doc-src/TutorialI/Recdef/document/Nested2.tex
 author wenzelm Wed Dec 06 21:10:40 2000 +0100 (2000-12-06) changeset 10617 adc0ed64a120 parent 10601 894f845c3dbf child 10645 175ccbd5415a permissions -rw-r--r--
updated;
 nipkow@9722  1 %  nipkow@9722  2 \begin{isabellebody}%  wenzelm@10267  3 \def\isabellecontext{Nested{\isadigit{2}}}%  nipkow@9690  4 %  nipkow@9690  5 \begin{isamarkuptext}%  nipkow@9690  6 \noindent  nipkow@9690  7 The termintion condition is easily proved by induction:%  nipkow@9690  8 \end{isamarkuptext}%  nipkow@9754  9 \isacommand{lemma}\ {\isacharbrackleft}simp{\isacharbrackright}{\isacharcolon}\ {\isachardoublequote}t\ {\isasymin}\ set\ ts\ {\isasymlongrightarrow}\ size\ t\ {\isacharless}\ Suc{\isacharparenleft}term{\isacharunderscore}list{\isacharunderscore}size\ ts{\isacharparenright}{\isachardoublequote}\isanewline  wenzelm@9698  10 \isacommand{by}{\isacharparenleft}induct{\isacharunderscore}tac\ ts{\isacharcomma}\ auto{\isacharparenright}%  nipkow@9690  11 \begin{isamarkuptext}%  nipkow@9690  12 \noindent  nipkow@9690  13 By making this theorem a simplification rule, \isacommand{recdef}  nipkow@9690  14 applies it automatically and the above definition of \isa{trev}  nipkow@9690  15 succeeds now. As a reward for our effort, we can now prove the desired  nipkow@9690  16 lemma directly. The key is the fact that we no longer need the verbose  nipkow@9690  17 induction schema for type \isa{term} but the simpler one arising from  nipkow@9690  18 \isa{trev}:%  nipkow@9690  19 \end{isamarkuptext}%  wenzelm@9698  20 \isacommand{lemma}\ {\isachardoublequote}trev{\isacharparenleft}trev\ t{\isacharparenright}\ {\isacharequal}\ t{\isachardoublequote}\isanewline  wenzelm@9698  21 \isacommand{apply}{\isacharparenleft}induct{\isacharunderscore}tac\ t\ rule{\isacharcolon}trev{\isachardot}induct{\isacharparenright}%  nipkow@9690  22 \begin{isamarkuptxt}%  nipkow@9690  23 \noindent  nipkow@9792  24 This leaves us with a trivial base case \isa{trev\ {\isacharparenleft}trev\ {\isacharparenleft}Var\ x{\isacharparenright}{\isacharparenright}\ {\isacharequal}\ Var\ x} and the step case  nipkow@9690  25 \begin{isabelle}%  nipkow@9834  26 \ \ \ \ \ {\isasymforall}t{\isachardot}\ t\ {\isasymin}\ set\ ts\ {\isasymlongrightarrow}\ trev\ {\isacharparenleft}trev\ t{\isacharparenright}\ {\isacharequal}\ t\ {\isasymLongrightarrow}\isanewline  nipkow@9834  27 \ \ \ \ \ trev\ {\isacharparenleft}trev\ {\isacharparenleft}App\ f\ ts{\isacharparenright}{\isacharparenright}\ {\isacharequal}\ App\ f\ ts%  wenzelm@9924  28 \end{isabelle}  nipkow@9690  29 both of which are solved by simplification:%  nipkow@9690  30 \end{isamarkuptxt}%  nipkow@9933  31 \isacommand{by}{\isacharparenleft}simp{\isacharunderscore}all\ add{\isacharcolon}rev{\isacharunderscore}map\ sym{\isacharbrackleft}OF\ map{\isacharunderscore}compose{\isacharbrackright}\ cong{\isacharcolon}map{\isacharunderscore}cong{\isacharparenright}%  nipkow@9690  32 \begin{isamarkuptext}%  nipkow@9690  33 \noindent  nipkow@9721  34 If the proof of the induction step mystifies you, we recommend to go through  nipkow@9754  35 the chain of simplification steps in detail; you will probably need the help of  nipkow@9933  36 \isa{trace{\isacharunderscore}simp}. Theorem \isa{map{\isacharunderscore}cong} is discussed below.  nipkow@9721  37 %\begin{quote}  nipkow@9721  38 %{term[display]"trev(trev(App f ts))"}\\  nipkow@9721  39 %{term[display]"App f (rev(map trev (rev(map trev ts))))"}\\  nipkow@9721  40 %{term[display]"App f (map trev (rev(rev(map trev ts))))"}\\  nipkow@9721  41 %{term[display]"App f (map trev (map trev ts))"}\\  nipkow@9721  42 %{term[display]"App f (map (trev o trev) ts)"}\\  nipkow@9721  43 %{term[display]"App f (map (%x. x) ts)"}\\  nipkow@9721  44 %{term[display]"App f ts"}  nipkow@9721  45 %\end{quote}  nipkow@9690  46 nipkow@9754  47 The above definition of \isa{trev} is superior to the one in  nipkow@9754  48 \S\ref{sec:nested-datatype} because it brings \isa{rev} into play, about  nipkow@9792  49 which already know a lot, in particular \isa{rev\ {\isacharparenleft}rev\ xs{\isacharparenright}\ {\isacharequal}\ xs}.  nipkow@9690  50 Thus this proof is a good example of an important principle:  nipkow@9690  51 \begin{quote}  nipkow@9690  52 \emph{Chose your definitions carefully\\  nipkow@9690  53 because they determine the complexity of your proofs.}  nipkow@9690  54 \end{quote}  nipkow@9690  55 nipkow@9721  56 Let us now return to the question of how \isacommand{recdef} can come up with  nipkow@9721  57 sensible termination conditions in the presence of higher-order functions  nipkow@9721  58 like \isa{map}. For a start, if nothing were known about \isa{map},  nipkow@9792  59 \isa{map\ trev\ ts} might apply \isa{trev} to arbitrary terms, and thus  nipkow@9792  60 \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  nipkow@9721  61 \isacommand{recdef} has been supplied with the congruence theorem  nipkow@9754  62 \isa{map{\isacharunderscore}cong}:  nipkow@9690  63 \begin{isabelle}%  wenzelm@10617  64 \ \ \ \ \ xs\ {\isacharequal}\ ys\ {\isasymLongrightarrow}\isanewline  wenzelm@10617  65 \ \ \ \ \ {\isacharparenleft}{\isasymAnd}x{\isachardot}\ x\ {\isasymin}\ set\ ys\ {\isasymLongrightarrow}\ f\ x\ {\isacharequal}\ g\ x{\isacharparenright}\ {\isasymLongrightarrow}\isanewline  wenzelm@10617  66 \ \ \ \ \ map\ f\ xs\ {\isacharequal}\ map\ g\ ys%  wenzelm@9924  67 \end{isabelle}  nipkow@9721  68 Its second premise expresses (indirectly) that the second argument of  nipkow@9721  69 \isa{map} is only applied to elements of its third argument. Congruence  nipkow@10212  70 rules for other higher-order functions on lists look very similar. If you get  nipkow@10212  71 into a situation where you need to supply \isacommand{recdef} with new  nipkow@10212  72 congruence rules, you can either append a hint locally  nipkow@9940  73 to the specific occurrence of \isacommand{recdef}%  nipkow@9940  74 \end{isamarkuptext}%  nipkow@10171  75 {\isacharparenleft}\isakeyword{hints}\ recdef{\isacharunderscore}cong{\isacharcolon}\ map{\isacharunderscore}cong{\isacharparenright}%  nipkow@9940  76 \begin{isamarkuptext}%  nipkow@9940  77 \noindent  nipkow@9940  78 or declare them globally  nipkow@9940  79 by giving them the \isa{recdef{\isacharunderscore}cong} attribute as in%  nipkow@9940  80 \end{isamarkuptext}%  nipkow@9940  81 \isacommand{declare}\ map{\isacharunderscore}cong{\isacharbrackleft}recdef{\isacharunderscore}cong{\isacharbrackright}%  nipkow@9940  82 \begin{isamarkuptext}%  nipkow@10171  83 Note that the \isa{cong} and \isa{recdef{\isacharunderscore}cong} attributes are  nipkow@9940  84 intentionally kept apart because they control different activities, namely  nipkow@10171  85 simplification and making recursive definitions.  nipkow@10171  86 % The local \isa{cong} in  nipkow@10171  87 % the hints section of \isacommand{recdef} is merely short for \isa{recdef{\isacharunderscore}cong}.  nipkow@9933  88 %The simplifier's congruence rules cannot be used by recdef.  nipkow@9933  89 %For example the weak congruence rules for if and case would prevent  nipkow@9933  90 %recdef from generating sensible termination conditions.%  nipkow@9690  91 \end{isamarkuptext}%  nipkow@9722  92 \end{isabellebody}%  nipkow@9690  93 %%% Local Variables:  nipkow@9690  94 %%% mode: latex  nipkow@9690  95 %%% TeX-master: "root"  nipkow@9690  96 %%% End: