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maps.tex
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% !TEX root=main.tex
%% Language %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\newmacro{O-Value}{
\begin{function}
\signature{\Value : \mathrm{Tasks} \times \mathrm{States} \rightharpoonup \mathrm{Values}} \\
\Value(\Edit v, \sigma) &=& v \\
\Value(\Enter \tau, \sigma) &=& \bot \\
\Value(\Update l, \sigma) &=& \sigma(l) \\
\Value(\Fail, \sigma) &=& \bot \\
\Value(t_1 \Then e_2, \sigma) &=& \bot \\
\Value(t_1 \Next e_2, \sigma) &=& \bot \\
\Value(t_1 \And t_2, \sigma) &=& \left\{
\begin{array}{ll}
\tuple{v_1, v_2} & \ \when\ \Value(t_1, \sigma) = v_1 \land \Value(t_2, \sigma) = v_2 \\
\bot & \ \otherwise
\end{array}
\right. \\
\Value(t_1 \Or t_2, \sigma) &=& \left\{
\begin{array}{ll}
v_1 & \ \when\ \Value(t_1, \sigma) = v_1 \\
v_2 & \ \when\ \Value(t_1, \sigma) = \bot \land \Value(t_2, \sigma) = v_2 \\
\obox{\tuple{v_1, v_2}}{\bot} & \ \otherwise
\end{array}
\right. \\
\Value(t_1 \Xor t_2, \sigma) &=& \bot
% \Value(u \At t, \sigma) &=& \Value(t, \sigma)
\end{function}
}
\newmacro{O-Value-Compact}{
\begin{function}
\signature{\Value : \mathrm{Tasks} \times \mathrm{States} \rightharpoonup \mathrm{Values}} \\
\Value(\Edit v, \sigma) &=& v \\
\Value(\Enter \tau, \sigma) &=& \bot \\
\Value(\Update l, \sigma) &=& s(l) \\
\Value(\Fail, \sigma) &=& \bot \\
\Value(t_1 \Then e_2, \sigma) &=& \bot \\
\Value(t_1 \Next e_2, \sigma) &=& \bot \\
\Value(t_1 \And t_2, \sigma) &=& \left\{
\begin{array}{@{}l@{\ }l@{}}
\tuple{v_1, v_2} & \when\ \Value(t_1, \sigma) = v_1 \\
& \multicolumn{1}{r@{}}{\mathbf{and\ } \Value(t_2, \sigma) = v_2} \\
\bot & \otherwise
\end{array}
\right. \\
\Value(t_1 \Or t_2, \sigma) &=& \left\{
\begin{array}{@{}l@{\ }l@{}}
v_1 & \when\ \Value(t_1, \sigma) = v_1 \\
v_2 & \when\ \Value(t_1, \sigma) = \bot \\
& \multicolumn{1}{r@{}}{\mathbf{and\ } \Value(t_2, \sigma) = v_2} \\
\obox{\tuple{v_1, v_2}}{\bot} & \otherwise
\end{array}
\right. \\
\Value(t_1 \Xor t_2, \sigma) &=& \bot
% \Value(u \At t, \sigma) &=& \Value(t, \sigma)
\end{function}
}
\newmacro{O-Value-Compacter}{
\begin{function}
\signature{\Value : \mathrm{Tasks} \times \mathrm{States} \rightharpoonup \mathrm{Values}} \\
\Value(\Edit v, \sigma) &=& v \\
\Value(\Enter \tau, \sigma) &=& \bot \\
\Value(\Update l, \sigma) &=& s(l) \\
\Value(\Fail, \sigma) &=& \bot \\
\Value(t_1 \Then e_2, \sigma) &=& \bot \\
\Value(t_1 \Next e_2, \sigma) &=& \bot \\
\Value(t_1 \And t_2, \sigma) &=& \tuple{v_1, v_2} \\
\inset{\when\ \Value(t_1, \sigma) = v_1 \land \Value(t_2, \sigma) = v_2} \\
&=& \bot \quad \otherwise \\
\Value(t_1 \Or t_2, \sigma) &=& v_1 \quad \when\ \Value(t_1, \sigma) = v_1 \\
&=& v_2 \\
\inset{\when\ \Value(t_1, \sigma) = \bot \land \Value(t_2, \sigma) = v_2} \\
&=& \bot \quad \otherwise \\
\Value(t_1 \Xor t_2, \sigma) &=& \bot
% \Value(u \At t, \sigma) &=& \Value(t, \sigma)
\end{function}
}
\newmacro{O-Inputs}{
\begin{function}
\signature{\Inputs : \mathrm{Tasks} \times \mathrm{States} \to \powerset(\mathrm{Inputs})} \\
\Inputs(\Edit v, \sigma) &=& \set{v':\tau, \Empty} \quad\where\ \Edit v : \Task\tau \\
\Inputs(\Enter \tau, \sigma) &=& \set{v':\tau} \\
\Inputs(\Update l, \sigma) &=& \obox{\set{v':\tau, \Empty}}{\set{v':\tau}} \quad\where\ \Update l : \Task\tau \\
\Inputs(\Fail, \sigma) &=& \nothing \\
\Inputs(t_1 \Then e_2, \sigma) &=& \Inputs(t_1, \sigma) \\
\Inputs(t_1 \Next e_2, \sigma) &=& \Inputs(t_1, \sigma) \cup \set{\Continue \mid \Value(t_1, \sigma) = v_1 \land e_2\ v_1, \sigma \normalise t_2, \sigma' \land \lnot\Failing(t_2, \sigma')} \\
\Inputs(t_1 \And t_2, \sigma) &=& \set{\First\ i \mid i \in \Inputs(t_1, \sigma)} \cup \set{\Second\ i \mid i \in \Inputs(t_2, \sigma)} \\
\Inputs(t_1 \Or t_2, \sigma) &=& \set{\First\ i \mid i \in \Inputs(t_1, \sigma)} \cup \set{\Second\ i \mid i \in \Inputs(t_2, \sigma)} \\
\Inputs(e_1 \Xor e_2, \sigma) &=& \set{\Left \mid e_1, \sigma \normalise t_1, \sigma' \land \lnot\Failing(t_1, \sigma')} \cup \set{\Right \mid e_2, \sigma \normalise t_2, \sigma' \land \lnot\Failing(t_2, \sigma')}
%\Inputs(u_1 \At t, \sigma) &=& \set{u_2 \At i\mid u_2 \At i \in \Inputs(t, \sigma)} \cup \set{u_1 \At i\mid i\in \Inputs(t, \sigma)}
\end{function}
}
\newmacro{O-Inputs-Compact}{
\begin{function}
\signature{\Inputs : \mathrm{Tasks} \times \mathrm{States} \to \powerset(\mathrm{Inputs})} \\
\Inputs(\Edit v, \sigma) &=& \set{v'\mid\nothing \infers v':\tau}\cup \set{\Empty} \quad\where\ \Edit v : \Task\tau \\
\Inputs(\Enter \tau, \sigma) &=& \set{v'\mid\nothing \infers v':\tau} \\
\Inputs(\Update l, \sigma) &=& \obox{\set{v'\mid\nothing \infers v':\tau}\cup \set{\Empty}}{\set{v'\mid\nothing \infers v':\tau}} \quad\where\ \Update l : \Task\tau \\
\Inputs(\Fail, \sigma) &=& \nothing \\
\Inputs(t_1 \Then e_2, \sigma) &=& \Inputs(t_1, \sigma) \\
\Inputs(t_1 \Next e_2, \sigma) &=& \Inputs(t_1, \sigma) \cup \{\Continue \mid \Value(t_1, \sigma) = v_1 \land \phantom{} \\
& & \quad e_2\ v_1, \sigma \normalise t_2, \sigma' \land \lnot\Failing(t_2, \sigma')\} \\
\Inputs(t_1 \And t_2, \sigma) &=& \set{\First\ i \mid i \in \Inputs(t_1, \sigma)} \cup \set{\Second\ i \mid i \in \Inputs(t_2, \sigma)} \\
\Inputs(t_1 \Or t_2, \sigma) &=& \set{\First\ i \mid i \in \Inputs(t_1, \sigma)} \cup \set{\Second\ i \mid i \in \Inputs(t_2, \sigma)} \\
\Inputs(e_1 \Xor e_2, \sigma) &=& \set{\Left \mid e_1, \sigma \normalise t_1, \sigma' \land \lnot\Failing(t_1, \sigma')} \cup \phantom{} \\
& & \set{\Right \mid e_2, \sigma \normalise t_2, \sigma' \land \lnot\Failing(t_2, \sigma')}
%\Inputs(u_1 \At t, \sigma) &=& \set{u_2 \At i\mid u_2 \At i \in \Inputs(t, \sigma)} \cup \set{u_1 \At i\mid i\in \Inputs(t, \sigma)}
\end{function}
}
\newmacro{O-Inputs-Compacter}{
\begin{function}
\signature{\Inputs : \mathrm{Tasks} \times \mathrm{States} \to \powerset(\mathrm{Inputs})} \\
\Inputs(\Edit v, \sigma) &=& \set{v':\tau, \Empty} \quad\where\ \Edit v : \Task\tau \\
\Inputs(\Enter \tau, \sigma) &=& \set{v':\tau} \\
\Inputs(\Update l, \sigma) &=& \obox{\set{v':\tau, \Empty}}{\set{v':\tau}} \quad\where\ \Update l : \Task\tau \\
\Inputs(\Fail, \sigma) &=& \nothing \\
\Inputs(t_1 \Then e_2, \sigma) &=& \Inputs(t_1, \sigma) \\
\Inputs(t_1 \Next e_2, \sigma) &=& \Inputs(t_1, \sigma) \cup \{\Continue \mid \Value(t_1, \sigma) = v_1 \land \phantom{} \\
& & \quad e_2\ v_1, \sigma \normalise t_2, \sigma' \land \lnot\Failing(t_2, \sigma')\} \\
\Inputs(t_1 \And t_2, \sigma) &=& \set{\First\ i \mid i \in \Inputs(t_1, \sigma)} \\
&\cup& \set{\Second\ i \mid i \in \Inputs(t_2, \sigma)} \\
\Inputs(t_1 \Or t_2, \sigma) &=& \set{\First\ i \mid i \in \Inputs(t_1, \sigma)} \\
&\cup& \set{\Second\ i \mid i \in \Inputs(t_2, \sigma)} \\
\Inputs(e_1 \Xor e_2, \sigma) &=& \set{\Pick \obox{\Right}{\Left} \mid e_1, \sigma \normalise t_1, \sigma' \land \lnot\Failing(t_1, \sigma')} \\
&\cup& \set{\Pick \Right \mid e_2, \sigma \normalise t_2, \sigma' \land \lnot\Failing(t_2, \sigma')} \\
% \Inputs(u_1 \At t, \sigma) &=& \set{u_2 \At i\mid u_2 \At i \in \Inputs(t, \sigma)} \\
% &\cup& \set{u_1 \At i\mid i\in \Inputs(t, \sigma)}
& &
\end{function}
}
\newmacro{O-Failing}{
\begin{function}
\signature{\Failing : \mathrm{Tasks} \times \mathrm{States} \to \mathrm{Booleans}} \\
\Failing(\Edit v, \sigma) &=& \False \\
\Failing(\Enter \tau, \sigma) &=& \False \\
\Failing(\Update l, \sigma) &=& \False \\
\Failing(\Fail, \sigma) &=& \True \\
\Failing(t_1 \Then e_2, \sigma) &=& \Failing(t_1, \sigma) \\
\Failing(t_1 \Next e_2, \sigma) &=& \Failing(t_1, \sigma) \\
\Failing(t_1 \And t_2, \sigma) &=& \Failing(t_1, \sigma) \land \Failing(t_2, \sigma) \\
\Failing(t_1 \Or t_2, \sigma) &=& \Failing(t_1, \sigma) \land \Failing(t_2, \sigma) \\
\Failing(e_1 \Xor e_2, \sigma) &=& \Failing(t_1, \sigma_1') \land \Failing(t_2, \sigma_2') \quad\where\ e_1, \sigma \normalise t_1, \sigma_1' \mathbf{\ and\ } e_2, \sigma \normalise t_2, \sigma_2'
% \Failing(u \At t, \sigma) &=& \Failing(t, \sigma)
\end{function}
}
\newmacro{O-Failing-Compact}{
\begin{function}
\signature{\Failing : \mathrm{Tasks} \times \mathrm{States} \to \mathrm{Booleans}} \\
\Failing(\Edit v, \sigma) &=& \False \\
\Failing(\Enter \tau, \sigma) &=& \False \\
\Failing(\Update l, \sigma) &=& \False \\
\Failing(\Fail, \sigma) &=& \True \\
\Failing(t_1 \Then e_2, \sigma) &=& \Failing(t_1, \sigma) \\
\Failing(t_1 \Next e_2, \sigma) &=& \Failing(t_1, \sigma) \\
\Failing(t_1 \And t_2, \sigma) &=& \Failing(t_1, \sigma) \land \Failing(t_2, \sigma) \\
\Failing(t_1 \Or t_2, \sigma) &=& \Failing(t_1, \sigma) \land \Failing(t_2, \sigma) \\
\Failing(e_1 \Xor e_2, \sigma) &=& \Failing(t_1, \sigma_1') \land \Failing(t_2, \sigma_2') \\
% \inset{\where\ e_1, \sigma \normalise t_1, \sigma_1' \mathbf{\ and\ } e_2, \sigma \normalise t_2, \sigma_2'} \\
& & \quad\where\ e_1, \sigma \normalise t_1, \sigma_1' \mathbf{\ and\ } e_2, \sigma \normalise t_2, \sigma_2'
% \Failing(u \At t, \sigma) &=& \Failing(t, \sigma)
\end{function}
}
\newmacro{O-Failing-Compacter}{
\begin{function}
\signature{\Failing : \mathrm{Tasks} \times \mathrm{States} \to \mathrm{Booleans}} \\
\Failing(\Edit v, \sigma) &=& \False \\
\Failing(\Enter \tau, \sigma) &=& \False \\
\Failing(\Update l, \sigma) &=& \False \\
\Failing(\Fail, \sigma) &=& \True \\
\Failing(t_1 \Then e_2, \sigma) &=& \Failing(t_1, \sigma) \\
\Failing(t_1 \Next e_2, \sigma) &=& \Failing(t_1, \sigma) \\
\Failing(t_1 \And t_2, \sigma) &=& \Failing(t_1, \sigma) \land \Failing(t_2, \sigma) \\
\Failing(t_1 \Or t_2, \sigma) &=& \Failing(t_1, \sigma) \land \Failing(t_2, \sigma) \\
\Failing(e_1 \Xor e_2, \sigma) &=& \Failing(t_1, \sigma_1') \land \Failing(t_2, \sigma_2') \\
\multicolumn{3}{R@{}}{\where\ e_1, \sigma \normalise t_1, \sigma_1'} \\
\multicolumn{3}{R@{}}{\mathbf{and\ } e_2, \sigma \normalise t_2, \sigma_2'} \\
% \Failing(u \At t, \sigma) &=& \Failing(t, \sigma) \\
& & \\
& & \\
& &
\end{function}
}
\newmacro{O-Watching}{
\begin{function}
\signature{\Watching : \mathrm{Tasks} \to \powerset(\mathrm{Locations})} \\
\Watching(\Edit v) &=& \nothing \\
\Watching(\Enter \tau) &=& \nothing \\
\Watching(\Update l) &=& \set{l} \\
\Watching(\Fail) &=& \nothing\\
\Watching(t_1 \Then e_2) &=& \Watching(t_1) \\
\Watching(t_1 \Next e_2) &=& \Watching(t_1) \\
\Watching(t_1 \And t_2) &=& \Watching(t_1) \cup \Watching(t_2) \\
\Watching(t_1 \Or t_2) &=& \Watching(t_1) \cup \Watching(t_2) \\
\Watching(e_1 \Xor e_2) &=& \nothing
% \Watching(u \At t) &=& \Watching(t)
\end{function}
}