Commit 7a2b2c6a authored by Ralf Jung's avatar Ralf Jung

docs: fix using the fixpoint for WP

parent 253d891a
Pipeline #2782 passed with stage
in 9 minutes and 19 seconds
......@@ -196,7 +196,7 @@
\newcommand{\namecl}[1]{{#1^{\kern0.2ex\uparrow}}}
%% various pieces of Syntax
\def\MU #1.{\mu #1.\spac}%
\def\MU #1.{\mu\spac #1.\spac}%
\def\Lam #1.{\lambda #1.\spac}%
\newcommand{\proves}{\vdash}
......
......@@ -113,13 +113,13 @@ Finally, we can define the core piece of the program logic, the assertion that r
We assume that everything making up the definition of the language, \ie values, expressions, states, the conversion functions, reduction relation and all their properties, are suitably reflected into the logic (\ie they are part of the signature $\Sig$).
\begin{align*}
\textdom{pre-wp}(\textdom{wp})(\mask, \expr, \pred) \eqdef{}&
(\Exists\val. \toval(\expr) = \val \land \pvs[\mask] \prop) \lor {}\\
&\Bigl(\toval(\expr) = \bot \land \All \state. \ownGhost{\gname_{\textmon{State}}}{\authfull \state} \vsW[\mask][\emptyset] {}\\
\textdom{wp} \eqdef{}& \MU \textdom{wp}. \Lam \mask, \expr, \pred. \\
& (\Exists\val. \toval(\expr) = \val \land \pvs[\mask] \prop) \lor {}\\
& \Bigl(\toval(\expr) = \bot \land \All \state. \ownGhost{\gname_{\textmon{State}}}{\authfull \state} \vsW[\mask][\emptyset] {}\\
&\qquad \red(\expr, \state) * \later\All \expr', \state', \bar\expr. (\expr, \state \step \expr', \state', \bar\expr) \vsW[\emptyset][\mask] {}\\
&\qquad\qquad \ownGhost{\gname_{\textmon{State}}}{\authfull \state'} * \textdom{wp}(\mask, \expr', \pred) * \Sep_{\expr'' \in \bar\expr} \textdom{wp}(\top, \expr'', \Lam \any. \TRUE)\Bigr) \\
% (* value case *)
\wpre\expr[\mask]{\Ret\val. \prop} \eqdef{}& (\MU \textdom{wp}. \textdom{pre-wp}(\textdom{wp}))(\mask, \expr, \Lam\val.\prop)
\wpre\expr[\mask]{\Ret\val. \prop} \eqdef{}& \textdom{wp}(\mask, \expr, \Lam\val.\prop)
\end{align*}
If we leave away the mask, we assume it to default to $\top$.
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