diff --git a/docs/ghost-state.tex b/docs/ghost-state.tex index 953a2fea746626d08472598cf0768ab57cf05625..b8d55d0fd6249636244263d094b789a78286630c 100644 --- a/docs/ghost-state.tex +++ b/docs/ghost-state.tex @@ -57,31 +57,7 @@ Persistence is preserved by conjunction, disjunction, separating conjunction as One of the troubles of working in a step-indexed logic is the later'' modality $\later$. It turns out that we can somewhat mitigate this trouble by working below the following \emph{except-0} modality: $\diamond \prop \eqdef \later\FALSE \lor \prop$ - -This modality is useful because there is a class of propositions which we call \emph{timeless} propositions, for which we have -$\timeless{\prop} \eqdef \later\prop \proves \diamond\prop$ -In other words, when working below the except-0 modality, we can \emph{strip - away} the later from timeless propositions. In fact, we can strip away later -from timeless propositions even when working under the later modality: -\begin{mathpar} - \inferH{later-timeless-strip}{\timeless{\prop} \and \prop \proves \later \propB} - {\later\prop \proves \later\propB} -\end{mathpar} -This rule looks different from the above ones, because we still do not have that -\begin{mathpar} - \inferH{later-fake-rule}{\timeless{\prop}} - {\later\prop \proves \prop} -\end{mathpar} -The proof of the former is $\later \prop \proves \diamond \prop = -\later\FALSE \lor \prop$, and then by straightforward disjunction elimination: -% Cut the second part if trivial. -\begin{mathpar} - \infer{\later\FALSE \proves \later \propB \and \prop \proves \later \propB} - {\later\FALSE \lor \prop \proves \propB} -\end{mathpar} - - -The following rules can be derived about except-0: +Except-0 satisfies the usual laws of a monadic'' modality (similar to, \eg the update modalities): \begin{mathpar} \inferH{ex0-mono} {\prop \proves \propB} @@ -106,6 +82,28 @@ The following rules can be derived about except-0: \diamond\later\prop &\proves& \later{\prop} \end{array} \end{mathpar} +In particular, from \ruleref{ex0-mono} and \ruleref{ex0-idem} we can derive a bind''-like elimination rule: +\begin{mathpar} + \inferH{ex0-elim} + {\prop \proves \diamond\propB} + {\diamond\prop \proves \diamond\propB} +\end{mathpar} + +This modality is useful because there is a class of propositions which we call \emph{timeless} propositions, for which we have +$\timeless{\prop} \eqdef \later\prop \proves \diamond\prop$ +In other words, when working below the except-0 modality, we can \emph{strip + away} the later from timeless propositions (using \ruleref{ex0-elim}): +\begin{mathpar} + \inferH{ex0-timeless-strip}{\timeless{\prop} \and \prop \proves \diamond\propB} + {\later\prop \proves \diamond\propB} +\end{mathpar} + + In fact, it turns out that we can strip away later from timeless propositions even when working under the later modality: +\begin{mathpar} + \inferH{later-timeless-strip}{\timeless{\prop} \and \prop \proves \later \propB} + {\later\prop \proves \later\propB} +\end{mathpar} +This follows from $\later \prop \proves \later\FALSE \lor \prop$, and then by straightforward disjunction elimination. The following rules identify the class of timeless propositions: \begin{mathparpagebreakable}