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\isamarkupsection{Concrete syntax \label{sec:concrete-syntax}%
}
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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}%
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\isamarkupsubsection{Infixes%
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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%
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\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}, which
would turn \isa{A\ {\isacharbrackleft}{\isacharplus}{\isacharbrackright}\ B\ {\isacharbrackleft}{\isacharplus}{\isacharbrackright}\ C} into \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%
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\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.%
\end{isamarkuptext}%
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