One step reduction

Recall that we introduced the $\beta$ rule

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

and said, informally, that we would permit this rule to be used in all ``contexts''. The $\beta$ rule is not the only basic rule possible. For instance, observe that the two terms $\lambda x.(M x)$ and $M$ are equivalent with respect to $\beta$ reduction--for any term $M'$, $(\lambda x.(M x)) M' \rightarrow _\beta M M'$. This can be formalized by a rule (normally called $\eta$) which says

$$\lambda x.(M x) \rightarrow _\eta M$$

Given a set of basic rules such as $\beta$, $\eta$, ..., we can inductively define a one step reduction that permits any of these basic rules to be used in any context. Let us denote one step reduction by $\rightarrow$. We define $\rightarrow$ through inference rules such as the following:

if $M \rightarrow _x M'$ for some basic rule $x$ such as $\beta$, $\eta$, ..., then $M \rightarrow M'$

Following the conventional notation used in logic, we present such a rule in the following form

$$\frac{M \rightarrow _x M'}{M \rightarrow M'}$$

Here is the complete set of rules defining $\rightarrow$:

$$\begin{array}{cccc} \displaystyle\frac{M \rightarrow _x M'}... ...& \displaystyle\frac{N \to N'}{MN \rightarrow MN'} \end{array}$$

Notice that all that we have done is to formalize the fact that we permit the basic reductions $\rightarrow _x$ within any subterm. In the ``calculations'' we have seen so far, we have already informally used this form of applying the $\beta$ rule.

In the discussion that follows, we shall dispense with the $\eta$ rule and assume that the only basic rule is the rule $\beta$.