Section DerivableRules_multi_impp.
Context {L: Language}
{minL: MinimunLanguage L}
{Gamma: ProofTheory L}
{minAX: MinimunAxiomatization L Gamma}.

(* TODO: maybe this one is also not useful now. *)
Lemma provable_multi_imp_split:
forall Phi1 Phi2 (xs: list expr) (y: expr),
Forall (Union _ Phi1 Phi2) xs ->
|-- multi_imp xs y ->
exists xs1 xs2,
Forall Phi1 xs1 /\
Forall Phi2 xs2 /\
|-- multi_imp xs1 (multi_imp xs2 y).