--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/Slides/Slides1.thy Thu Dec 06 15:11:21 2012 +0000
@@ -0,0 +1,669 @@
+(*<*)
+theory Slides1
+imports "../CpsG" "../ExtGG" "~~/src/HOL/Library/LaTeXsugar"
+begin
+
+notation (latex output)
+ set ("_") and
+ Cons ("_::/_" [66,65] 65)
+
+ML {*
+ open Printer;
+ show_question_marks_default := false;
+ *}
+
+notation (latex output)
+ Cons ("_::_" [78,77] 73) and
+ vt ("valid'_state") and
+ runing ("running") and
+ birthtime ("last'_set") and
+ If ("(\<^raw:\textrm{>if\<^raw:}> (_)/ \<^raw:\textrm{>then\<^raw:}> (_)/ \<^raw:\textrm{>else\<^raw:}> (_))" 10) and
+ Prc ("'(_, _')") and
+ holding ("holds") and
+ waiting ("waits") and
+ Th ("T") and
+ Cs ("C") and
+ readys ("ready") and
+ depend ("RAG") and
+ preced ("prec") and
+ cpreced ("cprec") and
+ dependents ("dependants") and
+ cp ("cprec") and
+ holdents ("resources") and
+ original_priority ("priority") and
+ DUMMY ("\<^raw:\mbox{$\_\!\_$}>")
+
+(*>*)
+
+
+
+text_raw {*
+ \renewcommand{\slidecaption}{Nanjing, P.R. China, 1 August 2012}
+ \newcommand{\bl}[1]{#1}
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+ \mode<presentation>{
+ \begin{frame}
+ \frametitle{%
+ \begin{tabular}{@ {}c@ {}}
+ \\[-3mm]
+ \Large Priority Inheritance Protocol \\[-3mm]
+ \Large Proved Correct \\[0mm]
+ \end{tabular}}
+
+ \begin{center}
+ \small Xingyuan Zhang \\
+ \small \mbox{PLA University of Science and Technology} \\
+ \small \mbox{Nanjing, China}
+ \end{center}
+
+ \begin{center}
+ \small joint work with \\
+ Christian Urban \\
+ Kings College, University of London, U.K.\\
+ Chunhan Wu \\
+ My Ph.D. student now working for Christian\\
+ \end{center}
+
+ \end{frame}}
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+*}
+
+text_raw {*
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+ \mode<presentation>{
+ \begin{frame}[c]
+ \frametitle{\large Prioirty Inheritance Protocol (PIP)}
+ \large
+
+ \begin{itemize}
+ \item Widely used in Real-Time OSs \pause
+ \item One solution of \textcolor{red}{`Priority Inversion'} \pause
+ \item A flawed manual correctness proof (1990)\pause
+ \begin{itemize} \large
+ \item {Notations with no precise definition}
+ \item {Resorts to intuitions}
+ \end{itemize} \pause
+ \item Formal treatments using model-checking \pause
+ \begin{itemize} \large
+ \item {Applicable to small size system models}
+ \item { Unhelpful for human understanding }
+ \end{itemize} \pause
+ \item Verification of PCP in PVS (2000)\pause
+ \begin{itemize} \large
+ \item {A related protocol}
+ \item {Priority Ceiling Protocol}
+ \end{itemize}
+ \end{itemize}
+
+ \end{frame}}
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+*}
+
+text_raw {*
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+ \mode<presentation>{
+ \begin{frame}[c]
+ \frametitle{Our Motivation}
+ \large
+
+ \begin{itemize}
+ \item Undergraduate OS course in our university \pause
+ \begin{itemize}
+ \item {\large Experiments using instrutional OSs }
+ \item {\large PINTOS (Stanford) is chosen }
+ \item {\large Core project: Implementing PIP in it}
+ \end{itemize} \pause
+ \item Understanding is crucial for the implemention \pause
+ \item Existing literature of little help \pause
+ \item Some mention the complication
+ \end{itemize}
+
+ \end{frame}}
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+*}
+
+
+text_raw {*
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+ \mode<presentation>{
+ \begin{frame}[c]
+ \frametitle{\mbox{Some excerpts}}
+
+ \begin{quote}
+ ``Priority inheritance is neither ef$\!$ficient nor reliable.
+ Implementations are either incomplete (and unreliable)
+ or surprisingly complex and intrusive.''
+ \end{quote}\medskip
+
+ \pause
+
+ \begin{quote}
+ ``I observed in the kernel code (to my disgust), the Linux
+ PIP implementation is a nightmare: extremely heavy weight,
+ involving maintenance of a full wait-for graph, and requiring
+ updates for a range of events, including priority changes and
+ interruptions of wait operations.''
+ \end{quote}
+
+ \end{frame}}
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+*}
+
+
+text_raw {*
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+ \mode<presentation>{
+ \begin{frame}[c]
+ \frametitle{Our Aims}
+ \large
+
+ \begin{itemize}
+ \item Formal specification at appropriate abstract level,
+ convenient for:
+ \begin{itemize} \large
+ \item Constructing interactive proofs
+ \item Clarifying the underlying ideas
+ \end{itemize} \pause
+ \item Theorems usable to guide implementation, critical point:
+ \begin{itemize} \large
+ \item Understanding the relationship with real OS code \pause
+ \item Not yet formalized
+ \end{itemize}
+ \end{itemize}
+
+ \end{frame}}
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+*}
+
+
+
+
+text_raw {*
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+ \mode<presentation>{
+ \begin{frame}[c]
+ \frametitle{Real-Time OSes}
+ \large
+
+ \begin{itemize}
+ \item Purpose: gurantee the most urgent task to be processed in time
+ \item Processes have priorities\\
+ \item Resources can be locked and unlocked
+ \end{itemize}
+
+ \end{frame}}
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+*}
+
+
+text_raw {*
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+ \mode<presentation>{
+ \begin{frame}[c]
+ \frametitle{Problem}
+ \Large
+
+ \begin{center}
+ \begin{tabular}{l}
+ \alert{H}igh-priority process\\[4mm]
+ \onslide<2->{\alert{M}edium-priority process}\\[4mm]
+ \alert{L}ow-priority process\\[4mm]
+ \end{tabular}
+ \end{center}
+
+ \onslide<3->{
+ \begin{itemize}
+ \item \alert{Priority Inversion} @{text "\<equiv>"} \alert{H $<$ L}
+ \item<4> avoid indefinite priority inversion
+ \end{itemize}}
+
+ \end{frame}}
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+*}
+
+
+text_raw {*
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+ \mode<presentation>{
+ \begin{frame}[c]
+ \frametitle{Priority Inversion}
+
+ \begin{center}
+ \includegraphics[scale=0.4]{PriorityInversion.png}
+ \end{center}
+
+ \end{frame}}
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+*}
+
+
+text_raw {*
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+ \mode<presentation>{
+ \begin{frame}[c]
+ \frametitle{Mars Pathfinder Mission 1997}
+ \Large
+
+ \begin{center}
+ \includegraphics[scale=0.2]{marspath1.png}
+ \includegraphics[scale=0.22]{marspath3.png}
+ \includegraphics[scale=0.4]{marsrover.png}
+ \end{center}
+
+ \end{frame}}
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+*}
+
+text_raw {*
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+ \mode<presentation>{
+ \begin{frame}[c]
+ \frametitle{Solution}
+ \Large
+
+ \alert{Priority Inheritance Protocol (PIP):}
+
+ \begin{center}
+ \begin{tabular}{l}
+ \alert{H}igh-priority process\\[4mm]
+ \textcolor{gray}{Medium-priority process}\\[4mm]
+ \alert{L}ow-priority process\\[21mm]
+ {\normalsize (temporarily raise its priority)}
+ \end{tabular}
+ \end{center}
+
+ \end{frame}}
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+*}
+
+
+
+
+text_raw {*
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+ \mode<presentation>{
+ \begin{frame}[c]
+ \frametitle{A Correctness ``Proof'' in 1990}
+ \Large
+
+ \begin{itemize}
+ \item a paper$^\star$
+ in 1990 ``proved'' the correctness of an algorithm for PIP\\[5mm]
+ \end{itemize}
+
+ \normalsize
+ \begin{quote}
+ \ldots{}after the thread (whose priority has been raised) completes its
+ critical section and releases the lock, it ``returns to its original
+ priority level''.
+ \end{quote}\bigskip
+
+ \small
+ $^\star$ in IEEE Transactions on Computers
+ \end{frame}}
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+*}
+
+text_raw {*
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+ \mode<presentation>{
+ \begin{frame}[c]
+ \frametitle{}
+ \Large
+
+ \begin{center}
+ \begin{tabular}{l}
+ \alert{H}igh-priority process 1\\[2mm]
+ \alert{H}igh-priority process 2\\[8mm]
+ \alert{L}ow-priority process
+ \end{tabular}
+ \end{center}
+
+ \onslide<2->{
+ \begin{itemize}
+ \item Solution: \\Return to highest \alert{remaining} priority
+ \end{itemize}}
+
+ \end{frame}}
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+*}
+
+
+text_raw {*
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+ \mode<presentation>{
+ \begin{frame}[c]
+ \frametitle{Event Abstraction}
+
+ \begin{itemize}\large
+ \item Use Inductive Approach of L. Paulson \pause
+ \item System is event-driven \pause
+ \item A \alert{state} is a list of events
+ \end{itemize}
+
+ \pause
+
+ \begin{center}
+ \includegraphics[scale=0.4]{EventAbstract.png}
+ \end{center}
+
+ \end{frame}}
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+*}
+
+text_raw {*
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+ \mode<presentation>{
+ \begin{frame}[c]
+ \frametitle{Events}
+ \Large
+
+ \begin{center}
+ \begin{tabular}{l}
+ Create \textcolor{gray}{thread priority}\\
+ Exit \textcolor{gray}{thread}\\
+ Set \textcolor{gray}{thread priority}\\
+ Lock \textcolor{gray}{thread cs}\\
+ Unlock \textcolor{gray}{thread cs}\\
+ \end{tabular}
+ \end{center}\medskip
+
+ \end{frame}}
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+*}
+
+text_raw {*
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+ \mode<presentation>{
+ \begin{frame}[c]
+ \frametitle{Precedences}
+ \large
+
+ \begin{center}
+ \begin{tabular}{l}
+ @{thm preced_def[where thread="th"]}
+ \end{tabular}
+ \end{center}
+
+
+
+ \end{frame}}
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+*}
+
+
+text_raw {*
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+ \mode<presentation>{
+ \begin{frame}[c]
+ \frametitle{RAGs}
+
+\begin{center}
+ \newcommand{\fnt}{\fontsize{7}{8}\selectfont}
+ \begin{tikzpicture}[scale=1]
+ %%\draw[step=2mm] (-3,2) grid (1,-1);
+
+ \node (A) at (0,0) [draw, rounded corners=1mm, rectangle, very thick] {@{text "th\<^isub>0"}};
+ \node (B) at (2,0) [draw, circle, very thick, inner sep=0.4mm] {@{text "cs\<^isub>1"}};
+ \node (C) at (4,0.7) [draw, rounded corners=1mm, rectangle, very thick] {@{text "th\<^isub>1"}};
+ \node (D) at (4,-0.7) [draw, rounded corners=1mm, rectangle, very thick] {@{text "th\<^isub>2"}};
+ \node (E) at (6,-0.7) [draw, circle, very thick, inner sep=0.4mm] {@{text "cs\<^isub>2"}};
+ \node (E1) at (6, 0.2) [draw, circle, very thick, inner sep=0.4mm] {@{text "cs\<^isub>3"}};
+ \node (F) at (8,-0.7) [draw, rounded corners=1mm, rectangle, very thick] {@{text "th\<^isub>3"}};
+
+ \draw [<-,line width=0.6mm] (A) to node [pos=0.54,sloped,above=-0.5mm] {\fnt{}holding} (B);
+ \draw [->,line width=0.6mm] (C) to node [pos=0.4,sloped,above=-0.5mm] {\fnt{}waiting} (B);
+ \draw [->,line width=0.6mm] (D) to node [pos=0.4,sloped,below=-0.5mm] {\fnt{}waiting} (B);
+ \draw [<-,line width=0.6mm] (D) to node [pos=0.54,sloped,below=-0.5mm] {\fnt{}holding} (E);
+ \draw [<-,line width=0.6mm] (D) to node [pos=0.54,sloped,above=-0.5mm] {\fnt{}holding} (E1);
+ \draw [->,line width=0.6mm] (F) to node [pos=0.45,sloped,below=-0.5mm] {\fnt{}waiting} (E);
+ \end{tikzpicture}
+ \end{center}\bigskip
+
+ \begin{center}
+ \begin{minipage}{0.8\linewidth}
+ \raggedleft
+ @{thm cs_depend_def}
+ \end{minipage}
+ \end{center}\pause
+
+
+
+ \end{frame}}
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+*}
+
+text_raw {*
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+ \mode<presentation>{
+ \begin{frame}[c]
+ \frametitle{Good Next Events}
+ %%\large
+
+ \begin{center}
+ @{thm[mode=Rule] thread_create[where thread=th]}\bigskip
+
+ @{thm[mode=Rule] thread_exit[where thread=th]}\bigskip
+
+ @{thm[mode=Rule] thread_set[where thread=th]}
+ \end{center}
+
+
+
+ \end{frame}}
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+*}
+
+text_raw {*
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+ \mode<presentation>{
+ \begin{frame}[c]
+ \frametitle{Good Next Events}
+ %%\large
+
+ \begin{center}
+ @{thm[mode=Rule] thread_P[where thread=th]}\bigskip
+
+ @{thm[mode=Rule] thread_V[where thread=th]}\bigskip
+ \end{center}
+
+
+
+ \end{frame}}
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+*}
+(*<*)
+context extend_highest_gen
+begin
+(*>*)
+text_raw {*
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+ \mode<presentation>{
+ \begin{frame}[c]
+ \frametitle{\mbox{\large Theorem: ``No indefinite priority inversion''}}
+
+ \pause
+
+ Theorem $^\star$: If th is the thread with the highest precedence in state
+ @{text "s"}: \pause
+ \begin{center}
+ \textcolor{red}{@{thm highest})}
+ \end{center}
+ \pause
+ and @{text "th"} is blocked by a thread @{text "th'"} in
+ a future state @{text "s'"} (with @{text "s' = t@s"}): \pause
+ \begin{center}
+ \textcolor{red}{@{text "th' \<in> running (t@s)"} and @{text "th' \<noteq> th"}} \pause
+ \end{center}
+ \fbox{ \hspace{1em} \pause
+ \begin{minipage}{0.95\textwidth}
+ \begin{itemize}
+ \item @{text "th'"} did not hold or wait for a resource in s:
+ \begin{center}
+ \textcolor{red}{@{text "\<not>detached s th'"}}
+ \end{center} \pause
+ \item @{text "th'"} is running with the precedence of @{text "th"}:
+ \begin{center}
+ \textcolor{red}{@{text "cp (t@s) th' = preced th s"}}
+ \end{center}
+ \end{itemize}
+ \end{minipage}}
+ \pause
+ \small
+ $^\star$ modulo some further assumptions\bigskip\pause
+ It does not matter which process gets a released lock.
+
+ \end{frame}}
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+*}
+
+text_raw {*
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+ \mode<presentation>{
+ \begin{frame}[t]
+ \frametitle{Implementation}
+
+ s $=$ current state; @{text "s'"} $=$ next state $=$ @{text "e#s"}\bigskip\bigskip
+
+ When @{text "e"} = \alert{Create th prio}, \alert{Exit th}
+
+ \begin{itemize}
+ \item @{text "RAG s' = RAG s"}
+ \item No precedence needs to be recomputed
+ \end{itemize}
+
+ \end{frame}}
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+*}
+
+text_raw {*
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+ \mode<presentation>{
+ \begin{frame}[t]
+ \frametitle{Implementation}
+
+ s $=$ current state; @{text "s'"} $=$ next state $=$ @{text "e#s"}\bigskip\bigskip
+
+
+ When @{text "e"} = \alert{Set th prio}
+
+ \begin{itemize}
+ \item @{text "RAG s' = RAG s"}
+ \item No precedence needs to be recomputed
+ \end{itemize}
+
+ \end{frame}}
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+*}
+
+text_raw {*
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+ \mode<presentation>{
+ \begin{frame}[t]
+ \frametitle{Implementation}
+
+ s $=$ current state; @{text "s'"} $=$ next state $=$ @{text "e#s"}\bigskip\bigskip
+
+ When @{text "e"} = \alert{Unlock th cs} where there is a thread to take over
+
+ \begin{itemize}
+ \item @{text "RAG s' = RAG s - {(C cs, T th), (T th', C cs)} \<union> {(C cs, T th')}"}
+ \item we have to recalculate the precedence of the direct descendants
+ \end{itemize}\bigskip
+
+ \pause
+
+ When @{text "e"} = \alert{Unlock th cs} where no thread takes over
+
+ \begin{itemize}
+ \item @{text "RAG s' = RAG s - {(C cs, T th)}"}
+ \item no recalculation of precedences
+ \end{itemize}
+
+
+ \end{frame}}
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+*}
+
+text_raw {*
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+ \mode<presentation>{
+ \begin{frame}[t]
+ \frametitle{Implementation}
+
+ s $=$ current state; @{text "s'"} $=$ next state $=$ @{text "e#s"}\bigskip\bigskip
+
+ When @{text "e"} = \alert{Lock th cs} where cs is not locked
+
+ \begin{itemize}
+ \item @{text "RAG s' = RAG s \<union> {(C cs, T th')}"}
+ \item no recalculation of precedences
+ \end{itemize}\bigskip
+
+ \pause
+
+ When @{text "e"} = \alert{Lock th cs} where cs is locked
+
+ \begin{itemize}
+ \item @{text "RAG s' = RAG s - {(T th, C cs)}"}
+ \item we have to recalculate the precedence of the descendants
+ \end{itemize}
+
+
+ \end{frame}}
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+*}
+
+
+text_raw {*
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+ \mode<presentation>{
+ \begin{frame}[c]
+ \frametitle{Conclusion}
+
+ \begin{itemize} \large
+ \item Aims fulfilled \medskip \pause
+ \item Alternative way \pause
+ \begin{itemize}
+ \item using Isabelle/HOL in OS code development \medskip
+ \item through the Inductive Approach
+ \end{itemize} \pause
+ \item Future research \pause
+ \begin{itemize}
+ \item scheduler in RT-Linux\medskip
+ \item multiprocessor case\medskip
+ \item other ``nails'' ? (networks, \ldots) \medskip \pause
+ \item Refinement to real code and relation between implementations
+ \end{itemize}
+ \end{itemize}
+ \end{frame}}
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+*}
+
+text_raw {*
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+ \mode<presentation>{
+ \begin{frame}[c]
+ \frametitle{Questions?}
+
+ \begin{itemize} \large
+ \item Thank you for listening!
+ \end{itemize}
+
+ \end{frame}}
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+*}
+
+
+(*<*)
+end
+end
+(*>*)
\ No newline at end of file