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     1.4 -\section{Introduction}
     1.5 -
     1.6 -Starting from Isabelle 2007, new facilities for recursive
     1.7 -function definitions~\cite{krauss2006} are available. They provide
     1.8 -better support for general recursive definitions than previous
     1.9 -packages.  But despite all tool support, function definitions can
    1.10 -sometimes be a difficult thing. 
    1.11 -
    1.12 -This tutorial is an example-guided introduction to the practical use
    1.13 -of the package and related tools. It should help you get started with
    1.14 -defining functions quickly. For the more difficult definitions we will
    1.15 -discuss what problems can arise, and how they can be solved.
    1.16 -
    1.17 -We assume that you have mastered the fundamentals of Isabelle/HOL
    1.18 -and are able to write basic specifications and proofs. To start out
    1.19 -with Isabelle in general, consult the Isabelle/HOL tutorial
    1.20 -\cite{isa-tutorial}.
    1.21 -
    1.22 -
    1.23 -
    1.24 -\paragraph{Structure of this tutorial.}
    1.25 -Section 2 introduces the syntax and basic operation of the \cmd{fun}
    1.26 -command, which provides full automation with reasonable default
    1.27 -behavior.  The impatient reader can stop after that
    1.28 -section, and consult the remaining sections only when needed.
    1.29 -Section 3 introduces the more verbose \cmd{function} command which
    1.30 -gives fine-grained control. This form should be used
    1.31 -whenever the short form fails.
    1.32 -After that we discuss more specialized issues:
    1.33 -termination, mutual, nested and higher-order recursion, partiality, pattern matching
    1.34 -and others.
    1.35 -
    1.36 -
    1.37 -\paragraph{Some background.}
    1.38 -Following the LCF tradition, the package is realized as a definitional
    1.39 -extension: Recursive definitions are internally transformed into a
    1.40 -non-recursive form, such that the function can be defined using
    1.41 -standard definition facilities. Then the recursive specification is
    1.42 -derived from the primitive definition.  This is a complex task, but it
    1.43 -is fully automated and mostly transparent to the user. Definitional
    1.44 -extensions are valuable because they are conservative by construction:
    1.45 -The \qt{new} concept of general wellfounded recursion is completely reduced
    1.46 -to existing principles.
    1.47 -
    1.48 -
    1.49 -
    1.50 -
    1.51 -The new \cmd{function} command, and its short form \cmd{fun} have mostly
    1.52 -replaced the traditional \cmd{recdef} command \cite{slind-tfl}. They solve
    1.53 -a few of technical issues around \cmd{recdef}, and allow definitions
    1.54 -which were not previously possible.
    1.55 -
    1.56 -
    1.57 -
    1.58 -