doc-src/TutorialI/Recdef/document/Nested2.tex
 author wenzelm Mon Sep 11 18:00:47 2000 +0200 (2000-09-11) changeset 9924 3370f6aa3200 parent 9834 109b11c4e77e child 9933 9feb1e0c4cb3 permissions -rw-r--r--
updated;
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{lemmas}\ {\isacharbrackleft}cong{\isacharbrackright}\ {\isacharequal}\ map{\isacharunderscore}cong\isanewline
21 \isacommand{lemma}\ {\isachardoublequote}trev{\isacharparenleft}trev\ t{\isacharparenright}\ {\isacharequal}\ t{\isachardoublequote}\isanewline
22 \isacommand{apply}{\isacharparenleft}induct{\isacharunderscore}tac\ t\ rule{\isacharcolon}trev{\isachardot}induct{\isacharparenright}%
23 \begin{isamarkuptxt}%
24 \noindent
25 This leaves us with a trivial base case \isa{trev\ {\isacharparenleft}trev\ {\isacharparenleft}Var\ x{\isacharparenright}{\isacharparenright}\ {\isacharequal}\ Var\ x} and the step case
26 \begin{isabelle}%
27 \ \ \ \ \ {\isasymforall}t{\isachardot}\ t\ {\isasymin}\ set\ ts\ {\isasymlongrightarrow}\ trev\ {\isacharparenleft}trev\ t{\isacharparenright}\ {\isacharequal}\ t\ {\isasymLongrightarrow}\isanewline
28 \ \ \ \ \ trev\ {\isacharparenleft}trev\ {\isacharparenleft}App\ f\ ts{\isacharparenright}{\isacharparenright}\ {\isacharequal}\ App\ f\ ts%
29 \end{isabelle}
30 both of which are solved by simplification:%
31 \end{isamarkuptxt}%
33 \begin{isamarkuptext}%
34 \noindent
35 If the proof of the induction step mystifies you, we recommend to go through
36 the chain of simplification steps in detail; you will probably need the help of
37 \isa{trace{\isacharunderscore}simp}.
38 %\begin{quote}
39 %{term[display]"trev(trev(App f ts))"}\\
40 %{term[display]"App f (rev(map trev (rev(map trev ts))))"}\\
41 %{term[display]"App f (map trev (rev(rev(map trev ts))))"}\\
42 %{term[display]"App f (map trev (map trev ts))"}\\
43 %{term[display]"App f (map (trev o trev) ts)"}\\
44 %{term[display]"App f (map (%x. x) ts)"}\\
45 %{term[display]"App f ts"}
46 %\end{quote}
48 The above definition of \isa{trev} is superior to the one in
49 \S\ref{sec:nested-datatype} because it brings \isa{rev} into play, about
50 which already know a lot, in particular \isa{rev\ {\isacharparenleft}rev\ xs{\isacharparenright}\ {\isacharequal}\ xs}.
51 Thus this proof is a good example of an important principle:
52 \begin{quote}
54 because they determine the complexity of your proofs.}
55 \end{quote}
57 Let us now return to the question of how \isacommand{recdef} can come up with
58 sensible termination conditions in the presence of higher-order functions
59 like \isa{map}. For a start, if nothing were known about \isa{map},
60 \isa{map\ trev\ ts} might apply \isa{trev} to arbitrary terms, and thus
61 \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
62 \isacommand{recdef} has been supplied with the congruence theorem
63 \isa{map{\isacharunderscore}cong}:
64 \begin{isabelle}%
65 \ \ \ \ \ {\isasymlbrakk}xs\ {\isacharequal}\ ys{\isacharsemicolon}\ {\isasymAnd}x{\isachardot}\ x\ {\isasymin}\ set\ ys\ {\isasymLongrightarrow}\ f\ x\ {\isacharequal}\ g\ x{\isasymrbrakk}\isanewline
66 \ \ \ \ \ {\isasymLongrightarrow}\ map\ f\ xs\ {\isacharequal}\ map\ g\ ys%
67 \end{isabelle}
68 Its second premise expresses (indirectly) that the second argument of
69 \isa{map} is only applied to elements of its third argument. Congruence
70 rules for other higher-order functions on lists would look very similar but
71 have not been proved yet because they were never needed. If you get into a
72 situation where you need to supply \isacommand{recdef} with new congruence
73 rules, you can either append the line
74 \begin{ttbox}
75 congs <congruence rules>
76 \end{ttbox}
77 to the specific occurrence of \isacommand{recdef} or declare them globally:
78 \begin{ttbox}
79 lemmas [????????] = <congruence rules>
80 \end{ttbox}
82 Note that \isacommand{recdef} feeds on exactly the same \emph{kind} of
83 congruence rules as the simplifier (\S\ref{sec:simp-cong}) but that
84 declaring a congruence rule for the simplifier does not make it
85 available to \isacommand{recdef}, and vice versa. This is intentional.%
86 \end{isamarkuptext}%
87 \end{isabellebody}%
88 %%% Local Variables:
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90 %%% TeX-master: "root"
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