Definition at_least (P cP: context -> Prop): Prop :=
  forall (Phi: context), cP Phi -> P Phi.

(* Specifications: *)
Lemma at_least_self: forall P, at_least P P.
Lemma at_least_left: forall P cP1 cP2, at_least P cP1 -> at_least P (Intersection _ cP1 cP2).
Lemma at_least_right: forall P cP1 cP2, at_least P cP2 -> at_least P (Intersection _ cP1 cP2).