diff --git a/docs/algebra.tex b/docs/algebra.tex
index 8f278074f4f9aacd3d1c5206cedda70c6c675fcc..6d738187e0f1c4c42484357cdc9ef04bb06b03fe 100644
--- a/docs/algebra.tex
+++ b/docs/algebra.tex
@@ -23,6 +23,8 @@ This definition varies slightly from the original one in~\cite{catlogic}.
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}
@@ -31,6 +33,7 @@ This definition varies slightly from the original one in~\cite{catlogic}.
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.
diff --git a/docs/constructions.tex b/docs/constructions.tex
index e7763440eddea9a1023c11e6d4b9859cbf36d5aa..d5849d53e2917a6937da71c99d5fa29997d67664 100644
--- a/docs/constructions.tex
+++ b/docs/constructions.tex
@@ -33,7 +33,7 @@ We start by defining the COFE of \emph{step-indexed propositions}: For every ste
\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}
@@ -149,34 +149,38 @@ We obtain the following frame-preserving updates:
{\osshot(\melt) \mupd \setComp{\osshot(\meltB)}{\meltB \in \meltsB}}
\end{mathpar}
-%TODO: These need syncing with Coq
-% \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}
-% 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}
+ \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}
+
+%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:
diff --git a/docs/iris.sty b/docs/iris.sty
index 644b63455f5460666c5af8c1820dedc5af3b45e4..3370db61803d4cf51099d725cfb1d498ab767657 100644
--- a/docs/iris.sty
+++ b/docs/iris.sty
@@ -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}
@@ -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}}
diff --git a/docs/logic.tex b/docs/logic.tex
index 1010939ad83c5f36baac6cfa0cd1fb905353c0ef..2c8a49da5615064e4faf17191d85438fc4dd2536 100644
--- a/docs/logic.tex
+++ b/docs/logic.tex
@@ -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 $\pvs[\term][\term] \prop$.
diff --git a/docs/model.tex b/docs/model.tex
index 341deb1a8cba17c36064ff5bf3eee666666e665c..d1af61a1eb409fc183f1b142d0d02aba703d00c1 100644
--- a/docs/model.tex
+++ b/docs/model.tex
@@ -3,10 +3,11 @@
The semantics closely follows the ideas laid out in~\cite{catlogic}.
\subsection{Generic model of base logic}
+\label{sec:upred-logic}
The base logic including equality, later, always, and a notion of ownership is defined on $\UPred(\monoid)$ for any CMRA $\monoid$.
-\typedsection{Interpretation of base assertions}{\Sem{\vctx \proves \term : \Prop} : \Sem{\vctx} \to \UPred(\monoid)}
+\typedsection{Interpretation of base assertions}{\Sem{\vctx \proves \term : \Prop} : \Sem{\vctx} \nfn \UPred(\monoid)}
Remember that $\UPred(\monoid)$ is isomorphic to $\monoid \monra \SProp$.
We are thus going to define the assertions as mapping CMRA elements to sets of step-indices.
@@ -42,410 +43,165 @@ We introduce an additional logical connective $\ownM\melt$, which will later be
\Sem{\vctx \proves \mval(\melt) : \Prop}_\gamma &\eqdef \Lam\any. \setComp{n}{\melt \in \mval_n} \\
\end{align*}
+For every definition, we have to show all the side-conditions: The maps have to be non-expansive and monotone.
-%\subsection{Iris model}
-% \subsection{Semantic structures: propositions}
-% \ralf{This needs to be synced with the Coq development again.}
+\subsection{Iris model}
-% \[
-% \begin{array}[t]{rcl}
-% % \protStatus &::=& \enabled \ALT \disabled \\[0.4em]
-% \textdom{Res} &\eqdef&
-% \{\, \rs = (\pres, \ghostRes) \mid
-% \pres \in \textdom{State} \uplus \{\munit\} \land \ghostRes \in \mcarp{\monoid} \,\} \\[0.5em]
-% (\pres, \ghostRes) \rtimes
-% (\pres', \ghostRes') &\eqdef&
-% \begin{cases}
-% (\pres, \ghostRes \mtimes \ghostRes') & \mbox{if $\pres' = \munit$ and $\ghostRes \mtimes \ghostRes' \neq \mzero$} \\
-% (\pres', \ghostRes \mtimes \ghostRes') & \mbox{if $\pres = \munit$ and $\ghostRes \mtimes \ghostRes' \neq \mzero$}
-% \end{cases}
-% \\[0.5em]
-% %
-% \rs \leq \rs' & \eqdef &
-% \Exists \rs''. \rs' = \rs \rtimes \rs''\\[1em]
-% %
-% \UPred(\textdom{Res}) &\eqdef&
-% \{\, p \subseteq \mathbb{N} \times \textdom{Res} \mid
-% \All (k,\rs) \in p.
-% \All j\leq k.
-% \All \rs' \geq \rs.
-% (j,\rs')\in p \,\}\\[0.5em]
-% \restr{p}{k} &\eqdef&
-% \{\, (j, \rs) \in p \mid j < k \,\}\\[0.5em]
-% p \nequiv{n} q & \eqdef & \restr{p}{n} = \restr{q}{n}\\[1em]
-% %
-% \textdom{PreProp} & \cong &
-% \latert\big( \textdom{World} \monra \UPred(\textdom{Res})
-% \big)\\[0.5em]
-% %
-% \textdom{World} & \eqdef &
-% \mathbb{N} \fpfn \textdom{PreProp}\\[0.5em]
-% %
-% w \nequiv{n} w' & \eqdef &
-% n = 0 \lor
-% \bigl(\dom(w) = \dom(w') \land \All i\in\dom(w). w(i) \nequiv{n} w'(i)\bigr)
-% \\[0.5em]
-% %
-% w \leq w' & \eqdef &
-% \dom(w) \subseteq \dom(w') \land \All i \in \dom(w). w(i) = w'(i)
-% \\[0.5em]
-% %
-% \textdom{Prop} & \eqdef & \textdom{World} \monra \UPred(\textdom{Res})
-% \end{array}
-% \]
-
-% For $p,q\in\UPred(\textdom{Res})$ with $p \nequiv{n} q$ defined
-% as above, $\UPred(\textdom{Res})$ is a
-% c.o.f.e.
-
-% $\textdom{Prop}$ is a c.o.f.e., which exists by America and Rutten's theorem~\cite{America-Rutten:JCSS89}.
-% We do not need to consider how the object is constructed.
-% We only need the isomorphism, given by maps
-% \begin{align*}
-% \wIso &: \latert \bigl(World \monra \UPred(\textdom{Res})\bigr) \to \textdom{PreProp} \\
-% \wIso^{-1} &: \textdom{PreProp} \to \latert \bigl(World \monra \UPred(\textdom{Res})\bigr)
-% \end{align*}
-% which are inverses to each other.
-% Note: this is an isomorphism in $\cal U$, i.e., $\wIso$ and
-% $\wIso^{-1}$ are both non-expansive.
-
-% $\textdom{World}$ is a c.o.f.e.\ with the family of equivalence
-% relations defined as shown above.
-
-% \subsection{Semantic structures: types and environments}
-
-% For a set $X$, write $\Delta X$ for the discrete c.o.f.e.\ with $x \nequiv{n}
-% x'$ iff $n = 0$ or $x = x'$
-% \[
-% \begin{array}[t]{@{}l@{\ }c@{\ }l@{}}
-% \Sem{\textsort{Unit}} &\eqdef& \Delta \{ \star \} \\
-% \Sem{\textsort{InvName}} &\eqdef& \Delta \mathbb{N} \\
-% \Sem{\textsort{InvMask}} &\eqdef& \Delta \pset{\mathbb{N}} \\
-% \Sem{\textsort{Monoid}} &\eqdef& \Delta |\monoid|
-% \end{array}
-% \qquad\qquad
-% \begin{array}[t]{@{}l@{\ }c@{\ }l@{}}
-% \Sem{\textsort{Val}} &\eqdef& \Delta \textdom{Val} \\
-% \Sem{\textsort{Exp}} &\eqdef& \Delta \textdom{Exp} \\
-% \Sem{\textsort{Ectx}} &\eqdef& \Delta \textdom{Ectx} \\
-% \Sem{\textsort{State}} &\eqdef& \Delta \textdom{State} \\
-% \end{array}
-% \qquad\qquad
-% \begin{array}[t]{@{}l@{\ }c@{\ }l@{}}
-% \Sem{\sort \times \sort'} &\eqdef& \Sem{\sort} \times \Sem{\sort} \\
-% \Sem{\sort \to \sort'} &\eqdef& \Sem{\sort} \to \Sem{\sort} \\
-% \Sem{\Prop} &\eqdef& \textdom{Prop} \\
-% \end{array}
-% \]
-
-% The balance of our signature $\Sig$ is interpreted as follows.
-% For each base type $\type$ not covered by the preceding table, we pick an object $X_\type$ in $\cal U$ and define
-% \[
-% \Sem{\type} \eqdef X_\type
-% \]
-% For each function symbol $\sigfn : \type_1, \dots, \type_n \to \type_{n+1} \in \SigFn$, we pick an arrow $\Sem{\sigfn} : \Sem{\type_1} \times \dots \times \Sem{\type_n} \to \Sem{\type_{n+1}}$ in $\cal U$.
-
-% An environment $\vctx$ is interpreted as the set of
-% maps $\rho$, with $\dom(\rho) = \dom(\vctx)$ and
-% $\rho(x)\in\Sem{\vctx(x)}$,
-% and
-% $\rho\nequiv{n} \rho' \iff n=0 \lor \bigl(\dom(\rho)=\dom(\rho') \land
-% \All x\in\dom(\rho). \rho(x) \nequiv{n} \rho'(x)\bigr)$.
-
-% \ralf{Re-check all the following definitions with the Coq development.}
-% %\typedsection{Validity}{valid : \pset{\textdom{Prop}} \in Sets}
-% %
-% %\begin{align*}
-% %valid(p) &\iff \All n \in \mathbb{N}. \All \rs \in \textdom{Res}. \All W \in \textdom{World}. (n, \rs) \in p(W)
-% %\end{align*}
-
-% \typedsection{Later modality}{\later : \textdom{Prop} \to \textdom{Prop} \in {\cal U}}
-
-% \begin{align*}
-% \later p &\eqdef \Lam W. \{\, (n + 1, r) \mid (n, r) \in p(W) \,\} \cup \{\, (0, r) \mid r \in \textdom{Res} \,\}
-% \end{align*}
-% \begin{lem}
-% $\later{}$ is well-defined: $\later {p}$ is a valid proposition (this amounts to showing non-expansiveness), and $\later{}$ itself is a \emph{contractive} map.
-% \end{lem}
-
-% \typedsection{Always modality}{\always{} : \textdom{Prop} \to \textdom{Prop} \in {\cal U}}
-
-% \begin{align*}
-% \always{p} \eqdef \Lam W. \{\, (n, r) \mid (n, \munit) \in p(W) \,\}
-% \end{align*}
-% \begin{lem}
-% $\always{}$ is well-defined: $\always{p}$ is a valid proposition (this amounts to showing non-expansiveness), and $\always{}$ itself is a non-expansive map.
-% \end{lem}
-
-% % PDS: p \Rightarrow q not defined.
-% %\begin{lem}\label{lem:always-impl-valid}
-% %\begin{align*}
-% %&\forall p, q \in \textdom{Prop}.~\\
-% %&\qquad
-% % (\forall n \in \mathbb{N}.~\forall \rs \in \textdom{Res}.~\forall W \in \textdom{World}.~(n, \rs) \in p(W) \Rightarrow (n, \rs) \in q(W)) \Leftrightarrow~valid(\always{(p \Rightarrow q)})
-% %\end{align*}
-% %\end{lem}
-
-% \typedsection{Invariant definition}{inv : \Delta(\mathbb{N}) \times \textdom{Prop} \to \textdom{Prop} \in {\cal U}}
-% \begin{align*}
-% \mathit{inv}(\iota, p) &\eqdef \Lam W. \{\, (n, r) \mid \iota\in\dom(W) \land W(\iota) \nequiv{n+1}_{\textdom{PreProp}} \wIso(p) \,\}
-% \end{align*}
-% \begin{lem}
-% $\mathit{inv}$ is well-defined: $\mathit{inv}(\iota, p)$ is a valid proposition (this amounts to showing non-expansiveness), and $\mathit{inv}$ itself is a non-expansive map.
-% \end{lem}
-
-% \typedsection{World satisfaction}{\wsat{-}{-}{-}{-} :
-% \textdom{State} \times
-% \pset{\mathbb{N}} \times
-% \textdom{Res} \times
-% \textdom{World} \to \psetdown{\mathbb{N}} \in {\cal U}}
-% \ralf{Make this Dave-compatible: Explicitly compose all the things in $s$}
-% \begin{align*}
-% \wsat{\state}{\mask}{\rs}{W} &=
-% \begin{aligned}[t]
-% \{\, n + 1 \in \mathbb{N} \mid &\Exists \rsB:\mathbb{N} \fpfn \textdom{Res}. (\rs \rtimes \rsB).\pres = \state \land{}\\
-% &\quad \All \iota \in \dom(W). \iota \in \dom(W) \leftrightarrow \iota \in \dom(\rsB) \land {}\\
-% &\quad\quad \iota \in \mask \ra (n, \rsB(\iota)) \in \wIso^{-1}(W(\iota))(W) \,\} \cup \{ 0 \}
-% \end{aligned}
-% \end{align*}
-% \begin{lem}\label{lem:fullsat-nonexpansive}
-% $\wsat{-}{-}{-}{-}$ is well-defined: It maps into $\psetdown{\mathbb{N}}$. (There is no need for it to be a non-expansive map, it doesn't itself live in $\cal U$.)
-% \end{lem}
-
-% \begin{lem}\label{lem:fullsat-weaken-mask}
-% \begin{align*}
-% \MoveEqLeft
-% \All \state \in \Delta(\textdom{State}).
-% \All \mask_1, \mask_2 \in \Delta(\pset{\mathbb{N}}).
-% \All \rs, \rsB \in \Delta(\textdom{Res}).
-% \All W \in \textdom{World}. \\&
-% \mask_1 \subseteq \mask_2 \implies (\wsat{\state}{\mask_2}{\rs}{W}) \subseteq (\wsat{\state}{\mask_1}{\rs}{W})
-% \end{align*}
-% \end{lem}
-
-% \begin{lem}\label{lem:nequal_ext_world}
-% \begin{align*}
-% &
-% \All n \in \mathbb{N}.
-% \All W_1, W_1', W_2 \in \textdom{World}.
-% W_1 \nequiv{n} W_2 \land W_1 \leq W_1' \implies \Exists W_2' \in \textdom{World}. W_1' \nequiv{n} W_2' \land W_2 \leq W_2'
-% \end{align*}
-% \end{lem}
-
-% \typedsection{Timeless}{\textit{timeless} : \textdom{Prop} \to \textdom{Prop}}
-
-% \begin{align*}
-% \textit{timeless}(p) \eqdef
-% \begin{aligned}[t]
-% \Lam W.
-% \{\, (n, r) &\mid \All W' \geq W. \All k \leq n. \All r' \in \textdom{Res}. \\
-% &\qquad
-% k > 0 \land (k - 1, r') \in p(W') \implies (k, r') \in p(W') \,\}
-% \end{aligned}
-% \end{align*}
-
-% \begin{lem}
-% \textit{timeless} is well-defined: \textit{timeless}(p) is a valid proposition, and \textit{timeless} itself is a non-expansive map.
-% \end{lem}
-
-% % PDS: \Ra undefined.
-% %\begin{lem}
-% %\begin{align*}
-% %&
-% % \All p \in \textdom{Prop}.
-% % \All \mask \in \pset{\mathbb{N}}.
-% %valid(\textit{timeless}(p) \Ra (\later p \vs[\mask][\mask] p))
-% %\end{align*}
-% %\end{lem}
-
-% \typedsection{View-shift}{\mathit{vs} : \Delta(\pset{\mathbb{N}}) \times \Delta(\pset{\mathbb{N}}) \times \textdom{Prop} \to \textdom{Prop} \in {\cal U}}
-% \begin{align*}
-% \mathit{vs}_{\mask_1}^{\mask_2}(q) &= \Lam W.
-% \begin{aligned}[t]
-% \{\, (n, \rs) &\mid \All W_F \geq W. \All \rs_F, \mask_F, \state. \All k \leq n.\\
-% &\qquad
-% k \in (\wsat{\state}{\mask_1 \cup \mask_F}{\rs \rtimes \rs_F}{W_F}) \land k > 0 \land \mask_F \sep (\mask_1 \cup \mask_2) \implies{} \\
-% &\qquad
-% \Exists W' \geq W_F. \Exists \rs'. k \in (\wsat{\state}{\mask_2 \cup \mask_F}{\rs' \rtimes \rs_F}{W'}) \land (k, \rs') \in q(W')
-% \,\}
-% \end{aligned}
-% \end{align*}
-% \begin{lem}
-% $\mathit{vs}$ is well-defined: $\mathit{vs}_{\mask_1}^{\mask_2}(q)$ is a valid proposition, and $\mathit{vs}$ is a non-expansive map.
-% \end{lem}
-
-
-% %\begin{lem}\label{lem:prim_view_shift_trans}
-% %\begin{align*}
-% %\MoveEqLeft
-% % \All \mask_1, \mask_2, \mask_3 \in \Delta(\pset{\mathbb{N}}).
-% % \All p, q \in \textdom{Prop}. \All W \in \textdom{World}.
-% % \All n \in \mathbb{N}.\\
-% %&
-% % \mask_2 \subseteq \mask_1 \cup \mask_3 \land
-% % \bigl(\All W' \geq W. \All r \in \textdom{Res}. \All k \leq n. (k, r) \in p(W') \implies (k, r) \in vs_{\mask_2}^{\mask_3}(q)(W')\bigr) \\
-% %&\qquad
-% % {}\implies \All r \in \textdom{Res}. (n, r) \in vs_{\mask_1}^{\mask_2}(p)(W) \implies (n, r) \in vs_{\mask_1}^{\mask_3}(q)(W)
-% %\end{align*}
-% %\end{lem}
-
-% % PDS: E_1 ==>> E_2 undefined.
-% %\begin{lem}
-% %\begin{align*}
-% %&
-% % \forall \mask_1, \mask_2, \mask_3 \in \Delta(\pset{\mathbb{N}}).~
-% % \forall p_1, p_2, p_3 \in \textdom{Prop}.~\\
-% %&\qquad
-% % \mask_2 \subseteq \mask_1 \cup \mask_3 \Rightarrow
-% % valid(((p_1 \vs[\mask_1][\mask_2] p_2) \land (p_2 \vs[\mask_2][\mask_3] p_3)) \Rightarrow (p_1 \vs[\mask_1][\mask_3] p_3))
-% %\end{align*}
-% %\end{lem}
-
-% %\begin{lem}
-% %\begin{align*}
-% %\MoveEqLeft
-% % \All \iota \in \mathbb{N}.
-% % \All p \in \textdom{Prop}.
-% % \All W \in \textdom{World}.
-% % \All \rs \in \textdom{Res}.
-% % \All n \in \mathbb{N}. \\
-% %&
-% % (n, \rs) \in inv(\iota, p)(W) \implies (n, \rs) \in vs_{\{ \iota \}}^{\emptyset}(\later p)(W)
-% %\end{align*}
-% %\end{lem}
-
-% % PDS: * undefined.
-% %\begin{lem}
-% %\begin{align*}
-% %&
-% % \forall \iota \in \mathbb{N}.~
-% % \forall p \in \textdom{Prop}.~
-% % \forall W \in \textdom{World}.~
-% % \forall \rs \in \textdom{Res}.~
-% % \forall n \in \mathbb{N}.~\\
-% %&\qquad
-% % (n, \rs) \in (inv(\iota, p) * \later p)(W) \Rightarrow (n, \rs) \in vs^{\{ \iota \}}_{\emptyset}(\top)(W)
-% %\end{align*}
-% %\end{lem}
-
-% % \begin{lem}
-% % \begin{align*}
-% % &
-% % \forall \mask_1, \mask_2 \in \Delta(\pset{\mathbb{N}}).~
-% % valid(\bot \vs[\mask_1][\mask_2] \bot)
-% % \end{align*}
-% % \end{lem}
-
-% % PDS: E_1 ==>> E_2 undefined.
-% %\begin{lem}
-% %\begin{align*}
-% %&
-% % \forall p, q \in \textdom{Prop}.~
-% % \forall \mask \in \pset{\mathbb{N}}.~
-% %valid(\always{(p \Rightarrow q)} \Rightarrow (p \vs[\mask][\mask] q))
-% %\end{align*}
-% %\end{lem}
+\paragraph{Semantic domain of assertions.}
+The first complicated task in building a model of full Iris is defining the semantic model of $\Prop$.
+We start by defining the functor that assembles the CMRAs we need to the global resource CMRA:
+\begin{align*}
+ \textdom{ResF}(\cofe) \eqdef{}& \record{\wld: \agm(\latert \cofe), \pres: \exm(\textdom{State}), \ghostRes: F(\cofe)}
+\end{align*}
+where $F$ is the user-chosen bifunctor from $\COFEs$ to $\CMRAs$.
+$\textdom{ResF}(\cofe)$ is a CMRA by lifting the individual CMRAs pointwise.
+Furthermore, if $F$ is locally contractive, then so is $\textdom{ResF}(-)$.
+
+Now we can write down the recursive domain equation:
+\[ \iPreProp \cong \UPred(\textdom{ResF}(\iPreProp)) \]
+$\iPreProp$ is a COFE, which exists by America and Rutten's theorem~\cite{America-Rutten:JCSS89}.
+We do not need to consider how the object is constructed.
+We only need the isomorphism, given by
+\begin{align*}
+ \Res &\eqdef \textdom{ResF}(\iPreProp) \\
+ \iProp &\eqdef \UPred(\Res) \\
+ \wIso &: \iProp \nfn \iPreProp \\
+ \wIso^{-1} &: \iPreProp \nfn \iProp
+\end{align*}
-% % PDS: E # E' and E_1 ==>> E_2 undefined.
-% %\begin{lem}
-% %\begin{align*}
-% %&
-% % \forall p_1, p_2, p_3 \in \textdom{Prop}.~
-% % \forall \mask_1, \mask_2, \mask \in \pset{\mathbb{N}}.~
-% %valid(\mask \sep \mask_1 \Ra \mask \sep \mask_2 \Ra (p_1 \vs[\mask_1][\mask_2] p_2) \Rightarrow (p_1 * p_3 \vs[\mask_1 \cup \mask][\mask_2 \cup \mask] p_2 * p_3))
-% %\end{align*}
-% %\end{lem}
+We then pick $\iProp$ as the interpretation of $\Prop$:
+\[ \Sem{\Prop} \eqdef \iProp \]
-% \typedsection{Weakest precondition}{\mathit{wp} : \Delta(\pset{\mathbb{N}}) \times \Delta(\textdom{Exp}) \times (\Delta(\textdom{Val}) \to \textdom{Prop}) \to \textdom{Prop} \in {\cal U}}
-% % \begin{align*}
-% % \mathit{wp}_\mask(\expr, q) &\eqdef \Lam W.
-% % \begin{aligned}[t]
-% % \{\, (n, \rs) &\mid \All W_F \geq W; k \leq n; \rs_F; \state; \mask_F \sep \mask. k > 0 \land k \in (\wsat{\state}{\mask \cup \mask_F}{\rs \rtimes \rs_F}{W_F}) \implies{}\\
-% % &\qquad
-% % (\expr \in \textdom{Val} \implies \Exists W' \geq W_F. \Exists \rs'. \\
-% % &\qquad\qquad
-% % k \in (\wsat{\state}{\mask \cup \mask_F}{\rs' \rtimes \rs_F}{W'}) \land (k, \rs') \in q(\expr)(W'))~\land \\
-% % &\qquad
-% % (\All\ectx,\expr_0,\expr'_0,\state'. \expr = \ectx[\expr_0] \land \cfg{\state}{\expr_0} \step \cfg{\state'}{\expr'_0} \implies \Exists W' \geq W_F. \Exists \rs'. \\
-% % &\qquad\qquad
-% % k - 1 \in (\wsat{\state'}{\mask \cup \mask_F}{\rs' \rtimes \rs_F}{W'}) \land (k-1, \rs') \in wp_\mask(\ectx[\expr_0'], q)(W'))~\land \\
-% % &\qquad
-% % (\All\ectx,\expr'. \expr = \ectx[\fork{\expr'}] \implies \Exists W' \geq W_F. \Exists \rs', \rs_1', \rs_2'. \\
-% % &\qquad\qquad
-% % k - 1 \in (\wsat{\state}{\mask \cup \mask_F}{\rs' \rtimes \rs_F}{W'}) \land \rs' = \rs_1' \rtimes \rs_2'~\land \\
-% % &\qquad\qquad
-% % (k-1, \rs_1') \in \mathit{wp}_\mask(\ectx[\textsf{fRet}], q)(W') \land
-% % (k-1, \rs_2') \in \mathit{wp}_\top(\expr', \Lam\any. \top)(W'))
-% % \,\}
-% % \end{aligned}
-% % \end{align*}
-% \begin{lem}
-% $\mathit{wp}$ is well-defined: $\mathit{wp}_{\mask}(\expr, q)$ is a valid proposition, and $\mathit{wp}$ is a non-expansive map. Besides, the dependency on the recursive occurrence is contractive, so $\mathit{wp}$ has a fixed-point.
-% \end{lem}
+\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.
-% \begin{lem}
-% $\mathit{wp}$ on values and non-mask-changing $\mathit{vs}$ agree:
-% \[ \mathit{wp}_\mask(\val, q) = \mathit{vs}_{\mask}^{\mask}(q \: \val) \]
-% \end{lem}
+\typedsection{World satisfaction}{\wsat{-}{-}{-} :
+ \Delta\textdom{State} \times
+ \Delta\pset{\mathbb{N}} \times
+ \textdom{Res} \nfn \SProp }
+\begin{align*}
+ \wsatpre(n, \mask, \state, \rss, \rs) & \eqdef \begin{inbox}[t]
+ \rs \in \mval_{n+1} \land \rs.\pres = \exinj(\sigma) \land
+ \dom(\rss) \subseteq \mask \cap \dom( \rs.\wld) \land {}\\
+ \All\iname \in \mask, \prop. \rs.\wld(\iname) \nequiv{n+1} \aginj(\latertinj(\wIso(\prop))) \Ra n \in \prop(\rss(\iname))
+ \end{inbox}\\
+ \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*}
-% \begin{align*}
-% \Sem{\vctx \proves x : \sort}_\gamma &= \gamma(x) \\
-% \Sem{\vctx \proves \sigfn(\term_1, \dots, \term_n) : \type_{n+1}}_\gamma &= \Sem{\sigfn}(\Sem{\vctx \proves \term_1 : \type_1}_\gamma, \dots, \Sem{\vctx \proves \term_n : \type_n}_\gamma) \ \WHEN \sigfn : \type_1, \dots, \type_n \to \type_{n+1} \in \SigFn \\
-% \Sem{\vctx \proves \Lam x. \term : \sort \to \sort'}_\gamma &=
-% \Lam v : \Sem{\sort}. \Sem{\vctx, x : \sort \proves \term : \sort'}_{\gamma[x \mapsto v]} \\
-% \Sem{\vctx \proves \term~\termB : \sort'}_\gamma &=
-% \Sem{\vctx \proves \term : \sort \to \sort'}_\gamma(\Sem{\vctx \proves \termB : \sort}_\gamma) \\
-% \Sem{\vctx \proves \unitval : \unitsort}_\gamma &= \star \\
-% \Sem{\vctx \proves (\term_1, \term_2) : \sort_1 \times \sort_2}_\gamma &= (\Sem{\vctx \proves \term_1 : \sort_1}_\gamma, \Sem{\vctx \proves \term_2 : \sort_2}_\gamma) \\
-% \Sem{\vctx \proves \pi_i~\term : \sort_1}_\gamma &= \pi_i(\Sem{\vctx \proves \term : \sort_1 \times \sort_2}_\gamma)
-% \end{align*}
-% %
-% \begin{align*}
-% \Sem{\vctx \proves \mzero : \textsort{Monoid}}_\gamma &= \mzero \\
-% \Sem{\vctx \proves \munit : \textsort{Monoid}}_\gamma &= \munit \\
-% \Sem{\vctx \proves \melt \mtimes \meltB : \textsort{Monoid}}_\gamma &=
-% \Sem{\vctx \proves \melt : \textsort{Monoid}}_\gamma \mtimes \Sem{\vctx \proves \meltB : \textsort{Monoid}}_\gamma
-% \end{align*}
-% %
-% \Sem{\vctx \proves \MU x. \pred : \sort \to \Prop}_\gamma &\eqdef
-% \mathit{fix}(\Lam v : \Sem{\sort \to \Prop}. \Sem{\vctx, x : \sort \to \Prop \proves \pred : \sort \to \Prop}_{\gamma[x \mapsto v]}) \\
+\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 {}\\&
+ \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}
- % \Sem{\vctx \proves \knowInv{\iname}{\prop} : \Prop}_\gamma &=
- % inv(\Sem{\vctx \proves \iname : \textsort{InvName}}_\gamma, \Sem{\vctx \proves \prop : \Prop}_\gamma) \\
- % \Sem{\vctx \proves \ownGGhost{\melt} : \Prop}_\gamma &=
- % \Lam W. \{\, (n, \rs) \mid \rs.\ghostRes \geq \Sem{\vctx \proves \melt : \textsort{Monoid}}_\gamma \,\} \\
- % \Sem{\vctx \proves \ownPhys{\state} : \Prop}_\gamma &=
- % \Lam W. \{\, (n, \rs) \mid \rs.\pres = \Sem{\vctx \proves \state : \textsort{State}}_\gamma \,\}
+$\textdom{wp}$ 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\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 {}&\\
+ &\qquad\qquad (\expr_\f = \bot \lor m \in \textdom{wp}(\top, \expr_\f, \Lam\any.\Lam\any.\mathbb{N})(\rsB_2))
+ \end{aligned}} \\
+ \textdom{wp}_\mask(\expr, \pred) &\eqdef \mathit{fix}(\textdom{pre-wp})(\mask, \expr, \pred)
+\end{align*}
-% %
-% \begin{align*}
-% \Sem{\vctx \proves \pvsA{\prop}{\mask_1}{\mask_2} : \Prop}_\gamma &=
-% \textdom{vs}^{\Sem{\vctx \proves \mask_2 : \textsort{InvMask}}_\gamma}_{\Sem{\vctx \proves \mask_1 : \textsort{InvMask}}_\gamma}(\Sem{\vctx \proves \prop : \Prop}_\gamma) \\
-% \Sem{\vctx \proves \dynA{\expr}{\pred}{\mask} : \Prop}_\gamma &=
-% \textdom{wp}_{\Sem{\vctx \proves \mask : \textsort{InvMask}}_\gamma}(\Sem{\vctx \proves \expr : \textsort{Exp}}_\gamma, \Sem{\vctx \proves \pred : \textsort{Val} \to \Prop}_\gamma) \\
-% \Sem{\vctx \proves \wtt{\timeless{\prop}}{\Prop}}_\gamma &=
-% \textdom{timeless}(\Sem{\vctx \proves \prop : \Prop}_\gamma)
-% \end{align*}
+\typedsection{Interpretation of program logic assertions}{\Sem{\vctx \proves \term : \Prop} : \Sem{\vctx} \nfn \iProp}
-% \typedsection{Interpretation of entailment}{\Sem{\vctx \mid \pfctx \proves \prop} : 2 \in \mathit{Sets}}
+\begin{align*}
+ \Sem{\vctx \proves \knowInv{\iname}{\prop} : \Prop}_\gamma &\eqdef \ownM{[\iname \mapsto \aginj(\latertinj(\wIso(\prop)))], \munit, \munit} \\
+ \Sem{\vctx \proves \ownGGhost{\melt} : \Prop}_\gamma &\eqdef \ownM{\munit, \munit, \melt} \\
+ \Sem{\vctx \proves \ownPhys{\state} : \Prop}_\gamma &\eqdef \ownM{\munit, \exinj(\state), \munit} \\
+ \Sem{\vctx \proves \pvs[\mask_1][\mask_2] \prop : \Prop}_\gamma &\eqdef
+ \textdom{pvs}^{\Sem{\vctx \proves \mask_2 : \textlog{InvMask}}_\gamma}_{\Sem{\vctx \proves \mask_1 : \textlog{InvMask}}_\gamma}(\Sem{\vctx \proves \prop : \Prop}_\gamma) \\
+ \Sem{\vctx \proves \wpre{\expr}[\mask]{\Ret\var.\prop} : \Prop}_\gamma &\eqdef
+ \textdom{wp}_{\Sem{\vctx \proves \mask : \textlog{InvMask}}_\gamma}(\Sem{\vctx \proves \expr : \textlog{Expr}}_\gamma, \Lam\val. \Sem{\vctx \proves \prop : \Prop}_{\gamma[\var\mapsto\val]})
+\end{align*}
-% \[
-% \Sem{\vctx \mid \pfctx \proves \propB} \eqdef
-% \begin{aligned}[t]
-% \MoveEqLeft
-% \forall n \in \mathbb{N}.\;
-% \forall W \in \textdom{World}.\;
-% \forall \rs \in \textdom{Res}.\;
-% \forall \gamma \in \Sem{\vctx},\;
-% \\&
-% \bigl(\All \propB \in \pfctx. (n, \rs) \in \Sem{\vctx \proves \propB : \Prop}_\gamma(W)\bigr)
-% \implies (n, \rs) \in \Sem{\vctx \proves \prop : \Prop}_\gamma(W)
-% \end{aligned}
-% \]
+\paragraph{Remaining semantic domains, and interpretation of non-assertion terms.}
+
+The remaining domains are interpreted as follows:
+\[
+\begin{array}[t]{@{}l@{\ }c@{\ }l@{}}
+\Sem{\textlog{InvName}} &\eqdef& \Delta \mathbb{N} \\
+\Sem{\textlog{InvMask}} &\eqdef& \Delta \pset{\mathbb{N}} \\
+\Sem{\textlog{M}} &\eqdef& F(\iProp)
+\end{array}
+\qquad\qquad
+\begin{array}[t]{@{}l@{\ }c@{\ }l@{}}
+\Sem{\textlog{Val}} &\eqdef& \Delta \textdom{Val} \\
+\Sem{\textlog{Expr}} &\eqdef& \Delta \textdom{Expr} \\
+\Sem{\textlog{State}} &\eqdef& \Delta \textdom{State} \\
+\end{array}
+\qquad\qquad
+\begin{array}[t]{@{}l@{\ }c@{\ }l@{}}
+\Sem{1} &\eqdef& \Delta \{ () \} \\
+\Sem{\type \times \type'} &\eqdef& \Sem{\type} \times \Sem{\type} \\
+\Sem{\type \to \type'} &\eqdef& \Sem{\type} \nfn \Sem{\type} \\
+\end{array}
+\]
+
+The balance of our signature $\Sig$ is interpreted as follows.
+For each base type $\type$ not covered by the preceding table, we pick an object $X_\type$ in $\cal U$ and define
+\[
+\Sem{\type} \eqdef X_\type
+\]
+For each function symbol $\sigfn : \type_1, \dots, \type_n \to \type_{n+1} \in \SigFn$, we pick a function $\Sem{\sigfn} : \Sem{\type_1} \times \dots \times \Sem{\type_n} \nfn \Sem{\type_{n+1}}$.
+
+\typedsection{Interpretation of non-propositional terms}{\Sem{\vctx \proves \term : \type} : \Sem{\vctx} \nfn \Sem{\type}}
+\begin{align*}
+ \Sem{\vctx \proves x : \type}_\gamma &\eqdef \gamma(x) \\
+ \Sem{\vctx \proves \sigfn(\term_1, \dots, \term_n) : \type_{n+1}}_\gamma &\eqdef \Sem{\sigfn}(\Sem{\vctx \proves \term_1 : \type_1}_\gamma, \dots, \Sem{\vctx \proves \term_n : \type_n}_\gamma) \\
+ \Sem{\vctx \proves \Lam \var:\type. \term : \type \to \type'}_\gamma &\eqdef
+ \Lam \termB : \Sem{\type}. \Sem{\vctx, \var : \type \proves \term : \type}_{\gamma[\var \mapsto \termB]} \\
+ \Sem{\vctx \proves \term(\termB) : \type'}_\gamma &\eqdef
+ \Sem{\vctx \proves \term : \type \to \type'}_\gamma(\Sem{\vctx \proves \termB : \type}_\gamma) \\
+ \Sem{\vctx \proves \MU \var:\type. \term : \type}_\gamma &\eqdef
+ \mathit{fix}(\Lam \termB : \Sem{\type}. \Sem{\vctx, x : \type \proves \term : \type}_{\gamma[x \mapsto \termB]}) \\
+ ~\\
+ \Sem{\vctx \proves () : 1}_\gamma &\eqdef () \\
+ \Sem{\vctx \proves (\term_1, \term_2) : \type_1 \times \type_2}_\gamma &\eqdef (\Sem{\vctx \proves \term_1 : \type_1}_\gamma, \Sem{\vctx \proves \term_2 : \type_2}_\gamma) \\
+ \Sem{\vctx \proves \pi_i(\term) : \type_i}_\gamma &\eqdef \pi_i(\Sem{\vctx \proves \term : \type_1 \times \type_2}_\gamma) \\
+ ~\\
+ \Sem{\vctx \proves \munit : \textlog{M}}_\gamma &\eqdef \munit \\
+ \Sem{\vctx \proves \mcore\melt : \textlog{M}}_\gamma &\eqdef \mcore{\Sem{\vctx \proves \melt : \textlog{M}}_\gamma} \\
+ \Sem{\vctx \proves \melt \mtimes \meltB : \textlog{M}}_\gamma &\eqdef
+ \Sem{\vctx \proves \melt : \textlog{M}}_\gamma \mtimes \Sem{\vctx \proves \meltB : \textlog{M}}_\gamma
+\end{align*}
+%
+
+An environment $\vctx$ is interpreted as the set of
+finite partial functions $\rho$, with $\dom(\rho) = \dom(\vctx)$ and
+$\rho(x)\in\Sem{\vctx(x)}$.
+
+\paragraph{Logical entailment.}
+We can now define \emph{semantic} logical entailment.
+
+\typedsection{Interpretation of entailment}{\Sem{\vctx \mid \pfctx \proves \prop} : 2}
+
+\[
+\Sem{\vctx \mid \pfctx \proves \propB} \eqdef
+\begin{aligned}[t]
+\MoveEqLeft
+\forall n \in \mathbb{N}.\;
+\forall \rs \in \textdom{Res}.\;
+\forall \gamma \in \Sem{\vctx},\;
+\\&
+\bigl(\All \propB \in \pfctx. n \in \Sem{\vctx \proves \propB : \Prop}_\gamma(\rs)\bigr)
+\Ra n \in \Sem{\vctx \proves \prop : \Prop}_\gamma(\rs)
+\end{aligned}
+\]
+
+The soundness statement of the logic reads
+\[ \vctx \mid \pfctx \proves \prop \Ra \Sem{\vctx \mid \pfctx \proves \prop} \]
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