in sensible way).
Moreover, the complexity of the code of \MATITA{} is greatly reduced with
-respect to \COQ. For instance, the API of the libraries of \MATITA{} comprise
+respect to \COQ. For instance, the API of the components of \MATITA{} comprise
989 functions, to be compared with the 4'286 functions of \COQ.
Finally, \MATITA{} has several innovatives features over \COQ{} that derive
we have took advantage of the research results and experiences previously
developed by others, comprising the authors of \COQ. Moreover, starting from
scratch, we have designed in advance the architecture and we have splitted
-the code in coherent minimally coupled libraries.
+the code in coherent minimally coupled components.
In the future we plan to exploit \MATITA{} as a test bench for new ideas and
-extensions. Keeping the single libraries and the whole architecture as
+extensions. Keeping the single components and the whole architecture as
simple as possible is thus crucial to foster future experiments and to
allow other developers to quickly understand our code and contribute.
%be able to contribute to \COQ{}'s code is quite steep and requires direct
%and frequent interactions with \COQ{} developers.
-\subsection{The system}
-
-DESCRIZIONE DEL SISTEMA DAL PUNTO DI VISTA ``UTENTE''\\
-ROBA CHE MANCA:
-\begin{itemize}
- \item scelta del sistema fondazionale
- \item sistema indipendente (da \COQ)
- \item compatibilit\`a con sistemi legacy
-\end{itemize}
-
-\begin{figure}[t]
- \begin{center}
- \includegraphics[width=0.95\textwidth]{matita-screenshot}
- \caption{\MATITA{} look and feel}
- \label{fig:screenshot}
- \end{center}
-\end{figure}
-
-\MATITA{} has a script based user interface. As can be seen in Fig.~... it is
-split in two main windows: on the left a textual widget is used to edit the
-script, on the right the list of open goal is shown using a \MATHML{} rendering
-widget. A distinguished part of the script (shaded in the screenshot) represent
-the commands already executed and can't be edited without undoing them. The
-remaining part can be freely edited and commands from that part can be executed
-moving down the execution point. An additional window --- the ``cicBrowser'' ---
-can be used to browse the library, including the proof being developed, and
-enable content based search on it. In the cicBrowser proofs are rendered in
-natural language, automatically generated from the low-level $\lambda$-terms
-using techniques inspired by \cite{natural,YANNTHESIS}.
-
-In the \MATITA{} philosophy the script is not relevant \emph{per se}, but is
-only seen as a convenient way to create mathematical objects. The universe of
-all these objects makes up the \HELM{} library, which is always completely
-visible to the user. The mathematical library is thus conceived as a global
-hypertext, where objects may freely reference each other. It is a duty of
-the system to guide the user through the relevant parts of the library.
-
-This methodological assumption has many important consequences
-which will be discussed in the next section.
-
-%on one side
-%it requires functionalities for the overall management of the library,
-%%%%%comprising efficient indexing techniques to retrieve and filter the
-%information;
-%on the other it introduces overloading in the use of
-%identifiers and mathematical notation, requiring sophisticated disambiguation
-%techniques for interpreting the user inputs.
-%In the next two sections we shall separately discuss the two previous
-%points.
-
-%In order to maximize accessibility mathematical objects are encoded in XML. (As%discussed in the introduction,) the modular architecture of \MATITA{} is
-%organized in components which work on data in this format. For instance the
-%rendering engine, which transform $\lambda$-terms encoded as XML document to
-%\MATHML{} Presentation documents, can be used apart from \MATITA{} to print ...
-%FINIRE
-
-A final section is devoted to some innovative aspects
-of the authoring system, such as a step by step tactical execution,
-content selection and copy-paste.
-
\section{Architecture}
\label{architettura}
\begin{figure}[ht]
\begin{center}
\includegraphics[width=0.9\textwidth]{librariesCluster.ps}
- \caption{\MATITA{} libraries}
+ \caption{\MATITA{} components}
\label{fig:libraries}
\end{center}
\end{figure}
%In particular, the \texttt{xml} \component{} is used to easily represent,
%parse and pretty-print XML files.
-\section{Library Management}
-\subsection{Indexing and searching}
+\section{The interface to the library}
+
+A proof assistant provides both an interface to interact with its library and
+an \emph{authoring} interface to develop new proofs and theories. According
+to its historical origins, \MATITA{} strives to provide innovative
+functionalities for the interaction with the library. It is more traditional
+in its script based authoring interface.
+
+In the remaining part of the paper we focus on the user view of \MATITA{}.
+This section is devoted to the aspects of the tool that arise from the
+document centric approach to the library. Sect.~\ref{authoring} describes
+the peculiarities of the authoring interface.
+
+
+The library of \MATITA{} comprises mathematical concepts (theorems,
+axioms, definitions) and notation. The concepts are authored sequentially
+using scripts that are (ordered) sequences of procedural commands.
+However, once they are produced we store them independently in the library.
+The only relation implicitly kept between the notions are the logical,
+acyclic dependencies among them. This way the library forms a global (and
+distributed) hypertext. Several useful operations can be implemented on the
+library only, regardless of the scripts. Examples of such operations
+implemented in \MATITA{} are: searching and browing (see Sect.~\ref{sec:index});
+disambiguation of content level terms (see Sect.~\ref{sec:disambiguation});
+automatic proof searching (see Sect.~\ref{sec:automation}).
+
+A 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:...}, 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 your
+development, you are 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:decompilazione}). Collaborative development and versioning
+is still under design.
-\subsection{Compilation and cleaning}
-\label{sec:libmanagement}
+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 mainteinance of the library.
+Indeed, once a notion is invalidated, the only way to restore it is to
+fix the possibly broken script that used to generate it.
+Moreover, during the authoring phase, scripts are a natural way to
+group notions together. They also constitute a less fine grained clustering
+of notions for invalidation.
-%
-%goals: consentire sviluppo di una librearia mantenendo integrita' referenziale e usando le teconologie nostre (quindi con metadati, XML, libreria visibile)
-%\subsubsection{Composition}
-%scripts.ma, .moo, XML, metadata
-%\subsubsection{Compilation}
-%analogie con compilazione classica dso.\\
-%granularita' differenti per uso interattivo e non
-%\paragraph{Batch}
-%- granularita' .ma/buri \\
-%-- motivazioni\\
-%- come si calcolano le dipendenze\\
-%- quando la si usa\\
-%- metodi (cc e clean)\\
-%- garanzie
-%\paragraph{Interactive}
-%- granularita' fine\\
-%-- motivazioni
-%\label{sec:libmanagement}
-%consistenza: integrita' referenziale
-%Goals: mantenere consistente la rappresentazione della libreria su
-%memoria persistente consentendo di compilare e pulire le compilation
-%unit (.ma).\\
-%Vincoli: dipendenze oggetti-oggetti e metadati-oggetti\\
-%Due livelli di gestione libreria, uno e' solo in fase interattiva dove la compilazione e' passo passo: \\
-%--- granularita' oggetto per matita interactive\\
-%--- granularita' baseuri (compilation unit) per la libreria\\
-%In entrmbi i casi ora:\\
-%--- matitaSync: add, remove, timetravel(facility-macro tra 2 stati)[obj]\\
-%--- matitaCleanLib: clean\_baseuri (che poi usa matitaSync a sua volta)[comp1]\\
-%Vincoli di add: typecheck ( ==$>$ tutto quello che usa sta in lib)\\
-%Vincoli di remove: \\
-%--- la remove di mSync non li controlla (ma sa cosa cancellare per ogni uri)\\
-%--- la clean\_baseuri calcola le dipendenze con i metadati (o anche i moo direi) e li rispetta\\
-%Undo di matita garantisce la consistenza a patto che l'history che tiene sia ok\\
-%Undo della lib (mClean) garantisce la consistenza (usando moo o Db).\\
+In the following sections we present in more details the functionalities
+of \MATITA{} related to library management and exploitation.
-The aim of this section is to describe the way \MATITA{}
-preserves the consistency and the availability of the library
-using the \WHELP{} technology, in response to the user addition or
-removal of mathematical objects.
-As already sketched in \ref{fully-spec} the output of the
-compilation of a script is split among two storage media, a
-classical filesystem and a relational database. The former is used to
-store the XML encoding of the objects defined in the script, the
-disambiguation aliases and the interpretation and notational convention defined,
-while the latter is used to store all the metadata needed by
-\WHELP{}.
-While the consistency of the data store in the two media has
-nothing to do with the nature of
-the content of the library and is thus uninteresting (but really
-tedious to implement and keep bug-free), there is a deeper
-notion of mathematical consistency we need to provide. Each object
-must reference only defined object (i.e. each proof must use only
-already proved theorems).
+\subsection{Indexing and searching}
-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, removing, modifying objects
-without loosing the feeling of an always visible and browsable
-library.
+\subsection{Disambiguation}
+\label{sec:disambiguation}
-\subsubsection{Compilation}
+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 typechecker component guarantees that if an object is well typed
-it depends only on well typed objects available in the library,
-that is exactly what we need to be sure that the logic consistency of
-the library is preserved. We have only to find the right order of
-compilation of the scripts that compose the user development.
+The key component of the translation is the generic disambiguation algorithm
+implemented in the \texttt{disambiguation} component of Fig.~\ref{fig:libraries}
+and presented in~\cite{disambiguation}. In this section we present how to use
+such an algorithm in the context of the development of a library of formalized
+mathematics. We will see that using multiple passes of the algorithm, varying
+some of its parameters, helps in keeping the input terse without sacrificing
+expressiveness.
-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 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{sec:disambaliases}).
+\subsubsection{Disambiguation aliases}
+\label{sec:disambaliases}
+Let's start with the definition of the ``strictly greater then'' notion over
+(Peano) natural numbers.
-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.
+\begin{grafite}
+include "nat/nat.ma".
+..
+definition gt: nat \to nat \to Prop \def
+ \lambda n, m. m < n.
+\end{grafite}
-Compilation is rather simple, and the only tricky case is when we want
-to compile again the same script, maybe after the removal of a
-theorem. Here the policy is simple: clean the output before recompiling.
-As we will see in the next section cleaning will ensure that
-there will be no theorems in the development that depends on the
-removed items.
+The \texttt{include} statement adds the requirement that the part of the library
+defining the notion of natural numbers should be defined before
+processing the what follows. Note indeed that the algorithm presented
+in~\cite{disambiguation} does not describe where interpretations for ambiguous
+expressions come from, since it is application-specific. As a first
+approximation, we will assume that in \MATITA{} they come from the library (i.e.
+all interpretations available in the library are used) and the \texttt{include}
+statements are used to ensure the availability of required library slices (see
+Sect.~\ref{sec:libmanagement}).
-\subsubsection{Cleaning}
+While processing the \texttt{gt} definition, \MATITA{} has to disambiguate two
+terms: its type and its body. Being available in the required library only one
+interpretation both for the unbound identifier \texttt{nat} and for the
+\OP{<} operator, and being the resulting partially specified term refinable,
+both type and body are easily disambiguated.
-With the term ``cleaning'' we mean the process of removing all the
-results of an object compilation. In order to keep the consistency of
-the library, cleaning an object requires the (recursive) cleaning
-of all the objects that depend on it (\emph{reverse dependencies}).
+Now suppose we have defined integers as signed natural numbers, and that we want
+to prove a theorem about an order relationship already defined on them (which of
+course overload the \OP{<} operator):
-The calculation of the reverse dependencies can be computed in two
-ways, using the relational database or using a simpler set of metadata
-that \MATITA{} saves in the filesystem as a result of compilation. The
-former technique is the same used by the \emph{Dependency Analyzer}
-described in \cite{zack-master} and really depends on a relational
-database.
+\begin{grafite}
+include "Z/z.ma".
+..
+theorem Zlt_compat:
+ \forall x, y, z. x < y \to y < z \to x < z.
+\end{grafite}
-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.
+Since integers are defined on top of natural numbers, the part of the library
+concerning the latters is available when disambiguating \texttt{Zlt\_compat}'s
+type. Thus, according to the disambiguation algorithm, two different partially
+specified terms could be associated to it. At first, this might not be seen as a
+problem, since the user is asked and can choose interactively which of the two
+she had in mind. However in the long run it has the drawbacks of inhibiting
+batch compilation of the library (a technique used in \MATITA{} for behind the
+scene compilation when needed, e.g. when an \texttt{include} is issued) and
+yields to poor user interaction (imagine how tedious would be to be asked for a
+choice each time you re-evaluate \texttt{Zlt\_compat}!).
-Cleaning guarantees that if an object is removed there are no dandling
-references to it, and that the part of the library still compiled is
-consistent. Since cleaning involves the removal of all the results of
-the compilation, metadata included, the library browsable trough the
-\WHELP{} technology is always kept up to date.
+For this reason we added to \MATITA{} the concept of \emph{disambiguation
+aliases}. Disambiguation aliases are one-to-many mappings from ambiguous
+expressions to partially specified terms, which are part of the runtime status
+of \MATITA. They can be provided by users with the \texttt{alias} statement, but
+are usually automatically added when evaluating \texttt{include} statements
+(\emph{implicit aliases}). Aliases implicitely inferred during disambiguation
+are remembered as well. Moreover, \MATITA{} does it best to ensure that terms
+which require interactive choice are saved in batch compilable format. Thus,
+after evaluating the above theorem the script will be changed to the following
+snippet (assuming that the interpretation of \OP{<} over integers has been
+choosed):
-\subsubsection{Batch vs Interactive}
-
-\MATITA{} includes an interactive graphical interface and a batch
-compiler (\MATITAC). Only the former is intended to be used directly by the
-user, the latter is automatically invoked when a
-part of the user development is required (for example issuing an
-\texttt{include} command) but not yet compiled.
-
-While they share the same engine for compilation and cleaning, they
-provide different granularity. The batch compiler is only able to
-compile a whole script and similarly to clean only a whole script
-(together with all the other scripts that rely on an object defined in
-it). The interactive interface is able to execute single steps of
-compilation, that may include the definition of an object, and
-similarly to undo single steps. Note that in the latter case there is
-no risk of introducing dangling references since the \MATITA{} user
-interface inhibit undoing a step which is not the last executed.
-
-\subsection{Automation}
-
-\subsection{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{User interface}
-
-\subsection{Disambiguation}
-\label{sec:disambiguation}
-
-Software applications that involve input of mathematical content should strive
-to require the user as less drift from informal mathematics as possible. We
-believe this to be a fundamental aspect of such applications user interfaces.
-Being that drift in general very large when inputing
-proofs~\cite{debrujinfactor}, in \MATITA{} we achieved good results for
-mathematical formulae which can be input using a \TeX-like encoding (the
-concrete syntax corresponding to presentation level terms) and are then
-translated (in multiple steps) to partially specified terms as sketched in
-Sect.~\ref{sec:contentintro}.
-
-The key component of the translation is the generic disambiguation algorithm
-implemented in the \texttt{disambiguation} component of Fig.~\ref{fig:libraries}
-and presented in~\cite{disambiguation}. In this section we present how to use
-such an algorithm in the context of the development of a library of formalized
-mathematics. We will see that using multiple passes of the algorithm, varying
-some of its parameters, helps in keeping the input terse without sacrificing
-expressiveness.
-
-\subsubsection{Disambiguation aliases}
-\label{sec:disambaliases}
-Let's start with the definition of the ``strictly greater then'' notion over
-(Peano) natural numbers.
-
-\begin{grafite}
-include "nat/nat.ma".
-..
-definition gt: nat \to nat \to Prop \def
- \lambda n, m. m < n.
-\end{grafite}
-
-The \texttt{include} statement adds the requirement that the part of the library
-defining the notion of natural numbers should be defined before
-processing the what follows. Note indeed that the algorithm presented
-in~\cite{disambiguation} does not describe where interpretations for ambiguous
-expressions come from, since it is application-specific. As a first
-approximation, we will assume that in \MATITA{} they come from the library (i.e.
-all interpretations available in the library are used) and the \texttt{include}
-statements are used to ensure the availability of required library slices (see
-Sect.~\ref{sec:libmanagement}).
-
-While processing the \texttt{gt} definition, \MATITA{} has to disambiguate two
-terms: its type and its body. Being available in the required library only one
-interpretation both for the unbound identifier \texttt{nat} and for the
-\OP{<} operator, and being the resulting partially specified term refinable,
-both type and body are easily disambiguated.
-
-Now suppose we have defined integers as signed natural numbers, and that we want
-to prove a theorem about an order relationship already defined on them (which of
-course overload the \OP{<} operator):
-
-\begin{grafite}
-include "Z/z.ma".
-..
-theorem Zlt_compat:
- \forall x, y, z. x < y \to y < z \to x < z.
-\end{grafite}
-
-Since integers are defined on top of natural numbers, the part of the library
-concerning the latters is available when disambiguating \texttt{Zlt\_compat}'s
-type. Thus, according to the disambiguation algorithm, two different partially
-specified terms could be associated to it. At first, this might not be seen as a
-problem, since the user is asked and can choose interactively which of the two
-she had in mind. However in the long run it has the drawbacks of inhibiting
-batch compilation of the library (a technique used in \MATITA{} for behind the
-scene compilation when needed, e.g. when an \texttt{include} is issued) and
-yields to poor user interaction (imagine how tedious would be to be asked for a
-choice each time you re-evaluate \texttt{Zlt\_compat}!).
-
-For this reason we added to \MATITA{} the concept of \emph{disambiguation
-aliases}. Disambiguation aliases are one-to-many mappings from ambiguous
-expressions to partially specified terms, which are part of the runtime status
-of \MATITA. They can be provided by users with the \texttt{alias} statement, but
-are usually automatically added when evaluating \texttt{include} statements
-(\emph{implicit aliases}). Aliases implicitely inferred during disambiguation
-are remembered as well. Moreover, \MATITA{} does it best to ensure that terms
-which require interactive choice are saved in batch compilable format. Thus,
-after evaluating the above theorem the script will be changed to the following
-snippet (assuming that the interpretation of \OP{<} over integers has been
-choosed):
-
-\begin{grafite}
-alias symbol "lt" = "integer 'less than'".
-theorem Zlt_compat:
- \forall x, y, z. x < y \to y < z \to x < z.
-\end{grafite}
+\begin{grafite}
+alias symbol "lt" = "integer 'less than'".
+theorem Zlt_compat:
+ \forall x, y, z. x < y \to y < z \to x < z.
+\end{grafite}
But how are disambiguation aliases used? Since they come from the parts of the
library explicitely included we may be tempted of using them as the only
\end{center}
\end{table}
+
+
+\subsection{Compilation and cleaning}
+\label{sec:libmanagement}
+
+%
+%goals: consentire sviluppo di una librearia mantenendo integrita' referenziale e usando le teconologie nostre (quindi con metadati, XML, libreria visibile)
+%\subsubsection{Composition}
+%scripts.ma, .moo, XML, metadata
+%\subsubsection{Compilation}
+%analogie con compilazione classica dso.\\
+%granularita' differenti per uso interattivo e non
+%\paragraph{Batch}
+%- granularita' .ma/buri \\
+%-- motivazioni\\
+%- come si calcolano le dipendenze\\
+%- quando la si usa\\
+%- metodi (cc e clean)\\
+%- garanzie
+%\paragraph{Interactive}
+%- granularita' fine\\
+%-- motivazioni
+%\label{sec:libmanagement}
+%consistenza: integrita' referenziale
+%Goals: mantenere consistente la rappresentazione della libreria su
+%memoria persistente consentendo di compilare e pulire le compilation
+%unit (.ma).\\
+%Vincoli: dipendenze oggetti-oggetti e metadati-oggetti\\
+%Due livelli di gestione libreria, uno e' solo in fase interattiva dove la compilazione e' passo passo: \\
+%--- granularita' oggetto per matita interactive\\
+%--- granularita' baseuri (compilation unit) per la libreria\\
+%In entrmbi i casi ora:\\
+%--- matitaSync: add, remove, timetravel(facility-macro tra 2 stati)[obj]\\
+%--- matitaCleanLib: clean\_baseuri (che poi usa matitaSync a sua volta)[comp1]\\
+%Vincoli di add: typecheck ( ==$>$ tutto quello che usa sta in lib)\\
+%Vincoli di remove: \\
+%--- la remove di mSync non li controlla (ma sa cosa cancellare per ogni uri)\\
+%--- la clean\_baseuri calcola le dipendenze con i metadati (o anche i moo direi) e li rispetta\\
+%Undo di matita garantisce la consistenza a patto che l'history che tiene sia ok\\
+%Undo della lib (mClean) garantisce la consistenza (usando moo o Db).\\
+
+The aim of this section is to describe the way \MATITA{}
+preserves the consistency and the availability of the library
+using the \WHELP{} technology, in response to the user addition or
+removal of mathematical objects.
+
+As already sketched in \ref{fully-spec} the output of the
+compilation of a script is split among two storage media, a
+classical filesystem and a relational database. The former is used to
+store the XML encoding of the objects defined in the script, the
+disambiguation aliases and the interpretation and notational convention defined,
+while the latter is used to store all the metadata needed by
+\WHELP{}.
+
+While the consistency of the data store in the two media has
+nothing to do with the nature of
+the content of the library and is thus uninteresting (but really
+tedious to implement and keep bug-free), there is a deeper
+notion of mathematical consistency we need to provide. Each object
+must reference only defined object (i.e. each proof must use only
+already proved theorems).
+
+We will focus on how \MATITA{} ensures the interesting kind
+of consistency during the formalization of a mathematical theory,
+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 typed objects available in the library,
+that is exactly what we need to be sure that the logic consistency of
+the library is preserved. We have only to find the right order of
+compilation of the scripts that compose the user development.
+
+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 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{sec:disambaliases}).
+
+The output of the compilation is immediately available to the user
+trough the \WHELP{} technology, since all metadata are stored in a
+user-specific area of the database where the search engine has read
+access, and all the automated tactics that operates on the whole
+library, like \AUTO, have full visibility of the newly defined objects.
+
+Compilation is rather simple, and the only tricky case is when we want
+to compile again the same script, maybe after the removal of a
+theorem. Here the policy is simple: clean the output before recompiling.
+As we will see in the next section cleaning will ensure that
+there will be no theorems in the development that depends on the
+removed items.
+
+\subsubsection{Cleaning}
+
+With the term ``cleaning'' we mean the process of removing all the
+results of an object compilation. In order to keep the consistency of
+the library, cleaning an object requires the (recursive) cleaning
+of all the objects that depend on it (\emph{reverse dependencies}).
+
+The calculation of the reverse dependencies can be computed in two
+ways, using the relational database or using a simpler set of metadata
+that \MATITA{} saves in the filesystem as a result of compilation. The
+former technique is the same used by the \emph{Dependency Analyzer}
+described in \cite{zack-master} and really depends on a relational
+database.
+
+The latter is a fall-back in case the database is not
+available.\footnote{Due to the complex deployment of a large piece of
+software like a database, it is a common practice for the \HELM{} team
+to use a shared remote database, that may be unavailable if the user
+workstation lacks network connectivity.} This facility has to be
+intended only as a fall-back, since the queries of the \WHELP{}
+technology depend require a working database.
+
+Cleaning guarantees that if an object is removed there are no dandling
+references to it, and that the part of the library still compiled is
+consistent. Since cleaning involves the removal of all the results of
+the compilation, metadata included, the library browsable trough the
+\WHELP{} technology is always kept up to date.
+
+\subsubsection{Batch vs Interactive}
+
+\MATITA{} includes an interactive graphical interface and a batch
+compiler (\MATITAC). Only the former is intended to be used directly by the
+user, the latter is automatically invoked when a
+part of the user development is required (for example issuing an
+\texttt{include} command) but not yet compiled.
+
+While they share the same engine for compilation and cleaning, they
+provide different granularity. The batch compiler is only able to
+compile a whole script and similarly to clean only a whole script
+(together with all the other scripts that rely on an object defined in
+it). The interactive interface is able to execute single steps of
+compilation, that may include the definition of an object, and
+similarly to undo single steps. Note that in the latter case there is
+no risk of introducing dangling references since the \MATITA{} user
+interface inhibit undoing a step which is not the last executed.
+
+\subsection{Automation}
+
+\subsection{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{The authoring interface}
+
+\begin{figure}[t]
+ \begin{center}
+ \includegraphics[width=0.95\textwidth]{matita-screenshot}
+ \caption{\MATITA{} look and feel}
+ \label{fig:screenshot}
+ \end{center}
+\end{figure}
+
+\MATITA{} has a script based user interface. As can be seen in Fig.~... it is
+split in two main windows: on the left a textual widget is used to edit the
+script, on the right the list of open goal is shown using a \MATHML{} rendering
+widget. A distinguished part of the script (shaded in the screenshot) represent
+the commands already executed and can't be edited without undoing them. The
+remaining part can be freely edited and commands from that part can be executed
+moving down the execution point. An additional window --- the ``cicBrowser'' ---
+can be used to browse the library, including the proof being developed, and
+enable content based search on it. In the cicBrowser proofs are rendered in
+natural language, automatically generated from the low-level $\lambda$-terms
+using techniques inspired by \cite{natural,YANNTHESIS}.
+
+%In the \MATITA{} philosophy the script is not relevant \emph{per se}, but is
+%only seen as a convenient way to create mathematical objects. The universe of
+%all these objects makes up the \HELM{} library, which is always completely
+%visible to the user. The mathematical library is thus conceived as a global
+%hypertext, where objects may freely reference each other. It is a duty of
+%the system to guide the user through the relevant parts of the library.
+
+%This methodological assumption has many important consequences
+%which will be discussed in the next section.
+
+%on one side
+%it requires functionalities for the overall management of the library,
+%%%%%comprising efficient indexing techniques to retrieve and filter the
+%information;
+%on the other it introduces overloading in the use of
+%identifiers and mathematical notation, requiring sophisticated disambiguation
+%techniques for interpreting the user inputs.
+%In the next two sections we shall separately discuss the two previous
+%points.
+
+%In order to maximize accessibility mathematical objects are encoded in XML. (As%discussed in the introduction,) the modular architecture of \MATITA{} is
+%organized in components which work on data in this format. For instance the
+%rendering engine, which transform $\lambda$-terms encoded as XML document to
+%\MATHML{} Presentation documents, can be used apart from \MATITA{} to print ...
+%FINIRE
+
+A final section is devoted to some innovative aspects
+of the authoring system, such as a step by step tactical execution,
+content selection and copy-paste.
+
\subsection{Patterns}
\subsubsection{Direct manipulation of terms}
\end{center}
\end{figure}
-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} enable 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
-$\forall~n:nat$ since the former denotes an application while the latter denotes
-an incomplete $\Pi$-binder. Once a (sub)term has been selected that way actions
-can be done on it like reductions or tactic applications.
-
-In our experience working with direct manipulation of terms is really effective
-and faster than retyping them. Nonetheless we need a way to encode term
-selections in scripts so that they can be batch compiled by \MATITAC. In
-\MATITA{} \emph{patterns} implement that encoding, being patterns the textual
-representations of \GTKMATHVIEW semantic selections.\NOTE{Zack:c'\`e scritto da
-qualche parte che l'utente non li deve necessariamente scrivere a mano, ma che
-pu\`o incollarli? Va scritto.}
+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} enable 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
+$\forall~n:nat$ since the former denotes an application while the
+latter denotes an incomplete $\Pi$-binder. Once a (sub)term has been
+selected that way actions can be done on it like reductions or tactic
+applications.
+
+In our experience working with direct manipulation of terms is really
+effective and faster than retyping them. Nonetheless we need a way to
+encode term selections in scripts so that they can be batch compiled
+by \MATITAC. In \MATITA{} \emph{patterns} implement that encoding,
+being patterns the textual representations of \GTKMATHVIEW{} semantic
+selections. \NOTE{Zack:c'\`e scritto da qualche parte che l'utente
+non li deve necessariamente scrivere a mano, ma che pu\`o incollarli?
+Va scritto.}
\subsubsection{Pattern syntax}
A pattern is composed of two terms: a $\NT{sequent\_path}$ and a
projects / helm.git / commitdiff