Add docs folder back

docs/tex/.gitignore

+*.pdf
+*.aux
+*.log
+*.out
+*.synctex.gz
+*.txss
+*.thm
+*.toc
+*.bbl
+*.blg
+*.bcf
+*.run.xml
+_*_.tex
+auto/*.el

docs/tex/atomic.tex

+\documentclass[10pt]{article}
+\usepackage{lmodern}
+\usepackage[T1]{fontenc}
+\usepackage[utf8]{inputenc}
+\usepackage{fontspec}
+\setmonofont{Source Code Pro}
+
+\newif\ifslow\slowfalse %\slowtrue
+\ifslow
+  \usepackage[english]{babel}
+  \usepackage[babel=true]{microtype}
+\fi
+\usepackage[top=1in, bottom=1in, left=1.25in, right=1.25in]{geometry}
+
+\usepackage[backend=biber]{biblatex}
+\bibliography{bib}
+
+\newcommand{\bdia}{\blacklozenge}
+\newcommand{\dia}{\Diamond}
+\newcommand{\injR}{\texttt{injR}}
+\newcommand{\injL}{\texttt{injL}}
+
+\input{setup}
+
+\begin{document}
+
+\title{\bfseries iris-atomic}
+\author{Zhen Zhang}
+\maketitle
+
+\section{sync.v}
+
+
+\[\begin{aligned}
+    \text{synced}(R, f', f) \eqdef
+    \All P, Q, x.
+      &\hoare{ R * P(x)}{f(x)}{ v.\,R * Q(x,v) } \ra \\
+      &\hoare{ P(x)}{f'(x)}{ v.\,Q(x,v) }
+  \end{aligned}\]
+
+\[
+  \text{is\_syncer}(R, s) \eqdef
+    \All f. \wpre{s(f)}{ f'.\, \text{synced}(R, f', f)} \]
+
+
+\[
+  mk\_syncer\_spec (mk\_syncer) :=
+    \All R.
+        \hoare{R}{mk\_syncer()}{ s.\,\always (is\_syncer(R, s)} \]
+
+
+\section{atomic.v}
+
+Logically atomic triple:
+
+\[  
+    \lahoare{g.\, \alpha(g)}{e}{v.\, \beta(g, v)}[E_i][E_o] \eqdef
+    \All P, Q.
+    \begin{aligned}
+          &P \vs[Eo][Ei] \Exists g, \alpha(g) * (\alpha(g) \vsW[Ei][Eo] P \land \All v. \beta(g, v) \vsW[Ei][Eo] Q(g, v)) \wand \\
+          &\hoare{P}{e}{v.\, \Exists g. Q(g, v)}
+    \end{aligned}\]
+
+\section{simple\_sync.v}
+
+\begin{verbatim}
+mk_sync :=
+    λ: <>,
+       let l := newlock() in
+       λ: f x,
+          acquire l;
+          let ret := f x in
+          release l;
+          ret.
+\end{verbatim}
+
+\section{atomic\_sync.v}
+
+Specialized logically atomic triple:
+
+\[ \lahoare{g.\, \ownGhost{\gname}{g^{1/2}} * \always \alpha(g)}
+           {f(x)}
+           {v.\, \Exists g'. \ownGhost{\gname}{g'^{1/2}} * \beta(x, g, g', v)}[E_i][E_o]\]
+
+\begin{verbatim}
+sync(mk_syncer) :=
+  λ: f_seq l,
+     let s := mk_syncer() in
+     s (f_seq l).
+\end{verbatim}
+
+\[seq\_spec(f, \phi, \alpha, \beta, E) \eqdef
+      \All l.
+         \hoare{\top}{f(l)}{f'.\,
+            \begin{aligned}
+            \pure &\All x, \Phi, g.\\
+                &\phi (l, g) * \always \alpha(x) *\\
+                &(\All v, g'. \phi(l, g') \wand \beta(x, g, g', v) \wand \pvs[E][E] \Phi(v))\\
+                &\proves \wpre{f'(x)}[E]{ \Phi }
+              \end{aligned}
+        }\]
+  
+\[\begin{aligned}
+      &\text{atomic\_spec}(mk\_syncer, f\_seq, l, \phi, \alpha, \beta, E_i) \eqdef\\
+      &\All g_0.
+        seq\_spec(f\_seq, \phi, \alpha, \beta, \top) \ra
+        mk\_syncer\_spec(mk\_syncer) \ra\\
+        &\phi(l, g_0)
+        \proves \wpre{sync(mk\_syncer, f\_seq, l)}{ f.\,
+          \Exists \gname. \ownGhost{\gname}{g_0^{1/2}} *
+          \All x. \always \lahoare{g.\, \ownGhost{\gname}{g^{1/2}} * \always \alpha(g)}{f(x)}
+                                  {v.\, \Exists g'. \ownGhost{\gname}{g'^{1/2}} * \beta(x, g, g', v)}[E_i][\top]}
+      \end{aligned} \]
+
+\[ (\Exists g. \phi(l, g') * \ownGhost{\gname}{g^{1/2}} * P x \]
+
+\section{treiber.v}
+
+\begin{verbatim}
+push s x :=
+  let hd := !s in
+  let s' := ref SOME (x, hd) in
+  if CAS s hd s'
+    then ()
+    else push s x.
+
+pop s :=
+  let hd := !s in
+  match !hd with
+  | SOME (x, hd') =>
+    if: CAS s hd hd'
+      then SOME x
+      else pop s
+  | NONE => NONE
+  end.
+
+iter hd f :=
+  match !hd with
+  | NONE => ()
+  | SOME (x, hd') => f x ; iter hd' f
+  end.
+
+\end{verbatim}
+
+Logiall atomic spec (version 1):
+
+\[ \lahoare{xs.\, stack(s, xs)}{push(s, x)}{stack(s, x::xs)}[heapN][\top]\]
+\[ \lahoare{xs.\, stack(s, xs)}{pop(s)}{v. \begin{split} (&\Exists x, xs'. v = SOME(x) * stack(s, xs')) \lor\\
+                                            (&v = NONE * xs = \emptyset * stack(s, \emptyset)) \end{split}}[heapN][\top]\]
+
+
+Logiall atomic spec (version 2):
+
+\[ \lahoare{hd, xs.\, s \mapsto hd * list(hd, xs)}{push(s, x)}{\Exists hd'. s \mapsto hd' * hd' \mapsto SOME(x, hd) * list(hd, xs)}[heapN][\top]\]
+\[ \lahoare{hd, xs.\, s \mapsto hd * list(hd, xs)}{pop(s)}{v.
+    \begin{split}
+      (&\Exists x, xs', hd'. v = SOME(x) * hd \mapsto SOME(x, hd') * s \mapsto hd' * list(hd', xs')) \lor\\
+      (&v = NONE * xs = \emptyset * hd \mapsto NONE)
+    \end{split}
+  }[heapN][\top]
+  \]
+
+
+\section{peritem.v}
+
+A crappy but working spec:
+
+\[f\_spec (\gname, xs, s, f, Rf, RI) \eqdef
+    \All x.
+      \hoare{x \in xs * \knowInv\iname{\Exists xs. stack'(\gname, xs, s) * RI} * Rf}{f(x)}{ v.\, v = () }.\]
+
+\[\begin{split}
+  iter\_spec(\gname, s, Rf, RI) \eqdef
+    &\All xs, hd, f.\\
+      &f\_spec(xs, s, f', Rf, RI) \ra\\
+      &\hoare{\knowInv\iname{\Exists xs. stack'(xs, s) * RI} * list'(\gname, hd, xs) * Rf}{iter(hd, f)}{ v.\, v = () * Rf}
+  \end{split}\]
+
+\[push\_spec (\gname, s, x, RI) \eqdef
+  \hoare{R(x) * \knowInv\iname{\Exists xs. stack'(xs, s) * RI}}{push(s, x)}{v.\, v = () * (\Exists hd. ev(\gname, hd, x))}\]
+
+\section{flat.v}
+
+\begin{verbatim}
+doOp :=
+  λ: p,
+     match !p with
+     | InjL (f, x) => p <- InjR (f x)
+     | InjR _ => ()
+     end.
+
+try_srv :=
+  λ: lk s,
+    if try_acquire lk
+      then let hd := !s in
+           iter hd doOp;
+           release lk
+      else ().
+
+loop p s lk :=
+    match !p with
+    | InjL _ =>
+        try_srv lk s;
+        loop p s lk
+    | InjR r => r
+    end.
+
+install :=
+  λ: f x s,
+     let p := ref (InjL (f, x)) in
+     push s p;
+     p.
+
+mk_flat :=
+  λ: <>,
+   let lk := newlock() in
+   let s := new_stack() in
+   λ: f x,
+      let p := install f x s in
+      let r := loop p s lk in
+      r.
+\end{verbatim}
+
+\[p \mapsto \injR(-)\]
+\[p \mapsto \texttt{injL}(f, x)\]
+\[p \mapsto \injR(y)\]
+\[\circ_i, \bullet_i, \dia_i, \bdia, \dia_i \circ_i, \dia_i \bullet_i, \bdia \bullet_i\]
+
+
+\begin{align*}
+     &\Exists y.          &&p \fmapsto[1/2] \injR(-) * \dia_i * \circ_i\\
+\lor &\Exists f, x, P, Q. &&p \fmapsto[1/2] \injL(f, x) * \ownGhost{\gname}{x^{1/2}} *
+                           P(x) * (\hoare{R * P(x)}{f(x)}{v.\,R * Q(x, v)}) * \gamma \mapstoprop Q(x) * \dia_i * \bullet_i\\
+\lor &\Exists x.          &&p \fmapsto[1/2] \injL(-, x) * \ownGhost{\gname}{x^{1/4}} * \bdia * \bullet_i\\
+\lor &\Exists x, y.       &&p \fmapsto[1/2] \injR(y) * \ownGhost{\gname}{x^{1/2}} * \gamma \mapstoprop Q(x) * Q(x, y) * \dia_i * \bullet_i
+\end{align*}
+
+\[\alpha = \alpha_a * \alpha_o, \alpha = \alpha_a' * \alpha_o\]
+\[\alpha = \alpha_o * \alpha_a, \alpha_a' * \alpha_o = \beta\]
+\[\alpha = \alpha_a * \alpha_o, \alpha = \alpha_a' * \alpha_o\]
+
+\end{document}

docs/tex/iris.sty

+\NeedsTeXFormat{LaTeX2e}[1999/12/01]
+\ProvidesPackage{iris}
+
+\RequirePackage{tikz}
+\RequirePackage{scalerel}
+\RequirePackage{array}
+\RequirePackage{dashbox}
+\RequirePackage{tensor}
+\RequirePackage{xparse}
+\RequirePackage{xstring}
+\RequirePackage{mathtools}
+
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+%% SETUP
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+
+\usetikzlibrary{shapes}
+%\usetikzlibrary{snakes}
+\usetikzlibrary{arrows}
+\usetikzlibrary{calc}
+\usetikzlibrary{arrows.meta}
+\tikzstyle{state}=[circle, draw, minimum size=1.2cm, align=center]
+\tikzstyle{trans}=[arrows={->[scale=1.4]}]
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+%% MATH SYMBOLS & NOTATION & IDENTIFIERS
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+\newcommand{\nat}{\mathbb{N}}
+
+\DeclareMathOperator*{\Sep}{\scalerel*{\ast}{\sum}}
+\newcommand*{\disj}[1][]{\mathrel{\#_{#1}}}
+\newcommand\pord{\sqsubseteq}
+\newcommand\dplus{\mathbin{+\kern-1.0ex+}}
+\newcommand{\upclose}{\mathord{\uparrow}}
+\newcommand{\ALT}{\ |\ }
+
+\newcommand{\spac}{\,} % a space
+
+\def\All #1.{\forall #1.\spac}%
+\def\Exists #1.{\exists #1.\spac}%
+\def\Ret #1.{#1.\spac}%
+
+\newcommand{\any}{{\rule[-.2ex]{1ex}{.4pt}}}%
+
+\newcommand{\judgment}[2][]{\paragraph{#1}\hspace{\stretch{1}}\fbox{$#2$}}
+
+\newcommand{\pfn}{\rightharpoonup}
+\newcommand\fpfn{\xrightharpoonup{\kern-0.25ex\textrm{fin}\kern-0.25ex}}
+\newcommand{\la}{\leftarrow}
+\newcommand{\ra}{\rightarrow}
+\newcommand{\Ra}{\Rightarrow}
+\newcommand{\Lra}{\Leftrightarrow}
+\newcommand\monra{\xrightarrow{\kern-0.15ex\textrm{mon}\kern-0.05ex}}
+\newcommand\nfn{\xrightarrow{\kern-0.15ex\textrm{ne}\kern-0.05ex}}
+\newcommand{\eqdef}{\triangleq}
+\newcommand{\bnfdef}{\vcentcolon\vcentcolon=}
+
+% \newcommand{\lor}{\wedge}
+\newcommand{\maybe}[1]{#1^?}
+
+\newcommand*\setComp[2]{\left\{#1\spac\middle|\spac#2\right\}}
+\newcommand*\set[1]{\left\{#1\right\}}
+\newcommand*\record[1]{\left\{\spac#1\spac\right\}}
+\newcommand*\recordComp[2]{\left\{\spac#1\spac\middle|\spac#2\spac\right\}}
+
+\newenvironment{inbox}[1][]{
+  \begin{array}[#1]{@{}l@{}}
+}{
+  \end{array}
+}
+
+\newcommand{\dom}{\mathrm{dom}}
+\newcommand{\cod}{\mathrm{cod}}
+\newcommand{\chain}{\mathrm{chain}}
+
+\newcommand{\pset}[1]{\wp(#1)} % Powerset
+\newcommand{\psetdown}[1]{\wp^\downarrow(#1)}
+\newcommand{\finpset}[1]{\wp^\mathrm{fin}(#1)}
+
+\newcommand{\Func}{F} % functor
+
+\newcommand{\subst}[3]{{#1}[{#3} / {#2}]}
+
+\newcommand{\mapinsert}[3]{#3[#1:=#2]}
+
+\newcommand{\nil}{\epsilon}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+%% MODEL-SPECIFIC SYMBOLS & NOTATION & IDENTIFIERS
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\newcommand{\textdom}[1]{\textit{#1}}
+
+\newcommand{\wIso}{\xi}
+
+\newcommand{\rs}{r}
+\newcommand{\rsB}{s}
+\newcommand{\rss}{R}
+
+\newcommand{\pres}{\pi}
+\newcommand{\wld}{w}
+\newcommand{\ghostRes}{g}
+
+%% Various pieces of syntax
+\newcommand{\wsat}[3]{#1 \models_{#2} #3}
+\newcommand{\wsatpre}{\textdom{pre-wsat}}
+
+\newcommand{\wtt}[2]{#1 : #2} % well-typed term
+
+\newcommand{\nequiv}[1]{\ensuremath{\mathrel{\stackrel{#1}{=}}}}
+\newcommand{\notnequiv}[1]{\ensuremath{\mathrel{\stackrel{#1}{\neq}}}}
+\newcommand{\nequivset}[2]{\ensuremath{\mathrel{\stackrel{#1}{=}_{#2}}}}
+\newcommand{\nequivB}[1]{\ensuremath{\mathrel{\stackrel{#1}{\equiv}}}}
+\newcommand{\latert}{\mathord{\blacktriangleright}}
+\newcommand{\latertinj}{\textlog{next}}
+
+\newcommand{\Sem}[1]{\llbracket #1 \rrbracket}
+
+\newcommand{\sembox}[1]{\hfill \normalfont \mbox{\fbox{\(#1\)}}}
+\newcommand{\typedsection}[2]{\subsubsection*{\rm\em #1 \sembox{#2}}}
+
+
+%% Some commonly used identifiers
+\newcommand{\SProp}{\textdom{SProp}}
+\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}}
+\newcommand{\State}{\textdom{State}}
+\newcommand{\Val}{\textdom{Val}}
+\newcommand{\Loc}{\textdom{Loc}}
+\newcommand{\Expr}{\textdom{Expr}}
+
+\newcommand{\cofe}{T}
+\newcommand{\cofeB}{U}
+\newcommand{\COFEs}{\mathcal{COFE}} % category of COFEs
+\newcommand{\iFunc}{\Sigma}
+\newcommand{\fix}{\textdom{fix}}
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+% CMRA (RESOURCE ALGEBRA) SYMBOLS & NOTATION & IDENTIFIERS
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\newcommand{\textmon}[1]{\textsc{#1}}
+
+\newcommand{\monoid}{M}
+\newcommand{\mval}{\mathcal{V}}
+
+\newcommand{\melt}{a}
+\newcommand{\meltB}{b}
+\newcommand{\meltC}{c}
+\newcommand{\melts}{A}
+\newcommand{\meltsB}{B}
+
+\newcommand{\f}{\mathrm{f}} % for "frame"
+
+\newcommand{\mcar}[1]{|#1|}
+\newcommand{\mcarp}[1]{\mcar{#1}^{+}}
+\newcommand{\munit}{\varepsilon}
+\newcommand{\mcore}[1]{{\mid}#1{\mid}} % using "|" here makes LaTeX diverge. WTF.
+\newcommand{\mnocore}\top
+\newcommand{\mtimes}{\mathbin{\cdot}}
+
+\newcommand{\mupd}{\rightsquigarrow}
+\newcommand{\mincl}[1][]{\ensuremath{\mathrel{\stackrel{#1}{\preccurlyeq}}}}
+
+\newcommand{\CMRAs}{\mathcal{CMRA}} % category of CMRAs
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+%% LOGIC SYMBOLS & NOTATION & IDENTIFIERS
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\newcommand{\textlog}[1]{\textsf{#1}}
+
+\newcommand{\Sig}{\mathcal{S}}
+\newcommand{\SigType}{\mathcal{T}}
+\newcommand{\SigFn}{\mathcal{F}}
+\newcommand{\SigAx}{\mathcal{A}}
+\newcommand{\sigtype}{T}
+\newcommand{\sigfn}{F}
+\newcommand{\sigax}{A}
+
+\newcommand{\type}{\tau}
+\newcommand{\typeB}{\sigma}
+
+\newcommand{\var}{x}
+\newcommand{\varB}{y}
+\newcommand{\varC}{z}
+
+\newcommand{\term}{t}
+\newcommand{\termB}{u}
+
+\newcommand{\vctx}{\Gamma}
+\newcommand{\pfctx}{\Theta}
+
+\newcommand{\prop}{P}
+\newcommand{\propB}{Q}
+\newcommand{\propC}{R}
+
+% pure propositions
+\newcommand{\pprop}{\phi}
+\newcommand{\ppropB}{\psi}
+
+\newcommand{\pred}{\varPhi}
+\newcommand{\predB}{\Psi}
+\newcommand{\predC}{\Zeta}
+
+\newcommand{\gname}{\gamma}
+\newcommand{\iname}{\iota}
+
+\newcommand{\mask}{\mathcal{E}}
+\newcommand{\namesp}{\mathcal{N}}
+\newcommand{\namecl}[1]{{#1^{\kern0.2ex\uparrow}}}
+
+%% various pieces of Syntax
+\def\MU #1.{\mu #1.\spac}%
+\def\Lam #1.{\lambda #1.\spac}%
+
+\newcommand{\proves}{\vdash}
+\newcommand{\provesCoq}{\mathrel{\vdash_{\mbox{\!\includegraphics[width=0.7em]{rooster.jpg}}}}}
+\newcommand{\provesIff}{\mathrel{\dashv\vdash}}
+
+\newcommand{\wand}{\mathrel{-\!\!*}}
+
+% oh my... I have to wrap the "-" in a \mathrm, otherwise all hell breaks lose...
+\newcommand{\fmapsto}[1][]{\xmapsto{#1}}
+\newcommand{\gmapsto}{\hookrightarrow}%
+\newcommand{\fgmapsto}[1][\mathrm{-}]{\xhookrightarrow{#1}}%
+
+\NewDocumentCommand\wpre{m O{} m}%
+  {\textlog{wp}_{#2}\spac#1\spac{\big\{#3\big\}}}
+
+\newcommand{\later}{\mathop{\triangleright}}
+\newcommand{\always}{\mathop{\Box}}
+\newcommand{\pure}{\mathop{\blacksquare}}
+
+%% Invariants and Ghost ownership
+% PDS: Was 0pt inner, 2pt outer.
+% \boxedassert [tikzoptions] contents [name]
+\tikzstyle{boxedassert_border} = [sharp corners,line width=0.2pt]
+\NewDocumentCommand \boxedassert {O{} m o}{%
+  \tikz[baseline=(m.base)]{
+    %   \node[rectangle, draw,inner sep=0.8pt,anchor=base,#1] (m) {${#2}\mathstrut$};
+    \node[rectangle,inner sep=0.8pt,outer sep=0.2pt,anchor=base] (m) {${\,#2\,}\mathstrut$};
+    \draw[#1,boxedassert_border] ($(m.south west) + (0,0.65pt)$) rectangle ($(m.north east) + (0, 0.7pt)$);
+  }\IfNoValueF{#3}{^{\,#3}}%
+}
+\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]{\lfloor\, #1\, \rfloor}
+
+%% View Shifts
+\NewDocumentCommand \vsGen {O{} m O{}}%
+  {\mathrel{%
+    \ifthenelse{\equal{#3}{}}{%
+      % Just one mask, or none
+      {#2}_{#1}%
+    }{%
+      % Two masks
+      \tensor*[^{#1}]{#2}{^{#3}}
+    }%
+  }}%
+\NewDocumentCommand \vs {O{} O{}} {\vsGen[#1]{\Rrightarrow}[#2]}
+\NewDocumentCommand \vsL {O{} O{}} {\vsGen[#1]{\Lleftarrow}[#2]}
+\NewDocumentCommand \vsE {O{} O{}} %
+  {\vsGen[#1]{\Lleftarrow\!\!\!\Rrightarrow}[#2]}
+\NewDocumentCommand \pvs {O{} O{}} {\mathord{\vsGen[#1]{{\mid\kern-0.4ex\Rrightarrow\kern-0.25ex}}[#2]\kern0.2ex}}
+
+\newcommand\vsWand{\kern0.1ex\tikz[baseline=(base),line width=0.375pt]{%
+  \draw (0, 0) -- (0.4, 0); \draw (0, -0.075) -- (0.28, -0.075); \draw (0, 0.075) -- (0.28, 0.075);
+  \node at (0.4, -0.235) (ast) {$\smash{\scaleto{\ast}{1.2em}}$};
+  \node at (0.4, -0.095) (base) {};
+}{\vphantom{\Rrightarrow}}\kern-1.2ex}
+\NewDocumentCommand \vsW {O{} O{}} {\vsGen[#1]{\vsWand}[#2]}
+
+% for now, the update modality looks like a pvs without masks.
+\NewDocumentCommand \upd {} {\mathop{\mid\kern-0.4ex\Rrightarrow\kern-0.25ex}}
+
+
+%% Hoare Triples
+\newcommand*{\hoaresizebox}[1]{%
+  \hbox{$\mathsurround=0pt{#1}\mathstrut$}}
+\newcommand*{\hoarescalebox}[2]{%
+  \hbox{\scalerel*[1ex]{#1}{#2}}}
+\newcommand{\triple}[5]{%
+  \setbox0=\hoaresizebox{{#3}{#5}}%
+  \setbox1=\hoarescalebox{#1}{\copy0}%
+  \setbox2=\hoarescalebox{#2}{\copy0}%
+  \copy1{#3}\copy2%
+  \; #4 \;%
+  \copy1{#5}\copy2}
+
+\newcommand{\bracket}[4][]{%
+  \setbox0=\hbox{$\mathsurround=0pt{#1}{#4}\mathstrut$}%
+  \scalerel*[1ex]{#2}{\copy0}%
+  {#4}%
+  \scalerel*[1ex]{#3}{\copy0}}
+% \curlybracket[other] x
+\newcommand{\curlybracket}[2][]{\bracket[{#1}]\{\}{#2}}
+\newcommand{\anglebracket}[2][]{\bracket[{#1}]\langle\rangle{#2}}
+
+\NewDocumentCommand \hoare {m m m O{}}{
+  \curlybracket{#1}\spac #2 \spac \curlybracket{#3}_{#4}%
+}
+
+\NewDocumentCommand \lahoare {m m m O{} O{}}{
+  \anglebracket{#1}\spac #2 \spac \anglebracket{#3}_{#4}^{#5}%
+}
+
+% \hoareV[t] pre c post [mask]
+\NewDocumentCommand \hoareV {O{c} m m m O{}}{
+    {\begin{aligned}[#1]
+    &\curlybracket{#2} \\
+    &\quad{#3} \\
+    &{\curlybracket{#4}}_{#5}
+    \end{aligned}}%
+}
+% \hoareHV[t] pre c post [mask]
+\NewDocumentCommand \hoareHV {O{c} m m m O{}}{
+  {\begin{aligned}[#1]
+  &\curlybracket{#2} \; {#3} \\
+  &{\curlybracket{#4}}_{#5}
+  \end{aligned}}%
+}
+
+
+%% Some commonly used identifiers
+\newcommand{\timeless}[1]{\textlog{timeless}(#1)}
+\newcommand{\persistent}[1]{\textlog{persistent}(#1)}
+\newcommand{\physatomic}[1]{\textlog{atomic}($#1$)}
+\newcommand{\infinite}{\textlog{infinite}}
+
+\newcommand{\Prop}{\textlog{Prop}}
+\newcommand{\Pred}{\textlog{Pred}}
+
+\newcommand{\TRUE}{\textlog{True}}
+\newcommand{\FALSE}{\textlog{False}}
+\newcommand{\TLam}{\Lambda\spac}%
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+% GENERIC LANGUAGE SYNTAX AND SEMANTICS
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\newcommand{\expr}{e}
+\newcommand{\val}{v}
+\newcommand{\valB}{w}
+\newcommand{\state}{\sigma}
+\newcommand{\step}{\ra}
+\newcommand{\hstep}{\ra_{\mathsf{h}}}
+\newcommand{\tpstep}{\ra_{\mathsf{tp}}}
+\newcommand{\lctx}{K}
+\newcommand{\Lctx}{\textdom{Ctx}}
+