### fix captialization

parent d96f252e
 \section{OFE and COFE constructions} \section{OFE and COFE Constructions} \subsection{Trivial pointwise lifting} \subsection{Trivial Pointwise Lifting} The (C)OFE structure on many types can be easily obtained by pointwise lifting of the structure of the components. This is what we do for option $\maybe\cofe$, product $(M_i)_{i \in I}$ (with $I$ some finite index set), sum $\cofe + \cofe'$ and finite partial functions $K \fpfn \monoid$ (with $K$ infinite countable). \subsection{Next (type-level later)} \subsection{Next (Type-Level Later)} Given a OFE $\cofe$, we define $\latert\cofe$ as follows (using a datatype-like notation to define the type): \begin{align*} ... ... @@ -51,7 +51,7 @@ connective. \clearpage \section{RA and CMRA constructions} \section{RA and CMRA Constructions} \subsection{Product} \label{sec:prodm} ... ... @@ -116,7 +116,7 @@ We can easily extend this to a full CMRA by defining a suitable core, namely \end{align*} Notice that this core is total, as the result always lies in $\maybe\monoid$ (rather than in $\maybe{\mathord{\maybe\monoid}}$). \subsection{Finite partial function} \subsection{Finite Partial Functions} \label{sec:fpfnm} Given some infinite countable $K$ and some CMRA $\monoid$, the set of finite partial functions $K \fpfn \monoid$ is equipped with a CMRA structure by lifting everything pointwise. ... ... @@ -308,7 +308,7 @@ We then obtain {\authfull \melt_1 , \authfrag \meltB_1 \mupd \authfull \melt_2 , \authfrag \meltB_2} \end{mathpar} \subsection{STS with tokens} \subsection{STS with Tokens} \label{sec:sts-cmra} 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. ... ...
 \section{Derived constructions} \section{Derived Constructions} \subsection{Non-atomic (thread-local'') invariants} \subsection{Non-Atomic (Thread-Local'') Invariants} Sometimes it is necessary to maintain invariants that we need to open non-atomically. Clearly, for this mechanism to be sound we need something that prevents us from opening the same invariant twice, something like the masks that avoid reentrancy on the normal'', atomic invariants. ... ...
 ... ... @@ -3,7 +3,7 @@ In this section we discuss some additional constructions that we define within and on top of the base logic. These are not extensions'' in the sense that they change the proof power of the logic, they just form useful derived principles. \subsection{Derived rules about base connectives} \subsection{Derived Rules about Base Connectives} We collect here some important and frequently used derived proof rules. \begin{mathparpagebreakable} \infer{} ... ... @@ -42,7 +42,7 @@ We collect here some important and frequently used derived proof rules. Noteworthy here is the fact that $\prop \proves \later\prop$ can be derived from Löb induction, and $\TRUE \proves \plainly\TRUE$ can be derived via $\plainly$ commuting with universal quantification ranging over the empty type $0$. \subsection{Persistent assertions} \subsection{Persistent Assertions} We call an assertion $\prop$ \emph{persistent} if $\prop \proves \always\prop$. These are assertions that don't own anything'', so we can (and will) treat them like normal'' intuitionistic assertions. ... ... @@ -52,7 +52,7 @@ Persistence is preserved by conjunction, disjunction, separating conjunction as \subsection{Timeless assertions and except-0} \subsection{Timeless Assertions and Except-0} One of the troubles of working in a step-indexed logic is the later'' modality $\later$. It turns out that we can somewhat mitigate this trouble by working below the following \emph{except-0} modality: ... ...
 ... ... @@ -37,7 +37,7 @@ A \emph{language} $\Lang$ consists of a set \Expr{} of \emph{expressions} (metav \end{enumerate} \end{defn} \subsection{Concurrent language} \subsection{Concurrent Language} For any language $\Lang$, we define the corresponding thread-pool semantics. ... ...
 \section{Model and semantics} \section{Model and Semantics} \label{sec:model} The semantics closely follows the ideas laid out in~\cite{catlogic}. ... ...
 \section{Logical paradoxes} \section{Logical Paradoxes} \newcommand{\starttoken}{\textsc{s}} \newcommand{\finishtoken}{\textsc{f}} In this section we provide proofs of some logical inconsistencies that arise when slight changes are made to the Iris logic. \subsection{Saved propositions without a later} \subsection{Saved Propositions without a Later} \label{sec:saved-prop-no-later} As a preparation for the proof about invariants in \Sref{app:section:invariants-without-a-later}, we show that omitting the later modality from a variant of \emph{saved propositions} leads to a contradiction. ... ... @@ -75,7 +75,7 @@ With this lemma in hand, the proof of \thmref{thm:counterexample-1} is simple. Together with the rule \ruleref{sprop-alloc} we thus derive $\upd \FALSE$. \end{proof} \subsection{Invariants without a later} \subsection{Invariants without a Later} \label{app:section:invariants-without-a-later} Now we come to the main paradox: if we remove the $\later$ from \ruleref{inv-open}, the logic becomes inconsistent. ... ...
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