Context {bSC: BasicSequentCalculus L Gamma}.

(* Specifications: *)
Lemma derivable_subset_preserved_subset_preserved: forall P,
derivable_subset_preserved P ->
subset_preserved P.
Lemma derivable_superset_preserved_superset_preserved: forall P,
derivable_superset_preserved P ->
superset_preserved P.
Lemma derivable_closed_element_derivable: forall (Phi: context),
derivable_closed Phi ->
(forall x: expr, Phi x <-> Phi |-- x).
Lemma can_derive_superset_preserved: forall x,
superset_preserved (can_derive x).
Lemma cannot_derive_subset_preserved: forall x,
subset_preserved (cannot_derive x).