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%BEGIN LATEX
\setheaders{Introduction}
%END LATEX
\chapter*{Introduction}
%HEVEA\cutname{introduction.html}
This document is the Reference Manual of version \coqversion{} of the \Coq\
proof assistant. A companion volume, the \Coq\ Tutorial, is provided
for the beginners. It is advised to read the Tutorial first.
A book~\cite{CoqArt} on practical uses of the \Coq{} system was published in 2004 and is a good support for both the beginner and
the advanced user.
%The system \Coq\ is designed to develop mathematical proofs. It can be
%used by mathematicians to develop mathematical theories and by
%computer scientists to write formal specifications,
The \Coq{} system is designed to develop mathematical proofs, and
especially to write formal specifications, programs and to verify that
programs are correct with respect to their specification. It provides
a specification language named \gallina. Terms of \gallina\ can
represent programs as well as properties of these programs and proofs
of these properties. Using the so-called \textit{Curry-Howard
isomorphism}, programs, properties and proofs are formalized in the
same language called \textit{Calculus of Inductive Constructions},
that is a $\lambda$-calculus with a rich type system. All logical
judgments in \Coq\ are typing judgments. The very heart of the Coq
system is the type-checking algorithm that checks the correctness of
proofs, in other words that checks that a program complies to its
specification. \Coq\ also provides an interactive proof assistant to
build proofs using specific programs called \textit{tactics}.
All services of the \Coq\ proof assistant are accessible by
interpretation of a command language called \textit{the vernacular}.
\Coq\ has an interactive mode in which commands are interpreted as the
user types them in from the keyboard and a compiled mode where
commands are processed from a file.
\begin{itemize}
\item The interactive mode may be used as a debugging mode in which
the user can develop his theories and proofs step by step,
backtracking if needed and so on. The interactive mode is run with
the {\tt coqtop} command from the operating system (which we shall
assume to be some variety of UNIX in the rest of this document).
\item The compiled mode acts as a proof checker taking a file
containing a whole development in order to ensure its correctness.
Moreover, \Coq's compiler provides an output file containing a
compact representation of its input. The compiled mode is run with
the {\tt coqc} command from the operating system.
\end{itemize}
These two modes are documented in Chapter~\ref{Addoc-coqc}.
Other modes of interaction with \Coq{} are possible: through an emacs
shell window, an emacs generic user-interface for proof assistant
({\ProofGeneral}~\cite{ProofGeneral}) or through a customized interface
(PCoq~\cite{Pcoq}). These facilities are not documented here. There
is also a \Coq{} Integrated Development Environment described in
Chapter~\ref{Addoc-coqide}.
\section*{How to read this book}
This is a Reference Manual, not a User Manual, so it is not made for a
continuous reading. However, it has some structure that is explained
below.
\begin{itemize}
\item The first part describes the specification language,
Gallina. Chapters~\ref{Gallina} and~\ref{Gallina-extension}
describe the concrete syntax as well as the meaning of programs,
theorems and proofs in the Calculus of Inductive
Constructions. Chapter~\ref{Theories} describes the standard library
of \Coq. Chapter~\ref{Cic} is a mathematical description of the
formalism. Chapter~\ref{chapter:Modules} describes the module system.
\item The second part describes the proof engine. It is divided in
five chapters. Chapter~\ref{Vernacular-commands} presents all
commands (we call them \emph{vernacular commands}) that are not
directly related to interactive proving: requests to the
environment, complete or partial evaluation, loading and compiling
files. How to start and stop proofs, do multiple proofs in parallel
is explained in Chapter~\ref{Proof-handling}. In
Chapter~\ref{Tactics}, all commands that realize one or more steps
of the proof are presented: we call them \emph{tactics}. The
language to combine these tactics into complex proof strategies is
given in Chapter~\ref{TacticLanguage}. Examples of tactics are
described in Chapter~\ref{Tactics-examples}.
%\item The third part describes how to extend the system in two ways:
% adding parsing and pretty-printing rules
% (Chapter~\ref{Addoc-syntax}) and writing new tactics
% (Chapter~\ref{TacticLanguage}).
\item The third part describes how to extend the syntax of \Coq. It
corresponds to the Chapter~\ref{Addoc-syntax}.
\item In the fourth part more practical tools are documented. First in
Chapter~\ref{Addoc-coqc}, the usage of \texttt{coqc} (batch mode)
and \texttt{coqtop} (interactive mode) with their options is
described. Then, in Chapter~\ref{Utilities},
various utilities that come with the \Coq\ distribution are
presented.
Finally, Chapter~\ref{Addoc-coqide} describes the \Coq{} integrated
development environment.
\item The fifth part documents a number of advanced features, including
coercions, canonical structures, typeclasses, program extraction, and
specialized solvers and tactics. See the table of contents for a complete
list.
\end{itemize}
At the end of the document, after the global index, the user can find
specific indexes for tactics, vernacular commands, and error
messages.
\section*{List of additional documentation}
This manual does not contain all the documentation the user may need
about \Coq{}. Various informations can be found in the following
documents:
\begin{description}
\item[Tutorial]
A companion volume to this reference manual, the \Coq{} Tutorial, is
aimed at gently introducing new users to developing proofs in \Coq{}
without assuming prior knowledge of type theory. In a second step, the
user can read also the tutorial on recursive types (document {\tt
RecTutorial.ps}).
\item[Installation] A text file INSTALL that comes with the sources
explains how to install \Coq{}.
\item[The \Coq{} standard library]
A commented version of sources of the \Coq{} standard library
(including only the specifications, the proofs are removed)
is given in the additional document {\tt Library.ps}.
\end{description}
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