From f1315c761dad2b71debeb0f3b81274ac5f6db6e7 Mon Sep 17 00:00:00 2001
From: Maxime Dénès
Date: Wed, 14 Mar 2018 11:23:15 +0100
Subject: [Sphinx] Move chapter 12 to new infrastructure

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-\chapter[Syntax extensions and interpretation scopes]{Syntax extensions and interpretation scopes\label{Addoc-syntax}}
-%HEVEA\cutname{syntax-extensions.html}
-
-In this chapter, we introduce advanced commands to modify the way
-{\Coq} parses and prints objects, i.e. the translations between the
-concrete and internal representations of terms and commands.
-
-The main commands to provide custom symbolic notations for terms are
-{\tt Notation} and {\tt Infix}. They are described in Section
-\ref{Notation}. There is also a variant of {\tt Notation} which does
-not modify the parser. This provides with a form of abbreviation and
-it is described in Section~\ref{Abbreviations}. It is sometimes
-expected that the same symbolic notation has different meanings in
-different contexts. To achieve this form of overloading, {\Coq} offers
-a notion of interpretation scope. This is described in
-Section~\ref{scopes}.
-
-The main command to provide custom notations for tactics is {\tt
-  Tactic Notation}. It is described in Section~\ref{Tactic-Notation}.
-
-% No need any more to remind this
-%% \Rem The commands {\tt Grammar}, {\tt Syntax} and {\tt Distfix} which
-%% were present for a while in {\Coq} are no longer available from {\Coq}
-%% version 8.0. The underlying AST structure is also no longer available.
-
-\section[Notations]{Notations\label{Notation}
-\comindex{Notation}}
-
-\subsection{Basic notations}
-
-A {\em notation} is a symbolic expression denoting some term
-or term pattern.
-
-A typical notation is the use of the infix symbol \verb=/\= to denote
-the logical conjunction (\texttt{and}). Such a notation is declared
-by
-
-\begin{coq_example*}
-Notation "A /\ B" := (and A B).
-\end{coq_example*}
-
-The expression \texttt{(and A B)} is the abbreviated term and the
-string \verb="A /\ B"= (called a {\em notation}) tells how it is 
-symbolically written.
-
-A notation is always surrounded by double quotes (except when the
-abbreviation has the form of an ordinary applicative expression; see \ref{Abbreviations}). The
-notation is composed of {\em tokens} separated by spaces.  Identifiers
-in the string (such as \texttt{A} and \texttt{B}) are the {\em
-parameters} of the notation. They must occur at least once each in the
-denoted term. The other elements of the string (such as \verb=/\=) are
-the {\em symbols}.
-
-An identifier can be used as a symbol but it must be surrounded by
-simple quotes to avoid the confusion with a parameter. Similarly,
-every symbol of at least 3 characters and starting with a simple quote
-must be quoted (then it starts by two single quotes). Here is an example.
-
-\begin{coq_example*}
-Notation "'IF' c1 'then' c2 'else' c3" := (IF_then_else c1 c2 c3).
-\end{coq_example*}
-
-%TODO quote the identifier when not in front, not a keyword, as in "x 'U' y" ?
-
-A notation binds a syntactic expression to a term. Unless the parser
-and pretty-printer of {\Coq} already know how to deal with the
-syntactic expression (see \ref{ReservedNotation}), explicit precedences and
-associativity rules have to be given.
-
-\Rem The right-hand side of a notation is interpreted at the time the
-notation is given. In particular, disambiguation of constants, implicit arguments (see
-Section~\ref{Implicit Arguments}), coercions (see
-Section~\ref{Coercions}), etc. are resolved at the time of the
-declaration of the notation.
-
-\subsection[Precedences and associativity]{Precedences and associativity\index{Precedences}
-\index{Associativity}}
-
-Mixing different symbolic notations in the same text may cause serious
-parsing ambiguity. To deal with the ambiguity of notations, {\Coq}
-uses precedence levels ranging from 0 to 100 (plus one extra level
-numbered 200) and associativity rules.
-
-Consider for example the new notation
-
-\begin{coq_example*}
-Notation "A \/ B" := (or A B).
-\end{coq_example*}
-
-Clearly, an expression such as {\tt forall A:Prop, True \verb=/\= A \verb=\/=
-A \verb=\/= False} is ambiguous. To tell the {\Coq} parser how to
-interpret the expression, a priority between the symbols \verb=/\= and
-\verb=\/= has to be given. Assume for instance that we want conjunction
-to bind more than disjunction. This is expressed by assigning a
-precedence level to each notation, knowing that a lower level binds
-more than a higher level.  Hence the level for disjunction must be
-higher than the level for conjunction.
-
-Since connectives are not tight articulation points of a text, it
-is reasonable to choose levels not so far from the highest level which
-is 100, for example 85 for disjunction and 80 for
-conjunction\footnote{which are the levels effectively chosen in the
-current implementation of {\Coq}}.
-
-Similarly, an associativity is needed to decide whether {\tt True \verb=/\=
-False \verb=/\= False} defaults to {\tt True \verb=/\= (False
-\verb=/\= False)} (right associativity) or to {\tt (True
-\verb=/\= False) \verb=/\= False} (left associativity). We may
-even consider that the expression is not well-formed and that
-parentheses are mandatory (this is a ``no associativity'')\footnote{
-{\Coq} accepts notations declared as no associative but the parser on
-which {\Coq} is built, namely {\camlpppp}, currently does not implement the
-no-associativity and replaces it by a left associativity; hence it is
-the same for {\Coq}: no-associativity is in fact left associativity}.
-We do not know of a special convention of the associativity of
-disjunction and conjunction, so let us apply for instance a right
-associativity (which is the choice of {\Coq}).
-
-Precedence levels and associativity rules of notations have to be
-given between parentheses in a list of modifiers that the
-\texttt{Notation} command understands. Here is how the previous
-examples refine.
-
-\begin{coq_example*}
-Notation "A /\ B" := (and A B) (at level 80, right associativity).
-Notation "A \/ B" := (or A B)  (at level 85, right associativity).
-\end{coq_example*}
-
-By default, a notation is considered non associative, but the
-precedence level is mandatory (except for special cases whose level is
-canonical). The level is either a number or the phrase {\tt next
-level} whose meaning is obvious. The list of levels already assigned
-is on Figure~\ref{init-notations}.
-
-\subsection{Complex notations}
-
-Notations can be made from arbitrarily complex symbols. One can for
-instance define prefix notations.
-
-\begin{coq_example*}
-Notation "~ x" := (not x) (at level 75, right associativity).
-\end{coq_example*}
-
-One can also define notations for incomplete terms, with the hole
-expected to be inferred at typing time.
-
-\begin{coq_example*}
-Notation "x = y" := (@eq _ x y) (at level 70, no associativity).
-\end{coq_example*}
-
-One can define {\em closed} notations whose both sides are symbols. In
-this case, the default precedence level for inner subexpression is
-200, and the default level for the notation itself is 0.
-
-\begin{coq_eval}
-Set Printing Depth 50.
-(********** The following is correct but produces **********)
-(**** an incompatibility with the reserved notation ********)
-\end{coq_eval}
-\begin{coq_example*}
-Notation "( x , y )" := (@pair _ _ x y).
-\end{coq_example*}
-
-One can also define notations for binders.
-
-\begin{coq_eval}
-Set Printing Depth 50.
-(********** The following is correct but produces **********)
-(**** an incompatibility with the reserved notation ********)
-\end{coq_eval}
-\begin{coq_example*}
-Notation "{ x : A  |  P }" := (sig A (fun x => P)).
-\end{coq_example*}
-
-In the last case though, there is a conflict with the notation for
-type casts. The notation for type casts, as shown by the command {\tt Print Grammar
-constr} is at level 100. To avoid \verb=x : A= being parsed as a type cast,
-it is necessary to put {\tt x} at a level below 100, typically 99. Hence, a
-correct definition is the following.
-
-\begin{coq_example*}
-Notation "{ x : A  |  P }" := (sig A (fun x => P)) (x at level 99).
-\end{coq_example*}
-
-%This change has retrospectively an effect on the notation for notation
-%{\tt "{ A } + { B }"}. For the sake of factorization, {\tt A} must be
-%put at level 99 too, which gives
-%
-%\begin{coq_example*}
-%Notation "{ A } + { B }" := (sumbool A B) (at level 0, A at level 99).
-%\end{coq_example*}
-
-More generally, it is required that notations are explicitly
-factorized on the left. See the next section for more about
-factorization.
-
-\subsection{Simple factorization rules}
-
-{\Coq} extensible parsing is performed by {\camlpppp} which is
-essentially a LL1 parser: it decides which notation to parse by
-looking tokens from left to right. Hence, some care has to be taken
-not to hide already existing rules by new rules. Some simple left
-factorization work has to be done. Here is an example.
-
-\begin{coq_eval}
-(********** The next rule for notation _ < _ < _  produces **********)
-(*** Error: Notation _ < _ < _ is already defined at level 70 ... ***)
-\end{coq_eval}
-\begin{coq_example*}
-Notation "x < y"     := (lt x y) (at level 70).
-Notation "x < y < z" := (x < y /\ y < z) (at level 70).
-\end{coq_example*}
-
-In order to factorize the left part of the rules, the subexpression
-referred by {\tt y} has to be at the same level in both rules. However
-the default behavior puts {\tt y} at the next level below 70
-in the first rule (no associativity is the default), and at the level
-200 in the second rule (level 200 is the default for inner expressions).
-To fix this, we need to force the parsing level of {\tt y},
-as follows.
-
-\begin{coq_example*}
-Notation "x < y"     := (lt x y) (at level 70).
-Notation "x < y < z" := (x < y /\ y < z) (at level 70, y at next level).
-\end{coq_example*}
-
-For the sake of factorization with {\Coq} predefined rules, simple
-rules have to be observed for notations starting with a symbol:
-e.g. rules starting with ``\{'' or ``('' should be put at level 0. The
-list of {\Coq} predefined notations can be found in Chapter~\ref{Theories}.
-
-The command to display the current state of the {\Coq} term parser is
-\comindex{Print Grammar constr}
-
-\begin{quote}
-\tt Print Grammar constr.
-\end{quote}
-
-\variant
-
-\comindex{Print Grammar pattern}
-{\tt Print Grammar pattern.}\\
-
-This displays the state of the subparser of patterns (the parser
-used in the grammar of the {\tt match} {\tt with} constructions).
-
-\subsection{Displaying symbolic notations}
-
-The command \texttt{Notation} has an effect both on the {\Coq} parser and
-on the {\Coq} printer. For example:
-
-\begin{coq_example}
-Check (and True True).
-\end{coq_example}
-
-However, printing, especially pretty-printing, also requires some
-care. We may want specific indentations, line breaks, alignment if on
-several lines, etc. For pretty-printing, {\Coq} relies on {\ocaml}
-formatting library, which provides indentation and automatic line
-breaks depending on page width by means of {\em formatting boxes}.
-
-The default printing of notations is rudimentary. For printing a
-notation, a formatting box is opened in such a way that if the
-notation and its arguments cannot fit on a single line, a line break
-is inserted before the symbols of the notation and the arguments on
-the next lines are aligned with the argument on the first line.
-
-A first simple control that a user can have on the printing of a
-notation is the insertion of spaces at some places of the
-notation. This is performed by adding extra spaces between the symbols
-and parameters: each extra space (other than the single space needed
-to separate the components) is interpreted as a space to be inserted
-by the printer. Here is an example showing how to add spaces around
-the bar of the notation.
-
-\begin{coq_example}
-Notation "{{ x : A  |  P }}" := (sig (fun x : A => P))
-  (at level 0, x at level 99).
-Check (sig (fun x : nat => x=x)).
-\end{coq_example}
-
-The second, more powerful control on printing is by using the {\tt
-format} modifier. Here is an example
-
-\begin{small}
-\begin{coq_example}
-Notation "'If' c1 'then' c2 'else' c3" := (IF_then_else c1 c2 c3)
-(at level 200, right associativity, format
-"'[v   ' 'If'  c1 '/' '[' 'then'  c2  ']' '/' '[' 'else'  c3 ']' ']'").
-\end{coq_example}
-\end{small}
-
-\begin{coq_example}
-Check
- (IF_then_else (IF_then_else True False True)
-   (IF_then_else True False True)
-   (IF_then_else True False True)).
-\end{coq_example}
-
-A {\em format} is an extension of the string denoting the notation with
-the possible following elements delimited by single quotes:
-
-\begin{itemize}
-\item extra spaces are translated into simple spaces
-\item tokens of the form \verb='/  '= are translated into breaking point,
-  in case a line break occurs, an indentation of the number of spaces
-  after the ``\verb=/='' is applied (2 spaces in the given example)
-\item token of the form \verb='//'= force writing on a new line
-\item well-bracketed pairs of tokens of the form \verb='[    '= and \verb=']'=
-  are translated into printing boxes; in case a line break occurs,
-  an extra indentation of the number of spaces given after the ``\verb=[=''
-  is applied (4 spaces in the example)
-\item well-bracketed pairs of tokens of the form \verb='[hv   '= and \verb=']'=
-  are translated into horizontal-orelse-vertical printing boxes; 
-  if the content of the box does not fit on a single line, then every breaking
-  point forces a newline and an extra  indentation of the number of spaces
-  given after the ``\verb=[='' is applied at the beginning of each newline
-  (3 spaces in the example)
-\item well-bracketed pairs of tokens of the form \verb='[v '= and
-  \verb=']'= are translated into vertical printing boxes; every
-  breaking point forces a newline, even if the line is large enough to
-  display the whole content of the box, and an extra indentation of the
-  number of spaces given after the ``\verb=[='' is applied at the beginning
-  of each newline
-\end{itemize}
-
-%Thus, for the previous example, we get
-%\footnote{The ``@'' is here to shunt
-%the notation "'IF' A 'then' B 'else' C" which is defined in {\Coq}
-%initial state}:
-
-Notations do not survive the end of sections. No typing of the denoted
-expression is performed at definition time. Type-checking is done only
-at the time of use of the notation.
-
-\Rem
-Sometimes, a notation is expected only for the parser.
-%(e.g. because
-%the underlying parser of {\Coq}, namely {\camlpppp}, is LL1 and some extra
-%rules are needed to circumvent the absence of factorization).
-To do so, the option {\tt only parsing} is allowed in the list of modifiers of
-\texttt{Notation}.
-
-Conversely, the {\tt only printing} can be used to declare
-that a notation should only be used for printing and should not declare a
-parsing rule. In particular, such notations do not modify the parser.
-
-\subsection{The \texttt{Infix} command
-\comindex{Infix}}
-
-The \texttt{Infix} command is a shortening for declaring notations of
-infix symbols. Its syntax is 
-
-\begin{quote}
-\noindent\texttt{Infix "{\symbolentry}" :=} {\term} {\tt (} \nelist{\em modifier}{,} {\tt )}.
-\end{quote}
-
-and it is equivalent to
-
-\begin{quote}
-\noindent\texttt{Notation "x {\symbolentry} y" := ({\term} x y)  (} \nelist{\em modifier}{,} {\tt )}.
-\end{quote}
-
-where {\tt x} and {\tt y} are fresh names. Here is an example.
-
-\begin{coq_example*}
-Infix "/\" := and (at level 80, right associativity).
-\end{coq_example*}
-
-\subsection{Reserving notations
-\label{ReservedNotation}
-\comindex{Reserved Notation}}
-
-A given notation may be used in different contexts. {\Coq} expects all
-uses of the notation to be defined at the same precedence and with the
-same associativity. To avoid giving the precedence and associativity
-every time, it is possible to declare a parsing rule in advance
-without giving its interpretation. Here is an example from the initial
-state of {\Coq}.
-
-\begin{coq_example}
-Reserved Notation "x = y" (at level 70, no associativity).
-\end{coq_example}
-
-Reserving a notation is also useful for simultaneously defining an
-inductive type or a recursive constant and a notation for it.
-
-\Rem The notations mentioned on Figure~\ref{init-notations} are
-reserved. Hence their precedence and associativity cannot be changed.
-
-\subsection{Simultaneous definition of terms and notations
-\comindex{Fixpoint {\ldots} where {\ldots}}
-\comindex{CoFixpoint {\ldots} where {\ldots}}
-\comindex{Inductive {\ldots} where {\ldots}}}
-
-Thanks to reserved notations, the inductive, co-inductive, record,
-recursive and corecursive definitions can benefit of customized
-notations. To do this, insert a {\tt where} notation clause after the
-definition of the (co)inductive type or (co)recursive term (or after
-the definition of each of them in case of mutual definitions). The
-exact syntax is given on Figure~\ref{notation-syntax} for inductive,
-co-inductive, recursive and corecursive definitions and on
-Figure~\ref{record-syntax} for records. Here are examples:
-
-\begin{coq_eval}
-Set Printing Depth 50.
-(********** The following is correct but produces an error **********)
-(********** because the symbol /\ is already bound **********)
-(**** Error: The conclusion of A -> B -> A /\ B is not valid *****)
-\end{coq_eval}
-
-\begin{coq_example*}
-Inductive and (A B:Prop) : Prop := conj : A -> B -> A /\ B 
-where "A /\ B" := (and A B).
-\end{coq_example*}
-
-\begin{coq_eval}
-Set Printing Depth 50.
-(********** The following is correct but produces an error **********)
-(********** because the symbol + is already bound **********)
-(**** Error: no recursive definition *****)
-\end{coq_eval}
-
-\begin{coq_example*}
-Fixpoint plus (n m:nat) {struct n} : nat :=
-  match n with
-  | O => m
-  | S p => S (p+m)
-  end
-where "n + m" := (plus n m).
-\end{coq_example*}
-
-\subsection{Displaying informations about notations
-\optindex{Printing Notations}}
-
-To deactivate the printing of all notations, use the command
-\begin{quote}
-\tt Unset Printing Notations.
-\end{quote}
-To reactivate it, use the command
-\begin{quote}
-\tt Set Printing Notations.
-\end{quote}
-The default is to use notations for printing terms wherever possible.
-
-\SeeAlso {\tt Set Printing All} in Section~\ref{SetPrintingAll}.
-
-\subsection{Locating notations
-\comindex{Locate}
-\label{LocateSymbol}}
-
-To know to which notations a given symbol belongs to, use the command
-\begin{quote}
-\tt Locate {\symbolentry}
-\end{quote}
-where symbol is any (composite) symbol surrounded by double quotes. To locate
-a particular notation, use a string where the variables of the
-notation are replaced by ``\_'' and where possible single quotes
-inserted around identifiers or tokens starting with a single quote are
-dropped.
-
-\Example
-\begin{coq_example}
-Locate "exists".
-Locate "exists _ .. _ , _".
-\end{coq_example}
-
-\SeeAlso Section \ref{Locate}.
-
-\begin{figure}
-\begin{small}
-\begin{centerframe}
-\begin{tabular}{lcl}
-{\sentence} & ::= & 
-   \zeroone{\tt Local} \texttt{Notation} {\str} \texttt{:=} {\term} 
-   \zeroone{\modifiers} \zeroone{:{\scope}} .\\
-  & $|$ & 
-   \zeroone{\tt Local} \texttt{Infix} {\str} \texttt{:=} {\qualid} 
-   \zeroone{\modifiers} \zeroone{:{\scope}} .\\
-  & $|$ & 
-   \zeroone{\tt Local} \texttt{Reserved Notation} {\str}
-   \zeroone{\modifiers} .\\
-  & $|$ & {\tt Inductive}
-   \nelist{{\inductivebody} \zeroone{\declnotation}}{with}{\tt .}\\
-  & $|$ & {\tt CoInductive}
-   \nelist{{\inductivebody} \zeroone{\declnotation}}{with}{\tt .}\\
-  & $|$ & {\tt Fixpoint}
-   \nelist{{\fixpointbody} \zeroone{\declnotation}}{with} {\tt .} \\
-  & $|$ & {\tt CoFixpoint}
-   \nelist{{\cofixpointbody} \zeroone{\declnotation}}{with} {\tt .} \\
-\\
-{\declnotation} & ::= & 
-  \zeroone{{\tt where} \nelist{{\str} {\tt :=} {\term} \zeroone{:{\scope}}}{\tt and}}.
-\\
-\\
-{\modifiers}
-  & ::= & {\tt at level} {\naturalnumber} \\
-  & $|$ & \nelist{\ident}{,} {\tt at level} {\naturalnumber} \zeroone{\binderinterp}\\
-  & $|$ & \nelist{\ident}{,} {\tt at next level} \zeroone{\binderinterp}\\
-  & $|$ & {\ident} {\binderinterp} \\
-  & $|$ & {\ident} {\tt ident} \\
-  & $|$ & {\ident} {\tt global} \\
-  & $|$ & {\ident} {\tt bigint} \\
-  & $|$ & {\ident} \zeroone{{\tt strict}} {\tt pattern} \zeroone{{\tt at level} {\naturalnumber}}\\
-  & $|$ & {\ident} {\tt binder} \\
-  & $|$ & {\ident} {\tt closed binder} \\
-  & $|$ & {\tt left associativity} \\
-  & $|$ & {\tt right associativity} \\
-  & $|$ & {\tt no associativity} \\
-  & $|$ & {\tt only parsing} \\
-  & $|$ & {\tt only printing} \\
-  & $|$ & {\tt format} {\str} \\
-\\
-\\
-{\binderinterp}
-  & ::= & {\tt as ident} \\
-  & $|$ & {\tt as pattern} \\
-  & $|$ & {\tt as strict pattern} \\
-\end{tabular}
-\end{centerframe}
-\end{small}
-\caption{Syntax of the variants of {\tt Notation}}
-\label{notation-syntax}
-\end{figure}
-
-\subsection{Notations and binders}
-
-Notations can include binders. This section lists
-different ways to deal with binders. For further examples, see also
-Section~\ref{RecursiveNotationsWithBinders}.
-
-\subsubsection{Binders bound in the notation and parsed as identifiers}
-
-Here is the basic example of a notation using a binder:
-
-\begin{coq_example*}
-Notation "'sigma' x : A , B" := (sigT (fun x : A => B))
-  (at level 200, x ident, A at level 200, right associativity).
-\end{coq_example*}
-
-The binding variables in the right-hand side that occur as a parameter
-of the notation (here {\tt x}) dynamically bind all the occurrences
-in their respective binding scope after instantiation of the
-parameters of the notation. This means that the term bound to {\tt B} can
-refer to the variable name bound to {\tt x} as shown in the following
-application of the notation:
-
-\begin{coq_example}
-Check sigma z : nat, z = 0.
-\end{coq_example}
-
-Notice the modifier {\tt x ident} in the declaration of the
-notation. It tells to parse {\tt x} as a single identifier.
-
-\subsubsection{Binders bound in the notation and parsed as patterns}
-
-In the same way as patterns can be used as binders, as in {\tt fun
-  '(x,y) => x+y} or {\tt fun '(existT \_ x \_) => x}, notations can be
-defined so that any pattern (in the sense of the entry {\pattern} of
-Figure~\ref{term-syntax-aux}) can be used in place of the
-binder. Here is an example:
-
-\begin{coq_eval}
-Reset Initial.
-\end{coq_eval}
-
-\begin{coq_example*}
-Notation "'subset' ' p , P " := (sig (fun p => P))
-  (at level 200, p pattern, format "'subset'  ' p ,  P").
-\end{coq_example*}
-
-\begin{coq_example}
-Check subset '(x,y), x+y=0.
-\end{coq_example}
-
-The modifier {\tt p pattern} in the declaration of the notation
-tells to parse $p$ as a pattern. Note that a single
-variable is both an identifier and a pattern, so, e.g., the following
-also works:
-
-% Note: we rely on the notation of the standard library which does not
-% print the expected output, so we hide the output.
-\begin{coq_example}
-Check subset 'x, x=0.
-\end{coq_example}
-
-If one wants to prevent such a notation to be used for printing when the
-pattern is reduced to a single identifier, one has to use instead
-the modifier {\tt p strict pattern}. For parsing, however, a {\tt
-  strict pattern} will continue to include the case of a
-variable. Here is an example showing the difference:
-
-\begin{coq_example*}
-Notation "'subset_bis' ' p , P" := (sig (fun p => P))
-  (at level 200, p strict pattern).
-Notation "'subset_bis' p , P " := (sig (fun p => P))
-  (at level 200, p ident).
-\end{coq_example*}
-
-\begin{coq_example}
-Check subset_bis 'x, x=0.
-\end{coq_example}
-
-The default level for a {\tt pattern} is 0. One can use a different level by
-using {\tt pattern at level} $n$ where the scale is the same as the one for
-terms (Figure~\ref{init-notations}).
-
-\subsubsection{Binders bound in the notation and parsed as terms}
-
-Sometimes, for the sake of factorization of rules, a binder has to be
-parsed as a term. This is typically the case for a notation such as
-the following:
-
-\begin{coq_eval}
-Set Printing Depth 50.
-(********** The following is correct but produces **********)
-(**** an incompatibility with the reserved notation ********)
-\end{coq_eval}
-\begin{coq_example*}
-Notation "{ x : A  |  P  }" := (sig (fun x : A => P))
-  (at level 0, x at level 99 as ident).
-\end{coq_example*}
-
-This is so because the grammar also contains rules starting with
-{\tt \{} and followed by a term, such as the rule for the notation
-  {\tt \{ A \} + \{ B \}} for the constant {\tt
-    sumbool}~(see Section~\ref{sumbool}).
-
-Then, in the rule, {\tt x ident} is replaced by {\tt x at level 99 as
-  ident} meaning that {\tt x} is parsed as a term at level 99 (as done
-in the notation for {\tt sumbool}), but that this term has actually to
-be an identifier.
-
-The notation {\tt \{ x | P \}} is already defined in the standard
-library with the {\tt as ident} modifier. We cannot redefine it but
-one can define an alternative notation, say {\tt \{ p such that P }\},
-using instead {\tt as pattern}.
-
-% Note, this conflicts with the default rule in the standard library, so
-% we don't show the
-\begin{coq_example*}
-Notation "{ p 'such' 'that' P }" := (sig (fun p => P))
-  (at level 0, p at level 99 as pattern).
-\end{coq_example*}
-
-Then, the following works:
-\begin{coq_example}
-Check {(x,y) such that x+y=0}.
-\end{coq_example}
-
-To enforce that the pattern should not be used for printing when it
-is just an identifier, one could have said {\tt p at level
-  99 as strict pattern}.
-
-Note also that in the absence of a {\tt as ident}, {\tt as strict
-  pattern} or {\tt as pattern} modifiers, the default is to consider
-subexpressions occurring in binding position and parsed as terms to be
-{\tt as ident}.
-
-\subsubsection{Binders not bound in the notation}
-\label{NotationsWithBinders}
-
-We can also have binders in the right-hand side of a notation which
-are not themselves bound in the notation. In this case, the binders
-are considered up to renaming of the internal binder. E.g., for the
-notation
-
-\begin{coq_example*}
-Notation "'exists_different' n" := (exists p:nat, p<>n) (at level 200).
-\end{coq_example*}
-the next command fails because {\tt p} does not bind in 
-the instance of {\tt n}.
-\begin{coq_eval}
-Set Printing Depth 50.
-\end{coq_eval}
-% (********** The following produces **********)
-% (**** The reference p was not found in the current environment ********)
-\begin{coq_example}
-Fail Check (exists_different p).
-\end{coq_example}
-
-\subsection{Notations with recursive patterns}
-\label{RecursiveNotations}
-
-A mechanism is provided for declaring elementary notations with
-recursive patterns. The basic example is:
-
-\begin{coq_example*}
-Notation "[ x ; .. ; y ]" := (cons x .. (cons y nil) ..).
-\end{coq_example*}
-
-On the right-hand side, an extra construction of the form {\tt ..} $t$
-{\tt ..} can be used. Notice that {\tt ..} is part of the {\Coq}
-syntax and it must not be confused with the three-dots notation
-$\ldots$ used in this manual to denote a sequence of arbitrary size.
-
-On the left-hand side, the part ``$x$ $s$ {\tt ..} $s$ $y$'' of the
-notation parses any number of time (but at least one time) a sequence
-of expressions separated by the sequence of tokens $s$ (in the
-example, $s$ is just ``{\tt ;}'').
-
-The right-hand side must contain a subterm of the form either
-$\phi(x,${\tt ..}  $\phi(y,t)$ {\tt ..}$)$ or $\phi(y,${\tt ..}
-$\phi(x,t)$ {\tt ..}$)$ where $\phi([~]_E,[~]_I)$, called the {\em
-  iterator} of the recursive notation is an arbitrary expression with
-distinguished placeholders and
-where $t$ is called the {\tt terminating expression} of the recursive
-notation. In the example, we choose the name s$x$ and $y$ but in
-practice they can of course be chosen arbitrarily. Note that the
-placeholder $[~]_I$ has to occur only once but the $[~]_E$ can occur
-several times.
-
-Parsing the notation produces a list of expressions which are used to
-fill the first placeholder of the iterating pattern which itself is
-repeatedly nested as many times as the length of the list, the second
-placeholder being the nesting point. In the innermost occurrence of the
-nested iterating pattern, the second placeholder is finally filled with the
-terminating expression.
-
-In the example above, the iterator $\phi([~]_E,[~]_I)$ is {\tt cons
-  $[~]_E$ $[~]_I$} and the terminating expression is {\tt nil}. Here are
-other examples:
-\begin{coq_example*}
-Notation "( x , y , .. , z )" := (pair .. (pair x y) .. z) (at level 0).
-Notation "[| t * ( x , y , .. , z ) ; ( a , b , .. , c )  * u |]" :=
-  (pair (pair .. (pair (pair t x) (pair t y)) .. (pair t z))
-        (pair .. (pair (pair a u) (pair b u)) .. (pair c u)))
-  (t at level 39).
-\end{coq_example*}
-
-Recursive patterns can occur several times on the right-hand side.
-Here is an example:
-
-\begin{coq_example*}
-Notation "[> a , .. , b <]" :=
-  (cons a .. (cons b nil) .., cons b .. (cons a nil) ..).
-\end{coq_example*}
-
-Notations with recursive patterns can be reserved like standard
-notations, they can also be declared within interpretation scopes (see
-section \ref{scopes}).
-
-\subsection{Notations with recursive patterns involving binders}
-\label{RecursiveNotationsWithBinders}
-
-Recursive notations can also be used with binders. The basic example is:
-
-\begin{coq_example*}
-Notation "'exists' x .. y , p" :=
-  (ex (fun x => .. (ex (fun y => p)) ..))
-  (at level 200, x binder, y binder, right associativity).
-\end{coq_example*}
-
-The principle is the same as in Section~\ref{RecursiveNotations}
-except that in the iterator $\phi([~]_E,[~]_I)$, the placeholder $[~]_E$ can
-also occur in position of the binding variable of a {\tt
-  fun} or a {\tt forall}.
-
-To specify that the part ``$x$ {\tt ..} $y$'' of the notation
-parses a sequence of binders, $x$ and $y$ must be marked as {\tt
-  binder} in the list of modifiers of the notation. The
-binders of the parsed sequence are used to fill the occurrences of the first
-placeholder of the iterating pattern which is repeatedly nested as many times
-as the number of binders generated. If ever the generalization
-operator {\tt `} (see Section~\ref{implicit-generalization}) is used
-in the binding list, the added binders are taken into account too.
-
-Binders parsing exist in two flavors. If $x$ and $y$ are marked as
-{\tt binder}, then a sequence such as {\tt a b c : T} will be accepted
-and interpreted as the sequence of binders {\tt (a:T) (b:T)
-  (c:T)}. For instance, in the notation above, the syntax {\tt exists
-  a b : nat, a = b} is valid.
-
-The variables $x$ and $y$ can also be marked as {\tt closed binder} in
-which case only well-bracketed binders of the form {\tt (a b c:T)} or
-{\tt \{a b c:T\}} etc. are accepted.
-
-With closed binders, the recursive sequence in the left-hand side can
-be of the more general form $x$ $s$ {\tt ..} $s$ $y$ where $s$ is an
-arbitrary sequence of tokens. With open binders though, $s$ has to be
-empty. Here is an example of recursive notation with closed binders:
-
-\begin{coq_example*}
-Notation "'mylet' f x .. y :=  t 'in' u":=
-  (let f := fun x => .. (fun y => t) .. in u)
-  (at level 200, x closed binder, y closed binder, right associativity).
-\end{coq_example*}
-
-A recursive pattern for binders can be used in position of a recursive
-pattern for terms. Here is an example:
-
-\begin{coq_example*}
-Notation "'FUNAPP' x .. y , f" :=
-  (fun x => .. (fun y => (.. (f x) ..) y ) ..)
-  (at level 200, x binder, y binder, right associativity).
-\end{coq_example*}
-
-If an occurrence of the $[~]_E$ is not in position of a binding
-variable but of a term, it is the name used in the binding which is
-used. Here is an example:
-
-\begin{coq_example*}
-Notation "'exists_non_null' x .. y  , P" :=
-  (ex (fun x => x <> 0 /\ .. (ex (fun y => y <> 0 /\ P)) ..))
-  (at level 200, x binder).
-\end{coq_example*}
-
-\subsection{Predefined entries}
-
-By default, sub-expressions are parsed as terms and the corresponding
-grammar entry is called {\tt constr}. However, one may sometimes want
-to restrict the syntax of terms in a notation. For instance, the
-following notation will accept to parse only global reference in
-position of {\tt x}:
-
-\begin{coq_example*}
-Notation "'apply' f a1 .. an" := (.. (f a1) .. an)
-  (at level 10, f global, a1, an at level 9).
-\end{coq_example*}
-
-In addition to {\tt global}, one can restrict the syntax of a
-sub-expression by using the entry names {\tt ident} or {\tt pattern}
-already seen in Section~\ref{NotationsWithBinders}, even when the
-corresponding expression is not used as a binder in the right-hand
-side. E.g.:
-
-\begin{coq_example*}
-Notation "'apply_id' f a1 .. an" := (.. (f a1) .. an)
-  (at level 10, f ident, a1, an at level 9).
-\end{coq_example*}
-
-\subsection{Summary}
-
-\paragraph{Syntax of notations}
-
-The different syntactic variants of the command \texttt{Notation} are
-given on Figure~\ref{notation-syntax}. The optional {\tt :{\scope}} is
-described in the Section~\ref{scopes}.
-
-\Rem No typing of the denoted expression is performed at definition
-time. Type-checking is done only at the time of use of the notation.
-
-\Rem Many examples of {\tt Notation} may be found in the files
-composing the initial state of {\Coq} (see directory {\tt
-\$COQLIB/theories/Init}).
-
-\Rem The notation \verb="{ x }"= has a special status in such a way
-that complex notations of the form \verb="x + { y }"= or
-\verb="x * { y }"= can be nested with correct precedences. Especially,
-every notation involving a pattern of the form \verb="{ x }"= is
-parsed as a notation where the pattern \verb="{ x }"= has been simply
-replaced by \verb="x"= and the curly brackets are parsed separately.
-E.g. \verb="y + { z }"= is not parsed as a term of the given form but
-as a term of the form \verb="y + z"= where \verb=z= has been parsed
-using the rule parsing \verb="{ x }"=. Especially, level and
-precedences for a rule including patterns of the form \verb="{ x }"=
-are relative not to the textual notation but to the notation where the
-curly brackets have been removed (e.g. the level and the associativity
-given to some notation, say \verb="{ y } & { z }"= in fact applies to
-the underlying \verb="{ x }"=-free rule which is \verb="y & z"=).
-
-\paragraph{Persistence of notations}
-
-Notations do not survive the end of sections. They survive modules
-unless the command {\tt Local Notation} is used instead of {\tt
-Notation}.
-
-\section[Interpretation scopes]{Interpretation scopes\index{Interpretation scopes}
-\label{scopes}}
-% Introduction
-
-An {\em interpretation scope} is a set of notations for terms with
-their interpretation. Interpretation scopes provide a weak,
-purely syntactical form of notation overloading: the same notation, for
-instance the infix symbol \verb=+=, can be used to denote distinct
-definitions of the additive operator. Depending on which interpretation
-scope is currently open, the interpretation is different.
-Interpretation scopes can include an interpretation for
-numerals and strings. However, this is only made possible at the
-{\ocaml} level.
-
-See Figure \ref{notation-syntax} for the syntax of notations including
-the possibility to declare them in a given scope.  Here is a typical
-example which declares the notation for conjunction in the scope {\tt
-type\_scope}.
-
-\begin{verbatim}
-Notation "A /\ B" := (and A B) : type_scope.
-\end{verbatim}
-
-\Rem A notation not defined in a scope is called a {\em lonely} notation.
-
-\subsection{Global interpretation rules for notations}
-
-At any time, the interpretation of a notation for term is done within
-a {\em stack} of interpretation scopes and lonely notations. In case a
-notation has several interpretations, the actual interpretation is the
-one defined by (or in) the more recently declared (or open) lonely
-notation (or interpretation scope) which defines this notation.
-Typically if a given notation is defined in some scope {\scope} but
-has also an interpretation not assigned to a scope, then, if {\scope}
-is open before the lonely interpretation is declared, then the lonely
-interpretation is used (and this is the case even if the
-interpretation of the notation in {\scope} is given after the lonely
-interpretation: otherwise said, only the order of lonely
-interpretations and opening of scopes matters, and not the declaration
-of interpretations within a scope).
-
-The initial state of {\Coq} declares three interpretation scopes and
-no lonely notations. These scopes, in opening order, are {\tt
-core\_scope}, {\tt type\_scope} and {\tt nat\_scope}.
-
-The command to add a scope to the interpretation scope stack is
-\comindex{Open Scope}
-\comindex{Close Scope}
-\begin{quote}
-{\tt Open Scope} {\scope}.
-\end{quote}
-It is also possible to remove a scope from the interpretation scope
-stack by using the command
-\begin{quote}
-{\tt Close Scope} {\scope}.
-\end{quote}
-Notice that this command does not only cancel the last {\tt Open Scope
-{\scope}} but all the invocations of it.
-
-\Rem {\tt Open Scope} and {\tt Close Scope} do not survive the end of
-sections where they occur. When defined outside of a section, they are
-exported to the modules that import the module where they occur.
-
-\begin{Variants}
-
-\item {\tt Local Open Scope} {\scope}.
-
-\item {\tt Local Close Scope} {\scope}.
-
-These variants are not exported to the modules that import the module
-where they occur, even if outside a section.
-
-\item {\tt Global Open Scope} {\scope}.
-
-\item {\tt Global Close Scope} {\scope}.
-
-These variants survive sections. They behave as if {\tt Global} were
-absent when not inside a section.
-
-\end{Variants}
-
-\subsection{Local interpretation rules for notations}
-
-In addition to the global rules of interpretation of notations, some
-ways to change the interpretation of subterms are available.
-
-\subsubsection{Local opening of an interpretation scope 
-\label{scopechange}
-\index{\%}
-\comindex{Delimit Scope}
-\comindex{Undelimit Scope}}
-
-It is possible to locally extend the interpretation scope stack using
-the syntax ({\term})\%{\delimkey} (or simply {\term}\%{\delimkey}
-for atomic terms), where {\delimkey} is a special identifier called
-{\em delimiting key} and bound to a given scope.
-
-In such a situation, the term {\term}, and all its subterms, are
-interpreted in the scope stack extended with the scope bound to
-{\delimkey}.
-
-To bind a delimiting key to a scope, use the command
-
-\begin{quote}
-\texttt{Delimit Scope} {\scope} \texttt{with} {\ident} 
-\end{quote}
-
-To remove a delimiting key of a scope, use the command
-
-\begin{quote}
-\texttt{Undelimit Scope} {\scope}
-\end{quote}
-
-\subsubsection{Binding arguments of a constant to an interpretation scope
-\comindex{Arguments}}
-
-It is possible to set in advance that some arguments of a given
-constant have to be interpreted in a given scope. The command is
-\begin{quote}
-{\tt Arguments} {\qualid} \nelist{\name {\tt \%}\scope}{}
-\end{quote}
-where the list is a prefix of the list of the arguments of {\qualid} eventually
-annotated with their {\scope}. Grouping round parentheses can be used to
-decorate multiple arguments with the same scope.  {\scope} can be either a scope
-name or its delimiting key. For example the following command puts the first two
-arguments of {\tt plus\_fct} in the scope delimited by the key {\tt F} ({\tt
-  Rfun\_scope}) and the last argument in the scope delimited by the key {\tt R}
-({\tt R\_scope}).
-
-\begin{coq_example*}
-Arguments plus_fct (f1 f2)%F x%R.
-\end{coq_example*}
-
-The {\tt Arguments} command accepts scopes decoration to all grouping
-parentheses. In the following example arguments {\tt A} and {\tt B} 
-are marked as maximally inserted implicit arguments and are
-put into the {\tt type\_scope} scope.
-
-\begin{coq_example*}
-Arguments respectful {A B}%type (R R')%signature _ _.
-\end{coq_example*}
-
-When interpreting a term, if some of the arguments of {\qualid} are
-built from a notation, then this notation is interpreted in the scope
-stack extended by the scope bound (if any) to this argument. The
-effect of the scope is limited to the argument itself. It does not propagate 
-to subterms but the subterms that, after interpretation of the
-notation, turn to be themselves arguments of a reference are
-interpreted accordingly to the arguments scopes bound to this reference.
-
-Arguments scopes can be cleared with the following command:
-
-\begin{quote}
-{\tt Arguments {\qualid} : clear scopes}
-\end{quote}
-
-Extra argument scopes, to be used in case of coercion to Funclass
-(see Chapter~\ref{Coercions-full}) or with a computed type,
-can be given with
-
-\begin{quote}
-{\tt Arguments} {\qualid} \nelist{\textunderscore {\tt \%} \scope}{} {\tt : extra scopes.}
-\end{quote}
-
-\begin{Variants}
-\item {\tt Global Arguments} {\qualid} \nelist{\name {\tt \%}\scope}{}
-
-This behaves like {\tt Arguments} {\qualid} \nelist{\name {\tt \%}\scope}{}
-but survives when a section is closed instead
-of stopping working at section closing. Without the {\tt Global} modifier,
-the effect of the command stops when the section it belongs to ends.
-
-\item {\tt Local Arguments} {\qualid} \nelist{\name {\tt \%}\scope}{}
-
-This behaves like {\tt Arguments} {\qualid} \nelist{\name {\tt \%}\scope}{}
-but does not survive modules and files.
-Without the {\tt Local} modifier, the effect of the command is
-visible from within other modules or files.
-
-\end{Variants}
-
-\SeeAlso The command to show the scopes bound to the arguments of a
-function is described in Section~\ref{About}.
-
-\subsubsection{Binding types of arguments to an interpretation scope}
-
-When an interpretation scope is naturally associated to a type
-(e.g. the scope of operations on the natural numbers), it may be
-convenient to bind it to this type. When a scope {\scope} is bound to
-a type {\type}, any new function defined later on gets its arguments
-of type {\type} interpreted by default in scope {\scope} (this default
-behavior can however be overwritten by explicitly using the command
-{\tt Arguments}).
-
-Whether the argument of a function has some type {\type} is determined
-statically. For instance, if {\tt f} is a polymorphic function of type
-{\tt forall X:Type, X -> X} and type {\tt t} is bound to a scope
-{\scope}, then {\tt a} of type {\tt t} in {\tt f~t~a} is not
-recognized as an argument to be interpreted in scope {\scope}.
-
-\comindex{Bind Scope}
-\label{bindscope}
-More generally, any coercion {\class} (see Chapter~\ref{Coercions-full}) can be
-bound to an interpretation scope. The command to do it is
-\begin{quote}
-{\tt Bind Scope} {\scope} \texttt{with} {\class}
-\end{quote}
-
-\Example
-\begin{coq_example}
-Parameter U : Set.
-Bind Scope U_scope with U.
-Parameter Uplus : U -> U -> U.
-Parameter P : forall T:Set, T -> U -> Prop.
-Parameter f : forall T:Set, T -> U.
-Infix "+" := Uplus : U_scope.
-Unset Printing Notations.
-Open Scope nat_scope. (* Define + on the nat as the default for + *)
-Check (fun x y1 y2 z t => P _ (x + t) ((f _ (y1 + y2) + z))).
-\end{coq_example}
-
-\Rem The scopes {\tt type\_scope} and {\tt function\_scope} also have a local effect on
-interpretation. See the next section.
-
-\SeeAlso The command to show the scopes bound to the arguments of a
-function is described in Section~\ref{About}.
-
-\Rem In notations, the subterms matching the identifiers of the
-notations are interpreted in the scope in which the identifiers
-occurred at the time of the declaration of the notation. Here is an
-example:
-
-\begin{coq_example}
-Parameter g : bool -> bool.
-Notation "@@" := true (only parsing) : bool_scope.
-Notation "@@" := false (only parsing): mybool_scope.
-
-(* Defining a notation while the argument of g is bound to bool_scope *)
-Bind Scope bool_scope with bool.
-Notation "# x #" := (g x) (at level 40).
-Check # @@ #.
-(* Rebinding the argument of g to mybool_scope has no effect on the notation *)
-Arguments g _%mybool_scope.
-Check # @@ #.
-(* But we can force the scope *)
-Delimit Scope mybool_scope with mybool.
-Check # @@%mybool #.
-\end{coq_example}
-
-\subsection[The {\tt type\_scope} interpretation scope]{The {\tt type\_scope} interpretation scope\index{type\_scope@\texttt{type\_scope}}}
-
-The scope {\tt type\_scope} has a special status. It is a primitive
-interpretation scope which is temporarily activated each time a
-subterm of an expression is expected to be a type.  It is delimited by
-the key {\tt type}, and bound to the coercion class {\tt Sortclass}. It is also
-used in certain situations where an expression is statically known to
-be a type, including the conclusion and the type of hypotheses within
-an {\tt Ltac} goal match (see Section~\ref{ltac-match-goal})
-the statement of a theorem, the type of
-a definition, the type of a binder, the domain and codomain of
-implication, the codomain of products, and more generally any type
-argument of a declared or defined constant.
-
-\subsection[The {\tt function\_scope} interpretation scope]{The {\tt function\_scope} interpretation scope\index{function\_scope@\texttt{function\_scope}}}
-
-The scope {\tt function\_scope} also has a special status. 
-It is temporarily activated each time the argument of a global reference is
-recognized to be a {\tt Funclass instance}, i.e., of type {\tt forall x:A, B} or {\tt A -> B}.
-
-\subsection{Interpretation scopes used in the standard library of {\Coq}}
-
-We give an overview of the scopes used in the standard library of
-{\Coq}. For a complete list of notations in each scope, use the
-commands {\tt Print Scopes} or {\tt Print Scope {\scope}}.
-
-\subsubsection{\tt type\_scope}
-
-This scope includes infix {\tt *} for product types and infix {\tt +} for
-sum types. It is delimited by key {\tt type}, and bound to the coercion class 
-{\tt Sortclass}, as described at \ref{bindscope}.
-
-\subsubsection{\tt nat\_scope}
-
-This scope includes the standard arithmetical operators and relations on
-type {\tt nat}. Positive numerals in this scope are mapped to their
-canonical representent built from {\tt O} and {\tt S}. The scope is
-delimited by key {\tt nat}, and bound to the type {\tt nat} (see \ref{bindscope}).
-
-\subsubsection{\tt N\_scope}
-
-This scope includes the standard arithmetical operators and relations on
-type {\tt N} (binary natural numbers). It is delimited by key {\tt N}
-and comes with an interpretation for numerals as closed term of type {\tt N}.
-
-\subsubsection{\tt Z\_scope}
-
-This scope includes the standard arithmetical operators and relations on
-type {\tt Z} (binary integer numbers). It is delimited by key {\tt Z} 
-and comes with an interpretation for numerals as closed term of type {\tt Z}.
-
-\subsubsection{\tt positive\_scope}
-
-This scope includes the standard arithmetical operators and relations on
-type {\tt positive} (binary strictly positive numbers). It is
-delimited by key {\tt positive} and comes with an interpretation for
-numerals as closed term of type {\tt positive}.
-
-\subsubsection{\tt Q\_scope}
-
-This scope includes the standard arithmetical operators and relations on
-type {\tt Q} (rational numbers defined as fractions of an integer and
-a strictly positive integer modulo the equality of the
-numerator-denominator cross-product). As for numerals, only $0$ and
-$1$ have an interpretation in scope {\tt Q\_scope} (their
-interpretations are $\frac{0}{1}$ and $\frac{1}{1}$ respectively).
-
-\subsubsection{\tt Qc\_scope}
-
-This scope includes the standard arithmetical operators and relations on the
-type {\tt Qc} of rational numbers defined as the type of irreducible
-fractions of an integer and a strictly positive integer.
-
-\subsubsection{\tt real\_scope}
-
-This scope includes the standard arithmetical operators and relations on
-type {\tt R} (axiomatic real numbers). It is delimited by key {\tt R}
-and comes with an interpretation for numerals using the {\tt IZR}
-morphism from binary integer numbers to {\tt R}.
-
-\subsubsection{\tt bool\_scope}
-
-This scope includes notations for the boolean operators. It is
-delimited by key {\tt bool}, and bound to the type {\tt bool} (see \ref{bindscope}).
-
-\subsubsection{\tt list\_scope}
-
-This scope includes notations for the list operators. It is
-delimited by key {\tt list}, and bound to the type {\tt list} (see \ref{bindscope}).
-
-\subsubsection{\tt function\_scope}
-
-This scope is delimited by the key {\tt function}, and bound to the coercion class {\tt Funclass}, 
-as described at \ref{bindscope}.
-
-\subsubsection{\tt core\_scope}
-
-This scope includes the notation for pairs. It is delimited by key {\tt core}.
-
-\subsubsection{\tt string\_scope}
-
-This scope includes notation for strings as elements of the type {\tt
-string}.  Special characters and escaping follow {\Coq} conventions
-on strings (see Section~\ref{strings}). Especially, there is no
-convention to visualize non printable characters of a string.  The
-file {\tt String.v} shows an example that contains quotes, a newline
-and a beep (i.e. the ASCII character of code 7).
-
-\subsubsection{\tt char\_scope}
-
-This scope includes interpretation for all strings of the form
-\verb!"!$c$\verb!"! where $c$ is an ASCII character, or of the form
-\verb!"!$nnn$\verb!"! where $nnn$ is a three-digits number (possibly
-with leading 0's), or of the form \verb!""""!. Their respective
-denotations are the ASCII code of $c$, the decimal ASCII code $nnn$,
-or the ASCII code of the character \verb!"! (i.e. the ASCII code
-34), all of them being represented in the type {\tt ascii}.
-
-\subsection{Displaying informations about scopes}
-
-\subsubsection{\tt Print Visibility\comindex{Print Visibility}}
-
-This displays the current stack of notations in scopes and lonely
-notations that is used to interpret a notation. The top of the stack
-is displayed last. Notations in scopes whose interpretation is hidden
-by the same notation in a more recently open scope are not
-displayed. Hence each notation is displayed only once.
-
-\variant
-
-{\tt Print Visibility {\scope}}\\
-
-This displays the current stack of notations in scopes and lonely
-notations assuming that {\scope} is pushed on top of the stack.  This
-is useful to know how a subterm locally occurring in the scope of
-{\scope} is interpreted.
-
-\subsubsection{\tt Print Scope {\scope}\comindex{Print Scope}}
-
-This displays all the notations defined in interpretation scope
-{\scope}.  It also displays the delimiting key if any and the class to
-which the scope is bound, if any.
-
-\subsubsection{\tt Print Scopes\comindex{Print Scopes}}
-
-This displays all the notations, delimiting keys and corresponding
-class of all the existing interpretation scopes.
-It also displays the lonely notations.
-
-\section[Abbreviations]{Abbreviations\index{Abbreviations}
-\label{Abbreviations}
-\comindex{Notation}}
-
-An {\em abbreviation} is a name, possibly applied to arguments, that
-denotes a (presumably) more complex expression. Here are examples:
-
-\begin{coq_eval}
-Require Import List.
-Require Import Relations.
-Set Printing Notations.
-\end{coq_eval}
-\begin{coq_example}
-Notation Nlist := (list nat).
-Check 1 :: 2 :: 3 :: nil.
-Notation reflexive R := (forall x, R x x).
-Check forall A:Prop, A <-> A.
-Check reflexive iff.
-\end{coq_example}
-
-An abbreviation expects no precedence nor associativity, since it
-is parsed as usual application. Abbreviations are used as
-much as possible by the {\Coq} printers unless the modifier
-\verb=(only parsing)= is given.
-
-Abbreviations are bound to an absolute name as an ordinary
-definition is, and they can be referred by qualified names too.
-
-Abbreviations are syntactic in the sense that they are bound to
-expressions which are not typed at the time of the definition of the
-abbreviation but at the time it is used. Especially, abbreviations can
-be bound to terms with holes (i.e. with ``\_''). The general syntax
-for abbreviations is
-\begin{quote}
-\zeroone{{\tt Local}} \texttt{Notation} {\ident} \sequence{\ident}{} \texttt{:=} {\term}
- \zeroone{{\tt (only parsing)}}~\verb=.=
-\end{quote}
-
-\Example
-\begin{coq_eval}
-Set Strict Implicit.
-Reset Initial.
-\end{coq_eval}
-\begin{coq_example}
-Definition explicit_id (A:Set) (a:A) := a.
-Notation id := (explicit_id _).
-Check (id 0).
-\end{coq_example}
-
-Abbreviations do not survive the end of sections. No typing of the denoted
-expression is performed at definition time. Type-checking is done only
-at the time of use of the abbreviation.
-
-%\Rem \index{Syntactic Definition} %
-%Abbreviations are similar to the {\em syntactic
-%definitions} available in versions of {\Coq} prior to version 8.0,
-%except that abbreviations are used for printing (unless the modifier
-%\verb=(only parsing)= is given) while syntactic definitions were not.
-
-\section{Tactic Notations
-\label{Tactic-Notation}
-\comindex{Tactic Notation}}
-
-Tactic notations allow to customize the syntax of the tactics of the
-tactic language.
-%% \footnote{Tactic notations are just a simplification of
-%% the {\tt Grammar tactic simple\_tactic} command that existed in
-%% versions prior to version 8.0.}
-Tactic notations obey the following syntax:
-\medskip
-
-\noindent
-\begin{tabular}{lcl}
-{\sentence} & ::= & \zeroone{\tt Local} \texttt{Tactic Notation} \zeroone{\taclevel} \sequence{\proditem}{} \\
-& & \texttt{:= {\tac} .}\\
-{\proditem} & ::= & {\str} $|$ {\tacargtype}{\tt ({\ident})} \\ 
-{\taclevel} & ::= & {\tt (at level} {\naturalnumber}{\tt )} \\
-{\tacargtype}\!\! & ::= &
-%{\tt preident} $|$
-{\tt ident} $|$
-{\tt simple\_intropattern} $|$
-{\tt reference} \\ & $|$ &
-{\tt hyp} $|$
-{\tt hyp\_list} $|$
-{\tt ne\_hyp\_list} \\ & $|$ &
-% {\tt quantified\_hypothesis} \\ & $|$ &
-{\tt constr} $|$ {\tt uconstr} $|$
-{\tt constr\_list} $|$
-{\tt ne\_constr\_list} \\ & $|$ &
-%{\tt castedopenconstr} $|$
-{\tt integer} $|$
-{\tt integer\_list} $|$
-{\tt ne\_integer\_list} \\ & $|$ &
-{\tt int\_or\_var} $|$
-{\tt int\_or\_var\_list} $|$
-{\tt ne\_int\_or\_var\_list} \\ & $|$ &
-{\tt tactic} $|$ {\tt tactic$n$} \qquad\mbox{(for $0\leq n\leq 5$)}
-
-\end{tabular}
-\medskip
-
-A tactic notation {\tt Tactic Notation {\taclevel}
-{\sequence{\proditem}{}} := {\tac}} extends the parser and
-pretty-printer of tactics with a new rule made of the list of
-production items. It then evaluates into the tactic expression
-{\tac}. For simple tactics, it is recommended to use a terminal
-symbol, i.e. a {\str}, for the first production item.  The tactic
-level indicates the parsing precedence of the tactic notation. This
-information is particularly relevant for notations of tacticals.
-Levels 0 to 5 are available (default is 0). 
-To know the parsing precedences of the
-existing tacticals, use the command
-\comindex{Print Grammar tactic}
- {\tt Print Grammar tactic.}
-
-Each type of tactic argument has a specific semantic regarding how it
-is parsed and how it is interpreted. The semantic is described in the
-following table. The last command gives examples of tactics which
-use the corresponding kind of argument.
-
-\medskip
-\noindent
-\begin{tabular}{l|l|l|l}
-Tactic argument type & parsed as & interpreted as & as in tactic \\
-\hline & & & \\
-{\tt\small ident} & identifier & a user-given name & {\tt intro} \\
-{\tt\small simple\_intropattern} & intro\_pattern & an intro\_pattern & {\tt intros}\\
-{\tt\small hyp} & identifier & an hypothesis defined in context & {\tt clear}\\
-%% quantified_hypothesis actually not supported
-%%{\tt\small quantified\_hypothesis} & identifier or integer & a named or non dep. hyp. of the goal & {\tt intros until}\\
-{\tt\small reference} & qualified identifier & a global reference of term & {\tt unfold}\\
-{\tt\small constr} & term & a term & {\tt exact} \\
-{\tt\small uconstr} & term & an untyped term & {\tt refine} \\
-%% castedopenconstr actually not supported
-%%{\tt\small castedopenconstr} & term & a term with its sign. of exist. var. & {\tt refine}\\
-{\tt\small integer} & integer & an integer &  \\
-{\tt\small int\_or\_var} & identifier or integer & an integer & {\tt do} \\
-{\tt\small tactic} & tactic at level 5 & a tactic &  \\
-{\tt\small tactic$n$} & tactic at level $n$ & a tactic & \\
-{\tt\small {\nterm{entry}}\_list} & list of {\nterm{entry}} & a list of how {\nterm{entry}} is interpreted & \\
-{\tt\small ne\_{\nterm{entry}}\_list} & non-empty list of {\nterm{entry}} & a list of how {\nterm{entry}} is interpreted& \\
-\end{tabular}
-
-\Rem In order to be bound in tactic definitions, each syntactic entry
-for argument type must include the case of simple {\ltac} identifier
-as part of what it parses. This is naturally the case for {\tt ident},
-{\tt simple\_intropattern}, {\tt reference}, {\tt constr}, ... but not
-for {\tt integer}. This is the reason for introducing a special entry
-{\tt int\_or\_var} which evaluates to integers only but which
-syntactically includes identifiers in order to be usable in tactic
-definitions.
-
-\Rem The {\tt {\nterm{entry}}\_list} and {\tt ne\_{\nterm{entry}}\_list}
-entries can be used in primitive tactics or in other notations at
-places where a list of the underlying entry can be used: {\nterm{entry}} is
-either {\tt\small constr}, {\tt\small hyp}, {\tt\small integer} or
-{\tt\small int\_or\_var}.
-
-Tactic notations do not survive the end of sections. They survive
-modules unless the command {\tt Local Tactic Notation} is used instead
-of {\tt Tactic Notation}.
-
-%%% Local Variables: 
-%%% mode: latex
-%%% TeX-master: "Reference-Manual"
-%%% End: 
-- 
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