(*<*)

theory Slides5

imports "~~/src/HOL/Library/LaTeXsugar" "Nominal"

begin

notation (latex output)

  set ("_") and

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

(*>*)

text_raw {*

  %% shallow, deep, and recursive binders

  %%

  %%\renewcommand{\slidecaption}{Cambridge, 8.~June 2010}

  %%\renewcommand{\slidecaption}{Uppsala, 3.~March 2011}

  \renewcommand{\slidecaption}{Saarbrücken, 31.~March 2011}

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

  \mode<presentation>{

  \begin{frame}<1>[t]

  \frametitle{%

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

  \\

  \LARGE General Bindings and\\ 

  \LARGE Alpha-Equivalence\\ 

  \LARGE in Nominal Isabelle\\[3mm] 

  \Large Or, Nominal Isabelle 2\\[-5mm]

  \end{tabular}}

  \begin{center}

  Christian Urban

  \end{center}

  \begin{center}

  joint work with {\bf Cezary Kaliszyk}\\[0mm] 

  \end{center}

  \end{frame}}

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

*}

text_raw {*

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

  \mode<presentation>{

  \begin{frame}<1-2>

  \frametitle{\begin{tabular}{c}Binding in Old Nominal\end{tabular}}

  \mbox{}\\[-6mm]

  \begin{itemize}

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

  \begin{center}

  Lam [a].(Var a)

  \end{center}\bigskip

  \item<2-> but representing 

  \begin{center}

  $\forall\{a_1,\ldots,a_n\}.\; T$ 

  \end{center}\medskip

  with single binders and reasoning about it is a \alert{\bf major} pain; 

  take my word for it!

  \end{itemize}

  \only<1>{

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

  \small

  for example\\

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

   & a $\fresh$ Lam [a]. t\\

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

   & Barendregt-style reasoning about bound variables\\

  \end{tabular}

  \end{textblock}}

  \end{frame}}

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

*}

text_raw {*

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

  \mode<presentation>{

  \begin{frame}<1-6>

  \frametitle{New Types in HOL}

   \begin{center}

  \begin{tikzpicture}[scale=1.5]

  %%%\draw[step=2mm] (-4,-1) grid (4,1);

  \onslide<2-4,6>{\draw[very thick] (0.7,0.4) circle (4.25mm);}

  \onslide<1-4,6>{\draw[rounded corners=1mm, very thick] ( 0.0,-0.8) rectangle ( 1.8, 0.9);}

  \onslide<3-5,6>{\draw[rounded corners=1mm, very thick] (-1.95,0.85) rectangle (-2.85,-0.05);}

  \onslide<3-4,6>{\draw (-2.0, 0.845) --  (0.7,0.845);}

  \onslide<3-4,6>{\draw (-2.0,-0.045)  -- (0.7,-0.045);}

  \onslide<4-4,6>{\alert{\draw ( 0.7, 0.4) node {\footnotesize\begin{tabular}{@ {}c@ {}}$\alpha$-\\[-1mm]classes\end{tabular}};}}

  \onslide<4-5,6>{\alert{\draw (-2.4, 0.4) node {\footnotesize\begin{tabular}{@ {}c@ {}}$\alpha$-eq.\\[-1mm]terms\end{tabular}};}}

  \onslide<1-4,6>{\draw (1.8, 0.48) node[right=-0.1mm]

    {\footnotesize\begin{tabular}{@ {}l@ {}}existing\\[-1mm] type\\ \onslide<4-4,6>{\alert{(sets of raw terms)}}\end{tabular}};}

  \onslide<2-4,6>{\draw (0.9, -0.35) node {\footnotesize\begin{tabular}{@ {}l@ {}}non-empty\\[-1mm]subset\end{tabular}};}

  \onslide<3-5,6>{\draw (-3.25, 0.55) node {\footnotesize\begin{tabular}{@ {}l@ {}}new\\[-1mm]type\end{tabular}};}

  \onslide<3-4,6>{\draw[<->, very thick] (-1.8, 0.3) -- (-0.1,0.3);}

  \onslide<3-4,6>{\draw (-0.95, 0.3) node[above=0mm] {\footnotesize{}isomorphism};}

  \onslide<6>{\draw[->, line width=2mm, red] (-1.0,-0.4) -- (0.35,0.16);}

  \end{tikzpicture}

  \end{center}

  \begin{center}

  \textcolor{red}{\large\bf\onslide<6>{define $\alpha$-equivalence}}

  \end{center}

  \end{frame}}

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

*}

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}

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

  \bigskip\smallskip\\

  \onslide<2->{%

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

  }\bigskip\smallskip\\

  \onslide<3->{%

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

  }\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

  \textcolor{blue}{\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}

  \textcolor{blue}{\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 \textcolor{gray}{(restriction)}\medskip

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

  must be the same \textcolor{gray}{(abstraction)}\medskip

  \item the order does matter \textcolor{gray}{(iterated single binders)}

  \end{itemize}

  \onslide<2->{

  \begin{center}

  \isacommand{bind (set+)}\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::}assns \only<2->{t::}trm

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

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

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

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

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

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

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

  \end{tabular}

  \end{frame}}

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

*}

text_raw {*

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

  \mode<presentation>{

  \begin{frame}<1-8>

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

  \mbox{}\\[-3mm]

  \begin{itemize}

  \item lets first look at pairs\bigskip\medskip

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

  $(as, x) \onslide<2->{\approx\!}\makebox[5mm][l]{\only<2-6>{${}_{\text{set}}$}%

           \only<7>{${}_{\text{\alert{list}}}$}%

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

           \,\onslide<2->{(bs,y)}$

  \end{tabular}}\bigskip

  \end{itemize}

  \only<1>{

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

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

   & \textcolor{blue}{$as$} is a set of names\ldots the binders\\

   & \textcolor{blue}{$x$} is the body (might be a tuple)\\

   & \textcolor{blue}{$\approx_{\text{set}}$} is where the cardinality 

  of the binders has to be the same\\

  \end{tabular}

  \end{textblock}}

  \only<4->{

  \begin{textblock}{12}(5,8)

  \textcolor{blue}{

  \begin{tabular}{ll@ {\hspace{1mm}}l}

  $\dn$ & \onslide<5->{$\exists \pi.\,$} & $\text{fv}(x) - as = \text{fv}(y) - bs$\\[1mm]

        & \onslide<5->{$\;\;\;\wedge$} & \onslide<5->{$\text{fv}(x) - as \fresh^* \pi$}\\[1mm]

        & \onslide<5->{$\;\;\;\wedge$} & \onslide<5->{$(\pi \act x) = y$}\\[1mm]

        & \only<6-7>{$\;\;\;\wedge$}\only<8>{\textcolor{gray}{\xout{$\;\;\;\wedge$}}} & 

          \only<6-7>{$\pi \act as = bs$}\only<8>{\textcolor{gray}{\xout{$\pi \act as = bs$}}}\\

  \end{tabular}}

  \end{textblock}}

  \only<7>{

  \begin{textblock}{7}(3,13.8)

  \footnotesize $^*$ $as$ and $bs$ are \alert{lists} of names 

  \end{textblock}}

  \end{frame}}

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

*}

text_raw {*

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

  \mode<presentation>{

  \begin{frame}<1-3>

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

  \mbox{}\\[-3mm]

  \begin{itemize}

  \item lets look at type-schemes:\medskip\medskip

  \begin{center}

  \textcolor{blue}{$(as, x) \approx\!\makebox[5mm][l]{${}_{\text{set}}$} (bs, y)$}

  \end{center}\medskip

  \onslide<2->{

  \begin{center}

  \textcolor{blue}{

  \begin{tabular}{l}

  $\text{fv}(x) = \{x\}$\\[1mm]

  $\text{fv}(T_1 \rightarrow T_2) = \text{fv}(T_1) \cup \text{fv}(T_2)$\\

  \end{tabular}}

  \end{center}}

  \end{itemize}

  \only<3->{

  \begin{textblock}{4}(0.3,12)

  \begin{tikzpicture}

  \draw (0,0) node[inner sep=1mm,fill=cream, ultra thick, draw=red, rounded corners=2mm] 

  {\tiny\color{darkgray}

  \begin{minipage}{3.4cm}\raggedright

  \begin{tabular}{r@ {\hspace{1mm}}l}

  \multicolumn{2}{@ {}l}{set+:}\\

  $\exists\pi.$ & $\text{fv}(x) - as = \text{fv}(y) - bs$\\

  $\wedge$ & $\text{fv}(x) - as \fresh^* \pi$\\

  $\wedge$ & $\pi \cdot x = y$\\

  \\

  \end{tabular}

  \end{minipage}};

  \end{tikzpicture}

  \end{textblock}}

  \only<3->{

  \begin{textblock}{4}(5.2,12)

  \begin{tikzpicture}

  \draw (0,0) node[inner sep=1mm,fill=cream, ultra thick, draw=red, rounded corners=2mm] 

  {\tiny\color{darkgray}

  \begin{minipage}{3.4cm}\raggedright

  \begin{tabular}{r@ {\hspace{1mm}}l}

  \multicolumn{2}{@ {}l}{set:}\\

  $\exists\pi.$ & $\text{fv}(x) - as = \text{fv}(y) - bs$\\

  $\wedge$ & $\text{fv}(x) - as \fresh^* \pi$\\

  $\wedge$ & $\pi \cdot x = y$\\

  $\wedge$ & $\pi \cdot as = bs$\\

  \end{tabular}

  \end{minipage}};

  \end{tikzpicture}

  \end{textblock}}

  \only<3->{

  \begin{textblock}{4}(10.2,12)

  \begin{tikzpicture}

  \draw (0,0) node[inner sep=1mm,fill=cream, ultra thick, draw=red, rounded corners=2mm] 

  {\tiny\color{darkgray}

  \begin{minipage}{3.4cm}\raggedright

  \begin{tabular}{r@ {\hspace{1mm}}l}

  \multicolumn{2}{@ {}l}{list:}\\

  $\exists\pi.$ & $\text{fv}(x) - as = \text{fv}(y) - bs$\\

  $\wedge$ & $\text{fv}(x) - as \fresh^* \pi$\\

  $\wedge$ & $\pi \cdot x = y$\\

  $\wedge$ & $\pi \cdot as = bs$\\

  \end{tabular}

  \end{minipage}};

  \end{tikzpicture}

  \end{textblock}}

  \end{frame}}

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

*}

text_raw {*

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

  \mode<presentation>{

  \begin{frame}<1-2>

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

  \mbox{}\\[-3mm]

  \begin{center}

  \textcolor{blue}{

  \only<1>{$(\{x, y\}, x \rightarrow y) \approx_? (\{x, y\}, y \rightarrow x)$}

  \only<2>{$([x, y], x \rightarrow y) \approx_? ([x, y], y \rightarrow x)$}}

  \end{center}

  \begin{itemize}

  \item \textcolor{blue}{$\approx_{\text{set+}}$, $\approx_{\text{set}}$% 

  \only<2>{, \alert{$\not\approx_{\text{list}}$}}}

  \end{itemize}

  \only<1->{

  \begin{textblock}{4}(0.3,12)

  \begin{tikzpicture}

  \draw (0,0) node[inner sep=1mm,fill=cream, ultra thick, draw=red, rounded corners=2mm] 

  {\tiny\color{darkgray}

  \begin{minipage}{3.4cm}\raggedright

  \begin{tabular}{r@ {\hspace{1mm}}l}

  \multicolumn{2}{@ {}l}{set+:}\\

  $\exists\pi.$ & $\text{fv}(x) - as = \text{fv}(y) - bs$\\

  $\wedge$ & $\text{fv}(x) - as \fresh^* \pi$\\

  $\wedge$ & $\pi \cdot x = y$\\

  \\

  \end{tabular}

  \end{minipage}};

  \end{tikzpicture}

  \end{textblock}}

  \only<1->{

  \begin{textblock}{4}(5.2,12)

  \begin{tikzpicture}

  \draw (0,0) node[inner sep=1mm,fill=cream, ultra thick, draw=red, rounded corners=2mm] 

  {\tiny\color{darkgray}

  \begin{minipage}{3.4cm}\raggedright

  \begin{tabular}{r@ {\hspace{1mm}}l}

  \multicolumn{2}{@ {}l}{set:}\\

  $\exists\pi.$ & $\text{fv}(x) - as = \text{fv}(y) - bs$\\

  $\wedge$ & $\text{fv}(x) - as \fresh^* \pi$\\

  $\wedge$ & $\pi \cdot x = y$\\

  $\wedge$ & $\pi \cdot as = bs$\\

  \end{tabular}

  \end{minipage}};

  \end{tikzpicture}

  \end{textblock}}

  \only<1->{

  \begin{textblock}{4}(10.2,12)

  \begin{tikzpicture}

  \draw (0,0) node[inner sep=1mm,fill=cream, ultra thick, draw=red, rounded corners=2mm] 

  {\tiny\color{darkgray}

  \begin{minipage}{3.4cm}\raggedright

  \begin{tabular}{r@ {\hspace{1mm}}l}

  \multicolumn{2}{@ {}l}{list:}\\

  $\exists\pi.$ & $\text{fv}(x) - as = \text{fv}(y) - bs$\\

  $\wedge$ & $\text{fv}(x) - as \fresh^* \pi$\\

  $\wedge$ & $\pi \cdot x = y$\\

  $\wedge$ & $\pi \cdot as = bs$\\

  \end{tabular}

  \end{minipage}};

  \end{tikzpicture}

  \end{textblock}}

  \end{frame}}

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

*}

text_raw {*

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

  \mode<presentation>{

  \begin{frame}<1-2>

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

  \mbox{}\\[-3mm]

  \begin{center}

  \textcolor{blue}{\only<1>{$(\{x\}, x) \approx_? (\{x, y\}, x)$}}

  \end{center}

  \begin{itemize}

  \item \textcolor{blue}{$\approx_{\text{set+}}$, $\not\approx_{\text{set}}$,

        $\not\approx_{\text{list}}$}

  \end{itemize}

  \only<1->{

  \begin{textblock}{4}(0.3,12)

  \begin{tikzpicture}

  \draw (0,0) node[inner sep=1mm,fill=cream, ultra thick, draw=red, rounded corners=2mm] 

  {\tiny\color{darkgray}

  \begin{minipage}{3.4cm}\raggedright

  \begin{tabular}{r@ {\hspace{1mm}}l}

  \multicolumn{2}{@ {}l}{set+:}\\

  $\exists\pi.$ & $\text{fv}(x) - as = \text{fv}(y) - bs$\\

  $\wedge$ & $\text{fv}(x) - as \fresh^* \pi$\\

  $\wedge$ & $\pi \cdot x = y$\\

  \\

  \end{tabular}

  \end{minipage}};

  \end{tikzpicture}

  \end{textblock}}

  \only<1->{

  \begin{textblock}{4}(5.2,12)

  \begin{tikzpicture}

  \draw (0,0) node[inner sep=1mm,fill=cream, ultra thick, draw=red, rounded corners=2mm] 

  {\tiny\color{darkgray}

  \begin{minipage}{3.4cm}\raggedright

  \begin{tabular}{r@ {\hspace{1mm}}l}

  \multicolumn{2}{@ {}l}{set:}\\

  $\exists\pi.$ & $\text{fv}(x) - as = \text{fv}(y) - bs$\\

  $\wedge$ & $\text{fv}(x) - as \fresh^* \pi$\\

  $\wedge$ & $\pi \cdot x = y$\\