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(*<*)theory Overloading2 imports Overloading1 begin(*>*)
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text{*
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Of course this is not the only possible definition of the two relations.
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Componentwise comparison of lists of equal length also makes sense. This time
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the elements of the list must also be of class @{text ordrel} to permit their
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comparison:
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*}
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instance list :: (ordrel)ordrel
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by intro_classes
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defs (overloaded)
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le_list_def: "xs <<= (ys::'a::ordrel list) \<equiv>
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size xs = size ys \<and> (\<forall>i<size xs. xs!i <<= ys!i)"
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text{*\noindent
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The infix function @{text"!"} yields the nth element of a list.
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\begin{warn}
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A type constructor can be instantiated in only one way to
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a given type class. For example, our two instantiations of @{text list} must
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reside in separate theories with disjoint scopes.
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\end{warn}
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*}
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subsubsection{*Predefined Overloading*}
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text{*
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HOL comes with a number of overloaded constants and corresponding classes.
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The most important ones are listed in Table~\ref{tab:overloading} in the appendix. They are
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defined on all numeric types and sometimes on other types as well, for example
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$-$ and @{text"\<le>"} on sets.
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In addition there is a special syntax for bounded quantifiers:
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\begin{center}
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\begin{tabular}{lcl}
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@{prop"\<forall>x \<le> y. P x"} & @{text"\<rightleftharpoons>"} & @{prop [source] "\<forall>x. x \<le> y \<longrightarrow> P x"} \\
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@{prop"\<exists>x \<le> y. P x"} & @{text"\<rightleftharpoons>"} & @{prop [source] "\<exists>x. x \<le> y \<and> P x"}
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\end{tabular}
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\end{center}
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And analogously for @{text"<"} instead of @{text"\<le>"}.
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*}(*<*)end(*>*)
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