@@ -70,22 +70,7 @@ Note that $\COFEs$ is cartesian closed.

This is a natural generalization of RAs over COFEs.

All operations have to be non-expansive, and the validity predicate $\mval$ can now also depend on the step-index.

\paragraph{The division operator $\mdiv$.}

One way to describe $\mdiv$ is to say that it extracts the witness from the extension order: If $\melt\leq\meltB$, then $\melt\mdiv\meltB$ computes the difference between the two elements (\ruleref{cmra-div-op}).

Otherwise, $\mdiv$ can have arbitrary behavior.

This means that, in classical logic, the division operator can be defined for any PCM using the axiom of choice, and it will trivially satisfy \ruleref{cmra-div-op}.

However, notice that the division operator also has to be \emph{non-expansive} --- so if the carrier $\monoid$ is equipped with a non-trivial $\nequiv{n}$, there is an additional proof obligation here.

This is crucial, for the following reason:

Considering that the extension order is defined using \emph{equality}, there is a natural notion of a \emph{step-indexed extension} order using the step-indexed equivalence of the underlying COFE:

One of the properties we would expect to hold is the usual correspondence between a step-indexed predicate and its non-step-indexed counterpart:

\[\All\melt, \meltB. \melt\leq\meltB\Lra(\All n. \melt\mincl{n}\meltB)\tagH{cmra-incl-limit}\]

The right-to-left direction here is trick.

For every $n$, we obtain a proof that $\melt\mincl{n}\meltB$.

From this, we could extract a sequence of witnesses $(\meltC_m)_{m}$, and we need to arrive at a single witness $\meltC$ showing that $\melt\leq\meltB$.

Without the division operator, there is no reason to believe that such a witness exists.

However, since we can use the division operator, and since we know that this operator is \emph{non-expansive}, we can pick $\meltC\eqdef\meltB\mdiv\melt$, and then we can prove that this is indeed the desired witness.

\ralf{The only reason we actually have division is that we are working constructively in an impredicative universe. This is pretty silly.}