(*<*)
theory Nested1 = Nested0:;
(*>*)
consts trev :: "('a,'b)term \<Rightarrow> ('a,'b)term";
text{*\noindent
Although the definition of @{term"trev"} is quite natural, we will have
overcome a minor difficulty in convincing Isabelle of is termination.
It is precisely this difficulty that is the \textit{raison d'\^etre} of
this subsection.
Defining @{term"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}:
*};
recdef trev "measure size"
"trev (Var x) = Var x"
"trev (App f ts) = App f (rev(map trev ts))";
text{*\noindent
Remember that function @{term"size"} is defined for each \isacommand{datatype}.
However, the definition does not succeed. Isabelle complains about an
unproved termination condition
@{term[display]"t : set ts --> size t < Suc (term_list_size ts)"}
where @{term"set"} returns the set of elements of a list
and @{text"term_list_size :: term list \<Rightarrow> nat"} is an auxiliary
function automatically defined by Isabelle
(when @{text"term"} was defined). First we have to understand why the
recursive call of @{term"trev"} underneath @{term"map"} leads to the above
condition. The reason is that \isacommand{recdef} ``knows'' that @{term"map"}
will apply @{term"trev"} only to elements of @{term"ts"}. Thus the above
condition expresses that the size of the argument @{term"t : set ts"} of any
recursive call of @{term"trev"} is strictly less than @{term"size(App f ts) =
Suc(term_list_size ts)"}. We will now prove the termination condition and
continue with our definition. Below we return to the question of how
\isacommand{recdef} ``knows'' about @{term"map"}.
*};
(*<*)
end;
(*>*)