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.
End ALPHA.
#[global]
Add Parametric Morphism
: proves with signature (eqProp ==> alpha_equiv ==> iff) as proves_alpha_comm.

Lemma cut A B Gamma
(PROVE1 : E.insert A Gamma \proves B)
(PROVE2 : Gamma \proves A)
: Gamma \proves B.