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\chapter{Isar Syntax Primitives}
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We give a complete reference of all basic syntactic entities underlying the
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Isabelle/Isar document syntax. Actual theory and proof commands will be
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introduced later on.
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\medskip
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In order to get started with writing well-formed Isabelle/Isar documents, the
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most important aspect to be noted is the difference of \emph{inner} versus
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\emph{outer} syntax. Inner syntax is that of Isabelle types and terms of the
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logic, while outer syntax is that of Isabelle/Isar theories (including
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proofs). As a general rule, inner syntax entities may occur only as
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\emph{atomic entities} within outer syntax. For example, the string
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\texttt{"x + y"} and identifier \texttt{z} are legal term specifications
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within a theory, while \texttt{x + y} is not.
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\begin{warn}
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Note that classic Isabelle theories used to fake parts of the inner syntax
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of types, with rather complicated rules when quotes may be omitted. Despite
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the minor drawback of requiring quotes more often, the syntax of
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Isabelle/Isar is much simpler and more robust in that respect.
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\end{warn}
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\medskip
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Isabelle/Isar input may contain any number of input termination characters
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``\texttt{;}'' (semicolon) to separate commands explicitly. This may be
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particularly useful in interactive shell sessions to make clear where the
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current command is intended to end. Otherwise, the interactive loop will
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continue until end-of-command in clearly indicated from the input syntax,
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e.g.\ encounter of the next command keyword.
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Advanced interfaces such as Proof~General \cite{proofgeneral} do not require
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explicit semicolons, the amount of input text is determined automatically by
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inspecting the Emacs text buffer. Also note that in the presentation of
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Isabelle/Isar documents, semicolons are omitted in any case to gain
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readability.
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\section{Lexical matters}\label{sec:lex-syntax}
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The Isabelle/Isar outer syntax provides token classes as presented below.
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Note that some of these coincide (by full intention) with the inner lexical
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syntax as presented in \cite{isabelle-ref}. These different levels of syntax
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should not be confused, though.
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%FIXME keyword, command
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\indexoutertoken{ident}\indexoutertoken{longident}\indexoutertoken{symident}
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\indexoutertoken{nat}\indexoutertoken{var}\indexoutertoken{typefree}
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\indexoutertoken{typevar}\indexoutertoken{string}\indexoutertoken{verbatim}
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\begin{matharray}{rcl}
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ident & = & letter~quasiletter^* \\
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longident & = & ident\verb,.,ident~\dots~ident \\
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symident & = & sym^+ ~|~ symbol \\
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nat & = & digit^+ \\
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var & = & \verb,?,ident ~|~ \verb,?,ident\verb,.,nat \\
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typefree & = & \verb,',ident \\
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typevar & = & \verb,?,typefree ~|~ \verb,?,typefree\verb,.,nat \\
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string & = & \verb,", ~\dots~ \verb,", \\
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verbatim & = & \verb,{*, ~\dots~ \verb,*}, \\
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\end{matharray}
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\begin{matharray}{rcl}
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letter & = & \verb,a, ~|~ \dots ~|~ \verb,z, ~|~ \verb,A, ~|~ \dots ~|~ \verb,Z, \\
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digit & = & \verb,0, ~|~ \dots ~|~ \verb,9, \\
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quasiletter & = & letter ~|~ digit ~|~ \verb,_, ~|~ \verb,', \\
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sym & = & \verb,!, ~|~ \verb,#, ~|~ \verb,$, ~|~ \verb,%, ~|~ \verb,&, ~|~ %$
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\verb,*, ~|~ \verb,+, ~|~ \verb,-, ~|~ \verb,/, ~|~ \verb,:, ~|~
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\verb,<, ~|~ \verb,=, ~|~ \verb,>, ~|~ \verb,?, ~|~ \mathtt{\at} ~|~ \\
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& & \verb,^, ~|~ \verb,_, ~|~ \verb,`, ~|~ \verb,|, ~|~ \verb,~, \\
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symbol & = & {\forall} ~|~ {\exists} ~|~ \dots
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\end{matharray}
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The syntax of \texttt{string} admits any characters, including newlines;
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``\verb|"|'' (double-quote) and ``\verb|\|'' (backslash) have to be escaped by
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a backslash; newlines need not be escaped. Note that ML-style control
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characters are \emph{not} supported. The body of \texttt{verbatim} may
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consist of any text not containing ``\verb|*}|''.
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Comments take the form \texttt{(*~\dots~*)} and may be
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nested\footnote{Proof~General may occasionally get confused by nested
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comments.}, just as in ML. Note that these are \emph{source} comments only,
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which are stripped after lexical analysis of the input. The Isar document
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syntax also provides \emph{formal comments} that are actually part of the text
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(see \S\ref{sec:comments}).
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\section{Common syntax entities}
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Subsequently, we introduce several basic syntactic entities, such as names,
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terms, and theorem specifications, which have been factored out of the actual
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Isar language elements to be described later.
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Note that some of the basic syntactic entities introduced below (e.g.\
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\railqtoken{name}) act much like tokens rather than plain nonterminals (e.g.\
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\railnonterm{sort}), especially for the sake of error messages. E.g.\ syntax
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elements such as $\CONSTS$ referring to \railqtoken{name} or \railqtoken{type}
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would really report a missing name or type rather than any of the constituent
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primitive tokens such as \railtoken{ident} or \railtoken{string}.
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\subsection{Names}
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Entity \railqtoken{name} usually refers to any name of types, constants,
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theorems etc.\ that are to be \emph{declared} or \emph{defined} (so qualified
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identifiers are excluded here). Quoted strings provide an escape for
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non-identifier names or those ruled out by outer syntax keywords (e.g.\
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\verb|"let"|). Already existing objects are usually referenced by
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\railqtoken{nameref}.
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\indexoutertoken{name}\indexoutertoken{parname}\indexoutertoken{nameref}
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\indexoutertoken{int}
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\begin{rail}
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name: ident | symident | string | nat
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;
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parname: '(' name ')'
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;
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nameref: name | longident
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;
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int: nat | '-' nat
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;
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\end{rail}
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\subsection{Comments}\label{sec:comments}
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Large chunks of plain \railqtoken{text} are usually given
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\railtoken{verbatim}, i.e.\ enclosed in \verb|{*|~\dots~\verb|*}|. For
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convenience, any of the smaller text units conforming to \railqtoken{nameref}
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are admitted as well. Almost any of the Isar commands may be annotated by
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marginal \railnonterm{comment} of the form \texttt{--} \railqtoken{text}.
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Note that the latter kind of comment is actually part of the language, while
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source level comments \verb|(*|~\dots~\verb|*)| are stripped at the lexical
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level. A few commands such as $\PROOFNAME$ admit additional markup with a
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``level of interest'': \texttt{\%} followed by an optional number $n$ (default
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$n = 1$) indicates that the respective part of the document becomes $n$ levels
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more obscure; \texttt{\%\%} means that interest drops by $\infty$ --- abandon
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every hope, who enter here.
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\indexoutertoken{text}\indexouternonterm{comment}\indexouternonterm{interest}
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\begin{rail}
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text: verbatim | nameref
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;
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comment: ('--' text +)
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;
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interest: percent nat? | ppercent
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;
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\end{rail}
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\subsection{Type classes, sorts and arities}
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Classes are specified by plain names. Sorts have a very simple inner syntax,
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which is either a single class name $c$ or a list $\{c@1, \dots, c@n\}$
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referring to the intersection of these classes. The syntax of type arities is
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given directly at the outer level.
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\indexouternonterm{sort}\indexouternonterm{arity}\indexouternonterm{simplearity}
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\indexouternonterm{classdecl}
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\begin{rail}
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classdecl: name ('<' (nameref + ','))?
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;
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sort: nameref
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;
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arity: ('(' (sort + ',') ')')? sort
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;
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simplearity: ('(' (sort + ',') ')')? nameref
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;
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\end{rail}
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\subsection{Types and terms}\label{sec:types-terms}
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The actual inner Isabelle syntax, that of types and terms of the logic, is far
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too sophisticated in order to be modelled explicitly at the outer theory
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level. Basically, any such entity has to be quoted to turn it into a single
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token (the parsing and type-checking is performed internally later). For
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convenience, a slightly more liberal convention is adopted: quotes may be
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omitted for any type or term that is already \emph{atomic} at the outer level.
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For example, one may write just \texttt{x} instead of \texttt{"x"}. Note that
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symbolic identifiers (e.g.\ \texttt{++} or $\forall$) are available as well,
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provided these are not superseded by commands or keywords (e.g.\ \texttt{+}).
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\indexoutertoken{type}\indexoutertoken{term}\indexoutertoken{prop}
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\begin{rail}
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type: nameref | typefree | typevar
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;
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term: nameref | var
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;
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prop: term
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;
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\end{rail}
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Positional instantiations are indicated by giving a sequence of terms, or the
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placeholder ``$\_$'' (underscore), which means to skip a position.
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\indexoutertoken{inst}\indexoutertoken{insts}
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\begin{rail}
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inst: underscore | term
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;
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insts: (inst *)
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;
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\end{rail}
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Type declarations and definitions usually refer to \railnonterm{typespec} on
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the left-hand side. This models basic type constructor application at the
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outer syntax level. Note that only plain postfix notation is available here,
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but no infixes.
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\indexouternonterm{typespec}
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\begin{rail}
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typespec: (() | typefree | '(' ( typefree + ',' ) ')') name
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;
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\end{rail}
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\subsection{Term patterns}\label{sec:term-pats}
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Assumptions and goal statements usually admit casual binding of schematic term
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variables by giving (optional) patterns of the form $\ISS{p@1\;\dots}{p@n}$.
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There are separate versions available for \railqtoken{term}s and
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\railqtoken{prop}s. The latter provides a $\CONCLNAME$ part with patterns
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referring the (atomic) conclusion of a rule.
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\indexouternonterm{termpat}\indexouternonterm{proppat}
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\begin{rail}
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termpat: '(' ('is' term +) ')'
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;
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proppat: '(' (('is' prop +) | 'concl' ('is' prop +) | ('is' prop +) 'concl' ('is' prop +)) ')'
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;
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\end{rail}
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\subsection{Mixfix annotations}
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Mixfix annotations specify concrete \emph{inner} syntax of Isabelle types and
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terms (see also \cite{isabelle-ref}). Some commands such as $\TYPES$ (see
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\S\ref{sec:types-pure}) admit infixes only, while $\CONSTS$ (see
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\S\ref{sec:consts}) and $\isarkeyword{syntax}$ (see \S\ref{sec:syn-trans})
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support the full range of general mixfixes and binders.
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\indexouternonterm{infix}\indexouternonterm{mixfix}
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\begin{rail}
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infix: '(' ('infixl' | 'infixr') string? nat ')'
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;
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mixfix: infix | '(' string prios? nat? ')' | '(' 'binder' string prios? nat ')'
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;
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prios: '[' (nat + ',') ']'
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;
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\end{rail}
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\subsection{Attributes and theorems}\label{sec:syn-att}
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Attributes (and proof methods, see \S\ref{sec:syn-meth}) have their own
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``semi-inner'' syntax, in the sense that input conforming to
|
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\railnonterm{args} below is parsed by the attribute a second time. The
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attribute argument specifications may be any sequence of atomic entities
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(identifiers, strings etc.), or properly bracketed argument lists. Below
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\railqtoken{atom} refers to any atomic entity, including any
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\railtoken{keyword} conforming to \railtoken{symident}.
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\indexoutertoken{atom}\indexouternonterm{args}\indexouternonterm{attributes}
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\begin{rail}
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atom: nameref | typefree | typevar | var | nat | keyword
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;
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arg: atom | '(' args ')' | '[' args ']'
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;
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args: arg *
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;
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attributes: '[' (nameref args * ',') ']'
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;
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\end{rail}
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Theorem specifications come in several flavors: \railnonterm{axmdecl} and
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\railnonterm{thmdecl} usually refer to axioms, assumptions or results of goal
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statements, while \railnonterm{thmdef} collects lists of existing theorems.
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Existing theorems are given by \railnonterm{thmref} and \railnonterm{thmrefs},
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the former requires an actual singleton result. Any of these theorem
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|
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specifications may include lists of attributes both on the left and right hand
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sides; attributes are applied to any immediately preceding theorem. If names
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|
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are omitted, the theorems are not stored within the theorem database of the
|
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theory or proof context; any given attributes are still applied, though.
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|
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\indexouternonterm{thmdecl}\indexouternonterm{axmdecl}
|
|
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\indexouternonterm{thmdef}\indexouternonterm{thmrefs}
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|
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\begin{rail}
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|
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axmdecl: name attributes? ':'
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;
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thmdecl: thmbind ':'
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|
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;
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|
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thmdef: thmbind '='
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|
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;
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|
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thmref: nameref attributes?
|
|
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;
|
|
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thmrefs: thmref +
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|
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;
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|
300 |
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9200
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thmbind: name attributes | name | attributes
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|
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;
|
|
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\end{rail}
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|
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|
|
305 |
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|
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\subsection{Proof methods}\label{sec:syn-meth}
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|
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|
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Proof methods are either basic ones, or expressions composed of methods via
|
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|
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``\texttt{,}'' (sequential composition), ``\texttt{|}'' (alternative choices),
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|
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``\texttt{?}'' (try), ``\texttt{+}'' (repeat at least once). In practice,
|
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proof methods are usually just a comma separated list of
|
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\railqtoken{nameref}~\railnonterm{args} specifications. Note that parentheses
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|
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may be dropped for single method specifications (with no arguments).
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|
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|
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\indexouternonterm{method}
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\begin{rail}
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method: (nameref | '(' methods ')') (() | '?' | '+')
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;
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methods: (nameref args | method) + (',' | '|')
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|
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;
|
|
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\end{rail}
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|
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8532
|
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Proper use of Isar proof methods does \emph{not} involve goal addressing.
|
|
324 |
Nevertheless, specifying goal ranges may occasionally come in handy in
|
|
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emulating tactic scripts. Note that $[n-]$ refers to all goals, starting from
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|
326 |
$n$. All goals may be specified by $[!]$, which is the same as $[1-]$.
|
8532
|
327 |
|
|
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\indexouternonterm{goalspec}
|
|
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\begin{rail}
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goalspec: '[' (nat '-' nat | nat '-' | nat | '!' ) ']'
|
8532
|
331 |
;
|
|
332 |
\end{rail}
|
|
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|
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9200
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\subsection{Antiquotations}\label{sec:antiq}
|
|
336 |
|
|
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The text body of formal comments (see also \S\ref{sec:comments}) may contain
|
|
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antiquotations of logical entities, such as theorems, terms and types, which
|
|
339 |
are to be presented in the final output produced by the Isabelle document
|
|
340 |
preparation system (see also \S\ref{sec:document-prep}).
|
|
341 |
|
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|
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Thus embedding of
|
|
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\texttt{{\at}{\ttlbrace}term[show_types]~"f(x)~=~a~+~x"{\ttrbrace}} within a
|
|
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text block would cause
|
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|
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\isa{(f{\isasymColon}'a~{\isasymRightarrow}~'a)~(x{\isasymColon}'a)~=~(a{\isasymColon}'a)~+~x}
|
|
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to appear in the final {\LaTeX} document.
|
|
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|
|
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Antiquotations do not only spare the author from tedious typing, but also
|
|
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achieve some degree of consistency-checking of informal explanations with
|
|
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formal developments, since well-formedness of terms and types with respect to
|
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|
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the current theory or proof context can be ensured. The $text$ antiquotation
|
|
352 |
is an exception to this rule: it prints an uninterpreted text argument that is
|
|
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not checked in any way.
|
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|
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|
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|
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\indexisarant{thm}\indexisarant{prop}\indexisarant{term}
|
|
356 |
\indexisarant{typ}\indexisarant{name}
|
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|
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\begin{rail}
|
|
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atsign lbrace antiquotation rbrace
|
|
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;
|
|
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|
|
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antiquotation:
|
|
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'thm' options thmrefs |
|
|
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'prop' options prop |
|
|
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'term' options term |
|
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|
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'typ' options type |
|
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|
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'text' options name
|
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|
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;
|
|
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options: '[' (option * ',') ']'
|
|
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;
|
|
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option: name | name '=' name
|
|
371 |
;
|
|
372 |
\end{rail}
|
|
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|
|
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Note that the syntax of antiquotations may \emph{not} include source comments
|
|
375 |
\texttt{(*~\dots~*)} or verbatim text \verb|{*|~\dots~\verb|*}|.
|
|
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|
|
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\medskip
|
|
378 |
|
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|
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The following options are available to tune the output. Note that some of
|
|
380 |
these coincide with ML flags of the same names (see also \cite{isabelle-ref}).
|
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|
381 |
\begin{descr}
|
9233
|
382 |
\item[$show_types = bool$ and $show_sorts = bool$] control printing of
|
9234
|
383 |
explicit type and sort constraints.
|
9233
|
384 |
\item[$long_names = bool$] forces names of types and constants etc.\ to be
|
|
385 |
printed in their fully qualified internal form.
|
|
386 |
\item[$eta_contract = bool$] prints terms in $\eta$-contracted form.
|
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|
387 |
\item[$display = bool$] indicates if the text is to be output as multi-line
|
|
388 |
``display material'', rather than a small piece of text without line breaks
|
|
389 |
(which is the default).
|
|
390 |
\item[$quotes = bool$] indicates if the output should be enclosed in double
|
|
391 |
quotes.
|
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|
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\item[$mode = name$] adds $name$ to the print mode to be used for presentation
|
|
393 |
(see also \cite{isabelle-ref}). Note that the standard setup for {\LaTeX}
|
|
394 |
output is already present by default, including the modes ``$latex$'',
|
|
395 |
``$xsymbols$'', ``$symbols$''.
|
9728
|
396 |
\item[$margin = nat$ and $indent = nat$] change the margin or indentation for
|
|
397 |
pretty printing of display material.
|
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|
398 |
\item[$source = bool$] prints the source text of the antiquotation arguments,
|
|
399 |
rather than the actual value. Note that this does not affect
|
|
400 |
well-formedness checks of $thm$, $term$, etc. (only the $text$ antiquotation
|
|
401 |
admits arbitrary output).
|
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|
402 |
\end{descr}
|
|
403 |
|
|
404 |
For boolean flags, ``$name = true$'' may be abbreviated as ``$name$''. All of
|
|
405 |
the above flags are disabled by default, unless changed from ML.
|
|
406 |
|
|
407 |
|
7046
|
408 |
%%% Local Variables:
|
|
409 |
%%% mode: latex
|
|
410 |
%%% TeX-master: "isar-ref"
|
|
411 |
%%% End:
|