Final lambda edits

Advanced/Lambda Calculus/parts/04 recursion.tex

@@ -1,4 +1,77 @@
 \section{Recursion}
+
+
+%\iftrue
+\iffalse
+
+	You now have a choice. Choose wisely --- there's no going back.
+
+	\begin{tcolorbox}[
+		breakable,
+		colback=white,
+		colframe=gray,
+		arc=0pt, outer arc=0pt
+	]
+		\raggedright
+		\textbf{Take the red pill:} You stay on this page and try to solve \ref{thechallenge}. \par
+		This will take a while, and it's very unlikely you'll finish before class ends.
+
+
+		\vspace{4mm}
+
+		I strongly prefer this option. It's not easy, but you'll be very happy if you solve it yourself.
+		This is a chance to build your own solution to a fundamental problem in this field, just as
+		Curry, Church, and Turing did when first developing the theory of lambda calculus.
+
+		- Mark
+	\end{tcolorbox}
+
+	\begin{tcolorbox}[
+		breakable,
+		colback=white,
+		colframe=gray,
+		arc=0pt, outer arc=0pt
+	]
+		\textbf{Take the blue pill:} You skip this problem and turn the page.
+		Half of the answer to \ref{thechallenge} will be free, and the rest will be
+		broken into smaller steps. This is how we usually learn out about interesting
+		mathematics, both in high school and in university.
+
+		\vspace{2mm}
+
+		This path isn't as rewarding as the one above, but it is well-paved
+		and easier to traverse.
+	\end{tcolorbox}
+
+	\problem{}<thechallenge>
+	Can you find a way to recursively call functions in lambda calculus? \par
+	Find a way to define a recursive factorial function. \par
+
+	\note{$A = (\lm a. A~a)$ doesn't count. You can't use a macro inside itself.}
+
+	\vfill
+	\pagebreak
+\fi
+
+
+
+
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+
 
 Say we want a function that computes the factorial of a positive integer. Here's one way we could define it:
 $$
@@ -30,7 +103,7 @@ They're just shorthand that simplifies notation.
 
 \problem{}
 Write an expression that resolves to itself. \par
-\note{Your answer should be quite short.}
+\hint{Your answer should be quite short.}
 
 \vspace{1ex}
 
@@ -45,12 +118,13 @@ $\Omega$ useless on its own, but it gives us a starting point for recursion. \pa
 	An uninspired mathematician might call the Mockingbird $\omega$, \say{little omega}.
 \end{solution}
 
-
 \vfill
 \pagebreak
 
 
 
+
+
 \definition{}
 This is the \textit{Y-combinator}. You may notice that it's just $\Omega$ put to work.
 $$
@@ -62,26 +136,27 @@ What does this thing do? \par
 Evaluate $Y f$.
 
 
-\vfill
 
-\definition{}
-We say $x$ is a \textit{fixed point} of a function $f$ if $f(x) = x$.
+%\vfill
 
-\problem{}
-Show that $Y F$ is a fixed point of $F$.
+%\definition{}
+%We say $x$ is a \textit{fixed point} of a function $f$ if $f(x) = x$.
 
-\vfill
+%\problem{}
+%Show that $Y F$ is a fixed point of $F$.
 
-\problem{}
-Let $\theta = (\lm xy . y(xxy))$ and $\Theta = \theta \theta$. \par
-Let $N = \Theta F$ for an arbitrary lambda expression $F$. \par
+%\vfill
 
-Show that $F N = N$.
+%\problem{}
+%Let $b = (\lm xy . y(xxy))$ and $B = b ~ b$. \par
+%Let $N = B F$ for an arbitrary lambda expression $F$. \par
 
-\vfill
+%Show that $F N = N$.
 
-\problem{Bonus}
-Find a fixed-point combinator that isn't $Y$ or $\Theta$.
+%\vfill
+
+%\problem{Bonus}
+%Find a fixed-point combinator that isn't $Y$ or $\Theta$.
 
 \vfill
 \pagebreak