signed_recursive_determinant_empty

Exact expanded first-order arithmetic statement

forall pb pc nb nc. (exists mdr_b_exact_empty mdr_c_exact_empty mdr_l_exact_empty mdr_i_exact_empty. ((forall mdr_i_exact_emptyh. (exists mdr_gap_exact_emptyhi. mdr_gap_exact_emptyhi + S (mdr_i_exact_emptyh) = (mdr_l_exact_empty)) -> exists mdr_d_exact_emptyh mdr_pb_exact_emptyh mdr_pc_exact_emptyh mdr_nb_exact_emptyh mdr_nc_exact_emptyh mdr_p_exact_emptyh mdr_n_exact_emptyh. ((exists mdr_z_exact_emptyhr. ((exists mdr_a_exact_emptyhrc mdr_b_exact_emptyhrc mdr_c_exact_emptyhrc mdr_e_exact_emptyhrc mdr_f_exact_emptyhrc. ((mdr_a_exact_emptyhrc = ((mdr_d_exact_emptyh) + (mdr_pb_exact_emptyh)) * S ((mdr_d_exact_emptyh) + (mdr_pb_exact_emptyh)) + ((mdr_pb_exact_emptyh) + (mdr_pb_exact_emptyh))) /\ ((mdr_b_exact_emptyhrc = ((mdr_pc_exact_emptyh) + (mdr_nb_exact_emptyh)) * S ((mdr_pc_exact_emptyh) + (mdr_nb_exact_emptyh)) + ((mdr_nb_exact_emptyh) + (mdr_nb_exact_emptyh))) /\ ((mdr_c_exact_emptyhrc = ((mdr_a_exact_emptyhrc) + (mdr_b_exact_emptyhrc)) * S ((mdr_a_exact_emptyhrc) + (mdr_b_exact_emptyhrc)) + ((mdr_b_exact_emptyhrc) + (mdr_b_exact_emptyhrc))) /\ ((mdr_e_exact_emptyhrc = ((mdr_p_exact_emptyh) + (mdr_n_exact_emptyh)) * S ((mdr_p_exact_emptyh) + (mdr_n_exact_emptyh)) + ((mdr_n_exact_emptyh) + (mdr_n_exact_emptyh))) /\ ((mdr_f_exact_emptyhrc = ((mdr_nc_exact_emptyh) + (mdr_e_exact_emptyhrc)) * S ((mdr_nc_exact_emptyh) + (mdr_e_exact_emptyhrc)) + ((mdr_e_exact_emptyhrc) + (mdr_e_exact_emptyhrc))) /\ ((mdr_z_exact_emptyhr) = ((mdr_c_exact_emptyhrc) + (mdr_f_exact_emptyhrc)) * S ((mdr_c_exact_emptyhrc) + (mdr_f_exact_emptyhrc)) + ((mdr_f_exact_emptyhrc) + (mdr_f_exact_emptyhrc))))))))) /\ (((exists ff_h_mdr_exact_emptyhrb. ff_h_mdr_exact_emptyhrb + S (mdr_z_exact_emptyhr) = S ((S (mdr_i_exact_emptyh)) * mdr_c_exact_empty)) /\ exists ff_q_mdr_exact_emptyhrb. mdr_b_exact_empty = ff_q_mdr_exact_emptyhrb * S ((S (mdr_i_exact_emptyh)) * mdr_c_exact_empty) + (mdr_z_exact_emptyhr))))) /\ (((((mdr_d_exact_emptyh) = 0) /\ (((mdr_p_exact_emptyh) = 1) /\ ((mdr_n_exact_emptyh) = 0))) \/ exists mdr_q_exact_emptyhs mdr_eb_exact_emptyhs mdr_ec_exact_emptyhs mdr_fb_exact_emptyhs mdr_fc_exact_emptyhs. (((mdr_d_exact_emptyh) = S (mdr_q_exact_emptyhs)) /\ ((forall mdr_j_exact_emptyhsc. (exists mdr_gap_exact_emptyhscj. mdr_gap_exact_emptyhscj + S (mdr_j_exact_emptyhsc) = (S (mdr_q_exact_emptyhs))) -> exists mdr_i_exact_emptyhsc mdr_up_exact_emptyhsc mdr_us_exact_emptyhsc mdr_un_exact_emptyhsc mdr_ut_exact_emptyhsc mdr_p_exact_emptyhsc mdr_n_exact_emptyhsc. ((exists mdr_gap_exact_emptyhsci. mdr_gap_exact_emptyhsci + S (mdr_i_exact_emptyhsc) = (mdr_i_exact_emptyh)) /\ ((exists mdr_z_exact_emptyhscr. ((exists mdr_a_exact_emptyhscrc mdr_b_exact_emptyhscrc mdr_c_exact_emptyhscrc mdr_e_exact_emptyhscrc mdr_f_exact_emptyhscrc. ((mdr_a_exact_emptyhscrc = ((mdr_q_exact_emptyhs) + (mdr_up_exact_emptyhsc)) * S ((mdr_q_exact_emptyhs) + (mdr_up_exact_emptyhsc)) + ((mdr_up_exact_emptyhsc) + (mdr_up_exact_emptyhsc))) /\ ((mdr_b_exact_emptyhscrc = ((mdr_us_exact_emptyhsc) + (mdr_un_exact_emptyhsc)) * S ((mdr_us_exact_emptyhsc) + (mdr_un_exact_emptyhsc)) + ((mdr_un_exact_emptyhsc) + (mdr_un_exact_emptyhsc))) /\ ((mdr_c_exact_emptyhscrc = ((mdr_a_exact_emptyhscrc) + (mdr_b_exact_emptyhscrc)) * S ((mdr_a_exact_emptyhscrc) + (mdr_b_exact_emptyhscrc)) + ((mdr_b_exact_emptyhscrc) + (mdr_b_exact_emptyhscrc))) /\ ((mdr_e_exact_emptyhscrc = ((mdr_p_exact_emptyhsc) + (mdr_n_exact_emptyhsc)) * S ((mdr_p_exact_emptyhsc) + (mdr_n_exact_emptyhsc)) + ((mdr_n_exact_emptyhsc) + (mdr_n_exact_emptyhsc))) /\ ((mdr_f_exact_emptyhscrc = ((mdr_ut_exact_emptyhsc) + (mdr_e_exact_emptyhscrc)) * S ((mdr_ut_exact_emptyhsc) + (mdr_e_exact_emptyhscrc)) + ((mdr_e_exact_emptyhscrc) + (mdr_e_exact_emptyhscrc))) /\ ((mdr_z_exact_emptyhscr) = ((mdr_c_exact_emptyhscrc) + (mdr_f_exact_emptyhscrc)) * S ((mdr_c_exact_emptyhscrc) + (mdr_f_exact_emptyhscrc)) + ((mdr_f_exact_emptyhscrc) + (mdr_f_exact_emptyhscrc))))))))) /\ (((exists ff_h_mdr_exact_emptyhscrb. ff_h_mdr_exact_emptyhscrb + S (mdr_z_exact_emptyhscr) = S ((S (mdr_i_exact_emptyhsc)) * mdr_c_exact_empty)) /\ exists ff_q_mdr_exact_emptyhscrb. mdr_b_exact_empty = ff_q_mdr_exact_emptyhscrb * S ((S (mdr_i_exact_emptyhsc)) * mdr_c_exact_empty) + (mdr_z_exact_emptyhscr))))) /\ ((((forall ff_index_mdm_prefix_mdr_exact_emptyhscm_positive. (exists ff_gap_mdm_lt_mdr_exact_emptyhscm_positive_index_bound. ff_gap_mdm_lt_mdr_exact_emptyhscm_positive_index_bound + S (ff_index_mdm_prefix_mdr_exact_emptyhscm_positive) = ((mdr_q_exact_emptyhs) * (mdr_q_exact_emptyhs))) -> exists ff_row_mdm_prefix_mdr_exact_emptyhscm_positive ff_column_mdm_prefix_mdr_exact_emptyhscm_positive ff_value_mdm_prefix_mdr_exact_emptyhscm_positive. (ff_index_mdm_prefix_mdr_exact_emptyhscm_positive = (mdr_q_exact_emptyhs) * ff_row_mdm_prefix_mdr_exact_emptyhscm_positive + ff_column_mdm_prefix_mdr_exact_emptyhscm_positive /\ ((exists ff_gap_mdm_lt_mdr_exact_emptyhscm_positive_column_bound. ff_gap_mdm_lt_mdr_exact_emptyhscm_positive_column_bound + S (ff_column_mdm_prefix_mdr_exact_emptyhscm_positive) = (mdr_q_exact_emptyhs)) /\ ((exists ff_row_mdm_cell_mdr_exact_emptyhscm_positive_cell ff_column_mdm_cell_mdr_exact_emptyhscm_positive_cell. (((((exists ff_gap_mdm_lt_mdr_exact_emptyhscm_positive_cell_row_before. ff_gap_mdm_lt_mdr_exact_emptyhscm_positive_cell_row_before + S (ff_row_mdm_prefix_mdr_exact_emptyhscm_positive) = (0)) /\ ff_row_mdm_cell_mdr_exact_emptyhscm_positive_cell = ff_row_mdm_prefix_mdr_exact_emptyhscm_positive) \/ ((exists ff_gap_mdm_le_mdr_exact_emptyhscm_positive_cell_row_after. ff_gap_mdm_le_mdr_exact_emptyhscm_positive_cell_row_after + (0) = (ff_row_mdm_prefix_mdr_exact_emptyhscm_positive)) /\ ff_row_mdm_cell_mdr_exact_emptyhscm_positive_cell = S ff_row_mdm_prefix_mdr_exact_emptyhscm_positive))) /\ (((((exists ff_gap_mdm_lt_mdr_exact_emptyhscm_positive_cell_column_before. ff_gap_mdm_lt_mdr_exact_emptyhscm_positive_cell_column_before + S (ff_column_mdm_prefix_mdr_exact_emptyhscm_positive) = (mdr_j_exact_emptyhsc)) /\ ff_column_mdm_cell_mdr_exact_emptyhscm_positive_cell = ff_column_mdm_prefix_mdr_exact_emptyhscm_positive) \/ ((exists ff_gap_mdm_le_mdr_exact_emptyhscm_positive_cell_column_after. ff_gap_mdm_le_mdr_exact_emptyhscm_positive_cell_column_after + (mdr_j_exact_emptyhsc) = (ff_column_mdm_prefix_mdr_exact_emptyhscm_positive)) /\ ff_column_mdm_cell_mdr_exact_emptyhscm_positive_cell = S ff_column_mdm_prefix_mdr_exact_emptyhscm_positive))) /\ (((exists ff_h_mdm_mdr_exact_emptyhscm_positive_cell_source. ff_h_mdm_mdr_exact_emptyhscm_positive_cell_source + S (ff_value_mdm_prefix_mdr_exact_emptyhscm_positive) = S ((S ((ff_row_mdm_cell_mdr_exact_emptyhscm_positive_cell) * (S (mdr_q_exact_emptyhs)) + (ff_column_mdm_cell_mdr_exact_emptyhscm_positive_cell))) * mdr_pc_exact_emptyh)) /\ exists ff_q_mdm_mdr_exact_emptyhscm_positive_cell_source. mdr_pb_exact_emptyh = ff_q_mdm_mdr_exact_emptyhscm_positive_cell_source * S ((S ((ff_row_mdm_cell_mdr_exact_emptyhscm_positive_cell) * (S (mdr_q_exact_emptyhs)) + (ff_column_mdm_cell_mdr_exact_emptyhscm_positive_cell))) * mdr_pc_exact_emptyh) + (ff_value_mdm_prefix_mdr_exact_emptyhscm_positive)))))) /\ (((exists ff_h_mdm_mdr_exact_emptyhscm_positive_target. ff_h_mdm_mdr_exact_emptyhscm_positive_target + S (ff_value_mdm_prefix_mdr_exact_emptyhscm_positive) = S ((S (ff_index_mdm_prefix_mdr_exact_emptyhscm_positive)) * mdr_us_exact_emptyhsc)) /\ exists ff_q_mdm_mdr_exact_emptyhscm_positive_target. mdr_up_exact_emptyhsc = ff_q_mdm_mdr_exact_emptyhscm_positive_target * S ((S (ff_index_mdm_prefix_mdr_exact_emptyhscm_positive)) * mdr_us_exact_emptyhsc) + (ff_value_mdm_prefix_mdr_exact_emptyhscm_positive))))))) /\ (forall ff_index_mdm_prefix_mdr_exact_emptyhscm_negative. (exists ff_gap_mdm_lt_mdr_exact_emptyhscm_negative_index_bound. ff_gap_mdm_lt_mdr_exact_emptyhscm_negative_index_bound + S (ff_index_mdm_prefix_mdr_exact_emptyhscm_negative) = ((mdr_q_exact_emptyhs) * (mdr_q_exact_emptyhs))) -> exists ff_row_mdm_prefix_mdr_exact_emptyhscm_negative ff_column_mdm_prefix_mdr_exact_emptyhscm_negative ff_value_mdm_prefix_mdr_exact_emptyhscm_negative. (ff_index_mdm_prefix_mdr_exact_emptyhscm_negative = (mdr_q_exact_emptyhs) * ff_row_mdm_prefix_mdr_exact_emptyhscm_negative + ff_column_mdm_prefix_mdr_exact_emptyhscm_negative /\ ((exists ff_gap_mdm_lt_mdr_exact_emptyhscm_negative_column_bound. ff_gap_mdm_lt_mdr_exact_emptyhscm_negative_column_bound + S (ff_column_mdm_prefix_mdr_exact_emptyhscm_negative) = (mdr_q_exact_emptyhs)) /\ ((exists ff_row_mdm_cell_mdr_exact_emptyhscm_negative_cell ff_column_mdm_cell_mdr_exact_emptyhscm_negative_cell. (((((exists ff_gap_mdm_lt_mdr_exact_emptyhscm_negative_cell_row_before. ff_gap_mdm_lt_mdr_exact_emptyhscm_negative_cell_row_before + S (ff_row_mdm_prefix_mdr_exact_emptyhscm_negative) = (0)) /\ ff_row_mdm_cell_mdr_exact_emptyhscm_negative_cell = ff_row_mdm_prefix_mdr_exact_emptyhscm_negative) \/ ((exists ff_gap_mdm_le_mdr_exact_emptyhscm_negative_cell_row_after. ff_gap_mdm_le_mdr_exact_emptyhscm_negative_cell_row_after + (0) = (ff_row_mdm_prefix_mdr_exact_emptyhscm_negative)) /\ ff_row_mdm_cell_mdr_exact_emptyhscm_negative_cell = S ff_row_mdm_prefix_mdr_exact_emptyhscm_negative))) /\ (((((exists ff_gap_mdm_lt_mdr_exact_emptyhscm_negative_cell_column_before. ff_gap_mdm_lt_mdr_exact_emptyhscm_negative_cell_column_before + S (ff_column_mdm_prefix_mdr_exact_emptyhscm_negative) = (mdr_j_exact_emptyhsc)) /\ ff_column_mdm_cell_mdr_exact_emptyhscm_negative_cell = ff_column_mdm_prefix_mdr_exact_emptyhscm_negative) \/ ((exists ff_gap_mdm_le_mdr_exact_emptyhscm_negative_cell_column_after. ff_gap_mdm_le_mdr_exact_emptyhscm_negative_cell_column_after + (mdr_j_exact_emptyhsc) = (ff_column_mdm_prefix_mdr_exact_emptyhscm_negative)) /\ ff_column_mdm_cell_mdr_exact_emptyhscm_negative_cell = S ff_column_mdm_prefix_mdr_exact_emptyhscm_negative))) /\ (((exists ff_h_mdm_mdr_exact_emptyhscm_negative_cell_source. ff_h_mdm_mdr_exact_emptyhscm_negative_cell_source + S (ff_value_mdm_prefix_mdr_exact_emptyhscm_negative) = S ((S ((ff_row_mdm_cell_mdr_exact_emptyhscm_negative_cell) * (S (mdr_q_exact_emptyhs)) + (ff_column_mdm_cell_mdr_exact_emptyhscm_negative_cell))) * mdr_nc_exact_emptyh)) /\ exists ff_q_mdm_mdr_exact_emptyhscm_negative_cell_source. mdr_nb_exact_emptyh = ff_q_mdm_mdr_exact_emptyhscm_negative_cell_source * S ((S ((ff_row_mdm_cell_mdr_exact_emptyhscm_negative_cell) * (S (mdr_q_exact_emptyhs)) + (ff_column_mdm_cell_mdr_exact_emptyhscm_negative_cell))) * mdr_nc_exact_emptyh) + (ff_value_mdm_prefix_mdr_exact_emptyhscm_negative)))))) /\ (((exists ff_h_mdm_mdr_exact_emptyhscm_negative_target. ff_h_mdm_mdr_exact_emptyhscm_negative_target + S (ff_value_mdm_prefix_mdr_exact_emptyhscm_negative) = S ((S (ff_index_mdm_prefix_mdr_exact_emptyhscm_negative)) * mdr_ut_exact_emptyhsc)) /\ exists ff_q_mdm_mdr_exact_emptyhscm_negative_target. mdr_un_exact_emptyhsc = ff_q_mdm_mdr_exact_emptyhscm_negative_target * S ((S (ff_index_mdm_prefix_mdr_exact_emptyhscm_negative)) * mdr_ut_exact_emptyhsc) + (ff_value_mdm_prefix_mdr_exact_emptyhscm_negative))))))))) /\ ((((exists ff_h_mdr_exact_emptyhscp. ff_h_mdr_exact_emptyhscp + S (mdr_p_exact_emptyhsc) = S ((S (mdr_j_exact_emptyhsc)) * mdr_ec_exact_emptyhs)) /\ exists ff_q_mdr_exact_emptyhscp. mdr_eb_exact_emptyhs = ff_q_mdr_exact_emptyhscp * S ((S (mdr_j_exact_emptyhsc)) * mdr_ec_exact_emptyhs) + (mdr_p_exact_emptyhsc))) /\ (((exists ff_h_mdr_exact_emptyhscn. ff_h_mdr_exact_emptyhscn + S (mdr_n_exact_emptyhsc) = S ((S (mdr_j_exact_emptyhsc)) * mdr_fc_exact_emptyhs)) /\ exists ff_q_mdr_exact_emptyhscn. mdr_fb_exact_emptyhs = ff_q_mdr_exact_emptyhscn * S ((S (mdr_j_exact_emptyhsc)) * mdr_fc_exact_emptyhs) + (mdr_n_exact_emptyhsc)))))))) /\ (exists ff_ub_mce_fold_mdr_exact_emptyhsf ff_uc_mce_fold_mdr_exact_emptyhsf ff_vb_mce_fold_mdr_exact_emptyhsf ff_vc_mce_fold_mdr_exact_emptyhsf. ((forall ff_index_mce_alternating_mdr_exact_emptyhsf_prefix. (exists ff_gap_mce_mdr_exact_emptyhsf_prefix_index. ff_gap_mce_mdr_exact_emptyhsf_prefix_index + S (ff_index_mce_alternating_mdr_exact_emptyhsf_prefix) = (S (mdr_q_exact_emptyhs))) -> exists ff_ap_mce_alternating_mdr_exact_emptyhsf_prefix ff_an_mce_alternating_mdr_exact_emptyhsf_prefix ff_bp_mce_alternating_mdr_exact_emptyhsf_prefix ff_bn_mce_alternating_mdr_exact_emptyhsf_prefix ff_p_mce_alternating_mdr_exact_emptyhsf_prefix ff_n_mce_alternating_mdr_exact_emptyhsf_prefix. ((((exists ff_h_mce_mdr_exact_emptyhsf_prefix_ap. ff_h_mce_mdr_exact_emptyhsf_prefix_ap + S (ff_ap_mce_alternating_mdr_exact_emptyhsf_prefix) = S ((S (ff_index_mce_alternating_mdr_exact_emptyhsf_prefix)) * mdr_pc_exact_emptyh)) /\ exists ff_q_mce_mdr_exact_emptyhsf_prefix_ap. mdr_pb_exact_emptyh = ff_q_mce_mdr_exact_emptyhsf_prefix_ap * S ((S (ff_index_mce_alternating_mdr_exact_emptyhsf_prefix)) * mdr_pc_exact_emptyh) + (ff_ap_mce_alternating_mdr_exact_emptyhsf_prefix))) /\ ((((exists ff_h_mce_mdr_exact_emptyhsf_prefix_an. ff_h_mce_mdr_exact_emptyhsf_prefix_an + S (ff_an_mce_alternating_mdr_exact_emptyhsf_prefix) = S ((S (ff_index_mce_alternating_mdr_exact_emptyhsf_prefix)) * mdr_nc_exact_emptyh)) /\ exists ff_q_mce_mdr_exact_emptyhsf_prefix_an. mdr_nb_exact_emptyh = ff_q_mce_mdr_exact_emptyhsf_prefix_an * S ((S (ff_index_mce_alternating_mdr_exact_emptyhsf_prefix)) * mdr_nc_exact_emptyh) + (ff_an_mce_alternating_mdr_exact_emptyhsf_prefix))) /\ ((((exists ff_h_mce_mdr_exact_emptyhsf_prefix_bp. ff_h_mce_mdr_exact_emptyhsf_prefix_bp + S (ff_bp_mce_alternating_mdr_exact_emptyhsf_prefix) = S ((S (ff_index_mce_alternating_mdr_exact_emptyhsf_prefix)) * mdr_ec_exact_emptyhs)) /\ exists ff_q_mce_mdr_exact_emptyhsf_prefix_bp. mdr_eb_exact_emptyhs = ff_q_mce_mdr_exact_emptyhsf_prefix_bp * S ((S (ff_index_mce_alternating_mdr_exact_emptyhsf_prefix)) * mdr_ec_exact_emptyhs) + (ff_bp_mce_alternating_mdr_exact_emptyhsf_prefix))) /\ ((((exists ff_h_mce_mdr_exact_emptyhsf_prefix_bn. ff_h_mce_mdr_exact_emptyhsf_prefix_bn + S (ff_bn_mce_alternating_mdr_exact_emptyhsf_prefix) = S ((S (ff_index_mce_alternating_mdr_exact_emptyhsf_prefix)) * mdr_fc_exact_emptyhs)) /\ exists ff_q_mce_mdr_exact_emptyhsf_prefix_bn. mdr_fb_exact_emptyhs = ff_q_mce_mdr_exact_emptyhsf_prefix_bn * S ((S (ff_index_mce_alternating_mdr_exact_emptyhsf_prefix)) * mdr_fc_exact_emptyhs) + (ff_bn_mce_alternating_mdr_exact_emptyhsf_prefix))) /\ ((((exists ff_h_mce_mdr_exact_emptyhsf_prefix_positive. ff_h_mce_mdr_exact_emptyhsf_prefix_positive + S (ff_p_mce_alternating_mdr_exact_emptyhsf_prefix) = S ((S (ff_index_mce_alternating_mdr_exact_emptyhsf_prefix)) * ff_uc_mce_fold_mdr_exact_emptyhsf)) /\ exists ff_q_mce_mdr_exact_emptyhsf_prefix_positive. ff_ub_mce_fold_mdr_exact_emptyhsf = ff_q_mce_mdr_exact_emptyhsf_prefix_positive * S ((S (ff_index_mce_alternating_mdr_exact_emptyhsf_prefix)) * ff_uc_mce_fold_mdr_exact_emptyhsf) + (ff_p_mce_alternating_mdr_exact_emptyhsf_prefix))) /\ ((((exists ff_h_mce_mdr_exact_emptyhsf_prefix_negative. ff_h_mce_mdr_exact_emptyhsf_prefix_negative + S (ff_n_mce_alternating_mdr_exact_emptyhsf_prefix) = S ((S (ff_index_mce_alternating_mdr_exact_emptyhsf_prefix)) * ff_vc_mce_fold_mdr_exact_emptyhsf)) /\ exists ff_q_mce_mdr_exact_emptyhsf_prefix_negative. ff_vb_mce_fold_mdr_exact_emptyhsf = ff_q_mce_mdr_exact_emptyhsf_prefix_negative * S ((S (ff_index_mce_alternating_mdr_exact_emptyhsf_prefix)) * ff_vc_mce_fold_mdr_exact_emptyhsf) + (ff_n_mce_alternating_mdr_exact_emptyhsf_prefix))) /\ (((exists ff_even_mce_term_mdr_exact_emptyhsf_prefix_term. ff_index_mce_alternating_mdr_exact_emptyhsf_prefix = 2 * ff_even_mce_term_mdr_exact_emptyhsf_prefix_term) /\ (ff_p_mce_alternating_mdr_exact_emptyhsf_prefix = (ff_ap_mce_alternating_mdr_exact_emptyhsf_prefix) * (ff_bp_mce_alternating_mdr_exact_emptyhsf_prefix) + (ff_an_mce_alternating_mdr_exact_emptyhsf_prefix) * (ff_bn_mce_alternating_mdr_exact_emptyhsf_prefix) /\ ff_n_mce_alternating_mdr_exact_emptyhsf_prefix = (ff_ap_mce_alternating_mdr_exact_emptyhsf_prefix) * (ff_bn_mce_alternating_mdr_exact_emptyhsf_prefix) + (ff_an_mce_alternating_mdr_exact_emptyhsf_prefix) * (ff_bp_mce_alternating_mdr_exact_emptyhsf_prefix))) \/ ((exists ff_odd_mce_term_mdr_exact_emptyhsf_prefix_term. ff_index_mce_alternating_mdr_exact_emptyhsf_prefix = 2 * ff_odd_mce_term_mdr_exact_emptyhsf_prefix_term + 1) /\ (ff_p_mce_alternating_mdr_exact_emptyhsf_prefix = (ff_ap_mce_alternating_mdr_exact_emptyhsf_prefix) * (ff_bn_mce_alternating_mdr_exact_emptyhsf_prefix) + (ff_an_mce_alternating_mdr_exact_emptyhsf_prefix) * (ff_bp_mce_alternating_mdr_exact_emptyhsf_prefix) /\ ff_n_mce_alternating_mdr_exact_emptyhsf_prefix = (ff_ap_mce_alternating_mdr_exact_emptyhsf_prefix) * (ff_bp_mce_alternating_mdr_exact_emptyhsf_prefix) + (ff_an_mce_alternating_mdr_exact_emptyhsf_prefix) * (ff_bn_mce_alternating_mdr_exact_emptyhsf_prefix))))))))))) /\ ((exists ff_u_mce_mdr_exact_emptyhsf_positive ff_v_mce_mdr_exact_emptyhsf_positive. ((((exists ff_h_mce_mdr_exact_emptyhsf_positive_start. ff_h_mce_mdr_exact_emptyhsf_positive_start + S (0) = S ((S (0)) * ff_v_mce_mdr_exact_emptyhsf_positive)) /\ exists ff_q_mce_mdr_exact_emptyhsf_positive_start. ff_u_mce_mdr_exact_emptyhsf_positive = ff_q_mce_mdr_exact_emptyhsf_positive_start * S ((S (0)) * ff_v_mce_mdr_exact_emptyhsf_positive) + (0))) /\ ((((exists ff_h_mce_mdr_exact_emptyhsf_positive_terminal. ff_h_mce_mdr_exact_emptyhsf_positive_terminal + S (mdr_p_exact_emptyh) = S ((S ((S (mdr_q_exact_emptyhs)))) * ff_v_mce_mdr_exact_emptyhsf_positive)) /\ exists ff_q_mce_mdr_exact_emptyhsf_positive_terminal. ff_u_mce_mdr_exact_emptyhsf_positive = ff_q_mce_mdr_exact_emptyhsf_positive_terminal * S ((S ((S (mdr_q_exact_emptyhs)))) * ff_v_mce_mdr_exact_emptyhsf_positive) + (mdr_p_exact_emptyh))) /\ forall ff_i_mce_mdr_exact_emptyhsf_positive. (exists ff_lt_mce_mdr_exact_emptyhsf_positive_bound. ff_lt_mce_mdr_exact_emptyhsf_positive_bound + S ff_i_mce_mdr_exact_emptyhsf_positive = (S (mdr_q_exact_emptyhs))) -> exists ff_a_mce_mdr_exact_emptyhsf_positive ff_r_mce_mdr_exact_emptyhsf_positive ff_s_mce_mdr_exact_emptyhsf_positive. ((((exists ff_h_mce_mdr_exact_emptyhsf_positive_summand. ff_h_mce_mdr_exact_emptyhsf_positive_summand + S (ff_a_mce_mdr_exact_emptyhsf_positive) = S ((S (ff_i_mce_mdr_exact_emptyhsf_positive)) * ff_uc_mce_fold_mdr_exact_emptyhsf)) /\ exists ff_q_mce_mdr_exact_emptyhsf_positive_summand. ff_ub_mce_fold_mdr_exact_emptyhsf = ff_q_mce_mdr_exact_emptyhsf_positive_summand * S ((S (ff_i_mce_mdr_exact_emptyhsf_positive)) * ff_uc_mce_fold_mdr_exact_emptyhsf) + (ff_a_mce_mdr_exact_emptyhsf_positive))) /\ ((((exists ff_h_mce_mdr_exact_emptyhsf_positive_partial. ff_h_mce_mdr_exact_emptyhsf_positive_partial + S (ff_r_mce_mdr_exact_emptyhsf_positive) = S ((S (ff_i_mce_mdr_exact_emptyhsf_positive)) * ff_v_mce_mdr_exact_emptyhsf_positive)) /\ exists ff_q_mce_mdr_exact_emptyhsf_positive_partial. ff_u_mce_mdr_exact_emptyhsf_positive = ff_q_mce_mdr_exact_emptyhsf_positive_partial * S ((S (ff_i_mce_mdr_exact_emptyhsf_positive)) * ff_v_mce_mdr_exact_emptyhsf_positive) + (ff_r_mce_mdr_exact_emptyhsf_positive))) /\ ((((exists ff_h_mce_mdr_exact_emptyhsf_positive_successor. ff_h_mce_mdr_exact_emptyhsf_positive_successor + S (ff_s_mce_mdr_exact_emptyhsf_positive) = S ((S (S ff_i_mce_mdr_exact_emptyhsf_positive)) * ff_v_mce_mdr_exact_emptyhsf_positive)) /\ exists ff_q_mce_mdr_exact_emptyhsf_positive_successor. ff_u_mce_mdr_exact_emptyhsf_positive = ff_q_mce_mdr_exact_emptyhsf_positive_successor * S ((S (S ff_i_mce_mdr_exact_emptyhsf_positive)) * ff_v_mce_mdr_exact_emptyhsf_positive) + (ff_s_mce_mdr_exact_emptyhsf_positive))) /\ ff_s_mce_mdr_exact_emptyhsf_positive = ff_r_mce_mdr_exact_emptyhsf_positive + ff_a_mce_mdr_exact_emptyhsf_positive)))))) /\ (exists ff_u_mce_mdr_exact_emptyhsf_negative ff_v_mce_mdr_exact_emptyhsf_negative. ((((exists ff_h_mce_mdr_exact_emptyhsf_negative_start. ff_h_mce_mdr_exact_emptyhsf_negative_start + S (0) = S ((S (0)) * ff_v_mce_mdr_exact_emptyhsf_negative)) /\ exists ff_q_mce_mdr_exact_emptyhsf_negative_start. ff_u_mce_mdr_exact_emptyhsf_negative = ff_q_mce_mdr_exact_emptyhsf_negative_start * S ((S (0)) * ff_v_mce_mdr_exact_emptyhsf_negative) + (0))) /\ ((((exists ff_h_mce_mdr_exact_emptyhsf_negative_terminal. ff_h_mce_mdr_exact_emptyhsf_negative_terminal + S (mdr_n_exact_emptyh) = S ((S ((S (mdr_q_exact_emptyhs)))) * ff_v_mce_mdr_exact_emptyhsf_negative)) /\ exists ff_q_mce_mdr_exact_emptyhsf_negative_terminal. ff_u_mce_mdr_exact_emptyhsf_negative = ff_q_mce_mdr_exact_emptyhsf_negative_terminal * S ((S ((S (mdr_q_exact_emptyhs)))) * ff_v_mce_mdr_exact_emptyhsf_negative) + (mdr_n_exact_emptyh))) /\ forall ff_i_mce_mdr_exact_emptyhsf_negative. (exists ff_lt_mce_mdr_exact_emptyhsf_negative_bound. ff_lt_mce_mdr_exact_emptyhsf_negative_bound + S ff_i_mce_mdr_exact_emptyhsf_negative = (S (mdr_q_exact_emptyhs))) -> exists ff_a_mce_mdr_exact_emptyhsf_negative ff_r_mce_mdr_exact_emptyhsf_negative ff_s_mce_mdr_exact_emptyhsf_negative. ((((exists ff_h_mce_mdr_exact_emptyhsf_negative_summand. ff_h_mce_mdr_exact_emptyhsf_negative_summand + S (ff_a_mce_mdr_exact_emptyhsf_negative) = S ((S (ff_i_mce_mdr_exact_emptyhsf_negative)) * ff_vc_mce_fold_mdr_exact_emptyhsf)) /\ exists ff_q_mce_mdr_exact_emptyhsf_negative_summand. ff_vb_mce_fold_mdr_exact_emptyhsf = ff_q_mce_mdr_exact_emptyhsf_negative_summand * S ((S (ff_i_mce_mdr_exact_emptyhsf_negative)) * ff_vc_mce_fold_mdr_exact_emptyhsf) + (ff_a_mce_mdr_exact_emptyhsf_negative))) /\ ((((exists ff_h_mce_mdr_exact_emptyhsf_negative_partial. ff_h_mce_mdr_exact_emptyhsf_negative_partial + S (ff_r_mce_mdr_exact_emptyhsf_negative) = S ((S (ff_i_mce_mdr_exact_emptyhsf_negative)) * ff_v_mce_mdr_exact_emptyhsf_negative)) /\ exists ff_q_mce_mdr_exact_emptyhsf_negative_partial. ff_u_mce_mdr_exact_emptyhsf_negative = ff_q_mce_mdr_exact_emptyhsf_negative_partial * S ((S (ff_i_mce_mdr_exact_emptyhsf_negative)) * ff_v_mce_mdr_exact_emptyhsf_negative) + (ff_r_mce_mdr_exact_emptyhsf_negative))) /\ ((((exists ff_h_mce_mdr_exact_emptyhsf_negative_successor. ff_h_mce_mdr_exact_emptyhsf_negative_successor + S (ff_s_mce_mdr_exact_emptyhsf_negative) = S ((S (S ff_i_mce_mdr_exact_emptyhsf_negative)) * ff_v_mce_mdr_exact_emptyhsf_negative)) /\ exists ff_q_mce_mdr_exact_emptyhsf_negative_successor. ff_u_mce_mdr_exact_emptyhsf_negative = ff_q_mce_mdr_exact_emptyhsf_negative_successor * S ((S (S ff_i_mce_mdr_exact_emptyhsf_negative)) * ff_v_mce_mdr_exact_emptyhsf_negative) + (ff_s_mce_mdr_exact_emptyhsf_negative))) /\ ff_s_mce_mdr_exact_emptyhsf_negative = ff_r_mce_mdr_exact_emptyhsf_negative + ff_a_mce_mdr_exact_emptyhsf_negative))))))))))))))) /\ ((exists mdr_gap_exact_emptyi. mdr_gap_exact_emptyi + S (mdr_i_exact_empty) = (mdr_l_exact_empty)) /\ (exists mdr_z_exact_emptyr. ((exists mdr_a_exact_emptyrc mdr_b_exact_emptyrc mdr_c_exact_emptyrc mdr_e_exact_emptyrc mdr_f_exact_emptyrc. ((mdr_a_exact_emptyrc = ((0) + (pb)) * S ((0) + (pb)) + ((pb) + (pb))) /\ ((mdr_b_exact_emptyrc = ((pc) + (nb)) * S ((pc) + (nb)) + ((nb) + (nb))) /\ ((mdr_c_exact_emptyrc = ((mdr_a_exact_emptyrc) + (mdr_b_exact_emptyrc)) * S ((mdr_a_exact_emptyrc) + (mdr_b_exact_emptyrc)) + ((mdr_b_exact_emptyrc) + (mdr_b_exact_emptyrc))) /\ ((mdr_e_exact_emptyrc = ((1) + (0)) * S ((1) + (0)) + ((0) + (0))) /\ ((mdr_f_exact_emptyrc = ((nc) + (mdr_e_exact_emptyrc)) * S ((nc) + (mdr_e_exact_emptyrc)) + ((mdr_e_exact_emptyrc) + (mdr_e_exact_emptyrc))) /\ ((mdr_z_exact_emptyr) = ((mdr_c_exact_emptyrc) + (mdr_f_exact_emptyrc)) * S ((mdr_c_exact_emptyrc) + (mdr_f_exact_emptyrc)) + ((mdr_f_exact_emptyrc) + (mdr_f_exact_emptyrc))))))))) /\ (((exists ff_h_mdr_exact_emptyrb. ff_h_mdr_exact_emptyrb + S (mdr_z_exact_emptyr) = S ((S (mdr_i_exact_empty)) * mdr_c_exact_empty)) /\ exists ff_q_mdr_exact_emptyrb. mdr_b_exact_empty = ff_q_mdr_exact_emptyrb * S ((S (mdr_i_exact_empty)) * mdr_c_exact_empty) + (mdr_z_exact_emptyr))))))))