Here is its syntax:
<blockquote>
-**Variables**: `x`, `y`, `z`, ...
+<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>
-**Abstract**: <code>(λa M)</code>
+<strong>Abstract</strong>: <code>(λa M)</code>
</blockquote>
-We'll tend to write <code>(λa M)</code> as just `( \a M )`.
+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>
-**Application**: `(M N)`
+<strong>Application</strong>: <code>(M N)</code>
</blockquote>
Some authors reserve the term "term" for just variables and abstracts. We won't participate in that convention; we'll probably just say "term" and "expression" indiscriminately for expressions of any of these three forms.
(x (\x x))
((\x (x x)) (\x (x x)))
-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:
+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)
The rule of beta-reduction permits a transition from that expression to the following:
- M {a:=N}
+ M [a:=N]
What this means is just `M`, with any *free occurrences* inside `M` of the variable `a` replaced with the term `N`.