Section CanonicalProperties.
Context {L: Language}
{Gamma: Derivable L}.
Definition derivable_closed : context -> Prop :=
fun Phi => forall x, derivable Phi x -> Phi x.
Definition derivable_superset_preserved (P: context -> Prop): Prop :=
forall Phi Psi, Included _ (derivable Phi) (derivable Psi) -> P Phi -> P Psi.
Definition derivable_subset_preserved (P: context -> Prop): Prop :=
forall Phi Psi, Included _ (derivable Phi) (derivable Psi) -> P Psi -> P Phi.
Context {bSC: BasicSequentCalculus L Gamma}.

Lemma derivable_closed_element_derivable: forall (Phi: context),
derivable_closed Phi ->
(forall x: expr, Phi x <-> Phi |-- x).