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doc-src/Ref/defining.tex

author | nipkow |

Thu, 07 Jul 1994 19:22:49 +0200 | |

changeset 452 | 395bbf6e55f9 |

parent 332 | 01b87a921967 |

child 711 | bb868a30e66f |

permissions | -rw-r--r-- |

changed priority of ::

%% $Id$ \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!