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\chapter{More about Types}

So far we have learned about a few basic types (for example \isa{bool} and
\isa{nat}), type abbreviations (\isacommand{types}) and recursive datatpes
(\isacommand{datatype}). This chapter will introduce the following more
advanced material:
\item More about basic types: numbers ({\S}\ref{sec:numbers}), pairs
  ({\S}\ref{sec:products}) and records ({\S}\ref{sec:records}), and how to reason
  about them.
\item Introducing your own types: how to introduce your own new types that
  cannot be constructed with any of the basic methods ({\S}\ref{sec:typedef}).
\item Type classes: how to specify and reason about axiomatic collections of
  types ({\S}\ref{sec:axclass}).

\section{Axiomatic type classes}
\index{axiomatic type class|(}

The programming language Haskell has popularized the notion of type classes.
Isabelle offers the related concept of an \textbf{axiomatic type class}.
Roughly speaking, an axiomatic type class is a type class with axioms, i.e.\ 
an axiomatic specification of a class of types. Thus we can talk about a type
$t$ being in a class $c$, which is written $\tau :: c$.  This is the case of
$\tau$ satisfies the axioms of $c$. Furthermore, type classes can be
organized in a hierarchy. Thus there is the notion of a class $d$ being a
\textbf{subclass} of a class $c$, written $d < c$. This is the case if all
axioms of $c$ are also provable in $d$. Let us now introduce these concepts
by means of a running example, ordering relations.




Finally we should remind our readers that \isa{Main} contains a much more
developed theory of orderings phrased in terms of the usual $\leq$ and
\isa{<}. It is recommended that, if possible, you base your own
ordering relations on this theory.

\index{axiomatic type class|)}