%

\begin{isabellebody}%

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

%

\isadelimtheory

\isanewline

\isanewline

%

\endisadelimtheory

%

\isatagtheory

\isacommand{theory}\isamarkupfalse%

\ Outer{\isacharunderscore}Syntax\isanewline

\isakeyword{imports}\ Pure\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}%

\isamarkuptrue%

%

\isadelimtheory

%

\endisadelimtheory

%

\isatagtheory

\isacommand{end}\isamarkupfalse%

%

\endisatagtheory

{\isafoldtheory}%

%

\isadelimtheory

%

\endisadelimtheory

\isanewline

\end{isabellebody}%

%%% Local Variables:

%%% mode: latex

%%% TeX-master: "root"

%%% End: