$\rightarrow ^*$-equivalence

We define an equivalence on lambda terms, which we shall write $\leftrightarrow$, based on the many step reduction relation $\rightarrow ^*$. Basically, $\leftrightarrow$ is the symmetric transitive closure of $\rightarrow ^*$. It can be defined inductively using inference rules in the same way that we defined $\rightarrow$ based on $\rightarrow _x$:

$$\begin{array}{ccc} \displaystyle\frac{M \rightarrow ^* N}{M... ...tarrow N, N \leftrightarrow P}{M \leftrightarrow P} \end{array}$$

We can do the same with any reflexive, transitive relation $R$-- the symmetric, transitive closure of $R$ defines an equivalence relation $\stackrel{{R}}{\leftrightarrow}$.