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;
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%
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\begin{isabellebody}%
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\def\isabellecontext{Nested{\isadigit{2}}}%
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%
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\begin{isamarkuptext}%
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\noindent
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The termintion condition is easily proved by induction:%
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\end{isamarkuptext}%
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\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
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\isacommand{by}{\isacharparenleft}induct{\isacharunderscore}tac\ ts{\isacharcomma}\ auto{\isacharparenright}%
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\begin{isamarkuptext}%
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\noindent
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By making this theorem a simplification rule, \isacommand{recdef}
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applies it automatically and the above definition of \isa{trev}
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succeeds now. As a reward for our effort, we can now prove the desired
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lemma directly. The key is the fact that we no longer need the verbose
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induction schema for type \isa{term} but the simpler one arising from
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\isa{trev}:%
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\end{isamarkuptext}%
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\isacommand{lemma}\ {\isachardoublequote}trev{\isacharparenleft}trev\ t{\isacharparenright}\ {\isacharequal}\ t{\isachardoublequote}\isanewline
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\isacommand{apply}{\isacharparenleft}induct{\isacharunderscore}tac\ t\ rule{\isacharcolon}trev{\isachardot}induct{\isacharparenright}%
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\begin{isamarkuptxt}%
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\noindent
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This leaves us with a trivial base case \isa{trev\ {\isacharparenleft}trev\ {\isacharparenleft}Var\ x{\isacharparenright}{\isacharparenright}\ {\isacharequal}\ Var\ x} and the step case
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\begin{isabelle}%
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\ \ \ \ \ {\isasymforall}t{\isachardot}\ t\ {\isasymin}\ set\ ts\ {\isasymlongrightarrow}\ trev\ {\isacharparenleft}trev\ t{\isacharparenright}\ {\isacharequal}\ t\ {\isasymLongrightarrow}\isanewline
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\ \ \ \ \ trev\ {\isacharparenleft}trev\ {\isacharparenleft}App\ f\ ts{\isacharparenright}{\isacharparenright}\ {\isacharequal}\ App\ f\ ts%
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\end{isabelle}
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both of which are solved by simplification:%
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\end{isamarkuptxt}%
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\isacommand{by}{\isacharparenleft}simp{\isacharunderscore}all\ add{\isacharcolon}rev{\isacharunderscore}map\ sym{\isacharbrackleft}OF\ map{\isacharunderscore}compose{\isacharbrackright}\ cong{\isacharcolon}map{\isacharunderscore}cong{\isacharparenright}%
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\begin{isamarkuptext}%
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\noindent
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If the proof of the induction step mystifies you, we recommend to go through
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the chain of simplification steps in detail; you will probably need the help of
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\isa{trace{\isacharunderscore}simp}. Theorem \isa{map{\isacharunderscore}cong} is discussed below.
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%\begin{quote}
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%{term[display]"trev(trev(App f ts))"}\\
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%{term[display]"App f (rev(map trev (rev(map trev ts))))"}\\
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%{term[display]"App f (map trev (rev(rev(map trev ts))))"}\\
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%{term[display]"App f (map trev (map trev ts))"}\\
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%{term[display]"App f (map (trev o trev) ts)"}\\
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%{term[display]"App f (map (%x. x) ts)"}\\
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%{term[display]"App f ts"}
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%\end{quote}
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The above definition of \isa{trev} is superior to the one in
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\S\ref{sec:nested-datatype} because it brings \isa{rev} into play, about
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which already know a lot, in particular \isa{rev\ {\isacharparenleft}rev\ xs{\isacharparenright}\ {\isacharequal}\ xs}.
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Thus this proof is a good example of an important principle:
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\begin{quote}
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\emph{Chose your definitions carefully\\
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because they determine the complexity of your proofs.}
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\end{quote}
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Let us now return to the question of how \isacommand{recdef} can come up with
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sensible termination conditions in the presence of higher-order functions
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like \isa{map}. For a start, if nothing were known about \isa{map},
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\isa{map\ trev\ ts} might apply \isa{trev} to arbitrary terms, and thus
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\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
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\isacommand{recdef} has been supplied with the congruence theorem
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\isa{map{\isacharunderscore}cong}:
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\begin{isabelle}%
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\ \ \ \ \ xs\ {\isacharequal}\ ys\ {\isasymLongrightarrow}\isanewline
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\ \ \ \ \ {\isacharparenleft}{\isasymAnd}x{\isachardot}\ x\ {\isasymin}\ set\ ys\ {\isasymLongrightarrow}\ f\ x\ {\isacharequal}\ g\ x{\isacharparenright}\ {\isasymLongrightarrow}\isanewline
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\ \ \ \ \ map\ f\ xs\ {\isacharequal}\ map\ g\ ys%
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\end{isabelle}
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Its second premise expresses (indirectly) that the second argument of
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\isa{map} is only applied to elements of its third argument. Congruence
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rules for other higher-order functions on lists look very similar. If you get
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into a situation where you need to supply \isacommand{recdef} with new
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congruence rules, you can either append a hint locally
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to the specific occurrence of \isacommand{recdef}%
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\end{isamarkuptext}%
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{\isacharparenleft}\isakeyword{hints}\ recdef{\isacharunderscore}cong{\isacharcolon}\ map{\isacharunderscore}cong{\isacharparenright}%
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\begin{isamarkuptext}%
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\noindent
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or declare them globally
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by giving them the \isa{recdef{\isacharunderscore}cong} attribute as in%
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\end{isamarkuptext}%
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\isacommand{declare}\ map{\isacharunderscore}cong{\isacharbrackleft}recdef{\isacharunderscore}cong{\isacharbrackright}%
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\begin{isamarkuptext}%
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Note that the \isa{cong} and \isa{recdef{\isacharunderscore}cong} attributes are
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intentionally kept apart because they control different activities, namely
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simplification and making recursive definitions.
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% The local \isa{cong} in
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% the hints section of \isacommand{recdef} is merely short for \isa{recdef{\isacharunderscore}cong}.
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%The simplifier's congruence rules cannot be used by recdef.
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%For example the weak congruence rules for if and case would prevent
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%recdef from generating sensible termination conditions.%
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\end{isamarkuptext}%
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\end{isabellebody}%
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%%% Local Variables:
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%%% mode: latex
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%%% TeX-master: "root"
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%%% End: