The lambda calculus we'll be focusing on for the first part of the course has no types. (Some prefer to say it instead has a single type---but if you say that, you have to say that functions from this type to this type also belong to this type. Which is weird.)
+Here is its syntax:
+
+<blockquote>
+<strong>Variables</strong>: <code>x</code>, <code>y</code>, <code>z</code>...
+</blockquote>
+
+Each variable is an expression. For any expressions M and N and variable a, the following are also expressions:
+
+<blockquote>
+<strong>Abstract</strong>: <code>(λa M)</code>
+</blockquote>
+
+We'll tend to write <code>(λa M)</code> as just `(\a M)`, so we don't have to write out the markup code for the <code>λ</code>. You can yourself write <code>(λa M)</code> or `(\a M)` or `(lambda a M)`.
+
+<blockquote>
+<strong>Application</strong>: <code>(M N)</code>
+</blockquote>
+
Some authors reserve the term "term" for just variables and abstracts. We'll probably just say "term" and "expression" indiscriminately for expressions of any of these three forms.
+<div>
Examples of expressions:
x
(\x (\y x))
(x (\x x))
((\x (x x)) (\x (x x)))
+</div>
The lambda calculus has an associated proof theory. For now, we can regard the
proof theory as having just one rule, called the rule of **beta-reduction** or