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%% $Id$


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%% THIS FILE IS COMMON TO ALL LOGIC MANUALS


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\chapter{Syntax definitions}


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The syntax of each logic is presented using a contextfree grammar.


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These grammars obey the following conventions:


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\begin{itemize}


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\item identifiers denote nonterminal symbols


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\item \texttt{typewriter} font denotes terminal symbols


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\item parentheses $(\ldots)$ express grouping


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


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can be repeated~0 or more times


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


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


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


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\end{itemize}


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


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


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


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


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


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as prefix and binding operators.


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


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an operator of lower priority unless brackets are used. Consider


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firstorder logic, where $\exists$ has lower priority than $\disj$,


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


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


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


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


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becomes syntactically invalid if the brackets are removed.


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

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


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


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syntax $\forall x.t$ to mean $All(\lambda x.t)$. We can also write $\forall


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x@1\ldots x@m.t$ to abbreviate $\forall x@1. \ldots \forall x@m.t$; this is


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possible for any constant provided that $\tau$ and $\tau'$ are the same type.

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The Hilbert description operator $\varepsilon x.P\,x$ has type $(\alpha\To


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bool)\To\alpha$ and normally binds only one variable.


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ZF's bounded quantifier $\forall x\in A.P(x)$ cannot be declared as a

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

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type constraints on bound variables, as in


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


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To avoid excess detail, the logic descriptions adopt a semiformal style.


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


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include their priorities. Grammar descriptions do not include numeric


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


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


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


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Each nonterminal symbol is associated with some Isabelle type. For


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example, the formulae of firstorder logic have type~$o$. Every


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


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


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


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


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separate possibility in the grammar for formulae.


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