Lemma st_produit_r_correct_wrt_sign_invar_with_offset : forall (s0 : state) (a : ad) (p : prec_list) (sigma : signature) (pa : pre_ad), state_correct_wrt_sign_with_offset s0 sigma pa -> state_correct_wrt_sign_with_offset (M1 prec_list a p) sigma pa -> state_correct_wrt_sign_with_offset (s_produit_r a p s0) sigma pa.