Skip to content
Snippets Groups Projects
Commit ccf8bc36 authored by Ralf Jung's avatar Ralf Jung
Browse files

editing

parent e85a1027
No related branches found
No related tags found
No related merge requests found
......@@ -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}
......
0% Loading or .
You are about to add 0 people to the discussion. Proceed with caution.
Finish editing this message first!
Please register or to comment