Local Open Scope logic_base.
Local Open Scope syntax.
Section Lindenbaum_Flat.
Context {L: Language}
{minL: MinimunLanguage L}
{pL: PropositionalLanguage L}
{sL: SeparationLanguage L}
{Gamma: ProofTheory L}
{SC: NormalSequentCalculus L Gamma}
{bSC: BasicSequentCalculus L Gamma}
{fwSC: FiniteWitnessedSequentCalculus L Gamma}
{minSC: MinimunSequentCalculus L Gamma}
{ipSC: IntuitionisticPropositionalSequentCalculus L Gamma}
{AX: NormalAxiomatization L Gamma}
{minAX: MinimunAxiomatization L Gamma}
{ipGamma: IntuitionisticPropositionalLogic L Gamma}
{sGamma: SeparationLogic L Gamma}.

(* Specifications: *)
Lemma Lindenbaum_preserves_context_sepcon_included_l: forall Phi2 Psi,
Lindenbaum_preserves (context_sepcon_included_l Phi2 Psi).
Lemma Lindenbaum_context_sepcon_included_l_ensures_derivable_closed: forall Phi Psi,
Lindenbaum_ensures (context_sepcon_included_l Phi Psi) derivable_closed.
Lemma Lindenbaum_context_sepcon_included_l_ensures_orp_witnessed: forall Phi Psi
(DC: derivable_closed Psi)
(OW: orp_witnessed Psi),
Lindenbaum_ensures (context_sepcon_included_l Phi Psi) orp_witnessed.
Lemma Lindenbaum_context_sepcon_included_l_ensures_consistent: forall Phi Psi
(CONSI: consistent Psi),
Lindenbaum_ensures (context_sepcon_included_l Phi Psi) consistent.