Current library: Alpha v34, 4,223 checked-use theorems; Stable remains 432. Historical first admissions, original proof editions, and non-admitted aliases are preserved. Exact original first-admission records.
Exact expanded first-order arithmetic statement
forall pb pc nb nc w. (((exists mdr_gap_zero_rows_rankrows_bound. mdr_gap_zero_rows_rankrows_bound + (0) = (0)) /\ ((exists mdr_gap_zero_rows_rankcolumns_bound. mdr_gap_zero_rows_rankcolumns_bound + (0) = (w)) /\ ((exists mdr_rb_zero_rows_rankwitness mdr_rc_zero_rows_rankwitness mdr_cb_zero_rows_rankwitness mdr_cc_zero_rows_rankwitness. (((((forall fom_index_mrf_zero_rows_rankwitnessminorrowsbound. (exists fom_gap_mrf_zero_rows_rankwitnessminorrowsbound_index_bound. fom_gap_mrf_zero_rows_rankwitnessminorrowsbound_index_bound + S (fom_index_mrf_zero_rows_rankwitnessminorrowsbound) = 0) -> exists fom_value_mrf_zero_rows_rankwitnessminorrowsbound. ((((exists fom_beta_height_mrf_zero_rows_rankwitnessminorrowsbound_entry. fom_beta_height_mrf_zero_rows_rankwitnessminorrowsbound_entry + S (fom_value_mrf_zero_rows_rankwitnessminorrowsbound) = S ((S (fom_index_mrf_zero_rows_rankwitnessminorrowsbound)) * mdr_rc_zero_rows_rankwitness)) /\ exists fom_beta_quotient_mrf_zero_rows_rankwitnessminorrowsbound_entry. mdr_rb_zero_rows_rankwitness = fom_beta_quotient_mrf_zero_rows_rankwitnessminorrowsbound_entry * S ((S (fom_index_mrf_zero_rows_rankwitnessminorrowsbound)) * mdr_rc_zero_rows_rankwitness) + (fom_value_mrf_zero_rows_rankwitnessminorrowsbound))) /\ (exists fom_gap_mrf_zero_rows_rankwitnessminorrowsbound_value_bound. fom_gap_mrf_zero_rows_rankwitnessminorrowsbound_value_bound + S (fom_value_mrf_zero_rows_rankwitnessminorrowsbound) = 0))) /\ (forall mdr_i_zero_rows_rankwitnessminorrowsdistinct mdr_j_zero_rows_rankwitnessminorrowsdistinct mdr_a_zero_rows_rankwitnessminorrowsdistinct. (exists mdr_gap_zero_rows_rankwitnessminorrowsdistincti. mdr_gap_zero_rows_rankwitnessminorrowsdistincti + S (mdr_i_zero_rows_rankwitnessminorrowsdistinct) = (0)) -> (exists mdr_gap_zero_rows_rankwitnessminorrowsdistinctj. mdr_gap_zero_rows_rankwitnessminorrowsdistinctj + S (mdr_j_zero_rows_rankwitnessminorrowsdistinct) = (0)) -> (((exists ff_h_mdr_zero_rows_rankwitnessminorrowsdistinctfirst. ff_h_mdr_zero_rows_rankwitnessminorrowsdistinctfirst + S (mdr_a_zero_rows_rankwitnessminorrowsdistinct) = S ((S (mdr_i_zero_rows_rankwitnessminorrowsdistinct)) * mdr_rc_zero_rows_rankwitness)) /\ exists ff_q_mdr_zero_rows_rankwitnessminorrowsdistinctfirst. mdr_rb_zero_rows_rankwitness = ff_q_mdr_zero_rows_rankwitnessminorrowsdistinctfirst * S ((S (mdr_i_zero_rows_rankwitnessminorrowsdistinct)) * mdr_rc_zero_rows_rankwitness) + (mdr_a_zero_rows_rankwitnessminorrowsdistinct))) -> (((exists ff_h_mdr_zero_rows_rankwitnessminorrowsdistinctsecond. ff_h_mdr_zero_rows_rankwitnessminorrowsdistinctsecond + S (mdr_a_zero_rows_rankwitnessminorrowsdistinct) = S ((S (mdr_j_zero_rows_rankwitnessminorrowsdistinct)) * mdr_rc_zero_rows_rankwitness)) /\ exists ff_q_mdr_zero_rows_rankwitnessminorrowsdistinctsecond. mdr_rb_zero_rows_rankwitness = ff_q_mdr_zero_rows_rankwitnessminorrowsdistinctsecond * S ((S (mdr_j_zero_rows_rankwitnessminorrowsdistinct)) * mdr_rc_zero_rows_rankwitness) + (mdr_a_zero_rows_rankwitnessminorrowsdistinct))) -> mdr_i_zero_rows_rankwitnessminorrowsdistinct = mdr_j_zero_rows_rankwitnessminorrowsdistinct))) /\ ((((forall fom_index_mrf_zero_rows_rankwitnessminorcolumnsbound. (exists fom_gap_mrf_zero_rows_rankwitnessminorcolumnsbound_index_bound. fom_gap_mrf_zero_rows_rankwitnessminorcolumnsbound_index_bound + S (fom_index_mrf_zero_rows_rankwitnessminorcolumnsbound) = 0) -> exists fom_value_mrf_zero_rows_rankwitnessminorcolumnsbound. ((((exists fom_beta_height_mrf_zero_rows_rankwitnessminorcolumnsbound_entry. fom_beta_height_mrf_zero_rows_rankwitnessminorcolumnsbound_entry + S (fom_value_mrf_zero_rows_rankwitnessminorcolumnsbound) = S ((S (fom_index_mrf_zero_rows_rankwitnessminorcolumnsbound)) * mdr_cc_zero_rows_rankwitness)) /\ exists fom_beta_quotient_mrf_zero_rows_rankwitnessminorcolumnsbound_entry. mdr_cb_zero_rows_rankwitness = fom_beta_quotient_mrf_zero_rows_rankwitnessminorcolumnsbound_entry * S ((S (fom_index_mrf_zero_rows_rankwitnessminorcolumnsbound)) * mdr_cc_zero_rows_rankwitness) + (fom_value_mrf_zero_rows_rankwitnessminorcolumnsbound))) /\ (exists fom_gap_mrf_zero_rows_rankwitnessminorcolumnsbound_value_bound. fom_gap_mrf_zero_rows_rankwitnessminorcolumnsbound_value_bound + S (fom_value_mrf_zero_rows_rankwitnessminorcolumnsbound) = w))) /\ (forall mdr_i_zero_rows_rankwitnessminorcolumnsdistinct mdr_j_zero_rows_rankwitnessminorcolumnsdistinct mdr_a_zero_rows_rankwitnessminorcolumnsdistinct. (exists mdr_gap_zero_rows_rankwitnessminorcolumnsdistincti. mdr_gap_zero_rows_rankwitnessminorcolumnsdistincti + S (mdr_i_zero_rows_rankwitnessminorcolumnsdistinct) = (0)) -> (exists mdr_gap_zero_rows_rankwitnessminorcolumnsdistinctj. mdr_gap_zero_rows_rankwitnessminorcolumnsdistinctj + S (mdr_j_zero_rows_rankwitnessminorcolumnsdistinct) = (0)) -> (((exists ff_h_mdr_zero_rows_rankwitnessminorcolumnsdistinctfirst. ff_h_mdr_zero_rows_rankwitnessminorcolumnsdistinctfirst + S (mdr_a_zero_rows_rankwitnessminorcolumnsdistinct) = S ((S (mdr_i_zero_rows_rankwitnessminorcolumnsdistinct)) * mdr_cc_zero_rows_rankwitness)) /\ exists ff_q_mdr_zero_rows_rankwitnessminorcolumnsdistinctfirst. mdr_cb_zero_rows_rankwitness = ff_q_mdr_zero_rows_rankwitnessminorcolumnsdistinctfirst * S ((S (mdr_i_zero_rows_rankwitnessminorcolumnsdistinct)) * mdr_cc_zero_rows_rankwitness) + (mdr_a_zero_rows_rankwitnessminorcolumnsdistinct))) -> (((exists ff_h_mdr_zero_rows_rankwitnessminorcolumnsdistinctsecond. ff_h_mdr_zero_rows_rankwitnessminorcolumnsdistinctsecond + S (mdr_a_zero_rows_rankwitnessminorcolumnsdistinct) = S ((S (mdr_j_zero_rows_rankwitnessminorcolumnsdistinct)) * mdr_cc_zero_rows_rankwitness)) /\ exists ff_q_mdr_zero_rows_rankwitnessminorcolumnsdistinctsecond. mdr_cb_zero_rows_rankwitness = ff_q_mdr_zero_rows_rankwitnessminorcolumnsdistinctsecond * S ((S (mdr_j_zero_rows_rankwitnessminorcolumnsdistinct)) * mdr_cc_zero_rows_rankwitness) + (mdr_a_zero_rows_rankwitnessminorcolumnsdistinct))) -> mdr_i_zero_rows_rankwitnessminorcolumnsdistinct = mdr_j_zero_rows_rankwitnessminorcolumnsdistinct))) /\ (exists mdr_p_zero_rows_rankwitnessminornonzero mdr_n_zero_rows_rankwitnessminornonzero. ((exists mdr_ub_zero_rows_rankwitnessminornonzeroevaluation mdr_uc_zero_rows_rankwitnessminornonzeroevaluation mdr_vb_zero_rows_rankwitnessminornonzeroevaluation mdr_vc_zero_rows_rankwitnessminornonzeroevaluation. ((((forall mdr_i_zero_rows_rankwitnessminornonzeroevaluationmatrixpositive. (exists mdr_gap_zero_rows_rankwitnessminornonzeroevaluationmatrixpositivebound. mdr_gap_zero_rows_rankwitnessminornonzeroevaluationmatrixpositivebound + S (mdr_i_zero_rows_rankwitnessminornonzeroevaluationmatrixpositive) = ((0) * (0))) -> exists mdr_a_zero_rows_rankwitnessminornonzeroevaluationmatrixpositive. (((exists mdr_r_zero_rows_rankwitnessminornonzeroevaluationmatrixpositivepoint mdr_s_zero_rows_rankwitnessminornonzeroevaluationmatrixpositivepoint mdr_u_zero_rows_rankwitnessminornonzeroevaluationmatrixpositivepoint mdr_v_zero_rows_rankwitnessminornonzeroevaluationmatrixpositivepoint. ((mdr_i_zero_rows_rankwitnessminornonzeroevaluationmatrixpositive = (0) * mdr_r_zero_rows_rankwitnessminornonzeroevaluationmatrixpositivepoint + mdr_s_zero_rows_rankwitnessminornonzeroevaluationmatrixpositivepoint) /\ ((exists mdr_gap_zero_rows_rankwitnessminornonzeroevaluationmatrixpositivepointcolumn. mdr_gap_zero_rows_rankwitnessminornonzeroevaluationmatrixpositivepointcolumn + S (mdr_s_zero_rows_rankwitnessminornonzeroevaluationmatrixpositivepoint) = (0)) /\ ((((exists ff_h_mdr_zero_rows_rankwitnessminornonzeroevaluationmatrixpositivepointrow_index. ff_h_mdr_zero_rows_rankwitnessminornonzeroevaluationmatrixpositivepointrow_index + S (mdr_u_zero_rows_rankwitnessminornonzeroevaluationmatrixpositivepoint) = S ((S (mdr_r_zero_rows_rankwitnessminornonzeroevaluationmatrixpositivepoint)) * mdr_rc_zero_rows_rankwitness)) /\ exists ff_q_mdr_zero_rows_rankwitnessminornonzeroevaluationmatrixpositivepointrow_index. mdr_rb_zero_rows_rankwitness = ff_q_mdr_zero_rows_rankwitnessminornonzeroevaluationmatrixpositivepointrow_index * S ((S (mdr_r_zero_rows_rankwitnessminornonzeroevaluationmatrixpositivepoint)) * mdr_rc_zero_rows_rankwitness) + (mdr_u_zero_rows_rankwitnessminornonzeroevaluationmatrixpositivepoint))) /\ ((((exists ff_h_mdr_zero_rows_rankwitnessminornonzeroevaluationmatrixpositivepointcolumn_index. ff_h_mdr_zero_rows_rankwitnessminornonzeroevaluationmatrixpositivepointcolumn_index + S (mdr_v_zero_rows_rankwitnessminornonzeroevaluationmatrixpositivepoint) = S ((S (mdr_s_zero_rows_rankwitnessminornonzeroevaluationmatrixpositivepoint)) * mdr_cc_zero_rows_rankwitness)) /\ exists ff_q_mdr_zero_rows_rankwitnessminornonzeroevaluationmatrixpositivepointcolumn_index. mdr_cb_zero_rows_rankwitness = ff_q_mdr_zero_rows_rankwitnessminornonzeroevaluationmatrixpositivepointcolumn_index * S ((S (mdr_s_zero_rows_rankwitnessminornonzeroevaluationmatrixpositivepoint)) * mdr_cc_zero_rows_rankwitness) + (mdr_v_zero_rows_rankwitnessminornonzeroevaluationmatrixpositivepoint))) /\ (((exists ff_h_mdr_zero_rows_rankwitnessminornonzeroevaluationmatrixpositivepointsource. ff_h_mdr_zero_rows_rankwitnessminornonzeroevaluationmatrixpositivepointsource + S (mdr_a_zero_rows_rankwitnessminornonzeroevaluationmatrixpositive) = S ((S ((mdr_u_zero_rows_rankwitnessminornonzeroevaluationmatrixpositivepoint) * (w) + (mdr_v_zero_rows_rankwitnessminornonzeroevaluationmatrixpositivepoint))) * pc)) /\ exists ff_q_mdr_zero_rows_rankwitnessminornonzeroevaluationmatrixpositivepointsource. pb = ff_q_mdr_zero_rows_rankwitnessminornonzeroevaluationmatrixpositivepointsource * S ((S ((mdr_u_zero_rows_rankwitnessminornonzeroevaluationmatrixpositivepoint) * (w) + (mdr_v_zero_rows_rankwitnessminornonzeroevaluationmatrixpositivepoint))) * pc) + (mdr_a_zero_rows_rankwitnessminornonzeroevaluationmatrixpositive)))))))) /\ (((exists ff_h_mdr_zero_rows_rankwitnessminornonzeroevaluationmatrixpositiveoutput. ff_h_mdr_zero_rows_rankwitnessminornonzeroevaluationmatrixpositiveoutput + S (mdr_a_zero_rows_rankwitnessminornonzeroevaluationmatrixpositive) = S ((S (mdr_i_zero_rows_rankwitnessminornonzeroevaluationmatrixpositive)) * mdr_uc_zero_rows_rankwitnessminornonzeroevaluation)) /\ exists ff_q_mdr_zero_rows_rankwitnessminornonzeroevaluationmatrixpositiveoutput. mdr_ub_zero_rows_rankwitnessminornonzeroevaluation = ff_q_mdr_zero_rows_rankwitnessminornonzeroevaluationmatrixpositiveoutput * S ((S (mdr_i_zero_rows_rankwitnessminornonzeroevaluationmatrixpositive)) * mdr_uc_zero_rows_rankwitnessminornonzeroevaluation) + (mdr_a_zero_rows_rankwitnessminornonzeroevaluationmatrixpositive)))))) /\ (forall mdr_i_zero_rows_rankwitnessminornonzeroevaluationmatrixnegative. (exists mdr_gap_zero_rows_rankwitnessminornonzeroevaluationmatrixnegativebound. mdr_gap_zero_rows_rankwitnessminornonzeroevaluationmatrixnegativebound + S (mdr_i_zero_rows_rankwitnessminornonzeroevaluationmatrixnegative) = ((0) * (0))) -> exists mdr_a_zero_rows_rankwitnessminornonzeroevaluationmatrixnegative. (((exists mdr_r_zero_rows_rankwitnessminornonzeroevaluationmatrixnegativepoint mdr_s_zero_rows_rankwitnessminornonzeroevaluationmatrixnegativepoint mdr_u_zero_rows_rankwitnessminornonzeroevaluationmatrixnegativepoint mdr_v_zero_rows_rankwitnessminornonzeroevaluationmatrixnegativepoint. ((mdr_i_zero_rows_rankwitnessminornonzeroevaluationmatrixnegative = (0) * mdr_r_zero_rows_rankwitnessminornonzeroevaluationmatrixnegativepoint + mdr_s_zero_rows_rankwitnessminornonzeroevaluationmatrixnegativepoint) /\ ((exists mdr_gap_zero_rows_rankwitnessminornonzeroevaluationmatrixnegativepointcolumn. mdr_gap_zero_rows_rankwitnessminornonzeroevaluationmatrixnegativepointcolumn + S (mdr_s_zero_rows_rankwitnessminornonzeroevaluationmatrixnegativepoint) = (0)) /\ ((((exists ff_h_mdr_zero_rows_rankwitnessminornonzeroevaluationmatrixnegativepointrow_index. ff_h_mdr_zero_rows_rankwitnessminornonzeroevaluationmatrixnegativepointrow_index + S (mdr_u_zero_rows_rankwitnessminornonzeroevaluationmatrixnegativepoint) = S ((S (mdr_r_zero_rows_rankwitnessminornonzeroevaluationmatrixnegativepoint)) * mdr_rc_zero_rows_rankwitness)) /\ exists ff_q_mdr_zero_rows_rankwitnessminornonzeroevaluationmatrixnegativepointrow_index. mdr_rb_zero_rows_rankwitness = ff_q_mdr_zero_rows_rankwitnessminornonzeroevaluationmatrixnegativepointrow_index * S ((S (mdr_r_zero_rows_rankwitnessminornonzeroevaluationmatrixnegativepoint)) * mdr_rc_zero_rows_rankwitness) + (mdr_u_zero_rows_rankwitnessminornonzeroevaluationmatrixnegativepoint))) /\ ((((exists ff_h_mdr_zero_rows_rankwitnessminornonzeroevaluationmatrixnegativepointcolumn_index. ff_h_mdr_zero_rows_rankwitnessminornonzeroevaluationmatrixnegativepointcolumn_index + S (mdr_v_zero_rows_rankwitnessminornonzeroevaluationmatrixnegativepoint) = S ((S (mdr_s_zero_rows_rankwitnessminornonzeroevaluationmatrixnegativepoint)) * mdr_cc_zero_rows_rankwitness)) /\ exists ff_q_mdr_zero_rows_rankwitnessminornonzeroevaluationmatrixnegativepointcolumn_index. mdr_cb_zero_rows_rankwitness = ff_q_mdr_zero_rows_rankwitnessminornonzeroevaluationmatrixnegativepointcolumn_index * S ((S (mdr_s_zero_rows_rankwitnessminornonzeroevaluationmatrixnegativepoint)) * mdr_cc_zero_rows_rankwitness) + (mdr_v_zero_rows_rankwitnessminornonzeroevaluationmatrixnegativepoint))) /\ (((exists ff_h_mdr_zero_rows_rankwitnessminornonzeroevaluationmatrixnegativepointsource. ff_h_mdr_zero_rows_rankwitnessminornonzeroevaluationmatrixnegativepointsource + S (mdr_a_zero_rows_rankwitnessminornonzeroevaluationmatrixnegative) = S ((S ((mdr_u_zero_rows_rankwitnessminornonzeroevaluationmatrixnegativepoint) * (w) + (mdr_v_zero_rows_rankwitnessminornonzeroevaluationmatrixnegativepoint))) * nc)) /\ exists ff_q_mdr_zero_rows_rankwitnessminornonzeroevaluationmatrixnegativepointsource. nb = ff_q_mdr_zero_rows_rankwitnessminornonzeroevaluationmatrixnegativepointsource * S ((S ((mdr_u_zero_rows_rankwitnessminornonzeroevaluationmatrixnegativepoint) * (w) + (mdr_v_zero_rows_rankwitnessminornonzeroevaluationmatrixnegativepoint))) * nc) + (mdr_a_zero_rows_rankwitnessminornonzeroevaluationmatrixnegative)))))))) /\ (((exists ff_h_mdr_zero_rows_rankwitnessminornonzeroevaluationmatrixnegativeoutput. ff_h_mdr_zero_rows_rankwitnessminornonzeroevaluationmatrixnegativeoutput + S (mdr_a_zero_rows_rankwitnessminornonzeroevaluationmatrixnegative) = S ((S (mdr_i_zero_rows_rankwitnessminornonzeroevaluationmatrixnegative)) * mdr_vc_zero_rows_rankwitnessminornonzeroevaluation)) /\ exists ff_q_mdr_zero_rows_rankwitnessminornonzeroevaluationmatrixnegativeoutput. mdr_vb_zero_rows_rankwitnessminornonzeroevaluation = ff_q_mdr_zero_rows_rankwitnessminornonzeroevaluationmatrixnegativeoutput * S ((S (mdr_i_zero_rows_rankwitnessminornonzeroevaluationmatrixnegative)) * mdr_vc_zero_rows_rankwitnessminornonzeroevaluation) + (mdr_a_zero_rows_rankwitnessminornonzeroevaluationmatrixnegative)))))))) /\ (exists mdr_b_zero_rows_rankwitnessminornonzeroevaluationdeterminant mdr_c_zero_rows_rankwitnessminornonzeroevaluationdeterminant mdr_l_zero_rows_rankwitnessminornonzeroevaluationdeterminant mdr_i_zero_rows_rankwitnessminornonzeroevaluationdeterminant. ((forall mdr_i_zero_rows_rankwitnessminornonzeroevaluationdeterminanth. (exists mdr_gap_zero_rows_rankwitnessminornonzeroevaluationdeterminanthi. mdr_gap_zero_rows_rankwitnessminornonzeroevaluationdeterminanthi + S (mdr_i_zero_rows_rankwitnessminornonzeroevaluationdeterminanth) = (mdr_l_zero_rows_rankwitnessminornonzeroevaluationdeterminant)) -> exists mdr_d_zero_rows_rankwitnessminornonzeroevaluationdeterminanth mdr_pb_zero_rows_rankwitnessminornonzeroevaluationdeterminanth mdr_pc_zero_rows_rankwitnessminornonzeroevaluationdeterminanth mdr_nb_zero_rows_rankwitnessminornonzeroevaluationdeterminanth mdr_nc_zero_rows_rankwitnessminornonzeroevaluationdeterminanth mdr_p_zero_rows_rankwitnessminornonzeroevaluationdeterminanth mdr_n_zero_rows_rankwitnessminornonzeroevaluationdeterminanth. ((exists mdr_z_zero_rows_rankwitnessminornonzeroevaluationdeterminanthr. ((exists mdr_a_zero_rows_rankwitnessminornonzeroevaluationdeterminanthrc mdr_b_zero_rows_rankwitnessminornonzeroevaluationdeterminanthrc mdr_c_zero_rows_rankwitnessminornonzeroevaluationdeterminanthrc mdr_e_zero_rows_rankwitnessminornonzeroevaluationdeterminanthrc mdr_f_zero_rows_rankwitnessminornonzeroevaluationdeterminanthrc. ((mdr_a_zero_rows_rankwitnessminornonzeroevaluationdeterminanthrc = ((mdr_d_zero_rows_rankwitnessminornonzeroevaluationdeterminanth) + (mdr_pb_zero_rows_rankwitnessminornonzeroevaluationdeterminanth)) * S ((mdr_d_zero_rows_rankwitnessminornonzeroevaluationdeterminanth) + (mdr_pb_zero_rows_rankwitnessminornonzeroevaluationdeterminanth)) + ((mdr_pb_zero_rows_rankwitnessminornonzeroevaluationdeterminanth) + (mdr_pb_zero_rows_rankwitnessminornonzeroevaluationdeterminanth))) /\ ((mdr_b_zero_rows_rankwitnessminornonzeroevaluationdeterminanthrc = ((mdr_pc_zero_rows_rankwitnessminornonzeroevaluationdeterminanth) + (mdr_nb_zero_rows_rankwitnessminornonzeroevaluationdeterminanth)) * S ((mdr_pc_zero_rows_rankwitnessminornonzeroevaluationdeterminanth) + (mdr_nb_zero_rows_rankwitnessminornonzeroevaluationdeterminanth)) + ((mdr_nb_zero_rows_rankwitnessminornonzeroevaluationdeterminanth) + (mdr_nb_zero_rows_rankwitnessminornonzeroevaluationdeterminanth))) /\ ((mdr_c_zero_rows_rankwitnessminornonzeroevaluationdeterminanthrc = ((mdr_a_zero_rows_rankwitnessminornonzeroevaluationdeterminanthrc) + (mdr_b_zero_rows_rankwitnessminornonzeroevaluationdeterminanthrc)) * S ((mdr_a_zero_rows_rankwitnessminornonzeroevaluationdeterminanthrc) + (mdr_b_zero_rows_rankwitnessminornonzeroevaluationdeterminanthrc)) + ((mdr_b_zero_rows_rankwitnessminornonzeroevaluationdeterminanthrc) + (mdr_b_zero_rows_rankwitnessminornonzeroevaluationdeterminanthrc))) /\ ((mdr_e_zero_rows_rankwitnessminornonzeroevaluationdeterminanthrc = ((mdr_p_zero_rows_rankwitnessminornonzeroevaluationdeterminanth) + (mdr_n_zero_rows_rankwitnessminornonzeroevaluationdeterminanth)) * S ((mdr_p_zero_rows_rankwitnessminornonzeroevaluationdeterminanth) + (mdr_n_zero_rows_rankwitnessminornonzeroevaluationdeterminanth)) + ((mdr_n_zero_rows_rankwitnessminornonzeroevaluationdeterminanth) + (mdr_n_zero_rows_rankwitnessminornonzeroevaluationdeterminanth))) /\ ((mdr_f_zero_rows_rankwitnessminornonzeroevaluationdeterminanthrc = ((mdr_nc_zero_rows_rankwitnessminornonzeroevaluationdeterminanth) + (mdr_e_zero_rows_rankwitnessminornonzeroevaluationdeterminanthrc)) * S ((mdr_nc_zero_rows_rankwitnessminornonzeroevaluationdeterminanth) + (mdr_e_zero_rows_rankwitnessminornonzeroevaluationdeterminanthrc)) + ((mdr_e_zero_rows_rankwitnessminornonzeroevaluationdeterminanthrc) + (mdr_e_zero_rows_rankwitnessminornonzeroevaluationdeterminanthrc))) /\ ((mdr_z_zero_rows_rankwitnessminornonzeroevaluationdeterminanthr) = ((mdr_c_zero_rows_rankwitnessminornonzeroevaluationdeterminanthrc) + (mdr_f_zero_rows_rankwitnessminornonzeroevaluationdeterminanthrc)) * S ((mdr_c_zero_rows_rankwitnessminornonzeroevaluationdeterminanthrc) + (mdr_f_zero_rows_rankwitnessminornonzeroevaluationdeterminanthrc)) + ((mdr_f_zero_rows_rankwitnessminornonzeroevaluationdeterminanthrc) + (mdr_f_zero_rows_rankwitnessminornonzeroevaluationdeterminanthrc))))))))) /\ (((exists ff_h_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthrb. ff_h_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthrb + S (mdr_z_zero_rows_rankwitnessminornonzeroevaluationdeterminanthr) = S ((S (mdr_i_zero_rows_rankwitnessminornonzeroevaluationdeterminanth)) * mdr_c_zero_rows_rankwitnessminornonzeroevaluationdeterminant)) /\ exists ff_q_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthrb. mdr_b_zero_rows_rankwitnessminornonzeroevaluationdeterminant = ff_q_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthrb * S ((S (mdr_i_zero_rows_rankwitnessminornonzeroevaluationdeterminanth)) * mdr_c_zero_rows_rankwitnessminornonzeroevaluationdeterminant) + (mdr_z_zero_rows_rankwitnessminornonzeroevaluationdeterminanthr))))) /\ (((((mdr_d_zero_rows_rankwitnessminornonzeroevaluationdeterminanth) = 0) /\ (((mdr_p_zero_rows_rankwitnessminornonzeroevaluationdeterminanth) = 1) /\ ((mdr_n_zero_rows_rankwitnessminornonzeroevaluationdeterminanth) = 0))) \/ exists mdr_q_zero_rows_rankwitnessminornonzeroevaluationdeterminanths mdr_eb_zero_rows_rankwitnessminornonzeroevaluationdeterminanths mdr_ec_zero_rows_rankwitnessminornonzeroevaluationdeterminanths mdr_fb_zero_rows_rankwitnessminornonzeroevaluationdeterminanths mdr_fc_zero_rows_rankwitnessminornonzeroevaluationdeterminanths. (((mdr_d_zero_rows_rankwitnessminornonzeroevaluationdeterminanth) = S (mdr_q_zero_rows_rankwitnessminornonzeroevaluationdeterminanths)) /\ ((forall mdr_j_zero_rows_rankwitnessminornonzeroevaluationdeterminanthsc. (exists mdr_gap_zero_rows_rankwitnessminornonzeroevaluationdeterminanthscj. mdr_gap_zero_rows_rankwitnessminornonzeroevaluationdeterminanthscj + S (mdr_j_zero_rows_rankwitnessminornonzeroevaluationdeterminanthsc) = (S (mdr_q_zero_rows_rankwitnessminornonzeroevaluationdeterminanths))) -> exists mdr_i_zero_rows_rankwitnessminornonzeroevaluationdeterminanthsc mdr_up_zero_rows_rankwitnessminornonzeroevaluationdeterminanthsc mdr_us_zero_rows_rankwitnessminornonzeroevaluationdeterminanthsc mdr_un_zero_rows_rankwitnessminornonzeroevaluationdeterminanthsc mdr_ut_zero_rows_rankwitnessminornonzeroevaluationdeterminanthsc mdr_p_zero_rows_rankwitnessminornonzeroevaluationdeterminanthsc mdr_n_zero_rows_rankwitnessminornonzeroevaluationdeterminanthsc. ((exists mdr_gap_zero_rows_rankwitnessminornonzeroevaluationdeterminanthsci. mdr_gap_zero_rows_rankwitnessminornonzeroevaluationdeterminanthsci + S (mdr_i_zero_rows_rankwitnessminornonzeroevaluationdeterminanthsc) = (mdr_i_zero_rows_rankwitnessminornonzeroevaluationdeterminanth)) /\ ((exists mdr_z_zero_rows_rankwitnessminornonzeroevaluationdeterminanthscr. ((exists mdr_a_zero_rows_rankwitnessminornonzeroevaluationdeterminanthscrc mdr_b_zero_rows_rankwitnessminornonzeroevaluationdeterminanthscrc mdr_c_zero_rows_rankwitnessminornonzeroevaluationdeterminanthscrc mdr_e_zero_rows_rankwitnessminornonzeroevaluationdeterminanthscrc mdr_f_zero_rows_rankwitnessminornonzeroevaluationdeterminanthscrc. ((mdr_a_zero_rows_rankwitnessminornonzeroevaluationdeterminanthscrc = ((mdr_q_zero_rows_rankwitnessminornonzeroevaluationdeterminanths) + (mdr_up_zero_rows_rankwitnessminornonzeroevaluationdeterminanthsc)) * S ((mdr_q_zero_rows_rankwitnessminornonzeroevaluationdeterminanths) + (mdr_up_zero_rows_rankwitnessminornonzeroevaluationdeterminanthsc)) + ((mdr_up_zero_rows_rankwitnessminornonzeroevaluationdeterminanthsc) + (mdr_up_zero_rows_rankwitnessminornonzeroevaluationdeterminanthsc))) /\ ((mdr_b_zero_rows_rankwitnessminornonzeroevaluationdeterminanthscrc = ((mdr_us_zero_rows_rankwitnessminornonzeroevaluationdeterminanthsc) + (mdr_un_zero_rows_rankwitnessminornonzeroevaluationdeterminanthsc)) * S ((mdr_us_zero_rows_rankwitnessminornonzeroevaluationdeterminanthsc) + (mdr_un_zero_rows_rankwitnessminornonzeroevaluationdeterminanthsc)) + ((mdr_un_zero_rows_rankwitnessminornonzeroevaluationdeterminanthsc) + (mdr_un_zero_rows_rankwitnessminornonzeroevaluationdeterminanthsc))) /\ ((mdr_c_zero_rows_rankwitnessminornonzeroevaluationdeterminanthscrc = ((mdr_a_zero_rows_rankwitnessminornonzeroevaluationdeterminanthscrc) + (mdr_b_zero_rows_rankwitnessminornonzeroevaluationdeterminanthscrc)) * S ((mdr_a_zero_rows_rankwitnessminornonzeroevaluationdeterminanthscrc) + (mdr_b_zero_rows_rankwitnessminornonzeroevaluationdeterminanthscrc)) + ((mdr_b_zero_rows_rankwitnessminornonzeroevaluationdeterminanthscrc) + (mdr_b_zero_rows_rankwitnessminornonzeroevaluationdeterminanthscrc))) /\ ((mdr_e_zero_rows_rankwitnessminornonzeroevaluationdeterminanthscrc = ((mdr_p_zero_rows_rankwitnessminornonzeroevaluationdeterminanthsc) + (mdr_n_zero_rows_rankwitnessminornonzeroevaluationdeterminanthsc)) * S ((mdr_p_zero_rows_rankwitnessminornonzeroevaluationdeterminanthsc) + (mdr_n_zero_rows_rankwitnessminornonzeroevaluationdeterminanthsc)) + ((mdr_n_zero_rows_rankwitnessminornonzeroevaluationdeterminanthsc) + (mdr_n_zero_rows_rankwitnessminornonzeroevaluationdeterminanthsc))) /\ ((mdr_f_zero_rows_rankwitnessminornonzeroevaluationdeterminanthscrc = ((mdr_ut_zero_rows_rankwitnessminornonzeroevaluationdeterminanthsc) + (mdr_e_zero_rows_rankwitnessminornonzeroevaluationdeterminanthscrc)) * S ((mdr_ut_zero_rows_rankwitnessminornonzeroevaluationdeterminanthsc) + (mdr_e_zero_rows_rankwitnessminornonzeroevaluationdeterminanthscrc)) + ((mdr_e_zero_rows_rankwitnessminornonzeroevaluationdeterminanthscrc) + (mdr_e_zero_rows_rankwitnessminornonzeroevaluationdeterminanthscrc))) /\ ((mdr_z_zero_rows_rankwitnessminornonzeroevaluationdeterminanthscr) = ((mdr_c_zero_rows_rankwitnessminornonzeroevaluationdeterminanthscrc) + (mdr_f_zero_rows_rankwitnessminornonzeroevaluationdeterminanthscrc)) * S ((mdr_c_zero_rows_rankwitnessminornonzeroevaluationdeterminanthscrc) + (mdr_f_zero_rows_rankwitnessminornonzeroevaluationdeterminanthscrc)) + ((mdr_f_zero_rows_rankwitnessminornonzeroevaluationdeterminanthscrc) + (mdr_f_zero_rows_rankwitnessminornonzeroevaluationdeterminanthscrc))))))))) /\ (((exists ff_h_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthscrb. ff_h_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthscrb + S (mdr_z_zero_rows_rankwitnessminornonzeroevaluationdeterminanthscr) = S ((S (mdr_i_zero_rows_rankwitnessminornonzeroevaluationdeterminanthsc)) * mdr_c_zero_rows_rankwitnessminornonzeroevaluationdeterminant)) /\ exists ff_q_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthscrb. mdr_b_zero_rows_rankwitnessminornonzeroevaluationdeterminant = ff_q_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthscrb * S ((S (mdr_i_zero_rows_rankwitnessminornonzeroevaluationdeterminanthsc)) * mdr_c_zero_rows_rankwitnessminornonzeroevaluationdeterminant) + (mdr_z_zero_rows_rankwitnessminornonzeroevaluationdeterminanthscr))))) /\ ((((forall ff_index_mdm_prefix_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthscm_positive. (exists ff_gap_mdm_lt_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthscm_positive_index_bound. ff_gap_mdm_lt_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthscm_positive_index_bound + S (ff_index_mdm_prefix_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthscm_positive) = ((mdr_q_zero_rows_rankwitnessminornonzeroevaluationdeterminanths) * (mdr_q_zero_rows_rankwitnessminornonzeroevaluationdeterminanths))) -> exists ff_row_mdm_prefix_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthscm_positive ff_column_mdm_prefix_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthscm_positive ff_value_mdm_prefix_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthscm_positive. (ff_index_mdm_prefix_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthscm_positive = (mdr_q_zero_rows_rankwitnessminornonzeroevaluationdeterminanths) * ff_row_mdm_prefix_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthscm_positive + ff_column_mdm_prefix_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthscm_positive /\ ((exists ff_gap_mdm_lt_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthscm_positive_column_bound. ff_gap_mdm_lt_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthscm_positive_column_bound + S (ff_column_mdm_prefix_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthscm_positive) = (mdr_q_zero_rows_rankwitnessminornonzeroevaluationdeterminanths)) /\ ((exists ff_row_mdm_cell_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthscm_positive_cell ff_column_mdm_cell_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthscm_positive_cell. (((((exists ff_gap_mdm_lt_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthscm_positive_cell_row_before. ff_gap_mdm_lt_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthscm_positive_cell_row_before + S (ff_row_mdm_prefix_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthscm_positive) = (0)) /\ ff_row_mdm_cell_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthscm_positive_cell = ff_row_mdm_prefix_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthscm_positive) \/ ((exists ff_gap_mdm_le_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthscm_positive_cell_row_after. ff_gap_mdm_le_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthscm_positive_cell_row_after + (0) = (ff_row_mdm_prefix_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthscm_positive)) /\ ff_row_mdm_cell_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthscm_positive_cell = S ff_row_mdm_prefix_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthscm_positive))) /\ (((((exists ff_gap_mdm_lt_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthscm_positive_cell_column_before. ff_gap_mdm_lt_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthscm_positive_cell_column_before + S (ff_column_mdm_prefix_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthscm_positive) = (mdr_j_zero_rows_rankwitnessminornonzeroevaluationdeterminanthsc)) /\ ff_column_mdm_cell_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthscm_positive_cell = ff_column_mdm_prefix_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthscm_positive) \/ ((exists ff_gap_mdm_le_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthscm_positive_cell_column_after. ff_gap_mdm_le_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthscm_positive_cell_column_after + (mdr_j_zero_rows_rankwitnessminornonzeroevaluationdeterminanthsc) = (ff_column_mdm_prefix_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthscm_positive)) /\ ff_column_mdm_cell_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthscm_positive_cell = S ff_column_mdm_prefix_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthscm_positive))) /\ (((exists ff_h_mdm_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthscm_positive_cell_source. ff_h_mdm_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthscm_positive_cell_source + S (ff_value_mdm_prefix_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthscm_positive) = S ((S ((ff_row_mdm_cell_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthscm_positive_cell) * (S (mdr_q_zero_rows_rankwitnessminornonzeroevaluationdeterminanths)) + (ff_column_mdm_cell_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthscm_positive_cell))) * mdr_pc_zero_rows_rankwitnessminornonzeroevaluationdeterminanth)) /\ exists ff_q_mdm_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthscm_positive_cell_source. mdr_pb_zero_rows_rankwitnessminornonzeroevaluationdeterminanth = ff_q_mdm_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthscm_positive_cell_source * S ((S ((ff_row_mdm_cell_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthscm_positive_cell) * (S (mdr_q_zero_rows_rankwitnessminornonzeroevaluationdeterminanths)) + (ff_column_mdm_cell_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthscm_positive_cell))) * mdr_pc_zero_rows_rankwitnessminornonzeroevaluationdeterminanth) + (ff_value_mdm_prefix_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthscm_positive)))))) /\ (((exists ff_h_mdm_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthscm_positive_target. ff_h_mdm_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthscm_positive_target + S (ff_value_mdm_prefix_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthscm_positive) = S ((S (ff_index_mdm_prefix_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthscm_positive)) * mdr_us_zero_rows_rankwitnessminornonzeroevaluationdeterminanthsc)) /\ exists ff_q_mdm_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthscm_positive_target. mdr_up_zero_rows_rankwitnessminornonzeroevaluationdeterminanthsc = ff_q_mdm_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthscm_positive_target * S ((S (ff_index_mdm_prefix_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthscm_positive)) * mdr_us_zero_rows_rankwitnessminornonzeroevaluationdeterminanthsc) + (ff_value_mdm_prefix_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthscm_positive))))))) /\ (forall ff_index_mdm_prefix_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthscm_negative. (exists ff_gap_mdm_lt_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthscm_negative_index_bound. ff_gap_mdm_lt_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthscm_negative_index_bound + S (ff_index_mdm_prefix_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthscm_negative) = ((mdr_q_zero_rows_rankwitnessminornonzeroevaluationdeterminanths) * (mdr_q_zero_rows_rankwitnessminornonzeroevaluationdeterminanths))) -> exists ff_row_mdm_prefix_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthscm_negative ff_column_mdm_prefix_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthscm_negative ff_value_mdm_prefix_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthscm_negative. (ff_index_mdm_prefix_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthscm_negative = (mdr_q_zero_rows_rankwitnessminornonzeroevaluationdeterminanths) * ff_row_mdm_prefix_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthscm_negative + ff_column_mdm_prefix_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthscm_negative /\ ((exists ff_gap_mdm_lt_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthscm_negative_column_bound. ff_gap_mdm_lt_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthscm_negative_column_bound + S (ff_column_mdm_prefix_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthscm_negative) = (mdr_q_zero_rows_rankwitnessminornonzeroevaluationdeterminanths)) /\ ((exists ff_row_mdm_cell_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthscm_negative_cell ff_column_mdm_cell_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthscm_negative_cell. (((((exists ff_gap_mdm_lt_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthscm_negative_cell_row_before. ff_gap_mdm_lt_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthscm_negative_cell_row_before + S (ff_row_mdm_prefix_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthscm_negative) = (0)) /\ ff_row_mdm_cell_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthscm_negative_cell = ff_row_mdm_prefix_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthscm_negative) \/ ((exists ff_gap_mdm_le_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthscm_negative_cell_row_after. ff_gap_mdm_le_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthscm_negative_cell_row_after + (0) = (ff_row_mdm_prefix_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthscm_negative)) /\ ff_row_mdm_cell_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthscm_negative_cell = S ff_row_mdm_prefix_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthscm_negative))) /\ (((((exists ff_gap_mdm_lt_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthscm_negative_cell_column_before. ff_gap_mdm_lt_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthscm_negative_cell_column_before + S (ff_column_mdm_prefix_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthscm_negative) = (mdr_j_zero_rows_rankwitnessminornonzeroevaluationdeterminanthsc)) /\ ff_column_mdm_cell_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthscm_negative_cell = ff_column_mdm_prefix_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthscm_negative) \/ ((exists ff_gap_mdm_le_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthscm_negative_cell_column_after. ff_gap_mdm_le_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthscm_negative_cell_column_after + (mdr_j_zero_rows_rankwitnessminornonzeroevaluationdeterminanthsc) = (ff_column_mdm_prefix_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthscm_negative)) /\ ff_column_mdm_cell_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthscm_negative_cell = S ff_column_mdm_prefix_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthscm_negative))) /\ (((exists ff_h_mdm_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthscm_negative_cell_source. ff_h_mdm_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthscm_negative_cell_source + S (ff_value_mdm_prefix_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthscm_negative) = S ((S ((ff_row_mdm_cell_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthscm_negative_cell) * (S (mdr_q_zero_rows_rankwitnessminornonzeroevaluationdeterminanths)) + (ff_column_mdm_cell_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthscm_negative_cell))) * mdr_nc_zero_rows_rankwitnessminornonzeroevaluationdeterminanth)) /\ exists ff_q_mdm_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthscm_negative_cell_source. mdr_nb_zero_rows_rankwitnessminornonzeroevaluationdeterminanth = ff_q_mdm_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthscm_negative_cell_source * S ((S ((ff_row_mdm_cell_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthscm_negative_cell) * (S (mdr_q_zero_rows_rankwitnessminornonzeroevaluationdeterminanths)) + (ff_column_mdm_cell_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthscm_negative_cell))) * mdr_nc_zero_rows_rankwitnessminornonzeroevaluationdeterminanth) + (ff_value_mdm_prefix_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthscm_negative)))))) /\ (((exists ff_h_mdm_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthscm_negative_target. ff_h_mdm_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthscm_negative_target + S (ff_value_mdm_prefix_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthscm_negative) = S ((S (ff_index_mdm_prefix_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthscm_negative)) * mdr_ut_zero_rows_rankwitnessminornonzeroevaluationdeterminanthsc)) /\ exists ff_q_mdm_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthscm_negative_target. mdr_un_zero_rows_rankwitnessminornonzeroevaluationdeterminanthsc = ff_q_mdm_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthscm_negative_target * S ((S (ff_index_mdm_prefix_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthscm_negative)) * mdr_ut_zero_rows_rankwitnessminornonzeroevaluationdeterminanthsc) + (ff_value_mdm_prefix_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthscm_negative))))))))) /\ ((((exists ff_h_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthscp. ff_h_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthscp + S (mdr_p_zero_rows_rankwitnessminornonzeroevaluationdeterminanthsc) = S ((S (mdr_j_zero_rows_rankwitnessminornonzeroevaluationdeterminanthsc)) * mdr_ec_zero_rows_rankwitnessminornonzeroevaluationdeterminanths)) /\ exists ff_q_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthscp. mdr_eb_zero_rows_rankwitnessminornonzeroevaluationdeterminanths = ff_q_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthscp * S ((S (mdr_j_zero_rows_rankwitnessminornonzeroevaluationdeterminanthsc)) * mdr_ec_zero_rows_rankwitnessminornonzeroevaluationdeterminanths) + (mdr_p_zero_rows_rankwitnessminornonzeroevaluationdeterminanthsc))) /\ (((exists ff_h_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthscn. ff_h_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthscn + S (mdr_n_zero_rows_rankwitnessminornonzeroevaluationdeterminanthsc) = S ((S (mdr_j_zero_rows_rankwitnessminornonzeroevaluationdeterminanthsc)) * mdr_fc_zero_rows_rankwitnessminornonzeroevaluationdeterminanths)) /\ exists ff_q_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthscn. mdr_fb_zero_rows_rankwitnessminornonzeroevaluationdeterminanths = ff_q_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthscn * S ((S (mdr_j_zero_rows_rankwitnessminornonzeroevaluationdeterminanthsc)) * mdr_fc_zero_rows_rankwitnessminornonzeroevaluationdeterminanths) + (mdr_n_zero_rows_rankwitnessminornonzeroevaluationdeterminanthsc)))))))) /\ (exists ff_ub_mce_fold_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthsf ff_uc_mce_fold_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthsf ff_vb_mce_fold_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthsf ff_vc_mce_fold_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthsf. ((forall ff_index_mce_alternating_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthsf_prefix. (exists ff_gap_mce_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthsf_prefix_index. ff_gap_mce_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthsf_prefix_index + S (ff_index_mce_alternating_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthsf_prefix) = (S (mdr_q_zero_rows_rankwitnessminornonzeroevaluationdeterminanths))) -> exists ff_ap_mce_alternating_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthsf_prefix ff_an_mce_alternating_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthsf_prefix ff_bp_mce_alternating_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthsf_prefix ff_bn_mce_alternating_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthsf_prefix ff_p_mce_alternating_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthsf_prefix ff_n_mce_alternating_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthsf_prefix. ((((exists ff_h_mce_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthsf_prefix_ap. ff_h_mce_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthsf_prefix_ap + S (ff_ap_mce_alternating_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthsf_prefix) = S ((S (ff_index_mce_alternating_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthsf_prefix)) * mdr_pc_zero_rows_rankwitnessminornonzeroevaluationdeterminanth)) /\ exists ff_q_mce_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthsf_prefix_ap. mdr_pb_zero_rows_rankwitnessminornonzeroevaluationdeterminanth = ff_q_mce_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthsf_prefix_ap * S ((S (ff_index_mce_alternating_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthsf_prefix)) * mdr_pc_zero_rows_rankwitnessminornonzeroevaluationdeterminanth) + (ff_ap_mce_alternating_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthsf_prefix))) /\ ((((exists ff_h_mce_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthsf_prefix_an. ff_h_mce_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthsf_prefix_an + S (ff_an_mce_alternating_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthsf_prefix) = S ((S (ff_index_mce_alternating_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthsf_prefix)) * mdr_nc_zero_rows_rankwitnessminornonzeroevaluationdeterminanth)) /\ exists ff_q_mce_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthsf_prefix_an. mdr_nb_zero_rows_rankwitnessminornonzeroevaluationdeterminanth = ff_q_mce_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthsf_prefix_an * S ((S (ff_index_mce_alternating_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthsf_prefix)) * mdr_nc_zero_rows_rankwitnessminornonzeroevaluationdeterminanth) + (ff_an_mce_alternating_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthsf_prefix))) /\ ((((exists ff_h_mce_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthsf_prefix_bp. ff_h_mce_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthsf_prefix_bp + S (ff_bp_mce_alternating_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthsf_prefix) = S ((S (ff_index_mce_alternating_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthsf_prefix)) * mdr_ec_zero_rows_rankwitnessminornonzeroevaluationdeterminanths)) /\ exists ff_q_mce_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthsf_prefix_bp. mdr_eb_zero_rows_rankwitnessminornonzeroevaluationdeterminanths = ff_q_mce_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthsf_prefix_bp * S ((S (ff_index_mce_alternating_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthsf_prefix)) * mdr_ec_zero_rows_rankwitnessminornonzeroevaluationdeterminanths) + (ff_bp_mce_alternating_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthsf_prefix))) /\ ((((exists ff_h_mce_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthsf_prefix_bn. ff_h_mce_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthsf_prefix_bn + S (ff_bn_mce_alternating_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthsf_prefix) = S ((S (ff_index_mce_alternating_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthsf_prefix)) * mdr_fc_zero_rows_rankwitnessminornonzeroevaluationdeterminanths)) /\ exists ff_q_mce_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthsf_prefix_bn. mdr_fb_zero_rows_rankwitnessminornonzeroevaluationdeterminanths = ff_q_mce_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthsf_prefix_bn * S ((S (ff_index_mce_alternating_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthsf_prefix)) * mdr_fc_zero_rows_rankwitnessminornonzeroevaluationdeterminanths) + (ff_bn_mce_alternating_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthsf_prefix))) /\ ((((exists ff_h_mce_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthsf_prefix_positive. ff_h_mce_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthsf_prefix_positive + S (ff_p_mce_alternating_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthsf_prefix) = S ((S (ff_index_mce_alternating_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthsf_prefix)) * ff_uc_mce_fold_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthsf)) /\ exists ff_q_mce_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthsf_prefix_positive. ff_ub_mce_fold_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthsf = ff_q_mce_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthsf_prefix_positive * S ((S (ff_index_mce_alternating_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthsf_prefix)) * ff_uc_mce_fold_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthsf) + (ff_p_mce_alternating_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthsf_prefix))) /\ ((((exists ff_h_mce_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthsf_prefix_negative. ff_h_mce_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthsf_prefix_negative + S (ff_n_mce_alternating_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthsf_prefix) = S ((S (ff_index_mce_alternating_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthsf_prefix)) * ff_vc_mce_fold_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthsf)) /\ exists ff_q_mce_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthsf_prefix_negative. ff_vb_mce_fold_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthsf = ff_q_mce_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthsf_prefix_negative * S ((S (ff_index_mce_alternating_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthsf_prefix)) * ff_vc_mce_fold_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthsf) + (ff_n_mce_alternating_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthsf_prefix))) /\ (((exists ff_even_mce_term_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthsf_prefix_term. ff_index_mce_alternating_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthsf_prefix = 2 * ff_even_mce_term_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthsf_prefix_term) /\ (ff_p_mce_alternating_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthsf_prefix = (ff_ap_mce_alternating_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthsf_prefix) * (ff_bp_mce_alternating_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthsf_prefix) + (ff_an_mce_alternating_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthsf_prefix) * (ff_bn_mce_alternating_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthsf_prefix) /\ ff_n_mce_alternating_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthsf_prefix = (ff_ap_mce_alternating_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthsf_prefix) * (ff_bn_mce_alternating_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthsf_prefix) + (ff_an_mce_alternating_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthsf_prefix) * (ff_bp_mce_alternating_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthsf_prefix))) \/ ((exists ff_odd_mce_term_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthsf_prefix_term. ff_index_mce_alternating_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthsf_prefix = 2 * ff_odd_mce_term_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthsf_prefix_term + 1) /\ (ff_p_mce_alternating_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthsf_prefix = (ff_ap_mce_alternating_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthsf_prefix) * (ff_bn_mce_alternating_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthsf_prefix) + (ff_an_mce_alternating_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthsf_prefix) * (ff_bp_mce_alternating_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthsf_prefix) /\ ff_n_mce_alternating_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthsf_prefix = (ff_ap_mce_alternating_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthsf_prefix) * (ff_bp_mce_alternating_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthsf_prefix) + (ff_an_mce_alternating_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthsf_prefix) * (ff_bn_mce_alternating_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthsf_prefix))))))))))) /\ ((exists ff_u_mce_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthsf_positive ff_v_mce_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthsf_positive. ((((exists ff_h_mce_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthsf_positive_start. ff_h_mce_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthsf_positive_start + S (0) = S ((S (0)) * ff_v_mce_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthsf_positive)) /\ exists ff_q_mce_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthsf_positive_start. ff_u_mce_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthsf_positive = ff_q_mce_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthsf_positive_start * S ((S (0)) * ff_v_mce_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthsf_positive) + (0))) /\ ((((exists ff_h_mce_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthsf_positive_terminal. ff_h_mce_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthsf_positive_terminal + S (mdr_p_zero_rows_rankwitnessminornonzeroevaluationdeterminanth) = S ((S ((S (mdr_q_zero_rows_rankwitnessminornonzeroevaluationdeterminanths)))) * ff_v_mce_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthsf_positive)) /\ exists ff_q_mce_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthsf_positive_terminal. ff_u_mce_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthsf_positive = ff_q_mce_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthsf_positive_terminal * S ((S ((S (mdr_q_zero_rows_rankwitnessminornonzeroevaluationdeterminanths)))) * ff_v_mce_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthsf_positive) + (mdr_p_zero_rows_rankwitnessminornonzeroevaluationdeterminanth))) /\ forall ff_i_mce_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthsf_positive. (exists ff_lt_mce_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthsf_positive_bound. ff_lt_mce_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthsf_positive_bound + S ff_i_mce_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthsf_positive = (S (mdr_q_zero_rows_rankwitnessminornonzeroevaluationdeterminanths))) -> exists ff_a_mce_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthsf_positive ff_r_mce_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthsf_positive ff_s_mce_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthsf_positive. ((((exists ff_h_mce_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthsf_positive_summand. ff_h_mce_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthsf_positive_summand + S (ff_a_mce_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthsf_positive) = S ((S (ff_i_mce_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthsf_positive)) * ff_uc_mce_fold_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthsf)) /\ exists ff_q_mce_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthsf_positive_summand. ff_ub_mce_fold_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthsf = ff_q_mce_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthsf_positive_summand * S ((S (ff_i_mce_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthsf_positive)) * ff_uc_mce_fold_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthsf) + (ff_a_mce_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthsf_positive))) /\ ((((exists ff_h_mce_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthsf_positive_partial. ff_h_mce_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthsf_positive_partial + S (ff_r_mce_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthsf_positive) = S ((S (ff_i_mce_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthsf_positive)) * ff_v_mce_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthsf_positive)) /\ exists ff_q_mce_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthsf_positive_partial. ff_u_mce_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthsf_positive = ff_q_mce_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthsf_positive_partial * S ((S (ff_i_mce_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthsf_positive)) * ff_v_mce_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthsf_positive) + (ff_r_mce_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthsf_positive))) /\ ((((exists ff_h_mce_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthsf_positive_successor. ff_h_mce_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthsf_positive_successor + S (ff_s_mce_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthsf_positive) = S ((S (S ff_i_mce_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthsf_positive)) * ff_v_mce_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthsf_positive)) /\ exists ff_q_mce_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthsf_positive_successor. ff_u_mce_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthsf_positive = ff_q_mce_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthsf_positive_successor * S ((S (S ff_i_mce_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthsf_positive)) * ff_v_mce_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthsf_positive) + (ff_s_mce_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthsf_positive))) /\ ff_s_mce_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthsf_positive = ff_r_mce_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthsf_positive + ff_a_mce_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthsf_positive)))))) /\ (exists ff_u_mce_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthsf_negative ff_v_mce_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthsf_negative. ((((exists ff_h_mce_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthsf_negative_start. ff_h_mce_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthsf_negative_start + S (0) = S ((S (0)) * ff_v_mce_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthsf_negative)) /\ exists ff_q_mce_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthsf_negative_start. ff_u_mce_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthsf_negative = ff_q_mce_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthsf_negative_start * S ((S (0)) * ff_v_mce_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthsf_negative) + (0))) /\ ((((exists ff_h_mce_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthsf_negative_terminal. ff_h_mce_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthsf_negative_terminal + S (mdr_n_zero_rows_rankwitnessminornonzeroevaluationdeterminanth) = S ((S ((S (mdr_q_zero_rows_rankwitnessminornonzeroevaluationdeterminanths)))) * ff_v_mce_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthsf_negative)) /\ exists ff_q_mce_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthsf_negative_terminal. ff_u_mce_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthsf_negative = ff_q_mce_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthsf_negative_terminal * S ((S ((S (mdr_q_zero_rows_rankwitnessminornonzeroevaluationdeterminanths)))) * ff_v_mce_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthsf_negative) + (mdr_n_zero_rows_rankwitnessminornonzeroevaluationdeterminanth))) /\ forall ff_i_mce_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthsf_negative. (exists ff_lt_mce_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthsf_negative_bound. ff_lt_mce_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthsf_negative_bound + S ff_i_mce_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthsf_negative = (S (mdr_q_zero_rows_rankwitnessminornonzeroevaluationdeterminanths))) -> exists ff_a_mce_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthsf_negative ff_r_mce_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthsf_negative ff_s_mce_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthsf_negative. ((((exists ff_h_mce_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthsf_negative_summand. ff_h_mce_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthsf_negative_summand + S (ff_a_mce_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthsf_negative) = S ((S (ff_i_mce_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthsf_negative)) * ff_vc_mce_fold_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthsf)) /\ exists ff_q_mce_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthsf_negative_summand. ff_vb_mce_fold_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthsf = ff_q_mce_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthsf_negative_summand * S ((S (ff_i_mce_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthsf_negative)) * ff_vc_mce_fold_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthsf) + (ff_a_mce_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthsf_negative))) /\ ((((exists ff_h_mce_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthsf_negative_partial. ff_h_mce_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthsf_negative_partial + S (ff_r_mce_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthsf_negative) = S ((S (ff_i_mce_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthsf_negative)) * ff_v_mce_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthsf_negative)) /\ exists ff_q_mce_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthsf_negative_partial. ff_u_mce_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthsf_negative = ff_q_mce_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthsf_negative_partial * S ((S (ff_i_mce_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthsf_negative)) * ff_v_mce_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthsf_negative) + (ff_r_mce_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthsf_negative))) /\ ((((exists ff_h_mce_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthsf_negative_successor. ff_h_mce_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthsf_negative_successor + S (ff_s_mce_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthsf_negative) = S ((S (S ff_i_mce_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthsf_negative)) * ff_v_mce_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthsf_negative)) /\ exists ff_q_mce_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthsf_negative_successor. ff_u_mce_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthsf_negative = ff_q_mce_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthsf_negative_successor * S ((S (S ff_i_mce_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthsf_negative)) * ff_v_mce_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthsf_negative) + (ff_s_mce_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthsf_negative))) /\ ff_s_mce_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthsf_negative = ff_r_mce_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthsf_negative + ff_a_mce_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminanthsf_negative))))))))))))))) /\ ((exists mdr_gap_zero_rows_rankwitnessminornonzeroevaluationdeterminanti. mdr_gap_zero_rows_rankwitnessminornonzeroevaluationdeterminanti + S (mdr_i_zero_rows_rankwitnessminornonzeroevaluationdeterminant) = (mdr_l_zero_rows_rankwitnessminornonzeroevaluationdeterminant)) /\ (exists mdr_z_zero_rows_rankwitnessminornonzeroevaluationdeterminantr. ((exists mdr_a_zero_rows_rankwitnessminornonzeroevaluationdeterminantrc mdr_b_zero_rows_rankwitnessminornonzeroevaluationdeterminantrc mdr_c_zero_rows_rankwitnessminornonzeroevaluationdeterminantrc mdr_e_zero_rows_rankwitnessminornonzeroevaluationdeterminantrc mdr_f_zero_rows_rankwitnessminornonzeroevaluationdeterminantrc. ((mdr_a_zero_rows_rankwitnessminornonzeroevaluationdeterminantrc = ((0) + (mdr_ub_zero_rows_rankwitnessminornonzeroevaluation)) * S ((0) + (mdr_ub_zero_rows_rankwitnessminornonzeroevaluation)) + ((mdr_ub_zero_rows_rankwitnessminornonzeroevaluation) + (mdr_ub_zero_rows_rankwitnessminornonzeroevaluation))) /\ ((mdr_b_zero_rows_rankwitnessminornonzeroevaluationdeterminantrc = ((mdr_uc_zero_rows_rankwitnessminornonzeroevaluation) + (mdr_vb_zero_rows_rankwitnessminornonzeroevaluation)) * S ((mdr_uc_zero_rows_rankwitnessminornonzeroevaluation) + (mdr_vb_zero_rows_rankwitnessminornonzeroevaluation)) + ((mdr_vb_zero_rows_rankwitnessminornonzeroevaluation) + (mdr_vb_zero_rows_rankwitnessminornonzeroevaluation))) /\ ((mdr_c_zero_rows_rankwitnessminornonzeroevaluationdeterminantrc = ((mdr_a_zero_rows_rankwitnessminornonzeroevaluationdeterminantrc) + (mdr_b_zero_rows_rankwitnessminornonzeroevaluationdeterminantrc)) * S ((mdr_a_zero_rows_rankwitnessminornonzeroevaluationdeterminantrc) + (mdr_b_zero_rows_rankwitnessminornonzeroevaluationdeterminantrc)) + ((mdr_b_zero_rows_rankwitnessminornonzeroevaluationdeterminantrc) + (mdr_b_zero_rows_rankwitnessminornonzeroevaluationdeterminantrc))) /\ ((mdr_e_zero_rows_rankwitnessminornonzeroevaluationdeterminantrc = ((mdr_p_zero_rows_rankwitnessminornonzero) + (mdr_n_zero_rows_rankwitnessminornonzero)) * S ((mdr_p_zero_rows_rankwitnessminornonzero) + (mdr_n_zero_rows_rankwitnessminornonzero)) + ((mdr_n_zero_rows_rankwitnessminornonzero) + (mdr_n_zero_rows_rankwitnessminornonzero))) /\ ((mdr_f_zero_rows_rankwitnessminornonzeroevaluationdeterminantrc = ((mdr_vc_zero_rows_rankwitnessminornonzeroevaluation) + (mdr_e_zero_rows_rankwitnessminornonzeroevaluationdeterminantrc)) * S ((mdr_vc_zero_rows_rankwitnessminornonzeroevaluation) + (mdr_e_zero_rows_rankwitnessminornonzeroevaluationdeterminantrc)) + ((mdr_e_zero_rows_rankwitnessminornonzeroevaluationdeterminantrc) + (mdr_e_zero_rows_rankwitnessminornonzeroevaluationdeterminantrc))) /\ ((mdr_z_zero_rows_rankwitnessminornonzeroevaluationdeterminantr) = ((mdr_c_zero_rows_rankwitnessminornonzeroevaluationdeterminantrc) + (mdr_f_zero_rows_rankwitnessminornonzeroevaluationdeterminantrc)) * S ((mdr_c_zero_rows_rankwitnessminornonzeroevaluationdeterminantrc) + (mdr_f_zero_rows_rankwitnessminornonzeroevaluationdeterminantrc)) + ((mdr_f_zero_rows_rankwitnessminornonzeroevaluationdeterminantrc) + (mdr_f_zero_rows_rankwitnessminornonzeroevaluationdeterminantrc))))))))) /\ (((exists ff_h_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminantrb. ff_h_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminantrb + S (mdr_z_zero_rows_rankwitnessminornonzeroevaluationdeterminantr) = S ((S (mdr_i_zero_rows_rankwitnessminornonzeroevaluationdeterminant)) * mdr_c_zero_rows_rankwitnessminornonzeroevaluationdeterminant)) /\ exists ff_q_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminantrb. mdr_b_zero_rows_rankwitnessminornonzeroevaluationdeterminant = ff_q_mdr_zero_rows_rankwitnessminornonzeroevaluationdeterminantrb * S ((S (mdr_i_zero_rows_rankwitnessminornonzeroevaluationdeterminant)) * mdr_c_zero_rows_rankwitnessminornonzeroevaluationdeterminant) + (mdr_z_zero_rows_rankwitnessminornonzeroevaluationdeterminantr)))))))))) /\ (~(mdr_p_zero_rows_rankwitnessminornonzero = mdr_n_zero_rows_rankwitnessminornonzero)))))))) /\ (forall mdr_q_zero_rows_rank. (exists mdr_gap_zero_rows_rankhigher. mdr_gap_zero_rows_rankhigher + S (0) = (mdr_q_zero_rows_rank)) -> (forall mdr_rb_zero_rows_rankzero mdr_rc_zero_rows_rankzero mdr_cb_zero_rows_rankzero mdr_cc_zero_rows_rankzero mdr_p_zero_rows_rankzero mdr_n_zero_rows_rankzero. (((forall fom_index_mrf_zero_rows_rankzerorowsbound. (exists fom_gap_mrf_zero_rows_rankzerorowsbound_index_bound. fom_gap_mrf_zero_rows_rankzerorowsbound_index_bound + S (fom_index_mrf_zero_rows_rankzerorowsbound) = mdr_q_zero_rows_rank) -> exists fom_value_mrf_zero_rows_rankzerorowsbound. ((((exists fom_beta_height_mrf_zero_rows_rankzerorowsbound_entry. fom_beta_height_mrf_zero_rows_rankzerorowsbound_entry + S (fom_value_mrf_zero_rows_rankzerorowsbound) = S ((S (fom_index_mrf_zero_rows_rankzerorowsbound)) * mdr_rc_zero_rows_rankzero)) /\ exists fom_beta_quotient_mrf_zero_rows_rankzerorowsbound_entry. mdr_rb_zero_rows_rankzero = fom_beta_quotient_mrf_zero_rows_rankzerorowsbound_entry * S ((S (fom_index_mrf_zero_rows_rankzerorowsbound)) * mdr_rc_zero_rows_rankzero) + (fom_value_mrf_zero_rows_rankzerorowsbound))) /\ (exists fom_gap_mrf_zero_rows_rankzerorowsbound_value_bound. fom_gap_mrf_zero_rows_rankzerorowsbound_value_bound + S (fom_value_mrf_zero_rows_rankzerorowsbound) = 0))) /\ (forall mdr_i_zero_rows_rankzerorowsdistinct mdr_j_zero_rows_rankzerorowsdistinct mdr_a_zero_rows_rankzerorowsdistinct. (exists mdr_gap_zero_rows_rankzerorowsdistincti. mdr_gap_zero_rows_rankzerorowsdistincti + S (mdr_i_zero_rows_rankzerorowsdistinct) = (mdr_q_zero_rows_rank)) -> (exists mdr_gap_zero_rows_rankzerorowsdistinctj. mdr_gap_zero_rows_rankzerorowsdistinctj + S (mdr_j_zero_rows_rankzerorowsdistinct) = (mdr_q_zero_rows_rank)) -> (((exists ff_h_mdr_zero_rows_rankzerorowsdistinctfirst. ff_h_mdr_zero_rows_rankzerorowsdistinctfirst + S (mdr_a_zero_rows_rankzerorowsdistinct) = S ((S (mdr_i_zero_rows_rankzerorowsdistinct)) * mdr_rc_zero_rows_rankzero)) /\ exists ff_q_mdr_zero_rows_rankzerorowsdistinctfirst. mdr_rb_zero_rows_rankzero = ff_q_mdr_zero_rows_rankzerorowsdistinctfirst * S ((S (mdr_i_zero_rows_rankzerorowsdistinct)) * mdr_rc_zero_rows_rankzero) + (mdr_a_zero_rows_rankzerorowsdistinct))) -> (((exists ff_h_mdr_zero_rows_rankzerorowsdistinctsecond. ff_h_mdr_zero_rows_rankzerorowsdistinctsecond + S (mdr_a_zero_rows_rankzerorowsdistinct) = S ((S (mdr_j_zero_rows_rankzerorowsdistinct)) * mdr_rc_zero_rows_rankzero)) /\ exists ff_q_mdr_zero_rows_rankzerorowsdistinctsecond. mdr_rb_zero_rows_rankzero = ff_q_mdr_zero_rows_rankzerorowsdistinctsecond * S ((S (mdr_j_zero_rows_rankzerorowsdistinct)) * mdr_rc_zero_rows_rankzero) + (mdr_a_zero_rows_rankzerorowsdistinct))) -> mdr_i_zero_rows_rankzerorowsdistinct = mdr_j_zero_rows_rankzerorowsdistinct))) -> (((forall fom_index_mrf_zero_rows_rankzerocolumnsbound. (exists fom_gap_mrf_zero_rows_rankzerocolumnsbound_index_bound. fom_gap_mrf_zero_rows_rankzerocolumnsbound_index_bound + S (fom_index_mrf_zero_rows_rankzerocolumnsbound) = mdr_q_zero_rows_rank) -> exists fom_value_mrf_zero_rows_rankzerocolumnsbound. ((((exists fom_beta_height_mrf_zero_rows_rankzerocolumnsbound_entry. fom_beta_height_mrf_zero_rows_rankzerocolumnsbound_entry + S (fom_value_mrf_zero_rows_rankzerocolumnsbound) = S ((S (fom_index_mrf_zero_rows_rankzerocolumnsbound)) * mdr_cc_zero_rows_rankzero)) /\ exists fom_beta_quotient_mrf_zero_rows_rankzerocolumnsbound_entry. mdr_cb_zero_rows_rankzero = fom_beta_quotient_mrf_zero_rows_rankzerocolumnsbound_entry * S ((S (fom_index_mrf_zero_rows_rankzerocolumnsbound)) * mdr_cc_zero_rows_rankzero) + (fom_value_mrf_zero_rows_rankzerocolumnsbound))) /\ (exists fom_gap_mrf_zero_rows_rankzerocolumnsbound_value_bound. fom_gap_mrf_zero_rows_rankzerocolumnsbound_value_bound + S (fom_value_mrf_zero_rows_rankzerocolumnsbound) = w))) /\ (forall mdr_i_zero_rows_rankzerocolumnsdistinct mdr_j_zero_rows_rankzerocolumnsdistinct mdr_a_zero_rows_rankzerocolumnsdistinct. (exists mdr_gap_zero_rows_rankzerocolumnsdistincti. mdr_gap_zero_rows_rankzerocolumnsdistincti + S (mdr_i_zero_rows_rankzerocolumnsdistinct) = (mdr_q_zero_rows_rank)) -> (exists mdr_gap_zero_rows_rankzerocolumnsdistinctj. mdr_gap_zero_rows_rankzerocolumnsdistinctj + S (mdr_j_zero_rows_rankzerocolumnsdistinct) = (mdr_q_zero_rows_rank)) -> (((exists ff_h_mdr_zero_rows_rankzerocolumnsdistinctfirst. ff_h_mdr_zero_rows_rankzerocolumnsdistinctfirst + S (mdr_a_zero_rows_rankzerocolumnsdistinct) = S ((S (mdr_i_zero_rows_rankzerocolumnsdistinct)) * mdr_cc_zero_rows_rankzero)) /\ exists ff_q_mdr_zero_rows_rankzerocolumnsdistinctfirst. mdr_cb_zero_rows_rankzero = ff_q_mdr_zero_rows_rankzerocolumnsdistinctfirst * S ((S (mdr_i_zero_rows_rankzerocolumnsdistinct)) * mdr_cc_zero_rows_rankzero) + (mdr_a_zero_rows_rankzerocolumnsdistinct))) -> (((exists ff_h_mdr_zero_rows_rankzerocolumnsdistinctsecond. ff_h_mdr_zero_rows_rankzerocolumnsdistinctsecond + S (mdr_a_zero_rows_rankzerocolumnsdistinct) = S ((S (mdr_j_zero_rows_rankzerocolumnsdistinct)) * mdr_cc_zero_rows_rankzero)) /\ exists ff_q_mdr_zero_rows_rankzerocolumnsdistinctsecond. mdr_cb_zero_rows_rankzero = ff_q_mdr_zero_rows_rankzerocolumnsdistinctsecond * S ((S (mdr_j_zero_rows_rankzerocolumnsdistinct)) * mdr_cc_zero_rows_rankzero) + (mdr_a_zero_rows_rankzerocolumnsdistinct))) -> mdr_i_zero_rows_rankzerocolumnsdistinct = mdr_j_zero_rows_rankzerocolumnsdistinct))) -> (exists mdr_ub_zero_rows_rankzeroevaluation mdr_uc_zero_rows_rankzeroevaluation mdr_vb_zero_rows_rankzeroevaluation mdr_vc_zero_rows_rankzeroevaluation. ((((forall mdr_i_zero_rows_rankzeroevaluationmatrixpositive. (exists mdr_gap_zero_rows_rankzeroevaluationmatrixpositivebound. mdr_gap_zero_rows_rankzeroevaluationmatrixpositivebound + S (mdr_i_zero_rows_rankzeroevaluationmatrixpositive) = ((mdr_q_zero_rows_rank) * (mdr_q_zero_rows_rank))) -> exists mdr_a_zero_rows_rankzeroevaluationmatrixpositive. (((exists mdr_r_zero_rows_rankzeroevaluationmatrixpositivepoint mdr_s_zero_rows_rankzeroevaluationmatrixpositivepoint mdr_u_zero_rows_rankzeroevaluationmatrixpositivepoint mdr_v_zero_rows_rankzeroevaluationmatrixpositivepoint. ((mdr_i_zero_rows_rankzeroevaluationmatrixpositive = (mdr_q_zero_rows_rank) * mdr_r_zero_rows_rankzeroevaluationmatrixpositivepoint + mdr_s_zero_rows_rankzeroevaluationmatrixpositivepoint) /\ ((exists mdr_gap_zero_rows_rankzeroevaluationmatrixpositivepointcolumn. mdr_gap_zero_rows_rankzeroevaluationmatrixpositivepointcolumn + S (mdr_s_zero_rows_rankzeroevaluationmatrixpositivepoint) = (mdr_q_zero_rows_rank)) /\ ((((exists ff_h_mdr_zero_rows_rankzeroevaluationmatrixpositivepointrow_index. ff_h_mdr_zero_rows_rankzeroevaluationmatrixpositivepointrow_index + S (mdr_u_zero_rows_rankzeroevaluationmatrixpositivepoint) = S ((S (mdr_r_zero_rows_rankzeroevaluationmatrixpositivepoint)) * mdr_rc_zero_rows_rankzero)) /\ exists ff_q_mdr_zero_rows_rankzeroevaluationmatrixpositivepointrow_index. mdr_rb_zero_rows_rankzero = ff_q_mdr_zero_rows_rankzeroevaluationmatrixpositivepointrow_index * S ((S (mdr_r_zero_rows_rankzeroevaluationmatrixpositivepoint)) * mdr_rc_zero_rows_rankzero) + (mdr_u_zero_rows_rankzeroevaluationmatrixpositivepoint))) /\ ((((exists ff_h_mdr_zero_rows_rankzeroevaluationmatrixpositivepointcolumn_index. ff_h_mdr_zero_rows_rankzeroevaluationmatrixpositivepointcolumn_index + S (mdr_v_zero_rows_rankzeroevaluationmatrixpositivepoint) = S ((S (mdr_s_zero_rows_rankzeroevaluationmatrixpositivepoint)) * mdr_cc_zero_rows_rankzero)) /\ exists ff_q_mdr_zero_rows_rankzeroevaluationmatrixpositivepointcolumn_index. mdr_cb_zero_rows_rankzero = ff_q_mdr_zero_rows_rankzeroevaluationmatrixpositivepointcolumn_index * S ((S (mdr_s_zero_rows_rankzeroevaluationmatrixpositivepoint)) * mdr_cc_zero_rows_rankzero) + (mdr_v_zero_rows_rankzeroevaluationmatrixpositivepoint))) /\ (((exists ff_h_mdr_zero_rows_rankzeroevaluationmatrixpositivepointsource. ff_h_mdr_zero_rows_rankzeroevaluationmatrixpositivepointsource + S (mdr_a_zero_rows_rankzeroevaluationmatrixpositive) = S ((S ((mdr_u_zero_rows_rankzeroevaluationmatrixpositivepoint) * (w) + (mdr_v_zero_rows_rankzeroevaluationmatrixpositivepoint))) * pc)) /\ exists ff_q_mdr_zero_rows_rankzeroevaluationmatrixpositivepointsource. pb = ff_q_mdr_zero_rows_rankzeroevaluationmatrixpositivepointsource * S ((S ((mdr_u_zero_rows_rankzeroevaluationmatrixpositivepoint) * (w) + (mdr_v_zero_rows_rankzeroevaluationmatrixpositivepoint))) * pc) + (mdr_a_zero_rows_rankzeroevaluationmatrixpositive)))))))) /\ (((exists ff_h_mdr_zero_rows_rankzeroevaluationmatrixpositiveoutput. ff_h_mdr_zero_rows_rankzeroevaluationmatrixpositiveoutput + S (mdr_a_zero_rows_rankzeroevaluationmatrixpositive) = S ((S (mdr_i_zero_rows_rankzeroevaluationmatrixpositive)) * mdr_uc_zero_rows_rankzeroevaluation)) /\ exists ff_q_mdr_zero_rows_rankzeroevaluationmatrixpositiveoutput. mdr_ub_zero_rows_rankzeroevaluation = ff_q_mdr_zero_rows_rankzeroevaluationmatrixpositiveoutput * S ((S (mdr_i_zero_rows_rankzeroevaluationmatrixpositive)) * mdr_uc_zero_rows_rankzeroevaluation) + (mdr_a_zero_rows_rankzeroevaluationmatrixpositive)))))) /\ (forall mdr_i_zero_rows_rankzeroevaluationmatrixnegative. (exists mdr_gap_zero_rows_rankzeroevaluationmatrixnegativebound. mdr_gap_zero_rows_rankzeroevaluationmatrixnegativebound + S (mdr_i_zero_rows_rankzeroevaluationmatrixnegative) = ((mdr_q_zero_rows_rank) * (mdr_q_zero_rows_rank))) -> exists mdr_a_zero_rows_rankzeroevaluationmatrixnegative. (((exists mdr_r_zero_rows_rankzeroevaluationmatrixnegativepoint mdr_s_zero_rows_rankzeroevaluationmatrixnegativepoint mdr_u_zero_rows_rankzeroevaluationmatrixnegativepoint mdr_v_zero_rows_rankzeroevaluationmatrixnegativepoint. ((mdr_i_zero_rows_rankzeroevaluationmatrixnegative = (mdr_q_zero_rows_rank) * mdr_r_zero_rows_rankzeroevaluationmatrixnegativepoint + mdr_s_zero_rows_rankzeroevaluationmatrixnegativepoint) /\ ((exists mdr_gap_zero_rows_rankzeroevaluationmatrixnegativepointcolumn. mdr_gap_zero_rows_rankzeroevaluationmatrixnegativepointcolumn + S (mdr_s_zero_rows_rankzeroevaluationmatrixnegativepoint) = (mdr_q_zero_rows_rank)) /\ ((((exists ff_h_mdr_zero_rows_rankzeroevaluationmatrixnegativepointrow_index. ff_h_mdr_zero_rows_rankzeroevaluationmatrixnegativepointrow_index + S (mdr_u_zero_rows_rankzeroevaluationmatrixnegativepoint) = S ((S (mdr_r_zero_rows_rankzeroevaluationmatrixnegativepoint)) * mdr_rc_zero_rows_rankzero)) /\ exists ff_q_mdr_zero_rows_rankzeroevaluationmatrixnegativepointrow_index. mdr_rb_zero_rows_rankzero = ff_q_mdr_zero_rows_rankzeroevaluationmatrixnegativepointrow_index * S ((S (mdr_r_zero_rows_rankzeroevaluationmatrixnegativepoint)) * mdr_rc_zero_rows_rankzero) + (mdr_u_zero_rows_rankzeroevaluationmatrixnegativepoint))) /\ ((((exists ff_h_mdr_zero_rows_rankzeroevaluationmatrixnegativepointcolumn_index. ff_h_mdr_zero_rows_rankzeroevaluationmatrixnegativepointcolumn_index + S (mdr_v_zero_rows_rankzeroevaluationmatrixnegativepoint) = S ((S (mdr_s_zero_rows_rankzeroevaluationmatrixnegativepoint)) * mdr_cc_zero_rows_rankzero)) /\ exists ff_q_mdr_zero_rows_rankzeroevaluationmatrixnegativepointcolumn_index. mdr_cb_zero_rows_rankzero = ff_q_mdr_zero_rows_rankzeroevaluationmatrixnegativepointcolumn_index * S ((S (mdr_s_zero_rows_rankzeroevaluationmatrixnegativepoint)) * mdr_cc_zero_rows_rankzero) + (mdr_v_zero_rows_rankzeroevaluationmatrixnegativepoint))) /\ (((exists ff_h_mdr_zero_rows_rankzeroevaluationmatrixnegativepointsource. ff_h_mdr_zero_rows_rankzeroevaluationmatrixnegativepointsource + S (mdr_a_zero_rows_rankzeroevaluationmatrixnegative) = S ((S ((mdr_u_zero_rows_rankzeroevaluationmatrixnegativepoint) * (w) + (mdr_v_zero_rows_rankzeroevaluationmatrixnegativepoint))) * nc)) /\ exists ff_q_mdr_zero_rows_rankzeroevaluationmatrixnegativepointsource. nb = ff_q_mdr_zero_rows_rankzeroevaluationmatrixnegativepointsource * S ((S ((mdr_u_zero_rows_rankzeroevaluationmatrixnegativepoint) * (w) + (mdr_v_zero_rows_rankzeroevaluationmatrixnegativepoint))) * nc) + (mdr_a_zero_rows_rankzeroevaluationmatrixnegative)))))))) /\ (((exists ff_h_mdr_zero_rows_rankzeroevaluationmatrixnegativeoutput. ff_h_mdr_zero_rows_rankzeroevaluationmatrixnegativeoutput + S (mdr_a_zero_rows_rankzeroevaluationmatrixnegative) = S ((S (mdr_i_zero_rows_rankzeroevaluationmatrixnegative)) * mdr_vc_zero_rows_rankzeroevaluation)) /\ exists ff_q_mdr_zero_rows_rankzeroevaluationmatrixnegativeoutput. mdr_vb_zero_rows_rankzeroevaluation = ff_q_mdr_zero_rows_rankzeroevaluationmatrixnegativeoutput * S ((S (mdr_i_zero_rows_rankzeroevaluationmatrixnegative)) * mdr_vc_zero_rows_rankzeroevaluation) + (mdr_a_zero_rows_rankzeroevaluationmatrixnegative)))))))) /\ (exists mdr_b_zero_rows_rankzeroevaluationdeterminant mdr_c_zero_rows_rankzeroevaluationdeterminant mdr_l_zero_rows_rankzeroevaluationdeterminant mdr_i_zero_rows_rankzeroevaluationdeterminant. ((forall mdr_i_zero_rows_rankzeroevaluationdeterminanth. (exists mdr_gap_zero_rows_rankzeroevaluationdeterminanthi. mdr_gap_zero_rows_rankzeroevaluationdeterminanthi + S (mdr_i_zero_rows_rankzeroevaluationdeterminanth) = (mdr_l_zero_rows_rankzeroevaluationdeterminant)) -> exists mdr_d_zero_rows_rankzeroevaluationdeterminanth mdr_pb_zero_rows_rankzeroevaluationdeterminanth mdr_pc_zero_rows_rankzeroevaluationdeterminanth mdr_nb_zero_rows_rankzeroevaluationdeterminanth mdr_nc_zero_rows_rankzeroevaluationdeterminanth mdr_p_zero_rows_rankzeroevaluationdeterminanth mdr_n_zero_rows_rankzeroevaluationdeterminanth. ((exists mdr_z_zero_rows_rankzeroevaluationdeterminanthr. ((exists mdr_a_zero_rows_rankzeroevaluationdeterminanthrc mdr_b_zero_rows_rankzeroevaluationdeterminanthrc mdr_c_zero_rows_rankzeroevaluationdeterminanthrc mdr_e_zero_rows_rankzeroevaluationdeterminanthrc mdr_f_zero_rows_rankzeroevaluationdeterminanthrc. ((mdr_a_zero_rows_rankzeroevaluationdeterminanthrc = ((mdr_d_zero_rows_rankzeroevaluationdeterminanth) + (mdr_pb_zero_rows_rankzeroevaluationdeterminanth)) * S ((mdr_d_zero_rows_rankzeroevaluationdeterminanth) + (mdr_pb_zero_rows_rankzeroevaluationdeterminanth)) + ((mdr_pb_zero_rows_rankzeroevaluationdeterminanth) + (mdr_pb_zero_rows_rankzeroevaluationdeterminanth))) /\ ((mdr_b_zero_rows_rankzeroevaluationdeterminanthrc = ((mdr_pc_zero_rows_rankzeroevaluationdeterminanth) + (mdr_nb_zero_rows_rankzeroevaluationdeterminanth)) * S ((mdr_pc_zero_rows_rankzeroevaluationdeterminanth) + (mdr_nb_zero_rows_rankzeroevaluationdeterminanth)) + ((mdr_nb_zero_rows_rankzeroevaluationdeterminanth) + (mdr_nb_zero_rows_rankzeroevaluationdeterminanth))) /\ ((mdr_c_zero_rows_rankzeroevaluationdeterminanthrc = ((mdr_a_zero_rows_rankzeroevaluationdeterminanthrc) + (mdr_b_zero_rows_rankzeroevaluationdeterminanthrc)) * S ((mdr_a_zero_rows_rankzeroevaluationdeterminanthrc) + (mdr_b_zero_rows_rankzeroevaluationdeterminanthrc)) + ((mdr_b_zero_rows_rankzeroevaluationdeterminanthrc) + (mdr_b_zero_rows_rankzeroevaluationdeterminanthrc))) /\ ((mdr_e_zero_rows_rankzeroevaluationdeterminanthrc = ((mdr_p_zero_rows_rankzeroevaluationdeterminanth) + (mdr_n_zero_rows_rankzeroevaluationdeterminanth)) * S ((mdr_p_zero_rows_rankzeroevaluationdeterminanth) + (mdr_n_zero_rows_rankzeroevaluationdeterminanth)) + ((mdr_n_zero_rows_rankzeroevaluationdeterminanth) + (mdr_n_zero_rows_rankzeroevaluationdeterminanth))) /\ ((mdr_f_zero_rows_rankzeroevaluationdeterminanthrc = ((mdr_nc_zero_rows_rankzeroevaluationdeterminanth) + (mdr_e_zero_rows_rankzeroevaluationdeterminanthrc)) * S ((mdr_nc_zero_rows_rankzeroevaluationdeterminanth) + (mdr_e_zero_rows_rankzeroevaluationdeterminanthrc)) + ((mdr_e_zero_rows_rankzeroevaluationdeterminanthrc) + (mdr_e_zero_rows_rankzeroevaluationdeterminanthrc))) /\ ((mdr_z_zero_rows_rankzeroevaluationdeterminanthr) = ((mdr_c_zero_rows_rankzeroevaluationdeterminanthrc) + (mdr_f_zero_rows_rankzeroevaluationdeterminanthrc)) * S ((mdr_c_zero_rows_rankzeroevaluationdeterminanthrc) + (mdr_f_zero_rows_rankzeroevaluationdeterminanthrc)) + ((mdr_f_zero_rows_rankzeroevaluationdeterminanthrc) + (mdr_f_zero_rows_rankzeroevaluationdeterminanthrc))))))))) /\ (((exists ff_h_mdr_zero_rows_rankzeroevaluationdeterminanthrb. ff_h_mdr_zero_rows_rankzeroevaluationdeterminanthrb + S (mdr_z_zero_rows_rankzeroevaluationdeterminanthr) = S ((S (mdr_i_zero_rows_rankzeroevaluationdeterminanth)) * mdr_c_zero_rows_rankzeroevaluationdeterminant)) /\ exists ff_q_mdr_zero_rows_rankzeroevaluationdeterminanthrb. mdr_b_zero_rows_rankzeroevaluationdeterminant = ff_q_mdr_zero_rows_rankzeroevaluationdeterminanthrb * S ((S (mdr_i_zero_rows_rankzeroevaluationdeterminanth)) * mdr_c_zero_rows_rankzeroevaluationdeterminant) + (mdr_z_zero_rows_rankzeroevaluationdeterminanthr))))) /\ (((((mdr_d_zero_rows_rankzeroevaluationdeterminanth) = 0) /\ (((mdr_p_zero_rows_rankzeroevaluationdeterminanth) = 1) /\ ((mdr_n_zero_rows_rankzeroevaluationdeterminanth) = 0))) \/ exists mdr_q_zero_rows_rankzeroevaluationdeterminanths mdr_eb_zero_rows_rankzeroevaluationdeterminanths mdr_ec_zero_rows_rankzeroevaluationdeterminanths mdr_fb_zero_rows_rankzeroevaluationdeterminanths mdr_fc_zero_rows_rankzeroevaluationdeterminanths. (((mdr_d_zero_rows_rankzeroevaluationdeterminanth) = S (mdr_q_zero_rows_rankzeroevaluationdeterminanths)) /\ ((forall mdr_j_zero_rows_rankzeroevaluationdeterminanthsc. (exists mdr_gap_zero_rows_rankzeroevaluationdeterminanthscj. mdr_gap_zero_rows_rankzeroevaluationdeterminanthscj + S (mdr_j_zero_rows_rankzeroevaluationdeterminanthsc) = (S (mdr_q_zero_rows_rankzeroevaluationdeterminanths))) -> exists mdr_i_zero_rows_rankzeroevaluationdeterminanthsc mdr_up_zero_rows_rankzeroevaluationdeterminanthsc mdr_us_zero_rows_rankzeroevaluationdeterminanthsc mdr_un_zero_rows_rankzeroevaluationdeterminanthsc mdr_ut_zero_rows_rankzeroevaluationdeterminanthsc mdr_p_zero_rows_rankzeroevaluationdeterminanthsc mdr_n_zero_rows_rankzeroevaluationdeterminanthsc. ((exists mdr_gap_zero_rows_rankzeroevaluationdeterminanthsci. mdr_gap_zero_rows_rankzeroevaluationdeterminanthsci + S (mdr_i_zero_rows_rankzeroevaluationdeterminanthsc) = (mdr_i_zero_rows_rankzeroevaluationdeterminanth)) /\ ((exists mdr_z_zero_rows_rankzeroevaluationdeterminanthscr. ((exists mdr_a_zero_rows_rankzeroevaluationdeterminanthscrc mdr_b_zero_rows_rankzeroevaluationdeterminanthscrc mdr_c_zero_rows_rankzeroevaluationdeterminanthscrc mdr_e_zero_rows_rankzeroevaluationdeterminanthscrc mdr_f_zero_rows_rankzeroevaluationdeterminanthscrc. ((mdr_a_zero_rows_rankzeroevaluationdeterminanthscrc = ((mdr_q_zero_rows_rankzeroevaluationdeterminanths) + (mdr_up_zero_rows_rankzeroevaluationdeterminanthsc)) * S ((mdr_q_zero_rows_rankzeroevaluationdeterminanths) + (mdr_up_zero_rows_rankzeroevaluationdeterminanthsc)) + ((mdr_up_zero_rows_rankzeroevaluationdeterminanthsc) + (mdr_up_zero_rows_rankzeroevaluationdeterminanthsc))) /\ ((mdr_b_zero_rows_rankzeroevaluationdeterminanthscrc = ((mdr_us_zero_rows_rankzeroevaluationdeterminanthsc) + (mdr_un_zero_rows_rankzeroevaluationdeterminanthsc)) * S ((mdr_us_zero_rows_rankzeroevaluationdeterminanthsc) + (mdr_un_zero_rows_rankzeroevaluationdeterminanthsc)) + ((mdr_un_zero_rows_rankzeroevaluationdeterminanthsc) + (mdr_un_zero_rows_rankzeroevaluationdeterminanthsc))) /\ ((mdr_c_zero_rows_rankzeroevaluationdeterminanthscrc = ((mdr_a_zero_rows_rankzeroevaluationdeterminanthscrc) + (mdr_b_zero_rows_rankzeroevaluationdeterminanthscrc)) * S ((mdr_a_zero_rows_rankzeroevaluationdeterminanthscrc) + (mdr_b_zero_rows_rankzeroevaluationdeterminanthscrc)) + ((mdr_b_zero_rows_rankzeroevaluationdeterminanthscrc) + (mdr_b_zero_rows_rankzeroevaluationdeterminanthscrc))) /\ ((mdr_e_zero_rows_rankzeroevaluationdeterminanthscrc = ((mdr_p_zero_rows_rankzeroevaluationdeterminanthsc) + (mdr_n_zero_rows_rankzeroevaluationdeterminanthsc)) * S ((mdr_p_zero_rows_rankzeroevaluationdeterminanthsc) + (mdr_n_zero_rows_rankzeroevaluationdeterminanthsc)) + ((mdr_n_zero_rows_rankzeroevaluationdeterminanthsc) + (mdr_n_zero_rows_rankzeroevaluationdeterminanthsc))) /\ ((mdr_f_zero_rows_rankzeroevaluationdeterminanthscrc = ((mdr_ut_zero_rows_rankzeroevaluationdeterminanthsc) + (mdr_e_zero_rows_rankzeroevaluationdeterminanthscrc)) * S ((mdr_ut_zero_rows_rankzeroevaluationdeterminanthsc) + (mdr_e_zero_rows_rankzeroevaluationdeterminanthscrc)) + ((mdr_e_zero_rows_rankzeroevaluationdeterminanthscrc) + (mdr_e_zero_rows_rankzeroevaluationdeterminanthscrc))) /\ ((mdr_z_zero_rows_rankzeroevaluationdeterminanthscr) = ((mdr_c_zero_rows_rankzeroevaluationdeterminanthscrc) + (mdr_f_zero_rows_rankzeroevaluationdeterminanthscrc)) * S ((mdr_c_zero_rows_rankzeroevaluationdeterminanthscrc) + (mdr_f_zero_rows_rankzeroevaluationdeterminanthscrc)) + ((mdr_f_zero_rows_rankzeroevaluationdeterminanthscrc) + (mdr_f_zero_rows_rankzeroevaluationdeterminanthscrc))))))))) /\ (((exists ff_h_mdr_zero_rows_rankzeroevaluationdeterminanthscrb. ff_h_mdr_zero_rows_rankzeroevaluationdeterminanthscrb + S (mdr_z_zero_rows_rankzeroevaluationdeterminanthscr) = S ((S (mdr_i_zero_rows_rankzeroevaluationdeterminanthsc)) * mdr_c_zero_rows_rankzeroevaluationdeterminant)) /\ exists ff_q_mdr_zero_rows_rankzeroevaluationdeterminanthscrb. mdr_b_zero_rows_rankzeroevaluationdeterminant = ff_q_mdr_zero_rows_rankzeroevaluationdeterminanthscrb * S ((S (mdr_i_zero_rows_rankzeroevaluationdeterminanthsc)) * mdr_c_zero_rows_rankzeroevaluationdeterminant) + (mdr_z_zero_rows_rankzeroevaluationdeterminanthscr))))) /\ ((((forall ff_index_mdm_prefix_mdr_zero_rows_rankzeroevaluationdeterminanthscm_positive. (exists ff_gap_mdm_lt_mdr_zero_rows_rankzeroevaluationdeterminanthscm_positive_index_bound. ff_gap_mdm_lt_mdr_zero_rows_rankzeroevaluationdeterminanthscm_positive_index_bound + S (ff_index_mdm_prefix_mdr_zero_rows_rankzeroevaluationdeterminanthscm_positive) = ((mdr_q_zero_rows_rankzeroevaluationdeterminanths) * (mdr_q_zero_rows_rankzeroevaluationdeterminanths))) -> exists ff_row_mdm_prefix_mdr_zero_rows_rankzeroevaluationdeterminanthscm_positive ff_column_mdm_prefix_mdr_zero_rows_rankzeroevaluationdeterminanthscm_positive ff_value_mdm_prefix_mdr_zero_rows_rankzeroevaluationdeterminanthscm_positive. (ff_index_mdm_prefix_mdr_zero_rows_rankzeroevaluationdeterminanthscm_positive = (mdr_q_zero_rows_rankzeroevaluationdeterminanths) * ff_row_mdm_prefix_mdr_zero_rows_rankzeroevaluationdeterminanthscm_positive + ff_column_mdm_prefix_mdr_zero_rows_rankzeroevaluationdeterminanthscm_positive /\ ((exists ff_gap_mdm_lt_mdr_zero_rows_rankzeroevaluationdeterminanthscm_positive_column_bound. ff_gap_mdm_lt_mdr_zero_rows_rankzeroevaluationdeterminanthscm_positive_column_bound + S (ff_column_mdm_prefix_mdr_zero_rows_rankzeroevaluationdeterminanthscm_positive) = (mdr_q_zero_rows_rankzeroevaluationdeterminanths)) /\ ((exists ff_row_mdm_cell_mdr_zero_rows_rankzeroevaluationdeterminanthscm_positive_cell ff_column_mdm_cell_mdr_zero_rows_rankzeroevaluationdeterminanthscm_positive_cell. (((((exists ff_gap_mdm_lt_mdr_zero_rows_rankzeroevaluationdeterminanthscm_positive_cell_row_before. ff_gap_mdm_lt_mdr_zero_rows_rankzeroevaluationdeterminanthscm_positive_cell_row_before + S (ff_row_mdm_prefix_mdr_zero_rows_rankzeroevaluationdeterminanthscm_positive) = (0)) /\ ff_row_mdm_cell_mdr_zero_rows_rankzeroevaluationdeterminanthscm_positive_cell = ff_row_mdm_prefix_mdr_zero_rows_rankzeroevaluationdeterminanthscm_positive) \/ ((exists ff_gap_mdm_le_mdr_zero_rows_rankzeroevaluationdeterminanthscm_positive_cell_row_after. ff_gap_mdm_le_mdr_zero_rows_rankzeroevaluationdeterminanthscm_positive_cell_row_after + (0) = (ff_row_mdm_prefix_mdr_zero_rows_rankzeroevaluationdeterminanthscm_positive)) /\ ff_row_mdm_cell_mdr_zero_rows_rankzeroevaluationdeterminanthscm_positive_cell = S ff_row_mdm_prefix_mdr_zero_rows_rankzeroevaluationdeterminanthscm_positive))) /\ (((((exists ff_gap_mdm_lt_mdr_zero_rows_rankzeroevaluationdeterminanthscm_positive_cell_column_before. ff_gap_mdm_lt_mdr_zero_rows_rankzeroevaluationdeterminanthscm_positive_cell_column_before + S (ff_column_mdm_prefix_mdr_zero_rows_rankzeroevaluationdeterminanthscm_positive) = (mdr_j_zero_rows_rankzeroevaluationdeterminanthsc)) /\ ff_column_mdm_cell_mdr_zero_rows_rankzeroevaluationdeterminanthscm_positive_cell = ff_column_mdm_prefix_mdr_zero_rows_rankzeroevaluationdeterminanthscm_positive) \/ ((exists ff_gap_mdm_le_mdr_zero_rows_rankzeroevaluationdeterminanthscm_positive_cell_column_after. ff_gap_mdm_le_mdr_zero_rows_rankzeroevaluationdeterminanthscm_positive_cell_column_after + (mdr_j_zero_rows_rankzeroevaluationdeterminanthsc) = (ff_column_mdm_prefix_mdr_zero_rows_rankzeroevaluationdeterminanthscm_positive)) /\ ff_column_mdm_cell_mdr_zero_rows_rankzeroevaluationdeterminanthscm_positive_cell = S ff_column_mdm_prefix_mdr_zero_rows_rankzeroevaluationdeterminanthscm_positive))) /\ (((exists ff_h_mdm_mdr_zero_rows_rankzeroevaluationdeterminanthscm_positive_cell_source. ff_h_mdm_mdr_zero_rows_rankzeroevaluationdeterminanthscm_positive_cell_source + S (ff_value_mdm_prefix_mdr_zero_rows_rankzeroevaluationdeterminanthscm_positive) = S ((S ((ff_row_mdm_cell_mdr_zero_rows_rankzeroevaluationdeterminanthscm_positive_cell) * (S (mdr_q_zero_rows_rankzeroevaluationdeterminanths)) + (ff_column_mdm_cell_mdr_zero_rows_rankzeroevaluationdeterminanthscm_positive_cell))) * mdr_pc_zero_rows_rankzeroevaluationdeterminanth)) /\ exists ff_q_mdm_mdr_zero_rows_rankzeroevaluationdeterminanthscm_positive_cell_source. mdr_pb_zero_rows_rankzeroevaluationdeterminanth = ff_q_mdm_mdr_zero_rows_rankzeroevaluationdeterminanthscm_positive_cell_source * S ((S ((ff_row_mdm_cell_mdr_zero_rows_rankzeroevaluationdeterminanthscm_positive_cell) * (S (mdr_q_zero_rows_rankzeroevaluationdeterminanths)) + (ff_column_mdm_cell_mdr_zero_rows_rankzeroevaluationdeterminanthscm_positive_cell))) * mdr_pc_zero_rows_rankzeroevaluationdeterminanth) + (ff_value_mdm_prefix_mdr_zero_rows_rankzeroevaluationdeterminanthscm_positive)))))) /\ (((exists ff_h_mdm_mdr_zero_rows_rankzeroevaluationdeterminanthscm_positive_target. ff_h_mdm_mdr_zero_rows_rankzeroevaluationdeterminanthscm_positive_target + S (ff_value_mdm_prefix_mdr_zero_rows_rankzeroevaluationdeterminanthscm_positive) = S ((S (ff_index_mdm_prefix_mdr_zero_rows_rankzeroevaluationdeterminanthscm_positive)) * mdr_us_zero_rows_rankzeroevaluationdeterminanthsc)) /\ exists ff_q_mdm_mdr_zero_rows_rankzeroevaluationdeterminanthscm_positive_target. mdr_up_zero_rows_rankzeroevaluationdeterminanthsc = ff_q_mdm_mdr_zero_rows_rankzeroevaluationdeterminanthscm_positive_target * S ((S (ff_index_mdm_prefix_mdr_zero_rows_rankzeroevaluationdeterminanthscm_positive)) * mdr_us_zero_rows_rankzeroevaluationdeterminanthsc) + (ff_value_mdm_prefix_mdr_zero_rows_rankzeroevaluationdeterminanthscm_positive))))))) /\ (forall ff_index_mdm_prefix_mdr_zero_rows_rankzeroevaluationdeterminanthscm_negative. (exists ff_gap_mdm_lt_mdr_zero_rows_rankzeroevaluationdeterminanthscm_negative_index_bound. ff_gap_mdm_lt_mdr_zero_rows_rankzeroevaluationdeterminanthscm_negative_index_bound + S (ff_index_mdm_prefix_mdr_zero_rows_rankzeroevaluationdeterminanthscm_negative) = ((mdr_q_zero_rows_rankzeroevaluationdeterminanths) * (mdr_q_zero_rows_rankzeroevaluationdeterminanths))) -> exists ff_row_mdm_prefix_mdr_zero_rows_rankzeroevaluationdeterminanthscm_negative ff_column_mdm_prefix_mdr_zero_rows_rankzeroevaluationdeterminanthscm_negative ff_value_mdm_prefix_mdr_zero_rows_rankzeroevaluationdeterminanthscm_negative. (ff_index_mdm_prefix_mdr_zero_rows_rankzeroevaluationdeterminanthscm_negative = (mdr_q_zero_rows_rankzeroevaluationdeterminanths) * ff_row_mdm_prefix_mdr_zero_rows_rankzeroevaluationdeterminanthscm_negative + ff_column_mdm_prefix_mdr_zero_rows_rankzeroevaluationdeterminanthscm_negative /\ ((exists ff_gap_mdm_lt_mdr_zero_rows_rankzeroevaluationdeterminanthscm_negative_column_bound. ff_gap_mdm_lt_mdr_zero_rows_rankzeroevaluationdeterminanthscm_negative_column_bound + S (ff_column_mdm_prefix_mdr_zero_rows_rankzeroevaluationdeterminanthscm_negative) = (mdr_q_zero_rows_rankzeroevaluationdeterminanths)) /\ ((exists ff_row_mdm_cell_mdr_zero_rows_rankzeroevaluationdeterminanthscm_negative_cell ff_column_mdm_cell_mdr_zero_rows_rankzeroevaluationdeterminanthscm_negative_cell. (((((exists ff_gap_mdm_lt_mdr_zero_rows_rankzeroevaluationdeterminanthscm_negative_cell_row_before. ff_gap_mdm_lt_mdr_zero_rows_rankzeroevaluationdeterminanthscm_negative_cell_row_before + S (ff_row_mdm_prefix_mdr_zero_rows_rankzeroevaluationdeterminanthscm_negative) = (0)) /\ ff_row_mdm_cell_mdr_zero_rows_rankzeroevaluationdeterminanthscm_negative_cell = ff_row_mdm_prefix_mdr_zero_rows_rankzeroevaluationdeterminanthscm_negative) \/ ((exists ff_gap_mdm_le_mdr_zero_rows_rankzeroevaluationdeterminanthscm_negative_cell_row_after. ff_gap_mdm_le_mdr_zero_rows_rankzeroevaluationdeterminanthscm_negative_cell_row_after + (0) = (ff_row_mdm_prefix_mdr_zero_rows_rankzeroevaluationdeterminanthscm_negative)) /\ ff_row_mdm_cell_mdr_zero_rows_rankzeroevaluationdeterminanthscm_negative_cell = S ff_row_mdm_prefix_mdr_zero_rows_rankzeroevaluationdeterminanthscm_negative))) /\ (((((exists ff_gap_mdm_lt_mdr_zero_rows_rankzeroevaluationdeterminanthscm_negative_cell_column_before. ff_gap_mdm_lt_mdr_zero_rows_rankzeroevaluationdeterminanthscm_negative_cell_column_before + S (ff_column_mdm_prefix_mdr_zero_rows_rankzeroevaluationdeterminanthscm_negative) = (mdr_j_zero_rows_rankzeroevaluationdeterminanthsc)) /\ ff_column_mdm_cell_mdr_zero_rows_rankzeroevaluationdeterminanthscm_negative_cell = ff_column_mdm_prefix_mdr_zero_rows_rankzeroevaluationdeterminanthscm_negative) \/ ((exists ff_gap_mdm_le_mdr_zero_rows_rankzeroevaluationdeterminanthscm_negative_cell_column_after. ff_gap_mdm_le_mdr_zero_rows_rankzeroevaluationdeterminanthscm_negative_cell_column_after + (mdr_j_zero_rows_rankzeroevaluationdeterminanthsc) = (ff_column_mdm_prefix_mdr_zero_rows_rankzeroevaluationdeterminanthscm_negative)) /\ ff_column_mdm_cell_mdr_zero_rows_rankzeroevaluationdeterminanthscm_negative_cell = S ff_column_mdm_prefix_mdr_zero_rows_rankzeroevaluationdeterminanthscm_negative))) /\ (((exists ff_h_mdm_mdr_zero_rows_rankzeroevaluationdeterminanthscm_negative_cell_source. ff_h_mdm_mdr_zero_rows_rankzeroevaluationdeterminanthscm_negative_cell_source + S (ff_value_mdm_prefix_mdr_zero_rows_rankzeroevaluationdeterminanthscm_negative) = S ((S ((ff_row_mdm_cell_mdr_zero_rows_rankzeroevaluationdeterminanthscm_negative_cell) * (S (mdr_q_zero_rows_rankzeroevaluationdeterminanths)) + (ff_column_mdm_cell_mdr_zero_rows_rankzeroevaluationdeterminanthscm_negative_cell))) * mdr_nc_zero_rows_rankzeroevaluationdeterminanth)) /\ exists ff_q_mdm_mdr_zero_rows_rankzeroevaluationdeterminanthscm_negative_cell_source. mdr_nb_zero_rows_rankzeroevaluationdeterminanth = ff_q_mdm_mdr_zero_rows_rankzeroevaluationdeterminanthscm_negative_cell_source * S ((S ((ff_row_mdm_cell_mdr_zero_rows_rankzeroevaluationdeterminanthscm_negative_cell) * (S (mdr_q_zero_rows_rankzeroevaluationdeterminanths)) + (ff_column_mdm_cell_mdr_zero_rows_rankzeroevaluationdeterminanthscm_negative_cell))) * mdr_nc_zero_rows_rankzeroevaluationdeterminanth) + (ff_value_mdm_prefix_mdr_zero_rows_rankzeroevaluationdeterminanthscm_negative)))))) /\ (((exists ff_h_mdm_mdr_zero_rows_rankzeroevaluationdeterminanthscm_negative_target. ff_h_mdm_mdr_zero_rows_rankzeroevaluationdeterminanthscm_negative_target + S (ff_value_mdm_prefix_mdr_zero_rows_rankzeroevaluationdeterminanthscm_negative) = S ((S (ff_index_mdm_prefix_mdr_zero_rows_rankzeroevaluationdeterminanthscm_negative)) * mdr_ut_zero_rows_rankzeroevaluationdeterminanthsc)) /\ exists ff_q_mdm_mdr_zero_rows_rankzeroevaluationdeterminanthscm_negative_target. mdr_un_zero_rows_rankzeroevaluationdeterminanthsc = ff_q_mdm_mdr_zero_rows_rankzeroevaluationdeterminanthscm_negative_target * S ((S (ff_index_mdm_prefix_mdr_zero_rows_rankzeroevaluationdeterminanthscm_negative)) * mdr_ut_zero_rows_rankzeroevaluationdeterminanthsc) + (ff_value_mdm_prefix_mdr_zero_rows_rankzeroevaluationdeterminanthscm_negative))))))))) /\ ((((exists ff_h_mdr_zero_rows_rankzeroevaluationdeterminanthscp. ff_h_mdr_zero_rows_rankzeroevaluationdeterminanthscp + S (mdr_p_zero_rows_rankzeroevaluationdeterminanthsc) = S ((S (mdr_j_zero_rows_rankzeroevaluationdeterminanthsc)) * mdr_ec_zero_rows_rankzeroevaluationdeterminanths)) /\ exists ff_q_mdr_zero_rows_rankzeroevaluationdeterminanthscp. mdr_eb_zero_rows_rankzeroevaluationdeterminanths = ff_q_mdr_zero_rows_rankzeroevaluationdeterminanthscp * S ((S (mdr_j_zero_rows_rankzeroevaluationdeterminanthsc)) * mdr_ec_zero_rows_rankzeroevaluationdeterminanths) + (mdr_p_zero_rows_rankzeroevaluationdeterminanthsc))) /\ (((exists ff_h_mdr_zero_rows_rankzeroevaluationdeterminanthscn. ff_h_mdr_zero_rows_rankzeroevaluationdeterminanthscn + S (mdr_n_zero_rows_rankzeroevaluationdeterminanthsc) = S ((S (mdr_j_zero_rows_rankzeroevaluationdeterminanthsc)) * mdr_fc_zero_rows_rankzeroevaluationdeterminanths)) /\ exists ff_q_mdr_zero_rows_rankzeroevaluationdeterminanthscn. mdr_fb_zero_rows_rankzeroevaluationdeterminanths = ff_q_mdr_zero_rows_rankzeroevaluationdeterminanthscn * S ((S (mdr_j_zero_rows_rankzeroevaluationdeterminanthsc)) * mdr_fc_zero_rows_rankzeroevaluationdeterminanths) + (mdr_n_zero_rows_rankzeroevaluationdeterminanthsc)))))))) /\ (exists ff_ub_mce_fold_mdr_zero_rows_rankzeroevaluationdeterminanthsf ff_uc_mce_fold_mdr_zero_rows_rankzeroevaluationdeterminanthsf ff_vb_mce_fold_mdr_zero_rows_rankzeroevaluationdeterminanthsf ff_vc_mce_fold_mdr_zero_rows_rankzeroevaluationdeterminanthsf. ((forall ff_index_mce_alternating_mdr_zero_rows_rankzeroevaluationdeterminanthsf_prefix. (exists ff_gap_mce_mdr_zero_rows_rankzeroevaluationdeterminanthsf_prefix_index. ff_gap_mce_mdr_zero_rows_rankzeroevaluationdeterminanthsf_prefix_index + S (ff_index_mce_alternating_mdr_zero_rows_rankzeroevaluationdeterminanthsf_prefix) = (S (mdr_q_zero_rows_rankzeroevaluationdeterminanths))) -> exists ff_ap_mce_alternating_mdr_zero_rows_rankzeroevaluationdeterminanthsf_prefix ff_an_mce_alternating_mdr_zero_rows_rankzeroevaluationdeterminanthsf_prefix ff_bp_mce_alternating_mdr_zero_rows_rankzeroevaluationdeterminanthsf_prefix ff_bn_mce_alternating_mdr_zero_rows_rankzeroevaluationdeterminanthsf_prefix ff_p_mce_alternating_mdr_zero_rows_rankzeroevaluationdeterminanthsf_prefix ff_n_mce_alternating_mdr_zero_rows_rankzeroevaluationdeterminanthsf_prefix. ((((exists ff_h_mce_mdr_zero_rows_rankzeroevaluationdeterminanthsf_prefix_ap. ff_h_mce_mdr_zero_rows_rankzeroevaluationdeterminanthsf_prefix_ap + S (ff_ap_mce_alternating_mdr_zero_rows_rankzeroevaluationdeterminanthsf_prefix) = S ((S (ff_index_mce_alternating_mdr_zero_rows_rankzeroevaluationdeterminanthsf_prefix)) * mdr_pc_zero_rows_rankzeroevaluationdeterminanth)) /\ exists ff_q_mce_mdr_zero_rows_rankzeroevaluationdeterminanthsf_prefix_ap. mdr_pb_zero_rows_rankzeroevaluationdeterminanth = ff_q_mce_mdr_zero_rows_rankzeroevaluationdeterminanthsf_prefix_ap * S ((S (ff_index_mce_alternating_mdr_zero_rows_rankzeroevaluationdeterminanthsf_prefix)) * mdr_pc_zero_rows_rankzeroevaluationdeterminanth) + (ff_ap_mce_alternating_mdr_zero_rows_rankzeroevaluationdeterminanthsf_prefix))) /\ ((((exists ff_h_mce_mdr_zero_rows_rankzeroevaluationdeterminanthsf_prefix_an. ff_h_mce_mdr_zero_rows_rankzeroevaluationdeterminanthsf_prefix_an + S (ff_an_mce_alternating_mdr_zero_rows_rankzeroevaluationdeterminanthsf_prefix) = S ((S (ff_index_mce_alternating_mdr_zero_rows_rankzeroevaluationdeterminanthsf_prefix)) * mdr_nc_zero_rows_rankzeroevaluationdeterminanth)) /\ exists ff_q_mce_mdr_zero_rows_rankzeroevaluationdeterminanthsf_prefix_an. mdr_nb_zero_rows_rankzeroevaluationdeterminanth = ff_q_mce_mdr_zero_rows_rankzeroevaluationdeterminanthsf_prefix_an * S ((S (ff_index_mce_alternating_mdr_zero_rows_rankzeroevaluationdeterminanthsf_prefix)) * mdr_nc_zero_rows_rankzeroevaluationdeterminanth) + (ff_an_mce_alternating_mdr_zero_rows_rankzeroevaluationdeterminanthsf_prefix))) /\ ((((exists ff_h_mce_mdr_zero_rows_rankzeroevaluationdeterminanthsf_prefix_bp. ff_h_mce_mdr_zero_rows_rankzeroevaluationdeterminanthsf_prefix_bp + S (ff_bp_mce_alternating_mdr_zero_rows_rankzeroevaluationdeterminanthsf_prefix) = S ((S (ff_index_mce_alternating_mdr_zero_rows_rankzeroevaluationdeterminanthsf_prefix)) * mdr_ec_zero_rows_rankzeroevaluationdeterminanths)) /\ exists ff_q_mce_mdr_zero_rows_rankzeroevaluationdeterminanthsf_prefix_bp. mdr_eb_zero_rows_rankzeroevaluationdeterminanths = ff_q_mce_mdr_zero_rows_rankzeroevaluationdeterminanthsf_prefix_bp * S ((S (ff_index_mce_alternating_mdr_zero_rows_rankzeroevaluationdeterminanthsf_prefix)) * mdr_ec_zero_rows_rankzeroevaluationdeterminanths) + (ff_bp_mce_alternating_mdr_zero_rows_rankzeroevaluationdeterminanthsf_prefix))) /\ ((((exists ff_h_mce_mdr_zero_rows_rankzeroevaluationdeterminanthsf_prefix_bn. ff_h_mce_mdr_zero_rows_rankzeroevaluationdeterminanthsf_prefix_bn + S (ff_bn_mce_alternating_mdr_zero_rows_rankzeroevaluationdeterminanthsf_prefix) = S ((S (ff_index_mce_alternating_mdr_zero_rows_rankzeroevaluationdeterminanthsf_prefix)) * mdr_fc_zero_rows_rankzeroevaluationdeterminanths)) /\ exists ff_q_mce_mdr_zero_rows_rankzeroevaluationdeterminanthsf_prefix_bn. mdr_fb_zero_rows_rankzeroevaluationdeterminanths = ff_q_mce_mdr_zero_rows_rankzeroevaluationdeterminanthsf_prefix_bn * S ((S (ff_index_mce_alternating_mdr_zero_rows_rankzeroevaluationdeterminanthsf_prefix)) * mdr_fc_zero_rows_rankzeroevaluationdeterminanths) + (ff_bn_mce_alternating_mdr_zero_rows_rankzeroevaluationdeterminanthsf_prefix))) /\ ((((exists ff_h_mce_mdr_zero_rows_rankzeroevaluationdeterminanthsf_prefix_positive. ff_h_mce_mdr_zero_rows_rankzeroevaluationdeterminanthsf_prefix_positive + S (ff_p_mce_alternating_mdr_zero_rows_rankzeroevaluationdeterminanthsf_prefix) = S ((S (ff_index_mce_alternating_mdr_zero_rows_rankzeroevaluationdeterminanthsf_prefix)) * ff_uc_mce_fold_mdr_zero_rows_rankzeroevaluationdeterminanthsf)) /\ exists ff_q_mce_mdr_zero_rows_rankzeroevaluationdeterminanthsf_prefix_positive. ff_ub_mce_fold_mdr_zero_rows_rankzeroevaluationdeterminanthsf = ff_q_mce_mdr_zero_rows_rankzeroevaluationdeterminanthsf_prefix_positive * S ((S (ff_index_mce_alternating_mdr_zero_rows_rankzeroevaluationdeterminanthsf_prefix)) * ff_uc_mce_fold_mdr_zero_rows_rankzeroevaluationdeterminanthsf) + (ff_p_mce_alternating_mdr_zero_rows_rankzeroevaluationdeterminanthsf_prefix))) /\ ((((exists ff_h_mce_mdr_zero_rows_rankzeroevaluationdeterminanthsf_prefix_negative. ff_h_mce_mdr_zero_rows_rankzeroevaluationdeterminanthsf_prefix_negative + S (ff_n_mce_alternating_mdr_zero_rows_rankzeroevaluationdeterminanthsf_prefix) = S ((S (ff_index_mce_alternating_mdr_zero_rows_rankzeroevaluationdeterminanthsf_prefix)) * ff_vc_mce_fold_mdr_zero_rows_rankzeroevaluationdeterminanthsf)) /\ exists ff_q_mce_mdr_zero_rows_rankzeroevaluationdeterminanthsf_prefix_negative. ff_vb_mce_fold_mdr_zero_rows_rankzeroevaluationdeterminanthsf = ff_q_mce_mdr_zero_rows_rankzeroevaluationdeterminanthsf_prefix_negative * S ((S (ff_index_mce_alternating_mdr_zero_rows_rankzeroevaluationdeterminanthsf_prefix)) * ff_vc_mce_fold_mdr_zero_rows_rankzeroevaluationdeterminanthsf) + (ff_n_mce_alternating_mdr_zero_rows_rankzeroevaluationdeterminanthsf_prefix))) /\ (((exists ff_even_mce_term_mdr_zero_rows_rankzeroevaluationdeterminanthsf_prefix_term. ff_index_mce_alternating_mdr_zero_rows_rankzeroevaluationdeterminanthsf_prefix = 2 * ff_even_mce_term_mdr_zero_rows_rankzeroevaluationdeterminanthsf_prefix_term) /\ (ff_p_mce_alternating_mdr_zero_rows_rankzeroevaluationdeterminanthsf_prefix = (ff_ap_mce_alternating_mdr_zero_rows_rankzeroevaluationdeterminanthsf_prefix) * (ff_bp_mce_alternating_mdr_zero_rows_rankzeroevaluationdeterminanthsf_prefix) + (ff_an_mce_alternating_mdr_zero_rows_rankzeroevaluationdeterminanthsf_prefix) * (ff_bn_mce_alternating_mdr_zero_rows_rankzeroevaluationdeterminanthsf_prefix) /\ ff_n_mce_alternating_mdr_zero_rows_rankzeroevaluationdeterminanthsf_prefix = (ff_ap_mce_alternating_mdr_zero_rows_rankzeroevaluationdeterminanthsf_prefix) * (ff_bn_mce_alternating_mdr_zero_rows_rankzeroevaluationdeterminanthsf_prefix) + (ff_an_mce_alternating_mdr_zero_rows_rankzeroevaluationdeterminanthsf_prefix) * (ff_bp_mce_alternating_mdr_zero_rows_rankzeroevaluationdeterminanthsf_prefix))) \/ ((exists ff_odd_mce_term_mdr_zero_rows_rankzeroevaluationdeterminanthsf_prefix_term. ff_index_mce_alternating_mdr_zero_rows_rankzeroevaluationdeterminanthsf_prefix = 2 * ff_odd_mce_term_mdr_zero_rows_rankzeroevaluationdeterminanthsf_prefix_term + 1) /\ (ff_p_mce_alternating_mdr_zero_rows_rankzeroevaluationdeterminanthsf_prefix = (ff_ap_mce_alternating_mdr_zero_rows_rankzeroevaluationdeterminanthsf_prefix) * (ff_bn_mce_alternating_mdr_zero_rows_rankzeroevaluationdeterminanthsf_prefix) + (ff_an_mce_alternating_mdr_zero_rows_rankzeroevaluationdeterminanthsf_prefix) * (ff_bp_mce_alternating_mdr_zero_rows_rankzeroevaluationdeterminanthsf_prefix) /\ ff_n_mce_alternating_mdr_zero_rows_rankzeroevaluationdeterminanthsf_prefix = (ff_ap_mce_alternating_mdr_zero_rows_rankzeroevaluationdeterminanthsf_prefix) * (ff_bp_mce_alternating_mdr_zero_rows_rankzeroevaluationdeterminanthsf_prefix) + (ff_an_mce_alternating_mdr_zero_rows_rankzeroevaluationdeterminanthsf_prefix) * (ff_bn_mce_alternating_mdr_zero_rows_rankzeroevaluationdeterminanthsf_prefix))))))))))) /\ ((exists ff_u_mce_mdr_zero_rows_rankzeroevaluationdeterminanthsf_positive ff_v_mce_mdr_zero_rows_rankzeroevaluationdeterminanthsf_positive. ((((exists ff_h_mce_mdr_zero_rows_rankzeroevaluationdeterminanthsf_positive_start. ff_h_mce_mdr_zero_rows_rankzeroevaluationdeterminanthsf_positive_start + S (0) = S ((S (0)) * ff_v_mce_mdr_zero_rows_rankzeroevaluationdeterminanthsf_positive)) /\ exists ff_q_mce_mdr_zero_rows_rankzeroevaluationdeterminanthsf_positive_start. ff_u_mce_mdr_zero_rows_rankzeroevaluationdeterminanthsf_positive = ff_q_mce_mdr_zero_rows_rankzeroevaluationdeterminanthsf_positive_start * S ((S (0)) * ff_v_mce_mdr_zero_rows_rankzeroevaluationdeterminanthsf_positive) + (0))) /\ ((((exists ff_h_mce_mdr_zero_rows_rankzeroevaluationdeterminanthsf_positive_terminal. ff_h_mce_mdr_zero_rows_rankzeroevaluationdeterminanthsf_positive_terminal + S (mdr_p_zero_rows_rankzeroevaluationdeterminanth) = S ((S ((S (mdr_q_zero_rows_rankzeroevaluationdeterminanths)))) * ff_v_mce_mdr_zero_rows_rankzeroevaluationdeterminanthsf_positive)) /\ exists ff_q_mce_mdr_zero_rows_rankzeroevaluationdeterminanthsf_positive_terminal. ff_u_mce_mdr_zero_rows_rankzeroevaluationdeterminanthsf_positive = ff_q_mce_mdr_zero_rows_rankzeroevaluationdeterminanthsf_positive_terminal * S ((S ((S (mdr_q_zero_rows_rankzeroevaluationdeterminanths)))) * ff_v_mce_mdr_zero_rows_rankzeroevaluationdeterminanthsf_positive) + (mdr_p_zero_rows_rankzeroevaluationdeterminanth))) /\ forall ff_i_mce_mdr_zero_rows_rankzeroevaluationdeterminanthsf_positive. (exists ff_lt_mce_mdr_zero_rows_rankzeroevaluationdeterminanthsf_positive_bound. ff_lt_mce_mdr_zero_rows_rankzeroevaluationdeterminanthsf_positive_bound + S ff_i_mce_mdr_zero_rows_rankzeroevaluationdeterminanthsf_positive = (S (mdr_q_zero_rows_rankzeroevaluationdeterminanths))) -> exists ff_a_mce_mdr_zero_rows_rankzeroevaluationdeterminanthsf_positive ff_r_mce_mdr_zero_rows_rankzeroevaluationdeterminanthsf_positive ff_s_mce_mdr_zero_rows_rankzeroevaluationdeterminanthsf_positive. ((((exists ff_h_mce_mdr_zero_rows_rankzeroevaluationdeterminanthsf_positive_summand. ff_h_mce_mdr_zero_rows_rankzeroevaluationdeterminanthsf_positive_summand + S (ff_a_mce_mdr_zero_rows_rankzeroevaluationdeterminanthsf_positive) = S ((S (ff_i_mce_mdr_zero_rows_rankzeroevaluationdeterminanthsf_positive)) * ff_uc_mce_fold_mdr_zero_rows_rankzeroevaluationdeterminanthsf)) /\ exists ff_q_mce_mdr_zero_rows_rankzeroevaluationdeterminanthsf_positive_summand. ff_ub_mce_fold_mdr_zero_rows_rankzeroevaluationdeterminanthsf = ff_q_mce_mdr_zero_rows_rankzeroevaluationdeterminanthsf_positive_summand * S ((S (ff_i_mce_mdr_zero_rows_rankzeroevaluationdeterminanthsf_positive)) * ff_uc_mce_fold_mdr_zero_rows_rankzeroevaluationdeterminanthsf) + (ff_a_mce_mdr_zero_rows_rankzeroevaluationdeterminanthsf_positive))) /\ ((((exists ff_h_mce_mdr_zero_rows_rankzeroevaluationdeterminanthsf_positive_partial. ff_h_mce_mdr_zero_rows_rankzeroevaluationdeterminanthsf_positive_partial + S (ff_r_mce_mdr_zero_rows_rankzeroevaluationdeterminanthsf_positive) = S ((S (ff_i_mce_mdr_zero_rows_rankzeroevaluationdeterminanthsf_positive)) * ff_v_mce_mdr_zero_rows_rankzeroevaluationdeterminanthsf_positive)) /\ exists ff_q_mce_mdr_zero_rows_rankzeroevaluationdeterminanthsf_positive_partial. ff_u_mce_mdr_zero_rows_rankzeroevaluationdeterminanthsf_positive = ff_q_mce_mdr_zero_rows_rankzeroevaluationdeterminanthsf_positive_partial * S ((S (ff_i_mce_mdr_zero_rows_rankzeroevaluationdeterminanthsf_positive)) * ff_v_mce_mdr_zero_rows_rankzeroevaluationdeterminanthsf_positive) + (ff_r_mce_mdr_zero_rows_rankzeroevaluationdeterminanthsf_positive))) /\ ((((exists ff_h_mce_mdr_zero_rows_rankzeroevaluationdeterminanthsf_positive_successor. ff_h_mce_mdr_zero_rows_rankzeroevaluationdeterminanthsf_positive_successor + S (ff_s_mce_mdr_zero_rows_rankzeroevaluationdeterminanthsf_positive) = S ((S (S ff_i_mce_mdr_zero_rows_rankzeroevaluationdeterminanthsf_positive)) * ff_v_mce_mdr_zero_rows_rankzeroevaluationdeterminanthsf_positive)) /\ exists ff_q_mce_mdr_zero_rows_rankzeroevaluationdeterminanthsf_positive_successor. ff_u_mce_mdr_zero_rows_rankzeroevaluationdeterminanthsf_positive = ff_q_mce_mdr_zero_rows_rankzeroevaluationdeterminanthsf_positive_successor * S ((S (S ff_i_mce_mdr_zero_rows_rankzeroevaluationdeterminanthsf_positive)) * ff_v_mce_mdr_zero_rows_rankzeroevaluationdeterminanthsf_positive) + (ff_s_mce_mdr_zero_rows_rankzeroevaluationdeterminanthsf_positive))) /\ ff_s_mce_mdr_zero_rows_rankzeroevaluationdeterminanthsf_positive = ff_r_mce_mdr_zero_rows_rankzeroevaluationdeterminanthsf_positive + ff_a_mce_mdr_zero_rows_rankzeroevaluationdeterminanthsf_positive)))))) /\ (exists ff_u_mce_mdr_zero_rows_rankzeroevaluationdeterminanthsf_negative ff_v_mce_mdr_zero_rows_rankzeroevaluationdeterminanthsf_negative. ((((exists ff_h_mce_mdr_zero_rows_rankzeroevaluationdeterminanthsf_negative_start. ff_h_mce_mdr_zero_rows_rankzeroevaluationdeterminanthsf_negative_start + S (0) = S ((S (0)) * ff_v_mce_mdr_zero_rows_rankzeroevaluationdeterminanthsf_negative)) /\ exists ff_q_mce_mdr_zero_rows_rankzeroevaluationdeterminanthsf_negative_start. ff_u_mce_mdr_zero_rows_rankzeroevaluationdeterminanthsf_negative = ff_q_mce_mdr_zero_rows_rankzeroevaluationdeterminanthsf_negative_start * S ((S (0)) * ff_v_mce_mdr_zero_rows_rankzeroevaluationdeterminanthsf_negative) + (0))) /\ ((((exists ff_h_mce_mdr_zero_rows_rankzeroevaluationdeterminanthsf_negative_terminal. ff_h_mce_mdr_zero_rows_rankzeroevaluationdeterminanthsf_negative_terminal + S (mdr_n_zero_rows_rankzeroevaluationdeterminanth) = S ((S ((S (mdr_q_zero_rows_rankzeroevaluationdeterminanths)))) * ff_v_mce_mdr_zero_rows_rankzeroevaluationdeterminanthsf_negative)) /\ exists ff_q_mce_mdr_zero_rows_rankzeroevaluationdeterminanthsf_negative_terminal. ff_u_mce_mdr_zero_rows_rankzeroevaluationdeterminanthsf_negative = ff_q_mce_mdr_zero_rows_rankzeroevaluationdeterminanthsf_negative_terminal * S ((S ((S (mdr_q_zero_rows_rankzeroevaluationdeterminanths)))) * ff_v_mce_mdr_zero_rows_rankzeroevaluationdeterminanthsf_negative) + (mdr_n_zero_rows_rankzeroevaluationdeterminanth))) /\ forall ff_i_mce_mdr_zero_rows_rankzeroevaluationdeterminanthsf_negative. (exists ff_lt_mce_mdr_zero_rows_rankzeroevaluationdeterminanthsf_negative_bound. ff_lt_mce_mdr_zero_rows_rankzeroevaluationdeterminanthsf_negative_bound + S ff_i_mce_mdr_zero_rows_rankzeroevaluationdeterminanthsf_negative = (S (mdr_q_zero_rows_rankzeroevaluationdeterminanths))) -> exists ff_a_mce_mdr_zero_rows_rankzeroevaluationdeterminanthsf_negative ff_r_mce_mdr_zero_rows_rankzeroevaluationdeterminanthsf_negative ff_s_mce_mdr_zero_rows_rankzeroevaluationdeterminanthsf_negative. ((((exists ff_h_mce_mdr_zero_rows_rankzeroevaluationdeterminanthsf_negative_summand. ff_h_mce_mdr_zero_rows_rankzeroevaluationdeterminanthsf_negative_summand + S (ff_a_mce_mdr_zero_rows_rankzeroevaluationdeterminanthsf_negative) = S ((S (ff_i_mce_mdr_zero_rows_rankzeroevaluationdeterminanthsf_negative)) * ff_vc_mce_fold_mdr_zero_rows_rankzeroevaluationdeterminanthsf)) /\ exists ff_q_mce_mdr_zero_rows_rankzeroevaluationdeterminanthsf_negative_summand. ff_vb_mce_fold_mdr_zero_rows_rankzeroevaluationdeterminanthsf = ff_q_mce_mdr_zero_rows_rankzeroevaluationdeterminanthsf_negative_summand * S ((S (ff_i_mce_mdr_zero_rows_rankzeroevaluationdeterminanthsf_negative)) * ff_vc_mce_fold_mdr_zero_rows_rankzeroevaluationdeterminanthsf) + (ff_a_mce_mdr_zero_rows_rankzeroevaluationdeterminanthsf_negative))) /\ ((((exists ff_h_mce_mdr_zero_rows_rankzeroevaluationdeterminanthsf_negative_partial. ff_h_mce_mdr_zero_rows_rankzeroevaluationdeterminanthsf_negative_partial + S (ff_r_mce_mdr_zero_rows_rankzeroevaluationdeterminanthsf_negative) = S ((S (ff_i_mce_mdr_zero_rows_rankzeroevaluationdeterminanthsf_negative)) * ff_v_mce_mdr_zero_rows_rankzeroevaluationdeterminanthsf_negative)) /\ exists ff_q_mce_mdr_zero_rows_rankzeroevaluationdeterminanthsf_negative_partial. ff_u_mce_mdr_zero_rows_rankzeroevaluationdeterminanthsf_negative = ff_q_mce_mdr_zero_rows_rankzeroevaluationdeterminanthsf_negative_partial * S ((S (ff_i_mce_mdr_zero_rows_rankzeroevaluationdeterminanthsf_negative)) * ff_v_mce_mdr_zero_rows_rankzeroevaluationdeterminanthsf_negative) + (ff_r_mce_mdr_zero_rows_rankzeroevaluationdeterminanthsf_negative))) /\ ((((exists ff_h_mce_mdr_zero_rows_rankzeroevaluationdeterminanthsf_negative_successor. ff_h_mce_mdr_zero_rows_rankzeroevaluationdeterminanthsf_negative_successor + S (ff_s_mce_mdr_zero_rows_rankzeroevaluationdeterminanthsf_negative) = S ((S (S ff_i_mce_mdr_zero_rows_rankzeroevaluationdeterminanthsf_negative)) * ff_v_mce_mdr_zero_rows_rankzeroevaluationdeterminanthsf_negative)) /\ exists ff_q_mce_mdr_zero_rows_rankzeroevaluationdeterminanthsf_negative_successor. ff_u_mce_mdr_zero_rows_rankzeroevaluationdeterminanthsf_negative = ff_q_mce_mdr_zero_rows_rankzeroevaluationdeterminanthsf_negative_successor * S ((S (S ff_i_mce_mdr_zero_rows_rankzeroevaluationdeterminanthsf_negative)) * ff_v_mce_mdr_zero_rows_rankzeroevaluationdeterminanthsf_negative) + (ff_s_mce_mdr_zero_rows_rankzeroevaluationdeterminanthsf_negative))) /\ ff_s_mce_mdr_zero_rows_rankzeroevaluationdeterminanthsf_negative = ff_r_mce_mdr_zero_rows_rankzeroevaluationdeterminanthsf_negative + ff_a_mce_mdr_zero_rows_rankzeroevaluationdeterminanthsf_negative))))))))))))))) /\ ((exists mdr_gap_zero_rows_rankzeroevaluationdeterminanti. mdr_gap_zero_rows_rankzeroevaluationdeterminanti + S (mdr_i_zero_rows_rankzeroevaluationdeterminant) = (mdr_l_zero_rows_rankzeroevaluationdeterminant)) /\ (exists mdr_z_zero_rows_rankzeroevaluationdeterminantr. ((exists mdr_a_zero_rows_rankzeroevaluationdeterminantrc mdr_b_zero_rows_rankzeroevaluationdeterminantrc mdr_c_zero_rows_rankzeroevaluationdeterminantrc mdr_e_zero_rows_rankzeroevaluationdeterminantrc mdr_f_zero_rows_rankzeroevaluationdeterminantrc. ((mdr_a_zero_rows_rankzeroevaluationdeterminantrc = ((mdr_q_zero_rows_rank) + (mdr_ub_zero_rows_rankzeroevaluation)) * S ((mdr_q_zero_rows_rank) + (mdr_ub_zero_rows_rankzeroevaluation)) + ((mdr_ub_zero_rows_rankzeroevaluation) + (mdr_ub_zero_rows_rankzeroevaluation))) /\ ((mdr_b_zero_rows_rankzeroevaluationdeterminantrc = ((mdr_uc_zero_rows_rankzeroevaluation) + (mdr_vb_zero_rows_rankzeroevaluation)) * S ((mdr_uc_zero_rows_rankzeroevaluation) + (mdr_vb_zero_rows_rankzeroevaluation)) + ((mdr_vb_zero_rows_rankzeroevaluation) + (mdr_vb_zero_rows_rankzeroevaluation))) /\ ((mdr_c_zero_rows_rankzeroevaluationdeterminantrc = ((mdr_a_zero_rows_rankzeroevaluationdeterminantrc) + (mdr_b_zero_rows_rankzeroevaluationdeterminantrc)) * S ((mdr_a_zero_rows_rankzeroevaluationdeterminantrc) + (mdr_b_zero_rows_rankzeroevaluationdeterminantrc)) + ((mdr_b_zero_rows_rankzeroevaluationdeterminantrc) + (mdr_b_zero_rows_rankzeroevaluationdeterminantrc))) /\ ((mdr_e_zero_rows_rankzeroevaluationdeterminantrc = ((mdr_p_zero_rows_rankzero) + (mdr_n_zero_rows_rankzero)) * S ((mdr_p_zero_rows_rankzero) + (mdr_n_zero_rows_rankzero)) + ((mdr_n_zero_rows_rankzero) + (mdr_n_zero_rows_rankzero))) /\ ((mdr_f_zero_rows_rankzeroevaluationdeterminantrc = ((mdr_vc_zero_rows_rankzeroevaluation) + (mdr_e_zero_rows_rankzeroevaluationdeterminantrc)) * S ((mdr_vc_zero_rows_rankzeroevaluation) + (mdr_e_zero_rows_rankzeroevaluationdeterminantrc)) + ((mdr_e_zero_rows_rankzeroevaluationdeterminantrc) + (mdr_e_zero_rows_rankzeroevaluationdeterminantrc))) /\ ((mdr_z_zero_rows_rankzeroevaluationdeterminantr) = ((mdr_c_zero_rows_rankzeroevaluationdeterminantrc) + (mdr_f_zero_rows_rankzeroevaluationdeterminantrc)) * S ((mdr_c_zero_rows_rankzeroevaluationdeterminantrc) + (mdr_f_zero_rows_rankzeroevaluationdeterminantrc)) + ((mdr_f_zero_rows_rankzeroevaluationdeterminantrc) + (mdr_f_zero_rows_rankzeroevaluationdeterminantrc))))))))) /\ (((exists ff_h_mdr_zero_rows_rankzeroevaluationdeterminantrb. ff_h_mdr_zero_rows_rankzeroevaluationdeterminantrb + S (mdr_z_zero_rows_rankzeroevaluationdeterminantr) = S ((S (mdr_i_zero_rows_rankzeroevaluationdeterminant)) * mdr_c_zero_rows_rankzeroevaluationdeterminant)) /\ exists ff_q_mdr_zero_rows_rankzeroevaluationdeterminantrb. mdr_b_zero_rows_rankzeroevaluationdeterminant = ff_q_mdr_zero_rows_rankzeroevaluationdeterminantrb * S ((S (mdr_i_zero_rows_rankzeroevaluationdeterminant)) * mdr_c_zero_rows_rankzeroevaluationdeterminant) + (mdr_z_zero_rows_rankzeroevaluationdeterminantr)))))))))) -> mdr_p_zero_rows_rankzero = mdr_n_zero_rows_rankzero))))))Constructive proof overview
Generated structural guide
Every zero-row rectangular matrix has rank zero, with the actual empty determinant supplying its nonzero zero-order minor.
The unchanged tactic script uses 2 declared prerequisites and contains 39 exact native proof lines.
Alpha v34 checked-use · first admitted v27 · independently kernel and Lean verified; not Stable
Proof neighborhood
Direct dependencies
DL005E rectangular_matrix_rank_certificate_exists le_zero Stable theorem; checked-use authorizedDirect dependents
none
Formal native tactic body
Dependencies are introduced as named hypotheses before line 1. Local theorem links identify exact declared prerequisites. This exact body belongs to a complete independently kernel-checked constructive proof bundle and has Alpha checked-use authority; it does not imply Stable membership.
Read the argument
Proof checkpoints
This is a reading aid, not a new proof or a proof-tree certificate. Checkpoint groups are consecutive commands, not inferred branch boundaries. Every step links to the preserved script.
Named ingredients (1)
01Fix variables and assumptionsL1–5
02Establish hrankL6–13
Establish this local claim before using it. It is not an additional assumption. The following proof commands apply rectangular matrix rank certificate exists.
- L6
have hrank : ∃ rank. RectangularMatrixRank(pb,pc,nb,nc,0,w,rank)Definitions: RectangularMatrixRank - L7
specialize rectangular_matrix_rank_certificate_exists (pb) - L8
specialize rectangular_matrix_rank_certificate_exists (pc) - L9
specialize rectangular_matrix_rank_certificate_exists (nb) - L10
specialize rectangular_matrix_rank_certificate_exists (nc) - L11
specialize rectangular_matrix_rank_certificate_exists (0) - L12
specialize rectangular_matrix_rank_certificate_exists (w) - L13
apply rectangular_matrix_rank_certificate_exists
03Separate the logical casesL14–15
04Establish hzeroL16–25
Establish this local claim before using it. It is not an additional assumption. The following proof commands apply le zero.
05Calculate and transport equalitiesL26–35
Carry out the recorded arithmetic or equality steps; inspect the exact commands for their direction and premises.
- L26
rewrite hzero at hrank_witness - L27
rewrite hzero at hrank_witness - L28
rewrite hzero at hrank_witness - L29
rewrite hzero at hrank_witness - L30
rewrite hzero at hrank_witness - L31
rewrite hzero at hrank_witness - L32
rewrite hzero at hrank_witness - L33
rewrite hzero at hrank_witness - L34
rewrite hzero at hrank_witness - L35
rewrite hzero at hrank_witness
06Calculate and transport equalitiesL36–38
07Use earlier factsL39–39
Instantiate or apply named facts and discharge the corresponding proof obligations.
- L39
exact hrank_witness
Original exact command ledger · 39 lines
- 0001
intro pb - 0002
intro pc - 0003
intro nb - 0004
intro nc - 0005
intro w - 0006
have hrank : exists rank. (((exists mdr_gap_zero_rows_existsrows_bound. mdr_gap_zero_rows_existsrows_bound + (rank) = (0)) /\ ((exists mdr_gap_zero_rows_existscolumns_bound. mdr_gap_zero_rows_existscolumns_bound + (rank) = (w)) /\ ((exists mdr_rb_zero_rows_existswitness mdr_rc_zero_rows_existswitness mdr_cb_zero_rows_existswitness mdr_cc_zero_rows_existswitness. (((((forall fom_index_mrf_zero_rows_existswitnessminorrowsbound. (exists fom_gap_mrf_zero_rows_existswitnessminorrowsbound_index_bound. fom_gap_mrf_zero_rows_existswitnessminorrowsbound_index_bound + S (fom_index_mrf_zero_rows_existswitnessminorrowsbound) = rank) -> exists fom_value_mrf_zero_rows_existswitnessminorrowsbound. ((((exists fom_beta_height_mrf_zero_rows_existswitnessminorrowsbound_entry. fom_beta_height_mrf_zero_rows_existswitnessminorrowsbound_entry + S (fom_value_mrf_zero_rows_existswitnessminorrowsbound) = S ((S (fom_index_mrf_zero_rows_existswitnessminorrowsbound)) * mdr_rc_zero_rows_existswitness)) /\ exists fom_beta_quotient_mrf_zero_rows_existswitnessminorrowsbound_entry. mdr_rb_zero_rows_existswitness = fom_beta_quotient_mrf_zero_rows_existswitnessminorrowsbound_entry * S ((S (fom_index_mrf_zero_rows_existswitnessminorrowsbound)) * mdr_rc_zero_rows_existswitness) + (fom_value_mrf_zero_rows_existswitnessminorrowsbound))) /\ (exists fom_gap_mrf_zero_rows_existswitnessminorrowsbound_value_bound. fom_gap_mrf_zero_rows_existswitnessminorrowsbound_value_bound + S (fom_value_mrf_zero_rows_existswitnessminorrowsbound) = 0))) /\ (forall mdr_i_zero_rows_existswitnessminorrowsdistinct mdr_j_zero_rows_existswitnessminorrowsdistinct mdr_a_zero_rows_existswitnessminorrowsdistinct. (exists mdr_gap_zero_rows_existswitnessminorrowsdistincti. mdr_gap_zero_rows_existswitnessminorrowsdistincti + S (mdr_i_zero_rows_existswitnessminorrowsdistinct) = (rank)) -> (exists mdr_gap_zero_rows_existswitnessminorrowsdistinctj. mdr_gap_zero_rows_existswitnessminorrowsdistinctj + S (mdr_j_zero_rows_existswitnessminorrowsdistinct) = (rank)) -> (((exists ff_h_mdr_zero_rows_existswitnessminorrowsdistinctfirst. ff_h_mdr_zero_rows_existswitnessminorrowsdistinctfirst + S (mdr_a_zero_rows_existswitnessminorrowsdistinct) = S ((S (mdr_i_zero_rows_existswitnessminorrowsdistinct)) * mdr_rc_zero_rows_existswitness)) /\ exists ff_q_mdr_zero_rows_existswitnessminorrowsdistinctfirst. mdr_rb_zero_rows_existswitness = ff_q_mdr_zero_rows_existswitnessminorrowsdistinctfirst * S ((S (mdr_i_zero_rows_existswitnessminorrowsdistinct)) * mdr_rc_zero_rows_existswitness) + (mdr_a_zero_rows_existswitnessminorrowsdistinct))) -> (((exists ff_h_mdr_zero_rows_existswitnessminorrowsdistinctsecond. ff_h_mdr_zero_rows_existswitnessminorrowsdistinctsecond + S (mdr_a_zero_rows_existswitnessminorrowsdistinct) = S ((S (mdr_j_zero_rows_existswitnessminorrowsdistinct)) * mdr_rc_zero_rows_existswitness)) /\ exists ff_q_mdr_zero_rows_existswitnessminorrowsdistinctsecond. mdr_rb_zero_rows_existswitness = ff_q_mdr_zero_rows_existswitnessminorrowsdistinctsecond * S ((S (mdr_j_zero_rows_existswitnessminorrowsdistinct)) * mdr_rc_zero_rows_existswitness) + (mdr_a_zero_rows_existswitnessminorrowsdistinct))) -> mdr_i_zero_rows_existswitnessminorrowsdistinct = mdr_j_zero_rows_existswitnessminorrowsdistinct))) /\ ((((forall fom_index_mrf_zero_rows_existswitnessminorcolumnsbound. (exists fom_gap_mrf_zero_rows_existswitnessminorcolumnsbound_index_bound. fom_gap_mrf_zero_rows_existswitnessminorcolumnsbound_index_bound + S (fom_index_mrf_zero_rows_existswitnessminorcolumnsbound) = rank) -> exists fom_value_mrf_zero_rows_existswitnessminorcolumnsbound. ((((exists fom_beta_height_mrf_zero_rows_existswitnessminorcolumnsbound_entry. fom_beta_height_mrf_zero_rows_existswitnessminorcolumnsbound_entry + S (fom_value_mrf_zero_rows_existswitnessminorcolumnsbound) = S ((S (fom_index_mrf_zero_rows_existswitnessminorcolumnsbound)) * mdr_cc_zero_rows_existswitness)) /\ exists fom_beta_quotient_mrf_zero_rows_existswitnessminorcolumnsbound_entry. mdr_cb_zero_rows_existswitness = fom_beta_quotient_mrf_zero_rows_existswitnessminorcolumnsbound_entry * S ((S (fom_index_mrf_zero_rows_existswitnessminorcolumnsbound)) * mdr_cc_zero_rows_existswitness) + (fom_value_mrf_zero_rows_existswitnessminorcolumnsbound))) /\ (exists fom_gap_mrf_zero_rows_existswitnessminorcolumnsbound_value_bound. fom_gap_mrf_zero_rows_existswitnessminorcolumnsbound_value_bound + S (fom_value_mrf_zero_rows_existswitnessminorcolumnsbound) = w))) /\ (forall mdr_i_zero_rows_existswitnessminorcolumnsdistinct mdr_j_zero_rows_existswitnessminorcolumnsdistinct mdr_a_zero_rows_existswitnessminorcolumnsdistinct. (exists mdr_gap_zero_rows_existswitnessminorcolumnsdistincti. mdr_gap_zero_rows_existswitnessminorcolumnsdistincti + S (mdr_i_zero_rows_existswitnessminorcolumnsdistinct) = (rank)) -> (exists mdr_gap_zero_rows_existswitnessminorcolumnsdistinctj. mdr_gap_zero_rows_existswitnessminorcolumnsdistinctj + S (mdr_j_zero_rows_existswitnessminorcolumnsdistinct) = (rank)) -> (((exists ff_h_mdr_zero_rows_existswitnessminorcolumnsdistinctfirst. ff_h_mdr_zero_rows_existswitnessminorcolumnsdistinctfirst + S (mdr_a_zero_rows_existswitnessminorcolumnsdistinct) = S ((S (mdr_i_zero_rows_existswitnessminorcolumnsdistinct)) * mdr_cc_zero_rows_existswitness)) /\ exists ff_q_mdr_zero_rows_existswitnessminorcolumnsdistinctfirst. mdr_cb_zero_rows_existswitness = ff_q_mdr_zero_rows_existswitnessminorcolumnsdistinctfirst * S ((S (mdr_i_zero_rows_existswitnessminorcolumnsdistinct)) * mdr_cc_zero_rows_existswitness) + (mdr_a_zero_rows_existswitnessminorcolumnsdistinct))) -> (((exists ff_h_mdr_zero_rows_existswitnessminorcolumnsdistinctsecond. ff_h_mdr_zero_rows_existswitnessminorcolumnsdistinctsecond + S (mdr_a_zero_rows_existswitnessminorcolumnsdistinct) = S ((S (mdr_j_zero_rows_existswitnessminorcolumnsdistinct)) * mdr_cc_zero_rows_existswitness)) /\ exists ff_q_mdr_zero_rows_existswitnessminorcolumnsdistinctsecond. mdr_cb_zero_rows_existswitness = ff_q_mdr_zero_rows_existswitnessminorcolumnsdistinctsecond * S ((S (mdr_j_zero_rows_existswitnessminorcolumnsdistinct)) * mdr_cc_zero_rows_existswitness) + (mdr_a_zero_rows_existswitnessminorcolumnsdistinct))) -> mdr_i_zero_rows_existswitnessminorcolumnsdistinct = mdr_j_zero_rows_existswitnessminorcolumnsdistinct))) /\ (exists mdr_p_zero_rows_existswitnessminornonzero mdr_n_zero_rows_existswitnessminornonzero. ((exists mdr_ub_zero_rows_existswitnessminornonzeroevaluation mdr_uc_zero_rows_existswitnessminornonzeroevaluation mdr_vb_zero_rows_existswitnessminornonzeroevaluation mdr_vc_zero_rows_existswitnessminornonzeroevaluation. ((((forall mdr_i_zero_rows_existswitnessminornonzeroevaluationmatrixpositive. (exists mdr_gap_zero_rows_existswitnessminornonzeroevaluationmatrixpositivebound. mdr_gap_zero_rows_existswitnessminornonzeroevaluationmatrixpositivebound + S (mdr_i_zero_rows_existswitnessminornonzeroevaluationmatrixpositive) = ((rank) * (rank))) -> exists mdr_a_zero_rows_existswitnessminornonzeroevaluationmatrixpositive. (((exists mdr_r_zero_rows_existswitnessminornonzeroevaluationmatrixpositivepoint mdr_s_zero_rows_existswitnessminornonzeroevaluationmatrixpositivepoint mdr_u_zero_rows_existswitnessminornonzeroevaluationmatrixpositivepoint mdr_v_zero_rows_existswitnessminornonzeroevaluationmatrixpositivepoint. ((mdr_i_zero_rows_existswitnessminornonzeroevaluationmatrixpositive = (rank) * mdr_r_zero_rows_existswitnessminornonzeroevaluationmatrixpositivepoint + mdr_s_zero_rows_existswitnessminornonzeroevaluationmatrixpositivepoint) /\ ((exists mdr_gap_zero_rows_existswitnessminornonzeroevaluationmatrixpositivepointcolumn. mdr_gap_zero_rows_existswitnessminornonzeroevaluationmatrixpositivepointcolumn + S (mdr_s_zero_rows_existswitnessminornonzeroevaluationmatrixpositivepoint) = (rank)) /\ ((((exists ff_h_mdr_zero_rows_existswitnessminornonzeroevaluationmatrixpositivepointrow_index. ff_h_mdr_zero_rows_existswitnessminornonzeroevaluationmatrixpositivepointrow_index + S (mdr_u_zero_rows_existswitnessminornonzeroevaluationmatrixpositivepoint) = S ((S (mdr_r_zero_rows_existswitnessminornonzeroevaluationmatrixpositivepoint)) * mdr_rc_zero_rows_existswitness)) /\ exists ff_q_mdr_zero_rows_existswitnessminornonzeroevaluationmatrixpositivepointrow_index. mdr_rb_zero_rows_existswitness = ff_q_mdr_zero_rows_existswitnessminornonzeroevaluationmatrixpositivepointrow_index * S ((S (mdr_r_zero_rows_existswitnessminornonzeroevaluationmatrixpositivepoint)) * mdr_rc_zero_rows_existswitness) + (mdr_u_zero_rows_existswitnessminornonzeroevaluationmatrixpositivepoint))) /\ ((((exists ff_h_mdr_zero_rows_existswitnessminornonzeroevaluationmatrixpositivepointcolumn_index. ff_h_mdr_zero_rows_existswitnessminornonzeroevaluationmatrixpositivepointcolumn_index + S (mdr_v_zero_rows_existswitnessminornonzeroevaluationmatrixpositivepoint) = S ((S (mdr_s_zero_rows_existswitnessminornonzeroevaluationmatrixpositivepoint)) * mdr_cc_zero_rows_existswitness)) /\ exists ff_q_mdr_zero_rows_existswitnessminornonzeroevaluationmatrixpositivepointcolumn_index. mdr_cb_zero_rows_existswitness = ff_q_mdr_zero_rows_existswitnessminornonzeroevaluationmatrixpositivepointcolumn_index * S ((S (mdr_s_zero_rows_existswitnessminornonzeroevaluationmatrixpositivepoint)) * mdr_cc_zero_rows_existswitness) + (mdr_v_zero_rows_existswitnessminornonzeroevaluationmatrixpositivepoint))) /\ (((exists ff_h_mdr_zero_rows_existswitnessminornonzeroevaluationmatrixpositivepointsource. ff_h_mdr_zero_rows_existswitnessminornonzeroevaluationmatrixpositivepointsource + S (mdr_a_zero_rows_existswitnessminornonzeroevaluationmatrixpositive) = S ((S ((mdr_u_zero_rows_existswitnessminornonzeroevaluationmatrixpositivepoint) * (w) + (mdr_v_zero_rows_existswitnessminornonzeroevaluationmatrixpositivepoint))) * pc)) /\ exists ff_q_mdr_zero_rows_existswitnessminornonzeroevaluationmatrixpositivepointsource. pb = ff_q_mdr_zero_rows_existswitnessminornonzeroevaluationmatrixpositivepointsource * S ((S ((mdr_u_zero_rows_existswitnessminornonzeroevaluationmatrixpositivepoint) * (w) + (mdr_v_zero_rows_existswitnessminornonzeroevaluationmatrixpositivepoint))) * pc) + (mdr_a_zero_rows_existswitnessminornonzeroevaluationmatrixpositive)))))))) /\ (((exists ff_h_mdr_zero_rows_existswitnessminornonzeroevaluationmatrixpositiveoutput. ff_h_mdr_zero_rows_existswitnessminornonzeroevaluationmatrixpositiveoutput + S (mdr_a_zero_rows_existswitnessminornonzeroevaluationmatrixpositive) = S ((S (mdr_i_zero_rows_existswitnessminornonzeroevaluationmatrixpositive)) * mdr_uc_zero_rows_existswitnessminornonzeroevaluation)) /\ exists ff_q_mdr_zero_rows_existswitnessminornonzeroevaluationmatrixpositiveoutput. mdr_ub_zero_rows_existswitnessminornonzeroevaluation = ff_q_mdr_zero_rows_existswitnessminornonzeroevaluationmatrixpositiveoutput * S ((S (mdr_i_zero_rows_existswitnessminornonzeroevaluationmatrixpositive)) * mdr_uc_zero_rows_existswitnessminornonzeroevaluation) + (mdr_a_zero_rows_existswitnessminornonzeroevaluationmatrixpositive)))))) /\ (forall mdr_i_zero_rows_existswitnessminornonzeroevaluationmatrixnegative. (exists mdr_gap_zero_rows_existswitnessminornonzeroevaluationmatrixnegativebound. mdr_gap_zero_rows_existswitnessminornonzeroevaluationmatrixnegativebound + S (mdr_i_zero_rows_existswitnessminornonzeroevaluationmatrixnegative) = ((rank) * (rank))) -> exists mdr_a_zero_rows_existswitnessminornonzeroevaluationmatrixnegative. (((exists mdr_r_zero_rows_existswitnessminornonzeroevaluationmatrixnegativepoint mdr_s_zero_rows_existswitnessminornonzeroevaluationmatrixnegativepoint mdr_u_zero_rows_existswitnessminornonzeroevaluationmatrixnegativepoint mdr_v_zero_rows_existswitnessminornonzeroevaluationmatrixnegativepoint. ((mdr_i_zero_rows_existswitnessminornonzeroevaluationmatrixnegative = (rank) * mdr_r_zero_rows_existswitnessminornonzeroevaluationmatrixnegativepoint + mdr_s_zero_rows_existswitnessminornonzeroevaluationmatrixnegativepoint) /\ ((exists mdr_gap_zero_rows_existswitnessminornonzeroevaluationmatrixnegativepointcolumn. mdr_gap_zero_rows_existswitnessminornonzeroevaluationmatrixnegativepointcolumn + S (mdr_s_zero_rows_existswitnessminornonzeroevaluationmatrixnegativepoint) = (rank)) /\ ((((exists ff_h_mdr_zero_rows_existswitnessminornonzeroevaluationmatrixnegativepointrow_index. ff_h_mdr_zero_rows_existswitnessminornonzeroevaluationmatrixnegativepointrow_index + S (mdr_u_zero_rows_existswitnessminornonzeroevaluationmatrixnegativepoint) = S ((S (mdr_r_zero_rows_existswitnessminornonzeroevaluationmatrixnegativepoint)) * mdr_rc_zero_rows_existswitness)) /\ exists ff_q_mdr_zero_rows_existswitnessminornonzeroevaluationmatrixnegativepointrow_index. mdr_rb_zero_rows_existswitness = ff_q_mdr_zero_rows_existswitnessminornonzeroevaluationmatrixnegativepointrow_index * S ((S (mdr_r_zero_rows_existswitnessminornonzeroevaluationmatrixnegativepoint)) * mdr_rc_zero_rows_existswitness) + (mdr_u_zero_rows_existswitnessminornonzeroevaluationmatrixnegativepoint))) /\ ((((exists ff_h_mdr_zero_rows_existswitnessminornonzeroevaluationmatrixnegativepointcolumn_index. ff_h_mdr_zero_rows_existswitnessminornonzeroevaluationmatrixnegativepointcolumn_index + S (mdr_v_zero_rows_existswitnessminornonzeroevaluationmatrixnegativepoint) = S ((S (mdr_s_zero_rows_existswitnessminornonzeroevaluationmatrixnegativepoint)) * mdr_cc_zero_rows_existswitness)) /\ exists ff_q_mdr_zero_rows_existswitnessminornonzeroevaluationmatrixnegativepointcolumn_index. mdr_cb_zero_rows_existswitness = ff_q_mdr_zero_rows_existswitnessminornonzeroevaluationmatrixnegativepointcolumn_index * S ((S (mdr_s_zero_rows_existswitnessminornonzeroevaluationmatrixnegativepoint)) * mdr_cc_zero_rows_existswitness) + (mdr_v_zero_rows_existswitnessminornonzeroevaluationmatrixnegativepoint))) /\ (((exists ff_h_mdr_zero_rows_existswitnessminornonzeroevaluationmatrixnegativepointsource. ff_h_mdr_zero_rows_existswitnessminornonzeroevaluationmatrixnegativepointsource + S (mdr_a_zero_rows_existswitnessminornonzeroevaluationmatrixnegative) = S ((S ((mdr_u_zero_rows_existswitnessminornonzeroevaluationmatrixnegativepoint) * (w) + (mdr_v_zero_rows_existswitnessminornonzeroevaluationmatrixnegativepoint))) * nc)) /\ exists ff_q_mdr_zero_rows_existswitnessminornonzeroevaluationmatrixnegativepointsource. nb = ff_q_mdr_zero_rows_existswitnessminornonzeroevaluationmatrixnegativepointsource * S ((S ((mdr_u_zero_rows_existswitnessminornonzeroevaluationmatrixnegativepoint) * (w) + (mdr_v_zero_rows_existswitnessminornonzeroevaluationmatrixnegativepoint))) * nc) + (mdr_a_zero_rows_existswitnessminornonzeroevaluationmatrixnegative)))))))) /\ (((exists ff_h_mdr_zero_rows_existswitnessminornonzeroevaluationmatrixnegativeoutput. ff_h_mdr_zero_rows_existswitnessminornonzeroevaluationmatrixnegativeoutput + S (mdr_a_zero_rows_existswitnessminornonzeroevaluationmatrixnegative) = S ((S (mdr_i_zero_rows_existswitnessminornonzeroevaluationmatrixnegative)) * mdr_vc_zero_rows_existswitnessminornonzeroevaluation)) /\ exists ff_q_mdr_zero_rows_existswitnessminornonzeroevaluationmatrixnegativeoutput. mdr_vb_zero_rows_existswitnessminornonzeroevaluation = ff_q_mdr_zero_rows_existswitnessminornonzeroevaluationmatrixnegativeoutput * S ((S (mdr_i_zero_rows_existswitnessminornonzeroevaluationmatrixnegative)) * mdr_vc_zero_rows_existswitnessminornonzeroevaluation) + (mdr_a_zero_rows_existswitnessminornonzeroevaluationmatrixnegative)))))))) /\ (exists mdr_b_zero_rows_existswitnessminornonzeroevaluationdeterminant mdr_c_zero_rows_existswitnessminornonzeroevaluationdeterminant mdr_l_zero_rows_existswitnessminornonzeroevaluationdeterminant mdr_i_zero_rows_existswitnessminornonzeroevaluationdeterminant. ((forall mdr_i_zero_rows_existswitnessminornonzeroevaluationdeterminanth. (exists mdr_gap_zero_rows_existswitnessminornonzeroevaluationdeterminanthi. mdr_gap_zero_rows_existswitnessminornonzeroevaluationdeterminanthi + S (mdr_i_zero_rows_existswitnessminornonzeroevaluationdeterminanth) = (mdr_l_zero_rows_existswitnessminornonzeroevaluationdeterminant)) -> exists mdr_d_zero_rows_existswitnessminornonzeroevaluationdeterminanth mdr_pb_zero_rows_existswitnessminornonzeroevaluationdeterminanth mdr_pc_zero_rows_existswitnessminornonzeroevaluationdeterminanth mdr_nb_zero_rows_existswitnessminornonzeroevaluationdeterminanth mdr_nc_zero_rows_existswitnessminornonzeroevaluationdeterminanth mdr_p_zero_rows_existswitnessminornonzeroevaluationdeterminanth mdr_n_zero_rows_existswitnessminornonzeroevaluationdeterminanth. ((exists mdr_z_zero_rows_existswitnessminornonzeroevaluationdeterminanthr. ((exists mdr_a_zero_rows_existswitnessminornonzeroevaluationdeterminanthrc mdr_b_zero_rows_existswitnessminornonzeroevaluationdeterminanthrc mdr_c_zero_rows_existswitnessminornonzeroevaluationdeterminanthrc mdr_e_zero_rows_existswitnessminornonzeroevaluationdeterminanthrc mdr_f_zero_rows_existswitnessminornonzeroevaluationdeterminanthrc. ((mdr_a_zero_rows_existswitnessminornonzeroevaluationdeterminanthrc = ((mdr_d_zero_rows_existswitnessminornonzeroevaluationdeterminanth) + (mdr_pb_zero_rows_existswitnessminornonzeroevaluationdeterminanth)) * S ((mdr_d_zero_rows_existswitnessminornonzeroevaluationdeterminanth) + (mdr_pb_zero_rows_existswitnessminornonzeroevaluationdeterminanth)) + ((mdr_pb_zero_rows_existswitnessminornonzeroevaluationdeterminanth) + (mdr_pb_zero_rows_existswitnessminornonzeroevaluationdeterminanth))) /\ ((mdr_b_zero_rows_existswitnessminornonzeroevaluationdeterminanthrc = ((mdr_pc_zero_rows_existswitnessminornonzeroevaluationdeterminanth) + (mdr_nb_zero_rows_existswitnessminornonzeroevaluationdeterminanth)) * S ((mdr_pc_zero_rows_existswitnessminornonzeroevaluationdeterminanth) + (mdr_nb_zero_rows_existswitnessminornonzeroevaluationdeterminanth)) + ((mdr_nb_zero_rows_existswitnessminornonzeroevaluationdeterminanth) + (mdr_nb_zero_rows_existswitnessminornonzeroevaluationdeterminanth))) /\ ((mdr_c_zero_rows_existswitnessminornonzeroevaluationdeterminanthrc = ((mdr_a_zero_rows_existswitnessminornonzeroevaluationdeterminanthrc) + (mdr_b_zero_rows_existswitnessminornonzeroevaluationdeterminanthrc)) * S ((mdr_a_zero_rows_existswitnessminornonzeroevaluationdeterminanthrc) + (mdr_b_zero_rows_existswitnessminornonzeroevaluationdeterminanthrc)) + ((mdr_b_zero_rows_existswitnessminornonzeroevaluationdeterminanthrc) + (mdr_b_zero_rows_existswitnessminornonzeroevaluationdeterminanthrc))) /\ ((mdr_e_zero_rows_existswitnessminornonzeroevaluationdeterminanthrc = ((mdr_p_zero_rows_existswitnessminornonzeroevaluationdeterminanth) + (mdr_n_zero_rows_existswitnessminornonzeroevaluationdeterminanth)) * S ((mdr_p_zero_rows_existswitnessminornonzeroevaluationdeterminanth) + (mdr_n_zero_rows_existswitnessminornonzeroevaluationdeterminanth)) + ((mdr_n_zero_rows_existswitnessminornonzeroevaluationdeterminanth) + (mdr_n_zero_rows_existswitnessminornonzeroevaluationdeterminanth))) /\ ((mdr_f_zero_rows_existswitnessminornonzeroevaluationdeterminanthrc = ((mdr_nc_zero_rows_existswitnessminornonzeroevaluationdeterminanth) + (mdr_e_zero_rows_existswitnessminornonzeroevaluationdeterminanthrc)) * S ((mdr_nc_zero_rows_existswitnessminornonzeroevaluationdeterminanth) + (mdr_e_zero_rows_existswitnessminornonzeroevaluationdeterminanthrc)) + ((mdr_e_zero_rows_existswitnessminornonzeroevaluationdeterminanthrc) + (mdr_e_zero_rows_existswitnessminornonzeroevaluationdeterminanthrc))) /\ ((mdr_z_zero_rows_existswitnessminornonzeroevaluationdeterminanthr) = ((mdr_c_zero_rows_existswitnessminornonzeroevaluationdeterminanthrc) + (mdr_f_zero_rows_existswitnessminornonzeroevaluationdeterminanthrc)) * S ((mdr_c_zero_rows_existswitnessminornonzeroevaluationdeterminanthrc) + (mdr_f_zero_rows_existswitnessminornonzeroevaluationdeterminanthrc)) + ((mdr_f_zero_rows_existswitnessminornonzeroevaluationdeterminanthrc) + (mdr_f_zero_rows_existswitnessminornonzeroevaluationdeterminanthrc))))))))) /\ (((exists ff_h_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthrb. ff_h_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthrb + S (mdr_z_zero_rows_existswitnessminornonzeroevaluationdeterminanthr) = S ((S (mdr_i_zero_rows_existswitnessminornonzeroevaluationdeterminanth)) * mdr_c_zero_rows_existswitnessminornonzeroevaluationdeterminant)) /\ exists ff_q_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthrb. mdr_b_zero_rows_existswitnessminornonzeroevaluationdeterminant = ff_q_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthrb * S ((S (mdr_i_zero_rows_existswitnessminornonzeroevaluationdeterminanth)) * mdr_c_zero_rows_existswitnessminornonzeroevaluationdeterminant) + (mdr_z_zero_rows_existswitnessminornonzeroevaluationdeterminanthr))))) /\ (((((mdr_d_zero_rows_existswitnessminornonzeroevaluationdeterminanth) = 0) /\ (((mdr_p_zero_rows_existswitnessminornonzeroevaluationdeterminanth) = 1) /\ ((mdr_n_zero_rows_existswitnessminornonzeroevaluationdeterminanth) = 0))) \/ exists mdr_q_zero_rows_existswitnessminornonzeroevaluationdeterminanths mdr_eb_zero_rows_existswitnessminornonzeroevaluationdeterminanths mdr_ec_zero_rows_existswitnessminornonzeroevaluationdeterminanths mdr_fb_zero_rows_existswitnessminornonzeroevaluationdeterminanths mdr_fc_zero_rows_existswitnessminornonzeroevaluationdeterminanths. (((mdr_d_zero_rows_existswitnessminornonzeroevaluationdeterminanth) = S (mdr_q_zero_rows_existswitnessminornonzeroevaluationdeterminanths)) /\ ((forall mdr_j_zero_rows_existswitnessminornonzeroevaluationdeterminanthsc. (exists mdr_gap_zero_rows_existswitnessminornonzeroevaluationdeterminanthscj. mdr_gap_zero_rows_existswitnessminornonzeroevaluationdeterminanthscj + S (mdr_j_zero_rows_existswitnessminornonzeroevaluationdeterminanthsc) = (S (mdr_q_zero_rows_existswitnessminornonzeroevaluationdeterminanths))) -> exists mdr_i_zero_rows_existswitnessminornonzeroevaluationdeterminanthsc mdr_up_zero_rows_existswitnessminornonzeroevaluationdeterminanthsc mdr_us_zero_rows_existswitnessminornonzeroevaluationdeterminanthsc mdr_un_zero_rows_existswitnessminornonzeroevaluationdeterminanthsc mdr_ut_zero_rows_existswitnessminornonzeroevaluationdeterminanthsc mdr_p_zero_rows_existswitnessminornonzeroevaluationdeterminanthsc mdr_n_zero_rows_existswitnessminornonzeroevaluationdeterminanthsc. ((exists mdr_gap_zero_rows_existswitnessminornonzeroevaluationdeterminanthsci. mdr_gap_zero_rows_existswitnessminornonzeroevaluationdeterminanthsci + S (mdr_i_zero_rows_existswitnessminornonzeroevaluationdeterminanthsc) = (mdr_i_zero_rows_existswitnessminornonzeroevaluationdeterminanth)) /\ ((exists mdr_z_zero_rows_existswitnessminornonzeroevaluationdeterminanthscr. ((exists mdr_a_zero_rows_existswitnessminornonzeroevaluationdeterminanthscrc mdr_b_zero_rows_existswitnessminornonzeroevaluationdeterminanthscrc mdr_c_zero_rows_existswitnessminornonzeroevaluationdeterminanthscrc mdr_e_zero_rows_existswitnessminornonzeroevaluationdeterminanthscrc mdr_f_zero_rows_existswitnessminornonzeroevaluationdeterminanthscrc. ((mdr_a_zero_rows_existswitnessminornonzeroevaluationdeterminanthscrc = ((mdr_q_zero_rows_existswitnessminornonzeroevaluationdeterminanths) + (mdr_up_zero_rows_existswitnessminornonzeroevaluationdeterminanthsc)) * S ((mdr_q_zero_rows_existswitnessminornonzeroevaluationdeterminanths) + (mdr_up_zero_rows_existswitnessminornonzeroevaluationdeterminanthsc)) + ((mdr_up_zero_rows_existswitnessminornonzeroevaluationdeterminanthsc) + (mdr_up_zero_rows_existswitnessminornonzeroevaluationdeterminanthsc))) /\ ((mdr_b_zero_rows_existswitnessminornonzeroevaluationdeterminanthscrc = ((mdr_us_zero_rows_existswitnessminornonzeroevaluationdeterminanthsc) + (mdr_un_zero_rows_existswitnessminornonzeroevaluationdeterminanthsc)) * S ((mdr_us_zero_rows_existswitnessminornonzeroevaluationdeterminanthsc) + (mdr_un_zero_rows_existswitnessminornonzeroevaluationdeterminanthsc)) + ((mdr_un_zero_rows_existswitnessminornonzeroevaluationdeterminanthsc) + (mdr_un_zero_rows_existswitnessminornonzeroevaluationdeterminanthsc))) /\ ((mdr_c_zero_rows_existswitnessminornonzeroevaluationdeterminanthscrc = ((mdr_a_zero_rows_existswitnessminornonzeroevaluationdeterminanthscrc) + (mdr_b_zero_rows_existswitnessminornonzeroevaluationdeterminanthscrc)) * S ((mdr_a_zero_rows_existswitnessminornonzeroevaluationdeterminanthscrc) + (mdr_b_zero_rows_existswitnessminornonzeroevaluationdeterminanthscrc)) + ((mdr_b_zero_rows_existswitnessminornonzeroevaluationdeterminanthscrc) + (mdr_b_zero_rows_existswitnessminornonzeroevaluationdeterminanthscrc))) /\ ((mdr_e_zero_rows_existswitnessminornonzeroevaluationdeterminanthscrc = ((mdr_p_zero_rows_existswitnessminornonzeroevaluationdeterminanthsc) + (mdr_n_zero_rows_existswitnessminornonzeroevaluationdeterminanthsc)) * S ((mdr_p_zero_rows_existswitnessminornonzeroevaluationdeterminanthsc) + (mdr_n_zero_rows_existswitnessminornonzeroevaluationdeterminanthsc)) + ((mdr_n_zero_rows_existswitnessminornonzeroevaluationdeterminanthsc) + (mdr_n_zero_rows_existswitnessminornonzeroevaluationdeterminanthsc))) /\ ((mdr_f_zero_rows_existswitnessminornonzeroevaluationdeterminanthscrc = ((mdr_ut_zero_rows_existswitnessminornonzeroevaluationdeterminanthsc) + (mdr_e_zero_rows_existswitnessminornonzeroevaluationdeterminanthscrc)) * S ((mdr_ut_zero_rows_existswitnessminornonzeroevaluationdeterminanthsc) + (mdr_e_zero_rows_existswitnessminornonzeroevaluationdeterminanthscrc)) + ((mdr_e_zero_rows_existswitnessminornonzeroevaluationdeterminanthscrc) + (mdr_e_zero_rows_existswitnessminornonzeroevaluationdeterminanthscrc))) /\ ((mdr_z_zero_rows_existswitnessminornonzeroevaluationdeterminanthscr) = ((mdr_c_zero_rows_existswitnessminornonzeroevaluationdeterminanthscrc) + (mdr_f_zero_rows_existswitnessminornonzeroevaluationdeterminanthscrc)) * S ((mdr_c_zero_rows_existswitnessminornonzeroevaluationdeterminanthscrc) + (mdr_f_zero_rows_existswitnessminornonzeroevaluationdeterminanthscrc)) + ((mdr_f_zero_rows_existswitnessminornonzeroevaluationdeterminanthscrc) + (mdr_f_zero_rows_existswitnessminornonzeroevaluationdeterminanthscrc))))))))) /\ (((exists ff_h_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthscrb. ff_h_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthscrb + S (mdr_z_zero_rows_existswitnessminornonzeroevaluationdeterminanthscr) = S ((S (mdr_i_zero_rows_existswitnessminornonzeroevaluationdeterminanthsc)) * mdr_c_zero_rows_existswitnessminornonzeroevaluationdeterminant)) /\ exists ff_q_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthscrb. mdr_b_zero_rows_existswitnessminornonzeroevaluationdeterminant = ff_q_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthscrb * S ((S (mdr_i_zero_rows_existswitnessminornonzeroevaluationdeterminanthsc)) * mdr_c_zero_rows_existswitnessminornonzeroevaluationdeterminant) + (mdr_z_zero_rows_existswitnessminornonzeroevaluationdeterminanthscr))))) /\ ((((forall ff_index_mdm_prefix_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthscm_positive. (exists ff_gap_mdm_lt_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthscm_positive_index_bound. ff_gap_mdm_lt_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthscm_positive_index_bound + S (ff_index_mdm_prefix_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthscm_positive) = ((mdr_q_zero_rows_existswitnessminornonzeroevaluationdeterminanths) * (mdr_q_zero_rows_existswitnessminornonzeroevaluationdeterminanths))) -> exists ff_row_mdm_prefix_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthscm_positive ff_column_mdm_prefix_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthscm_positive ff_value_mdm_prefix_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthscm_positive. (ff_index_mdm_prefix_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthscm_positive = (mdr_q_zero_rows_existswitnessminornonzeroevaluationdeterminanths) * ff_row_mdm_prefix_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthscm_positive + ff_column_mdm_prefix_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthscm_positive /\ ((exists ff_gap_mdm_lt_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthscm_positive_column_bound. ff_gap_mdm_lt_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthscm_positive_column_bound + S (ff_column_mdm_prefix_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthscm_positive) = (mdr_q_zero_rows_existswitnessminornonzeroevaluationdeterminanths)) /\ ((exists ff_row_mdm_cell_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthscm_positive_cell ff_column_mdm_cell_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthscm_positive_cell. (((((exists ff_gap_mdm_lt_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthscm_positive_cell_row_before. ff_gap_mdm_lt_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthscm_positive_cell_row_before + S (ff_row_mdm_prefix_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthscm_positive) = (0)) /\ ff_row_mdm_cell_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthscm_positive_cell = ff_row_mdm_prefix_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthscm_positive) \/ ((exists ff_gap_mdm_le_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthscm_positive_cell_row_after. ff_gap_mdm_le_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthscm_positive_cell_row_after + (0) = (ff_row_mdm_prefix_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthscm_positive)) /\ ff_row_mdm_cell_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthscm_positive_cell = S ff_row_mdm_prefix_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthscm_positive))) /\ (((((exists ff_gap_mdm_lt_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthscm_positive_cell_column_before. ff_gap_mdm_lt_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthscm_positive_cell_column_before + S (ff_column_mdm_prefix_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthscm_positive) = (mdr_j_zero_rows_existswitnessminornonzeroevaluationdeterminanthsc)) /\ ff_column_mdm_cell_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthscm_positive_cell = ff_column_mdm_prefix_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthscm_positive) \/ ((exists ff_gap_mdm_le_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthscm_positive_cell_column_after. ff_gap_mdm_le_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthscm_positive_cell_column_after + (mdr_j_zero_rows_existswitnessminornonzeroevaluationdeterminanthsc) = (ff_column_mdm_prefix_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthscm_positive)) /\ ff_column_mdm_cell_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthscm_positive_cell = S ff_column_mdm_prefix_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthscm_positive))) /\ (((exists ff_h_mdm_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthscm_positive_cell_source. ff_h_mdm_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthscm_positive_cell_source + S (ff_value_mdm_prefix_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthscm_positive) = S ((S ((ff_row_mdm_cell_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthscm_positive_cell) * (S (mdr_q_zero_rows_existswitnessminornonzeroevaluationdeterminanths)) + (ff_column_mdm_cell_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthscm_positive_cell))) * mdr_pc_zero_rows_existswitnessminornonzeroevaluationdeterminanth)) /\ exists ff_q_mdm_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthscm_positive_cell_source. mdr_pb_zero_rows_existswitnessminornonzeroevaluationdeterminanth = ff_q_mdm_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthscm_positive_cell_source * S ((S ((ff_row_mdm_cell_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthscm_positive_cell) * (S (mdr_q_zero_rows_existswitnessminornonzeroevaluationdeterminanths)) + (ff_column_mdm_cell_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthscm_positive_cell))) * mdr_pc_zero_rows_existswitnessminornonzeroevaluationdeterminanth) + (ff_value_mdm_prefix_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthscm_positive)))))) /\ (((exists ff_h_mdm_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthscm_positive_target. ff_h_mdm_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthscm_positive_target + S (ff_value_mdm_prefix_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthscm_positive) = S ((S (ff_index_mdm_prefix_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthscm_positive)) * mdr_us_zero_rows_existswitnessminornonzeroevaluationdeterminanthsc)) /\ exists ff_q_mdm_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthscm_positive_target. mdr_up_zero_rows_existswitnessminornonzeroevaluationdeterminanthsc = ff_q_mdm_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthscm_positive_target * S ((S (ff_index_mdm_prefix_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthscm_positive)) * mdr_us_zero_rows_existswitnessminornonzeroevaluationdeterminanthsc) + (ff_value_mdm_prefix_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthscm_positive))))))) /\ (forall ff_index_mdm_prefix_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthscm_negative. (exists ff_gap_mdm_lt_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthscm_negative_index_bound. ff_gap_mdm_lt_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthscm_negative_index_bound + S (ff_index_mdm_prefix_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthscm_negative) = ((mdr_q_zero_rows_existswitnessminornonzeroevaluationdeterminanths) * (mdr_q_zero_rows_existswitnessminornonzeroevaluationdeterminanths))) -> exists ff_row_mdm_prefix_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthscm_negative ff_column_mdm_prefix_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthscm_negative ff_value_mdm_prefix_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthscm_negative. (ff_index_mdm_prefix_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthscm_negative = (mdr_q_zero_rows_existswitnessminornonzeroevaluationdeterminanths) * ff_row_mdm_prefix_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthscm_negative + ff_column_mdm_prefix_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthscm_negative /\ ((exists ff_gap_mdm_lt_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthscm_negative_column_bound. ff_gap_mdm_lt_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthscm_negative_column_bound + S (ff_column_mdm_prefix_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthscm_negative) = (mdr_q_zero_rows_existswitnessminornonzeroevaluationdeterminanths)) /\ ((exists ff_row_mdm_cell_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthscm_negative_cell ff_column_mdm_cell_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthscm_negative_cell. (((((exists ff_gap_mdm_lt_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthscm_negative_cell_row_before. ff_gap_mdm_lt_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthscm_negative_cell_row_before + S (ff_row_mdm_prefix_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthscm_negative) = (0)) /\ ff_row_mdm_cell_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthscm_negative_cell = ff_row_mdm_prefix_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthscm_negative) \/ ((exists ff_gap_mdm_le_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthscm_negative_cell_row_after. ff_gap_mdm_le_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthscm_negative_cell_row_after + (0) = (ff_row_mdm_prefix_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthscm_negative)) /\ ff_row_mdm_cell_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthscm_negative_cell = S ff_row_mdm_prefix_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthscm_negative))) /\ (((((exists ff_gap_mdm_lt_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthscm_negative_cell_column_before. ff_gap_mdm_lt_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthscm_negative_cell_column_before + S (ff_column_mdm_prefix_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthscm_negative) = (mdr_j_zero_rows_existswitnessminornonzeroevaluationdeterminanthsc)) /\ ff_column_mdm_cell_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthscm_negative_cell = ff_column_mdm_prefix_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthscm_negative) \/ ((exists ff_gap_mdm_le_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthscm_negative_cell_column_after. ff_gap_mdm_le_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthscm_negative_cell_column_after + (mdr_j_zero_rows_existswitnessminornonzeroevaluationdeterminanthsc) = (ff_column_mdm_prefix_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthscm_negative)) /\ ff_column_mdm_cell_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthscm_negative_cell = S ff_column_mdm_prefix_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthscm_negative))) /\ (((exists ff_h_mdm_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthscm_negative_cell_source. ff_h_mdm_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthscm_negative_cell_source + S (ff_value_mdm_prefix_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthscm_negative) = S ((S ((ff_row_mdm_cell_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthscm_negative_cell) * (S (mdr_q_zero_rows_existswitnessminornonzeroevaluationdeterminanths)) + (ff_column_mdm_cell_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthscm_negative_cell))) * mdr_nc_zero_rows_existswitnessminornonzeroevaluationdeterminanth)) /\ exists ff_q_mdm_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthscm_negative_cell_source. mdr_nb_zero_rows_existswitnessminornonzeroevaluationdeterminanth = ff_q_mdm_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthscm_negative_cell_source * S ((S ((ff_row_mdm_cell_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthscm_negative_cell) * (S (mdr_q_zero_rows_existswitnessminornonzeroevaluationdeterminanths)) + (ff_column_mdm_cell_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthscm_negative_cell))) * mdr_nc_zero_rows_existswitnessminornonzeroevaluationdeterminanth) + (ff_value_mdm_prefix_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthscm_negative)))))) /\ (((exists ff_h_mdm_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthscm_negative_target. ff_h_mdm_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthscm_negative_target + S (ff_value_mdm_prefix_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthscm_negative) = S ((S (ff_index_mdm_prefix_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthscm_negative)) * mdr_ut_zero_rows_existswitnessminornonzeroevaluationdeterminanthsc)) /\ exists ff_q_mdm_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthscm_negative_target. mdr_un_zero_rows_existswitnessminornonzeroevaluationdeterminanthsc = ff_q_mdm_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthscm_negative_target * S ((S (ff_index_mdm_prefix_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthscm_negative)) * mdr_ut_zero_rows_existswitnessminornonzeroevaluationdeterminanthsc) + (ff_value_mdm_prefix_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthscm_negative))))))))) /\ ((((exists ff_h_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthscp. ff_h_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthscp + S (mdr_p_zero_rows_existswitnessminornonzeroevaluationdeterminanthsc) = S ((S (mdr_j_zero_rows_existswitnessminornonzeroevaluationdeterminanthsc)) * mdr_ec_zero_rows_existswitnessminornonzeroevaluationdeterminanths)) /\ exists ff_q_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthscp. mdr_eb_zero_rows_existswitnessminornonzeroevaluationdeterminanths = ff_q_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthscp * S ((S (mdr_j_zero_rows_existswitnessminornonzeroevaluationdeterminanthsc)) * mdr_ec_zero_rows_existswitnessminornonzeroevaluationdeterminanths) + (mdr_p_zero_rows_existswitnessminornonzeroevaluationdeterminanthsc))) /\ (((exists ff_h_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthscn. ff_h_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthscn + S (mdr_n_zero_rows_existswitnessminornonzeroevaluationdeterminanthsc) = S ((S (mdr_j_zero_rows_existswitnessminornonzeroevaluationdeterminanthsc)) * mdr_fc_zero_rows_existswitnessminornonzeroevaluationdeterminanths)) /\ exists ff_q_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthscn. mdr_fb_zero_rows_existswitnessminornonzeroevaluationdeterminanths = ff_q_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthscn * S ((S (mdr_j_zero_rows_existswitnessminornonzeroevaluationdeterminanthsc)) * mdr_fc_zero_rows_existswitnessminornonzeroevaluationdeterminanths) + (mdr_n_zero_rows_existswitnessminornonzeroevaluationdeterminanthsc)))))))) /\ (exists ff_ub_mce_fold_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthsf ff_uc_mce_fold_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthsf ff_vb_mce_fold_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthsf ff_vc_mce_fold_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthsf. ((forall ff_index_mce_alternating_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthsf_prefix. (exists ff_gap_mce_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthsf_prefix_index. ff_gap_mce_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthsf_prefix_index + S (ff_index_mce_alternating_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthsf_prefix) = (S (mdr_q_zero_rows_existswitnessminornonzeroevaluationdeterminanths))) -> exists ff_ap_mce_alternating_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthsf_prefix ff_an_mce_alternating_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthsf_prefix ff_bp_mce_alternating_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthsf_prefix ff_bn_mce_alternating_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthsf_prefix ff_p_mce_alternating_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthsf_prefix ff_n_mce_alternating_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthsf_prefix. ((((exists ff_h_mce_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthsf_prefix_ap. ff_h_mce_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthsf_prefix_ap + S (ff_ap_mce_alternating_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthsf_prefix) = S ((S (ff_index_mce_alternating_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthsf_prefix)) * mdr_pc_zero_rows_existswitnessminornonzeroevaluationdeterminanth)) /\ exists ff_q_mce_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthsf_prefix_ap. mdr_pb_zero_rows_existswitnessminornonzeroevaluationdeterminanth = ff_q_mce_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthsf_prefix_ap * S ((S (ff_index_mce_alternating_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthsf_prefix)) * mdr_pc_zero_rows_existswitnessminornonzeroevaluationdeterminanth) + (ff_ap_mce_alternating_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthsf_prefix))) /\ ((((exists ff_h_mce_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthsf_prefix_an. ff_h_mce_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthsf_prefix_an + S (ff_an_mce_alternating_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthsf_prefix) = S ((S (ff_index_mce_alternating_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthsf_prefix)) * mdr_nc_zero_rows_existswitnessminornonzeroevaluationdeterminanth)) /\ exists ff_q_mce_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthsf_prefix_an. mdr_nb_zero_rows_existswitnessminornonzeroevaluationdeterminanth = ff_q_mce_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthsf_prefix_an * S ((S (ff_index_mce_alternating_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthsf_prefix)) * mdr_nc_zero_rows_existswitnessminornonzeroevaluationdeterminanth) + (ff_an_mce_alternating_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthsf_prefix))) /\ ((((exists ff_h_mce_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthsf_prefix_bp. ff_h_mce_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthsf_prefix_bp + S (ff_bp_mce_alternating_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthsf_prefix) = S ((S (ff_index_mce_alternating_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthsf_prefix)) * mdr_ec_zero_rows_existswitnessminornonzeroevaluationdeterminanths)) /\ exists ff_q_mce_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthsf_prefix_bp. mdr_eb_zero_rows_existswitnessminornonzeroevaluationdeterminanths = ff_q_mce_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthsf_prefix_bp * S ((S (ff_index_mce_alternating_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthsf_prefix)) * mdr_ec_zero_rows_existswitnessminornonzeroevaluationdeterminanths) + (ff_bp_mce_alternating_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthsf_prefix))) /\ ((((exists ff_h_mce_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthsf_prefix_bn. ff_h_mce_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthsf_prefix_bn + S (ff_bn_mce_alternating_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthsf_prefix) = S ((S (ff_index_mce_alternating_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthsf_prefix)) * mdr_fc_zero_rows_existswitnessminornonzeroevaluationdeterminanths)) /\ exists ff_q_mce_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthsf_prefix_bn. mdr_fb_zero_rows_existswitnessminornonzeroevaluationdeterminanths = ff_q_mce_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthsf_prefix_bn * S ((S (ff_index_mce_alternating_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthsf_prefix)) * mdr_fc_zero_rows_existswitnessminornonzeroevaluationdeterminanths) + (ff_bn_mce_alternating_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthsf_prefix))) /\ ((((exists ff_h_mce_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthsf_prefix_positive. ff_h_mce_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthsf_prefix_positive + S (ff_p_mce_alternating_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthsf_prefix) = S ((S (ff_index_mce_alternating_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthsf_prefix)) * ff_uc_mce_fold_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthsf)) /\ exists ff_q_mce_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthsf_prefix_positive. ff_ub_mce_fold_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthsf = ff_q_mce_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthsf_prefix_positive * S ((S (ff_index_mce_alternating_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthsf_prefix)) * ff_uc_mce_fold_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthsf) + (ff_p_mce_alternating_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthsf_prefix))) /\ ((((exists ff_h_mce_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthsf_prefix_negative. ff_h_mce_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthsf_prefix_negative + S (ff_n_mce_alternating_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthsf_prefix) = S ((S (ff_index_mce_alternating_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthsf_prefix)) * ff_vc_mce_fold_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthsf)) /\ exists ff_q_mce_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthsf_prefix_negative. ff_vb_mce_fold_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthsf = ff_q_mce_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthsf_prefix_negative * S ((S (ff_index_mce_alternating_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthsf_prefix)) * ff_vc_mce_fold_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthsf) + (ff_n_mce_alternating_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthsf_prefix))) /\ (((exists ff_even_mce_term_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthsf_prefix_term. ff_index_mce_alternating_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthsf_prefix = 2 * ff_even_mce_term_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthsf_prefix_term) /\ (ff_p_mce_alternating_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthsf_prefix = (ff_ap_mce_alternating_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthsf_prefix) * (ff_bp_mce_alternating_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthsf_prefix) + (ff_an_mce_alternating_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthsf_prefix) * (ff_bn_mce_alternating_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthsf_prefix) /\ ff_n_mce_alternating_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthsf_prefix = (ff_ap_mce_alternating_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthsf_prefix) * (ff_bn_mce_alternating_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthsf_prefix) + (ff_an_mce_alternating_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthsf_prefix) * (ff_bp_mce_alternating_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthsf_prefix))) \/ ((exists ff_odd_mce_term_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthsf_prefix_term. ff_index_mce_alternating_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthsf_prefix = 2 * ff_odd_mce_term_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthsf_prefix_term + 1) /\ (ff_p_mce_alternating_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthsf_prefix = (ff_ap_mce_alternating_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthsf_prefix) * (ff_bn_mce_alternating_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthsf_prefix) + (ff_an_mce_alternating_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthsf_prefix) * (ff_bp_mce_alternating_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthsf_prefix) /\ ff_n_mce_alternating_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthsf_prefix = (ff_ap_mce_alternating_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthsf_prefix) * (ff_bp_mce_alternating_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthsf_prefix) + (ff_an_mce_alternating_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthsf_prefix) * (ff_bn_mce_alternating_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthsf_prefix))))))))))) /\ ((exists ff_u_mce_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthsf_positive ff_v_mce_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthsf_positive. ((((exists ff_h_mce_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthsf_positive_start. ff_h_mce_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthsf_positive_start + S (0) = S ((S (0)) * ff_v_mce_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthsf_positive)) /\ exists ff_q_mce_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthsf_positive_start. ff_u_mce_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthsf_positive = ff_q_mce_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthsf_positive_start * S ((S (0)) * ff_v_mce_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthsf_positive) + (0))) /\ ((((exists ff_h_mce_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthsf_positive_terminal. ff_h_mce_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthsf_positive_terminal + S (mdr_p_zero_rows_existswitnessminornonzeroevaluationdeterminanth) = S ((S ((S (mdr_q_zero_rows_existswitnessminornonzeroevaluationdeterminanths)))) * ff_v_mce_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthsf_positive)) /\ exists ff_q_mce_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthsf_positive_terminal. ff_u_mce_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthsf_positive = ff_q_mce_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthsf_positive_terminal * S ((S ((S (mdr_q_zero_rows_existswitnessminornonzeroevaluationdeterminanths)))) * ff_v_mce_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthsf_positive) + (mdr_p_zero_rows_existswitnessminornonzeroevaluationdeterminanth))) /\ forall ff_i_mce_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthsf_positive. (exists ff_lt_mce_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthsf_positive_bound. ff_lt_mce_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthsf_positive_bound + S ff_i_mce_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthsf_positive = (S (mdr_q_zero_rows_existswitnessminornonzeroevaluationdeterminanths))) -> exists ff_a_mce_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthsf_positive ff_r_mce_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthsf_positive ff_s_mce_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthsf_positive. ((((exists ff_h_mce_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthsf_positive_summand. ff_h_mce_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthsf_positive_summand + S (ff_a_mce_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthsf_positive) = S ((S (ff_i_mce_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthsf_positive)) * ff_uc_mce_fold_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthsf)) /\ exists ff_q_mce_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthsf_positive_summand. ff_ub_mce_fold_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthsf = ff_q_mce_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthsf_positive_summand * S ((S (ff_i_mce_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthsf_positive)) * ff_uc_mce_fold_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthsf) + (ff_a_mce_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthsf_positive))) /\ ((((exists ff_h_mce_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthsf_positive_partial. ff_h_mce_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthsf_positive_partial + S (ff_r_mce_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthsf_positive) = S ((S (ff_i_mce_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthsf_positive)) * ff_v_mce_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthsf_positive)) /\ exists ff_q_mce_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthsf_positive_partial. ff_u_mce_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthsf_positive = ff_q_mce_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthsf_positive_partial * S ((S (ff_i_mce_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthsf_positive)) * ff_v_mce_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthsf_positive) + (ff_r_mce_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthsf_positive))) /\ ((((exists ff_h_mce_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthsf_positive_successor. ff_h_mce_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthsf_positive_successor + S (ff_s_mce_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthsf_positive) = S ((S (S ff_i_mce_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthsf_positive)) * ff_v_mce_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthsf_positive)) /\ exists ff_q_mce_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthsf_positive_successor. ff_u_mce_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthsf_positive = ff_q_mce_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthsf_positive_successor * S ((S (S ff_i_mce_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthsf_positive)) * ff_v_mce_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthsf_positive) + (ff_s_mce_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthsf_positive))) /\ ff_s_mce_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthsf_positive = ff_r_mce_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthsf_positive + ff_a_mce_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthsf_positive)))))) /\ (exists ff_u_mce_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthsf_negative ff_v_mce_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthsf_negative. ((((exists ff_h_mce_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthsf_negative_start. ff_h_mce_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthsf_negative_start + S (0) = S ((S (0)) * ff_v_mce_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthsf_negative)) /\ exists ff_q_mce_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthsf_negative_start. ff_u_mce_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthsf_negative = ff_q_mce_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthsf_negative_start * S ((S (0)) * ff_v_mce_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthsf_negative) + (0))) /\ ((((exists ff_h_mce_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthsf_negative_terminal. ff_h_mce_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthsf_negative_terminal + S (mdr_n_zero_rows_existswitnessminornonzeroevaluationdeterminanth) = S ((S ((S (mdr_q_zero_rows_existswitnessminornonzeroevaluationdeterminanths)))) * ff_v_mce_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthsf_negative)) /\ exists ff_q_mce_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthsf_negative_terminal. ff_u_mce_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthsf_negative = ff_q_mce_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthsf_negative_terminal * S ((S ((S (mdr_q_zero_rows_existswitnessminornonzeroevaluationdeterminanths)))) * ff_v_mce_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthsf_negative) + (mdr_n_zero_rows_existswitnessminornonzeroevaluationdeterminanth))) /\ forall ff_i_mce_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthsf_negative. (exists ff_lt_mce_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthsf_negative_bound. ff_lt_mce_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthsf_negative_bound + S ff_i_mce_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthsf_negative = (S (mdr_q_zero_rows_existswitnessminornonzeroevaluationdeterminanths))) -> exists ff_a_mce_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthsf_negative ff_r_mce_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthsf_negative ff_s_mce_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthsf_negative. ((((exists ff_h_mce_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthsf_negative_summand. ff_h_mce_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthsf_negative_summand + S (ff_a_mce_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthsf_negative) = S ((S (ff_i_mce_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthsf_negative)) * ff_vc_mce_fold_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthsf)) /\ exists ff_q_mce_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthsf_negative_summand. ff_vb_mce_fold_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthsf = ff_q_mce_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthsf_negative_summand * S ((S (ff_i_mce_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthsf_negative)) * ff_vc_mce_fold_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthsf) + (ff_a_mce_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthsf_negative))) /\ ((((exists ff_h_mce_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthsf_negative_partial. ff_h_mce_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthsf_negative_partial + S (ff_r_mce_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthsf_negative) = S ((S (ff_i_mce_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthsf_negative)) * ff_v_mce_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthsf_negative)) /\ exists ff_q_mce_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthsf_negative_partial. ff_u_mce_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthsf_negative = ff_q_mce_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthsf_negative_partial * S ((S (ff_i_mce_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthsf_negative)) * ff_v_mce_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthsf_negative) + (ff_r_mce_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthsf_negative))) /\ ((((exists ff_h_mce_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthsf_negative_successor. ff_h_mce_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthsf_negative_successor + S (ff_s_mce_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthsf_negative) = S ((S (S ff_i_mce_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthsf_negative)) * ff_v_mce_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthsf_negative)) /\ exists ff_q_mce_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthsf_negative_successor. ff_u_mce_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthsf_negative = ff_q_mce_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthsf_negative_successor * S ((S (S ff_i_mce_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthsf_negative)) * ff_v_mce_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthsf_negative) + (ff_s_mce_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthsf_negative))) /\ ff_s_mce_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthsf_negative = ff_r_mce_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthsf_negative + ff_a_mce_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminanthsf_negative))))))))))))))) /\ ((exists mdr_gap_zero_rows_existswitnessminornonzeroevaluationdeterminanti. mdr_gap_zero_rows_existswitnessminornonzeroevaluationdeterminanti + S (mdr_i_zero_rows_existswitnessminornonzeroevaluationdeterminant) = (mdr_l_zero_rows_existswitnessminornonzeroevaluationdeterminant)) /\ (exists mdr_z_zero_rows_existswitnessminornonzeroevaluationdeterminantr. ((exists mdr_a_zero_rows_existswitnessminornonzeroevaluationdeterminantrc mdr_b_zero_rows_existswitnessminornonzeroevaluationdeterminantrc mdr_c_zero_rows_existswitnessminornonzeroevaluationdeterminantrc mdr_e_zero_rows_existswitnessminornonzeroevaluationdeterminantrc mdr_f_zero_rows_existswitnessminornonzeroevaluationdeterminantrc. ((mdr_a_zero_rows_existswitnessminornonzeroevaluationdeterminantrc = ((rank) + (mdr_ub_zero_rows_existswitnessminornonzeroevaluation)) * S ((rank) + (mdr_ub_zero_rows_existswitnessminornonzeroevaluation)) + ((mdr_ub_zero_rows_existswitnessminornonzeroevaluation) + (mdr_ub_zero_rows_existswitnessminornonzeroevaluation))) /\ ((mdr_b_zero_rows_existswitnessminornonzeroevaluationdeterminantrc = ((mdr_uc_zero_rows_existswitnessminornonzeroevaluation) + (mdr_vb_zero_rows_existswitnessminornonzeroevaluation)) * S ((mdr_uc_zero_rows_existswitnessminornonzeroevaluation) + (mdr_vb_zero_rows_existswitnessminornonzeroevaluation)) + ((mdr_vb_zero_rows_existswitnessminornonzeroevaluation) + (mdr_vb_zero_rows_existswitnessminornonzeroevaluation))) /\ ((mdr_c_zero_rows_existswitnessminornonzeroevaluationdeterminantrc = ((mdr_a_zero_rows_existswitnessminornonzeroevaluationdeterminantrc) + (mdr_b_zero_rows_existswitnessminornonzeroevaluationdeterminantrc)) * S ((mdr_a_zero_rows_existswitnessminornonzeroevaluationdeterminantrc) + (mdr_b_zero_rows_existswitnessminornonzeroevaluationdeterminantrc)) + ((mdr_b_zero_rows_existswitnessminornonzeroevaluationdeterminantrc) + (mdr_b_zero_rows_existswitnessminornonzeroevaluationdeterminantrc))) /\ ((mdr_e_zero_rows_existswitnessminornonzeroevaluationdeterminantrc = ((mdr_p_zero_rows_existswitnessminornonzero) + (mdr_n_zero_rows_existswitnessminornonzero)) * S ((mdr_p_zero_rows_existswitnessminornonzero) + (mdr_n_zero_rows_existswitnessminornonzero)) + ((mdr_n_zero_rows_existswitnessminornonzero) + (mdr_n_zero_rows_existswitnessminornonzero))) /\ ((mdr_f_zero_rows_existswitnessminornonzeroevaluationdeterminantrc = ((mdr_vc_zero_rows_existswitnessminornonzeroevaluation) + (mdr_e_zero_rows_existswitnessminornonzeroevaluationdeterminantrc)) * S ((mdr_vc_zero_rows_existswitnessminornonzeroevaluation) + (mdr_e_zero_rows_existswitnessminornonzeroevaluationdeterminantrc)) + ((mdr_e_zero_rows_existswitnessminornonzeroevaluationdeterminantrc) + (mdr_e_zero_rows_existswitnessminornonzeroevaluationdeterminantrc))) /\ ((mdr_z_zero_rows_existswitnessminornonzeroevaluationdeterminantr) = ((mdr_c_zero_rows_existswitnessminornonzeroevaluationdeterminantrc) + (mdr_f_zero_rows_existswitnessminornonzeroevaluationdeterminantrc)) * S ((mdr_c_zero_rows_existswitnessminornonzeroevaluationdeterminantrc) + (mdr_f_zero_rows_existswitnessminornonzeroevaluationdeterminantrc)) + ((mdr_f_zero_rows_existswitnessminornonzeroevaluationdeterminantrc) + (mdr_f_zero_rows_existswitnessminornonzeroevaluationdeterminantrc))))))))) /\ (((exists ff_h_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminantrb. ff_h_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminantrb + S (mdr_z_zero_rows_existswitnessminornonzeroevaluationdeterminantr) = S ((S (mdr_i_zero_rows_existswitnessminornonzeroevaluationdeterminant)) * mdr_c_zero_rows_existswitnessminornonzeroevaluationdeterminant)) /\ exists ff_q_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminantrb. mdr_b_zero_rows_existswitnessminornonzeroevaluationdeterminant = ff_q_mdr_zero_rows_existswitnessminornonzeroevaluationdeterminantrb * S ((S (mdr_i_zero_rows_existswitnessminornonzeroevaluationdeterminant)) * mdr_c_zero_rows_existswitnessminornonzeroevaluationdeterminant) + (mdr_z_zero_rows_existswitnessminornonzeroevaluationdeterminantr)))))))))) /\ (~(mdr_p_zero_rows_existswitnessminornonzero = mdr_n_zero_rows_existswitnessminornonzero)))))))) /\ (forall mdr_q_zero_rows_exists. (exists mdr_gap_zero_rows_existshigher. mdr_gap_zero_rows_existshigher + S (rank) = (mdr_q_zero_rows_exists)) -> (forall mdr_rb_zero_rows_existszero mdr_rc_zero_rows_existszero mdr_cb_zero_rows_existszero mdr_cc_zero_rows_existszero mdr_p_zero_rows_existszero mdr_n_zero_rows_existszero. (((forall fom_index_mrf_zero_rows_existszerorowsbound. (exists fom_gap_mrf_zero_rows_existszerorowsbound_index_bound. fom_gap_mrf_zero_rows_existszerorowsbound_index_bound + S (fom_index_mrf_zero_rows_existszerorowsbound) = mdr_q_zero_rows_exists) -> exists fom_value_mrf_zero_rows_existszerorowsbound. ((((exists fom_beta_height_mrf_zero_rows_existszerorowsbound_entry. fom_beta_height_mrf_zero_rows_existszerorowsbound_entry + S (fom_value_mrf_zero_rows_existszerorowsbound) = S ((S (fom_index_mrf_zero_rows_existszerorowsbound)) * mdr_rc_zero_rows_existszero)) /\ exists fom_beta_quotient_mrf_zero_rows_existszerorowsbound_entry. mdr_rb_zero_rows_existszero = fom_beta_quotient_mrf_zero_rows_existszerorowsbound_entry * S ((S (fom_index_mrf_zero_rows_existszerorowsbound)) * mdr_rc_zero_rows_existszero) + (fom_value_mrf_zero_rows_existszerorowsbound))) /\ (exists fom_gap_mrf_zero_rows_existszerorowsbound_value_bound. fom_gap_mrf_zero_rows_existszerorowsbound_value_bound + S (fom_value_mrf_zero_rows_existszerorowsbound) = 0))) /\ (forall mdr_i_zero_rows_existszerorowsdistinct mdr_j_zero_rows_existszerorowsdistinct mdr_a_zero_rows_existszerorowsdistinct. (exists mdr_gap_zero_rows_existszerorowsdistincti. mdr_gap_zero_rows_existszerorowsdistincti + S (mdr_i_zero_rows_existszerorowsdistinct) = (mdr_q_zero_rows_exists)) -> (exists mdr_gap_zero_rows_existszerorowsdistinctj. mdr_gap_zero_rows_existszerorowsdistinctj + S (mdr_j_zero_rows_existszerorowsdistinct) = (mdr_q_zero_rows_exists)) -> (((exists ff_h_mdr_zero_rows_existszerorowsdistinctfirst. ff_h_mdr_zero_rows_existszerorowsdistinctfirst + S (mdr_a_zero_rows_existszerorowsdistinct) = S ((S (mdr_i_zero_rows_existszerorowsdistinct)) * mdr_rc_zero_rows_existszero)) /\ exists ff_q_mdr_zero_rows_existszerorowsdistinctfirst. mdr_rb_zero_rows_existszero = ff_q_mdr_zero_rows_existszerorowsdistinctfirst * S ((S (mdr_i_zero_rows_existszerorowsdistinct)) * mdr_rc_zero_rows_existszero) + (mdr_a_zero_rows_existszerorowsdistinct))) -> (((exists ff_h_mdr_zero_rows_existszerorowsdistinctsecond. ff_h_mdr_zero_rows_existszerorowsdistinctsecond + S (mdr_a_zero_rows_existszerorowsdistinct) = S ((S (mdr_j_zero_rows_existszerorowsdistinct)) * mdr_rc_zero_rows_existszero)) /\ exists ff_q_mdr_zero_rows_existszerorowsdistinctsecond. mdr_rb_zero_rows_existszero = ff_q_mdr_zero_rows_existszerorowsdistinctsecond * S ((S (mdr_j_zero_rows_existszerorowsdistinct)) * mdr_rc_zero_rows_existszero) + (mdr_a_zero_rows_existszerorowsdistinct))) -> mdr_i_zero_rows_existszerorowsdistinct = mdr_j_zero_rows_existszerorowsdistinct))) -> (((forall fom_index_mrf_zero_rows_existszerocolumnsbound. (exists fom_gap_mrf_zero_rows_existszerocolumnsbound_index_bound. fom_gap_mrf_zero_rows_existszerocolumnsbound_index_bound + S (fom_index_mrf_zero_rows_existszerocolumnsbound) = mdr_q_zero_rows_exists) -> exists fom_value_mrf_zero_rows_existszerocolumnsbound. ((((exists fom_beta_height_mrf_zero_rows_existszerocolumnsbound_entry. fom_beta_height_mrf_zero_rows_existszerocolumnsbound_entry + S (fom_value_mrf_zero_rows_existszerocolumnsbound) = S ((S (fom_index_mrf_zero_rows_existszerocolumnsbound)) * mdr_cc_zero_rows_existszero)) /\ exists fom_beta_quotient_mrf_zero_rows_existszerocolumnsbound_entry. mdr_cb_zero_rows_existszero = fom_beta_quotient_mrf_zero_rows_existszerocolumnsbound_entry * S ((S (fom_index_mrf_zero_rows_existszerocolumnsbound)) * mdr_cc_zero_rows_existszero) + (fom_value_mrf_zero_rows_existszerocolumnsbound))) /\ (exists fom_gap_mrf_zero_rows_existszerocolumnsbound_value_bound. fom_gap_mrf_zero_rows_existszerocolumnsbound_value_bound + S (fom_value_mrf_zero_rows_existszerocolumnsbound) = w))) /\ (forall mdr_i_zero_rows_existszerocolumnsdistinct mdr_j_zero_rows_existszerocolumnsdistinct mdr_a_zero_rows_existszerocolumnsdistinct. (exists mdr_gap_zero_rows_existszerocolumnsdistincti. mdr_gap_zero_rows_existszerocolumnsdistincti + S (mdr_i_zero_rows_existszerocolumnsdistinct) = (mdr_q_zero_rows_exists)) -> (exists mdr_gap_zero_rows_existszerocolumnsdistinctj. mdr_gap_zero_rows_existszerocolumnsdistinctj + S (mdr_j_zero_rows_existszerocolumnsdistinct) = (mdr_q_zero_rows_exists)) -> (((exists ff_h_mdr_zero_rows_existszerocolumnsdistinctfirst. ff_h_mdr_zero_rows_existszerocolumnsdistinctfirst + S (mdr_a_zero_rows_existszerocolumnsdistinct) = S ((S (mdr_i_zero_rows_existszerocolumnsdistinct)) * mdr_cc_zero_rows_existszero)) /\ exists ff_q_mdr_zero_rows_existszerocolumnsdistinctfirst. mdr_cb_zero_rows_existszero = ff_q_mdr_zero_rows_existszerocolumnsdistinctfirst * S ((S (mdr_i_zero_rows_existszerocolumnsdistinct)) * mdr_cc_zero_rows_existszero) + (mdr_a_zero_rows_existszerocolumnsdistinct))) -> (((exists ff_h_mdr_zero_rows_existszerocolumnsdistinctsecond. ff_h_mdr_zero_rows_existszerocolumnsdistinctsecond + S (mdr_a_zero_rows_existszerocolumnsdistinct) = S ((S (mdr_j_zero_rows_existszerocolumnsdistinct)) * mdr_cc_zero_rows_existszero)) /\ exists ff_q_mdr_zero_rows_existszerocolumnsdistinctsecond. mdr_cb_zero_rows_existszero = ff_q_mdr_zero_rows_existszerocolumnsdistinctsecond * S ((S (mdr_j_zero_rows_existszerocolumnsdistinct)) * mdr_cc_zero_rows_existszero) + (mdr_a_zero_rows_existszerocolumnsdistinct))) -> mdr_i_zero_rows_existszerocolumnsdistinct = mdr_j_zero_rows_existszerocolumnsdistinct))) -> (exists mdr_ub_zero_rows_existszeroevaluation mdr_uc_zero_rows_existszeroevaluation mdr_vb_zero_rows_existszeroevaluation mdr_vc_zero_rows_existszeroevaluation. ((((forall mdr_i_zero_rows_existszeroevaluationmatrixpositive. (exists mdr_gap_zero_rows_existszeroevaluationmatrixpositivebound. mdr_gap_zero_rows_existszeroevaluationmatrixpositivebound + S (mdr_i_zero_rows_existszeroevaluationmatrixpositive) = ((mdr_q_zero_rows_exists) * (mdr_q_zero_rows_exists))) -> exists mdr_a_zero_rows_existszeroevaluationmatrixpositive. (((exists mdr_r_zero_rows_existszeroevaluationmatrixpositivepoint mdr_s_zero_rows_existszeroevaluationmatrixpositivepoint mdr_u_zero_rows_existszeroevaluationmatrixpositivepoint mdr_v_zero_rows_existszeroevaluationmatrixpositivepoint. ((mdr_i_zero_rows_existszeroevaluationmatrixpositive = (mdr_q_zero_rows_exists) * mdr_r_zero_rows_existszeroevaluationmatrixpositivepoint + mdr_s_zero_rows_existszeroevaluationmatrixpositivepoint) /\ ((exists mdr_gap_zero_rows_existszeroevaluationmatrixpositivepointcolumn. mdr_gap_zero_rows_existszeroevaluationmatrixpositivepointcolumn + S (mdr_s_zero_rows_existszeroevaluationmatrixpositivepoint) = (mdr_q_zero_rows_exists)) /\ ((((exists ff_h_mdr_zero_rows_existszeroevaluationmatrixpositivepointrow_index. ff_h_mdr_zero_rows_existszeroevaluationmatrixpositivepointrow_index + S (mdr_u_zero_rows_existszeroevaluationmatrixpositivepoint) = S ((S (mdr_r_zero_rows_existszeroevaluationmatrixpositivepoint)) * mdr_rc_zero_rows_existszero)) /\ exists ff_q_mdr_zero_rows_existszeroevaluationmatrixpositivepointrow_index. mdr_rb_zero_rows_existszero = ff_q_mdr_zero_rows_existszeroevaluationmatrixpositivepointrow_index * S ((S (mdr_r_zero_rows_existszeroevaluationmatrixpositivepoint)) * mdr_rc_zero_rows_existszero) + (mdr_u_zero_rows_existszeroevaluationmatrixpositivepoint))) /\ ((((exists ff_h_mdr_zero_rows_existszeroevaluationmatrixpositivepointcolumn_index. ff_h_mdr_zero_rows_existszeroevaluationmatrixpositivepointcolumn_index + S (mdr_v_zero_rows_existszeroevaluationmatrixpositivepoint) = S ((S (mdr_s_zero_rows_existszeroevaluationmatrixpositivepoint)) * mdr_cc_zero_rows_existszero)) /\ exists ff_q_mdr_zero_rows_existszeroevaluationmatrixpositivepointcolumn_index. mdr_cb_zero_rows_existszero = ff_q_mdr_zero_rows_existszeroevaluationmatrixpositivepointcolumn_index * S ((S (mdr_s_zero_rows_existszeroevaluationmatrixpositivepoint)) * mdr_cc_zero_rows_existszero) + (mdr_v_zero_rows_existszeroevaluationmatrixpositivepoint))) /\ (((exists ff_h_mdr_zero_rows_existszeroevaluationmatrixpositivepointsource. ff_h_mdr_zero_rows_existszeroevaluationmatrixpositivepointsource + S (mdr_a_zero_rows_existszeroevaluationmatrixpositive) = S ((S ((mdr_u_zero_rows_existszeroevaluationmatrixpositivepoint) * (w) + (mdr_v_zero_rows_existszeroevaluationmatrixpositivepoint))) * pc)) /\ exists ff_q_mdr_zero_rows_existszeroevaluationmatrixpositivepointsource. pb = ff_q_mdr_zero_rows_existszeroevaluationmatrixpositivepointsource * S ((S ((mdr_u_zero_rows_existszeroevaluationmatrixpositivepoint) * (w) + (mdr_v_zero_rows_existszeroevaluationmatrixpositivepoint))) * pc) + (mdr_a_zero_rows_existszeroevaluationmatrixpositive)))))))) /\ (((exists ff_h_mdr_zero_rows_existszeroevaluationmatrixpositiveoutput. ff_h_mdr_zero_rows_existszeroevaluationmatrixpositiveoutput + S (mdr_a_zero_rows_existszeroevaluationmatrixpositive) = S ((S (mdr_i_zero_rows_existszeroevaluationmatrixpositive)) * mdr_uc_zero_rows_existszeroevaluation)) /\ exists ff_q_mdr_zero_rows_existszeroevaluationmatrixpositiveoutput. mdr_ub_zero_rows_existszeroevaluation = ff_q_mdr_zero_rows_existszeroevaluationmatrixpositiveoutput * S ((S (mdr_i_zero_rows_existszeroevaluationmatrixpositive)) * mdr_uc_zero_rows_existszeroevaluation) + (mdr_a_zero_rows_existszeroevaluationmatrixpositive)))))) /\ (forall mdr_i_zero_rows_existszeroevaluationmatrixnegative. (exists mdr_gap_zero_rows_existszeroevaluationmatrixnegativebound. mdr_gap_zero_rows_existszeroevaluationmatrixnegativebound + S (mdr_i_zero_rows_existszeroevaluationmatrixnegative) = ((mdr_q_zero_rows_exists) * (mdr_q_zero_rows_exists))) -> exists mdr_a_zero_rows_existszeroevaluationmatrixnegative. (((exists mdr_r_zero_rows_existszeroevaluationmatrixnegativepoint mdr_s_zero_rows_existszeroevaluationmatrixnegativepoint mdr_u_zero_rows_existszeroevaluationmatrixnegativepoint mdr_v_zero_rows_existszeroevaluationmatrixnegativepoint. ((mdr_i_zero_rows_existszeroevaluationmatrixnegative = (mdr_q_zero_rows_exists) * mdr_r_zero_rows_existszeroevaluationmatrixnegativepoint + mdr_s_zero_rows_existszeroevaluationmatrixnegativepoint) /\ ((exists mdr_gap_zero_rows_existszeroevaluationmatrixnegativepointcolumn. mdr_gap_zero_rows_existszeroevaluationmatrixnegativepointcolumn + S (mdr_s_zero_rows_existszeroevaluationmatrixnegativepoint) = (mdr_q_zero_rows_exists)) /\ ((((exists ff_h_mdr_zero_rows_existszeroevaluationmatrixnegativepointrow_index. ff_h_mdr_zero_rows_existszeroevaluationmatrixnegativepointrow_index + S (mdr_u_zero_rows_existszeroevaluationmatrixnegativepoint) = S ((S (mdr_r_zero_rows_existszeroevaluationmatrixnegativepoint)) * mdr_rc_zero_rows_existszero)) /\ exists ff_q_mdr_zero_rows_existszeroevaluationmatrixnegativepointrow_index. mdr_rb_zero_rows_existszero = ff_q_mdr_zero_rows_existszeroevaluationmatrixnegativepointrow_index * S ((S (mdr_r_zero_rows_existszeroevaluationmatrixnegativepoint)) * mdr_rc_zero_rows_existszero) + (mdr_u_zero_rows_existszeroevaluationmatrixnegativepoint))) /\ ((((exists ff_h_mdr_zero_rows_existszeroevaluationmatrixnegativepointcolumn_index. ff_h_mdr_zero_rows_existszeroevaluationmatrixnegativepointcolumn_index + S (mdr_v_zero_rows_existszeroevaluationmatrixnegativepoint) = S ((S (mdr_s_zero_rows_existszeroevaluationmatrixnegativepoint)) * mdr_cc_zero_rows_existszero)) /\ exists ff_q_mdr_zero_rows_existszeroevaluationmatrixnegativepointcolumn_index. mdr_cb_zero_rows_existszero = ff_q_mdr_zero_rows_existszeroevaluationmatrixnegativepointcolumn_index * S ((S (mdr_s_zero_rows_existszeroevaluationmatrixnegativepoint)) * mdr_cc_zero_rows_existszero) + (mdr_v_zero_rows_existszeroevaluationmatrixnegativepoint))) /\ (((exists ff_h_mdr_zero_rows_existszeroevaluationmatrixnegativepointsource. ff_h_mdr_zero_rows_existszeroevaluationmatrixnegativepointsource + S (mdr_a_zero_rows_existszeroevaluationmatrixnegative) = S ((S ((mdr_u_zero_rows_existszeroevaluationmatrixnegativepoint) * (w) + (mdr_v_zero_rows_existszeroevaluationmatrixnegativepoint))) * nc)) /\ exists ff_q_mdr_zero_rows_existszeroevaluationmatrixnegativepointsource. nb = ff_q_mdr_zero_rows_existszeroevaluationmatrixnegativepointsource * S ((S ((mdr_u_zero_rows_existszeroevaluationmatrixnegativepoint) * (w) + (mdr_v_zero_rows_existszeroevaluationmatrixnegativepoint))) * nc) + (mdr_a_zero_rows_existszeroevaluationmatrixnegative)))))))) /\ (((exists ff_h_mdr_zero_rows_existszeroevaluationmatrixnegativeoutput. ff_h_mdr_zero_rows_existszeroevaluationmatrixnegativeoutput + S (mdr_a_zero_rows_existszeroevaluationmatrixnegative) = S ((S (mdr_i_zero_rows_existszeroevaluationmatrixnegative)) * mdr_vc_zero_rows_existszeroevaluation)) /\ exists ff_q_mdr_zero_rows_existszeroevaluationmatrixnegativeoutput. mdr_vb_zero_rows_existszeroevaluation = ff_q_mdr_zero_rows_existszeroevaluationmatrixnegativeoutput * S ((S (mdr_i_zero_rows_existszeroevaluationmatrixnegative)) * mdr_vc_zero_rows_existszeroevaluation) + (mdr_a_zero_rows_existszeroevaluationmatrixnegative)))))))) /\ (exists mdr_b_zero_rows_existszeroevaluationdeterminant mdr_c_zero_rows_existszeroevaluationdeterminant mdr_l_zero_rows_existszeroevaluationdeterminant mdr_i_zero_rows_existszeroevaluationdeterminant. ((forall mdr_i_zero_rows_existszeroevaluationdeterminanth. (exists mdr_gap_zero_rows_existszeroevaluationdeterminanthi. mdr_gap_zero_rows_existszeroevaluationdeterminanthi + S (mdr_i_zero_rows_existszeroevaluationdeterminanth) = (mdr_l_zero_rows_existszeroevaluationdeterminant)) -> exists mdr_d_zero_rows_existszeroevaluationdeterminanth mdr_pb_zero_rows_existszeroevaluationdeterminanth mdr_pc_zero_rows_existszeroevaluationdeterminanth mdr_nb_zero_rows_existszeroevaluationdeterminanth mdr_nc_zero_rows_existszeroevaluationdeterminanth mdr_p_zero_rows_existszeroevaluationdeterminanth mdr_n_zero_rows_existszeroevaluationdeterminanth. ((exists mdr_z_zero_rows_existszeroevaluationdeterminanthr. ((exists mdr_a_zero_rows_existszeroevaluationdeterminanthrc mdr_b_zero_rows_existszeroevaluationdeterminanthrc mdr_c_zero_rows_existszeroevaluationdeterminanthrc mdr_e_zero_rows_existszeroevaluationdeterminanthrc mdr_f_zero_rows_existszeroevaluationdeterminanthrc. ((mdr_a_zero_rows_existszeroevaluationdeterminanthrc = ((mdr_d_zero_rows_existszeroevaluationdeterminanth) + (mdr_pb_zero_rows_existszeroevaluationdeterminanth)) * S ((mdr_d_zero_rows_existszeroevaluationdeterminanth) + (mdr_pb_zero_rows_existszeroevaluationdeterminanth)) + ((mdr_pb_zero_rows_existszeroevaluationdeterminanth) + (mdr_pb_zero_rows_existszeroevaluationdeterminanth))) /\ ((mdr_b_zero_rows_existszeroevaluationdeterminanthrc = ((mdr_pc_zero_rows_existszeroevaluationdeterminanth) + (mdr_nb_zero_rows_existszeroevaluationdeterminanth)) * S ((mdr_pc_zero_rows_existszeroevaluationdeterminanth) + (mdr_nb_zero_rows_existszeroevaluationdeterminanth)) + ((mdr_nb_zero_rows_existszeroevaluationdeterminanth) + (mdr_nb_zero_rows_existszeroevaluationdeterminanth))) /\ ((mdr_c_zero_rows_existszeroevaluationdeterminanthrc = ((mdr_a_zero_rows_existszeroevaluationdeterminanthrc) + (mdr_b_zero_rows_existszeroevaluationdeterminanthrc)) * S ((mdr_a_zero_rows_existszeroevaluationdeterminanthrc) + (mdr_b_zero_rows_existszeroevaluationdeterminanthrc)) + ((mdr_b_zero_rows_existszeroevaluationdeterminanthrc) + (mdr_b_zero_rows_existszeroevaluationdeterminanthrc))) /\ ((mdr_e_zero_rows_existszeroevaluationdeterminanthrc = ((mdr_p_zero_rows_existszeroevaluationdeterminanth) + (mdr_n_zero_rows_existszeroevaluationdeterminanth)) * S ((mdr_p_zero_rows_existszeroevaluationdeterminanth) + (mdr_n_zero_rows_existszeroevaluationdeterminanth)) + ((mdr_n_zero_rows_existszeroevaluationdeterminanth) + (mdr_n_zero_rows_existszeroevaluationdeterminanth))) /\ ((mdr_f_zero_rows_existszeroevaluationdeterminanthrc = ((mdr_nc_zero_rows_existszeroevaluationdeterminanth) + (mdr_e_zero_rows_existszeroevaluationdeterminanthrc)) * S ((mdr_nc_zero_rows_existszeroevaluationdeterminanth) + (mdr_e_zero_rows_existszeroevaluationdeterminanthrc)) + ((mdr_e_zero_rows_existszeroevaluationdeterminanthrc) + (mdr_e_zero_rows_existszeroevaluationdeterminanthrc))) /\ ((mdr_z_zero_rows_existszeroevaluationdeterminanthr) = ((mdr_c_zero_rows_existszeroevaluationdeterminanthrc) + (mdr_f_zero_rows_existszeroevaluationdeterminanthrc)) * S ((mdr_c_zero_rows_existszeroevaluationdeterminanthrc) + (mdr_f_zero_rows_existszeroevaluationdeterminanthrc)) + ((mdr_f_zero_rows_existszeroevaluationdeterminanthrc) + (mdr_f_zero_rows_existszeroevaluationdeterminanthrc))))))))) /\ (((exists ff_h_mdr_zero_rows_existszeroevaluationdeterminanthrb. ff_h_mdr_zero_rows_existszeroevaluationdeterminanthrb + S (mdr_z_zero_rows_existszeroevaluationdeterminanthr) = S ((S (mdr_i_zero_rows_existszeroevaluationdeterminanth)) * mdr_c_zero_rows_existszeroevaluationdeterminant)) /\ exists ff_q_mdr_zero_rows_existszeroevaluationdeterminanthrb. mdr_b_zero_rows_existszeroevaluationdeterminant = ff_q_mdr_zero_rows_existszeroevaluationdeterminanthrb * S ((S (mdr_i_zero_rows_existszeroevaluationdeterminanth)) * mdr_c_zero_rows_existszeroevaluationdeterminant) + (mdr_z_zero_rows_existszeroevaluationdeterminanthr))))) /\ (((((mdr_d_zero_rows_existszeroevaluationdeterminanth) = 0) /\ (((mdr_p_zero_rows_existszeroevaluationdeterminanth) = 1) /\ ((mdr_n_zero_rows_existszeroevaluationdeterminanth) = 0))) \/ exists mdr_q_zero_rows_existszeroevaluationdeterminanths mdr_eb_zero_rows_existszeroevaluationdeterminanths mdr_ec_zero_rows_existszeroevaluationdeterminanths mdr_fb_zero_rows_existszeroevaluationdeterminanths mdr_fc_zero_rows_existszeroevaluationdeterminanths. (((mdr_d_zero_rows_existszeroevaluationdeterminanth) = S (mdr_q_zero_rows_existszeroevaluationdeterminanths)) /\ ((forall mdr_j_zero_rows_existszeroevaluationdeterminanthsc. (exists mdr_gap_zero_rows_existszeroevaluationdeterminanthscj. mdr_gap_zero_rows_existszeroevaluationdeterminanthscj + S (mdr_j_zero_rows_existszeroevaluationdeterminanthsc) = (S (mdr_q_zero_rows_existszeroevaluationdeterminanths))) -> exists mdr_i_zero_rows_existszeroevaluationdeterminanthsc mdr_up_zero_rows_existszeroevaluationdeterminanthsc mdr_us_zero_rows_existszeroevaluationdeterminanthsc mdr_un_zero_rows_existszeroevaluationdeterminanthsc mdr_ut_zero_rows_existszeroevaluationdeterminanthsc mdr_p_zero_rows_existszeroevaluationdeterminanthsc mdr_n_zero_rows_existszeroevaluationdeterminanthsc. ((exists mdr_gap_zero_rows_existszeroevaluationdeterminanthsci. mdr_gap_zero_rows_existszeroevaluationdeterminanthsci + S (mdr_i_zero_rows_existszeroevaluationdeterminanthsc) = (mdr_i_zero_rows_existszeroevaluationdeterminanth)) /\ ((exists mdr_z_zero_rows_existszeroevaluationdeterminanthscr. ((exists mdr_a_zero_rows_existszeroevaluationdeterminanthscrc mdr_b_zero_rows_existszeroevaluationdeterminanthscrc mdr_c_zero_rows_existszeroevaluationdeterminanthscrc mdr_e_zero_rows_existszeroevaluationdeterminanthscrc mdr_f_zero_rows_existszeroevaluationdeterminanthscrc. ((mdr_a_zero_rows_existszeroevaluationdeterminanthscrc = ((mdr_q_zero_rows_existszeroevaluationdeterminanths) + (mdr_up_zero_rows_existszeroevaluationdeterminanthsc)) * S ((mdr_q_zero_rows_existszeroevaluationdeterminanths) + (mdr_up_zero_rows_existszeroevaluationdeterminanthsc)) + ((mdr_up_zero_rows_existszeroevaluationdeterminanthsc) + (mdr_up_zero_rows_existszeroevaluationdeterminanthsc))) /\ ((mdr_b_zero_rows_existszeroevaluationdeterminanthscrc = ((mdr_us_zero_rows_existszeroevaluationdeterminanthsc) + (mdr_un_zero_rows_existszeroevaluationdeterminanthsc)) * S ((mdr_us_zero_rows_existszeroevaluationdeterminanthsc) + (mdr_un_zero_rows_existszeroevaluationdeterminanthsc)) + ((mdr_un_zero_rows_existszeroevaluationdeterminanthsc) + (mdr_un_zero_rows_existszeroevaluationdeterminanthsc))) /\ ((mdr_c_zero_rows_existszeroevaluationdeterminanthscrc = ((mdr_a_zero_rows_existszeroevaluationdeterminanthscrc) + (mdr_b_zero_rows_existszeroevaluationdeterminanthscrc)) * S ((mdr_a_zero_rows_existszeroevaluationdeterminanthscrc) + (mdr_b_zero_rows_existszeroevaluationdeterminanthscrc)) + ((mdr_b_zero_rows_existszeroevaluationdeterminanthscrc) + (mdr_b_zero_rows_existszeroevaluationdeterminanthscrc))) /\ ((mdr_e_zero_rows_existszeroevaluationdeterminanthscrc = ((mdr_p_zero_rows_existszeroevaluationdeterminanthsc) + (mdr_n_zero_rows_existszeroevaluationdeterminanthsc)) * S ((mdr_p_zero_rows_existszeroevaluationdeterminanthsc) + (mdr_n_zero_rows_existszeroevaluationdeterminanthsc)) + ((mdr_n_zero_rows_existszeroevaluationdeterminanthsc) + (mdr_n_zero_rows_existszeroevaluationdeterminanthsc))) /\ ((mdr_f_zero_rows_existszeroevaluationdeterminanthscrc = ((mdr_ut_zero_rows_existszeroevaluationdeterminanthsc) + (mdr_e_zero_rows_existszeroevaluationdeterminanthscrc)) * S ((mdr_ut_zero_rows_existszeroevaluationdeterminanthsc) + (mdr_e_zero_rows_existszeroevaluationdeterminanthscrc)) + ((mdr_e_zero_rows_existszeroevaluationdeterminanthscrc) + (mdr_e_zero_rows_existszeroevaluationdeterminanthscrc))) /\ ((mdr_z_zero_rows_existszeroevaluationdeterminanthscr) = ((mdr_c_zero_rows_existszeroevaluationdeterminanthscrc) + (mdr_f_zero_rows_existszeroevaluationdeterminanthscrc)) * S ((mdr_c_zero_rows_existszeroevaluationdeterminanthscrc) + (mdr_f_zero_rows_existszeroevaluationdeterminanthscrc)) + ((mdr_f_zero_rows_existszeroevaluationdeterminanthscrc) + (mdr_f_zero_rows_existszeroevaluationdeterminanthscrc))))))))) /\ (((exists ff_h_mdr_zero_rows_existszeroevaluationdeterminanthscrb. ff_h_mdr_zero_rows_existszeroevaluationdeterminanthscrb + S (mdr_z_zero_rows_existszeroevaluationdeterminanthscr) = S ((S (mdr_i_zero_rows_existszeroevaluationdeterminanthsc)) * mdr_c_zero_rows_existszeroevaluationdeterminant)) /\ exists ff_q_mdr_zero_rows_existszeroevaluationdeterminanthscrb. mdr_b_zero_rows_existszeroevaluationdeterminant = ff_q_mdr_zero_rows_existszeroevaluationdeterminanthscrb * S ((S (mdr_i_zero_rows_existszeroevaluationdeterminanthsc)) * mdr_c_zero_rows_existszeroevaluationdeterminant) + (mdr_z_zero_rows_existszeroevaluationdeterminanthscr))))) /\ ((((forall ff_index_mdm_prefix_mdr_zero_rows_existszeroevaluationdeterminanthscm_positive. (exists ff_gap_mdm_lt_mdr_zero_rows_existszeroevaluationdeterminanthscm_positive_index_bound. ff_gap_mdm_lt_mdr_zero_rows_existszeroevaluationdeterminanthscm_positive_index_bound + S (ff_index_mdm_prefix_mdr_zero_rows_existszeroevaluationdeterminanthscm_positive) = ((mdr_q_zero_rows_existszeroevaluationdeterminanths) * (mdr_q_zero_rows_existszeroevaluationdeterminanths))) -> exists ff_row_mdm_prefix_mdr_zero_rows_existszeroevaluationdeterminanthscm_positive ff_column_mdm_prefix_mdr_zero_rows_existszeroevaluationdeterminanthscm_positive ff_value_mdm_prefix_mdr_zero_rows_existszeroevaluationdeterminanthscm_positive. (ff_index_mdm_prefix_mdr_zero_rows_existszeroevaluationdeterminanthscm_positive = (mdr_q_zero_rows_existszeroevaluationdeterminanths) * ff_row_mdm_prefix_mdr_zero_rows_existszeroevaluationdeterminanthscm_positive + ff_column_mdm_prefix_mdr_zero_rows_existszeroevaluationdeterminanthscm_positive /\ ((exists ff_gap_mdm_lt_mdr_zero_rows_existszeroevaluationdeterminanthscm_positive_column_bound. ff_gap_mdm_lt_mdr_zero_rows_existszeroevaluationdeterminanthscm_positive_column_bound + S (ff_column_mdm_prefix_mdr_zero_rows_existszeroevaluationdeterminanthscm_positive) = (mdr_q_zero_rows_existszeroevaluationdeterminanths)) /\ ((exists ff_row_mdm_cell_mdr_zero_rows_existszeroevaluationdeterminanthscm_positive_cell ff_column_mdm_cell_mdr_zero_rows_existszeroevaluationdeterminanthscm_positive_cell. (((((exists ff_gap_mdm_lt_mdr_zero_rows_existszeroevaluationdeterminanthscm_positive_cell_row_before. ff_gap_mdm_lt_mdr_zero_rows_existszeroevaluationdeterminanthscm_positive_cell_row_before + S (ff_row_mdm_prefix_mdr_zero_rows_existszeroevaluationdeterminanthscm_positive) = (0)) /\ ff_row_mdm_cell_mdr_zero_rows_existszeroevaluationdeterminanthscm_positive_cell = ff_row_mdm_prefix_mdr_zero_rows_existszeroevaluationdeterminanthscm_positive) \/ ((exists ff_gap_mdm_le_mdr_zero_rows_existszeroevaluationdeterminanthscm_positive_cell_row_after. ff_gap_mdm_le_mdr_zero_rows_existszeroevaluationdeterminanthscm_positive_cell_row_after + (0) = (ff_row_mdm_prefix_mdr_zero_rows_existszeroevaluationdeterminanthscm_positive)) /\ ff_row_mdm_cell_mdr_zero_rows_existszeroevaluationdeterminanthscm_positive_cell = S ff_row_mdm_prefix_mdr_zero_rows_existszeroevaluationdeterminanthscm_positive))) /\ (((((exists ff_gap_mdm_lt_mdr_zero_rows_existszeroevaluationdeterminanthscm_positive_cell_column_before. ff_gap_mdm_lt_mdr_zero_rows_existszeroevaluationdeterminanthscm_positive_cell_column_before + S (ff_column_mdm_prefix_mdr_zero_rows_existszeroevaluationdeterminanthscm_positive) = (mdr_j_zero_rows_existszeroevaluationdeterminanthsc)) /\ ff_column_mdm_cell_mdr_zero_rows_existszeroevaluationdeterminanthscm_positive_cell = ff_column_mdm_prefix_mdr_zero_rows_existszeroevaluationdeterminanthscm_positive) \/ ((exists ff_gap_mdm_le_mdr_zero_rows_existszeroevaluationdeterminanthscm_positive_cell_column_after. ff_gap_mdm_le_mdr_zero_rows_existszeroevaluationdeterminanthscm_positive_cell_column_after + (mdr_j_zero_rows_existszeroevaluationdeterminanthsc) = (ff_column_mdm_prefix_mdr_zero_rows_existszeroevaluationdeterminanthscm_positive)) /\ ff_column_mdm_cell_mdr_zero_rows_existszeroevaluationdeterminanthscm_positive_cell = S ff_column_mdm_prefix_mdr_zero_rows_existszeroevaluationdeterminanthscm_positive))) /\ (((exists ff_h_mdm_mdr_zero_rows_existszeroevaluationdeterminanthscm_positive_cell_source. ff_h_mdm_mdr_zero_rows_existszeroevaluationdeterminanthscm_positive_cell_source + S (ff_value_mdm_prefix_mdr_zero_rows_existszeroevaluationdeterminanthscm_positive) = S ((S ((ff_row_mdm_cell_mdr_zero_rows_existszeroevaluationdeterminanthscm_positive_cell) * (S (mdr_q_zero_rows_existszeroevaluationdeterminanths)) + (ff_column_mdm_cell_mdr_zero_rows_existszeroevaluationdeterminanthscm_positive_cell))) * mdr_pc_zero_rows_existszeroevaluationdeterminanth)) /\ exists ff_q_mdm_mdr_zero_rows_existszeroevaluationdeterminanthscm_positive_cell_source. mdr_pb_zero_rows_existszeroevaluationdeterminanth = ff_q_mdm_mdr_zero_rows_existszeroevaluationdeterminanthscm_positive_cell_source * S ((S ((ff_row_mdm_cell_mdr_zero_rows_existszeroevaluationdeterminanthscm_positive_cell) * (S (mdr_q_zero_rows_existszeroevaluationdeterminanths)) + (ff_column_mdm_cell_mdr_zero_rows_existszeroevaluationdeterminanthscm_positive_cell))) * mdr_pc_zero_rows_existszeroevaluationdeterminanth) + (ff_value_mdm_prefix_mdr_zero_rows_existszeroevaluationdeterminanthscm_positive)))))) /\ (((exists ff_h_mdm_mdr_zero_rows_existszeroevaluationdeterminanthscm_positive_target. ff_h_mdm_mdr_zero_rows_existszeroevaluationdeterminanthscm_positive_target + S (ff_value_mdm_prefix_mdr_zero_rows_existszeroevaluationdeterminanthscm_positive) = S ((S (ff_index_mdm_prefix_mdr_zero_rows_existszeroevaluationdeterminanthscm_positive)) * mdr_us_zero_rows_existszeroevaluationdeterminanthsc)) /\ exists ff_q_mdm_mdr_zero_rows_existszeroevaluationdeterminanthscm_positive_target. mdr_up_zero_rows_existszeroevaluationdeterminanthsc = ff_q_mdm_mdr_zero_rows_existszeroevaluationdeterminanthscm_positive_target * S ((S (ff_index_mdm_prefix_mdr_zero_rows_existszeroevaluationdeterminanthscm_positive)) * mdr_us_zero_rows_existszeroevaluationdeterminanthsc) + (ff_value_mdm_prefix_mdr_zero_rows_existszeroevaluationdeterminanthscm_positive))))))) /\ (forall ff_index_mdm_prefix_mdr_zero_rows_existszeroevaluationdeterminanthscm_negative. (exists ff_gap_mdm_lt_mdr_zero_rows_existszeroevaluationdeterminanthscm_negative_index_bound. ff_gap_mdm_lt_mdr_zero_rows_existszeroevaluationdeterminanthscm_negative_index_bound + S (ff_index_mdm_prefix_mdr_zero_rows_existszeroevaluationdeterminanthscm_negative) = ((mdr_q_zero_rows_existszeroevaluationdeterminanths) * (mdr_q_zero_rows_existszeroevaluationdeterminanths))) -> exists ff_row_mdm_prefix_mdr_zero_rows_existszeroevaluationdeterminanthscm_negative ff_column_mdm_prefix_mdr_zero_rows_existszeroevaluationdeterminanthscm_negative ff_value_mdm_prefix_mdr_zero_rows_existszeroevaluationdeterminanthscm_negative. (ff_index_mdm_prefix_mdr_zero_rows_existszeroevaluationdeterminanthscm_negative = (mdr_q_zero_rows_existszeroevaluationdeterminanths) * ff_row_mdm_prefix_mdr_zero_rows_existszeroevaluationdeterminanthscm_negative + ff_column_mdm_prefix_mdr_zero_rows_existszeroevaluationdeterminanthscm_negative /\ ((exists ff_gap_mdm_lt_mdr_zero_rows_existszeroevaluationdeterminanthscm_negative_column_bound. ff_gap_mdm_lt_mdr_zero_rows_existszeroevaluationdeterminanthscm_negative_column_bound + S (ff_column_mdm_prefix_mdr_zero_rows_existszeroevaluationdeterminanthscm_negative) = (mdr_q_zero_rows_existszeroevaluationdeterminanths)) /\ ((exists ff_row_mdm_cell_mdr_zero_rows_existszeroevaluationdeterminanthscm_negative_cell ff_column_mdm_cell_mdr_zero_rows_existszeroevaluationdeterminanthscm_negative_cell. (((((exists ff_gap_mdm_lt_mdr_zero_rows_existszeroevaluationdeterminanthscm_negative_cell_row_before. ff_gap_mdm_lt_mdr_zero_rows_existszeroevaluationdeterminanthscm_negative_cell_row_before + S (ff_row_mdm_prefix_mdr_zero_rows_existszeroevaluationdeterminanthscm_negative) = (0)) /\ ff_row_mdm_cell_mdr_zero_rows_existszeroevaluationdeterminanthscm_negative_cell = ff_row_mdm_prefix_mdr_zero_rows_existszeroevaluationdeterminanthscm_negative) \/ ((exists ff_gap_mdm_le_mdr_zero_rows_existszeroevaluationdeterminanthscm_negative_cell_row_after. ff_gap_mdm_le_mdr_zero_rows_existszeroevaluationdeterminanthscm_negative_cell_row_after + (0) = (ff_row_mdm_prefix_mdr_zero_rows_existszeroevaluationdeterminanthscm_negative)) /\ ff_row_mdm_cell_mdr_zero_rows_existszeroevaluationdeterminanthscm_negative_cell = S ff_row_mdm_prefix_mdr_zero_rows_existszeroevaluationdeterminanthscm_negative))) /\ (((((exists ff_gap_mdm_lt_mdr_zero_rows_existszeroevaluationdeterminanthscm_negative_cell_column_before. ff_gap_mdm_lt_mdr_zero_rows_existszeroevaluationdeterminanthscm_negative_cell_column_before + S (ff_column_mdm_prefix_mdr_zero_rows_existszeroevaluationdeterminanthscm_negative) = (mdr_j_zero_rows_existszeroevaluationdeterminanthsc)) /\ ff_column_mdm_cell_mdr_zero_rows_existszeroevaluationdeterminanthscm_negative_cell = ff_column_mdm_prefix_mdr_zero_rows_existszeroevaluationdeterminanthscm_negative) \/ ((exists ff_gap_mdm_le_mdr_zero_rows_existszeroevaluationdeterminanthscm_negative_cell_column_after. ff_gap_mdm_le_mdr_zero_rows_existszeroevaluationdeterminanthscm_negative_cell_column_after + (mdr_j_zero_rows_existszeroevaluationdeterminanthsc) = (ff_column_mdm_prefix_mdr_zero_rows_existszeroevaluationdeterminanthscm_negative)) /\ ff_column_mdm_cell_mdr_zero_rows_existszeroevaluationdeterminanthscm_negative_cell = S ff_column_mdm_prefix_mdr_zero_rows_existszeroevaluationdeterminanthscm_negative))) /\ (((exists ff_h_mdm_mdr_zero_rows_existszeroevaluationdeterminanthscm_negative_cell_source. ff_h_mdm_mdr_zero_rows_existszeroevaluationdeterminanthscm_negative_cell_source + S (ff_value_mdm_prefix_mdr_zero_rows_existszeroevaluationdeterminanthscm_negative) = S ((S ((ff_row_mdm_cell_mdr_zero_rows_existszeroevaluationdeterminanthscm_negative_cell) * (S (mdr_q_zero_rows_existszeroevaluationdeterminanths)) + (ff_column_mdm_cell_mdr_zero_rows_existszeroevaluationdeterminanthscm_negative_cell))) * mdr_nc_zero_rows_existszeroevaluationdeterminanth)) /\ exists ff_q_mdm_mdr_zero_rows_existszeroevaluationdeterminanthscm_negative_cell_source. mdr_nb_zero_rows_existszeroevaluationdeterminanth = ff_q_mdm_mdr_zero_rows_existszeroevaluationdeterminanthscm_negative_cell_source * S ((S ((ff_row_mdm_cell_mdr_zero_rows_existszeroevaluationdeterminanthscm_negative_cell) * (S (mdr_q_zero_rows_existszeroevaluationdeterminanths)) + (ff_column_mdm_cell_mdr_zero_rows_existszeroevaluationdeterminanthscm_negative_cell))) * mdr_nc_zero_rows_existszeroevaluationdeterminanth) + (ff_value_mdm_prefix_mdr_zero_rows_existszeroevaluationdeterminanthscm_negative)))))) /\ (((exists ff_h_mdm_mdr_zero_rows_existszeroevaluationdeterminanthscm_negative_target. ff_h_mdm_mdr_zero_rows_existszeroevaluationdeterminanthscm_negative_target + S (ff_value_mdm_prefix_mdr_zero_rows_existszeroevaluationdeterminanthscm_negative) = S ((S (ff_index_mdm_prefix_mdr_zero_rows_existszeroevaluationdeterminanthscm_negative)) * mdr_ut_zero_rows_existszeroevaluationdeterminanthsc)) /\ exists ff_q_mdm_mdr_zero_rows_existszeroevaluationdeterminanthscm_negative_target. mdr_un_zero_rows_existszeroevaluationdeterminanthsc = ff_q_mdm_mdr_zero_rows_existszeroevaluationdeterminanthscm_negative_target * S ((S (ff_index_mdm_prefix_mdr_zero_rows_existszeroevaluationdeterminanthscm_negative)) * mdr_ut_zero_rows_existszeroevaluationdeterminanthsc) + (ff_value_mdm_prefix_mdr_zero_rows_existszeroevaluationdeterminanthscm_negative))))))))) /\ ((((exists ff_h_mdr_zero_rows_existszeroevaluationdeterminanthscp. ff_h_mdr_zero_rows_existszeroevaluationdeterminanthscp + S (mdr_p_zero_rows_existszeroevaluationdeterminanthsc) = S ((S (mdr_j_zero_rows_existszeroevaluationdeterminanthsc)) * mdr_ec_zero_rows_existszeroevaluationdeterminanths)) /\ exists ff_q_mdr_zero_rows_existszeroevaluationdeterminanthscp. mdr_eb_zero_rows_existszeroevaluationdeterminanths = ff_q_mdr_zero_rows_existszeroevaluationdeterminanthscp * S ((S (mdr_j_zero_rows_existszeroevaluationdeterminanthsc)) * mdr_ec_zero_rows_existszeroevaluationdeterminanths) + (mdr_p_zero_rows_existszeroevaluationdeterminanthsc))) /\ (((exists ff_h_mdr_zero_rows_existszeroevaluationdeterminanthscn. ff_h_mdr_zero_rows_existszeroevaluationdeterminanthscn + S (mdr_n_zero_rows_existszeroevaluationdeterminanthsc) = S ((S (mdr_j_zero_rows_existszeroevaluationdeterminanthsc)) * mdr_fc_zero_rows_existszeroevaluationdeterminanths)) /\ exists ff_q_mdr_zero_rows_existszeroevaluationdeterminanthscn. mdr_fb_zero_rows_existszeroevaluationdeterminanths = ff_q_mdr_zero_rows_existszeroevaluationdeterminanthscn * S ((S (mdr_j_zero_rows_existszeroevaluationdeterminanthsc)) * mdr_fc_zero_rows_existszeroevaluationdeterminanths) + (mdr_n_zero_rows_existszeroevaluationdeterminanthsc)))))))) /\ (exists ff_ub_mce_fold_mdr_zero_rows_existszeroevaluationdeterminanthsf ff_uc_mce_fold_mdr_zero_rows_existszeroevaluationdeterminanthsf ff_vb_mce_fold_mdr_zero_rows_existszeroevaluationdeterminanthsf ff_vc_mce_fold_mdr_zero_rows_existszeroevaluationdeterminanthsf. ((forall ff_index_mce_alternating_mdr_zero_rows_existszeroevaluationdeterminanthsf_prefix. (exists ff_gap_mce_mdr_zero_rows_existszeroevaluationdeterminanthsf_prefix_index. ff_gap_mce_mdr_zero_rows_existszeroevaluationdeterminanthsf_prefix_index + S (ff_index_mce_alternating_mdr_zero_rows_existszeroevaluationdeterminanthsf_prefix) = (S (mdr_q_zero_rows_existszeroevaluationdeterminanths))) -> exists ff_ap_mce_alternating_mdr_zero_rows_existszeroevaluationdeterminanthsf_prefix ff_an_mce_alternating_mdr_zero_rows_existszeroevaluationdeterminanthsf_prefix ff_bp_mce_alternating_mdr_zero_rows_existszeroevaluationdeterminanthsf_prefix ff_bn_mce_alternating_mdr_zero_rows_existszeroevaluationdeterminanthsf_prefix ff_p_mce_alternating_mdr_zero_rows_existszeroevaluationdeterminanthsf_prefix ff_n_mce_alternating_mdr_zero_rows_existszeroevaluationdeterminanthsf_prefix. ((((exists ff_h_mce_mdr_zero_rows_existszeroevaluationdeterminanthsf_prefix_ap. ff_h_mce_mdr_zero_rows_existszeroevaluationdeterminanthsf_prefix_ap + S (ff_ap_mce_alternating_mdr_zero_rows_existszeroevaluationdeterminanthsf_prefix) = S ((S (ff_index_mce_alternating_mdr_zero_rows_existszeroevaluationdeterminanthsf_prefix)) * mdr_pc_zero_rows_existszeroevaluationdeterminanth)) /\ exists ff_q_mce_mdr_zero_rows_existszeroevaluationdeterminanthsf_prefix_ap. mdr_pb_zero_rows_existszeroevaluationdeterminanth = ff_q_mce_mdr_zero_rows_existszeroevaluationdeterminanthsf_prefix_ap * S ((S (ff_index_mce_alternating_mdr_zero_rows_existszeroevaluationdeterminanthsf_prefix)) * mdr_pc_zero_rows_existszeroevaluationdeterminanth) + (ff_ap_mce_alternating_mdr_zero_rows_existszeroevaluationdeterminanthsf_prefix))) /\ ((((exists ff_h_mce_mdr_zero_rows_existszeroevaluationdeterminanthsf_prefix_an. ff_h_mce_mdr_zero_rows_existszeroevaluationdeterminanthsf_prefix_an + S (ff_an_mce_alternating_mdr_zero_rows_existszeroevaluationdeterminanthsf_prefix) = S ((S (ff_index_mce_alternating_mdr_zero_rows_existszeroevaluationdeterminanthsf_prefix)) * mdr_nc_zero_rows_existszeroevaluationdeterminanth)) /\ exists ff_q_mce_mdr_zero_rows_existszeroevaluationdeterminanthsf_prefix_an. mdr_nb_zero_rows_existszeroevaluationdeterminanth = ff_q_mce_mdr_zero_rows_existszeroevaluationdeterminanthsf_prefix_an * S ((S (ff_index_mce_alternating_mdr_zero_rows_existszeroevaluationdeterminanthsf_prefix)) * mdr_nc_zero_rows_existszeroevaluationdeterminanth) + (ff_an_mce_alternating_mdr_zero_rows_existszeroevaluationdeterminanthsf_prefix))) /\ ((((exists ff_h_mce_mdr_zero_rows_existszeroevaluationdeterminanthsf_prefix_bp. ff_h_mce_mdr_zero_rows_existszeroevaluationdeterminanthsf_prefix_bp + S (ff_bp_mce_alternating_mdr_zero_rows_existszeroevaluationdeterminanthsf_prefix) = S ((S (ff_index_mce_alternating_mdr_zero_rows_existszeroevaluationdeterminanthsf_prefix)) * mdr_ec_zero_rows_existszeroevaluationdeterminanths)) /\ exists ff_q_mce_mdr_zero_rows_existszeroevaluationdeterminanthsf_prefix_bp. mdr_eb_zero_rows_existszeroevaluationdeterminanths = ff_q_mce_mdr_zero_rows_existszeroevaluationdeterminanthsf_prefix_bp * S ((S (ff_index_mce_alternating_mdr_zero_rows_existszeroevaluationdeterminanthsf_prefix)) * mdr_ec_zero_rows_existszeroevaluationdeterminanths) + (ff_bp_mce_alternating_mdr_zero_rows_existszeroevaluationdeterminanthsf_prefix))) /\ ((((exists ff_h_mce_mdr_zero_rows_existszeroevaluationdeterminanthsf_prefix_bn. ff_h_mce_mdr_zero_rows_existszeroevaluationdeterminanthsf_prefix_bn + S (ff_bn_mce_alternating_mdr_zero_rows_existszeroevaluationdeterminanthsf_prefix) = S ((S (ff_index_mce_alternating_mdr_zero_rows_existszeroevaluationdeterminanthsf_prefix)) * mdr_fc_zero_rows_existszeroevaluationdeterminanths)) /\ exists ff_q_mce_mdr_zero_rows_existszeroevaluationdeterminanthsf_prefix_bn. mdr_fb_zero_rows_existszeroevaluationdeterminanths = ff_q_mce_mdr_zero_rows_existszeroevaluationdeterminanthsf_prefix_bn * S ((S (ff_index_mce_alternating_mdr_zero_rows_existszeroevaluationdeterminanthsf_prefix)) * mdr_fc_zero_rows_existszeroevaluationdeterminanths) + (ff_bn_mce_alternating_mdr_zero_rows_existszeroevaluationdeterminanthsf_prefix))) /\ ((((exists ff_h_mce_mdr_zero_rows_existszeroevaluationdeterminanthsf_prefix_positive. ff_h_mce_mdr_zero_rows_existszeroevaluationdeterminanthsf_prefix_positive + S (ff_p_mce_alternating_mdr_zero_rows_existszeroevaluationdeterminanthsf_prefix) = S ((S (ff_index_mce_alternating_mdr_zero_rows_existszeroevaluationdeterminanthsf_prefix)) * ff_uc_mce_fold_mdr_zero_rows_existszeroevaluationdeterminanthsf)) /\ exists ff_q_mce_mdr_zero_rows_existszeroevaluationdeterminanthsf_prefix_positive. ff_ub_mce_fold_mdr_zero_rows_existszeroevaluationdeterminanthsf = ff_q_mce_mdr_zero_rows_existszeroevaluationdeterminanthsf_prefix_positive * S ((S (ff_index_mce_alternating_mdr_zero_rows_existszeroevaluationdeterminanthsf_prefix)) * ff_uc_mce_fold_mdr_zero_rows_existszeroevaluationdeterminanthsf) + (ff_p_mce_alternating_mdr_zero_rows_existszeroevaluationdeterminanthsf_prefix))) /\ ((((exists ff_h_mce_mdr_zero_rows_existszeroevaluationdeterminanthsf_prefix_negative. ff_h_mce_mdr_zero_rows_existszeroevaluationdeterminanthsf_prefix_negative + S (ff_n_mce_alternating_mdr_zero_rows_existszeroevaluationdeterminanthsf_prefix) = S ((S (ff_index_mce_alternating_mdr_zero_rows_existszeroevaluationdeterminanthsf_prefix)) * ff_vc_mce_fold_mdr_zero_rows_existszeroevaluationdeterminanthsf)) /\ exists ff_q_mce_mdr_zero_rows_existszeroevaluationdeterminanthsf_prefix_negative. ff_vb_mce_fold_mdr_zero_rows_existszeroevaluationdeterminanthsf = ff_q_mce_mdr_zero_rows_existszeroevaluationdeterminanthsf_prefix_negative * S ((S (ff_index_mce_alternating_mdr_zero_rows_existszeroevaluationdeterminanthsf_prefix)) * ff_vc_mce_fold_mdr_zero_rows_existszeroevaluationdeterminanthsf) + (ff_n_mce_alternating_mdr_zero_rows_existszeroevaluationdeterminanthsf_prefix))) /\ (((exists ff_even_mce_term_mdr_zero_rows_existszeroevaluationdeterminanthsf_prefix_term. ff_index_mce_alternating_mdr_zero_rows_existszeroevaluationdeterminanthsf_prefix = 2 * ff_even_mce_term_mdr_zero_rows_existszeroevaluationdeterminanthsf_prefix_term) /\ (ff_p_mce_alternating_mdr_zero_rows_existszeroevaluationdeterminanthsf_prefix = (ff_ap_mce_alternating_mdr_zero_rows_existszeroevaluationdeterminanthsf_prefix) * (ff_bp_mce_alternating_mdr_zero_rows_existszeroevaluationdeterminanthsf_prefix) + (ff_an_mce_alternating_mdr_zero_rows_existszeroevaluationdeterminanthsf_prefix) * (ff_bn_mce_alternating_mdr_zero_rows_existszeroevaluationdeterminanthsf_prefix) /\ ff_n_mce_alternating_mdr_zero_rows_existszeroevaluationdeterminanthsf_prefix = (ff_ap_mce_alternating_mdr_zero_rows_existszeroevaluationdeterminanthsf_prefix) * (ff_bn_mce_alternating_mdr_zero_rows_existszeroevaluationdeterminanthsf_prefix) + (ff_an_mce_alternating_mdr_zero_rows_existszeroevaluationdeterminanthsf_prefix) * (ff_bp_mce_alternating_mdr_zero_rows_existszeroevaluationdeterminanthsf_prefix))) \/ ((exists ff_odd_mce_term_mdr_zero_rows_existszeroevaluationdeterminanthsf_prefix_term. ff_index_mce_alternating_mdr_zero_rows_existszeroevaluationdeterminanthsf_prefix = 2 * ff_odd_mce_term_mdr_zero_rows_existszeroevaluationdeterminanthsf_prefix_term + 1) /\ (ff_p_mce_alternating_mdr_zero_rows_existszeroevaluationdeterminanthsf_prefix = (ff_ap_mce_alternating_mdr_zero_rows_existszeroevaluationdeterminanthsf_prefix) * (ff_bn_mce_alternating_mdr_zero_rows_existszeroevaluationdeterminanthsf_prefix) + (ff_an_mce_alternating_mdr_zero_rows_existszeroevaluationdeterminanthsf_prefix) * (ff_bp_mce_alternating_mdr_zero_rows_existszeroevaluationdeterminanthsf_prefix) /\ ff_n_mce_alternating_mdr_zero_rows_existszeroevaluationdeterminanthsf_prefix = (ff_ap_mce_alternating_mdr_zero_rows_existszeroevaluationdeterminanthsf_prefix) * (ff_bp_mce_alternating_mdr_zero_rows_existszeroevaluationdeterminanthsf_prefix) + (ff_an_mce_alternating_mdr_zero_rows_existszeroevaluationdeterminanthsf_prefix) * (ff_bn_mce_alternating_mdr_zero_rows_existszeroevaluationdeterminanthsf_prefix))))))))))) /\ ((exists ff_u_mce_mdr_zero_rows_existszeroevaluationdeterminanthsf_positive ff_v_mce_mdr_zero_rows_existszeroevaluationdeterminanthsf_positive. ((((exists ff_h_mce_mdr_zero_rows_existszeroevaluationdeterminanthsf_positive_start. ff_h_mce_mdr_zero_rows_existszeroevaluationdeterminanthsf_positive_start + S (0) = S ((S (0)) * ff_v_mce_mdr_zero_rows_existszeroevaluationdeterminanthsf_positive)) /\ exists ff_q_mce_mdr_zero_rows_existszeroevaluationdeterminanthsf_positive_start. ff_u_mce_mdr_zero_rows_existszeroevaluationdeterminanthsf_positive = ff_q_mce_mdr_zero_rows_existszeroevaluationdeterminanthsf_positive_start * S ((S (0)) * ff_v_mce_mdr_zero_rows_existszeroevaluationdeterminanthsf_positive) + (0))) /\ ((((exists ff_h_mce_mdr_zero_rows_existszeroevaluationdeterminanthsf_positive_terminal. ff_h_mce_mdr_zero_rows_existszeroevaluationdeterminanthsf_positive_terminal + S (mdr_p_zero_rows_existszeroevaluationdeterminanth) = S ((S ((S (mdr_q_zero_rows_existszeroevaluationdeterminanths)))) * ff_v_mce_mdr_zero_rows_existszeroevaluationdeterminanthsf_positive)) /\ exists ff_q_mce_mdr_zero_rows_existszeroevaluationdeterminanthsf_positive_terminal. ff_u_mce_mdr_zero_rows_existszeroevaluationdeterminanthsf_positive = ff_q_mce_mdr_zero_rows_existszeroevaluationdeterminanthsf_positive_terminal * S ((S ((S (mdr_q_zero_rows_existszeroevaluationdeterminanths)))) * ff_v_mce_mdr_zero_rows_existszeroevaluationdeterminanthsf_positive) + (mdr_p_zero_rows_existszeroevaluationdeterminanth))) /\ forall ff_i_mce_mdr_zero_rows_existszeroevaluationdeterminanthsf_positive. (exists ff_lt_mce_mdr_zero_rows_existszeroevaluationdeterminanthsf_positive_bound. ff_lt_mce_mdr_zero_rows_existszeroevaluationdeterminanthsf_positive_bound + S ff_i_mce_mdr_zero_rows_existszeroevaluationdeterminanthsf_positive = (S (mdr_q_zero_rows_existszeroevaluationdeterminanths))) -> exists ff_a_mce_mdr_zero_rows_existszeroevaluationdeterminanthsf_positive ff_r_mce_mdr_zero_rows_existszeroevaluationdeterminanthsf_positive ff_s_mce_mdr_zero_rows_existszeroevaluationdeterminanthsf_positive. ((((exists ff_h_mce_mdr_zero_rows_existszeroevaluationdeterminanthsf_positive_summand. ff_h_mce_mdr_zero_rows_existszeroevaluationdeterminanthsf_positive_summand + S (ff_a_mce_mdr_zero_rows_existszeroevaluationdeterminanthsf_positive) = S ((S (ff_i_mce_mdr_zero_rows_existszeroevaluationdeterminanthsf_positive)) * ff_uc_mce_fold_mdr_zero_rows_existszeroevaluationdeterminanthsf)) /\ exists ff_q_mce_mdr_zero_rows_existszeroevaluationdeterminanthsf_positive_summand. ff_ub_mce_fold_mdr_zero_rows_existszeroevaluationdeterminanthsf = ff_q_mce_mdr_zero_rows_existszeroevaluationdeterminanthsf_positive_summand * S ((S (ff_i_mce_mdr_zero_rows_existszeroevaluationdeterminanthsf_positive)) * ff_uc_mce_fold_mdr_zero_rows_existszeroevaluationdeterminanthsf) + (ff_a_mce_mdr_zero_rows_existszeroevaluationdeterminanthsf_positive))) /\ ((((exists ff_h_mce_mdr_zero_rows_existszeroevaluationdeterminanthsf_positive_partial. ff_h_mce_mdr_zero_rows_existszeroevaluationdeterminanthsf_positive_partial + S (ff_r_mce_mdr_zero_rows_existszeroevaluationdeterminanthsf_positive) = S ((S (ff_i_mce_mdr_zero_rows_existszeroevaluationdeterminanthsf_positive)) * ff_v_mce_mdr_zero_rows_existszeroevaluationdeterminanthsf_positive)) /\ exists ff_q_mce_mdr_zero_rows_existszeroevaluationdeterminanthsf_positive_partial. ff_u_mce_mdr_zero_rows_existszeroevaluationdeterminanthsf_positive = ff_q_mce_mdr_zero_rows_existszeroevaluationdeterminanthsf_positive_partial * S ((S (ff_i_mce_mdr_zero_rows_existszeroevaluationdeterminanthsf_positive)) * ff_v_mce_mdr_zero_rows_existszeroevaluationdeterminanthsf_positive) + (ff_r_mce_mdr_zero_rows_existszeroevaluationdeterminanthsf_positive))) /\ ((((exists ff_h_mce_mdr_zero_rows_existszeroevaluationdeterminanthsf_positive_successor. ff_h_mce_mdr_zero_rows_existszeroevaluationdeterminanthsf_positive_successor + S (ff_s_mce_mdr_zero_rows_existszeroevaluationdeterminanthsf_positive) = S ((S (S ff_i_mce_mdr_zero_rows_existszeroevaluationdeterminanthsf_positive)) * ff_v_mce_mdr_zero_rows_existszeroevaluationdeterminanthsf_positive)) /\ exists ff_q_mce_mdr_zero_rows_existszeroevaluationdeterminanthsf_positive_successor. ff_u_mce_mdr_zero_rows_existszeroevaluationdeterminanthsf_positive = ff_q_mce_mdr_zero_rows_existszeroevaluationdeterminanthsf_positive_successor * S ((S (S ff_i_mce_mdr_zero_rows_existszeroevaluationdeterminanthsf_positive)) * ff_v_mce_mdr_zero_rows_existszeroevaluationdeterminanthsf_positive) + (ff_s_mce_mdr_zero_rows_existszeroevaluationdeterminanthsf_positive))) /\ ff_s_mce_mdr_zero_rows_existszeroevaluationdeterminanthsf_positive = ff_r_mce_mdr_zero_rows_existszeroevaluationdeterminanthsf_positive + ff_a_mce_mdr_zero_rows_existszeroevaluationdeterminanthsf_positive)))))) /\ (exists ff_u_mce_mdr_zero_rows_existszeroevaluationdeterminanthsf_negative ff_v_mce_mdr_zero_rows_existszeroevaluationdeterminanthsf_negative. ((((exists ff_h_mce_mdr_zero_rows_existszeroevaluationdeterminanthsf_negative_start. ff_h_mce_mdr_zero_rows_existszeroevaluationdeterminanthsf_negative_start + S (0) = S ((S (0)) * ff_v_mce_mdr_zero_rows_existszeroevaluationdeterminanthsf_negative)) /\ exists ff_q_mce_mdr_zero_rows_existszeroevaluationdeterminanthsf_negative_start. ff_u_mce_mdr_zero_rows_existszeroevaluationdeterminanthsf_negative = ff_q_mce_mdr_zero_rows_existszeroevaluationdeterminanthsf_negative_start * S ((S (0)) * ff_v_mce_mdr_zero_rows_existszeroevaluationdeterminanthsf_negative) + (0))) /\ ((((exists ff_h_mce_mdr_zero_rows_existszeroevaluationdeterminanthsf_negative_terminal. ff_h_mce_mdr_zero_rows_existszeroevaluationdeterminanthsf_negative_terminal + S (mdr_n_zero_rows_existszeroevaluationdeterminanth) = S ((S ((S (mdr_q_zero_rows_existszeroevaluationdeterminanths)))) * ff_v_mce_mdr_zero_rows_existszeroevaluationdeterminanthsf_negative)) /\ exists ff_q_mce_mdr_zero_rows_existszeroevaluationdeterminanthsf_negative_terminal. ff_u_mce_mdr_zero_rows_existszeroevaluationdeterminanthsf_negative = ff_q_mce_mdr_zero_rows_existszeroevaluationdeterminanthsf_negative_terminal * S ((S ((S (mdr_q_zero_rows_existszeroevaluationdeterminanths)))) * ff_v_mce_mdr_zero_rows_existszeroevaluationdeterminanthsf_negative) + (mdr_n_zero_rows_existszeroevaluationdeterminanth))) /\ forall ff_i_mce_mdr_zero_rows_existszeroevaluationdeterminanthsf_negative. (exists ff_lt_mce_mdr_zero_rows_existszeroevaluationdeterminanthsf_negative_bound. ff_lt_mce_mdr_zero_rows_existszeroevaluationdeterminanthsf_negative_bound + S ff_i_mce_mdr_zero_rows_existszeroevaluationdeterminanthsf_negative = (S (mdr_q_zero_rows_existszeroevaluationdeterminanths))) -> exists ff_a_mce_mdr_zero_rows_existszeroevaluationdeterminanthsf_negative ff_r_mce_mdr_zero_rows_existszeroevaluationdeterminanthsf_negative ff_s_mce_mdr_zero_rows_existszeroevaluationdeterminanthsf_negative. ((((exists ff_h_mce_mdr_zero_rows_existszeroevaluationdeterminanthsf_negative_summand. ff_h_mce_mdr_zero_rows_existszeroevaluationdeterminanthsf_negative_summand + S (ff_a_mce_mdr_zero_rows_existszeroevaluationdeterminanthsf_negative) = S ((S (ff_i_mce_mdr_zero_rows_existszeroevaluationdeterminanthsf_negative)) * ff_vc_mce_fold_mdr_zero_rows_existszeroevaluationdeterminanthsf)) /\ exists ff_q_mce_mdr_zero_rows_existszeroevaluationdeterminanthsf_negative_summand. ff_vb_mce_fold_mdr_zero_rows_existszeroevaluationdeterminanthsf = ff_q_mce_mdr_zero_rows_existszeroevaluationdeterminanthsf_negative_summand * S ((S (ff_i_mce_mdr_zero_rows_existszeroevaluationdeterminanthsf_negative)) * ff_vc_mce_fold_mdr_zero_rows_existszeroevaluationdeterminanthsf) + (ff_a_mce_mdr_zero_rows_existszeroevaluationdeterminanthsf_negative))) /\ ((((exists ff_h_mce_mdr_zero_rows_existszeroevaluationdeterminanthsf_negative_partial. ff_h_mce_mdr_zero_rows_existszeroevaluationdeterminanthsf_negative_partial + S (ff_r_mce_mdr_zero_rows_existszeroevaluationdeterminanthsf_negative) = S ((S (ff_i_mce_mdr_zero_rows_existszeroevaluationdeterminanthsf_negative)) * ff_v_mce_mdr_zero_rows_existszeroevaluationdeterminanthsf_negative)) /\ exists ff_q_mce_mdr_zero_rows_existszeroevaluationdeterminanthsf_negative_partial. ff_u_mce_mdr_zero_rows_existszeroevaluationdeterminanthsf_negative = ff_q_mce_mdr_zero_rows_existszeroevaluationdeterminanthsf_negative_partial * S ((S (ff_i_mce_mdr_zero_rows_existszeroevaluationdeterminanthsf_negative)) * ff_v_mce_mdr_zero_rows_existszeroevaluationdeterminanthsf_negative) + (ff_r_mce_mdr_zero_rows_existszeroevaluationdeterminanthsf_negative))) /\ ((((exists ff_h_mce_mdr_zero_rows_existszeroevaluationdeterminanthsf_negative_successor. ff_h_mce_mdr_zero_rows_existszeroevaluationdeterminanthsf_negative_successor + S (ff_s_mce_mdr_zero_rows_existszeroevaluationdeterminanthsf_negative) = S ((S (S ff_i_mce_mdr_zero_rows_existszeroevaluationdeterminanthsf_negative)) * ff_v_mce_mdr_zero_rows_existszeroevaluationdeterminanthsf_negative)) /\ exists ff_q_mce_mdr_zero_rows_existszeroevaluationdeterminanthsf_negative_successor. ff_u_mce_mdr_zero_rows_existszeroevaluationdeterminanthsf_negative = ff_q_mce_mdr_zero_rows_existszeroevaluationdeterminanthsf_negative_successor * S ((S (S ff_i_mce_mdr_zero_rows_existszeroevaluationdeterminanthsf_negative)) * ff_v_mce_mdr_zero_rows_existszeroevaluationdeterminanthsf_negative) + (ff_s_mce_mdr_zero_rows_existszeroevaluationdeterminanthsf_negative))) /\ ff_s_mce_mdr_zero_rows_existszeroevaluationdeterminanthsf_negative = ff_r_mce_mdr_zero_rows_existszeroevaluationdeterminanthsf_negative + ff_a_mce_mdr_zero_rows_existszeroevaluationdeterminanthsf_negative))))))))))))))) /\ ((exists mdr_gap_zero_rows_existszeroevaluationdeterminanti. mdr_gap_zero_rows_existszeroevaluationdeterminanti + S (mdr_i_zero_rows_existszeroevaluationdeterminant) = (mdr_l_zero_rows_existszeroevaluationdeterminant)) /\ (exists mdr_z_zero_rows_existszeroevaluationdeterminantr. ((exists mdr_a_zero_rows_existszeroevaluationdeterminantrc mdr_b_zero_rows_existszeroevaluationdeterminantrc mdr_c_zero_rows_existszeroevaluationdeterminantrc mdr_e_zero_rows_existszeroevaluationdeterminantrc mdr_f_zero_rows_existszeroevaluationdeterminantrc. ((mdr_a_zero_rows_existszeroevaluationdeterminantrc = ((mdr_q_zero_rows_exists) + (mdr_ub_zero_rows_existszeroevaluation)) * S ((mdr_q_zero_rows_exists) + (mdr_ub_zero_rows_existszeroevaluation)) + ((mdr_ub_zero_rows_existszeroevaluation) + (mdr_ub_zero_rows_existszeroevaluation))) /\ ((mdr_b_zero_rows_existszeroevaluationdeterminantrc = ((mdr_uc_zero_rows_existszeroevaluation) + (mdr_vb_zero_rows_existszeroevaluation)) * S ((mdr_uc_zero_rows_existszeroevaluation) + (mdr_vb_zero_rows_existszeroevaluation)) + ((mdr_vb_zero_rows_existszeroevaluation) + (mdr_vb_zero_rows_existszeroevaluation))) /\ ((mdr_c_zero_rows_existszeroevaluationdeterminantrc = ((mdr_a_zero_rows_existszeroevaluationdeterminantrc) + (mdr_b_zero_rows_existszeroevaluationdeterminantrc)) * S ((mdr_a_zero_rows_existszeroevaluationdeterminantrc) + (mdr_b_zero_rows_existszeroevaluationdeterminantrc)) + ((mdr_b_zero_rows_existszeroevaluationdeterminantrc) + (mdr_b_zero_rows_existszeroevaluationdeterminantrc))) /\ ((mdr_e_zero_rows_existszeroevaluationdeterminantrc = ((mdr_p_zero_rows_existszero) + (mdr_n_zero_rows_existszero)) * S ((mdr_p_zero_rows_existszero) + (mdr_n_zero_rows_existszero)) + ((mdr_n_zero_rows_existszero) + (mdr_n_zero_rows_existszero))) /\ ((mdr_f_zero_rows_existszeroevaluationdeterminantrc = ((mdr_vc_zero_rows_existszeroevaluation) + (mdr_e_zero_rows_existszeroevaluationdeterminantrc)) * S ((mdr_vc_zero_rows_existszeroevaluation) + (mdr_e_zero_rows_existszeroevaluationdeterminantrc)) + ((mdr_e_zero_rows_existszeroevaluationdeterminantrc) + (mdr_e_zero_rows_existszeroevaluationdeterminantrc))) /\ ((mdr_z_zero_rows_existszeroevaluationdeterminantr) = ((mdr_c_zero_rows_existszeroevaluationdeterminantrc) + (mdr_f_zero_rows_existszeroevaluationdeterminantrc)) * S ((mdr_c_zero_rows_existszeroevaluationdeterminantrc) + (mdr_f_zero_rows_existszeroevaluationdeterminantrc)) + ((mdr_f_zero_rows_existszeroevaluationdeterminantrc) + (mdr_f_zero_rows_existszeroevaluationdeterminantrc))))))))) /\ (((exists ff_h_mdr_zero_rows_existszeroevaluationdeterminantrb. ff_h_mdr_zero_rows_existszeroevaluationdeterminantrb + S (mdr_z_zero_rows_existszeroevaluationdeterminantr) = S ((S (mdr_i_zero_rows_existszeroevaluationdeterminant)) * mdr_c_zero_rows_existszeroevaluationdeterminant)) /\ exists ff_q_mdr_zero_rows_existszeroevaluationdeterminantrb. mdr_b_zero_rows_existszeroevaluationdeterminant = ff_q_mdr_zero_rows_existszeroevaluationdeterminantrb * S ((S (mdr_i_zero_rows_existszeroevaluationdeterminant)) * mdr_c_zero_rows_existszeroevaluationdeterminant) + (mdr_z_zero_rows_existszeroevaluationdeterminantr)))))))))) -> mdr_p_zero_rows_existszero = mdr_n_zero_rows_existszero)))))) - 0007
specialize rectangular_matrix_rank_certificate_exists (pb) - 0008
specialize rectangular_matrix_rank_certificate_exists (pc) - 0009
specialize rectangular_matrix_rank_certificate_exists (nb) - 0010
specialize rectangular_matrix_rank_certificate_exists (nc) - 0011
specialize rectangular_matrix_rank_certificate_exists (0) - 0012
specialize rectangular_matrix_rank_certificate_exists (w) - 0013
apply rectangular_matrix_rank_certificate_exists - 0014
cases hrank - 0015
cases hrank_witness - 0016
have hzero : x = 0 - 0017
specialize le_zero (x) - 0018
apply le_zero - 0019
exact hrank_witness_left - 0020
rewrite hzero at hrank_witness - 0021
rewrite hzero at hrank_witness - 0022
rewrite hzero at hrank_witness - 0023
rewrite hzero at hrank_witness - 0024
rewrite hzero at hrank_witness - 0025
rewrite hzero at hrank_witness - 0026
rewrite hzero at hrank_witness - 0027
rewrite hzero at hrank_witness - 0028
rewrite hzero at hrank_witness - 0029
rewrite hzero at hrank_witness - 0030
rewrite hzero at hrank_witness - 0031
rewrite hzero at hrank_witness - 0032
rewrite hzero at hrank_witness - 0033
rewrite hzero at hrank_witness - 0034
rewrite hzero at hrank_witness - 0035
rewrite hzero at hrank_witness - 0036
rewrite hzero at hrank_witness - 0037
rewrite hzero at hrank_witness - 0038
rewrite hzero at hrank_witness - 0039
exact hrank_witness