docs: describe the unit of a CMRA and how the logic demands one

docs/algebra.tex

@@ -112,6 +112,16 @@ This operation is needed to prove that $\later$ commutes with existential quanti
 \end{mathpar}
 (This assumes that the type $\type$ is non-empty.)
 
+\begin{defn}
+  An element $\munit$ of a CMRA $\monoid$ is called the \emph{unit} of $\monoid$ if it satisfies the following conditions:
+  \begin{enumerate}[itemsep=0pt]
+  \item $\munit$ is valid: \\ $\All n, \munit \in \mval_n$
+  \item $\munit$ is left-identity of the operation: \\
+    $\All \melt \in M. \munit \mtimes \melt = \melt$
+  \item $\munit$ is discrete
+  \end{enumerate}
+\end{defn}
+
 \begin{defn}
   It is possible to do a \emph{frame-preserving update} from $\melt \in \monoid$ to $\meltsB \subseteq \monoid$, written $\melt \mupd \meltsB$, if
   \[ \All n, \melt_f. \melt \mtimes \melt_f \in \mval_n \Ra \Exists \meltB \in \meltsB. \meltB \mtimes \melt_f \in \mval_n \]
@@ -124,7 +134,7 @@ Note that for RAs, this and the RA-based definition of a frame-preserving update
 
 \begin{defn}
   A function $f : M \to N$ between two CMRAs is \emph{monotone} if it satisfies the following conditions:
-  \begin{enumerate}
+  \begin{enumerate}[itemsep=0pt]
   \item $f$ is non-expansive
   \item $f$ preserves validity: \\
     $\All n, x \in M. x \in \mval_n \Ra f(x) \in \mval_n$

docs/iris.sty

@@ -129,6 +129,7 @@
 
 \newcommand{\mcar}[1]{|#1|}
 \newcommand{\mcarp}[1]{\mcar{#1}^{+}}
+\newcommand{\munit}{\varepsilon}
 \newcommand{\mcore}[1]{\lfloor#1\rfloor}
 \newcommand{\mtimes}{\mathbin{\cdot}}
 \newcommand{\mdiv}{\mathbin{\div}}

docs/logic.tex

@@ -26,7 +26,7 @@ It does not matter whether they fork off an arbitrary expression.
 
 \begin{defn}[Context]
   A function $\lctx : \textdom{Expr} \to \textdom{Expr}$ is a \emph{context} if the following conditions are satisfied:
-  \begin{enumerate}
+  \begin{enumerate}[itemsep=0pt]
   \item $\lctx$ does not turn non-values into values:\\
     $\All\expr. \toval(\expr) = \bot \Ra \toval(\lctx(\expr)) = \bot $
   \item One can perform reductions below $\lctx$:\\
@@ -64,8 +64,7 @@ For any language $\Lang$, we define the corresponding thread-pool semantics.
 To instantiate Iris, you need to define the following parameters:
 \begin{itemize}
 \item A language $\Lang$
-\item A locally contractive functor $\iFunc : \COFEs \to \CMRAs$ defining the ghost state
-  \ralf{$\iFunc$ also needs to have a single-unit.}
+\item A locally contractive functor $\iFunc : \COFEs \to \CMRAs$ defining the ghost state, such that for all COFEs $A$, the CMRA $\iFunc(A)$ has a unit
 \end{itemize}
 
 \noindent
@@ -107,6 +106,7 @@ Iris syntax is built up from a signature $\Sig$ and a countably infinite set $\t
       \pi_i\; \term \mid
       \Lam \var:\type.\term \mid
       \term(\term)  \mid
+      \munit \mid
       \mcore\term \mid
       \term \mtimes \term \mid
 \\&
@@ -213,6 +213,8 @@ In writing $\vctx, x:\type$, we presuppose that $x$ is not already declared in $
 	{\vctx \proves \wtt{\term}{\type \to \type'} \and \wtt{\termB}{\type}}
 	{\vctx \proves \wtt{\term(\termB)}{\type'}}
 %%% monoids
+\and
+        \infer{}{\vctx \proves \wtt\munit{\textlog{M}}}
 \and
 	\infer{\vctx \proves \wtt\melt{\textlog{M}}}{\vctx \proves \wtt{\mcore\melt}{\textlog{M}}}
 \and
@@ -416,7 +418,7 @@ This is entirely standard.
 \begin{mathpar}
 \begin{array}{rMcMl}
 \ownGGhost{\melt} * \ownGGhost{\meltB} &\Lra&  \ownGGhost{\melt \mtimes \meltB} \\
-%\TRUE &\Ra&  \ownGGhost{\munit}\\
+\TRUE &\Ra&  \ownGGhost{\munit}\\
 \ownGGhost{\melt} &\Ra& \melt \in \mval % * \ownGGhost{\melt}
 \end{array}
 \and

docs/setup.tex

@@ -22,7 +22,7 @@
 \usepackage{xcolor}  % for print version
 
 \usepackage{graphicx}
-
+\usepackage{enumitem}
 \usepackage{semantic}
 \usepackage{csquotes}