doc-src/TutorialI/Recdef/document/Nested2.tex
 author wenzelm Mon Sep 11 18:00:47 2000 +0200 (2000-09-11) changeset 9924 3370f6aa3200 parent 9834 109b11c4e77e child 9933 9feb1e0c4cb3 permissions -rw-r--r--
updated;
 nipkow@9722  1 %  nipkow@9722  2 \begin{isabellebody}%  wenzelm@9924  3 \def\isabellecontext{Nested2}%  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{lemmas}\ {\isacharbrackleft}cong{\isacharbrackright}\ {\isacharequal}\ map{\isacharunderscore}cong\isanewline  wenzelm@9698  21 \isacommand{lemma}\ {\isachardoublequote}trev{\isacharparenleft}trev\ t{\isacharparenright}\ {\isacharequal}\ t{\isachardoublequote}\isanewline  wenzelm@9698  22 \isacommand{apply}{\isacharparenleft}induct{\isacharunderscore}tac\ t\ rule{\isacharcolon}trev{\isachardot}induct{\isacharparenright}%  nipkow@9690  23 \begin{isamarkuptxt}%  nipkow@9690  24 \noindent  nipkow@9792  25 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  26 \begin{isabelle}%  nipkow@9834  27 \ \ \ \ \ {\isasymforall}t{\isachardot}\ t\ {\isasymin}\ set\ ts\ {\isasymlongrightarrow}\ trev\ {\isacharparenleft}trev\ t{\isacharparenright}\ {\isacharequal}\ t\ {\isasymLongrightarrow}\isanewline  nipkow@9834  28 \ \ \ \ \ trev\ {\isacharparenleft}trev\ {\isacharparenleft}App\ f\ ts{\isacharparenright}{\isacharparenright}\ {\isacharequal}\ App\ f\ ts%  wenzelm@9924  29 \end{isabelle}  nipkow@9690  30 both of which are solved by simplification:%  nipkow@9690  31 \end{isamarkuptxt}%  nipkow@9721  32 \isacommand{by}{\isacharparenleft}simp{\isacharunderscore}all\ add{\isacharcolon}rev{\isacharunderscore}map\ sym{\isacharbrackleft}OF\ map{\isacharunderscore}compose{\isacharbrackright}{\isacharparenright}%  nipkow@9690  33 \begin{isamarkuptext}%  nipkow@9690  34 \noindent  nipkow@9721  35 If the proof of the induction step mystifies you, we recommend to go through  nipkow@9754  36 the chain of simplification steps in detail; you will probably need the help of  nipkow@9754  37 \isa{trace{\isacharunderscore}simp}.  nipkow@9721  38 %\begin{quote}  nipkow@9721  39 %{term[display]"trev(trev(App f ts))"}\\  nipkow@9721  40 %{term[display]"App f (rev(map trev (rev(map trev ts))))"}\\  nipkow@9721  41 %{term[display]"App f (map trev (rev(rev(map trev ts))))"}\\  nipkow@9721  42 %{term[display]"App f (map trev (map trev ts))"}\\  nipkow@9721  43 %{term[display]"App f (map (trev o trev) ts)"}\\  nipkow@9721  44 %{term[display]"App f (map (%x. x) ts)"}\\  nipkow@9721  45 %{term[display]"App f ts"}  nipkow@9721  46 %\end{quote}  nipkow@9690  47 nipkow@9754  48 The above definition of \isa{trev} is superior to the one in  nipkow@9754  49 \S\ref{sec:nested-datatype} because it brings \isa{rev} into play, about  nipkow@9792  50 which already know a lot, in particular \isa{rev\ {\isacharparenleft}rev\ xs{\isacharparenright}\ {\isacharequal}\ xs}.  nipkow@9690  51 Thus this proof is a good example of an important principle:  nipkow@9690  52 \begin{quote}  nipkow@9690  53 \emph{Chose your definitions carefully\\  nipkow@9690  54 because they determine the complexity of your proofs.}  nipkow@9690  55 \end{quote}  nipkow@9690  56 nipkow@9721  57 Let us now return to the question of how \isacommand{recdef} can come up with  nipkow@9721  58 sensible termination conditions in the presence of higher-order functions  nipkow@9721  59 like \isa{map}. For a start, if nothing were known about \isa{map},  nipkow@9792  60 \isa{map\ trev\ ts} might apply \isa{trev} to arbitrary terms, and thus  nipkow@9792  61 \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  62 \isacommand{recdef} has been supplied with the congruence theorem  nipkow@9754  63 \isa{map{\isacharunderscore}cong}:  nipkow@9690  64 \begin{isabelle}%  nipkow@9834  65 \ \ \ \ \ {\isasymlbrakk}xs\ {\isacharequal}\ ys{\isacharsemicolon}\ {\isasymAnd}x{\isachardot}\ x\ {\isasymin}\ set\ ys\ {\isasymLongrightarrow}\ f\ x\ {\isacharequal}\ g\ x{\isasymrbrakk}\isanewline  nipkow@9834  66 \ \ \ \ \ {\isasymLongrightarrow}\ 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@9721  70 rules for other higher-order functions on lists would look very similar but  nipkow@9721  71 have not been proved yet because they were never needed. If you get into a  nipkow@9721  72 situation where you need to supply \isacommand{recdef} with new congruence  nipkow@9690  73 rules, you can either append the line  nipkow@9690  74 \begin{ttbox}  nipkow@9690  75 congs  nipkow@9690  76 \end{ttbox}  nipkow@9690  77 to the specific occurrence of \isacommand{recdef} or declare them globally:  nipkow@9690  78 \begin{ttbox}  nipkow@9690  79 lemmas [????????] =  nipkow@9690  80 \end{ttbox}  nipkow@9690  81 nipkow@9690  82 Note that \isacommand{recdef} feeds on exactly the same \emph{kind} of  nipkow@9690  83 congruence rules as the simplifier (\S\ref{sec:simp-cong}) but that  nipkow@9690  84 declaring a congruence rule for the simplifier does not make it  nipkow@9690  85 available to \isacommand{recdef}, and vice versa. This is intentional.%  nipkow@9690  86 \end{isamarkuptext}%  nipkow@9722  87 \end{isabellebody}%  nipkow@9690  88 %%% Local Variables:  nipkow@9690  89 %%% mode: latex  nipkow@9690  90 %%% TeX-master: "root"  nipkow@9690  91 %%% End: