2 %% THIS FILE IS COMMON TO ALL LOGIC MANUALS
4 \chapter{Syntax definitions}
5 The syntax of each logic is presented using a context-free grammar.
6 These grammars obey the following conventions:
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}
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.
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.
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 \[ \forall (x{::}\alpha) \; (y{::}\beta) \; z{::}\gamma. Q(x,y,z) \]
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.
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.