esop-satisfaction.tex

\begin{figure*}
\begin{mathpar}
\infer [M-Or]
{ \satisfies{\openv{}}{\prop{1}}\ \text{or}\  \satisfies{\openv{}}{\prop{2}}}
{ \satisfies{\openv{}}{\orprop{\prop{1}}{\prop{2}}}
                   }

\infer [M-Imp]
{ \satisfies{\openv{}}{\prop{1}}\ \text{implies}\ \satisfies{\openv{}}{\prop{2}}}
{ \satisfies{\openv{}}{\impprop{\prop{1}}{\prop{2}}}
                   }

\infer [M-And]
{ \satisfies{\openv{}}{\prop{1}}
\\ \satisfies{\openv{}}{\prop{2}}}
{ \satisfies{\openv{}}{\andprop{\prop{1}}{\prop{2}}}
                   }


\infer [M-Top]
{}
{ \satisfies{\openv{}}{\topprop{}}
                   }

                   \\

\infer [M-Type]
{ \judgement {} {\openv{}({\pth{\pathelem{}}{\x{}}})} {\t{}}{\filterset{\thenprop{\prop{}}}{\elseprop{\prop{}}}}{\object{}}}
{ \satisfies{\openv{}}{\isprop{\t{}}{\pth{\pathelem{}}{\x{}}}}
                   }

\infer [M-NotType]
{ \judgement {} {\openv{}({\pth{\pathelem{}}{\x{}}})} {\s{}}{\filterset{\thenprop{\prop{}}}{\elseprop{\prop{}}}}{\object{}}
\\\\
\text{there is no}\ \v{}\ \text{such that}\ \judgement{}{\v{}}{\t{}}{\filterset{\thenprop{\prop{1}}}{\elseprop{\prop{1}}}}{\object{1}}
\ \text{and}\ \judgement{}{\v{}}{\s{}}{\filterset{\thenprop{\prop{2}}}{\elseprop{\prop{2}}}}{\object{2}}
}
{ \satisfies{\openv{}}{\notprop{\t{}}{\pth{\pathelem{}}{\x{}}}}
                   }
\end{mathpar}
\caption{Satisfaction Relation}
\end{figure*}