doc-src/TutorialI/Recdef/document/Nested2.tex
author nipkow
Tue Sep 12 15:43:15 2000 +0200 (2000-09-12)
changeset 9933 9feb1e0c4cb3
parent 9924 3370f6aa3200
child 9940 102f2430cef9
permissions -rw-r--r--
*** empty log message ***
     1 %
     2 \begin{isabellebody}%
     3 \def\isabellecontext{Nested2}%
     4 %
     5 \begin{isamarkuptext}%
     6 \noindent
     7 The termintion condition is easily proved by induction:%
     8 \end{isamarkuptext}%
     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
    10 \isacommand{by}{\isacharparenleft}induct{\isacharunderscore}tac\ ts{\isacharcomma}\ auto{\isacharparenright}%
    11 \begin{isamarkuptext}%
    12 \noindent
    13 By making this theorem a simplification rule, \isacommand{recdef}
    14 applies it automatically and the above definition of \isa{trev}
    15 succeeds now. As a reward for our effort, we can now prove the desired
    16 lemma directly. The key is the fact that we no longer need the verbose
    17 induction schema for type \isa{term} but the simpler one arising from
    18 \isa{trev}:%
    19 \end{isamarkuptext}%
    20 \isacommand{lemma}\ {\isachardoublequote}trev{\isacharparenleft}trev\ t{\isacharparenright}\ {\isacharequal}\ t{\isachardoublequote}\isanewline
    21 \isacommand{apply}{\isacharparenleft}induct{\isacharunderscore}tac\ t\ rule{\isacharcolon}trev{\isachardot}induct{\isacharparenright}%
    22 \begin{isamarkuptxt}%
    23 \noindent
    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
    25 \begin{isabelle}%
    26 \ \ \ \ \ {\isasymforall}t{\isachardot}\ t\ {\isasymin}\ set\ ts\ {\isasymlongrightarrow}\ trev\ {\isacharparenleft}trev\ t{\isacharparenright}\ {\isacharequal}\ t\ {\isasymLongrightarrow}\isanewline
    27 \ \ \ \ \ trev\ {\isacharparenleft}trev\ {\isacharparenleft}App\ f\ ts{\isacharparenright}{\isacharparenright}\ {\isacharequal}\ App\ f\ ts%
    28 \end{isabelle}
    29 both of which are solved by simplification:%
    30 \end{isamarkuptxt}%
    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}%
    32 \begin{isamarkuptext}%
    33 \noindent
    34 If the proof of the induction step mystifies you, we recommend to go through
    35 the chain of simplification steps in detail; you will probably need the help of
    36 \isa{trace{\isacharunderscore}simp}. Theorem \isa{map{\isacharunderscore}cong} is discussed below.
    37 %\begin{quote}
    38 %{term[display]"trev(trev(App f ts))"}\\
    39 %{term[display]"App f (rev(map trev (rev(map trev ts))))"}\\
    40 %{term[display]"App f (map trev (rev(rev(map trev ts))))"}\\
    41 %{term[display]"App f (map trev (map trev ts))"}\\
    42 %{term[display]"App f (map (trev o trev) ts)"}\\
    43 %{term[display]"App f (map (%x. x) ts)"}\\
    44 %{term[display]"App f ts"}
    45 %\end{quote}
    46 
    47 The above definition of \isa{trev} is superior to the one in
    48 \S\ref{sec:nested-datatype} because it brings \isa{rev} into play, about
    49 which already know a lot, in particular \isa{rev\ {\isacharparenleft}rev\ xs{\isacharparenright}\ {\isacharequal}\ xs}.
    50 Thus this proof is a good example of an important principle:
    51 \begin{quote}
    52 \emph{Chose your definitions carefully\\
    53 because they determine the complexity of your proofs.}
    54 \end{quote}
    55 
    56 Let us now return to the question of how \isacommand{recdef} can come up with
    57 sensible termination conditions in the presence of higher-order functions
    58 like \isa{map}. For a start, if nothing were known about \isa{map},
    59 \isa{map\ trev\ ts} might apply \isa{trev} to arbitrary terms, and thus
    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
    61 \isacommand{recdef} has been supplied with the congruence theorem
    62 \isa{map{\isacharunderscore}cong}:
    63 \begin{isabelle}%
    64 \ \ \ \ \ {\isasymlbrakk}xs\ {\isacharequal}\ ys{\isacharsemicolon}\ {\isasymAnd}x{\isachardot}\ x\ {\isasymin}\ set\ ys\ {\isasymLongrightarrow}\ f\ x\ {\isacharequal}\ g\ x{\isasymrbrakk}\isanewline
    65 \ \ \ \ \ {\isasymLongrightarrow}\ map\ f\ xs\ {\isacharequal}\ map\ g\ ys%
    66 \end{isabelle}
    67 Its second premise expresses (indirectly) that the second argument of
    68 \isa{map} is only applied to elements of its third argument. Congruence
    69 rules for other higher-order functions on lists would look very similar but
    70 have not been proved yet because they were never needed. If you get into a
    71 situation where you need to supply \isacommand{recdef} with new congruence
    72 rules, you can either append the line
    73 \begin{ttbox}
    74 congs <congruence rules>
    75 \end{ttbox}
    76 to the specific occurrence of \isacommand{recdef} or declare them globally:
    77 \begin{ttbox}
    78 lemmas [????????] = <congruence rules>
    79 \end{ttbox}
    80 
    81 Note that \isacommand{recdef} feeds on exactly the same \emph{kind} of
    82 congruence rules as the simplifier (\S\ref{sec:simp-cong}) but that
    83 declaring a congruence rule for the simplifier does not make it
    84 available to \isacommand{recdef}, and vice versa. This is intentional.
    85 %The simplifier's congruence rules cannot be used by recdef.
    86 %For example the weak congruence rules for if and case would prevent
    87 %recdef from generating sensible termination conditions.%
    88 \end{isamarkuptext}%
    89 \end{isabellebody}%
    90 %%% Local Variables:
    91 %%% mode: latex
    92 %%% TeX-master: "root"
    93 %%% End: