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authorGravatar Matt Quinn <matt@mattjquinn.com>2017-12-11 07:28:23 +0300
committerGravatar Matt Quinn <matt@mattjquinn.com>2017-12-18 16:44:27 +0300
commitd1457ecc41713b8fff745a66cd6c8375f26f4fc9 (patch)
treef1d244c0900469368540f42a5f8b01277a4e168d /doc
parent359119ff1389d489454d2d55fc5d23e9e71c2daf (diff)
Removing the FAQ, which has been moved to the GitHub wiki for this
repository. Also removing FAQ-related build rules.
Diffstat (limited to 'doc')
-rw-r--r--doc/faq/FAQ.tex2713
-rw-r--r--doc/faq/axioms.fig131
-rw-r--r--doc/faq/fk.bib2221
-rw-r--r--doc/faq/hevea.sty78
-rw-r--r--doc/faq/interval_discr.v419
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-\RequirePackage{ifpdf}
-\ifpdf % si on est en pdflatex
-\documentclass[a4paper,pdftex]{article}
-\else
-\documentclass[a4paper]{article}
-\fi
-\pagestyle{plain}
-
-% yay les symboles
-\usepackage{textcomp}
-\usepackage{stmaryrd}
-\usepackage{amssymb}
-\usepackage{url}
-%\usepackage{multicol}
-\usepackage{hevea}
-\usepackage{fullpage}
-\usepackage[utf8]{inputenc}
-\usepackage[english]{babel}
-
-\ifpdf % si on est en pdflatex
- \usepackage[pdftex]{graphicx}
-\else
- \usepackage[dvips]{graphicx}
-\fi
-
-%\input{../macros.tex}
-
-% Making hevea happy
-%HEVEA \renewcommand{\textbar}{|}
-%HEVEA \renewcommand{\textunderscore}{\_}
-
-\def\Question#1{\stepcounter{question}\subsubsection{#1}}
-
-% version et date
-\def\faqversion{0.1}
-
-% les macros d'amour
-\def\Coq{\textsc{Coq}}
-\def\Why{\textsc{Why}}
-\def\Framac{\textsc{Frama-c}}
-\def\Krakatoa{\textsc{Krakatoa}}
-\def\Ltac{\textsc{Ltac}}
-\def\CoqIde{\textsc{CoqIde}}
-
-\newcommand{\coqtt}[1]{{\tt #1}}
-\newcommand{\coqimp}{{\mbox{\tt ->}}}
-\newcommand{\coqequiv}{{\mbox{\tt <->}}}
-
-
-% macro pour les tactics
-\def\split{{\tt split}}
-\def\assumption{{\tt assumption}}
-\def\auto{{\tt auto}}
-\def\trivial{{\tt trivial}}
-\def\tauto{{\tt tauto}}
-\def\left{{\tt left}}
-\def\right{{\tt right}}
-\def\decompose{{\tt decompose}}
-\def\intro{{\tt intro}}
-\def\intros{{\tt intros}}
-\def\field{{\tt field}}
-\def\ring{{\tt ring}}
-\def\apply{{\tt apply}}
-\def\exact{{\tt exact}}
-\def\cut{{\tt cut}}
-\def\assert{{\tt assert}}
-\def\solve{{\tt solve}}
-\def\idtac{{\tt idtac}}
-\def\fail{{\tt fail}}
-\def\existstac{{\tt exists}}
-\def\firstorder{{\tt firstorder}}
-\def\congruence{{\tt congruence}}
-\def\gb{{\tt gb}}
-\def\generalize{{\tt generalize}}
-\def\abstracttac{{\tt abstract}}
-\def\eapply{{\tt eapply}}
-\def\unfold{{\tt unfold}}
-\def\rewrite{{\tt rewrite}}
-\def\replace{{\tt replace}}
-\def\simpl{{\tt simpl}}
-\def\elim{{\tt elim}}
-\def\set{{\tt set}}
-\def\pose{{\tt pose}}
-\def\case{{\tt case}}
-\def\destruct{{\tt destruct}}
-\def\reflexivity{{\tt reflexivity}}
-\def\transitivity{{\tt transitivity}}
-\def\symmetry{{\tt symmetry}}
-\def\Focus{{\tt Focus}}
-\def\discriminate{{\tt discriminate}}
-\def\contradiction{{\tt contradiction}}
-\def\intuition{{\tt intuition}}
-\def\try{{\tt try}}
-\def\repeat{{\tt repeat}}
-\def\eauto{{\tt eauto}}
-\def\subst{{\tt subst}}
-\def\symmetryin{{\tt symmetryin}}
-\def\instantiate{{\tt instantiate}}
-\def\inversion{{\tt inversion}}
-\def\specialize{{\tt specialize}}
-\def\Defined{{\tt Defined}}
-\def\Qed{{\tt Qed}}
-\def\pattern{{\tt pattern}}
-\def\Type{{\tt Type}}
-\def\Prop{{\tt Prop}}
-\def\Set{{\tt Set}}
-
-
-\newcommand\vfile[2]{\ahref{#1}{\tt {#2}.v}}
-\urldef{\InitWf}\url
- {http://coq.inria.fr/library/Coq.Init.Wf.html}
-\urldef{\LogicBerardi}\url
- {http://coq.inria.fr/library/Coq.Logic.Berardi.html}
-\urldef{\LogicClassical}\url
- {http://coq.inria.fr/library/Coq.Logic.Classical.html}
-\urldef{\LogicClassicalFacts}\url
- {http://coq.inria.fr/library/Coq.Logic.ClassicalFacts.html}
-\urldef{\LogicClassicalDescription}\url
- {http://coq.inria.fr/library/Coq.Logic.ClassicalDescription.html}
-\urldef{\LogicProofIrrelevance}\url
- {http://coq.inria.fr/library/Coq.Logic.ProofIrrelevance.html}
-\urldef{\LogicEqdep}\url
- {http://coq.inria.fr/library/Coq.Logic.Eqdep.html}
-\urldef{\LogicEqdepDec}\url
- {http://coq.inria.fr/library/Coq.Logic.Eqdep_dec.html}
-
-
-
-
-\begin{document}
-\bibliographystyle{plain}
-\newcounter{question}
-\renewcommand{\thesubsubsection}{\arabic{question}}
-
-%%%%%%% Coq pour les nuls %%%%%%%
-
-\title{Coq Version 8.4 for the Clueless\\
- \large(\protect\ref{lastquestion}
- \ Hints)
-}
-\author{Pierre Castéran \and Hugo Herbelin \and Florent Kirchner \and Benjamin Monate \and Julien Narboux}
-\maketitle
-
-%%%%%%%
-
-\begin{abstract}
-This note intends to provide an easy way to get acquainted with the
-{\Coq} theorem prover. It tries to formulate appropriate answers
-to some of the questions any newcomers will face, and to give
-pointers to other references when possible.
-\end{abstract}
-
-%%%%%%%
-
-%\begin{multicols}{2}
-\tableofcontents
-%\end{multicols}
-
-%%%%%%%
-
-\newpage
-
-\section{Introduction}
-This FAQ is the sum of the questions that came to mind as we developed
-proofs in \Coq. Since we are singularly short-minded, we wrote the
-answers we found on bits of papers to have them at hand whenever the
-situation occurs again. This is pretty much the result of that: a
-collection of tips one can refer to when proofs become intricate. Yes,
-it means we won't take the blame for the shortcomings of this
-FAQ. But if you want to contribute and send in your own question and
-answers, feel free to write to us\ldots
-
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-
-\section{Presentation}
-
-\Question{What is {\Coq}?}\label{whatiscoq}
-The {\Coq} tool is a formal proof management system: a proof done with {\Coq} is mechanically checked by the machine.
-In particular, {\Coq} allows:
-\begin{itemize}
- \item the definition of mathematical objects and programming objects,
- \item to state mathematical theorems and software specifications,
- \item to interactively develop formal proofs of these theorems,
- \item to check these proofs by a small certification ``kernel''.
-\end{itemize}
-{\Coq} is based on a logical framework called ``Calculus of Inductive
-Constructions'' extended by a modular development system for theories.
-
-\Question{Did you really need to name it like that?}
-Some French computer scientists have a tradition of naming their
-software as animal species: Caml, Elan, Foc or Phox are examples
-of this tacit convention. In French, ``coq'' means rooster, and it
-sounds like the initials of the Calculus of Constructions CoC on which
-it is based.
-
-\Question{Is {\Coq} a theorem prover?}
-
-{\Coq} comes with decision and semi-decision procedures (
-propositional calculus, Presburger's arithmetic, ring and field
-simplification, resolution, ...) but the main style for proving
-theorems is interactively by using LCF-style tactics.
-
-
-\Question{What are the other theorem provers?}
-Many other theorem provers are available for use nowadays.
-Isabelle, HOL, HOL Light, Lego, Nuprl, PVS are examples of provers that are fairly similar
-to {\Coq} by the way they interact with the user. Other relatives of
-{\Coq} are ACL2, Agda/Alfa, Twelf, Kiv, Mizar, NqThm,
-\begin{htmlonly}%
-Omega\ldots
-\end{htmlonly}
-\begin{latexonly}%
-{$\Omega$}mega\ldots
-\end{latexonly}
-
-\Question{What do I have to trust when I see a proof checked by Coq?}
-
-You have to trust:
-
-\begin{description}
-\item[The theory behind Coq] The theory of {\Coq} version 8.0 is
-generally admitted to be consistent wrt Zermelo-Fraenkel set theory +
-inaccessible cardinals. Proofs of consistency of subsystems of the
-theory of Coq can be found in the literature.
-\item[The Coq kernel implementation] You have to trust that the
-implementation of the {\Coq} kernel mirrors the theory behind {\Coq}. The
-kernel is intentionally small to limit the risk of conceptual or
-accidental implementation bugs.
-\item[The Objective Caml compiler] The {\Coq} kernel is written using the
-Objective Caml language but it uses only the most standard features
-(no object, no label ...), so that it is highly improbable that an
-Objective Caml bug breaks the consistency of {\Coq} without breaking all
-other kinds of features of {\Coq} or of other software compiled with
-Objective Caml.
-\item[Your hardware] In theory, if your hardware does not work
-properly, it can accidentally be the case that False becomes
-provable. But it is more likely the case that the whole {\Coq} system
-will be unusable. You can check your proof using different computers
-if you feel the need to.
-\item[Your axioms] Your axioms must be consistent with the theory
-behind {\Coq}.
-\end{description}
-
-
-\Question{Where can I find information about the theory behind {\Coq}?}
-\begin{description}
-\item[The Calculus of Inductive Constructions] The
-\ahref{http://coq.inria.fr/doc/Reference-Manual006.html}{corresponding}
-chapter and the chapter on
-\ahref{http://coq.inria.fr/doc/Reference-Manual007.html}{modules} in
-the {\Coq} Reference Manual.
-\item[Type theory] A book~\cite{ProofsTypes} or some lecture
-notes~\cite{Types:Dowek}.
-\item[Inductive types]
-Christine Paulin-Mohring's habilitation thesis~\cite{Pau96b}.
-\item[Co-Inductive types]
-Eduardo Giménez' thesis~\cite{EGThese}.
-\item[Miscellaneous] A
-\ahref{http://coq.inria.fr/doc/biblio.html}{bibliography} about Coq
-\end{description}
-
-
-\Question{How can I use {\Coq} to prove programs?}
-
-You can either extract a program from a proof by using the extraction
-mechanism or use dedicated tools, such as
-\ahref{http://why3.lri.fr}{\Why},
-\ahref{http://krakatoa.lri.fr}{\Krakatoa},
-\ahref{http://frama-c.com}{\Framac}, to prove
-annotated programs written in other languages.
-
-%\Question{How many {\Coq} users are there?}
-%
-%An estimation is about 100 regular users.
-
-\Question{How old is {\Coq}?}
-
-The first implementation is from 1985 (it was named {\sf CoC} which is
-the acronym of the name of the logic it implemented: the Calculus of
-Constructions). The first official release of {\Coq} (version 4.10)
-was distributed in 1989.
-
-\Question{What are the \Coq-related tools?}
-
-There are graphical user interfaces:
-\begin{description}
-\item[Coqide] A GTK based GUI for \Coq.
-\item[Pcoq] A GUI for {\Coq} with proof by pointing and pretty printing.
-\item[coqwc] A tool similar to {\tt wc} to count lines in {\Coq} files.
-\item[Proof General] A emacs mode for {\Coq} and many other proof assistants.
-\item[ProofWeb] The ProofWeb online web interface for {\Coq} (and other proof assistants), with a focus on teaching.
-\item[ProverEditor] is an experimental Eclipse plugin with support for {\Coq}.
-\end{description}
-
-There are documentation and browsing tools:
-
-\begin{description}
-\item[coq-tex] A tool to insert {\Coq} examples within .tex files.
-\item[coqdoc] A documentation tool for \Coq.
-\item[coqgraph] A tool to generate a dependency graph from {\Coq} sources.
-\end{description}
-
-There are front-ends for specific languages:
-
-\begin{description}
-\item[Why] A back-end generator of verification conditions.
-\item[Krakatoa] A Java code certification tool that uses both {\Coq} and {\Why} to verify the soundness of implementations with regards to the specifications.
-\item[Caduceus] A C code certification tool that uses both {\Coq} and \Why.
-\item[Zenon] A first-order theorem prover.
-\item[Focal] The \ahref{http://focal.inria.fr}{Focal} project aims at building an environment to develop certified computer algebra libraries.
-\item[Concoqtion] is a dependently-typed extension of Objective Caml (and of MetaOCaml) with specifications expressed and proved in Coq.
-\item[Ynot] is an extension of Coq providing a "Hoare Type Theory" for specifying higher-order, imperative and concurrent programs.
-\item[Ott]is a tool to translate the descriptions of the syntax and semantics of programming languages to the syntax of Coq, or of other provers.
-\end{description}
-
-\Question{What are the high-level tactics of \Coq}
-
-\begin{itemize}
-\item Decision of quantifier-free Presburger's Arithmetic
-\item Simplification of expressions on rings and fields
-\item Decision of closed systems of equations
-\item Semi-decision of first-order logic
-\item Prolog-style proof search, possibly involving equalities
-\end{itemize}
-
-\Question{What are the main libraries available for \Coq}
-
-\begin{itemize}
-\item Basic Peano's arithmetic, binary integer numbers, rational numbers,
-\item Real analysis,
-\item Libraries for lists, boolean, maps, floating-point numbers,
-\item Libraries for relations, sets and constructive algebra,
-\item Geometry
-\end{itemize}
-
-
-\Question{What are the mathematical applications for {\Coq}?}
-
-{\Coq} is used for formalizing mathematical theories, for teaching,
-and for proving properties of algorithms or programs libraries.
-
-The largest mathematical formalization has been done at the University
-of Nijmegen (see the
-\ahref{http://c-corn.cs.ru.nl}{Constructive Coq
-Repository at Nijmegen}).
-
-A symbolic step has also been obtained by formalizing in full a proof
-of the Four Color Theorem.
-
-\Question{What are the industrial applications for {\Coq}?}
-
-{\Coq} is used e.g. to prove properties of the JavaCard system
-(especially by Schlumberger and Trusted Logic). It has
-also been used to formalize the semantics of the Lucid-Synchrone
-data-flow synchronous calculus used by Esterel-Technologies.
-
-\iffalse
-todo christine compilo lustre?
-\fi
-
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-
-\section{Documentation}
-
-\Question{Where can I find documentation about {\Coq}?}
-All the documentation about \Coq, from the reference manual~\cite{Coq:manual} to
-friendly tutorials~\cite{Coq:Tutorial} and documentation of the standard library, is available
-\ahref{http://coq.inria.fr/doc-eng.html}{online}.
-All these documents are viewable either in browsable HTML, or as
-downloadable postscripts.
-
-\Question{Where can I find this FAQ on the web?}
-
-This FAQ is available online at \ahref{http://coq.inria.fr/faq}{\url{http://coq.inria.fr/faq}}.
-
-\Question{How can I submit suggestions / improvements / additions for this FAQ?}
-
-This FAQ is unfinished (in the sense that there are some obvious
-sections that are missing). Please send contributions to Coq-Club.
-
-\Question{Is there any mailing list about {\Coq}?}
-The main {\Coq} mailing list is \url{coq-club@inria.fr}, which
-broadcasts questions and suggestions about the implementation, the
-logical formalism or proof developments. See
-\ahref{http://sympa.inria.fr/sympa/info/coq-club}{\url{http://sympa.inria.fr/sympa/info/coq-club}} for
-subscription. For bugs reports see question \ref{coqbug}.
-
-\Question{Where can I find an archive of the list?}
-The archives of the {\Coq} mailing list are available at
-\ahref{http://sympa.inria.fr/sympa/arc/coq-club}{\url{http://sympa.inria.fr/sympa/arc/coq-club}}.
-
-
-\Question{How can I be kept informed of new releases of {\Coq}?}
-
-New versions of {\Coq} are announced on the coq-club mailing list. If you only want to receive information about new releases, you can subscribe to {\Coq} on \ahref{http://freshmeat.net/projects/coq/}{\url{http://freshmeat.net/projects/coq/}}.
-
-
-\Question{Is there any book about {\Coq}?}
-
-The first book on \Coq, Yves Bertot and Pierre Castéran's Coq'Art has been published by Springer-Verlag in 2004:
-\begin{quote}
-``This book provides a pragmatic introduction to the development of
-proofs and certified programs using \Coq. With its large collection of
-examples and exercises it is an invaluable tool for researchers,
-students, and engineers interested in formal methods and the
-development of zero-default software.''
-\end{quote}
-
-\Question{Where can I find some {\Coq} examples?}
-
-There are examples in the manual~\cite{Coq:manual} and in the
-Coq'Art~\cite{Coq:coqart} exercises \ahref{\url{http://www.labri.fr/Perso/~casteran/CoqArt/index.html}}{\url{http://www.labri.fr/Perso/~casteran/CoqArt/index.html}}.
-You can also find large developments using
-{\Coq} in the {\Coq} user contributions:
-\ahref{http://coq.inria.fr/contribs}{\url{http://coq.inria.fr/contribs}}.
-
-\Question{How can I report a bug?}\label{coqbug}
-
-You can use the web interface accessible at \ahref{http://coq.inria.fr}{\url{http://coq.inria.fr}}, link ``contacts''.
-
-
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-
-\section{Installation}
-
-\Question{What is the license of {\Coq}?}
-{\Coq} is distributed under the GNU Lesser General License
-(LGPL).
-
-\Question{Where can I find the sources of {\Coq}?}
-The sources of {\Coq} can be found online in the tar.gz'ed packages
-(\ahref{http://coq.inria.fr}{\url{http://coq.inria.fr}}, link
-``download''). Development sources can be accessed at
-\ahref{http://coq.gforge.inria.fr/}{\url{http://coq.gforge.inria.fr/}}
-
-\Question{On which platform is {\Coq} available?}
-Compiled binaries are available for Linux, MacOS X, and Windows. The
-sources can be easily compiled on all platforms supporting Objective
-Caml.
-
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-
-\section{The logic of {\Coq}}
-
-\subsection{General}
-
-\Question{What is the logic of \Coq?}
-
-{\Coq} is based on an axiom-free type theory called
-the Calculus of Inductive Constructions (see Coquand \cite{CoHu86},
-Luo~\cite{Luo90}
-and Coquand--Paulin-Mohring \cite{CoPa89}). It includes higher-order
-functions and predicates, inductive and co-inductive datatypes and
-predicates, and a stratified hierarchy of sets.
-
-\Question{Is \Coq's logic intuitionistic or classical?}
-
-{\Coq}'s logic is modular. The core logic is intuitionistic
-(i.e. excluded-middle $A\vee\neg A$ is not granted by default). It can
-be extended to classical logic on demand by requiring an
-optional module stating $A\vee\neg A$.
-
-\Question{Can I define non-terminating programs in \Coq?}
-
-All programs in {\Coq} are terminating. Especially, loops
-must come with an evidence of their termination.
-
-Non-terminating programs can be simulated by passing around a
-bound on how long the program is allowed to run before dying.
-
-\Question{How is equational reasoning working in {\Coq}?}
-
- {\Coq} comes with an internal notion of computation called
-{\em conversion} (e.g. $(x+1)+y$ is internally equivalent to
-$(x+y)+1$; similarly applying argument $a$ to a function mapping $x$
-to some expression $t$ converts to the expression $t$ where $x$ is
-replaced by $a$). This notion of conversion (which is decidable
-because {\Coq} programs are terminating) covers a certain part of
-equational reasoning but is limited to sequential evaluation of
-expressions of (not necessarily closed) programs. Besides conversion,
-equations have to be treated by hand or using specialised tactics.
-
-\subsection{Axioms}
-
-\Question{What axioms can be safely added to {\Coq}?}
-
-There are a few typical useful axioms that are independent from the
-Calculus of Inductive Constructions and that are considered consistent with
-the theory of {\Coq}.
-Most of these axioms are stated in the directory {\tt Logic} of the
-standard library of {\Coq}. The most interesting ones are
-
-\begin{itemize}
-\item Excluded-middle: $\forall A:Prop, A \vee \neg A$
-\item Proof-irrelevance: $\forall A:Prop \forall p_1 p_2:A, p_1=p_2$
-\item Unicity of equality proofs (or equivalently Streicher's axiom $K$):
-$\forall A \forall x y:A \forall p_1 p_2:x=y, p_1=p_2$
-\item Hilbert's $\epsilon$ operator: if $A \neq \emptyset$, then there is $\epsilon_P$ such that $\exists x P(x) \rightarrow P(\epsilon_P)$
-\item Church's $\iota$ operator: if $A \neq \emptyset$, then there is $\iota_P$ such that $\exists! x P(x) \rightarrow P(\iota_P)$
-\item The axiom of unique choice: $\forall x \exists! y R(x,y) \rightarrow \exists f \forall x R(x,f(x))$
-\item The functional axiom of choice: $\forall x \exists y R(x,y) \rightarrow \exists f \forall x R(x,f(x))$
-\item Extensionality of predicates: $\forall P Q:A\rightarrow Prop, (\forall x, P(x) \leftrightarrow Q(x)) \rightarrow P=Q$
-\item Extensionality of functions: $\forall f g:A\rightarrow B, (\forall x, f(x)=g(x)) \rightarrow f=g$
-\end{itemize}
-
-Figure~\ref{fig:axioms} is a summary of the relative strength of these
-axioms, most proofs can be found in directory {\tt Logic} of the standard
-library. (Statements in boldface are the most ``interesting'' ones for
-Coq.) The justification of their validity relies on the interpretability
-in set theory.
-
-\begin{figure}[htbp]
-%HEVEA\imgsrc{axioms.png}
-%BEGIN LATEX
-\begin{center}
-\ifpdf % si on est en pdflatex
-\scalebox{0.65}{\input{axioms.pdf_t}}
-\else
-\scalebox{0.65}{\input{axioms.eps_t}}
-\fi
-\end{center}
-%END LATEX
-\caption{The dependency graph of axioms in the Calculus of Inductive Constructions}
-\label{fig:axioms}
-\end{figure}
-
-\Question{What standard axioms are inconsistent with {\Coq}?}
-
-The axiom of unique choice together with classical logic
-(e.g. excluded-middle) are inconsistent in the variant of the Calculus
-of Inductive Constructions where {\Set} is impredicative.
-
-As a consequence, the functional form of the axiom of choice and
-excluded-middle, or any form of the axiom of choice together with
-predicate extensionality are inconsistent in the {\Set}-impredicative
-version of the Calculus of Inductive Constructions.
-
-The main purpose of the \Set-predicative restriction of the Calculus
-of Inductive Constructions is precisely to accommodate these axioms
-which are quite standard in mathematical usage.
-
-The $\Set$-predicative system is commonly considered consistent by
-interpreting it in a standard set-theoretic boolean model, even with
-classical logic, axiom of choice and predicate extensionality added.
-
-\Question{What is Streicher's axiom $K$}
-\label{Streicher}
-
-Streicher's axiom $K$~\cite{HofStr98} is an axiom that asserts
-dependent elimination of reflexive equality proofs.
-
-\begin{coq_example*}
-Axiom Streicher_K :
- forall (A:Type) (x:A) (P: x=x -> Prop),
- P (eq_refl x) -> forall p: x=x, P p.
-\end{coq_example*}
-
-In the general case, axiom $K$ is an independent statement of the
-Calculus of Inductive Constructions. However, it is true on decidable
-domains (see file \vfile{\LogicEqdepDec}{Eqdep\_dec}). It is also
-trivially a consequence of proof-irrelevance (see
-\ref{proof-irrelevance}) hence of classical logic.
-
-Axiom $K$ is equivalent to {\em Uniqueness of Identity Proofs} \cite{HofStr98}
-
-\begin{coq_example*}
-Axiom UIP : forall (A:Set) (x y:A) (p1 p2: x=y), p1 = p2.
-\end{coq_example*}
-
-Axiom $K$ is also equivalent to {\em Uniqueness of Reflexive Identity Proofs} \cite{HofStr98}
-
-\begin{coq_example*}
-Axiom UIP_refl : forall (A:Set) (x:A) (p: x=x), p = eq_refl x.
-\end{coq_example*}
-
-Axiom $K$ is also equivalent to
-
-\begin{coq_example*}
-Axiom
- eq_rec_eq :
- forall (A:Set) (x:A) (P: A->Set) (p:P x) (h: x=x),
- p = eq_rect x P p x h.
-\end{coq_example*}
-
-It is also equivalent to the injectivity of dependent equality (dependent equality is itself equivalent to equality of dependent pairs).
-
-\begin{coq_example*}
-Inductive eq_dep (U:Set) (P:U -> Set) (p:U) (x:P p) :
-forall q:U, P q -> Prop :=
- eq_dep_intro : eq_dep U P p x p x.
-Axiom
- eq_dep_eq :
- forall (U:Set) (u:U) (P:U -> Set) (p1 p2:P u),
- eq_dep U P u p1 u p2 -> p1 = p2.
-\end{coq_example*}
-
-\Question{What is proof-irrelevance}
-\label{proof-irrelevance}
-
-A specificity of the Calculus of Inductive Constructions is to permit
-statements about proofs. This leads to the question of comparing two
-proofs of the same proposition. Identifying all proofs of the same
-proposition is called {\em proof-irrelevance}:
-$$
-\forall A:\Prop, \forall p q:A, p=q
-$$
-
-Proof-irrelevance (in {\Prop}) can be assumed without contradiction in
-{\Coq}. It expresses that only provability matters, whatever the exact
-form of the proof is. This is in harmony with the common purely
-logical interpretation of {\Prop}. Contrastingly, proof-irrelevance is
-inconsistent in {\Set} since there are types in {\Set}, such as the
-type of booleans, that provably have at least two distinct elements.
-
-Proof-irrelevance (in {\Prop}) is a consequence of classical logic
-(see proofs in file \vfile{\LogicClassical}{Classical} and
-\vfile{\LogicBerardi}{Berardi}). Proof-irrelevance is also a
-consequence of propositional extensionality (i.e. \coqtt{(A {\coqequiv} B)
-{\coqimp} A=B}, see the proof in file
-\vfile{\LogicClassicalFacts}{ClassicalFacts}).
-
-Proof-irrelevance directly implies Streicher's axiom $K$.
-
-\Question{What about functional extensionality?}
-
-Extensionality of functions is admittedly consistent with the
-Set-predicative Calculus of Inductive Constructions.
-
-%\begin{coq_example*}
-% Axiom extensionality : (A,B:Set)(f,g:(A->B))(x:A)(f x)=(g x)->f=g.
-%\end{coq_example*}
-
-Let {\tt A}, {\tt B} be types. To deal with extensionality on
-\verb=A->B= without relying on a general extensionality axiom,
-a possible approach is to define one's own extensional equality on
-\verb=A->B=.
-
-\begin{coq_eval}
-Variables A B : Set.
-\end{coq_eval}
-
-\begin{coq_example*}
-Definition ext_eq (f g: A->B) := forall x:A, f x = g x.
-\end{coq_example*}
-
-and to reason on \verb=A->B= as a setoid (see the Chapter on
-Setoids in the Reference Manual).
-
-\Question{Is {\Prop} impredicative?}
-
-Yes, the sort {\Prop} of propositions is {\em
-impredicative}. Otherwise said, a statement of the form $\forall
-A:Prop, P(A)$ can be instantiated by itself: if $\forall A:\Prop, P(A)$
-is provable, then $P(\forall A:\Prop, P(A))$ is.
-
-\Question{Is {\Set} impredicative?}
-
-No, the sort {\Set} lying at the bottom of the hierarchy of
-computational types is {\em predicative} in the basic {\Coq} system.
-This means that a family of types in {\Set}, e.g. $\forall A:\Set, A
-\rightarrow A$, is not a type in {\Set} and it cannot be applied on
-itself.
-
-However, the sort {\Set} was impredicative in the original versions of
-{\Coq}. For backward compatibility, or for experiments by
-knowledgeable users, the logic of {\Coq} can be set impredicative for
-{\Set} by calling {\Coq} with the option {\tt -impredicative-set}.
-
-{\Set} has been made predicative from version 8.0 of {\Coq}. The main
-reason is to interact smoothly with a classical mathematical world
-where both excluded-middle and the axiom of description are valid (see
-file \vfile{\LogicClassicalDescription}{ClassicalDescription} for a
-proof that excluded-middle and description implies the double negation
-of excluded-middle in {\Set} and file {\tt Hurkens\_Set.v} from the
-user contribution {\tt Paradoxes} at
-\ahref{http://coq.inria.fr/contribs}{\url{http://coq.inria.fr/contribs}}
-for a proof that impredicativity of {\Set} implies the simple negation
-of excluded-middle in {\Set}).
-
-\Question{Is {\Type} impredicative?}
-
-No, {\Type} is stratified. This is hidden for the
-user, but {\Coq} internally maintains a set of constraints ensuring
-stratification.
-
-If {\Type} were impredicative then it would be possible to encode
-Girard's systems $U-$ and $U$ in {\Coq} and it is known from Girard,
-Coquand, Hurkens and Miquel that systems $U-$ and $U$ are inconsistent
-[Girard 1972, Coquand 1991, Hurkens 1993, Miquel 2001]. This encoding
-can be found in file {\tt Logic/Hurkens.v} of {\Coq} standard library.
-
-For instance, when the user see {\tt $\forall$ X:Type, X->X : Type}, each
-occurrence of {\Type} is implicitly bound to a different level, say
-$\alpha$ and $\beta$ and the actual statement is {\tt
-forall X:Type($\alpha$), X->X : Type($\beta$)} with the constraint
-$\alpha<\beta$.
-
-When a statement violates a constraint, the message {\tt Universe
-inconsistency} appears. Example: {\tt fun (x:Type) (y:$\forall$ X:Type, X
-{\coqimp} X) => y x x}.
-
-\Question{I have two proofs of the same proposition. Can I prove they are equal?}
-
-In the base {\Coq} system, the answer is generally no. However, if
-classical logic is set, the answer is yes for propositions in {\Prop}.
-The answer is also yes if proof irrelevance holds (see question
-\ref{proof-irrelevance}).
-
-There are also ``simple enough'' propositions for which you can prove
-the equality without requiring any extra axioms. This is typically
-the case for propositions defined deterministically as a first-order
-inductive predicate on decidable sets. See for instance in question
-\ref{le-uniqueness} an axiom-free proof of the uniqueness of the proofs of
-the proposition {\tt le m n} (less or equal on {\tt nat}).
-
-% It is an ongoing work of research to natively include proof
-% irrelevance in {\Coq}.
-
-\Question{I have two proofs of an equality statement. Can I prove they are
-equal?}
-
- Yes, if equality is decidable on the domain considered (which
-is the case for {\tt nat}, {\tt bool}, etc): see {\Coq} file
-\verb=Eqdep_dec.v=). No otherwise, unless
-assuming Streicher's axiom $K$ (see \cite{HofStr98}) or a more general
-assumption such as proof-irrelevance (see \ref{proof-irrelevance}) or
-classical logic.
-
-All of these statements can be found in file \vfile{\LogicEqdep}{Eqdep}.
-
-\Question{Can I prove that the second components of equal dependent
-pairs are equal?}
-
- The answer is the same as for proofs of equality
-statements. It is provable if equality on the domain of the first
-component is decidable (look at \verb=inj_right_pair= from file
-\vfile{\LogicEqdepDec}{Eqdep\_dec}), but not provable in the general
-case. However, it is consistent (with the Calculus of Constructions)
-to assume it is true. The file \vfile{\LogicEqdep}{Eqdep} actually
-provides an axiom (equivalent to Streicher's axiom $K$) which entails
-the result (look at \verb=inj_pair2= in \vfile{\LogicEqdep}{Eqdep}).
-
-\subsection{Impredicativity}
-
-\Question{Why {\tt injection} does not work on impredicative {\tt Set}?}
-
- E.g. in this case (this occurs only in the {\tt Set}-impredicative
- variant of \Coq):
-
-\begin{coq_example*}
-Inductive I : Type :=
- intro : forall k:Set, k -> I.
-Lemma eq_jdef :
- forall x y:nat, intro _ x = intro _ y -> x = y.
-Proof.
- intros x y H; injection H.
-\end{coq_example*}
-
-\begin{coq_eval}
-Reset Initial.
-\end{coq_eval}
-
- Injectivity of constructors is restricted to predicative types. If
-injectivity on large inductive types were not restricted, we would be
-allowed to derive an inconsistency (e.g. following the lines of
-Burali-Forti paradox). The question remains open whether injectivity
-is consistent on some large inductive types not expressive enough to
-encode known paradoxes (such as type I above).
-
-
-\Question{What is a ``large inductive definition''?}
-
-An inductive definition in {\Prop} or {\Set} is called large
-if its constructors embed sets or propositions. As an example, here is
-a large inductive type:
-
-\begin{coq_example*}
-Inductive sigST (P:Set -> Set) : Type :=
- existST : forall X:Set, P X -> sigST P.
-\end{coq_example*}
-
-In the {\tt Set} impredicative variant of {\Coq}, large inductive
-definitions in {\tt Set} have restricted elimination schemes to
-prevent inconsistencies. Especially, projecting the set or the
-proposition content of a large inductive definition is forbidden. If
-it were allowed, it would be possible to encode e.g. Burali-Forti
-paradox \cite{Gir70,Coq85}.
-
-
-\Question{Is Coq's logic conservative over Coquand's Calculus of
-Constructions?}
-
-In the {\Set}-impredicative version of the Calculus of Inductive
-Constructions (CIC), there are two ways to interpret the Calculus of
-Constructions (CC) since the impredicative sort of CC can be
-interpreted either as {\Prop} or as {\Set}. In the {\Set}-predicative
-CIC, the impredicative sort of CC can only be interpreted as {\Prop}.
-
-If the impredicative sort of CC is interpreted as {\Set}, there is no
-conservativity of CIC over CC as the discrimination of
-constructors of inductive types in {\Set} transports to a
-discrimination of constructors of inductive types encoded
-impredicatively. Concretely, considering the impredicative encoding of
-Boolean, equality and falsity, we can prove the following CC statement
-DISCR in CIC which is not provable in CC, as CC has a
-``term-irrelevant'' model.
-
-\begin{coq_example*}
-Definition BOOL := forall X:Set, X -> X -> X.
-Definition TRUE : BOOL := fun X x1 x2 => x1.
-Definition FALSE : BOOL := fun X x1 x2 => x2.
-Definition EQBOOL (x1 x2:BOOL) := forall P:BOOL->Set, P x1 -> P x2.
-Definition BOT := forall X:Set, X.
-
-Definition BOOL2bool : BOOL -> bool := fun b => b bool true false.
-
-Theorem DISCR : EQBOOL TRUE FALSE -> BOT.
-intro X.
-assert (H : BOOL2bool TRUE = BOOL2bool FALSE).
-{ apply X. trivial. }
-discriminate H.
-Qed.
-\end{coq_example*}
-
-If the impredicative sort of CC is interpreted as {\Prop}, CIC is
-presumably conservative over CC. The general idea is that no
-proof-relevant information can flow from {\Prop} to {\Set}, even
-though singleton elimination can be used. Hence types in {\Set} should
-be smashable to the unit type and {\Set} and {\Type} themselves be
-mapped to {\Prop}.
-
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-\section{Talkin' with the Rooster}
-
-
-%%%%%%%
-\subsection{My goal is ..., how can I prove it?}
-
-
-\Question{My goal is a conjunction, how can I prove it?}
-
-Use some theorem or assumption or use the {\split} tactic.
-\begin{coq_example}
-Goal forall A B:Prop, A -> B -> A/\B.
-intros.
-split.
-assumption.
-assumption.
-\end{coq_example}
-\begin{coq_example*}
-Qed.
-\end{coq_example*}
-
-\Question{My goal contains a conjunction as an hypothesis, how can I use it?}
-
-If you want to decompose a hypothesis into several hypotheses, you can
-use the {\destruct} tactic:
-
-\begin{coq_example}
-Goal forall A B:Prop, A/\B -> B.
-intros.
-destruct H as [H1 H2].
-assumption.
-\end{coq_example}
-\begin{coq_example*}
-Qed.
-\end{coq_example*}
-
-You can also perform the destruction at the time of introduction:
-
-\begin{coq_example}
-Goal forall A B:Prop, A/\B -> B.
-intros A B [H1 H2].
-assumption.
-\end{coq_example}
-\begin{coq_example*}
-Qed.
-\end{coq_example*}
-
-\Question{My goal is a disjunction, how can I prove it?}
-
-You can prove the left part or the right part of the disjunction using
-{\left} or {\right} tactics. If you want to do a classical
-reasoning step, use the {\tt classic} axiom to prove the right part with the assumption
-that the left part of the disjunction is false.
-
-\begin{coq_example}
-Goal forall A B:Prop, A -> A\/B.
-intros.
-left.
-assumption.
-\end{coq_example}
-\begin{coq_example*}
-Qed.
-\end{coq_example*}
-
-An example using classical reasoning:
-
-\begin{coq_example}
-Require Import Classical.
-
-Ltac classical_right :=
-match goal with
-| _:_ |- ?X1 \/ _ => (elim (classic X1);intro;[left;trivial|right])
-end.
-
-Ltac classical_left :=
-match goal with
-| _:_ |- _ \/ ?X1 => (elim (classic X1);intro;[right;trivial|left])
-end.
-
-
-Goal forall A B:Prop, (~A -> B) -> A\/B.
-intros.
-classical_right.
-auto.
-\end{coq_example}
-\begin{coq_example*}
-Qed.
-\end{coq_example*}
-
-\Question{My goal is an universally quantified statement, how can I prove it?}
-
-Use some theorem or assumption or introduce the quantified variable in
-the context using the {\intro} tactic. If there are several
-variables you can use the {\intros} tactic. A good habit is to
-provide names for these variables: {\Coq} will do it anyway, but such
-automatic naming decreases legibility and robustness.
-
-
-\Question{My goal contains an universally quantified statement, how can I use it?}
-
-If the universally quantified assumption matches the goal you can
-use the {\apply} tactic. If it is an equation you can use the
-{\rewrite} tactic. Otherwise you can use the {\specialize} tactic
-to instantiate the quantified variables with terms. The variant
-{\tt assert(Ht := H t)} makes a copy of assumption {\tt H} before
-instantiating it.
-
-
-\Question{My goal is an existential, how can I prove it?}
-
-Use some theorem or assumption or exhibit the witness using the {\existstac} tactic.
-\begin{coq_example}
-Goal exists x:nat, forall y, x+y=y.
-exists 0.
-intros.
-auto.
-\end{coq_example}
-\begin{coq_example*}
-Qed.
-\end{coq_example*}
-
-
-\Question{My goal is solvable by some lemma, how can I prove it?}
-
-Just use the {\apply} tactic.
-
-\begin{coq_eval}
-Reset Initial.
-\end{coq_eval}
-
-\begin{coq_example}
-Lemma mylemma : forall x, x+0 = x.
-auto.
-Qed.
-
-Goal 3+0 = 3.
-apply mylemma.
-\end{coq_example}
-\begin{coq_example*}
-Qed.
-\end{coq_example*}
-
-
-
-\Question{My goal contains False as an hypothesis, how can I prove it?}
-
-You can use the {\contradiction} or {\intuition} tactics.
-
-
-\Question{My goal is an equality of two convertible terms, how can I prove it?}
-
-Just use the {\reflexivity} tactic.
-
-\begin{coq_example}
-Goal forall x, 0+x = x.
-intros.
-reflexivity.
-\end{coq_example}
-\begin{coq_example*}
-Qed.
-\end{coq_example*}
-
-\Question{My goal is a {\tt let x := a in ...}, how can I prove it?}
-
-Just use the {\intro} tactic.
-
-
-\Question{My goal is a {\tt let (a, ..., b) := c in}, how can I prove it?}
-
-Just use the {\destruct} c as (a,...,b) tactic.
-
-
-\Question{My goal contains some existential hypotheses, how can I use it?}
-
-As with conjunctive hypotheses, you can use the {\destruct} tactic or
-the {\intros} tactic to decompose them into several hypotheses.
-
-\begin{coq_example*}
-Require Import Arith.
-\end{coq_example*}
-\begin{coq_example}
-Goal forall x, (exists y, x * y = 1) -> x = 1.
-intros x [y H].
-apply mult_is_one in H.
-easy.
-\end{coq_example}
-\begin{coq_example*}
-Qed.
-\end{coq_example*}
-
-
-\Question{My goal is an equality, how can I swap the left and right hand terms?}
-
-Just use the {\symmetry} tactic.
-\begin{coq_example}
-Goal forall x y : nat, x=y -> y=x.
-intros.
-symmetry.
-assumption.
-\end{coq_example}
-\begin{coq_example*}
-Qed.
-\end{coq_example*}
-
-\Question{My hypothesis is an equality, how can I swap the left and right hand terms?}
-
-Just use the {\symmetryin} tactic.
-
-\begin{coq_example}
-Goal forall x y : nat, x=y -> y=x.
-intros.
-symmetry in H.
-assumption.
-\end{coq_example}
-\begin{coq_example*}
-Qed.
-\end{coq_example*}
-
-
-\Question{My goal is an equality, how can I prove it by transitivity?}
-
-Just use the {\transitivity} tactic.
-\begin{coq_example}
-Goal forall x y z : nat, x=y -> y=z -> x=z.
-intros.
-transitivity y.
-assumption.
-assumption.
-\end{coq_example}
-\begin{coq_example*}
-Qed.
-\end{coq_example*}
-
-
-\Question{My goal would be solvable using {\tt apply;assumption} if it would not create meta-variables, how can I prove it?}
-
-You can use {\tt eapply yourtheorem;eauto} but it won't work in all cases ! (for example if more than one hypothesis match one of the subgoals generated by \eapply) so you should rather use {\tt try solve [eapply yourtheorem;eauto]}, otherwise some metavariables may be incorrectly instantiated.
-
-\begin{coq_example}
-Lemma trans : forall x y z : nat, x=y -> y=z -> x=z.
-intros.
-transitivity y;assumption.
-Qed.
-
-Goal forall x y z : nat, x=y -> y=z -> x=z.
-intros.
-eapply trans;eauto.
-Qed.
-
-Goal forall x y z t : nat, x=y -> x=t -> y=z -> x=z.
-intros.
-eapply trans;eauto.
-Undo.
-eapply trans.
-apply H.
-auto.
-Qed.
-
-Goal forall x y z t : nat, x=y -> x=t -> y=z -> x=z.
-intros.
-eapply trans;eauto.
-Undo.
-try solve [eapply trans;eauto].
-eapply trans.
-apply H.
-auto.
-Qed.
-\end{coq_example}
-
-\Question{My goal is solvable by some lemma within a set of lemmas and I don't want to remember which one, how can I prove it?}
-
-You can use a what is called a hints' base.
-
-\begin{coq_example}
-Require Import ZArith.
-Require Ring.
-Local Open Scope Z_scope.
-Lemma toto1 : 1+1 = 2.
-ring.
-Qed.
-Lemma toto2 : 2+2 = 4.
-ring.
-Qed.
-Lemma toto3 : 2+1 = 3.
-ring.
-Qed.
-
-Hint Resolve toto1 toto2 toto3 : mybase.
-
-Goal 2+(1+1)=4.
-auto with mybase.
-Qed.
-\end{coq_example}
-
-
-\Question{My goal is one of the hypotheses, how can I prove it?}
-
-Use the {\assumption} tactic.
-
-\begin{coq_example}
-Goal 1=1 -> 1=1.
-intro.
-assumption.
-\end{coq_example}
-\begin{coq_example*}
-Qed.
-\end{coq_example*}
-
-
-\Question{My goal appears twice in the hypotheses and I want to choose which one is used, how can I do it?}
-
-Use the {\exact} tactic.
-\begin{coq_example}
-Goal 1=1 -> 1=1 -> 1=1.
-intros.
-exact H0.
-\end{coq_example}
-\begin{coq_example*}
-Qed.
-\end{coq_example*}
-
-\Question{What can be the difference between applying one hypothesis or another in the context of the last question?}
-
-From a proof point of view it is equivalent but if you want to extract
-a program from your proof, the two hypotheses can lead to different
-programs.
-
-
-\Question{My goal is a propositional tautology, how can I prove it?}
-
-Just use the {\tauto} tactic.
-\begin{coq_example}
-Goal forall A B:Prop, A-> (A\/B) /\ A.
-intros.
-tauto.
-\end{coq_example}
-\begin{coq_example*}
-Qed.
-\end{coq_example*}
-
-\Question{My goal is a first order formula, how can I prove it?}
-
-Just use the semi-decision tactic: \firstorder.
-
-\iffalse
-todo: demander un exemple à Pierre
-\fi
-
-\Question{My goal is solvable by a sequence of rewrites, how can I prove it?}
-
-Just use the {\congruence} tactic.
-\begin{coq_example}
-Goal forall a b c d e, a=d -> b=e -> c+b=d -> c+e=a.
-intros.
-congruence.
-\end{coq_example}
-\begin{coq_example*}
-Qed.
-\end{coq_example*}
-
-
-\Question{My goal is a disequality solvable by a sequence of rewrites, how can I prove it?}
-
-Just use the {\congruence} tactic.
-
-\begin{coq_example}
-Goal forall a b c d, a<>d -> b=a -> d=c+b -> b<>c+b.
-intros.
-congruence.
-\end{coq_example}
-\begin{coq_example*}
-Qed.
-\end{coq_example*}
-
-
-\Question{My goal is an equality on some ring (e.g. natural numbers), how can I prove it?}
-
-Just use the {\ring} tactic.
-
-\begin{coq_example}
-Require Import ZArith.
-Require Ring.
-Local Open Scope Z_scope.
-Goal forall a b : Z, (a+b)*(a+b) = a*a + 2*a*b + b*b.
-intros.
-ring.
-\end{coq_example}
-\begin{coq_example*}
-Qed.
-\end{coq_example*}
-
-\Question{My goal is an equality on some field (e.g. real numbers), how can I prove it?}
-
-Just use the {\field} tactic.
-
-\begin{coq_example}
-Require Import Reals.
-Require Ring.
-Local Open Scope R_scope.
-Goal forall a b : R, b*a<>0 -> (a/b) * (b/a) = 1.
-intros.
-field.
-split ; auto with real.
-\end{coq_example}
-\begin{coq_example*}
-Qed.
-\end{coq_example*}
-
-
-\Question{My goal is an inequality on integers in Presburger's arithmetic (an expression build from $+$, $-$, constants, and variables), how can I prove it?}
-
-
-\begin{coq_example}
-Require Import ZArith.
-Require Omega.
-Local Open Scope Z_scope.
-Goal forall a : Z, a>0 -> a+a > a.
-intros.
-omega.
-\end{coq_example}
-\begin{coq_example*}
-Qed.
-\end{coq_example*}
-
-
-\Question{My goal is an equation solvable using equational hypothesis on some ring (e.g. natural numbers), how can I prove it?}
-
-You need the {\gb} tactic (see Loïc Pottier's homepage).
-
-\subsection{Tactics usage}
-
-\Question{I want to state a fact that I will use later as an hypothesis, how can I do it?}
-
-If you want to use forward reasoning (first proving the fact and then
-using it) you just need to use the {\assert} tactic. If you want to use
-backward reasoning (proving your goal using an assumption and then
-proving the assumption) use the {\cut} tactic.
-
-\begin{coq_example}
-Goal forall A B C D : Prop, (A -> B) -> (B->C) -> A -> C.
-intros.
-assert (A->C).
-intro;apply H0;apply H;assumption.
-apply H2.
-assumption.
-\end{coq_example}
-\begin{coq_example*}
-Qed.
-\end{coq_example*}
-
-\begin{coq_example}
-Goal forall A B C D : Prop, (A -> B) -> (B->C) -> A -> C.
-intros.
-cut (A->C).
-intro.
-apply H2;assumption.
-intro;apply H0;apply H;assumption.
-\end{coq_example}
-\begin{coq_example*}
-Qed.
-\end{coq_example*}
-
-
-
-
-\Question{I want to state a fact that I will use later as an hypothesis and prove it later, how can I do it?}
-
-You can use {\cut} followed by {\intro} or you can use the following {\Ltac} command:
-\begin{verbatim}
-Ltac assert_later t := cut t;[intro|idtac].
-\end{verbatim}
-
-\Question{What is the difference between {\Qed} and {\Defined}?}
-
-These two commands perform type checking, but when {\Defined} is used the new definition is set as transparent, otherwise it is defined as opaque (see \ref{opaque}).
-
-
-\Question{How can I know what an automation tactic does in my example?}
-
-You can use its {\tt info} variant: info\_auto, info\_trivial, info\_eauto.
-
-\Question{Why {\auto} does not work? How can I fix it?}
-
-You can increase the depth of the proof search or add some lemmas in the base of hints.
-Perhaps you may need to use \eauto.
-
-\Question{What is {\eauto}?}
-
-This is the same tactic as \auto, but it relies on {\eapply} instead of \apply.
-
-\Question{How can I speed up {\auto}?}
-
-You can use \texttt{info\_}{\auto} to replace {\auto} by the tactics it generates.
-You can split your hint bases into smaller ones.
-
-
-\Question{What is the equivalent of {\tauto} for classical logic?}
-
-Currently there are no equivalent tactic for classical logic. You can use Gödel's ``not not'' translation.
-
-
-\Question{I want to replace some term with another in the goal, how can I do it?}
-
-If one of your hypothesis (say {\tt H}) states that the terms are equal you can use the {\rewrite} tactic. Otherwise you can use the {\replace} {\tt with} tactic.
-
-\Question{I want to replace some term with another in an hypothesis, how can I do it?}
-
-You can use the {\rewrite} {\tt in} tactic.
-
-\Question{I want to replace some symbol with its definition, how can I do it?}
-
-You can use the {\unfold} tactic.
-
-\Question{How can I reduce some term?}
-
-You can use the {\simpl} tactic.
-
-\Question{How can I declare a shortcut for some term?}
-
-You can use the {\set} or {\pose} tactics.
-
-\Question{How can I perform case analysis?}
-
-You can use the {\case} or {\destruct} tactics.
-
-\Question{How can I prevent the case tactic from losing information ?}
-
-You may want to use the (now standard) {\tt case\_eq} tactic. See the Coq'Art page 159.
-
-\Question{Why should I name my intros?}
-
-When you use the {\intro} tactic you don't have to give a name to your
-hypothesis. If you do so the name will be generated by {\Coq} but your
-scripts may be less robust. If you add some hypothesis to your theorem
-(or change their order), you will have to change your proof to adapt
-to the new names.
-
-\Question{How can I automatize the naming?}
-
-You can use the {\tt Show Intro.} or {\tt Show Intros.} commands to generate the names and use your editor to generate a fully named {\intro} tactic.
-This can be automatized within {\tt xemacs}.
-
-\begin{coq_example}
-Goal forall A B C : Prop, A -> B -> C -> A/\B/\C.
-Show Intros.
-(*
-A B C H H0
-H1
-*)
-intros A B C H H0 H1.
-repeat split;assumption.
-\end{coq_example}
-\begin{coq_example*}
-Qed.
-\end{coq_example*}
-
-\Question{I want to automatize the use of some tactic, how can I do it?}
-
-You need to use the {\tt proof with T} command and add {\ldots} at the
-end of your sentences.
-
-For instance:
-\begin{coq_example}
-Goal forall A B C : Prop, A -> B/\C -> A/\B/\C.
-Proof with assumption.
-intros.
-split...
-\end{coq_example}
-\begin{coq_example*}
-Qed.
-\end{coq_example*}
-
-\Question{I want to execute the {\texttt proof with} tactic only if it solves the goal, how can I do it?}
-
-You need to use the {\try} and {\solve} tactics. For instance:
-\begin{coq_example}
-Require Import ZArith.
-Require Ring.
-Local Open Scope Z_scope.
-Goal forall a b c : Z, a+b=b+a.
-Proof with try solve [ring].
-intros...
-\end{coq_example}
-\begin{coq_example*}
-Qed.
-\end{coq_example*}
-
-\Question{How can I do the opposite of the {\intro} tactic?}
-
-You can use the {\generalize} tactic.
-
-\begin{coq_example}
-Goal forall A B : Prop, A->B-> A/\B.
-intros.
-generalize H.
-intro.
-auto.
-\end{coq_example}
-\begin{coq_example*}
-Qed.
-\end{coq_example*}
-
-\Question{One of the hypothesis is an equality between a variable and some term, I want to get rid of this variable, how can I do it?}
-
-You can use the {\subst} tactic. This will rewrite the equality everywhere and clear the assumption.
-
-\Question{What can I do if I get ``{\tt generated subgoal term has metavariables in it }''?}
-
-You should use the {\eapply} tactic, this will generate some goals containing metavariables.
-
-\Question{How can I instantiate some metavariable?}
-
-Just use the {\instantiate} tactic.
-
-
-\Question{What is the use of the {\pattern} tactic?}
-
-The {\pattern} tactic transforms the current goal, performing
-beta-expansion on all the applications featuring this tactic's
-argument. For instance, if the current goal includes a subterm {\tt
-phi(t)}, then {\tt pattern t} transforms the subterm into {\tt (fun
-x:A => phi(x)) t}. This can be useful when {\apply} fails on matching,
-to abstract the appropriate terms.
-
-\Question{What is the difference between assert, cut and generalize?}
-
-PS: Notice for people that are interested in proof rendering that \assert
-and {\pose} (and \cut) are not rendered the same as {\generalize} (see the
-HELM experimental rendering tool at \ahref{http://helm.cs.unibo.it/library.html}{\url{http://helm.cs.unibo.it}}, link
-HELM, link COQ Online). Indeed {\generalize} builds a beta-expanded term
-while \assert, {\pose} and {\cut} uses a let-in.
-
-\begin{verbatim}
- (* Goal is T *)
- generalize (H1 H2).
- (* Goal is A->T *)
- ... a proof of A->T ...
-\end{verbatim}
-
-is rendered into something like
-\begin{verbatim}
- (h) ... the proof of A->T ...
- we proved A->T
- (h0) by (H1 H2) we proved A
- by (h h0) we proved T
-\end{verbatim}
-while
-\begin{verbatim}
- (* Goal is T *)
- assert q := (H1 H2).
- (* Goal is A *)
- ... a proof of A ...
- (* Goal is A |- T *)
- ... a proof of T ...
-\end{verbatim}
-is rendered into something like
-\begin{verbatim}
- (q) ... the proof of A ...
- we proved A
- ... the proof of T ...
- we proved T
-\end{verbatim}
-Otherwise said, {\generalize} is not rendered in a forward-reasoning way,
-while {\assert} is.
-
-\Question{What can I do if \Coq can not infer some implicit argument ?}
-
-You can state explicitly what this implicit argument is. See \ref{implicit}.
-
-\Question{How can I explicit some implicit argument ?}\label{implicit}
-
-Just use \texttt{A:=term} where \texttt{A} is the argument.
-
-For instance if you want to use the existence of ``nil'' on nat*nat lists:
-\begin{verbatim}
-exists (nil (A:=(nat*nat))).
-\end{verbatim}
-
-\iffalse
-\Question{Is there anyway to do pattern matching with dependent types?}
-
-todo
-\fi
-
-\subsection{Proof management}
-
-
-\Question{How can I change the order of the subgoals?}
-
-You can use the {\Focus} command to concentrate on some goal. When the goal is proved you will see the remaining goals.
-
-\Question{How can I change the order of the hypothesis?}
-
-You can use the {\tt Move ... after} command.
-
-\Question{How can I change the name of an hypothesis?}
-
-You can use the {\tt Rename ... into} command.
-
-\Question{How can I delete some hypothesis?}
-
-You can use the {\tt Clear} command.
-
-\Question{How can use a proof which is not finished?}
-
-You can use the {\tt Admitted} command to state your current proof as an axiom.
-You can use the {\tt give\_up} tactic to omit a portion of a proof.
-
-\Question{How can I state a conjecture?}
-
-You can use the {\tt Admitted} command to state your current proof as an axiom.
-
-\Question{What is the difference between a lemma, a fact and a theorem?}
-
-From {\Coq} point of view there are no difference. But some tools can
-have a different behavior when you use a lemma rather than a
-theorem. For instance {\tt coqdoc} will not generate documentation for
-the lemmas within your development.
-
-\Question{How can I organize my proofs?}
-
-You can organize your proofs using the section mechanism of \Coq. Have
-a look at the manual for further information.
-
-
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-\section{Inductive and Co-inductive types}
-
-\subsection{General}
-
-\Question{How can I prove that two constructors are different?}
-
-You can use the {\discriminate} tactic.
-
-\begin{coq_example}
-Inductive toto : Set := | C1 : toto | C2 : toto.
-Goal C1 <> C2.
-discriminate.
-\end{coq_example}
-\begin{coq_example*}
-Qed.
-\end{coq_example*}
-
-\Question{During an inductive proof, how to get rid of impossible cases of an inductive definition?}
-
-Use the {\inversion} tactic.
-
-
-\Question{How can I prove that 2 terms in an inductive set are equal? Or different?}
-
-Have a look at \coqtt{decide equality} and \coqtt{discriminate} in the \ahref{http://coq.inria.fr/doc/main.html}{Reference Manual}.
-
-\Question{Why is the proof of \coqtt{0+n=n} on natural numbers
-trivial but the proof of \coqtt{n+0=n} is not?}
-
- Since \coqtt{+} (\coqtt{plus}) on natural numbers is defined by analysis on its first argument
-
-\begin{coq_example}
-Print plus.
-\end{coq_example}
-
-{\noindent} The expression \coqtt{0+n} evaluates to \coqtt{n}. As {\Coq} reasons
-modulo evaluation of expressions, \coqtt{0+n} and \coqtt{n} are
-considered equal and the theorem \coqtt{0+n=n} is an instance of the
-reflexivity of equality. On the other side, \coqtt{n+0} does not
-evaluate to \coqtt{n} and a proof by induction on \coqtt{n} is
-necessary to trigger the evaluation of \coqtt{+}.
-
-\Question{Why is dependent elimination in Prop not
-available by default?}
-
-
-This is just because most of the time it is not needed. To derive a
-dependent elimination principle in {\tt Prop}, use the command {\tt Scheme} and
-apply the elimination scheme using the \verb=using= option of
-\verb=elim=, \verb=destruct= or \verb=induction=.
-
-
-\Question{Argh! I cannot write expressions like ``~{\tt if n <= p then p else n}~'', as in any programming language}
-\label{minmax}
-
-The short answer : You should use {\texttt le\_lt\_dec n p} instead.\\
-
-The long answer: That's right, you can't.
-If you type for instance the following ``definition'':
-\begin{coq_eval}
-Reset Initial.
-\end{coq_eval}
-\begin{coq_example}
-Fail Definition max (n p : nat) := if n <= p then p else n.
-\end{coq_example}
-
-As \Coq~ says, the term ``~\texttt{n <= p}~'' is a proposition, i.e. a
-statement that belongs to the mathematical world. There are many ways to
-prove such a proposition, either by some computation, or using some already
-proven theorems. For instance, proving $3-2 \leq 2^{45503}$ is very easy,
-using some theorems on arithmetical operations. If you compute both numbers
-before comparing them, you risk to use a lot of time and space.
-
-
-On the contrary, a function for computing the greatest of two natural numbers
-is an algorithm which, called on two natural numbers
-$n$ and $p$, determines whether $n\leq p$ or $p < n$.
-Such a function is a \emph{decision procedure} for the inequality of
- \texttt{nat}. The possibility of writing such a procedure comes
-directly from de decidability of the order $\leq$ on natural numbers.
-
-
-When you write a piece of code like
-``~\texttt{if n <= p then \dots{} else \dots}~''
-in a
-programming language like \emph{ML} or \emph{Java}, a call to such a
-decision procedure is generated. The decision procedure is in general
-a primitive function, written in a low-level language, in the correctness
-of which you have to trust.
-
-The standard Library of the system \emph{Coq} contains a
-(constructive) proof of decidability of the order $\leq$ on
-\texttt{nat} : the function \texttt{le\_lt\_dec} of
-the module \texttt{Compare\_dec} of library \texttt{Arith}.
-
-The following code shows how to define correctly \texttt{min} and
-\texttt{max}, and prove some properties of these functions.
-
-\begin{coq_example}
-Require Import Compare_dec.
-
-Definition max (n p : nat) := if le_lt_dec n p then p else n.
-
-Definition min (n p : nat) := if le_lt_dec n p then n else p.
-
-Eval compute in (min 4 7).
-
-Theorem min_plus_max : forall n p, min n p + max n p = n + p.
-Proof.
- intros n p;
- unfold min, max;
- case (le_lt_dec n p);
- simpl; auto with arith.
-Qed.
-
-Theorem max_equiv : forall n p, max n p = p <-> n <= p.
-Proof.
- unfold max; intros n p; case (le_lt_dec n p);simpl; auto.
- intuition auto with arith.
- split.
- intro e; rewrite e; auto with arith.
- intro H; absurd (p < p); eauto with arith.
-Qed.
-\end{coq_example}
-
-\Question{I wrote my own decision procedure for $\leq$, which
-is much faster than yours, but proving such theorems as
- \texttt{max\_equiv} seems to be quite difficult}
-
-Your code is probably the following one:
-
-\begin{coq_example}
-Fixpoint my_le_lt_dec (n p :nat) {struct n}: bool :=
- match n, p with 0, _ => true
- | S n', S p' => my_le_lt_dec n' p'
- | _ , _ => false
- end.
-
-Definition my_max (n p:nat) := if my_le_lt_dec n p then p else n.
-
-Definition my_min (n p:nat) := if my_le_lt_dec n p then n else p.
-\end{coq_example}
-
-
-For instance, the computation of \texttt{my\_max 567 321} is almost
-immediate, whereas one can't wait for the result of
-\texttt{max 56 32}, using \emph{Coq's} \texttt{le\_lt\_dec}.
-
-This is normal. Your definition is a simple recursive function which
-returns a boolean value. Coq's \texttt{le\_lt\_dec} is a \emph{certified
-function}, i.e. a complex object, able not only to tell whether $n\leq p$
-or $p<n$, but also of building a complete proof of the correct inequality.
-What make \texttt{le\_lt\_dec} inefficient for computing \texttt{min}
-and \texttt{max} is the building of a huge proof term.
-
-Nevertheless, \texttt{le\_lt\_dec} is very useful. Its type
-is a strong specification, using the
-\texttt{sumbool} type (look at the reference manual or chapter 9 of
-\cite{coqart}). Eliminations of the form
-``~\texttt{case (le\_lt\_dec n p)}~'' provide proofs of
-either $n \leq p$ or $p < n$, allowing easy proofs of some theorems as in
-question~\ref{minmax}. Unfortunately, this not the case of your
-\texttt{my\_le\_lt\_dec}, which returns a quite non-informative boolean
-value.
-
-
-\begin{coq_example}
-Check le_lt_dec.
-\end{coq_example}
-
-You should keep in mind that \texttt{le\_lt\_dec} is useful to build
-certified programs which need to compare natural numbers, and is not
-designed to compare quickly two numbers.
-
-Nevertheless, the \emph{extraction} of \texttt{le\_lt\_dec} towards
-\emph{OCaml} or \emph{Haskell}, is a reasonable program for comparing two
-natural numbers in Peano form in linear time.
-
-It is also possible to keep your boolean function as a decision procedure,
-but you have to establish yourself the relationship between \texttt{my\_le\_lt\_dec} and the propositions $n\leq p$ and $p<n$:
-
-\begin{coq_example*}
-Theorem my_le_lt_dec_true :
- forall n p, my_le_lt_dec n p = true <-> n <= p.
-
-Theorem my_le_lt_dec_false :
- forall n p, my_le_lt_dec n p = false <-> p < n.
-\end{coq_example*}
-
-
-\subsection{Recursion}
-
-\Question{Why can't I define a non terminating program?}
-
- Because otherwise the decidability of the type-checking
-algorithm (which involves evaluation of programs) is not ensured. On
-another side, if non terminating proofs were allowed, we could get a
-proof of {\tt False}:
-
-\begin{coq_example*}
-(* This is fortunately not allowed! *)
-Fixpoint InfiniteProof (n:nat) : False := InfiniteProof n.
-Theorem Paradox : False.
-Proof (InfiniteProof O).
-\end{coq_example*}
-
-
-\Question{Why only structurally well-founded loops are allowed?}
-
- The structural order on inductive types is a simple and
-powerful notion of termination. The consistency of the Calculus of
-Inductive Constructions relies on it and another consistency proof
-would have to be made for stronger termination arguments (such
-as the termination of the evaluation of CIC programs themselves!).
-
-In spite of this, all non-pathological termination orders can be mapped
-to a structural order. Tools to do this are provided in the file
-\vfile{\InitWf}{Wf} of the standard library of {\Coq}.
-
-\Question{How to define loops based on non structurally smaller
-recursive calls?}
-
- The procedure is as follows (we consider the definition of {\tt
-mergesort} as an example).
-
-\begin{itemize}
-
-\item Define the termination order, say {\tt R} on the type {\tt A} of
-the arguments of the loop.
-
-\begin{coq_eval}
-Reset Initial.
-Require Import List.
-\end{coq_eval}
-
-\begin{coq_example*}
-Definition R (a b:list nat) := length a < length b.
-\end{coq_example*}
-
-\item Prove that this order is well-founded (in fact that all elements in {\tt A} are accessible along {\tt R}).
-
-\begin{coq_example*}
-Lemma Rwf : well_founded R.
-\end{coq_example*}
-\begin{coq_eval}
-Admitted.
-\end{coq_eval}
-
-\item Define the step function (which needs proofs that recursive
-calls are on smaller arguments).
-
-\begin{coq_example*}
-Definition split (l : list nat)
- : {l1: list nat | R l1 l} * {l2 : list nat | R l2 l}.
-Admitted.
-Definition concat (l1 l2 : list nat) : list nat.
-Admitted.
-Definition merge_step (l : list nat) (f: forall l':list nat, R l' l -> list nat) :=
- let (lH1,lH2) := (split l) in
- let (l1,H1) := lH1 in
- let (l2,H2) := lH2 in
- concat (f l1 H1) (f l2 H2).
-\end{coq_example*}
-
-\item Define the recursive function by fixpoint on the step function.
-
-\begin{coq_example*}
-Definition merge := Fix Rwf (fun _ => list nat) merge_step.
-\end{coq_example*}
-
-\end{itemize}
-
-\Question{What is behind the accessibility and well-foundedness proofs?}
-
- Well-foundedness of some relation {\tt R} on some type {\tt A}
-is defined as the accessibility of all elements of {\tt A} along {\tt R}.
-
-\begin{coq_example}
-Print well_founded.
-Print Acc.
-\end{coq_example}
-
-The structure of the accessibility predicate is a well-founded tree
-branching at each node {\tt x} in {\tt A} along all the nodes {\tt x'}
-less than {\tt x} along {\tt R}. Any sequence of elements of {\tt A}
-decreasing along the order {\tt R} are branches in the accessibility
-tree. Hence any decreasing along {\tt R} is mapped into a structural
-decreasing in the accessibility tree of {\tt R}. This is emphasised in
-the definition of {\tt fix} which recurs not on its argument {\tt x:A}
-but on the accessibility of this argument along {\tt R}.
-
-See file \vfile{\InitWf}{Wf}.
-
-\Question{How to perform simultaneous double induction?}
-
- In general a (simultaneous) double induction is simply solved by an
-induction on the first hypothesis followed by an inversion over the
-second hypothesis. Here is an example
-
-\begin{coq_eval}
-Reset Initial.
-\end{coq_eval}
-
-\begin{coq_example}
-Inductive even : nat -> Prop :=
- | even_O : even 0
- | even_S : forall n:nat, even n -> even (S (S n)).
-
-Inductive odd : nat -> Prop :=
- | odd_SO : odd 1
- | odd_S : forall n:nat, odd n -> odd (S (S n)).
-
-Lemma not_even_and_odd : forall n:nat, even n -> odd n -> False.
-induction 1.
- inversion 1.
- inversion 1. apply IHeven; trivial.
-\end{coq_example}
-\begin{coq_example*}
-Qed.
-\end{coq_example*}
-
-In case the type of the second induction hypothesis is not
-dependent, {\tt inversion} can just be replaced by {\tt destruct}.
-
-\Question{How to define a function by simultaneous double recursion?}
-
- The same trick applies, you can even use the pattern-matching
-compilation algorithm to do the work for you. Here is an example:
-
-\begin{coq_example}
-Fixpoint minus (n m:nat) {struct n} : nat :=
- match n, m with
- | O, _ => 0
- | S k, O => S k
- | S k, S l => minus k l
- end.
-Print minus.
-\end{coq_example}
-
-In case of dependencies in the type of the induction objects
-$t_1$ and $t_2$, an extra argument stating $t_1=t_2$ must be given to
-the fixpoint definition
-
-\Question{How to perform nested and double induction?}
-
- To reason by nested (i.e. lexicographic) induction, just reason by
-induction on the successive components.
-
-\smallskip
-
-Double induction (or induction on pairs) is a restriction of the
-lexicographic induction. Here is an example of double induction.
-
-\begin{coq_example}
-Lemma nat_double_ind :
-forall P : nat -> nat -> Prop, P 0 0 ->
- (forall m n, P m n -> P m (S n)) ->
- (forall m n, P m n -> P (S m) n) ->
- forall m n, P m n.
-intros P H00 HmS HSn; induction m.
-(* case 0 *)
-induction n; [assumption | apply HmS; apply IHn].
-(* case Sm *)
-intro n; apply HSn; apply IHm.
-\end{coq_example}
-\begin{coq_example*}
-Qed.
-\end{coq_example*}
-
-\Question{How to define a function by nested recursion?}
-
- The same trick applies. Here is the example of Ackermann
-function.
-
-\begin{coq_example}
-Fixpoint ack (n:nat) : nat -> nat :=
- match n with
- | O => S
- | S n' =>
- (fix ack' (m:nat) : nat :=
- match m with
- | O => ack n' 1
- | S m' => ack n' (ack' m')
- end)
- end.
-\end{coq_example}
-
-
-\subsection{Co-inductive types}
-
-\Question{I have a cofixpoint $t:=F(t)$ and I want to prove $t=F(t)$. How to do it?}
-
-Just case-expand $F({\tt t})$ then complete by a trivial case analysis.
-Here is what it gives on e.g. the type of streams on naturals
-
-\begin{coq_eval}
-Set Implicit Arguments.
-\end{coq_eval}
-\begin{coq_example}
-CoInductive Stream (A:Set) : Set :=
- Cons : A -> Stream A -> Stream A.
-CoFixpoint nats (n:nat) : Stream nat := Cons n (nats (S n)).
-Lemma Stream_unfold :
- forall n:nat, nats n = Cons n (nats (S n)).
-Proof.
- intro;
- change (nats n = match nats n with
- | Cons x s => Cons x s
- end).
- case (nats n); reflexivity.
-\end{coq_example}
-\begin{coq_example*}
-Qed.
-\end{coq_example*}
-
-
-
-\section{Syntax and notations}
-
-\Question{I do not want to type ``forall'' because it is too long, what can I do?}
-
-You can define your own notation for forall:
-\begin{verbatim}
-Notation "fa x : t, P" := (forall x:t, P) (at level 200, x ident).
-\end{verbatim}
-or if your are using {\CoqIde} you can define a pretty symbol for for all and an input method (see \ref{forallcoqide}).
-
-
-
-\Question{How can I define a notation for square?}
-
-You can use for instance:
-\begin{verbatim}
-Notation "x ^2" := (Rmult x x) (at level 20).
-\end{verbatim}
-Note that you can not use:
-\begin{tt}
-Notation "x $^2$" := (Rmult x x) (at level 20).
-\end{tt}
-because ``$^2$'' is an iso-latin character. If you really want this kind of notation you should use UTF-8.
-
-
-\Question{Why ``no associativity'' and ``left associativity'' at the same level does not work?}
-
-Because we relie on Camlp4 for syntactical analysis and Camlp4 does not really
-implement no associativity. By default, non associative operators are defined
-as right associative.
-
-
-
-\Question{How can I know the associativity associated with a level?}
-
-You can do ``Print Grammar constr'', and decode the output from Camlp4, good luck !
-
-\section{Modules}
-
-
-
-
-%%%%%%%
-\section{\Ltac}
-
-\Question{What is {\Ltac}?}
-
-{\Ltac} is the tactic language for \Coq. It provides the user with a
-high-level ``toolbox'' for tactic creation.
-
-\Question{Is there any printing command in {\Ltac}?}
-
-You can use the {\idtac} tactic with a string argument. This string
-will be printed out. The same applies to the {\fail} tactic
-
-\Question{What is the syntax for let in {\Ltac}?}
-
-If $x_i$ are identifiers and $e_i$ and $expr$ are tactic expressions, then let reads:
-\begin{center}
-{\tt let $x_1$:=$e_1$ with $x_2$:=$e_2$\ldots with $x_n$:=$e_n$ in
-$expr$}.
-\end{center}
-Beware that if $expr$ is complex (i.e. features at least a sequence) parenthesis
-should be added around it. For example:
-\begin{coq_example}
-Ltac twoIntro := let x:=intro in (x;x).
-\end{coq_example}
-
-\Question{What is the syntax for pattern matching in {\Ltac}?}
-
-Pattern matching on a term $expr$ (non-linear first order unification)
-with patterns $p_i$ and tactic expressions $e_i$ reads:
-\begin{center}
-\hspace{10ex}
-{\tt match $expr$ with
-\hspace*{2ex}$p_1$ => $e_1$
-\hspace*{1ex}\textbar$p_2$ => $e_2$
-\hspace*{1ex}\ldots
-\hspace*{1ex}\textbar$p_n$ => $e_n$
-\hspace*{1ex}\textbar\ \textunderscore\ => $e_{n+1}$
-end.
-}
-\end{center}
-Underscore matches all terms.
-
-\Question{What is the semantics for ``match goal''?}
-
-The semantics of {\tt match goal} depends on whether it returns
-tactics or not. The {\tt match goal} expression matches the current
-goal against a series of patterns: {$hyp_1 {\ldots} hyp_n$ \textbar-
-$ccl$}. It uses a first-order unification algorithm and in case of
-success, if the right-hand-side is an expression, it tries to type it
-while if the right-hand-side is a tactic, it tries to apply it. If the
-typing or the tactic application fails, the {\tt match goal} tries all
-the possible combinations of $hyp_i$ before dropping the branch and
-moving to the next one. Underscore matches all terms.
-
-\Question{Why can't I use a ``match goal'' returning a tactic in a non
-tail-recursive position?}
-
-This is precisely because the semantics of {\tt match goal} is to
-apply the tactic on the right as soon as a pattern unifies what is
-meaningful only in tail-recursive uses.
-
-The semantics in non tail-recursive call could have been the one used
-for terms (i.e. fail if the tactic expression is not typable, but
-don't try to apply it). For uniformity of semantics though, this has
-been rejected.
-
-\Question{How can I generate a new name?}
-
-You can use the following syntax:
-{\tt let id:=fresh in \ldots}\\
-For example:
-\begin{coq_example}
-Ltac introIdGen := let id:=fresh in intro id.
-\end{coq_example}
-
-
-\iffalse
-\Question{How can I access the type of a term?}
-
-You can use typeof.
-todo
-\fi
-
-\iffalse
-\Question{How can I define static and dynamic code?}
-\fi
-
-\section{Tactics written in OCaml}
-
-\Question{Can you show me an example of a tactic written in OCaml?}
-
-Have a look at the skeleton ``Hello World'' tactic from the next question.
-You also have some examples of tactics written in OCaml in the ``plugins'' directory of {\Coq} sources.
-
-\Question{Is there a skeleton of OCaml tactic I can reuse somewhere?}
-
-The following steps describe how to write a simplistic ``Hello world'' OCaml
-tactic. This takes the form of a dynamically loadable OCaml module, which will
-be invoked from the Coq toplevel.
-\begin{enumerate}
-\item In the \verb+plugins+ directory of the Coq source location, create a
-directory \verb+hello+. Proceed to create a grammar and OCaml file, respectively
-\verb+plugins/hello/g_hello.ml4+ and \verb+plugins/hello/coq_hello.ml+,
-containing:
- \begin{itemize}
- \item in \verb+g_hello.ml4+:
-\begin{verbatim}
-(*i camlp4deps: "grammar/grammar.cma" i*)
-TACTIC EXTEND Hello
-| [ "hello" ] -> [ Coq_hello.printHello ]
-END
-\end{verbatim}
- \item in \verb+coq_hello.ml+:
-\begin{verbatim}
-let printHello gl =
-Tacticals.tclIDTAC_MESSAGE (Pp.str "Hello world") gl
- \end{verbatim}
- \end{itemize}
-\item Create a file \verb+plugins/hello/hello_plugin.mllib+, containing the
-names of the OCaml modules bundled in the dynamic library:
-\begin{verbatim}
-Coq_hello
-G_hello
-\end{verbatim}
-\item Append the following lines in \verb+plugins/plugins{byte,opt}.itarget+:
-\begin{itemize}
- \item in \verb+pluginsopt.itarget+:
-\begin{verbatim}
-hello/hello_plugin.cmxa
-\end{verbatim}
- \item in \verb+pluginsbyte.itarget+:
-\begin{verbatim}
-hello/hello_plugin.cma
-\end{verbatim}
-\end{itemize}
-\item In the root directory of the Coq source location, modify the file
-\verb+Makefile.common+:
- \begin{itemize}
- \item add \verb+hello+ to the \verb+SRCDIR+ definition (second argument of the
- \verb+addprefix+ function);
- \item in the section ``Object and Source files'', add \verb+HELLOCMA:=plugins/hello/hello_plugin.cma+;
- \item add \verb+$(HELLOCMA)+ to the \verb+PLUGINSCMA+ definition.
- \end{itemize}
-\item Modify the file \verb+Makefile.build+, adding in section ``3) plugins'' the
-line:
-\begin{verbatim}
-hello: $(HELLOCMA)
-\end{verbatim}
-\item From the command line, run \verb+make hello+, then \verb+make plugins/hello/hello_plugin.cmxs+.
-\end{enumerate}
-The call to the tactic \verb+hello+ from a Coq script has to be preceded by
-\verb+Declare ML Module "hello_plugin"+, which will load the dynamic object
-\verb+hello_plugin.cmxs+. For instance:
-\begin{verbatim}
-Declare ML Module "hello_plugin".
-Variable A:Prop.
-Goal A-> A.
-Proof.
-hello.
-auto.
-Qed.
-\end{verbatim}
-
-
-\section{Case studies}
-
-\iffalse
-\Question{How can I define vectors or lists of size n?}
-\fi
-
-
-\Question{How to prove that 2 sets are different?}
-
- You need to find a property true on one set and false on the
-other one. As an example we show how to prove that {\tt bool} and {\tt
-nat} are discriminable. As discrimination property we take the
-property to have no more than 2 elements.
-
-\begin{coq_example*}
-Theorem nat_bool_discr : bool <> nat.
-Proof.
- pose (discr :=
- fun X:Set =>
- ~ (forall a b:X, ~ (forall x:X, x <> a -> x <> b -> False))).
- intro Heq; assert (H: discr bool).
- intro H; apply (H true false); destruct x; auto.
- rewrite Heq in H; apply H; clear H.
- destruct a; destruct b as [|n]; intro H0; eauto.
- destruct n; [ apply (H0 2); discriminate | eauto ].
-Qed.
-\end{coq_example*}
-
-\Question{Is there an axiom-free proof of Streicher's axiom $K$ for
-the equality on {\tt nat}?}
-\label{K-nat}
-
-Yes, because equality is decidable on {\tt nat}. Here is the proof.
-
-\begin{coq_example*}
-Require Import Eqdep_dec.
-Require Import Peano_dec.
-Theorem K_nat :
- forall (x:nat) (P:x = x -> Prop), P (eq_refl x) -> forall p:x = x, P p.
-Proof.
-intros; apply K_dec_set with (p := p).
-apply eq_nat_dec.
-assumption.
-Qed.
-\end{coq_example*}
-
-Similarly, we have
-
-\begin{coq_example*}
-Theorem eq_rect_eq_nat :
- forall (p:nat) (Q:nat->Type) (x:Q p) (h:p=p), x = eq_rect p Q x p h.
-Proof.
-intros; apply K_nat with (p := h); reflexivity.
-Qed.
-\end{coq_example*}
-
-\Question{How to prove that two proofs of {\tt n<=m} on {\tt nat} are equal?}
-\label{le-uniqueness}
-
-This is provable without requiring any axiom because axiom $K$
-directly holds on {\tt nat}. Here is a proof using question \ref{K-nat}.
-
-\begin{coq_example*}
-Require Import Arith.
-Scheme le_ind' := Induction for le Sort Prop.
-Theorem le_uniqueness_proof : forall (n m : nat) (p q : n <= m), p = q.
-Proof.
-induction p using le_ind'; intro q.
- replace (le_n n) with
- (eq_rect _ (fun n0 => n <= n0) (le_n n) _ eq_refl).
- 2:reflexivity.
- generalize (eq_refl n).
- pattern n at 2 4 6 10, q; case q; [intro | intros m l e].
- rewrite <- eq_rect_eq_nat; trivial.
- contradiction (le_Sn_n m); rewrite <- e; assumption.
- replace (le_S n m p) with
- (eq_rect _ (fun n0 => n <= n0) (le_S n m p) _ eq_refl).
- 2:reflexivity.
- generalize (eq_refl (S m)).
- pattern (S m) at 1 3 4 6, q; case q; [intro Heq | intros m0 l HeqS].
- contradiction (le_Sn_n m); rewrite Heq; assumption.
- injection HeqS; intro Heq; generalize l HeqS.
- rewrite <- Heq; intros; rewrite <- eq_rect_eq_nat.
- rewrite (IHp l0); reflexivity.
-Qed.
-\end{coq_example*}
-
-\Question{How to exploit equalities on sets}
-
-To extract information from an equality on sets, you need to
-find a predicate of sets satisfied by the elements of the sets. As an
-example, let's consider the following theorem.
-
-\begin{coq_example*}
-Theorem interval_discr :
- forall m n:nat,
- {x : nat | x <= m} = {x : nat | x <= n} -> m = n.
-\end{coq_example*}
-
-We have a proof requiring the axiom of proof-irrelevance. We
-conjecture that proof-irrelevance can be circumvented by introducing a
-primitive definition of discrimination of the proofs of
-\verb!{x : nat | x <= m}!.
-
-\begin{latexonly}%
-The proof can be found in file {\tt interval$\_$discr.v} in this directory.
-%Here is the proof
-%\begin{small}
-%\begin{flushleft}
-%\begin{texttt}
-%\def_{\ifmmode\sb\else\subscr\fi}
-%\include{interval_discr.v}
-%%% WARNING semantics of \_ has changed !
-%\end{texttt}
-%$a\_b\_c$
-%\end{flushleft}
-%\end{small}
-\end{latexonly}%
-\begin{htmlonly}%
-\ahref{./interval_discr.v}{Here} is the proof.
-\end{htmlonly}
-
-\Question{I have a problem of dependent elimination on
-proofs, how to solve it?}
-
-\begin{coq_eval}
-Reset Initial.
-\end{coq_eval}
-
-\begin{coq_example*}
-Inductive Def1 : Set := c1 : Def1.
-Inductive DefProp : Def1 -> Prop :=
- c2 : forall d:Def1, DefProp d.
-Inductive Comb : Set :=
- c3 : forall d:Def1, DefProp d -> Comb.
-Lemma eq_comb :
- forall (d1 d1':Def1) (d2:DefProp d1) (d2':DefProp d1'),
- d1 = d1' -> c3 d1 d2 = c3 d1' d2'.
-\end{coq_example*}
-
- You need to derive the dependent elimination
-scheme for DefProp by hand using {\coqtt Scheme}.
-
-\begin{coq_eval}
-Abort.
-\end{coq_eval}
-
-\begin{coq_example*}
-Scheme DefProp_elim := Induction for DefProp Sort Prop.
-Lemma eq_comb :
- forall d1 d1':Def1,
- d1 = d1' ->
- forall (d2:DefProp d1) (d2':DefProp d1'), c3 d1 d2 = c3 d1' d2'.
-intros.
-destruct H.
-destruct d2 using DefProp_elim.
-destruct d2' using DefProp_elim.
-reflexivity.
-Qed.
-\end{coq_example*}
-
-
-\Question{And what if I want to prove the following?}
-
-\begin{coq_example*}
-Inductive natProp : nat -> Prop :=
- | p0 : natProp 0
- | pS : forall n:nat, natProp n -> natProp (S n).
-Inductive package : Set :=
- pack : forall n:nat, natProp n -> package.
-Lemma eq_pack :
- forall n n':nat,
- n = n' ->
- forall (np:natProp n) (np':natProp n'), pack n np = pack n' np'.
-\end{coq_example*}
-
-
-
-\begin{coq_eval}
-Abort.
-\end{coq_eval}
-\begin{coq_example*}
-Scheme natProp_elim := Induction for natProp Sort Prop.
-Definition pack_S : package -> package.
-destruct 1.
-apply (pack (S n)).
-apply pS; assumption.
-Defined.
-Lemma eq_pack :
- forall n n':nat,
- n = n' ->
- forall (np:natProp n) (np':natProp n'), pack n np = pack n' np'.
-intros n n' Heq np np'.
-generalize dependent n'.
-induction np using natProp_elim.
-induction np' using natProp_elim; intros; auto.
- discriminate Heq.
-induction np' using natProp_elim; intros; auto.
- discriminate Heq.
-change (pack_S (pack n np) = pack_S (pack n0 np')).
-apply (f_equal (A:=package)).
-apply IHnp.
-auto.
-Qed.
-\end{coq_example*}
-
-
-
-
-
-
-
-\section{Publishing tools}
-
-\Question{How can I generate some latex from my development?}
-
-You can use {\tt coqdoc}.
-
-\Question{How can I generate some HTML from my development?}
-
-You can use {\tt coqdoc}.
-
-\Question{How can I generate some dependency graph from my development?}
-
-You can use the tool \verb|coqgraph| developed by Philippe Audebaud in 2002.
-This tool transforms dependencies generated by \verb|coqdep| into 'dot' files which can be visualized using the Graphviz software (http://www.graphviz.org/).
-
-\Question{How can I cite some {\Coq} in my latex document?}
-
-You can use {\tt coq\_tex}.
-
-\Question{How can I cite the {\Coq} reference manual?}
-
-You can use this bibtex entry (to adapt to the appropriate version):
-\begin{verbatim}
-@manual{Coq:manual,
- author = {{Coq} {Development} {Team}, The},
- title = {The {Coq} Proof Assistant Reference Manual, version 8.7},
- month = Oct,
- year = {2017},
- url = {http://coq.inria.fr}
-}
-\end{verbatim}
-
-\Question{Where can I publish my developments in {\Coq}?}
-
-You can submit your developments as a user contribution to the {\Coq}
-development team. This ensures its liveness along the evolution and
-possible changes of {\Coq}.
-
-You can also submit your developments to the HELM/MoWGLI repository at
-the University of Bologna (see
-\ahref{http://mowgli.cs.unibo.it}{\url{http://mowgli.cs.unibo.it}}). For
-developments submitted in this database, it is possible to visualize
-the developments in natural language and execute various retrieving
-requests.
-
-\Question{How can I read my proof in natural language?}
-
-You can submit your proof to the HELM/MoWGLI repository and use the
-rendering tool provided by the server (see
-\ahref{http://mowgli.cs.unibo.it}{\url{http://mowgli.cs.unibo.it}}).
-
-\section{\CoqIde}
-
-\Question{What is {\CoqIde}?}
-
-{\CoqIde} is a gtk based GUI for \Coq.
-
-\Question{How to enable Emacs keybindings?}
-
-If in Gnome, run the gnome configuration editor (\texttt{gconf-editor})
-and set key \texttt{gtk-key-theme} to \texttt{Emacs} in the category
-\texttt{desktop/gnome/interface}.
-
-Otherwise, you need to find where the \verb#gtk-key-theme-name# option is located in
-your configuration, and set it to \texttt{Emacs}. Usually, it is in the
-\verb#$(HOME)/.gtkrc-2.0# file.
-
-
-%$ juste pour que la coloration emacs marche
-
-\Question{How to enable antialiased fonts?}
-
- Set the \verb#GDK_USE_XFT# variable to \verb#1#. This is by default
- with \verb#Gtk >= 2.2#. If some of your fonts are not available,
- set \verb#GDK_USE_XFT# to \verb#0#.
-
-\Question{How to use those Forall and Exists pretty symbols?}\label{forallcoqide}
- Thanks to the notation features in \Coq, you just need to insert these
-lines in your {\Coq} buffer:\\
-\begin{tt}
-Notation "$\forall$ x : t, P" := (forall x:t, P) (at level 200, x ident).
-\end{tt}\\
-\begin{tt}
-Notation "$\exists$ x : t, P" := (exists x:t, P) (at level 200, x ident).
-\end{tt}
-
-Copy/Paste of these lines from this file will not work outside of \CoqIde.
-You need to load a file containing these lines or to enter the $\forall$
-using an input method (see \ref{inputmeth}). To try it just use \verb#Require Import utf8# from inside
-\CoqIde.
-To enable these notations automatically start coqide with
-\begin{verbatim}
- coqide -l utf8
-\end{verbatim}
-In the ide subdir of {\Coq} library, you will find a sample utf8.v with some
-pretty simple notations.
-
-\Question{How to define an input method for non ASCII symbols?}\label{inputmeth}
-
-\begin{itemize}
-\item First solution: type \verb#<CONTROL><SHIFT>2200# to enter a forall in the script widow.
- 2200 is the hexadecimal code for forall in unicode charts and is encoded as
- in UTF-8.
- 2203 is for exists. See \ahref{http://www.unicode.org}{\url{http://www.unicode.org}} for more codes.
-\item Second solution: rebind \verb#<AltGr>a# to forall and \verb#<AltGr>e# to exists.
-
- Under X11, one can add those lines in the file ~/.xmodmaprc :
-
-\begin{verbatim}
-! forall
-keycode 24 = a A a A U2200 NoSymbol U2200 NoSymbol
-! exists
-keycode 26 = e E e E U2203 NoSymbol U2203 NoSymbol
-\end{verbatim}
-and then run xmodmap ~/.xmodmaprc.
-\end{itemize}
-
- Alternatively, you may use an input method editor such as SCIM or iBus.
-The latter offers a \LaTeX-like input method.
-
-\Question{How to customize the shortcuts for menus?}
- Two solutions are offered:
-\begin{itemize}
-\item Edit \verb+$XDG_CONFIG_HOME/coq/coqide.keys+ (which is usually \verb+$HOME/.config/coq/coqide.keys+) by hand or
-\item If your system supports it, from \CoqIde, you may select a menu entry and press the desired
- shortcut.
-\end{itemize}
-
-\Question{What encoding should I use? What is this $\backslash$x\{iiii\} in my file?}
- The encoding option is related to the way files are saved.
- Keep it as UTF-8 until it becomes important for you to exchange files
- with non UTF-8 aware applications.
- If you choose something else than UTF-8, then missing characters will
- be encoded by $\backslash$x\{....\} or $\backslash$x\{........\}
- where each dot is an hex. digit.
- The number between braces is the hexadecimal UNICODE index for the
- missing character.
-
-\Question{How to get rid of annoying unwanted automatic templates?}
-
-Some users may experiment problems with unwanted automatic
-templates while using Coqide. This is due to a change in the
-modifiers keys available through GTK. The straightest way to get
-rid of the problem is to edit by hand your coqiderc (either
-\verb|/home/<user>/.config/coq/coqiderc| under Linux, or \\
-\verb|C:\Documents and Settings\<user>\.config\coq\coqiderc| under Windows)
-and replace any occurrence of \texttt{MOD4} by \texttt{MOD1}.
-
-
-
-\section{Extraction}
-
-\Question{What is program extraction?}
-
-Program extraction consist in generating a program from a constructive proof.
-
-\Question{Which language can I extract to?}
-
-You can extract your programs to Objective Caml and Haskell.
-
-\Question{How can I extract an incomplete proof?}
-
-You can provide programs for your axioms.
-
-
-
-%%%%%%%
-\section{Glossary}
-
-\Question{Can you explain me what an evaluable constant is?}
-
-An evaluable constant is a constant which is unfoldable.
-
-\Question{What is a goal?}
-
-The goal is the statement to be proved.
-
-\Question{What is a meta variable?}
-
-A meta variable in {\Coq} represents a ``hole'', i.e. a part of a proof
-that is still unknown.
-
-\Question{What is Gallina?}
-
-Gallina is the specification language of \Coq. Complete documentation
-of this language can be found in the Reference Manual.
-
-\Question{What is The Vernacular?}
-
-It is the language of commands of Gallina i.e. definitions, lemmas, {\ldots}
-
-
-\Question{What is a dependent type?}
-
-A dependent type is a type which depends on some term. For instance
-``vector of size n'' is a dependent type representing all the vectors
-of size $n$. Its type depends on $n$
-
-\Question{What is a proof by reflection?}
-
-This is a proof generated by some computation which is done using the
-internal reduction of {\Coq} (not using the tactic language of {\Coq}
-(\Ltac) nor the implementation language for \Coq). An example of
-tactic using the reflection mechanism is the {\ring} tactic. The
-reflection method consist in reflecting a subset of {\Coq} language (for
-example the arithmetical expressions) into an object of the {\Coq}
-language itself (in this case an inductive type denoting arithmetical
-expressions). For more information see~\cite{howe,harrison,boutin}
-and the last chapter of the Coq'Art.
-
-\Question{What is intuitionistic logic?}
-
-This is any logic which does not assume that ``A or not A''.
-
-
-\Question{What is proof-irrelevance?}
-
-See question \ref{proof-irrelevance}
-
-
-\Question{What is the difference between opaque and transparent?}{\label{opaque}}
-
-Opaque definitions can not be unfolded but transparent ones can.
-
-
-\section{Troubleshooting}
-
-\Question{What can I do when {\tt Qed.} is slow?}
-
-Sometime you can use the {\abstracttac} tactic, which makes as if you had
-stated some local lemma, this speeds up the typing process.
-
-\Question{Why \texttt{Reset Initial.} does not work when using \texttt{coqc}?}
-
-The initial state corresponds to the state of \texttt{coqtop} when the interactive
-session began. It does not make sense in files to compile.
-
-
-\Question{What can I do if I get ``No more subgoals but non-instantiated existential variables''?}
-
-This means that {\eauto} or {\eapply} didn't instantiate an
-existential variable which eventually got erased by some computation.
-You may backtrack to the faulty occurrence of {\eauto} or {\eapply}
-and give the missing argument an explicit value. Alternatively, you
-can use the commands \texttt{Show Existentials.} and
-\texttt{Existential.} to display and instantiate the remaining
-existential variables.
-
-
-\begin{coq_example}
-Lemma example_show_existentials : forall a b c:nat, a=b -> b=c -> a=c.
-Proof.
-intros.
-eapply eq_trans.
-Show Existentials.
-eassumption.
-assumption.
-\end{coq_example}
-\begin{coq_example*}
-Qed.
-\end{coq_example*}
-
-
-\Question{What can I do if I get ``Cannot solve a second-order unification problem''?}
-
-You can help {\Coq} using the {\pattern} tactic.
-
-
-\Question{I copy-paste a term and {\Coq} says it is not convertible
- to the original term. Sometimes it even says the copied term is not
-well-typed.}
-
- This is probably due to invisible implicit information (implicit
-arguments, coercions and Cases annotations) in the printed term, which
-is not re-synthesised from the copied-pasted term in the same way as
-it is in the original term.
-
- Consider for instance {\tt (@eq Type True True)}. This term is
-printed as {\tt True=True} and re-parsed as {\tt (@eq Prop True
-True)}. The two terms are not convertible (hence they fool tactics
-like {\tt pattern}).
-
- There is currently no satisfactory answer to the problem. However,
-the command {\tt Set Printing All} is useful for diagnosing the
-problem.
-
- Due to coercions, one may even face type-checking errors. In some
-rare cases, the criterion to hide coercions is a bit too loose, which
-may result in a typing error message if the parser is not able to find
-again the missing coercion.
-
-
-
-\section{Conclusion and Farewell.}
-\label{ccl}
-
-\Question{What if my question isn't answered here?}
-\label{lastquestion}
-
-Don't panic \verb+:-)+. You can try the {\Coq} manual~\cite{Coq:manual} for a technical
-description of the prover. The Coq'Art~\cite{Coq:coqart} is the first
-book written on {\Coq} and provides a comprehensive review of the
-theorem prover as well as a number of example and exercises. Finally,
-the tutorial~\cite{Coq:Tutorial} provides a smooth introduction to
-theorem proving in \Coq.
-
-
-%%%%%%%
-\newpage
-\nocite{LaTeX:intro}
-\nocite{LaTeX:symb}
-\bibliography{fk}
-
-%%%%%%%
-\typeout{*********************************************}
-\typeout{********* That makes {\thequestion} questions **********}
-\typeout{*********************************************}
-
-\end{document}
diff --git a/doc/faq/axioms.fig b/doc/faq/axioms.fig
deleted file mode 100644
index 963178503..000000000
--- a/doc/faq/axioms.fig
+++ /dev/null
@@ -1,131 +0,0 @@
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diff --git a/doc/faq/fk.bib b/doc/faq/fk.bib
deleted file mode 100644
index 3410427de..000000000
--- a/doc/faq/fk.bib
+++ /dev/null
@@ -1,2221 +0,0 @@
-%%%%%%% FAQ %%%%%%%
-
-@book{ProofsTypes,
- Author="Girard, Jean-Yves and Yves Lafont and Paul Taylor",
- Title="Proofs and Types",
- Publisher="Cambrige Tracts in Theoretical Computer Science, Cambridge University Press",
- Year="1989"
-}
-
-@misc{Types:Dowek,
- author = "Gilles Dowek",
- title = "Th{\'e}orie des types",
- year = 2002,
- howpublished = "Lecture notes",
- url= "http://www.lix.polytechnique.fr/~dowek/Cours/theories_des_types.ps.gz"
-}
-
-@PHDTHESIS{EGThese,
- author = {Eduardo Giménez},
- title = {Un Calcul de Constructions Infinies et son application
-a la vérification de systèmes communicants},
- type = {thèse d'Université},
- school = {Ecole Normale Supérieure de Lyon},
- month = {December},
- year = {1996},
-}
-
-
-%%%%%%% Semantique %%%%%%%
-
-@misc{Sem:cours,
- author = "François Pottier",
- title = "{Typage et Programmation}",
- year = "2002",
- howpublished = "Lecture notes",
- note = "DEA PSPL"
-}
-
-@inproceedings{Sem:Dubois,
- author = {Catherine Dubois},
- editor = {Mark Aagaard and
- John Harrison},
- title = "{Proving ML Type Soundness Within Coq}",
- pages = {126-144},
- booktitle = {TPHOLs},
- publisher = {Springer},
- series = {Lecture Notes in Computer Science},
- volume = {1869},
- year = {2000},
- isbn = {3-540-67863-8},
- bibsource = {DBLP, http://dblp.uni-trier.de}
-}
-
-@techreport{Sem:Plotkin,
-author = {Gordon D. Plotkin},
-institution = {Aarhus University},
-number = {{DAIMI FN-19}},
-title = {{A structural approach to operational semantics}},
-year = {1981}
-}
-
-@article{Sem:RemyV98,
- author = "Didier R{\'e}my and J{\'e}r{\^o}me Vouillon",
- title = "Objective {ML}:
- An effective object-oriented extension to {ML}",
- journal = "Theory And Practice of Object Systems",
- year = 1998,
- volume = "4",
- number = "1",
- pages = "27--50",
- note = {A preliminary version appeared in the proceedings
- of the 24th ACM Conference on Principles
- of Programming Languages, 1997}
-}
-
-@book{Sem:Winskel,
- AUTHOR = {Winskel, Glynn},
- TITLE = {The Formal Semantics of Programming Languages},
- NOTE = {WIN g2 93:1 P-Ex},
- YEAR = {1993},
- PUBLISHER = {The MIT Press},
- SERIES = {Foundations of Computing},
- }
-
-@Article{Sem:WrightFelleisen,
- refkey = "C1210",
- title = "A Syntactic Approach to Type Soundness",
- author = "Andrew K. Wright and Matthias Felleisen",
- pages = "38--94",
- journal = "Information and Computation",
- month = "15~" # nov,
- year = "1994",
- volume = "115",
- number = "1"
-}
-
-@inproceedings{Sem:Nipkow-MOD,
- author={Tobias Nipkow},
- title={Jinja: Towards a Comprehensive Formal Semantics for a
- {J}ava-like Language},
- booktitle={Proc.\ Marktobderdorf Summer School 2003},
- publisher={IOS Press},editor={H. Schwichtenberg and K. Spies},
- year=2003,
- note={To appear}
-}
-
-%%%%%%% Coq %%%%%%%
-
-@book{Coq:coqart,
- title = "Interactive Theorem Proving and Program Development,
- Coq'Art: The Calculus of Inductive Constructions",
- author = "Yves Bertot and Pierre Castéran",
- publisher = "Springer Verlag",
- series = "Texts in Theoretical Computer Science. An
- EATCS series",
- year = 2004
-}
-
-@phdthesis{Coq:Del01,
- AUTHOR = "David Delahaye",
- TITLE = "Conception de langages pour décrire les preuves et les
- automatisations dans les outils d'aide à la preuve",
- SCHOOL = {Universit\'e Paris~6},
- YEAR = "2001",
- Type = {Th\`ese de Doctorat}
-}
-
-@techreport{Coq:gimenez-tut,
- author = "Eduardo Gim\'enez",
- title = "A Tutorial on Recursive Types in Coq",
- number = "RT-0221",
- pages = "42 p.",
- url = "citeseer.nj.nec.com/gimenez98tutorial.html" }
-
-@phdthesis{Coq:Mun97,
- AUTHOR = "César Mu{\~{n}}oz",
- TITLE = "Un calcul de substitutions pour la repr\'esentation
- de preuves partielles en th\'eorie de types",
- SCHOOL = {Universit\'e Paris~7},
- Number = {Unit\'e de recherche INRIA-Rocquencourt, TU-0488},
- YEAR = "1997",
- Note = {English version available as INRIA research report RR-3309},
- Type = {Th\`ese de Doctorat}
-}
-
-@PHDTHESIS{Coq:Filliatre99,
- AUTHOR = {J.-C. Filli\^atre},
- TITLE = {{Preuve de programmes imp\'eratifs en th\'eorie des types}},
- TYPE = {Th{\`e}se de Doctorat},
- SCHOOL = {Universit\'e Paris-Sud},
- YEAR = 1999,
- MONTH = {July},
-}
-
-@manual{Coq:Tutorial,
- AUTHOR = {G\'erard Huet and Gilles Kahn and Christine Paulin-Mohring},
- TITLE = {{The Coq Proof Assistant A Tutorial}},
- YEAR = 2004
-}
-
-%%%%%%% PVS %%%%%%%
-
-@manual{PVS:prover,
- title = "{PVS} Prover Guide",
- author = "N. Shankar and S. Owre and J. M. Rushby and D. W. J.
- Stringer-Calvert",
- month = sep,
- year = "1999",
- organization = "Computer Science Laboratory, SRI International",
- address = "Menlo Park, CA",
-}
-
-@techreport{PVS-Semantics:TR,
- TITLE = {The Formal Semantics of {PVS}},
- AUTHOR = {Sam Owre and Natarajan Shankar},
- NUMBER = {CR-1999-209321},
- INSTITUTION = {Computer Science Laboratory, SRI International},
- ADDRESS = {Menlo Park, CA},
- MONTH = may,
- YEAR = 1999,
-}
-
-@techreport{PVS-Tactics:DiVito,
- TITLE = {A {PVS} Prover Strategy Package for Common Manipulations},
- AUTHOR = {Ben L. Di Vito},
- NUMBER = {TM-2002-211647},
- INSTITUTION = {Langley Research Center},
- ADDRESS = {Hampton, VA},
- MONTH = apr,
- YEAR = 2002,
-}
-
-@misc{PVS-Tactics:cours,
- author = "César Muñoz",
- title = "Strategies in {PVS}",
- howpublished = "Lecture notes",
- note = "National Institute of Aerospace",
- year = 2002
-}
-
-@techreport{PVS-Tactics:field,
- author = "C. Mu{\~n}oz and M. Mayero",
- title = "Real Automation in the Field",
- institution = "ICASE-NASA Langley",
- number = "NASA/CR-2001-211271 Interim ICASE Report No. 39",
- month = "dec",
- year = "2001"
-}
-
-%%%%%%% Autres Prouveurs %%%%%%%
-
-@misc{ACL2:repNuPrl,
- author = "James L. Caldwell and John Cowles",
- title = "{Representing Nuprl Proof Objects in ACL2: toward a proof checker for Nuprl}",
- url = "http://www.cs.uwyo.edu/~jlc/papers/proof_checking.ps" }
-
-@inproceedings{Elan:ckl-strat,
- author = {H. Cirstea and C. Kirchner and L. Liquori},
- title = "{Rewrite Strategies in the Rewriting Calculus}",
- booktitle = {WRLA'02},
- publisher = "{Elsevier Science B.V.}",
- series = {Electronic Notes in Theoretical Computer Science},
- volume = {71},
- year = {2003},
-}
-
-@book{LCF:GMW,
- author = {M. Gordon and R. Milner and C. Wadsworth},
- publisher = {sv},
- series = {lncs},
- volume = 78,
- title = {Edinburgh {LCF}: A Mechanized Logic of Computation},
- year = 1979
-}
-
-%%%%%%% LaTeX %%%%%%%
-
-@manual{LaTeX:symb,
- title = "The Great, Big List of \LaTeX\ Symbols",
- author = "David Carlisle and Scott Pakin and Alexander Holt",
- month = feb,
- year = 2001,
-}
-
-@manual{LaTeX:intro,
- title = "The Not So Short Introduction to \LaTeX2e",
- author = "Tobias Oetiker",
- month = jan,
- year = 1999,
-}
-
-@MANUAL{CoqManualV7,
- AUTHOR = {{The {Coq} Development Team}},
- TITLE = {{The Coq Proof Assistant Reference Manual -- Version
- V7.1}},
- YEAR = {2001},
- MONTH = OCT,
- NOTE = {http://coq.inria.fr}
-}
-
-@MANUAL{CoqManual96,
- TITLE = {The {Coq Proof Assistant Reference Manual} Version 6.1},
- AUTHOR = {B. Barras and S. Boutin and C. Cornes and J. Courant and
- J.-C. Filli\^atre and
- H. Herbelin and G. Huet and P. Manoury and C. Mu{\~{n}}oz and
- C. Murthy and C. Parent and C. Paulin-Mohring and
- A. Sa{\"\i}bi and B. Werner},
- ORGANIZATION = {{INRIA-Rocquencourt}-{CNRS-ENS Lyon}},
- URL = {ftp://ftp.inria.fr/INRIA/coq/V6.1/doc/Reference-Manual.dvi.gz},
- YEAR = 1996,
- MONTH = DEC
-}
-
-@MANUAL{CoqTutorial99,
- AUTHOR = {G.~Huet and G.~Kahn and Ch.~Paulin-Mohring},
- TITLE = {The {\sf Coq} Proof Assistant - A tutorial - Version 6.3},
- MONTH = JUL,
- YEAR = {1999},
- ABSTRACT = {http://coq.inria.fr/doc/tutorial.html}
-}
-
-@MANUAL{CoqTutorialV7,
- AUTHOR = {G.~Huet and G.~Kahn and Ch.~Paulin-Mohring},
- TITLE = {The {\sf Coq} Proof Assistant - A tutorial - Version 7.1},
- MONTH = OCT,
- YEAR = {2001},
- NOTE = {http://coq.inria.fr}
-}
-
-@TECHREPORT{modelpa2000,
- AUTHOR = {B. Bérard and P. Castéran and E. Fleury and L. Fribourg
- and J.-F. Monin and C. Paulin and A. Petit and D. Rouillard},
- TITLE = {Automates temporisés CALIFE},
- INSTITUTION = {Calife},
- YEAR = 2000,
- URL = {http://www.loria.fr/projets/calife/WebCalifePublic/FOURNITURES/F1.1.ps.gz},
- TYPE = {Fourniture {F1.1}}
-}
-
-@TECHREPORT{CaFrPaRo2000,
- AUTHOR = {P. Castéran and E. Freund and C. Paulin and D. Rouillard},
- TITLE = {Bibliothèques Coq et Isabelle-HOL pour les systèmes de transitions et les p-automates},
- INSTITUTION = {Calife},
- YEAR = 2000,
- URL = {http://www.loria.fr/projets/calife/WebCalifePublic/FOURNITURES/F5.4.ps.gz},
- TYPE = {Fourniture {F5.4}}
-}
-
-@PROCEEDINGS{TPHOLs99,
- TITLE = {International Conference on
- Theorem Proving in Higher Order Logics (TPHOLs'99)},
- YEAR = 1999,
- EDITOR = {Y. Bertot and G. Dowek and C. Paulin-Mohring and L. Th{\'e}ry},
- SERIES = {Lecture Notes in Computer Science},
- MONTH = SEP,
- PUBLISHER = {{Sprin\-ger-Verlag}},
- ADDRESS = {Nice},
- TYPE_PUBLI = {editeur}
-}
-
-@INPROCEEDINGS{Pau01,
- AUTHOR = {Christine Paulin-Mohring},
- TITLE = {Modelisation of Timed Automata in {Coq}},
- BOOKTITLE = {Theoretical Aspects of Computer Software (TACS'2001)},
- PAGES = {298--315},
- YEAR = 2001,
- EDITOR = {N. Kobayashi and B. Pierce},
- VOLUME = 2215,
- SERIES = {Lecture Notes in Computer Science},
- PUBLISHER = {Springer-Verlag}
-}
-
-@PHDTHESIS{Moh89b,
- AUTHOR = {C. Paulin-Mohring},
- MONTH = JAN,
- SCHOOL = {{Paris 7}},
- TITLE = {Extraction de programmes dans le {Calcul des Constructions}},
- TYPE = {Thèse d'université},
- YEAR = {1989},
- URL = {http://www.lri.fr/~paulin/these.ps.gz}
-}
-
-@ARTICLE{HuMo92,
- AUTHOR = {G. Huet and C. Paulin-Mohring},
- EDITION = {INRIA},
- JOURNAL = {Courrier du CNRS - Informatique},
- TITLE = {Preuves et Construction de Programmes},
- YEAR = {1992},
- CATEGORY = {national}
-}
-
-@INPROCEEDINGS{LePa94,
- AUTHOR = {F. Leclerc and C. Paulin-Mohring},
- TITLE = {Programming with Streams in {Coq}. A case study : The Sieve of Eratosthenes},
- EDITOR = {H. Barendregt and T. Nipkow},
- VOLUME = 806,
- SERIES = {Lecture Notes in Computer Science},
- BOOKTITLE = {{Types for Proofs and Programs, Types' 93}},
- YEAR = 1994,
- PUBLISHER = {Springer-Verlag}
-}
-
-@INPROCEEDINGS{Moh86,
- AUTHOR = {C. Mohring},
- ADDRESS = {Cambridge, MA},
- BOOKTITLE = {Symposium on Logic in Computer Science},
- PUBLISHER = {IEEE Computer Society Press},
- TITLE = {Algorithm Development in the {Calculus of Constructions}},
- YEAR = {1986}
-}
-
-@INPROCEEDINGS{Moh89a,
- AUTHOR = {C. Paulin-Mohring},
- ADDRESS = {Austin},
- BOOKTITLE = {Sixteenth Annual ACM Symposium on Principles of Programming Languages},
- MONTH = JAN,
- PUBLISHER = {ACM},
- TITLE = {Extracting ${F}_{\omega}$'s programs from proofs in the {Calculus of Constructions}},
- YEAR = {1989}
-}
-
-@INCOLLECTION{Moh89c,
- AUTHOR = {C. Paulin-Mohring},
- TITLE = {{R\'ealisabilit\'e et extraction de programmes}},
- BOOKTITLE = {Logique et Informatique : une introduction},
- PUBLISHER = {INRIA},
- YEAR = 1991,
- EDITOR = {B. Courcelle},
- VOLUME = 8,
- SERIES = {Collection Didactique},
- PAGES = {163-180},
- CATEGORY = {national}
-}
-
-@INPROCEEDINGS{Moh93,
- AUTHOR = {C. Paulin-Mohring},
- BOOKTITLE = {Proceedings of the conference Typed Lambda Calculi a
-nd Applications},
- EDITOR = {M. Bezem and J.-F. Groote},
- INSTITUTION = {LIP-ENS Lyon},
- NOTE = {LIP research report 92-49},
- NUMBER = 664,
- SERIES = {Lecture Notes in Computer Science},
- TITLE = {{Inductive Definitions in the System {Coq} - Rules and Properties}},
- TYPE = {research report},
- YEAR = 1993
-}
-
-@ARTICLE{PaWe92,
- AUTHOR = {C. Paulin-Mohring and B. Werner},
- JOURNAL = {Journal of Symbolic Computation},
- TITLE = {{Synthesis of ML programs in the system Coq}},
- VOLUME = {15},
- YEAR = {1993},
- PAGES = {607--640}
-}
-
-@INPROCEEDINGS{Pau96,
- AUTHOR = {C. Paulin-Mohring},
- TITLE = {Circuits as streams in {Coq} : Verification of a sequential multiplier},
- BOOKTITLE = {Types for Proofs and Programs, TYPES'95},
- EDITOR = {S. Berardi and M. Coppo},
- SERIES = {Lecture Notes in Computer Science},
- YEAR = 1996,
- VOLUME = 1158
-}
-
-@PHDTHESIS{Pau96b,
- AUTHOR = {Christine Paulin-Mohring},
- TITLE = {Définitions Inductives en Théorie des Types d'Ordre Supérieur},
- SCHOOL = {Université Claude Bernard Lyon I},
- YEAR = 1996,
- MONTH = DEC,
- TYPE = {Habilitation à diriger les recherches},
- URL = {http://www.lri.fr/~paulin/habilitation.ps.gz}
-}
-
-@INPROCEEDINGS{PfPa89,
- AUTHOR = {F. Pfenning and C. Paulin-Mohring},
- BOOKTITLE = {Proceedings of Mathematical Foundations of Programming Semantics},
- NOTE = {technical report CMU-CS-89-209},
- PUBLISHER = {Springer-Verlag},
- SERIES = {Lecture Notes in Computer Science},
- VOLUME = 442,
- TITLE = {Inductively defined types in the {Calculus of Constructions}},
- YEAR = {1990}
-}
-
-@MISC{krakatoa02,
- AUTHOR = {Claude March\'e and Christine Paulin and Xavier Urbain},
- TITLE = {The \textsc{Krakatoa} proof tool},
- YEAR = 2002,
- NOTE = {\url{http://krakatoa.lri.fr/}}
-}
-
-@ARTICLE{marche03jlap,
- AUTHOR = {Claude March{\'e} and Christine Paulin-Mohring and Xavier Urbain},
- TITLE = {The \textsc{Krakatoa} Tool for Certification of \textsc{Java/JavaCard} Programs annotated in \textsc{JML}},
- JOURNAL = {Journal of Logic and Algebraic Programming},
- YEAR = 2003,
- NOTE = {To appear},
- URL = {http://krakatoa.lri.fr},
- TOPICS = {team}
-}
-@ARTICLE{marche04jlap,
- AUTHOR = {Claude March{\'e} and Christine Paulin-Mohring and Xavier Urbain},
- TITLE = {The \textsc{Krakatoa} Tool for Certification of \textsc{Java/JavaCard} Programs annotated in \textsc{JML}},
- JOURNAL = {Journal of Logic and Algebraic Programming},
- YEAR = 2004,
- VOLUME = 58,
- NUMBER = {1--2},
- PAGES = {89--106},
- URL = {http://krakatoa.lri.fr},
- TOPICS = {team}
-}
-
-@TECHREPORT{catano03deliv,
- AUTHOR = {N{\'e}stor Cata{\~n}o and Marek Gawkowski and
-Marieke Huisman and Bart Jacobs and Claude March{\'e} and Christine Paulin
-and Erik Poll and Nicole Rauch and Xavier Urbain},
- TITLE = {Logical Techniques for Applet Verification},
- INSTITUTION = {VerifiCard Project},
- YEAR = 2003,
- TYPE = {Deliverable},
- NUMBER = {5.2},
- TOPICS = {team},
- NOTE = {Available from \url{http://www.verificard.org}}
-}
-
-@TECHREPORT{kmu2002rr,
- AUTHOR = {Keiichirou Kusakari and Claude Marché and Xavier Urbain},
- TITLE = {Termination of Associative-Commutative Rewriting using Dependency Pairs Criteria},
- INSTITUTION = {LRI},
- YEAR = 2002,
- TYPE = {Research Report},
- NUMBER = 1304,
- TYPE_PUBLI = {interne},
- TOPICS = {team},
- NOTE = {\url{http://www.lri.fr/~urbain/textes/rr1304.ps.gz}},
- URL = {http://www.lri.fr/~urbain/textes/rr1304.ps.gz}
-}
-
-@ARTICLE{marche2004jsc,
- AUTHOR = {Claude March\'e and Xavier Urbain},
- TITLE = {Modular {\&} Incremental Proofs of {AC}-Termination},
- JOURNAL = {Journal of Symbolic Computation},
- YEAR = 2004,
- TOPICS = {team}
-}
-
-@INPROCEEDINGS{contejean03wst,
- AUTHOR = {Evelyne Contejean and Claude Marché and Benjamin Monate and Xavier Urbain},
- TITLE = {{Proving Termination of Rewriting with {\sc C\textit{i}ME}}},
- CROSSREF = {wst03},
- PAGES = {71--73},
- NOTE = {\url{http://cime.lri.fr/}},
- URL = {http://cime.lri.fr/},
- YEAR = 2003,
- TYPE_PUBLI = {icolcomlec},
- TOPICS = {team}
-}
-
-@TECHREPORT{contejean04rr,
- AUTHOR = {Evelyne Contejean and Claude March{\'e} and Ana-Paula Tom{\'a}s and Xavier Urbain},
- TITLE = {Mechanically proving termination using polynomial interpretations},
- INSTITUTION = {LRI},
- YEAR = {2004},
- TYPE = {Research Report},
- NUMBER = {1382},
- TYPE_PUBLI = {interne},
- TOPICS = {team},
- URL = {http://www.lri.fr/~urbain/textes/rr1382.ps.gz}
-}
-
-@UNPUBLISHED{duran_sub,
- AUTHOR = {Francisco Duran and Salvador Lucas and
- Claude {March\'e} and {Jos\'e} Meseguer and Xavier Urbain},
- TITLE = {Termination of Membership Equational Programs},
- NOTE = {Submitted}
-}
-
-@PROCEEDINGS{comon95lncs,
- TITLE = {Term Rewriting},
- BOOKTITLE = {Term Rewriting},
- TOPICS = {team, cclserver},
- YEAR = 1995,
- EDITOR = {Hubert Comon and Jean-Pierre Jouannaud},
- SERIES = {Lecture Notes in Computer Science},
- VOLUME = {909},
- PUBLISHER = {{Sprin\-ger-Verlag}},
- ORGANIZATION = {French Spring School of Theoretical Computer
- Science},
- TYPE_PUBLI = {editeur},
- CLEF_LABO = {CJ95}
-}
-
-@PROCEEDINGS{lics94,
- TITLE = {Proceedings of the Ninth Annual IEEE Symposium on Logic
- in Computer Science},
- BOOKTITLE = {Proceedings of the Ninth Annual IEEE Symposium on Logic
- in Computer Science},
- YEAR = 1994,
- MONTH = JUL,
- ADDRESS = {Paris, France},
- ORGANIZATION = {{IEEE} Comp. Soc. Press}
-}
-
-@PROCEEDINGS{rta91,
- TITLE = {4th International Conference on Rewriting Techniques and
- Applications},
- BOOKTITLE = {4th International Conference on Rewriting Techniques and
- Applications},
- EDITOR = {Ronald. V. Book},
- YEAR = 1991,
- MONTH = APR,
- ADDRESS = {Como, Italy},
- PUBLISHER = {{Sprin\-ger-Verlag}},
- SERIES = {Lecture Notes in Computer Science},
- VOLUME = 488
-}
-
-@PROCEEDINGS{rta96,
- TITLE = {7th International Conference on Rewriting Techniques and
- Applications},
- BOOKTITLE = {7th International Conference on Rewriting Techniques and
- Applications},
- EDITOR = {Harald Ganzinger},
- PUBLISHER = {{Sprin\-ger-Verlag}},
- YEAR = 1996,
- MONTH = JUL,
- ADDRESS = {New Brunswick, NJ, USA},
- SERIES = {Lecture Notes in Computer Science},
- VOLUME = 1103
-}
-
-@PROCEEDINGS{rta97,
- TITLE = {8th International Conference on Rewriting Techniques and
- Applications},
- BOOKTITLE = {8th International Conference on Rewriting Techniques and
- Applications},
- EDITOR = {Hubert Comon},
- PUBLISHER = {{Sprin\-ger-Verlag}},
- YEAR = 1997,
- MONTH = JUN,
- ADDRESS = {Barcelona, Spain},
- SERIES = {Lecture Notes in Computer Science},
- VOLUME = {1232}
-}
-
-@PROCEEDINGS{rta98,
- TITLE = {9th International Conference on Rewriting Techniques and
- Applications},
- BOOKTITLE = {9th International Conference on Rewriting Techniques and
- Applications},
- EDITOR = {Tobias Nipkow},
- PUBLISHER = {{Sprin\-ger-Verlag}},
- YEAR = 1998,
- MONTH = APR,
- ADDRESS = {Tsukuba, Japan},
- SERIES = {Lecture Notes in Computer Science},
- VOLUME = {1379}
-}
-
-@PROCEEDINGS{rta00,
- TITLE = {11th International Conference on Rewriting Techniques and Applications},
- BOOKTITLE = {11th International Conference on Rewriting Techniques and Applications},
- EDITOR = {Leo Bachmair},
- PUBLISHER = {{Sprin\-ger-Verlag}},
- SERIES = {Lecture Notes in Computer Science},
- VOLUME = 1833,
- MONTH = JUL,
- YEAR = 2000,
- ADDRESS = {Norwich, UK}
-}
-
-@PROCEEDINGS{srt95,
- TITLE = {Proceedings of the Conference on Symbolic Rewriting
- Techniques},
- BOOKTITLE = {Proceedings of the Conference on Symbolic Rewriting
- Techniques},
- YEAR = 1995,
- EDITOR = {Manuel Bronstein and Volker Weispfenning},
- ADDRESS = {Monte Verita, Switzerland}
-}
-
-@BOOK{comon01cclbook,
- BOOKTITLE = {Constraints in Computational Logics},
- TITLE = {Constraints in Computational Logics},
- EDITOR = {Hubert Comon and Claude March{\'e} and Ralf Treinen},
- YEAR = 2001,
- PUBLISHER = {{Sprin\-ger-Verlag}},
- SERIES = {Lecture Notes in Computer Science},
- VOLUME = 2002,
- TOPICS = {team},
- TYPE_PUBLI = {editeur}
-}
-
-@PROCEEDINGS{wst03,
- BOOKTITLE = {{Extended Abstracts of the 6th International Workshop on Termination, WST'03}},
- TITLE = {{Extended Abstracts of the 6th International Workshop on Termination, WST'03}},
- YEAR = {2003},
- EDITOR = {Albert Rubio},
- MONTH = JUN,
- NOTE = {Technical Report DSIC II/15/03, Universidad Politécnica de Valencia, Spain}
-}
-
-@INPROCEEDINGS{FilliatreLetouzey03,
- AUTHOR = {J.-C. Filli\^atre and P. Letouzey},
- TITLE = {{Functors for Proofs and Programs}},
- BOOKTITLE = {Proceedings of The European Symposium on Programming},
- YEAR = 2004,
- ADDRESS = {Barcelona, Spain},
- MONTH = {March 29-April 2},
- NOTE = {To appear},
- URL = {http://www.lri.fr/~filliatr/ftp/publis/fpp.ps.gz}
-}
-
-@TECHREPORT{Filliatre03,
- AUTHOR = {J.-C. Filli\^atre},
- TITLE = {{Why: a multi-language multi-prover verification tool}},
- INSTITUTION = {{LRI, Universit\'e Paris Sud}},
- TYPE = {{Research Report}},
- NUMBER = {1366},
- MONTH = {March},
- YEAR = 2003,
- URL = {http://www.lri.fr/~filliatr/ftp/publis/why-tool.ps.gz}
-}
-
-@ARTICLE{FilliatrePottier02,
- AUTHOR = {J.-C. Filli{\^a}tre and F. Pottier},
- TITLE = {{Producing All Ideals of a Forest, Functionally}},
- JOURNAL = {Journal of Functional Programming},
- VOLUME = 13,
- NUMBER = 5,
- PAGES = {945--956},
- MONTH = {September},
- YEAR = 2003,
- URL = {http://www.lri.fr/~filliatr/ftp/publis/kr-fp.ps.gz},
- ABSTRACT = {
- We present a functional implementation of Koda and Ruskey's
- algorithm for generating all ideals of a forest poset as a Gray
- code. Using a continuation-based approach, we give an extremely
- concise formulation of the algorithm's core. Then, in a number of
- steps, we derive a first-order version whose efficiency is
- comparable to a C implementation given by Knuth.}
-}
-
-@UNPUBLISHED{FORS01,
- AUTHOR = {J.-C. Filli{\^a}tre and S. Owre and H. Rue{\ss} and N. Shankar},
- TITLE = {Deciding Propositional Combinations of Equalities and Inequalities},
- NOTE = {Unpublished},
- MONTH = OCT,
- YEAR = 2001,
- URL = {http://www.lri.fr/~filliatr/ftp/publis/ics.ps},
- ABSTRACT = {
- We address the problem of combining individual decision procedures
- into a single decision procedure. Our combination approach is based
- on using the canonizer obtained from Shostak's combination algorithm
- for equality. We illustrate our approach with a combination
- algorithm for equality, disequality, arithmetic inequality, and
- propositional logic. Unlike the Nelson--Oppen combination where the
- processing of equalities is distributed across different closed
- decision procedures, our combination involves the centralized
- processing of equalities in a single procedure. The termination
- argument for the combination is based on that for Shostak's
- algorithm. We also give soundness and completeness arguments.}
-}
-
-@INPROCEEDINGS{ICS,
- AUTHOR = {J.-C. Filli{\^a}tre and S. Owre and H. Rue{\ss} and N. Shankar},
- TITLE = {{ICS: Integrated Canonization and Solving (Tool presentation)}},
- BOOKTITLE = {Proceedings of CAV'2001},
- EDITOR = {G. Berry and H. Comon and A. Finkel},
- PUBLISHER = {Springer-Verlag},
- SERIES = {Lecture Notes in Computer Science},
- VOLUME = 2102,
- PAGES = {246--249},
- YEAR = 2001
-}
-
-@INPROCEEDINGS{Filliatre01a,
- AUTHOR = {J.-C. Filli\^atre},
- TITLE = {La supériorité de l'ordre supérieur},
- BOOKTITLE = {Journées Francophones des Langages Applicatifs},
- PAGES = {15--26},
- MONTH = {Janvier},
- YEAR = 2002,
- ADDRESS = {Anglet, France},
- URL = {http://www.lri.fr/~filliatr/ftp/publis/sos.ps.gz},
- CODE = {http://www.lri.fr/~filliatr/ftp/ocaml/misc/koda-ruskey.ps},
- ABSTRACT = {
- Nous présentons ici une écriture fonctionnelle de l'algorithme de
- Koda-Ruskey, un algorithme pour engendrer une large famille
- de codes de Gray. En s'inspirant de techniques de programmation par
- continuation, nous aboutissons à un code de neuf lignes seulement,
- bien plus élégant que les implantations purement impératives
- proposées jusqu'ici, notamment par Knuth. Dans un second temps,
- nous montrons comment notre code peut être légèrement modifié pour
- aboutir à une version de complexité optimale.
- Notre implantation en Objective Caml rivalise d'efficacité avec les
- meilleurs codes C. Nous détaillons les calculs de complexité,
- un exercice intéressant en présence d'ordre supérieur et d'effets de
- bord combinés.}
-}
-
-@TECHREPORT{Filliatre00c,
- AUTHOR = {J.-C. Filli\^atre},
- TITLE = {{Design of a proof assistant: Coq version 7}},
- INSTITUTION = {{LRI, Universit\'e Paris Sud}},
- TYPE = {{Research Report}},
- NUMBER = {1369},
- MONTH = {October},
- YEAR = 2000,
- URL = {http://www.lri.fr/~filliatr/ftp/publis/coqv7.ps.gz},
- ABSTRACT = {
- We present the design and implementation of the new version of the
- Coq proof assistant. The main novelty is the isolation of the
- critical part of the system, which consists in a type checker for
- the Calculus of Inductive Constructions. This kernel is now
- completely independent of the rest of the system and has been
- rewritten in a purely functional way. This leads to greater clarity
- and safety, without compromising efficiency. It also opens the way to
- the ``bootstrap'' of the Coq system, where the kernel will be
- certified using Coq itself.}
-}
-
-@TECHREPORT{Filliatre00b,
- AUTHOR = {J.-C. Filli\^atre},
- TITLE = {{Hash consing in an ML framework}},
- INSTITUTION = {{LRI, Universit\'e Paris Sud}},
- TYPE = {{Research Report}},
- NUMBER = {1368},
- MONTH = {September},
- YEAR = 2000,
- URL = {http://www.lri.fr/~filliatr/ftp/publis/hash-consing.ps.gz},
- ABSTRACT = {
- Hash consing is a technique to share values that are structurally
- equal. Beyond the obvious advantage of saving memory blocks, hash
- consing may also be used to gain speed in several operations (like
- equality test) and data structures (like sets or maps) when sharing is
- maximal. However, physical adresses cannot be used directly for this
- purpose when the garbage collector is likely to move blocks
- underneath. We present an easy solution in such a framework, with
- many practical benefits.}
-}
-
-@MISC{ocamlweb,
- AUTHOR = {J.-C. Filli\^atre and C. March\'e},
- TITLE = {{ocamlweb, a literate programming tool for Objective Caml}},
- NOTE = {Available at \url{http://www.lri.fr/~filliatr/ocamlweb/}},
- URL = {http://www.lri.fr/~filliatr/ocamlweb/}
-}
-
-@ARTICLE{Filliatre00a,
- AUTHOR = {J.-C. Filli\^atre},
- TITLE = {{Verification of Non-Functional Programs
- using Interpretations in Type Theory}},
- JOURNAL = {Journal of Functional Programming},
- VOLUME = 13,
- NUMBER = 4,
- PAGES = {709--745},
- MONTH = {July},
- YEAR = 2003,
- NOTE = {English translation of~\cite{Filliatre99}.},
- URL = {http://www.lri.fr/~filliatr/ftp/publis/jphd.ps.gz},
- ABSTRACT = {We study the problem of certifying programs combining imperative and
- functional features within the general framework of type theory.
-
- Type theory constitutes a powerful specification language, which is
- naturally suited for the proof of purely functional programs. To
- deal with imperative programs, we propose a logical interpretation
- of an annotated program as a partial proof of its specification. The
- construction of the corresponding partial proof term is based on a
- static analysis of the effects of the program, and on the use of
- monads. The usual notion of monads is refined in order to account
- for the notion of effect. The missing subterms in the partial proof
- term are seen as proof obligations, whose actual proofs are left to
- the user. We show that the validity of those proof obligations
- implies the total correctness of the program.
- We also establish a result of partial completeness.
-
- This work has been implemented in the Coq proof assistant.
- It appears as a tactic taking an annotated program as argument and
- generating a set of proof obligations. Several nontrivial
- algorithms have been certified using this tactic.}
-}
-
-@ARTICLE{Filliatre99c,
- AUTHOR = {J.-C. Filli\^atre},
- TITLE = {{Formal Proof of a Program: Find}},
- JOURNAL = {Science of Computer Programming},
- YEAR = 2001,
- NOTE = {To appear},
- URL = {http://www.lri.fr/~filliatr/ftp/publis/find.ps.gz},
- ABSTRACT = {In 1971, C.~A.~R.~Hoare gave the proof of correctness and termination of a
- rather complex algorithm, in a paper entitled \emph{Proof of a
- program: Find}. It is a hand-made proof, where the
- program is given together with its formal specification and where
- each step is fully
- justified by a mathematical reasoning. We present here a formal
- proof of the same program in the system Coq, using the
- recent tactic of the system developed to establishing the total
- correctness of
- imperative programs. We follow Hoare's paper as close as
- possible, keeping the same program and the same specification. We
- show that we get exactly the same proof obligations, which are
- proved in a straightforward way, following the original paper.
- We also explain how more informal reasonings of Hoare's proof are
- formalized in the system Coq.
- This demonstrates the adequacy of the system Coq in the
- process of certifying imperative programs.}
-}
-
-@TECHREPORT{Filliatre99b,
- AUTHOR = {J.-C. Filli\^atre},
- TITLE = {{A theory of monads parameterized by effects}},
- INSTITUTION = {{LRI, Universit\'e Paris Sud}},
- TYPE = {{Research Report}},
- NUMBER = {1367},
- MONTH = {November},
- YEAR = 1999,
- URL = {http://www.lri.fr/~filliatr/ftp/publis/monads.ps.gz},
- ABSTRACT = {Monads were introduced in computer science to express the semantics
- of programs with computational effects, while type and effect
- inference was introduced to mark out those effects.
- In this article, we propose a combination of the notions of effects
- and monads, where the monadic operators are parameterized by effects.
- We establish some relationships between those generalized monads and
- the classical ones.
- Then we use a generalized monad to translate imperative programs
- into purely functional ones. We establish the correctness of that
- translation. This work has been put into practice in the Coq proof
- assistant to establish the correctness of imperative programs.}
-}
-
-@PHDTHESIS{Filliatre99,
- AUTHOR = {J.-C. Filli\^atre},
- TITLE = {{Preuve de programmes imp\'eratifs en th\'eorie des types}},
- TYPE = {Th{\`e}se de Doctorat},
- SCHOOL = {Universit\'e Paris-Sud},
- YEAR = 1999,
- MONTH = {July},
- URL = {http://www.lri.fr/~filliatr/ftp/publis/these.ps.gz},
- ABSTRACT = {Nous étudions le problème de la certification de programmes mêlant
- traits impératifs et fonctionnels dans le cadre de la théorie des
- types.
-
- La théorie des types constitue un puissant langage de spécification,
- naturellement adapté à la preuve de programmes purement
- fonctionnels. Pour y certifier également des programmes impératifs,
- nous commençons par exprimer leur sémantique de manière purement
- fonctionnelle. Cette traduction repose sur une analyse statique des
- effets de bord des programmes, et sur l'utilisation de la notion de
- monade, notion que nous raffinons en l'associant à la notion d'effet
- de manière générale. Nous montrons que cette traduction est
- sémantiquement correcte.
-
- Puis, à partir d'un programme annoté, nous construisons une preuve
- de sa spécification, traduite de manière fonctionnelle. Cette preuve
- est bâtie sur la traduction fonctionnelle précédemment
- introduite. Elle est presque toujours incomplète, les parties
- manquantes étant autant d'obligations de preuve qui seront laissées
- à la charge de l'utilisateur. Nous montrons que la validité de ces
- obligations entraîne la correction totale du programme.
-
- Nous avons implanté notre travail dans l'assistant de preuve
- Coq, avec lequel il est dès à présent distribué. Cette
- implantation se présente sous la forme d'une tactique prenant en
- argument un programme annoté et engendrant les obligations de
- preuve. Plusieurs algorithmes non triviaux ont été certifiés à
- l'aide de cet outil (Find, Quicksort, Heapsort, algorithme de
- Knuth-Morris-Pratt).}
-}
-
-@INPROCEEDINGS{FilliatreMagaud99,
- AUTHOR = {J.-C. Filli\^atre and N. Magaud},
- TITLE = {{Certification of sorting algorithms in the system Coq}},
- BOOKTITLE = {Theorem Proving in Higher Order Logics:
- Emerging Trends},
- YEAR = 1999,
- ABSTRACT = {We present the formal proofs of total correctness of three sorting
- algorithms in the system Coq, namely \textit{insertion sort},
- \textit{quicksort} and \textit{heapsort}. The implementations are
- imperative programs working in-place on a given array. Those
- developments demonstrate the usefulness of inductive types and higher-order
- logic in the process of software certification. They also
- show that the proof of rather complex algorithms may be done in a
- small amount of time --- only a few days for each development ---
- and without great difficulty.},
- URL = {http://www.lri.fr/~filliatr/ftp/publis/Filliatre-Magaud.ps.gz}
-}
-
-@INPROCEEDINGS{Filliatre98,
- AUTHOR = {J.-C. Filli\^atre},
- TITLE = {{Proof of Imperative Programs in Type Theory}},
- BOOKTITLE = {International Workshop, TYPES '98, Kloster Irsee, Germany},
- PUBLISHER = {Springer-Verlag},
- VOLUME = 1657,
- SERIES = {Lecture Notes in Computer Science},
- MONTH = MAR,
- YEAR = {1998},
- ABSTRACT = {We present a new approach to certifying imperative programs,
- in the context of Type Theory.
- The key is a functional translation of imperative programs, which is
- made possible by an analysis of their effects.
- On sequential imperative programs, we get the same proof
- obligations as those given by Floyd-Hoare logic,
- but our approach also includes functional constructions.
- As a side-effect, we propose a way to eradicate the use of auxiliary
- variables in specifications.
- This work has been implemented in the Coq Proof Assistant and applied
- on non-trivial examples.},
- URL = {http://www.lri.fr/~filliatr/ftp/publis/types98.ps.gz}
-}
-
-@TECHREPORT{Filliatre97,
- AUTHOR = {J.-C. Filli\^atre},
- INSTITUTION = {LIP - ENS Lyon},
- NUMBER = {97--04},
- TITLE = {{Finite Automata Theory in Coq:
- A constructive proof of Kleene's theorem}},
- TYPE = {Research Report},
- MONTH = {February},
- YEAR = {1997},
- ABSTRACT = {We describe here a development in the system Coq
- of a piece of Finite Automata Theory. The main result is the Kleene's
- theorem, expressing that regular expressions and finite automata
- define the same languages. From a constructive proof of this result,
- we automatically obtain a functional program that compiles any
- regular expression into a finite automata, which constitutes the main
- part of the implementation of {\tt grep}-like programs. This
- functional program is obtained by the automatic method of {\em
- extraction} which removes the logical parts of the proof to keep only
- its informative contents. Starting with an idea of what we would
- have written in ML, we write the specification and do the proofs in
- such a way that we obtain the expected program, which is therefore
- efficient.},
- URL = {ftp://ftp.ens-lyon.fr/pub/LIP/Rapports/RR/RR97/RR97-04.ps.Z}
-}
-
-@TECHREPORT{Filliatre95,
- AUTHOR = {J.-C. Filli\^atre},
- INSTITUTION = {LIP - ENS Lyon},
- NUMBER = {96--25},
- TITLE = {{A decision procedure for Direct Predicate
- Calculus: study and implementation in
- the Coq system}},
- TYPE = {Research Report},
- MONTH = {February},
- YEAR = {1995},
- ABSTRACT = {The paper of J. Ketonen and R. Weyhrauch \emph{A
- decidable fragment of Predicate Calculus} defines a decidable
- fragment of first-order predicate logic - Direct Predicate Calculus
- - as the subset which is provable in Gentzen sequent calculus
- without the contraction rule, and gives an effective decision
- procedure for it. This report is a detailed study of this
- procedure. We extend the decidability to non-prenex formulas. We
- prove that the intuitionnistic fragment is still decidable, with a
- refinement of the same procedure. An intuitionnistic version has
- been implemented in the Coq system using a translation into
- natural deduction.},
- URL = {ftp://ftp.ens-lyon.fr/pub/LIP/Rapports/RR/RR96/RR96-25.ps.Z}
-}
-
-@TECHREPORT{Filliatre94,
- AUTHOR = {J.-C. Filli\^atre},
- MONTH = {Juillet},
- INSTITUTION = {Ecole Normale Sup\'erieure},
- TITLE = {{Une proc\'edure de d\'ecision pour le Calcul des Pr\'edicats Direct~: \'etude et impl\'ementation dans le syst\`eme Coq}},
- TYPE = {Rapport de {DEA}},
- YEAR = {1994},
- URL = {ftp://ftp.lri.fr/LRI/articles/filliatr/memoire.dvi.gz}
-}
-
-@TECHREPORT{CourantFilliatre93,
- AUTHOR = {J. Courant et J.-C. Filli\^atre},
- MONTH = {Septembre},
- INSTITUTION = {Ecole Normale Sup\'erieure},
- TITLE = {{Formalisation de la th\'eorie des langages
- formels en Coq}},
- TYPE = {Rapport de ma\^{\i}trise},
- YEAR = {1993},
- URL = {http://www.ens-lyon.fr/~jcourant/stage_maitrise.dvi.gz},
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-}
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- editor = "J. Harrison and M. Aagaard",
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-@TechReport{harrison,
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- title = {Meta theory and Reflection in Theorem Proving:a Survey and Cri
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- year = {1988},
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- AUTHOR = {S. Boutin},
- MONTH = sep,
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- TITLE = {Certification d'un compilateur {ML en Coq}},
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- year = {1997}
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- VOLUME = {34},
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- YEAR = {1980}
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- TITLE = {{The Coq Proof Assistant User's Guide Version 5.8}},
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- AUTHOR = {C. Parent},
- INSTITUTION = {Ecole {Normale} {Sup\'erieure} de {Lyon}},
- MONTH = oct,
- NOTE = {Also in~\cite{Nijmegen93}},
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- TITLE = {Developing certified programs in the system {Coq}- {The} {Program} tactic},
- YEAR = {1993}
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- TITLE = {{Synth\`ese de preuves de programmes dans le Calcul des Constructions Inductives}},
- YEAR = {1995}
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- TITLE = {{The Calculus of Constructions. Documentation and user's guide, Version 4.10}},
- YEAR = {1989}
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- TITLE = {{Constructions : A Higher Order Proof System for Mechanizing Mathematics}},
- VOLUME = {203},
- YEAR = {1985}
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- YEAR = {1987}
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- AUTHOR = {Thierry Coquand and Gérard Huet},
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- NUMBER = {2/3},
- TITLE = {The {Calculus of Constructions}},
- VOLUME = {76},
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- AUTHOR = {Thierry Coquand and Christine Paulin-Mohring},
- BOOKTITLE = {Proceedings of Colog'88},
- EDITOR = {P. Martin-L\"of and G. Mints},
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- TITLE = {Inductively defined types},
- VOLUME = {417},
- YEAR = {1990}
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- TITLE = {{Implementing Mathematics with the Nuprl Proof Development System}},
- YEAR = {1986}
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- AUTHOR = {Thierry Coquand},
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- TITLE = {Une Th\'eorie des Constructions},
- YEAR = {1985}
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- AUTHOR = {Thierry Coquand},
- ADDRESS = {Cambridge, MA},
- BOOKTITLE = {Symposium on Logic in Computer Science},
- PUBLISHER = {IEEE Computer Society Press},
- TITLE = {{An Analysis of Girard's Paradox}},
- YEAR = {1986}
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- AUTHOR = {Thierry Coquand},
- BOOKTITLE = {Logic and Computer Science},
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- NOTE = {INRIA Research Report 1088, also in~\cite{CoC89}},
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- TITLE = {{Metamathematical Investigations of a Calculus of Constructions}},
- YEAR = {1990}
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- AUTHOR = {Thierry Coquand},
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- TITLE = {{A New Paradox in Type Theory}},
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- YEAR = {1991}
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- AUTHOR = {Thierry Coquand},
- TITLE = {{Pattern Matching with Dependent Types}},
- YEAR = {1992},
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- "{\sf TYPES99-SIsos.ps.gz}"
-}
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- url =
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- "{\sf LPAR2000-ltac.ps.gz}"
-}
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- YEAR = {1990}
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- AUTHOR = {G. Dowek},
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- TITLE = {L'Ind\'ecidabilit\'e du Filtrage du Troisi\`eme Ordre dans les Calculs avec Types D\'ependants ou Constructeurs de Types},
- VOLUME = {I, 312},
- YEAR = {1991}
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- AUTHOR = {G. Dowek},
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- PUBLISHER = SV,
- SERIES = LNCS,
- TITLE = {A Second Order Pattern Matching Algorithm in the Cube of Typed $\lambda$-calculi},
- VOLUME = {520},
- YEAR = {1991}
-}
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- AUTHOR = {G. Dowek},
- MONTH = dec,
- SCHOOL = {Universit\'e Paris 7},
- TITLE = {D\'emonstration automatique dans le Calcul des Constructions},
- YEAR = {1991}
-}
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- AUTHOR = {G. Dowek},
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- volume = 107,
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-}
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- AUTHOR = {G. Dowek},
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- VOLUME = {69},
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- YEAR = {1994}
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- AUTHOR = {P. Dybjer},
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- TITLE = {Inductive sets and families in {Martin-L{\"o}f's}
- Type Theory and their set-theoretic semantics: An inversion principle for {Martin-L\"of's} type theory},
- VOLUME = {14},
- YEAR = {1991}
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- AUTHOR = {J.-C. Filli\^atre},
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- YEAR = {1994}
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- AUTHOR = {J.-C. Filli\^atre},
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- TYPE = {Research report},
- NUMBER = {96--25},
- YEAR = {1995}
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- title = {Verification of Non-Functional Programs
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- pages = {709--745},
- month = jul,
- year = 2003,
- note = {[English translation of \cite{Filliatre99}]},
- url = {http://www.lri.fr/~filliatr/ftp/publis/jphd.ps.gz},
- topics = "team, lri",
- type_publi = "irevcomlec"
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- author = {J.-C. Filli\^atre},
- title = {Preuve de programmes imp\'eratifs en th\'eorie des types},
- type = {Th{\`e}se de Doctorat},
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- year = 1999,
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- author = {J.-C. Filli\^atre},
- title = {{Formal Proof of a Program: Find}},
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-}
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- Emerging Trends},
- year = 1999,
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- AUTHOR = {E. Fleury},
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- YEAR = {1990}
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- YEAR = {1994}
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- NUMBER = {1158},
- PAGES = {135-152},
- TITLE = {An application of co-Inductive types in Coq:
- verification of the Alternating Bit Protocol},
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- YEAR = {1995}
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- AUTHOR = {Jean-Yves Girard},
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- TITLE = {Une extension de l'interpr\'etation de {G\"odel} \`a l'analyse, et son application \`a l'\'elimination des coupures dans l'analyse et la th\'eorie des types},
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- AUTHOR = {Jean-Yves Girard},
- SCHOOL = {Universit\'e Paris~7},
- TITLE = {Interpr\'etation fonctionnelle et \'elimination des coupures de l'arithm\'etique d'ordre sup\'erieur},
- YEAR = {1972}
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- AUTHOR = {Jean-Yves Girard and Yves Lafont and Paul Taylor},
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- TITLE = {Proofs and Types},
- YEAR = {1989}
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- MONTH = sep,
- SCHOOL = {DEA IARFA, Ecole des Ponts et Chauss\'ees, Paris},
- TITLE = {{\'E}criture d'une tactique arithm\'etique pour le syst\`eme {\Coq}},
- YEAR = {1994}
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- AUTHOR = {Martin Hofmann and Thomas Streicher},
- TITLE = {The groupoid interpretation of type theory},
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- AUTHOR = {W.A. Howard},
- BOOKTITLE = {to H.B. Curry : Essays on Combinatory Logic, Lambda Calculus and Formalism.},
- EDITOR = {J.P. Seldin and J.R. Hindley},
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- YEAR = {1980}
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- volume = 249,
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- year = 1987,
- publisher = SV
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- AUTHOR = {G. Huet},
- BOOKTITLE = {Programming of Future Generation Computers},
- EDITOR = {K. Fuchi and M. Nivat},
- NOTE = {Also in \cite{Hue87tapsoft}},
- PUBLISHER = {Elsevier Science},
- TITLE = {Induction Principles Formalized in the {Calculus of Constructions}},
- YEAR = {1988}
-}
-
-
-
-@INPROCEEDINGS{Hue88,
- AUTHOR = {G. Huet},
- BOOKTITLE = {A perspective in Theoretical Computer Science. Commemorative Volume for Gift Siromoney},
- EDITOR = {R. Narasimhan},
- NOTE = {Also in~\cite{CoC89}},
- PUBLISHER = {World Scientific Publishing},
- TITLE = {{The Constructive Engine}},
- YEAR = {1989}
-}
-
-@BOOK{Hue89,
- EDITOR = {G. Huet},
- PUBLISHER = {Addison-Wesley},
- SERIES = {The UT Year of Programming Series},
- TITLE = {Logical Foundations of Functional Programming},
- YEAR = {1989}
-}
-
-@INPROCEEDINGS{Hue92,
- AUTHOR = {G. Huet},
- BOOKTITLE = {Proceedings of 12th FST/TCS Conference, New Delhi},
- PAGES = {229--240},
- PUBLISHER = SV,
- SERIES = LNCS,
- TITLE = {The Gallina Specification Language : A case study},
- VOLUME = {652},
- YEAR = {1992}
-}
-
-@ARTICLE{Hue94,
- AUTHOR = {G. Huet},
- JOURNAL = {J. Functional Programming},
- PAGES = {371--394},
- PUBLISHER = {Cambridge University Press},
- TITLE = {Residual theory in $\lambda$-calculus: a formal development},
- VOLUME = {4,3},
- YEAR = {1994}
-}
-
-@INCOLLECTION{HuetLevy79,
- AUTHOR = {G. Huet and J.-J. L\'{e}vy},
- TITLE = {Call by Need Computations in Non-Ambigous
-Linear Term Rewriting Systems},
- NOTE = {Also research report 359, INRIA, 1979},
- BOOKTITLE = {Computational Logic, Essays in Honor of
-Alan Robinson},
- EDITOR = {J.-L. Lassez and G. Plotkin},
- PUBLISHER = {The MIT press},
- YEAR = {1991}
-}
-
-@ARTICLE{KeWe84,
- AUTHOR = {J. Ketonen and R. Weyhrauch},
- JOURNAL = {Theoretical Computer Science},
- PAGES = {297--307},
- TITLE = {A decidable fragment of {P}redicate {C}alculus},
- VOLUME = {32},
- YEAR = {1984}
-}
-
-@BOOK{Kle52,
- AUTHOR = {S.C. Kleene},
- PUBLISHER = {North-Holland},
- SERIES = {Bibliotheca Mathematica},
- TITLE = {Introduction to Metamathematics},
- YEAR = {1952}
-}
-
-@BOOK{Kri90,
- AUTHOR = {J.-L. Krivine},
- PUBLISHER = {Masson},
- SERIES = {Etudes et recherche en informatique},
- TITLE = {Lambda-calcul {types et mod\`eles}},
- YEAR = {1990}
-}
-
-@BOOK{LE92,
- EDITOR = {G. Huet and G. Plotkin},
- PUBLISHER = {Cambridge University Press},
- TITLE = {Logical Environments},
- YEAR = {1992}
-}
-
-@BOOK{LF91,
- EDITOR = {G. Huet and G. Plotkin},
- PUBLISHER = {Cambridge University Press},
- TITLE = {Logical Frameworks},
- YEAR = {1991}
-}
-
-@ARTICLE{Laville91,
- AUTHOR = {A. Laville},
- TITLE = {Comparison of Priority Rules in Pattern
-Matching and Term Rewriting},
- JOURNAL = {Journal of Symbolic Computation},
- VOLUME = {11},
- PAGES = {321--347},
- YEAR = {1991}
-}
-
-@INPROCEEDINGS{LePa94,
- AUTHOR = {F. Leclerc and C. Paulin-Mohring},
- BOOKTITLE = {{Types for Proofs and Programs, Types' 93}},
- EDITOR = {H. Barendregt and T. Nipkow},
- PUBLISHER = SV,
- SERIES = {LNCS},
- TITLE = {{Programming with Streams in Coq. A case study : The Sieve of Eratosthenes}},
- VOLUME = {806},
- YEAR = {1994}
-}
-
-@TECHREPORT{Leroy90,
- AUTHOR = {X. Leroy},
- TITLE = {The {ZINC} experiment: an economical implementation
-of the {ML} language},
- INSTITUTION = {INRIA},
- NUMBER = {117},
- YEAR = {1990}
-}
-
-@INPROCEEDINGS{Let02,
- author = {P. Letouzey},
- title = {A New Extraction for Coq},
- booktitle = {Proceedings of the TYPES'2002 workshop},
- year = 2002,
- note = {to appear},
- url = {draft at \url{http://www.lri.fr/~letouzey/download/extraction2002.ps.gz}}
-}
-
-@BOOK{MaL84,
- AUTHOR = {{P. Martin-L\"of}},
- PUBLISHER = {Bibliopolis},
- SERIES = {Studies in Proof Theory},
- TITLE = {Intuitionistic Type Theory},
- YEAR = {1984}
-}
-
-@ARTICLE{MaSi94,
- AUTHOR = {P. Manoury and M. Simonot},
- JOURNAL = {TCS},
- TITLE = {Automatizing termination proof of recursively defined function},
- YEAR = {To appear}
-}
-
-@INPROCEEDINGS{Moh89a,
- AUTHOR = {Christine Paulin-Mohring},
- ADDRESS = {Austin},
- BOOKTITLE = {Sixteenth Annual ACM Symposium on Principles of Programming Languages},
- MONTH = jan,
- PUBLISHER = {ACM},
- TITLE = {Extracting ${F}_{\omega}$'s programs from proofs in the {Calculus of Constructions}},
- YEAR = {1989}
-}
-
-@PHDTHESIS{Moh89b,
- AUTHOR = {Christine Paulin-Mohring},
- MONTH = jan,
- SCHOOL = {{Universit\'e Paris 7}},
- TITLE = {Extraction de programmes dans le {Calcul des Constructions}},
- YEAR = {1989}
-}
-
-@INPROCEEDINGS{Moh93,
- AUTHOR = {Christine Paulin-Mohring},
- BOOKTITLE = {Proceedings of the conference Typed Lambda Calculi and Applications},
- EDITOR = {M. Bezem and J.-F. Groote},
- NOTE = {Also LIP research report 92-49, ENS Lyon},
- NUMBER = {664},
- PUBLISHER = SV,
- SERIES = {LNCS},
- TITLE = {{Inductive Definitions in the System Coq - Rules and Properties}},
- YEAR = {1993}
-}
-
-@BOOK{Moh97,
- AUTHOR = {Christine Paulin-Mohring},
- MONTH = jan,
- PUBLISHER = {{ENS Lyon}},
- TITLE = {{Le syst\`eme Coq. \mbox{Th\`ese d'habilitation}}},
- YEAR = {1997}
-}
-
-@MASTERSTHESIS{Mun94,
- AUTHOR = {C. Mu{\~n}oz},
- MONTH = sep,
- SCHOOL = {DEA d'Informatique Fondamentale, Universit\'e Paris 7},
- TITLE = {D\'emonstration automatique dans la logique propositionnelle intuitionniste},
- YEAR = {1994}
-}
-
-@PHDTHESIS{Mun97d,
- AUTHOR = "C. Mu{\~{n}}oz",
- TITLE = "Un calcul de substitutions pour la repr\'esentation
- de preuves partielles en th\'eorie de types",
- SCHOOL = {Universit\'e Paris 7},
- YEAR = "1997",
- Note = {Version en anglais disponible comme rapport de
- recherche INRIA RR-3309},
- Type = {Th\`ese de Doctorat}
-}
-
-@BOOK{NoPS90,
- AUTHOR = {B. {Nordstr\"om} and K. Peterson and J. Smith},
- BOOKTITLE = {Information Processing 83},
- PUBLISHER = {Oxford Science Publications},
- SERIES = {International Series of Monographs on Computer Science},
- TITLE = {Programming in {Martin-L\"of's} Type Theory},
- YEAR = {1990}
-}
-
-@ARTICLE{Nor88,
- AUTHOR = {B. {Nordstr\"om}},
- JOURNAL = {BIT},
- TITLE = {Terminating General Recursion},
- VOLUME = {28},
- YEAR = {1988}
-}
-
-@BOOK{Odi90,
- EDITOR = {P. Odifreddi},
- PUBLISHER = {Academic Press},
- TITLE = {Logic and Computer Science},
- YEAR = {1990}
-}
-
-@INPROCEEDINGS{PaMS92,
- AUTHOR = {M. Parigot and P. Manoury and M. Simonot},
- ADDRESS = {St. Petersburg, Russia},
- BOOKTITLE = {Logic Programming and automated reasoning},
- EDITOR = {A. Voronkov},
- MONTH = jul,
- NUMBER = {624},
- PUBLISHER = SV,
- SERIES = {LNCS},
- TITLE = {{ProPre : A Programming language with proofs}},
- YEAR = {1992}
-}
-
-@ARTICLE{PaWe92,
- AUTHOR = {Christine Paulin-Mohring and Benjamin Werner},
- JOURNAL = {Journal of Symbolic Computation},
- PAGES = {607--640},
- TITLE = {{Synthesis of ML programs in the system Coq}},
- VOLUME = {15},
- YEAR = {1993}
-}
-
-@ARTICLE{Par92,
- AUTHOR = {M. Parigot},
- JOURNAL = {Theoretical Computer Science},
- NUMBER = {2},
- PAGES = {335--356},
- TITLE = {{Recursive Programming with Proofs}},
- VOLUME = {94},
- YEAR = {1992}
-}
-
-@INPROCEEDINGS{Parent95b,
- AUTHOR = {C. Parent},
- BOOKTITLE = {{Mathematics of Program Construction'95}},
- PUBLISHER = SV,
- SERIES = {LNCS},
- TITLE = {{Synthesizing proofs from programs in
-the Calculus of Inductive Constructions}},
- VOLUME = {947},
- YEAR = {1995}
-}
-
-@INPROCEEDINGS{Prasad93,
- AUTHOR = {K.V. Prasad},
- BOOKTITLE = {{Proceedings of CONCUR'93}},
- PUBLISHER = SV,
- SERIES = {LNCS},
- TITLE = {{Programming with broadcasts}},
- VOLUME = {715},
- YEAR = {1993}
-}
-
-@BOOK{RC95,
- author = "di~Cosmo, R.",
- title = "Isomorphisms of Types: from $\lambda$-calculus to information
- retrieval and language design",
- series = "Progress in Theoretical Computer Science",
- publisher = "Birkhauser",
- year = "1995",
- note = "ISBN-0-8176-3763-X"
-}
-
-@TECHREPORT{Rou92,
- AUTHOR = {J. Rouyer},
- INSTITUTION = {INRIA},
- MONTH = nov,
- NUMBER = {1795},
- TITLE = {{D{\'e}veloppement de l'Algorithme d'Unification dans le Calcul des Constructions}},
- YEAR = {1992}
-}
-
-@TECHREPORT{Saibi94,
- AUTHOR = {A. Sa\"{\i}bi},
- INSTITUTION = {INRIA},
- MONTH = dec,
- NUMBER = {2345},
- TITLE = {{Axiomatization of a lambda-calculus with explicit-substitutions in the Coq System}},
- YEAR = {1994}
-}
-
-
-@MASTERSTHESIS{Ter92,
- AUTHOR = {D. Terrasse},
- MONTH = sep,
- SCHOOL = {IARFA},
- TITLE = {{Traduction de TYPOL en COQ. Application \`a Mini ML}},
- YEAR = {1992}
-}
-
-@TECHREPORT{ThBeKa92,
- AUTHOR = {L. Th\'ery and Y. Bertot and G. Kahn},
- INSTITUTION = {INRIA Sophia},
- MONTH = may,
- NUMBER = {1684},
- TITLE = {Real theorem provers deserve real user-interfaces},
- TYPE = {Research Report},
- YEAR = {1992}
-}
-
-@BOOK{TrDa89,
- AUTHOR = {A.S. Troelstra and D. van Dalen},
- PUBLISHER = {North-Holland},
- SERIES = {Studies in Logic and the foundations of Mathematics, volumes 121 and 123},
- TITLE = {Constructivism in Mathematics, an introduction},
- YEAR = {1988}
-}
-
-@PHDTHESIS{Wer94,
- AUTHOR = {B. Werner},
- SCHOOL = {Universit\'e Paris 7},
- TITLE = {Une th\'eorie des constructions inductives},
- TYPE = {Th\`ese de Doctorat},
- YEAR = {1994}
-}
-
-@PHDTHESIS{Bar99,
- AUTHOR = {B. Barras},
- SCHOOL = {Universit\'e Paris 7},
- TITLE = {Auto-validation d'un système de preuves avec familles inductives},
- TYPE = {Th\`ese de Doctorat},
- YEAR = {1999}
-}
-
-@UNPUBLISHED{ddr98,
- AUTHOR = {D. de Rauglaudre},
- TITLE = {Camlp4 version 1.07.2},
- YEAR = {1998},
- NOTE = {In Camlp4 distribution}
-}
-
-@ARTICLE{dowek93,
- AUTHOR = {G. Dowek},
- TITLE = {{A Complete Proof Synthesis Method for the Cube of Type Systems}},
- JOURNAL = {Journal Logic Computation},
- VOLUME = {3},
- NUMBER = {3},
- PAGES = {287--315},
- MONTH = {June},
- YEAR = {1993}
-}
-
-@INPROCEEDINGS{manoury94,
- AUTHOR = {P. Manoury},
- TITLE = {{A User's Friendly Syntax to Define
-Recursive Functions as Typed $\lambda-$Terms}},
- BOOKTITLE = {{Types for Proofs and Programs, TYPES'94}},
- SERIES = {LNCS},
- VOLUME = {996},
- MONTH = jun,
- YEAR = {1994}
-}
-
-@TECHREPORT{maranget94,
- AUTHOR = {L. Maranget},
- INSTITUTION = {INRIA},
- NUMBER = {2385},
- TITLE = {{Two Techniques for Compiling Lazy Pattern Matching}},
- YEAR = {1994}
-}
-
-@INPROCEEDINGS{puel-suarez90,
- AUTHOR = {L.Puel and A. Su\'arez},
- BOOKTITLE = {{Conference Lisp and Functional Programming}},
- SERIES = {ACM},
- PUBLISHER = SV,
- TITLE = {{Compiling Pattern Matching by Term
-Decomposition}},
- YEAR = {1990}
-}
-
-@MASTERSTHESIS{saidi94,
- AUTHOR = {H. Saidi},
- MONTH = sep,
- SCHOOL = {DEA d'Informatique Fondamentale, Universit\'e Paris 7},
- TITLE = {R\'esolution d'\'equations dans le syst\`eme T
- de G\"odel},
- YEAR = {1994}
-}
-
-@misc{streicher93semantical,
- author = "T. Streicher",
- title = "Semantical Investigations into Intensional Type Theory",
- note = "Habilitationsschrift, LMU Munchen.",
- year = "1993" }
-
-
-
-@Misc{Pcoq,
- author = {Lemme Team},
- title = {Pcoq a graphical user-interface for {Coq}},
- note = {\url{http://www-sop.inria.fr/lemme/pcoq/}}
-}
-
-
-@Misc{ProofGeneral,
- author = {David Aspinall},
- title = {Proof General},
- note = {\url{https://proofgeneral.github.io/}}
-}
-
-
-
-@Book{CoqArt,
- author = {Yves bertot and Pierre Castéran},
- title = {Coq'Art},
- publisher = {Springer-Verlag},
- year = 2004,
- note = {To appear}
-}
-
-@INCOLLECTION{wadler87,
- AUTHOR = {P. Wadler},
- TITLE = {Efficient Compilation of Pattern Matching},
- BOOKTITLE = {The Implementation of Functional Programming
-Languages},
- EDITOR = {S.L. Peyton Jones},
- PUBLISHER = {Prentice-Hall},
- YEAR = {1987}
-}
-
-
-@COMMENT{cross-references, must be at end}
-
-@BOOK{Bastad92,
- EDITOR = {B. Nordstr\"om and K. Petersson and G. Plotkin},
- PUBLISHER = {Available by ftp at site ftp.inria.fr},
- TITLE = {Proceedings of the 1992 Workshop on Types for Proofs and Programs},
- YEAR = {1992}
-}
-
-@BOOK{Nijmegen93,
- EDITOR = {H. Barendregt and T. Nipkow},
- PUBLISHER = SV,
- SERIES = LNCS,
- TITLE = {Types for Proofs and Programs},
- VOLUME = {806},
- YEAR = {1994}
-}
-
-@PHDTHESIS{Luo90,
- AUTHOR = {Z. Luo},
- TITLE = {An Extended Calculus of Constructions},
- SCHOOL = {University of Edinburgh},
- YEAR = {1990}
-}
diff --git a/doc/faq/hevea.sty b/doc/faq/hevea.sty
deleted file mode 100644
index 6d49aa8ce..000000000
--- a/doc/faq/hevea.sty
+++ /dev/null
@@ -1,78 +0,0 @@
-% hevea : hevea.sty
-% This is a very basic style file for latex document to be processed
-% with hevea. It contains definitions of LaTeX environment which are
-% processed in a special way by the translator.
-% Mostly :
-% - latexonly, not processed by hevea, processed by latex.
-% - htmlonly , the reverse.
-% - rawhtml, to include raw HTML in hevea output.
-% - toimage, to send text to the image file.
-% The package also provides hevea logos, html related commands (ahref
-% etc.), void cutting and image commands.
-\NeedsTeXFormat{LaTeX2e}
-\ProvidesPackage{hevea}[2002/01/11]
-\RequirePackage{comment}
-\newif\ifhevea\heveafalse
-\@ifundefined{ifimagen}{\newif\ifimagen\imagenfalse}
-\makeatletter%
-\newcommand{\heveasmup}[2]{%
-\raise #1\hbox{$\m@th$%
- \csname S@\f@size\endcsname
- \fontsize\sf@size 0%
- \math@fontsfalse\selectfont
-#2%
-}}%
-\DeclareRobustCommand{\hevea}{H\kern-.15em\heveasmup{.2ex}{E}\kern-.15emV\kern-.15em\heveasmup{.2ex}{E}\kern-.15emA}%
-\DeclareRobustCommand{\hacha}{H\kern-.15em\heveasmup{.2ex}{A}\kern-.15emC\kern-.1em\heveasmup{.2ex}{H}\kern-.15emA}%
-\DeclareRobustCommand{\html}{\protect\heveasmup{0.ex}{HTML}}
-%%%%%%%%% Hyperlinks hevea style
-\newcommand{\ahref}[2]{{#2}}
-\newcommand{\ahrefloc}[2]{{#2}}
-\newcommand{\aname}[2]{{#2}}
-\newcommand{\ahrefurl}[1]{\texttt{#1}}
-\newcommand{\footahref}[2]{#2\footnote{\texttt{#1}}}
-\newcommand{\mailto}[1]{\texttt{#1}}
-\newcommand{\imgsrc}[2][]{}
-\newcommand{\home}[1]{\protect\raisebox{-.75ex}{\char126}#1}
-\AtBeginDocument
-{\@ifundefined{url}
-{%url package is not loaded
-\let\url\ahref\let\oneurl\ahrefurl\let\footurl\footahref}
-{}}
-%% Void cutting instructions
-\newcounter{cuttingdepth}
-\newcommand{\tocnumber}{}
-\newcommand{\notocnumber}{}
-\newcommand{\cuttingunit}{}
-\newcommand{\cutdef}[2][]{}
-\newcommand{\cuthere}[2]{}
-\newcommand{\cutend}{}
-\newcommand{\htmlhead}[1]{}
-\newcommand{\htmlfoot}[1]{}
-\newcommand{\htmlprefix}[1]{}
-\newenvironment{cutflow}[1]{}{}
-\newcommand{\cutname}[1]{}
-\newcommand{\toplinks}[3]{}
-%%%% Html only
-\excludecomment{rawhtml}
-\newcommand{\rawhtmlinput}[1]{}
-\excludecomment{htmlonly}
-%%%% Latex only
-\newenvironment{latexonly}{}{}
-\newenvironment{verblatex}{}{}
-%%%% Image file stuff
-\def\toimage{\endgroup}
-\def\endtoimage{\begingroup\def\@currenvir{toimage}}
-\def\verbimage{\endgroup}
-\def\endverbimage{\begingroup\def\@currenvir{verbimage}}
-\newcommand{\imageflush}[1][]{}
-%%% Bgcolor definition
-\newsavebox{\@bgcolorbin}
-\newenvironment{bgcolor}[2][]
- {\newcommand{\@mycolor}{#2}\begin{lrbox}{\@bgcolorbin}\vbox\bgroup}
- {\egroup\end{lrbox}%
- \begin{flushleft}%
- \colorbox{\@mycolor}{\usebox{\@bgcolorbin}}%
- \end{flushleft}}
-%%% Postlude
-\makeatother
diff --git a/doc/faq/interval_discr.v b/doc/faq/interval_discr.v
deleted file mode 100644
index 671dc988a..000000000
--- a/doc/faq/interval_discr.v
+++ /dev/null
@@ -1,419 +0,0 @@
-(** Sketch of the proof of {p:nat|p<=n} = {p:nat|p<=m} -> n=m
-
- - preliminary results on the irrelevance of boundedness proofs
- - introduce the notion of finite cardinal |A|
- - prove that |{p:nat|p<=n}| = n
- - prove that |A| = n /\ |A| = m -> n = m if equality is decidable on A
- - prove that equality is decidable on A
- - conclude
-*)
-
-(** * Preliminary results on [nat] and [le] *)
-
-(** Proving axiom K on [nat] *)
-
-Require Import Eqdep_dec.
-Require Import Arith.
-
-Theorem eq_rect_eq_nat :
- forall (p:nat) (Q:nat->Type) (x:Q p) (h:p=p), x = eq_rect p Q x p h.
-Proof.
-intros.
-apply K_dec_set with (p := h).
-apply eq_nat_dec.
-reflexivity.
-Qed.
-
-(** Proving unicity of proofs of [(n<=m)%nat] *)
-
-Scheme le_ind' := Induction for le Sort Prop.
-
-Theorem le_uniqueness_proof : forall (n m : nat) (p q : n <= m), p = q.
-Proof.
-induction p using le_ind'; intro q.
- replace (le_n n) with
- (eq_rect _ (fun n0 => n <= n0) (le_n n) _ eq_refl).
- 2:reflexivity.
- generalize (eq_refl n).
- pattern n at 2 4 6 10, q; case q; [intro | intros m l e].
- rewrite <- eq_rect_eq_nat; trivial.
- contradiction (le_Sn_n m); rewrite <- e; assumption.
- replace (le_S n m p) with
- (eq_rect _ (fun n0 => n <= n0) (le_S n m p) _ eq_refl).
- 2:reflexivity.
- generalize (eq_refl (S m)).
- pattern (S m) at 1 3 4 6, q; case q; [intro Heq | intros m0 l HeqS].
- contradiction (le_Sn_n m); rewrite Heq; assumption.
- injection HeqS; intro Heq; generalize l HeqS.
- rewrite <- Heq; intros; rewrite <- eq_rect_eq_nat.
- rewrite (IHp l0); reflexivity.
-Qed.
-
-(** Proving irrelevance of boundedness proofs while building
- elements of interval *)
-
-Lemma dep_pair_intro :
- forall (n x y:nat) (Hx : x<=n) (Hy : y<=n), x=y ->
- exist (fun x => x <= n) x Hx = exist (fun x => x <= n) y Hy.
-Proof.
-intros n x y Hx Hy Heq.
-generalize Hy.
-rewrite <- Heq.
-intros.
-rewrite (le_uniqueness_proof x n Hx Hy0).
-reflexivity.
-Qed.
-
-(** * Proving that {p:nat|p<=n} = {p:nat|p<=m} -> n=m *)
-
-(** Definition of having finite cardinality [n+1] for a set [A] *)
-
-Definition card (A:Set) n :=
- exists f,
- (forall x:A, f x <= n) /\
- (forall x y:A, f x = f y -> x = y) /\
- (forall m, m <= n -> exists x:A, f x = m).
-
-Require Import Arith.
-
-(** Showing that the interval [0;n] has cardinality [n+1] *)
-
-Theorem card_interval : forall n, card {x:nat|x<=n} n.
-Proof.
-intro n.
-exists (fun x:{x:nat|x<=n} => proj1_sig x).
-split.
-(* bounded *)
-intro x; apply (proj2_sig x).
-split.
-(* injectivity *)
-intros (p,Hp) (q,Hq).
-simpl.
-intro Hpq.
-apply dep_pair_intro; assumption.
-(* surjectivity *)
-intros m Hmn.
-exists (exist (fun x : nat => x <= n) m Hmn).
-reflexivity.
-Qed.
-
-(** Showing that equality on the interval [0;n] is decidable *)
-
-Lemma interval_dec :
- forall n (x y : {m:nat|m<=n}), {x=y}+{x<>y}.
-Proof.
-intros n (p,Hp).
-induction p; intros ([|q],Hq).
-left.
- apply dep_pair_intro.
- reflexivity.
-right.
- intro H; discriminate H.
-right.
- intro H; discriminate H.
-assert (Hp' : p <= n).
- apply le_Sn_le; assumption.
-assert (Hq' : q <= n).
- apply le_Sn_le; assumption.
-destruct (IHp Hp' (exist (fun m => m <= n) q Hq'))
- as [Heq|Hneq].
-left.
- injection Heq; intro Heq'.
- apply dep_pair_intro.
- apply eq_S.
- assumption.
-right.
- intro HeqS.
- injection HeqS; intro Heq.
- apply Hneq.
- apply dep_pair_intro.
- assumption.
-Qed.
-
-(** Showing that the cardinality relation is functional on decidable sets *)
-
-Lemma card_inj_aux :
- forall (A:Type) f g n,
- (forall x:A, f x <= 0) ->
- (forall x y:A, f x = f y -> x = y) ->
- (forall m, m <= S n -> exists x:A, g x = m)
- -> False.
-Proof.
-intros A f g n Hfbound Hfinj Hgsurj.
-destruct (Hgsurj (S n) (le_n _)) as (x,Hx).
-destruct (Hgsurj n (le_S _ _ (le_n _))) as (x',Hx').
-assert (Hfx : 0 = f x).
-apply le_n_O_eq.
-apply Hfbound.
-assert (Hfx' : 0 = f x').
-apply le_n_O_eq.
-apply Hfbound.
-assert (x=x').
-apply Hfinj.
-rewrite <- Hfx.
-rewrite <- Hfx'.
-reflexivity.
-rewrite H in Hx.
-rewrite Hx' in Hx.
-apply (n_Sn _ Hx).
-Qed.
-
-(** For [dec_restrict], we use a lemma on the negation of equality
-that requires proof-irrelevance. It should be possible to avoid this
-lemma by generalizing over a first-order definition of [x<>y], say
-[neq] such that [{x=y}+{neq x y}] and [~(x=y /\ neq x y)]; for such
-[neq], unicity of proofs could be proven *)
-
- Require Import Classical.
- Lemma neq_dep_intro :
- forall (A:Set) (z x y:A) (p:x<>z) (q:y<>z), x=y ->
- exist (fun x => x <> z) x p = exist (fun x => x <> z) y q.
- Proof.
- intros A z x y p q Heq.
- generalize q; clear q; rewrite <- Heq; intro q.
- rewrite (proof_irrelevance _ p q); reflexivity.
- Qed.
-
-Lemma dec_restrict :
- forall (A:Set),
- (forall x y :A, {x=y}+{x<>y}) ->
- forall z (x y :{a:A|a<>z}), {x=y}+{x<>y}.
-Proof.
-intros A Hdec z (x,Hx) (y,Hy).
-destruct (Hdec x y) as [Heq|Hneq].
-left; apply neq_dep_intro; assumption.
-right; intro Heq; injection Heq; exact Hneq.
-Qed.
-
-Lemma pred_inj : forall n m,
- 0 <> n -> 0 <> m -> pred m = pred n -> m = n.
-Proof.
-destruct n.
-intros m H; destruct H; reflexivity.
-destruct m.
-intros _ H; destruct H; reflexivity.
-simpl; intros _ _ H.
-rewrite H.
-reflexivity.
-Qed.
-
-Lemma le_neq_lt : forall n m, n <= m -> n<>m -> n < m.
-Proof.
-intros n m Hle Hneq.
-destruct (le_lt_eq_dec n m Hle).
-assumption.
-contradiction.
-Qed.
-
-Lemma inj_restrict :
- forall (A:Set) (f:A->nat) x y z,
- (forall x y : A, f x = f y -> x = y)
- -> x <> z -> f y < f z -> f z <= f x
- -> pred (f x) = f y
- -> False.
-
-(* Search error sans le type de f !! *)
-Proof.
-intros A f x y z Hfinj Hneqx Hfy Hfx Heq.
-assert (f z <> f x).
- apply not_eq_sym.
- intro Heqf.
- apply Hneqx.
- apply Hfinj.
- assumption.
-assert (f x = S (f y)).
- assert (0 < f x).
- apply le_lt_trans with (f z).
- apply le_O_n.
- apply le_neq_lt; assumption.
- apply pred_inj.
- apply O_S.
- apply lt_O_neq; assumption.
- exact Heq.
-assert (f z <= f y).
-destruct (le_lt_or_eq _ _ Hfx).
- apply lt_n_Sm_le.
- rewrite <- H0.
- assumption.
- contradiction Hneqx.
- symmetry.
- apply Hfinj.
- assumption.
-contradiction (lt_not_le (f y) (f z)).
-Qed.
-
-Theorem card_inj : forall m n (A:Set),
- (forall x y :A, {x=y}+{x<>y}) ->
- card A m -> card A n -> m = n.
-Proof.
-induction m; destruct n;
-intros A Hdec
- (f,(Hfbound,(Hfinj,Hfsurj)))
- (g,(Hgbound,(Hginj,Hgsurj))).
-(* 0/0 *)
-reflexivity.
-(* 0/Sm *)
-destruct (card_inj_aux _ _ _ _ Hfbound Hfinj Hgsurj).
-(* Sn/0 *)
-destruct (card_inj_aux _ _ _ _ Hgbound Hginj Hfsurj).
-(* Sn/Sm *)
-destruct (Hgsurj (S n) (le_n _)) as (xSn,HSnx).
-rewrite IHm with (n:=n) (A := {x:A|x<>xSn}).
-reflexivity.
-(* decidability of eq on {x:A|x<>xSm} *)
-apply dec_restrict.
-assumption.
-(* cardinality of {x:A|x<>xSn} is m *)
-pose (f' := fun x' : {x:A|x<>xSn} =>
- let (x,Hneq) := x' in
- if le_lt_dec (f xSn) (f x)
- then pred (f x)
- else f x).
-exists f'.
-split.
-(* f' is bounded *)
-unfold f'.
-intros (x,_).
-destruct (le_lt_dec (f xSn) (f x)) as [Hle|Hge].
-change m with (pred (S m)).
-apply le_pred.
-apply Hfbound.
-apply le_S_n.
-apply le_trans with (f xSn).
-exact Hge.
-apply Hfbound.
-split.
-(* f' is injective *)
-unfold f'.
-intros (x,Hneqx) (y,Hneqy) Heqf'.
-destruct (le_lt_dec (f xSn) (f x)) as [Hlefx|Hgefx];
-destruct (le_lt_dec (f xSn) (f y)) as [Hlefy|Hgefy].
-(* f xSn <= f x et f xSn <= f y *)
-assert (Heq : x = y).
- apply Hfinj.
- assert (f xSn <> f y).
- apply not_eq_sym.
- intro Heqf.
- apply Hneqy.
- apply Hfinj.
- assumption.
- assert (0 < f y).
- apply le_lt_trans with (f xSn).
- apply le_O_n.
- apply le_neq_lt; assumption.
- assert (f xSn <> f x).
- apply not_eq_sym.
- intro Heqf.
- apply Hneqx.
- apply Hfinj.
- assumption.
- assert (0 < f x).
- apply le_lt_trans with (f xSn).
- apply le_O_n.
- apply le_neq_lt; assumption.
- apply pred_inj.
- apply lt_O_neq; assumption.
- apply lt_O_neq; assumption.
- assumption.
-apply neq_dep_intro; assumption.
-(* f y < f xSn <= f x *)
-destruct (inj_restrict A f x y xSn); assumption.
-(* f x < f xSn <= f y *)
-symmetry in Heqf'.
-destruct (inj_restrict A f y x xSn); assumption.
-(* f x < f xSn et f y < f xSn *)
-assert (Heq : x=y).
- apply Hfinj; assumption.
-apply neq_dep_intro; assumption.
-(* f' is surjective *)
-intros p Hlep.
-destruct (le_lt_dec (f xSn) p) as [Hle|Hlt].
-(* case f xSn <= p *)
-destruct (Hfsurj (S p) (le_n_S _ _ Hlep)) as (x,Hx).
-assert (Hneq : x <> xSn).
- intro Heqx.
- rewrite Heqx in Hx.
- rewrite Hx in Hle.
- apply le_Sn_n with p; assumption.
-exists (exist (fun a => a<>xSn) x Hneq).
-unfold f'.
-destruct (le_lt_dec (f xSn) (f x)) as [Hle'|Hlt'].
-rewrite Hx; reflexivity.
-rewrite Hx in Hlt'.
-contradiction (le_not_lt (f xSn) p).
-apply lt_trans with (S p).
-apply lt_n_Sn.
-assumption.
-(* case p < f xSn *)
-destruct (Hfsurj p (le_S _ _ Hlep)) as (x,Hx).
-assert (Hneq : x <> xSn).
- intro Heqx.
- rewrite Heqx in Hx.
- rewrite Hx in Hlt.
- apply (lt_irrefl p).
- assumption.
-exists (exist (fun a => a<>xSn) x Hneq).
-unfold f'.
-destruct (le_lt_dec (f xSn) (f x)) as [Hle'|Hlt'].
- rewrite Hx in Hle'.
- contradiction (lt_irrefl p).
- apply lt_le_trans with (f xSn); assumption.
- assumption.
-(* cardinality of {x:A|x<>xSn} is n *)
-pose (g' := fun x' : {x:A|x<>xSn} =>
- let (x,Hneq) := x' in
- if Hdec x xSn then 0 else g x).
-exists g'.
-split.
-(* g is bounded *)
-unfold g'.
-intros (x,_).
-destruct (Hdec x xSn) as [_|Hneq].
-apply le_O_n.
-assert (Hle_gx:=Hgbound x).
-destruct (le_lt_or_eq _ _ Hle_gx).
-apply lt_n_Sm_le.
-assumption.
-contradiction Hneq.
-apply Hginj.
-rewrite HSnx.
-assumption.
-split.
-(* g is injective *)
-unfold g'.
-intros (x,Hneqx) (y,Hneqy) Heqg'.
-destruct (Hdec x xSn) as [Heqx|_].
-contradiction Hneqx.
-destruct (Hdec y xSn) as [Heqy|_].
-contradiction Hneqy.
-assert (Heq : x=y).
- apply Hginj; assumption.
-apply neq_dep_intro; assumption.
-(* g is surjective *)
-intros p Hlep.
-destruct (Hgsurj p (le_S _ _ Hlep)) as (x,Hx).
-assert (Hneq : x<>xSn).
- intro Heq.
- rewrite Heq in Hx.
- rewrite Hx in HSnx.
- rewrite HSnx in Hlep.
- contradiction (le_Sn_n _ Hlep).
-exists (exist (fun a => a<>xSn) x Hneq).
-simpl.
-destruct (Hdec x xSn) as [Heqx|_].
-contradiction Hneq.
-assumption.
-Qed.
-
-(** Conclusion *)
-
-Theorem interval_discr :
- forall n m, {p:nat|p<=n} = {p:nat|p<=m} -> n=m.
-Proof.
-intros n m Heq.
-apply card_inj with (A := {p:nat|p<=n}).
-apply interval_dec.
-apply card_interval.
-rewrite Heq.
-apply card_interval.
-Qed.