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-\chapter[Calculus of Inductive Constructions]{Calculus of Inductive Constructions
-\label{Cic}
-\index{Cic@\textsc{CIC}}
-\index{pCic@p\textsc{CIC}}
-\index{Calculus of (Co)Inductive Constructions}}
-
-The underlying formal language of {\Coq} is a {\em Calculus of
-  Constructions} with {\em Inductive Definitions}.  It is presented in
-this chapter.  
-For {\Coq} version V7, this Calculus was known as the
-{\em Calculus of (Co)Inductive Constructions}\index{Calculus of
-  (Co)Inductive Constructions} (\iCIC\ in short).
-The underlying calculus of {\Coq} version V8.0 and up is a weaker
-  calculus where the sort \Set{} satisfies predicative rules. 
-We call this calculus the 
-{\em Predicative Calculus of (Co)Inductive
-  Constructions}\index{Predicative Calculus of
-  (Co)Inductive Constructions} (\pCIC\ in short).
-In Section~\ref{impredicativity} we give the extra-rules for \iCIC. A
-  compiling option of \Coq{} allows to type-check theories in this
-  extended system.
-
-In \CIC\, all objects have a {\em type}. There are types for functions (or
-programs), there are atomic types (especially datatypes)... but also
-types for proofs and types for the types themselves.
-Especially, any object handled in the formalism must belong to a
-type.  For instance, the statement {\it ``for all x, P''} is not
-allowed in type theory; you must say instead: {\it ``for all x
-belonging to T, P''}. The expression {\it ``x belonging to T''} is
-written {\it ``x:T''}. One also says: {\it ``x has type T''}.
-The terms of {\CIC} are detailed in Section~\ref{Terms}.
-
-In \CIC\, there is an internal reduction mechanism. In particular, it
-allows to decide if two programs are {\em intentionally} equal (one
-says {\em convertible}). Convertibility is presented in section 
-\ref{convertibility}.
-
-The remaining sections are concerned with the type-checking of terms.
-The beginner can skip them.
-
-The reader seeking a background on the Calculus of Inductive
-Constructions may read several papers. Giménez and Castéran~\cite{GimCas05} 
-provide
-an introduction to inductive and coinductive definitions in Coq. In
-their book~\cite{CoqArt}, Bertot and Castéran give a precise
-description of the \CIC{} based on numerous practical examples.
-Barras~\cite{Bar99}, Werner~\cite{Wer94} and
-Paulin-Mohring~\cite{Moh97} are the most recent theses dealing with
-Inductive Definitions. Coquand-Huet~\cite{CoHu85a,CoHu85b,CoHu86}
-introduces the Calculus of Constructions. Coquand-Paulin~\cite{CoPa89}
-extended this calculus to inductive definitions. The {\CIC} is a
-formulation of type theory including the possibility of inductive
-constructions, Barendregt~\cite{Bar91} studies the modern form of type
-theory.
-
-\section[The terms]{The terms\label{Terms}}
-
-In most type theories, one usually makes a syntactic distinction
-between types and terms. This is not the case for \CIC\ which defines
-both types and terms in the same syntactical structure. This is
-because the type-theory itself forces terms and types to be defined in
-a mutual recursive way and also because similar constructions can be
-applied to both terms and types and consequently can share the same
-syntactic structure.
-
-Consider for instance the $\ra$ constructor and assume \nat\ is the
-type of natural numbers. Then $\ra$ is used both to denote
-$\nat\ra\nat$ which is the type of functions from \nat\ to \nat, and
-to denote $\nat \ra \Prop$ which is the type of unary predicates over
-the natural numbers. Consider abstraction which builds functions. It
-serves to build ``ordinary'' functions as $\kw{fun}~x:\nat \Ra ({\tt mult} ~x~x)$ (assuming {\tt mult} is already defined) but may build also 
-predicates over the natural numbers. For instance $\kw{fun}~x:\nat \Ra
-(x=x)$ will
-represent a predicate $P$, informally written in mathematics
-$P(x)\equiv x=x$. If $P$ has type $\nat \ra \Prop$, $(P~x)$ is a
-proposition, furthermore $\kw{forall}~x:\nat,(P~x)$ will represent the type of
-functions which associate to each natural number $n$ an object of
-type $(P~n)$ and consequently represent proofs of the formula
-``$\forall x.P(x)$''.
-
-\subsection[Sorts]{Sorts\label{Sorts}
-\index{Sorts}}
-Types are seen as terms of the language and then should belong to
-another type. The type of a type is always a constant of the language
-called a {\em sort}.
-
-The two basic sorts in the language of \CIC\ are \Set\ and \Prop.
-
-The sort \Prop\ intends to be the type of logical propositions.  If
-$M$ is a logical proposition then it denotes a class, namely the class
-of terms representing proofs of $M$. An object $m$ belonging to $M$
-witnesses the fact that $M$ is true.  An object of type \Prop\ is
-called a {\em proposition}.
-
-The sort \Set\ intends to be the type of specifications. This includes
-programs and the usual sets such as booleans, naturals, lists
-etc.
-
-These sorts themselves can be manipulated as ordinary terms.
-Consequently sorts also should be given a type.  Because assuming
-simply that \Set\ has type \Set\ leads to an inconsistent theory, we
-have infinitely many sorts in the language of \CIC. These are, in
-addition to \Set\ and \Prop\, a hierarchy of universes \Type$(i)$
-for any integer $i$.  We call \Sort\ the set of sorts
-which is defined by:
-\[\Sort \equiv \{\Prop,\Set,\Type(i)| i \in \NN\} \]
-\index{Type@{\Type}}
-\index{Prop@{\Prop}}
-\index{Set@{\Set}}
-The sorts enjoy the following properties: {\Prop:\Type(0)}, {\Set:\Type(0)} and
-  {\Type$(i)$:\Type$(i+1)$}.
-
-The user will never mention explicitly the index $i$ when referring to
-the universe \Type$(i)$. One only writes \Type. The
-system itself generates for each instance of \Type\ a new
-index for the universe and checks that the constraints between these
-indexes can be solved. From the user point of view we consequently
-have {\sf Type :Type}.
-
-We shall make precise in the typing rules the constraints between the
-indexes. 
-
-\paragraph{Implementation issues}
-In practice, the {\Type} hierarchy is implemented using algebraic
-universes. An algebraic universe $u$ is either a variable (a qualified
-identifier with a number) or a successor of an algebraic universe (an
-expression $u+1$), or an upper bound of algebraic universes (an
-expression $max(u_1,...,u_n)$), or the base universe (the expression
-$0$) which corresponds, in the arity of sort-polymorphic inductive
-types, to the predicative sort {\Set}. A graph of constraints between
-the universe variables is maintained globally. To ensure the existence
-of a mapping of the universes to the positive integers, the graph of
-constraints must remain acyclic.  Typing expressions that violate the
-acyclicity of the graph of constraints results in a \errindex{Universe
-inconsistency} error (see also Section~\ref{PrintingUniverses}).
-
-\subsection{Constants}
-Besides the sorts, the language also contains constants denoting
-objects in the environment. These constants may denote previously
-defined objects but also objects related to inductive definitions
-(either the type itself or one of its constructors or destructors).
-
-\medskip\noindent {\bf Remark. } In other presentations of \CIC, 
-the inductive objects are not seen as
-external declarations but as first-class terms. Usually the
-definitions are also completely ignored.  This is a nice theoretical
-point of view but not so practical. An inductive definition is
-specified by a possibly huge set of declarations, clearly we want to
-share this specification among the various inductive objects and not
-to duplicate it. So the specification should exist somewhere and the
-various objects should refer to it.  We choose one more level of
-indirection where the objects are just represented as constants and
-the environment gives the information on the kind of object the
-constant refers to.
-
-\medskip
-Our inductive objects will be manipulated as constants declared in the
-environment. This roughly corresponds to the way they are actually
-implemented in the \Coq\ system. It is simple to map this presentation
-in a theory where inductive objects are represented by terms.
-
-\subsection{Terms}
-
-Terms are built from variables, global names, constructors,
-abstraction, application, local declarations bindings (``let-in''
-expressions) and product.
-
-From a syntactic point of view, types cannot be distinguished from terms,
-except that they cannot start by an abstraction, and that if a term is
-a sort or a product, it should be a type.
-
-More precisely the language of the {\em Calculus of Inductive
-  Constructions} is built from the following rules:
-
-\begin{enumerate}
-\item the sorts {\sf Set, Prop, Type} are terms.
-\item names for global constants of the environment are terms.
-\item variables are terms.
-\item if $x$ is a variable and $T$, $U$ are terms then $\forall~x:T,U$
-  ($\kw{forall}~x:T,U$ in \Coq{} concrete syntax) is a term. If $x$
-  occurs in $U$, $\forall~x:T,U$ reads as {\it ``for all x of type T,
-    U''}. As $U$ depends on $x$, one says that $\forall~x:T,U$ is a
-  {\em dependent product}. If $x$ doesn't occurs in $U$ then
-  $\forall~x:T,U$ reads as {\it ``if T then U''}. A non dependent
-  product can be written: $T \rightarrow U$.
-\item if $x$ is a variable and $T$, $U$ are terms then $\lb~x:T \mto U$
-  ($\kw{fun}~x:T\Ra U$ in \Coq{} concrete syntax) is a term. This is a
-  notation for the $\lambda$-abstraction of
-  $\lambda$-calculus\index{lambda-calculus@$\lambda$-calculus}
-  \cite{Bar81}. The term $\lb~x:T \mto U$ is a function which maps
-  elements of $T$ to $U$.
-\item if $T$ and $U$ are terms then $(T\ U)$ is a term  
- ($T~U$ in \Coq{} concrete syntax).  The term $(T\ 
-  U)$ reads as {\it ``T applied to U''}.
-\item if $x$ is a variable, and $T$, $U$ are terms then
-  $\kw{let}~x:=T~\kw{in}~U$ is a
-  term which denotes the term $U$ where the variable $x$ is locally
-  bound to $T$. This stands for the common ``let-in'' construction of
-  functional programs such as ML or Scheme.
-\end{enumerate}
-
-\paragraph{Notations.} Application associates to the left such that
-$(t~t_1\ldots t_n)$ represents $(\ldots (t~t_1)\ldots t_n)$. The
-products and arrows associate to the right such that $\forall~x:A,B\ra C\ra
-D$ represents $\forall~x:A,(B\ra (C\ra D))$.  One uses sometimes
-$\forall~x~y:A,B$ or
-$\lb~x~y:A\mto B$ to denote the abstraction or product of several variables
-of the same type. The equivalent formulation is $\forall~x:A, \forall y:A,B$ or
-$\lb~x:A \mto \lb y:A \mto B$
-
-\paragraph{Free variables.}
-The notion of free variables is defined as usual.  In the expressions
-$\lb~x:T\mto U$ and $\forall x:T, U$ the occurrences of $x$ in $U$
-are bound.  They are represented by de Bruijn indexes in the internal
-structure of terms.
-
-\paragraph[Substitution.]{Substitution.\index{Substitution}}
-The notion of substituting a term $t$ to free occurrences of a
-variable $x$ in a term $u$ is defined as usual. The resulting term
-is written $\subst{u}{x}{t}$.
-
-
-\section[Typed terms]{Typed terms\label{Typed-terms}}
-
-As objects of type theory, terms are subjected to {\em type
-discipline}. The well typing of a term depends on an environment which
-consists in a global environment (see below) and a local context.
-
-\paragraph{Local context.}
-A {\em local context} (or shortly context) is an ordered list of
-declarations of variables. The declaration of some variable $x$ is
-either an assumption, written $x:T$ ($T$ is a type) or a definition,
-written $x:=t:T$.  We use brackets to write contexts. A
-typical example is $[x:T;y:=u:U;z:V]$.  Notice that the variables
-declared in a context must be distinct. If $\Gamma$ declares some $x$,
-we write $x \in \Gamma$. By writing $(x:T) \in \Gamma$ we mean that
-either $x:T$ is an assumption in $\Gamma$ or that there exists some $t$ such
-that $x:=t:T$ is a definition in $\Gamma$. If $\Gamma$ defines some
-$x:=t:T$, we also write $(x:=t:T) \in \Gamma$.  Contexts must be
-themselves {\em well formed}.  For the rest of the chapter, the
-notation $\Gamma::(y:T)$ (resp. $\Gamma::(y:=t:T)$) denotes the context
-$\Gamma$ enriched with the declaration $y:T$ (resp. $y:=t:T$). The
-notation $[]$ denotes the empty context.  \index{Context}
-
-% Does not seem to be used further...
-% Si dans l'explication WF(E)[Gamma] concernant les constantes
-% definies ds un contexte
-
-We define the inclusion of two contexts $\Gamma$ and $\Delta$ (written
-as $\Gamma \subset \Delta$) as the property, for all variable $x$,
-type $T$ and term $t$, if $(x:T) \in \Gamma$ then $(x:T) \in \Delta$
-and if $(x:=t:T) \in \Gamma$ then $(x:=t:T) \in \Delta$.  
-%We write
-% $|\Delta|$ for the length of the context $\Delta$, that is for the number
-% of declarations (assumptions or definitions) in $\Delta$.
-
-A variable $x$ is said to be free in $\Gamma$ if $\Gamma$ contains a
-declaration $y:T$ such that $x$ is free in $T$.
-
-\paragraph[Environment.]{Environment.\index{Environment}}
-Because we are manipulating global declarations (constants and global
-assumptions), we also need to consider a global environment $E$.
-
-An environment is an ordered list of declarations of global
-names. Declarations are either assumptions or ``standard''
-definitions, that is abbreviations for well-formed terms
-but also definitions of inductive objects. In the latter
-case, an object in the environment will define one or more constants
-(that is types and constructors, see Section~\ref{Cic-inductive-definitions}).
-
-An assumption will be represented in the environment as
-\Assum{\Gamma}{c}{T} which means that $c$ is assumed of some type $T$
-well-defined in some context $\Gamma$. An (ordinary) definition will
-be represented in the environment as \Def{\Gamma}{c}{t}{T} which means
-that $c$ is a constant which is valid in some context $\Gamma$ whose
-value is $t$ and type is $T$.
-
-The rules for inductive definitions (see section
-\ref{Cic-inductive-definitions}) have to be considered as assumption
-rules to which the following definitions apply: if the name $c$ is
-declared in $E$, we write $c \in E$ and if $c:T$ or $c:=t:T$ is
-declared in $E$, we write $(c : T) \in E$.
-
-\paragraph[Typing rules.]{Typing rules.\label{Typing-rules}\index{Typing rules}}
-In the following, we assume $E$ is a valid environment w.r.t.
-inductive definitions.  We define simultaneously two
-judgments.  The first one \WTEG{t}{T} means the term $t$ is well-typed
-and has type $T$ in the environment $E$ and context $\Gamma$.  The
-second judgment \WFE{\Gamma} means that the environment $E$ is
-well-formed and the context $\Gamma$ is a valid context in this
-environment.  It also means a third property which makes sure that any
-constant in $E$ was defined in an environment which is included in
-$\Gamma$
-\footnote{This requirement could be relaxed if we instead introduced
-  an explicit mechanism for instantiating constants. At the external
-  level, the Coq engine works accordingly to this view that all the
-  definitions in the environment were built in a sub-context of the
-  current context.}.
-
-A term $t$ is well typed in an environment $E$ iff there exists a
-context $\Gamma$ and a term $T$ such that the judgment \WTEG{t}{T} can
-be derived from the following rules.
-\begin{description}
-\item[W-E] \inference{\WF{[]}{[]}}
-\item[W-S]  % Ce n'est pas vrai : x peut apparaitre plusieurs fois dans Gamma
-\inference{\frac{\WTEG{T}{s}~~~~s \in \Sort~~~~x \not\in
-      \Gamma           % \cup E
-      }
-      {\WFE{\Gamma::(x:T)}}~~~~~
-    \frac{\WTEG{t}{T}~~~~x \not\in
-      \Gamma           % \cup E
-     }{\WFE{\Gamma::(x:=t:T)}}}
-\item[Def] \inference{\frac{\WTEG{t}{T}~~~c \notin E \cup \Gamma}
-                      {\WF{E;\Def{\Gamma}{c}{t}{T}}{\Gamma}}}
-\item[Assum] \inference{\frac{\WTEG{T}{s}~~~~s \in \Sort~~~~c \notin E \cup \Gamma}
-                      {\WF{E;\Assum{\Gamma}{c}{T}}{\Gamma}}}
-\item[Ax] \index{Typing rules!Ax}
-\inference{\frac{\WFE{\Gamma}}{\WTEG{\Prop}{\Type(p)}}~~~~~
-\frac{\WFE{\Gamma}}{\WTEG{\Set}{\Type(q)}}}
-\inference{\frac{\WFE{\Gamma}~~~~i<j}{\WTEG{\Type(i)}{\Type(j)}}}
-\item[Var]\index{Typing rules!Var}
- \inference{\frac{ \WFE{\Gamma}~~~~~(x:T) \in \Gamma~~\mbox{or}~~(x:=t:T) \in \Gamma~\mbox{for some $t$}}{\WTEG{x}{T}}}
-\item[Const]  \index{Typing rules!Const}
-\inference{\frac{\WFE{\Gamma}~~~~(c:T) \in E~~\mbox{or}~~(c:=t:T) \in E~\mbox{for some $t$} }{\WTEG{c}{T}}}
-\item[Prod]  \index{Typing rules!Prod}
-\inference{\frac{\WTEG{T}{s}~~~~s \in \Sort~~~
-    \WTE{\Gamma::(x:T)}{U}{\Prop}}
-      { \WTEG{\forall~x:T,U}{\Prop}}} 
-\inference{\frac{\WTEG{T}{s}~~~~s \in\{\Prop, \Set\}~~~~~~
-    \WTE{\Gamma::(x:T)}{U}{\Set}}
-      { \WTEG{\forall~x:T,U}{\Set}}} 
-\inference{\frac{\WTEG{T}{\Type(i)}~~~~i\leq k~~~
-    \WTE{\Gamma::(x:T)}{U}{\Type(j)}~~~j \leq k}
-    {\WTEG{\forall~x:T,U}{\Type(k)}}}
-\item[Lam]\index{Typing rules!Lam} 
-\inference{\frac{\WTEG{\forall~x:T,U}{s}~~~~ \WTE{\Gamma::(x:T)}{t}{U}}
-        {\WTEG{\lb~x:T\mto t}{\forall x:T, U}}}
-\item[App]\index{Typing rules!App}
- \inference{\frac{\WTEG{t}{\forall~x:U,T}~~~~\WTEG{u}{U}}
-                 {\WTEG{(t\ u)}{\subst{T}{x}{u}}}}
-\item[Let]\index{Typing rules!Let} 
-\inference{\frac{\WTEG{t}{T}~~~~ \WTE{\Gamma::(x:=t:T)}{u}{U}}
-        {\WTEG{\kw{let}~x:=t~\kw{in}~u}{\subst{U}{x}{t}}}}
-\end{description}
-    
-\Rem We may have $\kw{let}~x:=t~\kw{in}~u$
-well-typed without having $((\lb~x:T\mto u)~t)$ well-typed (where
-$T$ is a type of $t$). This is because the value $t$ associated to $x$
-may be used in a conversion rule (see Section~\ref{conv-rules}).
-
-\section[Conversion rules]{Conversion rules\index{Conversion rules}
-\label{conv-rules}}
-\paragraph[$\beta$-reduction.]{$\beta$-reduction.\label{beta}\index{beta-reduction@$\beta$-reduction}}
-
-We want to be able to identify some terms as we can identify the
-application of a function to a given argument with its result. For
-instance the identity function over a given type $T$ can be written
-$\lb~x:T\mto x$. In any environment $E$ and context $\Gamma$, we want to identify any object $a$ (of type $T$) with the
-application $((\lb~x:T\mto x)~a)$. We define for this a {\em reduction} (or a
-{\em conversion}) rule we call $\beta$:
-\[ \WTEGRED{((\lb~x:T\mto
-  t)~u)}{\triangleright_{\beta}}{\subst{t}{x}{u}} \] 
-We say that $\subst{t}{x}{u}$ is the {\em $\beta$-contraction} of
-$((\lb~x:T\mto t)~u)$ and, conversely, that $((\lb~x:T\mto t)~u)$
-is the {\em $\beta$-expansion} of $\subst{t}{x}{u}$.
-
-According to $\beta$-reduction, terms of the {\em Calculus of
-  Inductive Constructions} enjoy some fundamental properties such as
-confluence, strong normalization, subject reduction. These results are
-theoretically of great importance but we will not detail them here and
-refer the interested reader to \cite{Coq85}.
-
-\paragraph[$\iota$-reduction.]{$\iota$-reduction.\label{iota}\index{iota-reduction@$\iota$-reduction}}
-A specific conversion rule is associated to the inductive objects in
-the environment.  We shall give later on (see Section~\ref{iotared}) the
-precise rules but it just says that a destructor applied to an object
-built from a constructor behaves as expected.  This reduction is
-called $\iota$-reduction and is more precisely studied in
-\cite{Moh93,Wer94}.
-
-
-\paragraph[$\delta$-reduction.]{$\delta$-reduction.\label{delta}\index{delta-reduction@$\delta$-reduction}}
-
-We may have defined variables in contexts or constants in the global
-environment. It is legal to identify such a reference with its value,
-that is to expand (or unfold) it into its value. This
-reduction is called $\delta$-reduction and shows as follows.
-
-$$\WTEGRED{x}{\triangleright_{\delta}}{t}~~~~~\mbox{if $(x:=t:T) \in \Gamma$}~~~~~~~~~\WTEGRED{c}{\triangleright_{\delta}}{t}~~~~~\mbox{if $(c:=t:T) \in E$}$$
-
-
-\paragraph[$\zeta$-reduction.]{$\zeta$-reduction.\label{zeta}\index{zeta-reduction@$\zeta$-reduction}}
-
-Coq allows also to remove local definitions occurring in terms by
-replacing the defined variable by its value. The declaration being
-destroyed, this reduction differs from $\delta$-reduction. It is
-called $\zeta$-reduction and shows as follows.
-
-$$\WTEGRED{\kw{let}~x:=u~\kw{in}~t}{\triangleright_{\zeta}}{\subst{t}{x}{u}}$$
-
-\paragraph[Convertibility.]{Convertibility.\label{convertibility}
-\index{beta-reduction@$\beta$-reduction}\index{iota-reduction@$\iota$-reduction}\index{delta-reduction@$\delta$-reduction}\index{zeta-reduction@$\zeta$-reduction}}
-
-Let us write $\WTEGRED{t}{\triangleright}{u}$ for the contextual closure of the relation $t$ reduces to $u$ in the environment $E$ and context $\Gamma$ with one of the previous reduction $\beta$, $\iota$, $\delta$ or $\zeta$.
-
-We say that two terms $t_1$ and $t_2$ are {\em convertible} (or {\em
-  equivalent)} in the environment $E$ and context $\Gamma$ iff there exists a term $u$ such that $\WTEGRED{t_1}{\triangleright \ldots \triangleright}{u}$
-and $\WTEGRED{t_2}{\triangleright \ldots \triangleright}{u}$.
-We then write $\WTEGCONV{t_1}{t_2}$.
-
-The convertibility relation allows to introduce a new typing rule
-which says that two convertible well-formed types have the same
-inhabitants.
-
-At the moment, we did not take into account one rule between universes
-which says that any term in a universe of index $i$ is also a term in
-the universe of index $i+1$. This property is included into the
-conversion rule by extending the equivalence relation of
-convertibility into an order inductively defined by:
-\begin{enumerate}
-\item if $\WTEGCONV{t}{u}$ then $\WTEGLECONV{t}{u}$,
-\item if $i \leq j$ then $\WTEGLECONV{\Type(i)}{\Type(j)}$,
-\item for any $i$, $\WTEGLECONV{\Prop}{\Type(i)}$,
-\item for any $i$, $\WTEGLECONV{\Set}{\Type(i)}$,
-\item $\WTEGLECONV{\Prop}{\Set}$,
-\item if $\WTEGCONV{T}{U}$ and $\WTELECONV{\Gamma::(x:T)}{T'}{U'}$ then $\WTEGLECONV{\forall~x:T,T'}{\forall~x:U,U'}$.
-\end{enumerate}
-
-The conversion rule is now exactly:
-
-\begin{description}\label{Conv}
-\item[Conv]\index{Typing rules!Conv}
- \inference{
-      \frac{\WTEG{U}{s}~~~~\WTEG{t}{T}~~~~\WTEGLECONV{T}{U}}{\WTEG{t}{U}}}
-  \end{description}
-
-
-\paragraph{$\eta$-conversion.
-\label{eta}
-\index{eta-conversion@$\eta$-conversion}
-\index{eta-reduction@$\eta$-reduction}}
-
-An other important rule is the $\eta$-conversion. It is to identify
-terms over a dummy abstraction of a variable followed by an
-application of this variable. Let $T$ be a type, $t$ be a term in
-which the variable $x$ doesn't occurs free. We have
-\[ \WTEGRED{\lb~x:T\mto (t\ x)}{\triangleright}{t} \]
-Indeed, as $x$ doesn't occur free in $t$, for any $u$ one
-applies to $\lb~x:T\mto (t\ x)$, it $\beta$-reduces to $(t\ u)$. So
-$\lb~x:T\mto (t\ x)$ and $t$ can be identified.
-
-\Rem The $\eta$-reduction is not taken into account in the
-convertibility rule of \Coq.
-
-\paragraph[Normal form.]{Normal form.\index{Normal form}\label{Normal-form}\label{Head-normal-form}\index{Head normal form}}
-A term which cannot be any more reduced is said to be in {\em normal
-  form}. There are several ways (or strategies) to apply the reduction
-rule. Among them, we have to mention the {\em head reduction} which
-will play an important role (see Chapter~\ref{Tactics}). Any term can
-be written as $\lb~x_1:T_1\mto \ldots \lb x_k:T_k \mto 
-(t_0\ t_1\ldots t_n)$ where
-$t_0$ is not an application. We say then that $t_0$ is the {\em head
-  of $t$}. If we assume that $t_0$ is $\lb~x:T\mto u_0$ then one step of
-$\beta$-head reduction of $t$ is:
-\[\lb~x_1:T_1\mto \ldots \lb x_k:T_k\mto (\lb~x:T\mto u_0\ t_1\ldots t_n) 
-~\triangleright ~ \lb~(x_1:T_1)\ldots(x_k:T_k)\mto 
-(\subst{u_0}{x}{t_1}\ t_2 \ldots t_n)\]
-Iterating the process of head reduction until the head of the reduced
-term is no more an abstraction leads to the {\em $\beta$-head normal
-  form} of $t$:
-\[ t \triangleright \ldots \triangleright
-\lb~x_1:T_1\mto \ldots\lb x_k:T_k\mto (v\ u_1
-\ldots u_m)\]
-where $v$ is not an abstraction (nor an application).  Note that the
-head normal form must not be confused with the normal form since some
-$u_i$ can be reducible.
-
-Similar notions of head-normal forms involving $\delta$, $\iota$ and $\zeta$
-reductions or any combination of those can also be defined.
-
-\section{Derived rules for environments}
-
-From the original rules of the type system, one can derive new rules
-which change the context of definition of objects in the environment.
-Because these rules correspond to elementary operations in the \Coq\ 
-engine used in the discharge mechanism at the end of a section, we
-state them explicitly.
-
-\paragraph{Mechanism of substitution.}
-
-One rule which can be proved valid, is to replace a term $c$ by its
-value in the environment. As we defined the substitution of a term for
-a variable in a term, one can define the substitution of a term for a
-constant. One easily extends this substitution to contexts and
-environments.
-
-\paragraph{Substitution Property:} 
-\inference{\frac{\WF{E;\Def{\Gamma}{c}{t}{T}; F}{\Delta}}
-           {\WF{E; \subst{F}{c}{t}}{\subst{\Delta}{c}{t}}}}
-
-
-\paragraph{Abstraction.}
-
-One can modify the context of definition of a constant $c$ by
-abstracting a constant with respect to the last variable $x$ of its
-defining context. For doing that, we need to check that the constants
-appearing in the body of the declaration do not depend on $x$, we need
-also to modify the reference to the constant $c$ in the environment
-and context by explicitly applying this constant to the variable $x$.
-Because of the rules for building environments and terms we know the
-variable $x$ is available at each stage where $c$ is mentioned.
-
-\paragraph{Abstracting property:} 
- \inference{\frac{\WF{E; \Def{\Gamma::(x:U)}{c}{t}{T};
-       F}{\Delta}~~~~\WFE{\Gamma}}
-           {\WF{E;\Def{\Gamma}{c}{\lb~x:U\mto t}{\forall~x:U,T};
-  \subst{F}{c}{(c~x)}}{\subst{\Delta}{c}{(c~x)}}}}
-
-\paragraph{Pruning the context.} 
-We said the judgment \WFE{\Gamma} means that the defining contexts of
-constants in $E$ are included in $\Gamma$. If one abstracts or
-substitutes the constants with the above rules then it may happen
-that the context $\Gamma$ is now bigger than the one needed for
-defining the constants in $E$. Because defining contexts are growing
-in $E$, the minimum context needed for defining the constants in $E$
-is the same as the one for the last constant. One can consequently
-derive the following property.
-
-\paragraph{Pruning property:}
-\inference{\frac{\WF{E; \Def{\Delta}{c}{t}{T}}{\Gamma}}
-                {\WF{E;\Def{\Delta}{c}{t}{T}}{\Delta}}}
-
-
-\section[Inductive Definitions]{Inductive Definitions\label{Cic-inductive-definitions}}
-
-A (possibly mutual) inductive definition is specified by giving the
-names and the type of the inductive sets or families to be
-defined and the names and types of the constructors of the inductive
-predicates.  An inductive declaration in the environment can
-consequently be represented with two contexts (one for inductive
-definitions, one for constructors).
-
-Stating the rules for inductive definitions in their general form
-needs quite tedious definitions. We shall try to give a concrete
-understanding of the rules by precising them on running examples.  We
-take as examples the type of natural numbers, the type of
-parameterized lists over a type $A$, the relation which states that
-a list has some given length and the mutual inductive definition of trees and
-forests. 
-
-\subsection{Representing an inductive definition}
-\subsubsection{Inductive definitions without parameters}
-As for constants, inductive definitions can be defined in a non-empty
-context. \\
-We write \NInd{\Gamma}{\Gamma_I}{\Gamma_C} an inductive
-definition valid in a context $\Gamma$, a
-context of definitions $\Gamma_I$ and a context of constructors
-$\Gamma_C$.
-\paragraph{Examples.}
-The inductive declaration for the type of natural numbers will be:
-\[\NInd{}{\nat:\Set}{\nO:\nat,\nS:\nat\ra\nat}\]
-In a context with a variable $A:\Set$, the lists of elements in $A$ are
-represented by:
-\[\NInd{A:\Set}{\List:\Set}{\Nil:\List,\cons : A \ra \List \ra
-  \List}\]
- Assuming 
-  $\Gamma_I$ is $[I_1:A_1;\ldots;I_k:A_k]$, and $\Gamma_C$ is
-  $[c_1:C_1;\ldots;c_n:C_n]$, the general typing rules are, 
-  for $1\leq j\leq k$ and $1\leq i\leq n$:
-
-\bigskip
-\inference{\frac{\NInd{\Gamma}{\Gamma_I}{\Gamma_C} \in E}{(I_j:A_j) \in E}}
-
-\inference{\frac{\NInd{\Gamma}{\Gamma_I}{\Gamma_C} \in E}{(c_i:C_i) \in E}}
-
-\subsubsection{Inductive definitions with parameters}
-
-We have to slightly complicate the representation above in order to handle
-the delicate problem of parameters. 
-Let us explain that on the example of \List. With the above definition,
-the type \List\ can only be used in an environment where we
-have a variable $A:\Set$. Generally one want to consider lists of
-elements in different types. For constants this is easily done by abstracting
-the value over the parameter. In the case of inductive definitions we
-have to handle the abstraction over several objects.
-
-One possible way to do that would be to define the type \List\
-inductively as being an inductive family of type $\Set\ra\Set$:
-\[\NInd{}{\List:\Set\ra\Set}{\Nil:(\forall A:\Set,\List~A),
-  \cons : (\forall A:\Set, A \ra \List~A \ra \List~A)}\]
-There are drawbacks to this point of view. The
-information which says that for any $A$, $(\List~A)$ is an inductively defined
-\Set\ has been lost.
-So we introduce two important definitions.
-
-\paragraph{Inductive parameters, real arguments.}
-An inductive definition $\NInd{\Gamma}{\Gamma_I}{\Gamma_C}$ admits 
-$r$ inductive parameters if each type of constructors $(c:C)$ in
-$\Gamma_C$ is such that 
-\[C\equiv \forall
-p_1:P_1,\ldots,\forall p_r:P_r,\forall a_1:A_1, \ldots \forall a_n:A_n,
-(I~p_1~\ldots p_r~t_1\ldots t_q)\]
-with $I$ one of the inductive definitions in $\Gamma_I$. 
-We say that $q$ is the number of real arguments of the constructor
-$c$. 
-\paragraph{Context of parameters.}
-If an inductive definition $\NInd{\Gamma}{\Gamma_I}{\Gamma_C}$ admits 
-$r$ inductive parameters, then there exists a context $\Gamma_P$ of
-size $r$, such that $\Gamma_P=[p_1:P_1;\ldots;p_r:P_r]$ and 
-if $(t:A) \in \Gamma_I,\Gamma_C$ then $A$ can be written as 
-$\forall p_1:P_1,\ldots \forall p_r:P_r,A'$. 
-We call $\Gamma_P$ the context of parameters of the inductive
-definition and use the notation $\forall \Gamma_P,A'$ for the term $A$.
-\paragraph{Remark.}
-If we have a term $t$ in an instance of an
-inductive definition $I$ which starts with a constructor $c$, then the
-$r$ first arguments of $c$ (the parameters) can be deduced from the
-type $T$ of $t$: these are exactly the $r$ first arguments of $I$ in
-the head normal form of $T$.
-\paragraph{Examples.}
-The \List{} definition has $1$ parameter:
-\[\NInd{}{\List:\Set\ra\Set}{\Nil:(\forall A:\Set, \List~A),
-  \cons : (\forall A:\Set, A \ra \List~A \ra \List~A)}\]
-This is also the case for this more complex definition where there is
-a recursive argument on a different instance of \List: 
-\[\NInd{}{\List:\Set\ra\Set}{\Nil:(\forall A:\Set, \List~A),
-  \cons : (\forall A:\Set, A \ra \List~(A \ra A) \ra \List~A)}\]
-But the following definition has $0$ parameters:
-\[\NInd{}{\List:\Set\ra\Set}{\Nil:(\forall A:\Set, \List~A),
-  \cons : (\forall A:\Set, A \ra \List~A \ra \List~(A*A))}\]
-
-%\footnote{
-%The interested reader may compare the above definition with the two 
-%following ones which have very different logical meaning:\\
-%$\NInd{}{\List:\Set}{\Nil:\List,\cons : (A:\Set)A
-%  \ra \List \ra \List}$ \\
-%$\NInd{}{\List:\Set\ra\Set}{\Nil:(A:\Set)(\List~A),\cons : (A:\Set)A
-%  \ra (\List~A\ra A) \ra (\List~A)}$.}
-\paragraph{Concrete syntax.}
-In the Coq system, the context of parameters is given explicitly
-after the name of the inductive definitions and is shared between the
-arities and the type of constructors.
-% The vernacular declaration of polymorphic trees and forests will be:\\
-% \begin{coq_example*}
-% Inductive Tree (A:Set) : Set := 
-%    Node : A -> Forest A -> Tree A
-% with Forest (A : Set) : Set := 
-%    Empty : Forest A
-%  | Cons  : Tree A -> Forest A -> Forest A
-% \end{coq_example*}
-% will correspond in our formalism to:
-% \[\NInd{}{{\tt Tree}:\Set\ra\Set;{\tt Forest}:\Set\ra \Set}
-%   {{\tt Node} : \forall A:\Set, A \ra {\tt Forest}~A \ra {\tt Tree}~A,
-%    {\tt Empty} : \forall A:\Set, {\tt Forest}~A,
-%    {\tt Cons} : \forall A:\Set, {\tt Tree}~A \ra {\tt Forest}~A \ra
-%    {\tt Forest}~A}\]
-We keep track in the syntax of the number of
-parameters. 
-
-Formally the representation of an inductive declaration
-will be 
-\Ind{\Gamma}{p}{\Gamma_I}{\Gamma_C} for an inductive
-definition valid in a context $\Gamma$ with $p$ parameters, a
-context of definitions $\Gamma_I$ and a context of constructors
-$\Gamma_C$.
-
-The definition \Ind{\Gamma}{p}{\Gamma_I}{\Gamma_C} will be
-well-formed exactly when \NInd{\Gamma}{\Gamma_I}{\Gamma_C} is and 
-when $p$ is (less or equal than) the number of parameters in
-\NInd{\Gamma}{\Gamma_I}{\Gamma_C}. 
-
-\paragraph{Examples}
-The declaration for parameterized lists is:
-\[\Ind{}{1}{\List:\Set\ra\Set}{\Nil:(\forall A:\Set,\List~A),\cons :
-  (\forall A:\Set, A \ra \List~A \ra   \List~A)}\]
-
-The declaration for the length of lists is:
-\[\Ind{}{1}{\Length:\forall A:\Set, (\List~A)\ra \nat\ra\Prop}
-      {\LNil:\forall A:\Set, \Length~A~(\Nil~A)~\nO,\\ 
-     \LCons :\forall A:\Set,\forall a:A, \forall l:(\List~A),\forall n:\nat, (\Length~A~l~n)\ra (\Length~A~(\cons~A~a~l)~(\nS~n))}\]
-
-The declaration for a mutual inductive definition of forests and trees is:
-\[\NInd{}{\tree:\Set,\forest:\Set}
-      {\\~~\node:\forest \ra \tree,
-       \emptyf:\forest,\consf:\tree \ra \forest \ra \forest\-}\]
-  
-These representations are the ones obtained as the result of the \Coq\ 
-declaration:
-\begin{coq_example*}
-Inductive nat : Set :=
-  | O : nat
-  | S : nat -> nat.
-Inductive list (A:Set) : Set :=
-  | nil : list A
-  | cons : A -> list A -> list A.
-\end{coq_example*}
-\begin{coq_example*}
-Inductive Length (A:Set) : list A -> nat -> Prop :=
-  | Lnil : Length A (nil A) O
-  | Lcons :
-      forall (a:A) (l:list A) (n:nat),
-        Length A l n -> Length A (cons A a l) (S n).
-Inductive tree : Set :=
-    node : forest -> tree
-with forest : Set :=
-  | emptyf : forest
-  | consf : tree -> forest -> forest.
-\end{coq_example*}
-% The inductive declaration in \Coq\ is slightly different from the one
-% we described theoretically. The difference is that in the type of
-% constructors the inductive definition is explicitly applied to the
-% parameters variables.
-The \Coq\ type-checker verifies that all
-parameters are applied in the correct manner in the conclusion of the
-type of each constructors~:
-
-In particular, the following definition will not be accepted because 
-there is an occurrence of \List\ which is not applied to the parameter
-variable in the conclusion of the type of {\tt cons'}:
-\begin{coq_eval}
-Set Printing Depth 50.
-(********** The following is not correct and should produce **********)
-(********* Error: The 1st argument of list' must be A in ... *********)
-\end{coq_eval}
-\begin{coq_example}
-Inductive list' (A:Set) : Set :=
-  | nil' : list' A
-  | cons' : A -> list' A -> list' (A*A).
-\end{coq_example}
-Since \Coq{} version 8.1, there is no restriction about parameters in
-the types of arguments of constructors. The following definition is
-valid:
-\begin{coq_example}
-Inductive list' (A:Set) : Set :=
-  | nil' : list' A
-  | cons' : A -> list' (A->A) -> list' A.
-\end{coq_example}
-
-
-\subsection{Types of inductive objects}
-We have to give the type of constants in an environment $E$ which
-contains an inductive declaration.
-
-\begin{description}
-\item[Ind-Const] Assuming 
-  $\Gamma_I$ is $[I_1:A_1;\ldots;I_k:A_k]$, and $\Gamma_C$ is
-  $[c_1:C_1;\ldots;c_n:C_n]$,
-   
-\inference{\frac{\Ind{\Gamma}{p}{\Gamma_I}{\Gamma_C} \in E
-    ~~j=1\ldots k}{(I_j:A_j) \in E}}
-
-\inference{\frac{\Ind{\Gamma}{p}{\Gamma_I}{\Gamma_C} \in E
-    ~~~~i=1.. n}
-   {(c_i:C_i) \in E}}
-\end{description}
-
-\paragraph{Example.}
-We have $(\List:\Set \ra \Set), (\cons:\forall~A:\Set,A\ra(\List~A)\ra
-(\List~A))$, \\ 
-$(\Length:\forall~A:\Set, (\List~A)\ra\nat\ra\Prop)$, $\tree:\Set$ and $\forest:\Set$.
-
-From now on, we write $\ListA$ instead of $(\List~A)$ and $\LengthA$
-for $(\Length~A)$.
-
-%\paragraph{Parameters.}
-%%The parameters introduce a distortion between the inside specification
-%%of the inductive declaration where parameters are supposed to be
-%%instantiated (this representation is appropriate for checking the
-%%correctness or deriving the destructor principle) and the outside
-%%typing rules where the inductive objects are seen as objects
-%%abstracted with respect to the parameters.
-
-%In the definition of \List\ or \Length\, $A$ is a parameter because
-%what is effectively inductively defined is $\ListA$ or $\LengthA$ for
-%a given $A$ which is constant in the type of constructors.  But when
-%we define $(\LengthA~l~n)$, $l$ and $n$ are not parameters because the
-%constructors manipulate different instances of this family.
-
-\subsection{Well-formed inductive definitions}
-We cannot accept any inductive declaration because some of them lead
-to inconsistent systems. We restrict ourselves to definitions which
-satisfy a syntactic criterion of positivity. Before giving the formal
-rules, we need a few definitions:
-
-\paragraph[Definitions]{Definitions\index{Positivity}\label{Positivity}}
-
-A type $T$ is an {\em arity of sort $s$}\index{Arity} if it converts
-to the sort $s$ or to a product $\forall~x:T,U$ with $U$ an arity
-of sort $s$. (For instance $A\ra \Set$ or $\forall~A:\Prop,A\ra
-\Prop$ are arities of sort respectively \Set\ and \Prop).  A {\em type
-  of constructor of $I$}\index{Type of constructor} is either a term
-$(I~t_1\ldots ~t_n)$ or $\fa x:T,C$ with $C$ recursively 
-a {\em type of constructor of $I$}.
-
-\smallskip
-
-The type of constructor $T$ will be said to {\em satisfy the positivity
-condition} for a constant $X$ in the following cases:
-
-\begin{itemize}
-\item $T=(X~t_1\ldots ~t_n)$ and $X$ does not occur free in
-any $t_i$
-\item $T=\forall~x:U,V$ and $X$ occurs only strictly positively in $U$ and
-the type $V$ satisfies the positivity condition for $X$
-\end{itemize}
-
-The constant $X$ {\em occurs strictly positively} in $T$ in the
-following cases:
-
-\begin{itemize}
-\item $X$ does not occur in $T$
-\item $T$ converts to $(X~t_1 \ldots ~t_n)$ and $X$ does not occur in
-  any of $t_i$
-\item $T$ converts to $\forall~x:U,V$ and $X$ does not occur in
-  type $U$ but occurs strictly positively in type $V$
-\item $T$ converts to $(I~a_1 \ldots ~a_m ~ t_1 \ldots ~t_p)$ where
-  $I$ is the name of an inductive declaration of the form
-  $\Ind{\Gamma}{m}{I:A}{c_1:\forall p_1:P_1,\ldots \forall
-    p_m:P_m,C_1;\ldots;c_n:\forall p_1:P_1,\ldots \forall
-    p_m:P_m,C_n}$ 
-  (in particular, it is not mutually defined and it has $m$
-  parameters) and $X$ does not occur in any of the $t_i$, and the
-  (instantiated) types of constructor $C_i\{p_j/a_j\}_{j=1\ldots m}$
-  of $I$ satisfy 
-  the nested positivity condition for $X$
-%\item more generally, when $T$ is not a type, $X$ occurs strictly
-%positively in $T[x:U]u$ if $X$ does not occur in $U$ but occurs
-%strictly positively in $u$
-\end{itemize}
-
-The type of constructor $T$ of $I$ {\em satisfies the nested
-positivity condition} for a constant $X$ in the following
-cases:
-
-\begin{itemize}
-\item $T=(I~b_1\ldots b_m~u_1\ldots ~u_{p})$, $I$ is an inductive
-  definition with $m$ parameters and $X$ does not occur in
-any $u_i$
-\item $T=\forall~x:U,V$ and $X$ occurs only strictly positively in $U$ and
-the type $V$ satisfies the nested positivity condition for $X$
-\end{itemize}
-
-\paragraph{Example}
-
-$X$ occurs strictly positively in $A\ra X$ or $X*A$ or $({\tt list}~
-X)$ but not in $X \ra A$ or $(X \ra A)\ra A$ nor $({\tt neg}~X)$
-assuming the notion of product and lists were already defined and {\tt
-  neg} is an inductive definition with declaration \Ind{}{A:\Set}{{\tt
-    neg}:\Set}{{\tt neg}:(A\ra{\tt False}) \ra {\tt neg}}.  Assuming
-$X$ has arity ${\tt nat \ra Prop}$ and {\tt ex} is the inductively
-defined existential quantifier, the occurrence of $X$ in ${\tt (ex~
-  nat~ \lb~n:nat\mto (X~ n))}$ is also strictly positive.
-
-\paragraph{Correctness rules.}
-We shall now describe the rules allowing the introduction of a new
-inductive definition.
-
-\begin{description}
-\item[W-Ind] Let $E$ be an environment and
-  $\Gamma,\Gamma_P,\Gamma_I,\Gamma_C$ are contexts such that
-  $\Gamma_I$ is $[I_1:\forall \Gamma_P,A_1;\ldots;I_k:\forall
-  \Gamma_P,A_k]$ and $\Gamma_C$ is 
-  $[c_1:\forall \Gamma_P,C_1;\ldots;c_n:\forall \Gamma_P,C_n]$. 
-\inference{
-  \frac{
-  (\WTE{\Gamma;\Gamma_P}{A_j}{s'_j})_{j=1\ldots  k}
-  ~~ (\WTE{\Gamma;\Gamma_I;\Gamma_P}{C_i}{s_{q_i}})_{i=1\ldots  n}
-}
-  {\WF{E;\Ind{\Gamma}{p}{\Gamma_I}{\Gamma_C}}{\Gamma}}}
-provided that the following side conditions hold:
-\begin{itemize}
-\item $k>0$ and all of $I_j$ and $c_i$ are distinct names for $j=1\ldots  k$ and $i=1\ldots  n$,
-\item $p$ is the number of parameters of \NInd{\Gamma}{\Gamma_I}{\Gamma_C}
-  and $\Gamma_P$ is the context of parameters, 
-\item for $j=1\ldots  k$ we have that $A_j$ is an arity of sort $s_j$ and $I_j
-  \notin \Gamma \cup E$,
-\item for $i=1\ldots  n$ we have that $C_i$ is a type of constructor of
-  $I_{q_i}$ which satisfies the positivity condition for $I_1 \ldots  I_k$
-  and $c_i \notin \Gamma \cup E$.
-\end{itemize}
-\end{description}
-One can remark that there is a constraint between the sort of the
-arity of the inductive type and the sort of the type of its
-constructors which will always be satisfied for the impredicative sort
-(\Prop) but may fail to define inductive definition 
-on sort \Set{} and generate constraints between universes for
-inductive definitions in the {\Type} hierarchy.
-
-\paragraph{Examples.}
-It is well known that existential quantifier can be encoded as an
-inductive definition.
-The following declaration introduces the second-order existential
-quantifier $\exists X.P(X)$.
-\begin{coq_example*}
-Inductive exProp (P:Prop->Prop) : Prop 
-  := exP_intro : forall X:Prop, P X -> exProp P.
-\end{coq_example*}
-The same definition on \Set{} is not allowed and fails~:
-\begin{coq_eval}
-(********** The following is not correct and should produce **********)
-(*** Error: Large non-propositional inductive types must be in Type***)
-\end{coq_eval}
-\begin{coq_example}
-Inductive exSet (P:Set->Prop) : Set 
-  := exS_intro : forall X:Set, P X -> exSet P.
-\end{coq_example}
-It is possible to declare the same inductive definition in the
-universe \Type. 
-The \texttt{exType} inductive definition has type  $(\Type_i \ra\Prop)\ra
-\Type_j$ with the constraint that the parameter \texttt{X} of \texttt{exT\_intro} has type $\Type_k$ with $k<j$ and $k\leq i$.
-\begin{coq_example*}
-Inductive exType (P:Type->Prop) : Type
-  := exT_intro : forall X:Type, P X -> exType P.
-\end{coq_example*}
-%We shall assume for the following definitions that, if necessary, we
-%annotated the type of constructors such that we know if the argument
-%is recursive or not.  We shall write the type $(x:_R T)C$ if it is 
-%a recursive argument and $(x:_P T)C$ if the argument is not recursive.
-
-\paragraph[Sort-polymorphism of inductive families.]{Sort-polymorphism of inductive families.\index{Sort-polymorphism of inductive families}}
-
-From {\Coq} version 8.1, inductive families declared in {\Type} are
-polymorphic over their arguments in {\Type}.
-
-If $A$ is an arity and $s$ a sort, we write $A_{/s}$ for the arity
-obtained from $A$ by replacing its sort with $s$. Especially, if $A$
-is well-typed in some environment and context, then $A_{/s}$ is typable
-by typability of all products in the Calculus of Inductive Constructions.
-The following typing rule is added to the theory.
-
-\begin{description}
-\item[Ind-Family] Let $\Ind{\Gamma}{p}{\Gamma_I}{\Gamma_C}$ be an
-  inductive definition. Let $\Gamma_P = [p_1:P_1;\ldots;p_{p}:P_{p}]$
-  be its context of parameters, $\Gamma_I = [I_1:\forall
-    \Gamma_P,A_1;\ldots;I_k:\forall \Gamma_P,A_k]$ its context of
-  definitions and $\Gamma_C = [c_1:\forall
-    \Gamma_P,C_1;\ldots;c_n:\forall \Gamma_P,C_n]$ its context of
-  constructors, with $c_i$ a constructor of $I_{q_i}$.
-
-  Let $m \leq p$ be the length of the longest prefix of parameters
-  such that the $m$ first arguments of all occurrences of all $I_j$ in
-  all $C_k$ (even the occurrences in the hypotheses of $C_k$) are
-  exactly applied to $p_1~\ldots~p_m$ ($m$ is the number of {\em
-    recursively uniform parameters} and the $p-m$ remaining parameters
-  are the {\em recursively non-uniform parameters}). Let $q_1$,
-  \ldots, $q_r$, with $0\leq r\leq m$, be a (possibly) partial
-  instantiation of the recursively uniform parameters of
-  $\Gamma_P$. We have:
-
-\inference{\frac
-{\left\{\begin{array}{l}
-\Ind{\Gamma}{p}{\Gamma_I}{\Gamma_C} \in E\\
-(E[\Gamma] \vdash q_l : P'_l)_{l=1\ldots r}\\
-(\WTEGLECONV{P'_l}{\subst{P_l}{p_u}{q_u}_{u=1\ldots l-1}})_{l=1\ldots r}\\
-1 \leq j \leq k
-\end{array}
-\right.}
-{E[\Gamma] \vdash (I_j\,q_1\,\ldots\,q_r:\forall [p_{r+1}:P_{r+1};\ldots;p_{p}:P_{p}], (A_j)_{/s_j})}
-}
-
-provided that the following side conditions hold:
-
-\begin{itemize}
-\item $\Gamma_{P'}$ is the context obtained from $\Gamma_P$ by
-replacing each $P_l$ that is an arity with $P'_l$ for $1\leq l \leq r$ (notice that
-$P_l$ arity implies $P'_l$ arity since $\WTEGLECONV{P'_l}{ \subst{P_l}{p_u}{q_u}_{u=1\ldots l-1}}$);
-\item there are sorts $s_i$, for $1 \leq i \leq k$ such that, for
- $\Gamma_{I'} = [I_1:\forall
-    \Gamma_{P'},(A_1)_{/s_1};\ldots;I_k:\forall \Gamma_{P'},(A_k)_{/s_k}]$
-we have $(\WTE{\Gamma;\Gamma_{I'};\Gamma_{P'}}{C_i}{s_{q_i}})_{i=1\ldots  n}$;
-\item the sorts are such that all eliminations, to {\Prop}, {\Set} and
-  $\Type(j)$, are allowed (see section~\ref{elimdep}).
-\end{itemize}
-\end{description}
-
-Notice that if $I_j\,q_1\,\ldots\,q_r$ is typable using the rules {\bf
-Ind-Const} and {\bf App}, then it is typable using the rule {\bf
-Ind-Family}. Conversely, the extended theory is not stronger than the
-theory without {\bf Ind-Family}. We get an equiconsistency result by
-mapping each $\Ind{\Gamma}{p}{\Gamma_I}{\Gamma_C}$ occurring into a
-given derivation into as many different inductive types and constructors
-as the number of different (partial) replacements of sorts, needed for
-this derivation, in the parameters that are arities (this is possible
-because $\Ind{\Gamma}{p}{\Gamma_I}{\Gamma_C}$ well-formed implies
-that $\Ind{\Gamma}{p}{\Gamma_{I'}}{\Gamma_{C'}}$ is well-formed and
-has the same allowed eliminations, where
-$\Gamma_{I'}$ is defined as above and $\Gamma_{C'} = [c_1:\forall
-\Gamma_{P'},C_1;\ldots;c_n:\forall \Gamma_{P'},C_n]$). That is,
-the changes in the types of each partial instance
-$q_1\,\ldots\,q_r$ can be characterized by the ordered sets of arity
-sorts among the types of parameters, and to each signature is
-associated a new inductive definition with fresh names. Conversion is
-preserved as any (partial) instance $I_j\,q_1\,\ldots\,q_r$ or
-$C_i\,q_1\,\ldots\,q_r$ is mapped to the names chosen in the specific
-instance of $\Ind{\Gamma}{p}{\Gamma_I}{\Gamma_C}$.
-
-\newcommand{\Single}{\mbox{\textsf{Set}}}
-
-In practice, the rule {\bf Ind-Family} is used by {\Coq} only when all the
-inductive types of the inductive definition are declared with an arity whose 
-sort is in the $\Type$
-hierarchy. Then, the polymorphism is over the parameters whose
-type is an arity of sort in the {\Type} hierarchy. 
-The sort $s_j$ are
-chosen canonically so that each $s_j$ is minimal with respect to the
-hierarchy ${\Prop}\subset{\Set_p}\subset\Type$ where $\Set_p$ is
-predicative {\Set}.
-%and ${\Prop_u}$ is the sort of small singleton
-%inductive types (i.e. of inductive types with one single constructor
-%and that contains either proofs or inhabitants of singleton types
-%only). 
-More precisely, an empty or small singleton inductive definition
-(i.e. an inductive definition of which all inductive types are
-singleton -- see paragraph~\ref{singleton}) is set in
-{\Prop}, a small non-singleton inductive family is set in {\Set} (even
-in case {\Set} is impredicative -- see Section~\ref{impredicativity}),
-and otherwise in the {\Type} hierarchy.
-% TODO: clarify the case of a partial application ??
-
-Note that the side-condition about allowed elimination sorts in the
-rule~{\bf Ind-Family} is just to avoid to recompute the allowed
-elimination sorts at each instance of a pattern-matching (see
-section~\ref{elimdep}).
-
-As an example, let us consider the following definition:
-\begin{coq_example*}
-Inductive option (A:Type) : Type := 
-| None : option A 
-| Some : A -> option A.
-\end{coq_example*}
-
-As the definition is set in the {\Type} hierarchy, it is used
-polymorphically over its parameters whose types are arities of a sort
-in the {\Type} hierarchy. Here, the parameter $A$ has this property,
-hence, if \texttt{option} is applied to a type in {\Set}, the result is
-in {\Set}. Note that if \texttt{option} is applied to a type in {\Prop},
-then, the result is not set in \texttt{Prop} but in \texttt{Set}
-still. This is because \texttt{option} is not a singleton type (see
-section~\ref{singleton}) and it would loose the elimination to {\Set} and
-{\Type} if set in {\Prop}.
-
-\begin{coq_example}
-Check (fun A:Set => option A).
-Check (fun A:Prop => option A).
-\end{coq_example}
-
-Here is another example.
-
-\begin{coq_example*}
-Inductive prod (A B:Type) : Type := pair : A -> B -> prod A B.
-\end{coq_example*}
-
-As \texttt{prod} is a singleton type, it will be in {\Prop} if applied
-twice to propositions, in {\Set} if applied twice to at least one type
-in {\Set} and none in {\Type}, and in {\Type} otherwise. In all cases,
-the three kind of eliminations schemes are allowed.
-
-\begin{coq_example}
-Check (fun A:Set => prod A).
-Check (fun A:Prop => prod A A).
-Check (fun (A:Prop) (B:Set) => prod A B).
-Check (fun (A:Type) (B:Prop) => prod A B).
-\end{coq_example}
-
-\subsection{Destructors}
-The specification of inductive definitions with arities and
-constructors is quite natural.  But we still have to say how to use an
-object in an inductive type.
-
-This problem is rather delicate. There are actually several different
-ways to do that. Some of them are logically equivalent but not always
-equivalent from the computational point of view or from the user point
-of view.
-
-From the computational point of view, we want to be able to define a
-function whose domain is an inductively defined type by using a
-combination of case analysis over the possible constructors of the
-object and recursion.
-
-Because we need to keep a consistent theory and also we prefer to keep
-a strongly normalizing reduction, we cannot accept any sort of
-recursion (even terminating). So the basic idea is to restrict
-ourselves to primitive recursive functions and functionals.
-
-For instance, assuming a parameter $A:\Set$ exists in the context, we
-want to build a function \length\ of type $\ListA\ra \nat$ which
-computes the length of the list, so such that $(\length~(\Nil~A)) = \nO$
-and $(\length~(\cons~A~a~l)) = (\nS~(\length~l))$.  We want these
-equalities to be recognized implicitly and taken into account in the
-conversion rule.
-
-From the logical point of view, we have built a type family by giving
-a set of constructors.  We want to capture the fact that we do not
-have any other way to build an object in this type. So when trying to
-prove a property $(P~m)$ for $m$ in an inductive definition it is
-enough to enumerate all the cases where $m$ starts with a different
-constructor.
-
-In case the inductive definition is effectively a recursive one, we
-want to capture the extra property that we have built the smallest
-fixed point of this recursive equation.  This says that we are only
-manipulating finite objects. This analysis provides induction
-principles.
-
-For instance, in order to prove $\forall l:\ListA,(\LengthA~l~(\length~l))$
-it is enough to prove:
-
-\noindent $(\LengthA~(\Nil~A)~(\length~(\Nil~A)))$ and
-
-\smallskip
-$\forall a:A, \forall l:\ListA, (\LengthA~l~(\length~l)) \ra
-(\LengthA~(\cons~A~a~l)~(\length~(\cons~A~a~l)))$.
-\smallskip
-
-\noindent which given the conversion equalities satisfied by \length\ is the
-same as proving:
-$(\LengthA~(\Nil~A)~\nO)$ and $\forall a:A, \forall l:\ListA, 
-(\LengthA~l~(\length~l)) \ra
-(\LengthA~(\cons~A~a~l)~(\nS~(\length~l)))$.
-
-One conceptually simple way to do that, following the basic scheme
-proposed by Martin-L\"of in his Intuitionistic Type Theory, is to
-introduce for each inductive definition an elimination operator. At
-the logical level it is a proof of the usual induction principle and
-at the computational level it implements a generic operator for doing
-primitive recursion over the structure.
-
-But this operator is rather tedious to implement and use. We choose in
-this version of Coq to factorize the operator for primitive recursion
-into two more primitive operations as was first suggested by Th. Coquand
-in~\cite{Coq92}.  One is the definition by pattern-matching. The second one is a definition by guarded fixpoints. 
-
-\subsubsection[The {\tt match\ldots with \ldots end} construction.]{The {\tt match\ldots with \ldots end} construction.\label{Caseexpr}
-\index{match@{\tt match\ldots with\ldots end}}}
-
-The basic idea of this destructor operation is that we have an object
-$m$ in an inductive type $I$ and we want to prove a property $(P~m)$
-which in general depends on $m$. For this, it is enough to prove the
-property for $m = (c_i~u_1\ldots  u_{p_i})$ for each constructor of $I$.
-
-The \Coq{} term for this proof will be written~:
-\[\kw{match}~m~\kw{with}~ (c_1~x_{11}~...~x_{1p_1}) \Ra f_1 ~|~\ldots~|~
-  (c_n~x_{n1}...x_{np_n}) \Ra f_n~ \kw{end}\]
-In this expression, if
-$m$ is a term built from a constructor $(c_i~u_1\ldots u_{p_i})$ then
-the expression will behave as it is specified with $i$-th branch and
-will reduce to $f_i$ where the $x_{i1}$\ldots $x_{ip_i}$ are replaced
-by the $u_1\ldots u_p$ according to the $\iota$-reduction.
-
-Actually, for type-checking a \kw{match\ldots with\ldots end}
-expression we also need to know the predicate $P$ to be proved by case
-analysis. In the general case where $I$ is an inductively defined
-$n$-ary relation, $P$ is a $n+1$-ary relation: the $n$ first arguments
-correspond to the arguments of $I$ (parameters excluded), and the last
-one corresponds to object $m$. \Coq{} can sometimes infer this
-predicate but sometimes not. The concrete syntax for describing this
-predicate uses the \kw{as\ldots in\ldots return} construction. For
-instance, let us assume that $I$ is an unary predicate with one
-parameter. The predicate is made explicit using the syntax~:
-\[\kw{match}~m~\kw{as}~ x~ \kw{in}~ I~\verb!_!~a~ \kw{return}~ (P~ x)
- ~\kw{with}~ (c_1~x_{11}~...~x_{1p_1}) \Ra f_1 ~|~\ldots~|~
-  (c_n~x_{n1}...x_{np_n}) \Ra f_n \kw{end}\]
-The \kw{as} part can be omitted if either the result type does not
-depend on $m$ (non-dependent elimination) or $m$ is a variable (in
-this case, the result type can depend on $m$). The \kw{in} part can be
-omitted if the result type does not depend on the arguments of
-$I$. Note that the arguments of $I$ corresponding to parameters
-\emph{must} be \verb!_!, because the result type is not generalized to
-all possible values of the parameters. The expression after \kw{in}
-must be seen as an \emph{inductive type pattern}. As a final remark,
-expansion of implicit arguments and notations apply to this pattern.
-
-For the purpose of presenting the inference rules, we use a more
-compact notation~:
-\[ \Case{(\lb a x \mto P)}{m}{ \lb x_{11}~...~x_{1p_1} \mto f_1 ~|~\ldots~|~
-  \lb x_{n1}...x_{np_n} \mto f_n}\]
-
-%% CP 06/06 Obsolete avec la nouvelle syntaxe et incompatible avec la
-%% presentation theorique qui suit
-% \paragraph{Non-dependent elimination.}
-%
-% When defining a function of codomain $C$ by case analysis over an
-% object in an inductive type $I$, we build an object of type $I
-% \ra C$. The minimality principle on an inductively defined logical
-% predicate $I$ of type $A \ra \Prop$ is often used to prove a property
-% $\forall x:A,(I~x)\ra (C~x)$.  These are particular cases of the dependent
-% principle that we stated before with a predicate which does not depend
-% explicitly on the object in the inductive definition.
-
-% For instance, a function testing whether a list is empty 
-% can be
-% defined as:
-% \[\kw{fun} l:\ListA \Ra \kw{match}~l~\kw{with}~ \Nil \Ra \true~
-% |~(\cons~a~m) \Ra \false \kw{end}\]
-% represented by
-% \[\lb~l:\ListA \mto\Case{\bool}{l}{\true~ |~ \lb a~m,~\false}\]
-%\noindent {\bf Remark. } 
-
-% In the system \Coq\ the expression above, can be
-% written without mentioning
-% the dummy abstraction:
-% \Case{\bool}{l}{\Nil~ \mbox{\tt =>}~\true~ |~ (\cons~a~m)~
-%  \mbox{\tt =>}~ \false}
-
-\paragraph[Allowed elimination sorts.]{Allowed elimination sorts.\index{Elimination sorts}}
-
-An important question for building the typing rule for \kw{match} is
-what can be the type of $P$ with respect to the type of the inductive
-definitions.
-
-We define now a relation \compat{I:A}{B} between an inductive
-definition $I$ of type $A$ and an arity $B$. This relation states that
-an object in the inductive definition $I$ can be eliminated for
-proving a property $P$ of type $B$.
-
-The case of inductive definitions in sorts \Set\ or \Type{} is simple.
-There is no restriction on the sort of the predicate to be
-eliminated. 
-
-\paragraph{Notations.}
-The \compat{I:A}{B} is defined as the smallest relation satisfying the
-following rules:
-We write \compat{I}{B} for \compat{I:A}{B} where $A$ is the type of
-$I$.
-
-\begin{description}
-\item[Prod] \inference{\frac{\compat{(I~x):A'}{B'}}
-                      {\compat{I:\forall x:A, A'}{\forall x:A, B'}}}
-\item[{\Set} \& \Type] \inference{\frac{
-    s_1 \in \{\Set,\Type(j)\}, 
-    s_2 \in \Sort}{\compat{I:s_1}{I\ra s_2}}}
-\end{description}
-
-The case of Inductive definitions of sort \Prop{} is a bit more
-complicated, because of our interpretation of this sort. The only
-harmless allowed elimination, is the one when predicate $P$ is also of
-sort \Prop.
-\begin{description}
-\item[\Prop] \inference{\compat{I:\Prop}{I\ra\Prop}}
-\end{description}
-\Prop{} is the type of logical propositions, the proofs of properties
-$P$ in \Prop{} could not be used for computation and are consequently
-ignored by the extraction mechanism.
-Assume $A$ and $B$ are two propositions, and the logical disjunction
-$A\vee B$ is defined inductively by~:
-\begin{coq_example*}
-Inductive or (A B:Prop) : Prop :=
-  lintro : A -> or A B | rintro : B -> or A B.
-\end{coq_example*}
-The following definition which computes a boolean value by case over
-the proof of \texttt{or A B} is not accepted~:
-\begin{coq_eval}
-(***************************************************************)
-(*** This example should fail with ``Incorrect elimination'' ***)
-\end{coq_eval}
-\begin{coq_example}
-Definition choice (A B: Prop) (x:or A B) := 
-   match x with lintro a => true | rintro b => false end.
-\end{coq_example}
-From the computational point of view, the structure of the proof of
-\texttt{(or A B)} in this term is needed for computing the boolean
-value.
-
-In general, if $I$ has type \Prop\ then $P$ cannot have type $I\ra
-\Set$, because it will mean to build an informative proof of type
-$(P~m)$ doing a case analysis over a non-computational object that
-will disappear in the extracted program.  But the other way is safe
-with respect to our interpretation we can have $I$ a computational
-object and $P$ a non-computational one, it just corresponds to proving
-a logical property of a computational object.
-
-% Also if $I$ is in one of the sorts \{\Prop, \Set\}, one cannot in
-% general allow an elimination over a bigger sort such as \Type.  But
-% this operation is safe whenever $I$ is a {\em small inductive} type,
-% which means that all the types of constructors of
-% $I$ are small with the following definition:\\
-% $(I~t_1\ldots t_s)$ is a {\em small type of constructor} and
-% $\forall~x:T,C$ is a small type of constructor if $C$ is and if $T$
-% has type \Prop\ or \Set.  \index{Small inductive type}
-
-% We call this particular elimination which gives the possibility to
-% compute a type by induction on the structure of a term, a {\em strong
-%   elimination}\index{Strong elimination}.
-
-In the same spirit, elimination on $P$ of type $I\ra
-\Type$ cannot be allowed because it trivially implies the elimination
-on $P$ of type $I\ra \Set$ by cumulativity. It also implies that there
-is two proofs of the same property which are provably different,
-contradicting the proof-irrelevance property which is sometimes a
-useful axiom~:
-\begin{coq_example}
-Axiom proof_irrelevance : forall (P : Prop) (x y : P), x=y.
-\end{coq_example}
-\begin{coq_eval}
-Reset proof_irrelevance.
-\end{coq_eval}
-The elimination of an inductive definition of type \Prop\ on a
-predicate $P$ of type $I\ra \Type$ leads to a paradox when applied to 
-impredicative inductive definition like the second-order existential
-quantifier \texttt{exProp} defined above, because it give access to
-the two projections on this type.
-
-%\paragraph{Warning: strong elimination}
-%\index{Elimination!Strong elimination}
-%In previous versions of Coq, for a small inductive definition, only the
-%non-informative strong elimination on \Type\ was allowed, because
-%strong elimination on \Typeset\ was not compatible with the current
-%extraction procedure. In this version, strong elimination on \Typeset\
-%is accepted but a dummy element is extracted from it and may generate
-%problems if extracted terms are explicitly used such as in the 
-%{\tt Program} tactic or when extracting ML programs.
-
-\paragraph[Empty and singleton elimination]{Empty and singleton elimination\label{singleton}
-\index{Elimination!Singleton elimination}
-\index{Elimination!Empty elimination}}
-
-There are special inductive definitions in \Prop\ for which more
-eliminations are allowed. 
-\begin{description}
-\item[\Prop-extended] 
-\inference{
-   \frac{I \mbox{~is an empty or singleton
-       definition}~~~s \in \Sort}{\compat{I:\Prop}{I\ra s}}
-}
-\end{description}
-
-% A {\em singleton definition} has always an informative content,
-% even if it is a proposition.
-
-A {\em singleton
-definition} has only one constructor and all the arguments of this
-constructor have type \Prop. In that case, there is a canonical
-way to interpret the informative extraction on an object in that type,
-such that the elimination on any sort $s$ is legal.  Typical examples are
-the conjunction of non-informative propositions and the equality. 
-If there is an hypothesis $h:a=b$ in the context, it can be used for
-rewriting not only in logical propositions but also in any type.
-% In that case, the term \verb!eq_rec! which was defined as an axiom, is
-% now a term of the calculus.
-\begin{coq_example}
-Print eq_rec.
-Extraction eq_rec.
-\end{coq_example}
-An empty definition has no constructors, in that case also,
-elimination on any sort is allowed.
-
-\paragraph{Type of branches.}
-Let $c$ be a term of type $C$, we assume $C$ is a type of constructor
-for an inductive definition $I$. Let $P$ be a term that represents the
-property to be proved.
-We assume $r$ is the number of parameters.
-
-We define a new type \CI{c:C}{P} which represents the type of the
-branch corresponding to the $c:C$ constructor.
-\[
-\begin{array}{ll}
-\CI{c:(I_i~p_1\ldots p_r\ t_1 \ldots t_p)}{P} &\equiv (P~t_1\ldots ~t_p~c) \\[2mm]
-\CI{c:\forall~x:T,C}{P} &\equiv \forall~x:T,\CI{(c~x):C}{P} 
-\end{array}
-\]
-We write \CI{c}{P} for \CI{c:C}{P} with $C$ the type of $c$.
-
-\paragraph{Examples.}
-For $\ListA$ the type of $P$ will be $\ListA\ra s$ for $s \in \Sort$. \\ 
-$ \CI{(\cons~A)}{P} \equiv
-\forall a:A, \forall l:\ListA,(P~(\cons~A~a~l))$.
-
-For $\LengthA$, the type of $P$ will be
-$\forall l:\ListA,\forall n:\nat, (\LengthA~l~n)\ra \Prop$ and the expression
-\CI{(\LCons~A)}{P} is defined as:\\ 
-$\forall a:A, \forall l:\ListA, \forall n:\nat, \forall
-h:(\LengthA~l~n), (P~(\cons~A~a~l)~(\nS~n)~(\LCons~A~a~l~n~l))$.\\ 
-If $P$ does not depend on its third argument, we find the more natural
-expression:\\ 
-$\forall a:A, \forall l:\ListA, \forall n:\nat,
-(\LengthA~l~n)\ra(P~(\cons~A~a~l)~(\nS~n))$.
-
-\paragraph{Typing rule.}
-
-Our very general destructor for inductive definition enjoys the
-following typing rule
-% , where we write 
-% \[
-% \Case{P}{c}{[x_{11}:T_{11}]\ldots[x_{1p_1}:T_{1p_1}]g_1\ldots
-%   [x_{n1}:T_{n1}]\ldots[x_{np_n}:T_{np_n}]g_n}
-% \]
-% for 
-% \[
-% \Case{P}{c}{(c_1~x_{11}~...~x_{1p_1}) \Ra g_1 ~|~\ldots~|~
-% (c_n~x_{n1}...x_{np_n}) \Ra g_n }
-% \]
-
-\begin{description}
-\item[match] \label{elimdep} \index{Typing rules!match}
-\inference{
-\frac{\WTEG{c}{(I~q_1\ldots  q_r~t_1\ldots  t_s)}~~
-  \WTEG{P}{B}~~\compat{(I~q_1\ldots  q_r)}{B}
- ~~
-(\WTEG{f_i}{\CI{(c_{p_i}~q_1\ldots  q_r)}{P}})_{i=1\ldots  l}}
-{\WTEG{\Case{P}{c}{f_1|\ldots  |f_l}}{(P\ t_1\ldots  t_s\ c)}}}%\\[3mm]
-
-provided $I$ is an inductive type in a declaration
-\Ind{\Delta}{r}{\Gamma_I}{\Gamma_C} with 
-$\Gamma_C = [c_1:C_1;\ldots;c_n:C_n]$ and $c_{p_1}\ldots  c_{p_l}$ are the
-only constructors of $I$.
-\end{description}
-
-\paragraph{Example.}
-For \List\ and \Length\ the typing rules for the {\tt match} expression
-are (writing just $t:M$ instead of \WTEG{t}{M}, the environment and
-context being the same in all the judgments).
-
-\[\frac{l:\ListA~~P:\ListA\ra s~~~f_1:(P~(\Nil~A))~~
-      f_2:\forall a:A, \forall l:\ListA, (P~(\cons~A~a~l))}
-    {\Case{P}{l}{f_1~|~f_2}:(P~l)}\]
-
-\[\frac{
-     \begin{array}[b]{@{}c@{}} 
-H:(\LengthA~L~N) \\ P:\forall l:\ListA, \forall n:\nat, (\LengthA~l~n)\ra
-  \Prop\\
- f_1:(P~(\Nil~A)~\nO~\LNil) \\
-   f_2:\forall a:A, \forall l:\ListA, \forall n:\nat, \forall
-  h:(\LengthA~l~n), (P~(\cons~A~a~n)~(\nS~n)~(\LCons~A~a~l~n~h)) 
-      \end{array}}
-    {\Case{P}{H}{f_1~|~f_2}:(P~L~N~H)}\]
-
-\paragraph[Definition of $\iota$-reduction.]{Definition of $\iota$-reduction.\label{iotared}
-\index{iota-reduction@$\iota$-reduction}}
-We still have to define the $\iota$-reduction in the general case.
-
-A $\iota$-redex is a term of the following form:
-\[\Case{P}{(c_{p_i}~q_1\ldots  q_r~a_1\ldots  a_m)}{f_1|\ldots |
-    f_l}\]
-with $c_{p_i}$ the $i$-th constructor of the inductive type $I$ with $r$
-parameters.
-
-The $\iota$-contraction of this term is $(f_i~a_1\ldots a_m)$ leading
-to the general reduction rule:
-\[ \Case{P}{(c_{p_i}~q_1\ldots  q_r~a_1\ldots  a_m)}{f_1|\ldots |
-    f_n} \triangleright_{\iota} (f_i~a_1\ldots a_m) \]
-
-\subsection[Fixpoint definitions]{Fixpoint definitions\label{Fix-term} \index{Fix@{\tt Fix}}}
-The second operator for elimination is fixpoint definition. 
-This fixpoint may involve several mutually recursive definitions.
-The basic concrete syntax for a recursive set of mutually recursive 
-declarations is (with $\Gamma_i$ contexts)~: 
-\[\kw{fix}~f_1 (\Gamma_1) :A_1:=t_1~\kw{with} \ldots \kw{with}~ f_n
-(\Gamma_n) :A_n:=t_n\]
-The terms are obtained by projections from this set of declarations
-and are written 
-\[\kw{fix}~f_1 (\Gamma_1) :A_1:=t_1~\kw{with} \ldots \kw{with}~ f_n
-(\Gamma_n) :A_n:=t_n~\kw{for}~f_i\]
-In the inference rules, we represent such a
-term by 
-\[\Fix{f_i}{f_1:A_1':=t_1' \ldots f_n:A_n':=t_n'}\]
-with $t_i'$ (resp. $A_i'$) representing the term $t_i$ abstracted
-(resp. generalized) with
-respect to the bindings in the context $\Gamma_i$, namely
-$t_i'=\lb \Gamma_i \mto t_i$ and $A_i'=\forall \Gamma_i, A_i$.
-
-\subsubsection{Typing rule}
-The typing rule is the expected one for a fixpoint.
-
-\begin{description}
-\item[Fix] \index{Typing rules!Fix}
-\inference{\frac{(\WTEG{A_i}{s_i})_{i=1\ldots n}~~~~
-            (\WTE{\Gamma,f_1:A_1,\ldots,f_n:A_n}{t_i}{A_i})_{i=1\ldots n}}
-         {\WTEG{\Fix{f_i}{f_1:A_1:=t_1 \ldots f_n:A_n:=t_n}}{A_i}}}
-\end{description}
-
-Any fixpoint definition cannot be accepted because non-normalizing terms
-will lead to proofs of absurdity.
-
-The basic scheme of recursion that should be allowed is the one needed for 
-defining primitive
-recursive functionals. In that case the fixpoint enjoys a special
-syntactic restriction, namely one of the arguments belongs to an
-inductive type, the function starts with a case analysis and recursive
-calls are done on variables coming from patterns and representing subterms.
-
-For instance in the case of natural numbers, a proof of the induction
-principle of type 
-\[\forall P:\nat\ra\Prop, (P~\nO)\ra(\forall n:\nat, (P~n)\ra(P~(\nS~n)))\ra
-\forall n:\nat, (P~n)\]
-can be represented by the term:
-\[\begin{array}{l}
-\lb P:\nat\ra\Prop\mto\lb f:(P~\nO)\mto \lb g:(\forall n:\nat,
-(P~n)\ra(P~(\nS~n))) \mto\\
-\Fix{h}{h:\forall n:\nat, (P~n):=\lb n:\nat\mto \Case{P}{n}{f~|~\lb
-    p:\nat\mto (g~p~(h~p))}}
-\end{array}
-\]
-
-Before accepting a fixpoint definition as being correctly typed, we
-check that the definition is ``guarded''. A precise analysis of this
-notion can be found in~\cite{Gim94}.
-
-The first stage is to precise on which argument the fixpoint will be
-decreasing. The type of this argument should be an inductive
-definition.
-
-For doing this the syntax of fixpoints is extended and becomes 
- \[\Fix{f_i}{f_1/k_1:A_1:=t_1 \ldots f_n/k_n:A_n:=t_n}\]
-where $k_i$ are positive integers.
-Each $A_i$ should be a type (reducible to a term) starting with at least
-$k_i$ products $\forall y_1:B_1,\ldots \forall y_{k_i}:B_{k_i}, A'_i$ 
-and $B_{k_i}$
-being an instance of an inductive definition.
-
-Now in the definition $t_i$, if $f_j$ occurs then it should be applied
-to at least $k_j$ arguments and the $k_j$-th argument should be
-syntactically recognized as structurally smaller than $y_{k_i}$
-
-
-The definition of being structurally smaller is a bit technical.
-One needs first to  define the notion of 
-{\em recursive arguments of a constructor}\index{Recursive arguments}.
-For an inductive definition \Ind{\Gamma}{r}{\Gamma_I}{\Gamma_C},
-the type of a constructor $c$ has the form
-$\forall p_1:P_1,\ldots \forall p_r:P_r, 
-\forall x_1:T_1, \ldots \forall x_r:T_r, (I_j~p_1\ldots 
-p_r~t_1\ldots  t_s)$ the recursive arguments will correspond to $T_i$ in
-which one of the $I_l$ occurs.
-
-
-The main rules for being structurally smaller are the following:\\
-Given a variable $y$ of type an inductive
-definition in a declaration 
-\Ind{\Gamma}{r}{\Gamma_I}{\Gamma_C}
-where $\Gamma_I$ is $[I_1:A_1;\ldots;I_k:A_k]$, and $\Gamma_C$ is
-  $[c_1:C_1;\ldots;c_n:C_n]$.
-The terms structurally smaller than $y$ are:
-\begin{itemize}
-\item $(t~u), \lb x:u \mto t$ when $t$ is structurally smaller than $y$ .
-\item \Case{P}{c}{f_1\ldots f_n} when each $f_i$ is structurally
-  smaller than $y$. \\
-  If $c$ is $y$ or is structurally smaller than $y$, its type is an inductive
-  definition $I_p$ part of the inductive
-  declaration corresponding to $y$. 
-  Each $f_i$ corresponds to a type of constructor $C_q \equiv
-  \forall p_1:P_1,\ldots,\forall p_r:P_r, \forall y_1:B_1, \ldots \forall y_k:B_k, (I~a_1\ldots a_k)$ 
-  and can consequently be
-  written $\lb y_1:B'_1\mto \ldots \lb y_k:B'_k\mto g_i$.
-  ($B'_i$ is obtained from $B_i$ by substituting parameters variables)
-  the variables $y_j$ occurring
-  in $g_i$ corresponding to recursive arguments $B_i$ (the ones in
-  which one of the $I_l$ occurs) are structurally smaller than $y$.
-\end{itemize}
-The following definitions are correct, we enter them using the
-{\tt Fixpoint} command as described in Section~\ref{Fixpoint} and show
-the internal representation.
-\begin{coq_example}
-Fixpoint plus (n m:nat) {struct n} : nat :=
-  match n with
-  | O => m
-  | S p => S (plus p m)
-  end.
-Print plus.
-Fixpoint lgth (A:Set) (l:list A) {struct l} : nat :=
-  match l with
-  | nil => O
-  | cons a l' => S (lgth A l')
-  end.
-Print lgth.
-Fixpoint sizet (t:tree) : nat := let (f) := t in S (sizef f)
- with sizef (f:forest) : nat :=
-  match f with
-  | emptyf => O
-  | consf t f => plus (sizet t) (sizef f)
-  end.
-Print sizet.
-\end{coq_example}
-
-
-\subsubsection[Reduction rule]{Reduction rule\index{iota-reduction@$\iota$-reduction}}
-Let $F$ be the set of declarations: $f_1/k_1:A_1:=t_1 \ldots
-f_n/k_n:A_n:=t_n$.
-The reduction for fixpoints is:
-\[ (\Fix{f_i}{F}~a_1\ldots
-a_{k_i}) \triangleright_{\iota} \substs{t_i}{f_k}{\Fix{f_k}{F}}{k=1\ldots n}
-~a_1\ldots a_{k_i}\]
-when $a_{k_i}$ starts with a constructor.
-This last restriction is needed in order to keep strong normalization
-and corresponds to the reduction for primitive recursive operators.
-
-We can illustrate this behavior on examples.
-\begin{coq_example}
-Goal forall n m:nat, plus (S n) m = S (plus n m).
-reflexivity.
-Abort.
-Goal forall f:forest, sizet (node f) = S (sizef f).
-reflexivity.
-Abort.
-\end{coq_example}
-But assuming the definition of a son function from \tree\ to \forest:
-\begin{coq_example}
-Definition sont (t:tree) : forest 
-   := let (f) := t in f.
-\end{coq_example}
-The following is not a conversion but can be proved after a case analysis.
-\begin{coq_eval}
-(******************************************************************)
-(** Error: Impossible to unify ....                              **)
-\end{coq_eval}
-\begin{coq_example}
-Goal forall t:tree, sizet t = S (sizef (sont t)).
-reflexivity. (** this one fails **)
-destruct t.
-reflexivity.
-\end{coq_example}
-\begin{coq_eval}
-Abort.
-\end{coq_eval}
-
-% La disparition de Program devrait rendre la construction Match obsolete
-% \subsubsection{The {\tt Match \ldots with \ldots end} expression}
-% \label{Matchexpr}
-% %\paragraph{A unary {\tt Match\ldots with \ldots end}.}
-% \index{Match...with...end@{\tt Match \ldots with \ldots end}}
-% The {\tt Match} operator which was a primitive notion in older
-% presentations of the Calculus of Inductive Constructions is now just a
-% macro definition which generates the good combination of {\tt Case}
-% and {\tt Fix} operators in order to generate an operator for primitive
-% recursive definitions. It always considers an inductive definition as 
-% a single inductive definition.
-
-% The following examples illustrates this feature.
-% \begin{coq_example}
-% Definition nat_pr : (C:Set)C->(nat->C->C)->nat->C 
-%   :=[C,x,g,n]Match n with x g end.
-% Print nat_pr.
-% \end{coq_example}
-% \begin{coq_example}
-% Definition forest_pr 
-%   : (C:Set)C->(tree->forest->C->C)->forest->C
-%   := [C,x,g,n]Match n with x g end.
-% \end{coq_example}
-
-% Cet exemple faisait error (HH le 12/12/96), j'ai change pour une
-% version plus simple
-%\begin{coq_example}
-%Definition forest_pr 
-%  : (P:forest->Set)(P emptyf)->((t:tree)(f:forest)(P f)->(P (consf t f)))
-%    ->(f:forest)(P f)
-%  := [C,x,g,n]Match n with x g end.
-%\end{coq_example}
-
-\subsubsection{Mutual induction}
-
-The principles of mutual induction can be automatically generated 
-using the {\tt Scheme} command described in Section~\ref{Scheme}.
-
-\section{Coinductive types}
-The implementation contains also coinductive definitions, which are
-types inhabited by infinite objects. 
-More information on coinductive definitions can be found
-in~\cite{Gimenez95b,Gim98,GimCas05}.
-%They are described in Chapter~\ref{Coinductives}.
-
-\section[\iCIC : the Calculus of Inductive Construction with
-  impredicative \Set]{\iCIC : the Calculus of Inductive Construction with
-  impredicative \Set\label{impredicativity}}
-
-\Coq{} can be used as a type-checker for \iCIC{}, the original 
-Calculus of Inductive Constructions with an impredicative sort \Set{}
-by using the compiler option \texttt{-impredicative-set}.
-
-For example, using the ordinary \texttt{coqtop} command, the following
-is rejected.
-\begin{coq_eval}
-(** This example should fail *******************************
-    Error: The term forall X:Set, X -> X has type Type
-    while it is expected to have type Set
-***)
-\end{coq_eval}
-\begin{coq_example}
-Definition id: Set := forall X:Set,X->X.
-\end{coq_example}
-while it will type-check, if one use instead the \texttt{coqtop
-  -impredicative-set} command.
-
-The major change in the theory concerns the rule for product formation
-in the sort \Set, which is extended to a domain in any sort~:
-\begin{description}
-\item [Prod]  \index{Typing rules!Prod (impredicative Set)}
-\inference{\frac{\WTEG{T}{s}~~~~s \in \Sort~~~~~~
-    \WTE{\Gamma::(x:T)}{U}{\Set}}
-      { \WTEG{\forall~x:T,U}{\Set}}} 
-\end{description}
-This extension has consequences on the inductive definitions which are
-allowed. 
-In the impredicative system, one can build so-called {\em large inductive
-  definitions} like the example of second-order existential
-quantifier (\texttt{exSet}).
-
-There should be restrictions on the eliminations which can be
-performed on such definitions. The eliminations rules in the
-impredicative system for sort \Set{} become~:
-\begin{description}
-\item[\Set] \inference{\frac{s \in
-      \{\Prop, \Set\}}{\compat{I:\Set}{I\ra s}}
-~~~~\frac{I \mbox{~is a small inductive definition}~~~~s \in
-      \{\Type(i)\}}
-         {\compat{I:\Set}{I\ra s}}}
-\end{description}
-     
-
-
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