Local Open Scope logic_base.
Local Open Scope syntax.
Section ContextProperties.
Context {L: Language}
{minL: MinimunLanguage L}
{pL: PropositionalLanguage L}
{Gamma: ProofTheory L}
{bSC: BasicSequentCalculus L Gamma}
{minSC: MinimunSequentCalculus L Gamma}
{ipSC: IntuitionisticPropositionalSequentCalculus L Gamma}
{cpSC: ClassicalPropositionalSequentCalculus L Gamma}.

(* Specifications: *)
Lemma classical_derivable_spec: forall (Phi: context) (x: expr),
Phi |-- x <-> ~ consistent (Union _ Phi (Singleton _ (~~ x))).
Lemma MCS_nonelement_inconsistent: forall (Phi: context),
maximal consistent Phi ->
(forall x: expr, ~ Phi x <-> Phi |-- x --> FF).
Lemma maximal_consistent_orp_witnessed: forall (Phi: context),
maximal consistent Phi -> orp_witnessed Phi.
Lemma MCS_andp_iff: forall (Phi: context),
maximal consistent Phi ->
(forall x y: expr, Phi (x && y) <-> (Phi x /\ Phi y)).
Lemma MCS_orp_iff: forall (Phi: context),
maximal consistent Phi ->
(forall x y: expr, Phi (x || y) <-> (Phi x \/ Phi y)).
Lemma MCS_impp_iff: forall (Phi: context),
maximal consistent Phi ->
(forall x y: expr, Phi (x --> y) <-> (Phi x -> Phi y)).
Lemma MCS_negp_iff: forall (Phi: context),
maximal consistent Phi ->
(forall x: expr, Phi (~~ x) <-> ~ Phi x).
Lemma DDCS_MCS: forall (Phi: context),
derivable_closed Phi ->
orp_witnessed Phi ->
consistent Phi ->
maximal consistent Phi.