The logic described in \Sref{sec:base-logic} works over an arbitrary CMRA $\monoid$ defining the structure of the resources.

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.

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.

Of course, $\Prop$ is just the syntactic type of assertions; for this to make sense we have to look at the semantics.

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

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:

From this, we construct the bifunctor defining the overall resources as follows:

\begin{align*}

\begin{align*}

...

@@ -29,7 +35,7 @@ We only need the isomorphism, given by

...

@@ -29,7 +35,7 @@ We only need the isomorphism, given by

Notice that $\iProp$ is the semantic model of assertions for the logic described in \Sref{sec:base-logic} with $\Res$:

Notice that $\iProp$ is the semantic model of assertions for the logic described in \Sref{sec:base-logic} with $\Res$:

\[\Sem{\Prop}\eqdef\iProp=\UPred(\Res)\]

\[\Sem{\Prop}\eqdef\iProp=\UPred(\Res)\]

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 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.

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: 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.}

\ralf{TODO: Show the rules for ownership in this world.}