I want to prove that = is symmetric in lambda calculus. ie. If $E=E'$ then $E'=E$.
From text I came across that if for instance
$$ E_1 \to_\beta E_2 \to_\beta E_3 $$
and $E \equiv E_1$ and $E' \equiv E_3$ then $E=E'$.
From this to prove that $E'=E$ so to prove that they are symmetric, I think somehow I need to show that
$$ E_3 \to E_2 \to E_1 $$
via either $\alpha$ or $\beta$ or $\eta$ conversion.
How can I prove that $E'=E$ so to prove that = is symmetric?
Thanks a bunch!
Here's my first try at this.
If $E=E'$ then by Leibnitz's law
$$ \lambda V.E = \lambda V.E' $$
If we apply $E^*$ to both since they are the same function
$$ EE^* = E'E^* $$
and
$$ E'E^* = EE^* $$
Hence, if $E=E'$ then $E'=E$.
$\square$