Local Open Scope logic_base.
Local Open Scope syntax.
Section Lindenbaum_Kripke.
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}
{minAX: MinimunAxiomatization L Gamma}
{ipGamma: IntuitionisticPropositionalLogic L Gamma}.

(* Specifications: *)
Lemma Lindenbaum_for_orp_witnessed: forall P,
Lindenbaum_preserves P ->
subset_preserved P ->
context_orp_captured P ->
Lindenbaum_ensures P derivable_closed ->
Lindenbaum_ensures P orp_witnessed.
Lemma Lindenbaum_cannot_derive_ensures_orp_witnessed {AX: NormalAxiomatization L Gamma}: forall x, Lindenbaum_ensures (cannot_derive x) orp_witnessed.
Lemma Lindenbaum_for_consistent: forall P,
Lindenbaum_preserves P ->
at_least consistent P ->
Lindenbaum_ensures P consistent.
Lemma Lindenbaum_cannot_derive_ensures_consistent {AX: NormalAxiomatization L Gamma}: forall x, Lindenbaum_ensures (cannot_derive x) consistent.
End Lindenbaum_Kripke.