--- /dev/null
+
+\section{Semantics}
+
+\subsection{Language}
+
+\[
+\begin{array}{rcll}
+ S & ::= & & \mbox{(\textbf{continuationals})}\\
+ & & \TACTIC{T} & \mbox{(tactic)}\\[2ex]
+ & | & \DOT & \mbox{(dot)} \\
+ & | & \SEMICOLON & \mbox{(semicolon)} \\
+ & | & \BRANCH & \mbox{(branch)} \\
+ & | & \SHIFT & \mbox{(shift)} \\
+ & | & \POS{i} & \mbox{(relative positioning)} \\
+ & | & \MERGE & \mbox{(merge)} \\[2ex]
+ & | & \FOCUS{g_1,\dots,g_n} & \mbox{(absolute positioning)} \\
+ & | & \UNFOCUS & \mbox{(unfocus)} \\[2ex]
+ & | & S ~ S & \mbox{(sequential composition)} \\[2ex]
+ T & : := & & \mbox{(\textbf{tactics})}\\
+ & & \SKIP & \mbox{(skip)} \\
+ & | & \mathtt{reflexivity} & \\
+ & | & \mathtt{apply}~t & \\
+ & | & \dots &
+\end{array}
+\]
+
+\subsection{Status}
+
+\[
+\begin{array}{rcll}
+ \xi & & & \mbox{(proof status)} \\
+ \mathit{goal} & & & \mbox{(proof goal)} \\[2ex]
+
+ \SWITCH & = & \OPEN~\mathit{goal} ~ | ~ \CLOSED~\mathit{goal} & \\
+ \mathit{locator} & = & \INT\times\SWITCH & \\
+ \mathit{tag} & = & \BRANCHTAG ~ | ~ \FOCUSTAG \\[2ex]
+
+ \Gamma & = & \mathit{locator}~\LIST & \mbox{(context)} \\
+ \tau & = & \mathit{locator}~\LIST & \mbox{(todo)} \\
+ \kappa & = & \mathit{locator}~\LIST & \mbox{(dot's future)} \\[2ex]
+
+ \mathit{stack} & = & (\Gamma\times\tau\times\kappa\times\mathit{tag})~\LIST
+ \\[2ex]
+
+ \mathit{status} & = & \xi\times\mathit{stack} \\
+\end{array}
+\]
+
+\paragraph{Utilities}
+\begin{itemize}
+ \item $\ZEROPOS([g_1;\cdots;g_n]) =
+ [\langle 0,\OPEN~g_1\rangle;\cdots;\langle 0,\OPEN~g_n\rangle]$
+ \item $\INITPOS([\langle i_1,s_1\rangle;\cdots;\langle i_n,s_n\rangle]) =
+ [\langle 1,s_1\rangle;\cdots;\langle n,s_n\rangle]$
+ \item $\ISFRESH(s) =
+ \left\{
+ \begin{array}{ll}
+ \mathit{true} & \mathrm{if} ~ s = \langle n, \OPEN~g\rangle\land n > 0 \\
+ \mathit{false} & \mathrm{otherwise} \\
+ \end{array}
+ \right.$
+ \item $\FILTEROPEN(\mathit{locs})=
+ \left\{
+ \begin{array}{ll}
+ [] & \mathrm{if}~\mathit{locs} = [] \\
+ \langle i,\OPEN~g\rangle :: \FILTEROPEN(\mathit{tl})
+ & \mathrm{if}~\mathit{locs} = \langle i,\OPEN~g\rangle :: \mathit{tl} \\
+ \FILTEROPEN(\mathit{tl})
+ & \mathrm{if}~\mathit{locs} = \mathit{hd} :: \mathit{tl} \\
+ \end{array}
+ \right.$
+ \item $\REMOVEGOALS(G,\mathit{locs}) =
+ \left\{
+ \begin{array}{ll}
+ [] & \mathrm{if}~\mathit{locs} = [] \\
+ \REMOVEGOALS(G,\mathit{tl})
+ & \mathrm{if}~\mathit{locs} = \langle i,\OPEN~g\rangle :: \mathit{tl}
+ \land g\in G\\
+ hd :: \REMOVEGOALS(G,\mathit{tl})
+ & \mathrm{if}~\mathit{locs} = \mathit{hd} :: \mathit{tl} \\
+ \end{array}
+ \right.$
+ \item $\DEEPCLOSE(G,S)$: (intuition) given a set of goals $G$ and a stack $S$
+ it returns a new stack $S'$ identical to the given one with the exceptions
+ that each locator whose goal is in $G$ is marked as closed in $\Gamma$ stack
+ components and removed from $\tau$ and $\kappa$ components.
+ \item $\GOALS(S)$: (inutition) return all goals appearing in whatever position
+ on a given stack $S$, appearing in an \OPEN{} switch.
+\end{itemize}
+
+\paragraph{Invariants}
+\begin{itemize}
+ \item $\forall~\mathrm{entry}~\ENTRY{\Gamma}{\tau}{\kappa}{t}, \forall s
+ \in\tau\cup\kappa, \exists g, s = \OPEN~g$ (each locator on the stack in
+ $\tau$ and $\kappa$ components has an \OPEN~switch).
+ \item Unless \FOCUS{} is used the stack contains no duplicate goals.
+ \item $\forall~\mathrm{locator}~l\in\Gamma \mbox{(with the exception of the
+ top-level $\Gamma$)}, \ISFRESH(l)$.
+\end{itemize}
+
+\subsection{Semantics}
+
+\[
+\begin{array}{rcll}
+ \SEMOP{\cdot} & : & C -> \mathit{status} -> \mathit{status} &
+ \mbox{(continuationals semantics)} \\
+ \TSEMOP{\cdot} & : & T -> \xi -> \SWITCH ->
+ \xi\times\GOAL~\LIST\times\GOAL~\LIST & \mbox{(tactics semantics)} \\
+\end{array}
+\]
+
+\[
+\begin{array}{rcl}
+ \mathit{apply\_tac} & : & T -> \xi -> \GOAL ->
+ \xi\times\GOAL~\LIST\times\GOAL~\LIST
+\end{array}
+\]
+
+\[
+\begin{array}{rlcc}
+ \TSEM{T}{\xi}{\OPEN~g} & = & \mathit{apply\_tac}(T,\xi,n) & T\neq\SKIP\\
+ \TSEM{\SKIP}{\xi}{\CLOSED~g} & = & \langle \xi, [], [g]\rangle &
+\end{array}
+\]
+
+\[
+\begin{array}{rcl}
+
+ \SEM{\TACTIC{T}}{\ENTRY{\GIN}{\tau}{\kappa}{t}::S}
+ & =
+ & \langle
+ \xi_n,
+ \ENTRY{\Gamma'}{\tau'}{\kappa'}{t}
+% \ENTRY{\ZEROPOS(G^o_n)}{\tau\setminus G^c_n}{\kappa\setminus G^o_n}{t}
+ :: \DEEPCLOSE(G^c_n,S)
+ \rangle
+ \\[1ex]
+ \multicolumn{3}{l}{\hspace{\sidecondlen}\mathit{where} ~ n\geq 1}
+ \\[1ex]
+ \multicolumn{3}{l}{\hspace{\sidecondlen}\mathit{and} ~
+ \Gamma' = \ZEROPOS(G^o_n)
+ \land \tau' = \REMOVEGOALS(G^c_n,\tau)
+ \land \kappa' = \REMOVEGOALS(G^o_n,\kappa)
+ }
+ \\[1ex]
+ \multicolumn{3}{l}{\hspace{\sidecondlen}\mathit{and} ~
+ \left\{
+ \begin{array}{rcll}
+ \langle\xi_0, G^o_0, G^c_0\rangle & = & \langle\xi, [], []\rangle \\
+ \langle\xi_{i+1}, G^o_{i+1}, G^c_{i+1}\rangle
+ & =
+ & \langle\xi_i, G^o_i, G^c_i\rangle
+ & l_{i+1}\in G^c_i \\
+ \langle\xi_{i+1}, G^o_{i+1}, G^c_{i+1}\rangle
+ & =
+ & \langle\xi, (G^o_i\setminus G^c)\cup G^o, G^c_i\cup G^c\rangle
+ & l_{i+1}\not\in G^c_i \\[1ex]
+ & & \mathit{where} ~ \langle\xi,G^o,G^c\rangle=\TSEM{T}{\xi_i}{l_{i+1}} \\
+ \end{array}
+ \right.
+ }
+ \\[6ex]
+
+ \SEM{~\DOT~}{\ENTRY{\Gamma}{\tau}{\kappa}{t}::S}
+ & =
+ & \langle \xi, \ENTRY{l_1}{\tau}{\GIN[2]\cup\kappa}{t}::S \rangle
+ \\[1ex]
+ & & \mathrm{where} ~ \FILTEROPEN(\Gamma)=\GIN \land n\geq 1
+ \\[2ex]
+
+ \SEM{~\DOT~}{\ENTRY{\Gamma}{\tau}{l::\kappa}{t}::S}
+ & =
+ & \langle \xi, \ENTRY{[l]}{\tau}{\kappa}{t}::S \rangle
+ \\[1ex]
+ & & \mathrm{where} ~ \FILTEROPEN(\Gamma)=[]
+ \\[2ex]
+
+ \SEM{~\SEMICOLON~}{S} & = & \langle \xi, S \rangle \\[1ex]
+
+ \SEM{~\BRANCH~}{\ENTRY{\GIN}{\tau}{\kappa}{t}::S}
+ \quad
+ & =
+ & \langle\xi, \ENTRY{[l_1']}{[]}{[]}{\BRANCHTAG}
+ ::\ENTRY{[l_2';\cdots;l_n']}{\tau}{\kappa}{t}::S
+ \\[1ex]
+ & & \mathrm{where} ~ n\geq 2 ~ \land ~ \INITPOS(\GIN)=[l_1';\cdots;l_n']
+ \\[2ex]
+
+ \SEM{~\SHIFT~}
+ {\ENTRY{\Gamma}{\tau}{\kappa}{\BRANCHTAG}::\ENTRY{\GIN}{\tau'}{\kappa'}{t'}
+ ::S}
+ & =
+ & \langle
+ \xi, \ENTRY{[l_1]}{\tau\cup\FILTEROPEN(\Gamma)}{[]}{\BRANCHTAG}
+ ::\ENTRY{\GIN[2]}{\tau'}{\kappa'}{t'}::S
+ \rangle
+ \\[1ex]
+ & & \mathrm{where} ~ n\geq 1
+ \\[2ex]
+
+ \SEM{~\POS{i}~}
+ {\ENTRY{[l]}{[]}{[]}{\BRANCHTAG}::\ENTRY{\Gamma'}{\tau'}{\kappa'}{t'}::S}
+ & =
+ & \langle \xi, \ENTRY{[l_i]}{[]}{[]}{\BRANCHTAG}
+ ::\ENTRY{l :: (\Gamma'\setminus [l_i])}{\tau'}{\kappa'}{t'}::S \rangle
+ \\[1ex]
+ & & \mathrm{where} ~ \langle i,l'\rangle = l_i\in \Gamma'~\land~\ISFRESH(l)
+ \\[2ex]
+
+ \SEM{~\POS{i}~}
+ {\ENTRY{\Gamma}{\tau}{\kappa}{\BRANCHTAG}
+ ::\ENTRY{\Gamma'}{\tau'}{\kappa'}{t'}::S}
+ & =
+ & \langle \xi, \ENTRY{[l_i]}{[]}{[]}{\BRANCHTAG}
+ ::\ENTRY{\Gamma'\setminus [l_i]}{\tau'\cup\FILTEROPEN(\Gamma)}{\kappa'}{t'}::S
+ \rangle
+ \\[1ex]
+ & & \mathrm{where} ~ \langle i, l'\rangle = l_i\in \Gamma'
+ \\[2ex]
+
+ \SEM{~\MERGE~}
+ {\ENTRY{\Gamma}{\tau}{\kappa}{\BRANCHTAG}::\ENTRY{\Gamma'}{\tau'}{\kappa'}{t'}
+ ::S}
+ & =
+ & \langle \xi,
+ \ENTRY{\tau\cup\FILTEROPEN(\Gamma)\cup\Gamma'\cup\kappa}{\tau'}{\kappa'}{t'}
+ :: S
+ \rangle
+ \\[2ex]
+
+ \SEM{\FOCUS{g_1,\dots,g_n}}{S}
+ & =
+ & \langle \xi, \ENTRY{\ZEROPOS([g_1;\cdots;g_n])}{[]}{[]}{\FOCUSTAG}
+ ::\DEEPCLOSE(S)
+ \rangle
+ \\[1ex]
+ & & \mathrm{where} ~
+ \forall i=1,\dots,n,~g_i\in\GOALS(S)
+ \\[2ex]
+
+ \SEM{\UNFOCUS}{\ENTRY{[]}{[]}{[]}{\FOCUSTAG}::S}
+ & =
+ & \langle \xi, S\rangle \\[2ex]
+
+\end{array}
+\]
+
projects / helm.git / blobdiff