### editing

parent e85a1027
 ... ... @@ -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} ... ...
Markdown is supported
0% or .
You are about to add 0 people to the discussion. Proceed with caution.
Finish editing this message first!