%% $Id$

 \chapter{Defining Logics} \label{Defining-Logics}

 This chapter explains how to define new formal systems --- in particular,

 their concrete syntax.  While Isabelle can be regarded as a theorem prover

 for set theory, higher-order logic or the sequent calculus, its

 distinguishing feature is support for the definition of new logics.

 Isabelle logics are hierarchies of theories, which are described and

 illustrated in

 \iflabelundefined{sec:defining-theories}{{\em Introduction to Isabelle}}%

 {\S\ref{sec:defining-theories}}.  That material, together with the theory

 files provided in the examples directories, should suffice for all simple

 applications.  The easiest way to define a new theory is by modifying a

 copy of an existing theory.

 This chapter documents the meta-logic syntax, mixfix declarations and

 pretty printing.  The extended examples in \S\ref{sec:min_logics}

 demonstrate the logical aspects of the definition of theories.

 \section{Priority grammars} \label{sec:priority_grammars}

 \index{priority grammars|(}

 A context-free grammar contains a set of {\bf nonterminal symbols}, a set of

 {\bf terminal symbols} and a set of {\bf productions}\index{productions}.

 Productions have the form ${A=\gamma}$, where $A$ is a nonterminal and

 $\gamma$ is a string of terminals and nonterminals.  One designated

 nonterminal is called the {\bf start symbol}.  The language defined by the

 grammar consists of all strings of terminals that can be derived from the

 start symbol by applying productions as rewrite rules.

 The syntax of an Isabelle logic is specified by a {\bf priority

   grammar}.\index{priorities} Each nonterminal is decorated by an integer

 priority, as in~$A^{(p)}$.  A nonterminal $A^{(p)}$ in a derivation may be

 rewritten using a production $A^{(q)} = \gamma$ only if~$p \le q$.  Any

 priority grammar can be translated into a normal context free grammar by

 introducing new nonterminals and productions.

 Formally, a set of context free productions $G$ induces a derivation

 relation $\longrightarrow@G$.  Let $\alpha$ and $\beta$ denote strings of

 terminal or nonterminal symbols.  Then

 \[ \alpha\, A^{(p)}\, \beta ~\longrightarrow@G~ \alpha\,\gamma\,\beta \]

 if and only if $G$ contains some production $A^{(q)}=\gamma$ for~$p \le q$.

 The following simple grammar for arithmetic expressions demonstrates how

 binding power and associativity of operators can be enforced by priorities.

 \begin{center}

 \begin{tabular}{rclr}

   $A^{(9)}$ & = & {\tt0} \\

   $A^{(9)}$ & = & {\tt(} $A^{(0)}$ {\tt)} \\

   $A^{(0)}$ & = & $A^{(0)}$ {\tt+} $A^{(1)}$ \\

   $A^{(2)}$ & = & $A^{(3)}$ {\tt*} $A^{(2)}$ \\

   $A^{(3)}$ & = & {\tt-} $A^{(3)}$

 \end{tabular}

 \end{center}

 The choice of priorities determines that {\tt -} binds tighter than {\tt *},

 which binds tighter than {\tt +}.  Furthermore {\tt +} associates to the

 left and {\tt *} to the right.

 For clarity, grammars obey these conventions:

 \begin{itemize}

 \item All priorities must lie between~0 and \ttindex{max_pri}, which is a

   some fixed integer.  Sometimes {\tt max_pri} is written as $\infty$.

 \item Priority 0 on the right-hand side and priority \ttindex{max_pri} on

   the left-hand side may be omitted.

 \item The production $A^{(p)} = \alpha$ is written as $A = \alpha~(p)$; the

   priority of the left-hand side actually appears in a column on the far

   right.

 \item Alternatives are separated by~$|$.

 \item Repetition is indicated by dots~(\dots) in an informal but obvious

   way.

 \end{itemize}

 Using these conventions and assuming $\infty=9$, the grammar

 takes the form

 \begin{center}

 \begin{tabular}{rclc}

 $A$ & = & {\tt0} & \hspace*{4em} \\

  & $|$ & {\tt(} $A$ {\tt)} \\

  & $|$ & $A$ {\tt+} $A^{(1)}$ & (0) \\

  & $|$ & $A^{(3)}$ {\tt*} $A^{(2)}$ & (2) \\

  & $|$ & {\tt-} $A^{(3)}$ & (3)

 \end{tabular}

 \end{center}

 \index{priority grammars|)}

 \begin{figure}

 \begin{center}

 \begin{tabular}{rclc}

 $any$ &=& $prop$ ~~$|$~~ $logic$ \\\\

 $prop$ &=& {\tt(} $prop$ {\tt)} \\

      &$|$& $prop^{(4)}$ {\tt::} $type$ & (3) \\

      &$|$& {\tt PROP} $aprop$ \\

      &$|$& $any^{(3)}$ {\tt ==} $any^{(2)}$ & (2) \\

      &$|$& $any^{(3)}$ {\tt =?=} $any^{(2)}$ & (2) \\

      &$|$& $prop^{(2)}$ {\tt ==>} $prop^{(1)}$ & (1) \\

      &$|$& {\tt[|} $prop$ {\tt;} \dots {\tt;} $prop$ {\tt|]} {\tt==>} $prop^{(1)}$ & (1) \\

      &$|$& {\tt!!} $idts$ {\tt.} $prop$ & (0) \\

      &$|$& {\tt OFCLASS} {\tt(} $type$ {\tt,} $logic$ {\tt)} \\\\

 $aprop$ &=& $id$ ~~$|$~~ $var$

     ~~$|$~~ $logic^{(\infty)}$ {\tt(} $any$ {\tt,} \dots {\tt,} $any$ {\tt)} \\\\

 $logic$ &=& {\tt(} $logic$ {\tt)} \\

       &$|$& $logic^{(4)}$ {\tt::} $type$ & (3) \\

       &$|$& $id$ ~~$|$~~ $var$

     ~~$|$~~ $logic^{(\infty)}$ {\tt(} $any$ {\tt,} \dots {\tt,} $any$ {\tt)} \\

       &$|$& {\tt \%} $idts$ {\tt.} $any$ & (0) \\\\

 $idts$ &=& $idt$ ~~$|$~~ $idt^{(1)}$ $idts$ \\\\

 $idt$ &=& $id$ ~~$|$~~ {\tt(} $idt$ {\tt)} \\

     &$|$& $id$ {\tt ::} $type$ & (0) \\\\

 $type$ &=& {\tt(} $type$ {\tt)} \\

      &$|$& $tid$ ~~$|$~~ $tvar$ ~~$|$~~ $tid$ {\tt::} $sort$

        ~~$|$~~ $tvar$ {\tt::} $sort$ \\

      &$|$& $id$ ~~$|$~~ $type^{(\infty)}$ $id$

                 ~~$|$~~ {\tt(} $type$ {\tt,} \dots {\tt,} $type$ {\tt)} $id$ \\

      &$|$& $type^{(1)}$ {\tt =>} $type$ & (0) \\

      &$|$& {\tt[}  $type$ {\tt,} \dots {\tt,} $type$ {\tt]} {\tt=>} $type$&(0) \\\\

 $sort$ &=& $id$ ~~$|$~~ {\tt\ttlbrace\ttrbrace}

                 ~~$|$~~ {\tt\ttlbrace} $id$ {\tt,} \dots {\tt,} $id$ {\tt\ttrbrace}

 \end{tabular}

 \index{*PROP symbol}

 \index{*== symbol}\index{*=?= symbol}\index{*==> symbol}

 \index{*:: symbol}\index{*=> symbol}

 \index{sort constraints}

 %the index command: a percent is permitted, but braces must match!

 \index{%@{\tt\%} symbol}

 \index{{}@{\tt\ttlbrace} symbol}\index{{}@{\tt\ttrbrace} symbol}

 \index{*[ symbol}\index{*] symbol}

 \index{*"!"! symbol}

 \index{*"["| symbol}

 \index{*"|"] symbol}

 \end{center}

 \caption{Meta-logic syntax}\label{fig:pure_gram}

 \end{figure}

 \section{The Pure syntax} \label{sec:basic_syntax}

 \index{syntax!Pure|(}

 At the root of all object-logics lies the theory \thydx{Pure}.  It

 contains, among many other things, the Pure syntax.  An informal account of

 this basic syntax (types, terms and formulae) appears in

 \iflabelundefined{sec:forward}{{\em Introduction to Isabelle}}%

 {\S\ref{sec:forward}}.  A more precise description using a priority grammar

 appears in Fig.\ts\ref{fig:pure_gram}.  It defines the following

 nonterminals:

 \begin{ttdescription}

   \item[\ndxbold{any}] denotes any term.

   \item[\ndxbold{prop}] denotes terms of type {\tt prop}.  These are formulae

     of the meta-logic.  Note that user constants of result type {\tt prop}

     (i.e.\ $c :: \ldots \To prop$) should always provide concrete syntax.

     Otherwise atomic propositions with head $c$ may be printed incorrectly.

   \item[\ndxbold{aprop}] denotes atomic propositions.

 %% FIXME huh!?

 %  These typically

 %  include the judgement forms of the object-logic; its definition

 %  introduces a meta-level predicate for each judgement form.

   \item[\ndxbold{logic}] denotes terms whose type belongs to class

     \cldx{logic}, excluding type \tydx{prop}.

   \item[\ndxbold{idts}] denotes a list of identifiers, possibly constrained

     by types.

   \item[\ndxbold{type}] denotes types of the meta-logic.

   \item[\ndxbold{sort}] denotes meta-level sorts.

 \end{ttdescription}

 \begin{warn}

   In {\tt idts}, note that \verb|x::nat y| is parsed as \verb|x::(nat y)|,

   treating {\tt y} like a type constructor applied to {\tt nat}.  The

   likely result is an error message.  To avoid this interpretation, use

   parentheses and write \verb|(x::nat) y|.

   \index{type constraints}\index{*:: symbol}

   Similarly, \verb|x::nat y::nat| is parsed as \verb|x::(nat y::nat)| and

   yields an error.  The correct form is \verb|(x::nat) (y::nat)|.

 \end{warn}

 \begin{warn}

   Type constraints bind very weakly. For example, \verb!x<y::nat! is normally

   parsed as \verb!(x<y)::nat!, unless \verb$<$ has priority of 3 or less, in

   which case the string is likely to be ambiguous. The correct form is

   \verb!x<(y::nat)!.

 \end{warn}

 Isabelle's representation of mathematical languages is based on the simply

 typed $\lambda$-calculus\index{lambda calc@$\lambda$-calculus}.  All user

 defined logical types, namely those of class \cldx{logic}, refer to the

 nonterminal {\tt logic}. Thus they are automatically equipped with a basic

 syntax of types, identifiers, variables, parentheses, $\lambda$-abstractions

 and applications.

 \subsection{Lexical matters}

 The parser does not process input strings directly.  It operates on token

 lists provided by Isabelle's \bfindex{lexer}.  There are two kinds of

 tokens: \bfindex{delimiters} and \bfindex{name tokens}.

 \index{reserved words}

 Delimiters can be regarded as reserved words of the syntax.  You can

 add new ones when extending theories.  In Fig.\ts\ref{fig:pure_gram} they

 appear in typewriter font, for example {\tt ==}, {\tt =?=} and

 {\tt PROP}\@.

 Name tokens have a predefined syntax.  The lexer distinguishes six disjoint

 classes of names: \rmindex{identifiers}, \rmindex{unknowns}, type

 identifiers\index{type identifiers}, type unknowns\index{type unknowns},

 \rmindex{numerals}, \rmindex{strings}. They are denoted by \ndxbold{id},

 \ndxbold{var}, \ndxbold{tid}, \ndxbold{tvar}, \ndxbold{xnum}, \ndxbold{xstr},

 respectively.  Typical examples are {\tt x}, {\tt ?x7}, {\tt 'a}, {\tt ?'a3},

 {\tt \#42}, {\tt ''foo bar''}. Here is the precise syntax:

 \begin{eqnarray*}

 id        & =   & letter~quasiletter^* \\

 var       & =   & \mbox{\tt ?}id ~~|~~ \mbox{\tt ?}id\mbox{\tt .}nat \\

 tid       & =   & \mbox{\tt '}id \\

 tvar      & =   & \mbox{\tt ?}tid ~~|~~

                   \mbox{\tt ?}tid\mbox{\tt .}nat \\

 xnum      & =   & \mbox{\tt \#}nat ~~|~~ \mbox{\tt \#\char`\~}nat \\

 xstr      & =   & \mbox{\tt ''\rm text\tt ''} \\[1ex]

 letter    & =   & \mbox{one of {\tt a}\dots {\tt z} {\tt A}\dots {\tt Z}} \\

 digit     & =   & \mbox{one of {\tt 0}\dots {\tt 9}} \\

 quasiletter & =  & letter ~~|~~ digit ~~|~~ \mbox{\tt _} ~~|~~ \mbox{\tt '} \\

 nat       & =   & digit^+

 \end{eqnarray*}

 The lexer repeatedly takes the maximal prefix of the input string that forms

 a valid token.  A maximal prefix that is both a delimiter and a name is

 treated as a delimiter.  Spaces, tabs, newlines and formfeeds are separators;

 they never occur within tokens, except those of class $xstr$.

 \medskip

 Delimiters need not be separated by white space.  For example, if {\tt -}

 is a delimiter but {\tt --} is not, then the string {\tt --} is treated as

 two consecutive occurrences of the token~{\tt -}.  In contrast, \ML\

 treats {\tt --} as a single symbolic name.  The consequence of Isabelle's

 more liberal scheme is that the same string may be parsed in different ways

 after extending the syntax: after adding {\tt --} as a delimiter, the input

 {\tt --} is treated as a single token.

 A \ndxbold{var} or \ndxbold{tvar} describes an unknown, which is internally

 a pair of base name and index (\ML\ type \mltydx{indexname}).  These

 components are either separated by a dot as in {\tt ?x.1} or {\tt ?x7.3} or

 run together as in {\tt ?x1}.  The latter form is possible if the base name

 does not end with digits.  If the index is 0, it may be dropped altogether:

 {\tt ?x} abbreviates both {\tt ?x0} and {\tt ?x.0}.

 Tokens of class $xnum$ or $xstr$ are not used by the meta-logic.

 Object-logics may provide numerals and string constants by adding appropriate

 productions and translation functions.

 \medskip

 Although name tokens are returned from the lexer rather than the parser, it

 is more logical to regard them as nonterminals.  Delimiters, however, are

 terminals; they are just syntactic sugar and contribute nothing to the

 abstract syntax tree.

 \subsection{*Inspecting the syntax}

 \begin{ttbox}

 syn_of              : theory -> Syntax.syntax

 print_syntax        : theory -> unit

 Syntax.print_syntax : Syntax.syntax -> unit

 Syntax.print_gram   : Syntax.syntax -> unit

 Syntax.print_trans  : Syntax.syntax -> unit

 \end{ttbox}

 The abstract type \mltydx{Syntax.syntax} allows manipulation of syntaxes

 in \ML.  You can display values of this type by calling the following

 functions:

 \begin{ttdescription}

 \item[\ttindexbold{syn_of} {\it thy}] returns the syntax of the Isabelle

   theory~{\it thy} as an \ML\ value.

 \item[\ttindexbold{print_syntax} $thy$] displays the syntax part of $thy$

   using {\tt Syntax.print_syntax}.

 \item[\ttindexbold{Syntax.print_syntax} {\it syn}] shows virtually all

   information contained in the syntax {\it syn}.  The displayed output can

   be large.  The following two functions are more selective.

 \item[\ttindexbold{Syntax.print_gram} {\it syn}] shows the grammar part

   of~{\it syn}, namely the lexicon, logical types and productions.  These are

   discussed below.

 \item[\ttindexbold{Syntax.print_trans} {\it syn}] shows the translation

   part of~{\it syn}, namely the constants, parse/print macros and

   parse/print translations.

 \end{ttdescription}

 Let us demonstrate these functions by inspecting Pure's syntax.  Even that

 is too verbose to display in full.

 \begin{ttbox}\index{*Pure theory}

 Syntax.print_syntax (syn_of Pure.thy);

 {\out lexicon: "!!" "\%" "(" ")" "," "." "::" ";" "==" "==>" \dots}

 {\out logtypes: fun itself}

 {\out prods:}

 {\out   type = tid  (1000)}

 {\out   type = tvar  (1000)}

 {\out   type = id  (1000)}

 {\out   type = tid "::" sort[0]  => "_ofsort" (1000)}

 {\out   type = tvar "::" sort[0]  => "_ofsort" (1000)}

 {\out   \vdots}

 \ttbreak

 {\out consts: "_K" "_appl" "_aprop" "_args" "_asms" "_bigimpl" \dots}

 {\out parse_ast_translation: "_appl" "_bigimpl" "_bracket"}

 {\out   "_idtyp" "_lambda" "_tapp" "_tappl"}

 {\out parse_rules:}

 {\out parse_translation: "!!" "_K" "_abs" "_aprop"}

 {\out print_translation: "all"}

 {\out print_rules:}

 {\out print_ast_translation: "==>" "_abs" "_idts" "fun"}

 \end{ttbox}

 As you can see, the output is divided into labelled sections.  The grammar

 is represented by {\tt lexicon}, {\tt logtypes} and {\tt prods}.  The rest

 refers to syntactic translations and macro expansion.  Here is an

 explanation of the various sections.

 \begin{description}

   \item[{\tt lexicon}] lists the delimiters used for lexical

     analysis.\index{delimiters}

   \item[{\tt logtypes}] lists the types that are regarded the same as {\tt

     logic} syntactically. Thus types of object-logics (e.g.\ {\tt nat}, say)

     will be automatically equipped with the standard syntax of

     $\lambda$-calculus.

   \item[{\tt prods}] lists the \rmindex{productions} of the priority grammar.

     The nonterminal $A^{(n)}$ is rendered in {\sc ascii} as {\tt $A$[$n$]}.

     Each delimiter is quoted.  Some productions are shown with {\tt =>} and

     an attached string.  These strings later become the heads of parse

     trees; they also play a vital role when terms are printed (see

     \S\ref{sec:asts}).

     Productions with no strings attached are called {\bf copy

       productions}\indexbold{productions!copy}.  Their right-hand side must

     have exactly one nonterminal symbol (or name token).  The parser does

     not create a new parse tree node for copy productions, but simply

     returns the parse tree of the right-hand symbol.

     If the right-hand side consists of a single nonterminal with no

     delimiters, then the copy production is called a {\bf chain

       production}.  Chain productions act as abbreviations:

     conceptually, they are removed from the grammar by adding new

     productions.  Priority information attached to chain productions is

     ignored; only the dummy value $-1$ is displayed.

   \item[{\tt consts}, {\tt parse_rules}, {\tt print_rules}]

     relate to macros (see \S\ref{sec:macros}).

   \item[{\tt parse_ast_translation}, {\tt print_ast_translation}]

     list sets of constants that invoke translation functions for abstract

     syntax trees.  Section \S\ref{sec:asts} below discusses this obscure

     matter.\index{constants!for translations}

   \item[{\tt parse_translation}, {\tt print_translation}] list sets

     of constants that invoke translation functions for terms (see

     \S\ref{sec:tr_funs}).

 \end{description}

 \index{syntax!Pure|)}

 \section{Mixfix declarations} \label{sec:mixfix}

 \index{mixfix declarations|(}

 When defining a theory, you declare new constants by giving their names,

 their type, and an optional {\bf mixfix annotation}.  Mixfix annotations

 allow you to extend Isabelle's basic $\lambda$-calculus syntax with

 readable notation.  They can express any context-free priority grammar.

 Isabelle syntax definitions are inspired by \OBJ~\cite{OBJ}; they are more

 general than the priority declarations of \ML\ and Prolog.

 A mixfix annotation defines a production of the priority grammar.  It

 describes the concrete syntax, the translation to abstract syntax, and the

 pretty printing.  Special case annotations provide a simple means of

 specifying infix operators and binders.

 %% FIXME remove

 %\subsection{Grammar productions}\label{sec:grammar}\index{productions}

 %

 %Let us examine the treatment of the production

 %\[ A^{(p)}= w@0\, A@1^{(p@1)}\, w@1\, A@2^{(p@2)}\, \ldots\,

 %                  A@n^{(p@n)}\, w@n. \]

 %Here $A@i^{(p@i)}$ is a nonterminal with priority~$p@i$ for $i=1$,

 %\ldots,~$n$, while $w@0$, \ldots,~$w@n$ are strings of terminals.

 %In the corresponding mixfix annotation, the priorities are given separately

 %as $[p@1,\ldots,p@n]$ and~$p$.  The nonterminal symbols are derived from

 %types~$\tau$, $\tau@1$, \ldots,~$\tau@n$ respectively, and the production's

 %effect on nonterminals is expressed as the function type

 %\[ [\tau@1, \ldots, \tau@n]\To \tau. \]

 %Finally, the template

 %\[ w@0  \;_\; w@1 \;_\; \ldots \;_\; w@n \]

 %describes the strings of terminals.

 \subsection{The general mixfix form}

 Here is a detailed account of mixfix declarations.  Suppose the following

 line occurs within a {\tt consts} or {\tt syntax} section of a {\tt .thy}

 file:

 \begin{center}

   {\tt $c$ ::\ "$\sigma$" ("$template$" $ps$ $p$)}

 \end{center}

 This constant declaration and mixfix annotation are interpreted as follows:

 \begin{itemize}\index{productions}

 \item The string {\tt $c$} is the name of the constant associated with the

   production; unless it is a valid identifier, it must be enclosed in

   quotes.  If $c$ is empty (given as~{\tt ""}) then this is a copy

   production.\index{productions!copy} Otherwise, parsing an instance of the

   phrase $template$ generates the \AST{} {\tt ("$c$" $a@1$ $\ldots$

     $a@n$)}, where $a@i$ is the \AST{} generated by parsing the $i$-th

   argument.

   \item The constant $c$, if non-empty, is declared to have type $\sigma$

     ({\tt consts} section only).

   \item The string $template$ specifies the right-hand side of

     the production.  It has the form

     \[ w@0 \;_\; w@1 \;_\; \ldots \;_\; w@n, \]

     where each occurrence of {\tt_} denotes an argument position and

     the~$w@i$ do not contain~{\tt _}.  (If you want a literal~{\tt _} in

     the concrete syntax, you must escape it as described below.)  The $w@i$

     may consist of \rmindex{delimiters}, spaces or

     \rmindex{pretty printing} annotations (see below).

   \item The type $\sigma$ specifies the production's nonterminal symbols

     (or name tokens).  If $template$ is of the form above then $\sigma$

     must be a function type with at least~$n$ argument positions, say

     $\sigma = [\tau@1, \dots, \tau@n] \To \tau$.  Nonterminal symbols are

     derived from the types $\tau@1$, \ldots,~$\tau@n$, $\tau$ as described

     below.  Any of these may be function types.

   \item The optional list~$ps$ may contain at most $n$ integers, say {\tt

       [$p@1$, $\ldots$, $p@m$]}, where $p@i$ is the minimal

     priority\indexbold{priorities} required of any phrase that may appear

     as the $i$-th argument.  Missing priorities default to~0.

   \item The integer $p$ is the priority of this production.  If omitted, it

     defaults to the maximal priority.

     Priorities range between 0 and \ttindexbold{max_pri} (= 1000).

 \end{itemize}

 %

 The resulting production is \[ A^{(p)}= w@0\, A@1^{(p@1)}\, w@1\,

 A@2^{(p@2)}\, \dots\, A@n^{(p@n)}\, w@n \] where $A$ and the $A@i$ are the

 nonterminals corresponding to the types $\tau$ and $\tau@i$ respectively.

 The nonterminal symbol associated with a type $(\ldots)ty$ is {\tt logic}, if

 this is a logical type (namely one of class {\tt logic} excluding {\tt

 prop}).  Otherwise it is $ty$ (note that only the outermost type constructor

 is taken into account).  Finally, the nonterminal of a type variable is {\tt

 any}.

 \begin{warn} 

   Theories must sometimes declare types for purely syntactic purposes ---

   merely playing the role of nonterminals. One example is \tydx{type}, the

   built-in type of types.  This is a `type of all types' in the syntactic

   sense only.  Do not declare such types under {\tt arities} as belonging to

   class {\tt logic}\index{*logic class}, for that would make them useless as

   separate nonterminal symbols.

 \end{warn}

 Associating nonterminals with types allows a constant's type to specify

 syntax as well.  We can declare the function~$f$ to have type $[\tau@1,

 \ldots, \tau@n]\To \tau$ and, through a mixfix annotation, specify the layout

 of the function's $n$ arguments.  The constant's name, in this case~$f$, will

 also serve as the label in the abstract syntax tree.

 You may also declare mixfix syntax without adding constants to the theory's

 signature, by using a {\tt syntax} section instead of {\tt consts}.  Thus a

 production need not map directly to a logical function (this typically

 requires additional syntactic translations, see also

 Chapter~\ref{chap:syntax}).

 \medskip 

 As a special case of the general mixfix declaration, the form 

 \begin{center}

   {\tt $c$ ::\ "$\sigma$" ("$template$")} 

 \end{center}

 specifies no priorities.  The resulting production puts no priority

 constraints on any of its arguments and has maximal priority itself.

 Omitting priorities in this manner is prone to syntactic ambiguities unless

 the production's right-hand side is fully bracketed, as in \verb|"if _ then _

 else _ fi"|.

 Omitting the mixfix annotation completely, as in {\tt $c$ ::\ "$\sigma$"},

 is sensible only if~$c$ is an identifier.  Otherwise you will be unable to

 write terms involving~$c$.

 \medskip 

 There is something artificial about the representation of productions as

 mixfix declarations, but it is convenient, particularly for simple theory

 extensions.

 \subsection{Example: arithmetic expressions}

 \index{examples!of mixfix declarations}

 This theory specification contains a {\tt syntax} section with mixfix

 declarations encoding the priority grammar from

 \S\ref{sec:priority_grammars}:

 \begin{ttbox}

 EXP = Pure +

 types

   exp

 syntax

   "0" :: "exp"                ("0"      9)

   "+" :: "[exp, exp] => exp"  ("_ + _"  [0, 1] 0)

   "*" :: "[exp, exp] => exp"  ("_ * _"  [3, 2] 2)

   "-" :: "exp => exp"         ("- _"    [3] 3)

 end

 \end{ttbox}

 If you put this into a file {\tt EXP.thy} and load it via {\tt use_thy"EXP"},

 you can run some tests:

 \begin{ttbox}

 val read_exp = Syntax.test_read (syn_of EXP.thy) "exp";

 {\out val it = fn : string -> unit}

 read_exp "0 * 0 * 0 * 0 + 0 + 0 + 0";

 {\out tokens: "0" "*" "0" "*" "0" "*" "0" "+" "0" "+" "0" "+" "0"}

 {\out raw: ("+" ("+" ("+" ("*" "0" ("*" "0" ("*" "0" "0"))) "0") "0") "0")}

 {\out \vdots}

 read_exp "0 + - 0 + 0";

 {\out tokens: "0" "+" "-" "0" "+" "0"}

 {\out raw: ("+" ("+" "0" ("-" "0")) "0")}

 {\out \vdots}

 \end{ttbox}

 The output of \ttindex{Syntax.test_read} includes the token list ({\tt

   tokens}) and the raw \AST{} directly derived from the parse tree,

 ignoring parse \AST{} translations.  The rest is tracing information

 provided by the macro expander (see \S\ref{sec:macros}).

 Executing {\tt Syntax.print_gram} reveals the productions derived from the

 above mixfix declarations (lots of additional information deleted):

 \begin{ttbox}

 Syntax.print_gram (syn_of EXP.thy);

 {\out exp = "0"  => "0" (9)}

 {\out exp = exp[0] "+" exp[1]  => "+" (0)}

 {\out exp = exp[3] "*" exp[2]  => "*" (2)}

 {\out exp = "-" exp[3]  => "-" (3)}

 \end{ttbox}

 \subsection{The mixfix template}

 Let us now take a closer look at the string $template$ appearing in mixfix

 annotations.  This string specifies a list of parsing and printing

 directives: delimiters\index{delimiters}, arguments, spaces, blocks of

 indentation and line breaks.  These are encoded by the following character

 sequences:

 \index{pretty printing|(}

 \begin{description}

 \item[~$d$~] is a delimiter, namely a non-empty sequence of characters

   other than the special characters {\tt _}, {\tt(}, {\tt)} and~{\tt/}.

   Even these characters may appear if escaped; this means preceding it with

   a~{\tt '} (single quote).  Thus you have to write {\tt ''} if you really

   want a single quote.  Delimiters may never contain spaces.

 \item[~{\tt_}~] is an argument position, which stands for a nonterminal symbol

   or name token.

 \item[~$s$~] is a non-empty sequence of spaces for printing.  This and the

   following specifications do not affect parsing at all.

 \item[~{\tt(}$n$~] opens a pretty printing block.  The optional number $n$

   specifies how much indentation to add when a line break occurs within the

   block.  If {\tt(} is not followed by digits, the indentation defaults

   to~0.

 \item[~{\tt)}~] closes a pretty printing block.

 \item[~{\tt//}~] forces a line break.

 \item[~{\tt/}$s$~] allows a line break.  Here $s$ stands for the string of

   spaces (zero or more) right after the {\tt /} character.  These spaces

   are printed if the break is not taken.

 \end{description}

 For example, the template {\tt"(_ +/ _)"} specifies an infix operator.

 There are two argument positions; the delimiter~{\tt+} is preceded by a

 space and followed by a space or line break; the entire phrase is a pretty

 printing block.  Other examples appear in Fig.\ts\ref{fig:set_trans} below.

 Isabelle's pretty printer resembles the one described in

 Paulson~\cite{paulson91}.

 \index{pretty printing|)}

 \subsection{Infixes}

 \indexbold{infixes}

 Infix operators associating to the left or right can be declared

 using {\tt infixl} or {\tt infixr}.

 Roughly speaking, the form {\tt $c$ ::\ "$\sigma$" (infixl $p$)}

 abbreviates the mixfix declarations

 \begin{ttbox}

 "op \(c\)" :: "\(\sigma\)"   ("(_ \(c\)/ _)" [\(p\), \(p+1\)] \(p\))

 "op \(c\)" :: "\(\sigma\)"   ("op \(c\)")

 \end{ttbox}

 and {\tt $c$ ::\ "$\sigma$" (infixr $p$)} abbreviates the mixfix declarations

 \begin{ttbox}

 "op \(c\)" :: "\(\sigma\)"   ("(_ \(c\)/ _)" [\(p+1\), \(p\)] \(p\))

 "op \(c\)" :: "\(\sigma\)"   ("op \(c\)")

 \end{ttbox}

 The infix operator is declared as a constant with the prefix {\tt op}.

 Thus, prefixing infixes with \sdx{op} makes them behave like ordinary

 function symbols, as in \ML.  Special characters occurring in~$c$ must be

 escaped, as in delimiters, using a single quote.

 \subsection{Binders}

 \indexbold{binders}

 \begingroup

 \def\Q{{\cal Q}}

 A {\bf binder} is a variable-binding construct such as a quantifier.  The

 constant declaration

 \begin{ttbox}

 \(c\) :: "\(\sigma\)"   (binder "\(\Q\)" \(p\))

 \end{ttbox}

 introduces a constant~$c$ of type~$\sigma$, which must have the form

 $(\tau@1 \To \tau@2) \To \tau@3$.  Its concrete syntax is $\Q~x.P$, where

 $x$ is a bound variable of type~$\tau@1$, the body~$P$ has type $\tau@2$

 and the whole term has type~$\tau@3$.  Special characters in $\Q$ must be

 escaped using a single quote.

 The declaration is expanded internally to something like

 \begin{ttbox}

 \(c\)    :: "(\(\tau@1\) => \(\tau@2\)) => \(\tau@3\)"

 "\(\Q\)"\hskip-3pt  :: "[idts, \(\tau@2\)] => \(\tau@3\)"   ("(3\(\Q\)_./ _)" \(p\))

 \end{ttbox}

 Here \ndx{idts} is the nonterminal symbol for a list of identifiers with

 \index{type constraints}

 optional type constraints (see Fig.\ts\ref{fig:pure_gram}).  The

 declaration also installs a parse translation\index{translations!parse}

 for~$\Q$ and a print translation\index{translations!print} for~$c$ to

 translate between the internal and external forms.

 A binder of type $(\sigma \To \tau) \To \tau$ can be nested by giving a

 list of variables.  The external form $\Q~x@1~x@2 \ldots x@n. P$

 corresponds to the internal form

 \[ c(\lambda x@1. c(\lambda x@2. \ldots c(\lambda x@n. P) \ldots)). \]

 \medskip

 For example, let us declare the quantifier~$\forall$:\index{quantifiers}

 \begin{ttbox}

 All :: "('a => o) => o"   (binder "ALL " 10)

 \end{ttbox}

 This lets us write $\forall x.P$ as either {\tt All(\%$x$.$P$)} or {\tt ALL

   $x$.$P$}.  When printing, Isabelle prefers the latter form, but must fall

 back on ${\tt All}(P)$ if $P$ is not an abstraction.  Both $P$ and {\tt ALL

   $x$.$P$} have type~$o$, the type of formulae, while the bound variable

 can be polymorphic.

 \endgroup

 \index{mixfix declarations|)}

 \section{Ambiguity of parsed expressions} \label{sec:ambiguity}

 \index{ambiguity!of parsed expressions}

 To keep the grammar small and allow common productions to be shared

 all logical types (except {\tt prop}) are internally represented

 by one nonterminal, namely {\tt logic}. This and omitted or too freely

 chosen priorities may lead to ways of parsing an expression that were

 not intended by the theory's maker. In most cases Isabelle is able to

 select one of multiple parse trees that an expression has lead

 to by checking which of them can be typed correctly. But this may not

 work in every case and always slows down parsing.

 The warning and error messages that can be produced during this process are

 as follows:

 If an ambiguity can be resolved by type inference this warning

 is shown to remind the user that parsing is (unnecessarily) slowed

 down:

 \begin{ttbox}

 {\out Warning: Ambiguous input "..."}

 {\out produces the following parse trees:}

 {\out ...}

 {\out Fortunately, only one parse tree is type correct.}

 {\out It helps (speed!) if you disambiguate your grammar or your input.}

 \end{ttbox}

 The following message is normally caused by using the same

 syntax in two different productions:

 \begin{ttbox}

 {\out Warning: Ambiguous input "..."}

 {\out produces the following parse trees:}

 {\out ...}

 {\out Error: More than one term is type correct:}

 {\out ...}

 \end{ttbox}

 Ambiguities occuring in syntax translation rules cannot be resolved by

 type inference because it is not necessary for these rules to be type

 correct. Therefore Isabelle always generates an error message and the

 ambiguity should be eliminated by changing the grammar or the rule.

 \section{Example: some minimal logics} \label{sec:min_logics}

 \index{examples!of logic definitions}

 This section presents some examples that have a simple syntax.  They

 demonstrate how to define new object-logics from scratch.

 First we must define how an object-logic syntax is embedded into the

 meta-logic.  Since all theorems must conform to the syntax for~\ndx{prop}

 (see Fig.\ts\ref{fig:pure_gram}), that syntax has to be extended with the

 object-level syntax.  Assume that the syntax of your object-logic defines a

 meta-type~\tydx{o} of formulae which refers to the nonterminal {\tt logic}.

 These formulae can now appear in axioms and theorems wherever \ndx{prop} does

 if you add the production

 \[ prop ~=~ logic. \]

 This is not supposed to be a copy production but an implicit coercion from

 formulae to propositions:

 \begin{ttbox}

 Base = Pure +

 types

   o

 arities

   o :: logic

 consts

   Trueprop :: "o => prop"   ("_" 5)

 end

 \end{ttbox}

 The constant \cdx{Trueprop} (the name is arbitrary) acts as an invisible

 coercion function.  Assuming this definition resides in a file {\tt Base.thy},

 you have to load it with the command {\tt use_thy "Base"}.

 One of the simplest nontrivial logics is {\bf minimal logic} of

 implication.  Its definition in Isabelle needs no advanced features but

 illustrates the overall mechanism nicely:

 \begin{ttbox}

 Hilbert = Base +

 consts

   "-->" :: "[o, o] => o"   (infixr 10)

 rules

   K     "P --> Q --> P"

   S     "(P --> Q --> R) --> (P --> Q) --> P --> R"

   MP    "[| P --> Q; P |] ==> Q"

 end

 \end{ttbox}

 After loading this definition from the file {\tt Hilbert.thy}, you can

 start to prove theorems in the logic:

 \begin{ttbox}

 goal Hilbert.thy "P --> P";

 {\out Level 0}

 {\out P --> P}

 {\out  1.  P --> P}

 \ttbreak

 by (resolve_tac [Hilbert.MP] 1);

 {\out Level 1}

 {\out P --> P}

 {\out  1.  ?P --> P --> P}

 {\out  2.  ?P}

 \ttbreak

 by (resolve_tac [Hilbert.MP] 1);

 {\out Level 2}

 {\out P --> P}

 {\out  1.  ?P1 --> ?P --> P --> P}

 {\out  2.  ?P1}

 {\out  3.  ?P}

 \ttbreak

 by (resolve_tac [Hilbert.S] 1);

 {\out Level 3}

 {\out P --> P}

 {\out  1.  P --> ?Q2 --> P}

 {\out  2.  P --> ?Q2}

 \ttbreak

 by (resolve_tac [Hilbert.K] 1);

 {\out Level 4}

 {\out P --> P}

 {\out  1.  P --> ?Q2}

 \ttbreak

 by (resolve_tac [Hilbert.K] 1);

 {\out Level 5}

 {\out P --> P}

 {\out No subgoals!}

 \end{ttbox}

 As we can see, this Hilbert-style formulation of minimal logic is easy to

 define but difficult to use.  The following natural deduction formulation is

 better:

 \begin{ttbox}

 MinI = Base +

 consts

   "-->" :: "[o, o] => o"   (infixr 10)

 rules

   impI  "(P ==> Q) ==> P --> Q"

   impE  "[| P --> Q; P |] ==> Q"

 end

 \end{ttbox}

 Note, however, that although the two systems are equivalent, this fact

 cannot be proved within Isabelle.  Axioms {\tt S} and {\tt K} can be

 derived in {\tt MinI} (exercise!), but {\tt impI} cannot be derived in {\tt

   Hilbert}.  The reason is that {\tt impI} is only an {\bf admissible} rule

 in {\tt Hilbert}, something that can only be shown by induction over all

 possible proofs in {\tt Hilbert}.

 We may easily extend minimal logic with falsity:

 \begin{ttbox}

 MinIF = MinI +

 consts

   False :: "o"

 rules

   FalseE "False ==> P"

 end

 \end{ttbox}

 On the other hand, we may wish to introduce conjunction only:

 \begin{ttbox}

 MinC = Base +

 consts

   "&" :: "[o, o] => o"   (infixr 30)

 \ttbreak

 rules

   conjI  "[| P; Q |] ==> P & Q"

   conjE1 "P & Q ==> P"

   conjE2 "P & Q ==> Q"

 end

 \end{ttbox}

 And if we want to have all three connectives together, we create and load a

 theory file consisting of a single line:\footnote{We can combine the

   theories without creating a theory file using the ML declaration

 \begin{ttbox}

 val MinIFC_thy = merge_theories(MinIF,MinC)

 \end{ttbox}

 \index{*merge_theories|fnote}}

 \begin{ttbox}

 MinIFC = MinIF + MinC

 \end{ttbox}

 Now we can prove mixed theorems like

 \begin{ttbox}

 goal MinIFC.thy "P & False --> Q";

 by (resolve_tac [MinI.impI] 1);

 by (dresolve_tac [MinC.conjE2] 1);

 by (eresolve_tac [MinIF.FalseE] 1);

 \end{ttbox}

 Try this as an exercise!