Commit ccf8bc36 authored by Ralf Jung's avatar Ralf Jung

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}
......
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