Work on data types chapter

tex/.vscode/ltex.dictionary.en-US.txt

@@ -2,3 +2,8 @@ Vatthauer
 mycase
 Coalgebras
 Coalgebra
+Milius
+FAU
+codomain
+surjective
+haskell

tex/.vscode/ltex.hiddenFalsePositives.en-US.txt

@@ -0,0 +1,7 @@
+{"rule":"MORFOLOGIK_RULE_EN_US","sentence":"^\\QThis is a summary of the course “Algebra of Programming” taught by Prof. Dr. Stefan Milius in the winter term 2023/2024 at the FAU Functions.\\E$"}
+{"rule":"MORFOLOGIK_RULE_EN_US","sentence":"^\\QFriedrich-Alexander-Universitity Erlangen-Nürnberg\\E$"}
+{"rule":"COMMA_PARENTHESIS_WHITESPACE","sentence":"^\\QThis is a summary of the course “Algebra of Programming” taught by Prof. Dr. Stefan Milius in the winter term 2023/2024 at the FAU .\\E$"}
+{"rule":"MORFOLOGIK_RULE_EN_US","sentence":"^\\QThis is a summary of the course “Algebra des Programmierens” taught by Prof. Dr. Stefan Milius in the winter term 2023/2024 at the FAU .\\E$"}
+{"rule":"MORFOLOGIK_RULE_EN_US","sentence":"^\\QFriedrich-Alexander-Universität Erlangen-Nürnberg\\E$"}
+{"rule":"COMMA_PARENTHESIS_WHITESPACE","sentence":"^\\QThis is a summary of the course “Algebra des Programmierens” taught by Prof. Dr. Stefan Milius in the winter term 2023/2024 at the FAU .\\E$"}
+{"rule":"MORFOLOGIK_RULE_EN_US","sentence":"^\\Q\\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q satisfies the following two rules law:natident Identity law:natfusion Fusion Lists.\\E$"}

tex/bib.bib

@@ -0,0 +1,17 @@
+@article{poll1999algebra,
+  title     = {Algebra of Programming by Richard Bird and Oege de Moor, Prentice Hall, 1996 (dated 1997).},
+  author    = {Poll, Erik and Thompson, Simon},
+  journal   = {Journal of Functional Programming},
+  volume    = {9},
+  number    = {3},
+  pages     = {347--354},
+  year      = {1999},
+  publisher = {Cambridge University Press}
+}
+
+@book{adamek1990abstract,
+  title     = {Abstract and concrete categories},
+  author    = {Ad{\'a}mek, Ji{\v{r}}{\'\i} and Herrlich, Horst and Strecker, George},
+  year      = {1990},
+  publisher = {Wiley-Interscience}
+}

tex/main.tex

@@ -4,8 +4,11 @@
 \usepackage[top=2cm,lmargin=1in,rmargin=1in,bottom=3cm,hmarginratio=1:1]{geometry}
 \usepackage{scrlayer-fancyhdr}
 \usepackage{extramarks}
+\usepackage[ngerman, english]{babel}
+\babeltags{german=ngerman}
 \usepackage{anyfontsize}
 \usepackage{unicode-math}
+\usepackage{amsmath}
 \usepackage[scale=.85]{noto-mono} % TODO find better unicode mono font
 \usepackage[final]{hyperref}
 \pagestyle{fancy}
@@ -34,11 +37,26 @@
 }
 %%%%
 
+%%%% Code listings
+\usepackage{minted}
+\setminted[haskell]{
+  linenos=true,
+  breaklines=true,
+  encoding=utf8,
+  fontsize=\small,
+  autogobble
+}
+\setmintedinline[haskell]{
+    style=bw
+}
+
+%%%%
+
 %%%% Math packages
 \usepackage{amsthm}
 \usepackage{thmtools}
 \usepackage{tikz}
-\usetikzlibrary{cd, quotes}
+\usetikzlibrary{cd, babel, quotes}
 \declaretheorem[name=Definition,style=definition,numberwithin=section]{definition}
 \declaretheorem[name=Example,style=definition,sibling=definition]{example}
 \declaretheorem[style=definition,numbered=no]{exercise}
@@ -52,6 +70,7 @@
 \declaretheorem[sibling=lemma]{proposition}
 %%%%
 
+%%%% href
 \makeatletter
 \hypersetup{
   pdfauthor={\@author},
@@ -60,6 +79,13 @@
   hidelinks,
 }
 \makeatother
+%%%%
+
+%%%% Bibliography
+\usepackage[style=ieee, sorting=ynt, language=british]{biblatex} % advanced citations, british to make dates DD-MM-YYYY
+\usepackage[english=british]{csquotes} % biblatex recommended to load this
+\addbibresource{bib.bib}
+%%%%
 
 %%%% Custom definitions: Commands and environments
 
@@ -114,4 +140,8 @@
 \include{sections/02_categories}
 \include{sections/03_constructions}
 %%
+
+\appendix
+\emergencystretch=1em
+\printbibliography[heading=bibintoc]{}
 \end{document}

tex/sections/01_introduction.tex

@@ -1,4 +1,182 @@
+% chktex-file 46
 \section{Introduction}
+This is a summary of the course ``Algebra des Programmierens'' taught by Prof.\ Dr.\ Stefan Milius in the winter term 2023/2024 at the FAU~\footnote{Friedrich-Alexander-Universität Erlangen-Nürnberg}.
+The course is based on~\cite{poll1999algebra} with~\cite{adamek1990abstract} as a reference for category theory.
+
+Goal of the course is to develop a mathematical theory for semantics of data types and their accompanying proof principles. The chosen environment is the field of category theory.
 
 \subsection{Functions}
-\subsection{Data Types}
+A function $f : X \rightarrow Y$ is a mapping from the set $X$ (the domain of $f$) to the set $Y$ (the codomain of $f$).
+More concretely $f$ is a relation $f \subseteq X \times Y$ which is
+\begin{itemize}
+  \item \emph{left-total}, i.e.\ for all $x \in X$ exists some $y \in Y$ such that $(x,y) \in f$;
+  \item \emph{right-unique}, i.e.\ any $(x,y),(x,y') \in f$ imply $y = y'$.
+\end{itemize}
+
+Often, one is also interested in the symmetrical properties, a function is called
+\begin{itemize}
+  \item \emph{injective} or \emph{left-unique} if for every $x,x' \in X$ the implication $f(x) = f(x') \rightarrow x = x'$ holds;
+  \item \emph{surjective} or \emph{right-total} if for every $y \in Y$ there exists an $x \in X$ such that $f(x) = y$;
+  \item \emph{bijective} if it is injective and surjective.
+\end{itemize}
+
+\begin{example}
+  \begin{enumerate}
+    \item The identity function $id_A : A \rightarrow A$, $id_A(x) = x$
+    \item The constant function $b! : A \rightarrow B$ for $b \in B$ defined by $b!(x) = b$
+    \item The inclusion function $i_A : A \rightarrow B$ for $A \subseteq B$ defined by $i_A(x) = x$
+    \item Constants $b : 1 \rightarrow B$, where $1 := {*}$. The function $b$ is in bijection with the set $B$.
+    \item Composition of function $f : A \rightarrow B, g : B \rightarrow C$ called $g \circ f : A \rightarrow C$ defined by $(g \circ f)(x) = g(f(x))$.
+    \item The empty function $¡ : \emptyset \rightarrow B$
+    \item The singleton function $! : A \rightarrow 1$
+  \end{enumerate}
+\end{example}
+
+\subsection{Data Types}
+
+Programs work with data that should ideally be organized in a useful manner.
+A useful representation for data in functional programming is by means of \emph{algebraic data types}.
+Some basic data types (written in Haskell notation) are
+\begin{minted}{haskell}
+  data Bool = True | False
+  data Nat  = Zero | Succ Nat
+\end{minted}
+
+These data types are declared by means of constructors, yielding concrete descriptions how inhabitants of these types are created.
+\emph{Parametric data types} are additionally parametrized by another data type, e.g.\
+\begin{minted}{haskell}
+  data Maybe a    = Nothing | Just a
+  data Either a b = Left a  | Right b
+  data List a     = Nil     | Cons a (List a)
+\end{minted}
+
+
+Such data types (parametric or non-parametric) usually come with a principle for defining functions called recursion and in richer type systems (e.g.\ in a dependently typed setting) with a principle for proving facts about recursive functions called induction.
+Equivalently, every function defined by recursion can be defined via a \emph{fold}-function which satisfies an identity and fusion law, which replace the induction principle. Let us now consider two examples of data types and illustrate this.
+
+\subsubsection{Natural Numbers}
+The type of natural numbers comes with a fold function $foldn : C \rightarrow (Nat \rightarrow C) \rightarrow Nat \rightarrow C$ for every $C$, defined by
+\begin{alignat*}{2}
+   & foldn\;c\;h\;zero     &  & = c                   \\
+   & foldn\;c\;h\;(suc\;n) &  & = h\;(foldn\;c\;h\;n)
+\end{alignat*}
+
+
+\begin{example}  Let us now consider some functions defined in terms of $foldn$.
+  \begin{itemize}
+    \item $iszero : Nat \rightarrow Bool$ is defined by
+          \[iszero = foldn\;true\;false!\]
+    \item $plus : Nat \rightarrow Nat \rightarrow Nat$ is defined by
+          \[plus = foldn\;id (\lambda f\;n. succ (f\;n)) \]
+  \end{itemize}
+\end{example}
+
+\begin{proposition}
+  $foldn$ satisfies the following two rules
+  \begin{enumerate}
+    \item \customlabel{law:natident}{\textbf{Identity}}: $foldn\;zero\;succ = id_{Nat}$
+    \item \customlabel{law:natfusion}{\textbf{Fusion}}: for all $c : C$, $h, h' : Nat
+            \rightarrow C$ and $k : C \rightarrow C'$ with $kc = c'$ and $kh = h'k$ follows $k \circ foldn\;c\;h = foldn\;c'\;h'$, or diagrammatically:
+          % https://q.uiver.app/#q=WzAsNSxbMiwwLCJDIl0sWzQsMCwiQyJdLFsyLDIsIkMnIl0sWzQsMiwiQyciXSxbMCwwLCIxIl0sWzQsMCwiYyJdLFswLDIsImsiXSxbNCwyLCJjJyIsMl0sWzAsMSwiaCIsMl0sWzIsMywiaCciLDJdLFsxLDMsImsiLDFdXQ==
+          \[
+            \begin{tikzcd}[ampersand replacement=\&]
+              1 \&\& C \&\& C \\
+              \\
+              \&\& {C'} \&\& {C'}
+              \arrow["c", from=1-1, to=1-3]
+              \arrow["k", from=1-3, to=3-3]
+              \arrow["{c'}"', from=1-1, to=3-3]
+              \arrow["h"', from=1-3, to=1-5]
+              \arrow["{h'}"', from=3-3, to=3-5]
+              \arrow["k"{description}, from=1-5, to=3-5]
+            \end{tikzcd}
+          \]
+          implies
+          % https://q.uiver.app/#q=WzAsMyxbMCwwLCJOYXQiXSxbMiwwLCJDIl0sWzIsMiwiQyciXSxbMSwyLCJrIl0sWzAsMSwiZm9sZG5cXDtjXFw7aCJdLFswLDIsImZvbGRuXFw7YydcXDtoJyIsMl1d
+          \[
+            \begin{tikzcd}[ampersand replacement=\&]
+              Nat \&\& C \\
+              \\
+              \&\& {C'}
+              \arrow["k", from=1-3, to=3-3]
+              \arrow["{foldn\;c\;h}", from=1-1, to=1-3]
+              \arrow["{foldn\;c'\;h'}"', from=1-1, to=3-3]
+            \end{tikzcd}
+          \]
+  \end{enumerate}
+\end{proposition}
+\begin{proof}
+  Both follow by induction over an argument $n : Nat$:
+  \begin{enumerate}
+    \item~\ref{law:natident}:
+          \begin{mycase}
+            \case{} $n = zero$
+            \[foldn\;zero\;succ\;zero = zero = id\;zero\]
+            \case{} $n = succ\;m$
+            \begin{alignat*}{1}
+              foldn\;zero\;succ\;(succ\;m) & = succ (foldn\;zero\;succ\;m)
+              \\&= succ\;m \tag{IH}
+              \\&= id (succ\;m)
+            \end{alignat*}
+          \end{mycase}
+    \item~\ref{law:natfusion}:
+          \begin{mycase}
+            \case{} $n = zero$
+            \[k(foldn\;c\;h\;zero) = k\;c = c' = foldn\;c'\;h'\;zero\]
+            \case{} $n = succ\;m$
+            \begin{alignat*}{1}
+              k(foldn\;c\;h\;(succ\;m)) & = k(h(foldn\;c\;h\;m))
+              \\&= h'(k(foldn\;c\;h\;m))
+              \\&= h'(foldn\;c'\;h'\;m) \tag{IH}
+              \\&= foldn\;c'\;h'\;(succ\;m)
+            \end{alignat*}
+          \end{mycase}
+  \end{enumerate}
+\end{proof}
+
+\begin{example}
+  The identity and fusion laws can in turn be used to prove the following induction principle:
+
+  For any predicate $p : Nat \rightarrow Bool$,
+  \begin{enumerate}
+    \item $p\;zero = true$ and
+    \item $p \circ succ = p$
+  \end{enumerate}
+  implies $p = true!$. This follows by % chktex 40
+  \begin{alignat*}{1}
+     & p
+    \\=\;&p \circ (foldn\;zero\;succ) \tag{\ref{law:natident}}
+    \\=\;&foldn\;true\;id \tag{\ref{law:natfusion}}
+    \\=\;&true! \circ (foldn\;zero\;succ) \tag{\ref{law:natfusion}}
+    \\=\;&true!. \tag{\ref{law:natident}}
+  \end{alignat*}
+
+  Where the first application of~\ref{law:natfusion} is justified, since the diagram
+  % https://q.uiver.app/#q=WzAsNSxbMiwwLCJOYXQiXSxbNCwwLCJOYXQiXSxbMiwyLCJCb29sIl0sWzQsMiwiQm9vbCJdLFswLDAsIjEiXSxbNCwwLCJ6ZXJvIl0sWzAsMSwic3VjYyJdLFsxLDMsInAiXSxbMCwyLCJwIl0sWzIsMywiaWQiLDFdLFs0LDIsInRydWUiLDJdXQ==
+  \[\begin{tikzcd}
+      1 && Nat && Nat \\
+      \\
+      && Bool && Bool
+      \arrow["zero", from=1-1, to=1-3]
+      \arrow["succ", from=1-3, to=1-5]
+      \arrow["p", from=1-5, to=3-5]
+      \arrow["p", from=1-3, to=3-3]
+      \arrow["id"{description}, from=3-3, to=3-5]
+      \arrow["true"', from=1-1, to=3-3]
+    \end{tikzcd}\]
+  commutes by the requisite properties of $p$, and the second application of~\ref{law:natfusion} is justified, since the diagram
+  % https://q.uiver.app/#q=WzAsNSxbMiwwLCJOYXQiXSxbNCwwLCJOYXQiXSxbMiwyLCJCb29sIl0sWzQsMiwiQm9vbCJdLFswLDAsIjEiXSxbNCwwLCJ6ZXJvIl0sWzAsMSwic3VjYyJdLFsxLDMsInRydWUhIl0sWzAsMiwidHJ1ZSEiXSxbMiwzLCJpZCIsMV0sWzQsMiwidHJ1ZSIsMl1d
+  \[\begin{tikzcd}
+      1 && Nat && Nat \\
+      \\
+      && Bool && Bool
+      \arrow["zero", from=1-1, to=1-3]
+      \arrow["succ", from=1-3, to=1-5]
+      \arrow["{true!}", from=1-5, to=3-5]
+      \arrow["{true!}", from=1-3, to=3-3]
+      \arrow["id"{description}, from=3-3, to=3-5]
+      \arrow["true"', from=1-1, to=3-3]
+    \end{tikzcd}\]
+  trivially commutes.
+\end{example}
+\subsubsection{Lists}

tex/sections/02_datatypes.tex

@@ -1 +0,0 @@
-\section{Functions and Data}