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     1 \chapter{More about Types}
     3 So far we have learned about a few basic types (for example \isa{bool} and
     4 \isa{nat}), type abbreviations (\isacommand{types}) and recursive datatpes
     5 (\isacommand{datatype}). This chapter will introduce the following more
     6 advanced material:
     7 \begin{itemize}
     8 \item More about basic types: numbers ({\S}\ref{sec:numbers}), pairs
     9   ({\S}\ref{sec:products}) and records ({\S}\ref{sec:records}), and how to reason
    10   about them.
    11 \item Introducing your own types: how to introduce your own new types that
    12   cannot be constructed with any of the basic methods ({\S}\ref{sec:typedef}).
    13 \item Type classes: how to specify and reason about axiomatic collections of
    14   types ({\S}\ref{sec:axclass}).
    15 \end{itemize}
    17 \section{Axiomatic type classes}
    18 \label{sec:axclass}
    19 \index{axiomatic type class|(}
    20 \index{*axclass|(}
    23 The programming language Haskell has popularized the notion of type classes.
    24 Isabelle offers the related concept of an \textbf{axiomatic type class}.
    25 Roughly speaking, an axiomatic type class is a type class with axioms, i.e.\ 
    26 an axiomatic specification of a class of types. Thus we can talk about a type
    27 $t$ being in a class $c$, which is written $\tau :: c$.  This is the case of
    28 $\tau$ satisfies the axioms of $c$. Furthermore, type classes can be
    29 organized in a hierarchy. Thus there is the notion of a class $d$ being a
    30 \textbf{subclass} of a class $c$, written $d < c$. This is the case if all
    31 axioms of $c$ are also provable in $d$. Let us now introduce these concepts
    32 by means of a running example, ordering relations.
    34 \subsection{Overloading}
    35 \label{sec:overloading}
    36 \index{overloading|(}
    38 \input{Types/document/Overloading0}
    39 \input{Types/document/Overloading1}
    40 \input{Types/document/Overloading}
    41 \input{Types/document/Overloading2}
    43 \index{overloading|)}
    45 Finally we should remind our readers that \isa{Main} contains a much more
    46 developed theory of orderings phrased in terms of the usual $\leq$ and
    47 \isa{<}. It is recommended that, if possible, you base your own
    48 ordering relations on this theory.
    50 \index{axiomatic type class|)}
    51 \index{*axclass|)}