The rule $\beta$

The basic rule for computing with the lambda calculus is called $\beta$ and is given by:

$$(\lambda x.M) M' \rightarrow _\beta M\{x \leftarrow M'\}$$

where $M\{x \leftarrow M'\}$ represents the effect of substituting all free occurrences of $x$ in $M$ uniformly by $M'$. (We shall formally define what we mean by a free occurrence of $x$ shortly.) The $\beta$ rule is one that we use unconciously all the time: If $f(x) = 2x^2 + 3x + 4$, then $f(7) = 2 \cdot 7^2 + 3 \cdot 7 + 4$ which is just $2x^2 + 3x + 4\{x \leftarrow 7\}$.

From the $\beta$ rule it is clear that an application $M M'$ is meaningful only if $M$ is of the form $\lambda x.M''$. Thus, though we can write terms like $x x$, we cannot do much with them.

Effectively, the $\beta$ rule is the only rule we need to set up the lambda calculus. For the moment, think of it as a rewriting rule for expressions that can be applied to any subterm in a lambda expression. We will formalize this later.