--- a/doc-src/TutorialI/Recdef/document/Nested2.tex Mon Aug 28 14:09:33 2000 +0200
+++ b/doc-src/TutorialI/Recdef/document/Nested2.tex Mon Aug 28 15:13:55 2000 +0200
@@ -4,8 +4,8 @@
\noindent
The termintion condition is easily proved by induction:%
\end{isamarkuptext}%
-\isacommand{lemma}\ [simp]:\ {"}t\ {\isasymin}\ set\ ts\ {\isasymlongrightarrow}\ size\ t\ <\ Suc(term\_size\ ts){"}\isanewline
-\isacommand{by}(induct\_tac\ ts,\ auto)%
+\isacommand{lemma}\ {\isacharbrackleft}simp{\isacharbrackright}{\isacharcolon}\ {\isachardoublequote}t\ {\isasymin}\ set\ ts\ {\isasymlongrightarrow}\ size\ t\ {\isacharless}\ Suc{\isacharparenleft}term{\isacharunderscore}size\ ts{\isacharparenright}{\isachardoublequote}\isanewline
+\isacommand{by}{\isacharparenleft}induct{\isacharunderscore}tac\ ts{\isacharcomma}\ auto{\isacharparenright}%
\begin{isamarkuptext}%
\noindent
By making this theorem a simplification rule, \isacommand{recdef}
@@ -15,30 +15,30 @@
induction schema for type \isa{term} but the simpler one arising from
\isa{trev}:%
\end{isamarkuptext}%
-\isacommand{lemmas}\ [cong]\ =\ map\_cong\isanewline
-\isacommand{lemma}\ {"}trev(trev\ t)\ =\ t{"}\isanewline
-\isacommand{apply}(induct\_tac\ t\ rule:trev.induct)%
+\isacommand{lemmas}\ {\isacharbrackleft}cong{\isacharbrackright}\ {\isacharequal}\ map{\isacharunderscore}cong\isanewline
+\isacommand{lemma}\ {\isachardoublequote}trev{\isacharparenleft}trev\ t{\isacharparenright}\ {\isacharequal}\ t{\isachardoublequote}\isanewline
+\isacommand{apply}{\isacharparenleft}induct{\isacharunderscore}tac\ t\ rule{\isacharcolon}trev{\isachardot}induct{\isacharparenright}%
\begin{isamarkuptxt}%
\noindent
-This leaves us with a trivial base case \isa{trev\ (trev\ (Var\ \mbox{x}))\ =\ Var\ \mbox{x}} and the step case
+This leaves us with a trivial base case \isa{trev\ {\isacharparenleft}trev\ {\isacharparenleft}Var\ \mbox{x}{\isacharparenright}{\isacharparenright}\ {\isacharequal}\ Var\ \mbox{x}} and the step case
\begin{quote}
\begin{isabelle}%
-{\isasymforall}\mbox{t}.\ \mbox{t}\ {\isasymin}\ set\ \mbox{ts}\ {\isasymlongrightarrow}\ trev\ (trev\ \mbox{t})\ =\ \mbox{t}\ {\isasymLongrightarrow}\isanewline
-trev\ (trev\ (App\ \mbox{f}\ \mbox{ts}))\ =\ App\ \mbox{f}\ \mbox{ts}
+{\isasymforall}\mbox{t}{\isachardot}\ \mbox{t}\ {\isasymin}\ set\ \mbox{ts}\ {\isasymlongrightarrow}\ trev\ {\isacharparenleft}trev\ \mbox{t}{\isacharparenright}\ {\isacharequal}\ \mbox{t}\ {\isasymLongrightarrow}\isanewline
+trev\ {\isacharparenleft}trev\ {\isacharparenleft}App\ \mbox{f}\ \mbox{ts}{\isacharparenright}{\isacharparenright}\ {\isacharequal}\ App\ \mbox{f}\ \mbox{ts}
\end{isabelle}%
\end{quote}
both of which are solved by simplification:%
\end{isamarkuptxt}%
-\isacommand{by}(simp\_all\ del:map\_compose\ add:sym[OF\ map\_compose]\ rev\_map)%
+\isacommand{by}{\isacharparenleft}simp{\isacharunderscore}all\ del{\isacharcolon}map{\isacharunderscore}compose\ add{\isacharcolon}sym{\isacharbrackleft}OF\ map{\isacharunderscore}compose{\isacharbrackright}\ rev{\isacharunderscore}map{\isacharparenright}%
\begin{isamarkuptext}%
\noindent
If this surprises you, see Datatype/Nested2......
The above definition of \isa{trev} is superior to the one in \S\ref{sec:nested-datatype}
because it brings \isa{rev} into play, about which already know a lot, in particular
-\isa{rev\ (rev\ \mbox{xs})\ =\ \mbox{xs}}.
+\isa{rev\ {\isacharparenleft}rev\ \mbox{xs}{\isacharparenright}\ {\isacharequal}\ \mbox{xs}}.
Thus this proof is a good example of an important principle:
\begin{quote}
\emph{Chose your definitions carefully\\
@@ -49,13 +49,13 @@
conditions in the presence of higher-order functions like \isa{map}. For a start, if nothing
were known about \isa{map}, \isa{map\ trev\ \mbox{ts}} might apply \isa{trev} to arbitrary terms,
and thus \isacommand{recdef} would try to prove the unprovable
-\isa{size\ \mbox{t}\ <\ Suc\ (term\_size\ \mbox{ts})}, without any assumption about \isa{t}.
+\isa{size\ \mbox{t}\ {\isacharless}\ Suc\ {\isacharparenleft}term{\isacharunderscore}size\ \mbox{ts}{\isacharparenright}}, without any assumption about \isa{t}.
Therefore \isacommand{recdef} has been supplied with the congruence theorem \isa{map\_cong}:
\begin{quote}
\begin{isabelle}%
-{\isasymlbrakk}\mbox{xs}\ =\ \mbox{ys};\ {\isasymAnd}\mbox{x}.\ \mbox{x}\ {\isasymin}\ set\ \mbox{ys}\ {\isasymLongrightarrow}\ \mbox{f}\ \mbox{x}\ =\ \mbox{g}\ \mbox{x}{\isasymrbrakk}\isanewline
-{\isasymLongrightarrow}\ map\ \mbox{f}\ \mbox{xs}\ =\ map\ \mbox{g}\ \mbox{ys}
+{\isasymlbrakk}\mbox{xs}\ {\isacharequal}\ \mbox{ys}{\isacharsemicolon}\ {\isasymAnd}\mbox{x}{\isachardot}\ \mbox{x}\ {\isasymin}\ set\ \mbox{ys}\ {\isasymLongrightarrow}\ \mbox{f}\ \mbox{x}\ {\isacharequal}\ \mbox{g}\ \mbox{x}{\isasymrbrakk}\isanewline
+{\isasymLongrightarrow}\ map\ \mbox{f}\ \mbox{xs}\ {\isacharequal}\ map\ \mbox{g}\ \mbox{ys}
\end{isabelle}%
\end{quote}