3 \def\isabellecontext{Nested{\isadigit{1}}}% |
3 \def\isabellecontext{Nested{\isadigit{1}}}% |
4 % |
4 % |
5 \begin{isamarkuptext}% |
5 \begin{isamarkuptext}% |
6 \noindent |
6 \noindent |
7 Although the definition of \isa{trev} is quite natural, we will have |
7 Although the definition of \isa{trev} is quite natural, we will have |
8 overcome a minor difficulty in convincing Isabelle of is termination. |
8 to overcome a minor difficulty in convincing Isabelle of its termination. |
9 It is precisely this difficulty that is the \textit{raison d'\^etre} of |
9 It is precisely this difficulty that is the \textit{raison d'\^etre} of |
10 this subsection. |
10 this subsection. |
11 |
11 |
12 Defining \isa{trev} by \isacommand{recdef} rather than \isacommand{primrec} |
12 Defining \isa{trev} by \isacommand{recdef} rather than \isacommand{primrec} |
13 simplifies matters because we are now free to use the recursion equation |
13 simplifies matters because we are now free to use the recursion equation |
25 \ \ \ \ \ t\ {\isasymin}\ set\ ts\ {\isasymlongrightarrow}\ size\ t\ {\isacharless}\ Suc\ {\isacharparenleft}term{\isacharunderscore}list{\isacharunderscore}size\ ts{\isacharparenright}% |
25 \ \ \ \ \ t\ {\isasymin}\ set\ ts\ {\isasymlongrightarrow}\ size\ t\ {\isacharless}\ Suc\ {\isacharparenleft}term{\isacharunderscore}list{\isacharunderscore}size\ ts{\isacharparenright}% |
26 \end{isabelle} |
26 \end{isabelle} |
27 where \isa{set} returns the set of elements of a list |
27 where \isa{set} returns the set of elements of a list |
28 and \isa{term{\isacharunderscore}list{\isacharunderscore}size\ {\isacharcolon}{\isacharcolon}\ term\ list\ {\isasymRightarrow}\ nat} is an auxiliary |
28 and \isa{term{\isacharunderscore}list{\isacharunderscore}size\ {\isacharcolon}{\isacharcolon}\ term\ list\ {\isasymRightarrow}\ nat} is an auxiliary |
29 function automatically defined by Isabelle |
29 function automatically defined by Isabelle |
30 (when \isa{term} was defined). First we have to understand why the |
30 (while processing the declaration of \isa{term}). First we have to understand why the |
31 recursive call of \isa{trev} underneath \isa{map} leads to the above |
31 recursive call of \isa{trev} underneath \isa{map} leads to the above |
32 condition. The reason is that \isacommand{recdef} ``knows'' that \isa{map} |
32 condition. The reason is that \isacommand{recdef} ``knows'' that \isa{map} |
33 will apply \isa{trev} only to elements of \isa{ts}. Thus the above |
33 will apply \isa{trev} only to elements of \isa{ts}. Thus the |
34 condition expresses that the size of the argument \isa{t\ {\isasymin}\ set\ ts} of any |
34 condition expresses that the size of the argument \isa{t\ {\isasymin}\ set\ ts} of any |
35 recursive call of \isa{trev} is strictly less than \isa{size\ {\isacharparenleft}App\ f\ ts{\isacharparenright}\ {\isacharequal}\ Suc\ {\isacharparenleft}term{\isacharunderscore}list{\isacharunderscore}size\ ts{\isacharparenright}}. We will now prove the termination condition and |
35 recursive call of \isa{trev} is strictly less than \isa{size\ {\isacharparenleft}App\ f\ ts{\isacharparenright}}, |
|
36 which equals \isa{Suc\ {\isacharparenleft}term{\isacharunderscore}list{\isacharunderscore}size\ ts{\isacharparenright}}. We will now prove the termination condition and |
36 continue with our definition. Below we return to the question of how |
37 continue with our definition. Below we return to the question of how |
37 \isacommand{recdef} ``knows'' about \isa{map}.% |
38 \isacommand{recdef} ``knows'' about \isa{map}.% |
38 \end{isamarkuptext}% |
39 \end{isamarkuptext}% |
39 \end{isabellebody}% |
40 \end{isabellebody}% |
40 %%% Local Variables: |
41 %%% Local Variables: |