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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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HOL's description operator $\varepsilon x.P\,x$ has type $(\alpha\To

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bool)\To\alpha$ and can bind only one variable, except when $\alpha$ is

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$bool$. 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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