Definition orp_witnessed: context -> Prop :=
  fun Phi => forall x y, Phi (orp x y) -> Phi x \/ Phi y.Definition context_orp_captured (P: context -> Prop): Prop :=
  forall Phi Psi, P (context_orp Phi Psi) -> P Phi \/ P Psi.

Lemma DCS_orp_iff: forall (Phi: context), derivable_closed Phi -> orp_witnessed Phi -> (forall x y: expr, Phi (x || y) <-> (Phi x \/ Phi y)).