What is beta equivalence?

In the script I am currently reading on the lambda calculus, beta equivalence is defined as this:


The $\beta$-equivalence $\equiv_\beta$ is the smallest equivalence that contains $\rightarrow_\beta$.

I have no idea what that means. Can someone explain it in simpler terms? Maybe with an example?

I need it for a lemma following from the Church-Russer theorem, saying


If M $\equiv_\beta$ N then there is a L with M $\twoheadrightarrow_\beta$ L and N $\twoheadrightarrow_\beta$ L.

$\to_\beta$ is the one-step relation between terms in the $\lambda$-calculus. This relation is neither reflexive, symmetric, or transitive. The equivalence relation $\equiv_\beta$ is the reflexive, symmetric, transitive closure of $\to_\beta$. This means

• If $M\to_\beta M'$ then $M\equiv_\beta M'$.

• For all terms $M$, $M\equiv_\beta M$ holds.

• If $M\equiv_\beta M'$, then $M'\equiv_\beta M$.

• If $M\equiv_\beta M'$ and $M'\equiv_\beta M''$, then $M\equiv_\beta M''$.

• $\equiv_\beta$ is the smallest relation satisfying conditions 1-4.

More constructively, first apply rules 1 and 2, then
repeat rules $3$ and $4$ over and over until they add no new elements to the relation.

StackExchange Computer Science Q#634, answer score: 21