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George Pirlea
Iris
Commits
a072e355
Commit
a072e355
authored
Mar 12, 2016
by
Ralf Jung
Browse files
docs: define RAs
parent
610698ec
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1
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docs/algebra.tex
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a072e355
...
...
@@ -52,7 +52,31 @@ Note that the composition of non-expansive (bi)functors is non-expansive, and th
\subsection
{
RA
}
\ralf
{
Copy this from the paper, when that one is more polished.
}
\begin{defn}
A
\emph
{
resource algebra
}
(RA) is a tuple
\\
$
(
\monoid
,
\mval
\subseteq
\monoid
,
\mcore
{
-
}
:
\monoid
\to
\monoid
,
(
\mtimes
)
:
\monoid
\times
\monoid
\to
\monoid
)
$
satisfying
\begin{align*}
\All
\melt
,
\meltB
,
\meltC
.
&
(
\melt
\mtimes
\meltB
)
\mtimes
\meltC
=
\melt
\mtimes
(
\meltB
\mtimes
\meltC
)
\tagH
{
ra-assoc
}
\\
\All
\melt
,
\meltB
.
&
\melt
\mtimes
\meltB
=
\meltB
\mtimes
\melt
\tagH
{
ra-comm
}
\\
\All
\melt
.
&
\mcore\melt
\mtimes
\melt
=
\melt
\tagH
{
ra-core-id
}
\\
\All
\melt
.
&
\mcore
{
\mcore\melt
}
=
\mcore\melt
\tagH
{
ra-core-idem
}
\\
\All
\melt
,
\meltB
.
&
\melt
\mincl
\meltB
\Ra
\mcore\melt
\mincl
\mcore\meltB
\tagH
{
ra-core-mono
}
\\
\All
\melt
,
\meltB
.
&
(
\melt
\mtimes
\meltB
)
\in
\mval
\Ra
\melt
\in
\mval
\tagH
{
ra-valid-op
}
\\
\text
{
where
}
\qquad
%\qquad\\
\melt
\mincl
\meltB
\eqdef
{}&
\Exists
\meltC
.
\meltB
=
\melt
\mtimes
\meltC
\tagH
{
ra-incl
}
\end{align*}
\end{defn}
\begin{defn}
It is possible to do a
\emph
{
frame-preserving update
}
from
$
\melt
\in
\monoid
$
to
$
\meltsB
\subseteq
\monoid
$
, written
$
\melt
\mupd
\meltsB
$
, if
\[
\All
\melt
_
f.
\melt
\mtimes
\melt
_
f
\in
\mval
\Ra
\Exists
\meltB
\in
\meltsB
.
\meltB
\mtimes
\melt
_
f
\in
\mval
\]
We further define
$
\melt
\mupd
\meltB
\eqdef
\melt
\mupd
\set\meltB
$
.
\end{defn}
\ralf
{
Copy the explanation from the paper, when that one is more polished.
}
\subsection
{
CMRA
}
...
...
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