Section ALPHA.
Let EMPTY_INTRO (ps : list (frm L)) : E.fromList ps \subseteq E.empty -> ps = [] :=
match ps with
| [] => fun _ => eq_refl
| p :: ps => fun SUBSET =>
match SUBSET p (or_introl eq_refl) with
end
end.

Lemma proves_alpha_comm_lemma (p : frm L) (q : frm L)
(ALPHA : p ≡ q)
: E.singleton p \proves q /\ E.singleton q \proves p.