454 lines
12 KiB
TeX
454 lines
12 KiB
TeX
\documentclass{beamer}
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\usepackage[utf8]{inputenc}
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\usepackage{formal-grammar}
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\usepackage[dvipsnames]{xcolor}
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\usepackage{mathpartir}
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\usepackage{stmaryrd}
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\newcommand{\metavariable}[1]{\textcolor{teal}{#1}}
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\newcommand{\typeterminal}[1]{\textcolor{brown}{#1}}
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\newcommand{\exprterminal}[1]{\textcolor{Sepia}{#1}}
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\title{A functional core calculus with ladder-types}
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%\date[ISPN ’80]{27th International Symposium of Prime Numbers}
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\author[Euclid]{Michael Sippel \texttt{michael.sippel@mailbox.tu-dresden.de}\inst{1}}
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\institute{\inst{1} Technische Universität Dresden}
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\usetheme{ccc}
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\begin{document}
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\begin{frame}
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\titlepage
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\end{frame}
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\begin{frame}
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\frametitle{Type Systems}
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\framesubtitle{Tension between Safety and Flexibility}
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\begin{itemize}
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\item<1-> Goal: eliminate undefined behaviour,
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facilitate abstract code
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\item<2-> simple types are overly restrictive,
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loss of low level control
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\item<3-|alert@3> polymorphism
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(regain flexibility in type-safe way)
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\begin{itemize}
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\item<4-> parametric polymorphism (generics)
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\item<4-> subtype polymorphism (inheritance / coercions)
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\end{itemize}
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\end{itemize}
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\end{frame}
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\begin{frame}{How to get fast code?}
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\framesubtitle{Requirements for a type system}
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\begin{itemize}
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\item<1-> algorithmic optimization
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\begin{itemize}
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\item see which methods are used most frequently
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\item switch structures to reduce complexity there
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\item identify special cases
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\end{itemize}
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\item<2-> optimize implementation detail:
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\begin{itemize}
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\item remove indirection
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\item bottleneck? memory!\\
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(plenty of Main-memory, but FAST memory is scarce\\
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\(\rightarrow\) improve cache \& bus efficiency with compact representations)
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\begin{itemize}
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\item<3-> Balance De-/Encode Overhead
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vs. Fetch-Overhead
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\item<4-> Synchronize locality in access order and memory layout.
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-> elements accessed in succession shall be close in memory.
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(e.g. AoS vs SoA)
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\item<5-> accounting for machine specific cache-sizes
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\item<6-> accounting for SIMD instructions
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\item<7-> interfacing with kernels running on accelerator devices
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\end{itemize}
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\end{itemize}
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\end{itemize}
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\end{frame}
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\begin{frame}{Abstract Concept - Multiple Representations}
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\framesubtitle{Requirements for a type system}
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\begin{itemize}
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\item
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\end{itemize}
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\end{frame}
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\begin{frame}{Coercions: (In-)Coherence with Transitivity}
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\begin{itemize}
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\item<1-> Common example: coerce \texttt{Int -> Real}
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(subsumption \(\mathbb{N} \subset \mathbb{R}\))
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\begin{itemize}
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\item \texttt{Int -> Real}
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\item \texttt{Int -> String}
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\item \texttt{Real -> String}
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\end{itemize}
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\item<2-> transitivity creates two extensionally \emph{different} coercion paths:
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\begin{itemize}
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\item \texttt{Int -> String} (3 -> "3")
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\item \texttt{Int -> Real -> String} (3 -> "3.0")
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\end{itemize}
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\item<3-> Subsumptive interpretation is misleading!
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\item<3-> \texttt{Int -> Real} is not as unambiguous as it might seem.
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\begin{itemize}
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\item value ranges (e.g. map int to unit interval [0,1])
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\item multiple quantization functions (linear, exponential, \dots)
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\end{itemize}
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\end{itemize}
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\end{frame}
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\begin{frame}{Sea of Types}
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\includegraphics[width=\textwidth]{sot-int.png}
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\end{frame}
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\begin{frame}{Sea of Types}
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\includegraphics[width=\textwidth]{sot-json.png}
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\end{frame}
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\begin{frame}{Ladder Types}
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\begin{block}{Intuition}
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Encode "represented-as" relation into type terms
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\end{block}
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\begin{block}{}
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\( \tau_1 \sim \tau_2 \) reads "\(\tau_1\) represented as \(\tau_2\)"
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\end{block}
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\begin{example}\(
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Color \sim sRGB
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\sim HSV \sim \langle \text{Vec3} \quad \mathbb{R}_{[0,1]}
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\sim \langle \text{QuantizeLinear} \quad 256 \rangle
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\sim \mathbb{Z}_{256} \sim \text{machine.UInt8} \sim \text{Byte}
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\rangle
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\sim \text{[Byte; 3]}
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\)\end{example}
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\end{frame}
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\begin{frame}{Extending System F}
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\small
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\begin{grammar}
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\firstcase{ T }{
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\metavariable{\sigma}
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}{Base Type}
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\otherform{
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\metavariable{\alpha}
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}{Type Variable}
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\otherform{
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\typeterminal{\forall} \metavariable{\alpha} \typeterminal{.} \nonterm{T}
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}{Universal Type}
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\otherform{
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\typeterminal{<} \nonterm{T} \quad \nonterm{T} \typeterminal{>}
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}{Specialized Type}
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\otherform{
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\nonterm{T} \quad \typeterminal{\rightarrow} \quad \nonterm{T}
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}{Function Type}
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\otherform{
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\nonterm{T} \quad \typeterminal{\rightarrow_\text{morph}} \quad \nonterm{T}
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}{Morphism Type}
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\otherform{
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\nonterm{T} \quad \typeterminal{\sim} \quad \nonterm{T}
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}{Ladder Type}
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$$\\$$
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\firstcase{ E
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% T_\selexpr
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}
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{ \metavariable{x}
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} {Variable}
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\otherform{
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\exprterminal{let} \quad \metavariable{x} \quad \exprterminal{=} \quad
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\nonterm{ E }
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\quad \exprterminal{in} \quad
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\nonterm{ E }
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}{Variable Binding}
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\otherform{
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$$ \exprterminal{\Lambda} \metavariable{\alpha}
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\quad \exprterminal{\mapsto} \quad $$
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\nonterm{ E }
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}{Type Abstraction}
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\otherform{
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$$ \exprterminal{\lambda} \metavariable{x} $$
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\exprterminal{:} \nonterm{ T }
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\quad $$\exprterminal{\mapsto}$$ \quad
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\nonterm{ E }
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}{Value Abstraction}
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\otherform{
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$$ \exprterminal{\lambda} \metavariable{x} $$
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\exprterminal{:} \nonterm{ T }
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\quad $$\exprterminal{\mapsto_\text{morph}}$$ \quad
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\nonterm{ E }
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}{Value Morphism}
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\otherform{
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\nonterm{ E }
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\quad
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\nonterm{ T }
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}{Type Application}
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\otherform{
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\nonterm{ E }
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\quad
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\nonterm{ E }
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}{Value Application}
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\otherform{
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\nonterm{ E }
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\quad
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\exprterminal{as}
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\quad
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\nonterm{ T }
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}{Ascension}
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\otherform{
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\nonterm{ E }
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\quad
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\exprterminal{des}
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\quad
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\nonterm{ T }
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}{Descension}
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\end{grammar}
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\end{frame}
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\begin{frame}{Morphism Graph}
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\begin{definition}[Morphism Paths]
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\label{def:pathrules}
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\begin{mathpar}
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\inferrule[M-Sub]{
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\metavariable{\tau} \leq \metavariable{\tau'}
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}{
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\Gamma \vdash \metavariable{\tau} \leadsto \metavariable{\tau'}
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}
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\inferrule[M-Single]{
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(\metavariable{h} : \metavariable{\tau} \typeterminal{\rightarrow_\text{morph}} \metavariable{\tau'}) \in \Gamma
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}{
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\Gamma \vdash \metavariable{\tau} \leadsto \metavariable{\tau'}
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}
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\inferrule[M-Chain]{
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\Gamma \vdash \metavariable{\tau} \leadsto \metavariable{\tau'}\\
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\Gamma \vdash \metavariable{\tau'} \leadsto \metavariable{\tau''}
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}{
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\Gamma \vdash \metavariable{\tau} \leadsto \metavariable{\tau''}
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}
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\inferrule[M-MapSeq]{
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\Gamma \vdash \metavariable{\tau} \leadsto \metavariable{\tau'}
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}{
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\Gamma \vdash
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\typeterminal{\langle\text{Seq } \metavariable{\tau}\rangle} \leadsto
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\typeterminal{\langle\text{Seq } \metavariable{\tau'}\rangle}
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}
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\end{mathpar}
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\end{definition}
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\end{frame}
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\begin{frame}{Morphism Graph}
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\begin{example}[Morphism Graph]
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\begin{mathpar}
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\small
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\text{Assume }
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\Gamma := \{\\
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\exprterminal{\text{degrees-to-turns}} : \typeterminal{\text{Angle}\sim\text{Degrees}\sim\mathbb{R} \quad \rightarrow_\text{morph} \quad \text{Angle}\sim\text{Turns}\sim\mathbb{R}},\\
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\exprterminal{\text{turns-to-radians}} : \typeterminal{\text{Angle}\sim\text{Turns}\sim\mathbb{R} \quad \rightarrow_\text{morph} \quad \text{Angle}\sim\text{Radians}\sim\mathbb{R}},\\
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\}.
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\end{mathpar}
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\end{example}
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\end{frame}
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\begin{frame}{Morphism Graph}
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\begin{example}[Morphism Graph]
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Then ..
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\small
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\begin{itemize}
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\item \(\Gamma \vdash \typeterminal{\text{Angle}\sim\text{Degrees}\sim\mathbb{R}} \leadsto \typeterminal{\mathbb{R}}\) (by \textsc{M-Sub})
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\item \(\Gamma \vdash \typeterminal{\text{Angle}\sim\text{Degrees}\sim\mathbb{R}} \leadsto \typeterminal{\text{Angle}\sim\text{Radians}\sim\mathbb{R}}\) (by \textsc{M-Chain})
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\item \(\Gamma \vdash \typeterminal{\langle\text{Seq }\text{Angle}\sim\text{Degrees}\sim\mathbb{R}\rangle} \leadsto \typeterminal{\langle\text{Seq }\text{Angle}\sim\text{Radians}\sim\mathbb{R}\rangle}\) (by \textsc{M-MapSeq})
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\end{itemize}
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\end{example}
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\end{frame}
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\begin{frame}{Typing}
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\begin{definition}[Typing Relation]
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\begin{mathpar}
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\inferrule[T-App]{
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\Gamma \vdash \metavariable{f} : \metavariable{\sigma} \typeterminal{\rightarrow} \metavariable{\tau}\\
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\Gamma \vdash \metavariable{a} : \metavariable{\sigma'}\\
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\Gamma \vdash \metavariable{\sigma'} \leadsto \metavariable{\sigma}
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}{
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\Gamma \vdash (\metavariable{f} \quad \metavariable{a}) : \metavariable{\tau}
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}
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\end{mathpar}
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\end{definition}
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\end{frame}
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\begin{frame}{Translation}
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\begin{definition}[Morphism Translation]
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\begin{mathpar}
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\small
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\Big{\llbracket}
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\inferrule[M-Sub]{
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\metavariable{\tau} \leq \metavariable{\tau'}
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}{
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\Gamma \vdash \metavariable{\tau} \leadsto \metavariable{\tau'}
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}
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\Big{\rrbracket} = \exprterminal{\lambda x:\metavariable{\tau} \mapsto x}
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\and
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\Big{\llbracket}
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\inferrule[M-Single]{
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(\metavariable{h} : \metavariable{\tau} \typeterminal{\rightarrow_\text{morph}} \metavariable{\tau'}) \in \Gamma
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}{
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\Gamma \vdash \metavariable{\tau} \leadsto \metavariable{\tau'}
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}
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\Big{\rrbracket} = \metavariable{h}
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\and
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\end{mathpar}
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\end{definition}
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\end{frame}
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\begin{frame}{Translation}
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\begin{definition}[Morphism Translation (2)]
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\begin{mathpar}
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\small
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\Big{\llbracket}
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\inferrule[M-Chain]{
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C_1 :: \Gamma \vdash \metavariable{\tau} \leadsto \metavariable{\tau'}\\
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C_2 :: \Gamma \vdash \metavariable{\tau'} \leadsto \metavariable{\tau''}
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}{
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\Gamma \vdash \metavariable{\tau} \leadsto \metavariable{\tau''}
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}
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\Big{\rrbracket} = \exprterminal{\lambda \text{x}:\metavariable{\tau} \mapsto}
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\Big{\llbracket} C_2 \Big{\rrbracket}
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\exprterminal{(}\Big{\llbracket} C_1 \Big{\rrbracket} \exprterminal{\text{x})}
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\and
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\Big{\llbracket}
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\inferrule[M-MapSeq]{
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C_1 :: \Gamma \vdash \metavariable{\tau} \leadsto \metavariable{\tau'}
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}{
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\Gamma \vdash
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\typeterminal{\langle\text{Seq } \metavariable{\tau}\rangle} \leadsto
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\typeterminal{\langle\text{Seq } \metavariable{\tau'}\rangle}
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}
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\Big{\rrbracket} = \exprterminal{\lambda \text{xs}:\typeterminal{\langle\text{Seq }\metavariable{\tau}\rangle} \mapsto}
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\exprterminal{( \text{map}} \Big{\llbracket} C_1 \Big{\rrbracket} \exprterminal{\text{xs})}
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\end{mathpar}
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\end{definition}
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\end{frame}
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\begin{frame}{Translation}
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\begin{definition}[Expression Translation]
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Translates a type-derivation tree into a fully expanded expression
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\begin{mathpar}
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\Big{\llbracket}
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\inferrule[T-App]{
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D_1 :: \Gamma \vdash \metavariable{f} : \metavariable{\sigma} \typeterminal{\rightarrow} \metavariable{\tau}\\
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D_2 :: \Gamma \vdash \metavariable{a} : \metavariable{\sigma'}\\\\
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C :: \Gamma \vdash \metavariable{\sigma'} \leadsto \metavariable{\sigma}
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}{
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\Gamma \vdash (\metavariable{f} \quad \metavariable{a}) : \metavariable{\tau}
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}
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\Big{\rrbracket} =
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\Big{\llbracket}D_1\Big{\rrbracket}
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\exprterminal{(}
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\Big{\llbracket}C\Big{\rrbracket}
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\Big{\llbracket}D_2\Big{\rrbracket}
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\exprterminal{)}
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\end{mathpar}
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\end{definition}
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\end{frame}
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\begin{frame}{Soundness}
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\begin{lemma}[Morphism generation has correct type]
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\begin{mathpar}
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C :: \Gamma \vdash \metavariable{\tau} \leadsto \metavariable{\tau'}\\
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\Rightarrow \Gamma \vdash \llbracket C \rrbracket : \metavariable{\tau} \typeterminal{\rightarrow_\text{morph}} \metavariable{\tau'}
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\end{mathpar}
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\end{lemma}
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\begin{lemma}[Expression Translation preserves typing]
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\begin{mathpar}
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D :: \Gamma \vdash \metavariable{e} : \metavariable{\tau}\\
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\Rightarrow \Gamma \vdash \llbracket D \rrbracket : \metavariable{\tau}
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\end{mathpar}
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\end{lemma}
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\end{frame}
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\begin{frame}{Soundness}
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\begin{theorem}[Preservation]
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\begin{mathpar} \forall \Gamma, \metavariable{e}, \metavariable{\tau},\\
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D :: \Gamma \vdash \metavariable{e} : \metavariable{\tau}\\
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\Rightarrow \llbracket D \rrbracket \rightarrow_{eval} \metavariable{e'} \\
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\Rightarrow \Gamma \vdash \metavariable{e'} : \metavariable{\tau}
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\end{mathpar}
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\end{theorem}
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\end{frame}
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\begin{frame}{Soundness}
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\begin{theorem}[Progress]
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\begin{mathpar} \forall \Gamma, \metavariable{e}, \metavariable{\tau},\\
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D :: \Gamma \vdash \metavariable{e} : \metavariable{\tau}\\
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\Rightarrow \llbracket D \rrbracket \text{ is a value} \\ \vee \\
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\exists \metavariable{e'} . \llbracket D \rrbracket \rightarrow_{eval} \metavariable{e'} \\
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\end{mathpar}
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\end{theorem}
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\end{frame}
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\begin{frame}{Summary}
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todo
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\end{frame}
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\end{document}
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