Local Open Scope logic_base.
Local Open Scope syntax.
Class DeMorganPropositionalLogic (L: Language) {minL: MinimunLanguage L} {pL: PropositionalLanguage L} (Gamma: ProofTheory L) {minAX: MinimunAxiomatization L Gamma} {ipGamma: IntuitionisticPropositionalLogic L Gamma} := {
weak_excluded_middle: forall x, |-- ~~ x || ~~ ~~ x
}.

Lemma MakeSequentCalculus_DeMorganPropositionalLogic {L: Language} {minL: MinimunLanguage L} {pL: PropositionalLanguage L} {Gamma: ProofTheory L} {minAX: MinimunAxiomatization L Gamma} (minAX': MinimunAxiomatization L (Build_AxiomaticProofTheory (@provable L Gamma))) (ipGamma: IntuitionisticPropositionalLogic L Gamma) {ipGamma': IntuitionisticPropositionalLogic L (Build_AxiomaticProofTheory (@provable L Gamma))} {dmpGamma: DeMorganPropositionalLogic L Gamma}:
Typeclass_Rewrite (exist (fun X: Prop => X) (IntuitionisticPropositionalLogic L (Build_AxiomaticProofTheory (@provable L Gamma))) ipGamma' :: nil) ->
forall (G: Prop) (l: list (sig (fun X: Prop => X))),
(forall
(dmpGamma: DeMorganPropositionalLogic L (Build_AxiomaticProofTheory (@provable L Gamma))),
OpaqueProp (Typeclass_Rewrite l -> G)) <->
OpaqueProp (Typeclass_Rewrite ((exist (fun X: Prop => X) (DeMorganPropositionalLogic L Gamma) dmpGamma) :: l) -> G).