Hint Rewrite <- @MakeSequentCalculus_IntuitionisticPropositionalLogic using (instantiate_must_succeed; apply Typeclass_Rewrite_I): AddSC.
Section DerivableRulesFromAxiomatization1.
Context {L: Language}
{minL: MinimunLanguage L}
{pL: PropositionalLanguage L}
{Gamma: ProofTheory L}
{minAX: MinimunAxiomatization L Gamma}
{ipGamma: IntuitionisticPropositionalLogic L Gamma}.

(* Specifications: *)
Lemma solve_andp_intros: forall x y: expr,
|-- x -> |-- y -> |-- x && y.
Lemma solve_andp_elim1: forall x y: expr,
|-- x && y -> |-- x.
Lemma solve_andp_elim2: forall x y: expr,
|-- x && y -> |-- y.
Lemma solve_impp_elim_left: forall x y: expr,
|-- y -> |-- x --> y.
Lemma solve_orp_impp: forall x y z: expr,
|-- x --> z -> |-- y --> z -> |-- x || y --> z.
Lemma solve_orp_intros1: forall x y: expr,
|-- x -> |-- x || y.
Lemma solve_orp_intros2: forall x y: expr,
|-- y -> |-- x || y.
Lemma solve_impp_andp: forall x y z: expr,
|-- x --> y -> |-- x --> z -> |-- x --> y && z.
Lemma double_negp_intros: forall (x: expr),
|-- x --> ~~ ~~ x.
Lemma provable_iffp_refl: forall (x: expr),
|-- x <--> x.
Lemma contrapositivePP: forall (x y: expr),
|-- (y --> x) --> ~~ x --> ~~ y.
Lemma contrapositivePN: forall (x y: expr),
|-- (y --> ~~ x) --> (x --> ~~ y).