-\documentclass[draft]{kluwer}
+\documentclass[]{kluwer}
\usepackage{color}
\usepackage{graphicx}
\usepackage{hyperref}
\newcommand{\MATITAC}{\texttt{matitac}}
\newcommand{\MATITADEP}{\texttt{matitadep}}
\newcommand{\MOWGLI}{MoWGLI}
+\newcommand{\MOWGLIIST}{IST-2001-33562}
\newcommand{\NAT}{\ensuremath{\mathit{nat}}}
\newcommand{\NATIND}{\mathit{nat\_ind}}
\newcommand{\NUPRL}{NuPRL}
\newcommand{\OCAML}{OCaml}
\newcommand{\PROP}{\mathit{Prop}}
\newcommand{\REF}[3]{\ensuremath{\mathit{Ref}_{#1}(#2,#3)}}
+\newcommand{\REWRITEHINT}{\textsc{RewriteHint}}
\newcommand{\TEXMACRO}[1]{\texttt{\char92 #1}}
\newcommand{\UWOBO}{UWOBO}
\newcommand{\GETTER}{Getter}
\fcolorbox{black}{gray}{\usebox{\tmpxyz}}
\end{center}}
-\bibliographystyle{alpha}
+\bibliographystyle{klunum}
\begin{document}
the system, focusing on its most distinctive and innovative
features.
-\subsection{Historical Perspective}
+\subsection{Historical perspective}
+
The origins of \MATITA{} go back to 1999. At the time we were mostly
interested to develop tools and techniques to enhance the accessibility
via Web of formal libraries of mathematics. Due to its dimension, the
have been also conducted with other systems, and notably
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
+European project \MOWGLIIST{} \MOWGLI~\cite{pechino}, mainly consisted in the
following steps:
\begin{itemize}
\item exporting the information from the internal representation of
(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
+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
standard way to interact with the system. Several procedural proof assistants
have either adopted or cloned Proof General as their main user interface.
The authoring interface of \MATITA{} is a clone of the Proof General interface.
+On the contrary, the interface to interact with the library is rather
+innovative and directly inspired by the Web interfaces to our Web servers.
-\TODO{item che seguono:}
-\begin{itemize}
- \item sistema indipendente (da \COQ)
- \item compatibilit\`a con sistemi legacy
-\end{itemize}
+\MATITA{} is backward compatible with the XML library of proof objects exported
+from \COQ{}, but, in order to test the actual usability of the system, we are
+also developing a new library of basic results from scratch.
\subsection{Relationship with \COQ{}}
However, the analogy essentially stops here and no code is shared by the
two systems.
-In a sense; we like to think of \MATITA{} as the way \COQ{} would
+In a sense, we like to think of \MATITA{} as the way \COQ{} would
look like if entirely rewritten from scratch: just to give an
idea, although \MATITA{} currently supports almost all functionalities of
\COQ{}, it links 60'000 lines of \OCAML{} code, against the 166'000 lines linked
The size and complexity improvements over \COQ{} must be understood
historically. \COQ{} is a quite old
-system whose development started 20 years ago.
-\NOTE{Zack: verificato su Coq'Art, \`e iniziato nel 1985-86}
-Since then
+system whose development started 20 years ago. Since then
several developers have took over the code and several new research ideas
that were not considered in the original architecture have been experimented
and integrated in the system. Moreover, there exists a lot of developments
\includegraphics[width=0.9\textwidth,height=0.8\textheight]{pics/libraries-clusters}
\caption[\MATITA{} components and related applications]{\MATITA{}
components and related applications, with thousands of line of
- codes (klocs).\strut}
+ codes (klocs)\strut}
\label{fig:libraries}
\end{center}
\end{figure}
Modules and \components{} provide coherent sets of functionalities
at different scales. Applications that require only a few functionalities
-depend on a restricted set of \components{}.
+depend on a restricted set of \components.
Only the proof assistant \MATITA{} and the \WHELP{} search engine are
applications meant to be used directly by the user. All the other applications
The \texttt{grafite} \component{} defines the abstract syntax tree (AST) for the
commands of the \MATITA{} proof assistant. Most of the commands are tactics.
Other commands are used to give definitions and axioms or to state theorems
-and lemmas. The \texttt{grafite\_engine} \component{} is the core of \MATITA{}.
+and lemmas. The \texttt{grafite\_engine} \component{} is the core of \MATITA.
It implements the semantics of each command in the grafite AST as a function
from status to status. It implements also an undo function to go back to
previous statuses.
The translation from partially specified terms to content level terms must
discriminate between terms used to represent proofs and terms used to represent
formulae. The firsts are translated to a content level representation of
-proof steps that can easily be rendered in natural language. The representation
+proof steps that can in turn easily be rendered in natural language
+using techniques inspired by~\cite{natural,YANNTHESIS}. The representation
adopted has greatly influenced the OMDoc~\cite{omdoc} proof format that is now
isomorphic to it. Terms that represent formulae are translated to \MATHML{}
Content formulae. \MATHML{} Content~\cite{mathml} is a W3C standard
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 De Brujin
-indexes. In Coq variables can be bound either by name or by position. A term
+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
\GETTER{} obtaining a document with fully specified terms. Then it translates
it to the presentation level passing through the content level. Finally
it returns the result document to be rendered by the user's
-browser.\NOTE{\TODO{manca la passata verso HTML}}
+browser.
The \components{} not yet described (\texttt{extlib}, \texttt{xml},
\texttt{logger}, \texttt{registry} and \texttt{utf8\_macros}) are
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{}.
+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.
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}).
+distributed) hypertext.
+
+\begin{figure}[!ht]
+ \begin{center}
+ \includegraphics[width=0.40\textwidth]{pics/cicbrowser-screenshot-browsing}
+ \hspace{0.05\textwidth}
+ \includegraphics[width=0.40\textwidth]{pics/cicbrowser-screenshot-query}
+ \caption{Browsing and searching the library\strut}
+ \label{fig:cicbrowser1}
+ \end{center}
+\end{figure}
+
+\begin{figure}[!ht]
+ \begin{center}
+ \includegraphics[width=0.70\textwidth]{pics/cicbrowser-screenshot-con}
+ \caption[Natural language rendering]{Natural language rendering of a theorem
+ from the library\strut}
+ \label{fig:cicbrowser2}
+ \end{center}
+\end{figure}
+
+Several useful operations can be implemented on the library only,
+regardless of the scripts. For instance, searching and browsing is
+implemented by the ``cicBrowser'' window available from the \MATITA{}
+GUI. Using it, the hierarchical structure of the library can be
+explored (on the left of Fig.~\ref{fig:cicbrowser1}), the natural
+language rendering of proofs can be inspected
+(Fig.~\ref{fig:cicbrowser2}), and content based searches on the
+library can be performed (on the right of Fig.~\ref{fig:cicbrowser1}).
+Available content based searches are described in
+Sect.~\ref{sec:indexing}. Other examples of library operations are
+disambiguation of content level terms (see
+Sect.~\ref{sec:disambiguation}) and automatic proof searching (see
+Sect.~\ref{sec:automation}).
The key requisite for the previous operations is that the library must
be fully accessible and in a logically consistent state. To preserve
\subsection{Indexing and searching}
\label{sec:indexing}
-\TODO{sezione sull'indicizzazione}
+The \MATITA{} system is first of all an interface between the user and
+the mathematical library. For this reason, it is important to be
+able to search and retrieve mathematical concepts in a quick and
+effective way, assuming as little knowledge as possible about the
+library. To this aim, \MATITA{} uses a sophisticated indexing mechanism
+for mathematical concepts, based on a rich metadata set that has been
+tuned along the European project \MOWGLIIST{} \MOWGLI. The metadata
+set, and the searching facilites built on top of them --- collected
+in the so called \WHELP{} search engine --- have been
+extensively described in~\cite{whelp}. Let us just recall here that
+the \WHELP{} metadata model is essentially based a single ternary relation
+\REF{p}{s}{t} stating that an object $s$ refers an object $t$ at a
+ given position $p$, where the position specify the place of the
+occurrence of $t$ inside $s$ (we currently work with a fixed set of
+positions, discriminating the hypothesis from the conclusion and
+outermost form innermost occurrences). This approach is extremely
+flexible, since extending the set of positions
+we may improve the granularity and the precision of our indexing technique,
+with no additional architectural impact.
+
+Every time a new mathematical concept is created and saved by the user it gets
+indexed, and becomes immediately visible in the library. Several
+interesting and innovative features of \MATITA{} described in the following
+sections rely in a direct or indirect way on its metadata system and
+the search features. Here, we shall just recall some of its most
+direct applications.
+
+A first, very simple but not negligeable feature is the check for duplicates.
+As soon as a theorem is stated, just before starting its proof,
+the library is searched
+to check that no other equivalent statement has been already proved
+(based on the pattern matching functionality of \WHELP); if this is the case,
+a warning is raised to the user. At present, the notion of equivalence
+adopted by \MATITA{} is convertibility, but we may imagine to weaken it
+in the future, covering for instance isomorphisms.
+
+Another useful \WHELP{} operation is \HINT; we may invoke this query
+at any moment during the authoring of a proof, resulting in the list
+of all theorems of the library which can be applied to the current
+goal. In practice, this is mostly used not really to discover what theorems
+can be applied to a given goal, but to actually retrieve a theorem that
+we wish to apply, but whose name we have forgotten.
+In fact, even if \MATITA{} adopts a semi-rigid naming convention for
+statements (see Sect.~\ref{sec:naming}) that greatly simplifies the effort
+of recalling names, the naming discipline remains one of the most
+annoying aspects of formal developments, and \HINT{} provides
+a very friendly solution.
+In the near feature, we expect to extend the \HINT{} operation to
+a \REWRITEHINT, resulting in all equational statements that
+can be applied to rewrite the current goal.
\subsection{Disambiguation}
\label{sec:disambiguation}
pass. In the third pass, called \emph{library-preferences}, all aliases
available in the library are considered.
-\TODO{rivedere questo periodo\ldots}
The rationale behind this choice is trying to respect user preferences in early
passes that complete quickly in case of failure; later passes are slower but
have more chances of success.
instances} pass).
Fresh instances lead to a non negligible performance loss (since the choice of
-an alias for one instances does not constraint the choice of the others). For
+an alias for one instance does not constraint the choice of the others). For
this reason we always attempt a fresh instances pass only after attempting a
non-fresh one.
\paragraph{One-shot preferences} Disambiguation preferecens as seen so far are
instance-independent. However, implicit preferences obtained as a result of a
disambiguation pass which uses fresh instances ought to be instance-dependent.
-Informally, the set of preferences that can be satisfied by the disambiguator on
+Informally, the set of preferences that can be respected by the disambiguator on
the theorem above is: ``the first instance of the \OP{<} symbol is over natural
numbers, while the second is on integer numbers''.
-Instance-depend preferences are meaningful only for the term whose
+Instance-dependent preferences are meaningful only for the term whose
disambiguation generated it. For this reason we call them \emph{one-shot
preferences} and \MATITA{} does not use them to disambiguate further terms in
the script.
choice about operator instances. Indeed, consider again
\texttt{lt\_to\_Zlt\_pos\_pos}, which can be disambiguated using fresh operator
instances. In case there exists a coercion from natural numbers to (positive)
-integers (which indeed does, it is the \texttt{pos} constructor itself), the
+integers (which indeed does), the
theorem can be disambiguated using twice that coercion on the left hand side of
the implication. The obtained partially specified term however would not
probably be the expected one, being a theorem which prove a trivial implication.
-For this reason we choose to always prefer fresh instances over implicit
-coercions, i.e. we always attempt disambiguation passes with fresh instances
+Motivated by this and similar examples we choose to always prefer fresh
+instances over implicit coercions, i.e. we always attempt disambiguation
+passes with fresh instances
and no implicit coercions before attempting passes with implicit coercions.
\subsubsection{Disambiguation passes}
\label{sec:disambpasses}
-According to the criteria described above in \MATITA{} we choose to perform the
-sequence of disambiguation passes depicted in Tab.~\ref{tab:disambpasses}. In
+According to the criteria described above, in \MATITA{} we perform the
+disambiguation passes depicted in Tab.~\ref{tab:disambpasses}. In
our experience that choice gives reasonable performance and minimize the need of
user interaction during the disambiguation.
\begin{table}[ht]
- \caption{Disambiguation passes sequence.\strut}
+ \caption{Disambiguation passes sequence\strut}
\label{tab:disambpasses}
\begin{center}
\begin{tabular}{c|c|c|c}
\end{center}
\end{table}
-\subsection{Generation and Invalidation}
+\subsection{Generation and invalidation}
\label{sec:libmanagement}
-The aim of this section is to describe the way \MATITA{}
-preserves the consistency and the availability of the library
-using the \WHELP{} technology, in response to the user alteration or
-removal of mathematical objects.
-
-As already sketched in Sect.~\ref{sec:fullyintro} what we generate
-from a script is split among two storage media, a
-classical filesystem and a relational database. The former is used to
-store the XML encoding of the objects defined in the script, the
-disambiguation aliases and the interpretation and notational convention defined,
-while the latter is used to store all the metadata needed by
-\WHELP{}.
-
-While the consistency of the data store in the two media has
-nothing to do with the nature of
-the content of the library and is thus uninteresting (but really
-tedious to implement and keep bug-free), there is a deeper
-notion of mathematical consistency we need to provide. Each object
-must reference only defined object (i.e. each proof must use only
-already proved theorems).
-
-We will focus on how \MATITA{} ensures the interesting kind
-of consistency during the formalization of a mathematical theory,
+%The aim of this section is to describe the way \MATITA{}
+%preserves the consistency and the availability of the library
+%using the \WHELP{} technology, in response to the user alteration or
+%removal of mathematical objects.
+%
+%As already sketched in Sect.~\ref{sec:fullyintro} what we generate
+%from a script is split among two storage media, a
+%classical filesystem and a relational database. The former is used to
+%store the XML encoding of the objects defined in the script, the
+%disambiguation aliases and the interpretation and notational convention defined,
+%while the latter is used to store all the metadata needed by
+%\WHELP.
+%
+%While the consistency of the data store in the two media has
+%nothing to do with the nature of
+%the content of the library and is thus uninteresting (but really
+%tedious to implement and keep bug-free), there is a deeper
+%notion of mathematical consistency we need to provide. Each object
+%must reference only defined object (i.e. each proof must use only
+%already proved theorems).
+
+In this section we will focus on how \MATITA{} ensures the library
+consistency during the formalization of a mathematical theory,
giving the user the freedom of adding, removing, modifying objects
without loosing the feeling of an always visible and browsable
library.
-\subsubsection{Compilation}
+\subsubsection{Invalidation}
+
+Invalidation (see Sect.~\ref{sec:library}) is implemented in two phases.
+
+The first one is the calculation of all the concepts that recursively
+depend on the ones we are invalidating. The calculation of the
+reverse dependencies can be computed using the relational database
+that stores metadata.
+This technique is the same used by the \emph{Dependency Analyzer}
+and is described in~\cite{zack-master}.
+
+The second phase is the removal of all the results of the generation,
+metadata included.
+
+\subsubsection{Regeneration}
-The typechecker component guarantees that if an object is well typed
-it depends only on well typed objects available in the library,
-that is exactly what we need to be sure that the logic consistency of
-the library is preserved. We have only to find the right order of
-compilation of the scripts that compose the user development.
+%The typechecker component guarantees that if an object is well typed
+%it depends only on well typed objects available in the library,
+%that is exactly what we need to be sure that the logic consistency of
+%the library is preserved.
+
+To regenerate an invalidated part of the library \MATITA{} re-executes
+the script files that produced the invalidated concepts. The main
+problem is to find a suitable order of execution of the scripts.
For this purpose we provide a tool called \MATITADEP{}
-that takes in input the list of files that compose the development and
+that takes in input the list of scripts that compose the development and
outputs their dependencies in a format suitable for the GNU \texttt{make} tool.
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
+script files can be found) and \MATITA{} will handle all the generation
related tasks, including dependencies calculation.
-To compute dependencies it is enough to look at the script files for
-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.
+To compute dependencies it is enough to look at the script files for
+disambiguation preferences declared or imported from other scripts
+(see \ref{sec:disambaliases}).
-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.
+Regenerating the content of a modified script file involves the preliminary
+invalidation of all its old content.
\subsubsection{Batch vs Interactive}
\MATITA{} includes an interactive authoring interface and a batch
-``compiler'' (\MATITAC). Only the former is intended to be used directly by the
-user, the latter is automatically invoked 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.
+``compiler'' (\MATITAC).
+
+Only the former is intended to be used directly by the
+user, the latter is automatically invoked by \MATITA{}
+to try to regenerate parts of the library previously invalidated.
+
+While they share the same engine for generation and invalidation, they
+provide different granularity. \MATITAC{} is only able to reexecute a
+whole script and similarly to invalidate the whole content of a script
+(together with all the other scripts that rely on an concept defined
+in it).
\subsection{Automation}
\label{sec:automation}
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
+\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
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}.
+advantage of the whole bidimensional mathematical notation. The foreground
+window is an instance of the cicBrowser used to render the proof being
+developed.
Note that the syntax used in the script view is \TeX-like, however Unicode is
fully supported so that mathematical glyphs can be input as such.
\begin{figure}[!ht]
\begin{center}
\includegraphics[width=0.95\textwidth]{pics/matita-screenshot}
- \caption{\MATITA{} look and feel.\strut}
+ \caption{Authoring interface\strut}
\label{fig:screenshot}
\end{center}
\end{figure}
\includegraphics[width=0.40\textwidth]{pics/matita-screenshot-selection}
\hspace{0.05\textwidth}
\raisebox{0.4cm}{\includegraphics[width=0.50\textwidth]{pics/matita-screenshot-href}}
- \caption{Semantic selection and hyperlinks.\strut}
+ \caption[Semantic selection and hyperlinks]{Semantic selection (on the left)
+ and hyperlinks (on the right)\strut}
\label{fig:directmanip}
\end{center}
\end{figure}
-\TODO{referenziarli e spostarli nella parte sulla libreria?}
-
-\begin{figure}[!ht]
- \begin{center}
- \includegraphics[width=0.30\textwidth]{pics/cicbrowser-screenshot-browsing}
- \hspace{0.02\textwidth}
- \includegraphics[width=0.30\textwidth]{pics/cicbrowser-screenshot-query}
- \hspace{0.02\textwidth}
- \includegraphics[width=0.30\textwidth]{pics/cicbrowser-screenshot-con}
- \caption{Browsing and searching the library.\strut}
- \label{fig:cicbrowser}
- \end{center}
-\end{figure}
-
\subsection{Patterns}
\label{sec:patterns}
help the users in writing concise and elegant patterns by hand.
\begin{table}
- \caption{Patterns concrete syntax.\strut}
+ \caption{Patterns concrete syntax\strut}
\label{tab:pathsyn}
\hrule
\[
%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\\
H: m + n = n}{
m=O
}
-\noindent
+
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
+
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 $eq$.
\begin{grafite}
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 locates $n$.
\begin{grafite}
change in H:(? ? (? ? %) %) with (O + n).
\end{grafite}
-\noindent
\subsubsection{Tactics supporting patterns}
+\TODO{Grazie ai pattern, rispetto a Coq noi abbiamo per esempio la possibilita' di fare riduzioni profonde!!!}
+
\TODO{mergiare con il successivo facendo notare che i patterns sono una
interfaccia comune per le tattiche}
where $H$ is $\beta$-expanded over the second $n$
occurrence.
-At this point, since Coq unification algorithm is essentially
+At this point, since \COQ{} unification algorithm is essentially
first-order, the application of an elimination principle (of the
form $\forall P.\forall x.(H~x)\to (P~x)$) will unify
$x$ with \texttt{n} and $P$ with \texttt{(fun n0 : nat => m + n = n0)}.
\subsection{Tacticals}
\label{sec:tinycals}
-There are mainly two kinds of languages used by proof assistants to recorder
-proofs: tactic based and declarative. We will not investigate the philosophy
-around the choice that many proof assistant made, \MATITA{} included, and we
-will not compare the two different approaches. We will describe the common
-issues of the tactic-based language approach and how \MATITA{} tries to solve
-them.
+%There are mainly two kinds of languages used by proof assistants to recorder
+%proofs: tactic based and declarative. We will not investigate the philosophy
+%around the choice that many proof assistant made, \MATITA{} included, and we
+%will not compare the two different approaches. We will describe the common
+%issues of the tactic-based language approach and how \MATITA{} tries to solve
+%them.
-\subsubsection{Tacticals overview}
+The procedural proof language implemented in \MATITA{} is pretty standard,
+with a notable exception for tacticals.
-Tacticals first appeared in LCF and can be seen as programming
-constructs, like looping, branching, error recovery or sequential composition.
-The following simple example shows three tacticals in action
-\begin{grafite}
-theorem trivial:
- \forall A,B:Prop.
- A = B \to ((A \to B) \land (B \to A)).
- intros (A B H).
- split; intro;
- [ rewrite < H. assumption.
- | rewrite > H. assumption.
- ]
-qed.
-\end{grafite}
+%\subsubsection{Tacticals overview}
+
+Tacticals first appeared in LCF as higher order tactics. They can be
+seen as control flow constructs, like looping, branching, error
+recovery or sequential composition.
-The first is ``\texttt{;}'' that combines the tactic \texttt{split}
-with \texttt{intro}, applying the latter to each goal opened by the
-former. Then we have ``\texttt{[}'' that branches on the goals (here
-we have two goals, the two sides of the logic and).
-The first goal $B$ (with $A$ in the context)
-is proved by the first sequence of tactics
-\texttt{rewrite} and \texttt{assumption}. Then we move to the second
-goal with the separator ``\texttt{|}''. The last tactical we see here
-is ``\texttt{.}'' that is a sequential composition that selects the
-first goal opened for the following tactic (instead of applying it to
-them all like ``\texttt{;}''). Note that usually ``\texttt{.}'' is
-not considered a tactical, but a sentence terminator (i.e. the
-delimiter of commands the proof assistant executes).
-
-Giving serious examples here is rather difficult, since they are hard
-to read without the interactive tool. To help the reader in
-understanding the following considerations we just give few common
-usage examples without a proof context.
+
+The following simple example
+shows a Coq script made of four dot-terminated commands
\begin{grafite}
- elim z; try assumption; [ ... | ... ].
- elim z; first [ assumption | reflexivity | id ].
+Theorem trivial:
+ forall A B:Prop,
+ A = B -> ((A -> B) /\ (B -> A)).
+ intros [A B H].
+ split; intro;
+ [ rewrite < H; assumption
+ | rewrite > H; assumption
+ ].
+Qed.
\end{grafite}
-The first example goes by induction on a term \texttt{z} and applies
-the tactic \texttt{assumption} to each opened goal eventually recovering if
-\texttt{assumption} fails. Here we are asking the system to close all
-trivial cases and then we branch on the remaining with ``\texttt{[}''.
-The second example goes again by induction on \texttt{z} and tries to
-close each opened goal first with \texttt{assumption}, if it fails it
-tries \texttt{reflexivity} and finally \texttt{id}
-that is the tactic that leaves the goal untouched without failing.
-
-Note that in the common implementation of tacticals both lines are
-compositions of tacticals and in particular they are a single
-statement (i.e. derived from the same non terminal entry of the
-grammar) ended with ``\texttt{.}''. As we will see later in \MATITA{}
-this is not true, since each atomic tactic or punctuation is considered
-a single statement.
+The third command is an application of the sequencing tactical
+``$\ldots$\texttt{;}$\ldots$'', that combines the tactic
+\texttt{split} with the application of the branching tactical
+``$\ldots$\texttt{;[}$\ldots$\texttt{|}$\ldots$\texttt{|}$\ldots$\texttt{]}''
+to other tactics and tacticals.
+
+The usual implementation of tacticals executes them atomically as any
+other command. In \MATITA{} thi is not true since each punctuation is
+executed as a single command.
+
+%The latter is applied to all the goals opened by \texttt{split}
+%
+%(here we have two goals, the two sides of the logic and). The first
+%goal $B$ (with $A$ in the context) is proved by the first sequence of
+%tactics \texttt{rewrite} and \texttt{assumption}. Then we move to the
+%second goal with the separator ``\texttt{|}''.
+%
+%Giving serious examples here is rather difficult, since they are hard
+%to read without the interactive tool. To help the reader in
+%understanding the following considerations we just give few common
+%usage examples without a proof context.
+%
+%\begin{grafite}
+% elim z; try assumption; [ ... | ... ].
+% elim z; first [ assumption | reflexivity | id ].
+%\end{grafite}
+%
+%The first example goes by induction on a term \texttt{z} and applies
+%the tactic \texttt{assumption} to each opened goal eventually recovering if
+%\texttt{assumption} fails. Here we are asking the system to close all
+%trivial cases and then we branch on the remaining with ``\texttt{[}''.
+%The second example goes again by induction on \texttt{z} and tries to
+%close each opened goal first with \texttt{assumption}, if it fails it
+%tries \texttt{reflexivity} and finally \texttt{id}
+%that is the tactic that leaves the goal untouched without failing.
+%
+%Note that in the common implementation of tacticals both lines are
+%compositions of tacticals and in particular they are a single
+%statement (i.e. derived from the same non terminal entry of the
+%grammar) ended with ``\texttt{.}''. As we will see later in \MATITA{}
+%this is not true, since each atomic tactic or punctuation is considered
+%a single statement.
\subsubsection{Common issues of tactic(als)-based proof languages}
We will examine the two main problems of tactic(als)-based proof script:
maintainability and readability.
-Huge libraries of formal mathematics have been developed, and backward
-compatibility is a really time consuming task. \\
-A real-life example in the history of \MATITA{} was the reordering of
-goals opened by a tactic application. We noticed that some tactics
-were not opening goals in the expected order. In particular the
-\texttt{elim} tactic on a term of an inductive type with constructors
-$c_1, \ldots, c_n$ used to open goals in order $g_1, g_n, g_{n-1}
-\ldots, g_2$. The library of \MATITA{} was still in an embryonic state
-but some theorems about integers were there. The inductive type of
-$\mathcal{Z}$ has three constructors: $zero$, $pos$ and $neg$. All the
-induction proofs on this type where written without tacticals and,
-obviously, considering the three induction cases in the wrong order.
-Fixing the behavior of the tactic broke the library and two days of
-work were needed to make it compile again. The whole time was spent in
-finding the list of tactics used to prove the third induction case and
-swap it with the list of tactics used to prove the second case. If
-the proofs was structured with the branch tactical this task could
-have been done automatically.
-
-From this experience we learned that the use of tacticals for
-structuring proofs gives some help but may have some drawbacks in
-proof script readability. We must highlight that proof scripts
-readability is poor by itself, but in conjunction with tacticals it
-can be nearly impossible. The main cause is the fact that in proof
-scripts there is no trace of what you are working on. It is not rare
-for two different theorems to have the same proof script (while the
-proof is completely different).\\
-Bad readability is not a big deal for the user while he is
-constructing the proof, but is considerably a problem when he tries to
-reread what he did or when he shows his work to someone else. The
-workaround commonly used to read a script is to execute it again
-step-by-step, so that you can see the proof goal changing and you can
-follow the proof steps. This works fine until you reach a tactical. A
-compound statement, made by some basic tactics glued with tacticals,
-is executed in a single step, while it obviously performs lot of proof
-steps. In the fist example of the previous section the whole branch
-over the two goals (respectively the left and right part of the logic
-and) result in a single step of execution. The workaround does not work
-anymore unless you de-structure on the fly the proof, putting some
-``\texttt{.}'' where you want the system to stop.\\
-
-Now we can understand the tradeoff between script readability and
-proof structuring with tacticals. Using tacticals helps in maintaining
-scripts, but makes it really hard to read them again, cause of the way
-they are executed.
-
-\MATITA{} uses a language of tactics and tacticals, but tries to avoid
-this tradeoff, alluring the user to write structured proof without
-making it impossible to read them again.
+%Huge libraries of formal mathematics have been developed, and backward
+%compatibility is a really time consuming task. \\
+%A real-life example in the history of \MATITA{} was the reordering of
+%goals opened by a tactic application. We noticed that some tactics
+%were not opening goals in the expected order. In particular the
+%\texttt{elim} tactic on a term of an inductive type with constructors
+%$c_1, \ldots, c_n$ used to open goals in order $g_1, g_n, g_{n-1}
+%\ldots, g_2$. The library of \MATITA{} was still in an embryonic state
+%but some theorems about integers were there. The inductive type of
+%$\mathcal{Z}$ has three constructors: $zero$, $pos$ and $neg$. All the
+%induction proofs on this type where written without tacticals and,
+%obviously, considering the three induction cases in the wrong order.
+%Fixing the behavior of the tactic broke the library and two days of
+%work were needed to make it compile again. The whole time was spent in
+%finding the list of tactics used to prove the third induction case and
+%swap it with the list of tactics used to prove the second case. If
+%the proofs was structured with the branch tactical this task could
+%have been done automatically.
+%
+%From this experience we learned that the use of tacticals for
+%structuring proofs gives some help but may have some drawbacks in
+%proof script readability.
+
+Tacticals are not only used to make scripts shorter by factoring out
+common cases and repeating commands. They are a primary way of making
+scripts more mainteable. Moreover, they also have the well-known
+role of structuring the proof.
+
+However, authoring a proof structured with tacticals is annoying.
+Consider for example a proof by induction, and imagine you
+are using one of the state of the art graphical interfaces for proof assistant
+like Proof General. After applying the induction principle you have to choose:
+immediately structure the proof or postpone the structuring.
+If you decide for the former you have to apply the branching tactical and write
+at once tactics for all the cases. Since the user does not even know the
+generated goals yet, she can only replace all the cases with the identity
+tactic and execute the command, just to receive feedback on the first
+goal. Then she has to go one step back to replace the first identity
+tactic with the wanted one and repeat the process until all the
+branches are closed.
+
+One could imagine that a structured script is simpler to understand.
+This is not the case.
+A proof script, being not declarative, is not meant to be read.
+However, the user has the need of explaining it to others.
+This is achieved by interactively re-playing the script to show each
+intermediate proof status. Tacticals make this operation uncomfortable.
+Indeed, a tactical is executed atomically, while it is obvious that it
+performs lot of smaller steps we are interested in.
+To show the intermediate steps, the proof must be de-structured on the
+fly, for example replacing ``\texttt{;}'' with ``\texttt{.}'' where
+possible.\\
+
+%Proof scripts
+%readability is poor by itself, but in conjunction with tacticals it
+%can be nearly impossible. The main cause is the fact that in proof
+%scripts there is no trace of what you are working on. It is not rare
+%for two different theorems to have the same proof script.\\
+%Bad readability is not a big deal for the user while he is
+%constructing the proof, but is considerably a problem when he tries to
+%reread what he did or when he shows his work to someone else. The
+%workaround commonly used to read a script is to execute it again
+%step-by-step, so that you can see the proof goal changing and you can
+%follow the proof steps. This works fine until you reach a tactical. A
+%compound statement, made by some basic tactics glued with tacticals,
+%is executed in a single step, while it obviously performs lot of proof
+%steps. In the fist example of the previous section the whole branch
+%over the two goals (respectively the left and right part of the logic
+%and) result in a single step of execution. The workaround does not work
+%anymore unless you de-structure on the fly the proof, putting some
+%``\texttt{.}'' where you want the system to stop.\\
+
+%Now we can understand the tradeoff between script readability and
+%proof structuring with tacticals. Using tacticals helps in maintaining
+%scripts, but makes it really hard to read them again, cause of the way
+%they are executed.
+
+\MATITA{} has a peculiar tacticals implementation that provides the
+same benefits as classical tacticals, while not burdening the user
+during proof authoring and re-playing.
+
+%\MATITA{} uses a language of tactics and tacticals, but tries to avoid
+%this tradeoff, alluring the user to write structured proof without
+%making it impossible to read them again.
\subsubsection{The \MATITA{} approach: Tinycals}
\begin{table}
- \caption{Concrete syntax of \MATITA{} tacticals.\strut}
+ \caption{Concrete syntax of tacticals\strut}
\label{tab:tacsyn}
\hrule
\[
\MATITA{} tacticals syntax is reported in Tab.~\ref{tab:tacsyn}.
While one would expect to find structured constructs like
$\verb+do+~n~\NT{tactic}$ the syntax allows pieces of tacticals to be written.
-This is essential for base idea behind \MATITA{} tacticals: step-by-step
+This is essential for the base idea behind \MATITA{} tacticals: step-by-step
execution.
The low-level tacticals implementation of \MATITA{} allows a step-by-step
execution of a tactical, that substantially means that a $\NT{block\_kind}$ is
-not executed as an atomic operation. This has two major benefits for the user,
-even being a so simple idea:
-\begin{description}
-\item[Proof structuring]
- is much easier. Consider for example a proof by induction, and imagine you
- are using classical tacticals in one of the state of the
- art graphical interfaces for proof assistant like Proof General or \COQIDE.
- After applying the induction principle you have to choose: structure
- the proof or not. If you decide for the former you have to branch with
- ``\texttt{[}'' and write tactics for all the cases separated by
- ``\texttt{|}'' and then close the tactical with ``\texttt{]}''.
- You can replace most of the cases by the identity tactic just to
- concentrate only on the first goal, but you will have to go one step back and
- one further every time you add something inside the tactical. Again this is
- caused by the one step execution of tacticals and by the fact that to modify
- the already executed script you have to undo one step.
- And if you are board of doing so, you will finish in giving up structuring
- the proof and write a plain list of tactics.\\
- With step-by-step tacticals you can apply the induction principle, and just
- open the branching tactical ``\texttt{[}''. Then you can interact with the
- system reaching a proof of the first case, without having to specify any
- tactic for the other goals. When you have proved all the induction cases, you
- close the branching tactical with ``\texttt{]}'' and you are done with a
- structured proof. \\
- While \MATITA{} tacticals help in structuring proofs they allow you to
- choose the amount of structure you want. There are no constraints imposed by
- the system, and if the user wants he can even write completely plain proofs.
+not executed as an atomic operation. This has major benefits for the
+user during proof structuring and re-playing.
+
+For instance, reconsider the previous example of a proof by induction.
+With step-by-step tacticals the user can apply the induction principle, and just
+open the branching tactical ``\texttt{[}''. Then she can interact with the
+system until the proof of the first case is terminated. After that
+``\texttt{|}'' is used to move to the next goal, until all goals are
+closed. After the last goal, the user closes the branching tactical with
+``\texttt{]}'' and is done with a structured proof. \\
+While \MATITA{} tacticals help in structuring proofs they allow you to
+choose the amount of structure you want. There are no constraints imposed by
+the system, and if the user wants he can even write completely plain proofs.
-\item[Rereading]
- is possible. Going on step by step shows exactly what is going on. Consider
- again a proof by induction, that starts applying the induction principle and
- suddenly branches with a ``\texttt{[}''. This clearly separates all the
- induction cases, but if the square brackets content is executed in one single
- step you completely loose the possibility of rereading it and you have to
- temporary remove the branching tactical to execute in a satisfying way the
- branches. Again, executing step-by-step is the way you would like to review
- the demonstration. Remember that understanding the proof from the script is
- not easy, and only the execution of tactics (and the resulting transformed
- goal) gives you the feeling of what is going on.
-\end{description}
+Re-playing a proof is also made simpler. There is no longer any need
+to destructure the proof on the fly since \MATITA{} executes each
+tactical not atomically.
+
+%\item[Rereading]
+% is possible. Going on step by step shows exactly what is going on. Consider
+% again a proof by induction, that starts applying the induction principle and
+% suddenly branches with a ``\texttt{[}''. This clearly separates all the
+% induction cases, but if the square brackets content is executed in one single
+% step you completely loose the possibility of rereading it and you have to
+% temporary remove the branching tactical to execute in a satisfying way the
+% branches. Again, executing step-by-step is the way you would like to review
+% the demonstration. Remember that understanding the proof from the script is
+% not easy, and only the execution of tactics (and the resulting transformed
+% goal) gives you the feeling of what is going on.
+%\end{description}
\section{Standard library}
\label{sec:stdlib}
\FILE{fermat\_little\_th.ma} \\
\FILE{totient.ma} & & \\
\end{tabular}
- \caption{Scripts on natural numbers in the standard library.\strut}
+ \caption{Scripts on natural numbers in the standard library\strut}
\label{tab:scripts}
\end{table}
\TODO{conclusioni}
\acknowledgements
-We would like to thank all the students that during the past
-five years collaborated in the \HELM{} project and contributed to
+We would like to thank all the people that during the past
+7 years collaborated in the \HELM{} project and contributed to
the development of \MATITA{}, and in particular
M.~Galat\`a, A.~Griggio, F.~Guidi, P.~Di~Lena, L.~Padovani, I.~Schena, M.~Selmi,
and V.~Tamburrelli.
\theendnotes
+\TODO{rivedere bibliografia, \'e un po' povera}
+
+\TODO{aggiungere entry per le coercion implicite}
+
\bibliography{matita}
\end{document}
-
projects / helm.git / blobdiff