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doc-src/Logics/syntax.tex

author | haftmann |

Tue, 13 Jul 2010 11:38:04 +0200 | |

changeset 37788 | 261c61fabc98 |

parent 14209 | 180cd69a5dbb |

child 42637 | 381fdcab0f36 |

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

corrected title

%% $Id$ %% 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. The Hilbert description operator $\varepsilon x.P\,x$ has type $(\alpha\To bool)\To\alpha$ and normally binds only one variable. 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.