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Date: Sat, 10 Jul 2010 15:57:24 +0100
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-\documentclass[11pt]{article}
-\title{A Tutorial on [Co-]Inductive Types in Coq}
-\author{Eduardo Gim\'enez\thanks{Eduardo.Gimenez@inria.fr},
-Pierre Cast\'eran\thanks{Pierre.Casteran@labri.fr}}
-\date{May 1998 --- \today}
-
-\usepackage{multirow}
-% \usepackage{aeguill}
-% \externaldocument{RefMan-gal.v}
-% \externaldocument{RefMan-ext.v}
-% \externaldocument{RefMan-tac.v}
-% \externaldocument{RefMan-oth}
-% \externaldocument{RefMan-tus.v}
-% \externaldocument{RefMan-syn.v}
-% \externaldocument{Extraction.v}
-\input{recmacros}
-\input{coqartmacros}
-\newcommand{\refmancite}[1]{{}}
-% \newcommand{\refmancite}[1]{\cite{coqrefman}}
-% \newcommand{\refmancite}[1]{\cite[#1] {]{coqrefman}}
-
-\usepackage[latin1]{inputenc}
-\usepackage[T1]{fontenc}
-\usepackage{makeidx}
-% \usepackage{multind}
-\usepackage{alltt}   
-\usepackage{verbatim}
-\usepackage{amssymb}
-\usepackage{amsmath}
-\usepackage{theorem}
-\usepackage[dvips]{epsfig}
-\usepackage{epic}
-\usepackage{eepic}
-% \usepackage{ecltree}
-\usepackage{moreverb} 
-\usepackage{color}
-\usepackage{pifont}
-\usepackage{xr}
-\usepackage{url}
-
-\usepackage{alltt}
-\renewcommand{\familydefault}{ptm}
-\renewcommand{\seriesdefault}{m}
-\renewcommand{\shapedefault}{n}
-\newtheorem{exercise}{Exercise}[section]	
-\makeindex
-\begin{document}
-\maketitle
-
-\begin{abstract}
-This document\footnote{The first versions of this document were entirely written by Eduardo Gimenez.
-Pierre Cast\'eran wrote the 2004 and 2006 revisions.} is an introduction to the definition and
-use of inductive and co-inductive  types in the {\coq} proof environment. It explains how types like natural numbers and infinite streams are defined
-in {\coq}, and the kind of proof techniques that can be used to reason
-about them (case analysis, induction, inversion of predicates,
-co-induction, etc).  Each technique is illustrated through an
-executable and self-contained {\coq} script.  
-\end{abstract}
-%\RRkeyword{Proof environments, recursive types.}
-%\makeRT
-
-\addtocontents{toc}{\protect \thispagestyle{empty}}
-\pagenumbering{arabic}
-
-\cleardoublepage
-\tableofcontents
-\clearpage
-
-\section{About this document}
-
-This document is an introduction to the definition and use of
-inductive and co-inductive  types in the {\coq} proof environment.  It was born from the
-notes written for the course about the version V5.10 of {\coq}, given
-by Eduardo Gimenez  at
-the Ecole Normale Sup\'erieure de Lyon in March 1996. This article is
-a revised and improved version of these notes for the version V8.0 of
-the system.
-
-
-We assume that the reader has some familiarity with the
-proofs-as-programs paradigm of Logic \cite{Coquand:metamathematical} and the generalities
-of the {\coq} system \cite{coqrefman}.  You would take a greater advantage of
-this document if you first read the general tutorial about {\coq} and
-{\coq}'s FAQ, both available on \cite{coqsite}.
-A text book \cite{coqart},  accompanied with a lot of
-examples and exercises \cite{Booksite}, presents a detailed description
-of the {\coq} system and its underlying
-formalism: the Calculus of Inductive Construction.
-Finally, the complete description of {\coq}  is given in the reference manual
-\cite{coqrefman}. Most of the tactics and commands we describe have
-several options, which we do not present exhaustively. 
-If some script herein uses a non described feature, please refer to
-the Reference Manual.
-
-
-If you are familiar with other proof environments
-based on type theory and the LCF style ---like PVS, LEGO, Isabelle,
-etc--- then you will find not difficulty to guess the unexplained
-details.
-
-The better way to read this document is to start up the {\coq} system,
-type by yourself the examples and exercises, and observe the
-behavior of the system. All the examples proposed in this tutorial
-can be downloaded from the same site as the present document. 
-
-
-The tutorial is organised as follows. The next section describes how
-inductive types are defined in {\coq}, and introduces some useful ones,
-like natural numbers, the empty type, the propositional equality type,
-and the logical connectives. Section \ref{CaseAnalysis} explains
-definitions by pattern-matching and their connection with the
-principle of case analysis. This principle is the most basic
-elimination rule associated with inductive or co-inductive types
- and follows a
-general scheme that we illustrate for some of the types introduced in
-Section \ref{Introduction}.  Section \ref{CaseTechniques} illustrates
-the pragmatics of this principle, showing different proof techniques
-based on it. Section \ref{StructuralInduction} introduces definitions
-by structural recursion and proofs by induction. 
-Section~\ref{CaseStudy} presents some elaborate techniques
-about dependent case analysis. Finally, Section
-\ref{CoInduction} is a brief introduction to co-inductive types
---i.e., types containing infinite objects-- and the principle of
-co-induction.
-
-
-Thanks to  Bruno Barras, Yves Bertot, Hugo Herbelin, Jean-Fran\c{c}ois Monin
-and Michel L\'evy for their help.
-
-\subsection*{Lexical conventions}
-The \texttt{typewriter} font is used to represent text
-input by the user, while the \textit{italic} font is used to represent
-the text output by the system as answers.  
-
-
-Moreover, the mathematical symbols \coqle{}, \coqdiff, \(\exists\),
-\(\forall\), \arrow{}, $\rightarrow{}$ \coqor{}, \coqand{}, and \funarrow{} 
-stand for the character strings \citecoq{<=}, \citecoq{<>},
-\citecoq{exists}, \citecoq{forall}, \citecoq{->}, \citecoq{<-},
-\texttt{\char'134/}, \texttt{/\char'134}, and \citecoq{=>},
-respectively.  For instance, the \coq{} statement
-%V8 A prendre
-% inclusion numero 1
-% traduction numero 1
-\begin{alltt}
-\hide{Open Scope nat_scope. Check (}forall A:Type,(exists x : A, forall (y:A), x <> y) -> 2 = 3\hide{).}
-\end{alltt}
-is written as follows in this tutorial:
-%V8 A prendre
-% inclusion numero 2
-% traduction numero 2
-\begin{alltt}
-\hide{Check (}{\prodsym}A:Type,(\exsym{}x:A, {\prodsym}y:A, x {\coqdiff} y) \arrow{} 2 = 3\hide{).}
-\end{alltt}
-
-When a fragment of \coq{} input text appears in the middle of
-regular text, we often place this fragment between double quotes
-``\dots.''  These double quotes do not belong to the \coq{} syntax.
-
-Finally, any
-string enclosed between \texttt{(*} and \texttt{*)} is a comment and
-is ignored by the \coq{} system.
-
-\section{Introducing Inductive Types} 
-\label{Introduction}
-
-Inductive types are types closed with respect to their introduction
-rules. These rules explain the most basic or \textsl{canonical} ways
-of constructing an element of the type.  In this sense, they
-characterize the recursive type. Different rules must be considered as
-introducing different objects. In order to fix ideas, let us introduce
-in {\coq} the most well-known example of a recursive type: the type of
-natural numbers.
-
-%V8 A prendre
-\begin{alltt}
-Inductive nat : Set := 
- | O : nat 
- | S : nat\arrow{}nat.
-\end{alltt}
-
-The definition of a recursive type has two main parts. First, we
-establish what kind of recursive type we will characterize (a set, in
-this case). Second, we present the introduction rules that define the
-type ({\Z} and {\SUCC}), also called its {\sl constructors}. The constructors
-{\Z} and {\SUCC} determine all the elements of this type. In other
-words, if $n\mbox{:}\nat$, then $n$ must have been introduced either
-by the rule {\Z} or by an application of the rule {\SUCC} to a
-previously constructed natural number. In this sense, we can say
-that {\nat} is \emph{closed}. On the contrary, the type
-$\Set$ is an {\it open} type, since we do not know {\it a priori} all
-the possible ways of introducing an object of type \texttt{Set}.
-
-After entering this command, the constants {\nat}, {\Z} and {\SUCC} are
-available in the current context. We can see their types using the
-\texttt{Check} command \refmancite{Section \ref{Check}}:
-
-%V8 A prendre 
-\begin{alltt}
-Check nat.
-\it{}nat : Set
-\tt{}Check O.
-\it{}O : nat
-\tt{}Check S.
-\it{}S : nat {\arrow} nat
-\end{alltt}
-
-Moreover, {\coq} adds to the context three constants named
- $\natind$, $\natrec$ and $\natrect$, which
- correspond to different principles of structural induction on
-natural numbers that {\coq} infers automatically from the definition.  We
-will come back to them in Section \ref{StructuralInduction}.
-
-
-In fact, the type of natural numbers as well as several useful
-theorems about them are already defined in the basic library of {\coq},
-so there is no need to introduce them.  Therefore, let us throw away
-our (re)definition of {\nat}, using the command \texttt{Reset}.
-
-%V8 A prendre
-\begin{alltt}
-Reset nat.
-Print nat.
-\it{}Inductive nat : Set :=  O : nat | S : nat \arrow{} nat
-For S: Argument scope is [nat_scope]
-\end{alltt}
-
-Notice that \coq{}'s \emph{interpretation scope} for natural numbers
-(called \texttt{nat\_scope}) 
-allows us to read and write natural numbers in decimal form (see \cite{coqrefman}). For instance, the constructor \texttt{O} can be read or written
-as the digit $0$, and the term ``~\texttt{S (S (S O))}~'' as $3$.
-
-%V8 A prendre
-\begin{alltt}
-Check O.
-\it 0 : nat.
-\tt
-Check (S (S (S O))).
-\it 3 : nat
-\end{alltt}
-
-Let us now take a look to some other
-recursive types contained in the standard library of {\coq}.
-
-\subsection{Lists}
-Lists are defined in library \citecoq{List}\footnote{Notice that in versions of 
-{\coq}
-prior to 8.1, the parameter $A$ had sort \citecoq{Set} instead of \citecoq{Type}; 
-the constant \citecoq{list} was thus of type \citecoq{Set\arrow{} Set}.}
-
-
-\begin{alltt}
-Require Import List.
-Print list.
-\it
-Inductive list (A : Type) : Type:=
-    nil : list A | cons : A {\arrow} list A {\arrow} list A
-For nil: Argument A is implicit
-For cons: Argument A is implicit
-For list: Argument scope is [type_scope]
-For nil: Argument scope is [type_scope]
-For cons: Argument scopes are [type_scope _ _]
-\end{alltt}
-
-In this definition, \citecoq{A} is a \emph{general parameter}, global
-to both constructors.
-This kind of definition allows us to build a whole family of
-inductive types, indexed over the sort \citecoq{Type}.
-This can be observed if we consider the type of identifiers
-\citecoq{list}, \citecoq{cons} and \citecoq{nil}.
-Notice the notation \citecoq{(A := \dots)} which must be used 
-when {\coq}'s type inference algorithm cannot infer the implicit
-parameter \citecoq{A}.
-\begin{alltt}
-Check list.
-\it list
-     : Type {\arrow} Type
-
-\tt Check (nil (A:=nat)).
-\it nil
-     : list nat
-
-\tt Check (nil (A:= nat {\arrow} nat)).
-\it nil
-     : list (nat {\arrow} nat)
-
-\tt Check (fun A: Type {\funarrow} (cons (A:=A))).
-\it fun A : Type {\funarrow} cons (A:=A)
-     : {\prodsym} A : Type, A {\arrow} list A {\arrow} list A
-
-\tt Check (cons 3 (cons 2 nil)).
-\it 3 :: 2 :: nil
-     : list nat
-
-\tt Check (nat :: bool ::nil).
-\it nat :: bool :: nil
-     : list Set
-
-\tt Check ((3<=4) :: True ::nil).
-\it (3<=4) :: True :: nil
-     : list Prop
-
-\tt Check (Prop::Set::nil).
-\it Prop::Set::nil
-     : list Type
-\end{alltt}
-
-\subsection{Vectors.}
-\label{vectors}
-
-Like \texttt{list}, \citecoq{vector} is a polymorphic type:
-if $A$ is a type, and $n$ a natural number, ``~\citecoq{vector $A$ $n$}~''
-is the type of vectors of elements of $A$ and size $n$.
-
-
-\begin{alltt}
-Require Import  Bvector.
-
-Print vector.
-\it
-Inductive vector (A : Type) : nat {\arrow} Type :=
-    Vnil : vector A 0
-  | Vcons : A {\arrow} {\prodsym} n : nat, vector A n {\arrow} vector A (S n)
-For vector: Argument scopes are [type_scope nat_scope]
-For Vnil: Argument scope is [type_scope]
-For Vcons: Argument scopes are [type_scope _ nat_scope _]
-\end{alltt}
-
-
-Remark the difference between the two parameters $A$ and $n$:
-The first one is a \textsl{general parameter}, global to all the
-introduction rules,while the second one is an \textsl{index}, which is
-instantiated differently in the introduction rules.
-Such types parameterized  by regular
-values are called \emph{dependent types}.
-
-\begin{alltt}
-Check (Vnil nat).
-\it Vnil nat
-     : vector nat 0
-
-\tt Check (fun (A:Type)(a:A){\funarrow} Vcons _ a _ (Vnil _)).
-\it fun (A : Type) (a : A) {\funarrow} Vcons A a 0 (Vnil A)
-     : {\prodsym} A : Type, A {\arrow} vector A 1
-
-
-\tt Check (Vcons _ 5 _ (Vcons _ 3 _ (Vnil _))).
-\it Vcons nat 5 1 (Vcons nat 3 0 (Vnil nat))
-     : vector nat 2
-\end{alltt}
-
-\subsection{The contradictory proposition.}
-Another example of an inductive type is the contradictory proposition.
-This type inhabits the universe of propositions, and has no element
-at all.
-%V8 A prendre
-\begin{alltt}
-Print False.
-\it{} Inductive False : Prop :=
-\end{alltt}
-
-\noindent Notice that no constructor is given in this definition.
-
-\subsection{The tautological proposition.}
-Similarly, the
-tautological proposition {\True} is defined as an inductive type
-with only one element {\I}:
-
-%V8 A prendre
-\begin{alltt}
-Print True.
-\it{}Inductive True : Prop :=  I : True
-\end{alltt}
-
-\subsection{Relations as inductive types.}
-Some relations can also be introduced in a smart way as an inductive family
-of propositions. Let us take as example the order $n \leq m$ on natural
-numbers, called \citecoq{le} in {\coq}.
- This relation is introduced through
-the following definition, quoted from the standard library\footnote{In the interpretation scope
-for Peano arithmetic:
-\citecoq{nat\_scope},  ``~\citecoq{n <= m}~'' is equivalent to 
-``~\citecoq{le n m}~'' .}:
-
-
-
-
-%V8 A prendre
-\begin{alltt}
-Print le. \it
-Inductive le (n:nat) : nat\arrow{}Prop := 
-|  le_n: n {\coqle} n 
-|  le_S: {\prodsym} m, n {\coqle} m \arrow{} n {\coqle} S m.
-\end{alltt}
-
-Notice that in this definition $n$ is a general parameter,
-while the second argument of \citecoq{le} is an index (see section
-~\ref{vectors}).
- This definition
-introduces the binary relation $n {\leq} m$ as the family of unary predicates
-``\textsl{to be greater or equal than a given $n$}'', parameterized by $n$.
-
-The introduction rules of this type can be seen as a sort of Prolog
-rules for proving that a given integer $n$ is less or equal than another one.
-In fact, an object of type $n{\leq} m$ is nothing but a proof 
-built up using the constructors \textsl{le\_n} and
-\textsl{le\_S} of this type.  As an example, let us construct
-a proof that zero is less or equal than three using {\coq}'s interactive
-proof mode.
-Such an object can be obtained applying three times the second
-introduction rule of \citecoq{le}, to a proof that zero is less or equal
-than itself,
-which is provided by the first constructor of \citecoq{le}:
-
-%V8 A prendre
-\begin{alltt}
-Theorem zero_leq_three: 0 {\coqle} 3.
-Proof.
-\it{} 1 subgoal
-
-============================
- 0 {\coqle} 3
-
-\tt{}Proof.
- constructor 2. 
-
-\it{} 1 subgoal
-============================
-  0 {\coqle} 2
-
-\tt{} constructor 2.  
-\it{} 1 subgoal
-============================
-  0 {\coqle} 1
-
-\tt{} constructor 2
-\it{} 1 subgoal
-============================
-  0 {\coqle} 0
-
-\tt{} constructor 1.
-
-\it{}Proof completed
-\tt{}Qed.
-\end{alltt}
-
-\noindent When
-the current goal is an inductive type, the tactic 
-``~\citecoq{constructor $i$}~'' \refmancite{Section \ref{constructor}} applies the $i$-th constructor in the
-definition of the type. We can take a look at the proof constructed
-using the command \texttt{Print}:
-
-%V8 A prendre
-\begin{alltt}
-Print Print zero_leq_three.
-\it{}zero_leq_three = 
-zero_leq_three = le_S 0 2 (le_S 0 1 (le_S 0 0 (le_n 0)))
-     : 0 {\coqle} 3
-\end{alltt}
-
-When the parameter $i$ is not supplied, the tactic \texttt{constructor}
-tries to apply ``~\texttt{constructor $1$}~'', ``~\texttt{constructor $2$}~'',\dots,
-``~\texttt{constructor $n$}~'' where $n$ is the number of constructors
-of the inductive type (2 in our example) of the conclusion of the goal.
-Our little proof can thus be obtained iterating the tactic
-\texttt{constructor} until it fails:
-
-%V8 A prendre
-\begin{alltt}
-Lemma zero_leq_three': 0 {\coqle} 3.
- repeat constructor.
-Qed.
-\end{alltt}
-
-Notice that the strict order on \texttt{nat}, called \citecoq{lt}
-is not inductively defined: the proposition $n<p$ (notation for \citecoq{lt $n$ $p$})
-is reducible to \citecoq{(S $n$) $\leq$ p}.
-
-\begin{alltt}
-Print lt.
-\it
-lt = fun n m : nat {\funarrow} S n {\coqle} m
-     : nat {\arrow} nat {\arrow} Prop
-\tt
-Lemma zero_lt_three : 0 < 3.
-Proof.
- repeat constructor. 
-Qed.
-
-Print zero_lt_three.
-\it zero_lt_three = le_S 1 2 (le_S 1 1 (le_n 1))
-     : 0 < 3
-\end{alltt}
-
-
-
-\subsection{About general parameters (\coq{} version $\geq$ 8.1)}
-\label{parameterstuff}
-
-Since version $8.1$, it is possible to write more compact inductive definitions
-than in earlier versions.
-
-Consider the following alternative definition of the relation $\leq$ on 
-type \citecoq{nat}:
-
-\begin{alltt}
-Inductive le'(n:nat):nat -> Prop :=
- | le'_n : le' n n
- | le'_S : forall p, le' (S n) p -> le' n p.
-
-Hint Constructors le'.
-\end{alltt}
-
-We notice that the type of the second constructor of \citecoq{le'}
-has an argument whose type is \citecoq{le' (S n) p}. 
-This constrasts with earlier versions 
-of {\coq}, in which a general parameter $a$ of an inductive
-type $I$ had to appear only in applications of the form $I\,\dots\,a$.
-
-Since version $8.1$, if $a$ is a general parameter of an inductive 
-type $I$, the type of an argument of a constructor of $I$ may be
-of the form  $I\,\dots\,t_a$ , where $t_a$ is any term.
-Notice that the final type of the constructors must be of the form
-$I\,\dots\,a$, since these constructors describe how to form 
-inhabitants of type $I\,\dots\,a$ (this is the role of parameter $a$).
-
-Another example of this new feature is {\coq}'s definition of accessibility
-(see Section~\ref{WellFoundedRecursion}), which has a general parameter
-$x$; the constructor for the predicate
-``$x$ is accessible'' takes an  argument of type ``$y$ is accessible''.
-
-
-
-In earlier versions of {\coq}, a relation like \citecoq{le'} would have to be
-defined without $n$ being a general parameter.
-
-\begin{alltt}
-Reset le'.
-
-Inductive le': nat-> nat -> Prop :=
- | le'_n : forall n, le' n n
- | le'_S : forall n p, le' (S n) p -> le' n p.
-\end{alltt}
-
-
-
-
-\subsection{The propositional equality type.} \label{equality}
-In {\coq}, the propositional equality between two inhabitants $a$ and
-$b$ of
-the same type $A$ ,
-noted $a=b$, is introduced as a family of recursive predicates
-``~\textsl{to be equal to $a$}~'',  parameterised by both $a$ and its type
-$A$. This family of types has only one introduction rule, which
-corresponds to reflexivity.
-Notice that the syntax ``\citecoq{$a$ = $b$}~'' is an abbreviation  
-for ``\citecoq{eq $a$  $b$}~'', and that the parameter $A$ is \emph{implicit},
-as it can be infered from $a$.
-%V8 A prendre
-\begin{alltt}
-Print eq.
-\it{} Inductive eq (A : Type) (x : A) : A \arrow{} Prop :=  
-    refl_equal : x = x
-For eq: Argument A is implicit
-For refl_equal: Argument A is implicit
-For eq: Argument scopes are [type_scope _ _]
-For refl_equal: Argument scopes are [type_scope _]
-\end{alltt}
-
-Notice also that  the first parameter $A$ of \texttt{eq} has type
-\texttt{Type}. The type system of {\coq} allows us to consider equality between 
-various kinds of terms: elements of a set, proofs, propositions,
-types, and so on.
-Look at \cite{coqrefman, coqart} to get more details on {\coq}'s type
-system, as well as implicit arguments and argument scopes.
-
-
-\begin{alltt}
-Lemma eq_3_3 : 2 + 1 = 3.
-Proof.
- reflexivity.
-Qed.
-
-Lemma eq_proof_proof : refl_equal (2*6) = refl_equal (3*4).
-Proof.
- reflexivity.
-Qed.
-
-Print eq_proof_proof.
-\it eq_proof_proof = 
-refl_equal (refl_equal (3 * 4))
-     : refl_equal (2 * 6) = refl_equal (3 * 4)
-\tt
-
-Lemma eq_lt_le : ( 2 < 4) = (3 {\coqle} 4).
-Proof.
- reflexivity.
-Qed.
-
-Lemma eq_nat_nat : nat = nat.
-Proof.
- reflexivity.
-Qed.
-
-Lemma eq_Set_Set : Set = Set.
-Proof.
- reflexivity.
-Qed.
-\end{alltt}
-
-\subsection{Logical connectives.} \label{LogicalConnectives}
-The conjunction and disjunction of two propositions are also examples
-of recursive types:
-
-\begin{alltt}
-Inductive or (A B : Prop) : Prop :=
-    or_introl : A \arrow{} A {\coqor} B | or_intror : B \arrow{} A {\coqor} B
-
-Inductive and (A B : Prop) : Prop :=  
-    conj : A \arrow{} B \arrow{} A {\coqand} B
-
-\end{alltt}
-
-The propositions $A$ and $B$ are general parameters of these
-connectives. Choosing different universes for 
-$A$ and $B$ and for the inductive type itself gives rise to different
-type constructors. For example, the type \textsl{sumbool} is a
-disjunction but with computational contents.
-
-\begin{alltt}
-Inductive sumbool (A B : Prop) : Set :=
-    left : A \arrow{} \{A\} + \{B\} | right : B \arrow{} \{A\} + \{B\}
-\end{alltt}
-
-
-
-This type --noted \texttt{\{$A$\}+\{$B$\}} in {\coq}-- can be used in {\coq}
-programs as a sort of boolean type, to check whether it is $A$ or $B$
-that is true.  The values ``~\citecoq{left $p$}~'' and
-``~\citecoq{right $q$}~'' replace the boolean values \textsl{true} and
-\textsl{false}, respectively. The advantage of this type over
-\textsl{bool} is that it makes available the proofs $p$ of $A$ or $q$
-of $B$, which could be necessary to construct a verification proof
-about the program.
-For instance, let us consider the certified program \citecoq{le\_lt\_dec}
-of the Standard Library.
-
-\begin{alltt}
-Require Import Compare_dec.
-Check le_lt_dec.
-\it
-le_lt_dec
-     : {\prodsym} n m : nat, \{n {\coqle} m\} + \{m < n\}
-
-\end{alltt}
-
-We use \citecoq{le\_lt\_dec} to build a function for computing
-the max of two natural numbers:
-
-\begin{alltt}
-Definition max (n p :nat) := match le_lt_dec n p with 
-                             | left _ {\funarrow} p
-                             | right _ {\funarrow} n
-                             end.
-\end{alltt}
-
-In the following proof, the case analysis on the term
-``~\citecoq{le\_lt\_dec n p}~'' gives us an access to proofs
-of $n\leq p$ in the first case, $p<n$ in the other.
-
-\begin{alltt}
-Theorem le_max : {\prodsym} n p, n {\coqle} p {\arrow} max n p = p.
-Proof.
- intros n p ; unfold max ; case (le_lt_dec n p); simpl.
-\it
-2 subgoals
-  
-  n : nat
-  p : nat
-  ============================
-   n {\coqle} p {\arrow} n {\coqle} p {\arrow} p = p
-
-subgoal 2 is:
- p < n {\arrow} n {\coqle} p {\arrow} n = p
-\tt
- trivial.
- intros; absurd (p < p); eauto with arith.
-Qed.
-\end{alltt}
-
-
- Once the program verified, the proofs are
-erased by the extraction procedure:
-
-\begin{alltt}
-Extraction max.
-\it
-(** val max : nat {\arrow} nat {\arrow} nat **)
-
-let max n p =
-  match le_lt_dec n p with
-    | Left {\arrow} p
-    | Right {\arrow} n
-\end{alltt}
-
-Another example of use of \citecoq{sumbool} is given  in Section
-\ref{WellFoundedRecursion}: the theorem \citecoq{eq\_nat\_dec} of
-library \citecoq{Coq.Arith.Peano\_dec} is used in an euclidean division
-algorithm.
-
-\subsection{The existential quantifier.}\label{ex-def}
-The existential quantifier is yet another example of a logical
-connective introduced as an inductive type.
-
-\begin{alltt}
-Inductive ex (A : Type) (P : A \arrow{} Prop) : Prop :=
-    ex_intro : {\prodsym} x : A, P x \arrow{} ex P
-\end{alltt}
-
-Notice that {\coq} uses the abreviation ``~\citecoq{\exsym\,$x$:$A$, $B$}~''
-for \linebreak ``~\citecoq{ex (fun $x$:$A$ \funarrow{} $B$)}~''.
-
-
-\noindent The former quantifier inhabits the universe of propositions.
-As for the conjunction and disjunction connectives, there is also another
-version of existential quantification inhabiting the universes $\Type_i$,
-which is written \texttt{sig $P$}. The syntax
-``~\citecoq{\{$x$:$A$ | $B$\}}~'' is an abreviation for ``~\citecoq{sig (fun $x$:$A$ {\funarrow} $B$)}~''.
-
-
-
-%\paragraph{The logical connectives.} Conjuction and disjuction are 
-%also introduced as recursive types:
-%\begin{alltt}
-%Print or.
-%\end{alltt}
-%begin{alltt}
-%Print and.
-%\end{alltt}
-
-
-\subsection{Mutually Dependent Definitions}
-\label{MutuallyDependent}
-
-Mutually dependent definitions of recursive types are also allowed in
-{\coq}.  A typical example of these kind of declaration is the
-introduction of the trees of unbounded (but finite) width:
-\label{Forest}
-\begin{alltt} 
-Inductive tree(A:Type)   : Type :=
-    node : A {\arrow} forest A \arrow{} tree A 
-with  forest (A: Set)   : Type := 
-    nochild  : forest A |
-    addchild : tree A \arrow{} forest A \arrow{} forest A.
-\end{alltt}
-\noindent  Yet another example of mutually dependent types are the
-predicates \texttt{even} and \texttt{odd} on natural numbers:
-\label{Even}
-\begin{alltt} 
-Inductive 
-  even    : nat\arrow{}Prop :=
-    evenO : even  O |
-    evenS : {\prodsym} n, odd n \arrow{} even (S n)
-with
-  odd    : nat\arrow{}Prop :=
-    oddS : {\prodsym} n, even n \arrow{} odd (S n).
-\end{alltt}
-
-\begin{alltt}
-Lemma odd_49 : odd (7 * 7).
- simpl; repeat constructor.
-Qed.
-\end{alltt}
-
-
-
-\section{Case Analysis and Pattern-matching}
-\label{CaseAnalysis}
-\subsection{Non-dependent Case Analysis}
-An \textsl{elimination rule} for the type $A$ is some way to use an
-object $a:A$ in order to define an object in some  type $B$. 
-A natural elimination rule for an inductive type is \emph{case analysis}.
-
-
-For instance, any value of type {\nat} is built using either \texttt{O} or \texttt{S}.
-Thus, a systematic way of building a value of type $B$ from any 
-value of type {\nat} is to associate to \texttt{O} a constant $t_O:B$ and
-to every term of the form ``~\texttt{S $p$}~'' a term $t_S:B$. The following
-construction has type $B$:
-\begin{alltt}
-match \(n\) return \(B\) with O \funarrow \(t\sb{O}\) | S p \funarrow \(t\sb{S}\) end
-\end{alltt}
-
-
-In most of the cases, {\coq} is able to infer the type $B$ of the object
-defined, so the ``\texttt{return $B$}'' part can be omitted.
-
-The computing rules associated with this construct are the expected ones 
-(the notation $t_S\{q/\texttt{p}\}$ stands for the substitution of $p$ by
-$q$ in $t_S$ :)
-
-\begin{eqnarray*}
-\texttt{match $O$ return $b$ with O {\funarrow} $t_O$ | S p {\funarrow} $t_S$ end} &\Longrightarrow& t_O\\
-\texttt{match $S\;q$  return $b$ with O {\funarrow} $t_O$ | S p {\funarrow} $t_S$ end}  &\Longrightarrow& t_S\{q/\texttt{p}\}
-\end{eqnarray*}
-
-
-\subsubsection{Example: the predecessor function.}\label{firstpred}
-An example of a definition by case analysis is the function which
-computes the predecessor of any given natural number:
-\begin{alltt}
-Definition pred (n:nat) := match n with
-                                   | O {\funarrow} O 
-                                   | S m {\funarrow} m 
-                           end.
-
-Eval simpl in pred 56.
-\it{}    = 55
-     : nat
-\tt
-Eval simpl in pred 0.
-\it{}    = 0
-     : nat
-
-\tt{}Eval simpl in fun p {\funarrow} pred (S p).
-\it{}     = fun p : nat {\funarrow} p
-     : nat {\arrow} nat
-\end{alltt}
-
-As in functional programming, tuples and wild-cards can be used in
-patterns \refmancite{Section \ref{ExtensionsOfCases}}. Such
-definitions are automatically compiled by {\coq} into an expression which
-may contain several nested case expressions. For example, the 
-exclusive \emph{or} on booleans can be defined as follows:
-\begin{alltt}
-Definition xorb (b1 b2:bool) :=
- match b1, b2 with 
- | false, true {\funarrow} true
- | true, false {\funarrow} true
- | _ , _       {\funarrow} false
- end.
-\end{alltt}
-
-This kind of definition is compiled in {\coq} as follows\footnote{{\coq} uses
-the conditional ``~\citecoq{if $b$ then $a$ else $b$}~'' as an abreviation to
-``~\citecoq{match $b$ with true \funarrow{} $a$ | false \funarrow{} $b$ end}~''.}:
-
-\begin{alltt}
-Print xorb.
-xorb = 
-fun b1 b2 : bool {\funarrow}
-if b1 then if b2 then false else true 
-      else if b2 then true else false
-     : bool {\arrow} bool {\arrow} bool
-\end{alltt}
-
-\subsection{Dependent Case Analysis}
-\label{DependentCase}
-
-For a pattern matching construct of the form
-``~\citecoq{match n with \dots end}~'' a more general typing rule
-is obtained considering that the type of the whole expression
-may also depend on \texttt{n}.
-  For instance, let us consider some function 
-$Q:\texttt{nat}\arrow{}\texttt{Type}$, and $n:\citecoq{nat}$.
-In order to build a term of type $Q\;n$, we can associate
-to the constructor \texttt{O} some term $t_O: Q\;\texttt{O}$ and to
-the pattern ``~\texttt{S p}~''  some term $t_S : Q\;(S\;p)$.
-Notice that the terms $t_O$ and $t_S$ do not have the same type.
-
-The syntax of the \emph{dependent case analysis} and its
-associated typing rule make precise how the resulting
-type depends on the argument of the pattern matching, and
-which constraint holds on the branches of the pattern matching:
-
-\label{Prod-sup-rule}
-\[
-\begin{array}[t]{l}
-Q: \texttt{nat}{\arrow}\texttt{Type}\quad{t_O}:{{Q\;\texttt{O}}}  \quad
-\smalljuge{p:\texttt{nat}}{t_p}{{Q\;(\texttt{S}\;p)}} \quad n:\texttt{nat} \\
-\hline
-{\texttt{match \(n\) as \(n\sb{0}\) return \(Q\;n\sb{0}\) with | O \funarrow \(t\sb{O}\) | S p \funarrow \(t\sb{S}\) end}}:{{Q\;n}}
-\end{array}
-\]
-
-
-The interest of this rule of \textsl{dependent} pattern-matching is
-that it can also be read as the following logical principle (when $Q$ has type \citecoq{nat\arrow{}Prop}
-by \texttt{Prop} in the type of $Q$): in order to prove
-that a property $Q$ holds for all $n$, it is sufficient to prove that
-$Q$ holds for {\Z} and that for all $p:\nat$, $Q$ holds for
-$(\SUCC\;p)$.  The former, non-dependent version of case analysis can
-be obtained from this latter rule just taking $Q$ as a constant
-function on $n$.
-
-Notice that destructuring $n$ into \citecoq{O} or ``~\citecoq{S p}~''
- doesn't
-make appear in the goal the equalities ``~$n=\citecoq{O}$~''
- and ``~$n=\citecoq{S p}$~''.
-They are ``internalized'' in the rules above (see section~\ref{inversion}.)
-
-\subsubsection{Example: strong specification of the predecessor function.}
-
-In Section~\ref{firstpred}, the predecessor function was defined directly
-as a function from \texttt{nat} to \texttt{nat}. It remains to prove
-that this function has some desired properties. Another way to proceed
-is to, first introduce a specification of what is the predecessor of a 
-natural number, under the form of a {\coq} type, then build an inhabitant 
-of this type: in other words, a realization of this specification. This way, the correctness
-of this realization is ensured by {\coq}'s type system.
-
-A reasonable specification for $\pred$ is to say that for all $n$
-there exists another $m$ such that either $m=n=0$, or $(\SUCC\;m)$
-is equal to  $n$. The function $\pred$ should be just the way to
-compute such an $m$. 
-
-\begin{alltt}
-Definition pred_spec (n:nat) := 
-   \{m:nat | n=0{\coqand} m=0 {\coqor} n = S m\}.
-  
-Definition  predecessor : {\prodsym} n:nat, pred_spec n.
- intro n; case n.
-\it{}  
-  n : nat
-  ============================
-   pred_spec 0
-
-\tt{} unfold pred_spec;exists 0;auto.
-\it{}
- =========================================
- {\prodsym} n0 : nat, pred_spec (S n0)
-\tt{}
- unfold pred_spec; intro n0; exists n0; auto.
-Defined.
-\end{alltt}
-
-If we print the term built by {\coq}, its dependent pattern-matching structure can be observed:
-
-\begin{alltt}
-predecessor =  fun n : nat {\funarrow}
-\textbf{match n as n0 return (pred_spec n0) with}
-\textbf{| O {\funarrow}}
-    exist (fun m : nat {\funarrow} 0 = 0 {\coqand} m = 0 {\coqor} 0 = S m) 0
-      (or_introl (0 = 1) 
-                 (conj (refl_equal 0) (refl_equal 0)))
-\textbf{| S n0 {\funarrow}}
-    exist (fun m : nat {\funarrow} S n0 = 0 {\coqand} m = 0 {\coqor} S n0 = S m) n0
-      (or_intror (S n0 = 0 {\coqand} n0 = 0) (refl_equal (S n0)))
-\textbf{end}  : {\prodsym} n : nat, \textbf{pred_spec n}
-\end{alltt}
-
-
-Notice that there are many variants to the pattern ``~\texttt{intros \dots; case \dots}~''. Look at for tactics
-``~\texttt{destruct}~'', ``~\texttt{intro \emph{pattern}}~'', etc. in 
-the reference manual and/or the book. 
-
-\noindent The command \texttt{Extraction} \refmancite{Section
-\ref{ExtractionIdent}} can be used to see the computational
-contents associated to the \emph{certified} function \texttt{predecessor}:
-\begin{alltt}
-Extraction predecessor.
-\it
-(** val predecessor : nat {\arrow} pred_spec **)
-
-let predecessor = function
-  | O {\arrow} O
-  | S n0 {\arrow} n0
-\end{alltt}
-
-
-\begin{exercise} \label{expand}
-Prove the following theorem:
-\begin{alltt}
-Theorem nat_expand : {\prodsym} n:nat, 
-      n = match n with 
-                  | 0 {\funarrow} 0 
-                  | S p {\funarrow} S p 
-          end.
-\end{alltt}
-\end{exercise}
-
-\subsection{Some Examples of Case Analysis}
-\label{CaseScheme}
-The reader will find in the Reference manual all details about
-typing case analysis (chapter 4: Calculus of Inductive Constructions,
-and chapter 15: Extended Pattern-Matching).
-
-The following commented examples will show the different situations to consider.
-
-
-%\subsubsection{General Scheme}
-
-%Case analysis is then the most basic elimination rule that {\coq}
-%provides for inductive  types.  This rule follows a general schema,
-%valid for any inductive type $I$. First, if $I$ has type
-%``~$\forall\,(z_1:A_1)\ldots(z_r:A_r),S$~'', with $S$ either $\Set$, $\Prop$ or
-%$\Type$, then a case expression on $p$ of type ``~$R\;a_1\ldots a_r$~''
-% inhabits ``~$Q\;a_1\ldots a_r\;p$~''. The types of the branches of the case expression
-%are obtained from the definition of the type in this way: if the type
-%of the $i$-th constructor $c_i$  of $R$ is 
-%``~$\forall\, (x_1:T_1)\ldots
-%(x_n:T_n),(R\;q_1\ldots q_r)$~'', then the  $i-th$ branch must have the
-%form ``~$c_i\; x_1\; \ldots \;x_n\;  \funarrow{}\; t_i$~'' where
-%$$(x_1:T_1),\ldots, (x_n:T_n) \vdash t_i : Q\;q_1\ldots q_r)$$
-% for non-dependent case
-%analysis, and $$(x_1:T_1)\ldots (x_n:T_n)\vdash t_i :Q\;q_1\ldots
-%q_r\;({c}_i\;x_1\;\ldots x_n)$$ for dependent one. In the
-%following section, we illustrate this general scheme for different
-%recursive types.
-%%\textbf{A vérifier}
-
-\subsubsection{The Empty Type}
-
-In a definition by case analysis, there is one branch for each
-introduction rule of the type. Hence, in a definition by case analysis
-on $p:\False$ there are no cases to be considered. In other words, the
-rule of (non-dependent) case analysis for the type $\False$ is 
-(for $s$ in  \texttt{Prop}, \texttt{Set} or \texttt{Type}):
-
-\begin{center}
-\snregla {\JM{Q}{s}\;\;\;\;\;
-          \JM{p}{\False}}
-         {\JM{\texttt{match $p$ return $Q$ with end}}{Q}}
-\end{center}
-
-As a corollary, if we could construct an object in $\False$, then it
-could be possible to define an object in any type.  The tactic
-\texttt{contradiction} \refmancite{Section \ref{Contradiction}}
-corresponds to the application of the elimination rule above.  It
-searches in the context for an absurd hypothesis (this is, a
-hypothesis whose type is $\False$) and then proves the goal by a case
-analysis of it.
-
-\begin{alltt}
-Theorem fromFalse : False \arrow{} 0=1.
-Proof.
- intro H. 
- contradiction.
-Qed.
-\end{alltt}
-
-
-In {\coq} the negation is defined as follows :
-
-\begin{alltt}
-Definition not (P:Prop) := P {\arrow} False
-\end{alltt}
-
-The proposition ``~\citecoq{not $A$}~'' is also written ``~$\neg A$~''.
-
-If $A$ and $B$ are propositions, $a$ is a proof of $A$ and
-$H$ is a proof of $\neg A$,
-the term ``~\citecoq{match $H\;a$ return $B$ with end}~'' is a proof term of
-$B$.
-Thus, if your goal is $B$ and you have some hypothesis $H:\neg A$,
-the tactic ``~\citecoq{case $H$}~'' generates a new subgoal with
-statement $A$, as shown by  the following example\footnote{Notice that
-$a\coqdiff b$ is just an abreviation for ``~\coqnot a= b~''}.
-
-\begin{alltt}
-Fact Nosense : 0 {\coqdiff} 0 {\arrow} 2 = 3.
-Proof.
-  intro H; case H.
-\it
-===========================
-  0 = 0
-\tt
-  reflexivity.
-Qed.
-\end{alltt}
-
-The tactic ``~\texttt{absurd $A$}~'' (where $A$ is any proposition), 
-is based on the same principle, but
-generates two subgoals: $A$ and $\neg A$, for solving $B$.
-
-\subsubsection{The Equality Type}
-
-Let $A:\Type$, $a$, $b$ of type $A$, and $\pi$ a proof of 
-$a=b$. Non dependent case analysis of $\pi$ allows us to
-associate to any proof  of ``~$Q\;a$~'' a proof of ``~$Q\;b$~'',
-where $Q:A\arrow{} s$ (where $s\in\{\Prop, \Set, \Type\}$).
-The following term is a proof  of ``~$Q\;a\, \arrow{}\, Q\;b$~''.
-
-\begin{alltt}
-fun H : Q a {\funarrow}
-  match \(\pi\) in (_ = y) return Q y with
-     refl_equal {\funarrow} H
-  end
-\end{alltt}
-Notice the header of the \texttt{match} construct.
-It expresses how the resulting type ``~\citecoq{Q y}~'' depends on 
-the \emph{type} of \texttt{p}.
-Notice also that in the pattern introduced by the keyword \texttt{in},
-the parameter \texttt{a} in the type ``~\texttt{a = y}~'' must be
-implicit, and replaced by a wildcard '\texttt{\_}'.
-
-
-Therefore, case analysis on a proof of the equality $a=b$
-amounts to replacing all the occurrences of the term $b$ with the term
-$a$ in the goal to be proven. Let us illustrate this through an
-example: the transitivity property of this equality. 
-\begin{alltt}
-Theorem trans : {\prodsym} n m p:nat, n=m \arrow{} m=p \arrow{} n=p.
-Proof.
- intros n m p eqnm.  
-\it{}  
-  n : nat
-  m : nat
-  p : nat
-  eqnm : n = m
-  ============================
-   m = p {\arrow} n = p
-\tt{} case eqnm.
-\it{}
-  n : nat
-  m : nat
-  p : nat
-  eqnm : n = m
-  ============================
-   n = p {\arrow} n = p
-\tt{} trivial.
-Qed.
-\end{alltt}
-
-%\noindent The case analysis on the hypothesis $H:n=m$ yields the
-%tautological subgoal $n=p\rightarrow n=p$, that is directly proven by
-%the tactic \texttt{Trivial}.
-
-\begin{exercise}
-Prove the symmetry property of  equality.
-\end{exercise}
-
-Instead of using \texttt{case}, we can use the tactic 
-\texttt{rewrite} \refmancite{Section \ref{Rewrite}}.  If $H$ is a proof
-of  $a=b$, then
-``~\citecoq{rewrite $H$}~''
- performs a case analysis on a proof of $b=a$, obtained by applying a
-symmetry theorem to $H$. This application of symmetry allows us to rewrite
-the equality from left to right, which looks more natural. An optional
-parameter (either \texttt{\arrow{}} or \texttt{$\leftarrow$}) can be used to precise
-in which sense the equality must be rewritten.  By default,
-``~\texttt{rewrite} $H$~'' corresponds to ``~\texttt{rewrite \arrow{}} $H$~''
-\begin{alltt}
-Lemma Rw :  {\prodsym} x y: nat, y = y * x {\arrow} y * x * x = y.
- intros x y e; do 2 rewrite <- e.
-\it
-1 subgoal
-  
-  x : nat
-  y : nat
-  e : y = y * x
-  ============================
-   y = y
-\tt
- reflexivity.
-Qed.
-\end{alltt}
-
-Notice that, if $H:a=b$, then the tactic ``~\texttt{rewrite $H$}~''
- replaces \textsl{all} the
-occurrences of $a$ by $b$. However, in certain situations we could be
-interested in rewriting some of the occurrences, but not all of them.
-This can be done using the tactic \texttt{pattern} \refmancite{Section
-\ref{Pattern}}.  Let us consider yet another example to
-illustrate this.
-
-Let us start with some simple theorems of arithmetic; two of them 
-are already proven in the Standard Library, the last is left as an exercise.
-
-\begin{alltt}
-\it
-mult_1_l
-     : {\prodsym} n : nat, 1 * n = n
-
-mult_plus_distr_r
-     : {\prodsym} n m p : nat, (n + m) * p = n * p + m * p
-
-mult_distr_S : {\prodsym} n p : nat, n * p + p = (S n)* p.
-\end{alltt}
-
-Let us now prove a simple result:
-
-\begin{alltt}
-Lemma four_n : {\prodsym} n:nat, n+n+n+n = 4*n.
-Proof.
- intro n;rewrite <- (mult_1_l n).
-\it
-  n : nat
-  ============================
-   1 * n + 1 * n + 1 * n + 1 * n = 4 * (1 * n)
-\end{alltt}
-
-We can see that the \texttt{rewrite} tactic call replaced \emph{all}
-the occurrences of \texttt{n} by the term ``~\citecoq{1 * n}~''.
-If we want to do the rewriting ony on the leftmost occurrence of
-\texttt{n}, we can mark this occurrence using the \texttt{pattern}
-tactic:
-
-
-\begin{alltt}
- Undo.
- intro n; pattern n at 1.
- \it
- n : nat
-  ============================
- (fun n0 : nat {\funarrow} n0 + n + n + n = 4 * n) n
-\end{alltt}
-Applying the tactic ``~\citecoq{pattern  n at 1}~'' allowed us
-to explicitly abstract the first occurrence of \texttt{n} from the
-goal, putting this goal under the form ``~\citecoq{$Q$ n}~'',
-thus pointing to \texttt{rewrite} the particular predicate on $n$
-that we search to prove. 
-
-
-\begin{alltt}
- rewrite <- mult_1_l.
-\it
-1 subgoal
-  
-  n : nat
-  ============================
-   1 * n + n + n + n = 4 * n
-\tt
- repeat rewrite   mult_distr_S.
-\it
-  n : nat
-  ============================
-   4 * n = 4 * n
-\tt
- trivial.
-Qed.
-\end{alltt}
-
-\subsubsection{The Predicate $n {\leq} m$}
-
-
-The last but one instance of the elimination schema that we will illustrate is
-case analysis for the predicate $n {\leq} m$:
-
-Let $n$ and $p$ be terms of type \citecoq{nat}, and $Q$ a predicate 
-of type $\citecoq{nat}\arrow{}\Prop$.
-If $H$ is a proof of ``~\texttt{n {\coqle} p}~'',
-$H_0$ a proof of ``~\texttt{$Q$  n}~'' and
-$H_S$ a proof of the statement ``~\citecoq{{\prodsym}m:nat, n {\coqle}  m {\arrow} Q (S m)}~'',
-then the term
-\begin{alltt}
-match H in (_ {\coqle} q) return (Q q) with
-    | le_n {\funarrow} H0
-    | le_S m Hm {\funarrow} HS m Hm
-end
-\end{alltt}
- is a proof term of ``~\citecoq{$Q$ $p$}~''.
-
-
-The two patterns of this \texttt{match} construct describe
-all possible forms of proofs of ``~\citecoq{n {\coqle} m}~'' (notice
-again that the general parameter \texttt{n} is implicit in
- the ``~\texttt{in \dots}~''
-clause and is absent from the match patterns.
-
-
-Notice that the choice of introducing some of the arguments of the
-predicate as being general parameters in its definition has
-consequences on the rule of case analysis that is derived. In
-particular, the type $Q$ of the object defined by the case expression
-only depends on the indexes of the predicate, and not on the general
-parameters.  In the definition of the predicate $\leq$, the first
-argument of this relation is a general parameter of the
-definition. Hence, the predicate $Q$ to be proven only depends on the
-second argument of the relation. In other words, the integer $n$ is
-also a general parameter of the rule of case analysis.
-
-An example of an application of this rule is the following theorem,
-showing that any integer greater or equal than $1$ is the successor of another
-natural number:
-
-\begin{alltt}
-Lemma predecessor_of_positive : 
- {\prodsym} n, 1 {\coqle} n {\arrow} {\exsym} p:nat, n = S p.
-Proof.
- intros n H;case H.
-\it
-  n : nat
-  H : 1 {\coqle} n
-  ============================
-   {\exsym} p : nat, 1 = S p
-\tt
-  exists 0; trivial.
-\it
-
-  n : nat
-  H : 1 {\coqle} n
-  ============================
-   {\prodsym} m : nat, 0 {\coqle} m {\arrow} {\exsym} p : nat, S m = S p
-\tt
-  intros m _  .
-  exists m.
-  trivial.
-Qed.
-\end{alltt}
-
-
-\subsubsection{Vectors}
-
-The \texttt{vector} polymorphic and dependent family of types will
-give an idea of the most general scheme of pattern-matching.
-
-For instance, let us define a function for computing the tail of
-any vector. Notice that we shall build a \emph{total} function,
-by considering that the tail of an empty vector is this vector itself.
-In that sense, it will be slightly different from the \texttt{Vtail}
-function of the Standard Library, which is defined only for vectors
-of type ``~\citecoq{vector $A$ (S $n$)}~''.
-
-The header of the function we want to build is the following:
-
-\begin{verbatim}
-Definition Vtail_total 
-   (A : Type) (n : nat) (v : vector A n) : vector A (pred n):=
-\end{verbatim}
-
-Since the branches will not have the same type
-(depending on the parameter \texttt{n}),
-the body of this function is a dependent pattern matching on 
-\citecoq{v}.
-So we will have :
-\begin{verbatim}
-match v in (vector _ n0) return (vector A (pred n0)) with
-\end{verbatim}
-
-The first branch  deals with the constructor \texttt{Vnil} and must
-return a value in ``~\citecoq{vector A (pred 0)}~'', convertible
-to ``~\citecoq{vector A 0}~''. So, we propose:
-\begin{alltt}
-| Vnil {\funarrow} Vnil A
-\end{alltt}
-
-The second branch considers a vector in ``~\citecoq{vector A (S n0)}~''
-of the form
-``~\citecoq{Vcons A n0 v0}~'', with ``~\citecoq{v0:vector A n0}~'',
-and must return a value of type ``~\citecoq{vector A (pred (S n0))}~'',
-which is convertible to ``~\citecoq{vector A n0}~''.
-This second branch is thus :
-\begin{alltt}
-| Vcons _ n0 v0 {\funarrow} v0
-\end{alltt}
-
-Here is the full definition:
-
-\begin{alltt}
-Definition Vtail_total 
-   (A : Type) (n : nat) (v : vector A n) : vector A (pred n):=
-match v in (vector _ n0) return (vector A (pred n0)) with
-| Vnil {\funarrow} Vnil A
-| Vcons _ n0 v0 {\funarrow} v0
-end.
-\end{alltt}
-
-
-\subsection{Case Analysis and Logical Paradoxes}
-
-In the previous section we have illustrated the general scheme for
-generating the rule of case analysis associated to some recursive type
-from the definition of the type. However, if the logical soundness is
-to be preserved, certain restrictions to this schema are
-necessary. This section provides a brief explanation of these
-restrictions.
-
-
-\subsubsection{The Positivity Condition}
-\label{postypes}
-
-In order to make sense of recursive types as types closed under their
-introduction rules, a constraint has to be imposed on the possible
-forms of such rules. This constraint, known as the
-\textsl{positivity condition}, is necessary to prevent the user from
-naively introducing some recursive types which would open the door to
-logical paradoxes.  An example of such a dangerous type is the
-``inductive type'' \citecoq{Lambda}, whose only constructor is 
-\citecoq{lambda} of type \citecoq{(Lambda\arrow False)\arrow Lambda}.
- Following the pattern
-given in Section \ref{CaseScheme}, the rule of (non dependent) case
-analysis for \citecoq{Lambda} would be the following:
-
-\begin{center}
-\snregla {\JM{Q}{\Prop}\;\;\;\;\;
-          \JM{p}{\texttt{Lambda}}\;\;\;\;\;
-          {h : {\texttt{Lambda}}\arrow\False\; \vdash\; t\,:\,Q}}
-         {\JM{\citecoq{match $p$ return $Q$ with lambda h {\funarrow} $t$  end}}{Q}}
-\end{center}
-
-In order to avoid paradoxes, it is impossible to construct
-the type \citecoq{Lambda} in {\coq}:
-
-\begin{alltt}
-Inductive Lambda : Set :=
-  lambda : (Lambda {\arrow} False) {\arrow} Lambda. 
-\it
-Error: Non strictly positive occurrence of "Lambda" in
- "(Lambda {\arrow} False) {\arrow} Lambda"
-\end{alltt}
-
-In order to explain this danger, we
-will declare some constants for simulating the construction of 
-\texttt{Lambda} as an inductive type.
-
-Let us open some section, and declare two variables, the first one for
-\texttt{Lambda}, the other for the constructor \texttt{lambda}.
-
-\begin{alltt}
-Section Paradox.
-Variable Lambda : Set.
-Variable lambda : (Lambda {\arrow} False) {\arrow}Lambda.
-\end{alltt}
-
-Since \texttt{Lambda} is not a truely inductive type, we can't use
-the \texttt{match} construct. Nevertheless, we can simulate it by a
-variable \texttt{matchL} such that the term 
-``~\citecoq{matchL $l$  $Q$ (fun $h$ : Lambda {\arrow} False {\funarrow} $t$)}~''
-should be understood  as 
-``~\citecoq{match $l$ return $Q$ with | lambda h {\funarrow} $t$)}~''
-
-
-\begin{alltt}
-Variable matchL : Lambda {\arrow} 
-                  {\prodsym} Q:Prop, ((Lambda {\arrow}False) {\arrow} Q) {\arrow}
-                  Q.
-\end{alltt}
-
->From these constants, it is possible to define application by case
-analysis. Then, through auto-application, the well-known looping term
-$(\lambda x.(x\;x)\;\lambda x.(x\;x))$ provides a proof of falsehood.
-
-\begin{alltt}
-Definition application (f x: Lambda) :False :=
-  matchL f False (fun h {\funarrow} h x).
-
-Definition Delta :  Lambda := 
-  lambda (fun x : Lambda {\funarrow} application x x).
-
-Definition loop : False := application Delta Delta.
-
-Theorem two_is_three : 2 = 3.
-Proof.
- elim loop.
-Qed.
-
-End Paradox.
-\end{alltt}
-
-\noindent This example can be seen as a formulation of Russell's
-paradox in type theory associating $(\textsl{application}\;x\;x)$ to the
-formula $x\not\in x$, and \textsl{Delta} to the set $\{ x \mid
-x\not\in x\}$. If \texttt{matchL} would satisfy the reduction rule
-associated to case analysis, that is, 
-$$ \citecoq{matchL (lambda $f$) $Q$ $h$} \Longrightarrow h\;f$$
-then the term \texttt{loop}
-would compute into itself.  This is not actually surprising, since the
-proof of the logical soundness of {\coq} strongly lays on the property
-that any well-typed term must terminate. Hence, non-termination is
-usually a synonymous of inconsistency.
-
-%\paragraph{} In this case, the construction of a non-terminating
-%program comes from the so-called \textsl{negative occurrence} of
-%$\Lambda$ in the type of the constructor $\lambda$. In order to be
-%admissible for {\coq}, all the occurrences of the recursive type in its
-%own introduction rules must be positive, in the sense on the following
-%definition:
-%
-%\begin{enumerate}
-%\item $R$ is positive in $(R\;\vec{t})$;
-%\item $R$ is positive in $(x: A)C$ if it does not 
-%occur in $A$ and $R$ is positive in $C$; 
-%\item if $P\equiv (\vec{x}:\vec{T})Q$, then $R$ is positive in $(P
-%\rightarrow C)$ if $R$ does not occur in $\vec{T}$, $R$ is positive
-%in $C$, and either
-%\begin{enumerate} 
-%\item $Q\equiv (R\;\vec{q})$ or 
-%\item $Q\equiv (J\;\vec{t})$, \label{relax}
-%      where $J$ is a recursive type, and for any term $t_i$ either : 
-%   \begin{enumerate}
-%      \item $R$ does not occur in $t_i$, or
-%      \item $t_i\equiv (z:\vec{Z})(R\;\vec{q})$, $R$ does not occur
-%       in $\vec{Z}$, $t_i$ instantiates a general
-%       parameter of $J$, and this parameter is positive in the 
-%       arguments of the constructors of $J$. 
-%   \end{enumerate}
-%\end{enumerate}
-%\end{enumerate}
-%\noindent Those types obtained by erasing option (\ref{relax}) in the
-%definition above are called \textsl{strictly positive} types.
-
-
-\subsubsection*{Remark} In this case, the construction of a non-terminating
-program comes from the so-called \textsl{negative occurrence} of
-\texttt{Lambda} in the argument of the constructor \texttt{lambda}. 
-
-The reader will find in the Reference Manual a complete formal 
-definition of the notions of \emph{positivity condition} and
-\emph{strict positivity} that an inductive definition must satisfy.
-
-
-%In order to be
-%admissible for {\coq}, the type $R$ must be positive in the types of the
-%arguments of its own introduction rules, in the sense on the following
-%definition:
-
-%\textbf{La définition du manuel de référence est plus complexe:
-%la recopier ou donner seulement des exemples?
-%}
-%\begin{enumerate}
-%\item $R$ is positive in $T$ if $R$ does not occur in $T$;
-%\item $R$ is positive in $(R\;\vec{t})$ if $R$ does not occur in $\vec{t}$;
-%\item $R$ is positive in $(x:A)C$ if it does not
-%      occur in $A$ and $R$ is positive in $C$; 
-%\item $R$ is positive in $(J\;\vec{t})$, \label{relax}
-%      if $J$ is a recursive type, and for any term $t_i$ either : 
-%   \begin{enumerate}
-%      \item $R$ does not occur in $t_i$, or
-%      \item $R$ is positive in $t_i$,  $t_i$ instantiates a general
-%       parameter of $J$, and this parameter is positive in the 
-%       arguments of the constructors of $J$. 
-%   \end{enumerate}
-%\end{enumerate}
-
-%\noindent When we can show that $R$ is positive without using the item
-%(\ref{relax}) of the definition above, then we say that $R$ is
-%\textsl{strictly positive}.
-
-%\textbf{Changer le discours sur les ordinaux}
-
-Notice that the positivity condition does not forbid us to
-put  functional recursive
-arguments in the constructors. 
-
-For instance, let us consider the type of infinitely branching trees,
-with labels in \texttt{Z}.
-\begin{alltt}
-Require Import ZArith.
-
-Inductive itree : Set :=
-| ileaf : itree
-| inode : Z {\arrow} (nat {\arrow} itree) {\arrow} itree.
-\end{alltt}
-
-In this representation, the $i$-th child of a tree 
-represented by ``~\texttt{inode $z$ $s$}~'' is obtained by applying
-the function $s$ to $i$.
-The following definitions show how to construct a tree with a single 
-node, a tree of height 1 and a tree of height 2:
-
-\begin{alltt}
-Definition isingle l := inode l (fun i {\funarrow} ileaf).
-
-Definition t1 := inode 0 (fun n {\funarrow} isingle (Z_of_nat n)).
-
-Definition t2 := 
- inode 0 
-      (fun n : nat {\funarrow} 
-                   inode (Z_of_nat n)
-                   (fun p {\funarrow} isingle (Z_of_nat (n*p)))).
-\end{alltt}
-
-
-Let us define a preorder on infinitely branching trees.
- In order to compare two non-leaf trees,
-it is necessary to compare each of their children 
- without taking care of the order in which they
-appear:
-
-\begin{alltt}
-Inductive itree_le : itree{\arrow} itree {\arrow} Prop :=
-  | le_leaf : {\prodsym} t, itree_le  ileaf t
-  | le_node : {\prodsym} l l' s s', 
-                Zle l l' {\arrow} 
-                ({\prodsym} i, {\exsym} j:nat, itree_le (s i) (s' j)){\arrow} 
-                itree_le  (inode  l s) (inode  l' s').
-
-\end{alltt}
-
-Notice that a call to the predicate \texttt{itree\_le} appears as
-a general parameter of the inductive type  \texttt{ex} (see Sect.\ref{ex-def}).
-This kind of definition is accepted by {\coq}, but may lead to some
-difficulties, since the induction principle automatically 
-generated by the system
-is not the most appropriate (see chapter 14 of~\cite{coqart} for a detailed
-explanation).
-
-
-The following definition, obtained by 
-skolemising the
-proposition \linebreak $\forall\, i,\exists\, j,(\texttt{itree\_le}\;(s\;i)\;(s'\;j))$ in
-the type of \texttt{itree\_le}, does not present this problem:
-
-
-\begin{alltt} 
-Inductive itree_le' : itree{\arrow} itree {\arrow} Prop :=
-  | le_leaf'  : {\prodsym} t, itree_le'  ileaf t
-  | le_node' : {\prodsym} l l' s s' g, 
-                  Zle l l' {\arrow}  
-                  ({\prodsym} i, itree_le' (s i) (s' (g i))) {\arrow} 
-                  itree_le'  (inode  l s) (inode  l' s').
-
-\end{alltt}
-\iffalse
-\begin{alltt}
-Lemma t1_le'_t2 :  itree_le' t1 t2.
-Proof.
- unfold t1, t2.
- constructor 2  with (fun i : nat {\funarrow} 2 * i).
- auto with zarith.
- unfold isingle;
- intro i ; constructor 2 with (fun i :nat {\funarrow} i).
- auto with zarith.
- constructor .
-Qed.
-\end{alltt}
-\fi
-
-%In general, strictly positive definitions are preferable to only
-%positive ones. The reason is that it is sometimes difficult to derive
-%structural induction combinators for the latter ones. Such combinators
-%are automatically generated for strictly positive types, but not for
-%the only positive ones. Nevertheless, sometimes non-strictly positive
-%definitions provide a smarter or shorter way of declaring a recursive
-%type.
-
-Another example is the type of trees 
- of unbounded width, in which a recursive subterm 
-\texttt{(ltree A)} instantiates the type  of polymorphic lists:
-
-\begin{alltt} 
-Require Import List.
-
-Inductive ltree  (A:Set) : Set :=  
-          lnode   : A {\arrow} list (ltree A) {\arrow} ltree A.
-\end{alltt}
-
-This declaration can be transformed  
-adding an extra type to the definition, as was done in Section
-\ref{MutuallyDependent}.
-
-
-\subsubsection{Impredicative Inductive Types}
-
-An inductive type $I$ inhabiting a universe $U$ is \textsl{predicative}
-if the introduction rules of $I$ do not make a universal
-quantification on a universe containing $U$. All the recursive types
-previously introduced are examples of predicative types. An example of
-an impredicative one is the following type:
-%\textsl{exT}, the dependent product
-%of a certain set (or proposition) $x$, and a proof of a property $P$
-%about $x$.
-
-%\begin{alltt} 
-%Print exT.
-%\end{alltt}
-%\textbf{ttention, EXT c'est ex!}
-%\begin{alltt}
-%Check (exists P:Prop, P {\arrow} not P).  
-%\end{alltt}
-
-%This type is useful for expressing existential quantification over
-%types, like ``there exists a proposition $x$ such that $(P\;x)$''
-%---written $(\textsl{EXT}\; x:Prop \mid (P\;x))$ in {\coq}. However,
-
-\begin{alltt}
-Inductive prop : Prop :=
- prop_intro : Prop {\arrow} prop.
-\end{alltt}
-
-Notice
-that the constructor of this type can be used to inject any
-proposition --even itself!-- into the type. 
-
-\begin{alltt}
-Check (prop_intro prop).\it
-prop_intro prop
-     : prop
-\end{alltt}
-
-A careless use of such a
-self-contained objects may lead to a variant of Burali-Forti's
-paradox. The construction of Burali-Forti's paradox is more
-complicated than Russel's one, so we will not describe it here, and
-point the interested reader to \cite{Bar98,Coq86}.
-
-
-Another example is the second order existential quantifier for propositions:
-
-\begin{alltt}
-Inductive ex_Prop  (P : Prop {\arrow} Prop) : Prop :=
-  exP_intro : {\prodsym} X : Prop, P X {\arrow} ex_Prop P.
-\end{alltt}
-
-%\begin{alltt}
-%(*
-%Check (match prop_inject with (prop_intro p _) {\funarrow} p end).
-
-%Error: Incorrect elimination of "prop_inject" in the inductive type
-% ex
-%The elimination predicate ""fun _ : prop {\funarrow} Prop" has type
-% "prop {\arrow} Type"
-%It should be one of :
-% "Prop"
-
-%Elimination of an inductive object of sort : "Prop"
-%is not allowed on a predicate in sort : "Type"
-%because non-informative objects may not construct informative ones.
-
-%*)
-%Print prop_inject.
-
-%(*
-%prop_inject = 
-%prop_inject = prop_intro prop (fun H : prop {\funarrow} H)
-%     : prop
-%*)
-%\end{alltt}
-
-% \textbf{Et par ça?
-%}
-
-Notice that predicativity on sort \citecoq{Set} forbids us to build
-the following definitions.
-
-
-\begin{alltt}
-Inductive aSet : Set :=
-  aSet_intro: Set {\arrow} aSet.
-
-\it{}User error: Large non-propositional inductive types must be in Type
-\tt 
-Inductive ex_Set  (P : Set {\arrow} Prop) : Set :=
-  exS_intro : {\prodsym} X : Set, P X {\arrow} ex_Set P.
-
-\it{}User error: Large non-propositional inductive types must be in Type
-\end{alltt}
-
-Nevertheless, one can define types like \citecoq{aSet} and \citecoq{ex\_Set}, as inhabitants of \citecoq{Type}.
-
-\begin{alltt}
-Inductive ex_Set  (P : Set {\arrow} Prop) : Type :=
-  exS_intro : {\prodsym} X : Set, P X {\arrow} ex_Set P.
-\end{alltt}
-
-In the following example, the inductive type \texttt{typ} can be defined,
-but the term associated with the interactive Definition of
-\citecoq{typ\_inject} is incompatible with {\coq}'s hierarchy of universes:
-
-
-\begin{alltt}
-Inductive  typ : Type := 
-  typ_intro : Type {\arrow} typ. 
-
-Definition typ_inject: typ.
- split; exact typ.
-\it Proof completed
-
-\tt{}Defined.
-\it Error: Universe Inconsistency.
-\tt
-Abort.
-\end{alltt}
-
-One possible way of avoiding this new source of paradoxes is to
-restrict the kind of eliminations by case analysis that can be done on
-impredicative types. In particular, projections on those universes
-equal or bigger than the one inhabited by the impredicative type must
-be forbidden \cite{Coq86}. A consequence of this restriction is that it
-is not possible to define the first projection of the type
-``~\citecoq{ex\_Prop $P$}~'':
-\begin{alltt}
-Check (fun (P:Prop{\arrow}Prop)(p: ex_Prop P) {\funarrow}
-      match p with exP_intro X HX {\funarrow} X end).
-\it
-Error:
-Incorrect elimination of "p" in the inductive type  
-"ex_Prop", the return type has sort "Type" while it should be 
-"Prop"
-
-Elimination of an inductive object of sort "Prop"
-is not allowed on a predicate in sort "Type"
-because proofs can be eliminated only to build proofs.
-\end{alltt}
-
-%In order to explain why, let us consider for example the following
-%impredicative type \texttt{ALambda}.
-%\begin{alltt} 
-%Inductive ALambda : Set := 
-%          alambda : (A:Set)(A\arrow{}False)\arrow{}ALambda.
-%
-%Definition Lambda : Set := ALambda.
-%Definition lambda : (ALambda\arrow{}False)\arrow{}ALambda := (alambda ALambda).
-%Lemma CaseAL : (Q:Prop)ALambda\arrow{}((ALambda\arrow{}False)\arrow{}Q)\arrow{}Q.
-%\end{alltt}
-%
-%This type contains all the elements of the dangerous type $\Lambda$
-%described at the beginning of this section.  Try to construct the
-%non-ending term $(\Delta\;\Delta)$ as an object of
-%\texttt{ALambda}. Why is it not possible?
-
-\subsubsection{Extraction Constraints}
-
-There is a final constraint on case analysis that is not motivated by
-the potential introduction of paradoxes, but for compatibility reasons
-with {\coq}'s extraction mechanism \refmancite{Appendix
-\ref{CamlHaskellExtraction}}. This mechanism is based on the
-classification of basic types into the universe $\Set$ of sets and the
-universe $\Prop$ of propositions.  The objects of a type in the
-universe $\Set$ are considered as relevant for computation
-purposes. The objects of a type in $\Prop$ are considered just as
-formalised comments, not necessary for execution. The extraction
-mechanism consists in erasing such formal comments in order to obtain
-an executable program. Hence, in general, it is not possible to define
-an object in a set (that should be kept by the extraction mechanism)
-by case analysis of a proof (which will be thrown away).
-
-Nevertheless, this general rule has an exception which is important in
-practice: if the definition proceeds by case analysis on a proof of a
-\textsl{singleton proposition} or an empty type (\emph{e.g.} \texttt{False}),
- then it is allowed. A singleton
-proposition is a non-recursive proposition with a single constructor
-$c$, all whose arguments are proofs. For example, the propositional
-equality and the conjunction of two propositions are examples of
-singleton propositions.
-
-%From the point of view of the extraction
-%mechanism, such types are isomorphic to a type containing a single
-%object $c$, so a definition $\Case{x}{c \Rightarrow b}$ is 
-%directly replaced by $b$ as an extra optimisation.
-
-\subsubsection{Strong Case Analysis on Proofs}
-
-One could consider allowing 
- to define a proposition $Q$ by case
-analysis on the proofs of another recursive proposition $R$. As we
-will see in Section \ref{Discrimination}, this would enable one to prove that
-different introduction rules of $R$ construct different
-objects. However, this property would be in contradiction with the principle
-of excluded middle of classical logic, because this principle entails
-that the proofs of a proposition cannot be distinguished. This
-principle is not provable in {\coq}, but it is frequently introduced by
-the users as an axiom, for reasoning in classical logic. For this
-reason, the definition of propositions by case analysis on proofs is
- not allowed in {\coq}.
-
-\begin{alltt}
-
-Definition comes_from_the_left (P Q:Prop)(H:P{\coqor}Q): Prop :=
- match H with
-         |  or_introl p {\funarrow} True 
-         |  or_intror q {\funarrow} False
- end.
-\it
-Error:
-Incorrect elimination of "H" in the inductive type  
-"or", the return type has sort "Type" while it should be 
-"Prop"
-
-Elimination of an inductive object of sort "Prop"
-is not allowed on a predicate in sort "Type"
-because proofs can be eliminated only to build proofs.
-
-\end{alltt}
-
-On the other hand, if we replace the proposition $P {\coqor} Q$ with
-the informative type $\{P\}+\{Q\}$, the elimination  is accepted:
-
-\begin{alltt}
-Definition comes_from_the_left_sumbool
-            (P Q:Prop)(x:\{P\} + \{Q\}): Prop :=
-  match x with
-         |  left  p {\funarrow} True 
-         |  right q {\funarrow} False
-  end.
-\end{alltt}
-
-
-\subsubsection{Summary of Constraints}
-
-To end with this section, the following table summarizes which
-universe $U_1$ may inhabit an object of type $Q$ defined by case
-analysis on $x:R$, depending on the universe $U_2$ inhabited by the
-inductive types $R$.\footnote{In the box indexed by $U_1=\citecoq{Type}$
-and $U_2=\citecoq{Set}$, the answer ``yes''  takes into account the 
-predicativity of sort \citecoq{Set}. If you are working with the
-option ``impredicative-set'', you must put in this box the
-condition ``if $R$ is predicative''.}
-
-
-\begin{center}
-%%% displease hevea less by using * in multirow rather than \LL
-\renewcommand{\multirowsetup}{\centering}
-%\newlength{\LL}
-%\settowidth{\LL}{$x : R : U_2$}
-\begin{tabular}{|c|c|c|c|c|}
-\hline
-\multirow{5}*{$x : R : U_2$} &
-\multicolumn{4}{|c|}{$Q : U_1$}\\
-\hline
-&                 &\textsl{Set}      & \textsl{Prop} & \textsl{Type}\\
-\cline{2-5}
-&\textsl{Set}     &  yes             &   yes         &  yes\\
-\cline{2-5}
-&\textsl{Prop}    & if $R$ singleton &   yes         &  no\\
-\cline{2-5}
-&\textsl{Type}    &  yes             &   yes         &  yes\\
-\hline
-\end{tabular}
-\end{center}
-
-\section{Some Proof Techniques Based on Case Analysis}
-\label{CaseTechniques}
-
-In this section we illustrate the use of case analysis as a proof
-principle, explaining the proof techniques behind three very useful
-{\coq} tactics, called \texttt{discriminate}, \texttt{injection} and
-\texttt{inversion}. 
-
-\subsection{Discrimination of introduction rules}
-\label{Discrimination}
-
-In the informal semantics of recursive types described in Section
-\ref{Introduction} it was said that each of the introduction rules of a
-recursive type is considered as being different from all the others. 
-It is possible to capture this fact inside the logical system using
-the propositional equality. We take as example the following theorem,
-stating that \textsl{O} constructs a natural number different 
-from any of those constructed with \texttt{S}. 
-
-\begin{alltt}
-Theorem S_is_not_O : {\prodsym} n, S n {\coqdiff} 0. 
-\end{alltt}
-
-In order to prove this theorem, we first define a proposition by case
-analysis on natural numbers, so that the proposition is true for {\Z}
-and false for any natural number constructed with {\SUCC}. This uses
-the empty and singleton type introduced in Sections \ref{Introduction}.
-
-\begin{alltt}
-Definition Is_zero (x:nat):= match x with 
-                                     | 0 {\funarrow} True  
-                                     | _ {\funarrow} False
-                             end.
-\end{alltt}
-
-\noindent Then, we prove the following lemma:
-
-\begin{alltt}
-Lemma O_is_zero : {\prodsym} m, m = 0 {\arrow} Is_zero m.
-Proof.
-  intros m H; subst m. 
-\it{}
-================
- Is_zero 0
-\tt{}
-simpl;trivial.
-Qed.
-\end{alltt}
-
-\noindent Finally, the proof of \texttt{S\_is\_not\_O} follows by the
-application of the previous lemma to $S\;n$.
-
-
-\begin{alltt}
-
- red; intros n Hn.
- \it{}   
-  n : nat
-  Hn : S n = 0
-  ============================
-   False \tt
-
- apply O_is_zero with (m := S n).
- assumption.
-Qed.
-\end{alltt}
-
-
-The tactic \texttt{discriminate} \refmancite{Section \ref{Discriminate}} is
-a special-purpose tactic for proving disequalities between two
-elements of a recursive type introduced by different constructors. It
-generalizes the proof method described here for natural numbers to any
-[co]-inductive type. This tactic is also capable of proving disequalities
-where the difference is not in the constructors at the head of the
-terms, but deeper inside them. For example, it can be used to prove
-the following theorem:
-
-\begin{alltt}
-Theorem disc2 : {\prodsym} n, S (S n) {\coqdiff} 1. 
-Proof.
- intros n Hn; discriminate.
-Qed.
-\end{alltt}
-
-When there is an assumption $H$ in the context stating a false
-equality $t_1=t_2$, \texttt{discriminate} solves the goal by first
-proving $(t_1\not =t_2)$ and then reasoning by absurdity with respect
-to $H$:
-
-\begin{alltt}
-Theorem disc3 : {\prodsym} n, S (S n) = 0 {\arrow} {\prodsym} Q:Prop, Q.
-Proof.
- intros n Hn Q.
- discriminate.
-Qed.
-\end{alltt}
-
-\noindent In this case, the proof proceeds by absurdity with respect
-to the false equality assumed, whose negation is proved by
-discrimination.
-
-\subsection{Injectiveness of introduction rules}
-
-Another useful property about recursive types is the
-\textsl{injectiveness} of introduction rules, i.e., that whenever two
-objects were built using the same introduction rule, then this rule
-should have been applied to the same element. This can be stated
-formally using the propositional equality:
-
-\begin{alltt}
-Theorem inj : {\prodsym} n m, S n = S m {\arrow} n = m.
-Proof.
-\end{alltt}
-
-\noindent This theorem is just a corollary of a lemma about the
-predecessor function:
-
-\begin{alltt}
- Lemma inj_pred : {\prodsym} n m, n = m {\arrow} pred n = pred m.
- Proof.
-  intros n m eq_n_m.
-  rewrite eq_n_m.
-  trivial.
- Qed.
-\end{alltt}
-\noindent Once this lemma is proven, the theorem follows directly
-from it:
-\begin{alltt}
- intros n m eq_Sn_Sm.
- apply inj_pred with (n:= S n) (m := S m); assumption.
-Qed.
-\end{alltt}
-
-This proof method is implemented by the tactic \texttt{injection}
-\refmancite{Section \ref{injection}}. This tactic is applied to
-a term $t$ of type ``~$c\;{t_1}\;\dots\;t_n = c\;t'_1\;\dots\;t'_n$~'', where $c$ is some constructor of
-an inductive type. The tactic \texttt{injection} is applied as deep as 
-possible to derive the equality of all pairs of subterms of $t_i$ and $t'_i$
-placed in the same position. All these equalities are put as antecedents 
-of the current goal.
-
-
-
-Like \texttt{discriminate}, the tactic \citecoq{injection} 
-can be also applied if $x$ does not
-occur in a direct sub-term, but somewhere deeper inside it. Its
-application may leave some trivial goals that can be easily solved
-using the tactic \texttt{trivial}.
-
-\begin{alltt}
-
- Lemma list_inject : {\prodsym} (A:Type)(a b :A)(l l':list A),
-             a :: b :: l = b :: a :: l' {\arrow} a = b {\coqand} l = l'.
-Proof.
- intros A a b l l' e.
-
-
-\it
-  e : a :: b :: l = b :: a :: l'
-  ============================
-   a = b {\coqand} l = l'
-\tt
- injection e.
-\it
-  ============================
-   l = l' {\arrow} b = a {\arrow} a = b {\arrow} a = b {\coqand} l = l'
-
-\tt{} auto.
-Qed.
-\end{alltt}
-
-\subsection{Inversion Techniques}\label{inversion}
-
-In section \ref{DependentCase}, we motivated the rule of dependent case
-analysis as a way of internalizing the informal equalities $n=O$ and
-$n=\SUCC\;p$ associated to each case. This internalisation
-consisted in instantiating $n$ with the corresponding term in the type
-of each branch. However, sometimes it could be better to internalise
-these equalities as extra hypotheses --for example, in order to use
-the tactics \texttt{rewrite}, \texttt{discriminate} or
-\texttt{injection} presented in the previous sections. This is
-frequently the case when the element analysed is denoted by a term
-which is not a variable, or when it is an object of a particular
-instance of a recursive family of types. Consider for example the
-following theorem:
-
-\begin{alltt}
-Theorem not_le_Sn_0 : {\prodsym} n:nat, ~ (S n {\coqle} 0).
-\end{alltt}
-
-\noindent Intuitively, this theorem should follow by case analysis on
-the hypothesis $H:(S\;n\;\leq\;\Z)$, because no introduction rule allows
-to instantiate the arguments of \citecoq{le} with respectively a successor
-and zero. However, there
-is no way of capturing this with the typing rule for case analysis
-presented in section \ref{Introduction}, because it does not take into
-account what particular instance of the family the type of $H$ is.
-Let us try it:
-\begin{alltt}
-Proof.
- red; intros n H; case H.
-\it 2 subgoals
-  
-  n : nat
-  H : S n {\coqle} 0
-  ============================
-   False
-
-subgoal 2 is:
- {\prodsym} m : nat, S n {\coqle} m {\arrow} False
-\tt
-Undo.
-\end{alltt}
-
-\noindent What is necessary here is to make available the equalities
-``~$\SUCC\;n = \Z$~'' and ``~$\SUCC\;m = \Z$~''
- as extra hypotheses of the
-branches, so that the goal can be solved using the
-\texttt{Discriminate} tactic. In order to obtain the desired
-equalities as hypotheses, let us prove an auxiliary lemma, that our
-theorem is a corollary of:
-
-\begin{alltt}
- Lemma not_le_Sn_0_with_constraints :
-  {\prodsym} n p , S n {\coqle} p {\arrow}  p = 0 {\arrow} False.
-  Proof.
-   intros n p H; case H .
-\it
-2 subgoals
-  
-  n : nat
-  p : nat
-  H : S n {\coqle} p
-  ============================
-   S n = 0 {\arrow} False
-
-subgoal 2 is:
- {\prodsym} m : nat, S n {\coqle} m {\arrow} S m = 0 {\arrow} False
-\tt
- intros;discriminate.
- intros;discriminate.
-Qed.
-\end{alltt}
-\noindent Our main theorem can now be solved by an application of this lemma:
-\begin{alltt}
-Show.
-\it
-2 subgoals
-  
-  n : nat
-  p : nat
-  H : S n {\coqle} p
-  ============================
-   S n = 0 {\arrow} False
-
-subgoal 2 is:
- {\prodsym} m : nat, S n {\coqle} m {\arrow} S m = 0 {\arrow} False
-\tt
- eapply not_le_Sn_0_with_constraints; eauto.
-Qed.
-\end{alltt}
-
-
-The general method to address such situations consists in changing the
-goal to be proven into an implication, introducing as preconditions
-the equalities needed to eliminate the cases that make no
-sense. This proof technique is implemented by the tactic
-\texttt{inversion} \refmancite{Section \ref{Inversion}}. In order
-to prove a goal $G\;\vec{q}$ from an object of type $R\;\vec{t}$,
-this tactic automatically generates a lemma $\forall, \vec{x}.
-(R\;\vec{x}) \rightarrow \vec{x}=\vec{t}\rightarrow \vec{B}\rightarrow
-(G\;\vec{q})$, where the list of propositions $\vec{B}$ correspond to
-the subgoals that cannot be directly proven using
-\texttt{discriminate}. This lemma can either be saved for later
-use, or generated interactively. In this latter case, the subgoals
-yielded by the tactic are the hypotheses $\vec{B}$ of the lemma. If the
-lemma has been stored, then the tactic \linebreak
- ``~\citecoq{inversion \dots using \dots}~'' can be
-used to apply it. 
-
-Let us show both techniques on our previous example:
-
-\subsubsection{Interactive mode}
-
-\begin{alltt}
-Theorem not_le_Sn_0' : {\prodsym} n:nat, ~ (S n {\coqle} 0).
-Proof.
- red; intros n H ; inversion H.
-Qed.
-\end{alltt}
-
-
-\subsubsection{Static mode}
-
-\begin{alltt}
-
-Derive Inversion le_Sn_0_inv with ({\prodsym} n :nat, S n {\coqle}  0).
-Theorem le_Sn_0'' : {\prodsym} n p : nat, ~ S n {\coqle} 0 .
-Proof.
- intros n p H; 
- inversion H using le_Sn_0_inv.
-Qed.
-\end{alltt}
-
-
-In the example above, all the cases are solved using discriminate, so
-there remains no subgoal to be proven (i.e. the list $\vec{B}$ is
-empty). Let us present a second example, where this list is not empty:
-
-
-\begin{alltt}
-TTheorem le_reverse_rules : 
-     {\prodsym} n m:nat, n {\coqle} m {\arrow} 
-                     n = m {\coqor}  
-                     {\exsym} p, n {\coqle}  p {\coqand} m = S p.
-Proof.
- intros n m H; inversion H.
-\it
-2 subgoals
-
-
-
-  
-  n : nat
-  m : nat
-  H : n {\coqle} m
-  H0 : n = m
-  ============================
-   m = m {\coqor} ({\exsym} p : nat, m {\coqle} p {\coqand} m = S p)
-
-subgoal 2 is:
- n = S m0 {\coqor} ({\exsym} p : nat, n {\coqle} p {\coqand} S m0 = S p)
-\tt
- left;trivial.
- right; exists m0; split; trivial.
-\it
-Proof completed
-\end{alltt}
-
-This example shows how this tactic can be used to ``reverse'' the
-introduction rules of a recursive type, deriving the possible premises
-that could lead to prove a given instance of the predicate. This is
-why these tactics are called \texttt{inversion} tactics: they go back
-from conclusions to premises.
-
-The hypotheses corresponding to the propositional equalities are not
-needed in this example, since the tactic does the necessary rewriting
-to solve the subgoals.  When the equalities are no longer needed after
-the inversion, it is better to use the tactic
-\texttt{Inversion\_clear}. This variant of the tactic clears from the
-context all the equalities introduced.
-
-\begin{alltt}
-Restart.
- intros n m H; inversion_clear H.
-\it
-\it
-  
-  n : nat
-  m : nat
-  ============================
-   m = m {\coqor} ({\exsym} p : nat, m {\coqle} p {\coqand} m = S p)
-\tt
- left;trivial.
-\it
-  n : nat
-  m : nat
-  m0 : nat
-  H0 : n {\coqle} m0
-  ============================
-   n = S m0 {\coqor} ({\exsym} p : nat, n {\coqle} p {\coqand} S m0 = S p)
-\tt
- right; exists m0; split; trivial.
-Qed.
-\end{alltt}
-
-
-%This proof technique works in most of the cases, but not always.  In
-%particular, it could not if the list $\vec{t}$ contains a term $t_j$
-%whose type $T$ depends on a previous term $t_i$, with $i<j$. Remark
-%that if this is the case, the propositional equality $x_j=t_j$ is not
-%well-typed, since $x_j:T(x_i)$ but $t_j:T(t_i)$, and both types are
-%not convertible (otherwise, the problem could be solved using the
-%tactic \texttt{Case}).
-
-
-
-\begin{exercise}
-Consider the following language of arithmetic expression, and 
-its operational semantics, described by a set of rewriting rules.
-%\textbf{J'ai enlevé une règle de commutativité de l'addition qui 
-%me paraissait bizarre du point de vue de la sémantique opérationnelle}
-
-\begin{alltt}
-Inductive ArithExp : Set :=
-   | Zero : ArithExp 
-   | Succ : ArithExp {\arrow} ArithExp
-   | Plus : ArithExp {\arrow} ArithExp {\arrow} ArithExp.
-
-Inductive RewriteRel : ArithExp {\arrow} ArithExp {\arrow} Prop :=
-  |  RewSucc  : {\prodsym} e1 e2 :ArithExp,
-                  RewriteRel e1 e2 {\arrow}
-                   RewriteRel (Succ e1) (Succ e2) 
-  |  RewPlus0 : {\prodsym} e:ArithExp,
-                  RewriteRel (Plus Zero e) e 
-  |  RewPlusS : {\prodsym} e1 e2:ArithExp,
-                  RewriteRel e1 e2 {\arrow}
-                  RewriteRel (Plus (Succ e1) e2) 
-                             (Succ (Plus e1 e2)).
-
-\end{alltt}
-\begin{enumerate}
-\item Prove that \texttt{Zero} cannot be rewritten any further.
-\item Prove that an expression of the form ``~$\texttt{Succ}\;e$~'' is always 
-rewritten
-into an expression of the same form.
-\end{enumerate}
-\end{exercise}
-
-%Theorem zeroNotCompute : (e:ArithExp)~(RewriteRel Zero e).
-%Intro e.
-%Red.
-%Intro H.
-%Inversion_clear H.
-%Defined.
-%Theorem evalPlus : 
-%  (e1,e2:ArithExp)
-%    (RewriteRel (Succ e1) e2)\arrow{}(EX e3 : ArithExp | e2=(Succ e3)).
-%Intros e1 e2 H.
-%Inversion_clear H.
-%Exists e3;Reflexivity.
-%Qed.
-
-
-\section{Inductive Types and Structural Induction} 
-\label{StructuralInduction}
-
-Elements of inductive types  are well-founded with
-respect to the structural order induced by the constructors of the
-type. In addition to case analysis, this extra hypothesis about
-well-foundedness justifies a stronger elimination rule for them, called
-\textsl{structural induction}.  This form of elimination consists in
-defining a value ``~$f\;x$~'' from some element $x$ of the inductive type
-$I$, assuming that values have been already associated in the same way
-to the sub-parts of $x$ of type $I$.
-
-
-Definitions by structural induction are expressed through the
-\texttt{Fixpoint} command \refmancite{Section
-\ref{Fixpoint}}. This command is quite close to the
-\texttt{let-rec} construction of functional programming languages.
-For example, the following definition introduces the addition of two
-natural numbers (already defined in the Standard Library:)
-
-\begin{alltt} 
-Fixpoint plus (n p:nat) \{struct n\} : nat :=
-  match n with
-          | 0 {\funarrow} p
-          | S m {\funarrow} S (plus m p)
- end.
-\end{alltt}
-
-The definition is by structural induction on the first argument of the
-function. This is indicated  by the ``~\citecoq{\{struct n\}}~''
-directive in the function's header\footnote{This directive is optional
-in the case of a function of a single argument}.
-  In
-order to be accepted, the definition must satisfy a syntactical
-condition, called the \textsl{guardedness condition}. Roughly
-speaking, this condition constrains the arguments of a recursive call
-to be pattern variables, issued from a case analysis of the formal
-argument of the function pointed by the \texttt{struct} directive.
- In the case of the
-function \texttt{plus}, the argument \texttt{m} in the recursive call is a
-pattern variable issued from a case analysis of \texttt{n}. Therefore, the
-definition is accepted.
-
-Notice that we could have defined the addition with structural induction 
-on its second argument:
-\begin{alltt} 
-Fixpoint plus' (n p:nat) \{struct p\} : nat :=
-    match p with
-          | 0 {\funarrow} n
-          | S q {\funarrow} S (plus' n q)
-    end.
-\end{alltt}
-
-%This notation is useful when defining a function whose decreasing
-%argument has a dependent type. As an example, consider the following
-%recursivly defined proof of the theorem
-%$(n,m:\texttt{nat})n<m \rightarrow (S\;n)<(S\;m)$:
-%\begin{alltt} 
-%Fixpoint lt_n_S [n,m:nat;p:(lt n m)] : (lt (S n) (S m)) :=
-% <[n0:nat](lt (S n) (S n0))>
-%     Cases p of
-%       lt_intro1        {\funarrow} (lt_intro1 (S n))
-%    | (lt_intro2 m1 p2) {\funarrow} (lt_intro2 (S n) (S m1) (lt_n_S n m1 p2))
-%     end.
-%\end{alltt}
-
-%The guardedness condition must be satisfied only by the last argument
-%of the enclosed list. For example, the following declaration is an
-%alternative way of defining addition:
-
-%\begin{alltt}
-%Reset add.
-%Fixpoint add [n:nat] : nat\arrow{}nat :=
-%  Cases n of 
-%       O    {\funarrow} [x:nat]x
-%    | (S m) {\funarrow} [x:nat](add m (S x))
-%  end.
-%\end{alltt}
-
-In the following definition of addition, 
-the second argument of {\tt plus{'}{'}} grows at each
-recursive call. However, as the first one always decreases, the
-definition is sound.
-\begin{alltt}
-Fixpoint plus'' (n p:nat) \{struct n\} : nat :=
- match n with
-          | 0 {\funarrow} p
-          | S m {\funarrow} plus'' m (S p)
- end.
-\end{alltt}
-
- Moreover, the argument in the recursive call
-could be a deeper component of $n$.  This is the case in the following
-definition of a boolean function determining whether a number is even
-or odd:
-
-\begin{alltt} 
-Fixpoint even_test (n:nat) : bool :=
-  match n 
-  with 0 {\funarrow}  true
-     | 1 {\funarrow}  false
-     | S (S p) {\funarrow} even_test p
-  end.
-\end{alltt}
-
-Mutually dependent definitions by structural induction are also
-allowed. For example, the previous function \textsl{even} could alternatively
-be defined using an auxiliary function \textsl{odd}:
-
-\begin{alltt}
-Reset even_test.
-
-
-
-Fixpoint even_test (n:nat) : bool :=
-  match n 
-  with 
-      | 0 {\funarrow}  true
-      | S p {\funarrow} odd_test p
-  end
-with odd_test (n:nat) : bool :=
-  match n
-  with 
-     | 0 {\funarrow} false
-     | S p {\funarrow} even_test p
- end.
-\end{alltt}
-
-%\begin{exercise}
-%Define a function by structural induction that computes the number of
-%nodes of a tree structure defined in page \pageref{Forest}.
-%\end{exercise}
-
-Definitions by structural induction are computed 
- only when they are applied, and the decreasing argument
-is a term having a constructor at the head. We can check this using
-the \texttt{Eval} command, which computes the normal form of a well
-typed term.
-
-\begin{alltt}
-Eval simpl in even_test.
-\it
-    = even_test
-     : nat {\arrow} bool
-\tt 
-Eval simpl in (fun x : nat {\funarrow} even x).
-\it
-     = fun x : nat {\funarrow} even x
-     : nat {\arrow} Prop
-\tt
-Eval simpl in (fun x : nat => plus 5 x).
-\it
-     =  fun x : nat {\funarrow} S (S (S (S (S x))))
-
-\tt
-Eval simpl in (fun x : nat {\funarrow} even_test (plus 5 x)).
-\it
-    = fun x : nat {\funarrow} odd_test x
-     : nat {\arrow} bool
-\tt
-Eval simpl in (fun x : nat {\funarrow} even_test (plus x 5)).
-\it
-    = fun x : nat {\funarrow} even_test (x + 5)
-     : nat {\arrow} bool
-\end{alltt}
- 
-
-%\begin{exercise}
-%Prove that the second definition of even satisfies the following
-%theorem:
-%\begin{verbatim}
-%Theorem unfold_even : 
-% (x:nat)
-%   (even x)= (Cases x of 
-%                 O        {\funarrow} true
-%              | (S O)     {\funarrow} false 
-%              | (S (S m)) {\funarrow} (even m)
-%              end).
-%\end{verbatim}
-%\end{exercise}
-
-\subsection{Proofs by Structural Induction}
-
-The principle of structural induction can be also used in order to
-define proofs, that is, to prove theorems. Let us call an
-\textsl{elimination combinator} any function that, given a predicate
-$P$, defines a proof of ``~$P\;x$~'' by structural induction on $x$.  In
-{\coq}, the principle of proof by induction on natural numbers is a
-particular case of an elimination combinator. The definition of this
-combinator depends on three general parameters: the predicate to be
-proven, the base case, and the inductive step:
-
-\begin{alltt}
-Section Principle_of_Induction.
-Variable    P               : nat {\arrow} Prop.
-Hypothesis  base_case       : P 0.
-Hypothesis  inductive_step  : {\prodsym} n:nat, P n {\arrow} P (S n).
-Fixpoint nat_ind  (n:nat)   : (P n) := 
-   match n return P n with
-          | 0 {\funarrow} base_case
-          | S m {\funarrow} inductive_step m (nat_ind m)
-   end. 
-
-End Principle_of_Induction.
-\end{alltt}
-
-As this proof principle is used very often, {\coq} automatically generates it
-when an inductive type is introduced.  Similar principles
-\texttt{nat\_rec} and \texttt{nat\_rect} for defining objects in the
-universes $\Set$ and $\Type$ are also automatically generated
-\footnote{In fact, whenever possible, {\coq} generates the
-principle \texttt{$I$\_rect}, then derives from it the
-weaker principles  \texttt{$I$\_ind} and \texttt{$I$\_rec}.
-If some principle has to be defined by hand, the user may try
-to build \texttt{$I$\_rect} (if possible). Thanks to {\coq}'s conversion
-rule, this principle can be used directly to build proofs and/or
-programs.}. The
-command \texttt{Scheme} \refmancite{Section \ref{Scheme}} can be
-used to generate an elimination combinator from certain parameters,
-like the universe that the defined objects must inhabit, whether the
-case analysis in the definitions must be dependent or not, etc. For
-example, it can be used to generate an elimination combinator for
-reasoning on even natural numbers from the mutually dependent
-predicates introduced in page \pageref{Even}. We do not display the
-combinators here by lack of space, but you can see them using the
-\texttt{Print} command.
-
-\begin{alltt}
-Scheme Even_induction := Minimality for even Sort Prop
-with   Odd_induction  := Minimality for odd  Sort Prop.
-\end{alltt}
-
-\begin{alltt}
-Theorem even_plus_four : {\prodsym} n:nat, even n {\arrow} even (4+n).
-Proof.
- intros n H.
- elim H using Even_induction with (P0 := fun n {\funarrow} odd (4+n));
- simpl;repeat constructor;assumption.
-Qed.
-\end{alltt}
-
-Another example of an elimination combinator is the principle 
-of double induction on natural numbers, introduced by the following
-definition:
-
-\begin{alltt}
-Section Principle_of_Double_Induction.
-Variable    P              : nat {\arrow} nat {\arrow}Prop.
-Hypothesis  base_case1     : {\prodsym} m:nat, P 0 m.
-Hypothesis  base_case2     : {\prodsym} n:nat, P (S n) 0.
-Hypothesis  inductive_step : {\prodsym} n m:nat, P n m {\arrow}
-                                       \,\, P (S n) (S m).
-
-Fixpoint nat_double_ind (n m:nat)\{struct n\} : P n m := 
- match n, m return P n m with 
- |     0 ,    x   {\funarrow}  base_case1 x 
- |  (S x),    0   {\funarrow} base_case2 x
- |  (S x), (S y) {\funarrow} inductive_step x y (nat_double_ind x y)
- end.
-End Principle_of_Double_Induction.
-\end{alltt}
-
-Changing the type of $P$ into $\nat\rightarrow\nat\rightarrow\Type$,
-another combinator for constructing 
-(certified) programs, \texttt{nat\_double\_rect}, can be defined in exactly the same way.
-This definition is left as an exercise.\label{natdoublerect}
-
-\iffalse
-\begin{alltt}
-Section Principle_of_Double_Recursion.
-Variable    P               : nat {\arrow} nat {\arrow} Type.
-Hypothesis  base_case1      : {\prodsym} x:nat, P 0 x.
-Hypothesis  base_case2      : {\prodsym} x:nat, P (S x) 0.
-Hypothesis  inductive_step   : {\prodsym} n m:nat, P n m {\arrow} P (S n) (S m).
-Fixpoint nat_double_rect (n m:nat)\{struct n\} : P n m := 
-  match n, m return P n m with 
-            0 ,     x   {\funarrow}  base_case1 x 
-         |  (S x),    0   {\funarrow} base_case2 x
-         |  (S x), (S y) {\funarrow} inductive_step x y (nat_double_rect x y)
-     end.
-End Principle_of_Double_Recursion.
-\end{alltt}
-\fi
-For instance the function computing the minimum of two natural
-numbers can be defined in the following way:
-
-\begin{alltt}
-Definition min : nat {\arrow} nat {\arrow} nat  := 
-  nat_double_rect (fun (x y:nat) {\funarrow} nat)
-                 (fun (x:nat) {\funarrow} 0)
-                 (fun (y:nat) {\funarrow} 0)
-                 (fun (x y r:nat) {\funarrow} S r).
-Eval compute in (min 5 8).
-\it
-= 5 : nat
-\end{alltt}
-
-
-%\begin{exercise}
-%
-%Define the combinator \texttt{nat\_double\_rec}, and apply it
-%to give another definition of \citecoq{le\_lt\_dec} (using the theorems
-%of the \texttt{Arith} library).
-%\end{exercise}
-
-\subsection{Using Elimination Combinators.} 
-The tactic \texttt{apply} can be used to apply one of these proof
-principles during the development of a proof. 
-
-\begin{alltt}
-Lemma not_circular : {\prodsym} n:nat, n {\coqdiff} S n.
-Proof.
- intro n.
- apply nat_ind with (P:= fun n {\funarrow} n {\coqdiff} S n).
-\it
-
-
-
-2 subgoals
-  
-  n : nat
-  ============================
-   0 {\coqdiff} 1
-
-
-subgoal 2 is:
- {\prodsym} n0 : nat, n0 {\coqdiff} S n0 {\arrow} S n0 {\coqdiff} S (S n0)
-
-\tt
- discriminate.
- red; intros n0 Hn0 eqn0Sn0;injection eqn0Sn0;trivial.
-Qed.
-\end{alltt}
-
-The tactic \texttt{elim} \refmancite{Section \ref{Elim}} is a
-refinement of \texttt{apply}, specially designed for the application
-of elimination combinators.  If $t$ is an object of an inductive type
-$I$, then ``~\citecoq{elim $t$}~'' tries to find an abstraction $P$ of the
-current goal $G$ such that $(P\;t)\equiv G$. Then it solves the goal
-applying ``~$I\texttt{\_ind}\;P$~'', where $I$\texttt{\_ind} is the
-combinator associated to $I$.  The different cases of the induction
-then appear as subgoals that remain to be solved.
-In the previous proof, the tactic call ``~\citecoq{apply nat\_ind with (P:= fun n {\funarrow} n {\coqdiff} S n)}~'' can simply be replaced with ``~\citecoq{elim n}~''.
-
-The option ``~\citecoq{\texttt{elim} $t$ \texttt{using} $C$}~''
- allows to use a
-derived combinator $C$ instead of the default one. Consider the
-following theorem, stating that equality is decidable on natural
-numbers:
-
-\label{iseqpage}
-\begin{alltt}
-Lemma eq_nat_dec : {\prodsym} n p:nat, \{n=p\}+\{n {\coqdiff} p\}.
-Proof.
- intros n p.
-\end{alltt}
-
-Let us prove this theorem using the combinator \texttt{nat\_double\_rect}
-of section~\ref{natdoublerect}. The example also illustrates how
-\texttt{elim} may sometimes fail in finding a suitable abstraction $P$
-of the goal. Note that if ``~\texttt{elim n}~''
- is used directly on the
-goal, the result is not the expected one.
-
-\vspace{12pt}
-
-%\pagebreak
-\begin{alltt}
- elim n using nat_double_rect.
-\it
-4 subgoals
-  
-  n : nat
-  p : nat
-  ============================
-   {\prodsym} x : nat, \{x = p\} + \{x {\coqdiff} p\}
-
-subgoal 2 is:
- nat {\arrow} \{0 = p\} + \{0 {\coqdiff} p\}
-
-subgoal 3 is:
- nat {\arrow} {\prodsym} m : nat, \{m = p\} + \{m {\coqdiff} p\} {\arrow} \{S m = p\} + \{S m {\coqdiff} p\}
-
-subgoal 4 is:
- nat
-\end{alltt}
-
-The four sub-goals obtained do not correspond to the premises that
-would be expected for the principle \texttt{nat\_double\_rec}. The
-problem comes from the fact that 
-this principle for eliminating $n$
-has a universally quantified formula as conclusion, which confuses
-\texttt{elim} about the right way of abstracting the goal. 
-
-%In effect, let us consider the type of the goal before the call to
-%\citecoq{elim}: ``~\citecoq{\{n = p\} + \{n {\coqdiff} p\}}~''.
-
-%Among all the abstractions that can be built by ``~\citecoq{elim n}~''
-%let us consider this one
-%$P=$\citecoq{fun n :nat {\funarrow} fun q : nat {\funarrow} {\{q= p\} + \{q {\coqdiff} p\}}}.
-%It is easy to verify that 
-%$P$ has type \citecoq{nat {\arrow} nat {\arrow} Set}, and that, if some 
-%$q:\citecoq{nat}$ is given, then $P\;q\;$ matches the current goal.
-%Then applying \citecoq{nat\_double\_rec} with $P$ generates
-%four goals, corresponding to 
- 
-
-
-
-Therefore,
-in this case the abstraction must be explicited using the 
-\texttt{pattern} tactic. Once the right abstraction is provided, the rest of
-the proof is immediate:
-
-\begin{alltt}
-Undo.
- pattern p,n.
-\it
-  n : nat
-  p : nat
-  ============================
-   (fun n0 n1 : nat {\funarrow} \{n1 = n0\} + \{n1 {\coqdiff} n0\}) p n
-\tt
- elim n using nat_double_rec.
-\it
-3 subgoals
-  
-  n : nat
-  p : nat
-  ============================
-   {\prodsym} x : nat, \{x = 0\} + \{x {\coqdiff} 0\}
-
-subgoal 2 is:
- {\prodsym} x : nat, \{0 = S x\} + \{0 {\coqdiff} S x\}
-subgoal 3 is:
- {\prodsym} n0 m : nat, \{m = n0\} + \{m {\coqdiff} n0\} {\arrow} \{S m = S n0\} + \{S m {\coqdiff} S n0\}
-
-\tt
- destruct x; auto.
- destruct x; auto.
- intros n0 m H; case H.
- intro eq; rewrite eq ; auto.
- intro neg; right; red ; injection 1; auto.
-Defined.
-\end{alltt}
-
-
-Notice that the tactic ``~\texttt{decide equality}~''
-\refmancite{Section\ref{DecideEquality}} generalises the proof
-above to a large class of inductive types.  It can be used for proving
-a proposition of the form 
-$\forall\,(x,y:R),\{x=y\}+\{x{\coqdiff}y\}$, where $R$ is an inductive datatype
-all whose constructors take informative arguments ---like for example
-the type {\nat}:
-
-\begin{alltt}
-Definition eq_nat_dec' : {\prodsym} n p:nat, \{n=p\} + \{n{\coqdiff}p\}.
- decide equality.
-Defined.
-\end{alltt}
-
-\begin{exercise}
-\begin{enumerate}
-\item Define a recursive  function of name \emph{nat2itree}
-that maps any natural number $n$ into an infinitely branching
-tree of height $n$.
-\item Provide an elimination combinator for these trees.
-\item Prove that the relation \citecoq{itree\_le} is a preorder 
-(i.e. reflexive and transitive).
-\end{enumerate}
-\end{exercise}
-
-\begin{exercise} \label{zeroton}
-Define the type of lists, and a predicate ``being an ordered list''
-using an inductive family. Then, define the function
-$(from\;n)=0::1\;\ldots\; n::\texttt{nil}$ and prove that it always generates an
-ordered list.
-\end{exercise}
-
-\begin{exercise}
-Prove that \citecoq{le' n p} and \citecoq{n $\leq$ p} are logically equivalent
-for all n and p. (\citecoq{le'} is defined in section \ref{parameterstuff}).
-\end{exercise}
-
-
-\subsection{Well-founded Recursion}
-\label{WellFoundedRecursion}
-
-Structural induction is a strong elimination rule for inductive types.
-This method can be used to define any function whose termination is
-a consequence of the well-foundedness of a certain order relation $R$ decreasing
-at each recursive call. What makes this principle so strong is the
-possibility of reasoning by structural induction on the proof that
-certain $R$ is well-founded.  In order to illustrate this we have
-first to introduce the predicate of accessibility.
-
-\begin{alltt}
-Print Acc.
-\it
-Inductive Acc (A : Type) (R : A {\arrow} A {\arrow} Prop) (x:A) : Prop :=
-    Acc_intro : ({\prodsym} y : A, R y x {\arrow} Acc R y) {\arrow} Acc R x
-For Acc: Argument A is implicit
-For Acc_intro: Arguments A, R are implicit
-
-\dots
-\end{alltt}
-
-\noindent This inductive predicate characterizes those elements $x$ of
-$A$ such that any descending $R$-chain $\ldots x_2\;R\;x_1\;R\;x$
-starting from $x$ is finite. A well-founded relation is a relation
-such that all the elements of $A$ are accessible.  
-\emph{Notice the use of parameter $x$ (see Section~\ref{parameterstuff}, page
-\pageref{parameterstuff}).}
-
-Consider now the problem of representing in {\coq} the following ML
-function $\textsl{div}(x,y)$ on natural numbers, which computes
-$\lceil\frac{x}{y}\rceil$ if $y>0$ and yields $x$ otherwise.
-
-\begin{verbatim}
-let rec div x y = 
-  if x = 0 then 0
-  else if y = 0 then x
-       else (div (x-y) y)+1;;
-\end{verbatim}
-
-
-The equality test on natural numbers can be implemented using the
-function \textsl{eq\_nat\_dec} that is defined page \pageref{iseqpage}. Giving $x$ and
-$y$, this function yields either the value $(\textsl{left}\;p)$ if
-there exists a proof $p:x=y$, or the value $(\textsl{right}\;q)$ if
-there exists $q:a\not = b$. The subtraction function is already
-defined in the library \citecoq{Minus}. 
-
-Hence, direct translation of the ML function \textsl{div} would be:
-
-\begin{alltt}
-Require Import Minus.
-
-Fixpoint div (x y:nat)\{struct x\}: nat :=
- if eq_nat_dec x 0 
-  then 0
-  else if eq_nat_dec y 0
-       then x
-       else S (div (x-y) y).
-
-\it Error:
-Recursive definition of div is ill-formed.
-In environment
-div : nat {\arrow} nat {\arrow} nat
-x : nat
-y : nat
-_ : x {\coqdiff} 0
-_ : y {\coqdiff} 0
-
-Recursive call to div has principal argument equal to
-"x - y"
-instead of a subterm of x
-\end{alltt}
-
-
-The program \texttt{div} is rejected by {\coq} because it does not verify
-the syntactical condition to ensure termination. In particular, the
-argument of the recursive call is not a pattern variable issued from a
-case analysis on $x$. 
-We would have the same problem if we had the directive
-``~\citecoq{\{struct y\}}~'' instead of ``~\citecoq{\{struct x\}}~''.
-However, we know that this program always
-stops. One way to justify its termination is to define it by
-structural induction on a proof that $x$ is accessible trough the
-relation $<$. Notice that any natural number $x$ is accessible
-for this relation. In order to do this, it is first necessary to prove
-some auxiliary lemmas, justifying that the first argument of
-\texttt{div} decreases at each recursive call.
-
-\begin{alltt}
-Lemma minus_smaller_S : {\prodsym} x y:nat, x - y < S x.
-Proof.
- intros x y; pattern y, x;
- elim x using nat_double_ind.
- destruct x0; auto with arith.
- simpl; auto with arith.
- simpl; auto with arith.
-Qed.
-
-
-Lemma minus_smaller_positive : 
- {\prodsym} x y:nat, x {\coqdiff}0 {\arrow} y {\coqdiff} 0 {\arrow}  x - y < x.
-Proof.
- destruct x; destruct y; 
- ( simpl;intros; apply minus_smaller || 
-   intros; absurd (0=0); auto).
-Qed.
-\end{alltt}
-
-\noindent The last two lemmas are necessary to prove that for any pair
-of positive natural numbers $x$ and $y$, if $x$ is accessible with
-respect to \citecoq{lt}, then so is $x-y$.
-
-\begin{alltt}
-Definition minus_decrease : {\prodsym} x y:nat, Acc lt x {\arrow} 
-                                         x {\coqdiff} 0 {\arrow} 
-                                         y {\coqdiff} 0 {\arrow}
-                                         Acc lt (x-y).
-Proof.
- intros x y H; case H.
- intros Hz posz posy. 
- apply Hz; apply minus_smaller_positive; assumption.
-Defined.
-\end{alltt}
-
-Let us take a look at the proof of the lemma \textsl{minus\_decrease}, since
-the way in which it has been proven is crucial for what follows.
-\begin{alltt}
-Print minus_decrease.
-\it
-minus_decrease = 
-fun (x y : nat) (H : Acc lt x) {\funarrow}
-match H in (Acc _ y0) return (y0 {\coqdiff} 0 {\arrow} y {\coqdiff} 0 {\arrow} Acc lt (y0 - y)) with
-| Acc_intro z Hz {\funarrow}
-    fun (posz : z {\coqdiff} 0) (posy : y {\coqdiff} 0) {\funarrow}
-    Hz (z - y) (minus_smaller_positive z y posz posy)
-end
-     : {\prodsym} x y : nat, Acc lt x {\arrow} x {\coqdiff} 0 {\arrow} y {\coqdiff} 0 {\arrow} Acc lt (x - y)
-
-\end{alltt}
-\noindent Notice that the function call 
-$(\texttt{minus\_decrease}\;n\;m\;H)$
-indeed yields an accessibility proof that is \textsl{structurally
-smaller} than its argument $H$, because it is (an application of) its
-recursive component $Hz$.  This enables to justify the following
-definition of \textsl{div\_aux}:
-
-\begin{alltt}
-Definition div_aux (x y:nat)(H: Acc lt x):nat.
- fix 3.
- intros.
- refine (if eq_nat_dec x 0 
-         then 0 
-         else if eq_nat_dec y 0 
-              then y
-              else div_aux (x-y) y _).
-\it
- div_aux : {\prodsym} x : nat, nat {\arrow} Acc lt x {\arrow} nat
-  x : nat
-  y : nat
-  H : Acc lt x
-  _ : x {\coqdiff} 0
-  _0 : y {\coqdiff} 0
-  ============================
-   Acc lt (x - y)
-
-\tt
- apply (minus_decrease x y H);auto. 
-Defined.
-\end{alltt}
-
-The main division function is easily defined, using the theorem
-\citecoq{lt\_wf} of the library \citecoq{Wf\_nat}. This theorem asserts that
-\citecoq{nat} is well founded w.r.t. \citecoq{lt}, thus any natural number
-is accessible.
-\begin{alltt}
-Definition div x y := div_aux x y (lt_wf x). 
-\end{alltt}
-
-Let us explain the proof above. In the definition of \citecoq{div\_aux},
-what decreases is not $x$ but the \textsl{proof} of the accessibility
-of $x$. The tactic ``~\texttt{fix 3}~'' is used to indicate that the proof
-proceeds by structural induction on the third argument of the theorem
---that is, on the accessibility proof. It also introduces a new
-hypothesis in the context, named as the current theorem, and with the
-same type as the goal. Then, the proof is refined with an incomplete
-proof term, containing a hole \texttt{\_}.  This hole corresponds to the proof
-of accessibility for $x-y$, and is filled up with the (smaller!)
-accessibility proof provided by the function \texttt{minus\_decrease}. 
-
-
-\noindent Let us take a look to the term \textsl{div\_aux} defined:
-
-\pagebreak
-\begin{alltt}
-Print div_aux.
-\it
-div_aux = 
-(fix div_aux (x y : nat) (H : Acc lt x) \{struct H\} : nat :=
-   match eq_nat_dec x 0 with
-   | left _ {\funarrow} 0
-   | right _ {\funarrow}
-       match eq_nat_dec y 0 with
-       | left _ {\funarrow} y
-       | right _0 {\funarrow} div_aux (x - y) y (minus_decrease x y H _ _0)
-       end
-   end)
-     : {\prodsym} x : nat, nat {\arrow} Acc lt x {\arrow} nat
-
-\end{alltt}
-
-If the non-informative parts from this proof --that is, the
-accessibility proof-- are erased, then we obtain exactly the program
-that we were looking for. 
-\begin{alltt}
-
-Extraction div.
-
-\it
-let div x y =
-  div_aux x y
-\tt
-
-Extraction div_aux.
-
-\it
-let rec div_aux x y =
-  match eq_nat_dec x O with
-    | Left {\arrow} O
-    | Right {\arrow}
-        (match eq_nat_dec y O with
-           | Left {\arrow} y
-           | Right {\arrow} div_aux (minus x y) y)
-\end{alltt}
-
-This methodology enables the representation
-of any program whose termination can be proved in {\coq}. Once the
-expected properties from this program have been verified, the
-justification of its termination can be thrown away, keeping just the
-desired computational behavior for it.
-
-\section{A case study in dependent elimination}\label{CaseStudy}
-
-Dependent types are very expressive, but ignoring some useful
-techniques can cause some problems to the beginner.
-Let us consider again the type of vectors (see section~\ref{vectors}).
-We want to prove a quite trivial property: the only value of type
-``~\citecoq{vector A 0}~'' is ``~\citecoq{Vnil $A$}~''.
-
-Our first naive attempt leads to a \emph{cul-de-sac}.
-\begin{alltt}
-Lemma vector0_is_vnil : 
-  {\prodsym} (A:Type)(v:vector A 0), v = Vnil A.
-Proof.
- intros A v;inversion v.
-\it
-1 subgoal
-  
-  A : Set
-  v : vector A 0
-  ============================
-   v = Vnil A
-\tt
-Abort.
-\end{alltt}
-
-Another attempt is to do a case analysis on a vector of any length
-$n$, under an explicit hypothesis $n=0$. The tactic 
-\texttt{discriminate} will help us to get rid of the case 
-$n=\texttt{S $p$}$. 
-Unfortunately, even the statement of our lemma is refused!
-
-\begin{alltt}
- Lemma vector0_is_vnil_aux : 
- {\prodsym} (A:Type)(n:nat)(v:vector A n), n = 0 {\arrow} v = Vnil A.
-
-\it
-Error: In environment
-A : Type
-n : nat
-v : vector A n
-e : n = 0
-The term "Vnil A" has type "vector A 0" while it is expected to have type
- "vector A n"
-\end{alltt}
-
-In effect, the equality ``~\citecoq{v = Vnil A}~'' is ill-typed and this is
-because the type ``~\citecoq{vector A n}~'' is not \emph{convertible}
-with ``~\citecoq{vector A 0}~''.
-
-This problem can be solved if we consider the heterogeneous 
-equality \citecoq{JMeq} \cite{conor:motive}
-which allows us to consider terms of different types, even if this
-equality can only be proven for terms in the same type.
-The axiom \citecoq{JMeq\_eq}, from the library \citecoq{JMeq} allows us to convert a
-heterogeneous equality to a standard one.
-
-\begin{alltt}
-Lemma vector0_is_vnil_aux : 
-   {\prodsym} (A:Type)(n:nat)(v:vector A n), 
-      n= 0 {\arrow} JMeq v (Vnil A).
-Proof.
- destruct v.
- auto.
- intro; discriminate.
-Qed.
-\end{alltt}
-
-Our property of vectors of null length can be easily proven:
-
-\begin{alltt}
-Lemma vector0_is_vnil : {\prodsym} (A:Type)(v:vector A 0), v = Vnil A.
- intros a v;apply JMeq_eq.
- apply vector0_is_vnil_aux.
- trivial.
-Qed.
-\end{alltt}
-
-It is interesting to look at another proof of 
-\citecoq{vector0\_is\_vnil}, which illustrates a technique developed
-and used by various people (consult in the \emph{Coq-club} mailing
-list archive the contributions by Yves Bertot, Pierre Letouzey, Laurent Théry,
-Jean Duprat, and Nicolas Magaud, Venanzio Capretta and Conor McBride).
-This technique is also used for unfolding  infinite list definitions
-(see chapter13 of~\cite{coqart}).
-Notice that this definition does not rely on any axiom (\emph{e.g.} \texttt{JMeq\_eq}).
-
-We first give a new definition of the identity on vectors. Before that,
-we make  the use of constructors and selectors lighter thanks to
-the implicit arguments feature:
-
-\begin{alltt}
-Implicit Arguments Vcons [A n].
-Implicit Arguments Vnil [A].
-Implicit Arguments Vhead [A n].
-Implicit Arguments Vtail [A n].
-
-Definition Vid : {\prodsym} (A : Type)(n:nat), vector A n {\arrow} vector A n.
-Proof.
- destruct n; intro v.
- exact Vnil.
- exact (Vcons (Vhead v) (Vtail v)).
-Defined.
-\end{alltt}
-
-
-Then we prove that \citecoq{Vid} is the identity on vectors:
-
-\begin{alltt}
-Lemma Vid_eq : {\prodsym} (n:nat) (A:Type)(v:vector A n), v=(Vid _ n v).
-Proof.
- destruct v.
-
-\it
-   A : Type
-  ============================
-   Vnil = Vid A 0 Vnil
-
-subgoal 2 is:
-  Vcons a v = Vid A (S n) (Vcons a v)
-\tt
- reflexivity.
- reflexivity.
-Defined.
-\end{alltt}
-
-Why defining a new identity function on vectors? The following
-dialogue shows that \citecoq{Vid} has some interesting computational
-properties:
-
-\begin{alltt}
-Eval simpl in (fun (A:Type)(v:vector A 0) {\funarrow} (Vid _ _ v)).
-\it = fun (A : Type) (_ : vector A 0) {\funarrow} Vnil
-     : {\prodsym} A : Type, vector A 0 {\arrow} vector A 0
-
-\end{alltt}
-
-Notice that the plain identity on vectors doesn't convert \citecoq{v}
-into \citecoq{Vnil}.
-\begin{alltt}
-Eval simpl in (fun (A:Type)(v:vector A 0) {\funarrow} v).
-\it = fun (A : Type) (v : vector A 0) {\funarrow} v
-     : {\prodsym} A : Type, vector A 0 {\arrow} vector A 0
-\end{alltt}
-
-Then we prove easily that any vector of length 0 is \citecoq{Vnil}:
-
-\begin{alltt}
-Theorem zero_nil : {\prodsym} A (v:vector A 0), v = Vnil.
-Proof.
- intros.
- change (Vnil (A:=A)) with (Vid _ 0 v). 
-\it
-1 subgoal
-  
-  A : Type
-  v : vector A 0
-  ============================
-   v = Vid A 0 v
-\tt
- apply Vid_eq.
-Defined.
-\end{alltt}
-
-A similar result can be proven about vectors of strictly positive
-length\footnote{As for \citecoq{Vid} and \citecoq{Vid\_eq}, this definition 
-is from Jean Duprat.}.
-
-\begin{alltt}
-
-
-Theorem decomp :
-  {\prodsym} (A : Type) (n : nat) (v : vector A (S n)),
-  v = Vcons (Vhead v) (Vtail v).
-Proof.
- intros.
- change (Vcons (Vhead v) (Vtail v)) with (Vid _  (S n) v).
-\it
- 1 subgoal
-  
-  A : Type
-  n : nat
-  v : vector A (S n)
-  ============================
-   v = Vid A (S n) v
-
-\tt{} apply Vid_eq.
-Defined.
-\end{alltt}
-
-
-Both lemmas: \citecoq{zero\_nil} and \citecoq{decomp},
-can be used to easily derive a double recursion principle
-on vectors of same length:
-
-
-\begin{alltt}
-Definition vector_double_rect : 
- {\prodsym} (A:Type) (P: {\prodsym} (n:nat),(vector A n){\arrow}(vector A n) {\arrow} Type),
-     P 0 Vnil Vnil {\arrow}
-     ({\prodsym} n (v1 v2 : vector A n) a b, P n v1 v2 {\arrow}
-          P (S n) (Vcons a v1) (Vcons  b v2)) {\arrow}
-     {\prodsym} n (v1 v2 : vector A n), P n v1 v2.
- induction n.
- intros; rewrite (zero_nil _ v1); rewrite (zero_nil _ v2).
- auto.
- intros v1 v2; rewrite (decomp _ _ v1);rewrite (decomp _ _ v2).
- apply X0; auto.
-Defined.
-\end{alltt}
-
-Notice that, due to the conversion rule of {\coq}'s type system,
-this function can be used directly with \citecoq{Prop} or \citecoq{Type}
-instead of type (thus it is useless to build 
-\citecoq{vector\_double\_ind} and \citecoq{vector\_double\_rec}) from scratch.
-
-We finish this example with showing how to define the bitwise
-\emph{or} on boolean vectors of the same length, 
-and proving a little property about this
-operation.
-
-\begin{alltt}
-Definition bitwise_or n v1 v2 : vector bool n :=
-   vector_double_rect 
-    bool 
-    (fun n v1 v2 {\funarrow} vector bool n)
-    Vnil
-    (fun n v1 v2 a b r {\funarrow} Vcons (orb a b) r) n v1 v2.
-\end{alltt}
-
-Let us  define recursively the $n$-th element of a vector. Notice
-that it must be a partial function, in case $n$ is greater or equal
-than the length of the vector. Since {\coq} only considers total
-functions, the function returns a value in an \emph{option} type.
-
-\begin{alltt}
-Fixpoint vector_nth (A:Type)(n:nat)(p:nat)(v:vector A p)
-                  \{struct v\}
-                  : option A :=
-  match n,v  with
-    _   , Vnil {\funarrow} None
-  | 0   , Vcons b  _ _ {\funarrow} Some b
-  | S n', Vcons _  p' v' {\funarrow} vector_nth A n'  p' v'
-  end.
-Implicit Arguments vector_nth [A p].
-\end{alltt}
-
-We can now prove --- using the double induction combinator ---
-a simple property relying \citecoq{vector\_nth} and \citecoq{bitwise\_or}:
-
-\begin{alltt}
-Lemma nth_bitwise :
-   {\prodsym} (n:nat) (v1 v2: vector bool n) i  a b,
-      vector_nth i v1 = Some a {\arrow}
-      vector_nth i v2 = Some b {\arrow}
-      vector_nth i (bitwise_or _ v1 v2) = Some (orb a b).
-Proof.
- intros  n v1 v2; pattern n,v1,v2.
- apply vector_double_rect.
- simpl.
- destruct i; discriminate 1.
- destruct i; simpl;auto.
- injection 1; injection 2;intros; subst a; subst b; auto.
-Qed.
-\end{alltt}
-
-
-\section{Co-inductive Types and Non-ending Constructions}
-\label{CoInduction}
-
-The objects of an inductive type are well-founded with respect to
-the constructors of the type. In other words, these objects are built
-by applying \emph{a finite number of times} the constructors of the type.
-Co-inductive types are obtained by relaxing this condition,
-and may contain non-well-founded objects \cite{EG96,EG95a}.  An
-example of a co-inductive type is the type of  infinite
-sequences formed with elements of type $A$, also called streams.  This
-type can be introduced through the following definition:
-
-\begin{alltt}
- CoInductive Stream (A: Type) :Type   := 
- | Cons : A\arrow{}Stream A\arrow{}Stream A.
-\end{alltt}
-
-If we are interested in finite or infinite sequences, we consider the type
-of \emph{lazy lists}:
-
-\begin{alltt}
-CoInductive LList (A: Type) : Type :=
- |  LNil : LList A
- |  LCons : A {\arrow} LList A {\arrow} LList A.
-\end{alltt}
-
-
-It is also possible to define  co-inductive types for the 
-trees with infinitely-many branches (see Chapter 13 of~\cite{coqart}).
-
-Structural induction is the way of expressing that inductive types
-only contain well-founded objects. Hence, this elimination principle
-is not valid for co-inductive types, and the only elimination rule for
-streams is case analysis.  This principle can be used, for example, to
-define the destructors \textsl{head} and \textsl{tail}.
-
-\begin{alltt}
- Definition head (A:Type)(s : Stream A) := 
-   match s with Cons a s' {\funarrow} a end.
-
- Definition tail (A : Type)(s : Stream A) :=
-      match s with Cons a s' {\funarrow} s' end.
-\end{alltt}
-
-Infinite objects are defined by means of (non-ending) methods of
-construction, like in lazy functional programming languages.  Such
-methods can be defined using the \texttt{CoFixpoint} command
-\refmancite{Section \ref{CoFixpoint}}. For example, the following
-definition introduces the infinite list $[a,a,a,\ldots]$:
-
-\begin{alltt}
- CoFixpoint repeat (A:Type)(a:A) : Stream A := 
-   Cons a (repeat a).
-\end{alltt}
-
-
-However, not every co-recursive definition is an admissible method of
-construction. Similarly to the case of structural induction, the
-definition must verify a \textsl{guardedness} condition to be
-accepted. This condition states that any recursive call in the
-definition must be protected --i.e, be an argument of-- some
-constructor, and only an argument of constructors \cite{EG94a}. The
-following definitions are examples of valid methods of construction:
-
-\begin{alltt}
-CoFixpoint iterate (A: Type)(f: A {\arrow} A)(a : A) : Stream A:=
-    Cons a (iterate f (f a)).
-
-CoFixpoint map 
-  (A B:Type)(f: A {\arrow} B)(s : Stream A) : Stream B:=
-  match s with Cons a tl {\funarrow} Cons (f a) (map f tl) end.
-\end{alltt}
-
-\begin{exercise}
-Define two different methods for constructing the stream which 
-infinitely alternates the values \citecoq{true} and \citecoq{false}.
-\end{exercise}
-\begin{exercise}
-Using the destructors \texttt{head} and \texttt{tail}, define a function
-which takes the n-th element of an infinite stream.
-\end{exercise}
-
-A non-ending method of construction is computed lazily. This means
-that its definition is unfolded only when the object that it
-introduces is eliminated, that is, when it appears as the argument of
-a case expression. We can check this using the command
-\texttt{Eval}.
-
-\begin{alltt}
-Eval simpl in (fun (A:Type)(a:A) {\funarrow} repeat a).
-\it  = fun (A : Type) (a : A) {\funarrow} repeat a
-     : {\prodsym} A : Type, A {\arrow} Stream A
-\tt
-Eval simpl in (fun (A:Type)(a:A) {\funarrow} head (repeat a)).
-\it  = fun (A : Type) (a : A) {\funarrow} a
-     : {\prodsym} A : Type, A {\arrow} A
-\end{alltt}
-
-%\begin{exercise}
-%Prove the following theorem:
-%\begin{verbatim}
-%Theorem expand_repeat : (a:A)(repeat a)=(Cons a (repeat a)). 
-%\end{verbatim}
-%Hint: Prove first the streams version of the lemma in exercise 
-%\ref{expand}.
-%\end{exercise}
-
-\subsection{Extensional Properties}
-
-Case analysis is also a valid proof principle for infinite
-objects. However, this principle is not sufficient to prove
-\textsl{extensional} properties, that is, properties concerning the
-whole infinite object \cite{EG95a}. A typical example of an
-extensional property is the predicate expressing that two streams have
-the same elements. In many cases, the minimal reflexive relation $a=b$
-that is used as equality for inductive types is too small to capture
-equality between streams. Consider for example the streams
-$\texttt{iterate}\;f\;(f\;x)$ and
-$(\texttt{map}\;f\;(\texttt{iterate}\;f\;x))$. Even though these two streams have
-the same elements, no finite expansion of their definitions lead to
-equal terms. In other words, in order to deal with extensional
-properties, it is necessary to construct infinite proofs. The type of
-infinite proofs of equality can be introduced as a co-inductive
-predicate, as follows:
-\begin{alltt}
-CoInductive EqSt (A: Type) : Stream A {\arrow} Stream A {\arrow} Prop :=
-  eqst : {\prodsym} s1 s2: Stream A,
-      head s1 = head s2 {\arrow}
-      EqSt (tail s1) (tail s2) {\arrow}
-      EqSt s1 s2.
-\end{alltt}
-
-It is possible to introduce proof principles for reasoning about
-infinite objects as combinators defined through
-\texttt{CoFixpoint}. However, oppositely to the case of inductive
-types, proof principles associated to co-inductive types are not
-elimination but \textsl{introduction} combinators. An example of such
-a combinator is Park's principle for proving the equality of two
-streams, usually called the \textsl{principle of co-induction}. It
-states that two streams are equal if they satisfy a
-\textit{bisimulation}.  A bisimulation is a binary relation $R$ such
-that any pair of streams $s_1$ ad $s_2$ satisfying $R$ have equal
-heads, and tails also satisfying $R$.  This principle is in fact a
-method for constructing an infinite proof:
-
-\begin{alltt}
-Section Parks_Principle.
-Variable A : Type.
-Variable    R      : Stream A {\arrow} Stream A {\arrow} Prop.
-Hypothesis  bisim1 : {\prodsym} s1 s2:Stream A, 
-                       R s1 s2 {\arrow} head s1 = head s2.
-
-Hypothesis  bisim2 : {\prodsym} s1 s2:Stream A, 
-                       R s1 s2 {\arrow} R (tail s1) (tail s2).
-
-CoFixpoint park_ppl     : 
- {\prodsym} s1 s2:Stream A, R s1 s2 {\arrow} EqSt s1 s2 :=
- fun s1 s2 (p : R s1 s2) {\funarrow}
-      eqst s1 s2 (bisim1 s1 s2 p) 
-                 (park_ppl (tail s1) 
-                           (tail s2) 
-                           (bisim2 s1 s2 p)).
-End Parks_Principle.
-\end{alltt}
-
-Let us use the principle of co-induction to prove the extensional
-equality mentioned above. 
-\begin{alltt}
-Theorem map_iterate : {\prodsym} (A:Type)(f:A{\arrow}A)(x:A),
-                       EqSt (iterate f (f x)) 
-                            (map f (iterate f x)).
-Proof.
- intros A f x.
- apply park_ppl with
-   (R:= fun s1 s2 {\funarrow}
-       {\exsym} x: A, s1 = iterate f (f x) {\coqand} 
-                s2 = map f (iterate f x)).
-
- intros s1 s2 (x0,(eqs1,eqs2));
-    rewrite eqs1; rewrite eqs2; reflexivity.
- intros s1 s2 (x0,(eqs1,eqs2)).
- exists (f x0);split;
-    [rewrite eqs1|rewrite eqs2]; reflexivity.
- exists x;split; reflexivity.
-Qed.
-\end{alltt}
-
-The use of Park's principle is sometimes annoying, because it requires
-to find an invariant relation and prove that it is indeed a
-bisimulation.  In many cases, a shorter proof can be obtained trying
-to construct an ad-hoc infinite proof, defined by a guarded
-declaration.  The tactic ``~``\texttt{Cofix $f$}~'' can be used to do
-that. Similarly to the tactic \texttt{fix} indicated in Section
-\ref{WellFoundedRecursion}, this tactic introduces an extra hypothesis
-$f$ into the context, whose type is the same as the current goal. Note
-that the applications of $f$ in the proof \textsl{must be guarded}. In
-order to prevent us from doing unguarded calls, we can define a tactic
-that always apply a constructor before using $f$ \refmancite{Chapter
-\ref{WritingTactics}} :
-
-\begin{alltt}
-Ltac infiniteproof f :=
-  cofix f; 
-  constructor; 
-  [clear f| simpl; try (apply f; clear f)].
-\end{alltt}
-
-
-In the example above, this tactic produces a much simpler proof
-that the former one:
-
-\begin{alltt}
-Theorem map_iterate' : {\prodsym} ((A:Type)f:A{\arrow}A)(x:A),
-                       EqSt (iterate f (f x))
-                            (map f (iterate f x)).
-Proof.
- infiniteproof map_iterate'.
- reflexivity.
-Qed.
-\end{alltt}
-
-\begin{exercise}
-Define a co-inductive type of name $Nat$ that contains non-standard 
-natural numbers --this is, verifying 
-
-$$\exists m  \in \mbox{\texttt{Nat}}, \forall\, n \in \mbox{\texttt{Nat}}, n<m$$.
-\end{exercise}
-
-\begin{exercise}
-Prove that the extensional equality of streams is an equivalence relation
-using Park's co-induction principle.
-\end{exercise}
-
-
-\begin{exercise}
-Provide a suitable definition of ``being an ordered list'' for infinite lists
-and define a principle for proving that an infinite list is ordered. Apply
-this method to the list $[0,1,\ldots ]$. Compare the result with 
-exercise \ref{zeroton}.
-\end{exercise}
-
-\subsection{About injection, discriminate, and inversion}
-Since co-inductive types are closed w.r.t. their constructors,
-the techniques shown in Section~\ref{CaseTechniques} work also 
-with these types.
-
-Let us consider the type of lazy lists, introduced on page~\pageref{CoInduction}.
-The following lemmas are straightforward applications
- of \texttt{discriminate} and \citecoq{injection}:
-
-\begin{alltt}
-Lemma Lnil_not_Lcons : {\prodsym} (A:Type)(a:A)(l:LList A),
-                               LNil {\coqdiff} (LCons a l).
-Proof.
- intros;discriminate.
-Qed.
-
-Lemma injection_demo : {\prodsym} (A:Type)(a b : A)(l l': LList A),
-                  LCons a (LCons b l) = LCons b (LCons a l') {\arrow}
-                  a = b {\coqand} l = l'.
-Proof.
- intros A a b l l' e; injection e; auto.
-Qed.
-
-\end{alltt}
-
-In order to show \citecoq{inversion} at work, let us define
-two predicates on lazy lists:
-
-\begin{alltt}
-Inductive Finite (A:Type) : LList A {\arrow} Prop :=
-|  Lnil_fin : Finite (LNil (A:=A))
-|  Lcons_fin : {\prodsym} a l, Finite l {\arrow} Finite (LCons a l).
-
-CoInductive Infinite  (A:Type) : LList A {\arrow} Prop :=
-| LCons_inf : {\prodsym} a l, Infinite l {\arrow} Infinite (LCons a l).
-\end{alltt}
-
-\noindent
-First, two easy theorems:
-\begin{alltt}
-Lemma LNil_not_Infinite : {\prodsym} (A:Type), ~ Infinite (LNil (A:=A)).
-Proof.
-  intros A H;inversion H.
-Qed.
-
-Lemma Finite_not_Infinite : {\prodsym} (A:Type)(l:LList A),
-                                Finite l {\arrow} ~ Infinite l.
-Proof.
- intros A l H; elim H.
- apply LNil_not_Infinite.
- intros a l0 F0 I0' I1.
- case I0'; inversion_clear I1.
- trivial.
-Qed.
-\end{alltt}
-
-
-On the other hand, the next proof uses the \citecoq{cofix} tactic. 
-Notice the destructuration of \citecoq{l}, which allows us to
-apply  the constructor \texttt{LCons\_inf}, thus  satisfying
- the guard condition: 
-\begin{alltt}
-Lemma Not_Finite_Infinite : {\prodsym} (A:Type)(l:LList A),
-                            ~ Finite l {\arrow} Infinite l.
-Proof.
- cofix H.
- destruct l.
- intro; 
-  absurd (Finite (LNil (A:=A)));
-  [auto|constructor].
-\it
-
-
-
-
-1 subgoal
-  
-  H : forall (A : Type) (l : LList A), ~ Finite l -> Infinite l
-  A : Type
-  a : A
-  l : LList A
-  H0 : ~ Finite (LCons a l)
-  ============================
-   Infinite l
-\end{alltt}
-At this point, one must not apply \citecoq{H}! . It would be possible
-to solve the current goal by an inversion of ``~\citecoq{Finite (LCons a l)}~'', but, since the guard condition would be violated, the user
-would get an error message after typing \citecoq{Qed}.
-In order to satisfy the guard condition, we apply the constructor of
-\citecoq{Infinite}, \emph{then} apply \citecoq{H}.
-
-\begin{alltt}
- constructor.
- apply H.
- red; intro H1;case H0.
- constructor.
- trivial.
-Qed.
-\end{alltt}
-
-
-
-
-The reader is invited to replay this proof and understand each of its steps.
-
-
-\bibliographystyle{abbrv}
-\bibliography{manbiblio,morebib}
-
-\end{document}
-
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