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