DL00A8

absolute_recursive_determinant_functional

The actual absolute determinant is unique across all recursive evaluation histories and sign orientations.

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. ∀ E. AbsoluteRecursiveDeterminant(ab,ac,bb,bc,d,D)AbsoluteRecursiveDeterminant(ab,ac,bb,bc,d,E) → D = E

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 E. (exists mdr_p_absolute_first mdr_n_absolute_first. ((exists mdr_b_absolute_firstevaluation mdr_c_absolute_firstevaluation mdr_l_absolute_firstevaluation mdr_i_absolute_firstevaluation. ((forall mdr_i_absolute_firstevaluationh. (exists mdr_gap_absolute_firstevaluationhi. mdr_gap_absolute_firstevaluationhi + S (mdr_i_absolute_firstevaluationh) = (mdr_l_absolute_firstevaluation)) -> exists mdr_d_absolute_firstevaluationh mdr_pb_absolute_firstevaluationh mdr_pc_absolute_firstevaluationh mdr_nb_absolute_firstevaluationh mdr_nc_absolute_firstevaluationh mdr_p_absolute_firstevaluationh mdr_n_absolute_firstevaluationh. ((exists mdr_z_absolute_firstevaluationhr. ((exists mdr_a_absolute_firstevaluationhrc mdr_b_absolute_firstevaluationhrc mdr_c_absolute_firstevaluationhrc mdr_e_absolute_firstevaluationhrc mdr_f_absolute_firstevaluationhrc. ((mdr_a_absolute_firstevaluationhrc = ((mdr_d_absolute_firstevaluationh) + (mdr_pb_absolute_firstevaluationh)) * S ((mdr_d_absolute_firstevaluationh) + (mdr_pb_absolute_firstevaluationh)) + ((mdr_pb_absolute_firstevaluationh) + (mdr_pb_absolute_firstevaluationh))) /\ ((mdr_b_absolute_firstevaluationhrc = ((mdr_pc_absolute_firstevaluationh) + (mdr_nb_absolute_firstevaluationh)) * S ((mdr_pc_absolute_firstevaluationh) + (mdr_nb_absolute_firstevaluationh)) + ((mdr_nb_absolute_firstevaluationh) + (mdr_nb_absolute_firstevaluationh))) /\ ((mdr_c_absolute_firstevaluationhrc = ((mdr_a_absolute_firstevaluationhrc) + (mdr_b_absolute_firstevaluationhrc)) * S ((mdr_a_absolute_firstevaluationhrc) + (mdr_b_absolute_firstevaluationhrc)) + ((mdr_b_absolute_firstevaluationhrc) + (mdr_b_absolute_firstevaluationhrc))) /\ ((mdr_e_absolute_firstevaluationhrc = ((mdr_p_absolute_firstevaluationh) + (mdr_n_absolute_firstevaluationh)) * S ((mdr_p_absolute_firstevaluationh) + (mdr_n_absolute_firstevaluationh)) + ((mdr_n_absolute_firstevaluationh) + (mdr_n_absolute_firstevaluationh))) /\ ((mdr_f_absolute_firstevaluationhrc = ((mdr_nc_absolute_firstevaluationh) + (mdr_e_absolute_firstevaluationhrc)) * S ((mdr_nc_absolute_firstevaluationh) + (mdr_e_absolute_firstevaluationhrc)) + ((mdr_e_absolute_firstevaluationhrc) + (mdr_e_absolute_firstevaluationhrc))) /\ ((mdr_z_absolute_firstevaluationhr) = ((mdr_c_absolute_firstevaluationhrc) + (mdr_f_absolute_firstevaluationhrc)) * S ((mdr_c_absolute_firstevaluationhrc) + (mdr_f_absolute_firstevaluationhrc)) + ((mdr_f_absolute_firstevaluationhrc) + (mdr_f_absolute_firstevaluationhrc))))))))) /\ (((exists ff_h_mdr_absolute_firstevaluationhrb. ff_h_mdr_absolute_firstevaluationhrb + S (mdr_z_absolute_firstevaluationhr) = S ((S (mdr_i_absolute_firstevaluationh)) * mdr_c_absolute_firstevaluation)) /\ exists ff_q_mdr_absolute_firstevaluationhrb. mdr_b_absolute_firstevaluation = ff_q_mdr_absolute_firstevaluationhrb * S ((S (mdr_i_absolute_firstevaluationh)) * mdr_c_absolute_firstevaluation) + (mdr_z_absolute_firstevaluationhr))))) /\ (((((mdr_d_absolute_firstevaluationh) = 0) /\ (((mdr_p_absolute_firstevaluationh) = 1) /\ ((mdr_n_absolute_firstevaluationh) = 0))) \/ exists mdr_q_absolute_firstevaluationhs mdr_eb_absolute_firstevaluationhs mdr_ec_absolute_firstevaluationhs mdr_fb_absolute_firstevaluationhs mdr_fc_absolute_firstevaluationhs. (((mdr_d_absolute_firstevaluationh) = S (mdr_q_absolute_firstevaluationhs)) /\ ((forall mdr_j_absolute_firstevaluationhsc. (exists mdr_gap_absolute_firstevaluationhscj. mdr_gap_absolute_firstevaluationhscj + S (mdr_j_absolute_firstevaluationhsc) = (S (mdr_q_absolute_firstevaluationhs))) -> exists mdr_i_absolute_firstevaluationhsc mdr_up_absolute_firstevaluationhsc mdr_us_absolute_firstevaluationhsc mdr_un_absolute_firstevaluationhsc mdr_ut_absolute_firstevaluationhsc mdr_p_absolute_firstevaluationhsc mdr_n_absolute_firstevaluationhsc. ((exists mdr_gap_absolute_firstevaluationhsci. mdr_gap_absolute_firstevaluationhsci + S (mdr_i_absolute_firstevaluationhsc) = (mdr_i_absolute_firstevaluationh)) /\ ((exists mdr_z_absolute_firstevaluationhscr. ((exists mdr_a_absolute_firstevaluationhscrc mdr_b_absolute_firstevaluationhscrc mdr_c_absolute_firstevaluationhscrc mdr_e_absolute_firstevaluationhscrc mdr_f_absolute_firstevaluationhscrc. ((mdr_a_absolute_firstevaluationhscrc = ((mdr_q_absolute_firstevaluationhs) + (mdr_up_absolute_firstevaluationhsc)) * S ((mdr_q_absolute_firstevaluationhs) + (mdr_up_absolute_firstevaluationhsc)) + ((mdr_up_absolute_firstevaluationhsc) + (mdr_up_absolute_firstevaluationhsc))) /\ ((mdr_b_absolute_firstevaluationhscrc = ((mdr_us_absolute_firstevaluationhsc) + (mdr_un_absolute_firstevaluationhsc)) * S ((mdr_us_absolute_firstevaluationhsc) + (mdr_un_absolute_firstevaluationhsc)) + ((mdr_un_absolute_firstevaluationhsc) + (mdr_un_absolute_firstevaluationhsc))) /\ ((mdr_c_absolute_firstevaluationhscrc = ((mdr_a_absolute_firstevaluationhscrc) + (mdr_b_absolute_firstevaluationhscrc)) * S ((mdr_a_absolute_firstevaluationhscrc) + (mdr_b_absolute_firstevaluationhscrc)) + ((mdr_b_absolute_firstevaluationhscrc) + (mdr_b_absolute_firstevaluationhscrc))) /\ ((mdr_e_absolute_firstevaluationhscrc = ((mdr_p_absolute_firstevaluationhsc) + (mdr_n_absolute_firstevaluationhsc)) * S ((mdr_p_absolute_firstevaluationhsc) + (mdr_n_absolute_firstevaluationhsc)) + ((mdr_n_absolute_firstevaluationhsc) + (mdr_n_absolute_firstevaluationhsc))) /\ ((mdr_f_absolute_firstevaluationhscrc = ((mdr_ut_absolute_firstevaluationhsc) + (mdr_e_absolute_firstevaluationhscrc)) * S ((mdr_ut_absolute_firstevaluationhsc) + (mdr_e_absolute_firstevaluationhscrc)) + ((mdr_e_absolute_firstevaluationhscrc) + (mdr_e_absolute_firstevaluationhscrc))) /\ ((mdr_z_absolute_firstevaluationhscr) = ((mdr_c_absolute_firstevaluationhscrc) + (mdr_f_absolute_firstevaluationhscrc)) * S ((mdr_c_absolute_firstevaluationhscrc) + (mdr_f_absolute_firstevaluationhscrc)) + ((mdr_f_absolute_firstevaluationhscrc) + (mdr_f_absolute_firstevaluationhscrc))))))))) /\ (((exists ff_h_mdr_absolute_firstevaluationhscrb. ff_h_mdr_absolute_firstevaluationhscrb + S (mdr_z_absolute_firstevaluationhscr) = S ((S (mdr_i_absolute_firstevaluationhsc)) * mdr_c_absolute_firstevaluation)) /\ exists ff_q_mdr_absolute_firstevaluationhscrb. mdr_b_absolute_firstevaluation = ff_q_mdr_absolute_firstevaluationhscrb * S ((S (mdr_i_absolute_firstevaluationhsc)) * mdr_c_absolute_firstevaluation) + (mdr_z_absolute_firstevaluationhscr))))) /\ ((((forall ff_index_mdm_prefix_mdr_absolute_firstevaluationhscm_positive. (exists ff_gap_mdm_lt_mdr_absolute_firstevaluationhscm_positive_index_bound. ff_gap_mdm_lt_mdr_absolute_firstevaluationhscm_positive_index_bound + S (ff_index_mdm_prefix_mdr_absolute_firstevaluationhscm_positive) = ((mdr_q_absolute_firstevaluationhs) * (mdr_q_absolute_firstevaluationhs))) -> exists ff_row_mdm_prefix_mdr_absolute_firstevaluationhscm_positive ff_column_mdm_prefix_mdr_absolute_firstevaluationhscm_positive ff_value_mdm_prefix_mdr_absolute_firstevaluationhscm_positive. (ff_index_mdm_prefix_mdr_absolute_firstevaluationhscm_positive = (mdr_q_absolute_firstevaluationhs) * ff_row_mdm_prefix_mdr_absolute_firstevaluationhscm_positive + ff_column_mdm_prefix_mdr_absolute_firstevaluationhscm_positive /\ ((exists ff_gap_mdm_lt_mdr_absolute_firstevaluationhscm_positive_column_bound. ff_gap_mdm_lt_mdr_absolute_firstevaluationhscm_positive_column_bound + S (ff_column_mdm_prefix_mdr_absolute_firstevaluationhscm_positive) = (mdr_q_absolute_firstevaluationhs)) /\ ((exists ff_row_mdm_cell_mdr_absolute_firstevaluationhscm_positive_cell ff_column_mdm_cell_mdr_absolute_firstevaluationhscm_positive_cell. (((((exists ff_gap_mdm_lt_mdr_absolute_firstevaluationhscm_positive_cell_row_before. ff_gap_mdm_lt_mdr_absolute_firstevaluationhscm_positive_cell_row_before + S (ff_row_mdm_prefix_mdr_absolute_firstevaluationhscm_positive) = (0)) /\ ff_row_mdm_cell_mdr_absolute_firstevaluationhscm_positive_cell = ff_row_mdm_prefix_mdr_absolute_firstevaluationhscm_positive) \/ ((exists ff_gap_mdm_le_mdr_absolute_firstevaluationhscm_positive_cell_row_after. ff_gap_mdm_le_mdr_absolute_firstevaluationhscm_positive_cell_row_after + (0) = (ff_row_mdm_prefix_mdr_absolute_firstevaluationhscm_positive)) /\ ff_row_mdm_cell_mdr_absolute_firstevaluationhscm_positive_cell = S ff_row_mdm_prefix_mdr_absolute_firstevaluationhscm_positive))) /\ (((((exists ff_gap_mdm_lt_mdr_absolute_firstevaluationhscm_positive_cell_column_before. ff_gap_mdm_lt_mdr_absolute_firstevaluationhscm_positive_cell_column_before + S (ff_column_mdm_prefix_mdr_absolute_firstevaluationhscm_positive) = (mdr_j_absolute_firstevaluationhsc)) /\ ff_column_mdm_cell_mdr_absolute_firstevaluationhscm_positive_cell = ff_column_mdm_prefix_mdr_absolute_firstevaluationhscm_positive) \/ ((exists ff_gap_mdm_le_mdr_absolute_firstevaluationhscm_positive_cell_column_after. ff_gap_mdm_le_mdr_absolute_firstevaluationhscm_positive_cell_column_after + (mdr_j_absolute_firstevaluationhsc) = (ff_column_mdm_prefix_mdr_absolute_firstevaluationhscm_positive)) /\ ff_column_mdm_cell_mdr_absolute_firstevaluationhscm_positive_cell = S ff_column_mdm_prefix_mdr_absolute_firstevaluationhscm_positive))) /\ (((exists ff_h_mdm_mdr_absolute_firstevaluationhscm_positive_cell_source. ff_h_mdm_mdr_absolute_firstevaluationhscm_positive_cell_source + S (ff_value_mdm_prefix_mdr_absolute_firstevaluationhscm_positive) = S ((S ((ff_row_mdm_cell_mdr_absolute_firstevaluationhscm_positive_cell) * (S (mdr_q_absolute_firstevaluationhs)) + (ff_column_mdm_cell_mdr_absolute_firstevaluationhscm_positive_cell))) * mdr_pc_absolute_firstevaluationh)) /\ exists ff_q_mdm_mdr_absolute_firstevaluationhscm_positive_cell_source. mdr_pb_absolute_firstevaluationh = ff_q_mdm_mdr_absolute_firstevaluationhscm_positive_cell_source * S ((S ((ff_row_mdm_cell_mdr_absolute_firstevaluationhscm_positive_cell) * (S (mdr_q_absolute_firstevaluationhs)) + (ff_column_mdm_cell_mdr_absolute_firstevaluationhscm_positive_cell))) * mdr_pc_absolute_firstevaluationh) + (ff_value_mdm_prefix_mdr_absolute_firstevaluationhscm_positive)))))) /\ (((exists ff_h_mdm_mdr_absolute_firstevaluationhscm_positive_target. ff_h_mdm_mdr_absolute_firstevaluationhscm_positive_target + S (ff_value_mdm_prefix_mdr_absolute_firstevaluationhscm_positive) = S ((S (ff_index_mdm_prefix_mdr_absolute_firstevaluationhscm_positive)) * mdr_us_absolute_firstevaluationhsc)) /\ exists ff_q_mdm_mdr_absolute_firstevaluationhscm_positive_target. mdr_up_absolute_firstevaluationhsc = ff_q_mdm_mdr_absolute_firstevaluationhscm_positive_target * S ((S (ff_index_mdm_prefix_mdr_absolute_firstevaluationhscm_positive)) * mdr_us_absolute_firstevaluationhsc) + (ff_value_mdm_prefix_mdr_absolute_firstevaluationhscm_positive))))))) /\ (forall ff_index_mdm_prefix_mdr_absolute_firstevaluationhscm_negative. (exists ff_gap_mdm_lt_mdr_absolute_firstevaluationhscm_negative_index_bound. ff_gap_mdm_lt_mdr_absolute_firstevaluationhscm_negative_index_bound + S (ff_index_mdm_prefix_mdr_absolute_firstevaluationhscm_negative) = ((mdr_q_absolute_firstevaluationhs) * (mdr_q_absolute_firstevaluationhs))) -> exists ff_row_mdm_prefix_mdr_absolute_firstevaluationhscm_negative ff_column_mdm_prefix_mdr_absolute_firstevaluationhscm_negative ff_value_mdm_prefix_mdr_absolute_firstevaluationhscm_negative. (ff_index_mdm_prefix_mdr_absolute_firstevaluationhscm_negative = (mdr_q_absolute_firstevaluationhs) * ff_row_mdm_prefix_mdr_absolute_firstevaluationhscm_negative + ff_column_mdm_prefix_mdr_absolute_firstevaluationhscm_negative /\ ((exists ff_gap_mdm_lt_mdr_absolute_firstevaluationhscm_negative_column_bound. ff_gap_mdm_lt_mdr_absolute_firstevaluationhscm_negative_column_bound + S (ff_column_mdm_prefix_mdr_absolute_firstevaluationhscm_negative) = (mdr_q_absolute_firstevaluationhs)) /\ ((exists ff_row_mdm_cell_mdr_absolute_firstevaluationhscm_negative_cell ff_column_mdm_cell_mdr_absolute_firstevaluationhscm_negative_cell. (((((exists ff_gap_mdm_lt_mdr_absolute_firstevaluationhscm_negative_cell_row_before. ff_gap_mdm_lt_mdr_absolute_firstevaluationhscm_negative_cell_row_before + S (ff_row_mdm_prefix_mdr_absolute_firstevaluationhscm_negative) = (0)) /\ ff_row_mdm_cell_mdr_absolute_firstevaluationhscm_negative_cell = ff_row_mdm_prefix_mdr_absolute_firstevaluationhscm_negative) \/ ((exists ff_gap_mdm_le_mdr_absolute_firstevaluationhscm_negative_cell_row_after. ff_gap_mdm_le_mdr_absolute_firstevaluationhscm_negative_cell_row_after + (0) = (ff_row_mdm_prefix_mdr_absolute_firstevaluationhscm_negative)) /\ ff_row_mdm_cell_mdr_absolute_firstevaluationhscm_negative_cell = S ff_row_mdm_prefix_mdr_absolute_firstevaluationhscm_negative))) /\ (((((exists ff_gap_mdm_lt_mdr_absolute_firstevaluationhscm_negative_cell_column_before. ff_gap_mdm_lt_mdr_absolute_firstevaluationhscm_negative_cell_column_before + S (ff_column_mdm_prefix_mdr_absolute_firstevaluationhscm_negative) = (mdr_j_absolute_firstevaluationhsc)) /\ ff_column_mdm_cell_mdr_absolute_firstevaluationhscm_negative_cell = ff_column_mdm_prefix_mdr_absolute_firstevaluationhscm_negative) \/ ((exists ff_gap_mdm_le_mdr_absolute_firstevaluationhscm_negative_cell_column_after. ff_gap_mdm_le_mdr_absolute_firstevaluationhscm_negative_cell_column_after + (mdr_j_absolute_firstevaluationhsc) = (ff_column_mdm_prefix_mdr_absolute_firstevaluationhscm_negative)) /\ ff_column_mdm_cell_mdr_absolute_firstevaluationhscm_negative_cell = S ff_column_mdm_prefix_mdr_absolute_firstevaluationhscm_negative))) /\ (((exists ff_h_mdm_mdr_absolute_firstevaluationhscm_negative_cell_source. ff_h_mdm_mdr_absolute_firstevaluationhscm_negative_cell_source + S (ff_value_mdm_prefix_mdr_absolute_firstevaluationhscm_negative) = S ((S ((ff_row_mdm_cell_mdr_absolute_firstevaluationhscm_negative_cell) * (S (mdr_q_absolute_firstevaluationhs)) + (ff_column_mdm_cell_mdr_absolute_firstevaluationhscm_negative_cell))) * mdr_nc_absolute_firstevaluationh)) /\ exists ff_q_mdm_mdr_absolute_firstevaluationhscm_negative_cell_source. mdr_nb_absolute_firstevaluationh = ff_q_mdm_mdr_absolute_firstevaluationhscm_negative_cell_source * S ((S ((ff_row_mdm_cell_mdr_absolute_firstevaluationhscm_negative_cell) * (S (mdr_q_absolute_firstevaluationhs)) + (ff_column_mdm_cell_mdr_absolute_firstevaluationhscm_negative_cell))) * mdr_nc_absolute_firstevaluationh) + (ff_value_mdm_prefix_mdr_absolute_firstevaluationhscm_negative)))))) /\ (((exists ff_h_mdm_mdr_absolute_firstevaluationhscm_negative_target. ff_h_mdm_mdr_absolute_firstevaluationhscm_negative_target + S (ff_value_mdm_prefix_mdr_absolute_firstevaluationhscm_negative) = S ((S (ff_index_mdm_prefix_mdr_absolute_firstevaluationhscm_negative)) * mdr_ut_absolute_firstevaluationhsc)) /\ exists ff_q_mdm_mdr_absolute_firstevaluationhscm_negative_target. mdr_un_absolute_firstevaluationhsc = ff_q_mdm_mdr_absolute_firstevaluationhscm_negative_target * S ((S (ff_index_mdm_prefix_mdr_absolute_firstevaluationhscm_negative)) * mdr_ut_absolute_firstevaluationhsc) + (ff_value_mdm_prefix_mdr_absolute_firstevaluationhscm_negative))))))))) /\ ((((exists ff_h_mdr_absolute_firstevaluationhscp. ff_h_mdr_absolute_firstevaluationhscp + S (mdr_p_absolute_firstevaluationhsc) = S ((S (mdr_j_absolute_firstevaluationhsc)) * mdr_ec_absolute_firstevaluationhs)) /\ exists ff_q_mdr_absolute_firstevaluationhscp. mdr_eb_absolute_firstevaluationhs = ff_q_mdr_absolute_firstevaluationhscp * S ((S (mdr_j_absolute_firstevaluationhsc)) * mdr_ec_absolute_firstevaluationhs) + (mdr_p_absolute_firstevaluationhsc))) /\ (((exists ff_h_mdr_absolute_firstevaluationhscn. ff_h_mdr_absolute_firstevaluationhscn + S (mdr_n_absolute_firstevaluationhsc) = S ((S (mdr_j_absolute_firstevaluationhsc)) * mdr_fc_absolute_firstevaluationhs)) /\ exists ff_q_mdr_absolute_firstevaluationhscn. mdr_fb_absolute_firstevaluationhs = ff_q_mdr_absolute_firstevaluationhscn * S ((S (mdr_j_absolute_firstevaluationhsc)) * mdr_fc_absolute_firstevaluationhs) + (mdr_n_absolute_firstevaluationhsc)))))))) /\ (exists ff_ub_mce_fold_mdr_absolute_firstevaluationhsf ff_uc_mce_fold_mdr_absolute_firstevaluationhsf ff_vb_mce_fold_mdr_absolute_firstevaluationhsf ff_vc_mce_fold_mdr_absolute_firstevaluationhsf. ((forall ff_index_mce_alternating_mdr_absolute_firstevaluationhsf_prefix. (exists ff_gap_mce_mdr_absolute_firstevaluationhsf_prefix_index. ff_gap_mce_mdr_absolute_firstevaluationhsf_prefix_index + S (ff_index_mce_alternating_mdr_absolute_firstevaluationhsf_prefix) = (S (mdr_q_absolute_firstevaluationhs))) -> exists ff_ap_mce_alternating_mdr_absolute_firstevaluationhsf_prefix ff_an_mce_alternating_mdr_absolute_firstevaluationhsf_prefix ff_bp_mce_alternating_mdr_absolute_firstevaluationhsf_prefix ff_bn_mce_alternating_mdr_absolute_firstevaluationhsf_prefix ff_p_mce_alternating_mdr_absolute_firstevaluationhsf_prefix ff_n_mce_alternating_mdr_absolute_firstevaluationhsf_prefix. ((((exists ff_h_mce_mdr_absolute_firstevaluationhsf_prefix_ap. ff_h_mce_mdr_absolute_firstevaluationhsf_prefix_ap + S (ff_ap_mce_alternating_mdr_absolute_firstevaluationhsf_prefix) = S ((S (ff_index_mce_alternating_mdr_absolute_firstevaluationhsf_prefix)) * mdr_pc_absolute_firstevaluationh)) /\ exists ff_q_mce_mdr_absolute_firstevaluationhsf_prefix_ap. mdr_pb_absolute_firstevaluationh = ff_q_mce_mdr_absolute_firstevaluationhsf_prefix_ap * S ((S (ff_index_mce_alternating_mdr_absolute_firstevaluationhsf_prefix)) * mdr_pc_absolute_firstevaluationh) + (ff_ap_mce_alternating_mdr_absolute_firstevaluationhsf_prefix))) /\ ((((exists ff_h_mce_mdr_absolute_firstevaluationhsf_prefix_an. ff_h_mce_mdr_absolute_firstevaluationhsf_prefix_an + S (ff_an_mce_alternating_mdr_absolute_firstevaluationhsf_prefix) = S ((S (ff_index_mce_alternating_mdr_absolute_firstevaluationhsf_prefix)) * mdr_nc_absolute_firstevaluationh)) /\ exists ff_q_mce_mdr_absolute_firstevaluationhsf_prefix_an. mdr_nb_absolute_firstevaluationh = ff_q_mce_mdr_absolute_firstevaluationhsf_prefix_an * S ((S (ff_index_mce_alternating_mdr_absolute_firstevaluationhsf_prefix)) * mdr_nc_absolute_firstevaluationh) + (ff_an_mce_alternating_mdr_absolute_firstevaluationhsf_prefix))) /\ ((((exists ff_h_mce_mdr_absolute_firstevaluationhsf_prefix_bp. ff_h_mce_mdr_absolute_firstevaluationhsf_prefix_bp + S (ff_bp_mce_alternating_mdr_absolute_firstevaluationhsf_prefix) = S ((S (ff_index_mce_alternating_mdr_absolute_firstevaluationhsf_prefix)) * mdr_ec_absolute_firstevaluationhs)) /\ exists ff_q_mce_mdr_absolute_firstevaluationhsf_prefix_bp. mdr_eb_absolute_firstevaluationhs = ff_q_mce_mdr_absolute_firstevaluationhsf_prefix_bp * S ((S (ff_index_mce_alternating_mdr_absolute_firstevaluationhsf_prefix)) * mdr_ec_absolute_firstevaluationhs) + (ff_bp_mce_alternating_mdr_absolute_firstevaluationhsf_prefix))) /\ ((((exists ff_h_mce_mdr_absolute_firstevaluationhsf_prefix_bn. ff_h_mce_mdr_absolute_firstevaluationhsf_prefix_bn + S (ff_bn_mce_alternating_mdr_absolute_firstevaluationhsf_prefix) = S ((S (ff_index_mce_alternating_mdr_absolute_firstevaluationhsf_prefix)) * mdr_fc_absolute_firstevaluationhs)) /\ exists ff_q_mce_mdr_absolute_firstevaluationhsf_prefix_bn. mdr_fb_absolute_firstevaluationhs = ff_q_mce_mdr_absolute_firstevaluationhsf_prefix_bn * S ((S (ff_index_mce_alternating_mdr_absolute_firstevaluationhsf_prefix)) * mdr_fc_absolute_firstevaluationhs) + (ff_bn_mce_alternating_mdr_absolute_firstevaluationhsf_prefix))) /\ ((((exists ff_h_mce_mdr_absolute_firstevaluationhsf_prefix_positive. ff_h_mce_mdr_absolute_firstevaluationhsf_prefix_positive + S (ff_p_mce_alternating_mdr_absolute_firstevaluationhsf_prefix) = S ((S (ff_index_mce_alternating_mdr_absolute_firstevaluationhsf_prefix)) * ff_uc_mce_fold_mdr_absolute_firstevaluationhsf)) /\ exists ff_q_mce_mdr_absolute_firstevaluationhsf_prefix_positive. ff_ub_mce_fold_mdr_absolute_firstevaluationhsf = ff_q_mce_mdr_absolute_firstevaluationhsf_prefix_positive * S ((S (ff_index_mce_alternating_mdr_absolute_firstevaluationhsf_prefix)) * ff_uc_mce_fold_mdr_absolute_firstevaluationhsf) + (ff_p_mce_alternating_mdr_absolute_firstevaluationhsf_prefix))) /\ ((((exists ff_h_mce_mdr_absolute_firstevaluationhsf_prefix_negative. ff_h_mce_mdr_absolute_firstevaluationhsf_prefix_negative + S (ff_n_mce_alternating_mdr_absolute_firstevaluationhsf_prefix) = S ((S (ff_index_mce_alternating_mdr_absolute_firstevaluationhsf_prefix)) * ff_vc_mce_fold_mdr_absolute_firstevaluationhsf)) /\ exists ff_q_mce_mdr_absolute_firstevaluationhsf_prefix_negative. ff_vb_mce_fold_mdr_absolute_firstevaluationhsf = ff_q_mce_mdr_absolute_firstevaluationhsf_prefix_negative * S ((S (ff_index_mce_alternating_mdr_absolute_firstevaluationhsf_prefix)) * ff_vc_mce_fold_mdr_absolute_firstevaluationhsf) + (ff_n_mce_alternating_mdr_absolute_firstevaluationhsf_prefix))) /\ (((exists ff_even_mce_term_mdr_absolute_firstevaluationhsf_prefix_term. ff_index_mce_alternating_mdr_absolute_firstevaluationhsf_prefix = 2 * ff_even_mce_term_mdr_absolute_firstevaluationhsf_prefix_term) /\ (ff_p_mce_alternating_mdr_absolute_firstevaluationhsf_prefix = (ff_ap_mce_alternating_mdr_absolute_firstevaluationhsf_prefix) * (ff_bp_mce_alternating_mdr_absolute_firstevaluationhsf_prefix) + (ff_an_mce_alternating_mdr_absolute_firstevaluationhsf_prefix) * (ff_bn_mce_alternating_mdr_absolute_firstevaluationhsf_prefix) /\ ff_n_mce_alternating_mdr_absolute_firstevaluationhsf_prefix = (ff_ap_mce_alternating_mdr_absolute_firstevaluationhsf_prefix) * (ff_bn_mce_alternating_mdr_absolute_firstevaluationhsf_prefix) + (ff_an_mce_alternating_mdr_absolute_firstevaluationhsf_prefix) * (ff_bp_mce_alternating_mdr_absolute_firstevaluationhsf_prefix))) \/ ((exists ff_odd_mce_term_mdr_absolute_firstevaluationhsf_prefix_term. ff_index_mce_alternating_mdr_absolute_firstevaluationhsf_prefix = 2 * ff_odd_mce_term_mdr_absolute_firstevaluationhsf_prefix_term + 1) /\ (ff_p_mce_alternating_mdr_absolute_firstevaluationhsf_prefix = (ff_ap_mce_alternating_mdr_absolute_firstevaluationhsf_prefix) * (ff_bn_mce_alternating_mdr_absolute_firstevaluationhsf_prefix) + (ff_an_mce_alternating_mdr_absolute_firstevaluationhsf_prefix) * (ff_bp_mce_alternating_mdr_absolute_firstevaluationhsf_prefix) /\ ff_n_mce_alternating_mdr_absolute_firstevaluationhsf_prefix = (ff_ap_mce_alternating_mdr_absolute_firstevaluationhsf_prefix) * (ff_bp_mce_alternating_mdr_absolute_firstevaluationhsf_prefix) + (ff_an_mce_alternating_mdr_absolute_firstevaluationhsf_prefix) * (ff_bn_mce_alternating_mdr_absolute_firstevaluationhsf_prefix))))))))))) /\ ((exists ff_u_mce_mdr_absolute_firstevaluationhsf_positive ff_v_mce_mdr_absolute_firstevaluationhsf_positive. ((((exists ff_h_mce_mdr_absolute_firstevaluationhsf_positive_start. ff_h_mce_mdr_absolute_firstevaluationhsf_positive_start + S (0) = S ((S (0)) * ff_v_mce_mdr_absolute_firstevaluationhsf_positive)) /\ exists ff_q_mce_mdr_absolute_firstevaluationhsf_positive_start. ff_u_mce_mdr_absolute_firstevaluationhsf_positive = ff_q_mce_mdr_absolute_firstevaluationhsf_positive_start * S ((S (0)) * ff_v_mce_mdr_absolute_firstevaluationhsf_positive) + (0))) /\ ((((exists ff_h_mce_mdr_absolute_firstevaluationhsf_positive_terminal. ff_h_mce_mdr_absolute_firstevaluationhsf_positive_terminal + S (mdr_p_absolute_firstevaluationh) = S ((S ((S (mdr_q_absolute_firstevaluationhs)))) * ff_v_mce_mdr_absolute_firstevaluationhsf_positive)) /\ exists ff_q_mce_mdr_absolute_firstevaluationhsf_positive_terminal. ff_u_mce_mdr_absolute_firstevaluationhsf_positive = ff_q_mce_mdr_absolute_firstevaluationhsf_positive_terminal * S ((S ((S (mdr_q_absolute_firstevaluationhs)))) * ff_v_mce_mdr_absolute_firstevaluationhsf_positive) + (mdr_p_absolute_firstevaluationh))) /\ forall ff_i_mce_mdr_absolute_firstevaluationhsf_positive. (exists ff_lt_mce_mdr_absolute_firstevaluationhsf_positive_bound. ff_lt_mce_mdr_absolute_firstevaluationhsf_positive_bound + S ff_i_mce_mdr_absolute_firstevaluationhsf_positive = (S (mdr_q_absolute_firstevaluationhs))) -> exists ff_a_mce_mdr_absolute_firstevaluationhsf_positive ff_r_mce_mdr_absolute_firstevaluationhsf_positive ff_s_mce_mdr_absolute_firstevaluationhsf_positive. ((((exists ff_h_mce_mdr_absolute_firstevaluationhsf_positive_summand. ff_h_mce_mdr_absolute_firstevaluationhsf_positive_summand + S (ff_a_mce_mdr_absolute_firstevaluationhsf_positive) = S ((S (ff_i_mce_mdr_absolute_firstevaluationhsf_positive)) * ff_uc_mce_fold_mdr_absolute_firstevaluationhsf)) /\ exists ff_q_mce_mdr_absolute_firstevaluationhsf_positive_summand. ff_ub_mce_fold_mdr_absolute_firstevaluationhsf = ff_q_mce_mdr_absolute_firstevaluationhsf_positive_summand * S ((S (ff_i_mce_mdr_absolute_firstevaluationhsf_positive)) * ff_uc_mce_fold_mdr_absolute_firstevaluationhsf) + (ff_a_mce_mdr_absolute_firstevaluationhsf_positive))) /\ ((((exists ff_h_mce_mdr_absolute_firstevaluationhsf_positive_partial. ff_h_mce_mdr_absolute_firstevaluationhsf_positive_partial + S (ff_r_mce_mdr_absolute_firstevaluationhsf_positive) = S ((S (ff_i_mce_mdr_absolute_firstevaluationhsf_positive)) * ff_v_mce_mdr_absolute_firstevaluationhsf_positive)) /\ exists ff_q_mce_mdr_absolute_firstevaluationhsf_positive_partial. ff_u_mce_mdr_absolute_firstevaluationhsf_positive = ff_q_mce_mdr_absolute_firstevaluationhsf_positive_partial * S ((S (ff_i_mce_mdr_absolute_firstevaluationhsf_positive)) * ff_v_mce_mdr_absolute_firstevaluationhsf_positive) + (ff_r_mce_mdr_absolute_firstevaluationhsf_positive))) /\ ((((exists ff_h_mce_mdr_absolute_firstevaluationhsf_positive_successor. ff_h_mce_mdr_absolute_firstevaluationhsf_positive_successor + S (ff_s_mce_mdr_absolute_firstevaluationhsf_positive) = S ((S (S ff_i_mce_mdr_absolute_firstevaluationhsf_positive)) * ff_v_mce_mdr_absolute_firstevaluationhsf_positive)) /\ exists ff_q_mce_mdr_absolute_firstevaluationhsf_positive_successor. ff_u_mce_mdr_absolute_firstevaluationhsf_positive = ff_q_mce_mdr_absolute_firstevaluationhsf_positive_successor * S ((S (S ff_i_mce_mdr_absolute_firstevaluationhsf_positive)) * ff_v_mce_mdr_absolute_firstevaluationhsf_positive) + (ff_s_mce_mdr_absolute_firstevaluationhsf_positive))) /\ ff_s_mce_mdr_absolute_firstevaluationhsf_positive = ff_r_mce_mdr_absolute_firstevaluationhsf_positive + ff_a_mce_mdr_absolute_firstevaluationhsf_positive)))))) /\ (exists ff_u_mce_mdr_absolute_firstevaluationhsf_negative ff_v_mce_mdr_absolute_firstevaluationhsf_negative. ((((exists ff_h_mce_mdr_absolute_firstevaluationhsf_negative_start. ff_h_mce_mdr_absolute_firstevaluationhsf_negative_start + S (0) = S ((S (0)) * ff_v_mce_mdr_absolute_firstevaluationhsf_negative)) /\ exists ff_q_mce_mdr_absolute_firstevaluationhsf_negative_start. ff_u_mce_mdr_absolute_firstevaluationhsf_negative = ff_q_mce_mdr_absolute_firstevaluationhsf_negative_start * S ((S (0)) * ff_v_mce_mdr_absolute_firstevaluationhsf_negative) + (0))) /\ ((((exists ff_h_mce_mdr_absolute_firstevaluationhsf_negative_terminal. ff_h_mce_mdr_absolute_firstevaluationhsf_negative_terminal + S (mdr_n_absolute_firstevaluationh) = S ((S ((S (mdr_q_absolute_firstevaluationhs)))) * ff_v_mce_mdr_absolute_firstevaluationhsf_negative)) /\ exists ff_q_mce_mdr_absolute_firstevaluationhsf_negative_terminal. ff_u_mce_mdr_absolute_firstevaluationhsf_negative = ff_q_mce_mdr_absolute_firstevaluationhsf_negative_terminal * S ((S ((S (mdr_q_absolute_firstevaluationhs)))) * ff_v_mce_mdr_absolute_firstevaluationhsf_negative) + (mdr_n_absolute_firstevaluationh))) /\ forall ff_i_mce_mdr_absolute_firstevaluationhsf_negative. (exists ff_lt_mce_mdr_absolute_firstevaluationhsf_negative_bound. ff_lt_mce_mdr_absolute_firstevaluationhsf_negative_bound + S ff_i_mce_mdr_absolute_firstevaluationhsf_negative = (S (mdr_q_absolute_firstevaluationhs))) -> exists ff_a_mce_mdr_absolute_firstevaluationhsf_negative ff_r_mce_mdr_absolute_firstevaluationhsf_negative ff_s_mce_mdr_absolute_firstevaluationhsf_negative. ((((exists ff_h_mce_mdr_absolute_firstevaluationhsf_negative_summand. ff_h_mce_mdr_absolute_firstevaluationhsf_negative_summand + S (ff_a_mce_mdr_absolute_firstevaluationhsf_negative) = S ((S (ff_i_mce_mdr_absolute_firstevaluationhsf_negative)) * ff_vc_mce_fold_mdr_absolute_firstevaluationhsf)) /\ exists ff_q_mce_mdr_absolute_firstevaluationhsf_negative_summand. ff_vb_mce_fold_mdr_absolute_firstevaluationhsf = ff_q_mce_mdr_absolute_firstevaluationhsf_negative_summand * S ((S (ff_i_mce_mdr_absolute_firstevaluationhsf_negative)) * ff_vc_mce_fold_mdr_absolute_firstevaluationhsf) + (ff_a_mce_mdr_absolute_firstevaluationhsf_negative))) /\ ((((exists ff_h_mce_mdr_absolute_firstevaluationhsf_negative_partial. ff_h_mce_mdr_absolute_firstevaluationhsf_negative_partial + S (ff_r_mce_mdr_absolute_firstevaluationhsf_negative) = S ((S (ff_i_mce_mdr_absolute_firstevaluationhsf_negative)) * ff_v_mce_mdr_absolute_firstevaluationhsf_negative)) /\ exists ff_q_mce_mdr_absolute_firstevaluationhsf_negative_partial. ff_u_mce_mdr_absolute_firstevaluationhsf_negative = ff_q_mce_mdr_absolute_firstevaluationhsf_negative_partial * S ((S (ff_i_mce_mdr_absolute_firstevaluationhsf_negative)) * ff_v_mce_mdr_absolute_firstevaluationhsf_negative) + (ff_r_mce_mdr_absolute_firstevaluationhsf_negative))) /\ ((((exists ff_h_mce_mdr_absolute_firstevaluationhsf_negative_successor. ff_h_mce_mdr_absolute_firstevaluationhsf_negative_successor + S (ff_s_mce_mdr_absolute_firstevaluationhsf_negative) = S ((S (S ff_i_mce_mdr_absolute_firstevaluationhsf_negative)) * ff_v_mce_mdr_absolute_firstevaluationhsf_negative)) /\ exists ff_q_mce_mdr_absolute_firstevaluationhsf_negative_successor. ff_u_mce_mdr_absolute_firstevaluationhsf_negative = ff_q_mce_mdr_absolute_firstevaluationhsf_negative_successor * S ((S (S ff_i_mce_mdr_absolute_firstevaluationhsf_negative)) * ff_v_mce_mdr_absolute_firstevaluationhsf_negative) + (ff_s_mce_mdr_absolute_firstevaluationhsf_negative))) /\ ff_s_mce_mdr_absolute_firstevaluationhsf_negative = ff_r_mce_mdr_absolute_firstevaluationhsf_negative + ff_a_mce_mdr_absolute_firstevaluationhsf_negative))))))))))))))) /\ ((exists mdr_gap_absolute_firstevaluationi. mdr_gap_absolute_firstevaluationi + S (mdr_i_absolute_firstevaluation) = (mdr_l_absolute_firstevaluation)) /\ (exists mdr_z_absolute_firstevaluationr. ((exists mdr_a_absolute_firstevaluationrc mdr_b_absolute_firstevaluationrc mdr_c_absolute_firstevaluationrc mdr_e_absolute_firstevaluationrc mdr_f_absolute_firstevaluationrc. ((mdr_a_absolute_firstevaluationrc = ((d) + (ab)) * S ((d) + (ab)) + ((ab) + (ab))) /\ ((mdr_b_absolute_firstevaluationrc = ((ac) + (bb)) * S ((ac) + (bb)) + ((bb) + (bb))) /\ ((mdr_c_absolute_firstevaluationrc = ((mdr_a_absolute_firstevaluationrc) + (mdr_b_absolute_firstevaluationrc)) * S ((mdr_a_absolute_firstevaluationrc) + (mdr_b_absolute_firstevaluationrc)) + ((mdr_b_absolute_firstevaluationrc) + (mdr_b_absolute_firstevaluationrc))) /\ ((mdr_e_absolute_firstevaluationrc = ((mdr_p_absolute_first) + (mdr_n_absolute_first)) * S ((mdr_p_absolute_first) + (mdr_n_absolute_first)) + ((mdr_n_absolute_first) + (mdr_n_absolute_first))) /\ ((mdr_f_absolute_firstevaluationrc = ((bc) + (mdr_e_absolute_firstevaluationrc)) * S ((bc) + (mdr_e_absolute_firstevaluationrc)) + ((mdr_e_absolute_firstevaluationrc) + (mdr_e_absolute_firstevaluationrc))) /\ ((mdr_z_absolute_firstevaluationr) = ((mdr_c_absolute_firstevaluationrc) + (mdr_f_absolute_firstevaluationrc)) * S ((mdr_c_absolute_firstevaluationrc) + (mdr_f_absolute_firstevaluationrc)) + ((mdr_f_absolute_firstevaluationrc) + (mdr_f_absolute_firstevaluationrc))))))))) /\ (((exists ff_h_mdr_absolute_firstevaluationrb. ff_h_mdr_absolute_firstevaluationrb + S (mdr_z_absolute_firstevaluationr) = S ((S (mdr_i_absolute_firstevaluation)) * mdr_c_absolute_firstevaluation)) /\ exists ff_q_mdr_absolute_firstevaluationrb. mdr_b_absolute_firstevaluation = ff_q_mdr_absolute_firstevaluationrb * S ((S (mdr_i_absolute_firstevaluation)) * mdr_c_absolute_firstevaluation) + (mdr_z_absolute_firstevaluationr)))))))) /\ (((mdr_p_absolute_first) = (mdr_n_absolute_first) + (D)) \/ ((mdr_n_absolute_first) = (mdr_p_absolute_first) + (D))))) -> (exists mdr_p_absolute_second mdr_n_absolute_second. ((exists mdr_b_absolute_secondevaluation mdr_c_absolute_secondevaluation mdr_l_absolute_secondevaluation mdr_i_absolute_secondevaluation. ((forall mdr_i_absolute_secondevaluationh. (exists mdr_gap_absolute_secondevaluationhi. mdr_gap_absolute_secondevaluationhi + S (mdr_i_absolute_secondevaluationh) = (mdr_l_absolute_secondevaluation)) -> exists mdr_d_absolute_secondevaluationh mdr_pb_absolute_secondevaluationh mdr_pc_absolute_secondevaluationh mdr_nb_absolute_secondevaluationh mdr_nc_absolute_secondevaluationh mdr_p_absolute_secondevaluationh mdr_n_absolute_secondevaluationh. ((exists mdr_z_absolute_secondevaluationhr. ((exists mdr_a_absolute_secondevaluationhrc mdr_b_absolute_secondevaluationhrc mdr_c_absolute_secondevaluationhrc mdr_e_absolute_secondevaluationhrc mdr_f_absolute_secondevaluationhrc. ((mdr_a_absolute_secondevaluationhrc = ((mdr_d_absolute_secondevaluationh) + (mdr_pb_absolute_secondevaluationh)) * S ((mdr_d_absolute_secondevaluationh) + (mdr_pb_absolute_secondevaluationh)) + ((mdr_pb_absolute_secondevaluationh) + (mdr_pb_absolute_secondevaluationh))) /\ ((mdr_b_absolute_secondevaluationhrc = ((mdr_pc_absolute_secondevaluationh) + (mdr_nb_absolute_secondevaluationh)) * S ((mdr_pc_absolute_secondevaluationh) + (mdr_nb_absolute_secondevaluationh)) + ((mdr_nb_absolute_secondevaluationh) + (mdr_nb_absolute_secondevaluationh))) /\ ((mdr_c_absolute_secondevaluationhrc = ((mdr_a_absolute_secondevaluationhrc) + (mdr_b_absolute_secondevaluationhrc)) * S ((mdr_a_absolute_secondevaluationhrc) + (mdr_b_absolute_secondevaluationhrc)) + ((mdr_b_absolute_secondevaluationhrc) + (mdr_b_absolute_secondevaluationhrc))) /\ ((mdr_e_absolute_secondevaluationhrc = ((mdr_p_absolute_secondevaluationh) + (mdr_n_absolute_secondevaluationh)) * S ((mdr_p_absolute_secondevaluationh) + (mdr_n_absolute_secondevaluationh)) + ((mdr_n_absolute_secondevaluationh) + (mdr_n_absolute_secondevaluationh))) /\ ((mdr_f_absolute_secondevaluationhrc = ((mdr_nc_absolute_secondevaluationh) + (mdr_e_absolute_secondevaluationhrc)) * S ((mdr_nc_absolute_secondevaluationh) + (mdr_e_absolute_secondevaluationhrc)) + ((mdr_e_absolute_secondevaluationhrc) + (mdr_e_absolute_secondevaluationhrc))) /\ ((mdr_z_absolute_secondevaluationhr) = ((mdr_c_absolute_secondevaluationhrc) + (mdr_f_absolute_secondevaluationhrc)) * S ((mdr_c_absolute_secondevaluationhrc) + (mdr_f_absolute_secondevaluationhrc)) + ((mdr_f_absolute_secondevaluationhrc) + (mdr_f_absolute_secondevaluationhrc))))))))) /\ (((exists ff_h_mdr_absolute_secondevaluationhrb. ff_h_mdr_absolute_secondevaluationhrb + S (mdr_z_absolute_secondevaluationhr) = S ((S (mdr_i_absolute_secondevaluationh)) * mdr_c_absolute_secondevaluation)) /\ exists ff_q_mdr_absolute_secondevaluationhrb. mdr_b_absolute_secondevaluation = ff_q_mdr_absolute_secondevaluationhrb * S ((S (mdr_i_absolute_secondevaluationh)) * mdr_c_absolute_secondevaluation) + (mdr_z_absolute_secondevaluationhr))))) /\ (((((mdr_d_absolute_secondevaluationh) = 0) /\ (((mdr_p_absolute_secondevaluationh) = 1) /\ ((mdr_n_absolute_secondevaluationh) = 0))) \/ exists mdr_q_absolute_secondevaluationhs mdr_eb_absolute_secondevaluationhs mdr_ec_absolute_secondevaluationhs mdr_fb_absolute_secondevaluationhs mdr_fc_absolute_secondevaluationhs. (((mdr_d_absolute_secondevaluationh) = S (mdr_q_absolute_secondevaluationhs)) /\ ((forall mdr_j_absolute_secondevaluationhsc. (exists mdr_gap_absolute_secondevaluationhscj. mdr_gap_absolute_secondevaluationhscj + S (mdr_j_absolute_secondevaluationhsc) = (S (mdr_q_absolute_secondevaluationhs))) -> exists mdr_i_absolute_secondevaluationhsc mdr_up_absolute_secondevaluationhsc mdr_us_absolute_secondevaluationhsc mdr_un_absolute_secondevaluationhsc mdr_ut_absolute_secondevaluationhsc mdr_p_absolute_secondevaluationhsc mdr_n_absolute_secondevaluationhsc. ((exists mdr_gap_absolute_secondevaluationhsci. mdr_gap_absolute_secondevaluationhsci + S (mdr_i_absolute_secondevaluationhsc) = (mdr_i_absolute_secondevaluationh)) /\ ((exists mdr_z_absolute_secondevaluationhscr. ((exists mdr_a_absolute_secondevaluationhscrc mdr_b_absolute_secondevaluationhscrc mdr_c_absolute_secondevaluationhscrc mdr_e_absolute_secondevaluationhscrc mdr_f_absolute_secondevaluationhscrc. ((mdr_a_absolute_secondevaluationhscrc = ((mdr_q_absolute_secondevaluationhs) + (mdr_up_absolute_secondevaluationhsc)) * S ((mdr_q_absolute_secondevaluationhs) + (mdr_up_absolute_secondevaluationhsc)) + ((mdr_up_absolute_secondevaluationhsc) + (mdr_up_absolute_secondevaluationhsc))) /\ ((mdr_b_absolute_secondevaluationhscrc = ((mdr_us_absolute_secondevaluationhsc) + (mdr_un_absolute_secondevaluationhsc)) * S ((mdr_us_absolute_secondevaluationhsc) + (mdr_un_absolute_secondevaluationhsc)) + ((mdr_un_absolute_secondevaluationhsc) + (mdr_un_absolute_secondevaluationhsc))) /\ ((mdr_c_absolute_secondevaluationhscrc = ((mdr_a_absolute_secondevaluationhscrc) + (mdr_b_absolute_secondevaluationhscrc)) * S ((mdr_a_absolute_secondevaluationhscrc) + (mdr_b_absolute_secondevaluationhscrc)) + ((mdr_b_absolute_secondevaluationhscrc) + (mdr_b_absolute_secondevaluationhscrc))) /\ ((mdr_e_absolute_secondevaluationhscrc = ((mdr_p_absolute_secondevaluationhsc) + (mdr_n_absolute_secondevaluationhsc)) * S ((mdr_p_absolute_secondevaluationhsc) + (mdr_n_absolute_secondevaluationhsc)) + ((mdr_n_absolute_secondevaluationhsc) + (mdr_n_absolute_secondevaluationhsc))) /\ ((mdr_f_absolute_secondevaluationhscrc = ((mdr_ut_absolute_secondevaluationhsc) + (mdr_e_absolute_secondevaluationhscrc)) * S ((mdr_ut_absolute_secondevaluationhsc) + (mdr_e_absolute_secondevaluationhscrc)) + ((mdr_e_absolute_secondevaluationhscrc) + (mdr_e_absolute_secondevaluationhscrc))) /\ ((mdr_z_absolute_secondevaluationhscr) = ((mdr_c_absolute_secondevaluationhscrc) + (mdr_f_absolute_secondevaluationhscrc)) * S ((mdr_c_absolute_secondevaluationhscrc) + (mdr_f_absolute_secondevaluationhscrc)) + ((mdr_f_absolute_secondevaluationhscrc) + (mdr_f_absolute_secondevaluationhscrc))))))))) /\ (((exists ff_h_mdr_absolute_secondevaluationhscrb. ff_h_mdr_absolute_secondevaluationhscrb + S (mdr_z_absolute_secondevaluationhscr) = S ((S (mdr_i_absolute_secondevaluationhsc)) * mdr_c_absolute_secondevaluation)) /\ exists ff_q_mdr_absolute_secondevaluationhscrb. mdr_b_absolute_secondevaluation = ff_q_mdr_absolute_secondevaluationhscrb * S ((S (mdr_i_absolute_secondevaluationhsc)) * mdr_c_absolute_secondevaluation) + (mdr_z_absolute_secondevaluationhscr))))) /\ ((((forall ff_index_mdm_prefix_mdr_absolute_secondevaluationhscm_positive. (exists ff_gap_mdm_lt_mdr_absolute_secondevaluationhscm_positive_index_bound. ff_gap_mdm_lt_mdr_absolute_secondevaluationhscm_positive_index_bound + S (ff_index_mdm_prefix_mdr_absolute_secondevaluationhscm_positive) = ((mdr_q_absolute_secondevaluationhs) * (mdr_q_absolute_secondevaluationhs))) -> exists ff_row_mdm_prefix_mdr_absolute_secondevaluationhscm_positive ff_column_mdm_prefix_mdr_absolute_secondevaluationhscm_positive ff_value_mdm_prefix_mdr_absolute_secondevaluationhscm_positive. (ff_index_mdm_prefix_mdr_absolute_secondevaluationhscm_positive = (mdr_q_absolute_secondevaluationhs) * ff_row_mdm_prefix_mdr_absolute_secondevaluationhscm_positive + ff_column_mdm_prefix_mdr_absolute_secondevaluationhscm_positive /\ ((exists ff_gap_mdm_lt_mdr_absolute_secondevaluationhscm_positive_column_bound. ff_gap_mdm_lt_mdr_absolute_secondevaluationhscm_positive_column_bound + S (ff_column_mdm_prefix_mdr_absolute_secondevaluationhscm_positive) = (mdr_q_absolute_secondevaluationhs)) /\ ((exists ff_row_mdm_cell_mdr_absolute_secondevaluationhscm_positive_cell ff_column_mdm_cell_mdr_absolute_secondevaluationhscm_positive_cell. (((((exists ff_gap_mdm_lt_mdr_absolute_secondevaluationhscm_positive_cell_row_before. ff_gap_mdm_lt_mdr_absolute_secondevaluationhscm_positive_cell_row_before + S (ff_row_mdm_prefix_mdr_absolute_secondevaluationhscm_positive) = (0)) /\ ff_row_mdm_cell_mdr_absolute_secondevaluationhscm_positive_cell = ff_row_mdm_prefix_mdr_absolute_secondevaluationhscm_positive) \/ ((exists ff_gap_mdm_le_mdr_absolute_secondevaluationhscm_positive_cell_row_after. ff_gap_mdm_le_mdr_absolute_secondevaluationhscm_positive_cell_row_after + (0) = (ff_row_mdm_prefix_mdr_absolute_secondevaluationhscm_positive)) /\ ff_row_mdm_cell_mdr_absolute_secondevaluationhscm_positive_cell = S ff_row_mdm_prefix_mdr_absolute_secondevaluationhscm_positive))) /\ (((((exists ff_gap_mdm_lt_mdr_absolute_secondevaluationhscm_positive_cell_column_before. ff_gap_mdm_lt_mdr_absolute_secondevaluationhscm_positive_cell_column_before + S (ff_column_mdm_prefix_mdr_absolute_secondevaluationhscm_positive) = (mdr_j_absolute_secondevaluationhsc)) /\ ff_column_mdm_cell_mdr_absolute_secondevaluationhscm_positive_cell = ff_column_mdm_prefix_mdr_absolute_secondevaluationhscm_positive) \/ ((exists ff_gap_mdm_le_mdr_absolute_secondevaluationhscm_positive_cell_column_after. ff_gap_mdm_le_mdr_absolute_secondevaluationhscm_positive_cell_column_after + (mdr_j_absolute_secondevaluationhsc) = (ff_column_mdm_prefix_mdr_absolute_secondevaluationhscm_positive)) /\ ff_column_mdm_cell_mdr_absolute_secondevaluationhscm_positive_cell = S ff_column_mdm_prefix_mdr_absolute_secondevaluationhscm_positive))) /\ (((exists ff_h_mdm_mdr_absolute_secondevaluationhscm_positive_cell_source. ff_h_mdm_mdr_absolute_secondevaluationhscm_positive_cell_source + S (ff_value_mdm_prefix_mdr_absolute_secondevaluationhscm_positive) = S ((S ((ff_row_mdm_cell_mdr_absolute_secondevaluationhscm_positive_cell) * (S (mdr_q_absolute_secondevaluationhs)) + (ff_column_mdm_cell_mdr_absolute_secondevaluationhscm_positive_cell))) * mdr_pc_absolute_secondevaluationh)) /\ exists ff_q_mdm_mdr_absolute_secondevaluationhscm_positive_cell_source. mdr_pb_absolute_secondevaluationh = ff_q_mdm_mdr_absolute_secondevaluationhscm_positive_cell_source * S ((S ((ff_row_mdm_cell_mdr_absolute_secondevaluationhscm_positive_cell) * (S (mdr_q_absolute_secondevaluationhs)) + (ff_column_mdm_cell_mdr_absolute_secondevaluationhscm_positive_cell))) * mdr_pc_absolute_secondevaluationh) + (ff_value_mdm_prefix_mdr_absolute_secondevaluationhscm_positive)))))) /\ (((exists ff_h_mdm_mdr_absolute_secondevaluationhscm_positive_target. ff_h_mdm_mdr_absolute_secondevaluationhscm_positive_target + S (ff_value_mdm_prefix_mdr_absolute_secondevaluationhscm_positive) = S ((S (ff_index_mdm_prefix_mdr_absolute_secondevaluationhscm_positive)) * mdr_us_absolute_secondevaluationhsc)) /\ exists ff_q_mdm_mdr_absolute_secondevaluationhscm_positive_target. mdr_up_absolute_secondevaluationhsc = ff_q_mdm_mdr_absolute_secondevaluationhscm_positive_target * S ((S (ff_index_mdm_prefix_mdr_absolute_secondevaluationhscm_positive)) * mdr_us_absolute_secondevaluationhsc) + (ff_value_mdm_prefix_mdr_absolute_secondevaluationhscm_positive))))))) /\ (forall ff_index_mdm_prefix_mdr_absolute_secondevaluationhscm_negative. (exists ff_gap_mdm_lt_mdr_absolute_secondevaluationhscm_negative_index_bound. ff_gap_mdm_lt_mdr_absolute_secondevaluationhscm_negative_index_bound + S (ff_index_mdm_prefix_mdr_absolute_secondevaluationhscm_negative) = ((mdr_q_absolute_secondevaluationhs) * (mdr_q_absolute_secondevaluationhs))) -> exists ff_row_mdm_prefix_mdr_absolute_secondevaluationhscm_negative ff_column_mdm_prefix_mdr_absolute_secondevaluationhscm_negative ff_value_mdm_prefix_mdr_absolute_secondevaluationhscm_negative. (ff_index_mdm_prefix_mdr_absolute_secondevaluationhscm_negative = (mdr_q_absolute_secondevaluationhs) * ff_row_mdm_prefix_mdr_absolute_secondevaluationhscm_negative + ff_column_mdm_prefix_mdr_absolute_secondevaluationhscm_negative /\ ((exists ff_gap_mdm_lt_mdr_absolute_secondevaluationhscm_negative_column_bound. ff_gap_mdm_lt_mdr_absolute_secondevaluationhscm_negative_column_bound + S (ff_column_mdm_prefix_mdr_absolute_secondevaluationhscm_negative) = (mdr_q_absolute_secondevaluationhs)) /\ ((exists ff_row_mdm_cell_mdr_absolute_secondevaluationhscm_negative_cell ff_column_mdm_cell_mdr_absolute_secondevaluationhscm_negative_cell. (((((exists ff_gap_mdm_lt_mdr_absolute_secondevaluationhscm_negative_cell_row_before. ff_gap_mdm_lt_mdr_absolute_secondevaluationhscm_negative_cell_row_before + S (ff_row_mdm_prefix_mdr_absolute_secondevaluationhscm_negative) = (0)) /\ ff_row_mdm_cell_mdr_absolute_secondevaluationhscm_negative_cell = ff_row_mdm_prefix_mdr_absolute_secondevaluationhscm_negative) \/ ((exists ff_gap_mdm_le_mdr_absolute_secondevaluationhscm_negative_cell_row_after. ff_gap_mdm_le_mdr_absolute_secondevaluationhscm_negative_cell_row_after + (0) = (ff_row_mdm_prefix_mdr_absolute_secondevaluationhscm_negative)) /\ ff_row_mdm_cell_mdr_absolute_secondevaluationhscm_negative_cell = S ff_row_mdm_prefix_mdr_absolute_secondevaluationhscm_negative))) /\ (((((exists ff_gap_mdm_lt_mdr_absolute_secondevaluationhscm_negative_cell_column_before. ff_gap_mdm_lt_mdr_absolute_secondevaluationhscm_negative_cell_column_before + S (ff_column_mdm_prefix_mdr_absolute_secondevaluationhscm_negative) = (mdr_j_absolute_secondevaluationhsc)) /\ ff_column_mdm_cell_mdr_absolute_secondevaluationhscm_negative_cell = ff_column_mdm_prefix_mdr_absolute_secondevaluationhscm_negative) \/ ((exists ff_gap_mdm_le_mdr_absolute_secondevaluationhscm_negative_cell_column_after. ff_gap_mdm_le_mdr_absolute_secondevaluationhscm_negative_cell_column_after + (mdr_j_absolute_secondevaluationhsc) = (ff_column_mdm_prefix_mdr_absolute_secondevaluationhscm_negative)) /\ ff_column_mdm_cell_mdr_absolute_secondevaluationhscm_negative_cell = S ff_column_mdm_prefix_mdr_absolute_secondevaluationhscm_negative))) /\ (((exists ff_h_mdm_mdr_absolute_secondevaluationhscm_negative_cell_source. ff_h_mdm_mdr_absolute_secondevaluationhscm_negative_cell_source + S (ff_value_mdm_prefix_mdr_absolute_secondevaluationhscm_negative) = S ((S ((ff_row_mdm_cell_mdr_absolute_secondevaluationhscm_negative_cell) * (S (mdr_q_absolute_secondevaluationhs)) + (ff_column_mdm_cell_mdr_absolute_secondevaluationhscm_negative_cell))) * mdr_nc_absolute_secondevaluationh)) /\ exists ff_q_mdm_mdr_absolute_secondevaluationhscm_negative_cell_source. mdr_nb_absolute_secondevaluationh = ff_q_mdm_mdr_absolute_secondevaluationhscm_negative_cell_source * S ((S ((ff_row_mdm_cell_mdr_absolute_secondevaluationhscm_negative_cell) * (S (mdr_q_absolute_secondevaluationhs)) + (ff_column_mdm_cell_mdr_absolute_secondevaluationhscm_negative_cell))) * mdr_nc_absolute_secondevaluationh) + (ff_value_mdm_prefix_mdr_absolute_secondevaluationhscm_negative)))))) /\ (((exists ff_h_mdm_mdr_absolute_secondevaluationhscm_negative_target. ff_h_mdm_mdr_absolute_secondevaluationhscm_negative_target + S (ff_value_mdm_prefix_mdr_absolute_secondevaluationhscm_negative) = S ((S (ff_index_mdm_prefix_mdr_absolute_secondevaluationhscm_negative)) * mdr_ut_absolute_secondevaluationhsc)) /\ exists ff_q_mdm_mdr_absolute_secondevaluationhscm_negative_target. mdr_un_absolute_secondevaluationhsc = ff_q_mdm_mdr_absolute_secondevaluationhscm_negative_target * S ((S (ff_index_mdm_prefix_mdr_absolute_secondevaluationhscm_negative)) * mdr_ut_absolute_secondevaluationhsc) + (ff_value_mdm_prefix_mdr_absolute_secondevaluationhscm_negative))))))))) /\ ((((exists ff_h_mdr_absolute_secondevaluationhscp. ff_h_mdr_absolute_secondevaluationhscp + S (mdr_p_absolute_secondevaluationhsc) = S ((S (mdr_j_absolute_secondevaluationhsc)) * mdr_ec_absolute_secondevaluationhs)) /\ exists ff_q_mdr_absolute_secondevaluationhscp. mdr_eb_absolute_secondevaluationhs = ff_q_mdr_absolute_secondevaluationhscp * S ((S (mdr_j_absolute_secondevaluationhsc)) * mdr_ec_absolute_secondevaluationhs) + (mdr_p_absolute_secondevaluationhsc))) /\ (((exists ff_h_mdr_absolute_secondevaluationhscn. ff_h_mdr_absolute_secondevaluationhscn + S (mdr_n_absolute_secondevaluationhsc) = S ((S (mdr_j_absolute_secondevaluationhsc)) * mdr_fc_absolute_secondevaluationhs)) /\ exists ff_q_mdr_absolute_secondevaluationhscn. mdr_fb_absolute_secondevaluationhs = ff_q_mdr_absolute_secondevaluationhscn * S ((S (mdr_j_absolute_secondevaluationhsc)) * mdr_fc_absolute_secondevaluationhs) + (mdr_n_absolute_secondevaluationhsc)))))))) /\ (exists ff_ub_mce_fold_mdr_absolute_secondevaluationhsf ff_uc_mce_fold_mdr_absolute_secondevaluationhsf ff_vb_mce_fold_mdr_absolute_secondevaluationhsf ff_vc_mce_fold_mdr_absolute_secondevaluationhsf. ((forall ff_index_mce_alternating_mdr_absolute_secondevaluationhsf_prefix. (exists ff_gap_mce_mdr_absolute_secondevaluationhsf_prefix_index. ff_gap_mce_mdr_absolute_secondevaluationhsf_prefix_index + S (ff_index_mce_alternating_mdr_absolute_secondevaluationhsf_prefix) = (S (mdr_q_absolute_secondevaluationhs))) -> exists ff_ap_mce_alternating_mdr_absolute_secondevaluationhsf_prefix ff_an_mce_alternating_mdr_absolute_secondevaluationhsf_prefix ff_bp_mce_alternating_mdr_absolute_secondevaluationhsf_prefix ff_bn_mce_alternating_mdr_absolute_secondevaluationhsf_prefix ff_p_mce_alternating_mdr_absolute_secondevaluationhsf_prefix ff_n_mce_alternating_mdr_absolute_secondevaluationhsf_prefix. ((((exists ff_h_mce_mdr_absolute_secondevaluationhsf_prefix_ap. ff_h_mce_mdr_absolute_secondevaluationhsf_prefix_ap + S (ff_ap_mce_alternating_mdr_absolute_secondevaluationhsf_prefix) = S ((S (ff_index_mce_alternating_mdr_absolute_secondevaluationhsf_prefix)) * mdr_pc_absolute_secondevaluationh)) /\ exists ff_q_mce_mdr_absolute_secondevaluationhsf_prefix_ap. mdr_pb_absolute_secondevaluationh = ff_q_mce_mdr_absolute_secondevaluationhsf_prefix_ap * S ((S (ff_index_mce_alternating_mdr_absolute_secondevaluationhsf_prefix)) * mdr_pc_absolute_secondevaluationh) + (ff_ap_mce_alternating_mdr_absolute_secondevaluationhsf_prefix))) /\ ((((exists ff_h_mce_mdr_absolute_secondevaluationhsf_prefix_an. ff_h_mce_mdr_absolute_secondevaluationhsf_prefix_an + S (ff_an_mce_alternating_mdr_absolute_secondevaluationhsf_prefix) = S ((S (ff_index_mce_alternating_mdr_absolute_secondevaluationhsf_prefix)) * mdr_nc_absolute_secondevaluationh)) /\ exists ff_q_mce_mdr_absolute_secondevaluationhsf_prefix_an. mdr_nb_absolute_secondevaluationh = ff_q_mce_mdr_absolute_secondevaluationhsf_prefix_an * S ((S (ff_index_mce_alternating_mdr_absolute_secondevaluationhsf_prefix)) * mdr_nc_absolute_secondevaluationh) + (ff_an_mce_alternating_mdr_absolute_secondevaluationhsf_prefix))) /\ ((((exists ff_h_mce_mdr_absolute_secondevaluationhsf_prefix_bp. ff_h_mce_mdr_absolute_secondevaluationhsf_prefix_bp + S (ff_bp_mce_alternating_mdr_absolute_secondevaluationhsf_prefix) = S ((S (ff_index_mce_alternating_mdr_absolute_secondevaluationhsf_prefix)) * mdr_ec_absolute_secondevaluationhs)) /\ exists ff_q_mce_mdr_absolute_secondevaluationhsf_prefix_bp. mdr_eb_absolute_secondevaluationhs = ff_q_mce_mdr_absolute_secondevaluationhsf_prefix_bp * S ((S (ff_index_mce_alternating_mdr_absolute_secondevaluationhsf_prefix)) * mdr_ec_absolute_secondevaluationhs) + (ff_bp_mce_alternating_mdr_absolute_secondevaluationhsf_prefix))) /\ ((((exists ff_h_mce_mdr_absolute_secondevaluationhsf_prefix_bn. ff_h_mce_mdr_absolute_secondevaluationhsf_prefix_bn + S (ff_bn_mce_alternating_mdr_absolute_secondevaluationhsf_prefix) = S ((S (ff_index_mce_alternating_mdr_absolute_secondevaluationhsf_prefix)) * mdr_fc_absolute_secondevaluationhs)) /\ exists ff_q_mce_mdr_absolute_secondevaluationhsf_prefix_bn. mdr_fb_absolute_secondevaluationhs = ff_q_mce_mdr_absolute_secondevaluationhsf_prefix_bn * S ((S (ff_index_mce_alternating_mdr_absolute_secondevaluationhsf_prefix)) * mdr_fc_absolute_secondevaluationhs) + (ff_bn_mce_alternating_mdr_absolute_secondevaluationhsf_prefix))) /\ ((((exists ff_h_mce_mdr_absolute_secondevaluationhsf_prefix_positive. ff_h_mce_mdr_absolute_secondevaluationhsf_prefix_positive + S (ff_p_mce_alternating_mdr_absolute_secondevaluationhsf_prefix) = S ((S (ff_index_mce_alternating_mdr_absolute_secondevaluationhsf_prefix)) * ff_uc_mce_fold_mdr_absolute_secondevaluationhsf)) /\ exists ff_q_mce_mdr_absolute_secondevaluationhsf_prefix_positive. ff_ub_mce_fold_mdr_absolute_secondevaluationhsf = ff_q_mce_mdr_absolute_secondevaluationhsf_prefix_positive * S ((S (ff_index_mce_alternating_mdr_absolute_secondevaluationhsf_prefix)) * ff_uc_mce_fold_mdr_absolute_secondevaluationhsf) + (ff_p_mce_alternating_mdr_absolute_secondevaluationhsf_prefix))) /\ ((((exists ff_h_mce_mdr_absolute_secondevaluationhsf_prefix_negative. ff_h_mce_mdr_absolute_secondevaluationhsf_prefix_negative + S (ff_n_mce_alternating_mdr_absolute_secondevaluationhsf_prefix) = S ((S (ff_index_mce_alternating_mdr_absolute_secondevaluationhsf_prefix)) * ff_vc_mce_fold_mdr_absolute_secondevaluationhsf)) /\ exists ff_q_mce_mdr_absolute_secondevaluationhsf_prefix_negative. ff_vb_mce_fold_mdr_absolute_secondevaluationhsf = ff_q_mce_mdr_absolute_secondevaluationhsf_prefix_negative * S ((S (ff_index_mce_alternating_mdr_absolute_secondevaluationhsf_prefix)) * ff_vc_mce_fold_mdr_absolute_secondevaluationhsf) + (ff_n_mce_alternating_mdr_absolute_secondevaluationhsf_prefix))) /\ (((exists ff_even_mce_term_mdr_absolute_secondevaluationhsf_prefix_term. ff_index_mce_alternating_mdr_absolute_secondevaluationhsf_prefix = 2 * ff_even_mce_term_mdr_absolute_secondevaluationhsf_prefix_term) /\ (ff_p_mce_alternating_mdr_absolute_secondevaluationhsf_prefix = (ff_ap_mce_alternating_mdr_absolute_secondevaluationhsf_prefix) * (ff_bp_mce_alternating_mdr_absolute_secondevaluationhsf_prefix) + (ff_an_mce_alternating_mdr_absolute_secondevaluationhsf_prefix) * (ff_bn_mce_alternating_mdr_absolute_secondevaluationhsf_prefix) /\ ff_n_mce_alternating_mdr_absolute_secondevaluationhsf_prefix = (ff_ap_mce_alternating_mdr_absolute_secondevaluationhsf_prefix) * (ff_bn_mce_alternating_mdr_absolute_secondevaluationhsf_prefix) + (ff_an_mce_alternating_mdr_absolute_secondevaluationhsf_prefix) * (ff_bp_mce_alternating_mdr_absolute_secondevaluationhsf_prefix))) \/ ((exists ff_odd_mce_term_mdr_absolute_secondevaluationhsf_prefix_term. ff_index_mce_alternating_mdr_absolute_secondevaluationhsf_prefix = 2 * ff_odd_mce_term_mdr_absolute_secondevaluationhsf_prefix_term + 1) /\ (ff_p_mce_alternating_mdr_absolute_secondevaluationhsf_prefix = (ff_ap_mce_alternating_mdr_absolute_secondevaluationhsf_prefix) * (ff_bn_mce_alternating_mdr_absolute_secondevaluationhsf_prefix) + (ff_an_mce_alternating_mdr_absolute_secondevaluationhsf_prefix) * (ff_bp_mce_alternating_mdr_absolute_secondevaluationhsf_prefix) /\ ff_n_mce_alternating_mdr_absolute_secondevaluationhsf_prefix = (ff_ap_mce_alternating_mdr_absolute_secondevaluationhsf_prefix) * (ff_bp_mce_alternating_mdr_absolute_secondevaluationhsf_prefix) + (ff_an_mce_alternating_mdr_absolute_secondevaluationhsf_prefix) * (ff_bn_mce_alternating_mdr_absolute_secondevaluationhsf_prefix))))))))))) /\ ((exists ff_u_mce_mdr_absolute_secondevaluationhsf_positive ff_v_mce_mdr_absolute_secondevaluationhsf_positive. ((((exists ff_h_mce_mdr_absolute_secondevaluationhsf_positive_start. ff_h_mce_mdr_absolute_secondevaluationhsf_positive_start + S (0) = S ((S (0)) * ff_v_mce_mdr_absolute_secondevaluationhsf_positive)) /\ exists ff_q_mce_mdr_absolute_secondevaluationhsf_positive_start. ff_u_mce_mdr_absolute_secondevaluationhsf_positive = ff_q_mce_mdr_absolute_secondevaluationhsf_positive_start * S ((S (0)) * ff_v_mce_mdr_absolute_secondevaluationhsf_positive) + (0))) /\ ((((exists ff_h_mce_mdr_absolute_secondevaluationhsf_positive_terminal. ff_h_mce_mdr_absolute_secondevaluationhsf_positive_terminal + S (mdr_p_absolute_secondevaluationh) = S ((S ((S (mdr_q_absolute_secondevaluationhs)))) * ff_v_mce_mdr_absolute_secondevaluationhsf_positive)) /\ exists ff_q_mce_mdr_absolute_secondevaluationhsf_positive_terminal. ff_u_mce_mdr_absolute_secondevaluationhsf_positive = ff_q_mce_mdr_absolute_secondevaluationhsf_positive_terminal * S ((S ((S (mdr_q_absolute_secondevaluationhs)))) * ff_v_mce_mdr_absolute_secondevaluationhsf_positive) + (mdr_p_absolute_secondevaluationh))) /\ forall ff_i_mce_mdr_absolute_secondevaluationhsf_positive. (exists ff_lt_mce_mdr_absolute_secondevaluationhsf_positive_bound. ff_lt_mce_mdr_absolute_secondevaluationhsf_positive_bound + S ff_i_mce_mdr_absolute_secondevaluationhsf_positive = (S (mdr_q_absolute_secondevaluationhs))) -> exists ff_a_mce_mdr_absolute_secondevaluationhsf_positive ff_r_mce_mdr_absolute_secondevaluationhsf_positive ff_s_mce_mdr_absolute_secondevaluationhsf_positive. ((((exists ff_h_mce_mdr_absolute_secondevaluationhsf_positive_summand. ff_h_mce_mdr_absolute_secondevaluationhsf_positive_summand + S (ff_a_mce_mdr_absolute_secondevaluationhsf_positive) = S ((S (ff_i_mce_mdr_absolute_secondevaluationhsf_positive)) * ff_uc_mce_fold_mdr_absolute_secondevaluationhsf)) /\ exists ff_q_mce_mdr_absolute_secondevaluationhsf_positive_summand. ff_ub_mce_fold_mdr_absolute_secondevaluationhsf = ff_q_mce_mdr_absolute_secondevaluationhsf_positive_summand * S ((S (ff_i_mce_mdr_absolute_secondevaluationhsf_positive)) * ff_uc_mce_fold_mdr_absolute_secondevaluationhsf) + (ff_a_mce_mdr_absolute_secondevaluationhsf_positive))) /\ ((((exists ff_h_mce_mdr_absolute_secondevaluationhsf_positive_partial. ff_h_mce_mdr_absolute_secondevaluationhsf_positive_partial + S (ff_r_mce_mdr_absolute_secondevaluationhsf_positive) = S ((S (ff_i_mce_mdr_absolute_secondevaluationhsf_positive)) * ff_v_mce_mdr_absolute_secondevaluationhsf_positive)) /\ exists ff_q_mce_mdr_absolute_secondevaluationhsf_positive_partial. ff_u_mce_mdr_absolute_secondevaluationhsf_positive = ff_q_mce_mdr_absolute_secondevaluationhsf_positive_partial * S ((S (ff_i_mce_mdr_absolute_secondevaluationhsf_positive)) * ff_v_mce_mdr_absolute_secondevaluationhsf_positive) + (ff_r_mce_mdr_absolute_secondevaluationhsf_positive))) /\ ((((exists ff_h_mce_mdr_absolute_secondevaluationhsf_positive_successor. ff_h_mce_mdr_absolute_secondevaluationhsf_positive_successor + S (ff_s_mce_mdr_absolute_secondevaluationhsf_positive) = S ((S (S ff_i_mce_mdr_absolute_secondevaluationhsf_positive)) * ff_v_mce_mdr_absolute_secondevaluationhsf_positive)) /\ exists ff_q_mce_mdr_absolute_secondevaluationhsf_positive_successor. ff_u_mce_mdr_absolute_secondevaluationhsf_positive = ff_q_mce_mdr_absolute_secondevaluationhsf_positive_successor * S ((S (S ff_i_mce_mdr_absolute_secondevaluationhsf_positive)) * ff_v_mce_mdr_absolute_secondevaluationhsf_positive) + (ff_s_mce_mdr_absolute_secondevaluationhsf_positive))) /\ ff_s_mce_mdr_absolute_secondevaluationhsf_positive = ff_r_mce_mdr_absolute_secondevaluationhsf_positive + ff_a_mce_mdr_absolute_secondevaluationhsf_positive)))))) /\ (exists ff_u_mce_mdr_absolute_secondevaluationhsf_negative ff_v_mce_mdr_absolute_secondevaluationhsf_negative. ((((exists ff_h_mce_mdr_absolute_secondevaluationhsf_negative_start. ff_h_mce_mdr_absolute_secondevaluationhsf_negative_start + S (0) = S ((S (0)) * ff_v_mce_mdr_absolute_secondevaluationhsf_negative)) /\ exists ff_q_mce_mdr_absolute_secondevaluationhsf_negative_start. ff_u_mce_mdr_absolute_secondevaluationhsf_negative = ff_q_mce_mdr_absolute_secondevaluationhsf_negative_start * S ((S (0)) * ff_v_mce_mdr_absolute_secondevaluationhsf_negative) + (0))) /\ ((((exists ff_h_mce_mdr_absolute_secondevaluationhsf_negative_terminal. ff_h_mce_mdr_absolute_secondevaluationhsf_negative_terminal + S (mdr_n_absolute_secondevaluationh) = S ((S ((S (mdr_q_absolute_secondevaluationhs)))) * ff_v_mce_mdr_absolute_secondevaluationhsf_negative)) /\ exists ff_q_mce_mdr_absolute_secondevaluationhsf_negative_terminal. ff_u_mce_mdr_absolute_secondevaluationhsf_negative = ff_q_mce_mdr_absolute_secondevaluationhsf_negative_terminal * S ((S ((S (mdr_q_absolute_secondevaluationhs)))) * ff_v_mce_mdr_absolute_secondevaluationhsf_negative) + (mdr_n_absolute_secondevaluationh))) /\ forall ff_i_mce_mdr_absolute_secondevaluationhsf_negative. (exists ff_lt_mce_mdr_absolute_secondevaluationhsf_negative_bound. ff_lt_mce_mdr_absolute_secondevaluationhsf_negative_bound + S ff_i_mce_mdr_absolute_secondevaluationhsf_negative = (S (mdr_q_absolute_secondevaluationhs))) -> exists ff_a_mce_mdr_absolute_secondevaluationhsf_negative ff_r_mce_mdr_absolute_secondevaluationhsf_negative ff_s_mce_mdr_absolute_secondevaluationhsf_negative. ((((exists ff_h_mce_mdr_absolute_secondevaluationhsf_negative_summand. ff_h_mce_mdr_absolute_secondevaluationhsf_negative_summand + S (ff_a_mce_mdr_absolute_secondevaluationhsf_negative) = S ((S (ff_i_mce_mdr_absolute_secondevaluationhsf_negative)) * ff_vc_mce_fold_mdr_absolute_secondevaluationhsf)) /\ exists ff_q_mce_mdr_absolute_secondevaluationhsf_negative_summand. ff_vb_mce_fold_mdr_absolute_secondevaluationhsf = ff_q_mce_mdr_absolute_secondevaluationhsf_negative_summand * S ((S (ff_i_mce_mdr_absolute_secondevaluationhsf_negative)) * ff_vc_mce_fold_mdr_absolute_secondevaluationhsf) + (ff_a_mce_mdr_absolute_secondevaluationhsf_negative))) /\ ((((exists ff_h_mce_mdr_absolute_secondevaluationhsf_negative_partial. ff_h_mce_mdr_absolute_secondevaluationhsf_negative_partial + S (ff_r_mce_mdr_absolute_secondevaluationhsf_negative) = S ((S (ff_i_mce_mdr_absolute_secondevaluationhsf_negative)) * ff_v_mce_mdr_absolute_secondevaluationhsf_negative)) /\ exists ff_q_mce_mdr_absolute_secondevaluationhsf_negative_partial. ff_u_mce_mdr_absolute_secondevaluationhsf_negative = ff_q_mce_mdr_absolute_secondevaluationhsf_negative_partial * S ((S (ff_i_mce_mdr_absolute_secondevaluationhsf_negative)) * ff_v_mce_mdr_absolute_secondevaluationhsf_negative) + (ff_r_mce_mdr_absolute_secondevaluationhsf_negative))) /\ ((((exists ff_h_mce_mdr_absolute_secondevaluationhsf_negative_successor. ff_h_mce_mdr_absolute_secondevaluationhsf_negative_successor + S (ff_s_mce_mdr_absolute_secondevaluationhsf_negative) = S ((S (S ff_i_mce_mdr_absolute_secondevaluationhsf_negative)) * ff_v_mce_mdr_absolute_secondevaluationhsf_negative)) /\ exists ff_q_mce_mdr_absolute_secondevaluationhsf_negative_successor. ff_u_mce_mdr_absolute_secondevaluationhsf_negative = ff_q_mce_mdr_absolute_secondevaluationhsf_negative_successor * S ((S (S ff_i_mce_mdr_absolute_secondevaluationhsf_negative)) * ff_v_mce_mdr_absolute_secondevaluationhsf_negative) + (ff_s_mce_mdr_absolute_secondevaluationhsf_negative))) /\ ff_s_mce_mdr_absolute_secondevaluationhsf_negative = ff_r_mce_mdr_absolute_secondevaluationhsf_negative + ff_a_mce_mdr_absolute_secondevaluationhsf_negative))))))))))))))) /\ ((exists mdr_gap_absolute_secondevaluationi. mdr_gap_absolute_secondevaluationi + S (mdr_i_absolute_secondevaluation) = (mdr_l_absolute_secondevaluation)) /\ (exists mdr_z_absolute_secondevaluationr. ((exists mdr_a_absolute_secondevaluationrc mdr_b_absolute_secondevaluationrc mdr_c_absolute_secondevaluationrc mdr_e_absolute_secondevaluationrc mdr_f_absolute_secondevaluationrc. ((mdr_a_absolute_secondevaluationrc = ((d) + (ab)) * S ((d) + (ab)) + ((ab) + (ab))) /\ ((mdr_b_absolute_secondevaluationrc = ((ac) + (bb)) * S ((ac) + (bb)) + ((bb) + (bb))) /\ ((mdr_c_absolute_secondevaluationrc = ((mdr_a_absolute_secondevaluationrc) + (mdr_b_absolute_secondevaluationrc)) * S ((mdr_a_absolute_secondevaluationrc) + (mdr_b_absolute_secondevaluationrc)) + ((mdr_b_absolute_secondevaluationrc) + (mdr_b_absolute_secondevaluationrc))) /\ ((mdr_e_absolute_secondevaluationrc = ((mdr_p_absolute_second) + (mdr_n_absolute_second)) * S ((mdr_p_absolute_second) + (mdr_n_absolute_second)) + ((mdr_n_absolute_second) + (mdr_n_absolute_second))) /\ ((mdr_f_absolute_secondevaluationrc = ((bc) + (mdr_e_absolute_secondevaluationrc)) * S ((bc) + (mdr_e_absolute_secondevaluationrc)) + ((mdr_e_absolute_secondevaluationrc) + (mdr_e_absolute_secondevaluationrc))) /\ ((mdr_z_absolute_secondevaluationr) = ((mdr_c_absolute_secondevaluationrc) + (mdr_f_absolute_secondevaluationrc)) * S ((mdr_c_absolute_secondevaluationrc) + (mdr_f_absolute_secondevaluationrc)) + ((mdr_f_absolute_secondevaluationrc) + (mdr_f_absolute_secondevaluationrc))))))))) /\ (((exists ff_h_mdr_absolute_secondevaluationrb. ff_h_mdr_absolute_secondevaluationrb + S (mdr_z_absolute_secondevaluationr) = S ((S (mdr_i_absolute_secondevaluation)) * mdr_c_absolute_secondevaluation)) /\ exists ff_q_mdr_absolute_secondevaluationrb. mdr_b_absolute_secondevaluation = ff_q_mdr_absolute_secondevaluationrb * S ((S (mdr_i_absolute_secondevaluation)) * mdr_c_absolute_secondevaluation) + (mdr_z_absolute_secondevaluationr)))))))) /\ (((mdr_p_absolute_second) = (mdr_n_absolute_second) + (E)) \/ ((mdr_n_absolute_second) = (mdr_p_absolute_second) + (E))))) -> D = E

Complete tactic proof in conservative notation

All 40 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

40 script commands · 8 reading checkpoints · 1 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 (2)
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 casesL10–15

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

  1. L10
    cases hfirst
  2. L11
    cases hfirst_witness
  3. L12
    cases hfirst_witness_witness
  4. L13
    cases hsecond
  5. L14
    cases hsecond_witness
  6. L15
    cases hsecond_witness_witness
03Establish hvaluesL16–25

Establish this local claim before using it. It is not an additional assumption.

  1. L16
    have hvalues : x = x2 /\ x1 = x3
  2. L17
    specialize signed_recursive_determinant_functional (ab)
  3. L18
    specialize signed_recursive_determinant_functional (ac)
  4. L19
    specialize signed_recursive_determinant_functional (bb)
  5. L20
    specialize signed_recursive_determinant_functional (bc)
  6. L21
    specialize signed_recursive_determinant_functional (d)
  7. L22
    specialize signed_recursive_determinant_functional (x)
  8. L23
    specialize signed_recursive_determinant_functional (x1)
  9. L24
    specialize signed_recursive_determinant_functional (x2)
  10. L25
    specialize signed_recursive_determinant_functional (x3)
04Use earlier factsL26–28

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

  1. L26
    apply signed_recursive_determinant_functional
  2. L27
    exact hfirst_witness_witness_left
  3. L28
    exact hsecond_witness_witness_left
05Separate the logical casesL29–29

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

  1. L29
    cases hvalues
06Use earlier factsL30–34

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

  1. L30
    specialize matrix_lattice_absolute_difference_functional (x2)
  2. L31
    specialize matrix_lattice_absolute_difference_functional (x3)
  3. L32
    specialize matrix_lattice_absolute_difference_functional (D)
  4. L33
    specialize matrix_lattice_absolute_difference_functional (E)
  5. L34
    apply matrix_lattice_absolute_difference_functional
07Calculate and transport equalitiesL35–38

Carry out the recorded arithmetic or equality steps; inspect the exact commands for their direction and premises.

  1. L35
    rewrite hvalues_left at hfirst_witness_witness_right
  2. L36
    rewrite hvalues_left at hfirst_witness_witness_right
  3. L37
    rewrite hvalues_right at hfirst_witness_witness_right
  4. L38
    rewrite hvalues_right at hfirst_witness_witness_right
08Use earlier factsL39–40

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

  1. L39
    exact hfirst_witness_witness_right
  2. L40
    exact hsecond_witness_witness_right

Library-wide reading audit

Original defined command ledger · 40 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_witness
  12. 0012cases hfirst_witness_witness
  13. 0013cases hsecond
  14. 0014cases hsecond_witness
  15. 0015cases hsecond_witness_witness
  16. 0016have hvalues : x = x2 /\ x1 = x3
  17. 0017specialize signed_recursive_determinant_functional (ab)
  18. 0018specialize signed_recursive_determinant_functional (ac)
  19. 0019specialize signed_recursive_determinant_functional (bb)
  20. 0020specialize signed_recursive_determinant_functional (bc)
  21. 0021specialize signed_recursive_determinant_functional (d)
  22. 0022specialize signed_recursive_determinant_functional (x)
  23. 0023specialize signed_recursive_determinant_functional (x1)
  24. 0024specialize signed_recursive_determinant_functional (x2)
  25. 0025specialize signed_recursive_determinant_functional (x3)
  26. 0026apply signed_recursive_determinant_functional
  27. 0027exact hfirst_witness_witness_left
  28. 0028exact hsecond_witness_witness_left
  29. 0029cases hvalues
  30. 0030specialize matrix_lattice_absolute_difference_functional (x2)
  31. 0031specialize matrix_lattice_absolute_difference_functional (x3)
  32. 0032specialize matrix_lattice_absolute_difference_functional (D)
  33. 0033specialize matrix_lattice_absolute_difference_functional (E)
  34. 0034apply matrix_lattice_absolute_difference_functional
  35. 0035rewrite hvalues_left at hfirst_witness_witness_right
  36. 0036rewrite hvalues_left at hfirst_witness_witness_right
  37. 0037rewrite hvalues_right at hfirst_witness_witness_right
  38. 0038rewrite hvalues_right at hfirst_witness_witness_right
  39. 0039exact hfirst_witness_witness_right
  40. 0040exact hsecond_witness_witness_right