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--
*** empty log message ***
     1 \begin{isabelle}%
     2 %
     3 \begin{isamarkuptext}%
     4 \noindent
     5 The termintion condition is easily proved by induction:%
     6 \end{isamarkuptext}%
     7 \isacommand{lemma}\ {\isacharbrackleft}simp{\isacharbrackright}{\isacharcolon}\ {\isachardoublequote}t\ {\isasymin}\ set\ ts\ {\isasymlongrightarrow}\ size\ t\ {\isacharless}\ Suc{\isacharparenleft}term{\isacharunderscore}size\ ts{\isacharparenright}{\isachardoublequote}\isanewline
     8 \isacommand{by}{\isacharparenleft}induct{\isacharunderscore}tac\ ts{\isacharcomma}\ auto{\isacharparenright}%
     9 \begin{isamarkuptext}%
    10 \noindent
    11 By making this theorem a simplification rule, \isacommand{recdef}
    12 applies it automatically and the above definition of \isa{trev}
    13 succeeds now. As a reward for our effort, we can now prove the desired
    14 lemma directly. The key is the fact that we no longer need the verbose
    15 induction schema for type \isa{term} but the simpler one arising from
    16 \isa{trev}:%
    17 \end{isamarkuptext}%
    18 \isacommand{lemmas}\ {\isacharbrackleft}cong{\isacharbrackright}\ {\isacharequal}\ map{\isacharunderscore}cong\isanewline
    19 \isacommand{lemma}\ {\isachardoublequote}trev{\isacharparenleft}trev\ t{\isacharparenright}\ {\isacharequal}\ t{\isachardoublequote}\isanewline
    20 \isacommand{apply}{\isacharparenleft}induct{\isacharunderscore}tac\ t\ rule{\isacharcolon}trev{\isachardot}induct{\isacharparenright}%
    21 \begin{isamarkuptxt}%
    22 \noindent
    23 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
    24 \begin{quote}
    26 \begin{isabelle}%
    27 {\isasymforall}\mbox{t}{\isachardot}\ \mbox{t}\ {\isasymin}\ set\ \mbox{ts}\ {\isasymlongrightarrow}\ trev\ {\isacharparenleft}trev\ \mbox{t}{\isacharparenright}\ {\isacharequal}\ \mbox{t}\ {\isasymLongrightarrow}\isanewline
    28 trev\ {\isacharparenleft}trev\ {\isacharparenleft}App\ \mbox{f}\ \mbox{ts}{\isacharparenright}{\isacharparenright}\ {\isacharequal}\ App\ \mbox{f}\ \mbox{ts}
    29 \end{isabelle}%
    31 \end{quote}
    32 both of which are solved by simplification:%
    33 \end{isamarkuptxt}%
    34 \isacommand{by}{\isacharparenleft}simp{\isacharunderscore}all\ add{\isacharcolon}rev{\isacharunderscore}map\ sym{\isacharbrackleft}OF\ map{\isacharunderscore}compose{\isacharbrackright}{\isacharparenright}%
    35 \begin{isamarkuptext}%
    36 \noindent
    37 If the proof of the induction step mystifies you, we recommend to go through
    38 the chain of simplification steps in detail, probably with the help of
    39 \isa{trace\_simp}.
    40 %\begin{quote}
    41 %{term[display]"trev(trev(App f ts))"}\\
    42 %{term[display]"App f (rev(map trev (rev(map trev ts))))"}\\
    43 %{term[display]"App f (map trev (rev(rev(map trev ts))))"}\\
    44 %{term[display]"App f (map trev (map trev ts))"}\\
    45 %{term[display]"App f (map (trev o trev) ts)"}\\
    46 %{term[display]"App f (map (%x. x) ts)"}\\
    47 %{term[display]"App f ts"}
    48 %\end{quote}
    50 The above definition of \isa{trev} is superior to the one in \S\ref{sec:nested-datatype}
    51 because it brings \isa{rev} into play, about which already know a lot, in particular
    52 \isa{rev\ {\isacharparenleft}rev\ \mbox{xs}{\isacharparenright}\ {\isacharequal}\ \mbox{xs}}.
    53 Thus this proof is a good example of an important principle:
    54 \begin{quote}
    55 \emph{Chose your definitions carefully\\
    56 because they determine the complexity of your proofs.}
    57 \end{quote}
    59 Let us now return to the question of how \isacommand{recdef} can come up with
    60 sensible termination conditions in the presence of higher-order functions
    61 like \isa{map}. For a start, if nothing were known about \isa{map},
    62 \isa{map\ trev\ \mbox{ts}} might apply \isa{trev} to arbitrary terms, and thus
    63 \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
    64 \isacommand{recdef} has been supplied with the congruence theorem
    65 \isa{map\_cong}:
    66 \begin{quote}
    68 \begin{isabelle}%
    69 {\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
    70 {\isasymLongrightarrow}\ map\ \mbox{f}\ \mbox{xs}\ {\isacharequal}\ map\ \mbox{g}\ \mbox{ys}
    71 \end{isabelle}%
    73 \end{quote}
    74 Its second premise expresses (indirectly) that the second argument of
    75 \isa{map} is only applied to elements of its third argument. Congruence
    76 rules for other higher-order functions on lists would look very similar but
    77 have not been proved yet because they were never needed. If you get into a
    78 situation where you need to supply \isacommand{recdef} with new congruence
    79 rules, you can either append the line
    80 \begin{ttbox}
    81 congs <congruence rules>
    82 \end{ttbox}
    83 to the specific occurrence of \isacommand{recdef} or declare them globally:
    84 \begin{ttbox}
    85 lemmas [????????] = <congruence rules>
    86 \end{ttbox}
    88 Note that \isacommand{recdef} feeds on exactly the same \emph{kind} of
    89 congruence rules as the simplifier (\S\ref{sec:simp-cong}) but that
    90 declaring a congruence rule for the simplifier does not make it
    91 available to \isacommand{recdef}, and vice versa. This is intentional.%
    92 \end{isamarkuptext}%
    93 \end{isabelle}%
    94 %%% Local Variables:
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    96 %%% TeX-master: "root"
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