diff -r 3ed58bbcf4bd -r 332347b9b942 doc-src/TutorialI/Sets/sets.tex --- a/doc-src/TutorialI/Sets/sets.tex Mon Jul 16 13:14:19 2001 +0200 +++ b/doc-src/TutorialI/Sets/sets.tex Tue Jul 17 13:46:21 2001 +0200 @@ -324,7 +324,7 @@ \rulenamedx{UN_E} \end{isabelle} % -The following built-in syntax translation (see \S\ref{sec:def-translations}) +The following built-in syntax translation (see {\S}\ref{sec:def-translations}) lets us express the union over a \emph{type}: \begin{isabelle} \ \ \ \ \ @@ -368,7 +368,7 @@ Isabelle uses logical equivalences such as those above in automatic proof. Unions, intersections and so forth are not simply replaced by their definitions. Instead, membership tests are simplified. For example, $x\in -A\cup B$ is replaced by $x\in A\vee x\in B$. +A\cup B$ is replaced by $x\in A\lor x\in B$. The internal form of a comprehension involves the constant \cdx{Collect}, @@ -742,7 +742,7 @@ \subsection{The Reflexive and Transitive Closure} -\index{closure!reflexive and transitive|(}% +\index{reflexive and transitive closure|(}% The \textbf{reflexive and transitive closure} of the relation~\isa{r} is written with a postfix syntax. In ASCII we write \isa{r\isacharcircum*} and in @@ -871,7 +871,7 @@ \isa{auto}, \isa{fast} and \isa{best}. Section~\ref{sec:products} will discuss proof techniques for ordered pairs in more detail. \end{warn} -\index{relations|)}\index{closure!reflexive and transitive|)} +\index{relations|)}\index{reflexive and transitive closure|)} \section{Well-Founded Relations and Induction} @@ -902,7 +902,7 @@ complex recursive function definition or induction. The induction rule returned by \isacommand{recdef} is good enough for most purposes. We use an explicit -well-founded induction only in \S\ref{sec:CTL-revisited}. +well-founded induction only in {\S}\ref{sec:CTL-revisited}. \end{warn} Isabelle/HOL declares \cdx{less_than} as a relation object, @@ -960,11 +960,11 @@ \end{isabelle} These constructions can be used in a -\textbf{recdef} declaration (\S\ref{sec:recdef-simplification}) to define +\textbf{recdef} declaration ({\S}\ref{sec:recdef-simplification}) to define the well-founded relation used to prove termination. The \bfindex{multiset ordering}, useful for hard termination proofs, is -available in the Library. Baader and Nipkow \cite[\S2.5]{Baader-Nipkow} +available in the Library. Baader and Nipkow \cite[{\S}2.5]{Baader-Nipkow} discuss it. \medskip @@ -1014,7 +1014,7 @@ \begin{warn} Casual readers should skip the rest of this section. We use fixed point -operators only in \S\ref{sec:VMC}. +operators only in {\S}\ref{sec:VMC}. \end{warn} The theory applies only to monotonic functions.\index{monotone functions|bold} @@ -1065,7 +1065,8 @@ gfp\ f% \rulename{coinduct} \end{isabelle} -A \bfindex{bisimulation} is perhaps the best-known concept defined as a +A \textbf{bisimulation}\index{bisimulations} +is perhaps the best-known concept defined as a greatest fixed point. Exhibiting a bisimulation to prove the equality of two agents in a process algebra is an example of coinduction. The coinduction rule can be strengthened in various ways; see