# HG changeset patch
# User Christian Urban <christian dot urban at kcl dot ac dot uk>
# Date 1384253462 0
# Node ID 4e8482e505904c0d7ce36254a2f902a0dd62837b
# Parent  10526c967679f39d0196ae582b65fa2a01b53c69
more slides

diff -r 10526c967679 -r 4e8482e50590 programs/prove2.scala
--- a/programs/prove2.scala	Tue Nov 12 09:57:22 2013 +0000
+++ b/programs/prove2.scala	Tue Nov 12 10:51:02 2013 +0000
@@ -37,8 +37,8 @@
 
 val Gamma: Set[Form] = 
   Set( (Admin says Del) -> Del,
-       (Admin says ((Bob says Del) -> Del)),
-       (Bob says Del) )
+       Admin says ((Bob says Del) -> Del),
+       Bob says Del )
 
 val goal = Gamma |- Del // request: provable or not?
 
@@ -48,23 +48,23 @@
 
 def prove(j: Judgement, sc: () => Unit) : Unit = {
   if (j.lhs.contains(j.rhs))  sc ()   // Axiom rule 
-  else prove1(j.lhs, j.rhs, sc) 
+  else prove1(j, sc) 
 }
 
-def prove1(lhs: Set[Form], rhs: Form, sc: () => Unit) : Unit = 
-  rhs match {
+def prove1(j: Judgement, sc: () => Unit) : Unit = 
+  j.rhs match {
     case True => sc ()
     case False => ()
-    case Imp(f1, f2) => prove(lhs + f1 |- f2, sc) 
-    case Says(p, f1) => prove(lhs |- f1, sc) 
+    case Imp(f1, f2) => prove(j.lhs + f1 |- f2, sc) 
+    case Says(p, f1) => prove(j.lhs |- f1, sc) 
     case Or(f1, f2) => 
-      { prove(lhs |- f1, sc);
-        prove(lhs |- f2, sc) }
+      { prove(j.lhs |- f1, sc);
+        prove(j.lhs |- f2, sc) }
     case And(f1, f2) => 
-      prove(lhs |- f1, 
-            () => prove(lhs |- f2, sc))
-    case _ => { for ((f, lhs_rest) <- partitions(lhs))
-                  prove2(f, lhs_rest, rhs, sc) }
+      prove(j.lhs |- f1, 
+            () => prove(j.lhs |- f2, sc))
+    case _ => { for ((f, lhs_rest) <- partitions(j.lhs))
+                  prove2(f, lhs_rest, j.rhs, sc) }
   }
 
 def prove2(f: Form, lhs_rest: Set[Form], rhs: Form, sc: () => Unit) : Unit = 
diff -r 10526c967679 -r 4e8482e50590 slides/slides06.pdf
Binary file slides/slides06.pdf has changed
diff -r 10526c967679 -r 4e8482e50590 slides/slides06.tex
--- a/slides/slides06.tex	Tue Nov 12 09:57:22 2013 +0000
+++ b/slides/slides06.tex	Tue Nov 12 10:51:02 2013 +0000
@@ -16,6 +16,7 @@
 \usetikzlibrary{calc}
 \usetikzlibrary{shapes}
 \usepackage{graphicx} 
+\setmonofont[Scale=MatchLowercase]{Consolas}
 
 \isabellestyle{rm}
 \renewcommand{\isastyle}{\rm}%
@@ -106,6 +107,7 @@
         \stepcounter{row}
 }
 
+\newcommand{\dn}{\stackrel{\mbox{\scriptsize def}}{=}}% for definitions
 
 % beamer stuff 
 \renewcommand{\slidecaption}{APP 06, King's College London, 12 November 2013}
@@ -246,31 +248,6 @@
 \end{frame}}
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  
 
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-  \mode<presentation>{
-  \begin{frame}[c]
-  \frametitle{Sending Messages}
-
-  \begin{itemize}
-  \item Alice sends a message \bl{$m$}
-  \begin{center}
-  \bl{Alice says $m$}
-  \end{center}\medskip\pause
-
-  \item Alice sends an encrypted message \bl{$m$}\\ (with key \bl{$K$})
-  \begin{center}
-  \bl{Alice says $\{m\}_K$}
-  \end{center}\medskip\pause
-
-  \item Decryption of Alice's message\smallskip
-  \begin{center}
-  \bl{\mbox{\infer{\Gamma \vdash \text{Alice}\;\text{says}\;m}
-              {\Gamma \vdash \text{Alice}\;\text{says}\;\{m\}_K & \Gamma \vdash \text{Alice}\,\text{says}\,K}}}
-  \end{center}
-  \end{itemize}
-
-  \end{frame}}
-  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \mode<presentation>{
@@ -294,6 +271,33 @@
 \end{frame}}
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+  \mode<presentation>{
+  \begin{frame}[c]
+  \frametitle{Sending Messages}
+
+  \begin{itemize}
+  \item Alice sends a message \bl{$m$}
+  \begin{center}
+  \bl{Alice says $m$}
+  \end{center}\medskip\pause
+
+  \item Alice sends an encrypted message \bl{$m$} with key \bl{$K$} 
+  (\bl{$\{m\}_K \dn K \Rightarrow m$})
+  \begin{center}
+  \bl{Alice says $\{m\}_K$}
+  \end{center}\medskip\pause
+
+  \item Decryption of Alice's message\smallskip
+  \begin{center}
+  \bl{\mbox{\infer{\Gamma \vdash \text{Alice}\;\text{says}\;m}
+              {\Gamma \vdash \text{Alice}\;\text{says}\;\{m\}_K & \Gamma \vdash \text{Alice}\,\text{says}\,K}}}
+  \end{center}
+  \end{itemize}
+
+  \end{frame}}
+  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
+
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \mode<presentation>{
@@ -758,37 +762,11 @@
 
 \end{frame}}
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%        
-     
-     
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-  \mode<presentation>{
-  \begin{frame}[c]
-  \frametitle{}
-
-  Recall the following scenario:
-
-  \begin{itemize}
-  \item If \textcolor{blue}{Admin} says that \textcolor{blue}{\isa{file\isaliteral{5C3C5E697375623E}{} {}}} 
-  should be deleted, then this file must be deleted.
-  \item \textcolor{blue}{Admin} trusts \textcolor{blue}{Bob} to decide whether 
-  \textcolor{blue}{\isa{file\isaliteral{5C3C5E697375623E}{}}} should be deleted.
-  \item \textcolor{blue}{Bob} wants to delete \textcolor{blue}{\isa{file\isaliteral{5C3C5E697375623E}{}}}.
-  \end{itemize}\bigskip
-
-  \small
-  \textcolor{blue}{\isa{{\isaliteral{5C3C47616D6D613E}{\isasymGamma}}\ {\isaliteral{3D}{\isacharequal}}}\small\begin{tabular}{l}
-  \isa{{\isaliteral{28}{\isacharparenleft}}Admin\ says\ del{\isaliteral{5F}{\isacharunderscore}}file\isaliteral{5C3C5E697375623E}{}{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ del{\isaliteral{5F}{\isacharunderscore}}file\isaliteral{5C3C5E697375623E}{}},\\
-  \isa{{\isaliteral{28}{\isacharparenleft}}Admin\ says\ {\isaliteral{28}{\isacharparenleft}}{\isaliteral{28}{\isacharparenleft}}Bob\ says\ del{\isaliteral{5F}{\isacharunderscore}}file\isaliteral{5C3C5E697375623E}{}{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ del{\isaliteral{5F}{\isacharunderscore}}file\isaliteral{5C3C5E697375623E}{}{\isaliteral{29}{\isacharparenright}}{\isaliteral{29}{\isacharparenright}}},\\
-  \isa{Bob\ says\ del{\isaliteral{5F}{\isacharunderscore}}file\isaliteral{5C3C5E697375623E}{}}\\
-  \end{tabular}}\medskip
-
-  \textcolor{blue}{\isa{{\isaliteral{5C3C47616D6D613E}{\isasymGamma}}\ {\isaliteral{5C3C7475726E7374696C653E}{\isasymturnstile}}\ del{\isaliteral{5F}{\isacharunderscore}}file\isaliteral{5C3C5E697375623E}{}}}
-  \end{frame}}
-  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \mode<presentation>{
 \begin{frame}[c]
+\frametitle{Program}
 
 How to prove \bl{$\Gamma \vdash F$}?\bigskip\bigskip
 
@@ -848,9 +826,12 @@
 \end{frame}}
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  
 
+
+     
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \mode<presentation>{
 \begin{frame}[t]
+\frametitle{Program: \texttt{prove2}}
 
 I want to prove \bl{$\Gamma \vdash \text{Pred}$}\bigskip\bigskip\pause
 
@@ -861,7 +842,7 @@
 \bl{$\Gamma \vdash F_2$}
 \end{center}\bigskip\pause
 
-\item So better I try to prove \bl{$\Gamma \vdash \text{Pred}$} with the additional assumption
+\item So I am able to try to prove \bl{$\Gamma \vdash \text{Pred}$} with the additional assumption
 \bl{$F_2$}.\bigskip
 \begin{center}
 \bl{$F_2, \Gamma \vdash \text{Pred}$}
@@ -875,8 +856,34 @@
 
 
 \end{frame}}
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  
-\newcommand{\dn}{\stackrel{\mbox{\scriptsize def}}{=}}% for definitions
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%       
+     
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+  \mode<presentation>{
+  \begin{frame}[c]
+  \frametitle{}
+
+  Recall the following scenario:
+
+  \begin{itemize}
+  \item If \textcolor{blue}{Admin} says that \textcolor{blue}{\isa{file\isaliteral{5C3C5E697375623E}{} {}}} 
+  should be deleted, then this file must be deleted.
+  \item \textcolor{blue}{Admin} trusts \textcolor{blue}{Bob} to decide whether 
+  \textcolor{blue}{\isa{file\isaliteral{5C3C5E697375623E}{}}} should be deleted.
+  \item \textcolor{blue}{Bob} wants to delete \textcolor{blue}{\isa{file\isaliteral{5C3C5E697375623E}{}}}.
+  \end{itemize}\bigskip
+
+  \small
+  \textcolor{blue}{\isa{{\isaliteral{5C3C47616D6D613E}{\isasymGamma}}\ {\isaliteral{3D}{\isacharequal}}}\small\begin{tabular}{l}
+  \isa{{\isaliteral{28}{\isacharparenleft}}Admin\ says\ del{\isaliteral{5F}{\isacharunderscore}}file\isaliteral{5C3C5E697375623E}{}{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ del{\isaliteral{5F}{\isacharunderscore}}file\isaliteral{5C3C5E697375623E}{}},\\
+  \isa{{\isaliteral{28}{\isacharparenleft}}Admin\ says\ {\isaliteral{28}{\isacharparenleft}}{\isaliteral{28}{\isacharparenleft}}Bob\ says\ del{\isaliteral{5F}{\isacharunderscore}}file\isaliteral{5C3C5E697375623E}{}{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ del{\isaliteral{5F}{\isacharunderscore}}file\isaliteral{5C3C5E697375623E}{}{\isaliteral{29}{\isacharparenright}}{\isaliteral{29}{\isacharparenright}}},\\
+  \isa{Bob\ says\ del{\isaliteral{5F}{\isacharunderscore}}file\isaliteral{5C3C5E697375623E}{}}\\
+  \end{tabular}}\medskip
+
+  \textcolor{blue}{\isa{{\isaliteral{5C3C47616D6D613E}{\isasymGamma}}\ {\isaliteral{5C3C7475726E7374696C653E}{\isasymturnstile}}\ del{\isaliteral{5F}{\isacharunderscore}}file\isaliteral{5C3C5E697375623E}{}}}
+  \end{frame}}
+  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \mode<presentation>{
 \begin{frame}[c]
@@ -889,15 +896,7 @@
 \bl{\infer{\Gamma \vdash F}{\Gamma \vdash P\,\text{controls}\, F & \Gamma \vdash P\,\text{says}\,F}}
 \end{center}
 
-\item \bl{$P$} speaks for \bl{$Q$}\smallskip\\
-\bl{$P \mapsto Q \,\dn\, \forall F. (P\,\text{says}\, F) \Rightarrow (Q \,\text{says}\,F)$}\medskip
 
-\begin{center}
-\bl{\infer{\Gamma \vdash Q\,\text{says}\,F}{\Gamma \vdash P\mapsto Q & \Gamma \vdash P\,\text{says}\,F}}
-\medskip\\
-\bl{\infer{\Gamma \vdash P\,\text{controls}\,F}{\Gamma \vdash P\mapsto Q & \Gamma \vdash Q\,\text{controls}\,F}}\\
-
-\end{center}
 \end{itemize}
 
 \end{frame}}
@@ -905,25 +904,6 @@
 
 
 
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-\mode<presentation>{
-\begin{frame}[c]
-\frametitle{Protocol Specifications}
-
-The Needham-Schroeder Protocol:
-
-\begin{center}
-\begin{tabular}{@ {\hspace{-7mm}}l@{\hspace{2mm}}r@ {\hspace{1mm}}l}
-Message 1 & \bl{$A \rightarrow S :$} & \bl{$A, B, N_A$}\\
-Message 2 & \bl{$S \rightarrow A :$} & \bl{$\{N_A, B, K_{AB},\{K_{AB}, A\}_{K_{BS}} \}_{K_{AS}}$}\\
-Message 3 & \bl{$A \rightarrow B :$} & \bl{$\{K_{AB}, A\}_{K_{BS}} $}\\
-Message 4 & \bl{$B \rightarrow A :$} & \bl{$\{N_B\}_{K_{AB}}$}\\
-Message 5 & \bl{$A \rightarrow B :$} & \bl{$\{N_B-1\}_{K_{AB}}$}\\
-\end{tabular}
-\end{center}
-
-\end{frame}}
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%   
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \mode<presentation>{
@@ -945,31 +925,245 @@
 \end{frame}}
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%   
 
- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+   %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+  \mode<presentation>{
+  \begin{frame}[c]
+  \frametitle{Sending Rule}
+
+  \bl{\begin{center}
+  \mbox{\infer{\Gamma \vdash Q \;\text{says}\; F}
+              {\Gamma \vdash P \;\text{says}\; F & \Gamma \vdash P \;\text{sends}\; Q : F}}
+  \end{center}}\bigskip\pause
+  
+  \bl{$P \,\text{sends}\, Q : F \dn$}\\
+  \hspace{6mm}\bl{$(P \,\text{says}\, F) \Rightarrow (Q \,\text{says}\, F)$}
+
+  \end{frame}}
+  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%     
+  
+    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
   \mode<presentation>{
   \begin{frame}[c]
-  \frametitle{Sending Messages}
+  \frametitle{Trusted Third Party}
+
+  \begin{center}
+  \bl{\begin{tabular}{l}
+  $A$ sends $S$ : $\text{Connect}(A,B)$\\  
+  \bl{$S \,\text{says}\, (\text{Connect}(A,B) \Rightarrow$}\\ 
+  \hspace{2.5cm}\bl{$\{K_{AB}\}_{K_{AS}} \wedge 
+  \{\{K_{AB}\}_{K_{BS}}\}_{K_{AS}})$}\\
+ $S$ sends $A$ : $\{K_{AB}\}_{K_{AS}}$ \bl{$\wedge$} $\{\{K_{AB}\}_{K_{BS}}\}_{K_{AS}}$\\
+  $A$ sends $B$ : $\{K_{AB}\}_{K_{BS}}$\\
+  $A$ sends $B$ : $\{m\}_{K_{AB}}$
+  \end{tabular}}
+  \end{center}\bigskip\pause
+  
+  
+  \bl{$\Gamma \vdash B \,\text{says} \, m$}?
+  \end{frame}}
+  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%      
+
+   %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+  \mode<presentation>{
+  \begin{frame}[c]
+  \frametitle{Public/Private Keys}
 
   \begin{itemize}
-  \item Alice sends a message \bl{$m$}
-  \begin{center}
-  \bl{Alice says $m$}
-  \end{center}\medskip\pause
-
-  \item Alice sends an encrypted message \bl{$m$}\\ (with key \bl{$K$})
-  \begin{center}
-  \bl{Alice says $\{m\}_K$}
-  \end{center}\medskip\pause
-
-  \item Decryption of Alice's message\smallskip
+  \item Bob has a private and public key: \bl{$K_{Bob}^{pub}$}, \bl{$K_{Bob}^{priv}$}\bigskip
   \begin{center}
   \bl{\mbox{\infer{\Gamma \vdash \text{Alice}\;\text{says}\;m}
-              {\Gamma \vdash \text{Alice}\;\text{says}\;\{m\}_K & \Gamma \vdash \text{Alice}\,\text{says}\,K}}}
+              {\Gamma \vdash \text{Alice}\;\text{says}\;\{m\}_{K_{Bob}^{pub}} & 
+               \Gamma \vdash K_{Bob}^{priv}}}}
+  \end{center}\bigskip\pause
+
+  \item this is {\bf not} a derived rule! 
+  \end{itemize}
+
+  \end{frame}}
+  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%      
+  
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+  \mode<presentation>{
+  \begin{frame}[c]
+  \frametitle{Security Levels}
+  \small
+
+  \begin{itemize}
+  \item Top secret (\bl{$T\!S$})
+  \item Secret (\bl{$S$})
+  \item Public (\bl{$P$})
+  \end{itemize}
+
+  \begin{center}
+  \bl{$slev(P) < slev(S) < slev(T\!S)$}\pause
   \end{center}
+
+  \begin{itemize}
+  \item Bob has a clearance for ``secret''
+  \item Bob can read documents that are public or sectret, but not top secret
   \end{itemize}
 
   \end{frame}}
-  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
+  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+%
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+  \mode<presentation>{
+  \begin{frame}[c]
+  \frametitle{Reading a File}
+
+  \bl{\begin{center}
+  \begin{tabular}{c}
+  \begin{tabular}{@ {}l@ {}}
+  \only<2->{\textcolor{red}{$slev($File$)$ $<$ $slev($Bob$)$ $\Rightarrow$}}\\ 
+  \only<2->{\hspace{3cm}}Bob controls Permitted $($File, read$)$\\
+  Bob says Permitted $($File, read$)$\only<2->{\\}
+  \only<2>{\textcolor{red}{$slev($File$)$ $<$ $slev($Bob$)$}}%
+  \only<3>{\textcolor{red}{$slev($File$)$ $=$ $P$}\\}%
+  \only<3>{\textcolor{red}{$slev($Bob$)$ $=$ $S$}\\}%
+  \only<3>{\textcolor{red}{$slev(P)$ $<$ $slev(S)$}\\}%
+  \end{tabular}\\
+  \hline
+  Permitted $($File, read$)$
+  \end{tabular}
+  \end{center}}
+
+  \end{frame}}
+  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+%
+
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+  \mode<presentation>{
+  \begin{frame}[c]
+  \frametitle{Substitution Rule}
+  \small
+  
+  \bl{\begin{center}
+  \begin{tabular}{c}
+  $\Gamma \vdash slev(P) = l_1$ \hspace{4mm} $\Gamma \vdash slev(Q) = l_2$
+  \hspace{4mm} $\Gamma \vdash l_1 < l_2$\\\hline
+  $\Gamma \vdash slev(P) < slev(Q)$
+  \end{tabular}
+  \end{center}}\bigskip\pause
+
+  \begin{itemize}
+  \item \bl{$slev($Bob$)$ $=$ $S$}
+  \item \bl{$slev($File$)$ $=$ $P$}
+  \item \bl{$slev(P) < slev(S)$}
+  \end{itemize}
+
+  \end{frame}}
+  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+%
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+  \mode<presentation>{
+  \begin{frame}[c]
+  \frametitle{Reading a File}
+
+  \bl{\begin{center}
+  \begin{tabular}{c}
+  \begin{tabular}{@ {}l@ {}}
+  $slev($File$)$ $<$ $slev($Bob$)$ $\Rightarrow$\\ 
+  \hspace{3cm}Bob controls Permitted $($File, read$)$\\
+  Bob says Permitted $($File, read$)$\\
+  $slev($File$)$ $=$ $P$\\
+  $slev($Bob$)$ $=$ $T\!S$\\
+  \only<1>{\textcolor{red}{$?$}}%
+  \only<2>{\textcolor{red}{$slev(P) < slev(S)$}\\}%
+  \only<2>{\textcolor{red}{$slev(S) < slev(T\!S)$}}%
+  \end{tabular}\\
+  \hline
+  Permitted $($File, read$)$
+  \end{tabular}
+  \end{center}}
+
+  \end{frame}}
+  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+%
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+  \mode<presentation>{
+  \begin{frame}[c]
+  \frametitle{Transitivity Rule}
+  \small
+  
+  \bl{\begin{center}
+  \begin{tabular}{c}
+  $\Gamma \vdash l_1 < l_2$ 
+  \hspace{4mm} $\Gamma \vdash l_2 < l_3$\\\hline
+  $\Gamma \vdash l_1 < l_3$
+  \end{tabular}
+  \end{center}}\bigskip
+
+  \begin{itemize}
+  \item \bl{$slev(P) < slev (S)$}
+  \item \bl{$slev(S) < slev (T\!S)$}
+  \item[] \bl{$slev(P) < slev (T\!S)$}
+  \end{itemize}
+
+  \end{frame}}
+  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+%
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+  \mode<presentation>{
+  \begin{frame}[c]
+  \frametitle{Reading Files}
+
+  \begin{itemize}
+  \item Access policy for reading
+  \end{itemize}
+
+  \bl{\begin{center}
+  \begin{tabular}{c}
+  \begin{tabular}{@ {}l@ {}}
+  $\forall f.\;slev(f)$ \only<1>{$<$}\only<2>{\textcolor{red}{$\le$}} $slev($Bob$)$ $\Rightarrow$\\ 
+  \hspace{3cm}Bob controls Permitted $(f$, read$)$\\
+  Bob says Permitted $($File, read$)$\\
+  $slev($File$)$ $=$ \only<1>{$P$}\only<2>{\textcolor{red}{$T\!S$}}\\
+  $slev($Bob$)$ $=$ $T\!S$\\
+  $slev(P) < slev(S)$\\
+  $slev(S) < slev(T\!S)$
+  \end{tabular}\\
+  \hline
+  Permitted $($File, read$)$
+  \end{tabular}
+  \end{center}}
+
+  \end{frame}}
+  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+%
+
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+  \mode<presentation>{
+  \begin{frame}[c]
+  \frametitle{Writing Files}
+
+  \begin{itemize}
+  \item Access policy for \underline{writing}
+  \end{itemize}
+
+  \bl{\begin{center}
+  \begin{tabular}{c}
+  \begin{tabular}{@ {}l@ {}}
+  $\forall f.\;slev($Bob$)$ $\le$ $slev(f)$ $\Rightarrow$\\ 
+  \hspace{3cm}Bob controls Permitted $(f$, write$)$\\
+  Bob says Permitted $($File, write$)$\\
+  $slev($File$)$ $=$ $T\!S$\\
+  $slev($Bob$)$ $=$ $S$\\
+  $slev(P) < slev(S)$\\
+  $slev(S) < slev(T\!S)$
+  \end{tabular}\\
+  \hline
+  Permitted $($File, write$)$
+  \end{tabular}
+  \end{center}}
+
+  \end{frame}}
+  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+%
+
+  
+\end{document}
   
  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
   \mode<presentation>{