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\isamarkupsection{Concrete syntax \label{sec:concrete-syntax}%

}

\isamarkuptrue%

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\begin{isamarkuptext}%

Concerning Isabelle's ``inner'' language of simply-typed \isa{{\isasymlambda}}-calculus, the core concept of Isabelle's elaborate infrastructure

  for concrete syntax is that of general \emph{mixfix

  annotations}\index{mixfix annotations|bold}.  Associated with any

  kind of name and type declaration, mixfixes give rise both to

  grammar productions for the parser and output templates for the

  pretty printer.

  In full generality, the whole affair of parser and pretty printer

  configuration is rather subtle.  Any syntax specifications given by

  end-users need to interact properly with the existing setup of

  Isabelle/Pure and Isabelle/HOL; see \cite{isabelle-ref} for further

  details.  It is particularly important to get the precedence of new

  syntactic constructs right, avoiding ambiguities with existing

  elements.

  \medskip Subsequently we introduce a few simple declaration forms

  that already cover the most common situations fairly well.%

\end{isamarkuptext}%

\isamarkuptrue%

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\isamarkupsubsection{Infixes%

}

\isamarkuptrue%

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\begin{isamarkuptext}%

Syntax annotations may be included wherever constants are declared

  directly or indirectly, including \isacommand{consts},

  \isacommand{constdefs}, or \isacommand{datatype} (for the

  constructor operations).  Type-constructors may be annotated as

  well, although this is less frequently encountered in practice

  (\isa{{\isacharasterisk}} and \isa{{\isacharplus}} types may come to mind).

  Infix declarations\index{infix annotations|bold} provide a useful

  special case of mixfixes, where users need not care about the full

  details of priorities, nesting, spacing, etc.  The subsequent

  example of the exclusive-or operation on boolean values illustrates

  typical infix declarations.%

\end{isamarkuptext}%

\isamarkuptrue%

\isacommand{constdefs}\isanewline

\ \ xor\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequote}bool\ {\isasymRightarrow}\ bool\ {\isasymRightarrow}\ bool{\isachardoublequote}\ \ \ \ {\isacharparenleft}\isakeyword{infixl}\ {\isachardoublequote}{\isacharbrackleft}{\isacharplus}{\isacharbrackright}{\isachardoublequote}\ {\isadigit{6}}{\isadigit{0}}{\isacharparenright}\isanewline

\ \ {\isachardoublequote}A\ {\isacharbrackleft}{\isacharplus}{\isacharbrackright}\ B\ {\isasymequiv}\ {\isacharparenleft}A\ {\isasymand}\ {\isasymnot}\ B{\isacharparenright}\ {\isasymor}\ {\isacharparenleft}{\isasymnot}\ A\ {\isasymand}\ B{\isacharparenright}{\isachardoublequote}\isamarkupfalse%

%

\begin{isamarkuptext}%

Any curried function with at least two arguments may be associated

  with infix syntax: \isa{xor\ A\ B} and \isa{A\ {\isacharbrackleft}{\isacharplus}{\isacharbrackright}\ B} refer to

  the same expression internally.  In partial applications with less

  than two operands there is a special notation with \isa{op} prefix:

  \isa{xor} without arguments is represented as \isa{op\ {\isacharbrackleft}{\isacharplus}{\isacharbrackright}};

  combined with plain prefix application this turns \isa{xor\ A}

  into \isa{op\ {\isacharbrackleft}{\isacharplus}{\isacharbrackright}\ A}.

  \medskip The string \isa{{\isachardoublequote}{\isacharbrackleft}{\isacharplus}{\isacharbrackright}{\isachardoublequote}} in the above declaration

  refers to the bit of concrete syntax to represent the operator,

  while the number \isa{{\isadigit{6}}{\isadigit{0}}} determines the precedence of the whole

  construct.

  As it happens, Isabelle/HOL already spends many popular combinations

  of ASCII symbols for its own use, including both \isa{{\isacharplus}} and

  \isa{{\isacharplus}{\isacharplus}}.  Slightly more awkward combinations like the present

  \isa{{\isacharbrackleft}{\isacharplus}{\isacharbrackright}} tend to be available for user extensions.  The current

  arrangement of inner syntax may be inspected via

  \commdx{print\protect\_syntax}, albeit its output is enormous.

  Operator precedence also needs some special considerations.  The

  admissible range is 0--1000.  Very low or high priorities are

  basically reserved for the meta-logic.  Syntax of Isabelle/HOL

  mainly uses the range of 10--100: the equality infix \isa{{\isacharequal}} is

  centered at 50, logical connectives (like \isa{{\isasymor}} and \isa{{\isasymand}}) are below 50, and algebraic ones (like \isa{{\isacharplus}} and \isa{{\isacharasterisk}}) above 50.  User syntax should strive to coexist with common

  HOL forms, or use the mostly unused range 100--900.

  \medskip The keyword \isakeyword{infixl} specifies an operator that

  is nested to the \emph{left}: in iterated applications the more

  complex expression appears on the left-hand side: \isa{A\ {\isacharbrackleft}{\isacharplus}{\isacharbrackright}\ B\ {\isacharbrackleft}{\isacharplus}{\isacharbrackright}\ C} stands for \isa{{\isacharparenleft}A\ {\isacharbrackleft}{\isacharplus}{\isacharbrackright}\ B{\isacharparenright}\ {\isacharbrackleft}{\isacharplus}{\isacharbrackright}\ C}.  Similarly,

  \isakeyword{infixr} refers to nesting to the \emph{right}, reading

  \isa{A\ {\isacharbrackleft}{\isacharplus}{\isacharbrackright}\ B\ {\isacharbrackleft}{\isacharplus}{\isacharbrackright}\ C} as \isa{A\ {\isacharbrackleft}{\isacharplus}{\isacharbrackright}\ {\isacharparenleft}B\ {\isacharbrackleft}{\isacharplus}{\isacharbrackright}\ C{\isacharparenright}}.  In contrast,

  a \emph{non-oriented} declaration via \isakeyword{infix} would

  always demand explicit parentheses.

  Many binary operations observe the associative law, so the exact

  grouping does not matter.  Nevertheless, formal statements need be

  given in a particular format, associativity needs to be treated

  explicitly within the logic.  Exclusive-or is happens to be

  associative, as shown below.%

\end{isamarkuptext}%

\isamarkuptrue%

\isacommand{lemma}\ xor{\isacharunderscore}assoc{\isacharcolon}\ {\isachardoublequote}{\isacharparenleft}A\ {\isacharbrackleft}{\isacharplus}{\isacharbrackright}\ B{\isacharparenright}\ {\isacharbrackleft}{\isacharplus}{\isacharbrackright}\ C\ {\isacharequal}\ A\ {\isacharbrackleft}{\isacharplus}{\isacharbrackright}\ {\isacharparenleft}B\ {\isacharbrackleft}{\isacharplus}{\isacharbrackright}\ C{\isacharparenright}{\isachardoublequote}\isanewline

\ \ \isamarkupfalse%

\isacommand{by}\ {\isacharparenleft}auto\ simp\ add{\isacharcolon}\ xor{\isacharunderscore}def{\isacharparenright}\isamarkupfalse%

%

\begin{isamarkuptext}%

Such rules may be used in simplification to regroup nested

  expressions as required.  Note that the system would actually print

  the above statement as \isa{A\ {\isacharbrackleft}{\isacharplus}{\isacharbrackright}\ B\ {\isacharbrackleft}{\isacharplus}{\isacharbrackright}\ C\ {\isacharequal}\ A\ {\isacharbrackleft}{\isacharplus}{\isacharbrackright}\ {\isacharparenleft}B\ {\isacharbrackleft}{\isacharplus}{\isacharbrackright}\ C{\isacharparenright}}

  (due to nesting to the left).  We have preferred to give the fully

  parenthesized form in the text for clarity.  Only in rare situations

  one may consider to force parentheses by use of non-oriented infix

  syntax; equality would probably be a typical candidate.%

\end{isamarkuptext}%

\isamarkuptrue%

%

\isamarkupsubsection{Mathematical symbols \label{sec:thy-present-symbols}%

}

\isamarkuptrue%

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\begin{isamarkuptext}%

Concrete syntax based on plain ASCII characters has its inherent

  limitations.  Rich mathematical notation demands a larger repertoire

  of symbols.  Several standards of extended character sets have been

  proposed over decades, but none has become universally available so

  far, not even Unicode\index{Unicode}.

  Isabelle supports a generic notion of

  \emph{symbols}\index{symbols|bold} as the smallest entities of

  source text, without referring to internal encodings.  Such

  ``generalized characters'' may be of one of the following three

  kinds:

  \begin{enumerate}

  \item Traditional 7-bit ASCII characters.

  \item Named symbols: \verb,\,\verb,<,$ident$\verb,>, (or

  \verb,\\,\verb,<,$ident$\verb,>,).

  \item Named control symbols: \verb,\,\verb,<^,$ident$\verb,>, (or

  \verb,\\,\verb,<^,$ident$\verb,>,).

  \end{enumerate}

  Here $ident$ may be any identifier according to the usual Isabelle

  conventions.  This results in an infinite store of symbols, whose

  interpretation is left to further front-end tools.  For example, the

  \verb,\,\verb,<forall>, symbol of Isabelle is really displayed as

  $\forall$ --- both by the user-interface of Proof~General + X-Symbol

  and the Isabelle document processor (see \S\ref{FIXME}).

  A list of standard Isabelle symbols is given in

  \cite[appendix~A]{isabelle-sys}.  Users may introduce their own

  interpretation of further symbols by configuring the appropriate

  front-end tool accordingly, e.g.\ defining appropriate {\LaTeX}

  macros for document preparation.  There are also a few predefined

  control symbols, such as \verb,\,\verb,<^sub>, and

  \verb,\,\verb,<^sup>, for sub- and superscript of the subsequent

  (printable) symbol, respectively.

  \medskip The following version of our \isa{xor} definition uses a

  standard Isabelle symbol to achieve typographically pleasing output.%

\end{isamarkuptext}%

\isamarkuptrue%

\isamarkupfalse%

\isamarkupfalse%

\isacommand{constdefs}\isanewline

\ \ xor\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequote}bool\ {\isasymRightarrow}\ bool\ {\isasymRightarrow}\ bool{\isachardoublequote}\ \ \ \ {\isacharparenleft}\isakeyword{infixl}\ {\isachardoublequote}{\isasymoplus}{\isachardoublequote}\ {\isadigit{6}}{\isadigit{0}}{\isacharparenright}\isanewline

\ \ {\isachardoublequote}A\ {\isasymoplus}\ B\ {\isasymequiv}\ {\isacharparenleft}A\ {\isasymand}\ {\isasymnot}\ B{\isacharparenright}\ {\isasymor}\ {\isacharparenleft}{\isasymnot}\ A\ {\isasymand}\ B{\isacharparenright}{\isachardoublequote}\isamarkupfalse%

\isamarkupfalse%

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\begin{isamarkuptext}%

The X-Symbol package within Proof~General provides several input

  methods to enter \isa{{\isasymoplus}} in the text.  If all fails one may just

  type \verb,\,\verb,<oplus>, by hand; the display is adapted

  immediately after continuing further input.

  \medskip A slightly more refined scheme is to provide alternative

  syntax via the \emph{print mode}\index{print mode} concept of

  Isabelle (see also \cite{isabelle-ref}).  By convention, the mode

  ``$xsymbols$'' is enabled whenever X-Symbol is active.  Consider the

  following hybrid declaration of \isa{xor}.%

\end{isamarkuptext}%

\isamarkuptrue%

\isamarkupfalse%

\isamarkupfalse%

\isacommand{constdefs}\isanewline

\ \ xor\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequote}bool\ {\isasymRightarrow}\ bool\ {\isasymRightarrow}\ bool{\isachardoublequote}\ \ \ \ {\isacharparenleft}\isakeyword{infixl}\ {\isachardoublequote}{\isacharbrackleft}{\isacharplus}{\isacharbrackright}{\isasymignore}{\isachardoublequote}\ {\isadigit{6}}{\isadigit{0}}{\isacharparenright}\isanewline

\ \ {\isachardoublequote}A\ {\isacharbrackleft}{\isacharplus}{\isacharbrackright}{\isasymignore}\ B\ {\isasymequiv}\ {\isacharparenleft}A\ {\isasymand}\ {\isasymnot}\ B{\isacharparenright}\ {\isasymor}\ {\isacharparenleft}{\isasymnot}\ A\ {\isasymand}\ B{\isacharparenright}{\isachardoublequote}\isanewline

\isanewline

\isamarkupfalse%

\isacommand{syntax}\ {\isacharparenleft}xsymbols{\isacharparenright}\isanewline

\ \ xor\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequote}bool\ {\isasymRightarrow}\ bool\ {\isasymRightarrow}\ bool{\isachardoublequote}\ \ \ \ {\isacharparenleft}\isakeyword{infixl}\ {\isachardoublequote}{\isasymoplus}{\isasymignore}{\isachardoublequote}\ {\isadigit{6}}{\isadigit{0}}{\isacharparenright}\isamarkupfalse%

\isamarkupfalse%

%

\begin{isamarkuptext}%

Here the \commdx{syntax} command acts like \isakeyword{consts}, but

  without declaring a logical constant, and with an optional print

  mode specification.  Note that the type declaration given here

  merely serves for syntactic purposes, and is not checked for

  consistency with the real constant.

  \medskip Now we may write either \isa{{\isacharbrackleft}{\isacharplus}{\isacharbrackright}} or \isa{{\isasymoplus}} in

  input, while output uses the nicer syntax of $xsymbols$, provided

  that print mode is presently active.  This scheme is particularly

  useful for interactive development, with the user typing plain ASCII

  text, but gaining improved visual feedback from the system (say in

  current goal output).

  \begin{warn}

  Using alternative syntax declarations easily results in varying

  versions of input sources.  Isabelle provides no systematic way to

  convert alternative expressions back and forth.  Print modes only

  affect situations where formal entities are pretty printed by the

  Isabelle process (e.g.\ output of terms and types), but not the

  original theory text.

  \end{warn}

  \medskip The following variant makes the alternative \isa{{\isasymoplus}}

  notation only available for output.  Thus we may enforce input

  sources to refer to plain ASCII only.%

\end{isamarkuptext}%

\isamarkuptrue%

\isacommand{syntax}\ {\isacharparenleft}xsymbols\ \isakeyword{output}{\isacharparenright}\isanewline

\ \ xor\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequote}bool\ {\isasymRightarrow}\ bool\ {\isasymRightarrow}\ bool{\isachardoublequote}\ \ \ \ {\isacharparenleft}\isakeyword{infixl}\ {\isachardoublequote}{\isasymoplus}{\isasymignore}{\isachardoublequote}\ {\isadigit{6}}{\isadigit{0}}{\isacharparenright}\isamarkupfalse%

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\isamarkupsubsection{Prefixes%

}

\isamarkuptrue%

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\begin{isamarkuptext}%

Prefix syntax annotations\index{prefix annotation|bold} are just a

  very degenerate of the general mixfix form \cite{isabelle-ref},

  without any template arguments or priorities --- just some piece of

  literal syntax.

  The following example illustrates this idea idea by associating

  common symbols with the constructors of a currency datatype.%

\end{isamarkuptext}%

\isamarkuptrue%

\isacommand{datatype}\ currency\ {\isacharequal}\isanewline

\ \ \ \ Euro\ nat\ \ \ \ {\isacharparenleft}{\isachardoublequote}{\isasymeuro}{\isachardoublequote}{\isacharparenright}\isanewline

\ \ {\isacharbar}\ Pounds\ nat\ \ {\isacharparenleft}{\isachardoublequote}{\isasympounds}{\isachardoublequote}{\isacharparenright}\isanewline

\ \ {\isacharbar}\ Yen\ nat\ \ \ \ \ {\isacharparenleft}{\isachardoublequote}{\isasymyen}{\isachardoublequote}{\isacharparenright}\isanewline

\ \ {\isacharbar}\ Dollar\ nat\ \ {\isacharparenleft}{\isachardoublequote}{\isachardollar}{\isachardoublequote}{\isacharparenright}\isamarkupfalse%

%

\begin{isamarkuptext}%

Here the degenerate mixfix annotations on the rightmost column

  happen to consist of a single Isabelle symbol each:

  \verb,\,\verb,<euro>,, \verb,\,\verb,<pounds>,,

  \verb,\,\verb,<yen>,, and \verb,$,.

  Recall that a constructor like \isa{Euro} actually is a function

  \isa{nat\ {\isasymRightarrow}\ currency}.  An expression like \isa{Euro\ {\isadigit{1}}{\isadigit{0}}} will

  be printed as \isa{{\isasymeuro}\ {\isadigit{1}}{\isadigit{0}}}.  Merely the head of the application is

  subject to our trivial concrete syntax; this form is sufficient to

  achieve fair conformance to EU~Commission standards of currency

  notation.

  \medskip Certainly, the same idea of prefix syntax also works for

  \isakeyword{consts}, \isakeyword{constdefs} etc.  For example, we

  might introduce a (slightly unrealistic) function to calculate an

  abstract currency value, by cases on the datatype constructors and

  fixed exchange rates.%

\end{isamarkuptext}%

\isamarkuptrue%

\isacommand{consts}\isanewline

\ \ currency\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequote}currency\ {\isasymRightarrow}\ nat{\isachardoublequote}\ \ \ \ {\isacharparenleft}{\isachardoublequote}{\isasymcurrency}{\isachardoublequote}{\isacharparenright}\isamarkupfalse%

%

\begin{isamarkuptext}%

\noindent The funny symbol encountered here is that of

  \verb,\<currency>,.%

\end{isamarkuptext}%

\isamarkuptrue%

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\isamarkupsubsection{Syntax translations \label{sec:def-translations}%

}

\isamarkuptrue%

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\begin{isamarkuptext}%

FIXME

\index{syntax translations|(}%

\index{translations@\isacommand {translations} (command)|(}

Isabelle offers an additional definitional facility,

\textbf{syntax translations}.

They resemble macros: upon parsing, the defined concept is immediately

replaced by its definition.  This effect is reversed upon printing.  For example,

the symbol \isa{{\isasymnoteq}} is defined via a syntax translation:%

\end{isamarkuptext}%

\isamarkuptrue%

\isacommand{translations}\ {\isachardoublequote}x\ {\isasymnoteq}\ y{\isachardoublequote}\ {\isasymrightleftharpoons}\ {\isachardoublequote}{\isasymnot}{\isacharparenleft}x\ {\isacharequal}\ y{\isacharparenright}{\isachardoublequote}\isamarkupfalse%

%

\begin{isamarkuptext}%

\index{$IsaEqTrans@\isasymrightleftharpoons}

\noindent

Internally, \isa{{\isasymnoteq}} never appears.

In addition to \isa{{\isasymrightleftharpoons}} there are

\isa{{\isasymrightharpoonup}}\index{$IsaEqTrans1@\isasymrightharpoonup}

and \isa{{\isasymleftharpoondown}}\index{$IsaEqTrans2@\isasymleftharpoondown}

for uni-directional translations, which only affect

parsing or printing.  This tutorial will not cover the details of

translations.  We have mentioned the concept merely because it

crops up occasionally; a number of HOL's built-in constructs are defined

via translations.  Translations are preferable to definitions when the new 

concept is a trivial variation on an existing one.  For example, we

don't need to derive new theorems about \isa{{\isasymnoteq}}, since existing theorems

about \isa{{\isacharequal}} still apply.%

\index{syntax translations|)}%

\index{translations@\isacommand {translations} (command)|)}%

\end{isamarkuptext}%

\isamarkuptrue%

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\isamarkupsection{Document preparation%

}

\isamarkuptrue%

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\isamarkupsubsection{Batch-mode sessions%

}

\isamarkuptrue%

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\isamarkupsubsection{{\LaTeX} macros%

}

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\isamarkupsubsubsection{Structure markup%

}

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\isamarkupsubsubsection{Symbols and characters%

}

\isamarkuptrue%

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\begin{isamarkuptext}%

FIXME%

\end{isamarkuptext}%

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