\contentsline {section}{\numberline {1}Introduction}{1}
\contentsline {section}{\numberline {2}Fixedpoint operators}{2}
\contentsline {section}{\numberline {3}Elements of an inductive or co-inductive definition}{2}
\contentsline {subsection}{\numberline {3.1}The form of the introduction rules}{2}
\contentsline {subsection}{\numberline {3.2}The fixedpoint definitions}{3}
\contentsline {subsection}{\numberline {3.3}Mutual recursion}{4}
\contentsline {subsection}{\numberline {3.4}Proving the introduction rules}{4}
\contentsline {subsection}{\numberline {3.5}The elimination rule}{4}
\contentsline {section}{\numberline {4}Induction and co-induction rules}{5}
\contentsline {subsection}{\numberline {4.1}The basic induction rule}{5}
\contentsline {subsection}{\numberline {4.2}Mutual induction}{5}
\contentsline {subsection}{\numberline {4.3}Co-induction}{6}
\contentsline {section}{\numberline {5}Examples of inductive and co-inductive definitions}{6}
\contentsline {subsection}{\numberline {5.1}The finite set operator}{7}
\contentsline {subsection}{\numberline {5.2}Lists of $n$ elements}{7}
\contentsline {subsection}{\numberline {5.3}A demonstration of {\ptt mk\unhbox \voidb@x \kern .06em \vbox {\hrule width.3em}cases}}{9}
\contentsline {subsection}{\numberline {5.4}A co-inductive definition: bisimulations on lazy lists}{9}
\contentsline {subsection}{\numberline {5.5}The accessible part of a relation}{10}
\contentsline {subsection}{\numberline {5.6}The primitive recursive functions}{11}
\contentsline {section}{\numberline {6}Datatypes and co-datatypes}{13}
\contentsline {subsection}{\numberline {6.1}Constructors and their domain}{13}
\contentsline {subsection}{\numberline {6.2}The case analysis operator}{14}
\contentsline {subsection}{\numberline {6.3}Example: lists and lazy lists}{15}
\contentsline {subsection}{\numberline {6.4}Example: mutual recursion}{16}
\contentsline {subsection}{\numberline {6.5}A four-constructor datatype}{18}
\contentsline {subsection}{\numberline {6.6}Proving freeness theorems}{19}
\contentsline {section}{\numberline {7}Conclusions and future work}{20}
\contentsline {section}{\numberline {A}Inductive and co-inductive definitions: users guide}{22}
\contentsline {subsection}{\numberline {A.1}The result structure}{22}
\contentsline {subsection}{\numberline {A.2}The argument structure}{23}
\contentsline {section}{\numberline {B}Datatype and co-datatype definitions: users guide}{24}
\contentsline {subsection}{\numberline {B.1}The result structure}{24}
\contentsline {subsection}{\numberline {B.2}The argument structure}{24}