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 r. (((exists mdr_gap_zero_columns_rankrows_bound. mdr_gap_zero_columns_rankrows_bound + (0) = (r)) /\ ((exists mdr_gap_zero_columns_rankcolumns_bound. mdr_gap_zero_columns_rankcolumns_bound + (0) = (0)) /\ ((exists mdr_rb_zero_columns_rankwitness mdr_rc_zero_columns_rankwitness mdr_cb_zero_columns_rankwitness mdr_cc_zero_columns_rankwitness. (((((forall fom_index_mrf_zero_columns_rankwitnessminorrowsbound. (exists fom_gap_mrf_zero_columns_rankwitnessminorrowsbound_index_bound. fom_gap_mrf_zero_columns_rankwitnessminorrowsbound_index_bound + S (fom_index_mrf_zero_columns_rankwitnessminorrowsbound) = 0) -> exists fom_value_mrf_zero_columns_rankwitnessminorrowsbound. ((((exists fom_beta_height_mrf_zero_columns_rankwitnessminorrowsbound_entry. fom_beta_height_mrf_zero_columns_rankwitnessminorrowsbound_entry + S (fom_value_mrf_zero_columns_rankwitnessminorrowsbound) = S ((S (fom_index_mrf_zero_columns_rankwitnessminorrowsbound)) * mdr_rc_zero_columns_rankwitness)) /\ exists fom_beta_quotient_mrf_zero_columns_rankwitnessminorrowsbound_entry. mdr_rb_zero_columns_rankwitness = fom_beta_quotient_mrf_zero_columns_rankwitnessminorrowsbound_entry * S ((S (fom_index_mrf_zero_columns_rankwitnessminorrowsbound)) * mdr_rc_zero_columns_rankwitness) + (fom_value_mrf_zero_columns_rankwitnessminorrowsbound))) /\ (exists fom_gap_mrf_zero_columns_rankwitnessminorrowsbound_value_bound. fom_gap_mrf_zero_columns_rankwitnessminorrowsbound_value_bound + S (fom_value_mrf_zero_columns_rankwitnessminorrowsbound) = r))) /\ (forall mdr_i_zero_columns_rankwitnessminorrowsdistinct mdr_j_zero_columns_rankwitnessminorrowsdistinct mdr_a_zero_columns_rankwitnessminorrowsdistinct. (exists mdr_gap_zero_columns_rankwitnessminorrowsdistincti. mdr_gap_zero_columns_rankwitnessminorrowsdistincti + S (mdr_i_zero_columns_rankwitnessminorrowsdistinct) = (0)) -> (exists mdr_gap_zero_columns_rankwitnessminorrowsdistinctj. mdr_gap_zero_columns_rankwitnessminorrowsdistinctj + S (mdr_j_zero_columns_rankwitnessminorrowsdistinct) = (0)) -> (((exists ff_h_mdr_zero_columns_rankwitnessminorrowsdistinctfirst. ff_h_mdr_zero_columns_rankwitnessminorrowsdistinctfirst + S (mdr_a_zero_columns_rankwitnessminorrowsdistinct) = S ((S (mdr_i_zero_columns_rankwitnessminorrowsdistinct)) * mdr_rc_zero_columns_rankwitness)) /\ exists ff_q_mdr_zero_columns_rankwitnessminorrowsdistinctfirst. mdr_rb_zero_columns_rankwitness = ff_q_mdr_zero_columns_rankwitnessminorrowsdistinctfirst * S ((S (mdr_i_zero_columns_rankwitnessminorrowsdistinct)) * mdr_rc_zero_columns_rankwitness) + (mdr_a_zero_columns_rankwitnessminorrowsdistinct))) -> (((exists ff_h_mdr_zero_columns_rankwitnessminorrowsdistinctsecond. ff_h_mdr_zero_columns_rankwitnessminorrowsdistinctsecond + S (mdr_a_zero_columns_rankwitnessminorrowsdistinct) = S ((S (mdr_j_zero_columns_rankwitnessminorrowsdistinct)) * mdr_rc_zero_columns_rankwitness)) /\ exists ff_q_mdr_zero_columns_rankwitnessminorrowsdistinctsecond. mdr_rb_zero_columns_rankwitness = ff_q_mdr_zero_columns_rankwitnessminorrowsdistinctsecond * S ((S (mdr_j_zero_columns_rankwitnessminorrowsdistinct)) * mdr_rc_zero_columns_rankwitness) + (mdr_a_zero_columns_rankwitnessminorrowsdistinct))) -> mdr_i_zero_columns_rankwitnessminorrowsdistinct = mdr_j_zero_columns_rankwitnessminorrowsdistinct))) /\ ((((forall fom_index_mrf_zero_columns_rankwitnessminorcolumnsbound. (exists fom_gap_mrf_zero_columns_rankwitnessminorcolumnsbound_index_bound. fom_gap_mrf_zero_columns_rankwitnessminorcolumnsbound_index_bound + S (fom_index_mrf_zero_columns_rankwitnessminorcolumnsbound) = 0) -> exists fom_value_mrf_zero_columns_rankwitnessminorcolumnsbound. ((((exists fom_beta_height_mrf_zero_columns_rankwitnessminorcolumnsbound_entry. fom_beta_height_mrf_zero_columns_rankwitnessminorcolumnsbound_entry + S (fom_value_mrf_zero_columns_rankwitnessminorcolumnsbound) = S ((S (fom_index_mrf_zero_columns_rankwitnessminorcolumnsbound)) * mdr_cc_zero_columns_rankwitness)) /\ exists fom_beta_quotient_mrf_zero_columns_rankwitnessminorcolumnsbound_entry. mdr_cb_zero_columns_rankwitness = fom_beta_quotient_mrf_zero_columns_rankwitnessminorcolumnsbound_entry * S ((S (fom_index_mrf_zero_columns_rankwitnessminorcolumnsbound)) * mdr_cc_zero_columns_rankwitness) + (fom_value_mrf_zero_columns_rankwitnessminorcolumnsbound))) /\ (exists fom_gap_mrf_zero_columns_rankwitnessminorcolumnsbound_value_bound. fom_gap_mrf_zero_columns_rankwitnessminorcolumnsbound_value_bound + S (fom_value_mrf_zero_columns_rankwitnessminorcolumnsbound) = 0))) /\ (forall mdr_i_zero_columns_rankwitnessminorcolumnsdistinct mdr_j_zero_columns_rankwitnessminorcolumnsdistinct mdr_a_zero_columns_rankwitnessminorcolumnsdistinct. (exists mdr_gap_zero_columns_rankwitnessminorcolumnsdistincti. mdr_gap_zero_columns_rankwitnessminorcolumnsdistincti + S (mdr_i_zero_columns_rankwitnessminorcolumnsdistinct) = (0)) -> (exists mdr_gap_zero_columns_rankwitnessminorcolumnsdistinctj. mdr_gap_zero_columns_rankwitnessminorcolumnsdistinctj + S (mdr_j_zero_columns_rankwitnessminorcolumnsdistinct) = (0)) -> (((exists ff_h_mdr_zero_columns_rankwitnessminorcolumnsdistinctfirst. ff_h_mdr_zero_columns_rankwitnessminorcolumnsdistinctfirst + S (mdr_a_zero_columns_rankwitnessminorcolumnsdistinct) = S ((S (mdr_i_zero_columns_rankwitnessminorcolumnsdistinct)) * mdr_cc_zero_columns_rankwitness)) /\ exists ff_q_mdr_zero_columns_rankwitnessminorcolumnsdistinctfirst. mdr_cb_zero_columns_rankwitness = ff_q_mdr_zero_columns_rankwitnessminorcolumnsdistinctfirst * S ((S (mdr_i_zero_columns_rankwitnessminorcolumnsdistinct)) * mdr_cc_zero_columns_rankwitness) + (mdr_a_zero_columns_rankwitnessminorcolumnsdistinct))) -> (((exists ff_h_mdr_zero_columns_rankwitnessminorcolumnsdistinctsecond. ff_h_mdr_zero_columns_rankwitnessminorcolumnsdistinctsecond + S (mdr_a_zero_columns_rankwitnessminorcolumnsdistinct) = S ((S (mdr_j_zero_columns_rankwitnessminorcolumnsdistinct)) * mdr_cc_zero_columns_rankwitness)) /\ exists ff_q_mdr_zero_columns_rankwitnessminorcolumnsdistinctsecond. mdr_cb_zero_columns_rankwitness = ff_q_mdr_zero_columns_rankwitnessminorcolumnsdistinctsecond * S ((S (mdr_j_zero_columns_rankwitnessminorcolumnsdistinct)) * mdr_cc_zero_columns_rankwitness) + (mdr_a_zero_columns_rankwitnessminorcolumnsdistinct))) -> mdr_i_zero_columns_rankwitnessminorcolumnsdistinct = mdr_j_zero_columns_rankwitnessminorcolumnsdistinct))) /\ (exists mdr_p_zero_columns_rankwitnessminornonzero mdr_n_zero_columns_rankwitnessminornonzero. ((exists mdr_ub_zero_columns_rankwitnessminornonzeroevaluation mdr_uc_zero_columns_rankwitnessminornonzeroevaluation mdr_vb_zero_columns_rankwitnessminornonzeroevaluation mdr_vc_zero_columns_rankwitnessminornonzeroevaluation. ((((forall mdr_i_zero_columns_rankwitnessminornonzeroevaluationmatrixpositive. (exists mdr_gap_zero_columns_rankwitnessminornonzeroevaluationmatrixpositivebound. mdr_gap_zero_columns_rankwitnessminornonzeroevaluationmatrixpositivebound + S (mdr_i_zero_columns_rankwitnessminornonzeroevaluationmatrixpositive) = ((0) * (0))) -> exists mdr_a_zero_columns_rankwitnessminornonzeroevaluationmatrixpositive. (((exists mdr_r_zero_columns_rankwitnessminornonzeroevaluationmatrixpositivepoint mdr_s_zero_columns_rankwitnessminornonzeroevaluationmatrixpositivepoint mdr_u_zero_columns_rankwitnessminornonzeroevaluationmatrixpositivepoint mdr_v_zero_columns_rankwitnessminornonzeroevaluationmatrixpositivepoint. ((mdr_i_zero_columns_rankwitnessminornonzeroevaluationmatrixpositive = (0) * mdr_r_zero_columns_rankwitnessminornonzeroevaluationmatrixpositivepoint + mdr_s_zero_columns_rankwitnessminornonzeroevaluationmatrixpositivepoint) /\ ((exists mdr_gap_zero_columns_rankwitnessminornonzeroevaluationmatrixpositivepointcolumn. mdr_gap_zero_columns_rankwitnessminornonzeroevaluationmatrixpositivepointcolumn + S (mdr_s_zero_columns_rankwitnessminornonzeroevaluationmatrixpositivepoint) = (0)) /\ ((((exists ff_h_mdr_zero_columns_rankwitnessminornonzeroevaluationmatrixpositivepointrow_index. ff_h_mdr_zero_columns_rankwitnessminornonzeroevaluationmatrixpositivepointrow_index + S (mdr_u_zero_columns_rankwitnessminornonzeroevaluationmatrixpositivepoint) = S ((S (mdr_r_zero_columns_rankwitnessminornonzeroevaluationmatrixpositivepoint)) * mdr_rc_zero_columns_rankwitness)) /\ exists ff_q_mdr_zero_columns_rankwitnessminornonzeroevaluationmatrixpositivepointrow_index. mdr_rb_zero_columns_rankwitness = ff_q_mdr_zero_columns_rankwitnessminornonzeroevaluationmatrixpositivepointrow_index * S ((S (mdr_r_zero_columns_rankwitnessminornonzeroevaluationmatrixpositivepoint)) * mdr_rc_zero_columns_rankwitness) + (mdr_u_zero_columns_rankwitnessminornonzeroevaluationmatrixpositivepoint))) /\ ((((exists ff_h_mdr_zero_columns_rankwitnessminornonzeroevaluationmatrixpositivepointcolumn_index. ff_h_mdr_zero_columns_rankwitnessminornonzeroevaluationmatrixpositivepointcolumn_index + S (mdr_v_zero_columns_rankwitnessminornonzeroevaluationmatrixpositivepoint) = S ((S (mdr_s_zero_columns_rankwitnessminornonzeroevaluationmatrixpositivepoint)) * mdr_cc_zero_columns_rankwitness)) /\ exists ff_q_mdr_zero_columns_rankwitnessminornonzeroevaluationmatrixpositivepointcolumn_index. mdr_cb_zero_columns_rankwitness = ff_q_mdr_zero_columns_rankwitnessminornonzeroevaluationmatrixpositivepointcolumn_index * S ((S (mdr_s_zero_columns_rankwitnessminornonzeroevaluationmatrixpositivepoint)) * mdr_cc_zero_columns_rankwitness) + (mdr_v_zero_columns_rankwitnessminornonzeroevaluationmatrixpositivepoint))) /\ (((exists ff_h_mdr_zero_columns_rankwitnessminornonzeroevaluationmatrixpositivepointsource. ff_h_mdr_zero_columns_rankwitnessminornonzeroevaluationmatrixpositivepointsource + S (mdr_a_zero_columns_rankwitnessminornonzeroevaluationmatrixpositive) = S ((S ((mdr_u_zero_columns_rankwitnessminornonzeroevaluationmatrixpositivepoint) * (0) + (mdr_v_zero_columns_rankwitnessminornonzeroevaluationmatrixpositivepoint))) * pc)) /\ exists ff_q_mdr_zero_columns_rankwitnessminornonzeroevaluationmatrixpositivepointsource. pb = ff_q_mdr_zero_columns_rankwitnessminornonzeroevaluationmatrixpositivepointsource * S ((S ((mdr_u_zero_columns_rankwitnessminornonzeroevaluationmatrixpositivepoint) * (0) + (mdr_v_zero_columns_rankwitnessminornonzeroevaluationmatrixpositivepoint))) * pc) + (mdr_a_zero_columns_rankwitnessminornonzeroevaluationmatrixpositive)))))))) /\ (((exists ff_h_mdr_zero_columns_rankwitnessminornonzeroevaluationmatrixpositiveoutput. ff_h_mdr_zero_columns_rankwitnessminornonzeroevaluationmatrixpositiveoutput + S (mdr_a_zero_columns_rankwitnessminornonzeroevaluationmatrixpositive) = S ((S (mdr_i_zero_columns_rankwitnessminornonzeroevaluationmatrixpositive)) * mdr_uc_zero_columns_rankwitnessminornonzeroevaluation)) /\ exists ff_q_mdr_zero_columns_rankwitnessminornonzeroevaluationmatrixpositiveoutput. mdr_ub_zero_columns_rankwitnessminornonzeroevaluation = ff_q_mdr_zero_columns_rankwitnessminornonzeroevaluationmatrixpositiveoutput * S ((S (mdr_i_zero_columns_rankwitnessminornonzeroevaluationmatrixpositive)) * mdr_uc_zero_columns_rankwitnessminornonzeroevaluation) + (mdr_a_zero_columns_rankwitnessminornonzeroevaluationmatrixpositive)))))) /\ (forall mdr_i_zero_columns_rankwitnessminornonzeroevaluationmatrixnegative. (exists mdr_gap_zero_columns_rankwitnessminornonzeroevaluationmatrixnegativebound. mdr_gap_zero_columns_rankwitnessminornonzeroevaluationmatrixnegativebound + S (mdr_i_zero_columns_rankwitnessminornonzeroevaluationmatrixnegative) = ((0) * (0))) -> exists mdr_a_zero_columns_rankwitnessminornonzeroevaluationmatrixnegative. (((exists mdr_r_zero_columns_rankwitnessminornonzeroevaluationmatrixnegativepoint mdr_s_zero_columns_rankwitnessminornonzeroevaluationmatrixnegativepoint mdr_u_zero_columns_rankwitnessminornonzeroevaluationmatrixnegativepoint mdr_v_zero_columns_rankwitnessminornonzeroevaluationmatrixnegativepoint. ((mdr_i_zero_columns_rankwitnessminornonzeroevaluationmatrixnegative = (0) * mdr_r_zero_columns_rankwitnessminornonzeroevaluationmatrixnegativepoint + mdr_s_zero_columns_rankwitnessminornonzeroevaluationmatrixnegativepoint) /\ ((exists mdr_gap_zero_columns_rankwitnessminornonzeroevaluationmatrixnegativepointcolumn. mdr_gap_zero_columns_rankwitnessminornonzeroevaluationmatrixnegativepointcolumn + S (mdr_s_zero_columns_rankwitnessminornonzeroevaluationmatrixnegativepoint) = (0)) /\ ((((exists ff_h_mdr_zero_columns_rankwitnessminornonzeroevaluationmatrixnegativepointrow_index. ff_h_mdr_zero_columns_rankwitnessminornonzeroevaluationmatrixnegativepointrow_index + S (mdr_u_zero_columns_rankwitnessminornonzeroevaluationmatrixnegativepoint) = S ((S (mdr_r_zero_columns_rankwitnessminornonzeroevaluationmatrixnegativepoint)) * mdr_rc_zero_columns_rankwitness)) /\ exists ff_q_mdr_zero_columns_rankwitnessminornonzeroevaluationmatrixnegativepointrow_index. mdr_rb_zero_columns_rankwitness = ff_q_mdr_zero_columns_rankwitnessminornonzeroevaluationmatrixnegativepointrow_index * S ((S (mdr_r_zero_columns_rankwitnessminornonzeroevaluationmatrixnegativepoint)) * mdr_rc_zero_columns_rankwitness) + (mdr_u_zero_columns_rankwitnessminornonzeroevaluationmatrixnegativepoint))) /\ ((((exists ff_h_mdr_zero_columns_rankwitnessminornonzeroevaluationmatrixnegativepointcolumn_index. ff_h_mdr_zero_columns_rankwitnessminornonzeroevaluationmatrixnegativepointcolumn_index + S (mdr_v_zero_columns_rankwitnessminornonzeroevaluationmatrixnegativepoint) = S ((S (mdr_s_zero_columns_rankwitnessminornonzeroevaluationmatrixnegativepoint)) * mdr_cc_zero_columns_rankwitness)) /\ exists ff_q_mdr_zero_columns_rankwitnessminornonzeroevaluationmatrixnegativepointcolumn_index. mdr_cb_zero_columns_rankwitness = ff_q_mdr_zero_columns_rankwitnessminornonzeroevaluationmatrixnegativepointcolumn_index * S ((S (mdr_s_zero_columns_rankwitnessminornonzeroevaluationmatrixnegativepoint)) * mdr_cc_zero_columns_rankwitness) + (mdr_v_zero_columns_rankwitnessminornonzeroevaluationmatrixnegativepoint))) /\ (((exists ff_h_mdr_zero_columns_rankwitnessminornonzeroevaluationmatrixnegativepointsource. ff_h_mdr_zero_columns_rankwitnessminornonzeroevaluationmatrixnegativepointsource + S (mdr_a_zero_columns_rankwitnessminornonzeroevaluationmatrixnegative) = S ((S ((mdr_u_zero_columns_rankwitnessminornonzeroevaluationmatrixnegativepoint) * (0) + (mdr_v_zero_columns_rankwitnessminornonzeroevaluationmatrixnegativepoint))) * nc)) /\ exists ff_q_mdr_zero_columns_rankwitnessminornonzeroevaluationmatrixnegativepointsource. nb = ff_q_mdr_zero_columns_rankwitnessminornonzeroevaluationmatrixnegativepointsource * S ((S ((mdr_u_zero_columns_rankwitnessminornonzeroevaluationmatrixnegativepoint) * (0) + (mdr_v_zero_columns_rankwitnessminornonzeroevaluationmatrixnegativepoint))) * nc) + (mdr_a_zero_columns_rankwitnessminornonzeroevaluationmatrixnegative)))))))) /\ (((exists ff_h_mdr_zero_columns_rankwitnessminornonzeroevaluationmatrixnegativeoutput. ff_h_mdr_zero_columns_rankwitnessminornonzeroevaluationmatrixnegativeoutput + S (mdr_a_zero_columns_rankwitnessminornonzeroevaluationmatrixnegative) = S ((S (mdr_i_zero_columns_rankwitnessminornonzeroevaluationmatrixnegative)) * mdr_vc_zero_columns_rankwitnessminornonzeroevaluation)) /\ exists ff_q_mdr_zero_columns_rankwitnessminornonzeroevaluationmatrixnegativeoutput. mdr_vb_zero_columns_rankwitnessminornonzeroevaluation = ff_q_mdr_zero_columns_rankwitnessminornonzeroevaluationmatrixnegativeoutput * S ((S (mdr_i_zero_columns_rankwitnessminornonzeroevaluationmatrixnegative)) * mdr_vc_zero_columns_rankwitnessminornonzeroevaluation) + (mdr_a_zero_columns_rankwitnessminornonzeroevaluationmatrixnegative)))))))) /\ (exists mdr_b_zero_columns_rankwitnessminornonzeroevaluationdeterminant mdr_c_zero_columns_rankwitnessminornonzeroevaluationdeterminant mdr_l_zero_columns_rankwitnessminornonzeroevaluationdeterminant mdr_i_zero_columns_rankwitnessminornonzeroevaluationdeterminant. ((forall mdr_i_zero_columns_rankwitnessminornonzeroevaluationdeterminanth. (exists mdr_gap_zero_columns_rankwitnessminornonzeroevaluationdeterminanthi. mdr_gap_zero_columns_rankwitnessminornonzeroevaluationdeterminanthi + S (mdr_i_zero_columns_rankwitnessminornonzeroevaluationdeterminanth) = (mdr_l_zero_columns_rankwitnessminornonzeroevaluationdeterminant)) -> exists mdr_d_zero_columns_rankwitnessminornonzeroevaluationdeterminanth mdr_pb_zero_columns_rankwitnessminornonzeroevaluationdeterminanth mdr_pc_zero_columns_rankwitnessminornonzeroevaluationdeterminanth mdr_nb_zero_columns_rankwitnessminornonzeroevaluationdeterminanth mdr_nc_zero_columns_rankwitnessminornonzeroevaluationdeterminanth mdr_p_zero_columns_rankwitnessminornonzeroevaluationdeterminanth mdr_n_zero_columns_rankwitnessminornonzeroevaluationdeterminanth. ((exists mdr_z_zero_columns_rankwitnessminornonzeroevaluationdeterminanthr. ((exists mdr_a_zero_columns_rankwitnessminornonzeroevaluationdeterminanthrc mdr_b_zero_columns_rankwitnessminornonzeroevaluationdeterminanthrc mdr_c_zero_columns_rankwitnessminornonzeroevaluationdeterminanthrc mdr_e_zero_columns_rankwitnessminornonzeroevaluationdeterminanthrc mdr_f_zero_columns_rankwitnessminornonzeroevaluationdeterminanthrc. ((mdr_a_zero_columns_rankwitnessminornonzeroevaluationdeterminanthrc = ((mdr_d_zero_columns_rankwitnessminornonzeroevaluationdeterminanth) + (mdr_pb_zero_columns_rankwitnessminornonzeroevaluationdeterminanth)) * S ((mdr_d_zero_columns_rankwitnessminornonzeroevaluationdeterminanth) + (mdr_pb_zero_columns_rankwitnessminornonzeroevaluationdeterminanth)) + ((mdr_pb_zero_columns_rankwitnessminornonzeroevaluationdeterminanth) + (mdr_pb_zero_columns_rankwitnessminornonzeroevaluationdeterminanth))) /\ ((mdr_b_zero_columns_rankwitnessminornonzeroevaluationdeterminanthrc = ((mdr_pc_zero_columns_rankwitnessminornonzeroevaluationdeterminanth) + (mdr_nb_zero_columns_rankwitnessminornonzeroevaluationdeterminanth)) * S ((mdr_pc_zero_columns_rankwitnessminornonzeroevaluationdeterminanth) + (mdr_nb_zero_columns_rankwitnessminornonzeroevaluationdeterminanth)) + ((mdr_nb_zero_columns_rankwitnessminornonzeroevaluationdeterminanth) + (mdr_nb_zero_columns_rankwitnessminornonzeroevaluationdeterminanth))) /\ ((mdr_c_zero_columns_rankwitnessminornonzeroevaluationdeterminanthrc = ((mdr_a_zero_columns_rankwitnessminornonzeroevaluationdeterminanthrc) + (mdr_b_zero_columns_rankwitnessminornonzeroevaluationdeterminanthrc)) * S ((mdr_a_zero_columns_rankwitnessminornonzeroevaluationdeterminanthrc) + (mdr_b_zero_columns_rankwitnessminornonzeroevaluationdeterminanthrc)) + ((mdr_b_zero_columns_rankwitnessminornonzeroevaluationdeterminanthrc) + (mdr_b_zero_columns_rankwitnessminornonzeroevaluationdeterminanthrc))) /\ ((mdr_e_zero_columns_rankwitnessminornonzeroevaluationdeterminanthrc = ((mdr_p_zero_columns_rankwitnessminornonzeroevaluationdeterminanth) + (mdr_n_zero_columns_rankwitnessminornonzeroevaluationdeterminanth)) * S ((mdr_p_zero_columns_rankwitnessminornonzeroevaluationdeterminanth) + (mdr_n_zero_columns_rankwitnessminornonzeroevaluationdeterminanth)) + ((mdr_n_zero_columns_rankwitnessminornonzeroevaluationdeterminanth) + (mdr_n_zero_columns_rankwitnessminornonzeroevaluationdeterminanth))) /\ ((mdr_f_zero_columns_rankwitnessminornonzeroevaluationdeterminanthrc = ((mdr_nc_zero_columns_rankwitnessminornonzeroevaluationdeterminanth) + (mdr_e_zero_columns_rankwitnessminornonzeroevaluationdeterminanthrc)) * S ((mdr_nc_zero_columns_rankwitnessminornonzeroevaluationdeterminanth) + (mdr_e_zero_columns_rankwitnessminornonzeroevaluationdeterminanthrc)) + ((mdr_e_zero_columns_rankwitnessminornonzeroevaluationdeterminanthrc) + (mdr_e_zero_columns_rankwitnessminornonzeroevaluationdeterminanthrc))) /\ ((mdr_z_zero_columns_rankwitnessminornonzeroevaluationdeterminanthr) = ((mdr_c_zero_columns_rankwitnessminornonzeroevaluationdeterminanthrc) + (mdr_f_zero_columns_rankwitnessminornonzeroevaluationdeterminanthrc)) * S ((mdr_c_zero_columns_rankwitnessminornonzeroevaluationdeterminanthrc) + (mdr_f_zero_columns_rankwitnessminornonzeroevaluationdeterminanthrc)) + ((mdr_f_zero_columns_rankwitnessminornonzeroevaluationdeterminanthrc) + (mdr_f_zero_columns_rankwitnessminornonzeroevaluationdeterminanthrc))))))))) /\ (((exists ff_h_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthrb. ff_h_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthrb + S (mdr_z_zero_columns_rankwitnessminornonzeroevaluationdeterminanthr) = S ((S (mdr_i_zero_columns_rankwitnessminornonzeroevaluationdeterminanth)) * mdr_c_zero_columns_rankwitnessminornonzeroevaluationdeterminant)) /\ exists ff_q_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthrb. mdr_b_zero_columns_rankwitnessminornonzeroevaluationdeterminant = ff_q_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthrb * S ((S (mdr_i_zero_columns_rankwitnessminornonzeroevaluationdeterminanth)) * mdr_c_zero_columns_rankwitnessminornonzeroevaluationdeterminant) + (mdr_z_zero_columns_rankwitnessminornonzeroevaluationdeterminanthr))))) /\ (((((mdr_d_zero_columns_rankwitnessminornonzeroevaluationdeterminanth) = 0) /\ (((mdr_p_zero_columns_rankwitnessminornonzeroevaluationdeterminanth) = 1) /\ ((mdr_n_zero_columns_rankwitnessminornonzeroevaluationdeterminanth) = 0))) \/ exists mdr_q_zero_columns_rankwitnessminornonzeroevaluationdeterminanths mdr_eb_zero_columns_rankwitnessminornonzeroevaluationdeterminanths mdr_ec_zero_columns_rankwitnessminornonzeroevaluationdeterminanths mdr_fb_zero_columns_rankwitnessminornonzeroevaluationdeterminanths mdr_fc_zero_columns_rankwitnessminornonzeroevaluationdeterminanths. (((mdr_d_zero_columns_rankwitnessminornonzeroevaluationdeterminanth) = S (mdr_q_zero_columns_rankwitnessminornonzeroevaluationdeterminanths)) /\ ((forall mdr_j_zero_columns_rankwitnessminornonzeroevaluationdeterminanthsc. (exists mdr_gap_zero_columns_rankwitnessminornonzeroevaluationdeterminanthscj. mdr_gap_zero_columns_rankwitnessminornonzeroevaluationdeterminanthscj + S (mdr_j_zero_columns_rankwitnessminornonzeroevaluationdeterminanthsc) = (S (mdr_q_zero_columns_rankwitnessminornonzeroevaluationdeterminanths))) -> exists mdr_i_zero_columns_rankwitnessminornonzeroevaluationdeterminanthsc mdr_up_zero_columns_rankwitnessminornonzeroevaluationdeterminanthsc mdr_us_zero_columns_rankwitnessminornonzeroevaluationdeterminanthsc mdr_un_zero_columns_rankwitnessminornonzeroevaluationdeterminanthsc mdr_ut_zero_columns_rankwitnessminornonzeroevaluationdeterminanthsc mdr_p_zero_columns_rankwitnessminornonzeroevaluationdeterminanthsc mdr_n_zero_columns_rankwitnessminornonzeroevaluationdeterminanthsc. ((exists mdr_gap_zero_columns_rankwitnessminornonzeroevaluationdeterminanthsci. mdr_gap_zero_columns_rankwitnessminornonzeroevaluationdeterminanthsci + S (mdr_i_zero_columns_rankwitnessminornonzeroevaluationdeterminanthsc) = (mdr_i_zero_columns_rankwitnessminornonzeroevaluationdeterminanth)) /\ ((exists mdr_z_zero_columns_rankwitnessminornonzeroevaluationdeterminanthscr. ((exists mdr_a_zero_columns_rankwitnessminornonzeroevaluationdeterminanthscrc mdr_b_zero_columns_rankwitnessminornonzeroevaluationdeterminanthscrc mdr_c_zero_columns_rankwitnessminornonzeroevaluationdeterminanthscrc mdr_e_zero_columns_rankwitnessminornonzeroevaluationdeterminanthscrc mdr_f_zero_columns_rankwitnessminornonzeroevaluationdeterminanthscrc. ((mdr_a_zero_columns_rankwitnessminornonzeroevaluationdeterminanthscrc = ((mdr_q_zero_columns_rankwitnessminornonzeroevaluationdeterminanths) + (mdr_up_zero_columns_rankwitnessminornonzeroevaluationdeterminanthsc)) * S ((mdr_q_zero_columns_rankwitnessminornonzeroevaluationdeterminanths) + (mdr_up_zero_columns_rankwitnessminornonzeroevaluationdeterminanthsc)) + ((mdr_up_zero_columns_rankwitnessminornonzeroevaluationdeterminanthsc) + (mdr_up_zero_columns_rankwitnessminornonzeroevaluationdeterminanthsc))) /\ ((mdr_b_zero_columns_rankwitnessminornonzeroevaluationdeterminanthscrc = ((mdr_us_zero_columns_rankwitnessminornonzeroevaluationdeterminanthsc) + (mdr_un_zero_columns_rankwitnessminornonzeroevaluationdeterminanthsc)) * S ((mdr_us_zero_columns_rankwitnessminornonzeroevaluationdeterminanthsc) + (mdr_un_zero_columns_rankwitnessminornonzeroevaluationdeterminanthsc)) + ((mdr_un_zero_columns_rankwitnessminornonzeroevaluationdeterminanthsc) + (mdr_un_zero_columns_rankwitnessminornonzeroevaluationdeterminanthsc))) /\ ((mdr_c_zero_columns_rankwitnessminornonzeroevaluationdeterminanthscrc = ((mdr_a_zero_columns_rankwitnessminornonzeroevaluationdeterminanthscrc) + (mdr_b_zero_columns_rankwitnessminornonzeroevaluationdeterminanthscrc)) * S ((mdr_a_zero_columns_rankwitnessminornonzeroevaluationdeterminanthscrc) + (mdr_b_zero_columns_rankwitnessminornonzeroevaluationdeterminanthscrc)) + ((mdr_b_zero_columns_rankwitnessminornonzeroevaluationdeterminanthscrc) + (mdr_b_zero_columns_rankwitnessminornonzeroevaluationdeterminanthscrc))) /\ ((mdr_e_zero_columns_rankwitnessminornonzeroevaluationdeterminanthscrc = ((mdr_p_zero_columns_rankwitnessminornonzeroevaluationdeterminanthsc) + (mdr_n_zero_columns_rankwitnessminornonzeroevaluationdeterminanthsc)) * S ((mdr_p_zero_columns_rankwitnessminornonzeroevaluationdeterminanthsc) + (mdr_n_zero_columns_rankwitnessminornonzeroevaluationdeterminanthsc)) + ((mdr_n_zero_columns_rankwitnessminornonzeroevaluationdeterminanthsc) + (mdr_n_zero_columns_rankwitnessminornonzeroevaluationdeterminanthsc))) /\ ((mdr_f_zero_columns_rankwitnessminornonzeroevaluationdeterminanthscrc = ((mdr_ut_zero_columns_rankwitnessminornonzeroevaluationdeterminanthsc) + (mdr_e_zero_columns_rankwitnessminornonzeroevaluationdeterminanthscrc)) * S ((mdr_ut_zero_columns_rankwitnessminornonzeroevaluationdeterminanthsc) + (mdr_e_zero_columns_rankwitnessminornonzeroevaluationdeterminanthscrc)) + ((mdr_e_zero_columns_rankwitnessminornonzeroevaluationdeterminanthscrc) + (mdr_e_zero_columns_rankwitnessminornonzeroevaluationdeterminanthscrc))) /\ ((mdr_z_zero_columns_rankwitnessminornonzeroevaluationdeterminanthscr) = ((mdr_c_zero_columns_rankwitnessminornonzeroevaluationdeterminanthscrc) + (mdr_f_zero_columns_rankwitnessminornonzeroevaluationdeterminanthscrc)) * S ((mdr_c_zero_columns_rankwitnessminornonzeroevaluationdeterminanthscrc) + (mdr_f_zero_columns_rankwitnessminornonzeroevaluationdeterminanthscrc)) + ((mdr_f_zero_columns_rankwitnessminornonzeroevaluationdeterminanthscrc) + (mdr_f_zero_columns_rankwitnessminornonzeroevaluationdeterminanthscrc))))))))) /\ (((exists ff_h_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthscrb. ff_h_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthscrb + S (mdr_z_zero_columns_rankwitnessminornonzeroevaluationdeterminanthscr) = S ((S (mdr_i_zero_columns_rankwitnessminornonzeroevaluationdeterminanthsc)) * mdr_c_zero_columns_rankwitnessminornonzeroevaluationdeterminant)) /\ exists ff_q_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthscrb. mdr_b_zero_columns_rankwitnessminornonzeroevaluationdeterminant = ff_q_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthscrb * S ((S (mdr_i_zero_columns_rankwitnessminornonzeroevaluationdeterminanthsc)) * mdr_c_zero_columns_rankwitnessminornonzeroevaluationdeterminant) + (mdr_z_zero_columns_rankwitnessminornonzeroevaluationdeterminanthscr))))) /\ ((((forall ff_index_mdm_prefix_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthscm_positive. (exists ff_gap_mdm_lt_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthscm_positive_index_bound. ff_gap_mdm_lt_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthscm_positive_index_bound + S (ff_index_mdm_prefix_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthscm_positive) = ((mdr_q_zero_columns_rankwitnessminornonzeroevaluationdeterminanths) * (mdr_q_zero_columns_rankwitnessminornonzeroevaluationdeterminanths))) -> exists ff_row_mdm_prefix_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthscm_positive ff_column_mdm_prefix_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthscm_positive ff_value_mdm_prefix_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthscm_positive. (ff_index_mdm_prefix_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthscm_positive = (mdr_q_zero_columns_rankwitnessminornonzeroevaluationdeterminanths) * ff_row_mdm_prefix_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthscm_positive + ff_column_mdm_prefix_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthscm_positive /\ ((exists ff_gap_mdm_lt_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthscm_positive_column_bound. ff_gap_mdm_lt_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthscm_positive_column_bound + S (ff_column_mdm_prefix_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthscm_positive) = (mdr_q_zero_columns_rankwitnessminornonzeroevaluationdeterminanths)) /\ ((exists ff_row_mdm_cell_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthscm_positive_cell ff_column_mdm_cell_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthscm_positive_cell. (((((exists ff_gap_mdm_lt_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthscm_positive_cell_row_before. ff_gap_mdm_lt_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthscm_positive_cell_row_before + S (ff_row_mdm_prefix_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthscm_positive) = (0)) /\ ff_row_mdm_cell_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthscm_positive_cell = ff_row_mdm_prefix_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthscm_positive) \/ ((exists ff_gap_mdm_le_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthscm_positive_cell_row_after. ff_gap_mdm_le_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthscm_positive_cell_row_after + (0) = (ff_row_mdm_prefix_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthscm_positive)) /\ ff_row_mdm_cell_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthscm_positive_cell = S ff_row_mdm_prefix_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthscm_positive))) /\ (((((exists ff_gap_mdm_lt_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthscm_positive_cell_column_before. ff_gap_mdm_lt_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthscm_positive_cell_column_before + S (ff_column_mdm_prefix_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthscm_positive) = (mdr_j_zero_columns_rankwitnessminornonzeroevaluationdeterminanthsc)) /\ ff_column_mdm_cell_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthscm_positive_cell = ff_column_mdm_prefix_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthscm_positive) \/ ((exists ff_gap_mdm_le_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthscm_positive_cell_column_after. ff_gap_mdm_le_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthscm_positive_cell_column_after + (mdr_j_zero_columns_rankwitnessminornonzeroevaluationdeterminanthsc) = (ff_column_mdm_prefix_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthscm_positive)) /\ ff_column_mdm_cell_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthscm_positive_cell = S ff_column_mdm_prefix_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthscm_positive))) /\ (((exists ff_h_mdm_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthscm_positive_cell_source. ff_h_mdm_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthscm_positive_cell_source + S (ff_value_mdm_prefix_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthscm_positive) = S ((S ((ff_row_mdm_cell_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthscm_positive_cell) * (S (mdr_q_zero_columns_rankwitnessminornonzeroevaluationdeterminanths)) + (ff_column_mdm_cell_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthscm_positive_cell))) * mdr_pc_zero_columns_rankwitnessminornonzeroevaluationdeterminanth)) /\ exists ff_q_mdm_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthscm_positive_cell_source. mdr_pb_zero_columns_rankwitnessminornonzeroevaluationdeterminanth = ff_q_mdm_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthscm_positive_cell_source * S ((S ((ff_row_mdm_cell_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthscm_positive_cell) * (S (mdr_q_zero_columns_rankwitnessminornonzeroevaluationdeterminanths)) + (ff_column_mdm_cell_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthscm_positive_cell))) * mdr_pc_zero_columns_rankwitnessminornonzeroevaluationdeterminanth) + (ff_value_mdm_prefix_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthscm_positive)))))) /\ (((exists ff_h_mdm_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthscm_positive_target. ff_h_mdm_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthscm_positive_target + S (ff_value_mdm_prefix_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthscm_positive) = S ((S (ff_index_mdm_prefix_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthscm_positive)) * mdr_us_zero_columns_rankwitnessminornonzeroevaluationdeterminanthsc)) /\ exists ff_q_mdm_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthscm_positive_target. mdr_up_zero_columns_rankwitnessminornonzeroevaluationdeterminanthsc = ff_q_mdm_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthscm_positive_target * S ((S (ff_index_mdm_prefix_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthscm_positive)) * mdr_us_zero_columns_rankwitnessminornonzeroevaluationdeterminanthsc) + (ff_value_mdm_prefix_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthscm_positive))))))) /\ (forall ff_index_mdm_prefix_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthscm_negative. (exists ff_gap_mdm_lt_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthscm_negative_index_bound. ff_gap_mdm_lt_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthscm_negative_index_bound + S (ff_index_mdm_prefix_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthscm_negative) = ((mdr_q_zero_columns_rankwitnessminornonzeroevaluationdeterminanths) * (mdr_q_zero_columns_rankwitnessminornonzeroevaluationdeterminanths))) -> exists ff_row_mdm_prefix_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthscm_negative ff_column_mdm_prefix_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthscm_negative ff_value_mdm_prefix_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthscm_negative. (ff_index_mdm_prefix_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthscm_negative = (mdr_q_zero_columns_rankwitnessminornonzeroevaluationdeterminanths) * ff_row_mdm_prefix_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthscm_negative + ff_column_mdm_prefix_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthscm_negative /\ ((exists ff_gap_mdm_lt_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthscm_negative_column_bound. ff_gap_mdm_lt_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthscm_negative_column_bound + S (ff_column_mdm_prefix_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthscm_negative) = (mdr_q_zero_columns_rankwitnessminornonzeroevaluationdeterminanths)) /\ ((exists ff_row_mdm_cell_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthscm_negative_cell ff_column_mdm_cell_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthscm_negative_cell. (((((exists ff_gap_mdm_lt_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthscm_negative_cell_row_before. ff_gap_mdm_lt_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthscm_negative_cell_row_before + S (ff_row_mdm_prefix_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthscm_negative) = (0)) /\ ff_row_mdm_cell_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthscm_negative_cell = ff_row_mdm_prefix_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthscm_negative) \/ ((exists ff_gap_mdm_le_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthscm_negative_cell_row_after. ff_gap_mdm_le_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthscm_negative_cell_row_after + (0) = (ff_row_mdm_prefix_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthscm_negative)) /\ ff_row_mdm_cell_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthscm_negative_cell = S ff_row_mdm_prefix_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthscm_negative))) /\ (((((exists ff_gap_mdm_lt_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthscm_negative_cell_column_before. ff_gap_mdm_lt_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthscm_negative_cell_column_before + S (ff_column_mdm_prefix_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthscm_negative) = (mdr_j_zero_columns_rankwitnessminornonzeroevaluationdeterminanthsc)) /\ ff_column_mdm_cell_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthscm_negative_cell = ff_column_mdm_prefix_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthscm_negative) \/ ((exists ff_gap_mdm_le_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthscm_negative_cell_column_after. ff_gap_mdm_le_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthscm_negative_cell_column_after + (mdr_j_zero_columns_rankwitnessminornonzeroevaluationdeterminanthsc) = (ff_column_mdm_prefix_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthscm_negative)) /\ ff_column_mdm_cell_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthscm_negative_cell = S ff_column_mdm_prefix_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthscm_negative))) /\ (((exists ff_h_mdm_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthscm_negative_cell_source. ff_h_mdm_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthscm_negative_cell_source + S (ff_value_mdm_prefix_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthscm_negative) = S ((S ((ff_row_mdm_cell_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthscm_negative_cell) * (S (mdr_q_zero_columns_rankwitnessminornonzeroevaluationdeterminanths)) + (ff_column_mdm_cell_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthscm_negative_cell))) * mdr_nc_zero_columns_rankwitnessminornonzeroevaluationdeterminanth)) /\ exists ff_q_mdm_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthscm_negative_cell_source. mdr_nb_zero_columns_rankwitnessminornonzeroevaluationdeterminanth = ff_q_mdm_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthscm_negative_cell_source * S ((S ((ff_row_mdm_cell_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthscm_negative_cell) * (S (mdr_q_zero_columns_rankwitnessminornonzeroevaluationdeterminanths)) + (ff_column_mdm_cell_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthscm_negative_cell))) * mdr_nc_zero_columns_rankwitnessminornonzeroevaluationdeterminanth) + (ff_value_mdm_prefix_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthscm_negative)))))) /\ (((exists ff_h_mdm_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthscm_negative_target. ff_h_mdm_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthscm_negative_target + S (ff_value_mdm_prefix_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthscm_negative) = S ((S (ff_index_mdm_prefix_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthscm_negative)) * mdr_ut_zero_columns_rankwitnessminornonzeroevaluationdeterminanthsc)) /\ exists ff_q_mdm_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthscm_negative_target. mdr_un_zero_columns_rankwitnessminornonzeroevaluationdeterminanthsc = ff_q_mdm_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthscm_negative_target * S ((S (ff_index_mdm_prefix_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthscm_negative)) * mdr_ut_zero_columns_rankwitnessminornonzeroevaluationdeterminanthsc) + (ff_value_mdm_prefix_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthscm_negative))))))))) /\ ((((exists ff_h_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthscp. ff_h_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthscp + S (mdr_p_zero_columns_rankwitnessminornonzeroevaluationdeterminanthsc) = S ((S (mdr_j_zero_columns_rankwitnessminornonzeroevaluationdeterminanthsc)) * mdr_ec_zero_columns_rankwitnessminornonzeroevaluationdeterminanths)) /\ exists ff_q_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthscp. mdr_eb_zero_columns_rankwitnessminornonzeroevaluationdeterminanths = ff_q_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthscp * S ((S (mdr_j_zero_columns_rankwitnessminornonzeroevaluationdeterminanthsc)) * mdr_ec_zero_columns_rankwitnessminornonzeroevaluationdeterminanths) + (mdr_p_zero_columns_rankwitnessminornonzeroevaluationdeterminanthsc))) /\ (((exists ff_h_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthscn. ff_h_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthscn + S (mdr_n_zero_columns_rankwitnessminornonzeroevaluationdeterminanthsc) = S ((S (mdr_j_zero_columns_rankwitnessminornonzeroevaluationdeterminanthsc)) * mdr_fc_zero_columns_rankwitnessminornonzeroevaluationdeterminanths)) /\ exists ff_q_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthscn. mdr_fb_zero_columns_rankwitnessminornonzeroevaluationdeterminanths = ff_q_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthscn * S ((S (mdr_j_zero_columns_rankwitnessminornonzeroevaluationdeterminanthsc)) * mdr_fc_zero_columns_rankwitnessminornonzeroevaluationdeterminanths) + (mdr_n_zero_columns_rankwitnessminornonzeroevaluationdeterminanthsc)))))))) /\ (exists ff_ub_mce_fold_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthsf ff_uc_mce_fold_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthsf ff_vb_mce_fold_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthsf ff_vc_mce_fold_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthsf. ((forall ff_index_mce_alternating_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthsf_prefix. (exists ff_gap_mce_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthsf_prefix_index. ff_gap_mce_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthsf_prefix_index + S (ff_index_mce_alternating_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthsf_prefix) = (S (mdr_q_zero_columns_rankwitnessminornonzeroevaluationdeterminanths))) -> exists ff_ap_mce_alternating_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthsf_prefix ff_an_mce_alternating_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthsf_prefix ff_bp_mce_alternating_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthsf_prefix ff_bn_mce_alternating_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthsf_prefix ff_p_mce_alternating_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthsf_prefix ff_n_mce_alternating_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthsf_prefix. ((((exists ff_h_mce_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthsf_prefix_ap. ff_h_mce_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthsf_prefix_ap + S (ff_ap_mce_alternating_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthsf_prefix) = S ((S (ff_index_mce_alternating_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthsf_prefix)) * mdr_pc_zero_columns_rankwitnessminornonzeroevaluationdeterminanth)) /\ exists ff_q_mce_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthsf_prefix_ap. mdr_pb_zero_columns_rankwitnessminornonzeroevaluationdeterminanth = ff_q_mce_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthsf_prefix_ap * S ((S (ff_index_mce_alternating_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthsf_prefix)) * mdr_pc_zero_columns_rankwitnessminornonzeroevaluationdeterminanth) + (ff_ap_mce_alternating_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthsf_prefix))) /\ ((((exists ff_h_mce_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthsf_prefix_an. ff_h_mce_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthsf_prefix_an + S (ff_an_mce_alternating_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthsf_prefix) = S ((S (ff_index_mce_alternating_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthsf_prefix)) * mdr_nc_zero_columns_rankwitnessminornonzeroevaluationdeterminanth)) /\ exists ff_q_mce_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthsf_prefix_an. mdr_nb_zero_columns_rankwitnessminornonzeroevaluationdeterminanth = ff_q_mce_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthsf_prefix_an * S ((S (ff_index_mce_alternating_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthsf_prefix)) * mdr_nc_zero_columns_rankwitnessminornonzeroevaluationdeterminanth) + (ff_an_mce_alternating_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthsf_prefix))) /\ ((((exists ff_h_mce_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthsf_prefix_bp. ff_h_mce_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthsf_prefix_bp + S (ff_bp_mce_alternating_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthsf_prefix) = S ((S (ff_index_mce_alternating_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthsf_prefix)) * mdr_ec_zero_columns_rankwitnessminornonzeroevaluationdeterminanths)) /\ exists ff_q_mce_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthsf_prefix_bp. mdr_eb_zero_columns_rankwitnessminornonzeroevaluationdeterminanths = ff_q_mce_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthsf_prefix_bp * S ((S (ff_index_mce_alternating_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthsf_prefix)) * mdr_ec_zero_columns_rankwitnessminornonzeroevaluationdeterminanths) + (ff_bp_mce_alternating_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthsf_prefix))) /\ ((((exists ff_h_mce_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthsf_prefix_bn. ff_h_mce_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthsf_prefix_bn + S (ff_bn_mce_alternating_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthsf_prefix) = S ((S (ff_index_mce_alternating_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthsf_prefix)) * mdr_fc_zero_columns_rankwitnessminornonzeroevaluationdeterminanths)) /\ exists ff_q_mce_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthsf_prefix_bn. mdr_fb_zero_columns_rankwitnessminornonzeroevaluationdeterminanths = ff_q_mce_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthsf_prefix_bn * S ((S (ff_index_mce_alternating_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthsf_prefix)) * mdr_fc_zero_columns_rankwitnessminornonzeroevaluationdeterminanths) + (ff_bn_mce_alternating_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthsf_prefix))) /\ ((((exists ff_h_mce_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthsf_prefix_positive. ff_h_mce_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthsf_prefix_positive + S (ff_p_mce_alternating_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthsf_prefix) = S ((S (ff_index_mce_alternating_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthsf_prefix)) * ff_uc_mce_fold_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthsf)) /\ exists ff_q_mce_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthsf_prefix_positive. ff_ub_mce_fold_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthsf = ff_q_mce_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthsf_prefix_positive * S ((S (ff_index_mce_alternating_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthsf_prefix)) * ff_uc_mce_fold_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthsf) + (ff_p_mce_alternating_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthsf_prefix))) /\ ((((exists ff_h_mce_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthsf_prefix_negative. ff_h_mce_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthsf_prefix_negative + S (ff_n_mce_alternating_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthsf_prefix) = S ((S (ff_index_mce_alternating_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthsf_prefix)) * ff_vc_mce_fold_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthsf)) /\ exists ff_q_mce_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthsf_prefix_negative. ff_vb_mce_fold_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthsf = ff_q_mce_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthsf_prefix_negative * S ((S (ff_index_mce_alternating_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthsf_prefix)) * ff_vc_mce_fold_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthsf) + (ff_n_mce_alternating_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthsf_prefix))) /\ (((exists ff_even_mce_term_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthsf_prefix_term. ff_index_mce_alternating_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthsf_prefix = 2 * ff_even_mce_term_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthsf_prefix_term) /\ (ff_p_mce_alternating_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthsf_prefix = (ff_ap_mce_alternating_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthsf_prefix) * (ff_bp_mce_alternating_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthsf_prefix) + (ff_an_mce_alternating_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthsf_prefix) * (ff_bn_mce_alternating_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthsf_prefix) /\ ff_n_mce_alternating_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthsf_prefix = (ff_ap_mce_alternating_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthsf_prefix) * (ff_bn_mce_alternating_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthsf_prefix) + (ff_an_mce_alternating_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthsf_prefix) * (ff_bp_mce_alternating_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthsf_prefix))) \/ ((exists ff_odd_mce_term_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthsf_prefix_term. ff_index_mce_alternating_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthsf_prefix = 2 * ff_odd_mce_term_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthsf_prefix_term + 1) /\ (ff_p_mce_alternating_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthsf_prefix = (ff_ap_mce_alternating_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthsf_prefix) * (ff_bn_mce_alternating_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthsf_prefix) + (ff_an_mce_alternating_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthsf_prefix) * (ff_bp_mce_alternating_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthsf_prefix) /\ ff_n_mce_alternating_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthsf_prefix = (ff_ap_mce_alternating_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthsf_prefix) * (ff_bp_mce_alternating_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthsf_prefix) + (ff_an_mce_alternating_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthsf_prefix) * (ff_bn_mce_alternating_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthsf_prefix))))))))))) /\ ((exists ff_u_mce_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthsf_positive ff_v_mce_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthsf_positive. ((((exists ff_h_mce_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthsf_positive_start. ff_h_mce_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthsf_positive_start + S (0) = S ((S (0)) * ff_v_mce_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthsf_positive)) /\ exists ff_q_mce_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthsf_positive_start. ff_u_mce_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthsf_positive = ff_q_mce_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthsf_positive_start * S ((S (0)) * ff_v_mce_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthsf_positive) + (0))) /\ ((((exists ff_h_mce_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthsf_positive_terminal. ff_h_mce_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthsf_positive_terminal + S (mdr_p_zero_columns_rankwitnessminornonzeroevaluationdeterminanth) = S ((S ((S (mdr_q_zero_columns_rankwitnessminornonzeroevaluationdeterminanths)))) * ff_v_mce_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthsf_positive)) /\ exists ff_q_mce_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthsf_positive_terminal. ff_u_mce_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthsf_positive = ff_q_mce_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthsf_positive_terminal * S ((S ((S (mdr_q_zero_columns_rankwitnessminornonzeroevaluationdeterminanths)))) * ff_v_mce_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthsf_positive) + (mdr_p_zero_columns_rankwitnessminornonzeroevaluationdeterminanth))) /\ forall ff_i_mce_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthsf_positive. (exists ff_lt_mce_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthsf_positive_bound. ff_lt_mce_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthsf_positive_bound + S ff_i_mce_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthsf_positive = (S (mdr_q_zero_columns_rankwitnessminornonzeroevaluationdeterminanths))) -> exists ff_a_mce_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthsf_positive ff_r_mce_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthsf_positive ff_s_mce_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthsf_positive. ((((exists ff_h_mce_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthsf_positive_summand. ff_h_mce_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthsf_positive_summand + S (ff_a_mce_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthsf_positive) = S ((S (ff_i_mce_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthsf_positive)) * ff_uc_mce_fold_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthsf)) /\ exists ff_q_mce_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthsf_positive_summand. ff_ub_mce_fold_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthsf = ff_q_mce_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthsf_positive_summand * S ((S (ff_i_mce_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthsf_positive)) * ff_uc_mce_fold_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthsf) + (ff_a_mce_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthsf_positive))) /\ ((((exists ff_h_mce_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthsf_positive_partial. ff_h_mce_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthsf_positive_partial + S (ff_r_mce_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthsf_positive) = S ((S (ff_i_mce_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthsf_positive)) * ff_v_mce_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthsf_positive)) /\ exists ff_q_mce_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthsf_positive_partial. ff_u_mce_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthsf_positive = ff_q_mce_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthsf_positive_partial * S ((S (ff_i_mce_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthsf_positive)) * ff_v_mce_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthsf_positive) + (ff_r_mce_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthsf_positive))) /\ ((((exists ff_h_mce_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthsf_positive_successor. ff_h_mce_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthsf_positive_successor + S (ff_s_mce_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthsf_positive) = S ((S (S ff_i_mce_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthsf_positive)) * ff_v_mce_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthsf_positive)) /\ exists ff_q_mce_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthsf_positive_successor. ff_u_mce_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthsf_positive = ff_q_mce_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthsf_positive_successor * S ((S (S ff_i_mce_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthsf_positive)) * ff_v_mce_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthsf_positive) + (ff_s_mce_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthsf_positive))) /\ ff_s_mce_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthsf_positive = ff_r_mce_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthsf_positive + ff_a_mce_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthsf_positive)))))) /\ (exists ff_u_mce_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthsf_negative ff_v_mce_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthsf_negative. ((((exists ff_h_mce_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthsf_negative_start. ff_h_mce_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthsf_negative_start + S (0) = S ((S (0)) * ff_v_mce_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthsf_negative)) /\ exists ff_q_mce_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthsf_negative_start. ff_u_mce_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthsf_negative = ff_q_mce_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthsf_negative_start * S ((S (0)) * ff_v_mce_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthsf_negative) + (0))) /\ ((((exists ff_h_mce_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthsf_negative_terminal. ff_h_mce_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthsf_negative_terminal + S (mdr_n_zero_columns_rankwitnessminornonzeroevaluationdeterminanth) = S ((S ((S (mdr_q_zero_columns_rankwitnessminornonzeroevaluationdeterminanths)))) * ff_v_mce_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthsf_negative)) /\ exists ff_q_mce_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthsf_negative_terminal. ff_u_mce_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthsf_negative = ff_q_mce_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthsf_negative_terminal * S ((S ((S (mdr_q_zero_columns_rankwitnessminornonzeroevaluationdeterminanths)))) * ff_v_mce_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthsf_negative) + (mdr_n_zero_columns_rankwitnessminornonzeroevaluationdeterminanth))) /\ forall ff_i_mce_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthsf_negative. (exists ff_lt_mce_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthsf_negative_bound. ff_lt_mce_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthsf_negative_bound + S ff_i_mce_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthsf_negative = (S (mdr_q_zero_columns_rankwitnessminornonzeroevaluationdeterminanths))) -> exists ff_a_mce_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthsf_negative ff_r_mce_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthsf_negative ff_s_mce_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthsf_negative. ((((exists ff_h_mce_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthsf_negative_summand. ff_h_mce_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthsf_negative_summand + S (ff_a_mce_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthsf_negative) = S ((S (ff_i_mce_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthsf_negative)) * ff_vc_mce_fold_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthsf)) /\ exists ff_q_mce_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthsf_negative_summand. ff_vb_mce_fold_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthsf = ff_q_mce_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthsf_negative_summand * S ((S (ff_i_mce_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthsf_negative)) * ff_vc_mce_fold_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthsf) + (ff_a_mce_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthsf_negative))) /\ ((((exists ff_h_mce_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthsf_negative_partial. ff_h_mce_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthsf_negative_partial + S (ff_r_mce_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthsf_negative) = S ((S (ff_i_mce_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthsf_negative)) * ff_v_mce_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthsf_negative)) /\ exists ff_q_mce_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthsf_negative_partial. ff_u_mce_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthsf_negative = ff_q_mce_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthsf_negative_partial * S ((S (ff_i_mce_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthsf_negative)) * ff_v_mce_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthsf_negative) + (ff_r_mce_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthsf_negative))) /\ ((((exists ff_h_mce_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthsf_negative_successor. ff_h_mce_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthsf_negative_successor + S (ff_s_mce_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthsf_negative) = S ((S (S ff_i_mce_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthsf_negative)) * ff_v_mce_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthsf_negative)) /\ exists ff_q_mce_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthsf_negative_successor. ff_u_mce_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthsf_negative = ff_q_mce_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthsf_negative_successor * S ((S (S ff_i_mce_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthsf_negative)) * ff_v_mce_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthsf_negative) + (ff_s_mce_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthsf_negative))) /\ ff_s_mce_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthsf_negative = ff_r_mce_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthsf_negative + ff_a_mce_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminanthsf_negative))))))))))))))) /\ ((exists mdr_gap_zero_columns_rankwitnessminornonzeroevaluationdeterminanti. mdr_gap_zero_columns_rankwitnessminornonzeroevaluationdeterminanti + S (mdr_i_zero_columns_rankwitnessminornonzeroevaluationdeterminant) = (mdr_l_zero_columns_rankwitnessminornonzeroevaluationdeterminant)) /\ (exists mdr_z_zero_columns_rankwitnessminornonzeroevaluationdeterminantr. ((exists mdr_a_zero_columns_rankwitnessminornonzeroevaluationdeterminantrc mdr_b_zero_columns_rankwitnessminornonzeroevaluationdeterminantrc mdr_c_zero_columns_rankwitnessminornonzeroevaluationdeterminantrc mdr_e_zero_columns_rankwitnessminornonzeroevaluationdeterminantrc mdr_f_zero_columns_rankwitnessminornonzeroevaluationdeterminantrc. ((mdr_a_zero_columns_rankwitnessminornonzeroevaluationdeterminantrc = ((0) + (mdr_ub_zero_columns_rankwitnessminornonzeroevaluation)) * S ((0) + (mdr_ub_zero_columns_rankwitnessminornonzeroevaluation)) + ((mdr_ub_zero_columns_rankwitnessminornonzeroevaluation) + (mdr_ub_zero_columns_rankwitnessminornonzeroevaluation))) /\ ((mdr_b_zero_columns_rankwitnessminornonzeroevaluationdeterminantrc = ((mdr_uc_zero_columns_rankwitnessminornonzeroevaluation) + (mdr_vb_zero_columns_rankwitnessminornonzeroevaluation)) * S ((mdr_uc_zero_columns_rankwitnessminornonzeroevaluation) + (mdr_vb_zero_columns_rankwitnessminornonzeroevaluation)) + ((mdr_vb_zero_columns_rankwitnessminornonzeroevaluation) + (mdr_vb_zero_columns_rankwitnessminornonzeroevaluation))) /\ ((mdr_c_zero_columns_rankwitnessminornonzeroevaluationdeterminantrc = ((mdr_a_zero_columns_rankwitnessminornonzeroevaluationdeterminantrc) + (mdr_b_zero_columns_rankwitnessminornonzeroevaluationdeterminantrc)) * S ((mdr_a_zero_columns_rankwitnessminornonzeroevaluationdeterminantrc) + (mdr_b_zero_columns_rankwitnessminornonzeroevaluationdeterminantrc)) + ((mdr_b_zero_columns_rankwitnessminornonzeroevaluationdeterminantrc) + (mdr_b_zero_columns_rankwitnessminornonzeroevaluationdeterminantrc))) /\ ((mdr_e_zero_columns_rankwitnessminornonzeroevaluationdeterminantrc = ((mdr_p_zero_columns_rankwitnessminornonzero) + (mdr_n_zero_columns_rankwitnessminornonzero)) * S ((mdr_p_zero_columns_rankwitnessminornonzero) + (mdr_n_zero_columns_rankwitnessminornonzero)) + ((mdr_n_zero_columns_rankwitnessminornonzero) + (mdr_n_zero_columns_rankwitnessminornonzero))) /\ ((mdr_f_zero_columns_rankwitnessminornonzeroevaluationdeterminantrc = ((mdr_vc_zero_columns_rankwitnessminornonzeroevaluation) + (mdr_e_zero_columns_rankwitnessminornonzeroevaluationdeterminantrc)) * S ((mdr_vc_zero_columns_rankwitnessminornonzeroevaluation) + (mdr_e_zero_columns_rankwitnessminornonzeroevaluationdeterminantrc)) + ((mdr_e_zero_columns_rankwitnessminornonzeroevaluationdeterminantrc) + (mdr_e_zero_columns_rankwitnessminornonzeroevaluationdeterminantrc))) /\ ((mdr_z_zero_columns_rankwitnessminornonzeroevaluationdeterminantr) = ((mdr_c_zero_columns_rankwitnessminornonzeroevaluationdeterminantrc) + (mdr_f_zero_columns_rankwitnessminornonzeroevaluationdeterminantrc)) * S ((mdr_c_zero_columns_rankwitnessminornonzeroevaluationdeterminantrc) + (mdr_f_zero_columns_rankwitnessminornonzeroevaluationdeterminantrc)) + ((mdr_f_zero_columns_rankwitnessminornonzeroevaluationdeterminantrc) + (mdr_f_zero_columns_rankwitnessminornonzeroevaluationdeterminantrc))))))))) /\ (((exists ff_h_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminantrb. ff_h_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminantrb + S (mdr_z_zero_columns_rankwitnessminornonzeroevaluationdeterminantr) = S ((S (mdr_i_zero_columns_rankwitnessminornonzeroevaluationdeterminant)) * mdr_c_zero_columns_rankwitnessminornonzeroevaluationdeterminant)) /\ exists ff_q_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminantrb. mdr_b_zero_columns_rankwitnessminornonzeroevaluationdeterminant = ff_q_mdr_zero_columns_rankwitnessminornonzeroevaluationdeterminantrb * S ((S (mdr_i_zero_columns_rankwitnessminornonzeroevaluationdeterminant)) * mdr_c_zero_columns_rankwitnessminornonzeroevaluationdeterminant) + (mdr_z_zero_columns_rankwitnessminornonzeroevaluationdeterminantr)))))))))) /\ (~(mdr_p_zero_columns_rankwitnessminornonzero = mdr_n_zero_columns_rankwitnessminornonzero)))))))) /\ (forall mdr_q_zero_columns_rank. (exists mdr_gap_zero_columns_rankhigher. mdr_gap_zero_columns_rankhigher + S (0) = (mdr_q_zero_columns_rank)) -> (forall mdr_rb_zero_columns_rankzero mdr_rc_zero_columns_rankzero mdr_cb_zero_columns_rankzero mdr_cc_zero_columns_rankzero mdr_p_zero_columns_rankzero mdr_n_zero_columns_rankzero. (((forall fom_index_mrf_zero_columns_rankzerorowsbound. (exists fom_gap_mrf_zero_columns_rankzerorowsbound_index_bound. fom_gap_mrf_zero_columns_rankzerorowsbound_index_bound + S (fom_index_mrf_zero_columns_rankzerorowsbound) = mdr_q_zero_columns_rank) -> exists fom_value_mrf_zero_columns_rankzerorowsbound. ((((exists fom_beta_height_mrf_zero_columns_rankzerorowsbound_entry. fom_beta_height_mrf_zero_columns_rankzerorowsbound_entry + S (fom_value_mrf_zero_columns_rankzerorowsbound) = S ((S (fom_index_mrf_zero_columns_rankzerorowsbound)) * mdr_rc_zero_columns_rankzero)) /\ exists fom_beta_quotient_mrf_zero_columns_rankzerorowsbound_entry. mdr_rb_zero_columns_rankzero = fom_beta_quotient_mrf_zero_columns_rankzerorowsbound_entry * S ((S (fom_index_mrf_zero_columns_rankzerorowsbound)) * mdr_rc_zero_columns_rankzero) + (fom_value_mrf_zero_columns_rankzerorowsbound))) /\ (exists fom_gap_mrf_zero_columns_rankzerorowsbound_value_bound. fom_gap_mrf_zero_columns_rankzerorowsbound_value_bound + S (fom_value_mrf_zero_columns_rankzerorowsbound) = r))) /\ (forall mdr_i_zero_columns_rankzerorowsdistinct mdr_j_zero_columns_rankzerorowsdistinct mdr_a_zero_columns_rankzerorowsdistinct. (exists mdr_gap_zero_columns_rankzerorowsdistincti. mdr_gap_zero_columns_rankzerorowsdistincti + S (mdr_i_zero_columns_rankzerorowsdistinct) = (mdr_q_zero_columns_rank)) -> (exists mdr_gap_zero_columns_rankzerorowsdistinctj. mdr_gap_zero_columns_rankzerorowsdistinctj + S (mdr_j_zero_columns_rankzerorowsdistinct) = (mdr_q_zero_columns_rank)) -> (((exists ff_h_mdr_zero_columns_rankzerorowsdistinctfirst. ff_h_mdr_zero_columns_rankzerorowsdistinctfirst + S (mdr_a_zero_columns_rankzerorowsdistinct) = S ((S (mdr_i_zero_columns_rankzerorowsdistinct)) * mdr_rc_zero_columns_rankzero)) /\ exists ff_q_mdr_zero_columns_rankzerorowsdistinctfirst. mdr_rb_zero_columns_rankzero = ff_q_mdr_zero_columns_rankzerorowsdistinctfirst * S ((S (mdr_i_zero_columns_rankzerorowsdistinct)) * mdr_rc_zero_columns_rankzero) + (mdr_a_zero_columns_rankzerorowsdistinct))) -> (((exists ff_h_mdr_zero_columns_rankzerorowsdistinctsecond. ff_h_mdr_zero_columns_rankzerorowsdistinctsecond + S (mdr_a_zero_columns_rankzerorowsdistinct) = S ((S (mdr_j_zero_columns_rankzerorowsdistinct)) * mdr_rc_zero_columns_rankzero)) /\ exists ff_q_mdr_zero_columns_rankzerorowsdistinctsecond. mdr_rb_zero_columns_rankzero = ff_q_mdr_zero_columns_rankzerorowsdistinctsecond * S ((S (mdr_j_zero_columns_rankzerorowsdistinct)) * mdr_rc_zero_columns_rankzero) + (mdr_a_zero_columns_rankzerorowsdistinct))) -> mdr_i_zero_columns_rankzerorowsdistinct = mdr_j_zero_columns_rankzerorowsdistinct))) -> (((forall fom_index_mrf_zero_columns_rankzerocolumnsbound. (exists fom_gap_mrf_zero_columns_rankzerocolumnsbound_index_bound. fom_gap_mrf_zero_columns_rankzerocolumnsbound_index_bound + S (fom_index_mrf_zero_columns_rankzerocolumnsbound) = mdr_q_zero_columns_rank) -> exists fom_value_mrf_zero_columns_rankzerocolumnsbound. ((((exists fom_beta_height_mrf_zero_columns_rankzerocolumnsbound_entry. fom_beta_height_mrf_zero_columns_rankzerocolumnsbound_entry + S (fom_value_mrf_zero_columns_rankzerocolumnsbound) = S ((S (fom_index_mrf_zero_columns_rankzerocolumnsbound)) * mdr_cc_zero_columns_rankzero)) /\ exists fom_beta_quotient_mrf_zero_columns_rankzerocolumnsbound_entry. mdr_cb_zero_columns_rankzero = fom_beta_quotient_mrf_zero_columns_rankzerocolumnsbound_entry * S ((S (fom_index_mrf_zero_columns_rankzerocolumnsbound)) * mdr_cc_zero_columns_rankzero) + (fom_value_mrf_zero_columns_rankzerocolumnsbound))) /\ (exists fom_gap_mrf_zero_columns_rankzerocolumnsbound_value_bound. fom_gap_mrf_zero_columns_rankzerocolumnsbound_value_bound + S (fom_value_mrf_zero_columns_rankzerocolumnsbound) = 0))) /\ (forall mdr_i_zero_columns_rankzerocolumnsdistinct mdr_j_zero_columns_rankzerocolumnsdistinct mdr_a_zero_columns_rankzerocolumnsdistinct. (exists mdr_gap_zero_columns_rankzerocolumnsdistincti. mdr_gap_zero_columns_rankzerocolumnsdistincti + S (mdr_i_zero_columns_rankzerocolumnsdistinct) = (mdr_q_zero_columns_rank)) -> (exists mdr_gap_zero_columns_rankzerocolumnsdistinctj. mdr_gap_zero_columns_rankzerocolumnsdistinctj + S (mdr_j_zero_columns_rankzerocolumnsdistinct) = (mdr_q_zero_columns_rank)) -> (((exists ff_h_mdr_zero_columns_rankzerocolumnsdistinctfirst. ff_h_mdr_zero_columns_rankzerocolumnsdistinctfirst + S (mdr_a_zero_columns_rankzerocolumnsdistinct) = S ((S (mdr_i_zero_columns_rankzerocolumnsdistinct)) * mdr_cc_zero_columns_rankzero)) /\ exists ff_q_mdr_zero_columns_rankzerocolumnsdistinctfirst. mdr_cb_zero_columns_rankzero = ff_q_mdr_zero_columns_rankzerocolumnsdistinctfirst * S ((S (mdr_i_zero_columns_rankzerocolumnsdistinct)) * mdr_cc_zero_columns_rankzero) + (mdr_a_zero_columns_rankzerocolumnsdistinct))) -> (((exists ff_h_mdr_zero_columns_rankzerocolumnsdistinctsecond. ff_h_mdr_zero_columns_rankzerocolumnsdistinctsecond + S (mdr_a_zero_columns_rankzerocolumnsdistinct) = S ((S (mdr_j_zero_columns_rankzerocolumnsdistinct)) * mdr_cc_zero_columns_rankzero)) /\ exists ff_q_mdr_zero_columns_rankzerocolumnsdistinctsecond. mdr_cb_zero_columns_rankzero = ff_q_mdr_zero_columns_rankzerocolumnsdistinctsecond * S ((S (mdr_j_zero_columns_rankzerocolumnsdistinct)) * mdr_cc_zero_columns_rankzero) + (mdr_a_zero_columns_rankzerocolumnsdistinct))) -> mdr_i_zero_columns_rankzerocolumnsdistinct = mdr_j_zero_columns_rankzerocolumnsdistinct))) -> (exists mdr_ub_zero_columns_rankzeroevaluation mdr_uc_zero_columns_rankzeroevaluation mdr_vb_zero_columns_rankzeroevaluation mdr_vc_zero_columns_rankzeroevaluation. ((((forall mdr_i_zero_columns_rankzeroevaluationmatrixpositive. (exists mdr_gap_zero_columns_rankzeroevaluationmatrixpositivebound. mdr_gap_zero_columns_rankzeroevaluationmatrixpositivebound + S (mdr_i_zero_columns_rankzeroevaluationmatrixpositive) = ((mdr_q_zero_columns_rank) * (mdr_q_zero_columns_rank))) -> exists mdr_a_zero_columns_rankzeroevaluationmatrixpositive. (((exists mdr_r_zero_columns_rankzeroevaluationmatrixpositivepoint mdr_s_zero_columns_rankzeroevaluationmatrixpositivepoint mdr_u_zero_columns_rankzeroevaluationmatrixpositivepoint mdr_v_zero_columns_rankzeroevaluationmatrixpositivepoint. ((mdr_i_zero_columns_rankzeroevaluationmatrixpositive = (mdr_q_zero_columns_rank) * mdr_r_zero_columns_rankzeroevaluationmatrixpositivepoint + mdr_s_zero_columns_rankzeroevaluationmatrixpositivepoint) /\ ((exists mdr_gap_zero_columns_rankzeroevaluationmatrixpositivepointcolumn. mdr_gap_zero_columns_rankzeroevaluationmatrixpositivepointcolumn + S (mdr_s_zero_columns_rankzeroevaluationmatrixpositivepoint) = (mdr_q_zero_columns_rank)) /\ ((((exists ff_h_mdr_zero_columns_rankzeroevaluationmatrixpositivepointrow_index. ff_h_mdr_zero_columns_rankzeroevaluationmatrixpositivepointrow_index + S (mdr_u_zero_columns_rankzeroevaluationmatrixpositivepoint) = S ((S (mdr_r_zero_columns_rankzeroevaluationmatrixpositivepoint)) * mdr_rc_zero_columns_rankzero)) /\ exists ff_q_mdr_zero_columns_rankzeroevaluationmatrixpositivepointrow_index. mdr_rb_zero_columns_rankzero = ff_q_mdr_zero_columns_rankzeroevaluationmatrixpositivepointrow_index * S ((S (mdr_r_zero_columns_rankzeroevaluationmatrixpositivepoint)) * mdr_rc_zero_columns_rankzero) + (mdr_u_zero_columns_rankzeroevaluationmatrixpositivepoint))) /\ ((((exists ff_h_mdr_zero_columns_rankzeroevaluationmatrixpositivepointcolumn_index. ff_h_mdr_zero_columns_rankzeroevaluationmatrixpositivepointcolumn_index + S (mdr_v_zero_columns_rankzeroevaluationmatrixpositivepoint) = S ((S (mdr_s_zero_columns_rankzeroevaluationmatrixpositivepoint)) * mdr_cc_zero_columns_rankzero)) /\ exists ff_q_mdr_zero_columns_rankzeroevaluationmatrixpositivepointcolumn_index. mdr_cb_zero_columns_rankzero = ff_q_mdr_zero_columns_rankzeroevaluationmatrixpositivepointcolumn_index * S ((S (mdr_s_zero_columns_rankzeroevaluationmatrixpositivepoint)) * mdr_cc_zero_columns_rankzero) + (mdr_v_zero_columns_rankzeroevaluationmatrixpositivepoint))) /\ (((exists ff_h_mdr_zero_columns_rankzeroevaluationmatrixpositivepointsource. ff_h_mdr_zero_columns_rankzeroevaluationmatrixpositivepointsource + S (mdr_a_zero_columns_rankzeroevaluationmatrixpositive) = S ((S ((mdr_u_zero_columns_rankzeroevaluationmatrixpositivepoint) * (0) + (mdr_v_zero_columns_rankzeroevaluationmatrixpositivepoint))) * pc)) /\ exists ff_q_mdr_zero_columns_rankzeroevaluationmatrixpositivepointsource. pb = ff_q_mdr_zero_columns_rankzeroevaluationmatrixpositivepointsource * S ((S ((mdr_u_zero_columns_rankzeroevaluationmatrixpositivepoint) * (0) + (mdr_v_zero_columns_rankzeroevaluationmatrixpositivepoint))) * pc) + (mdr_a_zero_columns_rankzeroevaluationmatrixpositive)))))))) /\ (((exists ff_h_mdr_zero_columns_rankzeroevaluationmatrixpositiveoutput. ff_h_mdr_zero_columns_rankzeroevaluationmatrixpositiveoutput + S (mdr_a_zero_columns_rankzeroevaluationmatrixpositive) = S ((S (mdr_i_zero_columns_rankzeroevaluationmatrixpositive)) * mdr_uc_zero_columns_rankzeroevaluation)) /\ exists ff_q_mdr_zero_columns_rankzeroevaluationmatrixpositiveoutput. mdr_ub_zero_columns_rankzeroevaluation = ff_q_mdr_zero_columns_rankzeroevaluationmatrixpositiveoutput * S ((S (mdr_i_zero_columns_rankzeroevaluationmatrixpositive)) * mdr_uc_zero_columns_rankzeroevaluation) + (mdr_a_zero_columns_rankzeroevaluationmatrixpositive)))))) /\ (forall mdr_i_zero_columns_rankzeroevaluationmatrixnegative. (exists mdr_gap_zero_columns_rankzeroevaluationmatrixnegativebound. mdr_gap_zero_columns_rankzeroevaluationmatrixnegativebound + S (mdr_i_zero_columns_rankzeroevaluationmatrixnegative) = ((mdr_q_zero_columns_rank) * (mdr_q_zero_columns_rank))) -> exists mdr_a_zero_columns_rankzeroevaluationmatrixnegative. (((exists mdr_r_zero_columns_rankzeroevaluationmatrixnegativepoint mdr_s_zero_columns_rankzeroevaluationmatrixnegativepoint mdr_u_zero_columns_rankzeroevaluationmatrixnegativepoint mdr_v_zero_columns_rankzeroevaluationmatrixnegativepoint. ((mdr_i_zero_columns_rankzeroevaluationmatrixnegative = (mdr_q_zero_columns_rank) * mdr_r_zero_columns_rankzeroevaluationmatrixnegativepoint + mdr_s_zero_columns_rankzeroevaluationmatrixnegativepoint) /\ ((exists mdr_gap_zero_columns_rankzeroevaluationmatrixnegativepointcolumn. mdr_gap_zero_columns_rankzeroevaluationmatrixnegativepointcolumn + S (mdr_s_zero_columns_rankzeroevaluationmatrixnegativepoint) = (mdr_q_zero_columns_rank)) /\ ((((exists ff_h_mdr_zero_columns_rankzeroevaluationmatrixnegativepointrow_index. ff_h_mdr_zero_columns_rankzeroevaluationmatrixnegativepointrow_index + S (mdr_u_zero_columns_rankzeroevaluationmatrixnegativepoint) = S ((S (mdr_r_zero_columns_rankzeroevaluationmatrixnegativepoint)) * mdr_rc_zero_columns_rankzero)) /\ exists ff_q_mdr_zero_columns_rankzeroevaluationmatrixnegativepointrow_index. mdr_rb_zero_columns_rankzero = ff_q_mdr_zero_columns_rankzeroevaluationmatrixnegativepointrow_index * S ((S (mdr_r_zero_columns_rankzeroevaluationmatrixnegativepoint)) * mdr_rc_zero_columns_rankzero) + (mdr_u_zero_columns_rankzeroevaluationmatrixnegativepoint))) /\ ((((exists ff_h_mdr_zero_columns_rankzeroevaluationmatrixnegativepointcolumn_index. ff_h_mdr_zero_columns_rankzeroevaluationmatrixnegativepointcolumn_index + S (mdr_v_zero_columns_rankzeroevaluationmatrixnegativepoint) = S ((S (mdr_s_zero_columns_rankzeroevaluationmatrixnegativepoint)) * mdr_cc_zero_columns_rankzero)) /\ exists ff_q_mdr_zero_columns_rankzeroevaluationmatrixnegativepointcolumn_index. mdr_cb_zero_columns_rankzero = ff_q_mdr_zero_columns_rankzeroevaluationmatrixnegativepointcolumn_index * S ((S (mdr_s_zero_columns_rankzeroevaluationmatrixnegativepoint)) * mdr_cc_zero_columns_rankzero) + (mdr_v_zero_columns_rankzeroevaluationmatrixnegativepoint))) /\ (((exists ff_h_mdr_zero_columns_rankzeroevaluationmatrixnegativepointsource. ff_h_mdr_zero_columns_rankzeroevaluationmatrixnegativepointsource + S (mdr_a_zero_columns_rankzeroevaluationmatrixnegative) = S ((S ((mdr_u_zero_columns_rankzeroevaluationmatrixnegativepoint) * (0) + (mdr_v_zero_columns_rankzeroevaluationmatrixnegativepoint))) * nc)) /\ exists ff_q_mdr_zero_columns_rankzeroevaluationmatrixnegativepointsource. nb = ff_q_mdr_zero_columns_rankzeroevaluationmatrixnegativepointsource * S ((S ((mdr_u_zero_columns_rankzeroevaluationmatrixnegativepoint) * (0) + (mdr_v_zero_columns_rankzeroevaluationmatrixnegativepoint))) * nc) + (mdr_a_zero_columns_rankzeroevaluationmatrixnegative)))))))) /\ (((exists ff_h_mdr_zero_columns_rankzeroevaluationmatrixnegativeoutput. ff_h_mdr_zero_columns_rankzeroevaluationmatrixnegativeoutput + S (mdr_a_zero_columns_rankzeroevaluationmatrixnegative) = S ((S (mdr_i_zero_columns_rankzeroevaluationmatrixnegative)) * mdr_vc_zero_columns_rankzeroevaluation)) /\ exists ff_q_mdr_zero_columns_rankzeroevaluationmatrixnegativeoutput. mdr_vb_zero_columns_rankzeroevaluation = ff_q_mdr_zero_columns_rankzeroevaluationmatrixnegativeoutput * S ((S (mdr_i_zero_columns_rankzeroevaluationmatrixnegative)) * mdr_vc_zero_columns_rankzeroevaluation) + (mdr_a_zero_columns_rankzeroevaluationmatrixnegative)))))))) /\ (exists mdr_b_zero_columns_rankzeroevaluationdeterminant mdr_c_zero_columns_rankzeroevaluationdeterminant mdr_l_zero_columns_rankzeroevaluationdeterminant mdr_i_zero_columns_rankzeroevaluationdeterminant. ((forall mdr_i_zero_columns_rankzeroevaluationdeterminanth. (exists mdr_gap_zero_columns_rankzeroevaluationdeterminanthi. mdr_gap_zero_columns_rankzeroevaluationdeterminanthi + S (mdr_i_zero_columns_rankzeroevaluationdeterminanth) = (mdr_l_zero_columns_rankzeroevaluationdeterminant)) -> exists mdr_d_zero_columns_rankzeroevaluationdeterminanth mdr_pb_zero_columns_rankzeroevaluationdeterminanth mdr_pc_zero_columns_rankzeroevaluationdeterminanth mdr_nb_zero_columns_rankzeroevaluationdeterminanth mdr_nc_zero_columns_rankzeroevaluationdeterminanth mdr_p_zero_columns_rankzeroevaluationdeterminanth mdr_n_zero_columns_rankzeroevaluationdeterminanth. ((exists mdr_z_zero_columns_rankzeroevaluationdeterminanthr. ((exists mdr_a_zero_columns_rankzeroevaluationdeterminanthrc mdr_b_zero_columns_rankzeroevaluationdeterminanthrc mdr_c_zero_columns_rankzeroevaluationdeterminanthrc mdr_e_zero_columns_rankzeroevaluationdeterminanthrc mdr_f_zero_columns_rankzeroevaluationdeterminanthrc. ((mdr_a_zero_columns_rankzeroevaluationdeterminanthrc = ((mdr_d_zero_columns_rankzeroevaluationdeterminanth) + (mdr_pb_zero_columns_rankzeroevaluationdeterminanth)) * S ((mdr_d_zero_columns_rankzeroevaluationdeterminanth) + (mdr_pb_zero_columns_rankzeroevaluationdeterminanth)) + ((mdr_pb_zero_columns_rankzeroevaluationdeterminanth) + (mdr_pb_zero_columns_rankzeroevaluationdeterminanth))) /\ ((mdr_b_zero_columns_rankzeroevaluationdeterminanthrc = ((mdr_pc_zero_columns_rankzeroevaluationdeterminanth) + (mdr_nb_zero_columns_rankzeroevaluationdeterminanth)) * S ((mdr_pc_zero_columns_rankzeroevaluationdeterminanth) + (mdr_nb_zero_columns_rankzeroevaluationdeterminanth)) + ((mdr_nb_zero_columns_rankzeroevaluationdeterminanth) + (mdr_nb_zero_columns_rankzeroevaluationdeterminanth))) /\ ((mdr_c_zero_columns_rankzeroevaluationdeterminanthrc = ((mdr_a_zero_columns_rankzeroevaluationdeterminanthrc) + (mdr_b_zero_columns_rankzeroevaluationdeterminanthrc)) * S ((mdr_a_zero_columns_rankzeroevaluationdeterminanthrc) + (mdr_b_zero_columns_rankzeroevaluationdeterminanthrc)) + ((mdr_b_zero_columns_rankzeroevaluationdeterminanthrc) + (mdr_b_zero_columns_rankzeroevaluationdeterminanthrc))) /\ ((mdr_e_zero_columns_rankzeroevaluationdeterminanthrc = ((mdr_p_zero_columns_rankzeroevaluationdeterminanth) + (mdr_n_zero_columns_rankzeroevaluationdeterminanth)) * S ((mdr_p_zero_columns_rankzeroevaluationdeterminanth) + (mdr_n_zero_columns_rankzeroevaluationdeterminanth)) + ((mdr_n_zero_columns_rankzeroevaluationdeterminanth) + (mdr_n_zero_columns_rankzeroevaluationdeterminanth))) /\ ((mdr_f_zero_columns_rankzeroevaluationdeterminanthrc = ((mdr_nc_zero_columns_rankzeroevaluationdeterminanth) + (mdr_e_zero_columns_rankzeroevaluationdeterminanthrc)) * S ((mdr_nc_zero_columns_rankzeroevaluationdeterminanth) + (mdr_e_zero_columns_rankzeroevaluationdeterminanthrc)) + ((mdr_e_zero_columns_rankzeroevaluationdeterminanthrc) + (mdr_e_zero_columns_rankzeroevaluationdeterminanthrc))) /\ ((mdr_z_zero_columns_rankzeroevaluationdeterminanthr) = ((mdr_c_zero_columns_rankzeroevaluationdeterminanthrc) + (mdr_f_zero_columns_rankzeroevaluationdeterminanthrc)) * S ((mdr_c_zero_columns_rankzeroevaluationdeterminanthrc) + (mdr_f_zero_columns_rankzeroevaluationdeterminanthrc)) + ((mdr_f_zero_columns_rankzeroevaluationdeterminanthrc) + (mdr_f_zero_columns_rankzeroevaluationdeterminanthrc))))))))) /\ (((exists ff_h_mdr_zero_columns_rankzeroevaluationdeterminanthrb. ff_h_mdr_zero_columns_rankzeroevaluationdeterminanthrb + S (mdr_z_zero_columns_rankzeroevaluationdeterminanthr) = S ((S (mdr_i_zero_columns_rankzeroevaluationdeterminanth)) * mdr_c_zero_columns_rankzeroevaluationdeterminant)) /\ exists ff_q_mdr_zero_columns_rankzeroevaluationdeterminanthrb. mdr_b_zero_columns_rankzeroevaluationdeterminant = ff_q_mdr_zero_columns_rankzeroevaluationdeterminanthrb * S ((S (mdr_i_zero_columns_rankzeroevaluationdeterminanth)) * mdr_c_zero_columns_rankzeroevaluationdeterminant) + (mdr_z_zero_columns_rankzeroevaluationdeterminanthr))))) /\ (((((mdr_d_zero_columns_rankzeroevaluationdeterminanth) = 0) /\ (((mdr_p_zero_columns_rankzeroevaluationdeterminanth) = 1) /\ ((mdr_n_zero_columns_rankzeroevaluationdeterminanth) = 0))) \/ exists mdr_q_zero_columns_rankzeroevaluationdeterminanths mdr_eb_zero_columns_rankzeroevaluationdeterminanths mdr_ec_zero_columns_rankzeroevaluationdeterminanths mdr_fb_zero_columns_rankzeroevaluationdeterminanths mdr_fc_zero_columns_rankzeroevaluationdeterminanths. (((mdr_d_zero_columns_rankzeroevaluationdeterminanth) = S (mdr_q_zero_columns_rankzeroevaluationdeterminanths)) /\ ((forall mdr_j_zero_columns_rankzeroevaluationdeterminanthsc. (exists mdr_gap_zero_columns_rankzeroevaluationdeterminanthscj. mdr_gap_zero_columns_rankzeroevaluationdeterminanthscj + S (mdr_j_zero_columns_rankzeroevaluationdeterminanthsc) = (S (mdr_q_zero_columns_rankzeroevaluationdeterminanths))) -> exists mdr_i_zero_columns_rankzeroevaluationdeterminanthsc mdr_up_zero_columns_rankzeroevaluationdeterminanthsc mdr_us_zero_columns_rankzeroevaluationdeterminanthsc mdr_un_zero_columns_rankzeroevaluationdeterminanthsc mdr_ut_zero_columns_rankzeroevaluationdeterminanthsc mdr_p_zero_columns_rankzeroevaluationdeterminanthsc mdr_n_zero_columns_rankzeroevaluationdeterminanthsc. ((exists mdr_gap_zero_columns_rankzeroevaluationdeterminanthsci. mdr_gap_zero_columns_rankzeroevaluationdeterminanthsci + S (mdr_i_zero_columns_rankzeroevaluationdeterminanthsc) = (mdr_i_zero_columns_rankzeroevaluationdeterminanth)) /\ ((exists mdr_z_zero_columns_rankzeroevaluationdeterminanthscr. ((exists mdr_a_zero_columns_rankzeroevaluationdeterminanthscrc mdr_b_zero_columns_rankzeroevaluationdeterminanthscrc mdr_c_zero_columns_rankzeroevaluationdeterminanthscrc mdr_e_zero_columns_rankzeroevaluationdeterminanthscrc mdr_f_zero_columns_rankzeroevaluationdeterminanthscrc. ((mdr_a_zero_columns_rankzeroevaluationdeterminanthscrc = ((mdr_q_zero_columns_rankzeroevaluationdeterminanths) + (mdr_up_zero_columns_rankzeroevaluationdeterminanthsc)) * S ((mdr_q_zero_columns_rankzeroevaluationdeterminanths) + (mdr_up_zero_columns_rankzeroevaluationdeterminanthsc)) + ((mdr_up_zero_columns_rankzeroevaluationdeterminanthsc) + (mdr_up_zero_columns_rankzeroevaluationdeterminanthsc))) /\ ((mdr_b_zero_columns_rankzeroevaluationdeterminanthscrc = ((mdr_us_zero_columns_rankzeroevaluationdeterminanthsc) + (mdr_un_zero_columns_rankzeroevaluationdeterminanthsc)) * S ((mdr_us_zero_columns_rankzeroevaluationdeterminanthsc) + (mdr_un_zero_columns_rankzeroevaluationdeterminanthsc)) + ((mdr_un_zero_columns_rankzeroevaluationdeterminanthsc) + (mdr_un_zero_columns_rankzeroevaluationdeterminanthsc))) /\ ((mdr_c_zero_columns_rankzeroevaluationdeterminanthscrc = ((mdr_a_zero_columns_rankzeroevaluationdeterminanthscrc) + (mdr_b_zero_columns_rankzeroevaluationdeterminanthscrc)) * S ((mdr_a_zero_columns_rankzeroevaluationdeterminanthscrc) + (mdr_b_zero_columns_rankzeroevaluationdeterminanthscrc)) + ((mdr_b_zero_columns_rankzeroevaluationdeterminanthscrc) + (mdr_b_zero_columns_rankzeroevaluationdeterminanthscrc))) /\ ((mdr_e_zero_columns_rankzeroevaluationdeterminanthscrc = ((mdr_p_zero_columns_rankzeroevaluationdeterminanthsc) + (mdr_n_zero_columns_rankzeroevaluationdeterminanthsc)) * S ((mdr_p_zero_columns_rankzeroevaluationdeterminanthsc) + (mdr_n_zero_columns_rankzeroevaluationdeterminanthsc)) + ((mdr_n_zero_columns_rankzeroevaluationdeterminanthsc) + (mdr_n_zero_columns_rankzeroevaluationdeterminanthsc))) /\ ((mdr_f_zero_columns_rankzeroevaluationdeterminanthscrc = ((mdr_ut_zero_columns_rankzeroevaluationdeterminanthsc) + (mdr_e_zero_columns_rankzeroevaluationdeterminanthscrc)) * S ((mdr_ut_zero_columns_rankzeroevaluationdeterminanthsc) + (mdr_e_zero_columns_rankzeroevaluationdeterminanthscrc)) + ((mdr_e_zero_columns_rankzeroevaluationdeterminanthscrc) + (mdr_e_zero_columns_rankzeroevaluationdeterminanthscrc))) /\ ((mdr_z_zero_columns_rankzeroevaluationdeterminanthscr) = ((mdr_c_zero_columns_rankzeroevaluationdeterminanthscrc) + (mdr_f_zero_columns_rankzeroevaluationdeterminanthscrc)) * S ((mdr_c_zero_columns_rankzeroevaluationdeterminanthscrc) + (mdr_f_zero_columns_rankzeroevaluationdeterminanthscrc)) + ((mdr_f_zero_columns_rankzeroevaluationdeterminanthscrc) + (mdr_f_zero_columns_rankzeroevaluationdeterminanthscrc))))))))) /\ (((exists ff_h_mdr_zero_columns_rankzeroevaluationdeterminanthscrb. ff_h_mdr_zero_columns_rankzeroevaluationdeterminanthscrb + S (mdr_z_zero_columns_rankzeroevaluationdeterminanthscr) = S ((S (mdr_i_zero_columns_rankzeroevaluationdeterminanthsc)) * mdr_c_zero_columns_rankzeroevaluationdeterminant)) /\ exists ff_q_mdr_zero_columns_rankzeroevaluationdeterminanthscrb. mdr_b_zero_columns_rankzeroevaluationdeterminant = ff_q_mdr_zero_columns_rankzeroevaluationdeterminanthscrb * S ((S (mdr_i_zero_columns_rankzeroevaluationdeterminanthsc)) * mdr_c_zero_columns_rankzeroevaluationdeterminant) + (mdr_z_zero_columns_rankzeroevaluationdeterminanthscr))))) /\ ((((forall ff_index_mdm_prefix_mdr_zero_columns_rankzeroevaluationdeterminanthscm_positive. (exists ff_gap_mdm_lt_mdr_zero_columns_rankzeroevaluationdeterminanthscm_positive_index_bound. ff_gap_mdm_lt_mdr_zero_columns_rankzeroevaluationdeterminanthscm_positive_index_bound + S (ff_index_mdm_prefix_mdr_zero_columns_rankzeroevaluationdeterminanthscm_positive) = ((mdr_q_zero_columns_rankzeroevaluationdeterminanths) * (mdr_q_zero_columns_rankzeroevaluationdeterminanths))) -> exists ff_row_mdm_prefix_mdr_zero_columns_rankzeroevaluationdeterminanthscm_positive ff_column_mdm_prefix_mdr_zero_columns_rankzeroevaluationdeterminanthscm_positive ff_value_mdm_prefix_mdr_zero_columns_rankzeroevaluationdeterminanthscm_positive. (ff_index_mdm_prefix_mdr_zero_columns_rankzeroevaluationdeterminanthscm_positive = (mdr_q_zero_columns_rankzeroevaluationdeterminanths) * ff_row_mdm_prefix_mdr_zero_columns_rankzeroevaluationdeterminanthscm_positive + ff_column_mdm_prefix_mdr_zero_columns_rankzeroevaluationdeterminanthscm_positive /\ ((exists ff_gap_mdm_lt_mdr_zero_columns_rankzeroevaluationdeterminanthscm_positive_column_bound. ff_gap_mdm_lt_mdr_zero_columns_rankzeroevaluationdeterminanthscm_positive_column_bound + S (ff_column_mdm_prefix_mdr_zero_columns_rankzeroevaluationdeterminanthscm_positive) = (mdr_q_zero_columns_rankzeroevaluationdeterminanths)) /\ ((exists ff_row_mdm_cell_mdr_zero_columns_rankzeroevaluationdeterminanthscm_positive_cell ff_column_mdm_cell_mdr_zero_columns_rankzeroevaluationdeterminanthscm_positive_cell. (((((exists ff_gap_mdm_lt_mdr_zero_columns_rankzeroevaluationdeterminanthscm_positive_cell_row_before. ff_gap_mdm_lt_mdr_zero_columns_rankzeroevaluationdeterminanthscm_positive_cell_row_before + S (ff_row_mdm_prefix_mdr_zero_columns_rankzeroevaluationdeterminanthscm_positive) = (0)) /\ ff_row_mdm_cell_mdr_zero_columns_rankzeroevaluationdeterminanthscm_positive_cell = ff_row_mdm_prefix_mdr_zero_columns_rankzeroevaluationdeterminanthscm_positive) \/ ((exists ff_gap_mdm_le_mdr_zero_columns_rankzeroevaluationdeterminanthscm_positive_cell_row_after. ff_gap_mdm_le_mdr_zero_columns_rankzeroevaluationdeterminanthscm_positive_cell_row_after + (0) = (ff_row_mdm_prefix_mdr_zero_columns_rankzeroevaluationdeterminanthscm_positive)) /\ ff_row_mdm_cell_mdr_zero_columns_rankzeroevaluationdeterminanthscm_positive_cell = S ff_row_mdm_prefix_mdr_zero_columns_rankzeroevaluationdeterminanthscm_positive))) /\ (((((exists ff_gap_mdm_lt_mdr_zero_columns_rankzeroevaluationdeterminanthscm_positive_cell_column_before. ff_gap_mdm_lt_mdr_zero_columns_rankzeroevaluationdeterminanthscm_positive_cell_column_before + S (ff_column_mdm_prefix_mdr_zero_columns_rankzeroevaluationdeterminanthscm_positive) = (mdr_j_zero_columns_rankzeroevaluationdeterminanthsc)) /\ ff_column_mdm_cell_mdr_zero_columns_rankzeroevaluationdeterminanthscm_positive_cell = ff_column_mdm_prefix_mdr_zero_columns_rankzeroevaluationdeterminanthscm_positive) \/ ((exists ff_gap_mdm_le_mdr_zero_columns_rankzeroevaluationdeterminanthscm_positive_cell_column_after. ff_gap_mdm_le_mdr_zero_columns_rankzeroevaluationdeterminanthscm_positive_cell_column_after + (mdr_j_zero_columns_rankzeroevaluationdeterminanthsc) = (ff_column_mdm_prefix_mdr_zero_columns_rankzeroevaluationdeterminanthscm_positive)) /\ ff_column_mdm_cell_mdr_zero_columns_rankzeroevaluationdeterminanthscm_positive_cell = S ff_column_mdm_prefix_mdr_zero_columns_rankzeroevaluationdeterminanthscm_positive))) /\ (((exists ff_h_mdm_mdr_zero_columns_rankzeroevaluationdeterminanthscm_positive_cell_source. ff_h_mdm_mdr_zero_columns_rankzeroevaluationdeterminanthscm_positive_cell_source + S (ff_value_mdm_prefix_mdr_zero_columns_rankzeroevaluationdeterminanthscm_positive) = S ((S ((ff_row_mdm_cell_mdr_zero_columns_rankzeroevaluationdeterminanthscm_positive_cell) * (S (mdr_q_zero_columns_rankzeroevaluationdeterminanths)) + (ff_column_mdm_cell_mdr_zero_columns_rankzeroevaluationdeterminanthscm_positive_cell))) * mdr_pc_zero_columns_rankzeroevaluationdeterminanth)) /\ exists ff_q_mdm_mdr_zero_columns_rankzeroevaluationdeterminanthscm_positive_cell_source. mdr_pb_zero_columns_rankzeroevaluationdeterminanth = ff_q_mdm_mdr_zero_columns_rankzeroevaluationdeterminanthscm_positive_cell_source * S ((S ((ff_row_mdm_cell_mdr_zero_columns_rankzeroevaluationdeterminanthscm_positive_cell) * (S (mdr_q_zero_columns_rankzeroevaluationdeterminanths)) + (ff_column_mdm_cell_mdr_zero_columns_rankzeroevaluationdeterminanthscm_positive_cell))) * mdr_pc_zero_columns_rankzeroevaluationdeterminanth) + (ff_value_mdm_prefix_mdr_zero_columns_rankzeroevaluationdeterminanthscm_positive)))))) /\ (((exists ff_h_mdm_mdr_zero_columns_rankzeroevaluationdeterminanthscm_positive_target. ff_h_mdm_mdr_zero_columns_rankzeroevaluationdeterminanthscm_positive_target + S (ff_value_mdm_prefix_mdr_zero_columns_rankzeroevaluationdeterminanthscm_positive) = S ((S (ff_index_mdm_prefix_mdr_zero_columns_rankzeroevaluationdeterminanthscm_positive)) * mdr_us_zero_columns_rankzeroevaluationdeterminanthsc)) /\ exists ff_q_mdm_mdr_zero_columns_rankzeroevaluationdeterminanthscm_positive_target. mdr_up_zero_columns_rankzeroevaluationdeterminanthsc = ff_q_mdm_mdr_zero_columns_rankzeroevaluationdeterminanthscm_positive_target * S ((S (ff_index_mdm_prefix_mdr_zero_columns_rankzeroevaluationdeterminanthscm_positive)) * mdr_us_zero_columns_rankzeroevaluationdeterminanthsc) + (ff_value_mdm_prefix_mdr_zero_columns_rankzeroevaluationdeterminanthscm_positive))))))) /\ (forall ff_index_mdm_prefix_mdr_zero_columns_rankzeroevaluationdeterminanthscm_negative. (exists ff_gap_mdm_lt_mdr_zero_columns_rankzeroevaluationdeterminanthscm_negative_index_bound. ff_gap_mdm_lt_mdr_zero_columns_rankzeroevaluationdeterminanthscm_negative_index_bound + S (ff_index_mdm_prefix_mdr_zero_columns_rankzeroevaluationdeterminanthscm_negative) = ((mdr_q_zero_columns_rankzeroevaluationdeterminanths) * (mdr_q_zero_columns_rankzeroevaluationdeterminanths))) -> exists ff_row_mdm_prefix_mdr_zero_columns_rankzeroevaluationdeterminanthscm_negative ff_column_mdm_prefix_mdr_zero_columns_rankzeroevaluationdeterminanthscm_negative ff_value_mdm_prefix_mdr_zero_columns_rankzeroevaluationdeterminanthscm_negative. (ff_index_mdm_prefix_mdr_zero_columns_rankzeroevaluationdeterminanthscm_negative = (mdr_q_zero_columns_rankzeroevaluationdeterminanths) * ff_row_mdm_prefix_mdr_zero_columns_rankzeroevaluationdeterminanthscm_negative + ff_column_mdm_prefix_mdr_zero_columns_rankzeroevaluationdeterminanthscm_negative /\ ((exists ff_gap_mdm_lt_mdr_zero_columns_rankzeroevaluationdeterminanthscm_negative_column_bound. ff_gap_mdm_lt_mdr_zero_columns_rankzeroevaluationdeterminanthscm_negative_column_bound + S (ff_column_mdm_prefix_mdr_zero_columns_rankzeroevaluationdeterminanthscm_negative) = (mdr_q_zero_columns_rankzeroevaluationdeterminanths)) /\ ((exists ff_row_mdm_cell_mdr_zero_columns_rankzeroevaluationdeterminanthscm_negative_cell ff_column_mdm_cell_mdr_zero_columns_rankzeroevaluationdeterminanthscm_negative_cell. (((((exists ff_gap_mdm_lt_mdr_zero_columns_rankzeroevaluationdeterminanthscm_negative_cell_row_before. ff_gap_mdm_lt_mdr_zero_columns_rankzeroevaluationdeterminanthscm_negative_cell_row_before + S (ff_row_mdm_prefix_mdr_zero_columns_rankzeroevaluationdeterminanthscm_negative) = (0)) /\ ff_row_mdm_cell_mdr_zero_columns_rankzeroevaluationdeterminanthscm_negative_cell = ff_row_mdm_prefix_mdr_zero_columns_rankzeroevaluationdeterminanthscm_negative) \/ ((exists ff_gap_mdm_le_mdr_zero_columns_rankzeroevaluationdeterminanthscm_negative_cell_row_after. ff_gap_mdm_le_mdr_zero_columns_rankzeroevaluationdeterminanthscm_negative_cell_row_after + (0) = (ff_row_mdm_prefix_mdr_zero_columns_rankzeroevaluationdeterminanthscm_negative)) /\ ff_row_mdm_cell_mdr_zero_columns_rankzeroevaluationdeterminanthscm_negative_cell = S ff_row_mdm_prefix_mdr_zero_columns_rankzeroevaluationdeterminanthscm_negative))) /\ (((((exists ff_gap_mdm_lt_mdr_zero_columns_rankzeroevaluationdeterminanthscm_negative_cell_column_before. ff_gap_mdm_lt_mdr_zero_columns_rankzeroevaluationdeterminanthscm_negative_cell_column_before + S (ff_column_mdm_prefix_mdr_zero_columns_rankzeroevaluationdeterminanthscm_negative) = (mdr_j_zero_columns_rankzeroevaluationdeterminanthsc)) /\ ff_column_mdm_cell_mdr_zero_columns_rankzeroevaluationdeterminanthscm_negative_cell = ff_column_mdm_prefix_mdr_zero_columns_rankzeroevaluationdeterminanthscm_negative) \/ ((exists ff_gap_mdm_le_mdr_zero_columns_rankzeroevaluationdeterminanthscm_negative_cell_column_after. ff_gap_mdm_le_mdr_zero_columns_rankzeroevaluationdeterminanthscm_negative_cell_column_after + (mdr_j_zero_columns_rankzeroevaluationdeterminanthsc) = (ff_column_mdm_prefix_mdr_zero_columns_rankzeroevaluationdeterminanthscm_negative)) /\ ff_column_mdm_cell_mdr_zero_columns_rankzeroevaluationdeterminanthscm_negative_cell = S ff_column_mdm_prefix_mdr_zero_columns_rankzeroevaluationdeterminanthscm_negative))) /\ (((exists ff_h_mdm_mdr_zero_columns_rankzeroevaluationdeterminanthscm_negative_cell_source. ff_h_mdm_mdr_zero_columns_rankzeroevaluationdeterminanthscm_negative_cell_source + S (ff_value_mdm_prefix_mdr_zero_columns_rankzeroevaluationdeterminanthscm_negative) = S ((S ((ff_row_mdm_cell_mdr_zero_columns_rankzeroevaluationdeterminanthscm_negative_cell) * (S (mdr_q_zero_columns_rankzeroevaluationdeterminanths)) + (ff_column_mdm_cell_mdr_zero_columns_rankzeroevaluationdeterminanthscm_negative_cell))) * mdr_nc_zero_columns_rankzeroevaluationdeterminanth)) /\ exists ff_q_mdm_mdr_zero_columns_rankzeroevaluationdeterminanthscm_negative_cell_source. mdr_nb_zero_columns_rankzeroevaluationdeterminanth = ff_q_mdm_mdr_zero_columns_rankzeroevaluationdeterminanthscm_negative_cell_source * S ((S ((ff_row_mdm_cell_mdr_zero_columns_rankzeroevaluationdeterminanthscm_negative_cell) * (S (mdr_q_zero_columns_rankzeroevaluationdeterminanths)) + (ff_column_mdm_cell_mdr_zero_columns_rankzeroevaluationdeterminanthscm_negative_cell))) * mdr_nc_zero_columns_rankzeroevaluationdeterminanth) + (ff_value_mdm_prefix_mdr_zero_columns_rankzeroevaluationdeterminanthscm_negative)))))) /\ (((exists ff_h_mdm_mdr_zero_columns_rankzeroevaluationdeterminanthscm_negative_target. ff_h_mdm_mdr_zero_columns_rankzeroevaluationdeterminanthscm_negative_target + S (ff_value_mdm_prefix_mdr_zero_columns_rankzeroevaluationdeterminanthscm_negative) = S ((S (ff_index_mdm_prefix_mdr_zero_columns_rankzeroevaluationdeterminanthscm_negative)) * mdr_ut_zero_columns_rankzeroevaluationdeterminanthsc)) /\ exists ff_q_mdm_mdr_zero_columns_rankzeroevaluationdeterminanthscm_negative_target. mdr_un_zero_columns_rankzeroevaluationdeterminanthsc = ff_q_mdm_mdr_zero_columns_rankzeroevaluationdeterminanthscm_negative_target * S ((S (ff_index_mdm_prefix_mdr_zero_columns_rankzeroevaluationdeterminanthscm_negative)) * mdr_ut_zero_columns_rankzeroevaluationdeterminanthsc) + (ff_value_mdm_prefix_mdr_zero_columns_rankzeroevaluationdeterminanthscm_negative))))))))) /\ ((((exists ff_h_mdr_zero_columns_rankzeroevaluationdeterminanthscp. ff_h_mdr_zero_columns_rankzeroevaluationdeterminanthscp + S (mdr_p_zero_columns_rankzeroevaluationdeterminanthsc) = S ((S (mdr_j_zero_columns_rankzeroevaluationdeterminanthsc)) * mdr_ec_zero_columns_rankzeroevaluationdeterminanths)) /\ exists ff_q_mdr_zero_columns_rankzeroevaluationdeterminanthscp. mdr_eb_zero_columns_rankzeroevaluationdeterminanths = ff_q_mdr_zero_columns_rankzeroevaluationdeterminanthscp * S ((S (mdr_j_zero_columns_rankzeroevaluationdeterminanthsc)) * mdr_ec_zero_columns_rankzeroevaluationdeterminanths) + (mdr_p_zero_columns_rankzeroevaluationdeterminanthsc))) /\ (((exists ff_h_mdr_zero_columns_rankzeroevaluationdeterminanthscn. ff_h_mdr_zero_columns_rankzeroevaluationdeterminanthscn + S (mdr_n_zero_columns_rankzeroevaluationdeterminanthsc) = S ((S (mdr_j_zero_columns_rankzeroevaluationdeterminanthsc)) * mdr_fc_zero_columns_rankzeroevaluationdeterminanths)) /\ exists ff_q_mdr_zero_columns_rankzeroevaluationdeterminanthscn. mdr_fb_zero_columns_rankzeroevaluationdeterminanths = ff_q_mdr_zero_columns_rankzeroevaluationdeterminanthscn * S ((S (mdr_j_zero_columns_rankzeroevaluationdeterminanthsc)) * mdr_fc_zero_columns_rankzeroevaluationdeterminanths) + (mdr_n_zero_columns_rankzeroevaluationdeterminanthsc)))))))) /\ (exists ff_ub_mce_fold_mdr_zero_columns_rankzeroevaluationdeterminanthsf ff_uc_mce_fold_mdr_zero_columns_rankzeroevaluationdeterminanthsf ff_vb_mce_fold_mdr_zero_columns_rankzeroevaluationdeterminanthsf ff_vc_mce_fold_mdr_zero_columns_rankzeroevaluationdeterminanthsf. ((forall ff_index_mce_alternating_mdr_zero_columns_rankzeroevaluationdeterminanthsf_prefix. (exists ff_gap_mce_mdr_zero_columns_rankzeroevaluationdeterminanthsf_prefix_index. ff_gap_mce_mdr_zero_columns_rankzeroevaluationdeterminanthsf_prefix_index + S (ff_index_mce_alternating_mdr_zero_columns_rankzeroevaluationdeterminanthsf_prefix) = (S (mdr_q_zero_columns_rankzeroevaluationdeterminanths))) -> exists ff_ap_mce_alternating_mdr_zero_columns_rankzeroevaluationdeterminanthsf_prefix ff_an_mce_alternating_mdr_zero_columns_rankzeroevaluationdeterminanthsf_prefix ff_bp_mce_alternating_mdr_zero_columns_rankzeroevaluationdeterminanthsf_prefix ff_bn_mce_alternating_mdr_zero_columns_rankzeroevaluationdeterminanthsf_prefix ff_p_mce_alternating_mdr_zero_columns_rankzeroevaluationdeterminanthsf_prefix ff_n_mce_alternating_mdr_zero_columns_rankzeroevaluationdeterminanthsf_prefix. ((((exists ff_h_mce_mdr_zero_columns_rankzeroevaluationdeterminanthsf_prefix_ap. ff_h_mce_mdr_zero_columns_rankzeroevaluationdeterminanthsf_prefix_ap + S (ff_ap_mce_alternating_mdr_zero_columns_rankzeroevaluationdeterminanthsf_prefix) = S ((S (ff_index_mce_alternating_mdr_zero_columns_rankzeroevaluationdeterminanthsf_prefix)) * mdr_pc_zero_columns_rankzeroevaluationdeterminanth)) /\ exists ff_q_mce_mdr_zero_columns_rankzeroevaluationdeterminanthsf_prefix_ap. mdr_pb_zero_columns_rankzeroevaluationdeterminanth = ff_q_mce_mdr_zero_columns_rankzeroevaluationdeterminanthsf_prefix_ap * S ((S (ff_index_mce_alternating_mdr_zero_columns_rankzeroevaluationdeterminanthsf_prefix)) * mdr_pc_zero_columns_rankzeroevaluationdeterminanth) + (ff_ap_mce_alternating_mdr_zero_columns_rankzeroevaluationdeterminanthsf_prefix))) /\ ((((exists ff_h_mce_mdr_zero_columns_rankzeroevaluationdeterminanthsf_prefix_an. ff_h_mce_mdr_zero_columns_rankzeroevaluationdeterminanthsf_prefix_an + S (ff_an_mce_alternating_mdr_zero_columns_rankzeroevaluationdeterminanthsf_prefix) = S ((S (ff_index_mce_alternating_mdr_zero_columns_rankzeroevaluationdeterminanthsf_prefix)) * mdr_nc_zero_columns_rankzeroevaluationdeterminanth)) /\ exists ff_q_mce_mdr_zero_columns_rankzeroevaluationdeterminanthsf_prefix_an. mdr_nb_zero_columns_rankzeroevaluationdeterminanth = ff_q_mce_mdr_zero_columns_rankzeroevaluationdeterminanthsf_prefix_an * S ((S (ff_index_mce_alternating_mdr_zero_columns_rankzeroevaluationdeterminanthsf_prefix)) * mdr_nc_zero_columns_rankzeroevaluationdeterminanth) + (ff_an_mce_alternating_mdr_zero_columns_rankzeroevaluationdeterminanthsf_prefix))) /\ ((((exists ff_h_mce_mdr_zero_columns_rankzeroevaluationdeterminanthsf_prefix_bp. ff_h_mce_mdr_zero_columns_rankzeroevaluationdeterminanthsf_prefix_bp + S (ff_bp_mce_alternating_mdr_zero_columns_rankzeroevaluationdeterminanthsf_prefix) = S ((S (ff_index_mce_alternating_mdr_zero_columns_rankzeroevaluationdeterminanthsf_prefix)) * mdr_ec_zero_columns_rankzeroevaluationdeterminanths)) /\ exists ff_q_mce_mdr_zero_columns_rankzeroevaluationdeterminanthsf_prefix_bp. mdr_eb_zero_columns_rankzeroevaluationdeterminanths = ff_q_mce_mdr_zero_columns_rankzeroevaluationdeterminanthsf_prefix_bp * S ((S (ff_index_mce_alternating_mdr_zero_columns_rankzeroevaluationdeterminanthsf_prefix)) * mdr_ec_zero_columns_rankzeroevaluationdeterminanths) + (ff_bp_mce_alternating_mdr_zero_columns_rankzeroevaluationdeterminanthsf_prefix))) /\ ((((exists ff_h_mce_mdr_zero_columns_rankzeroevaluationdeterminanthsf_prefix_bn. ff_h_mce_mdr_zero_columns_rankzeroevaluationdeterminanthsf_prefix_bn + S (ff_bn_mce_alternating_mdr_zero_columns_rankzeroevaluationdeterminanthsf_prefix) = S ((S (ff_index_mce_alternating_mdr_zero_columns_rankzeroevaluationdeterminanthsf_prefix)) * mdr_fc_zero_columns_rankzeroevaluationdeterminanths)) /\ exists ff_q_mce_mdr_zero_columns_rankzeroevaluationdeterminanthsf_prefix_bn. mdr_fb_zero_columns_rankzeroevaluationdeterminanths = ff_q_mce_mdr_zero_columns_rankzeroevaluationdeterminanthsf_prefix_bn * S ((S (ff_index_mce_alternating_mdr_zero_columns_rankzeroevaluationdeterminanthsf_prefix)) * mdr_fc_zero_columns_rankzeroevaluationdeterminanths) + (ff_bn_mce_alternating_mdr_zero_columns_rankzeroevaluationdeterminanthsf_prefix))) /\ ((((exists ff_h_mce_mdr_zero_columns_rankzeroevaluationdeterminanthsf_prefix_positive. ff_h_mce_mdr_zero_columns_rankzeroevaluationdeterminanthsf_prefix_positive + S (ff_p_mce_alternating_mdr_zero_columns_rankzeroevaluationdeterminanthsf_prefix) = S ((S (ff_index_mce_alternating_mdr_zero_columns_rankzeroevaluationdeterminanthsf_prefix)) * ff_uc_mce_fold_mdr_zero_columns_rankzeroevaluationdeterminanthsf)) /\ exists ff_q_mce_mdr_zero_columns_rankzeroevaluationdeterminanthsf_prefix_positive. ff_ub_mce_fold_mdr_zero_columns_rankzeroevaluationdeterminanthsf = ff_q_mce_mdr_zero_columns_rankzeroevaluationdeterminanthsf_prefix_positive * S ((S (ff_index_mce_alternating_mdr_zero_columns_rankzeroevaluationdeterminanthsf_prefix)) * ff_uc_mce_fold_mdr_zero_columns_rankzeroevaluationdeterminanthsf) + (ff_p_mce_alternating_mdr_zero_columns_rankzeroevaluationdeterminanthsf_prefix))) /\ ((((exists ff_h_mce_mdr_zero_columns_rankzeroevaluationdeterminanthsf_prefix_negative. ff_h_mce_mdr_zero_columns_rankzeroevaluationdeterminanthsf_prefix_negative + S (ff_n_mce_alternating_mdr_zero_columns_rankzeroevaluationdeterminanthsf_prefix) = S ((S (ff_index_mce_alternating_mdr_zero_columns_rankzeroevaluationdeterminanthsf_prefix)) * ff_vc_mce_fold_mdr_zero_columns_rankzeroevaluationdeterminanthsf)) /\ exists ff_q_mce_mdr_zero_columns_rankzeroevaluationdeterminanthsf_prefix_negative. ff_vb_mce_fold_mdr_zero_columns_rankzeroevaluationdeterminanthsf = ff_q_mce_mdr_zero_columns_rankzeroevaluationdeterminanthsf_prefix_negative * S ((S (ff_index_mce_alternating_mdr_zero_columns_rankzeroevaluationdeterminanthsf_prefix)) * ff_vc_mce_fold_mdr_zero_columns_rankzeroevaluationdeterminanthsf) + (ff_n_mce_alternating_mdr_zero_columns_rankzeroevaluationdeterminanthsf_prefix))) /\ (((exists ff_even_mce_term_mdr_zero_columns_rankzeroevaluationdeterminanthsf_prefix_term. ff_index_mce_alternating_mdr_zero_columns_rankzeroevaluationdeterminanthsf_prefix = 2 * ff_even_mce_term_mdr_zero_columns_rankzeroevaluationdeterminanthsf_prefix_term) /\ (ff_p_mce_alternating_mdr_zero_columns_rankzeroevaluationdeterminanthsf_prefix = (ff_ap_mce_alternating_mdr_zero_columns_rankzeroevaluationdeterminanthsf_prefix) * (ff_bp_mce_alternating_mdr_zero_columns_rankzeroevaluationdeterminanthsf_prefix) + (ff_an_mce_alternating_mdr_zero_columns_rankzeroevaluationdeterminanthsf_prefix) * (ff_bn_mce_alternating_mdr_zero_columns_rankzeroevaluationdeterminanthsf_prefix) /\ ff_n_mce_alternating_mdr_zero_columns_rankzeroevaluationdeterminanthsf_prefix = (ff_ap_mce_alternating_mdr_zero_columns_rankzeroevaluationdeterminanthsf_prefix) * (ff_bn_mce_alternating_mdr_zero_columns_rankzeroevaluationdeterminanthsf_prefix) + (ff_an_mce_alternating_mdr_zero_columns_rankzeroevaluationdeterminanthsf_prefix) * (ff_bp_mce_alternating_mdr_zero_columns_rankzeroevaluationdeterminanthsf_prefix))) \/ ((exists ff_odd_mce_term_mdr_zero_columns_rankzeroevaluationdeterminanthsf_prefix_term. ff_index_mce_alternating_mdr_zero_columns_rankzeroevaluationdeterminanthsf_prefix = 2 * ff_odd_mce_term_mdr_zero_columns_rankzeroevaluationdeterminanthsf_prefix_term + 1) /\ (ff_p_mce_alternating_mdr_zero_columns_rankzeroevaluationdeterminanthsf_prefix = (ff_ap_mce_alternating_mdr_zero_columns_rankzeroevaluationdeterminanthsf_prefix) * (ff_bn_mce_alternating_mdr_zero_columns_rankzeroevaluationdeterminanthsf_prefix) + (ff_an_mce_alternating_mdr_zero_columns_rankzeroevaluationdeterminanthsf_prefix) * (ff_bp_mce_alternating_mdr_zero_columns_rankzeroevaluationdeterminanthsf_prefix) /\ ff_n_mce_alternating_mdr_zero_columns_rankzeroevaluationdeterminanthsf_prefix = (ff_ap_mce_alternating_mdr_zero_columns_rankzeroevaluationdeterminanthsf_prefix) * (ff_bp_mce_alternating_mdr_zero_columns_rankzeroevaluationdeterminanthsf_prefix) + (ff_an_mce_alternating_mdr_zero_columns_rankzeroevaluationdeterminanthsf_prefix) * (ff_bn_mce_alternating_mdr_zero_columns_rankzeroevaluationdeterminanthsf_prefix))))))))))) /\ ((exists ff_u_mce_mdr_zero_columns_rankzeroevaluationdeterminanthsf_positive ff_v_mce_mdr_zero_columns_rankzeroevaluationdeterminanthsf_positive. ((((exists ff_h_mce_mdr_zero_columns_rankzeroevaluationdeterminanthsf_positive_start. ff_h_mce_mdr_zero_columns_rankzeroevaluationdeterminanthsf_positive_start + S (0) = S ((S (0)) * ff_v_mce_mdr_zero_columns_rankzeroevaluationdeterminanthsf_positive)) /\ exists ff_q_mce_mdr_zero_columns_rankzeroevaluationdeterminanthsf_positive_start. ff_u_mce_mdr_zero_columns_rankzeroevaluationdeterminanthsf_positive = ff_q_mce_mdr_zero_columns_rankzeroevaluationdeterminanthsf_positive_start * S ((S (0)) * ff_v_mce_mdr_zero_columns_rankzeroevaluationdeterminanthsf_positive) + (0))) /\ ((((exists ff_h_mce_mdr_zero_columns_rankzeroevaluationdeterminanthsf_positive_terminal. ff_h_mce_mdr_zero_columns_rankzeroevaluationdeterminanthsf_positive_terminal + S (mdr_p_zero_columns_rankzeroevaluationdeterminanth) = S ((S ((S (mdr_q_zero_columns_rankzeroevaluationdeterminanths)))) * ff_v_mce_mdr_zero_columns_rankzeroevaluationdeterminanthsf_positive)) /\ exists ff_q_mce_mdr_zero_columns_rankzeroevaluationdeterminanthsf_positive_terminal. ff_u_mce_mdr_zero_columns_rankzeroevaluationdeterminanthsf_positive = ff_q_mce_mdr_zero_columns_rankzeroevaluationdeterminanthsf_positive_terminal * S ((S ((S (mdr_q_zero_columns_rankzeroevaluationdeterminanths)))) * ff_v_mce_mdr_zero_columns_rankzeroevaluationdeterminanthsf_positive) + (mdr_p_zero_columns_rankzeroevaluationdeterminanth))) /\ forall ff_i_mce_mdr_zero_columns_rankzeroevaluationdeterminanthsf_positive. (exists ff_lt_mce_mdr_zero_columns_rankzeroevaluationdeterminanthsf_positive_bound. ff_lt_mce_mdr_zero_columns_rankzeroevaluationdeterminanthsf_positive_bound + S ff_i_mce_mdr_zero_columns_rankzeroevaluationdeterminanthsf_positive = (S (mdr_q_zero_columns_rankzeroevaluationdeterminanths))) -> exists ff_a_mce_mdr_zero_columns_rankzeroevaluationdeterminanthsf_positive ff_r_mce_mdr_zero_columns_rankzeroevaluationdeterminanthsf_positive ff_s_mce_mdr_zero_columns_rankzeroevaluationdeterminanthsf_positive. ((((exists ff_h_mce_mdr_zero_columns_rankzeroevaluationdeterminanthsf_positive_summand. ff_h_mce_mdr_zero_columns_rankzeroevaluationdeterminanthsf_positive_summand + S (ff_a_mce_mdr_zero_columns_rankzeroevaluationdeterminanthsf_positive) = S ((S (ff_i_mce_mdr_zero_columns_rankzeroevaluationdeterminanthsf_positive)) * ff_uc_mce_fold_mdr_zero_columns_rankzeroevaluationdeterminanthsf)) /\ exists ff_q_mce_mdr_zero_columns_rankzeroevaluationdeterminanthsf_positive_summand. ff_ub_mce_fold_mdr_zero_columns_rankzeroevaluationdeterminanthsf = ff_q_mce_mdr_zero_columns_rankzeroevaluationdeterminanthsf_positive_summand * S ((S (ff_i_mce_mdr_zero_columns_rankzeroevaluationdeterminanthsf_positive)) * ff_uc_mce_fold_mdr_zero_columns_rankzeroevaluationdeterminanthsf) + (ff_a_mce_mdr_zero_columns_rankzeroevaluationdeterminanthsf_positive))) /\ ((((exists ff_h_mce_mdr_zero_columns_rankzeroevaluationdeterminanthsf_positive_partial. ff_h_mce_mdr_zero_columns_rankzeroevaluationdeterminanthsf_positive_partial + S (ff_r_mce_mdr_zero_columns_rankzeroevaluationdeterminanthsf_positive) = S ((S (ff_i_mce_mdr_zero_columns_rankzeroevaluationdeterminanthsf_positive)) * ff_v_mce_mdr_zero_columns_rankzeroevaluationdeterminanthsf_positive)) /\ exists ff_q_mce_mdr_zero_columns_rankzeroevaluationdeterminanthsf_positive_partial. ff_u_mce_mdr_zero_columns_rankzeroevaluationdeterminanthsf_positive = ff_q_mce_mdr_zero_columns_rankzeroevaluationdeterminanthsf_positive_partial * S ((S (ff_i_mce_mdr_zero_columns_rankzeroevaluationdeterminanthsf_positive)) * ff_v_mce_mdr_zero_columns_rankzeroevaluationdeterminanthsf_positive) + (ff_r_mce_mdr_zero_columns_rankzeroevaluationdeterminanthsf_positive))) /\ ((((exists ff_h_mce_mdr_zero_columns_rankzeroevaluationdeterminanthsf_positive_successor. ff_h_mce_mdr_zero_columns_rankzeroevaluationdeterminanthsf_positive_successor + S (ff_s_mce_mdr_zero_columns_rankzeroevaluationdeterminanthsf_positive) = S ((S (S ff_i_mce_mdr_zero_columns_rankzeroevaluationdeterminanthsf_positive)) * ff_v_mce_mdr_zero_columns_rankzeroevaluationdeterminanthsf_positive)) /\ exists ff_q_mce_mdr_zero_columns_rankzeroevaluationdeterminanthsf_positive_successor. ff_u_mce_mdr_zero_columns_rankzeroevaluationdeterminanthsf_positive = ff_q_mce_mdr_zero_columns_rankzeroevaluationdeterminanthsf_positive_successor * S ((S (S ff_i_mce_mdr_zero_columns_rankzeroevaluationdeterminanthsf_positive)) * ff_v_mce_mdr_zero_columns_rankzeroevaluationdeterminanthsf_positive) + (ff_s_mce_mdr_zero_columns_rankzeroevaluationdeterminanthsf_positive))) /\ ff_s_mce_mdr_zero_columns_rankzeroevaluationdeterminanthsf_positive = ff_r_mce_mdr_zero_columns_rankzeroevaluationdeterminanthsf_positive + ff_a_mce_mdr_zero_columns_rankzeroevaluationdeterminanthsf_positive)))))) /\ (exists ff_u_mce_mdr_zero_columns_rankzeroevaluationdeterminanthsf_negative ff_v_mce_mdr_zero_columns_rankzeroevaluationdeterminanthsf_negative. ((((exists ff_h_mce_mdr_zero_columns_rankzeroevaluationdeterminanthsf_negative_start. ff_h_mce_mdr_zero_columns_rankzeroevaluationdeterminanthsf_negative_start + S (0) = S ((S (0)) * ff_v_mce_mdr_zero_columns_rankzeroevaluationdeterminanthsf_negative)) /\ exists ff_q_mce_mdr_zero_columns_rankzeroevaluationdeterminanthsf_negative_start. ff_u_mce_mdr_zero_columns_rankzeroevaluationdeterminanthsf_negative = ff_q_mce_mdr_zero_columns_rankzeroevaluationdeterminanthsf_negative_start * S ((S (0)) * ff_v_mce_mdr_zero_columns_rankzeroevaluationdeterminanthsf_negative) + (0))) /\ ((((exists ff_h_mce_mdr_zero_columns_rankzeroevaluationdeterminanthsf_negative_terminal. ff_h_mce_mdr_zero_columns_rankzeroevaluationdeterminanthsf_negative_terminal + S (mdr_n_zero_columns_rankzeroevaluationdeterminanth) = S ((S ((S (mdr_q_zero_columns_rankzeroevaluationdeterminanths)))) * ff_v_mce_mdr_zero_columns_rankzeroevaluationdeterminanthsf_negative)) /\ exists ff_q_mce_mdr_zero_columns_rankzeroevaluationdeterminanthsf_negative_terminal. ff_u_mce_mdr_zero_columns_rankzeroevaluationdeterminanthsf_negative = ff_q_mce_mdr_zero_columns_rankzeroevaluationdeterminanthsf_negative_terminal * S ((S ((S (mdr_q_zero_columns_rankzeroevaluationdeterminanths)))) * ff_v_mce_mdr_zero_columns_rankzeroevaluationdeterminanthsf_negative) + (mdr_n_zero_columns_rankzeroevaluationdeterminanth))) /\ forall ff_i_mce_mdr_zero_columns_rankzeroevaluationdeterminanthsf_negative. (exists ff_lt_mce_mdr_zero_columns_rankzeroevaluationdeterminanthsf_negative_bound. ff_lt_mce_mdr_zero_columns_rankzeroevaluationdeterminanthsf_negative_bound + S ff_i_mce_mdr_zero_columns_rankzeroevaluationdeterminanthsf_negative = (S (mdr_q_zero_columns_rankzeroevaluationdeterminanths))) -> exists ff_a_mce_mdr_zero_columns_rankzeroevaluationdeterminanthsf_negative ff_r_mce_mdr_zero_columns_rankzeroevaluationdeterminanthsf_negative ff_s_mce_mdr_zero_columns_rankzeroevaluationdeterminanthsf_negative. ((((exists ff_h_mce_mdr_zero_columns_rankzeroevaluationdeterminanthsf_negative_summand. ff_h_mce_mdr_zero_columns_rankzeroevaluationdeterminanthsf_negative_summand + S (ff_a_mce_mdr_zero_columns_rankzeroevaluationdeterminanthsf_negative) = S ((S (ff_i_mce_mdr_zero_columns_rankzeroevaluationdeterminanthsf_negative)) * ff_vc_mce_fold_mdr_zero_columns_rankzeroevaluationdeterminanthsf)) /\ exists ff_q_mce_mdr_zero_columns_rankzeroevaluationdeterminanthsf_negative_summand. ff_vb_mce_fold_mdr_zero_columns_rankzeroevaluationdeterminanthsf = ff_q_mce_mdr_zero_columns_rankzeroevaluationdeterminanthsf_negative_summand * S ((S (ff_i_mce_mdr_zero_columns_rankzeroevaluationdeterminanthsf_negative)) * ff_vc_mce_fold_mdr_zero_columns_rankzeroevaluationdeterminanthsf) + (ff_a_mce_mdr_zero_columns_rankzeroevaluationdeterminanthsf_negative))) /\ ((((exists ff_h_mce_mdr_zero_columns_rankzeroevaluationdeterminanthsf_negative_partial. ff_h_mce_mdr_zero_columns_rankzeroevaluationdeterminanthsf_negative_partial + S (ff_r_mce_mdr_zero_columns_rankzeroevaluationdeterminanthsf_negative) = S ((S (ff_i_mce_mdr_zero_columns_rankzeroevaluationdeterminanthsf_negative)) * ff_v_mce_mdr_zero_columns_rankzeroevaluationdeterminanthsf_negative)) /\ exists ff_q_mce_mdr_zero_columns_rankzeroevaluationdeterminanthsf_negative_partial. ff_u_mce_mdr_zero_columns_rankzeroevaluationdeterminanthsf_negative = ff_q_mce_mdr_zero_columns_rankzeroevaluationdeterminanthsf_negative_partial * S ((S (ff_i_mce_mdr_zero_columns_rankzeroevaluationdeterminanthsf_negative)) * ff_v_mce_mdr_zero_columns_rankzeroevaluationdeterminanthsf_negative) + (ff_r_mce_mdr_zero_columns_rankzeroevaluationdeterminanthsf_negative))) /\ ((((exists ff_h_mce_mdr_zero_columns_rankzeroevaluationdeterminanthsf_negative_successor. ff_h_mce_mdr_zero_columns_rankzeroevaluationdeterminanthsf_negative_successor + S (ff_s_mce_mdr_zero_columns_rankzeroevaluationdeterminanthsf_negative) = S ((S (S ff_i_mce_mdr_zero_columns_rankzeroevaluationdeterminanthsf_negative)) * ff_v_mce_mdr_zero_columns_rankzeroevaluationdeterminanthsf_negative)) /\ exists ff_q_mce_mdr_zero_columns_rankzeroevaluationdeterminanthsf_negative_successor. ff_u_mce_mdr_zero_columns_rankzeroevaluationdeterminanthsf_negative = ff_q_mce_mdr_zero_columns_rankzeroevaluationdeterminanthsf_negative_successor * S ((S (S ff_i_mce_mdr_zero_columns_rankzeroevaluationdeterminanthsf_negative)) * ff_v_mce_mdr_zero_columns_rankzeroevaluationdeterminanthsf_negative) + (ff_s_mce_mdr_zero_columns_rankzeroevaluationdeterminanthsf_negative))) /\ ff_s_mce_mdr_zero_columns_rankzeroevaluationdeterminanthsf_negative = ff_r_mce_mdr_zero_columns_rankzeroevaluationdeterminanthsf_negative + ff_a_mce_mdr_zero_columns_rankzeroevaluationdeterminanthsf_negative))))))))))))))) /\ ((exists mdr_gap_zero_columns_rankzeroevaluationdeterminanti. mdr_gap_zero_columns_rankzeroevaluationdeterminanti + S (mdr_i_zero_columns_rankzeroevaluationdeterminant) = (mdr_l_zero_columns_rankzeroevaluationdeterminant)) /\ (exists mdr_z_zero_columns_rankzeroevaluationdeterminantr. ((exists mdr_a_zero_columns_rankzeroevaluationdeterminantrc mdr_b_zero_columns_rankzeroevaluationdeterminantrc mdr_c_zero_columns_rankzeroevaluationdeterminantrc mdr_e_zero_columns_rankzeroevaluationdeterminantrc mdr_f_zero_columns_rankzeroevaluationdeterminantrc. ((mdr_a_zero_columns_rankzeroevaluationdeterminantrc = ((mdr_q_zero_columns_rank) + (mdr_ub_zero_columns_rankzeroevaluation)) * S ((mdr_q_zero_columns_rank) + (mdr_ub_zero_columns_rankzeroevaluation)) + ((mdr_ub_zero_columns_rankzeroevaluation) + (mdr_ub_zero_columns_rankzeroevaluation))) /\ ((mdr_b_zero_columns_rankzeroevaluationdeterminantrc = ((mdr_uc_zero_columns_rankzeroevaluation) + (mdr_vb_zero_columns_rankzeroevaluation)) * S ((mdr_uc_zero_columns_rankzeroevaluation) + (mdr_vb_zero_columns_rankzeroevaluation)) + ((mdr_vb_zero_columns_rankzeroevaluation) + (mdr_vb_zero_columns_rankzeroevaluation))) /\ ((mdr_c_zero_columns_rankzeroevaluationdeterminantrc = ((mdr_a_zero_columns_rankzeroevaluationdeterminantrc) + (mdr_b_zero_columns_rankzeroevaluationdeterminantrc)) * S ((mdr_a_zero_columns_rankzeroevaluationdeterminantrc) + (mdr_b_zero_columns_rankzeroevaluationdeterminantrc)) + ((mdr_b_zero_columns_rankzeroevaluationdeterminantrc) + (mdr_b_zero_columns_rankzeroevaluationdeterminantrc))) /\ ((mdr_e_zero_columns_rankzeroevaluationdeterminantrc = ((mdr_p_zero_columns_rankzero) + (mdr_n_zero_columns_rankzero)) * S ((mdr_p_zero_columns_rankzero) + (mdr_n_zero_columns_rankzero)) + ((mdr_n_zero_columns_rankzero) + (mdr_n_zero_columns_rankzero))) /\ ((mdr_f_zero_columns_rankzeroevaluationdeterminantrc = ((mdr_vc_zero_columns_rankzeroevaluation) + (mdr_e_zero_columns_rankzeroevaluationdeterminantrc)) * S ((mdr_vc_zero_columns_rankzeroevaluation) + (mdr_e_zero_columns_rankzeroevaluationdeterminantrc)) + ((mdr_e_zero_columns_rankzeroevaluationdeterminantrc) + (mdr_e_zero_columns_rankzeroevaluationdeterminantrc))) /\ ((mdr_z_zero_columns_rankzeroevaluationdeterminantr) = ((mdr_c_zero_columns_rankzeroevaluationdeterminantrc) + (mdr_f_zero_columns_rankzeroevaluationdeterminantrc)) * S ((mdr_c_zero_columns_rankzeroevaluationdeterminantrc) + (mdr_f_zero_columns_rankzeroevaluationdeterminantrc)) + ((mdr_f_zero_columns_rankzeroevaluationdeterminantrc) + (mdr_f_zero_columns_rankzeroevaluationdeterminantrc))))))))) /\ (((exists ff_h_mdr_zero_columns_rankzeroevaluationdeterminantrb. ff_h_mdr_zero_columns_rankzeroevaluationdeterminantrb + S (mdr_z_zero_columns_rankzeroevaluationdeterminantr) = S ((S (mdr_i_zero_columns_rankzeroevaluationdeterminant)) * mdr_c_zero_columns_rankzeroevaluationdeterminant)) /\ exists ff_q_mdr_zero_columns_rankzeroevaluationdeterminantrb. mdr_b_zero_columns_rankzeroevaluationdeterminant = ff_q_mdr_zero_columns_rankzeroevaluationdeterminantrb * S ((S (mdr_i_zero_columns_rankzeroevaluationdeterminant)) * mdr_c_zero_columns_rankzeroevaluationdeterminant) + (mdr_z_zero_columns_rankzeroevaluationdeterminantr)))))))))) -> mdr_p_zero_columns_rankzero = mdr_n_zero_columns_rankzero))))))Constructive proof overview
Generated structural guide
Every zero-column rectangular matrix has rank zero, including the zero-by-zero matrix, without a positivity premise on either dimension.
The unchanged tactic script uses 2 declared prerequisites and contains 40 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,r,0,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 (r) - L12
specialize rectangular_matrix_rank_certificate_exists (0) - L13
apply rectangular_matrix_rank_certificate_exists
03Separate the logical casesL14–16
04Establish hzeroL17–26
Establish this local claim before using it. It is not an additional assumption. The following proof commands apply le zero.
05Calculate and transport equalitiesL27–36
Carry out the recorded arithmetic or equality steps; inspect the exact commands for their direction and premises.
- 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 - L36
rewrite hzero at hrank_witness
06Calculate and transport equalitiesL37–39
07Use earlier factsL40–40
Instantiate or apply named facts and discharge the corresponding proof obligations.
- L40
exact hrank_witness
Original exact command ledger · 40 lines
- 0001
intro pb - 0002
intro pc - 0003
intro nb - 0004
intro nc - 0005
intro r - 0006
have hrank : exists rank. (((exists mdr_gap_zero_columns_existsrows_bound. mdr_gap_zero_columns_existsrows_bound + (rank) = (r)) /\ ((exists mdr_gap_zero_columns_existscolumns_bound. mdr_gap_zero_columns_existscolumns_bound + (rank) = (0)) /\ ((exists mdr_rb_zero_columns_existswitness mdr_rc_zero_columns_existswitness mdr_cb_zero_columns_existswitness mdr_cc_zero_columns_existswitness. (((((forall fom_index_mrf_zero_columns_existswitnessminorrowsbound. (exists fom_gap_mrf_zero_columns_existswitnessminorrowsbound_index_bound. fom_gap_mrf_zero_columns_existswitnessminorrowsbound_index_bound + S (fom_index_mrf_zero_columns_existswitnessminorrowsbound) = rank) -> exists fom_value_mrf_zero_columns_existswitnessminorrowsbound. ((((exists fom_beta_height_mrf_zero_columns_existswitnessminorrowsbound_entry. fom_beta_height_mrf_zero_columns_existswitnessminorrowsbound_entry + S (fom_value_mrf_zero_columns_existswitnessminorrowsbound) = S ((S (fom_index_mrf_zero_columns_existswitnessminorrowsbound)) * mdr_rc_zero_columns_existswitness)) /\ exists fom_beta_quotient_mrf_zero_columns_existswitnessminorrowsbound_entry. mdr_rb_zero_columns_existswitness = fom_beta_quotient_mrf_zero_columns_existswitnessminorrowsbound_entry * S ((S (fom_index_mrf_zero_columns_existswitnessminorrowsbound)) * mdr_rc_zero_columns_existswitness) + (fom_value_mrf_zero_columns_existswitnessminorrowsbound))) /\ (exists fom_gap_mrf_zero_columns_existswitnessminorrowsbound_value_bound. fom_gap_mrf_zero_columns_existswitnessminorrowsbound_value_bound + S (fom_value_mrf_zero_columns_existswitnessminorrowsbound) = r))) /\ (forall mdr_i_zero_columns_existswitnessminorrowsdistinct mdr_j_zero_columns_existswitnessminorrowsdistinct mdr_a_zero_columns_existswitnessminorrowsdistinct. (exists mdr_gap_zero_columns_existswitnessminorrowsdistincti. mdr_gap_zero_columns_existswitnessminorrowsdistincti + S (mdr_i_zero_columns_existswitnessminorrowsdistinct) = (rank)) -> (exists mdr_gap_zero_columns_existswitnessminorrowsdistinctj. mdr_gap_zero_columns_existswitnessminorrowsdistinctj + S (mdr_j_zero_columns_existswitnessminorrowsdistinct) = (rank)) -> (((exists ff_h_mdr_zero_columns_existswitnessminorrowsdistinctfirst. ff_h_mdr_zero_columns_existswitnessminorrowsdistinctfirst + S (mdr_a_zero_columns_existswitnessminorrowsdistinct) = S ((S (mdr_i_zero_columns_existswitnessminorrowsdistinct)) * mdr_rc_zero_columns_existswitness)) /\ exists ff_q_mdr_zero_columns_existswitnessminorrowsdistinctfirst. mdr_rb_zero_columns_existswitness = ff_q_mdr_zero_columns_existswitnessminorrowsdistinctfirst * S ((S (mdr_i_zero_columns_existswitnessminorrowsdistinct)) * mdr_rc_zero_columns_existswitness) + (mdr_a_zero_columns_existswitnessminorrowsdistinct))) -> (((exists ff_h_mdr_zero_columns_existswitnessminorrowsdistinctsecond. ff_h_mdr_zero_columns_existswitnessminorrowsdistinctsecond + S (mdr_a_zero_columns_existswitnessminorrowsdistinct) = S ((S (mdr_j_zero_columns_existswitnessminorrowsdistinct)) * mdr_rc_zero_columns_existswitness)) /\ exists ff_q_mdr_zero_columns_existswitnessminorrowsdistinctsecond. mdr_rb_zero_columns_existswitness = ff_q_mdr_zero_columns_existswitnessminorrowsdistinctsecond * S ((S (mdr_j_zero_columns_existswitnessminorrowsdistinct)) * mdr_rc_zero_columns_existswitness) + (mdr_a_zero_columns_existswitnessminorrowsdistinct))) -> mdr_i_zero_columns_existswitnessminorrowsdistinct = mdr_j_zero_columns_existswitnessminorrowsdistinct))) /\ ((((forall fom_index_mrf_zero_columns_existswitnessminorcolumnsbound. (exists fom_gap_mrf_zero_columns_existswitnessminorcolumnsbound_index_bound. fom_gap_mrf_zero_columns_existswitnessminorcolumnsbound_index_bound + S (fom_index_mrf_zero_columns_existswitnessminorcolumnsbound) = rank) -> exists fom_value_mrf_zero_columns_existswitnessminorcolumnsbound. ((((exists fom_beta_height_mrf_zero_columns_existswitnessminorcolumnsbound_entry. fom_beta_height_mrf_zero_columns_existswitnessminorcolumnsbound_entry + S (fom_value_mrf_zero_columns_existswitnessminorcolumnsbound) = S ((S (fom_index_mrf_zero_columns_existswitnessminorcolumnsbound)) * mdr_cc_zero_columns_existswitness)) /\ exists fom_beta_quotient_mrf_zero_columns_existswitnessminorcolumnsbound_entry. mdr_cb_zero_columns_existswitness = fom_beta_quotient_mrf_zero_columns_existswitnessminorcolumnsbound_entry * S ((S (fom_index_mrf_zero_columns_existswitnessminorcolumnsbound)) * mdr_cc_zero_columns_existswitness) + (fom_value_mrf_zero_columns_existswitnessminorcolumnsbound))) /\ (exists fom_gap_mrf_zero_columns_existswitnessminorcolumnsbound_value_bound. fom_gap_mrf_zero_columns_existswitnessminorcolumnsbound_value_bound + S (fom_value_mrf_zero_columns_existswitnessminorcolumnsbound) = 0))) /\ (forall mdr_i_zero_columns_existswitnessminorcolumnsdistinct mdr_j_zero_columns_existswitnessminorcolumnsdistinct mdr_a_zero_columns_existswitnessminorcolumnsdistinct. (exists mdr_gap_zero_columns_existswitnessminorcolumnsdistincti. mdr_gap_zero_columns_existswitnessminorcolumnsdistincti + S (mdr_i_zero_columns_existswitnessminorcolumnsdistinct) = (rank)) -> (exists mdr_gap_zero_columns_existswitnessminorcolumnsdistinctj. mdr_gap_zero_columns_existswitnessminorcolumnsdistinctj + S (mdr_j_zero_columns_existswitnessminorcolumnsdistinct) = (rank)) -> (((exists ff_h_mdr_zero_columns_existswitnessminorcolumnsdistinctfirst. ff_h_mdr_zero_columns_existswitnessminorcolumnsdistinctfirst + S (mdr_a_zero_columns_existswitnessminorcolumnsdistinct) = S ((S (mdr_i_zero_columns_existswitnessminorcolumnsdistinct)) * mdr_cc_zero_columns_existswitness)) /\ exists ff_q_mdr_zero_columns_existswitnessminorcolumnsdistinctfirst. mdr_cb_zero_columns_existswitness = ff_q_mdr_zero_columns_existswitnessminorcolumnsdistinctfirst * S ((S (mdr_i_zero_columns_existswitnessminorcolumnsdistinct)) * mdr_cc_zero_columns_existswitness) + (mdr_a_zero_columns_existswitnessminorcolumnsdistinct))) -> (((exists ff_h_mdr_zero_columns_existswitnessminorcolumnsdistinctsecond. ff_h_mdr_zero_columns_existswitnessminorcolumnsdistinctsecond + S (mdr_a_zero_columns_existswitnessminorcolumnsdistinct) = S ((S (mdr_j_zero_columns_existswitnessminorcolumnsdistinct)) * mdr_cc_zero_columns_existswitness)) /\ exists ff_q_mdr_zero_columns_existswitnessminorcolumnsdistinctsecond. mdr_cb_zero_columns_existswitness = ff_q_mdr_zero_columns_existswitnessminorcolumnsdistinctsecond * S ((S (mdr_j_zero_columns_existswitnessminorcolumnsdistinct)) * mdr_cc_zero_columns_existswitness) + (mdr_a_zero_columns_existswitnessminorcolumnsdistinct))) -> mdr_i_zero_columns_existswitnessminorcolumnsdistinct = mdr_j_zero_columns_existswitnessminorcolumnsdistinct))) /\ (exists mdr_p_zero_columns_existswitnessminornonzero mdr_n_zero_columns_existswitnessminornonzero. ((exists mdr_ub_zero_columns_existswitnessminornonzeroevaluation mdr_uc_zero_columns_existswitnessminornonzeroevaluation mdr_vb_zero_columns_existswitnessminornonzeroevaluation mdr_vc_zero_columns_existswitnessminornonzeroevaluation. ((((forall mdr_i_zero_columns_existswitnessminornonzeroevaluationmatrixpositive. (exists mdr_gap_zero_columns_existswitnessminornonzeroevaluationmatrixpositivebound. mdr_gap_zero_columns_existswitnessminornonzeroevaluationmatrixpositivebound + S (mdr_i_zero_columns_existswitnessminornonzeroevaluationmatrixpositive) = ((rank) * (rank))) -> exists mdr_a_zero_columns_existswitnessminornonzeroevaluationmatrixpositive. (((exists mdr_r_zero_columns_existswitnessminornonzeroevaluationmatrixpositivepoint mdr_s_zero_columns_existswitnessminornonzeroevaluationmatrixpositivepoint mdr_u_zero_columns_existswitnessminornonzeroevaluationmatrixpositivepoint mdr_v_zero_columns_existswitnessminornonzeroevaluationmatrixpositivepoint. ((mdr_i_zero_columns_existswitnessminornonzeroevaluationmatrixpositive = (rank) * mdr_r_zero_columns_existswitnessminornonzeroevaluationmatrixpositivepoint + mdr_s_zero_columns_existswitnessminornonzeroevaluationmatrixpositivepoint) /\ ((exists mdr_gap_zero_columns_existswitnessminornonzeroevaluationmatrixpositivepointcolumn. mdr_gap_zero_columns_existswitnessminornonzeroevaluationmatrixpositivepointcolumn + S (mdr_s_zero_columns_existswitnessminornonzeroevaluationmatrixpositivepoint) = (rank)) /\ ((((exists ff_h_mdr_zero_columns_existswitnessminornonzeroevaluationmatrixpositivepointrow_index. ff_h_mdr_zero_columns_existswitnessminornonzeroevaluationmatrixpositivepointrow_index + S (mdr_u_zero_columns_existswitnessminornonzeroevaluationmatrixpositivepoint) = S ((S (mdr_r_zero_columns_existswitnessminornonzeroevaluationmatrixpositivepoint)) * mdr_rc_zero_columns_existswitness)) /\ exists ff_q_mdr_zero_columns_existswitnessminornonzeroevaluationmatrixpositivepointrow_index. mdr_rb_zero_columns_existswitness = ff_q_mdr_zero_columns_existswitnessminornonzeroevaluationmatrixpositivepointrow_index * S ((S (mdr_r_zero_columns_existswitnessminornonzeroevaluationmatrixpositivepoint)) * mdr_rc_zero_columns_existswitness) + (mdr_u_zero_columns_existswitnessminornonzeroevaluationmatrixpositivepoint))) /\ ((((exists ff_h_mdr_zero_columns_existswitnessminornonzeroevaluationmatrixpositivepointcolumn_index. ff_h_mdr_zero_columns_existswitnessminornonzeroevaluationmatrixpositivepointcolumn_index + S (mdr_v_zero_columns_existswitnessminornonzeroevaluationmatrixpositivepoint) = S ((S (mdr_s_zero_columns_existswitnessminornonzeroevaluationmatrixpositivepoint)) * mdr_cc_zero_columns_existswitness)) /\ exists ff_q_mdr_zero_columns_existswitnessminornonzeroevaluationmatrixpositivepointcolumn_index. mdr_cb_zero_columns_existswitness = ff_q_mdr_zero_columns_existswitnessminornonzeroevaluationmatrixpositivepointcolumn_index * S ((S (mdr_s_zero_columns_existswitnessminornonzeroevaluationmatrixpositivepoint)) * mdr_cc_zero_columns_existswitness) + (mdr_v_zero_columns_existswitnessminornonzeroevaluationmatrixpositivepoint))) /\ (((exists ff_h_mdr_zero_columns_existswitnessminornonzeroevaluationmatrixpositivepointsource. ff_h_mdr_zero_columns_existswitnessminornonzeroevaluationmatrixpositivepointsource + S (mdr_a_zero_columns_existswitnessminornonzeroevaluationmatrixpositive) = S ((S ((mdr_u_zero_columns_existswitnessminornonzeroevaluationmatrixpositivepoint) * (0) + (mdr_v_zero_columns_existswitnessminornonzeroevaluationmatrixpositivepoint))) * pc)) /\ exists ff_q_mdr_zero_columns_existswitnessminornonzeroevaluationmatrixpositivepointsource. pb = ff_q_mdr_zero_columns_existswitnessminornonzeroevaluationmatrixpositivepointsource * S ((S ((mdr_u_zero_columns_existswitnessminornonzeroevaluationmatrixpositivepoint) * (0) + (mdr_v_zero_columns_existswitnessminornonzeroevaluationmatrixpositivepoint))) * pc) + (mdr_a_zero_columns_existswitnessminornonzeroevaluationmatrixpositive)))))))) /\ (((exists ff_h_mdr_zero_columns_existswitnessminornonzeroevaluationmatrixpositiveoutput. ff_h_mdr_zero_columns_existswitnessminornonzeroevaluationmatrixpositiveoutput + S (mdr_a_zero_columns_existswitnessminornonzeroevaluationmatrixpositive) = S ((S (mdr_i_zero_columns_existswitnessminornonzeroevaluationmatrixpositive)) * mdr_uc_zero_columns_existswitnessminornonzeroevaluation)) /\ exists ff_q_mdr_zero_columns_existswitnessminornonzeroevaluationmatrixpositiveoutput. mdr_ub_zero_columns_existswitnessminornonzeroevaluation = ff_q_mdr_zero_columns_existswitnessminornonzeroevaluationmatrixpositiveoutput * S ((S (mdr_i_zero_columns_existswitnessminornonzeroevaluationmatrixpositive)) * mdr_uc_zero_columns_existswitnessminornonzeroevaluation) + (mdr_a_zero_columns_existswitnessminornonzeroevaluationmatrixpositive)))))) /\ (forall mdr_i_zero_columns_existswitnessminornonzeroevaluationmatrixnegative. (exists mdr_gap_zero_columns_existswitnessminornonzeroevaluationmatrixnegativebound. mdr_gap_zero_columns_existswitnessminornonzeroevaluationmatrixnegativebound + S (mdr_i_zero_columns_existswitnessminornonzeroevaluationmatrixnegative) = ((rank) * (rank))) -> exists mdr_a_zero_columns_existswitnessminornonzeroevaluationmatrixnegative. (((exists mdr_r_zero_columns_existswitnessminornonzeroevaluationmatrixnegativepoint mdr_s_zero_columns_existswitnessminornonzeroevaluationmatrixnegativepoint mdr_u_zero_columns_existswitnessminornonzeroevaluationmatrixnegativepoint mdr_v_zero_columns_existswitnessminornonzeroevaluationmatrixnegativepoint. ((mdr_i_zero_columns_existswitnessminornonzeroevaluationmatrixnegative = (rank) * mdr_r_zero_columns_existswitnessminornonzeroevaluationmatrixnegativepoint + mdr_s_zero_columns_existswitnessminornonzeroevaluationmatrixnegativepoint) /\ ((exists mdr_gap_zero_columns_existswitnessminornonzeroevaluationmatrixnegativepointcolumn. mdr_gap_zero_columns_existswitnessminornonzeroevaluationmatrixnegativepointcolumn + S (mdr_s_zero_columns_existswitnessminornonzeroevaluationmatrixnegativepoint) = (rank)) /\ ((((exists ff_h_mdr_zero_columns_existswitnessminornonzeroevaluationmatrixnegativepointrow_index. ff_h_mdr_zero_columns_existswitnessminornonzeroevaluationmatrixnegativepointrow_index + S (mdr_u_zero_columns_existswitnessminornonzeroevaluationmatrixnegativepoint) = S ((S (mdr_r_zero_columns_existswitnessminornonzeroevaluationmatrixnegativepoint)) * mdr_rc_zero_columns_existswitness)) /\ exists ff_q_mdr_zero_columns_existswitnessminornonzeroevaluationmatrixnegativepointrow_index. mdr_rb_zero_columns_existswitness = ff_q_mdr_zero_columns_existswitnessminornonzeroevaluationmatrixnegativepointrow_index * S ((S (mdr_r_zero_columns_existswitnessminornonzeroevaluationmatrixnegativepoint)) * mdr_rc_zero_columns_existswitness) + (mdr_u_zero_columns_existswitnessminornonzeroevaluationmatrixnegativepoint))) /\ ((((exists ff_h_mdr_zero_columns_existswitnessminornonzeroevaluationmatrixnegativepointcolumn_index. ff_h_mdr_zero_columns_existswitnessminornonzeroevaluationmatrixnegativepointcolumn_index + S (mdr_v_zero_columns_existswitnessminornonzeroevaluationmatrixnegativepoint) = S ((S (mdr_s_zero_columns_existswitnessminornonzeroevaluationmatrixnegativepoint)) * mdr_cc_zero_columns_existswitness)) /\ exists ff_q_mdr_zero_columns_existswitnessminornonzeroevaluationmatrixnegativepointcolumn_index. mdr_cb_zero_columns_existswitness = ff_q_mdr_zero_columns_existswitnessminornonzeroevaluationmatrixnegativepointcolumn_index * S ((S (mdr_s_zero_columns_existswitnessminornonzeroevaluationmatrixnegativepoint)) * mdr_cc_zero_columns_existswitness) + (mdr_v_zero_columns_existswitnessminornonzeroevaluationmatrixnegativepoint))) /\ (((exists ff_h_mdr_zero_columns_existswitnessminornonzeroevaluationmatrixnegativepointsource. ff_h_mdr_zero_columns_existswitnessminornonzeroevaluationmatrixnegativepointsource + S (mdr_a_zero_columns_existswitnessminornonzeroevaluationmatrixnegative) = S ((S ((mdr_u_zero_columns_existswitnessminornonzeroevaluationmatrixnegativepoint) * (0) + (mdr_v_zero_columns_existswitnessminornonzeroevaluationmatrixnegativepoint))) * nc)) /\ exists ff_q_mdr_zero_columns_existswitnessminornonzeroevaluationmatrixnegativepointsource. nb = ff_q_mdr_zero_columns_existswitnessminornonzeroevaluationmatrixnegativepointsource * S ((S ((mdr_u_zero_columns_existswitnessminornonzeroevaluationmatrixnegativepoint) * (0) + (mdr_v_zero_columns_existswitnessminornonzeroevaluationmatrixnegativepoint))) * nc) + (mdr_a_zero_columns_existswitnessminornonzeroevaluationmatrixnegative)))))))) /\ (((exists ff_h_mdr_zero_columns_existswitnessminornonzeroevaluationmatrixnegativeoutput. ff_h_mdr_zero_columns_existswitnessminornonzeroevaluationmatrixnegativeoutput + S (mdr_a_zero_columns_existswitnessminornonzeroevaluationmatrixnegative) = S ((S (mdr_i_zero_columns_existswitnessminornonzeroevaluationmatrixnegative)) * mdr_vc_zero_columns_existswitnessminornonzeroevaluation)) /\ exists ff_q_mdr_zero_columns_existswitnessminornonzeroevaluationmatrixnegativeoutput. mdr_vb_zero_columns_existswitnessminornonzeroevaluation = ff_q_mdr_zero_columns_existswitnessminornonzeroevaluationmatrixnegativeoutput * S ((S (mdr_i_zero_columns_existswitnessminornonzeroevaluationmatrixnegative)) * mdr_vc_zero_columns_existswitnessminornonzeroevaluation) + (mdr_a_zero_columns_existswitnessminornonzeroevaluationmatrixnegative)))))))) /\ (exists mdr_b_zero_columns_existswitnessminornonzeroevaluationdeterminant mdr_c_zero_columns_existswitnessminornonzeroevaluationdeterminant mdr_l_zero_columns_existswitnessminornonzeroevaluationdeterminant mdr_i_zero_columns_existswitnessminornonzeroevaluationdeterminant. ((forall mdr_i_zero_columns_existswitnessminornonzeroevaluationdeterminanth. (exists mdr_gap_zero_columns_existswitnessminornonzeroevaluationdeterminanthi. mdr_gap_zero_columns_existswitnessminornonzeroevaluationdeterminanthi + S (mdr_i_zero_columns_existswitnessminornonzeroevaluationdeterminanth) = (mdr_l_zero_columns_existswitnessminornonzeroevaluationdeterminant)) -> exists mdr_d_zero_columns_existswitnessminornonzeroevaluationdeterminanth mdr_pb_zero_columns_existswitnessminornonzeroevaluationdeterminanth mdr_pc_zero_columns_existswitnessminornonzeroevaluationdeterminanth mdr_nb_zero_columns_existswitnessminornonzeroevaluationdeterminanth mdr_nc_zero_columns_existswitnessminornonzeroevaluationdeterminanth mdr_p_zero_columns_existswitnessminornonzeroevaluationdeterminanth mdr_n_zero_columns_existswitnessminornonzeroevaluationdeterminanth. ((exists mdr_z_zero_columns_existswitnessminornonzeroevaluationdeterminanthr. ((exists mdr_a_zero_columns_existswitnessminornonzeroevaluationdeterminanthrc mdr_b_zero_columns_existswitnessminornonzeroevaluationdeterminanthrc mdr_c_zero_columns_existswitnessminornonzeroevaluationdeterminanthrc mdr_e_zero_columns_existswitnessminornonzeroevaluationdeterminanthrc mdr_f_zero_columns_existswitnessminornonzeroevaluationdeterminanthrc. ((mdr_a_zero_columns_existswitnessminornonzeroevaluationdeterminanthrc = ((mdr_d_zero_columns_existswitnessminornonzeroevaluationdeterminanth) + (mdr_pb_zero_columns_existswitnessminornonzeroevaluationdeterminanth)) * S ((mdr_d_zero_columns_existswitnessminornonzeroevaluationdeterminanth) + (mdr_pb_zero_columns_existswitnessminornonzeroevaluationdeterminanth)) + ((mdr_pb_zero_columns_existswitnessminornonzeroevaluationdeterminanth) + (mdr_pb_zero_columns_existswitnessminornonzeroevaluationdeterminanth))) /\ ((mdr_b_zero_columns_existswitnessminornonzeroevaluationdeterminanthrc = ((mdr_pc_zero_columns_existswitnessminornonzeroevaluationdeterminanth) + (mdr_nb_zero_columns_existswitnessminornonzeroevaluationdeterminanth)) * S ((mdr_pc_zero_columns_existswitnessminornonzeroevaluationdeterminanth) + (mdr_nb_zero_columns_existswitnessminornonzeroevaluationdeterminanth)) + ((mdr_nb_zero_columns_existswitnessminornonzeroevaluationdeterminanth) + (mdr_nb_zero_columns_existswitnessminornonzeroevaluationdeterminanth))) /\ ((mdr_c_zero_columns_existswitnessminornonzeroevaluationdeterminanthrc = ((mdr_a_zero_columns_existswitnessminornonzeroevaluationdeterminanthrc) + (mdr_b_zero_columns_existswitnessminornonzeroevaluationdeterminanthrc)) * S ((mdr_a_zero_columns_existswitnessminornonzeroevaluationdeterminanthrc) + (mdr_b_zero_columns_existswitnessminornonzeroevaluationdeterminanthrc)) + ((mdr_b_zero_columns_existswitnessminornonzeroevaluationdeterminanthrc) + (mdr_b_zero_columns_existswitnessminornonzeroevaluationdeterminanthrc))) /\ ((mdr_e_zero_columns_existswitnessminornonzeroevaluationdeterminanthrc = ((mdr_p_zero_columns_existswitnessminornonzeroevaluationdeterminanth) + (mdr_n_zero_columns_existswitnessminornonzeroevaluationdeterminanth)) * S ((mdr_p_zero_columns_existswitnessminornonzeroevaluationdeterminanth) + (mdr_n_zero_columns_existswitnessminornonzeroevaluationdeterminanth)) + ((mdr_n_zero_columns_existswitnessminornonzeroevaluationdeterminanth) + (mdr_n_zero_columns_existswitnessminornonzeroevaluationdeterminanth))) /\ ((mdr_f_zero_columns_existswitnessminornonzeroevaluationdeterminanthrc = ((mdr_nc_zero_columns_existswitnessminornonzeroevaluationdeterminanth) + (mdr_e_zero_columns_existswitnessminornonzeroevaluationdeterminanthrc)) * S ((mdr_nc_zero_columns_existswitnessminornonzeroevaluationdeterminanth) + (mdr_e_zero_columns_existswitnessminornonzeroevaluationdeterminanthrc)) + ((mdr_e_zero_columns_existswitnessminornonzeroevaluationdeterminanthrc) + (mdr_e_zero_columns_existswitnessminornonzeroevaluationdeterminanthrc))) /\ ((mdr_z_zero_columns_existswitnessminornonzeroevaluationdeterminanthr) = ((mdr_c_zero_columns_existswitnessminornonzeroevaluationdeterminanthrc) + (mdr_f_zero_columns_existswitnessminornonzeroevaluationdeterminanthrc)) * S ((mdr_c_zero_columns_existswitnessminornonzeroevaluationdeterminanthrc) + (mdr_f_zero_columns_existswitnessminornonzeroevaluationdeterminanthrc)) + ((mdr_f_zero_columns_existswitnessminornonzeroevaluationdeterminanthrc) + (mdr_f_zero_columns_existswitnessminornonzeroevaluationdeterminanthrc))))))))) /\ (((exists ff_h_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthrb. ff_h_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthrb + S (mdr_z_zero_columns_existswitnessminornonzeroevaluationdeterminanthr) = S ((S (mdr_i_zero_columns_existswitnessminornonzeroevaluationdeterminanth)) * mdr_c_zero_columns_existswitnessminornonzeroevaluationdeterminant)) /\ exists ff_q_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthrb. mdr_b_zero_columns_existswitnessminornonzeroevaluationdeterminant = ff_q_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthrb * S ((S (mdr_i_zero_columns_existswitnessminornonzeroevaluationdeterminanth)) * mdr_c_zero_columns_existswitnessminornonzeroevaluationdeterminant) + (mdr_z_zero_columns_existswitnessminornonzeroevaluationdeterminanthr))))) /\ (((((mdr_d_zero_columns_existswitnessminornonzeroevaluationdeterminanth) = 0) /\ (((mdr_p_zero_columns_existswitnessminornonzeroevaluationdeterminanth) = 1) /\ ((mdr_n_zero_columns_existswitnessminornonzeroevaluationdeterminanth) = 0))) \/ exists mdr_q_zero_columns_existswitnessminornonzeroevaluationdeterminanths mdr_eb_zero_columns_existswitnessminornonzeroevaluationdeterminanths mdr_ec_zero_columns_existswitnessminornonzeroevaluationdeterminanths mdr_fb_zero_columns_existswitnessminornonzeroevaluationdeterminanths mdr_fc_zero_columns_existswitnessminornonzeroevaluationdeterminanths. (((mdr_d_zero_columns_existswitnessminornonzeroevaluationdeterminanth) = S (mdr_q_zero_columns_existswitnessminornonzeroevaluationdeterminanths)) /\ ((forall mdr_j_zero_columns_existswitnessminornonzeroevaluationdeterminanthsc. (exists mdr_gap_zero_columns_existswitnessminornonzeroevaluationdeterminanthscj. mdr_gap_zero_columns_existswitnessminornonzeroevaluationdeterminanthscj + S (mdr_j_zero_columns_existswitnessminornonzeroevaluationdeterminanthsc) = (S (mdr_q_zero_columns_existswitnessminornonzeroevaluationdeterminanths))) -> exists mdr_i_zero_columns_existswitnessminornonzeroevaluationdeterminanthsc mdr_up_zero_columns_existswitnessminornonzeroevaluationdeterminanthsc mdr_us_zero_columns_existswitnessminornonzeroevaluationdeterminanthsc mdr_un_zero_columns_existswitnessminornonzeroevaluationdeterminanthsc mdr_ut_zero_columns_existswitnessminornonzeroevaluationdeterminanthsc mdr_p_zero_columns_existswitnessminornonzeroevaluationdeterminanthsc mdr_n_zero_columns_existswitnessminornonzeroevaluationdeterminanthsc. ((exists mdr_gap_zero_columns_existswitnessminornonzeroevaluationdeterminanthsci. mdr_gap_zero_columns_existswitnessminornonzeroevaluationdeterminanthsci + S (mdr_i_zero_columns_existswitnessminornonzeroevaluationdeterminanthsc) = (mdr_i_zero_columns_existswitnessminornonzeroevaluationdeterminanth)) /\ ((exists mdr_z_zero_columns_existswitnessminornonzeroevaluationdeterminanthscr. ((exists mdr_a_zero_columns_existswitnessminornonzeroevaluationdeterminanthscrc mdr_b_zero_columns_existswitnessminornonzeroevaluationdeterminanthscrc mdr_c_zero_columns_existswitnessminornonzeroevaluationdeterminanthscrc mdr_e_zero_columns_existswitnessminornonzeroevaluationdeterminanthscrc mdr_f_zero_columns_existswitnessminornonzeroevaluationdeterminanthscrc. ((mdr_a_zero_columns_existswitnessminornonzeroevaluationdeterminanthscrc = ((mdr_q_zero_columns_existswitnessminornonzeroevaluationdeterminanths) + (mdr_up_zero_columns_existswitnessminornonzeroevaluationdeterminanthsc)) * S ((mdr_q_zero_columns_existswitnessminornonzeroevaluationdeterminanths) + (mdr_up_zero_columns_existswitnessminornonzeroevaluationdeterminanthsc)) + ((mdr_up_zero_columns_existswitnessminornonzeroevaluationdeterminanthsc) + (mdr_up_zero_columns_existswitnessminornonzeroevaluationdeterminanthsc))) /\ ((mdr_b_zero_columns_existswitnessminornonzeroevaluationdeterminanthscrc = ((mdr_us_zero_columns_existswitnessminornonzeroevaluationdeterminanthsc) + (mdr_un_zero_columns_existswitnessminornonzeroevaluationdeterminanthsc)) * S ((mdr_us_zero_columns_existswitnessminornonzeroevaluationdeterminanthsc) + (mdr_un_zero_columns_existswitnessminornonzeroevaluationdeterminanthsc)) + ((mdr_un_zero_columns_existswitnessminornonzeroevaluationdeterminanthsc) + (mdr_un_zero_columns_existswitnessminornonzeroevaluationdeterminanthsc))) /\ ((mdr_c_zero_columns_existswitnessminornonzeroevaluationdeterminanthscrc = ((mdr_a_zero_columns_existswitnessminornonzeroevaluationdeterminanthscrc) + (mdr_b_zero_columns_existswitnessminornonzeroevaluationdeterminanthscrc)) * S ((mdr_a_zero_columns_existswitnessminornonzeroevaluationdeterminanthscrc) + (mdr_b_zero_columns_existswitnessminornonzeroevaluationdeterminanthscrc)) + ((mdr_b_zero_columns_existswitnessminornonzeroevaluationdeterminanthscrc) + (mdr_b_zero_columns_existswitnessminornonzeroevaluationdeterminanthscrc))) /\ ((mdr_e_zero_columns_existswitnessminornonzeroevaluationdeterminanthscrc = ((mdr_p_zero_columns_existswitnessminornonzeroevaluationdeterminanthsc) + (mdr_n_zero_columns_existswitnessminornonzeroevaluationdeterminanthsc)) * S ((mdr_p_zero_columns_existswitnessminornonzeroevaluationdeterminanthsc) + (mdr_n_zero_columns_existswitnessminornonzeroevaluationdeterminanthsc)) + ((mdr_n_zero_columns_existswitnessminornonzeroevaluationdeterminanthsc) + (mdr_n_zero_columns_existswitnessminornonzeroevaluationdeterminanthsc))) /\ ((mdr_f_zero_columns_existswitnessminornonzeroevaluationdeterminanthscrc = ((mdr_ut_zero_columns_existswitnessminornonzeroevaluationdeterminanthsc) + (mdr_e_zero_columns_existswitnessminornonzeroevaluationdeterminanthscrc)) * S ((mdr_ut_zero_columns_existswitnessminornonzeroevaluationdeterminanthsc) + (mdr_e_zero_columns_existswitnessminornonzeroevaluationdeterminanthscrc)) + ((mdr_e_zero_columns_existswitnessminornonzeroevaluationdeterminanthscrc) + (mdr_e_zero_columns_existswitnessminornonzeroevaluationdeterminanthscrc))) /\ ((mdr_z_zero_columns_existswitnessminornonzeroevaluationdeterminanthscr) = ((mdr_c_zero_columns_existswitnessminornonzeroevaluationdeterminanthscrc) + (mdr_f_zero_columns_existswitnessminornonzeroevaluationdeterminanthscrc)) * S ((mdr_c_zero_columns_existswitnessminornonzeroevaluationdeterminanthscrc) + (mdr_f_zero_columns_existswitnessminornonzeroevaluationdeterminanthscrc)) + ((mdr_f_zero_columns_existswitnessminornonzeroevaluationdeterminanthscrc) + (mdr_f_zero_columns_existswitnessminornonzeroevaluationdeterminanthscrc))))))))) /\ (((exists ff_h_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthscrb. ff_h_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthscrb + S (mdr_z_zero_columns_existswitnessminornonzeroevaluationdeterminanthscr) = S ((S (mdr_i_zero_columns_existswitnessminornonzeroevaluationdeterminanthsc)) * mdr_c_zero_columns_existswitnessminornonzeroevaluationdeterminant)) /\ exists ff_q_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthscrb. mdr_b_zero_columns_existswitnessminornonzeroevaluationdeterminant = ff_q_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthscrb * S ((S (mdr_i_zero_columns_existswitnessminornonzeroevaluationdeterminanthsc)) * mdr_c_zero_columns_existswitnessminornonzeroevaluationdeterminant) + (mdr_z_zero_columns_existswitnessminornonzeroevaluationdeterminanthscr))))) /\ ((((forall ff_index_mdm_prefix_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthscm_positive. (exists ff_gap_mdm_lt_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthscm_positive_index_bound. ff_gap_mdm_lt_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthscm_positive_index_bound + S (ff_index_mdm_prefix_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthscm_positive) = ((mdr_q_zero_columns_existswitnessminornonzeroevaluationdeterminanths) * (mdr_q_zero_columns_existswitnessminornonzeroevaluationdeterminanths))) -> exists ff_row_mdm_prefix_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthscm_positive ff_column_mdm_prefix_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthscm_positive ff_value_mdm_prefix_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthscm_positive. (ff_index_mdm_prefix_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthscm_positive = (mdr_q_zero_columns_existswitnessminornonzeroevaluationdeterminanths) * ff_row_mdm_prefix_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthscm_positive + ff_column_mdm_prefix_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthscm_positive /\ ((exists ff_gap_mdm_lt_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthscm_positive_column_bound. ff_gap_mdm_lt_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthscm_positive_column_bound + S (ff_column_mdm_prefix_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthscm_positive) = (mdr_q_zero_columns_existswitnessminornonzeroevaluationdeterminanths)) /\ ((exists ff_row_mdm_cell_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthscm_positive_cell ff_column_mdm_cell_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthscm_positive_cell. (((((exists ff_gap_mdm_lt_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthscm_positive_cell_row_before. ff_gap_mdm_lt_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthscm_positive_cell_row_before + S (ff_row_mdm_prefix_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthscm_positive) = (0)) /\ ff_row_mdm_cell_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthscm_positive_cell = ff_row_mdm_prefix_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthscm_positive) \/ ((exists ff_gap_mdm_le_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthscm_positive_cell_row_after. ff_gap_mdm_le_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthscm_positive_cell_row_after + (0) = (ff_row_mdm_prefix_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthscm_positive)) /\ ff_row_mdm_cell_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthscm_positive_cell = S ff_row_mdm_prefix_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthscm_positive))) /\ (((((exists ff_gap_mdm_lt_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthscm_positive_cell_column_before. ff_gap_mdm_lt_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthscm_positive_cell_column_before + S (ff_column_mdm_prefix_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthscm_positive) = (mdr_j_zero_columns_existswitnessminornonzeroevaluationdeterminanthsc)) /\ ff_column_mdm_cell_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthscm_positive_cell = ff_column_mdm_prefix_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthscm_positive) \/ ((exists ff_gap_mdm_le_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthscm_positive_cell_column_after. ff_gap_mdm_le_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthscm_positive_cell_column_after + (mdr_j_zero_columns_existswitnessminornonzeroevaluationdeterminanthsc) = (ff_column_mdm_prefix_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthscm_positive)) /\ ff_column_mdm_cell_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthscm_positive_cell = S ff_column_mdm_prefix_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthscm_positive))) /\ (((exists ff_h_mdm_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthscm_positive_cell_source. ff_h_mdm_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthscm_positive_cell_source + S (ff_value_mdm_prefix_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthscm_positive) = S ((S ((ff_row_mdm_cell_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthscm_positive_cell) * (S (mdr_q_zero_columns_existswitnessminornonzeroevaluationdeterminanths)) + (ff_column_mdm_cell_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthscm_positive_cell))) * mdr_pc_zero_columns_existswitnessminornonzeroevaluationdeterminanth)) /\ exists ff_q_mdm_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthscm_positive_cell_source. mdr_pb_zero_columns_existswitnessminornonzeroevaluationdeterminanth = ff_q_mdm_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthscm_positive_cell_source * S ((S ((ff_row_mdm_cell_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthscm_positive_cell) * (S (mdr_q_zero_columns_existswitnessminornonzeroevaluationdeterminanths)) + (ff_column_mdm_cell_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthscm_positive_cell))) * mdr_pc_zero_columns_existswitnessminornonzeroevaluationdeterminanth) + (ff_value_mdm_prefix_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthscm_positive)))))) /\ (((exists ff_h_mdm_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthscm_positive_target. ff_h_mdm_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthscm_positive_target + S (ff_value_mdm_prefix_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthscm_positive) = S ((S (ff_index_mdm_prefix_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthscm_positive)) * mdr_us_zero_columns_existswitnessminornonzeroevaluationdeterminanthsc)) /\ exists ff_q_mdm_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthscm_positive_target. mdr_up_zero_columns_existswitnessminornonzeroevaluationdeterminanthsc = ff_q_mdm_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthscm_positive_target * S ((S (ff_index_mdm_prefix_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthscm_positive)) * mdr_us_zero_columns_existswitnessminornonzeroevaluationdeterminanthsc) + (ff_value_mdm_prefix_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthscm_positive))))))) /\ (forall ff_index_mdm_prefix_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthscm_negative. (exists ff_gap_mdm_lt_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthscm_negative_index_bound. ff_gap_mdm_lt_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthscm_negative_index_bound + S (ff_index_mdm_prefix_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthscm_negative) = ((mdr_q_zero_columns_existswitnessminornonzeroevaluationdeterminanths) * (mdr_q_zero_columns_existswitnessminornonzeroevaluationdeterminanths))) -> exists ff_row_mdm_prefix_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthscm_negative ff_column_mdm_prefix_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthscm_negative ff_value_mdm_prefix_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthscm_negative. (ff_index_mdm_prefix_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthscm_negative = (mdr_q_zero_columns_existswitnessminornonzeroevaluationdeterminanths) * ff_row_mdm_prefix_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthscm_negative + ff_column_mdm_prefix_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthscm_negative /\ ((exists ff_gap_mdm_lt_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthscm_negative_column_bound. ff_gap_mdm_lt_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthscm_negative_column_bound + S (ff_column_mdm_prefix_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthscm_negative) = (mdr_q_zero_columns_existswitnessminornonzeroevaluationdeterminanths)) /\ ((exists ff_row_mdm_cell_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthscm_negative_cell ff_column_mdm_cell_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthscm_negative_cell. (((((exists ff_gap_mdm_lt_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthscm_negative_cell_row_before. ff_gap_mdm_lt_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthscm_negative_cell_row_before + S (ff_row_mdm_prefix_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthscm_negative) = (0)) /\ ff_row_mdm_cell_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthscm_negative_cell = ff_row_mdm_prefix_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthscm_negative) \/ ((exists ff_gap_mdm_le_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthscm_negative_cell_row_after. ff_gap_mdm_le_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthscm_negative_cell_row_after + (0) = (ff_row_mdm_prefix_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthscm_negative)) /\ ff_row_mdm_cell_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthscm_negative_cell = S ff_row_mdm_prefix_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthscm_negative))) /\ (((((exists ff_gap_mdm_lt_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthscm_negative_cell_column_before. ff_gap_mdm_lt_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthscm_negative_cell_column_before + S (ff_column_mdm_prefix_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthscm_negative) = (mdr_j_zero_columns_existswitnessminornonzeroevaluationdeterminanthsc)) /\ ff_column_mdm_cell_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthscm_negative_cell = ff_column_mdm_prefix_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthscm_negative) \/ ((exists ff_gap_mdm_le_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthscm_negative_cell_column_after. ff_gap_mdm_le_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthscm_negative_cell_column_after + (mdr_j_zero_columns_existswitnessminornonzeroevaluationdeterminanthsc) = (ff_column_mdm_prefix_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthscm_negative)) /\ ff_column_mdm_cell_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthscm_negative_cell = S ff_column_mdm_prefix_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthscm_negative))) /\ (((exists ff_h_mdm_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthscm_negative_cell_source. ff_h_mdm_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthscm_negative_cell_source + S (ff_value_mdm_prefix_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthscm_negative) = S ((S ((ff_row_mdm_cell_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthscm_negative_cell) * (S (mdr_q_zero_columns_existswitnessminornonzeroevaluationdeterminanths)) + (ff_column_mdm_cell_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthscm_negative_cell))) * mdr_nc_zero_columns_existswitnessminornonzeroevaluationdeterminanth)) /\ exists ff_q_mdm_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthscm_negative_cell_source. mdr_nb_zero_columns_existswitnessminornonzeroevaluationdeterminanth = ff_q_mdm_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthscm_negative_cell_source * S ((S ((ff_row_mdm_cell_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthscm_negative_cell) * (S (mdr_q_zero_columns_existswitnessminornonzeroevaluationdeterminanths)) + (ff_column_mdm_cell_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthscm_negative_cell))) * mdr_nc_zero_columns_existswitnessminornonzeroevaluationdeterminanth) + (ff_value_mdm_prefix_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthscm_negative)))))) /\ (((exists ff_h_mdm_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthscm_negative_target. ff_h_mdm_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthscm_negative_target + S (ff_value_mdm_prefix_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthscm_negative) = S ((S (ff_index_mdm_prefix_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthscm_negative)) * mdr_ut_zero_columns_existswitnessminornonzeroevaluationdeterminanthsc)) /\ exists ff_q_mdm_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthscm_negative_target. mdr_un_zero_columns_existswitnessminornonzeroevaluationdeterminanthsc = ff_q_mdm_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthscm_negative_target * S ((S (ff_index_mdm_prefix_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthscm_negative)) * mdr_ut_zero_columns_existswitnessminornonzeroevaluationdeterminanthsc) + (ff_value_mdm_prefix_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthscm_negative))))))))) /\ ((((exists ff_h_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthscp. ff_h_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthscp + S (mdr_p_zero_columns_existswitnessminornonzeroevaluationdeterminanthsc) = S ((S (mdr_j_zero_columns_existswitnessminornonzeroevaluationdeterminanthsc)) * mdr_ec_zero_columns_existswitnessminornonzeroevaluationdeterminanths)) /\ exists ff_q_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthscp. mdr_eb_zero_columns_existswitnessminornonzeroevaluationdeterminanths = ff_q_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthscp * S ((S (mdr_j_zero_columns_existswitnessminornonzeroevaluationdeterminanthsc)) * mdr_ec_zero_columns_existswitnessminornonzeroevaluationdeterminanths) + (mdr_p_zero_columns_existswitnessminornonzeroevaluationdeterminanthsc))) /\ (((exists ff_h_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthscn. ff_h_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthscn + S (mdr_n_zero_columns_existswitnessminornonzeroevaluationdeterminanthsc) = S ((S (mdr_j_zero_columns_existswitnessminornonzeroevaluationdeterminanthsc)) * mdr_fc_zero_columns_existswitnessminornonzeroevaluationdeterminanths)) /\ exists ff_q_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthscn. mdr_fb_zero_columns_existswitnessminornonzeroevaluationdeterminanths = ff_q_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthscn * S ((S (mdr_j_zero_columns_existswitnessminornonzeroevaluationdeterminanthsc)) * mdr_fc_zero_columns_existswitnessminornonzeroevaluationdeterminanths) + (mdr_n_zero_columns_existswitnessminornonzeroevaluationdeterminanthsc)))))))) /\ (exists ff_ub_mce_fold_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthsf ff_uc_mce_fold_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthsf ff_vb_mce_fold_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthsf ff_vc_mce_fold_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthsf. ((forall ff_index_mce_alternating_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthsf_prefix. (exists ff_gap_mce_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthsf_prefix_index. ff_gap_mce_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthsf_prefix_index + S (ff_index_mce_alternating_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthsf_prefix) = (S (mdr_q_zero_columns_existswitnessminornonzeroevaluationdeterminanths))) -> exists ff_ap_mce_alternating_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthsf_prefix ff_an_mce_alternating_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthsf_prefix ff_bp_mce_alternating_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthsf_prefix ff_bn_mce_alternating_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthsf_prefix ff_p_mce_alternating_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthsf_prefix ff_n_mce_alternating_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthsf_prefix. ((((exists ff_h_mce_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthsf_prefix_ap. ff_h_mce_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthsf_prefix_ap + S (ff_ap_mce_alternating_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthsf_prefix) = S ((S (ff_index_mce_alternating_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthsf_prefix)) * mdr_pc_zero_columns_existswitnessminornonzeroevaluationdeterminanth)) /\ exists ff_q_mce_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthsf_prefix_ap. mdr_pb_zero_columns_existswitnessminornonzeroevaluationdeterminanth = ff_q_mce_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthsf_prefix_ap * S ((S (ff_index_mce_alternating_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthsf_prefix)) * mdr_pc_zero_columns_existswitnessminornonzeroevaluationdeterminanth) + (ff_ap_mce_alternating_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthsf_prefix))) /\ ((((exists ff_h_mce_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthsf_prefix_an. ff_h_mce_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthsf_prefix_an + S (ff_an_mce_alternating_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthsf_prefix) = S ((S (ff_index_mce_alternating_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthsf_prefix)) * mdr_nc_zero_columns_existswitnessminornonzeroevaluationdeterminanth)) /\ exists ff_q_mce_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthsf_prefix_an. mdr_nb_zero_columns_existswitnessminornonzeroevaluationdeterminanth = ff_q_mce_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthsf_prefix_an * S ((S (ff_index_mce_alternating_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthsf_prefix)) * mdr_nc_zero_columns_existswitnessminornonzeroevaluationdeterminanth) + (ff_an_mce_alternating_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthsf_prefix))) /\ ((((exists ff_h_mce_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthsf_prefix_bp. ff_h_mce_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthsf_prefix_bp + S (ff_bp_mce_alternating_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthsf_prefix) = S ((S (ff_index_mce_alternating_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthsf_prefix)) * mdr_ec_zero_columns_existswitnessminornonzeroevaluationdeterminanths)) /\ exists ff_q_mce_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthsf_prefix_bp. mdr_eb_zero_columns_existswitnessminornonzeroevaluationdeterminanths = ff_q_mce_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthsf_prefix_bp * S ((S (ff_index_mce_alternating_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthsf_prefix)) * mdr_ec_zero_columns_existswitnessminornonzeroevaluationdeterminanths) + (ff_bp_mce_alternating_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthsf_prefix))) /\ ((((exists ff_h_mce_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthsf_prefix_bn. ff_h_mce_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthsf_prefix_bn + S (ff_bn_mce_alternating_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthsf_prefix) = S ((S (ff_index_mce_alternating_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthsf_prefix)) * mdr_fc_zero_columns_existswitnessminornonzeroevaluationdeterminanths)) /\ exists ff_q_mce_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthsf_prefix_bn. mdr_fb_zero_columns_existswitnessminornonzeroevaluationdeterminanths = ff_q_mce_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthsf_prefix_bn * S ((S (ff_index_mce_alternating_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthsf_prefix)) * mdr_fc_zero_columns_existswitnessminornonzeroevaluationdeterminanths) + (ff_bn_mce_alternating_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthsf_prefix))) /\ ((((exists ff_h_mce_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthsf_prefix_positive. ff_h_mce_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthsf_prefix_positive + S (ff_p_mce_alternating_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthsf_prefix) = S ((S (ff_index_mce_alternating_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthsf_prefix)) * ff_uc_mce_fold_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthsf)) /\ exists ff_q_mce_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthsf_prefix_positive. ff_ub_mce_fold_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthsf = ff_q_mce_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthsf_prefix_positive * S ((S (ff_index_mce_alternating_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthsf_prefix)) * ff_uc_mce_fold_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthsf) + (ff_p_mce_alternating_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthsf_prefix))) /\ ((((exists ff_h_mce_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthsf_prefix_negative. ff_h_mce_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthsf_prefix_negative + S (ff_n_mce_alternating_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthsf_prefix) = S ((S (ff_index_mce_alternating_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthsf_prefix)) * ff_vc_mce_fold_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthsf)) /\ exists ff_q_mce_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthsf_prefix_negative. ff_vb_mce_fold_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthsf = ff_q_mce_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthsf_prefix_negative * S ((S (ff_index_mce_alternating_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthsf_prefix)) * ff_vc_mce_fold_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthsf) + (ff_n_mce_alternating_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthsf_prefix))) /\ (((exists ff_even_mce_term_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthsf_prefix_term. ff_index_mce_alternating_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthsf_prefix = 2 * ff_even_mce_term_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthsf_prefix_term) /\ (ff_p_mce_alternating_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthsf_prefix = (ff_ap_mce_alternating_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthsf_prefix) * (ff_bp_mce_alternating_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthsf_prefix) + (ff_an_mce_alternating_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthsf_prefix) * (ff_bn_mce_alternating_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthsf_prefix) /\ ff_n_mce_alternating_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthsf_prefix = (ff_ap_mce_alternating_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthsf_prefix) * (ff_bn_mce_alternating_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthsf_prefix) + (ff_an_mce_alternating_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthsf_prefix) * (ff_bp_mce_alternating_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthsf_prefix))) \/ ((exists ff_odd_mce_term_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthsf_prefix_term. ff_index_mce_alternating_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthsf_prefix = 2 * ff_odd_mce_term_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthsf_prefix_term + 1) /\ (ff_p_mce_alternating_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthsf_prefix = (ff_ap_mce_alternating_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthsf_prefix) * (ff_bn_mce_alternating_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthsf_prefix) + (ff_an_mce_alternating_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthsf_prefix) * (ff_bp_mce_alternating_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthsf_prefix) /\ ff_n_mce_alternating_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthsf_prefix = (ff_ap_mce_alternating_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthsf_prefix) * (ff_bp_mce_alternating_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthsf_prefix) + (ff_an_mce_alternating_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthsf_prefix) * (ff_bn_mce_alternating_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthsf_prefix))))))))))) /\ ((exists ff_u_mce_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthsf_positive ff_v_mce_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthsf_positive. ((((exists ff_h_mce_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthsf_positive_start. ff_h_mce_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthsf_positive_start + S (0) = S ((S (0)) * ff_v_mce_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthsf_positive)) /\ exists ff_q_mce_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthsf_positive_start. ff_u_mce_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthsf_positive = ff_q_mce_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthsf_positive_start * S ((S (0)) * ff_v_mce_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthsf_positive) + (0))) /\ ((((exists ff_h_mce_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthsf_positive_terminal. ff_h_mce_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthsf_positive_terminal + S (mdr_p_zero_columns_existswitnessminornonzeroevaluationdeterminanth) = S ((S ((S (mdr_q_zero_columns_existswitnessminornonzeroevaluationdeterminanths)))) * ff_v_mce_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthsf_positive)) /\ exists ff_q_mce_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthsf_positive_terminal. ff_u_mce_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthsf_positive = ff_q_mce_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthsf_positive_terminal * S ((S ((S (mdr_q_zero_columns_existswitnessminornonzeroevaluationdeterminanths)))) * ff_v_mce_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthsf_positive) + (mdr_p_zero_columns_existswitnessminornonzeroevaluationdeterminanth))) /\ forall ff_i_mce_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthsf_positive. (exists ff_lt_mce_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthsf_positive_bound. ff_lt_mce_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthsf_positive_bound + S ff_i_mce_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthsf_positive = (S (mdr_q_zero_columns_existswitnessminornonzeroevaluationdeterminanths))) -> exists ff_a_mce_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthsf_positive ff_r_mce_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthsf_positive ff_s_mce_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthsf_positive. ((((exists ff_h_mce_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthsf_positive_summand. ff_h_mce_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthsf_positive_summand + S (ff_a_mce_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthsf_positive) = S ((S (ff_i_mce_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthsf_positive)) * ff_uc_mce_fold_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthsf)) /\ exists ff_q_mce_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthsf_positive_summand. ff_ub_mce_fold_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthsf = ff_q_mce_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthsf_positive_summand * S ((S (ff_i_mce_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthsf_positive)) * ff_uc_mce_fold_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthsf) + (ff_a_mce_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthsf_positive))) /\ ((((exists ff_h_mce_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthsf_positive_partial. ff_h_mce_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthsf_positive_partial + S (ff_r_mce_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthsf_positive) = S ((S (ff_i_mce_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthsf_positive)) * ff_v_mce_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthsf_positive)) /\ exists ff_q_mce_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthsf_positive_partial. ff_u_mce_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthsf_positive = ff_q_mce_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthsf_positive_partial * S ((S (ff_i_mce_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthsf_positive)) * ff_v_mce_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthsf_positive) + (ff_r_mce_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthsf_positive))) /\ ((((exists ff_h_mce_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthsf_positive_successor. ff_h_mce_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthsf_positive_successor + S (ff_s_mce_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthsf_positive) = S ((S (S ff_i_mce_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthsf_positive)) * ff_v_mce_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthsf_positive)) /\ exists ff_q_mce_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthsf_positive_successor. ff_u_mce_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthsf_positive = ff_q_mce_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthsf_positive_successor * S ((S (S ff_i_mce_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthsf_positive)) * ff_v_mce_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthsf_positive) + (ff_s_mce_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthsf_positive))) /\ ff_s_mce_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthsf_positive = ff_r_mce_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthsf_positive + ff_a_mce_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthsf_positive)))))) /\ (exists ff_u_mce_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthsf_negative ff_v_mce_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthsf_negative. ((((exists ff_h_mce_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthsf_negative_start. ff_h_mce_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthsf_negative_start + S (0) = S ((S (0)) * ff_v_mce_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthsf_negative)) /\ exists ff_q_mce_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthsf_negative_start. ff_u_mce_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthsf_negative = ff_q_mce_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthsf_negative_start * S ((S (0)) * ff_v_mce_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthsf_negative) + (0))) /\ ((((exists ff_h_mce_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthsf_negative_terminal. ff_h_mce_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthsf_negative_terminal + S (mdr_n_zero_columns_existswitnessminornonzeroevaluationdeterminanth) = S ((S ((S (mdr_q_zero_columns_existswitnessminornonzeroevaluationdeterminanths)))) * ff_v_mce_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthsf_negative)) /\ exists ff_q_mce_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthsf_negative_terminal. ff_u_mce_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthsf_negative = ff_q_mce_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthsf_negative_terminal * S ((S ((S (mdr_q_zero_columns_existswitnessminornonzeroevaluationdeterminanths)))) * ff_v_mce_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthsf_negative) + (mdr_n_zero_columns_existswitnessminornonzeroevaluationdeterminanth))) /\ forall ff_i_mce_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthsf_negative. (exists ff_lt_mce_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthsf_negative_bound. ff_lt_mce_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthsf_negative_bound + S ff_i_mce_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthsf_negative = (S (mdr_q_zero_columns_existswitnessminornonzeroevaluationdeterminanths))) -> exists ff_a_mce_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthsf_negative ff_r_mce_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthsf_negative ff_s_mce_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthsf_negative. ((((exists ff_h_mce_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthsf_negative_summand. ff_h_mce_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthsf_negative_summand + S (ff_a_mce_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthsf_negative) = S ((S (ff_i_mce_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthsf_negative)) * ff_vc_mce_fold_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthsf)) /\ exists ff_q_mce_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthsf_negative_summand. ff_vb_mce_fold_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthsf = ff_q_mce_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthsf_negative_summand * S ((S (ff_i_mce_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthsf_negative)) * ff_vc_mce_fold_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthsf) + (ff_a_mce_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthsf_negative))) /\ ((((exists ff_h_mce_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthsf_negative_partial. ff_h_mce_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthsf_negative_partial + S (ff_r_mce_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthsf_negative) = S ((S (ff_i_mce_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthsf_negative)) * ff_v_mce_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthsf_negative)) /\ exists ff_q_mce_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthsf_negative_partial. ff_u_mce_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthsf_negative = ff_q_mce_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthsf_negative_partial * S ((S (ff_i_mce_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthsf_negative)) * ff_v_mce_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthsf_negative) + (ff_r_mce_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthsf_negative))) /\ ((((exists ff_h_mce_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthsf_negative_successor. ff_h_mce_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthsf_negative_successor + S (ff_s_mce_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthsf_negative) = S ((S (S ff_i_mce_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthsf_negative)) * ff_v_mce_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthsf_negative)) /\ exists ff_q_mce_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthsf_negative_successor. ff_u_mce_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthsf_negative = ff_q_mce_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthsf_negative_successor * S ((S (S ff_i_mce_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthsf_negative)) * ff_v_mce_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthsf_negative) + (ff_s_mce_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthsf_negative))) /\ ff_s_mce_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthsf_negative = ff_r_mce_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthsf_negative + ff_a_mce_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminanthsf_negative))))))))))))))) /\ ((exists mdr_gap_zero_columns_existswitnessminornonzeroevaluationdeterminanti. mdr_gap_zero_columns_existswitnessminornonzeroevaluationdeterminanti + S (mdr_i_zero_columns_existswitnessminornonzeroevaluationdeterminant) = (mdr_l_zero_columns_existswitnessminornonzeroevaluationdeterminant)) /\ (exists mdr_z_zero_columns_existswitnessminornonzeroevaluationdeterminantr. ((exists mdr_a_zero_columns_existswitnessminornonzeroevaluationdeterminantrc mdr_b_zero_columns_existswitnessminornonzeroevaluationdeterminantrc mdr_c_zero_columns_existswitnessminornonzeroevaluationdeterminantrc mdr_e_zero_columns_existswitnessminornonzeroevaluationdeterminantrc mdr_f_zero_columns_existswitnessminornonzeroevaluationdeterminantrc. ((mdr_a_zero_columns_existswitnessminornonzeroevaluationdeterminantrc = ((rank) + (mdr_ub_zero_columns_existswitnessminornonzeroevaluation)) * S ((rank) + (mdr_ub_zero_columns_existswitnessminornonzeroevaluation)) + ((mdr_ub_zero_columns_existswitnessminornonzeroevaluation) + (mdr_ub_zero_columns_existswitnessminornonzeroevaluation))) /\ ((mdr_b_zero_columns_existswitnessminornonzeroevaluationdeterminantrc = ((mdr_uc_zero_columns_existswitnessminornonzeroevaluation) + (mdr_vb_zero_columns_existswitnessminornonzeroevaluation)) * S ((mdr_uc_zero_columns_existswitnessminornonzeroevaluation) + (mdr_vb_zero_columns_existswitnessminornonzeroevaluation)) + ((mdr_vb_zero_columns_existswitnessminornonzeroevaluation) + (mdr_vb_zero_columns_existswitnessminornonzeroevaluation))) /\ ((mdr_c_zero_columns_existswitnessminornonzeroevaluationdeterminantrc = ((mdr_a_zero_columns_existswitnessminornonzeroevaluationdeterminantrc) + (mdr_b_zero_columns_existswitnessminornonzeroevaluationdeterminantrc)) * S ((mdr_a_zero_columns_existswitnessminornonzeroevaluationdeterminantrc) + (mdr_b_zero_columns_existswitnessminornonzeroevaluationdeterminantrc)) + ((mdr_b_zero_columns_existswitnessminornonzeroevaluationdeterminantrc) + (mdr_b_zero_columns_existswitnessminornonzeroevaluationdeterminantrc))) /\ ((mdr_e_zero_columns_existswitnessminornonzeroevaluationdeterminantrc = ((mdr_p_zero_columns_existswitnessminornonzero) + (mdr_n_zero_columns_existswitnessminornonzero)) * S ((mdr_p_zero_columns_existswitnessminornonzero) + (mdr_n_zero_columns_existswitnessminornonzero)) + ((mdr_n_zero_columns_existswitnessminornonzero) + (mdr_n_zero_columns_existswitnessminornonzero))) /\ ((mdr_f_zero_columns_existswitnessminornonzeroevaluationdeterminantrc = ((mdr_vc_zero_columns_existswitnessminornonzeroevaluation) + (mdr_e_zero_columns_existswitnessminornonzeroevaluationdeterminantrc)) * S ((mdr_vc_zero_columns_existswitnessminornonzeroevaluation) + (mdr_e_zero_columns_existswitnessminornonzeroevaluationdeterminantrc)) + ((mdr_e_zero_columns_existswitnessminornonzeroevaluationdeterminantrc) + (mdr_e_zero_columns_existswitnessminornonzeroevaluationdeterminantrc))) /\ ((mdr_z_zero_columns_existswitnessminornonzeroevaluationdeterminantr) = ((mdr_c_zero_columns_existswitnessminornonzeroevaluationdeterminantrc) + (mdr_f_zero_columns_existswitnessminornonzeroevaluationdeterminantrc)) * S ((mdr_c_zero_columns_existswitnessminornonzeroevaluationdeterminantrc) + (mdr_f_zero_columns_existswitnessminornonzeroevaluationdeterminantrc)) + ((mdr_f_zero_columns_existswitnessminornonzeroevaluationdeterminantrc) + (mdr_f_zero_columns_existswitnessminornonzeroevaluationdeterminantrc))))))))) /\ (((exists ff_h_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminantrb. ff_h_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminantrb + S (mdr_z_zero_columns_existswitnessminornonzeroevaluationdeterminantr) = S ((S (mdr_i_zero_columns_existswitnessminornonzeroevaluationdeterminant)) * mdr_c_zero_columns_existswitnessminornonzeroevaluationdeterminant)) /\ exists ff_q_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminantrb. mdr_b_zero_columns_existswitnessminornonzeroevaluationdeterminant = ff_q_mdr_zero_columns_existswitnessminornonzeroevaluationdeterminantrb * S ((S (mdr_i_zero_columns_existswitnessminornonzeroevaluationdeterminant)) * mdr_c_zero_columns_existswitnessminornonzeroevaluationdeterminant) + (mdr_z_zero_columns_existswitnessminornonzeroevaluationdeterminantr)))))))))) /\ (~(mdr_p_zero_columns_existswitnessminornonzero = mdr_n_zero_columns_existswitnessminornonzero)))))))) /\ (forall mdr_q_zero_columns_exists. (exists mdr_gap_zero_columns_existshigher. mdr_gap_zero_columns_existshigher + S (rank) = (mdr_q_zero_columns_exists)) -> (forall mdr_rb_zero_columns_existszero mdr_rc_zero_columns_existszero mdr_cb_zero_columns_existszero mdr_cc_zero_columns_existszero mdr_p_zero_columns_existszero mdr_n_zero_columns_existszero. (((forall fom_index_mrf_zero_columns_existszerorowsbound. (exists fom_gap_mrf_zero_columns_existszerorowsbound_index_bound. fom_gap_mrf_zero_columns_existszerorowsbound_index_bound + S (fom_index_mrf_zero_columns_existszerorowsbound) = mdr_q_zero_columns_exists) -> exists fom_value_mrf_zero_columns_existszerorowsbound. ((((exists fom_beta_height_mrf_zero_columns_existszerorowsbound_entry. fom_beta_height_mrf_zero_columns_existszerorowsbound_entry + S (fom_value_mrf_zero_columns_existszerorowsbound) = S ((S (fom_index_mrf_zero_columns_existszerorowsbound)) * mdr_rc_zero_columns_existszero)) /\ exists fom_beta_quotient_mrf_zero_columns_existszerorowsbound_entry. mdr_rb_zero_columns_existszero = fom_beta_quotient_mrf_zero_columns_existszerorowsbound_entry * S ((S (fom_index_mrf_zero_columns_existszerorowsbound)) * mdr_rc_zero_columns_existszero) + (fom_value_mrf_zero_columns_existszerorowsbound))) /\ (exists fom_gap_mrf_zero_columns_existszerorowsbound_value_bound. fom_gap_mrf_zero_columns_existszerorowsbound_value_bound + S (fom_value_mrf_zero_columns_existszerorowsbound) = r))) /\ (forall mdr_i_zero_columns_existszerorowsdistinct mdr_j_zero_columns_existszerorowsdistinct mdr_a_zero_columns_existszerorowsdistinct. (exists mdr_gap_zero_columns_existszerorowsdistincti. mdr_gap_zero_columns_existszerorowsdistincti + S (mdr_i_zero_columns_existszerorowsdistinct) = (mdr_q_zero_columns_exists)) -> (exists mdr_gap_zero_columns_existszerorowsdistinctj. mdr_gap_zero_columns_existszerorowsdistinctj + S (mdr_j_zero_columns_existszerorowsdistinct) = (mdr_q_zero_columns_exists)) -> (((exists ff_h_mdr_zero_columns_existszerorowsdistinctfirst. ff_h_mdr_zero_columns_existszerorowsdistinctfirst + S (mdr_a_zero_columns_existszerorowsdistinct) = S ((S (mdr_i_zero_columns_existszerorowsdistinct)) * mdr_rc_zero_columns_existszero)) /\ exists ff_q_mdr_zero_columns_existszerorowsdistinctfirst. mdr_rb_zero_columns_existszero = ff_q_mdr_zero_columns_existszerorowsdistinctfirst * S ((S (mdr_i_zero_columns_existszerorowsdistinct)) * mdr_rc_zero_columns_existszero) + (mdr_a_zero_columns_existszerorowsdistinct))) -> (((exists ff_h_mdr_zero_columns_existszerorowsdistinctsecond. ff_h_mdr_zero_columns_existszerorowsdistinctsecond + S (mdr_a_zero_columns_existszerorowsdistinct) = S ((S (mdr_j_zero_columns_existszerorowsdistinct)) * mdr_rc_zero_columns_existszero)) /\ exists ff_q_mdr_zero_columns_existszerorowsdistinctsecond. mdr_rb_zero_columns_existszero = ff_q_mdr_zero_columns_existszerorowsdistinctsecond * S ((S (mdr_j_zero_columns_existszerorowsdistinct)) * mdr_rc_zero_columns_existszero) + (mdr_a_zero_columns_existszerorowsdistinct))) -> mdr_i_zero_columns_existszerorowsdistinct = mdr_j_zero_columns_existszerorowsdistinct))) -> (((forall fom_index_mrf_zero_columns_existszerocolumnsbound. (exists fom_gap_mrf_zero_columns_existszerocolumnsbound_index_bound. fom_gap_mrf_zero_columns_existszerocolumnsbound_index_bound + S (fom_index_mrf_zero_columns_existszerocolumnsbound) = mdr_q_zero_columns_exists) -> exists fom_value_mrf_zero_columns_existszerocolumnsbound. ((((exists fom_beta_height_mrf_zero_columns_existszerocolumnsbound_entry. fom_beta_height_mrf_zero_columns_existszerocolumnsbound_entry + S (fom_value_mrf_zero_columns_existszerocolumnsbound) = S ((S (fom_index_mrf_zero_columns_existszerocolumnsbound)) * mdr_cc_zero_columns_existszero)) /\ exists fom_beta_quotient_mrf_zero_columns_existszerocolumnsbound_entry. mdr_cb_zero_columns_existszero = fom_beta_quotient_mrf_zero_columns_existszerocolumnsbound_entry * S ((S (fom_index_mrf_zero_columns_existszerocolumnsbound)) * mdr_cc_zero_columns_existszero) + (fom_value_mrf_zero_columns_existszerocolumnsbound))) /\ (exists fom_gap_mrf_zero_columns_existszerocolumnsbound_value_bound. fom_gap_mrf_zero_columns_existszerocolumnsbound_value_bound + S (fom_value_mrf_zero_columns_existszerocolumnsbound) = 0))) /\ (forall mdr_i_zero_columns_existszerocolumnsdistinct mdr_j_zero_columns_existszerocolumnsdistinct mdr_a_zero_columns_existszerocolumnsdistinct. (exists mdr_gap_zero_columns_existszerocolumnsdistincti. mdr_gap_zero_columns_existszerocolumnsdistincti + S (mdr_i_zero_columns_existszerocolumnsdistinct) = (mdr_q_zero_columns_exists)) -> (exists mdr_gap_zero_columns_existszerocolumnsdistinctj. mdr_gap_zero_columns_existszerocolumnsdistinctj + S (mdr_j_zero_columns_existszerocolumnsdistinct) = (mdr_q_zero_columns_exists)) -> (((exists ff_h_mdr_zero_columns_existszerocolumnsdistinctfirst. ff_h_mdr_zero_columns_existszerocolumnsdistinctfirst + S (mdr_a_zero_columns_existszerocolumnsdistinct) = S ((S (mdr_i_zero_columns_existszerocolumnsdistinct)) * mdr_cc_zero_columns_existszero)) /\ exists ff_q_mdr_zero_columns_existszerocolumnsdistinctfirst. mdr_cb_zero_columns_existszero = ff_q_mdr_zero_columns_existszerocolumnsdistinctfirst * S ((S (mdr_i_zero_columns_existszerocolumnsdistinct)) * mdr_cc_zero_columns_existszero) + (mdr_a_zero_columns_existszerocolumnsdistinct))) -> (((exists ff_h_mdr_zero_columns_existszerocolumnsdistinctsecond. ff_h_mdr_zero_columns_existszerocolumnsdistinctsecond + S (mdr_a_zero_columns_existszerocolumnsdistinct) = S ((S (mdr_j_zero_columns_existszerocolumnsdistinct)) * mdr_cc_zero_columns_existszero)) /\ exists ff_q_mdr_zero_columns_existszerocolumnsdistinctsecond. mdr_cb_zero_columns_existszero = ff_q_mdr_zero_columns_existszerocolumnsdistinctsecond * S ((S (mdr_j_zero_columns_existszerocolumnsdistinct)) * mdr_cc_zero_columns_existszero) + (mdr_a_zero_columns_existszerocolumnsdistinct))) -> mdr_i_zero_columns_existszerocolumnsdistinct = mdr_j_zero_columns_existszerocolumnsdistinct))) -> (exists mdr_ub_zero_columns_existszeroevaluation mdr_uc_zero_columns_existszeroevaluation mdr_vb_zero_columns_existszeroevaluation mdr_vc_zero_columns_existszeroevaluation. ((((forall mdr_i_zero_columns_existszeroevaluationmatrixpositive. (exists mdr_gap_zero_columns_existszeroevaluationmatrixpositivebound. mdr_gap_zero_columns_existszeroevaluationmatrixpositivebound + S (mdr_i_zero_columns_existszeroevaluationmatrixpositive) = ((mdr_q_zero_columns_exists) * (mdr_q_zero_columns_exists))) -> exists mdr_a_zero_columns_existszeroevaluationmatrixpositive. (((exists mdr_r_zero_columns_existszeroevaluationmatrixpositivepoint mdr_s_zero_columns_existszeroevaluationmatrixpositivepoint mdr_u_zero_columns_existszeroevaluationmatrixpositivepoint mdr_v_zero_columns_existszeroevaluationmatrixpositivepoint. ((mdr_i_zero_columns_existszeroevaluationmatrixpositive = (mdr_q_zero_columns_exists) * mdr_r_zero_columns_existszeroevaluationmatrixpositivepoint + mdr_s_zero_columns_existszeroevaluationmatrixpositivepoint) /\ ((exists mdr_gap_zero_columns_existszeroevaluationmatrixpositivepointcolumn. mdr_gap_zero_columns_existszeroevaluationmatrixpositivepointcolumn + S (mdr_s_zero_columns_existszeroevaluationmatrixpositivepoint) = (mdr_q_zero_columns_exists)) /\ ((((exists ff_h_mdr_zero_columns_existszeroevaluationmatrixpositivepointrow_index. ff_h_mdr_zero_columns_existszeroevaluationmatrixpositivepointrow_index + S (mdr_u_zero_columns_existszeroevaluationmatrixpositivepoint) = S ((S (mdr_r_zero_columns_existszeroevaluationmatrixpositivepoint)) * mdr_rc_zero_columns_existszero)) /\ exists ff_q_mdr_zero_columns_existszeroevaluationmatrixpositivepointrow_index. mdr_rb_zero_columns_existszero = ff_q_mdr_zero_columns_existszeroevaluationmatrixpositivepointrow_index * S ((S (mdr_r_zero_columns_existszeroevaluationmatrixpositivepoint)) * mdr_rc_zero_columns_existszero) + (mdr_u_zero_columns_existszeroevaluationmatrixpositivepoint))) /\ ((((exists ff_h_mdr_zero_columns_existszeroevaluationmatrixpositivepointcolumn_index. ff_h_mdr_zero_columns_existszeroevaluationmatrixpositivepointcolumn_index + S (mdr_v_zero_columns_existszeroevaluationmatrixpositivepoint) = S ((S (mdr_s_zero_columns_existszeroevaluationmatrixpositivepoint)) * mdr_cc_zero_columns_existszero)) /\ exists ff_q_mdr_zero_columns_existszeroevaluationmatrixpositivepointcolumn_index. mdr_cb_zero_columns_existszero = ff_q_mdr_zero_columns_existszeroevaluationmatrixpositivepointcolumn_index * S ((S (mdr_s_zero_columns_existszeroevaluationmatrixpositivepoint)) * mdr_cc_zero_columns_existszero) + (mdr_v_zero_columns_existszeroevaluationmatrixpositivepoint))) /\ (((exists ff_h_mdr_zero_columns_existszeroevaluationmatrixpositivepointsource. ff_h_mdr_zero_columns_existszeroevaluationmatrixpositivepointsource + S (mdr_a_zero_columns_existszeroevaluationmatrixpositive) = S ((S ((mdr_u_zero_columns_existszeroevaluationmatrixpositivepoint) * (0) + (mdr_v_zero_columns_existszeroevaluationmatrixpositivepoint))) * pc)) /\ exists ff_q_mdr_zero_columns_existszeroevaluationmatrixpositivepointsource. pb = ff_q_mdr_zero_columns_existszeroevaluationmatrixpositivepointsource * S ((S ((mdr_u_zero_columns_existszeroevaluationmatrixpositivepoint) * (0) + (mdr_v_zero_columns_existszeroevaluationmatrixpositivepoint))) * pc) + (mdr_a_zero_columns_existszeroevaluationmatrixpositive)))))))) /\ (((exists ff_h_mdr_zero_columns_existszeroevaluationmatrixpositiveoutput. ff_h_mdr_zero_columns_existszeroevaluationmatrixpositiveoutput + S (mdr_a_zero_columns_existszeroevaluationmatrixpositive) = S ((S (mdr_i_zero_columns_existszeroevaluationmatrixpositive)) * mdr_uc_zero_columns_existszeroevaluation)) /\ exists ff_q_mdr_zero_columns_existszeroevaluationmatrixpositiveoutput. mdr_ub_zero_columns_existszeroevaluation = ff_q_mdr_zero_columns_existszeroevaluationmatrixpositiveoutput * S ((S (mdr_i_zero_columns_existszeroevaluationmatrixpositive)) * mdr_uc_zero_columns_existszeroevaluation) + (mdr_a_zero_columns_existszeroevaluationmatrixpositive)))))) /\ (forall mdr_i_zero_columns_existszeroevaluationmatrixnegative. (exists mdr_gap_zero_columns_existszeroevaluationmatrixnegativebound. mdr_gap_zero_columns_existszeroevaluationmatrixnegativebound + S (mdr_i_zero_columns_existszeroevaluationmatrixnegative) = ((mdr_q_zero_columns_exists) * (mdr_q_zero_columns_exists))) -> exists mdr_a_zero_columns_existszeroevaluationmatrixnegative. (((exists mdr_r_zero_columns_existszeroevaluationmatrixnegativepoint mdr_s_zero_columns_existszeroevaluationmatrixnegativepoint mdr_u_zero_columns_existszeroevaluationmatrixnegativepoint mdr_v_zero_columns_existszeroevaluationmatrixnegativepoint. ((mdr_i_zero_columns_existszeroevaluationmatrixnegative = (mdr_q_zero_columns_exists) * mdr_r_zero_columns_existszeroevaluationmatrixnegativepoint + mdr_s_zero_columns_existszeroevaluationmatrixnegativepoint) /\ ((exists mdr_gap_zero_columns_existszeroevaluationmatrixnegativepointcolumn. mdr_gap_zero_columns_existszeroevaluationmatrixnegativepointcolumn + S (mdr_s_zero_columns_existszeroevaluationmatrixnegativepoint) = (mdr_q_zero_columns_exists)) /\ ((((exists ff_h_mdr_zero_columns_existszeroevaluationmatrixnegativepointrow_index. ff_h_mdr_zero_columns_existszeroevaluationmatrixnegativepointrow_index + S (mdr_u_zero_columns_existszeroevaluationmatrixnegativepoint) = S ((S (mdr_r_zero_columns_existszeroevaluationmatrixnegativepoint)) * mdr_rc_zero_columns_existszero)) /\ exists ff_q_mdr_zero_columns_existszeroevaluationmatrixnegativepointrow_index. mdr_rb_zero_columns_existszero = ff_q_mdr_zero_columns_existszeroevaluationmatrixnegativepointrow_index * S ((S (mdr_r_zero_columns_existszeroevaluationmatrixnegativepoint)) * mdr_rc_zero_columns_existszero) + (mdr_u_zero_columns_existszeroevaluationmatrixnegativepoint))) /\ ((((exists ff_h_mdr_zero_columns_existszeroevaluationmatrixnegativepointcolumn_index. ff_h_mdr_zero_columns_existszeroevaluationmatrixnegativepointcolumn_index + S (mdr_v_zero_columns_existszeroevaluationmatrixnegativepoint) = S ((S (mdr_s_zero_columns_existszeroevaluationmatrixnegativepoint)) * mdr_cc_zero_columns_existszero)) /\ exists ff_q_mdr_zero_columns_existszeroevaluationmatrixnegativepointcolumn_index. mdr_cb_zero_columns_existszero = ff_q_mdr_zero_columns_existszeroevaluationmatrixnegativepointcolumn_index * S ((S (mdr_s_zero_columns_existszeroevaluationmatrixnegativepoint)) * mdr_cc_zero_columns_existszero) + (mdr_v_zero_columns_existszeroevaluationmatrixnegativepoint))) /\ (((exists ff_h_mdr_zero_columns_existszeroevaluationmatrixnegativepointsource. ff_h_mdr_zero_columns_existszeroevaluationmatrixnegativepointsource + S (mdr_a_zero_columns_existszeroevaluationmatrixnegative) = S ((S ((mdr_u_zero_columns_existszeroevaluationmatrixnegativepoint) * (0) + (mdr_v_zero_columns_existszeroevaluationmatrixnegativepoint))) * nc)) /\ exists ff_q_mdr_zero_columns_existszeroevaluationmatrixnegativepointsource. nb = ff_q_mdr_zero_columns_existszeroevaluationmatrixnegativepointsource * S ((S ((mdr_u_zero_columns_existszeroevaluationmatrixnegativepoint) * (0) + (mdr_v_zero_columns_existszeroevaluationmatrixnegativepoint))) * nc) + (mdr_a_zero_columns_existszeroevaluationmatrixnegative)))))))) /\ (((exists ff_h_mdr_zero_columns_existszeroevaluationmatrixnegativeoutput. ff_h_mdr_zero_columns_existszeroevaluationmatrixnegativeoutput + S (mdr_a_zero_columns_existszeroevaluationmatrixnegative) = S ((S (mdr_i_zero_columns_existszeroevaluationmatrixnegative)) * mdr_vc_zero_columns_existszeroevaluation)) /\ exists ff_q_mdr_zero_columns_existszeroevaluationmatrixnegativeoutput. mdr_vb_zero_columns_existszeroevaluation = ff_q_mdr_zero_columns_existszeroevaluationmatrixnegativeoutput * S ((S (mdr_i_zero_columns_existszeroevaluationmatrixnegative)) * mdr_vc_zero_columns_existszeroevaluation) + (mdr_a_zero_columns_existszeroevaluationmatrixnegative)))))))) /\ (exists mdr_b_zero_columns_existszeroevaluationdeterminant mdr_c_zero_columns_existszeroevaluationdeterminant mdr_l_zero_columns_existszeroevaluationdeterminant mdr_i_zero_columns_existszeroevaluationdeterminant. ((forall mdr_i_zero_columns_existszeroevaluationdeterminanth. (exists mdr_gap_zero_columns_existszeroevaluationdeterminanthi. mdr_gap_zero_columns_existszeroevaluationdeterminanthi + S (mdr_i_zero_columns_existszeroevaluationdeterminanth) = (mdr_l_zero_columns_existszeroevaluationdeterminant)) -> exists mdr_d_zero_columns_existszeroevaluationdeterminanth mdr_pb_zero_columns_existszeroevaluationdeterminanth mdr_pc_zero_columns_existszeroevaluationdeterminanth mdr_nb_zero_columns_existszeroevaluationdeterminanth mdr_nc_zero_columns_existszeroevaluationdeterminanth mdr_p_zero_columns_existszeroevaluationdeterminanth mdr_n_zero_columns_existszeroevaluationdeterminanth. ((exists mdr_z_zero_columns_existszeroevaluationdeterminanthr. ((exists mdr_a_zero_columns_existszeroevaluationdeterminanthrc mdr_b_zero_columns_existszeroevaluationdeterminanthrc mdr_c_zero_columns_existszeroevaluationdeterminanthrc mdr_e_zero_columns_existszeroevaluationdeterminanthrc mdr_f_zero_columns_existszeroevaluationdeterminanthrc. ((mdr_a_zero_columns_existszeroevaluationdeterminanthrc = ((mdr_d_zero_columns_existszeroevaluationdeterminanth) + (mdr_pb_zero_columns_existszeroevaluationdeterminanth)) * S ((mdr_d_zero_columns_existszeroevaluationdeterminanth) + (mdr_pb_zero_columns_existszeroevaluationdeterminanth)) + ((mdr_pb_zero_columns_existszeroevaluationdeterminanth) + (mdr_pb_zero_columns_existszeroevaluationdeterminanth))) /\ ((mdr_b_zero_columns_existszeroevaluationdeterminanthrc = ((mdr_pc_zero_columns_existszeroevaluationdeterminanth) + (mdr_nb_zero_columns_existszeroevaluationdeterminanth)) * S ((mdr_pc_zero_columns_existszeroevaluationdeterminanth) + (mdr_nb_zero_columns_existszeroevaluationdeterminanth)) + ((mdr_nb_zero_columns_existszeroevaluationdeterminanth) + (mdr_nb_zero_columns_existszeroevaluationdeterminanth))) /\ ((mdr_c_zero_columns_existszeroevaluationdeterminanthrc = ((mdr_a_zero_columns_existszeroevaluationdeterminanthrc) + (mdr_b_zero_columns_existszeroevaluationdeterminanthrc)) * S ((mdr_a_zero_columns_existszeroevaluationdeterminanthrc) + (mdr_b_zero_columns_existszeroevaluationdeterminanthrc)) + ((mdr_b_zero_columns_existszeroevaluationdeterminanthrc) + (mdr_b_zero_columns_existszeroevaluationdeterminanthrc))) /\ ((mdr_e_zero_columns_existszeroevaluationdeterminanthrc = ((mdr_p_zero_columns_existszeroevaluationdeterminanth) + (mdr_n_zero_columns_existszeroevaluationdeterminanth)) * S ((mdr_p_zero_columns_existszeroevaluationdeterminanth) + (mdr_n_zero_columns_existszeroevaluationdeterminanth)) + ((mdr_n_zero_columns_existszeroevaluationdeterminanth) + (mdr_n_zero_columns_existszeroevaluationdeterminanth))) /\ ((mdr_f_zero_columns_existszeroevaluationdeterminanthrc = ((mdr_nc_zero_columns_existszeroevaluationdeterminanth) + (mdr_e_zero_columns_existszeroevaluationdeterminanthrc)) * S ((mdr_nc_zero_columns_existszeroevaluationdeterminanth) + (mdr_e_zero_columns_existszeroevaluationdeterminanthrc)) + ((mdr_e_zero_columns_existszeroevaluationdeterminanthrc) + (mdr_e_zero_columns_existszeroevaluationdeterminanthrc))) /\ ((mdr_z_zero_columns_existszeroevaluationdeterminanthr) = ((mdr_c_zero_columns_existszeroevaluationdeterminanthrc) + (mdr_f_zero_columns_existszeroevaluationdeterminanthrc)) * S ((mdr_c_zero_columns_existszeroevaluationdeterminanthrc) + (mdr_f_zero_columns_existszeroevaluationdeterminanthrc)) + ((mdr_f_zero_columns_existszeroevaluationdeterminanthrc) + (mdr_f_zero_columns_existszeroevaluationdeterminanthrc))))))))) /\ (((exists ff_h_mdr_zero_columns_existszeroevaluationdeterminanthrb. ff_h_mdr_zero_columns_existszeroevaluationdeterminanthrb + S (mdr_z_zero_columns_existszeroevaluationdeterminanthr) = S ((S (mdr_i_zero_columns_existszeroevaluationdeterminanth)) * mdr_c_zero_columns_existszeroevaluationdeterminant)) /\ exists ff_q_mdr_zero_columns_existszeroevaluationdeterminanthrb. mdr_b_zero_columns_existszeroevaluationdeterminant = ff_q_mdr_zero_columns_existszeroevaluationdeterminanthrb * S ((S (mdr_i_zero_columns_existszeroevaluationdeterminanth)) * mdr_c_zero_columns_existszeroevaluationdeterminant) + (mdr_z_zero_columns_existszeroevaluationdeterminanthr))))) /\ (((((mdr_d_zero_columns_existszeroevaluationdeterminanth) = 0) /\ (((mdr_p_zero_columns_existszeroevaluationdeterminanth) = 1) /\ ((mdr_n_zero_columns_existszeroevaluationdeterminanth) = 0))) \/ exists mdr_q_zero_columns_existszeroevaluationdeterminanths mdr_eb_zero_columns_existszeroevaluationdeterminanths mdr_ec_zero_columns_existszeroevaluationdeterminanths mdr_fb_zero_columns_existszeroevaluationdeterminanths mdr_fc_zero_columns_existszeroevaluationdeterminanths. (((mdr_d_zero_columns_existszeroevaluationdeterminanth) = S (mdr_q_zero_columns_existszeroevaluationdeterminanths)) /\ ((forall mdr_j_zero_columns_existszeroevaluationdeterminanthsc. (exists mdr_gap_zero_columns_existszeroevaluationdeterminanthscj. mdr_gap_zero_columns_existszeroevaluationdeterminanthscj + S (mdr_j_zero_columns_existszeroevaluationdeterminanthsc) = (S (mdr_q_zero_columns_existszeroevaluationdeterminanths))) -> exists mdr_i_zero_columns_existszeroevaluationdeterminanthsc mdr_up_zero_columns_existszeroevaluationdeterminanthsc mdr_us_zero_columns_existszeroevaluationdeterminanthsc mdr_un_zero_columns_existszeroevaluationdeterminanthsc mdr_ut_zero_columns_existszeroevaluationdeterminanthsc mdr_p_zero_columns_existszeroevaluationdeterminanthsc mdr_n_zero_columns_existszeroevaluationdeterminanthsc. ((exists mdr_gap_zero_columns_existszeroevaluationdeterminanthsci. mdr_gap_zero_columns_existszeroevaluationdeterminanthsci + S (mdr_i_zero_columns_existszeroevaluationdeterminanthsc) = (mdr_i_zero_columns_existszeroevaluationdeterminanth)) /\ ((exists mdr_z_zero_columns_existszeroevaluationdeterminanthscr. ((exists mdr_a_zero_columns_existszeroevaluationdeterminanthscrc mdr_b_zero_columns_existszeroevaluationdeterminanthscrc mdr_c_zero_columns_existszeroevaluationdeterminanthscrc mdr_e_zero_columns_existszeroevaluationdeterminanthscrc mdr_f_zero_columns_existszeroevaluationdeterminanthscrc. ((mdr_a_zero_columns_existszeroevaluationdeterminanthscrc = ((mdr_q_zero_columns_existszeroevaluationdeterminanths) + (mdr_up_zero_columns_existszeroevaluationdeterminanthsc)) * S ((mdr_q_zero_columns_existszeroevaluationdeterminanths) + (mdr_up_zero_columns_existszeroevaluationdeterminanthsc)) + ((mdr_up_zero_columns_existszeroevaluationdeterminanthsc) + (mdr_up_zero_columns_existszeroevaluationdeterminanthsc))) /\ ((mdr_b_zero_columns_existszeroevaluationdeterminanthscrc = ((mdr_us_zero_columns_existszeroevaluationdeterminanthsc) + (mdr_un_zero_columns_existszeroevaluationdeterminanthsc)) * S ((mdr_us_zero_columns_existszeroevaluationdeterminanthsc) + (mdr_un_zero_columns_existszeroevaluationdeterminanthsc)) + ((mdr_un_zero_columns_existszeroevaluationdeterminanthsc) + (mdr_un_zero_columns_existszeroevaluationdeterminanthsc))) /\ ((mdr_c_zero_columns_existszeroevaluationdeterminanthscrc = ((mdr_a_zero_columns_existszeroevaluationdeterminanthscrc) + (mdr_b_zero_columns_existszeroevaluationdeterminanthscrc)) * S ((mdr_a_zero_columns_existszeroevaluationdeterminanthscrc) + (mdr_b_zero_columns_existszeroevaluationdeterminanthscrc)) + ((mdr_b_zero_columns_existszeroevaluationdeterminanthscrc) + (mdr_b_zero_columns_existszeroevaluationdeterminanthscrc))) /\ ((mdr_e_zero_columns_existszeroevaluationdeterminanthscrc = ((mdr_p_zero_columns_existszeroevaluationdeterminanthsc) + (mdr_n_zero_columns_existszeroevaluationdeterminanthsc)) * S ((mdr_p_zero_columns_existszeroevaluationdeterminanthsc) + (mdr_n_zero_columns_existszeroevaluationdeterminanthsc)) + ((mdr_n_zero_columns_existszeroevaluationdeterminanthsc) + (mdr_n_zero_columns_existszeroevaluationdeterminanthsc))) /\ ((mdr_f_zero_columns_existszeroevaluationdeterminanthscrc = ((mdr_ut_zero_columns_existszeroevaluationdeterminanthsc) + (mdr_e_zero_columns_existszeroevaluationdeterminanthscrc)) * S ((mdr_ut_zero_columns_existszeroevaluationdeterminanthsc) + (mdr_e_zero_columns_existszeroevaluationdeterminanthscrc)) + ((mdr_e_zero_columns_existszeroevaluationdeterminanthscrc) + (mdr_e_zero_columns_existszeroevaluationdeterminanthscrc))) /\ ((mdr_z_zero_columns_existszeroevaluationdeterminanthscr) = ((mdr_c_zero_columns_existszeroevaluationdeterminanthscrc) + (mdr_f_zero_columns_existszeroevaluationdeterminanthscrc)) * S ((mdr_c_zero_columns_existszeroevaluationdeterminanthscrc) + (mdr_f_zero_columns_existszeroevaluationdeterminanthscrc)) + ((mdr_f_zero_columns_existszeroevaluationdeterminanthscrc) + (mdr_f_zero_columns_existszeroevaluationdeterminanthscrc))))))))) /\ (((exists ff_h_mdr_zero_columns_existszeroevaluationdeterminanthscrb. ff_h_mdr_zero_columns_existszeroevaluationdeterminanthscrb + S (mdr_z_zero_columns_existszeroevaluationdeterminanthscr) = S ((S (mdr_i_zero_columns_existszeroevaluationdeterminanthsc)) * mdr_c_zero_columns_existszeroevaluationdeterminant)) /\ exists ff_q_mdr_zero_columns_existszeroevaluationdeterminanthscrb. mdr_b_zero_columns_existszeroevaluationdeterminant = ff_q_mdr_zero_columns_existszeroevaluationdeterminanthscrb * S ((S (mdr_i_zero_columns_existszeroevaluationdeterminanthsc)) * mdr_c_zero_columns_existszeroevaluationdeterminant) + (mdr_z_zero_columns_existszeroevaluationdeterminanthscr))))) /\ ((((forall ff_index_mdm_prefix_mdr_zero_columns_existszeroevaluationdeterminanthscm_positive. (exists ff_gap_mdm_lt_mdr_zero_columns_existszeroevaluationdeterminanthscm_positive_index_bound. ff_gap_mdm_lt_mdr_zero_columns_existszeroevaluationdeterminanthscm_positive_index_bound + S (ff_index_mdm_prefix_mdr_zero_columns_existszeroevaluationdeterminanthscm_positive) = ((mdr_q_zero_columns_existszeroevaluationdeterminanths) * (mdr_q_zero_columns_existszeroevaluationdeterminanths))) -> exists ff_row_mdm_prefix_mdr_zero_columns_existszeroevaluationdeterminanthscm_positive ff_column_mdm_prefix_mdr_zero_columns_existszeroevaluationdeterminanthscm_positive ff_value_mdm_prefix_mdr_zero_columns_existszeroevaluationdeterminanthscm_positive. (ff_index_mdm_prefix_mdr_zero_columns_existszeroevaluationdeterminanthscm_positive = (mdr_q_zero_columns_existszeroevaluationdeterminanths) * ff_row_mdm_prefix_mdr_zero_columns_existszeroevaluationdeterminanthscm_positive + ff_column_mdm_prefix_mdr_zero_columns_existszeroevaluationdeterminanthscm_positive /\ ((exists ff_gap_mdm_lt_mdr_zero_columns_existszeroevaluationdeterminanthscm_positive_column_bound. ff_gap_mdm_lt_mdr_zero_columns_existszeroevaluationdeterminanthscm_positive_column_bound + S (ff_column_mdm_prefix_mdr_zero_columns_existszeroevaluationdeterminanthscm_positive) = (mdr_q_zero_columns_existszeroevaluationdeterminanths)) /\ ((exists ff_row_mdm_cell_mdr_zero_columns_existszeroevaluationdeterminanthscm_positive_cell ff_column_mdm_cell_mdr_zero_columns_existszeroevaluationdeterminanthscm_positive_cell. (((((exists ff_gap_mdm_lt_mdr_zero_columns_existszeroevaluationdeterminanthscm_positive_cell_row_before. ff_gap_mdm_lt_mdr_zero_columns_existszeroevaluationdeterminanthscm_positive_cell_row_before + S (ff_row_mdm_prefix_mdr_zero_columns_existszeroevaluationdeterminanthscm_positive) = (0)) /\ ff_row_mdm_cell_mdr_zero_columns_existszeroevaluationdeterminanthscm_positive_cell = ff_row_mdm_prefix_mdr_zero_columns_existszeroevaluationdeterminanthscm_positive) \/ ((exists ff_gap_mdm_le_mdr_zero_columns_existszeroevaluationdeterminanthscm_positive_cell_row_after. ff_gap_mdm_le_mdr_zero_columns_existszeroevaluationdeterminanthscm_positive_cell_row_after + (0) = (ff_row_mdm_prefix_mdr_zero_columns_existszeroevaluationdeterminanthscm_positive)) /\ ff_row_mdm_cell_mdr_zero_columns_existszeroevaluationdeterminanthscm_positive_cell = S ff_row_mdm_prefix_mdr_zero_columns_existszeroevaluationdeterminanthscm_positive))) /\ (((((exists ff_gap_mdm_lt_mdr_zero_columns_existszeroevaluationdeterminanthscm_positive_cell_column_before. ff_gap_mdm_lt_mdr_zero_columns_existszeroevaluationdeterminanthscm_positive_cell_column_before + S (ff_column_mdm_prefix_mdr_zero_columns_existszeroevaluationdeterminanthscm_positive) = (mdr_j_zero_columns_existszeroevaluationdeterminanthsc)) /\ ff_column_mdm_cell_mdr_zero_columns_existszeroevaluationdeterminanthscm_positive_cell = ff_column_mdm_prefix_mdr_zero_columns_existszeroevaluationdeterminanthscm_positive) \/ ((exists ff_gap_mdm_le_mdr_zero_columns_existszeroevaluationdeterminanthscm_positive_cell_column_after. ff_gap_mdm_le_mdr_zero_columns_existszeroevaluationdeterminanthscm_positive_cell_column_after + (mdr_j_zero_columns_existszeroevaluationdeterminanthsc) = (ff_column_mdm_prefix_mdr_zero_columns_existszeroevaluationdeterminanthscm_positive)) /\ ff_column_mdm_cell_mdr_zero_columns_existszeroevaluationdeterminanthscm_positive_cell = S ff_column_mdm_prefix_mdr_zero_columns_existszeroevaluationdeterminanthscm_positive))) /\ (((exists ff_h_mdm_mdr_zero_columns_existszeroevaluationdeterminanthscm_positive_cell_source. ff_h_mdm_mdr_zero_columns_existszeroevaluationdeterminanthscm_positive_cell_source + S (ff_value_mdm_prefix_mdr_zero_columns_existszeroevaluationdeterminanthscm_positive) = S ((S ((ff_row_mdm_cell_mdr_zero_columns_existszeroevaluationdeterminanthscm_positive_cell) * (S (mdr_q_zero_columns_existszeroevaluationdeterminanths)) + (ff_column_mdm_cell_mdr_zero_columns_existszeroevaluationdeterminanthscm_positive_cell))) * mdr_pc_zero_columns_existszeroevaluationdeterminanth)) /\ exists ff_q_mdm_mdr_zero_columns_existszeroevaluationdeterminanthscm_positive_cell_source. mdr_pb_zero_columns_existszeroevaluationdeterminanth = ff_q_mdm_mdr_zero_columns_existszeroevaluationdeterminanthscm_positive_cell_source * S ((S ((ff_row_mdm_cell_mdr_zero_columns_existszeroevaluationdeterminanthscm_positive_cell) * (S (mdr_q_zero_columns_existszeroevaluationdeterminanths)) + (ff_column_mdm_cell_mdr_zero_columns_existszeroevaluationdeterminanthscm_positive_cell))) * mdr_pc_zero_columns_existszeroevaluationdeterminanth) + (ff_value_mdm_prefix_mdr_zero_columns_existszeroevaluationdeterminanthscm_positive)))))) /\ (((exists ff_h_mdm_mdr_zero_columns_existszeroevaluationdeterminanthscm_positive_target. ff_h_mdm_mdr_zero_columns_existszeroevaluationdeterminanthscm_positive_target + S (ff_value_mdm_prefix_mdr_zero_columns_existszeroevaluationdeterminanthscm_positive) = S ((S (ff_index_mdm_prefix_mdr_zero_columns_existszeroevaluationdeterminanthscm_positive)) * mdr_us_zero_columns_existszeroevaluationdeterminanthsc)) /\ exists ff_q_mdm_mdr_zero_columns_existszeroevaluationdeterminanthscm_positive_target. mdr_up_zero_columns_existszeroevaluationdeterminanthsc = ff_q_mdm_mdr_zero_columns_existszeroevaluationdeterminanthscm_positive_target * S ((S (ff_index_mdm_prefix_mdr_zero_columns_existszeroevaluationdeterminanthscm_positive)) * mdr_us_zero_columns_existszeroevaluationdeterminanthsc) + (ff_value_mdm_prefix_mdr_zero_columns_existszeroevaluationdeterminanthscm_positive))))))) /\ (forall ff_index_mdm_prefix_mdr_zero_columns_existszeroevaluationdeterminanthscm_negative. (exists ff_gap_mdm_lt_mdr_zero_columns_existszeroevaluationdeterminanthscm_negative_index_bound. ff_gap_mdm_lt_mdr_zero_columns_existszeroevaluationdeterminanthscm_negative_index_bound + S (ff_index_mdm_prefix_mdr_zero_columns_existszeroevaluationdeterminanthscm_negative) = ((mdr_q_zero_columns_existszeroevaluationdeterminanths) * (mdr_q_zero_columns_existszeroevaluationdeterminanths))) -> exists ff_row_mdm_prefix_mdr_zero_columns_existszeroevaluationdeterminanthscm_negative ff_column_mdm_prefix_mdr_zero_columns_existszeroevaluationdeterminanthscm_negative ff_value_mdm_prefix_mdr_zero_columns_existszeroevaluationdeterminanthscm_negative. (ff_index_mdm_prefix_mdr_zero_columns_existszeroevaluationdeterminanthscm_negative = (mdr_q_zero_columns_existszeroevaluationdeterminanths) * ff_row_mdm_prefix_mdr_zero_columns_existszeroevaluationdeterminanthscm_negative + ff_column_mdm_prefix_mdr_zero_columns_existszeroevaluationdeterminanthscm_negative /\ ((exists ff_gap_mdm_lt_mdr_zero_columns_existszeroevaluationdeterminanthscm_negative_column_bound. ff_gap_mdm_lt_mdr_zero_columns_existszeroevaluationdeterminanthscm_negative_column_bound + S (ff_column_mdm_prefix_mdr_zero_columns_existszeroevaluationdeterminanthscm_negative) = (mdr_q_zero_columns_existszeroevaluationdeterminanths)) /\ ((exists ff_row_mdm_cell_mdr_zero_columns_existszeroevaluationdeterminanthscm_negative_cell ff_column_mdm_cell_mdr_zero_columns_existszeroevaluationdeterminanthscm_negative_cell. (((((exists ff_gap_mdm_lt_mdr_zero_columns_existszeroevaluationdeterminanthscm_negative_cell_row_before. ff_gap_mdm_lt_mdr_zero_columns_existszeroevaluationdeterminanthscm_negative_cell_row_before + S (ff_row_mdm_prefix_mdr_zero_columns_existszeroevaluationdeterminanthscm_negative) = (0)) /\ ff_row_mdm_cell_mdr_zero_columns_existszeroevaluationdeterminanthscm_negative_cell = ff_row_mdm_prefix_mdr_zero_columns_existszeroevaluationdeterminanthscm_negative) \/ ((exists ff_gap_mdm_le_mdr_zero_columns_existszeroevaluationdeterminanthscm_negative_cell_row_after. ff_gap_mdm_le_mdr_zero_columns_existszeroevaluationdeterminanthscm_negative_cell_row_after + (0) = (ff_row_mdm_prefix_mdr_zero_columns_existszeroevaluationdeterminanthscm_negative)) /\ ff_row_mdm_cell_mdr_zero_columns_existszeroevaluationdeterminanthscm_negative_cell = S ff_row_mdm_prefix_mdr_zero_columns_existszeroevaluationdeterminanthscm_negative))) /\ (((((exists ff_gap_mdm_lt_mdr_zero_columns_existszeroevaluationdeterminanthscm_negative_cell_column_before. ff_gap_mdm_lt_mdr_zero_columns_existszeroevaluationdeterminanthscm_negative_cell_column_before + S (ff_column_mdm_prefix_mdr_zero_columns_existszeroevaluationdeterminanthscm_negative) = (mdr_j_zero_columns_existszeroevaluationdeterminanthsc)) /\ ff_column_mdm_cell_mdr_zero_columns_existszeroevaluationdeterminanthscm_negative_cell = ff_column_mdm_prefix_mdr_zero_columns_existszeroevaluationdeterminanthscm_negative) \/ ((exists ff_gap_mdm_le_mdr_zero_columns_existszeroevaluationdeterminanthscm_negative_cell_column_after. ff_gap_mdm_le_mdr_zero_columns_existszeroevaluationdeterminanthscm_negative_cell_column_after + (mdr_j_zero_columns_existszeroevaluationdeterminanthsc) = (ff_column_mdm_prefix_mdr_zero_columns_existszeroevaluationdeterminanthscm_negative)) /\ ff_column_mdm_cell_mdr_zero_columns_existszeroevaluationdeterminanthscm_negative_cell = S ff_column_mdm_prefix_mdr_zero_columns_existszeroevaluationdeterminanthscm_negative))) /\ (((exists ff_h_mdm_mdr_zero_columns_existszeroevaluationdeterminanthscm_negative_cell_source. ff_h_mdm_mdr_zero_columns_existszeroevaluationdeterminanthscm_negative_cell_source + S (ff_value_mdm_prefix_mdr_zero_columns_existszeroevaluationdeterminanthscm_negative) = S ((S ((ff_row_mdm_cell_mdr_zero_columns_existszeroevaluationdeterminanthscm_negative_cell) * (S (mdr_q_zero_columns_existszeroevaluationdeterminanths)) + (ff_column_mdm_cell_mdr_zero_columns_existszeroevaluationdeterminanthscm_negative_cell))) * mdr_nc_zero_columns_existszeroevaluationdeterminanth)) /\ exists ff_q_mdm_mdr_zero_columns_existszeroevaluationdeterminanthscm_negative_cell_source. mdr_nb_zero_columns_existszeroevaluationdeterminanth = ff_q_mdm_mdr_zero_columns_existszeroevaluationdeterminanthscm_negative_cell_source * S ((S ((ff_row_mdm_cell_mdr_zero_columns_existszeroevaluationdeterminanthscm_negative_cell) * (S (mdr_q_zero_columns_existszeroevaluationdeterminanths)) + (ff_column_mdm_cell_mdr_zero_columns_existszeroevaluationdeterminanthscm_negative_cell))) * mdr_nc_zero_columns_existszeroevaluationdeterminanth) + (ff_value_mdm_prefix_mdr_zero_columns_existszeroevaluationdeterminanthscm_negative)))))) /\ (((exists ff_h_mdm_mdr_zero_columns_existszeroevaluationdeterminanthscm_negative_target. ff_h_mdm_mdr_zero_columns_existszeroevaluationdeterminanthscm_negative_target + S (ff_value_mdm_prefix_mdr_zero_columns_existszeroevaluationdeterminanthscm_negative) = S ((S (ff_index_mdm_prefix_mdr_zero_columns_existszeroevaluationdeterminanthscm_negative)) * mdr_ut_zero_columns_existszeroevaluationdeterminanthsc)) /\ exists ff_q_mdm_mdr_zero_columns_existszeroevaluationdeterminanthscm_negative_target. mdr_un_zero_columns_existszeroevaluationdeterminanthsc = ff_q_mdm_mdr_zero_columns_existszeroevaluationdeterminanthscm_negative_target * S ((S (ff_index_mdm_prefix_mdr_zero_columns_existszeroevaluationdeterminanthscm_negative)) * mdr_ut_zero_columns_existszeroevaluationdeterminanthsc) + (ff_value_mdm_prefix_mdr_zero_columns_existszeroevaluationdeterminanthscm_negative))))))))) /\ ((((exists ff_h_mdr_zero_columns_existszeroevaluationdeterminanthscp. ff_h_mdr_zero_columns_existszeroevaluationdeterminanthscp + S (mdr_p_zero_columns_existszeroevaluationdeterminanthsc) = S ((S (mdr_j_zero_columns_existszeroevaluationdeterminanthsc)) * mdr_ec_zero_columns_existszeroevaluationdeterminanths)) /\ exists ff_q_mdr_zero_columns_existszeroevaluationdeterminanthscp. mdr_eb_zero_columns_existszeroevaluationdeterminanths = ff_q_mdr_zero_columns_existszeroevaluationdeterminanthscp * S ((S (mdr_j_zero_columns_existszeroevaluationdeterminanthsc)) * mdr_ec_zero_columns_existszeroevaluationdeterminanths) + (mdr_p_zero_columns_existszeroevaluationdeterminanthsc))) /\ (((exists ff_h_mdr_zero_columns_existszeroevaluationdeterminanthscn. ff_h_mdr_zero_columns_existszeroevaluationdeterminanthscn + S (mdr_n_zero_columns_existszeroevaluationdeterminanthsc) = S ((S (mdr_j_zero_columns_existszeroevaluationdeterminanthsc)) * mdr_fc_zero_columns_existszeroevaluationdeterminanths)) /\ exists ff_q_mdr_zero_columns_existszeroevaluationdeterminanthscn. mdr_fb_zero_columns_existszeroevaluationdeterminanths = ff_q_mdr_zero_columns_existszeroevaluationdeterminanthscn * S ((S (mdr_j_zero_columns_existszeroevaluationdeterminanthsc)) * mdr_fc_zero_columns_existszeroevaluationdeterminanths) + (mdr_n_zero_columns_existszeroevaluationdeterminanthsc)))))))) /\ (exists ff_ub_mce_fold_mdr_zero_columns_existszeroevaluationdeterminanthsf ff_uc_mce_fold_mdr_zero_columns_existszeroevaluationdeterminanthsf ff_vb_mce_fold_mdr_zero_columns_existszeroevaluationdeterminanthsf ff_vc_mce_fold_mdr_zero_columns_existszeroevaluationdeterminanthsf. ((forall ff_index_mce_alternating_mdr_zero_columns_existszeroevaluationdeterminanthsf_prefix. (exists ff_gap_mce_mdr_zero_columns_existszeroevaluationdeterminanthsf_prefix_index. ff_gap_mce_mdr_zero_columns_existszeroevaluationdeterminanthsf_prefix_index + S (ff_index_mce_alternating_mdr_zero_columns_existszeroevaluationdeterminanthsf_prefix) = (S (mdr_q_zero_columns_existszeroevaluationdeterminanths))) -> exists ff_ap_mce_alternating_mdr_zero_columns_existszeroevaluationdeterminanthsf_prefix ff_an_mce_alternating_mdr_zero_columns_existszeroevaluationdeterminanthsf_prefix ff_bp_mce_alternating_mdr_zero_columns_existszeroevaluationdeterminanthsf_prefix ff_bn_mce_alternating_mdr_zero_columns_existszeroevaluationdeterminanthsf_prefix ff_p_mce_alternating_mdr_zero_columns_existszeroevaluationdeterminanthsf_prefix ff_n_mce_alternating_mdr_zero_columns_existszeroevaluationdeterminanthsf_prefix. ((((exists ff_h_mce_mdr_zero_columns_existszeroevaluationdeterminanthsf_prefix_ap. ff_h_mce_mdr_zero_columns_existszeroevaluationdeterminanthsf_prefix_ap + S (ff_ap_mce_alternating_mdr_zero_columns_existszeroevaluationdeterminanthsf_prefix) = S ((S (ff_index_mce_alternating_mdr_zero_columns_existszeroevaluationdeterminanthsf_prefix)) * mdr_pc_zero_columns_existszeroevaluationdeterminanth)) /\ exists ff_q_mce_mdr_zero_columns_existszeroevaluationdeterminanthsf_prefix_ap. mdr_pb_zero_columns_existszeroevaluationdeterminanth = ff_q_mce_mdr_zero_columns_existszeroevaluationdeterminanthsf_prefix_ap * S ((S (ff_index_mce_alternating_mdr_zero_columns_existszeroevaluationdeterminanthsf_prefix)) * mdr_pc_zero_columns_existszeroevaluationdeterminanth) + (ff_ap_mce_alternating_mdr_zero_columns_existszeroevaluationdeterminanthsf_prefix))) /\ ((((exists ff_h_mce_mdr_zero_columns_existszeroevaluationdeterminanthsf_prefix_an. ff_h_mce_mdr_zero_columns_existszeroevaluationdeterminanthsf_prefix_an + S (ff_an_mce_alternating_mdr_zero_columns_existszeroevaluationdeterminanthsf_prefix) = S ((S (ff_index_mce_alternating_mdr_zero_columns_existszeroevaluationdeterminanthsf_prefix)) * mdr_nc_zero_columns_existszeroevaluationdeterminanth)) /\ exists ff_q_mce_mdr_zero_columns_existszeroevaluationdeterminanthsf_prefix_an. mdr_nb_zero_columns_existszeroevaluationdeterminanth = ff_q_mce_mdr_zero_columns_existszeroevaluationdeterminanthsf_prefix_an * S ((S (ff_index_mce_alternating_mdr_zero_columns_existszeroevaluationdeterminanthsf_prefix)) * mdr_nc_zero_columns_existszeroevaluationdeterminanth) + (ff_an_mce_alternating_mdr_zero_columns_existszeroevaluationdeterminanthsf_prefix))) /\ ((((exists ff_h_mce_mdr_zero_columns_existszeroevaluationdeterminanthsf_prefix_bp. ff_h_mce_mdr_zero_columns_existszeroevaluationdeterminanthsf_prefix_bp + S (ff_bp_mce_alternating_mdr_zero_columns_existszeroevaluationdeterminanthsf_prefix) = S ((S (ff_index_mce_alternating_mdr_zero_columns_existszeroevaluationdeterminanthsf_prefix)) * mdr_ec_zero_columns_existszeroevaluationdeterminanths)) /\ exists ff_q_mce_mdr_zero_columns_existszeroevaluationdeterminanthsf_prefix_bp. mdr_eb_zero_columns_existszeroevaluationdeterminanths = ff_q_mce_mdr_zero_columns_existszeroevaluationdeterminanthsf_prefix_bp * S ((S (ff_index_mce_alternating_mdr_zero_columns_existszeroevaluationdeterminanthsf_prefix)) * mdr_ec_zero_columns_existszeroevaluationdeterminanths) + (ff_bp_mce_alternating_mdr_zero_columns_existszeroevaluationdeterminanthsf_prefix))) /\ ((((exists ff_h_mce_mdr_zero_columns_existszeroevaluationdeterminanthsf_prefix_bn. ff_h_mce_mdr_zero_columns_existszeroevaluationdeterminanthsf_prefix_bn + S (ff_bn_mce_alternating_mdr_zero_columns_existszeroevaluationdeterminanthsf_prefix) = S ((S (ff_index_mce_alternating_mdr_zero_columns_existszeroevaluationdeterminanthsf_prefix)) * mdr_fc_zero_columns_existszeroevaluationdeterminanths)) /\ exists ff_q_mce_mdr_zero_columns_existszeroevaluationdeterminanthsf_prefix_bn. mdr_fb_zero_columns_existszeroevaluationdeterminanths = ff_q_mce_mdr_zero_columns_existszeroevaluationdeterminanthsf_prefix_bn * S ((S (ff_index_mce_alternating_mdr_zero_columns_existszeroevaluationdeterminanthsf_prefix)) * mdr_fc_zero_columns_existszeroevaluationdeterminanths) + (ff_bn_mce_alternating_mdr_zero_columns_existszeroevaluationdeterminanthsf_prefix))) /\ ((((exists ff_h_mce_mdr_zero_columns_existszeroevaluationdeterminanthsf_prefix_positive. ff_h_mce_mdr_zero_columns_existszeroevaluationdeterminanthsf_prefix_positive + S (ff_p_mce_alternating_mdr_zero_columns_existszeroevaluationdeterminanthsf_prefix) = S ((S (ff_index_mce_alternating_mdr_zero_columns_existszeroevaluationdeterminanthsf_prefix)) * ff_uc_mce_fold_mdr_zero_columns_existszeroevaluationdeterminanthsf)) /\ exists ff_q_mce_mdr_zero_columns_existszeroevaluationdeterminanthsf_prefix_positive. ff_ub_mce_fold_mdr_zero_columns_existszeroevaluationdeterminanthsf = ff_q_mce_mdr_zero_columns_existszeroevaluationdeterminanthsf_prefix_positive * S ((S (ff_index_mce_alternating_mdr_zero_columns_existszeroevaluationdeterminanthsf_prefix)) * ff_uc_mce_fold_mdr_zero_columns_existszeroevaluationdeterminanthsf) + (ff_p_mce_alternating_mdr_zero_columns_existszeroevaluationdeterminanthsf_prefix))) /\ ((((exists ff_h_mce_mdr_zero_columns_existszeroevaluationdeterminanthsf_prefix_negative. ff_h_mce_mdr_zero_columns_existszeroevaluationdeterminanthsf_prefix_negative + S (ff_n_mce_alternating_mdr_zero_columns_existszeroevaluationdeterminanthsf_prefix) = S ((S (ff_index_mce_alternating_mdr_zero_columns_existszeroevaluationdeterminanthsf_prefix)) * ff_vc_mce_fold_mdr_zero_columns_existszeroevaluationdeterminanthsf)) /\ exists ff_q_mce_mdr_zero_columns_existszeroevaluationdeterminanthsf_prefix_negative. ff_vb_mce_fold_mdr_zero_columns_existszeroevaluationdeterminanthsf = ff_q_mce_mdr_zero_columns_existszeroevaluationdeterminanthsf_prefix_negative * S ((S (ff_index_mce_alternating_mdr_zero_columns_existszeroevaluationdeterminanthsf_prefix)) * ff_vc_mce_fold_mdr_zero_columns_existszeroevaluationdeterminanthsf) + (ff_n_mce_alternating_mdr_zero_columns_existszeroevaluationdeterminanthsf_prefix))) /\ (((exists ff_even_mce_term_mdr_zero_columns_existszeroevaluationdeterminanthsf_prefix_term. ff_index_mce_alternating_mdr_zero_columns_existszeroevaluationdeterminanthsf_prefix = 2 * ff_even_mce_term_mdr_zero_columns_existszeroevaluationdeterminanthsf_prefix_term) /\ (ff_p_mce_alternating_mdr_zero_columns_existszeroevaluationdeterminanthsf_prefix = (ff_ap_mce_alternating_mdr_zero_columns_existszeroevaluationdeterminanthsf_prefix) * (ff_bp_mce_alternating_mdr_zero_columns_existszeroevaluationdeterminanthsf_prefix) + (ff_an_mce_alternating_mdr_zero_columns_existszeroevaluationdeterminanthsf_prefix) * (ff_bn_mce_alternating_mdr_zero_columns_existszeroevaluationdeterminanthsf_prefix) /\ ff_n_mce_alternating_mdr_zero_columns_existszeroevaluationdeterminanthsf_prefix = (ff_ap_mce_alternating_mdr_zero_columns_existszeroevaluationdeterminanthsf_prefix) * (ff_bn_mce_alternating_mdr_zero_columns_existszeroevaluationdeterminanthsf_prefix) + (ff_an_mce_alternating_mdr_zero_columns_existszeroevaluationdeterminanthsf_prefix) * (ff_bp_mce_alternating_mdr_zero_columns_existszeroevaluationdeterminanthsf_prefix))) \/ ((exists ff_odd_mce_term_mdr_zero_columns_existszeroevaluationdeterminanthsf_prefix_term. ff_index_mce_alternating_mdr_zero_columns_existszeroevaluationdeterminanthsf_prefix = 2 * ff_odd_mce_term_mdr_zero_columns_existszeroevaluationdeterminanthsf_prefix_term + 1) /\ (ff_p_mce_alternating_mdr_zero_columns_existszeroevaluationdeterminanthsf_prefix = (ff_ap_mce_alternating_mdr_zero_columns_existszeroevaluationdeterminanthsf_prefix) * (ff_bn_mce_alternating_mdr_zero_columns_existszeroevaluationdeterminanthsf_prefix) + (ff_an_mce_alternating_mdr_zero_columns_existszeroevaluationdeterminanthsf_prefix) * (ff_bp_mce_alternating_mdr_zero_columns_existszeroevaluationdeterminanthsf_prefix) /\ ff_n_mce_alternating_mdr_zero_columns_existszeroevaluationdeterminanthsf_prefix = (ff_ap_mce_alternating_mdr_zero_columns_existszeroevaluationdeterminanthsf_prefix) * (ff_bp_mce_alternating_mdr_zero_columns_existszeroevaluationdeterminanthsf_prefix) + (ff_an_mce_alternating_mdr_zero_columns_existszeroevaluationdeterminanthsf_prefix) * (ff_bn_mce_alternating_mdr_zero_columns_existszeroevaluationdeterminanthsf_prefix))))))))))) /\ ((exists ff_u_mce_mdr_zero_columns_existszeroevaluationdeterminanthsf_positive ff_v_mce_mdr_zero_columns_existszeroevaluationdeterminanthsf_positive. ((((exists ff_h_mce_mdr_zero_columns_existszeroevaluationdeterminanthsf_positive_start. ff_h_mce_mdr_zero_columns_existszeroevaluationdeterminanthsf_positive_start + S (0) = S ((S (0)) * ff_v_mce_mdr_zero_columns_existszeroevaluationdeterminanthsf_positive)) /\ exists ff_q_mce_mdr_zero_columns_existszeroevaluationdeterminanthsf_positive_start. ff_u_mce_mdr_zero_columns_existszeroevaluationdeterminanthsf_positive = ff_q_mce_mdr_zero_columns_existszeroevaluationdeterminanthsf_positive_start * S ((S (0)) * ff_v_mce_mdr_zero_columns_existszeroevaluationdeterminanthsf_positive) + (0))) /\ ((((exists ff_h_mce_mdr_zero_columns_existszeroevaluationdeterminanthsf_positive_terminal. ff_h_mce_mdr_zero_columns_existszeroevaluationdeterminanthsf_positive_terminal + S (mdr_p_zero_columns_existszeroevaluationdeterminanth) = S ((S ((S (mdr_q_zero_columns_existszeroevaluationdeterminanths)))) * ff_v_mce_mdr_zero_columns_existszeroevaluationdeterminanthsf_positive)) /\ exists ff_q_mce_mdr_zero_columns_existszeroevaluationdeterminanthsf_positive_terminal. ff_u_mce_mdr_zero_columns_existszeroevaluationdeterminanthsf_positive = ff_q_mce_mdr_zero_columns_existszeroevaluationdeterminanthsf_positive_terminal * S ((S ((S (mdr_q_zero_columns_existszeroevaluationdeterminanths)))) * ff_v_mce_mdr_zero_columns_existszeroevaluationdeterminanthsf_positive) + (mdr_p_zero_columns_existszeroevaluationdeterminanth))) /\ forall ff_i_mce_mdr_zero_columns_existszeroevaluationdeterminanthsf_positive. (exists ff_lt_mce_mdr_zero_columns_existszeroevaluationdeterminanthsf_positive_bound. ff_lt_mce_mdr_zero_columns_existszeroevaluationdeterminanthsf_positive_bound + S ff_i_mce_mdr_zero_columns_existszeroevaluationdeterminanthsf_positive = (S (mdr_q_zero_columns_existszeroevaluationdeterminanths))) -> exists ff_a_mce_mdr_zero_columns_existszeroevaluationdeterminanthsf_positive ff_r_mce_mdr_zero_columns_existszeroevaluationdeterminanthsf_positive ff_s_mce_mdr_zero_columns_existszeroevaluationdeterminanthsf_positive. ((((exists ff_h_mce_mdr_zero_columns_existszeroevaluationdeterminanthsf_positive_summand. ff_h_mce_mdr_zero_columns_existszeroevaluationdeterminanthsf_positive_summand + S (ff_a_mce_mdr_zero_columns_existszeroevaluationdeterminanthsf_positive) = S ((S (ff_i_mce_mdr_zero_columns_existszeroevaluationdeterminanthsf_positive)) * ff_uc_mce_fold_mdr_zero_columns_existszeroevaluationdeterminanthsf)) /\ exists ff_q_mce_mdr_zero_columns_existszeroevaluationdeterminanthsf_positive_summand. ff_ub_mce_fold_mdr_zero_columns_existszeroevaluationdeterminanthsf = ff_q_mce_mdr_zero_columns_existszeroevaluationdeterminanthsf_positive_summand * S ((S (ff_i_mce_mdr_zero_columns_existszeroevaluationdeterminanthsf_positive)) * ff_uc_mce_fold_mdr_zero_columns_existszeroevaluationdeterminanthsf) + (ff_a_mce_mdr_zero_columns_existszeroevaluationdeterminanthsf_positive))) /\ ((((exists ff_h_mce_mdr_zero_columns_existszeroevaluationdeterminanthsf_positive_partial. ff_h_mce_mdr_zero_columns_existszeroevaluationdeterminanthsf_positive_partial + S (ff_r_mce_mdr_zero_columns_existszeroevaluationdeterminanthsf_positive) = S ((S (ff_i_mce_mdr_zero_columns_existszeroevaluationdeterminanthsf_positive)) * ff_v_mce_mdr_zero_columns_existszeroevaluationdeterminanthsf_positive)) /\ exists ff_q_mce_mdr_zero_columns_existszeroevaluationdeterminanthsf_positive_partial. ff_u_mce_mdr_zero_columns_existszeroevaluationdeterminanthsf_positive = ff_q_mce_mdr_zero_columns_existszeroevaluationdeterminanthsf_positive_partial * S ((S (ff_i_mce_mdr_zero_columns_existszeroevaluationdeterminanthsf_positive)) * ff_v_mce_mdr_zero_columns_existszeroevaluationdeterminanthsf_positive) + (ff_r_mce_mdr_zero_columns_existszeroevaluationdeterminanthsf_positive))) /\ ((((exists ff_h_mce_mdr_zero_columns_existszeroevaluationdeterminanthsf_positive_successor. ff_h_mce_mdr_zero_columns_existszeroevaluationdeterminanthsf_positive_successor + S (ff_s_mce_mdr_zero_columns_existszeroevaluationdeterminanthsf_positive) = S ((S (S ff_i_mce_mdr_zero_columns_existszeroevaluationdeterminanthsf_positive)) * ff_v_mce_mdr_zero_columns_existszeroevaluationdeterminanthsf_positive)) /\ exists ff_q_mce_mdr_zero_columns_existszeroevaluationdeterminanthsf_positive_successor. ff_u_mce_mdr_zero_columns_existszeroevaluationdeterminanthsf_positive = ff_q_mce_mdr_zero_columns_existszeroevaluationdeterminanthsf_positive_successor * S ((S (S ff_i_mce_mdr_zero_columns_existszeroevaluationdeterminanthsf_positive)) * ff_v_mce_mdr_zero_columns_existszeroevaluationdeterminanthsf_positive) + (ff_s_mce_mdr_zero_columns_existszeroevaluationdeterminanthsf_positive))) /\ ff_s_mce_mdr_zero_columns_existszeroevaluationdeterminanthsf_positive = ff_r_mce_mdr_zero_columns_existszeroevaluationdeterminanthsf_positive + ff_a_mce_mdr_zero_columns_existszeroevaluationdeterminanthsf_positive)))))) /\ (exists ff_u_mce_mdr_zero_columns_existszeroevaluationdeterminanthsf_negative ff_v_mce_mdr_zero_columns_existszeroevaluationdeterminanthsf_negative. ((((exists ff_h_mce_mdr_zero_columns_existszeroevaluationdeterminanthsf_negative_start. ff_h_mce_mdr_zero_columns_existszeroevaluationdeterminanthsf_negative_start + S (0) = S ((S (0)) * ff_v_mce_mdr_zero_columns_existszeroevaluationdeterminanthsf_negative)) /\ exists ff_q_mce_mdr_zero_columns_existszeroevaluationdeterminanthsf_negative_start. ff_u_mce_mdr_zero_columns_existszeroevaluationdeterminanthsf_negative = ff_q_mce_mdr_zero_columns_existszeroevaluationdeterminanthsf_negative_start * S ((S (0)) * ff_v_mce_mdr_zero_columns_existszeroevaluationdeterminanthsf_negative) + (0))) /\ ((((exists ff_h_mce_mdr_zero_columns_existszeroevaluationdeterminanthsf_negative_terminal. ff_h_mce_mdr_zero_columns_existszeroevaluationdeterminanthsf_negative_terminal + S (mdr_n_zero_columns_existszeroevaluationdeterminanth) = S ((S ((S (mdr_q_zero_columns_existszeroevaluationdeterminanths)))) * ff_v_mce_mdr_zero_columns_existszeroevaluationdeterminanthsf_negative)) /\ exists ff_q_mce_mdr_zero_columns_existszeroevaluationdeterminanthsf_negative_terminal. ff_u_mce_mdr_zero_columns_existszeroevaluationdeterminanthsf_negative = ff_q_mce_mdr_zero_columns_existszeroevaluationdeterminanthsf_negative_terminal * S ((S ((S (mdr_q_zero_columns_existszeroevaluationdeterminanths)))) * ff_v_mce_mdr_zero_columns_existszeroevaluationdeterminanthsf_negative) + (mdr_n_zero_columns_existszeroevaluationdeterminanth))) /\ forall ff_i_mce_mdr_zero_columns_existszeroevaluationdeterminanthsf_negative. (exists ff_lt_mce_mdr_zero_columns_existszeroevaluationdeterminanthsf_negative_bound. ff_lt_mce_mdr_zero_columns_existszeroevaluationdeterminanthsf_negative_bound + S ff_i_mce_mdr_zero_columns_existszeroevaluationdeterminanthsf_negative = (S (mdr_q_zero_columns_existszeroevaluationdeterminanths))) -> exists ff_a_mce_mdr_zero_columns_existszeroevaluationdeterminanthsf_negative ff_r_mce_mdr_zero_columns_existszeroevaluationdeterminanthsf_negative ff_s_mce_mdr_zero_columns_existszeroevaluationdeterminanthsf_negative. ((((exists ff_h_mce_mdr_zero_columns_existszeroevaluationdeterminanthsf_negative_summand. ff_h_mce_mdr_zero_columns_existszeroevaluationdeterminanthsf_negative_summand + S (ff_a_mce_mdr_zero_columns_existszeroevaluationdeterminanthsf_negative) = S ((S (ff_i_mce_mdr_zero_columns_existszeroevaluationdeterminanthsf_negative)) * ff_vc_mce_fold_mdr_zero_columns_existszeroevaluationdeterminanthsf)) /\ exists ff_q_mce_mdr_zero_columns_existszeroevaluationdeterminanthsf_negative_summand. ff_vb_mce_fold_mdr_zero_columns_existszeroevaluationdeterminanthsf = ff_q_mce_mdr_zero_columns_existszeroevaluationdeterminanthsf_negative_summand * S ((S (ff_i_mce_mdr_zero_columns_existszeroevaluationdeterminanthsf_negative)) * ff_vc_mce_fold_mdr_zero_columns_existszeroevaluationdeterminanthsf) + (ff_a_mce_mdr_zero_columns_existszeroevaluationdeterminanthsf_negative))) /\ ((((exists ff_h_mce_mdr_zero_columns_existszeroevaluationdeterminanthsf_negative_partial. ff_h_mce_mdr_zero_columns_existszeroevaluationdeterminanthsf_negative_partial + S (ff_r_mce_mdr_zero_columns_existszeroevaluationdeterminanthsf_negative) = S ((S (ff_i_mce_mdr_zero_columns_existszeroevaluationdeterminanthsf_negative)) * ff_v_mce_mdr_zero_columns_existszeroevaluationdeterminanthsf_negative)) /\ exists ff_q_mce_mdr_zero_columns_existszeroevaluationdeterminanthsf_negative_partial. ff_u_mce_mdr_zero_columns_existszeroevaluationdeterminanthsf_negative = ff_q_mce_mdr_zero_columns_existszeroevaluationdeterminanthsf_negative_partial * S ((S (ff_i_mce_mdr_zero_columns_existszeroevaluationdeterminanthsf_negative)) * ff_v_mce_mdr_zero_columns_existszeroevaluationdeterminanthsf_negative) + (ff_r_mce_mdr_zero_columns_existszeroevaluationdeterminanthsf_negative))) /\ ((((exists ff_h_mce_mdr_zero_columns_existszeroevaluationdeterminanthsf_negative_successor. ff_h_mce_mdr_zero_columns_existszeroevaluationdeterminanthsf_negative_successor + S (ff_s_mce_mdr_zero_columns_existszeroevaluationdeterminanthsf_negative) = S ((S (S ff_i_mce_mdr_zero_columns_existszeroevaluationdeterminanthsf_negative)) * ff_v_mce_mdr_zero_columns_existszeroevaluationdeterminanthsf_negative)) /\ exists ff_q_mce_mdr_zero_columns_existszeroevaluationdeterminanthsf_negative_successor. ff_u_mce_mdr_zero_columns_existszeroevaluationdeterminanthsf_negative = ff_q_mce_mdr_zero_columns_existszeroevaluationdeterminanthsf_negative_successor * S ((S (S ff_i_mce_mdr_zero_columns_existszeroevaluationdeterminanthsf_negative)) * ff_v_mce_mdr_zero_columns_existszeroevaluationdeterminanthsf_negative) + (ff_s_mce_mdr_zero_columns_existszeroevaluationdeterminanthsf_negative))) /\ ff_s_mce_mdr_zero_columns_existszeroevaluationdeterminanthsf_negative = ff_r_mce_mdr_zero_columns_existszeroevaluationdeterminanthsf_negative + ff_a_mce_mdr_zero_columns_existszeroevaluationdeterminanthsf_negative))))))))))))))) /\ ((exists mdr_gap_zero_columns_existszeroevaluationdeterminanti. mdr_gap_zero_columns_existszeroevaluationdeterminanti + S (mdr_i_zero_columns_existszeroevaluationdeterminant) = (mdr_l_zero_columns_existszeroevaluationdeterminant)) /\ (exists mdr_z_zero_columns_existszeroevaluationdeterminantr. ((exists mdr_a_zero_columns_existszeroevaluationdeterminantrc mdr_b_zero_columns_existszeroevaluationdeterminantrc mdr_c_zero_columns_existszeroevaluationdeterminantrc mdr_e_zero_columns_existszeroevaluationdeterminantrc mdr_f_zero_columns_existszeroevaluationdeterminantrc. ((mdr_a_zero_columns_existszeroevaluationdeterminantrc = ((mdr_q_zero_columns_exists) + (mdr_ub_zero_columns_existszeroevaluation)) * S ((mdr_q_zero_columns_exists) + (mdr_ub_zero_columns_existszeroevaluation)) + ((mdr_ub_zero_columns_existszeroevaluation) + (mdr_ub_zero_columns_existszeroevaluation))) /\ ((mdr_b_zero_columns_existszeroevaluationdeterminantrc = ((mdr_uc_zero_columns_existszeroevaluation) + (mdr_vb_zero_columns_existszeroevaluation)) * S ((mdr_uc_zero_columns_existszeroevaluation) + (mdr_vb_zero_columns_existszeroevaluation)) + ((mdr_vb_zero_columns_existszeroevaluation) + (mdr_vb_zero_columns_existszeroevaluation))) /\ ((mdr_c_zero_columns_existszeroevaluationdeterminantrc = ((mdr_a_zero_columns_existszeroevaluationdeterminantrc) + (mdr_b_zero_columns_existszeroevaluationdeterminantrc)) * S ((mdr_a_zero_columns_existszeroevaluationdeterminantrc) + (mdr_b_zero_columns_existszeroevaluationdeterminantrc)) + ((mdr_b_zero_columns_existszeroevaluationdeterminantrc) + (mdr_b_zero_columns_existszeroevaluationdeterminantrc))) /\ ((mdr_e_zero_columns_existszeroevaluationdeterminantrc = ((mdr_p_zero_columns_existszero) + (mdr_n_zero_columns_existszero)) * S ((mdr_p_zero_columns_existszero) + (mdr_n_zero_columns_existszero)) + ((mdr_n_zero_columns_existszero) + (mdr_n_zero_columns_existszero))) /\ ((mdr_f_zero_columns_existszeroevaluationdeterminantrc = ((mdr_vc_zero_columns_existszeroevaluation) + (mdr_e_zero_columns_existszeroevaluationdeterminantrc)) * S ((mdr_vc_zero_columns_existszeroevaluation) + (mdr_e_zero_columns_existszeroevaluationdeterminantrc)) + ((mdr_e_zero_columns_existszeroevaluationdeterminantrc) + (mdr_e_zero_columns_existszeroevaluationdeterminantrc))) /\ ((mdr_z_zero_columns_existszeroevaluationdeterminantr) = ((mdr_c_zero_columns_existszeroevaluationdeterminantrc) + (mdr_f_zero_columns_existszeroevaluationdeterminantrc)) * S ((mdr_c_zero_columns_existszeroevaluationdeterminantrc) + (mdr_f_zero_columns_existszeroevaluationdeterminantrc)) + ((mdr_f_zero_columns_existszeroevaluationdeterminantrc) + (mdr_f_zero_columns_existszeroevaluationdeterminantrc))))))))) /\ (((exists ff_h_mdr_zero_columns_existszeroevaluationdeterminantrb. ff_h_mdr_zero_columns_existszeroevaluationdeterminantrb + S (mdr_z_zero_columns_existszeroevaluationdeterminantr) = S ((S (mdr_i_zero_columns_existszeroevaluationdeterminant)) * mdr_c_zero_columns_existszeroevaluationdeterminant)) /\ exists ff_q_mdr_zero_columns_existszeroevaluationdeterminantrb. mdr_b_zero_columns_existszeroevaluationdeterminant = ff_q_mdr_zero_columns_existszeroevaluationdeterminantrb * S ((S (mdr_i_zero_columns_existszeroevaluationdeterminant)) * mdr_c_zero_columns_existszeroevaluationdeterminant) + (mdr_z_zero_columns_existszeroevaluationdeterminantr)))))))))) -> mdr_p_zero_columns_existszero = mdr_n_zero_columns_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 (r) - 0012
specialize rectangular_matrix_rank_certificate_exists (0) - 0013
apply rectangular_matrix_rank_certificate_exists - 0014
cases hrank - 0015
cases hrank_witness - 0016
cases hrank_witness_right - 0017
have hzero : x = 0 - 0018
specialize le_zero (x) - 0019
apply le_zero - 0020
exact hrank_witness_right_left - 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
rewrite hzero at hrank_witness - 0040
exact hrank_witness