doc-src/Logics/syntax.tex
author paulson
Fri Sep 26 11:04:21 2003 +0200 (2003-09-26)
changeset 14209 180cd69a5dbb
parent 9695 ec7d7f877712
child 42637 381fdcab0f36
permissions -rw-r--r--
tweak
     1 %% $Id$
     2 %% THIS FILE IS COMMON TO ALL LOGIC MANUALS
     3 
     4 \chapter{Syntax definitions}
     5 The syntax of each logic is presented using a context-free grammar.
     6 These grammars obey the following conventions:
     7 \begin{itemize}
     8 \item identifiers denote nonterminal symbols
     9 \item \texttt{typewriter} font denotes terminal symbols
    10 \item parentheses $(\ldots)$ express grouping
    11 \item constructs followed by a Kleene star, such as $id^*$ and $(\ldots)^*$
    12 can be repeated~0 or more times 
    13 \item alternatives are separated by a vertical bar,~$|$
    14 \item the symbol for alphanumeric identifiers is~{\it id\/} 
    15 \item the symbol for scheme variables is~{\it var}
    16 \end{itemize}
    17 To reduce the number of nonterminals and grammar rules required, Isabelle's
    18 syntax module employs {\bf priorities},\index{priorities} or precedences.
    19 Each grammar rule is given by a mixfix declaration, which has a priority,
    20 and each argument place has a priority.  This general approach handles
    21 infix operators that associate either to the left or to the right, as well
    22 as prefix and binding operators.
    23 
    24 In a syntactically valid expression, an operator's arguments never involve
    25 an operator of lower priority unless brackets are used.  Consider
    26 first-order logic, where $\exists$ has lower priority than $\disj$,
    27 which has lower priority than $\conj$.  There, $P\conj Q \disj R$
    28 abbreviates $(P\conj Q) \disj R$ rather than $P\conj (Q\disj R)$.  Also,
    29 $\exists x.P\disj Q$ abbreviates $\exists x.(P\disj Q)$ rather than
    30 $(\exists x.P)\disj Q$.  Note especially that $P\disj(\exists x.Q)$
    31 becomes syntactically invalid if the brackets are removed.
    32 
    33 A {\bf binder} is a symbol associated with a constant of type
    34 $(\sigma\To\tau)\To\tau'$.  For instance, we may declare~$\forall$ as a binder
    35 for the constant~$All$, which has type $(\alpha\To o)\To o$.  This defines the
    36 syntax $\forall x.t$ to mean $All(\lambda x.t)$.  We can also write $\forall
    37 x@1\ldots x@m.t$ to abbreviate $\forall x@1.  \ldots \forall x@m.t$; this is
    38 possible for any constant provided that $\tau$ and $\tau'$ are the same type.
    39 The Hilbert description operator $\varepsilon x.P\,x$ has type $(\alpha\To
    40 bool)\To\alpha$ and normally binds only one variable.  
    41 ZF's bounded quantifier $\forall x\in A.P(x)$ cannot be declared as a
    42 binder because it has type $[i, i\To o]\To o$.  The syntax for binders allows
    43 type constraints on bound variables, as in
    44 \[ \forall (x{::}\alpha) \; (y{::}\beta) \; z{::}\gamma. Q(x,y,z) \]
    45 
    46 To avoid excess detail, the logic descriptions adopt a semi-formal style.
    47 Infix operators and binding operators are listed in separate tables, which
    48 include their priorities.  Grammar descriptions do not include numeric
    49 priorities; instead, the rules appear in order of decreasing priority.
    50 This should suffice for most purposes; for full details, please consult the
    51 actual syntax definitions in the {\tt.thy} files.
    52 
    53 Each nonterminal symbol is associated with some Isabelle type.  For
    54 example, the formulae of first-order logic have type~$o$.  Every
    55 Isabelle expression of type~$o$ is therefore a formula.  These include
    56 atomic formulae such as $P$, where $P$ is a variable of type~$o$, and more
    57 generally expressions such as $P(t,u)$, where $P$, $t$ and~$u$ have
    58 suitable types.  Therefore, `expression of type~$o$' is listed as a
    59 separate possibility in the grammar for formulae.
    60 
    61