doc-src/TutorialI/Recdef/document/Nested1.tex

move lemmas from Nat_Numeral to Int.thy and Num.thy

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\begin{isamarkuptext}%
\noindent
Although the definition of \isa{trev} below is quite natural, we will have
to overcome a minor difficulty in convincing Isabelle of its termination.
It is precisely this difficulty that is the \textit{raison d'\^etre} of
this subsection.

Defining \isa{trev} by \isacommand{recdef} rather than \isacommand{primrec}
simplifies matters because we are now free to use the recursion equation
suggested at the end of \S\ref{sec:nested-datatype}:%
\end{isamarkuptext}%
\isamarkuptrue%
\isacommand{recdef}\isamarkupfalse%
\ trev\ {\isachardoublequoteopen}measure\ size{\isachardoublequoteclose}\isanewline
\ {\isachardoublequoteopen}trev\ {\isacharparenleft}Var\ x{\isacharparenright}\ \ \ \ {\isacharequal}\ Var\ x{\isachardoublequoteclose}\isanewline
\ {\isachardoublequoteopen}trev\ {\isacharparenleft}App\ f\ ts{\isacharparenright}\ {\isacharequal}\ App\ f\ {\isacharparenleft}rev{\isacharparenleft}map\ trev\ ts{\isacharparenright}{\isacharparenright}{\isachardoublequoteclose}%
\begin{isamarkuptext}%
\noindent
Remember that function \isa{size} is defined for each \isacommand{datatype}.
However, the definition does not succeed. Isabelle complains about an
unproved termination condition
\begin{isabelle}%
\ \ \ \ \ t\ {\isasymin}\ set\ ts\ {\isasymlongrightarrow}\ size\ t\ {\isacharless}\ Suc\ {\isacharparenleft}term{\isacharunderscore}list{\isacharunderscore}size\ ts{\isacharparenright}%
\end{isabelle}
where \isa{set} returns the set of elements of a list
and \isa{term{\isacharunderscore}list{\isacharunderscore}size\ {\isacharcolon}{\isacharcolon}\ term\ list\ {\isasymRightarrow}\ nat} is an auxiliary
function automatically defined by Isabelle
(while processing the declaration of \isa{term}).  Why does the
recursive call of \isa{trev} lead to this
condition?  Because \isacommand{recdef} knows that \isa{map}
will apply \isa{trev} only to elements of \isa{ts}. Thus the 
condition expresses that the size of the argument \isa{t\ {\isasymin}\ set\ ts} of any
recursive call of \isa{trev} is strictly less than \isa{size\ {\isacharparenleft}App\ f\ ts{\isacharparenright}},
which equals \isa{Suc\ {\isacharparenleft}term{\isacharunderscore}list{\isacharunderscore}size\ ts{\isacharparenright}}.  We will now prove the termination condition and
continue with our definition.  Below we return to the question of how
\isacommand{recdef} knows about \isa{map}.

The termination condition is easily proved by induction:%
\end{isamarkuptext}%
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