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.

Lemma not_derivable_superset_preserved_derivable_subset_preserved: forall P,
derivable_superset_preserved P ->
derivable_subset_preserved (fun X => ~ P X).