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doc-src/Logics/syntax.tex

author | haftmann |

Wed Dec 27 19:10:06 2006 +0100 (2006-12-27) | |

changeset 21915 | 4e63c55f4cb4 |

parent 14209 | 180cd69a5dbb |

child 42637 | 381fdcab0f36 |

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

different handling of type variable names

1 %% $Id$

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:

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.

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.