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%% THIS FILE IS COMMON TO ALL LOGIC MANUALS

\chapter{Syntax definitions}

The syntax of each logic is presented using a context-free grammar.

These grammars obey the following conventions:

\begin{itemize}

\item identifiers denote nonterminal symbols

\item \texttt{typewriter} font denotes terminal symbols

\item parentheses $(\ldots)$ express grouping

\item constructs followed by a Kleene star, such as $id^*$ and $(\ldots)^*$

can be repeated~0 or more times 

\item alternatives are separated by a vertical bar,~$|$

\item the symbol for alphanumeric identifiers is~{\it id\/} 

\item the symbol for scheme variables is~{\it var}

\end{itemize}

To reduce the number of nonterminals and grammar rules required, Isabelle's

syntax module employs {\bf priorities},\index{priorities} or precedences.

Each grammar rule is given by a mixfix declaration, which has a priority,

and each argument place has a priority.  This general approach handles

infix operators that associate either to the left or to the right, as well

as prefix and binding operators.

In a syntactically valid expression, an operator's arguments never involve

an operator of lower priority unless brackets are used.  Consider

first-order logic, where $\exists$ has lower priority than $\disj$,

which has lower priority than $\conj$.  There, $P\conj Q \disj R$

abbreviates $(P\conj Q) \disj R$ rather than $P\conj (Q\disj R)$.  Also,

$\exists x.P\disj Q$ abbreviates $\exists x.(P\disj Q)$ rather than

$(\exists x.P)\disj Q$.  Note especially that $P\disj(\exists x.Q)$

becomes syntactically invalid if the brackets are removed.

A {\bf binder} is a symbol associated with a constant of type

$(\sigma\To\tau)\To\tau'$.  For instance, we may declare~$\forall$ as

a binder for the constant~$All$, which has type $(\alpha\To o)\To o$.

This defines the syntax $\forall x.t$ to mean $All(\lambda x.t)$.  We

can also write $\forall x@1\ldots x@m.t$ to abbreviate $\forall x@1.

\ldots \forall x@m.t$; this is possible for any constant provided that

$\tau$ and $\tau'$ are the same type.  \HOL's description operator

$\varepsilon x.P\,x$ has type $(\alpha\To bool)\To\alpha$ and can bind

only one variable, except when $\alpha$ is $bool$.  \ZF's bounded

quantifier $\forall x\in A.P(x)$ cannot be declared as a binder

because it has type $[i, i\To o]\To o$.  The syntax for binders allows

type constraints on bound variables, as in

\[ \forall (x{::}\alpha) \; (y{::}\beta) \; z{::}\gamma. Q(x,y,z) \]

To avoid excess detail, the logic descriptions adopt a semi-formal style.

Infix operators and binding operators are listed in separate tables, which

include their priorities.  Grammar descriptions do not include numeric

priorities; instead, the rules appear in order of decreasing priority.

This should suffice for most purposes; for full details, please consult the

actual syntax definitions in the {\tt.thy} files.

Each nonterminal symbol is associated with some Isabelle type.  For

example, the formulae of first-order logic have type~$o$.  Every

Isabelle expression of type~$o$ is therefore a formula.  These include

atomic formulae such as $P$, where $P$ is a variable of type~$o$, and more

generally expressions such as $P(t,u)$, where $P$, $t$ and~$u$ have

suitable types.  Therefore, `expression of type~$o$' is listed as a

separate possibility in the grammar for formulae.