--- a/doc-src/Exercises/0304/a1/generated/a1.tex Fri Apr 29 18:13:28 2005 +0200
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,108 +0,0 @@
-%
-\begin{isabellebody}%
-\def\isabellecontext{a{\isadigit{1}}}%
-\isamarkupfalse%
-%
-\isamarkupsubsection{List functions%
-}
-\isamarkuptrue%
-%
-\begin{isamarkuptext}%
-Define a function \isa{sum}, which computes the sum of
-elements of a list of natural numbers.%
-\end{isamarkuptext}%
-\isamarkuptrue%
-\ \ sum\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequote}nat\ list\ {\isasymRightarrow}\ nat{\isachardoublequote}\isamarkupfalse%
-%
-\begin{isamarkuptext}%
-Then, define a function \isa{flatten} which flattens a list
-of lists by appending the member lists.%
-\end{isamarkuptext}%
-\isamarkuptrue%
-\ \ flatten\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequote}{\isacharprime}a\ list\ list\ {\isasymRightarrow}\ {\isacharprime}a\ list{\isachardoublequote}\isamarkupfalse%
-%
-\begin{isamarkuptext}%
-Test your function by applying them to the following example lists:%
-\end{isamarkuptext}%
-\isamarkuptrue%
-\isacommand{lemma}\ {\isachardoublequote}sum\ {\isacharbrackleft}{\isadigit{2}}{\isacharcolon}{\isacharcolon}nat{\isacharcomma}\ {\isadigit{4}}{\isacharcomma}\ {\isadigit{8}}{\isacharbrackright}\ {\isacharequal}\ x{\isachardoublequote}\isamarkupfalse%
-\isanewline
-\isamarkupfalse%
-\isacommand{lemma}\ {\isachardoublequote}flatten\ {\isacharbrackleft}{\isacharbrackleft}{\isadigit{2}}{\isacharcolon}{\isacharcolon}nat{\isacharcomma}\ {\isadigit{3}}{\isacharbrackright}{\isacharcomma}\ {\isacharbrackleft}{\isadigit{4}}{\isacharcomma}\ {\isadigit{5}}{\isacharbrackright}{\isacharcomma}\ {\isacharbrackleft}{\isadigit{7}}{\isacharcomma}\ {\isadigit{9}}{\isacharbrackright}{\isacharbrackright}\ {\isacharequal}\ x{\isachardoublequote}\isamarkupfalse%
-\isamarkupfalse%
-%
-\begin{isamarkuptext}%
-Prove the following statements, or give a counterexample:%
-\end{isamarkuptext}%
-\isamarkuptrue%
-\isacommand{lemma}\ {\isachardoublequote}length\ {\isacharparenleft}flatten\ xs{\isacharparenright}\ {\isacharequal}\ sum\ {\isacharparenleft}map\ length\ xs{\isacharparenright}{\isachardoublequote}\isamarkupfalse%
-\isanewline
-\isamarkupfalse%
-\isacommand{lemma}\ sum{\isacharunderscore}append{\isacharcolon}\ {\isachardoublequote}sum\ {\isacharparenleft}xs\ {\isacharat}\ ys{\isacharparenright}\ {\isacharequal}\ sum\ xs\ {\isacharplus}\ sum\ ys{\isachardoublequote}\isamarkupfalse%
-\isanewline
-\isamarkupfalse%
-\isacommand{lemma}\ flatten{\isacharunderscore}append{\isacharcolon}\ {\isachardoublequote}flatten\ {\isacharparenleft}xs\ {\isacharat}\ ys{\isacharparenright}\ {\isacharequal}\ flatten\ xs\ {\isacharat}\ flatten\ ys{\isachardoublequote}\isamarkupfalse%
-\isanewline
-\isamarkupfalse%
-\isacommand{lemma}\ {\isachardoublequote}flatten\ {\isacharparenleft}map\ rev\ {\isacharparenleft}rev\ xs{\isacharparenright}{\isacharparenright}\ {\isacharequal}\ rev\ {\isacharparenleft}flatten\ xs{\isacharparenright}{\isachardoublequote}\isamarkupfalse%
-\isanewline
-\isamarkupfalse%
-\isacommand{lemma}\ {\isachardoublequote}flatten\ {\isacharparenleft}rev\ {\isacharparenleft}map\ rev\ xs{\isacharparenright}{\isacharparenright}\ {\isacharequal}\ rev\ {\isacharparenleft}flatten\ xs{\isacharparenright}{\isachardoublequote}\isamarkupfalse%
-\isanewline
-\isamarkupfalse%
-\isacommand{lemma}\ {\isachardoublequote}list{\isacharunderscore}all\ {\isacharparenleft}list{\isacharunderscore}all\ P{\isacharparenright}\ xs\ {\isacharequal}\ list{\isacharunderscore}all\ P\ {\isacharparenleft}flatten\ xs{\isacharparenright}{\isachardoublequote}\isamarkupfalse%
-\isanewline
-\isamarkupfalse%
-\isacommand{lemma}\ {\isachardoublequote}flatten\ {\isacharparenleft}rev\ xs{\isacharparenright}\ {\isacharequal}\ flatten\ xs{\isachardoublequote}\isamarkupfalse%
-\isanewline
-\isamarkupfalse%
-\isacommand{lemma}\ {\isachardoublequote}sum\ {\isacharparenleft}rev\ xs{\isacharparenright}\ {\isacharequal}\ sum\ xs{\isachardoublequote}\isamarkupfalse%
-\isamarkupfalse%
-%
-\begin{isamarkuptext}%
-Find a predicate \isa{P} which satisfies%
-\end{isamarkuptext}%
-\isamarkuptrue%
-\isacommand{lemma}\ {\isachardoublequote}list{\isacharunderscore}all\ P\ xs\ {\isasymlongrightarrow}\ length\ xs\ {\isasymle}\ sum\ xs{\isachardoublequote}\isamarkupfalse%
-\isamarkupfalse%
-%
-\begin{isamarkuptext}%
-Define, by means of primitive recursion, a function \isa{exists} which checks whether an element satisfying a given property is
-contained in the list:%
-\end{isamarkuptext}%
-\isamarkuptrue%
-\ \ list{\isacharunderscore}exists\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequote}{\isacharparenleft}{\isacharprime}a\ {\isasymRightarrow}\ bool{\isacharparenright}\ {\isasymRightarrow}\ {\isacharparenleft}{\isacharprime}a\ list\ {\isasymRightarrow}\ bool{\isacharparenright}{\isachardoublequote}\isamarkupfalse%
-%
-\begin{isamarkuptext}%
-Test your function on the following examples:%
-\end{isamarkuptext}%
-\isamarkuptrue%
-\isacommand{lemma}\ {\isachardoublequote}list{\isacharunderscore}exists\ {\isacharparenleft}{\isasymlambda}\ n{\isachardot}\ n\ {\isacharless}\ {\isadigit{3}}{\isacharparenright}\ {\isacharbrackleft}{\isadigit{4}}{\isacharcolon}{\isacharcolon}nat{\isacharcomma}\ {\isadigit{3}}{\isacharcomma}\ {\isadigit{7}}{\isacharbrackright}\ {\isacharequal}\ b{\isachardoublequote}\isamarkupfalse%
-\isanewline
-\isamarkupfalse%
-\isacommand{lemma}\ {\isachardoublequote}list{\isacharunderscore}exists\ {\isacharparenleft}{\isasymlambda}\ n{\isachardot}\ n\ {\isacharless}\ {\isadigit{4}}{\isacharparenright}\ {\isacharbrackleft}{\isadigit{4}}{\isacharcolon}{\isacharcolon}nat{\isacharcomma}\ {\isadigit{3}}{\isacharcomma}\ {\isadigit{7}}{\isacharbrackright}\ {\isacharequal}\ b{\isachardoublequote}\isamarkupfalse%
-\isamarkupfalse%
-%
-\begin{isamarkuptext}%
-Prove the following statements:%
-\end{isamarkuptext}%
-\isamarkuptrue%
-\isacommand{lemma}\ list{\isacharunderscore}exists{\isacharunderscore}append{\isacharcolon}\ \isanewline
-\ \ {\isachardoublequote}list{\isacharunderscore}exists\ P\ {\isacharparenleft}xs\ {\isacharat}\ ys{\isacharparenright}\ {\isacharequal}\ {\isacharparenleft}list{\isacharunderscore}exists\ P\ xs\ {\isasymor}\ list{\isacharunderscore}exists\ P\ ys{\isacharparenright}{\isachardoublequote}\isamarkupfalse%
-\isanewline
-\isamarkupfalse%
-\isacommand{lemma}\ {\isachardoublequote}list{\isacharunderscore}exists\ {\isacharparenleft}list{\isacharunderscore}exists\ P{\isacharparenright}\ xs\ {\isacharequal}\ list{\isacharunderscore}exists\ P\ {\isacharparenleft}flatten\ xs{\isacharparenright}{\isachardoublequote}\isamarkupfalse%
-\isamarkupfalse%
-%
-\begin{isamarkuptext}%
-You could have defined \isa{list{\isacharunderscore}exists} only with the aid of
-\isa{list{\isacharunderscore}all}. Do this now, i.e. define a function \isa{list{\isacharunderscore}exists{\isadigit{2}}} and show that it is equivalent to \isa{list{\isacharunderscore}exists}.%
-\end{isamarkuptext}%
-\isamarkuptrue%
-\isanewline
-\isamarkupfalse%
-\end{isabellebody}%
-%%% Local Variables:
-%%% mode: latex
-%%% TeX-master: "root"
-%%% End: