Cleanup

Advanced/Lambda Calculus/parts/03 numbers.tex

@@ -1,6 +1,6 @@
 \section{Numbers}
 
-Since the only objects we have in $\lm$-calculus are functions, it's natural to think of quantities as \textit{adverbs} (once, twice, thrice,...) rather than \textit{nouns} (one, two, three ...) 
+Since the only objects we have in $\lm$ calculus are functions, it's natural to think of quantities as \textit{adverbs} (once, twice, thrice,...) rather than \textit{nouns} (one, two, three ...) 
 
 \vspace{1ex}
 
@@ -9,7 +9,7 @@ Here's our \textit{don't} function: given a function and an input, don't apply t
 $$
 	0 = \lm fa.a
 $$
-If you look closely, you'll find that $0$ is $\alpha$-equivalent to the false function $F$.
+If you look closely, you'll find that $0$ is equivalent to the false function $F$.
 
 \problem{}
 Write $1$, $2$, and $3$. We will call these \textit{Church numerals}.\hspace{-0.5ex}\footnotemark{} \\
@@ -29,16 +29,8 @@ Write $1$, $2$, and $3$. We will call these \textit{Church numerals}.\hspace{-0.
 
 	\vspace{1ex}
 
-	The round parentheses are \textit{essential}. Our lambda calculus is left-associative!
-
-	\vspace{2ex}
-
-	Also, notice that $1$ is very similar to $I$:
-	$$
-		I~\heartsuit~\star = 1~\heartsuit~\star
-	$$
-
-	Zero is false, and one is the identity. Isn't that wonderful?
+	The round parentheses are \textit{essential}. Our lambda calculus is left-associative! \par
+	Also, note that zero is false and one is the (two-variable) identity.
 \end{solution}
 
 \vfill
@@ -111,7 +103,6 @@ Define a function ADD that adds two Church numerals.
 \vfill
 
 \problem{}
-Adding is nice, but we can do better. \par
 Design a function MULT that multiplies two numbers. \par
 \hint{The easy solution uses ADD, the elegant one doesn't. Find both!}
 
@@ -139,10 +130,9 @@ $NZ$ does the opposite. $Z$ and $NZ$ should look fairly similar.
 \vfill
 
 \problem{}
-Data structures will be useful. Design an expression PAIR that constructs two-value tuples. \par
+Design an expression PAIR that constructs two-value tuples. \par
 For example, say $A = \text{PAIR}~1~2$. Then, \par
-$(A~T)$ should reduce to $1$ \par
-$(A~F)$ should reduce to $2$
+$(A~T)$ should reduce to $1$ and $(A~F)$ should reduce to $2$.
 
 \begin{solution}
 	$\text{PAIR} = \lm ab . \lm i . (i~a~b) = \lm abi.iab$
@@ -150,7 +140,7 @@ $(A~F)$ should reduce to $2$
 
 \vfill
 From now on, I'll write (PAIR $A$ $B$) as $\langle A,B \rangle$. \par
-Like currying, this is only notation. The underlying $\lm$ logic remains the same.
+Like currying, this is only notation. The underlying logic remains the same.
 
 \pagebreak
 
@@ -160,7 +150,7 @@ It does exactly what it says on the tin:
 
 \vspace{1ex}
 
-Given an input pair, it should shift its right argument left, then add one. \par
+Given an input pair, it should shift its second argument left, then add one. \par
 $H~\langle 0, 1 \rangle$ should reduce to $\langle 1, 2\rangle$ \par
 $H~\langle 1, 2 \rangle$ should reduce to $\langle 2, 3\rangle$ \par
 $H~\langle 10, 4 \rangle$ should reduce to $\langle 4, 5\rangle$
@@ -176,7 +166,7 @@ $H~\langle 10, 4 \rangle$ should reduce to $\langle 4, 5\rangle$
 
 \vfill
 
-\problem{}
+\problem{}<decrement>
 Design a function $D$ that un-does $S$. That means \par
 $D(1) = 0$, $D(2) = 1$, etc. $D(0)$ should be zero. \par
 \hint{$H$ will help you make an elegant solution.}