Commit dbdd25ba authored by Ralf Jung's avatar Ralf Jung

docs: define plainly

parent 19649432
......@@ -63,12 +63,13 @@ Below, $\melt$ ranges over $\monoid$ and $i$ ranges over $\set{1,2}$.
%\\&
\ownM{\term} \mid \mval(\term) \mid
\always\prop \mid
\plainly\prop \mid
{\later\prop} \mid
\upd \prop
\end{align*}
Recursive predicates must be \emph{guarded}: in $\MU \var. \term$, the variable $\var$ can only appear under the later $\later$ modality.
Note that the modalities $\upd$, $\always$ and $\later$ bind more tightly than $*$, $\wand$, $\land$, $\lor$, and $\Ra$.
Note that the modalities $\upd$, $\always$, $\plainly$ and $\later$ bind more tightly than $*$, $\wand$, $\land$, $\lor$, and $\Ra$.
\paragraph{Variable conventions.}
......@@ -175,6 +176,9 @@ In writing $\vctx, x:\type$, we presuppose that $x$ is not already declared in $
\and
\infer{\vctx \proves \wtt{\prop}{\Prop}}
{\vctx \proves \wtt{\always\prop}{\Prop}}
\and
\infer{\vctx \proves \wtt{\prop}{\Prop}}
{\vctx \proves \wtt{\plainly\prop}{\Prop}}
\and
\infer{\vctx \proves \wtt{\prop}{\Prop}}
{\vctx \proves \wtt{\later\prop}{\Prop}}
......
......@@ -260,6 +260,7 @@
\newcommand{\later}{\mathop{{\triangleright}}}
\newcommand*{\lateropt}[1]{\mathop{{\later}^{#1}}}
\newcommand{\always}{\mathop{\Box}}
\newcommand{\plainly}{\mathop{\blacksquare}}
%% Invariants and Ghost ownership
% PDS: Was 0pt inner, 2pt outer.
......
......@@ -54,10 +54,11 @@ We are thus going to define the assertions as mapping CMRA elements to sets of s
\Lam \melt. \setComp{n}{\begin{aligned}
\All m, \meltB.& m \leq n \land \melt\mtimes\meltB \in \mval_m \Ra {} \\
& m \in \Sem{\vctx \proves \prop : \Prop}_\gamma(\meltB) \Ra {}\\& m \in \Sem{\vctx \proves \propB : \Prop}_\gamma(\melt\mtimes\meltB)\end{aligned}} \\
\Sem{\vctx \proves \always{\prop} : \Prop}_\gamma &\eqdef \Lam\melt. \Sem{\vctx \proves \prop : \Prop}_\gamma(\mcore\melt) \\
\Sem{\vctx \proves \later{\prop} : \Prop}_\gamma &\eqdef \Lam\melt. \setComp{n}{n = 0 \lor n-1 \in \Sem{\vctx \proves \prop : \Prop}_\gamma(\melt)}\\
\Sem{\vctx \proves \ownM{\term} : \Prop}_\gamma &\eqdef \Lam\meltB. \setComp{n}{\Sem{\vctx \proves \term : \textlog{M}}_\gamma \mincl[n] \meltB} \\
\Sem{\vctx \proves \mval(\term) : \Prop}_\gamma &\eqdef \Lam\any. \setComp{n}{\Sem{\vctx \proves \term : \textlog{M}}_\gamma \in \mval_n} \\
\Sem{\vctx \proves \always{\prop} : \Prop}_\gamma &\eqdef \Lam\melt. \Sem{\vctx \proves \prop : \Prop}_\gamma(\mcore\melt) \\
\Sem{\vctx \proves \plainly{\prop} : \Prop}_\gamma &\eqdef \Lam\melt. \Sem{\vctx \proves \prop : \Prop}_\gamma(\munit) \\
\Sem{\vctx \proves \later{\prop} : \Prop}_\gamma &\eqdef \Lam\melt. \setComp{n}{n = 0 \lor n-1 \in \Sem{\vctx \proves \prop : \Prop}_\gamma(\melt)}\\
\Sem{\vctx \proves \upd\prop : \Prop}_\gamma &\eqdef \Lam\melt. \setComp{n}{\begin{aligned}
\All m, \melt'. & m \leq n \land (\melt \mtimes \melt') \in \mval_m \Ra {}\\& \Exists \meltB. (\meltB \mtimes \melt') \in \mval_m \land m \in \Sem{\vctx \proves \prop :\Prop}_\gamma(\meltB)
\end{aligned}
......
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