%% $Id$

\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}

$prop$ &=& {\tt PROP} $aprop$ ~~$|$~~ {\tt(} $prop$ {\tt)} \\

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

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

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

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

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

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

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

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

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

    &$|$& $fun^{(\infty)}$ {\tt(} $logic$ {\tt,} \dots {\tt,} $logic$ {\tt)} \\

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

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

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

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

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

$type$ &=& $tid$ ~~$|$~~ $tvar$ ~~$|$~~ $tid$ {\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)\\

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

$sort$ &=& $id$ ~~$|$~~ {\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{prop}] denotes terms of type {\tt prop}.  These are formulae

  of the meta-logic.

\item[\ndxbold{aprop}] denotes atomic propositions.  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}.  As the syntax is extended by new object-logics, more

  productions for {\tt logic} are added automatically (see below).

  \item[\ndxbold{fun}] denotes terms potentially of function type.

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

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

    by types.

\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}

\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, $\lambda$-abstractions and

applications.

More precisely, for each type constructor $ty$ with arity $(\vec{s})c$,

where $c$ is a subclass of \cldx{logic}, several productions are added:

\begin{center}

\begin{tabular}{rclc}

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

  &$|$& $fun^{(\infty)}$ {\tt(} $logic$ {\tt,} \dots {\tt,} $logic$ {\tt)}\\

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

$logic$ &=& $ty$

\end{tabular}

\end{center}

\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 4 or less, in

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

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

\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 four

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

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

They are denoted by \ndxbold{id}, \ndxbold{var}, \ndxbold{tid},

\ndxbold{tvar}, respectively.  Typical examples are {\tt x}, {\tt ?x7},

{\tt 'a}, {\tt ?'a3}.  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 \\[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*}

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}.

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 and newlines are separators; they

never occur within tokens.

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.

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

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{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, roots 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 roots: logic type fun prop}

{\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 roots} 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 roots}] lists the grammar's nonterminal symbols.  You must

    name the desired root when calling lower level functions or specifying

    macros.  Higher level functions usually expect a type and derive the

    actual root as described in~\S\ref{sec:grammar}.

  \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, binders and so forth.

\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 identified with

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.

A simple type is typically declared for each nonterminal symbol.  In

first-order logic, type~$i$ stands for terms and~$o$ for formulae.  Only

the outermost type constructor is taken into account.  For example, any

type of the form $\sigma list$ stands for a list;  productions may refer

to the symbol {\tt list} and will apply to lists of any type.

The symbol associated with a type is called its {\bf root} since it may

serve as the root of a parse tree.  Precisely, the root of $(\tau@1, \dots,

\tau@n)ty$ is $ty$, where $\tau@1$, \ldots, $\tau@n$ are types and $ty$ is

a type constructor.  Type infixes are a special case of this; in

particular, the root of $\tau@1 \To \tau@2$ is {\tt fun}.  Finally, the

root of a type variable is {\tt logic}; general productions might

refer to this nonterminal.

Identifying 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.  There

are two exceptions to this treatment of constants:

\begin{enumerate}\index{constants!syntactic}

  \item A production need not map directly to a logical function.  In this

    case, you must declare a constant whose purpose is purely syntactic.

    By convention such constants begin with the symbol~{\tt\at}, 

    ensuring that they can never be written in formulae.

  \item A copy production has no associated constant.\index{productions!copy}

\end{enumerate}

There is something artificial about this representation of productions,

but it is convenient, particularly for simple theory extensions.

\subsection{The general mixfix form}

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

line occurs within the {\tt consts} 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 $\sigma$.

  \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

    above.  Any of these may be function types; the corresponding root is

    then \tydx{fun}.

  \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 declaration {\tt $c$ ::\ "$\sigma$" ("$template$")} 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 will introduce 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~$c$.

\begin{warn}

  Theories must sometimes declare types for purely syntactic purposes.  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 allow their use in arbitrary Isabelle

  expressions~(\S\ref{logical-types}).

\end{warn}

\subsection{Example: arithmetic expressions}

\index{examples!of mixfix declarations}

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

declarations encoding the priority grammar from

\S\ref{sec:priority_grammars}:

\begin{ttbox}

EXP = Pure +

types

  exp

arities

  exp :: logic

consts

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

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

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

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

end

\end{ttbox}

The {\tt arities} declaration causes {\tt exp} to be added as a new root.

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 our 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 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 constant declarations

\begin{ttbox}

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

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

\end{ttbox}

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

\begin{ttbox}

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

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

\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.

The expanded forms above would be actually illegal in a {\tt .thy} file

because they declare the constant \hbox{\tt"op \(c\)"} twice.

\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

\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{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 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

nonterminal symbol~\ndx{o} of formulae.  These formulae can now appear in

axioms and theorems wherever \ndx{prop} does if you add the production

\[ prop ~=~ o. \]

This is not a copy production but a 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!