(*<*)

theory Slides2

imports "LaTeXsugar" "Nominal"

begin

notation (latex output)

  set ("_") and

  Cons  ("_::/_" [66,65] 65) 

(*>*)

text_raw {*

  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

  \mode<presentation>{

  \begin{frame}<1>[t]

  \frametitle{%

  \begin{tabular}{@ {\hspace{-3mm}}c@ {}}

  \\

  \end{tabular}}

  \begin{center}

  \end{center}

  \end{frame}}

  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%     

*}

text_raw {*

  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

  \mode<presentation>{

  \begin{frame}<1-2>

  \mbox{}\\[-3mm]

  \begin{itemize}

  \item sorted atoms and sort-respecting permutations\medskip

  \onslide<2->{

  \item[] in old Nominal: atoms have \underline{dif\!ferent} type\medskip

  \begin{center}

  \begin{tabular}{c@ {\hspace{7mm}}c}

  $[]\;\act\;c \dn c$ &

  $(a\;b)\!::\!\pi\;\act\;c \dn$ 

  $\begin{cases}

  b & \text{if}\; \pi \act c = a\\

  a & \text{if}\; \pi \act c = b\\

  \pi \act c & \text{otherwise}

  \end{cases}$

  \end{tabular}

  \end{center}}

  \end{itemize}

  \end{frame}}

  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%     

*}

text_raw {*

  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

  \mode<presentation>{

  \begin{frame}<1>

  \frametitle{\begin{tabular}{c}Problems\end{tabular}}

  \mbox{}\\[-3mm]

  \begin{itemize}

  \item @{text "_ \<bullet> _ :: \<alpha> perm \<Rightarrow> \<beta> \<Rightarrow> \<beta>"}\bigskip

  \item @{text "supp _ :: \<beta> \<Rightarrow> \<alpha> set"}

  \begin{center}

  $\text{finite} (\text{supp}\;x)_{\,\alpha_1\,\text{set}}$ \ldots 

  $\text{finite} (\text{supp}\;x)_{\,\alpha_n\,\text{set}}$

  \end{center}\bigskip

  \item $\forall \pi_{\alpha_1} \ldots \pi_{\alpha_n}\;.\; P$\bigskip

  \item type-classes

  \end{itemize}

  \end{frame}}

  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%     

*}

text_raw {*

  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

  \mode<presentation>{

  \begin{frame}<1-4>

  \frametitle{\begin{tabular}{c}Our New Solution\end{tabular}}

  \mbox{}\\[-3mm]

*}

datatype atom = Atom string nat

text_raw {*

  \mbox{}\bigskip

  \begin{itemize}

  \item<2-> permutations are (restricted) bijective functions from @{text "atom \<Rightarrow> atom"}

     \begin{itemize}

     \item sort-respecting \hspace{5mm}($\forall a.\;\text{sort}(f a) = \text{sort}(a)$)

     \item finite domain \hspace{5mm}($\text{finite} \{a.\;f a \not= a\}$)

     \end{itemize}\medskip

  \item<3-> swappings:

     \small

     \[

     \begin{array}{l@ {\hspace{1mm}}l}

     (a\;b) \dn & \text{if}\;\text{sort}(a) = \text{sort}(b)\\

        & \text{then}\;\lambda  c. \text{if}\;a = c\;\text{then}\;b\;\text{else}\;

          \text{if}\;b = c\;\text{then}\;a\;\text{else}\;c\\

        & \text{else}\;\only<3>{\mbox{\textcolor{red}{\bf ?}}}\only<4->{\text{id}}

     \end{array}

     \]

  \end{itemize}

  \end{frame}}

  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%     

*}

text_raw {*

  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

  \mode<presentation>{

  \begin{frame}<1-6>

  \frametitle{\begin{tabular}{c}\LARGE{}A Smoother Nominal Theory\end{tabular}}

  \mbox{}\\[-3mm]

  \begin{itemize}

  \item<1-> $(a\;b) = (b\;a) \onslide<3->{= (a\;c) + (b\;c) + (a\;c)}$\bigskip  

  \item<2-> permutations are an instance of group\_add\\ $0$, $\pi_1 + \pi_2$, $- \pi$\bigskip

  \item<5-> $\_\;\act\;\_ :: \text{perm} \Rightarrow \alpha \Rightarrow \alpha$\medskip

   \begin{itemize}

   \item $0\;\act\;x = x$\\

   \item $(\pi_1 + \pi_2)\;\act\;x = \pi_1\;\act\;(\pi_2\;\act\;x)$

   \end{itemize}

   \small

   \onslide<6->{$\text{finite}(\text{supp}\;x)$, $\forall \pi. P$}

  \end{itemize}

  \only<4>{

  \begin{textblock}{6}(2.5,11)

  \begin{tikzpicture}

  \draw (0,0) node[inner sep=3mm,fill=cream, ultra thick, draw=red, rounded corners=2mm] 

  {\normalsize\color{darkgray}

  \begin{minipage}{8cm}\raggedright

  This is slightly odd, since in general: 

  \begin{center}$\pi_1 + \pi_2 \alert{\not=} \pi_2 + \pi_1$\end{center}

  \end{minipage}};

  \end{tikzpicture}

  \end{textblock}}

  \end{frame}}

  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%     

*}

text_raw {*

  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

  \mode<presentation>{

  \begin{frame}<1-3>

  \frametitle{\begin{tabular}{c}Very Few Snatches\end{tabular}}

  \mbox{}\\[-3mm]

  \begin{itemize}

  \item \underline{concrete} atoms:

  \end{itemize}

*}

(*<*)

consts sort :: "atom \<Rightarrow> string"

(*>*)

typedef name = "{a :: atom. sort a = ''name''}" (*<*)sorry(*>*)

typedef ident = "{a :: atom. sort a = ''ident''}" (*<*)sorry(*>*)

text_raw {*

  \mbox{}\bigskip\bigskip

  \begin{itemize}

  \item<2-> there is an overloaded  function \underline{atom}, which injects concrete 

  atoms into generic ones\medskip 

  \begin{center}

  \begin{tabular}{l}

  $\text{atom}(a) \fresh x$\\

  $(a \leftrightarrow b) \dn (\text{atom}(a)\;\;\text{atom}(b))$

  \end{tabular}

  \end{center}\bigskip\medskip

  \onslide<3->

  {I would like to have $a \fresh x$, $(a\; b)$, \ldots}

  \end{itemize}

  \end{frame}}

  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%     

*}

text_raw {*

  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

  \mode<presentation>{

  \begin{frame}<1-2>[c]

  \frametitle{\begin{tabular}{c}\LARGE{}End of Part I\end{tabular}}

  \mbox{}\\[-3mm]

  \begin{itemize}

  \item the formalised version of the nominal theory is now much nicer to 

  work with (sorts are occasionally explicit)\bigskip

  \item permutations: ``be as abstract as you can'' (group\_add is a slight oddity)\bigskip

  \item allow sort-disrespecting swappings\onslide<2->{: just define them as the identity}

  \end{itemize}

  \end{frame}}

  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%     

*}

(*<*)

end

(*>*)