(*<*)theory Overloading2 = Overloading1:(*>*)

text{*

Of course this is not the only possible definition of the two relations.

Componentwise comparison of lists of equal length also makes sense. This time

the elements of the list must also be of class @{text ordrel} to permit their

comparison:

*}

instance list :: (ordrel)ordrel

by intro_classes

defs (overloaded)

le_list_def: "xs <<= (ys::'a::ordrel list) \<equiv>

              size xs = size ys \<and> (\<forall>i<size xs. xs!i <<= ys!i)"

text{*\noindent

The infix function @{text"!"} yields the nth element of a list.

*}

subsubsection{*Predefined Overloading*}

text{*

HOL comes with a number of overloaded constants and corresponding classes.

The most important ones are listed in Table~\ref{tab:overloading}. They are

defined on all numeric types and sometimes on other types as well, for example

@{text"-"}, @{text"\<le>"} and @{text"<"} on sets.

\begin{table}[htbp]

\begin{center}

\begin{tabular}{lll}

Constant & Type & Syntax \\

\hline

@{term 0} & @{text "'a::zero"} \\

@{text"+"} & @{text "('a::plus) \<Rightarrow> 'a \<Rightarrow> 'a"} & (infixl 65) \\

@{text"-"} & @{text "('a::minus) \<Rightarrow> 'a \<Rightarrow> 'a"} &  (infixl 65) \\

@{text"-"} & @{text "('a::minus) \<Rightarrow> 'a"} \\

@{text"*"} & @{text "('a::times) \<Rightarrow> 'a \<Rightarrow> 'a"} & (infixl 70) \\

@{text div} & @{text "('a::div) \<Rightarrow> 'a \<Rightarrow> 'a"} & (infixl 70) \\

@{text mod} & @{text "('a::div) \<Rightarrow> 'a \<Rightarrow> 'a"} & (infixl 70) \\

@{text dvd} & @{text "('a::times) \<Rightarrow> 'a \<Rightarrow> bool"} & (infixl 50) \\

@{text"/"}  & @{text "('a::inverse) \<Rightarrow> 'a \<Rightarrow> 'a"} & (infixl 70) \\

@{text"^"} & @{text "('a::power) \<Rightarrow> nat \<Rightarrow> 'a"} & (infixr 80) \\

@{term abs} &  @{text "('a::minus) \<Rightarrow> 'a"} & ${\mid} x {\mid}$\\

@{text"\<le>"} & @{text "('a::ord) \<Rightarrow> 'a \<Rightarrow> bool"} & (infixl 50) \\

@{text"<"} & @{text "('a::ord) \<Rightarrow> 'a \<Rightarrow> bool"} & (infixl 50) \\

@{term min} &  @{text "('a::ord) \<Rightarrow> 'a \<Rightarrow> 'a"} \\

@{term max} &  @{text "('a::ord) \<Rightarrow> 'a \<Rightarrow> 'a"} \\

@{term Least} & @{text "('a::ord \<Rightarrow> bool) \<Rightarrow> 'a"} &

@{text LEAST}$~x.~P$

\end{tabular}

\caption{Overloaded Constants in HOL}

\label{tab:overloading}

\end{center}

\end{table}

In addition there is a special input syntax for bounded quantifiers:

\begin{center}

\begin{tabular}{lcl}

@{text"\<forall>x \<le> y. P x"} & @{text"\<equiv>"} & @{prop"\<forall>x. x \<le> y \<longrightarrow> P x"} \\

@{text"\<exists>x \<le> y. P x"} & @{text"\<equiv>"} & @{prop"\<exists>x. x \<le> y \<and> P x"}

\end{tabular}

\end{center}

And analogously for @{text"<"} instead of @{text"\<le>"}.

The form on the left is translated into the one on the right upon input but it is not

translated back upon output.

*}(*<*)end(*>*)