(*<*)

theory Slides2

imports "LaTeXsugar" "Nominal"

begin

notation (latex output)

  set ("_") and

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

(*>*)

text_raw {*

  \renewcommand{\slidecaption}{Edinburgh, 11.~July 2010}

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

  \mode<presentation>{

  \begin{frame}<1>[t]

  \frametitle{%

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

  \\

  \LARGE Proof Pearl:\\[-0mm]

  \LARGE A New Foundation for\\[-2mm] 

  \LARGE Nominal Isabelle\\[12mm]

  \end{tabular}}

  \begin{center}

  Brian Huf\!fman and {\bf Christian Urban}\\[0mm] 

  \end{center}

  \end{frame}}

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

*}

text_raw {*

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

  \mode<presentation>{

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

  \frametitle{Nominal Isabelle}

  \begin{itemize}

  \item \ldots is a definitional extension of Isabelle/HOL

  (let-polymorphism and type classes)\medskip

  \item \ldots provides a convenient reasoning infrastructure for

  terms involving binders (e.g.~lambda calculus, variable convention)\medskip

  \item<2-> \ldots mainly used to find errors in my own 

  (published) paper proofs and in those of others \alert{\bf ;o)}

  \end{itemize}

  \end{frame}}

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

*}

text_raw {*

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

  \mode<presentation>{

  \begin{frame}<1-3>[c]

  \frametitle{Nominal Theory}

  \ldots by Pitts; at its core are:\bigskip

  \begin{itemize}

  \item sorted atoms and 

  \item sort-respecting permutations

  \end{itemize}

  \onslide<2->{

  \begin{textblock}{8}(4,11)

  \onslide<3->{$\text{inv\_of\_}\pi \,\act\, ($}\onslide<2->{$\pi \,\act\, x$}%

  \onslide<3->{$) \,=\, x$}

  \end{textblock}}

  \end{frame}}

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

*}

text_raw {*

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

  \mode<presentation>{

  \begin{frame}<1-4>[t]

  \frametitle{The ``Old Way''}

  \begin{itemize}

  \item sorted atoms\\ \alert{$\quad\mapsto$ separate types}\; (``copies'' of nat)\bigskip 

  \item sort-respecting permutations\\ \alert{$\quad\mapsto$ lists of pairs of atoms (list swappings)}

  \onslide<2->{

  \begin{center} 

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

  $\text{[]} \,\act\, c = c$ &

  $\swap{a}{b}\!::\!\pi \,\act\, c =

  \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}

  \only<3>{

  \begin{textblock}{14}(1,12.5)

  \alert{The big benefit:} the type system takes care of the sort-respecting requirement.

  \end{textblock}}

  \only<4>{

  \begin{textblock}{14}(1,12.5)

  \alert{A small benefit:} permutation composition is \alert{list append} 

  and permutation inversion is \alert{list reversal}.

  \end{textblock}

  }

  \end{frame}}

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

*}

text_raw {*

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

  \mode<presentation>{

  \begin{frame}<1-4>

  \frametitle{Problems}

  \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 

  \onslide<3>{\small\hspace{5mm}can only have \alert{\bf one} type parameter}

  \begin{itemize}

  \item<2-> $\text{[]}\,\act\, x = x$

  \item<2-> $(\pi_1 @  \pi_2) \,\act \, x = \pi_1 \,\act\,(\pi_2 \,\act\, x)$

  \item<2-> if $\pi_1 \sim \pi_2$ then $\pi_1 \,\act\, x = \pi_2 \,\act\, x$

  \item<2-> if $\pi_1$, $\pi_2$ have dif$\!$f.~type, then

    $\pi_1 \act (\pi_2 \act x) = \pi_2 \act (\pi_1 \act x)$

  \end{itemize}

  \end{itemize}

  \only<4->{

  \begin{textblock}{9}(3,7)\begin{tikzpicture}

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

  {\normalsize\color{darkgray}

  \begin{quote}

  \begin{itemize}

  \item \alert{lots} of ML-code 

  \item \alert{not} pretty 

  \item \alert{not a {\bf proof pearl}} :o( 

  \end{itemize}

  \end{quote}};

  \end{tikzpicture}

  \end{textblock}}

  \end{frame}}

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

*}

text_raw {*

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

  \mode<presentation>{

  \begin{frame}<1-4>

  \frametitle{A Better Way}

*}

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}($\,\forall a.\;\text{sort}(\pi a) = \text{sort}(a)$)

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

     \end{itemize}\medskip

  \item<3-> \alert{What about {\bf 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>{\raisebox{-5mm}{\textcolor{red}{\huge\bf $?$}}}

          \only<4->{\text{\alert{\bf id}}}

     \end{array}

     \]

  \end{itemize}

  \only<2>{

  \begin{textblock}{7}(4,11)

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

  \end{textblock}}

  \end{frame}}

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

*}

text_raw {*

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

  \mode<presentation>{

  \begin{frame}<1-7>

  \frametitle{A Smoother Nominal Theory}

  From there it is essentially plain sailing:\bigskip

  \begin{itemize}

  \item<2-> $(a\;b) = (b\;a) \onslide<4->{= (a\;c) + (b\;c) + (a\;c)}$\bigskip  

  \item<3-> permutations are an instance of Isabelle's\\  

  group\_add ($0$, $\pi_1 + \pi_2$, $- \pi$)\bigskip

  \item<6-> $\_\;\act\;\_ :: \text{perm} \Rightarrow \alpha \Rightarrow \alpha$\medskip

   \begin{itemize}

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

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

   \end{itemize}\medskip

   \onslide<7->{\alert{$\;\mapsto\;$}only one type class needed, $\text{finite}(\text{supp}\;x)$,\\

   \hspace{9mm}$\forall \pi. P$}

  \end{itemize}

  \only<5>{

  \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>[c]

  \frametitle{One Snatch}

*}

datatype \<iota>atom = \<iota>Atom string nat

text_raw {*

  \isanewline\isanewline

  \begin{itemize}

  \item You like to get the advantages of the old way back: you 

  \alert{cannot mix} atoms of dif$\!$ferent sort:\bigskip

  \small

  e.g.~LF-objects:

  \begin{center}

  $\quad M ::= c \mid x \mid \lambda x\!:\!A. M \mid M_1\;M_2$

  \end{center}

  \end{itemize}

  \end{frame}}

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

*}

text_raw {*

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

  \mode<presentation>{

  \begin{frame}<1-2>

  \frametitle{Our Solution}

  \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 they are a ``subtype'' of the generic atom type

  \item there is an overloaded  function \alert{\bf 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}

  \pause

  One would like to have $a \fresh x$, $\swap{a}{b}$, \ldots

  \end{itemize}

  \end{frame}}

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

*}

text_raw {*

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

  \mode<presentation>{

  \begin{frame}<1-4>

  \frametitle{Sorts Reloaded}

*}

datatype atom\<iota> = Atom\<iota> string nat

text_raw {*

  \isanewline\isanewline

  \pause

  \alert{Problem}: HOL-binders or Church-style lambda-terms

  \begin{center}

  $\lambda x_\alpha.\, x_\alpha\;x_\beta$

  \end{center}

  \pause

  \isanewline\isanewline

\isacommand{datatype} ty = TVar string $\mid$ ty $\rightarrow$ ty\\

\isacommand{datatype} var = Var name ty\\

\pause

$(x \leftrightarrow y) \,\act\, (x_\alpha, x_\beta) = (y_\alpha, y_\beta)$

  \end{frame}}

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

*}

text_raw {*

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

  \mode<presentation>{

  \begin{frame}<1-2>

  \frametitle{Non-Working Solution}

Instead of\isanewline\isanewline

*}

    datatype atom\<iota>\<iota> = Atom\<iota>\<iota> string nat

text_raw {*

\isanewline\isanewline

have

\isanewline\isanewline

*}

    datatype 'a atom\<iota>\<iota>\<iota> = Atom\<iota>\<iota>\<iota> 'a nat

text_raw {*

  \pause

  \isanewline\isanewline

  But then

  \begin{center}

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

  \end{center}

  \end{frame}}

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

*}

(*<*)

hide_const sort

(*>*)

text_raw {*

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

  \mode<presentation>{

  \begin{frame}<1-4>

  \frametitle{A Working Solution}

*}

    datatype sort = Sort string "sort list"

    datatype atom\<iota>\<iota>\<iota>\<iota> = Atom\<iota>\<iota>\<iota>\<iota> sort nat

(*<*)

consts sort_ty :: "nat \<Rightarrow> sort"

(*>*)

text_raw {*

  \pause

  \isanewline

  {\small\begin{tabular}{rcl}

  @{text "sort_ty (TVar x)"}  & @{text "\<equiv>"} & @{text "Sort ''TVar'' [Sort x []]"} \\

  @{text "sort_ty (\<tau>\<^isub>1 \<rightarrow> \<tau>\<^isub>2)"} & @{text "\<equiv>"} & @{text "Sort ''Fun'' [sort_ty \<tau>\<^isub>1, sort_ty \<tau>\<^isub>2]"}\\

  \end{tabular}}

  \pause

  \isanewline\isanewline

  \isacommand{typedef} @{text "var = {a :: atom. sort a \<in> range sort_ty}"}

  \pause

  \isanewline\isanewline

  \small

  \begin{minipage}{12cm}

  @{text "Var x \<tau> \<equiv> \<lceil> Atom (sort_ty \<tau>) x \<rceil>"}\isanewline

  @{text "(Var x \<tau> \<leftrightarrow> Var y \<tau>) \<bullet> Var x \<tau> = Var y \<tau>"}\\