Section IntuitionisticDerivedRules.
Context {L: Language}
{minL: MinimumLanguage L}
{pL: PropositionalLanguage L}
{mL: ModalLanguage L}
{Gamma: Provable L}
{minAX: MinimumAxiomatization L Gamma}
{ipAX: IntuitionisticPropositionalLogic L Gamma}
{KmAX: SystemK L Gamma}.

Lemma impp_diamondp: forall x y, |-- boxp (x --> y) --> (diamondp x --> diamondp y).
(* Lemma derivable_impp_diamondp: forall Phi x y, Phi |-- boxp (x --> y) --> (diamondp x --> diamondp y).  Lemma deduction_impp_diamondp: forall Phi x y, Phi |-- boxp (x --> y) -> Phi |-- (diamondp x --> diamondp y). *)
Lemma diamondp_andp: forall x y, |-- diamondp (x && y) --> (diamondp x && diamondp y).