\usepackage{color}
\usepackage{fancyvrb}
\usepackage[show]{ed}
-\usepackage{floatflt}
\definecolor{gray}{gray}{0.85}
%\newcommand{\logo}[3]{
\newcommand{\components}{components}
\newcommand{\AUTO}{\textsc{Auto}}
+\newcommand{\BOXML}{BoxML}
\newcommand{\COQ}{Coq}
\newcommand{\COQIDE}{CoqIde}
\newcommand{\ELIM}{\textsc{Elim}}
\newcommand{\GDOME}{Gdome}
+\newcommand{\GTKMATHVIEW}{\textsc{GtkMathView}}
\newcommand{\HELM}{Helm}
\newcommand{\HINT}{\textsc{Hint}}
\newcommand{\IN}{\ensuremath{\dN}}
\newcommand{\LIBXSLT}{LibXSLT}
\newcommand{\LOCATE}{\textsc{Locate}}
\newcommand{\MATCH}{\textsc{Match}}
+\newcommand{\MATHML}{MathML}
\newcommand{\MATITA}{Matita}
\newcommand{\MATITAC}{\texttt{matitac}}
\newcommand{\MATITADEP}{\texttt{matitadep}}
\newcommand{\TACTIC}[1]{\ensuremath{\mathtt{tactic}~#1}}
\definecolor{gray}{gray}{0.85} % 1 -> white; 0 -> black
-\newcommand{\NT}[1]{\langle\mathit{#1}\rangle}
+\newcommand{\NT}[1]{\ensuremath{\langle\mathit{#1}\rangle}}
\newcommand{\URI}[1]{\texttt{#1}}
\newcommand{\OP}[1]{``\texttt{#1}''}
-%{\end{SaveVerbatim}\setlength{\fboxrule}{.5mm}\setlength{\fboxsep}{2mm}%
\newenvironment{grafite}{\VerbatimEnvironment
\begin{SaveVerbatim}{boxtmp}}%
{\end{SaveVerbatim}\setlength{\fboxsep}{3mm}%
{}
\newcommand{\ASSIGNEDTO}[1]{\textbf{Assigned to:} #1}
\newcommand{\FILE}[1]{\texttt{#1}}
-% \newcommand{\NOTE}[1]{\ifodd \arabic{page} \else \hspace{-2cm}\fi\ednote{#1}}
-\newcommand{\NOTE}[1]{\ednote{#1}{foo}}
+\newcommand{\NOTE}[1]{\ednote{#1}{}}
\newcommand{\TODO}[1]{\textbf{TODO: #1}}
\newcounter{pass}
\end{opening}
\tableofcontents
+\listoffigures
\section{Introduction}
\label{sec:intro}
\cite{mkm-helm} at the University of Bologna, under the direction of
Prof.~Asperti. \\
The paper describes the overall architecture of
-the system, focusing on its most distintive and innovative
+the system, focusing on its most distinctive and innovative
features.
\subsection{Historical Perspective}
interested to develop tools and techniques to enhance the accessibility
via Web of formal libraries of mathematics. Due to its dimension, the
library of the \COQ~\cite{CoqManual} proof assistant (of the order of 35'000 theorems)
-was choosed as a privileged test bench for our work, although experiments
+was chosen as a privileged test bench for our work, although experiments
have been also conducted with other systems, and notably
-with \NUPRL{}\cite{nuprl-book}.
+with \NUPRL~\cite{nuprl-book}.
The work, mostly performed in the framework of the recently concluded
European project IST-33562 \MOWGLI{}~\cite{pechino}, mainly consisted in the
following steps:
has to be build;
\item developing indexing and searching techniques supporting semantic
queries to the library;
-%these efforts gave birth to our \WHELP{}
-%search engine, described in~\cite{whelp};
\item developing languages and tools for a high-quality notational
rendering of mathematical information\footnote{We have been
-active in the MathML Working group since 1999.};
-%and developed inside
-%\HELM{} a MathML-compliant widget for the GTK graphical environment
-%which can be integrated in any application.
+active in the \MATHML{} Working group since 1999.};
\end{itemize}
According to our content-centric commitment, the library exported from
means of a Web interface that orchestrates the Web services.
The Web services and the other tools have been implemented as front-ends
-to a set of software libraries, collectively called the \HELM{} libraries.
+to a set of software components, collectively called the \HELM{} components.
At the end of the \MOWGLI{} project we already disposed of the following
-tools and software libraries:
+tools and software components:
\begin{itemize}
\item XML specifications for the Calculus of Inductive Constructions,
-with libraries for parsing and saving mathematical objects in such a format
+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
partially specified terms, used by the disambiguating parser;
\item complex transformation algorithms for proof rendering in natural
language \cite{remathematization};
-\item an innovative, MathML-compliant rendering widget for the GTK
+\item an innovative, \MATHML-compliant rendering widget for the GTK
graphical environment\cite{padovani}, supporting
high-quality bidimensional
rendering, and semantic selection, i.e. the possibility to select semantically
-meaningful rendering expressions, and to past the respective content into
+meaningful rendering expressions, and to paste the respective content into
a different text area.
\end{itemize}
Starting from all this, developing our own proof assistant was not
overall management of the library, integrating everything into a
single system. \MATITA{} is the result of this effort.
+\subsection{The system}
+
+\MATITA{} is a proof assistant (also called interactive theorem prover).
+It is based on the Calculus of (Co)Inductive Constructions
+(CIC)~\cite{Werner} that is a dependently typed lambda-calculus \`a la
+Church enriched with primitive inductive and co-inductive data types.
+Via the Curry-Howard isomorphism, the calculus can be seen as a very
+rich higher order logic and proofs can be simply represented and
+stored as lambda-terms. \COQ{} and Lego are other systems that adopt
+(variations of) CIC as their foundation.
+
+The proof language of \MATITA{} is procedural, in the tradition of the LCF
+theorem prover. Coq, NuPRL, PVS, Isabelle are all examples of others systems
+whose proof language is procedural. Traditionally, in a procedural system
+the user interacts only with the \emph{script}, while proof terms are internal
+records kept by the system. On the contrary, in \MATITA{} proof terms are
+praised as declarative versions of the proof. With this role, they are the
+primary mean of communication of proofs (once rendered to natural language
+for human audiences).
+
+The user interfaces now adopted by all the proof assistants based on a
+procedural proof language have been inspired by the CtCoq pioneering
+system~\cite{ctcoq1}. One successful incarnation of the ideas introduced
+by CtCoq is the Proof General generic interface~\cite{proofgeneral},
+that has set a sort of
+standard way to interact with the system. Several procedural proof assistants
+have either adopted or cloned Proof General as their main user interface.
+The authoring interface of \MATITA{} is a clone of the Proof General interface.
+
+\begin{itemize}
+ \item scelta del sistema fondazional.
+ \item sistema indipendente (da \COQ)
+ \item compatibilit\`a con sistemi legacy
+\end{itemize}
+
\subsection{Relationship with \COQ{}}
At first sight, \MATITA{} looks as (and partly is) a \COQ{} clone. This is
more the effect of the circumstances of its creation described
above than the result of a deliberate design. In particular, we
(essentially) share the same foundational dialect of \COQ{} (the
-Calculus of (Co)Inductive Constructions), the same implementative
+Calculus of (Co)Inductive Constructions), the same implementation
language (\OCAML{}), and the same (script based) authoring philosophy.
However, the analogy essentially stops here and no code is shared by the
two systems.
idea, although \MATITA{} currently supports almost all functionalities of
\COQ{}, it links 60'000 lines of \OCAML{} code, against the 166'000 lines linked
by \COQ{} (and we are convinced that, starting from scratch again,
-we could furtherly reduce our code in sensible way).
+we could reduce our code even further in sensible way).
Moreover, the complexity of the code of \MATITA{} is greatly reduced with
-respect to \COQ. For instance, the API of the libraries of \MATITA{} comprise
+respect to \COQ. For instance, the API of the components of \MATITA{} comprise
989 functions, to be compared with the 4'286 functions of \COQ.
-Finally, \MATITA{} has several innovatives features over \COQ{} that derive
+FINQUI SPELL CHECKATO
+
+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.
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{figure}[!ht]
\begin{center}
- \includegraphics[width=0.9\textwidth]{librariesCluster.ps}
- \caption{\MATITA{} libraries}
+ \includegraphics[width=0.9\textwidth,height=0.8\textheight]{libraries-clusters}
+ \caption[\MATITA{} components and related applications]{\MATITA{}
+ components and related applications, with thousands of line of
+ codes (klocs)}
\label{fig:libraries}
\end{center}
\end{figure}
Fig.~\ref{fig:libraries} shows the architecture of the \emph{\components}
(circle nodes) and \emph{applications} (squared nodes) developed in the HELM
-project.
+project. Each node is annotated with the number of lines of source code
+(comprising comments).
Applications and \components{} depend over other \components{} forming a
directed acyclic graph (DAG). Each \component{} can be decomposed in
For those \components{} whose functionalities are also provided by the
aforementioned Web services, it is also possible to link stub code that
forwards the request to a remote Web service. For instance, the Getter
-is just a wrapper to the \GETTER \component{} that allows the
+is just a wrapper to the \GETTER{} \component{} that allows the
\component{} to be used as a Web service. \MATITA{} can directly link the code
-of the \GETTER \component, or it can use a stub library with the same
+of the \GETTER{} \component, or it can use a stub library with the same
API that forwards every request to the Getter.
To better understand the architecture of \MATITA{} and the role of each
content level terms; presentation level terms.
\subsection{Fully specified terms}
-\label{fully-spec}
+\label{sec:fullyspec}
+
\emph{Fully specified terms} are CIC terms where no information is
missing or left implicit. A fully specified term should be well-typed.
The mathematical notions (axioms, definitions, theorems) that are stored
Terms may reference other mathematical notions in the library.
One commitment of our project is that the library should be physically
- distributed. The \GETTER \component{} manages the distribution,
+ distributed. The \GETTER{} \component{} manages the distribution,
providing a mapping from logical names (URIs) to the physical location
of a notion (an URL). The \texttt{urimanager} \component{} provides the URI
data type and several utility functions over URIs. The
in Sect.~\ref{sec:libmanagement}.
\subsection{Partially specified terms}
+\label{sec:partspec}
+
\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
formulae. The firsts are translated to a content level representation of
proof steps that can easily be rendered in natural language. The representation
adopted has greatly influenced the OMDoc~\cite{omdoc} proof format that is now
-isomorphic to it. Terms that represent formulae are translated to MathML
-Content formulae. MathML Content~\cite{mathml} is a W3C standard
+isomorphic to it. Terms that represent formulae are translated to \MATHML{}
+Content formulae. \MATHML{} Content~\cite{mathml} is a W3C standard
for the representation of content level formulae in an XML extensible format.
The translation to content level is implemented in the
interpretations. An interpretation is correct if the partially specified term
obtained using the interpretation is refinable.
-In the last section we have described the semantics of a command as a
+In Sect.~\ref{sec:partspec} the last section we described the semantics of a
+command as a
function from status to status. We also suggested that the formulae in a
command are encoded as partially specified terms. However, consider the
command ``\texttt{replace} $x$ \texttt{with} $y^2$''. Until the occurrence
context of the occurrence $x$ it must replace.
The elegant solution we have implemented consists in representing terms
-in a command as function from a context to a partially refined term. The
+in a command as functions from a context to a partially refined term. The
function is obtained by partially applying our disambiguation function to
the content term to be disambiguated. Our solution should be compared with
-the one adopted in the Coq system (where ambiguity is only relative to
-DeBrujin indexes). In Coq variables can be bound either by name or by
-position. This makes more complex every operation over terms (i.e. according
-to our architecture every module that depends on \texttt{cic}). Moreover,
-this solution cannot cope with other forms of ambiguity (as the meaning
-of the $~^2$ exponent in the previous example that depends on the context).
+the one adopted in the Coq system, where ambiguity is only relative to De Brujin
+indexes. In Coq variables can be bound either by name or by position. A term
+occurring in a command has all its variables bound by name to avoid the need of
+a context during disambiguation. Moreover, this makes more complex every
+operation over terms (i.e. according to our architecture every module that
+depends on \texttt{cic}) since the code must deal consistently with both kinds
+of binding. Also, this solution cannot cope with other forms of ambiguity (as
+the context dependent meaning of the exponent in the previous example).
\subsection{Presentation level terms}
single day, but they stick to a set of symbols that is more or less fixed.
The fact that the presentation language is finite allows the definition of
-standard languages. In particular, for formulae we have adopt MathML
+standard languages. In particular, for formulae we have adopt \MATHML{}
Presentation~\cite{mathml} that is an XML dialect standardized by the W3C. To
visually
represent proofs it is enough to embed formulae in plain text enriched with
formatting boxes. Since the language of formatting boxes is very simple,
many equivalent specifications exist and we have adopted our own, called
-BoxML.
+\BOXML.
The \texttt{content\_pres} \component{} contains the implementation of the
translation from content level terms to presentation level terms. The
rendering of presentation level terms is left to the application that uses
the \component. However, in the \texttt{hgdome} \component{} we provide a few
-utility functions to build a \GDOME~\cite{gdome2} MathML+BoxML tree from our
+utility functions to build a \GDOME~\cite{gdome2} \MATHML+\BOXML{} tree from our
presentation
-level terms. \GDOME{} MathML+BoxML trees can be rendered by the GtkMathView
+level terms. \GDOME{} \MATHML+\BOXML{} trees can be rendered by the
+\GTKMATHVIEW{}
widget developed by Luca Padovani \cite{padovani}. The widget is
particularly interesting since it allows to implement \emph{semantic
selection}.
against the \texttt{grafite\_parser} \components{}
since they provide an interface to the user. In both cases the formulae
written by the user are parsed using the \texttt{content\_pres} \component{} and
-then disambiguated using the \texttt{cic\_disambiguation} \component.
-However, only \MATITA{} is linked against the \texttt{grafite\_engine} and
-\texttt{tactics} components since \WHELP{} can only execute those ASTs that
-correspond to queries (implemented in the \texttt{whelp} component).
+then disambiguated using the \texttt{cic\_disambiguation} \component. However,
+only \MATITA{} is linked against the \texttt{grafite\_engine} and
+\texttt{tactics} components (summing up to a total of 11'200 lines of code)
+since \WHELP{} can only execute those ASTs that correspond to queries
+(implemented in the \texttt{whelp} component).
The \UWOBO{} Web service wraps the \texttt{content\_pres} \component,
providing a rendering service for the documents in the distributed library.
%In particular, the \texttt{xml} \component{} is used to easily represent,
%parse and pretty-print XML files.
-\section{Library Management}
-
-\subsection{Indexing and searching}
-
-
-\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
+\section{The interface to the library}
+\label{sec:library}
+
+A proof assistant provides both an interface to interact with its library and
+an \emph{authoring} interface to develop new proofs and theories. According
+to its historical origins, \MATITA{} strives to provide innovative
+functionalities for the interaction with the library. It is more traditional
+in its script based authoring interface.
+
+In the remaining part of the paper we focus on the user view of \MATITA{}.
+This section is devoted to the aspects of the tool that arise from the
+document centric approach to the library. Sect.~\ref{sec:authoring} describes
+the peculiarities of the authoring interface.
+
+The library of \MATITA{} comprises mathematical concepts (theorems,
+axioms, definitions) and notation. The concepts are authored sequentially
+using scripts that are (ordered) sequences of procedural commands.
+However, once they are produced we store them independently in the library.
+The only relation implicitly kept between the notions are the logical,
+acyclic dependencies among them. This way the library forms a global (and
+distributed) hypertext. Several useful operations can be implemented on the
+library only, regardless of the scripts. Examples of such operations
+implemented in \MATITA{} are: searching and browsing (see Sect.~\ref{sec:indexing});
+disambiguation of content level terms (see Sect.~\ref{sec:disambiguation});
+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.
-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.
+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 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.
+
+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.
-\section{User interface}
+\subsection{Indexing and searching}
+\label{sec:indexing}
\subsection{Disambiguation}
\label{sec:disambiguation}
\subsubsection{Disambiguation aliases}
\label{sec:disambaliases}
-Let's start with the definition of the ``strictly greater then'' notion over
+Let us start with the definition of the ``strictly greater then'' notion over
(Peano) natural numbers.
\begin{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
+processing 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.
the alias for \OP{<} points to the integer version of the operator, no
refinable partially specified term matching the term could be found.
-For this reason we choosed to attempt \emph{multiple disambiguation passes}. A
-first pass attempt to disambiguate using the last available disambiguation
-aliases (\emph{mono aliases} pass), in case of failure the next pass try again
-the disambiguation forgetting the aliases and using the whole library to
+For this reason we chose to attempt \emph{multiple disambiguation passes}. A
+first pass attempts to disambiguate using the last available disambiguation
+aliases (\emph{mono aliases} pass); in case of failure the next pass tries
+disambiguation again forgetting the aliases and using the whole library to
retrieve interpretation for ambiguous expressions (\emph{library aliases} pass).
Since the latter pass may lead to too many choices we intertwined an additional
pass among the two which use as interpretations all the aliases coming for
\subsubsection{Operator instances}
-Let's suppose now we want to define a theorem relating ordering relations on
+Let us suppose now we want to define a theorem relating ordering relations on
natural and integer numbers. The way we would like to write such a theorem (as
we can read it in the \MATITA{} standard library) is:
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
+aliases} and \MATITA{} does not use it to disambiguate further terms down in the
script.
\subsubsection{Implicit coercions}
-Let's now consider a (rather hypothetical) theorem about derivation:
+Let us now consider a theorem about derivation:
\begin{grafite}
theorem power_deriv:
and suppose there exists a \texttt{R \TEXMACRO{to} nat \TEXMACRO{to} R}
interpretation for \OP{\^}, and a real number interpretation for \OP{*}.
-Mathematichians would write the term that way since it is well known that the
+Mathematichans would write the term that way since it is well known that the
natural number \texttt{n} could be ``injected'' in \IR{} and considered a real
number for the purpose of real multiplication. The refiner of \MATITA{} supports
\emph{implicit coercions} for this reason: given as input the above content
(assuming it has been defined as such of course).
Nonetheless coercions are not always desirable. For example, in disambiguating
-\texttt{\TEXMACRO{forall} x: nat. n < n + 1} we don't want the term which uses
+\texttt{\TEXMACRO{forall} x: nat. n < n + 1} we do not want the term which uses
two coercions from \texttt{nat} to \texttt{R} around \OP{<} arguments to show up
among the possible partially specified term choices. For this reason in
\MATITA{} we always try first a disambiguation pass which require the refiner
\subsubsection{Disambiguation passes}
According to the criteria described above in \MATITA{} we choose to perform the
-sequence of disambiguation passes depicted in Tab.~\ref{tab:disambpasses}. In
-our experience that choice implements a good trade off among disambiguation time
-and admitted ambiguity in terms input by users.
+sequence of disambiguation passes depicted in Tab.~\ref{tab:disambpasses}. In
+our experience that choice gives reasonable performance and minimize the need of
+user interaction during the disambiguation.
\begin{table}[ht]
\caption{Sequence of disambiguation passes used in \MATITA.\strut}
\end{center}
\end{table}
-\subsection{Patterns}
+
+
+\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 Sec.~\ref{sec:fullyspec} 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).
+
+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 authoring 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}
+\label{sec: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}
+\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, also implemented around \GTKMATHVIEW, is called
+``cicBrowser''. It is used to browse the library, including the proof being
+developed, and enable content based search over it. Proofs are rendered in
+natural language, automatically generated from the low-level lambda-terms,
+using techniques inspired by \cite{natural,YANNTHESIS} and already described
+in~\cite{remathematization}.
+
+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]{matita-screenshot}
+ \caption{\MATITA{} look and feel}
+ \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}
+
+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}[t]
\begin{center}
\includegraphics[width=0.40\textwidth]{matita-screenshot-selection}
\hspace{0.05\textwidth}
- \includegraphics[width=0.50\textwidth]{matita-screenshot-href}
+ \raisebox{0.4cm}{\includegraphics[width=0.50\textwidth]{matita-screenshot-href}}
\caption{Semantic selection and hyperlinks}
- \label{fig:semselection}
+ \label{fig:directmanip}
\end{center}
\end{figure}
-serve una intro che almeno cita il widget (per i patterns) e che fa
-il resoconto delle cose che abbiamo e che non descriviamo,
-sottolineando che abbiamo qualcosa da dire sui pattern e sui
-tattichini.\\
-Patterns are the textual counterpart of the MathML widget graphical
-selection.
-\MATITA{} benefits of a graphical interface and a powerful MathML rendering
-widget that allows the user to select pieces of the sequent he is working
-on. While this is an extremely intuitive way for the user to
-restrict the application of tactics, for example, to some subterms of the
-conclusion or some hypothesis, the way this action is recorded to the text
-script is not obvious.\\
-In \MATITA{} this issue is addressed by patterns.
+\subsection{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}
-A pattern is composed of two terms: a $\NT{sequent\_path}$ and a
-$\NT{wanted}$.
-The former mocks-up a sequent, discharging unwanted subterms with $?$ and
-selecting the interesting parts with the placeholder $\%$.
-The latter is a term that lives in the context of the placeholders.
-
-The concrete syntax is reported in table \ref{tab:pathsyn}
-\NOTE{uso nomi diversi dalla grammatica ma che hanno + senso}
+
+Patterns are composed of two parts: \NT{sequent\_path} and
+\NT{wanted}; their concrete syntax is reported in table
+\ref{tab:pathsyn}.
+
+\NT{sequent\_path} mocks-up a sequent, discharging unwanted subterms
+with $?$ and selecting the interesting parts with the placeholder
+$\%$. \NT{wanted} is a term that lives in the context of the
+placeholders.
+
+Textual patterns produced from a graphical selection are made of the
+\NT{sequent\_path} only. Such patterns can represent every selection,
+but are quite verbose. The \NT{wanted} part of the syntax is meant to
+help the users in writing concise and elegant patterns by hand.
+
\begin{table}
- \caption{\label{tab:pathsyn} Concrete syntax of \MATITA{} patterns.\strut}
+ \caption{\label{tab:pathsyn} Patterns concrete syntax.\strut}
\hrule
\[
\begin{array}{@{}rcll@{}}
\NT{pattern} &
- ::= & [~\verb+in match+~\NT{wanted}~]~[~\verb+in+~\NT{sequent\_path}~] & \\
+ ::= & [~\verb+in+~\NT{sequent\_path}~]~[~\verb+match+~\NT{wanted}~] & \\
\NT{sequent\_path} &
::= & \{~\NT{ident}~[~\verb+:+~\NT{multipath}~]~\}~
[~\verb+\vdash+~\NT{multipath}~] & \\
- \NT{wanted} & ::= & \NT{term} & \\
\NT{multipath} & ::= & \NT{term\_with\_placeholders} & \\
+ \NT{wanted} & ::= & \NT{term} & \\
\end{array}
\]
\hrule
\end{table}
-\subsubsection{How patterns work}
-Patterns mimic the user's selection in two steps. The first one
-selects roots (subterms) of the sequent, using the
-$\NT{sequent\_path}$, while the second
-one searches the $\NT{wanted}$ term starting from these roots. Both are
-optional steps, and by convention the empty pattern selects the whole
-conclusion.
+\subsubsection{Pattern evaluation}
+
+Patterns are evaluated in two phases. The first selects roots
+(subterms) of the sequent, using the $\NT{sequent\_path}$, while the
+second searches the $\NT{wanted}$ term starting from these roots.
+% Both are optional steps, and by convention the empty pattern selects
+% the whole conclusion.
\begin{description}
\item[Phase 1]
the source of an arrow and the head of the application that is
found in the arrow target.
- The first phase selects not only terms (roots of subterms) but also
- their context that will be eventually used in the second phase.
+ The first phase not only selects terms (roots of subterms) but
+ determines also their context that will be eventually used in the
+ second phase.
\item[Phase 2]
- plays a role only if the $[~\verb+in match+~\NT{wanted}~]$
+ plays a role only if the $[~\verb+match+~\NT{wanted}~]$
part is specified. From the first phase we have some terms, that we
will see as subterm roots, and their context. For each of these
contexts the $\NT{wanted}$ term is disambiguated in it and the
- corresponding root is searched for a subterm $\alpha$-equivalent to
+ corresponding root is searched for a subterm that can be unified to
$\NT{wanted}$. The result of this search is the selection the
pattern represents.
\end{description}
-\noindent
-Since the first step is equipotent to the composition of the two
-steps, the system uses it to represent each visual selection.
-The second step is only meant for the
-experienced user that writes patterns by hand, since it really
-helps in writing concise patterns as we will see in the
-following examples.
-
\subsubsection{Examples}
-To explain how the first step works let's give an example. Consider
-you want to prove the uniqueness of the identity element $0$ for natural
-sum, and that you can relay on the previously demonstrated left
-injectivity of the sum, that is $inj\_plus\_l:\forall x,y,z.x+y=z+y \to x =z$.
-Typing
-\begin{grafite}
-theorem valid_name: \forall n,m. m + n = n \to m = O.
- intros (n m H).
-\end{grafite}
-\noindent
-leads you to the following sequent
+%To explain how the first phase works let us give an example. Consider
+%you want to prove the uniqueness of the identity element $0$ for natural
+%sum, and that you can rely on the previously demonstrated left
+%injectivity of the sum, that is $inj\_plus\_l:\forall x,y,z.x+y=z+y \to x =z$.
+%Typing
+%\begin{grafite}
+%theorem valid_name: \forall n,m. m + n = n \to m = O.
+% intros (n m H).
+%\end{grafite}
+%\noindent
+Consider the following sequent
\sequent{
n:nat\\
m:nat\\
m=O
}
\noindent
-where you want to change the right part of the equivalence of the $H$
-hypothesis with $O + n$ and then use $inj\_plus\_l$ to prove $m=O$.
+To change the right part of the equivalence of the $H$
+hypothesis with $O + n$ the user selects and pastes it as the pattern
+in the following statement.
\begin{grafite}
change in H:(? ? ? %) with (O + n).
\end{grafite}
\noindent
-This pattern, that is a simple instance of the $\NT{sequent\_path}$
-grammar entry, acts on $H$ that has type (without notation) $(eq~nat~(m+n)~n)$
-and discharges the head of the application and the first two arguments with a
-$?$ and selects the last argument with $\%$. The syntax may seem uncomfortable,
-but the user can simply select with the mouse the right part of the equivalence
-and left to the system the burden of writing down in the script file the
-corresponding pattern with $?$ and $\%$ in the right place (that is not
-trivial, expecially where implicit arguments are hidden by the notation, like
-the type $nat$ in this example).
-
-Changing all the occurrences of $n$ in the hypothesis $H$ with $O+n$
-works too and can be done, by the experienced user, writing directly
-a simpler pattern that uses the second phase.
+To understand the pattern (or produce it by hand) the user should be
+aware that the notation $m+n=n$ hides the term $(eq~nat~(m+n)~n)$, so
+that the pattern selects only the third argument of $eq$.
+
+The experienced user may also write by hand a concise pattern
+to change at once all the occcurrences of $n$ in the hypothesis $H$:
\begin{grafite}
- change in match n in H with (O + n).
+ change in H match n with (O + n).
\end{grafite}
\noindent
In this case the $\NT{sequent\_path}$ selects the whole $H$, while
-the second phase searches the wanted $n$ inside it by
-$\alpha$-equivalence. The resulting
-equivalence will be $m+(O+n)=O+n$ since the second phase found two
-occurrences of $n$ in $H$ and the tactic changed both.
+the second phase locates $n$.
-Just for completeness the second pattern is equivalent to the
-following one, that is less readable but uses only the first phase.
+The latter pattern is equivalent to the following one, that the system
+can automatically generate from the selection.
\begin{grafite}
change in H:(? ? (? ? %) %) with (O + n).
\end{grafite}
\noindent
\subsubsection{Tactics supporting patterns}
+MERGIARE CON IL SUCCESSIVO FACENDO NOTARE CHE I PATTERNS SONO UNA
+INTERFACCIA OCMUNE PER LE TATTICHE
+
In \MATITA{} all the tactics that can be restricted to subterm of the working
sequent accept the pattern syntax. In particular these tactics are: simplify,
change, fold, unfold, generalize, replace and rewrite.
con una pattern\_of(select(pattern))}
\subsubsection{Comparison with \COQ{}}
-\COQ{} has a two diffrent ways of restricting the application of tactis to
+\COQ{} has a two different ways of restricting the application of tactis to
subterms of the sequent, both relaying on the same special syntax to identify
a term occurrence.
-The first way is to use this special syntax to specify directly to the
-tactic the occurrnces of a wanted term that should be affected, while
-the second is to prepare the sequent with another tactic called
-pattern and the apply the real tactic. Note that the choice is not
+The first way is to use this special syntax to tell the
+tactic what occurrences of a wanted term should be affected.
+The second is to prepare the sequent with another tactic called
+pattern and then apply the real tactic. Note that the choice is not
left to the user, since some tactics needs the sequent to be prepared
with pattern and do not accept directly this special syntax.
\end{grafite}
\noindent
meaning that in the hypothesis $H$ the $n$ we want to change is the
-second we encounter proceeding from left toright.
+second we encounter proceeding from left to right.
The tactic pattern computes a
$\beta$-expansion of a part of the sequent with respect to some
is executed in a single step, while it obviously performs lot of proof
steps. In the fist example of the previous section the whole branch
over the two goals (respectively the left and right part of the logic
-and) result in a single step of execution. The workaround doesn't work
+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.\\
\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.
projects / helm.git / blobdiff