%

 \begin{isabellebody}%

 \def\isabellecontext{syntax}%

 %

 \isadelimtheory

 \isanewline

 \isanewline

 %

 \endisadelimtheory

 %

 \isatagtheory

 \isacommand{theory}\isamarkupfalse%

 \ {\isachardoublequoteopen}syntax{\isachardoublequoteclose}\isanewline

 \isakeyword{imports}\ CPure\isanewline

 \isakeyword{begin}%

 \endisatagtheory

 {\isafoldtheory}%

 %

 \isadelimtheory

 %

 \endisadelimtheory

 %

 \isamarkupchapter{Syntax primitives%

 }

 \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:lex-syntax}%

 }

 \isamarkuptrue%

 %

 \begin{isamarkuptext}%

 The Isabelle/Isar outer syntax provides token classes as presented

   below; most of these coincide with the inner lexical syntax as

   presented in \cite{isabelle-ref}.

   \begin{matharray}{rcl}

     \indexdef{}{syntax}{ident}\hypertarget{syntax.ident}{\hyperlink{syntax.ident}{\mbox{\isa{ident}}}} & = & letter\,quasiletter^* \\

     \indexdef{}{syntax}{longident}\hypertarget{syntax.longident}{\hyperlink{syntax.longident}{\mbox{\isa{longident}}}} & = & ident (\verb,.,ident)^+ \\

     \indexdef{}{syntax}{symident}\hypertarget{syntax.symident}{\hyperlink{syntax.symident}{\mbox{\isa{symident}}}} & = & sym^+ ~|~ \verb,\,\verb,<,ident\verb,>, \\

     \indexdef{}{syntax}{nat}\hypertarget{syntax.nat}{\hyperlink{syntax.nat}{\mbox{\isa{nat}}}} & = & digit^+ \\

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

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

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

     \indexdef{}{syntax}{string}\hypertarget{syntax.string}{\hyperlink{syntax.string}{\mbox{\isa{string}}}} & = & \verb,", ~\dots~ \verb,", \\

     \indexdef{}{syntax}{altstring}\hypertarget{syntax.altstring}{\hyperlink{syntax.altstring}{\mbox{\isa{altstring}}}} & = & \backquote ~\dots~ \backquote \\

     \indexdef{}{syntax}{verbatim}\hypertarget{syntax.verbatim}{\hyperlink{syntax.verbatim}{\mbox{\isa{verbatim}}}} & = & \verb,{*, ~\dots~ \verb,*,\verb,}, \\[1ex]

     letter & = & latin ~|~ \verb,\,\verb,<,latin\verb,>, ~|~ \verb,\,\verb,<,latin\,latin\verb,>, ~|~ greek ~|~ \\

            &   & \verb,\<^isub>, ~|~ \verb,\<^isup>, \\

     quasiletter & = & letter ~|~ digit ~|~ \verb,_, ~|~ \verb,', \\

     latin & = & \verb,a, ~|~ \dots ~|~ \verb,z, ~|~ \verb,A, ~|~ \dots ~|~ \verb,Z, \\

     digit & = & \verb,0, ~|~ \dots ~|~ \verb,9, \\

     sym & = & \verb,!, ~|~ \verb,#, ~|~ \verb,$, ~|~ \verb,%, ~|~ \verb,&, ~|~

      \verb,*, ~|~ \verb,+, ~|~ \verb,-, ~|~ \verb,/, ~|~ \\

     & & \verb,<, ~|~ \verb,=, ~|~ \verb,>, ~|~ \verb,?, ~|~ \texttt{\at} ~|~

     \verb,^, ~|~ \verb,_, ~|~ \verb,|, ~|~ \verb,~, \\

     greek & = & \verb,\<alpha>, ~|~ \verb,\<beta>, ~|~ \verb,\<gamma>, ~|~ \verb,\<delta>, ~| \\

           &   & \verb,\<epsilon>, ~|~ \verb,\<zeta>, ~|~ \verb,\<eta>, ~|~ \verb,\<theta>, ~| \\

           &   & \verb,\<iota>, ~|~ \verb,\<kappa>, ~|~ \verb,\<mu>, ~|~ \verb,\<nu>, ~| \\

           &   & \verb,\<xi>, ~|~ \verb,\<pi>, ~|~ \verb,\<rho>, ~|~ \verb,\<sigma>, ~|~ \verb,\<tau>, ~| \\

           &   & \verb,\<upsilon>, ~|~ \verb,\<phi>, ~|~ \verb,\<chi>, ~|~ \verb,\<psi>, ~| \\

           &   & \verb,\<omega>, ~|~ \verb,\<Gamma>, ~|~ \verb,\<Delta>, ~|~ \verb,\<Theta>, ~| \\

           &   & \verb,\<Lambda>, ~|~ \verb,\<Xi>, ~|~ \verb,\<Pi>, ~|~ \verb,\<Sigma>, ~| \\

           &   & \verb,\<Upsilon>, ~|~ \verb,\<Phi>, ~|~ \verb,\<Psi>, ~|~ \verb,\<Omega>, \\

   \end{matharray}

   The syntax of \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

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

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

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

   convenient inclusion of quotes without further escapes.  The greek

   letters do \emph{not} include \verb|\<lambda>|, which is already used

   differently in the meta-logic.

   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 \cite[appendix~A]{isabelle-sys}.

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

   tools might prevent this.  Note that this form indicates source

   comments only, which are stripped after lexical analysis of the

   input.  The Isar document syntax also provides formal comments that

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

 \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{Mixfix annotations%

 }

 \isamarkuptrue%

 %

 \begin{isamarkuptext}%

 Mixfix annotations specify concrete \emph{inner} syntax of Isabelle

   types and terms.  Some commands such as \hyperlink{command.types}{\mbox{\isa{\isacommand{types}}}} (see

   \secref{sec:types-pure}) admit infixes only, while \hyperlink{command.consts}{\mbox{\isa{\isacommand{consts}}}} (see \secref{sec:consts}) and \hyperlink{command.syntax}{\mbox{\isa{\isacommand{syntax}}}} (see

   \secref{sec:syn-trans}) support the full range of general mixfixes

   and binders.

   \indexouternonterm{infix}\indexouternonterm{mixfix}\indexouternonterm{structmixfix}

   \begin{rail}

     infix: '(' ('infix' | 'infixl' | 'infixr') string? nat ')'

     ;

     mixfix: infix | '(' string prios? nat? ')' | '(' 'binder' string prios? nat ')'

     ;

     structmixfix: mixfix | '(' 'structure' ')'

     ;

     prios: '[' (nat + ',') ']'

     ;

   \end{rail}

   Here the \railtok{string} specifications refer to the actual mixfix

   template (see also \cite{isabelle-ref}), which may include literal

   text, spacing, blocks, and arguments (denoted by ``\isa{{\isacharunderscore}}''); the

   special symbol ``\verb|\<index>|'' (printed as ``\isa{{\isachardoublequote}{\isasymindex}{\isachardoublequote}}'')

   represents an index argument that specifies an implicit structure

   reference (see also \secref{sec:locale}).  Infix and binder

   declarations provide common abbreviations for particular mixfix

   declarations.  So in practice, mixfix templates mostly degenerate to

   literal text for concrete syntax, such as ``\verb|++|'' for

   an infix symbol, or ``\verb|++|\isa{{\isachardoublequote}{\isasymindex}{\isachardoublequote}}'' for an infix of

   an implicit structure.%

 \end{isamarkuptext}%

 \isamarkuptrue%

 %

 \isamarkupsubsection{Proof methods \label{sec:syn-meth}%

 }

 \isamarkuptrue%

 %

 \begin{isamarkuptext}%

 Proof methods are either basic ones, or expressions composed of

   methods via ``\verb|,|'' (sequential composition),

   ``\verb||\verb,|,\verb||'' (alternative choices), ``\verb|?|'' 

   (try), ``\verb|+|'' (repeat at least once), ``\verb|[|\isa{n}\verb|]|'' (restriction to first \isa{n}

   sub-goals, with default \isa{{\isachardoublequote}n\ {\isacharequal}\ {\isadigit{1}}{\isachardoublequote}}).  In practice, proof

   methods are usually just a comma separated list of

   \railqtok{nameref}~\railnonterm{args} specifications.  Note that

   parentheses may be dropped for single method specifications (with no

   arguments).

   \indexouternonterm{method}

   \begin{rail}

     method: (nameref | '(' methods ')') (() | '?' | '+' | '[' nat? ']')

     ;

     methods: (nameref args | method) + (',' | '|')

     ;

   \end{rail}

   Proper Isar proof methods do \emph{not} admit arbitrary goal

   addressing, but refer either to the first sub-goal or all sub-goals

   uniformly.  The goal restriction operator ``\isa{{\isachardoublequote}{\isacharbrackleft}n{\isacharbrackright}{\isachardoublequote}}''

   evaluates a method expression within a sandbox consisting of the

   first \isa{n} sub-goals (which need to exist).  For example, the

   method ``\isa{{\isachardoublequote}simp{\isacharunderscore}all{\isacharbrackleft}{\isadigit{3}}{\isacharbrackright}{\isachardoublequote}}'' simplifies the first three

   sub-goals, while ``\isa{{\isachardoublequote}{\isacharparenleft}rule\ foo{\isacharcomma}\ simp{\isacharunderscore}all{\isacharparenright}{\isacharbrackleft}{\isacharbrackright}{\isachardoublequote}}'' simplifies all

   new goals that emerge from applying rule \isa{{\isachardoublequote}foo{\isachardoublequote}} to the

   originally first one.

   Improper methods, notably tactic emulations, offer a separate

   low-level goal addressing scheme as explicit argument to the

   individual tactic being involved.  Here ``\isa{{\isachardoublequote}{\isacharbrackleft}{\isacharbang}{\isacharbrackright}{\isachardoublequote}}'' refers to

   all goals, and ``\isa{{\isachardoublequote}{\isacharbrackleft}n{\isacharminus}{\isacharbrackright}{\isachardoublequote}}'' to all goals starting from \isa{{\isachardoublequote}n{\isachardoublequote}}.

   \indexouternonterm{goalspec}

   \begin{rail}

     goalspec: '[' (nat '-' nat | nat '-' | nat | '!' ) ']'

     ;

   \end{rail}%

 \end{isamarkuptext}%

 \isamarkuptrue%

 %

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

 }

 \isamarkuptrue%

 %

 \begin{isamarkuptext}%

 Attributes (and proof methods, see \secref{sec:syn-meth}) 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}}} in \secref{sec:pure-meth-att}).

   \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%

 %

 \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{Antiquotations \label{sec:antiq}%

 }

 \isamarkuptrue%

 %

 \begin{isamarkuptext}%

 \begin{matharray}{rcl}

     \indexdef{}{antiquotation}{theory}\hypertarget{antiquotation.theory}{\hyperlink{antiquotation.theory}{\mbox{\isa{theory}}}} & : & \isarantiq \\

     \indexdef{}{antiquotation}{thm}\hypertarget{antiquotation.thm}{\hyperlink{antiquotation.thm}{\mbox{\isa{thm}}}} & : & \isarantiq \\

     \indexdef{}{antiquotation}{prop}\hypertarget{antiquotation.prop}{\hyperlink{antiquotation.prop}{\mbox{\isa{prop}}}} & : & \isarantiq \\

     \indexdef{}{antiquotation}{term}\hypertarget{antiquotation.term}{\hyperlink{antiquotation.term}{\mbox{\isa{term}}}} & : & \isarantiq \\

     \indexdef{}{antiquotation}{const}\hypertarget{antiquotation.const}{\hyperlink{antiquotation.const}{\mbox{\isa{const}}}} & : & \isarantiq \\

     \indexdef{}{antiquotation}{abbrev}\hypertarget{antiquotation.abbrev}{\hyperlink{antiquotation.abbrev}{\mbox{\isa{abbrev}}}} & : & \isarantiq \\

     \indexdef{}{antiquotation}{typeof}\hypertarget{antiquotation.typeof}{\hyperlink{antiquotation.typeof}{\mbox{\isa{typeof}}}} & : & \isarantiq \\

     \indexdef{}{antiquotation}{typ}\hypertarget{antiquotation.typ}{\hyperlink{antiquotation.typ}{\mbox{\isa{typ}}}} & : & \isarantiq \\

     \indexdef{}{antiquotation}{thm\_style}\hypertarget{antiquotation.thm_style}{\hyperlink{antiquotation.thm_style}{\mbox{\isa{thm{\isacharunderscore}style}}}} & : & \isarantiq \\

     \indexdef{}{antiquotation}{term\_style}\hypertarget{antiquotation.term_style}{\hyperlink{antiquotation.term_style}{\mbox{\isa{term{\isacharunderscore}style}}}} & : & \isarantiq \\

     \indexdef{}{antiquotation}{text}\hypertarget{antiquotation.text}{\hyperlink{antiquotation.text}{\mbox{\isa{text}}}} & : & \isarantiq \\

     \indexdef{}{antiquotation}{goals}\hypertarget{antiquotation.goals}{\hyperlink{antiquotation.goals}{\mbox{\isa{goals}}}} & : & \isarantiq \\

     \indexdef{}{antiquotation}{subgoals}\hypertarget{antiquotation.subgoals}{\hyperlink{antiquotation.subgoals}{\mbox{\isa{subgoals}}}} & : & \isarantiq \\

     \indexdef{}{antiquotation}{prf}\hypertarget{antiquotation.prf}{\hyperlink{antiquotation.prf}{\mbox{\isa{prf}}}} & : & \isarantiq \\

     \indexdef{}{antiquotation}{full\_prf}\hypertarget{antiquotation.full_prf}{\hyperlink{antiquotation.full_prf}{\mbox{\isa{full{\isacharunderscore}prf}}}} & : & \isarantiq \\

     \indexdef{}{antiquotation}{ML}\hypertarget{antiquotation.ML}{\hyperlink{antiquotation.ML}{\mbox{\isa{ML}}}} & : & \isarantiq \\

     \indexdef{}{antiquotation}{ML\_type}\hypertarget{antiquotation.ML_type}{\hyperlink{antiquotation.ML_type}{\mbox{\isa{ML{\isacharunderscore}type}}}} & : & \isarantiq \\

     \indexdef{}{antiquotation}{ML\_struct}\hypertarget{antiquotation.ML_struct}{\hyperlink{antiquotation.ML_struct}{\mbox{\isa{ML{\isacharunderscore}struct}}}} & : & \isarantiq \\

   \end{matharray}

   The text body of formal comments (see also \secref{sec:comments})

   may contain antiquotations of logical entities, such as theorems,

   terms and types, which are to be presented in the final output

   produced by the Isabelle document preparation system (see also

   \secref{sec:document-prep}).

   Thus embedding of ``\isa{{\isachardoublequote}{\isacharat}{\isacharbraceleft}term\ {\isacharbrackleft}show{\isacharunderscore}types{\isacharbrackright}\ {\isachardoublequote}f\ x\ {\isacharequal}\ a\ {\isacharplus}\ x{\isachardoublequote}{\isacharbraceright}{\isachardoublequote}}''

   within a text block would cause

   \isa{{\isacharparenleft}f{\isasymColon}{\isacharprime}a\ {\isasymRightarrow}\ {\isacharprime}a{\isacharparenright}\ {\isacharparenleft}x{\isasymColon}{\isacharprime}a{\isacharparenright}\ {\isacharequal}\ {\isacharparenleft}a{\isasymColon}{\isacharprime}a{\isacharparenright}\ {\isacharplus}\ x} to appear in the final {\LaTeX} document.  Also note that theorem

   antiquotations may involve attributes as well.  For example,

   \isa{{\isachardoublequote}{\isacharat}{\isacharbraceleft}thm\ sym\ {\isacharbrackleft}no{\isacharunderscore}vars{\isacharbrackright}{\isacharbraceright}{\isachardoublequote}} would print the theorem's

   statement where all schematic variables have been replaced by fixed

   ones, which are easier to read.

   \begin{rail}

     atsign lbrace antiquotation rbrace

     ;

     antiquotation:

       'theory' options name |

       'thm' options thmrefs |

       'prop' options prop |

       'term' options term |

       'const' options term |

       'abbrev' options term |

       'typeof' options term |

       'typ' options type |

       'thm\_style' options name thmref |

       'term\_style' options name term |

       'text' options name |

       'goals' options |

       'subgoals' options |

       'prf' options thmrefs |

       'full\_prf' options thmrefs |

       'ML' options name |

       'ML\_type' options name |

       'ML\_struct' options name

     ;

     options: '[' (option * ',') ']'

     ;

     option: name | name '=' name

     ;

   \end{rail}

   Note that the syntax of antiquotations may \emph{not} include source

   comments \verb|(*|~\isa{{\isachardoublequote}{\isasymdots}{\isachardoublequote}}~\verb|*)| or verbatim

   text \verb|{|\verb|*|~\isa{{\isachardoublequote}{\isasymdots}{\isachardoublequote}}~\verb|*|\verb|}|.

   \begin{descr}

   \item [\isa{{\isachardoublequote}{\isacharat}{\isacharbraceleft}theory\ A{\isacharbraceright}{\isachardoublequote}}] prints the name \isa{{\isachardoublequote}A{\isachardoublequote}}, which is

   guaranteed to refer to a valid ancestor theory in the current

   context.

   \item [\isa{{\isachardoublequote}{\isacharat}{\isacharbraceleft}thm\ a\isactrlsub {\isadigit{1}}\ {\isasymdots}\ a\isactrlsub n{\isacharbraceright}{\isachardoublequote}}] prints theorems

   \isa{{\isachardoublequote}a\isactrlsub {\isadigit{1}}\ {\isasymdots}\ a\isactrlsub n{\isachardoublequote}}.  Note that attribute specifications

   may be included as well (see also \secref{sec:syn-att}); the

   \indexref{}{attribute}{no\_vars}\hyperlink{attribute.no_vars}{\mbox{\isa{no{\isacharunderscore}vars}}} rule (see \secref{sec:misc-meth-att}) would

   be particularly useful to suppress printing of schematic variables.

   \item [\isa{{\isachardoublequote}{\isacharat}{\isacharbraceleft}prop\ {\isasymphi}{\isacharbraceright}{\isachardoublequote}}] prints a well-typed proposition \isa{{\isachardoublequote}{\isasymphi}{\isachardoublequote}}.

   \item [\isa{{\isachardoublequote}{\isacharat}{\isacharbraceleft}term\ t{\isacharbraceright}{\isachardoublequote}}] prints a well-typed term \isa{{\isachardoublequote}t{\isachardoublequote}}.

   \item [\isa{{\isachardoublequote}{\isacharat}{\isacharbraceleft}const\ c{\isacharbraceright}{\isachardoublequote}}] prints a logical or syntactic constant

   \isa{{\isachardoublequote}c{\isachardoublequote}}.

   \item [\isa{{\isachardoublequote}{\isacharat}{\isacharbraceleft}abbrev\ c\ x\isactrlsub {\isadigit{1}}\ {\isasymdots}\ x\isactrlsub n{\isacharbraceright}{\isachardoublequote}}] prints a constant

   abbreviation \isa{{\isachardoublequote}c\ x\isactrlsub {\isadigit{1}}\ {\isasymdots}\ x\isactrlsub n\ {\isasymequiv}\ rhs{\isachardoublequote}} as defined in

   the current context.

   \item [\isa{{\isachardoublequote}{\isacharat}{\isacharbraceleft}typeof\ t{\isacharbraceright}{\isachardoublequote}}] prints the type of a well-typed term

   \isa{{\isachardoublequote}t{\isachardoublequote}}.

   \item [\isa{{\isachardoublequote}{\isacharat}{\isacharbraceleft}typ\ {\isasymtau}{\isacharbraceright}{\isachardoublequote}}] prints a well-formed type \isa{{\isachardoublequote}{\isasymtau}{\isachardoublequote}}.

   \item [\isa{{\isachardoublequote}{\isacharat}{\isacharbraceleft}thm{\isacharunderscore}style\ s\ a{\isacharbraceright}{\isachardoublequote}}] prints theorem \isa{a},

   previously applying a style \isa{s} to it (see below).

   \item [\isa{{\isachardoublequote}{\isacharat}{\isacharbraceleft}term{\isacharunderscore}style\ s\ t{\isacharbraceright}{\isachardoublequote}}] prints a well-typed term \isa{t} after applying a style \isa{s} to it (see below).

   \item [\isa{{\isachardoublequote}{\isacharat}{\isacharbraceleft}text\ s{\isacharbraceright}{\isachardoublequote}}] prints uninterpreted source text \isa{s}.  This is particularly useful to print portions of text according

   to the Isabelle {\LaTeX} output style, without demanding

   well-formedness (e.g.\ small pieces of terms that should not be

   parsed or type-checked yet).

   \item [\isa{{\isachardoublequote}{\isacharat}{\isacharbraceleft}goals{\isacharbraceright}{\isachardoublequote}}] prints the current \emph{dynamic} goal

   state.  This is mainly for support of tactic-emulation scripts

   within Isar --- presentation of goal states does not conform to

   actual human-readable proof documents.

   Please do not include goal states into document output unless you

   really know what you are doing!

   \item [\isa{{\isachardoublequote}{\isacharat}{\isacharbraceleft}subgoals{\isacharbraceright}{\isachardoublequote}}] is similar to \isa{{\isachardoublequote}{\isacharat}{\isacharbraceleft}goals{\isacharbraceright}{\isachardoublequote}}, but

   does not print the main goal.

   \item [\isa{{\isachardoublequote}{\isacharat}{\isacharbraceleft}prf\ a\isactrlsub {\isadigit{1}}\ {\isasymdots}\ a\isactrlsub n{\isacharbraceright}{\isachardoublequote}}] prints the (compact)

   proof terms corresponding to the theorems \isa{{\isachardoublequote}a\isactrlsub {\isadigit{1}}\ {\isasymdots}\ a\isactrlsub n{\isachardoublequote}}. Note that this requires proof terms to be switched on

   for the current object logic (see the ``Proof terms'' section of the

   Isabelle reference manual for information on how to do this).

   \item [\isa{{\isachardoublequote}{\isacharat}{\isacharbraceleft}full{\isacharunderscore}prf\ a\isactrlsub {\isadigit{1}}\ {\isasymdots}\ a\isactrlsub n{\isacharbraceright}{\isachardoublequote}}] is like \isa{{\isachardoublequote}{\isacharat}{\isacharbraceleft}prf\ a\isactrlsub {\isadigit{1}}\ {\isasymdots}\ a\isactrlsub n{\isacharbraceright}{\isachardoublequote}}, but displays the full proof terms,

   i.e.\ also displays information omitted in the compact proof term,

   which is denoted by ``\isa{{\isacharunderscore}}'' placeholders there.

   \item [\isa{{\isachardoublequote}{\isacharat}{\isacharbraceleft}ML\ s{\isacharbraceright}{\isachardoublequote}}, \isa{{\isachardoublequote}{\isacharat}{\isacharbraceleft}ML{\isacharunderscore}type\ s{\isacharbraceright}{\isachardoublequote}}, and \isa{{\isachardoublequote}{\isacharat}{\isacharbraceleft}ML{\isacharunderscore}struct\ s{\isacharbraceright}{\isachardoublequote}}] check text \isa{s} as ML value, type, and

   structure, respectively.  The source is displayed verbatim.

   \end{descr}

   \medskip The following standard styles for use with \isa{thm{\isacharunderscore}style} and \isa{term{\isacharunderscore}style} are available:

   \begin{descr}

   \item [\isa{lhs}] extracts the first argument of any application

   form with at least two arguments -- typically meta-level or

   object-level equality, or any other binary relation.

   \item [\isa{rhs}] is like \isa{lhs}, but extracts the second

   argument.

   \item [\isa{{\isachardoublequote}concl{\isachardoublequote}}] extracts the conclusion \isa{C} from a rule

   in Horn-clause normal form \isa{{\isachardoublequote}A\isactrlsub {\isadigit{1}}\ {\isasymLongrightarrow}\ {\isasymdots}\ A\isactrlsub n\ {\isasymLongrightarrow}\ C{\isachardoublequote}}.

   \item [\isa{{\isachardoublequote}prem{\isadigit{1}}{\isachardoublequote}}, \dots, \isa{{\isachardoublequote}prem{\isadigit{9}}{\isachardoublequote}}] extract premise

   number \isa{{\isachardoublequote}{\isadigit{1}}{\isacharcomma}\ {\isasymdots}{\isacharcomma}\ {\isadigit{9}}{\isachardoublequote}}, respectively, from from a rule in

   Horn-clause normal form \isa{{\isachardoublequote}A\isactrlsub {\isadigit{1}}\ {\isasymLongrightarrow}\ {\isasymdots}\ A\isactrlsub n\ {\isasymLongrightarrow}\ C{\isachardoublequote}}

   \end{descr}

   \medskip

   The following options are available to tune the output.  Note that most of

   these coincide with ML flags of the same names (see also \cite{isabelle-ref}).

   \begin{descr}

   \item[\isa{{\isachardoublequote}show{\isacharunderscore}types\ {\isacharequal}\ bool{\isachardoublequote}} and \isa{{\isachardoublequote}show{\isacharunderscore}sorts\ {\isacharequal}\ bool{\isachardoublequote}}]

   control printing of explicit type and sort constraints.

   \item[\isa{{\isachardoublequote}show{\isacharunderscore}structs\ {\isacharequal}\ bool{\isachardoublequote}}] controls printing of implicit

   structures.

   \item[\isa{{\isachardoublequote}long{\isacharunderscore}names\ {\isacharequal}\ bool{\isachardoublequote}}] forces names of types and

   constants etc.\ to be printed in their fully qualified internal

   form.

   \item[\isa{{\isachardoublequote}short{\isacharunderscore}names\ {\isacharequal}\ bool{\isachardoublequote}}] forces names of types and

   constants etc.\ to be printed unqualified.  Note that internalizing

   the output again in the current context may well yield a different

   result.

   \item[\isa{{\isachardoublequote}unique{\isacharunderscore}names\ {\isacharequal}\ bool{\isachardoublequote}}] determines whether the printed

   version of qualified names should be made sufficiently long to avoid

   overlap with names declared further back.  Set to \isa{false} for

   more concise output.

   \item[\isa{{\isachardoublequote}eta{\isacharunderscore}contract\ {\isacharequal}\ bool{\isachardoublequote}}] prints terms in \isa{{\isasymeta}}-contracted form.

   \item[\isa{{\isachardoublequote}display\ {\isacharequal}\ bool{\isachardoublequote}}] indicates if the text is to be

   output as multi-line ``display material'', rather than a small piece

   of text without line breaks (which is the default).

   \item[\isa{{\isachardoublequote}break\ {\isacharequal}\ bool{\isachardoublequote}}] controls line breaks in non-display

   material.

   \item[\isa{{\isachardoublequote}quotes\ {\isacharequal}\ bool{\isachardoublequote}}] indicates if the output should be

   enclosed in double quotes.

   \item[\isa{{\isachardoublequote}mode\ {\isacharequal}\ name{\isachardoublequote}}] adds \isa{name} to the print mode to

   be used for presentation (see also \cite{isabelle-ref}).  Note that

   the standard setup for {\LaTeX} output is already present by

   default, including the modes \isa{latex} and \isa{xsymbols}.

   \item[\isa{{\isachardoublequote}margin\ {\isacharequal}\ nat{\isachardoublequote}} and \isa{{\isachardoublequote}indent\ {\isacharequal}\ nat{\isachardoublequote}}] change the

   margin or indentation for pretty printing of display material.

   \item[\isa{{\isachardoublequote}source\ {\isacharequal}\ bool{\isachardoublequote}}] prints the source text of the

   antiquotation arguments, rather than the actual value.  Note that

   this does not affect well-formedness checks of \hyperlink{antiquotation.thm}{\mbox{\isa{thm}}}, \hyperlink{antiquotation.term}{\mbox{\isa{term}}}, etc. (only the \hyperlink{antiquotation.text}{\mbox{\isa{text}}} antiquotation admits arbitrary output).

   \item[\isa{{\isachardoublequote}goals{\isacharunderscore}limit\ {\isacharequal}\ nat{\isachardoublequote}}] determines the maximum number of

   goals to be printed.

   \item[\isa{{\isachardoublequote}locale\ {\isacharequal}\ name{\isachardoublequote}}] specifies an alternative locale

   context used for evaluating and printing the subsequent argument.

   \end{descr}

   For boolean flags, ``\isa{{\isachardoublequote}name\ {\isacharequal}\ true{\isachardoublequote}}'' may be abbreviated as

   ``\isa{name}''.  All of the above flags are disabled by default,

   unless changed from ML.

   \medskip Note that antiquotations do not only spare the author from

   tedious typing of logical entities, but also achieve some degree of

   consistency-checking of informal explanations with formal

   developments: well-formedness of terms and types with respect to the

   current theory or proof context is ensured here.%

 \end{isamarkuptext}%

 \isamarkuptrue%

 %

 \isamarkupsubsection{Tagged commands \label{sec:tags}%

 }

 \isamarkuptrue%

 %

 \begin{isamarkuptext}%

 Each Isabelle/Isar command may be decorated by presentation tags:

   \indexouternonterm{tags}

   \begin{rail}

     tags: ( tag * )

     ;

     tag: '\%' (ident | string)

   \end{rail}

   The tags \isa{{\isachardoublequote}theory{\isachardoublequote}}, \isa{{\isachardoublequote}proof{\isachardoublequote}}, \isa{{\isachardoublequote}ML{\isachardoublequote}} are already

   pre-declared for certain classes of commands:

  \medskip

   \begin{tabular}{ll}

     \isa{{\isachardoublequote}theory{\isachardoublequote}} & theory begin/end \\

     \isa{{\isachardoublequote}proof{\isachardoublequote}} & all proof commands \\

     \isa{{\isachardoublequote}ML{\isachardoublequote}} & all commands involving ML code \\

   \end{tabular}

   \medskip The Isabelle document preparation system (see also

   \cite{isabelle-sys}) allows tagged command regions to be presented

   specifically, e.g.\ to fold proof texts, or drop parts of the text

   completely.

   For example ``\hyperlink{command.by}{\mbox{\isa{\isacommand{by}}}}~\isa{{\isachardoublequote}{\isacharpercent}invisible\ auto{\isachardoublequote}}'' would

   cause that piece of proof to be treated as \isa{invisible} instead

   of \isa{{\isachardoublequote}proof{\isachardoublequote}} (the default), which may be either show or hidden

   depending on the document setup.  In contrast, ``\hyperlink{command.by}{\mbox{\isa{\isacommand{by}}}}~\isa{{\isachardoublequote}{\isacharpercent}visible\ auto{\isachardoublequote}}'' would force this text to be shown

   invariably.

   Explicit tag specifications within a proof apply to all subsequent

   commands of the same level of nesting.  For example, ``\hyperlink{command.proof}{\mbox{\isa{\isacommand{proof}}}}~\isa{{\isachardoublequote}{\isacharpercent}visible\ {\isasymdots}{\isachardoublequote}}~\hyperlink{command.qed}{\mbox{\isa{\isacommand{qed}}}}'' would force the

   whole sub-proof to be typeset as \isa{visible} (unless some of its

   parts are tagged differently).%

 \end{isamarkuptext}%

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 \isadelimtheory

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 \endisadelimtheory

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 \isatagtheory

 \isacommand{end}\isamarkupfalse%

 %

 \endisatagtheory

 {\isafoldtheory}%

 %

 \isadelimtheory

 %

 \endisadelimtheory

 \isanewline

 \end{isabellebody}%

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