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\begin{frontmatter}
\title{Teaching logic using a state-of-the-art \\ proof assistant}
\author{Cezary Kaliszyk}
\author{Freek Wiedijk}
\address{Radboud Universiteit Nijmegen, The Netherlands}
\author{Maxim Hendriks}
\author{Femke van Raamsdonk}
\address{Vrije Universiteit Amsterdam, The Netherlands}
\thanks[emails]{Email: \texttt{\normalshape $\{$cek,freek$\}$@cs.ru.nl $\{$mhendri,femke$\}$@few.vu.nl}}
\thanks[surf]{This research was funded by SURF project `Web-deductie voor het onderwijs in for\-meel denken'.}
\begin{abstract}
This article describes the system {\textsc{ProofWeb}}
that is currently being developed in Nijmegen
and Amsterdam for
teaching logic to undergraduate computer science students.
This system is based on the higher order proof assistant Coq,
and is made available to the students through an interactive web interface.
Part of this system will be a large database of logic problems.
This database will also hold the solutions of the students.
This means that the students do not need to install anything to be able
to use the system (not even a browser plug-in), and that the teachers
will be able to centrally track progress of the students.
The system makes the full power of Coq available to the students,
but simultaneously presents the logic problems in a way that
is customary in undergraduate logic courses.
Both styles of presenting natural deduction proofs (Gentzen-style `tree view'
and Fitch-style `box view') are supported.
Part of the system is a parser that indicates whether the students
used the automation of Coq to solve their problems or that they
solved it themselves using only the inference rules of the logic.
For these inference rules dedicated tactics for Coq have been developed.
The system has already been used in a type theory
course, and is currently being further developed in the first year logic course
of computer science in Nijmegen.
\end{abstract}
\begin{keyword}
Logic Education, Proof Assistants, Coq, Web Interface,
AJAX, DOM, Natural Deduction, Gentzen, Fitch
\end{keyword}
\end{frontmatter}
\section{Introduction}
\subsection{Motivation}
At every university, part of the undergraduate computer science curriculum
is an introductory course that teaches
the rules of propositional and predicate logic.
At the Radboud Universiteit (RU) in Nijmegen
this course is taught in the first year and is called `Beweren en Bewijzen'
(Dutch for `Stating and Proving').
At the Vrije Universiteit (VU) in Amsterdam this
course is taught in the second year and is called `Inleiding
Logica' (`Introduction to Logic').
Almost all computer science curricula will have similar undergraduate courses.
For learning this kind of elementary mathematical logic
it is crucial to make many exercises.
Those exercises can of course be made in the traditional way,
using pen and paper.
The student is completely on his own, and in practice
it often happens that proofs that are almost-but-not-completely-right
are produced.
Alternatively, they can be made using some computer program,
which guides the student through the development of a completely
correct proof.
A disadvantage of the computerized way of practicing mathematical
logic is that a student often will be able to finish proofs by random
experimentation with the commands of the system (accidentally hitting a solution),
without really having understood how the proof works.
Of course, a combination of the two styles of practicing
formal proofs seems to be the best option.
So computer assistance for learning to construct
derivations in mathematical logic is desirable.
Currently the most popular program that is used
for this kind of `computer-assisted logic teaching'
is a system called Jape~\cite{jape}, developed at the university of Oxford.
Besides exercises there is also the issue of examination.
It would be good if the student has the opportunity to do
at any moment a (part of the) logic exam by logging in to
the system and be presented with a set of exercises from
a database that have to be solved within a certain time.
This may require human supervision to prevent cheating.
We did not yet work on this, but just mention it as a
possible interesting application of computer-assisted logic teaching.
\subsection{Our contribution}
This paper describes a system, currently named {\textsc{ProofWeb}},
that is in development at the RU
in Nijmegen and at the VU in Amsterdam.
This system is much like Jape
(it might be considered to be an `improved Jape-clone').
The two main innovations that our system offers over other similar
systems are:
\begin{itemize}
\item
The system makes the students work on a centralized server that
has to be accessed through a web interface.
The proof assistant that the students use will not run
on their computer, but instead will run on the server.
A first advantage is flexibility.
The web interface is extremely light: the student will not need
to install anything to be able to use it, not even a plug-in.
When designing our system we tried to make it as low-threshold and
non-threatening as possible.
The student can work from any internet-connection at any time.
A second advantage is that the student does not need to worry
about version problems with the software or the exercises.
Since everything is on the same centralized server,
the students have at any time the right version of the
software, exercises, and possibly solutions to exercises
available, and moreover the teachers know at any time
the current status of the work of the students.
\item
The system makes use of a state-of-the-art proof assistant,
namely Coq~\cite{coqman}, and not of a `toy' system.
Coq has been in development since 1984 at the INRIA institute in France.
It is based on a type theoretical system called \emph{the Calculus of Inductive
Constructions}.
It has been implemented in the dialect of the ML programming language called Objective Caml,
and has been used for the formal verification of many proofs, both from mathematics
and from computer science.
The most impressive verification using Coq is the verification of the
proof of the Four Colour Theorem by Georges Gonthier~\cite{gon:06}.
Another important verification has been the development of a verified
C compiler by Xavier Leroy and others~\cite{cminor}.
The choice for a state-of-the-art proof assistant fell on Coq
because both at the RU and at the VU it is already used in
research and teaching.
An advantage of using a state-of-the-art proof assistant is
again flexibility.
The same interface can be used (possibly adapted) for teaching
more advanced courses in logic or concerning the use of the proof assistant.
\end{itemize}
The system {\textsc{ProofWeb}} comes equipped with two more products.
\begin{itemize}
\item
A large collection of logic exercises.
The exercises range from very easy to very difficult,
and will be graded for their difficulty.
The exercise set is sufficiently large (presently over
200 exercises) so the student will not soon run out of practice material.
More about the exercise set can be found in Section~\ref{sec:exercise-set}.
\item
Course notes, with a basic presentation of propositional and predicate logic,
and a description of how to use the system {\textsc{ProofWeb}}.
We want the presentation of the proofs in the system to be
identical to the presentation of the proofs in the textbook.
Therefore we develop both the `Gentzen-style' and the `Fitch-style'
natural deduction variants.
The course notes are still under development.
\end{itemize}
\subsection{Related work}
There are already numerous systems for doing logic by computer, of which Jape
is the best known.
A relatively comprehensive list
is maintained by Hans van Ditmarsch~\cite{courselist}.
Of course many of these system are quite similar to our system
(as well as to each other.)
For instance, quite a number of these systems are already web-based.
The distinctive features of our system
are the use of a serious proof assistant,
together with a \emph{centralized} `web application' architecture.
The work of the students remains on the web server, can be saved
and loaded back in, and the progress of the student is at all
times available both to the student, the teacher and the system (i.e., the
system has at all times an accurate `user model' of the abilities of the student).
\subsection{Contents}
In the rest of the paper we present both our project
and the current state of the system that we are building.
We start with a short description of our project in Section~\ref{sec:structure},
and discuss our experiences so far in Section~\ref{sec:experience}.
Next, in Section~\ref{sec:architecture} we present the architecture
of the interface.
Section~\ref{sec:deduction} is concerned with
the supporting infrastructure of tactics and exercises,
and Section~\ref{sec:exercise-set} with the presentation
of the collection of exercises.
Finally, in Section~\ref{sec:outlook} we give an outlook on
future work and work that is currently in progress.
\section{Structure of the project} \label{sec:structure}
The project of developing {\textsc{ProofWeb}}
is financed by the SURF foundation~\cite{surf}
(the Dutch organization for computers in academic teaching)
and runs in the period fall 2006 till fall 2007 (three semesters).
Cezary Kaliszyk is
employed for a full year at half time to program the system,
while Maxim Hendriks is employed for half a year at full time
to develop the educational materials
(the database of problems and the course notes),
as well as to evaluate the educational success of the project.
We identified the following nine sub-tasks, called `work packages':
\begin{enumerate}
\item the database of the system,
\item Coq tactics that exactly correspond to the rules the logic,
\item graphical representations for the proofs,
\item checking a Coq file against an exercise,
\item a large set of logic problems,
\item course notes that explain the system,
\item using the system in actual courses,
\item dissemination of the results of the project,
\item evaluation of the project.
\end{enumerate}
\section{Experience so far} \label{sec:experience}
The system {\textsc{ProofWeb}} is used in the following advanced courses:
\begin{enumerate}
\item
In fall 2006:
the course `Logical Verification'~\cite{lv} at the VU,
taught by Femke van Raamsdonk.
This is a computer science master's course
about the type theory of the Coq system.
The course is meant for more mature students but also
recapitulates some undergraduate logic.
It is therefore suitable for testing a first version of {\textsc{ProofWeb}}.
Natural deduction is taught in Gentzen style,
that is, proofs have a tree-like structure,
and grow upward from the conclusion of the proof.
\item
In spring 2007:
the course `Type Theory' at the RU,
taught by Freek Wiedijk and Milad Niqui.
This course
is also a master's level course about the type theory of the Coq system,
and corresponds to the Logical Verification course at the VU.
\item
In spring 2007:
the course `Type Theory and Proof Assistants' in the
`Master Class Logic 2006-2007',
taught by Herman Geuvers and Bas Spitters.
This course is similar to the previous one,
but is not exclusively aimed at students of the RU
but at master's students from all over the Netherlands.
\end{enumerate}
Moreover, {\textsc{ProofWeb}} is or will be used in the following introductory courses:
\begin{enumerate}
\item
In spring 2007:
the course `Beweren en Bewijzen'~\cite{bb} at the RU,
taught by Hanno Wupper and Erik Barendsen.
This is a computer science undergraduate course in logic,
with natural deduction in Gentzen style.
\item
In fall 2007:
the course `Inleiding Logica'~\cite{il} at the VU,
taught by Roel de Vrijer.
This is a computer science undergraduate course in logic,
with natural deduction in Fitch style (cf.~Section \ref{sec:outlook}), that is,
proofs have a structure of nested boxes, which structure a
sequential list of proof steps.
Another name for this kind of proofs is `flag-style proofs',
because often the assumptions of a subproof are written in the shape of `flags'.
\end{enumerate}
The course `Logical Verification' at the VU in fall 2006
was a first opportunity to test the system.
About 25 students followed and completed the course.
They were all mature (graduate) students,
very well able to deal with a system that was still in beta.
A part of the course consists of learning type theory and Coq
via basic (undergraduate) logic exercises, which were done
using the system {\textsc{ProofWeb}}.
We learned the following from the use of the system
{\textsc{ProofWeb}} in this course.
Initially we did not
have a dedicated server, so it was running on one of the group servers
of the research group in Nijmegen
on a non-standard port.
One of the issues was, that the web-proxy at the
VU did not allow the students to access pages running on
non-standard ports, so they were required to turn the proxy off.
One of the assignments involves program extraction.
Of course we did not allow running the
extracted programs on the server,
and therefore a mechanism allowing the students to obtain
the extracted program was implemented.
The efficiency of the server turned out not to be a problem.
At peak times the
twenty-five students were able to use about 400Mb memory and a fraction of a
CPU.
This might be thanks to the fact that the students were not using
tactics that involve automation.
During this course there was not yet support for visualizing proofs.
Instead the students had to do their proofs using the customary
Coq proof style,
which consists of building a tactic script using
the standard Coq tactics.
This was not problematic, since one of the aims of the course is to learn Coq.
The second course in which {\textsc{ProofWeb}} is used is
the course `Type Theory' in spring 2007 at the RU.
The first half of this course is basically
an accelerated clone of the `Logical Verification'
course.
As it turned out that initially there were only very few students who
wanted to follow this course,
it was decided that there would be no lectures, and
that the students just would be given the course notes
of `Logical Verification' together with access
to the server.
They then would work on their own, with an opportunity
to call for help if needed.
It turns out that this worked unexpectedly well.
The students just studied the lecture notes and did the exercises of the
course.
And even without much pressure on them in the form of requiring them to meet
deadlines, they managed to keep on schedule reasonably well.
The only thing that at some point confused them (after which a lecture
was organized to make things clear) was the part of the course that did
not correspond to Coq work:
derivations in Pure Type Systems.
All in all our experience so far is that the system {\textsc{ProofWeb}}
seems to work very well in teaching.
Indeed, hardly any students used
more traditional Coq interfaces like Proof General or
CoqIDE.
The courses so far are more advanced ones,
so it remains to be seen whether {\textsc{ProofWeb}}
also works well for larger numbers of undergraduate students,
but we are optimistic about that.
In addition, as of
May 2007, the progress on all of the nine work packages seems to be well on
target.
\section{Architecture of the interface} \label{sec:architecture}
% \section{Web interface to Coq}\label{product}
In this section we shortly describe the architecture of the interface to Coq
used in {\textsc{ProofWeb}}.
The interface is an implementation of an architecture for creating
responsive web interfaces for proof assistants \cite{cek-uitp}. It combines
the current web development technologies with the functionality of local
interfaces for proof assistants to create an interface that behaves like a
local one, but is available completely with just a web browser (no Java, Flash
or plugins are required).
To obtain this it uses the \emph{asynchronous \textrm{DOM} modification}
technology (sometimes referred to as \emph{AJAX} or \emph{Web Application}).
This technique is a combination of three available web technologies:
\textrm{JavaScript} - a scripting programming language interpreted by the web
browsers, \emph{Document Object Model} (\emph{DOM}) - a way of referring to
sub elements of a web page that allows modification of the page on the fly
creating dynamic elements and \textrm{XmlHttp} - an \textrm{API} available to
client side scripts, that allows requesting information from the web server
without reloading the page.
The technique consists in creating a web page that captures events on the
client side and processes them without reloading the page. Events that require
information from the server send the data in asynchronous \textrm{XmlHttp}
requests and modify the web page in place. Other events are processed only
locally.
{\textsc{ProofWeb}} uses an implementation of this architecture that is used to
create a web interface for proof assistants.
The server keeps prover sessions
for all users on the server (see Figure \ref{fig:arch}) and the clients are
presented with an interface that is completely available in a web-browser but
resembles and is comparably responsive to a local interface.
The architecture described in \cite{cek-uitp} was designed as a
publicly available web service.
Using it for teaching required the creation of groups of logins for particular courses.
The students are allowed to access only their own files via the web interface,
and teachers of particular courses have access to the directories of the
students of these courses.
\begin{figure}[htpb]
\begin{center}
\includegraphics[width=.6\textwidth]{proofweb}
\caption{{\textsc{ProofWeb}}{} architecture.}
\label{fig:arch}
\end{center}
\end{figure}
An example of the use of the interface in the `Logical Verification'
course can be seen in Figure~\ref{fig:lv}.
\begin{figure}[htpb]
\begin{center}
\includegraphics[width=.99\textwidth]{lv}
\caption{{\textsc{ProofWeb}}{} in the `Logical Verification' course.}
\label{fig:lv}
\end{center}
\end{figure}
\section{Natural deduction for first-order logic} \label{sec:deduction}
This section is concerned with natural deduction proofs for first-order logic in `Gentzen style', where a proof is a tree.
\subsection{Tactics}
A first aim is to
enable students of logic courses to construct derivations that correspond exactly
to the derivations in the presentation of natural deduction that they use.
Because in principle the full power of Coq is available, this means that
we had to write tactics (in effect, to dumb Coq down) to match the traditional logic rules. What then arose was that, whereas in a Coq proof one can look at a hypothesis and eliminate it, ending up in a new proof state, traditional natural deduction offers no such jumps. So we naturally arrived at a set of backward working tactics: every proposition (the current goal) is deduced from another proposition (the new goal) using a deduction rule. The display style that fits most naturally to this kind of proof is a proof tree (for flag-style proofs see Section \ref{sec:outlook}).
This imposes a relatively strict way of working. The proof trees have to be constructed from `bottom to top'. On the one hand, this makes the construction of a deduction more difficult than on paper, because there is no possibility of building snippets of the proof in a forward way, using what is known from the hypotheses and their consequences. But on the other hand, the method forces the student to ponder the general structure of the proof before deciding by what step he will eventually end up with the current proposition. And the imposed rigidity is congenial with the aim of a logic course to encourage rigorous analytical thinking. Moreover, it becomes very clear where ingenuity comes in, such as with the disjunction elimination rule. The student is supposed to prove some proposition $C$. It is a creative step to find a disjunction $A \vee B$, prove this, and also prove that $C$ follows from both $A$ and $B$ separately. The same goes for the introduction and elimination of negation.
As an example we present the tactic for disjunction elimination,
which gives a good impression of the way additional tactics are implemented:
\begin{lstlisting}
Ltac dis_el X H1 H2 :=
match X with
| ( _ \/ _ ) =>
assert X;
[ idtac |
match goal with
| x : X |- _ =>
elim x; [intro H1 | intro H2]; clear x
end
]
| _ => fail "The first argument is not a disjunction"
end.
\end{lstlisting}
If the current goal is $C$, the tactic \verb#dis_el (A \/ B) G H#
will create the following three new goals:
\begin{enumerate}
\item $A \vee B$;
\item $C$, but now with the extra assumption $A$ with name (or proof, if viewed constructively) $G$;
\item $C$, but now with the extra assumption $B$ with name (or proof, if viewed constructively) $H$.
\end{enumerate}
Also, the tactic gives a nice and understandable error message.
All the tactics have been given a name by using three letters of the connective's name and indicating whether the tactic implements an introduction rule or an elimination rule (and if necessary, if that is a left or a right variant). We give a small example of a proof with our set of tactics, and hope it speaks for itself:
\begin{verbatim}
Theorem example : ((A \/ B) /\ ~A) -> B.
Proof.
imp_in z.
dis_el (A \/ B).
con_ell (~A).
ass z.
imp_in y.
neg_el A.
con_elr (A \/ B).
ass z.
ass y.
imp_in x.
ass x.
Qed.
\end{verbatim}
\subsection{Visualization}
A second aim
is a visual presentation of proofs as in Jape.
This meant requesting the proof information from Coq and
converting it to a graph format. Coq internally keeps a proof state. This
proof state is a recursive {\textsc{OCaml}} structure, that holds a goal, a rule which
allows to obtain this goal from the subgoals, and the subgoals themselves. It
is not just a tree structure, since a rule can be a compound rule that
contains another proof state. Tactics and tacticals modify the proof state.
Coq includes commands that allow inspecting the proof state. \texttt{Show}
allows the user to see a non-current goal, \texttt{Show Tree} shows the
succession of conclusions, hypotheses and tactics used to obtain the current
goal and \texttt{Show Proof} displays the CIC term (possibly with holes).
The output of these commands was not sufficient to build a natural deduction tree for the proof. We added a new command \texttt{Dump Tree} to Coq that allows exporting the whole proof state in an XML format. An example of the output of the
\texttt{Dump Tree} command for a very simple Coq proof:
\begin{alltt}
\end{alltt}
\noindent
We modified {\textsc{ProofWeb}}{} to be able to parse XML trees dumped by Coq and
generate natural deduction diagrams (see Figure~\ref{fig:natural}). Those
diagrams may be requested by the user's browser in special query
requests. The diagrams are displayed in a separate frame in the
interface along with the usual Coq proof state. If the user switches on
the display of the diagrams, the client side requests them when no
text is being processed.
\begin{figure}[htb]
\centering
\hspace{-1em}\fbox{\includegraphics[width=.99\textwidth]{natural}}\hspace{-1em}
\caption{A natural deduction tree as seen on the webpage (cropped screenshot).}
\label{fig:natural}
\end{figure}
\section{The exercise set} \label{sec:exercise-set}
Also part of the project is the development of
a set of exercises for the students.
For a particular course,
a number of exercises assigned to the students.
It is desirable that {\textsc{ProofWeb}} can be used as a
complete course environment.
So when a student logs in via the web interface
as participant to a specific course, he is able to see the list of all the
assigned tasks (see Figure~\ref{fig:tasklist}).
Every task has a certain status.
The status can be one of the following:
\begin{itemize}
\item Not touched --- When a particular exercise has not been opened, or has
been opened but has not been saved.
\item Does not compile --- When the file has been edited and saved, but is not a
correct Coq file. It can be because of real errors or because proofs are
missing.
\item Incorrect --- The students are supposed to modify the given file only in
designated places and to use only a set of allowed tactics. If the student
uses a non-allowed too powerful tactic or just removes a task from the file
it is marked as incorrect.
\item Correct --- Passed the verification by our tool.
\end{itemize}
\begin{figure}[htpb]
\begin{center}
\includegraphics[width=.6\textwidth]{tasklist}
\caption{Tasks assigned to students and their status.}
\label{fig:tasklist}
\end{center}
\end{figure}
\noindent
The verification tool lexes the original task and the student's solution in
parallel. The original solution includes placeholders that are valid Coq
comments. Those placeholders mean that a particular place needs to contain a
valid Coq term or a valid proof. For proofs the kind of proof determines the
set of allowed tactics. For proofs and terms of given types the automatic
verification is enough. However, there are tasks where students are required
to give a definition of a particular object in type theory. For this kind of
tasks manual verification by a teaching assistant of a course is required.
\section{Outlook} \label{sec:outlook}
At the moment of writing, the project {\textsc{ProofWeb}} is more or less half way, so this paper reports on work in progress.
In this section we first discuss a main issue we are currently
working on:
adding the possibility of using the system for `Fitch-style' natural
deduction derivations.
We then briefly comment on further points of current and future work.
\subsection{Fitch-style deductions}
The most important improvement is to add the possibility to use the system for so-called \emph{Fitch-style} natural deduction proofs.\footnote{This style of proof was initially developed by Stanis{\l}aw Ja\'skowski in 1934 and perfected by Frederic Brenton Fitch in 1952.} Fitch-style proofs have the graphical advantage over Gentzen-style proofs of being linear (as opposed to having a branching tree structure), which makes them more convenient to display for large proofs, like the ones constructed by the students for final assignments. Another name for these kind of proofs is \emph{flag-style proofs}, because the assumptions of a subproof are often written in the shape of `flags'.
\begin{figure}[htb]
\begin{center}
\fbox{\includegraphics[width=.6\textwidth]{fitch}}
\caption{A Fitch-style deduction as currently displayed by the system (implementation in progress).}
\label{fig:fitch}
\end{center}
\end{figure}
We are working on having the system display Fitch-style deductions. A basic version of this has already been implemented (see Figure~\ref{fig:fitch}), but needs further development. A number of issues arise. When a tactic
creates a number of assumptions, should these be kept in one flag or should the system create multiple flags?
Also, Fitch-style deductions may include repetitions of assertions assumed by flags. Most of these repetitions are redundant. However, not repeating assumptions immediately before they are used leads to very unreadable proofs.
What should be done? We are currently looking at adapting the approach presented in~\cite{herman-fitch} to Coq tactics.
\subsection{Future work}
Some of the other issues that currently are being worked on are:
\begin{itemize}
\item
The course notes that are to accompany the system.
These are still in a rudimentary stage.
\item
There is no separate web interface yet for the teacher to manage the student logins and the set of exercises for the
course, nor to inspect the work of the students. At the moment this is only possible by logging on to the server
through an ssh connection, and then listing and editing files manually. Clearly, a proper web interface for this is necessary.
\item
The deduction trees are currently rendered in an HTML IFrames, and can be
optionally opened in a separate browser window to allow easy printing as
PostScript or PDF. However students may need to use the trees in texts, and
for that a dedicated {\TeX} or image rendering of the trees could be
implemented.
\item
The interface uses some web technologies that are not implemented in the same
way in all browsers. It includes a small layer that is supposed to abstract
over incompatible functionalities. Currently this works well with Mozilla
compatible browsers (Firefox, Galeon, Epiphany, Netscape, \ldots).
Also, some effort has been made to make the system work reasonably well
with the most common versions of Internet Explorer.
However, the compatibility of the system with most common web browsers is
something that will need further attention.
\item
At the moment there is hardly any documentation of how to
install and maintain the server.
Our server currently is available to everyone who wants to
experiment with our system, but
there is no good guide available that explains how to
install a server of his own.
Because the server is still very much in a constant state of
flux, documenting the installation and maintenance processes is
at this moment not yet reasonable.
However, in the final phase of the project it will be important
to also create this kind of documentation.
\item
With the current version of the system a log of each interaction
of each student session is already stored on the server.
Using these logs, it is possible to develop software for `replaying'
such a student session (possibly speeded up or slowed down).
We are currently discussing whether it is useful to develop
such an extension of the system.
\item
The system was designed in a way to be used in standard university courses.
It might be useful to create a more complete online environment that would
include introductory explanations and adaptive user profiles, therefore
allowing students to learn logic without teacher interaction.
\end{itemize}
\subsection{Beyond the project}
If the development of {\textsc{ProofWeb}} is finished, a possibility is
to integrate it
with a system that supports the development
of more serious proofs with the Coq system.
One of the other projects that currently is being pursued in Nijmegen
is the creation of a so-called `math wiki' \cite{mathwiki}.
Here, traditional wiki technology is integrated
with the same Coq front end that our system is based on.
\subsection{Using the system}
We think that it is important that our system is
experimented with (and hopefully someday frequently used)
by as many people as possible.
For this reason, we currently offer the use of
our system to anyone on the internet, even without any registration.
The {\textsc{ProofWeb}} system can be tried at
\begin{center}
\url{http://prover.cs.ru.nl}
\end{center}
\bibliographystyle{entcs}
\bibliography{biblio}
\end{document}