docs: address Lars' remaining comments

docs/algebra.tex

@@ -211,7 +211,7 @@ Furthermore, discrete CMRAs can be turned into RAs by ignoring their COFE struct
 \end{defn}
 Note that every object/arrow in $\CMRAs$ is also an object/arrow of $\COFEs$.
 The notion of a locally non-expansive (or contractive) bifunctor naturally generalizes to bifunctors between these categories.
-
+\ralf{Discuss how we probably have a commuting square of functors between Set, RA, CMRA, COFE.}
 
 %%% Local Variables: 
 %%% mode: latex

docs/constructions.tex

@@ -3,7 +3,7 @@
 
 \subsection{Next (type-level later)}
 
-Given a COFE $\cofe$, we define $\latert\cofe$ as follows:
+Given a COFE $\cofe$, we define $\latert\cofe$ as follows (using a datatype-like notation to define the type):
 \begin{align*}
   \latert\cofe \eqdef{}& \latertinj(x:\cofe) \\
   \latertinj(x) \nequiv{n} \latertinj(y) \eqdef{}& n = 0 \lor x \nequiv{n-1} y
@@ -62,7 +62,7 @@ Frame-preserving updates on the $M_i$ lift to the product:
 \subsection{Sum}
 \label{sec:summ}
 
-The \emph{sum CMRA} $\monoid_1 \csumm \monoid_2$ for any CMRAs $\monoid_1$ and $\monoid_2$ is defined as:
+The \emph{sum CMRA} $\monoid_1 \csumm \monoid_2$ for any CMRAs $\monoid_1$ and $\monoid_2$ is defined as (again, we use a datatype-like notation):
 \begin{align*}
   \monoid_1 \csumm \monoid_2 \eqdef{}& \cinl(\melt_1:\monoid_1) \mid \cinr(\melt_2:\monoid_2) \mid \bot \\
   \mval_n \eqdef{}& \setComp{\cinl(\melt_1)\!}{\!\melt_1 \in \mval'_n}
@@ -104,36 +104,35 @@ We obtain the following frame-preserving updates:
   {\emptyset \mupd \setComp{[\gname \mapsto \melt]}{\gname \in K}}
 
   \inferH{fpfn-update}
-  {\melt \mupd \meltsB}
+  {\melt \mupd_\monoid \meltsB}
   {f[i \mapsto \melt] \mupd \setComp{ f[i \mapsto \meltB]}{\meltB \in \meltsB}}
 \end{mathpar}
-Remember that $\mval$ is the set of elements of a CMRA that are valid at \emph{all} step-indices.
+Above, $\mval$ refers to the validity of $\monoid$.
 
 $K \fpfn (-)$ is a locally non-expansive functor from $\CMRAs$ to $\CMRAs$.
 
 \subsection{Agreement}
 
 Given some COFE $\cofe$, we define $\agm(\cofe)$ as follows:
-\newcommand{\aginjc}{\mathrm{c}} % the "c" field of an agreement element
-\newcommand{\aginjV}{\mathrm{V}} % the "V" field of an agreement element
 \begin{align*}
-  \agm(\cofe) \eqdef{}& \record{\aginjc : \mathbb{N} \to \cofe , \aginjV : \SProp} \\
-  & \text{quotiented by} \\
-  \melt \equiv \meltB \eqdef{}& \melt.\aginjV = \meltB.\aginjV \land \All n. n \in \melt.\aginjV \Ra \melt.\aginjc(n) \nequiv{n} \meltB.\aginjc(n) \\
-  \melt \nequiv{n} \meltB \eqdef{}& (\All m \leq n. m \in \melt.\aginjV \Lra m \in \meltB.\aginjV) \land (\All m \leq n. m \in \melt.\aginjV \Ra \melt.\aginjc(m) \nequiv{m} \meltB.\aginjc(m)) \\
-  \mval_n \eqdef{}& \setComp{\melt \in \agm(\cofe)}{ n \in \melt.\aginjV \land \All m \leq n. \melt.\aginjc(n) \nequiv{m} \melt.\aginjc(m) } \\
+  \agm(\cofe) \eqdef{}& \set{(c, V) \in (\mathbb{N} \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)  \\
+%    \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) } \\
   \mcore\melt \eqdef{}& \melt \\
-  \melt \mtimes \meltB \eqdef{}& (\melt.\aginjc, \setComp{n}{n \in \melt.\aginjV \land n \in \meltB.\aginjV \land \melt \nequiv{n} \meltB })
+  \melt \mtimes \meltB \eqdef{}& \left(\melt.c, \setComp{n}{n \in \melt.V \land n \in \meltB.V \land \melt \nequiv{n} \meltB }\right)
 \end{align*}
-Note that the carrier $\agm(\cofe)$ is a \emph{record} consisting of the two fields $\aginjc$ and $\aginjV$.
+%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 $\COFEs$ to $\CMRAs$.
 
-You can think of the $\aginjc$ as a \emph{chain} of elements of $\cofe$ that has to converge only for $n \in \aginjV$ steps.
+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 $\aginjV$ of the limit is the limit of the $\aginjV$ of the chain.
+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 $\aginjV = \mathbb{N}$ we have a chain of $\cofe$ that we can take a limit of; if the $\aginjV$ is smaller, the chain ``cancels'', \ie stops converging as we reach indices $n \notin \aginjV$.
+Only if $V = \mathbb{N}$ 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:

docs/logic.tex

@@ -134,7 +134,7 @@ Recursive predicates must be \emph{guarded}: in $\MU \var. \term$, the variable
 Note that $\always$ and $\later$ bind more tightly than $*$, $\wand$, $\land$, $\lor$, and $\Ra$.
 We will write $\pvs[\term] \prop$ for $\pvs[\term][\term] \prop$.
 If we omit the mask, then it is $\top$ for weakest precondition $\wpre\expr{\Ret\var.\prop}$ and $\emptyset$ for primitive view shifts $\pvs \prop$.
-%FIXME $\top$ is not a term in the logic. Neither is any of the operations on masks that we use in the rules for weakestpre.
+\ralf{$\top$ is not a term in the logic. Neither is any of the operations on masks that we use in the rules for weakestpre.}
 
 Some propositions are \emph{timeless}, which intuitively means that step-indexing does not affect them.
 This is a \emph{meta-level} assertion about propositions, defined as follows:
@@ -382,15 +382,17 @@ This is entirely standard.
   {\vctx\mid\pfctx \proves \exists \var: \type.\; \prop \\
    \vctx,\var : \type\mid\pfctx , \prop \proves \propB}
   {\vctx\mid\pfctx \proves \propB}
-\and
-\infer[$\lambda$]
-  {}
-  {\pfctx \proves (\Lam\var: \type. \prop)(\term) =_{\type\to\type'} \prop[\term/\var]}
-\and
-\infer[$\mu$]
-  {}
-  {\pfctx \proves \mu\var: \type. \prop =_{\type} \prop[\mu\var: \type. \prop/\var]}
+% \and
+% \infer[$\lambda$]
+%   {}
+%   {\pfctx \proves (\Lam\var: \type. \prop)(\term) =_{\type\to\type'} \prop[\term/\var]}
+% \and
+% \infer[$\mu$]
+%   {}
+%   {\pfctx \proves \mu\var: \type. \prop =_{\type} \prop[\mu\var: \type. \prop/\var]}
 \end{mathparpagebreakable}
+Furthermore, we have the usual $\eta$ and $\beta$ laws for projections, $\lambda$ and $\mu$.
+
 
 \paragraph{Laws of (affine) bunched implications.}
 \begin{mathpar}
@@ -449,6 +451,7 @@ This is entirely standard.
   \later{(\prop * \propB)} &\provesIff& \later\prop * \later\propB
 \end{array}
 \end{mathpar}
+A type $\type$ being \emph{inhabited} means that $ \proves \wtt{\term}{\type}$ is derivable for some $\term$.
 
 \begin{mathpar}
 \infer