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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