--- a/doc-src/TutorialI/Ifexpr/document/Ifexpr.tex Tue Aug 16 13:42:21 2005 +0200
+++ b/doc-src/TutorialI/Ifexpr/document/Ifexpr.tex Tue Aug 16 13:42:23 2005 +0200
@@ -1,7 +1,20 @@
%
\begin{isabellebody}%
\def\isabellecontext{Ifexpr}%
-\isamarkupfalse%
+%
+\isadelimtheory
+%
+\endisadelimtheory
+%
+\isatagtheory
+%
+\endisatagtheory
+{\isafoldtheory}%
+%
+\isadelimtheory
+%
+\endisadelimtheory
+\isamarkuptrue%
%
\isamarkupsubsection{Case Study: Boolean Expressions%
}
@@ -24,9 +37,9 @@
constants by negation and conjunction. The following datatype serves exactly
that purpose:%
\end{isamarkuptext}%
-\isamarkuptrue%
+\isamarkupfalse%
\isacommand{datatype}\ boolex\ {\isacharequal}\ Const\ bool\ {\isacharbar}\ Var\ nat\ {\isacharbar}\ Neg\ boolex\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharbar}\ And\ boolex\ boolex\isamarkupfalse%
+\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharbar}\ And\ boolex\ boolex\isamarkuptrue%
%
\begin{isamarkuptext}%
\noindent
@@ -43,14 +56,14 @@
\emph{environment} of type \isa{nat\ {\isasymRightarrow}\ bool}, which maps variables to their
values:%
\end{isamarkuptext}%
-\isamarkuptrue%
+\isamarkupfalse%
\isacommand{consts}\ value\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequote}boolex\ {\isasymRightarrow}\ {\isacharparenleft}nat\ {\isasymRightarrow}\ bool{\isacharparenright}\ {\isasymRightarrow}\ bool{\isachardoublequote}\isanewline
\isamarkupfalse%
\isacommand{primrec}\isanewline
{\isachardoublequote}value\ {\isacharparenleft}Const\ b{\isacharparenright}\ env\ {\isacharequal}\ b{\isachardoublequote}\isanewline
{\isachardoublequote}value\ {\isacharparenleft}Var\ x{\isacharparenright}\ \ \ env\ {\isacharequal}\ env\ x{\isachardoublequote}\isanewline
{\isachardoublequote}value\ {\isacharparenleft}Neg\ b{\isacharparenright}\ \ \ env\ {\isacharequal}\ {\isacharparenleft}{\isasymnot}\ value\ b\ env{\isacharparenright}{\isachardoublequote}\isanewline
-{\isachardoublequote}value\ {\isacharparenleft}And\ b\ c{\isacharparenright}\ env\ {\isacharequal}\ {\isacharparenleft}value\ b\ env\ {\isasymand}\ value\ c\ env{\isacharparenright}{\isachardoublequote}\isamarkupfalse%
+{\isachardoublequote}value\ {\isacharparenleft}And\ b\ c{\isacharparenright}\ env\ {\isacharequal}\ {\isacharparenleft}value\ b\ env\ {\isasymand}\ value\ c\ env{\isacharparenright}{\isachardoublequote}\isamarkuptrue%
%
\begin{isamarkuptext}%
\noindent
@@ -61,21 +74,21 @@
from constants (\isa{CIF}), variables (\isa{VIF}) and conditionals
(\isa{IF}):%
\end{isamarkuptext}%
-\isamarkuptrue%
-\isacommand{datatype}\ ifex\ {\isacharequal}\ CIF\ bool\ {\isacharbar}\ VIF\ nat\ {\isacharbar}\ IF\ ifex\ ifex\ ifex\isamarkupfalse%
+\isamarkupfalse%
+\isacommand{datatype}\ ifex\ {\isacharequal}\ CIF\ bool\ {\isacharbar}\ VIF\ nat\ {\isacharbar}\ IF\ ifex\ ifex\ ifex\isamarkuptrue%
%
\begin{isamarkuptext}%
\noindent
The evaluation of If-expressions proceeds as for \isa{boolex}:%
\end{isamarkuptext}%
-\isamarkuptrue%
+\isamarkupfalse%
\isacommand{consts}\ valif\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequote}ifex\ {\isasymRightarrow}\ {\isacharparenleft}nat\ {\isasymRightarrow}\ bool{\isacharparenright}\ {\isasymRightarrow}\ bool{\isachardoublequote}\isanewline
\isamarkupfalse%
\isacommand{primrec}\isanewline
{\isachardoublequote}valif\ {\isacharparenleft}CIF\ b{\isacharparenright}\ \ \ \ env\ {\isacharequal}\ b{\isachardoublequote}\isanewline
{\isachardoublequote}valif\ {\isacharparenleft}VIF\ x{\isacharparenright}\ \ \ \ env\ {\isacharequal}\ env\ x{\isachardoublequote}\isanewline
{\isachardoublequote}valif\ {\isacharparenleft}IF\ b\ t\ e{\isacharparenright}\ env\ {\isacharequal}\ {\isacharparenleft}if\ valif\ b\ env\ then\ valif\ t\ env\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ else\ valif\ e\ env{\isacharparenright}{\isachardoublequote}\isamarkupfalse%
+\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ else\ valif\ e\ env{\isacharparenright}{\isachardoublequote}\isamarkuptrue%
%
\begin{isamarkuptext}%
\subsubsection{Converting Boolean and If-Expressions}
@@ -84,33 +97,46 @@
formulae, whereas \isa{ifex} is designed for efficiency. It is easy to
translate from \isa{boolex} into \isa{ifex}:%
\end{isamarkuptext}%
-\isamarkuptrue%
+\isamarkupfalse%
\isacommand{consts}\ bool{\isadigit{2}}if\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequote}boolex\ {\isasymRightarrow}\ ifex{\isachardoublequote}\isanewline
\isamarkupfalse%
\isacommand{primrec}\isanewline
{\isachardoublequote}bool{\isadigit{2}}if\ {\isacharparenleft}Const\ b{\isacharparenright}\ {\isacharequal}\ CIF\ b{\isachardoublequote}\isanewline
{\isachardoublequote}bool{\isadigit{2}}if\ {\isacharparenleft}Var\ x{\isacharparenright}\ \ \ {\isacharequal}\ VIF\ x{\isachardoublequote}\isanewline
{\isachardoublequote}bool{\isadigit{2}}if\ {\isacharparenleft}Neg\ b{\isacharparenright}\ \ \ {\isacharequal}\ IF\ {\isacharparenleft}bool{\isadigit{2}}if\ b{\isacharparenright}\ {\isacharparenleft}CIF\ False{\isacharparenright}\ {\isacharparenleft}CIF\ True{\isacharparenright}{\isachardoublequote}\isanewline
-{\isachardoublequote}bool{\isadigit{2}}if\ {\isacharparenleft}And\ b\ c{\isacharparenright}\ {\isacharequal}\ IF\ {\isacharparenleft}bool{\isadigit{2}}if\ b{\isacharparenright}\ {\isacharparenleft}bool{\isadigit{2}}if\ c{\isacharparenright}\ {\isacharparenleft}CIF\ False{\isacharparenright}{\isachardoublequote}\isamarkupfalse%
+{\isachardoublequote}bool{\isadigit{2}}if\ {\isacharparenleft}And\ b\ c{\isacharparenright}\ {\isacharequal}\ IF\ {\isacharparenleft}bool{\isadigit{2}}if\ b{\isacharparenright}\ {\isacharparenleft}bool{\isadigit{2}}if\ c{\isacharparenright}\ {\isacharparenleft}CIF\ False{\isacharparenright}{\isachardoublequote}\isamarkuptrue%
%
\begin{isamarkuptext}%
\noindent
At last, we have something we can verify: that \isa{bool{\isadigit{2}}if} preserves the
value of its argument:%
\end{isamarkuptext}%
+\isamarkupfalse%
+\isacommand{lemma}\ {\isachardoublequote}valif\ {\isacharparenleft}bool{\isadigit{2}}if\ b{\isacharparenright}\ env\ {\isacharequal}\ value\ b\ env{\isachardoublequote}%
+\isadelimproof
+%
+\endisadelimproof
+%
+\isatagproof
\isamarkuptrue%
-\isacommand{lemma}\ {\isachardoublequote}valif\ {\isacharparenleft}bool{\isadigit{2}}if\ b{\isacharparenright}\ env\ {\isacharequal}\ value\ b\ env{\isachardoublequote}\isamarkupfalse%
%
\begin{isamarkuptxt}%
\noindent
The proof is canonical:%
\end{isamarkuptxt}%
-\isamarkuptrue%
+\isamarkupfalse%
\isacommand{apply}{\isacharparenleft}induct{\isacharunderscore}tac\ b{\isacharparenright}\isanewline
\isamarkupfalse%
\isacommand{apply}{\isacharparenleft}auto{\isacharparenright}\isanewline
\isamarkupfalse%
-\isacommand{done}\isamarkupfalse%
+\isacommand{done}%
+\endisatagproof
+{\isafoldproof}%
+%
+\isadelimproof
+%
+\endisadelimproof
+\isamarkuptrue%
%
\begin{isamarkuptext}%
\noindent
@@ -124,7 +150,7 @@
\isa{IF\ b\ {\isacharparenleft}IF\ x\ z\ u{\isacharparenright}\ {\isacharparenleft}IF\ y\ z\ u{\isacharparenright}}, which has the same value. The following
primitive recursive functions perform this task:%
\end{isamarkuptext}%
-\isamarkuptrue%
+\isamarkupfalse%
\isacommand{consts}\ normif\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequote}ifex\ {\isasymRightarrow}\ ifex\ {\isasymRightarrow}\ ifex\ {\isasymRightarrow}\ ifex{\isachardoublequote}\isanewline
\isamarkupfalse%
\isacommand{primrec}\isanewline
@@ -138,7 +164,7 @@
\isacommand{primrec}\isanewline
{\isachardoublequote}norm\ {\isacharparenleft}CIF\ b{\isacharparenright}\ \ \ \ {\isacharequal}\ CIF\ b{\isachardoublequote}\isanewline
{\isachardoublequote}norm\ {\isacharparenleft}VIF\ x{\isacharparenright}\ \ \ \ {\isacharequal}\ VIF\ x{\isachardoublequote}\isanewline
-{\isachardoublequote}norm\ {\isacharparenleft}IF\ b\ t\ e{\isacharparenright}\ {\isacharequal}\ normif\ b\ {\isacharparenleft}norm\ t{\isacharparenright}\ {\isacharparenleft}norm\ e{\isacharparenright}{\isachardoublequote}\isamarkupfalse%
+{\isachardoublequote}norm\ {\isacharparenleft}IF\ b\ t\ e{\isacharparenright}\ {\isacharequal}\ normif\ b\ {\isacharparenleft}norm\ t{\isacharparenright}\ {\isacharparenleft}norm\ e{\isacharparenright}{\isachardoublequote}\isamarkuptrue%
%
\begin{isamarkuptext}%
\noindent
@@ -146,23 +172,56 @@
intuitive understanding. Fortunately, Isabelle can help us to verify that the
transformation preserves the value of the expression:%
\end{isamarkuptext}%
+\isamarkupfalse%
+\isacommand{theorem}\ {\isachardoublequote}valif\ {\isacharparenleft}norm\ b{\isacharparenright}\ env\ {\isacharequal}\ valif\ b\ env{\isachardoublequote}%
+\isadelimproof
+%
+\endisadelimproof
+%
+\isatagproof
+%
+\endisatagproof
+{\isafoldproof}%
+%
+\isadelimproof
+%
+\endisadelimproof
\isamarkuptrue%
-\isacommand{theorem}\ {\isachardoublequote}valif\ {\isacharparenleft}norm\ b{\isacharparenright}\ env\ {\isacharequal}\ valif\ b\ env{\isachardoublequote}\isamarkupfalse%
-\isamarkupfalse%
%
\begin{isamarkuptext}%
\noindent
The proof is canonical, provided we first show the following simplification
lemma, which also helps to understand what \isa{normif} does:%
\end{isamarkuptext}%
-\isamarkuptrue%
+\isamarkupfalse%
\isacommand{lemma}\ {\isacharbrackleft}simp{\isacharbrackright}{\isacharcolon}\isanewline
-\ \ {\isachardoublequote}{\isasymforall}t\ e{\isachardot}\ valif\ {\isacharparenleft}normif\ b\ t\ e{\isacharparenright}\ env\ {\isacharequal}\ valif\ {\isacharparenleft}IF\ b\ t\ e{\isacharparenright}\ env{\isachardoublequote}\isamarkupfalse%
-\isamarkupfalse%
-\isamarkupfalse%
-\isamarkupfalse%
-\isamarkupfalse%
-\isamarkupfalse%
+\ \ {\isachardoublequote}{\isasymforall}t\ e{\isachardot}\ valif\ {\isacharparenleft}normif\ b\ t\ e{\isacharparenright}\ env\ {\isacharequal}\ valif\ {\isacharparenleft}IF\ b\ t\ e{\isacharparenright}\ env{\isachardoublequote}%
+\isadelimproof
+%
+\endisadelimproof
+%
+\isatagproof
+%
+\endisatagproof
+{\isafoldproof}%
+%
+\isadelimproof
+%
+\endisadelimproof
+%
+\isadelimproof
+%
+\endisadelimproof
+%
+\isatagproof
+%
+\endisatagproof
+{\isafoldproof}%
+%
+\isadelimproof
+%
+\endisadelimproof
+\isamarkuptrue%
%
\begin{isamarkuptext}%
\noindent
@@ -172,27 +231,48 @@
But how can we be sure that \isa{norm} really produces a normal form in
the above sense? We define a function that tests If-expressions for normality:%
\end{isamarkuptext}%
-\isamarkuptrue%
+\isamarkupfalse%
\isacommand{consts}\ normal\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequote}ifex\ {\isasymRightarrow}\ bool{\isachardoublequote}\isanewline
\isamarkupfalse%
\isacommand{primrec}\isanewline
{\isachardoublequote}normal{\isacharparenleft}CIF\ b{\isacharparenright}\ {\isacharequal}\ True{\isachardoublequote}\isanewline
{\isachardoublequote}normal{\isacharparenleft}VIF\ x{\isacharparenright}\ {\isacharequal}\ True{\isachardoublequote}\isanewline
{\isachardoublequote}normal{\isacharparenleft}IF\ b\ t\ e{\isacharparenright}\ {\isacharequal}\ {\isacharparenleft}normal\ t\ {\isasymand}\ normal\ e\ {\isasymand}\isanewline
-\ \ \ \ \ {\isacharparenleft}case\ b\ of\ CIF\ b\ {\isasymRightarrow}\ True\ {\isacharbar}\ VIF\ x\ {\isasymRightarrow}\ True\ {\isacharbar}\ IF\ x\ y\ z\ {\isasymRightarrow}\ False{\isacharparenright}{\isacharparenright}{\isachardoublequote}\isamarkupfalse%
+\ \ \ \ \ {\isacharparenleft}case\ b\ of\ CIF\ b\ {\isasymRightarrow}\ True\ {\isacharbar}\ VIF\ x\ {\isasymRightarrow}\ True\ {\isacharbar}\ IF\ x\ y\ z\ {\isasymRightarrow}\ False{\isacharparenright}{\isacharparenright}{\isachardoublequote}\isamarkuptrue%
%
\begin{isamarkuptext}%
\noindent
Now we prove \isa{normal\ {\isacharparenleft}norm\ b{\isacharparenright}}. Of course, this requires a lemma about
normality of \isa{normif}:%
\end{isamarkuptext}%
+\isamarkupfalse%
+\isacommand{lemma}\ {\isacharbrackleft}simp{\isacharbrackright}{\isacharcolon}\ {\isachardoublequote}{\isasymforall}t\ e{\isachardot}\ normal{\isacharparenleft}normif\ b\ t\ e{\isacharparenright}\ {\isacharequal}\ {\isacharparenleft}normal\ t\ {\isasymand}\ normal\ e{\isacharparenright}{\isachardoublequote}%
+\isadelimproof
+%
+\endisadelimproof
+%
+\isatagproof
+%
+\endisatagproof
+{\isafoldproof}%
+%
+\isadelimproof
+%
+\endisadelimproof
+%
+\isadelimproof
+%
+\endisadelimproof
+%
+\isatagproof
+%
+\endisatagproof
+{\isafoldproof}%
+%
+\isadelimproof
+%
+\endisadelimproof
\isamarkuptrue%
-\isacommand{lemma}\ {\isacharbrackleft}simp{\isacharbrackright}{\isacharcolon}\ {\isachardoublequote}{\isasymforall}t\ e{\isachardot}\ normal{\isacharparenleft}normif\ b\ t\ e{\isacharparenright}\ {\isacharequal}\ {\isacharparenleft}normal\ t\ {\isasymand}\ normal\ e{\isacharparenright}{\isachardoublequote}\isamarkupfalse%
-\isamarkupfalse%
-\isamarkupfalse%
-\isamarkupfalse%
-\isamarkupfalse%
-\isamarkupfalse%
%
\begin{isamarkuptext}%
\medskip
@@ -211,26 +291,71 @@
\end{exercise}
\index{boolean expressions example|)}%
\end{isamarkuptext}%
-\isamarkuptrue%
-\isamarkupfalse%
-\isamarkupfalse%
-\isamarkupfalse%
-\isamarkupfalse%
-\isamarkupfalse%
-\isamarkupfalse%
-\isamarkupfalse%
-\isamarkupfalse%
-\isamarkupfalse%
-\isamarkupfalse%
-\isamarkupfalse%
-\isamarkupfalse%
-\isamarkupfalse%
-\isamarkupfalse%
-\isamarkupfalse%
-\isamarkupfalse%
-\isamarkupfalse%
-\isamarkupfalse%
-\isamarkupfalse%
+%
+\isadelimproof
+%
+\endisadelimproof
+%
+\isatagproof
+%
+\endisatagproof
+{\isafoldproof}%
+%
+\isadelimproof
+%
+\endisadelimproof
+%
+\isadelimproof
+%
+\endisadelimproof
+%
+\isatagproof
+%
+\endisatagproof
+{\isafoldproof}%
+%
+\isadelimproof
+%
+\endisadelimproof
+%
+\isadelimproof
+%
+\endisadelimproof
+%
+\isatagproof
+%
+\endisatagproof
+{\isafoldproof}%
+%
+\isadelimproof
+%
+\endisadelimproof
+%
+\isadelimproof
+%
+\endisadelimproof
+%
+\isatagproof
+%
+\endisatagproof
+{\isafoldproof}%
+%
+\isadelimproof
+%
+\endisadelimproof
+%
+\isadelimtheory
+%
+\endisadelimtheory
+%
+\isatagtheory
+%
+\endisatagtheory
+{\isafoldtheory}%
+%
+\isadelimtheory
+%
+\endisadelimtheory
\end{isabellebody}%
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