Lemma union_mpl_correct_wrt_sign_invar_0 : forall (s : state) (a : ad) (p : prec_list) (pa : pre_ad) (sigma : signature), state_correct_wrt_sign_with_offset s sigma pa -> state_correct_wrt_sign_with_offset (M1 prec_list a p) sigma pa -> state_correct_wrt_sign_with_offset (union_mpl_0 a p s) sigma pa.