funcprog: Added missing week 2, assignment 4 to repository.

funcprog/week2/Makefile

+TEXFLAGS := -halt-on-error -interaction=nonstopmode -file-line-error
+
+all: ass4.pdf
+
+clean:
+	rm *.out *.toc *.aux *.log *.pdf
+
+%.pdf: %.tex
+	pdflatex $(TEXFLAGS) $< | grep -i ".*:[0-9]*:.*\|warning" || true

funcprog/week2/ass4.tex

+\documentclass[10pt,a4paper]{article}
+
+\usepackage[english]{babel}
+\usepackage[utf8]{inputenc}
+\usepackage{amsmath,hyperref,graphicx,booktabs,float}
+
+% Paragraph indentation
+\setlength{\parindent}{0pt}
+\setlength{\parskip}{1ex plus 0.5ex minus 0.2ex}
+
+\title{Functional programming: week 2, assignment 4}
+\author{Sander Mathijs van Veen (6167969; smvv@kompiler.org)}
+
+\begin{document}
+
+\maketitle
+\tableofcontents
+
+\newcommand{\s}{\hspace{.3em}}
+\newcommand{\la}{\lambda a}
+\newcommand{\lb}{\lambda b}
+\newcommand{\lp}{\lambda p}
+\newcommand{\laq}{\lambda q}
+\newcommand{\lu}{\lambda u}
+\newcommand{\lv}{\lambda v}
+\newcommand{\lx}{\lambda x}
+\newcommand{\ly}{\lambda y}
+\newcommand{\tb}[1]{\textbf{#1}}
+\newcommand{\ra}{\rightarrow_\alpha}
+\newcommand{\rb}{\rightarrow_\beta}
+\newcommand{\ea}{\equiv_\alpha}
+
+\newcommand{\true}{\la.\lb.a}
+\newcommand{\false}{\la.\lb.b}
+\newcommand{\truea}{\lu.\lv.u}
+\newcommand{\falsea}{\lu.\lv.v}
+
+\section{Negation}
+\label{sec:Negation}
+
+Negation inverts the value of a boolean: \texttt{true} becomes \texttt{false},
+and \texttt{false} becomes \texttt{true}. In lambda calculi, negation can be
+expressed as the $\lambda$-term $\neg x = \lb.\lx.\ly.((b \s y) \s x)$. By
+applying $\alpha$-conversions and $\beta$-reductions, we can prove this
+$\lambda$-term. First, $\neg \tb{true} \equiv \tb{false}$:
+\begin{align*}
+    \neg \tb{true} & = (\lb.\lx.\ly.((b \s y) \s x) \s \la.\lb.a) \\
+                   & \rb \lx.\ly.((\la.\lb.a \s y) \s x) \\
+                   & \rb \lx.\ly.(\lb.y \s x) \\
+                   & \rb \lx.\ly.y \\
+                   & \ea \tb{false}
+\end{align*}
+And now $\neg \tb{false} \equiv \tb{true}$:
+\begin{align*}
+    \neg \tb{true} & = (\lb.\lx.\ly.((b \s y) \s x) \s \la.\lb.b) \\
+                   & \rb \lx.\ly.((\la.\lb.b \s y) \s x) \\
+                   & \rb \lx.\ly.(\lb.b \s x) \\
+                   & \rb \lx.\ly.x \\
+                   & \ea \tb{false}
+\end{align*}
+
+\pagebreak
+
+\section{Disjunction}
+\label{sec:Disjunction}
+
+Disjunction results in \tb{true} whenever one or more of its operands are
+\tb{true}. Thus, only iff all operands are \tb{false}, the disjunction will
+return \tb{false}.
+
+\begin{align*}
+    \tb{true} \s\vee\s \tb{true} & = (\lp.(\laq.((p \s p) \s q) \s \true) \s \true) \\
+    & \rb (\laq.((\true \s \true) \s q) \s \true) \\
+    & \rb ((\true \s \true) \s \true) \\
+    & \ra ((\true \s \truea) \s \true) \\
+    & \rb (\lb.\truea \s \true) \\
+    & \rb \truea \\
+    & \ea \tb{true} \\
+\end{align*}
+\begin{align*}
+    \tb{true} \s\vee\s \tb{false} & = (\lp.(\laq.((p \s p) \s q) \s \false) \s \true) \\
+    & \rb (\laq.((\true \s \true) \s q) \s \false) \\
+    & \rb ((\true \s \true) \s \false) \\
+    & \ra ((\true \s \truea) \s \false) \\
+    & \rb (\lb.\truea \s \false) \\
+    & \rb \truea \\
+    & \ea \tb{true} \\
+\end{align*}
+\begin{align*}
+    \tb{false} \s\vee\s \tb{true} & = (\lp.(\laq.((p \s p) \s q) \s \true) \s \false) \\
+    & \rb (\laq.((\false \s \false) \s q) \s \true) \\
+    & \rb ((\false \s \false) \s \true) \\
+    & \rb (\lb.b \s \true) \\
+    & \rb \true \\
+    & \ea \tb{true} \\
+\end{align*}
+\begin{align*}
+    \tb{false} \s\vee\s \tb{false} & = (\lp.(\laq.((p \s p) \s q) \s \false) \s \false) \\
+    & \rb (\laq.((\false \s \false) \s q) \s \false) \\
+    & \rb ((\false \s \false) \s \false) \\
+    & \rb (\lb.b \s \false) \\
+    & \rb \false \\
+    & \ea \tb{false} \\
+\end{align*}
+
+\pagebreak
+
+\section{Conjunction}
+\label{sec:Conjunction}
+
+Conjunction results in \tb{true} whenever all of its operands are \tb{true}.
+Thus, only iff one or more of operands are \tb{false}, the disjunction will
+return \tb{false}.
+
+\begin{align*}
+    \tb{true} \s\wedge\s \tb{true} & = (\lp.(\laq.((p \s q) \s p) \s \true) \s \true) \\
+    & \rb (\laq.((\true \s q) \s \true) \s \true) \\
+    & \rb ((\true \s \true) \s \true) \\
+    & \ra ((\true \s \truea) \s \true) \\
+    & \rb (\lb.\truea \s \true) \\
+    & \rb \truea \\
+    & \ea \tb{true} \\
+\end{align*}
+\begin{align*}
+    \tb{true} \s\wedge\s \tb{false} & = (\lp.(\laq.((p \s q) \s p) \s \false) \s \true) \\
+    & \rb (\laq.((\true \s q) \s \true) \s \false) \\
+    & \rb ((\true \s \false) \s \true) \\
+    & \ra ((\true \s \falsea) \s \true) \\
+    & \rb (\lb.\falsea \s \true) \\
+    & \rb \falsea \\
+    & \ea \tb{false} \\
+\end{align*}
+\begin{align*}
+    \tb{false} \s\wedge\s \tb{true} & = (\lp.(\laq.((p \s q) \s p) \s \true) \s \false) \\
+    & \rb (\laq.((\false \s q) \s \false) \s \true) \\
+    & \rb ((\false \s \true) \s \false) \\
+    & \rb (\lb.b \s \false) \\
+    & \rb \false \\
+    & \ea \tb{false} \\
+\end{align*}
+\begin{align*}
+    \tb{false} \s\wedge\s \tb{false} & = (\lp.(\laq.((p \s q) \s p) \s \false) \s \false) \\
+    & \rb (\laq.((\false \s q) \s \false) \s \false) \\
+    & \rb ((\false \s \false) \s \false) \\
+    & \rb (\lb.b \s \false) \\
+    & \rb \false \\
+    & \ea \tb{false} \\
+\end{align*}
+
+\end{document}