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 \def\isabellecontext{Outer{\isacharunderscore}Syntax}%

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 \ Outer{\isacharunderscore}Syntax\isanewline

 \isakeyword{imports}\ Main\isanewline

 \isakeyword{begin}%

 \endisatagtheory

 {\isafoldtheory}%

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 \isadelimtheory

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 \endisadelimtheory

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 \isamarkupchapter{Outer syntax%

 }

 \isamarkuptrue%

 %

 \begin{isamarkuptext}%

 The rather generic framework of Isabelle/Isar syntax emerges from

   three main syntactic categories: \emph{commands} of the top-level

   Isar engine (covering theory and proof elements), \emph{methods} for

   general goal refinements (analogous to traditional ``tactics''), and

   \emph{attributes} for operations on facts (within a certain

   context).  Subsequently we give a reference of basic syntactic

   entities underlying Isabelle/Isar syntax in a bottom-up manner.

   Concrete theory and proof language elements will be introduced later

   on.

   \medskip In order to get started with writing well-formed

   Isabelle/Isar documents, the most important aspect to be noted is

   the difference of \emph{inner} versus \emph{outer} syntax.  Inner

   syntax is that of Isabelle types and terms of the logic, while outer

   syntax is that of Isabelle/Isar theory sources (specifications and

   proofs).  As a general rule, inner syntax entities may occur only as

   \emph{atomic entities} within outer syntax.  For example, the string

   \verb|"x + y"| and identifier \verb|z| are legal term

   specifications within a theory, while \verb|x + y| without

   quotes is not.

   Printed theory documents usually omit quotes to gain readability

   (this is a matter of {\LaTeX} macro setup, say via \verb|\isabellestyle|, see also \cite{isabelle-sys}).  Experienced

   users of Isabelle/Isar may easily reconstruct the lost technical

   information, while mere readers need not care about quotes at all.

   \medskip Isabelle/Isar input may contain any number of input

   termination characters ``\verb|;|'' (semicolon) to separate

   commands explicitly.  This is particularly useful in interactive

   shell sessions to make clear where the current command is intended

   to end.  Otherwise, the interpreter loop will continue to issue a

   secondary prompt ``\verb|#|'' until an end-of-command is

   clearly recognized from the input syntax, e.g.\ encounter of the

   next command keyword.

   More advanced interfaces such as Proof~General \cite{proofgeneral}

   do not require explicit semicolons, the amount of input text is

   determined automatically by inspecting the present content of the

   Emacs text buffer.  In the printed presentation of Isabelle/Isar

   documents semicolons are omitted altogether for readability.

   \begin{warn}

     Proof~General requires certain syntax classification tables in

     order to achieve properly synchronized interaction with the

     Isabelle/Isar process.  These tables need to be consistent with

     the Isabelle version and particular logic image to be used in a

     running session (common object-logics may well change the outer

     syntax).  The standard setup should work correctly with any of the

     ``official'' logic images derived from Isabelle/HOL (including

     HOLCF etc.).  Users of alternative logics may need to tell

     Proof~General explicitly, e.g.\ by giving an option \verb|-k ZF|

     (in conjunction with \verb|-l ZF|, to specify the default

     logic image).  Note that option \verb|-L| does both

     of this at the same time.

   \end{warn}%

 \end{isamarkuptext}%

 \isamarkuptrue%

 %

 \isamarkupsection{Lexical matters \label{sec:outer-lex}%

 }

 \isamarkuptrue%

 %

 \begin{isamarkuptext}%

 The outer lexical syntax consists of three main categories of

   syntax tokens:

   \begin{enumerate}

   \item \emph{major keywords} --- the command names that are available

   in the present logic session;

   \item \emph{minor keywords} --- additional literal tokens required

   by the syntax of commands;

   \item \emph{named tokens} --- various categories of identifiers etc.

   \end{enumerate}

   Major keywords and minor keywords are guaranteed to be disjoint.

   This helps user-interfaces to determine the overall structure of a

   theory text, without knowing the full details of command syntax.

   Internally, there is some additional information about the kind of

   major keywords, which approximates the command type (theory command,

   proof command etc.).

   Keywords override named tokens.  For example, the presence of a

   command called \verb|term| inhibits the identifier \verb|term|, but the string \verb|"term"| can be used instead.

   By convention, the outer syntax always allows quoted strings in

   addition to identifiers, wherever a named entity is expected.

   When tokenizing a given input sequence, the lexer repeatedly takes

   the longest prefix of the input that forms a valid token.  Spaces,

   tabs, newlines and formfeeds between tokens serve as explicit

   separators.

   \medskip The categories for named tokens are defined once and for

   all as follows.

   \begin{center}

   \begin{supertabular}{rcl}

     \indexdef{}{syntax}{ident}\hypertarget{syntax.ident}{\hyperlink{syntax.ident}{\mbox{\isa{ident}}}} & = & \isa{{\isachardoublequote}letter\ quasiletter\isactrlsup {\isacharasterisk}{\isachardoublequote}} \\

     \indexdef{}{syntax}{longident}\hypertarget{syntax.longident}{\hyperlink{syntax.longident}{\mbox{\isa{longident}}}} & = & \isa{{\isachardoublequote}ident{\isacharparenleft}{\isachardoublequote}}\verb|.|\isa{{\isachardoublequote}ident{\isacharparenright}\isactrlsup {\isacharplus}{\isachardoublequote}} \\

     \indexdef{}{syntax}{symident}\hypertarget{syntax.symident}{\hyperlink{syntax.symident}{\mbox{\isa{symident}}}} & = & \isa{{\isachardoublequote}sym\isactrlsup {\isacharplus}\ \ {\isacharbar}\ \ {\isachardoublequote}}\verb|\|\verb|<|\isa{ident}\verb|>| \\

     \indexdef{}{syntax}{nat}\hypertarget{syntax.nat}{\hyperlink{syntax.nat}{\mbox{\isa{nat}}}} & = & \isa{{\isachardoublequote}digit\isactrlsup {\isacharplus}{\isachardoublequote}} \\

     \indexdef{}{syntax}{var}\hypertarget{syntax.var}{\hyperlink{syntax.var}{\mbox{\isa{var}}}} & = & \verb|?|\isa{{\isachardoublequote}ident\ \ {\isacharbar}\ \ {\isachardoublequote}}\verb|?|\isa{ident}\verb|.|\isa{nat} \\

     \indexdef{}{syntax}{typefree}\hypertarget{syntax.typefree}{\hyperlink{syntax.typefree}{\mbox{\isa{typefree}}}} & = & \verb|'|\isa{ident} \\

     \indexdef{}{syntax}{typevar}\hypertarget{syntax.typevar}{\hyperlink{syntax.typevar}{\mbox{\isa{typevar}}}} & = & \verb|?|\isa{{\isachardoublequote}typefree\ \ {\isacharbar}\ \ {\isachardoublequote}}\verb|?|\isa{typefree}\verb|.|\isa{nat} \\

     \indexdef{}{syntax}{string}\hypertarget{syntax.string}{\hyperlink{syntax.string}{\mbox{\isa{string}}}} & = & \verb|"| \isa{{\isachardoublequote}{\isasymdots}{\isachardoublequote}} \verb|"| \\

     \indexdef{}{syntax}{altstring}\hypertarget{syntax.altstring}{\hyperlink{syntax.altstring}{\mbox{\isa{altstring}}}} & = & \verb|`| \isa{{\isachardoublequote}{\isasymdots}{\isachardoublequote}} \verb|`| \\

     \indexdef{}{syntax}{verbatim}\hypertarget{syntax.verbatim}{\hyperlink{syntax.verbatim}{\mbox{\isa{verbatim}}}} & = & \verb|{*| \isa{{\isachardoublequote}{\isasymdots}{\isachardoublequote}} \verb|*|\verb|}| \\[1ex]

     \isa{letter} & = & \isa{{\isachardoublequote}latin\ \ {\isacharbar}\ \ {\isachardoublequote}}\verb|\|\verb|<|\isa{latin}\verb|>|\isa{{\isachardoublequote}\ \ {\isacharbar}\ \ {\isachardoublequote}}\verb|\|\verb|<|\isa{{\isachardoublequote}latin\ latin{\isachardoublequote}}\verb|>|\isa{{\isachardoublequote}\ \ {\isacharbar}\ \ greek\ \ {\isacharbar}{\isachardoublequote}} \\

           &   & \verb|\<^isub>|\isa{{\isachardoublequote}\ \ {\isacharbar}\ \ {\isachardoublequote}}\verb|\<^isup>| \\

     \isa{quasiletter} & = & \isa{{\isachardoublequote}letter\ \ {\isacharbar}\ \ digit\ \ {\isacharbar}\ \ {\isachardoublequote}}\verb|_|\isa{{\isachardoublequote}\ \ {\isacharbar}\ \ {\isachardoublequote}}\verb|'| \\

     \isa{latin} & = & \verb|a|\isa{{\isachardoublequote}\ \ {\isacharbar}\ {\isasymdots}\ {\isacharbar}\ \ {\isachardoublequote}}\verb|z|\isa{{\isachardoublequote}\ \ {\isacharbar}\ \ {\isachardoublequote}}\verb|A|\isa{{\isachardoublequote}\ \ {\isacharbar}\ \ {\isasymdots}\ {\isacharbar}\ \ {\isachardoublequote}}\verb|Z| \\

     \isa{digit} & = & \verb|0|\isa{{\isachardoublequote}\ \ {\isacharbar}\ \ {\isasymdots}\ {\isacharbar}\ \ {\isachardoublequote}}\verb|9| \\

     \isa{sym} & = & \verb|!|\isa{{\isachardoublequote}\ \ {\isacharbar}\ \ {\isachardoublequote}}\verb|#|\isa{{\isachardoublequote}\ \ {\isacharbar}\ \ {\isachardoublequote}}\verb|$|\isa{{\isachardoublequote}\ \ {\isacharbar}\ \ {\isachardoublequote}}\verb|%|\isa{{\isachardoublequote}\ \ {\isacharbar}\ \ {\isachardoublequote}}\verb|&|\isa{{\isachardoublequote}\ \ {\isacharbar}\ \ {\isachardoublequote}}\verb|*|\isa{{\isachardoublequote}\ \ {\isacharbar}\ \ {\isachardoublequote}}\verb|+|\isa{{\isachardoublequote}\ \ {\isacharbar}\ \ {\isachardoublequote}}\verb|-|\isa{{\isachardoublequote}\ \ {\isacharbar}\ \ {\isachardoublequote}}\verb|/|\isa{{\isachardoublequote}\ \ {\isacharbar}{\isachardoublequote}} \\

     & & \verb|<|\isa{{\isachardoublequote}\ \ {\isacharbar}\ \ {\isachardoublequote}}\verb|=|\isa{{\isachardoublequote}\ \ {\isacharbar}\ \ {\isachardoublequote}}\verb|>|\isa{{\isachardoublequote}\ \ {\isacharbar}\ \ {\isachardoublequote}}\verb|?|\isa{{\isachardoublequote}\ \ {\isacharbar}\ \ {\isachardoublequote}}\verb|@|\isa{{\isachardoublequote}\ \ {\isacharbar}\ \ {\isachardoublequote}}\verb|^|\isa{{\isachardoublequote}\ \ {\isacharbar}\ \ {\isachardoublequote}}\verb|_|\isa{{\isachardoublequote}\ \ {\isacharbar}\ \ {\isachardoublequote}}\verb||\verb,|,\verb||\isa{{\isachardoublequote}\ \ {\isacharbar}\ \ {\isachardoublequote}}\verb|~| \\

     \isa{greek} & = & \verb|\<alpha>|\isa{{\isachardoublequote}\ \ {\isacharbar}\ \ {\isachardoublequote}}\verb|\<beta>|\isa{{\isachardoublequote}\ \ {\isacharbar}\ \ {\isachardoublequote}}\verb|\<gamma>|\isa{{\isachardoublequote}\ \ {\isacharbar}\ \ {\isachardoublequote}}\verb|\<delta>|\isa{{\isachardoublequote}\ \ {\isacharbar}{\isachardoublequote}} \\

           &   & \verb|\<epsilon>|\isa{{\isachardoublequote}\ \ {\isacharbar}\ \ {\isachardoublequote}}\verb|\<zeta>|\isa{{\isachardoublequote}\ \ {\isacharbar}\ \ {\isachardoublequote}}\verb|\<eta>|\isa{{\isachardoublequote}\ \ {\isacharbar}\ \ {\isachardoublequote}}\verb|\<theta>|\isa{{\isachardoublequote}\ \ {\isacharbar}{\isachardoublequote}} \\

           &   & \verb|\<iota>|\isa{{\isachardoublequote}\ \ {\isacharbar}\ \ {\isachardoublequote}}\verb|\<kappa>|\isa{{\isachardoublequote}\ \ {\isacharbar}\ \ {\isachardoublequote}}\verb|\<mu>|\isa{{\isachardoublequote}\ \ {\isacharbar}\ \ {\isachardoublequote}}\verb|\<nu>|\isa{{\isachardoublequote}\ \ {\isacharbar}{\isachardoublequote}} \\

           &   & \verb|\<xi>|\isa{{\isachardoublequote}\ \ {\isacharbar}\ \ {\isachardoublequote}}\verb|\<pi>|\isa{{\isachardoublequote}\ \ {\isacharbar}\ \ {\isachardoublequote}}\verb|\<rho>|\isa{{\isachardoublequote}\ \ {\isacharbar}\ \ {\isachardoublequote}}\verb|\<sigma>|\isa{{\isachardoublequote}\ \ {\isacharbar}\ \ {\isachardoublequote}}\verb|\<tau>|\isa{{\isachardoublequote}\ \ {\isacharbar}{\isachardoublequote}} \\

           &   & \verb|\<upsilon>|\isa{{\isachardoublequote}\ \ {\isacharbar}\ \ {\isachardoublequote}}\verb|\<phi>|\isa{{\isachardoublequote}\ \ {\isacharbar}\ \ {\isachardoublequote}}\verb|\<chi>|\isa{{\isachardoublequote}\ \ {\isacharbar}\ \ {\isachardoublequote}}\verb|\<psi>|\isa{{\isachardoublequote}\ \ {\isacharbar}{\isachardoublequote}} \\

           &   & \verb|\<omega>|\isa{{\isachardoublequote}\ \ {\isacharbar}\ \ {\isachardoublequote}}\verb|\<Gamma>|\isa{{\isachardoublequote}\ \ {\isacharbar}\ \ {\isachardoublequote}}\verb|\<Delta>|\isa{{\isachardoublequote}\ \ {\isacharbar}\ \ {\isachardoublequote}}\verb|\<Theta>|\isa{{\isachardoublequote}\ \ {\isacharbar}{\isachardoublequote}} \\

           &   & \verb|\<Lambda>|\isa{{\isachardoublequote}\ \ {\isacharbar}\ \ {\isachardoublequote}}\verb|\<Xi>|\isa{{\isachardoublequote}\ \ {\isacharbar}\ \ {\isachardoublequote}}\verb|\<Pi>|\isa{{\isachardoublequote}\ \ {\isacharbar}\ \ {\isachardoublequote}}\verb|\<Sigma>|\isa{{\isachardoublequote}\ \ {\isacharbar}{\isachardoublequote}} \\

           &   & \verb|\<Upsilon>|\isa{{\isachardoublequote}\ \ {\isacharbar}\ \ {\isachardoublequote}}\verb|\<Phi>|\isa{{\isachardoublequote}\ \ {\isacharbar}\ \ {\isachardoublequote}}\verb|\<Psi>|\isa{{\isachardoublequote}\ \ {\isacharbar}\ \ {\isachardoublequote}}\verb|\<Omega>| \\

   \end{supertabular}

   \end{center}

   A \indexref{}{syntax}{var}\hyperlink{syntax.var}{\mbox{\isa{var}}} or \indexref{}{syntax}{typevar}\hyperlink{syntax.typevar}{\mbox{\isa{typevar}}} describes an unknown,

   which is internally a pair of base name and index (ML type \verb|indexname|).  These components are either separated by a dot as in

   \isa{{\isachardoublequote}{\isacharquery}x{\isachardot}{\isadigit{1}}{\isachardoublequote}} or \isa{{\isachardoublequote}{\isacharquery}x{\isadigit{7}}{\isachardot}{\isadigit{3}}{\isachardoublequote}} or run together as in \isa{{\isachardoublequote}{\isacharquery}x{\isadigit{1}}{\isachardoublequote}}.  The latter form is possible if the base name does not end

   with digits.  If the index is 0, it may be dropped altogether:

   \isa{{\isachardoublequote}{\isacharquery}x{\isachardoublequote}} and \isa{{\isachardoublequote}{\isacharquery}x{\isadigit{0}}{\isachardoublequote}} and \isa{{\isachardoublequote}{\isacharquery}x{\isachardot}{\isadigit{0}}{\isachardoublequote}} all refer to the

   same unknown, with basename \isa{{\isachardoublequote}x{\isachardoublequote}} and index 0.

   The syntax of \indexref{}{syntax}{string}\hyperlink{syntax.string}{\mbox{\isa{string}}} admits any characters, including

   newlines; ``\verb|"|'' (double-quote) and ``\verb|\|'' (backslash) need to be escaped by a backslash; arbitrary

   character codes may be specified as ``\verb|\|\isa{ddd}'',

   with three decimal digits.  Alternative strings according to

   \indexref{}{syntax}{altstring}\hyperlink{syntax.altstring}{\mbox{\isa{altstring}}} are analogous, using single back-quotes

   instead.

   The body of \indexref{}{syntax}{verbatim}\hyperlink{syntax.verbatim}{\mbox{\isa{verbatim}}} may consist of any text not

   containing ``\verb|*|\verb|}|''; this allows

   convenient inclusion of quotes without further escapes.  There is no

   way to escape ``\verb|*|\verb|}|''.  If the quoted

   text is {\LaTeX} source, one may usually add some blank or comment

   to avoid the critical character sequence.

   Source comments take the form \verb|(*|~\isa{{\isachardoublequote}{\isasymdots}{\isachardoublequote}}~\verb|*)| and may be nested, although the user-interface

   might prevent this.  Note that this form indicates source comments

   only, which are stripped after lexical analysis of the input.  The

   Isar syntax also provides proper \emph{document comments} that are

   considered as part of the text (see \secref{sec:comments}).

   Common mathematical symbols such as \isa{{\isasymforall}} are represented in

   Isabelle as \verb|\<forall>|.  There are infinitely many Isabelle

   symbols like this, although proper presentation is left to front-end

   tools such as {\LaTeX} or Proof~General with the X-Symbol package.

   A list of standard Isabelle symbols that work well with these tools

   is given in \appref{app:symbols}.  Note that \verb|\<lambda>| does

   not belong to the \isa{letter} category, since it is already used

   differently in the Pure term language.%

 \end{isamarkuptext}%

 \isamarkuptrue%

 %

 \isamarkupsection{Common syntax entities%

 }

 \isamarkuptrue%

 %

 \begin{isamarkuptext}%

 We now introduce several basic syntactic entities, such as names,

   terms, and theorem specifications, which are factored out of the

   actual Isar language elements to be described later.%

 \end{isamarkuptext}%

 \isamarkuptrue%

 %

 \isamarkupsubsection{Names%

 }

 \isamarkuptrue%

 %

 \begin{isamarkuptext}%

 Entity \railqtok{name} usually refers to any name of types,

   constants, theorems etc.\ that are to be \emph{declared} or

   \emph{defined} (so qualified identifiers are excluded here).  Quoted

   strings provide an escape for non-identifier names or those ruled

   out by outer syntax keywords (e.g.\ quoted \verb|"let"|).

   Already existing objects are usually referenced by

   \railqtok{nameref}.

   \indexoutertoken{name}\indexoutertoken{parname}\indexoutertoken{nameref}

   \indexoutertoken{int}

   \begin{rail}

     name: ident | symident | string | nat

     ;

     parname: '(' name ')'

     ;

     nameref: name | longident

     ;

     int: nat | '-' nat

     ;

   \end{rail}%

 \end{isamarkuptext}%

 \isamarkuptrue%

 %

 \isamarkupsubsection{Comments \label{sec:comments}%

 }

 \isamarkuptrue%

 %

 \begin{isamarkuptext}%

 Large chunks of plain \railqtok{text} are usually given

   \railtok{verbatim}, i.e.\ enclosed in \verb|{|\verb|*|~\isa{{\isachardoublequote}{\isasymdots}{\isachardoublequote}}~\verb|*|\verb|}|.  For convenience,

   any of the smaller text units conforming to \railqtok{nameref} are

   admitted as well.  A marginal \railnonterm{comment} is of the form

   \verb|--| \railqtok{text}.  Any number of these may occur

   within Isabelle/Isar commands.

   \indexoutertoken{text}\indexouternonterm{comment}

   \begin{rail}

     text: verbatim | nameref

     ;

     comment: '--' text

     ;

   \end{rail}%

 \end{isamarkuptext}%

 \isamarkuptrue%

 %

 \isamarkupsubsection{Type classes, sorts and arities%

 }

 \isamarkuptrue%

 %

 \begin{isamarkuptext}%

 Classes are specified by plain names.  Sorts have a very simple

   inner syntax, which is either a single class name \isa{c} or a

   list \isa{{\isachardoublequote}{\isacharbraceleft}c\isactrlsub {\isadigit{1}}{\isacharcomma}\ {\isasymdots}{\isacharcomma}\ c\isactrlsub n{\isacharbraceright}{\isachardoublequote}} referring to the

   intersection of these classes.  The syntax of type arities is given

   directly at the outer level.

   \indexouternonterm{sort}\indexouternonterm{arity}

   \indexouternonterm{classdecl}

   \begin{rail}

     classdecl: name (('<' | subseteq) (nameref + ','))?

     ;

     sort: nameref

     ;

     arity: ('(' (sort + ',') ')')? sort

     ;

   \end{rail}%

 \end{isamarkuptext}%

 \isamarkuptrue%

 %

 \isamarkupsubsection{Types and terms \label{sec:types-terms}%

 }

 \isamarkuptrue%

 %

 \begin{isamarkuptext}%

 The actual inner Isabelle syntax, that of types and terms of the

   logic, is far too sophisticated in order to be modelled explicitly

   at the outer theory level.  Basically, any such entity has to be

   quoted to turn it into a single token (the parsing and type-checking

   is performed internally later).  For convenience, a slightly more

   liberal convention is adopted: quotes may be omitted for any type or

   term that is already atomic at the outer level.  For example, one

   may just write \verb|x| instead of quoted \verb|"x"|.

   Note that symbolic identifiers (e.g.\ \verb|++| or \isa{{\isachardoublequote}{\isasymforall}{\isachardoublequote}} are available as well, provided these have not been superseded

   by commands or other keywords already (such as \verb|=| or

   \verb|+|).

   \indexoutertoken{type}\indexoutertoken{term}\indexoutertoken{prop}

   \begin{rail}

     type: nameref | typefree | typevar

     ;

     term: nameref | var

     ;

     prop: term

     ;

   \end{rail}

   Positional instantiations are indicated by giving a sequence of

   terms, or the placeholder ``\isa{{\isacharunderscore}}'' (underscore), which means to

   skip a position.

   \indexoutertoken{inst}\indexoutertoken{insts}

   \begin{rail}

     inst: underscore | term

     ;

     insts: (inst *)

     ;

   \end{rail}

   Type declarations and definitions usually refer to

   \railnonterm{typespec} on the left-hand side.  This models basic

   type constructor application at the outer syntax level.  Note that

   only plain postfix notation is available here, but no infixes.

   \indexouternonterm{typespec}

   \begin{rail}

     typespec: (() | typefree | '(' ( typefree + ',' ) ')') name

     ;

   \end{rail}%

 \end{isamarkuptext}%

 \isamarkuptrue%

 %

 \isamarkupsubsection{Term patterns and declarations \label{sec:term-decls}%

 }

 \isamarkuptrue%

 %

 \begin{isamarkuptext}%

 Wherever explicit propositions (or term fragments) occur in a proof

   text, casual binding of schematic term variables may be given

   specified via patterns of the form ``\isa{{\isachardoublequote}{\isacharparenleft}{\isasymIS}\ p\isactrlsub {\isadigit{1}}\ {\isasymdots}\ p\isactrlsub n{\isacharparenright}{\isachardoublequote}}''.  This works both for \railqtok{term} and \railqtok{prop}.

   \indexouternonterm{termpat}\indexouternonterm{proppat}

   \begin{rail}

     termpat: '(' ('is' term +) ')'

     ;

     proppat: '(' ('is' prop +) ')'

     ;

   \end{rail}

   \medskip Declarations of local variables \isa{{\isachardoublequote}x\ {\isacharcolon}{\isacharcolon}\ {\isasymtau}{\isachardoublequote}} and

   logical propositions \isa{{\isachardoublequote}a\ {\isacharcolon}\ {\isasymphi}{\isachardoublequote}} represent different views on

   the same principle of introducing a local scope.  In practice, one

   may usually omit the typing of \railnonterm{vars} (due to

   type-inference), and the naming of propositions (due to implicit

   references of current facts).  In any case, Isar proof elements

   usually admit to introduce multiple such items simultaneously.

   \indexouternonterm{vars}\indexouternonterm{props}

   \begin{rail}

     vars: (name+) ('::' type)?

     ;

     props: thmdecl? (prop proppat? +)

     ;

   \end{rail}

   The treatment of multiple declarations corresponds to the

   complementary focus of \railnonterm{vars} versus

   \railnonterm{props}.  In ``\isa{{\isachardoublequote}x\isactrlsub {\isadigit{1}}\ {\isasymdots}\ x\isactrlsub n\ {\isacharcolon}{\isacharcolon}\ {\isasymtau}{\isachardoublequote}}''

   the typing refers to all variables, while in \isa{{\isachardoublequote}a{\isacharcolon}\ {\isasymphi}\isactrlsub {\isadigit{1}}\ {\isasymdots}\ {\isasymphi}\isactrlsub n{\isachardoublequote}} the naming refers to all propositions collectively.

   Isar language elements that refer to \railnonterm{vars} or

   \railnonterm{props} typically admit separate typings or namings via

   another level of iteration, with explicit \indexref{}{keyword}{and}\hyperlink{keyword.and}{\mbox{\isa{\isakeyword{and}}}}

   separators; e.g.\ see \hyperlink{command.fix}{\mbox{\isa{\isacommand{fix}}}} and \hyperlink{command.assume}{\mbox{\isa{\isacommand{assume}}}} in

   \secref{sec:proof-context}.%

 \end{isamarkuptext}%

 \isamarkuptrue%

 %

 \isamarkupsubsection{Attributes and theorems \label{sec:syn-att}%

 }

 \isamarkuptrue%

 %

 \begin{isamarkuptext}%

 Attributes have their own ``semi-inner'' syntax, in the sense

   that input conforming to \railnonterm{args} below is parsed by the

   attribute a second time.  The attribute argument specifications may

   be any sequence of atomic entities (identifiers, strings etc.), or

   properly bracketed argument lists.  Below \railqtok{atom} refers to

   any atomic entity, including any \railtok{keyword} conforming to

   \railtok{symident}.

   \indexoutertoken{atom}\indexouternonterm{args}\indexouternonterm{attributes}

   \begin{rail}

     atom: nameref | typefree | typevar | var | nat | keyword

     ;

     arg: atom | '(' args ')' | '[' args ']'

     ;

     args: arg *

     ;

     attributes: '[' (nameref args * ',') ']'

     ;

   \end{rail}

   Theorem specifications come in several flavors:

   \railnonterm{axmdecl} and \railnonterm{thmdecl} usually refer to

   axioms, assumptions or results of goal statements, while

   \railnonterm{thmdef} collects lists of existing theorems.  Existing

   theorems are given by \railnonterm{thmref} and

   \railnonterm{thmrefs}, the former requires an actual singleton

   result.

   There are three forms of theorem references:

   \begin{enumerate}

   \item named facts \isa{{\isachardoublequote}a{\isachardoublequote}},

   \item selections from named facts \isa{{\isachardoublequote}a{\isacharparenleft}i{\isacharparenright}{\isachardoublequote}} or \isa{{\isachardoublequote}a{\isacharparenleft}j\ {\isacharminus}\ k{\isacharparenright}{\isachardoublequote}},

   \item literal fact propositions using \indexref{}{syntax}{altstring}\hyperlink{syntax.altstring}{\mbox{\isa{altstring}}} syntax

   \verb|`|\isa{{\isachardoublequote}{\isasymphi}{\isachardoublequote}}\verb|`| (see also method

   \indexref{}{method}{fact}\hyperlink{method.fact}{\mbox{\isa{fact}}}).

   \end{enumerate}

   Any kind of theorem specification may include lists of attributes

   both on the left and right hand sides; attributes are applied to any

   immediately preceding fact.  If names are omitted, the theorems are

   not stored within the theorem database of the theory or proof

   context, but any given attributes are applied nonetheless.

   An extra pair of brackets around attributes (like ``\isa{{\isachardoublequote}{\isacharbrackleft}{\isacharbrackleft}simproc\ a{\isacharbrackright}{\isacharbrackright}{\isachardoublequote}}'') abbreviates a theorem reference involving an

   internal dummy fact, which will be ignored later on.  So only the

   effect of the attribute on the background context will persist.

   This form of in-place declarations is particularly useful with

   commands like \hyperlink{command.declare}{\mbox{\isa{\isacommand{declare}}}} and \hyperlink{command.using}{\mbox{\isa{\isacommand{using}}}}.

   \indexouternonterm{axmdecl}\indexouternonterm{thmdecl}

   \indexouternonterm{thmdef}\indexouternonterm{thmref}

   \indexouternonterm{thmrefs}\indexouternonterm{selection}

   \begin{rail}

     axmdecl: name attributes? ':'

     ;

     thmdecl: thmbind ':'

     ;

     thmdef: thmbind '='

     ;

     thmref: (nameref selection? | altstring) attributes? | '[' attributes ']'

     ;

     thmrefs: thmref +

     ;

     thmbind: name attributes | name | attributes

     ;

     selection: '(' ((nat | nat '-' nat?) + ',') ')'

     ;

   \end{rail}%

 \end{isamarkuptext}%

 \isamarkuptrue%

 %

 \isadelimtheory

 %

 \endisadelimtheory

 %

 \isatagtheory

 \isacommand{end}\isamarkupfalse%

 %

 \endisatagtheory

 {\isafoldtheory}%

 %

 \isadelimtheory

 %

 \endisadelimtheory

 \isanewline

 \end{isabellebody}%

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