--- a/doc-src/TutorialI/Recdef/document/simplification.tex Wed Oct 11 09:09:06 2000 +0200
+++ b/doc-src/TutorialI/Recdef/document/simplification.tex Wed Oct 11 10:44:42 2000 +0200
@@ -12,13 +12,13 @@
\end{isamarkuptext}%
\isacommand{consts}\ gcd\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequote}nat{\isasymtimes}nat\ {\isasymRightarrow}\ nat{\isachardoublequote}\isanewline
\isacommand{recdef}\ gcd\ {\isachardoublequote}measure\ {\isacharparenleft}{\isasymlambda}{\isacharparenleft}m{\isacharcomma}n{\isacharparenright}{\isachardot}n{\isacharparenright}{\isachardoublequote}\isanewline
-\ \ {\isachardoublequote}gcd\ {\isacharparenleft}m{\isacharcomma}\ n{\isacharparenright}\ {\isacharequal}\ {\isacharparenleft}if\ n{\isacharequal}\isadigit{0}\ then\ m\ else\ gcd{\isacharparenleft}n{\isacharcomma}\ m\ mod\ n{\isacharparenright}{\isacharparenright}{\isachardoublequote}%
+\ \ {\isachardoublequote}gcd\ {\isacharparenleft}m{\isacharcomma}\ n{\isacharparenright}\ {\isacharequal}\ {\isacharparenleft}if\ n{\isacharequal}{\isadigit{0}}\ then\ m\ else\ gcd{\isacharparenleft}n{\isacharcomma}\ m\ mod\ n{\isacharparenright}{\isacharparenright}{\isachardoublequote}%
\begin{isamarkuptext}%
\noindent
According to the measure function, the second argument should decrease with
each recursive call. The resulting termination condition
\begin{isabelle}%
-\ \ \ \ \ n\ {\isasymnoteq}\ \isadigit{0}\ {\isasymLongrightarrow}\ m\ mod\ n\ {\isacharless}\ n%
+\ \ \ \ \ n\ {\isasymnoteq}\ {\isadigit{0}}\ {\isasymLongrightarrow}\ m\ mod\ n\ {\isacharless}\ n%
\end{isabelle}
is provded automatically because it is already present as a lemma in the
arithmetic library. Thus the recursion equation becomes a simplification
@@ -33,11 +33,11 @@
\end{isabelle}
in one step to
\begin{isabelle}%
-\ \ \ \ \ {\isacharparenleft}if\ n\ {\isacharequal}\ \isadigit{0}\ then\ m\ else\ gcd\ {\isacharparenleft}n{\isacharcomma}\ m\ mod\ n{\isacharparenright}{\isacharparenright}\ {\isacharequal}\ k%
+\ \ \ \ \ {\isacharparenleft}if\ n\ {\isacharequal}\ {\isadigit{0}}\ then\ m\ else\ gcd\ {\isacharparenleft}n{\isacharcomma}\ m\ mod\ n{\isacharparenright}{\isacharparenright}\ {\isacharequal}\ k%
\end{isabelle}
where the condition cannot be reduced further, and splitting leads to
\begin{isabelle}%
-\ \ \ \ \ {\isacharparenleft}n\ {\isacharequal}\ \isadigit{0}\ {\isasymlongrightarrow}\ m\ {\isacharequal}\ k{\isacharparenright}\ {\isasymand}\ {\isacharparenleft}n\ {\isasymnoteq}\ \isadigit{0}\ {\isasymlongrightarrow}\ gcd\ {\isacharparenleft}n{\isacharcomma}\ m\ mod\ n{\isacharparenright}\ {\isacharequal}\ k{\isacharparenright}%
+\ \ \ \ \ {\isacharparenleft}n\ {\isacharequal}\ {\isadigit{0}}\ {\isasymlongrightarrow}\ m\ {\isacharequal}\ k{\isacharparenright}\ {\isasymand}\ {\isacharparenleft}n\ {\isasymnoteq}\ {\isadigit{0}}\ {\isasymlongrightarrow}\ gcd\ {\isacharparenleft}n{\isacharcomma}\ m\ mod\ n{\isacharparenright}\ {\isacharequal}\ k{\isacharparenright}%
\end{isabelle}
Since the recursive call \isa{gcd\ {\isacharparenleft}n{\isacharcomma}\ m\ mod\ n{\isacharparenright}} is no longer protected by
an \isa{if}, it is unfolded again, which leads to an infinite chain of
@@ -54,22 +54,22 @@
rather than \isa{if} on the right. In the case of \isa{gcd} the
following alternative definition suggests itself:%
\end{isamarkuptext}%
-\isacommand{consts}\ gcd\isadigit{1}\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequote}nat{\isasymtimes}nat\ {\isasymRightarrow}\ nat{\isachardoublequote}\isanewline
-\isacommand{recdef}\ gcd\isadigit{1}\ {\isachardoublequote}measure\ {\isacharparenleft}{\isasymlambda}{\isacharparenleft}m{\isacharcomma}n{\isacharparenright}{\isachardot}n{\isacharparenright}{\isachardoublequote}\isanewline
-\ \ {\isachardoublequote}gcd\isadigit{1}\ {\isacharparenleft}m{\isacharcomma}\ \isadigit{0}{\isacharparenright}\ {\isacharequal}\ m{\isachardoublequote}\isanewline
-\ \ {\isachardoublequote}gcd\isadigit{1}\ {\isacharparenleft}m{\isacharcomma}\ n{\isacharparenright}\ {\isacharequal}\ gcd\isadigit{1}{\isacharparenleft}n{\isacharcomma}\ m\ mod\ n{\isacharparenright}{\isachardoublequote}%
+\isacommand{consts}\ gcd{\isadigit{1}}\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequote}nat{\isasymtimes}nat\ {\isasymRightarrow}\ nat{\isachardoublequote}\isanewline
+\isacommand{recdef}\ gcd{\isadigit{1}}\ {\isachardoublequote}measure\ {\isacharparenleft}{\isasymlambda}{\isacharparenleft}m{\isacharcomma}n{\isacharparenright}{\isachardot}n{\isacharparenright}{\isachardoublequote}\isanewline
+\ \ {\isachardoublequote}gcd{\isadigit{1}}\ {\isacharparenleft}m{\isacharcomma}\ {\isadigit{0}}{\isacharparenright}\ {\isacharequal}\ m{\isachardoublequote}\isanewline
+\ \ {\isachardoublequote}gcd{\isadigit{1}}\ {\isacharparenleft}m{\isacharcomma}\ n{\isacharparenright}\ {\isacharequal}\ gcd{\isadigit{1}}{\isacharparenleft}n{\isacharcomma}\ m\ mod\ n{\isacharparenright}{\isachardoublequote}%
\begin{isamarkuptext}%
\noindent
Note that the order of equations is important and hides the side condition
-\isa{n\ {\isasymnoteq}\ \isadigit{0}}. Unfortunately, in general the case distinction
+\isa{n\ {\isasymnoteq}\ {\isadigit{0}}}. Unfortunately, in general the case distinction
may not be expressible by pattern matching.
A very simple alternative is to replace \isa{if} by \isa{case}, which
is also available for \isa{bool} but is not split automatically:%
\end{isamarkuptext}%
-\isacommand{consts}\ gcd\isadigit{2}\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequote}nat{\isasymtimes}nat\ {\isasymRightarrow}\ nat{\isachardoublequote}\isanewline
-\isacommand{recdef}\ gcd\isadigit{2}\ {\isachardoublequote}measure\ {\isacharparenleft}{\isasymlambda}{\isacharparenleft}m{\isacharcomma}n{\isacharparenright}{\isachardot}n{\isacharparenright}{\isachardoublequote}\isanewline
-\ \ {\isachardoublequote}gcd\isadigit{2}{\isacharparenleft}m{\isacharcomma}n{\isacharparenright}\ {\isacharequal}\ {\isacharparenleft}case\ n{\isacharequal}\isadigit{0}\ of\ True\ {\isasymRightarrow}\ m\ {\isacharbar}\ False\ {\isasymRightarrow}\ gcd\isadigit{2}{\isacharparenleft}n{\isacharcomma}m\ mod\ n{\isacharparenright}{\isacharparenright}{\isachardoublequote}%
+\isacommand{consts}\ gcd{\isadigit{2}}\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequote}nat{\isasymtimes}nat\ {\isasymRightarrow}\ nat{\isachardoublequote}\isanewline
+\isacommand{recdef}\ gcd{\isadigit{2}}\ {\isachardoublequote}measure\ {\isacharparenleft}{\isasymlambda}{\isacharparenleft}m{\isacharcomma}n{\isacharparenright}{\isachardot}n{\isacharparenright}{\isachardoublequote}\isanewline
+\ \ {\isachardoublequote}gcd{\isadigit{2}}{\isacharparenleft}m{\isacharcomma}n{\isacharparenright}\ {\isacharequal}\ {\isacharparenleft}case\ n{\isacharequal}{\isadigit{0}}\ of\ True\ {\isasymRightarrow}\ m\ {\isacharbar}\ False\ {\isasymRightarrow}\ gcd{\isadigit{2}}{\isacharparenleft}n{\isacharcomma}m\ mod\ n{\isacharparenright}{\isacharparenright}{\isachardoublequote}%
\begin{isamarkuptext}%
\noindent
In fact, this is probably the neatest solution next to pattern matching.
@@ -77,10 +77,10 @@
A final alternative is to replace the offending simplification rules by
derived conditional ones. For \isa{gcd} it means we have to prove%
\end{isamarkuptext}%
-\isacommand{lemma}\ {\isacharbrackleft}simp{\isacharbrackright}{\isacharcolon}\ {\isachardoublequote}gcd\ {\isacharparenleft}m{\isacharcomma}\ \isadigit{0}{\isacharparenright}\ {\isacharequal}\ m{\isachardoublequote}\isanewline
+\isacommand{lemma}\ {\isacharbrackleft}simp{\isacharbrackright}{\isacharcolon}\ {\isachardoublequote}gcd\ {\isacharparenleft}m{\isacharcomma}\ {\isadigit{0}}{\isacharparenright}\ {\isacharequal}\ m{\isachardoublequote}\isanewline
\isacommand{apply}{\isacharparenleft}simp{\isacharparenright}\isanewline
\isacommand{done}\isanewline
-\isacommand{lemma}\ {\isacharbrackleft}simp{\isacharbrackright}{\isacharcolon}\ {\isachardoublequote}n\ {\isasymnoteq}\ \isadigit{0}\ {\isasymLongrightarrow}\ gcd{\isacharparenleft}m{\isacharcomma}\ n{\isacharparenright}\ {\isacharequal}\ gcd{\isacharparenleft}n{\isacharcomma}\ m\ mod\ n{\isacharparenright}{\isachardoublequote}\isanewline
+\isacommand{lemma}\ {\isacharbrackleft}simp{\isacharbrackright}{\isacharcolon}\ {\isachardoublequote}n\ {\isasymnoteq}\ {\isadigit{0}}\ {\isasymLongrightarrow}\ gcd{\isacharparenleft}m{\isacharcomma}\ n{\isacharparenright}\ {\isacharequal}\ gcd{\isacharparenleft}n{\isacharcomma}\ m\ mod\ n{\isacharparenright}{\isachardoublequote}\isanewline
\isacommand{apply}{\isacharparenleft}simp{\isacharparenright}\isanewline
\isacommand{done}%
\begin{isamarkuptext}%