\large General Bindings in Isabelle\\[15mm]

  \begin{frame}<1-6>

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

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

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

  \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}}

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

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

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

  atoms into generic ones\medskip 

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

  work with (sorts are occasionally explicit)\bigskip

  \mbox{}\\[-6mm]

  \item old Nominal provided a reasoning infrastructure for single binders\medskip

  \item<2-> but representing 

  \end{center}\medskip

  with single binders is a \alert{major} pain; take my word for it!

  \only<1>{

  \begin{textblock}{6}(1.5,11)

  \small

  for example\\

  \begin{tabular}{l@ {\hspace{2mm}}l}

  \pgfuseshading{smallspherered} & a $\fresh$ Lam [a]. t\\

  \pgfuseshading{smallspherered} & Lam [a]. (Var a) \alert{$=$} Lam [b]. (Var b)\\

  \end{tabular}

  \end{textblock}}

text_raw {*

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

  \mode<presentation>{

  \begin{frame}<1-4>

  \frametitle{\begin{tabular}{c}Binding Sets of Names\end{tabular}}

  \mbox{}\\[-3mm]

  \begin{itemize}

  \item binding sets of names has some interesting properties:\medskip

  \begin{center}

  \begin{tabular}{l}

  $\forall\{x, y\}.\, x \rightarrow y \;\;\approx_\alpha\;\; \forall\{y, x\}.\, y \rightarrow x$

  \bigskip\smallskip\\

  \onslide<2->{%

  $\forall\{x, y\}.\, x \rightarrow y \;\;\not\approx_\alpha\;\; \forall\{z\}.\, z \rightarrow z$

  }\bigskip\smallskip\\

  \onslide<3->{%

  $\forall\{x\}.\, x \rightarrow y \;\;\approx_\alpha\;\; \forall\{x, \alert{z}\}.\, x \rightarrow y$

  }\medskip\\

  \onslide<3->{\hspace{4cm}\small provided $z$ is fresh for the type}

  \end{tabular}

  \end{center}

  \end{itemize}

  \begin{textblock}{8}(2,14.5)

  \footnotesize $^*$ $x$, $y$, $z$ are assumed to be distinct

  \end{textblock}

  \only<4>{

  \begin{textblock}{6}(2.5,4)

  \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

  For type-schemes the order of bound names does not matter, and

  alpha-equivalence is preserved under \alert{vacuous} binders.

  \end{minipage}};

  \end{tikzpicture}

  \end{textblock}}

  \end{frame}}

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

*}

text_raw {*

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

  \mode<presentation>{

  \begin{frame}<1-3>

  \frametitle{\begin{tabular}{c}Other Binding Modes\end{tabular}}

  \mbox{}\\[-3mm]

  \begin{itemize}

  \item alpha-equivalence being preserved under vacuous binders is \underline{not} always

  wanted:\bigskip\bigskip\normalsize

  \begin{tabular}{@ {\hspace{-8mm}}l}

  $\text{let}\;x = 3\;\text{and}\;y = 2\;\text{in}\;x - y\;\text{end}$\medskip\\

  \onslide<2->{$\;\;\;\only<2>{\approx_\alpha}\only<3>{\alert{\not\approx_\alpha}}

   \text{let}\;y = 2\;\text{and}\;x = 3\only<3->{\alert{\;\text{and}

    \;z = \text{loop}}}\;\text{in}\;x - y\;\text{end}$}

  \end{tabular}

  \end{itemize}

  \end{frame}}

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

*}

text_raw {*

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

  \mode<presentation>{

  \begin{frame}<1>

  \frametitle{\begin{tabular}{c}\LARGE{}Even Another Binding Mode\end{tabular}}

  \mbox{}\\[-3mm]

  \begin{itemize}

  \item sometimes one wants to abstract more than one name, but the order \underline{does} matter\bigskip

  \begin{center}

  \begin{tabular}{@ {\hspace{-8mm}}l}

  $\text{let}\;(x, y) = (3, 2)\;\text{in}\;x - y\;\text{end}$\medskip\\

  $\;\;\;\not\approx_\alpha

   \text{let}\;(y, x) = (3, 2)\;\text{in}\;x - y\;\text{end}$

  \end{tabular}

  \end{center}

  \end{itemize}

  \end{frame}}

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

*}

text_raw {*

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

  \mode<presentation>{

  \begin{frame}<1-2>

  \frametitle{\begin{tabular}{c}\LARGE{}Three Binding Modes\end{tabular}}

  \mbox{}\\[-3mm]

  \begin{itemize}

  \item the order does not matter and alpha-equivelence is preserved under

  vacuous binders (restriction)\medskip

  \item the order does not matter, but the cardinality of the binders 

  must be the same (abstraction)\medskip

  \item the order does matter

  \end{itemize}

  \onslide<2->{

  \begin{center}

  \isacommand{bind\_res}\hspace{6mm}

  \isacommand{bind\_set}\hspace{6mm}

  \isacommand{bind}

  \end{center}}

  \end{frame}}

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

*}

text_raw {*

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

  \mode<presentation>{

  \begin{frame}<1-3>

  \frametitle{\begin{tabular}{c}Specification of Binding\end{tabular}}

  \mbox{}\\[-6mm]

  \mbox{}\hspace{10mm}

  \begin{tabular}{ll}

  \multicolumn{2}{l}{\isacommand{nominal\_datatype} trm $=$}\\

  \hspace{5mm}\phantom{$|$} Var name\\

  \hspace{5mm}$|$ App trm trm\\

  \hspace{5mm}$|$ Lam \only<2->{x::}name \only<2->{t::}trm

  & \onslide<2->{\isacommand{bind} x \isacommand{in} t}\\

  \hspace{5mm}$|$ Let \only<2->{as::}assn \only<2->{t::}trm

  & \onslide<2->{\isacommand{bind} bn(as) \isacommand{in} t}\\

  \multicolumn{2}{l}{\isacommand{and} assn $=$}\\

  \multicolumn{2}{l}{\hspace{5mm}\phantom{$|$} ANil}\\

  \multicolumn{2}{l}{\hspace{5mm}$|$ ACons name trm assn}\\

  \multicolumn{2}{l}{\onslide<3->{\isacommand{binder} bn \isacommand{where}}}\\

  \multicolumn{2}{l}{\onslide<3->{\hspace{5mm}\phantom{$|$} bn(ANil) $=$ $\varnothing$}}\\

  \multicolumn{2}{l}{\onslide<3->{\hspace{5mm}$|$ bn(ACons a t as) $=$ $\{$a$\}$ $\cup$ bn(as)}}\\

  \end{tabular}

  \end{frame}}

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

*}

text_raw {*

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

  \mode<presentation>{

  \begin{frame}<1-4>

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

  \mbox{}\\[-3mm]

  \begin{itemize}

  \item this way of specifying binding is pretty much stolen from 

  Ott\onslide<2->{, \alert{\bf but} with adjustments:}\medskip

  \begin{itemize}

  \item<2-> Ott allows specifications like\smallskip

  \begin{center}

  $t ::= t\;t\; |\;\lambda x.t$

  \end{center}\medskip

  \item<3-> whether something is bound can depend on other bound things\smallskip

  \begin{center}

  Foo $(\lambda x. t)\; s$ 

  \end{center}\medskip

  \onslide<4->{this might make sense for ``raw'' terms, but not at all 

  for $\alpha$-equated terms}

  \end{itemize}

  \end{itemize}

  \end{frame}}

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

*}

text_raw {*

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

  \mode<presentation>{

  \begin{frame}<1>

  \frametitle{\begin{tabular}{c}Alpha-Equivalence\end{tabular}}

  \mbox{}\\[-3mm]

  \begin{itemize}

  \item in old Nominal we represented single binders as partial functions:\bigskip

  \begin{center}

  \begin{tabular}{l}

  Lam [$a$].$t$ $\;\dn$\\[2mm]

  \;\;\;\;$\lambda b.$\;$\text{if}\;a = b\;\text{then}\;t\;\text{else}$\\

  \phantom{\;\;\;\;$\lambda b.$\;\;\;}$\text{if}\;b \fresh t\;

  \text{then}\;(a\;b)\act t\;\text{else}\;\text{error}$ 

  \end{tabular}

  \end{center}

  \end{itemize}

  \begin{textblock}{10}(2,14)

  \footnotesize $^*$ alpha-equality coincides with equality on functions

  \end{textblock}

  \end{frame}}

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

*}

text_raw {*

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

  \mode<presentation>{

  \begin{frame}<1-9>

  \frametitle{\begin{tabular}{c}Alpha-Equivalence\end{tabular}}

  \mbox{}\\[-3mm]

  \begin{itemize}

  \item lets first look at pairs\bigskip\medskip

  \begin{tabular}{@ {\hspace{1cm}}l}

  $(as, x) \onslide<2->{\approx\!}\makebox[0mm][l]{\only<2-7>{${}_{\text{set}}$}%

           \only<8>{${}_{\text{\alert{list}}}$}%

           \only<9>{${}_{\text{\alert{res}}}$}}%

           \onslide<3->{^{R,\text{fv}}}\,\onslide<2->{(bs,y)}$

  \end{tabular}\bigskip

  \end{itemize}

  \only<1>{

  \begin{textblock}{8}(3,8.5)

  \begin{tabular}{l@ {\hspace{2mm}}p{8cm}}

  \pgfuseshading{smallspherered} & $as$ is a set of atoms\ldots the binders\\

  \pgfuseshading{smallspherered} & $x$ is the body\\

  \pgfuseshading{smallspherered} & $\approx_{\text{set}}$ is where the cardinality 

  of the binders has to be the same\\