DL00AC

positive_determinant_matrix_data_functional

Alpha v34 independently verified · alpha_closed; checked-use authorized; not Stable

The positive absolute determinant in the nondegenerate square-matrix data is uniquely determined by the actual matrix.

Current library: Alpha v34, 4,223 checked-use theorems; Stable remains 432. Historical first admissions, original proof editions, and non-admitted aliases are preserved. Exact original first-admission records.

Exact expanded first-order arithmetic statement

forall ab ac bb bc d D E. (((~(d = 0)) /\ ((~(D = 0)) /\ (exists mdr_p_first_positive_dataabsolute mdr_n_first_positive_dataabsolute. ((exists mdr_b_first_positive_dataabsoluteevaluation mdr_c_first_positive_dataabsoluteevaluation mdr_l_first_positive_dataabsoluteevaluation mdr_i_first_positive_dataabsoluteevaluation. ((forall mdr_i_first_positive_dataabsoluteevaluationh. (exists mdr_gap_first_positive_dataabsoluteevaluationhi. mdr_gap_first_positive_dataabsoluteevaluationhi + S (mdr_i_first_positive_dataabsoluteevaluationh) = (mdr_l_first_positive_dataabsoluteevaluation)) -> exists mdr_d_first_positive_dataabsoluteevaluationh mdr_pb_first_positive_dataabsoluteevaluationh mdr_pc_first_positive_dataabsoluteevaluationh mdr_nb_first_positive_dataabsoluteevaluationh mdr_nc_first_positive_dataabsoluteevaluationh mdr_p_first_positive_dataabsoluteevaluationh mdr_n_first_positive_dataabsoluteevaluationh. ((exists mdr_z_first_positive_dataabsoluteevaluationhr. ((exists mdr_a_first_positive_dataabsoluteevaluationhrc mdr_b_first_positive_dataabsoluteevaluationhrc mdr_c_first_positive_dataabsoluteevaluationhrc mdr_e_first_positive_dataabsoluteevaluationhrc mdr_f_first_positive_dataabsoluteevaluationhrc. ((mdr_a_first_positive_dataabsoluteevaluationhrc = ((mdr_d_first_positive_dataabsoluteevaluationh) + (mdr_pb_first_positive_dataabsoluteevaluationh)) * S ((mdr_d_first_positive_dataabsoluteevaluationh) + (mdr_pb_first_positive_dataabsoluteevaluationh)) + ((mdr_pb_first_positive_dataabsoluteevaluationh) + (mdr_pb_first_positive_dataabsoluteevaluationh))) /\ ((mdr_b_first_positive_dataabsoluteevaluationhrc = ((mdr_pc_first_positive_dataabsoluteevaluationh) + (mdr_nb_first_positive_dataabsoluteevaluationh)) * S ((mdr_pc_first_positive_dataabsoluteevaluationh) + (mdr_nb_first_positive_dataabsoluteevaluationh)) + ((mdr_nb_first_positive_dataabsoluteevaluationh) + (mdr_nb_first_positive_dataabsoluteevaluationh))) /\ ((mdr_c_first_positive_dataabsoluteevaluationhrc = ((mdr_a_first_positive_dataabsoluteevaluationhrc) + (mdr_b_first_positive_dataabsoluteevaluationhrc)) * S ((mdr_a_first_positive_dataabsoluteevaluationhrc) + (mdr_b_first_positive_dataabsoluteevaluationhrc)) + ((mdr_b_first_positive_dataabsoluteevaluationhrc) + (mdr_b_first_positive_dataabsoluteevaluationhrc))) /\ ((mdr_e_first_positive_dataabsoluteevaluationhrc = ((mdr_p_first_positive_dataabsoluteevaluationh) + (mdr_n_first_positive_dataabsoluteevaluationh)) * S ((mdr_p_first_positive_dataabsoluteevaluationh) + (mdr_n_first_positive_dataabsoluteevaluationh)) + ((mdr_n_first_positive_dataabsoluteevaluationh) + (mdr_n_first_positive_dataabsoluteevaluationh))) /\ ((mdr_f_first_positive_dataabsoluteevaluationhrc = ((mdr_nc_first_positive_dataabsoluteevaluationh) + (mdr_e_first_positive_dataabsoluteevaluationhrc)) * S ((mdr_nc_first_positive_dataabsoluteevaluationh) + (mdr_e_first_positive_dataabsoluteevaluationhrc)) + ((mdr_e_first_positive_dataabsoluteevaluationhrc) + (mdr_e_first_positive_dataabsoluteevaluationhrc))) /\ ((mdr_z_first_positive_dataabsoluteevaluationhr) = ((mdr_c_first_positive_dataabsoluteevaluationhrc) + (mdr_f_first_positive_dataabsoluteevaluationhrc)) * S ((mdr_c_first_positive_dataabsoluteevaluationhrc) + (mdr_f_first_positive_dataabsoluteevaluationhrc)) + ((mdr_f_first_positive_dataabsoluteevaluationhrc) + (mdr_f_first_positive_dataabsoluteevaluationhrc))))))))) /\ (((exists ff_h_mdr_first_positive_dataabsoluteevaluationhrb. ff_h_mdr_first_positive_dataabsoluteevaluationhrb + S (mdr_z_first_positive_dataabsoluteevaluationhr) = S ((S (mdr_i_first_positive_dataabsoluteevaluationh)) * mdr_c_first_positive_dataabsoluteevaluation)) /\ exists ff_q_mdr_first_positive_dataabsoluteevaluationhrb. mdr_b_first_positive_dataabsoluteevaluation = ff_q_mdr_first_positive_dataabsoluteevaluationhrb * S ((S (mdr_i_first_positive_dataabsoluteevaluationh)) * mdr_c_first_positive_dataabsoluteevaluation) + (mdr_z_first_positive_dataabsoluteevaluationhr))))) /\ (((((mdr_d_first_positive_dataabsoluteevaluationh) = 0) /\ (((mdr_p_first_positive_dataabsoluteevaluationh) = 1) /\ ((mdr_n_first_positive_dataabsoluteevaluationh) = 0))) \/ exists mdr_q_first_positive_dataabsoluteevaluationhs mdr_eb_first_positive_dataabsoluteevaluationhs mdr_ec_first_positive_dataabsoluteevaluationhs mdr_fb_first_positive_dataabsoluteevaluationhs mdr_fc_first_positive_dataabsoluteevaluationhs. (((mdr_d_first_positive_dataabsoluteevaluationh) = S (mdr_q_first_positive_dataabsoluteevaluationhs)) /\ ((forall mdr_j_first_positive_dataabsoluteevaluationhsc. (exists mdr_gap_first_positive_dataabsoluteevaluationhscj. mdr_gap_first_positive_dataabsoluteevaluationhscj + S (mdr_j_first_positive_dataabsoluteevaluationhsc) = (S (mdr_q_first_positive_dataabsoluteevaluationhs))) -> exists mdr_i_first_positive_dataabsoluteevaluationhsc mdr_up_first_positive_dataabsoluteevaluationhsc mdr_us_first_positive_dataabsoluteevaluationhsc mdr_un_first_positive_dataabsoluteevaluationhsc mdr_ut_first_positive_dataabsoluteevaluationhsc mdr_p_first_positive_dataabsoluteevaluationhsc mdr_n_first_positive_dataabsoluteevaluationhsc. ((exists mdr_gap_first_positive_dataabsoluteevaluationhsci. mdr_gap_first_positive_dataabsoluteevaluationhsci + S (mdr_i_first_positive_dataabsoluteevaluationhsc) = (mdr_i_first_positive_dataabsoluteevaluationh)) /\ ((exists mdr_z_first_positive_dataabsoluteevaluationhscr. ((exists mdr_a_first_positive_dataabsoluteevaluationhscrc mdr_b_first_positive_dataabsoluteevaluationhscrc mdr_c_first_positive_dataabsoluteevaluationhscrc mdr_e_first_positive_dataabsoluteevaluationhscrc mdr_f_first_positive_dataabsoluteevaluationhscrc. ((mdr_a_first_positive_dataabsoluteevaluationhscrc = ((mdr_q_first_positive_dataabsoluteevaluationhs) + (mdr_up_first_positive_dataabsoluteevaluationhsc)) * S ((mdr_q_first_positive_dataabsoluteevaluationhs) + (mdr_up_first_positive_dataabsoluteevaluationhsc)) + ((mdr_up_first_positive_dataabsoluteevaluationhsc) + (mdr_up_first_positive_dataabsoluteevaluationhsc))) /\ ((mdr_b_first_positive_dataabsoluteevaluationhscrc = ((mdr_us_first_positive_dataabsoluteevaluationhsc) + (mdr_un_first_positive_dataabsoluteevaluationhsc)) * S ((mdr_us_first_positive_dataabsoluteevaluationhsc) + (mdr_un_first_positive_dataabsoluteevaluationhsc)) + ((mdr_un_first_positive_dataabsoluteevaluationhsc) + (mdr_un_first_positive_dataabsoluteevaluationhsc))) /\ ((mdr_c_first_positive_dataabsoluteevaluationhscrc = ((mdr_a_first_positive_dataabsoluteevaluationhscrc) + (mdr_b_first_positive_dataabsoluteevaluationhscrc)) * S ((mdr_a_first_positive_dataabsoluteevaluationhscrc) + (mdr_b_first_positive_dataabsoluteevaluationhscrc)) + ((mdr_b_first_positive_dataabsoluteevaluationhscrc) + (mdr_b_first_positive_dataabsoluteevaluationhscrc))) /\ ((mdr_e_first_positive_dataabsoluteevaluationhscrc = ((mdr_p_first_positive_dataabsoluteevaluationhsc) + (mdr_n_first_positive_dataabsoluteevaluationhsc)) * S ((mdr_p_first_positive_dataabsoluteevaluationhsc) + (mdr_n_first_positive_dataabsoluteevaluationhsc)) + ((mdr_n_first_positive_dataabsoluteevaluationhsc) + (mdr_n_first_positive_dataabsoluteevaluationhsc))) /\ ((mdr_f_first_positive_dataabsoluteevaluationhscrc = ((mdr_ut_first_positive_dataabsoluteevaluationhsc) + (mdr_e_first_positive_dataabsoluteevaluationhscrc)) * S ((mdr_ut_first_positive_dataabsoluteevaluationhsc) + (mdr_e_first_positive_dataabsoluteevaluationhscrc)) + ((mdr_e_first_positive_dataabsoluteevaluationhscrc) + (mdr_e_first_positive_dataabsoluteevaluationhscrc))) /\ ((mdr_z_first_positive_dataabsoluteevaluationhscr) = ((mdr_c_first_positive_dataabsoluteevaluationhscrc) + (mdr_f_first_positive_dataabsoluteevaluationhscrc)) * S ((mdr_c_first_positive_dataabsoluteevaluationhscrc) + (mdr_f_first_positive_dataabsoluteevaluationhscrc)) + ((mdr_f_first_positive_dataabsoluteevaluationhscrc) + (mdr_f_first_positive_dataabsoluteevaluationhscrc))))))))) /\ (((exists ff_h_mdr_first_positive_dataabsoluteevaluationhscrb. ff_h_mdr_first_positive_dataabsoluteevaluationhscrb + S (mdr_z_first_positive_dataabsoluteevaluationhscr) = S ((S (mdr_i_first_positive_dataabsoluteevaluationhsc)) * mdr_c_first_positive_dataabsoluteevaluation)) /\ exists ff_q_mdr_first_positive_dataabsoluteevaluationhscrb. mdr_b_first_positive_dataabsoluteevaluation = ff_q_mdr_first_positive_dataabsoluteevaluationhscrb * S ((S (mdr_i_first_positive_dataabsoluteevaluationhsc)) * mdr_c_first_positive_dataabsoluteevaluation) + (mdr_z_first_positive_dataabsoluteevaluationhscr))))) /\ ((((forall ff_index_mdm_prefix_mdr_first_positive_dataabsoluteevaluationhscm_positive. (exists ff_gap_mdm_lt_mdr_first_positive_dataabsoluteevaluationhscm_positive_index_bound. ff_gap_mdm_lt_mdr_first_positive_dataabsoluteevaluationhscm_positive_index_bound + S (ff_index_mdm_prefix_mdr_first_positive_dataabsoluteevaluationhscm_positive) = ((mdr_q_first_positive_dataabsoluteevaluationhs) * (mdr_q_first_positive_dataabsoluteevaluationhs))) -> exists ff_row_mdm_prefix_mdr_first_positive_dataabsoluteevaluationhscm_positive ff_column_mdm_prefix_mdr_first_positive_dataabsoluteevaluationhscm_positive ff_value_mdm_prefix_mdr_first_positive_dataabsoluteevaluationhscm_positive. (ff_index_mdm_prefix_mdr_first_positive_dataabsoluteevaluationhscm_positive = (mdr_q_first_positive_dataabsoluteevaluationhs) * ff_row_mdm_prefix_mdr_first_positive_dataabsoluteevaluationhscm_positive + ff_column_mdm_prefix_mdr_first_positive_dataabsoluteevaluationhscm_positive /\ ((exists ff_gap_mdm_lt_mdr_first_positive_dataabsoluteevaluationhscm_positive_column_bound. ff_gap_mdm_lt_mdr_first_positive_dataabsoluteevaluationhscm_positive_column_bound + S (ff_column_mdm_prefix_mdr_first_positive_dataabsoluteevaluationhscm_positive) = (mdr_q_first_positive_dataabsoluteevaluationhs)) /\ ((exists ff_row_mdm_cell_mdr_first_positive_dataabsoluteevaluationhscm_positive_cell ff_column_mdm_cell_mdr_first_positive_dataabsoluteevaluationhscm_positive_cell. (((((exists ff_gap_mdm_lt_mdr_first_positive_dataabsoluteevaluationhscm_positive_cell_row_before. ff_gap_mdm_lt_mdr_first_positive_dataabsoluteevaluationhscm_positive_cell_row_before + S (ff_row_mdm_prefix_mdr_first_positive_dataabsoluteevaluationhscm_positive) = (0)) /\ ff_row_mdm_cell_mdr_first_positive_dataabsoluteevaluationhscm_positive_cell = ff_row_mdm_prefix_mdr_first_positive_dataabsoluteevaluationhscm_positive) \/ ((exists ff_gap_mdm_le_mdr_first_positive_dataabsoluteevaluationhscm_positive_cell_row_after. ff_gap_mdm_le_mdr_first_positive_dataabsoluteevaluationhscm_positive_cell_row_after + (0) = (ff_row_mdm_prefix_mdr_first_positive_dataabsoluteevaluationhscm_positive)) /\ ff_row_mdm_cell_mdr_first_positive_dataabsoluteevaluationhscm_positive_cell = S ff_row_mdm_prefix_mdr_first_positive_dataabsoluteevaluationhscm_positive))) /\ (((((exists ff_gap_mdm_lt_mdr_first_positive_dataabsoluteevaluationhscm_positive_cell_column_before. ff_gap_mdm_lt_mdr_first_positive_dataabsoluteevaluationhscm_positive_cell_column_before + S (ff_column_mdm_prefix_mdr_first_positive_dataabsoluteevaluationhscm_positive) = (mdr_j_first_positive_dataabsoluteevaluationhsc)) /\ ff_column_mdm_cell_mdr_first_positive_dataabsoluteevaluationhscm_positive_cell = ff_column_mdm_prefix_mdr_first_positive_dataabsoluteevaluationhscm_positive) \/ ((exists ff_gap_mdm_le_mdr_first_positive_dataabsoluteevaluationhscm_positive_cell_column_after. ff_gap_mdm_le_mdr_first_positive_dataabsoluteevaluationhscm_positive_cell_column_after + (mdr_j_first_positive_dataabsoluteevaluationhsc) = (ff_column_mdm_prefix_mdr_first_positive_dataabsoluteevaluationhscm_positive)) /\ ff_column_mdm_cell_mdr_first_positive_dataabsoluteevaluationhscm_positive_cell = S ff_column_mdm_prefix_mdr_first_positive_dataabsoluteevaluationhscm_positive))) /\ (((exists ff_h_mdm_mdr_first_positive_dataabsoluteevaluationhscm_positive_cell_source. ff_h_mdm_mdr_first_positive_dataabsoluteevaluationhscm_positive_cell_source + S (ff_value_mdm_prefix_mdr_first_positive_dataabsoluteevaluationhscm_positive) = S ((S ((ff_row_mdm_cell_mdr_first_positive_dataabsoluteevaluationhscm_positive_cell) * (S (mdr_q_first_positive_dataabsoluteevaluationhs)) + (ff_column_mdm_cell_mdr_first_positive_dataabsoluteevaluationhscm_positive_cell))) * mdr_pc_first_positive_dataabsoluteevaluationh)) /\ exists ff_q_mdm_mdr_first_positive_dataabsoluteevaluationhscm_positive_cell_source. mdr_pb_first_positive_dataabsoluteevaluationh = ff_q_mdm_mdr_first_positive_dataabsoluteevaluationhscm_positive_cell_source * S ((S ((ff_row_mdm_cell_mdr_first_positive_dataabsoluteevaluationhscm_positive_cell) * (S (mdr_q_first_positive_dataabsoluteevaluationhs)) + (ff_column_mdm_cell_mdr_first_positive_dataabsoluteevaluationhscm_positive_cell))) * mdr_pc_first_positive_dataabsoluteevaluationh) + (ff_value_mdm_prefix_mdr_first_positive_dataabsoluteevaluationhscm_positive)))))) /\ (((exists ff_h_mdm_mdr_first_positive_dataabsoluteevaluationhscm_positive_target. ff_h_mdm_mdr_first_positive_dataabsoluteevaluationhscm_positive_target + S (ff_value_mdm_prefix_mdr_first_positive_dataabsoluteevaluationhscm_positive) = S ((S (ff_index_mdm_prefix_mdr_first_positive_dataabsoluteevaluationhscm_positive)) * mdr_us_first_positive_dataabsoluteevaluationhsc)) /\ exists ff_q_mdm_mdr_first_positive_dataabsoluteevaluationhscm_positive_target. mdr_up_first_positive_dataabsoluteevaluationhsc = ff_q_mdm_mdr_first_positive_dataabsoluteevaluationhscm_positive_target * S ((S (ff_index_mdm_prefix_mdr_first_positive_dataabsoluteevaluationhscm_positive)) * mdr_us_first_positive_dataabsoluteevaluationhsc) + (ff_value_mdm_prefix_mdr_first_positive_dataabsoluteevaluationhscm_positive))))))) /\ (forall ff_index_mdm_prefix_mdr_first_positive_dataabsoluteevaluationhscm_negative. (exists ff_gap_mdm_lt_mdr_first_positive_dataabsoluteevaluationhscm_negative_index_bound. ff_gap_mdm_lt_mdr_first_positive_dataabsoluteevaluationhscm_negative_index_bound + S (ff_index_mdm_prefix_mdr_first_positive_dataabsoluteevaluationhscm_negative) = ((mdr_q_first_positive_dataabsoluteevaluationhs) * (mdr_q_first_positive_dataabsoluteevaluationhs))) -> exists ff_row_mdm_prefix_mdr_first_positive_dataabsoluteevaluationhscm_negative ff_column_mdm_prefix_mdr_first_positive_dataabsoluteevaluationhscm_negative ff_value_mdm_prefix_mdr_first_positive_dataabsoluteevaluationhscm_negative. (ff_index_mdm_prefix_mdr_first_positive_dataabsoluteevaluationhscm_negative = (mdr_q_first_positive_dataabsoluteevaluationhs) * ff_row_mdm_prefix_mdr_first_positive_dataabsoluteevaluationhscm_negative + ff_column_mdm_prefix_mdr_first_positive_dataabsoluteevaluationhscm_negative /\ ((exists ff_gap_mdm_lt_mdr_first_positive_dataabsoluteevaluationhscm_negative_column_bound. ff_gap_mdm_lt_mdr_first_positive_dataabsoluteevaluationhscm_negative_column_bound + S (ff_column_mdm_prefix_mdr_first_positive_dataabsoluteevaluationhscm_negative) = (mdr_q_first_positive_dataabsoluteevaluationhs)) /\ ((exists ff_row_mdm_cell_mdr_first_positive_dataabsoluteevaluationhscm_negative_cell ff_column_mdm_cell_mdr_first_positive_dataabsoluteevaluationhscm_negative_cell. (((((exists ff_gap_mdm_lt_mdr_first_positive_dataabsoluteevaluationhscm_negative_cell_row_before. ff_gap_mdm_lt_mdr_first_positive_dataabsoluteevaluationhscm_negative_cell_row_before + S (ff_row_mdm_prefix_mdr_first_positive_dataabsoluteevaluationhscm_negative) = (0)) /\ ff_row_mdm_cell_mdr_first_positive_dataabsoluteevaluationhscm_negative_cell = ff_row_mdm_prefix_mdr_first_positive_dataabsoluteevaluationhscm_negative) \/ ((exists ff_gap_mdm_le_mdr_first_positive_dataabsoluteevaluationhscm_negative_cell_row_after. ff_gap_mdm_le_mdr_first_positive_dataabsoluteevaluationhscm_negative_cell_row_after + (0) = (ff_row_mdm_prefix_mdr_first_positive_dataabsoluteevaluationhscm_negative)) /\ ff_row_mdm_cell_mdr_first_positive_dataabsoluteevaluationhscm_negative_cell = S ff_row_mdm_prefix_mdr_first_positive_dataabsoluteevaluationhscm_negative))) /\ (((((exists ff_gap_mdm_lt_mdr_first_positive_dataabsoluteevaluationhscm_negative_cell_column_before. ff_gap_mdm_lt_mdr_first_positive_dataabsoluteevaluationhscm_negative_cell_column_before + S (ff_column_mdm_prefix_mdr_first_positive_dataabsoluteevaluationhscm_negative) = (mdr_j_first_positive_dataabsoluteevaluationhsc)) /\ ff_column_mdm_cell_mdr_first_positive_dataabsoluteevaluationhscm_negative_cell = ff_column_mdm_prefix_mdr_first_positive_dataabsoluteevaluationhscm_negative) \/ ((exists ff_gap_mdm_le_mdr_first_positive_dataabsoluteevaluationhscm_negative_cell_column_after. ff_gap_mdm_le_mdr_first_positive_dataabsoluteevaluationhscm_negative_cell_column_after + (mdr_j_first_positive_dataabsoluteevaluationhsc) = (ff_column_mdm_prefix_mdr_first_positive_dataabsoluteevaluationhscm_negative)) /\ ff_column_mdm_cell_mdr_first_positive_dataabsoluteevaluationhscm_negative_cell = S ff_column_mdm_prefix_mdr_first_positive_dataabsoluteevaluationhscm_negative))) /\ (((exists ff_h_mdm_mdr_first_positive_dataabsoluteevaluationhscm_negative_cell_source. ff_h_mdm_mdr_first_positive_dataabsoluteevaluationhscm_negative_cell_source + S (ff_value_mdm_prefix_mdr_first_positive_dataabsoluteevaluationhscm_negative) = S ((S ((ff_row_mdm_cell_mdr_first_positive_dataabsoluteevaluationhscm_negative_cell) * (S (mdr_q_first_positive_dataabsoluteevaluationhs)) + (ff_column_mdm_cell_mdr_first_positive_dataabsoluteevaluationhscm_negative_cell))) * mdr_nc_first_positive_dataabsoluteevaluationh)) /\ exists ff_q_mdm_mdr_first_positive_dataabsoluteevaluationhscm_negative_cell_source. mdr_nb_first_positive_dataabsoluteevaluationh = ff_q_mdm_mdr_first_positive_dataabsoluteevaluationhscm_negative_cell_source * S ((S ((ff_row_mdm_cell_mdr_first_positive_dataabsoluteevaluationhscm_negative_cell) * (S (mdr_q_first_positive_dataabsoluteevaluationhs)) + (ff_column_mdm_cell_mdr_first_positive_dataabsoluteevaluationhscm_negative_cell))) * mdr_nc_first_positive_dataabsoluteevaluationh) + (ff_value_mdm_prefix_mdr_first_positive_dataabsoluteevaluationhscm_negative)))))) /\ (((exists ff_h_mdm_mdr_first_positive_dataabsoluteevaluationhscm_negative_target. ff_h_mdm_mdr_first_positive_dataabsoluteevaluationhscm_negative_target + S (ff_value_mdm_prefix_mdr_first_positive_dataabsoluteevaluationhscm_negative) = S ((S (ff_index_mdm_prefix_mdr_first_positive_dataabsoluteevaluationhscm_negative)) * mdr_ut_first_positive_dataabsoluteevaluationhsc)) /\ exists ff_q_mdm_mdr_first_positive_dataabsoluteevaluationhscm_negative_target. mdr_un_first_positive_dataabsoluteevaluationhsc = ff_q_mdm_mdr_first_positive_dataabsoluteevaluationhscm_negative_target * S ((S (ff_index_mdm_prefix_mdr_first_positive_dataabsoluteevaluationhscm_negative)) * mdr_ut_first_positive_dataabsoluteevaluationhsc) + (ff_value_mdm_prefix_mdr_first_positive_dataabsoluteevaluationhscm_negative))))))))) /\ ((((exists ff_h_mdr_first_positive_dataabsoluteevaluationhscp. ff_h_mdr_first_positive_dataabsoluteevaluationhscp + S (mdr_p_first_positive_dataabsoluteevaluationhsc) = S ((S (mdr_j_first_positive_dataabsoluteevaluationhsc)) * mdr_ec_first_positive_dataabsoluteevaluationhs)) /\ exists ff_q_mdr_first_positive_dataabsoluteevaluationhscp. mdr_eb_first_positive_dataabsoluteevaluationhs = ff_q_mdr_first_positive_dataabsoluteevaluationhscp * S ((S (mdr_j_first_positive_dataabsoluteevaluationhsc)) * mdr_ec_first_positive_dataabsoluteevaluationhs) + (mdr_p_first_positive_dataabsoluteevaluationhsc))) /\ (((exists ff_h_mdr_first_positive_dataabsoluteevaluationhscn. ff_h_mdr_first_positive_dataabsoluteevaluationhscn + S (mdr_n_first_positive_dataabsoluteevaluationhsc) = S ((S (mdr_j_first_positive_dataabsoluteevaluationhsc)) * mdr_fc_first_positive_dataabsoluteevaluationhs)) /\ exists ff_q_mdr_first_positive_dataabsoluteevaluationhscn. mdr_fb_first_positive_dataabsoluteevaluationhs = ff_q_mdr_first_positive_dataabsoluteevaluationhscn * S ((S (mdr_j_first_positive_dataabsoluteevaluationhsc)) * mdr_fc_first_positive_dataabsoluteevaluationhs) + (mdr_n_first_positive_dataabsoluteevaluationhsc)))))))) /\ (exists ff_ub_mce_fold_mdr_first_positive_dataabsoluteevaluationhsf ff_uc_mce_fold_mdr_first_positive_dataabsoluteevaluationhsf ff_vb_mce_fold_mdr_first_positive_dataabsoluteevaluationhsf ff_vc_mce_fold_mdr_first_positive_dataabsoluteevaluationhsf. ((forall ff_index_mce_alternating_mdr_first_positive_dataabsoluteevaluationhsf_prefix. (exists ff_gap_mce_mdr_first_positive_dataabsoluteevaluationhsf_prefix_index. ff_gap_mce_mdr_first_positive_dataabsoluteevaluationhsf_prefix_index + S (ff_index_mce_alternating_mdr_first_positive_dataabsoluteevaluationhsf_prefix) = (S (mdr_q_first_positive_dataabsoluteevaluationhs))) -> exists ff_ap_mce_alternating_mdr_first_positive_dataabsoluteevaluationhsf_prefix ff_an_mce_alternating_mdr_first_positive_dataabsoluteevaluationhsf_prefix ff_bp_mce_alternating_mdr_first_positive_dataabsoluteevaluationhsf_prefix ff_bn_mce_alternating_mdr_first_positive_dataabsoluteevaluationhsf_prefix ff_p_mce_alternating_mdr_first_positive_dataabsoluteevaluationhsf_prefix ff_n_mce_alternating_mdr_first_positive_dataabsoluteevaluationhsf_prefix. ((((exists ff_h_mce_mdr_first_positive_dataabsoluteevaluationhsf_prefix_ap. ff_h_mce_mdr_first_positive_dataabsoluteevaluationhsf_prefix_ap + S (ff_ap_mce_alternating_mdr_first_positive_dataabsoluteevaluationhsf_prefix) = S ((S (ff_index_mce_alternating_mdr_first_positive_dataabsoluteevaluationhsf_prefix)) * mdr_pc_first_positive_dataabsoluteevaluationh)) /\ exists ff_q_mce_mdr_first_positive_dataabsoluteevaluationhsf_prefix_ap. mdr_pb_first_positive_dataabsoluteevaluationh = ff_q_mce_mdr_first_positive_dataabsoluteevaluationhsf_prefix_ap * S ((S (ff_index_mce_alternating_mdr_first_positive_dataabsoluteevaluationhsf_prefix)) * mdr_pc_first_positive_dataabsoluteevaluationh) + (ff_ap_mce_alternating_mdr_first_positive_dataabsoluteevaluationhsf_prefix))) /\ ((((exists ff_h_mce_mdr_first_positive_dataabsoluteevaluationhsf_prefix_an. ff_h_mce_mdr_first_positive_dataabsoluteevaluationhsf_prefix_an + S (ff_an_mce_alternating_mdr_first_positive_dataabsoluteevaluationhsf_prefix) = S ((S (ff_index_mce_alternating_mdr_first_positive_dataabsoluteevaluationhsf_prefix)) * mdr_nc_first_positive_dataabsoluteevaluationh)) /\ exists ff_q_mce_mdr_first_positive_dataabsoluteevaluationhsf_prefix_an. mdr_nb_first_positive_dataabsoluteevaluationh = ff_q_mce_mdr_first_positive_dataabsoluteevaluationhsf_prefix_an * S ((S (ff_index_mce_alternating_mdr_first_positive_dataabsoluteevaluationhsf_prefix)) * mdr_nc_first_positive_dataabsoluteevaluationh) + (ff_an_mce_alternating_mdr_first_positive_dataabsoluteevaluationhsf_prefix))) /\ ((((exists ff_h_mce_mdr_first_positive_dataabsoluteevaluationhsf_prefix_bp. ff_h_mce_mdr_first_positive_dataabsoluteevaluationhsf_prefix_bp + S (ff_bp_mce_alternating_mdr_first_positive_dataabsoluteevaluationhsf_prefix) = S ((S (ff_index_mce_alternating_mdr_first_positive_dataabsoluteevaluationhsf_prefix)) * mdr_ec_first_positive_dataabsoluteevaluationhs)) /\ exists ff_q_mce_mdr_first_positive_dataabsoluteevaluationhsf_prefix_bp. mdr_eb_first_positive_dataabsoluteevaluationhs = ff_q_mce_mdr_first_positive_dataabsoluteevaluationhsf_prefix_bp * S ((S (ff_index_mce_alternating_mdr_first_positive_dataabsoluteevaluationhsf_prefix)) * mdr_ec_first_positive_dataabsoluteevaluationhs) + (ff_bp_mce_alternating_mdr_first_positive_dataabsoluteevaluationhsf_prefix))) /\ ((((exists ff_h_mce_mdr_first_positive_dataabsoluteevaluationhsf_prefix_bn. ff_h_mce_mdr_first_positive_dataabsoluteevaluationhsf_prefix_bn + S (ff_bn_mce_alternating_mdr_first_positive_dataabsoluteevaluationhsf_prefix) = S ((S (ff_index_mce_alternating_mdr_first_positive_dataabsoluteevaluationhsf_prefix)) * mdr_fc_first_positive_dataabsoluteevaluationhs)) /\ exists ff_q_mce_mdr_first_positive_dataabsoluteevaluationhsf_prefix_bn. mdr_fb_first_positive_dataabsoluteevaluationhs = ff_q_mce_mdr_first_positive_dataabsoluteevaluationhsf_prefix_bn * S ((S (ff_index_mce_alternating_mdr_first_positive_dataabsoluteevaluationhsf_prefix)) * mdr_fc_first_positive_dataabsoluteevaluationhs) + (ff_bn_mce_alternating_mdr_first_positive_dataabsoluteevaluationhsf_prefix))) /\ ((((exists ff_h_mce_mdr_first_positive_dataabsoluteevaluationhsf_prefix_positive. ff_h_mce_mdr_first_positive_dataabsoluteevaluationhsf_prefix_positive + S (ff_p_mce_alternating_mdr_first_positive_dataabsoluteevaluationhsf_prefix) = S ((S (ff_index_mce_alternating_mdr_first_positive_dataabsoluteevaluationhsf_prefix)) * ff_uc_mce_fold_mdr_first_positive_dataabsoluteevaluationhsf)) /\ exists ff_q_mce_mdr_first_positive_dataabsoluteevaluationhsf_prefix_positive. ff_ub_mce_fold_mdr_first_positive_dataabsoluteevaluationhsf = ff_q_mce_mdr_first_positive_dataabsoluteevaluationhsf_prefix_positive * S ((S (ff_index_mce_alternating_mdr_first_positive_dataabsoluteevaluationhsf_prefix)) * ff_uc_mce_fold_mdr_first_positive_dataabsoluteevaluationhsf) + (ff_p_mce_alternating_mdr_first_positive_dataabsoluteevaluationhsf_prefix))) /\ ((((exists ff_h_mce_mdr_first_positive_dataabsoluteevaluationhsf_prefix_negative. ff_h_mce_mdr_first_positive_dataabsoluteevaluationhsf_prefix_negative + S (ff_n_mce_alternating_mdr_first_positive_dataabsoluteevaluationhsf_prefix) = S ((S (ff_index_mce_alternating_mdr_first_positive_dataabsoluteevaluationhsf_prefix)) * ff_vc_mce_fold_mdr_first_positive_dataabsoluteevaluationhsf)) /\ exists ff_q_mce_mdr_first_positive_dataabsoluteevaluationhsf_prefix_negative. ff_vb_mce_fold_mdr_first_positive_dataabsoluteevaluationhsf = ff_q_mce_mdr_first_positive_dataabsoluteevaluationhsf_prefix_negative * S ((S (ff_index_mce_alternating_mdr_first_positive_dataabsoluteevaluationhsf_prefix)) * ff_vc_mce_fold_mdr_first_positive_dataabsoluteevaluationhsf) + (ff_n_mce_alternating_mdr_first_positive_dataabsoluteevaluationhsf_prefix))) /\ (((exists ff_even_mce_term_mdr_first_positive_dataabsoluteevaluationhsf_prefix_term. ff_index_mce_alternating_mdr_first_positive_dataabsoluteevaluationhsf_prefix = 2 * ff_even_mce_term_mdr_first_positive_dataabsoluteevaluationhsf_prefix_term) /\ (ff_p_mce_alternating_mdr_first_positive_dataabsoluteevaluationhsf_prefix = (ff_ap_mce_alternating_mdr_first_positive_dataabsoluteevaluationhsf_prefix) * (ff_bp_mce_alternating_mdr_first_positive_dataabsoluteevaluationhsf_prefix) + (ff_an_mce_alternating_mdr_first_positive_dataabsoluteevaluationhsf_prefix) * (ff_bn_mce_alternating_mdr_first_positive_dataabsoluteevaluationhsf_prefix) /\ ff_n_mce_alternating_mdr_first_positive_dataabsoluteevaluationhsf_prefix = (ff_ap_mce_alternating_mdr_first_positive_dataabsoluteevaluationhsf_prefix) * (ff_bn_mce_alternating_mdr_first_positive_dataabsoluteevaluationhsf_prefix) + (ff_an_mce_alternating_mdr_first_positive_dataabsoluteevaluationhsf_prefix) * (ff_bp_mce_alternating_mdr_first_positive_dataabsoluteevaluationhsf_prefix))) \/ ((exists ff_odd_mce_term_mdr_first_positive_dataabsoluteevaluationhsf_prefix_term. ff_index_mce_alternating_mdr_first_positive_dataabsoluteevaluationhsf_prefix = 2 * ff_odd_mce_term_mdr_first_positive_dataabsoluteevaluationhsf_prefix_term + 1) /\ (ff_p_mce_alternating_mdr_first_positive_dataabsoluteevaluationhsf_prefix = (ff_ap_mce_alternating_mdr_first_positive_dataabsoluteevaluationhsf_prefix) * (ff_bn_mce_alternating_mdr_first_positive_dataabsoluteevaluationhsf_prefix) + (ff_an_mce_alternating_mdr_first_positive_dataabsoluteevaluationhsf_prefix) * (ff_bp_mce_alternating_mdr_first_positive_dataabsoluteevaluationhsf_prefix) /\ ff_n_mce_alternating_mdr_first_positive_dataabsoluteevaluationhsf_prefix = (ff_ap_mce_alternating_mdr_first_positive_dataabsoluteevaluationhsf_prefix) * (ff_bp_mce_alternating_mdr_first_positive_dataabsoluteevaluationhsf_prefix) + (ff_an_mce_alternating_mdr_first_positive_dataabsoluteevaluationhsf_prefix) * (ff_bn_mce_alternating_mdr_first_positive_dataabsoluteevaluationhsf_prefix))))))))))) /\ ((exists ff_u_mce_mdr_first_positive_dataabsoluteevaluationhsf_positive ff_v_mce_mdr_first_positive_dataabsoluteevaluationhsf_positive. ((((exists ff_h_mce_mdr_first_positive_dataabsoluteevaluationhsf_positive_start. ff_h_mce_mdr_first_positive_dataabsoluteevaluationhsf_positive_start + S (0) = S ((S (0)) * ff_v_mce_mdr_first_positive_dataabsoluteevaluationhsf_positive)) /\ exists ff_q_mce_mdr_first_positive_dataabsoluteevaluationhsf_positive_start. ff_u_mce_mdr_first_positive_dataabsoluteevaluationhsf_positive = ff_q_mce_mdr_first_positive_dataabsoluteevaluationhsf_positive_start * S ((S (0)) * ff_v_mce_mdr_first_positive_dataabsoluteevaluationhsf_positive) + (0))) /\ ((((exists ff_h_mce_mdr_first_positive_dataabsoluteevaluationhsf_positive_terminal. ff_h_mce_mdr_first_positive_dataabsoluteevaluationhsf_positive_terminal + S (mdr_p_first_positive_dataabsoluteevaluationh) = S ((S ((S (mdr_q_first_positive_dataabsoluteevaluationhs)))) * ff_v_mce_mdr_first_positive_dataabsoluteevaluationhsf_positive)) /\ exists ff_q_mce_mdr_first_positive_dataabsoluteevaluationhsf_positive_terminal. ff_u_mce_mdr_first_positive_dataabsoluteevaluationhsf_positive = ff_q_mce_mdr_first_positive_dataabsoluteevaluationhsf_positive_terminal * S ((S ((S (mdr_q_first_positive_dataabsoluteevaluationhs)))) * ff_v_mce_mdr_first_positive_dataabsoluteevaluationhsf_positive) + (mdr_p_first_positive_dataabsoluteevaluationh))) /\ forall ff_i_mce_mdr_first_positive_dataabsoluteevaluationhsf_positive. (exists ff_lt_mce_mdr_first_positive_dataabsoluteevaluationhsf_positive_bound. ff_lt_mce_mdr_first_positive_dataabsoluteevaluationhsf_positive_bound + S ff_i_mce_mdr_first_positive_dataabsoluteevaluationhsf_positive = (S (mdr_q_first_positive_dataabsoluteevaluationhs))) -> exists ff_a_mce_mdr_first_positive_dataabsoluteevaluationhsf_positive ff_r_mce_mdr_first_positive_dataabsoluteevaluationhsf_positive ff_s_mce_mdr_first_positive_dataabsoluteevaluationhsf_positive. ((((exists ff_h_mce_mdr_first_positive_dataabsoluteevaluationhsf_positive_summand. ff_h_mce_mdr_first_positive_dataabsoluteevaluationhsf_positive_summand + S (ff_a_mce_mdr_first_positive_dataabsoluteevaluationhsf_positive) = S ((S (ff_i_mce_mdr_first_positive_dataabsoluteevaluationhsf_positive)) * ff_uc_mce_fold_mdr_first_positive_dataabsoluteevaluationhsf)) /\ exists ff_q_mce_mdr_first_positive_dataabsoluteevaluationhsf_positive_summand. ff_ub_mce_fold_mdr_first_positive_dataabsoluteevaluationhsf = ff_q_mce_mdr_first_positive_dataabsoluteevaluationhsf_positive_summand * S ((S (ff_i_mce_mdr_first_positive_dataabsoluteevaluationhsf_positive)) * ff_uc_mce_fold_mdr_first_positive_dataabsoluteevaluationhsf) + (ff_a_mce_mdr_first_positive_dataabsoluteevaluationhsf_positive))) /\ ((((exists ff_h_mce_mdr_first_positive_dataabsoluteevaluationhsf_positive_partial. ff_h_mce_mdr_first_positive_dataabsoluteevaluationhsf_positive_partial + S (ff_r_mce_mdr_first_positive_dataabsoluteevaluationhsf_positive) = S ((S (ff_i_mce_mdr_first_positive_dataabsoluteevaluationhsf_positive)) * ff_v_mce_mdr_first_positive_dataabsoluteevaluationhsf_positive)) /\ exists ff_q_mce_mdr_first_positive_dataabsoluteevaluationhsf_positive_partial. ff_u_mce_mdr_first_positive_dataabsoluteevaluationhsf_positive = ff_q_mce_mdr_first_positive_dataabsoluteevaluationhsf_positive_partial * S ((S (ff_i_mce_mdr_first_positive_dataabsoluteevaluationhsf_positive)) * ff_v_mce_mdr_first_positive_dataabsoluteevaluationhsf_positive) + (ff_r_mce_mdr_first_positive_dataabsoluteevaluationhsf_positive))) /\ ((((exists ff_h_mce_mdr_first_positive_dataabsoluteevaluationhsf_positive_successor. ff_h_mce_mdr_first_positive_dataabsoluteevaluationhsf_positive_successor + S (ff_s_mce_mdr_first_positive_dataabsoluteevaluationhsf_positive) = S ((S (S ff_i_mce_mdr_first_positive_dataabsoluteevaluationhsf_positive)) * ff_v_mce_mdr_first_positive_dataabsoluteevaluationhsf_positive)) /\ exists ff_q_mce_mdr_first_positive_dataabsoluteevaluationhsf_positive_successor. ff_u_mce_mdr_first_positive_dataabsoluteevaluationhsf_positive = ff_q_mce_mdr_first_positive_dataabsoluteevaluationhsf_positive_successor * S ((S (S ff_i_mce_mdr_first_positive_dataabsoluteevaluationhsf_positive)) * ff_v_mce_mdr_first_positive_dataabsoluteevaluationhsf_positive) + (ff_s_mce_mdr_first_positive_dataabsoluteevaluationhsf_positive))) /\ ff_s_mce_mdr_first_positive_dataabsoluteevaluationhsf_positive = ff_r_mce_mdr_first_positive_dataabsoluteevaluationhsf_positive + ff_a_mce_mdr_first_positive_dataabsoluteevaluationhsf_positive)))))) /\ (exists ff_u_mce_mdr_first_positive_dataabsoluteevaluationhsf_negative ff_v_mce_mdr_first_positive_dataabsoluteevaluationhsf_negative. ((((exists ff_h_mce_mdr_first_positive_dataabsoluteevaluationhsf_negative_start. ff_h_mce_mdr_first_positive_dataabsoluteevaluationhsf_negative_start + S (0) = S ((S (0)) * ff_v_mce_mdr_first_positive_dataabsoluteevaluationhsf_negative)) /\ exists ff_q_mce_mdr_first_positive_dataabsoluteevaluationhsf_negative_start. ff_u_mce_mdr_first_positive_dataabsoluteevaluationhsf_negative = ff_q_mce_mdr_first_positive_dataabsoluteevaluationhsf_negative_start * S ((S (0)) * ff_v_mce_mdr_first_positive_dataabsoluteevaluationhsf_negative) + (0))) /\ ((((exists ff_h_mce_mdr_first_positive_dataabsoluteevaluationhsf_negative_terminal. ff_h_mce_mdr_first_positive_dataabsoluteevaluationhsf_negative_terminal + S (mdr_n_first_positive_dataabsoluteevaluationh) = S ((S ((S (mdr_q_first_positive_dataabsoluteevaluationhs)))) * ff_v_mce_mdr_first_positive_dataabsoluteevaluationhsf_negative)) /\ exists ff_q_mce_mdr_first_positive_dataabsoluteevaluationhsf_negative_terminal. ff_u_mce_mdr_first_positive_dataabsoluteevaluationhsf_negative = ff_q_mce_mdr_first_positive_dataabsoluteevaluationhsf_negative_terminal * S ((S ((S (mdr_q_first_positive_dataabsoluteevaluationhs)))) * ff_v_mce_mdr_first_positive_dataabsoluteevaluationhsf_negative) + (mdr_n_first_positive_dataabsoluteevaluationh))) /\ forall ff_i_mce_mdr_first_positive_dataabsoluteevaluationhsf_negative. (exists ff_lt_mce_mdr_first_positive_dataabsoluteevaluationhsf_negative_bound. ff_lt_mce_mdr_first_positive_dataabsoluteevaluationhsf_negative_bound + S ff_i_mce_mdr_first_positive_dataabsoluteevaluationhsf_negative = (S (mdr_q_first_positive_dataabsoluteevaluationhs))) -> exists ff_a_mce_mdr_first_positive_dataabsoluteevaluationhsf_negative ff_r_mce_mdr_first_positive_dataabsoluteevaluationhsf_negative ff_s_mce_mdr_first_positive_dataabsoluteevaluationhsf_negative. ((((exists ff_h_mce_mdr_first_positive_dataabsoluteevaluationhsf_negative_summand. ff_h_mce_mdr_first_positive_dataabsoluteevaluationhsf_negative_summand + S (ff_a_mce_mdr_first_positive_dataabsoluteevaluationhsf_negative) = S ((S (ff_i_mce_mdr_first_positive_dataabsoluteevaluationhsf_negative)) * ff_vc_mce_fold_mdr_first_positive_dataabsoluteevaluationhsf)) /\ exists ff_q_mce_mdr_first_positive_dataabsoluteevaluationhsf_negative_summand. ff_vb_mce_fold_mdr_first_positive_dataabsoluteevaluationhsf = ff_q_mce_mdr_first_positive_dataabsoluteevaluationhsf_negative_summand * S ((S (ff_i_mce_mdr_first_positive_dataabsoluteevaluationhsf_negative)) * ff_vc_mce_fold_mdr_first_positive_dataabsoluteevaluationhsf) + (ff_a_mce_mdr_first_positive_dataabsoluteevaluationhsf_negative))) /\ ((((exists ff_h_mce_mdr_first_positive_dataabsoluteevaluationhsf_negative_partial. ff_h_mce_mdr_first_positive_dataabsoluteevaluationhsf_negative_partial + S (ff_r_mce_mdr_first_positive_dataabsoluteevaluationhsf_negative) = S ((S (ff_i_mce_mdr_first_positive_dataabsoluteevaluationhsf_negative)) * ff_v_mce_mdr_first_positive_dataabsoluteevaluationhsf_negative)) /\ exists ff_q_mce_mdr_first_positive_dataabsoluteevaluationhsf_negative_partial. ff_u_mce_mdr_first_positive_dataabsoluteevaluationhsf_negative = ff_q_mce_mdr_first_positive_dataabsoluteevaluationhsf_negative_partial * S ((S (ff_i_mce_mdr_first_positive_dataabsoluteevaluationhsf_negative)) * ff_v_mce_mdr_first_positive_dataabsoluteevaluationhsf_negative) + (ff_r_mce_mdr_first_positive_dataabsoluteevaluationhsf_negative))) /\ ((((exists ff_h_mce_mdr_first_positive_dataabsoluteevaluationhsf_negative_successor. ff_h_mce_mdr_first_positive_dataabsoluteevaluationhsf_negative_successor + S (ff_s_mce_mdr_first_positive_dataabsoluteevaluationhsf_negative) = S ((S (S ff_i_mce_mdr_first_positive_dataabsoluteevaluationhsf_negative)) * ff_v_mce_mdr_first_positive_dataabsoluteevaluationhsf_negative)) /\ exists ff_q_mce_mdr_first_positive_dataabsoluteevaluationhsf_negative_successor. ff_u_mce_mdr_first_positive_dataabsoluteevaluationhsf_negative = ff_q_mce_mdr_first_positive_dataabsoluteevaluationhsf_negative_successor * S ((S (S ff_i_mce_mdr_first_positive_dataabsoluteevaluationhsf_negative)) * ff_v_mce_mdr_first_positive_dataabsoluteevaluationhsf_negative) + (ff_s_mce_mdr_first_positive_dataabsoluteevaluationhsf_negative))) /\ ff_s_mce_mdr_first_positive_dataabsoluteevaluationhsf_negative = ff_r_mce_mdr_first_positive_dataabsoluteevaluationhsf_negative + ff_a_mce_mdr_first_positive_dataabsoluteevaluationhsf_negative))))))))))))))) /\ ((exists mdr_gap_first_positive_dataabsoluteevaluationi. mdr_gap_first_positive_dataabsoluteevaluationi + S (mdr_i_first_positive_dataabsoluteevaluation) = (mdr_l_first_positive_dataabsoluteevaluation)) /\ (exists mdr_z_first_positive_dataabsoluteevaluationr. ((exists mdr_a_first_positive_dataabsoluteevaluationrc mdr_b_first_positive_dataabsoluteevaluationrc mdr_c_first_positive_dataabsoluteevaluationrc mdr_e_first_positive_dataabsoluteevaluationrc mdr_f_first_positive_dataabsoluteevaluationrc. ((mdr_a_first_positive_dataabsoluteevaluationrc = ((d) + (ab)) * S ((d) + (ab)) + ((ab) + (ab))) /\ ((mdr_b_first_positive_dataabsoluteevaluationrc = ((ac) + (bb)) * S ((ac) + (bb)) + ((bb) + (bb))) /\ ((mdr_c_first_positive_dataabsoluteevaluationrc = ((mdr_a_first_positive_dataabsoluteevaluationrc) + (mdr_b_first_positive_dataabsoluteevaluationrc)) * S ((mdr_a_first_positive_dataabsoluteevaluationrc) + (mdr_b_first_positive_dataabsoluteevaluationrc)) + ((mdr_b_first_positive_dataabsoluteevaluationrc) + (mdr_b_first_positive_dataabsoluteevaluationrc))) /\ ((mdr_e_first_positive_dataabsoluteevaluationrc = ((mdr_p_first_positive_dataabsolute) + (mdr_n_first_positive_dataabsolute)) * S ((mdr_p_first_positive_dataabsolute) + (mdr_n_first_positive_dataabsolute)) + ((mdr_n_first_positive_dataabsolute) + (mdr_n_first_positive_dataabsolute))) /\ ((mdr_f_first_positive_dataabsoluteevaluationrc = ((bc) + (mdr_e_first_positive_dataabsoluteevaluationrc)) * S ((bc) + (mdr_e_first_positive_dataabsoluteevaluationrc)) + ((mdr_e_first_positive_dataabsoluteevaluationrc) + (mdr_e_first_positive_dataabsoluteevaluationrc))) /\ ((mdr_z_first_positive_dataabsoluteevaluationr) = ((mdr_c_first_positive_dataabsoluteevaluationrc) + (mdr_f_first_positive_dataabsoluteevaluationrc)) * S ((mdr_c_first_positive_dataabsoluteevaluationrc) + (mdr_f_first_positive_dataabsoluteevaluationrc)) + ((mdr_f_first_positive_dataabsoluteevaluationrc) + (mdr_f_first_positive_dataabsoluteevaluationrc))))))))) /\ (((exists ff_h_mdr_first_positive_dataabsoluteevaluationrb. ff_h_mdr_first_positive_dataabsoluteevaluationrb + S (mdr_z_first_positive_dataabsoluteevaluationr) = S ((S (mdr_i_first_positive_dataabsoluteevaluation)) * mdr_c_first_positive_dataabsoluteevaluation)) /\ exists ff_q_mdr_first_positive_dataabsoluteevaluationrb. mdr_b_first_positive_dataabsoluteevaluation = ff_q_mdr_first_positive_dataabsoluteevaluationrb * S ((S (mdr_i_first_positive_dataabsoluteevaluation)) * mdr_c_first_positive_dataabsoluteevaluation) + (mdr_z_first_positive_dataabsoluteevaluationr)))))))) /\ (((mdr_p_first_positive_dataabsolute) = (mdr_n_first_positive_dataabsolute) + (D)) \/ ((mdr_n_first_positive_dataabsolute) = (mdr_p_first_positive_dataabsolute) + (D)))))))) -> (((~(d = 0)) /\ ((~(E = 0)) /\ (exists mdr_p_second_positive_dataabsolute mdr_n_second_positive_dataabsolute. ((exists mdr_b_second_positive_dataabsoluteevaluation mdr_c_second_positive_dataabsoluteevaluation mdr_l_second_positive_dataabsoluteevaluation mdr_i_second_positive_dataabsoluteevaluation. ((forall mdr_i_second_positive_dataabsoluteevaluationh. (exists mdr_gap_second_positive_dataabsoluteevaluationhi. mdr_gap_second_positive_dataabsoluteevaluationhi + S (mdr_i_second_positive_dataabsoluteevaluationh) = (mdr_l_second_positive_dataabsoluteevaluation)) -> exists mdr_d_second_positive_dataabsoluteevaluationh mdr_pb_second_positive_dataabsoluteevaluationh mdr_pc_second_positive_dataabsoluteevaluationh mdr_nb_second_positive_dataabsoluteevaluationh mdr_nc_second_positive_dataabsoluteevaluationh mdr_p_second_positive_dataabsoluteevaluationh mdr_n_second_positive_dataabsoluteevaluationh. ((exists mdr_z_second_positive_dataabsoluteevaluationhr. ((exists mdr_a_second_positive_dataabsoluteevaluationhrc mdr_b_second_positive_dataabsoluteevaluationhrc mdr_c_second_positive_dataabsoluteevaluationhrc mdr_e_second_positive_dataabsoluteevaluationhrc mdr_f_second_positive_dataabsoluteevaluationhrc. ((mdr_a_second_positive_dataabsoluteevaluationhrc = ((mdr_d_second_positive_dataabsoluteevaluationh) + (mdr_pb_second_positive_dataabsoluteevaluationh)) * S ((mdr_d_second_positive_dataabsoluteevaluationh) + (mdr_pb_second_positive_dataabsoluteevaluationh)) + ((mdr_pb_second_positive_dataabsoluteevaluationh) + (mdr_pb_second_positive_dataabsoluteevaluationh))) /\ ((mdr_b_second_positive_dataabsoluteevaluationhrc = ((mdr_pc_second_positive_dataabsoluteevaluationh) + (mdr_nb_second_positive_dataabsoluteevaluationh)) * S ((mdr_pc_second_positive_dataabsoluteevaluationh) + (mdr_nb_second_positive_dataabsoluteevaluationh)) + ((mdr_nb_second_positive_dataabsoluteevaluationh) + (mdr_nb_second_positive_dataabsoluteevaluationh))) /\ ((mdr_c_second_positive_dataabsoluteevaluationhrc = ((mdr_a_second_positive_dataabsoluteevaluationhrc) + (mdr_b_second_positive_dataabsoluteevaluationhrc)) * S ((mdr_a_second_positive_dataabsoluteevaluationhrc) + (mdr_b_second_positive_dataabsoluteevaluationhrc)) + ((mdr_b_second_positive_dataabsoluteevaluationhrc) + (mdr_b_second_positive_dataabsoluteevaluationhrc))) /\ ((mdr_e_second_positive_dataabsoluteevaluationhrc = ((mdr_p_second_positive_dataabsoluteevaluationh) + (mdr_n_second_positive_dataabsoluteevaluationh)) * S ((mdr_p_second_positive_dataabsoluteevaluationh) + (mdr_n_second_positive_dataabsoluteevaluationh)) + ((mdr_n_second_positive_dataabsoluteevaluationh) + (mdr_n_second_positive_dataabsoluteevaluationh))) /\ ((mdr_f_second_positive_dataabsoluteevaluationhrc = ((mdr_nc_second_positive_dataabsoluteevaluationh) + (mdr_e_second_positive_dataabsoluteevaluationhrc)) * S ((mdr_nc_second_positive_dataabsoluteevaluationh) + (mdr_e_second_positive_dataabsoluteevaluationhrc)) + ((mdr_e_second_positive_dataabsoluteevaluationhrc) + (mdr_e_second_positive_dataabsoluteevaluationhrc))) /\ ((mdr_z_second_positive_dataabsoluteevaluationhr) = ((mdr_c_second_positive_dataabsoluteevaluationhrc) + (mdr_f_second_positive_dataabsoluteevaluationhrc)) * S ((mdr_c_second_positive_dataabsoluteevaluationhrc) + (mdr_f_second_positive_dataabsoluteevaluationhrc)) + ((mdr_f_second_positive_dataabsoluteevaluationhrc) + (mdr_f_second_positive_dataabsoluteevaluationhrc))))))))) /\ (((exists ff_h_mdr_second_positive_dataabsoluteevaluationhrb. ff_h_mdr_second_positive_dataabsoluteevaluationhrb + S (mdr_z_second_positive_dataabsoluteevaluationhr) = S ((S (mdr_i_second_positive_dataabsoluteevaluationh)) * mdr_c_second_positive_dataabsoluteevaluation)) /\ exists ff_q_mdr_second_positive_dataabsoluteevaluationhrb. mdr_b_second_positive_dataabsoluteevaluation = ff_q_mdr_second_positive_dataabsoluteevaluationhrb * S ((S (mdr_i_second_positive_dataabsoluteevaluationh)) * mdr_c_second_positive_dataabsoluteevaluation) + (mdr_z_second_positive_dataabsoluteevaluationhr))))) /\ (((((mdr_d_second_positive_dataabsoluteevaluationh) = 0) /\ (((mdr_p_second_positive_dataabsoluteevaluationh) = 1) /\ ((mdr_n_second_positive_dataabsoluteevaluationh) = 0))) \/ exists mdr_q_second_positive_dataabsoluteevaluationhs mdr_eb_second_positive_dataabsoluteevaluationhs mdr_ec_second_positive_dataabsoluteevaluationhs mdr_fb_second_positive_dataabsoluteevaluationhs mdr_fc_second_positive_dataabsoluteevaluationhs. (((mdr_d_second_positive_dataabsoluteevaluationh) = S (mdr_q_second_positive_dataabsoluteevaluationhs)) /\ ((forall mdr_j_second_positive_dataabsoluteevaluationhsc. (exists mdr_gap_second_positive_dataabsoluteevaluationhscj. mdr_gap_second_positive_dataabsoluteevaluationhscj + S (mdr_j_second_positive_dataabsoluteevaluationhsc) = (S (mdr_q_second_positive_dataabsoluteevaluationhs))) -> exists mdr_i_second_positive_dataabsoluteevaluationhsc mdr_up_second_positive_dataabsoluteevaluationhsc mdr_us_second_positive_dataabsoluteevaluationhsc mdr_un_second_positive_dataabsoluteevaluationhsc mdr_ut_second_positive_dataabsoluteevaluationhsc mdr_p_second_positive_dataabsoluteevaluationhsc mdr_n_second_positive_dataabsoluteevaluationhsc. ((exists mdr_gap_second_positive_dataabsoluteevaluationhsci. mdr_gap_second_positive_dataabsoluteevaluationhsci + S (mdr_i_second_positive_dataabsoluteevaluationhsc) = (mdr_i_second_positive_dataabsoluteevaluationh)) /\ ((exists mdr_z_second_positive_dataabsoluteevaluationhscr. ((exists mdr_a_second_positive_dataabsoluteevaluationhscrc mdr_b_second_positive_dataabsoluteevaluationhscrc mdr_c_second_positive_dataabsoluteevaluationhscrc mdr_e_second_positive_dataabsoluteevaluationhscrc mdr_f_second_positive_dataabsoluteevaluationhscrc. ((mdr_a_second_positive_dataabsoluteevaluationhscrc = ((mdr_q_second_positive_dataabsoluteevaluationhs) + (mdr_up_second_positive_dataabsoluteevaluationhsc)) * S ((mdr_q_second_positive_dataabsoluteevaluationhs) + (mdr_up_second_positive_dataabsoluteevaluationhsc)) + ((mdr_up_second_positive_dataabsoluteevaluationhsc) + (mdr_up_second_positive_dataabsoluteevaluationhsc))) /\ ((mdr_b_second_positive_dataabsoluteevaluationhscrc = ((mdr_us_second_positive_dataabsoluteevaluationhsc) + (mdr_un_second_positive_dataabsoluteevaluationhsc)) * S ((mdr_us_second_positive_dataabsoluteevaluationhsc) + (mdr_un_second_positive_dataabsoluteevaluationhsc)) + ((mdr_un_second_positive_dataabsoluteevaluationhsc) + (mdr_un_second_positive_dataabsoluteevaluationhsc))) /\ ((mdr_c_second_positive_dataabsoluteevaluationhscrc = ((mdr_a_second_positive_dataabsoluteevaluationhscrc) + (mdr_b_second_positive_dataabsoluteevaluationhscrc)) * S ((mdr_a_second_positive_dataabsoluteevaluationhscrc) + (mdr_b_second_positive_dataabsoluteevaluationhscrc)) + ((mdr_b_second_positive_dataabsoluteevaluationhscrc) + (mdr_b_second_positive_dataabsoluteevaluationhscrc))) /\ ((mdr_e_second_positive_dataabsoluteevaluationhscrc = ((mdr_p_second_positive_dataabsoluteevaluationhsc) + (mdr_n_second_positive_dataabsoluteevaluationhsc)) * S ((mdr_p_second_positive_dataabsoluteevaluationhsc) + (mdr_n_second_positive_dataabsoluteevaluationhsc)) + ((mdr_n_second_positive_dataabsoluteevaluationhsc) + (mdr_n_second_positive_dataabsoluteevaluationhsc))) /\ ((mdr_f_second_positive_dataabsoluteevaluationhscrc = ((mdr_ut_second_positive_dataabsoluteevaluationhsc) + (mdr_e_second_positive_dataabsoluteevaluationhscrc)) * S ((mdr_ut_second_positive_dataabsoluteevaluationhsc) + (mdr_e_second_positive_dataabsoluteevaluationhscrc)) + ((mdr_e_second_positive_dataabsoluteevaluationhscrc) + (mdr_e_second_positive_dataabsoluteevaluationhscrc))) /\ ((mdr_z_second_positive_dataabsoluteevaluationhscr) = ((mdr_c_second_positive_dataabsoluteevaluationhscrc) + (mdr_f_second_positive_dataabsoluteevaluationhscrc)) * S ((mdr_c_second_positive_dataabsoluteevaluationhscrc) + (mdr_f_second_positive_dataabsoluteevaluationhscrc)) + ((mdr_f_second_positive_dataabsoluteevaluationhscrc) + (mdr_f_second_positive_dataabsoluteevaluationhscrc))))))))) /\ (((exists ff_h_mdr_second_positive_dataabsoluteevaluationhscrb. ff_h_mdr_second_positive_dataabsoluteevaluationhscrb + S (mdr_z_second_positive_dataabsoluteevaluationhscr) = S ((S (mdr_i_second_positive_dataabsoluteevaluationhsc)) * mdr_c_second_positive_dataabsoluteevaluation)) /\ exists ff_q_mdr_second_positive_dataabsoluteevaluationhscrb. mdr_b_second_positive_dataabsoluteevaluation = ff_q_mdr_second_positive_dataabsoluteevaluationhscrb * S ((S (mdr_i_second_positive_dataabsoluteevaluationhsc)) * mdr_c_second_positive_dataabsoluteevaluation) + (mdr_z_second_positive_dataabsoluteevaluationhscr))))) /\ ((((forall ff_index_mdm_prefix_mdr_second_positive_dataabsoluteevaluationhscm_positive. (exists ff_gap_mdm_lt_mdr_second_positive_dataabsoluteevaluationhscm_positive_index_bound. ff_gap_mdm_lt_mdr_second_positive_dataabsoluteevaluationhscm_positive_index_bound + S (ff_index_mdm_prefix_mdr_second_positive_dataabsoluteevaluationhscm_positive) = ((mdr_q_second_positive_dataabsoluteevaluationhs) * (mdr_q_second_positive_dataabsoluteevaluationhs))) -> exists ff_row_mdm_prefix_mdr_second_positive_dataabsoluteevaluationhscm_positive ff_column_mdm_prefix_mdr_second_positive_dataabsoluteevaluationhscm_positive ff_value_mdm_prefix_mdr_second_positive_dataabsoluteevaluationhscm_positive. (ff_index_mdm_prefix_mdr_second_positive_dataabsoluteevaluationhscm_positive = (mdr_q_second_positive_dataabsoluteevaluationhs) * ff_row_mdm_prefix_mdr_second_positive_dataabsoluteevaluationhscm_positive + ff_column_mdm_prefix_mdr_second_positive_dataabsoluteevaluationhscm_positive /\ ((exists ff_gap_mdm_lt_mdr_second_positive_dataabsoluteevaluationhscm_positive_column_bound. ff_gap_mdm_lt_mdr_second_positive_dataabsoluteevaluationhscm_positive_column_bound + S (ff_column_mdm_prefix_mdr_second_positive_dataabsoluteevaluationhscm_positive) = (mdr_q_second_positive_dataabsoluteevaluationhs)) /\ ((exists ff_row_mdm_cell_mdr_second_positive_dataabsoluteevaluationhscm_positive_cell ff_column_mdm_cell_mdr_second_positive_dataabsoluteevaluationhscm_positive_cell. (((((exists ff_gap_mdm_lt_mdr_second_positive_dataabsoluteevaluationhscm_positive_cell_row_before. ff_gap_mdm_lt_mdr_second_positive_dataabsoluteevaluationhscm_positive_cell_row_before + S (ff_row_mdm_prefix_mdr_second_positive_dataabsoluteevaluationhscm_positive) = (0)) /\ ff_row_mdm_cell_mdr_second_positive_dataabsoluteevaluationhscm_positive_cell = ff_row_mdm_prefix_mdr_second_positive_dataabsoluteevaluationhscm_positive) \/ ((exists ff_gap_mdm_le_mdr_second_positive_dataabsoluteevaluationhscm_positive_cell_row_after. ff_gap_mdm_le_mdr_second_positive_dataabsoluteevaluationhscm_positive_cell_row_after + (0) = (ff_row_mdm_prefix_mdr_second_positive_dataabsoluteevaluationhscm_positive)) /\ ff_row_mdm_cell_mdr_second_positive_dataabsoluteevaluationhscm_positive_cell = S ff_row_mdm_prefix_mdr_second_positive_dataabsoluteevaluationhscm_positive))) /\ (((((exists ff_gap_mdm_lt_mdr_second_positive_dataabsoluteevaluationhscm_positive_cell_column_before. ff_gap_mdm_lt_mdr_second_positive_dataabsoluteevaluationhscm_positive_cell_column_before + S (ff_column_mdm_prefix_mdr_second_positive_dataabsoluteevaluationhscm_positive) = (mdr_j_second_positive_dataabsoluteevaluationhsc)) /\ ff_column_mdm_cell_mdr_second_positive_dataabsoluteevaluationhscm_positive_cell = ff_column_mdm_prefix_mdr_second_positive_dataabsoluteevaluationhscm_positive) \/ ((exists ff_gap_mdm_le_mdr_second_positive_dataabsoluteevaluationhscm_positive_cell_column_after. ff_gap_mdm_le_mdr_second_positive_dataabsoluteevaluationhscm_positive_cell_column_after + (mdr_j_second_positive_dataabsoluteevaluationhsc) = (ff_column_mdm_prefix_mdr_second_positive_dataabsoluteevaluationhscm_positive)) /\ ff_column_mdm_cell_mdr_second_positive_dataabsoluteevaluationhscm_positive_cell = S ff_column_mdm_prefix_mdr_second_positive_dataabsoluteevaluationhscm_positive))) /\ (((exists ff_h_mdm_mdr_second_positive_dataabsoluteevaluationhscm_positive_cell_source. ff_h_mdm_mdr_second_positive_dataabsoluteevaluationhscm_positive_cell_source + S (ff_value_mdm_prefix_mdr_second_positive_dataabsoluteevaluationhscm_positive) = S ((S ((ff_row_mdm_cell_mdr_second_positive_dataabsoluteevaluationhscm_positive_cell) * (S (mdr_q_second_positive_dataabsoluteevaluationhs)) + (ff_column_mdm_cell_mdr_second_positive_dataabsoluteevaluationhscm_positive_cell))) * mdr_pc_second_positive_dataabsoluteevaluationh)) /\ exists ff_q_mdm_mdr_second_positive_dataabsoluteevaluationhscm_positive_cell_source. mdr_pb_second_positive_dataabsoluteevaluationh = ff_q_mdm_mdr_second_positive_dataabsoluteevaluationhscm_positive_cell_source * S ((S ((ff_row_mdm_cell_mdr_second_positive_dataabsoluteevaluationhscm_positive_cell) * (S (mdr_q_second_positive_dataabsoluteevaluationhs)) + (ff_column_mdm_cell_mdr_second_positive_dataabsoluteevaluationhscm_positive_cell))) * mdr_pc_second_positive_dataabsoluteevaluationh) + (ff_value_mdm_prefix_mdr_second_positive_dataabsoluteevaluationhscm_positive)))))) /\ (((exists ff_h_mdm_mdr_second_positive_dataabsoluteevaluationhscm_positive_target. ff_h_mdm_mdr_second_positive_dataabsoluteevaluationhscm_positive_target + S (ff_value_mdm_prefix_mdr_second_positive_dataabsoluteevaluationhscm_positive) = S ((S (ff_index_mdm_prefix_mdr_second_positive_dataabsoluteevaluationhscm_positive)) * mdr_us_second_positive_dataabsoluteevaluationhsc)) /\ exists ff_q_mdm_mdr_second_positive_dataabsoluteevaluationhscm_positive_target. mdr_up_second_positive_dataabsoluteevaluationhsc = ff_q_mdm_mdr_second_positive_dataabsoluteevaluationhscm_positive_target * S ((S (ff_index_mdm_prefix_mdr_second_positive_dataabsoluteevaluationhscm_positive)) * mdr_us_second_positive_dataabsoluteevaluationhsc) + (ff_value_mdm_prefix_mdr_second_positive_dataabsoluteevaluationhscm_positive))))))) /\ (forall ff_index_mdm_prefix_mdr_second_positive_dataabsoluteevaluationhscm_negative. (exists ff_gap_mdm_lt_mdr_second_positive_dataabsoluteevaluationhscm_negative_index_bound. ff_gap_mdm_lt_mdr_second_positive_dataabsoluteevaluationhscm_negative_index_bound + S (ff_index_mdm_prefix_mdr_second_positive_dataabsoluteevaluationhscm_negative) = ((mdr_q_second_positive_dataabsoluteevaluationhs) * (mdr_q_second_positive_dataabsoluteevaluationhs))) -> exists ff_row_mdm_prefix_mdr_second_positive_dataabsoluteevaluationhscm_negative ff_column_mdm_prefix_mdr_second_positive_dataabsoluteevaluationhscm_negative ff_value_mdm_prefix_mdr_second_positive_dataabsoluteevaluationhscm_negative. (ff_index_mdm_prefix_mdr_second_positive_dataabsoluteevaluationhscm_negative = (mdr_q_second_positive_dataabsoluteevaluationhs) * ff_row_mdm_prefix_mdr_second_positive_dataabsoluteevaluationhscm_negative + ff_column_mdm_prefix_mdr_second_positive_dataabsoluteevaluationhscm_negative /\ ((exists ff_gap_mdm_lt_mdr_second_positive_dataabsoluteevaluationhscm_negative_column_bound. ff_gap_mdm_lt_mdr_second_positive_dataabsoluteevaluationhscm_negative_column_bound + S (ff_column_mdm_prefix_mdr_second_positive_dataabsoluteevaluationhscm_negative) = (mdr_q_second_positive_dataabsoluteevaluationhs)) /\ ((exists ff_row_mdm_cell_mdr_second_positive_dataabsoluteevaluationhscm_negative_cell ff_column_mdm_cell_mdr_second_positive_dataabsoluteevaluationhscm_negative_cell. (((((exists ff_gap_mdm_lt_mdr_second_positive_dataabsoluteevaluationhscm_negative_cell_row_before. ff_gap_mdm_lt_mdr_second_positive_dataabsoluteevaluationhscm_negative_cell_row_before + S (ff_row_mdm_prefix_mdr_second_positive_dataabsoluteevaluationhscm_negative) = (0)) /\ ff_row_mdm_cell_mdr_second_positive_dataabsoluteevaluationhscm_negative_cell = ff_row_mdm_prefix_mdr_second_positive_dataabsoluteevaluationhscm_negative) \/ ((exists ff_gap_mdm_le_mdr_second_positive_dataabsoluteevaluationhscm_negative_cell_row_after. ff_gap_mdm_le_mdr_second_positive_dataabsoluteevaluationhscm_negative_cell_row_after + (0) = (ff_row_mdm_prefix_mdr_second_positive_dataabsoluteevaluationhscm_negative)) /\ ff_row_mdm_cell_mdr_second_positive_dataabsoluteevaluationhscm_negative_cell = S ff_row_mdm_prefix_mdr_second_positive_dataabsoluteevaluationhscm_negative))) /\ (((((exists ff_gap_mdm_lt_mdr_second_positive_dataabsoluteevaluationhscm_negative_cell_column_before. ff_gap_mdm_lt_mdr_second_positive_dataabsoluteevaluationhscm_negative_cell_column_before + S (ff_column_mdm_prefix_mdr_second_positive_dataabsoluteevaluationhscm_negative) = (mdr_j_second_positive_dataabsoluteevaluationhsc)) /\ ff_column_mdm_cell_mdr_second_positive_dataabsoluteevaluationhscm_negative_cell = ff_column_mdm_prefix_mdr_second_positive_dataabsoluteevaluationhscm_negative) \/ ((exists ff_gap_mdm_le_mdr_second_positive_dataabsoluteevaluationhscm_negative_cell_column_after. ff_gap_mdm_le_mdr_second_positive_dataabsoluteevaluationhscm_negative_cell_column_after + (mdr_j_second_positive_dataabsoluteevaluationhsc) = (ff_column_mdm_prefix_mdr_second_positive_dataabsoluteevaluationhscm_negative)) /\ ff_column_mdm_cell_mdr_second_positive_dataabsoluteevaluationhscm_negative_cell = S ff_column_mdm_prefix_mdr_second_positive_dataabsoluteevaluationhscm_negative))) /\ (((exists ff_h_mdm_mdr_second_positive_dataabsoluteevaluationhscm_negative_cell_source. ff_h_mdm_mdr_second_positive_dataabsoluteevaluationhscm_negative_cell_source + S (ff_value_mdm_prefix_mdr_second_positive_dataabsoluteevaluationhscm_negative) = S ((S ((ff_row_mdm_cell_mdr_second_positive_dataabsoluteevaluationhscm_negative_cell) * (S (mdr_q_second_positive_dataabsoluteevaluationhs)) + (ff_column_mdm_cell_mdr_second_positive_dataabsoluteevaluationhscm_negative_cell))) * mdr_nc_second_positive_dataabsoluteevaluationh)) /\ exists ff_q_mdm_mdr_second_positive_dataabsoluteevaluationhscm_negative_cell_source. mdr_nb_second_positive_dataabsoluteevaluationh = ff_q_mdm_mdr_second_positive_dataabsoluteevaluationhscm_negative_cell_source * S ((S ((ff_row_mdm_cell_mdr_second_positive_dataabsoluteevaluationhscm_negative_cell) * (S (mdr_q_second_positive_dataabsoluteevaluationhs)) + (ff_column_mdm_cell_mdr_second_positive_dataabsoluteevaluationhscm_negative_cell))) * mdr_nc_second_positive_dataabsoluteevaluationh) + (ff_value_mdm_prefix_mdr_second_positive_dataabsoluteevaluationhscm_negative)))))) /\ (((exists ff_h_mdm_mdr_second_positive_dataabsoluteevaluationhscm_negative_target. ff_h_mdm_mdr_second_positive_dataabsoluteevaluationhscm_negative_target + S (ff_value_mdm_prefix_mdr_second_positive_dataabsoluteevaluationhscm_negative) = S ((S (ff_index_mdm_prefix_mdr_second_positive_dataabsoluteevaluationhscm_negative)) * mdr_ut_second_positive_dataabsoluteevaluationhsc)) /\ exists ff_q_mdm_mdr_second_positive_dataabsoluteevaluationhscm_negative_target. mdr_un_second_positive_dataabsoluteevaluationhsc = ff_q_mdm_mdr_second_positive_dataabsoluteevaluationhscm_negative_target * S ((S (ff_index_mdm_prefix_mdr_second_positive_dataabsoluteevaluationhscm_negative)) * mdr_ut_second_positive_dataabsoluteevaluationhsc) + (ff_value_mdm_prefix_mdr_second_positive_dataabsoluteevaluationhscm_negative))))))))) /\ ((((exists ff_h_mdr_second_positive_dataabsoluteevaluationhscp. ff_h_mdr_second_positive_dataabsoluteevaluationhscp + S (mdr_p_second_positive_dataabsoluteevaluationhsc) = S ((S (mdr_j_second_positive_dataabsoluteevaluationhsc)) * mdr_ec_second_positive_dataabsoluteevaluationhs)) /\ exists ff_q_mdr_second_positive_dataabsoluteevaluationhscp. mdr_eb_second_positive_dataabsoluteevaluationhs = ff_q_mdr_second_positive_dataabsoluteevaluationhscp * S ((S (mdr_j_second_positive_dataabsoluteevaluationhsc)) * mdr_ec_second_positive_dataabsoluteevaluationhs) + (mdr_p_second_positive_dataabsoluteevaluationhsc))) /\ (((exists ff_h_mdr_second_positive_dataabsoluteevaluationhscn. ff_h_mdr_second_positive_dataabsoluteevaluationhscn + S (mdr_n_second_positive_dataabsoluteevaluationhsc) = S ((S (mdr_j_second_positive_dataabsoluteevaluationhsc)) * mdr_fc_second_positive_dataabsoluteevaluationhs)) /\ exists ff_q_mdr_second_positive_dataabsoluteevaluationhscn. mdr_fb_second_positive_dataabsoluteevaluationhs = ff_q_mdr_second_positive_dataabsoluteevaluationhscn * S ((S (mdr_j_second_positive_dataabsoluteevaluationhsc)) * mdr_fc_second_positive_dataabsoluteevaluationhs) + (mdr_n_second_positive_dataabsoluteevaluationhsc)))))))) /\ (exists ff_ub_mce_fold_mdr_second_positive_dataabsoluteevaluationhsf ff_uc_mce_fold_mdr_second_positive_dataabsoluteevaluationhsf ff_vb_mce_fold_mdr_second_positive_dataabsoluteevaluationhsf ff_vc_mce_fold_mdr_second_positive_dataabsoluteevaluationhsf. ((forall ff_index_mce_alternating_mdr_second_positive_dataabsoluteevaluationhsf_prefix. (exists ff_gap_mce_mdr_second_positive_dataabsoluteevaluationhsf_prefix_index. ff_gap_mce_mdr_second_positive_dataabsoluteevaluationhsf_prefix_index + S (ff_index_mce_alternating_mdr_second_positive_dataabsoluteevaluationhsf_prefix) = (S (mdr_q_second_positive_dataabsoluteevaluationhs))) -> exists ff_ap_mce_alternating_mdr_second_positive_dataabsoluteevaluationhsf_prefix ff_an_mce_alternating_mdr_second_positive_dataabsoluteevaluationhsf_prefix ff_bp_mce_alternating_mdr_second_positive_dataabsoluteevaluationhsf_prefix ff_bn_mce_alternating_mdr_second_positive_dataabsoluteevaluationhsf_prefix ff_p_mce_alternating_mdr_second_positive_dataabsoluteevaluationhsf_prefix ff_n_mce_alternating_mdr_second_positive_dataabsoluteevaluationhsf_prefix. ((((exists ff_h_mce_mdr_second_positive_dataabsoluteevaluationhsf_prefix_ap. ff_h_mce_mdr_second_positive_dataabsoluteevaluationhsf_prefix_ap + S (ff_ap_mce_alternating_mdr_second_positive_dataabsoluteevaluationhsf_prefix) = S ((S (ff_index_mce_alternating_mdr_second_positive_dataabsoluteevaluationhsf_prefix)) * mdr_pc_second_positive_dataabsoluteevaluationh)) /\ exists ff_q_mce_mdr_second_positive_dataabsoluteevaluationhsf_prefix_ap. mdr_pb_second_positive_dataabsoluteevaluationh = ff_q_mce_mdr_second_positive_dataabsoluteevaluationhsf_prefix_ap * S ((S (ff_index_mce_alternating_mdr_second_positive_dataabsoluteevaluationhsf_prefix)) * mdr_pc_second_positive_dataabsoluteevaluationh) + (ff_ap_mce_alternating_mdr_second_positive_dataabsoluteevaluationhsf_prefix))) /\ ((((exists ff_h_mce_mdr_second_positive_dataabsoluteevaluationhsf_prefix_an. ff_h_mce_mdr_second_positive_dataabsoluteevaluationhsf_prefix_an + S (ff_an_mce_alternating_mdr_second_positive_dataabsoluteevaluationhsf_prefix) = S ((S (ff_index_mce_alternating_mdr_second_positive_dataabsoluteevaluationhsf_prefix)) * mdr_nc_second_positive_dataabsoluteevaluationh)) /\ exists ff_q_mce_mdr_second_positive_dataabsoluteevaluationhsf_prefix_an. mdr_nb_second_positive_dataabsoluteevaluationh = ff_q_mce_mdr_second_positive_dataabsoluteevaluationhsf_prefix_an * S ((S (ff_index_mce_alternating_mdr_second_positive_dataabsoluteevaluationhsf_prefix)) * mdr_nc_second_positive_dataabsoluteevaluationh) + (ff_an_mce_alternating_mdr_second_positive_dataabsoluteevaluationhsf_prefix))) /\ ((((exists ff_h_mce_mdr_second_positive_dataabsoluteevaluationhsf_prefix_bp. ff_h_mce_mdr_second_positive_dataabsoluteevaluationhsf_prefix_bp + S (ff_bp_mce_alternating_mdr_second_positive_dataabsoluteevaluationhsf_prefix) = S ((S (ff_index_mce_alternating_mdr_second_positive_dataabsoluteevaluationhsf_prefix)) * mdr_ec_second_positive_dataabsoluteevaluationhs)) /\ exists ff_q_mce_mdr_second_positive_dataabsoluteevaluationhsf_prefix_bp. mdr_eb_second_positive_dataabsoluteevaluationhs = ff_q_mce_mdr_second_positive_dataabsoluteevaluationhsf_prefix_bp * S ((S (ff_index_mce_alternating_mdr_second_positive_dataabsoluteevaluationhsf_prefix)) * mdr_ec_second_positive_dataabsoluteevaluationhs) + (ff_bp_mce_alternating_mdr_second_positive_dataabsoluteevaluationhsf_prefix))) /\ ((((exists ff_h_mce_mdr_second_positive_dataabsoluteevaluationhsf_prefix_bn. ff_h_mce_mdr_second_positive_dataabsoluteevaluationhsf_prefix_bn + S (ff_bn_mce_alternating_mdr_second_positive_dataabsoluteevaluationhsf_prefix) = S ((S (ff_index_mce_alternating_mdr_second_positive_dataabsoluteevaluationhsf_prefix)) * mdr_fc_second_positive_dataabsoluteevaluationhs)) /\ exists ff_q_mce_mdr_second_positive_dataabsoluteevaluationhsf_prefix_bn. mdr_fb_second_positive_dataabsoluteevaluationhs = ff_q_mce_mdr_second_positive_dataabsoluteevaluationhsf_prefix_bn * S ((S (ff_index_mce_alternating_mdr_second_positive_dataabsoluteevaluationhsf_prefix)) * mdr_fc_second_positive_dataabsoluteevaluationhs) + (ff_bn_mce_alternating_mdr_second_positive_dataabsoluteevaluationhsf_prefix))) /\ ((((exists ff_h_mce_mdr_second_positive_dataabsoluteevaluationhsf_prefix_positive. ff_h_mce_mdr_second_positive_dataabsoluteevaluationhsf_prefix_positive + S (ff_p_mce_alternating_mdr_second_positive_dataabsoluteevaluationhsf_prefix) = S ((S (ff_index_mce_alternating_mdr_second_positive_dataabsoluteevaluationhsf_prefix)) * ff_uc_mce_fold_mdr_second_positive_dataabsoluteevaluationhsf)) /\ exists ff_q_mce_mdr_second_positive_dataabsoluteevaluationhsf_prefix_positive. ff_ub_mce_fold_mdr_second_positive_dataabsoluteevaluationhsf = ff_q_mce_mdr_second_positive_dataabsoluteevaluationhsf_prefix_positive * S ((S (ff_index_mce_alternating_mdr_second_positive_dataabsoluteevaluationhsf_prefix)) * ff_uc_mce_fold_mdr_second_positive_dataabsoluteevaluationhsf) + (ff_p_mce_alternating_mdr_second_positive_dataabsoluteevaluationhsf_prefix))) /\ ((((exists ff_h_mce_mdr_second_positive_dataabsoluteevaluationhsf_prefix_negative. ff_h_mce_mdr_second_positive_dataabsoluteevaluationhsf_prefix_negative + S (ff_n_mce_alternating_mdr_second_positive_dataabsoluteevaluationhsf_prefix) = S ((S (ff_index_mce_alternating_mdr_second_positive_dataabsoluteevaluationhsf_prefix)) * ff_vc_mce_fold_mdr_second_positive_dataabsoluteevaluationhsf)) /\ exists ff_q_mce_mdr_second_positive_dataabsoluteevaluationhsf_prefix_negative. ff_vb_mce_fold_mdr_second_positive_dataabsoluteevaluationhsf = ff_q_mce_mdr_second_positive_dataabsoluteevaluationhsf_prefix_negative * S ((S (ff_index_mce_alternating_mdr_second_positive_dataabsoluteevaluationhsf_prefix)) * ff_vc_mce_fold_mdr_second_positive_dataabsoluteevaluationhsf) + (ff_n_mce_alternating_mdr_second_positive_dataabsoluteevaluationhsf_prefix))) /\ (((exists ff_even_mce_term_mdr_second_positive_dataabsoluteevaluationhsf_prefix_term. ff_index_mce_alternating_mdr_second_positive_dataabsoluteevaluationhsf_prefix = 2 * ff_even_mce_term_mdr_second_positive_dataabsoluteevaluationhsf_prefix_term) /\ (ff_p_mce_alternating_mdr_second_positive_dataabsoluteevaluationhsf_prefix = (ff_ap_mce_alternating_mdr_second_positive_dataabsoluteevaluationhsf_prefix) * (ff_bp_mce_alternating_mdr_second_positive_dataabsoluteevaluationhsf_prefix) + (ff_an_mce_alternating_mdr_second_positive_dataabsoluteevaluationhsf_prefix) * (ff_bn_mce_alternating_mdr_second_positive_dataabsoluteevaluationhsf_prefix) /\ ff_n_mce_alternating_mdr_second_positive_dataabsoluteevaluationhsf_prefix = (ff_ap_mce_alternating_mdr_second_positive_dataabsoluteevaluationhsf_prefix) * (ff_bn_mce_alternating_mdr_second_positive_dataabsoluteevaluationhsf_prefix) + (ff_an_mce_alternating_mdr_second_positive_dataabsoluteevaluationhsf_prefix) * (ff_bp_mce_alternating_mdr_second_positive_dataabsoluteevaluationhsf_prefix))) \/ ((exists ff_odd_mce_term_mdr_second_positive_dataabsoluteevaluationhsf_prefix_term. ff_index_mce_alternating_mdr_second_positive_dataabsoluteevaluationhsf_prefix = 2 * ff_odd_mce_term_mdr_second_positive_dataabsoluteevaluationhsf_prefix_term + 1) /\ (ff_p_mce_alternating_mdr_second_positive_dataabsoluteevaluationhsf_prefix = (ff_ap_mce_alternating_mdr_second_positive_dataabsoluteevaluationhsf_prefix) * (ff_bn_mce_alternating_mdr_second_positive_dataabsoluteevaluationhsf_prefix) + (ff_an_mce_alternating_mdr_second_positive_dataabsoluteevaluationhsf_prefix) * (ff_bp_mce_alternating_mdr_second_positive_dataabsoluteevaluationhsf_prefix) /\ ff_n_mce_alternating_mdr_second_positive_dataabsoluteevaluationhsf_prefix = (ff_ap_mce_alternating_mdr_second_positive_dataabsoluteevaluationhsf_prefix) * (ff_bp_mce_alternating_mdr_second_positive_dataabsoluteevaluationhsf_prefix) + (ff_an_mce_alternating_mdr_second_positive_dataabsoluteevaluationhsf_prefix) * (ff_bn_mce_alternating_mdr_second_positive_dataabsoluteevaluationhsf_prefix))))))))))) /\ ((exists ff_u_mce_mdr_second_positive_dataabsoluteevaluationhsf_positive ff_v_mce_mdr_second_positive_dataabsoluteevaluationhsf_positive. ((((exists ff_h_mce_mdr_second_positive_dataabsoluteevaluationhsf_positive_start. ff_h_mce_mdr_second_positive_dataabsoluteevaluationhsf_positive_start + S (0) = S ((S (0)) * ff_v_mce_mdr_second_positive_dataabsoluteevaluationhsf_positive)) /\ exists ff_q_mce_mdr_second_positive_dataabsoluteevaluationhsf_positive_start. ff_u_mce_mdr_second_positive_dataabsoluteevaluationhsf_positive = ff_q_mce_mdr_second_positive_dataabsoluteevaluationhsf_positive_start * S ((S (0)) * ff_v_mce_mdr_second_positive_dataabsoluteevaluationhsf_positive) + (0))) /\ ((((exists ff_h_mce_mdr_second_positive_dataabsoluteevaluationhsf_positive_terminal. ff_h_mce_mdr_second_positive_dataabsoluteevaluationhsf_positive_terminal + S (mdr_p_second_positive_dataabsoluteevaluationh) = S ((S ((S (mdr_q_second_positive_dataabsoluteevaluationhs)))) * ff_v_mce_mdr_second_positive_dataabsoluteevaluationhsf_positive)) /\ exists ff_q_mce_mdr_second_positive_dataabsoluteevaluationhsf_positive_terminal. ff_u_mce_mdr_second_positive_dataabsoluteevaluationhsf_positive = ff_q_mce_mdr_second_positive_dataabsoluteevaluationhsf_positive_terminal * S ((S ((S (mdr_q_second_positive_dataabsoluteevaluationhs)))) * ff_v_mce_mdr_second_positive_dataabsoluteevaluationhsf_positive) + (mdr_p_second_positive_dataabsoluteevaluationh))) /\ forall ff_i_mce_mdr_second_positive_dataabsoluteevaluationhsf_positive. (exists ff_lt_mce_mdr_second_positive_dataabsoluteevaluationhsf_positive_bound. ff_lt_mce_mdr_second_positive_dataabsoluteevaluationhsf_positive_bound + S ff_i_mce_mdr_second_positive_dataabsoluteevaluationhsf_positive = (S (mdr_q_second_positive_dataabsoluteevaluationhs))) -> exists ff_a_mce_mdr_second_positive_dataabsoluteevaluationhsf_positive ff_r_mce_mdr_second_positive_dataabsoluteevaluationhsf_positive ff_s_mce_mdr_second_positive_dataabsoluteevaluationhsf_positive. ((((exists ff_h_mce_mdr_second_positive_dataabsoluteevaluationhsf_positive_summand. ff_h_mce_mdr_second_positive_dataabsoluteevaluationhsf_positive_summand + S (ff_a_mce_mdr_second_positive_dataabsoluteevaluationhsf_positive) = S ((S (ff_i_mce_mdr_second_positive_dataabsoluteevaluationhsf_positive)) * ff_uc_mce_fold_mdr_second_positive_dataabsoluteevaluationhsf)) /\ exists ff_q_mce_mdr_second_positive_dataabsoluteevaluationhsf_positive_summand. ff_ub_mce_fold_mdr_second_positive_dataabsoluteevaluationhsf = ff_q_mce_mdr_second_positive_dataabsoluteevaluationhsf_positive_summand * S ((S (ff_i_mce_mdr_second_positive_dataabsoluteevaluationhsf_positive)) * ff_uc_mce_fold_mdr_second_positive_dataabsoluteevaluationhsf) + (ff_a_mce_mdr_second_positive_dataabsoluteevaluationhsf_positive))) /\ ((((exists ff_h_mce_mdr_second_positive_dataabsoluteevaluationhsf_positive_partial. ff_h_mce_mdr_second_positive_dataabsoluteevaluationhsf_positive_partial + S (ff_r_mce_mdr_second_positive_dataabsoluteevaluationhsf_positive) = S ((S (ff_i_mce_mdr_second_positive_dataabsoluteevaluationhsf_positive)) * ff_v_mce_mdr_second_positive_dataabsoluteevaluationhsf_positive)) /\ exists ff_q_mce_mdr_second_positive_dataabsoluteevaluationhsf_positive_partial. ff_u_mce_mdr_second_positive_dataabsoluteevaluationhsf_positive = ff_q_mce_mdr_second_positive_dataabsoluteevaluationhsf_positive_partial * S ((S (ff_i_mce_mdr_second_positive_dataabsoluteevaluationhsf_positive)) * ff_v_mce_mdr_second_positive_dataabsoluteevaluationhsf_positive) + (ff_r_mce_mdr_second_positive_dataabsoluteevaluationhsf_positive))) /\ ((((exists ff_h_mce_mdr_second_positive_dataabsoluteevaluationhsf_positive_successor. ff_h_mce_mdr_second_positive_dataabsoluteevaluationhsf_positive_successor + S (ff_s_mce_mdr_second_positive_dataabsoluteevaluationhsf_positive) = S ((S (S ff_i_mce_mdr_second_positive_dataabsoluteevaluationhsf_positive)) * ff_v_mce_mdr_second_positive_dataabsoluteevaluationhsf_positive)) /\ exists ff_q_mce_mdr_second_positive_dataabsoluteevaluationhsf_positive_successor. ff_u_mce_mdr_second_positive_dataabsoluteevaluationhsf_positive = ff_q_mce_mdr_second_positive_dataabsoluteevaluationhsf_positive_successor * S ((S (S ff_i_mce_mdr_second_positive_dataabsoluteevaluationhsf_positive)) * ff_v_mce_mdr_second_positive_dataabsoluteevaluationhsf_positive) + (ff_s_mce_mdr_second_positive_dataabsoluteevaluationhsf_positive))) /\ ff_s_mce_mdr_second_positive_dataabsoluteevaluationhsf_positive = ff_r_mce_mdr_second_positive_dataabsoluteevaluationhsf_positive + ff_a_mce_mdr_second_positive_dataabsoluteevaluationhsf_positive)))))) /\ (exists ff_u_mce_mdr_second_positive_dataabsoluteevaluationhsf_negative ff_v_mce_mdr_second_positive_dataabsoluteevaluationhsf_negative. ((((exists ff_h_mce_mdr_second_positive_dataabsoluteevaluationhsf_negative_start. ff_h_mce_mdr_second_positive_dataabsoluteevaluationhsf_negative_start + S (0) = S ((S (0)) * ff_v_mce_mdr_second_positive_dataabsoluteevaluationhsf_negative)) /\ exists ff_q_mce_mdr_second_positive_dataabsoluteevaluationhsf_negative_start. ff_u_mce_mdr_second_positive_dataabsoluteevaluationhsf_negative = ff_q_mce_mdr_second_positive_dataabsoluteevaluationhsf_negative_start * S ((S (0)) * ff_v_mce_mdr_second_positive_dataabsoluteevaluationhsf_negative) + (0))) /\ ((((exists ff_h_mce_mdr_second_positive_dataabsoluteevaluationhsf_negative_terminal. ff_h_mce_mdr_second_positive_dataabsoluteevaluationhsf_negative_terminal + S (mdr_n_second_positive_dataabsoluteevaluationh) = S ((S ((S (mdr_q_second_positive_dataabsoluteevaluationhs)))) * ff_v_mce_mdr_second_positive_dataabsoluteevaluationhsf_negative)) /\ exists ff_q_mce_mdr_second_positive_dataabsoluteevaluationhsf_negative_terminal. ff_u_mce_mdr_second_positive_dataabsoluteevaluationhsf_negative = ff_q_mce_mdr_second_positive_dataabsoluteevaluationhsf_negative_terminal * S ((S ((S (mdr_q_second_positive_dataabsoluteevaluationhs)))) * ff_v_mce_mdr_second_positive_dataabsoluteevaluationhsf_negative) + (mdr_n_second_positive_dataabsoluteevaluationh))) /\ forall ff_i_mce_mdr_second_positive_dataabsoluteevaluationhsf_negative. (exists ff_lt_mce_mdr_second_positive_dataabsoluteevaluationhsf_negative_bound. ff_lt_mce_mdr_second_positive_dataabsoluteevaluationhsf_negative_bound + S ff_i_mce_mdr_second_positive_dataabsoluteevaluationhsf_negative = (S (mdr_q_second_positive_dataabsoluteevaluationhs))) -> exists ff_a_mce_mdr_second_positive_dataabsoluteevaluationhsf_negative ff_r_mce_mdr_second_positive_dataabsoluteevaluationhsf_negative ff_s_mce_mdr_second_positive_dataabsoluteevaluationhsf_negative. ((((exists ff_h_mce_mdr_second_positive_dataabsoluteevaluationhsf_negative_summand. ff_h_mce_mdr_second_positive_dataabsoluteevaluationhsf_negative_summand + S (ff_a_mce_mdr_second_positive_dataabsoluteevaluationhsf_negative) = S ((S (ff_i_mce_mdr_second_positive_dataabsoluteevaluationhsf_negative)) * ff_vc_mce_fold_mdr_second_positive_dataabsoluteevaluationhsf)) /\ exists ff_q_mce_mdr_second_positive_dataabsoluteevaluationhsf_negative_summand. ff_vb_mce_fold_mdr_second_positive_dataabsoluteevaluationhsf = ff_q_mce_mdr_second_positive_dataabsoluteevaluationhsf_negative_summand * S ((S (ff_i_mce_mdr_second_positive_dataabsoluteevaluationhsf_negative)) * ff_vc_mce_fold_mdr_second_positive_dataabsoluteevaluationhsf) + (ff_a_mce_mdr_second_positive_dataabsoluteevaluationhsf_negative))) /\ ((((exists ff_h_mce_mdr_second_positive_dataabsoluteevaluationhsf_negative_partial. ff_h_mce_mdr_second_positive_dataabsoluteevaluationhsf_negative_partial + S (ff_r_mce_mdr_second_positive_dataabsoluteevaluationhsf_negative) = S ((S (ff_i_mce_mdr_second_positive_dataabsoluteevaluationhsf_negative)) * ff_v_mce_mdr_second_positive_dataabsoluteevaluationhsf_negative)) /\ exists ff_q_mce_mdr_second_positive_dataabsoluteevaluationhsf_negative_partial. ff_u_mce_mdr_second_positive_dataabsoluteevaluationhsf_negative = ff_q_mce_mdr_second_positive_dataabsoluteevaluationhsf_negative_partial * S ((S (ff_i_mce_mdr_second_positive_dataabsoluteevaluationhsf_negative)) * ff_v_mce_mdr_second_positive_dataabsoluteevaluationhsf_negative) + (ff_r_mce_mdr_second_positive_dataabsoluteevaluationhsf_negative))) /\ ((((exists ff_h_mce_mdr_second_positive_dataabsoluteevaluationhsf_negative_successor. ff_h_mce_mdr_second_positive_dataabsoluteevaluationhsf_negative_successor + S (ff_s_mce_mdr_second_positive_dataabsoluteevaluationhsf_negative) = S ((S (S ff_i_mce_mdr_second_positive_dataabsoluteevaluationhsf_negative)) * ff_v_mce_mdr_second_positive_dataabsoluteevaluationhsf_negative)) /\ exists ff_q_mce_mdr_second_positive_dataabsoluteevaluationhsf_negative_successor. ff_u_mce_mdr_second_positive_dataabsoluteevaluationhsf_negative = ff_q_mce_mdr_second_positive_dataabsoluteevaluationhsf_negative_successor * S ((S (S ff_i_mce_mdr_second_positive_dataabsoluteevaluationhsf_negative)) * ff_v_mce_mdr_second_positive_dataabsoluteevaluationhsf_negative) + (ff_s_mce_mdr_second_positive_dataabsoluteevaluationhsf_negative))) /\ ff_s_mce_mdr_second_positive_dataabsoluteevaluationhsf_negative = ff_r_mce_mdr_second_positive_dataabsoluteevaluationhsf_negative + ff_a_mce_mdr_second_positive_dataabsoluteevaluationhsf_negative))))))))))))))) /\ ((exists mdr_gap_second_positive_dataabsoluteevaluationi. mdr_gap_second_positive_dataabsoluteevaluationi + S (mdr_i_second_positive_dataabsoluteevaluation) = (mdr_l_second_positive_dataabsoluteevaluation)) /\ (exists mdr_z_second_positive_dataabsoluteevaluationr. ((exists mdr_a_second_positive_dataabsoluteevaluationrc mdr_b_second_positive_dataabsoluteevaluationrc mdr_c_second_positive_dataabsoluteevaluationrc mdr_e_second_positive_dataabsoluteevaluationrc mdr_f_second_positive_dataabsoluteevaluationrc. ((mdr_a_second_positive_dataabsoluteevaluationrc = ((d) + (ab)) * S ((d) + (ab)) + ((ab) + (ab))) /\ ((mdr_b_second_positive_dataabsoluteevaluationrc = ((ac) + (bb)) * S ((ac) + (bb)) + ((bb) + (bb))) /\ ((mdr_c_second_positive_dataabsoluteevaluationrc = ((mdr_a_second_positive_dataabsoluteevaluationrc) + (mdr_b_second_positive_dataabsoluteevaluationrc)) * S ((mdr_a_second_positive_dataabsoluteevaluationrc) + (mdr_b_second_positive_dataabsoluteevaluationrc)) + ((mdr_b_second_positive_dataabsoluteevaluationrc) + (mdr_b_second_positive_dataabsoluteevaluationrc))) /\ ((mdr_e_second_positive_dataabsoluteevaluationrc = ((mdr_p_second_positive_dataabsolute) + (mdr_n_second_positive_dataabsolute)) * S ((mdr_p_second_positive_dataabsolute) + (mdr_n_second_positive_dataabsolute)) + ((mdr_n_second_positive_dataabsolute) + (mdr_n_second_positive_dataabsolute))) /\ ((mdr_f_second_positive_dataabsoluteevaluationrc = ((bc) + (mdr_e_second_positive_dataabsoluteevaluationrc)) * S ((bc) + (mdr_e_second_positive_dataabsoluteevaluationrc)) + ((mdr_e_second_positive_dataabsoluteevaluationrc) + (mdr_e_second_positive_dataabsoluteevaluationrc))) /\ ((mdr_z_second_positive_dataabsoluteevaluationr) = ((mdr_c_second_positive_dataabsoluteevaluationrc) + (mdr_f_second_positive_dataabsoluteevaluationrc)) * S ((mdr_c_second_positive_dataabsoluteevaluationrc) + (mdr_f_second_positive_dataabsoluteevaluationrc)) + ((mdr_f_second_positive_dataabsoluteevaluationrc) + (mdr_f_second_positive_dataabsoluteevaluationrc))))))))) /\ (((exists ff_h_mdr_second_positive_dataabsoluteevaluationrb. ff_h_mdr_second_positive_dataabsoluteevaluationrb + S (mdr_z_second_positive_dataabsoluteevaluationr) = S ((S (mdr_i_second_positive_dataabsoluteevaluation)) * mdr_c_second_positive_dataabsoluteevaluation)) /\ exists ff_q_mdr_second_positive_dataabsoluteevaluationrb. mdr_b_second_positive_dataabsoluteevaluation = ff_q_mdr_second_positive_dataabsoluteevaluationrb * S ((S (mdr_i_second_positive_dataabsoluteevaluation)) * mdr_c_second_positive_dataabsoluteevaluation) + (mdr_z_second_positive_dataabsoluteevaluationr)))))))) /\ (((mdr_p_second_positive_dataabsolute) = (mdr_n_second_positive_dataabsolute) + (E)) \/ ((mdr_n_second_positive_dataabsolute) = (mdr_p_second_positive_dataabsolute) + (E)))))))) -> D = E

Constructive proof overview

Generated structural guide

The positive absolute determinant in the nondegenerate square-matrix data is uniquely determined by the actual matrix.

The unchanged tactic script uses 1 declared prerequisite and contains 23 exact native proof lines.

Alpha v34 checked-use · first admitted v27 · independently kernel and Lean verified; not Stable

Proof neighborhood

Direct dependencies

Direct dependents

Formal native tactic body

Dependencies are introduced as named hypotheses before line 1. Local theorem links identify exact declared prerequisites. This exact body belongs to a complete independently kernel-checked constructive proof bundle and has Alpha checked-use authority; it does not imply Stable membership.

Read the argument

Proof checkpoints

23 script commands · 3 reading checkpoints · 0 local claims

This is a reading aid, not a new proof or a proof-tree certificate. Checkpoint groups are consecutive commands, not inferred branch boundaries. Every step links to the preserved script.

Named ingredients (1)
01Fix variables and assumptionsL1–9

Work with arbitrary variables or the premises of the current implication.

  1. L1
    intro ab
  2. L2
    intro ac
  3. L3
    intro bb
  4. L4
    intro bc
  5. L5
    intro d
  6. L6
    intro D
  7. L7
    intro E
  8. L8
    intro hfirst
  9. L9
    intro hsecond
02Separate the logical casesL10–13

Follow the explicit conjunction, disjunction, witness, or contradiction step recorded below.

  1. L10
    cases hfirst
  2. L11
    cases hfirst_right
  3. L12
    cases hsecond
  4. L13
    cases hsecond_right
03Use earlier factsL14–23

Instantiate or apply named facts and discharge the corresponding proof obligations.

  1. L14
    specialize absolute_recursive_determinant_functional (ab)
  2. L15
    specialize absolute_recursive_determinant_functional (ac)
  3. L16
    specialize absolute_recursive_determinant_functional (bb)
  4. L17
    specialize absolute_recursive_determinant_functional (bc)
  5. L18
    specialize absolute_recursive_determinant_functional (d)
  6. L19
    specialize absolute_recursive_determinant_functional (D)
  7. L20
    specialize absolute_recursive_determinant_functional (E)
  8. L21
    apply absolute_recursive_determinant_functional
  9. L22
    exact hfirst_right_right
  10. L23
    exact hsecond_right_right

Library-wide reading audit

Original exact command ledger · 23 lines
  1. 0001intro ab
  2. 0002intro ac
  3. 0003intro bb
  4. 0004intro bc
  5. 0005intro d
  6. 0006intro D
  7. 0007intro E
  8. 0008intro hfirst
  9. 0009intro hsecond
  10. 0010cases hfirst
  11. 0011cases hfirst_right
  12. 0012cases hsecond
  13. 0013cases hsecond_right
  14. 0014specialize absolute_recursive_determinant_functional (ab)
  15. 0015specialize absolute_recursive_determinant_functional (ac)
  16. 0016specialize absolute_recursive_determinant_functional (bb)
  17. 0017specialize absolute_recursive_determinant_functional (bc)
  18. 0018specialize absolute_recursive_determinant_functional (d)
  19. 0019specialize absolute_recursive_determinant_functional (D)
  20. 0020specialize absolute_recursive_determinant_functional (E)
  21. 0021apply absolute_recursive_determinant_functional
  22. 0022exact hfirst_right_right
  23. 0023exact hsecond_right_right