The base logic is parameterized by an arbitrary CMRA $\monoid$ having a unit.

By \lemref{lem:cmra-unit-total-core}, this means that the core of $\monoid$ is a total function, so we will treat it as such in the following.

This defines the structure of resources that can be owned.

As usual for higher-order logics, you can furthermore pick a \emph{signature}$\Sig=(\SigType, \SigFn, \SigAx)$ to add more types, symbols and axioms to the language.

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 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.}

A \emph{language}$\Lang$ consists of a set \textdom{Expr} of \emph{expressions} (metavariable $\expr$), a set \textdom{Val} of \emph{values} (metavariable $\val$), and a set \textdom{State} of \emph{states} (metvariable $\state$) such that

\begin{itemize}

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@@ -58,6 +59,7 @@ For any language $\Lang$, we define the corresponding thread-pool semantics.

\clearpage

\section{Program Logic}

\label{sec:program-logic}

\ralf{TODO: Right now, this is a dump of all the things that moved out of the base...}

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@@ -243,32 +245,7 @@ Notice that this is stronger than saying that the thread pool can reduce; we act

\subsection{Iris model}

\paragraph{Semantic domain of assertions.}

The first complicated task in building a model of full Iris is defining the semantic model of $\Prop$.

We start by defining the functor that assembles the CMRAs we need to the global resource CMRA: