Definition context_sepcon_included_l (Phi2 Psi: context): context -> Prop :=
  fun Phi1 => Included _ (context_sepcon Phi1 Phi2) Psi.Definition context_sepcon_included_r (Phi1 Psi: context): context -> Prop :=
  fun Phi2 => Included _ (context_sepcon Phi1 Phi2) Psi.

(* Specifications: *)
Lemma context_sepcon_consistent_rev_left: forall (Phi1 Phi2 Psi: context), Included _ (context_sepcon Phi1 Phi2) Psi -> consistent Psi -> consistent Phi1.
Lemma context_sepcon_included_l_derivable_subset_preserved: forall Phi2 Psi, derivable_subset_preserved (context_sepcon_included_l Phi2 Psi).
Lemma context_sepcon_included_l_subset_preserved: forall Phi2 Psi, subset_preserved (context_sepcon_included_l Phi2 Psi).
Lemma context_sepcon_included_l_finite_captured: forall Phi2 Psi, finite_captured (context_sepcon_included_l Phi2 Psi).
Lemma context_sepcon_included_l_context_orp_captured: forall Phi2 Psi (DC: derivable_closed Psi) (OW: orp_witnessed Psi), context_orp_captured (context_sepcon_included_l Phi2 Psi).
Lemma context_sepcon_included_equiv: forall Phi Psi, derivable_closed Psi -> Same_set _ (context_sepcon_included_l Phi Psi) (context_sepcon_included_r Phi Psi).