Lemma invariant1_inductive : forall (c : Config) (t : class_trans c), legal c -> sigma_send_table (st c) = (sigma_receive_table (rt c) + sigma_weight (bm c))%Z -> sigma_send_table (st (transition c t)) = (sigma_receive_table (rt (transition c t)) + sigma_weight (bm (transition c t)))%Z.