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).