### docs: fix some typos

parent 98e51974
 ... @@ -45,7 +45,7 @@ The reason that contractive functions are interesting is that for every contract ... @@ -45,7 +45,7 @@ The reason that contractive functions are interesting is that for every contract The category $\COFEs$ consists of COFEs as objects, and non-expansive functions as arrows. The category $\COFEs$ consists of COFEs as objects, and non-expansive functions as arrows. \end{defn} \end{defn} Note that $\COFEs$ is cartesian closed: Note that $\COFEs$ is cartesian closed. In particular: \begin{defn} \begin{defn} Given two COFEs $\cofe$ and $\cofeB$, the set of non-expansive functions $\set{f : \cofe \nfn \cofeB}$ is itself a COFE with Given two COFEs $\cofe$ and $\cofeB$, the set of non-expansive functions $\set{f : \cofe \nfn \cofeB}$ is itself a COFE with \begin{align*} \begin{align*} ... ...
 ... @@ -91,7 +91,7 @@ Crucially, the second rule allows us to \emph{swap} the side'' of the sum that ... @@ -91,7 +91,7 @@ Crucially, the second rule allows us to \emph{swap} the side'' of the sum that \subsection{Finite partial function} \subsection{Finite partial function} \label{sec:fpfnm} \label{sec:fpfnm} Given some countable $K$ and some CMRA $\monoid$, the set of finite partial functions $K \fpfn \monoid$ is equipped with a COFE and CMRA structure by lifting everything pointwise. Given some infinite countable $K$ and some CMRA $\monoid$, the set of finite partial functions $K \fpfn \monoid$ is equipped with a COFE and CMRA structure by lifting everything pointwise. We obtain the following frame-preserving updates: We obtain the following frame-preserving updates: \begin{mathpar} \begin{mathpar} ... @@ -149,7 +149,7 @@ There are no interesting frame-preserving updates for $\agm(\cofe)$, but we can ... @@ -149,7 +149,7 @@ There are no interesting frame-preserving updates for $\agm(\cofe)$, but we can \subsection{Exclusive CMRA} \subsection{Exclusive CMRA} Given a cofe $\cofe$, we define a CMRA $\exm(\cofe)$ such that at most one $x \in \cofe$ can be owned: Given a COFE $\cofe$, we define a CMRA $\exm(\cofe)$ such that at most one $x \in \cofe$ can be owned: \begin{align*} \begin{align*} \exm(\cofe) \eqdef{}& \exinj(\cofe) + \bot \\ \exm(\cofe) \eqdef{}& \exinj(\cofe) + \bot \\ \mval_n \eqdef{}& \setComp{\melt\in\exm(\cofe)}{\melt \neq \bot} \mval_n \eqdef{}& \setComp{\melt\in\exm(\cofe)}{\melt \neq \bot} ... @@ -375,7 +375,7 @@ We obtain the following frame-preserving update: ... @@ -375,7 +375,7 @@ We obtain the following frame-preserving update: \subsection{STS with tokens} \subsection{STS with tokens} \label{sec:stsmon} \label{sec:stsmon} Given a state-transition system~(STS, \ie a directed graph) $(\STSS, {\stsstep} \subseteq \STSS \times \STSS)$, a set of tokens $\STST$, and a labeling $\STSL: \STSS \ra \wp(\STST)$ of \emph{protocol-owned} tokens for each state, we construct a monoid modeling an authoritative current state and permitting transitions given a \emph{bound} on the current state and a set of \emph{locally-owned} tokens. Given a state-transition system~(STS, \ie a directed graph) $(\STSS, {\stsstep} \subseteq \STSS \times \STSS)$, a set of tokens $\STST$, and a labeling $\STSL: \STSS \ra \wp(\STST)$ of \emph{protocol-owned} tokens for each state, we construct an RA modeling an authoritative current state and permitting transitions given a \emph{bound} on the current state and a set of \emph{locally-owned} tokens. The construction follows the idea of STSs as described in CaReSL \cite{caresl}. The construction follows the idea of STSs as described in CaReSL \cite{caresl}. We first lift the transition relation to $\STSS \times \wp(\STST)$ (implementing a \emph{law of token conservation}) and define a stepping relation for the \emph{frame} of a given token set: We first lift the transition relation to $\STSS \times \wp(\STST)$ (implementing a \emph{law of token conservation}) and define a stepping relation for the \emph{frame} of a given token set: ... ...
 \section{Derived proof rules and other constructions} \section{Derived proof rules and other constructions} We will below abuse notation, using the \emph{term} meta-variables like $\val$ to range over (bound) \emph{variables} of the corresponding type.. We will below abuse notation, using the \emph{term} meta-variables like $\val$ to range over (bound) \emph{variables} of the corresponding type. We omit type annotations in binders and equality, when the type is clear from context. We omit type annotations in binders and equality, when the type is clear from context. We assume that the signature $\Sig$ embeds all the meta-level concepts we use, and their properties, into the logic. We assume that the signature $\Sig$ embeds all the meta-level concepts we use, and their properties, into the logic. (The Coq formalization is a \emph{shallow embedding} of the logic, so we have direct access to all meta-level notions within the logic anyways.) (The Coq formalization is a \emph{shallow embedding} of the logic, so we have direct access to all meta-level notions within the logic anyways.) ... ...
 ... @@ -7,7 +7,7 @@ A \emph{language} $\Lang$ consists of a set \textdom{Expr} of \emph{expressions} ... @@ -7,7 +7,7 @@ A \emph{language} $\Lang$ consists of a set \textdom{Expr} of \emph{expressions} \end{mathpar} \end{mathpar} \item There exists a \emph{primitive reduction relation} $(-,- \step -,-,-) \subseteq \textdom{Expr} \times \textdom{State} \times \textdom{Expr} \times \textdom{State} \times (\textdom{Expr} \uplus \set{\bot})$ \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 $\expr_1, \state_1 \step \expr_2, \state_2, \bot$. \\ 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 $\expr_2$, a \emph{new thread} $\expr_\f$ is forked off. \item All values are stuck: \item All values are stuck: $\expr, \_ \step \_, \_, \_ \Ra \toval(\expr) = \bot$ $\expr, \_ \step \_, \_, \_ \Ra \toval(\expr) = \bot$ \end{itemize} \end{itemize} ... @@ -40,7 +40,7 @@ For any language $\Lang$, we define the corresponding thread-pool semantics. ... @@ -40,7 +40,7 @@ For any language $\Lang$, we define the corresponding thread-pool semantics. \paragraph{Machine syntax} \paragraph{Machine syntax} $\[ \tpool \in \textdom{ThreadPool} \eqdef \bigcup_n \textdom{Exp}^n \tpool \in \textdom{ThreadPool} \eqdef \bigcup_n \textdom{Expr}^n$ \] \judgment[Machine reduction]{\cfg{\tpool}{\state} \step \judgment[Machine reduction]{\cfg{\tpool}{\state} \step ... @@ -48,12 +48,12 @@ For any language $\Lang$, we define the corresponding thread-pool semantics. ... @@ -48,12 +48,12 @@ For any language $\Lang$, we define the corresponding thread-pool semantics. \begin{mathpar} \begin{mathpar} \infer \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_1] \dplus \tpool'}{\state_1} \step \cfg{\tpool \dplus [\expr_2] \dplus \tpool' \dplus [\expr_\f]}{\state'}} \cfg{\tpool \dplus [\expr_2] \dplus \tpool' \dplus [\expr_\f]}{\state_2}} \and\infer \and\infer {\expr_1, \state_1 \step \expr_2, \state_2} {\expr_1, \state_1 \step \expr_2, \state_2} {\cfg{\tpool \dplus [\expr_1] \dplus \tpool'}{\state} \step {\cfg{\tpool \dplus [\expr_1] \dplus \tpool'}{\state_1} \step \cfg{\tpool \dplus [\expr_2] \dplus \tpool'}{\state'}} \cfg{\tpool \dplus [\expr_2] \dplus \tpool'}{\state_2}} \end{mathpar} \end{mathpar} \clearpage \clearpage ... @@ -595,15 +595,16 @@ A type $\type$ being \emph{inhabited} means that $\proves \wtt{\term}{\type}$ i ... @@ -595,15 +595,16 @@ A type $\type$ being \emph{inhabited} means that $\proves \wtt{\term}{\type}$ i {\mask_2 \subseteq \mask_1 \and {\mask_2 \subseteq \mask_1 \and \toval(\expr_1) = \bot} \toval(\expr_1) = \bot} { {\begin{inbox} % for some crazy reason, LaTeX is actually sensitive to the space between the "{ {" here and the "} }" below... { {\begin{inbox} % for some crazy reason, LaTeX is actually sensitive to the space between the "{ {" here and the "} }" below... ~~\pvs[\mask_1][\mask_2] \Exists \state_1. \red(\expr_1,\state_1) \land \later\ownPhys{\state_1} * \later\All \expr_2, \state_2, \expr_\f. (\expr_1, \state_1 \step \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] \Exists \state_1. \red(\expr_1,\state_1) \land \later\ownPhys{\state_1} * {}\\\qquad\qquad\qquad \later\All \expr_2, \state_2, \expr_\f. \left( (\expr_1, \state_1 \step \expr_2, \state_2, \expr_\f) \land \ownPhys{\state_2} \right) \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}} } \end{inbox}} } \\\\ \infer[wp-lift-pure-step] \infer[wp-lift-pure-step] {\toval(\expr_1) = \bot \and {\toval(\expr_1) = \bot \and \All \state_1. \red(\expr_1, \state_1) \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 } \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 } {\later\All \state, \expr_2, \expr_\f. (\expr_1,\state \step \expr_2, \state,\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}} {\later\All \state, \expr_2, \expr_\f. (\expr_1,\state \step \expr_2, \state,\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} \end{mathpar} Notice that primitive view shifts cover everything to their right, \ie $\pvs \prop * \propB \eqdef \pvs (\prop * \propB)$. 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). ... ...
 ... @@ -26,7 +26,7 @@ We introduce an additional logical connective $\ownM\melt$, which will later be ... @@ -26,7 +26,7 @@ We introduce an additional logical connective $\ownM\melt$, which will later be \Sem{\vctx \proves \prop \Ra \propB : \Prop}_\gamma &\eqdef \Sem{\vctx \proves \prop \Ra \propB : \Prop}_\gamma &\eqdef \Lam \melt. \setComp{n}{\begin{aligned} \Lam \melt. \setComp{n}{\begin{aligned} \All m, \meltB.& m \leq n \land \melt \mincl \meltB \land \meltB \in \mval_m \Ra {} \\ \All m, \meltB.& m \leq n \land \melt \mincl \meltB \land \meltB \in \mval_m \Ra {} \\ & m \in \Sem{\vctx \proves \prop : \Prop}_\gamma(\melt) \Ra {}\\& m \in \Sem{\vctx \proves \propB : \Prop}_\gamma(\melt)\end{aligned}}\\ & m \in \Sem{\vctx \proves \prop : \Prop}_\gamma(\meltB) \Ra {}\\& m \in \Sem{\vctx \proves \propB : \Prop}_\gamma(\meltB)\end{aligned}}\\ \Sem{\vctx \proves \All x : \type. \prop : \Prop}_\gamma &\eqdef \Sem{\vctx \proves \All x : \type. \prop : \Prop}_\gamma &\eqdef \Lam \melt. \setComp{n}{ \All v \in \Sem{\type}. n \in \Sem{\vctx, x : \type \proves \prop : \Prop}_{\gamma[x \mapsto v]}(\melt) } \\ \Lam \melt. \setComp{n}{ \All v \in \Sem{\type}. n \in \Sem{\vctx, x : \type \proves \prop : \Prop}_{\gamma[x \mapsto v]}(\melt) } \\ \Sem{\vctx \proves \Exists x : \type. \prop : \Prop}_\gamma &\eqdef \Sem{\vctx \proves \Exists x : \type. \prop : \Prop}_\gamma &\eqdef ... @@ -58,7 +58,7 @@ Above, $\maybe\monoid$ is the monoid obtained by adding a unit to $\monoid$. ... @@ -58,7 +58,7 @@ Above, $\maybe\monoid$ is the monoid obtained by adding a unit to $\monoid$. (It's not a coincidence that we used the same notation for the range of the core; it's the same type either way: $\monoid + 1$.) (It's not a coincidence that we used the same notation for the range of the core; it's the same type either way: $\monoid + 1$.) Remember that $\iFunc$ is the user-chosen bifunctor from $\COFEs$ to $\CMRAs$ (see~\Sref{sec:logic}). Remember that $\iFunc$ is the user-chosen bifunctor from $\COFEs$ to $\CMRAs$ (see~\Sref{sec:logic}). $\textdom{ResF}(\cofe^\op, \cofe)$ is a CMRA by lifting the individual CMRAs pointwise. $\textdom{ResF}(\cofe^\op, \cofe)$ is a CMRA by lifting the individual CMRAs pointwise. Furthermore, if $F$ is locally contractive, then so is $\textdom{ResF}$. Furthermore, since $\Sigma$ is locally contractive, so is $\textdom{ResF}$. Now we can write down the recursive domain equation: Now we can write down the recursive domain equation: $\iPreProp \cong \UPred(\textdom{ResF}(\iPreProp, \iPreProp))$ $\iPreProp \cong \UPred(\textdom{ResF}(\iPreProp, \iPreProp))$ ... @@ -187,10 +187,10 @@ $\rho(x)\in\Sem{\vctx(x)}$. ... @@ -187,10 +187,10 @@ $\rho(x)\in\Sem{\vctx(x)}$. \paragraph{Logical entailment.} \paragraph{Logical entailment.} We can now define \emph{semantic} logical entailment. We can now define \emph{semantic} logical entailment. \typedsection{Interpretation of entailment}{\Sem{\vctx \mid \pfctx \proves \prop} : 2} \typedsection{Interpretation of entailment}{\Sem{\vctx \mid \pfctx \proves \prop} : \mProp} \[ \[ \Sem{\vctx \mid \pfctx \proves \propB} \eqdef \Sem{\vctx \mid \pfctx \proves \prop} \eqdef \begin{aligned}[t] \begin{aligned}[t] \MoveEqLeft \MoveEqLeft \forall n \in \mathbb{N}.\; \forall n \in \mathbb{N}.\; ... ...
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