Commit 3ca48414 authored by Robbert Krebbers's avatar Robbert Krebbers

Merge branch 'master' of gitlab.mpi-sws.org:FP/iris-coq

parents d71cbe0f 04d3ee68
......@@ -2,6 +2,7 @@
\subsection{COFE}
This definition varies slightly from the original one in~\cite{catlogic}.
\begin{defn}[Chain]
Given some set $\cofe$ and an indexed family $({\nequiv{n}} \subseteq \cofe \times \cofe)_{n \in \mathbb{N}}$ of equivalence relations, a \emph{chain} is a function $c : \mathbb{N} \to \cofe$ such that $\All n, m. n \leq m \Ra c (m) \nequiv{n} c (n)$.
\end{defn}
......@@ -22,6 +23,8 @@
An element $x \in \cofe$ of a COFE is called \emph{discrete} if
\[ \All y \in \cofe. x \nequiv{0} y \Ra x = y\]
A COFE $A$ is called \emph{discrete} if all its elements are discrete.
For a set $X$, we write $\Delta X$ for the discrete COFE with $x \nequiv{n} x' \eqdef x = x'$
\end{defn}
\begin{defn}
......@@ -30,6 +33,7 @@
It is \emph{contractive} if
\[ \All n, x \in \cofe, y \in \cofe. (\All m < n. x \nequiv{m} y) \Ra f(x) \nequiv{n} f(x) \]
\end{defn}
The reason that contractive functions are interesting is that for every contractive $f : \cofe \to \cofe$ with $\cofe$ inhabited, there exists a fixed-point $\fix(f)$ such that $\fix(f) = f(\fix(f))$.
\begin{defn}
The category $\COFEs$ consists of COFEs as objects, and non-expansive functions as arrows.
......@@ -52,7 +56,31 @@ Note that the composition of non-expansive (bi)functors is non-expansive, and th
\subsection{RA}
\ralf{Copy this from the paper, when that one is more polished.}
\begin{defn}
A \emph{resource algebra} (RA) is a tuple \\
$(\monoid, \mval \subseteq \monoid, \mcore{-}:
\monoid \to \monoid, (\mtimes) : \monoid \times \monoid \to \monoid)$ satisfying
\begin{align*}
\All \melt, \meltB, \meltC.& (\melt \mtimes \meltB) \mtimes \meltC = \melt \mtimes (\meltB \mtimes \meltC) \tagH{ra-assoc} \\
\All \melt, \meltB.& \melt \mtimes \meltB = \meltB \mtimes \melt \tagH{ra-comm} \\
\All \melt.& \mcore\melt \mtimes \melt = \melt \tagH{ra-core-id} \\
\All \melt.& \mcore{\mcore\melt} = \mcore\melt \tagH{ra-core-idem} \\
\All \melt, \meltB.& \melt \mincl \meltB \Ra \mcore\melt \mincl \mcore\meltB \tagH{ra-core-mono} \\
\All \melt, \meltB.& (\melt \mtimes \meltB) \in \mval \Ra \melt \in \mval \tagH{ra-valid-op} \\
\text{where}\qquad %\qquad\\
\melt \mincl \meltB \eqdef{}& \Exists \meltC. \meltB = \melt \mtimes \meltC \tagH{ra-incl}
\end{align*}
\end{defn}
\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
\[ \All \melt_\f. \melt \mtimes \melt_\f \in \mval \Ra \Exists \meltB \in \meltsB. \meltB \mtimes \melt_\f \in \mval \]
We further define $\melt \mupd \meltB \eqdef \melt \mupd \set\meltB$.
\end{defn}
\ralf{Copy the explanation from the paper, when that one is more polished.}
\subsection{CMRA}
......@@ -70,7 +98,8 @@ Note that the composition of non-expansive (bi)functors is non-expansive, and th
\All n, \melt, \meltB_1, \meltB_2.& \omit\rlap{$\melt \in \mval_n \land \melt \nequiv{n} \meltB_1 \mtimes \meltB_2 \Ra {}$} \\
&\Exists \meltC_1, \meltC_2. \melt = \meltC_1 \mtimes \meltC_2 \land \meltC_1 \nequiv{n} \meltB_1 \land \meltC_2 \nequiv{n} \meltB_2 \tagH{cmra-extend} \\
\text{where}\qquad\qquad\\
\melt \mincl \meltB \eqdef{}& \Exists \meltC. \meltB = \melt \mtimes \meltC \tagH{cmra-incl}
\melt \mincl \meltB \eqdef{}& \Exists \meltC. \meltB = \melt \mtimes \meltC \tagH{cmra-incl}\\
\melt \mincl[n] \meltB \eqdef{}& \Exists \meltC. \meltB \nequiv{n} \melt \mtimes \meltC \tagH{cmra-inclN}
\end{align*}
\end{defn}
......@@ -117,7 +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}
......
......@@ -25,7 +25,7 @@ where $\mProp$ is the set of meta-level propositions, \eg Coq's \texttt{Prop}.
$\UPred(-)$ is a locally non-expansive functor from $\CMRAs$ to $\COFEs$.
One way to understand this definition is to re-write it a little.
We start by defining the COFE of \emph{step-indexed propositions}:
We start by defining the COFE of \emph{step-indexed propositions}: For every step-index, we proposition either holds or does not hold.
\begin{align*}
\SProp \eqdef{}& \psetdown{\mathbb{N}} \\
\eqdef{}& \setComp{\prop \in \pset{\mathbb{N}}}{ \All n, m. n \geq m \Ra n \in \prop \Ra m \in \prop } \\
......@@ -33,7 +33,7 @@ We start by defining the COFE of \emph{step-indexed propositions}:
\end{align*}
Now we can rewrite $\UPred(\monoid)$ as monotone step-indexed predicates over $\monoid$, where the definition of a ``monotone'' function here is a little funny.
\begin{align*}
\UPred(\monoid) \approx{}& \monoid \monra \SProp \\
\UPred(\monoid) \cong{}& \monoid \monra \SProp \\
\eqdef{}& \setComp{\pred: \monoid \nfn \SProp}{\All n, m, x, y. n \in \pred(x) \land x \mincl y \land m \leq n \land y \in \mval_m \Ra m \in \pred(y)}
\end{align*}
The reason we chose the first definition is that it is easier to work with in Coq.
......@@ -77,35 +77,35 @@ $K \fpfn (-)$ is a locally non-expansive functor from $\CMRAs$ to $\CMRAs$.
\subsection{Agreement}
Given some COFE $\cofe$, we define $\agm(\cofe)$ as follows:
\newcommand{\agc}{\mathrm{c}} % the "c" field of an agreement element
\newcommand{\agV}{\mathrm{V}} % the "V" field of an agreement element
\newcommand{\aginjc}{\mathrm{c}} % the "c" field of an agreement element
\newcommand{\aginjV}{\mathrm{V}} % the "V" field of an agreement element
\begin{align*}
\agm(\cofe) \eqdef{}& \record{\agc : \mathbb{N} \to \cofe , \agV : \SProp} \\
\agm(\cofe) \eqdef{}& \record{\aginjc : \mathbb{N} \to \cofe , \aginjV : \SProp} \\
& \text{quotiented by} \\
\melt \equiv \meltB \eqdef{}& \melt.\agV = \meltB.\agV \land \All n. n \in \melt.\agV \Ra \melt.\agc(n) \nequiv{n} \meltB.\agc(n) \\
\melt \nequiv{n} \meltB \eqdef{}& (\All m \leq n. m \in \melt.\agV \Lra m \in \meltB.\agV) \land (\All m \leq n. m \in \melt.\agV \Ra \melt.\agc(m) \nequiv{m} \meltB.\agc(m)) \\
\mval_n \eqdef{}& \setComp{\melt \in \monoid}{ n \in \melt.\agV \land \All m \leq n. \melt.\agc(n) \nequiv{m} \melt.\agc(m) } \\
\melt \equiv \meltB \eqdef{}& \melt.\aginjV = \meltB.\aginjV \land \All n. n \in \melt.\aginjV \Ra \melt.\aginjc(n) \nequiv{n} \meltB.\aginjc(n) \\
\melt \nequiv{n} \meltB \eqdef{}& (\All m \leq n. m \in \melt.\aginjV \Lra m \in \meltB.\aginjV) \land (\All m \leq n. m \in \melt.\aginjV \Ra \melt.\aginjc(m) \nequiv{m} \meltB.\aginjc(m)) \\
\mval_n \eqdef{}& \setComp{\melt \in \monoid}{ n \in \melt.\aginjV \land \All m \leq n. \melt.\aginjc(n) \nequiv{m} \melt.\aginjc(m) } \\
\mcore\melt \eqdef{}& \melt \\
\melt \mtimes \meltB \eqdef{}& (\melt.\agc, \setComp{n}{n \in \melt.\agV \land n \in \meltB.\agV \land \melt \nequiv{n} \meltB })
\melt \mtimes \meltB \eqdef{}& (\melt.\aginjc, \setComp{n}{n \in \melt.\aginjV \land n \in \meltB.\aginjV \land \melt \nequiv{n} \meltB })
\end{align*}
$\agm(-)$ is a locally non-expansive functor from $\COFEs$ to $\CMRAs$.
You can think of the $\agc$ as a \emph{chain} of elements of $\cofe$ that has to converge only for $n \in \agV$ steps.
You can think of the $\aginjc$ as a \emph{chain} of elements of $\cofe$ that has to converge only for $n \in \aginjV$ steps.
The reason we store a chain, rather than a single element, is that $\agm(\cofe)$ needs to be a COFE itself, so we need to be able to give a limit for every chain of $\agm(\cofe)$.
However, given such a chain, we cannot constructively define its limit: Clearly, the $\agV$ of the limit is the limit of the $\agV$ of the chain.
However, given such a chain, we cannot constructively define its limit: Clearly, the $\aginjV$ of the limit is the limit of the $\aginjV$ of the chain.
But what to pick for the actual data, for the element of $\cofe$?
Only if $\agV = \mathbb{N}$ we have a chain of $\cofe$ that we can take a limit of; if the $\agV$ is smaller, the chain ``cancels'', \ie stops converging as we reach indices $n \notin \agV$.
Only if $\aginjV = \mathbb{N}$ we have a chain of $\cofe$ that we can take a limit of; if the $\aginjV$ is smaller, the chain ``cancels'', \ie stops converging as we reach indices $n \notin \aginjV$.
To mitigate this, we apply the usual construction to close a set; we go from elements of $\cofe$ to chains of $\cofe$.
We define an injection $\ag$ into $\agm(\cofe)$ as follows:
\[ \ag(x) \eqdef \record{\mathrm c \eqdef \Lam \any. x, \mathrm V \eqdef \mathbb{N}} \]
We define an injection $\aginj$ into $\agm(\cofe)$ as follows:
\[ \aginj(x) \eqdef \record{\mathrm c \eqdef \Lam \any. x, \mathrm V \eqdef \mathbb{N}} \]
There are no interesting frame-preserving updates for $\agm(\cofe)$, but we can show the following:
\begin{mathpar}
\axiomH{ag-val}{\ag(x) \in \mval_n}
\axiomH{ag-val}{\aginj(x) \in \mval_n}
\axiomH{ag-dup}{\ag(x) = \ag(x)\mtimes\ag(x)}
\axiomH{ag-dup}{\aginj(x) = \aginj(x)\mtimes\aginj(x)}
\axiomH{ag-agree}{\ag(x) \mtimes \ag(y) \in \mval_n \Ra x \nequiv{n} y}
\axiomH{ag-agree}{\aginj(x) \mtimes \aginj(y) \in \mval_n \Ra x \nequiv{n} y}
\end{mathpar}
\subsection{One-shot}
......@@ -115,17 +115,17 @@ Given some CMRA $\monoid$, we define $\oneshotm(\monoid)$ as follows:
\begin{align*}
\oneshotm(\monoid) \eqdef{}& \ospending + \osshot(\monoid) + \munit + \bot \\
\mval_n \eqdef{}& \set{\ospending, \munit} \cup \setComp{\osshot(\melt)}{\melt \in \mval_n}
\end{align*}
\begin{align*}
\mcore{\ospending} \eqdef{}& \munit & \mcore{\osshot(\melt)} \eqdef{}& \mcore\melt \\
\mcore{\munit} \eqdef{}& \munit & \mcore{\bot} \eqdef{}& \bot
\end{align*}
\begin{align*}
\\%\end{align*}
%\begin{align*}
\osshot(\melt) \mtimes \osshot(\meltB) \eqdef{}& \osshot(\melt \mtimes \meltB) \\
\munit \mtimes \ospending \eqdef{}& \ospending \mtimes \munit \eqdef \ospending \\
\munit \mtimes \osshot(\melt) \eqdef{}& \osshot(\melt) \mtimes \munit \eqdef \osshot(\melt)
\end{align*}
\end{align*}%
The remaining cases of composition go to $\bot$.
\begin{align*}
\mcore{\ospending} \eqdef{}& \munit & \mcore{\osshot(\melt)} \eqdef{}& \mcore\melt \\
\mcore{\munit} \eqdef{}& \munit & \mcore{\bot} \eqdef{}& \bot
\end{align*}
The step-indexed equivalence is inductively defined as follows:
\begin{mathpar}
\axiom{\ospending \nequiv{n} \ospending}
......@@ -149,33 +149,38 @@ We obtain the following frame-preserving updates:
{\osshot(\melt) \mupd \setComp{\osshot(\meltB)}{\meltB \in \meltsB}}
\end{mathpar}
% \subsection{Exclusive monoid}
\subsection{Exclusive CMRA}
% Given a set $X$, we define a monoid such that at most one $x \in X$ can be owned.
% Let $\exm{X}$ be the monoid with carrier $X \uplus \{ \munit \}$ and multiplication
% \[
% \melt \cdot \meltB \;\eqdef\;
% \begin{cases}
% \melt & \mbox{if } \meltB = \munit \\
% \meltB & \mbox{if } \melt = \munit
% \end{cases}
% \]
Given a cofe $\cofe$, we define a CMRA $\exm(\cofe)$ such that at most one $x \in \cofe$ can be owned:
\begin{align*}
\exm(\cofe) \eqdef{}& \exinj(\cofe) + \munit + \bot \\
\mval_n \eqdef{}& \setComp{\melt\in\exm(\cofe)}{\melt \neq \bot} \\
\munit \mtimes \exinj(x) \eqdef{}& \exinj(x) \mtimes \munit \eqdef \exinj(x)
\end{align*}
The remaining cases of composition go to $\bot$.
\begin{align*}
\mcore{\exinj(x)} \eqdef{}& \munit & \mcore{\munit} \eqdef{}& \munit &
\mcore{\bot} \eqdef{}& \bot
\end{align*}
The step-indexed equivalence is inductively defined as follows:
\begin{mathpar}
\infer{x \nequiv{n} y}{\exinj(x) \nequiv{n} \exinj(y)}
% The frame-preserving update
% \begin{mathpar}
% \inferH{ExUpd}
% {x \in X}
% {x \mupd \melt}
% \end{mathpar}
% is easily shown, as the only possible frame for $x$ is $\munit$.
\axiom{\munit \nequiv{n} \munit}
\axiom{\bot \nequiv{n} \bot}
\end{mathpar}
$\exm(-)$ is a locally non-expansive functor from $\COFEs$ to $\CMRAs$.
We obtain the following frame-preserving update:
\begin{mathpar}
\inferH{ex-update}{}
{\exinj(x) \mupd \exinj(y)}
\end{mathpar}
% Exclusive monoids are cancellative.
% \begin{proof}[Proof of cancellativity]
% If $\melt_f = \munit$, then the statement is trivial.
% If $\melt_f \neq \munit$, then we must have $\melt = \meltB = \munit$, as otherwise one of the two products would be $\mzero$.
% \end{proof}
%TODO: These need syncing with Coq
% \subsection{Finite Powerset Monoid}
% Given an infinite set $X$, we define a monoid $\textmon{PowFin}$ with carrier $\mathcal{P}^{\textrm{fin}}(X)$ as follows:
......@@ -190,16 +195,16 @@ We obtain the following frame-preserving updates:
% \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}
......@@ -233,20 +238,20 @@ We obtain the following frame-preserving updates:
% \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}
......@@ -307,7 +312,7 @@ We obtain the following frame-preserving updates:
% 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'
......@@ -373,8 +378,6 @@ We obtain the following frame-preserving updates:
% \subsection{STS with tokens monoid}
% \label{sec:stsmon}
% \ralf{This needs syncing with the Coq development.}
% Given a state-transition system~(STS) $(\STSS, \ra)$, a set of tokens $\STSS$, and a labeling $\STSL: \STSS \ra \mathcal{P}(\STST)$ 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.
% The construction follows the idea of STSs as described in CaReSL \cite{caresl}.
......@@ -389,11 +392,11 @@ We obtain the following frame-preserving updates:
% 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}
......@@ -415,8 +418,8 @@ We obtain the following frame-preserving updates:
% {(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}
......
......@@ -205,9 +205,64 @@ The following rules can be derived for Hoare triples.
\end{mathparpagebreakable}
\paragraph{Lifting of operational semantics.}
We can derive some specialized forms of the lifting axioms for the operational semantics, as well as some forms that involve view shifts and Hoare triples.
We can derive some specialized forms of the lifting axioms for the operational semantics.
\begin{mathparpagebreakable}
\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}}
\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}}
\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}}
\end{mathparpagebreakable}
\ralf{Add these.}
Furthermore, we derive some forms that directly involve view shifts and Hoare triples.
\begin{mathparpagebreakable}
\infer[ht-lift-step]
{\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) \\\\
\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]}
{ \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) \\\\
\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] }
\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]}
{ \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' \\\\
\hoare{\prop}{\expr_2}{\Ret\val.\propB}[\mask_1] \and
\hoare{\prop'}{\expr_\f}{\Ret\any. \TRUE}[\top]}
{ \hoare{\later(\prop*\prop')}{\expr_1}{\Ret\val.\propB}[\mask_1] }
\end{mathparpagebreakable}
\subsection{Global functor and ghost ownership}
......@@ -282,6 +337,7 @@ We can now derive the following rules for this derived form of the invariant ass
{\knowInv\namesp\prop \proves \propB \vs[\mask] \propC}
\end{mathpar}
% TODO: These need syncing with Coq
% \subsection{STSs with interpretation}\label{sec:stsinterp}
% Building on \Sref{sec:stsmon}, after constructing the monoid $\STSMon{\STSS}$ for a particular STS, we can use an invariant to tie an interpretation, $\pred : \STSS \to \Prop$, to the STS's current state, recovering CaReSL-style reasoning~\cite{caresl}.
......@@ -382,21 +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)}
......
......@@ -86,13 +86,15 @@
\newcommand{\rs}{r}
\newcommand{\rsB}{s}
\newcommand{\rss}{R}
\newcommand{\pres}{\pi}
\newcommand{\wld}{w}
\newcommand{\ghostRes}{g}
%% Various pieces of syntax
\newcommand{\wsat}[4]{#1 \models_{#2} #3; #4}
\newcommand{\wsat}[3]{#1 \models_{#2} #3}
\newcommand{\wsatpre}{\textdom{pre-wsat}}
\newcommand{\wtt}[2]{#1 : #2} % well-typed term
......@@ -114,6 +116,7 @@
\newcommand{\UPred}{\textdom{UPred}}
\newcommand{\mProp}{\textdom{Prop}} % meta-level prop
\newcommand{\iProp}{\textdom{iProp}}
\newcommand{\iPreProp}{\textdom{iPreProp}}
\newcommand{\Wld}{\textdom{Wld}}
\newcommand{\Res}{\textdom{Res}}
......@@ -121,6 +124,7 @@
\newcommand{\cofeB}{U}
\newcommand{\COFEs}{\mathcal{U}} % category of COFEs
\newcommand{\iFunc}{\Sigma}
\newcommand{\fix}{\textdom{fix}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% CMRA (RESOURCE ALGEBRA) SYMBOLS & NOTATION & IDENTIFIERS
......@@ -136,6 +140,8 @@
\newcommand{\melts}{A}
\newcommand{\meltsB}{B}
\newcommand{\f}{\mathrm{f}} % for "frame"
\newcommand{\mcar}[1]{|#1|}
\newcommand{\mcarp}[1]{\mcar{#1}^{+}}
\newcommand{\munit}{\varepsilon}
......@@ -221,7 +227,7 @@
\newcommand*{\knowInv}[2]{\boxedassert{#2}[#1]}
\newcommand*{\ownGhost}[2]{\boxedassert[densely dashed]{#2}[#1]}
\newcommand*{\ownGGhost}[1]{\boxedassert[densely dashed]{#1}}
\newcommand{\ownM}[1]{\textlog{Own}(#1)}
\newcommand{\ownPhys}[1]{\textlog{Phy}(#1)}
%% View Shifts
......@@ -286,7 +292,7 @@
%% Some commonly used identifiers
\newcommand{\timeless}[1]{\textlog{timeless}(#1)}
\newcommand{\physatomic}[1]{\textlog{$#1$ phys.\ atomic}}
\newcommand{\physatomic}[1]{\textlog{atomic}($#1$)}
\newcommand{\infinite}{\textlog{infinite}}
\newcommand{\Prop}{\textlog{Prop}}
......@@ -321,13 +327,14 @@
% Agreement
\newcommand{\agm}{\ensuremath{\textmon{Ag}}}
\newcommand{\ag}{\textlog{ag}}
\newcommand{\aginj}{\textlog{ag}}
% Fraction
\newcommand{\fracm}{\ensuremath{\textmon{Frac}}}
% Exclusive
\newcommand{\exm}{\ensuremath{\textmon{Ex}}}
\newcommand{\exinj}{\textlog{ex}}
% Auth
\newcommand{\authm}{\textmon{Auth}}
......
......@@ -33,8 +33,8 @@
\endgroup\clearpage\begingroup
\input{logic}
\endgroup\clearpage\begingroup
%\input{model}
%\endgroup\clearpage\begingroup
\input{model}
\endgroup\clearpage\begingroup
\input{derived}
\endgroup\clearpage\begingroup
\printbibliography
......
......@@ -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 $\expr_1, \state_1 \step \expr_2, \state_2, \expr'$ indicates that, when $\expr_1$ reduces to $\expr$, a \emph{new thread} $\expr'$ 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'. \atomic(\expr_1) \land \expr_1, \state_1 \step \expr_2, \state_2, \expr' \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
\[ \Exists \expr_2, \state_2, \expr'. \expr,\state \step \expr_2,\state_2,\expr' \]
\[ \Exists \expr_2, \state_2, \expr_\f. \expr,\state \step \expr_2,\state_2,\expr_\f \]
\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'. \expr_1, \state_1 \step \expr_2,\state_2,\expr' \Ra \lctx(\expr_1), \state_1 \step \lctx(\expr_2),\state_2,\expr' $
$\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'. \toval(\expr_1') = \bot \land \lctx(\expr_1'), \state_1 \step \expr_2,\state_2,\expr' \Ra \Exists\expr_2'. \expr_2 = \lctx(\expr_2') \land \expr_1', \state_1 \step \expr_2',\state_2,\expr' $
$\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' \and \expr' \neq ()}
{\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']}{\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
......@@ -124,7 +124,7 @@ Iris syntax is built up from a signature $\Sig$ and a countably infinite set $\t
\prop * \prop \mid
\prop \wand \prop \mid
\\&
\MU \var:\type. \pred \mid
\MU \var:\type. \term \mid
\Exists \var:\type. \prop \mid
\All \var:\type. \prop \mid
\\&
......@@ -136,7 +136,7 @@ Iris syntax is built up from a signature $\Sig$ and a countably infinite set $\t
\pvs[\term][\term] \prop\mid
\wpre{\term}[\term]{\Ret\var.\term}
\end{align*}
Recursive predicates must be \emph{guarded}: in $\MU \var. \pred$, the variable $\var$ can only appear under the later $\later$ modality.
Recursive predicates must be \emph{guarded}: in $\MU \var. \term$, the variable $\var$ can only appear under the later $\later$ modality.
Note that $\always$ and $\later$ bind more tightly than $*$, $\wand$, $\land$, $\lor$, and $\Ra$.
We will write $\pvs[\term] \prop$ for $\pvs[\term][\term] \prop$.
......@@ -170,8 +170,6 @@ We introduce additional metavariables ranging over terms and generally let the c
\]
\paragraph{Variable conventions.}
We often abuse notation, using the preceding \emph{term} meta-variables to range over (bound) \emph{variables}.
We omit type annotations in binders, when the type is clear from context.
We assume that, if a term occurs multiple times in a rule, its free variables are exactly those binders which are available at every occurrence.
......@@ -538,7 +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}
......@@ -596,17 +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'. \expr_1,\state_1 \step \expr_2,\state_2,\expr' \Ra \pred(\expr_2,\state_2,\expr')}
{\pvs[\mask_1][\mask_2] \later\ownPhys{\state_1} * \later\All \expr_2, \state_2, \expr'. \pred(\expr_2, \state_2, \expr') \land \ownPhys{\state_2} \wand \pvs[\mask_2][\mask_1] \wpre{\expr_2}[\mask_1]{\Ret\var.\prop} * \wpre{\expr'}[\top]{\Ret\var.\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)}
{ {\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}
\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'. \expr_1,\state_1 \step \expr_2,\state_2,\expr' \Ra \state_1 = \state_2 \land \pred(\expr_2,\expr')}
{\later\All \expr_2, \expr'. \pred(\expr_2, \expr') \wand \wpre{\expr_2}[\mask_1]{\Ret\var.\prop} * \wpre{\expr'}[\top]{\Ret\var.\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 $\wpre{\expr'}[\mask]{\Ret\var.\prop} \eqdef \TRUE$ if $\expr' = \bot$ (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}
......
\section{Model and semantics}
\ralf{What also needs to be done here: Define uPred and its later function; define black later; define the resource CMRA}
The semantics closely follows the ideas laid out in~\cite{catlogic}.
We just repeat some of the most important definitions here.
An \emph{ordered family of equivalence relations} (o.f.e.\@) is a pair
$(X,(\nequiv{n})_{n\in\mathbb{N}})$, with $X$ a set, and each $\nequiv{n}$
an equivalence relation over $X$ satisfying
\begin{itemize}
\item $\All x,x'. x \nequiv{0} x',$
\item $\All x,x',n. x \nequiv{n+1} x' \implies x \nequiv{n} x',$
\item $\All x,x'. (\All n. x\nequiv{n} x') \implies x = x'.$
\end{itemize}
\a
Let $(X,(\nequivset{n}{X})_{n\in\mathbb{N}})$ and
$(Y,(\nequivset{n}{Y})_{n\in\mathbb{N}})$ be o.f.e.'s. A function $f:
X\to Y$ is \emph{non-expansive} if, for all $x$, $x'$ and $n$,
\[
x \nequivset{n}{X} x' \implies
fx \nequivset{n}{Y} f x'.
\]
Let $(X,(\nequiv{n})_{n\in\mathbb{N}})$ be an o.f.e.
A sequence $(x_i)_{i\in\mathbb{N}}$ of elements in $X$ is a
\emph{chain} (aka \emph{Cauchy sequence}) if