helm/ocaml/tactics/doc/main.tex

   1 \documentclass[a4paper,draft]{article}
   2
   3 \usepackage{a4wide}
   4 \usepackage{pifont}
   5 \usepackage{semantic}
   6 \usepackage{stmaryrd}
   7
   8 \newcommand{\MATITA}{\ding{46}\textsf{\textbf{Matita}}}
   9
  10 \title{Continuationals semantics for \MATITA}
  11 \author{Claudio Sacerdoti Coen \quad Enrico Tassi \quad Stefano Zacchiroli \\
  12 \small Department of Computer Science, University of Bologna \\
  13 \small Mura Anteo Zamboni, 7 -- 40127 Bologna, ITALY \\
  14 \small \{\texttt{sacerdot}, \texttt{tassi}, \texttt{zacchiro}\}\texttt{@cs.unibo.it}}
  15
  16 \newcommand{\DOT}{\mbox{\textbf{.}}}
  17 \newcommand{\SEMICOLON}{\mbox{\textbf{;}}}
  18 \newcommand{\BRANCH}{\mbox{\textbf{[}}}
  19 \newcommand{\SHIFT}{\mbox{\textbf{\textbar}}}
  20 \newcommand{\MERGE}{\mbox{\textbf{]}}}
  21 \newcommand{\APPLY}[1]{\ensuremath{\mathtt{apply}~\mathit{#1}}}
  22 \newcommand{\SKIP}{\ensuremath{\mathtt{skip}}}
  23 \newcommand{\TACTICAL}[1]{\ensuremath{\mathtt{tactical}~\mathit{#1}}}
  24 \newcommand{\SELECT}[2]{\ensuremath{\mathtt{select} ~ #1 ~ #2}}
  25
  26 \newcommand{\GOAL}{\ensuremath{\mathit{goal}}}
  27 \newcommand{\GOALSWITCH}{\ensuremath{\mathit{goal\_switch}}}
  28 \newcommand{\LIST}{\ensuremath{\mathtt{list}}}
  29 \newcommand{\BOOL}{\ensuremath{\mathtt{bool}}}
  30 \newcommand{\STACK}{\ensuremath{\mathtt{stack}}}
  31 \newcommand{\OPEN}[1]{\ensuremath{\mathtt{Open}~#1}}
  32 \newcommand{\CLOSED}[1]{\ensuremath{\mathtt{Closed}~#1}}
  33
  34 \newcommand{\SEMOP}[1]{|[#1|]}
  35 \newcommand{\TSEMOP}[1]{{}_t|[#1|]}
  36 \newcommand{\SEM}[5]{\SEMOP{#1}_{#2,#3,#4,#5}}
  37 \newcommand{\TSEM}[3]{\TSEMOP{#1}_{#2,#3}}
  38
  39 \newcommand{\FUN}[2]{\ensuremath{\mathit{#1}(#2)}}
  40 \newcommand{\FILTERFUN}[2]{\FUN{filter}{#1,#2}}
  41 \newcommand{\MAPFUN}[2]{\FUN{map}{#1,#2}}
  42 \newcommand{\DEEPMAPFUN}[2]{\FUN{deep\_map}{#1,#2}}
  43 \newcommand{\ISOPENFUN}{\ensuremath{\mathit{is\_open}}}
  44 \newcommand{\CLOSEFUN}{\ensuremath{\mathit{close}}}
  45 \newcommand{\GOALOFFUN}{\ensuremath{\mathit{goal\_of}}}
  46 \newcommand{\DEEPCLOSEFUN}[2]{\FUN{deep\_close}{#1,#2}}
  47
  48 \begin{document}
  49 \maketitle
  50
  51 \section{Foo}
  52
  53 \subsection{Language}
  54
  55 \[
  56 \begin{array}{rcll}
  57  C & ::= & & \mbox{(\textbf{continuationals})} \\
  58    &     & C ~ \DOT & \mbox{(dot)} \\
  59    &  |  & C ~ \SEMICOLON ~ C & \mbox{(semicolon)} \\
  60    &  |  & \BRANCH & \mbox{(branch)} \\
  61    &  |  & \SHIFT & \mbox{(shift)} \\
  62    &  |  & \MERGE & \mbox{(merge)} \\
  63    &  |  & \SELECT{n_1,\dots,n_k}{C} & \mbox{(select)} \\
  64    &  |  & \TACTICAL{T} & \mbox{(tactical)} \\[2ex]
  65
  66  T & ::= & & \mbox{(\textbf{tacticals})} \\
  67    &     & \APPLY{tac} & \mbox{(tactic application)} \\
  68    &  |  & \SKIP & \mbox{(closed goal skipping)} \\
  69 \end{array}
  70 \]
  71
  72 \subsection{Status}
  73
  74 \[
  75 \begin{array}{rcll}
  76  \mathit{status} & = & \xi\times\Gamma\times\tau\times\kappa & \\
  77  \xi & = & \langle n,\alpha\rangle~\LIST& \mbox{(metasenv)} \\
  78  \Gamma & = & \GOALSWITCH~\LIST~\STACK & \mbox{(context)} \\
  79  \GOALSWITCH & = & \OPEN{n} \quad | \quad \CLOSED{n} & \\
  80  \tau & = & \GOAL~\LIST~\STACK & \mbox{(todo)} \\
  81  \kappa & = & \GOAL~\LIST~\STACK & \mbox{(dot continuations)} \\
  82 \end{array}
  83 \]
  84
  85 \subsection{Semantics}
  86
  87 \[
  88 \begin{array}{rcll}
  89  \SEMOP{\cdot} & : & C -> \mathit{status} -> \mathit{status} &
  90   \mbox{(continuationals semantics)} \\
  91  \TSEMOP{\cdot} & : & T -> \xi -> \GOALSWITCH ->
  92   \xi\times\GOAL~\LIST\times\GOAL~\LIST & \mbox{(tacticals semantics)} \\
  93 \end{array}
  94 \]
  95
  96 \[
  97 \begin{array}{rcl}
  98  \mathit{filter} & : & (\alpha->\BOOL)->\alpha~\LIST->\alpha~\LIST \\
  99  \mathit{map} & : & (\alpha->\beta)->\alpha~\LIST->\beta~\LIST \\
 100  \mathit{deep\_map} & :
 101   & (\alpha->\beta)->\alpha~\LIST~\STACK->\beta~\LIST~\STACK \\
 102  \in & : & \alpha->\alpha~\LIST->\BOOL \\
 103  \cup & : & \alpha~\LIST->\alpha~\LIST->\alpha~\LIST \\
 104  \setminus & : & \alpha~\LIST->\alpha~\LIST->\alpha~\LIST \\
 105 \end{array}
 106 \]
 107
 108 \[
 109 \begin{array}{rcl}
 110  \mathit{apply\_tac} & : & \mathit{tac} -> \xi -> \GOAL ->
 111   \xi\times\GOAL~\LIST\times\GOAL~\LIST \\[1ex]
 112 %  \ISCLOSEDFUN & : & \GOALSWITCH -> \BOOL \\
 113  \ISOPENFUN & : & \GOALSWITCH -> \BOOL \\
 114  \CLOSEFUN & : & \GOALSWITCH -> \GOALSWITCH \\[1ex]
 115 %  \OPENFUN & : & \GOAL -> \GOALSWITCH \\[1ex]
 116  \GOALOFFUN & : & \GOALSWITCH -> \GOAL \\[1ex]
 117  \mathit{deep\_close} & :
 118  & \GOAL~\LIST -> \GOALSWITCH~\STACK -> \GOALSWITCH~\STACK \\[1ex]
 119 \end{array}
 120 \]
 121
 122 \[
 123 \begin{array}{rcl}
 124  \DEEPCLOSEFUN{G}{\Gamma}
 125  & =
 126  & \MAPFUN{\mathit{fold}
 127   (\lambda g.\lambda acc.
 128    \mathit{if}~\GOALOFFUN(g)\in G~\land\ISOPENFUN(g)~
 129    \mathit{then}~[]~
 130    \mathit{else}~[g]\cup\mathit{acc})
 131    []}{\Gamma}
 132 \end{array}
 133 \]
 134
 135 \[
 136 \begin{array}{rlcc}
 137  \TSEM{\APPLY{tac}}{\xi}{\OPEN{n}} & =
 138  & \mathit{apply\_tac}(\mathit{tac},\xi,n) & \\
 139  \TSEM{\SKIP}{\xi}{\CLOSED{n}} & = & \langle \xi, [], [n]\rangle & \\
 140 \end{array}
 141 \]
 142
 143 \[
 144 \begin{array}{rcl}
 145  \SEM{\TACTICAL{T}}{\xi}{[g_1;\dots;g_n]::\Gamma}{\tau}{\kappa}
 146  \quad (n\geq 1)
 147  & =
 148  & \langle\xi_n,
 149  \MAPFUN{\mathtt{Open}}{G^O_n}::\DEEPCLOSEFUN{G^C_n}{\Gamma},
 150    \tau\setminus G^C_n,\kappa\setminus G^O_n\rangle
 151  \\[1ex]
 152
 153  \multicolumn{3}{l}{\hspace{2em}\mathit{where} ~
 154  \left\{
 155  \begin{array}{rcll}
 156   \langle\xi_0, G^O_0, G^C_0\rangle & = & \langle\xi, [], []\rangle \\
 157   \langle\xi_{i+1}, G^O_{i+1}, G^C_{i+1}\rangle
 158   & =
 159   & \langle\xi_i, G^O_i, G^C_i\rangle
 160   & g_{i+1}\in G^C_i \\
 161   \langle\xi_{i+1}, G^O_{i+1}, G^C_{i+1}\rangle
 162   & =
 163   & \langle\xi, (G^O_i\setminus G^C)\cup G^O, G^C_i\cup G^C\rangle
 164   & g_{i+1}\not\in G^C_i \\
 165   & & \mathit{where} ~ \langle\xi,G^O,G^C\rangle=\TSEM{T}{\xi_i}{g_{i+1}}
 166   \\
 167  \end{array}
 168  \right.
 169  }
 170  \\[2ex]
 171
 172  \SEM{C_1 ~ \SEMICOLON ~ C_2}{\xi}{\Gamma}{\tau}{\kappa}
 173  & =
 174  & \SEM{C_2}{\xi'}{\Gamma'}{\tau'}{\kappa'}
 175  \\[1ex]
 176
 177  & & \mathit{where} ~
 178  \langle\xi',\Gamma',\tau',\kappa',\rangle =
 179  \SEM{C_1}{\xi}{\Gamma}{\tau}{\kappa}
 180  \\[2ex]
 181
 182  \SEM{C~\DOT~}{\xi}{\Gamma}{\tau}{\kappa}
 183  & =
 184  & \langle \xi'', \Gamma'', \tau'', \kappa'' \rangle
 185  \\[1ex]
 186
 187  \multicolumn{3}{l}{\hspace{2em}\mathit{where} ~
 188   \langle \xi',G::\Gamma',\tau',K::\kappa' \rangle
 189    = \SEM{C}{\xi}{\Gamma}{\tau}{\kappa}}
 190  \\[1ex]
 191
 192  \multicolumn{3}{l}{\hspace{2em}\mathit{and} ~
 193  \langle \xi'', \Gamma'', \tau'', \kappa'' \rangle =
 194  \left\{
 195  \begin{array}{ll}
 196   \langle \xi', [g]::\Gamma', \tau', (\MAPFUN{\GOALOFFUN}{G'}\cup K)::\kappa'
 197   \rangle
 198   & \FILTERFUN{\ISOPENFUN}{G} = g::G'
 199   \\
 200   \langle \xi', [\mathtt{Open}~n]::\Gamma', \tau', K'::\kappa' \rangle
 201   & \FILTERFUN{\ISOPENFUN}{G} = []~\land~K=n::K'
 202  \end{array}
 203  \right.}
 204  \\[2ex]
 205
 206  \SEM{~\BRANCH~}{\xi}{[g_1;\dots;g_n]::\Gamma}{\tau}{\kappa}
 207  \quad (n\geq 2)
 208  & =
 209  & \langle\xi, [g_1]::[g_2;\dots;g_n]::\Gamma, []::\tau, []::\kappa\rangle
 210  \\[2ex]
 211
 212  \SEM{~\SHIFT~}{\xi}{G::[g_i;\dots;g_n]::\Gamma}{T::\tau}{K::\kappa}
 213  & =
 214  & \langle\xi,
 215    [g_i]::[g_{i+1};\dots;g_n]::\Gamma,
 216    \tau'::\tau,
 217    []::\kappa\rangle
 218  \\[1ex]
 219
 220  & & \mathit{where} ~
 221    \tau' = T\cup\MAPFUN{\GOALOFFUN}{\FILTERFUN{\ISOPENFUN}{G}}\cup K
 222  \\[2ex]
 223
 224  \SEM{~\MERGE~}{\xi}{G::[g_i;\dots;g_n]::\Gamma}{T::\tau}{K::\kappa}
 225  & =
 226  & \langle \xi, \Gamma'::\Gamma, \tau, \kappa \rangle
 227  \\[1ex]
 228
 229  & & \mathit{where} ~
 230    \Gamma' =
 231     T \cup\FILTERFUN{\ISOPENFUN}{G}
 232      \cup[g_i;\dots;g_n]
 233      \cup\MAPFUN{\mathtt{Open}}{K}
 234  \\[2ex]
 235
 236  \SEM{\SELECT{n_1,\dots,n_k}{C}}{\xi}{\Gamma}{\tau}{\kappa}
 237  & =
 238  & \langle \xi',\Gamma',\tau',\kappa' \rangle
 239  \\[1ex]
 240
 241  & & \mathit{where} ~
 242  \forall ~ i=1,\dots,k,~\exists ~ \alpha_i,~\langle n_i,\alpha_i\rangle \in \xi
 243  \\[1ex]
 244
 245  & & \mathit{and} ~
 246  \langle \xi', []::\Gamma', []::\tau', []::\kappa' \rangle
 247  = \SEM{C}{\xi}{\MAPFUN{\mathtt{Open}}{[n_1;\dots;n_k]}::\Gamma}{[]::\tau}{[]::\kappa}
 248  \\[2ex]
 249
 250 \end{array}
 251 \]
 252
 253 \end{document}
 254