--- a/doc-src/TutorialI/Recdef/document/termination.tex Sun Oct 21 19:48:19 2001 +0200
+++ b/doc-src/TutorialI/Recdef/document/termination.tex Sun Oct 21 19:49:29 2001 +0200
@@ -1,6 +1,7 @@
%
\begin{isabellebody}%
\def\isabellecontext{termination}%
+\isamarkupfalse%
%
\begin{isamarkuptext}%
When a function~$f$ is defined via \isacommand{recdef}, Isabelle tries to prove
@@ -16,9 +17,12 @@
Isabelle may fail to prove the termination condition for some
recursive call. Let us try the following artificial function:%
\end{isamarkuptext}%
+\isamarkuptrue%
\isacommand{consts}\ f\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequote}nat{\isasymtimes}nat\ {\isasymRightarrow}\ nat{\isachardoublequote}\isanewline
+\isamarkupfalse%
\isacommand{recdef}\ f\ {\isachardoublequote}measure{\isacharparenleft}{\isasymlambda}{\isacharparenleft}x{\isacharcomma}y{\isacharparenright}{\isachardot}\ x{\isacharminus}y{\isacharparenright}{\isachardoublequote}\isanewline
-\ \ {\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}%
+\ \ {\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}\isamarkupfalse%
+%
\begin{isamarkuptext}%
\noindent
Isabelle prints a
@@ -27,14 +31,19 @@
have to prove it as a separate lemma before you attempt the definition
of your function once more. In our case the required lemma is the obvious one:%
\end{isamarkuptext}%
-\isacommand{lemma}\ termi{\isacharunderscore}lem{\isacharcolon}\ {\isachardoublequote}{\isasymnot}\ x\ {\isasymle}\ y\ {\isasymLongrightarrow}\ x\ {\isacharminus}\ Suc\ y\ {\isacharless}\ x\ {\isacharminus}\ y{\isachardoublequote}%
+\isamarkuptrue%
+\isacommand{lemma}\ termi{\isacharunderscore}lem{\isacharcolon}\ {\isachardoublequote}{\isasymnot}\ x\ {\isasymle}\ y\ {\isasymLongrightarrow}\ x\ {\isacharminus}\ Suc\ y\ {\isacharless}\ x\ {\isacharminus}\ y{\isachardoublequote}\isamarkupfalse%
+%
\begin{isamarkuptxt}%
\noindent
It was not proved automatically because of the awkward behaviour of subtraction
on type \isa{nat}. This requires more arithmetic than is tried by default:%
\end{isamarkuptxt}%
+\isamarkuptrue%
\isacommand{apply}{\isacharparenleft}arith{\isacharparenright}\isanewline
-\isacommand{done}%
+\isamarkupfalse%
+\isacommand{done}\isamarkupfalse%
+%
\begin{isamarkuptext}%
\noindent
Because \isacommand{recdef}'s termination prover involves simplification,
@@ -42,19 +51,26 @@
says to use \isa{termi{\isacharunderscore}lem} as
a simplification rule.%
\end{isamarkuptext}%
+\isamarkuptrue%
\isacommand{consts}\ g\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequote}nat{\isasymtimes}nat\ {\isasymRightarrow}\ nat{\isachardoublequote}\isanewline
+\isamarkupfalse%
\isacommand{recdef}\ g\ {\isachardoublequote}measure{\isacharparenleft}{\isasymlambda}{\isacharparenleft}x{\isacharcomma}y{\isacharparenright}{\isachardot}\ x{\isacharminus}y{\isacharparenright}{\isachardoublequote}\isanewline
\ \ {\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
-{\isacharparenleft}\isakeyword{hints}\ recdef{\isacharunderscore}simp{\isacharcolon}\ termi{\isacharunderscore}lem{\isacharparenright}%
+{\isacharparenleft}\isakeyword{hints}\ recdef{\isacharunderscore}simp{\isacharcolon}\ termi{\isacharunderscore}lem{\isacharparenright}\isamarkupfalse%
+%
\begin{isamarkuptext}%
\noindent
This time everything works fine. Now \isa{g{\isachardot}simps} contains precisely
the stated recursion equation for \isa{g}, which has been stored as a
simplification rule. Thus we can automatically prove results such as this one:%
\end{isamarkuptext}%
+\isamarkuptrue%
\isacommand{theorem}\ {\isachardoublequote}g{\isacharparenleft}{\isadigit{1}}{\isacharcomma}{\isadigit{0}}{\isacharparenright}\ {\isacharequal}\ g{\isacharparenleft}{\isadigit{1}}{\isacharcomma}{\isadigit{1}}{\isacharparenright}{\isachardoublequote}\isanewline
+\isamarkupfalse%
\isacommand{apply}{\isacharparenleft}simp{\isacharparenright}\isanewline
-\isacommand{done}%
+\isamarkupfalse%
+\isacommand{done}\isamarkupfalse%
+%
\begin{isamarkuptext}%
\noindent
More exciting theorems require induction, which is discussed below.
@@ -79,6 +95,8 @@
up front.
\REMARK{FIXME, with one exception: nested recursion.}%
\end{isamarkuptext}%
+\isamarkuptrue%
+\isamarkupfalse%
\end{isabellebody}%
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