Commit 82aa84a7 authored by Ralf Jung's avatar Ralf Jung

describe frac RA

parent a007ecf5
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......@@ -173,7 +173,7 @@ There are no interesting frame-preserving updates for $\agm(\cofe)$, but we can
Given an OFE $\cofe$, we define a camera $\exm(\cofe)$ such that at most one $x \in \cofe$ can be owned:
\begin{align*}
\exm(\cofe) \eqdef{}& \exinj(\cofe) \mid \mundef \\
\mval(\melt) \eqdef{}& \setComp{n}{\melt \neq \mundef}
\mval(\melt) \eqdef{}& \setComp{n}{\melt \notnequiv{n} \mundef}
\end{align*}
All cases of composition go to $\mundef$.
\begin{align*}
......@@ -196,7 +196,25 @@ We obtain the following frame-preserving update:
{\exinj(x) \mupd \exinj(y)}
\end{mathpar}
\subsection{Fractions}
We define an RA structure on the rational numbers in $(0, 1]$ as follows:
\begin{align*}
\fracm \eqdef{}& \fracinj(\mathbb{Q} \cap (0, 1]) \mid \mundef \\
\mvalFull(\melt) \eqdef{}& \melt \neq \mundef \\
\fracinj(x) \mtimes \fracinj(y) \eqdef{}& \fracinj(x + y) \quad \text{if $x + y \leq 1$} \\
\mcore{\fracinj(x)} \eqdef{}& \bot \\
\mcore{\mundef} \eqdef{}& \mundef
\end{align*}
All remaining cases of composition go to $\mundef$.
Frequently, we will write just $x$ instead of $\fracinj(x)$.
The most important property of this RA is that $1$ has no frame.
This is useful in combination with \ruleref{sum-swap}, and also when used with pairs:
\begin{mathpar}
\inferH{pair-frac-change}{}
{(1, a) \mupd (1, b)}
\end{mathpar}
%TODO: These need syncing with Coq
% \subsection{Finite Powerset Monoid}
......@@ -227,53 +245,6 @@ We obtain the following frame-preserving update:
% \end{proof}
% \subsection{Fractional monoid}
% \label{sec:fracm}
% Given a monoid $M$, we define a monoid representing fractional ownership of some piece $\melt \in M$.
% The idea is to preserve all the frame-preserving update that $M$ could have, while additionally being able to do \emph{any} update if we own the full state (as determined by the fraction being $1$).
% Let $\fracm{M}$ be the monoid with carrier $(((0, 1] \cap \mathbb{Q}) \times M) \uplus \{\munit\}$ and multiplication
% \begin{align*}
% (q, a) \mtimes (q', a') &\eqdef (q + q', a \mtimes a') \qquad \mbox{if $q+q'\le 1$} \\
% (q, a) \mtimes \munit &\eqdef (q,a) \\
% \munit \mtimes (q,a) &\eqdef (q,a).
% \end{align*}
% We get the following frame-preserving update.
% \begin{mathpar}
% \inferH{FracUpdFull}
% {a, b \in M}
% {(1, a) \mupd (1, b)}
% \and\inferH{FracUpdLocal}
% {a \mupd_M B}
% {(q, a) \mupd \{q\} \times B}
% \end{mathpar}
% \begin{proof}[Proof of \ruleref{FracUpdFull}]
% Assume some $f \sep (1, a)$. This can only be $f = \munit$, so showing $f \sep (1, b)$ is trivial.
% \end{proof}
% \begin{proof}[Proof of \ruleref{FracUpdLocal}]
% Assume some $f \sep (q, a)$. If $f = \munit$, then $f \sep (q, b)$ is trivial for any $b \in B$. Just pick the one we obtain by choosing $\munit_M$ as the frame for $a$.
% In the interesting case, we have $f = (q_\f, a_\f)$.
% Obtain $b$ such that $b \in B \land b \sep a_\f$.
% Then $(q, b) \sep f$, and we are done.
% \end{proof}
% $\fracm{M}$ is cancellative if $M$ is cancellative.
% \begin{proof}[Proof of cancellativitiy]
% If $\melt_\f = \munit$, we are trivially done.
% So let $\melt_\f = (q_\f, \melt_\f')$.
% If $\melt = \munit$, then $\meltB = \munit$ as otherwise the fractions could not match up.
% Again, we are trivially done.
% Similar so for $\meltB = \munit$.
% So let $\melt = (q_a, \melt')$ and $\meltB = (q_b, \meltB')$.
% We have $(q_\f + q_a, \melt_\f' \mtimes \melt') = (q_\f + q_b, \melt_\f' \mtimes \meltB')$.
% We have to show $q_a = q_b$ and $\melt' = \meltB'$.
% The first is trivial, the second follows from cancellativitiy of $M$.
% \end{proof}
\subsection{Authoritative}
\label{sec:auth-camera}
......
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