Lambda edits

Advanced/Lambda Calculus/parts/05 challenges.tex

@@ -5,34 +5,19 @@ Do \ref{Yfac} first, then finish the rest in any order.
 \problem{}<Yfac>
 Design a recursive factorial function using $Y$.
 
+\begin{solution}
+	$\text{FAC} = \lm yn.[Z~n][1][\text{MULT}~n~(y~(\text{D}~n))]$
+
+	To compute the factorial of 5, evaluate $(\text{Y}~\text{FAC}~5)$.
+\end{solution}
+
 \vfill
 
 \problem{}
 Design a non-recursive factorial function. \par
 \note{This one is 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}. \par
-Lists have a defined length, so you should be able to tell when you're on the last element.
-
-\problem{}
-Solve \ref{decrement} without using $H$.
-
-\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]
@@ -47,12 +32,55 @@ Write a MOD $a$ $b$ function that reduces to the remainder of $a \div b$.
 	\text{FAC} = \lm n .
 	\bigl( n~\text{FAC}_0~\langle n, 1 \rangle \bigr)
 	$
+\end{solution}
 
-	\linehack{}
+\vfill
 
-	\textbf{Lists:}
-	\vspace{3ex}
 
+
+\problem{}
+Solve \ref{decrement} without using $H$.
+
+\begin{solution}
+	One solution is below. Can you figure out how it works? \par
+
+	\vspace{1ex}
+
+	$
+	D_0 =
+	\lm p . \Bigl[p~T\Bigr]
+	\Bigl\langle
+		F ~,~ p~F
+	\Bigr\rangle
+	\Bigl\langle
+		F
+		~,~
+		\bigl\langle
+			p~F~T ~,~ ( (p~F~T)~(P~F~F) )
+		\bigr\rangle
+	\Bigr\rangle
+	$
+
+	\vspace{1ex}
+
+	$
+	D = \lm nfa .
+	\Bigl(
+		n D_0 \Bigl\langle T, \langle f, a \rangle \Bigr\rangle
+	\Bigr)~F~F
+	$
+\end{solution}
+
+\vfill
+
+
+
+\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}. \par
+Lists have a defined length, so you should be able to tell when you're on the last element.
+
+\begin{solution}
 	One possible implementation is
 	$\Bigl\langle
 		\langle \text{is last} ~,~ \text{item} \rangle
@@ -75,8 +103,27 @@ Write a MOD $a$ $b$ function that reduces to the remainder of $a \div b$.
 	This will break if $n$ is out of range.
 \end{solution}
 
+\vfill
+\pagebreak
+
+\problem{}
+Write a lambda expression that represents the Fibonacci function: \par
+$f(0) = 1$, $f(1) = 1$, $f(n + 2) = f(n + 1) + f(n)$.
+
+\vfill
+
+\problem{}
+Write a lambda expression that evaluates to $T$ if a number $n$ is prime, and to $F$ otherwise.
+
+\vfill
+
+\problem{}
+Write a MOD $a$ $b$ function that reduces to the remainder of $a \div b$.
+
 \vfill
 
 \problem{Bonus}
 Play with \textit{Lamb}, an automatic lambda expression evaluator. \par
-\url{https://git.betalupi.com/Mark/lamb}
+\url{https://git.betalupi.com/Mark/lamb}
+
+\pagebreak