model.tex

\section{Model and semantics}

The semantics closely follows the ideas laid out in~\cite{catlogic}.

\subsection{Generic model of base 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)}
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.

We introduce an additional logical connective $\ownM\melt$, which will later be used to encode all of $\knowInv\iname\prop$, $\ownGGhost\melt$ and $\ownPhys\state$.

\begin{align*}
	\Sem{\vctx \proves t =_\type u : \Prop}_\gamma &\eqdef
	\Lam \any. \setComp{n}{\Sem{\vctx \proves t : \type}_\gamma \nequiv{n} \Sem{\vctx \proves u : \type}_\gamma} \\
	\Sem{\vctx \proves \FALSE : \Prop}_\gamma &\eqdef \Lam \any. \emptyset \\
	\Sem{\vctx \proves \TRUE : \Prop}_\gamma &\eqdef \Lam \any. \mathbb{N} \\
	\Sem{\vctx \proves \prop \land \propB : \Prop}_\gamma &\eqdef
	\Lam \melt. \Sem{\vctx \proves \prop : \Prop}_\gamma(\melt) \cap \Sem{\vctx \proves \propB : \Prop}_\gamma(\melt) \\
	\Sem{\vctx \proves \prop \lor \propB : \Prop}_\gamma &\eqdef
	\Lam \melt. \Sem{\vctx \proves \prop : \Prop}_\gamma(\melt) \cup \Sem{\vctx \proves \propB : \Prop}_\gamma(\melt) \\
	\Sem{\vctx \proves \prop \Ra \propB : \Prop}_\gamma &\eqdef
	\Lam \melt. \setComp{n}{\begin{aligned}
            \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}}\\
	\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) } \\
	\Sem{\vctx \proves \Exists x : \type. \prop : \Prop}_\gamma &\eqdef
        \Lam \melt. \setComp{n}{ \Exists v \in \Sem{\type}. n \in \Sem{\vctx, x : \type \proves \prop : \Prop}_{\gamma[x \mapsto v]}(\melt) } \\
  ~\\
	\Sem{\vctx \proves \always{\prop} : \Prop}_\gamma &\eqdef \Lam\melt. \Sem{\vctx \proves \prop : \Prop}_\gamma(\mcore\melt) \\
	\Sem{\vctx \proves \later{\prop} : \Prop}_\gamma &\eqdef \Lam\melt. \setComp{n}{n = 0 \lor n-1 \in \Sem{\vctx \proves \prop : \Prop}_\gamma(\melt)}\\
	\Sem{\vctx \proves \prop * \propB : \Prop}_\gamma &\eqdef \Lam\melt. \setComp{n}{\begin{aligned}\Exists \meltB_1, \meltB_2. &\melt \nequiv{n} \meltB_1 \mtimes \meltB_2 \land {}\\& n \in \Sem{\vctx \proves \prop : \Prop}_\gamma(\meltB_1) \land n \in \Sem{\vctx \proves \propB : \Prop}_\gamma(\meltB_2)\end{aligned}}
\\
	\Sem{\vctx \proves \prop \wand \propB : \Prop}_\gamma &\eqdef
	\Lam \melt. \setComp{n}{\begin{aligned}
            \All m, \meltB.& m \leq n \land  \melt\mtimes\meltB \in \mval_m \Ra {} \\
            & m \in \Sem{\vctx \proves \prop : \Prop}_\gamma(\meltB) \Ra {}\\& m \in \Sem{\vctx \proves \propB : \Prop}_\gamma(\melt\mtimes\meltB)\end{aligned}} \\
        \Sem{\vctx \proves \ownM{\melt} : \Prop}_\gamma &\eqdef \Lam\meltB. \setComp{n}{\melt \mincl[n] \meltB}  \\
        \Sem{\vctx \proves \mval(\melt) : \Prop}_\gamma &\eqdef \Lam\any. \setComp{n}{\melt \in \mval_n} \\
\end{align*}


%\subsection{Iris model}

% \subsection{Semantic structures: propositions}
% \ralf{This needs to be synced with the Coq development again.}

% \[
% \begin{array}[t]{rcl}
% %  \protStatus &::=& \enabled \ALT \disabled \\[0.4em]
% \textdom{Res} &\eqdef&
% \{\, \rs = (\pres, \ghostRes) \mid
% \pres \in \textdom{State} \uplus \{\munit\} \land \ghostRes \in \mcarp{\monoid} \,\} \\[0.5em]
% (\pres, \ghostRes) \rtimes
% (\pres', \ghostRes') &\eqdef&
% \begin{cases}
% (\pres, \ghostRes \mtimes \ghostRes')  & \mbox{if $\pres' = \munit$ and $\ghostRes \mtimes \ghostRes' \neq \mzero$} \\
% (\pres', \ghostRes \mtimes \ghostRes') & \mbox{if $\pres = \munit$ and $\ghostRes \mtimes \ghostRes' \neq \mzero$}
% \end{cases}
% \\[0.5em]
% %
% \rs \leq \rs' & \eqdef &
% \Exists \rs''. \rs' = \rs \rtimes \rs''\\[1em]
% %
% \UPred(\textdom{Res}) &\eqdef& 
% \{\, p \subseteq \mathbb{N} \times \textdom{Res} \mid
% \All (k,\rs) \in p.
% \All j\leq k.
% \All \rs' \geq \rs.
% (j,\rs')\in p \,\}\\[0.5em]
% \restr{p}{k} &\eqdef& 
% \{\, (j, \rs) \in p \mid j < k \,\}\\[0.5em]
% p \nequiv{n} q & \eqdef & \restr{p}{n} = \restr{q}{n}\\[1em]
% %
% \textdom{PreProp} & \cong  &
% \latert\big( \textdom{World} \monra \UPred(\textdom{Res})
% \big)\\[0.5em]
% %
% \textdom{World} & \eqdef &
% \mathbb{N} \fpfn \textdom{PreProp}\\[0.5em]
% %
% w \nequiv{n} w' & \eqdef & 
% n = 0 \lor
% \bigl(\dom(w) = \dom(w') \land \All i\in\dom(w). w(i) \nequiv{n} w'(i)\bigr)
% \\[0.5em]
% %
% w \leq w' & \eqdef & 
% \dom(w) \subseteq \dom(w') \land \All i \in \dom(w). w(i) = w'(i) 
% \\[0.5em]
% %
% \textdom{Prop} & \eqdef & \textdom{World} \monra \UPred(\textdom{Res})
% \end{array}
% \]

% 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}

% % 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}

% \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}

% \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}

% \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]}) \\


	% \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 \,\}


% %
% \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 entailment}{\Sem{\vctx \mid \pfctx \proves \prop} : 2 \in \mathit{Sets}}

% \[
% \Sem{\vctx \mid \pfctx \proves \propB} \eqdef
% \begin{aligned}[t]
% \MoveEqLeft
% \forall n \in \mathbb{N}.\;
% \forall W \in \textdom{World}.\;
% \forall \rs \in \textdom{Res}.\; 
% \forall \gamma \in \Sem{\vctx},\;
% \\&
% \bigl(\All \propB \in \pfctx. (n, \rs) \in \Sem{\vctx \proves \propB : \Prop}_\gamma(W)\bigr)
% \implies (n, \rs) \in \Sem{\vctx \proves \prop : \Prop}_\gamma(W)
% \end{aligned}
% \]

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