work on slides

beamer/lc-beamer/Makefile

@@ -9,7 +9,7 @@ EXAMPLE_PDF  = example.pdf
 
 CACHE_DIR   := $(shell pwd)/.latex-cache
 
-COMPILE_TEX := latexmk -lualatex -output-directory="$(CACHE_DIR)"
+COMPILE_TEX := latexmk -lualatex  -latexoption="-shell-escape"  -output-directory="$(CACHE_DIR)"
 
 .PHONY: install uninstall example all clean
 

beamer/lc-beamer/example.tex

@@ -6,6 +6,16 @@
 \usepackage[dvipsnames]{xcolor}
 \usepackage{mathpartir}
 \usepackage{stmaryrd}
+%\usepackage{minted}
+%\usemintedstyle{tango}
+
+
+\usepackage{listings}% http://ctan.org/pkg/listings
+\lstset{
+  basicstyle=\ttfamily,
+  mathescape
+}
+
 
 \newcommand{\metavariable}[1]{\textcolor{teal}{#1}}
 \newcommand{\typeterminal}[1]{\textcolor{brown}{#1}}
@@ -28,9 +38,11 @@
 \titlepage
 \end{frame}
 
+
+
 \begin{frame}
   \frametitle{Type Systems}
-  \framesubtitle{Tension between Safety and Flexibility}
+  \framesubtitle{Safety <> Flexibility}
 
   \begin{itemize}
   \item<1-> Goal: eliminate undefined behaviour,
@@ -47,46 +59,213 @@
   \end{itemize}
 \end{frame}
 
-\begin{frame}{How to get fast code?}
+
-  \framesubtitle{Requirements for a type system}
+\begin{frame}[t, fragile]{Type Systems}
+\framesubtitle{Safety <> Flexibility}
+
+\begin{lstlisting}
+	/* this is an angle in degrees */
+	double hue = 156.4;
+\end{lstlisting}
+\end{frame}
+
+
+
+
+\begin{frame}{Performance Checklist}
+	\framesubtitle{(singlethreaded)}
+  \begin{enumerate}
+  \item<1-> algorithmic
     \begin{itemize}
-  \item<1-> algorithmic optimization
+     \item choose appropriate data structures\\
-    \begin{itemize}
+			(reduce complexity in frequent cases)
-     \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:
+ \item<2-> implementation detail:
    \begin{itemize}
-    \item remove indirection
+	\item optimize memory bottleneck
-    \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
+       \item improve cache \& bus efficiency with compact representations
-	         vs. Fetch-Overhead
+  	   \item<3-> Synchronize locality in access order and memory layout.
+       (elements accessed in succession shall be close in memory,
+	     AoS vs SoA)
 
-       \item<4-> Synchronize locality in access order and memory layout.
+	   \item<4-> \(\rightarrow\) De-/Encode Overhead
-       -> elements accessed in succession shall be close in memory.
+	         vs. Fetch-Overhead,\\
-	        (e.g. AoS vs SoA)
+       		account for machine specific cache-sizes
+	\end{itemize}
+	\item
 
-       \item<5-> accounting for machine specific cache-sizes
+  \item<6-> use SIMD instructions
-       \item<6-> accounting for SIMD instructions
   \item<7-> interfacing with kernels running on accelerator devices
  \end{itemize}
+\end{enumerate}
+\end{frame}
+
+
+
+\begin{frame}[t, fragile]{Example - Image Processing}
+\begin{lstlisting}
+image-read :: FilePath -> [Color]
+image-write :: FilePath -> [Color] -> ()
+
+image-mix :: $\mathbb{R}$ -> [Color] -> [Color] -> [Color]
+image-saturate :: $\mathbb{R}$ -> [Color] -> [Color]
+
+let im1 = image-read "in1.png"
+let im2 = image-read "in2.png"
+let im3 = image-mix 0.5
+		im1
+		(image-saturate 0.8 im2)		
+
+image-write "out.png" im3
+\end{lstlisting}
+\end{frame}
+
+
+\begin{frame}{Example - Image Processing}
+\framesubtitle{More Types}
+
+	\begin{itemize}
+		\item \texttt{ColorRGBu8}
+		\item \texttt{ColorRGBf32}
+		\item \texttt{ColorHSVu8}
+		\item \texttt{ColorHSVf32}
 	\end{itemize}
-  \end{itemize}
+\end{frame}
+
+
+\begin{frame}[t, fragile]{Example - Image Processing}
+\framesubtitle{More Types}
+
+\begin{lstlisting}
+image-read :: FilePath -> [ColorRGBu8]
+image-write :: FilePath -> [ColorRGBu8] -> ()
+
+image-mix :: f32 -> [ColorRGBf32] -> [ColorRGBf32] -> [ColorRGBf32]
+image-saturate:: f32->[ColorHSVf32]->[ColorHSVf32]
+
+let im1 = image-read "in1.png"
+let im2 = image-read "in2.png"
+let im3 = image-mix 0.5
+		im1
+		(image-saturate 0.8 im2)		
+
+image-write "out.png" im3
+\end{lstlisting}
+\end{frame}
+
+
+\begin{frame}[t, fragile]{Example - Image Processing}
+
+\framesubtitle{AoS vs SoA}
+
+\begin{verbatim}
+[cacheline][cacheline][cacheline]
+\end{verbatim}
+
+\begin{verbatim}
+[RGBRGBRG][BRGBRGBR][GBRGBRGB]
+\end{verbatim}
+
+\begin{verbatim}
+[HSVHSVHS][VHSVHSVH][SVHSVHSV]
+\end{verbatim}
+
+\begin{verbatim}
+[HHHHHHHH][SSSSSSSS][VVVVVVVV]
+\end{verbatim}
+
 \end{frame}
 
 \begin{frame}{Abstract Concept - Multiple Representations}
-\framesubtitle{Requirements for a type system}
+  \framesubtitle{OpenGL Image Formats}
+
   \begin{itemize}
-    \item 
+    \item \texttt{GL\_R3\_G3\_B2}
+    \item \texttt{GL\_RGB10\_A2}
+	\item \texttt{GL\_R11F\_G11F\_B10F}
+	\item \texttt{GL\_SRGB8\_ALPHA8}
+	\item \ldots
   \end{itemize}
 \end{frame}
 
+
+\begin{frame}{Sea of Types}
+\centering
+\includegraphics[width=0.5\textwidth]{sea-of-types/target/sea-of-types-0-bytes.png}
+\end{frame}
+
+\begin{frame}{Sea of Types}
+\centering
+\includegraphics[width=0.5\textwidth]{sea-of-types/target/sea-of-types-1-ieee754.png}
+\end{frame}
+
+\begin{frame}{Sea of Types}
+\centering
+\includegraphics[height=0.5\textheight]{sea-of-types/target/sea-of-types-2-real.png}
+\end{frame}
+
+\begin{frame}{Sea of Types}
+\centering
+\includegraphics[height=0.5\textheight]{sea-of-types/target/sea-of-types-3-real.png}
+\end{frame}
+
+\begin{frame}{Sea of Types}
+\centering
+\includegraphics[height=0.8\textheight]{sea-of-types/target/sea-of-types-4-degrees.png}
+\end{frame}
+
+\begin{frame}{Sea of Types}
+\centering
+\includegraphics[height=0.8\textheight]{sea-of-types/target/sea-of-types-5-turns.png}
+\end{frame}
+
+\begin{frame}{Sea of Types}
+\centering
+\includegraphics[height=0.8\textheight]{sea-of-types/target/sea-of-types-6-radians.png}
+\end{frame}
+
+\begin{frame}{Sea of Types}
+\centering
+\includegraphics[height=0.8\textheight]{sea-of-types/target/sea-of-types-7-angle.png}
+\end{frame}
+
+\begin{frame}{Sea of Types}
+\centering
+\includegraphics[height=0.8\textheight]{sea-of-types/target/sea-of-types-8-hue.png}
+\end{frame}
+
+\begin{frame}{Sea of Types}
+\centering
+\includegraphics[height=0.8\textheight]{sea-of-types/target/sea-of-types-9-u8.png}
+\end{frame}
+
+\begin{frame}{Sea of Types}
+\centering
+\includegraphics[width=\textwidth]{sea-of-types/target/sea-of-types-10-u8.png}
+\end{frame}
+
+
+
+
+\begin{frame}{Solutions (I): Traits / Typeclasses / Interfaces}
+  \begin{itemize}
+	\item todo   
+  \end{itemize}
+\end{frame}
+
+
+\begin{frame}{Solutions (II): Wrapper structs / newtype}
+  \begin{itemize}
+	\item todo
+  \end{itemize}
+\end{frame}
+
+
+
 \begin{frame}{Coercions: (In-)Coherence with Transitivity}
   \begin{itemize}
     \item<1-> Common example: coerce  \texttt{Int -> Real}
@@ -114,15 +293,16 @@
   \end{itemize}
 \end{frame}
 
-\begin{frame}{Sea of Types}
-\includegraphics[width=\textwidth]{sot-int.png}
-\end{frame}
 
-\begin{frame}{Sea of Types}
+\begin{frame}{Related Work}
-\includegraphics[width=\textwidth]{sot-json.png}
+	\begin{itemize}
+	\item TODO
+	\item cite
+	\item some
+	\item stuff
+	\end{itemize}
 \end{frame}
 
-
 \begin{frame}{Ladder Types}
   \begin{block}{Intuition}
     Encode "represented-as" relation into type terms
@@ -142,6 +322,38 @@
   \)\end{example}
 \end{frame}
 
+\begin{frame}{Morphisms}
+  \begin{block}{Intuition}
+	Transform between semantically equivalent representations of the same abstract concept
+  \end{block}
+
+  \begin{block}{Type}
+    \( \tau \sim \tau_1 \rightarrow_\text{morph} \tau \sim \tau_2 \)
+  \end{block}
+
+  \begin{example}\(
+	 Angle \sim Degrees \sim \mathbb{R}
+	\rightarrow_\text{morph}
+	 Angle \sim Radians \sim \mathbb{R}
+  \)\end{example}
+\end{frame}
+
+\begin{frame}{Morphisms}
+\framesubtitle{Coherence}
+  \begin{block}{Coherence Condition}
+	Assume\\
+	\(\metavariable{m_\sigma} : \typeterminal{\metavariable{\sigma} \sim \metavariable{\sigma_1}  \rightarrow_\text{morph}  \metavariable{\sigma} \sim \metavariable{\sigma_2}}\)\\
+	\(\metavariable{m_\tau} : \typeterminal{\metavariable{\tau} \sim \metavariable{\tau_1}  \rightarrow_\text{morph}  \metavariable{\tau} \sim \metavariable{\tau_2}}\)\\
+	\(\metavariable{f_1} : \typeterminal{\metavariable{\sigma} \sim \metavariable{\sigma_1}  \rightarrow  \metavariable{\tau} \sim \metavariable{\tau_1} }\)\\
+	\(\metavariable{f_2} : \typeterminal{\metavariable{\sigma} \sim \metavariable{\sigma_2}  \rightarrow  \metavariable{\tau} \sim \metavariable{\tau_2} }\)\\
+	then it holds that:\\
+	\(\forall x:\typeterminal{\metavariable{\sigma}\sim\metavariable{\sigma_1}},\quad
+	\metavariable{m_\tau}(\metavariable{f_1} x) = \metavariable{f_2} (\metavariable{m_\sigma} x) \)
+  \end{block}
+\end{frame}
+
+
+
 \begin{frame}{Extending System F}
 \small
 \begin{grammar}
@@ -238,10 +450,10 @@ $$\\$$
 }{Descension}
 
 \end{grammar}
-
-  
 \end{frame}
 
+
+
 \begin{frame}{Morphism Graph}
 
 \begin{definition}[Morphism Paths]
@@ -278,8 +490,8 @@ $$\\$$
 \end{definition}  
 \end{frame}
 
-\begin{frame}{Morphism Graph}
 
+\begin{frame}{Morphism Graph}
 \begin{example}[Morphism Graph]
 \begin{mathpar}
 \small
@@ -290,12 +502,11 @@ $$\\$$
 \}.
 \end{mathpar}
 \end{example}
-
 \end{frame}
 
 
-\begin{frame}{Morphism Graph}
 
+\begin{frame}{Morphism Graph}
 \begin{example}[Morphism Graph]
 Then ..
 \small
@@ -305,23 +516,9 @@ Then ..
 \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]
@@ -382,8 +579,46 @@ Then ..
 \end{definition}
 \end{frame}
 
-\begin{frame}{Translation}
 
+\begin{frame}{Translation}
+\begin{example}[Morphism Translation]
+\begin{mathpar}
+\small
+\Big{\llbracket}
+	\inferrule[M-Chain]{
+		\ldots
+	}{
+		\Gamma \vdash \typeterminal{\langle\text{Seq}\quad\text{Angle}\sim\text{Degrees}\sim\mathbb{R}\rangle}\\
+		\leadsto \typeterminal{\langle\text{Seq}\quad\text{Angle}\sim\text{Radians}\sim\mathbb{R}\rangle}
+	}
+\Big{\rrbracket} =
+\exprterminal{\lambda \text{xs}:\typeterminal{\langle\text{Seq}\quad \text{Angle}\sim\text{Degrees}\sim\mathbb{R} \rangle}}
+\newline
+\exprterminal{\mapsto (map \text{ }(\lambda \text{x}:\metavariable{\tau} \mapsto \text{turns-to-radians } (\text{degrees-to-turns } x)) \text{ } xs)	}
+\end{mathpar}
+\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}[Expression Translation]
 Translates a type-derivation tree into a fully expanded expression
 
@@ -424,8 +659,10 @@ D :: \Gamma \vdash \metavariable{e} : \metavariable{\tau}\\
 \end{lemma}
 \end{frame}
 
-\begin{frame}{Soundness}
 
+
+
+\begin{frame}{Soundness}
 \begin{theorem}[Preservation]
 \begin{mathpar} \forall \Gamma, \metavariable{e}, \metavariable{\tau},\\
 D :: \Gamma \vdash \metavariable{e} : \metavariable{\tau}\\
@@ -436,19 +673,26 @@ D :: \Gamma \vdash \metavariable{e} : \metavariable{\tau}\\
 \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 \\
+\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}
 

beamer/lc-beamer/sea-of-types/makefile

@@ -0,0 +1,8 @@
+
+TARGETS=$(shell find -name '*.dot' | sed -E 's~\.dot~\.png~')
+
+all: $(TARGETS)
+
+%.png: %.dot
+	dot -Tpng $< -o target/$@
+

beamer/lc-beamer/sea-of-types/sea-of-types-0-bytes.dot

@@ -0,0 +1,6 @@
+
+digraph SeaOfTypes {
+    Byte4 [label="[Byte; 4]", fontsize=24, shape=plaintext];
+}
+
+

beamer/lc-beamer/sea-of-types/sea-of-types-1-ieee754.dot

@@ -0,0 +1,9 @@
+
+digraph SeaOfTypes {
+
+    Byte4 [label="[Byte; 4]", fontsize=24, shape=plaintext];
+    IEEE754single [label="IEEE-754.single", fontsize=24, shape=plaintext];
+    IEEE754single -> Byte4
+}
+
+

beamer/lc-beamer/sea-of-types/sea-of-types-10-u8.dot

@@ -0,0 +1,69 @@
+
+digraph SeaOfTypes {
+
+    Byte1 [label="[Byte; 1]", fontsize=10, shape=plaintext, color=lightgrey];
+    Byte4 [label="[Byte; 4]", fontsize=10, shape=plaintext, color=lightgrey];
+    Byte8 [label="[Byte; 8]", fontsize=10, shape=plaintext, color=lightgrey];
+
+    IEEE754single [label="IEEE-754.single", fontsize=10, shape=plaintext, color=lightgrey];
+    IEEE754double [label="IEEE-754.double", fontsize=10, shape=plaintext, color=lightgrey];
+
+    Real [label="ℝ", fontsize=20, shape=plaintext];
+
+    Degrees [label="Degrees", fontsize=18, shape=plaintext];
+    Turns [label="Turns", fontsize=18, shape=plaintext];
+    Radians [label="Radians", fontsize=18, shape=plaintext];
+
+    Angle [label="Angle", fontsize=20, shape=plaintext];
+    RealUnit [label="ℝ_[0,1]", fontsize=24, shape=plaintext];    
+    HSV [label="HSV", fontsize=20, shape=plaintext];
+
+    U8 [label="machine.UInt8", fontsize=18, shape=plaintext];
+    Z256 [label="ℤ_256", fontsize=20, shape=plaintext];
+
+    Real360 [label="ℝ_[0,360)", fontsize=18, shape=plaintext];
+    Real1 [label="ℝ_[0,1)", fontsize=18, shape=plaintext];
+    Real2pi [label="ℝ_[0,2π)", fontsize=18, shape=plaintext];
+
+    QuantizedLinear [label="<QuantizedLinear[0,1] 256>", fontsize=20, shape=plaintext];
+    QuantizedLinear1 [label="<QuantizedLinear[0,1) 256>", fontsize=20, shape=plaintext];
+    QuantizedLinear360 [label="<QuantizedLinear[0,360) 256>", fontsize=20, shape=plaintext];
+    QuantizedLinear2pi [label="<QuantizedLinear[0,2π) 256>", fontsize=20, shape=plaintext];
+
+    IEEE754single -> Byte4
+    IEEE754double -> Byte8
+
+    Real -> IEEE754single
+    Real -> IEEE754double
+
+    Real360 -> Real
+    Real1 -> Real
+    Real2pi -> Real
+
+    Real360 -> QuantizedLinear360
+    Real1 -> QuantizedLinear1
+    Real2pi -> QuantizedLinear2pi
+
+    Degrees -> Real360
+    Turns -> Real1
+    Radians -> Real2pi
+    RealUnit -> Real
+
+    RealUnit -> QuantizedLinear
+    QuantizedLinear -> Z256
+    QuantizedLinear1 -> Z256
+    QuantizedLinear360 -> Z256
+    QuantizedLinear2pi -> Z256
+    Z256 -> U8
+    U8 -> Byte1
+
+    Angle -> Degrees
+    Angle -> Turns
+    Angle -> Radians
+
+    HSV -> Angle [label="hue"]
+    HSV -> RealUnit [label="saturation"]
+    HSV -> RealUnit [label="value"]
+}
+
+

beamer/lc-beamer/sea-of-types/sea-of-types-2-real.dot

@@ -0,0 +1,11 @@
+
+digraph SeaOfTypes {
+
+    Byte4 [label="[Byte; 4]", fontsize=24, shape=plaintext];
+    IEEE754single [label="IEEE-754.single", fontsize=24, shape=plaintext];
+    Real [label="ℝ", fontsize=28, shape=plaintext];
+
+    IEEE754single -> Byte4
+    Real -> IEEE754single
+}
+

beamer/lc-beamer/sea-of-types/sea-of-types-3-real.dot

@@ -0,0 +1,20 @@
+
+digraph SeaOfTypes {
+
+    Byte4 [label="[Byte; 4]", fontsize=24, shape=plaintext];
+    Byte8 [label="[Byte; 8]", fontsize=24, shape=plaintext];
+
+    IEEE754single [label="IEEE-754.single", fontsize=24, shape=plaintext];
+    IEEE754double [label="IEEE-754.double", fontsize=24, shape=plaintext];
+
+    Real [label="ℝ", fontsize=28, shape=plaintext];
+
+
+    IEEE754single -> Byte4
+    IEEE754double -> Byte8
+
+    Real -> IEEE754single
+    Real -> IEEE754double
+}
+
+

beamer/lc-beamer/sea-of-types/sea-of-types-4-degrees.dot

@@ -0,0 +1,24 @@
+
+digraph SeaOfTypes {
+
+    Byte4 [label="[Byte; 4]", fontsize=12, shape=plaintext, color=lightgrey];
+    Byte8 [label="[Byte; 8]", fontsize=12, shape=plaintext, color=lightgrey];
+
+    IEEE754single [label="IEEE-754.single", fontsize=12, shape=plaintext, color=lightgrey];
+    IEEE754double [label="IEEE-754.double", fontsize=12, shape=plaintext, color=lightgrey];
+
+    Real [label="ℝ", fontsize=28, shape=plaintext];
+
+    Degrees [label="Degrees", fontsize=24, shape=plaintext];
+
+
+    IEEE754single -> Byte4
+    IEEE754double -> Byte8
+
+    Real -> IEEE754single
+    Real -> IEEE754double
+
+    Degrees -> Real
+}
+
+

beamer/lc-beamer/sea-of-types/sea-of-types-5-turns.dot

@@ -0,0 +1,25 @@
+
+digraph SeaOfTypes {
+
+    Byte4 [label="[Byte; 4]", fontsize=12, shape=plaintext, color=lightgrey];
+    Byte8 [label="[Byte; 8]", fontsize=12, shape=plaintext, color=lightgrey];
+
+    IEEE754single [label="IEEE-754.single", fontsize=12, shape=plaintext, color=lightgrey];
+    IEEE754double [label="IEEE-754.double", fontsize=12, shape=plaintext, color=lightgrey];
+
+    Real [label="ℝ", fontsize=28, shape=plaintext];
+
+    Degrees [label="Degrees", fontsize=24, shape=plaintext];
+    Turns [label="Turns", fontsize=24, shape=plaintext];
+
+    IEEE754single -> Byte4
+    IEEE754double -> Byte8
+
+    Real -> IEEE754single
+    Real -> IEEE754double
+
+    Degrees -> Real
+    Turns -> Real
+}
+
+

beamer/lc-beamer/sea-of-types/sea-of-types-6-radians.dot

@@ -0,0 +1,27 @@
+
+digraph SeaOfTypes {
+
+    Byte4 [label="[Byte; 4]", fontsize=12, shape=plaintext, color=lightgrey];
+    Byte8 [label="[Byte; 8]", fontsize=12, shape=plaintext, color=lightgrey];
+
+    IEEE754single [label="IEEE-754.single", fontsize=12, shape=plaintext, color=lightgrey];
+    IEEE754double [label="IEEE-754.double", fontsize=12, shape=plaintext, color=lightgrey];
+
+    Real [label="ℝ", fontsize=28, shape=plaintext];
+
+    Degrees [label="Degrees", fontsize=24, shape=plaintext];
+    Turns [label="Turns", fontsize=24, shape=plaintext];
+    Radians [label="Radians", fontsize=24, shape=plaintext];
+
+    IEEE754single -> Byte4
+    IEEE754double -> Byte8
+
+    Real -> IEEE754single
+    Real -> IEEE754double
+
+    Degrees -> Real
+    Turns -> Real
+    Radians -> Real
+}
+
+

beamer/lc-beamer/sea-of-types/sea-of-types-7-angle.dot

@@ -0,0 +1,33 @@
+
+digraph SeaOfTypes {
+
+    Byte4 [label="[Byte; 4]", fontsize=10, shape=plaintext, color=lightgrey];
+    Byte8 [label="[Byte; 8]", fontsize=10, shape=plaintext, color=lightgrey];
+
+    IEEE754single [label="IEEE-754.single", fontsize=10, shape=plaintext, color=lightgrey];
+    IEEE754double [label="IEEE-754.double", fontsize=10, shape=plaintext, color=lightgrey];
+
+    Real [label="ℝ", fontsize=24, shape=plaintext];
+
+    Degrees [label="Degrees", fontsize=20, shape=plaintext];
+    Turns [label="Turns", fontsize=20, shape=plaintext];
+    Radians [label="Radians", fontsize=20, shape=plaintext];
+    
+    Angle [label="Angle", fontsize=24, shape=plaintext, color=brown];
+
+    IEEE754single -> Byte4
+    IEEE754double -> Byte8
+
+    Real -> IEEE754single
+    Real -> IEEE754double
+
+    Degrees -> Real
+    Turns -> Real
+    Radians -> Real
+
+    Angle -> Degrees
+    Angle -> Turns
+    Angle -> Radians
+}
+
+

beamer/lc-beamer/sea-of-types/sea-of-types-8-hue.dot

@@ -0,0 +1,40 @@
+
+digraph SeaOfTypes {
+
+    Byte4 [label="[Byte; 4]", fontsize=10, shape=plaintext, color=lightgrey];
+    Byte8 [label="[Byte; 8]", fontsize=10, shape=plaintext, color=lightgrey];
+
+    IEEE754single [label="IEEE-754.single", fontsize=10, shape=plaintext, color=lightgrey];
+    IEEE754double [label="IEEE-754.double", fontsize=10, shape=plaintext, color=lightgrey];
+
+    Real [label="ℝ", fontsize=20, shape=plaintext];
+
+    Degrees [label="Degrees", fontsize=18, shape=plaintext];
+    Turns [label="Turns", fontsize=18, shape=plaintext];
+    Radians [label="Radians", fontsize=18, shape=plaintext];
+    
+    Angle [label="Angle", fontsize=20, shape=plaintext];
+    RealUnit [label="ℝ_[0,1]", fontsize=24, shape=plaintext];    
+    HSV [label="HSV", fontsize=24, shape=plaintext];
+
+    IEEE754single -> Byte4
+    IEEE754double -> Byte8
+
+    Real -> IEEE754single
+    Real -> IEEE754double
+
+    Degrees -> Real
+    Turns -> Real
+    Radians -> Real
+    RealUnit -> Real
+
+    Angle -> Degrees
+    Angle -> Turns
+    Angle -> Radians
+
+    HSV -> Angle [label="hue", textcolor=blue]
+    HSV -> RealUnit [label="saturation", textcolor=blue]
+    HSV -> RealUnit [label="value", textcolor=blue]
+}
+
+

beamer/lc-beamer/sea-of-types/sea-of-types-9-u8.dot

@@ -0,0 +1,50 @@
+
+digraph SeaOfTypes {
+
+    Byte1 [label="[Byte; 1]", fontsize=10, shape=plaintext, color=lightgrey];
+    Byte4 [label="[Byte; 4]", fontsize=10, shape=plaintext, color=lightgrey];
+    Byte8 [label="[Byte; 8]", fontsize=10, shape=plaintext, color=lightgrey];
+
+    IEEE754single [label="IEEE-754.single", fontsize=10, shape=plaintext, color=lightgrey];
+    IEEE754double [label="IEEE-754.double", fontsize=10, shape=plaintext, color=lightgrey];
+
+    Real [label="ℝ", fontsize=20, shape=plaintext];
+
+    Degrees [label="Degrees", fontsize=18, shape=plaintext];
+    Turns [label="Turns", fontsize=18, shape=plaintext];
+    Radians [label="Radians", fontsize=18, shape=plaintext];
+    
+    Angle [label="Angle", fontsize=20, shape=plaintext];
+    RealUnit [label="ℝ_[0,1]", fontsize=24, shape=plaintext];    
+    HSV [label="HSV", fontsize=20, shape=plaintext];
+
+    U8 [label="machine.UInt8", fontsize=18, shape=plaintext];
+    Z256 [label="ℤ_256", fontsize=20, shape=plaintext];
+    QuantizedLinear [label="<QuantizedLinear[0,1] 256>", fontsize=20, shape=plaintext];
+
+    IEEE754single -> Byte4
+    IEEE754double -> Byte8
+
+    Real -> IEEE754single
+    Real -> IEEE754double
+
+    Degrees -> Real
+    Turns -> Real
+    Radians -> Real
+    RealUnit -> Real
+
+    RealUnit -> QuantizedLinear
+    QuantizedLinear -> Z256
+    Z256 -> U8
+    U8 -> Byte1
+
+    Angle -> Degrees
+    Angle -> Turns
+    Angle -> Radians
+
+    HSV -> Angle [label="hue"]
+    HSV -> RealUnit [label="saturation"]
+    HSV -> RealUnit [label="value"]
+}
+
+