docs: update agreement construction

CHANGELOG.md

@@ -12,11 +12,11 @@ Coq development, but not every API-breaking change is listed.  Changes marked
   updates.  Weakestpre is defined inside the logic, and invariants and view
   shifts with masks are also coded up inside Iris.  Adequacy of weakestpre
   is proven in the logic.
-* [#] Use OFEs instead of COFEs everywhere.  COFEs are only used for solving the
+* Use OFEs instead of COFEs everywhere.  COFEs are only used for solving the
   recursive domain equation.  As a consequence, CMRAs no longer need a proof
   of completeness.
   (The old `cofeT` is provided by `algebra.deprecated`.)
-* [#] Implement a new agreement construction.  Unlike the old one, this one
+* Implement a new agreement construction.  Unlike the old one, this one
   preserves discreteness.
 * Renaming and moving things around: uPred and the rest of the base logic are
   in `base_logic`, while `program_logic` is for everything involving the

docs/constructions.tex

@@ -141,35 +141,28 @@ $K \fpfn (-)$ is a locally non-expansive functor from $\CMRAs$ to $\CMRAs$.
 
 Given some OFE $\cofe$, we define the CMRA $\agm(\cofe)$ as follows:
 \begin{align*}
-  \agm(\cofe) \eqdef{}& \set{(c, V) \in (\nat \to \cofe) \times \SProp}/\ {\sim} \\[-0.2em]
-  \textnormal{where }& \melt \sim \meltB \eqdef{} \melt.V = \meltB.V \land 
-    \All n. n \in \melt.V \Ra \melt.c(n) \nequiv{n} \meltB.c(n)  \\
+  \agm(\cofe) \eqdef{}& \setComp{\melt \in \finpset\cofe}{c \neq \emptyset} /\ {\sim} \\[-0.2em]
+  \melt \nequiv{n} \meltB \eqdef{}& (\All x \in \melt. \Exists y \in \meltB. x \nequiv{n} y) \land (\All y \in \meltB. \Exists x \in \melt. x \nequiv{n} y) \\
+  \textnormal{where }& \melt \sim \meltB \eqdef{} \All n. \melt \nequiv{n} \meltB  \\
+~\\
 %    \All n \in {\melt.V}.\, \melt.x \nequiv{n} \meltB.x \\
-  \melt \nequiv{n} \meltB \eqdef{}& (\All m \leq n. m \in \melt.V \Lra m \in \meltB.V) \land (\All m \leq n. m \in \melt.V \Ra \melt.c(m) \nequiv{m} \meltB.c(m)) \\
-  \mval_n \eqdef{}& \setComp{\melt \in \agm(\cofe)}{ n \in \melt.V \land \All m \leq n. \melt.c(n) \nequiv{m} \melt.c(m) } \\
+  \mval_n \eqdef{}& \setComp{\melt \in \agm(\cofe)}{ \All x, y \in \melt. x \nequiv{n} y } \\
   \mcore\melt \eqdef{}& \melt \\
-  \melt \mtimes \meltB \eqdef{}& \left(\melt.c, \setComp{n}{n \in \melt.V \land n \in \meltB.V \land \melt \nequiv{n} \meltB }\right)
+  \melt \mtimes \meltB \eqdef{}& \melt \cup \meltB
 \end{align*}
 %Note that the carrier $\agm(\cofe)$ is a \emph{record} consisting of the two fields $c$ and $V$.
 
 $\agm(-)$ is a locally non-expansive functor from $\OFEs$ to $\CMRAs$.
 
-You can think of the $c$ as a \emph{chain} of elements of $\cofe$ that has to converge only for $n \in V$ steps.
-The reason we store a chain, rather than a single element, is that $\agm(\cofe)$ needs to be a COFE itself, so we need to be able to give a limit for every chain of $\agm(\cofe)$.
-However, given such a chain, we cannot constructively define its limit: Clearly, the $V$ of the limit is the limit of the $V$ of the chain.
-But what to pick for the actual data, for the element of $\cofe$?
-Only if $V = \nat$ we have a chain of $\cofe$ that we can take a limit of; if the $V$ is smaller, the chain ``cancels'', \ie stops converging as we reach indices $n \notin V$.
-To mitigate this, we apply the usual construction to close a set; we go from elements of $\cofe$ to chains of $\cofe$.
-
-We define an injection $\aginj$ into $\agm(\cofe)$ as follows:
-\[ \aginj(x) \eqdef \record{\mathrm c \eqdef \Lam \any. x, \mathrm V \eqdef \nat} \]
+We define a non-expansive injection $\aginj$ into $\agm(\cofe)$ as follows:
+\[ \aginj(x) \eqdef \set{x} \]
 There are no interesting frame-preserving updates for $\agm(\cofe)$, but we can show the following:
 \begin{mathpar}
   \axiomH{ag-val}{\aginj(x) \in \mval_n}
 
   \axiomH{ag-dup}{\aginj(x) = \aginj(x)\mtimes\aginj(x)}
   
-  \axiomH{ag-agree}{\aginj(x) \mtimes \aginj(y) \in \mval_n \Ra x \nequiv{n} y}
+  \axiomH{ag-agree}{\aginj(x) \mtimes \aginj(y) \in \mval_n \Lra x \nequiv{n} y}
 \end{mathpar}