Local Open Scope logic_base.
Section ContextProperty.
Context {L: Language}
{Gamma: ProofTheory L}
{bSC: BasicSequentCalculus L Gamma}.

(* Specifications: *)
Lemma Included_derivable: forall (Phi: context),
Included _ Phi (derivable Phi).
Lemma derivable_mono: forall (Phi Psi: context),
Included _ Phi Psi ->
Included _ (derivable Phi) (derivable Psi).
Lemma derivable_derivable: forall (Phi: context),
Included _ (derivable (derivable Phi)) (derivable Phi).
Lemma derivable_derivable_Same_set: forall (Phi: context),
Same_set _ (derivable (derivable Phi)) (derivable Phi).
Lemma maximal_consistent_spec :
forall Phi, maximal consistent Phi <-> consistent Phi /\ forall x, consistent (Union _ Phi (Singleton _ x)) -> Phi x.
Lemma maximal_consistent_derivable_closed:
forall (Phi: context),
maximal consistent Phi ->
derivable_closed Phi.
Lemma MCS_element_derivable:
forall(Phi: context),
maximal consistent Phi ->
(forall x: expr, Phi x <-> Phi |-- x).