Examples of expressions:
<blockquote><code>
-x<p>
-(y x)<p>
-(x x)<p>
-(\x y)<p>
-(\x x)<p>
-(\x (\y x))<p>
-(x (\x x))<p>
-((\x (x x)) (\x (x x)))<p>
+x
+(y x)
+(x x)
+(\x y)
+(\x x)
+(\x (\y x))
+(x (\x x))
+((\x (x x)) (\x (x x)))
</code></blockquote>
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
"beta-contraction". Suppose you have some expression of the form:
- ((\a M) N)
+ ((lambda a M) N)
that is, an application of an abstract to some other expression. This compound form is called a **redex**, meaning it's a "beta-reducible expression." `(\a M)` is called the **head** of the redex; `N` is called the **argument**, and `M` is called the **body**.