Commit ce91f292 authored by Ralf Jung's avatar Ralf Jung

docs: plainly rules

parent dbdd25ba
......@@ -307,7 +307,30 @@ Furthermore, we have the usual $\eta$ and $\beta$ laws for projections, $\lambda
{\prop \proves \propB \wand \propC}
\end{mathpar}
\paragraph{Laws for the always modality.}
\paragraph{Laws for the plainness modality.}
\begin{mathpar}
\infer[$\plainly$-mono]
{\prop \proves \propB}
{\plainly{\prop} \proves \plainly{\propB}}
\and
\infer[$\plainly$-E]{}
{\plainly\prop \proves \always\prop}
\and
\begin{array}[c]{rMcMl}
\TRUE &\proves& \plainly{\TRUE} \\
(\plainly P \Ra \plainly Q) &\proves& \plainly (\plainly P \Ra Q)
\end{array}
\and
\begin{array}[c]{rMcMl}
\plainly{\prop} &\proves& \plainly\plainly\prop \\
\All x. \plainly{\prop} &\proves& \plainly{\All x. \prop} \\
\plainly{\Exists x. \prop} &\proves& \Exists x. \plainly{\prop}
\end{array}
\and
\infer[PropExt]{}{\plainly ( ( P \Ra Q) \land (Q \Ra P ) ) \proves P =_{\Prop} Q}
\end{mathpar}
\paragraph{Laws for the persistence modality.}
\begin{mathpar}
\infer[$\always$-mono]
{\prop \proves \propB}
......@@ -317,9 +340,8 @@ Furthermore, we have the usual $\eta$ and $\beta$ laws for projections, $\lambda
{\always\prop \proves \prop}
\and
\begin{array}[c]{rMcMl}
\TRUE &\proves& \always{\TRUE} \\
\always{(\prop \land \propB)} &\proves& \always{(\prop * \propB)} \\
\always{\prop} \land \propB &\proves& \always{\prop} * \propB
\always{\prop} \land \propB &\proves& \always{\prop} * \propB \\
(\plainly P \Ra \always Q) &\proves& \always (\plainly P \Ra Q)
\end{array}
\and
\begin{array}[c]{rMcMl}
......@@ -348,7 +370,8 @@ Furthermore, we have the usual $\eta$ and $\beta$ laws for projections, $\lambda
\and
\begin{array}[c]{rMcMl}
\later{(\prop * \propB)} &\provesIff& \later\prop * \later\propB \\
\always{\later\prop} &\provesIff& \later\always{\prop}
\always{\later\prop} &\provesIff& \later\always{\prop} \\
\plainly{\later\prop} &\provesIff& \later\plainly{\prop}
\end{array}
\end{mathpar}
......@@ -397,6 +420,10 @@ Furthermore, we have the usual $\eta$ and $\beta$ laws for projections, $\lambda
\inferH{upd-update}
{\melt \mupd \meltsB}
{\ownM\melt \proves \upd \Exists\meltB\in\meltsB. \ownM\meltB}
\inferH{upd-plainly}
{}
{\upd\plainly\prop \proves \prop}
\end{mathpar}
The premise in \ruleref{upd-update} is a \emph{meta-level} side-condition that has to be proven about $a$ and $B$.
%\ralf{Trouble is, we don't actually have $\in$ inside the logic...}
......
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