doc-src/TutorialI/Recdef/document/Nested2.tex
author nipkow
Tue Aug 29 15:13:10 2000 +0200 (2000-08-29)
changeset 9721 7e51c9f3d5a0
parent 9719 c753196599f9
child 9722 a5f86aed785b
permissions -rw-r--r--
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\begin{isabelle}%
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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}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{lemmas}\ {\isacharbrackleft}cong{\isacharbrackright}\ {\isacharequal}\ map{\isacharunderscore}cong\isanewline
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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\ \mbox{x}{\isacharparenright}{\isacharparenright}\ {\isacharequal}\ Var\ \mbox{x}} and the step case
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\begin{quote}
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\begin{isabelle}%
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{\isasymforall}\mbox{t}{\isachardot}\ \mbox{t}\ {\isasymin}\ set\ \mbox{ts}\ {\isasymlongrightarrow}\ trev\ {\isacharparenleft}trev\ \mbox{t}{\isacharparenright}\ {\isacharequal}\ \mbox{t}\ {\isasymLongrightarrow}\isanewline
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trev\ {\isacharparenleft}trev\ {\isacharparenleft}App\ \mbox{f}\ \mbox{ts}{\isacharparenright}{\isacharparenright}\ {\isacharequal}\ App\ \mbox{f}\ \mbox{ts}
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\end{isabelle}%
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\end{quote}
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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}{\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, probably with the help of
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\isa{trace\_simp}.
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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 \S\ref{sec:nested-datatype}
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because it brings \isa{rev} into play, about which already know a lot, in particular
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\isa{rev\ {\isacharparenleft}rev\ \mbox{xs}{\isacharparenright}\ {\isacharequal}\ \mbox{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\ \mbox{ts}} might apply \isa{trev} to arbitrary terms, and thus
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\isacommand{recdef} would try to prove the unprovable \isa{size\ \mbox{t}\ {\isacharless}\ Suc\ {\isacharparenleft}term{\isacharunderscore}size\ \mbox{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\_cong}:
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\begin{quote}
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\begin{isabelle}%
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{\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
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{\isasymLongrightarrow}\ map\ \mbox{f}\ \mbox{xs}\ {\isacharequal}\ map\ \mbox{g}\ \mbox{ys}
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\end{isabelle}%
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\end{quote}
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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 would look very similar but
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have not been proved yet because they were never needed. If you get into a
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situation where you need to supply \isacommand{recdef} with new congruence
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rules, you can either append the line
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\begin{ttbox}
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congs <congruence rules>
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\end{ttbox}
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to the specific occurrence of \isacommand{recdef} or declare them globally:
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\begin{ttbox}
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lemmas [????????] = <congruence rules>
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\end{ttbox}
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Note that \isacommand{recdef} feeds on exactly the same \emph{kind} of
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congruence rules as the simplifier (\S\ref{sec:simp-cong}) but that
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declaring a congruence rule for the simplifier does not make it
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available to \isacommand{recdef}, and vice versa. This is intentional.%
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\end{isamarkuptext}%
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\end{isabelle}%
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%%% Local Variables:
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%%% mode: latex
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%%% TeX-master: "root"
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%%% End: