(*<*)

theory Slides5

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

begin

notation (latex output)

  set ("_") and

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

(*>*)

text_raw {*

  %%\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@ {}}

  \\

  \huge General Bindings and Alpha-Equivalence in Nominal Isabelle\\[-2mm] 

  \large Or, Nominal 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}<1-4>

  \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 \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::}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) $=$ []}}\\

  \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}<1-5>

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

  \mbox{}\\[-3mm]

  \begin{itemize}

  \item this way of specifying binding is inspired by 

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

  \only<2>{

  \begin{itemize}

  \item Ott allows specifications like\smallskip

  \begin{center}

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

  \end{center}

  \end{itemize}}

  \only<3-4>{

  \begin{itemize}

  \item whether something is bound can depend in Ott on other bound things\smallskip

  \begin{center}

  \begin{tikzpicture}

  \node (A) at (-0.5,1) {Foo $(\lambda y. \lambda x. t)$};

  \node (B) at ( 1.1,1) {$s$};

  \onslide<4>{\node (C) at (0.5,0) {$\{y, x\}$};}

  \onslide<4>{\draw[->,red,line width=1mm] (A) -- (C);}

  \onslide<4>{\draw[->,red,line width=1mm] (C) -- (B);}

  \end{tikzpicture}

  \end{center}

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

  for $\alpha$-equated terms}

  \end{itemize}}

  \only<5>{

  \begin{itemize}

  \item we allow multiple ``binders'' and ``bodies''\smallskip

  \begin{center}

  \begin{tabular}{l}

  \isacommand{bind} a b c \ldots \isacommand{in} x y z \ldots\\

  \isacommand{bind (set)} a b c \ldots \isacommand{in} x y z \ldots\\

  \isacommand{bind (set+)} a b c \ldots \isacommand{in} x y z \ldots

  \end{tabular}

  \end{center}\bigskip\medskip

  the reason is that with our definition of $\alpha$-equivalence\medskip

  \begin{center}

  \begin{tabular}{l}

  \isacommand{bind (set+)} as \isacommand{in} x y $\not\Leftrightarrow$\\ 

  \hspace{8mm}\isacommand{bind (set+)} as \isacommand{in} x, \isacommand{bind (set+)} as \isacommand{in} y

  \end{tabular}

  \end{center}\medskip

  same with \isacommand{bind (set)}

  \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 the old Nominal Isabelle, we represented single binders as partial functions:\bigskip

  \begin{center}

  \begin{tabular}{l}

  Lam [$a$].\,$t$ $\;{^\text{``}}\!\dn{}\!^{\text{''}}$\\[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->

  \frametitle{\begin{tabular}{c}New Design\end{tabular}}

  \mbox{}\\[4mm]

  \begin{center}

  \begin{tikzpicture}

  {\draw (0,0) node[inner sep=3mm, ultra thick, draw=fg, rounded corners=2mm]

  (A) {\begin{minipage}{1.1cm}bind.\\spec.\end{minipage}};}

  {\draw (3,0) node[inner sep=3mm, ultra thick, draw=fg, rounded corners=2mm]

  (B) {\begin{minipage}{1.1cm}raw\\terms\end{minipage}};}

  \alt<2>

  {\draw (6,0) node[inner sep=3mm, ultra thick, draw=red, rounded corners=2mm]

  (C) {\textcolor{red}{\begin{minipage}{1.1cm}$\alpha$-\\equiv.\end{minipage}}};}

  {\draw (6,0) node[inner sep=3mm, ultra thick, draw=fg, rounded corners=2mm]

  (C) {\begin{minipage}{1.1cm}$\alpha$-\\equiv.\end{minipage}};}

  {\draw (0,-3) node[inner sep=3mm, ultra thick, draw=fg, rounded corners=2mm]

  (D) {\begin{minipage}{1.1cm}quot.\\type\end{minipage}};}

  {\draw (3,-3) node[inner sep=3mm, ultra thick, draw=fg, rounded corners=2mm]

  (E) {\begin{minipage}{1.1cm}lift\\thms\end{minipage}};}

  {\draw (6,-3) node[inner sep=3mm, ultra thick, draw=fg, rounded corners=2mm]

  (F) {\begin{minipage}{1.1cm}add.\\thms\end{minipage}};}

  \draw[->,fg!50,line width=1mm] (A) -- (B);

  \draw[->,fg!50,line width=1mm] (B) -- (C);

  \draw[->,fg!50,line width=1mm, line join=round, rounded corners=2mm] 

  (C) -- (8,0) -- (8,-1.5) -- (-2,-1.5) -- (-2,-3) -- (D);

  \draw[->,fg!50,line width=1mm] (D) -- (E);

  \draw[->,fg!50,line width=1mm] (E) -- (F);

  \end{tikzpicture}

  \end{center}

  \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

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

  $(as, x) \onslide<2->{\approx\!}\makebox[0mm][l]{\only<2-6>{${}_{\text{set}}$}%

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

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

           \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}}

   & $as$ is a set of names\ldots the binders\\

   & $x$ is the body (might be a tuple)\\

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

  of the binders has to be the same\\

  \end{tabular}

  \end{textblock}}

  \only<4->{

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

  \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->{$(\pi \act x)\;R\;y$}\\[1mm]

        & \onslide<6-7>{$\;\;\;\wedge$} & \onslide<6-7>{$\pi \act as = bs$}\\

  \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}

  $(as, x) \approx\!\makebox[0mm][l]{${}_{\text{set}}$}\only<1>{{}^{R,\text{fv}}}\only<2->{{}^{\alert{=},\alert{\text{fv}}}} (bs, y)$

  \end{center}\medskip

  \onslide<2->{

  \begin{center}

  \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}

  \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 $\approx_{\text{set+}}$, $\approx_{\text{set}}$% 

  \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:}\\