From 0816b3de1b6a724897447df833d158e8c384e292 Mon Sep 17 00:00:00 2001
From: Ralf Jung <jung@mpi-sws.org>
Date: Sat, 12 Mar 2016 15:19:13 +0100
Subject: [PATCH] docs: typos, nits
---
docs/algebra.tex | 4 ++--
docs/constructions.tex | 32 +++++++++++++++---------------
docs/derived.tex | 44 +++++++++++++++++++++---------------------
docs/logic.tex | 26 ++++++++++++-------------
docs/model.tex | 10 +++++-----
5 files changed, 58 insertions(+), 58 deletions(-)
diff --git a/docs/algebra.tex b/docs/algebra.tex
index 6d738187e..7d18bfab1 100644
--- a/docs/algebra.tex
+++ b/docs/algebra.tex
@@ -74,7 +74,7 @@ Note that the composition of non-expansive (bi)functors is non-expansive, and th
\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
- \[ \All \melt_f. \melt \mtimes \melt_f \in \mval \Ra \Exists \meltB \in \meltsB. \meltB \mtimes \melt_f \in \mval \]
+ \[ \All \melt_\f. \melt \mtimes \melt_\f \in \mval \Ra \Exists \meltB \in \meltsB. \meltB \mtimes \melt_\f \in \mval \]
We further define $\melt \mupd \meltB \eqdef \melt \mupd \set\meltB$.
\end{defn}
@@ -146,7 +146,7 @@ This operation is needed to prove that $\later$ commutes with existential quanti
\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
- \[ \All n, \melt_f. \melt \mtimes \melt_f \in \mval_n \Ra \Exists \meltB \in \meltsB. \meltB \mtimes \melt_f \in \mval_n \]
+ \[ \All n, \melt_\f. \melt \mtimes \melt_\f \in \mval_n \Ra \Exists \meltB \in \meltsB. \meltB \mtimes \melt_\f \in \mval_n \]
We further define $\melt \mupd \meltB \eqdef \melt \mupd \set\meltB$.
\end{defn}
diff --git a/docs/constructions.tex b/docs/constructions.tex
index d5849d53e..dc3b643b3 100644
--- a/docs/constructions.tex
+++ b/docs/constructions.tex
@@ -195,16 +195,16 @@ We obtain the following frame-preserving update:
% \end{mathpar}
% \begin{proof}[Proof of \ruleref{PowFinUpd}]
-% Assume some frame $\melt_f \sep \emptyset$. Since $\melt_f$ is finite and $X$ is infinite, there exists an $x \notin \melt_f$.
+% Assume some frame $\melt_\f \sep \emptyset$. Since $\melt_\f$ is finite and $X$ is infinite, there exists an $x \notin \melt_\f$.
% Pick that for the result.
% \end{proof}
% The powerset monoids is cancellative.
% \begin{proof}[Proof of cancellativity]
-% Let $\melt_f \mtimes \melt = \melt_f \mtimes \meltB \neq \mzero$.
-% So we have $\melt_f \sep \melt$ and $\melt_f \sep \meltB$, and we have to show $\melt = \meltB$.
-% Assume $x \in \melt$. Hence $x \in \melt_f \mtimes \melt$ and thus $x \in \melt_f \mtimes \meltB$.
-% By disjointness, $x \notin \melt_f$ and hence $x \in meltB$.
+% Let $\melt_\f \mtimes \melt = \melt_\f \mtimes \meltB \neq \mzero$.
+% So we have $\melt_\f \sep \melt$ and $\melt_\f \sep \meltB$, and we have to show $\melt = \meltB$.
+% Assume $x \in \melt$. Hence $x \in \melt_\f \mtimes \melt$ and thus $x \in \melt_\f \mtimes \meltB$.
+% By disjointness, $x \notin \melt_\f$ and hence $x \in meltB$.
% The other direction works the same way.
% \end{proof}
@@ -238,20 +238,20 @@ We obtain the following frame-preserving update:
% \begin{proof}[Proof of \ruleref{FracUpdLocal}]
% Assume some $f \sep (q, a)$. If $f = \munit$, then $f \sep (q, b)$ is trivial for any $b \in B$. Just pick the one we obtain by choosing $\munit_M$ as the frame for $a$.
-% In the interesting case, we have $f = (q_f, a_f)$.
-% Obtain $b$ such that $b \in B \land b \sep a_f$.
+% In the interesting case, we have $f = (q_\f, a_\f)$.
+% Obtain $b$ such that $b \in B \land b \sep a_\f$.
% Then $(q, b) \sep f$, and we are done.
% \end{proof}
% $\fracm{M}$ is cancellative if $M$ is cancellative.
% \begin{proof}[Proof of cancellativitiy]
-% If $\melt_f = \munit$, we are trivially done.
-% So let $\melt_f = (q_f, \melt_f')$.
+% If $\melt_\f = \munit$, we are trivially done.
+% So let $\melt_\f = (q_\f, \melt_\f')$.
% If $\melt = \munit$, then $\meltB = \munit$ as otherwise the fractions could not match up.
% Again, we are trivially done.
% Similar so for $\meltB = \munit$.
% So let $\melt = (q_a, \melt')$ and $\meltB = (q_b, \meltB')$.
-% We have $(q_f + q_a, \melt_f' \mtimes \melt') = (q_f + q_b, \melt_f' \mtimes \meltB')$.
+% We have $(q_\f + q_a, \melt_\f' \mtimes \melt') = (q_\f + q_b, \melt_\f' \mtimes \meltB')$.
% We have to show $q_a = q_b$ and $\melt' = \meltB'$.
% The first is trivial, the second follows from cancellativitiy of $M$.
% \end{proof}
@@ -312,7 +312,7 @@ We obtain the following frame-preserving update:
% The frame-preserving update involves a rather unwieldy side-condition:
% \begin{mathpar}
% \inferH{AuthUpd}{
-% \All\melt_f\in\mcar{\monoid}. \melt\sep\meltB \land \melt\mtimes\melt_f \le \meltB\mtimes\melt_f \Ra \melt'\mtimes\melt_f \le \melt'\mtimes\meltB \and
+% \All\melt_\f\in\mcar{\monoid}. \melt\sep\meltB \land \melt\mtimes\melt_\f \le \meltB\mtimes\melt_\f \Ra \melt'\mtimes\melt_\f \le \melt'\mtimes\meltB \and
% \melt' \sep \meltB
% }{
% \authfull \melt \mtimes \meltB, \authfrag \melt \mupd \authfull \melt' \mtimes \meltB, \authfrag \melt'
@@ -392,11 +392,11 @@ We obtain the following frame-preserving update:
% We have
% \begin{quote}
% If $(s, T) \ra (s', T')$\\
-% and $T_f \sep (T \uplus \STSL(s))$,\\
-% then $\textsf{frame}(s, T_f) \ra \textsf{frame}(s', T_f)$.
+% and $T_\f \sep (T \uplus \STSL(s))$,\\
+% then $\textsf{frame}(s, T_\f) \ra \textsf{frame}(s', T_\f)$.
% \end{quote}
% \begin{proof}
-% This follows directly by framing the tokens in $\STST \setminus (T_f \uplus T \uplus \STSL(s))$ around the given transition, which yields $(s, \STST \setminus (T_f \uplus \STSL{T}(s))) \ra (s', T' \uplus (\STST \setminus (T_f \uplus T \uplus \STSL{T}(s))))$.
+% This follows directly by framing the tokens in $\STST \setminus (T_\f \uplus T \uplus \STSL(s))$ around the given transition, which yields $(s, \STST \setminus (T_\f \uplus \STSL{T}(s))) \ra (s', T' \uplus (\STST \setminus (T_\f \uplus T \uplus \STSL{T}(s))))$.
% This is exactly what we have to show, since we know $\STSL(s) \uplus T = \STSL(s') \uplus T'$.
% \end{proof}
@@ -418,8 +418,8 @@ We obtain the following frame-preserving update:
% {(s, S, T) \mupd (s', \upclose(\{s'\}, T'), T')}
% \end{mathpar}
% \begin{proof}[Proof of \ruleref{StsStep}]
-% Assume some upwards-closed $S_f, T_f$ (the frame cannot be authoritative) s.t.\ $s \in S_f$ and $T_f \sep (T \uplus \STSL(s))$. We have to show that this frame combines with our final monoid element, which is the case if $s' \in S_f$ and $T_f \sep T'$.
-% By upward-closedness, it suffices to show $\textsf{frame}(s, T_f) \ststrans \textsf{frame}(s', T_f)$.
+% Assume some upwards-closed $S_\f, T_\f$ (the frame cannot be authoritative) s.t.\ $s \in S_\f$ and $T_\f \sep (T \uplus \STSL(s))$. We have to show that this frame combines with our final monoid element, which is the case if $s' \in S_\f$ and $T_\f \sep T'$.
+% By upward-closedness, it suffices to show $\textsf{frame}(s, T_\f) \ststrans \textsf{frame}(s', T_\f)$.
% This follows by induction on the path $(s, T) \ststrans (s', T')$, and using the lemma proven above for each step.
% \end{proof}
diff --git a/docs/derived.tex b/docs/derived.tex
index 34b94a536..55a0c06fe 100644
--- a/docs/derived.tex
+++ b/docs/derived.tex
@@ -210,20 +210,20 @@ We can derive some specialized forms of the lifting axioms for the operational s
\infer[wp-lift-atomic-step]
{\atomic(\expr_1) \and
\red(\expr_1, \state_1) \and
- \All \expr_2, \state_2, \expr_f. \expr_1,\state_1 \step \expr_2,\state_2,\expr_f \Ra \pred(\expr_2,\state_2,\expr_f)}
- {\later\ownPhys{\state_1} * \later\All \val_2, \state_2, \expr_f. \pred(\ofval(\val), \state_2, \expr_f) \land \ownPhys{\state_2} \wand \prop[\val_2/\var] * \wpre{\expr_f}[\top]{\Ret\any.\TRUE} \proves \wpre{\expr_1}[\mask_1]{\Ret\var.\prop}}
+ \All \expr_2, \state_2, \expr_\f. \expr_1,\state_1 \step \expr_2,\state_2,\expr_\f \Ra \pred(\expr_2,\state_2,\expr_\f)}
+ {\later\ownPhys{\state_1} * \later\All \val_2, \state_2, \expr_\f. \pred(\ofval(\val), \state_2, \expr_\f) \land \ownPhys{\state_2} \wand \prop[\val_2/\var] * \wpre{\expr_\f}[\top]{\Ret\any.\TRUE} \proves \wpre{\expr_1}[\mask_1]{\Ret\var.\prop}}
\infer[wp-lift-atomic-det-step]
{\atomic(\expr_1) \and
\red(\expr_1, \state_1) \and
- \All \expr'_2, \state'_2, \expr_f'. \expr_1,\state_1 \step \expr_2,\state_2,\expr_f \Ra \state_2 = \state_2' \land \toval(\expr_2') = \val_2 \land \expr_f = \expr_f'}
- {\later\ownPhys{\state_1} * \later(\ownPhys{\state_2} \wand \prop[\val_2/\var] * \wpre{\expr_f}[\top]{\Ret\any.\TRUE}) \proves \wpre{\expr_1}[\mask_1]{\Ret\var.\prop}}
+ \All \expr'_2, \state'_2, \expr_\f'. \expr_1,\state_1 \step \expr_2,\state_2,\expr_\f \Ra \state_2 = \state_2' \land \toval(\expr_2') = \val_2 \land \expr_\f = \expr_\f'}
+ {\later\ownPhys{\state_1} * \later(\ownPhys{\state_2} \wand \prop[\val_2/\var] * \wpre{\expr_\f}[\top]{\Ret\any.\TRUE}) \proves \wpre{\expr_1}[\mask_1]{\Ret\var.\prop}}
\infer[wp-lift-pure-det-step]
{\toval(\expr_1) = \bot \and
\All \state_1. \red(\expr_1, \state_1) \and
- \All \state_1, \expr_2', \state_2, \expr_f'. \expr_1,\state_1 \step \expr_2,\state_2,\expr_f \Ra \state_1 = \state_2 \land \expr_2 = \expr_2' \land \expr_f = \expr_f'}
- {\later ( \wpre{\expr_2}[\mask_1]{\Ret\var.\prop} * \wpre{\expr_f}[\top]{\Ret\any.\TRUE}) \proves \wpre{\expr_1}[\mask_1]{\Ret\var.\prop}}
+ \All \state_1, \expr_2', \state_2, \expr_\f'. \expr_1,\state_1 \step \expr_2,\state_2,\expr_\f \Ra \state_1 = \state_2 \land \expr_2 = \expr_2' \land \expr_\f = \expr_\f'}
+ {\later ( \wpre{\expr_2}[\mask_1]{\Ret\var.\prop} * \wpre{\expr_\f}[\top]{\Ret\any.\TRUE}) \proves \wpre{\expr_1}[\mask_1]{\Ret\var.\prop}}
\end{mathparpagebreakable}
Furthermore, we derive some forms that directly involve view shifts and Hoare triples.
@@ -232,35 +232,35 @@ Furthermore, we derive some forms that directly involve view shifts and Hoare tr
{\mask_2 \subseteq \mask_1 \and
\toval(\expr_1) = \bot \and
\red(\expr_1, \state_1) \and
- \All \expr_2, \state_2, \expr_f. \expr_1,\state_1 \step \expr_2,\state_2,\expr_f \Ra \pred(\expr_2,\state_2,\expr_f) \\\\
+ \All \expr_2, \state_2, \expr_\f. \expr_1,\state_1 \step \expr_2,\state_2,\expr_\f \Ra \pred(\expr_2,\state_2,\expr_\f) \\\\
\prop \vs[\mask_1][\mask_2] \later\ownPhys{\state_1} * \later\prop' \and
- \All \expr_2, \state_2, \expr_f. \pred(\expr_2, \state_2, \expr_f) * \ownPhys{\state_2} * \prop' \vs[\mask_2][\mask_1] \propB_1 * \propB_2 \\\\
- \All \expr_2, \state_2, \expr_f. \hoare{\propB_1}{\expr_2}{\Ret\val.\propC}[\mask_1] \and
- \All \expr_2, \state_2, \expr_f. \hoare{\propB_2}{\expr_f}{\Ret\any. \TRUE}[\top]}
+ \All \expr_2, \state_2, \expr_\f. \pred(\expr_2, \state_2, \expr_\f) * \ownPhys{\state_2} * \prop' \vs[\mask_2][\mask_1] \propB_1 * \propB_2 \\\\
+ \All \expr_2, \state_2, \expr_\f. \hoare{\propB_1}{\expr_2}{\Ret\val.\propC}[\mask_1] \and
+ \All \expr_2, \state_2, \expr_\f. \hoare{\propB_2}{\expr_\f}{\Ret\any. \TRUE}[\top]}
{ \hoare\prop{\expr_1}{\Ret\val.\propC}[\mask_1] }
\infer[ht-lift-atomic-step]
{\atomic(\expr_1) \and
\red(\expr_1, \state_1) \and
- \All \expr_2, \state_2, \expr_f. \expr_1,\state_1 \step \expr_2,\state_2,\expr_f \Ra \pred(\expr_2,\state_2,\expr_f) \\\\
+ \All \expr_2, \state_2, \expr_\f. \expr_1,\state_1 \step \expr_2,\state_2,\expr_\f \Ra \pred(\expr_2,\state_2,\expr_\f) \\\\
\prop \vs[\mask_1][\mask_2] \later\ownPhys{\state_1} * \later\prop' \and
- \All \expr_2, \state_2, \expr_f. \hoare{\pred(\expr_2,\state_2,\expr_f) * \prop}{\expr_f}{\Ret\any. \TRUE}[\top]}
- { \hoare{\later\ownPhys{\state_1} * \later\prop}{\expr_1}{\Ret\val.\Exists \state_2, \expr_f. \ownPhys{\state_2} * \pred(\ofval(\expr_2),\state_2,\expr_f)}[\mask_1] }
+ \All \expr_2, \state_2, \expr_\f. \hoare{\pred(\expr_2,\state_2,\expr_\f) * \prop}{\expr_\f}{\Ret\any. \TRUE}[\top]}
+ { \hoare{\later\ownPhys{\state_1} * \later\prop}{\expr_1}{\Ret\val.\Exists \state_2, \expr_\f. \ownPhys{\state_2} * \pred(\ofval(\expr_2),\state_2,\expr_\f)}[\mask_1] }
\infer[ht-lift-pure-step]
{\toval(\expr_1) = \bot \and
\All\state_1. \red(\expr_1, \state_1) \and
- \All \state_1, \expr_2, \state_2, \expr_f. \expr_1,\state_1 \step \expr_2,\state_2,\expr_f \Ra \state_1 = \state_2 \land \pred(\expr_2,\expr_f) \\\\
- \All \expr_2, \expr_f. \hoare{\pred(\expr_2,\expr_f) * \prop}{\expr_2}{\Ret\val.\propB}[\mask_1] \and
- \All \expr_2, \expr_f. \hoare{\pred(\expr_2,\expr_f) * \prop'}{\expr_f}{\Ret\any. \TRUE}[\top]}
+ \All \state_1, \expr_2, \state_2, \expr_\f. \expr_1,\state_1 \step \expr_2,\state_2,\expr_\f \Ra \state_1 = \state_2 \land \pred(\expr_2,\expr_\f) \\\\
+ \All \expr_2, \expr_\f. \hoare{\pred(\expr_2,\expr_\f) * \prop}{\expr_2}{\Ret\val.\propB}[\mask_1] \and
+ \All \expr_2, \expr_\f. \hoare{\pred(\expr_2,\expr_\f) * \prop'}{\expr_\f}{\Ret\any. \TRUE}[\top]}
{ \hoare{\later(\prop*\prop')}{\expr_1}{\Ret\val.\propB}[\mask_1] }
\infer[ht-lift-pure-det-step]
{\toval(\expr_1) = \bot \and
\All\state_1. \red(\expr_1, \state_1) \and
- \All \state_1, \expr_2', \state_2, \expr_f'. \expr_1,\state_1 \step \expr_2,\state_2,\expr_f \Ra \state_1 = \state_2 \land \expr_2 = \expr_2' \land \expr_f = \expr_f' \\\\
+ \All \state_1, \expr_2', \state_2, \expr_\f'. \expr_1,\state_1 \step \expr_2,\state_2,\expr_\f \Ra \state_1 = \state_2 \land \expr_2 = \expr_2' \land \expr_\f = \expr_\f' \\\\
\hoare{\prop}{\expr_2}{\Ret\val.\propB}[\mask_1] \and
- \hoare{\prop'}{\expr_f}{\Ret\any. \TRUE}[\top]}
+ \hoare{\prop'}{\expr_\f}{\Ret\any. \TRUE}[\top]}
{ \hoare{\later(\prop*\prop')}{\expr_1}{\Ret\val.\propB}[\mask_1] }
\end{mathparpagebreakable}
@@ -438,21 +438,21 @@ We can now derive the following rules for this derived form of the invariant ass
% {\later\pred_\bot(a) \vs[\mask] \exists \iname \in \mask, \gname.\; \Auth(M, \pred, \gname, \iname) * \ownGhost{\gname}{\authfrag a : \auth{M}}}
% \and
% \axiomH{AuthOpen}
-% {\Auth(M, \pred, \gname, \iname) \vdash \ownGhost{\gname}{\authfrag \melt : \auth{M}} \vsE[\{\iname\}][\emptyset] \exists \melt_f.\; \later\pred_\bot(\melt \mtimes \melt_f) * \ownGhost{\gname}{\authfull \melt \mtimes \melt_f, \authfrag a:\auth{M}}}
+% {\Auth(M, \pred, \gname, \iname) \vdash \ownGhost{\gname}{\authfrag \melt : \auth{M}} \vsE[\{\iname\}][\emptyset] \exists \melt_\f.\; \later\pred_\bot(\melt \mtimes \melt_\f) * \ownGhost{\gname}{\authfull \melt \mtimes \melt_\f, \authfrag a:\auth{M}}}
% \and
% \axiomH{AuthClose}
-% {\Auth(M, \pred, \gname, \iname) \vdash \later\pred_\bot(\meltB \mtimes \melt_f) * \ownGhost{\gname}{\authfull a \mtimes \melt_f, \authfrag a:\auth{M}} \vs[\emptyset][\{\iname\}] \ownGhost{\gname}{\authfrag \meltB : \auth{M}} }
+% {\Auth(M, \pred, \gname, \iname) \vdash \later\pred_\bot(\meltB \mtimes \melt_\f) * \ownGhost{\gname}{\authfull a \mtimes \melt_\f, \authfrag a:\auth{M}} \vs[\emptyset][\{\iname\}] \ownGhost{\gname}{\authfrag \meltB : \auth{M}} }
% \end{mathpar}
% These view shifts in turn can be used to prove variants of the invariant rules:
% \begin{mathpar}
% \inferH{Auth}
-% {\forall \melt_f.\; \hoare{\later\pred_\bot(a \mtimes \melt_f) * P}{\expr}{\Ret\val. \exists \meltB.\; \later\pred_\bot(\meltB\mtimes \melt_f) * Q}[\mask]
+% {\forall \melt_\f.\; \hoare{\later\pred_\bot(a \mtimes \melt_\f) * P}{\expr}{\Ret\val. \exists \meltB.\; \later\pred_\bot(\meltB\mtimes \melt_\f) * Q}[\mask]
% \and \physatomic{\expr}}
% {\Auth(M, \pred, \gname, \iname) \vdash \hoare{\ownGhost{\gname}{\authfrag a:\auth{M}} * P}{\expr}{\Ret\val. \exists \meltB.\; \ownGhost{\gname}{\authfrag \meltB:\auth{M}} * Q}[\mask \uplus \{\iname\}]}
% \and
% \inferH{VSAuth}
-% {\forall \melt_f.\; \later\pred_\bot(a \mtimes \melt_f) * P \vs[\mask_1][\mask_2] \exists \meltB.\; \later\pred_\bot(\meltB \mtimes \melt_f) * Q(\meltB)}
+% {\forall \melt_\f.\; \later\pred_\bot(a \mtimes \melt_\f) * P \vs[\mask_1][\mask_2] \exists \meltB.\; \later\pred_\bot(\meltB \mtimes \melt_\f) * Q(\meltB)}
% {\Auth(M, \pred, \gname, \iname) \vdash
% \ownGhost{\gname}{\authfrag a:\auth{M}} * P \vs[\mask_1 \uplus \{\iname\}][\mask_2 \uplus \{\iname\}]
% \exists \meltB.\; \ownGhost{\gname}{\authfrag \meltB:\auth{M}} * Q(\meltB)}
diff --git a/docs/logic.tex b/docs/logic.tex
index 2c8a49da5..53ba97a9b 100644
--- a/docs/logic.tex
+++ b/docs/logic.tex
@@ -7,7 +7,7 @@ A \emph{language} $\Lang$ consists of a set \textdom{Expr} of \emph{expressions}
\end{mathpar}
\item There exists a \emph{primitive reduction relation} \[(-,- \step -,-,-) \subseteq \textdom{Expr} \times \textdom{State} \times \textdom{Expr} \times \textdom{State} \times (\textdom{Expr} \uplus \set{\bot})\]
We will write $\expr_1, \state_1 \step \expr_2, \state_2$ for $\expr_1, \state_1 \step \expr_2, \state_2, \bot$. \\
- A reduction $\expr_1, \state_1 \step \expr_2, \state_2, \expr_f$ indicates that, when $\expr_1$ reduces to $\expr$, a \emph{new thread} $\expr_f$ is forked off.
+ A reduction $\expr_1, \state_1 \step \expr_2, \state_2, \expr_\f$ indicates that, when $\expr_1$ reduces to $\expr$, a \emph{new thread} $\expr_\f$ is forked off.
\item All values are stuck:
\[ \expr, \_ \step \_, \_, \_ \Ra \toval(\expr) = \bot \]
\item There is a predicate defining \emph{atomic} expressions satisfying
@@ -16,7 +16,7 @@ A \emph{language} $\Lang$ consists of a set \textdom{Expr} of \emph{expressions}
{\All\expr. \atomic(\expr) \Ra \toval(\expr) = \bot} \and
{{
\begin{inbox}
-\All\expr_1, \state_1, \expr_2, \state_2, \expr_f. \atomic(\expr_1) \land \expr_1, \state_1 \step \expr_2, \state_2, \expr_f \Ra {}\\\qquad\qquad\qquad\quad~~ \Exists \val_2. \toval(\expr_2) = \val_2
+\All\expr_1, \state_1, \expr_2, \state_2, \expr_\f. \atomic(\expr_1) \land \expr_1, \state_1 \step \expr_2, \state_2, \expr_\f \Ra {}\\\qquad\qquad\qquad\quad~~ \Exists \val_2. \toval(\expr_2) = \val_2
\end{inbox}
}}
\end{mathpar}
@@ -26,7 +26,7 @@ It does not matter whether they fork off an arbitrary expression.
\begin{defn}
An expression $\expr$ and state $\state$ are \emph{reducible} (written $\red(\expr, \state)$) if
- \[ \Exists \expr_2, \state_2, \expr_f. \expr,\state \step \expr_2,\state_2,\expr_f \]
+ \[ \Exists \expr_2, \state_2, \expr_\f. \expr,\state \step \expr_2,\state_2,\expr_\f \]
\end{defn}
\begin{defn}[Context]
@@ -35,9 +35,9 @@ It does not matter whether they fork off an arbitrary expression.
\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 $\lctx$:\\
- $\All \expr_1, \state_1, \expr_2, \state_2, \expr_f. \expr_1, \state_1 \step \expr_2,\state_2,\expr_f \Ra \lctx(\expr_1), \state_1 \step \lctx(\expr_2),\state_2,\expr_f $
+ $\All \expr_1, \state_1, \expr_2, \state_2, \expr_\f. \expr_1, \state_1 \step \expr_2,\state_2,\expr_\f \Ra \lctx(\expr_1), \state_1 \step \lctx(\expr_2),\state_2,\expr_\f $
\item Reductions stay below $\lctx$ until there is a value in the hole:\\
- $\All \expr_1', \state_1, \expr_2, \state_2, \expr_f. \toval(\expr_1') = \bot \land \lctx(\expr_1'), \state_1 \step \expr_2,\state_2,\expr_f \Ra \Exists\expr_2'. \expr_2 = \lctx(\expr_2') \land \expr_1', \state_1 \step \expr_2',\state_2,\expr_f $
+ $\All \expr_1', \state_1, \expr_2, \state_2, \expr_\f. \toval(\expr_1') = \bot \land \lctx(\expr_1'), \state_1 \step \expr_2,\state_2,\expr_\f \Ra \Exists\expr_2'. \expr_2 = \lctx(\expr_2') \land \expr_1', \state_1 \step \expr_2',\state_2,\expr_\f $
\end{enumerate}
\end{defn}
@@ -54,9 +54,9 @@ For any language $\Lang$, we define the corresponding thread-pool semantics.
\cfg{\tpool'}{\state'}}
\begin{mathpar}
\infer
- {\expr_1, \state_1 \step \expr_2, \state_2, \expr_f \and \expr_f \neq \bot}
+ {\expr_1, \state_1 \step \expr_2, \state_2, \expr_\f \and \expr_\f \neq \bot}
{\cfg{\tpool \dplus [\expr_1] \dplus \tpool'}{\state} \step
- \cfg{\tpool \dplus [\expr_2] \dplus \tpool' \dplus [\expr_f]}{\state'}}
+ \cfg{\tpool \dplus [\expr_2] \dplus \tpool' \dplus [\expr_\f]}{\state'}}
\and\infer
{\expr_1, \state_1 \step \expr_2, \state_2}
{\cfg{\tpool \dplus [\expr_1] \dplus \tpool'}{\state} \step
@@ -536,7 +536,7 @@ This is entirely standard.
{\pvs[\mask_1][\mask_2] \pvs[\mask_2][\mask_3] \prop \proves \pvs[\mask_1][\mask_3] \prop}
\infer[pvs-mask-frame]
-{}{\pvs[\mask_1][\mask_2] \prop \proves \pvs[\mask_1 \uplus \mask_f][\mask_2 \uplus \mask_f] \prop}
+{}{\pvs[\mask_1][\mask_2] \prop \proves \pvs[\mask_1 \uplus \mask_\f][\mask_2 \uplus \mask_\f] \prop}
\infer[pvs-frame]
{}{\propB * \pvs[\mask_1][\mask_2]\prop \proves \pvs[\mask_1][\mask_2] \propB * \prop}
@@ -594,19 +594,19 @@ This is entirely standard.
{\mask_2 \subseteq \mask_1 \and
\toval(\expr_1) = \bot \and
\red(\expr_1, \state_1) \and
- \All \expr_2, \state_2, \expr_f. \expr_1,\state_1 \step \expr_2,\state_2,\expr_f \Ra \pred(\expr_2,\state_2,\expr_f)}
+ \All \expr_2, \state_2, \expr_\f. \expr_1,\state_1 \step \expr_2,\state_2,\expr_\f \Ra \pred(\expr_2,\state_2,\expr_\f)}
{ {\begin{inbox} % for some crazy reason, LaTeX is actually sensitive to the space between the "{ {" here and the "} }" below...
- ~~\pvs[\mask_1][\mask_2] \later\ownPhys{\state_1} * \later\All \expr_2, \state_2, \expr_f. \pred(\expr_2, \state_2, \expr_f) \land {}\\\qquad\qquad\qquad\qquad\qquad \ownPhys{\state_2} \wand \pvs[\mask_2][\mask_1] \wpre{\expr_2}[\mask_1]{\Ret\var.\prop} * \wpre{\expr_f}[\top]{\Ret\any.\TRUE} {}\\\proves \wpre{\expr_1}[\mask_1]{\Ret\var.\prop}
+ ~~\pvs[\mask_1][\mask_2] \later\ownPhys{\state_1} * \later\All \expr_2, \state_2, \expr_\f. \pred(\expr_2, \state_2, \expr_\f) \land {}\\\qquad\qquad\qquad\qquad\qquad \ownPhys{\state_2} \wand \pvs[\mask_2][\mask_1] \wpre{\expr_2}[\mask_1]{\Ret\var.\prop} * \wpre{\expr_\f}[\top]{\Ret\any.\TRUE} {}\\\proves \wpre{\expr_1}[\mask_1]{\Ret\var.\prop}
\end{inbox}} }
\infer[wp-lift-pure-step]
{\toval(\expr_1) = \bot \and
\All \state_1. \red(\expr_1, \state_1) \and
- \All \state_1, \expr_2, \state_2, \expr_f. \expr_1,\state_1 \step \expr_2,\state_2,\expr_f \Ra \state_1 = \state_2 \land \pred(\expr_2,\expr_f)}
- {\later\All \expr_2, \expr_f. \pred(\expr_2, \expr_f) \Ra \wpre{\expr_2}[\mask_1]{\Ret\var.\prop} * \wpre{\expr_f}[\top]{\Ret\any.\TRUE} \proves \wpre{\expr_1}[\mask_1]{\Ret\var.\prop}}
+ \All \state_1, \expr_2, \state_2, \expr_\f. \expr_1,\state_1 \step \expr_2,\state_2,\expr_\f \Ra \state_1 = \state_2 \land \pred(\expr_2,\expr_\f)}
+ {\later\All \expr_2, \expr_\f. \pred(\expr_2, \expr_\f) \Ra \wpre{\expr_2}[\mask_1]{\Ret\var.\prop} * \wpre{\expr_\f}[\top]{\Ret\any.\TRUE} \proves \wpre{\expr_1}[\mask_1]{\Ret\var.\prop}}
\end{mathpar}
-Here we define $\wpre{\expr_f}[\mask]{\Ret\var.\prop} \eqdef \TRUE$ if $\expr_f = \bot$ (remember that our stepping relation can, but does not have to, define a forked-off expression).
+Here we define $\wpre{\expr_\f}[\mask]{\Ret\var.\prop} \eqdef \TRUE$ if $\expr_\f = \bot$ (remember that our stepping relation can, but does not have to, define a forked-off expression).
\subsection{Adequacy}
diff --git a/docs/model.tex b/docs/model.tex
index d1af61a1e..ea2a89bdc 100644
--- a/docs/model.tex
+++ b/docs/model.tex
@@ -76,7 +76,7 @@ We then pick $\iProp$ as the interpretation of $\Prop$:
\paragraph{Interpretation of assertions.}
$\iProp$ is a $\UPred$, and hence the definitions from \Sref{sec:upred-logic} apply.
-We only have to define the missing connectives, the most interesting bits being are primitive view shifts and weakest preconditions.
+We only have to define the missing connectives, the most interesting bits being primitive view shifts and weakest preconditions.
\typedsection{World satisfaction}{\wsat{-}{-}{-} :
\Delta\textdom{State} \times
@@ -91,20 +91,20 @@ We only have to define the missing connectives, the most interesting bits being
\wsat{\state}{\mask}{\rs} &\eqdef \set{0}\cup\setComp{n+1}{\Exists \rss : \mathbb{N} \fpfn \textdom{Res}. \wsatpre(n, \mask, \state, \rss, \rs \mtimes \prod_\iname \rss(\iname))}
\end{align*}
-\typedsection{Primitive view-shift}{\mathit{pvs} : \Delta(\pset{\mathbb{N}}) \times \Delta(\pset{\mathbb{N}}) \times \iProp \nfn \iProp}
+\typedsection{Primitive view-shift}{\mathit{pvs}_{-}^{-}(-) : \Delta(\pset{\mathbb{N}}) \times \Delta(\pset{\mathbb{N}}) \times \iProp \nfn \iProp}
\begin{align*}
\mathit{pvs}_{\mask_1}^{\mask_2}(\prop) &= \Lam \rs. \setComp{n}{\begin{aligned}
- \All \rs_\f, m, \mask_\f, \state.& 0 < m \leq n \land (\mask_1 \cup \mask_2) \sep \mask_f \land k \in \wsat\state{\mask_1 \cup \mask_\f}{\rs \mtimes \rs_\f} \Ra {}\\&
+ \All \rs_\f, m, \mask_\f, \state.& 0 < m \leq n \land (\mask_1 \cup \mask_2) \sep \mask_\f \land k \in \wsat\state{\mask_1 \cup \mask_\f}{\rs \mtimes \rs_\f} \Ra {}\\&
\Exists \rsB. k \in \prop(\rsB) \land k \in \wsat\state{\mask_2 \cup \mask_\f}{\rsB \mtimes \rs_\f}
\end{aligned}}
\end{align*}
-\typedsection{Weakest precondition}{\mathit{wp} : \Delta(\pset{\mathbb{N}}) \times \Delta(\textdom{Exp}) \times (\Delta(\textdom{Val}) \nfn \iProp) \nfn \iProp}
+\typedsection{Weakest precondition}{\mathit{wp}_{-}(-, -) : \Delta(\pset{\mathbb{N}}) \times \Delta(\textdom{Exp}) \times (\Delta(\textdom{Val}) \nfn \iProp) \nfn \iProp}
$\textdom{wp}$ is defined as the fixed-point of a contractive function.
\begin{align*}
\textdom{pre-wp}(\textdom{wp})(\mask, \expr, \pred) &\eqdef \Lam\rs. \setComp{n}{\begin{aligned}
- \All &\rs_\f, m, \mask_f, \state. 0 \leq m < n \land \mask \sep \mask_\f \land m+1 \in \wsat\state{\mask \cup \mask_\f}{\rs \mtimes \rs_\f} \Ra {}\\
+ \All &\rs_\f, m, \mask_\f, \state. 0 \leq m < n \land \mask \sep \mask_\f \land m+1 \in \wsat\state{\mask \cup \mask_\f}{\rs \mtimes \rs_\f} \Ra {}\\
&(\All\val. \toval(\expr) = \val \Ra \Exists \rsB. m+1 \in \prop(\rs') \land m+1 \in \wsat\state{\mask \cup \mask_\f}{\rs' \mtimes \rs_\f}) \land {}\\
&(\toval(\expr) = \bot \land 0 < m \Ra \red(\expr, \state) \land \All \expr_2, \state_2, \expr_\f. \expr,\state \step \expr_2,\state_2,\expr_\f \Ra {}\\
&\qquad \Exists \rsB_1, \rsB_2. m \in \wsat\state{\mask \cup \mask_\f}{\rs' \mtimes \rs_\f} \land m \in \textdom{wp}(\mask, \expr_2, \pred)(\rsB_1) \land {}&\\
--
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