Added Crypto & Lambda handouts

Advanced/Lambda Calculus/parts/05 challenges.tex

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+\documentclass[../main.tex]{subfiles}
+
+\begin{document}
+
+	\section{Challenges}
+
+	Do \ref{Yfac} first, then finish the rest in any order. \\
+	Have fun!
+
+	\problem{}<Yfac>
+	Design a recursive factorial function using $Y$. \\
+
+	\vfill
+
+	\problem{}
+	Design a non-recursive factorial function. \\
+	\note{This one is a lot easier than \ref{Yfac}, but I don't think it will help you solve it.}
+
+	\problem{}
+	Using pairs, make a \say{list} data structure. Define a GET function, so that $\text{GET}~L~n$ reduces to the nth item in the list. $\text{GET}~L~0$ should give the first item in the list, and $\text{GET}~L~1$, the \textit{second}. \\
+	Lists have a defined length, so you should be able to tell when you're on the last element.
+
+	\problem{}
+	Write POW $a$ $b$, which raises $b$ to the $a$th power.
+
+	\problem{}
+	Write a MOD $a$ $b$ function that reduces to the remainder of $a \div b$.
+
+	\begin{solution}
+		\textbf{Factorial with recursion:}
+		\vspace{3ex}
+
+		$\text{FAC} = \lm yn.[Z~n][1][\text{MULT}~n~(y~(\text{D}~n))]$
+
+		\linehack{}
+
+		\textbf{Factorial without recursion:}
+		\vspace{3ex}
+
+		$\text{FAC}_0 = \lm p .
+		\Bigl\langle~~
+		\Bigl[D~(p~t)\Bigr]
+		~,~
+		\Bigl[\text{MULT}~(p~T)~(p~F)\Bigr]
+		~~\Bigr\rangle
+		$
+
+		\vspace{2ex}
+
+		$
+		\text{FAC} = \lm n .
+		\bigl( n~\text{FAC}_0~\langle n, 1 \rangle \bigr)
+		$
+
+		\linehack{}
+
+		\textbf{Lists:}
+		\vspace{3ex}
+
+		One possible implementation is
+		$\Bigl\langle
+			\langle \text{is last} ~,~ \text{item} \rangle
+			~,~
+			\text{next}...
+		\Bigr\rangle$, where:
+
+		\vspace{1ex}
+
+		\say{is last} is a boolean, true iff this is the last item in the list. \\
+		\say{item} is the thing you're storing \\
+		\say{next...} is another one of these list fragments. \\
+
+		It doesn't matter what \say{next} is in the last list fragment. A dedicated \say{is last} slot allows us to store ANY function in this list.
+
+		\vspace{1ex}
+
+		Here, $\text{GET} = \lm nL.[(n~L~F)~T~F$] \\
+
+		This will break if $n$ is out of range.
+	\end{solution}
+
+	\vfill
+
+	\problem{Bonus}
+	Play with \textit{Lamb}, an automatic lambda expression evaluator. \\
+	\url{https://git.betalupi.com/Mark/lamb}
+\end{document}