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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@@ -401,13 +454,14 @@ The premise in \ruleref{upd-update} is a \emph{meta-level} side-condition that h
The consistency statement of the logic reads as follows: For any $n$, we have
\begin{align*}
\lnot(\TRUE\proves (\upd\later)^n\spac\FALSE)
\lnot(\TRUE\proves (\later)^n\spac\FALSE)
\end{align*}
where $(\upd\later)^n$ is short for $\upd\later$ being nested $n$ times.
where $(\later)^n$ is short for $\later$ being nested $n$ times.
The reason we want a stronger consistency than the usual $\lnot(\TRUE\proves\FALSE)$ is our modalities: it should be impossible to derive a contradiction below the modalities.
For $\always$, this follows from the elimination rule, but the other two modalities do not have an elimination rule.
Hence we declare that it is impossible to derive a contradiction below any combination of these two modalities.
For $\always$ and $\plainly$, this follows from the elimination rules.
For updates, we use the fact that $\upd\FALSE\proves\upd\plainly\FALSE\proves\FALSE$.
However, there is no elimination rule for $\later$, so we declare that it is impossible to derive a contradiction below any number of laters.
@@ -35,8 +35,13 @@ We collect here some important and frequently used derived proof rules.
\infer{}
{\prop\proves\later\prop}
\infer{}
{\TRUE\proves\plainly\TRUE}
\end{mathparpagebreakable}
Noteworthy here is the fact that $\prop\proves\later\prop$ can be derived from Löb induction, and $\TRUE\proves\plainly\TRUE$ can be derived via $\plainly$ commuting with universal quantification ranging over the empty type $0$.
\subsection{Persistent assertions}
We call an assertion $\prop$\emph{persistent} if $\prop\proves\always\prop$.
These are assertions that ``don't own anything'', so we can (and will) treat them like ``normal'' intuitionistic assertions.
\All m, \melt'. & m \leq n \land (\melt\mtimes\melt') \in\mval_m \Ra{}\\&\Exists\meltB. (\meltB\mtimes\melt') \in\mval_m \land m \in\Sem{\vctx\proves\prop :\Prop}_\gamma(\meltB)
\end{aligned}
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@@ -79,9 +82,15 @@ For every definition, we have to show all the side-conditions: The maps have to
