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-\chapter[Calculus of Inductive Constructions]{Calculus of Inductive Constructions
-\label{Cic}
-\index{Cic@\textsc{CIC}}
-\index{Calculus of Inductive Constructions}}
-%HEVEA\cutname{cic.html}
-
-The underlying formal language of {\Coq} is a {\em Calculus of
-Inductive Constructions} (\CIC) whose inference rules are presented in
-this chapter. The history of this formalism as well as pointers to related work
-are provided in a separate chapter; see {\em Credits}.
-
-\section[The terms]{The terms\label{Terms}}
-
-The expressions of the {\CIC} are {\em terms} and all terms 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, universal quantification is relative to a type and
-takes the form {\it ``for all x
-of type T, P''}. The expression {\it ``x of type T''} is
-written {\it ``x:T''}. Informally, {\it ``x:T''} can be thought as
-{\it ``x belongs to T''}.
-
-The types of types are {\em sorts}. Types and sorts are themselves
-terms so that terms, types and sorts are all components of a common
-syntactic language of terms which is described in
-Section~\ref{cic:terms} but, first, we describe sorts.
-
-\subsection[Sorts]{Sorts\label{Sorts}
-\index{Sorts}}
-All sorts have a type and there is an infinite well-founded
-typing hierarchy of sorts whose base sorts are {\Prop} and {\Set}.
-
-The sort {\Prop} intends to be the type of logical propositions. If
-$M$ is a logical proposition then it denotes the class of terms
-representing proofs of $M$. An object $m$ belonging to $M$ witnesses
-the fact that $M$ is provable. An object of type {\Prop} is called a
-proposition.
-
-The sort {\Set} intends to be the type of small sets. This includes data
-types such as booleans and naturals, but also products, subsets, and
-function types over these data types.
-
-{\Prop} and {\Set} themselves can be manipulated as ordinary
-terms. Consequently they also have a type. Because assuming simply
-that {\Set} has type {\Set} leads to an inconsistent theory~\cite{Coq86}, the
-language of {\CIC} has infinitely many sorts. There are, in addition
-to {\Set} and {\Prop} a hierarchy of universes {\Type$(i)$} for any
-integer $i$.
-
-Like {\Set}, all of the sorts {\Type$(i)$} contain small sets such as
-booleans, natural numbers, as well as products, subsets and function
-types over small sets. But, unlike {\Set}, they also contain large
-sets, namely the sorts {\Set} and {\Type$(j)$} for $j<i$, and all
-products, subsets and function types over these sorts.
-
-Formally, we call {\Sort} the set of sorts which is defined by:
-\index{Type@{\Type}}%
-\index{Prop@{\Prop}}%
-\index{Set@{\Set}}%
-\[\Sort \equiv \{\Prop,\Set,\Type(i)\;|\; i \in \NN\} \]
-Their properties, such as:
-{\Prop:\Type$(1)$}, {\Set:\Type$(1)$}, and {\Type$(i)$:\Type$(i+1)$},
-are defined in Section~\ref{subtyping-rules}.
-
-The user does not have to 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 {\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
-{\em algebraic universes}\index{algebraic universe}.
-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 template polymorphic inductive
-types (see Section \ref{Template-polymorphism}),
-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}).
-
-%% HH: This looks to me more like source of confusion than helpful
-
-%% \subsection{Constants}
-
-%% Constants refers to
-%% objects in the global 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}
-\label{cic:terms}
-
-Terms are built from sorts, variables, constants,
-%constructors, inductive types,
-abstractions, applications, local definitions,
-%case analysis, fixpoints, cofixpoints
-and products.
-From a syntactic point of view, types cannot be distinguished from terms,
-except that they cannot start by an abstraction or a constructor.
-More precisely the language of the {\em Calculus of Inductive
-  Constructions} is built from the following rules.
-%
-\begin{enumerate}
-\item the sorts {\Set}, {\Prop}, ${\Type(i)}$ are terms.
-\item variables, hereafter ranged over by letters $x$, $y$, etc., are terms
-\item constants, hereafter ranged over by letters $c$, $d$, etc.,  are terms.
-%\item constructors, hereafter ranged over by letter $C$, are terms.
-%\item inductive types, hereafter ranged over by letter $I$, are terms.
-\item\index{products} 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 $x$ does not occur in $U$ then
-  $\forall~x:T,U$ reads as {\it ``if T then U''}. A {\em non dependent
-  product} can be written: $T \ra U$.
-\item if $x$ is a variable and $T$, $u$ are terms then $\lb x:T \mto u$
-  ($\kw{fun}~x:T~ {\tt =>}~ u$ 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 the expression $u$.
-\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 $t$, $T$ and $u$ are terms then
-  $\kw{let}~x:=t:T~\kw{in}~u$ is a
-  term which denotes the term $u$ where the variable $x$ is locally
-  bound to $t$ of type $T$. This stands for the common ``let-in''
-  construction of functional programs such as ML or Scheme.
-%\item case ...
-%\item fixpoint ...
-%\item cofixpoint ...
-\end{enumerate}
-
-\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.
-
-\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 $\subst{u}{x}{t}$.
-
-\paragraph[The logical vs programming readings.]{The logical vs programming readings.}
-
-The constructions of the {\CIC} can be used to express both logical
-and programming notions, accordingly to the Curry-Howard
-correspondence between proofs and programs, and between propositions
-and types~\cite{Cur58,How80,Bru72}.
-
-For instance, let us assume that \nat\ is the type of natural numbers
-with zero element written $0$ and that ${\tt True}$ is the always true
-proposition.  Then $\ra$ is used both to denote $\nat\ra\nat$ which is
-the type of functions from \nat\ to \nat, to denote ${\tt True}\ra{\tt
-  True}$ which is an implicative proposition, to denote $\nat \ra
-\Prop$ which is the type of unary predicates over the natural numbers,
-etc.
-
-Let us assume that ${\tt mult}$ is a function of type $\nat\ra\nat\ra
-\nat$ and ${\tt eqnat}$ a predicate of type $\nat\ra\nat\ra \Prop$.
-The $\lambda$-abstraction can serve to build ``ordinary'' functions as
-in $\lambda x:\nat.({\tt mult}~x~x)$ (i.e. $\kw{fun}~x:\nat ~{\tt =>}~
-{\tt mult} ~x~x$ in {\Coq} notation) but may build also predicates
-over the natural numbers. For instance $\lambda x:\nat.({\tt eqnat}~
-x~0)$ (i.e. $\kw{fun}~x:\nat ~{\tt =>}~ {\tt eqnat}~ x~0$ in {\Coq}
-notation) will represent the predicate of one variable $x$ which
-asserts the equality of $x$ with $0$. This predicate has type $\nat
-\ra \Prop$ and it can be applied to any expression of type ${\nat}$,
-say $t$, to give an object $P~t$ of type \Prop, namely 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 the type of proofs of the formula
-``$\forall x.\,P(x)$''.
-
-\section[Typing rules]{Typing rules\label{Typed-terms}}
-
-As objects of type theory, terms are subjected to {\em type
-discipline}. The well typing of a term depends on
-a global environment and a local context.
-
-\paragraph{Local context.\index{Local context}}
-A {\em local context} is an ordered list of
-{\em local declarations\index{declaration!local}} of names which we call {\em variables\index{variable}}.
-The declaration of some variable $x$ is
-either a {\em local assumption\index{assumption!local}}, written $x:T$ ($T$ is a type) or a {\em local definition\index{definition!local}},
-written $x:=t:T$.  We use brackets to write local contexts. A
-typical example is $[x:T;y:=u:U;z:V]$.  Notice that the variables
-declared in a local 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 $(x:=t:T) \in \Gamma$.
-For the rest of the chapter, the $\Gamma::(y:T)$ denotes the local context
-$\Gamma$ enriched with the local assumption $y:T$.
-Similarly, $\Gamma::(y:=t:T)$ denotes the local context
-$\Gamma$ enriched with the local definition $(y:=t:T)$.
-The notation $[]$ denotes the empty local context.
-By $\Gamma_1; \Gamma_2$ we mean concatenation of the local context $\Gamma_1$
-and the local context $\Gamma_2$.
-
-% 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 local 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 $\Delta$.
-
-\paragraph[Global environment.]{Global environment.\index{Global environment}}
-%Because we are manipulating global declarations (global constants and global
-%assumptions), we also need to consider a global environment $E$.
-
-A {\em global environment} is an ordered list of {\em global declarations\index{declaration!global}}.
-Global declarations are either {\em global assumptions\index{assumption!global}} or {\em global
-definitions\index{definition!global}}, but also declarations of inductive objects. Inductive objects themselves declare both inductive or coinductive types and constructors
-(see Section~\ref{Cic-inductive-definitions}).
-
-A {\em global assumption} will be represented in the global environment as
-$(c:T)$ which assumes the name $c$ to be of some type $T$.
-A {\em global definition} will
-be represented in the global environment as $c:=t:T$ which defines
-the name $c$ to have value $t$ and type $T$.
-We shall call such names {\em constants}.
-For the rest of the chapter, the $E;c:T$ denotes the global environment
-$E$ enriched with the global assumption $c:T$.
-Similarly, $E;c:=t:T$ denotes the global environment
-$E$ enriched with the global definition $(c:=t:T)$.
-
-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 define simultaneously two
-judgments.  The first one \WTEG{t}{T} means the term $t$ is well-typed
-and has type $T$ in the global environment $E$ and local context $\Gamma$.  The
-second judgment \WFE{\Gamma} means that the global environment $E$ is
-well-formed and the local context $\Gamma$ is a valid local context in this
-global environment.
-% HH: This looks to me complicated. I think it would be better to talk
-% about ``discharge'' as a transformation of global environments,
-% rather than as keeping a local context next to global constants.
-%
-%%  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 local sub-context of the
-%%  current local context.}.
-
-A term $t$ is well typed in a global environment $E$ iff there exists a
-local context $\Gamma$ and a term $T$ such that the judgment \WTEG{t}{T} can
-be derived from the following rules.
-\begin{description}
-\item[W-Empty] \inference{\WF{[]}{}}
-\item[W-Local-Assum]  % 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)}}}
-\item[W-Local-Def]
-\inference{\frac{\WTEG{t}{T}~~~~x \not\in \Gamma % \cup E
-     }{\WFE{\Gamma::(x:=t:T)}}}
-\item[W-Global-Assum] \inference{\frac{\WTE{}{T}{s}~~~~s \in \Sort~~~~c \notin E}
-                      {\WF{E;c:T}{}}}
-\item[W-Global-Def] \inference{\frac{\WTE{}{t}{T}~~~c \notin E}
-                      {\WF{E;c:=t:T}{}}}
-\item[Ax-Prop] \index{Typing rules!Ax-Prop}
-\inference{\frac{\WFE{\Gamma}}{\WTEG{\Prop}{\Type(1)}}}
-\item[Ax-Set] \index{Typing rules!Ax-Set}
-\inference{\frac{\WFE{\Gamma}}{\WTEG{\Set}{\Type(1)}}}
-\item[Ax-Type] \index{Typing rules!Ax-Type}
-\inference{\frac{\WFE{\Gamma}}{\WTEG{\Type(i)}{\Type(i+1)}}}
-\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-Prop]  \index{Typing rules!Prod-Prop}
-\inference{\frac{\WTEG{T}{s}~~~~s \in \Sort~~~
-    \WTE{\Gamma::(x:T)}{U}{\Prop}}
-      { \WTEG{\forall~x:T,U}{\Prop}}} 
-\item[Prod-Set]  \index{Typing rules!Prod-Set}
-\inference{\frac{\WTEG{T}{s}~~~~s \in\{\Prop, \Set\}~~~~~~
-    \WTE{\Gamma::(x:T)}{U}{\Set}}
-      { \WTEG{\forall~x:T,U}{\Set}}} 
-\item[Prod-Type]  \index{Typing rules!Prod-Type}
-\inference{\frac{\WTEG{T}{\Type(i)}~~~~
-    \WTE{\Gamma::(x:T)}{U}{\Type(i)}}
-    {\WTEG{\forall~x:T,U}{\Type(i)}}}
-\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{\letin{x}{t:T}{u}}{\subst{U}{x}{t}}}}
-\end{description}
-
-\Rem Prod$_1$ and Prod$_2$ typing-rules make sense if we consider the semantic
-difference between {\Prop} and {\Set}:
-\begin{itemize}
-  \item All values of a type that has a sort {\Set} are extractable.
-  \item No values of a type that has a sort {\Prop} are extractable.
-\end{itemize}
-
-\Rem We may have $\kw{let}~x:=t:T~\kw{in}~u$
-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}}
-
-In \CIC, there is an internal reduction mechanism. In particular, it
-can decide if two programs are {\em intentionally} equal (one
-says {\em convertible}). Convertibility is described in this section.
-
-\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 global environment $E$ and local 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 global 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 variables defined in local contexts or constants defined 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{$\eta$-expansion.%
-\label{eta}%
-\index{eta-expansion@$\eta$-expansion}%
-%\index{eta-reduction@$\eta$-reduction}
-}%
-Another important concept is $\eta$-expansion. It is legal to identify any
-term $t$ of functional type $\forall x:T, U$ with its so-called
-$\eta$-expansion $\lb x:T\mto (t\ x)$ for $x$ an arbitrary variable
-name fresh in $t$.
-
-\Rem We deliberately do not define $\eta$-reduction:
-\begin{latexonly}%
-  $$\lb x:T\mto (t\ x)\not\triangleright_\eta\hskip.3em t$$
-\end{latexonly}%
-\begin{htmlonly}
-  $$\lb x:T\mto (t\ x)~\not\triangleright_\eta~t$$
-\end{htmlonly}
-This is because, in general, the type of $t$ need not to be convertible to the type of $\lb x:T\mto (t\ x)$.
-E.g., if we take $f$ such that:
-\begin{latexonly}%
-  $$f\hskip.5em:\hskip.5em\forall x:Type(2),Type(1)$$
-\end{latexonly}%
-\begin{htmlonly}
-  $$f~:~\forall x:Type(2),Type(1)$$
-\end{htmlonly}
-then
-\begin{latexonly}%
-  $$\lb x:Type(1),(f\, x)\hskip.5em:\hskip.5em\forall x:Type(1),Type(1)$$
-\end{latexonly}%
-\begin{htmlonly}
-  $$\lb x:Type(1),(f\, x)~:~\forall x:Type(1),Type(1)$$
-\end{htmlonly}
-We could not allow
-\begin{latexonly}%
-  $$\lb x:Type(1),(f\,x)\hskip.4em\not\triangleright_\eta\hskip.6em f$$
-\end{latexonly}%
-\begin{htmlonly}
-  $$\lb x:Type(1),(f\,x)~\not\triangleright_\eta~f$$
-\end{htmlonly}
-because the type of the reduced term $\forall x:Type(2),Type(1)$
-would not be convertible to the type of the original term $\forall x:Type(1),Type(1)$.
-
-\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 global environment $E$ and local 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
-  $\beta\iota\delta\zeta\eta$-convertible}, or simply {\em
-  convertible}, or {\em equivalent}, in the global environment $E$ and
-local context $\Gamma$ iff there exist terms $u_1$ and $u_2$ such that
-$\WTEGRED{t_1}{\triangleright \ldots \triangleright}{u_1}$ and
-$\WTEGRED{t_2}{\triangleright \ldots \triangleright}{u_2}$ and either
-$u_1$ and $u_2$ are identical, or they are convertible up to
-$\eta$-expansion, i.e. $u_1$ is $\lb x:T\mto u'_1$ and $u_2\,x$ is
-recursively convertible to $u'_1$, or, symmetrically, $u_2$ is $\lb
-x:T\mto u'_2$ and $u_1\,x$ is recursively convertible to $u'_2$.  We
-then write $\WTEGCONV{t_1}{t_2}$.
-
-Apart from this we consider two instances of polymorphic and cumulative (see Chapter~\ref{Universes-full}) inductive types (see below)
-convertible $\WTEGCONV{t\ w_1 \dots w_m}{t\ w_1' \dots w_m'}$ if we have subtypings (see below) in both directions, i.e.,
-$\WTEGLECONV{t\ w_1 \dots w_m}{t\ w_1' \dots w_m'}$ and $\WTEGLECONV{t\ w_1' \dots w_m'}{t\ w_1 \dots w_m}$.
-Furthermore, we consider $\WTEGCONV{c\ v_1 \dots v_m}{c'\ v_1' \dots v_m'}$ convertible if $\WTEGCONV{v_i}{v_i'}$
-and we have that $c$ and $c'$ are the same constructors of different instances the same inductive types (differing only in universe levels)
-such that $\WTEG{c\ v_1 \dots v_m}{t\ w_1 \dots w_m}$ and $\WTEG{c'\ v_1' \dots v_m'}{t'\ w_1' \dots w_m'}$ and we have $\WTEGCONV{t\ w_1 \dots w_m}{t\ w_1' \dots w_m'}$.
-
-The convertibility relation allows introducing a new typing rule
-which says that two convertible well-formed types have the same
-inhabitants.
-
-\section[Subtyping rules]{Subtyping rules\index{Subtyping rules}
-\label{subtyping-rules}}
-
-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 is the {\em cumulativity} rule of
-{\CIC}). This property extends the equivalence relation of
-convertibility into a {\em subtyping} relation 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 $i$, $\WTEGLECONV{\Set}{\Type(i)}$,
-\item $\WTEGLECONV{\Prop}{\Set}$, hence, by transitivity,
-  $\WTEGLECONV{\Prop}{\Type(i)}$, for any $i$
-\item if $\WTEGCONV{T}{U}$ and $\WTELECONV{\Gamma::(x:T)}{T'}{U'}$ then $\WTEGLECONV{\forall~x:T, T'}{\forall~x:U, U'}$.
-\item if $\Ind{}{p}{\Gamma_I}{\Gamma_C}$ is a universe polymorphic and cumulative (see Chapter~\ref{Universes-full})
-  inductive type (see below) and $(t : \forall\Gamma_P,\forall\Gamma_{\mathit{Arr}(t)}, \Sort)\in\Gamma_I$
-  and $(t' : \forall\Gamma_P',\forall\Gamma_{\mathit{Arr}(t)}', \Sort')\in\Gamma_I$
-  are two different instances of \emph{the same} inductive type (differing only in universe levels) with constructors
-  \[[c_1: \forall\Gamma_P,\forall T_{1,1} \dots T_{1,n_1},t\ v_{1,1} \dots v_{1,m}; \dots; c_k: \forall\Gamma_P,\forall T_{k, 1} \dots T_{k,n_k},t\ v_{n,1}\dots v_{n,m}]\]
-  and
-  \[[c_1: \forall\Gamma_P',\forall T_{1,1}' \dots T_{1,n_1}',t'\ v_{1,1}' \dots v_{1,m}'; \dots; c_k: \forall\Gamma_P',\forall T_{k, 1}' \dots T_{k,n_k}',t\ v_{n,1}'\dots v_{n,m}']\]
-  respectively then $\WTEGLECONV{t\ w_1 \dots w_m}{t\ w_1' \dots w_m'}$ (notice that $t$ and $t'$ are both fully applied, i.e., they have a sort as a type)
-  if $\WTEGCONV{w_i}{w_i'}$ for $1 \le i \le m$ and we have
-  \[ \WTEGLECONV{T_{i,j}}{T_{i,j}'} \text{ and } \WTEGLECONV{A_i}{A_i'}\]
-  where $\Gamma_{\mathit{Arr}(t)} = [a_1 : A_1; a_1 : A_l]$ and $\Gamma_{\mathit{Arr}(t)} = [a_1 : A_1'; a_1 : A_l']$.
-\end{enumerate}
-
-The conversion rule up to subtyping 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[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
-rules. 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[Inductive definitions]{Inductive Definitions\label{Cic-inductive-definitions}}
-
-% Here we assume that the reader knows what is an inductive definition.
-
-Formally, we can represent any {\em inductive definition\index{definition!inductive}} as \Ind{}{p}{\Gamma_I}{\Gamma_C} where:
-\begin{itemize}
-  \item $\Gamma_I$ determines the names and types of inductive types;
-  \item $\Gamma_C$ determines the names and types of constructors of these inductive types;
-  \item $p$ determines the number of parameters of these inductive types.
-\end{itemize}
-These inductive definitions, together with global assumptions and global definitions, then form the global environment.
-%
-Additionally, for any $p$ there always exists $\Gamma_P=[a_1:A_1;\dots;a_p:A_p]$
-such that each $T$ in $(t:T)\in\Gamma_I\cup\Gamma_C$ can be written as:
-$\forall\Gamma_P, T^\prime$ where $\Gamma_P$ is called the {\em context of parameters\index{context of parameters}}.
-Furthermore, we must have that each $T$ in $(t:T)\in\Gamma_I$ can be written as:
-$\forall\Gamma_P,\forall\Gamma_{\mathit{Arr}(t)}, \Sort$ where $\Gamma_{\mathit{Arr}(t)}$ is called the
-{\em Arity} of the inductive type\index{arity of inductive type} $t$ and
-$\Sort$ is called the sort of the inductive type $t$.
-
-\paragraph{Examples}
-
-    \newcommand\ind[3]{$\mathsf{Ind}~[#1]\left(\hskip-.4em
-    \begin{array}{r@{\mathrm{~:=~}}l}
-      #2 & #3 \\
-    \end{array}
-    \hskip-.4em
-    \right)$}
-    \def\colon{@{\hskip.5em:\hskip.5em}}
-
-The declaration for parameterized lists is:
-\begin{latexonly}
-    \vskip.5em
-
-    \ind{1}{[\List:\Set\ra\Set]}{\left[\begin{array}{r@{:}l}
-                                      \Nil & \forall A:\Set,\List~A \\
-                                      \cons & \forall A:\Set, A \ra \List~A \ra \List~A
-                                   \end{array}
-                             \right]}
-    \vskip.5em
-\end{latexonly}
-\begin{rawhtml}<pre><table style="border-spacing:0">
-  <tr style="vertical-align:middle">
-    <td style="width:10pt;text-align:center;font-family:sans-serif;font-style:italic">Ind</td>
-    <td style="width:20pt;text-align:center">[1]</td>
-    <td style="width:5pt;text-align:center">⎛<br>⎝</td>
-    <td style="width:120pt;text-align:center">[ <span style="font-family:monospace">list : Set → Set</span> ]</td>
-    <td style="width:20pt;text-align:center;font-family:monospace">:=</td>
-    <td style="width:10pt;text-align:center">⎡<br>⎣</td>
-    <td>
-      <table style="border-spacing:0">
-        <tr>
-          <td style="width:20pt;text-align:right;font-family:monospace">nil</td>
-          <td style="width:20pt;text-align:center;font-family:monospace">:=</td>
-          <td style="text-align:left;font-family:monospace">∀A : Set, list A</td>
-        </tr>
-        <tr>
-          <td style="width:20pt;text-align:right;font-family:monospace">cons</td>
-          <td style="width:20pt;text-align:center;font-family:monospace">:=</td>
-          <td style="text-align:left;font-family:monospace">∀A : Set, A → list A → list A</td>
-        </tr>
-      </table>
-    </td>
-    <td style="width:10pt;text-align:center">⎤<br>⎦</td>
-    <td style="width:5pt;text-align:center">⎞<br>⎠</td>
-  </tr>
-</table></pre>
-\end{rawhtml}
-\noindent which corresponds to the result of the \Coq\ declaration:
-\begin{coq_example*}
-Inductive list (A:Set) : Set :=
-  | nil : list A
-  | cons : A -> list A -> list A.
-\end{coq_example*}
-
-\noindent The declaration for a mutual inductive definition of {\tree} and {\forest} is:
-\begin{latexonly}
-    \vskip.5em
-\ind{~}{\left[\begin{array}{r@{:}l}\tree&\Set\\\forest&\Set\end{array}\right]}
-    {\left[\begin{array}{r@{:}l}
-             \node & \forest \ra \tree\\
-             \emptyf & \forest\\
-             \consf & \tree \ra \forest \ra \forest\\
-                       \end{array}\right]}
-    \vskip.5em
-\end{latexonly}
-\begin{rawhtml}<pre><table style="border-spacing:0">
-  <tr style="vertical-align:middle">
-    <td style="width:10pt;text-align:center;font-family:sans-serif;font-style:italic">Ind</td>
-    <td style="width:20pt;text-align:center">[1]</td>
-    <td style="width:5pt;text-align:center">⎛<br>⎜<br>⎝</td>
-    <td style="width:10pt;text-align:center">⎡<br>⎣</td>
-    <td>
-      <table style="border-spacing:0">
-        <tr>
-          <td style="width:20pt;text-align:right;font-family:monospace">tree</td>
-          <td style="width:20pt;text-align:center;font-family:monospace">:</td>
-          <td style="text-align:left;font-family:monospace">Set</td>
-        </tr>
-        <tr>
-          <td style="width:20pt;text-align:right;font-family:monospace">forest</td>
-          <td style="width:20pt;text-align:center;font-family:monospace">:</td>
-          <td style="text-align:left;font-family:monospace">Set</td>
-        </tr>
-      </table>
-    </td>
-    <td style="width:10pt;text-align:center">⎤<br>⎦</td>
-    <td style="width:20pt;text-align:center;font-family:monospace">:=</td>
-    <td style="width:10pt;text-align:center">⎡<br>⎢<br>⎣</td>
-    <td>
-      <table style="border-spacing:0">
-        <tr>
-          <td style="width:20pt;text-align:right;font-family:monospace">node</td>
-          <td style="width:20pt;text-align:center;font-family:monospace">:</td>
-          <td style="text-align:left;font-family:monospace">forest → tree</td>
-        </tr>
-        <tr>
-          <td style="width:20pt;text-align:right;font-family:monospace">emptyf</td>
-          <td style="width:20pt;text-align:center;font-family:monospace">:</td>
-          <td style="text-align:left;font-family:monospace">forest</td>
-        </tr>
-        <tr>
-          <td style="width:20pt;text-align:right;font-family:monospace">consf</td>
-          <td style="width:20pt;text-align:center;font-family:monospace">:</td>
-          <td style="text-align:left;font-family:monospace">tree → forest → forest</td>
-        </tr>
-      </table>
-    </td>
-    <td style="width:10pt;text-align:center">⎤<br>⎥<br>⎦</td>
-    <td style="width:5pt;text-align:center">⎞<br>⎟<br>⎠</td>
-  </tr>
-</table></pre>
-\end{rawhtml}
-\noindent which corresponds to the result of the \Coq\
-declaration:
-\begin{coq_example*}
-Inductive tree : Set :=
-    node : forest -> tree
-with forest : Set :=
-  | emptyf : forest
-  | consf : tree -> forest -> forest.
-\end{coq_example*}
-
-\noindent The declaration for a mutual inductive definition of {\even} and {\odd} is:
-\begin{latexonly}
-	\newcommand\GammaI{\left[\begin{array}{r@{:}l}
-			    \even & \nat\ra\Prop \\
-			    \odd & \nat\ra\Prop
-                      \end{array}
-                \right]}
-		\newcommand\GammaC{\left[\begin{array}{r@{:}l}
-			    \evenO & \even~\nO \\
-			    \evenS & \forall n : \nat, \odd~n \ra \even~(\nS~n)\\
-			    \oddS & \forall n : \nat, \even~n \ra \odd~(\nS~n)
-                      \end{array}
-                \right]}
-    \vskip.5em
-    \ind{1}{\GammaI}{\GammaC}
-    \vskip.5em
-\end{latexonly}
-\begin{rawhtml}<pre><table style="border-spacing:0">
-  <tr style="vertical-align:middle">
-    <td style="width:10pt;text-align:center;font-family:sans-serif;font-style:italic">Ind</td>
-    <td style="width:20pt;text-align:center">[1]</td>
-    <td style="width:5pt;text-align:center">⎛<br>⎜<br>⎝</td>
-    <td style="width:10pt;text-align:center">⎡<br>⎣</td>
-    <td>
-      <table style="border-spacing:0">
-        <tr>
-          <td style="width:20pt;text-align:right;font-family:monospace">even</td>
-          <td style="width:20pt;text-align:center;font-family:monospace">:</td>
-          <td style="text-align:left;font-family:monospace">nat → Prop</td>
-        </tr>
-        <tr>
-          <td style="width:20pt;text-align:right;font-family:monospace">odd</td>
-          <td style="width:20pt;text-align:center;font-family:monospace">:</td>
-          <td style="text-align:left;font-family:monospace">nat → Prop</td>
-        </tr>
-      </table>
-    </td>
-    <td style="width:10pt;text-align:center">⎤<br>⎦</td>
-    <td style="width:20pt;text-align:center;font-family:monospace">:=</td>
-    <td style="width:10pt;text-align:center">⎡<br>⎢<br>⎣</td>
-    <td>
-      <table style="border-spacing:0">
-        <tr>
-          <td style="width:20pt;text-align:right;font-family:monospace">even_O</td>
-          <td style="width:20pt;text-align:center;font-family:monospace">:</td>
-          <td style="text-align:left;font-family:monospace">even O</td>
-        </tr>
-        <tr>
-          <td style="width:20pt;text-align:right;font-family:monospace">even_S</td>
-          <td style="width:20pt;text-align:center;font-family:monospace">:</td>
-          <td style="text-align:left;font-family:monospace">∀n : nat, odd n → even (S n)</td>
-        </tr>
-        <tr>
-          <td style="width:20pt;text-align:right;font-family:monospace">odd_S</td>
-          <td style="width:20pt;text-align:center;font-family:monospace">:</td>
-          <td style="text-align:left;font-family:monospace">∀n : nat, even n → odd (S n)</td>
-        </tr>
-      </table>
-    </td>
-    <td style="width:10pt;text-align:center">⎤<br>⎥<br>⎦</td>
-    <td style="width:5pt;text-align:center">⎞<br>⎟<br>⎠</td>
-  </tr>
-</table></pre>
-\end{rawhtml}
-\noindent which corresponds to the result of the \Coq\
-declaration:
-\begin{coq_example*}
-Inductive even : nat -> Prop :=
-  | even_O : even 0
-  | even_S : forall n, odd n -> even (S n)
-with odd : nat -> Prop :=
-  | odd_S : forall n, even n -> odd (S n).
-\end{coq_example*}
-
-\subsection{Types of inductive objects}
-We have to give the type of constants in a global environment $E$ which
-contains an inductive declaration.
-
-\begin{description}
-\item[Ind]  \index{Typing rules!Ind}
-  \inference{\frac{\WFE{\Gamma}~~~~~~~~\Ind{}{p}{\Gamma_I}{\Gamma_C} \in E~~~~~~~~(a:A)\in\Gamma_I}{\WTEG{a}{A}}}
-\item[Constr]  \index{Typing rules!Constr}
-  \inference{\frac{\WFE{\Gamma}~~~~~~~~\Ind{}{p}{\Gamma_I}{\Gamma_C} \in E~~~~~~~~(c:C)\in\Gamma_C}{\WTEG{c}{C}}}
-\end{description}
-
-\begin{latexonly}%
-\paragraph{Example.}
-Provided that our environment $E$ contains inductive definitions we showed before,
-these two inference rules above enable us to conclude that:
-\vskip.5em
-\newcommand\prefix{E[\Gamma]\vdash\hskip.25em}
-$\begin{array}{@{}l}
-   \prefix\even : \nat\ra\Prop\\
-   \prefix\odd : \nat\ra\Prop\\
-   \prefix\evenO : \even~\nO\\
-   \prefix\evenS : \forall~n:\nat, \odd~n \ra \even~(\nS~n)\\
-   \prefix\oddS : \forall~n:\nat, \even~n \ra \odd~(\nS~n)
- \end{array}$
-\end{latexonly}%
-
-%\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 \haslength\, $A$ is a parameter because
-%what is effectively inductively defined is $\ListA$ or $\haslengthA$ for
-%a given $A$ which is constant in the type of constructors.  But when
-%we define $(\haslengthA~l~n)$, $l$ 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[Definition]{Definition\index{Arity}\label{Arity}}
-A type $T$ is an {\em arity of sort $s$} if it converts
-to the sort $s$ or to a product $\forall~x:T,U$ with $U$ an arity
-of sort $s$.
-
-\paragraph[Examples]{Examples}
-$A\ra \Set$ is an arity of sort $\Set$.
-$\forall~A:\Prop,A\ra \Prop$ is an arity of sort \Prop.
-
-\paragraph[Definition]{Definition}
-A type $T$ is an {\em arity} if there is a $s\in\Sort$
-such that $T$ is an arity of sort $s$.
-
-\paragraph[Examples]{Examples}
-$A\ra \Set$ and $\forall~A:\Prop,A\ra \Prop$ are arities.
-
-\paragraph[Definition]{Definition\index{type of constructor}}
-We say that $T$ is a {\em type of constructor of $I$\index{type of constructor}}
-in one of the following two cases:
-\begin{itemize}
-  \item $T$ is $(I~t_1\ldots ~t_n)$
-  \item $T$ is $\forall x:U,T^\prime$ where $T^\prime$ is also a type of constructor of $I$
-\end{itemize}
-
-\paragraph[Examples]{Examples}
-$\nat$ and $\nat\ra\nat$ are types of constructors of $\nat$.\\
-$\forall A:\Type,\List~A$ and $\forall A:\Type,A\ra\List~A\ra\List~A$ are constructors of $\List$.
-
-\paragraph[Definition]{Definition\index{Positivity}\label{Positivity}}
-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}
-
-\newcommand\vv{\textSFxi}    % │
-\newcommand\hh{\textSFx}     %   ─
-\newcommand\vh{\textSFviii}  % ├
-\newcommand\hv{\textSFii}    %   └
-\newlength\framecharacterwidth
-\settowidth\framecharacterwidth{\hh}
-\newcommand\ws{\hbox{}\hskip\the\framecharacterwidth}
-\newcommand\ruleref[1]{\hskip.25em\dots\hskip.2em{\em (bullet #1)}}
-\newcommand{\NatTree}{\mbox{\textsf{nattree}}}
-\newcommand{\NatTreeA}{\mbox{\textsf{nattree}}~\ensuremath{A}}
-\newcommand{\cnode}{\mbox{\textsf{node}}}
-\newcommand{\cleaf}{\mbox{\textsf{leaf}}}
-
-\noindent For instance, if one considers the following variant of a tree type branching over the natural numbers
-
-\begin{verbatim}
-Inductive nattree (A:Type) : Type :=
- | leaf : nattree A
- | node : A -> (nat -> nattree A) -> nattree A
-\end{verbatim}
-
-\begin{latexonly}
-\noindent Then every instantiated constructor of $\NatTreeA$ satisfies the nested positivity condition for $\NatTree$\\
-\noindent
-\ws\ws\vv\\
-\ws\ws\vh\hh\ws concerning type $\NatTreeA$ of constructor $\cleaf$:\\
-\ws\ws\vv\ws\ws\ws\ws Type $\NatTreeA$ of constructor $\cleaf$ satisfies the positivity condition for $\NatTree$\\
-\ws\ws\vv\ws\ws\ws\ws because $\NatTree$ does not appear in any (real) arguments of the type of that constructor\\
-\ws\ws\vv\ws\ws\ws\ws (primarily because $\NatTree$ does not have any (real) arguments)\ruleref1\\
-\ws\ws\vv\\
-\ws\ws\hv\hh\ws concerning type $\forall~A\ra(\NN\ra\NatTreeA)\ra\NatTreeA$ of constructor $\cnode$:\\
-    \ws\ws\ws\ws\ws\ws\ws Type $\forall~A:\Type,A\ra(\NN\ra\NatTreeA)\ra\NatTreeA$ of constructor $\cnode$\\
-\ws\ws\ws\ws\ws\ws\ws satisfies the positivity condition for $\NatTree$ because:\\
-\ws\ws\ws\ws\ws\ws\ws\vv\\
-\ws\ws\ws\ws\ws\ws\ws\vh\hh\ws $\NatTree$ occurs only strictly positively in $\Type$\ruleref1\\
-\ws\ws\ws\ws\ws\ws\ws\vv\\
-\ws\ws\ws\ws\ws\ws\ws\vh\hh\ws $\NatTree$ occurs only strictly positively in $A$\ruleref1\\
-\ws\ws\ws\ws\ws\ws\ws\vv\\
-    \ws\ws\ws\ws\ws\ws\ws\vh\hh\ws $\NatTree$ occurs only strictly positively in $\NN\ra\NatTreeA$\ruleref{3+2}\\
-\ws\ws\ws\ws\ws\ws\ws\vv\\
-\ws\ws\ws\ws\ws\ws\ws\hv\hh\ws $\NatTree$ satisfies the positivity condition for $\NatTreeA$\ruleref1
-\end{latexonly}
-\begin{rawhtml}
-<pre>
-<span style="font-family:serif">Then every instantiated constructor of <span style="font-family:monospace">nattree A</span> satisfies the nested positivity condition for <span style="font-family:monospace">nattree</span></span>
-  │
-  ├─ <span style="font-family:serif">concerning type <span style="font-family:monospace">nattree A</span> of constructor <span style="font-family:monospace">nil</span>:</span>
-  │    <span style="font-family:serif">Type <span style="font-family:monospace">nattree A</span> of constructor <span style="font-family:monospace">nil</span> satisfies the positivity condition for <span style="font-family:monospace">nattree</span></span>
-  │    <span style="font-family:serif">because <span style="font-family:monospace">nattree</span> does not appear in any (real) arguments of the type of that constructor</span>
-  │    <span style="font-family:serif">(primarily because nattree does not have any (real) arguments) ... <span style="font-style:italic">(bullet 1)</span></span>
-  │
-  ╰─ <span style="font-family:serif">concerning type <span style="font-family:monospace">∀ A → (nat → nattree A) → nattree A</span> of constructor <span style="font-family:monospace">cons</span>:</span>
-       <span style="font-family:serif">Type <span style="font-family:monospace">∀ A : Type, A → (nat → nattree A) → nattree A</span> of constructor <span style="font-family:monospace">cons</span></span>
-       <span style="font-family:serif">satisfies the positivity condition for <span style="font-family:monospace">nattree</span> because:</span>
-       │
-       ├─ <span style="font-family:serif"><span style="font-family:monospace">nattree</span> occurs only strictly positively in <span style="font-family:monospace">Type</span> ... <span style="font-style:italic">(bullet 1)</span></span>
-       │
-       ├─ <span style="font-family:serif"><span style="font-family:monospace">nattree</span> occurs only strictly positively in <span style="font-family:monospace">A</span> ... <span style="font-style:italic">(bullet 1)</span></span>
-       │
-       ├─ <span style="font-family:serif"><span style="font-family:monospace">nattree</span> occurs only strictly positively in <span style="font-family:monospace">nat → nattree A</span> ... <span style="font-style:italic">(bullet 3+2)</span></span>
-       │
-       ╰─ <span style="font-family:serif"><span style="font-family:monospace">nattree</span> satisfies the positivity condition for <span style="font-family:monospace">nattree A</span> ... <span style="font-style:italic">(bullet 1)</span></span>
-</pre>
-\end{rawhtml}
-
-\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 a global environment and
-  $\Gamma_P,\Gamma_I,\Gamma_C$ 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_P}{A_j}{s'_j})_{j=1\ldots  k}
-  ~~~~~~~~ (\WTE{\Gamma_I;\Gamma_P}{C_i}{s_{q_i}})_{i=1\ldots  n}
-}
-  {\WF{E;\Ind{}{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_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 $I_j
-  \notin E$,
-\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:
-% (********** The following is not correct and should produce **********)
-% (*** Error: Large non-propositional inductive types must be in Type***)
-\begin{coq_example}
-Fail 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[Template polymorphism.]{Template polymorphism.\index{Template polymorphism}}
-\label{Template-polymorphism}
-
-Inductive types declared in {\Type} are
-polymorphic over their arguments in {\Type}.
-If $A$ is an arity of some sort and $s$ is 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 global environment and local 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 $\Ind{}{p}{\Gamma_I}{\Gamma_C}$ 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{}{p}{\Gamma_I}{\Gamma_C} \in E\\
-(E[] \vdash q_l : P'_l)_{l=1\ldots r}\\
-(\WTELECONV{}{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[] \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 $\WTELECONV{}{P'_l}{ \subst{P_l}{p_u}{q_u}_{u=1\ldots l-1}}$);
-\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 $(\WTE{\Gamma_{I'};\Gamma_{P'}}{C_i}{s_{q_i}})_{i=1\ldots  n}$;
-\item the sorts $s_i$ are such that all eliminations, to {\Prop}, {\Set} and
-  $\Type(j)$, are allowed (see Section~\ref{allowedeleminationofsorts}).
-\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 $\Ind{}{p}{\Gamma_I}{\Gamma_C}$ 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 $\Ind{}{p}{\Gamma_I}{\Gamma_C}$ well-formed implies
-that $\Ind{}{p}{\Gamma_{I'}}{\Gamma_{C'}}$ 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 $\Ind{}{p}{\Gamma_I}{\Gamma_C}$.
-
-\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 type is set in {\Set} (even
-in case {\Set} is impredicative -- see Section~\ref{impredicativity}),
-and otherwise in the {\Type} hierarchy.
-
-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 lose 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}
-
-\Rem Template polymorphism used to be called ``sort-polymorphism of
-inductive types'' before universe polymorphism (see
-Chapter~\ref{Universes-full}) was introduced.
-
-\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 local 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 about an object $m$ 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 $\forall l:\ListA,(\haslengthA~l~(\length~l))$
-it is enough to prove:
-%
-\begin{itemize}
-  \item $(\haslengthA~(\Nil~A)~(\length~(\Nil~A)))$
-  \item $\forall a:A, \forall l:\ListA, (\haslengthA~l~(\length~l)) \ra\\
-        \ra (\haslengthA~(\cons~A~a~l)~(\length~(\cons~A~a~l)))$
-\end{itemize}
-%
-which given the conversion equalities satisfied by \length\ is the
-same as proving:
-%
-\begin{itemize}
-  \item $(\haslengthA~(\Nil~A)~\nO)$
-  \item $\forall a:A, \forall l:\ListA, (\haslengthA~l~(\length~l)) \ra\\
-        \ra (\haslengthA~(\cons~A~a~l)~(\nS~(\length~l)))$
-\end{itemize}
-%
-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 operator is that we have an object
-$m$ in an inductive type $I$ and we want to prove a property
-which possibly 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
-$m$ eventually happens to evaluate to $(c_i~u_1\ldots u_{p_i})$ then
-the expression will behave as specified in its $i$-th branch and
-it will reduce to $f_i$ where the $x_{i1}$\ldots $x_{ip_i}$ are replaced
-by the $u_1\ldots u_{p_i}$ 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 predicate over $n+1$ arguments: the $n$ first ones
-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 and one argument. The predicate is made explicit using the syntax:
-\[\kw{match}~m~\kw{as}~ x~ \kw{in}~ I~\verb!_!~a~ \kw{return}~ P
- ~\kw{with}~ (c_1~x_{11}~...~x_{1p_1}) \Ra f_1 ~|~\ldots~|~
-  (c_n~x_{n1}~...~x_{np_n}) \Ra f_n \kw{end}\]
-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, $m$ can occur in $P$ where it is considered a bound variable).
-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 other arguments of $I$
-(sometimes called indices in the literature)
-% NOTE: e.g. http://www.qatar.cmu.edu/~sacchini/papers/types08.pdf
-have to be variables
-($a$ above) and these variables can occur in $P$.
-The expression after \kw{in}
-must be seen as an \emph{inductive type pattern}. Notice that
-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}}
-\label{allowedeleminationofsorts}
-
-An important question for building the typing rule for \kw{match} is
-what can be the type of $\lb a x \mto P$ with respect to the type of $m$. If
-$m:I$ and
-$I:A$ and
-$\lb a x \mto P : B$
-then by \compat{I:A}{B} we mean that one can use $\lb a x \mto P$ with $m$ in the above
-match-construct.
-
-\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
-$I$.
-
-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. 
-%
-\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 :=
-  or_introl : A -> or A B | or_intror : 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:
-% (***************************************************************)
-% (*** This example should fail with ``Incorrect elimination'' ***)
-\begin{coq_example}
-Fail Definition choice (A B: Prop) (x:or A B) :=
-  match x with or_introl _ _ a => true | or_intror _ _ 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
-are 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 local 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_eval}
-Require Extraction.
-\end{coq_eval}
-\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 type $I$. Let $P$ be a term that represents the
-property to be proved.
-We assume $r$ is the number of parameters and $p$ is the number of arguments.
-
-We define a new type \CI{c:C}{P} which represents the type of the
-branch corresponding to the $c:C$ constructor.
-\[
-\begin{array}{ll}
-\CI{c:(I~p_1\ldots p_r\ t_1 \ldots t_p)}{P} &\equiv (P~t_1\ldots ~t_p~c) \\[2mm]
-\CI{c:\forall~x:T,C}{P} &\equiv \forall~x:T,\CI{(c~x):C}{P} 
-\end{array}
-\]
-We write \CI{c}{P} for \CI{c:C}{P} with $C$ the type of $c$.
-
-\paragraph{Example.}
-The following term in concrete syntax:
-\begin{verbatim}
-match t as l return P' with
-| nil _ => t1
-| cons _ hd tl => t2
-end
-\end{verbatim}
-can be represented in abstract syntax as $$\Case{P}{t}{f_1\,|\,f_2}$$
-where
-\begin{eqnarray*}
-  P & = & \lambda~l~.~P^\prime\\
-  f_1 & = & t_1\\
-  f_2 & = & \lambda~(hd:\nat)~.~\lambda~(tl:\List~\nat)~.~t_2
-\end{eqnarray*}
-According to the definition:
-\begin{latexonly}\vskip.5em\noindent\end{latexonly}%
-\begin{htmlonly}
-
-\end{htmlonly}
-$ \CI{(\Nil~\nat)}{P} \equiv \CI{(\Nil~\nat) : (\List~\nat)}{P} \equiv (P~(\Nil~\nat))$
-\begin{latexonly}\vskip.5em\noindent\end{latexonly}%
-\begin{htmlonly}
-
-\end{htmlonly}
-$ \CI{(\cons~\nat)}{P}
-  \equiv\CI{(\cons~\nat) : (\nat\ra\List~\nat\ra\List~\nat)}{P} \equiv\\
-  \equiv\forall n:\nat, \CI{(\cons~\nat~n) : \List~\nat\ra\List~\nat)}{P} \equiv\\
-  \equiv\forall n:\nat, \forall l:\List~\nat, \CI{(\cons~\nat~n~l) : \List~\nat)}{P} \equiv\\
-\equiv\forall n:\nat, \forall l:\List~\nat,(P~(\cons~\nat~n~l))$.
-\begin{latexonly}\vskip.5em\noindent\end{latexonly}%
-\begin{htmlonly}
-
-\end{htmlonly}
-Given some $P$, then \CI{(\Nil~\nat)}{P} represents the expected type of $f_1$, and
-\CI{(\cons~\nat)}{P} represents the expected type of $f_2$.
-
-\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 definition
-\Ind{}{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.}
-
-Below is a typing rule for the term shown in the previous example:
-\inference{
-  \frac{%
-    \WTEG{t}{(\List~\nat)}~~~~%
-    \WTEG{P}{B}~~~~%
-    \compat{(\List~\nat)}{B}~~~~%
-    \WTEG{f_1}{\CI{(\Nil~\nat)}{P}}~~~~%
-    \WTEG{f_2}{\CI{(\cons~\nat)}{P}}%
-  }
-{\WTEG{\Case{P}{t}{f_1|f_2}}{(P~t)}}}
-
-\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
-allow 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 $k_i$ represents the index of pararameter of $f_i$, on which $f_i$ is decreasing.
-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 $B_{k_i}$ an is unductive type.
-
-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{}{r}{\Gamma_I}{\Gamma_C},
-if the type of a constructor $c$ has the form
-$\forall p_1:P_1,\ldots \forall p_r:P_r, 
-\forall x_1:T_1, \ldots \forall x_r:T_r, (I_j~p_1\ldots 
-p_r~t_1\ldots  t_s)$, then 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{}{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 $(t~u)$ and $\lb x:u \mto t$ 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.
-%
-The following reductions are now possible:
-\def\plus{\mathsf{plus}}
-\def\tri{\triangleright_\iota}
-\begin{eqnarray*}
-  \plus~(\nS~(\nS~\nO))~(\nS~\nO) & \tri & \nS~(\plus~(\nS~\nO)~(\nS~\nO))\\
-                                  & \tri & \nS~(\nS~(\plus~\nO~(\nS~\nO)))\\
-                                  & \tri & \nS~(\nS~(\nS~\nO))\\
-\end{eqnarray*}
-
-% 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{Admissible rules for global environments}
-
-From the original rules of the type system, one can show the
-admissibility of rules which change the local context of definition of
-objects in the global environment. We show here the admissible rules
-that are used used in the discharge mechanism at the end of a section.
-
-% This is obsolete: Abstraction over defined constants actually uses a
-% let-in since there are let-ins in Coq
-
-%% \paragraph{Mechanism of substitution.}
-
-%% One rule which can be proved valid, is to replace a term $c$ by its
-%% value in the global 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 local contexts and global
-%% environments.
-
-%% \paragraph{Substitution Property:}
-%% \inference{\frac{\WF{E;c:=t:T; E'}{\Gamma}}
-%%            {\WF{E; \subst{E'}{c}{t}}{\subst{\Gamma}{c}{t}}}}
-
-\paragraph{Abstraction.}
-
-One can modify a global declaration by generalizing it over a
-previously assumed constant $c$. For doing that, we need to modify the
-reference to the global declaration in the subsequent global
-environment and local context by explicitly applying this constant to
-the constant $c'$.
-
-Below, if $\Gamma$ is a context of the form
-$[y_1:A_1;\ldots;y_n:A_n]$, we write $\forall
-x:U,\subst{\Gamma}{c}{x}$ to mean
-$[y_1:\forall~x:U,\subst{A_1}{c}{x};\ldots;y_n:\forall~x:U,\subst{A_n}{c}{x}]$
-and
-$\subst{E}{|\Gamma|}{|\Gamma|c}$.
-to mean the parallel substitution
-$\subst{\subst{E}{y_1}{(y_1~c)}\ldots}{y_n}{(y_n~c)}$.
-
-\paragraph{First abstracting property:}
- \inference{\frac{\WF{E;c:U;E';c':=t:T;E''}{\Gamma}}
-                 {\WF{E;c:U;E';c':=\lb x:U\mto \subst{t}{c}{x}:\forall~x:U,\subst{T}{c}{x};
-                      \subst{E''}{c'}{(c'~c)}}{\subst{\Gamma}{c}{(c~c')}}}}
-
- \inference{\frac{\WF{E;c:U;E';c':T;E''}{\Gamma}}
-                 {\WF{E;c:U;E';c':\forall~x:U,\subst{T}{c}{x};
-                      \subst{E''}{c'}{(c'~c)}}{\subst{\Gamma}{c}{(c~c')}}}}
-
- \inference{\frac{\WF{E;c:U;E';\Ind{}{p}{\Gamma_I}{\Gamma_C};E''}{\Gamma}}
-                 {\WFTWOLINES{E;c:U;E';\Ind{}{p+1}{\forall x:U,\subst{\Gamma_I}{c}{x}}{\forall x:U,\subst{\Gamma_C}{c}{x}};\subst{E''}{|\Gamma_I,\Gamma_C|}{|\Gamma_I,\Gamma_C|~c}}{\subst{\Gamma}{|\Gamma_I,\Gamma_C|}{|\Gamma_I,\Gamma_C|~c}}}}
-%
-One can similarly modify a global declaration by generalizing it over
-a previously defined constant~$c'$.  Below, if $\Gamma$ is a context
-of the form $[y_1:A_1;\ldots;y_n:A_n]$, we write $
-\subst{\Gamma}{c}{u}$ to mean
-$[y_1:\subst{A_1}{c}{u};\ldots;y_n:\subst{A_n}{c}{u}]$.
-
-\paragraph{Second abstracting property:}
- \inference{\frac{\WF{E;c:=u:U;E';c':=t:T;E''}{\Gamma}}
-           {\WF{E;c:=u:U;E';c':=(\letin{x}{u:U}{\subst{t}{c}{x}}):\subst{T}{c}{u};E''}{\Gamma}}}
-
- \inference{\frac{\WF{E;c:=u:U;E';c':T;E''}{\Gamma}}
-           {\WF{E;c:=u:U;E';c':\subst{T}{c}{u};E''}{\Gamma}}}
-
- \inference{\frac{\WF{E;c:=u:U;E';\Ind{}{p}{\Gamma_I}{\Gamma_C};E''}{\Gamma}}
-                 {\WF{E;c:=u:U;E';\Ind{}{p}{\subst{\Gamma_I}{c}{u}}{\subst{\Gamma_C}{c}{u}};E''}{\Gamma}}}
-
-\paragraph{Pruning the local context.}
-If one abstracts or substitutes constants with the above rules then it
-may happen that some declared or defined constant does not occur any
-more in the subsequent global environment and in the local context. One can
-consequently derive the following property.
-
-\paragraph{First pruning property:}
-\inference{\frac{\WF{E;c:U;E'}{\Gamma} \qquad c \mbox{ does not occur in $E'$ and $\Gamma$}}
-                {\WF{E;E'}{\Gamma}}}
- 
-\paragraph{Second pruning property:}
-\inference{\frac{\WF{E;c:=u:U;E'}{\Gamma} \qquad c \mbox{ does not occur in $E'$ and $\Gamma$}}
-                {\WF{E;E'}{\Gamma}}}
- 
-\section{Co-inductive types}
-The implementation contains also co-inductive definitions, which are
-types inhabited by infinite objects. 
-More information on co-inductive definitions can be found
-in~\cite{Gimenez95b,Gim98,GimCas05}.
-%They are described in Chapter~\ref{Co-inductives}.
-
-\section[The Calculus of Inductive Construction with
-  impredicative \Set]{The Calculus of Inductive Construction with
-  impredicative \Set\label{impredicativity}}
-
-\Coq{} can be used as a type-checker for the
-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.
-% (** This example should fail *******************************
-%     Error: The term forall X:Set, X -> X has type Type
-%     while it is expected to have type Set ***)
-\begin{coq_example}
-Fail Definition id: Set := forall X:Set,X->X.
-\end{coq_example}
-while it will type-check, if one uses 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}
-
-%%% Local Variables: 
-%%% mode: latex
-%%% TeX-master: "Reference-Manual"
-%%% End: 
-
-
diff --git a/doc/refman/Reference-Manual.tex b/doc/refman/Reference-Manual.tex
index c7a98fce4d..ec36304a64 100644
--- a/doc/refman/Reference-Manual.tex
+++ b/doc/refman/Reference-Manual.tex
@@ -96,7 +96,6 @@ Options A and B of the licence are {\em not} elected.}
 %END LATEX
 \include{RefMan-gal.v}% Gallina
 \include{RefMan-lib.v}% The coq library
-\include{RefMan-cic.v}% The Calculus of Constructions
 \include{RefMan-modr}% The module system