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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"