doc-src/TutorialI/Recdef/document/Nested2.tex
author nipkow
Wed Aug 30 18:09:20 2000 +0200 (2000-08-30)
changeset 9754 a123a64cadeb
parent 9722 a5f86aed785b
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permissions -rw-r--r--
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     1 %
     2 \begin{isabellebody}%
     3 %
     4 \begin{isamarkuptext}%
     5 \noindent
     6 The termintion condition is easily proved by induction:%
     7 \end{isamarkuptext}%
     8 \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
     9 \isacommand{by}{\isacharparenleft}induct{\isacharunderscore}tac\ ts{\isacharcomma}\ auto{\isacharparenright}%
    10 \begin{isamarkuptext}%
    11 \noindent
    12 By making this theorem a simplification rule, \isacommand{recdef}
    13 applies it automatically and the above definition of \isa{trev}
    14 succeeds now. As a reward for our effort, we can now prove the desired
    15 lemma directly. The key is the fact that we no longer need the verbose
    16 induction schema for type \isa{term} but the simpler one arising from
    17 \isa{trev}:%
    18 \end{isamarkuptext}%
    19 \isacommand{lemmas}\ {\isacharbrackleft}cong{\isacharbrackright}\ {\isacharequal}\ map{\isacharunderscore}cong\isanewline
    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\ \mbox{x}{\isacharparenright}{\isacharparenright}\ {\isacharequal}\ Var\ \mbox{x}} and the step case
    25 \begin{quote}
    26 
    27 \begin{isabelle}%
    28 {\isasymforall}\mbox{t}{\isachardot}\ \mbox{t}\ {\isasymin}\ set\ \mbox{ts}\ {\isasymlongrightarrow}\ trev\ {\isacharparenleft}trev\ \mbox{t}{\isacharparenright}\ {\isacharequal}\ \mbox{t}\ {\isasymLongrightarrow}\isanewline
    29 trev\ {\isacharparenleft}trev\ {\isacharparenleft}App\ \mbox{f}\ \mbox{ts}{\isacharparenright}{\isacharparenright}\ {\isacharequal}\ App\ \mbox{f}\ \mbox{ts}
    30 \end{isabelle}%
    31 
    32 \end{quote}
    33 both of which are solved by simplification:%
    34 \end{isamarkuptxt}%
    35 \isacommand{by}{\isacharparenleft}simp{\isacharunderscore}all\ add{\isacharcolon}rev{\isacharunderscore}map\ sym{\isacharbrackleft}OF\ map{\isacharunderscore}compose{\isacharbrackright}{\isacharparenright}%
    36 \begin{isamarkuptext}%
    37 \noindent
    38 If the proof of the induction step mystifies you, we recommend to go through
    39 the chain of simplification steps in detail; you will probably need the help of
    40 \isa{trace{\isacharunderscore}simp}.
    41 %\begin{quote}
    42 %{term[display]"trev(trev(App f ts))"}\\
    43 %{term[display]"App f (rev(map trev (rev(map trev ts))))"}\\
    44 %{term[display]"App f (map trev (rev(rev(map trev ts))))"}\\
    45 %{term[display]"App f (map trev (map trev ts))"}\\
    46 %{term[display]"App f (map (trev o trev) ts)"}\\
    47 %{term[display]"App f (map (%x. x) ts)"}\\
    48 %{term[display]"App f ts"}
    49 %\end{quote}
    50 
    51 The above definition of \isa{trev} is superior to the one in
    52 \S\ref{sec:nested-datatype} because it brings \isa{rev} into play, about
    53 which already know a lot, in particular \isa{rev\ {\isacharparenleft}rev\ \mbox{xs}{\isacharparenright}\ {\isacharequal}\ \mbox{xs}}.
    54 Thus this proof is a good example of an important principle:
    55 \begin{quote}
    56 \emph{Chose your definitions carefully\\
    57 because they determine the complexity of your proofs.}
    58 \end{quote}
    59 
    60 Let us now return to the question of how \isacommand{recdef} can come up with
    61 sensible termination conditions in the presence of higher-order functions
    62 like \isa{map}. For a start, if nothing were known about \isa{map},
    63 \isa{map\ trev\ \mbox{ts}} might apply \isa{trev} to arbitrary terms, and thus
    64 \isacommand{recdef} would try to prove the unprovable \isa{size\ \mbox{t}\ {\isacharless}\ Suc\ {\isacharparenleft}term{\isacharunderscore}list{\isacharunderscore}size\ \mbox{ts}{\isacharparenright}}, without any assumption about \isa{\mbox{t}}.  Therefore
    65 \isacommand{recdef} has been supplied with the congruence theorem
    66 \isa{map{\isacharunderscore}cong}:
    67 \begin{quote}
    68 
    69 \begin{isabelle}%
    70 {\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
    71 {\isasymLongrightarrow}\ map\ \mbox{f}\ \mbox{xs}\ {\isacharequal}\ map\ \mbox{g}\ \mbox{ys}
    72 \end{isabelle}%
    73 
    74 \end{quote}
    75 Its second premise expresses (indirectly) that the second argument of
    76 \isa{map} is only applied to elements of its third argument. Congruence
    77 rules for other higher-order functions on lists would look very similar but
    78 have not been proved yet because they were never needed. If you get into a
    79 situation where you need to supply \isacommand{recdef} with new congruence
    80 rules, you can either append the line
    81 \begin{ttbox}
    82 congs <congruence rules>
    83 \end{ttbox}
    84 to the specific occurrence of \isacommand{recdef} or declare them globally:
    85 \begin{ttbox}
    86 lemmas [????????] = <congruence rules>
    87 \end{ttbox}
    88 
    89 Note that \isacommand{recdef} feeds on exactly the same \emph{kind} of
    90 congruence rules as the simplifier (\S\ref{sec:simp-cong}) but that
    91 declaring a congruence rule for the simplifier does not make it
    92 available to \isacommand{recdef}, and vice versa. This is intentional.%
    93 \end{isamarkuptext}%
    94 \end{isabellebody}%
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