9722
|
1 |
%
|
|
2 |
\begin{isabellebody}%
|
9924
|
3 |
\def\isabellecontext{termination}%
|
8749
|
4 |
%
|
|
5 |
\begin{isamarkuptext}%
|
|
6 |
When a function is defined via \isacommand{recdef}, Isabelle tries to prove
|
|
7 |
its termination with the help of the user-supplied measure. All of the above
|
|
8 |
examples are simple enough that Isabelle can prove automatically that the
|
8771
|
9 |
measure of the argument goes down in each recursive call. As a result,
|
9792
|
10 |
$f$\isa{{\isachardot}simps} will contain the defining equations (or variants derived
|
|
11 |
from them) as theorems. For example, look (via \isacommand{thm}) at
|
10187
|
12 |
\isa{sep{\isachardot}simps} and \isa{sep{\isadigit{1}}{\isachardot}simps} to see that they define
|
9792
|
13 |
the same function. What is more, those equations are automatically declared as
|
8749
|
14 |
simplification rules.
|
|
15 |
|
|
16 |
In general, Isabelle may not be able to prove all termination condition
|
|
17 |
(there is one for each recursive call) automatically. For example,
|
|
18 |
termination of the following artificial function%
|
|
19 |
\end{isamarkuptext}%
|
9933
|
20 |
\isacommand{consts}\ f\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequote}nat{\isasymtimes}nat\ {\isasymRightarrow}\ nat{\isachardoublequote}\isanewline
|
9674
|
21 |
\isacommand{recdef}\ f\ {\isachardoublequote}measure{\isacharparenleft}{\isasymlambda}{\isacharparenleft}x{\isacharcomma}y{\isacharparenright}{\isachardot}\ x{\isacharminus}y{\isacharparenright}{\isachardoublequote}\isanewline
|
10187
|
22 |
\ \ {\isachardoublequote}f{\isacharparenleft}x{\isacharcomma}y{\isacharparenright}\ {\isacharequal}\ {\isacharparenleft}if\ x\ {\isasymle}\ y\ then\ x\ else\ f{\isacharparenleft}x{\isacharcomma}y{\isacharplus}{\isadigit{1}}{\isacharparenright}{\isacharparenright}{\isachardoublequote}%
|
8749
|
23 |
\begin{isamarkuptext}%
|
|
24 |
\noindent
|
|
25 |
is not proved automatically (although maybe it should be). Isabelle prints a
|
|
26 |
kind of error message showing you what it was unable to prove. You will then
|
|
27 |
have to prove it as a separate lemma before you attempt the definition
|
|
28 |
of your function once more. In our case the required lemma is the obvious one:%
|
|
29 |
\end{isamarkuptext}%
|
9933
|
30 |
\isacommand{lemma}\ termi{\isacharunderscore}lem{\isacharcolon}\ {\isachardoublequote}{\isasymnot}\ x\ {\isasymle}\ y\ {\isasymLongrightarrow}\ x\ {\isacharminus}\ Suc\ y\ {\isacharless}\ x\ {\isacharminus}\ y{\isachardoublequote}%
|
8749
|
31 |
\begin{isamarkuptxt}%
|
|
32 |
\noindent
|
9792
|
33 |
It was not proved automatically because of the special nature of \isa{{\isacharminus}}
|
8749
|
34 |
on \isa{nat}. This requires more arithmetic than is tried by default:%
|
|
35 |
\end{isamarkuptxt}%
|
10171
|
36 |
\isacommand{apply}{\isacharparenleft}arith{\isacharparenright}\isanewline
|
|
37 |
\isacommand{done}%
|
8749
|
38 |
\begin{isamarkuptext}%
|
|
39 |
\noindent
|
8771
|
40 |
Because \isacommand{recdef}'s termination prover involves simplification,
|
9933
|
41 |
we include with our second attempt the hint to use \isa{termi{\isacharunderscore}lem} as
|
|
42 |
a simplification rule:%
|
8749
|
43 |
\end{isamarkuptext}%
|
9933
|
44 |
\isacommand{consts}\ g\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequote}nat{\isasymtimes}nat\ {\isasymRightarrow}\ nat{\isachardoublequote}\isanewline
|
9674
|
45 |
\isacommand{recdef}\ g\ {\isachardoublequote}measure{\isacharparenleft}{\isasymlambda}{\isacharparenleft}x{\isacharcomma}y{\isacharparenright}{\isachardot}\ x{\isacharminus}y{\isacharparenright}{\isachardoublequote}\isanewline
|
10187
|
46 |
\ \ {\isachardoublequote}g{\isacharparenleft}x{\isacharcomma}y{\isacharparenright}\ {\isacharequal}\ {\isacharparenleft}if\ x\ {\isasymle}\ y\ then\ x\ else\ g{\isacharparenleft}x{\isacharcomma}y{\isacharplus}{\isadigit{1}}{\isacharparenright}{\isacharparenright}{\isachardoublequote}\isanewline
|
9992
|
47 |
{\isacharparenleft}\isakeyword{hints}\ recdef{\isacharunderscore}simp{\isacharcolon}\ termi{\isacharunderscore}lem{\isacharparenright}%
|
8749
|
48 |
\begin{isamarkuptext}%
|
|
49 |
\noindent
|
9792
|
50 |
This time everything works fine. Now \isa{g{\isachardot}simps} contains precisely
|
|
51 |
the stated recursion equation for \isa{g} and they are simplification
|
8749
|
52 |
rules. Thus we can automatically prove%
|
|
53 |
\end{isamarkuptext}%
|
10187
|
54 |
\isacommand{theorem}\ {\isachardoublequote}g{\isacharparenleft}{\isadigit{1}}{\isacharcomma}{\isadigit{0}}{\isacharparenright}\ {\isacharequal}\ g{\isacharparenleft}{\isadigit{1}}{\isacharcomma}{\isadigit{1}}{\isacharparenright}{\isachardoublequote}\isanewline
|
10171
|
55 |
\isacommand{apply}{\isacharparenleft}simp{\isacharparenright}\isanewline
|
|
56 |
\isacommand{done}%
|
8749
|
57 |
\begin{isamarkuptext}%
|
|
58 |
\noindent
|
|
59 |
More exciting theorems require induction, which is discussed below.
|
|
60 |
|
9933
|
61 |
If the termination proof requires a new lemma that is of general use, you can
|
|
62 |
turn it permanently into a simplification rule, in which case the above
|
|
63 |
\isacommand{hint} is not necessary. But our \isa{termi{\isacharunderscore}lem} is not
|
|
64 |
sufficiently general to warrant this distinction.
|
|
65 |
|
8749
|
66 |
The attentive reader may wonder why we chose to call our function \isa{g}
|
|
67 |
rather than \isa{f} the second time around. The reason is that, despite
|
|
68 |
the failed termination proof, the definition of \isa{f} did not
|
9792
|
69 |
fail, and thus we could not define it a second time. However, all theorems
|
|
70 |
about \isa{f}, for example \isa{f{\isachardot}simps}, carry as a precondition
|
|
71 |
the unproved termination condition. Moreover, the theorems
|
|
72 |
\isa{f{\isachardot}simps} are not simplification rules. However, this mechanism
|
|
73 |
allows a delayed proof of termination: instead of proving
|
|
74 |
\isa{termi{\isacharunderscore}lem} up front, we could prove
|
8749
|
75 |
it later on and then use it to remove the preconditions from the theorems
|
|
76 |
about \isa{f}. In most cases this is more cumbersome than proving things
|
9792
|
77 |
up front.
|
10186
|
78 |
%FIXME, with one exception: nested recursion.%
|
8749
|
79 |
\end{isamarkuptext}%
|
9722
|
80 |
\end{isabellebody}%
|
9145
|
81 |
%%% Local Variables:
|
|
82 |
%%% mode: latex
|
|
83 |
%%% TeX-master: "root"
|
|
84 |
%%% End:
|