\begin{isabelle}%
%
\begin{isamarkuptext}%
Here is a simple example, the Fibonacci function:%
\end{isamarkuptext}%
\isacommand{consts}\ fib\ ::\ {"}nat\ {\isasymRightarrow}\ nat{"}\isanewline
\isacommand{recdef}\ fib\ {"}measure({\isasymlambda}n.\ n){"}\isanewline
\ \ {"}fib\ 0\ =\ 0{"}\isanewline
\ \ {"}fib\ 1\ =\ 1{"}\isanewline
\ \ {"}fib\ (Suc(Suc\ x))\ =\ fib\ x\ +\ fib\ (Suc\ x){"}%
\begin{isamarkuptext}%
\noindent
The definition of \isa{fib} is accompanied by a \bfindex{measure function}
\isa{{\isasymlambda}n.\ n} which maps the argument of \isa{fib} to a
natural number. The requirement is that in each equation the measure of the
argument on the left-hand side is strictly greater than the measure of the
argument of each recursive call. In the case of \isa{fib} this is
obviously true because the measure function is the identity and
\isa{Suc\ (Suc\ x)} is strictly greater than both \isa{x} and
\isa{Suc\ x}.

Slightly more interesting is the insertion of a fixed element
between any two elements of a list:%
\end{isamarkuptext}%
\isacommand{consts}\ sep\ ::\ {"}'a\ *\ 'a\ list\ {\isasymRightarrow}\ 'a\ list{"}\isanewline
\isacommand{recdef}\ sep\ {"}measure\ ({\isasymlambda}(a,xs).\ length\ xs){"}\isanewline
\ \ {"}sep(a,\ [])\ \ \ \ \ =\ []{"}\isanewline
\ \ {"}sep(a,\ [x])\ \ \ \ =\ [x]{"}\isanewline
\ \ {"}sep(a,\ x\#y\#zs)\ =\ x\ \#\ a\ \#\ sep(a,y\#zs){"}%
\begin{isamarkuptext}%
\noindent
This time the measure is the length of the list, which decreases with the
recursive call; the first component of the argument tuple is irrelevant.

Pattern matching need not be exhaustive:%
\end{isamarkuptext}%
\isacommand{consts}\ last\ ::\ {"}'a\ list\ {\isasymRightarrow}\ 'a{"}\isanewline
\isacommand{recdef}\ last\ {"}measure\ ({\isasymlambda}xs.\ length\ xs){"}\isanewline
\ \ {"}last\ [x]\ \ \ \ \ \ =\ x{"}\isanewline
\ \ {"}last\ (x\#y\#zs)\ =\ last\ (y\#zs){"}%
\begin{isamarkuptext}%
Overlapping patterns are disambiguated by taking the order of equations into
account, just as in functional programming:%
\end{isamarkuptext}%
\isacommand{consts}\ sep1\ ::\ {"}'a\ *\ 'a\ list\ {\isasymRightarrow}\ 'a\ list{"}\isanewline
\isacommand{recdef}\ sep1\ {"}measure\ ({\isasymlambda}(a,xs).\ length\ xs){"}\isanewline
\ \ {"}sep1(a,\ x\#y\#zs)\ =\ x\ \#\ a\ \#\ sep1(a,y\#zs){"}\isanewline
\ \ {"}sep1(a,\ xs)\ \ \ \ \ =\ xs{"}%
\begin{isamarkuptext}%
\noindent
This defines exactly the same function as \isa{sep} above, i.e.\
\isa{sep1 = sep}.

\begin{warn}
  \isacommand{recdef} only takes the first argument of a (curried)
  recursive function into account. This means both the termination measure
  and pattern matching can only use that first argument. In general, you will
  therefore have to combine several arguments into a tuple. In case only one
  argument is relevant for termination, you can also rearrange the order of
  arguments as in the following definition:
\end{warn}%
\end{isamarkuptext}%
\isacommand{consts}\ sep2\ ::\ {"}'a\ list\ {\isasymRightarrow}\ 'a\ {\isasymRightarrow}\ 'a\ list{"}\isanewline
\isacommand{recdef}\ sep2\ {"}measure\ length{"}\isanewline
\ \ {"}sep2\ (x\#y\#zs)\ =\ ({\isasymlambda}a.\ x\ \#\ a\ \#\ sep2\ zs\ a){"}\isanewline
\ \ {"}sep2\ xs\ \ \ \ \ \ \ =\ ({\isasymlambda}a.\ xs){"}%
\begin{isamarkuptext}%
Because of its pattern-matching syntax, \isacommand{recdef} is also useful
for the definition of non-recursive functions:%
\end{isamarkuptext}%
\isacommand{consts}\ swap12\ ::\ {"}'a\ list\ {\isasymRightarrow}\ 'a\ list{"}\isanewline
\isacommand{recdef}\ swap12\ {"}{\isabraceleft}{\isabraceright}{"}\isanewline
\ \ {"}swap12\ (x\#y\#zs)\ =\ y\#x\#zs{"}\isanewline
\ \ {"}swap12\ zs\ \ \ \ \ \ \ =\ zs{"}%
\begin{isamarkuptext}%
\noindent
For non-recursive functions the termination measure degenerates to the empty
set \isa{\{\}}.%
\end{isamarkuptext}%
\end{isabelle}%
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