--- a/doc-src/TutorialI/Recdef/document/Nested2.tex Tue Aug 28 13:15:15 2012 +0200
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,152 +0,0 @@
-%
-\begin{isabellebody}%
-\def\isabellecontext{Nested{\isadigit{2}}}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\isatagtheory
-%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-\isanewline
-%
-\endisadelimtheory
-\isacommand{lemma}\isamarkupfalse%
-\ {\isacharbrackleft}simp{\isacharbrackright}{\isacharcolon}\ {\isachardoublequoteopen}t\ {\isasymin}\ set\ ts\ {\isasymlongrightarrow}\ size\ t\ {\isacharless}\ Suc{\isacharparenleft}term{\isacharunderscore}list{\isacharunderscore}size\ ts{\isacharparenright}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-{\isacharparenleft}induct{\isacharunderscore}tac\ ts{\isacharcomma}\ auto{\isacharparenright}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\begin{isamarkuptext}%
-\noindent
-By making this theorem a simplification rule, \isacommand{recdef}
-applies it automatically and the definition of \isa{trev}
-succeeds now. As a reward for our effort, we can now prove the desired
-lemma directly. We no longer need the verbose
-induction schema for type \isa{term} and can use the simpler one arising from
-\isa{trev}:%
-\end{isamarkuptext}%
-\isamarkuptrue%
-\isacommand{lemma}\isamarkupfalse%
-\ {\isachardoublequoteopen}trev{\isacharparenleft}trev\ t{\isacharparenright}\ {\isacharequal}\ t{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}induct{\isacharunderscore}tac\ t\ rule{\isacharcolon}\ trev{\isachardot}induct{\isacharparenright}%
-\begin{isamarkuptxt}%
-\begin{isabelle}%
-\ {\isadigit{1}}{\isachardot}\ {\isasymAnd}x{\isachardot}\ trev\ {\isacharparenleft}trev\ {\isacharparenleft}Var\ x{\isacharparenright}{\isacharparenright}\ {\isacharequal}\ Var\ x\isanewline
-\ {\isadigit{2}}{\isachardot}\ {\isasymAnd}f\ ts{\isachardot}\isanewline
-\isaindent{\ {\isadigit{2}}{\isachardot}\ \ \ \ }{\isasymforall}x{\isachardot}\ x\ {\isasymin}\ set\ ts\ {\isasymlongrightarrow}\ trev\ {\isacharparenleft}trev\ x{\isacharparenright}\ {\isacharequal}\ x\ {\isasymLongrightarrow}\isanewline
-\isaindent{\ {\isadigit{2}}{\isachardot}\ \ \ \ }trev\ {\isacharparenleft}trev\ {\isacharparenleft}App\ f\ ts{\isacharparenright}{\isacharparenright}\ {\isacharequal}\ App\ f\ ts%
-\end{isabelle}
-Both the base case and the induction step fall to simplification:%
-\end{isamarkuptxt}%
-\isamarkuptrue%
-\isacommand{by}\isamarkupfalse%
-{\isacharparenleft}simp{\isacharunderscore}all\ add{\isacharcolon}\ rev{\isacharunderscore}map\ sym{\isacharbrackleft}OF\ map{\isacharunderscore}compose{\isacharbrackright}\ cong{\isacharcolon}\ map{\isacharunderscore}cong{\isacharparenright}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\begin{isamarkuptext}%
-\noindent
-If the proof of the induction step mystifies you, we recommend that you go through
-the chain of simplification steps in detail; you will probably need the help of
-\isa{trace{\isacharunderscore}simp}. Theorem \isa{map{\isacharunderscore}cong} is discussed below.
-%\begin{quote}
-%{term[display]"trev(trev(App f ts))"}\\
-%{term[display]"App f (rev(map trev (rev(map trev ts))))"}\\
-%{term[display]"App f (map trev (rev(rev(map trev ts))))"}\\
-%{term[display]"App f (map trev (map trev ts))"}\\
-%{term[display]"App f (map (trev o trev) ts)"}\\
-%{term[display]"App f (map (%x. x) ts)"}\\
-%{term[display]"App f ts"}
-%\end{quote}
-
-The definition of \isa{trev} above is superior to the one in
-\S\ref{sec:nested-datatype} because it uses \isa{rev}
-and lets us use existing facts such as \hbox{\isa{rev\ {\isacharparenleft}rev\ xs{\isacharparenright}\ {\isacharequal}\ xs}}.
-Thus this proof is a good example of an important principle:
-\begin{quote}
-\emph{Chose your definitions carefully\\
-because they determine the complexity of your proofs.}
-\end{quote}
-
-Let us now return to the question of how \isacommand{recdef} can come up with
-sensible termination conditions in the presence of higher-order functions
-like \isa{map}. For a start, if nothing were known about \isa{map}, then
-\isa{map\ trev\ ts} might apply \isa{trev} to arbitrary terms, and thus
-\isacommand{recdef} would try to prove the unprovable \isa{size\ t\ {\isacharless}\ Suc\ {\isacharparenleft}term{\isacharunderscore}list{\isacharunderscore}size\ ts{\isacharparenright}}, without any assumption about \isa{t}. Therefore
-\isacommand{recdef} has been supplied with the congruence theorem
-\isa{map{\isacharunderscore}cong}:
-\begin{isabelle}%
-\ \ \ \ \ {\isasymlbrakk}xs\ {\isacharequal}\ ys{\isacharsemicolon}\ {\isasymAnd}x{\isachardot}\ x\ {\isasymin}\ set\ ys\ {\isasymLongrightarrow}\ f\ x\ {\isacharequal}\ g\ x{\isasymrbrakk}\isanewline
-\isaindent{\ \ \ \ \ }{\isasymLongrightarrow}\ map\ f\ xs\ {\isacharequal}\ map\ g\ ys%
-\end{isabelle}
-Its second premise expresses that in \isa{map\ f\ xs},
-function \isa{f} is only applied to elements of list \isa{xs}. Congruence
-rules for other higher-order functions on lists are similar. If you get
-into a situation where you need to supply \isacommand{recdef} with new
-congruence rules, you can append a hint after the end of
-the recursion equations:\cmmdx{hints}%
-\end{isamarkuptext}%
-\isamarkuptrue%
-{\isacharparenleft}\isakeyword{hints}\ recdef{\isacharunderscore}cong{\isacharcolon}\ map{\isacharunderscore}cong{\isacharparenright}%
-\begin{isamarkuptext}%
-\noindent
-Or you can declare them globally
-by giving them the \attrdx{recdef_cong} attribute:%
-\end{isamarkuptext}%
-\isamarkuptrue%
-\isacommand{declare}\isamarkupfalse%
-\ map{\isacharunderscore}cong{\isacharbrackleft}recdef{\isacharunderscore}cong{\isacharbrackright}%
-\begin{isamarkuptext}%
-The \isa{cong} and \isa{recdef{\isacharunderscore}cong} attributes are
-intentionally kept apart because they control different activities, namely
-simplification and making recursive definitions.
-%The simplifier's congruence rules cannot be used by recdef.
-%For example the weak congruence rules for if and case would prevent
-%recdef from generating sensible termination conditions.%
-\end{isamarkuptext}%
-\isamarkuptrue%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\isatagtheory
-%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-\end{isabellebody}%
-%%% Local Variables:
-%%% mode: latex
-%%% TeX-master: "root"
-%%% End: