End CanonicalProperties.
Section ContextProperties.
Context {L: Language}
{Gamma: ProofTheory L}.
Definition can_derive (x: expr): context -> Prop :=
fun Phi => Phi |-- x.
Definition cannot_derive (x: expr): context -> Prop :=
fun Phi => ~ Phi |-- x.

(* Specifications: *)
Lemma can_derive_derivable_superset_preserved: forall x,
derivable_superset_preserved (can_derive x).
Lemma cannot_derive_derivable_subset_preserved: forall x,
derivable_subset_preserved (cannot_derive x).