1 \documentclass[a4paper,draft]{article}
8 \newcommand{\MATITA}{\ding{46}\textsf{\textbf{Matita}}}
10 \title{Continuationals semantics for \MATITA}
11 \author{Enrico Tassi \qquad Stefano Zacchiroli \\
12 \small Department of Computer Science, University of Bologna \\
13 \small Mura Anteo Zamboni, 7 -- 40127 Bologna, ITALY \\
14 \small \{\texttt{tassi}, \texttt{zacchiro}\}\texttt{@cs.unibo.it}}
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{\GOTO}[2]{\ensuremath{\mathtt{goal} ~ #1 ~ #2}}
26 \newcommand{\GOAL}{\ensuremath{\mathit{goal}}}
27 \newcommand{\GOALSWITCH}{\ensuremath{\mathit{goal\_switch}}}
28 \newcommand{\LIST}{\ensuremath{\mathtt{list}}}
29 \newcommand{\STACK}{\ensuremath{\mathtt{stack}}}
30 \newcommand{\OPEN}[1]{\ensuremath{\mathtt{Open}~#1}}
31 \newcommand{\CLOSED}[1]{\ensuremath{\mathtt{Closed}~#1}}
33 \newcommand{\SEMOP}[1]{|[#1|]}
34 \newcommand{\TSEMOP}[1]{{}_t|[#1|]}
35 \newcommand{\SEM}[5]{\SEMOP{#1}_{#2,#3,#4,#5}}
36 \newcommand{\TSEM}[3]{\TSEMOP{#1}_{#2,#3}}
38 \newcommand{\FUN}[2]{\ensuremath{\mathit{#1}(#2)}}
39 \newcommand{\FILTEROPEN}[1]{\FUN{filter\_open}{#1}}
40 \newcommand{\MAPOPEN}[1]{\FUN{map\_open}{#1}}
41 \newcommand{\DEEPCLOSE}[1]{\FUN{deep\_close}{#1}}
52 C & ::= & & \mbox{(\textbf{continuationals})} \\
53 & & \DOT & \mbox{(dot)} \\
54 & | & C ~ \SEMICOLON ~ C & \mbox{(semicolon)} \\
55 & | & \BRANCH & \mbox{(branch)} \\
56 & | & \SHIFT & \mbox{(shift)} \\
57 & | & \MERGE & \mbox{(merge)} \\
58 & | & \GOTO{n}{C} & \mbox{(goto)} \\
59 & | & \TACTICAL{T} & \mbox{(tactical)} \\[2ex]
61 T & ::= & & \mbox{(\textbf{tacticals})} \\
62 & & \APPLY{tac} & \mbox{(tactic application)} \\
63 & | & \SKIP & \mbox{(closed goal skipping)} \\
71 \mathit{status} & = & \xi\times\Gamma\times\tau\times\kappa & \\
72 \xi & & & \mbox{(metasenv)} \\
73 \Gamma & = & \GOALSWITCH~\LIST~\STACK & \mbox{(context)} \\
74 \GOALSWITCH & = & \OPEN{n} \quad | \quad \CLOSED{n} & \\
75 \tau & = & \GOAL~\LIST~\STACK & \mbox{(todo)} \\
76 \kappa & = & \GOAL~\LIST~\STACK & \mbox{(dot continuations)} \\
80 \subsection{Semantics}
84 \SEMOP{\cdot} & : & C -> \mathit{status} -> \mathit{status} &
85 \mbox{(continuationals semantics)} \\
86 \TSEMOP{\cdot} & : & T -> \xi -> \GOALSWITCH ->
87 \GOAL~\LIST \times \GOAL~\LIST & \mbox{(tacticals semantics)} \\
93 \mathit{apply\_tac} & : & \mathit{tac} -> \xi -> \GOAL ->
94 \GOAL~\LIST\times\GOAL~\LIST \\
95 \mathit{map\_open} & : & \GOAL~\LIST -> \GOALSWITCH~\LIST \\
96 \mathit{filter\_open} & : & \GOALSWITCH~\LIST -> \GOAL~\LIST \\
97 \mathit{deep\_close} & :
98 & \GOAL~\LIST -> \GOALSWITCH~\LIST~\STACK -> \GOALSWITCH~\LIST~\STACK \\
99 \cup & : & \alpha~\LIST -> \alpha~\LIST -> \alpha~\LIST\\
100 \setminus & : & \alpha~\LIST -> \alpha~\LIST -> \alpha~\LIST\\
106 \TSEM{\APPLY{tac}}{\xi}{\OPEN{n}} & =
107 & \mathit{apply\_tac}(\mathit{tac},\xi,n) & \\
108 \TSEM{\SKIP}{\xi}{\CLOSED{n}} & = & \langle [], [n]\rangle & \\
114 \SEM{\TACTICAL{T}}{\xi}{[g_1;\dots;g_n]::\Gamma}{\tau}{\kappa}
116 & \langle\xi_n,\MAPOPEN{G^O_n}::\DEEPCLOSE{G^C_n,\Gamma},\tau,\kappa\rangle
119 \multicolumn{3}{l}{\hspace{2em}\mathit{where} ~
122 \langle\xi_0, G^O_0, G^C_0\rangle & = & \langle\xi, [], []\rangle \\
123 \langle\xi_{i+1}, G^O_{i+1}, G^C_{i+1}\rangle
125 & \langle\xi_i, G^O_i, G^C_i\rangle
126 & g_{i+1}\in G^C_i \\
127 \langle\xi_{i+1}, G^O_{i+1}, G^C_{i+1}\rangle
129 & \langle\xi, (G^O_i\setminus G^C)\cup G^O, G^C_i\cup G^C\rangle
130 & g_{i+1}\not\in G^C_i \\
131 & & \mathit{where} ~ \langle\xi,G^O,G^C\rangle=\TSEM{T}{\xi_i}{g_{i+1}}
138 \SEM{C_1 ~ \SEMICOLON ~ C_2}{\xi}{\Gamma}{\tau}{\kappa}
140 & \SEM{C_2}{\xi'}{\Gamma'}{\tau'}{\kappa'}
144 \langle\xi',\Gamma',\tau',\kappa',\rangle =
145 \SEM{C_1}{\xi}{\Gamma}{\tau}{\kappa}
148 \SEM{~\DOT~}{\xi}{(g::G)::\Gamma}{\tau}{K::\kappa}
150 & \langle\xi, [g]::\Gamma, \tau,
151 (\FILTEROPEN{G}\cup K)::\kappa\rangle
154 \SEM{~\DOT~}{\xi}{[]::\Gamma}{\tau}{(n::N)::\kappa}
156 & \langle\xi, [\OPEN{n}]::\Gamma, \tau, N::\kappa\rangle
159 \SEM{~\BRANCH~}{\xi}{[g_1;\dots;g_n]::\Gamma}{\tau}{\kappa}
161 & \langle\xi, [g_1]::[g_2;\dots;g_n]::\Gamma, []::\tau, []::\kappa\rangle
164 \SEM{~\SHIFT~}{\xi}{G::[g_i;\dots;g_n]::\Gamma}{T::\tau}{K::\kappa}
167 [g_i]::[g_{i+1};\dots;g_n]::\Gamma,
168 (T\cup\FILTEROPEN{G}\cup K)::\tau,
172 \SEM{~\MERGE~}{\xi}{G::[g_i;\dots;g_n]::\Gamma}{T::\tau}{K::\kappa}
175 (T\cup G\cup[g_i;\dots;g_n]\cup\MAPOPEN{K})::\Gamma,
180 \SEM{\GOTO{n}{C}}{\xi}{\Gamma}{\tau}{\kappa}
182 & \langle\xi',\Gamma',\tau\setminus[n],\kappa\setminus[n]\rangle
186 \langle \xi', []::\Gamma', []::\tau, []::\kappa\rangle
187 = \SEM{C}{\xi}{[\OPEN{n}]::\Gamma}{[]::\tau}{[]::\kappa}