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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 +