End AddAxiomatization.
Hint Rewrite <- @MakeAxiomatization_ClassicalPropositionalSequentCalculus using first [typeclasses eauto | instantiate (1 := _); instantiate (1 := _); instantiate (1 := _); instantiate_must_succeed; apply Typeclass_Rewrite_I]: AddAX.
Section DerivableRulesFromAxiomatization1.
Context {L: Language}
{minL: MinimunLanguage L}
{pL: PropositionalLanguage L}
{Gamma: ProofTheory L}
{minAX: MinimunAxiomatization L Gamma}
{ipGamma: IntuitionisticPropositionalLogic L Gamma}
{cpGamma: ClassicalPropositionalLogic L Gamma}.

(* Specifications: *)
Lemma double_negp_elim: forall (x: expr),
|-- ~~ (~~ x) --> x.
Lemma double_negp: forall (x: expr),
|-- ~~ (~~ x) <--> x.