crossslink chapter 10

003_lambda_calculus.md

@@ -37,6 +37,22 @@ names:
 - **Lam** - A lambda abstraction
 - **App** - An application
 
+A lambda term is said to bind its variable. For example the lambda here binds
+$x$. In mathematics we would typically write:
+
+$$
+f(x) = e
+$$
+
+Using the lambda calculus notation we write:
+
+$$
+f = \lambda x. e
+$$
+
+In other words, $\lambda x.e$ is a function that takes a variable $x$ and
+returns $e$.
+
 $$
 \begin{aligned}
 e :=\ & x            & \trule{Var} \\
@@ -52,13 +68,6 @@ Standard ML, etc. The variation we will discuss first is known as **untyped
 lambda calculus**, by contrast later we will discuss the **typed lambda
 calculus** which is an extension thereof.
 
-A lambda abstraction is said to bind its variable.
-For example the lambda here binds $x$.
-
-$$
-\lambda x. e
-$$
-
 There are several syntactical conventions that we will adopt when writing lambda
 expressions. Application of multiple expressions associates to the left.
 
@@ -149,8 +158,8 @@ This fact is a useful sanity check when testing an implementation of the lambda
 
 **Omega Combinator**
 
-An important degenerate case that we'll test is the omega combinator which applies a single argument to
-itself.
+An important degenerate case that we'll test is the omega combinator which
+applies a single argument to itself.
 
 $$
 \omega = \lambda x. x x \\
@@ -446,6 +455,13 @@ $$
 \end{aligned}
 $$
 
+$$
+n! = n (n-1)!
+$$
+
+$$
+\textbf{fac}\ n = \textbf{R}(\textbf{fac}) = \textbf{R}(\textbf{R}(\textbf{fac}))
+$$
 
 For fun one can prove that the Y-combinator can be expressed in terms of the S
 and K combinators.

009_datatypes.md

@@ -35,8 +35,8 @@ Likewise for the product there are two function ``fst`` and ``snd`` which are
 *projections* which de construct products.
 
 ```haskell
-fst :: (a,b) -> a
-snd :: (a,b) -> b
+fst :: a * b -> a
+snd :: a * b -> b
 ```
 
 Once a language is endowed with the capacity to write a single product or a

README.md

@@ -37,7 +37,7 @@ Releases
 * [Chapter 7: Hindley-Milner Inference](http://dev.stephendiehl.com/fun/006_hindley_milner.html)
 * [Chapter 8: Design of ProtoHaskell](http://dev.stephendiehl.com/fun/007_path.html)
 * [Chapter 9: Extended Parser](http://dev.stephendiehl.com/fun/008_extended_parser.html)
-* Chapter 10: Custom Datatypes
+* [Chapter 10: Custom Datatypes](http://dev.stephendiehl.com/fun/009_datatypes.html)
 * Chapter 11: Renamer
 * Chapter 12: Pattern Matching & Desugaring
 * Chapter 13: System-F