doc-src/TutorialI/Recdef/document/Nested2.tex
changeset 9721 7e51c9f3d5a0
parent 9719 c753196599f9
child 9722 a5f86aed785b
     1.1 --- a/doc-src/TutorialI/Recdef/document/Nested2.tex	Tue Aug 29 12:28:48 2000 +0200
     1.2 +++ b/doc-src/TutorialI/Recdef/document/Nested2.tex	Tue Aug 29 15:13:10 2000 +0200
     1.3 @@ -1,5 +1,4 @@
     1.4 -%
     1.5 -\begin{isabellebody}%
     1.6 +\begin{isabelle}%
     1.7  %
     1.8  \begin{isamarkuptext}%
     1.9  \noindent
    1.10 @@ -32,10 +31,21 @@
    1.11  \end{quote}
    1.12  both of which are solved by simplification:%
    1.13  \end{isamarkuptxt}%
    1.14 -\isacommand{by}{\isacharparenleft}simp{\isacharunderscore}all\ del{\isacharcolon}map{\isacharunderscore}compose\ add{\isacharcolon}sym{\isacharbrackleft}OF\ map{\isacharunderscore}compose{\isacharbrackright}\ rev{\isacharunderscore}map{\isacharparenright}%
    1.15 +\isacommand{by}{\isacharparenleft}simp{\isacharunderscore}all\ add{\isacharcolon}rev{\isacharunderscore}map\ sym{\isacharbrackleft}OF\ map{\isacharunderscore}compose{\isacharbrackright}{\isacharparenright}%
    1.16  \begin{isamarkuptext}%
    1.17  \noindent
    1.18 -If this surprises you, see Datatype/Nested2......
    1.19 +If the proof of the induction step mystifies you, we recommend to go through
    1.20 +the chain of simplification steps in detail, probably with the help of
    1.21 +\isa{trace\_simp}.
    1.22 +%\begin{quote}
    1.23 +%{term[display]"trev(trev(App f ts))"}\\
    1.24 +%{term[display]"App f (rev(map trev (rev(map trev ts))))"}\\
    1.25 +%{term[display]"App f (map trev (rev(rev(map trev ts))))"}\\
    1.26 +%{term[display]"App f (map trev (map trev ts))"}\\
    1.27 +%{term[display]"App f (map (trev o trev) ts)"}\\
    1.28 +%{term[display]"App f (map (%x. x) ts)"}\\
    1.29 +%{term[display]"App f ts"}
    1.30 +%\end{quote}
    1.31  
    1.32  The above definition of \isa{trev} is superior to the one in \S\ref{sec:nested-datatype}
    1.33  because it brings \isa{rev} into play, about which already know a lot, in particular
    1.34 @@ -46,12 +56,13 @@
    1.35  because they determine the complexity of your proofs.}
    1.36  \end{quote}
    1.37  
    1.38 -Let us now return to the question of how \isacommand{recdef} can come up with sensible termination
    1.39 -conditions in the presence of higher-order functions like \isa{map}. For a start, if nothing
    1.40 -were known about \isa{map}, \isa{map\ trev\ \mbox{ts}} might apply \isa{trev} to arbitrary terms,
    1.41 -and thus \isacommand{recdef} would try to prove the unprovable
    1.42 -\isa{size\ \mbox{t}\ {\isacharless}\ Suc\ {\isacharparenleft}term{\isacharunderscore}size\ \mbox{ts}{\isacharparenright}}, without any assumption about \isa{t}.
    1.43 -Therefore \isacommand{recdef} has been supplied with the congruence theorem \isa{map\_cong}: 
    1.44 +Let us now return to the question of how \isacommand{recdef} can come up with
    1.45 +sensible termination conditions in the presence of higher-order functions
    1.46 +like \isa{map}. For a start, if nothing were known about \isa{map},
    1.47 +\isa{map\ trev\ \mbox{ts}} might apply \isa{trev} to arbitrary terms, and thus
    1.48 +\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
    1.49 +\isacommand{recdef} has been supplied with the congruence theorem
    1.50 +\isa{map\_cong}:
    1.51  \begin{quote}
    1.52  
    1.53  \begin{isabelle}%
    1.54 @@ -60,10 +71,11 @@
    1.55  \end{isabelle}%
    1.56  
    1.57  \end{quote}
    1.58 -Its second premise expresses (indirectly) that the second argument of \isa{map} is only applied
    1.59 -to elements of its third argument. Congruence rules for other higher-order functions on lists would
    1.60 -look very similar but have not been proved yet because they were never needed.
    1.61 -If you get into a situation where you need to supply \isacommand{recdef} with new congruence
    1.62 +Its second premise expresses (indirectly) that the second argument of
    1.63 +\isa{map} is only applied to elements of its third argument. Congruence
    1.64 +rules for other higher-order functions on lists would look very similar but
    1.65 +have not been proved yet because they were never needed. If you get into a
    1.66 +situation where you need to supply \isacommand{recdef} with new congruence
    1.67  rules, you can either append the line
    1.68  \begin{ttbox}
    1.69  congs <congruence rules>
    1.70 @@ -78,7 +90,7 @@
    1.71  declaring a congruence rule for the simplifier does not make it
    1.72  available to \isacommand{recdef}, and vice versa. This is intentional.%
    1.73  \end{isamarkuptext}%
    1.74 -\end{isabellebody}%
    1.75 +\end{isabelle}%
    1.76  %%% Local Variables:
    1.77  %%% mode: latex
    1.78  %%% TeX-master: "root"