DL00AB

positive_determinant_matrix_data_nonzero

Nondegenerate square-matrix data contains an actual nonzero full determinant witness, not only a positivity label.

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

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.

This branch proves the finite determinant/rank/span substrate. It does not claim Smith or Hermite normal form, lattice index equals determinant, determinant multiplicativity, lattice reduction, or geometry-of-numbers theorems.

Exact theorem in conservative defined notation

∀ ab. ∀ ac. ∀ bb. ∀ bc. ∀ d. ∀ D. PositiveDeterminantMatrixData(ab,ac,bb,bc,d,D) → ∃ x. ∃ y. SignedRecursiveDeterminant(ab,ac,bb,bc,d,x,y) ∧ ¬x = y

Every linked abbreviation expands hygienically to the identical original native formula.

Definition DAG

Actual proof prerequisites

Original expanded first-order statement
forall ab ac bb bc d D. (((~(d = 0)) /\ ((~(D = 0)) /\ (exists mdr_p_positive_dataabsolute mdr_n_positive_dataabsolute. ((exists mdr_b_positive_dataabsoluteevaluation mdr_c_positive_dataabsoluteevaluation mdr_l_positive_dataabsoluteevaluation mdr_i_positive_dataabsoluteevaluation. ((forall mdr_i_positive_dataabsoluteevaluationh. (exists mdr_gap_positive_dataabsoluteevaluationhi. mdr_gap_positive_dataabsoluteevaluationhi + S (mdr_i_positive_dataabsoluteevaluationh) = (mdr_l_positive_dataabsoluteevaluation)) -> exists mdr_d_positive_dataabsoluteevaluationh mdr_pb_positive_dataabsoluteevaluationh mdr_pc_positive_dataabsoluteevaluationh mdr_nb_positive_dataabsoluteevaluationh mdr_nc_positive_dataabsoluteevaluationh mdr_p_positive_dataabsoluteevaluationh mdr_n_positive_dataabsoluteevaluationh. ((exists mdr_z_positive_dataabsoluteevaluationhr. ((exists mdr_a_positive_dataabsoluteevaluationhrc mdr_b_positive_dataabsoluteevaluationhrc mdr_c_positive_dataabsoluteevaluationhrc mdr_e_positive_dataabsoluteevaluationhrc mdr_f_positive_dataabsoluteevaluationhrc. ((mdr_a_positive_dataabsoluteevaluationhrc = ((mdr_d_positive_dataabsoluteevaluationh) + (mdr_pb_positive_dataabsoluteevaluationh)) * S ((mdr_d_positive_dataabsoluteevaluationh) + (mdr_pb_positive_dataabsoluteevaluationh)) + ((mdr_pb_positive_dataabsoluteevaluationh) + (mdr_pb_positive_dataabsoluteevaluationh))) /\ ((mdr_b_positive_dataabsoluteevaluationhrc = ((mdr_pc_positive_dataabsoluteevaluationh) + (mdr_nb_positive_dataabsoluteevaluationh)) * S ((mdr_pc_positive_dataabsoluteevaluationh) + (mdr_nb_positive_dataabsoluteevaluationh)) + ((mdr_nb_positive_dataabsoluteevaluationh) + (mdr_nb_positive_dataabsoluteevaluationh))) /\ ((mdr_c_positive_dataabsoluteevaluationhrc = ((mdr_a_positive_dataabsoluteevaluationhrc) + (mdr_b_positive_dataabsoluteevaluationhrc)) * S ((mdr_a_positive_dataabsoluteevaluationhrc) + (mdr_b_positive_dataabsoluteevaluationhrc)) + ((mdr_b_positive_dataabsoluteevaluationhrc) + (mdr_b_positive_dataabsoluteevaluationhrc))) /\ ((mdr_e_positive_dataabsoluteevaluationhrc = ((mdr_p_positive_dataabsoluteevaluationh) + (mdr_n_positive_dataabsoluteevaluationh)) * S ((mdr_p_positive_dataabsoluteevaluationh) + (mdr_n_positive_dataabsoluteevaluationh)) + ((mdr_n_positive_dataabsoluteevaluationh) + (mdr_n_positive_dataabsoluteevaluationh))) /\ ((mdr_f_positive_dataabsoluteevaluationhrc = ((mdr_nc_positive_dataabsoluteevaluationh) + (mdr_e_positive_dataabsoluteevaluationhrc)) * S ((mdr_nc_positive_dataabsoluteevaluationh) + (mdr_e_positive_dataabsoluteevaluationhrc)) + ((mdr_e_positive_dataabsoluteevaluationhrc) + (mdr_e_positive_dataabsoluteevaluationhrc))) /\ ((mdr_z_positive_dataabsoluteevaluationhr) = ((mdr_c_positive_dataabsoluteevaluationhrc) + (mdr_f_positive_dataabsoluteevaluationhrc)) * S ((mdr_c_positive_dataabsoluteevaluationhrc) + (mdr_f_positive_dataabsoluteevaluationhrc)) + ((mdr_f_positive_dataabsoluteevaluationhrc) + (mdr_f_positive_dataabsoluteevaluationhrc))))))))) /\ (((exists ff_h_mdr_positive_dataabsoluteevaluationhrb. ff_h_mdr_positive_dataabsoluteevaluationhrb + S (mdr_z_positive_dataabsoluteevaluationhr) = S ((S (mdr_i_positive_dataabsoluteevaluationh)) * mdr_c_positive_dataabsoluteevaluation)) /\ exists ff_q_mdr_positive_dataabsoluteevaluationhrb. mdr_b_positive_dataabsoluteevaluation = ff_q_mdr_positive_dataabsoluteevaluationhrb * S ((S (mdr_i_positive_dataabsoluteevaluationh)) * mdr_c_positive_dataabsoluteevaluation) + (mdr_z_positive_dataabsoluteevaluationhr))))) /\ (((((mdr_d_positive_dataabsoluteevaluationh) = 0) /\ (((mdr_p_positive_dataabsoluteevaluationh) = 1) /\ ((mdr_n_positive_dataabsoluteevaluationh) = 0))) \/ exists mdr_q_positive_dataabsoluteevaluationhs mdr_eb_positive_dataabsoluteevaluationhs mdr_ec_positive_dataabsoluteevaluationhs mdr_fb_positive_dataabsoluteevaluationhs mdr_fc_positive_dataabsoluteevaluationhs. (((mdr_d_positive_dataabsoluteevaluationh) = S (mdr_q_positive_dataabsoluteevaluationhs)) /\ ((forall mdr_j_positive_dataabsoluteevaluationhsc. (exists mdr_gap_positive_dataabsoluteevaluationhscj. mdr_gap_positive_dataabsoluteevaluationhscj + S (mdr_j_positive_dataabsoluteevaluationhsc) = (S (mdr_q_positive_dataabsoluteevaluationhs))) -> exists mdr_i_positive_dataabsoluteevaluationhsc mdr_up_positive_dataabsoluteevaluationhsc mdr_us_positive_dataabsoluteevaluationhsc mdr_un_positive_dataabsoluteevaluationhsc mdr_ut_positive_dataabsoluteevaluationhsc mdr_p_positive_dataabsoluteevaluationhsc mdr_n_positive_dataabsoluteevaluationhsc. ((exists mdr_gap_positive_dataabsoluteevaluationhsci. mdr_gap_positive_dataabsoluteevaluationhsci + S (mdr_i_positive_dataabsoluteevaluationhsc) = (mdr_i_positive_dataabsoluteevaluationh)) /\ ((exists mdr_z_positive_dataabsoluteevaluationhscr. ((exists mdr_a_positive_dataabsoluteevaluationhscrc mdr_b_positive_dataabsoluteevaluationhscrc mdr_c_positive_dataabsoluteevaluationhscrc mdr_e_positive_dataabsoluteevaluationhscrc mdr_f_positive_dataabsoluteevaluationhscrc. ((mdr_a_positive_dataabsoluteevaluationhscrc = ((mdr_q_positive_dataabsoluteevaluationhs) + (mdr_up_positive_dataabsoluteevaluationhsc)) * S ((mdr_q_positive_dataabsoluteevaluationhs) + (mdr_up_positive_dataabsoluteevaluationhsc)) + ((mdr_up_positive_dataabsoluteevaluationhsc) + (mdr_up_positive_dataabsoluteevaluationhsc))) /\ ((mdr_b_positive_dataabsoluteevaluationhscrc = ((mdr_us_positive_dataabsoluteevaluationhsc) + (mdr_un_positive_dataabsoluteevaluationhsc)) * S ((mdr_us_positive_dataabsoluteevaluationhsc) + (mdr_un_positive_dataabsoluteevaluationhsc)) + ((mdr_un_positive_dataabsoluteevaluationhsc) + (mdr_un_positive_dataabsoluteevaluationhsc))) /\ ((mdr_c_positive_dataabsoluteevaluationhscrc = ((mdr_a_positive_dataabsoluteevaluationhscrc) + (mdr_b_positive_dataabsoluteevaluationhscrc)) * S ((mdr_a_positive_dataabsoluteevaluationhscrc) + (mdr_b_positive_dataabsoluteevaluationhscrc)) + ((mdr_b_positive_dataabsoluteevaluationhscrc) + (mdr_b_positive_dataabsoluteevaluationhscrc))) /\ ((mdr_e_positive_dataabsoluteevaluationhscrc = ((mdr_p_positive_dataabsoluteevaluationhsc) + (mdr_n_positive_dataabsoluteevaluationhsc)) * S ((mdr_p_positive_dataabsoluteevaluationhsc) + (mdr_n_positive_dataabsoluteevaluationhsc)) + ((mdr_n_positive_dataabsoluteevaluationhsc) + (mdr_n_positive_dataabsoluteevaluationhsc))) /\ ((mdr_f_positive_dataabsoluteevaluationhscrc = ((mdr_ut_positive_dataabsoluteevaluationhsc) + (mdr_e_positive_dataabsoluteevaluationhscrc)) * S ((mdr_ut_positive_dataabsoluteevaluationhsc) + (mdr_e_positive_dataabsoluteevaluationhscrc)) + ((mdr_e_positive_dataabsoluteevaluationhscrc) + (mdr_e_positive_dataabsoluteevaluationhscrc))) /\ ((mdr_z_positive_dataabsoluteevaluationhscr) = ((mdr_c_positive_dataabsoluteevaluationhscrc) + (mdr_f_positive_dataabsoluteevaluationhscrc)) * S ((mdr_c_positive_dataabsoluteevaluationhscrc) + (mdr_f_positive_dataabsoluteevaluationhscrc)) + ((mdr_f_positive_dataabsoluteevaluationhscrc) + (mdr_f_positive_dataabsoluteevaluationhscrc))))))))) /\ (((exists ff_h_mdr_positive_dataabsoluteevaluationhscrb. ff_h_mdr_positive_dataabsoluteevaluationhscrb + S (mdr_z_positive_dataabsoluteevaluationhscr) = S ((S (mdr_i_positive_dataabsoluteevaluationhsc)) * mdr_c_positive_dataabsoluteevaluation)) /\ exists ff_q_mdr_positive_dataabsoluteevaluationhscrb. mdr_b_positive_dataabsoluteevaluation = ff_q_mdr_positive_dataabsoluteevaluationhscrb * S ((S (mdr_i_positive_dataabsoluteevaluationhsc)) * mdr_c_positive_dataabsoluteevaluation) + (mdr_z_positive_dataabsoluteevaluationhscr))))) /\ ((((forall ff_index_mdm_prefix_mdr_positive_dataabsoluteevaluationhscm_positive. (exists ff_gap_mdm_lt_mdr_positive_dataabsoluteevaluationhscm_positive_index_bound. ff_gap_mdm_lt_mdr_positive_dataabsoluteevaluationhscm_positive_index_bound + S (ff_index_mdm_prefix_mdr_positive_dataabsoluteevaluationhscm_positive) = ((mdr_q_positive_dataabsoluteevaluationhs) * (mdr_q_positive_dataabsoluteevaluationhs))) -> exists ff_row_mdm_prefix_mdr_positive_dataabsoluteevaluationhscm_positive ff_column_mdm_prefix_mdr_positive_dataabsoluteevaluationhscm_positive ff_value_mdm_prefix_mdr_positive_dataabsoluteevaluationhscm_positive. (ff_index_mdm_prefix_mdr_positive_dataabsoluteevaluationhscm_positive = (mdr_q_positive_dataabsoluteevaluationhs) * ff_row_mdm_prefix_mdr_positive_dataabsoluteevaluationhscm_positive + ff_column_mdm_prefix_mdr_positive_dataabsoluteevaluationhscm_positive /\ ((exists ff_gap_mdm_lt_mdr_positive_dataabsoluteevaluationhscm_positive_column_bound. ff_gap_mdm_lt_mdr_positive_dataabsoluteevaluationhscm_positive_column_bound + S (ff_column_mdm_prefix_mdr_positive_dataabsoluteevaluationhscm_positive) = (mdr_q_positive_dataabsoluteevaluationhs)) /\ ((exists ff_row_mdm_cell_mdr_positive_dataabsoluteevaluationhscm_positive_cell ff_column_mdm_cell_mdr_positive_dataabsoluteevaluationhscm_positive_cell. (((((exists ff_gap_mdm_lt_mdr_positive_dataabsoluteevaluationhscm_positive_cell_row_before. ff_gap_mdm_lt_mdr_positive_dataabsoluteevaluationhscm_positive_cell_row_before + S (ff_row_mdm_prefix_mdr_positive_dataabsoluteevaluationhscm_positive) = (0)) /\ ff_row_mdm_cell_mdr_positive_dataabsoluteevaluationhscm_positive_cell = ff_row_mdm_prefix_mdr_positive_dataabsoluteevaluationhscm_positive) \/ ((exists ff_gap_mdm_le_mdr_positive_dataabsoluteevaluationhscm_positive_cell_row_after. ff_gap_mdm_le_mdr_positive_dataabsoluteevaluationhscm_positive_cell_row_after + (0) = (ff_row_mdm_prefix_mdr_positive_dataabsoluteevaluationhscm_positive)) /\ ff_row_mdm_cell_mdr_positive_dataabsoluteevaluationhscm_positive_cell = S ff_row_mdm_prefix_mdr_positive_dataabsoluteevaluationhscm_positive))) /\ (((((exists ff_gap_mdm_lt_mdr_positive_dataabsoluteevaluationhscm_positive_cell_column_before. ff_gap_mdm_lt_mdr_positive_dataabsoluteevaluationhscm_positive_cell_column_before + S (ff_column_mdm_prefix_mdr_positive_dataabsoluteevaluationhscm_positive) = (mdr_j_positive_dataabsoluteevaluationhsc)) /\ ff_column_mdm_cell_mdr_positive_dataabsoluteevaluationhscm_positive_cell = ff_column_mdm_prefix_mdr_positive_dataabsoluteevaluationhscm_positive) \/ ((exists ff_gap_mdm_le_mdr_positive_dataabsoluteevaluationhscm_positive_cell_column_after. ff_gap_mdm_le_mdr_positive_dataabsoluteevaluationhscm_positive_cell_column_after + (mdr_j_positive_dataabsoluteevaluationhsc) = (ff_column_mdm_prefix_mdr_positive_dataabsoluteevaluationhscm_positive)) /\ ff_column_mdm_cell_mdr_positive_dataabsoluteevaluationhscm_positive_cell = S ff_column_mdm_prefix_mdr_positive_dataabsoluteevaluationhscm_positive))) /\ (((exists ff_h_mdm_mdr_positive_dataabsoluteevaluationhscm_positive_cell_source. ff_h_mdm_mdr_positive_dataabsoluteevaluationhscm_positive_cell_source + S (ff_value_mdm_prefix_mdr_positive_dataabsoluteevaluationhscm_positive) = S ((S ((ff_row_mdm_cell_mdr_positive_dataabsoluteevaluationhscm_positive_cell) * (S (mdr_q_positive_dataabsoluteevaluationhs)) + (ff_column_mdm_cell_mdr_positive_dataabsoluteevaluationhscm_positive_cell))) * mdr_pc_positive_dataabsoluteevaluationh)) /\ exists ff_q_mdm_mdr_positive_dataabsoluteevaluationhscm_positive_cell_source. mdr_pb_positive_dataabsoluteevaluationh = ff_q_mdm_mdr_positive_dataabsoluteevaluationhscm_positive_cell_source * S ((S ((ff_row_mdm_cell_mdr_positive_dataabsoluteevaluationhscm_positive_cell) * (S (mdr_q_positive_dataabsoluteevaluationhs)) + (ff_column_mdm_cell_mdr_positive_dataabsoluteevaluationhscm_positive_cell))) * mdr_pc_positive_dataabsoluteevaluationh) + (ff_value_mdm_prefix_mdr_positive_dataabsoluteevaluationhscm_positive)))))) /\ (((exists ff_h_mdm_mdr_positive_dataabsoluteevaluationhscm_positive_target. ff_h_mdm_mdr_positive_dataabsoluteevaluationhscm_positive_target + S (ff_value_mdm_prefix_mdr_positive_dataabsoluteevaluationhscm_positive) = S ((S (ff_index_mdm_prefix_mdr_positive_dataabsoluteevaluationhscm_positive)) * mdr_us_positive_dataabsoluteevaluationhsc)) /\ exists ff_q_mdm_mdr_positive_dataabsoluteevaluationhscm_positive_target. mdr_up_positive_dataabsoluteevaluationhsc = ff_q_mdm_mdr_positive_dataabsoluteevaluationhscm_positive_target * S ((S (ff_index_mdm_prefix_mdr_positive_dataabsoluteevaluationhscm_positive)) * mdr_us_positive_dataabsoluteevaluationhsc) + (ff_value_mdm_prefix_mdr_positive_dataabsoluteevaluationhscm_positive))))))) /\ (forall ff_index_mdm_prefix_mdr_positive_dataabsoluteevaluationhscm_negative. (exists ff_gap_mdm_lt_mdr_positive_dataabsoluteevaluationhscm_negative_index_bound. ff_gap_mdm_lt_mdr_positive_dataabsoluteevaluationhscm_negative_index_bound + S (ff_index_mdm_prefix_mdr_positive_dataabsoluteevaluationhscm_negative) = ((mdr_q_positive_dataabsoluteevaluationhs) * (mdr_q_positive_dataabsoluteevaluationhs))) -> exists ff_row_mdm_prefix_mdr_positive_dataabsoluteevaluationhscm_negative ff_column_mdm_prefix_mdr_positive_dataabsoluteevaluationhscm_negative ff_value_mdm_prefix_mdr_positive_dataabsoluteevaluationhscm_negative. (ff_index_mdm_prefix_mdr_positive_dataabsoluteevaluationhscm_negative = (mdr_q_positive_dataabsoluteevaluationhs) * ff_row_mdm_prefix_mdr_positive_dataabsoluteevaluationhscm_negative + ff_column_mdm_prefix_mdr_positive_dataabsoluteevaluationhscm_negative /\ ((exists ff_gap_mdm_lt_mdr_positive_dataabsoluteevaluationhscm_negative_column_bound. ff_gap_mdm_lt_mdr_positive_dataabsoluteevaluationhscm_negative_column_bound + S (ff_column_mdm_prefix_mdr_positive_dataabsoluteevaluationhscm_negative) = (mdr_q_positive_dataabsoluteevaluationhs)) /\ ((exists ff_row_mdm_cell_mdr_positive_dataabsoluteevaluationhscm_negative_cell ff_column_mdm_cell_mdr_positive_dataabsoluteevaluationhscm_negative_cell. (((((exists ff_gap_mdm_lt_mdr_positive_dataabsoluteevaluationhscm_negative_cell_row_before. ff_gap_mdm_lt_mdr_positive_dataabsoluteevaluationhscm_negative_cell_row_before + S (ff_row_mdm_prefix_mdr_positive_dataabsoluteevaluationhscm_negative) = (0)) /\ ff_row_mdm_cell_mdr_positive_dataabsoluteevaluationhscm_negative_cell = ff_row_mdm_prefix_mdr_positive_dataabsoluteevaluationhscm_negative) \/ ((exists ff_gap_mdm_le_mdr_positive_dataabsoluteevaluationhscm_negative_cell_row_after. ff_gap_mdm_le_mdr_positive_dataabsoluteevaluationhscm_negative_cell_row_after + (0) = (ff_row_mdm_prefix_mdr_positive_dataabsoluteevaluationhscm_negative)) /\ ff_row_mdm_cell_mdr_positive_dataabsoluteevaluationhscm_negative_cell = S ff_row_mdm_prefix_mdr_positive_dataabsoluteevaluationhscm_negative))) /\ (((((exists ff_gap_mdm_lt_mdr_positive_dataabsoluteevaluationhscm_negative_cell_column_before. ff_gap_mdm_lt_mdr_positive_dataabsoluteevaluationhscm_negative_cell_column_before + S (ff_column_mdm_prefix_mdr_positive_dataabsoluteevaluationhscm_negative) = (mdr_j_positive_dataabsoluteevaluationhsc)) /\ ff_column_mdm_cell_mdr_positive_dataabsoluteevaluationhscm_negative_cell = ff_column_mdm_prefix_mdr_positive_dataabsoluteevaluationhscm_negative) \/ ((exists ff_gap_mdm_le_mdr_positive_dataabsoluteevaluationhscm_negative_cell_column_after. ff_gap_mdm_le_mdr_positive_dataabsoluteevaluationhscm_negative_cell_column_after + (mdr_j_positive_dataabsoluteevaluationhsc) = (ff_column_mdm_prefix_mdr_positive_dataabsoluteevaluationhscm_negative)) /\ ff_column_mdm_cell_mdr_positive_dataabsoluteevaluationhscm_negative_cell = S ff_column_mdm_prefix_mdr_positive_dataabsoluteevaluationhscm_negative))) /\ (((exists ff_h_mdm_mdr_positive_dataabsoluteevaluationhscm_negative_cell_source. ff_h_mdm_mdr_positive_dataabsoluteevaluationhscm_negative_cell_source + S (ff_value_mdm_prefix_mdr_positive_dataabsoluteevaluationhscm_negative) = S ((S ((ff_row_mdm_cell_mdr_positive_dataabsoluteevaluationhscm_negative_cell) * (S (mdr_q_positive_dataabsoluteevaluationhs)) + (ff_column_mdm_cell_mdr_positive_dataabsoluteevaluationhscm_negative_cell))) * mdr_nc_positive_dataabsoluteevaluationh)) /\ exists ff_q_mdm_mdr_positive_dataabsoluteevaluationhscm_negative_cell_source. mdr_nb_positive_dataabsoluteevaluationh = ff_q_mdm_mdr_positive_dataabsoluteevaluationhscm_negative_cell_source * S ((S ((ff_row_mdm_cell_mdr_positive_dataabsoluteevaluationhscm_negative_cell) * (S (mdr_q_positive_dataabsoluteevaluationhs)) + (ff_column_mdm_cell_mdr_positive_dataabsoluteevaluationhscm_negative_cell))) * mdr_nc_positive_dataabsoluteevaluationh) + (ff_value_mdm_prefix_mdr_positive_dataabsoluteevaluationhscm_negative)))))) /\ (((exists ff_h_mdm_mdr_positive_dataabsoluteevaluationhscm_negative_target. ff_h_mdm_mdr_positive_dataabsoluteevaluationhscm_negative_target + S (ff_value_mdm_prefix_mdr_positive_dataabsoluteevaluationhscm_negative) = S ((S (ff_index_mdm_prefix_mdr_positive_dataabsoluteevaluationhscm_negative)) * mdr_ut_positive_dataabsoluteevaluationhsc)) /\ exists ff_q_mdm_mdr_positive_dataabsoluteevaluationhscm_negative_target. mdr_un_positive_dataabsoluteevaluationhsc = ff_q_mdm_mdr_positive_dataabsoluteevaluationhscm_negative_target * S ((S (ff_index_mdm_prefix_mdr_positive_dataabsoluteevaluationhscm_negative)) * mdr_ut_positive_dataabsoluteevaluationhsc) + (ff_value_mdm_prefix_mdr_positive_dataabsoluteevaluationhscm_negative))))))))) /\ ((((exists ff_h_mdr_positive_dataabsoluteevaluationhscp. ff_h_mdr_positive_dataabsoluteevaluationhscp + S (mdr_p_positive_dataabsoluteevaluationhsc) = S ((S (mdr_j_positive_dataabsoluteevaluationhsc)) * mdr_ec_positive_dataabsoluteevaluationhs)) /\ exists ff_q_mdr_positive_dataabsoluteevaluationhscp. mdr_eb_positive_dataabsoluteevaluationhs = ff_q_mdr_positive_dataabsoluteevaluationhscp * S ((S (mdr_j_positive_dataabsoluteevaluationhsc)) * mdr_ec_positive_dataabsoluteevaluationhs) + (mdr_p_positive_dataabsoluteevaluationhsc))) /\ (((exists ff_h_mdr_positive_dataabsoluteevaluationhscn. ff_h_mdr_positive_dataabsoluteevaluationhscn + S (mdr_n_positive_dataabsoluteevaluationhsc) = S ((S (mdr_j_positive_dataabsoluteevaluationhsc)) * mdr_fc_positive_dataabsoluteevaluationhs)) /\ exists ff_q_mdr_positive_dataabsoluteevaluationhscn. mdr_fb_positive_dataabsoluteevaluationhs = ff_q_mdr_positive_dataabsoluteevaluationhscn * S ((S (mdr_j_positive_dataabsoluteevaluationhsc)) * mdr_fc_positive_dataabsoluteevaluationhs) + (mdr_n_positive_dataabsoluteevaluationhsc)))))))) /\ (exists ff_ub_mce_fold_mdr_positive_dataabsoluteevaluationhsf ff_uc_mce_fold_mdr_positive_dataabsoluteevaluationhsf ff_vb_mce_fold_mdr_positive_dataabsoluteevaluationhsf ff_vc_mce_fold_mdr_positive_dataabsoluteevaluationhsf. ((forall ff_index_mce_alternating_mdr_positive_dataabsoluteevaluationhsf_prefix. (exists ff_gap_mce_mdr_positive_dataabsoluteevaluationhsf_prefix_index. ff_gap_mce_mdr_positive_dataabsoluteevaluationhsf_prefix_index + S (ff_index_mce_alternating_mdr_positive_dataabsoluteevaluationhsf_prefix) = (S (mdr_q_positive_dataabsoluteevaluationhs))) -> exists ff_ap_mce_alternating_mdr_positive_dataabsoluteevaluationhsf_prefix ff_an_mce_alternating_mdr_positive_dataabsoluteevaluationhsf_prefix ff_bp_mce_alternating_mdr_positive_dataabsoluteevaluationhsf_prefix ff_bn_mce_alternating_mdr_positive_dataabsoluteevaluationhsf_prefix ff_p_mce_alternating_mdr_positive_dataabsoluteevaluationhsf_prefix ff_n_mce_alternating_mdr_positive_dataabsoluteevaluationhsf_prefix. ((((exists ff_h_mce_mdr_positive_dataabsoluteevaluationhsf_prefix_ap. ff_h_mce_mdr_positive_dataabsoluteevaluationhsf_prefix_ap + S (ff_ap_mce_alternating_mdr_positive_dataabsoluteevaluationhsf_prefix) = S ((S (ff_index_mce_alternating_mdr_positive_dataabsoluteevaluationhsf_prefix)) * mdr_pc_positive_dataabsoluteevaluationh)) /\ exists ff_q_mce_mdr_positive_dataabsoluteevaluationhsf_prefix_ap. mdr_pb_positive_dataabsoluteevaluationh = ff_q_mce_mdr_positive_dataabsoluteevaluationhsf_prefix_ap * S ((S (ff_index_mce_alternating_mdr_positive_dataabsoluteevaluationhsf_prefix)) * mdr_pc_positive_dataabsoluteevaluationh) + (ff_ap_mce_alternating_mdr_positive_dataabsoluteevaluationhsf_prefix))) /\ ((((exists ff_h_mce_mdr_positive_dataabsoluteevaluationhsf_prefix_an. ff_h_mce_mdr_positive_dataabsoluteevaluationhsf_prefix_an + S (ff_an_mce_alternating_mdr_positive_dataabsoluteevaluationhsf_prefix) = S ((S (ff_index_mce_alternating_mdr_positive_dataabsoluteevaluationhsf_prefix)) * mdr_nc_positive_dataabsoluteevaluationh)) /\ exists ff_q_mce_mdr_positive_dataabsoluteevaluationhsf_prefix_an. mdr_nb_positive_dataabsoluteevaluationh = ff_q_mce_mdr_positive_dataabsoluteevaluationhsf_prefix_an * S ((S (ff_index_mce_alternating_mdr_positive_dataabsoluteevaluationhsf_prefix)) * mdr_nc_positive_dataabsoluteevaluationh) + (ff_an_mce_alternating_mdr_positive_dataabsoluteevaluationhsf_prefix))) /\ ((((exists ff_h_mce_mdr_positive_dataabsoluteevaluationhsf_prefix_bp. ff_h_mce_mdr_positive_dataabsoluteevaluationhsf_prefix_bp + S (ff_bp_mce_alternating_mdr_positive_dataabsoluteevaluationhsf_prefix) = S ((S (ff_index_mce_alternating_mdr_positive_dataabsoluteevaluationhsf_prefix)) * mdr_ec_positive_dataabsoluteevaluationhs)) /\ exists ff_q_mce_mdr_positive_dataabsoluteevaluationhsf_prefix_bp. mdr_eb_positive_dataabsoluteevaluationhs = ff_q_mce_mdr_positive_dataabsoluteevaluationhsf_prefix_bp * S ((S (ff_index_mce_alternating_mdr_positive_dataabsoluteevaluationhsf_prefix)) * mdr_ec_positive_dataabsoluteevaluationhs) + (ff_bp_mce_alternating_mdr_positive_dataabsoluteevaluationhsf_prefix))) /\ ((((exists ff_h_mce_mdr_positive_dataabsoluteevaluationhsf_prefix_bn. ff_h_mce_mdr_positive_dataabsoluteevaluationhsf_prefix_bn + S (ff_bn_mce_alternating_mdr_positive_dataabsoluteevaluationhsf_prefix) = S ((S (ff_index_mce_alternating_mdr_positive_dataabsoluteevaluationhsf_prefix)) * mdr_fc_positive_dataabsoluteevaluationhs)) /\ exists ff_q_mce_mdr_positive_dataabsoluteevaluationhsf_prefix_bn. mdr_fb_positive_dataabsoluteevaluationhs = ff_q_mce_mdr_positive_dataabsoluteevaluationhsf_prefix_bn * S ((S (ff_index_mce_alternating_mdr_positive_dataabsoluteevaluationhsf_prefix)) * mdr_fc_positive_dataabsoluteevaluationhs) + (ff_bn_mce_alternating_mdr_positive_dataabsoluteevaluationhsf_prefix))) /\ ((((exists ff_h_mce_mdr_positive_dataabsoluteevaluationhsf_prefix_positive. ff_h_mce_mdr_positive_dataabsoluteevaluationhsf_prefix_positive + S (ff_p_mce_alternating_mdr_positive_dataabsoluteevaluationhsf_prefix) = S ((S (ff_index_mce_alternating_mdr_positive_dataabsoluteevaluationhsf_prefix)) * ff_uc_mce_fold_mdr_positive_dataabsoluteevaluationhsf)) /\ exists ff_q_mce_mdr_positive_dataabsoluteevaluationhsf_prefix_positive. ff_ub_mce_fold_mdr_positive_dataabsoluteevaluationhsf = ff_q_mce_mdr_positive_dataabsoluteevaluationhsf_prefix_positive * S ((S (ff_index_mce_alternating_mdr_positive_dataabsoluteevaluationhsf_prefix)) * ff_uc_mce_fold_mdr_positive_dataabsoluteevaluationhsf) + (ff_p_mce_alternating_mdr_positive_dataabsoluteevaluationhsf_prefix))) /\ ((((exists ff_h_mce_mdr_positive_dataabsoluteevaluationhsf_prefix_negative. ff_h_mce_mdr_positive_dataabsoluteevaluationhsf_prefix_negative + S (ff_n_mce_alternating_mdr_positive_dataabsoluteevaluationhsf_prefix) = S ((S (ff_index_mce_alternating_mdr_positive_dataabsoluteevaluationhsf_prefix)) * ff_vc_mce_fold_mdr_positive_dataabsoluteevaluationhsf)) /\ exists ff_q_mce_mdr_positive_dataabsoluteevaluationhsf_prefix_negative. ff_vb_mce_fold_mdr_positive_dataabsoluteevaluationhsf = ff_q_mce_mdr_positive_dataabsoluteevaluationhsf_prefix_negative * S ((S (ff_index_mce_alternating_mdr_positive_dataabsoluteevaluationhsf_prefix)) * ff_vc_mce_fold_mdr_positive_dataabsoluteevaluationhsf) + (ff_n_mce_alternating_mdr_positive_dataabsoluteevaluationhsf_prefix))) /\ (((exists ff_even_mce_term_mdr_positive_dataabsoluteevaluationhsf_prefix_term. ff_index_mce_alternating_mdr_positive_dataabsoluteevaluationhsf_prefix = 2 * ff_even_mce_term_mdr_positive_dataabsoluteevaluationhsf_prefix_term) /\ (ff_p_mce_alternating_mdr_positive_dataabsoluteevaluationhsf_prefix = (ff_ap_mce_alternating_mdr_positive_dataabsoluteevaluationhsf_prefix) * (ff_bp_mce_alternating_mdr_positive_dataabsoluteevaluationhsf_prefix) + (ff_an_mce_alternating_mdr_positive_dataabsoluteevaluationhsf_prefix) * (ff_bn_mce_alternating_mdr_positive_dataabsoluteevaluationhsf_prefix) /\ ff_n_mce_alternating_mdr_positive_dataabsoluteevaluationhsf_prefix = (ff_ap_mce_alternating_mdr_positive_dataabsoluteevaluationhsf_prefix) * (ff_bn_mce_alternating_mdr_positive_dataabsoluteevaluationhsf_prefix) + (ff_an_mce_alternating_mdr_positive_dataabsoluteevaluationhsf_prefix) * (ff_bp_mce_alternating_mdr_positive_dataabsoluteevaluationhsf_prefix))) \/ ((exists ff_odd_mce_term_mdr_positive_dataabsoluteevaluationhsf_prefix_term. ff_index_mce_alternating_mdr_positive_dataabsoluteevaluationhsf_prefix = 2 * ff_odd_mce_term_mdr_positive_dataabsoluteevaluationhsf_prefix_term + 1) /\ (ff_p_mce_alternating_mdr_positive_dataabsoluteevaluationhsf_prefix = (ff_ap_mce_alternating_mdr_positive_dataabsoluteevaluationhsf_prefix) * (ff_bn_mce_alternating_mdr_positive_dataabsoluteevaluationhsf_prefix) + (ff_an_mce_alternating_mdr_positive_dataabsoluteevaluationhsf_prefix) * (ff_bp_mce_alternating_mdr_positive_dataabsoluteevaluationhsf_prefix) /\ ff_n_mce_alternating_mdr_positive_dataabsoluteevaluationhsf_prefix = (ff_ap_mce_alternating_mdr_positive_dataabsoluteevaluationhsf_prefix) * (ff_bp_mce_alternating_mdr_positive_dataabsoluteevaluationhsf_prefix) + (ff_an_mce_alternating_mdr_positive_dataabsoluteevaluationhsf_prefix) * (ff_bn_mce_alternating_mdr_positive_dataabsoluteevaluationhsf_prefix))))))))))) /\ ((exists ff_u_mce_mdr_positive_dataabsoluteevaluationhsf_positive ff_v_mce_mdr_positive_dataabsoluteevaluationhsf_positive. ((((exists ff_h_mce_mdr_positive_dataabsoluteevaluationhsf_positive_start. ff_h_mce_mdr_positive_dataabsoluteevaluationhsf_positive_start + S (0) = S ((S (0)) * ff_v_mce_mdr_positive_dataabsoluteevaluationhsf_positive)) /\ exists ff_q_mce_mdr_positive_dataabsoluteevaluationhsf_positive_start. ff_u_mce_mdr_positive_dataabsoluteevaluationhsf_positive = ff_q_mce_mdr_positive_dataabsoluteevaluationhsf_positive_start * S ((S (0)) * ff_v_mce_mdr_positive_dataabsoluteevaluationhsf_positive) + (0))) /\ ((((exists ff_h_mce_mdr_positive_dataabsoluteevaluationhsf_positive_terminal. ff_h_mce_mdr_positive_dataabsoluteevaluationhsf_positive_terminal + S (mdr_p_positive_dataabsoluteevaluationh) = S ((S ((S (mdr_q_positive_dataabsoluteevaluationhs)))) * ff_v_mce_mdr_positive_dataabsoluteevaluationhsf_positive)) /\ exists ff_q_mce_mdr_positive_dataabsoluteevaluationhsf_positive_terminal. ff_u_mce_mdr_positive_dataabsoluteevaluationhsf_positive = ff_q_mce_mdr_positive_dataabsoluteevaluationhsf_positive_terminal * S ((S ((S (mdr_q_positive_dataabsoluteevaluationhs)))) * ff_v_mce_mdr_positive_dataabsoluteevaluationhsf_positive) + (mdr_p_positive_dataabsoluteevaluationh))) /\ forall ff_i_mce_mdr_positive_dataabsoluteevaluationhsf_positive. (exists ff_lt_mce_mdr_positive_dataabsoluteevaluationhsf_positive_bound. ff_lt_mce_mdr_positive_dataabsoluteevaluationhsf_positive_bound + S ff_i_mce_mdr_positive_dataabsoluteevaluationhsf_positive = (S (mdr_q_positive_dataabsoluteevaluationhs))) -> exists ff_a_mce_mdr_positive_dataabsoluteevaluationhsf_positive ff_r_mce_mdr_positive_dataabsoluteevaluationhsf_positive ff_s_mce_mdr_positive_dataabsoluteevaluationhsf_positive. ((((exists ff_h_mce_mdr_positive_dataabsoluteevaluationhsf_positive_summand. ff_h_mce_mdr_positive_dataabsoluteevaluationhsf_positive_summand + S (ff_a_mce_mdr_positive_dataabsoluteevaluationhsf_positive) = S ((S (ff_i_mce_mdr_positive_dataabsoluteevaluationhsf_positive)) * ff_uc_mce_fold_mdr_positive_dataabsoluteevaluationhsf)) /\ exists ff_q_mce_mdr_positive_dataabsoluteevaluationhsf_positive_summand. ff_ub_mce_fold_mdr_positive_dataabsoluteevaluationhsf = ff_q_mce_mdr_positive_dataabsoluteevaluationhsf_positive_summand * S ((S (ff_i_mce_mdr_positive_dataabsoluteevaluationhsf_positive)) * ff_uc_mce_fold_mdr_positive_dataabsoluteevaluationhsf) + (ff_a_mce_mdr_positive_dataabsoluteevaluationhsf_positive))) /\ ((((exists ff_h_mce_mdr_positive_dataabsoluteevaluationhsf_positive_partial. ff_h_mce_mdr_positive_dataabsoluteevaluationhsf_positive_partial + S (ff_r_mce_mdr_positive_dataabsoluteevaluationhsf_positive) = S ((S (ff_i_mce_mdr_positive_dataabsoluteevaluationhsf_positive)) * ff_v_mce_mdr_positive_dataabsoluteevaluationhsf_positive)) /\ exists ff_q_mce_mdr_positive_dataabsoluteevaluationhsf_positive_partial. ff_u_mce_mdr_positive_dataabsoluteevaluationhsf_positive = ff_q_mce_mdr_positive_dataabsoluteevaluationhsf_positive_partial * S ((S (ff_i_mce_mdr_positive_dataabsoluteevaluationhsf_positive)) * ff_v_mce_mdr_positive_dataabsoluteevaluationhsf_positive) + (ff_r_mce_mdr_positive_dataabsoluteevaluationhsf_positive))) /\ ((((exists ff_h_mce_mdr_positive_dataabsoluteevaluationhsf_positive_successor. ff_h_mce_mdr_positive_dataabsoluteevaluationhsf_positive_successor + S (ff_s_mce_mdr_positive_dataabsoluteevaluationhsf_positive) = S ((S (S ff_i_mce_mdr_positive_dataabsoluteevaluationhsf_positive)) * ff_v_mce_mdr_positive_dataabsoluteevaluationhsf_positive)) /\ exists ff_q_mce_mdr_positive_dataabsoluteevaluationhsf_positive_successor. ff_u_mce_mdr_positive_dataabsoluteevaluationhsf_positive = ff_q_mce_mdr_positive_dataabsoluteevaluationhsf_positive_successor * S ((S (S ff_i_mce_mdr_positive_dataabsoluteevaluationhsf_positive)) * ff_v_mce_mdr_positive_dataabsoluteevaluationhsf_positive) + (ff_s_mce_mdr_positive_dataabsoluteevaluationhsf_positive))) /\ ff_s_mce_mdr_positive_dataabsoluteevaluationhsf_positive = ff_r_mce_mdr_positive_dataabsoluteevaluationhsf_positive + ff_a_mce_mdr_positive_dataabsoluteevaluationhsf_positive)))))) /\ (exists ff_u_mce_mdr_positive_dataabsoluteevaluationhsf_negative ff_v_mce_mdr_positive_dataabsoluteevaluationhsf_negative. ((((exists ff_h_mce_mdr_positive_dataabsoluteevaluationhsf_negative_start. ff_h_mce_mdr_positive_dataabsoluteevaluationhsf_negative_start + S (0) = S ((S (0)) * ff_v_mce_mdr_positive_dataabsoluteevaluationhsf_negative)) /\ exists ff_q_mce_mdr_positive_dataabsoluteevaluationhsf_negative_start. ff_u_mce_mdr_positive_dataabsoluteevaluationhsf_negative = ff_q_mce_mdr_positive_dataabsoluteevaluationhsf_negative_start * S ((S (0)) * ff_v_mce_mdr_positive_dataabsoluteevaluationhsf_negative) + (0))) /\ ((((exists ff_h_mce_mdr_positive_dataabsoluteevaluationhsf_negative_terminal. ff_h_mce_mdr_positive_dataabsoluteevaluationhsf_negative_terminal + S (mdr_n_positive_dataabsoluteevaluationh) = S ((S ((S (mdr_q_positive_dataabsoluteevaluationhs)))) * ff_v_mce_mdr_positive_dataabsoluteevaluationhsf_negative)) /\ exists ff_q_mce_mdr_positive_dataabsoluteevaluationhsf_negative_terminal. ff_u_mce_mdr_positive_dataabsoluteevaluationhsf_negative = ff_q_mce_mdr_positive_dataabsoluteevaluationhsf_negative_terminal * S ((S ((S (mdr_q_positive_dataabsoluteevaluationhs)))) * ff_v_mce_mdr_positive_dataabsoluteevaluationhsf_negative) + (mdr_n_positive_dataabsoluteevaluationh))) /\ forall ff_i_mce_mdr_positive_dataabsoluteevaluationhsf_negative. (exists ff_lt_mce_mdr_positive_dataabsoluteevaluationhsf_negative_bound. ff_lt_mce_mdr_positive_dataabsoluteevaluationhsf_negative_bound + S ff_i_mce_mdr_positive_dataabsoluteevaluationhsf_negative = (S (mdr_q_positive_dataabsoluteevaluationhs))) -> exists ff_a_mce_mdr_positive_dataabsoluteevaluationhsf_negative ff_r_mce_mdr_positive_dataabsoluteevaluationhsf_negative ff_s_mce_mdr_positive_dataabsoluteevaluationhsf_negative. ((((exists ff_h_mce_mdr_positive_dataabsoluteevaluationhsf_negative_summand. ff_h_mce_mdr_positive_dataabsoluteevaluationhsf_negative_summand + S (ff_a_mce_mdr_positive_dataabsoluteevaluationhsf_negative) = S ((S (ff_i_mce_mdr_positive_dataabsoluteevaluationhsf_negative)) * ff_vc_mce_fold_mdr_positive_dataabsoluteevaluationhsf)) /\ exists ff_q_mce_mdr_positive_dataabsoluteevaluationhsf_negative_summand. ff_vb_mce_fold_mdr_positive_dataabsoluteevaluationhsf = ff_q_mce_mdr_positive_dataabsoluteevaluationhsf_negative_summand * S ((S (ff_i_mce_mdr_positive_dataabsoluteevaluationhsf_negative)) * ff_vc_mce_fold_mdr_positive_dataabsoluteevaluationhsf) + (ff_a_mce_mdr_positive_dataabsoluteevaluationhsf_negative))) /\ ((((exists ff_h_mce_mdr_positive_dataabsoluteevaluationhsf_negative_partial. ff_h_mce_mdr_positive_dataabsoluteevaluationhsf_negative_partial + S (ff_r_mce_mdr_positive_dataabsoluteevaluationhsf_negative) = S ((S (ff_i_mce_mdr_positive_dataabsoluteevaluationhsf_negative)) * ff_v_mce_mdr_positive_dataabsoluteevaluationhsf_negative)) /\ exists ff_q_mce_mdr_positive_dataabsoluteevaluationhsf_negative_partial. ff_u_mce_mdr_positive_dataabsoluteevaluationhsf_negative = ff_q_mce_mdr_positive_dataabsoluteevaluationhsf_negative_partial * S ((S (ff_i_mce_mdr_positive_dataabsoluteevaluationhsf_negative)) * ff_v_mce_mdr_positive_dataabsoluteevaluationhsf_negative) + (ff_r_mce_mdr_positive_dataabsoluteevaluationhsf_negative))) /\ ((((exists ff_h_mce_mdr_positive_dataabsoluteevaluationhsf_negative_successor. ff_h_mce_mdr_positive_dataabsoluteevaluationhsf_negative_successor + S (ff_s_mce_mdr_positive_dataabsoluteevaluationhsf_negative) = S ((S (S ff_i_mce_mdr_positive_dataabsoluteevaluationhsf_negative)) * ff_v_mce_mdr_positive_dataabsoluteevaluationhsf_negative)) /\ exists ff_q_mce_mdr_positive_dataabsoluteevaluationhsf_negative_successor. ff_u_mce_mdr_positive_dataabsoluteevaluationhsf_negative = ff_q_mce_mdr_positive_dataabsoluteevaluationhsf_negative_successor * S ((S (S ff_i_mce_mdr_positive_dataabsoluteevaluationhsf_negative)) * ff_v_mce_mdr_positive_dataabsoluteevaluationhsf_negative) + (ff_s_mce_mdr_positive_dataabsoluteevaluationhsf_negative))) /\ ff_s_mce_mdr_positive_dataabsoluteevaluationhsf_negative = ff_r_mce_mdr_positive_dataabsoluteevaluationhsf_negative + ff_a_mce_mdr_positive_dataabsoluteevaluationhsf_negative))))))))))))))) /\ ((exists mdr_gap_positive_dataabsoluteevaluationi. mdr_gap_positive_dataabsoluteevaluationi + S (mdr_i_positive_dataabsoluteevaluation) = (mdr_l_positive_dataabsoluteevaluation)) /\ (exists mdr_z_positive_dataabsoluteevaluationr. ((exists mdr_a_positive_dataabsoluteevaluationrc mdr_b_positive_dataabsoluteevaluationrc mdr_c_positive_dataabsoluteevaluationrc mdr_e_positive_dataabsoluteevaluationrc mdr_f_positive_dataabsoluteevaluationrc. ((mdr_a_positive_dataabsoluteevaluationrc = ((d) + (ab)) * S ((d) + (ab)) + ((ab) + (ab))) /\ ((mdr_b_positive_dataabsoluteevaluationrc = ((ac) + (bb)) * S ((ac) + (bb)) + ((bb) + (bb))) /\ ((mdr_c_positive_dataabsoluteevaluationrc = ((mdr_a_positive_dataabsoluteevaluationrc) + (mdr_b_positive_dataabsoluteevaluationrc)) * S ((mdr_a_positive_dataabsoluteevaluationrc) + (mdr_b_positive_dataabsoluteevaluationrc)) + ((mdr_b_positive_dataabsoluteevaluationrc) + (mdr_b_positive_dataabsoluteevaluationrc))) /\ ((mdr_e_positive_dataabsoluteevaluationrc = ((mdr_p_positive_dataabsolute) + (mdr_n_positive_dataabsolute)) * S ((mdr_p_positive_dataabsolute) + (mdr_n_positive_dataabsolute)) + ((mdr_n_positive_dataabsolute) + (mdr_n_positive_dataabsolute))) /\ ((mdr_f_positive_dataabsoluteevaluationrc = ((bc) + (mdr_e_positive_dataabsoluteevaluationrc)) * S ((bc) + (mdr_e_positive_dataabsoluteevaluationrc)) + ((mdr_e_positive_dataabsoluteevaluationrc) + (mdr_e_positive_dataabsoluteevaluationrc))) /\ ((mdr_z_positive_dataabsoluteevaluationr) = ((mdr_c_positive_dataabsoluteevaluationrc) + (mdr_f_positive_dataabsoluteevaluationrc)) * S ((mdr_c_positive_dataabsoluteevaluationrc) + (mdr_f_positive_dataabsoluteevaluationrc)) + ((mdr_f_positive_dataabsoluteevaluationrc) + (mdr_f_positive_dataabsoluteevaluationrc))))))))) /\ (((exists ff_h_mdr_positive_dataabsoluteevaluationrb. ff_h_mdr_positive_dataabsoluteevaluationrb + S (mdr_z_positive_dataabsoluteevaluationr) = S ((S (mdr_i_positive_dataabsoluteevaluation)) * mdr_c_positive_dataabsoluteevaluation)) /\ exists ff_q_mdr_positive_dataabsoluteevaluationrb. mdr_b_positive_dataabsoluteevaluation = ff_q_mdr_positive_dataabsoluteevaluationrb * S ((S (mdr_i_positive_dataabsoluteevaluation)) * mdr_c_positive_dataabsoluteevaluation) + (mdr_z_positive_dataabsoluteevaluationr)))))))) /\ (((mdr_p_positive_dataabsolute) = (mdr_n_positive_dataabsolute) + (D)) \/ ((mdr_n_positive_dataabsolute) = (mdr_p_positive_dataabsolute) + (D)))))))) -> exists p n. (((exists mdr_b_positive_actual_det mdr_c_positive_actual_det mdr_l_positive_actual_det mdr_i_positive_actual_det. ((forall mdr_i_positive_actual_deth. (exists mdr_gap_positive_actual_dethi. mdr_gap_positive_actual_dethi + S (mdr_i_positive_actual_deth) = (mdr_l_positive_actual_det)) -> exists mdr_d_positive_actual_deth mdr_pb_positive_actual_deth mdr_pc_positive_actual_deth mdr_nb_positive_actual_deth mdr_nc_positive_actual_deth mdr_p_positive_actual_deth mdr_n_positive_actual_deth. ((exists mdr_z_positive_actual_dethr. ((exists mdr_a_positive_actual_dethrc mdr_b_positive_actual_dethrc mdr_c_positive_actual_dethrc mdr_e_positive_actual_dethrc mdr_f_positive_actual_dethrc. ((mdr_a_positive_actual_dethrc = ((mdr_d_positive_actual_deth) + (mdr_pb_positive_actual_deth)) * S ((mdr_d_positive_actual_deth) + (mdr_pb_positive_actual_deth)) + ((mdr_pb_positive_actual_deth) + (mdr_pb_positive_actual_deth))) /\ ((mdr_b_positive_actual_dethrc = ((mdr_pc_positive_actual_deth) + (mdr_nb_positive_actual_deth)) * S ((mdr_pc_positive_actual_deth) + (mdr_nb_positive_actual_deth)) + ((mdr_nb_positive_actual_deth) + (mdr_nb_positive_actual_deth))) /\ ((mdr_c_positive_actual_dethrc = ((mdr_a_positive_actual_dethrc) + (mdr_b_positive_actual_dethrc)) * S ((mdr_a_positive_actual_dethrc) + (mdr_b_positive_actual_dethrc)) + ((mdr_b_positive_actual_dethrc) + (mdr_b_positive_actual_dethrc))) /\ ((mdr_e_positive_actual_dethrc = ((mdr_p_positive_actual_deth) + (mdr_n_positive_actual_deth)) * S ((mdr_p_positive_actual_deth) + (mdr_n_positive_actual_deth)) + ((mdr_n_positive_actual_deth) + (mdr_n_positive_actual_deth))) /\ ((mdr_f_positive_actual_dethrc = ((mdr_nc_positive_actual_deth) + (mdr_e_positive_actual_dethrc)) * S ((mdr_nc_positive_actual_deth) + (mdr_e_positive_actual_dethrc)) + ((mdr_e_positive_actual_dethrc) + (mdr_e_positive_actual_dethrc))) /\ ((mdr_z_positive_actual_dethr) = ((mdr_c_positive_actual_dethrc) + (mdr_f_positive_actual_dethrc)) * S ((mdr_c_positive_actual_dethrc) + (mdr_f_positive_actual_dethrc)) + ((mdr_f_positive_actual_dethrc) + (mdr_f_positive_actual_dethrc))))))))) /\ (((exists ff_h_mdr_positive_actual_dethrb. ff_h_mdr_positive_actual_dethrb + S (mdr_z_positive_actual_dethr) = S ((S (mdr_i_positive_actual_deth)) * mdr_c_positive_actual_det)) /\ exists ff_q_mdr_positive_actual_dethrb. mdr_b_positive_actual_det = ff_q_mdr_positive_actual_dethrb * S ((S (mdr_i_positive_actual_deth)) * mdr_c_positive_actual_det) + (mdr_z_positive_actual_dethr))))) /\ (((((mdr_d_positive_actual_deth) = 0) /\ (((mdr_p_positive_actual_deth) = 1) /\ ((mdr_n_positive_actual_deth) = 0))) \/ exists mdr_q_positive_actual_deths mdr_eb_positive_actual_deths mdr_ec_positive_actual_deths mdr_fb_positive_actual_deths mdr_fc_positive_actual_deths. (((mdr_d_positive_actual_deth) = S (mdr_q_positive_actual_deths)) /\ ((forall mdr_j_positive_actual_dethsc. (exists mdr_gap_positive_actual_dethscj. mdr_gap_positive_actual_dethscj + S (mdr_j_positive_actual_dethsc) = (S (mdr_q_positive_actual_deths))) -> exists mdr_i_positive_actual_dethsc mdr_up_positive_actual_dethsc mdr_us_positive_actual_dethsc mdr_un_positive_actual_dethsc mdr_ut_positive_actual_dethsc mdr_p_positive_actual_dethsc mdr_n_positive_actual_dethsc. ((exists mdr_gap_positive_actual_dethsci. mdr_gap_positive_actual_dethsci + S (mdr_i_positive_actual_dethsc) = (mdr_i_positive_actual_deth)) /\ ((exists mdr_z_positive_actual_dethscr. ((exists mdr_a_positive_actual_dethscrc mdr_b_positive_actual_dethscrc mdr_c_positive_actual_dethscrc mdr_e_positive_actual_dethscrc mdr_f_positive_actual_dethscrc. ((mdr_a_positive_actual_dethscrc = ((mdr_q_positive_actual_deths) + (mdr_up_positive_actual_dethsc)) * S ((mdr_q_positive_actual_deths) + (mdr_up_positive_actual_dethsc)) + ((mdr_up_positive_actual_dethsc) + (mdr_up_positive_actual_dethsc))) /\ ((mdr_b_positive_actual_dethscrc = ((mdr_us_positive_actual_dethsc) + (mdr_un_positive_actual_dethsc)) * S ((mdr_us_positive_actual_dethsc) + (mdr_un_positive_actual_dethsc)) + ((mdr_un_positive_actual_dethsc) + (mdr_un_positive_actual_dethsc))) /\ ((mdr_c_positive_actual_dethscrc = ((mdr_a_positive_actual_dethscrc) + (mdr_b_positive_actual_dethscrc)) * S ((mdr_a_positive_actual_dethscrc) + (mdr_b_positive_actual_dethscrc)) + ((mdr_b_positive_actual_dethscrc) + (mdr_b_positive_actual_dethscrc))) /\ ((mdr_e_positive_actual_dethscrc = ((mdr_p_positive_actual_dethsc) + (mdr_n_positive_actual_dethsc)) * S ((mdr_p_positive_actual_dethsc) + (mdr_n_positive_actual_dethsc)) + ((mdr_n_positive_actual_dethsc) + (mdr_n_positive_actual_dethsc))) /\ ((mdr_f_positive_actual_dethscrc = ((mdr_ut_positive_actual_dethsc) + (mdr_e_positive_actual_dethscrc)) * S ((mdr_ut_positive_actual_dethsc) + (mdr_e_positive_actual_dethscrc)) + ((mdr_e_positive_actual_dethscrc) + (mdr_e_positive_actual_dethscrc))) /\ ((mdr_z_positive_actual_dethscr) = ((mdr_c_positive_actual_dethscrc) + (mdr_f_positive_actual_dethscrc)) * S ((mdr_c_positive_actual_dethscrc) + (mdr_f_positive_actual_dethscrc)) + ((mdr_f_positive_actual_dethscrc) + (mdr_f_positive_actual_dethscrc))))))))) /\ (((exists ff_h_mdr_positive_actual_dethscrb. ff_h_mdr_positive_actual_dethscrb + S (mdr_z_positive_actual_dethscr) = S ((S (mdr_i_positive_actual_dethsc)) * mdr_c_positive_actual_det)) /\ exists ff_q_mdr_positive_actual_dethscrb. mdr_b_positive_actual_det = ff_q_mdr_positive_actual_dethscrb * S ((S (mdr_i_positive_actual_dethsc)) * mdr_c_positive_actual_det) + (mdr_z_positive_actual_dethscr))))) /\ ((((forall ff_index_mdm_prefix_mdr_positive_actual_dethscm_positive. (exists ff_gap_mdm_lt_mdr_positive_actual_dethscm_positive_index_bound. ff_gap_mdm_lt_mdr_positive_actual_dethscm_positive_index_bound + S (ff_index_mdm_prefix_mdr_positive_actual_dethscm_positive) = ((mdr_q_positive_actual_deths) * (mdr_q_positive_actual_deths))) -> exists ff_row_mdm_prefix_mdr_positive_actual_dethscm_positive ff_column_mdm_prefix_mdr_positive_actual_dethscm_positive ff_value_mdm_prefix_mdr_positive_actual_dethscm_positive. (ff_index_mdm_prefix_mdr_positive_actual_dethscm_positive = (mdr_q_positive_actual_deths) * ff_row_mdm_prefix_mdr_positive_actual_dethscm_positive + ff_column_mdm_prefix_mdr_positive_actual_dethscm_positive /\ ((exists ff_gap_mdm_lt_mdr_positive_actual_dethscm_positive_column_bound. ff_gap_mdm_lt_mdr_positive_actual_dethscm_positive_column_bound + S (ff_column_mdm_prefix_mdr_positive_actual_dethscm_positive) = (mdr_q_positive_actual_deths)) /\ ((exists ff_row_mdm_cell_mdr_positive_actual_dethscm_positive_cell ff_column_mdm_cell_mdr_positive_actual_dethscm_positive_cell. (((((exists ff_gap_mdm_lt_mdr_positive_actual_dethscm_positive_cell_row_before. ff_gap_mdm_lt_mdr_positive_actual_dethscm_positive_cell_row_before + S (ff_row_mdm_prefix_mdr_positive_actual_dethscm_positive) = (0)) /\ ff_row_mdm_cell_mdr_positive_actual_dethscm_positive_cell = ff_row_mdm_prefix_mdr_positive_actual_dethscm_positive) \/ ((exists ff_gap_mdm_le_mdr_positive_actual_dethscm_positive_cell_row_after. ff_gap_mdm_le_mdr_positive_actual_dethscm_positive_cell_row_after + (0) = (ff_row_mdm_prefix_mdr_positive_actual_dethscm_positive)) /\ ff_row_mdm_cell_mdr_positive_actual_dethscm_positive_cell = S ff_row_mdm_prefix_mdr_positive_actual_dethscm_positive))) /\ (((((exists ff_gap_mdm_lt_mdr_positive_actual_dethscm_positive_cell_column_before. ff_gap_mdm_lt_mdr_positive_actual_dethscm_positive_cell_column_before + S (ff_column_mdm_prefix_mdr_positive_actual_dethscm_positive) = (mdr_j_positive_actual_dethsc)) /\ ff_column_mdm_cell_mdr_positive_actual_dethscm_positive_cell = ff_column_mdm_prefix_mdr_positive_actual_dethscm_positive) \/ ((exists ff_gap_mdm_le_mdr_positive_actual_dethscm_positive_cell_column_after. ff_gap_mdm_le_mdr_positive_actual_dethscm_positive_cell_column_after + (mdr_j_positive_actual_dethsc) = (ff_column_mdm_prefix_mdr_positive_actual_dethscm_positive)) /\ ff_column_mdm_cell_mdr_positive_actual_dethscm_positive_cell = S ff_column_mdm_prefix_mdr_positive_actual_dethscm_positive))) /\ (((exists ff_h_mdm_mdr_positive_actual_dethscm_positive_cell_source. ff_h_mdm_mdr_positive_actual_dethscm_positive_cell_source + S (ff_value_mdm_prefix_mdr_positive_actual_dethscm_positive) = S ((S ((ff_row_mdm_cell_mdr_positive_actual_dethscm_positive_cell) * (S (mdr_q_positive_actual_deths)) + (ff_column_mdm_cell_mdr_positive_actual_dethscm_positive_cell))) * mdr_pc_positive_actual_deth)) /\ exists ff_q_mdm_mdr_positive_actual_dethscm_positive_cell_source. mdr_pb_positive_actual_deth = ff_q_mdm_mdr_positive_actual_dethscm_positive_cell_source * S ((S ((ff_row_mdm_cell_mdr_positive_actual_dethscm_positive_cell) * (S (mdr_q_positive_actual_deths)) + (ff_column_mdm_cell_mdr_positive_actual_dethscm_positive_cell))) * mdr_pc_positive_actual_deth) + (ff_value_mdm_prefix_mdr_positive_actual_dethscm_positive)))))) /\ (((exists ff_h_mdm_mdr_positive_actual_dethscm_positive_target. ff_h_mdm_mdr_positive_actual_dethscm_positive_target + S (ff_value_mdm_prefix_mdr_positive_actual_dethscm_positive) = S ((S (ff_index_mdm_prefix_mdr_positive_actual_dethscm_positive)) * mdr_us_positive_actual_dethsc)) /\ exists ff_q_mdm_mdr_positive_actual_dethscm_positive_target. mdr_up_positive_actual_dethsc = ff_q_mdm_mdr_positive_actual_dethscm_positive_target * S ((S (ff_index_mdm_prefix_mdr_positive_actual_dethscm_positive)) * mdr_us_positive_actual_dethsc) + (ff_value_mdm_prefix_mdr_positive_actual_dethscm_positive))))))) /\ (forall ff_index_mdm_prefix_mdr_positive_actual_dethscm_negative. (exists ff_gap_mdm_lt_mdr_positive_actual_dethscm_negative_index_bound. ff_gap_mdm_lt_mdr_positive_actual_dethscm_negative_index_bound + S (ff_index_mdm_prefix_mdr_positive_actual_dethscm_negative) = ((mdr_q_positive_actual_deths) * (mdr_q_positive_actual_deths))) -> exists ff_row_mdm_prefix_mdr_positive_actual_dethscm_negative ff_column_mdm_prefix_mdr_positive_actual_dethscm_negative ff_value_mdm_prefix_mdr_positive_actual_dethscm_negative. (ff_index_mdm_prefix_mdr_positive_actual_dethscm_negative = (mdr_q_positive_actual_deths) * ff_row_mdm_prefix_mdr_positive_actual_dethscm_negative + ff_column_mdm_prefix_mdr_positive_actual_dethscm_negative /\ ((exists ff_gap_mdm_lt_mdr_positive_actual_dethscm_negative_column_bound. ff_gap_mdm_lt_mdr_positive_actual_dethscm_negative_column_bound + S (ff_column_mdm_prefix_mdr_positive_actual_dethscm_negative) = (mdr_q_positive_actual_deths)) /\ ((exists ff_row_mdm_cell_mdr_positive_actual_dethscm_negative_cell ff_column_mdm_cell_mdr_positive_actual_dethscm_negative_cell. (((((exists ff_gap_mdm_lt_mdr_positive_actual_dethscm_negative_cell_row_before. ff_gap_mdm_lt_mdr_positive_actual_dethscm_negative_cell_row_before + S (ff_row_mdm_prefix_mdr_positive_actual_dethscm_negative) = (0)) /\ ff_row_mdm_cell_mdr_positive_actual_dethscm_negative_cell = ff_row_mdm_prefix_mdr_positive_actual_dethscm_negative) \/ ((exists ff_gap_mdm_le_mdr_positive_actual_dethscm_negative_cell_row_after. ff_gap_mdm_le_mdr_positive_actual_dethscm_negative_cell_row_after + (0) = (ff_row_mdm_prefix_mdr_positive_actual_dethscm_negative)) /\ ff_row_mdm_cell_mdr_positive_actual_dethscm_negative_cell = S ff_row_mdm_prefix_mdr_positive_actual_dethscm_negative))) /\ (((((exists ff_gap_mdm_lt_mdr_positive_actual_dethscm_negative_cell_column_before. ff_gap_mdm_lt_mdr_positive_actual_dethscm_negative_cell_column_before + S (ff_column_mdm_prefix_mdr_positive_actual_dethscm_negative) = (mdr_j_positive_actual_dethsc)) /\ ff_column_mdm_cell_mdr_positive_actual_dethscm_negative_cell = ff_column_mdm_prefix_mdr_positive_actual_dethscm_negative) \/ ((exists ff_gap_mdm_le_mdr_positive_actual_dethscm_negative_cell_column_after. ff_gap_mdm_le_mdr_positive_actual_dethscm_negative_cell_column_after + (mdr_j_positive_actual_dethsc) = (ff_column_mdm_prefix_mdr_positive_actual_dethscm_negative)) /\ ff_column_mdm_cell_mdr_positive_actual_dethscm_negative_cell = S ff_column_mdm_prefix_mdr_positive_actual_dethscm_negative))) /\ (((exists ff_h_mdm_mdr_positive_actual_dethscm_negative_cell_source. ff_h_mdm_mdr_positive_actual_dethscm_negative_cell_source + S (ff_value_mdm_prefix_mdr_positive_actual_dethscm_negative) = S ((S ((ff_row_mdm_cell_mdr_positive_actual_dethscm_negative_cell) * (S (mdr_q_positive_actual_deths)) + (ff_column_mdm_cell_mdr_positive_actual_dethscm_negative_cell))) * mdr_nc_positive_actual_deth)) /\ exists ff_q_mdm_mdr_positive_actual_dethscm_negative_cell_source. mdr_nb_positive_actual_deth = ff_q_mdm_mdr_positive_actual_dethscm_negative_cell_source * S ((S ((ff_row_mdm_cell_mdr_positive_actual_dethscm_negative_cell) * (S (mdr_q_positive_actual_deths)) + (ff_column_mdm_cell_mdr_positive_actual_dethscm_negative_cell))) * mdr_nc_positive_actual_deth) + (ff_value_mdm_prefix_mdr_positive_actual_dethscm_negative)))))) /\ (((exists ff_h_mdm_mdr_positive_actual_dethscm_negative_target. ff_h_mdm_mdr_positive_actual_dethscm_negative_target + S (ff_value_mdm_prefix_mdr_positive_actual_dethscm_negative) = S ((S (ff_index_mdm_prefix_mdr_positive_actual_dethscm_negative)) * mdr_ut_positive_actual_dethsc)) /\ exists ff_q_mdm_mdr_positive_actual_dethscm_negative_target. mdr_un_positive_actual_dethsc = ff_q_mdm_mdr_positive_actual_dethscm_negative_target * S ((S (ff_index_mdm_prefix_mdr_positive_actual_dethscm_negative)) * mdr_ut_positive_actual_dethsc) + (ff_value_mdm_prefix_mdr_positive_actual_dethscm_negative))))))))) /\ ((((exists ff_h_mdr_positive_actual_dethscp. ff_h_mdr_positive_actual_dethscp + S (mdr_p_positive_actual_dethsc) = S ((S (mdr_j_positive_actual_dethsc)) * mdr_ec_positive_actual_deths)) /\ exists ff_q_mdr_positive_actual_dethscp. mdr_eb_positive_actual_deths = ff_q_mdr_positive_actual_dethscp * S ((S (mdr_j_positive_actual_dethsc)) * mdr_ec_positive_actual_deths) + (mdr_p_positive_actual_dethsc))) /\ (((exists ff_h_mdr_positive_actual_dethscn. ff_h_mdr_positive_actual_dethscn + S (mdr_n_positive_actual_dethsc) = S ((S (mdr_j_positive_actual_dethsc)) * mdr_fc_positive_actual_deths)) /\ exists ff_q_mdr_positive_actual_dethscn. mdr_fb_positive_actual_deths = ff_q_mdr_positive_actual_dethscn * S ((S (mdr_j_positive_actual_dethsc)) * mdr_fc_positive_actual_deths) + (mdr_n_positive_actual_dethsc)))))))) /\ (exists ff_ub_mce_fold_mdr_positive_actual_dethsf ff_uc_mce_fold_mdr_positive_actual_dethsf ff_vb_mce_fold_mdr_positive_actual_dethsf ff_vc_mce_fold_mdr_positive_actual_dethsf. ((forall ff_index_mce_alternating_mdr_positive_actual_dethsf_prefix. (exists ff_gap_mce_mdr_positive_actual_dethsf_prefix_index. ff_gap_mce_mdr_positive_actual_dethsf_prefix_index + S (ff_index_mce_alternating_mdr_positive_actual_dethsf_prefix) = (S (mdr_q_positive_actual_deths))) -> exists ff_ap_mce_alternating_mdr_positive_actual_dethsf_prefix ff_an_mce_alternating_mdr_positive_actual_dethsf_prefix ff_bp_mce_alternating_mdr_positive_actual_dethsf_prefix ff_bn_mce_alternating_mdr_positive_actual_dethsf_prefix ff_p_mce_alternating_mdr_positive_actual_dethsf_prefix ff_n_mce_alternating_mdr_positive_actual_dethsf_prefix. ((((exists ff_h_mce_mdr_positive_actual_dethsf_prefix_ap. ff_h_mce_mdr_positive_actual_dethsf_prefix_ap + S (ff_ap_mce_alternating_mdr_positive_actual_dethsf_prefix) = S ((S (ff_index_mce_alternating_mdr_positive_actual_dethsf_prefix)) * mdr_pc_positive_actual_deth)) /\ exists ff_q_mce_mdr_positive_actual_dethsf_prefix_ap. mdr_pb_positive_actual_deth = ff_q_mce_mdr_positive_actual_dethsf_prefix_ap * S ((S (ff_index_mce_alternating_mdr_positive_actual_dethsf_prefix)) * mdr_pc_positive_actual_deth) + (ff_ap_mce_alternating_mdr_positive_actual_dethsf_prefix))) /\ ((((exists ff_h_mce_mdr_positive_actual_dethsf_prefix_an. ff_h_mce_mdr_positive_actual_dethsf_prefix_an + S (ff_an_mce_alternating_mdr_positive_actual_dethsf_prefix) = S ((S (ff_index_mce_alternating_mdr_positive_actual_dethsf_prefix)) * mdr_nc_positive_actual_deth)) /\ exists ff_q_mce_mdr_positive_actual_dethsf_prefix_an. mdr_nb_positive_actual_deth = ff_q_mce_mdr_positive_actual_dethsf_prefix_an * S ((S (ff_index_mce_alternating_mdr_positive_actual_dethsf_prefix)) * mdr_nc_positive_actual_deth) + (ff_an_mce_alternating_mdr_positive_actual_dethsf_prefix))) /\ ((((exists ff_h_mce_mdr_positive_actual_dethsf_prefix_bp. ff_h_mce_mdr_positive_actual_dethsf_prefix_bp + S (ff_bp_mce_alternating_mdr_positive_actual_dethsf_prefix) = S ((S (ff_index_mce_alternating_mdr_positive_actual_dethsf_prefix)) * mdr_ec_positive_actual_deths)) /\ exists ff_q_mce_mdr_positive_actual_dethsf_prefix_bp. mdr_eb_positive_actual_deths = ff_q_mce_mdr_positive_actual_dethsf_prefix_bp * S ((S (ff_index_mce_alternating_mdr_positive_actual_dethsf_prefix)) * mdr_ec_positive_actual_deths) + (ff_bp_mce_alternating_mdr_positive_actual_dethsf_prefix))) /\ ((((exists ff_h_mce_mdr_positive_actual_dethsf_prefix_bn. ff_h_mce_mdr_positive_actual_dethsf_prefix_bn + S (ff_bn_mce_alternating_mdr_positive_actual_dethsf_prefix) = S ((S (ff_index_mce_alternating_mdr_positive_actual_dethsf_prefix)) * mdr_fc_positive_actual_deths)) /\ exists ff_q_mce_mdr_positive_actual_dethsf_prefix_bn. mdr_fb_positive_actual_deths = ff_q_mce_mdr_positive_actual_dethsf_prefix_bn * S ((S (ff_index_mce_alternating_mdr_positive_actual_dethsf_prefix)) * mdr_fc_positive_actual_deths) + (ff_bn_mce_alternating_mdr_positive_actual_dethsf_prefix))) /\ ((((exists ff_h_mce_mdr_positive_actual_dethsf_prefix_positive. ff_h_mce_mdr_positive_actual_dethsf_prefix_positive + S (ff_p_mce_alternating_mdr_positive_actual_dethsf_prefix) = S ((S (ff_index_mce_alternating_mdr_positive_actual_dethsf_prefix)) * ff_uc_mce_fold_mdr_positive_actual_dethsf)) /\ exists ff_q_mce_mdr_positive_actual_dethsf_prefix_positive. ff_ub_mce_fold_mdr_positive_actual_dethsf = ff_q_mce_mdr_positive_actual_dethsf_prefix_positive * S ((S (ff_index_mce_alternating_mdr_positive_actual_dethsf_prefix)) * ff_uc_mce_fold_mdr_positive_actual_dethsf) + (ff_p_mce_alternating_mdr_positive_actual_dethsf_prefix))) /\ ((((exists ff_h_mce_mdr_positive_actual_dethsf_prefix_negative. ff_h_mce_mdr_positive_actual_dethsf_prefix_negative + S (ff_n_mce_alternating_mdr_positive_actual_dethsf_prefix) = S ((S (ff_index_mce_alternating_mdr_positive_actual_dethsf_prefix)) * ff_vc_mce_fold_mdr_positive_actual_dethsf)) /\ exists ff_q_mce_mdr_positive_actual_dethsf_prefix_negative. ff_vb_mce_fold_mdr_positive_actual_dethsf = ff_q_mce_mdr_positive_actual_dethsf_prefix_negative * S ((S (ff_index_mce_alternating_mdr_positive_actual_dethsf_prefix)) * ff_vc_mce_fold_mdr_positive_actual_dethsf) + (ff_n_mce_alternating_mdr_positive_actual_dethsf_prefix))) /\ (((exists ff_even_mce_term_mdr_positive_actual_dethsf_prefix_term. ff_index_mce_alternating_mdr_positive_actual_dethsf_prefix = 2 * ff_even_mce_term_mdr_positive_actual_dethsf_prefix_term) /\ (ff_p_mce_alternating_mdr_positive_actual_dethsf_prefix = (ff_ap_mce_alternating_mdr_positive_actual_dethsf_prefix) * (ff_bp_mce_alternating_mdr_positive_actual_dethsf_prefix) + (ff_an_mce_alternating_mdr_positive_actual_dethsf_prefix) * (ff_bn_mce_alternating_mdr_positive_actual_dethsf_prefix) /\ ff_n_mce_alternating_mdr_positive_actual_dethsf_prefix = (ff_ap_mce_alternating_mdr_positive_actual_dethsf_prefix) * (ff_bn_mce_alternating_mdr_positive_actual_dethsf_prefix) + (ff_an_mce_alternating_mdr_positive_actual_dethsf_prefix) * (ff_bp_mce_alternating_mdr_positive_actual_dethsf_prefix))) \/ ((exists ff_odd_mce_term_mdr_positive_actual_dethsf_prefix_term. ff_index_mce_alternating_mdr_positive_actual_dethsf_prefix = 2 * ff_odd_mce_term_mdr_positive_actual_dethsf_prefix_term + 1) /\ (ff_p_mce_alternating_mdr_positive_actual_dethsf_prefix = (ff_ap_mce_alternating_mdr_positive_actual_dethsf_prefix) * (ff_bn_mce_alternating_mdr_positive_actual_dethsf_prefix) + (ff_an_mce_alternating_mdr_positive_actual_dethsf_prefix) * (ff_bp_mce_alternating_mdr_positive_actual_dethsf_prefix) /\ ff_n_mce_alternating_mdr_positive_actual_dethsf_prefix = (ff_ap_mce_alternating_mdr_positive_actual_dethsf_prefix) * (ff_bp_mce_alternating_mdr_positive_actual_dethsf_prefix) + (ff_an_mce_alternating_mdr_positive_actual_dethsf_prefix) * (ff_bn_mce_alternating_mdr_positive_actual_dethsf_prefix))))))))))) /\ ((exists ff_u_mce_mdr_positive_actual_dethsf_positive ff_v_mce_mdr_positive_actual_dethsf_positive. ((((exists ff_h_mce_mdr_positive_actual_dethsf_positive_start. ff_h_mce_mdr_positive_actual_dethsf_positive_start + S (0) = S ((S (0)) * ff_v_mce_mdr_positive_actual_dethsf_positive)) /\ exists ff_q_mce_mdr_positive_actual_dethsf_positive_start. ff_u_mce_mdr_positive_actual_dethsf_positive = ff_q_mce_mdr_positive_actual_dethsf_positive_start * S ((S (0)) * ff_v_mce_mdr_positive_actual_dethsf_positive) + (0))) /\ ((((exists ff_h_mce_mdr_positive_actual_dethsf_positive_terminal. ff_h_mce_mdr_positive_actual_dethsf_positive_terminal + S (mdr_p_positive_actual_deth) = S ((S ((S (mdr_q_positive_actual_deths)))) * ff_v_mce_mdr_positive_actual_dethsf_positive)) /\ exists ff_q_mce_mdr_positive_actual_dethsf_positive_terminal. ff_u_mce_mdr_positive_actual_dethsf_positive = ff_q_mce_mdr_positive_actual_dethsf_positive_terminal * S ((S ((S (mdr_q_positive_actual_deths)))) * ff_v_mce_mdr_positive_actual_dethsf_positive) + (mdr_p_positive_actual_deth))) /\ forall ff_i_mce_mdr_positive_actual_dethsf_positive. (exists ff_lt_mce_mdr_positive_actual_dethsf_positive_bound. ff_lt_mce_mdr_positive_actual_dethsf_positive_bound + S ff_i_mce_mdr_positive_actual_dethsf_positive = (S (mdr_q_positive_actual_deths))) -> exists ff_a_mce_mdr_positive_actual_dethsf_positive ff_r_mce_mdr_positive_actual_dethsf_positive ff_s_mce_mdr_positive_actual_dethsf_positive. ((((exists ff_h_mce_mdr_positive_actual_dethsf_positive_summand. ff_h_mce_mdr_positive_actual_dethsf_positive_summand + S (ff_a_mce_mdr_positive_actual_dethsf_positive) = S ((S (ff_i_mce_mdr_positive_actual_dethsf_positive)) * ff_uc_mce_fold_mdr_positive_actual_dethsf)) /\ exists ff_q_mce_mdr_positive_actual_dethsf_positive_summand. ff_ub_mce_fold_mdr_positive_actual_dethsf = ff_q_mce_mdr_positive_actual_dethsf_positive_summand * S ((S (ff_i_mce_mdr_positive_actual_dethsf_positive)) * ff_uc_mce_fold_mdr_positive_actual_dethsf) + (ff_a_mce_mdr_positive_actual_dethsf_positive))) /\ ((((exists ff_h_mce_mdr_positive_actual_dethsf_positive_partial. ff_h_mce_mdr_positive_actual_dethsf_positive_partial + S (ff_r_mce_mdr_positive_actual_dethsf_positive) = S ((S (ff_i_mce_mdr_positive_actual_dethsf_positive)) * ff_v_mce_mdr_positive_actual_dethsf_positive)) /\ exists ff_q_mce_mdr_positive_actual_dethsf_positive_partial. ff_u_mce_mdr_positive_actual_dethsf_positive = ff_q_mce_mdr_positive_actual_dethsf_positive_partial * S ((S (ff_i_mce_mdr_positive_actual_dethsf_positive)) * ff_v_mce_mdr_positive_actual_dethsf_positive) + (ff_r_mce_mdr_positive_actual_dethsf_positive))) /\ ((((exists ff_h_mce_mdr_positive_actual_dethsf_positive_successor. ff_h_mce_mdr_positive_actual_dethsf_positive_successor + S (ff_s_mce_mdr_positive_actual_dethsf_positive) = S ((S (S ff_i_mce_mdr_positive_actual_dethsf_positive)) * ff_v_mce_mdr_positive_actual_dethsf_positive)) /\ exists ff_q_mce_mdr_positive_actual_dethsf_positive_successor. ff_u_mce_mdr_positive_actual_dethsf_positive = ff_q_mce_mdr_positive_actual_dethsf_positive_successor * S ((S (S ff_i_mce_mdr_positive_actual_dethsf_positive)) * ff_v_mce_mdr_positive_actual_dethsf_positive) + (ff_s_mce_mdr_positive_actual_dethsf_positive))) /\ ff_s_mce_mdr_positive_actual_dethsf_positive = ff_r_mce_mdr_positive_actual_dethsf_positive + ff_a_mce_mdr_positive_actual_dethsf_positive)))))) /\ (exists ff_u_mce_mdr_positive_actual_dethsf_negative ff_v_mce_mdr_positive_actual_dethsf_negative. ((((exists ff_h_mce_mdr_positive_actual_dethsf_negative_start. ff_h_mce_mdr_positive_actual_dethsf_negative_start + S (0) = S ((S (0)) * ff_v_mce_mdr_positive_actual_dethsf_negative)) /\ exists ff_q_mce_mdr_positive_actual_dethsf_negative_start. ff_u_mce_mdr_positive_actual_dethsf_negative = ff_q_mce_mdr_positive_actual_dethsf_negative_start * S ((S (0)) * ff_v_mce_mdr_positive_actual_dethsf_negative) + (0))) /\ ((((exists ff_h_mce_mdr_positive_actual_dethsf_negative_terminal. ff_h_mce_mdr_positive_actual_dethsf_negative_terminal + S (mdr_n_positive_actual_deth) = S ((S ((S (mdr_q_positive_actual_deths)))) * ff_v_mce_mdr_positive_actual_dethsf_negative)) /\ exists ff_q_mce_mdr_positive_actual_dethsf_negative_terminal. ff_u_mce_mdr_positive_actual_dethsf_negative = ff_q_mce_mdr_positive_actual_dethsf_negative_terminal * S ((S ((S (mdr_q_positive_actual_deths)))) * ff_v_mce_mdr_positive_actual_dethsf_negative) + (mdr_n_positive_actual_deth))) /\ forall ff_i_mce_mdr_positive_actual_dethsf_negative. (exists ff_lt_mce_mdr_positive_actual_dethsf_negative_bound. ff_lt_mce_mdr_positive_actual_dethsf_negative_bound + S ff_i_mce_mdr_positive_actual_dethsf_negative = (S (mdr_q_positive_actual_deths))) -> exists ff_a_mce_mdr_positive_actual_dethsf_negative ff_r_mce_mdr_positive_actual_dethsf_negative ff_s_mce_mdr_positive_actual_dethsf_negative. ((((exists ff_h_mce_mdr_positive_actual_dethsf_negative_summand. ff_h_mce_mdr_positive_actual_dethsf_negative_summand + S (ff_a_mce_mdr_positive_actual_dethsf_negative) = S ((S (ff_i_mce_mdr_positive_actual_dethsf_negative)) * ff_vc_mce_fold_mdr_positive_actual_dethsf)) /\ exists ff_q_mce_mdr_positive_actual_dethsf_negative_summand. ff_vb_mce_fold_mdr_positive_actual_dethsf = ff_q_mce_mdr_positive_actual_dethsf_negative_summand * S ((S (ff_i_mce_mdr_positive_actual_dethsf_negative)) * ff_vc_mce_fold_mdr_positive_actual_dethsf) + (ff_a_mce_mdr_positive_actual_dethsf_negative))) /\ ((((exists ff_h_mce_mdr_positive_actual_dethsf_negative_partial. ff_h_mce_mdr_positive_actual_dethsf_negative_partial + S (ff_r_mce_mdr_positive_actual_dethsf_negative) = S ((S (ff_i_mce_mdr_positive_actual_dethsf_negative)) * ff_v_mce_mdr_positive_actual_dethsf_negative)) /\ exists ff_q_mce_mdr_positive_actual_dethsf_negative_partial. ff_u_mce_mdr_positive_actual_dethsf_negative = ff_q_mce_mdr_positive_actual_dethsf_negative_partial * S ((S (ff_i_mce_mdr_positive_actual_dethsf_negative)) * ff_v_mce_mdr_positive_actual_dethsf_negative) + (ff_r_mce_mdr_positive_actual_dethsf_negative))) /\ ((((exists ff_h_mce_mdr_positive_actual_dethsf_negative_successor. ff_h_mce_mdr_positive_actual_dethsf_negative_successor + S (ff_s_mce_mdr_positive_actual_dethsf_negative) = S ((S (S ff_i_mce_mdr_positive_actual_dethsf_negative)) * ff_v_mce_mdr_positive_actual_dethsf_negative)) /\ exists ff_q_mce_mdr_positive_actual_dethsf_negative_successor. ff_u_mce_mdr_positive_actual_dethsf_negative = ff_q_mce_mdr_positive_actual_dethsf_negative_successor * S ((S (S ff_i_mce_mdr_positive_actual_dethsf_negative)) * ff_v_mce_mdr_positive_actual_dethsf_negative) + (ff_s_mce_mdr_positive_actual_dethsf_negative))) /\ ff_s_mce_mdr_positive_actual_dethsf_negative = ff_r_mce_mdr_positive_actual_dethsf_negative + ff_a_mce_mdr_positive_actual_dethsf_negative))))))))))))))) /\ ((exists mdr_gap_positive_actual_deti. mdr_gap_positive_actual_deti + S (mdr_i_positive_actual_det) = (mdr_l_positive_actual_det)) /\ (exists mdr_z_positive_actual_detr. ((exists mdr_a_positive_actual_detrc mdr_b_positive_actual_detrc mdr_c_positive_actual_detrc mdr_e_positive_actual_detrc mdr_f_positive_actual_detrc. ((mdr_a_positive_actual_detrc = ((d) + (ab)) * S ((d) + (ab)) + ((ab) + (ab))) /\ ((mdr_b_positive_actual_detrc = ((ac) + (bb)) * S ((ac) + (bb)) + ((bb) + (bb))) /\ ((mdr_c_positive_actual_detrc = ((mdr_a_positive_actual_detrc) + (mdr_b_positive_actual_detrc)) * S ((mdr_a_positive_actual_detrc) + (mdr_b_positive_actual_detrc)) + ((mdr_b_positive_actual_detrc) + (mdr_b_positive_actual_detrc))) /\ ((mdr_e_positive_actual_detrc = ((p) + (n)) * S ((p) + (n)) + ((n) + (n))) /\ ((mdr_f_positive_actual_detrc = ((bc) + (mdr_e_positive_actual_detrc)) * S ((bc) + (mdr_e_positive_actual_detrc)) + ((mdr_e_positive_actual_detrc) + (mdr_e_positive_actual_detrc))) /\ ((mdr_z_positive_actual_detr) = ((mdr_c_positive_actual_detrc) + (mdr_f_positive_actual_detrc)) * S ((mdr_c_positive_actual_detrc) + (mdr_f_positive_actual_detrc)) + ((mdr_f_positive_actual_detrc) + (mdr_f_positive_actual_detrc))))))))) /\ (((exists ff_h_mdr_positive_actual_detrb. ff_h_mdr_positive_actual_detrb + S (mdr_z_positive_actual_detr) = S ((S (mdr_i_positive_actual_det)) * mdr_c_positive_actual_det)) /\ exists ff_q_mdr_positive_actual_detrb. mdr_b_positive_actual_det = ff_q_mdr_positive_actual_detrb * S ((S (mdr_i_positive_actual_det)) * mdr_c_positive_actual_det) + (mdr_z_positive_actual_detr)))))))) /\ (~(p = n))))

Complete tactic proof in conservative notation

All 24 original proof lines are preserved. Only local proposition formulas are abbreviated; every abbreviation has an exact binder-safe expansion check. The linked exact edition contains the unchanged replay script.

Read the argument

Proof checkpoints

24 script commands · 7 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.

Definition notation is shown below. Open the paired exact edition for the original native formulas. Source pairing is not a new equivalence certificate.

Named ingredients (1)
01Fix variables and assumptionsL1–7

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 hdata
02Separate the logical casesL8–12

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

  1. L8
    cases hdata
  2. L9
    cases hdata_right
  3. L10
    cases hdata_right_right
  4. L11
    cases hdata_right_right_witness
  5. L12
    cases hdata_right_right_witness_witness
03Construct an explicit witnessL13–14

Supply the displayed value, then prove that it has the required property.

  1. L13
    exists x
  2. L14
    exists x1
04Separate the logical casesL15–15

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

  1. L15
    split
05Use earlier factsL16–16

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

  1. L16
    exact hdata_right_right_witness_witness_left
06Fix variables and assumptionsL17–17

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

  1. L17
    intro hzero
07Use earlier factsL18–24

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

  1. L18
    specialize matrix_lattice_pair_nonzero_of_absolute (x)
  2. L19
    specialize matrix_lattice_pair_nonzero_of_absolute (x1)
  3. L20
    specialize matrix_lattice_pair_nonzero_of_absolute (D)
  4. L21
    apply matrix_lattice_pair_nonzero_of_absolute
  5. L22
    exact hdata_right_left
  6. L23
    exact hdata_right_right_witness_witness_right
  7. L24
    exact hzero

Library-wide reading audit

Original defined command ledger · 24 lines
  1. 0001intro ab
  2. 0002intro ac
  3. 0003intro bb
  4. 0004intro bc
  5. 0005intro d
  6. 0006intro D
  7. 0007intro hdata
  8. 0008cases hdata
  9. 0009cases hdata_right
  10. 0010cases hdata_right_right
  11. 0011cases hdata_right_right_witness
  12. 0012cases hdata_right_right_witness_witness
  13. 0013exists x
  14. 0014exists x1
  15. 0015split
  16. 0016exact hdata_right_right_witness_witness_left
  17. 0017intro hzero
  18. 0018specialize matrix_lattice_pair_nonzero_of_absolute (x)
  19. 0019specialize matrix_lattice_pair_nonzero_of_absolute (x1)
  20. 0020specialize matrix_lattice_pair_nonzero_of_absolute (D)
  21. 0021apply matrix_lattice_pair_nonzero_of_absolute
  22. 0022exact hdata_right_left
  23. 0023exact hdata_right_right_witness_witness_right
  24. 0024exact hzero