Current library: Alpha v34, 4,223 checked-use theorems; Stable remains 432. Historical first admissions, original proof editions, and non-admitted aliases are preserved. Exact original first-admission records.
This branch proves the finite determinant/rank/span substrate. It does not claim Smith or Hermite normal form, lattice index equals determinant, determinant multiplicativity, lattice reduction, or geometry-of-numbers theorems.
Exact theorem in conservative defined notation
∀ ab. ∀ ac. ∀ bb. ∀ bc. ∀ eb. ∀ ec. ∀ fb. ∀ fc. ∀ r. ∀ w. ∀ rank. IntegerMatrixEntrywiseEqual(ab,ac,bb,bc,eb,ec,fb,fc,r,w) → RectangularMatrixRank(ab,ac,bb,bc,r,w,rank) → RectangularMatrixRank(eb,ec,fb,fc,r,w,rank)Every linked abbreviation expands hygienically to the identical original native formula.
Definition DAG
Actual proof prerequisites
Original expanded first-order statement
forall ab ac bb bc eb ec fb fc r w rank. (forall ics_index_rank_parent_equality ics_value0_rank_parent_equality ics_value1_rank_parent_equality ics_value2_rank_parent_equality ics_value3_rank_parent_equality. (exists ics_gap_rank_parent_equality_bound. ics_gap_rank_parent_equality_bound + S (ics_index_rank_parent_equality) = ((r) * (w))) -> (((exists fs_h_ics_rank_parent_equality_at0. fs_h_ics_rank_parent_equality_at0 + S (ics_value0_rank_parent_equality) = S ((S (ics_index_rank_parent_equality)) * ac)) /\ exists fs_q_ics_rank_parent_equality_at0. ab = fs_q_ics_rank_parent_equality_at0 * S ((S (ics_index_rank_parent_equality)) * ac) + (ics_value0_rank_parent_equality))) -> (((exists fs_h_ics_rank_parent_equality_at1. fs_h_ics_rank_parent_equality_at1 + S (ics_value1_rank_parent_equality) = S ((S (ics_index_rank_parent_equality)) * bc)) /\ exists fs_q_ics_rank_parent_equality_at1. bb = fs_q_ics_rank_parent_equality_at1 * S ((S (ics_index_rank_parent_equality)) * bc) + (ics_value1_rank_parent_equality))) -> (((exists fs_h_ics_rank_parent_equality_at2. fs_h_ics_rank_parent_equality_at2 + S (ics_value2_rank_parent_equality) = S ((S (ics_index_rank_parent_equality)) * ec)) /\ exists fs_q_ics_rank_parent_equality_at2. eb = fs_q_ics_rank_parent_equality_at2 * S ((S (ics_index_rank_parent_equality)) * ec) + (ics_value2_rank_parent_equality))) -> (((exists fs_h_ics_rank_parent_equality_at3. fs_h_ics_rank_parent_equality_at3 + S (ics_value3_rank_parent_equality) = S ((S (ics_index_rank_parent_equality)) * fc)) /\ exists fs_q_ics_rank_parent_equality_at3. fb = fs_q_ics_rank_parent_equality_at3 * S ((S (ics_index_rank_parent_equality)) * fc) + (ics_value3_rank_parent_equality))) -> ics_value0_rank_parent_equality + ics_value3_rank_parent_equality = ics_value2_rank_parent_equality + ics_value1_rank_parent_equality) -> (((exists mdr_gap_first_integer_rankrows_bound. mdr_gap_first_integer_rankrows_bound + (rank) = (r)) /\ ((exists mdr_gap_first_integer_rankcolumns_bound. mdr_gap_first_integer_rankcolumns_bound + (rank) = (w)) /\ ((exists mdr_rb_first_integer_rankwitness mdr_rc_first_integer_rankwitness mdr_cb_first_integer_rankwitness mdr_cc_first_integer_rankwitness. (((((forall fom_index_mrf_first_integer_rankwitnessminorrowsbound. (exists fom_gap_mrf_first_integer_rankwitnessminorrowsbound_index_bound. fom_gap_mrf_first_integer_rankwitnessminorrowsbound_index_bound + S (fom_index_mrf_first_integer_rankwitnessminorrowsbound) = rank) -> exists fom_value_mrf_first_integer_rankwitnessminorrowsbound. ((((exists fom_beta_height_mrf_first_integer_rankwitnessminorrowsbound_entry. fom_beta_height_mrf_first_integer_rankwitnessminorrowsbound_entry + S (fom_value_mrf_first_integer_rankwitnessminorrowsbound) = S ((S (fom_index_mrf_first_integer_rankwitnessminorrowsbound)) * mdr_rc_first_integer_rankwitness)) /\ exists fom_beta_quotient_mrf_first_integer_rankwitnessminorrowsbound_entry. mdr_rb_first_integer_rankwitness = fom_beta_quotient_mrf_first_integer_rankwitnessminorrowsbound_entry * S ((S (fom_index_mrf_first_integer_rankwitnessminorrowsbound)) * mdr_rc_first_integer_rankwitness) + (fom_value_mrf_first_integer_rankwitnessminorrowsbound))) /\ (exists fom_gap_mrf_first_integer_rankwitnessminorrowsbound_value_bound. fom_gap_mrf_first_integer_rankwitnessminorrowsbound_value_bound + S (fom_value_mrf_first_integer_rankwitnessminorrowsbound) = r))) /\ (forall mdr_i_first_integer_rankwitnessminorrowsdistinct mdr_j_first_integer_rankwitnessminorrowsdistinct mdr_a_first_integer_rankwitnessminorrowsdistinct. (exists mdr_gap_first_integer_rankwitnessminorrowsdistincti. mdr_gap_first_integer_rankwitnessminorrowsdistincti + S (mdr_i_first_integer_rankwitnessminorrowsdistinct) = (rank)) -> (exists mdr_gap_first_integer_rankwitnessminorrowsdistinctj. mdr_gap_first_integer_rankwitnessminorrowsdistinctj + S (mdr_j_first_integer_rankwitnessminorrowsdistinct) = (rank)) -> (((exists ff_h_mdr_first_integer_rankwitnessminorrowsdistinctfirst. ff_h_mdr_first_integer_rankwitnessminorrowsdistinctfirst + S (mdr_a_first_integer_rankwitnessminorrowsdistinct) = S ((S (mdr_i_first_integer_rankwitnessminorrowsdistinct)) * mdr_rc_first_integer_rankwitness)) /\ exists ff_q_mdr_first_integer_rankwitnessminorrowsdistinctfirst. mdr_rb_first_integer_rankwitness = ff_q_mdr_first_integer_rankwitnessminorrowsdistinctfirst * S ((S (mdr_i_first_integer_rankwitnessminorrowsdistinct)) * mdr_rc_first_integer_rankwitness) + (mdr_a_first_integer_rankwitnessminorrowsdistinct))) -> (((exists ff_h_mdr_first_integer_rankwitnessminorrowsdistinctsecond. ff_h_mdr_first_integer_rankwitnessminorrowsdistinctsecond + S (mdr_a_first_integer_rankwitnessminorrowsdistinct) = S ((S (mdr_j_first_integer_rankwitnessminorrowsdistinct)) * mdr_rc_first_integer_rankwitness)) /\ exists ff_q_mdr_first_integer_rankwitnessminorrowsdistinctsecond. mdr_rb_first_integer_rankwitness = ff_q_mdr_first_integer_rankwitnessminorrowsdistinctsecond * S ((S (mdr_j_first_integer_rankwitnessminorrowsdistinct)) * mdr_rc_first_integer_rankwitness) + (mdr_a_first_integer_rankwitnessminorrowsdistinct))) -> mdr_i_first_integer_rankwitnessminorrowsdistinct = mdr_j_first_integer_rankwitnessminorrowsdistinct))) /\ ((((forall fom_index_mrf_first_integer_rankwitnessminorcolumnsbound. (exists fom_gap_mrf_first_integer_rankwitnessminorcolumnsbound_index_bound. fom_gap_mrf_first_integer_rankwitnessminorcolumnsbound_index_bound + S (fom_index_mrf_first_integer_rankwitnessminorcolumnsbound) = rank) -> exists fom_value_mrf_first_integer_rankwitnessminorcolumnsbound. ((((exists fom_beta_height_mrf_first_integer_rankwitnessminorcolumnsbound_entry. fom_beta_height_mrf_first_integer_rankwitnessminorcolumnsbound_entry + S (fom_value_mrf_first_integer_rankwitnessminorcolumnsbound) = S ((S (fom_index_mrf_first_integer_rankwitnessminorcolumnsbound)) * mdr_cc_first_integer_rankwitness)) /\ exists fom_beta_quotient_mrf_first_integer_rankwitnessminorcolumnsbound_entry. mdr_cb_first_integer_rankwitness = fom_beta_quotient_mrf_first_integer_rankwitnessminorcolumnsbound_entry * S ((S (fom_index_mrf_first_integer_rankwitnessminorcolumnsbound)) * mdr_cc_first_integer_rankwitness) + (fom_value_mrf_first_integer_rankwitnessminorcolumnsbound))) /\ (exists fom_gap_mrf_first_integer_rankwitnessminorcolumnsbound_value_bound. fom_gap_mrf_first_integer_rankwitnessminorcolumnsbound_value_bound + S (fom_value_mrf_first_integer_rankwitnessminorcolumnsbound) = w))) /\ (forall mdr_i_first_integer_rankwitnessminorcolumnsdistinct mdr_j_first_integer_rankwitnessminorcolumnsdistinct mdr_a_first_integer_rankwitnessminorcolumnsdistinct. (exists mdr_gap_first_integer_rankwitnessminorcolumnsdistincti. mdr_gap_first_integer_rankwitnessminorcolumnsdistincti + S (mdr_i_first_integer_rankwitnessminorcolumnsdistinct) = (rank)) -> (exists mdr_gap_first_integer_rankwitnessminorcolumnsdistinctj. mdr_gap_first_integer_rankwitnessminorcolumnsdistinctj + S (mdr_j_first_integer_rankwitnessminorcolumnsdistinct) = (rank)) -> (((exists ff_h_mdr_first_integer_rankwitnessminorcolumnsdistinctfirst. ff_h_mdr_first_integer_rankwitnessminorcolumnsdistinctfirst + S (mdr_a_first_integer_rankwitnessminorcolumnsdistinct) = S ((S (mdr_i_first_integer_rankwitnessminorcolumnsdistinct)) * mdr_cc_first_integer_rankwitness)) /\ exists ff_q_mdr_first_integer_rankwitnessminorcolumnsdistinctfirst. mdr_cb_first_integer_rankwitness = ff_q_mdr_first_integer_rankwitnessminorcolumnsdistinctfirst * S ((S (mdr_i_first_integer_rankwitnessminorcolumnsdistinct)) * mdr_cc_first_integer_rankwitness) + (mdr_a_first_integer_rankwitnessminorcolumnsdistinct))) -> (((exists ff_h_mdr_first_integer_rankwitnessminorcolumnsdistinctsecond. ff_h_mdr_first_integer_rankwitnessminorcolumnsdistinctsecond + S (mdr_a_first_integer_rankwitnessminorcolumnsdistinct) = S ((S (mdr_j_first_integer_rankwitnessminorcolumnsdistinct)) * mdr_cc_first_integer_rankwitness)) /\ exists ff_q_mdr_first_integer_rankwitnessminorcolumnsdistinctsecond. mdr_cb_first_integer_rankwitness = ff_q_mdr_first_integer_rankwitnessminorcolumnsdistinctsecond * S ((S (mdr_j_first_integer_rankwitnessminorcolumnsdistinct)) * mdr_cc_first_integer_rankwitness) + (mdr_a_first_integer_rankwitnessminorcolumnsdistinct))) -> mdr_i_first_integer_rankwitnessminorcolumnsdistinct = mdr_j_first_integer_rankwitnessminorcolumnsdistinct))) /\ (exists mdr_p_first_integer_rankwitnessminornonzero mdr_n_first_integer_rankwitnessminornonzero. ((exists mdr_ub_first_integer_rankwitnessminornonzeroevaluation mdr_uc_first_integer_rankwitnessminornonzeroevaluation mdr_vb_first_integer_rankwitnessminornonzeroevaluation mdr_vc_first_integer_rankwitnessminornonzeroevaluation. ((((forall mdr_i_first_integer_rankwitnessminornonzeroevaluationmatrixpositive. (exists mdr_gap_first_integer_rankwitnessminornonzeroevaluationmatrixpositivebound. mdr_gap_first_integer_rankwitnessminornonzeroevaluationmatrixpositivebound + S (mdr_i_first_integer_rankwitnessminornonzeroevaluationmatrixpositive) = ((rank) * (rank))) -> exists mdr_a_first_integer_rankwitnessminornonzeroevaluationmatrixpositive. (((exists mdr_r_first_integer_rankwitnessminornonzeroevaluationmatrixpositivepoint mdr_s_first_integer_rankwitnessminornonzeroevaluationmatrixpositivepoint mdr_u_first_integer_rankwitnessminornonzeroevaluationmatrixpositivepoint mdr_v_first_integer_rankwitnessminornonzeroevaluationmatrixpositivepoint. ((mdr_i_first_integer_rankwitnessminornonzeroevaluationmatrixpositive = (rank) * mdr_r_first_integer_rankwitnessminornonzeroevaluationmatrixpositivepoint + mdr_s_first_integer_rankwitnessminornonzeroevaluationmatrixpositivepoint) /\ ((exists mdr_gap_first_integer_rankwitnessminornonzeroevaluationmatrixpositivepointcolumn. mdr_gap_first_integer_rankwitnessminornonzeroevaluationmatrixpositivepointcolumn + S (mdr_s_first_integer_rankwitnessminornonzeroevaluationmatrixpositivepoint) = (rank)) /\ ((((exists ff_h_mdr_first_integer_rankwitnessminornonzeroevaluationmatrixpositivepointrow_index. ff_h_mdr_first_integer_rankwitnessminornonzeroevaluationmatrixpositivepointrow_index + S (mdr_u_first_integer_rankwitnessminornonzeroevaluationmatrixpositivepoint) = S ((S (mdr_r_first_integer_rankwitnessminornonzeroevaluationmatrixpositivepoint)) * mdr_rc_first_integer_rankwitness)) /\ exists ff_q_mdr_first_integer_rankwitnessminornonzeroevaluationmatrixpositivepointrow_index. mdr_rb_first_integer_rankwitness = ff_q_mdr_first_integer_rankwitnessminornonzeroevaluationmatrixpositivepointrow_index * S ((S (mdr_r_first_integer_rankwitnessminornonzeroevaluationmatrixpositivepoint)) * mdr_rc_first_integer_rankwitness) + (mdr_u_first_integer_rankwitnessminornonzeroevaluationmatrixpositivepoint))) /\ ((((exists ff_h_mdr_first_integer_rankwitnessminornonzeroevaluationmatrixpositivepointcolumn_index. ff_h_mdr_first_integer_rankwitnessminornonzeroevaluationmatrixpositivepointcolumn_index + S (mdr_v_first_integer_rankwitnessminornonzeroevaluationmatrixpositivepoint) = S ((S (mdr_s_first_integer_rankwitnessminornonzeroevaluationmatrixpositivepoint)) * mdr_cc_first_integer_rankwitness)) /\ exists ff_q_mdr_first_integer_rankwitnessminornonzeroevaluationmatrixpositivepointcolumn_index. mdr_cb_first_integer_rankwitness = ff_q_mdr_first_integer_rankwitnessminornonzeroevaluationmatrixpositivepointcolumn_index * S ((S (mdr_s_first_integer_rankwitnessminornonzeroevaluationmatrixpositivepoint)) * mdr_cc_first_integer_rankwitness) + (mdr_v_first_integer_rankwitnessminornonzeroevaluationmatrixpositivepoint))) /\ (((exists ff_h_mdr_first_integer_rankwitnessminornonzeroevaluationmatrixpositivepointsource. ff_h_mdr_first_integer_rankwitnessminornonzeroevaluationmatrixpositivepointsource + S (mdr_a_first_integer_rankwitnessminornonzeroevaluationmatrixpositive) = S ((S ((mdr_u_first_integer_rankwitnessminornonzeroevaluationmatrixpositivepoint) * (w) + (mdr_v_first_integer_rankwitnessminornonzeroevaluationmatrixpositivepoint))) * ac)) /\ exists ff_q_mdr_first_integer_rankwitnessminornonzeroevaluationmatrixpositivepointsource. ab = ff_q_mdr_first_integer_rankwitnessminornonzeroevaluationmatrixpositivepointsource * S ((S ((mdr_u_first_integer_rankwitnessminornonzeroevaluationmatrixpositivepoint) * (w) + (mdr_v_first_integer_rankwitnessminornonzeroevaluationmatrixpositivepoint))) * ac) + (mdr_a_first_integer_rankwitnessminornonzeroevaluationmatrixpositive)))))))) /\ (((exists ff_h_mdr_first_integer_rankwitnessminornonzeroevaluationmatrixpositiveoutput. ff_h_mdr_first_integer_rankwitnessminornonzeroevaluationmatrixpositiveoutput + S (mdr_a_first_integer_rankwitnessminornonzeroevaluationmatrixpositive) = S ((S (mdr_i_first_integer_rankwitnessminornonzeroevaluationmatrixpositive)) * mdr_uc_first_integer_rankwitnessminornonzeroevaluation)) /\ exists ff_q_mdr_first_integer_rankwitnessminornonzeroevaluationmatrixpositiveoutput. mdr_ub_first_integer_rankwitnessminornonzeroevaluation = ff_q_mdr_first_integer_rankwitnessminornonzeroevaluationmatrixpositiveoutput * S ((S (mdr_i_first_integer_rankwitnessminornonzeroevaluationmatrixpositive)) * mdr_uc_first_integer_rankwitnessminornonzeroevaluation) + (mdr_a_first_integer_rankwitnessminornonzeroevaluationmatrixpositive)))))) /\ (forall mdr_i_first_integer_rankwitnessminornonzeroevaluationmatrixnegative. (exists mdr_gap_first_integer_rankwitnessminornonzeroevaluationmatrixnegativebound. mdr_gap_first_integer_rankwitnessminornonzeroevaluationmatrixnegativebound + S (mdr_i_first_integer_rankwitnessminornonzeroevaluationmatrixnegative) = ((rank) * (rank))) -> exists mdr_a_first_integer_rankwitnessminornonzeroevaluationmatrixnegative. (((exists mdr_r_first_integer_rankwitnessminornonzeroevaluationmatrixnegativepoint mdr_s_first_integer_rankwitnessminornonzeroevaluationmatrixnegativepoint mdr_u_first_integer_rankwitnessminornonzeroevaluationmatrixnegativepoint mdr_v_first_integer_rankwitnessminornonzeroevaluationmatrixnegativepoint. ((mdr_i_first_integer_rankwitnessminornonzeroevaluationmatrixnegative = (rank) * mdr_r_first_integer_rankwitnessminornonzeroevaluationmatrixnegativepoint + mdr_s_first_integer_rankwitnessminornonzeroevaluationmatrixnegativepoint) /\ ((exists mdr_gap_first_integer_rankwitnessminornonzeroevaluationmatrixnegativepointcolumn. mdr_gap_first_integer_rankwitnessminornonzeroevaluationmatrixnegativepointcolumn + S (mdr_s_first_integer_rankwitnessminornonzeroevaluationmatrixnegativepoint) = (rank)) /\ ((((exists ff_h_mdr_first_integer_rankwitnessminornonzeroevaluationmatrixnegativepointrow_index. ff_h_mdr_first_integer_rankwitnessminornonzeroevaluationmatrixnegativepointrow_index + S (mdr_u_first_integer_rankwitnessminornonzeroevaluationmatrixnegativepoint) = S ((S (mdr_r_first_integer_rankwitnessminornonzeroevaluationmatrixnegativepoint)) * mdr_rc_first_integer_rankwitness)) /\ exists ff_q_mdr_first_integer_rankwitnessminornonzeroevaluationmatrixnegativepointrow_index. mdr_rb_first_integer_rankwitness = ff_q_mdr_first_integer_rankwitnessminornonzeroevaluationmatrixnegativepointrow_index * S ((S (mdr_r_first_integer_rankwitnessminornonzeroevaluationmatrixnegativepoint)) * mdr_rc_first_integer_rankwitness) + (mdr_u_first_integer_rankwitnessminornonzeroevaluationmatrixnegativepoint))) /\ ((((exists ff_h_mdr_first_integer_rankwitnessminornonzeroevaluationmatrixnegativepointcolumn_index. ff_h_mdr_first_integer_rankwitnessminornonzeroevaluationmatrixnegativepointcolumn_index + S (mdr_v_first_integer_rankwitnessminornonzeroevaluationmatrixnegativepoint) = S ((S (mdr_s_first_integer_rankwitnessminornonzeroevaluationmatrixnegativepoint)) * mdr_cc_first_integer_rankwitness)) /\ exists ff_q_mdr_first_integer_rankwitnessminornonzeroevaluationmatrixnegativepointcolumn_index. mdr_cb_first_integer_rankwitness = ff_q_mdr_first_integer_rankwitnessminornonzeroevaluationmatrixnegativepointcolumn_index * S ((S (mdr_s_first_integer_rankwitnessminornonzeroevaluationmatrixnegativepoint)) * mdr_cc_first_integer_rankwitness) + (mdr_v_first_integer_rankwitnessminornonzeroevaluationmatrixnegativepoint))) /\ (((exists ff_h_mdr_first_integer_rankwitnessminornonzeroevaluationmatrixnegativepointsource. ff_h_mdr_first_integer_rankwitnessminornonzeroevaluationmatrixnegativepointsource + S (mdr_a_first_integer_rankwitnessminornonzeroevaluationmatrixnegative) = S ((S ((mdr_u_first_integer_rankwitnessminornonzeroevaluationmatrixnegativepoint) * (w) + (mdr_v_first_integer_rankwitnessminornonzeroevaluationmatrixnegativepoint))) * bc)) /\ exists ff_q_mdr_first_integer_rankwitnessminornonzeroevaluationmatrixnegativepointsource. bb = ff_q_mdr_first_integer_rankwitnessminornonzeroevaluationmatrixnegativepointsource * S ((S ((mdr_u_first_integer_rankwitnessminornonzeroevaluationmatrixnegativepoint) * (w) + (mdr_v_first_integer_rankwitnessminornonzeroevaluationmatrixnegativepoint))) * bc) + (mdr_a_first_integer_rankwitnessminornonzeroevaluationmatrixnegative)))))))) /\ (((exists ff_h_mdr_first_integer_rankwitnessminornonzeroevaluationmatrixnegativeoutput. ff_h_mdr_first_integer_rankwitnessminornonzeroevaluationmatrixnegativeoutput + S (mdr_a_first_integer_rankwitnessminornonzeroevaluationmatrixnegative) = S ((S (mdr_i_first_integer_rankwitnessminornonzeroevaluationmatrixnegative)) * mdr_vc_first_integer_rankwitnessminornonzeroevaluation)) /\ exists ff_q_mdr_first_integer_rankwitnessminornonzeroevaluationmatrixnegativeoutput. mdr_vb_first_integer_rankwitnessminornonzeroevaluation = ff_q_mdr_first_integer_rankwitnessminornonzeroevaluationmatrixnegativeoutput * S ((S (mdr_i_first_integer_rankwitnessminornonzeroevaluationmatrixnegative)) * mdr_vc_first_integer_rankwitnessminornonzeroevaluation) + (mdr_a_first_integer_rankwitnessminornonzeroevaluationmatrixnegative)))))))) /\ (exists mdr_b_first_integer_rankwitnessminornonzeroevaluationdeterminant mdr_c_first_integer_rankwitnessminornonzeroevaluationdeterminant mdr_l_first_integer_rankwitnessminornonzeroevaluationdeterminant mdr_i_first_integer_rankwitnessminornonzeroevaluationdeterminant. ((forall mdr_i_first_integer_rankwitnessminornonzeroevaluationdeterminanth. (exists mdr_gap_first_integer_rankwitnessminornonzeroevaluationdeterminanthi. mdr_gap_first_integer_rankwitnessminornonzeroevaluationdeterminanthi + S (mdr_i_first_integer_rankwitnessminornonzeroevaluationdeterminanth) = (mdr_l_first_integer_rankwitnessminornonzeroevaluationdeterminant)) -> exists mdr_d_first_integer_rankwitnessminornonzeroevaluationdeterminanth mdr_pb_first_integer_rankwitnessminornonzeroevaluationdeterminanth mdr_pc_first_integer_rankwitnessminornonzeroevaluationdeterminanth mdr_nb_first_integer_rankwitnessminornonzeroevaluationdeterminanth mdr_nc_first_integer_rankwitnessminornonzeroevaluationdeterminanth mdr_p_first_integer_rankwitnessminornonzeroevaluationdeterminanth mdr_n_first_integer_rankwitnessminornonzeroevaluationdeterminanth. ((exists mdr_z_first_integer_rankwitnessminornonzeroevaluationdeterminanthr. ((exists mdr_a_first_integer_rankwitnessminornonzeroevaluationdeterminanthrc mdr_b_first_integer_rankwitnessminornonzeroevaluationdeterminanthrc mdr_c_first_integer_rankwitnessminornonzeroevaluationdeterminanthrc mdr_e_first_integer_rankwitnessminornonzeroevaluationdeterminanthrc mdr_f_first_integer_rankwitnessminornonzeroevaluationdeterminanthrc. ((mdr_a_first_integer_rankwitnessminornonzeroevaluationdeterminanthrc = ((mdr_d_first_integer_rankwitnessminornonzeroevaluationdeterminanth) + (mdr_pb_first_integer_rankwitnessminornonzeroevaluationdeterminanth)) * S ((mdr_d_first_integer_rankwitnessminornonzeroevaluationdeterminanth) + (mdr_pb_first_integer_rankwitnessminornonzeroevaluationdeterminanth)) + ((mdr_pb_first_integer_rankwitnessminornonzeroevaluationdeterminanth) + (mdr_pb_first_integer_rankwitnessminornonzeroevaluationdeterminanth))) /\ ((mdr_b_first_integer_rankwitnessminornonzeroevaluationdeterminanthrc = ((mdr_pc_first_integer_rankwitnessminornonzeroevaluationdeterminanth) + (mdr_nb_first_integer_rankwitnessminornonzeroevaluationdeterminanth)) * S ((mdr_pc_first_integer_rankwitnessminornonzeroevaluationdeterminanth) + (mdr_nb_first_integer_rankwitnessminornonzeroevaluationdeterminanth)) + ((mdr_nb_first_integer_rankwitnessminornonzeroevaluationdeterminanth) + (mdr_nb_first_integer_rankwitnessminornonzeroevaluationdeterminanth))) /\ ((mdr_c_first_integer_rankwitnessminornonzeroevaluationdeterminanthrc = ((mdr_a_first_integer_rankwitnessminornonzeroevaluationdeterminanthrc) + (mdr_b_first_integer_rankwitnessminornonzeroevaluationdeterminanthrc)) * S ((mdr_a_first_integer_rankwitnessminornonzeroevaluationdeterminanthrc) + (mdr_b_first_integer_rankwitnessminornonzeroevaluationdeterminanthrc)) + ((mdr_b_first_integer_rankwitnessminornonzeroevaluationdeterminanthrc) + (mdr_b_first_integer_rankwitnessminornonzeroevaluationdeterminanthrc))) /\ ((mdr_e_first_integer_rankwitnessminornonzeroevaluationdeterminanthrc = ((mdr_p_first_integer_rankwitnessminornonzeroevaluationdeterminanth) + (mdr_n_first_integer_rankwitnessminornonzeroevaluationdeterminanth)) * S ((mdr_p_first_integer_rankwitnessminornonzeroevaluationdeterminanth) + (mdr_n_first_integer_rankwitnessminornonzeroevaluationdeterminanth)) + ((mdr_n_first_integer_rankwitnessminornonzeroevaluationdeterminanth) + (mdr_n_first_integer_rankwitnessminornonzeroevaluationdeterminanth))) /\ ((mdr_f_first_integer_rankwitnessminornonzeroevaluationdeterminanthrc = ((mdr_nc_first_integer_rankwitnessminornonzeroevaluationdeterminanth) + (mdr_e_first_integer_rankwitnessminornonzeroevaluationdeterminanthrc)) * S ((mdr_nc_first_integer_rankwitnessminornonzeroevaluationdeterminanth) + (mdr_e_first_integer_rankwitnessminornonzeroevaluationdeterminanthrc)) + ((mdr_e_first_integer_rankwitnessminornonzeroevaluationdeterminanthrc) + (mdr_e_first_integer_rankwitnessminornonzeroevaluationdeterminanthrc))) /\ ((mdr_z_first_integer_rankwitnessminornonzeroevaluationdeterminanthr) = ((mdr_c_first_integer_rankwitnessminornonzeroevaluationdeterminanthrc) + (mdr_f_first_integer_rankwitnessminornonzeroevaluationdeterminanthrc)) * S ((mdr_c_first_integer_rankwitnessminornonzeroevaluationdeterminanthrc) + (mdr_f_first_integer_rankwitnessminornonzeroevaluationdeterminanthrc)) + ((mdr_f_first_integer_rankwitnessminornonzeroevaluationdeterminanthrc) + (mdr_f_first_integer_rankwitnessminornonzeroevaluationdeterminanthrc))))))))) /\ (((exists ff_h_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthrb. ff_h_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthrb + S (mdr_z_first_integer_rankwitnessminornonzeroevaluationdeterminanthr) = S ((S (mdr_i_first_integer_rankwitnessminornonzeroevaluationdeterminanth)) * mdr_c_first_integer_rankwitnessminornonzeroevaluationdeterminant)) /\ exists ff_q_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthrb. mdr_b_first_integer_rankwitnessminornonzeroevaluationdeterminant = ff_q_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthrb * S ((S (mdr_i_first_integer_rankwitnessminornonzeroevaluationdeterminanth)) * mdr_c_first_integer_rankwitnessminornonzeroevaluationdeterminant) + (mdr_z_first_integer_rankwitnessminornonzeroevaluationdeterminanthr))))) /\ (((((mdr_d_first_integer_rankwitnessminornonzeroevaluationdeterminanth) = 0) /\ (((mdr_p_first_integer_rankwitnessminornonzeroevaluationdeterminanth) = 1) /\ ((mdr_n_first_integer_rankwitnessminornonzeroevaluationdeterminanth) = 0))) \/ exists mdr_q_first_integer_rankwitnessminornonzeroevaluationdeterminanths mdr_eb_first_integer_rankwitnessminornonzeroevaluationdeterminanths mdr_ec_first_integer_rankwitnessminornonzeroevaluationdeterminanths mdr_fb_first_integer_rankwitnessminornonzeroevaluationdeterminanths mdr_fc_first_integer_rankwitnessminornonzeroevaluationdeterminanths. (((mdr_d_first_integer_rankwitnessminornonzeroevaluationdeterminanth) = S (mdr_q_first_integer_rankwitnessminornonzeroevaluationdeterminanths)) /\ ((forall mdr_j_first_integer_rankwitnessminornonzeroevaluationdeterminanthsc. (exists mdr_gap_first_integer_rankwitnessminornonzeroevaluationdeterminanthscj. mdr_gap_first_integer_rankwitnessminornonzeroevaluationdeterminanthscj + S (mdr_j_first_integer_rankwitnessminornonzeroevaluationdeterminanthsc) = (S (mdr_q_first_integer_rankwitnessminornonzeroevaluationdeterminanths))) -> exists mdr_i_first_integer_rankwitnessminornonzeroevaluationdeterminanthsc mdr_up_first_integer_rankwitnessminornonzeroevaluationdeterminanthsc mdr_us_first_integer_rankwitnessminornonzeroevaluationdeterminanthsc mdr_un_first_integer_rankwitnessminornonzeroevaluationdeterminanthsc mdr_ut_first_integer_rankwitnessminornonzeroevaluationdeterminanthsc mdr_p_first_integer_rankwitnessminornonzeroevaluationdeterminanthsc mdr_n_first_integer_rankwitnessminornonzeroevaluationdeterminanthsc. ((exists mdr_gap_first_integer_rankwitnessminornonzeroevaluationdeterminanthsci. mdr_gap_first_integer_rankwitnessminornonzeroevaluationdeterminanthsci + S (mdr_i_first_integer_rankwitnessminornonzeroevaluationdeterminanthsc) = (mdr_i_first_integer_rankwitnessminornonzeroevaluationdeterminanth)) /\ ((exists mdr_z_first_integer_rankwitnessminornonzeroevaluationdeterminanthscr. ((exists mdr_a_first_integer_rankwitnessminornonzeroevaluationdeterminanthscrc mdr_b_first_integer_rankwitnessminornonzeroevaluationdeterminanthscrc mdr_c_first_integer_rankwitnessminornonzeroevaluationdeterminanthscrc mdr_e_first_integer_rankwitnessminornonzeroevaluationdeterminanthscrc mdr_f_first_integer_rankwitnessminornonzeroevaluationdeterminanthscrc. ((mdr_a_first_integer_rankwitnessminornonzeroevaluationdeterminanthscrc = ((mdr_q_first_integer_rankwitnessminornonzeroevaluationdeterminanths) + (mdr_up_first_integer_rankwitnessminornonzeroevaluationdeterminanthsc)) * S ((mdr_q_first_integer_rankwitnessminornonzeroevaluationdeterminanths) + (mdr_up_first_integer_rankwitnessminornonzeroevaluationdeterminanthsc)) + ((mdr_up_first_integer_rankwitnessminornonzeroevaluationdeterminanthsc) + (mdr_up_first_integer_rankwitnessminornonzeroevaluationdeterminanthsc))) /\ ((mdr_b_first_integer_rankwitnessminornonzeroevaluationdeterminanthscrc = ((mdr_us_first_integer_rankwitnessminornonzeroevaluationdeterminanthsc) + (mdr_un_first_integer_rankwitnessminornonzeroevaluationdeterminanthsc)) * S ((mdr_us_first_integer_rankwitnessminornonzeroevaluationdeterminanthsc) + (mdr_un_first_integer_rankwitnessminornonzeroevaluationdeterminanthsc)) + ((mdr_un_first_integer_rankwitnessminornonzeroevaluationdeterminanthsc) + (mdr_un_first_integer_rankwitnessminornonzeroevaluationdeterminanthsc))) /\ ((mdr_c_first_integer_rankwitnessminornonzeroevaluationdeterminanthscrc = ((mdr_a_first_integer_rankwitnessminornonzeroevaluationdeterminanthscrc) + (mdr_b_first_integer_rankwitnessminornonzeroevaluationdeterminanthscrc)) * S ((mdr_a_first_integer_rankwitnessminornonzeroevaluationdeterminanthscrc) + (mdr_b_first_integer_rankwitnessminornonzeroevaluationdeterminanthscrc)) + ((mdr_b_first_integer_rankwitnessminornonzeroevaluationdeterminanthscrc) + (mdr_b_first_integer_rankwitnessminornonzeroevaluationdeterminanthscrc))) /\ ((mdr_e_first_integer_rankwitnessminornonzeroevaluationdeterminanthscrc = ((mdr_p_first_integer_rankwitnessminornonzeroevaluationdeterminanthsc) + (mdr_n_first_integer_rankwitnessminornonzeroevaluationdeterminanthsc)) * S ((mdr_p_first_integer_rankwitnessminornonzeroevaluationdeterminanthsc) + (mdr_n_first_integer_rankwitnessminornonzeroevaluationdeterminanthsc)) + ((mdr_n_first_integer_rankwitnessminornonzeroevaluationdeterminanthsc) + (mdr_n_first_integer_rankwitnessminornonzeroevaluationdeterminanthsc))) /\ ((mdr_f_first_integer_rankwitnessminornonzeroevaluationdeterminanthscrc = ((mdr_ut_first_integer_rankwitnessminornonzeroevaluationdeterminanthsc) + (mdr_e_first_integer_rankwitnessminornonzeroevaluationdeterminanthscrc)) * S ((mdr_ut_first_integer_rankwitnessminornonzeroevaluationdeterminanthsc) + (mdr_e_first_integer_rankwitnessminornonzeroevaluationdeterminanthscrc)) + ((mdr_e_first_integer_rankwitnessminornonzeroevaluationdeterminanthscrc) + (mdr_e_first_integer_rankwitnessminornonzeroevaluationdeterminanthscrc))) /\ ((mdr_z_first_integer_rankwitnessminornonzeroevaluationdeterminanthscr) = ((mdr_c_first_integer_rankwitnessminornonzeroevaluationdeterminanthscrc) + (mdr_f_first_integer_rankwitnessminornonzeroevaluationdeterminanthscrc)) * S ((mdr_c_first_integer_rankwitnessminornonzeroevaluationdeterminanthscrc) + (mdr_f_first_integer_rankwitnessminornonzeroevaluationdeterminanthscrc)) + ((mdr_f_first_integer_rankwitnessminornonzeroevaluationdeterminanthscrc) + (mdr_f_first_integer_rankwitnessminornonzeroevaluationdeterminanthscrc))))))))) /\ (((exists ff_h_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthscrb. ff_h_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthscrb + S (mdr_z_first_integer_rankwitnessminornonzeroevaluationdeterminanthscr) = S ((S (mdr_i_first_integer_rankwitnessminornonzeroevaluationdeterminanthsc)) * mdr_c_first_integer_rankwitnessminornonzeroevaluationdeterminant)) /\ exists ff_q_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthscrb. mdr_b_first_integer_rankwitnessminornonzeroevaluationdeterminant = ff_q_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthscrb * S ((S (mdr_i_first_integer_rankwitnessminornonzeroevaluationdeterminanthsc)) * mdr_c_first_integer_rankwitnessminornonzeroevaluationdeterminant) + (mdr_z_first_integer_rankwitnessminornonzeroevaluationdeterminanthscr))))) /\ ((((forall ff_index_mdm_prefix_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthscm_positive. (exists ff_gap_mdm_lt_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthscm_positive_index_bound. ff_gap_mdm_lt_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthscm_positive_index_bound + S (ff_index_mdm_prefix_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthscm_positive) = ((mdr_q_first_integer_rankwitnessminornonzeroevaluationdeterminanths) * (mdr_q_first_integer_rankwitnessminornonzeroevaluationdeterminanths))) -> exists ff_row_mdm_prefix_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthscm_positive ff_column_mdm_prefix_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthscm_positive ff_value_mdm_prefix_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthscm_positive. (ff_index_mdm_prefix_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthscm_positive = (mdr_q_first_integer_rankwitnessminornonzeroevaluationdeterminanths) * ff_row_mdm_prefix_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthscm_positive + ff_column_mdm_prefix_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthscm_positive /\ ((exists ff_gap_mdm_lt_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthscm_positive_column_bound. ff_gap_mdm_lt_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthscm_positive_column_bound + S (ff_column_mdm_prefix_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthscm_positive) = (mdr_q_first_integer_rankwitnessminornonzeroevaluationdeterminanths)) /\ ((exists ff_row_mdm_cell_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthscm_positive_cell ff_column_mdm_cell_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthscm_positive_cell. (((((exists ff_gap_mdm_lt_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthscm_positive_cell_row_before. ff_gap_mdm_lt_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthscm_positive_cell_row_before + S (ff_row_mdm_prefix_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthscm_positive) = (0)) /\ ff_row_mdm_cell_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthscm_positive_cell = ff_row_mdm_prefix_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthscm_positive) \/ ((exists ff_gap_mdm_le_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthscm_positive_cell_row_after. ff_gap_mdm_le_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthscm_positive_cell_row_after + (0) = (ff_row_mdm_prefix_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthscm_positive)) /\ ff_row_mdm_cell_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthscm_positive_cell = S ff_row_mdm_prefix_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthscm_positive))) /\ (((((exists ff_gap_mdm_lt_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthscm_positive_cell_column_before. ff_gap_mdm_lt_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthscm_positive_cell_column_before + S (ff_column_mdm_prefix_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthscm_positive) = (mdr_j_first_integer_rankwitnessminornonzeroevaluationdeterminanthsc)) /\ ff_column_mdm_cell_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthscm_positive_cell = ff_column_mdm_prefix_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthscm_positive) \/ ((exists ff_gap_mdm_le_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthscm_positive_cell_column_after. ff_gap_mdm_le_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthscm_positive_cell_column_after + (mdr_j_first_integer_rankwitnessminornonzeroevaluationdeterminanthsc) = (ff_column_mdm_prefix_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthscm_positive)) /\ ff_column_mdm_cell_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthscm_positive_cell = S ff_column_mdm_prefix_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthscm_positive))) /\ (((exists ff_h_mdm_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthscm_positive_cell_source. ff_h_mdm_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthscm_positive_cell_source + S (ff_value_mdm_prefix_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthscm_positive) = S ((S ((ff_row_mdm_cell_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthscm_positive_cell) * (S (mdr_q_first_integer_rankwitnessminornonzeroevaluationdeterminanths)) + (ff_column_mdm_cell_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthscm_positive_cell))) * mdr_pc_first_integer_rankwitnessminornonzeroevaluationdeterminanth)) /\ exists ff_q_mdm_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthscm_positive_cell_source. mdr_pb_first_integer_rankwitnessminornonzeroevaluationdeterminanth = ff_q_mdm_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthscm_positive_cell_source * S ((S ((ff_row_mdm_cell_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthscm_positive_cell) * (S (mdr_q_first_integer_rankwitnessminornonzeroevaluationdeterminanths)) + (ff_column_mdm_cell_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthscm_positive_cell))) * mdr_pc_first_integer_rankwitnessminornonzeroevaluationdeterminanth) + (ff_value_mdm_prefix_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthscm_positive)))))) /\ (((exists ff_h_mdm_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthscm_positive_target. ff_h_mdm_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthscm_positive_target + S (ff_value_mdm_prefix_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthscm_positive) = S ((S (ff_index_mdm_prefix_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthscm_positive)) * mdr_us_first_integer_rankwitnessminornonzeroevaluationdeterminanthsc)) /\ exists ff_q_mdm_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthscm_positive_target. mdr_up_first_integer_rankwitnessminornonzeroevaluationdeterminanthsc = ff_q_mdm_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthscm_positive_target * S ((S (ff_index_mdm_prefix_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthscm_positive)) * mdr_us_first_integer_rankwitnessminornonzeroevaluationdeterminanthsc) + (ff_value_mdm_prefix_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthscm_positive))))))) /\ (forall ff_index_mdm_prefix_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthscm_negative. (exists ff_gap_mdm_lt_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthscm_negative_index_bound. ff_gap_mdm_lt_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthscm_negative_index_bound + S (ff_index_mdm_prefix_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthscm_negative) = ((mdr_q_first_integer_rankwitnessminornonzeroevaluationdeterminanths) * (mdr_q_first_integer_rankwitnessminornonzeroevaluationdeterminanths))) -> exists ff_row_mdm_prefix_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthscm_negative ff_column_mdm_prefix_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthscm_negative ff_value_mdm_prefix_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthscm_negative. (ff_index_mdm_prefix_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthscm_negative = (mdr_q_first_integer_rankwitnessminornonzeroevaluationdeterminanths) * ff_row_mdm_prefix_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthscm_negative + ff_column_mdm_prefix_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthscm_negative /\ ((exists ff_gap_mdm_lt_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthscm_negative_column_bound. ff_gap_mdm_lt_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthscm_negative_column_bound + S (ff_column_mdm_prefix_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthscm_negative) = (mdr_q_first_integer_rankwitnessminornonzeroevaluationdeterminanths)) /\ ((exists ff_row_mdm_cell_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthscm_negative_cell ff_column_mdm_cell_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthscm_negative_cell. (((((exists ff_gap_mdm_lt_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthscm_negative_cell_row_before. ff_gap_mdm_lt_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthscm_negative_cell_row_before + S (ff_row_mdm_prefix_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthscm_negative) = (0)) /\ ff_row_mdm_cell_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthscm_negative_cell = ff_row_mdm_prefix_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthscm_negative) \/ ((exists ff_gap_mdm_le_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthscm_negative_cell_row_after. ff_gap_mdm_le_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthscm_negative_cell_row_after + (0) = (ff_row_mdm_prefix_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthscm_negative)) /\ ff_row_mdm_cell_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthscm_negative_cell = S ff_row_mdm_prefix_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthscm_negative))) /\ (((((exists ff_gap_mdm_lt_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthscm_negative_cell_column_before. ff_gap_mdm_lt_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthscm_negative_cell_column_before + S (ff_column_mdm_prefix_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthscm_negative) = (mdr_j_first_integer_rankwitnessminornonzeroevaluationdeterminanthsc)) /\ ff_column_mdm_cell_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthscm_negative_cell = ff_column_mdm_prefix_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthscm_negative) \/ ((exists ff_gap_mdm_le_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthscm_negative_cell_column_after. ff_gap_mdm_le_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthscm_negative_cell_column_after + (mdr_j_first_integer_rankwitnessminornonzeroevaluationdeterminanthsc) = (ff_column_mdm_prefix_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthscm_negative)) /\ ff_column_mdm_cell_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthscm_negative_cell = S ff_column_mdm_prefix_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthscm_negative))) /\ (((exists ff_h_mdm_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthscm_negative_cell_source. ff_h_mdm_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthscm_negative_cell_source + S (ff_value_mdm_prefix_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthscm_negative) = S ((S ((ff_row_mdm_cell_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthscm_negative_cell) * (S (mdr_q_first_integer_rankwitnessminornonzeroevaluationdeterminanths)) + (ff_column_mdm_cell_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthscm_negative_cell))) * mdr_nc_first_integer_rankwitnessminornonzeroevaluationdeterminanth)) /\ exists ff_q_mdm_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthscm_negative_cell_source. mdr_nb_first_integer_rankwitnessminornonzeroevaluationdeterminanth = ff_q_mdm_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthscm_negative_cell_source * S ((S ((ff_row_mdm_cell_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthscm_negative_cell) * (S (mdr_q_first_integer_rankwitnessminornonzeroevaluationdeterminanths)) + (ff_column_mdm_cell_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthscm_negative_cell))) * mdr_nc_first_integer_rankwitnessminornonzeroevaluationdeterminanth) + (ff_value_mdm_prefix_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthscm_negative)))))) /\ (((exists ff_h_mdm_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthscm_negative_target. ff_h_mdm_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthscm_negative_target + S (ff_value_mdm_prefix_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthscm_negative) = S ((S (ff_index_mdm_prefix_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthscm_negative)) * mdr_ut_first_integer_rankwitnessminornonzeroevaluationdeterminanthsc)) /\ exists ff_q_mdm_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthscm_negative_target. mdr_un_first_integer_rankwitnessminornonzeroevaluationdeterminanthsc = ff_q_mdm_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthscm_negative_target * S ((S (ff_index_mdm_prefix_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthscm_negative)) * mdr_ut_first_integer_rankwitnessminornonzeroevaluationdeterminanthsc) + (ff_value_mdm_prefix_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthscm_negative))))))))) /\ ((((exists ff_h_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthscp. ff_h_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthscp + S (mdr_p_first_integer_rankwitnessminornonzeroevaluationdeterminanthsc) = S ((S (mdr_j_first_integer_rankwitnessminornonzeroevaluationdeterminanthsc)) * mdr_ec_first_integer_rankwitnessminornonzeroevaluationdeterminanths)) /\ exists ff_q_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthscp. mdr_eb_first_integer_rankwitnessminornonzeroevaluationdeterminanths = ff_q_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthscp * S ((S (mdr_j_first_integer_rankwitnessminornonzeroevaluationdeterminanthsc)) * mdr_ec_first_integer_rankwitnessminornonzeroevaluationdeterminanths) + (mdr_p_first_integer_rankwitnessminornonzeroevaluationdeterminanthsc))) /\ (((exists ff_h_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthscn. ff_h_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthscn + S (mdr_n_first_integer_rankwitnessminornonzeroevaluationdeterminanthsc) = S ((S (mdr_j_first_integer_rankwitnessminornonzeroevaluationdeterminanthsc)) * mdr_fc_first_integer_rankwitnessminornonzeroevaluationdeterminanths)) /\ exists ff_q_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthscn. mdr_fb_first_integer_rankwitnessminornonzeroevaluationdeterminanths = ff_q_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthscn * S ((S (mdr_j_first_integer_rankwitnessminornonzeroevaluationdeterminanthsc)) * mdr_fc_first_integer_rankwitnessminornonzeroevaluationdeterminanths) + (mdr_n_first_integer_rankwitnessminornonzeroevaluationdeterminanthsc)))))))) /\ (exists ff_ub_mce_fold_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf ff_uc_mce_fold_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf ff_vb_mce_fold_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf ff_vc_mce_fold_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf. ((forall ff_index_mce_alternating_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix. (exists ff_gap_mce_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix_index. ff_gap_mce_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix_index + S (ff_index_mce_alternating_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix) = (S (mdr_q_first_integer_rankwitnessminornonzeroevaluationdeterminanths))) -> exists ff_ap_mce_alternating_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix ff_an_mce_alternating_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix ff_bp_mce_alternating_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix ff_bn_mce_alternating_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix ff_p_mce_alternating_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix ff_n_mce_alternating_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix. ((((exists ff_h_mce_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix_ap. ff_h_mce_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix_ap + S (ff_ap_mce_alternating_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix) = S ((S (ff_index_mce_alternating_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix)) * mdr_pc_first_integer_rankwitnessminornonzeroevaluationdeterminanth)) /\ exists ff_q_mce_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix_ap. mdr_pb_first_integer_rankwitnessminornonzeroevaluationdeterminanth = ff_q_mce_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix_ap * S ((S (ff_index_mce_alternating_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix)) * mdr_pc_first_integer_rankwitnessminornonzeroevaluationdeterminanth) + (ff_ap_mce_alternating_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix))) /\ ((((exists ff_h_mce_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix_an. ff_h_mce_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix_an + S (ff_an_mce_alternating_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix) = S ((S (ff_index_mce_alternating_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix)) * mdr_nc_first_integer_rankwitnessminornonzeroevaluationdeterminanth)) /\ exists ff_q_mce_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix_an. mdr_nb_first_integer_rankwitnessminornonzeroevaluationdeterminanth = ff_q_mce_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix_an * S ((S (ff_index_mce_alternating_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix)) * mdr_nc_first_integer_rankwitnessminornonzeroevaluationdeterminanth) + (ff_an_mce_alternating_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix))) /\ ((((exists ff_h_mce_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix_bp. ff_h_mce_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix_bp + S (ff_bp_mce_alternating_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix) = S ((S (ff_index_mce_alternating_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix)) * mdr_ec_first_integer_rankwitnessminornonzeroevaluationdeterminanths)) /\ exists ff_q_mce_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix_bp. mdr_eb_first_integer_rankwitnessminornonzeroevaluationdeterminanths = ff_q_mce_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix_bp * S ((S (ff_index_mce_alternating_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix)) * mdr_ec_first_integer_rankwitnessminornonzeroevaluationdeterminanths) + (ff_bp_mce_alternating_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix))) /\ ((((exists ff_h_mce_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix_bn. ff_h_mce_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix_bn + S (ff_bn_mce_alternating_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix) = S ((S (ff_index_mce_alternating_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix)) * mdr_fc_first_integer_rankwitnessminornonzeroevaluationdeterminanths)) /\ exists ff_q_mce_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix_bn. mdr_fb_first_integer_rankwitnessminornonzeroevaluationdeterminanths = ff_q_mce_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix_bn * S ((S (ff_index_mce_alternating_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix)) * mdr_fc_first_integer_rankwitnessminornonzeroevaluationdeterminanths) + (ff_bn_mce_alternating_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix))) /\ ((((exists ff_h_mce_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix_positive. ff_h_mce_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix_positive + S (ff_p_mce_alternating_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix) = S ((S (ff_index_mce_alternating_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix)) * ff_uc_mce_fold_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf)) /\ exists ff_q_mce_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix_positive. ff_ub_mce_fold_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf = ff_q_mce_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix_positive * S ((S (ff_index_mce_alternating_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix)) * ff_uc_mce_fold_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf) + (ff_p_mce_alternating_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix))) /\ ((((exists ff_h_mce_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix_negative. ff_h_mce_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix_negative + S (ff_n_mce_alternating_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix) = S ((S (ff_index_mce_alternating_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix)) * ff_vc_mce_fold_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf)) /\ exists ff_q_mce_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix_negative. ff_vb_mce_fold_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf = ff_q_mce_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix_negative * S ((S (ff_index_mce_alternating_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix)) * ff_vc_mce_fold_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf) + (ff_n_mce_alternating_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix))) /\ (((exists ff_even_mce_term_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix_term. ff_index_mce_alternating_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix = 2 * ff_even_mce_term_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix_term) /\ (ff_p_mce_alternating_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix = (ff_ap_mce_alternating_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix) * (ff_bp_mce_alternating_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix) + (ff_an_mce_alternating_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix) * (ff_bn_mce_alternating_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix) /\ ff_n_mce_alternating_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix = (ff_ap_mce_alternating_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix) * (ff_bn_mce_alternating_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix) + (ff_an_mce_alternating_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix) * (ff_bp_mce_alternating_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix))) \/ ((exists ff_odd_mce_term_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix_term. ff_index_mce_alternating_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix = 2 * ff_odd_mce_term_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix_term + 1) /\ (ff_p_mce_alternating_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix = (ff_ap_mce_alternating_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix) * (ff_bn_mce_alternating_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix) + (ff_an_mce_alternating_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix) * (ff_bp_mce_alternating_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix) /\ ff_n_mce_alternating_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix = (ff_ap_mce_alternating_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix) * (ff_bp_mce_alternating_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix) + (ff_an_mce_alternating_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix) * (ff_bn_mce_alternating_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix))))))))))) /\ ((exists ff_u_mce_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_positive ff_v_mce_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_positive. ((((exists ff_h_mce_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_positive_start. ff_h_mce_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_positive_start + S (0) = S ((S (0)) * ff_v_mce_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_positive)) /\ exists ff_q_mce_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_positive_start. ff_u_mce_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_positive = ff_q_mce_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_positive_start * S ((S (0)) * ff_v_mce_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_positive) + (0))) /\ ((((exists ff_h_mce_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_positive_terminal. ff_h_mce_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_positive_terminal + S (mdr_p_first_integer_rankwitnessminornonzeroevaluationdeterminanth) = S ((S ((S (mdr_q_first_integer_rankwitnessminornonzeroevaluationdeterminanths)))) * ff_v_mce_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_positive)) /\ exists ff_q_mce_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_positive_terminal. ff_u_mce_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_positive = ff_q_mce_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_positive_terminal * S ((S ((S (mdr_q_first_integer_rankwitnessminornonzeroevaluationdeterminanths)))) * ff_v_mce_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_positive) + (mdr_p_first_integer_rankwitnessminornonzeroevaluationdeterminanth))) /\ forall ff_i_mce_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_positive. (exists ff_lt_mce_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_positive_bound. ff_lt_mce_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_positive_bound + S ff_i_mce_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_positive = (S (mdr_q_first_integer_rankwitnessminornonzeroevaluationdeterminanths))) -> exists ff_a_mce_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_positive ff_r_mce_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_positive ff_s_mce_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_positive. ((((exists ff_h_mce_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_positive_summand. ff_h_mce_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_positive_summand + S (ff_a_mce_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_positive) = S ((S (ff_i_mce_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_positive)) * ff_uc_mce_fold_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf)) /\ exists ff_q_mce_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_positive_summand. ff_ub_mce_fold_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf = ff_q_mce_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_positive_summand * S ((S (ff_i_mce_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_positive)) * ff_uc_mce_fold_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf) + (ff_a_mce_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_positive))) /\ ((((exists ff_h_mce_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_positive_partial. ff_h_mce_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_positive_partial + S (ff_r_mce_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_positive) = S ((S (ff_i_mce_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_positive)) * ff_v_mce_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_positive)) /\ exists ff_q_mce_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_positive_partial. ff_u_mce_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_positive = ff_q_mce_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_positive_partial * S ((S (ff_i_mce_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_positive)) * ff_v_mce_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_positive) + (ff_r_mce_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_positive))) /\ ((((exists ff_h_mce_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_positive_successor. ff_h_mce_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_positive_successor + S (ff_s_mce_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_positive) = S ((S (S ff_i_mce_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_positive)) * ff_v_mce_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_positive)) /\ exists ff_q_mce_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_positive_successor. ff_u_mce_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_positive = ff_q_mce_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_positive_successor * S ((S (S ff_i_mce_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_positive)) * ff_v_mce_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_positive) + (ff_s_mce_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_positive))) /\ ff_s_mce_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_positive = ff_r_mce_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_positive + ff_a_mce_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_positive)))))) /\ (exists ff_u_mce_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_negative ff_v_mce_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_negative. ((((exists ff_h_mce_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_negative_start. ff_h_mce_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_negative_start + S (0) = S ((S (0)) * ff_v_mce_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_negative)) /\ exists ff_q_mce_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_negative_start. ff_u_mce_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_negative = ff_q_mce_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_negative_start * S ((S (0)) * ff_v_mce_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_negative) + (0))) /\ ((((exists ff_h_mce_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_negative_terminal. ff_h_mce_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_negative_terminal + S (mdr_n_first_integer_rankwitnessminornonzeroevaluationdeterminanth) = S ((S ((S (mdr_q_first_integer_rankwitnessminornonzeroevaluationdeterminanths)))) * ff_v_mce_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_negative)) /\ exists ff_q_mce_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_negative_terminal. ff_u_mce_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_negative = ff_q_mce_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_negative_terminal * S ((S ((S (mdr_q_first_integer_rankwitnessminornonzeroevaluationdeterminanths)))) * ff_v_mce_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_negative) + (mdr_n_first_integer_rankwitnessminornonzeroevaluationdeterminanth))) /\ forall ff_i_mce_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_negative. (exists ff_lt_mce_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_negative_bound. ff_lt_mce_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_negative_bound + S ff_i_mce_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_negative = (S (mdr_q_first_integer_rankwitnessminornonzeroevaluationdeterminanths))) -> exists ff_a_mce_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_negative ff_r_mce_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_negative ff_s_mce_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_negative. ((((exists ff_h_mce_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_negative_summand. ff_h_mce_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_negative_summand + S (ff_a_mce_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_negative) = S ((S (ff_i_mce_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_negative)) * ff_vc_mce_fold_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf)) /\ exists ff_q_mce_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_negative_summand. ff_vb_mce_fold_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf = ff_q_mce_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_negative_summand * S ((S (ff_i_mce_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_negative)) * ff_vc_mce_fold_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf) + (ff_a_mce_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_negative))) /\ ((((exists ff_h_mce_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_negative_partial. ff_h_mce_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_negative_partial + S (ff_r_mce_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_negative) = S ((S (ff_i_mce_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_negative)) * ff_v_mce_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_negative)) /\ exists ff_q_mce_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_negative_partial. ff_u_mce_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_negative = ff_q_mce_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_negative_partial * S ((S (ff_i_mce_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_negative)) * ff_v_mce_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_negative) + (ff_r_mce_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_negative))) /\ ((((exists ff_h_mce_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_negative_successor. ff_h_mce_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_negative_successor + S (ff_s_mce_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_negative) = S ((S (S ff_i_mce_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_negative)) * ff_v_mce_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_negative)) /\ exists ff_q_mce_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_negative_successor. ff_u_mce_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_negative = ff_q_mce_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_negative_successor * S ((S (S ff_i_mce_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_negative)) * ff_v_mce_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_negative) + (ff_s_mce_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_negative))) /\ ff_s_mce_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_negative = ff_r_mce_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_negative + ff_a_mce_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminanthsf_negative))))))))))))))) /\ ((exists mdr_gap_first_integer_rankwitnessminornonzeroevaluationdeterminanti. mdr_gap_first_integer_rankwitnessminornonzeroevaluationdeterminanti + S (mdr_i_first_integer_rankwitnessminornonzeroevaluationdeterminant) = (mdr_l_first_integer_rankwitnessminornonzeroevaluationdeterminant)) /\ (exists mdr_z_first_integer_rankwitnessminornonzeroevaluationdeterminantr. ((exists mdr_a_first_integer_rankwitnessminornonzeroevaluationdeterminantrc mdr_b_first_integer_rankwitnessminornonzeroevaluationdeterminantrc mdr_c_first_integer_rankwitnessminornonzeroevaluationdeterminantrc mdr_e_first_integer_rankwitnessminornonzeroevaluationdeterminantrc mdr_f_first_integer_rankwitnessminornonzeroevaluationdeterminantrc. ((mdr_a_first_integer_rankwitnessminornonzeroevaluationdeterminantrc = ((rank) + (mdr_ub_first_integer_rankwitnessminornonzeroevaluation)) * S ((rank) + (mdr_ub_first_integer_rankwitnessminornonzeroevaluation)) + ((mdr_ub_first_integer_rankwitnessminornonzeroevaluation) + (mdr_ub_first_integer_rankwitnessminornonzeroevaluation))) /\ ((mdr_b_first_integer_rankwitnessminornonzeroevaluationdeterminantrc = ((mdr_uc_first_integer_rankwitnessminornonzeroevaluation) + (mdr_vb_first_integer_rankwitnessminornonzeroevaluation)) * S ((mdr_uc_first_integer_rankwitnessminornonzeroevaluation) + (mdr_vb_first_integer_rankwitnessminornonzeroevaluation)) + ((mdr_vb_first_integer_rankwitnessminornonzeroevaluation) + (mdr_vb_first_integer_rankwitnessminornonzeroevaluation))) /\ ((mdr_c_first_integer_rankwitnessminornonzeroevaluationdeterminantrc = ((mdr_a_first_integer_rankwitnessminornonzeroevaluationdeterminantrc) + (mdr_b_first_integer_rankwitnessminornonzeroevaluationdeterminantrc)) * S ((mdr_a_first_integer_rankwitnessminornonzeroevaluationdeterminantrc) + (mdr_b_first_integer_rankwitnessminornonzeroevaluationdeterminantrc)) + ((mdr_b_first_integer_rankwitnessminornonzeroevaluationdeterminantrc) + (mdr_b_first_integer_rankwitnessminornonzeroevaluationdeterminantrc))) /\ ((mdr_e_first_integer_rankwitnessminornonzeroevaluationdeterminantrc = ((mdr_p_first_integer_rankwitnessminornonzero) + (mdr_n_first_integer_rankwitnessminornonzero)) * S ((mdr_p_first_integer_rankwitnessminornonzero) + (mdr_n_first_integer_rankwitnessminornonzero)) + ((mdr_n_first_integer_rankwitnessminornonzero) + (mdr_n_first_integer_rankwitnessminornonzero))) /\ ((mdr_f_first_integer_rankwitnessminornonzeroevaluationdeterminantrc = ((mdr_vc_first_integer_rankwitnessminornonzeroevaluation) + (mdr_e_first_integer_rankwitnessminornonzeroevaluationdeterminantrc)) * S ((mdr_vc_first_integer_rankwitnessminornonzeroevaluation) + (mdr_e_first_integer_rankwitnessminornonzeroevaluationdeterminantrc)) + ((mdr_e_first_integer_rankwitnessminornonzeroevaluationdeterminantrc) + (mdr_e_first_integer_rankwitnessminornonzeroevaluationdeterminantrc))) /\ ((mdr_z_first_integer_rankwitnessminornonzeroevaluationdeterminantr) = ((mdr_c_first_integer_rankwitnessminornonzeroevaluationdeterminantrc) + (mdr_f_first_integer_rankwitnessminornonzeroevaluationdeterminantrc)) * S ((mdr_c_first_integer_rankwitnessminornonzeroevaluationdeterminantrc) + (mdr_f_first_integer_rankwitnessminornonzeroevaluationdeterminantrc)) + ((mdr_f_first_integer_rankwitnessminornonzeroevaluationdeterminantrc) + (mdr_f_first_integer_rankwitnessminornonzeroevaluationdeterminantrc))))))))) /\ (((exists ff_h_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminantrb. ff_h_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminantrb + S (mdr_z_first_integer_rankwitnessminornonzeroevaluationdeterminantr) = S ((S (mdr_i_first_integer_rankwitnessminornonzeroevaluationdeterminant)) * mdr_c_first_integer_rankwitnessminornonzeroevaluationdeterminant)) /\ exists ff_q_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminantrb. mdr_b_first_integer_rankwitnessminornonzeroevaluationdeterminant = ff_q_mdr_first_integer_rankwitnessminornonzeroevaluationdeterminantrb * S ((S (mdr_i_first_integer_rankwitnessminornonzeroevaluationdeterminant)) * mdr_c_first_integer_rankwitnessminornonzeroevaluationdeterminant) + (mdr_z_first_integer_rankwitnessminornonzeroevaluationdeterminantr)))))))))) /\ (~(mdr_p_first_integer_rankwitnessminornonzero = mdr_n_first_integer_rankwitnessminornonzero)))))))) /\ (forall mdr_q_first_integer_rank. (exists mdr_gap_first_integer_rankhigher. mdr_gap_first_integer_rankhigher + S (rank) = (mdr_q_first_integer_rank)) -> (forall mdr_rb_first_integer_rankzero mdr_rc_first_integer_rankzero mdr_cb_first_integer_rankzero mdr_cc_first_integer_rankzero mdr_p_first_integer_rankzero mdr_n_first_integer_rankzero. (((forall fom_index_mrf_first_integer_rankzerorowsbound. (exists fom_gap_mrf_first_integer_rankzerorowsbound_index_bound. fom_gap_mrf_first_integer_rankzerorowsbound_index_bound + S (fom_index_mrf_first_integer_rankzerorowsbound) = mdr_q_first_integer_rank) -> exists fom_value_mrf_first_integer_rankzerorowsbound. ((((exists fom_beta_height_mrf_first_integer_rankzerorowsbound_entry. fom_beta_height_mrf_first_integer_rankzerorowsbound_entry + S (fom_value_mrf_first_integer_rankzerorowsbound) = S ((S (fom_index_mrf_first_integer_rankzerorowsbound)) * mdr_rc_first_integer_rankzero)) /\ exists fom_beta_quotient_mrf_first_integer_rankzerorowsbound_entry. mdr_rb_first_integer_rankzero = fom_beta_quotient_mrf_first_integer_rankzerorowsbound_entry * S ((S (fom_index_mrf_first_integer_rankzerorowsbound)) * mdr_rc_first_integer_rankzero) + (fom_value_mrf_first_integer_rankzerorowsbound))) /\ (exists fom_gap_mrf_first_integer_rankzerorowsbound_value_bound. fom_gap_mrf_first_integer_rankzerorowsbound_value_bound + S (fom_value_mrf_first_integer_rankzerorowsbound) = r))) /\ (forall mdr_i_first_integer_rankzerorowsdistinct mdr_j_first_integer_rankzerorowsdistinct mdr_a_first_integer_rankzerorowsdistinct. (exists mdr_gap_first_integer_rankzerorowsdistincti. mdr_gap_first_integer_rankzerorowsdistincti + S (mdr_i_first_integer_rankzerorowsdistinct) = (mdr_q_first_integer_rank)) -> (exists mdr_gap_first_integer_rankzerorowsdistinctj. mdr_gap_first_integer_rankzerorowsdistinctj + S (mdr_j_first_integer_rankzerorowsdistinct) = (mdr_q_first_integer_rank)) -> (((exists ff_h_mdr_first_integer_rankzerorowsdistinctfirst. ff_h_mdr_first_integer_rankzerorowsdistinctfirst + S (mdr_a_first_integer_rankzerorowsdistinct) = S ((S (mdr_i_first_integer_rankzerorowsdistinct)) * mdr_rc_first_integer_rankzero)) /\ exists ff_q_mdr_first_integer_rankzerorowsdistinctfirst. mdr_rb_first_integer_rankzero = ff_q_mdr_first_integer_rankzerorowsdistinctfirst * S ((S (mdr_i_first_integer_rankzerorowsdistinct)) * mdr_rc_first_integer_rankzero) + (mdr_a_first_integer_rankzerorowsdistinct))) -> (((exists ff_h_mdr_first_integer_rankzerorowsdistinctsecond. ff_h_mdr_first_integer_rankzerorowsdistinctsecond + S (mdr_a_first_integer_rankzerorowsdistinct) = S ((S (mdr_j_first_integer_rankzerorowsdistinct)) * mdr_rc_first_integer_rankzero)) /\ exists ff_q_mdr_first_integer_rankzerorowsdistinctsecond. mdr_rb_first_integer_rankzero = ff_q_mdr_first_integer_rankzerorowsdistinctsecond * S ((S (mdr_j_first_integer_rankzerorowsdistinct)) * mdr_rc_first_integer_rankzero) + (mdr_a_first_integer_rankzerorowsdistinct))) -> mdr_i_first_integer_rankzerorowsdistinct = mdr_j_first_integer_rankzerorowsdistinct))) -> (((forall fom_index_mrf_first_integer_rankzerocolumnsbound. (exists fom_gap_mrf_first_integer_rankzerocolumnsbound_index_bound. fom_gap_mrf_first_integer_rankzerocolumnsbound_index_bound + S (fom_index_mrf_first_integer_rankzerocolumnsbound) = mdr_q_first_integer_rank) -> exists fom_value_mrf_first_integer_rankzerocolumnsbound. ((((exists fom_beta_height_mrf_first_integer_rankzerocolumnsbound_entry. fom_beta_height_mrf_first_integer_rankzerocolumnsbound_entry + S (fom_value_mrf_first_integer_rankzerocolumnsbound) = S ((S (fom_index_mrf_first_integer_rankzerocolumnsbound)) * mdr_cc_first_integer_rankzero)) /\ exists fom_beta_quotient_mrf_first_integer_rankzerocolumnsbound_entry. mdr_cb_first_integer_rankzero = fom_beta_quotient_mrf_first_integer_rankzerocolumnsbound_entry * S ((S (fom_index_mrf_first_integer_rankzerocolumnsbound)) * mdr_cc_first_integer_rankzero) + (fom_value_mrf_first_integer_rankzerocolumnsbound))) /\ (exists fom_gap_mrf_first_integer_rankzerocolumnsbound_value_bound. fom_gap_mrf_first_integer_rankzerocolumnsbound_value_bound + S (fom_value_mrf_first_integer_rankzerocolumnsbound) = w))) /\ (forall mdr_i_first_integer_rankzerocolumnsdistinct mdr_j_first_integer_rankzerocolumnsdistinct mdr_a_first_integer_rankzerocolumnsdistinct. (exists mdr_gap_first_integer_rankzerocolumnsdistincti. mdr_gap_first_integer_rankzerocolumnsdistincti + S (mdr_i_first_integer_rankzerocolumnsdistinct) = (mdr_q_first_integer_rank)) -> (exists mdr_gap_first_integer_rankzerocolumnsdistinctj. mdr_gap_first_integer_rankzerocolumnsdistinctj + S (mdr_j_first_integer_rankzerocolumnsdistinct) = (mdr_q_first_integer_rank)) -> (((exists ff_h_mdr_first_integer_rankzerocolumnsdistinctfirst. ff_h_mdr_first_integer_rankzerocolumnsdistinctfirst + S (mdr_a_first_integer_rankzerocolumnsdistinct) = S ((S (mdr_i_first_integer_rankzerocolumnsdistinct)) * mdr_cc_first_integer_rankzero)) /\ exists ff_q_mdr_first_integer_rankzerocolumnsdistinctfirst. mdr_cb_first_integer_rankzero = ff_q_mdr_first_integer_rankzerocolumnsdistinctfirst * S ((S (mdr_i_first_integer_rankzerocolumnsdistinct)) * mdr_cc_first_integer_rankzero) + (mdr_a_first_integer_rankzerocolumnsdistinct))) -> (((exists ff_h_mdr_first_integer_rankzerocolumnsdistinctsecond. ff_h_mdr_first_integer_rankzerocolumnsdistinctsecond + S (mdr_a_first_integer_rankzerocolumnsdistinct) = S ((S (mdr_j_first_integer_rankzerocolumnsdistinct)) * mdr_cc_first_integer_rankzero)) /\ exists ff_q_mdr_first_integer_rankzerocolumnsdistinctsecond. mdr_cb_first_integer_rankzero = ff_q_mdr_first_integer_rankzerocolumnsdistinctsecond * S ((S (mdr_j_first_integer_rankzerocolumnsdistinct)) * mdr_cc_first_integer_rankzero) + (mdr_a_first_integer_rankzerocolumnsdistinct))) -> mdr_i_first_integer_rankzerocolumnsdistinct = mdr_j_first_integer_rankzerocolumnsdistinct))) -> (exists mdr_ub_first_integer_rankzeroevaluation mdr_uc_first_integer_rankzeroevaluation mdr_vb_first_integer_rankzeroevaluation mdr_vc_first_integer_rankzeroevaluation. ((((forall mdr_i_first_integer_rankzeroevaluationmatrixpositive. (exists mdr_gap_first_integer_rankzeroevaluationmatrixpositivebound. mdr_gap_first_integer_rankzeroevaluationmatrixpositivebound + S (mdr_i_first_integer_rankzeroevaluationmatrixpositive) = ((mdr_q_first_integer_rank) * (mdr_q_first_integer_rank))) -> exists mdr_a_first_integer_rankzeroevaluationmatrixpositive. (((exists mdr_r_first_integer_rankzeroevaluationmatrixpositivepoint mdr_s_first_integer_rankzeroevaluationmatrixpositivepoint mdr_u_first_integer_rankzeroevaluationmatrixpositivepoint mdr_v_first_integer_rankzeroevaluationmatrixpositivepoint. ((mdr_i_first_integer_rankzeroevaluationmatrixpositive = (mdr_q_first_integer_rank) * mdr_r_first_integer_rankzeroevaluationmatrixpositivepoint + mdr_s_first_integer_rankzeroevaluationmatrixpositivepoint) /\ ((exists mdr_gap_first_integer_rankzeroevaluationmatrixpositivepointcolumn. mdr_gap_first_integer_rankzeroevaluationmatrixpositivepointcolumn + S (mdr_s_first_integer_rankzeroevaluationmatrixpositivepoint) = (mdr_q_first_integer_rank)) /\ ((((exists ff_h_mdr_first_integer_rankzeroevaluationmatrixpositivepointrow_index. ff_h_mdr_first_integer_rankzeroevaluationmatrixpositivepointrow_index + S (mdr_u_first_integer_rankzeroevaluationmatrixpositivepoint) = S ((S (mdr_r_first_integer_rankzeroevaluationmatrixpositivepoint)) * mdr_rc_first_integer_rankzero)) /\ exists ff_q_mdr_first_integer_rankzeroevaluationmatrixpositivepointrow_index. mdr_rb_first_integer_rankzero = ff_q_mdr_first_integer_rankzeroevaluationmatrixpositivepointrow_index * S ((S (mdr_r_first_integer_rankzeroevaluationmatrixpositivepoint)) * mdr_rc_first_integer_rankzero) + (mdr_u_first_integer_rankzeroevaluationmatrixpositivepoint))) /\ ((((exists ff_h_mdr_first_integer_rankzeroevaluationmatrixpositivepointcolumn_index. ff_h_mdr_first_integer_rankzeroevaluationmatrixpositivepointcolumn_index + S (mdr_v_first_integer_rankzeroevaluationmatrixpositivepoint) = S ((S (mdr_s_first_integer_rankzeroevaluationmatrixpositivepoint)) * mdr_cc_first_integer_rankzero)) /\ exists ff_q_mdr_first_integer_rankzeroevaluationmatrixpositivepointcolumn_index. mdr_cb_first_integer_rankzero = ff_q_mdr_first_integer_rankzeroevaluationmatrixpositivepointcolumn_index * S ((S (mdr_s_first_integer_rankzeroevaluationmatrixpositivepoint)) * mdr_cc_first_integer_rankzero) + (mdr_v_first_integer_rankzeroevaluationmatrixpositivepoint))) /\ (((exists ff_h_mdr_first_integer_rankzeroevaluationmatrixpositivepointsource. ff_h_mdr_first_integer_rankzeroevaluationmatrixpositivepointsource + S (mdr_a_first_integer_rankzeroevaluationmatrixpositive) = S ((S ((mdr_u_first_integer_rankzeroevaluationmatrixpositivepoint) * (w) + (mdr_v_first_integer_rankzeroevaluationmatrixpositivepoint))) * ac)) /\ exists ff_q_mdr_first_integer_rankzeroevaluationmatrixpositivepointsource. ab = ff_q_mdr_first_integer_rankzeroevaluationmatrixpositivepointsource * S ((S ((mdr_u_first_integer_rankzeroevaluationmatrixpositivepoint) * (w) + (mdr_v_first_integer_rankzeroevaluationmatrixpositivepoint))) * ac) + (mdr_a_first_integer_rankzeroevaluationmatrixpositive)))))))) /\ (((exists ff_h_mdr_first_integer_rankzeroevaluationmatrixpositiveoutput. ff_h_mdr_first_integer_rankzeroevaluationmatrixpositiveoutput + S (mdr_a_first_integer_rankzeroevaluationmatrixpositive) = S ((S (mdr_i_first_integer_rankzeroevaluationmatrixpositive)) * mdr_uc_first_integer_rankzeroevaluation)) /\ exists ff_q_mdr_first_integer_rankzeroevaluationmatrixpositiveoutput. mdr_ub_first_integer_rankzeroevaluation = ff_q_mdr_first_integer_rankzeroevaluationmatrixpositiveoutput * S ((S (mdr_i_first_integer_rankzeroevaluationmatrixpositive)) * mdr_uc_first_integer_rankzeroevaluation) + (mdr_a_first_integer_rankzeroevaluationmatrixpositive)))))) /\ (forall mdr_i_first_integer_rankzeroevaluationmatrixnegative. (exists mdr_gap_first_integer_rankzeroevaluationmatrixnegativebound. mdr_gap_first_integer_rankzeroevaluationmatrixnegativebound + S (mdr_i_first_integer_rankzeroevaluationmatrixnegative) = ((mdr_q_first_integer_rank) * (mdr_q_first_integer_rank))) -> exists mdr_a_first_integer_rankzeroevaluationmatrixnegative. (((exists mdr_r_first_integer_rankzeroevaluationmatrixnegativepoint mdr_s_first_integer_rankzeroevaluationmatrixnegativepoint mdr_u_first_integer_rankzeroevaluationmatrixnegativepoint mdr_v_first_integer_rankzeroevaluationmatrixnegativepoint. ((mdr_i_first_integer_rankzeroevaluationmatrixnegative = (mdr_q_first_integer_rank) * mdr_r_first_integer_rankzeroevaluationmatrixnegativepoint + mdr_s_first_integer_rankzeroevaluationmatrixnegativepoint) /\ ((exists mdr_gap_first_integer_rankzeroevaluationmatrixnegativepointcolumn. mdr_gap_first_integer_rankzeroevaluationmatrixnegativepointcolumn + S (mdr_s_first_integer_rankzeroevaluationmatrixnegativepoint) = (mdr_q_first_integer_rank)) /\ ((((exists ff_h_mdr_first_integer_rankzeroevaluationmatrixnegativepointrow_index. ff_h_mdr_first_integer_rankzeroevaluationmatrixnegativepointrow_index + S (mdr_u_first_integer_rankzeroevaluationmatrixnegativepoint) = S ((S (mdr_r_first_integer_rankzeroevaluationmatrixnegativepoint)) * mdr_rc_first_integer_rankzero)) /\ exists ff_q_mdr_first_integer_rankzeroevaluationmatrixnegativepointrow_index. mdr_rb_first_integer_rankzero = ff_q_mdr_first_integer_rankzeroevaluationmatrixnegativepointrow_index * S ((S (mdr_r_first_integer_rankzeroevaluationmatrixnegativepoint)) * mdr_rc_first_integer_rankzero) + (mdr_u_first_integer_rankzeroevaluationmatrixnegativepoint))) /\ ((((exists ff_h_mdr_first_integer_rankzeroevaluationmatrixnegativepointcolumn_index. ff_h_mdr_first_integer_rankzeroevaluationmatrixnegativepointcolumn_index + S (mdr_v_first_integer_rankzeroevaluationmatrixnegativepoint) = S ((S (mdr_s_first_integer_rankzeroevaluationmatrixnegativepoint)) * mdr_cc_first_integer_rankzero)) /\ exists ff_q_mdr_first_integer_rankzeroevaluationmatrixnegativepointcolumn_index. mdr_cb_first_integer_rankzero = ff_q_mdr_first_integer_rankzeroevaluationmatrixnegativepointcolumn_index * S ((S (mdr_s_first_integer_rankzeroevaluationmatrixnegativepoint)) * mdr_cc_first_integer_rankzero) + (mdr_v_first_integer_rankzeroevaluationmatrixnegativepoint))) /\ (((exists ff_h_mdr_first_integer_rankzeroevaluationmatrixnegativepointsource. ff_h_mdr_first_integer_rankzeroevaluationmatrixnegativepointsource + S (mdr_a_first_integer_rankzeroevaluationmatrixnegative) = S ((S ((mdr_u_first_integer_rankzeroevaluationmatrixnegativepoint) * (w) + (mdr_v_first_integer_rankzeroevaluationmatrixnegativepoint))) * bc)) /\ exists ff_q_mdr_first_integer_rankzeroevaluationmatrixnegativepointsource. bb = ff_q_mdr_first_integer_rankzeroevaluationmatrixnegativepointsource * S ((S ((mdr_u_first_integer_rankzeroevaluationmatrixnegativepoint) * (w) + (mdr_v_first_integer_rankzeroevaluationmatrixnegativepoint))) * bc) + (mdr_a_first_integer_rankzeroevaluationmatrixnegative)))))))) /\ (((exists ff_h_mdr_first_integer_rankzeroevaluationmatrixnegativeoutput. ff_h_mdr_first_integer_rankzeroevaluationmatrixnegativeoutput + S (mdr_a_first_integer_rankzeroevaluationmatrixnegative) = S ((S (mdr_i_first_integer_rankzeroevaluationmatrixnegative)) * mdr_vc_first_integer_rankzeroevaluation)) /\ exists ff_q_mdr_first_integer_rankzeroevaluationmatrixnegativeoutput. mdr_vb_first_integer_rankzeroevaluation = ff_q_mdr_first_integer_rankzeroevaluationmatrixnegativeoutput * S ((S (mdr_i_first_integer_rankzeroevaluationmatrixnegative)) * mdr_vc_first_integer_rankzeroevaluation) + (mdr_a_first_integer_rankzeroevaluationmatrixnegative)))))))) /\ (exists mdr_b_first_integer_rankzeroevaluationdeterminant mdr_c_first_integer_rankzeroevaluationdeterminant mdr_l_first_integer_rankzeroevaluationdeterminant mdr_i_first_integer_rankzeroevaluationdeterminant. ((forall mdr_i_first_integer_rankzeroevaluationdeterminanth. (exists mdr_gap_first_integer_rankzeroevaluationdeterminanthi. mdr_gap_first_integer_rankzeroevaluationdeterminanthi + S (mdr_i_first_integer_rankzeroevaluationdeterminanth) = (mdr_l_first_integer_rankzeroevaluationdeterminant)) -> exists mdr_d_first_integer_rankzeroevaluationdeterminanth mdr_pb_first_integer_rankzeroevaluationdeterminanth mdr_pc_first_integer_rankzeroevaluationdeterminanth mdr_nb_first_integer_rankzeroevaluationdeterminanth mdr_nc_first_integer_rankzeroevaluationdeterminanth mdr_p_first_integer_rankzeroevaluationdeterminanth mdr_n_first_integer_rankzeroevaluationdeterminanth. ((exists mdr_z_first_integer_rankzeroevaluationdeterminanthr. ((exists mdr_a_first_integer_rankzeroevaluationdeterminanthrc mdr_b_first_integer_rankzeroevaluationdeterminanthrc mdr_c_first_integer_rankzeroevaluationdeterminanthrc mdr_e_first_integer_rankzeroevaluationdeterminanthrc mdr_f_first_integer_rankzeroevaluationdeterminanthrc. ((mdr_a_first_integer_rankzeroevaluationdeterminanthrc = ((mdr_d_first_integer_rankzeroevaluationdeterminanth) + (mdr_pb_first_integer_rankzeroevaluationdeterminanth)) * S ((mdr_d_first_integer_rankzeroevaluationdeterminanth) + (mdr_pb_first_integer_rankzeroevaluationdeterminanth)) + ((mdr_pb_first_integer_rankzeroevaluationdeterminanth) + (mdr_pb_first_integer_rankzeroevaluationdeterminanth))) /\ ((mdr_b_first_integer_rankzeroevaluationdeterminanthrc = ((mdr_pc_first_integer_rankzeroevaluationdeterminanth) + (mdr_nb_first_integer_rankzeroevaluationdeterminanth)) * S ((mdr_pc_first_integer_rankzeroevaluationdeterminanth) + (mdr_nb_first_integer_rankzeroevaluationdeterminanth)) + ((mdr_nb_first_integer_rankzeroevaluationdeterminanth) + (mdr_nb_first_integer_rankzeroevaluationdeterminanth))) /\ ((mdr_c_first_integer_rankzeroevaluationdeterminanthrc = ((mdr_a_first_integer_rankzeroevaluationdeterminanthrc) + (mdr_b_first_integer_rankzeroevaluationdeterminanthrc)) * S ((mdr_a_first_integer_rankzeroevaluationdeterminanthrc) + (mdr_b_first_integer_rankzeroevaluationdeterminanthrc)) + ((mdr_b_first_integer_rankzeroevaluationdeterminanthrc) + (mdr_b_first_integer_rankzeroevaluationdeterminanthrc))) /\ ((mdr_e_first_integer_rankzeroevaluationdeterminanthrc = ((mdr_p_first_integer_rankzeroevaluationdeterminanth) + (mdr_n_first_integer_rankzeroevaluationdeterminanth)) * S ((mdr_p_first_integer_rankzeroevaluationdeterminanth) + (mdr_n_first_integer_rankzeroevaluationdeterminanth)) + ((mdr_n_first_integer_rankzeroevaluationdeterminanth) + (mdr_n_first_integer_rankzeroevaluationdeterminanth))) /\ ((mdr_f_first_integer_rankzeroevaluationdeterminanthrc = ((mdr_nc_first_integer_rankzeroevaluationdeterminanth) + (mdr_e_first_integer_rankzeroevaluationdeterminanthrc)) * S ((mdr_nc_first_integer_rankzeroevaluationdeterminanth) + (mdr_e_first_integer_rankzeroevaluationdeterminanthrc)) + ((mdr_e_first_integer_rankzeroevaluationdeterminanthrc) + (mdr_e_first_integer_rankzeroevaluationdeterminanthrc))) /\ ((mdr_z_first_integer_rankzeroevaluationdeterminanthr) = ((mdr_c_first_integer_rankzeroevaluationdeterminanthrc) + (mdr_f_first_integer_rankzeroevaluationdeterminanthrc)) * S ((mdr_c_first_integer_rankzeroevaluationdeterminanthrc) + (mdr_f_first_integer_rankzeroevaluationdeterminanthrc)) + ((mdr_f_first_integer_rankzeroevaluationdeterminanthrc) + (mdr_f_first_integer_rankzeroevaluationdeterminanthrc))))))))) /\ (((exists ff_h_mdr_first_integer_rankzeroevaluationdeterminanthrb. ff_h_mdr_first_integer_rankzeroevaluationdeterminanthrb + S (mdr_z_first_integer_rankzeroevaluationdeterminanthr) = S ((S (mdr_i_first_integer_rankzeroevaluationdeterminanth)) * mdr_c_first_integer_rankzeroevaluationdeterminant)) /\ exists ff_q_mdr_first_integer_rankzeroevaluationdeterminanthrb. mdr_b_first_integer_rankzeroevaluationdeterminant = ff_q_mdr_first_integer_rankzeroevaluationdeterminanthrb * S ((S (mdr_i_first_integer_rankzeroevaluationdeterminanth)) * mdr_c_first_integer_rankzeroevaluationdeterminant) + (mdr_z_first_integer_rankzeroevaluationdeterminanthr))))) /\ (((((mdr_d_first_integer_rankzeroevaluationdeterminanth) = 0) /\ (((mdr_p_first_integer_rankzeroevaluationdeterminanth) = 1) /\ ((mdr_n_first_integer_rankzeroevaluationdeterminanth) = 0))) \/ exists mdr_q_first_integer_rankzeroevaluationdeterminanths mdr_eb_first_integer_rankzeroevaluationdeterminanths mdr_ec_first_integer_rankzeroevaluationdeterminanths mdr_fb_first_integer_rankzeroevaluationdeterminanths mdr_fc_first_integer_rankzeroevaluationdeterminanths. (((mdr_d_first_integer_rankzeroevaluationdeterminanth) = S (mdr_q_first_integer_rankzeroevaluationdeterminanths)) /\ ((forall mdr_j_first_integer_rankzeroevaluationdeterminanthsc. (exists mdr_gap_first_integer_rankzeroevaluationdeterminanthscj. mdr_gap_first_integer_rankzeroevaluationdeterminanthscj + S (mdr_j_first_integer_rankzeroevaluationdeterminanthsc) = (S (mdr_q_first_integer_rankzeroevaluationdeterminanths))) -> exists mdr_i_first_integer_rankzeroevaluationdeterminanthsc mdr_up_first_integer_rankzeroevaluationdeterminanthsc mdr_us_first_integer_rankzeroevaluationdeterminanthsc mdr_un_first_integer_rankzeroevaluationdeterminanthsc mdr_ut_first_integer_rankzeroevaluationdeterminanthsc mdr_p_first_integer_rankzeroevaluationdeterminanthsc mdr_n_first_integer_rankzeroevaluationdeterminanthsc. ((exists mdr_gap_first_integer_rankzeroevaluationdeterminanthsci. mdr_gap_first_integer_rankzeroevaluationdeterminanthsci + S (mdr_i_first_integer_rankzeroevaluationdeterminanthsc) = (mdr_i_first_integer_rankzeroevaluationdeterminanth)) /\ ((exists mdr_z_first_integer_rankzeroevaluationdeterminanthscr. ((exists mdr_a_first_integer_rankzeroevaluationdeterminanthscrc mdr_b_first_integer_rankzeroevaluationdeterminanthscrc mdr_c_first_integer_rankzeroevaluationdeterminanthscrc mdr_e_first_integer_rankzeroevaluationdeterminanthscrc mdr_f_first_integer_rankzeroevaluationdeterminanthscrc. ((mdr_a_first_integer_rankzeroevaluationdeterminanthscrc = ((mdr_q_first_integer_rankzeroevaluationdeterminanths) + (mdr_up_first_integer_rankzeroevaluationdeterminanthsc)) * S ((mdr_q_first_integer_rankzeroevaluationdeterminanths) + (mdr_up_first_integer_rankzeroevaluationdeterminanthsc)) + ((mdr_up_first_integer_rankzeroevaluationdeterminanthsc) + (mdr_up_first_integer_rankzeroevaluationdeterminanthsc))) /\ ((mdr_b_first_integer_rankzeroevaluationdeterminanthscrc = ((mdr_us_first_integer_rankzeroevaluationdeterminanthsc) + (mdr_un_first_integer_rankzeroevaluationdeterminanthsc)) * S ((mdr_us_first_integer_rankzeroevaluationdeterminanthsc) + (mdr_un_first_integer_rankzeroevaluationdeterminanthsc)) + ((mdr_un_first_integer_rankzeroevaluationdeterminanthsc) + (mdr_un_first_integer_rankzeroevaluationdeterminanthsc))) /\ ((mdr_c_first_integer_rankzeroevaluationdeterminanthscrc = ((mdr_a_first_integer_rankzeroevaluationdeterminanthscrc) + (mdr_b_first_integer_rankzeroevaluationdeterminanthscrc)) * S ((mdr_a_first_integer_rankzeroevaluationdeterminanthscrc) + (mdr_b_first_integer_rankzeroevaluationdeterminanthscrc)) + ((mdr_b_first_integer_rankzeroevaluationdeterminanthscrc) + (mdr_b_first_integer_rankzeroevaluationdeterminanthscrc))) /\ ((mdr_e_first_integer_rankzeroevaluationdeterminanthscrc = ((mdr_p_first_integer_rankzeroevaluationdeterminanthsc) + (mdr_n_first_integer_rankzeroevaluationdeterminanthsc)) * S ((mdr_p_first_integer_rankzeroevaluationdeterminanthsc) + (mdr_n_first_integer_rankzeroevaluationdeterminanthsc)) + ((mdr_n_first_integer_rankzeroevaluationdeterminanthsc) + (mdr_n_first_integer_rankzeroevaluationdeterminanthsc))) /\ ((mdr_f_first_integer_rankzeroevaluationdeterminanthscrc = ((mdr_ut_first_integer_rankzeroevaluationdeterminanthsc) + (mdr_e_first_integer_rankzeroevaluationdeterminanthscrc)) * S ((mdr_ut_first_integer_rankzeroevaluationdeterminanthsc) + (mdr_e_first_integer_rankzeroevaluationdeterminanthscrc)) + ((mdr_e_first_integer_rankzeroevaluationdeterminanthscrc) + (mdr_e_first_integer_rankzeroevaluationdeterminanthscrc))) /\ ((mdr_z_first_integer_rankzeroevaluationdeterminanthscr) = ((mdr_c_first_integer_rankzeroevaluationdeterminanthscrc) + (mdr_f_first_integer_rankzeroevaluationdeterminanthscrc)) * S ((mdr_c_first_integer_rankzeroevaluationdeterminanthscrc) + (mdr_f_first_integer_rankzeroevaluationdeterminanthscrc)) + ((mdr_f_first_integer_rankzeroevaluationdeterminanthscrc) + (mdr_f_first_integer_rankzeroevaluationdeterminanthscrc))))))))) /\ (((exists ff_h_mdr_first_integer_rankzeroevaluationdeterminanthscrb. ff_h_mdr_first_integer_rankzeroevaluationdeterminanthscrb + S (mdr_z_first_integer_rankzeroevaluationdeterminanthscr) = S ((S (mdr_i_first_integer_rankzeroevaluationdeterminanthsc)) * mdr_c_first_integer_rankzeroevaluationdeterminant)) /\ exists ff_q_mdr_first_integer_rankzeroevaluationdeterminanthscrb. mdr_b_first_integer_rankzeroevaluationdeterminant = ff_q_mdr_first_integer_rankzeroevaluationdeterminanthscrb * S ((S (mdr_i_first_integer_rankzeroevaluationdeterminanthsc)) * mdr_c_first_integer_rankzeroevaluationdeterminant) + (mdr_z_first_integer_rankzeroevaluationdeterminanthscr))))) /\ ((((forall ff_index_mdm_prefix_mdr_first_integer_rankzeroevaluationdeterminanthscm_positive. (exists ff_gap_mdm_lt_mdr_first_integer_rankzeroevaluationdeterminanthscm_positive_index_bound. ff_gap_mdm_lt_mdr_first_integer_rankzeroevaluationdeterminanthscm_positive_index_bound + S (ff_index_mdm_prefix_mdr_first_integer_rankzeroevaluationdeterminanthscm_positive) = ((mdr_q_first_integer_rankzeroevaluationdeterminanths) * (mdr_q_first_integer_rankzeroevaluationdeterminanths))) -> exists ff_row_mdm_prefix_mdr_first_integer_rankzeroevaluationdeterminanthscm_positive ff_column_mdm_prefix_mdr_first_integer_rankzeroevaluationdeterminanthscm_positive ff_value_mdm_prefix_mdr_first_integer_rankzeroevaluationdeterminanthscm_positive. (ff_index_mdm_prefix_mdr_first_integer_rankzeroevaluationdeterminanthscm_positive = (mdr_q_first_integer_rankzeroevaluationdeterminanths) * ff_row_mdm_prefix_mdr_first_integer_rankzeroevaluationdeterminanthscm_positive + ff_column_mdm_prefix_mdr_first_integer_rankzeroevaluationdeterminanthscm_positive /\ ((exists ff_gap_mdm_lt_mdr_first_integer_rankzeroevaluationdeterminanthscm_positive_column_bound. ff_gap_mdm_lt_mdr_first_integer_rankzeroevaluationdeterminanthscm_positive_column_bound + S (ff_column_mdm_prefix_mdr_first_integer_rankzeroevaluationdeterminanthscm_positive) = (mdr_q_first_integer_rankzeroevaluationdeterminanths)) /\ ((exists ff_row_mdm_cell_mdr_first_integer_rankzeroevaluationdeterminanthscm_positive_cell ff_column_mdm_cell_mdr_first_integer_rankzeroevaluationdeterminanthscm_positive_cell. (((((exists ff_gap_mdm_lt_mdr_first_integer_rankzeroevaluationdeterminanthscm_positive_cell_row_before. ff_gap_mdm_lt_mdr_first_integer_rankzeroevaluationdeterminanthscm_positive_cell_row_before + S (ff_row_mdm_prefix_mdr_first_integer_rankzeroevaluationdeterminanthscm_positive) = (0)) /\ ff_row_mdm_cell_mdr_first_integer_rankzeroevaluationdeterminanthscm_positive_cell = ff_row_mdm_prefix_mdr_first_integer_rankzeroevaluationdeterminanthscm_positive) \/ ((exists ff_gap_mdm_le_mdr_first_integer_rankzeroevaluationdeterminanthscm_positive_cell_row_after. ff_gap_mdm_le_mdr_first_integer_rankzeroevaluationdeterminanthscm_positive_cell_row_after + (0) = (ff_row_mdm_prefix_mdr_first_integer_rankzeroevaluationdeterminanthscm_positive)) /\ ff_row_mdm_cell_mdr_first_integer_rankzeroevaluationdeterminanthscm_positive_cell = S ff_row_mdm_prefix_mdr_first_integer_rankzeroevaluationdeterminanthscm_positive))) /\ (((((exists ff_gap_mdm_lt_mdr_first_integer_rankzeroevaluationdeterminanthscm_positive_cell_column_before. ff_gap_mdm_lt_mdr_first_integer_rankzeroevaluationdeterminanthscm_positive_cell_column_before + S (ff_column_mdm_prefix_mdr_first_integer_rankzeroevaluationdeterminanthscm_positive) = (mdr_j_first_integer_rankzeroevaluationdeterminanthsc)) /\ ff_column_mdm_cell_mdr_first_integer_rankzeroevaluationdeterminanthscm_positive_cell = ff_column_mdm_prefix_mdr_first_integer_rankzeroevaluationdeterminanthscm_positive) \/ ((exists ff_gap_mdm_le_mdr_first_integer_rankzeroevaluationdeterminanthscm_positive_cell_column_after. ff_gap_mdm_le_mdr_first_integer_rankzeroevaluationdeterminanthscm_positive_cell_column_after + (mdr_j_first_integer_rankzeroevaluationdeterminanthsc) = (ff_column_mdm_prefix_mdr_first_integer_rankzeroevaluationdeterminanthscm_positive)) /\ ff_column_mdm_cell_mdr_first_integer_rankzeroevaluationdeterminanthscm_positive_cell = S ff_column_mdm_prefix_mdr_first_integer_rankzeroevaluationdeterminanthscm_positive))) /\ (((exists ff_h_mdm_mdr_first_integer_rankzeroevaluationdeterminanthscm_positive_cell_source. ff_h_mdm_mdr_first_integer_rankzeroevaluationdeterminanthscm_positive_cell_source + S (ff_value_mdm_prefix_mdr_first_integer_rankzeroevaluationdeterminanthscm_positive) = S ((S ((ff_row_mdm_cell_mdr_first_integer_rankzeroevaluationdeterminanthscm_positive_cell) * (S (mdr_q_first_integer_rankzeroevaluationdeterminanths)) + (ff_column_mdm_cell_mdr_first_integer_rankzeroevaluationdeterminanthscm_positive_cell))) * mdr_pc_first_integer_rankzeroevaluationdeterminanth)) /\ exists ff_q_mdm_mdr_first_integer_rankzeroevaluationdeterminanthscm_positive_cell_source. mdr_pb_first_integer_rankzeroevaluationdeterminanth = ff_q_mdm_mdr_first_integer_rankzeroevaluationdeterminanthscm_positive_cell_source * S ((S ((ff_row_mdm_cell_mdr_first_integer_rankzeroevaluationdeterminanthscm_positive_cell) * (S (mdr_q_first_integer_rankzeroevaluationdeterminanths)) + (ff_column_mdm_cell_mdr_first_integer_rankzeroevaluationdeterminanthscm_positive_cell))) * mdr_pc_first_integer_rankzeroevaluationdeterminanth) + (ff_value_mdm_prefix_mdr_first_integer_rankzeroevaluationdeterminanthscm_positive)))))) /\ (((exists ff_h_mdm_mdr_first_integer_rankzeroevaluationdeterminanthscm_positive_target. ff_h_mdm_mdr_first_integer_rankzeroevaluationdeterminanthscm_positive_target + S (ff_value_mdm_prefix_mdr_first_integer_rankzeroevaluationdeterminanthscm_positive) = S ((S (ff_index_mdm_prefix_mdr_first_integer_rankzeroevaluationdeterminanthscm_positive)) * mdr_us_first_integer_rankzeroevaluationdeterminanthsc)) /\ exists ff_q_mdm_mdr_first_integer_rankzeroevaluationdeterminanthscm_positive_target. mdr_up_first_integer_rankzeroevaluationdeterminanthsc = ff_q_mdm_mdr_first_integer_rankzeroevaluationdeterminanthscm_positive_target * S ((S (ff_index_mdm_prefix_mdr_first_integer_rankzeroevaluationdeterminanthscm_positive)) * mdr_us_first_integer_rankzeroevaluationdeterminanthsc) + (ff_value_mdm_prefix_mdr_first_integer_rankzeroevaluationdeterminanthscm_positive))))))) /\ (forall ff_index_mdm_prefix_mdr_first_integer_rankzeroevaluationdeterminanthscm_negative. (exists ff_gap_mdm_lt_mdr_first_integer_rankzeroevaluationdeterminanthscm_negative_index_bound. ff_gap_mdm_lt_mdr_first_integer_rankzeroevaluationdeterminanthscm_negative_index_bound + S (ff_index_mdm_prefix_mdr_first_integer_rankzeroevaluationdeterminanthscm_negative) = ((mdr_q_first_integer_rankzeroevaluationdeterminanths) * (mdr_q_first_integer_rankzeroevaluationdeterminanths))) -> exists ff_row_mdm_prefix_mdr_first_integer_rankzeroevaluationdeterminanthscm_negative ff_column_mdm_prefix_mdr_first_integer_rankzeroevaluationdeterminanthscm_negative ff_value_mdm_prefix_mdr_first_integer_rankzeroevaluationdeterminanthscm_negative. (ff_index_mdm_prefix_mdr_first_integer_rankzeroevaluationdeterminanthscm_negative = (mdr_q_first_integer_rankzeroevaluationdeterminanths) * ff_row_mdm_prefix_mdr_first_integer_rankzeroevaluationdeterminanthscm_negative + ff_column_mdm_prefix_mdr_first_integer_rankzeroevaluationdeterminanthscm_negative /\ ((exists ff_gap_mdm_lt_mdr_first_integer_rankzeroevaluationdeterminanthscm_negative_column_bound. ff_gap_mdm_lt_mdr_first_integer_rankzeroevaluationdeterminanthscm_negative_column_bound + S (ff_column_mdm_prefix_mdr_first_integer_rankzeroevaluationdeterminanthscm_negative) = (mdr_q_first_integer_rankzeroevaluationdeterminanths)) /\ ((exists ff_row_mdm_cell_mdr_first_integer_rankzeroevaluationdeterminanthscm_negative_cell ff_column_mdm_cell_mdr_first_integer_rankzeroevaluationdeterminanthscm_negative_cell. (((((exists ff_gap_mdm_lt_mdr_first_integer_rankzeroevaluationdeterminanthscm_negative_cell_row_before. ff_gap_mdm_lt_mdr_first_integer_rankzeroevaluationdeterminanthscm_negative_cell_row_before + S (ff_row_mdm_prefix_mdr_first_integer_rankzeroevaluationdeterminanthscm_negative) = (0)) /\ ff_row_mdm_cell_mdr_first_integer_rankzeroevaluationdeterminanthscm_negative_cell = ff_row_mdm_prefix_mdr_first_integer_rankzeroevaluationdeterminanthscm_negative) \/ ((exists ff_gap_mdm_le_mdr_first_integer_rankzeroevaluationdeterminanthscm_negative_cell_row_after. ff_gap_mdm_le_mdr_first_integer_rankzeroevaluationdeterminanthscm_negative_cell_row_after + (0) = (ff_row_mdm_prefix_mdr_first_integer_rankzeroevaluationdeterminanthscm_negative)) /\ ff_row_mdm_cell_mdr_first_integer_rankzeroevaluationdeterminanthscm_negative_cell = S ff_row_mdm_prefix_mdr_first_integer_rankzeroevaluationdeterminanthscm_negative))) /\ (((((exists ff_gap_mdm_lt_mdr_first_integer_rankzeroevaluationdeterminanthscm_negative_cell_column_before. ff_gap_mdm_lt_mdr_first_integer_rankzeroevaluationdeterminanthscm_negative_cell_column_before + S (ff_column_mdm_prefix_mdr_first_integer_rankzeroevaluationdeterminanthscm_negative) = (mdr_j_first_integer_rankzeroevaluationdeterminanthsc)) /\ ff_column_mdm_cell_mdr_first_integer_rankzeroevaluationdeterminanthscm_negative_cell = ff_column_mdm_prefix_mdr_first_integer_rankzeroevaluationdeterminanthscm_negative) \/ ((exists ff_gap_mdm_le_mdr_first_integer_rankzeroevaluationdeterminanthscm_negative_cell_column_after. ff_gap_mdm_le_mdr_first_integer_rankzeroevaluationdeterminanthscm_negative_cell_column_after + (mdr_j_first_integer_rankzeroevaluationdeterminanthsc) = (ff_column_mdm_prefix_mdr_first_integer_rankzeroevaluationdeterminanthscm_negative)) /\ ff_column_mdm_cell_mdr_first_integer_rankzeroevaluationdeterminanthscm_negative_cell = S ff_column_mdm_prefix_mdr_first_integer_rankzeroevaluationdeterminanthscm_negative))) /\ (((exists ff_h_mdm_mdr_first_integer_rankzeroevaluationdeterminanthscm_negative_cell_source. ff_h_mdm_mdr_first_integer_rankzeroevaluationdeterminanthscm_negative_cell_source + S (ff_value_mdm_prefix_mdr_first_integer_rankzeroevaluationdeterminanthscm_negative) = S ((S ((ff_row_mdm_cell_mdr_first_integer_rankzeroevaluationdeterminanthscm_negative_cell) * (S (mdr_q_first_integer_rankzeroevaluationdeterminanths)) + (ff_column_mdm_cell_mdr_first_integer_rankzeroevaluationdeterminanthscm_negative_cell))) * mdr_nc_first_integer_rankzeroevaluationdeterminanth)) /\ exists ff_q_mdm_mdr_first_integer_rankzeroevaluationdeterminanthscm_negative_cell_source. mdr_nb_first_integer_rankzeroevaluationdeterminanth = ff_q_mdm_mdr_first_integer_rankzeroevaluationdeterminanthscm_negative_cell_source * S ((S ((ff_row_mdm_cell_mdr_first_integer_rankzeroevaluationdeterminanthscm_negative_cell) * (S (mdr_q_first_integer_rankzeroevaluationdeterminanths)) + (ff_column_mdm_cell_mdr_first_integer_rankzeroevaluationdeterminanthscm_negative_cell))) * mdr_nc_first_integer_rankzeroevaluationdeterminanth) + (ff_value_mdm_prefix_mdr_first_integer_rankzeroevaluationdeterminanthscm_negative)))))) /\ (((exists ff_h_mdm_mdr_first_integer_rankzeroevaluationdeterminanthscm_negative_target. ff_h_mdm_mdr_first_integer_rankzeroevaluationdeterminanthscm_negative_target + S (ff_value_mdm_prefix_mdr_first_integer_rankzeroevaluationdeterminanthscm_negative) = S ((S (ff_index_mdm_prefix_mdr_first_integer_rankzeroevaluationdeterminanthscm_negative)) * mdr_ut_first_integer_rankzeroevaluationdeterminanthsc)) /\ exists ff_q_mdm_mdr_first_integer_rankzeroevaluationdeterminanthscm_negative_target. mdr_un_first_integer_rankzeroevaluationdeterminanthsc = ff_q_mdm_mdr_first_integer_rankzeroevaluationdeterminanthscm_negative_target * S ((S (ff_index_mdm_prefix_mdr_first_integer_rankzeroevaluationdeterminanthscm_negative)) * mdr_ut_first_integer_rankzeroevaluationdeterminanthsc) + (ff_value_mdm_prefix_mdr_first_integer_rankzeroevaluationdeterminanthscm_negative))))))))) /\ ((((exists ff_h_mdr_first_integer_rankzeroevaluationdeterminanthscp. ff_h_mdr_first_integer_rankzeroevaluationdeterminanthscp + S (mdr_p_first_integer_rankzeroevaluationdeterminanthsc) = S ((S (mdr_j_first_integer_rankzeroevaluationdeterminanthsc)) * mdr_ec_first_integer_rankzeroevaluationdeterminanths)) /\ exists ff_q_mdr_first_integer_rankzeroevaluationdeterminanthscp. mdr_eb_first_integer_rankzeroevaluationdeterminanths = ff_q_mdr_first_integer_rankzeroevaluationdeterminanthscp * S ((S (mdr_j_first_integer_rankzeroevaluationdeterminanthsc)) * mdr_ec_first_integer_rankzeroevaluationdeterminanths) + (mdr_p_first_integer_rankzeroevaluationdeterminanthsc))) /\ (((exists ff_h_mdr_first_integer_rankzeroevaluationdeterminanthscn. ff_h_mdr_first_integer_rankzeroevaluationdeterminanthscn + S (mdr_n_first_integer_rankzeroevaluationdeterminanthsc) = S ((S (mdr_j_first_integer_rankzeroevaluationdeterminanthsc)) * mdr_fc_first_integer_rankzeroevaluationdeterminanths)) /\ exists ff_q_mdr_first_integer_rankzeroevaluationdeterminanthscn. mdr_fb_first_integer_rankzeroevaluationdeterminanths = ff_q_mdr_first_integer_rankzeroevaluationdeterminanthscn * S ((S (mdr_j_first_integer_rankzeroevaluationdeterminanthsc)) * mdr_fc_first_integer_rankzeroevaluationdeterminanths) + (mdr_n_first_integer_rankzeroevaluationdeterminanthsc)))))))) /\ (exists ff_ub_mce_fold_mdr_first_integer_rankzeroevaluationdeterminanthsf ff_uc_mce_fold_mdr_first_integer_rankzeroevaluationdeterminanthsf ff_vb_mce_fold_mdr_first_integer_rankzeroevaluationdeterminanthsf ff_vc_mce_fold_mdr_first_integer_rankzeroevaluationdeterminanthsf. ((forall ff_index_mce_alternating_mdr_first_integer_rankzeroevaluationdeterminanthsf_prefix. (exists ff_gap_mce_mdr_first_integer_rankzeroevaluationdeterminanthsf_prefix_index. ff_gap_mce_mdr_first_integer_rankzeroevaluationdeterminanthsf_prefix_index + S (ff_index_mce_alternating_mdr_first_integer_rankzeroevaluationdeterminanthsf_prefix) = (S (mdr_q_first_integer_rankzeroevaluationdeterminanths))) -> exists ff_ap_mce_alternating_mdr_first_integer_rankzeroevaluationdeterminanthsf_prefix ff_an_mce_alternating_mdr_first_integer_rankzeroevaluationdeterminanthsf_prefix ff_bp_mce_alternating_mdr_first_integer_rankzeroevaluationdeterminanthsf_prefix ff_bn_mce_alternating_mdr_first_integer_rankzeroevaluationdeterminanthsf_prefix ff_p_mce_alternating_mdr_first_integer_rankzeroevaluationdeterminanthsf_prefix ff_n_mce_alternating_mdr_first_integer_rankzeroevaluationdeterminanthsf_prefix. ((((exists ff_h_mce_mdr_first_integer_rankzeroevaluationdeterminanthsf_prefix_ap. ff_h_mce_mdr_first_integer_rankzeroevaluationdeterminanthsf_prefix_ap + S (ff_ap_mce_alternating_mdr_first_integer_rankzeroevaluationdeterminanthsf_prefix) = S ((S (ff_index_mce_alternating_mdr_first_integer_rankzeroevaluationdeterminanthsf_prefix)) * mdr_pc_first_integer_rankzeroevaluationdeterminanth)) /\ exists ff_q_mce_mdr_first_integer_rankzeroevaluationdeterminanthsf_prefix_ap. mdr_pb_first_integer_rankzeroevaluationdeterminanth = ff_q_mce_mdr_first_integer_rankzeroevaluationdeterminanthsf_prefix_ap * S ((S (ff_index_mce_alternating_mdr_first_integer_rankzeroevaluationdeterminanthsf_prefix)) * mdr_pc_first_integer_rankzeroevaluationdeterminanth) + (ff_ap_mce_alternating_mdr_first_integer_rankzeroevaluationdeterminanthsf_prefix))) /\ ((((exists ff_h_mce_mdr_first_integer_rankzeroevaluationdeterminanthsf_prefix_an. ff_h_mce_mdr_first_integer_rankzeroevaluationdeterminanthsf_prefix_an + S (ff_an_mce_alternating_mdr_first_integer_rankzeroevaluationdeterminanthsf_prefix) = S ((S (ff_index_mce_alternating_mdr_first_integer_rankzeroevaluationdeterminanthsf_prefix)) * mdr_nc_first_integer_rankzeroevaluationdeterminanth)) /\ exists ff_q_mce_mdr_first_integer_rankzeroevaluationdeterminanthsf_prefix_an. mdr_nb_first_integer_rankzeroevaluationdeterminanth = ff_q_mce_mdr_first_integer_rankzeroevaluationdeterminanthsf_prefix_an * S ((S (ff_index_mce_alternating_mdr_first_integer_rankzeroevaluationdeterminanthsf_prefix)) * mdr_nc_first_integer_rankzeroevaluationdeterminanth) + (ff_an_mce_alternating_mdr_first_integer_rankzeroevaluationdeterminanthsf_prefix))) /\ ((((exists ff_h_mce_mdr_first_integer_rankzeroevaluationdeterminanthsf_prefix_bp. ff_h_mce_mdr_first_integer_rankzeroevaluationdeterminanthsf_prefix_bp + S (ff_bp_mce_alternating_mdr_first_integer_rankzeroevaluationdeterminanthsf_prefix) = S ((S (ff_index_mce_alternating_mdr_first_integer_rankzeroevaluationdeterminanthsf_prefix)) * mdr_ec_first_integer_rankzeroevaluationdeterminanths)) /\ exists ff_q_mce_mdr_first_integer_rankzeroevaluationdeterminanthsf_prefix_bp. mdr_eb_first_integer_rankzeroevaluationdeterminanths = ff_q_mce_mdr_first_integer_rankzeroevaluationdeterminanthsf_prefix_bp * S ((S (ff_index_mce_alternating_mdr_first_integer_rankzeroevaluationdeterminanthsf_prefix)) * mdr_ec_first_integer_rankzeroevaluationdeterminanths) + (ff_bp_mce_alternating_mdr_first_integer_rankzeroevaluationdeterminanthsf_prefix))) /\ ((((exists ff_h_mce_mdr_first_integer_rankzeroevaluationdeterminanthsf_prefix_bn. ff_h_mce_mdr_first_integer_rankzeroevaluationdeterminanthsf_prefix_bn + S (ff_bn_mce_alternating_mdr_first_integer_rankzeroevaluationdeterminanthsf_prefix) = S ((S (ff_index_mce_alternating_mdr_first_integer_rankzeroevaluationdeterminanthsf_prefix)) * mdr_fc_first_integer_rankzeroevaluationdeterminanths)) /\ exists ff_q_mce_mdr_first_integer_rankzeroevaluationdeterminanthsf_prefix_bn. mdr_fb_first_integer_rankzeroevaluationdeterminanths = ff_q_mce_mdr_first_integer_rankzeroevaluationdeterminanthsf_prefix_bn * S ((S (ff_index_mce_alternating_mdr_first_integer_rankzeroevaluationdeterminanthsf_prefix)) * mdr_fc_first_integer_rankzeroevaluationdeterminanths) + (ff_bn_mce_alternating_mdr_first_integer_rankzeroevaluationdeterminanthsf_prefix))) /\ ((((exists ff_h_mce_mdr_first_integer_rankzeroevaluationdeterminanthsf_prefix_positive. ff_h_mce_mdr_first_integer_rankzeroevaluationdeterminanthsf_prefix_positive + S (ff_p_mce_alternating_mdr_first_integer_rankzeroevaluationdeterminanthsf_prefix) = S ((S (ff_index_mce_alternating_mdr_first_integer_rankzeroevaluationdeterminanthsf_prefix)) * ff_uc_mce_fold_mdr_first_integer_rankzeroevaluationdeterminanthsf)) /\ exists ff_q_mce_mdr_first_integer_rankzeroevaluationdeterminanthsf_prefix_positive. ff_ub_mce_fold_mdr_first_integer_rankzeroevaluationdeterminanthsf = ff_q_mce_mdr_first_integer_rankzeroevaluationdeterminanthsf_prefix_positive * S ((S (ff_index_mce_alternating_mdr_first_integer_rankzeroevaluationdeterminanthsf_prefix)) * ff_uc_mce_fold_mdr_first_integer_rankzeroevaluationdeterminanthsf) + (ff_p_mce_alternating_mdr_first_integer_rankzeroevaluationdeterminanthsf_prefix))) /\ ((((exists ff_h_mce_mdr_first_integer_rankzeroevaluationdeterminanthsf_prefix_negative. ff_h_mce_mdr_first_integer_rankzeroevaluationdeterminanthsf_prefix_negative + S (ff_n_mce_alternating_mdr_first_integer_rankzeroevaluationdeterminanthsf_prefix) = S ((S (ff_index_mce_alternating_mdr_first_integer_rankzeroevaluationdeterminanthsf_prefix)) * ff_vc_mce_fold_mdr_first_integer_rankzeroevaluationdeterminanthsf)) /\ exists ff_q_mce_mdr_first_integer_rankzeroevaluationdeterminanthsf_prefix_negative. ff_vb_mce_fold_mdr_first_integer_rankzeroevaluationdeterminanthsf = ff_q_mce_mdr_first_integer_rankzeroevaluationdeterminanthsf_prefix_negative * S ((S (ff_index_mce_alternating_mdr_first_integer_rankzeroevaluationdeterminanthsf_prefix)) * ff_vc_mce_fold_mdr_first_integer_rankzeroevaluationdeterminanthsf) + (ff_n_mce_alternating_mdr_first_integer_rankzeroevaluationdeterminanthsf_prefix))) /\ (((exists ff_even_mce_term_mdr_first_integer_rankzeroevaluationdeterminanthsf_prefix_term. ff_index_mce_alternating_mdr_first_integer_rankzeroevaluationdeterminanthsf_prefix = 2 * ff_even_mce_term_mdr_first_integer_rankzeroevaluationdeterminanthsf_prefix_term) /\ (ff_p_mce_alternating_mdr_first_integer_rankzeroevaluationdeterminanthsf_prefix = (ff_ap_mce_alternating_mdr_first_integer_rankzeroevaluationdeterminanthsf_prefix) * (ff_bp_mce_alternating_mdr_first_integer_rankzeroevaluationdeterminanthsf_prefix) + (ff_an_mce_alternating_mdr_first_integer_rankzeroevaluationdeterminanthsf_prefix) * (ff_bn_mce_alternating_mdr_first_integer_rankzeroevaluationdeterminanthsf_prefix) /\ ff_n_mce_alternating_mdr_first_integer_rankzeroevaluationdeterminanthsf_prefix = (ff_ap_mce_alternating_mdr_first_integer_rankzeroevaluationdeterminanthsf_prefix) * (ff_bn_mce_alternating_mdr_first_integer_rankzeroevaluationdeterminanthsf_prefix) + (ff_an_mce_alternating_mdr_first_integer_rankzeroevaluationdeterminanthsf_prefix) * (ff_bp_mce_alternating_mdr_first_integer_rankzeroevaluationdeterminanthsf_prefix))) \/ ((exists ff_odd_mce_term_mdr_first_integer_rankzeroevaluationdeterminanthsf_prefix_term. ff_index_mce_alternating_mdr_first_integer_rankzeroevaluationdeterminanthsf_prefix = 2 * ff_odd_mce_term_mdr_first_integer_rankzeroevaluationdeterminanthsf_prefix_term + 1) /\ (ff_p_mce_alternating_mdr_first_integer_rankzeroevaluationdeterminanthsf_prefix = (ff_ap_mce_alternating_mdr_first_integer_rankzeroevaluationdeterminanthsf_prefix) * (ff_bn_mce_alternating_mdr_first_integer_rankzeroevaluationdeterminanthsf_prefix) + (ff_an_mce_alternating_mdr_first_integer_rankzeroevaluationdeterminanthsf_prefix) * (ff_bp_mce_alternating_mdr_first_integer_rankzeroevaluationdeterminanthsf_prefix) /\ ff_n_mce_alternating_mdr_first_integer_rankzeroevaluationdeterminanthsf_prefix = (ff_ap_mce_alternating_mdr_first_integer_rankzeroevaluationdeterminanthsf_prefix) * (ff_bp_mce_alternating_mdr_first_integer_rankzeroevaluationdeterminanthsf_prefix) + (ff_an_mce_alternating_mdr_first_integer_rankzeroevaluationdeterminanthsf_prefix) * (ff_bn_mce_alternating_mdr_first_integer_rankzeroevaluationdeterminanthsf_prefix))))))))))) /\ ((exists ff_u_mce_mdr_first_integer_rankzeroevaluationdeterminanthsf_positive ff_v_mce_mdr_first_integer_rankzeroevaluationdeterminanthsf_positive. ((((exists ff_h_mce_mdr_first_integer_rankzeroevaluationdeterminanthsf_positive_start. ff_h_mce_mdr_first_integer_rankzeroevaluationdeterminanthsf_positive_start + S (0) = S ((S (0)) * ff_v_mce_mdr_first_integer_rankzeroevaluationdeterminanthsf_positive)) /\ exists ff_q_mce_mdr_first_integer_rankzeroevaluationdeterminanthsf_positive_start. ff_u_mce_mdr_first_integer_rankzeroevaluationdeterminanthsf_positive = ff_q_mce_mdr_first_integer_rankzeroevaluationdeterminanthsf_positive_start * S ((S (0)) * ff_v_mce_mdr_first_integer_rankzeroevaluationdeterminanthsf_positive) + (0))) /\ ((((exists ff_h_mce_mdr_first_integer_rankzeroevaluationdeterminanthsf_positive_terminal. ff_h_mce_mdr_first_integer_rankzeroevaluationdeterminanthsf_positive_terminal + S (mdr_p_first_integer_rankzeroevaluationdeterminanth) = S ((S ((S (mdr_q_first_integer_rankzeroevaluationdeterminanths)))) * ff_v_mce_mdr_first_integer_rankzeroevaluationdeterminanthsf_positive)) /\ exists ff_q_mce_mdr_first_integer_rankzeroevaluationdeterminanthsf_positive_terminal. ff_u_mce_mdr_first_integer_rankzeroevaluationdeterminanthsf_positive = ff_q_mce_mdr_first_integer_rankzeroevaluationdeterminanthsf_positive_terminal * S ((S ((S (mdr_q_first_integer_rankzeroevaluationdeterminanths)))) * ff_v_mce_mdr_first_integer_rankzeroevaluationdeterminanthsf_positive) + (mdr_p_first_integer_rankzeroevaluationdeterminanth))) /\ forall ff_i_mce_mdr_first_integer_rankzeroevaluationdeterminanthsf_positive. (exists ff_lt_mce_mdr_first_integer_rankzeroevaluationdeterminanthsf_positive_bound. ff_lt_mce_mdr_first_integer_rankzeroevaluationdeterminanthsf_positive_bound + S ff_i_mce_mdr_first_integer_rankzeroevaluationdeterminanthsf_positive = (S (mdr_q_first_integer_rankzeroevaluationdeterminanths))) -> exists ff_a_mce_mdr_first_integer_rankzeroevaluationdeterminanthsf_positive ff_r_mce_mdr_first_integer_rankzeroevaluationdeterminanthsf_positive ff_s_mce_mdr_first_integer_rankzeroevaluationdeterminanthsf_positive. ((((exists ff_h_mce_mdr_first_integer_rankzeroevaluationdeterminanthsf_positive_summand. ff_h_mce_mdr_first_integer_rankzeroevaluationdeterminanthsf_positive_summand + S (ff_a_mce_mdr_first_integer_rankzeroevaluationdeterminanthsf_positive) = S ((S (ff_i_mce_mdr_first_integer_rankzeroevaluationdeterminanthsf_positive)) * ff_uc_mce_fold_mdr_first_integer_rankzeroevaluationdeterminanthsf)) /\ exists ff_q_mce_mdr_first_integer_rankzeroevaluationdeterminanthsf_positive_summand. ff_ub_mce_fold_mdr_first_integer_rankzeroevaluationdeterminanthsf = ff_q_mce_mdr_first_integer_rankzeroevaluationdeterminanthsf_positive_summand * S ((S (ff_i_mce_mdr_first_integer_rankzeroevaluationdeterminanthsf_positive)) * ff_uc_mce_fold_mdr_first_integer_rankzeroevaluationdeterminanthsf) + (ff_a_mce_mdr_first_integer_rankzeroevaluationdeterminanthsf_positive))) /\ ((((exists ff_h_mce_mdr_first_integer_rankzeroevaluationdeterminanthsf_positive_partial. ff_h_mce_mdr_first_integer_rankzeroevaluationdeterminanthsf_positive_partial + S (ff_r_mce_mdr_first_integer_rankzeroevaluationdeterminanthsf_positive) = S ((S (ff_i_mce_mdr_first_integer_rankzeroevaluationdeterminanthsf_positive)) * ff_v_mce_mdr_first_integer_rankzeroevaluationdeterminanthsf_positive)) /\ exists ff_q_mce_mdr_first_integer_rankzeroevaluationdeterminanthsf_positive_partial. ff_u_mce_mdr_first_integer_rankzeroevaluationdeterminanthsf_positive = ff_q_mce_mdr_first_integer_rankzeroevaluationdeterminanthsf_positive_partial * S ((S (ff_i_mce_mdr_first_integer_rankzeroevaluationdeterminanthsf_positive)) * ff_v_mce_mdr_first_integer_rankzeroevaluationdeterminanthsf_positive) + (ff_r_mce_mdr_first_integer_rankzeroevaluationdeterminanthsf_positive))) /\ ((((exists ff_h_mce_mdr_first_integer_rankzeroevaluationdeterminanthsf_positive_successor. ff_h_mce_mdr_first_integer_rankzeroevaluationdeterminanthsf_positive_successor + S (ff_s_mce_mdr_first_integer_rankzeroevaluationdeterminanthsf_positive) = S ((S (S ff_i_mce_mdr_first_integer_rankzeroevaluationdeterminanthsf_positive)) * ff_v_mce_mdr_first_integer_rankzeroevaluationdeterminanthsf_positive)) /\ exists ff_q_mce_mdr_first_integer_rankzeroevaluationdeterminanthsf_positive_successor. ff_u_mce_mdr_first_integer_rankzeroevaluationdeterminanthsf_positive = ff_q_mce_mdr_first_integer_rankzeroevaluationdeterminanthsf_positive_successor * S ((S (S ff_i_mce_mdr_first_integer_rankzeroevaluationdeterminanthsf_positive)) * ff_v_mce_mdr_first_integer_rankzeroevaluationdeterminanthsf_positive) + (ff_s_mce_mdr_first_integer_rankzeroevaluationdeterminanthsf_positive))) /\ ff_s_mce_mdr_first_integer_rankzeroevaluationdeterminanthsf_positive = ff_r_mce_mdr_first_integer_rankzeroevaluationdeterminanthsf_positive + ff_a_mce_mdr_first_integer_rankzeroevaluationdeterminanthsf_positive)))))) /\ (exists ff_u_mce_mdr_first_integer_rankzeroevaluationdeterminanthsf_negative ff_v_mce_mdr_first_integer_rankzeroevaluationdeterminanthsf_negative. ((((exists ff_h_mce_mdr_first_integer_rankzeroevaluationdeterminanthsf_negative_start. ff_h_mce_mdr_first_integer_rankzeroevaluationdeterminanthsf_negative_start + S (0) = S ((S (0)) * ff_v_mce_mdr_first_integer_rankzeroevaluationdeterminanthsf_negative)) /\ exists ff_q_mce_mdr_first_integer_rankzeroevaluationdeterminanthsf_negative_start. ff_u_mce_mdr_first_integer_rankzeroevaluationdeterminanthsf_negative = ff_q_mce_mdr_first_integer_rankzeroevaluationdeterminanthsf_negative_start * S ((S (0)) * ff_v_mce_mdr_first_integer_rankzeroevaluationdeterminanthsf_negative) + (0))) /\ ((((exists ff_h_mce_mdr_first_integer_rankzeroevaluationdeterminanthsf_negative_terminal. ff_h_mce_mdr_first_integer_rankzeroevaluationdeterminanthsf_negative_terminal + S (mdr_n_first_integer_rankzeroevaluationdeterminanth) = S ((S ((S (mdr_q_first_integer_rankzeroevaluationdeterminanths)))) * ff_v_mce_mdr_first_integer_rankzeroevaluationdeterminanthsf_negative)) /\ exists ff_q_mce_mdr_first_integer_rankzeroevaluationdeterminanthsf_negative_terminal. ff_u_mce_mdr_first_integer_rankzeroevaluationdeterminanthsf_negative = ff_q_mce_mdr_first_integer_rankzeroevaluationdeterminanthsf_negative_terminal * S ((S ((S (mdr_q_first_integer_rankzeroevaluationdeterminanths)))) * ff_v_mce_mdr_first_integer_rankzeroevaluationdeterminanthsf_negative) + (mdr_n_first_integer_rankzeroevaluationdeterminanth))) /\ forall ff_i_mce_mdr_first_integer_rankzeroevaluationdeterminanthsf_negative. (exists ff_lt_mce_mdr_first_integer_rankzeroevaluationdeterminanthsf_negative_bound. ff_lt_mce_mdr_first_integer_rankzeroevaluationdeterminanthsf_negative_bound + S ff_i_mce_mdr_first_integer_rankzeroevaluationdeterminanthsf_negative = (S (mdr_q_first_integer_rankzeroevaluationdeterminanths))) -> exists ff_a_mce_mdr_first_integer_rankzeroevaluationdeterminanthsf_negative ff_r_mce_mdr_first_integer_rankzeroevaluationdeterminanthsf_negative ff_s_mce_mdr_first_integer_rankzeroevaluationdeterminanthsf_negative. ((((exists ff_h_mce_mdr_first_integer_rankzeroevaluationdeterminanthsf_negative_summand. ff_h_mce_mdr_first_integer_rankzeroevaluationdeterminanthsf_negative_summand + S (ff_a_mce_mdr_first_integer_rankzeroevaluationdeterminanthsf_negative) = S ((S (ff_i_mce_mdr_first_integer_rankzeroevaluationdeterminanthsf_negative)) * ff_vc_mce_fold_mdr_first_integer_rankzeroevaluationdeterminanthsf)) /\ exists ff_q_mce_mdr_first_integer_rankzeroevaluationdeterminanthsf_negative_summand. ff_vb_mce_fold_mdr_first_integer_rankzeroevaluationdeterminanthsf = ff_q_mce_mdr_first_integer_rankzeroevaluationdeterminanthsf_negative_summand * S ((S (ff_i_mce_mdr_first_integer_rankzeroevaluationdeterminanthsf_negative)) * ff_vc_mce_fold_mdr_first_integer_rankzeroevaluationdeterminanthsf) + (ff_a_mce_mdr_first_integer_rankzeroevaluationdeterminanthsf_negative))) /\ ((((exists ff_h_mce_mdr_first_integer_rankzeroevaluationdeterminanthsf_negative_partial. ff_h_mce_mdr_first_integer_rankzeroevaluationdeterminanthsf_negative_partial + S (ff_r_mce_mdr_first_integer_rankzeroevaluationdeterminanthsf_negative) = S ((S (ff_i_mce_mdr_first_integer_rankzeroevaluationdeterminanthsf_negative)) * ff_v_mce_mdr_first_integer_rankzeroevaluationdeterminanthsf_negative)) /\ exists ff_q_mce_mdr_first_integer_rankzeroevaluationdeterminanthsf_negative_partial. ff_u_mce_mdr_first_integer_rankzeroevaluationdeterminanthsf_negative = ff_q_mce_mdr_first_integer_rankzeroevaluationdeterminanthsf_negative_partial * S ((S (ff_i_mce_mdr_first_integer_rankzeroevaluationdeterminanthsf_negative)) * ff_v_mce_mdr_first_integer_rankzeroevaluationdeterminanthsf_negative) + (ff_r_mce_mdr_first_integer_rankzeroevaluationdeterminanthsf_negative))) /\ ((((exists ff_h_mce_mdr_first_integer_rankzeroevaluationdeterminanthsf_negative_successor. ff_h_mce_mdr_first_integer_rankzeroevaluationdeterminanthsf_negative_successor + S (ff_s_mce_mdr_first_integer_rankzeroevaluationdeterminanthsf_negative) = S ((S (S ff_i_mce_mdr_first_integer_rankzeroevaluationdeterminanthsf_negative)) * ff_v_mce_mdr_first_integer_rankzeroevaluationdeterminanthsf_negative)) /\ exists ff_q_mce_mdr_first_integer_rankzeroevaluationdeterminanthsf_negative_successor. ff_u_mce_mdr_first_integer_rankzeroevaluationdeterminanthsf_negative = ff_q_mce_mdr_first_integer_rankzeroevaluationdeterminanthsf_negative_successor * S ((S (S ff_i_mce_mdr_first_integer_rankzeroevaluationdeterminanthsf_negative)) * ff_v_mce_mdr_first_integer_rankzeroevaluationdeterminanthsf_negative) + (ff_s_mce_mdr_first_integer_rankzeroevaluationdeterminanthsf_negative))) /\ ff_s_mce_mdr_first_integer_rankzeroevaluationdeterminanthsf_negative = ff_r_mce_mdr_first_integer_rankzeroevaluationdeterminanthsf_negative + ff_a_mce_mdr_first_integer_rankzeroevaluationdeterminanthsf_negative))))))))))))))) /\ ((exists mdr_gap_first_integer_rankzeroevaluationdeterminanti. mdr_gap_first_integer_rankzeroevaluationdeterminanti + S (mdr_i_first_integer_rankzeroevaluationdeterminant) = (mdr_l_first_integer_rankzeroevaluationdeterminant)) /\ (exists mdr_z_first_integer_rankzeroevaluationdeterminantr. ((exists mdr_a_first_integer_rankzeroevaluationdeterminantrc mdr_b_first_integer_rankzeroevaluationdeterminantrc mdr_c_first_integer_rankzeroevaluationdeterminantrc mdr_e_first_integer_rankzeroevaluationdeterminantrc mdr_f_first_integer_rankzeroevaluationdeterminantrc. ((mdr_a_first_integer_rankzeroevaluationdeterminantrc = ((mdr_q_first_integer_rank) + (mdr_ub_first_integer_rankzeroevaluation)) * S ((mdr_q_first_integer_rank) + (mdr_ub_first_integer_rankzeroevaluation)) + ((mdr_ub_first_integer_rankzeroevaluation) + (mdr_ub_first_integer_rankzeroevaluation))) /\ ((mdr_b_first_integer_rankzeroevaluationdeterminantrc = ((mdr_uc_first_integer_rankzeroevaluation) + (mdr_vb_first_integer_rankzeroevaluation)) * S ((mdr_uc_first_integer_rankzeroevaluation) + (mdr_vb_first_integer_rankzeroevaluation)) + ((mdr_vb_first_integer_rankzeroevaluation) + (mdr_vb_first_integer_rankzeroevaluation))) /\ ((mdr_c_first_integer_rankzeroevaluationdeterminantrc = ((mdr_a_first_integer_rankzeroevaluationdeterminantrc) + (mdr_b_first_integer_rankzeroevaluationdeterminantrc)) * S ((mdr_a_first_integer_rankzeroevaluationdeterminantrc) + (mdr_b_first_integer_rankzeroevaluationdeterminantrc)) + ((mdr_b_first_integer_rankzeroevaluationdeterminantrc) + (mdr_b_first_integer_rankzeroevaluationdeterminantrc))) /\ ((mdr_e_first_integer_rankzeroevaluationdeterminantrc = ((mdr_p_first_integer_rankzero) + (mdr_n_first_integer_rankzero)) * S ((mdr_p_first_integer_rankzero) + (mdr_n_first_integer_rankzero)) + ((mdr_n_first_integer_rankzero) + (mdr_n_first_integer_rankzero))) /\ ((mdr_f_first_integer_rankzeroevaluationdeterminantrc = ((mdr_vc_first_integer_rankzeroevaluation) + (mdr_e_first_integer_rankzeroevaluationdeterminantrc)) * S ((mdr_vc_first_integer_rankzeroevaluation) + (mdr_e_first_integer_rankzeroevaluationdeterminantrc)) + ((mdr_e_first_integer_rankzeroevaluationdeterminantrc) + (mdr_e_first_integer_rankzeroevaluationdeterminantrc))) /\ ((mdr_z_first_integer_rankzeroevaluationdeterminantr) = ((mdr_c_first_integer_rankzeroevaluationdeterminantrc) + (mdr_f_first_integer_rankzeroevaluationdeterminantrc)) * S ((mdr_c_first_integer_rankzeroevaluationdeterminantrc) + (mdr_f_first_integer_rankzeroevaluationdeterminantrc)) + ((mdr_f_first_integer_rankzeroevaluationdeterminantrc) + (mdr_f_first_integer_rankzeroevaluationdeterminantrc))))))))) /\ (((exists ff_h_mdr_first_integer_rankzeroevaluationdeterminantrb. ff_h_mdr_first_integer_rankzeroevaluationdeterminantrb + S (mdr_z_first_integer_rankzeroevaluationdeterminantr) = S ((S (mdr_i_first_integer_rankzeroevaluationdeterminant)) * mdr_c_first_integer_rankzeroevaluationdeterminant)) /\ exists ff_q_mdr_first_integer_rankzeroevaluationdeterminantrb. mdr_b_first_integer_rankzeroevaluationdeterminant = ff_q_mdr_first_integer_rankzeroevaluationdeterminantrb * S ((S (mdr_i_first_integer_rankzeroevaluationdeterminant)) * mdr_c_first_integer_rankzeroevaluationdeterminant) + (mdr_z_first_integer_rankzeroevaluationdeterminantr)))))))))) -> mdr_p_first_integer_rankzero = mdr_n_first_integer_rankzero)))))) -> (((exists mdr_gap_second_integer_rankrows_bound. mdr_gap_second_integer_rankrows_bound + (rank) = (r)) /\ ((exists mdr_gap_second_integer_rankcolumns_bound. mdr_gap_second_integer_rankcolumns_bound + (rank) = (w)) /\ ((exists mdr_rb_second_integer_rankwitness mdr_rc_second_integer_rankwitness mdr_cb_second_integer_rankwitness mdr_cc_second_integer_rankwitness. (((((forall fom_index_mrf_second_integer_rankwitnessminorrowsbound. (exists fom_gap_mrf_second_integer_rankwitnessminorrowsbound_index_bound. fom_gap_mrf_second_integer_rankwitnessminorrowsbound_index_bound + S (fom_index_mrf_second_integer_rankwitnessminorrowsbound) = rank) -> exists fom_value_mrf_second_integer_rankwitnessminorrowsbound. ((((exists fom_beta_height_mrf_second_integer_rankwitnessminorrowsbound_entry. fom_beta_height_mrf_second_integer_rankwitnessminorrowsbound_entry + S (fom_value_mrf_second_integer_rankwitnessminorrowsbound) = S ((S (fom_index_mrf_second_integer_rankwitnessminorrowsbound)) * mdr_rc_second_integer_rankwitness)) /\ exists fom_beta_quotient_mrf_second_integer_rankwitnessminorrowsbound_entry. mdr_rb_second_integer_rankwitness = fom_beta_quotient_mrf_second_integer_rankwitnessminorrowsbound_entry * S ((S (fom_index_mrf_second_integer_rankwitnessminorrowsbound)) * mdr_rc_second_integer_rankwitness) + (fom_value_mrf_second_integer_rankwitnessminorrowsbound))) /\ (exists fom_gap_mrf_second_integer_rankwitnessminorrowsbound_value_bound. fom_gap_mrf_second_integer_rankwitnessminorrowsbound_value_bound + S (fom_value_mrf_second_integer_rankwitnessminorrowsbound) = r))) /\ (forall mdr_i_second_integer_rankwitnessminorrowsdistinct mdr_j_second_integer_rankwitnessminorrowsdistinct mdr_a_second_integer_rankwitnessminorrowsdistinct. (exists mdr_gap_second_integer_rankwitnessminorrowsdistincti. mdr_gap_second_integer_rankwitnessminorrowsdistincti + S (mdr_i_second_integer_rankwitnessminorrowsdistinct) = (rank)) -> (exists mdr_gap_second_integer_rankwitnessminorrowsdistinctj. mdr_gap_second_integer_rankwitnessminorrowsdistinctj + S (mdr_j_second_integer_rankwitnessminorrowsdistinct) = (rank)) -> (((exists ff_h_mdr_second_integer_rankwitnessminorrowsdistinctfirst. ff_h_mdr_second_integer_rankwitnessminorrowsdistinctfirst + S (mdr_a_second_integer_rankwitnessminorrowsdistinct) = S ((S (mdr_i_second_integer_rankwitnessminorrowsdistinct)) * mdr_rc_second_integer_rankwitness)) /\ exists ff_q_mdr_second_integer_rankwitnessminorrowsdistinctfirst. mdr_rb_second_integer_rankwitness = ff_q_mdr_second_integer_rankwitnessminorrowsdistinctfirst * S ((S (mdr_i_second_integer_rankwitnessminorrowsdistinct)) * mdr_rc_second_integer_rankwitness) + (mdr_a_second_integer_rankwitnessminorrowsdistinct))) -> (((exists ff_h_mdr_second_integer_rankwitnessminorrowsdistinctsecond. ff_h_mdr_second_integer_rankwitnessminorrowsdistinctsecond + S (mdr_a_second_integer_rankwitnessminorrowsdistinct) = S ((S (mdr_j_second_integer_rankwitnessminorrowsdistinct)) * mdr_rc_second_integer_rankwitness)) /\ exists ff_q_mdr_second_integer_rankwitnessminorrowsdistinctsecond. mdr_rb_second_integer_rankwitness = ff_q_mdr_second_integer_rankwitnessminorrowsdistinctsecond * S ((S (mdr_j_second_integer_rankwitnessminorrowsdistinct)) * mdr_rc_second_integer_rankwitness) + (mdr_a_second_integer_rankwitnessminorrowsdistinct))) -> mdr_i_second_integer_rankwitnessminorrowsdistinct = mdr_j_second_integer_rankwitnessminorrowsdistinct))) /\ ((((forall fom_index_mrf_second_integer_rankwitnessminorcolumnsbound. (exists fom_gap_mrf_second_integer_rankwitnessminorcolumnsbound_index_bound. fom_gap_mrf_second_integer_rankwitnessminorcolumnsbound_index_bound + S (fom_index_mrf_second_integer_rankwitnessminorcolumnsbound) = rank) -> exists fom_value_mrf_second_integer_rankwitnessminorcolumnsbound. ((((exists fom_beta_height_mrf_second_integer_rankwitnessminorcolumnsbound_entry. fom_beta_height_mrf_second_integer_rankwitnessminorcolumnsbound_entry + S (fom_value_mrf_second_integer_rankwitnessminorcolumnsbound) = S ((S (fom_index_mrf_second_integer_rankwitnessminorcolumnsbound)) * mdr_cc_second_integer_rankwitness)) /\ exists fom_beta_quotient_mrf_second_integer_rankwitnessminorcolumnsbound_entry. mdr_cb_second_integer_rankwitness = fom_beta_quotient_mrf_second_integer_rankwitnessminorcolumnsbound_entry * S ((S (fom_index_mrf_second_integer_rankwitnessminorcolumnsbound)) * mdr_cc_second_integer_rankwitness) + (fom_value_mrf_second_integer_rankwitnessminorcolumnsbound))) /\ (exists fom_gap_mrf_second_integer_rankwitnessminorcolumnsbound_value_bound. fom_gap_mrf_second_integer_rankwitnessminorcolumnsbound_value_bound + S (fom_value_mrf_second_integer_rankwitnessminorcolumnsbound) = w))) /\ (forall mdr_i_second_integer_rankwitnessminorcolumnsdistinct mdr_j_second_integer_rankwitnessminorcolumnsdistinct mdr_a_second_integer_rankwitnessminorcolumnsdistinct. (exists mdr_gap_second_integer_rankwitnessminorcolumnsdistincti. mdr_gap_second_integer_rankwitnessminorcolumnsdistincti + S (mdr_i_second_integer_rankwitnessminorcolumnsdistinct) = (rank)) -> (exists mdr_gap_second_integer_rankwitnessminorcolumnsdistinctj. mdr_gap_second_integer_rankwitnessminorcolumnsdistinctj + S (mdr_j_second_integer_rankwitnessminorcolumnsdistinct) = (rank)) -> (((exists ff_h_mdr_second_integer_rankwitnessminorcolumnsdistinctfirst. ff_h_mdr_second_integer_rankwitnessminorcolumnsdistinctfirst + S (mdr_a_second_integer_rankwitnessminorcolumnsdistinct) = S ((S (mdr_i_second_integer_rankwitnessminorcolumnsdistinct)) * mdr_cc_second_integer_rankwitness)) /\ exists ff_q_mdr_second_integer_rankwitnessminorcolumnsdistinctfirst. mdr_cb_second_integer_rankwitness = ff_q_mdr_second_integer_rankwitnessminorcolumnsdistinctfirst * S ((S (mdr_i_second_integer_rankwitnessminorcolumnsdistinct)) * mdr_cc_second_integer_rankwitness) + (mdr_a_second_integer_rankwitnessminorcolumnsdistinct))) -> (((exists ff_h_mdr_second_integer_rankwitnessminorcolumnsdistinctsecond. ff_h_mdr_second_integer_rankwitnessminorcolumnsdistinctsecond + S (mdr_a_second_integer_rankwitnessminorcolumnsdistinct) = S ((S (mdr_j_second_integer_rankwitnessminorcolumnsdistinct)) * mdr_cc_second_integer_rankwitness)) /\ exists ff_q_mdr_second_integer_rankwitnessminorcolumnsdistinctsecond. mdr_cb_second_integer_rankwitness = ff_q_mdr_second_integer_rankwitnessminorcolumnsdistinctsecond * S ((S (mdr_j_second_integer_rankwitnessminorcolumnsdistinct)) * mdr_cc_second_integer_rankwitness) + (mdr_a_second_integer_rankwitnessminorcolumnsdistinct))) -> mdr_i_second_integer_rankwitnessminorcolumnsdistinct = mdr_j_second_integer_rankwitnessminorcolumnsdistinct))) /\ (exists mdr_p_second_integer_rankwitnessminornonzero mdr_n_second_integer_rankwitnessminornonzero. ((exists mdr_ub_second_integer_rankwitnessminornonzeroevaluation mdr_uc_second_integer_rankwitnessminornonzeroevaluation mdr_vb_second_integer_rankwitnessminornonzeroevaluation mdr_vc_second_integer_rankwitnessminornonzeroevaluation. ((((forall mdr_i_second_integer_rankwitnessminornonzeroevaluationmatrixpositive. (exists mdr_gap_second_integer_rankwitnessminornonzeroevaluationmatrixpositivebound. mdr_gap_second_integer_rankwitnessminornonzeroevaluationmatrixpositivebound + S (mdr_i_second_integer_rankwitnessminornonzeroevaluationmatrixpositive) = ((rank) * (rank))) -> exists mdr_a_second_integer_rankwitnessminornonzeroevaluationmatrixpositive. (((exists mdr_r_second_integer_rankwitnessminornonzeroevaluationmatrixpositivepoint mdr_s_second_integer_rankwitnessminornonzeroevaluationmatrixpositivepoint mdr_u_second_integer_rankwitnessminornonzeroevaluationmatrixpositivepoint mdr_v_second_integer_rankwitnessminornonzeroevaluationmatrixpositivepoint. ((mdr_i_second_integer_rankwitnessminornonzeroevaluationmatrixpositive = (rank) * mdr_r_second_integer_rankwitnessminornonzeroevaluationmatrixpositivepoint + mdr_s_second_integer_rankwitnessminornonzeroevaluationmatrixpositivepoint) /\ ((exists mdr_gap_second_integer_rankwitnessminornonzeroevaluationmatrixpositivepointcolumn. mdr_gap_second_integer_rankwitnessminornonzeroevaluationmatrixpositivepointcolumn + S (mdr_s_second_integer_rankwitnessminornonzeroevaluationmatrixpositivepoint) = (rank)) /\ ((((exists ff_h_mdr_second_integer_rankwitnessminornonzeroevaluationmatrixpositivepointrow_index. ff_h_mdr_second_integer_rankwitnessminornonzeroevaluationmatrixpositivepointrow_index + S (mdr_u_second_integer_rankwitnessminornonzeroevaluationmatrixpositivepoint) = S ((S (mdr_r_second_integer_rankwitnessminornonzeroevaluationmatrixpositivepoint)) * mdr_rc_second_integer_rankwitness)) /\ exists ff_q_mdr_second_integer_rankwitnessminornonzeroevaluationmatrixpositivepointrow_index. mdr_rb_second_integer_rankwitness = ff_q_mdr_second_integer_rankwitnessminornonzeroevaluationmatrixpositivepointrow_index * S ((S (mdr_r_second_integer_rankwitnessminornonzeroevaluationmatrixpositivepoint)) * mdr_rc_second_integer_rankwitness) + (mdr_u_second_integer_rankwitnessminornonzeroevaluationmatrixpositivepoint))) /\ ((((exists ff_h_mdr_second_integer_rankwitnessminornonzeroevaluationmatrixpositivepointcolumn_index. ff_h_mdr_second_integer_rankwitnessminornonzeroevaluationmatrixpositivepointcolumn_index + S (mdr_v_second_integer_rankwitnessminornonzeroevaluationmatrixpositivepoint) = S ((S (mdr_s_second_integer_rankwitnessminornonzeroevaluationmatrixpositivepoint)) * mdr_cc_second_integer_rankwitness)) /\ exists ff_q_mdr_second_integer_rankwitnessminornonzeroevaluationmatrixpositivepointcolumn_index. mdr_cb_second_integer_rankwitness = ff_q_mdr_second_integer_rankwitnessminornonzeroevaluationmatrixpositivepointcolumn_index * S ((S (mdr_s_second_integer_rankwitnessminornonzeroevaluationmatrixpositivepoint)) * mdr_cc_second_integer_rankwitness) + (mdr_v_second_integer_rankwitnessminornonzeroevaluationmatrixpositivepoint))) /\ (((exists ff_h_mdr_second_integer_rankwitnessminornonzeroevaluationmatrixpositivepointsource. ff_h_mdr_second_integer_rankwitnessminornonzeroevaluationmatrixpositivepointsource + S (mdr_a_second_integer_rankwitnessminornonzeroevaluationmatrixpositive) = S ((S ((mdr_u_second_integer_rankwitnessminornonzeroevaluationmatrixpositivepoint) * (w) + (mdr_v_second_integer_rankwitnessminornonzeroevaluationmatrixpositivepoint))) * ec)) /\ exists ff_q_mdr_second_integer_rankwitnessminornonzeroevaluationmatrixpositivepointsource. eb = ff_q_mdr_second_integer_rankwitnessminornonzeroevaluationmatrixpositivepointsource * S ((S ((mdr_u_second_integer_rankwitnessminornonzeroevaluationmatrixpositivepoint) * (w) + (mdr_v_second_integer_rankwitnessminornonzeroevaluationmatrixpositivepoint))) * ec) + (mdr_a_second_integer_rankwitnessminornonzeroevaluationmatrixpositive)))))))) /\ (((exists ff_h_mdr_second_integer_rankwitnessminornonzeroevaluationmatrixpositiveoutput. ff_h_mdr_second_integer_rankwitnessminornonzeroevaluationmatrixpositiveoutput + S (mdr_a_second_integer_rankwitnessminornonzeroevaluationmatrixpositive) = S ((S (mdr_i_second_integer_rankwitnessminornonzeroevaluationmatrixpositive)) * mdr_uc_second_integer_rankwitnessminornonzeroevaluation)) /\ exists ff_q_mdr_second_integer_rankwitnessminornonzeroevaluationmatrixpositiveoutput. mdr_ub_second_integer_rankwitnessminornonzeroevaluation = ff_q_mdr_second_integer_rankwitnessminornonzeroevaluationmatrixpositiveoutput * S ((S (mdr_i_second_integer_rankwitnessminornonzeroevaluationmatrixpositive)) * mdr_uc_second_integer_rankwitnessminornonzeroevaluation) + (mdr_a_second_integer_rankwitnessminornonzeroevaluationmatrixpositive)))))) /\ (forall mdr_i_second_integer_rankwitnessminornonzeroevaluationmatrixnegative. (exists mdr_gap_second_integer_rankwitnessminornonzeroevaluationmatrixnegativebound. mdr_gap_second_integer_rankwitnessminornonzeroevaluationmatrixnegativebound + S (mdr_i_second_integer_rankwitnessminornonzeroevaluationmatrixnegative) = ((rank) * (rank))) -> exists mdr_a_second_integer_rankwitnessminornonzeroevaluationmatrixnegative. (((exists mdr_r_second_integer_rankwitnessminornonzeroevaluationmatrixnegativepoint mdr_s_second_integer_rankwitnessminornonzeroevaluationmatrixnegativepoint mdr_u_second_integer_rankwitnessminornonzeroevaluationmatrixnegativepoint mdr_v_second_integer_rankwitnessminornonzeroevaluationmatrixnegativepoint. ((mdr_i_second_integer_rankwitnessminornonzeroevaluationmatrixnegative = (rank) * mdr_r_second_integer_rankwitnessminornonzeroevaluationmatrixnegativepoint + mdr_s_second_integer_rankwitnessminornonzeroevaluationmatrixnegativepoint) /\ ((exists mdr_gap_second_integer_rankwitnessminornonzeroevaluationmatrixnegativepointcolumn. mdr_gap_second_integer_rankwitnessminornonzeroevaluationmatrixnegativepointcolumn + S (mdr_s_second_integer_rankwitnessminornonzeroevaluationmatrixnegativepoint) = (rank)) /\ ((((exists ff_h_mdr_second_integer_rankwitnessminornonzeroevaluationmatrixnegativepointrow_index. ff_h_mdr_second_integer_rankwitnessminornonzeroevaluationmatrixnegativepointrow_index + S (mdr_u_second_integer_rankwitnessminornonzeroevaluationmatrixnegativepoint) = S ((S (mdr_r_second_integer_rankwitnessminornonzeroevaluationmatrixnegativepoint)) * mdr_rc_second_integer_rankwitness)) /\ exists ff_q_mdr_second_integer_rankwitnessminornonzeroevaluationmatrixnegativepointrow_index. mdr_rb_second_integer_rankwitness = ff_q_mdr_second_integer_rankwitnessminornonzeroevaluationmatrixnegativepointrow_index * S ((S (mdr_r_second_integer_rankwitnessminornonzeroevaluationmatrixnegativepoint)) * mdr_rc_second_integer_rankwitness) + (mdr_u_second_integer_rankwitnessminornonzeroevaluationmatrixnegativepoint))) /\ ((((exists ff_h_mdr_second_integer_rankwitnessminornonzeroevaluationmatrixnegativepointcolumn_index. ff_h_mdr_second_integer_rankwitnessminornonzeroevaluationmatrixnegativepointcolumn_index + S (mdr_v_second_integer_rankwitnessminornonzeroevaluationmatrixnegativepoint) = S ((S (mdr_s_second_integer_rankwitnessminornonzeroevaluationmatrixnegativepoint)) * mdr_cc_second_integer_rankwitness)) /\ exists ff_q_mdr_second_integer_rankwitnessminornonzeroevaluationmatrixnegativepointcolumn_index. mdr_cb_second_integer_rankwitness = ff_q_mdr_second_integer_rankwitnessminornonzeroevaluationmatrixnegativepointcolumn_index * S ((S (mdr_s_second_integer_rankwitnessminornonzeroevaluationmatrixnegativepoint)) * mdr_cc_second_integer_rankwitness) + (mdr_v_second_integer_rankwitnessminornonzeroevaluationmatrixnegativepoint))) /\ (((exists ff_h_mdr_second_integer_rankwitnessminornonzeroevaluationmatrixnegativepointsource. ff_h_mdr_second_integer_rankwitnessminornonzeroevaluationmatrixnegativepointsource + S (mdr_a_second_integer_rankwitnessminornonzeroevaluationmatrixnegative) = S ((S ((mdr_u_second_integer_rankwitnessminornonzeroevaluationmatrixnegativepoint) * (w) + (mdr_v_second_integer_rankwitnessminornonzeroevaluationmatrixnegativepoint))) * fc)) /\ exists ff_q_mdr_second_integer_rankwitnessminornonzeroevaluationmatrixnegativepointsource. fb = ff_q_mdr_second_integer_rankwitnessminornonzeroevaluationmatrixnegativepointsource * S ((S ((mdr_u_second_integer_rankwitnessminornonzeroevaluationmatrixnegativepoint) * (w) + (mdr_v_second_integer_rankwitnessminornonzeroevaluationmatrixnegativepoint))) * fc) + (mdr_a_second_integer_rankwitnessminornonzeroevaluationmatrixnegative)))))))) /\ (((exists ff_h_mdr_second_integer_rankwitnessminornonzeroevaluationmatrixnegativeoutput. ff_h_mdr_second_integer_rankwitnessminornonzeroevaluationmatrixnegativeoutput + S (mdr_a_second_integer_rankwitnessminornonzeroevaluationmatrixnegative) = S ((S (mdr_i_second_integer_rankwitnessminornonzeroevaluationmatrixnegative)) * mdr_vc_second_integer_rankwitnessminornonzeroevaluation)) /\ exists ff_q_mdr_second_integer_rankwitnessminornonzeroevaluationmatrixnegativeoutput. mdr_vb_second_integer_rankwitnessminornonzeroevaluation = ff_q_mdr_second_integer_rankwitnessminornonzeroevaluationmatrixnegativeoutput * S ((S (mdr_i_second_integer_rankwitnessminornonzeroevaluationmatrixnegative)) * mdr_vc_second_integer_rankwitnessminornonzeroevaluation) + (mdr_a_second_integer_rankwitnessminornonzeroevaluationmatrixnegative)))))))) /\ (exists mdr_b_second_integer_rankwitnessminornonzeroevaluationdeterminant mdr_c_second_integer_rankwitnessminornonzeroevaluationdeterminant mdr_l_second_integer_rankwitnessminornonzeroevaluationdeterminant mdr_i_second_integer_rankwitnessminornonzeroevaluationdeterminant. ((forall mdr_i_second_integer_rankwitnessminornonzeroevaluationdeterminanth. (exists mdr_gap_second_integer_rankwitnessminornonzeroevaluationdeterminanthi. mdr_gap_second_integer_rankwitnessminornonzeroevaluationdeterminanthi + S (mdr_i_second_integer_rankwitnessminornonzeroevaluationdeterminanth) = (mdr_l_second_integer_rankwitnessminornonzeroevaluationdeterminant)) -> exists mdr_d_second_integer_rankwitnessminornonzeroevaluationdeterminanth mdr_pb_second_integer_rankwitnessminornonzeroevaluationdeterminanth mdr_pc_second_integer_rankwitnessminornonzeroevaluationdeterminanth mdr_nb_second_integer_rankwitnessminornonzeroevaluationdeterminanth mdr_nc_second_integer_rankwitnessminornonzeroevaluationdeterminanth mdr_p_second_integer_rankwitnessminornonzeroevaluationdeterminanth mdr_n_second_integer_rankwitnessminornonzeroevaluationdeterminanth. ((exists mdr_z_second_integer_rankwitnessminornonzeroevaluationdeterminanthr. ((exists mdr_a_second_integer_rankwitnessminornonzeroevaluationdeterminanthrc mdr_b_second_integer_rankwitnessminornonzeroevaluationdeterminanthrc mdr_c_second_integer_rankwitnessminornonzeroevaluationdeterminanthrc mdr_e_second_integer_rankwitnessminornonzeroevaluationdeterminanthrc mdr_f_second_integer_rankwitnessminornonzeroevaluationdeterminanthrc. ((mdr_a_second_integer_rankwitnessminornonzeroevaluationdeterminanthrc = ((mdr_d_second_integer_rankwitnessminornonzeroevaluationdeterminanth) + (mdr_pb_second_integer_rankwitnessminornonzeroevaluationdeterminanth)) * S ((mdr_d_second_integer_rankwitnessminornonzeroevaluationdeterminanth) + (mdr_pb_second_integer_rankwitnessminornonzeroevaluationdeterminanth)) + ((mdr_pb_second_integer_rankwitnessminornonzeroevaluationdeterminanth) + (mdr_pb_second_integer_rankwitnessminornonzeroevaluationdeterminanth))) /\ ((mdr_b_second_integer_rankwitnessminornonzeroevaluationdeterminanthrc = ((mdr_pc_second_integer_rankwitnessminornonzeroevaluationdeterminanth) + (mdr_nb_second_integer_rankwitnessminornonzeroevaluationdeterminanth)) * S ((mdr_pc_second_integer_rankwitnessminornonzeroevaluationdeterminanth) + (mdr_nb_second_integer_rankwitnessminornonzeroevaluationdeterminanth)) + ((mdr_nb_second_integer_rankwitnessminornonzeroevaluationdeterminanth) + (mdr_nb_second_integer_rankwitnessminornonzeroevaluationdeterminanth))) /\ ((mdr_c_second_integer_rankwitnessminornonzeroevaluationdeterminanthrc = ((mdr_a_second_integer_rankwitnessminornonzeroevaluationdeterminanthrc) + (mdr_b_second_integer_rankwitnessminornonzeroevaluationdeterminanthrc)) * S ((mdr_a_second_integer_rankwitnessminornonzeroevaluationdeterminanthrc) + (mdr_b_second_integer_rankwitnessminornonzeroevaluationdeterminanthrc)) + ((mdr_b_second_integer_rankwitnessminornonzeroevaluationdeterminanthrc) + (mdr_b_second_integer_rankwitnessminornonzeroevaluationdeterminanthrc))) /\ ((mdr_e_second_integer_rankwitnessminornonzeroevaluationdeterminanthrc = ((mdr_p_second_integer_rankwitnessminornonzeroevaluationdeterminanth) + (mdr_n_second_integer_rankwitnessminornonzeroevaluationdeterminanth)) * S ((mdr_p_second_integer_rankwitnessminornonzeroevaluationdeterminanth) + (mdr_n_second_integer_rankwitnessminornonzeroevaluationdeterminanth)) + ((mdr_n_second_integer_rankwitnessminornonzeroevaluationdeterminanth) + (mdr_n_second_integer_rankwitnessminornonzeroevaluationdeterminanth))) /\ ((mdr_f_second_integer_rankwitnessminornonzeroevaluationdeterminanthrc = ((mdr_nc_second_integer_rankwitnessminornonzeroevaluationdeterminanth) + (mdr_e_second_integer_rankwitnessminornonzeroevaluationdeterminanthrc)) * S ((mdr_nc_second_integer_rankwitnessminornonzeroevaluationdeterminanth) + (mdr_e_second_integer_rankwitnessminornonzeroevaluationdeterminanthrc)) + ((mdr_e_second_integer_rankwitnessminornonzeroevaluationdeterminanthrc) + (mdr_e_second_integer_rankwitnessminornonzeroevaluationdeterminanthrc))) /\ ((mdr_z_second_integer_rankwitnessminornonzeroevaluationdeterminanthr) = ((mdr_c_second_integer_rankwitnessminornonzeroevaluationdeterminanthrc) + (mdr_f_second_integer_rankwitnessminornonzeroevaluationdeterminanthrc)) * S ((mdr_c_second_integer_rankwitnessminornonzeroevaluationdeterminanthrc) + (mdr_f_second_integer_rankwitnessminornonzeroevaluationdeterminanthrc)) + ((mdr_f_second_integer_rankwitnessminornonzeroevaluationdeterminanthrc) + (mdr_f_second_integer_rankwitnessminornonzeroevaluationdeterminanthrc))))))))) /\ (((exists ff_h_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthrb. ff_h_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthrb + S (mdr_z_second_integer_rankwitnessminornonzeroevaluationdeterminanthr) = S ((S (mdr_i_second_integer_rankwitnessminornonzeroevaluationdeterminanth)) * mdr_c_second_integer_rankwitnessminornonzeroevaluationdeterminant)) /\ exists ff_q_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthrb. mdr_b_second_integer_rankwitnessminornonzeroevaluationdeterminant = ff_q_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthrb * S ((S (mdr_i_second_integer_rankwitnessminornonzeroevaluationdeterminanth)) * mdr_c_second_integer_rankwitnessminornonzeroevaluationdeterminant) + (mdr_z_second_integer_rankwitnessminornonzeroevaluationdeterminanthr))))) /\ (((((mdr_d_second_integer_rankwitnessminornonzeroevaluationdeterminanth) = 0) /\ (((mdr_p_second_integer_rankwitnessminornonzeroevaluationdeterminanth) = 1) /\ ((mdr_n_second_integer_rankwitnessminornonzeroevaluationdeterminanth) = 0))) \/ exists mdr_q_second_integer_rankwitnessminornonzeroevaluationdeterminanths mdr_eb_second_integer_rankwitnessminornonzeroevaluationdeterminanths mdr_ec_second_integer_rankwitnessminornonzeroevaluationdeterminanths mdr_fb_second_integer_rankwitnessminornonzeroevaluationdeterminanths mdr_fc_second_integer_rankwitnessminornonzeroevaluationdeterminanths. (((mdr_d_second_integer_rankwitnessminornonzeroevaluationdeterminanth) = S (mdr_q_second_integer_rankwitnessminornonzeroevaluationdeterminanths)) /\ ((forall mdr_j_second_integer_rankwitnessminornonzeroevaluationdeterminanthsc. (exists mdr_gap_second_integer_rankwitnessminornonzeroevaluationdeterminanthscj. mdr_gap_second_integer_rankwitnessminornonzeroevaluationdeterminanthscj + S (mdr_j_second_integer_rankwitnessminornonzeroevaluationdeterminanthsc) = (S (mdr_q_second_integer_rankwitnessminornonzeroevaluationdeterminanths))) -> exists mdr_i_second_integer_rankwitnessminornonzeroevaluationdeterminanthsc mdr_up_second_integer_rankwitnessminornonzeroevaluationdeterminanthsc mdr_us_second_integer_rankwitnessminornonzeroevaluationdeterminanthsc mdr_un_second_integer_rankwitnessminornonzeroevaluationdeterminanthsc mdr_ut_second_integer_rankwitnessminornonzeroevaluationdeterminanthsc mdr_p_second_integer_rankwitnessminornonzeroevaluationdeterminanthsc mdr_n_second_integer_rankwitnessminornonzeroevaluationdeterminanthsc. ((exists mdr_gap_second_integer_rankwitnessminornonzeroevaluationdeterminanthsci. mdr_gap_second_integer_rankwitnessminornonzeroevaluationdeterminanthsci + S (mdr_i_second_integer_rankwitnessminornonzeroevaluationdeterminanthsc) = (mdr_i_second_integer_rankwitnessminornonzeroevaluationdeterminanth)) /\ ((exists mdr_z_second_integer_rankwitnessminornonzeroevaluationdeterminanthscr. ((exists mdr_a_second_integer_rankwitnessminornonzeroevaluationdeterminanthscrc mdr_b_second_integer_rankwitnessminornonzeroevaluationdeterminanthscrc mdr_c_second_integer_rankwitnessminornonzeroevaluationdeterminanthscrc mdr_e_second_integer_rankwitnessminornonzeroevaluationdeterminanthscrc mdr_f_second_integer_rankwitnessminornonzeroevaluationdeterminanthscrc. ((mdr_a_second_integer_rankwitnessminornonzeroevaluationdeterminanthscrc = ((mdr_q_second_integer_rankwitnessminornonzeroevaluationdeterminanths) + (mdr_up_second_integer_rankwitnessminornonzeroevaluationdeterminanthsc)) * S ((mdr_q_second_integer_rankwitnessminornonzeroevaluationdeterminanths) + (mdr_up_second_integer_rankwitnessminornonzeroevaluationdeterminanthsc)) + ((mdr_up_second_integer_rankwitnessminornonzeroevaluationdeterminanthsc) + (mdr_up_second_integer_rankwitnessminornonzeroevaluationdeterminanthsc))) /\ ((mdr_b_second_integer_rankwitnessminornonzeroevaluationdeterminanthscrc = ((mdr_us_second_integer_rankwitnessminornonzeroevaluationdeterminanthsc) + (mdr_un_second_integer_rankwitnessminornonzeroevaluationdeterminanthsc)) * S ((mdr_us_second_integer_rankwitnessminornonzeroevaluationdeterminanthsc) + (mdr_un_second_integer_rankwitnessminornonzeroevaluationdeterminanthsc)) + ((mdr_un_second_integer_rankwitnessminornonzeroevaluationdeterminanthsc) + (mdr_un_second_integer_rankwitnessminornonzeroevaluationdeterminanthsc))) /\ ((mdr_c_second_integer_rankwitnessminornonzeroevaluationdeterminanthscrc = ((mdr_a_second_integer_rankwitnessminornonzeroevaluationdeterminanthscrc) + (mdr_b_second_integer_rankwitnessminornonzeroevaluationdeterminanthscrc)) * S ((mdr_a_second_integer_rankwitnessminornonzeroevaluationdeterminanthscrc) + (mdr_b_second_integer_rankwitnessminornonzeroevaluationdeterminanthscrc)) + ((mdr_b_second_integer_rankwitnessminornonzeroevaluationdeterminanthscrc) + (mdr_b_second_integer_rankwitnessminornonzeroevaluationdeterminanthscrc))) /\ ((mdr_e_second_integer_rankwitnessminornonzeroevaluationdeterminanthscrc = ((mdr_p_second_integer_rankwitnessminornonzeroevaluationdeterminanthsc) + (mdr_n_second_integer_rankwitnessminornonzeroevaluationdeterminanthsc)) * S ((mdr_p_second_integer_rankwitnessminornonzeroevaluationdeterminanthsc) + (mdr_n_second_integer_rankwitnessminornonzeroevaluationdeterminanthsc)) + ((mdr_n_second_integer_rankwitnessminornonzeroevaluationdeterminanthsc) + (mdr_n_second_integer_rankwitnessminornonzeroevaluationdeterminanthsc))) /\ ((mdr_f_second_integer_rankwitnessminornonzeroevaluationdeterminanthscrc = ((mdr_ut_second_integer_rankwitnessminornonzeroevaluationdeterminanthsc) + (mdr_e_second_integer_rankwitnessminornonzeroevaluationdeterminanthscrc)) * S ((mdr_ut_second_integer_rankwitnessminornonzeroevaluationdeterminanthsc) + (mdr_e_second_integer_rankwitnessminornonzeroevaluationdeterminanthscrc)) + ((mdr_e_second_integer_rankwitnessminornonzeroevaluationdeterminanthscrc) + (mdr_e_second_integer_rankwitnessminornonzeroevaluationdeterminanthscrc))) /\ ((mdr_z_second_integer_rankwitnessminornonzeroevaluationdeterminanthscr) = ((mdr_c_second_integer_rankwitnessminornonzeroevaluationdeterminanthscrc) + (mdr_f_second_integer_rankwitnessminornonzeroevaluationdeterminanthscrc)) * S ((mdr_c_second_integer_rankwitnessminornonzeroevaluationdeterminanthscrc) + (mdr_f_second_integer_rankwitnessminornonzeroevaluationdeterminanthscrc)) + ((mdr_f_second_integer_rankwitnessminornonzeroevaluationdeterminanthscrc) + (mdr_f_second_integer_rankwitnessminornonzeroevaluationdeterminanthscrc))))))))) /\ (((exists ff_h_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthscrb. ff_h_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthscrb + S (mdr_z_second_integer_rankwitnessminornonzeroevaluationdeterminanthscr) = S ((S (mdr_i_second_integer_rankwitnessminornonzeroevaluationdeterminanthsc)) * mdr_c_second_integer_rankwitnessminornonzeroevaluationdeterminant)) /\ exists ff_q_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthscrb. mdr_b_second_integer_rankwitnessminornonzeroevaluationdeterminant = ff_q_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthscrb * S ((S (mdr_i_second_integer_rankwitnessminornonzeroevaluationdeterminanthsc)) * mdr_c_second_integer_rankwitnessminornonzeroevaluationdeterminant) + (mdr_z_second_integer_rankwitnessminornonzeroevaluationdeterminanthscr))))) /\ ((((forall ff_index_mdm_prefix_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthscm_positive. (exists ff_gap_mdm_lt_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthscm_positive_index_bound. ff_gap_mdm_lt_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthscm_positive_index_bound + S (ff_index_mdm_prefix_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthscm_positive) = ((mdr_q_second_integer_rankwitnessminornonzeroevaluationdeterminanths) * (mdr_q_second_integer_rankwitnessminornonzeroevaluationdeterminanths))) -> exists ff_row_mdm_prefix_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthscm_positive ff_column_mdm_prefix_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthscm_positive ff_value_mdm_prefix_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthscm_positive. (ff_index_mdm_prefix_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthscm_positive = (mdr_q_second_integer_rankwitnessminornonzeroevaluationdeterminanths) * ff_row_mdm_prefix_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthscm_positive + ff_column_mdm_prefix_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthscm_positive /\ ((exists ff_gap_mdm_lt_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthscm_positive_column_bound. ff_gap_mdm_lt_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthscm_positive_column_bound + S (ff_column_mdm_prefix_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthscm_positive) = (mdr_q_second_integer_rankwitnessminornonzeroevaluationdeterminanths)) /\ ((exists ff_row_mdm_cell_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthscm_positive_cell ff_column_mdm_cell_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthscm_positive_cell. (((((exists ff_gap_mdm_lt_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthscm_positive_cell_row_before. ff_gap_mdm_lt_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthscm_positive_cell_row_before + S (ff_row_mdm_prefix_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthscm_positive) = (0)) /\ ff_row_mdm_cell_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthscm_positive_cell = ff_row_mdm_prefix_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthscm_positive) \/ ((exists ff_gap_mdm_le_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthscm_positive_cell_row_after. ff_gap_mdm_le_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthscm_positive_cell_row_after + (0) = (ff_row_mdm_prefix_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthscm_positive)) /\ ff_row_mdm_cell_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthscm_positive_cell = S ff_row_mdm_prefix_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthscm_positive))) /\ (((((exists ff_gap_mdm_lt_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthscm_positive_cell_column_before. ff_gap_mdm_lt_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthscm_positive_cell_column_before + S (ff_column_mdm_prefix_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthscm_positive) = (mdr_j_second_integer_rankwitnessminornonzeroevaluationdeterminanthsc)) /\ ff_column_mdm_cell_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthscm_positive_cell = ff_column_mdm_prefix_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthscm_positive) \/ ((exists ff_gap_mdm_le_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthscm_positive_cell_column_after. ff_gap_mdm_le_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthscm_positive_cell_column_after + (mdr_j_second_integer_rankwitnessminornonzeroevaluationdeterminanthsc) = (ff_column_mdm_prefix_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthscm_positive)) /\ ff_column_mdm_cell_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthscm_positive_cell = S ff_column_mdm_prefix_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthscm_positive))) /\ (((exists ff_h_mdm_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthscm_positive_cell_source. ff_h_mdm_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthscm_positive_cell_source + S (ff_value_mdm_prefix_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthscm_positive) = S ((S ((ff_row_mdm_cell_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthscm_positive_cell) * (S (mdr_q_second_integer_rankwitnessminornonzeroevaluationdeterminanths)) + (ff_column_mdm_cell_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthscm_positive_cell))) * mdr_pc_second_integer_rankwitnessminornonzeroevaluationdeterminanth)) /\ exists ff_q_mdm_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthscm_positive_cell_source. mdr_pb_second_integer_rankwitnessminornonzeroevaluationdeterminanth = ff_q_mdm_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthscm_positive_cell_source * S ((S ((ff_row_mdm_cell_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthscm_positive_cell) * (S (mdr_q_second_integer_rankwitnessminornonzeroevaluationdeterminanths)) + (ff_column_mdm_cell_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthscm_positive_cell))) * mdr_pc_second_integer_rankwitnessminornonzeroevaluationdeterminanth) + (ff_value_mdm_prefix_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthscm_positive)))))) /\ (((exists ff_h_mdm_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthscm_positive_target. ff_h_mdm_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthscm_positive_target + S (ff_value_mdm_prefix_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthscm_positive) = S ((S (ff_index_mdm_prefix_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthscm_positive)) * mdr_us_second_integer_rankwitnessminornonzeroevaluationdeterminanthsc)) /\ exists ff_q_mdm_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthscm_positive_target. mdr_up_second_integer_rankwitnessminornonzeroevaluationdeterminanthsc = ff_q_mdm_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthscm_positive_target * S ((S (ff_index_mdm_prefix_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthscm_positive)) * mdr_us_second_integer_rankwitnessminornonzeroevaluationdeterminanthsc) + (ff_value_mdm_prefix_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthscm_positive))))))) /\ (forall ff_index_mdm_prefix_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthscm_negative. (exists ff_gap_mdm_lt_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthscm_negative_index_bound. ff_gap_mdm_lt_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthscm_negative_index_bound + S (ff_index_mdm_prefix_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthscm_negative) = ((mdr_q_second_integer_rankwitnessminornonzeroevaluationdeterminanths) * (mdr_q_second_integer_rankwitnessminornonzeroevaluationdeterminanths))) -> exists ff_row_mdm_prefix_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthscm_negative ff_column_mdm_prefix_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthscm_negative ff_value_mdm_prefix_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthscm_negative. (ff_index_mdm_prefix_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthscm_negative = (mdr_q_second_integer_rankwitnessminornonzeroevaluationdeterminanths) * ff_row_mdm_prefix_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthscm_negative + ff_column_mdm_prefix_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthscm_negative /\ ((exists ff_gap_mdm_lt_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthscm_negative_column_bound. ff_gap_mdm_lt_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthscm_negative_column_bound + S (ff_column_mdm_prefix_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthscm_negative) = (mdr_q_second_integer_rankwitnessminornonzeroevaluationdeterminanths)) /\ ((exists ff_row_mdm_cell_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthscm_negative_cell ff_column_mdm_cell_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthscm_negative_cell. (((((exists ff_gap_mdm_lt_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthscm_negative_cell_row_before. ff_gap_mdm_lt_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthscm_negative_cell_row_before + S (ff_row_mdm_prefix_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthscm_negative) = (0)) /\ ff_row_mdm_cell_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthscm_negative_cell = ff_row_mdm_prefix_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthscm_negative) \/ ((exists ff_gap_mdm_le_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthscm_negative_cell_row_after. ff_gap_mdm_le_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthscm_negative_cell_row_after + (0) = (ff_row_mdm_prefix_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthscm_negative)) /\ ff_row_mdm_cell_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthscm_negative_cell = S ff_row_mdm_prefix_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthscm_negative))) /\ (((((exists ff_gap_mdm_lt_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthscm_negative_cell_column_before. ff_gap_mdm_lt_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthscm_negative_cell_column_before + S (ff_column_mdm_prefix_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthscm_negative) = (mdr_j_second_integer_rankwitnessminornonzeroevaluationdeterminanthsc)) /\ ff_column_mdm_cell_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthscm_negative_cell = ff_column_mdm_prefix_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthscm_negative) \/ ((exists ff_gap_mdm_le_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthscm_negative_cell_column_after. ff_gap_mdm_le_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthscm_negative_cell_column_after + (mdr_j_second_integer_rankwitnessminornonzeroevaluationdeterminanthsc) = (ff_column_mdm_prefix_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthscm_negative)) /\ ff_column_mdm_cell_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthscm_negative_cell = S ff_column_mdm_prefix_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthscm_negative))) /\ (((exists ff_h_mdm_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthscm_negative_cell_source. ff_h_mdm_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthscm_negative_cell_source + S (ff_value_mdm_prefix_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthscm_negative) = S ((S ((ff_row_mdm_cell_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthscm_negative_cell) * (S (mdr_q_second_integer_rankwitnessminornonzeroevaluationdeterminanths)) + (ff_column_mdm_cell_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthscm_negative_cell))) * mdr_nc_second_integer_rankwitnessminornonzeroevaluationdeterminanth)) /\ exists ff_q_mdm_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthscm_negative_cell_source. mdr_nb_second_integer_rankwitnessminornonzeroevaluationdeterminanth = ff_q_mdm_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthscm_negative_cell_source * S ((S ((ff_row_mdm_cell_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthscm_negative_cell) * (S (mdr_q_second_integer_rankwitnessminornonzeroevaluationdeterminanths)) + (ff_column_mdm_cell_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthscm_negative_cell))) * mdr_nc_second_integer_rankwitnessminornonzeroevaluationdeterminanth) + (ff_value_mdm_prefix_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthscm_negative)))))) /\ (((exists ff_h_mdm_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthscm_negative_target. ff_h_mdm_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthscm_negative_target + S (ff_value_mdm_prefix_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthscm_negative) = S ((S (ff_index_mdm_prefix_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthscm_negative)) * mdr_ut_second_integer_rankwitnessminornonzeroevaluationdeterminanthsc)) /\ exists ff_q_mdm_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthscm_negative_target. mdr_un_second_integer_rankwitnessminornonzeroevaluationdeterminanthsc = ff_q_mdm_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthscm_negative_target * S ((S (ff_index_mdm_prefix_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthscm_negative)) * mdr_ut_second_integer_rankwitnessminornonzeroevaluationdeterminanthsc) + (ff_value_mdm_prefix_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthscm_negative))))))))) /\ ((((exists ff_h_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthscp. ff_h_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthscp + S (mdr_p_second_integer_rankwitnessminornonzeroevaluationdeterminanthsc) = S ((S (mdr_j_second_integer_rankwitnessminornonzeroevaluationdeterminanthsc)) * mdr_ec_second_integer_rankwitnessminornonzeroevaluationdeterminanths)) /\ exists ff_q_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthscp. mdr_eb_second_integer_rankwitnessminornonzeroevaluationdeterminanths = ff_q_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthscp * S ((S (mdr_j_second_integer_rankwitnessminornonzeroevaluationdeterminanthsc)) * mdr_ec_second_integer_rankwitnessminornonzeroevaluationdeterminanths) + (mdr_p_second_integer_rankwitnessminornonzeroevaluationdeterminanthsc))) /\ (((exists ff_h_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthscn. ff_h_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthscn + S (mdr_n_second_integer_rankwitnessminornonzeroevaluationdeterminanthsc) = S ((S (mdr_j_second_integer_rankwitnessminornonzeroevaluationdeterminanthsc)) * mdr_fc_second_integer_rankwitnessminornonzeroevaluationdeterminanths)) /\ exists ff_q_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthscn. mdr_fb_second_integer_rankwitnessminornonzeroevaluationdeterminanths = ff_q_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthscn * S ((S (mdr_j_second_integer_rankwitnessminornonzeroevaluationdeterminanthsc)) * mdr_fc_second_integer_rankwitnessminornonzeroevaluationdeterminanths) + (mdr_n_second_integer_rankwitnessminornonzeroevaluationdeterminanthsc)))))))) /\ (exists ff_ub_mce_fold_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf ff_uc_mce_fold_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf ff_vb_mce_fold_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf ff_vc_mce_fold_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf. ((forall ff_index_mce_alternating_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix. (exists ff_gap_mce_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix_index. ff_gap_mce_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix_index + S (ff_index_mce_alternating_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix) = (S (mdr_q_second_integer_rankwitnessminornonzeroevaluationdeterminanths))) -> exists ff_ap_mce_alternating_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix ff_an_mce_alternating_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix ff_bp_mce_alternating_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix ff_bn_mce_alternating_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix ff_p_mce_alternating_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix ff_n_mce_alternating_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix. ((((exists ff_h_mce_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix_ap. ff_h_mce_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix_ap + S (ff_ap_mce_alternating_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix) = S ((S (ff_index_mce_alternating_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix)) * mdr_pc_second_integer_rankwitnessminornonzeroevaluationdeterminanth)) /\ exists ff_q_mce_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix_ap. mdr_pb_second_integer_rankwitnessminornonzeroevaluationdeterminanth = ff_q_mce_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix_ap * S ((S (ff_index_mce_alternating_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix)) * mdr_pc_second_integer_rankwitnessminornonzeroevaluationdeterminanth) + (ff_ap_mce_alternating_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix))) /\ ((((exists ff_h_mce_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix_an. ff_h_mce_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix_an + S (ff_an_mce_alternating_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix) = S ((S (ff_index_mce_alternating_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix)) * mdr_nc_second_integer_rankwitnessminornonzeroevaluationdeterminanth)) /\ exists ff_q_mce_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix_an. mdr_nb_second_integer_rankwitnessminornonzeroevaluationdeterminanth = ff_q_mce_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix_an * S ((S (ff_index_mce_alternating_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix)) * mdr_nc_second_integer_rankwitnessminornonzeroevaluationdeterminanth) + (ff_an_mce_alternating_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix))) /\ ((((exists ff_h_mce_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix_bp. ff_h_mce_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix_bp + S (ff_bp_mce_alternating_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix) = S ((S (ff_index_mce_alternating_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix)) * mdr_ec_second_integer_rankwitnessminornonzeroevaluationdeterminanths)) /\ exists ff_q_mce_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix_bp. mdr_eb_second_integer_rankwitnessminornonzeroevaluationdeterminanths = ff_q_mce_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix_bp * S ((S (ff_index_mce_alternating_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix)) * mdr_ec_second_integer_rankwitnessminornonzeroevaluationdeterminanths) + (ff_bp_mce_alternating_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix))) /\ ((((exists ff_h_mce_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix_bn. ff_h_mce_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix_bn + S (ff_bn_mce_alternating_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix) = S ((S (ff_index_mce_alternating_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix)) * mdr_fc_second_integer_rankwitnessminornonzeroevaluationdeterminanths)) /\ exists ff_q_mce_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix_bn. mdr_fb_second_integer_rankwitnessminornonzeroevaluationdeterminanths = ff_q_mce_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix_bn * S ((S (ff_index_mce_alternating_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix)) * mdr_fc_second_integer_rankwitnessminornonzeroevaluationdeterminanths) + (ff_bn_mce_alternating_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix))) /\ ((((exists ff_h_mce_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix_positive. ff_h_mce_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix_positive + S (ff_p_mce_alternating_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix) = S ((S (ff_index_mce_alternating_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix)) * ff_uc_mce_fold_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf)) /\ exists ff_q_mce_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix_positive. ff_ub_mce_fold_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf = ff_q_mce_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix_positive * S ((S (ff_index_mce_alternating_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix)) * ff_uc_mce_fold_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf) + (ff_p_mce_alternating_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix))) /\ ((((exists ff_h_mce_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix_negative. ff_h_mce_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix_negative + S (ff_n_mce_alternating_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix) = S ((S (ff_index_mce_alternating_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix)) * ff_vc_mce_fold_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf)) /\ exists ff_q_mce_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix_negative. ff_vb_mce_fold_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf = ff_q_mce_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix_negative * S ((S (ff_index_mce_alternating_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix)) * ff_vc_mce_fold_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf) + (ff_n_mce_alternating_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix))) /\ (((exists ff_even_mce_term_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix_term. ff_index_mce_alternating_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix = 2 * ff_even_mce_term_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix_term) /\ (ff_p_mce_alternating_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix = (ff_ap_mce_alternating_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix) * (ff_bp_mce_alternating_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix) + (ff_an_mce_alternating_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix) * (ff_bn_mce_alternating_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix) /\ ff_n_mce_alternating_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix = (ff_ap_mce_alternating_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix) * (ff_bn_mce_alternating_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix) + (ff_an_mce_alternating_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix) * (ff_bp_mce_alternating_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix))) \/ ((exists ff_odd_mce_term_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix_term. ff_index_mce_alternating_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix = 2 * ff_odd_mce_term_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix_term + 1) /\ (ff_p_mce_alternating_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix = (ff_ap_mce_alternating_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix) * (ff_bn_mce_alternating_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix) + (ff_an_mce_alternating_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix) * (ff_bp_mce_alternating_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix) /\ ff_n_mce_alternating_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix = (ff_ap_mce_alternating_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix) * (ff_bp_mce_alternating_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix) + (ff_an_mce_alternating_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix) * (ff_bn_mce_alternating_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_prefix))))))))))) /\ ((exists ff_u_mce_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_positive ff_v_mce_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_positive. ((((exists ff_h_mce_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_positive_start. ff_h_mce_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_positive_start + S (0) = S ((S (0)) * ff_v_mce_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_positive)) /\ exists ff_q_mce_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_positive_start. ff_u_mce_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_positive = ff_q_mce_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_positive_start * S ((S (0)) * ff_v_mce_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_positive) + (0))) /\ ((((exists ff_h_mce_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_positive_terminal. ff_h_mce_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_positive_terminal + S (mdr_p_second_integer_rankwitnessminornonzeroevaluationdeterminanth) = S ((S ((S (mdr_q_second_integer_rankwitnessminornonzeroevaluationdeterminanths)))) * ff_v_mce_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_positive)) /\ exists ff_q_mce_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_positive_terminal. ff_u_mce_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_positive = ff_q_mce_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_positive_terminal * S ((S ((S (mdr_q_second_integer_rankwitnessminornonzeroevaluationdeterminanths)))) * ff_v_mce_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_positive) + (mdr_p_second_integer_rankwitnessminornonzeroevaluationdeterminanth))) /\ forall ff_i_mce_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_positive. (exists ff_lt_mce_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_positive_bound. ff_lt_mce_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_positive_bound + S ff_i_mce_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_positive = (S (mdr_q_second_integer_rankwitnessminornonzeroevaluationdeterminanths))) -> exists ff_a_mce_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_positive ff_r_mce_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_positive ff_s_mce_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_positive. ((((exists ff_h_mce_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_positive_summand. ff_h_mce_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_positive_summand + S (ff_a_mce_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_positive) = S ((S (ff_i_mce_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_positive)) * ff_uc_mce_fold_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf)) /\ exists ff_q_mce_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_positive_summand. ff_ub_mce_fold_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf = ff_q_mce_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_positive_summand * S ((S (ff_i_mce_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_positive)) * ff_uc_mce_fold_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf) + (ff_a_mce_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_positive))) /\ ((((exists ff_h_mce_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_positive_partial. ff_h_mce_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_positive_partial + S (ff_r_mce_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_positive) = S ((S (ff_i_mce_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_positive)) * ff_v_mce_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_positive)) /\ exists ff_q_mce_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_positive_partial. ff_u_mce_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_positive = ff_q_mce_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_positive_partial * S ((S (ff_i_mce_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_positive)) * ff_v_mce_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_positive) + (ff_r_mce_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_positive))) /\ ((((exists ff_h_mce_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_positive_successor. ff_h_mce_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_positive_successor + S (ff_s_mce_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_positive) = S ((S (S ff_i_mce_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_positive)) * ff_v_mce_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_positive)) /\ exists ff_q_mce_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_positive_successor. ff_u_mce_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_positive = ff_q_mce_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_positive_successor * S ((S (S ff_i_mce_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_positive)) * ff_v_mce_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_positive) + (ff_s_mce_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_positive))) /\ ff_s_mce_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_positive = ff_r_mce_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_positive + ff_a_mce_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_positive)))))) /\ (exists ff_u_mce_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_negative ff_v_mce_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_negative. ((((exists ff_h_mce_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_negative_start. ff_h_mce_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_negative_start + S (0) = S ((S (0)) * ff_v_mce_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_negative)) /\ exists ff_q_mce_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_negative_start. ff_u_mce_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_negative = ff_q_mce_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_negative_start * S ((S (0)) * ff_v_mce_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_negative) + (0))) /\ ((((exists ff_h_mce_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_negative_terminal. ff_h_mce_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_negative_terminal + S (mdr_n_second_integer_rankwitnessminornonzeroevaluationdeterminanth) = S ((S ((S (mdr_q_second_integer_rankwitnessminornonzeroevaluationdeterminanths)))) * ff_v_mce_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_negative)) /\ exists ff_q_mce_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_negative_terminal. ff_u_mce_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_negative = ff_q_mce_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_negative_terminal * S ((S ((S (mdr_q_second_integer_rankwitnessminornonzeroevaluationdeterminanths)))) * ff_v_mce_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_negative) + (mdr_n_second_integer_rankwitnessminornonzeroevaluationdeterminanth))) /\ forall ff_i_mce_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_negative. (exists ff_lt_mce_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_negative_bound. ff_lt_mce_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_negative_bound + S ff_i_mce_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_negative = (S (mdr_q_second_integer_rankwitnessminornonzeroevaluationdeterminanths))) -> exists ff_a_mce_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_negative ff_r_mce_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_negative ff_s_mce_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_negative. ((((exists ff_h_mce_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_negative_summand. ff_h_mce_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_negative_summand + S (ff_a_mce_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_negative) = S ((S (ff_i_mce_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_negative)) * ff_vc_mce_fold_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf)) /\ exists ff_q_mce_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_negative_summand. ff_vb_mce_fold_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf = ff_q_mce_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_negative_summand * S ((S (ff_i_mce_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_negative)) * ff_vc_mce_fold_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf) + (ff_a_mce_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_negative))) /\ ((((exists ff_h_mce_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_negative_partial. ff_h_mce_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_negative_partial + S (ff_r_mce_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_negative) = S ((S (ff_i_mce_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_negative)) * ff_v_mce_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_negative)) /\ exists ff_q_mce_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_negative_partial. ff_u_mce_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_negative = ff_q_mce_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_negative_partial * S ((S (ff_i_mce_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_negative)) * ff_v_mce_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_negative) + (ff_r_mce_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_negative))) /\ ((((exists ff_h_mce_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_negative_successor. ff_h_mce_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_negative_successor + S (ff_s_mce_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_negative) = S ((S (S ff_i_mce_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_negative)) * ff_v_mce_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_negative)) /\ exists ff_q_mce_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_negative_successor. ff_u_mce_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_negative = ff_q_mce_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_negative_successor * S ((S (S ff_i_mce_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_negative)) * ff_v_mce_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_negative) + (ff_s_mce_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_negative))) /\ ff_s_mce_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_negative = ff_r_mce_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_negative + ff_a_mce_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminanthsf_negative))))))))))))))) /\ ((exists mdr_gap_second_integer_rankwitnessminornonzeroevaluationdeterminanti. mdr_gap_second_integer_rankwitnessminornonzeroevaluationdeterminanti + S (mdr_i_second_integer_rankwitnessminornonzeroevaluationdeterminant) = (mdr_l_second_integer_rankwitnessminornonzeroevaluationdeterminant)) /\ (exists mdr_z_second_integer_rankwitnessminornonzeroevaluationdeterminantr. ((exists mdr_a_second_integer_rankwitnessminornonzeroevaluationdeterminantrc mdr_b_second_integer_rankwitnessminornonzeroevaluationdeterminantrc mdr_c_second_integer_rankwitnessminornonzeroevaluationdeterminantrc mdr_e_second_integer_rankwitnessminornonzeroevaluationdeterminantrc mdr_f_second_integer_rankwitnessminornonzeroevaluationdeterminantrc. ((mdr_a_second_integer_rankwitnessminornonzeroevaluationdeterminantrc = ((rank) + (mdr_ub_second_integer_rankwitnessminornonzeroevaluation)) * S ((rank) + (mdr_ub_second_integer_rankwitnessminornonzeroevaluation)) + ((mdr_ub_second_integer_rankwitnessminornonzeroevaluation) + (mdr_ub_second_integer_rankwitnessminornonzeroevaluation))) /\ ((mdr_b_second_integer_rankwitnessminornonzeroevaluationdeterminantrc = ((mdr_uc_second_integer_rankwitnessminornonzeroevaluation) + (mdr_vb_second_integer_rankwitnessminornonzeroevaluation)) * S ((mdr_uc_second_integer_rankwitnessminornonzeroevaluation) + (mdr_vb_second_integer_rankwitnessminornonzeroevaluation)) + ((mdr_vb_second_integer_rankwitnessminornonzeroevaluation) + (mdr_vb_second_integer_rankwitnessminornonzeroevaluation))) /\ ((mdr_c_second_integer_rankwitnessminornonzeroevaluationdeterminantrc = ((mdr_a_second_integer_rankwitnessminornonzeroevaluationdeterminantrc) + (mdr_b_second_integer_rankwitnessminornonzeroevaluationdeterminantrc)) * S ((mdr_a_second_integer_rankwitnessminornonzeroevaluationdeterminantrc) + (mdr_b_second_integer_rankwitnessminornonzeroevaluationdeterminantrc)) + ((mdr_b_second_integer_rankwitnessminornonzeroevaluationdeterminantrc) + (mdr_b_second_integer_rankwitnessminornonzeroevaluationdeterminantrc))) /\ ((mdr_e_second_integer_rankwitnessminornonzeroevaluationdeterminantrc = ((mdr_p_second_integer_rankwitnessminornonzero) + (mdr_n_second_integer_rankwitnessminornonzero)) * S ((mdr_p_second_integer_rankwitnessminornonzero) + (mdr_n_second_integer_rankwitnessminornonzero)) + ((mdr_n_second_integer_rankwitnessminornonzero) + (mdr_n_second_integer_rankwitnessminornonzero))) /\ ((mdr_f_second_integer_rankwitnessminornonzeroevaluationdeterminantrc = ((mdr_vc_second_integer_rankwitnessminornonzeroevaluation) + (mdr_e_second_integer_rankwitnessminornonzeroevaluationdeterminantrc)) * S ((mdr_vc_second_integer_rankwitnessminornonzeroevaluation) + (mdr_e_second_integer_rankwitnessminornonzeroevaluationdeterminantrc)) + ((mdr_e_second_integer_rankwitnessminornonzeroevaluationdeterminantrc) + (mdr_e_second_integer_rankwitnessminornonzeroevaluationdeterminantrc))) /\ ((mdr_z_second_integer_rankwitnessminornonzeroevaluationdeterminantr) = ((mdr_c_second_integer_rankwitnessminornonzeroevaluationdeterminantrc) + (mdr_f_second_integer_rankwitnessminornonzeroevaluationdeterminantrc)) * S ((mdr_c_second_integer_rankwitnessminornonzeroevaluationdeterminantrc) + (mdr_f_second_integer_rankwitnessminornonzeroevaluationdeterminantrc)) + ((mdr_f_second_integer_rankwitnessminornonzeroevaluationdeterminantrc) + (mdr_f_second_integer_rankwitnessminornonzeroevaluationdeterminantrc))))))))) /\ (((exists ff_h_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminantrb. ff_h_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminantrb + S (mdr_z_second_integer_rankwitnessminornonzeroevaluationdeterminantr) = S ((S (mdr_i_second_integer_rankwitnessminornonzeroevaluationdeterminant)) * mdr_c_second_integer_rankwitnessminornonzeroevaluationdeterminant)) /\ exists ff_q_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminantrb. mdr_b_second_integer_rankwitnessminornonzeroevaluationdeterminant = ff_q_mdr_second_integer_rankwitnessminornonzeroevaluationdeterminantrb * S ((S (mdr_i_second_integer_rankwitnessminornonzeroevaluationdeterminant)) * mdr_c_second_integer_rankwitnessminornonzeroevaluationdeterminant) + (mdr_z_second_integer_rankwitnessminornonzeroevaluationdeterminantr)))))))))) /\ (~(mdr_p_second_integer_rankwitnessminornonzero = mdr_n_second_integer_rankwitnessminornonzero)))))))) /\ (forall mdr_q_second_integer_rank. (exists mdr_gap_second_integer_rankhigher. mdr_gap_second_integer_rankhigher + S (rank) = (mdr_q_second_integer_rank)) -> (forall mdr_rb_second_integer_rankzero mdr_rc_second_integer_rankzero mdr_cb_second_integer_rankzero mdr_cc_second_integer_rankzero mdr_p_second_integer_rankzero mdr_n_second_integer_rankzero. (((forall fom_index_mrf_second_integer_rankzerorowsbound. (exists fom_gap_mrf_second_integer_rankzerorowsbound_index_bound. fom_gap_mrf_second_integer_rankzerorowsbound_index_bound + S (fom_index_mrf_second_integer_rankzerorowsbound) = mdr_q_second_integer_rank) -> exists fom_value_mrf_second_integer_rankzerorowsbound. ((((exists fom_beta_height_mrf_second_integer_rankzerorowsbound_entry. fom_beta_height_mrf_second_integer_rankzerorowsbound_entry + S (fom_value_mrf_second_integer_rankzerorowsbound) = S ((S (fom_index_mrf_second_integer_rankzerorowsbound)) * mdr_rc_second_integer_rankzero)) /\ exists fom_beta_quotient_mrf_second_integer_rankzerorowsbound_entry. mdr_rb_second_integer_rankzero = fom_beta_quotient_mrf_second_integer_rankzerorowsbound_entry * S ((S (fom_index_mrf_second_integer_rankzerorowsbound)) * mdr_rc_second_integer_rankzero) + (fom_value_mrf_second_integer_rankzerorowsbound))) /\ (exists fom_gap_mrf_second_integer_rankzerorowsbound_value_bound. fom_gap_mrf_second_integer_rankzerorowsbound_value_bound + S (fom_value_mrf_second_integer_rankzerorowsbound) = r))) /\ (forall mdr_i_second_integer_rankzerorowsdistinct mdr_j_second_integer_rankzerorowsdistinct mdr_a_second_integer_rankzerorowsdistinct. (exists mdr_gap_second_integer_rankzerorowsdistincti. mdr_gap_second_integer_rankzerorowsdistincti + S (mdr_i_second_integer_rankzerorowsdistinct) = (mdr_q_second_integer_rank)) -> (exists mdr_gap_second_integer_rankzerorowsdistinctj. mdr_gap_second_integer_rankzerorowsdistinctj + S (mdr_j_second_integer_rankzerorowsdistinct) = (mdr_q_second_integer_rank)) -> (((exists ff_h_mdr_second_integer_rankzerorowsdistinctfirst. ff_h_mdr_second_integer_rankzerorowsdistinctfirst + S (mdr_a_second_integer_rankzerorowsdistinct) = S ((S (mdr_i_second_integer_rankzerorowsdistinct)) * mdr_rc_second_integer_rankzero)) /\ exists ff_q_mdr_second_integer_rankzerorowsdistinctfirst. mdr_rb_second_integer_rankzero = ff_q_mdr_second_integer_rankzerorowsdistinctfirst * S ((S (mdr_i_second_integer_rankzerorowsdistinct)) * mdr_rc_second_integer_rankzero) + (mdr_a_second_integer_rankzerorowsdistinct))) -> (((exists ff_h_mdr_second_integer_rankzerorowsdistinctsecond. ff_h_mdr_second_integer_rankzerorowsdistinctsecond + S (mdr_a_second_integer_rankzerorowsdistinct) = S ((S (mdr_j_second_integer_rankzerorowsdistinct)) * mdr_rc_second_integer_rankzero)) /\ exists ff_q_mdr_second_integer_rankzerorowsdistinctsecond. mdr_rb_second_integer_rankzero = ff_q_mdr_second_integer_rankzerorowsdistinctsecond * S ((S (mdr_j_second_integer_rankzerorowsdistinct)) * mdr_rc_second_integer_rankzero) + (mdr_a_second_integer_rankzerorowsdistinct))) -> mdr_i_second_integer_rankzerorowsdistinct = mdr_j_second_integer_rankzerorowsdistinct))) -> (((forall fom_index_mrf_second_integer_rankzerocolumnsbound. (exists fom_gap_mrf_second_integer_rankzerocolumnsbound_index_bound. fom_gap_mrf_second_integer_rankzerocolumnsbound_index_bound + S (fom_index_mrf_second_integer_rankzerocolumnsbound) = mdr_q_second_integer_rank) -> exists fom_value_mrf_second_integer_rankzerocolumnsbound. ((((exists fom_beta_height_mrf_second_integer_rankzerocolumnsbound_entry. fom_beta_height_mrf_second_integer_rankzerocolumnsbound_entry + S (fom_value_mrf_second_integer_rankzerocolumnsbound) = S ((S (fom_index_mrf_second_integer_rankzerocolumnsbound)) * mdr_cc_second_integer_rankzero)) /\ exists fom_beta_quotient_mrf_second_integer_rankzerocolumnsbound_entry. mdr_cb_second_integer_rankzero = fom_beta_quotient_mrf_second_integer_rankzerocolumnsbound_entry * S ((S (fom_index_mrf_second_integer_rankzerocolumnsbound)) * mdr_cc_second_integer_rankzero) + (fom_value_mrf_second_integer_rankzerocolumnsbound))) /\ (exists fom_gap_mrf_second_integer_rankzerocolumnsbound_value_bound. fom_gap_mrf_second_integer_rankzerocolumnsbound_value_bound + S (fom_value_mrf_second_integer_rankzerocolumnsbound) = w))) /\ (forall mdr_i_second_integer_rankzerocolumnsdistinct mdr_j_second_integer_rankzerocolumnsdistinct mdr_a_second_integer_rankzerocolumnsdistinct. (exists mdr_gap_second_integer_rankzerocolumnsdistincti. mdr_gap_second_integer_rankzerocolumnsdistincti + S (mdr_i_second_integer_rankzerocolumnsdistinct) = (mdr_q_second_integer_rank)) -> (exists mdr_gap_second_integer_rankzerocolumnsdistinctj. mdr_gap_second_integer_rankzerocolumnsdistinctj + S (mdr_j_second_integer_rankzerocolumnsdistinct) = (mdr_q_second_integer_rank)) -> (((exists ff_h_mdr_second_integer_rankzerocolumnsdistinctfirst. ff_h_mdr_second_integer_rankzerocolumnsdistinctfirst + S (mdr_a_second_integer_rankzerocolumnsdistinct) = S ((S (mdr_i_second_integer_rankzerocolumnsdistinct)) * mdr_cc_second_integer_rankzero)) /\ exists ff_q_mdr_second_integer_rankzerocolumnsdistinctfirst. mdr_cb_second_integer_rankzero = ff_q_mdr_second_integer_rankzerocolumnsdistinctfirst * S ((S (mdr_i_second_integer_rankzerocolumnsdistinct)) * mdr_cc_second_integer_rankzero) + (mdr_a_second_integer_rankzerocolumnsdistinct))) -> (((exists ff_h_mdr_second_integer_rankzerocolumnsdistinctsecond. ff_h_mdr_second_integer_rankzerocolumnsdistinctsecond + S (mdr_a_second_integer_rankzerocolumnsdistinct) = S ((S (mdr_j_second_integer_rankzerocolumnsdistinct)) * mdr_cc_second_integer_rankzero)) /\ exists ff_q_mdr_second_integer_rankzerocolumnsdistinctsecond. mdr_cb_second_integer_rankzero = ff_q_mdr_second_integer_rankzerocolumnsdistinctsecond * S ((S (mdr_j_second_integer_rankzerocolumnsdistinct)) * mdr_cc_second_integer_rankzero) + (mdr_a_second_integer_rankzerocolumnsdistinct))) -> mdr_i_second_integer_rankzerocolumnsdistinct = mdr_j_second_integer_rankzerocolumnsdistinct))) -> (exists mdr_ub_second_integer_rankzeroevaluation mdr_uc_second_integer_rankzeroevaluation mdr_vb_second_integer_rankzeroevaluation mdr_vc_second_integer_rankzeroevaluation. ((((forall mdr_i_second_integer_rankzeroevaluationmatrixpositive. (exists mdr_gap_second_integer_rankzeroevaluationmatrixpositivebound. mdr_gap_second_integer_rankzeroevaluationmatrixpositivebound + S (mdr_i_second_integer_rankzeroevaluationmatrixpositive) = ((mdr_q_second_integer_rank) * (mdr_q_second_integer_rank))) -> exists mdr_a_second_integer_rankzeroevaluationmatrixpositive. (((exists mdr_r_second_integer_rankzeroevaluationmatrixpositivepoint mdr_s_second_integer_rankzeroevaluationmatrixpositivepoint mdr_u_second_integer_rankzeroevaluationmatrixpositivepoint mdr_v_second_integer_rankzeroevaluationmatrixpositivepoint. ((mdr_i_second_integer_rankzeroevaluationmatrixpositive = (mdr_q_second_integer_rank) * mdr_r_second_integer_rankzeroevaluationmatrixpositivepoint + mdr_s_second_integer_rankzeroevaluationmatrixpositivepoint) /\ ((exists mdr_gap_second_integer_rankzeroevaluationmatrixpositivepointcolumn. mdr_gap_second_integer_rankzeroevaluationmatrixpositivepointcolumn + S (mdr_s_second_integer_rankzeroevaluationmatrixpositivepoint) = (mdr_q_second_integer_rank)) /\ ((((exists ff_h_mdr_second_integer_rankzeroevaluationmatrixpositivepointrow_index. ff_h_mdr_second_integer_rankzeroevaluationmatrixpositivepointrow_index + S (mdr_u_second_integer_rankzeroevaluationmatrixpositivepoint) = S ((S (mdr_r_second_integer_rankzeroevaluationmatrixpositivepoint)) * mdr_rc_second_integer_rankzero)) /\ exists ff_q_mdr_second_integer_rankzeroevaluationmatrixpositivepointrow_index. mdr_rb_second_integer_rankzero = ff_q_mdr_second_integer_rankzeroevaluationmatrixpositivepointrow_index * S ((S (mdr_r_second_integer_rankzeroevaluationmatrixpositivepoint)) * mdr_rc_second_integer_rankzero) + (mdr_u_second_integer_rankzeroevaluationmatrixpositivepoint))) /\ ((((exists ff_h_mdr_second_integer_rankzeroevaluationmatrixpositivepointcolumn_index. ff_h_mdr_second_integer_rankzeroevaluationmatrixpositivepointcolumn_index + S (mdr_v_second_integer_rankzeroevaluationmatrixpositivepoint) = S ((S (mdr_s_second_integer_rankzeroevaluationmatrixpositivepoint)) * mdr_cc_second_integer_rankzero)) /\ exists ff_q_mdr_second_integer_rankzeroevaluationmatrixpositivepointcolumn_index. mdr_cb_second_integer_rankzero = ff_q_mdr_second_integer_rankzeroevaluationmatrixpositivepointcolumn_index * S ((S (mdr_s_second_integer_rankzeroevaluationmatrixpositivepoint)) * mdr_cc_second_integer_rankzero) + (mdr_v_second_integer_rankzeroevaluationmatrixpositivepoint))) /\ (((exists ff_h_mdr_second_integer_rankzeroevaluationmatrixpositivepointsource. ff_h_mdr_second_integer_rankzeroevaluationmatrixpositivepointsource + S (mdr_a_second_integer_rankzeroevaluationmatrixpositive) = S ((S ((mdr_u_second_integer_rankzeroevaluationmatrixpositivepoint) * (w) + (mdr_v_second_integer_rankzeroevaluationmatrixpositivepoint))) * ec)) /\ exists ff_q_mdr_second_integer_rankzeroevaluationmatrixpositivepointsource. eb = ff_q_mdr_second_integer_rankzeroevaluationmatrixpositivepointsource * S ((S ((mdr_u_second_integer_rankzeroevaluationmatrixpositivepoint) * (w) + (mdr_v_second_integer_rankzeroevaluationmatrixpositivepoint))) * ec) + (mdr_a_second_integer_rankzeroevaluationmatrixpositive)))))))) /\ (((exists ff_h_mdr_second_integer_rankzeroevaluationmatrixpositiveoutput. ff_h_mdr_second_integer_rankzeroevaluationmatrixpositiveoutput + S (mdr_a_second_integer_rankzeroevaluationmatrixpositive) = S ((S (mdr_i_second_integer_rankzeroevaluationmatrixpositive)) * mdr_uc_second_integer_rankzeroevaluation)) /\ exists ff_q_mdr_second_integer_rankzeroevaluationmatrixpositiveoutput. mdr_ub_second_integer_rankzeroevaluation = ff_q_mdr_second_integer_rankzeroevaluationmatrixpositiveoutput * S ((S (mdr_i_second_integer_rankzeroevaluationmatrixpositive)) * mdr_uc_second_integer_rankzeroevaluation) + (mdr_a_second_integer_rankzeroevaluationmatrixpositive)))))) /\ (forall mdr_i_second_integer_rankzeroevaluationmatrixnegative. (exists mdr_gap_second_integer_rankzeroevaluationmatrixnegativebound. mdr_gap_second_integer_rankzeroevaluationmatrixnegativebound + S (mdr_i_second_integer_rankzeroevaluationmatrixnegative) = ((mdr_q_second_integer_rank) * (mdr_q_second_integer_rank))) -> exists mdr_a_second_integer_rankzeroevaluationmatrixnegative. (((exists mdr_r_second_integer_rankzeroevaluationmatrixnegativepoint mdr_s_second_integer_rankzeroevaluationmatrixnegativepoint mdr_u_second_integer_rankzeroevaluationmatrixnegativepoint mdr_v_second_integer_rankzeroevaluationmatrixnegativepoint. ((mdr_i_second_integer_rankzeroevaluationmatrixnegative = (mdr_q_second_integer_rank) * mdr_r_second_integer_rankzeroevaluationmatrixnegativepoint + mdr_s_second_integer_rankzeroevaluationmatrixnegativepoint) /\ ((exists mdr_gap_second_integer_rankzeroevaluationmatrixnegativepointcolumn. mdr_gap_second_integer_rankzeroevaluationmatrixnegativepointcolumn + S (mdr_s_second_integer_rankzeroevaluationmatrixnegativepoint) = (mdr_q_second_integer_rank)) /\ ((((exists ff_h_mdr_second_integer_rankzeroevaluationmatrixnegativepointrow_index. ff_h_mdr_second_integer_rankzeroevaluationmatrixnegativepointrow_index + S (mdr_u_second_integer_rankzeroevaluationmatrixnegativepoint) = S ((S (mdr_r_second_integer_rankzeroevaluationmatrixnegativepoint)) * mdr_rc_second_integer_rankzero)) /\ exists ff_q_mdr_second_integer_rankzeroevaluationmatrixnegativepointrow_index. mdr_rb_second_integer_rankzero = ff_q_mdr_second_integer_rankzeroevaluationmatrixnegativepointrow_index * S ((S (mdr_r_second_integer_rankzeroevaluationmatrixnegativepoint)) * mdr_rc_second_integer_rankzero) + (mdr_u_second_integer_rankzeroevaluationmatrixnegativepoint))) /\ ((((exists ff_h_mdr_second_integer_rankzeroevaluationmatrixnegativepointcolumn_index. ff_h_mdr_second_integer_rankzeroevaluationmatrixnegativepointcolumn_index + S (mdr_v_second_integer_rankzeroevaluationmatrixnegativepoint) = S ((S (mdr_s_second_integer_rankzeroevaluationmatrixnegativepoint)) * mdr_cc_second_integer_rankzero)) /\ exists ff_q_mdr_second_integer_rankzeroevaluationmatrixnegativepointcolumn_index. mdr_cb_second_integer_rankzero = ff_q_mdr_second_integer_rankzeroevaluationmatrixnegativepointcolumn_index * S ((S (mdr_s_second_integer_rankzeroevaluationmatrixnegativepoint)) * mdr_cc_second_integer_rankzero) + (mdr_v_second_integer_rankzeroevaluationmatrixnegativepoint))) /\ (((exists ff_h_mdr_second_integer_rankzeroevaluationmatrixnegativepointsource. ff_h_mdr_second_integer_rankzeroevaluationmatrixnegativepointsource + S (mdr_a_second_integer_rankzeroevaluationmatrixnegative) = S ((S ((mdr_u_second_integer_rankzeroevaluationmatrixnegativepoint) * (w) + (mdr_v_second_integer_rankzeroevaluationmatrixnegativepoint))) * fc)) /\ exists ff_q_mdr_second_integer_rankzeroevaluationmatrixnegativepointsource. fb = ff_q_mdr_second_integer_rankzeroevaluationmatrixnegativepointsource * S ((S ((mdr_u_second_integer_rankzeroevaluationmatrixnegativepoint) * (w) + (mdr_v_second_integer_rankzeroevaluationmatrixnegativepoint))) * fc) + (mdr_a_second_integer_rankzeroevaluationmatrixnegative)))))))) /\ (((exists ff_h_mdr_second_integer_rankzeroevaluationmatrixnegativeoutput. ff_h_mdr_second_integer_rankzeroevaluationmatrixnegativeoutput + S (mdr_a_second_integer_rankzeroevaluationmatrixnegative) = S ((S (mdr_i_second_integer_rankzeroevaluationmatrixnegative)) * mdr_vc_second_integer_rankzeroevaluation)) /\ exists ff_q_mdr_second_integer_rankzeroevaluationmatrixnegativeoutput. mdr_vb_second_integer_rankzeroevaluation = ff_q_mdr_second_integer_rankzeroevaluationmatrixnegativeoutput * S ((S (mdr_i_second_integer_rankzeroevaluationmatrixnegative)) * mdr_vc_second_integer_rankzeroevaluation) + (mdr_a_second_integer_rankzeroevaluationmatrixnegative)))))))) /\ (exists mdr_b_second_integer_rankzeroevaluationdeterminant mdr_c_second_integer_rankzeroevaluationdeterminant mdr_l_second_integer_rankzeroevaluationdeterminant mdr_i_second_integer_rankzeroevaluationdeterminant. ((forall mdr_i_second_integer_rankzeroevaluationdeterminanth. (exists mdr_gap_second_integer_rankzeroevaluationdeterminanthi. mdr_gap_second_integer_rankzeroevaluationdeterminanthi + S (mdr_i_second_integer_rankzeroevaluationdeterminanth) = (mdr_l_second_integer_rankzeroevaluationdeterminant)) -> exists mdr_d_second_integer_rankzeroevaluationdeterminanth mdr_pb_second_integer_rankzeroevaluationdeterminanth mdr_pc_second_integer_rankzeroevaluationdeterminanth mdr_nb_second_integer_rankzeroevaluationdeterminanth mdr_nc_second_integer_rankzeroevaluationdeterminanth mdr_p_second_integer_rankzeroevaluationdeterminanth mdr_n_second_integer_rankzeroevaluationdeterminanth. ((exists mdr_z_second_integer_rankzeroevaluationdeterminanthr. ((exists mdr_a_second_integer_rankzeroevaluationdeterminanthrc mdr_b_second_integer_rankzeroevaluationdeterminanthrc mdr_c_second_integer_rankzeroevaluationdeterminanthrc mdr_e_second_integer_rankzeroevaluationdeterminanthrc mdr_f_second_integer_rankzeroevaluationdeterminanthrc. ((mdr_a_second_integer_rankzeroevaluationdeterminanthrc = ((mdr_d_second_integer_rankzeroevaluationdeterminanth) + (mdr_pb_second_integer_rankzeroevaluationdeterminanth)) * S ((mdr_d_second_integer_rankzeroevaluationdeterminanth) + (mdr_pb_second_integer_rankzeroevaluationdeterminanth)) + ((mdr_pb_second_integer_rankzeroevaluationdeterminanth) + (mdr_pb_second_integer_rankzeroevaluationdeterminanth))) /\ ((mdr_b_second_integer_rankzeroevaluationdeterminanthrc = ((mdr_pc_second_integer_rankzeroevaluationdeterminanth) + (mdr_nb_second_integer_rankzeroevaluationdeterminanth)) * S ((mdr_pc_second_integer_rankzeroevaluationdeterminanth) + (mdr_nb_second_integer_rankzeroevaluationdeterminanth)) + ((mdr_nb_second_integer_rankzeroevaluationdeterminanth) + (mdr_nb_second_integer_rankzeroevaluationdeterminanth))) /\ ((mdr_c_second_integer_rankzeroevaluationdeterminanthrc = ((mdr_a_second_integer_rankzeroevaluationdeterminanthrc) + (mdr_b_second_integer_rankzeroevaluationdeterminanthrc)) * S ((mdr_a_second_integer_rankzeroevaluationdeterminanthrc) + (mdr_b_second_integer_rankzeroevaluationdeterminanthrc)) + ((mdr_b_second_integer_rankzeroevaluationdeterminanthrc) + (mdr_b_second_integer_rankzeroevaluationdeterminanthrc))) /\ ((mdr_e_second_integer_rankzeroevaluationdeterminanthrc = ((mdr_p_second_integer_rankzeroevaluationdeterminanth) + (mdr_n_second_integer_rankzeroevaluationdeterminanth)) * S ((mdr_p_second_integer_rankzeroevaluationdeterminanth) + (mdr_n_second_integer_rankzeroevaluationdeterminanth)) + ((mdr_n_second_integer_rankzeroevaluationdeterminanth) + (mdr_n_second_integer_rankzeroevaluationdeterminanth))) /\ ((mdr_f_second_integer_rankzeroevaluationdeterminanthrc = ((mdr_nc_second_integer_rankzeroevaluationdeterminanth) + (mdr_e_second_integer_rankzeroevaluationdeterminanthrc)) * S ((mdr_nc_second_integer_rankzeroevaluationdeterminanth) + (mdr_e_second_integer_rankzeroevaluationdeterminanthrc)) + ((mdr_e_second_integer_rankzeroevaluationdeterminanthrc) + (mdr_e_second_integer_rankzeroevaluationdeterminanthrc))) /\ ((mdr_z_second_integer_rankzeroevaluationdeterminanthr) = ((mdr_c_second_integer_rankzeroevaluationdeterminanthrc) + (mdr_f_second_integer_rankzeroevaluationdeterminanthrc)) * S ((mdr_c_second_integer_rankzeroevaluationdeterminanthrc) + (mdr_f_second_integer_rankzeroevaluationdeterminanthrc)) + ((mdr_f_second_integer_rankzeroevaluationdeterminanthrc) + (mdr_f_second_integer_rankzeroevaluationdeterminanthrc))))))))) /\ (((exists ff_h_mdr_second_integer_rankzeroevaluationdeterminanthrb. ff_h_mdr_second_integer_rankzeroevaluationdeterminanthrb + S (mdr_z_second_integer_rankzeroevaluationdeterminanthr) = S ((S (mdr_i_second_integer_rankzeroevaluationdeterminanth)) * mdr_c_second_integer_rankzeroevaluationdeterminant)) /\ exists ff_q_mdr_second_integer_rankzeroevaluationdeterminanthrb. mdr_b_second_integer_rankzeroevaluationdeterminant = ff_q_mdr_second_integer_rankzeroevaluationdeterminanthrb * S ((S (mdr_i_second_integer_rankzeroevaluationdeterminanth)) * mdr_c_second_integer_rankzeroevaluationdeterminant) + (mdr_z_second_integer_rankzeroevaluationdeterminanthr))))) /\ (((((mdr_d_second_integer_rankzeroevaluationdeterminanth) = 0) /\ (((mdr_p_second_integer_rankzeroevaluationdeterminanth) = 1) /\ ((mdr_n_second_integer_rankzeroevaluationdeterminanth) = 0))) \/ exists mdr_q_second_integer_rankzeroevaluationdeterminanths mdr_eb_second_integer_rankzeroevaluationdeterminanths mdr_ec_second_integer_rankzeroevaluationdeterminanths mdr_fb_second_integer_rankzeroevaluationdeterminanths mdr_fc_second_integer_rankzeroevaluationdeterminanths. (((mdr_d_second_integer_rankzeroevaluationdeterminanth) = S (mdr_q_second_integer_rankzeroevaluationdeterminanths)) /\ ((forall mdr_j_second_integer_rankzeroevaluationdeterminanthsc. (exists mdr_gap_second_integer_rankzeroevaluationdeterminanthscj. mdr_gap_second_integer_rankzeroevaluationdeterminanthscj + S (mdr_j_second_integer_rankzeroevaluationdeterminanthsc) = (S (mdr_q_second_integer_rankzeroevaluationdeterminanths))) -> exists mdr_i_second_integer_rankzeroevaluationdeterminanthsc mdr_up_second_integer_rankzeroevaluationdeterminanthsc mdr_us_second_integer_rankzeroevaluationdeterminanthsc mdr_un_second_integer_rankzeroevaluationdeterminanthsc mdr_ut_second_integer_rankzeroevaluationdeterminanthsc mdr_p_second_integer_rankzeroevaluationdeterminanthsc mdr_n_second_integer_rankzeroevaluationdeterminanthsc. ((exists mdr_gap_second_integer_rankzeroevaluationdeterminanthsci. mdr_gap_second_integer_rankzeroevaluationdeterminanthsci + S (mdr_i_second_integer_rankzeroevaluationdeterminanthsc) = (mdr_i_second_integer_rankzeroevaluationdeterminanth)) /\ ((exists mdr_z_second_integer_rankzeroevaluationdeterminanthscr. ((exists mdr_a_second_integer_rankzeroevaluationdeterminanthscrc mdr_b_second_integer_rankzeroevaluationdeterminanthscrc mdr_c_second_integer_rankzeroevaluationdeterminanthscrc mdr_e_second_integer_rankzeroevaluationdeterminanthscrc mdr_f_second_integer_rankzeroevaluationdeterminanthscrc. ((mdr_a_second_integer_rankzeroevaluationdeterminanthscrc = ((mdr_q_second_integer_rankzeroevaluationdeterminanths) + (mdr_up_second_integer_rankzeroevaluationdeterminanthsc)) * S ((mdr_q_second_integer_rankzeroevaluationdeterminanths) + (mdr_up_second_integer_rankzeroevaluationdeterminanthsc)) + ((mdr_up_second_integer_rankzeroevaluationdeterminanthsc) + (mdr_up_second_integer_rankzeroevaluationdeterminanthsc))) /\ ((mdr_b_second_integer_rankzeroevaluationdeterminanthscrc = ((mdr_us_second_integer_rankzeroevaluationdeterminanthsc) + (mdr_un_second_integer_rankzeroevaluationdeterminanthsc)) * S ((mdr_us_second_integer_rankzeroevaluationdeterminanthsc) + (mdr_un_second_integer_rankzeroevaluationdeterminanthsc)) + ((mdr_un_second_integer_rankzeroevaluationdeterminanthsc) + (mdr_un_second_integer_rankzeroevaluationdeterminanthsc))) /\ ((mdr_c_second_integer_rankzeroevaluationdeterminanthscrc = ((mdr_a_second_integer_rankzeroevaluationdeterminanthscrc) + (mdr_b_second_integer_rankzeroevaluationdeterminanthscrc)) * S ((mdr_a_second_integer_rankzeroevaluationdeterminanthscrc) + (mdr_b_second_integer_rankzeroevaluationdeterminanthscrc)) + ((mdr_b_second_integer_rankzeroevaluationdeterminanthscrc) + (mdr_b_second_integer_rankzeroevaluationdeterminanthscrc))) /\ ((mdr_e_second_integer_rankzeroevaluationdeterminanthscrc = ((mdr_p_second_integer_rankzeroevaluationdeterminanthsc) + (mdr_n_second_integer_rankzeroevaluationdeterminanthsc)) * S ((mdr_p_second_integer_rankzeroevaluationdeterminanthsc) + (mdr_n_second_integer_rankzeroevaluationdeterminanthsc)) + ((mdr_n_second_integer_rankzeroevaluationdeterminanthsc) + (mdr_n_second_integer_rankzeroevaluationdeterminanthsc))) /\ ((mdr_f_second_integer_rankzeroevaluationdeterminanthscrc = ((mdr_ut_second_integer_rankzeroevaluationdeterminanthsc) + (mdr_e_second_integer_rankzeroevaluationdeterminanthscrc)) * S ((mdr_ut_second_integer_rankzeroevaluationdeterminanthsc) + (mdr_e_second_integer_rankzeroevaluationdeterminanthscrc)) + ((mdr_e_second_integer_rankzeroevaluationdeterminanthscrc) + (mdr_e_second_integer_rankzeroevaluationdeterminanthscrc))) /\ ((mdr_z_second_integer_rankzeroevaluationdeterminanthscr) = ((mdr_c_second_integer_rankzeroevaluationdeterminanthscrc) + (mdr_f_second_integer_rankzeroevaluationdeterminanthscrc)) * S ((mdr_c_second_integer_rankzeroevaluationdeterminanthscrc) + (mdr_f_second_integer_rankzeroevaluationdeterminanthscrc)) + ((mdr_f_second_integer_rankzeroevaluationdeterminanthscrc) + (mdr_f_second_integer_rankzeroevaluationdeterminanthscrc))))))))) /\ (((exists ff_h_mdr_second_integer_rankzeroevaluationdeterminanthscrb. ff_h_mdr_second_integer_rankzeroevaluationdeterminanthscrb + S (mdr_z_second_integer_rankzeroevaluationdeterminanthscr) = S ((S (mdr_i_second_integer_rankzeroevaluationdeterminanthsc)) * mdr_c_second_integer_rankzeroevaluationdeterminant)) /\ exists ff_q_mdr_second_integer_rankzeroevaluationdeterminanthscrb. mdr_b_second_integer_rankzeroevaluationdeterminant = ff_q_mdr_second_integer_rankzeroevaluationdeterminanthscrb * S ((S (mdr_i_second_integer_rankzeroevaluationdeterminanthsc)) * mdr_c_second_integer_rankzeroevaluationdeterminant) + (mdr_z_second_integer_rankzeroevaluationdeterminanthscr))))) /\ ((((forall ff_index_mdm_prefix_mdr_second_integer_rankzeroevaluationdeterminanthscm_positive. (exists ff_gap_mdm_lt_mdr_second_integer_rankzeroevaluationdeterminanthscm_positive_index_bound. ff_gap_mdm_lt_mdr_second_integer_rankzeroevaluationdeterminanthscm_positive_index_bound + S (ff_index_mdm_prefix_mdr_second_integer_rankzeroevaluationdeterminanthscm_positive) = ((mdr_q_second_integer_rankzeroevaluationdeterminanths) * (mdr_q_second_integer_rankzeroevaluationdeterminanths))) -> exists ff_row_mdm_prefix_mdr_second_integer_rankzeroevaluationdeterminanthscm_positive ff_column_mdm_prefix_mdr_second_integer_rankzeroevaluationdeterminanthscm_positive ff_value_mdm_prefix_mdr_second_integer_rankzeroevaluationdeterminanthscm_positive. (ff_index_mdm_prefix_mdr_second_integer_rankzeroevaluationdeterminanthscm_positive = (mdr_q_second_integer_rankzeroevaluationdeterminanths) * ff_row_mdm_prefix_mdr_second_integer_rankzeroevaluationdeterminanthscm_positive + ff_column_mdm_prefix_mdr_second_integer_rankzeroevaluationdeterminanthscm_positive /\ ((exists ff_gap_mdm_lt_mdr_second_integer_rankzeroevaluationdeterminanthscm_positive_column_bound. ff_gap_mdm_lt_mdr_second_integer_rankzeroevaluationdeterminanthscm_positive_column_bound + S (ff_column_mdm_prefix_mdr_second_integer_rankzeroevaluationdeterminanthscm_positive) = (mdr_q_second_integer_rankzeroevaluationdeterminanths)) /\ ((exists ff_row_mdm_cell_mdr_second_integer_rankzeroevaluationdeterminanthscm_positive_cell ff_column_mdm_cell_mdr_second_integer_rankzeroevaluationdeterminanthscm_positive_cell. (((((exists ff_gap_mdm_lt_mdr_second_integer_rankzeroevaluationdeterminanthscm_positive_cell_row_before. ff_gap_mdm_lt_mdr_second_integer_rankzeroevaluationdeterminanthscm_positive_cell_row_before + S (ff_row_mdm_prefix_mdr_second_integer_rankzeroevaluationdeterminanthscm_positive) = (0)) /\ ff_row_mdm_cell_mdr_second_integer_rankzeroevaluationdeterminanthscm_positive_cell = ff_row_mdm_prefix_mdr_second_integer_rankzeroevaluationdeterminanthscm_positive) \/ ((exists ff_gap_mdm_le_mdr_second_integer_rankzeroevaluationdeterminanthscm_positive_cell_row_after. ff_gap_mdm_le_mdr_second_integer_rankzeroevaluationdeterminanthscm_positive_cell_row_after + (0) = (ff_row_mdm_prefix_mdr_second_integer_rankzeroevaluationdeterminanthscm_positive)) /\ ff_row_mdm_cell_mdr_second_integer_rankzeroevaluationdeterminanthscm_positive_cell = S ff_row_mdm_prefix_mdr_second_integer_rankzeroevaluationdeterminanthscm_positive))) /\ (((((exists ff_gap_mdm_lt_mdr_second_integer_rankzeroevaluationdeterminanthscm_positive_cell_column_before. ff_gap_mdm_lt_mdr_second_integer_rankzeroevaluationdeterminanthscm_positive_cell_column_before + S (ff_column_mdm_prefix_mdr_second_integer_rankzeroevaluationdeterminanthscm_positive) = (mdr_j_second_integer_rankzeroevaluationdeterminanthsc)) /\ ff_column_mdm_cell_mdr_second_integer_rankzeroevaluationdeterminanthscm_positive_cell = ff_column_mdm_prefix_mdr_second_integer_rankzeroevaluationdeterminanthscm_positive) \/ ((exists ff_gap_mdm_le_mdr_second_integer_rankzeroevaluationdeterminanthscm_positive_cell_column_after. ff_gap_mdm_le_mdr_second_integer_rankzeroevaluationdeterminanthscm_positive_cell_column_after + (mdr_j_second_integer_rankzeroevaluationdeterminanthsc) = (ff_column_mdm_prefix_mdr_second_integer_rankzeroevaluationdeterminanthscm_positive)) /\ ff_column_mdm_cell_mdr_second_integer_rankzeroevaluationdeterminanthscm_positive_cell = S ff_column_mdm_prefix_mdr_second_integer_rankzeroevaluationdeterminanthscm_positive))) /\ (((exists ff_h_mdm_mdr_second_integer_rankzeroevaluationdeterminanthscm_positive_cell_source. ff_h_mdm_mdr_second_integer_rankzeroevaluationdeterminanthscm_positive_cell_source + S (ff_value_mdm_prefix_mdr_second_integer_rankzeroevaluationdeterminanthscm_positive) = S ((S ((ff_row_mdm_cell_mdr_second_integer_rankzeroevaluationdeterminanthscm_positive_cell) * (S (mdr_q_second_integer_rankzeroevaluationdeterminanths)) + (ff_column_mdm_cell_mdr_second_integer_rankzeroevaluationdeterminanthscm_positive_cell))) * mdr_pc_second_integer_rankzeroevaluationdeterminanth)) /\ exists ff_q_mdm_mdr_second_integer_rankzeroevaluationdeterminanthscm_positive_cell_source. mdr_pb_second_integer_rankzeroevaluationdeterminanth = ff_q_mdm_mdr_second_integer_rankzeroevaluationdeterminanthscm_positive_cell_source * S ((S ((ff_row_mdm_cell_mdr_second_integer_rankzeroevaluationdeterminanthscm_positive_cell) * (S (mdr_q_second_integer_rankzeroevaluationdeterminanths)) + (ff_column_mdm_cell_mdr_second_integer_rankzeroevaluationdeterminanthscm_positive_cell))) * mdr_pc_second_integer_rankzeroevaluationdeterminanth) + (ff_value_mdm_prefix_mdr_second_integer_rankzeroevaluationdeterminanthscm_positive)))))) /\ (((exists ff_h_mdm_mdr_second_integer_rankzeroevaluationdeterminanthscm_positive_target. ff_h_mdm_mdr_second_integer_rankzeroevaluationdeterminanthscm_positive_target + S (ff_value_mdm_prefix_mdr_second_integer_rankzeroevaluationdeterminanthscm_positive) = S ((S (ff_index_mdm_prefix_mdr_second_integer_rankzeroevaluationdeterminanthscm_positive)) * mdr_us_second_integer_rankzeroevaluationdeterminanthsc)) /\ exists ff_q_mdm_mdr_second_integer_rankzeroevaluationdeterminanthscm_positive_target. mdr_up_second_integer_rankzeroevaluationdeterminanthsc = ff_q_mdm_mdr_second_integer_rankzeroevaluationdeterminanthscm_positive_target * S ((S (ff_index_mdm_prefix_mdr_second_integer_rankzeroevaluationdeterminanthscm_positive)) * mdr_us_second_integer_rankzeroevaluationdeterminanthsc) + (ff_value_mdm_prefix_mdr_second_integer_rankzeroevaluationdeterminanthscm_positive))))))) /\ (forall ff_index_mdm_prefix_mdr_second_integer_rankzeroevaluationdeterminanthscm_negative. (exists ff_gap_mdm_lt_mdr_second_integer_rankzeroevaluationdeterminanthscm_negative_index_bound. ff_gap_mdm_lt_mdr_second_integer_rankzeroevaluationdeterminanthscm_negative_index_bound + S (ff_index_mdm_prefix_mdr_second_integer_rankzeroevaluationdeterminanthscm_negative) = ((mdr_q_second_integer_rankzeroevaluationdeterminanths) * (mdr_q_second_integer_rankzeroevaluationdeterminanths))) -> exists ff_row_mdm_prefix_mdr_second_integer_rankzeroevaluationdeterminanthscm_negative ff_column_mdm_prefix_mdr_second_integer_rankzeroevaluationdeterminanthscm_negative ff_value_mdm_prefix_mdr_second_integer_rankzeroevaluationdeterminanthscm_negative. (ff_index_mdm_prefix_mdr_second_integer_rankzeroevaluationdeterminanthscm_negative = (mdr_q_second_integer_rankzeroevaluationdeterminanths) * ff_row_mdm_prefix_mdr_second_integer_rankzeroevaluationdeterminanthscm_negative + ff_column_mdm_prefix_mdr_second_integer_rankzeroevaluationdeterminanthscm_negative /\ ((exists ff_gap_mdm_lt_mdr_second_integer_rankzeroevaluationdeterminanthscm_negative_column_bound. ff_gap_mdm_lt_mdr_second_integer_rankzeroevaluationdeterminanthscm_negative_column_bound + S (ff_column_mdm_prefix_mdr_second_integer_rankzeroevaluationdeterminanthscm_negative) = (mdr_q_second_integer_rankzeroevaluationdeterminanths)) /\ ((exists ff_row_mdm_cell_mdr_second_integer_rankzeroevaluationdeterminanthscm_negative_cell ff_column_mdm_cell_mdr_second_integer_rankzeroevaluationdeterminanthscm_negative_cell. (((((exists ff_gap_mdm_lt_mdr_second_integer_rankzeroevaluationdeterminanthscm_negative_cell_row_before. ff_gap_mdm_lt_mdr_second_integer_rankzeroevaluationdeterminanthscm_negative_cell_row_before + S (ff_row_mdm_prefix_mdr_second_integer_rankzeroevaluationdeterminanthscm_negative) = (0)) /\ ff_row_mdm_cell_mdr_second_integer_rankzeroevaluationdeterminanthscm_negative_cell = ff_row_mdm_prefix_mdr_second_integer_rankzeroevaluationdeterminanthscm_negative) \/ ((exists ff_gap_mdm_le_mdr_second_integer_rankzeroevaluationdeterminanthscm_negative_cell_row_after. ff_gap_mdm_le_mdr_second_integer_rankzeroevaluationdeterminanthscm_negative_cell_row_after + (0) = (ff_row_mdm_prefix_mdr_second_integer_rankzeroevaluationdeterminanthscm_negative)) /\ ff_row_mdm_cell_mdr_second_integer_rankzeroevaluationdeterminanthscm_negative_cell = S ff_row_mdm_prefix_mdr_second_integer_rankzeroevaluationdeterminanthscm_negative))) /\ (((((exists ff_gap_mdm_lt_mdr_second_integer_rankzeroevaluationdeterminanthscm_negative_cell_column_before. ff_gap_mdm_lt_mdr_second_integer_rankzeroevaluationdeterminanthscm_negative_cell_column_before + S (ff_column_mdm_prefix_mdr_second_integer_rankzeroevaluationdeterminanthscm_negative) = (mdr_j_second_integer_rankzeroevaluationdeterminanthsc)) /\ ff_column_mdm_cell_mdr_second_integer_rankzeroevaluationdeterminanthscm_negative_cell = ff_column_mdm_prefix_mdr_second_integer_rankzeroevaluationdeterminanthscm_negative) \/ ((exists ff_gap_mdm_le_mdr_second_integer_rankzeroevaluationdeterminanthscm_negative_cell_column_after. ff_gap_mdm_le_mdr_second_integer_rankzeroevaluationdeterminanthscm_negative_cell_column_after + (mdr_j_second_integer_rankzeroevaluationdeterminanthsc) = (ff_column_mdm_prefix_mdr_second_integer_rankzeroevaluationdeterminanthscm_negative)) /\ ff_column_mdm_cell_mdr_second_integer_rankzeroevaluationdeterminanthscm_negative_cell = S ff_column_mdm_prefix_mdr_second_integer_rankzeroevaluationdeterminanthscm_negative))) /\ (((exists ff_h_mdm_mdr_second_integer_rankzeroevaluationdeterminanthscm_negative_cell_source. ff_h_mdm_mdr_second_integer_rankzeroevaluationdeterminanthscm_negative_cell_source + S (ff_value_mdm_prefix_mdr_second_integer_rankzeroevaluationdeterminanthscm_negative) = S ((S ((ff_row_mdm_cell_mdr_second_integer_rankzeroevaluationdeterminanthscm_negative_cell) * (S (mdr_q_second_integer_rankzeroevaluationdeterminanths)) + (ff_column_mdm_cell_mdr_second_integer_rankzeroevaluationdeterminanthscm_negative_cell))) * mdr_nc_second_integer_rankzeroevaluationdeterminanth)) /\ exists ff_q_mdm_mdr_second_integer_rankzeroevaluationdeterminanthscm_negative_cell_source. mdr_nb_second_integer_rankzeroevaluationdeterminanth = ff_q_mdm_mdr_second_integer_rankzeroevaluationdeterminanthscm_negative_cell_source * S ((S ((ff_row_mdm_cell_mdr_second_integer_rankzeroevaluationdeterminanthscm_negative_cell) * (S (mdr_q_second_integer_rankzeroevaluationdeterminanths)) + (ff_column_mdm_cell_mdr_second_integer_rankzeroevaluationdeterminanthscm_negative_cell))) * mdr_nc_second_integer_rankzeroevaluationdeterminanth) + (ff_value_mdm_prefix_mdr_second_integer_rankzeroevaluationdeterminanthscm_negative)))))) /\ (((exists ff_h_mdm_mdr_second_integer_rankzeroevaluationdeterminanthscm_negative_target. ff_h_mdm_mdr_second_integer_rankzeroevaluationdeterminanthscm_negative_target + S (ff_value_mdm_prefix_mdr_second_integer_rankzeroevaluationdeterminanthscm_negative) = S ((S (ff_index_mdm_prefix_mdr_second_integer_rankzeroevaluationdeterminanthscm_negative)) * mdr_ut_second_integer_rankzeroevaluationdeterminanthsc)) /\ exists ff_q_mdm_mdr_second_integer_rankzeroevaluationdeterminanthscm_negative_target. mdr_un_second_integer_rankzeroevaluationdeterminanthsc = ff_q_mdm_mdr_second_integer_rankzeroevaluationdeterminanthscm_negative_target * S ((S (ff_index_mdm_prefix_mdr_second_integer_rankzeroevaluationdeterminanthscm_negative)) * mdr_ut_second_integer_rankzeroevaluationdeterminanthsc) + (ff_value_mdm_prefix_mdr_second_integer_rankzeroevaluationdeterminanthscm_negative))))))))) /\ ((((exists ff_h_mdr_second_integer_rankzeroevaluationdeterminanthscp. ff_h_mdr_second_integer_rankzeroevaluationdeterminanthscp + S (mdr_p_second_integer_rankzeroevaluationdeterminanthsc) = S ((S (mdr_j_second_integer_rankzeroevaluationdeterminanthsc)) * mdr_ec_second_integer_rankzeroevaluationdeterminanths)) /\ exists ff_q_mdr_second_integer_rankzeroevaluationdeterminanthscp. mdr_eb_second_integer_rankzeroevaluationdeterminanths = ff_q_mdr_second_integer_rankzeroevaluationdeterminanthscp * S ((S (mdr_j_second_integer_rankzeroevaluationdeterminanthsc)) * mdr_ec_second_integer_rankzeroevaluationdeterminanths) + (mdr_p_second_integer_rankzeroevaluationdeterminanthsc))) /\ (((exists ff_h_mdr_second_integer_rankzeroevaluationdeterminanthscn. ff_h_mdr_second_integer_rankzeroevaluationdeterminanthscn + S (mdr_n_second_integer_rankzeroevaluationdeterminanthsc) = S ((S (mdr_j_second_integer_rankzeroevaluationdeterminanthsc)) * mdr_fc_second_integer_rankzeroevaluationdeterminanths)) /\ exists ff_q_mdr_second_integer_rankzeroevaluationdeterminanthscn. mdr_fb_second_integer_rankzeroevaluationdeterminanths = ff_q_mdr_second_integer_rankzeroevaluationdeterminanthscn * S ((S (mdr_j_second_integer_rankzeroevaluationdeterminanthsc)) * mdr_fc_second_integer_rankzeroevaluationdeterminanths) + (mdr_n_second_integer_rankzeroevaluationdeterminanthsc)))))))) /\ (exists ff_ub_mce_fold_mdr_second_integer_rankzeroevaluationdeterminanthsf ff_uc_mce_fold_mdr_second_integer_rankzeroevaluationdeterminanthsf ff_vb_mce_fold_mdr_second_integer_rankzeroevaluationdeterminanthsf ff_vc_mce_fold_mdr_second_integer_rankzeroevaluationdeterminanthsf. ((forall ff_index_mce_alternating_mdr_second_integer_rankzeroevaluationdeterminanthsf_prefix. (exists ff_gap_mce_mdr_second_integer_rankzeroevaluationdeterminanthsf_prefix_index. ff_gap_mce_mdr_second_integer_rankzeroevaluationdeterminanthsf_prefix_index + S (ff_index_mce_alternating_mdr_second_integer_rankzeroevaluationdeterminanthsf_prefix) = (S (mdr_q_second_integer_rankzeroevaluationdeterminanths))) -> exists ff_ap_mce_alternating_mdr_second_integer_rankzeroevaluationdeterminanthsf_prefix ff_an_mce_alternating_mdr_second_integer_rankzeroevaluationdeterminanthsf_prefix ff_bp_mce_alternating_mdr_second_integer_rankzeroevaluationdeterminanthsf_prefix ff_bn_mce_alternating_mdr_second_integer_rankzeroevaluationdeterminanthsf_prefix ff_p_mce_alternating_mdr_second_integer_rankzeroevaluationdeterminanthsf_prefix ff_n_mce_alternating_mdr_second_integer_rankzeroevaluationdeterminanthsf_prefix. ((((exists ff_h_mce_mdr_second_integer_rankzeroevaluationdeterminanthsf_prefix_ap. ff_h_mce_mdr_second_integer_rankzeroevaluationdeterminanthsf_prefix_ap + S (ff_ap_mce_alternating_mdr_second_integer_rankzeroevaluationdeterminanthsf_prefix) = S ((S (ff_index_mce_alternating_mdr_second_integer_rankzeroevaluationdeterminanthsf_prefix)) * mdr_pc_second_integer_rankzeroevaluationdeterminanth)) /\ exists ff_q_mce_mdr_second_integer_rankzeroevaluationdeterminanthsf_prefix_ap. mdr_pb_second_integer_rankzeroevaluationdeterminanth = ff_q_mce_mdr_second_integer_rankzeroevaluationdeterminanthsf_prefix_ap * S ((S (ff_index_mce_alternating_mdr_second_integer_rankzeroevaluationdeterminanthsf_prefix)) * mdr_pc_second_integer_rankzeroevaluationdeterminanth) + (ff_ap_mce_alternating_mdr_second_integer_rankzeroevaluationdeterminanthsf_prefix))) /\ ((((exists ff_h_mce_mdr_second_integer_rankzeroevaluationdeterminanthsf_prefix_an. ff_h_mce_mdr_second_integer_rankzeroevaluationdeterminanthsf_prefix_an + S (ff_an_mce_alternating_mdr_second_integer_rankzeroevaluationdeterminanthsf_prefix) = S ((S (ff_index_mce_alternating_mdr_second_integer_rankzeroevaluationdeterminanthsf_prefix)) * mdr_nc_second_integer_rankzeroevaluationdeterminanth)) /\ exists ff_q_mce_mdr_second_integer_rankzeroevaluationdeterminanthsf_prefix_an. mdr_nb_second_integer_rankzeroevaluationdeterminanth = ff_q_mce_mdr_second_integer_rankzeroevaluationdeterminanthsf_prefix_an * S ((S (ff_index_mce_alternating_mdr_second_integer_rankzeroevaluationdeterminanthsf_prefix)) * mdr_nc_second_integer_rankzeroevaluationdeterminanth) + (ff_an_mce_alternating_mdr_second_integer_rankzeroevaluationdeterminanthsf_prefix))) /\ ((((exists ff_h_mce_mdr_second_integer_rankzeroevaluationdeterminanthsf_prefix_bp. ff_h_mce_mdr_second_integer_rankzeroevaluationdeterminanthsf_prefix_bp + S (ff_bp_mce_alternating_mdr_second_integer_rankzeroevaluationdeterminanthsf_prefix) = S ((S (ff_index_mce_alternating_mdr_second_integer_rankzeroevaluationdeterminanthsf_prefix)) * mdr_ec_second_integer_rankzeroevaluationdeterminanths)) /\ exists ff_q_mce_mdr_second_integer_rankzeroevaluationdeterminanthsf_prefix_bp. mdr_eb_second_integer_rankzeroevaluationdeterminanths = ff_q_mce_mdr_second_integer_rankzeroevaluationdeterminanthsf_prefix_bp * S ((S (ff_index_mce_alternating_mdr_second_integer_rankzeroevaluationdeterminanthsf_prefix)) * mdr_ec_second_integer_rankzeroevaluationdeterminanths) + (ff_bp_mce_alternating_mdr_second_integer_rankzeroevaluationdeterminanthsf_prefix))) /\ ((((exists ff_h_mce_mdr_second_integer_rankzeroevaluationdeterminanthsf_prefix_bn. ff_h_mce_mdr_second_integer_rankzeroevaluationdeterminanthsf_prefix_bn + S (ff_bn_mce_alternating_mdr_second_integer_rankzeroevaluationdeterminanthsf_prefix) = S ((S (ff_index_mce_alternating_mdr_second_integer_rankzeroevaluationdeterminanthsf_prefix)) * mdr_fc_second_integer_rankzeroevaluationdeterminanths)) /\ exists ff_q_mce_mdr_second_integer_rankzeroevaluationdeterminanthsf_prefix_bn. mdr_fb_second_integer_rankzeroevaluationdeterminanths = ff_q_mce_mdr_second_integer_rankzeroevaluationdeterminanthsf_prefix_bn * S ((S (ff_index_mce_alternating_mdr_second_integer_rankzeroevaluationdeterminanthsf_prefix)) * mdr_fc_second_integer_rankzeroevaluationdeterminanths) + (ff_bn_mce_alternating_mdr_second_integer_rankzeroevaluationdeterminanthsf_prefix))) /\ ((((exists ff_h_mce_mdr_second_integer_rankzeroevaluationdeterminanthsf_prefix_positive. ff_h_mce_mdr_second_integer_rankzeroevaluationdeterminanthsf_prefix_positive + S (ff_p_mce_alternating_mdr_second_integer_rankzeroevaluationdeterminanthsf_prefix) = S ((S (ff_index_mce_alternating_mdr_second_integer_rankzeroevaluationdeterminanthsf_prefix)) * ff_uc_mce_fold_mdr_second_integer_rankzeroevaluationdeterminanthsf)) /\ exists ff_q_mce_mdr_second_integer_rankzeroevaluationdeterminanthsf_prefix_positive. ff_ub_mce_fold_mdr_second_integer_rankzeroevaluationdeterminanthsf = ff_q_mce_mdr_second_integer_rankzeroevaluationdeterminanthsf_prefix_positive * S ((S (ff_index_mce_alternating_mdr_second_integer_rankzeroevaluationdeterminanthsf_prefix)) * ff_uc_mce_fold_mdr_second_integer_rankzeroevaluationdeterminanthsf) + (ff_p_mce_alternating_mdr_second_integer_rankzeroevaluationdeterminanthsf_prefix))) /\ ((((exists ff_h_mce_mdr_second_integer_rankzeroevaluationdeterminanthsf_prefix_negative. ff_h_mce_mdr_second_integer_rankzeroevaluationdeterminanthsf_prefix_negative + S (ff_n_mce_alternating_mdr_second_integer_rankzeroevaluationdeterminanthsf_prefix) = S ((S (ff_index_mce_alternating_mdr_second_integer_rankzeroevaluationdeterminanthsf_prefix)) * ff_vc_mce_fold_mdr_second_integer_rankzeroevaluationdeterminanthsf)) /\ exists ff_q_mce_mdr_second_integer_rankzeroevaluationdeterminanthsf_prefix_negative. ff_vb_mce_fold_mdr_second_integer_rankzeroevaluationdeterminanthsf = ff_q_mce_mdr_second_integer_rankzeroevaluationdeterminanthsf_prefix_negative * S ((S (ff_index_mce_alternating_mdr_second_integer_rankzeroevaluationdeterminanthsf_prefix)) * ff_vc_mce_fold_mdr_second_integer_rankzeroevaluationdeterminanthsf) + (ff_n_mce_alternating_mdr_second_integer_rankzeroevaluationdeterminanthsf_prefix))) /\ (((exists ff_even_mce_term_mdr_second_integer_rankzeroevaluationdeterminanthsf_prefix_term. ff_index_mce_alternating_mdr_second_integer_rankzeroevaluationdeterminanthsf_prefix = 2 * ff_even_mce_term_mdr_second_integer_rankzeroevaluationdeterminanthsf_prefix_term) /\ (ff_p_mce_alternating_mdr_second_integer_rankzeroevaluationdeterminanthsf_prefix = (ff_ap_mce_alternating_mdr_second_integer_rankzeroevaluationdeterminanthsf_prefix) * (ff_bp_mce_alternating_mdr_second_integer_rankzeroevaluationdeterminanthsf_prefix) + (ff_an_mce_alternating_mdr_second_integer_rankzeroevaluationdeterminanthsf_prefix) * (ff_bn_mce_alternating_mdr_second_integer_rankzeroevaluationdeterminanthsf_prefix) /\ ff_n_mce_alternating_mdr_second_integer_rankzeroevaluationdeterminanthsf_prefix = (ff_ap_mce_alternating_mdr_second_integer_rankzeroevaluationdeterminanthsf_prefix) * (ff_bn_mce_alternating_mdr_second_integer_rankzeroevaluationdeterminanthsf_prefix) + (ff_an_mce_alternating_mdr_second_integer_rankzeroevaluationdeterminanthsf_prefix) * (ff_bp_mce_alternating_mdr_second_integer_rankzeroevaluationdeterminanthsf_prefix))) \/ ((exists ff_odd_mce_term_mdr_second_integer_rankzeroevaluationdeterminanthsf_prefix_term. ff_index_mce_alternating_mdr_second_integer_rankzeroevaluationdeterminanthsf_prefix = 2 * ff_odd_mce_term_mdr_second_integer_rankzeroevaluationdeterminanthsf_prefix_term + 1) /\ (ff_p_mce_alternating_mdr_second_integer_rankzeroevaluationdeterminanthsf_prefix = (ff_ap_mce_alternating_mdr_second_integer_rankzeroevaluationdeterminanthsf_prefix) * (ff_bn_mce_alternating_mdr_second_integer_rankzeroevaluationdeterminanthsf_prefix) + (ff_an_mce_alternating_mdr_second_integer_rankzeroevaluationdeterminanthsf_prefix) * (ff_bp_mce_alternating_mdr_second_integer_rankzeroevaluationdeterminanthsf_prefix) /\ ff_n_mce_alternating_mdr_second_integer_rankzeroevaluationdeterminanthsf_prefix = (ff_ap_mce_alternating_mdr_second_integer_rankzeroevaluationdeterminanthsf_prefix) * (ff_bp_mce_alternating_mdr_second_integer_rankzeroevaluationdeterminanthsf_prefix) + (ff_an_mce_alternating_mdr_second_integer_rankzeroevaluationdeterminanthsf_prefix) * (ff_bn_mce_alternating_mdr_second_integer_rankzeroevaluationdeterminanthsf_prefix))))))))))) /\ ((exists ff_u_mce_mdr_second_integer_rankzeroevaluationdeterminanthsf_positive ff_v_mce_mdr_second_integer_rankzeroevaluationdeterminanthsf_positive. ((((exists ff_h_mce_mdr_second_integer_rankzeroevaluationdeterminanthsf_positive_start. ff_h_mce_mdr_second_integer_rankzeroevaluationdeterminanthsf_positive_start + S (0) = S ((S (0)) * ff_v_mce_mdr_second_integer_rankzeroevaluationdeterminanthsf_positive)) /\ exists ff_q_mce_mdr_second_integer_rankzeroevaluationdeterminanthsf_positive_start. ff_u_mce_mdr_second_integer_rankzeroevaluationdeterminanthsf_positive = ff_q_mce_mdr_second_integer_rankzeroevaluationdeterminanthsf_positive_start * S ((S (0)) * ff_v_mce_mdr_second_integer_rankzeroevaluationdeterminanthsf_positive) + (0))) /\ ((((exists ff_h_mce_mdr_second_integer_rankzeroevaluationdeterminanthsf_positive_terminal. ff_h_mce_mdr_second_integer_rankzeroevaluationdeterminanthsf_positive_terminal + S (mdr_p_second_integer_rankzeroevaluationdeterminanth) = S ((S ((S (mdr_q_second_integer_rankzeroevaluationdeterminanths)))) * ff_v_mce_mdr_second_integer_rankzeroevaluationdeterminanthsf_positive)) /\ exists ff_q_mce_mdr_second_integer_rankzeroevaluationdeterminanthsf_positive_terminal. ff_u_mce_mdr_second_integer_rankzeroevaluationdeterminanthsf_positive = ff_q_mce_mdr_second_integer_rankzeroevaluationdeterminanthsf_positive_terminal * S ((S ((S (mdr_q_second_integer_rankzeroevaluationdeterminanths)))) * ff_v_mce_mdr_second_integer_rankzeroevaluationdeterminanthsf_positive) + (mdr_p_second_integer_rankzeroevaluationdeterminanth))) /\ forall ff_i_mce_mdr_second_integer_rankzeroevaluationdeterminanthsf_positive. (exists ff_lt_mce_mdr_second_integer_rankzeroevaluationdeterminanthsf_positive_bound. ff_lt_mce_mdr_second_integer_rankzeroevaluationdeterminanthsf_positive_bound + S ff_i_mce_mdr_second_integer_rankzeroevaluationdeterminanthsf_positive = (S (mdr_q_second_integer_rankzeroevaluationdeterminanths))) -> exists ff_a_mce_mdr_second_integer_rankzeroevaluationdeterminanthsf_positive ff_r_mce_mdr_second_integer_rankzeroevaluationdeterminanthsf_positive ff_s_mce_mdr_second_integer_rankzeroevaluationdeterminanthsf_positive. ((((exists ff_h_mce_mdr_second_integer_rankzeroevaluationdeterminanthsf_positive_summand. ff_h_mce_mdr_second_integer_rankzeroevaluationdeterminanthsf_positive_summand + S (ff_a_mce_mdr_second_integer_rankzeroevaluationdeterminanthsf_positive) = S ((S (ff_i_mce_mdr_second_integer_rankzeroevaluationdeterminanthsf_positive)) * ff_uc_mce_fold_mdr_second_integer_rankzeroevaluationdeterminanthsf)) /\ exists ff_q_mce_mdr_second_integer_rankzeroevaluationdeterminanthsf_positive_summand. ff_ub_mce_fold_mdr_second_integer_rankzeroevaluationdeterminanthsf = ff_q_mce_mdr_second_integer_rankzeroevaluationdeterminanthsf_positive_summand * S ((S (ff_i_mce_mdr_second_integer_rankzeroevaluationdeterminanthsf_positive)) * ff_uc_mce_fold_mdr_second_integer_rankzeroevaluationdeterminanthsf) + (ff_a_mce_mdr_second_integer_rankzeroevaluationdeterminanthsf_positive))) /\ ((((exists ff_h_mce_mdr_second_integer_rankzeroevaluationdeterminanthsf_positive_partial. ff_h_mce_mdr_second_integer_rankzeroevaluationdeterminanthsf_positive_partial + S (ff_r_mce_mdr_second_integer_rankzeroevaluationdeterminanthsf_positive) = S ((S (ff_i_mce_mdr_second_integer_rankzeroevaluationdeterminanthsf_positive)) * ff_v_mce_mdr_second_integer_rankzeroevaluationdeterminanthsf_positive)) /\ exists ff_q_mce_mdr_second_integer_rankzeroevaluationdeterminanthsf_positive_partial. ff_u_mce_mdr_second_integer_rankzeroevaluationdeterminanthsf_positive = ff_q_mce_mdr_second_integer_rankzeroevaluationdeterminanthsf_positive_partial * S ((S (ff_i_mce_mdr_second_integer_rankzeroevaluationdeterminanthsf_positive)) * ff_v_mce_mdr_second_integer_rankzeroevaluationdeterminanthsf_positive) + (ff_r_mce_mdr_second_integer_rankzeroevaluationdeterminanthsf_positive))) /\ ((((exists ff_h_mce_mdr_second_integer_rankzeroevaluationdeterminanthsf_positive_successor. ff_h_mce_mdr_second_integer_rankzeroevaluationdeterminanthsf_positive_successor + S (ff_s_mce_mdr_second_integer_rankzeroevaluationdeterminanthsf_positive) = S ((S (S ff_i_mce_mdr_second_integer_rankzeroevaluationdeterminanthsf_positive)) * ff_v_mce_mdr_second_integer_rankzeroevaluationdeterminanthsf_positive)) /\ exists ff_q_mce_mdr_second_integer_rankzeroevaluationdeterminanthsf_positive_successor. ff_u_mce_mdr_second_integer_rankzeroevaluationdeterminanthsf_positive = ff_q_mce_mdr_second_integer_rankzeroevaluationdeterminanthsf_positive_successor * S ((S (S ff_i_mce_mdr_second_integer_rankzeroevaluationdeterminanthsf_positive)) * ff_v_mce_mdr_second_integer_rankzeroevaluationdeterminanthsf_positive) + (ff_s_mce_mdr_second_integer_rankzeroevaluationdeterminanthsf_positive))) /\ ff_s_mce_mdr_second_integer_rankzeroevaluationdeterminanthsf_positive = ff_r_mce_mdr_second_integer_rankzeroevaluationdeterminanthsf_positive + ff_a_mce_mdr_second_integer_rankzeroevaluationdeterminanthsf_positive)))))) /\ (exists ff_u_mce_mdr_second_integer_rankzeroevaluationdeterminanthsf_negative ff_v_mce_mdr_second_integer_rankzeroevaluationdeterminanthsf_negative. ((((exists ff_h_mce_mdr_second_integer_rankzeroevaluationdeterminanthsf_negative_start. ff_h_mce_mdr_second_integer_rankzeroevaluationdeterminanthsf_negative_start + S (0) = S ((S (0)) * ff_v_mce_mdr_second_integer_rankzeroevaluationdeterminanthsf_negative)) /\ exists ff_q_mce_mdr_second_integer_rankzeroevaluationdeterminanthsf_negative_start. ff_u_mce_mdr_second_integer_rankzeroevaluationdeterminanthsf_negative = ff_q_mce_mdr_second_integer_rankzeroevaluationdeterminanthsf_negative_start * S ((S (0)) * ff_v_mce_mdr_second_integer_rankzeroevaluationdeterminanthsf_negative) + (0))) /\ ((((exists ff_h_mce_mdr_second_integer_rankzeroevaluationdeterminanthsf_negative_terminal. ff_h_mce_mdr_second_integer_rankzeroevaluationdeterminanthsf_negative_terminal + S (mdr_n_second_integer_rankzeroevaluationdeterminanth) = S ((S ((S (mdr_q_second_integer_rankzeroevaluationdeterminanths)))) * ff_v_mce_mdr_second_integer_rankzeroevaluationdeterminanthsf_negative)) /\ exists ff_q_mce_mdr_second_integer_rankzeroevaluationdeterminanthsf_negative_terminal. ff_u_mce_mdr_second_integer_rankzeroevaluationdeterminanthsf_negative = ff_q_mce_mdr_second_integer_rankzeroevaluationdeterminanthsf_negative_terminal * S ((S ((S (mdr_q_second_integer_rankzeroevaluationdeterminanths)))) * ff_v_mce_mdr_second_integer_rankzeroevaluationdeterminanthsf_negative) + (mdr_n_second_integer_rankzeroevaluationdeterminanth))) /\ forall ff_i_mce_mdr_second_integer_rankzeroevaluationdeterminanthsf_negative. (exists ff_lt_mce_mdr_second_integer_rankzeroevaluationdeterminanthsf_negative_bound. ff_lt_mce_mdr_second_integer_rankzeroevaluationdeterminanthsf_negative_bound + S ff_i_mce_mdr_second_integer_rankzeroevaluationdeterminanthsf_negative = (S (mdr_q_second_integer_rankzeroevaluationdeterminanths))) -> exists ff_a_mce_mdr_second_integer_rankzeroevaluationdeterminanthsf_negative ff_r_mce_mdr_second_integer_rankzeroevaluationdeterminanthsf_negative ff_s_mce_mdr_second_integer_rankzeroevaluationdeterminanthsf_negative. ((((exists ff_h_mce_mdr_second_integer_rankzeroevaluationdeterminanthsf_negative_summand. ff_h_mce_mdr_second_integer_rankzeroevaluationdeterminanthsf_negative_summand + S (ff_a_mce_mdr_second_integer_rankzeroevaluationdeterminanthsf_negative) = S ((S (ff_i_mce_mdr_second_integer_rankzeroevaluationdeterminanthsf_negative)) * ff_vc_mce_fold_mdr_second_integer_rankzeroevaluationdeterminanthsf)) /\ exists ff_q_mce_mdr_second_integer_rankzeroevaluationdeterminanthsf_negative_summand. ff_vb_mce_fold_mdr_second_integer_rankzeroevaluationdeterminanthsf = ff_q_mce_mdr_second_integer_rankzeroevaluationdeterminanthsf_negative_summand * S ((S (ff_i_mce_mdr_second_integer_rankzeroevaluationdeterminanthsf_negative)) * ff_vc_mce_fold_mdr_second_integer_rankzeroevaluationdeterminanthsf) + (ff_a_mce_mdr_second_integer_rankzeroevaluationdeterminanthsf_negative))) /\ ((((exists ff_h_mce_mdr_second_integer_rankzeroevaluationdeterminanthsf_negative_partial. ff_h_mce_mdr_second_integer_rankzeroevaluationdeterminanthsf_negative_partial + S (ff_r_mce_mdr_second_integer_rankzeroevaluationdeterminanthsf_negative) = S ((S (ff_i_mce_mdr_second_integer_rankzeroevaluationdeterminanthsf_negative)) * ff_v_mce_mdr_second_integer_rankzeroevaluationdeterminanthsf_negative)) /\ exists ff_q_mce_mdr_second_integer_rankzeroevaluationdeterminanthsf_negative_partial. ff_u_mce_mdr_second_integer_rankzeroevaluationdeterminanthsf_negative = ff_q_mce_mdr_second_integer_rankzeroevaluationdeterminanthsf_negative_partial * S ((S (ff_i_mce_mdr_second_integer_rankzeroevaluationdeterminanthsf_negative)) * ff_v_mce_mdr_second_integer_rankzeroevaluationdeterminanthsf_negative) + (ff_r_mce_mdr_second_integer_rankzeroevaluationdeterminanthsf_negative))) /\ ((((exists ff_h_mce_mdr_second_integer_rankzeroevaluationdeterminanthsf_negative_successor. ff_h_mce_mdr_second_integer_rankzeroevaluationdeterminanthsf_negative_successor + S (ff_s_mce_mdr_second_integer_rankzeroevaluationdeterminanthsf_negative) = S ((S (S ff_i_mce_mdr_second_integer_rankzeroevaluationdeterminanthsf_negative)) * ff_v_mce_mdr_second_integer_rankzeroevaluationdeterminanthsf_negative)) /\ exists ff_q_mce_mdr_second_integer_rankzeroevaluationdeterminanthsf_negative_successor. ff_u_mce_mdr_second_integer_rankzeroevaluationdeterminanthsf_negative = ff_q_mce_mdr_second_integer_rankzeroevaluationdeterminanthsf_negative_successor * S ((S (S ff_i_mce_mdr_second_integer_rankzeroevaluationdeterminanthsf_negative)) * ff_v_mce_mdr_second_integer_rankzeroevaluationdeterminanthsf_negative) + (ff_s_mce_mdr_second_integer_rankzeroevaluationdeterminanthsf_negative))) /\ ff_s_mce_mdr_second_integer_rankzeroevaluationdeterminanthsf_negative = ff_r_mce_mdr_second_integer_rankzeroevaluationdeterminanthsf_negative + ff_a_mce_mdr_second_integer_rankzeroevaluationdeterminanthsf_negative))))))))))))))) /\ ((exists mdr_gap_second_integer_rankzeroevaluationdeterminanti. mdr_gap_second_integer_rankzeroevaluationdeterminanti + S (mdr_i_second_integer_rankzeroevaluationdeterminant) = (mdr_l_second_integer_rankzeroevaluationdeterminant)) /\ (exists mdr_z_second_integer_rankzeroevaluationdeterminantr. ((exists mdr_a_second_integer_rankzeroevaluationdeterminantrc mdr_b_second_integer_rankzeroevaluationdeterminantrc mdr_c_second_integer_rankzeroevaluationdeterminantrc mdr_e_second_integer_rankzeroevaluationdeterminantrc mdr_f_second_integer_rankzeroevaluationdeterminantrc. ((mdr_a_second_integer_rankzeroevaluationdeterminantrc = ((mdr_q_second_integer_rank) + (mdr_ub_second_integer_rankzeroevaluation)) * S ((mdr_q_second_integer_rank) + (mdr_ub_second_integer_rankzeroevaluation)) + ((mdr_ub_second_integer_rankzeroevaluation) + (mdr_ub_second_integer_rankzeroevaluation))) /\ ((mdr_b_second_integer_rankzeroevaluationdeterminantrc = ((mdr_uc_second_integer_rankzeroevaluation) + (mdr_vb_second_integer_rankzeroevaluation)) * S ((mdr_uc_second_integer_rankzeroevaluation) + (mdr_vb_second_integer_rankzeroevaluation)) + ((mdr_vb_second_integer_rankzeroevaluation) + (mdr_vb_second_integer_rankzeroevaluation))) /\ ((mdr_c_second_integer_rankzeroevaluationdeterminantrc = ((mdr_a_second_integer_rankzeroevaluationdeterminantrc) + (mdr_b_second_integer_rankzeroevaluationdeterminantrc)) * S ((mdr_a_second_integer_rankzeroevaluationdeterminantrc) + (mdr_b_second_integer_rankzeroevaluationdeterminantrc)) + ((mdr_b_second_integer_rankzeroevaluationdeterminantrc) + (mdr_b_second_integer_rankzeroevaluationdeterminantrc))) /\ ((mdr_e_second_integer_rankzeroevaluationdeterminantrc = ((mdr_p_second_integer_rankzero) + (mdr_n_second_integer_rankzero)) * S ((mdr_p_second_integer_rankzero) + (mdr_n_second_integer_rankzero)) + ((mdr_n_second_integer_rankzero) + (mdr_n_second_integer_rankzero))) /\ ((mdr_f_second_integer_rankzeroevaluationdeterminantrc = ((mdr_vc_second_integer_rankzeroevaluation) + (mdr_e_second_integer_rankzeroevaluationdeterminantrc)) * S ((mdr_vc_second_integer_rankzeroevaluation) + (mdr_e_second_integer_rankzeroevaluationdeterminantrc)) + ((mdr_e_second_integer_rankzeroevaluationdeterminantrc) + (mdr_e_second_integer_rankzeroevaluationdeterminantrc))) /\ ((mdr_z_second_integer_rankzeroevaluationdeterminantr) = ((mdr_c_second_integer_rankzeroevaluationdeterminantrc) + (mdr_f_second_integer_rankzeroevaluationdeterminantrc)) * S ((mdr_c_second_integer_rankzeroevaluationdeterminantrc) + (mdr_f_second_integer_rankzeroevaluationdeterminantrc)) + ((mdr_f_second_integer_rankzeroevaluationdeterminantrc) + (mdr_f_second_integer_rankzeroevaluationdeterminantrc))))))))) /\ (((exists ff_h_mdr_second_integer_rankzeroevaluationdeterminantrb. ff_h_mdr_second_integer_rankzeroevaluationdeterminantrb + S (mdr_z_second_integer_rankzeroevaluationdeterminantr) = S ((S (mdr_i_second_integer_rankzeroevaluationdeterminant)) * mdr_c_second_integer_rankzeroevaluationdeterminant)) /\ exists ff_q_mdr_second_integer_rankzeroevaluationdeterminantrb. mdr_b_second_integer_rankzeroevaluationdeterminant = ff_q_mdr_second_integer_rankzeroevaluationdeterminantrb * S ((S (mdr_i_second_integer_rankzeroevaluationdeterminant)) * mdr_c_second_integer_rankzeroevaluationdeterminant) + (mdr_z_second_integer_rankzeroevaluationdeterminantr)))))))))) -> mdr_p_second_integer_rankzero = mdr_n_second_integer_rankzero))))))Complete tactic proof in conservative notation
All 53 original proof lines are preserved. Only local proposition formulas are abbreviated; every abbreviation has an exact binder-safe expansion check. The linked exact edition contains the unchanged replay script.
Read the argument
Proof checkpoints
53 script commands · 12 reading checkpoints · 0 local claims
This is a reading aid, not a new proof or a proof-tree certificate. Checkpoint groups are consecutive commands, not inferred branch boundaries. Every step links to the preserved script.
Definition notation is shown below. Open the paired exact edition for the original native formulas. Source pairing is not a new equivalence certificate.
Named ingredients (2)
01Fix variables and assumptionsL1–10
02Fix variables and assumptionsL11–13
03Separate the logical casesL14–17
04Use earlier factsL18–18
Instantiate or apply named facts and discharge the corresponding proof obligations.
- L18
exact hrank_left
05Separate the logical casesL19–19
Follow the explicit conjunction, disjunction, witness, or contradiction step recorded below.
- L19
split
06Use earlier factsL20–20
Instantiate or apply named facts and discharge the corresponding proof obligations.
- L20
exact hrank_right_left
07Separate the logical casesL21–21
Follow the explicit conjunction, disjunction, witness, or contradiction step recorded below.
- L21
split
08Use earlier factsL22–31
Instantiate or apply named facts and discharge the corresponding proof obligations.
- L22
specialize matrix_integer_nonzero_minor_transport (ab) - L23
specialize matrix_integer_nonzero_minor_transport (ac) - L24
specialize matrix_integer_nonzero_minor_transport (bb) - L25
specialize matrix_integer_nonzero_minor_transport (bc) - L26
specialize matrix_integer_nonzero_minor_transport (eb) - L27
specialize matrix_integer_nonzero_minor_transport (ec) - L28
specialize matrix_integer_nonzero_minor_transport (fb) - L29
specialize matrix_integer_nonzero_minor_transport (fc) - L30
specialize matrix_integer_nonzero_minor_transport (r) - L31
specialize matrix_integer_nonzero_minor_transport (w)
09Use earlier factsL32–35
10Fix variables and assumptionsL36–37
11Use earlier factsL38–47
Instantiate or apply named facts and discharge the corresponding proof obligations.
- L38
specialize matrix_integer_all_minors_zero_transport (ab) - L39
specialize matrix_integer_all_minors_zero_transport (ac) - L40
specialize matrix_integer_all_minors_zero_transport (bb) - L41
specialize matrix_integer_all_minors_zero_transport (bc) - L42
specialize matrix_integer_all_minors_zero_transport (eb) - L43
specialize matrix_integer_all_minors_zero_transport (ec) - L44
specialize matrix_integer_all_minors_zero_transport (fb) - L45
specialize matrix_integer_all_minors_zero_transport (fc) - L46
specialize matrix_integer_all_minors_zero_transport (r) - L47
specialize matrix_integer_all_minors_zero_transport (w)
12Use earlier factsL48–53
Original defined command ledger · 53 lines
- 0001
intro ab - 0002
intro ac - 0003
intro bb - 0004
intro bc - 0005
intro eb - 0006
intro ec - 0007
intro fb - 0008
intro fc - 0009
intro r - 0010
intro w - 0011
intro rank - 0012
intro hequal - 0013
intro hrank - 0014
cases hrank - 0015
cases hrank_right - 0016
cases hrank_right_right - 0017
split - 0018
exact hrank_left - 0019
split - 0020
exact hrank_right_left - 0021
split - 0022
specialize matrix_integer_nonzero_minor_transport (ab) - 0023
specialize matrix_integer_nonzero_minor_transport (ac) - 0024
specialize matrix_integer_nonzero_minor_transport (bb) - 0025
specialize matrix_integer_nonzero_minor_transport (bc) - 0026
specialize matrix_integer_nonzero_minor_transport (eb) - 0027
specialize matrix_integer_nonzero_minor_transport (ec) - 0028
specialize matrix_integer_nonzero_minor_transport (fb) - 0029
specialize matrix_integer_nonzero_minor_transport (fc) - 0030
specialize matrix_integer_nonzero_minor_transport (r) - 0031
specialize matrix_integer_nonzero_minor_transport (w) - 0032
specialize matrix_integer_nonzero_minor_transport (rank) - 0033
apply matrix_integer_nonzero_minor_transport - 0034
exact hequal - 0035
exact hrank_right_right_left - 0036
intro q - 0037
intro hq - 0038
specialize matrix_integer_all_minors_zero_transport (ab) - 0039
specialize matrix_integer_all_minors_zero_transport (ac) - 0040
specialize matrix_integer_all_minors_zero_transport (bb) - 0041
specialize matrix_integer_all_minors_zero_transport (bc) - 0042
specialize matrix_integer_all_minors_zero_transport (eb) - 0043
specialize matrix_integer_all_minors_zero_transport (ec) - 0044
specialize matrix_integer_all_minors_zero_transport (fb) - 0045
specialize matrix_integer_all_minors_zero_transport (fc) - 0046
specialize matrix_integer_all_minors_zero_transport (r) - 0047
specialize matrix_integer_all_minors_zero_transport (w) - 0048
specialize matrix_integer_all_minors_zero_transport (q) - 0049
apply matrix_integer_all_minors_zero_transport - 0050
exact hequal - 0051
specialize hrank_right_right_right (q) - 0052
apply hrank_right_right_right - 0053
exact hq