rectangular_matrix_rank_integer_transport

Exact expanded first-order arithmetic statement

forall ab ac bb bc eb ec fb fc r w rank. (forall ics_index_rank_parent_equality ics_value0_rank_parent_equality ics_value1_rank_parent_equality ics_value2_rank_parent_equality ics_value3_rank_parent_equality. (exists ics_gap_rank_parent_equality_bound. ics_gap_rank_parent_equality_bound + S (ics_index_rank_parent_equality) = ((r) * (w))) -> (((exists fs_h_ics_rank_parent_equality_at0. fs_h_ics_rank_parent_equality_at0 + S (ics_value0_rank_parent_equality) = S ((S (ics_index_rank_parent_equality)) * ac)) /\ exists fs_q_ics_rank_parent_equality_at0. ab = fs_q_ics_rank_parent_equality_at0 * S ((S (ics_index_rank_parent_equality)) * ac) + (ics_value0_rank_parent_equality))) -> (((exists fs_h_ics_rank_parent_equality_at1. fs_h_ics_rank_parent_equality_at1 + S (ics_value1_rank_parent_equality) = S ((S (ics_index_rank_parent_equality)) * bc)) /\ exists fs_q_ics_rank_parent_equality_at1. bb = fs_q_ics_rank_parent_equality_at1 * S ((S (ics_index_rank_parent_equality)) * bc) + (ics_value1_rank_parent_equality))) -> (((exists fs_h_ics_rank_parent_equality_at2. fs_h_ics_rank_parent_equality_at2 + S (ics_value2_rank_parent_equality) = S ((S (ics_index_rank_parent_equality)) * ec)) /\ exists fs_q_ics_rank_parent_equality_at2. eb = fs_q_ics_rank_parent_equality_at2 * S ((S (ics_index_rank_parent_equality)) * ec) + (ics_value2_rank_parent_equality))) -> (((exists fs_h_ics_rank_parent_equality_at3. fs_h_ics_rank_parent_equality_at3 + S (ics_value3_rank_parent_equality) = S ((S (ics_index_rank_parent_equality)) * fc)) /\ exists fs_q_ics_rank_parent_equality_at3. fb = fs_q_ics_rank_parent_equality_at3 * S ((S (ics_index_rank_parent_equality)) * fc) + (ics_value3_rank_parent_equality))) -> ics_value0_rank_parent_equality + ics_value3_rank_parent_equality = ics_value2_rank_parent_equality + ics_value1_rank_parent_equality) -> (((exists mdr_gap_first_integer_rankrows_bound. mdr_gap_first_integer_rankrows_bound + (rank) = (r)) /\ ((exists mdr_gap_first_integer_rankcolumns_bound. mdr_gap_first_integer_rankcolumns_bound + (rank) = (w)) /\ ((exists mdr_rb_first_integer_rankwitness mdr_rc_first_integer_rankwitness mdr_cb_first_integer_rankwitness mdr_cc_first_integer_rankwitness. (((((forall fom_index_mrf_first_integer_rankwitnessminorrowsbound. (exists fom_gap_mrf_first_integer_rankwitnessminorrowsbound_index_bound. fom_gap_mrf_first_integer_rankwitnessminorrowsbound_index_bound + S (fom_index_mrf_first_integer_rankwitnessminorrowsbound) = rank) -> exists fom_value_mrf_first_integer_rankwitnessminorrowsbound. ((((exists fom_beta_height_mrf_first_integer_rankwitnessminorrowsbound_entry. fom_beta_height_mrf_first_integer_rankwitnessminorrowsbound_entry + S (fom_value_mrf_first_integer_rankwitnessminorrowsbound) = S ((S (fom_index_mrf_first_integer_rankwitnessminorrowsbound)) * mdr_rc_first_integer_rankwitness)) /\ exists fom_beta_quotient_mrf_first_integer_rankwitnessminorrowsbound_entry. mdr_rb_first_integer_rankwitness = fom_beta_quotient_mrf_first_integer_rankwitnessminorrowsbound_entry * S ((S (fom_index_mrf_first_integer_rankwitnessminorrowsbound)) * mdr_rc_first_integer_rankwitness) + (fom_value_mrf_first_integer_rankwitnessminorrowsbound))) /\ (exists fom_gap_mrf_first_integer_rankwitnessminorrowsbound_value_bound. fom_gap_mrf_first_integer_rankwitnessminorrowsbound_value_bound + S (fom_value_mrf_first_integer_rankwitnessminorrowsbound) = r))) /\ (forall mdr_i_first_integer_rankwitnessminorrowsdistinct mdr_j_first_integer_rankwitnessminorrowsdistinct mdr_a_first_integer_rankwitnessminorrowsdistinct. (exists mdr_gap_first_integer_rankwitnessminorrowsdistincti. mdr_gap_first_integer_rankwitnessminorrowsdistincti + S (mdr_i_first_integer_rankwitnessminorrowsdistinct) = (rank)) -> (exists mdr_gap_first_integer_rankwitnessminorrowsdistinctj. mdr_gap_first_integer_rankwitnessminorrowsdistinctj + S (mdr_j_first_integer_rankwitnessminorrowsdistinct) = (rank)) -> (((exists ff_h_mdr_first_integer_rankwitnessminorrowsdistinctfirst. ff_h_mdr_first_integer_rankwitnessminorrowsdistinctfirst + S (mdr_a_first_integer_rankwitnessminorrowsdistinct) = S ((S (mdr_i_first_integer_rankwitnessminorrowsdistinct)) * mdr_rc_first_integer_rankwitness)) /\ exists ff_q_mdr_first_integer_rankwitnessminorrowsdistinctfirst. mdr_rb_first_integer_rankwitness = ff_q_mdr_first_integer_rankwitnessminorrowsdistinctfirst * S ((S (mdr_i_first_integer_rankwitnessminorrowsdistinct)) * mdr_rc_first_integer_rankwitness) + (mdr_a_first_integer_rankwitnessminorrowsdistinct))) -> (((exists ff_h_mdr_first_integer_rankwitnessminorrowsdistinctsecond. ff_h_mdr_first_integer_rankwitnessminorrowsdistinctsecond + S (mdr_a_first_integer_rankwitnessminorrowsdistinct) = S ((S (mdr_j_first_integer_rankwitnessminorrowsdistinct)) * mdr_rc_first_integer_rankwitness)) /\ exists ff_q_mdr_first_integer_rankwitnessminorrowsdistinctsecond. mdr_rb_first_integer_rankwitness = ff_q_mdr_first_integer_rankwitnessminorrowsdistinctsecond * S ((S (mdr_j_first_integer_rankwitnessminorrowsdistinct)) * mdr_rc_first_integer_rankwitness) + (mdr_a_first_integer_rankwitnessminorrowsdistinct))) -> mdr_i_first_integer_rankwitnessminorrowsdistinct = mdr_j_first_integer_rankwitnessminorrowsdistinct))) /\ ((((forall fom_index_mrf_first_integer_rankwitnessminorcolumnsbound. (exists fom_gap_mrf_first_integer_rankwitnessminorcolumnsbound_index_bound. fom_gap_mrf_first_integer_rankwitnessminorcolumnsbound_index_bound + S (fom_index_mrf_first_integer_rankwitnessminorcolumnsbound) = rank) -> exists fom_value_mrf_first_integer_rankwitnessminorcolumnsbound. ((((exists fom_beta_height_mrf_first_integer_rankwitnessminorcolumnsbound_entry. fom_beta_height_mrf_first_integer_rankwitnessminorcolumnsbound_entry + S (fom_value_mrf_first_integer_rankwitnessminorcolumnsbound) = S ((S (fom_index_mrf_first_integer_rankwitnessminorcolumnsbound)) * mdr_cc_first_integer_rankwitness)) /\ exists fom_beta_quotient_mrf_first_integer_rankwitnessminorcolumnsbound_entry. mdr_cb_first_integer_rankwitness = fom_beta_quotient_mrf_first_integer_rankwitnessminorcolumnsbound_entry * S ((S (fom_index_mrf_first_integer_rankwitnessminorcolumnsbound)) * mdr_cc_first_integer_rankwitness) + (fom_value_mrf_first_integer_rankwitnessminorcolumnsbound))) /\ (exists fom_gap_mrf_first_integer_rankwitnessminorcolumnsbound_value_bound. fom_gap_mrf_first_integer_rankwitnessminorcolumnsbound_value_bound + S (fom_value_mrf_first_integer_rankwitnessminorcolumnsbound) = w))) /\ (forall mdr_i_first_integer_rankwitnessminorcolumnsdistinct mdr_j_first_integer_rankwitnessminorcolumnsdistinct mdr_a_first_integer_rankwitnessminorcolumnsdistinct. (exists mdr_gap_first_integer_rankwitnessminorcolumnsdistincti. mdr_gap_first_integer_rankwitnessminorcolumnsdistincti + S (mdr_i_first_integer_rankwitnessminorcolumnsdistinct) = (rank)) -> (exists mdr_gap_first_integer_rankwitnessminorcolumnsdistinctj. mdr_gap_first_integer_rankwitnessminorcolumnsdistinctj + S (mdr_j_first_integer_rankwitnessminorcolumnsdistinct) = (rank)) -> (((exists ff_h_mdr_first_integer_rankwitnessminorcolumnsdistinctfirst. ff_h_mdr_first_integer_rankwitnessminorcolumnsdistinctfirst + S (mdr_a_first_integer_rankwitnessminorcolumnsdistinct) = S ((S (mdr_i_first_integer_rankwitnessminorcolumnsdistinct)) * mdr_cc_first_integer_rankwitness)) /\ exists ff_q_mdr_first_integer_rankwitnessminorcolumnsdistinctfirst. mdr_cb_first_integer_rankwitness = ff_q_mdr_first_integer_rankwitnessminorcolumnsdistinctfirst * S ((S (mdr_i_first_integer_rankwitnessminorcolumnsdistinct)) * mdr_cc_first_integer_rankwitness) + (mdr_a_first_integer_rankwitnessminorcolumnsdistinct))) -> (((exists ff_h_mdr_first_integer_rankwitnessminorcolumnsdistinctsecond. ff_h_mdr_first_integer_rankwitnessminorcolumnsdistinctsecond + S (mdr_a_first_integer_rankwitnessminorcolumnsdistinct) = S ((S (mdr_j_first_integer_rankwitnessminorcolumnsdistinct)) * mdr_cc_first_integer_rankwitness)) /\ exists ff_q_mdr_first_integer_rankwitnessminorcolumnsdistinctsecond. mdr_cb_first_integer_rankwitness = ff_q_mdr_first_integer_rankwitnessminorcolumnsdistinctsecond * S ((S (mdr_j_first_integer_rankwitnessminorcolumnsdistinct)) * mdr_cc_first_integer_rankwitness) + (mdr_a_first_integer_rankwitnessminorcolumnsdistinct))) -> mdr_i_first_integer_rankwitnessminorcolumnsdistinct = mdr_j_first_integer_rankwitnessminorcolumnsdistinct))) /\ (exists mdr_p_first_integer_rankwitnessminornonzero mdr_n_first_integer_rankwitnessminornonzero. ((exists mdr_ub_first_integer_rankwitnessminornonzeroevaluation mdr_uc_first_integer_rankwitnessminornonzeroevaluation mdr_vb_first_integer_rankwitnessminornonzeroevaluation mdr_vc_first_integer_rankwitnessminornonzeroevaluation. ((((forall mdr_i_first_integer_rankwitnessminornonzeroevaluationmatrixpositive. (exists mdr_gap_first_integer_rankwitnessminornonzeroevaluationmatrixpositivebound. mdr_gap_first_integer_rankwitnessminornonzeroevaluationmatrixpositivebound + S (mdr_i_first_integer_rankwitnessminornonzeroevaluationmatrixpositive) = ((rank) * (rank))) -> exists mdr_a_first_integer_rankwitnessminornonzeroevaluationmatrixpositive. (((exists mdr_r_first_integer_rankwitnessminornonzeroevaluationmatrixpositivepoint mdr_s_first_integer_rankwitnessminornonzeroevaluationmatrixpositivepoint mdr_u_first_integer_rankwitnessminornonzeroevaluationmatrixpositivepoint mdr_v_first_integer_rankwitnessminornonzeroevaluationmatrixpositivepoint. ((mdr_i_first_integer_rankwitnessminornonzeroevaluationmatrixpositive = (rank) * mdr_r_first_integer_rankwitnessminornonzeroevaluationmatrixpositivepoint + mdr_s_first_integer_rankwitnessminornonzeroevaluationmatrixpositivepoint) /\ ((exists mdr_gap_first_integer_rankwitnessminornonzeroevaluationmatrixpositivepointcolumn. mdr_gap_first_integer_rankwitnessminornonzeroevaluationmatrixpositivepointcolumn + S (mdr_s_first_integer_rankwitnessminornonzeroevaluationmatrixpositivepoint) = (rank)) /\ ((((exists ff_h_mdr_first_integer_rankwitnessminornonzeroevaluationmatrixpositivepointrow_index. ff_h_mdr_first_integer_rankwitnessminornonzeroevaluationmatrixpositivepointrow_index + S (mdr_u_first_integer_rankwitnessminornonzeroevaluationmatrixpositivepoint) = S ((S (mdr_r_first_integer_rankwitnessminornonzeroevaluationmatrixpositivepoint)) * mdr_rc_first_integer_rankwitness)) /\ exists ff_q_mdr_first_integer_rankwitnessminornonzeroevaluationmatrixpositivepointrow_index. mdr_rb_first_integer_rankwitness = ff_q_mdr_first_integer_rankwitnessminornonzeroevaluationmatrixpositivepointrow_index * S ((S (mdr_r_first_integer_rankwitnessminornonzeroevaluationmatrixpositivepoint)) * mdr_rc_first_integer_rankwitness) + (mdr_u_first_integer_rankwitnessminornonzeroevaluationmatrixpositivepoint))) /\ ((((exists ff_h_mdr_first_integer_rankwitnessminornonzeroevaluationmatrixpositivepointcolumn_index. ff_h_mdr_first_integer_rankwitnessminornonzeroevaluationmatrixpositivepointcolumn_index + S (mdr_v_first_integer_rankwitnessminornonzeroevaluationmatrixpositivepoint) = S ((S (mdr_s_first_integer_rankwitnessminornonzeroevaluationmatrixpositivepoint)) * mdr_cc_first_integer_rankwitness)) /\ exists ff_q_mdr_first_integer_rankwitnessminornonzeroevaluationmatrixpositivepointcolumn_index. mdr_cb_first_integer_rankwitness = ff_q_mdr_first_integer_rankwitnessminornonzeroevaluationmatrixpositivepointcolumn_index * S ((S (mdr_s_first_integer_rankwitnessminornonzeroevaluationmatrixpositivepoint)) * mdr_cc_first_integer_rankwitness) + (mdr_v_first_integer_rankwitnessminornonzeroevaluationmatrixpositivepoint))) /\ (((exists ff_h_mdr_first_integer_rankwitnessminornonzeroevaluationmatrixpositivepointsource. ff_h_mdr_first_integer_rankwitnessminornonzeroevaluationmatrixpositivepointsource + S (mdr_a_first_integer_rankwitnessminornonzeroevaluationmatrixpositive) = S ((S ((mdr_u_first_integer_rankwitnessminornonzeroevaluationmatrixpositivepoint) * (w) + (mdr_v_first_integer_rankwitnessminornonzeroevaluationmatrixpositivepoint))) * ac)) /\ exists ff_q_mdr_first_integer_rankwitnessminornonzeroevaluationmatrixpositivepointsource. ab = ff_q_mdr_first_integer_rankwitnessminornonzeroevaluationmatrixpositivepointsource * S ((S ((mdr_u_first_integer_rankwitnessminornonzeroevaluationmatrixpositivepoint) * (w) + (mdr_v_first_integer_rankwitnessminornonzeroevaluationmatrixpositivepoint))) * ac) + (mdr_a_first_integer_rankwitnessminornonzeroevaluationmatrixpositive)))))))) /\ (((exists ff_h_mdr_first_integer_rankwitnessminornonzeroevaluationmatrixpositiveoutput. ff_h_mdr_first_integer_rankwitnessminornonzeroevaluationmatrixpositiveoutput + S (mdr_a_first_integer_rankwitnessminornonzeroevaluationmatrixpositive) = S ((S (mdr_i_first_integer_rankwitnessminornonzeroevaluationmatrixpositive)) * mdr_uc_first_integer_rankwitnessminornonzeroevaluation)) /\ exists ff_q_mdr_first_integer_rankwitnessminornonzeroevaluationmatrixpositiveoutput. mdr_ub_first_integer_rankwitnessminornonzeroevaluation = ff_q_mdr_first_integer_rankwitnessminornonzeroevaluationmatrixpositiveoutput * S ((S (mdr_i_first_integer_rankwitnessminornonzeroevaluationmatrixpositive)) * mdr_uc_first_integer_rankwitnessminornonzeroevaluation) + (mdr_a_first_integer_rankwitnessminornonzeroevaluationmatrixpositive)))))) /\ (forall mdr_i_first_integer_rankwitnessminornonzeroevaluationmatrixnegative. (exists mdr_gap_first_integer_rankwitnessminornonzeroevaluationmatrixnegativebound. mdr_gap_first_integer_rankwitnessminornonzeroevaluationmatrixnegativebound + S (mdr_i_first_integer_rankwitnessminornonzeroevaluationmatrixnegative) = ((rank) * (rank))) -> exists mdr_a_first_integer_rankwitnessminornonzeroevaluationmatrixnegative. (((exists mdr_r_first_integer_rankwitnessminornonzeroevaluationmatrixnegativepoint mdr_s_first_integer_rankwitnessminornonzeroevaluationmatrixnegativepoint mdr_u_first_integer_rankwitnessminornonzeroevaluationmatrixnegativepoint mdr_v_first_integer_rankwitnessminornonzeroevaluationmatrixnegativepoint. ((mdr_i_first_integer_rankwitnessminornonzeroevaluationmatrixnegative = (rank) * mdr_r_first_integer_rankwitnessminornonzeroevaluationmatrixnegativepoint + mdr_s_first_integer_rankwitnessminornonzeroevaluationmatrixnegativepoint) /\ ((exists mdr_gap_first_integer_rankwitnessminornonzeroevaluationmatrixnegativepointcolumn. mdr_gap_first_integer_rankwitnessminornonzeroevaluationmatrixnegativepointcolumn + S (mdr_s_first_integer_rankwitnessminornonzeroevaluationmatrixnegativepoint) = (rank)) /\ ((((exists ff_h_mdr_first_integer_rankwitnessminornonzeroevaluationmatrixnegativepointrow_index. ff_h_mdr_first_integer_rankwitnessminornonzeroevaluationmatrixnegativepointrow_index + S (mdr_u_first_integer_rankwitnessminornonzeroevaluationmatrixnegativepoint) = S ((S (mdr_r_first_integer_rankwitnessminornonzeroevaluationmatrixnegativepoint)) * mdr_rc_first_integer_rankwitness)) /\ exists ff_q_mdr_first_integer_rankwitnessminornonzeroevaluationmatrixnegativepointrow_index. mdr_rb_first_integer_rankwitness = ff_q_mdr_first_integer_rankwitnessminornonzeroevaluationmatrixnegativepointrow_index * S ((S (mdr_r_first_integer_rankwitnessminornonzeroevaluationmatrixnegativepoint)) * mdr_rc_first_integer_rankwitness) + (mdr_u_first_integer_rankwitnessminornonzeroevaluationmatrixnegativepoint))) /\ ((((exists ff_h_mdr_first_integer_rankwitnessminornonzeroevaluationmatrixnegativepointcolumn_index. ff_h_mdr_first_integer_rankwitnessminornonzeroevaluationmatrixnegativepointcolumn_index + S (mdr_v_first_integer_rankwitnessminornonzeroevaluationmatrixnegativepoint) = S ((S (mdr_s_first_integer_rankwitnessminornonzeroevaluationmatrixnegativepoint)) * mdr_cc_first_integer_rankwitness)) /\ exists ff_q_mdr_first_integer_rankwitnessminornonzeroevaluationmatrixnegativepointcolumn_index. mdr_cb_first_integer_rankwitness = ff_q_mdr_first_integer_rankwitnessminornonzeroevaluationmatrixnegativepointcolumn_index * S ((S (mdr_s_first_integer_rankwitnessminornonzeroevaluationmatrixnegativepoint)) * mdr_cc_first_integer_rankwitness) + (mdr_v_first_integer_rankwitnessminornonzeroevaluationmatrixnegativepoint))) /\ (((exists ff_h_mdr_first_integer_rankwitnessminornonzeroevaluationmatrixnegativepointsource. ff_h_mdr_first_integer_rankwitnessminornonzeroevaluationmatrixnegativepointsource + S (mdr_a_first_integer_rankwitnessminornonzeroevaluationmatrixnegative) = S ((S ((mdr_u_first_integer_rankwitnessminornonzeroevaluationmatrixnegativepoint) * (w) + (mdr_v_first_integer_rankwitnessminornonzeroevaluationmatrixnegativepoint))) * bc)) /\ exists ff_q_mdr_first_integer_rankwitnessminornonzeroevaluationmatrixnegativepointsource. bb = ff_q_mdr_first_integer_rankwitnessminornonzeroevaluationmatrixnegativepointsource * S ((S ((mdr_u_first_integer_rankwitnessminornonzeroevaluationmatrixnegativepoint) * (w) + (mdr_v_first_integer_rankwitnessminornonzeroevaluationmatrixnegativepoint))) * bc) + (mdr_a_first_integer_rankwitnessminornonzeroevaluationmatrixnegative)))))))) /\ (((exists ff_h_mdr_first_integer_rankwitnessminornonzeroevaluationmatrixnegativeoutput. ff_h_mdr_first_integer_rankwitnessminornonzeroevaluationmatrixnegativeoutput + S (mdr_a_first_integer_rankwitnessminornonzeroevaluationmatrixnegative) = S ((S (mdr_i_first_integer_rankwitnessminornonzeroevaluationmatrixnegative)) * mdr_vc_first_integer_rankwitnessminornonzeroevaluation)) /\ exists ff_q_mdr_first_integer_rankwitnessminornonzeroevaluationmatrixnegativeoutput. mdr_vb_first_integer_rankwitnessminornonzeroevaluation = ff_q_mdr_first_integer_rankwitnessminornonzeroevaluationmatrixnegativeoutput * S ((S (mdr_i_first_integer_rankwitnessminornonzeroevaluationmatrixnegative)) * mdr_vc_first_integer_rankwitnessminornonzeroevaluation) + (mdr_a_first_integer_rankwitnessminornonzeroevaluationmatrixnegative)))))))) /\ (exists mdr_b_first_integer_rankwitnessminornonzeroevaluationdeterminant mdr_c_first_integer_rankwitnessminornonzeroevaluationdeterminant mdr_l_first_integer_rankwitnessminornonzeroevaluationdeterminant mdr_i_first_integer_rankwitnessminornonzeroevaluationdeterminant. ((forall mdr_i_first_integer_rankwitnessminornonzeroevaluationdeterminanth. (exists mdr_gap_first_integer_rankwitnessminornonzeroevaluationdeterminanthi. mdr_gap_first_integer_rankwitnessminornonzeroevaluationdeterminanthi + S (mdr_i_first_integer_rankwitnessminornonzeroevaluationdeterminanth) = (mdr_l_first_integer_rankwitnessminornonzeroevaluationdeterminant)) -> exists mdr_d_first_integer_rankwitnessminornonzeroevaluationdeterminanth mdr_pb_first_integer_rankwitnessminornonzeroevaluationdeterminanth mdr_pc_first_integer_rankwitnessminornonzeroevaluationdeterminanth mdr_nb_first_integer_rankwitnessminornonzeroevaluationdeterminanth mdr_nc_first_integer_rankwitnessminornonzeroevaluationdeterminanth mdr_p_first_integer_rankwitnessminornonzeroevaluationdeterminanth mdr_n_first_integer_rankwitnessminornonzeroevaluationdeterminanth. ((exists mdr_z_first_integer_rankwitnessminornonzeroevaluationdeterminanthr. ((exists mdr_a_first_integer_rankwitnessminornonzeroevaluationdeterminanthrc mdr_b_first_integer_rankwitnessminornonzeroevaluationdeterminanthrc mdr_c_first_integer_rankwitnessminornonzeroevaluationdeterminanthrc mdr_e_first_integer_rankwitnessminornonzeroevaluationdeterminanthrc mdr_f_first_integer_rankwitnessminornonzeroevaluationdeterminanthrc. ((mdr_a_first_integer_rankwitnessminornonzeroevaluationdeterminanthrc = ((mdr_d_first_integer_rankwitnessminornonzeroevaluationdeterminanth) + (mdr_pb_first_integer_rankwitnessminornonzeroevaluationdeterminanth)) * S ((mdr_d_first_integer_rankwitnessminornonzeroevaluationdeterminanth) + (mdr_pb_first_integer_rankwitnessminornonzeroevaluationdeterminanth)) + ((mdr_pb_first_integer_rankwitnessminornonzeroevaluationdeterminanth) + (mdr_pb_first_integer_rankwitnessminornonzeroevaluationdeterminanth))) /\ ((mdr_b_first_integer_rankwitnessminornonzeroevaluationdeterminanthrc = ((mdr_pc_first_integer_rankwitnessminornonzeroevaluationdeterminanth) + (mdr_nb_first_integer_rankwitnessminornonzeroevaluationdeterminanth)) * S ((mdr_pc_first_integer_rankwitnessminornonzeroevaluationdeterminanth) + (mdr_nb_first_integer_rankwitnessminornonzeroevaluationdeterminanth)) + ((mdr_nb_first_integer_rankwitnessminornonzeroevaluationdeterminanth) + (mdr_nb_first_integer_rankwitnessminornonzeroevaluationdeterminanth))) /\ ((mdr_c_first_integer_rankwitnessminornonzeroevaluationdeterminanthrc = ((mdr_a_first_integer_rankwitnessminornonzeroevaluationdeterminanthrc) + (mdr_b_first_integer_rankwitnessminornonzeroevaluationdeterminanthrc)) * S ((mdr_a_first_integer_rankwitnessminornonzeroevaluationdeterminanthrc) + (mdr_b_first_integer_rankwitnessminornonzeroevaluationdeterminanthrc)) + ((mdr_b_first_integer_rankwitnessminornonzeroevaluationdeterminanthrc) + (mdr_b_first_integer_rankwitnessminornonzeroevaluationdeterminanthrc))) /\ ((mdr_e_first_integer_rankwitnessminornonzeroevaluationdeterminanthrc = ((mdr_p_first_integer_rankwitnessminornonzeroevaluationdeterminanth) + (mdr_n_first_integer_rankwitnessminornonzeroevaluationdeterminanth)) * S ((mdr_p_first_integer_rankwitnessminornonzeroevaluationdeterminanth) + (mdr_n_first_integer_rankwitnessminornonzeroevaluationdeterminanth)) + ((mdr_n_first_integer_rankwitnessminornonzeroevaluationdeterminanth) + (mdr_n_first_integer_rankwitnessminornonzeroevaluationdeterminanth))) /\ ((mdr_f_first_integer_rankwitnessminornonzeroevaluationdeterminanthrc = ((mdr_nc_first_integer_rankwitnessminornonzeroevaluationdeterminanth) + (mdr_e_first_integer_rankwitnessminornonzeroevaluationdeterminanthrc)) * S ((mdr_nc_first_integer_rankwitnessminornonzeroevaluationdeterminanth) + (mdr_e_first_integer_rankwitnessminornonzeroevaluationdeterminanthrc)) + ((mdr_e_first_integer_rankwitnessminornonzeroevaluationdeterminanthrc) + (mdr_e_first_integer_rankwitnessminornonzeroevaluationdeterminanthrc))) /\ ((mdr_z_first_integer_rankwitnessminornonzeroevaluationdeterminanthr) = ((mdr_c_first_integer_rankwitnessminornonzeroevaluationdeterminanthrc) + (mdr_f_first_integer_rankwitnessminornonzeroevaluationdeterminanthrc)) * S ((mdr_c_first_integer_rankwitnessminornonzeroevaluationdeterminanthrc) + (mdr_f_first_integer_rankwitnessminornonzeroevaluationdeterminanthrc)) + ((mdr_f_first_integer_rankwitnessminornonzeroevaluationdeterminanthrc) + (mdr_f_first_integer_rankwitnessminornonzeroevaluationdeterminanthrc))))))))) /\ (((exists ff_h_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthrb. ff_h_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthrb + S (mdr_z_first_integer_rankwitnessminornonzeroevaluationdeterminanthr) = S ((S (mdr_i_first_integer_rankwitnessminornonzeroevaluationdeterminanth)) * mdr_c_first_integer_rankwitnessminornonzeroevaluationdeterminant)) /\ exists ff_q_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthrb. mdr_b_first_integer_rankwitnessminornonzeroevaluationdeterminant = ff_q_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthrb * S ((S (mdr_i_first_integer_rankwitnessminornonzeroevaluationdeterminanth)) * mdr_c_first_integer_rankwitnessminornonzeroevaluationdeterminant) + (mdr_z_first_integer_rankwitnessminornonzeroevaluationdeterminanthr))))) /\ (((((mdr_d_first_integer_rankwitnessminornonzeroevaluationdeterminanth) = 0) /\ (((mdr_p_first_integer_rankwitnessminornonzeroevaluationdeterminanth) = 1) /\ ((mdr_n_first_integer_rankwitnessminornonzeroevaluationdeterminanth) = 0))) \/ exists mdr_q_first_integer_rankwitnessminornonzeroevaluationdeterminanths mdr_eb_first_integer_rankwitnessminornonzeroevaluationdeterminanths mdr_ec_first_integer_rankwitnessminornonzeroevaluationdeterminanths mdr_fb_first_integer_rankwitnessminornonzeroevaluationdeterminanths mdr_fc_first_integer_rankwitnessminornonzeroevaluationdeterminanths. (((mdr_d_first_integer_rankwitnessminornonzeroevaluationdeterminanth) = S (mdr_q_first_integer_rankwitnessminornonzeroevaluationdeterminanths)) /\ ((forall mdr_j_first_integer_rankwitnessminornonzeroevaluationdeterminanthsc. (exists mdr_gap_first_integer_rankwitnessminornonzeroevaluationdeterminanthscj. mdr_gap_first_integer_rankwitnessminornonzeroevaluationdeterminanthscj + S (mdr_j_first_integer_rankwitnessminornonzeroevaluationdeterminanthsc) = (S (mdr_q_first_integer_rankwitnessminornonzeroevaluationdeterminanths))) -> exists mdr_i_first_integer_rankwitnessminornonzeroevaluationdeterminanthsc mdr_up_first_integer_rankwitnessminornonzeroevaluationdeterminanthsc mdr_us_first_integer_rankwitnessminornonzeroevaluationdeterminanthsc mdr_un_first_integer_rankwitnessminornonzeroevaluationdeterminanthsc mdr_ut_first_integer_rankwitnessminornonzeroevaluationdeterminanthsc mdr_p_first_integer_rankwitnessminornonzeroevaluationdeterminanthsc mdr_n_first_integer_rankwitnessminornonzeroevaluationdeterminanthsc. ((exists mdr_gap_first_integer_rankwitnessminornonzeroevaluationdeterminanthsci. mdr_gap_first_integer_rankwitnessminornonzeroevaluationdeterminanthsci + S (mdr_i_first_integer_rankwitnessminornonzeroevaluationdeterminanthsc) = (mdr_i_first_integer_rankwitnessminornonzeroevaluationdeterminanth)) /\ ((exists mdr_z_first_integer_rankwitnessminornonzeroevaluationdeterminanthscr. ((exists mdr_a_first_integer_rankwitnessminornonzeroevaluationdeterminanthscrc mdr_b_first_integer_rankwitnessminornonzeroevaluationdeterminanthscrc mdr_c_first_integer_rankwitnessminornonzeroevaluationdeterminanthscrc mdr_e_first_integer_rankwitnessminornonzeroevaluationdeterminanthscrc mdr_f_first_integer_rankwitnessminornonzeroevaluationdeterminanthscrc. ((mdr_a_first_integer_rankwitnessminornonzeroevaluationdeterminanthscrc = ((mdr_q_first_integer_rankwitnessminornonzeroevaluationdeterminanths) + (mdr_up_first_integer_rankwitnessminornonzeroevaluationdeterminanthsc)) * S ((mdr_q_first_integer_rankwitnessminornonzeroevaluationdeterminanths) + (mdr_up_first_integer_rankwitnessminornonzeroevaluationdeterminanthsc)) + ((mdr_up_first_integer_rankwitnessminornonzeroevaluationdeterminanthsc) + (mdr_up_first_integer_rankwitnessminornonzeroevaluationdeterminanthsc))) /\ ((mdr_b_first_integer_rankwitnessminornonzeroevaluationdeterminanthscrc = ((mdr_us_first_integer_rankwitnessminornonzeroevaluationdeterminanthsc) + (mdr_un_first_integer_rankwitnessminornonzeroevaluationdeterminanthsc)) * S ((mdr_us_first_integer_rankwitnessminornonzeroevaluationdeterminanthsc) + (mdr_un_first_integer_rankwitnessminornonzeroevaluationdeterminanthsc)) + ((mdr_un_first_integer_rankwitnessminornonzeroevaluationdeterminanthsc) + (mdr_un_first_integer_rankwitnessminornonzeroevaluationdeterminanthsc))) /\ ((mdr_c_first_integer_rankwitnessminornonzeroevaluationdeterminanthscrc = ((mdr_a_first_integer_rankwitnessminornonzeroevaluationdeterminanthscrc) + (mdr_b_first_integer_rankwitnessminornonzeroevaluationdeterminanthscrc)) * S ((mdr_a_first_integer_rankwitnessminornonzeroevaluationdeterminanthscrc) + (mdr_b_first_integer_rankwitnessminornonzeroevaluationdeterminanthscrc)) + ((mdr_b_first_integer_rankwitnessminornonzeroevaluationdeterminanthscrc) + (mdr_b_first_integer_rankwitnessminornonzeroevaluationdeterminanthscrc))) /\ ((mdr_e_first_integer_rankwitnessminornonzeroevaluationdeterminanthscrc = ((mdr_p_first_integer_rankwitnessminornonzeroevaluationdeterminanthsc) + (mdr_n_first_integer_rankwitnessminornonzeroevaluationdeterminanthsc)) * S ((mdr_p_first_integer_rankwitnessminornonzeroevaluationdeterminanthsc) + (mdr_n_first_integer_rankwitnessminornonzeroevaluationdeterminanthsc)) + ((mdr_n_first_integer_rankwitnessminornonzeroevaluationdeterminanthsc) + (mdr_n_first_integer_rankwitnessminornonzeroevaluationdeterminanthsc))) /\ ((mdr_f_first_integer_rankwitnessminornonzeroevaluationdeterminanthscrc = ((mdr_ut_first_integer_rankwitnessminornonzeroevaluationdeterminanthsc) + (mdr_e_first_integer_rankwitnessminornonzeroevaluationdeterminanthscrc)) * S ((mdr_ut_first_integer_rankwitnessminornonzeroevaluationdeterminanthsc) + (mdr_e_first_integer_rankwitnessminornonzeroevaluationdeterminanthscrc)) + ((mdr_e_first_integer_rankwitnessminornonzeroevaluationdeterminanthscrc) + (mdr_e_first_integer_rankwitnessminornonzeroevaluationdeterminanthscrc))) /\ ((mdr_z_first_integer_rankwitnessminornonzeroevaluationdeterminanthscr) = ((mdr_c_first_integer_rankwitnessminornonzeroevaluationdeterminanthscrc) + (mdr_f_first_integer_rankwitnessminornonzeroevaluationdeterminanthscrc)) * S ((mdr_c_first_integer_rankwitnessminornonzeroevaluationdeterminanthscrc) + (mdr_f_first_integer_rankwitnessminornonzeroevaluationdeterminanthscrc)) + ((mdr_f_first_integer_rankwitnessminornonzeroevaluationdeterminanthscrc) + (mdr_f_first_integer_rankwitnessminornonzeroevaluationdeterminanthscrc))))))))) /\ (((exists ff_h_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthscrb. ff_h_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthscrb + S (mdr_z_first_integer_rankwitnessminornonzeroevaluationdeterminanthscr) = S ((S (mdr_i_first_integer_rankwitnessminornonzeroevaluationdeterminanthsc)) * mdr_c_first_integer_rankwitnessminornonzeroevaluationdeterminant)) /\ exists ff_q_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthscrb. mdr_b_first_integer_rankwitnessminornonzeroevaluationdeterminant = ff_q_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthscrb * S ((S (mdr_i_first_integer_rankwitnessminornonzeroevaluationdeterminanthsc)) * mdr_c_first_integer_rankwitnessminornonzeroevaluationdeterminant) + (mdr_z_first_integer_rankwitnessminornonzeroevaluationdeterminanthscr))))) /\ ((((forall ff_index_mdm_prefix_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthscm_positive. (exists ff_gap_mdm_lt_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthscm_positive_index_bound. ff_gap_mdm_lt_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthscm_positive_index_bound + S (ff_index_mdm_prefix_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthscm_positive) = ((mdr_q_first_integer_rankwitnessminornonzeroevaluationdeterminanths) * (mdr_q_first_integer_rankwitnessminornonzeroevaluationdeterminanths))) -> exists ff_row_mdm_prefix_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthscm_positive ff_column_mdm_prefix_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthscm_positive ff_value_mdm_prefix_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthscm_positive. (ff_index_mdm_prefix_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthscm_positive = (mdr_q_first_integer_rankwitnessminornonzeroevaluationdeterminanths) * ff_row_mdm_prefix_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthscm_positive + ff_column_mdm_prefix_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthscm_positive /\ ((exists ff_gap_mdm_lt_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthscm_positive_column_bound. ff_gap_mdm_lt_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthscm_positive_column_bound + S (ff_column_mdm_prefix_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthscm_positive) = (mdr_q_first_integer_rankwitnessminornonzeroevaluationdeterminanths)) /\ ((exists ff_row_mdm_cell_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthscm_positive_cell ff_column_mdm_cell_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthscm_positive_cell. (((((exists ff_gap_mdm_lt_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthscm_positive_cell_row_before. ff_gap_mdm_lt_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthscm_positive_cell_row_before + S (ff_row_mdm_prefix_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthscm_positive) = (0)) /\ ff_row_mdm_cell_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthscm_positive_cell = ff_row_mdm_prefix_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthscm_positive) \/ ((exists ff_gap_mdm_le_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthscm_positive_cell_row_after. ff_gap_mdm_le_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthscm_positive_cell_row_after + (0) = (ff_row_mdm_prefix_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthscm_positive)) /\ ff_row_mdm_cell_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthscm_positive_cell = S ff_row_mdm_prefix_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthscm_positive))) /\ (((((exists ff_gap_mdm_lt_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthscm_positive_cell_column_before. ff_gap_mdm_lt_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthscm_positive_cell_column_before + S (ff_column_mdm_prefix_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthscm_positive) = (mdr_j_first_integer_rankwitnessminornonzeroevaluationdeterminanthsc)) /\ ff_column_mdm_cell_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthscm_positive_cell = ff_column_mdm_prefix_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthscm_positive) \/ ((exists ff_gap_mdm_le_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthscm_positive_cell_column_after. ff_gap_mdm_le_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthscm_positive_cell_column_after + (mdr_j_first_integer_rankwitnessminornonzeroevaluationdeterminanthsc) = (ff_column_mdm_prefix_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthscm_positive)) /\ ff_column_mdm_cell_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthscm_positive_cell = S ff_column_mdm_prefix_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthscm_positive))) /\ (((exists ff_h_mdm_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthscm_positive_cell_source. ff_h_mdm_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthscm_positive_cell_source + S (ff_value_mdm_prefix_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthscm_positive) = S ((S ((ff_row_mdm_cell_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthscm_positive_cell) * (S (mdr_q_first_integer_rankwitnessminornonzeroevaluationdeterminanths)) + (ff_column_mdm_cell_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthscm_positive_cell))) * mdr_pc_first_integer_rankwitnessminornonzeroevaluationdeterminanth)) /\ exists ff_q_mdm_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthscm_positive_cell_source. mdr_pb_first_integer_rankwitnessminornonzeroevaluationdeterminanth = ff_q_mdm_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthscm_positive_cell_source * S ((S ((ff_row_mdm_cell_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthscm_positive_cell) * (S (mdr_q_first_integer_rankwitnessminornonzeroevaluationdeterminanths)) + (ff_column_mdm_cell_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthscm_positive_cell))) * mdr_pc_first_integer_rankwitnessminornonzeroevaluationdeterminanth) + (ff_value_mdm_prefix_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthscm_positive)))))) /\ (((exists ff_h_mdm_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthscm_positive_target. ff_h_mdm_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthscm_positive_target + S (ff_value_mdm_prefix_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthscm_positive) = S ((S (ff_index_mdm_prefix_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthscm_positive)) * mdr_us_first_integer_rankwitnessminornonzeroevaluationdeterminanthsc)) /\ exists ff_q_mdm_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthscm_positive_target. mdr_up_first_integer_rankwitnessminornonzeroevaluationdeterminanthsc = ff_q_mdm_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthscm_positive_target * S ((S (ff_index_mdm_prefix_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthscm_positive)) * mdr_us_first_integer_rankwitnessminornonzeroevaluationdeterminanthsc) + (ff_value_mdm_prefix_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthscm_positive))))))) /\ (forall ff_index_mdm_prefix_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthscm_negative. (exists ff_gap_mdm_lt_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthscm_negative_index_bound. ff_gap_mdm_lt_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthscm_negative_index_bound + S (ff_index_mdm_prefix_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthscm_negative) = ((mdr_q_first_integer_rankwitnessminornonzeroevaluationdeterminanths) * (mdr_q_first_integer_rankwitnessminornonzeroevaluationdeterminanths))) -> exists ff_row_mdm_prefix_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthscm_negative ff_column_mdm_prefix_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthscm_negative ff_value_mdm_prefix_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthscm_negative. (ff_index_mdm_prefix_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthscm_negative = (mdr_q_first_integer_rankwitnessminornonzeroevaluationdeterminanths) * ff_row_mdm_prefix_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthscm_negative + ff_column_mdm_prefix_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthscm_negative /\ ((exists ff_gap_mdm_lt_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthscm_negative_column_bound. ff_gap_mdm_lt_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthscm_negative_column_bound + S (ff_column_mdm_prefix_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthscm_negative) = (mdr_q_first_integer_rankwitnessminornonzeroevaluationdeterminanths)) /\ ((exists ff_row_mdm_cell_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthscm_negative_cell ff_column_mdm_cell_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthscm_negative_cell. (((((exists ff_gap_mdm_lt_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthscm_negative_cell_row_before. ff_gap_mdm_lt_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthscm_negative_cell_row_before + S (ff_row_mdm_prefix_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthscm_negative) = (0)) /\ ff_row_mdm_cell_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthscm_negative_cell = ff_row_mdm_prefix_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthscm_negative) \/ ((exists ff_gap_mdm_le_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthscm_negative_cell_row_after. ff_gap_mdm_le_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthscm_negative_cell_row_after + (0) = (ff_row_mdm_prefix_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthscm_negative)) /\ ff_row_mdm_cell_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthscm_negative_cell = S ff_row_mdm_prefix_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthscm_negative))) /\ (((((exists ff_gap_mdm_lt_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthscm_negative_cell_column_before. ff_gap_mdm_lt_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthscm_negative_cell_column_before + S (ff_column_mdm_prefix_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthscm_negative) = (mdr_j_first_integer_rankwitnessminornonzeroevaluationdeterminanthsc)) /\ ff_column_mdm_cell_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthscm_negative_cell = ff_column_mdm_prefix_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthscm_negative) \/ ((exists ff_gap_mdm_le_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthscm_negative_cell_column_after. ff_gap_mdm_le_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthscm_negative_cell_column_after + (mdr_j_first_integer_rankwitnessminornonzeroevaluationdeterminanthsc) = (ff_column_mdm_prefix_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthscm_negative)) /\ ff_column_mdm_cell_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthscm_negative_cell = S ff_column_mdm_prefix_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthscm_negative))) /\ (((exists ff_h_mdm_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthscm_negative_cell_source. ff_h_mdm_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthscm_negative_cell_source + S (ff_value_mdm_prefix_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthscm_negative) = S ((S ((ff_row_mdm_cell_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthscm_negative_cell) * (S (mdr_q_first_integer_rankwitnessminornonzeroevaluationdeterminanths)) + (ff_column_mdm_cell_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthscm_negative_cell))) * mdr_nc_first_integer_rankwitnessminornonzeroevaluationdeterminanth)) /\ exists ff_q_mdm_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthscm_negative_cell_source. mdr_nb_first_integer_rankwitnessminornonzeroevaluationdeterminanth = ff_q_mdm_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthscm_negative_cell_source * S ((S ((ff_row_mdm_cell_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthscm_negative_cell) * (S (mdr_q_first_integer_rankwitnessminornonzeroevaluationdeterminanths)) + (ff_column_mdm_cell_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthscm_negative_cell))) * mdr_nc_first_integer_rankwitnessminornonzeroevaluationdeterminanth) + (ff_value_mdm_prefix_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthscm_negative)))))) /\ (((exists ff_h_mdm_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthscm_negative_target. ff_h_mdm_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthscm_negative_target + S (ff_value_mdm_prefix_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthscm_negative) = S ((S (ff_index_mdm_prefix_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthscm_negative)) * mdr_ut_first_integer_rankwitnessminornonzeroevaluationdeterminanthsc)) /\ exists ff_q_mdm_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthscm_negative_target. mdr_un_first_integer_rankwitnessminornonzeroevaluationdeterminanthsc = ff_q_mdm_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthscm_negative_target * S ((S (ff_index_mdm_prefix_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthscm_negative)) * mdr_ut_first_integer_rankwitnessminornonzeroevaluationdeterminanthsc) + (ff_value_mdm_prefix_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthscm_negative))))))))) /\ ((((exists ff_h_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthscp. ff_h_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthscp + S (mdr_p_first_integer_rankwitnessminornonzeroevaluationdeterminanthsc) = S ((S (mdr_j_first_integer_rankwitnessminornonzeroevaluationdeterminanthsc)) * mdr_ec_first_integer_rankwitnessminornonzeroevaluationdeterminanths)) /\ exists ff_q_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthscp. mdr_eb_first_integer_rankwitnessminornonzeroevaluationdeterminanths = ff_q_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthscp * S ((S (mdr_j_first_integer_rankwitnessminornonzeroevaluationdeterminanthsc)) * mdr_ec_first_integer_rankwitnessminornonzeroevaluationdeterminanths) + (mdr_p_first_integer_rankwitnessminornonzeroevaluationdeterminanthsc))) /\ (((exists ff_h_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthscn. ff_h_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthscn + S (mdr_n_first_integer_rankwitnessminornonzeroevaluationdeterminanthsc) = S ((S (mdr_j_first_integer_rankwitnessminornonzeroevaluationdeterminanthsc)) * mdr_fc_first_integer_rankwitnessminornonzeroevaluationdeterminanths)) /\ exists ff_q_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthscn. mdr_fb_first_integer_rankwitnessminornonzeroevaluationdeterminanths = ff_q_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthscn * S ((S (mdr_j_first_integer_rankwitnessminornonzeroevaluationdeterminanthsc)) * mdr_fc_first_integer_rankwitnessminornonzeroevaluationdeterminanths) + (mdr_n_first_integer_rankwitnessminornonzeroevaluationdeterminanthsc)))))))) /\ (exists ff_ub_mce_fold_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf ff_uc_mce_fold_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf ff_vb_mce_fold_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf ff_vc_mce_fold_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf. ((forall ff_index_mce_alternating_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix. (exists ff_gap_mce_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix_index. ff_gap_mce_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix_index + S (ff_index_mce_alternating_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix) = (S (mdr_q_first_integer_rankwitnessminornonzeroevaluationdeterminanths))) -> exists ff_ap_mce_alternating_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix ff_an_mce_alternating_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix ff_bp_mce_alternating_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix ff_bn_mce_alternating_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix ff_p_mce_alternating_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix ff_n_mce_alternating_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix. ((((exists ff_h_mce_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix_ap. ff_h_mce_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix_ap + S (ff_ap_mce_alternating_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix) = S ((S (ff_index_mce_alternating_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix)) * mdr_pc_first_integer_rankwitnessminornonzeroevaluationdeterminanth)) /\ exists ff_q_mce_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix_ap. mdr_pb_first_integer_rankwitnessminornonzeroevaluationdeterminanth = ff_q_mce_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix_ap * S ((S (ff_index_mce_alternating_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix)) * mdr_pc_first_integer_rankwitnessminornonzeroevaluationdeterminanth) + (ff_ap_mce_alternating_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix))) /\ ((((exists ff_h_mce_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix_an. ff_h_mce_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix_an + S (ff_an_mce_alternating_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix) = S ((S (ff_index_mce_alternating_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix)) * mdr_nc_first_integer_rankwitnessminornonzeroevaluationdeterminanth)) /\ exists ff_q_mce_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix_an. mdr_nb_first_integer_rankwitnessminornonzeroevaluationdeterminanth = ff_q_mce_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix_an * S ((S (ff_index_mce_alternating_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix)) * mdr_nc_first_integer_rankwitnessminornonzeroevaluationdeterminanth) + (ff_an_mce_alternating_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix))) /\ ((((exists ff_h_mce_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix_bp. ff_h_mce_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix_bp + S (ff_bp_mce_alternating_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix) = S ((S (ff_index_mce_alternating_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix)) * mdr_ec_first_integer_rankwitnessminornonzeroevaluationdeterminanths)) /\ exists ff_q_mce_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix_bp. mdr_eb_first_integer_rankwitnessminornonzeroevaluationdeterminanths = ff_q_mce_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix_bp * S ((S (ff_index_mce_alternating_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix)) * mdr_ec_first_integer_rankwitnessminornonzeroevaluationdeterminanths) + (ff_bp_mce_alternating_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix))) /\ ((((exists ff_h_mce_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix_bn. ff_h_mce_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix_bn + S (ff_bn_mce_alternating_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix) = S ((S (ff_index_mce_alternating_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix)) * mdr_fc_first_integer_rankwitnessminornonzeroevaluationdeterminanths)) /\ exists ff_q_mce_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix_bn. mdr_fb_first_integer_rankwitnessminornonzeroevaluationdeterminanths = ff_q_mce_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix_bn * S ((S (ff_index_mce_alternating_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix)) * mdr_fc_first_integer_rankwitnessminornonzeroevaluationdeterminanths) + (ff_bn_mce_alternating_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix))) /\ ((((exists ff_h_mce_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix_positive. ff_h_mce_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix_positive + S (ff_p_mce_alternating_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix) = S ((S (ff_index_mce_alternating_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix)) * ff_uc_mce_fold_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf)) /\ exists ff_q_mce_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix_positive. ff_ub_mce_fold_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf = ff_q_mce_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix_positive * S ((S (ff_index_mce_alternating_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix)) * ff_uc_mce_fold_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf) + (ff_p_mce_alternating_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix))) /\ ((((exists ff_h_mce_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix_negative. ff_h_mce_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix_negative + S (ff_n_mce_alternating_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix) = S ((S (ff_index_mce_alternating_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix)) * ff_vc_mce_fold_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf)) /\ exists ff_q_mce_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix_negative. ff_vb_mce_fold_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf = ff_q_mce_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix_negative * S ((S (ff_index_mce_alternating_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix)) * ff_vc_mce_fold_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf) + (ff_n_mce_alternating_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix))) /\ (((exists ff_even_mce_term_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix_term. ff_index_mce_alternating_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix = 2 * ff_even_mce_term_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix_term) /\ (ff_p_mce_alternating_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix = (ff_ap_mce_alternating_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix) * (ff_bp_mce_alternating_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix) + (ff_an_mce_alternating_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix) * (ff_bn_mce_alternating_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix) /\ ff_n_mce_alternating_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix = (ff_ap_mce_alternating_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix) * (ff_bn_mce_alternating_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix) + (ff_an_mce_alternating_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix) * (ff_bp_mce_alternating_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix))) \/ ((exists ff_odd_mce_term_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix_term. ff_index_mce_alternating_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix = 2 * ff_odd_mce_term_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix_term + 1) /\ (ff_p_mce_alternating_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix = (ff_ap_mce_alternating_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix) * (ff_bn_mce_alternating_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix) + (ff_an_mce_alternating_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix) * (ff_bp_mce_alternating_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix) /\ ff_n_mce_alternating_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix = (ff_ap_mce_alternating_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix) * (ff_bp_mce_alternating_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix) + (ff_an_mce_alternating_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix) * (ff_bn_mce_alternating_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix))))))))))) /\ ((exists ff_u_mce_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_positive ff_v_mce_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_positive. ((((exists ff_h_mce_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_positive_start. ff_h_mce_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_positive_start + S (0) = S ((S (0)) * ff_v_mce_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_positive)) /\ exists ff_q_mce_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_positive_start. ff_u_mce_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_positive = ff_q_mce_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_positive_start * S ((S (0)) * ff_v_mce_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_positive) + (0))) /\ ((((exists ff_h_mce_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_positive_terminal. ff_h_mce_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_positive_terminal + S (mdr_p_first_integer_rankwitnessminornonzeroevaluationdeterminanth) = S ((S ((S (mdr_q_first_integer_rankwitnessminornonzeroevaluationdeterminanths)))) * ff_v_mce_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_positive)) /\ exists ff_q_mce_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_positive_terminal. ff_u_mce_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_positive = ff_q_mce_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_positive_terminal * S ((S ((S (mdr_q_first_integer_rankwitnessminornonzeroevaluationdeterminanths)))) * ff_v_mce_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_positive) + (mdr_p_first_integer_rankwitnessminornonzeroevaluationdeterminanth))) /\ forall ff_i_mce_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_positive. (exists ff_lt_mce_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_positive_bound. ff_lt_mce_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_positive_bound + S ff_i_mce_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_positive = (S (mdr_q_first_integer_rankwitnessminornonzeroevaluationdeterminanths))) -> exists ff_a_mce_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_positive ff_r_mce_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_positive ff_s_mce_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_positive. ((((exists ff_h_mce_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_positive_summand. ff_h_mce_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_positive_summand + S (ff_a_mce_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_positive) = S ((S (ff_i_mce_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_positive)) * ff_uc_mce_fold_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf)) /\ exists ff_q_mce_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_positive_summand. ff_ub_mce_fold_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf = ff_q_mce_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_positive_summand * S ((S (ff_i_mce_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_positive)) * ff_uc_mce_fold_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf) + (ff_a_mce_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_positive))) /\ ((((exists ff_h_mce_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_positive_partial. ff_h_mce_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_positive_partial + S (ff_r_mce_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_positive) = S ((S (ff_i_mce_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_positive)) * ff_v_mce_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_positive)) /\ exists ff_q_mce_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_positive_partial. ff_u_mce_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_positive = ff_q_mce_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_positive_partial * S ((S (ff_i_mce_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_positive)) * ff_v_mce_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_positive) + (ff_r_mce_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_positive))) /\ ((((exists ff_h_mce_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_positive_successor. ff_h_mce_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_positive_successor + S (ff_s_mce_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_positive) = S ((S (S ff_i_mce_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_positive)) * ff_v_mce_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_positive)) /\ exists ff_q_mce_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_positive_successor. ff_u_mce_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_positive = ff_q_mce_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_positive_successor * S ((S (S ff_i_mce_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_positive)) * ff_v_mce_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_positive) + (ff_s_mce_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_positive))) /\ ff_s_mce_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_positive = ff_r_mce_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_positive + ff_a_mce_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_positive)))))) /\ (exists ff_u_mce_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_negative ff_v_mce_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_negative. ((((exists ff_h_mce_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_negative_start. ff_h_mce_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_negative_start + S (0) = S ((S (0)) * ff_v_mce_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_negative)) /\ exists ff_q_mce_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_negative_start. ff_u_mce_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_negative = ff_q_mce_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_negative_start * S ((S (0)) * ff_v_mce_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_negative) + (0))) /\ ((((exists ff_h_mce_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_negative_terminal. ff_h_mce_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_negative_terminal + S (mdr_n_first_integer_rankwitnessminornonzeroevaluationdeterminanth) = S ((S ((S (mdr_q_first_integer_rankwitnessminornonzeroevaluationdeterminanths)))) * ff_v_mce_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_negative)) /\ exists ff_q_mce_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_negative_terminal. ff_u_mce_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_negative = ff_q_mce_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_negative_terminal * S ((S ((S (mdr_q_first_integer_rankwitnessminornonzeroevaluationdeterminanths)))) * ff_v_mce_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_negative) + (mdr_n_first_integer_rankwitnessminornonzeroevaluationdeterminanth))) /\ forall ff_i_mce_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_negative. (exists ff_lt_mce_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_negative_bound. ff_lt_mce_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_negative_bound + S ff_i_mce_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_negative = (S (mdr_q_first_integer_rankwitnessminornonzeroevaluationdeterminanths))) -> exists ff_a_mce_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_negative ff_r_mce_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_negative ff_s_mce_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_negative. ((((exists ff_h_mce_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_negative_summand. ff_h_mce_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_negative_summand + S (ff_a_mce_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_negative) = S ((S (ff_i_mce_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_negative)) * ff_vc_mce_fold_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf)) /\ exists ff_q_mce_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_negative_summand. ff_vb_mce_fold_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf = ff_q_mce_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_negative_summand * S ((S (ff_i_mce_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_negative)) * ff_vc_mce_fold_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf) + (ff_a_mce_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_negative))) /\ ((((exists ff_h_mce_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_negative_partial. ff_h_mce_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_negative_partial + S (ff_r_mce_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_negative) = S ((S (ff_i_mce_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_negative)) * ff_v_mce_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_negative)) /\ exists ff_q_mce_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_negative_partial. ff_u_mce_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_negative = ff_q_mce_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_negative_partial * S ((S (ff_i_mce_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_negative)) * ff_v_mce_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_negative) + (ff_r_mce_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_negative))) /\ ((((exists ff_h_mce_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_negative_successor. ff_h_mce_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_negative_successor + S (ff_s_mce_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_negative) = S ((S (S ff_i_mce_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_negative)) * ff_v_mce_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_negative)) /\ exists ff_q_mce_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_negative_successor. ff_u_mce_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_negative = ff_q_mce_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_negative_successor * S ((S (S ff_i_mce_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_negative)) * ff_v_mce_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_negative) + (ff_s_mce_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_negative))) /\ ff_s_mce_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_negative = ff_r_mce_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_negative + ff_a_mce_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_negative))))))))))))))) /\ ((exists mdr_gap_first_integer_rankwitnessminornonzeroevaluationdeterminanti. mdr_gap_first_integer_rankwitnessminornonzeroevaluationdeterminanti + S (mdr_i_first_integer_rankwitnessminornonzeroevaluationdeterminant) = (mdr_l_first_integer_rankwitnessminornonzeroevaluationdeterminant)) /\ (exists mdr_z_first_integer_rankwitnessminornonzeroevaluationdeterminantr. ((exists mdr_a_first_integer_rankwitnessminornonzeroevaluationdeterminantrc mdr_b_first_integer_rankwitnessminornonzeroevaluationdeterminantrc mdr_c_first_integer_rankwitnessminornonzeroevaluationdeterminantrc mdr_e_first_integer_rankwitnessminornonzeroevaluationdeterminantrc mdr_f_first_integer_rankwitnessminornonzeroevaluationdeterminantrc. ((mdr_a_first_integer_rankwitnessminornonzeroevaluationdeterminantrc = ((rank) + (mdr_ub_first_integer_rankwitnessminornonzeroevaluation)) * S ((rank) + (mdr_ub_first_integer_rankwitnessminornonzeroevaluation)) + ((mdr_ub_first_integer_rankwitnessminornonzeroevaluation) + (mdr_ub_first_integer_rankwitnessminornonzeroevaluation))) /\ ((mdr_b_first_integer_rankwitnessminornonzeroevaluationdeterminantrc = ((mdr_uc_first_integer_rankwitnessminornonzeroevaluation) + (mdr_vb_first_integer_rankwitnessminornonzeroevaluation)) * S ((mdr_uc_first_integer_rankwitnessminornonzeroevaluation) + (mdr_vb_first_integer_rankwitnessminornonzeroevaluation)) + ((mdr_vb_first_integer_rankwitnessminornonzeroevaluation) + (mdr_vb_first_integer_rankwitnessminornonzeroevaluation))) /\ ((mdr_c_first_integer_rankwitnessminornonzeroevaluationdeterminantrc = ((mdr_a_first_integer_rankwitnessminornonzeroevaluationdeterminantrc) + (mdr_b_first_integer_rankwitnessminornonzeroevaluationdeterminantrc)) * S ((mdr_a_first_integer_rankwitnessminornonzeroevaluationdeterminantrc) + (mdr_b_first_integer_rankwitnessminornonzeroevaluationdeterminantrc)) + ((mdr_b_first_integer_rankwitnessminornonzeroevaluationdeterminantrc) + (mdr_b_first_integer_rankwitnessminornonzeroevaluationdeterminantrc))) /\ ((mdr_e_first_integer_rankwitnessminornonzeroevaluationdeterminantrc = ((mdr_p_first_integer_rankwitnessminornonzero) + (mdr_n_first_integer_rankwitnessminornonzero)) * S ((mdr_p_first_integer_rankwitnessminornonzero) + (mdr_n_first_integer_rankwitnessminornonzero)) + ((mdr_n_first_integer_rankwitnessminornonzero) + (mdr_n_first_integer_rankwitnessminornonzero))) /\ ((mdr_f_first_integer_rankwitnessminornonzeroevaluationdeterminantrc = ((mdr_vc_first_integer_rankwitnessminornonzeroevaluation) + (mdr_e_first_integer_rankwitnessminornonzeroevaluationdeterminantrc)) * S ((mdr_vc_first_integer_rankwitnessminornonzeroevaluation) + (mdr_e_first_integer_rankwitnessminornonzeroevaluationdeterminantrc)) + ((mdr_e_first_integer_rankwitnessminornonzeroevaluationdeterminantrc) + (mdr_e_first_integer_rankwitnessminornonzeroevaluationdeterminantrc))) /\ ((mdr_z_first_integer_rankwitnessminornonzeroevaluationdeterminantr) = ((mdr_c_first_integer_rankwitnessminornonzeroevaluationdeterminantrc) + (mdr_f_first_integer_rankwitnessminornonzeroevaluationdeterminantrc)) * S ((mdr_c_first_integer_rankwitnessminornonzeroevaluationdeterminantrc) + (mdr_f_first_integer_rankwitnessminornonzeroevaluationdeterminantrc)) + ((mdr_f_first_integer_rankwitnessminornonzeroevaluationdeterminantrc) + (mdr_f_first_integer_rankwitnessminornonzeroevaluationdeterminantrc))))))))) /\ (((exists ff_h_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminantrb. ff_h_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminantrb + S (mdr_z_first_integer_rankwitnessminornonzeroevaluationdeterminantr) = S ((S (mdr_i_first_integer_rankwitnessminornonzeroevaluationdeterminant)) * mdr_c_first_integer_rankwitnessminornonzeroevaluationdeterminant)) /\ exists ff_q_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminantrb. mdr_b_first_integer_rankwitnessminornonzeroevaluationdeterminant = ff_q_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminantrb * S ((S (mdr_i_first_integer_rankwitnessminornonzeroevaluationdeterminant)) * mdr_c_first_integer_rankwitnessminornonzeroevaluationdeterminant) + (mdr_z_first_integer_rankwitnessminornonzeroevaluationdeterminantr)))))))))) /\ (~(mdr_p_first_integer_rankwitnessminornonzero = mdr_n_first_integer_rankwitnessminornonzero)))))))) /\ (forall mdr_q_first_integer_rank. (exists mdr_gap_first_integer_rankhigher. mdr_gap_first_integer_rankhigher + S (rank) = (mdr_q_first_integer_rank)) -> (forall mdr_rb_first_integer_rankzero mdr_rc_first_integer_rankzero mdr_cb_first_integer_rankzero mdr_cc_first_integer_rankzero mdr_p_first_integer_rankzero mdr_n_first_integer_rankzero. (((forall fom_index_mrf_first_integer_rankzerorowsbound. (exists fom_gap_mrf_first_integer_rankzerorowsbound_index_bound. fom_gap_mrf_first_integer_rankzerorowsbound_index_bound + S (fom_index_mrf_first_integer_rankzerorowsbound) = mdr_q_first_integer_rank) -> exists fom_value_mrf_first_integer_rankzerorowsbound. ((((exists fom_beta_height_mrf_first_integer_rankzerorowsbound_entry. fom_beta_height_mrf_first_integer_rankzerorowsbound_entry + S (fom_value_mrf_first_integer_rankzerorowsbound) = S ((S (fom_index_mrf_first_integer_rankzerorowsbound)) * mdr_rc_first_integer_rankzero)) /\ exists fom_beta_quotient_mrf_first_integer_rankzerorowsbound_entry. mdr_rb_first_integer_rankzero = fom_beta_quotient_mrf_first_integer_rankzerorowsbound_entry * S ((S (fom_index_mrf_first_integer_rankzerorowsbound)) * mdr_rc_first_integer_rankzero) + (fom_value_mrf_first_integer_rankzerorowsbound))) /\ (exists fom_gap_mrf_first_integer_rankzerorowsbound_value_bound. fom_gap_mrf_first_integer_rankzerorowsbound_value_bound + S (fom_value_mrf_first_integer_rankzerorowsbound) = r))) /\ (forall mdr_i_first_integer_rankzerorowsdistinct mdr_j_first_integer_rankzerorowsdistinct mdr_a_first_integer_rankzerorowsdistinct. (exists mdr_gap_first_integer_rankzerorowsdistincti. mdr_gap_first_integer_rankzerorowsdistincti + S (mdr_i_first_integer_rankzerorowsdistinct) = (mdr_q_first_integer_rank)) -> (exists mdr_gap_first_integer_rankzerorowsdistinctj. mdr_gap_first_integer_rankzerorowsdistinctj + S (mdr_j_first_integer_rankzerorowsdistinct) = (mdr_q_first_integer_rank)) -> (((exists ff_h_mdr_first_integer_rankzerorowsdistinctfirst. ff_h_mdr_first_integer_rankzerorowsdistinctfirst + S (mdr_a_first_integer_rankzerorowsdistinct) = S ((S (mdr_i_first_integer_rankzerorowsdistinct)) * mdr_rc_first_integer_rankzero)) /\ exists ff_q_mdr_first_integer_rankzerorowsdistinctfirst. mdr_rb_first_integer_rankzero = ff_q_mdr_first_integer_rankzerorowsdistinctfirst * S ((S (mdr_i_first_integer_rankzerorowsdistinct)) * mdr_rc_first_integer_rankzero) + (mdr_a_first_integer_rankzerorowsdistinct))) -> (((exists ff_h_mdr_first_integer_rankzerorowsdistinctsecond. ff_h_mdr_first_integer_rankzerorowsdistinctsecond + S (mdr_a_first_integer_rankzerorowsdistinct) = S ((S (mdr_j_first_integer_rankzerorowsdistinct)) * mdr_rc_first_integer_rankzero)) /\ exists ff_q_mdr_first_integer_rankzerorowsdistinctsecond. mdr_rb_first_integer_rankzero = ff_q_mdr_first_integer_rankzerorowsdistinctsecond * S ((S (mdr_j_first_integer_rankzerorowsdistinct)) * mdr_rc_first_integer_rankzero) + (mdr_a_first_integer_rankzerorowsdistinct))) -> mdr_i_first_integer_rankzerorowsdistinct = mdr_j_first_integer_rankzerorowsdistinct))) -> (((forall fom_index_mrf_first_integer_rankzerocolumnsbound. (exists fom_gap_mrf_first_integer_rankzerocolumnsbound_index_bound. fom_gap_mrf_first_integer_rankzerocolumnsbound_index_bound + S (fom_index_mrf_first_integer_rankzerocolumnsbound) = mdr_q_first_integer_rank) -> exists fom_value_mrf_first_integer_rankzerocolumnsbound. ((((exists fom_beta_height_mrf_first_integer_rankzerocolumnsbound_entry. fom_beta_height_mrf_first_integer_rankzerocolumnsbound_entry + S (fom_value_mrf_first_integer_rankzerocolumnsbound) = S ((S (fom_index_mrf_first_integer_rankzerocolumnsbound)) * mdr_cc_first_integer_rankzero)) /\ exists fom_beta_quotient_mrf_first_integer_rankzerocolumnsbound_entry. mdr_cb_first_integer_rankzero = fom_beta_quotient_mrf_first_integer_rankzerocolumnsbound_entry * S ((S (fom_index_mrf_first_integer_rankzerocolumnsbound)) * mdr_cc_first_integer_rankzero) + (fom_value_mrf_first_integer_rankzerocolumnsbound))) /\ (exists fom_gap_mrf_first_integer_rankzerocolumnsbound_value_bound. fom_gap_mrf_first_integer_rankzerocolumnsbound_value_bound + S (fom_value_mrf_first_integer_rankzerocolumnsbound) = w))) /\ (forall mdr_i_first_integer_rankzerocolumnsdistinct mdr_j_first_integer_rankzerocolumnsdistinct mdr_a_first_integer_rankzerocolumnsdistinct. (exists mdr_gap_first_integer_rankzerocolumnsdistincti. mdr_gap_first_integer_rankzerocolumnsdistincti + S (mdr_i_first_integer_rankzerocolumnsdistinct) = (mdr_q_first_integer_rank)) -> (exists mdr_gap_first_integer_rankzerocolumnsdistinctj. mdr_gap_first_integer_rankzerocolumnsdistinctj + S (mdr_j_first_integer_rankzerocolumnsdistinct) = (mdr_q_first_integer_rank)) -> (((exists ff_h_mdr_first_integer_rankzerocolumnsdistinctfirst. ff_h_mdr_first_integer_rankzerocolumnsdistinctfirst + S (mdr_a_first_integer_rankzerocolumnsdistinct) = S ((S (mdr_i_first_integer_rankzerocolumnsdistinct)) * mdr_cc_first_integer_rankzero)) /\ exists ff_q_mdr_first_integer_rankzerocolumnsdistinctfirst. mdr_cb_first_integer_rankzero = ff_q_mdr_first_integer_rankzerocolumnsdistinctfirst * S ((S (mdr_i_first_integer_rankzerocolumnsdistinct)) * mdr_cc_first_integer_rankzero) + (mdr_a_first_integer_rankzerocolumnsdistinct))) -> (((exists ff_h_mdr_first_integer_rankzerocolumnsdistinctsecond. ff_h_mdr_first_integer_rankzerocolumnsdistinctsecond + S (mdr_a_first_integer_rankzerocolumnsdistinct) = S ((S (mdr_j_first_integer_rankzerocolumnsdistinct)) * mdr_cc_first_integer_rankzero)) /\ exists ff_q_mdr_first_integer_rankzerocolumnsdistinctsecond. mdr_cb_first_integer_rankzero = ff_q_mdr_first_integer_rankzerocolumnsdistinctsecond * S ((S (mdr_j_first_integer_rankzerocolumnsdistinct)) * mdr_cc_first_integer_rankzero) + (mdr_a_first_integer_rankzerocolumnsdistinct))) -> mdr_i_first_integer_rankzerocolumnsdistinct = mdr_j_first_integer_rankzerocolumnsdistinct))) -> (exists mdr_ub_first_integer_rankzeroevaluation mdr_uc_first_integer_rankzeroevaluation mdr_vb_first_integer_rankzeroevaluation mdr_vc_first_integer_rankzeroevaluation. ((((forall mdr_i_first_integer_rankzeroevaluationmatrixpositive. (exists mdr_gap_first_integer_rankzeroevaluationmatrixpositivebound. mdr_gap_first_integer_rankzeroevaluationmatrixpositivebound + S (mdr_i_first_integer_rankzeroevaluationmatrixpositive) = ((mdr_q_first_integer_rank) * (mdr_q_first_integer_rank))) -> exists mdr_a_first_integer_rankzeroevaluationmatrixpositive. (((exists mdr_r_first_integer_rankzeroevaluationmatrixpositivepoint mdr_s_first_integer_rankzeroevaluationmatrixpositivepoint mdr_u_first_integer_rankzeroevaluationmatrixpositivepoint mdr_v_first_integer_rankzeroevaluationmatrixpositivepoint. ((mdr_i_first_integer_rankzeroevaluationmatrixpositive = (mdr_q_first_integer_rank) * mdr_r_first_integer_rankzeroevaluationmatrixpositivepoint + mdr_s_first_integer_rankzeroevaluationmatrixpositivepoint) /\ ((exists mdr_gap_first_integer_rankzeroevaluationmatrixpositivepointcolumn. mdr_gap_first_integer_rankzeroevaluationmatrixpositivepointcolumn + S (mdr_s_first_integer_rankzeroevaluationmatrixpositivepoint) = (mdr_q_first_integer_rank)) /\ ((((exists ff_h_mdr_first_integer_rankzeroevaluationmatrixpositivepointrow_index. ff_h_mdr_first_integer_rankzeroevaluationmatrixpositivepointrow_index + S (mdr_u_first_integer_rankzeroevaluationmatrixpositivepoint) = S ((S (mdr_r_first_integer_rankzeroevaluationmatrixpositivepoint)) * mdr_rc_first_integer_rankzero)) /\ exists ff_q_mdr_first_integer_rankzeroevaluationmatrixpositivepointrow_index. mdr_rb_first_integer_rankzero = ff_q_mdr_first_integer_rankzeroevaluationmatrixpositivepointrow_index * S ((S (mdr_r_first_integer_rankzeroevaluationmatrixpositivepoint)) * mdr_rc_first_integer_rankzero) + (mdr_u_first_integer_rankzeroevaluationmatrixpositivepoint))) /\ ((((exists ff_h_mdr_first_integer_rankzeroevaluationmatrixpositivepointcolumn_index. ff_h_mdr_first_integer_rankzeroevaluationmatrixpositivepointcolumn_index + S (mdr_v_first_integer_rankzeroevaluationmatrixpositivepoint) = S ((S (mdr_s_first_integer_rankzeroevaluationmatrixpositivepoint)) * mdr_cc_first_integer_rankzero)) /\ exists ff_q_mdr_first_integer_rankzeroevaluationmatrixpositivepointcolumn_index. mdr_cb_first_integer_rankzero = ff_q_mdr_first_integer_rankzeroevaluationmatrixpositivepointcolumn_index * S ((S (mdr_s_first_integer_rankzeroevaluationmatrixpositivepoint)) * mdr_cc_first_integer_rankzero) + (mdr_v_first_integer_rankzeroevaluationmatrixpositivepoint))) /\ (((exists ff_h_mdr_first_integer_rankzeroevaluationmatrixpositivepointsource. ff_h_mdr_first_integer_rankzeroevaluationmatrixpositivepointsource + S (mdr_a_first_integer_rankzeroevaluationmatrixpositive) = S ((S ((mdr_u_first_integer_rankzeroevaluationmatrixpositivepoint) * (w) + (mdr_v_first_integer_rankzeroevaluationmatrixpositivepoint))) * ac)) /\ exists ff_q_mdr_first_integer_rankzeroevaluationmatrixpositivepointsource. ab = ff_q_mdr_first_integer_rankzeroevaluationmatrixpositivepointsource * S ((S ((mdr_u_first_integer_rankzeroevaluationmatrixpositivepoint) * (w) + (mdr_v_first_integer_rankzeroevaluationmatrixpositivepoint))) * ac) + (mdr_a_first_integer_rankzeroevaluationmatrixpositive)))))))) /\ (((exists ff_h_mdr_first_integer_rankzeroevaluationmatrixpositiveoutput. ff_h_mdr_first_integer_rankzeroevaluationmatrixpositiveoutput + S (mdr_a_first_integer_rankzeroevaluationmatrixpositive) = S ((S (mdr_i_first_integer_rankzeroevaluationmatrixpositive)) * mdr_uc_first_integer_rankzeroevaluation)) /\ exists ff_q_mdr_first_integer_rankzeroevaluationmatrixpositiveoutput. mdr_ub_first_integer_rankzeroevaluation = ff_q_mdr_first_integer_rankzeroevaluationmatrixpositiveoutput * S ((S (mdr_i_first_integer_rankzeroevaluationmatrixpositive)) * mdr_uc_first_integer_rankzeroevaluation) + (mdr_a_first_integer_rankzeroevaluationmatrixpositive)))))) /\ (forall mdr_i_first_integer_rankzeroevaluationmatrixnegative. (exists mdr_gap_first_integer_rankzeroevaluationmatrixnegativebound. mdr_gap_first_integer_rankzeroevaluationmatrixnegativebound + S (mdr_i_first_integer_rankzeroevaluationmatrixnegative) = ((mdr_q_first_integer_rank) * (mdr_q_first_integer_rank))) -> exists mdr_a_first_integer_rankzeroevaluationmatrixnegative. (((exists mdr_r_first_integer_rankzeroevaluationmatrixnegativepoint mdr_s_first_integer_rankzeroevaluationmatrixnegativepoint mdr_u_first_integer_rankzeroevaluationmatrixnegativepoint mdr_v_first_integer_rankzeroevaluationmatrixnegativepoint. ((mdr_i_first_integer_rankzeroevaluationmatrixnegative = (mdr_q_first_integer_rank) * mdr_r_first_integer_rankzeroevaluationmatrixnegativepoint + mdr_s_first_integer_rankzeroevaluationmatrixnegativepoint) /\ ((exists mdr_gap_first_integer_rankzeroevaluationmatrixnegativepointcolumn. mdr_gap_first_integer_rankzeroevaluationmatrixnegativepointcolumn + S (mdr_s_first_integer_rankzeroevaluationmatrixnegativepoint) = (mdr_q_first_integer_rank)) /\ ((((exists ff_h_mdr_first_integer_rankzeroevaluationmatrixnegativepointrow_index. ff_h_mdr_first_integer_rankzeroevaluationmatrixnegativepointrow_index + S (mdr_u_first_integer_rankzeroevaluationmatrixnegativepoint) = S ((S (mdr_r_first_integer_rankzeroevaluationmatrixnegativepoint)) * mdr_rc_first_integer_rankzero)) /\ exists ff_q_mdr_first_integer_rankzeroevaluationmatrixnegativepointrow_index. mdr_rb_first_integer_rankzero = ff_q_mdr_first_integer_rankzeroevaluationmatrixnegativepointrow_index * S ((S (mdr_r_first_integer_rankzeroevaluationmatrixnegativepoint)) * mdr_rc_first_integer_rankzero) + (mdr_u_first_integer_rankzeroevaluationmatrixnegativepoint))) /\ ((((exists ff_h_mdr_first_integer_rankzeroevaluationmatrixnegativepointcolumn_index. ff_h_mdr_first_integer_rankzeroevaluationmatrixnegativepointcolumn_index + S (mdr_v_first_integer_rankzeroevaluationmatrixnegativepoint) = S ((S (mdr_s_first_integer_rankzeroevaluationmatrixnegativepoint)) * mdr_cc_first_integer_rankzero)) /\ exists ff_q_mdr_first_integer_rankzeroevaluationmatrixnegativepointcolumn_index. mdr_cb_first_integer_rankzero = ff_q_mdr_first_integer_rankzeroevaluationmatrixnegativepointcolumn_index * S ((S (mdr_s_first_integer_rankzeroevaluationmatrixnegativepoint)) * mdr_cc_first_integer_rankzero) + (mdr_v_first_integer_rankzeroevaluationmatrixnegativepoint))) /\ (((exists ff_h_mdr_first_integer_rankzeroevaluationmatrixnegativepointsource. ff_h_mdr_first_integer_rankzeroevaluationmatrixnegativepointsource + S (mdr_a_first_integer_rankzeroevaluationmatrixnegative) = S ((S ((mdr_u_first_integer_rankzeroevaluationmatrixnegativepoint) * (w) + (mdr_v_first_integer_rankzeroevaluationmatrixnegativepoint))) * bc)) /\ exists ff_q_mdr_first_integer_rankzeroevaluationmatrixnegativepointsource. bb = ff_q_mdr_first_integer_rankzeroevaluationmatrixnegativepointsource * S ((S ((mdr_u_first_integer_rankzeroevaluationmatrixnegativepoint) * (w) + (mdr_v_first_integer_rankzeroevaluationmatrixnegativepoint))) * bc) + (mdr_a_first_integer_rankzeroevaluationmatrixnegative)))))))) /\ (((exists ff_h_mdr_first_integer_rankzeroevaluationmatrixnegativeoutput. ff_h_mdr_first_integer_rankzeroevaluationmatrixnegativeoutput + S (mdr_a_first_integer_rankzeroevaluationmatrixnegative) = S ((S (mdr_i_first_integer_rankzeroevaluationmatrixnegative)) * mdr_vc_first_integer_rankzeroevaluation)) /\ exists ff_q_mdr_first_integer_rankzeroevaluationmatrixnegativeoutput. mdr_vb_first_integer_rankzeroevaluation = ff_q_mdr_first_integer_rankzeroevaluationmatrixnegativeoutput * S ((S (mdr_i_first_integer_rankzeroevaluationmatrixnegative)) * mdr_vc_first_integer_rankzeroevaluation) + (mdr_a_first_integer_rankzeroevaluationmatrixnegative)))))))) /\ (exists mdr_b_first_integer_rankzeroevaluationdeterminant mdr_c_first_integer_rankzeroevaluationdeterminant mdr_l_first_integer_rankzeroevaluationdeterminant mdr_i_first_integer_rankzeroevaluationdeterminant. ((forall mdr_i_first_integer_rankzeroevaluationdeterminanth. (exists mdr_gap_first_integer_rankzeroevaluationdeterminanthi. mdr_gap_first_integer_rankzeroevaluationdeterminanthi + S (mdr_i_first_integer_rankzeroevaluationdeterminanth) = (mdr_l_first_integer_rankzeroevaluationdeterminant)) -> exists mdr_d_first_integer_rankzeroevaluationdeterminanth mdr_pb_first_integer_rankzeroevaluationdeterminanth mdr_pc_first_integer_rankzeroevaluationdeterminanth mdr_nb_first_integer_rankzeroevaluationdeterminanth mdr_nc_first_integer_rankzeroevaluationdeterminanth mdr_p_first_integer_rankzeroevaluationdeterminanth mdr_n_first_integer_rankzeroevaluationdeterminanth. ((exists mdr_z_first_integer_rankzeroevaluationdeterminanthr. ((exists mdr_a_first_integer_rankzeroevaluationdeterminanthrc mdr_b_first_integer_rankzeroevaluationdeterminanthrc mdr_c_first_integer_rankzeroevaluationdeterminanthrc mdr_e_first_integer_rankzeroevaluationdeterminanthrc mdr_f_first_integer_rankzeroevaluationdeterminanthrc. ((mdr_a_first_integer_rankzeroevaluationdeterminanthrc = ((mdr_d_first_integer_rankzeroevaluationdeterminanth) + (mdr_pb_first_integer_rankzeroevaluationdeterminanth)) * S ((mdr_d_first_integer_rankzeroevaluationdeterminanth) + (mdr_pb_first_integer_rankzeroevaluationdeterminanth)) + ((mdr_pb_first_integer_rankzeroevaluationdeterminanth) + (mdr_pb_first_integer_rankzeroevaluationdeterminanth))) /\ ((mdr_b_first_integer_rankzeroevaluationdeterminanthrc = ((mdr_pc_first_integer_rankzeroevaluationdeterminanth) + (mdr_nb_first_integer_rankzeroevaluationdeterminanth)) * S ((mdr_pc_first_integer_rankzeroevaluationdeterminanth) + (mdr_nb_first_integer_rankzeroevaluationdeterminanth)) + ((mdr_nb_first_integer_rankzeroevaluationdeterminanth) + (mdr_nb_first_integer_rankzeroevaluationdeterminanth))) /\ ((mdr_c_first_integer_rankzeroevaluationdeterminanthrc = ((mdr_a_first_integer_rankzeroevaluationdeterminanthrc) + (mdr_b_first_integer_rankzeroevaluationdeterminanthrc)) * S ((mdr_a_first_integer_rankzeroevaluationdeterminanthrc) + (mdr_b_first_integer_rankzeroevaluationdeterminanthrc)) + ((mdr_b_first_integer_rankzeroevaluationdeterminanthrc) + (mdr_b_first_integer_rankzeroevaluationdeterminanthrc))) /\ ((mdr_e_first_integer_rankzeroevaluationdeterminanthrc = ((mdr_p_first_integer_rankzeroevaluationdeterminanth) + (mdr_n_first_integer_rankzeroevaluationdeterminanth)) * S ((mdr_p_first_integer_rankzeroevaluationdeterminanth) + (mdr_n_first_integer_rankzeroevaluationdeterminanth)) + ((mdr_n_first_integer_rankzeroevaluationdeterminanth) + (mdr_n_first_integer_rankzeroevaluationdeterminanth))) /\ ((mdr_f_first_integer_rankzeroevaluationdeterminanthrc = ((mdr_nc_first_integer_rankzeroevaluationdeterminanth) + (mdr_e_first_integer_rankzeroevaluationdeterminanthrc)) * S ((mdr_nc_first_integer_rankzeroevaluationdeterminanth) + (mdr_e_first_integer_rankzeroevaluationdeterminanthrc)) + ((mdr_e_first_integer_rankzeroevaluationdeterminanthrc) + (mdr_e_first_integer_rankzeroevaluationdeterminanthrc))) /\ ((mdr_z_first_integer_rankzeroevaluationdeterminanthr) = ((mdr_c_first_integer_rankzeroevaluationdeterminanthrc) + (mdr_f_first_integer_rankzeroevaluationdeterminanthrc)) * S ((mdr_c_first_integer_rankzeroevaluationdeterminanthrc) + (mdr_f_first_integer_rankzeroevaluationdeterminanthrc)) + ((mdr_f_first_integer_rankzeroevaluationdeterminanthrc) + (mdr_f_first_integer_rankzeroevaluationdeterminanthrc))))))))) /\ (((exists ff_h_mdr_first_integer_rankzeroevaluationdeterminanthrb. ff_h_mdr_first_integer_rankzeroevaluationdeterminanthrb + S (mdr_z_first_integer_rankzeroevaluationdeterminanthr) = S ((S (mdr_i_first_integer_rankzeroevaluationdeterminanth)) * mdr_c_first_integer_rankzeroevaluationdeterminant)) /\ exists ff_q_mdr_first_integer_rankzeroevaluationdeterminanthrb. mdr_b_first_integer_rankzeroevaluationdeterminant = ff_q_mdr_first_integer_rankzeroevaluationdeterminanthrb * S ((S (mdr_i_first_integer_rankzeroevaluationdeterminanth)) * mdr_c_first_integer_rankzeroevaluationdeterminant) + (mdr_z_first_integer_rankzeroevaluationdeterminanthr))))) /\ (((((mdr_d_first_integer_rankzeroevaluationdeterminanth) = 0) /\ (((mdr_p_first_integer_rankzeroevaluationdeterminanth) = 1) /\ ((mdr_n_first_integer_rankzeroevaluationdeterminanth) = 0))) \/ exists mdr_q_first_integer_rankzeroevaluationdeterminanths mdr_eb_first_integer_rankzeroevaluationdeterminanths mdr_ec_first_integer_rankzeroevaluationdeterminanths mdr_fb_first_integer_rankzeroevaluationdeterminanths mdr_fc_first_integer_rankzeroevaluationdeterminanths. (((mdr_d_first_integer_rankzeroevaluationdeterminanth) = S (mdr_q_first_integer_rankzeroevaluationdeterminanths)) /\ ((forall mdr_j_first_integer_rankzeroevaluationdeterminanthsc. (exists mdr_gap_first_integer_rankzeroevaluationdeterminanthscj. mdr_gap_first_integer_rankzeroevaluationdeterminanthscj + S (mdr_j_first_integer_rankzeroevaluationdeterminanthsc) = (S (mdr_q_first_integer_rankzeroevaluationdeterminanths))) -> exists mdr_i_first_integer_rankzeroevaluationdeterminanthsc mdr_up_first_integer_rankzeroevaluationdeterminanthsc mdr_us_first_integer_rankzeroevaluationdeterminanthsc mdr_un_first_integer_rankzeroevaluationdeterminanthsc mdr_ut_first_integer_rankzeroevaluationdeterminanthsc mdr_p_first_integer_rankzeroevaluationdeterminanthsc mdr_n_first_integer_rankzeroevaluationdeterminanthsc. ((exists mdr_gap_first_integer_rankzeroevaluationdeterminanthsci. mdr_gap_first_integer_rankzeroevaluationdeterminanthsci + S (mdr_i_first_integer_rankzeroevaluationdeterminanthsc) = (mdr_i_first_integer_rankzeroevaluationdeterminanth)) /\ ((exists mdr_z_first_integer_rankzeroevaluationdeterminanthscr. ((exists mdr_a_first_integer_rankzeroevaluationdeterminanthscrc mdr_b_first_integer_rankzeroevaluationdeterminanthscrc mdr_c_first_integer_rankzeroevaluationdeterminanthscrc mdr_e_first_integer_rankzeroevaluationdeterminanthscrc mdr_f_first_integer_rankzeroevaluationdeterminanthscrc. ((mdr_a_first_integer_rankzeroevaluationdeterminanthscrc = ((mdr_q_first_integer_rankzeroevaluationdeterminanths) + (mdr_up_first_integer_rankzeroevaluationdeterminanthsc)) * S ((mdr_q_first_integer_rankzeroevaluationdeterminanths) + (mdr_up_first_integer_rankzeroevaluationdeterminanthsc)) + ((mdr_up_first_integer_rankzeroevaluationdeterminanthsc) + (mdr_up_first_integer_rankzeroevaluationdeterminanthsc))) /\ ((mdr_b_first_integer_rankzeroevaluationdeterminanthscrc = ((mdr_us_first_integer_rankzeroevaluationdeterminanthsc) + (mdr_un_first_integer_rankzeroevaluationdeterminanthsc)) * S ((mdr_us_first_integer_rankzeroevaluationdeterminanthsc) + (mdr_un_first_integer_rankzeroevaluationdeterminanthsc)) + ((mdr_un_first_integer_rankzeroevaluationdeterminanthsc) + (mdr_un_first_integer_rankzeroevaluationdeterminanthsc))) /\ ((mdr_c_first_integer_rankzeroevaluationdeterminanthscrc = ((mdr_a_first_integer_rankzeroevaluationdeterminanthscrc) + (mdr_b_first_integer_rankzeroevaluationdeterminanthscrc)) * S ((mdr_a_first_integer_rankzeroevaluationdeterminanthscrc) + (mdr_b_first_integer_rankzeroevaluationdeterminanthscrc)) + ((mdr_b_first_integer_rankzeroevaluationdeterminanthscrc) + (mdr_b_first_integer_rankzeroevaluationdeterminanthscrc))) /\ ((mdr_e_first_integer_rankzeroevaluationdeterminanthscrc = ((mdr_p_first_integer_rankzeroevaluationdeterminanthsc) + (mdr_n_first_integer_rankzeroevaluationdeterminanthsc)) * S ((mdr_p_first_integer_rankzeroevaluationdeterminanthsc) + (mdr_n_first_integer_rankzeroevaluationdeterminanthsc)) + ((mdr_n_first_integer_rankzeroevaluationdeterminanthsc) + (mdr_n_first_integer_rankzeroevaluationdeterminanthsc))) /\ ((mdr_f_first_integer_rankzeroevaluationdeterminanthscrc = ((mdr_ut_first_integer_rankzeroevaluationdeterminanthsc) + (mdr_e_first_integer_rankzeroevaluationdeterminanthscrc)) * S ((mdr_ut_first_integer_rankzeroevaluationdeterminanthsc) + (mdr_e_first_integer_rankzeroevaluationdeterminanthscrc)) + ((mdr_e_first_integer_rankzeroevaluationdeterminanthscrc) + (mdr_e_first_integer_rankzeroevaluationdeterminanthscrc))) /\ ((mdr_z_first_integer_rankzeroevaluationdeterminanthscr) = ((mdr_c_first_integer_rankzeroevaluationdeterminanthscrc) + (mdr_f_first_integer_rankzeroevaluationdeterminanthscrc)) * S ((mdr_c_first_integer_rankzeroevaluationdeterminanthscrc) + (mdr_f_first_integer_rankzeroevaluationdeterminanthscrc)) + ((mdr_f_first_integer_rankzeroevaluationdeterminanthscrc) + (mdr_f_first_integer_rankzeroevaluationdeterminanthscrc))))))))) /\ (((exists ff_h_mdr_first_integer_rankzeroevaluationdeterminanthscrb. ff_h_mdr_first_integer_rankzeroevaluationdeterminanthscrb + S (mdr_z_first_integer_rankzeroevaluationdeterminanthscr) = S ((S (mdr_i_first_integer_rankzeroevaluationdeterminanthsc)) * mdr_c_first_integer_rankzeroevaluationdeterminant)) /\ exists ff_q_mdr_first_integer_rankzeroevaluationdeterminanthscrb. mdr_b_first_integer_rankzeroevaluationdeterminant = ff_q_mdr_first_integer_rankzeroevaluationdeterminanthscrb * S ((S (mdr_i_first_integer_rankzeroevaluationdeterminanthsc)) * mdr_c_first_integer_rankzeroevaluationdeterminant) + (mdr_z_first_integer_rankzeroevaluationdeterminanthscr))))) /\ ((((forall ff_index_mdm_prefix_mdr_first_integer_rankzeroevaluationdeterminanthscm_positive. (exists ff_gap_mdm_lt_mdr_first_integer_rankzeroevaluationdeterminanthscm_positive_index_bound. ff_gap_mdm_lt_mdr_first_integer_rankzeroevaluationdeterminanthscm_positive_index_bound + S (ff_index_mdm_prefix_mdr_first_integer_rankzeroevaluationdeterminanthscm_positive) = ((mdr_q_first_integer_rankzeroevaluationdeterminanths) * (mdr_q_first_integer_rankzeroevaluationdeterminanths))) -> exists ff_row_mdm_prefix_mdr_first_integer_rankzeroevaluationdeterminanthscm_positive ff_column_mdm_prefix_mdr_first_integer_rankzeroevaluationdeterminanthscm_positive ff_value_mdm_prefix_mdr_first_integer_rankzeroevaluationdeterminanthscm_positive. (ff_index_mdm_prefix_mdr_first_integer_rankzeroevaluationdeterminanthscm_positive = (mdr_q_first_integer_rankzeroevaluationdeterminanths) * ff_row_mdm_prefix_mdr_first_integer_rankzeroevaluationdeterminanthscm_positive + ff_column_mdm_prefix_mdr_first_integer_rankzeroevaluationdeterminanthscm_positive /\ ((exists ff_gap_mdm_lt_mdr_first_integer_rankzeroevaluationdeterminanthscm_positive_column_bound. ff_gap_mdm_lt_mdr_first_integer_rankzeroevaluationdeterminanthscm_positive_column_bound + S (ff_column_mdm_prefix_mdr_first_integer_rankzeroevaluationdeterminanthscm_positive) = (mdr_q_first_integer_rankzeroevaluationdeterminanths)) /\ ((exists ff_row_mdm_cell_mdr_first_integer_rankzeroevaluationdeterminanthscm_positive_cell ff_column_mdm_cell_mdr_first_integer_rankzeroevaluationdeterminanthscm_positive_cell. (((((exists ff_gap_mdm_lt_mdr_first_integer_rankzeroevaluationdeterminanthscm_positive_cell_row_before. ff_gap_mdm_lt_mdr_first_integer_rankzeroevaluationdeterminanthscm_positive_cell_row_before + S (ff_row_mdm_prefix_mdr_first_integer_rankzeroevaluationdeterminanthscm_positive) = (0)) /\ ff_row_mdm_cell_mdr_first_integer_rankzeroevaluationdeterminanthscm_positive_cell = ff_row_mdm_prefix_mdr_first_integer_rankzeroevaluationdeterminanthscm_positive) \/ ((exists ff_gap_mdm_le_mdr_first_integer_rankzeroevaluationdeterminanthscm_positive_cell_row_after. ff_gap_mdm_le_mdr_first_integer_rankzeroevaluationdeterminanthscm_positive_cell_row_after + (0) = (ff_row_mdm_prefix_mdr_first_integer_rankzeroevaluationdeterminanthscm_positive)) /\ ff_row_mdm_cell_mdr_first_integer_rankzeroevaluationdeterminanthscm_positive_cell = S ff_row_mdm_prefix_mdr_first_integer_rankzeroevaluationdeterminanthscm_positive))) /\ (((((exists ff_gap_mdm_lt_mdr_first_integer_rankzeroevaluationdeterminanthscm_positive_cell_column_before. ff_gap_mdm_lt_mdr_first_integer_rankzeroevaluationdeterminanthscm_positive_cell_column_before + S (ff_column_mdm_prefix_mdr_first_integer_rankzeroevaluationdeterminanthscm_positive) = (mdr_j_first_integer_rankzeroevaluationdeterminanthsc)) /\ ff_column_mdm_cell_mdr_first_integer_rankzeroevaluationdeterminanthscm_positive_cell = ff_column_mdm_prefix_mdr_first_integer_rankzeroevaluationdeterminanthscm_positive) \/ ((exists ff_gap_mdm_le_mdr_first_integer_rankzeroevaluationdeterminanthscm_positive_cell_column_after. ff_gap_mdm_le_mdr_first_integer_rankzeroevaluationdeterminanthscm_positive_cell_column_after + (mdr_j_first_integer_rankzeroevaluationdeterminanthsc) = (ff_column_mdm_prefix_mdr_first_integer_rankzeroevaluationdeterminanthscm_positive)) /\ ff_column_mdm_cell_mdr_first_integer_rankzeroevaluationdeterminanthscm_positive_cell = S ff_column_mdm_prefix_mdr_first_integer_rankzeroevaluationdeterminanthscm_positive))) /\ (((exists ff_h_mdm_mdr_first_integer_rankzeroevaluationdeterminanthscm_positive_cell_source. ff_h_mdm_mdr_first_integer_rankzeroevaluationdeterminanthscm_positive_cell_source + S (ff_value_mdm_prefix_mdr_first_integer_rankzeroevaluationdeterminanthscm_positive) = S ((S ((ff_row_mdm_cell_mdr_first_integer_rankzeroevaluationdeterminanthscm_positive_cell) * (S (mdr_q_first_integer_rankzeroevaluationdeterminanths)) + (ff_column_mdm_cell_mdr_first_integer_rankzeroevaluationdeterminanthscm_positive_cell))) * mdr_pc_first_integer_rankzeroevaluationdeterminanth)) /\ exists ff_q_mdm_mdr_first_integer_rankzeroevaluationdeterminanthscm_positive_cell_source. mdr_pb_first_integer_rankzeroevaluationdeterminanth = ff_q_mdm_mdr_first_integer_rankzeroevaluationdeterminanthscm_positive_cell_source * S ((S ((ff_row_mdm_cell_mdr_first_integer_rankzeroevaluationdeterminanthscm_positive_cell) * (S (mdr_q_first_integer_rankzeroevaluationdeterminanths)) + (ff_column_mdm_cell_mdr_first_integer_rankzeroevaluationdeterminanthscm_positive_cell))) * mdr_pc_first_integer_rankzeroevaluationdeterminanth) + (ff_value_mdm_prefix_mdr_first_integer_rankzeroevaluationdeterminanthscm_positive)))))) /\ (((exists ff_h_mdm_mdr_first_integer_rankzeroevaluationdeterminanthscm_positive_target. ff_h_mdm_mdr_first_integer_rankzeroevaluationdeterminanthscm_positive_target + S (ff_value_mdm_prefix_mdr_first_integer_rankzeroevaluationdeterminanthscm_positive) = S ((S (ff_index_mdm_prefix_mdr_first_integer_rankzeroevaluationdeterminanthscm_positive)) * mdr_us_first_integer_rankzeroevaluationdeterminanthsc)) /\ exists ff_q_mdm_mdr_first_integer_rankzeroevaluationdeterminanthscm_positive_target. mdr_up_first_integer_rankzeroevaluationdeterminanthsc = ff_q_mdm_mdr_first_integer_rankzeroevaluationdeterminanthscm_positive_target * S ((S (ff_index_mdm_prefix_mdr_first_integer_rankzeroevaluationdeterminanthscm_positive)) * mdr_us_first_integer_rankzeroevaluationdeterminanthsc) + (ff_value_mdm_prefix_mdr_first_integer_rankzeroevaluationdeterminanthscm_positive))))))) /\ (forall ff_index_mdm_prefix_mdr_first_integer_rankzeroevaluationdeterminanthscm_negative. (exists ff_gap_mdm_lt_mdr_first_integer_rankzeroevaluationdeterminanthscm_negative_index_bound. ff_gap_mdm_lt_mdr_first_integer_rankzeroevaluationdeterminanthscm_negative_index_bound + S (ff_index_mdm_prefix_mdr_first_integer_rankzeroevaluationdeterminanthscm_negative) = ((mdr_q_first_integer_rankzeroevaluationdeterminanths) * (mdr_q_first_integer_rankzeroevaluationdeterminanths))) -> exists ff_row_mdm_prefix_mdr_first_integer_rankzeroevaluationdeterminanthscm_negative ff_column_mdm_prefix_mdr_first_integer_rankzeroevaluationdeterminanthscm_negative ff_value_mdm_prefix_mdr_first_integer_rankzeroevaluationdeterminanthscm_negative. (ff_index_mdm_prefix_mdr_first_integer_rankzeroevaluationdeterminanthscm_negative = (mdr_q_first_integer_rankzeroevaluationdeterminanths) * ff_row_mdm_prefix_mdr_first_integer_rankzeroevaluationdeterminanthscm_negative + ff_column_mdm_prefix_mdr_first_integer_rankzeroevaluationdeterminanthscm_negative /\ ((exists ff_gap_mdm_lt_mdr_first_integer_rankzeroevaluationdeterminanthscm_negative_column_bound. ff_gap_mdm_lt_mdr_first_integer_rankzeroevaluationdeterminanthscm_negative_column_bound + S (ff_column_mdm_prefix_mdr_first_integer_rankzeroevaluationdeterminanthscm_negative) = (mdr_q_first_integer_rankzeroevaluationdeterminanths)) /\ ((exists ff_row_mdm_cell_mdr_first_integer_rankzeroevaluationdeterminanthscm_negative_cell ff_column_mdm_cell_mdr_first_integer_rankzeroevaluationdeterminanthscm_negative_cell. (((((exists ff_gap_mdm_lt_mdr_first_integer_rankzeroevaluationdeterminanthscm_negative_cell_row_before. ff_gap_mdm_lt_mdr_first_integer_rankzeroevaluationdeterminanthscm_negative_cell_row_before + S (ff_row_mdm_prefix_mdr_first_integer_rankzeroevaluationdeterminanthscm_negative) = (0)) /\ ff_row_mdm_cell_mdr_first_integer_rankzeroevaluationdeterminanthscm_negative_cell = ff_row_mdm_prefix_mdr_first_integer_rankzeroevaluationdeterminanthscm_negative) \/ ((exists ff_gap_mdm_le_mdr_first_integer_rankzeroevaluationdeterminanthscm_negative_cell_row_after. ff_gap_mdm_le_mdr_first_integer_rankzeroevaluationdeterminanthscm_negative_cell_row_after + (0) = (ff_row_mdm_prefix_mdr_first_integer_rankzeroevaluationdeterminanthscm_negative)) /\ ff_row_mdm_cell_mdr_first_integer_rankzeroevaluationdeterminanthscm_negative_cell = S ff_row_mdm_prefix_mdr_first_integer_rankzeroevaluationdeterminanthscm_negative))) /\ (((((exists ff_gap_mdm_lt_mdr_first_integer_rankzeroevaluationdeterminanthscm_negative_cell_column_before. ff_gap_mdm_lt_mdr_first_integer_rankzeroevaluationdeterminanthscm_negative_cell_column_before + S (ff_column_mdm_prefix_mdr_first_integer_rankzeroevaluationdeterminanthscm_negative) = (mdr_j_first_integer_rankzeroevaluationdeterminanthsc)) /\ ff_column_mdm_cell_mdr_first_integer_rankzeroevaluationdeterminanthscm_negative_cell = ff_column_mdm_prefix_mdr_first_integer_rankzeroevaluationdeterminanthscm_negative) \/ ((exists ff_gap_mdm_le_mdr_first_integer_rankzeroevaluationdeterminanthscm_negative_cell_column_after. ff_gap_mdm_le_mdr_first_integer_rankzeroevaluationdeterminanthscm_negative_cell_column_after + (mdr_j_first_integer_rankzeroevaluationdeterminanthsc) = (ff_column_mdm_prefix_mdr_first_integer_rankzeroevaluationdeterminanthscm_negative)) /\ ff_column_mdm_cell_mdr_first_integer_rankzeroevaluationdeterminanthscm_negative_cell = S ff_column_mdm_prefix_mdr_first_integer_rankzeroevaluationdeterminanthscm_negative))) /\ (((exists ff_h_mdm_mdr_first_integer_rankzeroevaluationdeterminanthscm_negative_cell_source. ff_h_mdm_mdr_first_integer_rankzeroevaluationdeterminanthscm_negative_cell_source + S (ff_value_mdm_prefix_mdr_first_integer_rankzeroevaluationdeterminanthscm_negative) = S ((S ((ff_row_mdm_cell_mdr_first_integer_rankzeroevaluationdeterminanthscm_negative_cell) * (S (mdr_q_first_integer_rankzeroevaluationdeterminanths)) + (ff_column_mdm_cell_mdr_first_integer_rankzeroevaluationdeterminanthscm_negative_cell))) * mdr_nc_first_integer_rankzeroevaluationdeterminanth)) /\ exists ff_q_mdm_mdr_first_integer_rankzeroevaluationdeterminanthscm_negative_cell_source. mdr_nb_first_integer_rankzeroevaluationdeterminanth = ff_q_mdm_mdr_first_integer_rankzeroevaluationdeterminanthscm_negative_cell_source * S ((S ((ff_row_mdm_cell_mdr_first_integer_rankzeroevaluationdeterminanthscm_negative_cell) * (S (mdr_q_first_integer_rankzeroevaluationdeterminanths)) + (ff_column_mdm_cell_mdr_first_integer_rankzeroevaluationdeterminanthscm_negative_cell))) * mdr_nc_first_integer_rankzeroevaluationdeterminanth) + (ff_value_mdm_prefix_mdr_first_integer_rankzeroevaluationdeterminanthscm_negative)))))) /\ (((exists ff_h_mdm_mdr_first_integer_rankzeroevaluationdeterminanthscm_negative_target. ff_h_mdm_mdr_first_integer_rankzeroevaluationdeterminanthscm_negative_target + S (ff_value_mdm_prefix_mdr_first_integer_rankzeroevaluationdeterminanthscm_negative) = S ((S (ff_index_mdm_prefix_mdr_first_integer_rankzeroevaluationdeterminanthscm_negative)) * mdr_ut_first_integer_rankzeroevaluationdeterminanthsc)) /\ exists ff_q_mdm_mdr_first_integer_rankzeroevaluationdeterminanthscm_negative_target. mdr_un_first_integer_rankzeroevaluationdeterminanthsc = ff_q_mdm_mdr_first_integer_rankzeroevaluationdeterminanthscm_negative_target * S ((S (ff_index_mdm_prefix_mdr_first_integer_rankzeroevaluationdeterminanthscm_negative)) * mdr_ut_first_integer_rankzeroevaluationdeterminanthsc) + (ff_value_mdm_prefix_mdr_first_integer_rankzeroevaluationdeterminanthscm_negative))))))))) /\ ((((exists ff_h_mdr_first_integer_rankzeroevaluationdeterminanthscp. ff_h_mdr_first_integer_rankzeroevaluationdeterminanthscp + S (mdr_p_first_integer_rankzeroevaluationdeterminanthsc) = S ((S (mdr_j_first_integer_rankzeroevaluationdeterminanthsc)) * mdr_ec_first_integer_rankzeroevaluationdeterminanths)) /\ exists ff_q_mdr_first_integer_rankzeroevaluationdeterminanthscp. mdr_eb_first_integer_rankzeroevaluationdeterminanths = ff_q_mdr_first_integer_rankzeroevaluationdeterminanthscp * S ((S (mdr_j_first_integer_rankzeroevaluationdeterminanthsc)) * mdr_ec_first_integer_rankzeroevaluationdeterminanths) + (mdr_p_first_integer_rankzeroevaluationdeterminanthsc))) /\ (((exists ff_h_mdr_first_integer_rankzeroevaluationdeterminanthscn. ff_h_mdr_first_integer_rankzeroevaluationdeterminanthscn + S (mdr_n_first_integer_rankzeroevaluationdeterminanthsc) = S ((S (mdr_j_first_integer_rankzeroevaluationdeterminanthsc)) * mdr_fc_first_integer_rankzeroevaluationdeterminanths)) /\ exists ff_q_mdr_first_integer_rankzeroevaluationdeterminanthscn. mdr_fb_first_integer_rankzeroevaluationdeterminanths = ff_q_mdr_first_integer_rankzeroevaluationdeterminanthscn * S ((S (mdr_j_first_integer_rankzeroevaluationdeterminanthsc)) * mdr_fc_first_integer_rankzeroevaluationdeterminanths) + (mdr_n_first_integer_rankzeroevaluationdeterminanthsc)))))))) /\ (exists ff_ub_mce_fold_mdr_first_integer_rankzeroevaluationdeterminanthsf ff_uc_mce_fold_mdr_first_integer_rankzeroevaluationdeterminanthsf ff_vb_mce_fold_mdr_first_integer_rankzeroevaluationdeterminanthsf ff_vc_mce_fold_mdr_first_integer_rankzeroevaluationdeterminanthsf. ((forall ff_index_mce_alternating_mdr_first_integer_rankzeroevaluationdeterminanthsf_prefix. (exists ff_gap_mce_mdr_first_integer_rankzeroevaluationdeterminanthsf_prefix_index. ff_gap_mce_mdr_first_integer_rankzeroevaluationdeterminanthsf_prefix_index + S (ff_index_mce_alternating_mdr_first_integer_rankzeroevaluationdeterminanthsf_prefix) = (S (mdr_q_first_integer_rankzeroevaluationdeterminanths))) -> exists ff_ap_mce_alternating_mdr_first_integer_rankzeroevaluationdeterminanthsf_prefix ff_an_mce_alternating_mdr_first_integer_rankzeroevaluationdeterminanthsf_prefix ff_bp_mce_alternating_mdr_first_integer_rankzeroevaluationdeterminanthsf_prefix ff_bn_mce_alternating_mdr_first_integer_rankzeroevaluationdeterminanthsf_prefix ff_p_mce_alternating_mdr_first_integer_rankzeroevaluationdeterminanthsf_prefix ff_n_mce_alternating_mdr_first_integer_rankzeroevaluationdeterminanthsf_prefix. ((((exists ff_h_mce_mdr_first_integer_rankzeroevaluationdeterminanthsf_prefix_ap. ff_h_mce_mdr_first_integer_rankzeroevaluationdeterminanthsf_prefix_ap + S (ff_ap_mce_alternating_mdr_first_integer_rankzeroevaluationdeterminanthsf_prefix) = S ((S (ff_index_mce_alternating_mdr_first_integer_rankzeroevaluationdeterminanthsf_prefix)) * mdr_pc_first_integer_rankzeroevaluationdeterminanth)) /\ exists ff_q_mce_mdr_first_integer_rankzeroevaluationdeterminanthsf_prefix_ap. mdr_pb_first_integer_rankzeroevaluationdeterminanth = ff_q_mce_mdr_first_integer_rankzeroevaluationdeterminanthsf_prefix_ap * S ((S (ff_index_mce_alternating_mdr_first_integer_rankzeroevaluationdeterminanthsf_prefix)) * mdr_pc_first_integer_rankzeroevaluationdeterminanth) + (ff_ap_mce_alternating_mdr_first_integer_rankzeroevaluationdeterminanthsf_prefix))) /\ ((((exists ff_h_mce_mdr_first_integer_rankzeroevaluationdeterminanthsf_prefix_an. ff_h_mce_mdr_first_integer_rankzeroevaluationdeterminanthsf_prefix_an + S (ff_an_mce_alternating_mdr_first_integer_rankzeroevaluationdeterminanthsf_prefix) = S ((S (ff_index_mce_alternating_mdr_first_integer_rankzeroevaluationdeterminanthsf_prefix)) * mdr_nc_first_integer_rankzeroevaluationdeterminanth)) /\ exists ff_q_mce_mdr_first_integer_rankzeroevaluationdeterminanthsf_prefix_an. mdr_nb_first_integer_rankzeroevaluationdeterminanth = ff_q_mce_mdr_first_integer_rankzeroevaluationdeterminanthsf_prefix_an * S ((S (ff_index_mce_alternating_mdr_first_integer_rankzeroevaluationdeterminanthsf_prefix)) * mdr_nc_first_integer_rankzeroevaluationdeterminanth) + (ff_an_mce_alternating_mdr_first_integer_rankzeroevaluationdeterminanthsf_prefix))) /\ ((((exists ff_h_mce_mdr_first_integer_rankzeroevaluationdeterminanthsf_prefix_bp. ff_h_mce_mdr_first_integer_rankzeroevaluationdeterminanthsf_prefix_bp + S (ff_bp_mce_alternating_mdr_first_integer_rankzeroevaluationdeterminanthsf_prefix) = S ((S (ff_index_mce_alternating_mdr_first_integer_rankzeroevaluationdeterminanthsf_prefix)) * mdr_ec_first_integer_rankzeroevaluationdeterminanths)) /\ exists ff_q_mce_mdr_first_integer_rankzeroevaluationdeterminanthsf_prefix_bp. mdr_eb_first_integer_rankzeroevaluationdeterminanths = ff_q_mce_mdr_first_integer_rankzeroevaluationdeterminanthsf_prefix_bp * S ((S (ff_index_mce_alternating_mdr_first_integer_rankzeroevaluationdeterminanthsf_prefix)) * mdr_ec_first_integer_rankzeroevaluationdeterminanths) + (ff_bp_mce_alternating_mdr_first_integer_rankzeroevaluationdeterminanthsf_prefix))) /\ ((((exists ff_h_mce_mdr_first_integer_rankzeroevaluationdeterminanthsf_prefix_bn. ff_h_mce_mdr_first_integer_rankzeroevaluationdeterminanthsf_prefix_bn + S (ff_bn_mce_alternating_mdr_first_integer_rankzeroevaluationdeterminanthsf_prefix) = S ((S (ff_index_mce_alternating_mdr_first_integer_rankzeroevaluationdeterminanthsf_prefix)) * mdr_fc_first_integer_rankzeroevaluationdeterminanths)) /\ exists ff_q_mce_mdr_first_integer_rankzeroevaluationdeterminanthsf_prefix_bn. mdr_fb_first_integer_rankzeroevaluationdeterminanths = ff_q_mce_mdr_first_integer_rankzeroevaluationdeterminanthsf_prefix_bn * S ((S (ff_index_mce_alternating_mdr_first_integer_rankzeroevaluationdeterminanthsf_prefix)) * mdr_fc_first_integer_rankzeroevaluationdeterminanths) + (ff_bn_mce_alternating_mdr_first_integer_rankzeroevaluationdeterminanthsf_prefix))) /\ ((((exists ff_h_mce_mdr_first_integer_rankzeroevaluationdeterminanthsf_prefix_positive. ff_h_mce_mdr_first_integer_rankzeroevaluationdeterminanthsf_prefix_positive + S (ff_p_mce_alternating_mdr_first_integer_rankzeroevaluationdeterminanthsf_prefix) = S ((S (ff_index_mce_alternating_mdr_first_integer_rankzeroevaluationdeterminanthsf_prefix)) * ff_uc_mce_fold_mdr_first_integer_rankzeroevaluationdeterminanthsf)) /\ exists ff_q_mce_mdr_first_integer_rankzeroevaluationdeterminanthsf_prefix_positive. ff_ub_mce_fold_mdr_first_integer_rankzeroevaluationdeterminanthsf = ff_q_mce_mdr_first_integer_rankzeroevaluationdeterminanthsf_prefix_positive * S ((S (ff_index_mce_alternating_mdr_first_integer_rankzeroevaluationdeterminanthsf_prefix)) * ff_uc_mce_fold_mdr_first_integer_rankzeroevaluationdeterminanthsf) + (ff_p_mce_alternating_mdr_first_integer_rankzeroevaluationdeterminanthsf_prefix))) /\ ((((exists ff_h_mce_mdr_first_integer_rankzeroevaluationdeterminanthsf_prefix_negative. ff_h_mce_mdr_first_integer_rankzeroevaluationdeterminanthsf_prefix_negative + S (ff_n_mce_alternating_mdr_first_integer_rankzeroevaluationdeterminanthsf_prefix) = S ((S (ff_index_mce_alternating_mdr_first_integer_rankzeroevaluationdeterminanthsf_prefix)) * ff_vc_mce_fold_mdr_first_integer_rankzeroevaluationdeterminanthsf)) /\ exists ff_q_mce_mdr_first_integer_rankzeroevaluationdeterminanthsf_prefix_negative. ff_vb_mce_fold_mdr_first_integer_rankzeroevaluationdeterminanthsf = ff_q_mce_mdr_first_integer_rankzeroevaluationdeterminanthsf_prefix_negative * S ((S (ff_index_mce_alternating_mdr_first_integer_rankzeroevaluationdeterminanthsf_prefix)) * ff_vc_mce_fold_mdr_first_integer_rankzeroevaluationdeterminanthsf) + (ff_n_mce_alternating_mdr_first_integer_rankzeroevaluationdeterminanthsf_prefix))) /\ (((exists ff_even_mce_term_mdr_first_integer_rankzeroevaluationdeterminanthsf_prefix_term. ff_index_mce_alternating_mdr_first_integer_rankzeroevaluationdeterminanthsf_prefix = 2 * ff_even_mce_term_mdr_first_integer_rankzeroevaluationdeterminanthsf_prefix_term) /\ (ff_p_mce_alternating_mdr_first_integer_rankzeroevaluationdeterminanthsf_prefix = (ff_ap_mce_alternating_mdr_first_integer_rankzeroevaluationdeterminanthsf_prefix) * (ff_bp_mce_alternating_mdr_first_integer_rankzeroevaluationdeterminanthsf_prefix) + (ff_an_mce_alternating_mdr_first_integer_rankzeroevaluationdeterminanthsf_prefix) * (ff_bn_mce_alternating_mdr_first_integer_rankzeroevaluationdeterminanthsf_prefix) /\ ff_n_mce_alternating_mdr_first_integer_rankzeroevaluationdeterminanthsf_prefix = (ff_ap_mce_alternating_mdr_first_integer_rankzeroevaluationdeterminanthsf_prefix) * (ff_bn_mce_alternating_mdr_first_integer_rankzeroevaluationdeterminanthsf_prefix) + (ff_an_mce_alternating_mdr_first_integer_rankzeroevaluationdeterminanthsf_prefix) * (ff_bp_mce_alternating_mdr_first_integer_rankzeroevaluationdeterminanthsf_prefix))) \/ ((exists ff_odd_mce_term_mdr_first_integer_rankzeroevaluationdeterminanthsf_prefix_term. ff_index_mce_alternating_mdr_first_integer_rankzeroevaluationdeterminanthsf_prefix = 2 * ff_odd_mce_term_mdr_first_integer_rankzeroevaluationdeterminanthsf_prefix_term + 1) /\ (ff_p_mce_alternating_mdr_first_integer_rankzeroevaluationdeterminanthsf_prefix = (ff_ap_mce_alternating_mdr_first_integer_rankzeroevaluationdeterminanthsf_prefix) * (ff_bn_mce_alternating_mdr_first_integer_rankzeroevaluationdeterminanthsf_prefix) + (ff_an_mce_alternating_mdr_first_integer_rankzeroevaluationdeterminanthsf_prefix) * (ff_bp_mce_alternating_mdr_first_integer_rankzeroevaluationdeterminanthsf_prefix) /\ ff_n_mce_alternating_mdr_first_integer_rankzeroevaluationdeterminanthsf_prefix = (ff_ap_mce_alternating_mdr_first_integer_rankzeroevaluationdeterminanthsf_prefix) * (ff_bp_mce_alternating_mdr_first_integer_rankzeroevaluationdeterminanthsf_prefix) + (ff_an_mce_alternating_mdr_first_integer_rankzeroevaluationdeterminanthsf_prefix) * (ff_bn_mce_alternating_mdr_first_integer_rankzeroevaluationdeterminanthsf_prefix))))))))))) /\ ((exists ff_u_mce_mdr_first_integer_rankzeroevaluationdeterminanthsf_positive ff_v_mce_mdr_first_integer_rankzeroevaluationdeterminanthsf_positive. ((((exists ff_h_mce_mdr_first_integer_rankzeroevaluationdeterminanthsf_positive_start. ff_h_mce_mdr_first_integer_rankzeroevaluationdeterminanthsf_positive_start + S (0) = S ((S (0)) * ff_v_mce_mdr_first_integer_rankzeroevaluationdeterminanthsf_positive)) /\ exists ff_q_mce_mdr_first_integer_rankzeroevaluationdeterminanthsf_positive_start. ff_u_mce_mdr_first_integer_rankzeroevaluationdeterminanthsf_positive = ff_q_mce_mdr_first_integer_rankzeroevaluationdeterminanthsf_positive_start * S ((S (0)) * ff_v_mce_mdr_first_integer_rankzeroevaluationdeterminanthsf_positive) + (0))) /\ ((((exists ff_h_mce_mdr_first_integer_rankzeroevaluationdeterminanthsf_positive_terminal. ff_h_mce_mdr_first_integer_rankzeroevaluationdeterminanthsf_positive_terminal + S (mdr_p_first_integer_rankzeroevaluationdeterminanth) = S ((S ((S (mdr_q_first_integer_rankzeroevaluationdeterminanths)))) * ff_v_mce_mdr_first_integer_rankzeroevaluationdeterminanthsf_positive)) /\ exists ff_q_mce_mdr_first_integer_rankzeroevaluationdeterminanthsf_positive_terminal. ff_u_mce_mdr_first_integer_rankzeroevaluationdeterminanthsf_positive = ff_q_mce_mdr_first_integer_rankzeroevaluationdeterminanthsf_positive_terminal * S ((S ((S (mdr_q_first_integer_rankzeroevaluationdeterminanths)))) * ff_v_mce_mdr_first_integer_rankzeroevaluationdeterminanthsf_positive) + (mdr_p_first_integer_rankzeroevaluationdeterminanth))) /\ forall ff_i_mce_mdr_first_integer_rankzeroevaluationdeterminanthsf_positive. (exists ff_lt_mce_mdr_first_integer_rankzeroevaluationdeterminanthsf_positive_bound. ff_lt_mce_mdr_first_integer_rankzeroevaluationdeterminanthsf_positive_bound + S ff_i_mce_mdr_first_integer_rankzeroevaluationdeterminanthsf_positive = (S (mdr_q_first_integer_rankzeroevaluationdeterminanths))) -> exists ff_a_mce_mdr_first_integer_rankzeroevaluationdeterminanthsf_positive ff_r_mce_mdr_first_integer_rankzeroevaluationdeterminanthsf_positive ff_s_mce_mdr_first_integer_rankzeroevaluationdeterminanthsf_positive. ((((exists ff_h_mce_mdr_first_integer_rankzeroevaluationdeterminanthsf_positive_summand. ff_h_mce_mdr_first_integer_rankzeroevaluationdeterminanthsf_positive_summand + S (ff_a_mce_mdr_first_integer_rankzeroevaluationdeterminanthsf_positive) = S ((S (ff_i_mce_mdr_first_integer_rankzeroevaluationdeterminanthsf_positive)) * ff_uc_mce_fold_mdr_first_integer_rankzeroevaluationdeterminanthsf)) /\ exists ff_q_mce_mdr_first_integer_rankzeroevaluationdeterminanthsf_positive_summand. ff_ub_mce_fold_mdr_first_integer_rankzeroevaluationdeterminanthsf = ff_q_mce_mdr_first_integer_rankzeroevaluationdeterminanthsf_positive_summand * S ((S (ff_i_mce_mdr_first_integer_rankzeroevaluationdeterminanthsf_positive)) * ff_uc_mce_fold_mdr_first_integer_rankzeroevaluationdeterminanthsf) + (ff_a_mce_mdr_first_integer_rankzeroevaluationdeterminanthsf_positive))) /\ ((((exists ff_h_mce_mdr_first_integer_rankzeroevaluationdeterminanthsf_positive_partial. ff_h_mce_mdr_first_integer_rankzeroevaluationdeterminanthsf_positive_partial + S (ff_r_mce_mdr_first_integer_rankzeroevaluationdeterminanthsf_positive) = S ((S (ff_i_mce_mdr_first_integer_rankzeroevaluationdeterminanthsf_positive)) * ff_v_mce_mdr_first_integer_rankzeroevaluationdeterminanthsf_positive)) /\ exists ff_q_mce_mdr_first_integer_rankzeroevaluationdeterminanthsf_positive_partial. ff_u_mce_mdr_first_integer_rankzeroevaluationdeterminanthsf_positive = ff_q_mce_mdr_first_integer_rankzeroevaluationdeterminanthsf_positive_partial * S ((S (ff_i_mce_mdr_first_integer_rankzeroevaluationdeterminanthsf_positive)) * ff_v_mce_mdr_first_integer_rankzeroevaluationdeterminanthsf_positive) + (ff_r_mce_mdr_first_integer_rankzeroevaluationdeterminanthsf_positive))) /\ ((((exists ff_h_mce_mdr_first_integer_rankzeroevaluationdeterminanthsf_positive_successor. ff_h_mce_mdr_first_integer_rankzeroevaluationdeterminanthsf_positive_successor + S (ff_s_mce_mdr_first_integer_rankzeroevaluationdeterminanthsf_positive) = S ((S (S ff_i_mce_mdr_first_integer_rankzeroevaluationdeterminanthsf_positive)) * ff_v_mce_mdr_first_integer_rankzeroevaluationdeterminanthsf_positive)) /\ exists ff_q_mce_mdr_first_integer_rankzeroevaluationdeterminanthsf_positive_successor. ff_u_mce_mdr_first_integer_rankzeroevaluationdeterminanthsf_positive = ff_q_mce_mdr_first_integer_rankzeroevaluationdeterminanthsf_positive_successor * S ((S (S ff_i_mce_mdr_first_integer_rankzeroevaluationdeterminanthsf_positive)) * ff_v_mce_mdr_first_integer_rankzeroevaluationdeterminanthsf_positive) + (ff_s_mce_mdr_first_integer_rankzeroevaluationdeterminanthsf_positive))) /\ ff_s_mce_mdr_first_integer_rankzeroevaluationdeterminanthsf_positive = ff_r_mce_mdr_first_integer_rankzeroevaluationdeterminanthsf_positive + ff_a_mce_mdr_first_integer_rankzeroevaluationdeterminanthsf_positive)))))) /\ (exists ff_u_mce_mdr_first_integer_rankzeroevaluationdeterminanthsf_negative ff_v_mce_mdr_first_integer_rankzeroevaluationdeterminanthsf_negative. ((((exists ff_h_mce_mdr_first_integer_rankzeroevaluationdeterminanthsf_negative_start. ff_h_mce_mdr_first_integer_rankzeroevaluationdeterminanthsf_negative_start + S (0) = S ((S (0)) * ff_v_mce_mdr_first_integer_rankzeroevaluationdeterminanthsf_negative)) /\ exists ff_q_mce_mdr_first_integer_rankzeroevaluationdeterminanthsf_negative_start. ff_u_mce_mdr_first_integer_rankzeroevaluationdeterminanthsf_negative = ff_q_mce_mdr_first_integer_rankzeroevaluationdeterminanthsf_negative_start * S ((S (0)) * ff_v_mce_mdr_first_integer_rankzeroevaluationdeterminanthsf_negative) + (0))) /\ ((((exists ff_h_mce_mdr_first_integer_rankzeroevaluationdeterminanthsf_negative_terminal. ff_h_mce_mdr_first_integer_rankzeroevaluationdeterminanthsf_negative_terminal + S (mdr_n_first_integer_rankzeroevaluationdeterminanth) = S ((S ((S (mdr_q_first_integer_rankzeroevaluationdeterminanths)))) * ff_v_mce_mdr_first_integer_rankzeroevaluationdeterminanthsf_negative)) /\ exists ff_q_mce_mdr_first_integer_rankzeroevaluationdeterminanthsf_negative_terminal. ff_u_mce_mdr_first_integer_rankzeroevaluationdeterminanthsf_negative = ff_q_mce_mdr_first_integer_rankzeroevaluationdeterminanthsf_negative_terminal * S ((S ((S (mdr_q_first_integer_rankzeroevaluationdeterminanths)))) * ff_v_mce_mdr_first_integer_rankzeroevaluationdeterminanthsf_negative) + (mdr_n_first_integer_rankzeroevaluationdeterminanth))) /\ forall ff_i_mce_mdr_first_integer_rankzeroevaluationdeterminanthsf_negative. (exists ff_lt_mce_mdr_first_integer_rankzeroevaluationdeterminanthsf_negative_bound. ff_lt_mce_mdr_first_integer_rankzeroevaluationdeterminanthsf_negative_bound + S ff_i_mce_mdr_first_integer_rankzeroevaluationdeterminanthsf_negative = (S (mdr_q_first_integer_rankzeroevaluationdeterminanths))) -> exists ff_a_mce_mdr_first_integer_rankzeroevaluationdeterminanthsf_negative ff_r_mce_mdr_first_integer_rankzeroevaluationdeterminanthsf_negative ff_s_mce_mdr_first_integer_rankzeroevaluationdeterminanthsf_negative. ((((exists ff_h_mce_mdr_first_integer_rankzeroevaluationdeterminanthsf_negative_summand. ff_h_mce_mdr_first_integer_rankzeroevaluationdeterminanthsf_negative_summand + S (ff_a_mce_mdr_first_integer_rankzeroevaluationdeterminanthsf_negative) = S ((S (ff_i_mce_mdr_first_integer_rankzeroevaluationdeterminanthsf_negative)) * ff_vc_mce_fold_mdr_first_integer_rankzeroevaluationdeterminanthsf)) /\ exists ff_q_mce_mdr_first_integer_rankzeroevaluationdeterminanthsf_negative_summand. ff_vb_mce_fold_mdr_first_integer_rankzeroevaluationdeterminanthsf = ff_q_mce_mdr_first_integer_rankzeroevaluationdeterminanthsf_negative_summand * S ((S (ff_i_mce_mdr_first_integer_rankzeroevaluationdeterminanthsf_negative)) * ff_vc_mce_fold_mdr_first_integer_rankzeroevaluationdeterminanthsf) + (ff_a_mce_mdr_first_integer_rankzeroevaluationdeterminanthsf_negative))) /\ ((((exists ff_h_mce_mdr_first_integer_rankzeroevaluationdeterminanthsf_negative_partial. ff_h_mce_mdr_first_integer_rankzeroevaluationdeterminanthsf_negative_partial + S (ff_r_mce_mdr_first_integer_rankzeroevaluationdeterminanthsf_negative) = S ((S (ff_i_mce_mdr_first_integer_rankzeroevaluationdeterminanthsf_negative)) * ff_v_mce_mdr_first_integer_rankzeroevaluationdeterminanthsf_negative)) /\ exists ff_q_mce_mdr_first_integer_rankzeroevaluationdeterminanthsf_negative_partial. ff_u_mce_mdr_first_integer_rankzeroevaluationdeterminanthsf_negative = ff_q_mce_mdr_first_integer_rankzeroevaluationdeterminanthsf_negative_partial * S ((S (ff_i_mce_mdr_first_integer_rankzeroevaluationdeterminanthsf_negative)) * ff_v_mce_mdr_first_integer_rankzeroevaluationdeterminanthsf_negative) + (ff_r_mce_mdr_first_integer_rankzeroevaluationdeterminanthsf_negative))) /\ ((((exists ff_h_mce_mdr_first_integer_rankzeroevaluationdeterminanthsf_negative_successor. ff_h_mce_mdr_first_integer_rankzeroevaluationdeterminanthsf_negative_successor + S (ff_s_mce_mdr_first_integer_rankzeroevaluationdeterminanthsf_negative) = S ((S (S ff_i_mce_mdr_first_integer_rankzeroevaluationdeterminanthsf_negative)) * ff_v_mce_mdr_first_integer_rankzeroevaluationdeterminanthsf_negative)) /\ exists ff_q_mce_mdr_first_integer_rankzeroevaluationdeterminanthsf_negative_successor. ff_u_mce_mdr_first_integer_rankzeroevaluationdeterminanthsf_negative = ff_q_mce_mdr_first_integer_rankzeroevaluationdeterminanthsf_negative_successor * S ((S (S ff_i_mce_mdr_first_integer_rankzeroevaluationdeterminanthsf_negative)) * ff_v_mce_mdr_first_integer_rankzeroevaluationdeterminanthsf_negative) + (ff_s_mce_mdr_first_integer_rankzeroevaluationdeterminanthsf_negative))) /\ ff_s_mce_mdr_first_integer_rankzeroevaluationdeterminanthsf_negative = ff_r_mce_mdr_first_integer_rankzeroevaluationdeterminanthsf_negative + ff_a_mce_mdr_first_integer_rankzeroevaluationdeterminanthsf_negative))))))))))))))) /\ ((exists mdr_gap_first_integer_rankzeroevaluationdeterminanti. mdr_gap_first_integer_rankzeroevaluationdeterminanti + S (mdr_i_first_integer_rankzeroevaluationdeterminant) = (mdr_l_first_integer_rankzeroevaluationdeterminant)) /\ (exists mdr_z_first_integer_rankzeroevaluationdeterminantr. ((exists mdr_a_first_integer_rankzeroevaluationdeterminantrc mdr_b_first_integer_rankzeroevaluationdeterminantrc mdr_c_first_integer_rankzeroevaluationdeterminantrc mdr_e_first_integer_rankzeroevaluationdeterminantrc mdr_f_first_integer_rankzeroevaluationdeterminantrc. ((mdr_a_first_integer_rankzeroevaluationdeterminantrc = ((mdr_q_first_integer_rank) + (mdr_ub_first_integer_rankzeroevaluation)) * S ((mdr_q_first_integer_rank) + (mdr_ub_first_integer_rankzeroevaluation)) + ((mdr_ub_first_integer_rankzeroevaluation) + (mdr_ub_first_integer_rankzeroevaluation))) /\ ((mdr_b_first_integer_rankzeroevaluationdeterminantrc = ((mdr_uc_first_integer_rankzeroevaluation) + (mdr_vb_first_integer_rankzeroevaluation)) * S ((mdr_uc_first_integer_rankzeroevaluation) + (mdr_vb_first_integer_rankzeroevaluation)) + ((mdr_vb_first_integer_rankzeroevaluation) + (mdr_vb_first_integer_rankzeroevaluation))) /\ ((mdr_c_first_integer_rankzeroevaluationdeterminantrc = ((mdr_a_first_integer_rankzeroevaluationdeterminantrc) + (mdr_b_first_integer_rankzeroevaluationdeterminantrc)) * S ((mdr_a_first_integer_rankzeroevaluationdeterminantrc) + (mdr_b_first_integer_rankzeroevaluationdeterminantrc)) + ((mdr_b_first_integer_rankzeroevaluationdeterminantrc) + (mdr_b_first_integer_rankzeroevaluationdeterminantrc))) /\ ((mdr_e_first_integer_rankzeroevaluationdeterminantrc = ((mdr_p_first_integer_rankzero) + (mdr_n_first_integer_rankzero)) * S ((mdr_p_first_integer_rankzero) + (mdr_n_first_integer_rankzero)) + ((mdr_n_first_integer_rankzero) + (mdr_n_first_integer_rankzero))) /\ ((mdr_f_first_integer_rankzeroevaluationdeterminantrc = ((mdr_vc_first_integer_rankzeroevaluation) + (mdr_e_first_integer_rankzeroevaluationdeterminantrc)) * S ((mdr_vc_first_integer_rankzeroevaluation) + (mdr_e_first_integer_rankzeroevaluationdeterminantrc)) + ((mdr_e_first_integer_rankzeroevaluationdeterminantrc) + (mdr_e_first_integer_rankzeroevaluationdeterminantrc))) /\ ((mdr_z_first_integer_rankzeroevaluationdeterminantr) = ((mdr_c_first_integer_rankzeroevaluationdeterminantrc) + (mdr_f_first_integer_rankzeroevaluationdeterminantrc)) * S ((mdr_c_first_integer_rankzeroevaluationdeterminantrc) + (mdr_f_first_integer_rankzeroevaluationdeterminantrc)) + ((mdr_f_first_integer_rankzeroevaluationdeterminantrc) + (mdr_f_first_integer_rankzeroevaluationdeterminantrc))))))))) /\ (((exists ff_h_mdr_first_integer_rankzeroevaluationdeterminantrb. ff_h_mdr_first_integer_rankzeroevaluationdeterminantrb + S (mdr_z_first_integer_rankzeroevaluationdeterminantr) = S ((S (mdr_i_first_integer_rankzeroevaluationdeterminant)) * mdr_c_first_integer_rankzeroevaluationdeterminant)) /\ exists ff_q_mdr_first_integer_rankzeroevaluationdeterminantrb. mdr_b_first_integer_rankzeroevaluationdeterminant = ff_q_mdr_first_integer_rankzeroevaluationdeterminantrb * S ((S (mdr_i_first_integer_rankzeroevaluationdeterminant)) * mdr_c_first_integer_rankzeroevaluationdeterminant) + (mdr_z_first_integer_rankzeroevaluationdeterminantr)))))))))) -> mdr_p_first_integer_rankzero = mdr_n_first_integer_rankzero)))))) -> (((exists mdr_gap_second_integer_rankrows_bound. mdr_gap_second_integer_rankrows_bound + (rank) = (r)) /\ ((exists mdr_gap_second_integer_rankcolumns_bound. mdr_gap_second_integer_rankcolumns_bound + (rank) = (w)) /\ ((exists mdr_rb_second_integer_rankwitness mdr_rc_second_integer_rankwitness mdr_cb_second_integer_rankwitness mdr_cc_second_integer_rankwitness. (((((forall fom_index_mrf_second_integer_rankwitnessminorrowsbound. (exists fom_gap_mrf_second_integer_rankwitnessminorrowsbound_index_bound. fom_gap_mrf_second_integer_rankwitnessminorrowsbound_index_bound + S (fom_index_mrf_second_integer_rankwitnessminorrowsbound) = rank) -> exists fom_value_mrf_second_integer_rankwitnessminorrowsbound. ((((exists fom_beta_height_mrf_second_integer_rankwitnessminorrowsbound_entry. fom_beta_height_mrf_second_integer_rankwitnessminorrowsbound_entry + S (fom_value_mrf_second_integer_rankwitnessminorrowsbound) = S ((S (fom_index_mrf_second_integer_rankwitnessminorrowsbound)) * mdr_rc_second_integer_rankwitness)) /\ exists fom_beta_quotient_mrf_second_integer_rankwitnessminorrowsbound_entry. mdr_rb_second_integer_rankwitness = fom_beta_quotient_mrf_second_integer_rankwitnessminorrowsbound_entry * S ((S (fom_index_mrf_second_integer_rankwitnessminorrowsbound)) * mdr_rc_second_integer_rankwitness) + (fom_value_mrf_second_integer_rankwitnessminorrowsbound))) /\ (exists fom_gap_mrf_second_integer_rankwitnessminorrowsbound_value_bound. fom_gap_mrf_second_integer_rankwitnessminorrowsbound_value_bound + S (fom_value_mrf_second_integer_rankwitnessminorrowsbound) = r))) /\ (forall mdr_i_second_integer_rankwitnessminorrowsdistinct mdr_j_second_integer_rankwitnessminorrowsdistinct mdr_a_second_integer_rankwitnessminorrowsdistinct. (exists mdr_gap_second_integer_rankwitnessminorrowsdistincti. mdr_gap_second_integer_rankwitnessminorrowsdistincti + S (mdr_i_second_integer_rankwitnessminorrowsdistinct) = (rank)) -> (exists mdr_gap_second_integer_rankwitnessminorrowsdistinctj. mdr_gap_second_integer_rankwitnessminorrowsdistinctj + S (mdr_j_second_integer_rankwitnessminorrowsdistinct) = (rank)) -> (((exists ff_h_mdr_second_integer_rankwitnessminorrowsdistinctfirst. ff_h_mdr_second_integer_rankwitnessminorrowsdistinctfirst + S (mdr_a_second_integer_rankwitnessminorrowsdistinct) = S ((S (mdr_i_second_integer_rankwitnessminorrowsdistinct)) * mdr_rc_second_integer_rankwitness)) /\ exists ff_q_mdr_second_integer_rankwitnessminorrowsdistinctfirst. mdr_rb_second_integer_rankwitness = ff_q_mdr_second_integer_rankwitnessminorrowsdistinctfirst * S ((S (mdr_i_second_integer_rankwitnessminorrowsdistinct)) * mdr_rc_second_integer_rankwitness) + (mdr_a_second_integer_rankwitnessminorrowsdistinct))) -> (((exists ff_h_mdr_second_integer_rankwitnessminorrowsdistinctsecond. ff_h_mdr_second_integer_rankwitnessminorrowsdistinctsecond + S (mdr_a_second_integer_rankwitnessminorrowsdistinct) = S ((S (mdr_j_second_integer_rankwitnessminorrowsdistinct)) * mdr_rc_second_integer_rankwitness)) /\ exists ff_q_mdr_second_integer_rankwitnessminorrowsdistinctsecond. mdr_rb_second_integer_rankwitness = ff_q_mdr_second_integer_rankwitnessminorrowsdistinctsecond * S ((S (mdr_j_second_integer_rankwitnessminorrowsdistinct)) * mdr_rc_second_integer_rankwitness) + (mdr_a_second_integer_rankwitnessminorrowsdistinct))) -> mdr_i_second_integer_rankwitnessminorrowsdistinct = mdr_j_second_integer_rankwitnessminorrowsdistinct))) /\ ((((forall fom_index_mrf_second_integer_rankwitnessminorcolumnsbound. (exists fom_gap_mrf_second_integer_rankwitnessminorcolumnsbound_index_bound. fom_gap_mrf_second_integer_rankwitnessminorcolumnsbound_index_bound + S (fom_index_mrf_second_integer_rankwitnessminorcolumnsbound) = rank) -> exists fom_value_mrf_second_integer_rankwitnessminorcolumnsbound. ((((exists fom_beta_height_mrf_second_integer_rankwitnessminorcolumnsbound_entry. fom_beta_height_mrf_second_integer_rankwitnessminorcolumnsbound_entry + S (fom_value_mrf_second_integer_rankwitnessminorcolumnsbound) = S ((S (fom_index_mrf_second_integer_rankwitnessminorcolumnsbound)) * mdr_cc_second_integer_rankwitness)) /\ exists fom_beta_quotient_mrf_second_integer_rankwitnessminorcolumnsbound_entry. mdr_cb_second_integer_rankwitness = fom_beta_quotient_mrf_second_integer_rankwitnessminorcolumnsbound_entry * S ((S (fom_index_mrf_second_integer_rankwitnessminorcolumnsbound)) * mdr_cc_second_integer_rankwitness) + (fom_value_mrf_second_integer_rankwitnessminorcolumnsbound))) /\ (exists fom_gap_mrf_second_integer_rankwitnessminorcolumnsbound_value_bound. fom_gap_mrf_second_integer_rankwitnessminorcolumnsbound_value_bound + S (fom_value_mrf_second_integer_rankwitnessminorcolumnsbound) = w))) /\ (forall mdr_i_second_integer_rankwitnessminorcolumnsdistinct mdr_j_second_integer_rankwitnessminorcolumnsdistinct mdr_a_second_integer_rankwitnessminorcolumnsdistinct. (exists mdr_gap_second_integer_rankwitnessminorcolumnsdistincti. mdr_gap_second_integer_rankwitnessminorcolumnsdistincti + S (mdr_i_second_integer_rankwitnessminorcolumnsdistinct) = (rank)) -> (exists mdr_gap_second_integer_rankwitnessminorcolumnsdistinctj. mdr_gap_second_integer_rankwitnessminorcolumnsdistinctj + S (mdr_j_second_integer_rankwitnessminorcolumnsdistinct) = (rank)) -> (((exists ff_h_mdr_second_integer_rankwitnessminorcolumnsdistinctfirst. ff_h_mdr_second_integer_rankwitnessminorcolumnsdistinctfirst + S (mdr_a_second_integer_rankwitnessminorcolumnsdistinct) = S ((S (mdr_i_second_integer_rankwitnessminorcolumnsdistinct)) * mdr_cc_second_integer_rankwitness)) /\ exists ff_q_mdr_second_integer_rankwitnessminorcolumnsdistinctfirst. mdr_cb_second_integer_rankwitness = ff_q_mdr_second_integer_rankwitnessminorcolumnsdistinctfirst * S ((S (mdr_i_second_integer_rankwitnessminorcolumnsdistinct)) * mdr_cc_second_integer_rankwitness) + (mdr_a_second_integer_rankwitnessminorcolumnsdistinct))) -> (((exists ff_h_mdr_second_integer_rankwitnessminorcolumnsdistinctsecond. ff_h_mdr_second_integer_rankwitnessminorcolumnsdistinctsecond + S (mdr_a_second_integer_rankwitnessminorcolumnsdistinct) = S ((S (mdr_j_second_integer_rankwitnessminorcolumnsdistinct)) * mdr_cc_second_integer_rankwitness)) /\ exists ff_q_mdr_second_integer_rankwitnessminorcolumnsdistinctsecond. mdr_cb_second_integer_rankwitness = ff_q_mdr_second_integer_rankwitnessminorcolumnsdistinctsecond * S ((S (mdr_j_second_integer_rankwitnessminorcolumnsdistinct)) * mdr_cc_second_integer_rankwitness) + (mdr_a_second_integer_rankwitnessminorcolumnsdistinct))) -> mdr_i_second_integer_rankwitnessminorcolumnsdistinct = mdr_j_second_integer_rankwitnessminorcolumnsdistinct))) /\ (exists mdr_p_second_integer_rankwitnessminornonzero mdr_n_second_integer_rankwitnessminornonzero. ((exists mdr_ub_second_integer_rankwitnessminornonzeroevaluation mdr_uc_second_integer_rankwitnessminornonzeroevaluation mdr_vb_second_integer_rankwitnessminornonzeroevaluation mdr_vc_second_integer_rankwitnessminornonzeroevaluation. ((((forall mdr_i_second_integer_rankwitnessminornonzeroevaluationmatrixpositive. (exists mdr_gap_second_integer_rankwitnessminornonzeroevaluationmatrixpositivebound. mdr_gap_second_integer_rankwitnessminornonzeroevaluationmatrixpositivebound + S (mdr_i_second_integer_rankwitnessminornonzeroevaluationmatrixpositive) = ((rank) * (rank))) -> exists mdr_a_second_integer_rankwitnessminornonzeroevaluationmatrixpositive. (((exists mdr_r_second_integer_rankwitnessminornonzeroevaluationmatrixpositivepoint mdr_s_second_integer_rankwitnessminornonzeroevaluationmatrixpositivepoint mdr_u_second_integer_rankwitnessminornonzeroevaluationmatrixpositivepoint mdr_v_second_integer_rankwitnessminornonzeroevaluationmatrixpositivepoint. ((mdr_i_second_integer_rankwitnessminornonzeroevaluationmatrixpositive = (rank) * mdr_r_second_integer_rankwitnessminornonzeroevaluationmatrixpositivepoint + mdr_s_second_integer_rankwitnessminornonzeroevaluationmatrixpositivepoint) /\ ((exists mdr_gap_second_integer_rankwitnessminornonzeroevaluationmatrixpositivepointcolumn. mdr_gap_second_integer_rankwitnessminornonzeroevaluationmatrixpositivepointcolumn + S (mdr_s_second_integer_rankwitnessminornonzeroevaluationmatrixpositivepoint) = (rank)) /\ ((((exists ff_h_mdr_second_integer_rankwitnessminornonzeroevaluationmatrixpositivepointrow_index. ff_h_mdr_second_integer_rankwitnessminornonzeroevaluationmatrixpositivepointrow_index + S (mdr_u_second_integer_rankwitnessminornonzeroevaluationmatrixpositivepoint) = S ((S (mdr_r_second_integer_rankwitnessminornonzeroevaluationmatrixpositivepoint)) * mdr_rc_second_integer_rankwitness)) /\ exists ff_q_mdr_second_integer_rankwitnessminornonzeroevaluationmatrixpositivepointrow_index. mdr_rb_second_integer_rankwitness = ff_q_mdr_second_integer_rankwitnessminornonzeroevaluationmatrixpositivepointrow_index * S ((S (mdr_r_second_integer_rankwitnessminornonzeroevaluationmatrixpositivepoint)) * mdr_rc_second_integer_rankwitness) + (mdr_u_second_integer_rankwitnessminornonzeroevaluationmatrixpositivepoint))) /\ ((((exists ff_h_mdr_second_integer_rankwitnessminornonzeroevaluationmatrixpositivepointcolumn_index. ff_h_mdr_second_integer_rankwitnessminornonzeroevaluationmatrixpositivepointcolumn_index + S (mdr_v_second_integer_rankwitnessminornonzeroevaluationmatrixpositivepoint) = S ((S (mdr_s_second_integer_rankwitnessminornonzeroevaluationmatrixpositivepoint)) * mdr_cc_second_integer_rankwitness)) /\ exists ff_q_mdr_second_integer_rankwitnessminornonzeroevaluationmatrixpositivepointcolumn_index. mdr_cb_second_integer_rankwitness = ff_q_mdr_second_integer_rankwitnessminornonzeroevaluationmatrixpositivepointcolumn_index * S ((S (mdr_s_second_integer_rankwitnessminornonzeroevaluationmatrixpositivepoint)) * mdr_cc_second_integer_rankwitness) + (mdr_v_second_integer_rankwitnessminornonzeroevaluationmatrixpositivepoint))) /\ (((exists ff_h_mdr_second_integer_rankwitnessminornonzeroevaluationmatrixpositivepointsource. ff_h_mdr_second_integer_rankwitnessminornonzeroevaluationmatrixpositivepointsource + S (mdr_a_second_integer_rankwitnessminornonzeroevaluationmatrixpositive) = S ((S ((mdr_u_second_integer_rankwitnessminornonzeroevaluationmatrixpositivepoint) * (w) + (mdr_v_second_integer_rankwitnessminornonzeroevaluationmatrixpositivepoint))) * ec)) /\ exists ff_q_mdr_second_integer_rankwitnessminornonzeroevaluationmatrixpositivepointsource. eb = ff_q_mdr_second_integer_rankwitnessminornonzeroevaluationmatrixpositivepointsource * S ((S ((mdr_u_second_integer_rankwitnessminornonzeroevaluationmatrixpositivepoint) * (w) + (mdr_v_second_integer_rankwitnessminornonzeroevaluationmatrixpositivepoint))) * ec) + (mdr_a_second_integer_rankwitnessminornonzeroevaluationmatrixpositive)))))))) /\ (((exists ff_h_mdr_second_integer_rankwitnessminornonzeroevaluationmatrixpositiveoutput. ff_h_mdr_second_integer_rankwitnessminornonzeroevaluationmatrixpositiveoutput + S (mdr_a_second_integer_rankwitnessminornonzeroevaluationmatrixpositive) = S ((S (mdr_i_second_integer_rankwitnessminornonzeroevaluationmatrixpositive)) * mdr_uc_second_integer_rankwitnessminornonzeroevaluation)) /\ exists ff_q_mdr_second_integer_rankwitnessminornonzeroevaluationmatrixpositiveoutput. mdr_ub_second_integer_rankwitnessminornonzeroevaluation = ff_q_mdr_second_integer_rankwitnessminornonzeroevaluationmatrixpositiveoutput * S ((S (mdr_i_second_integer_rankwitnessminornonzeroevaluationmatrixpositive)) * mdr_uc_second_integer_rankwitnessminornonzeroevaluation) + (mdr_a_second_integer_rankwitnessminornonzeroevaluationmatrixpositive)))))) /\ (forall mdr_i_second_integer_rankwitnessminornonzeroevaluationmatrixnegative. (exists mdr_gap_second_integer_rankwitnessminornonzeroevaluationmatrixnegativebound. mdr_gap_second_integer_rankwitnessminornonzeroevaluationmatrixnegativebound + S (mdr_i_second_integer_rankwitnessminornonzeroevaluationmatrixnegative) = ((rank) * (rank))) -> exists mdr_a_second_integer_rankwitnessminornonzeroevaluationmatrixnegative. (((exists mdr_r_second_integer_rankwitnessminornonzeroevaluationmatrixnegativepoint mdr_s_second_integer_rankwitnessminornonzeroevaluationmatrixnegativepoint mdr_u_second_integer_rankwitnessminornonzeroevaluationmatrixnegativepoint mdr_v_second_integer_rankwitnessminornonzeroevaluationmatrixnegativepoint. ((mdr_i_second_integer_rankwitnessminornonzeroevaluationmatrixnegative = (rank) * mdr_r_second_integer_rankwitnessminornonzeroevaluationmatrixnegativepoint + mdr_s_second_integer_rankwitnessminornonzeroevaluationmatrixnegativepoint) /\ ((exists mdr_gap_second_integer_rankwitnessminornonzeroevaluationmatrixnegativepointcolumn. mdr_gap_second_integer_rankwitnessminornonzeroevaluationmatrixnegativepointcolumn + S (mdr_s_second_integer_rankwitnessminornonzeroevaluationmatrixnegativepoint) = (rank)) /\ ((((exists ff_h_mdr_second_integer_rankwitnessminornonzeroevaluationmatrixnegativepointrow_index. ff_h_mdr_second_integer_rankwitnessminornonzeroevaluationmatrixnegativepointrow_index + S (mdr_u_second_integer_rankwitnessminornonzeroevaluationmatrixnegativepoint) = S ((S (mdr_r_second_integer_rankwitnessminornonzeroevaluationmatrixnegativepoint)) * mdr_rc_second_integer_rankwitness)) /\ exists ff_q_mdr_second_integer_rankwitnessminornonzeroevaluationmatrixnegativepointrow_index. mdr_rb_second_integer_rankwitness = ff_q_mdr_second_integer_rankwitnessminornonzeroevaluationmatrixnegativepointrow_index * S ((S (mdr_r_second_integer_rankwitnessminornonzeroevaluationmatrixnegativepoint)) * mdr_rc_second_integer_rankwitness) + (mdr_u_second_integer_rankwitnessminornonzeroevaluationmatrixnegativepoint))) /\ ((((exists ff_h_mdr_second_integer_rankwitnessminornonzeroevaluationmatrixnegativepointcolumn_index. ff_h_mdr_second_integer_rankwitnessminornonzeroevaluationmatrixnegativepointcolumn_index + S (mdr_v_second_integer_rankwitnessminornonzeroevaluationmatrixnegativepoint) = S ((S (mdr_s_second_integer_rankwitnessminornonzeroevaluationmatrixnegativepoint)) * mdr_cc_second_integer_rankwitness)) /\ exists ff_q_mdr_second_integer_rankwitnessminornonzeroevaluationmatrixnegativepointcolumn_index. mdr_cb_second_integer_rankwitness = ff_q_mdr_second_integer_rankwitnessminornonzeroevaluationmatrixnegativepointcolumn_index * S ((S (mdr_s_second_integer_rankwitnessminornonzeroevaluationmatrixnegativepoint)) * mdr_cc_second_integer_rankwitness) + (mdr_v_second_integer_rankwitnessminornonzeroevaluationmatrixnegativepoint))) /\ (((exists ff_h_mdr_second_integer_rankwitnessminornonzeroevaluationmatrixnegativepointsource. ff_h_mdr_second_integer_rankwitnessminornonzeroevaluationmatrixnegativepointsource + S (mdr_a_second_integer_rankwitnessminornonzeroevaluationmatrixnegative) = S ((S ((mdr_u_second_integer_rankwitnessminornonzeroevaluationmatrixnegativepoint) * (w) + (mdr_v_second_integer_rankwitnessminornonzeroevaluationmatrixnegativepoint))) * fc)) /\ exists ff_q_mdr_second_integer_rankwitnessminornonzeroevaluationmatrixnegativepointsource. fb = ff_q_mdr_second_integer_rankwitnessminornonzeroevaluationmatrixnegativepointsource * S ((S ((mdr_u_second_integer_rankwitnessminornonzeroevaluationmatrixnegativepoint) * (w) + (mdr_v_second_integer_rankwitnessminornonzeroevaluationmatrixnegativepoint))) * fc) + (mdr_a_second_integer_rankwitnessminornonzeroevaluationmatrixnegative)))))))) /\ (((exists ff_h_mdr_second_integer_rankwitnessminornonzeroevaluationmatrixnegativeoutput. ff_h_mdr_second_integer_rankwitnessminornonzeroevaluationmatrixnegativeoutput + S (mdr_a_second_integer_rankwitnessminornonzeroevaluationmatrixnegative) = S ((S (mdr_i_second_integer_rankwitnessminornonzeroevaluationmatrixnegative)) * mdr_vc_second_integer_rankwitnessminornonzeroevaluation)) /\ exists ff_q_mdr_second_integer_rankwitnessminornonzeroevaluationmatrixnegativeoutput. mdr_vb_second_integer_rankwitnessminornonzeroevaluation = ff_q_mdr_second_integer_rankwitnessminornonzeroevaluationmatrixnegativeoutput * S ((S (mdr_i_second_integer_rankwitnessminornonzeroevaluationmatrixnegative)) * mdr_vc_second_integer_rankwitnessminornonzeroevaluation) + (mdr_a_second_integer_rankwitnessminornonzeroevaluationmatrixnegative)))))))) /\ (exists mdr_b_second_integer_rankwitnessminornonzeroevaluationdeterminant mdr_c_second_integer_rankwitnessminornonzeroevaluationdeterminant mdr_l_second_integer_rankwitnessminornonzeroevaluationdeterminant mdr_i_second_integer_rankwitnessminornonzeroevaluationdeterminant. ((forall mdr_i_second_integer_rankwitnessminornonzeroevaluationdeterminanth. (exists mdr_gap_second_integer_rankwitnessminornonzeroevaluationdeterminanthi. mdr_gap_second_integer_rankwitnessminornonzeroevaluationdeterminanthi + S (mdr_i_second_integer_rankwitnessminornonzeroevaluationdeterminanth) = (mdr_l_second_integer_rankwitnessminornonzeroevaluationdeterminant)) -> exists mdr_d_second_integer_rankwitnessminornonzeroevaluationdeterminanth mdr_pb_second_integer_rankwitnessminornonzeroevaluationdeterminanth mdr_pc_second_integer_rankwitnessminornonzeroevaluationdeterminanth mdr_nb_second_integer_rankwitnessminornonzeroevaluationdeterminanth mdr_nc_second_integer_rankwitnessminornonzeroevaluationdeterminanth mdr_p_second_integer_rankwitnessminornonzeroevaluationdeterminanth mdr_n_second_integer_rankwitnessminornonzeroevaluationdeterminanth. ((exists mdr_z_second_integer_rankwitnessminornonzeroevaluationdeterminanthr. ((exists mdr_a_second_integer_rankwitnessminornonzeroevaluationdeterminanthrc mdr_b_second_integer_rankwitnessminornonzeroevaluationdeterminanthrc mdr_c_second_integer_rankwitnessminornonzeroevaluationdeterminanthrc mdr_e_second_integer_rankwitnessminornonzeroevaluationdeterminanthrc mdr_f_second_integer_rankwitnessminornonzeroevaluationdeterminanthrc. ((mdr_a_second_integer_rankwitnessminornonzeroevaluationdeterminanthrc = ((mdr_d_second_integer_rankwitnessminornonzeroevaluationdeterminanth) + (mdr_pb_second_integer_rankwitnessminornonzeroevaluationdeterminanth)) * S ((mdr_d_second_integer_rankwitnessminornonzeroevaluationdeterminanth) + (mdr_pb_second_integer_rankwitnessminornonzeroevaluationdeterminanth)) + ((mdr_pb_second_integer_rankwitnessminornonzeroevaluationdeterminanth) + (mdr_pb_second_integer_rankwitnessminornonzeroevaluationdeterminanth))) /\ ((mdr_b_second_integer_rankwitnessminornonzeroevaluationdeterminanthrc = ((mdr_pc_second_integer_rankwitnessminornonzeroevaluationdeterminanth) + (mdr_nb_second_integer_rankwitnessminornonzeroevaluationdeterminanth)) * S ((mdr_pc_second_integer_rankwitnessminornonzeroevaluationdeterminanth) + (mdr_nb_second_integer_rankwitnessminornonzeroevaluationdeterminanth)) + ((mdr_nb_second_integer_rankwitnessminornonzeroevaluationdeterminanth) + (mdr_nb_second_integer_rankwitnessminornonzeroevaluationdeterminanth))) /\ ((mdr_c_second_integer_rankwitnessminornonzeroevaluationdeterminanthrc = ((mdr_a_second_integer_rankwitnessminornonzeroevaluationdeterminanthrc) + (mdr_b_second_integer_rankwitnessminornonzeroevaluationdeterminanthrc)) * S ((mdr_a_second_integer_rankwitnessminornonzeroevaluationdeterminanthrc) + (mdr_b_second_integer_rankwitnessminornonzeroevaluationdeterminanthrc)) + ((mdr_b_second_integer_rankwitnessminornonzeroevaluationdeterminanthrc) + (mdr_b_second_integer_rankwitnessminornonzeroevaluationdeterminanthrc))) /\ ((mdr_e_second_integer_rankwitnessminornonzeroevaluationdeterminanthrc = ((mdr_p_second_integer_rankwitnessminornonzeroevaluationdeterminanth) + (mdr_n_second_integer_rankwitnessminornonzeroevaluationdeterminanth)) * S ((mdr_p_second_integer_rankwitnessminornonzeroevaluationdeterminanth) + (mdr_n_second_integer_rankwitnessminornonzeroevaluationdeterminanth)) + ((mdr_n_second_integer_rankwitnessminornonzeroevaluationdeterminanth) + (mdr_n_second_integer_rankwitnessminornonzeroevaluationdeterminanth))) /\ ((mdr_f_second_integer_rankwitnessminornonzeroevaluationdeterminanthrc = ((mdr_nc_second_integer_rankwitnessminornonzeroevaluationdeterminanth) + (mdr_e_second_integer_rankwitnessminornonzeroevaluationdeterminanthrc)) * S ((mdr_nc_second_integer_rankwitnessminornonzeroevaluationdeterminanth) + (mdr_e_second_integer_rankwitnessminornonzeroevaluationdeterminanthrc)) + ((mdr_e_second_integer_rankwitnessminornonzeroevaluationdeterminanthrc) + (mdr_e_second_integer_rankwitnessminornonzeroevaluationdeterminanthrc))) /\ ((mdr_z_second_integer_rankwitnessminornonzeroevaluationdeterminanthr) = ((mdr_c_second_integer_rankwitnessminornonzeroevaluationdeterminanthrc) + (mdr_f_second_integer_rankwitnessminornonzeroevaluationdeterminanthrc)) * S ((mdr_c_second_integer_rankwitnessminornonzeroevaluationdeterminanthrc) + (mdr_f_second_integer_rankwitnessminornonzeroevaluationdeterminanthrc)) + ((mdr_f_second_integer_rankwitnessminornonzeroevaluationdeterminanthrc) + (mdr_f_second_integer_rankwitnessminornonzeroevaluationdeterminanthrc))))))))) /\ (((exists ff_h_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthrb. ff_h_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthrb + S (mdr_z_second_integer_rankwitnessminornonzeroevaluationdeterminanthr) = S ((S (mdr_i_second_integer_rankwitnessminornonzeroevaluationdeterminanth)) * mdr_c_second_integer_rankwitnessminornonzeroevaluationdeterminant)) /\ exists ff_q_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthrb. mdr_b_second_integer_rankwitnessminornonzeroevaluationdeterminant = ff_q_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthrb * S ((S (mdr_i_second_integer_rankwitnessminornonzeroevaluationdeterminanth)) * mdr_c_second_integer_rankwitnessminornonzeroevaluationdeterminant) + (mdr_z_second_integer_rankwitnessminornonzeroevaluationdeterminanthr))))) /\ (((((mdr_d_second_integer_rankwitnessminornonzeroevaluationdeterminanth) = 0) /\ (((mdr_p_second_integer_rankwitnessminornonzeroevaluationdeterminanth) = 1) /\ ((mdr_n_second_integer_rankwitnessminornonzeroevaluationdeterminanth) = 0))) \/ exists mdr_q_second_integer_rankwitnessminornonzeroevaluationdeterminanths mdr_eb_second_integer_rankwitnessminornonzeroevaluationdeterminanths mdr_ec_second_integer_rankwitnessminornonzeroevaluationdeterminanths mdr_fb_second_integer_rankwitnessminornonzeroevaluationdeterminanths mdr_fc_second_integer_rankwitnessminornonzeroevaluationdeterminanths. (((mdr_d_second_integer_rankwitnessminornonzeroevaluationdeterminanth) = S (mdr_q_second_integer_rankwitnessminornonzeroevaluationdeterminanths)) /\ ((forall mdr_j_second_integer_rankwitnessminornonzeroevaluationdeterminanthsc. (exists mdr_gap_second_integer_rankwitnessminornonzeroevaluationdeterminanthscj. mdr_gap_second_integer_rankwitnessminornonzeroevaluationdeterminanthscj + S (mdr_j_second_integer_rankwitnessminornonzeroevaluationdeterminanthsc) = (S (mdr_q_second_integer_rankwitnessminornonzeroevaluationdeterminanths))) -> exists mdr_i_second_integer_rankwitnessminornonzeroevaluationdeterminanthsc mdr_up_second_integer_rankwitnessminornonzeroevaluationdeterminanthsc mdr_us_second_integer_rankwitnessminornonzeroevaluationdeterminanthsc mdr_un_second_integer_rankwitnessminornonzeroevaluationdeterminanthsc mdr_ut_second_integer_rankwitnessminornonzeroevaluationdeterminanthsc mdr_p_second_integer_rankwitnessminornonzeroevaluationdeterminanthsc mdr_n_second_integer_rankwitnessminornonzeroevaluationdeterminanthsc. ((exists mdr_gap_second_integer_rankwitnessminornonzeroevaluationdeterminanthsci. mdr_gap_second_integer_rankwitnessminornonzeroevaluationdeterminanthsci + S (mdr_i_second_integer_rankwitnessminornonzeroevaluationdeterminanthsc) = (mdr_i_second_integer_rankwitnessminornonzeroevaluationdeterminanth)) /\ ((exists mdr_z_second_integer_rankwitnessminornonzeroevaluationdeterminanthscr. ((exists mdr_a_second_integer_rankwitnessminornonzeroevaluationdeterminanthscrc mdr_b_second_integer_rankwitnessminornonzeroevaluationdeterminanthscrc mdr_c_second_integer_rankwitnessminornonzeroevaluationdeterminanthscrc mdr_e_second_integer_rankwitnessminornonzeroevaluationdeterminanthscrc mdr_f_second_integer_rankwitnessminornonzeroevaluationdeterminanthscrc. ((mdr_a_second_integer_rankwitnessminornonzeroevaluationdeterminanthscrc = ((mdr_q_second_integer_rankwitnessminornonzeroevaluationdeterminanths) + (mdr_up_second_integer_rankwitnessminornonzeroevaluationdeterminanthsc)) * S ((mdr_q_second_integer_rankwitnessminornonzeroevaluationdeterminanths) + (mdr_up_second_integer_rankwitnessminornonzeroevaluationdeterminanthsc)) + ((mdr_up_second_integer_rankwitnessminornonzeroevaluationdeterminanthsc) + (mdr_up_second_integer_rankwitnessminornonzeroevaluationdeterminanthsc))) /\ ((mdr_b_second_integer_rankwitnessminornonzeroevaluationdeterminanthscrc = ((mdr_us_second_integer_rankwitnessminornonzeroevaluationdeterminanthsc) + (mdr_un_second_integer_rankwitnessminornonzeroevaluationdeterminanthsc)) * S ((mdr_us_second_integer_rankwitnessminornonzeroevaluationdeterminanthsc) + (mdr_un_second_integer_rankwitnessminornonzeroevaluationdeterminanthsc)) + ((mdr_un_second_integer_rankwitnessminornonzeroevaluationdeterminanthsc) + (mdr_un_second_integer_rankwitnessminornonzeroevaluationdeterminanthsc))) /\ ((mdr_c_second_integer_rankwitnessminornonzeroevaluationdeterminanthscrc = ((mdr_a_second_integer_rankwitnessminornonzeroevaluationdeterminanthscrc) + (mdr_b_second_integer_rankwitnessminornonzeroevaluationdeterminanthscrc)) * S ((mdr_a_second_integer_rankwitnessminornonzeroevaluationdeterminanthscrc) + (mdr_b_second_integer_rankwitnessminornonzeroevaluationdeterminanthscrc)) + ((mdr_b_second_integer_rankwitnessminornonzeroevaluationdeterminanthscrc) + (mdr_b_second_integer_rankwitnessminornonzeroevaluationdeterminanthscrc))) /\ ((mdr_e_second_integer_rankwitnessminornonzeroevaluationdeterminanthscrc = ((mdr_p_second_integer_rankwitnessminornonzeroevaluationdeterminanthsc) + (mdr_n_second_integer_rankwitnessminornonzeroevaluationdeterminanthsc)) * S ((mdr_p_second_integer_rankwitnessminornonzeroevaluationdeterminanthsc) + (mdr_n_second_integer_rankwitnessminornonzeroevaluationdeterminanthsc)) + ((mdr_n_second_integer_rankwitnessminornonzeroevaluationdeterminanthsc) + (mdr_n_second_integer_rankwitnessminornonzeroevaluationdeterminanthsc))) /\ ((mdr_f_second_integer_rankwitnessminornonzeroevaluationdeterminanthscrc = ((mdr_ut_second_integer_rankwitnessminornonzeroevaluationdeterminanthsc) + (mdr_e_second_integer_rankwitnessminornonzeroevaluationdeterminanthscrc)) * S ((mdr_ut_second_integer_rankwitnessminornonzeroevaluationdeterminanthsc) + (mdr_e_second_integer_rankwitnessminornonzeroevaluationdeterminanthscrc)) + ((mdr_e_second_integer_rankwitnessminornonzeroevaluationdeterminanthscrc) + (mdr_e_second_integer_rankwitnessminornonzeroevaluationdeterminanthscrc))) /\ ((mdr_z_second_integer_rankwitnessminornonzeroevaluationdeterminanthscr) = ((mdr_c_second_integer_rankwitnessminornonzeroevaluationdeterminanthscrc) + (mdr_f_second_integer_rankwitnessminornonzeroevaluationdeterminanthscrc)) * S ((mdr_c_second_integer_rankwitnessminornonzeroevaluationdeterminanthscrc) + (mdr_f_second_integer_rankwitnessminornonzeroevaluationdeterminanthscrc)) + ((mdr_f_second_integer_rankwitnessminornonzeroevaluationdeterminanthscrc) + (mdr_f_second_integer_rankwitnessminornonzeroevaluationdeterminanthscrc))))))))) /\ (((exists ff_h_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthscrb. ff_h_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthscrb + S (mdr_z_second_integer_rankwitnessminornonzeroevaluationdeterminanthscr) = S ((S (mdr_i_second_integer_rankwitnessminornonzeroevaluationdeterminanthsc)) * mdr_c_second_integer_rankwitnessminornonzeroevaluationdeterminant)) /\ exists ff_q_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthscrb. mdr_b_second_integer_rankwitnessminornonzeroevaluationdeterminant = ff_q_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthscrb * S ((S (mdr_i_second_integer_rankwitnessminornonzeroevaluationdeterminanthsc)) * mdr_c_second_integer_rankwitnessminornonzeroevaluationdeterminant) + (mdr_z_second_integer_rankwitnessminornonzeroevaluationdeterminanthscr))))) /\ ((((forall ff_index_mdm_prefix_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthscm_positive. (exists ff_gap_mdm_lt_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthscm_positive_index_bound. ff_gap_mdm_lt_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthscm_positive_index_bound + S (ff_index_mdm_prefix_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthscm_positive) = ((mdr_q_second_integer_rankwitnessminornonzeroevaluationdeterminanths) * (mdr_q_second_integer_rankwitnessminornonzeroevaluationdeterminanths))) -> exists ff_row_mdm_prefix_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthscm_positive ff_column_mdm_prefix_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthscm_positive ff_value_mdm_prefix_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthscm_positive. (ff_index_mdm_prefix_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthscm_positive = (mdr_q_second_integer_rankwitnessminornonzeroevaluationdeterminanths) * ff_row_mdm_prefix_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthscm_positive + ff_column_mdm_prefix_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthscm_positive /\ ((exists ff_gap_mdm_lt_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthscm_positive_column_bound. ff_gap_mdm_lt_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthscm_positive_column_bound + S (ff_column_mdm_prefix_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthscm_positive) = (mdr_q_second_integer_rankwitnessminornonzeroevaluationdeterminanths)) /\ ((exists ff_row_mdm_cell_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthscm_positive_cell ff_column_mdm_cell_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthscm_positive_cell. (((((exists ff_gap_mdm_lt_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthscm_positive_cell_row_before. ff_gap_mdm_lt_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthscm_positive_cell_row_before + S (ff_row_mdm_prefix_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthscm_positive) = (0)) /\ ff_row_mdm_cell_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthscm_positive_cell = ff_row_mdm_prefix_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthscm_positive) \/ ((exists ff_gap_mdm_le_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthscm_positive_cell_row_after. ff_gap_mdm_le_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthscm_positive_cell_row_after + (0) = (ff_row_mdm_prefix_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthscm_positive)) /\ ff_row_mdm_cell_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthscm_positive_cell = S ff_row_mdm_prefix_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthscm_positive))) /\ (((((exists ff_gap_mdm_lt_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthscm_positive_cell_column_before. ff_gap_mdm_lt_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthscm_positive_cell_column_before + S (ff_column_mdm_prefix_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthscm_positive) = (mdr_j_second_integer_rankwitnessminornonzeroevaluationdeterminanthsc)) /\ ff_column_mdm_cell_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthscm_positive_cell = ff_column_mdm_prefix_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthscm_positive) \/ ((exists ff_gap_mdm_le_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthscm_positive_cell_column_after. ff_gap_mdm_le_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthscm_positive_cell_column_after + (mdr_j_second_integer_rankwitnessminornonzeroevaluationdeterminanthsc) = (ff_column_mdm_prefix_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthscm_positive)) /\ ff_column_mdm_cell_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthscm_positive_cell = S ff_column_mdm_prefix_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthscm_positive))) /\ (((exists ff_h_mdm_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthscm_positive_cell_source. ff_h_mdm_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthscm_positive_cell_source + S (ff_value_mdm_prefix_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthscm_positive) = S ((S ((ff_row_mdm_cell_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthscm_positive_cell) * (S (mdr_q_second_integer_rankwitnessminornonzeroevaluationdeterminanths)) + (ff_column_mdm_cell_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthscm_positive_cell))) * mdr_pc_second_integer_rankwitnessminornonzeroevaluationdeterminanth)) /\ exists ff_q_mdm_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthscm_positive_cell_source. mdr_pb_second_integer_rankwitnessminornonzeroevaluationdeterminanth = ff_q_mdm_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthscm_positive_cell_source * S ((S ((ff_row_mdm_cell_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthscm_positive_cell) * (S (mdr_q_second_integer_rankwitnessminornonzeroevaluationdeterminanths)) + (ff_column_mdm_cell_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthscm_positive_cell))) * mdr_pc_second_integer_rankwitnessminornonzeroevaluationdeterminanth) + (ff_value_mdm_prefix_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthscm_positive)))))) /\ (((exists ff_h_mdm_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthscm_positive_target. ff_h_mdm_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthscm_positive_target + S (ff_value_mdm_prefix_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthscm_positive) = S ((S (ff_index_mdm_prefix_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthscm_positive)) * mdr_us_second_integer_rankwitnessminornonzeroevaluationdeterminanthsc)) /\ exists ff_q_mdm_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthscm_positive_target. mdr_up_second_integer_rankwitnessminornonzeroevaluationdeterminanthsc = ff_q_mdm_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthscm_positive_target * S ((S (ff_index_mdm_prefix_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthscm_positive)) * mdr_us_second_integer_rankwitnessminornonzeroevaluationdeterminanthsc) + (ff_value_mdm_prefix_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthscm_positive))))))) /\ (forall ff_index_mdm_prefix_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthscm_negative. (exists ff_gap_mdm_lt_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthscm_negative_index_bound. ff_gap_mdm_lt_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthscm_negative_index_bound + S (ff_index_mdm_prefix_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthscm_negative) = ((mdr_q_second_integer_rankwitnessminornonzeroevaluationdeterminanths) * (mdr_q_second_integer_rankwitnessminornonzeroevaluationdeterminanths))) -> exists ff_row_mdm_prefix_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthscm_negative ff_column_mdm_prefix_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthscm_negative ff_value_mdm_prefix_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthscm_negative. (ff_index_mdm_prefix_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthscm_negative = (mdr_q_second_integer_rankwitnessminornonzeroevaluationdeterminanths) * ff_row_mdm_prefix_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthscm_negative + ff_column_mdm_prefix_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthscm_negative /\ ((exists ff_gap_mdm_lt_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthscm_negative_column_bound. ff_gap_mdm_lt_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthscm_negative_column_bound + S (ff_column_mdm_prefix_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthscm_negative) = (mdr_q_second_integer_rankwitnessminornonzeroevaluationdeterminanths)) /\ ((exists ff_row_mdm_cell_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthscm_negative_cell ff_column_mdm_cell_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthscm_negative_cell. (((((exists ff_gap_mdm_lt_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthscm_negative_cell_row_before. ff_gap_mdm_lt_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthscm_negative_cell_row_before + S (ff_row_mdm_prefix_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthscm_negative) = (0)) /\ ff_row_mdm_cell_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthscm_negative_cell = ff_row_mdm_prefix_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthscm_negative) \/ ((exists ff_gap_mdm_le_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthscm_negative_cell_row_after. ff_gap_mdm_le_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthscm_negative_cell_row_after + (0) = (ff_row_mdm_prefix_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthscm_negative)) /\ ff_row_mdm_cell_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthscm_negative_cell = S ff_row_mdm_prefix_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthscm_negative))) /\ (((((exists ff_gap_mdm_lt_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthscm_negative_cell_column_before. ff_gap_mdm_lt_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthscm_negative_cell_column_before + S (ff_column_mdm_prefix_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthscm_negative) = (mdr_j_second_integer_rankwitnessminornonzeroevaluationdeterminanthsc)) /\ ff_column_mdm_cell_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthscm_negative_cell = ff_column_mdm_prefix_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthscm_negative) \/ ((exists ff_gap_mdm_le_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthscm_negative_cell_column_after. ff_gap_mdm_le_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthscm_negative_cell_column_after + (mdr_j_second_integer_rankwitnessminornonzeroevaluationdeterminanthsc) = (ff_column_mdm_prefix_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthscm_negative)) /\ ff_column_mdm_cell_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthscm_negative_cell = S ff_column_mdm_prefix_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthscm_negative))) /\ (((exists ff_h_mdm_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthscm_negative_cell_source. ff_h_mdm_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthscm_negative_cell_source + S (ff_value_mdm_prefix_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthscm_negative) = S ((S ((ff_row_mdm_cell_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthscm_negative_cell) * (S (mdr_q_second_integer_rankwitnessminornonzeroevaluationdeterminanths)) + (ff_column_mdm_cell_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthscm_negative_cell))) * mdr_nc_second_integer_rankwitnessminornonzeroevaluationdeterminanth)) /\ exists ff_q_mdm_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthscm_negative_cell_source. mdr_nb_second_integer_rankwitnessminornonzeroevaluationdeterminanth = ff_q_mdm_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthscm_negative_cell_source * S ((S ((ff_row_mdm_cell_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthscm_negative_cell) * (S (mdr_q_second_integer_rankwitnessminornonzeroevaluationdeterminanths)) + (ff_column_mdm_cell_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthscm_negative_cell))) * mdr_nc_second_integer_rankwitnessminornonzeroevaluationdeterminanth) + (ff_value_mdm_prefix_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthscm_negative)))))) /\ (((exists ff_h_mdm_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthscm_negative_target. ff_h_mdm_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthscm_negative_target + S (ff_value_mdm_prefix_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthscm_negative) = S ((S (ff_index_mdm_prefix_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthscm_negative)) * mdr_ut_second_integer_rankwitnessminornonzeroevaluationdeterminanthsc)) /\ exists ff_q_mdm_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthscm_negative_target. mdr_un_second_integer_rankwitnessminornonzeroevaluationdeterminanthsc = ff_q_mdm_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthscm_negative_target * S ((S (ff_index_mdm_prefix_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthscm_negative)) * mdr_ut_second_integer_rankwitnessminornonzeroevaluationdeterminanthsc) + (ff_value_mdm_prefix_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthscm_negative))))))))) /\ ((((exists ff_h_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthscp. ff_h_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthscp + S (mdr_p_second_integer_rankwitnessminornonzeroevaluationdeterminanthsc) = S ((S (mdr_j_second_integer_rankwitnessminornonzeroevaluationdeterminanthsc)) * mdr_ec_second_integer_rankwitnessminornonzeroevaluationdeterminanths)) /\ exists ff_q_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthscp. mdr_eb_second_integer_rankwitnessminornonzeroevaluationdeterminanths = ff_q_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthscp * S ((S (mdr_j_second_integer_rankwitnessminornonzeroevaluationdeterminanthsc)) * mdr_ec_second_integer_rankwitnessminornonzeroevaluationdeterminanths) + (mdr_p_second_integer_rankwitnessminornonzeroevaluationdeterminanthsc))) /\ (((exists ff_h_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthscn. ff_h_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthscn + S (mdr_n_second_integer_rankwitnessminornonzeroevaluationdeterminanthsc) = S ((S (mdr_j_second_integer_rankwitnessminornonzeroevaluationdeterminanthsc)) * mdr_fc_second_integer_rankwitnessminornonzeroevaluationdeterminanths)) /\ exists ff_q_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthscn. mdr_fb_second_integer_rankwitnessminornonzeroevaluationdeterminanths = ff_q_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthscn * S ((S (mdr_j_second_integer_rankwitnessminornonzeroevaluationdeterminanthsc)) * mdr_fc_second_integer_rankwitnessminornonzeroevaluationdeterminanths) + (mdr_n_second_integer_rankwitnessminornonzeroevaluationdeterminanthsc)))))))) /\ (exists ff_ub_mce_fold_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf ff_uc_mce_fold_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf ff_vb_mce_fold_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf ff_vc_mce_fold_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf. ((forall ff_index_mce_alternating_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix. (exists ff_gap_mce_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix_index. ff_gap_mce_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix_index + S (ff_index_mce_alternating_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix) = (S (mdr_q_second_integer_rankwitnessminornonzeroevaluationdeterminanths))) -> exists ff_ap_mce_alternating_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix ff_an_mce_alternating_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix ff_bp_mce_alternating_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix ff_bn_mce_alternating_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix ff_p_mce_alternating_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix ff_n_mce_alternating_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix. ((((exists ff_h_mce_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix_ap. ff_h_mce_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix_ap + S (ff_ap_mce_alternating_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix) = S ((S (ff_index_mce_alternating_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix)) * mdr_pc_second_integer_rankwitnessminornonzeroevaluationdeterminanth)) /\ exists ff_q_mce_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix_ap. mdr_pb_second_integer_rankwitnessminornonzeroevaluationdeterminanth = ff_q_mce_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix_ap * S ((S (ff_index_mce_alternating_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix)) * mdr_pc_second_integer_rankwitnessminornonzeroevaluationdeterminanth) + (ff_ap_mce_alternating_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix))) /\ ((((exists ff_h_mce_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix_an. ff_h_mce_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix_an + S (ff_an_mce_alternating_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix) = S ((S (ff_index_mce_alternating_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix)) * mdr_nc_second_integer_rankwitnessminornonzeroevaluationdeterminanth)) /\ exists ff_q_mce_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix_an. mdr_nb_second_integer_rankwitnessminornonzeroevaluationdeterminanth = ff_q_mce_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix_an * S ((S (ff_index_mce_alternating_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix)) * mdr_nc_second_integer_rankwitnessminornonzeroevaluationdeterminanth) + (ff_an_mce_alternating_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix))) /\ ((((exists ff_h_mce_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix_bp. ff_h_mce_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix_bp + S (ff_bp_mce_alternating_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix) = S ((S (ff_index_mce_alternating_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix)) * mdr_ec_second_integer_rankwitnessminornonzeroevaluationdeterminanths)) /\ exists ff_q_mce_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix_bp. mdr_eb_second_integer_rankwitnessminornonzeroevaluationdeterminanths = ff_q_mce_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix_bp * S ((S (ff_index_mce_alternating_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix)) * mdr_ec_second_integer_rankwitnessminornonzeroevaluationdeterminanths) + (ff_bp_mce_alternating_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix))) /\ ((((exists ff_h_mce_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix_bn. ff_h_mce_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix_bn + S (ff_bn_mce_alternating_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix) = S ((S (ff_index_mce_alternating_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix)) * mdr_fc_second_integer_rankwitnessminornonzeroevaluationdeterminanths)) /\ exists ff_q_mce_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix_bn. mdr_fb_second_integer_rankwitnessminornonzeroevaluationdeterminanths = ff_q_mce_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix_bn * S ((S (ff_index_mce_alternating_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix)) * mdr_fc_second_integer_rankwitnessminornonzeroevaluationdeterminanths) + (ff_bn_mce_alternating_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix))) /\ ((((exists ff_h_mce_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix_positive. ff_h_mce_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix_positive + S (ff_p_mce_alternating_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix) = S ((S (ff_index_mce_alternating_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix)) * ff_uc_mce_fold_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf)) /\ exists ff_q_mce_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix_positive. ff_ub_mce_fold_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf = ff_q_mce_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix_positive * S ((S (ff_index_mce_alternating_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix)) * ff_uc_mce_fold_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf) + (ff_p_mce_alternating_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix))) /\ ((((exists ff_h_mce_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix_negative. ff_h_mce_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix_negative + S (ff_n_mce_alternating_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix) = S ((S (ff_index_mce_alternating_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix)) * ff_vc_mce_fold_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf)) /\ exists ff_q_mce_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix_negative. ff_vb_mce_fold_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf = ff_q_mce_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix_negative * S ((S (ff_index_mce_alternating_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix)) * ff_vc_mce_fold_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf) + (ff_n_mce_alternating_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix))) /\ (((exists ff_even_mce_term_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix_term. ff_index_mce_alternating_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix = 2 * ff_even_mce_term_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix_term) /\ (ff_p_mce_alternating_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix = (ff_ap_mce_alternating_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix) * (ff_bp_mce_alternating_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix) + (ff_an_mce_alternating_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix) * (ff_bn_mce_alternating_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix) /\ ff_n_mce_alternating_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix = (ff_ap_mce_alternating_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix) * (ff_bn_mce_alternating_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix) + (ff_an_mce_alternating_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix) * (ff_bp_mce_alternating_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix))) \/ ((exists ff_odd_mce_term_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix_term. ff_index_mce_alternating_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix = 2 * ff_odd_mce_term_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix_term + 1) /\ (ff_p_mce_alternating_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix = (ff_ap_mce_alternating_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix) * (ff_bn_mce_alternating_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix) + (ff_an_mce_alternating_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix) * (ff_bp_mce_alternating_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix) /\ ff_n_mce_alternating_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix = (ff_ap_mce_alternating_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix) * (ff_bp_mce_alternating_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix) + (ff_an_mce_alternating_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix) * (ff_bn_mce_alternating_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix))))))))))) /\ ((exists ff_u_mce_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_positive ff_v_mce_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_positive. ((((exists ff_h_mce_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_positive_start. ff_h_mce_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_positive_start + S (0) = S ((S (0)) * ff_v_mce_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_positive)) /\ exists ff_q_mce_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_positive_start. ff_u_mce_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_positive = ff_q_mce_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_positive_start * S ((S (0)) * ff_v_mce_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_positive) + (0))) /\ ((((exists ff_h_mce_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_positive_terminal. ff_h_mce_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_positive_terminal + S (mdr_p_second_integer_rankwitnessminornonzeroevaluationdeterminanth) = S ((S ((S (mdr_q_second_integer_rankwitnessminornonzeroevaluationdeterminanths)))) * ff_v_mce_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_positive)) /\ exists ff_q_mce_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_positive_terminal. ff_u_mce_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_positive = ff_q_mce_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_positive_terminal * S ((S ((S (mdr_q_second_integer_rankwitnessminornonzeroevaluationdeterminanths)))) * ff_v_mce_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_positive) + (mdr_p_second_integer_rankwitnessminornonzeroevaluationdeterminanth))) /\ forall ff_i_mce_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_positive. (exists ff_lt_mce_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_positive_bound. ff_lt_mce_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_positive_bound + S ff_i_mce_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_positive = (S (mdr_q_second_integer_rankwitnessminornonzeroevaluationdeterminanths))) -> exists ff_a_mce_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_positive ff_r_mce_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_positive ff_s_mce_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_positive. ((((exists ff_h_mce_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_positive_summand. ff_h_mce_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_positive_summand + S (ff_a_mce_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_positive) = S ((S (ff_i_mce_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_positive)) * ff_uc_mce_fold_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf)) /\ exists ff_q_mce_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_positive_summand. ff_ub_mce_fold_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf = ff_q_mce_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_positive_summand * S ((S (ff_i_mce_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_positive)) * ff_uc_mce_fold_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf) + (ff_a_mce_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_positive))) /\ ((((exists ff_h_mce_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_positive_partial. ff_h_mce_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_positive_partial + S (ff_r_mce_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_positive) = S ((S (ff_i_mce_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_positive)) * ff_v_mce_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_positive)) /\ exists ff_q_mce_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_positive_partial. ff_u_mce_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_positive = ff_q_mce_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_positive_partial * S ((S (ff_i_mce_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_positive)) * ff_v_mce_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_positive) + (ff_r_mce_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_positive))) /\ ((((exists ff_h_mce_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_positive_successor. ff_h_mce_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_positive_successor + S (ff_s_mce_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_positive) = S ((S (S ff_i_mce_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_positive)) * ff_v_mce_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_positive)) /\ exists ff_q_mce_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_positive_successor. ff_u_mce_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_positive = ff_q_mce_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_positive_successor * S ((S (S ff_i_mce_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_positive)) * ff_v_mce_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_positive) + (ff_s_mce_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_positive))) /\ ff_s_mce_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_positive = ff_r_mce_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_positive + ff_a_mce_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_positive)))))) /\ (exists ff_u_mce_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_negative ff_v_mce_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_negative. ((((exists ff_h_mce_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_negative_start. ff_h_mce_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_negative_start + S (0) = S ((S (0)) * ff_v_mce_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_negative)) /\ exists ff_q_mce_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_negative_start. ff_u_mce_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_negative = ff_q_mce_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_negative_start * S ((S (0)) * ff_v_mce_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_negative) + (0))) /\ ((((exists ff_h_mce_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_negative_terminal. ff_h_mce_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_negative_terminal + S (mdr_n_second_integer_rankwitnessminornonzeroevaluationdeterminanth) = S ((S ((S (mdr_q_second_integer_rankwitnessminornonzeroevaluationdeterminanths)))) * ff_v_mce_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_negative)) /\ exists ff_q_mce_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_negative_terminal. ff_u_mce_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_negative = ff_q_mce_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_negative_terminal * S ((S ((S (mdr_q_second_integer_rankwitnessminornonzeroevaluationdeterminanths)))) * ff_v_mce_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_negative) + (mdr_n_second_integer_rankwitnessminornonzeroevaluationdeterminanth))) /\ forall ff_i_mce_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_negative. (exists ff_lt_mce_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_negative_bound. ff_lt_mce_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_negative_bound + S ff_i_mce_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_negative = (S (mdr_q_second_integer_rankwitnessminornonzeroevaluationdeterminanths))) -> exists ff_a_mce_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_negative ff_r_mce_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_negative ff_s_mce_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_negative. ((((exists ff_h_mce_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_negative_summand. ff_h_mce_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_negative_summand + S (ff_a_mce_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_negative) = S ((S (ff_i_mce_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_negative)) * ff_vc_mce_fold_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf)) /\ exists ff_q_mce_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_negative_summand. ff_vb_mce_fold_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf = ff_q_mce_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_negative_summand * S ((S (ff_i_mce_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_negative)) * ff_vc_mce_fold_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf) + (ff_a_mce_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_negative))) /\ ((((exists ff_h_mce_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_negative_partial. ff_h_mce_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_negative_partial + S (ff_r_mce_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_negative) = S ((S (ff_i_mce_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_negative)) * ff_v_mce_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_negative)) /\ exists ff_q_mce_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_negative_partial. ff_u_mce_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_negative = ff_q_mce_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_negative_partial * S ((S (ff_i_mce_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_negative)) * ff_v_mce_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_negative) + (ff_r_mce_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_negative))) /\ ((((exists ff_h_mce_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_negative_successor. ff_h_mce_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_negative_successor + S (ff_s_mce_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_negative) = S ((S (S ff_i_mce_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_negative)) * ff_v_mce_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_negative)) /\ exists ff_q_mce_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_negative_successor. ff_u_mce_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_negative = ff_q_mce_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_negative_successor * S ((S (S ff_i_mce_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_negative)) * ff_v_mce_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_negative) + (ff_s_mce_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_negative))) /\ ff_s_mce_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_negative = ff_r_mce_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_negative + ff_a_mce_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_negative))))))))))))))) /\ ((exists mdr_gap_second_integer_rankwitnessminornonzeroevaluationdeterminanti. mdr_gap_second_integer_rankwitnessminornonzeroevaluationdeterminanti + S (mdr_i_second_integer_rankwitnessminornonzeroevaluationdeterminant) = (mdr_l_second_integer_rankwitnessminornonzeroevaluationdeterminant)) /\ (exists mdr_z_second_integer_rankwitnessminornonzeroevaluationdeterminantr. ((exists mdr_a_second_integer_rankwitnessminornonzeroevaluationdeterminantrc mdr_b_second_integer_rankwitnessminornonzeroevaluationdeterminantrc mdr_c_second_integer_rankwitnessminornonzeroevaluationdeterminantrc mdr_e_second_integer_rankwitnessminornonzeroevaluationdeterminantrc mdr_f_second_integer_rankwitnessminornonzeroevaluationdeterminantrc. ((mdr_a_second_integer_rankwitnessminornonzeroevaluationdeterminantrc = ((rank) + (mdr_ub_second_integer_rankwitnessminornonzeroevaluation)) * S ((rank) + (mdr_ub_second_integer_rankwitnessminornonzeroevaluation)) + ((mdr_ub_second_integer_rankwitnessminornonzeroevaluation) + (mdr_ub_second_integer_rankwitnessminornonzeroevaluation))) /\ ((mdr_b_second_integer_rankwitnessminornonzeroevaluationdeterminantrc = ((mdr_uc_second_integer_rankwitnessminornonzeroevaluation) + (mdr_vb_second_integer_rankwitnessminornonzeroevaluation)) * S ((mdr_uc_second_integer_rankwitnessminornonzeroevaluation) + (mdr_vb_second_integer_rankwitnessminornonzeroevaluation)) + ((mdr_vb_second_integer_rankwitnessminornonzeroevaluation) + (mdr_vb_second_integer_rankwitnessminornonzeroevaluation))) /\ ((mdr_c_second_integer_rankwitnessminornonzeroevaluationdeterminantrc = ((mdr_a_second_integer_rankwitnessminornonzeroevaluationdeterminantrc) + (mdr_b_second_integer_rankwitnessminornonzeroevaluationdeterminantrc)) * S ((mdr_a_second_integer_rankwitnessminornonzeroevaluationdeterminantrc) + (mdr_b_second_integer_rankwitnessminornonzeroevaluationdeterminantrc)) + ((mdr_b_second_integer_rankwitnessminornonzeroevaluationdeterminantrc) + (mdr_b_second_integer_rankwitnessminornonzeroevaluationdeterminantrc))) /\ ((mdr_e_second_integer_rankwitnessminornonzeroevaluationdeterminantrc = ((mdr_p_second_integer_rankwitnessminornonzero) + (mdr_n_second_integer_rankwitnessminornonzero)) * S ((mdr_p_second_integer_rankwitnessminornonzero) + (mdr_n_second_integer_rankwitnessminornonzero)) + ((mdr_n_second_integer_rankwitnessminornonzero) + (mdr_n_second_integer_rankwitnessminornonzero))) /\ ((mdr_f_second_integer_rankwitnessminornonzeroevaluationdeterminantrc = ((mdr_vc_second_integer_rankwitnessminornonzeroevaluation) + (mdr_e_second_integer_rankwitnessminornonzeroevaluationdeterminantrc)) * S ((mdr_vc_second_integer_rankwitnessminornonzeroevaluation) + (mdr_e_second_integer_rankwitnessminornonzeroevaluationdeterminantrc)) + ((mdr_e_second_integer_rankwitnessminornonzeroevaluationdeterminantrc) + (mdr_e_second_integer_rankwitnessminornonzeroevaluationdeterminantrc))) /\ ((mdr_z_second_integer_rankwitnessminornonzeroevaluationdeterminantr) = ((mdr_c_second_integer_rankwitnessminornonzeroevaluationdeterminantrc) + (mdr_f_second_integer_rankwitnessminornonzeroevaluationdeterminantrc)) * S ((mdr_c_second_integer_rankwitnessminornonzeroevaluationdeterminantrc) + (mdr_f_second_integer_rankwitnessminornonzeroevaluationdeterminantrc)) + ((mdr_f_second_integer_rankwitnessminornonzeroevaluationdeterminantrc) + (mdr_f_second_integer_rankwitnessminornonzeroevaluationdeterminantrc))))))))) /\ (((exists ff_h_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminantrb. ff_h_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminantrb + S (mdr_z_second_integer_rankwitnessminornonzeroevaluationdeterminantr) = S ((S (mdr_i_second_integer_rankwitnessminornonzeroevaluationdeterminant)) * mdr_c_second_integer_rankwitnessminornonzeroevaluationdeterminant)) /\ exists ff_q_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminantrb. mdr_b_second_integer_rankwitnessminornonzeroevaluationdeterminant = ff_q_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminantrb * S ((S (mdr_i_second_integer_rankwitnessminornonzeroevaluationdeterminant)) * mdr_c_second_integer_rankwitnessminornonzeroevaluationdeterminant) + (mdr_z_second_integer_rankwitnessminornonzeroevaluationdeterminantr)))))))))) /\ (~(mdr_p_second_integer_rankwitnessminornonzero = mdr_n_second_integer_rankwitnessminornonzero)))))))) /\ (forall mdr_q_second_integer_rank. (exists mdr_gap_second_integer_rankhigher. mdr_gap_second_integer_rankhigher + S (rank) = (mdr_q_second_integer_rank)) -> (forall mdr_rb_second_integer_rankzero mdr_rc_second_integer_rankzero mdr_cb_second_integer_rankzero mdr_cc_second_integer_rankzero mdr_p_second_integer_rankzero mdr_n_second_integer_rankzero. (((forall fom_index_mrf_second_integer_rankzerorowsbound. (exists fom_gap_mrf_second_integer_rankzerorowsbound_index_bound. fom_gap_mrf_second_integer_rankzerorowsbound_index_bound + S (fom_index_mrf_second_integer_rankzerorowsbound) = mdr_q_second_integer_rank) -> exists fom_value_mrf_second_integer_rankzerorowsbound. ((((exists fom_beta_height_mrf_second_integer_rankzerorowsbound_entry. fom_beta_height_mrf_second_integer_rankzerorowsbound_entry + S (fom_value_mrf_second_integer_rankzerorowsbound) = S ((S (fom_index_mrf_second_integer_rankzerorowsbound)) * mdr_rc_second_integer_rankzero)) /\ exists fom_beta_quotient_mrf_second_integer_rankzerorowsbound_entry. mdr_rb_second_integer_rankzero = fom_beta_quotient_mrf_second_integer_rankzerorowsbound_entry * S ((S (fom_index_mrf_second_integer_rankzerorowsbound)) * mdr_rc_second_integer_rankzero) + (fom_value_mrf_second_integer_rankzerorowsbound))) /\ (exists fom_gap_mrf_second_integer_rankzerorowsbound_value_bound. fom_gap_mrf_second_integer_rankzerorowsbound_value_bound + S (fom_value_mrf_second_integer_rankzerorowsbound) = r))) /\ (forall mdr_i_second_integer_rankzerorowsdistinct mdr_j_second_integer_rankzerorowsdistinct mdr_a_second_integer_rankzerorowsdistinct. (exists mdr_gap_second_integer_rankzerorowsdistincti. mdr_gap_second_integer_rankzerorowsdistincti + S (mdr_i_second_integer_rankzerorowsdistinct) = (mdr_q_second_integer_rank)) -> (exists mdr_gap_second_integer_rankzerorowsdistinctj. mdr_gap_second_integer_rankzerorowsdistinctj + S (mdr_j_second_integer_rankzerorowsdistinct) = (mdr_q_second_integer_rank)) -> (((exists ff_h_mdr_second_integer_rankzerorowsdistinctfirst. ff_h_mdr_second_integer_rankzerorowsdistinctfirst + S (mdr_a_second_integer_rankzerorowsdistinct) = S ((S (mdr_i_second_integer_rankzerorowsdistinct)) * mdr_rc_second_integer_rankzero)) /\ exists ff_q_mdr_second_integer_rankzerorowsdistinctfirst. mdr_rb_second_integer_rankzero = ff_q_mdr_second_integer_rankzerorowsdistinctfirst * S ((S (mdr_i_second_integer_rankzerorowsdistinct)) * mdr_rc_second_integer_rankzero) + (mdr_a_second_integer_rankzerorowsdistinct))) -> (((exists ff_h_mdr_second_integer_rankzerorowsdistinctsecond. ff_h_mdr_second_integer_rankzerorowsdistinctsecond + S (mdr_a_second_integer_rankzerorowsdistinct) = S ((S (mdr_j_second_integer_rankzerorowsdistinct)) * mdr_rc_second_integer_rankzero)) /\ exists ff_q_mdr_second_integer_rankzerorowsdistinctsecond. mdr_rb_second_integer_rankzero = ff_q_mdr_second_integer_rankzerorowsdistinctsecond * S ((S (mdr_j_second_integer_rankzerorowsdistinct)) * mdr_rc_second_integer_rankzero) + (mdr_a_second_integer_rankzerorowsdistinct))) -> mdr_i_second_integer_rankzerorowsdistinct = mdr_j_second_integer_rankzerorowsdistinct))) -> (((forall fom_index_mrf_second_integer_rankzerocolumnsbound. (exists fom_gap_mrf_second_integer_rankzerocolumnsbound_index_bound. fom_gap_mrf_second_integer_rankzerocolumnsbound_index_bound + S (fom_index_mrf_second_integer_rankzerocolumnsbound) = mdr_q_second_integer_rank) -> exists fom_value_mrf_second_integer_rankzerocolumnsbound. ((((exists fom_beta_height_mrf_second_integer_rankzerocolumnsbound_entry. fom_beta_height_mrf_second_integer_rankzerocolumnsbound_entry + S (fom_value_mrf_second_integer_rankzerocolumnsbound) = S ((S (fom_index_mrf_second_integer_rankzerocolumnsbound)) * mdr_cc_second_integer_rankzero)) /\ exists fom_beta_quotient_mrf_second_integer_rankzerocolumnsbound_entry. mdr_cb_second_integer_rankzero = fom_beta_quotient_mrf_second_integer_rankzerocolumnsbound_entry * S ((S (fom_index_mrf_second_integer_rankzerocolumnsbound)) * mdr_cc_second_integer_rankzero) + (fom_value_mrf_second_integer_rankzerocolumnsbound))) /\ (exists fom_gap_mrf_second_integer_rankzerocolumnsbound_value_bound. fom_gap_mrf_second_integer_rankzerocolumnsbound_value_bound + S (fom_value_mrf_second_integer_rankzerocolumnsbound) = w))) /\ (forall mdr_i_second_integer_rankzerocolumnsdistinct mdr_j_second_integer_rankzerocolumnsdistinct mdr_a_second_integer_rankzerocolumnsdistinct. (exists mdr_gap_second_integer_rankzerocolumnsdistincti. mdr_gap_second_integer_rankzerocolumnsdistincti + S (mdr_i_second_integer_rankzerocolumnsdistinct) = (mdr_q_second_integer_rank)) -> (exists mdr_gap_second_integer_rankzerocolumnsdistinctj. mdr_gap_second_integer_rankzerocolumnsdistinctj + S (mdr_j_second_integer_rankzerocolumnsdistinct) = (mdr_q_second_integer_rank)) -> (((exists ff_h_mdr_second_integer_rankzerocolumnsdistinctfirst. ff_h_mdr_second_integer_rankzerocolumnsdistinctfirst + S (mdr_a_second_integer_rankzerocolumnsdistinct) = S ((S (mdr_i_second_integer_rankzerocolumnsdistinct)) * mdr_cc_second_integer_rankzero)) /\ exists ff_q_mdr_second_integer_rankzerocolumnsdistinctfirst. mdr_cb_second_integer_rankzero = ff_q_mdr_second_integer_rankzerocolumnsdistinctfirst * S ((S (mdr_i_second_integer_rankzerocolumnsdistinct)) * mdr_cc_second_integer_rankzero) + (mdr_a_second_integer_rankzerocolumnsdistinct))) -> (((exists ff_h_mdr_second_integer_rankzerocolumnsdistinctsecond. ff_h_mdr_second_integer_rankzerocolumnsdistinctsecond + S (mdr_a_second_integer_rankzerocolumnsdistinct) = S ((S (mdr_j_second_integer_rankzerocolumnsdistinct)) * mdr_cc_second_integer_rankzero)) /\ exists ff_q_mdr_second_integer_rankzerocolumnsdistinctsecond. mdr_cb_second_integer_rankzero = ff_q_mdr_second_integer_rankzerocolumnsdistinctsecond * S ((S (mdr_j_second_integer_rankzerocolumnsdistinct)) * mdr_cc_second_integer_rankzero) + (mdr_a_second_integer_rankzerocolumnsdistinct))) -> mdr_i_second_integer_rankzerocolumnsdistinct = mdr_j_second_integer_rankzerocolumnsdistinct))) -> (exists mdr_ub_second_integer_rankzeroevaluation mdr_uc_second_integer_rankzeroevaluation mdr_vb_second_integer_rankzeroevaluation mdr_vc_second_integer_rankzeroevaluation. ((((forall mdr_i_second_integer_rankzeroevaluationmatrixpositive. (exists mdr_gap_second_integer_rankzeroevaluationmatrixpositivebound. mdr_gap_second_integer_rankzeroevaluationmatrixpositivebound + S (mdr_i_second_integer_rankzeroevaluationmatrixpositive) = ((mdr_q_second_integer_rank) * (mdr_q_second_integer_rank))) -> exists mdr_a_second_integer_rankzeroevaluationmatrixpositive. (((exists mdr_r_second_integer_rankzeroevaluationmatrixpositivepoint mdr_s_second_integer_rankzeroevaluationmatrixpositivepoint mdr_u_second_integer_rankzeroevaluationmatrixpositivepoint mdr_v_second_integer_rankzeroevaluationmatrixpositivepoint. ((mdr_i_second_integer_rankzeroevaluationmatrixpositive = (mdr_q_second_integer_rank) * mdr_r_second_integer_rankzeroevaluationmatrixpositivepoint + mdr_s_second_integer_rankzeroevaluationmatrixpositivepoint) /\ ((exists mdr_gap_second_integer_rankzeroevaluationmatrixpositivepointcolumn. mdr_gap_second_integer_rankzeroevaluationmatrixpositivepointcolumn + S (mdr_s_second_integer_rankzeroevaluationmatrixpositivepoint) = (mdr_q_second_integer_rank)) /\ ((((exists ff_h_mdr_second_integer_rankzeroevaluationmatrixpositivepointrow_index. ff_h_mdr_second_integer_rankzeroevaluationmatrixpositivepointrow_index + S (mdr_u_second_integer_rankzeroevaluationmatrixpositivepoint) = S ((S (mdr_r_second_integer_rankzeroevaluationmatrixpositivepoint)) * mdr_rc_second_integer_rankzero)) /\ exists ff_q_mdr_second_integer_rankzeroevaluationmatrixpositivepointrow_index. mdr_rb_second_integer_rankzero = ff_q_mdr_second_integer_rankzeroevaluationmatrixpositivepointrow_index * S ((S (mdr_r_second_integer_rankzeroevaluationmatrixpositivepoint)) * mdr_rc_second_integer_rankzero) + (mdr_u_second_integer_rankzeroevaluationmatrixpositivepoint))) /\ ((((exists ff_h_mdr_second_integer_rankzeroevaluationmatrixpositivepointcolumn_index. ff_h_mdr_second_integer_rankzeroevaluationmatrixpositivepointcolumn_index + S (mdr_v_second_integer_rankzeroevaluationmatrixpositivepoint) = S ((S (mdr_s_second_integer_rankzeroevaluationmatrixpositivepoint)) * mdr_cc_second_integer_rankzero)) /\ exists ff_q_mdr_second_integer_rankzeroevaluationmatrixpositivepointcolumn_index. mdr_cb_second_integer_rankzero = ff_q_mdr_second_integer_rankzeroevaluationmatrixpositivepointcolumn_index * S ((S (mdr_s_second_integer_rankzeroevaluationmatrixpositivepoint)) * mdr_cc_second_integer_rankzero) + (mdr_v_second_integer_rankzeroevaluationmatrixpositivepoint))) /\ (((exists ff_h_mdr_second_integer_rankzeroevaluationmatrixpositivepointsource. ff_h_mdr_second_integer_rankzeroevaluationmatrixpositivepointsource + S (mdr_a_second_integer_rankzeroevaluationmatrixpositive) = S ((S ((mdr_u_second_integer_rankzeroevaluationmatrixpositivepoint) * (w) + (mdr_v_second_integer_rankzeroevaluationmatrixpositivepoint))) * ec)) /\ exists ff_q_mdr_second_integer_rankzeroevaluationmatrixpositivepointsource. eb = ff_q_mdr_second_integer_rankzeroevaluationmatrixpositivepointsource * S ((S ((mdr_u_second_integer_rankzeroevaluationmatrixpositivepoint) * (w) + (mdr_v_second_integer_rankzeroevaluationmatrixpositivepoint))) * ec) + (mdr_a_second_integer_rankzeroevaluationmatrixpositive)))))))) /\ (((exists ff_h_mdr_second_integer_rankzeroevaluationmatrixpositiveoutput. ff_h_mdr_second_integer_rankzeroevaluationmatrixpositiveoutput + S (mdr_a_second_integer_rankzeroevaluationmatrixpositive) = S ((S (mdr_i_second_integer_rankzeroevaluationmatrixpositive)) * mdr_uc_second_integer_rankzeroevaluation)) /\ exists ff_q_mdr_second_integer_rankzeroevaluationmatrixpositiveoutput. mdr_ub_second_integer_rankzeroevaluation = ff_q_mdr_second_integer_rankzeroevaluationmatrixpositiveoutput * S ((S (mdr_i_second_integer_rankzeroevaluationmatrixpositive)) * mdr_uc_second_integer_rankzeroevaluation) + (mdr_a_second_integer_rankzeroevaluationmatrixpositive)))))) /\ (forall mdr_i_second_integer_rankzeroevaluationmatrixnegative. (exists mdr_gap_second_integer_rankzeroevaluationmatrixnegativebound. mdr_gap_second_integer_rankzeroevaluationmatrixnegativebound + S (mdr_i_second_integer_rankzeroevaluationmatrixnegative) = ((mdr_q_second_integer_rank) * (mdr_q_second_integer_rank))) -> exists mdr_a_second_integer_rankzeroevaluationmatrixnegative. (((exists mdr_r_second_integer_rankzeroevaluationmatrixnegativepoint mdr_s_second_integer_rankzeroevaluationmatrixnegativepoint mdr_u_second_integer_rankzeroevaluationmatrixnegativepoint mdr_v_second_integer_rankzeroevaluationmatrixnegativepoint. ((mdr_i_second_integer_rankzeroevaluationmatrixnegative = (mdr_q_second_integer_rank) * mdr_r_second_integer_rankzeroevaluationmatrixnegativepoint + mdr_s_second_integer_rankzeroevaluationmatrixnegativepoint) /\ ((exists mdr_gap_second_integer_rankzeroevaluationmatrixnegativepointcolumn. mdr_gap_second_integer_rankzeroevaluationmatrixnegativepointcolumn + S (mdr_s_second_integer_rankzeroevaluationmatrixnegativepoint) = (mdr_q_second_integer_rank)) /\ ((((exists ff_h_mdr_second_integer_rankzeroevaluationmatrixnegativepointrow_index. ff_h_mdr_second_integer_rankzeroevaluationmatrixnegativepointrow_index + S (mdr_u_second_integer_rankzeroevaluationmatrixnegativepoint) = S ((S (mdr_r_second_integer_rankzeroevaluationmatrixnegativepoint)) * mdr_rc_second_integer_rankzero)) /\ exists ff_q_mdr_second_integer_rankzeroevaluationmatrixnegativepointrow_index. mdr_rb_second_integer_rankzero = ff_q_mdr_second_integer_rankzeroevaluationmatrixnegativepointrow_index * S ((S (mdr_r_second_integer_rankzeroevaluationmatrixnegativepoint)) * mdr_rc_second_integer_rankzero) + (mdr_u_second_integer_rankzeroevaluationmatrixnegativepoint))) /\ ((((exists ff_h_mdr_second_integer_rankzeroevaluationmatrixnegativepointcolumn_index. ff_h_mdr_second_integer_rankzeroevaluationmatrixnegativepointcolumn_index + S (mdr_v_second_integer_rankzeroevaluationmatrixnegativepoint) = S ((S (mdr_s_second_integer_rankzeroevaluationmatrixnegativepoint)) * mdr_cc_second_integer_rankzero)) /\ exists ff_q_mdr_second_integer_rankzeroevaluationmatrixnegativepointcolumn_index. mdr_cb_second_integer_rankzero = ff_q_mdr_second_integer_rankzeroevaluationmatrixnegativepointcolumn_index * S ((S (mdr_s_second_integer_rankzeroevaluationmatrixnegativepoint)) * mdr_cc_second_integer_rankzero) + (mdr_v_second_integer_rankzeroevaluationmatrixnegativepoint))) /\ (((exists ff_h_mdr_second_integer_rankzeroevaluationmatrixnegativepointsource. ff_h_mdr_second_integer_rankzeroevaluationmatrixnegativepointsource + S (mdr_a_second_integer_rankzeroevaluationmatrixnegative) = S ((S ((mdr_u_second_integer_rankzeroevaluationmatrixnegativepoint) * (w) + (mdr_v_second_integer_rankzeroevaluationmatrixnegativepoint))) * fc)) /\ exists ff_q_mdr_second_integer_rankzeroevaluationmatrixnegativepointsource. fb = ff_q_mdr_second_integer_rankzeroevaluationmatrixnegativepointsource * S ((S ((mdr_u_second_integer_rankzeroevaluationmatrixnegativepoint) * (w) + (mdr_v_second_integer_rankzeroevaluationmatrixnegativepoint))) * fc) + (mdr_a_second_integer_rankzeroevaluationmatrixnegative)))))))) /\ (((exists ff_h_mdr_second_integer_rankzeroevaluationmatrixnegativeoutput. ff_h_mdr_second_integer_rankzeroevaluationmatrixnegativeoutput + S (mdr_a_second_integer_rankzeroevaluationmatrixnegative) = S ((S (mdr_i_second_integer_rankzeroevaluationmatrixnegative)) * mdr_vc_second_integer_rankzeroevaluation)) /\ exists ff_q_mdr_second_integer_rankzeroevaluationmatrixnegativeoutput. mdr_vb_second_integer_rankzeroevaluation = ff_q_mdr_second_integer_rankzeroevaluationmatrixnegativeoutput * S ((S (mdr_i_second_integer_rankzeroevaluationmatrixnegative)) * mdr_vc_second_integer_rankzeroevaluation) + (mdr_a_second_integer_rankzeroevaluationmatrixnegative)))))))) /\ (exists mdr_b_second_integer_rankzeroevaluationdeterminant mdr_c_second_integer_rankzeroevaluationdeterminant mdr_l_second_integer_rankzeroevaluationdeterminant mdr_i_second_integer_rankzeroevaluationdeterminant. ((forall mdr_i_second_integer_rankzeroevaluationdeterminanth. (exists mdr_gap_second_integer_rankzeroevaluationdeterminanthi. mdr_gap_second_integer_rankzeroevaluationdeterminanthi + S (mdr_i_second_integer_rankzeroevaluationdeterminanth) = (mdr_l_second_integer_rankzeroevaluationdeterminant)) -> exists mdr_d_second_integer_rankzeroevaluationdeterminanth mdr_pb_second_integer_rankzeroevaluationdeterminanth mdr_pc_second_integer_rankzeroevaluationdeterminanth mdr_nb_second_integer_rankzeroevaluationdeterminanth mdr_nc_second_integer_rankzeroevaluationdeterminanth mdr_p_second_integer_rankzeroevaluationdeterminanth mdr_n_second_integer_rankzeroevaluationdeterminanth. ((exists mdr_z_second_integer_rankzeroevaluationdeterminanthr. ((exists mdr_a_second_integer_rankzeroevaluationdeterminanthrc mdr_b_second_integer_rankzeroevaluationdeterminanthrc mdr_c_second_integer_rankzeroevaluationdeterminanthrc mdr_e_second_integer_rankzeroevaluationdeterminanthrc mdr_f_second_integer_rankzeroevaluationdeterminanthrc. ((mdr_a_second_integer_rankzeroevaluationdeterminanthrc = ((mdr_d_second_integer_rankzeroevaluationdeterminanth) + (mdr_pb_second_integer_rankzeroevaluationdeterminanth)) * S ((mdr_d_second_integer_rankzeroevaluationdeterminanth) + (mdr_pb_second_integer_rankzeroevaluationdeterminanth)) + ((mdr_pb_second_integer_rankzeroevaluationdeterminanth) + (mdr_pb_second_integer_rankzeroevaluationdeterminanth))) /\ ((mdr_b_second_integer_rankzeroevaluationdeterminanthrc = ((mdr_pc_second_integer_rankzeroevaluationdeterminanth) + (mdr_nb_second_integer_rankzeroevaluationdeterminanth)) * S ((mdr_pc_second_integer_rankzeroevaluationdeterminanth) + (mdr_nb_second_integer_rankzeroevaluationdeterminanth)) + ((mdr_nb_second_integer_rankzeroevaluationdeterminanth) + (mdr_nb_second_integer_rankzeroevaluationdeterminanth))) /\ ((mdr_c_second_integer_rankzeroevaluationdeterminanthrc = ((mdr_a_second_integer_rankzeroevaluationdeterminanthrc) + (mdr_b_second_integer_rankzeroevaluationdeterminanthrc)) * S ((mdr_a_second_integer_rankzeroevaluationdeterminanthrc) + (mdr_b_second_integer_rankzeroevaluationdeterminanthrc)) + ((mdr_b_second_integer_rankzeroevaluationdeterminanthrc) + (mdr_b_second_integer_rankzeroevaluationdeterminanthrc))) /\ ((mdr_e_second_integer_rankzeroevaluationdeterminanthrc = ((mdr_p_second_integer_rankzeroevaluationdeterminanth) + (mdr_n_second_integer_rankzeroevaluationdeterminanth)) * S ((mdr_p_second_integer_rankzeroevaluationdeterminanth) + (mdr_n_second_integer_rankzeroevaluationdeterminanth)) + ((mdr_n_second_integer_rankzeroevaluationdeterminanth) + (mdr_n_second_integer_rankzeroevaluationdeterminanth))) /\ ((mdr_f_second_integer_rankzeroevaluationdeterminanthrc = ((mdr_nc_second_integer_rankzeroevaluationdeterminanth) + (mdr_e_second_integer_rankzeroevaluationdeterminanthrc)) * S ((mdr_nc_second_integer_rankzeroevaluationdeterminanth) + (mdr_e_second_integer_rankzeroevaluationdeterminanthrc)) + ((mdr_e_second_integer_rankzeroevaluationdeterminanthrc) + (mdr_e_second_integer_rankzeroevaluationdeterminanthrc))) /\ ((mdr_z_second_integer_rankzeroevaluationdeterminanthr) = ((mdr_c_second_integer_rankzeroevaluationdeterminanthrc) + (mdr_f_second_integer_rankzeroevaluationdeterminanthrc)) * S ((mdr_c_second_integer_rankzeroevaluationdeterminanthrc) + (mdr_f_second_integer_rankzeroevaluationdeterminanthrc)) + ((mdr_f_second_integer_rankzeroevaluationdeterminanthrc) + (mdr_f_second_integer_rankzeroevaluationdeterminanthrc))))))))) /\ (((exists ff_h_mdr_second_integer_rankzeroevaluationdeterminanthrb. ff_h_mdr_second_integer_rankzeroevaluationdeterminanthrb + S (mdr_z_second_integer_rankzeroevaluationdeterminanthr) = S ((S (mdr_i_second_integer_rankzeroevaluationdeterminanth)) * mdr_c_second_integer_rankzeroevaluationdeterminant)) /\ exists ff_q_mdr_second_integer_rankzeroevaluationdeterminanthrb. mdr_b_second_integer_rankzeroevaluationdeterminant = ff_q_mdr_second_integer_rankzeroevaluationdeterminanthrb * S ((S (mdr_i_second_integer_rankzeroevaluationdeterminanth)) * mdr_c_second_integer_rankzeroevaluationdeterminant) + (mdr_z_second_integer_rankzeroevaluationdeterminanthr))))) /\ (((((mdr_d_second_integer_rankzeroevaluationdeterminanth) = 0) /\ (((mdr_p_second_integer_rankzeroevaluationdeterminanth) = 1) /\ ((mdr_n_second_integer_rankzeroevaluationdeterminanth) = 0))) \/ exists mdr_q_second_integer_rankzeroevaluationdeterminanths mdr_eb_second_integer_rankzeroevaluationdeterminanths mdr_ec_second_integer_rankzeroevaluationdeterminanths mdr_fb_second_integer_rankzeroevaluationdeterminanths mdr_fc_second_integer_rankzeroevaluationdeterminanths. (((mdr_d_second_integer_rankzeroevaluationdeterminanth) = S (mdr_q_second_integer_rankzeroevaluationdeterminanths)) /\ ((forall mdr_j_second_integer_rankzeroevaluationdeterminanthsc. (exists mdr_gap_second_integer_rankzeroevaluationdeterminanthscj. mdr_gap_second_integer_rankzeroevaluationdeterminanthscj + S (mdr_j_second_integer_rankzeroevaluationdeterminanthsc) = (S (mdr_q_second_integer_rankzeroevaluationdeterminanths))) -> exists mdr_i_second_integer_rankzeroevaluationdeterminanthsc mdr_up_second_integer_rankzeroevaluationdeterminanthsc mdr_us_second_integer_rankzeroevaluationdeterminanthsc mdr_un_second_integer_rankzeroevaluationdeterminanthsc mdr_ut_second_integer_rankzeroevaluationdeterminanthsc mdr_p_second_integer_rankzeroevaluationdeterminanthsc mdr_n_second_integer_rankzeroevaluationdeterminanthsc. ((exists mdr_gap_second_integer_rankzeroevaluationdeterminanthsci. mdr_gap_second_integer_rankzeroevaluationdeterminanthsci + S (mdr_i_second_integer_rankzeroevaluationdeterminanthsc) = (mdr_i_second_integer_rankzeroevaluationdeterminanth)) /\ ((exists mdr_z_second_integer_rankzeroevaluationdeterminanthscr. ((exists mdr_a_second_integer_rankzeroevaluationdeterminanthscrc mdr_b_second_integer_rankzeroevaluationdeterminanthscrc mdr_c_second_integer_rankzeroevaluationdeterminanthscrc mdr_e_second_integer_rankzeroevaluationdeterminanthscrc mdr_f_second_integer_rankzeroevaluationdeterminanthscrc. ((mdr_a_second_integer_rankzeroevaluationdeterminanthscrc = ((mdr_q_second_integer_rankzeroevaluationdeterminanths) + (mdr_up_second_integer_rankzeroevaluationdeterminanthsc)) * S ((mdr_q_second_integer_rankzeroevaluationdeterminanths) + (mdr_up_second_integer_rankzeroevaluationdeterminanthsc)) + ((mdr_up_second_integer_rankzeroevaluationdeterminanthsc) + (mdr_up_second_integer_rankzeroevaluationdeterminanthsc))) /\ ((mdr_b_second_integer_rankzeroevaluationdeterminanthscrc = ((mdr_us_second_integer_rankzeroevaluationdeterminanthsc) + (mdr_un_second_integer_rankzeroevaluationdeterminanthsc)) * S ((mdr_us_second_integer_rankzeroevaluationdeterminanthsc) + (mdr_un_second_integer_rankzeroevaluationdeterminanthsc)) + ((mdr_un_second_integer_rankzeroevaluationdeterminanthsc) + (mdr_un_second_integer_rankzeroevaluationdeterminanthsc))) /\ ((mdr_c_second_integer_rankzeroevaluationdeterminanthscrc = ((mdr_a_second_integer_rankzeroevaluationdeterminanthscrc) + (mdr_b_second_integer_rankzeroevaluationdeterminanthscrc)) * S ((mdr_a_second_integer_rankzeroevaluationdeterminanthscrc) + (mdr_b_second_integer_rankzeroevaluationdeterminanthscrc)) + ((mdr_b_second_integer_rankzeroevaluationdeterminanthscrc) + (mdr_b_second_integer_rankzeroevaluationdeterminanthscrc))) /\ ((mdr_e_second_integer_rankzeroevaluationdeterminanthscrc = ((mdr_p_second_integer_rankzeroevaluationdeterminanthsc) + (mdr_n_second_integer_rankzeroevaluationdeterminanthsc)) * S ((mdr_p_second_integer_rankzeroevaluationdeterminanthsc) + (mdr_n_second_integer_rankzeroevaluationdeterminanthsc)) + ((mdr_n_second_integer_rankzeroevaluationdeterminanthsc) + (mdr_n_second_integer_rankzeroevaluationdeterminanthsc))) /\ ((mdr_f_second_integer_rankzeroevaluationdeterminanthscrc = ((mdr_ut_second_integer_rankzeroevaluationdeterminanthsc) + (mdr_e_second_integer_rankzeroevaluationdeterminanthscrc)) * S ((mdr_ut_second_integer_rankzeroevaluationdeterminanthsc) + (mdr_e_second_integer_rankzeroevaluationdeterminanthscrc)) + ((mdr_e_second_integer_rankzeroevaluationdeterminanthscrc) + (mdr_e_second_integer_rankzeroevaluationdeterminanthscrc))) /\ ((mdr_z_second_integer_rankzeroevaluationdeterminanthscr) = ((mdr_c_second_integer_rankzeroevaluationdeterminanthscrc) + (mdr_f_second_integer_rankzeroevaluationdeterminanthscrc)) * S ((mdr_c_second_integer_rankzeroevaluationdeterminanthscrc) + (mdr_f_second_integer_rankzeroevaluationdeterminanthscrc)) + ((mdr_f_second_integer_rankzeroevaluationdeterminanthscrc) + (mdr_f_second_integer_rankzeroevaluationdeterminanthscrc))))))))) /\ (((exists ff_h_mdr_second_integer_rankzeroevaluationdeterminanthscrb. ff_h_mdr_second_integer_rankzeroevaluationdeterminanthscrb + S (mdr_z_second_integer_rankzeroevaluationdeterminanthscr) = S ((S (mdr_i_second_integer_rankzeroevaluationdeterminanthsc)) * mdr_c_second_integer_rankzeroevaluationdeterminant)) /\ exists ff_q_mdr_second_integer_rankzeroevaluationdeterminanthscrb. mdr_b_second_integer_rankzeroevaluationdeterminant = ff_q_mdr_second_integer_rankzeroevaluationdeterminanthscrb * S ((S (mdr_i_second_integer_rankzeroevaluationdeterminanthsc)) * mdr_c_second_integer_rankzeroevaluationdeterminant) + (mdr_z_second_integer_rankzeroevaluationdeterminanthscr))))) /\ ((((forall ff_index_mdm_prefix_mdr_second_integer_rankzeroevaluationdeterminanthscm_positive. (exists ff_gap_mdm_lt_mdr_second_integer_rankzeroevaluationdeterminanthscm_positive_index_bound. ff_gap_mdm_lt_mdr_second_integer_rankzeroevaluationdeterminanthscm_positive_index_bound + S (ff_index_mdm_prefix_mdr_second_integer_rankzeroevaluationdeterminanthscm_positive) = ((mdr_q_second_integer_rankzeroevaluationdeterminanths) * (mdr_q_second_integer_rankzeroevaluationdeterminanths))) -> exists ff_row_mdm_prefix_mdr_second_integer_rankzeroevaluationdeterminanthscm_positive ff_column_mdm_prefix_mdr_second_integer_rankzeroevaluationdeterminanthscm_positive ff_value_mdm_prefix_mdr_second_integer_rankzeroevaluationdeterminanthscm_positive. (ff_index_mdm_prefix_mdr_second_integer_rankzeroevaluationdeterminanthscm_positive = (mdr_q_second_integer_rankzeroevaluationdeterminanths) * ff_row_mdm_prefix_mdr_second_integer_rankzeroevaluationdeterminanthscm_positive + ff_column_mdm_prefix_mdr_second_integer_rankzeroevaluationdeterminanthscm_positive /\ ((exists ff_gap_mdm_lt_mdr_second_integer_rankzeroevaluationdeterminanthscm_positive_column_bound. ff_gap_mdm_lt_mdr_second_integer_rankzeroevaluationdeterminanthscm_positive_column_bound + S (ff_column_mdm_prefix_mdr_second_integer_rankzeroevaluationdeterminanthscm_positive) = (mdr_q_second_integer_rankzeroevaluationdeterminanths)) /\ ((exists ff_row_mdm_cell_mdr_second_integer_rankzeroevaluationdeterminanthscm_positive_cell ff_column_mdm_cell_mdr_second_integer_rankzeroevaluationdeterminanthscm_positive_cell. (((((exists ff_gap_mdm_lt_mdr_second_integer_rankzeroevaluationdeterminanthscm_positive_cell_row_before. ff_gap_mdm_lt_mdr_second_integer_rankzeroevaluationdeterminanthscm_positive_cell_row_before + S (ff_row_mdm_prefix_mdr_second_integer_rankzeroevaluationdeterminanthscm_positive) = (0)) /\ ff_row_mdm_cell_mdr_second_integer_rankzeroevaluationdeterminanthscm_positive_cell = ff_row_mdm_prefix_mdr_second_integer_rankzeroevaluationdeterminanthscm_positive) \/ ((exists ff_gap_mdm_le_mdr_second_integer_rankzeroevaluationdeterminanthscm_positive_cell_row_after. ff_gap_mdm_le_mdr_second_integer_rankzeroevaluationdeterminanthscm_positive_cell_row_after + (0) = (ff_row_mdm_prefix_mdr_second_integer_rankzeroevaluationdeterminanthscm_positive)) /\ ff_row_mdm_cell_mdr_second_integer_rankzeroevaluationdeterminanthscm_positive_cell = S ff_row_mdm_prefix_mdr_second_integer_rankzeroevaluationdeterminanthscm_positive))) /\ (((((exists ff_gap_mdm_lt_mdr_second_integer_rankzeroevaluationdeterminanthscm_positive_cell_column_before. ff_gap_mdm_lt_mdr_second_integer_rankzeroevaluationdeterminanthscm_positive_cell_column_before + S (ff_column_mdm_prefix_mdr_second_integer_rankzeroevaluationdeterminanthscm_positive) = (mdr_j_second_integer_rankzeroevaluationdeterminanthsc)) /\ ff_column_mdm_cell_mdr_second_integer_rankzeroevaluationdeterminanthscm_positive_cell = ff_column_mdm_prefix_mdr_second_integer_rankzeroevaluationdeterminanthscm_positive) \/ ((exists ff_gap_mdm_le_mdr_second_integer_rankzeroevaluationdeterminanthscm_positive_cell_column_after. ff_gap_mdm_le_mdr_second_integer_rankzeroevaluationdeterminanthscm_positive_cell_column_after + (mdr_j_second_integer_rankzeroevaluationdeterminanthsc) = (ff_column_mdm_prefix_mdr_second_integer_rankzeroevaluationdeterminanthscm_positive)) /\ ff_column_mdm_cell_mdr_second_integer_rankzeroevaluationdeterminanthscm_positive_cell = S ff_column_mdm_prefix_mdr_second_integer_rankzeroevaluationdeterminanthscm_positive))) /\ (((exists ff_h_mdm_mdr_second_integer_rankzeroevaluationdeterminanthscm_positive_cell_source. ff_h_mdm_mdr_second_integer_rankzeroevaluationdeterminanthscm_positive_cell_source + S (ff_value_mdm_prefix_mdr_second_integer_rankzeroevaluationdeterminanthscm_positive) = S ((S ((ff_row_mdm_cell_mdr_second_integer_rankzeroevaluationdeterminanthscm_positive_cell) * (S (mdr_q_second_integer_rankzeroevaluationdeterminanths)) + (ff_column_mdm_cell_mdr_second_integer_rankzeroevaluationdeterminanthscm_positive_cell))) * mdr_pc_second_integer_rankzeroevaluationdeterminanth)) /\ exists ff_q_mdm_mdr_second_integer_rankzeroevaluationdeterminanthscm_positive_cell_source. mdr_pb_second_integer_rankzeroevaluationdeterminanth = ff_q_mdm_mdr_second_integer_rankzeroevaluationdeterminanthscm_positive_cell_source * S ((S ((ff_row_mdm_cell_mdr_second_integer_rankzeroevaluationdeterminanthscm_positive_cell) * (S (mdr_q_second_integer_rankzeroevaluationdeterminanths)) + (ff_column_mdm_cell_mdr_second_integer_rankzeroevaluationdeterminanthscm_positive_cell))) * mdr_pc_second_integer_rankzeroevaluationdeterminanth) + (ff_value_mdm_prefix_mdr_second_integer_rankzeroevaluationdeterminanthscm_positive)))))) /\ (((exists ff_h_mdm_mdr_second_integer_rankzeroevaluationdeterminanthscm_positive_target. ff_h_mdm_mdr_second_integer_rankzeroevaluationdeterminanthscm_positive_target + S (ff_value_mdm_prefix_mdr_second_integer_rankzeroevaluationdeterminanthscm_positive) = S ((S (ff_index_mdm_prefix_mdr_second_integer_rankzeroevaluationdeterminanthscm_positive)) * mdr_us_second_integer_rankzeroevaluationdeterminanthsc)) /\ exists ff_q_mdm_mdr_second_integer_rankzeroevaluationdeterminanthscm_positive_target. mdr_up_second_integer_rankzeroevaluationdeterminanthsc = ff_q_mdm_mdr_second_integer_rankzeroevaluationdeterminanthscm_positive_target * S ((S (ff_index_mdm_prefix_mdr_second_integer_rankzeroevaluationdeterminanthscm_positive)) * mdr_us_second_integer_rankzeroevaluationdeterminanthsc) + (ff_value_mdm_prefix_mdr_second_integer_rankzeroevaluationdeterminanthscm_positive))))))) /\ (forall ff_index_mdm_prefix_mdr_second_integer_rankzeroevaluationdeterminanthscm_negative. (exists ff_gap_mdm_lt_mdr_second_integer_rankzeroevaluationdeterminanthscm_negative_index_bound. ff_gap_mdm_lt_mdr_second_integer_rankzeroevaluationdeterminanthscm_negative_index_bound + S (ff_index_mdm_prefix_mdr_second_integer_rankzeroevaluationdeterminanthscm_negative) = ((mdr_q_second_integer_rankzeroevaluationdeterminanths) * (mdr_q_second_integer_rankzeroevaluationdeterminanths))) -> exists ff_row_mdm_prefix_mdr_second_integer_rankzeroevaluationdeterminanthscm_negative ff_column_mdm_prefix_mdr_second_integer_rankzeroevaluationdeterminanthscm_negative ff_value_mdm_prefix_mdr_second_integer_rankzeroevaluationdeterminanthscm_negative. (ff_index_mdm_prefix_mdr_second_integer_rankzeroevaluationdeterminanthscm_negative = (mdr_q_second_integer_rankzeroevaluationdeterminanths) * ff_row_mdm_prefix_mdr_second_integer_rankzeroevaluationdeterminanthscm_negative + ff_column_mdm_prefix_mdr_second_integer_rankzeroevaluationdeterminanthscm_negative /\ ((exists ff_gap_mdm_lt_mdr_second_integer_rankzeroevaluationdeterminanthscm_negative_column_bound. ff_gap_mdm_lt_mdr_second_integer_rankzeroevaluationdeterminanthscm_negative_column_bound + S (ff_column_mdm_prefix_mdr_second_integer_rankzeroevaluationdeterminanthscm_negative) = (mdr_q_second_integer_rankzeroevaluationdeterminanths)) /\ ((exists ff_row_mdm_cell_mdr_second_integer_rankzeroevaluationdeterminanthscm_negative_cell ff_column_mdm_cell_mdr_second_integer_rankzeroevaluationdeterminanthscm_negative_cell. (((((exists ff_gap_mdm_lt_mdr_second_integer_rankzeroevaluationdeterminanthscm_negative_cell_row_before. ff_gap_mdm_lt_mdr_second_integer_rankzeroevaluationdeterminanthscm_negative_cell_row_before + S (ff_row_mdm_prefix_mdr_second_integer_rankzeroevaluationdeterminanthscm_negative) = (0)) /\ ff_row_mdm_cell_mdr_second_integer_rankzeroevaluationdeterminanthscm_negative_cell = ff_row_mdm_prefix_mdr_second_integer_rankzeroevaluationdeterminanthscm_negative) \/ ((exists ff_gap_mdm_le_mdr_second_integer_rankzeroevaluationdeterminanthscm_negative_cell_row_after. ff_gap_mdm_le_mdr_second_integer_rankzeroevaluationdeterminanthscm_negative_cell_row_after + (0) = (ff_row_mdm_prefix_mdr_second_integer_rankzeroevaluationdeterminanthscm_negative)) /\ ff_row_mdm_cell_mdr_second_integer_rankzeroevaluationdeterminanthscm_negative_cell = S ff_row_mdm_prefix_mdr_second_integer_rankzeroevaluationdeterminanthscm_negative))) /\ (((((exists ff_gap_mdm_lt_mdr_second_integer_rankzeroevaluationdeterminanthscm_negative_cell_column_before. ff_gap_mdm_lt_mdr_second_integer_rankzeroevaluationdeterminanthscm_negative_cell_column_before + S (ff_column_mdm_prefix_mdr_second_integer_rankzeroevaluationdeterminanthscm_negative) = (mdr_j_second_integer_rankzeroevaluationdeterminanthsc)) /\ ff_column_mdm_cell_mdr_second_integer_rankzeroevaluationdeterminanthscm_negative_cell = ff_column_mdm_prefix_mdr_second_integer_rankzeroevaluationdeterminanthscm_negative) \/ ((exists ff_gap_mdm_le_mdr_second_integer_rankzeroevaluationdeterminanthscm_negative_cell_column_after. ff_gap_mdm_le_mdr_second_integer_rankzeroevaluationdeterminanthscm_negative_cell_column_after + (mdr_j_second_integer_rankzeroevaluationdeterminanthsc) = (ff_column_mdm_prefix_mdr_second_integer_rankzeroevaluationdeterminanthscm_negative)) /\ ff_column_mdm_cell_mdr_second_integer_rankzeroevaluationdeterminanthscm_negative_cell = S ff_column_mdm_prefix_mdr_second_integer_rankzeroevaluationdeterminanthscm_negative))) /\ (((exists ff_h_mdm_mdr_second_integer_rankzeroevaluationdeterminanthscm_negative_cell_source. ff_h_mdm_mdr_second_integer_rankzeroevaluationdeterminanthscm_negative_cell_source + S (ff_value_mdm_prefix_mdr_second_integer_rankzeroevaluationdeterminanthscm_negative) = S ((S ((ff_row_mdm_cell_mdr_second_integer_rankzeroevaluationdeterminanthscm_negative_cell) * (S (mdr_q_second_integer_rankzeroevaluationdeterminanths)) + (ff_column_mdm_cell_mdr_second_integer_rankzeroevaluationdeterminanthscm_negative_cell))) * mdr_nc_second_integer_rankzeroevaluationdeterminanth)) /\ exists ff_q_mdm_mdr_second_integer_rankzeroevaluationdeterminanthscm_negative_cell_source. mdr_nb_second_integer_rankzeroevaluationdeterminanth = ff_q_mdm_mdr_second_integer_rankzeroevaluationdeterminanthscm_negative_cell_source * S ((S ((ff_row_mdm_cell_mdr_second_integer_rankzeroevaluationdeterminanthscm_negative_cell) * (S (mdr_q_second_integer_rankzeroevaluationdeterminanths)) + (ff_column_mdm_cell_mdr_second_integer_rankzeroevaluationdeterminanthscm_negative_cell))) * mdr_nc_second_integer_rankzeroevaluationdeterminanth) + (ff_value_mdm_prefix_mdr_second_integer_rankzeroevaluationdeterminanthscm_negative)))))) /\ (((exists ff_h_mdm_mdr_second_integer_rankzeroevaluationdeterminanthscm_negative_target. ff_h_mdm_mdr_second_integer_rankzeroevaluationdeterminanthscm_negative_target + S (ff_value_mdm_prefix_mdr_second_integer_rankzeroevaluationdeterminanthscm_negative) = S ((S (ff_index_mdm_prefix_mdr_second_integer_rankzeroevaluationdeterminanthscm_negative)) * mdr_ut_second_integer_rankzeroevaluationdeterminanthsc)) /\ exists ff_q_mdm_mdr_second_integer_rankzeroevaluationdeterminanthscm_negative_target. mdr_un_second_integer_rankzeroevaluationdeterminanthsc = ff_q_mdm_mdr_second_integer_rankzeroevaluationdeterminanthscm_negative_target * S ((S (ff_index_mdm_prefix_mdr_second_integer_rankzeroevaluationdeterminanthscm_negative)) * mdr_ut_second_integer_rankzeroevaluationdeterminanthsc) + (ff_value_mdm_prefix_mdr_second_integer_rankzeroevaluationdeterminanthscm_negative))))))))) /\ ((((exists ff_h_mdr_second_integer_rankzeroevaluationdeterminanthscp. ff_h_mdr_second_integer_rankzeroevaluationdeterminanthscp + S (mdr_p_second_integer_rankzeroevaluationdeterminanthsc) = S ((S (mdr_j_second_integer_rankzeroevaluationdeterminanthsc)) * mdr_ec_second_integer_rankzeroevaluationdeterminanths)) /\ exists ff_q_mdr_second_integer_rankzeroevaluationdeterminanthscp. mdr_eb_second_integer_rankzeroevaluationdeterminanths = ff_q_mdr_second_integer_rankzeroevaluationdeterminanthscp * S ((S (mdr_j_second_integer_rankzeroevaluationdeterminanthsc)) * mdr_ec_second_integer_rankzeroevaluationdeterminanths) + (mdr_p_second_integer_rankzeroevaluationdeterminanthsc))) /\ (((exists ff_h_mdr_second_integer_rankzeroevaluationdeterminanthscn. ff_h_mdr_second_integer_rankzeroevaluationdeterminanthscn + S (mdr_n_second_integer_rankzeroevaluationdeterminanthsc) = S ((S (mdr_j_second_integer_rankzeroevaluationdeterminanthsc)) * mdr_fc_second_integer_rankzeroevaluationdeterminanths)) /\ exists ff_q_mdr_second_integer_rankzeroevaluationdeterminanthscn. mdr_fb_second_integer_rankzeroevaluationdeterminanths = ff_q_mdr_second_integer_rankzeroevaluationdeterminanthscn * S ((S (mdr_j_second_integer_rankzeroevaluationdeterminanthsc)) * mdr_fc_second_integer_rankzeroevaluationdeterminanths) + (mdr_n_second_integer_rankzeroevaluationdeterminanthsc)))))))) /\ (exists ff_ub_mce_fold_mdr_second_integer_rankzeroevaluationdeterminanthsf ff_uc_mce_fold_mdr_second_integer_rankzeroevaluationdeterminanthsf ff_vb_mce_fold_mdr_second_integer_rankzeroevaluationdeterminanthsf ff_vc_mce_fold_mdr_second_integer_rankzeroevaluationdeterminanthsf. ((forall ff_index_mce_alternating_mdr_second_integer_rankzeroevaluationdeterminanthsf_prefix. (exists ff_gap_mce_mdr_second_integer_rankzeroevaluationdeterminanthsf_prefix_index. ff_gap_mce_mdr_second_integer_rankzeroevaluationdeterminanthsf_prefix_index + S (ff_index_mce_alternating_mdr_second_integer_rankzeroevaluationdeterminanthsf_prefix) = (S (mdr_q_second_integer_rankzeroevaluationdeterminanths))) -> exists ff_ap_mce_alternating_mdr_second_integer_rankzeroevaluationdeterminanthsf_prefix ff_an_mce_alternating_mdr_second_integer_rankzeroevaluationdeterminanthsf_prefix ff_bp_mce_alternating_mdr_second_integer_rankzeroevaluationdeterminanthsf_prefix ff_bn_mce_alternating_mdr_second_integer_rankzeroevaluationdeterminanthsf_prefix ff_p_mce_alternating_mdr_second_integer_rankzeroevaluationdeterminanthsf_prefix ff_n_mce_alternating_mdr_second_integer_rankzeroevaluationdeterminanthsf_prefix. ((((exists ff_h_mce_mdr_second_integer_rankzeroevaluationdeterminanthsf_prefix_ap. ff_h_mce_mdr_second_integer_rankzeroevaluationdeterminanthsf_prefix_ap + S (ff_ap_mce_alternating_mdr_second_integer_rankzeroevaluationdeterminanthsf_prefix) = S ((S (ff_index_mce_alternating_mdr_second_integer_rankzeroevaluationdeterminanthsf_prefix)) * mdr_pc_second_integer_rankzeroevaluationdeterminanth)) /\ exists ff_q_mce_mdr_second_integer_rankzeroevaluationdeterminanthsf_prefix_ap. mdr_pb_second_integer_rankzeroevaluationdeterminanth = ff_q_mce_mdr_second_integer_rankzeroevaluationdeterminanthsf_prefix_ap * S ((S (ff_index_mce_alternating_mdr_second_integer_rankzeroevaluationdeterminanthsf_prefix)) * mdr_pc_second_integer_rankzeroevaluationdeterminanth) + (ff_ap_mce_alternating_mdr_second_integer_rankzeroevaluationdeterminanthsf_prefix))) /\ ((((exists ff_h_mce_mdr_second_integer_rankzeroevaluationdeterminanthsf_prefix_an. ff_h_mce_mdr_second_integer_rankzeroevaluationdeterminanthsf_prefix_an + S (ff_an_mce_alternating_mdr_second_integer_rankzeroevaluationdeterminanthsf_prefix) = S ((S (ff_index_mce_alternating_mdr_second_integer_rankzeroevaluationdeterminanthsf_prefix)) * mdr_nc_second_integer_rankzeroevaluationdeterminanth)) /\ exists ff_q_mce_mdr_second_integer_rankzeroevaluationdeterminanthsf_prefix_an. mdr_nb_second_integer_rankzeroevaluationdeterminanth = ff_q_mce_mdr_second_integer_rankzeroevaluationdeterminanthsf_prefix_an * S ((S (ff_index_mce_alternating_mdr_second_integer_rankzeroevaluationdeterminanthsf_prefix)) * mdr_nc_second_integer_rankzeroevaluationdeterminanth) + (ff_an_mce_alternating_mdr_second_integer_rankzeroevaluationdeterminanthsf_prefix))) /\ ((((exists ff_h_mce_mdr_second_integer_rankzeroevaluationdeterminanthsf_prefix_bp. ff_h_mce_mdr_second_integer_rankzeroevaluationdeterminanthsf_prefix_bp + S (ff_bp_mce_alternating_mdr_second_integer_rankzeroevaluationdeterminanthsf_prefix) = S ((S (ff_index_mce_alternating_mdr_second_integer_rankzeroevaluationdeterminanthsf_prefix)) * mdr_ec_second_integer_rankzeroevaluationdeterminanths)) /\ exists ff_q_mce_mdr_second_integer_rankzeroevaluationdeterminanthsf_prefix_bp. mdr_eb_second_integer_rankzeroevaluationdeterminanths = ff_q_mce_mdr_second_integer_rankzeroevaluationdeterminanthsf_prefix_bp * S ((S (ff_index_mce_alternating_mdr_second_integer_rankzeroevaluationdeterminanthsf_prefix)) * mdr_ec_second_integer_rankzeroevaluationdeterminanths) + (ff_bp_mce_alternating_mdr_second_integer_rankzeroevaluationdeterminanthsf_prefix))) /\ ((((exists ff_h_mce_mdr_second_integer_rankzeroevaluationdeterminanthsf_prefix_bn. ff_h_mce_mdr_second_integer_rankzeroevaluationdeterminanthsf_prefix_bn + S (ff_bn_mce_alternating_mdr_second_integer_rankzeroevaluationdeterminanthsf_prefix) = S ((S (ff_index_mce_alternating_mdr_second_integer_rankzeroevaluationdeterminanthsf_prefix)) * mdr_fc_second_integer_rankzeroevaluationdeterminanths)) /\ exists ff_q_mce_mdr_second_integer_rankzeroevaluationdeterminanthsf_prefix_bn. mdr_fb_second_integer_rankzeroevaluationdeterminanths = ff_q_mce_mdr_second_integer_rankzeroevaluationdeterminanthsf_prefix_bn * S ((S (ff_index_mce_alternating_mdr_second_integer_rankzeroevaluationdeterminanthsf_prefix)) * mdr_fc_second_integer_rankzeroevaluationdeterminanths) + (ff_bn_mce_alternating_mdr_second_integer_rankzeroevaluationdeterminanthsf_prefix))) /\ ((((exists ff_h_mce_mdr_second_integer_rankzeroevaluationdeterminanthsf_prefix_positive. ff_h_mce_mdr_second_integer_rankzeroevaluationdeterminanthsf_prefix_positive + S (ff_p_mce_alternating_mdr_second_integer_rankzeroevaluationdeterminanthsf_prefix) = S ((S (ff_index_mce_alternating_mdr_second_integer_rankzeroevaluationdeterminanthsf_prefix)) * ff_uc_mce_fold_mdr_second_integer_rankzeroevaluationdeterminanthsf)) /\ exists ff_q_mce_mdr_second_integer_rankzeroevaluationdeterminanthsf_prefix_positive. ff_ub_mce_fold_mdr_second_integer_rankzeroevaluationdeterminanthsf = ff_q_mce_mdr_second_integer_rankzeroevaluationdeterminanthsf_prefix_positive * S ((S (ff_index_mce_alternating_mdr_second_integer_rankzeroevaluationdeterminanthsf_prefix)) * ff_uc_mce_fold_mdr_second_integer_rankzeroevaluationdeterminanthsf) + (ff_p_mce_alternating_mdr_second_integer_rankzeroevaluationdeterminanthsf_prefix))) /\ ((((exists ff_h_mce_mdr_second_integer_rankzeroevaluationdeterminanthsf_prefix_negative. ff_h_mce_mdr_second_integer_rankzeroevaluationdeterminanthsf_prefix_negative + S (ff_n_mce_alternating_mdr_second_integer_rankzeroevaluationdeterminanthsf_prefix) = S ((S (ff_index_mce_alternating_mdr_second_integer_rankzeroevaluationdeterminanthsf_prefix)) * ff_vc_mce_fold_mdr_second_integer_rankzeroevaluationdeterminanthsf)) /\ exists ff_q_mce_mdr_second_integer_rankzeroevaluationdeterminanthsf_prefix_negative. ff_vb_mce_fold_mdr_second_integer_rankzeroevaluationdeterminanthsf = ff_q_mce_mdr_second_integer_rankzeroevaluationdeterminanthsf_prefix_negative * S ((S (ff_index_mce_alternating_mdr_second_integer_rankzeroevaluationdeterminanthsf_prefix)) * ff_vc_mce_fold_mdr_second_integer_rankzeroevaluationdeterminanthsf) + (ff_n_mce_alternating_mdr_second_integer_rankzeroevaluationdeterminanthsf_prefix))) /\ (((exists ff_even_mce_term_mdr_second_integer_rankzeroevaluationdeterminanthsf_prefix_term. ff_index_mce_alternating_mdr_second_integer_rankzeroevaluationdeterminanthsf_prefix = 2 * ff_even_mce_term_mdr_second_integer_rankzeroevaluationdeterminanthsf_prefix_term) /\ (ff_p_mce_alternating_mdr_second_integer_rankzeroevaluationdeterminanthsf_prefix = (ff_ap_mce_alternating_mdr_second_integer_rankzeroevaluationdeterminanthsf_prefix) * (ff_bp_mce_alternating_mdr_second_integer_rankzeroevaluationdeterminanthsf_prefix) + (ff_an_mce_alternating_mdr_second_integer_rankzeroevaluationdeterminanthsf_prefix) * (ff_bn_mce_alternating_mdr_second_integer_rankzeroevaluationdeterminanthsf_prefix) /\ ff_n_mce_alternating_mdr_second_integer_rankzeroevaluationdeterminanthsf_prefix = (ff_ap_mce_alternating_mdr_second_integer_rankzeroevaluationdeterminanthsf_prefix) * (ff_bn_mce_alternating_mdr_second_integer_rankzeroevaluationdeterminanthsf_prefix) + (ff_an_mce_alternating_mdr_second_integer_rankzeroevaluationdeterminanthsf_prefix) * (ff_bp_mce_alternating_mdr_second_integer_rankzeroevaluationdeterminanthsf_prefix))) \/ ((exists ff_odd_mce_term_mdr_second_integer_rankzeroevaluationdeterminanthsf_prefix_term. ff_index_mce_alternating_mdr_second_integer_rankzeroevaluationdeterminanthsf_prefix = 2 * ff_odd_mce_term_mdr_second_integer_rankzeroevaluationdeterminanthsf_prefix_term + 1) /\ (ff_p_mce_alternating_mdr_second_integer_rankzeroevaluationdeterminanthsf_prefix = (ff_ap_mce_alternating_mdr_second_integer_rankzeroevaluationdeterminanthsf_prefix) * (ff_bn_mce_alternating_mdr_second_integer_rankzeroevaluationdeterminanthsf_prefix) + (ff_an_mce_alternating_mdr_second_integer_rankzeroevaluationdeterminanthsf_prefix) * (ff_bp_mce_alternating_mdr_second_integer_rankzeroevaluationdeterminanthsf_prefix) /\ ff_n_mce_alternating_mdr_second_integer_rankzeroevaluationdeterminanthsf_prefix = (ff_ap_mce_alternating_mdr_second_integer_rankzeroevaluationdeterminanthsf_prefix) * (ff_bp_mce_alternating_mdr_second_integer_rankzeroevaluationdeterminanthsf_prefix) + (ff_an_mce_alternating_mdr_second_integer_rankzeroevaluationdeterminanthsf_prefix) * (ff_bn_mce_alternating_mdr_second_integer_rankzeroevaluationdeterminanthsf_prefix))))))))))) /\ ((exists ff_u_mce_mdr_second_integer_rankzeroevaluationdeterminanthsf_positive ff_v_mce_mdr_second_integer_rankzeroevaluationdeterminanthsf_positive. ((((exists ff_h_mce_mdr_second_integer_rankzeroevaluationdeterminanthsf_positive_start. ff_h_mce_mdr_second_integer_rankzeroevaluationdeterminanthsf_positive_start + S (0) = S ((S (0)) * ff_v_mce_mdr_second_integer_rankzeroevaluationdeterminanthsf_positive)) /\ exists ff_q_mce_mdr_second_integer_rankzeroevaluationdeterminanthsf_positive_start. ff_u_mce_mdr_second_integer_rankzeroevaluationdeterminanthsf_positive = ff_q_mce_mdr_second_integer_rankzeroevaluationdeterminanthsf_positive_start * S ((S (0)) * ff_v_mce_mdr_second_integer_rankzeroevaluationdeterminanthsf_positive) + (0))) /\ ((((exists ff_h_mce_mdr_second_integer_rankzeroevaluationdeterminanthsf_positive_terminal. ff_h_mce_mdr_second_integer_rankzeroevaluationdeterminanthsf_positive_terminal + S (mdr_p_second_integer_rankzeroevaluationdeterminanth) = S ((S ((S (mdr_q_second_integer_rankzeroevaluationdeterminanths)))) * ff_v_mce_mdr_second_integer_rankzeroevaluationdeterminanthsf_positive)) /\ exists ff_q_mce_mdr_second_integer_rankzeroevaluationdeterminanthsf_positive_terminal. ff_u_mce_mdr_second_integer_rankzeroevaluationdeterminanthsf_positive = ff_q_mce_mdr_second_integer_rankzeroevaluationdeterminanthsf_positive_terminal * S ((S ((S (mdr_q_second_integer_rankzeroevaluationdeterminanths)))) * ff_v_mce_mdr_second_integer_rankzeroevaluationdeterminanthsf_positive) + (mdr_p_second_integer_rankzeroevaluationdeterminanth))) /\ forall ff_i_mce_mdr_second_integer_rankzeroevaluationdeterminanthsf_positive. (exists ff_lt_mce_mdr_second_integer_rankzeroevaluationdeterminanthsf_positive_bound. ff_lt_mce_mdr_second_integer_rankzeroevaluationdeterminanthsf_positive_bound + S ff_i_mce_mdr_second_integer_rankzeroevaluationdeterminanthsf_positive = (S (mdr_q_second_integer_rankzeroevaluationdeterminanths))) -> exists ff_a_mce_mdr_second_integer_rankzeroevaluationdeterminanthsf_positive ff_r_mce_mdr_second_integer_rankzeroevaluationdeterminanthsf_positive ff_s_mce_mdr_second_integer_rankzeroevaluationdeterminanthsf_positive. ((((exists ff_h_mce_mdr_second_integer_rankzeroevaluationdeterminanthsf_positive_summand. ff_h_mce_mdr_second_integer_rankzeroevaluationdeterminanthsf_positive_summand + S (ff_a_mce_mdr_second_integer_rankzeroevaluationdeterminanthsf_positive) = S ((S (ff_i_mce_mdr_second_integer_rankzeroevaluationdeterminanthsf_positive)) * ff_uc_mce_fold_mdr_second_integer_rankzeroevaluationdeterminanthsf)) /\ exists ff_q_mce_mdr_second_integer_rankzeroevaluationdeterminanthsf_positive_summand. ff_ub_mce_fold_mdr_second_integer_rankzeroevaluationdeterminanthsf = ff_q_mce_mdr_second_integer_rankzeroevaluationdeterminanthsf_positive_summand * S ((S (ff_i_mce_mdr_second_integer_rankzeroevaluationdeterminanthsf_positive)) * ff_uc_mce_fold_mdr_second_integer_rankzeroevaluationdeterminanthsf) + (ff_a_mce_mdr_second_integer_rankzeroevaluationdeterminanthsf_positive))) /\ ((((exists ff_h_mce_mdr_second_integer_rankzeroevaluationdeterminanthsf_positive_partial. ff_h_mce_mdr_second_integer_rankzeroevaluationdeterminanthsf_positive_partial + S (ff_r_mce_mdr_second_integer_rankzeroevaluationdeterminanthsf_positive) = S ((S (ff_i_mce_mdr_second_integer_rankzeroevaluationdeterminanthsf_positive)) * ff_v_mce_mdr_second_integer_rankzeroevaluationdeterminanthsf_positive)) /\ exists ff_q_mce_mdr_second_integer_rankzeroevaluationdeterminanthsf_positive_partial. ff_u_mce_mdr_second_integer_rankzeroevaluationdeterminanthsf_positive = ff_q_mce_mdr_second_integer_rankzeroevaluationdeterminanthsf_positive_partial * S ((S (ff_i_mce_mdr_second_integer_rankzeroevaluationdeterminanthsf_positive)) * ff_v_mce_mdr_second_integer_rankzeroevaluationdeterminanthsf_positive) + (ff_r_mce_mdr_second_integer_rankzeroevaluationdeterminanthsf_positive))) /\ ((((exists ff_h_mce_mdr_second_integer_rankzeroevaluationdeterminanthsf_positive_successor. ff_h_mce_mdr_second_integer_rankzeroevaluationdeterminanthsf_positive_successor + S (ff_s_mce_mdr_second_integer_rankzeroevaluationdeterminanthsf_positive) = S ((S (S ff_i_mce_mdr_second_integer_rankzeroevaluationdeterminanthsf_positive)) * ff_v_mce_mdr_second_integer_rankzeroevaluationdeterminanthsf_positive)) /\ exists ff_q_mce_mdr_second_integer_rankzeroevaluationdeterminanthsf_positive_successor. ff_u_mce_mdr_second_integer_rankzeroevaluationdeterminanthsf_positive = ff_q_mce_mdr_second_integer_rankzeroevaluationdeterminanthsf_positive_successor * S ((S (S ff_i_mce_mdr_second_integer_rankzeroevaluationdeterminanthsf_positive)) * ff_v_mce_mdr_second_integer_rankzeroevaluationdeterminanthsf_positive) + (ff_s_mce_mdr_second_integer_rankzeroevaluationdeterminanthsf_positive))) /\ ff_s_mce_mdr_second_integer_rankzeroevaluationdeterminanthsf_positive = ff_r_mce_mdr_second_integer_rankzeroevaluationdeterminanthsf_positive + ff_a_mce_mdr_second_integer_rankzeroevaluationdeterminanthsf_positive)))))) /\ (exists ff_u_mce_mdr_second_integer_rankzeroevaluationdeterminanthsf_negative ff_v_mce_mdr_second_integer_rankzeroevaluationdeterminanthsf_negative. ((((exists ff_h_mce_mdr_second_integer_rankzeroevaluationdeterminanthsf_negative_start. ff_h_mce_mdr_second_integer_rankzeroevaluationdeterminanthsf_negative_start + S (0) = S ((S (0)) * ff_v_mce_mdr_second_integer_rankzeroevaluationdeterminanthsf_negative)) /\ exists ff_q_mce_mdr_second_integer_rankzeroevaluationdeterminanthsf_negative_start. ff_u_mce_mdr_second_integer_rankzeroevaluationdeterminanthsf_negative = ff_q_mce_mdr_second_integer_rankzeroevaluationdeterminanthsf_negative_start * S ((S (0)) * ff_v_mce_mdr_second_integer_rankzeroevaluationdeterminanthsf_negative) + (0))) /\ ((((exists ff_h_mce_mdr_second_integer_rankzeroevaluationdeterminanthsf_negative_terminal. ff_h_mce_mdr_second_integer_rankzeroevaluationdeterminanthsf_negative_terminal + S (mdr_n_second_integer_rankzeroevaluationdeterminanth) = S ((S ((S (mdr_q_second_integer_rankzeroevaluationdeterminanths)))) * ff_v_mce_mdr_second_integer_rankzeroevaluationdeterminanthsf_negative)) /\ exists ff_q_mce_mdr_second_integer_rankzeroevaluationdeterminanthsf_negative_terminal. ff_u_mce_mdr_second_integer_rankzeroevaluationdeterminanthsf_negative = ff_q_mce_mdr_second_integer_rankzeroevaluationdeterminanthsf_negative_terminal * S ((S ((S (mdr_q_second_integer_rankzeroevaluationdeterminanths)))) * ff_v_mce_mdr_second_integer_rankzeroevaluationdeterminanthsf_negative) + (mdr_n_second_integer_rankzeroevaluationdeterminanth))) /\ forall ff_i_mce_mdr_second_integer_rankzeroevaluationdeterminanthsf_negative. (exists ff_lt_mce_mdr_second_integer_rankzeroevaluationdeterminanthsf_negative_bound. ff_lt_mce_mdr_second_integer_rankzeroevaluationdeterminanthsf_negative_bound + S ff_i_mce_mdr_second_integer_rankzeroevaluationdeterminanthsf_negative = (S (mdr_q_second_integer_rankzeroevaluationdeterminanths))) -> exists ff_a_mce_mdr_second_integer_rankzeroevaluationdeterminanthsf_negative ff_r_mce_mdr_second_integer_rankzeroevaluationdeterminanthsf_negative ff_s_mce_mdr_second_integer_rankzeroevaluationdeterminanthsf_negative. ((((exists ff_h_mce_mdr_second_integer_rankzeroevaluationdeterminanthsf_negative_summand. ff_h_mce_mdr_second_integer_rankzeroevaluationdeterminanthsf_negative_summand + S (ff_a_mce_mdr_second_integer_rankzeroevaluationdeterminanthsf_negative) = S ((S (ff_i_mce_mdr_second_integer_rankzeroevaluationdeterminanthsf_negative)) * ff_vc_mce_fold_mdr_second_integer_rankzeroevaluationdeterminanthsf)) /\ exists ff_q_mce_mdr_second_integer_rankzeroevaluationdeterminanthsf_negative_summand. ff_vb_mce_fold_mdr_second_integer_rankzeroevaluationdeterminanthsf = ff_q_mce_mdr_second_integer_rankzeroevaluationdeterminanthsf_negative_summand * S ((S (ff_i_mce_mdr_second_integer_rankzeroevaluationdeterminanthsf_negative)) * ff_vc_mce_fold_mdr_second_integer_rankzeroevaluationdeterminanthsf) + (ff_a_mce_mdr_second_integer_rankzeroevaluationdeterminanthsf_negative))) /\ ((((exists ff_h_mce_mdr_second_integer_rankzeroevaluationdeterminanthsf_negative_partial. ff_h_mce_mdr_second_integer_rankzeroevaluationdeterminanthsf_negative_partial + S (ff_r_mce_mdr_second_integer_rankzeroevaluationdeterminanthsf_negative) = S ((S (ff_i_mce_mdr_second_integer_rankzeroevaluationdeterminanthsf_negative)) * ff_v_mce_mdr_second_integer_rankzeroevaluationdeterminanthsf_negative)) /\ exists ff_q_mce_mdr_second_integer_rankzeroevaluationdeterminanthsf_negative_partial. ff_u_mce_mdr_second_integer_rankzeroevaluationdeterminanthsf_negative = ff_q_mce_mdr_second_integer_rankzeroevaluationdeterminanthsf_negative_partial * S ((S (ff_i_mce_mdr_second_integer_rankzeroevaluationdeterminanthsf_negative)) * ff_v_mce_mdr_second_integer_rankzeroevaluationdeterminanthsf_negative) + (ff_r_mce_mdr_second_integer_rankzeroevaluationdeterminanthsf_negative))) /\ ((((exists ff_h_mce_mdr_second_integer_rankzeroevaluationdeterminanthsf_negative_successor. ff_h_mce_mdr_second_integer_rankzeroevaluationdeterminanthsf_negative_successor + S (ff_s_mce_mdr_second_integer_rankzeroevaluationdeterminanthsf_negative) = S ((S (S ff_i_mce_mdr_second_integer_rankzeroevaluationdeterminanthsf_negative)) * ff_v_mce_mdr_second_integer_rankzeroevaluationdeterminanthsf_negative)) /\ exists ff_q_mce_mdr_second_integer_rankzeroevaluationdeterminanthsf_negative_successor. ff_u_mce_mdr_second_integer_rankzeroevaluationdeterminanthsf_negative = ff_q_mce_mdr_second_integer_rankzeroevaluationdeterminanthsf_negative_successor * S ((S (S ff_i_mce_mdr_second_integer_rankzeroevaluationdeterminanthsf_negative)) * ff_v_mce_mdr_second_integer_rankzeroevaluationdeterminanthsf_negative) + (ff_s_mce_mdr_second_integer_rankzeroevaluationdeterminanthsf_negative))) /\ ff_s_mce_mdr_second_integer_rankzeroevaluationdeterminanthsf_negative = ff_r_mce_mdr_second_integer_rankzeroevaluationdeterminanthsf_negative + ff_a_mce_mdr_second_integer_rankzeroevaluationdeterminanthsf_negative))))))))))))))) /\ ((exists mdr_gap_second_integer_rankzeroevaluationdeterminanti. mdr_gap_second_integer_rankzeroevaluationdeterminanti + S (mdr_i_second_integer_rankzeroevaluationdeterminant) = (mdr_l_second_integer_rankzeroevaluationdeterminant)) /\ (exists mdr_z_second_integer_rankzeroevaluationdeterminantr. ((exists mdr_a_second_integer_rankzeroevaluationdeterminantrc mdr_b_second_integer_rankzeroevaluationdeterminantrc mdr_c_second_integer_rankzeroevaluationdeterminantrc mdr_e_second_integer_rankzeroevaluationdeterminantrc mdr_f_second_integer_rankzeroevaluationdeterminantrc. ((mdr_a_second_integer_rankzeroevaluationdeterminantrc = ((mdr_q_second_integer_rank) + (mdr_ub_second_integer_rankzeroevaluation)) * S ((mdr_q_second_integer_rank) + (mdr_ub_second_integer_rankzeroevaluation)) + ((mdr_ub_second_integer_rankzeroevaluation) + (mdr_ub_second_integer_rankzeroevaluation))) /\ ((mdr_b_second_integer_rankzeroevaluationdeterminantrc = ((mdr_uc_second_integer_rankzeroevaluation) + (mdr_vb_second_integer_rankzeroevaluation)) * S ((mdr_uc_second_integer_rankzeroevaluation) + (mdr_vb_second_integer_rankzeroevaluation)) + ((mdr_vb_second_integer_rankzeroevaluation) + (mdr_vb_second_integer_rankzeroevaluation))) /\ ((mdr_c_second_integer_rankzeroevaluationdeterminantrc = ((mdr_a_second_integer_rankzeroevaluationdeterminantrc) + (mdr_b_second_integer_rankzeroevaluationdeterminantrc)) * S ((mdr_a_second_integer_rankzeroevaluationdeterminantrc) + (mdr_b_second_integer_rankzeroevaluationdeterminantrc)) + ((mdr_b_second_integer_rankzeroevaluationdeterminantrc) + (mdr_b_second_integer_rankzeroevaluationdeterminantrc))) /\ ((mdr_e_second_integer_rankzeroevaluationdeterminantrc = ((mdr_p_second_integer_rankzero) + (mdr_n_second_integer_rankzero)) * S ((mdr_p_second_integer_rankzero) + (mdr_n_second_integer_rankzero)) + ((mdr_n_second_integer_rankzero) + (mdr_n_second_integer_rankzero))) /\ ((mdr_f_second_integer_rankzeroevaluationdeterminantrc = ((mdr_vc_second_integer_rankzeroevaluation) + (mdr_e_second_integer_rankzeroevaluationdeterminantrc)) * S ((mdr_vc_second_integer_rankzeroevaluation) + (mdr_e_second_integer_rankzeroevaluationdeterminantrc)) + ((mdr_e_second_integer_rankzeroevaluationdeterminantrc) + (mdr_e_second_integer_rankzeroevaluationdeterminantrc))) /\ ((mdr_z_second_integer_rankzeroevaluationdeterminantr) = ((mdr_c_second_integer_rankzeroevaluationdeterminantrc) + (mdr_f_second_integer_rankzeroevaluationdeterminantrc)) * S ((mdr_c_second_integer_rankzeroevaluationdeterminantrc) + (mdr_f_second_integer_rankzeroevaluationdeterminantrc)) + ((mdr_f_second_integer_rankzeroevaluationdeterminantrc) + (mdr_f_second_integer_rankzeroevaluationdeterminantrc))))))))) /\ (((exists ff_h_mdr_second_integer_rankzeroevaluationdeterminantrb. ff_h_mdr_second_integer_rankzeroevaluationdeterminantrb + S (mdr_z_second_integer_rankzeroevaluationdeterminantr) = S ((S (mdr_i_second_integer_rankzeroevaluationdeterminant)) * mdr_c_second_integer_rankzeroevaluationdeterminant)) /\ exists ff_q_mdr_second_integer_rankzeroevaluationdeterminantrb. mdr_b_second_integer_rankzeroevaluationdeterminant = ff_q_mdr_second_integer_rankzeroevaluationdeterminantrb * S ((S (mdr_i_second_integer_rankzeroevaluationdeterminant)) * mdr_c_second_integer_rankzeroevaluationdeterminant) + (mdr_z_second_integer_rankzeroevaluationdeterminantr)))))))))) -> mdr_p_second_integer_rankzero = mdr_n_second_integer_rankzero))))))