\( % Latex Preamble definitions here (mostly usepackage) \usepackage% %[dvipsnames] {xcolor} % make sure this is before the loading font packages \newcommand\hmmax{0} \newcommand\bmmax{0} \usepackage{amssymb} \usepackage{amsmath} \usepackage{mathtools} \usepackage %[dvipsnames] {graphicx} \usepackage{float} %\usepackage[numbers]{natbib} \usepackage[document]{ragged2e} % % enumitem doesn't seem to work with beamer %\usepackage[inline]{enumitem} \usepackage{wrapfig} \usepackage{stackrel} % extensible arrows \usepackage{extpfeil} % \usepackage{trfrac} \usepackage{amsthm} \usepackage{tikz} \usepackage{tikz-cd} \usepackage{tikz} \usepackage{tikz-qtree} \usetikzlibrary{automata, positioning, arrows, shapes.geometric} \usepackage{turnstile} \usepackage{comment} %https://tex.stackexchange.com/questions/21334/is-there-a-package-that-has-the-clockwise-gapped-circle-arrow-in-it % \usepackage{mathbx} \usepackage{datetime} \usepackage{datetime2} %% Also See %% http://u.cs.biu.ac.il/~tsaban/Pdf/LaTeXCommonErrs.pdf %% for general tips \usepackage{listings} \usepackage{subfigure} \usepackage{bm} \usepackage{amsfonts} %% - also included by amssymb \usepackage{mathpazo} %% - because the OP uses mathpazo, optional %\usepackage{tufte-latex} \usepackage{comment} \usepackage{mathtools} \usepackage{bussproofs} \usepackage{hyperref} %\usepackage{cleveref} \)

\( %% Your math definitions here % \newcommand{\alphaequiv}{{\underset{\raise 0.7em\alpha}{=}}} \newcommand{\yields}{\Rightarrow} \newcommand{\derives}{\overset{*}{\yields}} \newcommand{\alphaequiv}{=_{\alpha}} \newcommand{\tto}[2]{{\overset{#1}{\underset{#2}{\longrightarrow}}}} \newcommand{\transitsto}[2]{{\overset{#1}{\underset{#2}{\longrightarrow}}}} \newcommand{\xtransitsto}[2]{{\underset{#2}{\xrightarrow{#1}}}} \newcommand{\xtransitsfrom}[2]{{\underset{#2}{\xleftarrow{#1}}}} \newcommand{\xto}[2]{{\xtransitsto{#1}{#2}}} \newcommand{\xfrom}[2]{{\xtransitsfrom{#1}{#2}}} \newcommand{\xreaches}[2]{{\underset{#2}{\xtwoheadrightarrow{#1}}}} \newcommand{\reaches}[2]{{\underset{#2}{\xtwoheadrightarrow{#1}}}} %\newcommand{\reaches}[2]{{\overset{#1}{\underset{#2}{\twoheadrightarrow}}}} %\newcommand{\goesto}[2]{\transitsto{#1}{#2}} %\newcommand{\betareducesto}{{\underset{\beta}{\rightarrow}}} \newcommand{\betareducesto}{\rightarrow_{\beta}} %\newcommand{\etareducesto}{{\underset{\eta}{\rightarrow}}} \newcommand{\etareducesto}{\rightarrow_{\eta}} %\newcommand{\betaetareducesto}{{\underset{\beta\ \eta}{\rightarrow}}} \newcommand{\betaetareducesto}{\rightarrow_{\beta\eta}} \newcommand{\preducesto}{\rhd} \newcommand{\psimplifiesto}{\stackrel{\scriptstyle{*}}{\rhd}} \newcommand{\lreducesto}{\rightsquigarrow} \newcommand{\lsimplifiesto}{\stackrel{\scriptstyle{*}}{\lreducesto}} \newcommand{\rewritesto}{\hookrightarrow} \newcommand{\goesto}[1]{\stackrel{#1}{\rightarrow}} \newcommand{\xgoesto}[1]{\xrightarrow{#1}} \newcommand{\reducesto}{\stackrel{}{\rightarrow}} \newcommand{\simplifiesto}{\stackrel{\scriptstyle{*}}{\rightarrow}} \newcommand{\connected}[1]{\stackrel{#1}{\leftrightarrow}} \newcommand{\joins}{\downarrow} \newcommand{\evaluatesto}{\Longrightarrow} %\newcommand{\lit}[1]{\hbox{\sf{#1}}} \newcommand{\lit}[1]{{\sf{#1}}} \newcommand{\true}{\lit{true}} \newcommand{\false}{\lit{false}} \def\Z{\mbox{${\mathbb Z}$}} \def\N{\mbox{${\mathbb N}$}} \def\P{\mbox{${\mathbb P}$}} \def\R{\mbox{${\mathbb R}$}} \def\T{\mbox{${\mathbb T}$}} \newcommand{\Rp}{{\mathbb{R}}^+} \def\Bool{\mbox{${\mathbb B}$}} \def\Q{\mbox{${\mathbb Q}$}} \def\sA{\mbox{${\cal A}$}} \def\sB{\mbox{${\cal B}$}} \def\sC{\mbox{${\cal C}$}} \def\sD{\mbox{${\cal D}$}} \def\sF{\mbox{${\cal F}$}} \def\sG{\mbox{${\cal G}$}} \def\sL{\mbox{${\cal L}$}} \def\sP{\mbox{${\cal P}$}} \def\sM{\mbox{${\cal M}$}} \def\sN{\mbox{${\cal N}$}} \def\sR{\mbox{${\cal R}$}} \def\sS{\mbox{${\cal S}$}} \def\sO{\mbox{${\cal O}$}} \def\sT{\mbox{${\cal T}$}} \def\sU{\mbox{${\cal U}$}} \def\th{\mbox{$\widetilde{h}$}} \def\tg{\mbox{$\widetilde{g}$}} \def\tP{\mbox{$\widetilde{P}$}} \def\norm{\mbox{$\parallel$}} \def\osum{${{\bigcirc}}\!\!\!\!{\rm s}~$} \def\pf{\noindent {\bf Proof}~~} \def\exec{\mathit{exec}} \def\Act{\mathit{A\!ct}} \def\Traces{\mathit{Traces}} \def\Spec{\mathit{Spec}} \def\uns{\mathit{unless}} \def\ens{\mathit{ensures}} \def\lto{\mathit{leads\!\!-\!\!to}} \def\a{\alpha} \def\b{\beta} \def\c{\gamma} \def\d{\delta} \def\sP{\mbox{${\cal P}$}} \def\sM{\mbox{${\cal M}$}} \def\sA{\mbox{${\cal A}$}} \def\sB{\mbox{${\cal B}$}} \def\sC{\mbox{${\cal C}$}} \def\sI{\mbox{${\cal I}$}} \def\sS{\mbox{${\cal S}$}} \def\sD{\mbox{${\cal D}$}} \def\sF{\mbox{${\cal F}$}} \def\sG{\mbox{${\cal G}$}} \def\sR{\mbox{${\cal R}$}} \def\tg{\mbox{$\widetilde{g}$}} \def\ta{\mbox{$\widetilde{a}$}} \def\tb{\mbox{$\widetilde{b}$}} \def\tc{\mbox{$\widetilde{c}$}} \def\tx{\mbox{$\widetilde{x}$}} \def\ty{\mbox{$\widetilde{y}$}} \def\tz{\mbox{$\widetilde{z}$}} \def\tI{\mbox{$\widetilde{I}$}} \def\norm{\mbox{$\parallel$}} \def\sL{\mbox{${\cal L}$}} \def\sM{\mbox{${\cal M}$}} \def\sN{\mbox{${\cal N}$}} \def\th{\mbox{$\widetilde{h}$}} \def\tg{\mbox{$\widetilde{g}$}} \def\tP{\mbox{$\widetilde{P}$}} \def\norm{\mbox{$\parallel$}} \def\to{\rightarrow} \def\ov{\overline} \def\gets{\leftarrow} \def\too{\longrightarrow} \def\To{\Rightarrow} %\def\points{\mapsto} %\def\yields{\mapsto^{*}} \def\un{\underline} \def\vep{$\varepsilon$} \def\ep{$\epsilon$} \def\tri{$\bigtriangleup$} \def\Fi{$F^{\infty}$} \def\Di{\Delta^{\infty}} \def\ebox\Box \def\emp{\emptyset} \def\leadsto{\rightharpoondown^{*}} \newcommand{\benum}{\begin{enumerate}} \newcommand{\eenum}{\end{enumerate}} \newcommand{\bdes}{\begin{description}} \newcommand{\edes}{\end{description}} \newcommand{\bt}{\begin{theorem}} \newcommand{\et}{\end{theorem}} \newcommand{\bl}{\begin{lemma}} \newcommand{\el}{\end{lemma}} % \newcommand{\bp}{\begin{prop}} % \newcommand{\ep}{\end{prop}} \newcommand{\bd}{\begin{defn}} \newcommand{\ed}{\end{defn}} \newcommand{\brem}{\begin{remark}} \newcommand{\erem}{\end{remark}} \newcommand{\bxr}{\begin{exercise}} \newcommand{\exr}{\end{exercise}} \newcommand{\bxm}{\begin{example}} \newcommand{\exm}{\end{example}} \newcommand{\beqa}{\begin{eqnarray*}} \newcommand{\eeqa}{\end{eqnarray*}} \newcommand{\bc}{\begin{center}} \newcommand{\ec}{\end{center}} \newcommand{\bcent}{\begin{center}} \newcommand{\ecent}{\end{center}} \newcommand{\la}{\langle} \newcommand{\ra}{\rangle} \newcommand{\bcor}{\begin{corollary}} \newcommand{\ecor}{\end{corollary}} \newcommand{\bds}{\begin{defns}} \newcommand{\eds}{\end{defns}} \newcommand{\brems}{\begin{remarks}} \newcommand{\erems}{\end{remarks}} \newcommand{\bxrs}{\begin{exercises}} \newcommand{\exrs}{\end{exercises}} \newcommand{\bxms}{\begin{examples}} \newcommand{\exms}{\end{examples}} \newcommand{\bfig}{\begin{figure}} \newcommand{\efig}{\end{figure}} \newcommand{\set}[1]{\{#1\}} \newcommand{\pair}[1]{\langle #1\rangle} \newcommand{\tuple}[1]{\langle #1\rangle} \newcommand{\size}[1]{| #1 |} \newcommand{\union}{\cup} \newcommand{\Union}{\bigcup} \newcommand{\intersection}{\cap} \newcommand{\Intersection}{\bigcap} \newcommand{\B}{\textbf{B}} %\newcommand{\be}[2]{\begin{equation} \label{#1} \tag{#2} \end{equation}} \newcommand{\abs}[1]{{\lvert}#1{\rvert}} \newcommand{\id}[1]{\mathit{#1}} \newcommand{\pfun}{\rightharpoonup} %\newcommand{\ra}[1]{\kern-1.5ex\xrightarrow{\ \ #1\ \ }\phantom{}\kern-1.5ex} %\newcommand{\ras}[1]{\kern-1.5ex\xrightarrow{\ \ \smash{#1}\ \ }\phantom{}\kern-1.5ex} \newcommand{\da}[1]{\bigg\downarrow\raise.5ex\rlap{\scriptstyle#1}} \newcommand{\ua}[1]{\bigg\uparrow\raise.5ex\rlap{\scriptstyle#1}} % \newcommand{\lift}[1]{#1_{\bot}} \newcommand{\signal}[1]{\tilde{#1}} \newcommand{\ida}{\stackrel{{\sf def}}{=}} \newcommand{\eqn}{\doteq} \newcommand{\deduce}[1]{\sststile{#1}{}} %% These don't sit very well with MathJax %% so we don't plan to use theorem like environments %% in org documents. %% instead we plan to use headings with %% 1. property drawers with a CLASS property identifying %% the environment %% 2. A tag with the same name as the CLASS property %% In LaTeX export, these turn into (sub)sections. %% See http://u.cs.biu.ac.il/~tsaban/Pdf/LaTeXCommonErrs.pdf %% \newtheorem{prop}[thm]{Proposition} %% \theoremstyle{plain}%default %% \newtheorem{theorem}{Theorem}[section] %% \newtheorem{lemma}{Lemma}[section] %% \newtheorem{corollary}{Corollary}[section] %% \newtheorem{definition}{Definition}[section] %% \newtheorem{remark}{Remark}[section] %% \newtheorem{example}{Example}[section] %% \newtheorem{exercise}{Exercise}[section] \newcommand{\less}[1]{#1_{<}} \newcommand{\pfn}{\rightharpoonup} \newcommand{\ffn}{\stackrel{{\sf fin}}{\rightharpoonup}} \newcommand{\stkout}[1]{\ifmmode\text{\sout{\ensuremath{#1}}}\else\sout{#1}\fi} % Caution: Not supported by MathJax! % ---------------------------------- % \DeclareMathSymbol{\shortminus}{\mathbin}{AMSa}{"39} % \usepackage{amsfonts} %% <- also included by amssymb % \DeclareMathSymbol{\shortminus}{\mathbin}{AMSa}{"39} \usepackage{mathpazo} %% <- because the OP uses mathpazo, optional \newcommand{\mbf}[1]{\mathbf{#1}} \newcommand{\floor}[1]{\left\lfloor #1 \right\rfloor} \newcommand{\ceil}[1]{\left\lceil #1 \right\rceil} \newcommand{\rel}{\twoheadrightarrow} \newcommand{\map}{\rightarrow} %\newcommand{\fixed}{\boldsymbol{\circlearrowleft}} \newcommand{\terminal}{\not\xto{}{}} \newcommand{\fixed}{\bm\circlearrowleft} \newcommand{\imp}{\rightarrow} \newcommand{\dimp}{\leftrightarrow} % double implication \newcommand{\lequiv}{\Longleftrightarrow} % logical equivalence \newcommand{\limplies}{\Rightarrow} \newcommand{\lxor}{\veebar} \)

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POPL Lecture notes: PARLANG – a parallel interlude