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\section{Model and semantics}

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\ralf{What also needs to be done here: Define uPred and its later function; define black later; define the resource CMRA}

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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}
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\a
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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
\[
\All k. \Exists n. \All i,j\geq n. x_i \nequiv{k} x_j.
\]
A \emph{limit} of a chain $(x_i)_{i\in\mathbb{N}}$ is an element
$x\in X$ such that
\[
\All n. \Exists k. \All i\geq k. x_i \nequiv{n} x.
\]
An o.f.e.\ $(X,(\nequiv{n})_{n\in\mathbb{N}})$ is \emph{complete} 
if all chains have a limit.
A complete o.f.e.\ is called a c.o.f.e.\ (pronounced ``coffee'').
When the family of equivalence relations is clear from context we
simply
write $X$ for a c.o.f.e.\ $(X,(\nequiv{n})_{n\in\mathbb{N}})$.


Let $\cal U$ be the category of c.o.f.e.'s and nonexpansive maps.

Products and function spaces are defined as follows.
For c.o.f.e.'s $(X,(\nequivset{n}{X})_{n\in\mathbb{N}})$ and
$(Y,(\nequivset{n}{Y})_{n\in\mathbb{N}})$, their product 
is 
$(X\times Y, (\nequiv{n})_{n\in\mathbb{N}}),$
where
\[
(x,y) \nequiv{n} (x',y') \iff
x \nequiv{n} x' \land
y \nequiv{n} y'.
\]
The function space is
\[
(\{\, f : X\to Y \mid f \text{ is non-expansive}\,\}, (\nequiv{n})_{n\in\mathbb{N}}),
\]
where
\[
f \nequiv{n} g \iff
\All x. f(x)  \nequiv{n}  g(x).
\]

For a c.o.f.e.\ $(X,(\nequiv{n}_{n\in\mathbb{N}}))$, 
$\latert (X,(\nequiv{n}_{n\in\mathbb{N}}))$ is the c.o.f.e.\@
$(X,(\nequivB{n}_{n\in\mathbb{N}}))$,  where
\[
x \nequivB{n} x' \iff \begin{cases}
\top	&\IF n=0 \\
x \nequiv{n-1} x' &\IF n>0
\end{cases}
\]

(Sidenote: $\latert$ extends to a functor on $\cal U$ by the identity
action on morphisms).


\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&
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\{\, \rs = (\pres, \ghostRes) \mid
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\pres \in \textdom{State} \uplus \{\munit\} \land \ghostRes \in \mcarp{\monoid} \,\} \\[0.5em]
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(\pres, \ghostRes) \rtimes
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(\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]
%
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\rs \leq \rs' & \eqdef &
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\Exists \rs''. \rs' = \rs \rtimes \rs''\\[1em]
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%
\UPred(\textdom{Res}) &\eqdef& 
\{\, p \subseteq \mathbb{N} \times \textdom{Res} \mid
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\All (k,\rs) \in p.
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\All j\leq k.
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\All \rs' \geq \rs.
(j,\rs')\in p \,\}\\[0.5em]
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\restr{p}{k} &\eqdef& 
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\{\, (j, \rs) \in p \mid j < k \,\}\\[0.5em]
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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 $x = x'$
\[
\begin{array}[t]{@{}l@{\ }c@{\ }l@{}}
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\Sem{\textsort{Unit}} &\eqdef& \Delta \{ \star \} \\
\Sem{\textsort{InvName}} &\eqdef& \Delta \mathbb{N}  \\
\Sem{\textsort{InvMask}} &\eqdef& \Delta \pset{\mathbb{N}} \\
\Sem{\textsort{Monoid}} &\eqdef& \Delta |\monoid|
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\end{array}
\qquad\qquad
\begin{array}[t]{@{}l@{\ }c@{\ }l@{}}
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\Sem{\textsort{Val}} &\eqdef& \Delta \textdom{Val} \\
\Sem{\textsort{Exp}} &\eqdef& \Delta \textdom{Exp} \\
\Sem{\textsort{Ectx}} &\eqdef& \Delta \textdom{Ectx} \\
\Sem{\textsort{State}} &\eqdef& \Delta \textdom{State} \\
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\end{array}
\qquad\qquad
\begin{array}[t]{@{}l@{\ }c@{\ }l@{}}
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\Sem{\sort \times \sort'} &\eqdef& \Sem{\sort} \times \Sem{\sort} \\
\Sem{\sort \to \sort'} &\eqdef& \Sem{\sort} \to \Sem{\sort} \\
\Sem{\Prop} &\eqdef& \textdom{Prop} \\
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\end{array}
\]

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The balance of our signature $\Sig$ is interpreted as follows.
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For each base type $\type$ not covered by the preceding table, we pick an object $X_\type$ in $\cal U$ and define
\[
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\Sem{\type} \eqdef X_\type
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\]
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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 $\cal U$.
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An environment $\vctx$ is interpreted as the set of
maps $\rho$, with $\dom(\rho) = \dom(\vctx)$ and
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$\rho(x)\in\Sem{\vctx(x)}$,
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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*}
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%valid(p) &\iff \All n \in \mathbb{N}. \All \rs \in \textdom{Res}. \All W \in \textdom{World}. (n, \rs) \in p(W)
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%\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
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%  (\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)})
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%\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}{\fullSat{-}{-}{-}{-} : 
	\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*}
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	\fullSat{\state}{\mask}{\rs}{W} &=
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	\begin{aligned}[t]
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		\{\, n + 1 \in \mathbb{N} \mid &\Exists  \rsB:\mathbb{N} \fpfn \textdom{Res}. (\rs \rtimes \rsB).\pres = \state \land{}\\
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		&\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 \}
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	\end{aligned}
\end{align*}
\begin{lem}\label{lem:fullsat-nonexpansive}
	$\fullSat{-}{-}{-}{-}$ 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}}).
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		\All \rs, \rsB \in \Delta(\textdom{Res}).
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		\All W \in \textdom{World}. \\&
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		\mask_1 \subseteq \mask_2 \implies (\fullSat{\state}{\mask_2}{\rs}{W}) \subseteq (\fullSat{\state}{\mask_1}{\rs}{W})
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	\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]
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		\{\, (n, \rs) &\mid \All W_F \geq W. \All \rs_F, \mask_F, \state. \All k \leq n.\\
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		&\qquad 
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		k \in (\fullSat{\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{} \\
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		&\qquad
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		\Exists W' \geq W_F. \Exists \rs'. k \in (\fullSat{\state}{\mask_2 \cup \mask_F}{\rs' \rtimes \rs_F}{W'}) \land (k, \rs') \in q(W')
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		\,\}
	\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}.
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%  \All \rs \in \textdom{Res}.
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%  \All n \in \mathbb{N}. \\
%&
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%  (n, \rs) \in inv(\iota, p)(W) \implies (n, \rs) \in vs_{\{ \iota \}}^{\emptyset}(\later p)(W)
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%\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}.~
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%  \forall \rs \in \textdom{Res}.~
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%  \forall n \in \mathbb{N}.~\\
%&\qquad
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%  (n, \rs) \in (inv(\iota, p) * \later p)(W) \Rightarrow (n, \rs) \in vs^{\{ \iota \}}_{\emptyset}(\top)(W)
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%\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}}

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% \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 (\fullSat{\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 (\fullSat{\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 (\fullSat{\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 (\fullSat{\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*}
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\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}

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\typedsection{Interpretation of terms}{\Sem{\vctx \proves \term : \sort} : \Sem{\vctx} \to \Sem{\sort} \in {\cal U}}
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%A term $\vctx \proves \term : \sort$ is interpreted as a non-expansive map from $\Sem{\vctx}$ to $\Sem{\sort}$.
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\begin{align*}
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	\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) \\
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	\Sem{\vctx \proves \unitval : \unitsort}_\gamma &= \star \\
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	\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)
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\end{align*}
%
\begin{align*}
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	\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
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\end{align*}
%
\begin{align*}
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	\Sem{\vctx \proves t =_\sort u : \Prop}_\gamma &=
	\Lam W. \{\, (n, r) \mid \Sem{\vctx \proves t : \sort}_\gamma \nequiv{n+1} \Sem{\vctx \proves u : \sort}_\gamma \,\} \\
	\Sem{\vctx \proves \FALSE : \Prop}_\gamma &= \Lam W. \emptyset \\
	\Sem{\vctx \proves \TRUE : \Prop}_\gamma &= \Lam W. \mathbb{N} \times \textdom{Res} \\
	\Sem{\vctx \proves P \land Q : \Prop}_\gamma &=
	\Lam W. \Sem{\vctx \proves P : \Prop}_\gamma(W) \cap \Sem{\vctx \proves Q : \Prop}_\gamma(W) \\
	\Sem{\vctx \proves P \lor Q : \Prop}_\gamma &=
	\Lam W. \Sem{\vctx \proves P : \Prop}_\gamma(W) \cup \Sem{\vctx \proves Q : \Prop}_\gamma(W) \\
	\Sem{\vctx \proves P \Ra Q : \Prop}_\gamma &=
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	\Lam W. \begin{aligned}[t]
		\{\, (n, r) &\mid \All n' \leq n. \All W' \geq W. \All r' \geq r. \\
		&\qquad
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		(n', r') \in \Sem{\vctx \proves P : \Prop}_\gamma(W')~ \\
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		&\qquad 
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		\implies (n', r') \in \Sem{\vctx \proves Q : \Prop}_\gamma(W') \,\}
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	\end{aligned} \\
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	\Sem{\vctx \proves \All x : \sort. P : \Prop}_\gamma &=
	\Lam W. \{\, (n, r) \mid \All v \in \Sem{\sort}. (n, r) \in \Sem{\vctx, x : \sort \proves P : \Prop}_{\gamma[x \mapsto v]}(W) \,\} \\
	\Sem{\vctx \proves \Exists x : \sort. P : \Prop}_\gamma &=
	\Lam W. \{\, (n, r) \mid \Exists v \in \Sem{\sort}. (n, r) \in \Sem{\vctx, x : \sort \proves P : \Prop}_{\gamma[x \mapsto v]}(W) \,\}
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\end{align*}
%
\begin{align*}
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	\Sem{\vctx \proves \always{\prop} : \Prop}_\gamma &= \always{\Sem{\vctx \proves \prop : \Prop}_\gamma} \\
	\Sem{\vctx \proves \later{\prop} : \Prop}_\gamma &= \later \Sem{\vctx \proves \prop : \Prop}_\gamma\\
	\Sem{\vctx \proves \MU x. \pred : \sort \to \Prop}_\gamma &=
	\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 \prop * \propB : \Prop}_\gamma &=
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	\begin{aligned}[t]
		\Lam W. \{\, (n, r) &\mid \Exists r_1, r_2. r = r_1 \bullet r_2 \land{} \\
		&\qquad
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		(n, r_1) \in \Sem{\vctx \proves \prop : \Prop}_\gamma \land{} \\
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		&\qquad
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		(n, r_2) \in \Sem{\vctx \proves \propB : \Prop}_\gamma \,\}
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	\end{aligned} \\
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	\Sem{\vctx \proves \prop \wand \propB : \Prop}_\gamma &=
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	\begin{aligned}[t]
		\Lam W. \{\, (n, r) &\mid \All n' \leq n. \All W' \geq W. \All r'. \\
		&\qquad
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		(n', r') \in \Sem{\vctx \proves \prop : \Prop}_\gamma(W') \land r \sep r' \\
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		&\qquad
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		\implies (n', r \bullet r') \in \Sem{\vctx \proves \propB : \Prop}_\gamma(W')
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		\}
	\end{aligned} \\
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	\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 \,\}
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\end{align*}
%
\begin{align*}
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	\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)
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\end{align*}

\typedsection{Interpretation of entailment}{\Sem{\vctx \mid \pfctx \proves \prop} : 2 \in \mathit{Sets}}

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