%% $Id$

\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}

\subsection{Logical types and default syntax}\label{logical-types}

\index{lambda calc@$\lambda$-calculus}

Isabelle's representation of mathematical languages is based on the

simply typed $\lambda$-calculus.  All logical types, namely those of

class \cldx{logic}, are automatically equipped with a basic syntax of

types, identifiers, variables, parentheses, $\lambda$-abstraction and

application.

\begin{warn}

  Isabelle combines the syntaxes for all types of class \cldx{logic} by

  mapping all those types to the single nonterminal $logic$.  Thus all

  productions of $logic$, in particular $id$, $var$ etc, become available.

\end{warn}

\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.

\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~$c$.

\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}

Note that because {\tt exp} is not of class {\tt logic}, it has been retained

as a separate nonterminal. This also entails that the syntax does not provide

for identifiers or paranthesized expressions. Normally you would also want to

add the declaration {\tt arities exp :: logic} and use {\tt consts} instead

of {\tt syntax}. Try this as an exercise and study the changes in the

grammar.

\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.  Furthermore, a~{\tt '} followed by a space separates

  delimiters without extra white space being added for printing.

\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 "\(\Q\)" [\(pb\)] \(p\))

\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~$\tau@3$. The optional integer $pb$

specifies the body's priority, by default~$p$.  Special characters

in $\Q$ must be escaped using a single quote.

The declaration is expanded internally to something like

\begin{ttbox}

\(c\)\hskip3pt    :: (\(\tau@1\) => \(\tau@2\)) => \(\tau@3\)

"\(\Q\)"  :: [idts, \(\tau@2\)] => \(\tau@3\)   ("(3\(\Q\)_./ _)" [0, \(pb\)] \(p\))

\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 the following

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

slowed down. In cases where it's not easily possible to eliminate the

ambiguity the frequency of the warning can be controlled by changing

the value of {\tt Syntax.ambiguity_level} which has type {\tt int

ref}. Its default value is 1 and by increasing it one can control how

many parse trees are necessary to generate the warning.

\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!