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Jonas Kastberg
iris
Commits
ccf8bc36
Commit
ccf8bc36
authored
Apr 29, 2019
by
Ralf Jung
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docs/ghost-state.tex
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ccf8bc36
...
...
@@ -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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