Local Open Scope logic_base.
Local Open Scope syntax.
Section Lindenbaum_Trivial.
Context {L: Language}
{minL: MinimunLanguage L}
{pL: PropositionalLanguage L}
{Gamma: ProofTheory L}
{SC: NormalSequentCalculus L Gamma}
{bSC: BasicSequentCalculus L Gamma}
{fwSC: FiniteWitnessedSequentCalculus L Gamma}
{minSC: MinimunSequentCalculus L Gamma}
{ipSC: IntuitionisticPropositionalSequentCalculus L Gamma}
{cpSC: ClassicalPropositionalSequentCalculus L Gamma}
{minAX: MinimunAxiomatization L Gamma}
{ipGamma: IntuitionisticPropositionalLogic L Gamma}.

(* Specifications: *)
Lemma Lindenbaum_for_max_consistent: forall P,
Lindenbaum_ensures P derivable_closed ->
Lindenbaum_ensures P orp_witnessed ->
Lindenbaum_ensures P consistent ->
Lindenbaum_ensures P (maximal consistent).
Lemma Lindenbaum_cannot_derive_ensures_max_consistent {AX: NormalAxiomatization L Gamma}: forall x, Lindenbaum_ensures (cannot_derive x) (maximal consistent).
End Lindenbaum_Trivial.