ghost-state.tex

\section{Composeable Dynamic Higher-Order Resources}
\label{sec:hogs}

The logic described in \Sref{sec:base-logic} works over an arbitrary CMRA $\monoid$ defining the structure of the resources.
It turns out that we can generalize this further and permit picking CMRAs ``$\iFunc(\Prop)$'' that depend on the structure of assertions themselves.
Of course, $\Prop$ is just the syntactic type of assertions; for this to make sense we have to look at the semantics.

Furthermore, there is a composeability problem with the given logic: if we have one proof performed with CMRA $\monoid_1$, and another proof carried out with a \emph{different} CMRA $\monoid_2$, then the two proofs are actually carried out in two entirely separate logics and hence cannot be combined.

The purpose of this section is to describe how we solve these two issues.

\subsection{Multiple CMRAs and Higher-Order Resources}

To instantiate the logic, you pick a \emph{family} of locally contractive bifunctors $(\iFunc_i : \COFEs \to \CMRAs)_{i \in \mathcal{I}}$.

From this, we construct the bifunctor defining the overall resources as follows:
\begin{align*}
  \textdom{ResF}(\cofe^\op, \cofe) \eqdef{}& \prod_{i \in \mathcal I} \mathbb{N} \fpfn \iFunc_i(\cofe^\op, \cofe)
\end{align*}
$\textdom{ResF}(\cofe^\op, \cofe)$ is a CMRA by lifting the individual CMRAs pointwise, and it has a unit (using the empty finite partial functions).
Furthermore, since the $\iFunc_i$ are locally contractive, so is $\textdom{ResF}$.

Now we can write down the recursive domain equation:
\[ \iPreProp \cong \UPred(\textdom{ResF}(\iPreProp, \iPreProp)) \]
$\iPreProp$ is a COFE defined as the fixed-point of a locally contractive bifunctor.
This fixed-point exists and is unique by America and Rutten's theorem~\cite{America-Rutten:JCSS89,birkedal:metric-space}.
We do not need to consider how the object is constructed. 
We only need the isomorphism, given by
\begin{align*}
  \Res &\eqdef \textdom{ResF}(\iPreProp, \iPreProp) \\
  \iProp &\eqdef \UPred(\Res) \\
	\wIso &: \iProp \nfn \iPreProp \\
	\wIso^{-1} &: \iPreProp \nfn \iProp
\end{align*}

Notice that $\iProp$ is the semantic model of assertions for  the logic described in \Sref{sec:base-logic} with $\Res$:
\[ \Sem{\Prop} \eqdef \iProp = \UPred(\Res) \]
We thus obtain all the rules of \Sref{sec:base-logic}, and furthermore, we can use the maps $\wIso$ and $\wIso^{-1}$ \emph{in the logic} to convert between logical assertions $\Sem\Prop$ and the domain $\iPreProp$ which is used in the construction of $\Res$ -- so from elements of $\iPreProp$, we can construct elements of $\Sem{\textlog M}$, which are the elements that can be owned in our logic.

\ralf{TODO: Describe the pattern of only assuming some elements of the index family to indicate a particular functor.}
\ralf{TODO: Show the rules for ownership in this world.}




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