initial beamer

beamer/lc-beamer/Makefile

@@ -0,0 +1,41 @@
+PACKAGE_STY = $(wildcard *.sty)
+PACKAGE_IMG = $(wildcard ccc_img/*)
+
+DESTDIR     ?= $(shell kpsewhich -var-value=TEXMFHOME)
+INSTALL_DIR  = $(DESTDIR)/tex/latex/ccc
+
+EXAMPLE_SRC  = example.tex
+EXAMPLE_PDF  = example.pdf
+
+CACHE_DIR   := $(shell pwd)/.latex-cache
+
+COMPILE_TEX := latexmk -lualatex -output-directory="$(CACHE_DIR)"
+
+.PHONY: install uninstall example all clean
+
+all: example
+
+example: $(EXAMPLE_PDF)
+
+install: $(PACKAGE_STY)
+	@mkdir -p "$(INSTALL_DIR)"
+	@cp $(PACKAGE_STY) "$(INSTALL_DIR)"
+	@mkdir -p "$(INSTALL_DIR)/ccc_img"
+	@cp $(PACKAGE_IMG) "$(INSTALL_DIR)/ccc_img"
+
+uninstall:
+	@rm -f $(addprefix "$(INSTALL_DIR)"/, $(PACKAGE_STY))
+	@rm -f $(addprefix "$(INSTALL_DIR)"/, $(PACKAGE_IMG))
+	@rmdir "$(INSTALL_DIR)/ccc_img"
+	@rmdir "$(INSTALL_DIR)"
+
+clean:
+	@rm -rf "$(CACHE_DIR)"
+	@rm -f "$(EXAMPLE_PDF)"
+
+$(EXAMPLE_PDF) : $(EXAMPLE_SRC)
+	@$(COMPILE_TEX) $(EXAMPLE_SRC)
+	@cp "$(CACHE_DIR)/$(EXAMPLE_PDF)" .
+
+
+

beamer/lc-beamer/beamercolorthemeccc.sty

@@ -0,0 +1,31 @@
+\mode<presentation>
+
+\definecolor{darkblue}{HTML}{265E87}
+\definecolor{darkteal}{RGB}{0,180,141}
+\definecolor{orange}{RGB}{231,123,20}
+\definecolor{gray}{rgb}{0.929,0.929,0.929}
+
+% Settings
+\setbeamercolor*{title page header}{fg=darkblue}
+\setbeamercolor*{author}{fg=darkblue}
+\setbeamercolor*{institute}{fg=darkblue}
+\setbeamercolor*{date}{fg=darkblue}
+\setbeamercolor*{frametitle}{fg=darkblue}
+\setbeamercolor*{framesubtitle}{fg=darkblue}
+\setbeamercolor*{pagenum}{fg=darkblue}
+\setbeamercolor*{alerted text}{fg=orange}
+\setbeamercolor*{item}{fg=darkblue}
+\setbeamercolor*{subitem}{fg=darkteal}
+\setbeamercolor*{subsubitem}{fg=orange}
+\setbeamercolor*{block title}{fg=darkblue}
+\setbeamercolor*{block title alerted}{fg=orange}
+\setbeamercolor*{block title example}{fg=darkteal}
+\setbeamercolor*{block body}{fg=black}
+\setbeamercolor*{bibliography entry title}{fg=black}
+\setbeamercolor*{bibliography entry author}{fg=darkblue}
+\setbeamercolor*{bibliography entry location}{fg=black}
+\setbeamercolor*{bibliography entry note}{fg=black}
+\setbeamercolor*{section in toc}{fg=darkblue}
+\setbeamercolor*{subsection in toc}{fg=darkteal}
+
+\mode<all>

beamer/lc-beamer/beamerfontthemeccc.sty

@@ -0,0 +1,11 @@
+\mode<presentation>
+
+\setbeamerfont{title}{size=\Large,series=\bfseries}
+\setbeamerfont{subtitle}{series=\bfseries}
+\setbeamerfont{frametitle}{size=\large, series=\bfseries}
+\setbeamerfont{framesubtitle}{size=\small, series=\bfseries}
+\setbeamerfont{pagenum}{size=\footnotesize}
+\setbeamerfont{block body}{size=\normalsize}
+\setbeamerfont{footnote}{size=\scriptsize}
+
+\mode<all>

beamer/lc-beamer/beamerinnerthemeccc.sty

@@ -0,0 +1,182 @@
+\mode<presentation>
+
+\def\beamer@andinst{\\[0.5em]}
+
+\setbeamertemplate{background}{
+  \begin{tikzpicture}
+    \useasboundingbox (0,0) rectangle(\the\paperwidth,\the\paperheight);
+    % the header
+    \fill[gray, anchor=north west] (0,\the\paperheight) rectangle(\the\paperwidth,\paperheight-1.4cm);
+    % ccc logo on the left (only on title page)
+    \ifnum\thepage=1\relax%
+    \node[anchor=west, inner sep=0] (header) at (0.25cm,\paperheight-0.7cm) {
+          \includegraphics[width=90px]{ccc_img/cclogo.pdf}};
+    \fi
+    % cfaed logo on the right (only on title page or if cfaedlogo is set)
+    \ifbeamer@cfaedlogo
+    \node[anchor=east, inner sep=0] (header) at (\paperwidth-0.25cm,\paperheight-0.7cm) {
+      \includegraphics[width=60px]{ccc_img/cfaed.png}};
+    \else
+    \ifnum\thepage=1\relax%
+    \node[anchor=east, inner sep=0] (header) at (\paperwidth-0.25cm,\paperheight-0.7cm) {
+          \includegraphics[width=60px]{ccc_img/cfaed.png}};
+    \fi
+    \fi
+    % the footer
+    \node[anchor=south west, inner sep=0] (footer) at (0,0) {
+      \ifnum\thepage=1\relax%
+      \includegraphics[width=\paperwidth]{ccc_img/titlefooter.png}
+      \fi
+      \includegraphics[width=\paperwidth]{ccc_img/footer.png}};
+    \ifnum\thepage>1\relax%
+    % page number on the left
+    \node[anchor=west, inner sep=0] (pagenum) at (0.2,0.55) {
+      \usebeamerfont{pagenum}\usebeamercolor[fg]{pagenum}\insertframenumber};
+    % page ccc logo on the right
+    \ifbeamer@ccclogo
+    \node[anchor=east, inner sep=0] (pagenum) at (\pagewidth-0.2,0.55) {
+      \includegraphics[width=50pt]{ccc_img/cclogo.pdf}};
+    \fi
+    \fi
+  \end{tikzpicture}
+}
+
+\defbeamertemplate*{title page}{ccc}[1][]{
+  \vskip1cm%
+  \begin{beamercolorbox}[wd=\textwidth,#1]{title page header}
+    \usebeamerfont{title}\inserttitle\par%
+    \medskip
+    \usebeamerfont{subtitle}\insertsubtitle\par%
+  \end{beamercolorbox}%
+  \vskip0.5cm%
+  \begin{beamercolorbox}[wd=\textwidth,#1]{author}
+    \usebeamerfont{author}\insertauthor%
+  \end{beamercolorbox}
+  \vskip0.5cm%
+  \begin{beamercolorbox}[wd=\textwidth,#1]{institute}
+    \usebeamerfont{institute}\insertinstitute%
+  \end{beamercolorbox}
+  \vskip0.5cm%
+  \begin{beamercolorbox}[wd=\textwidth,#1]{date}
+    \usebeamerfont{date}\insertdate%
+  \end{beamercolorbox}
+  \vfill
+}
+
+% Frame title
+\defbeamertemplate*{frametitle}{ccc}[1][]
+{
+  \begin{tikzpicture}
+    \useasboundingbox (0,0) rectangle(\textwidth,1.4cm);
+    \ifx\insertframesubtitle\@empty%
+    \node[anchor=west, inner sep=0, text width=\textwidth, align=left] at (0,0.7){
+      \usebeamerfont{frametitle}\insertframetitle};
+    \else
+    \node[anchor=west, inner sep=0, text width=\textwidth, align=left] at (0,0.7){
+      \usebeamerfont{frametitle}\insertframetitle\\
+      \usebeamerfont{framesubtitle}\insertframesubtitle};
+    \fi
+  \end{tikzpicture}
+}
+
+\defbeamertemplate*{block begin}{ccc}
+{
+  \par\vskip\medskipamount%
+  \usebeamercolor{block title}
+  \begin{tikzpicture}
+    \node [
+      draw=fg, inner sep=1ex,
+      text width=\textwidth-2ex-1pt,
+      minimum width=\textwidth-1pt,
+      rounded corners
+    ] (BOXCONTENT) \bgroup%
+    \ifx\insertblocktitle\@empty\else\par\fi
+    \usebeamerfont{block body}
+    \usebeamercolor[fg]{block body}
+}
+
+\defbeamertemplate*{block end}{ccc}
+{
+  \egroup;
+  \usebeamercolor{block title}
+  \ifx\insertblocktitle\@empty
+  \else{
+    \node[ fill=bg, anchor=west, text=fg, rounded corners ] at
+      ([shift={(5pt,0)}]BOXCONTENT.north west) {\insertblocktitle};
+  }
+  \fi
+  \end{tikzpicture}
+  \vskip\smallskipamount
+}
+
+\defbeamertemplate*{block example begin}{ccc}
+{
+  \par\vskip\medskipamount%
+  \usebeamercolor{block title example}
+  \begin{tikzpicture}
+    \node [
+      draw=fg, inner sep=1ex,
+      text width=\textwidth-2ex-1pt,
+      minimum width=\textwidth-1pt,
+      rounded corners
+    ] (BOXCONTENT) \bgroup%
+    \ifx\insertblocktitle\@empty\else\par\fi
+    \usebeamerfont{block body}
+    \usebeamercolor[fg]{block body}
+}
+
+\defbeamertemplate*{block example end}{ccc}
+{
+  \egroup;
+  \usebeamercolor{block title example}
+  \ifx\insertblocktitle\@empty
+  \else%
+  {
+    \node [ fill=bg, anchor=west, text=fg, rounded corners]
+      at ([shift={(5pt,0)}]BOXCONTENT.north west) {\insertblocktitle};
+  }
+  \fi
+  \end{tikzpicture}
+  \vskip\smallskipamount
+}
+
+\defbeamertemplate*{block alerted begin}{ccc}
+{
+  \par\vskip\medskipamount%
+  \usebeamercolor{block title alerted}
+  \begin{tikzpicture}
+    \node [
+      draw=fg, inner sep=1ex,
+      text width=\textwidth-2ex-1pt,
+      minimum width=\textwidth-1pt,
+      rounded corners
+    ] (BOXCONTENT) \bgroup%
+    \ifx\insertblocktitle\@empty\else\par\fi
+    \usebeamerfont{block body}
+    \usebeamercolor[fg]{block body}
+}
+
+\defbeamertemplate*{block alerted end}{ccc}
+{
+  \egroup;
+  \usebeamercolor{block title alerted}
+  \ifx\insertblocktitle\@empty
+  \else%
+  {
+    \node [fill=bg, anchor=west, text=fg, rounded corners] at
+      ([shift={(5pt,0)}]BOXCONTENT.north west) {\insertblocktitle};
+  }
+  \fi
+  \end{tikzpicture}
+  \vskip\smallskipamount
+}
+
+% Items
+\setbeamertemplate{itemize items}{\Squarepipe}
+
+\setbeamertemplate{footline}{\vspace{4mm}}
+
+\setbeamertemplate{section in toc}[sections numbered]
+\setbeamertemplate{subsection in toc}[subsections numbered]
+
+\mode<all>

beamer/lc-beamer/beamerthemeccc.sty

@@ -0,0 +1,31 @@
+\newif\ifbeamer@cfaedlogo
+\beamer@cfaedlogofalse
+\DeclareOptionBeamer{cfaedlogo}{\beamer@cfaedlogotrue}
+\ProcessOptionsBeamer
+
+\newif\ifbeamer@ccclogo
+\beamer@ccclogotrue
+\DeclareOptionBeamer{noccclogo}{\beamer@ccclogofalse}
+\ProcessOptionsBeamer
+
+\mode<presentation>
+
+% Requirement
+\RequirePackage{tikz}
+\usetikzlibrary{calc}
+\RequirePackage{fontspec}
+\RequirePackage{marvosym}
+
+% Settings
+\useinnertheme{ccc}
+\useoutertheme{ccc}
+\usecolortheme{ccc}
+\usefonttheme{ccc}
+
+% Set the font
+\setsansfont{Open Sans}
+
+% disable navigation symbols
+\setbeamertemplate{navigation symbols}{}
+
+\mode<all>

beamer/lc-beamer/example.tex

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