1 \documentclass[a4paper,draft]{article}
8 \newcommand{\MATITA}{\ding{46}\textsf{\textbf{Matita}}}
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}}
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}}
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}}
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}}
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}}
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]
66 T & ::= & & \mbox{(\textbf{tacticals})} \\
67 & & \APPLY{tac} & \mbox{(tactic application)} \\
68 & | & \SKIP & \mbox{(closed goal skipping)} \\
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)} \\
85 \subsection{Semantics}
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)} \\
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 \\
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]
124 \DEEPCLOSEFUN{G}{\Gamma}
126 & \MAPFUN{\mathit{fold}
127 (\lambda g.\lambda acc.
128 \mathit{if}~\GOALOFFUN(g)\in G~\land\ISOPENFUN(g)~
130 \mathit{else}~[g]\cup\mathit{acc})
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 & \\
145 \SEM{\TACTICAL{T}}{\xi}{[g_1;\dots;g_n]::\Gamma}{\tau}{\kappa}
149 \MAPFUN{\mathtt{Open}}{G^O_n}::\DEEPCLOSEFUN{G^C_n}{\Gamma},
150 \tau\setminus G^C_n,\kappa\setminus G^O_n\rangle
153 \multicolumn{3}{l}{\hspace{2em}\mathit{where} ~
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
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
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}}
172 \SEM{C_1 ~ \SEMICOLON ~ C_2}{\xi}{\Gamma}{\tau}{\kappa}
174 & \SEM{C_2}{\xi'}{\Gamma'}{\tau'}{\kappa'}
178 \langle\xi',\Gamma',\tau',\kappa',\rangle =
179 \SEM{C_1}{\xi}{\Gamma}{\tau}{\kappa}
182 \SEM{C~\DOT~}{\xi}{\Gamma}{\tau}{\kappa}
184 & \langle \xi'', \Gamma'', \tau'', \kappa'' \rangle
187 \multicolumn{3}{l}{\hspace{2em}\mathit{where} ~
188 \langle \xi',G::\Gamma',\tau',K::\kappa' \rangle
189 = \SEM{C}{\xi}{\Gamma}{\tau}{\kappa}}
192 \multicolumn{3}{l}{\hspace{2em}\mathit{and} ~
193 \langle \xi'', \Gamma'', \tau'', \kappa'' \rangle =
196 \langle \xi', [g]::\Gamma', \tau', (\MAPFUN{\GOALOFFUN}{G'}\cup K)::\kappa'
198 & \FILTERFUN{\ISOPENFUN}{G} = g::G'
200 \langle \xi', [\mathtt{Open}~n]::\Gamma', \tau', K'::\kappa' \rangle
201 & \FILTERFUN{\ISOPENFUN}{G} = []~\land~K=n::K'
206 \SEM{~\BRANCH~}{\xi}{[g_1;\dots;g_n]::\Gamma}{\tau}{\kappa}
209 & \langle\xi, [g_1]::[g_2;\dots;g_n]::\Gamma, []::\tau, []::\kappa\rangle
212 \SEM{~\SHIFT~}{\xi}{G::[g_i;\dots;g_n]::\Gamma}{T::\tau}{K::\kappa}
215 [g_i]::[g_{i+1};\dots;g_n]::\Gamma,
221 \tau' = T\cup\MAPFUN{\GOALOFFUN}{\FILTERFUN{\ISOPENFUN}{G}}\cup K
224 \SEM{~\MERGE~}{\xi}{G::[g_i;\dots;g_n]::\Gamma}{T::\tau}{K::\kappa}
226 & \langle \xi, \Gamma'::\Gamma, \tau, \kappa \rangle
231 T \cup\FILTERFUN{\ISOPENFUN}{G}
233 \cup\MAPFUN{\mathtt{Open}}{K}
236 \SEM{\SELECT{n_1,\dots,n_k}{C}}{\xi}{\Gamma}{\tau}{\kappa}
238 & \langle \xi',\Gamma',\tau',\kappa' \rangle
242 \forall ~ i=1,\dots,k,~\exists ~ \alpha_i,~\langle n_i,\alpha_i\rangle \in \xi
246 \langle \xi', []::\Gamma', []::\tau', []::\kappa' \rangle
247 = \SEM{C}{\xi}{\MAPFUN{\mathtt{Open}}{[n_1;\dots;n_k]}::\Gamma}{[]::\tau}{[]::\kappa}