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diff --git a/doc/sphinx/user-extensions/syntax-extensions.rst b/doc/sphinx/user-extensions/syntax-extensions.rst new file mode 100644 index 000000000..836753db1 --- /dev/null +++ b/doc/sphinx/user-extensions/syntax-extensions.rst @@ -0,0 +1,1431 @@ +\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: |