doc-src/Logics/syntax.tex
author wenzelm
Mon Aug 28 13:52:38 2000 +0200 (2000-08-28)
changeset 9695 ec7d7f877712
parent 6120 f40d61cd6b32
child 14209 180cd69a5dbb
permissions -rw-r--r--
proper setup of iman.sty/extra.sty/ttbox.sty;
paulson@6120
     1
%% $Id$
paulson@6120
     2
%% THIS FILE IS COMMON TO ALL LOGIC MANUALS
paulson@6120
     3
paulson@6120
     4
\chapter{Syntax definitions}
paulson@6120
     5
The syntax of each logic is presented using a context-free grammar.
paulson@6120
     6
These grammars obey the following conventions:
paulson@6120
     7
\begin{itemize}
paulson@6120
     8
\item identifiers denote nonterminal symbols
paulson@6120
     9
\item \texttt{typewriter} font denotes terminal symbols
paulson@6120
    10
\item parentheses $(\ldots)$ express grouping
paulson@6120
    11
\item constructs followed by a Kleene star, such as $id^*$ and $(\ldots)^*$
paulson@6120
    12
can be repeated~0 or more times 
paulson@6120
    13
\item alternatives are separated by a vertical bar,~$|$
paulson@6120
    14
\item the symbol for alphanumeric identifiers is~{\it id\/} 
paulson@6120
    15
\item the symbol for scheme variables is~{\it var}
paulson@6120
    16
\end{itemize}
paulson@6120
    17
To reduce the number of nonterminals and grammar rules required, Isabelle's
paulson@6120
    18
syntax module employs {\bf priorities},\index{priorities} or precedences.
paulson@6120
    19
Each grammar rule is given by a mixfix declaration, which has a priority,
paulson@6120
    20
and each argument place has a priority.  This general approach handles
paulson@6120
    21
infix operators that associate either to the left or to the right, as well
paulson@6120
    22
as prefix and binding operators.
paulson@6120
    23
paulson@6120
    24
In a syntactically valid expression, an operator's arguments never involve
paulson@6120
    25
an operator of lower priority unless brackets are used.  Consider
paulson@6120
    26
first-order logic, where $\exists$ has lower priority than $\disj$,
paulson@6120
    27
which has lower priority than $\conj$.  There, $P\conj Q \disj R$
paulson@6120
    28
abbreviates $(P\conj Q) \disj R$ rather than $P\conj (Q\disj R)$.  Also,
paulson@6120
    29
$\exists x.P\disj Q$ abbreviates $\exists x.(P\disj Q)$ rather than
paulson@6120
    30
$(\exists x.P)\disj Q$.  Note especially that $P\disj(\exists x.Q)$
paulson@6120
    31
becomes syntactically invalid if the brackets are removed.
paulson@6120
    32
paulson@6120
    33
A {\bf binder} is a symbol associated with a constant of type
wenzelm@9695
    34
$(\sigma\To\tau)\To\tau'$.  For instance, we may declare~$\forall$ as a binder
wenzelm@9695
    35
for the constant~$All$, which has type $(\alpha\To o)\To o$.  This defines the
wenzelm@9695
    36
syntax $\forall x.t$ to mean $All(\lambda x.t)$.  We can also write $\forall
wenzelm@9695
    37
x@1\ldots x@m.t$ to abbreviate $\forall x@1.  \ldots \forall x@m.t$; this is
wenzelm@9695
    38
possible for any constant provided that $\tau$ and $\tau'$ are the same type.
wenzelm@9695
    39
HOL's description operator $\varepsilon x.P\,x$ has type $(\alpha\To
wenzelm@9695
    40
bool)\To\alpha$ and can bind only one variable, except when $\alpha$ is
wenzelm@9695
    41
$bool$.  ZF's bounded quantifier $\forall x\in A.P(x)$ cannot be declared as a
wenzelm@9695
    42
binder because it has type $[i, i\To o]\To o$.  The syntax for binders allows
paulson@6120
    43
type constraints on bound variables, as in
paulson@6120
    44
\[ \forall (x{::}\alpha) \; (y{::}\beta) \; z{::}\gamma. Q(x,y,z) \]
paulson@6120
    45
paulson@6120
    46
To avoid excess detail, the logic descriptions adopt a semi-formal style.
paulson@6120
    47
Infix operators and binding operators are listed in separate tables, which
paulson@6120
    48
include their priorities.  Grammar descriptions do not include numeric
paulson@6120
    49
priorities; instead, the rules appear in order of decreasing priority.
paulson@6120
    50
This should suffice for most purposes; for full details, please consult the
paulson@6120
    51
actual syntax definitions in the {\tt.thy} files.
paulson@6120
    52
paulson@6120
    53
Each nonterminal symbol is associated with some Isabelle type.  For
paulson@6120
    54
example, the formulae of first-order logic have type~$o$.  Every
paulson@6120
    55
Isabelle expression of type~$o$ is therefore a formula.  These include
paulson@6120
    56
atomic formulae such as $P$, where $P$ is a variable of type~$o$, and more
paulson@6120
    57
generally expressions such as $P(t,u)$, where $P$, $t$ and~$u$ have
paulson@6120
    58
suitable types.  Therefore, `expression of type~$o$' is listed as a
paulson@6120
    59
separate possibility in the grammar for formulae.
paulson@6120
    60
paulson@6120
    61