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. ∀ other. 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,other) → rank = otherEvery 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 other. (forall ics_index_rank_integer_equality ics_value0_rank_integer_equality ics_value1_rank_integer_equality ics_value2_rank_integer_equality ics_value3_rank_integer_equality. (exists ics_gap_rank_integer_equality_bound. ics_gap_rank_integer_equality_bound + S (ics_index_rank_integer_equality) = ((r) * (w))) -> (((exists fs_h_ics_rank_integer_equality_at0. fs_h_ics_rank_integer_equality_at0 + S (ics_value0_rank_integer_equality) = S ((S (ics_index_rank_integer_equality)) * ac)) /\ exists fs_q_ics_rank_integer_equality_at0. ab = fs_q_ics_rank_integer_equality_at0 * S ((S (ics_index_rank_integer_equality)) * ac) + (ics_value0_rank_integer_equality))) -> (((exists fs_h_ics_rank_integer_equality_at1. fs_h_ics_rank_integer_equality_at1 + S (ics_value1_rank_integer_equality) = S ((S (ics_index_rank_integer_equality)) * bc)) /\ exists fs_q_ics_rank_integer_equality_at1. bb = fs_q_ics_rank_integer_equality_at1 * S ((S (ics_index_rank_integer_equality)) * bc) + (ics_value1_rank_integer_equality))) -> (((exists fs_h_ics_rank_integer_equality_at2. fs_h_ics_rank_integer_equality_at2 + S (ics_value2_rank_integer_equality) = S ((S (ics_index_rank_integer_equality)) * ec)) /\ exists fs_q_ics_rank_integer_equality_at2. eb = fs_q_ics_rank_integer_equality_at2 * S ((S (ics_index_rank_integer_equality)) * ec) + (ics_value2_rank_integer_equality))) -> (((exists fs_h_ics_rank_integer_equality_at3. fs_h_ics_rank_integer_equality_at3 + S (ics_value3_rank_integer_equality) = S ((S (ics_index_rank_integer_equality)) * fc)) /\ exists fs_q_ics_rank_integer_equality_at3. fb = fs_q_ics_rank_integer_equality_at3 * S ((S (ics_index_rank_integer_equality)) * fc) + (ics_value3_rank_integer_equality))) -> ics_value0_rank_integer_equality + ics_value3_rank_integer_equality = ics_value2_rank_integer_equality + ics_value1_rank_integer_equality) -> (((exists mdr_gap_rank_integer_firstrows_bound. mdr_gap_rank_integer_firstrows_bound + (rank) = (r)) /\ ((exists mdr_gap_rank_integer_firstcolumns_bound. mdr_gap_rank_integer_firstcolumns_bound + (rank) = (w)) /\ ((exists mdr_rb_rank_integer_firstwitness mdr_rc_rank_integer_firstwitness mdr_cb_rank_integer_firstwitness mdr_cc_rank_integer_firstwitness. (((((forall fom_index_mrf_rank_integer_firstwitnessminorrowsbound. (exists fom_gap_mrf_rank_integer_firstwitnessminorrowsbound_index_bound. fom_gap_mrf_rank_integer_firstwitnessminorrowsbound_index_bound + S (fom_index_mrf_rank_integer_firstwitnessminorrowsbound) = rank) -> exists fom_value_mrf_rank_integer_firstwitnessminorrowsbound. ((((exists fom_beta_height_mrf_rank_integer_firstwitnessminorrowsbound_entry. fom_beta_height_mrf_rank_integer_firstwitnessminorrowsbound_entry + S (fom_value_mrf_rank_integer_firstwitnessminorrowsbound) = S ((S (fom_index_mrf_rank_integer_firstwitnessminorrowsbound)) * mdr_rc_rank_integer_firstwitness)) /\ exists fom_beta_quotient_mrf_rank_integer_firstwitnessminorrowsbound_entry. mdr_rb_rank_integer_firstwitness = fom_beta_quotient_mrf_rank_integer_firstwitnessminorrowsbound_entry * S ((S (fom_index_mrf_rank_integer_firstwitnessminorrowsbound)) * mdr_rc_rank_integer_firstwitness) + (fom_value_mrf_rank_integer_firstwitnessminorrowsbound))) /\ (exists fom_gap_mrf_rank_integer_firstwitnessminorrowsbound_value_bound. fom_gap_mrf_rank_integer_firstwitnessminorrowsbound_value_bound + S (fom_value_mrf_rank_integer_firstwitnessminorrowsbound) = r))) /\ (forall mdr_i_rank_integer_firstwitnessminorrowsdistinct mdr_j_rank_integer_firstwitnessminorrowsdistinct mdr_a_rank_integer_firstwitnessminorrowsdistinct. (exists mdr_gap_rank_integer_firstwitnessminorrowsdistincti. mdr_gap_rank_integer_firstwitnessminorrowsdistincti + S (mdr_i_rank_integer_firstwitnessminorrowsdistinct) = (rank)) -> (exists mdr_gap_rank_integer_firstwitnessminorrowsdistinctj. mdr_gap_rank_integer_firstwitnessminorrowsdistinctj + S (mdr_j_rank_integer_firstwitnessminorrowsdistinct) = (rank)) -> (((exists ff_h_mdr_rank_integer_firstwitnessminorrowsdistinctfirst. ff_h_mdr_rank_integer_firstwitnessminorrowsdistinctfirst + S (mdr_a_rank_integer_firstwitnessminorrowsdistinct) = S ((S (mdr_i_rank_integer_firstwitnessminorrowsdistinct)) * mdr_rc_rank_integer_firstwitness)) /\ exists ff_q_mdr_rank_integer_firstwitnessminorrowsdistinctfirst. mdr_rb_rank_integer_firstwitness = ff_q_mdr_rank_integer_firstwitnessminorrowsdistinctfirst * S ((S (mdr_i_rank_integer_firstwitnessminorrowsdistinct)) * mdr_rc_rank_integer_firstwitness) + (mdr_a_rank_integer_firstwitnessminorrowsdistinct))) -> (((exists ff_h_mdr_rank_integer_firstwitnessminorrowsdistinctsecond. ff_h_mdr_rank_integer_firstwitnessminorrowsdistinctsecond + S (mdr_a_rank_integer_firstwitnessminorrowsdistinct) = S ((S (mdr_j_rank_integer_firstwitnessminorrowsdistinct)) * mdr_rc_rank_integer_firstwitness)) /\ exists ff_q_mdr_rank_integer_firstwitnessminorrowsdistinctsecond. mdr_rb_rank_integer_firstwitness = ff_q_mdr_rank_integer_firstwitnessminorrowsdistinctsecond * S ((S (mdr_j_rank_integer_firstwitnessminorrowsdistinct)) * mdr_rc_rank_integer_firstwitness) + (mdr_a_rank_integer_firstwitnessminorrowsdistinct))) -> mdr_i_rank_integer_firstwitnessminorrowsdistinct = mdr_j_rank_integer_firstwitnessminorrowsdistinct))) /\ ((((forall fom_index_mrf_rank_integer_firstwitnessminorcolumnsbound. (exists fom_gap_mrf_rank_integer_firstwitnessminorcolumnsbound_index_bound. fom_gap_mrf_rank_integer_firstwitnessminorcolumnsbound_index_bound + S (fom_index_mrf_rank_integer_firstwitnessminorcolumnsbound) = rank) -> exists fom_value_mrf_rank_integer_firstwitnessminorcolumnsbound. ((((exists fom_beta_height_mrf_rank_integer_firstwitnessminorcolumnsbound_entry. fom_beta_height_mrf_rank_integer_firstwitnessminorcolumnsbound_entry + S (fom_value_mrf_rank_integer_firstwitnessminorcolumnsbound) = S ((S (fom_index_mrf_rank_integer_firstwitnessminorcolumnsbound)) * mdr_cc_rank_integer_firstwitness)) /\ exists fom_beta_quotient_mrf_rank_integer_firstwitnessminorcolumnsbound_entry. mdr_cb_rank_integer_firstwitness = fom_beta_quotient_mrf_rank_integer_firstwitnessminorcolumnsbound_entry * S ((S (fom_index_mrf_rank_integer_firstwitnessminorcolumnsbound)) * mdr_cc_rank_integer_firstwitness) + (fom_value_mrf_rank_integer_firstwitnessminorcolumnsbound))) /\ (exists fom_gap_mrf_rank_integer_firstwitnessminorcolumnsbound_value_bound. fom_gap_mrf_rank_integer_firstwitnessminorcolumnsbound_value_bound + S (fom_value_mrf_rank_integer_firstwitnessminorcolumnsbound) = w))) /\ (forall mdr_i_rank_integer_firstwitnessminorcolumnsdistinct mdr_j_rank_integer_firstwitnessminorcolumnsdistinct mdr_a_rank_integer_firstwitnessminorcolumnsdistinct. (exists mdr_gap_rank_integer_firstwitnessminorcolumnsdistincti. mdr_gap_rank_integer_firstwitnessminorcolumnsdistincti + S (mdr_i_rank_integer_firstwitnessminorcolumnsdistinct) = (rank)) -> (exists mdr_gap_rank_integer_firstwitnessminorcolumnsdistinctj. mdr_gap_rank_integer_firstwitnessminorcolumnsdistinctj + S (mdr_j_rank_integer_firstwitnessminorcolumnsdistinct) = (rank)) -> (((exists ff_h_mdr_rank_integer_firstwitnessminorcolumnsdistinctfirst. ff_h_mdr_rank_integer_firstwitnessminorcolumnsdistinctfirst + S (mdr_a_rank_integer_firstwitnessminorcolumnsdistinct) = S ((S (mdr_i_rank_integer_firstwitnessminorcolumnsdistinct)) * mdr_cc_rank_integer_firstwitness)) /\ exists ff_q_mdr_rank_integer_firstwitnessminorcolumnsdistinctfirst. mdr_cb_rank_integer_firstwitness = ff_q_mdr_rank_integer_firstwitnessminorcolumnsdistinctfirst * S ((S (mdr_i_rank_integer_firstwitnessminorcolumnsdistinct)) * mdr_cc_rank_integer_firstwitness) + (mdr_a_rank_integer_firstwitnessminorcolumnsdistinct))) -> (((exists ff_h_mdr_rank_integer_firstwitnessminorcolumnsdistinctsecond. ff_h_mdr_rank_integer_firstwitnessminorcolumnsdistinctsecond + S (mdr_a_rank_integer_firstwitnessminorcolumnsdistinct) = S ((S (mdr_j_rank_integer_firstwitnessminorcolumnsdistinct)) * mdr_cc_rank_integer_firstwitness)) /\ exists ff_q_mdr_rank_integer_firstwitnessminorcolumnsdistinctsecond. mdr_cb_rank_integer_firstwitness = ff_q_mdr_rank_integer_firstwitnessminorcolumnsdistinctsecond * S ((S (mdr_j_rank_integer_firstwitnessminorcolumnsdistinct)) * mdr_cc_rank_integer_firstwitness) + (mdr_a_rank_integer_firstwitnessminorcolumnsdistinct))) -> mdr_i_rank_integer_firstwitnessminorcolumnsdistinct = mdr_j_rank_integer_firstwitnessminorcolumnsdistinct))) /\ (exists mdr_p_rank_integer_firstwitnessminornonzero mdr_n_rank_integer_firstwitnessminornonzero. ((exists mdr_ub_rank_integer_firstwitnessminornonzeroevaluation mdr_uc_rank_integer_firstwitnessminornonzeroevaluation mdr_vb_rank_integer_firstwitnessminornonzeroevaluation mdr_vc_rank_integer_firstwitnessminornonzeroevaluation. ((((forall mdr_i_rank_integer_firstwitnessminornonzeroevaluationmatrixpositive. (exists mdr_gap_rank_integer_firstwitnessminornonzeroevaluationmatrixpositivebound. mdr_gap_rank_integer_firstwitnessminornonzeroevaluationmatrixpositivebound + S (mdr_i_rank_integer_firstwitnessminornonzeroevaluationmatrixpositive) = ((rank) * (rank))) -> exists mdr_a_rank_integer_firstwitnessminornonzeroevaluationmatrixpositive. (((exists mdr_r_rank_integer_firstwitnessminornonzeroevaluationmatrixpositivepoint mdr_s_rank_integer_firstwitnessminornonzeroevaluationmatrixpositivepoint mdr_u_rank_integer_firstwitnessminornonzeroevaluationmatrixpositivepoint mdr_v_rank_integer_firstwitnessminornonzeroevaluationmatrixpositivepoint. ((mdr_i_rank_integer_firstwitnessminornonzeroevaluationmatrixpositive = (rank) * mdr_r_rank_integer_firstwitnessminornonzeroevaluationmatrixpositivepoint + mdr_s_rank_integer_firstwitnessminornonzeroevaluationmatrixpositivepoint) /\ ((exists mdr_gap_rank_integer_firstwitnessminornonzeroevaluationmatrixpositivepointcolumn. mdr_gap_rank_integer_firstwitnessminornonzeroevaluationmatrixpositivepointcolumn + S (mdr_s_rank_integer_firstwitnessminornonzeroevaluationmatrixpositivepoint) = (rank)) /\ ((((exists ff_h_mdr_rank_integer_firstwitnessminornonzeroevaluationmatrixpositivepointrow_index. ff_h_mdr_rank_integer_firstwitnessminornonzeroevaluationmatrixpositivepointrow_index + S (mdr_u_rank_integer_firstwitnessminornonzeroevaluationmatrixpositivepoint) = S ((S (mdr_r_rank_integer_firstwitnessminornonzeroevaluationmatrixpositivepoint)) * mdr_rc_rank_integer_firstwitness)) /\ exists ff_q_mdr_rank_integer_firstwitnessminornonzeroevaluationmatrixpositivepointrow_index. mdr_rb_rank_integer_firstwitness = ff_q_mdr_rank_integer_firstwitnessminornonzeroevaluationmatrixpositivepointrow_index * S ((S (mdr_r_rank_integer_firstwitnessminornonzeroevaluationmatrixpositivepoint)) * mdr_rc_rank_integer_firstwitness) + (mdr_u_rank_integer_firstwitnessminornonzeroevaluationmatrixpositivepoint))) /\ ((((exists ff_h_mdr_rank_integer_firstwitnessminornonzeroevaluationmatrixpositivepointcolumn_index. ff_h_mdr_rank_integer_firstwitnessminornonzeroevaluationmatrixpositivepointcolumn_index + S (mdr_v_rank_integer_firstwitnessminornonzeroevaluationmatrixpositivepoint) = S ((S (mdr_s_rank_integer_firstwitnessminornonzeroevaluationmatrixpositivepoint)) * mdr_cc_rank_integer_firstwitness)) /\ exists ff_q_mdr_rank_integer_firstwitnessminornonzeroevaluationmatrixpositivepointcolumn_index. mdr_cb_rank_integer_firstwitness = ff_q_mdr_rank_integer_firstwitnessminornonzeroevaluationmatrixpositivepointcolumn_index * S ((S (mdr_s_rank_integer_firstwitnessminornonzeroevaluationmatrixpositivepoint)) * mdr_cc_rank_integer_firstwitness) + (mdr_v_rank_integer_firstwitnessminornonzeroevaluationmatrixpositivepoint))) /\ (((exists ff_h_mdr_rank_integer_firstwitnessminornonzeroevaluationmatrixpositivepointsource. ff_h_mdr_rank_integer_firstwitnessminornonzeroevaluationmatrixpositivepointsource + S (mdr_a_rank_integer_firstwitnessminornonzeroevaluationmatrixpositive) = S ((S ((mdr_u_rank_integer_firstwitnessminornonzeroevaluationmatrixpositivepoint) * (w) + (mdr_v_rank_integer_firstwitnessminornonzeroevaluationmatrixpositivepoint))) * ac)) /\ exists ff_q_mdr_rank_integer_firstwitnessminornonzeroevaluationmatrixpositivepointsource. ab = ff_q_mdr_rank_integer_firstwitnessminornonzeroevaluationmatrixpositivepointsource * S ((S ((mdr_u_rank_integer_firstwitnessminornonzeroevaluationmatrixpositivepoint) * (w) + (mdr_v_rank_integer_firstwitnessminornonzeroevaluationmatrixpositivepoint))) * ac) + (mdr_a_rank_integer_firstwitnessminornonzeroevaluationmatrixpositive)))))))) /\ (((exists ff_h_mdr_rank_integer_firstwitnessminornonzeroevaluationmatrixpositiveoutput. ff_h_mdr_rank_integer_firstwitnessminornonzeroevaluationmatrixpositiveoutput + S (mdr_a_rank_integer_firstwitnessminornonzeroevaluationmatrixpositive) = S ((S (mdr_i_rank_integer_firstwitnessminornonzeroevaluationmatrixpositive)) * mdr_uc_rank_integer_firstwitnessminornonzeroevaluation)) /\ exists ff_q_mdr_rank_integer_firstwitnessminornonzeroevaluationmatrixpositiveoutput. mdr_ub_rank_integer_firstwitnessminornonzeroevaluation = ff_q_mdr_rank_integer_firstwitnessminornonzeroevaluationmatrixpositiveoutput * S ((S (mdr_i_rank_integer_firstwitnessminornonzeroevaluationmatrixpositive)) * mdr_uc_rank_integer_firstwitnessminornonzeroevaluation) + (mdr_a_rank_integer_firstwitnessminornonzeroevaluationmatrixpositive)))))) /\ (forall mdr_i_rank_integer_firstwitnessminornonzeroevaluationmatrixnegative. (exists mdr_gap_rank_integer_firstwitnessminornonzeroevaluationmatrixnegativebound. mdr_gap_rank_integer_firstwitnessminornonzeroevaluationmatrixnegativebound + S (mdr_i_rank_integer_firstwitnessminornonzeroevaluationmatrixnegative) = ((rank) * (rank))) -> exists mdr_a_rank_integer_firstwitnessminornonzeroevaluationmatrixnegative. (((exists mdr_r_rank_integer_firstwitnessminornonzeroevaluationmatrixnegativepoint mdr_s_rank_integer_firstwitnessminornonzeroevaluationmatrixnegativepoint mdr_u_rank_integer_firstwitnessminornonzeroevaluationmatrixnegativepoint mdr_v_rank_integer_firstwitnessminornonzeroevaluationmatrixnegativepoint. ((mdr_i_rank_integer_firstwitnessminornonzeroevaluationmatrixnegative = (rank) * mdr_r_rank_integer_firstwitnessminornonzeroevaluationmatrixnegativepoint + mdr_s_rank_integer_firstwitnessminornonzeroevaluationmatrixnegativepoint) /\ ((exists mdr_gap_rank_integer_firstwitnessminornonzeroevaluationmatrixnegativepointcolumn. mdr_gap_rank_integer_firstwitnessminornonzeroevaluationmatrixnegativepointcolumn + S (mdr_s_rank_integer_firstwitnessminornonzeroevaluationmatrixnegativepoint) = (rank)) /\ ((((exists ff_h_mdr_rank_integer_firstwitnessminornonzeroevaluationmatrixnegativepointrow_index. ff_h_mdr_rank_integer_firstwitnessminornonzeroevaluationmatrixnegativepointrow_index + S (mdr_u_rank_integer_firstwitnessminornonzeroevaluationmatrixnegativepoint) = S ((S (mdr_r_rank_integer_firstwitnessminornonzeroevaluationmatrixnegativepoint)) * mdr_rc_rank_integer_firstwitness)) /\ exists ff_q_mdr_rank_integer_firstwitnessminornonzeroevaluationmatrixnegativepointrow_index. mdr_rb_rank_integer_firstwitness = ff_q_mdr_rank_integer_firstwitnessminornonzeroevaluationmatrixnegativepointrow_index * S ((S (mdr_r_rank_integer_firstwitnessminornonzeroevaluationmatrixnegativepoint)) * mdr_rc_rank_integer_firstwitness) + (mdr_u_rank_integer_firstwitnessminornonzeroevaluationmatrixnegativepoint))) /\ ((((exists ff_h_mdr_rank_integer_firstwitnessminornonzeroevaluationmatrixnegativepointcolumn_index. ff_h_mdr_rank_integer_firstwitnessminornonzeroevaluationmatrixnegativepointcolumn_index + S (mdr_v_rank_integer_firstwitnessminornonzeroevaluationmatrixnegativepoint) = S ((S (mdr_s_rank_integer_firstwitnessminornonzeroevaluationmatrixnegativepoint)) * mdr_cc_rank_integer_firstwitness)) /\ exists ff_q_mdr_rank_integer_firstwitnessminornonzeroevaluationmatrixnegativepointcolumn_index. mdr_cb_rank_integer_firstwitness = ff_q_mdr_rank_integer_firstwitnessminornonzeroevaluationmatrixnegativepointcolumn_index * S ((S (mdr_s_rank_integer_firstwitnessminornonzeroevaluationmatrixnegativepoint)) * mdr_cc_rank_integer_firstwitness) + (mdr_v_rank_integer_firstwitnessminornonzeroevaluationmatrixnegativepoint))) /\ (((exists ff_h_mdr_rank_integer_firstwitnessminornonzeroevaluationmatrixnegativepointsource. ff_h_mdr_rank_integer_firstwitnessminornonzeroevaluationmatrixnegativepointsource + S (mdr_a_rank_integer_firstwitnessminornonzeroevaluationmatrixnegative) = S ((S ((mdr_u_rank_integer_firstwitnessminornonzeroevaluationmatrixnegativepoint) * (w) + (mdr_v_rank_integer_firstwitnessminornonzeroevaluationmatrixnegativepoint))) * bc)) /\ exists ff_q_mdr_rank_integer_firstwitnessminornonzeroevaluationmatrixnegativepointsource. bb = ff_q_mdr_rank_integer_firstwitnessminornonzeroevaluationmatrixnegativepointsource * S ((S ((mdr_u_rank_integer_firstwitnessminornonzeroevaluationmatrixnegativepoint) * (w) + (mdr_v_rank_integer_firstwitnessminornonzeroevaluationmatrixnegativepoint))) * bc) + (mdr_a_rank_integer_firstwitnessminornonzeroevaluationmatrixnegative)))))))) /\ (((exists ff_h_mdr_rank_integer_firstwitnessminornonzeroevaluationmatrixnegativeoutput. ff_h_mdr_rank_integer_firstwitnessminornonzeroevaluationmatrixnegativeoutput + S (mdr_a_rank_integer_firstwitnessminornonzeroevaluationmatrixnegative) = S ((S (mdr_i_rank_integer_firstwitnessminornonzeroevaluationmatrixnegative)) * mdr_vc_rank_integer_firstwitnessminornonzeroevaluation)) /\ exists ff_q_mdr_rank_integer_firstwitnessminornonzeroevaluationmatrixnegativeoutput. mdr_vb_rank_integer_firstwitnessminornonzeroevaluation = ff_q_mdr_rank_integer_firstwitnessminornonzeroevaluationmatrixnegativeoutput * S ((S (mdr_i_rank_integer_firstwitnessminornonzeroevaluationmatrixnegative)) * mdr_vc_rank_integer_firstwitnessminornonzeroevaluation) + (mdr_a_rank_integer_firstwitnessminornonzeroevaluationmatrixnegative)))))))) /\ (exists mdr_b_rank_integer_firstwitnessminornonzeroevaluationdeterminant mdr_c_rank_integer_firstwitnessminornonzeroevaluationdeterminant mdr_l_rank_integer_firstwitnessminornonzeroevaluationdeterminant mdr_i_rank_integer_firstwitnessminornonzeroevaluationdeterminant. ((forall mdr_i_rank_integer_firstwitnessminornonzeroevaluationdeterminanth. (exists mdr_gap_rank_integer_firstwitnessminornonzeroevaluationdeterminanthi. mdr_gap_rank_integer_firstwitnessminornonzeroevaluationdeterminanthi + S (mdr_i_rank_integer_firstwitnessminornonzeroevaluationdeterminanth) = (mdr_l_rank_integer_firstwitnessminornonzeroevaluationdeterminant)) -> exists mdr_d_rank_integer_firstwitnessminornonzeroevaluationdeterminanth mdr_pb_rank_integer_firstwitnessminornonzeroevaluationdeterminanth mdr_pc_rank_integer_firstwitnessminornonzeroevaluationdeterminanth mdr_nb_rank_integer_firstwitnessminornonzeroevaluationdeterminanth mdr_nc_rank_integer_firstwitnessminornonzeroevaluationdeterminanth mdr_p_rank_integer_firstwitnessminornonzeroevaluationdeterminanth mdr_n_rank_integer_firstwitnessminornonzeroevaluationdeterminanth. ((exists mdr_z_rank_integer_firstwitnessminornonzeroevaluationdeterminanthr. ((exists mdr_a_rank_integer_firstwitnessminornonzeroevaluationdeterminanthrc mdr_b_rank_integer_firstwitnessminornonzeroevaluationdeterminanthrc mdr_c_rank_integer_firstwitnessminornonzeroevaluationdeterminanthrc mdr_e_rank_integer_firstwitnessminornonzeroevaluationdeterminanthrc mdr_f_rank_integer_firstwitnessminornonzeroevaluationdeterminanthrc. ((mdr_a_rank_integer_firstwitnessminornonzeroevaluationdeterminanthrc = ((mdr_d_rank_integer_firstwitnessminornonzeroevaluationdeterminanth) + (mdr_pb_rank_integer_firstwitnessminornonzeroevaluationdeterminanth)) * S ((mdr_d_rank_integer_firstwitnessminornonzeroevaluationdeterminanth) + (mdr_pb_rank_integer_firstwitnessminornonzeroevaluationdeterminanth)) + ((mdr_pb_rank_integer_firstwitnessminornonzeroevaluationdeterminanth) + (mdr_pb_rank_integer_firstwitnessminornonzeroevaluationdeterminanth))) /\ ((mdr_b_rank_integer_firstwitnessminornonzeroevaluationdeterminanthrc = ((mdr_pc_rank_integer_firstwitnessminornonzeroevaluationdeterminanth) + (mdr_nb_rank_integer_firstwitnessminornonzeroevaluationdeterminanth)) * S ((mdr_pc_rank_integer_firstwitnessminornonzeroevaluationdeterminanth) + (mdr_nb_rank_integer_firstwitnessminornonzeroevaluationdeterminanth)) + ((mdr_nb_rank_integer_firstwitnessminornonzeroevaluationdeterminanth) + (mdr_nb_rank_integer_firstwitnessminornonzeroevaluationdeterminanth))) /\ ((mdr_c_rank_integer_firstwitnessminornonzeroevaluationdeterminanthrc = ((mdr_a_rank_integer_firstwitnessminornonzeroevaluationdeterminanthrc) + (mdr_b_rank_integer_firstwitnessminornonzeroevaluationdeterminanthrc)) * S ((mdr_a_rank_integer_firstwitnessminornonzeroevaluationdeterminanthrc) + (mdr_b_rank_integer_firstwitnessminornonzeroevaluationdeterminanthrc)) + ((mdr_b_rank_integer_firstwitnessminornonzeroevaluationdeterminanthrc) + (mdr_b_rank_integer_firstwitnessminornonzeroevaluationdeterminanthrc))) /\ ((mdr_e_rank_integer_firstwitnessminornonzeroevaluationdeterminanthrc = ((mdr_p_rank_integer_firstwitnessminornonzeroevaluationdeterminanth) + (mdr_n_rank_integer_firstwitnessminornonzeroevaluationdeterminanth)) * S ((mdr_p_rank_integer_firstwitnessminornonzeroevaluationdeterminanth) + (mdr_n_rank_integer_firstwitnessminornonzeroevaluationdeterminanth)) + ((mdr_n_rank_integer_firstwitnessminornonzeroevaluationdeterminanth) + (mdr_n_rank_integer_firstwitnessminornonzeroevaluationdeterminanth))) /\ ((mdr_f_rank_integer_firstwitnessminornonzeroevaluationdeterminanthrc = ((mdr_nc_rank_integer_firstwitnessminornonzeroevaluationdeterminanth) + (mdr_e_rank_integer_firstwitnessminornonzeroevaluationdeterminanthrc)) * S ((mdr_nc_rank_integer_firstwitnessminornonzeroevaluationdeterminanth) + (mdr_e_rank_integer_firstwitnessminornonzeroevaluationdeterminanthrc)) + ((mdr_e_rank_integer_firstwitnessminornonzeroevaluationdeterminanthrc) + (mdr_e_rank_integer_firstwitnessminornonzeroevaluationdeterminanthrc))) /\ ((mdr_z_rank_integer_firstwitnessminornonzeroevaluationdeterminanthr) = ((mdr_c_rank_integer_firstwitnessminornonzeroevaluationdeterminanthrc) + (mdr_f_rank_integer_firstwitnessminornonzeroevaluationdeterminanthrc)) * S ((mdr_c_rank_integer_firstwitnessminornonzeroevaluationdeterminanthrc) + (mdr_f_rank_integer_firstwitnessminornonzeroevaluationdeterminanthrc)) + ((mdr_f_rank_integer_firstwitnessminornonzeroevaluationdeterminanthrc) + (mdr_f_rank_integer_firstwitnessminornonzeroevaluationdeterminanthrc))))))))) /\ (((exists ff_h_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthrb. ff_h_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthrb + S (mdr_z_rank_integer_firstwitnessminornonzeroevaluationdeterminanthr) = S ((S (mdr_i_rank_integer_firstwitnessminornonzeroevaluationdeterminanth)) * mdr_c_rank_integer_firstwitnessminornonzeroevaluationdeterminant)) /\ exists ff_q_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthrb. mdr_b_rank_integer_firstwitnessminornonzeroevaluationdeterminant = ff_q_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthrb * S ((S (mdr_i_rank_integer_firstwitnessminornonzeroevaluationdeterminanth)) * mdr_c_rank_integer_firstwitnessminornonzeroevaluationdeterminant) + (mdr_z_rank_integer_firstwitnessminornonzeroevaluationdeterminanthr))))) /\ (((((mdr_d_rank_integer_firstwitnessminornonzeroevaluationdeterminanth) = 0) /\ (((mdr_p_rank_integer_firstwitnessminornonzeroevaluationdeterminanth) = 1) /\ ((mdr_n_rank_integer_firstwitnessminornonzeroevaluationdeterminanth) = 0))) \/ exists mdr_q_rank_integer_firstwitnessminornonzeroevaluationdeterminanths mdr_eb_rank_integer_firstwitnessminornonzeroevaluationdeterminanths mdr_ec_rank_integer_firstwitnessminornonzeroevaluationdeterminanths mdr_fb_rank_integer_firstwitnessminornonzeroevaluationdeterminanths mdr_fc_rank_integer_firstwitnessminornonzeroevaluationdeterminanths. (((mdr_d_rank_integer_firstwitnessminornonzeroevaluationdeterminanth) = S (mdr_q_rank_integer_firstwitnessminornonzeroevaluationdeterminanths)) /\ ((forall mdr_j_rank_integer_firstwitnessminornonzeroevaluationdeterminanthsc. (exists mdr_gap_rank_integer_firstwitnessminornonzeroevaluationdeterminanthscj. mdr_gap_rank_integer_firstwitnessminornonzeroevaluationdeterminanthscj + S (mdr_j_rank_integer_firstwitnessminornonzeroevaluationdeterminanthsc) = (S (mdr_q_rank_integer_firstwitnessminornonzeroevaluationdeterminanths))) -> exists mdr_i_rank_integer_firstwitnessminornonzeroevaluationdeterminanthsc mdr_up_rank_integer_firstwitnessminornonzeroevaluationdeterminanthsc mdr_us_rank_integer_firstwitnessminornonzeroevaluationdeterminanthsc mdr_un_rank_integer_firstwitnessminornonzeroevaluationdeterminanthsc mdr_ut_rank_integer_firstwitnessminornonzeroevaluationdeterminanthsc mdr_p_rank_integer_firstwitnessminornonzeroevaluationdeterminanthsc mdr_n_rank_integer_firstwitnessminornonzeroevaluationdeterminanthsc. ((exists mdr_gap_rank_integer_firstwitnessminornonzeroevaluationdeterminanthsci. mdr_gap_rank_integer_firstwitnessminornonzeroevaluationdeterminanthsci + S (mdr_i_rank_integer_firstwitnessminornonzeroevaluationdeterminanthsc) = (mdr_i_rank_integer_firstwitnessminornonzeroevaluationdeterminanth)) /\ ((exists mdr_z_rank_integer_firstwitnessminornonzeroevaluationdeterminanthscr. ((exists mdr_a_rank_integer_firstwitnessminornonzeroevaluationdeterminanthscrc mdr_b_rank_integer_firstwitnessminornonzeroevaluationdeterminanthscrc mdr_c_rank_integer_firstwitnessminornonzeroevaluationdeterminanthscrc mdr_e_rank_integer_firstwitnessminornonzeroevaluationdeterminanthscrc mdr_f_rank_integer_firstwitnessminornonzeroevaluationdeterminanthscrc. ((mdr_a_rank_integer_firstwitnessminornonzeroevaluationdeterminanthscrc = ((mdr_q_rank_integer_firstwitnessminornonzeroevaluationdeterminanths) + (mdr_up_rank_integer_firstwitnessminornonzeroevaluationdeterminanthsc)) * S ((mdr_q_rank_integer_firstwitnessminornonzeroevaluationdeterminanths) + (mdr_up_rank_integer_firstwitnessminornonzeroevaluationdeterminanthsc)) + ((mdr_up_rank_integer_firstwitnessminornonzeroevaluationdeterminanthsc) + (mdr_up_rank_integer_firstwitnessminornonzeroevaluationdeterminanthsc))) /\ ((mdr_b_rank_integer_firstwitnessminornonzeroevaluationdeterminanthscrc = ((mdr_us_rank_integer_firstwitnessminornonzeroevaluationdeterminanthsc) + (mdr_un_rank_integer_firstwitnessminornonzeroevaluationdeterminanthsc)) * S ((mdr_us_rank_integer_firstwitnessminornonzeroevaluationdeterminanthsc) + (mdr_un_rank_integer_firstwitnessminornonzeroevaluationdeterminanthsc)) + ((mdr_un_rank_integer_firstwitnessminornonzeroevaluationdeterminanthsc) + (mdr_un_rank_integer_firstwitnessminornonzeroevaluationdeterminanthsc))) /\ ((mdr_c_rank_integer_firstwitnessminornonzeroevaluationdeterminanthscrc = ((mdr_a_rank_integer_firstwitnessminornonzeroevaluationdeterminanthscrc) + (mdr_b_rank_integer_firstwitnessminornonzeroevaluationdeterminanthscrc)) * S ((mdr_a_rank_integer_firstwitnessminornonzeroevaluationdeterminanthscrc) + (mdr_b_rank_integer_firstwitnessminornonzeroevaluationdeterminanthscrc)) + ((mdr_b_rank_integer_firstwitnessminornonzeroevaluationdeterminanthscrc) + (mdr_b_rank_integer_firstwitnessminornonzeroevaluationdeterminanthscrc))) /\ ((mdr_e_rank_integer_firstwitnessminornonzeroevaluationdeterminanthscrc = ((mdr_p_rank_integer_firstwitnessminornonzeroevaluationdeterminanthsc) + (mdr_n_rank_integer_firstwitnessminornonzeroevaluationdeterminanthsc)) * S ((mdr_p_rank_integer_firstwitnessminornonzeroevaluationdeterminanthsc) + (mdr_n_rank_integer_firstwitnessminornonzeroevaluationdeterminanthsc)) + ((mdr_n_rank_integer_firstwitnessminornonzeroevaluationdeterminanthsc) + (mdr_n_rank_integer_firstwitnessminornonzeroevaluationdeterminanthsc))) /\ ((mdr_f_rank_integer_firstwitnessminornonzeroevaluationdeterminanthscrc = ((mdr_ut_rank_integer_firstwitnessminornonzeroevaluationdeterminanthsc) + (mdr_e_rank_integer_firstwitnessminornonzeroevaluationdeterminanthscrc)) * S ((mdr_ut_rank_integer_firstwitnessminornonzeroevaluationdeterminanthsc) + (mdr_e_rank_integer_firstwitnessminornonzeroevaluationdeterminanthscrc)) + ((mdr_e_rank_integer_firstwitnessminornonzeroevaluationdeterminanthscrc) + (mdr_e_rank_integer_firstwitnessminornonzeroevaluationdeterminanthscrc))) /\ ((mdr_z_rank_integer_firstwitnessminornonzeroevaluationdeterminanthscr) = ((mdr_c_rank_integer_firstwitnessminornonzeroevaluationdeterminanthscrc) + (mdr_f_rank_integer_firstwitnessminornonzeroevaluationdeterminanthscrc)) * S ((mdr_c_rank_integer_firstwitnessminornonzeroevaluationdeterminanthscrc) + (mdr_f_rank_integer_firstwitnessminornonzeroevaluationdeterminanthscrc)) + ((mdr_f_rank_integer_firstwitnessminornonzeroevaluationdeterminanthscrc) + (mdr_f_rank_integer_firstwitnessminornonzeroevaluationdeterminanthscrc))))))))) /\ (((exists ff_h_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthscrb. ff_h_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthscrb + S (mdr_z_rank_integer_firstwitnessminornonzeroevaluationdeterminanthscr) = S ((S (mdr_i_rank_integer_firstwitnessminornonzeroevaluationdeterminanthsc)) * mdr_c_rank_integer_firstwitnessminornonzeroevaluationdeterminant)) /\ exists ff_q_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthscrb. mdr_b_rank_integer_firstwitnessminornonzeroevaluationdeterminant = ff_q_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthscrb * S ((S (mdr_i_rank_integer_firstwitnessminornonzeroevaluationdeterminanthsc)) * mdr_c_rank_integer_firstwitnessminornonzeroevaluationdeterminant) + (mdr_z_rank_integer_firstwitnessminornonzeroevaluationdeterminanthscr))))) /\ ((((forall ff_index_mdm_prefix_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthscm_positive. (exists ff_gap_mdm_lt_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthscm_positive_index_bound. ff_gap_mdm_lt_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthscm_positive_index_bound + S (ff_index_mdm_prefix_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthscm_positive) = ((mdr_q_rank_integer_firstwitnessminornonzeroevaluationdeterminanths) * (mdr_q_rank_integer_firstwitnessminornonzeroevaluationdeterminanths))) -> exists ff_row_mdm_prefix_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthscm_positive ff_column_mdm_prefix_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthscm_positive ff_value_mdm_prefix_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthscm_positive. (ff_index_mdm_prefix_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthscm_positive = (mdr_q_rank_integer_firstwitnessminornonzeroevaluationdeterminanths) * ff_row_mdm_prefix_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthscm_positive + ff_column_mdm_prefix_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthscm_positive /\ ((exists ff_gap_mdm_lt_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthscm_positive_column_bound. ff_gap_mdm_lt_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthscm_positive_column_bound + S (ff_column_mdm_prefix_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthscm_positive) = (mdr_q_rank_integer_firstwitnessminornonzeroevaluationdeterminanths)) /\ ((exists ff_row_mdm_cell_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthscm_positive_cell ff_column_mdm_cell_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthscm_positive_cell. (((((exists ff_gap_mdm_lt_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthscm_positive_cell_row_before. ff_gap_mdm_lt_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthscm_positive_cell_row_before + S (ff_row_mdm_prefix_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthscm_positive) = (0)) /\ ff_row_mdm_cell_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthscm_positive_cell = ff_row_mdm_prefix_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthscm_positive) \/ ((exists ff_gap_mdm_le_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthscm_positive_cell_row_after. ff_gap_mdm_le_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthscm_positive_cell_row_after + (0) = (ff_row_mdm_prefix_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthscm_positive)) /\ ff_row_mdm_cell_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthscm_positive_cell = S ff_row_mdm_prefix_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthscm_positive))) /\ (((((exists ff_gap_mdm_lt_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthscm_positive_cell_column_before. ff_gap_mdm_lt_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthscm_positive_cell_column_before + S (ff_column_mdm_prefix_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthscm_positive) = (mdr_j_rank_integer_firstwitnessminornonzeroevaluationdeterminanthsc)) /\ ff_column_mdm_cell_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthscm_positive_cell = ff_column_mdm_prefix_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthscm_positive) \/ ((exists ff_gap_mdm_le_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthscm_positive_cell_column_after. ff_gap_mdm_le_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthscm_positive_cell_column_after + (mdr_j_rank_integer_firstwitnessminornonzeroevaluationdeterminanthsc) = (ff_column_mdm_prefix_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthscm_positive)) /\ ff_column_mdm_cell_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthscm_positive_cell = S ff_column_mdm_prefix_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthscm_positive))) /\ (((exists ff_h_mdm_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthscm_positive_cell_source. ff_h_mdm_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthscm_positive_cell_source + S (ff_value_mdm_prefix_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthscm_positive) = S ((S ((ff_row_mdm_cell_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthscm_positive_cell) * (S (mdr_q_rank_integer_firstwitnessminornonzeroevaluationdeterminanths)) + (ff_column_mdm_cell_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthscm_positive_cell))) * mdr_pc_rank_integer_firstwitnessminornonzeroevaluationdeterminanth)) /\ exists ff_q_mdm_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthscm_positive_cell_source. mdr_pb_rank_integer_firstwitnessminornonzeroevaluationdeterminanth = ff_q_mdm_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthscm_positive_cell_source * S ((S ((ff_row_mdm_cell_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthscm_positive_cell) * (S (mdr_q_rank_integer_firstwitnessminornonzeroevaluationdeterminanths)) + (ff_column_mdm_cell_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthscm_positive_cell))) * mdr_pc_rank_integer_firstwitnessminornonzeroevaluationdeterminanth) + (ff_value_mdm_prefix_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthscm_positive)))))) /\ (((exists ff_h_mdm_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthscm_positive_target. ff_h_mdm_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthscm_positive_target + S (ff_value_mdm_prefix_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthscm_positive) = S ((S (ff_index_mdm_prefix_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthscm_positive)) * mdr_us_rank_integer_firstwitnessminornonzeroevaluationdeterminanthsc)) /\ exists ff_q_mdm_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthscm_positive_target. mdr_up_rank_integer_firstwitnessminornonzeroevaluationdeterminanthsc = ff_q_mdm_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthscm_positive_target * S ((S (ff_index_mdm_prefix_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthscm_positive)) * mdr_us_rank_integer_firstwitnessminornonzeroevaluationdeterminanthsc) + (ff_value_mdm_prefix_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthscm_positive))))))) /\ (forall ff_index_mdm_prefix_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthscm_negative. (exists ff_gap_mdm_lt_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthscm_negative_index_bound. ff_gap_mdm_lt_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthscm_negative_index_bound + S (ff_index_mdm_prefix_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthscm_negative) = ((mdr_q_rank_integer_firstwitnessminornonzeroevaluationdeterminanths) * (mdr_q_rank_integer_firstwitnessminornonzeroevaluationdeterminanths))) -> exists ff_row_mdm_prefix_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthscm_negative ff_column_mdm_prefix_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthscm_negative ff_value_mdm_prefix_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthscm_negative. (ff_index_mdm_prefix_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthscm_negative = (mdr_q_rank_integer_firstwitnessminornonzeroevaluationdeterminanths) * ff_row_mdm_prefix_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthscm_negative + ff_column_mdm_prefix_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthscm_negative /\ ((exists ff_gap_mdm_lt_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthscm_negative_column_bound. ff_gap_mdm_lt_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthscm_negative_column_bound + S (ff_column_mdm_prefix_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthscm_negative) = (mdr_q_rank_integer_firstwitnessminornonzeroevaluationdeterminanths)) /\ ((exists ff_row_mdm_cell_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthscm_negative_cell ff_column_mdm_cell_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthscm_negative_cell. (((((exists ff_gap_mdm_lt_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthscm_negative_cell_row_before. ff_gap_mdm_lt_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthscm_negative_cell_row_before + S (ff_row_mdm_prefix_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthscm_negative) = (0)) /\ ff_row_mdm_cell_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthscm_negative_cell = ff_row_mdm_prefix_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthscm_negative) \/ ((exists ff_gap_mdm_le_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthscm_negative_cell_row_after. ff_gap_mdm_le_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthscm_negative_cell_row_after + (0) = (ff_row_mdm_prefix_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthscm_negative)) /\ ff_row_mdm_cell_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthscm_negative_cell = S ff_row_mdm_prefix_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthscm_negative))) /\ (((((exists ff_gap_mdm_lt_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthscm_negative_cell_column_before. ff_gap_mdm_lt_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthscm_negative_cell_column_before + S (ff_column_mdm_prefix_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthscm_negative) = (mdr_j_rank_integer_firstwitnessminornonzeroevaluationdeterminanthsc)) /\ ff_column_mdm_cell_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthscm_negative_cell = ff_column_mdm_prefix_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthscm_negative) \/ ((exists ff_gap_mdm_le_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthscm_negative_cell_column_after. ff_gap_mdm_le_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthscm_negative_cell_column_after + (mdr_j_rank_integer_firstwitnessminornonzeroevaluationdeterminanthsc) = (ff_column_mdm_prefix_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthscm_negative)) /\ ff_column_mdm_cell_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthscm_negative_cell = S ff_column_mdm_prefix_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthscm_negative))) /\ (((exists ff_h_mdm_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthscm_negative_cell_source. ff_h_mdm_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthscm_negative_cell_source + S (ff_value_mdm_prefix_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthscm_negative) = S ((S ((ff_row_mdm_cell_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthscm_negative_cell) * (S (mdr_q_rank_integer_firstwitnessminornonzeroevaluationdeterminanths)) + (ff_column_mdm_cell_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthscm_negative_cell))) * mdr_nc_rank_integer_firstwitnessminornonzeroevaluationdeterminanth)) /\ exists ff_q_mdm_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthscm_negative_cell_source. mdr_nb_rank_integer_firstwitnessminornonzeroevaluationdeterminanth = ff_q_mdm_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthscm_negative_cell_source * S ((S ((ff_row_mdm_cell_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthscm_negative_cell) * (S (mdr_q_rank_integer_firstwitnessminornonzeroevaluationdeterminanths)) + (ff_column_mdm_cell_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthscm_negative_cell))) * mdr_nc_rank_integer_firstwitnessminornonzeroevaluationdeterminanth) + (ff_value_mdm_prefix_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthscm_negative)))))) /\ (((exists ff_h_mdm_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthscm_negative_target. ff_h_mdm_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthscm_negative_target + S (ff_value_mdm_prefix_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthscm_negative) = S ((S (ff_index_mdm_prefix_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthscm_negative)) * mdr_ut_rank_integer_firstwitnessminornonzeroevaluationdeterminanthsc)) /\ exists ff_q_mdm_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthscm_negative_target. mdr_un_rank_integer_firstwitnessminornonzeroevaluationdeterminanthsc = ff_q_mdm_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthscm_negative_target * S ((S (ff_index_mdm_prefix_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthscm_negative)) * mdr_ut_rank_integer_firstwitnessminornonzeroevaluationdeterminanthsc) + (ff_value_mdm_prefix_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthscm_negative))))))))) /\ ((((exists ff_h_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthscp. ff_h_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthscp + S (mdr_p_rank_integer_firstwitnessminornonzeroevaluationdeterminanthsc) = S ((S (mdr_j_rank_integer_firstwitnessminornonzeroevaluationdeterminanthsc)) * mdr_ec_rank_integer_firstwitnessminornonzeroevaluationdeterminanths)) /\ exists ff_q_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthscp. mdr_eb_rank_integer_firstwitnessminornonzeroevaluationdeterminanths = ff_q_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthscp * S ((S (mdr_j_rank_integer_firstwitnessminornonzeroevaluationdeterminanthsc)) * mdr_ec_rank_integer_firstwitnessminornonzeroevaluationdeterminanths) + (mdr_p_rank_integer_firstwitnessminornonzeroevaluationdeterminanthsc))) /\ (((exists ff_h_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthscn. ff_h_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthscn + S (mdr_n_rank_integer_firstwitnessminornonzeroevaluationdeterminanthsc) = S ((S (mdr_j_rank_integer_firstwitnessminornonzeroevaluationdeterminanthsc)) * mdr_fc_rank_integer_firstwitnessminornonzeroevaluationdeterminanths)) /\ exists ff_q_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthscn. mdr_fb_rank_integer_firstwitnessminornonzeroevaluationdeterminanths = ff_q_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthscn * S ((S (mdr_j_rank_integer_firstwitnessminornonzeroevaluationdeterminanthsc)) * mdr_fc_rank_integer_firstwitnessminornonzeroevaluationdeterminanths) + (mdr_n_rank_integer_firstwitnessminornonzeroevaluationdeterminanthsc)))))))) /\ (exists ff_ub_mce_fold_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthsf ff_uc_mce_fold_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthsf ff_vb_mce_fold_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthsf ff_vc_mce_fold_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthsf. ((forall ff_index_mce_alternating_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthsf_prefix. (exists ff_gap_mce_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthsf_prefix_index. ff_gap_mce_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthsf_prefix_index + S (ff_index_mce_alternating_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthsf_prefix) = (S (mdr_q_rank_integer_firstwitnessminornonzeroevaluationdeterminanths))) -> exists ff_ap_mce_alternating_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthsf_prefix ff_an_mce_alternating_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthsf_prefix ff_bp_mce_alternating_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthsf_prefix ff_bn_mce_alternating_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthsf_prefix ff_p_mce_alternating_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthsf_prefix ff_n_mce_alternating_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthsf_prefix. ((((exists ff_h_mce_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthsf_prefix_ap. ff_h_mce_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthsf_prefix_ap + S (ff_ap_mce_alternating_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthsf_prefix) = S ((S (ff_index_mce_alternating_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthsf_prefix)) * mdr_pc_rank_integer_firstwitnessminornonzeroevaluationdeterminanth)) /\ exists ff_q_mce_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthsf_prefix_ap. mdr_pb_rank_integer_firstwitnessminornonzeroevaluationdeterminanth = ff_q_mce_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthsf_prefix_ap * S ((S (ff_index_mce_alternating_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthsf_prefix)) * mdr_pc_rank_integer_firstwitnessminornonzeroevaluationdeterminanth) + (ff_ap_mce_alternating_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthsf_prefix))) /\ ((((exists ff_h_mce_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthsf_prefix_an. ff_h_mce_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthsf_prefix_an + S (ff_an_mce_alternating_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthsf_prefix) = S ((S (ff_index_mce_alternating_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthsf_prefix)) * mdr_nc_rank_integer_firstwitnessminornonzeroevaluationdeterminanth)) /\ exists ff_q_mce_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthsf_prefix_an. mdr_nb_rank_integer_firstwitnessminornonzeroevaluationdeterminanth = ff_q_mce_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthsf_prefix_an * S ((S (ff_index_mce_alternating_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthsf_prefix)) * mdr_nc_rank_integer_firstwitnessminornonzeroevaluationdeterminanth) + (ff_an_mce_alternating_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthsf_prefix))) /\ ((((exists ff_h_mce_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthsf_prefix_bp. ff_h_mce_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthsf_prefix_bp + S (ff_bp_mce_alternating_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthsf_prefix) = S ((S (ff_index_mce_alternating_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthsf_prefix)) * mdr_ec_rank_integer_firstwitnessminornonzeroevaluationdeterminanths)) /\ exists ff_q_mce_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthsf_prefix_bp. mdr_eb_rank_integer_firstwitnessminornonzeroevaluationdeterminanths = ff_q_mce_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthsf_prefix_bp * S ((S (ff_index_mce_alternating_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthsf_prefix)) * mdr_ec_rank_integer_firstwitnessminornonzeroevaluationdeterminanths) + (ff_bp_mce_alternating_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthsf_prefix))) /\ ((((exists ff_h_mce_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthsf_prefix_bn. ff_h_mce_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthsf_prefix_bn + S (ff_bn_mce_alternating_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthsf_prefix) = S ((S (ff_index_mce_alternating_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthsf_prefix)) * mdr_fc_rank_integer_firstwitnessminornonzeroevaluationdeterminanths)) /\ exists ff_q_mce_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthsf_prefix_bn. mdr_fb_rank_integer_firstwitnessminornonzeroevaluationdeterminanths = ff_q_mce_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthsf_prefix_bn * S ((S (ff_index_mce_alternating_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthsf_prefix)) * mdr_fc_rank_integer_firstwitnessminornonzeroevaluationdeterminanths) + (ff_bn_mce_alternating_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthsf_prefix))) /\ ((((exists ff_h_mce_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthsf_prefix_positive. ff_h_mce_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthsf_prefix_positive + S (ff_p_mce_alternating_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthsf_prefix) = S ((S (ff_index_mce_alternating_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthsf_prefix)) * ff_uc_mce_fold_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthsf)) /\ exists ff_q_mce_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthsf_prefix_positive. ff_ub_mce_fold_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthsf = ff_q_mce_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthsf_prefix_positive * S ((S (ff_index_mce_alternating_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthsf_prefix)) * ff_uc_mce_fold_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthsf) + (ff_p_mce_alternating_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthsf_prefix))) /\ ((((exists ff_h_mce_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthsf_prefix_negative. ff_h_mce_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthsf_prefix_negative + S (ff_n_mce_alternating_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthsf_prefix) = S ((S (ff_index_mce_alternating_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthsf_prefix)) * ff_vc_mce_fold_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthsf)) /\ exists ff_q_mce_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthsf_prefix_negative. ff_vb_mce_fold_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthsf = ff_q_mce_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthsf_prefix_negative * S ((S (ff_index_mce_alternating_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthsf_prefix)) * ff_vc_mce_fold_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthsf) + (ff_n_mce_alternating_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthsf_prefix))) /\ (((exists ff_even_mce_term_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthsf_prefix_term. ff_index_mce_alternating_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthsf_prefix = 2 * ff_even_mce_term_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthsf_prefix_term) /\ (ff_p_mce_alternating_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthsf_prefix = (ff_ap_mce_alternating_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthsf_prefix) * (ff_bp_mce_alternating_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthsf_prefix) + (ff_an_mce_alternating_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthsf_prefix) * (ff_bn_mce_alternating_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthsf_prefix) /\ ff_n_mce_alternating_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthsf_prefix = (ff_ap_mce_alternating_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthsf_prefix) * (ff_bn_mce_alternating_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthsf_prefix) + (ff_an_mce_alternating_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthsf_prefix) * (ff_bp_mce_alternating_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthsf_prefix))) \/ ((exists ff_odd_mce_term_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthsf_prefix_term. ff_index_mce_alternating_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthsf_prefix = 2 * ff_odd_mce_term_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthsf_prefix_term + 1) /\ (ff_p_mce_alternating_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthsf_prefix = (ff_ap_mce_alternating_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthsf_prefix) * (ff_bn_mce_alternating_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthsf_prefix) + (ff_an_mce_alternating_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthsf_prefix) * (ff_bp_mce_alternating_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthsf_prefix) /\ ff_n_mce_alternating_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthsf_prefix = (ff_ap_mce_alternating_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthsf_prefix) * (ff_bp_mce_alternating_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthsf_prefix) + (ff_an_mce_alternating_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthsf_prefix) * (ff_bn_mce_alternating_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthsf_prefix))))))))))) /\ ((exists ff_u_mce_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthsf_positive ff_v_mce_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthsf_positive. ((((exists ff_h_mce_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthsf_positive_start. ff_h_mce_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthsf_positive_start + S (0) = S ((S (0)) * ff_v_mce_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthsf_positive)) /\ exists ff_q_mce_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthsf_positive_start. ff_u_mce_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthsf_positive = ff_q_mce_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthsf_positive_start * S ((S (0)) * ff_v_mce_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthsf_positive) + (0))) /\ ((((exists ff_h_mce_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthsf_positive_terminal. ff_h_mce_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthsf_positive_terminal + S (mdr_p_rank_integer_firstwitnessminornonzeroevaluationdeterminanth) = S ((S ((S (mdr_q_rank_integer_firstwitnessminornonzeroevaluationdeterminanths)))) * ff_v_mce_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthsf_positive)) /\ exists ff_q_mce_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthsf_positive_terminal. ff_u_mce_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthsf_positive = ff_q_mce_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthsf_positive_terminal * S ((S ((S (mdr_q_rank_integer_firstwitnessminornonzeroevaluationdeterminanths)))) * ff_v_mce_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthsf_positive) + (mdr_p_rank_integer_firstwitnessminornonzeroevaluationdeterminanth))) /\ forall ff_i_mce_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthsf_positive. (exists ff_lt_mce_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthsf_positive_bound. ff_lt_mce_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthsf_positive_bound + S ff_i_mce_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthsf_positive = (S (mdr_q_rank_integer_firstwitnessminornonzeroevaluationdeterminanths))) -> exists ff_a_mce_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthsf_positive ff_r_mce_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthsf_positive ff_s_mce_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthsf_positive. ((((exists ff_h_mce_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthsf_positive_summand. ff_h_mce_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthsf_positive_summand + S (ff_a_mce_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthsf_positive) = S ((S (ff_i_mce_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthsf_positive)) * ff_uc_mce_fold_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthsf)) /\ exists ff_q_mce_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthsf_positive_summand. ff_ub_mce_fold_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthsf = ff_q_mce_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthsf_positive_summand * S ((S (ff_i_mce_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthsf_positive)) * ff_uc_mce_fold_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthsf) + (ff_a_mce_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthsf_positive))) /\ ((((exists ff_h_mce_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthsf_positive_partial. ff_h_mce_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthsf_positive_partial + S (ff_r_mce_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthsf_positive) = S ((S (ff_i_mce_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthsf_positive)) * ff_v_mce_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthsf_positive)) /\ exists ff_q_mce_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthsf_positive_partial. ff_u_mce_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthsf_positive = ff_q_mce_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthsf_positive_partial * S ((S (ff_i_mce_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthsf_positive)) * ff_v_mce_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthsf_positive) + (ff_r_mce_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthsf_positive))) /\ ((((exists ff_h_mce_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthsf_positive_successor. ff_h_mce_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthsf_positive_successor + S (ff_s_mce_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthsf_positive) = S ((S (S ff_i_mce_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthsf_positive)) * ff_v_mce_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthsf_positive)) /\ exists ff_q_mce_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthsf_positive_successor. ff_u_mce_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthsf_positive = ff_q_mce_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthsf_positive_successor * S ((S (S ff_i_mce_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthsf_positive)) * ff_v_mce_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthsf_positive) + (ff_s_mce_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthsf_positive))) /\ ff_s_mce_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthsf_positive = ff_r_mce_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthsf_positive + ff_a_mce_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthsf_positive)))))) /\ (exists ff_u_mce_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthsf_negative ff_v_mce_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthsf_negative. ((((exists ff_h_mce_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthsf_negative_start. ff_h_mce_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthsf_negative_start + S (0) = S ((S (0)) * ff_v_mce_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthsf_negative)) /\ exists ff_q_mce_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthsf_negative_start. ff_u_mce_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthsf_negative = ff_q_mce_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthsf_negative_start * S ((S (0)) * ff_v_mce_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthsf_negative) + (0))) /\ ((((exists ff_h_mce_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthsf_negative_terminal. ff_h_mce_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthsf_negative_terminal + S (mdr_n_rank_integer_firstwitnessminornonzeroevaluationdeterminanth) = S ((S ((S (mdr_q_rank_integer_firstwitnessminornonzeroevaluationdeterminanths)))) * ff_v_mce_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthsf_negative)) /\ exists ff_q_mce_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthsf_negative_terminal. ff_u_mce_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthsf_negative = ff_q_mce_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthsf_negative_terminal * S ((S ((S (mdr_q_rank_integer_firstwitnessminornonzeroevaluationdeterminanths)))) * ff_v_mce_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthsf_negative) + (mdr_n_rank_integer_firstwitnessminornonzeroevaluationdeterminanth))) /\ forall ff_i_mce_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthsf_negative. (exists ff_lt_mce_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthsf_negative_bound. ff_lt_mce_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthsf_negative_bound + S ff_i_mce_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthsf_negative = (S (mdr_q_rank_integer_firstwitnessminornonzeroevaluationdeterminanths))) -> exists ff_a_mce_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthsf_negative ff_r_mce_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthsf_negative ff_s_mce_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthsf_negative. ((((exists ff_h_mce_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthsf_negative_summand. ff_h_mce_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthsf_negative_summand + S (ff_a_mce_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthsf_negative) = S ((S (ff_i_mce_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthsf_negative)) * ff_vc_mce_fold_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthsf)) /\ exists ff_q_mce_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthsf_negative_summand. ff_vb_mce_fold_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthsf = ff_q_mce_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthsf_negative_summand * S ((S (ff_i_mce_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthsf_negative)) * ff_vc_mce_fold_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthsf) + (ff_a_mce_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthsf_negative))) /\ ((((exists ff_h_mce_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthsf_negative_partial. ff_h_mce_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthsf_negative_partial + S (ff_r_mce_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthsf_negative) = S ((S (ff_i_mce_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthsf_negative)) * ff_v_mce_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthsf_negative)) /\ exists ff_q_mce_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthsf_negative_partial. ff_u_mce_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthsf_negative = ff_q_mce_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthsf_negative_partial * S ((S (ff_i_mce_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthsf_negative)) * ff_v_mce_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthsf_negative) + (ff_r_mce_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthsf_negative))) /\ ((((exists ff_h_mce_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthsf_negative_successor. ff_h_mce_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthsf_negative_successor + S (ff_s_mce_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthsf_negative) = S ((S (S ff_i_mce_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthsf_negative)) * ff_v_mce_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthsf_negative)) /\ exists ff_q_mce_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthsf_negative_successor. ff_u_mce_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthsf_negative = ff_q_mce_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthsf_negative_successor * S ((S (S ff_i_mce_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthsf_negative)) * ff_v_mce_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthsf_negative) + (ff_s_mce_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthsf_negative))) /\ ff_s_mce_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthsf_negative = ff_r_mce_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthsf_negative + ff_a_mce_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminanthsf_negative))))))))))))))) /\ ((exists mdr_gap_rank_integer_firstwitnessminornonzeroevaluationdeterminanti. mdr_gap_rank_integer_firstwitnessminornonzeroevaluationdeterminanti + S (mdr_i_rank_integer_firstwitnessminornonzeroevaluationdeterminant) = (mdr_l_rank_integer_firstwitnessminornonzeroevaluationdeterminant)) /\ (exists mdr_z_rank_integer_firstwitnessminornonzeroevaluationdeterminantr. ((exists mdr_a_rank_integer_firstwitnessminornonzeroevaluationdeterminantrc mdr_b_rank_integer_firstwitnessminornonzeroevaluationdeterminantrc mdr_c_rank_integer_firstwitnessminornonzeroevaluationdeterminantrc mdr_e_rank_integer_firstwitnessminornonzeroevaluationdeterminantrc mdr_f_rank_integer_firstwitnessminornonzeroevaluationdeterminantrc. ((mdr_a_rank_integer_firstwitnessminornonzeroevaluationdeterminantrc = ((rank) + (mdr_ub_rank_integer_firstwitnessminornonzeroevaluation)) * S ((rank) + (mdr_ub_rank_integer_firstwitnessminornonzeroevaluation)) + ((mdr_ub_rank_integer_firstwitnessminornonzeroevaluation) + (mdr_ub_rank_integer_firstwitnessminornonzeroevaluation))) /\ ((mdr_b_rank_integer_firstwitnessminornonzeroevaluationdeterminantrc = ((mdr_uc_rank_integer_firstwitnessminornonzeroevaluation) + (mdr_vb_rank_integer_firstwitnessminornonzeroevaluation)) * S ((mdr_uc_rank_integer_firstwitnessminornonzeroevaluation) + (mdr_vb_rank_integer_firstwitnessminornonzeroevaluation)) + ((mdr_vb_rank_integer_firstwitnessminornonzeroevaluation) + (mdr_vb_rank_integer_firstwitnessminornonzeroevaluation))) /\ ((mdr_c_rank_integer_firstwitnessminornonzeroevaluationdeterminantrc = ((mdr_a_rank_integer_firstwitnessminornonzeroevaluationdeterminantrc) + (mdr_b_rank_integer_firstwitnessminornonzeroevaluationdeterminantrc)) * S ((mdr_a_rank_integer_firstwitnessminornonzeroevaluationdeterminantrc) + (mdr_b_rank_integer_firstwitnessminornonzeroevaluationdeterminantrc)) + ((mdr_b_rank_integer_firstwitnessminornonzeroevaluationdeterminantrc) + (mdr_b_rank_integer_firstwitnessminornonzeroevaluationdeterminantrc))) /\ ((mdr_e_rank_integer_firstwitnessminornonzeroevaluationdeterminantrc = ((mdr_p_rank_integer_firstwitnessminornonzero) + (mdr_n_rank_integer_firstwitnessminornonzero)) * S ((mdr_p_rank_integer_firstwitnessminornonzero) + (mdr_n_rank_integer_firstwitnessminornonzero)) + ((mdr_n_rank_integer_firstwitnessminornonzero) + (mdr_n_rank_integer_firstwitnessminornonzero))) /\ ((mdr_f_rank_integer_firstwitnessminornonzeroevaluationdeterminantrc = ((mdr_vc_rank_integer_firstwitnessminornonzeroevaluation) + (mdr_e_rank_integer_firstwitnessminornonzeroevaluationdeterminantrc)) * S ((mdr_vc_rank_integer_firstwitnessminornonzeroevaluation) + (mdr_e_rank_integer_firstwitnessminornonzeroevaluationdeterminantrc)) + ((mdr_e_rank_integer_firstwitnessminornonzeroevaluationdeterminantrc) + (mdr_e_rank_integer_firstwitnessminornonzeroevaluationdeterminantrc))) /\ ((mdr_z_rank_integer_firstwitnessminornonzeroevaluationdeterminantr) = ((mdr_c_rank_integer_firstwitnessminornonzeroevaluationdeterminantrc) + (mdr_f_rank_integer_firstwitnessminornonzeroevaluationdeterminantrc)) * S ((mdr_c_rank_integer_firstwitnessminornonzeroevaluationdeterminantrc) + (mdr_f_rank_integer_firstwitnessminornonzeroevaluationdeterminantrc)) + ((mdr_f_rank_integer_firstwitnessminornonzeroevaluationdeterminantrc) + (mdr_f_rank_integer_firstwitnessminornonzeroevaluationdeterminantrc))))))))) /\ (((exists ff_h_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminantrb. ff_h_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminantrb + S (mdr_z_rank_integer_firstwitnessminornonzeroevaluationdeterminantr) = S ((S (mdr_i_rank_integer_firstwitnessminornonzeroevaluationdeterminant)) * mdr_c_rank_integer_firstwitnessminornonzeroevaluationdeterminant)) /\ exists ff_q_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminantrb. mdr_b_rank_integer_firstwitnessminornonzeroevaluationdeterminant = ff_q_mdr_rank_integer_firstwitnessminornonzeroevaluationdeterminantrb * S ((S (mdr_i_rank_integer_firstwitnessminornonzeroevaluationdeterminant)) * mdr_c_rank_integer_firstwitnessminornonzeroevaluationdeterminant) + (mdr_z_rank_integer_firstwitnessminornonzeroevaluationdeterminantr)))))))))) /\ (~(mdr_p_rank_integer_firstwitnessminornonzero = mdr_n_rank_integer_firstwitnessminornonzero)))))))) /\ (forall mdr_q_rank_integer_first. (exists mdr_gap_rank_integer_firsthigher. mdr_gap_rank_integer_firsthigher + S (rank) = (mdr_q_rank_integer_first)) -> (forall mdr_rb_rank_integer_firstzero mdr_rc_rank_integer_firstzero mdr_cb_rank_integer_firstzero mdr_cc_rank_integer_firstzero mdr_p_rank_integer_firstzero mdr_n_rank_integer_firstzero. (((forall fom_index_mrf_rank_integer_firstzerorowsbound. (exists fom_gap_mrf_rank_integer_firstzerorowsbound_index_bound. fom_gap_mrf_rank_integer_firstzerorowsbound_index_bound + S (fom_index_mrf_rank_integer_firstzerorowsbound) = mdr_q_rank_integer_first) -> exists fom_value_mrf_rank_integer_firstzerorowsbound. ((((exists fom_beta_height_mrf_rank_integer_firstzerorowsbound_entry. fom_beta_height_mrf_rank_integer_firstzerorowsbound_entry + S (fom_value_mrf_rank_integer_firstzerorowsbound) = S ((S (fom_index_mrf_rank_integer_firstzerorowsbound)) * mdr_rc_rank_integer_firstzero)) /\ exists fom_beta_quotient_mrf_rank_integer_firstzerorowsbound_entry. mdr_rb_rank_integer_firstzero = fom_beta_quotient_mrf_rank_integer_firstzerorowsbound_entry * S ((S (fom_index_mrf_rank_integer_firstzerorowsbound)) * mdr_rc_rank_integer_firstzero) + (fom_value_mrf_rank_integer_firstzerorowsbound))) /\ (exists fom_gap_mrf_rank_integer_firstzerorowsbound_value_bound. fom_gap_mrf_rank_integer_firstzerorowsbound_value_bound + S (fom_value_mrf_rank_integer_firstzerorowsbound) = r))) /\ (forall mdr_i_rank_integer_firstzerorowsdistinct mdr_j_rank_integer_firstzerorowsdistinct mdr_a_rank_integer_firstzerorowsdistinct. (exists mdr_gap_rank_integer_firstzerorowsdistincti. mdr_gap_rank_integer_firstzerorowsdistincti + S (mdr_i_rank_integer_firstzerorowsdistinct) = (mdr_q_rank_integer_first)) -> (exists mdr_gap_rank_integer_firstzerorowsdistinctj. mdr_gap_rank_integer_firstzerorowsdistinctj + S (mdr_j_rank_integer_firstzerorowsdistinct) = (mdr_q_rank_integer_first)) -> (((exists ff_h_mdr_rank_integer_firstzerorowsdistinctfirst. ff_h_mdr_rank_integer_firstzerorowsdistinctfirst + S (mdr_a_rank_integer_firstzerorowsdistinct) = S ((S (mdr_i_rank_integer_firstzerorowsdistinct)) * mdr_rc_rank_integer_firstzero)) /\ exists ff_q_mdr_rank_integer_firstzerorowsdistinctfirst. mdr_rb_rank_integer_firstzero = ff_q_mdr_rank_integer_firstzerorowsdistinctfirst * S ((S (mdr_i_rank_integer_firstzerorowsdistinct)) * mdr_rc_rank_integer_firstzero) + (mdr_a_rank_integer_firstzerorowsdistinct))) -> (((exists ff_h_mdr_rank_integer_firstzerorowsdistinctsecond. ff_h_mdr_rank_integer_firstzerorowsdistinctsecond + S (mdr_a_rank_integer_firstzerorowsdistinct) = S ((S (mdr_j_rank_integer_firstzerorowsdistinct)) * mdr_rc_rank_integer_firstzero)) /\ exists ff_q_mdr_rank_integer_firstzerorowsdistinctsecond. mdr_rb_rank_integer_firstzero = ff_q_mdr_rank_integer_firstzerorowsdistinctsecond * S ((S (mdr_j_rank_integer_firstzerorowsdistinct)) * mdr_rc_rank_integer_firstzero) + (mdr_a_rank_integer_firstzerorowsdistinct))) -> mdr_i_rank_integer_firstzerorowsdistinct = mdr_j_rank_integer_firstzerorowsdistinct))) -> (((forall fom_index_mrf_rank_integer_firstzerocolumnsbound. (exists fom_gap_mrf_rank_integer_firstzerocolumnsbound_index_bound. fom_gap_mrf_rank_integer_firstzerocolumnsbound_index_bound + S (fom_index_mrf_rank_integer_firstzerocolumnsbound) = mdr_q_rank_integer_first) -> exists fom_value_mrf_rank_integer_firstzerocolumnsbound. ((((exists fom_beta_height_mrf_rank_integer_firstzerocolumnsbound_entry. fom_beta_height_mrf_rank_integer_firstzerocolumnsbound_entry + S (fom_value_mrf_rank_integer_firstzerocolumnsbound) = S ((S (fom_index_mrf_rank_integer_firstzerocolumnsbound)) * mdr_cc_rank_integer_firstzero)) /\ exists fom_beta_quotient_mrf_rank_integer_firstzerocolumnsbound_entry. mdr_cb_rank_integer_firstzero = fom_beta_quotient_mrf_rank_integer_firstzerocolumnsbound_entry * S ((S (fom_index_mrf_rank_integer_firstzerocolumnsbound)) * mdr_cc_rank_integer_firstzero) + (fom_value_mrf_rank_integer_firstzerocolumnsbound))) /\ (exists fom_gap_mrf_rank_integer_firstzerocolumnsbound_value_bound. fom_gap_mrf_rank_integer_firstzerocolumnsbound_value_bound + S (fom_value_mrf_rank_integer_firstzerocolumnsbound) = w))) /\ (forall mdr_i_rank_integer_firstzerocolumnsdistinct mdr_j_rank_integer_firstzerocolumnsdistinct mdr_a_rank_integer_firstzerocolumnsdistinct. (exists mdr_gap_rank_integer_firstzerocolumnsdistincti. mdr_gap_rank_integer_firstzerocolumnsdistincti + S (mdr_i_rank_integer_firstzerocolumnsdistinct) = (mdr_q_rank_integer_first)) -> (exists mdr_gap_rank_integer_firstzerocolumnsdistinctj. mdr_gap_rank_integer_firstzerocolumnsdistinctj + S (mdr_j_rank_integer_firstzerocolumnsdistinct) = (mdr_q_rank_integer_first)) -> (((exists ff_h_mdr_rank_integer_firstzerocolumnsdistinctfirst. ff_h_mdr_rank_integer_firstzerocolumnsdistinctfirst + S (mdr_a_rank_integer_firstzerocolumnsdistinct) = S ((S (mdr_i_rank_integer_firstzerocolumnsdistinct)) * mdr_cc_rank_integer_firstzero)) /\ exists ff_q_mdr_rank_integer_firstzerocolumnsdistinctfirst. mdr_cb_rank_integer_firstzero = ff_q_mdr_rank_integer_firstzerocolumnsdistinctfirst * S ((S (mdr_i_rank_integer_firstzerocolumnsdistinct)) * mdr_cc_rank_integer_firstzero) + (mdr_a_rank_integer_firstzerocolumnsdistinct))) -> (((exists ff_h_mdr_rank_integer_firstzerocolumnsdistinctsecond. ff_h_mdr_rank_integer_firstzerocolumnsdistinctsecond + S (mdr_a_rank_integer_firstzerocolumnsdistinct) = S ((S (mdr_j_rank_integer_firstzerocolumnsdistinct)) * mdr_cc_rank_integer_firstzero)) /\ exists ff_q_mdr_rank_integer_firstzerocolumnsdistinctsecond. mdr_cb_rank_integer_firstzero = ff_q_mdr_rank_integer_firstzerocolumnsdistinctsecond * S ((S (mdr_j_rank_integer_firstzerocolumnsdistinct)) * mdr_cc_rank_integer_firstzero) + (mdr_a_rank_integer_firstzerocolumnsdistinct))) -> mdr_i_rank_integer_firstzerocolumnsdistinct = mdr_j_rank_integer_firstzerocolumnsdistinct))) -> (exists mdr_ub_rank_integer_firstzeroevaluation mdr_uc_rank_integer_firstzeroevaluation mdr_vb_rank_integer_firstzeroevaluation mdr_vc_rank_integer_firstzeroevaluation. ((((forall mdr_i_rank_integer_firstzeroevaluationmatrixpositive. (exists mdr_gap_rank_integer_firstzeroevaluationmatrixpositivebound. mdr_gap_rank_integer_firstzeroevaluationmatrixpositivebound + S (mdr_i_rank_integer_firstzeroevaluationmatrixpositive) = ((mdr_q_rank_integer_first) * (mdr_q_rank_integer_first))) -> exists mdr_a_rank_integer_firstzeroevaluationmatrixpositive. (((exists mdr_r_rank_integer_firstzeroevaluationmatrixpositivepoint mdr_s_rank_integer_firstzeroevaluationmatrixpositivepoint mdr_u_rank_integer_firstzeroevaluationmatrixpositivepoint mdr_v_rank_integer_firstzeroevaluationmatrixpositivepoint. ((mdr_i_rank_integer_firstzeroevaluationmatrixpositive = (mdr_q_rank_integer_first) * mdr_r_rank_integer_firstzeroevaluationmatrixpositivepoint + mdr_s_rank_integer_firstzeroevaluationmatrixpositivepoint) /\ ((exists mdr_gap_rank_integer_firstzeroevaluationmatrixpositivepointcolumn. mdr_gap_rank_integer_firstzeroevaluationmatrixpositivepointcolumn + S (mdr_s_rank_integer_firstzeroevaluationmatrixpositivepoint) = (mdr_q_rank_integer_first)) /\ ((((exists ff_h_mdr_rank_integer_firstzeroevaluationmatrixpositivepointrow_index. ff_h_mdr_rank_integer_firstzeroevaluationmatrixpositivepointrow_index + S (mdr_u_rank_integer_firstzeroevaluationmatrixpositivepoint) = S ((S (mdr_r_rank_integer_firstzeroevaluationmatrixpositivepoint)) * mdr_rc_rank_integer_firstzero)) /\ exists ff_q_mdr_rank_integer_firstzeroevaluationmatrixpositivepointrow_index. mdr_rb_rank_integer_firstzero = ff_q_mdr_rank_integer_firstzeroevaluationmatrixpositivepointrow_index * S ((S (mdr_r_rank_integer_firstzeroevaluationmatrixpositivepoint)) * mdr_rc_rank_integer_firstzero) + (mdr_u_rank_integer_firstzeroevaluationmatrixpositivepoint))) /\ ((((exists ff_h_mdr_rank_integer_firstzeroevaluationmatrixpositivepointcolumn_index. ff_h_mdr_rank_integer_firstzeroevaluationmatrixpositivepointcolumn_index + S (mdr_v_rank_integer_firstzeroevaluationmatrixpositivepoint) = S ((S (mdr_s_rank_integer_firstzeroevaluationmatrixpositivepoint)) * mdr_cc_rank_integer_firstzero)) /\ exists ff_q_mdr_rank_integer_firstzeroevaluationmatrixpositivepointcolumn_index. mdr_cb_rank_integer_firstzero = ff_q_mdr_rank_integer_firstzeroevaluationmatrixpositivepointcolumn_index * S ((S (mdr_s_rank_integer_firstzeroevaluationmatrixpositivepoint)) * mdr_cc_rank_integer_firstzero) + (mdr_v_rank_integer_firstzeroevaluationmatrixpositivepoint))) /\ (((exists ff_h_mdr_rank_integer_firstzeroevaluationmatrixpositivepointsource. ff_h_mdr_rank_integer_firstzeroevaluationmatrixpositivepointsource + S (mdr_a_rank_integer_firstzeroevaluationmatrixpositive) = S ((S ((mdr_u_rank_integer_firstzeroevaluationmatrixpositivepoint) * (w) + (mdr_v_rank_integer_firstzeroevaluationmatrixpositivepoint))) * ac)) /\ exists ff_q_mdr_rank_integer_firstzeroevaluationmatrixpositivepointsource. ab = ff_q_mdr_rank_integer_firstzeroevaluationmatrixpositivepointsource * S ((S ((mdr_u_rank_integer_firstzeroevaluationmatrixpositivepoint) * (w) + (mdr_v_rank_integer_firstzeroevaluationmatrixpositivepoint))) * ac) + (mdr_a_rank_integer_firstzeroevaluationmatrixpositive)))))))) /\ (((exists ff_h_mdr_rank_integer_firstzeroevaluationmatrixpositiveoutput. ff_h_mdr_rank_integer_firstzeroevaluationmatrixpositiveoutput + S (mdr_a_rank_integer_firstzeroevaluationmatrixpositive) = S ((S (mdr_i_rank_integer_firstzeroevaluationmatrixpositive)) * mdr_uc_rank_integer_firstzeroevaluation)) /\ exists ff_q_mdr_rank_integer_firstzeroevaluationmatrixpositiveoutput. mdr_ub_rank_integer_firstzeroevaluation = ff_q_mdr_rank_integer_firstzeroevaluationmatrixpositiveoutput * S ((S (mdr_i_rank_integer_firstzeroevaluationmatrixpositive)) * mdr_uc_rank_integer_firstzeroevaluation) + (mdr_a_rank_integer_firstzeroevaluationmatrixpositive)))))) /\ (forall mdr_i_rank_integer_firstzeroevaluationmatrixnegative. (exists mdr_gap_rank_integer_firstzeroevaluationmatrixnegativebound. mdr_gap_rank_integer_firstzeroevaluationmatrixnegativebound + S (mdr_i_rank_integer_firstzeroevaluationmatrixnegative) = ((mdr_q_rank_integer_first) * (mdr_q_rank_integer_first))) -> exists mdr_a_rank_integer_firstzeroevaluationmatrixnegative. (((exists mdr_r_rank_integer_firstzeroevaluationmatrixnegativepoint mdr_s_rank_integer_firstzeroevaluationmatrixnegativepoint mdr_u_rank_integer_firstzeroevaluationmatrixnegativepoint mdr_v_rank_integer_firstzeroevaluationmatrixnegativepoint. ((mdr_i_rank_integer_firstzeroevaluationmatrixnegative = (mdr_q_rank_integer_first) * mdr_r_rank_integer_firstzeroevaluationmatrixnegativepoint + mdr_s_rank_integer_firstzeroevaluationmatrixnegativepoint) /\ ((exists mdr_gap_rank_integer_firstzeroevaluationmatrixnegativepointcolumn. mdr_gap_rank_integer_firstzeroevaluationmatrixnegativepointcolumn + S (mdr_s_rank_integer_firstzeroevaluationmatrixnegativepoint) = (mdr_q_rank_integer_first)) /\ ((((exists ff_h_mdr_rank_integer_firstzeroevaluationmatrixnegativepointrow_index. ff_h_mdr_rank_integer_firstzeroevaluationmatrixnegativepointrow_index + S (mdr_u_rank_integer_firstzeroevaluationmatrixnegativepoint) = S ((S (mdr_r_rank_integer_firstzeroevaluationmatrixnegativepoint)) * mdr_rc_rank_integer_firstzero)) /\ exists ff_q_mdr_rank_integer_firstzeroevaluationmatrixnegativepointrow_index. mdr_rb_rank_integer_firstzero = ff_q_mdr_rank_integer_firstzeroevaluationmatrixnegativepointrow_index * S ((S (mdr_r_rank_integer_firstzeroevaluationmatrixnegativepoint)) * mdr_rc_rank_integer_firstzero) + (mdr_u_rank_integer_firstzeroevaluationmatrixnegativepoint))) /\ ((((exists ff_h_mdr_rank_integer_firstzeroevaluationmatrixnegativepointcolumn_index. ff_h_mdr_rank_integer_firstzeroevaluationmatrixnegativepointcolumn_index + S (mdr_v_rank_integer_firstzeroevaluationmatrixnegativepoint) = S ((S (mdr_s_rank_integer_firstzeroevaluationmatrixnegativepoint)) * mdr_cc_rank_integer_firstzero)) /\ exists ff_q_mdr_rank_integer_firstzeroevaluationmatrixnegativepointcolumn_index. mdr_cb_rank_integer_firstzero = ff_q_mdr_rank_integer_firstzeroevaluationmatrixnegativepointcolumn_index * S ((S (mdr_s_rank_integer_firstzeroevaluationmatrixnegativepoint)) * mdr_cc_rank_integer_firstzero) + (mdr_v_rank_integer_firstzeroevaluationmatrixnegativepoint))) /\ (((exists ff_h_mdr_rank_integer_firstzeroevaluationmatrixnegativepointsource. ff_h_mdr_rank_integer_firstzeroevaluationmatrixnegativepointsource + S (mdr_a_rank_integer_firstzeroevaluationmatrixnegative) = S ((S ((mdr_u_rank_integer_firstzeroevaluationmatrixnegativepoint) * (w) + (mdr_v_rank_integer_firstzeroevaluationmatrixnegativepoint))) * bc)) /\ exists ff_q_mdr_rank_integer_firstzeroevaluationmatrixnegativepointsource. bb = ff_q_mdr_rank_integer_firstzeroevaluationmatrixnegativepointsource * S ((S ((mdr_u_rank_integer_firstzeroevaluationmatrixnegativepoint) * (w) + (mdr_v_rank_integer_firstzeroevaluationmatrixnegativepoint))) * bc) + (mdr_a_rank_integer_firstzeroevaluationmatrixnegative)))))))) /\ (((exists ff_h_mdr_rank_integer_firstzeroevaluationmatrixnegativeoutput. ff_h_mdr_rank_integer_firstzeroevaluationmatrixnegativeoutput + S (mdr_a_rank_integer_firstzeroevaluationmatrixnegative) = S ((S (mdr_i_rank_integer_firstzeroevaluationmatrixnegative)) * mdr_vc_rank_integer_firstzeroevaluation)) /\ exists ff_q_mdr_rank_integer_firstzeroevaluationmatrixnegativeoutput. mdr_vb_rank_integer_firstzeroevaluation = ff_q_mdr_rank_integer_firstzeroevaluationmatrixnegativeoutput * S ((S (mdr_i_rank_integer_firstzeroevaluationmatrixnegative)) * mdr_vc_rank_integer_firstzeroevaluation) + (mdr_a_rank_integer_firstzeroevaluationmatrixnegative)))))))) /\ (exists mdr_b_rank_integer_firstzeroevaluationdeterminant mdr_c_rank_integer_firstzeroevaluationdeterminant mdr_l_rank_integer_firstzeroevaluationdeterminant mdr_i_rank_integer_firstzeroevaluationdeterminant. ((forall mdr_i_rank_integer_firstzeroevaluationdeterminanth. (exists mdr_gap_rank_integer_firstzeroevaluationdeterminanthi. mdr_gap_rank_integer_firstzeroevaluationdeterminanthi + S (mdr_i_rank_integer_firstzeroevaluationdeterminanth) = (mdr_l_rank_integer_firstzeroevaluationdeterminant)) -> exists mdr_d_rank_integer_firstzeroevaluationdeterminanth mdr_pb_rank_integer_firstzeroevaluationdeterminanth mdr_pc_rank_integer_firstzeroevaluationdeterminanth mdr_nb_rank_integer_firstzeroevaluationdeterminanth mdr_nc_rank_integer_firstzeroevaluationdeterminanth mdr_p_rank_integer_firstzeroevaluationdeterminanth mdr_n_rank_integer_firstzeroevaluationdeterminanth. ((exists mdr_z_rank_integer_firstzeroevaluationdeterminanthr. ((exists mdr_a_rank_integer_firstzeroevaluationdeterminanthrc mdr_b_rank_integer_firstzeroevaluationdeterminanthrc mdr_c_rank_integer_firstzeroevaluationdeterminanthrc mdr_e_rank_integer_firstzeroevaluationdeterminanthrc mdr_f_rank_integer_firstzeroevaluationdeterminanthrc. ((mdr_a_rank_integer_firstzeroevaluationdeterminanthrc = ((mdr_d_rank_integer_firstzeroevaluationdeterminanth) + (mdr_pb_rank_integer_firstzeroevaluationdeterminanth)) * S ((mdr_d_rank_integer_firstzeroevaluationdeterminanth) + (mdr_pb_rank_integer_firstzeroevaluationdeterminanth)) + ((mdr_pb_rank_integer_firstzeroevaluationdeterminanth) + (mdr_pb_rank_integer_firstzeroevaluationdeterminanth))) /\ ((mdr_b_rank_integer_firstzeroevaluationdeterminanthrc = ((mdr_pc_rank_integer_firstzeroevaluationdeterminanth) + (mdr_nb_rank_integer_firstzeroevaluationdeterminanth)) * S ((mdr_pc_rank_integer_firstzeroevaluationdeterminanth) + (mdr_nb_rank_integer_firstzeroevaluationdeterminanth)) + ((mdr_nb_rank_integer_firstzeroevaluationdeterminanth) + (mdr_nb_rank_integer_firstzeroevaluationdeterminanth))) /\ ((mdr_c_rank_integer_firstzeroevaluationdeterminanthrc = ((mdr_a_rank_integer_firstzeroevaluationdeterminanthrc) + (mdr_b_rank_integer_firstzeroevaluationdeterminanthrc)) * S ((mdr_a_rank_integer_firstzeroevaluationdeterminanthrc) + (mdr_b_rank_integer_firstzeroevaluationdeterminanthrc)) + ((mdr_b_rank_integer_firstzeroevaluationdeterminanthrc) + (mdr_b_rank_integer_firstzeroevaluationdeterminanthrc))) /\ ((mdr_e_rank_integer_firstzeroevaluationdeterminanthrc = ((mdr_p_rank_integer_firstzeroevaluationdeterminanth) + (mdr_n_rank_integer_firstzeroevaluationdeterminanth)) * S ((mdr_p_rank_integer_firstzeroevaluationdeterminanth) + (mdr_n_rank_integer_firstzeroevaluationdeterminanth)) + ((mdr_n_rank_integer_firstzeroevaluationdeterminanth) + (mdr_n_rank_integer_firstzeroevaluationdeterminanth))) /\ ((mdr_f_rank_integer_firstzeroevaluationdeterminanthrc = ((mdr_nc_rank_integer_firstzeroevaluationdeterminanth) + (mdr_e_rank_integer_firstzeroevaluationdeterminanthrc)) * S ((mdr_nc_rank_integer_firstzeroevaluationdeterminanth) + (mdr_e_rank_integer_firstzeroevaluationdeterminanthrc)) + ((mdr_e_rank_integer_firstzeroevaluationdeterminanthrc) + (mdr_e_rank_integer_firstzeroevaluationdeterminanthrc))) /\ ((mdr_z_rank_integer_firstzeroevaluationdeterminanthr) = ((mdr_c_rank_integer_firstzeroevaluationdeterminanthrc) + (mdr_f_rank_integer_firstzeroevaluationdeterminanthrc)) * S ((mdr_c_rank_integer_firstzeroevaluationdeterminanthrc) + (mdr_f_rank_integer_firstzeroevaluationdeterminanthrc)) + ((mdr_f_rank_integer_firstzeroevaluationdeterminanthrc) + (mdr_f_rank_integer_firstzeroevaluationdeterminanthrc))))))))) /\ (((exists ff_h_mdr_rank_integer_firstzeroevaluationdeterminanthrb. ff_h_mdr_rank_integer_firstzeroevaluationdeterminanthrb + S (mdr_z_rank_integer_firstzeroevaluationdeterminanthr) = S ((S (mdr_i_rank_integer_firstzeroevaluationdeterminanth)) * mdr_c_rank_integer_firstzeroevaluationdeterminant)) /\ exists ff_q_mdr_rank_integer_firstzeroevaluationdeterminanthrb. mdr_b_rank_integer_firstzeroevaluationdeterminant = ff_q_mdr_rank_integer_firstzeroevaluationdeterminanthrb * S ((S (mdr_i_rank_integer_firstzeroevaluationdeterminanth)) * mdr_c_rank_integer_firstzeroevaluationdeterminant) + (mdr_z_rank_integer_firstzeroevaluationdeterminanthr))))) /\ (((((mdr_d_rank_integer_firstzeroevaluationdeterminanth) = 0) /\ (((mdr_p_rank_integer_firstzeroevaluationdeterminanth) = 1) /\ ((mdr_n_rank_integer_firstzeroevaluationdeterminanth) = 0))) \/ exists mdr_q_rank_integer_firstzeroevaluationdeterminanths mdr_eb_rank_integer_firstzeroevaluationdeterminanths mdr_ec_rank_integer_firstzeroevaluationdeterminanths mdr_fb_rank_integer_firstzeroevaluationdeterminanths mdr_fc_rank_integer_firstzeroevaluationdeterminanths. (((mdr_d_rank_integer_firstzeroevaluationdeterminanth) = S (mdr_q_rank_integer_firstzeroevaluationdeterminanths)) /\ ((forall mdr_j_rank_integer_firstzeroevaluationdeterminanthsc. (exists mdr_gap_rank_integer_firstzeroevaluationdeterminanthscj. mdr_gap_rank_integer_firstzeroevaluationdeterminanthscj + S (mdr_j_rank_integer_firstzeroevaluationdeterminanthsc) = (S (mdr_q_rank_integer_firstzeroevaluationdeterminanths))) -> exists mdr_i_rank_integer_firstzeroevaluationdeterminanthsc mdr_up_rank_integer_firstzeroevaluationdeterminanthsc mdr_us_rank_integer_firstzeroevaluationdeterminanthsc mdr_un_rank_integer_firstzeroevaluationdeterminanthsc mdr_ut_rank_integer_firstzeroevaluationdeterminanthsc mdr_p_rank_integer_firstzeroevaluationdeterminanthsc mdr_n_rank_integer_firstzeroevaluationdeterminanthsc. ((exists mdr_gap_rank_integer_firstzeroevaluationdeterminanthsci. mdr_gap_rank_integer_firstzeroevaluationdeterminanthsci + S (mdr_i_rank_integer_firstzeroevaluationdeterminanthsc) = (mdr_i_rank_integer_firstzeroevaluationdeterminanth)) /\ ((exists mdr_z_rank_integer_firstzeroevaluationdeterminanthscr. ((exists mdr_a_rank_integer_firstzeroevaluationdeterminanthscrc mdr_b_rank_integer_firstzeroevaluationdeterminanthscrc mdr_c_rank_integer_firstzeroevaluationdeterminanthscrc mdr_e_rank_integer_firstzeroevaluationdeterminanthscrc mdr_f_rank_integer_firstzeroevaluationdeterminanthscrc. ((mdr_a_rank_integer_firstzeroevaluationdeterminanthscrc = ((mdr_q_rank_integer_firstzeroevaluationdeterminanths) + (mdr_up_rank_integer_firstzeroevaluationdeterminanthsc)) * S ((mdr_q_rank_integer_firstzeroevaluationdeterminanths) + (mdr_up_rank_integer_firstzeroevaluationdeterminanthsc)) + ((mdr_up_rank_integer_firstzeroevaluationdeterminanthsc) + (mdr_up_rank_integer_firstzeroevaluationdeterminanthsc))) /\ ((mdr_b_rank_integer_firstzeroevaluationdeterminanthscrc = ((mdr_us_rank_integer_firstzeroevaluationdeterminanthsc) + (mdr_un_rank_integer_firstzeroevaluationdeterminanthsc)) * S ((mdr_us_rank_integer_firstzeroevaluationdeterminanthsc) + (mdr_un_rank_integer_firstzeroevaluationdeterminanthsc)) + ((mdr_un_rank_integer_firstzeroevaluationdeterminanthsc) + (mdr_un_rank_integer_firstzeroevaluationdeterminanthsc))) /\ ((mdr_c_rank_integer_firstzeroevaluationdeterminanthscrc = ((mdr_a_rank_integer_firstzeroevaluationdeterminanthscrc) + (mdr_b_rank_integer_firstzeroevaluationdeterminanthscrc)) * S ((mdr_a_rank_integer_firstzeroevaluationdeterminanthscrc) + (mdr_b_rank_integer_firstzeroevaluationdeterminanthscrc)) + ((mdr_b_rank_integer_firstzeroevaluationdeterminanthscrc) + (mdr_b_rank_integer_firstzeroevaluationdeterminanthscrc))) /\ ((mdr_e_rank_integer_firstzeroevaluationdeterminanthscrc = ((mdr_p_rank_integer_firstzeroevaluationdeterminanthsc) + (mdr_n_rank_integer_firstzeroevaluationdeterminanthsc)) * S ((mdr_p_rank_integer_firstzeroevaluationdeterminanthsc) + (mdr_n_rank_integer_firstzeroevaluationdeterminanthsc)) + ((mdr_n_rank_integer_firstzeroevaluationdeterminanthsc) + (mdr_n_rank_integer_firstzeroevaluationdeterminanthsc))) /\ ((mdr_f_rank_integer_firstzeroevaluationdeterminanthscrc = ((mdr_ut_rank_integer_firstzeroevaluationdeterminanthsc) + (mdr_e_rank_integer_firstzeroevaluationdeterminanthscrc)) * S ((mdr_ut_rank_integer_firstzeroevaluationdeterminanthsc) + (mdr_e_rank_integer_firstzeroevaluationdeterminanthscrc)) + ((mdr_e_rank_integer_firstzeroevaluationdeterminanthscrc) + (mdr_e_rank_integer_firstzeroevaluationdeterminanthscrc))) /\ ((mdr_z_rank_integer_firstzeroevaluationdeterminanthscr) = ((mdr_c_rank_integer_firstzeroevaluationdeterminanthscrc) + (mdr_f_rank_integer_firstzeroevaluationdeterminanthscrc)) * S ((mdr_c_rank_integer_firstzeroevaluationdeterminanthscrc) + (mdr_f_rank_integer_firstzeroevaluationdeterminanthscrc)) + ((mdr_f_rank_integer_firstzeroevaluationdeterminanthscrc) + (mdr_f_rank_integer_firstzeroevaluationdeterminanthscrc))))))))) /\ (((exists ff_h_mdr_rank_integer_firstzeroevaluationdeterminanthscrb. ff_h_mdr_rank_integer_firstzeroevaluationdeterminanthscrb + S (mdr_z_rank_integer_firstzeroevaluationdeterminanthscr) = S ((S (mdr_i_rank_integer_firstzeroevaluationdeterminanthsc)) * mdr_c_rank_integer_firstzeroevaluationdeterminant)) /\ exists ff_q_mdr_rank_integer_firstzeroevaluationdeterminanthscrb. mdr_b_rank_integer_firstzeroevaluationdeterminant = ff_q_mdr_rank_integer_firstzeroevaluationdeterminanthscrb * S ((S (mdr_i_rank_integer_firstzeroevaluationdeterminanthsc)) * mdr_c_rank_integer_firstzeroevaluationdeterminant) + (mdr_z_rank_integer_firstzeroevaluationdeterminanthscr))))) /\ ((((forall ff_index_mdm_prefix_mdr_rank_integer_firstzeroevaluationdeterminanthscm_positive. (exists ff_gap_mdm_lt_mdr_rank_integer_firstzeroevaluationdeterminanthscm_positive_index_bound. ff_gap_mdm_lt_mdr_rank_integer_firstzeroevaluationdeterminanthscm_positive_index_bound + S (ff_index_mdm_prefix_mdr_rank_integer_firstzeroevaluationdeterminanthscm_positive) = ((mdr_q_rank_integer_firstzeroevaluationdeterminanths) * (mdr_q_rank_integer_firstzeroevaluationdeterminanths))) -> exists ff_row_mdm_prefix_mdr_rank_integer_firstzeroevaluationdeterminanthscm_positive ff_column_mdm_prefix_mdr_rank_integer_firstzeroevaluationdeterminanthscm_positive ff_value_mdm_prefix_mdr_rank_integer_firstzeroevaluationdeterminanthscm_positive. (ff_index_mdm_prefix_mdr_rank_integer_firstzeroevaluationdeterminanthscm_positive = (mdr_q_rank_integer_firstzeroevaluationdeterminanths) * ff_row_mdm_prefix_mdr_rank_integer_firstzeroevaluationdeterminanthscm_positive + ff_column_mdm_prefix_mdr_rank_integer_firstzeroevaluationdeterminanthscm_positive /\ ((exists ff_gap_mdm_lt_mdr_rank_integer_firstzeroevaluationdeterminanthscm_positive_column_bound. ff_gap_mdm_lt_mdr_rank_integer_firstzeroevaluationdeterminanthscm_positive_column_bound + S (ff_column_mdm_prefix_mdr_rank_integer_firstzeroevaluationdeterminanthscm_positive) = (mdr_q_rank_integer_firstzeroevaluationdeterminanths)) /\ ((exists ff_row_mdm_cell_mdr_rank_integer_firstzeroevaluationdeterminanthscm_positive_cell ff_column_mdm_cell_mdr_rank_integer_firstzeroevaluationdeterminanthscm_positive_cell. (((((exists ff_gap_mdm_lt_mdr_rank_integer_firstzeroevaluationdeterminanthscm_positive_cell_row_before. ff_gap_mdm_lt_mdr_rank_integer_firstzeroevaluationdeterminanthscm_positive_cell_row_before + S (ff_row_mdm_prefix_mdr_rank_integer_firstzeroevaluationdeterminanthscm_positive) = (0)) /\ ff_row_mdm_cell_mdr_rank_integer_firstzeroevaluationdeterminanthscm_positive_cell = ff_row_mdm_prefix_mdr_rank_integer_firstzeroevaluationdeterminanthscm_positive) \/ ((exists ff_gap_mdm_le_mdr_rank_integer_firstzeroevaluationdeterminanthscm_positive_cell_row_after. ff_gap_mdm_le_mdr_rank_integer_firstzeroevaluationdeterminanthscm_positive_cell_row_after + (0) = (ff_row_mdm_prefix_mdr_rank_integer_firstzeroevaluationdeterminanthscm_positive)) /\ ff_row_mdm_cell_mdr_rank_integer_firstzeroevaluationdeterminanthscm_positive_cell = S ff_row_mdm_prefix_mdr_rank_integer_firstzeroevaluationdeterminanthscm_positive))) /\ (((((exists ff_gap_mdm_lt_mdr_rank_integer_firstzeroevaluationdeterminanthscm_positive_cell_column_before. ff_gap_mdm_lt_mdr_rank_integer_firstzeroevaluationdeterminanthscm_positive_cell_column_before + S (ff_column_mdm_prefix_mdr_rank_integer_firstzeroevaluationdeterminanthscm_positive) = (mdr_j_rank_integer_firstzeroevaluationdeterminanthsc)) /\ ff_column_mdm_cell_mdr_rank_integer_firstzeroevaluationdeterminanthscm_positive_cell = ff_column_mdm_prefix_mdr_rank_integer_firstzeroevaluationdeterminanthscm_positive) \/ ((exists ff_gap_mdm_le_mdr_rank_integer_firstzeroevaluationdeterminanthscm_positive_cell_column_after. ff_gap_mdm_le_mdr_rank_integer_firstzeroevaluationdeterminanthscm_positive_cell_column_after + (mdr_j_rank_integer_firstzeroevaluationdeterminanthsc) = (ff_column_mdm_prefix_mdr_rank_integer_firstzeroevaluationdeterminanthscm_positive)) /\ ff_column_mdm_cell_mdr_rank_integer_firstzeroevaluationdeterminanthscm_positive_cell = S ff_column_mdm_prefix_mdr_rank_integer_firstzeroevaluationdeterminanthscm_positive))) /\ (((exists ff_h_mdm_mdr_rank_integer_firstzeroevaluationdeterminanthscm_positive_cell_source. ff_h_mdm_mdr_rank_integer_firstzeroevaluationdeterminanthscm_positive_cell_source + S (ff_value_mdm_prefix_mdr_rank_integer_firstzeroevaluationdeterminanthscm_positive) = S ((S ((ff_row_mdm_cell_mdr_rank_integer_firstzeroevaluationdeterminanthscm_positive_cell) * (S (mdr_q_rank_integer_firstzeroevaluationdeterminanths)) + (ff_column_mdm_cell_mdr_rank_integer_firstzeroevaluationdeterminanthscm_positive_cell))) * mdr_pc_rank_integer_firstzeroevaluationdeterminanth)) /\ exists ff_q_mdm_mdr_rank_integer_firstzeroevaluationdeterminanthscm_positive_cell_source. mdr_pb_rank_integer_firstzeroevaluationdeterminanth = ff_q_mdm_mdr_rank_integer_firstzeroevaluationdeterminanthscm_positive_cell_source * S ((S ((ff_row_mdm_cell_mdr_rank_integer_firstzeroevaluationdeterminanthscm_positive_cell) * (S (mdr_q_rank_integer_firstzeroevaluationdeterminanths)) + (ff_column_mdm_cell_mdr_rank_integer_firstzeroevaluationdeterminanthscm_positive_cell))) * mdr_pc_rank_integer_firstzeroevaluationdeterminanth) + (ff_value_mdm_prefix_mdr_rank_integer_firstzeroevaluationdeterminanthscm_positive)))))) /\ (((exists ff_h_mdm_mdr_rank_integer_firstzeroevaluationdeterminanthscm_positive_target. ff_h_mdm_mdr_rank_integer_firstzeroevaluationdeterminanthscm_positive_target + S (ff_value_mdm_prefix_mdr_rank_integer_firstzeroevaluationdeterminanthscm_positive) = S ((S (ff_index_mdm_prefix_mdr_rank_integer_firstzeroevaluationdeterminanthscm_positive)) * mdr_us_rank_integer_firstzeroevaluationdeterminanthsc)) /\ exists ff_q_mdm_mdr_rank_integer_firstzeroevaluationdeterminanthscm_positive_target. mdr_up_rank_integer_firstzeroevaluationdeterminanthsc = ff_q_mdm_mdr_rank_integer_firstzeroevaluationdeterminanthscm_positive_target * S ((S (ff_index_mdm_prefix_mdr_rank_integer_firstzeroevaluationdeterminanthscm_positive)) * mdr_us_rank_integer_firstzeroevaluationdeterminanthsc) + (ff_value_mdm_prefix_mdr_rank_integer_firstzeroevaluationdeterminanthscm_positive))))))) /\ (forall ff_index_mdm_prefix_mdr_rank_integer_firstzeroevaluationdeterminanthscm_negative. (exists ff_gap_mdm_lt_mdr_rank_integer_firstzeroevaluationdeterminanthscm_negative_index_bound. ff_gap_mdm_lt_mdr_rank_integer_firstzeroevaluationdeterminanthscm_negative_index_bound + S (ff_index_mdm_prefix_mdr_rank_integer_firstzeroevaluationdeterminanthscm_negative) = ((mdr_q_rank_integer_firstzeroevaluationdeterminanths) * (mdr_q_rank_integer_firstzeroevaluationdeterminanths))) -> exists ff_row_mdm_prefix_mdr_rank_integer_firstzeroevaluationdeterminanthscm_negative ff_column_mdm_prefix_mdr_rank_integer_firstzeroevaluationdeterminanthscm_negative ff_value_mdm_prefix_mdr_rank_integer_firstzeroevaluationdeterminanthscm_negative. (ff_index_mdm_prefix_mdr_rank_integer_firstzeroevaluationdeterminanthscm_negative = (mdr_q_rank_integer_firstzeroevaluationdeterminanths) * ff_row_mdm_prefix_mdr_rank_integer_firstzeroevaluationdeterminanthscm_negative + ff_column_mdm_prefix_mdr_rank_integer_firstzeroevaluationdeterminanthscm_negative /\ ((exists ff_gap_mdm_lt_mdr_rank_integer_firstzeroevaluationdeterminanthscm_negative_column_bound. ff_gap_mdm_lt_mdr_rank_integer_firstzeroevaluationdeterminanthscm_negative_column_bound + S (ff_column_mdm_prefix_mdr_rank_integer_firstzeroevaluationdeterminanthscm_negative) = (mdr_q_rank_integer_firstzeroevaluationdeterminanths)) /\ ((exists ff_row_mdm_cell_mdr_rank_integer_firstzeroevaluationdeterminanthscm_negative_cell ff_column_mdm_cell_mdr_rank_integer_firstzeroevaluationdeterminanthscm_negative_cell. (((((exists ff_gap_mdm_lt_mdr_rank_integer_firstzeroevaluationdeterminanthscm_negative_cell_row_before. ff_gap_mdm_lt_mdr_rank_integer_firstzeroevaluationdeterminanthscm_negative_cell_row_before + S (ff_row_mdm_prefix_mdr_rank_integer_firstzeroevaluationdeterminanthscm_negative) = (0)) /\ ff_row_mdm_cell_mdr_rank_integer_firstzeroevaluationdeterminanthscm_negative_cell = ff_row_mdm_prefix_mdr_rank_integer_firstzeroevaluationdeterminanthscm_negative) \/ ((exists ff_gap_mdm_le_mdr_rank_integer_firstzeroevaluationdeterminanthscm_negative_cell_row_after. ff_gap_mdm_le_mdr_rank_integer_firstzeroevaluationdeterminanthscm_negative_cell_row_after + (0) = (ff_row_mdm_prefix_mdr_rank_integer_firstzeroevaluationdeterminanthscm_negative)) /\ ff_row_mdm_cell_mdr_rank_integer_firstzeroevaluationdeterminanthscm_negative_cell = S ff_row_mdm_prefix_mdr_rank_integer_firstzeroevaluationdeterminanthscm_negative))) /\ (((((exists ff_gap_mdm_lt_mdr_rank_integer_firstzeroevaluationdeterminanthscm_negative_cell_column_before. ff_gap_mdm_lt_mdr_rank_integer_firstzeroevaluationdeterminanthscm_negative_cell_column_before + S (ff_column_mdm_prefix_mdr_rank_integer_firstzeroevaluationdeterminanthscm_negative) = (mdr_j_rank_integer_firstzeroevaluationdeterminanthsc)) /\ ff_column_mdm_cell_mdr_rank_integer_firstzeroevaluationdeterminanthscm_negative_cell = ff_column_mdm_prefix_mdr_rank_integer_firstzeroevaluationdeterminanthscm_negative) \/ ((exists ff_gap_mdm_le_mdr_rank_integer_firstzeroevaluationdeterminanthscm_negative_cell_column_after. ff_gap_mdm_le_mdr_rank_integer_firstzeroevaluationdeterminanthscm_negative_cell_column_after + (mdr_j_rank_integer_firstzeroevaluationdeterminanthsc) = (ff_column_mdm_prefix_mdr_rank_integer_firstzeroevaluationdeterminanthscm_negative)) /\ ff_column_mdm_cell_mdr_rank_integer_firstzeroevaluationdeterminanthscm_negative_cell = S ff_column_mdm_prefix_mdr_rank_integer_firstzeroevaluationdeterminanthscm_negative))) /\ (((exists ff_h_mdm_mdr_rank_integer_firstzeroevaluationdeterminanthscm_negative_cell_source. ff_h_mdm_mdr_rank_integer_firstzeroevaluationdeterminanthscm_negative_cell_source + S (ff_value_mdm_prefix_mdr_rank_integer_firstzeroevaluationdeterminanthscm_negative) = S ((S ((ff_row_mdm_cell_mdr_rank_integer_firstzeroevaluationdeterminanthscm_negative_cell) * (S (mdr_q_rank_integer_firstzeroevaluationdeterminanths)) + (ff_column_mdm_cell_mdr_rank_integer_firstzeroevaluationdeterminanthscm_negative_cell))) * mdr_nc_rank_integer_firstzeroevaluationdeterminanth)) /\ exists ff_q_mdm_mdr_rank_integer_firstzeroevaluationdeterminanthscm_negative_cell_source. mdr_nb_rank_integer_firstzeroevaluationdeterminanth = ff_q_mdm_mdr_rank_integer_firstzeroevaluationdeterminanthscm_negative_cell_source * S ((S ((ff_row_mdm_cell_mdr_rank_integer_firstzeroevaluationdeterminanthscm_negative_cell) * (S (mdr_q_rank_integer_firstzeroevaluationdeterminanths)) + (ff_column_mdm_cell_mdr_rank_integer_firstzeroevaluationdeterminanthscm_negative_cell))) * mdr_nc_rank_integer_firstzeroevaluationdeterminanth) + (ff_value_mdm_prefix_mdr_rank_integer_firstzeroevaluationdeterminanthscm_negative)))))) /\ (((exists ff_h_mdm_mdr_rank_integer_firstzeroevaluationdeterminanthscm_negative_target. ff_h_mdm_mdr_rank_integer_firstzeroevaluationdeterminanthscm_negative_target + S (ff_value_mdm_prefix_mdr_rank_integer_firstzeroevaluationdeterminanthscm_negative) = S ((S (ff_index_mdm_prefix_mdr_rank_integer_firstzeroevaluationdeterminanthscm_negative)) * mdr_ut_rank_integer_firstzeroevaluationdeterminanthsc)) /\ exists ff_q_mdm_mdr_rank_integer_firstzeroevaluationdeterminanthscm_negative_target. mdr_un_rank_integer_firstzeroevaluationdeterminanthsc = ff_q_mdm_mdr_rank_integer_firstzeroevaluationdeterminanthscm_negative_target * S ((S (ff_index_mdm_prefix_mdr_rank_integer_firstzeroevaluationdeterminanthscm_negative)) * mdr_ut_rank_integer_firstzeroevaluationdeterminanthsc) + (ff_value_mdm_prefix_mdr_rank_integer_firstzeroevaluationdeterminanthscm_negative))))))))) /\ ((((exists ff_h_mdr_rank_integer_firstzeroevaluationdeterminanthscp. ff_h_mdr_rank_integer_firstzeroevaluationdeterminanthscp + S (mdr_p_rank_integer_firstzeroevaluationdeterminanthsc) = S ((S (mdr_j_rank_integer_firstzeroevaluationdeterminanthsc)) * mdr_ec_rank_integer_firstzeroevaluationdeterminanths)) /\ exists ff_q_mdr_rank_integer_firstzeroevaluationdeterminanthscp. mdr_eb_rank_integer_firstzeroevaluationdeterminanths = ff_q_mdr_rank_integer_firstzeroevaluationdeterminanthscp * S ((S (mdr_j_rank_integer_firstzeroevaluationdeterminanthsc)) * mdr_ec_rank_integer_firstzeroevaluationdeterminanths) + (mdr_p_rank_integer_firstzeroevaluationdeterminanthsc))) /\ (((exists ff_h_mdr_rank_integer_firstzeroevaluationdeterminanthscn. ff_h_mdr_rank_integer_firstzeroevaluationdeterminanthscn + S (mdr_n_rank_integer_firstzeroevaluationdeterminanthsc) = S ((S (mdr_j_rank_integer_firstzeroevaluationdeterminanthsc)) * mdr_fc_rank_integer_firstzeroevaluationdeterminanths)) /\ exists ff_q_mdr_rank_integer_firstzeroevaluationdeterminanthscn. mdr_fb_rank_integer_firstzeroevaluationdeterminanths = ff_q_mdr_rank_integer_firstzeroevaluationdeterminanthscn * S ((S (mdr_j_rank_integer_firstzeroevaluationdeterminanthsc)) * mdr_fc_rank_integer_firstzeroevaluationdeterminanths) + (mdr_n_rank_integer_firstzeroevaluationdeterminanthsc)))))))) /\ (exists ff_ub_mce_fold_mdr_rank_integer_firstzeroevaluationdeterminanthsf ff_uc_mce_fold_mdr_rank_integer_firstzeroevaluationdeterminanthsf ff_vb_mce_fold_mdr_rank_integer_firstzeroevaluationdeterminanthsf ff_vc_mce_fold_mdr_rank_integer_firstzeroevaluationdeterminanthsf. ((forall ff_index_mce_alternating_mdr_rank_integer_firstzeroevaluationdeterminanthsf_prefix. (exists ff_gap_mce_mdr_rank_integer_firstzeroevaluationdeterminanthsf_prefix_index. ff_gap_mce_mdr_rank_integer_firstzeroevaluationdeterminanthsf_prefix_index + S (ff_index_mce_alternating_mdr_rank_integer_firstzeroevaluationdeterminanthsf_prefix) = (S (mdr_q_rank_integer_firstzeroevaluationdeterminanths))) -> exists ff_ap_mce_alternating_mdr_rank_integer_firstzeroevaluationdeterminanthsf_prefix ff_an_mce_alternating_mdr_rank_integer_firstzeroevaluationdeterminanthsf_prefix ff_bp_mce_alternating_mdr_rank_integer_firstzeroevaluationdeterminanthsf_prefix ff_bn_mce_alternating_mdr_rank_integer_firstzeroevaluationdeterminanthsf_prefix ff_p_mce_alternating_mdr_rank_integer_firstzeroevaluationdeterminanthsf_prefix ff_n_mce_alternating_mdr_rank_integer_firstzeroevaluationdeterminanthsf_prefix. ((((exists ff_h_mce_mdr_rank_integer_firstzeroevaluationdeterminanthsf_prefix_ap. ff_h_mce_mdr_rank_integer_firstzeroevaluationdeterminanthsf_prefix_ap + S (ff_ap_mce_alternating_mdr_rank_integer_firstzeroevaluationdeterminanthsf_prefix) = S ((S (ff_index_mce_alternating_mdr_rank_integer_firstzeroevaluationdeterminanthsf_prefix)) * mdr_pc_rank_integer_firstzeroevaluationdeterminanth)) /\ exists ff_q_mce_mdr_rank_integer_firstzeroevaluationdeterminanthsf_prefix_ap. mdr_pb_rank_integer_firstzeroevaluationdeterminanth = ff_q_mce_mdr_rank_integer_firstzeroevaluationdeterminanthsf_prefix_ap * S ((S (ff_index_mce_alternating_mdr_rank_integer_firstzeroevaluationdeterminanthsf_prefix)) * mdr_pc_rank_integer_firstzeroevaluationdeterminanth) + (ff_ap_mce_alternating_mdr_rank_integer_firstzeroevaluationdeterminanthsf_prefix))) /\ ((((exists ff_h_mce_mdr_rank_integer_firstzeroevaluationdeterminanthsf_prefix_an. ff_h_mce_mdr_rank_integer_firstzeroevaluationdeterminanthsf_prefix_an + S (ff_an_mce_alternating_mdr_rank_integer_firstzeroevaluationdeterminanthsf_prefix) = S ((S (ff_index_mce_alternating_mdr_rank_integer_firstzeroevaluationdeterminanthsf_prefix)) * mdr_nc_rank_integer_firstzeroevaluationdeterminanth)) /\ exists ff_q_mce_mdr_rank_integer_firstzeroevaluationdeterminanthsf_prefix_an. mdr_nb_rank_integer_firstzeroevaluationdeterminanth = ff_q_mce_mdr_rank_integer_firstzeroevaluationdeterminanthsf_prefix_an * S ((S (ff_index_mce_alternating_mdr_rank_integer_firstzeroevaluationdeterminanthsf_prefix)) * mdr_nc_rank_integer_firstzeroevaluationdeterminanth) + (ff_an_mce_alternating_mdr_rank_integer_firstzeroevaluationdeterminanthsf_prefix))) /\ ((((exists ff_h_mce_mdr_rank_integer_firstzeroevaluationdeterminanthsf_prefix_bp. ff_h_mce_mdr_rank_integer_firstzeroevaluationdeterminanthsf_prefix_bp + S (ff_bp_mce_alternating_mdr_rank_integer_firstzeroevaluationdeterminanthsf_prefix) = S ((S (ff_index_mce_alternating_mdr_rank_integer_firstzeroevaluationdeterminanthsf_prefix)) * mdr_ec_rank_integer_firstzeroevaluationdeterminanths)) /\ exists ff_q_mce_mdr_rank_integer_firstzeroevaluationdeterminanthsf_prefix_bp. mdr_eb_rank_integer_firstzeroevaluationdeterminanths = ff_q_mce_mdr_rank_integer_firstzeroevaluationdeterminanthsf_prefix_bp * S ((S (ff_index_mce_alternating_mdr_rank_integer_firstzeroevaluationdeterminanthsf_prefix)) * mdr_ec_rank_integer_firstzeroevaluationdeterminanths) + (ff_bp_mce_alternating_mdr_rank_integer_firstzeroevaluationdeterminanthsf_prefix))) /\ ((((exists ff_h_mce_mdr_rank_integer_firstzeroevaluationdeterminanthsf_prefix_bn. ff_h_mce_mdr_rank_integer_firstzeroevaluationdeterminanthsf_prefix_bn + S (ff_bn_mce_alternating_mdr_rank_integer_firstzeroevaluationdeterminanthsf_prefix) = S ((S (ff_index_mce_alternating_mdr_rank_integer_firstzeroevaluationdeterminanthsf_prefix)) * mdr_fc_rank_integer_firstzeroevaluationdeterminanths)) /\ exists ff_q_mce_mdr_rank_integer_firstzeroevaluationdeterminanthsf_prefix_bn. mdr_fb_rank_integer_firstzeroevaluationdeterminanths = ff_q_mce_mdr_rank_integer_firstzeroevaluationdeterminanthsf_prefix_bn * S ((S (ff_index_mce_alternating_mdr_rank_integer_firstzeroevaluationdeterminanthsf_prefix)) * mdr_fc_rank_integer_firstzeroevaluationdeterminanths) + (ff_bn_mce_alternating_mdr_rank_integer_firstzeroevaluationdeterminanthsf_prefix))) /\ ((((exists ff_h_mce_mdr_rank_integer_firstzeroevaluationdeterminanthsf_prefix_positive. ff_h_mce_mdr_rank_integer_firstzeroevaluationdeterminanthsf_prefix_positive + S (ff_p_mce_alternating_mdr_rank_integer_firstzeroevaluationdeterminanthsf_prefix) = S ((S (ff_index_mce_alternating_mdr_rank_integer_firstzeroevaluationdeterminanthsf_prefix)) * ff_uc_mce_fold_mdr_rank_integer_firstzeroevaluationdeterminanthsf)) /\ exists ff_q_mce_mdr_rank_integer_firstzeroevaluationdeterminanthsf_prefix_positive. ff_ub_mce_fold_mdr_rank_integer_firstzeroevaluationdeterminanthsf = ff_q_mce_mdr_rank_integer_firstzeroevaluationdeterminanthsf_prefix_positive * S ((S (ff_index_mce_alternating_mdr_rank_integer_firstzeroevaluationdeterminanthsf_prefix)) * ff_uc_mce_fold_mdr_rank_integer_firstzeroevaluationdeterminanthsf) + (ff_p_mce_alternating_mdr_rank_integer_firstzeroevaluationdeterminanthsf_prefix))) /\ ((((exists ff_h_mce_mdr_rank_integer_firstzeroevaluationdeterminanthsf_prefix_negative. ff_h_mce_mdr_rank_integer_firstzeroevaluationdeterminanthsf_prefix_negative + S (ff_n_mce_alternating_mdr_rank_integer_firstzeroevaluationdeterminanthsf_prefix) = S ((S (ff_index_mce_alternating_mdr_rank_integer_firstzeroevaluationdeterminanthsf_prefix)) * ff_vc_mce_fold_mdr_rank_integer_firstzeroevaluationdeterminanthsf)) /\ exists ff_q_mce_mdr_rank_integer_firstzeroevaluationdeterminanthsf_prefix_negative. ff_vb_mce_fold_mdr_rank_integer_firstzeroevaluationdeterminanthsf = ff_q_mce_mdr_rank_integer_firstzeroevaluationdeterminanthsf_prefix_negative * S ((S (ff_index_mce_alternating_mdr_rank_integer_firstzeroevaluationdeterminanthsf_prefix)) * ff_vc_mce_fold_mdr_rank_integer_firstzeroevaluationdeterminanthsf) + (ff_n_mce_alternating_mdr_rank_integer_firstzeroevaluationdeterminanthsf_prefix))) /\ (((exists ff_even_mce_term_mdr_rank_integer_firstzeroevaluationdeterminanthsf_prefix_term. ff_index_mce_alternating_mdr_rank_integer_firstzeroevaluationdeterminanthsf_prefix = 2 * ff_even_mce_term_mdr_rank_integer_firstzeroevaluationdeterminanthsf_prefix_term) /\ (ff_p_mce_alternating_mdr_rank_integer_firstzeroevaluationdeterminanthsf_prefix = (ff_ap_mce_alternating_mdr_rank_integer_firstzeroevaluationdeterminanthsf_prefix) * (ff_bp_mce_alternating_mdr_rank_integer_firstzeroevaluationdeterminanthsf_prefix) + (ff_an_mce_alternating_mdr_rank_integer_firstzeroevaluationdeterminanthsf_prefix) * (ff_bn_mce_alternating_mdr_rank_integer_firstzeroevaluationdeterminanthsf_prefix) /\ ff_n_mce_alternating_mdr_rank_integer_firstzeroevaluationdeterminanthsf_prefix = (ff_ap_mce_alternating_mdr_rank_integer_firstzeroevaluationdeterminanthsf_prefix) * (ff_bn_mce_alternating_mdr_rank_integer_firstzeroevaluationdeterminanthsf_prefix) + (ff_an_mce_alternating_mdr_rank_integer_firstzeroevaluationdeterminanthsf_prefix) * (ff_bp_mce_alternating_mdr_rank_integer_firstzeroevaluationdeterminanthsf_prefix))) \/ ((exists ff_odd_mce_term_mdr_rank_integer_firstzeroevaluationdeterminanthsf_prefix_term. ff_index_mce_alternating_mdr_rank_integer_firstzeroevaluationdeterminanthsf_prefix = 2 * ff_odd_mce_term_mdr_rank_integer_firstzeroevaluationdeterminanthsf_prefix_term + 1) /\ (ff_p_mce_alternating_mdr_rank_integer_firstzeroevaluationdeterminanthsf_prefix = (ff_ap_mce_alternating_mdr_rank_integer_firstzeroevaluationdeterminanthsf_prefix) * (ff_bn_mce_alternating_mdr_rank_integer_firstzeroevaluationdeterminanthsf_prefix) + (ff_an_mce_alternating_mdr_rank_integer_firstzeroevaluationdeterminanthsf_prefix) * (ff_bp_mce_alternating_mdr_rank_integer_firstzeroevaluationdeterminanthsf_prefix) /\ ff_n_mce_alternating_mdr_rank_integer_firstzeroevaluationdeterminanthsf_prefix = (ff_ap_mce_alternating_mdr_rank_integer_firstzeroevaluationdeterminanthsf_prefix) * (ff_bp_mce_alternating_mdr_rank_integer_firstzeroevaluationdeterminanthsf_prefix) + (ff_an_mce_alternating_mdr_rank_integer_firstzeroevaluationdeterminanthsf_prefix) * (ff_bn_mce_alternating_mdr_rank_integer_firstzeroevaluationdeterminanthsf_prefix))))))))))) /\ ((exists ff_u_mce_mdr_rank_integer_firstzeroevaluationdeterminanthsf_positive ff_v_mce_mdr_rank_integer_firstzeroevaluationdeterminanthsf_positive. ((((exists ff_h_mce_mdr_rank_integer_firstzeroevaluationdeterminanthsf_positive_start. ff_h_mce_mdr_rank_integer_firstzeroevaluationdeterminanthsf_positive_start + S (0) = S ((S (0)) * ff_v_mce_mdr_rank_integer_firstzeroevaluationdeterminanthsf_positive)) /\ exists ff_q_mce_mdr_rank_integer_firstzeroevaluationdeterminanthsf_positive_start. ff_u_mce_mdr_rank_integer_firstzeroevaluationdeterminanthsf_positive = ff_q_mce_mdr_rank_integer_firstzeroevaluationdeterminanthsf_positive_start * S ((S (0)) * ff_v_mce_mdr_rank_integer_firstzeroevaluationdeterminanthsf_positive) + (0))) /\ ((((exists ff_h_mce_mdr_rank_integer_firstzeroevaluationdeterminanthsf_positive_terminal. ff_h_mce_mdr_rank_integer_firstzeroevaluationdeterminanthsf_positive_terminal + S (mdr_p_rank_integer_firstzeroevaluationdeterminanth) = S ((S ((S (mdr_q_rank_integer_firstzeroevaluationdeterminanths)))) * ff_v_mce_mdr_rank_integer_firstzeroevaluationdeterminanthsf_positive)) /\ exists ff_q_mce_mdr_rank_integer_firstzeroevaluationdeterminanthsf_positive_terminal. ff_u_mce_mdr_rank_integer_firstzeroevaluationdeterminanthsf_positive = ff_q_mce_mdr_rank_integer_firstzeroevaluationdeterminanthsf_positive_terminal * S ((S ((S (mdr_q_rank_integer_firstzeroevaluationdeterminanths)))) * ff_v_mce_mdr_rank_integer_firstzeroevaluationdeterminanthsf_positive) + (mdr_p_rank_integer_firstzeroevaluationdeterminanth))) /\ forall ff_i_mce_mdr_rank_integer_firstzeroevaluationdeterminanthsf_positive. (exists ff_lt_mce_mdr_rank_integer_firstzeroevaluationdeterminanthsf_positive_bound. ff_lt_mce_mdr_rank_integer_firstzeroevaluationdeterminanthsf_positive_bound + S ff_i_mce_mdr_rank_integer_firstzeroevaluationdeterminanthsf_positive = (S (mdr_q_rank_integer_firstzeroevaluationdeterminanths))) -> exists ff_a_mce_mdr_rank_integer_firstzeroevaluationdeterminanthsf_positive ff_r_mce_mdr_rank_integer_firstzeroevaluationdeterminanthsf_positive ff_s_mce_mdr_rank_integer_firstzeroevaluationdeterminanthsf_positive. ((((exists ff_h_mce_mdr_rank_integer_firstzeroevaluationdeterminanthsf_positive_summand. ff_h_mce_mdr_rank_integer_firstzeroevaluationdeterminanthsf_positive_summand + S (ff_a_mce_mdr_rank_integer_firstzeroevaluationdeterminanthsf_positive) = S ((S (ff_i_mce_mdr_rank_integer_firstzeroevaluationdeterminanthsf_positive)) * ff_uc_mce_fold_mdr_rank_integer_firstzeroevaluationdeterminanthsf)) /\ exists ff_q_mce_mdr_rank_integer_firstzeroevaluationdeterminanthsf_positive_summand. ff_ub_mce_fold_mdr_rank_integer_firstzeroevaluationdeterminanthsf = ff_q_mce_mdr_rank_integer_firstzeroevaluationdeterminanthsf_positive_summand * S ((S (ff_i_mce_mdr_rank_integer_firstzeroevaluationdeterminanthsf_positive)) * ff_uc_mce_fold_mdr_rank_integer_firstzeroevaluationdeterminanthsf) + (ff_a_mce_mdr_rank_integer_firstzeroevaluationdeterminanthsf_positive))) /\ ((((exists ff_h_mce_mdr_rank_integer_firstzeroevaluationdeterminanthsf_positive_partial. ff_h_mce_mdr_rank_integer_firstzeroevaluationdeterminanthsf_positive_partial + S (ff_r_mce_mdr_rank_integer_firstzeroevaluationdeterminanthsf_positive) = S ((S (ff_i_mce_mdr_rank_integer_firstzeroevaluationdeterminanthsf_positive)) * ff_v_mce_mdr_rank_integer_firstzeroevaluationdeterminanthsf_positive)) /\ exists ff_q_mce_mdr_rank_integer_firstzeroevaluationdeterminanthsf_positive_partial. ff_u_mce_mdr_rank_integer_firstzeroevaluationdeterminanthsf_positive = ff_q_mce_mdr_rank_integer_firstzeroevaluationdeterminanthsf_positive_partial * S ((S (ff_i_mce_mdr_rank_integer_firstzeroevaluationdeterminanthsf_positive)) * ff_v_mce_mdr_rank_integer_firstzeroevaluationdeterminanthsf_positive) + (ff_r_mce_mdr_rank_integer_firstzeroevaluationdeterminanthsf_positive))) /\ ((((exists ff_h_mce_mdr_rank_integer_firstzeroevaluationdeterminanthsf_positive_successor. ff_h_mce_mdr_rank_integer_firstzeroevaluationdeterminanthsf_positive_successor + S (ff_s_mce_mdr_rank_integer_firstzeroevaluationdeterminanthsf_positive) = S ((S (S ff_i_mce_mdr_rank_integer_firstzeroevaluationdeterminanthsf_positive)) * ff_v_mce_mdr_rank_integer_firstzeroevaluationdeterminanthsf_positive)) /\ exists ff_q_mce_mdr_rank_integer_firstzeroevaluationdeterminanthsf_positive_successor. ff_u_mce_mdr_rank_integer_firstzeroevaluationdeterminanthsf_positive = ff_q_mce_mdr_rank_integer_firstzeroevaluationdeterminanthsf_positive_successor * S ((S (S ff_i_mce_mdr_rank_integer_firstzeroevaluationdeterminanthsf_positive)) * ff_v_mce_mdr_rank_integer_firstzeroevaluationdeterminanthsf_positive) + (ff_s_mce_mdr_rank_integer_firstzeroevaluationdeterminanthsf_positive))) /\ ff_s_mce_mdr_rank_integer_firstzeroevaluationdeterminanthsf_positive = ff_r_mce_mdr_rank_integer_firstzeroevaluationdeterminanthsf_positive + ff_a_mce_mdr_rank_integer_firstzeroevaluationdeterminanthsf_positive)))))) /\ (exists ff_u_mce_mdr_rank_integer_firstzeroevaluationdeterminanthsf_negative ff_v_mce_mdr_rank_integer_firstzeroevaluationdeterminanthsf_negative. ((((exists ff_h_mce_mdr_rank_integer_firstzeroevaluationdeterminanthsf_negative_start. ff_h_mce_mdr_rank_integer_firstzeroevaluationdeterminanthsf_negative_start + S (0) = S ((S (0)) * ff_v_mce_mdr_rank_integer_firstzeroevaluationdeterminanthsf_negative)) /\ exists ff_q_mce_mdr_rank_integer_firstzeroevaluationdeterminanthsf_negative_start. ff_u_mce_mdr_rank_integer_firstzeroevaluationdeterminanthsf_negative = ff_q_mce_mdr_rank_integer_firstzeroevaluationdeterminanthsf_negative_start * S ((S (0)) * ff_v_mce_mdr_rank_integer_firstzeroevaluationdeterminanthsf_negative) + (0))) /\ ((((exists ff_h_mce_mdr_rank_integer_firstzeroevaluationdeterminanthsf_negative_terminal. ff_h_mce_mdr_rank_integer_firstzeroevaluationdeterminanthsf_negative_terminal + S (mdr_n_rank_integer_firstzeroevaluationdeterminanth) = S ((S ((S (mdr_q_rank_integer_firstzeroevaluationdeterminanths)))) * ff_v_mce_mdr_rank_integer_firstzeroevaluationdeterminanthsf_negative)) /\ exists ff_q_mce_mdr_rank_integer_firstzeroevaluationdeterminanthsf_negative_terminal. ff_u_mce_mdr_rank_integer_firstzeroevaluationdeterminanthsf_negative = ff_q_mce_mdr_rank_integer_firstzeroevaluationdeterminanthsf_negative_terminal * S ((S ((S (mdr_q_rank_integer_firstzeroevaluationdeterminanths)))) * ff_v_mce_mdr_rank_integer_firstzeroevaluationdeterminanthsf_negative) + (mdr_n_rank_integer_firstzeroevaluationdeterminanth))) /\ forall ff_i_mce_mdr_rank_integer_firstzeroevaluationdeterminanthsf_negative. (exists ff_lt_mce_mdr_rank_integer_firstzeroevaluationdeterminanthsf_negative_bound. ff_lt_mce_mdr_rank_integer_firstzeroevaluationdeterminanthsf_negative_bound + S ff_i_mce_mdr_rank_integer_firstzeroevaluationdeterminanthsf_negative = (S (mdr_q_rank_integer_firstzeroevaluationdeterminanths))) -> exists ff_a_mce_mdr_rank_integer_firstzeroevaluationdeterminanthsf_negative ff_r_mce_mdr_rank_integer_firstzeroevaluationdeterminanthsf_negative ff_s_mce_mdr_rank_integer_firstzeroevaluationdeterminanthsf_negative. ((((exists ff_h_mce_mdr_rank_integer_firstzeroevaluationdeterminanthsf_negative_summand. ff_h_mce_mdr_rank_integer_firstzeroevaluationdeterminanthsf_negative_summand + S (ff_a_mce_mdr_rank_integer_firstzeroevaluationdeterminanthsf_negative) = S ((S (ff_i_mce_mdr_rank_integer_firstzeroevaluationdeterminanthsf_negative)) * ff_vc_mce_fold_mdr_rank_integer_firstzeroevaluationdeterminanthsf)) /\ exists ff_q_mce_mdr_rank_integer_firstzeroevaluationdeterminanthsf_negative_summand. ff_vb_mce_fold_mdr_rank_integer_firstzeroevaluationdeterminanthsf = ff_q_mce_mdr_rank_integer_firstzeroevaluationdeterminanthsf_negative_summand * S ((S (ff_i_mce_mdr_rank_integer_firstzeroevaluationdeterminanthsf_negative)) * ff_vc_mce_fold_mdr_rank_integer_firstzeroevaluationdeterminanthsf) + (ff_a_mce_mdr_rank_integer_firstzeroevaluationdeterminanthsf_negative))) /\ ((((exists ff_h_mce_mdr_rank_integer_firstzeroevaluationdeterminanthsf_negative_partial. ff_h_mce_mdr_rank_integer_firstzeroevaluationdeterminanthsf_negative_partial + S (ff_r_mce_mdr_rank_integer_firstzeroevaluationdeterminanthsf_negative) = S ((S (ff_i_mce_mdr_rank_integer_firstzeroevaluationdeterminanthsf_negative)) * ff_v_mce_mdr_rank_integer_firstzeroevaluationdeterminanthsf_negative)) /\ exists ff_q_mce_mdr_rank_integer_firstzeroevaluationdeterminanthsf_negative_partial. ff_u_mce_mdr_rank_integer_firstzeroevaluationdeterminanthsf_negative = ff_q_mce_mdr_rank_integer_firstzeroevaluationdeterminanthsf_negative_partial * S ((S (ff_i_mce_mdr_rank_integer_firstzeroevaluationdeterminanthsf_negative)) * ff_v_mce_mdr_rank_integer_firstzeroevaluationdeterminanthsf_negative) + (ff_r_mce_mdr_rank_integer_firstzeroevaluationdeterminanthsf_negative))) /\ ((((exists ff_h_mce_mdr_rank_integer_firstzeroevaluationdeterminanthsf_negative_successor. ff_h_mce_mdr_rank_integer_firstzeroevaluationdeterminanthsf_negative_successor + S (ff_s_mce_mdr_rank_integer_firstzeroevaluationdeterminanthsf_negative) = S ((S (S ff_i_mce_mdr_rank_integer_firstzeroevaluationdeterminanthsf_negative)) * ff_v_mce_mdr_rank_integer_firstzeroevaluationdeterminanthsf_negative)) /\ exists ff_q_mce_mdr_rank_integer_firstzeroevaluationdeterminanthsf_negative_successor. ff_u_mce_mdr_rank_integer_firstzeroevaluationdeterminanthsf_negative = ff_q_mce_mdr_rank_integer_firstzeroevaluationdeterminanthsf_negative_successor * S ((S (S ff_i_mce_mdr_rank_integer_firstzeroevaluationdeterminanthsf_negative)) * ff_v_mce_mdr_rank_integer_firstzeroevaluationdeterminanthsf_negative) + (ff_s_mce_mdr_rank_integer_firstzeroevaluationdeterminanthsf_negative))) /\ ff_s_mce_mdr_rank_integer_firstzeroevaluationdeterminanthsf_negative = ff_r_mce_mdr_rank_integer_firstzeroevaluationdeterminanthsf_negative + ff_a_mce_mdr_rank_integer_firstzeroevaluationdeterminanthsf_negative))))))))))))))) /\ ((exists mdr_gap_rank_integer_firstzeroevaluationdeterminanti. mdr_gap_rank_integer_firstzeroevaluationdeterminanti + S (mdr_i_rank_integer_firstzeroevaluationdeterminant) = (mdr_l_rank_integer_firstzeroevaluationdeterminant)) /\ (exists mdr_z_rank_integer_firstzeroevaluationdeterminantr. ((exists mdr_a_rank_integer_firstzeroevaluationdeterminantrc mdr_b_rank_integer_firstzeroevaluationdeterminantrc mdr_c_rank_integer_firstzeroevaluationdeterminantrc mdr_e_rank_integer_firstzeroevaluationdeterminantrc mdr_f_rank_integer_firstzeroevaluationdeterminantrc. ((mdr_a_rank_integer_firstzeroevaluationdeterminantrc = ((mdr_q_rank_integer_first) + (mdr_ub_rank_integer_firstzeroevaluation)) * S ((mdr_q_rank_integer_first) + (mdr_ub_rank_integer_firstzeroevaluation)) + ((mdr_ub_rank_integer_firstzeroevaluation) + (mdr_ub_rank_integer_firstzeroevaluation))) /\ ((mdr_b_rank_integer_firstzeroevaluationdeterminantrc = ((mdr_uc_rank_integer_firstzeroevaluation) + (mdr_vb_rank_integer_firstzeroevaluation)) * S ((mdr_uc_rank_integer_firstzeroevaluation) + (mdr_vb_rank_integer_firstzeroevaluation)) + ((mdr_vb_rank_integer_firstzeroevaluation) + (mdr_vb_rank_integer_firstzeroevaluation))) /\ ((mdr_c_rank_integer_firstzeroevaluationdeterminantrc = ((mdr_a_rank_integer_firstzeroevaluationdeterminantrc) + (mdr_b_rank_integer_firstzeroevaluationdeterminantrc)) * S ((mdr_a_rank_integer_firstzeroevaluationdeterminantrc) + (mdr_b_rank_integer_firstzeroevaluationdeterminantrc)) + ((mdr_b_rank_integer_firstzeroevaluationdeterminantrc) + (mdr_b_rank_integer_firstzeroevaluationdeterminantrc))) /\ ((mdr_e_rank_integer_firstzeroevaluationdeterminantrc = ((mdr_p_rank_integer_firstzero) + (mdr_n_rank_integer_firstzero)) * S ((mdr_p_rank_integer_firstzero) + (mdr_n_rank_integer_firstzero)) + ((mdr_n_rank_integer_firstzero) + (mdr_n_rank_integer_firstzero))) /\ ((mdr_f_rank_integer_firstzeroevaluationdeterminantrc = ((mdr_vc_rank_integer_firstzeroevaluation) + (mdr_e_rank_integer_firstzeroevaluationdeterminantrc)) * S ((mdr_vc_rank_integer_firstzeroevaluation) + (mdr_e_rank_integer_firstzeroevaluationdeterminantrc)) + ((mdr_e_rank_integer_firstzeroevaluationdeterminantrc) + (mdr_e_rank_integer_firstzeroevaluationdeterminantrc))) /\ ((mdr_z_rank_integer_firstzeroevaluationdeterminantr) = ((mdr_c_rank_integer_firstzeroevaluationdeterminantrc) + (mdr_f_rank_integer_firstzeroevaluationdeterminantrc)) * S ((mdr_c_rank_integer_firstzeroevaluationdeterminantrc) + (mdr_f_rank_integer_firstzeroevaluationdeterminantrc)) + ((mdr_f_rank_integer_firstzeroevaluationdeterminantrc) + (mdr_f_rank_integer_firstzeroevaluationdeterminantrc))))))))) /\ (((exists ff_h_mdr_rank_integer_firstzeroevaluationdeterminantrb. ff_h_mdr_rank_integer_firstzeroevaluationdeterminantrb + S (mdr_z_rank_integer_firstzeroevaluationdeterminantr) = S ((S (mdr_i_rank_integer_firstzeroevaluationdeterminant)) * mdr_c_rank_integer_firstzeroevaluationdeterminant)) /\ exists ff_q_mdr_rank_integer_firstzeroevaluationdeterminantrb. mdr_b_rank_integer_firstzeroevaluationdeterminant = ff_q_mdr_rank_integer_firstzeroevaluationdeterminantrb * S ((S (mdr_i_rank_integer_firstzeroevaluationdeterminant)) * mdr_c_rank_integer_firstzeroevaluationdeterminant) + (mdr_z_rank_integer_firstzeroevaluationdeterminantr)))))))))) -> mdr_p_rank_integer_firstzero = mdr_n_rank_integer_firstzero)))))) -> (((exists mdr_gap_rank_integer_secondrows_bound. mdr_gap_rank_integer_secondrows_bound + (other) = (r)) /\ ((exists mdr_gap_rank_integer_secondcolumns_bound. mdr_gap_rank_integer_secondcolumns_bound + (other) = (w)) /\ ((exists mdr_rb_rank_integer_secondwitness mdr_rc_rank_integer_secondwitness mdr_cb_rank_integer_secondwitness mdr_cc_rank_integer_secondwitness. (((((forall fom_index_mrf_rank_integer_secondwitnessminorrowsbound. (exists fom_gap_mrf_rank_integer_secondwitnessminorrowsbound_index_bound. fom_gap_mrf_rank_integer_secondwitnessminorrowsbound_index_bound + S (fom_index_mrf_rank_integer_secondwitnessminorrowsbound) = other) -> exists fom_value_mrf_rank_integer_secondwitnessminorrowsbound. ((((exists fom_beta_height_mrf_rank_integer_secondwitnessminorrowsbound_entry. fom_beta_height_mrf_rank_integer_secondwitnessminorrowsbound_entry + S (fom_value_mrf_rank_integer_secondwitnessminorrowsbound) = S ((S (fom_index_mrf_rank_integer_secondwitnessminorrowsbound)) * mdr_rc_rank_integer_secondwitness)) /\ exists fom_beta_quotient_mrf_rank_integer_secondwitnessminorrowsbound_entry. mdr_rb_rank_integer_secondwitness = fom_beta_quotient_mrf_rank_integer_secondwitnessminorrowsbound_entry * S ((S (fom_index_mrf_rank_integer_secondwitnessminorrowsbound)) * mdr_rc_rank_integer_secondwitness) + (fom_value_mrf_rank_integer_secondwitnessminorrowsbound))) /\ (exists fom_gap_mrf_rank_integer_secondwitnessminorrowsbound_value_bound. fom_gap_mrf_rank_integer_secondwitnessminorrowsbound_value_bound + S (fom_value_mrf_rank_integer_secondwitnessminorrowsbound) = r))) /\ (forall mdr_i_rank_integer_secondwitnessminorrowsdistinct mdr_j_rank_integer_secondwitnessminorrowsdistinct mdr_a_rank_integer_secondwitnessminorrowsdistinct. (exists mdr_gap_rank_integer_secondwitnessminorrowsdistincti. mdr_gap_rank_integer_secondwitnessminorrowsdistincti + S (mdr_i_rank_integer_secondwitnessminorrowsdistinct) = (other)) -> (exists mdr_gap_rank_integer_secondwitnessminorrowsdistinctj. mdr_gap_rank_integer_secondwitnessminorrowsdistinctj + S (mdr_j_rank_integer_secondwitnessminorrowsdistinct) = (other)) -> (((exists ff_h_mdr_rank_integer_secondwitnessminorrowsdistinctfirst. ff_h_mdr_rank_integer_secondwitnessminorrowsdistinctfirst + S (mdr_a_rank_integer_secondwitnessminorrowsdistinct) = S ((S (mdr_i_rank_integer_secondwitnessminorrowsdistinct)) * mdr_rc_rank_integer_secondwitness)) /\ exists ff_q_mdr_rank_integer_secondwitnessminorrowsdistinctfirst. mdr_rb_rank_integer_secondwitness = ff_q_mdr_rank_integer_secondwitnessminorrowsdistinctfirst * S ((S (mdr_i_rank_integer_secondwitnessminorrowsdistinct)) * mdr_rc_rank_integer_secondwitness) + (mdr_a_rank_integer_secondwitnessminorrowsdistinct))) -> (((exists ff_h_mdr_rank_integer_secondwitnessminorrowsdistinctsecond. ff_h_mdr_rank_integer_secondwitnessminorrowsdistinctsecond + S (mdr_a_rank_integer_secondwitnessminorrowsdistinct) = S ((S (mdr_j_rank_integer_secondwitnessminorrowsdistinct)) * mdr_rc_rank_integer_secondwitness)) /\ exists ff_q_mdr_rank_integer_secondwitnessminorrowsdistinctsecond. mdr_rb_rank_integer_secondwitness = ff_q_mdr_rank_integer_secondwitnessminorrowsdistinctsecond * S ((S (mdr_j_rank_integer_secondwitnessminorrowsdistinct)) * mdr_rc_rank_integer_secondwitness) + (mdr_a_rank_integer_secondwitnessminorrowsdistinct))) -> mdr_i_rank_integer_secondwitnessminorrowsdistinct = mdr_j_rank_integer_secondwitnessminorrowsdistinct))) /\ ((((forall fom_index_mrf_rank_integer_secondwitnessminorcolumnsbound. (exists fom_gap_mrf_rank_integer_secondwitnessminorcolumnsbound_index_bound. fom_gap_mrf_rank_integer_secondwitnessminorcolumnsbound_index_bound + S (fom_index_mrf_rank_integer_secondwitnessminorcolumnsbound) = other) -> exists fom_value_mrf_rank_integer_secondwitnessminorcolumnsbound. ((((exists fom_beta_height_mrf_rank_integer_secondwitnessminorcolumnsbound_entry. fom_beta_height_mrf_rank_integer_secondwitnessminorcolumnsbound_entry + S (fom_value_mrf_rank_integer_secondwitnessminorcolumnsbound) = S ((S (fom_index_mrf_rank_integer_secondwitnessminorcolumnsbound)) * mdr_cc_rank_integer_secondwitness)) /\ exists fom_beta_quotient_mrf_rank_integer_secondwitnessminorcolumnsbound_entry. mdr_cb_rank_integer_secondwitness = fom_beta_quotient_mrf_rank_integer_secondwitnessminorcolumnsbound_entry * S ((S (fom_index_mrf_rank_integer_secondwitnessminorcolumnsbound)) * mdr_cc_rank_integer_secondwitness) + (fom_value_mrf_rank_integer_secondwitnessminorcolumnsbound))) /\ (exists fom_gap_mrf_rank_integer_secondwitnessminorcolumnsbound_value_bound. fom_gap_mrf_rank_integer_secondwitnessminorcolumnsbound_value_bound + S (fom_value_mrf_rank_integer_secondwitnessminorcolumnsbound) = w))) /\ (forall mdr_i_rank_integer_secondwitnessminorcolumnsdistinct mdr_j_rank_integer_secondwitnessminorcolumnsdistinct mdr_a_rank_integer_secondwitnessminorcolumnsdistinct. (exists mdr_gap_rank_integer_secondwitnessminorcolumnsdistincti. mdr_gap_rank_integer_secondwitnessminorcolumnsdistincti + S (mdr_i_rank_integer_secondwitnessminorcolumnsdistinct) = (other)) -> (exists mdr_gap_rank_integer_secondwitnessminorcolumnsdistinctj. mdr_gap_rank_integer_secondwitnessminorcolumnsdistinctj + S (mdr_j_rank_integer_secondwitnessminorcolumnsdistinct) = (other)) -> (((exists ff_h_mdr_rank_integer_secondwitnessminorcolumnsdistinctfirst. ff_h_mdr_rank_integer_secondwitnessminorcolumnsdistinctfirst + S (mdr_a_rank_integer_secondwitnessminorcolumnsdistinct) = S ((S (mdr_i_rank_integer_secondwitnessminorcolumnsdistinct)) * mdr_cc_rank_integer_secondwitness)) /\ exists ff_q_mdr_rank_integer_secondwitnessminorcolumnsdistinctfirst. mdr_cb_rank_integer_secondwitness = ff_q_mdr_rank_integer_secondwitnessminorcolumnsdistinctfirst * S ((S (mdr_i_rank_integer_secondwitnessminorcolumnsdistinct)) * mdr_cc_rank_integer_secondwitness) + (mdr_a_rank_integer_secondwitnessminorcolumnsdistinct))) -> (((exists ff_h_mdr_rank_integer_secondwitnessminorcolumnsdistinctsecond. ff_h_mdr_rank_integer_secondwitnessminorcolumnsdistinctsecond + S (mdr_a_rank_integer_secondwitnessminorcolumnsdistinct) = S ((S (mdr_j_rank_integer_secondwitnessminorcolumnsdistinct)) * mdr_cc_rank_integer_secondwitness)) /\ exists ff_q_mdr_rank_integer_secondwitnessminorcolumnsdistinctsecond. mdr_cb_rank_integer_secondwitness = ff_q_mdr_rank_integer_secondwitnessminorcolumnsdistinctsecond * S ((S (mdr_j_rank_integer_secondwitnessminorcolumnsdistinct)) * mdr_cc_rank_integer_secondwitness) + (mdr_a_rank_integer_secondwitnessminorcolumnsdistinct))) -> mdr_i_rank_integer_secondwitnessminorcolumnsdistinct = mdr_j_rank_integer_secondwitnessminorcolumnsdistinct))) /\ (exists mdr_p_rank_integer_secondwitnessminornonzero mdr_n_rank_integer_secondwitnessminornonzero. ((exists mdr_ub_rank_integer_secondwitnessminornonzeroevaluation mdr_uc_rank_integer_secondwitnessminornonzeroevaluation mdr_vb_rank_integer_secondwitnessminornonzeroevaluation mdr_vc_rank_integer_secondwitnessminornonzeroevaluation. ((((forall mdr_i_rank_integer_secondwitnessminornonzeroevaluationmatrixpositive. (exists mdr_gap_rank_integer_secondwitnessminornonzeroevaluationmatrixpositivebound. mdr_gap_rank_integer_secondwitnessminornonzeroevaluationmatrixpositivebound + S (mdr_i_rank_integer_secondwitnessminornonzeroevaluationmatrixpositive) = ((other) * (other))) -> exists mdr_a_rank_integer_secondwitnessminornonzeroevaluationmatrixpositive. (((exists mdr_r_rank_integer_secondwitnessminornonzeroevaluationmatrixpositivepoint mdr_s_rank_integer_secondwitnessminornonzeroevaluationmatrixpositivepoint mdr_u_rank_integer_secondwitnessminornonzeroevaluationmatrixpositivepoint mdr_v_rank_integer_secondwitnessminornonzeroevaluationmatrixpositivepoint. ((mdr_i_rank_integer_secondwitnessminornonzeroevaluationmatrixpositive = (other) * mdr_r_rank_integer_secondwitnessminornonzeroevaluationmatrixpositivepoint + mdr_s_rank_integer_secondwitnessminornonzeroevaluationmatrixpositivepoint) /\ ((exists mdr_gap_rank_integer_secondwitnessminornonzeroevaluationmatrixpositivepointcolumn. mdr_gap_rank_integer_secondwitnessminornonzeroevaluationmatrixpositivepointcolumn + S (mdr_s_rank_integer_secondwitnessminornonzeroevaluationmatrixpositivepoint) = (other)) /\ ((((exists ff_h_mdr_rank_integer_secondwitnessminornonzeroevaluationmatrixpositivepointrow_index. ff_h_mdr_rank_integer_secondwitnessminornonzeroevaluationmatrixpositivepointrow_index + S (mdr_u_rank_integer_secondwitnessminornonzeroevaluationmatrixpositivepoint) = S ((S (mdr_r_rank_integer_secondwitnessminornonzeroevaluationmatrixpositivepoint)) * mdr_rc_rank_integer_secondwitness)) /\ exists ff_q_mdr_rank_integer_secondwitnessminornonzeroevaluationmatrixpositivepointrow_index. mdr_rb_rank_integer_secondwitness = ff_q_mdr_rank_integer_secondwitnessminornonzeroevaluationmatrixpositivepointrow_index * S ((S (mdr_r_rank_integer_secondwitnessminornonzeroevaluationmatrixpositivepoint)) * mdr_rc_rank_integer_secondwitness) + (mdr_u_rank_integer_secondwitnessminornonzeroevaluationmatrixpositivepoint))) /\ ((((exists ff_h_mdr_rank_integer_secondwitnessminornonzeroevaluationmatrixpositivepointcolumn_index. ff_h_mdr_rank_integer_secondwitnessminornonzeroevaluationmatrixpositivepointcolumn_index + S (mdr_v_rank_integer_secondwitnessminornonzeroevaluationmatrixpositivepoint) = S ((S (mdr_s_rank_integer_secondwitnessminornonzeroevaluationmatrixpositivepoint)) * mdr_cc_rank_integer_secondwitness)) /\ exists ff_q_mdr_rank_integer_secondwitnessminornonzeroevaluationmatrixpositivepointcolumn_index. mdr_cb_rank_integer_secondwitness = ff_q_mdr_rank_integer_secondwitnessminornonzeroevaluationmatrixpositivepointcolumn_index * S ((S (mdr_s_rank_integer_secondwitnessminornonzeroevaluationmatrixpositivepoint)) * mdr_cc_rank_integer_secondwitness) + (mdr_v_rank_integer_secondwitnessminornonzeroevaluationmatrixpositivepoint))) /\ (((exists ff_h_mdr_rank_integer_secondwitnessminornonzeroevaluationmatrixpositivepointsource. ff_h_mdr_rank_integer_secondwitnessminornonzeroevaluationmatrixpositivepointsource + S (mdr_a_rank_integer_secondwitnessminornonzeroevaluationmatrixpositive) = S ((S ((mdr_u_rank_integer_secondwitnessminornonzeroevaluationmatrixpositivepoint) * (w) + (mdr_v_rank_integer_secondwitnessminornonzeroevaluationmatrixpositivepoint))) * ec)) /\ exists ff_q_mdr_rank_integer_secondwitnessminornonzeroevaluationmatrixpositivepointsource. eb = ff_q_mdr_rank_integer_secondwitnessminornonzeroevaluationmatrixpositivepointsource * S ((S ((mdr_u_rank_integer_secondwitnessminornonzeroevaluationmatrixpositivepoint) * (w) + (mdr_v_rank_integer_secondwitnessminornonzeroevaluationmatrixpositivepoint))) * ec) + (mdr_a_rank_integer_secondwitnessminornonzeroevaluationmatrixpositive)))))))) /\ (((exists ff_h_mdr_rank_integer_secondwitnessminornonzeroevaluationmatrixpositiveoutput. ff_h_mdr_rank_integer_secondwitnessminornonzeroevaluationmatrixpositiveoutput + S (mdr_a_rank_integer_secondwitnessminornonzeroevaluationmatrixpositive) = S ((S (mdr_i_rank_integer_secondwitnessminornonzeroevaluationmatrixpositive)) * mdr_uc_rank_integer_secondwitnessminornonzeroevaluation)) /\ exists ff_q_mdr_rank_integer_secondwitnessminornonzeroevaluationmatrixpositiveoutput. mdr_ub_rank_integer_secondwitnessminornonzeroevaluation = ff_q_mdr_rank_integer_secondwitnessminornonzeroevaluationmatrixpositiveoutput * S ((S (mdr_i_rank_integer_secondwitnessminornonzeroevaluationmatrixpositive)) * mdr_uc_rank_integer_secondwitnessminornonzeroevaluation) + (mdr_a_rank_integer_secondwitnessminornonzeroevaluationmatrixpositive)))))) /\ (forall mdr_i_rank_integer_secondwitnessminornonzeroevaluationmatrixnegative. (exists mdr_gap_rank_integer_secondwitnessminornonzeroevaluationmatrixnegativebound. mdr_gap_rank_integer_secondwitnessminornonzeroevaluationmatrixnegativebound + S (mdr_i_rank_integer_secondwitnessminornonzeroevaluationmatrixnegative) = ((other) * (other))) -> exists mdr_a_rank_integer_secondwitnessminornonzeroevaluationmatrixnegative. (((exists mdr_r_rank_integer_secondwitnessminornonzeroevaluationmatrixnegativepoint mdr_s_rank_integer_secondwitnessminornonzeroevaluationmatrixnegativepoint mdr_u_rank_integer_secondwitnessminornonzeroevaluationmatrixnegativepoint mdr_v_rank_integer_secondwitnessminornonzeroevaluationmatrixnegativepoint. ((mdr_i_rank_integer_secondwitnessminornonzeroevaluationmatrixnegative = (other) * mdr_r_rank_integer_secondwitnessminornonzeroevaluationmatrixnegativepoint + mdr_s_rank_integer_secondwitnessminornonzeroevaluationmatrixnegativepoint) /\ ((exists mdr_gap_rank_integer_secondwitnessminornonzeroevaluationmatrixnegativepointcolumn. mdr_gap_rank_integer_secondwitnessminornonzeroevaluationmatrixnegativepointcolumn + S (mdr_s_rank_integer_secondwitnessminornonzeroevaluationmatrixnegativepoint) = (other)) /\ ((((exists ff_h_mdr_rank_integer_secondwitnessminornonzeroevaluationmatrixnegativepointrow_index. ff_h_mdr_rank_integer_secondwitnessminornonzeroevaluationmatrixnegativepointrow_index + S (mdr_u_rank_integer_secondwitnessminornonzeroevaluationmatrixnegativepoint) = S ((S (mdr_r_rank_integer_secondwitnessminornonzeroevaluationmatrixnegativepoint)) * mdr_rc_rank_integer_secondwitness)) /\ exists ff_q_mdr_rank_integer_secondwitnessminornonzeroevaluationmatrixnegativepointrow_index. mdr_rb_rank_integer_secondwitness = ff_q_mdr_rank_integer_secondwitnessminornonzeroevaluationmatrixnegativepointrow_index * S ((S (mdr_r_rank_integer_secondwitnessminornonzeroevaluationmatrixnegativepoint)) * mdr_rc_rank_integer_secondwitness) + (mdr_u_rank_integer_secondwitnessminornonzeroevaluationmatrixnegativepoint))) /\ ((((exists ff_h_mdr_rank_integer_secondwitnessminornonzeroevaluationmatrixnegativepointcolumn_index. ff_h_mdr_rank_integer_secondwitnessminornonzeroevaluationmatrixnegativepointcolumn_index + S (mdr_v_rank_integer_secondwitnessminornonzeroevaluationmatrixnegativepoint) = S ((S (mdr_s_rank_integer_secondwitnessminornonzeroevaluationmatrixnegativepoint)) * mdr_cc_rank_integer_secondwitness)) /\ exists ff_q_mdr_rank_integer_secondwitnessminornonzeroevaluationmatrixnegativepointcolumn_index. mdr_cb_rank_integer_secondwitness = ff_q_mdr_rank_integer_secondwitnessminornonzeroevaluationmatrixnegativepointcolumn_index * S ((S (mdr_s_rank_integer_secondwitnessminornonzeroevaluationmatrixnegativepoint)) * mdr_cc_rank_integer_secondwitness) + (mdr_v_rank_integer_secondwitnessminornonzeroevaluationmatrixnegativepoint))) /\ (((exists ff_h_mdr_rank_integer_secondwitnessminornonzeroevaluationmatrixnegativepointsource. ff_h_mdr_rank_integer_secondwitnessminornonzeroevaluationmatrixnegativepointsource + S (mdr_a_rank_integer_secondwitnessminornonzeroevaluationmatrixnegative) = S ((S ((mdr_u_rank_integer_secondwitnessminornonzeroevaluationmatrixnegativepoint) * (w) + (mdr_v_rank_integer_secondwitnessminornonzeroevaluationmatrixnegativepoint))) * fc)) /\ exists ff_q_mdr_rank_integer_secondwitnessminornonzeroevaluationmatrixnegativepointsource. fb = ff_q_mdr_rank_integer_secondwitnessminornonzeroevaluationmatrixnegativepointsource * S ((S ((mdr_u_rank_integer_secondwitnessminornonzeroevaluationmatrixnegativepoint) * (w) + (mdr_v_rank_integer_secondwitnessminornonzeroevaluationmatrixnegativepoint))) * fc) + (mdr_a_rank_integer_secondwitnessminornonzeroevaluationmatrixnegative)))))))) /\ (((exists ff_h_mdr_rank_integer_secondwitnessminornonzeroevaluationmatrixnegativeoutput. ff_h_mdr_rank_integer_secondwitnessminornonzeroevaluationmatrixnegativeoutput + S (mdr_a_rank_integer_secondwitnessminornonzeroevaluationmatrixnegative) = S ((S (mdr_i_rank_integer_secondwitnessminornonzeroevaluationmatrixnegative)) * mdr_vc_rank_integer_secondwitnessminornonzeroevaluation)) /\ exists ff_q_mdr_rank_integer_secondwitnessminornonzeroevaluationmatrixnegativeoutput. mdr_vb_rank_integer_secondwitnessminornonzeroevaluation = ff_q_mdr_rank_integer_secondwitnessminornonzeroevaluationmatrixnegativeoutput * S ((S (mdr_i_rank_integer_secondwitnessminornonzeroevaluationmatrixnegative)) * mdr_vc_rank_integer_secondwitnessminornonzeroevaluation) + (mdr_a_rank_integer_secondwitnessminornonzeroevaluationmatrixnegative)))))))) /\ (exists mdr_b_rank_integer_secondwitnessminornonzeroevaluationdeterminant mdr_c_rank_integer_secondwitnessminornonzeroevaluationdeterminant mdr_l_rank_integer_secondwitnessminornonzeroevaluationdeterminant mdr_i_rank_integer_secondwitnessminornonzeroevaluationdeterminant. ((forall mdr_i_rank_integer_secondwitnessminornonzeroevaluationdeterminanth. (exists mdr_gap_rank_integer_secondwitnessminornonzeroevaluationdeterminanthi. mdr_gap_rank_integer_secondwitnessminornonzeroevaluationdeterminanthi + S (mdr_i_rank_integer_secondwitnessminornonzeroevaluationdeterminanth) = (mdr_l_rank_integer_secondwitnessminornonzeroevaluationdeterminant)) -> exists mdr_d_rank_integer_secondwitnessminornonzeroevaluationdeterminanth mdr_pb_rank_integer_secondwitnessminornonzeroevaluationdeterminanth mdr_pc_rank_integer_secondwitnessminornonzeroevaluationdeterminanth mdr_nb_rank_integer_secondwitnessminornonzeroevaluationdeterminanth mdr_nc_rank_integer_secondwitnessminornonzeroevaluationdeterminanth mdr_p_rank_integer_secondwitnessminornonzeroevaluationdeterminanth mdr_n_rank_integer_secondwitnessminornonzeroevaluationdeterminanth. ((exists mdr_z_rank_integer_secondwitnessminornonzeroevaluationdeterminanthr. ((exists mdr_a_rank_integer_secondwitnessminornonzeroevaluationdeterminanthrc mdr_b_rank_integer_secondwitnessminornonzeroevaluationdeterminanthrc mdr_c_rank_integer_secondwitnessminornonzeroevaluationdeterminanthrc mdr_e_rank_integer_secondwitnessminornonzeroevaluationdeterminanthrc mdr_f_rank_integer_secondwitnessminornonzeroevaluationdeterminanthrc. ((mdr_a_rank_integer_secondwitnessminornonzeroevaluationdeterminanthrc = ((mdr_d_rank_integer_secondwitnessminornonzeroevaluationdeterminanth) + (mdr_pb_rank_integer_secondwitnessminornonzeroevaluationdeterminanth)) * S ((mdr_d_rank_integer_secondwitnessminornonzeroevaluationdeterminanth) + (mdr_pb_rank_integer_secondwitnessminornonzeroevaluationdeterminanth)) + ((mdr_pb_rank_integer_secondwitnessminornonzeroevaluationdeterminanth) + (mdr_pb_rank_integer_secondwitnessminornonzeroevaluationdeterminanth))) /\ ((mdr_b_rank_integer_secondwitnessminornonzeroevaluationdeterminanthrc = ((mdr_pc_rank_integer_secondwitnessminornonzeroevaluationdeterminanth) + (mdr_nb_rank_integer_secondwitnessminornonzeroevaluationdeterminanth)) * S ((mdr_pc_rank_integer_secondwitnessminornonzeroevaluationdeterminanth) + (mdr_nb_rank_integer_secondwitnessminornonzeroevaluationdeterminanth)) + ((mdr_nb_rank_integer_secondwitnessminornonzeroevaluationdeterminanth) + (mdr_nb_rank_integer_secondwitnessminornonzeroevaluationdeterminanth))) /\ ((mdr_c_rank_integer_secondwitnessminornonzeroevaluationdeterminanthrc = ((mdr_a_rank_integer_secondwitnessminornonzeroevaluationdeterminanthrc) + (mdr_b_rank_integer_secondwitnessminornonzeroevaluationdeterminanthrc)) * S ((mdr_a_rank_integer_secondwitnessminornonzeroevaluationdeterminanthrc) + (mdr_b_rank_integer_secondwitnessminornonzeroevaluationdeterminanthrc)) + ((mdr_b_rank_integer_secondwitnessminornonzeroevaluationdeterminanthrc) + (mdr_b_rank_integer_secondwitnessminornonzeroevaluationdeterminanthrc))) /\ ((mdr_e_rank_integer_secondwitnessminornonzeroevaluationdeterminanthrc = ((mdr_p_rank_integer_secondwitnessminornonzeroevaluationdeterminanth) + (mdr_n_rank_integer_secondwitnessminornonzeroevaluationdeterminanth)) * S ((mdr_p_rank_integer_secondwitnessminornonzeroevaluationdeterminanth) + (mdr_n_rank_integer_secondwitnessminornonzeroevaluationdeterminanth)) + ((mdr_n_rank_integer_secondwitnessminornonzeroevaluationdeterminanth) + (mdr_n_rank_integer_secondwitnessminornonzeroevaluationdeterminanth))) /\ ((mdr_f_rank_integer_secondwitnessminornonzeroevaluationdeterminanthrc = ((mdr_nc_rank_integer_secondwitnessminornonzeroevaluationdeterminanth) + (mdr_e_rank_integer_secondwitnessminornonzeroevaluationdeterminanthrc)) * S ((mdr_nc_rank_integer_secondwitnessminornonzeroevaluationdeterminanth) + (mdr_e_rank_integer_secondwitnessminornonzeroevaluationdeterminanthrc)) + ((mdr_e_rank_integer_secondwitnessminornonzeroevaluationdeterminanthrc) + (mdr_e_rank_integer_secondwitnessminornonzeroevaluationdeterminanthrc))) /\ ((mdr_z_rank_integer_secondwitnessminornonzeroevaluationdeterminanthr) = ((mdr_c_rank_integer_secondwitnessminornonzeroevaluationdeterminanthrc) + (mdr_f_rank_integer_secondwitnessminornonzeroevaluationdeterminanthrc)) * S ((mdr_c_rank_integer_secondwitnessminornonzeroevaluationdeterminanthrc) + (mdr_f_rank_integer_secondwitnessminornonzeroevaluationdeterminanthrc)) + ((mdr_f_rank_integer_secondwitnessminornonzeroevaluationdeterminanthrc) + (mdr_f_rank_integer_secondwitnessminornonzeroevaluationdeterminanthrc))))))))) /\ (((exists ff_h_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthrb. ff_h_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthrb + S (mdr_z_rank_integer_secondwitnessminornonzeroevaluationdeterminanthr) = S ((S (mdr_i_rank_integer_secondwitnessminornonzeroevaluationdeterminanth)) * mdr_c_rank_integer_secondwitnessminornonzeroevaluationdeterminant)) /\ exists ff_q_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthrb. mdr_b_rank_integer_secondwitnessminornonzeroevaluationdeterminant = ff_q_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthrb * S ((S (mdr_i_rank_integer_secondwitnessminornonzeroevaluationdeterminanth)) * mdr_c_rank_integer_secondwitnessminornonzeroevaluationdeterminant) + (mdr_z_rank_integer_secondwitnessminornonzeroevaluationdeterminanthr))))) /\ (((((mdr_d_rank_integer_secondwitnessminornonzeroevaluationdeterminanth) = 0) /\ (((mdr_p_rank_integer_secondwitnessminornonzeroevaluationdeterminanth) = 1) /\ ((mdr_n_rank_integer_secondwitnessminornonzeroevaluationdeterminanth) = 0))) \/ exists mdr_q_rank_integer_secondwitnessminornonzeroevaluationdeterminanths mdr_eb_rank_integer_secondwitnessminornonzeroevaluationdeterminanths mdr_ec_rank_integer_secondwitnessminornonzeroevaluationdeterminanths mdr_fb_rank_integer_secondwitnessminornonzeroevaluationdeterminanths mdr_fc_rank_integer_secondwitnessminornonzeroevaluationdeterminanths. (((mdr_d_rank_integer_secondwitnessminornonzeroevaluationdeterminanth) = S (mdr_q_rank_integer_secondwitnessminornonzeroevaluationdeterminanths)) /\ ((forall mdr_j_rank_integer_secondwitnessminornonzeroevaluationdeterminanthsc. (exists mdr_gap_rank_integer_secondwitnessminornonzeroevaluationdeterminanthscj. mdr_gap_rank_integer_secondwitnessminornonzeroevaluationdeterminanthscj + S (mdr_j_rank_integer_secondwitnessminornonzeroevaluationdeterminanthsc) = (S (mdr_q_rank_integer_secondwitnessminornonzeroevaluationdeterminanths))) -> exists mdr_i_rank_integer_secondwitnessminornonzeroevaluationdeterminanthsc mdr_up_rank_integer_secondwitnessminornonzeroevaluationdeterminanthsc mdr_us_rank_integer_secondwitnessminornonzeroevaluationdeterminanthsc mdr_un_rank_integer_secondwitnessminornonzeroevaluationdeterminanthsc mdr_ut_rank_integer_secondwitnessminornonzeroevaluationdeterminanthsc mdr_p_rank_integer_secondwitnessminornonzeroevaluationdeterminanthsc mdr_n_rank_integer_secondwitnessminornonzeroevaluationdeterminanthsc. ((exists mdr_gap_rank_integer_secondwitnessminornonzeroevaluationdeterminanthsci. mdr_gap_rank_integer_secondwitnessminornonzeroevaluationdeterminanthsci + S (mdr_i_rank_integer_secondwitnessminornonzeroevaluationdeterminanthsc) = (mdr_i_rank_integer_secondwitnessminornonzeroevaluationdeterminanth)) /\ ((exists mdr_z_rank_integer_secondwitnessminornonzeroevaluationdeterminanthscr. ((exists mdr_a_rank_integer_secondwitnessminornonzeroevaluationdeterminanthscrc mdr_b_rank_integer_secondwitnessminornonzeroevaluationdeterminanthscrc mdr_c_rank_integer_secondwitnessminornonzeroevaluationdeterminanthscrc mdr_e_rank_integer_secondwitnessminornonzeroevaluationdeterminanthscrc mdr_f_rank_integer_secondwitnessminornonzeroevaluationdeterminanthscrc. ((mdr_a_rank_integer_secondwitnessminornonzeroevaluationdeterminanthscrc = ((mdr_q_rank_integer_secondwitnessminornonzeroevaluationdeterminanths) + (mdr_up_rank_integer_secondwitnessminornonzeroevaluationdeterminanthsc)) * S ((mdr_q_rank_integer_secondwitnessminornonzeroevaluationdeterminanths) + (mdr_up_rank_integer_secondwitnessminornonzeroevaluationdeterminanthsc)) + ((mdr_up_rank_integer_secondwitnessminornonzeroevaluationdeterminanthsc) + (mdr_up_rank_integer_secondwitnessminornonzeroevaluationdeterminanthsc))) /\ ((mdr_b_rank_integer_secondwitnessminornonzeroevaluationdeterminanthscrc = ((mdr_us_rank_integer_secondwitnessminornonzeroevaluationdeterminanthsc) + (mdr_un_rank_integer_secondwitnessminornonzeroevaluationdeterminanthsc)) * S ((mdr_us_rank_integer_secondwitnessminornonzeroevaluationdeterminanthsc) + (mdr_un_rank_integer_secondwitnessminornonzeroevaluationdeterminanthsc)) + ((mdr_un_rank_integer_secondwitnessminornonzeroevaluationdeterminanthsc) + (mdr_un_rank_integer_secondwitnessminornonzeroevaluationdeterminanthsc))) /\ ((mdr_c_rank_integer_secondwitnessminornonzeroevaluationdeterminanthscrc = ((mdr_a_rank_integer_secondwitnessminornonzeroevaluationdeterminanthscrc) + (mdr_b_rank_integer_secondwitnessminornonzeroevaluationdeterminanthscrc)) * S ((mdr_a_rank_integer_secondwitnessminornonzeroevaluationdeterminanthscrc) + (mdr_b_rank_integer_secondwitnessminornonzeroevaluationdeterminanthscrc)) + ((mdr_b_rank_integer_secondwitnessminornonzeroevaluationdeterminanthscrc) + (mdr_b_rank_integer_secondwitnessminornonzeroevaluationdeterminanthscrc))) /\ ((mdr_e_rank_integer_secondwitnessminornonzeroevaluationdeterminanthscrc = ((mdr_p_rank_integer_secondwitnessminornonzeroevaluationdeterminanthsc) + (mdr_n_rank_integer_secondwitnessminornonzeroevaluationdeterminanthsc)) * S ((mdr_p_rank_integer_secondwitnessminornonzeroevaluationdeterminanthsc) + (mdr_n_rank_integer_secondwitnessminornonzeroevaluationdeterminanthsc)) + ((mdr_n_rank_integer_secondwitnessminornonzeroevaluationdeterminanthsc) + (mdr_n_rank_integer_secondwitnessminornonzeroevaluationdeterminanthsc))) /\ ((mdr_f_rank_integer_secondwitnessminornonzeroevaluationdeterminanthscrc = ((mdr_ut_rank_integer_secondwitnessminornonzeroevaluationdeterminanthsc) + (mdr_e_rank_integer_secondwitnessminornonzeroevaluationdeterminanthscrc)) * S ((mdr_ut_rank_integer_secondwitnessminornonzeroevaluationdeterminanthsc) + (mdr_e_rank_integer_secondwitnessminornonzeroevaluationdeterminanthscrc)) + ((mdr_e_rank_integer_secondwitnessminornonzeroevaluationdeterminanthscrc) + (mdr_e_rank_integer_secondwitnessminornonzeroevaluationdeterminanthscrc))) /\ ((mdr_z_rank_integer_secondwitnessminornonzeroevaluationdeterminanthscr) = ((mdr_c_rank_integer_secondwitnessminornonzeroevaluationdeterminanthscrc) + (mdr_f_rank_integer_secondwitnessminornonzeroevaluationdeterminanthscrc)) * S ((mdr_c_rank_integer_secondwitnessminornonzeroevaluationdeterminanthscrc) + (mdr_f_rank_integer_secondwitnessminornonzeroevaluationdeterminanthscrc)) + ((mdr_f_rank_integer_secondwitnessminornonzeroevaluationdeterminanthscrc) + (mdr_f_rank_integer_secondwitnessminornonzeroevaluationdeterminanthscrc))))))))) /\ (((exists ff_h_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthscrb. ff_h_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthscrb + S (mdr_z_rank_integer_secondwitnessminornonzeroevaluationdeterminanthscr) = S ((S (mdr_i_rank_integer_secondwitnessminornonzeroevaluationdeterminanthsc)) * mdr_c_rank_integer_secondwitnessminornonzeroevaluationdeterminant)) /\ exists ff_q_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthscrb. mdr_b_rank_integer_secondwitnessminornonzeroevaluationdeterminant = ff_q_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthscrb * S ((S (mdr_i_rank_integer_secondwitnessminornonzeroevaluationdeterminanthsc)) * mdr_c_rank_integer_secondwitnessminornonzeroevaluationdeterminant) + (mdr_z_rank_integer_secondwitnessminornonzeroevaluationdeterminanthscr))))) /\ ((((forall ff_index_mdm_prefix_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthscm_positive. (exists ff_gap_mdm_lt_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthscm_positive_index_bound. ff_gap_mdm_lt_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthscm_positive_index_bound + S (ff_index_mdm_prefix_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthscm_positive) = ((mdr_q_rank_integer_secondwitnessminornonzeroevaluationdeterminanths) * (mdr_q_rank_integer_secondwitnessminornonzeroevaluationdeterminanths))) -> exists ff_row_mdm_prefix_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthscm_positive ff_column_mdm_prefix_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthscm_positive ff_value_mdm_prefix_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthscm_positive. (ff_index_mdm_prefix_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthscm_positive = (mdr_q_rank_integer_secondwitnessminornonzeroevaluationdeterminanths) * ff_row_mdm_prefix_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthscm_positive + ff_column_mdm_prefix_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthscm_positive /\ ((exists ff_gap_mdm_lt_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthscm_positive_column_bound. ff_gap_mdm_lt_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthscm_positive_column_bound + S (ff_column_mdm_prefix_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthscm_positive) = (mdr_q_rank_integer_secondwitnessminornonzeroevaluationdeterminanths)) /\ ((exists ff_row_mdm_cell_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthscm_positive_cell ff_column_mdm_cell_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthscm_positive_cell. (((((exists ff_gap_mdm_lt_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthscm_positive_cell_row_before. ff_gap_mdm_lt_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthscm_positive_cell_row_before + S (ff_row_mdm_prefix_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthscm_positive) = (0)) /\ ff_row_mdm_cell_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthscm_positive_cell = ff_row_mdm_prefix_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthscm_positive) \/ ((exists ff_gap_mdm_le_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthscm_positive_cell_row_after. ff_gap_mdm_le_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthscm_positive_cell_row_after + (0) = (ff_row_mdm_prefix_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthscm_positive)) /\ ff_row_mdm_cell_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthscm_positive_cell = S ff_row_mdm_prefix_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthscm_positive))) /\ (((((exists ff_gap_mdm_lt_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthscm_positive_cell_column_before. ff_gap_mdm_lt_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthscm_positive_cell_column_before + S (ff_column_mdm_prefix_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthscm_positive) = (mdr_j_rank_integer_secondwitnessminornonzeroevaluationdeterminanthsc)) /\ ff_column_mdm_cell_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthscm_positive_cell = ff_column_mdm_prefix_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthscm_positive) \/ ((exists ff_gap_mdm_le_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthscm_positive_cell_column_after. ff_gap_mdm_le_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthscm_positive_cell_column_after + (mdr_j_rank_integer_secondwitnessminornonzeroevaluationdeterminanthsc) = (ff_column_mdm_prefix_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthscm_positive)) /\ ff_column_mdm_cell_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthscm_positive_cell = S ff_column_mdm_prefix_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthscm_positive))) /\ (((exists ff_h_mdm_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthscm_positive_cell_source. ff_h_mdm_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthscm_positive_cell_source + S (ff_value_mdm_prefix_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthscm_positive) = S ((S ((ff_row_mdm_cell_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthscm_positive_cell) * (S (mdr_q_rank_integer_secondwitnessminornonzeroevaluationdeterminanths)) + (ff_column_mdm_cell_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthscm_positive_cell))) * mdr_pc_rank_integer_secondwitnessminornonzeroevaluationdeterminanth)) /\ exists ff_q_mdm_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthscm_positive_cell_source. mdr_pb_rank_integer_secondwitnessminornonzeroevaluationdeterminanth = ff_q_mdm_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthscm_positive_cell_source * S ((S ((ff_row_mdm_cell_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthscm_positive_cell) * (S (mdr_q_rank_integer_secondwitnessminornonzeroevaluationdeterminanths)) + (ff_column_mdm_cell_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthscm_positive_cell))) * mdr_pc_rank_integer_secondwitnessminornonzeroevaluationdeterminanth) + (ff_value_mdm_prefix_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthscm_positive)))))) /\ (((exists ff_h_mdm_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthscm_positive_target. ff_h_mdm_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthscm_positive_target + S (ff_value_mdm_prefix_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthscm_positive) = S ((S (ff_index_mdm_prefix_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthscm_positive)) * mdr_us_rank_integer_secondwitnessminornonzeroevaluationdeterminanthsc)) /\ exists ff_q_mdm_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthscm_positive_target. mdr_up_rank_integer_secondwitnessminornonzeroevaluationdeterminanthsc = ff_q_mdm_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthscm_positive_target * S ((S (ff_index_mdm_prefix_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthscm_positive)) * mdr_us_rank_integer_secondwitnessminornonzeroevaluationdeterminanthsc) + (ff_value_mdm_prefix_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthscm_positive))))))) /\ (forall ff_index_mdm_prefix_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthscm_negative. (exists ff_gap_mdm_lt_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthscm_negative_index_bound. ff_gap_mdm_lt_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthscm_negative_index_bound + S (ff_index_mdm_prefix_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthscm_negative) = ((mdr_q_rank_integer_secondwitnessminornonzeroevaluationdeterminanths) * (mdr_q_rank_integer_secondwitnessminornonzeroevaluationdeterminanths))) -> exists ff_row_mdm_prefix_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthscm_negative ff_column_mdm_prefix_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthscm_negative ff_value_mdm_prefix_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthscm_negative. (ff_index_mdm_prefix_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthscm_negative = (mdr_q_rank_integer_secondwitnessminornonzeroevaluationdeterminanths) * ff_row_mdm_prefix_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthscm_negative + ff_column_mdm_prefix_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthscm_negative /\ ((exists ff_gap_mdm_lt_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthscm_negative_column_bound. ff_gap_mdm_lt_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthscm_negative_column_bound + S (ff_column_mdm_prefix_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthscm_negative) = (mdr_q_rank_integer_secondwitnessminornonzeroevaluationdeterminanths)) /\ ((exists ff_row_mdm_cell_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthscm_negative_cell ff_column_mdm_cell_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthscm_negative_cell. (((((exists ff_gap_mdm_lt_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthscm_negative_cell_row_before. ff_gap_mdm_lt_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthscm_negative_cell_row_before + S (ff_row_mdm_prefix_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthscm_negative) = (0)) /\ ff_row_mdm_cell_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthscm_negative_cell = ff_row_mdm_prefix_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthscm_negative) \/ ((exists ff_gap_mdm_le_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthscm_negative_cell_row_after. ff_gap_mdm_le_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthscm_negative_cell_row_after + (0) = (ff_row_mdm_prefix_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthscm_negative)) /\ ff_row_mdm_cell_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthscm_negative_cell = S ff_row_mdm_prefix_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthscm_negative))) /\ (((((exists ff_gap_mdm_lt_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthscm_negative_cell_column_before. ff_gap_mdm_lt_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthscm_negative_cell_column_before + S (ff_column_mdm_prefix_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthscm_negative) = (mdr_j_rank_integer_secondwitnessminornonzeroevaluationdeterminanthsc)) /\ ff_column_mdm_cell_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthscm_negative_cell = ff_column_mdm_prefix_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthscm_negative) \/ ((exists ff_gap_mdm_le_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthscm_negative_cell_column_after. ff_gap_mdm_le_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthscm_negative_cell_column_after + (mdr_j_rank_integer_secondwitnessminornonzeroevaluationdeterminanthsc) = (ff_column_mdm_prefix_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthscm_negative)) /\ ff_column_mdm_cell_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthscm_negative_cell = S ff_column_mdm_prefix_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthscm_negative))) /\ (((exists ff_h_mdm_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthscm_negative_cell_source. ff_h_mdm_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthscm_negative_cell_source + S (ff_value_mdm_prefix_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthscm_negative) = S ((S ((ff_row_mdm_cell_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthscm_negative_cell) * (S (mdr_q_rank_integer_secondwitnessminornonzeroevaluationdeterminanths)) + (ff_column_mdm_cell_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthscm_negative_cell))) * mdr_nc_rank_integer_secondwitnessminornonzeroevaluationdeterminanth)) /\ exists ff_q_mdm_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthscm_negative_cell_source. mdr_nb_rank_integer_secondwitnessminornonzeroevaluationdeterminanth = ff_q_mdm_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthscm_negative_cell_source * S ((S ((ff_row_mdm_cell_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthscm_negative_cell) * (S (mdr_q_rank_integer_secondwitnessminornonzeroevaluationdeterminanths)) + (ff_column_mdm_cell_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthscm_negative_cell))) * mdr_nc_rank_integer_secondwitnessminornonzeroevaluationdeterminanth) + (ff_value_mdm_prefix_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthscm_negative)))))) /\ (((exists ff_h_mdm_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthscm_negative_target. ff_h_mdm_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthscm_negative_target + S (ff_value_mdm_prefix_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthscm_negative) = S ((S (ff_index_mdm_prefix_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthscm_negative)) * mdr_ut_rank_integer_secondwitnessminornonzeroevaluationdeterminanthsc)) /\ exists ff_q_mdm_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthscm_negative_target. mdr_un_rank_integer_secondwitnessminornonzeroevaluationdeterminanthsc = ff_q_mdm_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthscm_negative_target * S ((S (ff_index_mdm_prefix_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthscm_negative)) * mdr_ut_rank_integer_secondwitnessminornonzeroevaluationdeterminanthsc) + (ff_value_mdm_prefix_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthscm_negative))))))))) /\ ((((exists ff_h_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthscp. ff_h_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthscp + S (mdr_p_rank_integer_secondwitnessminornonzeroevaluationdeterminanthsc) = S ((S (mdr_j_rank_integer_secondwitnessminornonzeroevaluationdeterminanthsc)) * mdr_ec_rank_integer_secondwitnessminornonzeroevaluationdeterminanths)) /\ exists ff_q_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthscp. mdr_eb_rank_integer_secondwitnessminornonzeroevaluationdeterminanths = ff_q_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthscp * S ((S (mdr_j_rank_integer_secondwitnessminornonzeroevaluationdeterminanthsc)) * mdr_ec_rank_integer_secondwitnessminornonzeroevaluationdeterminanths) + (mdr_p_rank_integer_secondwitnessminornonzeroevaluationdeterminanthsc))) /\ (((exists ff_h_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthscn. ff_h_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthscn + S (mdr_n_rank_integer_secondwitnessminornonzeroevaluationdeterminanthsc) = S ((S (mdr_j_rank_integer_secondwitnessminornonzeroevaluationdeterminanthsc)) * mdr_fc_rank_integer_secondwitnessminornonzeroevaluationdeterminanths)) /\ exists ff_q_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthscn. mdr_fb_rank_integer_secondwitnessminornonzeroevaluationdeterminanths = ff_q_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthscn * S ((S (mdr_j_rank_integer_secondwitnessminornonzeroevaluationdeterminanthsc)) * mdr_fc_rank_integer_secondwitnessminornonzeroevaluationdeterminanths) + (mdr_n_rank_integer_secondwitnessminornonzeroevaluationdeterminanthsc)))))))) /\ (exists ff_ub_mce_fold_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthsf ff_uc_mce_fold_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthsf ff_vb_mce_fold_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthsf ff_vc_mce_fold_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthsf. ((forall ff_index_mce_alternating_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthsf_prefix. (exists ff_gap_mce_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthsf_prefix_index. ff_gap_mce_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthsf_prefix_index + S (ff_index_mce_alternating_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthsf_prefix) = (S (mdr_q_rank_integer_secondwitnessminornonzeroevaluationdeterminanths))) -> exists ff_ap_mce_alternating_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthsf_prefix ff_an_mce_alternating_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthsf_prefix ff_bp_mce_alternating_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthsf_prefix ff_bn_mce_alternating_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthsf_prefix ff_p_mce_alternating_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthsf_prefix ff_n_mce_alternating_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthsf_prefix. ((((exists ff_h_mce_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthsf_prefix_ap. ff_h_mce_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthsf_prefix_ap + S (ff_ap_mce_alternating_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthsf_prefix) = S ((S (ff_index_mce_alternating_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthsf_prefix)) * mdr_pc_rank_integer_secondwitnessminornonzeroevaluationdeterminanth)) /\ exists ff_q_mce_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthsf_prefix_ap. mdr_pb_rank_integer_secondwitnessminornonzeroevaluationdeterminanth = ff_q_mce_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthsf_prefix_ap * S ((S (ff_index_mce_alternating_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthsf_prefix)) * mdr_pc_rank_integer_secondwitnessminornonzeroevaluationdeterminanth) + (ff_ap_mce_alternating_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthsf_prefix))) /\ ((((exists ff_h_mce_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthsf_prefix_an. ff_h_mce_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthsf_prefix_an + S (ff_an_mce_alternating_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthsf_prefix) = S ((S (ff_index_mce_alternating_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthsf_prefix)) * mdr_nc_rank_integer_secondwitnessminornonzeroevaluationdeterminanth)) /\ exists ff_q_mce_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthsf_prefix_an. mdr_nb_rank_integer_secondwitnessminornonzeroevaluationdeterminanth = ff_q_mce_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthsf_prefix_an * S ((S (ff_index_mce_alternating_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthsf_prefix)) * mdr_nc_rank_integer_secondwitnessminornonzeroevaluationdeterminanth) + (ff_an_mce_alternating_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthsf_prefix))) /\ ((((exists ff_h_mce_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthsf_prefix_bp. ff_h_mce_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthsf_prefix_bp + S (ff_bp_mce_alternating_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthsf_prefix) = S ((S (ff_index_mce_alternating_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthsf_prefix)) * mdr_ec_rank_integer_secondwitnessminornonzeroevaluationdeterminanths)) /\ exists ff_q_mce_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthsf_prefix_bp. mdr_eb_rank_integer_secondwitnessminornonzeroevaluationdeterminanths = ff_q_mce_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthsf_prefix_bp * S ((S (ff_index_mce_alternating_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthsf_prefix)) * mdr_ec_rank_integer_secondwitnessminornonzeroevaluationdeterminanths) + (ff_bp_mce_alternating_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthsf_prefix))) /\ ((((exists ff_h_mce_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthsf_prefix_bn. ff_h_mce_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthsf_prefix_bn + S (ff_bn_mce_alternating_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthsf_prefix) = S ((S (ff_index_mce_alternating_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthsf_prefix)) * mdr_fc_rank_integer_secondwitnessminornonzeroevaluationdeterminanths)) /\ exists ff_q_mce_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthsf_prefix_bn. mdr_fb_rank_integer_secondwitnessminornonzeroevaluationdeterminanths = ff_q_mce_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthsf_prefix_bn * S ((S (ff_index_mce_alternating_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthsf_prefix)) * mdr_fc_rank_integer_secondwitnessminornonzeroevaluationdeterminanths) + (ff_bn_mce_alternating_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthsf_prefix))) /\ ((((exists ff_h_mce_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthsf_prefix_positive. ff_h_mce_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthsf_prefix_positive + S (ff_p_mce_alternating_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthsf_prefix) = S ((S (ff_index_mce_alternating_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthsf_prefix)) * ff_uc_mce_fold_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthsf)) /\ exists ff_q_mce_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthsf_prefix_positive. ff_ub_mce_fold_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthsf = ff_q_mce_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthsf_prefix_positive * S ((S (ff_index_mce_alternating_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthsf_prefix)) * ff_uc_mce_fold_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthsf) + (ff_p_mce_alternating_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthsf_prefix))) /\ ((((exists ff_h_mce_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthsf_prefix_negative. ff_h_mce_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthsf_prefix_negative + S (ff_n_mce_alternating_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthsf_prefix) = S ((S (ff_index_mce_alternating_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthsf_prefix)) * ff_vc_mce_fold_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthsf)) /\ exists ff_q_mce_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthsf_prefix_negative. ff_vb_mce_fold_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthsf = ff_q_mce_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthsf_prefix_negative * S ((S (ff_index_mce_alternating_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthsf_prefix)) * ff_vc_mce_fold_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthsf) + (ff_n_mce_alternating_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthsf_prefix))) /\ (((exists ff_even_mce_term_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthsf_prefix_term. ff_index_mce_alternating_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthsf_prefix = 2 * ff_even_mce_term_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthsf_prefix_term) /\ (ff_p_mce_alternating_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthsf_prefix = (ff_ap_mce_alternating_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthsf_prefix) * (ff_bp_mce_alternating_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthsf_prefix) + (ff_an_mce_alternating_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthsf_prefix) * (ff_bn_mce_alternating_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthsf_prefix) /\ ff_n_mce_alternating_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthsf_prefix = (ff_ap_mce_alternating_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthsf_prefix) * (ff_bn_mce_alternating_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthsf_prefix) + (ff_an_mce_alternating_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthsf_prefix) * (ff_bp_mce_alternating_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthsf_prefix))) \/ ((exists ff_odd_mce_term_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthsf_prefix_term. ff_index_mce_alternating_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthsf_prefix = 2 * ff_odd_mce_term_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthsf_prefix_term + 1) /\ (ff_p_mce_alternating_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthsf_prefix = (ff_ap_mce_alternating_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthsf_prefix) * (ff_bn_mce_alternating_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthsf_prefix) + (ff_an_mce_alternating_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthsf_prefix) * (ff_bp_mce_alternating_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthsf_prefix) /\ ff_n_mce_alternating_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthsf_prefix = (ff_ap_mce_alternating_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthsf_prefix) * (ff_bp_mce_alternating_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthsf_prefix) + (ff_an_mce_alternating_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthsf_prefix) * (ff_bn_mce_alternating_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthsf_prefix))))))))))) /\ ((exists ff_u_mce_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthsf_positive ff_v_mce_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthsf_positive. ((((exists ff_h_mce_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthsf_positive_start. ff_h_mce_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthsf_positive_start + S (0) = S ((S (0)) * ff_v_mce_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthsf_positive)) /\ exists ff_q_mce_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthsf_positive_start. ff_u_mce_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthsf_positive = ff_q_mce_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthsf_positive_start * S ((S (0)) * ff_v_mce_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthsf_positive) + (0))) /\ ((((exists ff_h_mce_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthsf_positive_terminal. ff_h_mce_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthsf_positive_terminal + S (mdr_p_rank_integer_secondwitnessminornonzeroevaluationdeterminanth) = S ((S ((S (mdr_q_rank_integer_secondwitnessminornonzeroevaluationdeterminanths)))) * ff_v_mce_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthsf_positive)) /\ exists ff_q_mce_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthsf_positive_terminal. ff_u_mce_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthsf_positive = ff_q_mce_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthsf_positive_terminal * S ((S ((S (mdr_q_rank_integer_secondwitnessminornonzeroevaluationdeterminanths)))) * ff_v_mce_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthsf_positive) + (mdr_p_rank_integer_secondwitnessminornonzeroevaluationdeterminanth))) /\ forall ff_i_mce_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthsf_positive. (exists ff_lt_mce_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthsf_positive_bound. ff_lt_mce_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthsf_positive_bound + S ff_i_mce_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthsf_positive = (S (mdr_q_rank_integer_secondwitnessminornonzeroevaluationdeterminanths))) -> exists ff_a_mce_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthsf_positive ff_r_mce_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthsf_positive ff_s_mce_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthsf_positive. ((((exists ff_h_mce_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthsf_positive_summand. ff_h_mce_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthsf_positive_summand + S (ff_a_mce_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthsf_positive) = S ((S (ff_i_mce_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthsf_positive)) * ff_uc_mce_fold_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthsf)) /\ exists ff_q_mce_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthsf_positive_summand. ff_ub_mce_fold_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthsf = ff_q_mce_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthsf_positive_summand * S ((S (ff_i_mce_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthsf_positive)) * ff_uc_mce_fold_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthsf) + (ff_a_mce_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthsf_positive))) /\ ((((exists ff_h_mce_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthsf_positive_partial. ff_h_mce_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthsf_positive_partial + S (ff_r_mce_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthsf_positive) = S ((S (ff_i_mce_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthsf_positive)) * ff_v_mce_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthsf_positive)) /\ exists ff_q_mce_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthsf_positive_partial. ff_u_mce_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthsf_positive = ff_q_mce_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthsf_positive_partial * S ((S (ff_i_mce_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthsf_positive)) * ff_v_mce_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthsf_positive) + (ff_r_mce_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthsf_positive))) /\ ((((exists ff_h_mce_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthsf_positive_successor. ff_h_mce_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthsf_positive_successor + S (ff_s_mce_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthsf_positive) = S ((S (S ff_i_mce_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthsf_positive)) * ff_v_mce_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthsf_positive)) /\ exists ff_q_mce_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthsf_positive_successor. ff_u_mce_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthsf_positive = ff_q_mce_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthsf_positive_successor * S ((S (S ff_i_mce_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthsf_positive)) * ff_v_mce_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthsf_positive) + (ff_s_mce_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthsf_positive))) /\ ff_s_mce_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthsf_positive = ff_r_mce_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthsf_positive + ff_a_mce_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthsf_positive)))))) /\ (exists ff_u_mce_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthsf_negative ff_v_mce_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthsf_negative. ((((exists ff_h_mce_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthsf_negative_start. ff_h_mce_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthsf_negative_start + S (0) = S ((S (0)) * ff_v_mce_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthsf_negative)) /\ exists ff_q_mce_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthsf_negative_start. ff_u_mce_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthsf_negative = ff_q_mce_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthsf_negative_start * S ((S (0)) * ff_v_mce_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthsf_negative) + (0))) /\ ((((exists ff_h_mce_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthsf_negative_terminal. ff_h_mce_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthsf_negative_terminal + S (mdr_n_rank_integer_secondwitnessminornonzeroevaluationdeterminanth) = S ((S ((S (mdr_q_rank_integer_secondwitnessminornonzeroevaluationdeterminanths)))) * ff_v_mce_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthsf_negative)) /\ exists ff_q_mce_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthsf_negative_terminal. ff_u_mce_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthsf_negative = ff_q_mce_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthsf_negative_terminal * S ((S ((S (mdr_q_rank_integer_secondwitnessminornonzeroevaluationdeterminanths)))) * ff_v_mce_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthsf_negative) + (mdr_n_rank_integer_secondwitnessminornonzeroevaluationdeterminanth))) /\ forall ff_i_mce_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthsf_negative. (exists ff_lt_mce_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthsf_negative_bound. ff_lt_mce_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthsf_negative_bound + S ff_i_mce_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthsf_negative = (S (mdr_q_rank_integer_secondwitnessminornonzeroevaluationdeterminanths))) -> exists ff_a_mce_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthsf_negative ff_r_mce_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthsf_negative ff_s_mce_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthsf_negative. ((((exists ff_h_mce_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthsf_negative_summand. ff_h_mce_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthsf_negative_summand + S (ff_a_mce_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthsf_negative) = S ((S (ff_i_mce_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthsf_negative)) * ff_vc_mce_fold_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthsf)) /\ exists ff_q_mce_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthsf_negative_summand. ff_vb_mce_fold_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthsf = ff_q_mce_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthsf_negative_summand * S ((S (ff_i_mce_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthsf_negative)) * ff_vc_mce_fold_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthsf) + (ff_a_mce_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthsf_negative))) /\ ((((exists ff_h_mce_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthsf_negative_partial. ff_h_mce_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthsf_negative_partial + S (ff_r_mce_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthsf_negative) = S ((S (ff_i_mce_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthsf_negative)) * ff_v_mce_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthsf_negative)) /\ exists ff_q_mce_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthsf_negative_partial. ff_u_mce_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthsf_negative = ff_q_mce_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthsf_negative_partial * S ((S (ff_i_mce_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthsf_negative)) * ff_v_mce_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthsf_negative) + (ff_r_mce_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthsf_negative))) /\ ((((exists ff_h_mce_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthsf_negative_successor. ff_h_mce_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthsf_negative_successor + S (ff_s_mce_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthsf_negative) = S ((S (S ff_i_mce_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthsf_negative)) * ff_v_mce_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthsf_negative)) /\ exists ff_q_mce_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthsf_negative_successor. ff_u_mce_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthsf_negative = ff_q_mce_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthsf_negative_successor * S ((S (S ff_i_mce_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthsf_negative)) * ff_v_mce_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthsf_negative) + (ff_s_mce_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthsf_negative))) /\ ff_s_mce_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthsf_negative = ff_r_mce_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthsf_negative + ff_a_mce_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminanthsf_negative))))))))))))))) /\ ((exists mdr_gap_rank_integer_secondwitnessminornonzeroevaluationdeterminanti. mdr_gap_rank_integer_secondwitnessminornonzeroevaluationdeterminanti + S (mdr_i_rank_integer_secondwitnessminornonzeroevaluationdeterminant) = (mdr_l_rank_integer_secondwitnessminornonzeroevaluationdeterminant)) /\ (exists mdr_z_rank_integer_secondwitnessminornonzeroevaluationdeterminantr. ((exists mdr_a_rank_integer_secondwitnessminornonzeroevaluationdeterminantrc mdr_b_rank_integer_secondwitnessminornonzeroevaluationdeterminantrc mdr_c_rank_integer_secondwitnessminornonzeroevaluationdeterminantrc mdr_e_rank_integer_secondwitnessminornonzeroevaluationdeterminantrc mdr_f_rank_integer_secondwitnessminornonzeroevaluationdeterminantrc. ((mdr_a_rank_integer_secondwitnessminornonzeroevaluationdeterminantrc = ((other) + (mdr_ub_rank_integer_secondwitnessminornonzeroevaluation)) * S ((other) + (mdr_ub_rank_integer_secondwitnessminornonzeroevaluation)) + ((mdr_ub_rank_integer_secondwitnessminornonzeroevaluation) + (mdr_ub_rank_integer_secondwitnessminornonzeroevaluation))) /\ ((mdr_b_rank_integer_secondwitnessminornonzeroevaluationdeterminantrc = ((mdr_uc_rank_integer_secondwitnessminornonzeroevaluation) + (mdr_vb_rank_integer_secondwitnessminornonzeroevaluation)) * S ((mdr_uc_rank_integer_secondwitnessminornonzeroevaluation) + (mdr_vb_rank_integer_secondwitnessminornonzeroevaluation)) + ((mdr_vb_rank_integer_secondwitnessminornonzeroevaluation) + (mdr_vb_rank_integer_secondwitnessminornonzeroevaluation))) /\ ((mdr_c_rank_integer_secondwitnessminornonzeroevaluationdeterminantrc = ((mdr_a_rank_integer_secondwitnessminornonzeroevaluationdeterminantrc) + (mdr_b_rank_integer_secondwitnessminornonzeroevaluationdeterminantrc)) * S ((mdr_a_rank_integer_secondwitnessminornonzeroevaluationdeterminantrc) + (mdr_b_rank_integer_secondwitnessminornonzeroevaluationdeterminantrc)) + ((mdr_b_rank_integer_secondwitnessminornonzeroevaluationdeterminantrc) + (mdr_b_rank_integer_secondwitnessminornonzeroevaluationdeterminantrc))) /\ ((mdr_e_rank_integer_secondwitnessminornonzeroevaluationdeterminantrc = ((mdr_p_rank_integer_secondwitnessminornonzero) + (mdr_n_rank_integer_secondwitnessminornonzero)) * S ((mdr_p_rank_integer_secondwitnessminornonzero) + (mdr_n_rank_integer_secondwitnessminornonzero)) + ((mdr_n_rank_integer_secondwitnessminornonzero) + (mdr_n_rank_integer_secondwitnessminornonzero))) /\ ((mdr_f_rank_integer_secondwitnessminornonzeroevaluationdeterminantrc = ((mdr_vc_rank_integer_secondwitnessminornonzeroevaluation) + (mdr_e_rank_integer_secondwitnessminornonzeroevaluationdeterminantrc)) * S ((mdr_vc_rank_integer_secondwitnessminornonzeroevaluation) + (mdr_e_rank_integer_secondwitnessminornonzeroevaluationdeterminantrc)) + ((mdr_e_rank_integer_secondwitnessminornonzeroevaluationdeterminantrc) + (mdr_e_rank_integer_secondwitnessminornonzeroevaluationdeterminantrc))) /\ ((mdr_z_rank_integer_secondwitnessminornonzeroevaluationdeterminantr) = ((mdr_c_rank_integer_secondwitnessminornonzeroevaluationdeterminantrc) + (mdr_f_rank_integer_secondwitnessminornonzeroevaluationdeterminantrc)) * S ((mdr_c_rank_integer_secondwitnessminornonzeroevaluationdeterminantrc) + (mdr_f_rank_integer_secondwitnessminornonzeroevaluationdeterminantrc)) + ((mdr_f_rank_integer_secondwitnessminornonzeroevaluationdeterminantrc) + (mdr_f_rank_integer_secondwitnessminornonzeroevaluationdeterminantrc))))))))) /\ (((exists ff_h_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminantrb. ff_h_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminantrb + S (mdr_z_rank_integer_secondwitnessminornonzeroevaluationdeterminantr) = S ((S (mdr_i_rank_integer_secondwitnessminornonzeroevaluationdeterminant)) * mdr_c_rank_integer_secondwitnessminornonzeroevaluationdeterminant)) /\ exists ff_q_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminantrb. mdr_b_rank_integer_secondwitnessminornonzeroevaluationdeterminant = ff_q_mdr_rank_integer_secondwitnessminornonzeroevaluationdeterminantrb * S ((S (mdr_i_rank_integer_secondwitnessminornonzeroevaluationdeterminant)) * mdr_c_rank_integer_secondwitnessminornonzeroevaluationdeterminant) + (mdr_z_rank_integer_secondwitnessminornonzeroevaluationdeterminantr)))))))))) /\ (~(mdr_p_rank_integer_secondwitnessminornonzero = mdr_n_rank_integer_secondwitnessminornonzero)))))))) /\ (forall mdr_q_rank_integer_second. (exists mdr_gap_rank_integer_secondhigher. mdr_gap_rank_integer_secondhigher + S (other) = (mdr_q_rank_integer_second)) -> (forall mdr_rb_rank_integer_secondzero mdr_rc_rank_integer_secondzero mdr_cb_rank_integer_secondzero mdr_cc_rank_integer_secondzero mdr_p_rank_integer_secondzero mdr_n_rank_integer_secondzero. (((forall fom_index_mrf_rank_integer_secondzerorowsbound. (exists fom_gap_mrf_rank_integer_secondzerorowsbound_index_bound. fom_gap_mrf_rank_integer_secondzerorowsbound_index_bound + S (fom_index_mrf_rank_integer_secondzerorowsbound) = mdr_q_rank_integer_second) -> exists fom_value_mrf_rank_integer_secondzerorowsbound. ((((exists fom_beta_height_mrf_rank_integer_secondzerorowsbound_entry. fom_beta_height_mrf_rank_integer_secondzerorowsbound_entry + S (fom_value_mrf_rank_integer_secondzerorowsbound) = S ((S (fom_index_mrf_rank_integer_secondzerorowsbound)) * mdr_rc_rank_integer_secondzero)) /\ exists fom_beta_quotient_mrf_rank_integer_secondzerorowsbound_entry. mdr_rb_rank_integer_secondzero = fom_beta_quotient_mrf_rank_integer_secondzerorowsbound_entry * S ((S (fom_index_mrf_rank_integer_secondzerorowsbound)) * mdr_rc_rank_integer_secondzero) + (fom_value_mrf_rank_integer_secondzerorowsbound))) /\ (exists fom_gap_mrf_rank_integer_secondzerorowsbound_value_bound. fom_gap_mrf_rank_integer_secondzerorowsbound_value_bound + S (fom_value_mrf_rank_integer_secondzerorowsbound) = r))) /\ (forall mdr_i_rank_integer_secondzerorowsdistinct mdr_j_rank_integer_secondzerorowsdistinct mdr_a_rank_integer_secondzerorowsdistinct. (exists mdr_gap_rank_integer_secondzerorowsdistincti. mdr_gap_rank_integer_secondzerorowsdistincti + S (mdr_i_rank_integer_secondzerorowsdistinct) = (mdr_q_rank_integer_second)) -> (exists mdr_gap_rank_integer_secondzerorowsdistinctj. mdr_gap_rank_integer_secondzerorowsdistinctj + S (mdr_j_rank_integer_secondzerorowsdistinct) = (mdr_q_rank_integer_second)) -> (((exists ff_h_mdr_rank_integer_secondzerorowsdistinctfirst. ff_h_mdr_rank_integer_secondzerorowsdistinctfirst + S (mdr_a_rank_integer_secondzerorowsdistinct) = S ((S (mdr_i_rank_integer_secondzerorowsdistinct)) * mdr_rc_rank_integer_secondzero)) /\ exists ff_q_mdr_rank_integer_secondzerorowsdistinctfirst. mdr_rb_rank_integer_secondzero = ff_q_mdr_rank_integer_secondzerorowsdistinctfirst * S ((S (mdr_i_rank_integer_secondzerorowsdistinct)) * mdr_rc_rank_integer_secondzero) + (mdr_a_rank_integer_secondzerorowsdistinct))) -> (((exists ff_h_mdr_rank_integer_secondzerorowsdistinctsecond. ff_h_mdr_rank_integer_secondzerorowsdistinctsecond + S (mdr_a_rank_integer_secondzerorowsdistinct) = S ((S (mdr_j_rank_integer_secondzerorowsdistinct)) * mdr_rc_rank_integer_secondzero)) /\ exists ff_q_mdr_rank_integer_secondzerorowsdistinctsecond. mdr_rb_rank_integer_secondzero = ff_q_mdr_rank_integer_secondzerorowsdistinctsecond * S ((S (mdr_j_rank_integer_secondzerorowsdistinct)) * mdr_rc_rank_integer_secondzero) + (mdr_a_rank_integer_secondzerorowsdistinct))) -> mdr_i_rank_integer_secondzerorowsdistinct = mdr_j_rank_integer_secondzerorowsdistinct))) -> (((forall fom_index_mrf_rank_integer_secondzerocolumnsbound. (exists fom_gap_mrf_rank_integer_secondzerocolumnsbound_index_bound. fom_gap_mrf_rank_integer_secondzerocolumnsbound_index_bound + S (fom_index_mrf_rank_integer_secondzerocolumnsbound) = mdr_q_rank_integer_second) -> exists fom_value_mrf_rank_integer_secondzerocolumnsbound. ((((exists fom_beta_height_mrf_rank_integer_secondzerocolumnsbound_entry. fom_beta_height_mrf_rank_integer_secondzerocolumnsbound_entry + S (fom_value_mrf_rank_integer_secondzerocolumnsbound) = S ((S (fom_index_mrf_rank_integer_secondzerocolumnsbound)) * mdr_cc_rank_integer_secondzero)) /\ exists fom_beta_quotient_mrf_rank_integer_secondzerocolumnsbound_entry. mdr_cb_rank_integer_secondzero = fom_beta_quotient_mrf_rank_integer_secondzerocolumnsbound_entry * S ((S (fom_index_mrf_rank_integer_secondzerocolumnsbound)) * mdr_cc_rank_integer_secondzero) + (fom_value_mrf_rank_integer_secondzerocolumnsbound))) /\ (exists fom_gap_mrf_rank_integer_secondzerocolumnsbound_value_bound. fom_gap_mrf_rank_integer_secondzerocolumnsbound_value_bound + S (fom_value_mrf_rank_integer_secondzerocolumnsbound) = w))) /\ (forall mdr_i_rank_integer_secondzerocolumnsdistinct mdr_j_rank_integer_secondzerocolumnsdistinct mdr_a_rank_integer_secondzerocolumnsdistinct. (exists mdr_gap_rank_integer_secondzerocolumnsdistincti. mdr_gap_rank_integer_secondzerocolumnsdistincti + S (mdr_i_rank_integer_secondzerocolumnsdistinct) = (mdr_q_rank_integer_second)) -> (exists mdr_gap_rank_integer_secondzerocolumnsdistinctj. mdr_gap_rank_integer_secondzerocolumnsdistinctj + S (mdr_j_rank_integer_secondzerocolumnsdistinct) = (mdr_q_rank_integer_second)) -> (((exists ff_h_mdr_rank_integer_secondzerocolumnsdistinctfirst. ff_h_mdr_rank_integer_secondzerocolumnsdistinctfirst + S (mdr_a_rank_integer_secondzerocolumnsdistinct) = S ((S (mdr_i_rank_integer_secondzerocolumnsdistinct)) * mdr_cc_rank_integer_secondzero)) /\ exists ff_q_mdr_rank_integer_secondzerocolumnsdistinctfirst. mdr_cb_rank_integer_secondzero = ff_q_mdr_rank_integer_secondzerocolumnsdistinctfirst * S ((S (mdr_i_rank_integer_secondzerocolumnsdistinct)) * mdr_cc_rank_integer_secondzero) + (mdr_a_rank_integer_secondzerocolumnsdistinct))) -> (((exists ff_h_mdr_rank_integer_secondzerocolumnsdistinctsecond. ff_h_mdr_rank_integer_secondzerocolumnsdistinctsecond + S (mdr_a_rank_integer_secondzerocolumnsdistinct) = S ((S (mdr_j_rank_integer_secondzerocolumnsdistinct)) * mdr_cc_rank_integer_secondzero)) /\ exists ff_q_mdr_rank_integer_secondzerocolumnsdistinctsecond. mdr_cb_rank_integer_secondzero = ff_q_mdr_rank_integer_secondzerocolumnsdistinctsecond * S ((S (mdr_j_rank_integer_secondzerocolumnsdistinct)) * mdr_cc_rank_integer_secondzero) + (mdr_a_rank_integer_secondzerocolumnsdistinct))) -> mdr_i_rank_integer_secondzerocolumnsdistinct = mdr_j_rank_integer_secondzerocolumnsdistinct))) -> (exists mdr_ub_rank_integer_secondzeroevaluation mdr_uc_rank_integer_secondzeroevaluation mdr_vb_rank_integer_secondzeroevaluation mdr_vc_rank_integer_secondzeroevaluation. ((((forall mdr_i_rank_integer_secondzeroevaluationmatrixpositive. (exists mdr_gap_rank_integer_secondzeroevaluationmatrixpositivebound. mdr_gap_rank_integer_secondzeroevaluationmatrixpositivebound + S (mdr_i_rank_integer_secondzeroevaluationmatrixpositive) = ((mdr_q_rank_integer_second) * (mdr_q_rank_integer_second))) -> exists mdr_a_rank_integer_secondzeroevaluationmatrixpositive. (((exists mdr_r_rank_integer_secondzeroevaluationmatrixpositivepoint mdr_s_rank_integer_secondzeroevaluationmatrixpositivepoint mdr_u_rank_integer_secondzeroevaluationmatrixpositivepoint mdr_v_rank_integer_secondzeroevaluationmatrixpositivepoint. ((mdr_i_rank_integer_secondzeroevaluationmatrixpositive = (mdr_q_rank_integer_second) * mdr_r_rank_integer_secondzeroevaluationmatrixpositivepoint + mdr_s_rank_integer_secondzeroevaluationmatrixpositivepoint) /\ ((exists mdr_gap_rank_integer_secondzeroevaluationmatrixpositivepointcolumn. mdr_gap_rank_integer_secondzeroevaluationmatrixpositivepointcolumn + S (mdr_s_rank_integer_secondzeroevaluationmatrixpositivepoint) = (mdr_q_rank_integer_second)) /\ ((((exists ff_h_mdr_rank_integer_secondzeroevaluationmatrixpositivepointrow_index. ff_h_mdr_rank_integer_secondzeroevaluationmatrixpositivepointrow_index + S (mdr_u_rank_integer_secondzeroevaluationmatrixpositivepoint) = S ((S (mdr_r_rank_integer_secondzeroevaluationmatrixpositivepoint)) * mdr_rc_rank_integer_secondzero)) /\ exists ff_q_mdr_rank_integer_secondzeroevaluationmatrixpositivepointrow_index. mdr_rb_rank_integer_secondzero = ff_q_mdr_rank_integer_secondzeroevaluationmatrixpositivepointrow_index * S ((S (mdr_r_rank_integer_secondzeroevaluationmatrixpositivepoint)) * mdr_rc_rank_integer_secondzero) + (mdr_u_rank_integer_secondzeroevaluationmatrixpositivepoint))) /\ ((((exists ff_h_mdr_rank_integer_secondzeroevaluationmatrixpositivepointcolumn_index. ff_h_mdr_rank_integer_secondzeroevaluationmatrixpositivepointcolumn_index + S (mdr_v_rank_integer_secondzeroevaluationmatrixpositivepoint) = S ((S (mdr_s_rank_integer_secondzeroevaluationmatrixpositivepoint)) * mdr_cc_rank_integer_secondzero)) /\ exists ff_q_mdr_rank_integer_secondzeroevaluationmatrixpositivepointcolumn_index. mdr_cb_rank_integer_secondzero = ff_q_mdr_rank_integer_secondzeroevaluationmatrixpositivepointcolumn_index * S ((S (mdr_s_rank_integer_secondzeroevaluationmatrixpositivepoint)) * mdr_cc_rank_integer_secondzero) + (mdr_v_rank_integer_secondzeroevaluationmatrixpositivepoint))) /\ (((exists ff_h_mdr_rank_integer_secondzeroevaluationmatrixpositivepointsource. ff_h_mdr_rank_integer_secondzeroevaluationmatrixpositivepointsource + S (mdr_a_rank_integer_secondzeroevaluationmatrixpositive) = S ((S ((mdr_u_rank_integer_secondzeroevaluationmatrixpositivepoint) * (w) + (mdr_v_rank_integer_secondzeroevaluationmatrixpositivepoint))) * ec)) /\ exists ff_q_mdr_rank_integer_secondzeroevaluationmatrixpositivepointsource. eb = ff_q_mdr_rank_integer_secondzeroevaluationmatrixpositivepointsource * S ((S ((mdr_u_rank_integer_secondzeroevaluationmatrixpositivepoint) * (w) + (mdr_v_rank_integer_secondzeroevaluationmatrixpositivepoint))) * ec) + (mdr_a_rank_integer_secondzeroevaluationmatrixpositive)))))))) /\ (((exists ff_h_mdr_rank_integer_secondzeroevaluationmatrixpositiveoutput. ff_h_mdr_rank_integer_secondzeroevaluationmatrixpositiveoutput + S (mdr_a_rank_integer_secondzeroevaluationmatrixpositive) = S ((S (mdr_i_rank_integer_secondzeroevaluationmatrixpositive)) * mdr_uc_rank_integer_secondzeroevaluation)) /\ exists ff_q_mdr_rank_integer_secondzeroevaluationmatrixpositiveoutput. mdr_ub_rank_integer_secondzeroevaluation = ff_q_mdr_rank_integer_secondzeroevaluationmatrixpositiveoutput * S ((S (mdr_i_rank_integer_secondzeroevaluationmatrixpositive)) * mdr_uc_rank_integer_secondzeroevaluation) + (mdr_a_rank_integer_secondzeroevaluationmatrixpositive)))))) /\ (forall mdr_i_rank_integer_secondzeroevaluationmatrixnegative. (exists mdr_gap_rank_integer_secondzeroevaluationmatrixnegativebound. mdr_gap_rank_integer_secondzeroevaluationmatrixnegativebound + S (mdr_i_rank_integer_secondzeroevaluationmatrixnegative) = ((mdr_q_rank_integer_second) * (mdr_q_rank_integer_second))) -> exists mdr_a_rank_integer_secondzeroevaluationmatrixnegative. (((exists mdr_r_rank_integer_secondzeroevaluationmatrixnegativepoint mdr_s_rank_integer_secondzeroevaluationmatrixnegativepoint mdr_u_rank_integer_secondzeroevaluationmatrixnegativepoint mdr_v_rank_integer_secondzeroevaluationmatrixnegativepoint. ((mdr_i_rank_integer_secondzeroevaluationmatrixnegative = (mdr_q_rank_integer_second) * mdr_r_rank_integer_secondzeroevaluationmatrixnegativepoint + mdr_s_rank_integer_secondzeroevaluationmatrixnegativepoint) /\ ((exists mdr_gap_rank_integer_secondzeroevaluationmatrixnegativepointcolumn. mdr_gap_rank_integer_secondzeroevaluationmatrixnegativepointcolumn + S (mdr_s_rank_integer_secondzeroevaluationmatrixnegativepoint) = (mdr_q_rank_integer_second)) /\ ((((exists ff_h_mdr_rank_integer_secondzeroevaluationmatrixnegativepointrow_index. ff_h_mdr_rank_integer_secondzeroevaluationmatrixnegativepointrow_index + S (mdr_u_rank_integer_secondzeroevaluationmatrixnegativepoint) = S ((S (mdr_r_rank_integer_secondzeroevaluationmatrixnegativepoint)) * mdr_rc_rank_integer_secondzero)) /\ exists ff_q_mdr_rank_integer_secondzeroevaluationmatrixnegativepointrow_index. mdr_rb_rank_integer_secondzero = ff_q_mdr_rank_integer_secondzeroevaluationmatrixnegativepointrow_index * S ((S (mdr_r_rank_integer_secondzeroevaluationmatrixnegativepoint)) * mdr_rc_rank_integer_secondzero) + (mdr_u_rank_integer_secondzeroevaluationmatrixnegativepoint))) /\ ((((exists ff_h_mdr_rank_integer_secondzeroevaluationmatrixnegativepointcolumn_index. ff_h_mdr_rank_integer_secondzeroevaluationmatrixnegativepointcolumn_index + S (mdr_v_rank_integer_secondzeroevaluationmatrixnegativepoint) = S ((S (mdr_s_rank_integer_secondzeroevaluationmatrixnegativepoint)) * mdr_cc_rank_integer_secondzero)) /\ exists ff_q_mdr_rank_integer_secondzeroevaluationmatrixnegativepointcolumn_index. mdr_cb_rank_integer_secondzero = ff_q_mdr_rank_integer_secondzeroevaluationmatrixnegativepointcolumn_index * S ((S (mdr_s_rank_integer_secondzeroevaluationmatrixnegativepoint)) * mdr_cc_rank_integer_secondzero) + (mdr_v_rank_integer_secondzeroevaluationmatrixnegativepoint))) /\ (((exists ff_h_mdr_rank_integer_secondzeroevaluationmatrixnegativepointsource. ff_h_mdr_rank_integer_secondzeroevaluationmatrixnegativepointsource + S (mdr_a_rank_integer_secondzeroevaluationmatrixnegative) = S ((S ((mdr_u_rank_integer_secondzeroevaluationmatrixnegativepoint) * (w) + (mdr_v_rank_integer_secondzeroevaluationmatrixnegativepoint))) * fc)) /\ exists ff_q_mdr_rank_integer_secondzeroevaluationmatrixnegativepointsource. fb = ff_q_mdr_rank_integer_secondzeroevaluationmatrixnegativepointsource * S ((S ((mdr_u_rank_integer_secondzeroevaluationmatrixnegativepoint) * (w) + (mdr_v_rank_integer_secondzeroevaluationmatrixnegativepoint))) * fc) + (mdr_a_rank_integer_secondzeroevaluationmatrixnegative)))))))) /\ (((exists ff_h_mdr_rank_integer_secondzeroevaluationmatrixnegativeoutput. ff_h_mdr_rank_integer_secondzeroevaluationmatrixnegativeoutput + S (mdr_a_rank_integer_secondzeroevaluationmatrixnegative) = S ((S (mdr_i_rank_integer_secondzeroevaluationmatrixnegative)) * mdr_vc_rank_integer_secondzeroevaluation)) /\ exists ff_q_mdr_rank_integer_secondzeroevaluationmatrixnegativeoutput. mdr_vb_rank_integer_secondzeroevaluation = ff_q_mdr_rank_integer_secondzeroevaluationmatrixnegativeoutput * S ((S (mdr_i_rank_integer_secondzeroevaluationmatrixnegative)) * mdr_vc_rank_integer_secondzeroevaluation) + (mdr_a_rank_integer_secondzeroevaluationmatrixnegative)))))))) /\ (exists mdr_b_rank_integer_secondzeroevaluationdeterminant mdr_c_rank_integer_secondzeroevaluationdeterminant mdr_l_rank_integer_secondzeroevaluationdeterminant mdr_i_rank_integer_secondzeroevaluationdeterminant. ((forall mdr_i_rank_integer_secondzeroevaluationdeterminanth. (exists mdr_gap_rank_integer_secondzeroevaluationdeterminanthi. mdr_gap_rank_integer_secondzeroevaluationdeterminanthi + S (mdr_i_rank_integer_secondzeroevaluationdeterminanth) = (mdr_l_rank_integer_secondzeroevaluationdeterminant)) -> exists mdr_d_rank_integer_secondzeroevaluationdeterminanth mdr_pb_rank_integer_secondzeroevaluationdeterminanth mdr_pc_rank_integer_secondzeroevaluationdeterminanth mdr_nb_rank_integer_secondzeroevaluationdeterminanth mdr_nc_rank_integer_secondzeroevaluationdeterminanth mdr_p_rank_integer_secondzeroevaluationdeterminanth mdr_n_rank_integer_secondzeroevaluationdeterminanth. ((exists mdr_z_rank_integer_secondzeroevaluationdeterminanthr. ((exists mdr_a_rank_integer_secondzeroevaluationdeterminanthrc mdr_b_rank_integer_secondzeroevaluationdeterminanthrc mdr_c_rank_integer_secondzeroevaluationdeterminanthrc mdr_e_rank_integer_secondzeroevaluationdeterminanthrc mdr_f_rank_integer_secondzeroevaluationdeterminanthrc. ((mdr_a_rank_integer_secondzeroevaluationdeterminanthrc = ((mdr_d_rank_integer_secondzeroevaluationdeterminanth) + (mdr_pb_rank_integer_secondzeroevaluationdeterminanth)) * S ((mdr_d_rank_integer_secondzeroevaluationdeterminanth) + (mdr_pb_rank_integer_secondzeroevaluationdeterminanth)) + ((mdr_pb_rank_integer_secondzeroevaluationdeterminanth) + (mdr_pb_rank_integer_secondzeroevaluationdeterminanth))) /\ ((mdr_b_rank_integer_secondzeroevaluationdeterminanthrc = ((mdr_pc_rank_integer_secondzeroevaluationdeterminanth) + (mdr_nb_rank_integer_secondzeroevaluationdeterminanth)) * S ((mdr_pc_rank_integer_secondzeroevaluationdeterminanth) + (mdr_nb_rank_integer_secondzeroevaluationdeterminanth)) + ((mdr_nb_rank_integer_secondzeroevaluationdeterminanth) + (mdr_nb_rank_integer_secondzeroevaluationdeterminanth))) /\ ((mdr_c_rank_integer_secondzeroevaluationdeterminanthrc = ((mdr_a_rank_integer_secondzeroevaluationdeterminanthrc) + (mdr_b_rank_integer_secondzeroevaluationdeterminanthrc)) * S ((mdr_a_rank_integer_secondzeroevaluationdeterminanthrc) + (mdr_b_rank_integer_secondzeroevaluationdeterminanthrc)) + ((mdr_b_rank_integer_secondzeroevaluationdeterminanthrc) + (mdr_b_rank_integer_secondzeroevaluationdeterminanthrc))) /\ ((mdr_e_rank_integer_secondzeroevaluationdeterminanthrc = ((mdr_p_rank_integer_secondzeroevaluationdeterminanth) + (mdr_n_rank_integer_secondzeroevaluationdeterminanth)) * S ((mdr_p_rank_integer_secondzeroevaluationdeterminanth) + (mdr_n_rank_integer_secondzeroevaluationdeterminanth)) + ((mdr_n_rank_integer_secondzeroevaluationdeterminanth) + (mdr_n_rank_integer_secondzeroevaluationdeterminanth))) /\ ((mdr_f_rank_integer_secondzeroevaluationdeterminanthrc = ((mdr_nc_rank_integer_secondzeroevaluationdeterminanth) + (mdr_e_rank_integer_secondzeroevaluationdeterminanthrc)) * S ((mdr_nc_rank_integer_secondzeroevaluationdeterminanth) + (mdr_e_rank_integer_secondzeroevaluationdeterminanthrc)) + ((mdr_e_rank_integer_secondzeroevaluationdeterminanthrc) + (mdr_e_rank_integer_secondzeroevaluationdeterminanthrc))) /\ ((mdr_z_rank_integer_secondzeroevaluationdeterminanthr) = ((mdr_c_rank_integer_secondzeroevaluationdeterminanthrc) + (mdr_f_rank_integer_secondzeroevaluationdeterminanthrc)) * S ((mdr_c_rank_integer_secondzeroevaluationdeterminanthrc) + (mdr_f_rank_integer_secondzeroevaluationdeterminanthrc)) + ((mdr_f_rank_integer_secondzeroevaluationdeterminanthrc) + (mdr_f_rank_integer_secondzeroevaluationdeterminanthrc))))))))) /\ (((exists ff_h_mdr_rank_integer_secondzeroevaluationdeterminanthrb. ff_h_mdr_rank_integer_secondzeroevaluationdeterminanthrb + S (mdr_z_rank_integer_secondzeroevaluationdeterminanthr) = S ((S (mdr_i_rank_integer_secondzeroevaluationdeterminanth)) * mdr_c_rank_integer_secondzeroevaluationdeterminant)) /\ exists ff_q_mdr_rank_integer_secondzeroevaluationdeterminanthrb. mdr_b_rank_integer_secondzeroevaluationdeterminant = ff_q_mdr_rank_integer_secondzeroevaluationdeterminanthrb * S ((S (mdr_i_rank_integer_secondzeroevaluationdeterminanth)) * mdr_c_rank_integer_secondzeroevaluationdeterminant) + (mdr_z_rank_integer_secondzeroevaluationdeterminanthr))))) /\ (((((mdr_d_rank_integer_secondzeroevaluationdeterminanth) = 0) /\ (((mdr_p_rank_integer_secondzeroevaluationdeterminanth) = 1) /\ ((mdr_n_rank_integer_secondzeroevaluationdeterminanth) = 0))) \/ exists mdr_q_rank_integer_secondzeroevaluationdeterminanths mdr_eb_rank_integer_secondzeroevaluationdeterminanths mdr_ec_rank_integer_secondzeroevaluationdeterminanths mdr_fb_rank_integer_secondzeroevaluationdeterminanths mdr_fc_rank_integer_secondzeroevaluationdeterminanths. (((mdr_d_rank_integer_secondzeroevaluationdeterminanth) = S (mdr_q_rank_integer_secondzeroevaluationdeterminanths)) /\ ((forall mdr_j_rank_integer_secondzeroevaluationdeterminanthsc. (exists mdr_gap_rank_integer_secondzeroevaluationdeterminanthscj. mdr_gap_rank_integer_secondzeroevaluationdeterminanthscj + S (mdr_j_rank_integer_secondzeroevaluationdeterminanthsc) = (S (mdr_q_rank_integer_secondzeroevaluationdeterminanths))) -> exists mdr_i_rank_integer_secondzeroevaluationdeterminanthsc mdr_up_rank_integer_secondzeroevaluationdeterminanthsc mdr_us_rank_integer_secondzeroevaluationdeterminanthsc mdr_un_rank_integer_secondzeroevaluationdeterminanthsc mdr_ut_rank_integer_secondzeroevaluationdeterminanthsc mdr_p_rank_integer_secondzeroevaluationdeterminanthsc mdr_n_rank_integer_secondzeroevaluationdeterminanthsc. ((exists mdr_gap_rank_integer_secondzeroevaluationdeterminanthsci. mdr_gap_rank_integer_secondzeroevaluationdeterminanthsci + S (mdr_i_rank_integer_secondzeroevaluationdeterminanthsc) = (mdr_i_rank_integer_secondzeroevaluationdeterminanth)) /\ ((exists mdr_z_rank_integer_secondzeroevaluationdeterminanthscr. ((exists mdr_a_rank_integer_secondzeroevaluationdeterminanthscrc mdr_b_rank_integer_secondzeroevaluationdeterminanthscrc mdr_c_rank_integer_secondzeroevaluationdeterminanthscrc mdr_e_rank_integer_secondzeroevaluationdeterminanthscrc mdr_f_rank_integer_secondzeroevaluationdeterminanthscrc. ((mdr_a_rank_integer_secondzeroevaluationdeterminanthscrc = ((mdr_q_rank_integer_secondzeroevaluationdeterminanths) + (mdr_up_rank_integer_secondzeroevaluationdeterminanthsc)) * S ((mdr_q_rank_integer_secondzeroevaluationdeterminanths) + (mdr_up_rank_integer_secondzeroevaluationdeterminanthsc)) + ((mdr_up_rank_integer_secondzeroevaluationdeterminanthsc) + (mdr_up_rank_integer_secondzeroevaluationdeterminanthsc))) /\ ((mdr_b_rank_integer_secondzeroevaluationdeterminanthscrc = ((mdr_us_rank_integer_secondzeroevaluationdeterminanthsc) + (mdr_un_rank_integer_secondzeroevaluationdeterminanthsc)) * S ((mdr_us_rank_integer_secondzeroevaluationdeterminanthsc) + (mdr_un_rank_integer_secondzeroevaluationdeterminanthsc)) + ((mdr_un_rank_integer_secondzeroevaluationdeterminanthsc) + (mdr_un_rank_integer_secondzeroevaluationdeterminanthsc))) /\ ((mdr_c_rank_integer_secondzeroevaluationdeterminanthscrc = ((mdr_a_rank_integer_secondzeroevaluationdeterminanthscrc) + (mdr_b_rank_integer_secondzeroevaluationdeterminanthscrc)) * S ((mdr_a_rank_integer_secondzeroevaluationdeterminanthscrc) + (mdr_b_rank_integer_secondzeroevaluationdeterminanthscrc)) + ((mdr_b_rank_integer_secondzeroevaluationdeterminanthscrc) + (mdr_b_rank_integer_secondzeroevaluationdeterminanthscrc))) /\ ((mdr_e_rank_integer_secondzeroevaluationdeterminanthscrc = ((mdr_p_rank_integer_secondzeroevaluationdeterminanthsc) + (mdr_n_rank_integer_secondzeroevaluationdeterminanthsc)) * S ((mdr_p_rank_integer_secondzeroevaluationdeterminanthsc) + (mdr_n_rank_integer_secondzeroevaluationdeterminanthsc)) + ((mdr_n_rank_integer_secondzeroevaluationdeterminanthsc) + (mdr_n_rank_integer_secondzeroevaluationdeterminanthsc))) /\ ((mdr_f_rank_integer_secondzeroevaluationdeterminanthscrc = ((mdr_ut_rank_integer_secondzeroevaluationdeterminanthsc) + (mdr_e_rank_integer_secondzeroevaluationdeterminanthscrc)) * S ((mdr_ut_rank_integer_secondzeroevaluationdeterminanthsc) + (mdr_e_rank_integer_secondzeroevaluationdeterminanthscrc)) + ((mdr_e_rank_integer_secondzeroevaluationdeterminanthscrc) + (mdr_e_rank_integer_secondzeroevaluationdeterminanthscrc))) /\ ((mdr_z_rank_integer_secondzeroevaluationdeterminanthscr) = ((mdr_c_rank_integer_secondzeroevaluationdeterminanthscrc) + (mdr_f_rank_integer_secondzeroevaluationdeterminanthscrc)) * S ((mdr_c_rank_integer_secondzeroevaluationdeterminanthscrc) + (mdr_f_rank_integer_secondzeroevaluationdeterminanthscrc)) + ((mdr_f_rank_integer_secondzeroevaluationdeterminanthscrc) + (mdr_f_rank_integer_secondzeroevaluationdeterminanthscrc))))))))) /\ (((exists ff_h_mdr_rank_integer_secondzeroevaluationdeterminanthscrb. ff_h_mdr_rank_integer_secondzeroevaluationdeterminanthscrb + S (mdr_z_rank_integer_secondzeroevaluationdeterminanthscr) = S ((S (mdr_i_rank_integer_secondzeroevaluationdeterminanthsc)) * mdr_c_rank_integer_secondzeroevaluationdeterminant)) /\ exists ff_q_mdr_rank_integer_secondzeroevaluationdeterminanthscrb. mdr_b_rank_integer_secondzeroevaluationdeterminant = ff_q_mdr_rank_integer_secondzeroevaluationdeterminanthscrb * S ((S (mdr_i_rank_integer_secondzeroevaluationdeterminanthsc)) * mdr_c_rank_integer_secondzeroevaluationdeterminant) + (mdr_z_rank_integer_secondzeroevaluationdeterminanthscr))))) /\ ((((forall ff_index_mdm_prefix_mdr_rank_integer_secondzeroevaluationdeterminanthscm_positive. (exists ff_gap_mdm_lt_mdr_rank_integer_secondzeroevaluationdeterminanthscm_positive_index_bound. ff_gap_mdm_lt_mdr_rank_integer_secondzeroevaluationdeterminanthscm_positive_index_bound + S (ff_index_mdm_prefix_mdr_rank_integer_secondzeroevaluationdeterminanthscm_positive) = ((mdr_q_rank_integer_secondzeroevaluationdeterminanths) * (mdr_q_rank_integer_secondzeroevaluationdeterminanths))) -> exists ff_row_mdm_prefix_mdr_rank_integer_secondzeroevaluationdeterminanthscm_positive ff_column_mdm_prefix_mdr_rank_integer_secondzeroevaluationdeterminanthscm_positive ff_value_mdm_prefix_mdr_rank_integer_secondzeroevaluationdeterminanthscm_positive. (ff_index_mdm_prefix_mdr_rank_integer_secondzeroevaluationdeterminanthscm_positive = (mdr_q_rank_integer_secondzeroevaluationdeterminanths) * ff_row_mdm_prefix_mdr_rank_integer_secondzeroevaluationdeterminanthscm_positive + ff_column_mdm_prefix_mdr_rank_integer_secondzeroevaluationdeterminanthscm_positive /\ ((exists ff_gap_mdm_lt_mdr_rank_integer_secondzeroevaluationdeterminanthscm_positive_column_bound. ff_gap_mdm_lt_mdr_rank_integer_secondzeroevaluationdeterminanthscm_positive_column_bound + S (ff_column_mdm_prefix_mdr_rank_integer_secondzeroevaluationdeterminanthscm_positive) = (mdr_q_rank_integer_secondzeroevaluationdeterminanths)) /\ ((exists ff_row_mdm_cell_mdr_rank_integer_secondzeroevaluationdeterminanthscm_positive_cell ff_column_mdm_cell_mdr_rank_integer_secondzeroevaluationdeterminanthscm_positive_cell. (((((exists ff_gap_mdm_lt_mdr_rank_integer_secondzeroevaluationdeterminanthscm_positive_cell_row_before. ff_gap_mdm_lt_mdr_rank_integer_secondzeroevaluationdeterminanthscm_positive_cell_row_before + S (ff_row_mdm_prefix_mdr_rank_integer_secondzeroevaluationdeterminanthscm_positive) = (0)) /\ ff_row_mdm_cell_mdr_rank_integer_secondzeroevaluationdeterminanthscm_positive_cell = ff_row_mdm_prefix_mdr_rank_integer_secondzeroevaluationdeterminanthscm_positive) \/ ((exists ff_gap_mdm_le_mdr_rank_integer_secondzeroevaluationdeterminanthscm_positive_cell_row_after. ff_gap_mdm_le_mdr_rank_integer_secondzeroevaluationdeterminanthscm_positive_cell_row_after + (0) = (ff_row_mdm_prefix_mdr_rank_integer_secondzeroevaluationdeterminanthscm_positive)) /\ ff_row_mdm_cell_mdr_rank_integer_secondzeroevaluationdeterminanthscm_positive_cell = S ff_row_mdm_prefix_mdr_rank_integer_secondzeroevaluationdeterminanthscm_positive))) /\ (((((exists ff_gap_mdm_lt_mdr_rank_integer_secondzeroevaluationdeterminanthscm_positive_cell_column_before. ff_gap_mdm_lt_mdr_rank_integer_secondzeroevaluationdeterminanthscm_positive_cell_column_before + S (ff_column_mdm_prefix_mdr_rank_integer_secondzeroevaluationdeterminanthscm_positive) = (mdr_j_rank_integer_secondzeroevaluationdeterminanthsc)) /\ ff_column_mdm_cell_mdr_rank_integer_secondzeroevaluationdeterminanthscm_positive_cell = ff_column_mdm_prefix_mdr_rank_integer_secondzeroevaluationdeterminanthscm_positive) \/ ((exists ff_gap_mdm_le_mdr_rank_integer_secondzeroevaluationdeterminanthscm_positive_cell_column_after. ff_gap_mdm_le_mdr_rank_integer_secondzeroevaluationdeterminanthscm_positive_cell_column_after + (mdr_j_rank_integer_secondzeroevaluationdeterminanthsc) = (ff_column_mdm_prefix_mdr_rank_integer_secondzeroevaluationdeterminanthscm_positive)) /\ ff_column_mdm_cell_mdr_rank_integer_secondzeroevaluationdeterminanthscm_positive_cell = S ff_column_mdm_prefix_mdr_rank_integer_secondzeroevaluationdeterminanthscm_positive))) /\ (((exists ff_h_mdm_mdr_rank_integer_secondzeroevaluationdeterminanthscm_positive_cell_source. ff_h_mdm_mdr_rank_integer_secondzeroevaluationdeterminanthscm_positive_cell_source + S (ff_value_mdm_prefix_mdr_rank_integer_secondzeroevaluationdeterminanthscm_positive) = S ((S ((ff_row_mdm_cell_mdr_rank_integer_secondzeroevaluationdeterminanthscm_positive_cell) * (S (mdr_q_rank_integer_secondzeroevaluationdeterminanths)) + (ff_column_mdm_cell_mdr_rank_integer_secondzeroevaluationdeterminanthscm_positive_cell))) * mdr_pc_rank_integer_secondzeroevaluationdeterminanth)) /\ exists ff_q_mdm_mdr_rank_integer_secondzeroevaluationdeterminanthscm_positive_cell_source. mdr_pb_rank_integer_secondzeroevaluationdeterminanth = ff_q_mdm_mdr_rank_integer_secondzeroevaluationdeterminanthscm_positive_cell_source * S ((S ((ff_row_mdm_cell_mdr_rank_integer_secondzeroevaluationdeterminanthscm_positive_cell) * (S (mdr_q_rank_integer_secondzeroevaluationdeterminanths)) + (ff_column_mdm_cell_mdr_rank_integer_secondzeroevaluationdeterminanthscm_positive_cell))) * mdr_pc_rank_integer_secondzeroevaluationdeterminanth) + (ff_value_mdm_prefix_mdr_rank_integer_secondzeroevaluationdeterminanthscm_positive)))))) /\ (((exists ff_h_mdm_mdr_rank_integer_secondzeroevaluationdeterminanthscm_positive_target. ff_h_mdm_mdr_rank_integer_secondzeroevaluationdeterminanthscm_positive_target + S (ff_value_mdm_prefix_mdr_rank_integer_secondzeroevaluationdeterminanthscm_positive) = S ((S (ff_index_mdm_prefix_mdr_rank_integer_secondzeroevaluationdeterminanthscm_positive)) * mdr_us_rank_integer_secondzeroevaluationdeterminanthsc)) /\ exists ff_q_mdm_mdr_rank_integer_secondzeroevaluationdeterminanthscm_positive_target. mdr_up_rank_integer_secondzeroevaluationdeterminanthsc = ff_q_mdm_mdr_rank_integer_secondzeroevaluationdeterminanthscm_positive_target * S ((S (ff_index_mdm_prefix_mdr_rank_integer_secondzeroevaluationdeterminanthscm_positive)) * mdr_us_rank_integer_secondzeroevaluationdeterminanthsc) + (ff_value_mdm_prefix_mdr_rank_integer_secondzeroevaluationdeterminanthscm_positive))))))) /\ (forall ff_index_mdm_prefix_mdr_rank_integer_secondzeroevaluationdeterminanthscm_negative. (exists ff_gap_mdm_lt_mdr_rank_integer_secondzeroevaluationdeterminanthscm_negative_index_bound. ff_gap_mdm_lt_mdr_rank_integer_secondzeroevaluationdeterminanthscm_negative_index_bound + S (ff_index_mdm_prefix_mdr_rank_integer_secondzeroevaluationdeterminanthscm_negative) = ((mdr_q_rank_integer_secondzeroevaluationdeterminanths) * (mdr_q_rank_integer_secondzeroevaluationdeterminanths))) -> exists ff_row_mdm_prefix_mdr_rank_integer_secondzeroevaluationdeterminanthscm_negative ff_column_mdm_prefix_mdr_rank_integer_secondzeroevaluationdeterminanthscm_negative ff_value_mdm_prefix_mdr_rank_integer_secondzeroevaluationdeterminanthscm_negative. (ff_index_mdm_prefix_mdr_rank_integer_secondzeroevaluationdeterminanthscm_negative = (mdr_q_rank_integer_secondzeroevaluationdeterminanths) * ff_row_mdm_prefix_mdr_rank_integer_secondzeroevaluationdeterminanthscm_negative + ff_column_mdm_prefix_mdr_rank_integer_secondzeroevaluationdeterminanthscm_negative /\ ((exists ff_gap_mdm_lt_mdr_rank_integer_secondzeroevaluationdeterminanthscm_negative_column_bound. ff_gap_mdm_lt_mdr_rank_integer_secondzeroevaluationdeterminanthscm_negative_column_bound + S (ff_column_mdm_prefix_mdr_rank_integer_secondzeroevaluationdeterminanthscm_negative) = (mdr_q_rank_integer_secondzeroevaluationdeterminanths)) /\ ((exists ff_row_mdm_cell_mdr_rank_integer_secondzeroevaluationdeterminanthscm_negative_cell ff_column_mdm_cell_mdr_rank_integer_secondzeroevaluationdeterminanthscm_negative_cell. (((((exists ff_gap_mdm_lt_mdr_rank_integer_secondzeroevaluationdeterminanthscm_negative_cell_row_before. ff_gap_mdm_lt_mdr_rank_integer_secondzeroevaluationdeterminanthscm_negative_cell_row_before + S (ff_row_mdm_prefix_mdr_rank_integer_secondzeroevaluationdeterminanthscm_negative) = (0)) /\ ff_row_mdm_cell_mdr_rank_integer_secondzeroevaluationdeterminanthscm_negative_cell = ff_row_mdm_prefix_mdr_rank_integer_secondzeroevaluationdeterminanthscm_negative) \/ ((exists ff_gap_mdm_le_mdr_rank_integer_secondzeroevaluationdeterminanthscm_negative_cell_row_after. ff_gap_mdm_le_mdr_rank_integer_secondzeroevaluationdeterminanthscm_negative_cell_row_after + (0) = (ff_row_mdm_prefix_mdr_rank_integer_secondzeroevaluationdeterminanthscm_negative)) /\ ff_row_mdm_cell_mdr_rank_integer_secondzeroevaluationdeterminanthscm_negative_cell = S ff_row_mdm_prefix_mdr_rank_integer_secondzeroevaluationdeterminanthscm_negative))) /\ (((((exists ff_gap_mdm_lt_mdr_rank_integer_secondzeroevaluationdeterminanthscm_negative_cell_column_before. ff_gap_mdm_lt_mdr_rank_integer_secondzeroevaluationdeterminanthscm_negative_cell_column_before + S (ff_column_mdm_prefix_mdr_rank_integer_secondzeroevaluationdeterminanthscm_negative) = (mdr_j_rank_integer_secondzeroevaluationdeterminanthsc)) /\ ff_column_mdm_cell_mdr_rank_integer_secondzeroevaluationdeterminanthscm_negative_cell = ff_column_mdm_prefix_mdr_rank_integer_secondzeroevaluationdeterminanthscm_negative) \/ ((exists ff_gap_mdm_le_mdr_rank_integer_secondzeroevaluationdeterminanthscm_negative_cell_column_after. ff_gap_mdm_le_mdr_rank_integer_secondzeroevaluationdeterminanthscm_negative_cell_column_after + (mdr_j_rank_integer_secondzeroevaluationdeterminanthsc) = (ff_column_mdm_prefix_mdr_rank_integer_secondzeroevaluationdeterminanthscm_negative)) /\ ff_column_mdm_cell_mdr_rank_integer_secondzeroevaluationdeterminanthscm_negative_cell = S ff_column_mdm_prefix_mdr_rank_integer_secondzeroevaluationdeterminanthscm_negative))) /\ (((exists ff_h_mdm_mdr_rank_integer_secondzeroevaluationdeterminanthscm_negative_cell_source. ff_h_mdm_mdr_rank_integer_secondzeroevaluationdeterminanthscm_negative_cell_source + S (ff_value_mdm_prefix_mdr_rank_integer_secondzeroevaluationdeterminanthscm_negative) = S ((S ((ff_row_mdm_cell_mdr_rank_integer_secondzeroevaluationdeterminanthscm_negative_cell) * (S (mdr_q_rank_integer_secondzeroevaluationdeterminanths)) + (ff_column_mdm_cell_mdr_rank_integer_secondzeroevaluationdeterminanthscm_negative_cell))) * mdr_nc_rank_integer_secondzeroevaluationdeterminanth)) /\ exists ff_q_mdm_mdr_rank_integer_secondzeroevaluationdeterminanthscm_negative_cell_source. mdr_nb_rank_integer_secondzeroevaluationdeterminanth = ff_q_mdm_mdr_rank_integer_secondzeroevaluationdeterminanthscm_negative_cell_source * S ((S ((ff_row_mdm_cell_mdr_rank_integer_secondzeroevaluationdeterminanthscm_negative_cell) * (S (mdr_q_rank_integer_secondzeroevaluationdeterminanths)) + (ff_column_mdm_cell_mdr_rank_integer_secondzeroevaluationdeterminanthscm_negative_cell))) * mdr_nc_rank_integer_secondzeroevaluationdeterminanth) + (ff_value_mdm_prefix_mdr_rank_integer_secondzeroevaluationdeterminanthscm_negative)))))) /\ (((exists ff_h_mdm_mdr_rank_integer_secondzeroevaluationdeterminanthscm_negative_target. ff_h_mdm_mdr_rank_integer_secondzeroevaluationdeterminanthscm_negative_target + S (ff_value_mdm_prefix_mdr_rank_integer_secondzeroevaluationdeterminanthscm_negative) = S ((S (ff_index_mdm_prefix_mdr_rank_integer_secondzeroevaluationdeterminanthscm_negative)) * mdr_ut_rank_integer_secondzeroevaluationdeterminanthsc)) /\ exists ff_q_mdm_mdr_rank_integer_secondzeroevaluationdeterminanthscm_negative_target. mdr_un_rank_integer_secondzeroevaluationdeterminanthsc = ff_q_mdm_mdr_rank_integer_secondzeroevaluationdeterminanthscm_negative_target * S ((S (ff_index_mdm_prefix_mdr_rank_integer_secondzeroevaluationdeterminanthscm_negative)) * mdr_ut_rank_integer_secondzeroevaluationdeterminanthsc) + (ff_value_mdm_prefix_mdr_rank_integer_secondzeroevaluationdeterminanthscm_negative))))))))) /\ ((((exists ff_h_mdr_rank_integer_secondzeroevaluationdeterminanthscp. ff_h_mdr_rank_integer_secondzeroevaluationdeterminanthscp + S (mdr_p_rank_integer_secondzeroevaluationdeterminanthsc) = S ((S (mdr_j_rank_integer_secondzeroevaluationdeterminanthsc)) * mdr_ec_rank_integer_secondzeroevaluationdeterminanths)) /\ exists ff_q_mdr_rank_integer_secondzeroevaluationdeterminanthscp. mdr_eb_rank_integer_secondzeroevaluationdeterminanths = ff_q_mdr_rank_integer_secondzeroevaluationdeterminanthscp * S ((S (mdr_j_rank_integer_secondzeroevaluationdeterminanthsc)) * mdr_ec_rank_integer_secondzeroevaluationdeterminanths) + (mdr_p_rank_integer_secondzeroevaluationdeterminanthsc))) /\ (((exists ff_h_mdr_rank_integer_secondzeroevaluationdeterminanthscn. ff_h_mdr_rank_integer_secondzeroevaluationdeterminanthscn + S (mdr_n_rank_integer_secondzeroevaluationdeterminanthsc) = S ((S (mdr_j_rank_integer_secondzeroevaluationdeterminanthsc)) * mdr_fc_rank_integer_secondzeroevaluationdeterminanths)) /\ exists ff_q_mdr_rank_integer_secondzeroevaluationdeterminanthscn. mdr_fb_rank_integer_secondzeroevaluationdeterminanths = ff_q_mdr_rank_integer_secondzeroevaluationdeterminanthscn * S ((S (mdr_j_rank_integer_secondzeroevaluationdeterminanthsc)) * mdr_fc_rank_integer_secondzeroevaluationdeterminanths) + (mdr_n_rank_integer_secondzeroevaluationdeterminanthsc)))))))) /\ (exists ff_ub_mce_fold_mdr_rank_integer_secondzeroevaluationdeterminanthsf ff_uc_mce_fold_mdr_rank_integer_secondzeroevaluationdeterminanthsf ff_vb_mce_fold_mdr_rank_integer_secondzeroevaluationdeterminanthsf ff_vc_mce_fold_mdr_rank_integer_secondzeroevaluationdeterminanthsf. ((forall ff_index_mce_alternating_mdr_rank_integer_secondzeroevaluationdeterminanthsf_prefix. (exists ff_gap_mce_mdr_rank_integer_secondzeroevaluationdeterminanthsf_prefix_index. ff_gap_mce_mdr_rank_integer_secondzeroevaluationdeterminanthsf_prefix_index + S (ff_index_mce_alternating_mdr_rank_integer_secondzeroevaluationdeterminanthsf_prefix) = (S (mdr_q_rank_integer_secondzeroevaluationdeterminanths))) -> exists ff_ap_mce_alternating_mdr_rank_integer_secondzeroevaluationdeterminanthsf_prefix ff_an_mce_alternating_mdr_rank_integer_secondzeroevaluationdeterminanthsf_prefix ff_bp_mce_alternating_mdr_rank_integer_secondzeroevaluationdeterminanthsf_prefix ff_bn_mce_alternating_mdr_rank_integer_secondzeroevaluationdeterminanthsf_prefix ff_p_mce_alternating_mdr_rank_integer_secondzeroevaluationdeterminanthsf_prefix ff_n_mce_alternating_mdr_rank_integer_secondzeroevaluationdeterminanthsf_prefix. ((((exists ff_h_mce_mdr_rank_integer_secondzeroevaluationdeterminanthsf_prefix_ap. ff_h_mce_mdr_rank_integer_secondzeroevaluationdeterminanthsf_prefix_ap + S (ff_ap_mce_alternating_mdr_rank_integer_secondzeroevaluationdeterminanthsf_prefix) = S ((S (ff_index_mce_alternating_mdr_rank_integer_secondzeroevaluationdeterminanthsf_prefix)) * mdr_pc_rank_integer_secondzeroevaluationdeterminanth)) /\ exists ff_q_mce_mdr_rank_integer_secondzeroevaluationdeterminanthsf_prefix_ap. mdr_pb_rank_integer_secondzeroevaluationdeterminanth = ff_q_mce_mdr_rank_integer_secondzeroevaluationdeterminanthsf_prefix_ap * S ((S (ff_index_mce_alternating_mdr_rank_integer_secondzeroevaluationdeterminanthsf_prefix)) * mdr_pc_rank_integer_secondzeroevaluationdeterminanth) + (ff_ap_mce_alternating_mdr_rank_integer_secondzeroevaluationdeterminanthsf_prefix))) /\ ((((exists ff_h_mce_mdr_rank_integer_secondzeroevaluationdeterminanthsf_prefix_an. ff_h_mce_mdr_rank_integer_secondzeroevaluationdeterminanthsf_prefix_an + S (ff_an_mce_alternating_mdr_rank_integer_secondzeroevaluationdeterminanthsf_prefix) = S ((S (ff_index_mce_alternating_mdr_rank_integer_secondzeroevaluationdeterminanthsf_prefix)) * mdr_nc_rank_integer_secondzeroevaluationdeterminanth)) /\ exists ff_q_mce_mdr_rank_integer_secondzeroevaluationdeterminanthsf_prefix_an. mdr_nb_rank_integer_secondzeroevaluationdeterminanth = ff_q_mce_mdr_rank_integer_secondzeroevaluationdeterminanthsf_prefix_an * S ((S (ff_index_mce_alternating_mdr_rank_integer_secondzeroevaluationdeterminanthsf_prefix)) * mdr_nc_rank_integer_secondzeroevaluationdeterminanth) + (ff_an_mce_alternating_mdr_rank_integer_secondzeroevaluationdeterminanthsf_prefix))) /\ ((((exists ff_h_mce_mdr_rank_integer_secondzeroevaluationdeterminanthsf_prefix_bp. ff_h_mce_mdr_rank_integer_secondzeroevaluationdeterminanthsf_prefix_bp + S (ff_bp_mce_alternating_mdr_rank_integer_secondzeroevaluationdeterminanthsf_prefix) = S ((S (ff_index_mce_alternating_mdr_rank_integer_secondzeroevaluationdeterminanthsf_prefix)) * mdr_ec_rank_integer_secondzeroevaluationdeterminanths)) /\ exists ff_q_mce_mdr_rank_integer_secondzeroevaluationdeterminanthsf_prefix_bp. mdr_eb_rank_integer_secondzeroevaluationdeterminanths = ff_q_mce_mdr_rank_integer_secondzeroevaluationdeterminanthsf_prefix_bp * S ((S (ff_index_mce_alternating_mdr_rank_integer_secondzeroevaluationdeterminanthsf_prefix)) * mdr_ec_rank_integer_secondzeroevaluationdeterminanths) + (ff_bp_mce_alternating_mdr_rank_integer_secondzeroevaluationdeterminanthsf_prefix))) /\ ((((exists ff_h_mce_mdr_rank_integer_secondzeroevaluationdeterminanthsf_prefix_bn. ff_h_mce_mdr_rank_integer_secondzeroevaluationdeterminanthsf_prefix_bn + S (ff_bn_mce_alternating_mdr_rank_integer_secondzeroevaluationdeterminanthsf_prefix) = S ((S (ff_index_mce_alternating_mdr_rank_integer_secondzeroevaluationdeterminanthsf_prefix)) * mdr_fc_rank_integer_secondzeroevaluationdeterminanths)) /\ exists ff_q_mce_mdr_rank_integer_secondzeroevaluationdeterminanthsf_prefix_bn. mdr_fb_rank_integer_secondzeroevaluationdeterminanths = ff_q_mce_mdr_rank_integer_secondzeroevaluationdeterminanthsf_prefix_bn * S ((S (ff_index_mce_alternating_mdr_rank_integer_secondzeroevaluationdeterminanthsf_prefix)) * mdr_fc_rank_integer_secondzeroevaluationdeterminanths) + (ff_bn_mce_alternating_mdr_rank_integer_secondzeroevaluationdeterminanthsf_prefix))) /\ ((((exists ff_h_mce_mdr_rank_integer_secondzeroevaluationdeterminanthsf_prefix_positive. ff_h_mce_mdr_rank_integer_secondzeroevaluationdeterminanthsf_prefix_positive + S (ff_p_mce_alternating_mdr_rank_integer_secondzeroevaluationdeterminanthsf_prefix) = S ((S (ff_index_mce_alternating_mdr_rank_integer_secondzeroevaluationdeterminanthsf_prefix)) * ff_uc_mce_fold_mdr_rank_integer_secondzeroevaluationdeterminanthsf)) /\ exists ff_q_mce_mdr_rank_integer_secondzeroevaluationdeterminanthsf_prefix_positive. ff_ub_mce_fold_mdr_rank_integer_secondzeroevaluationdeterminanthsf = ff_q_mce_mdr_rank_integer_secondzeroevaluationdeterminanthsf_prefix_positive * S ((S (ff_index_mce_alternating_mdr_rank_integer_secondzeroevaluationdeterminanthsf_prefix)) * ff_uc_mce_fold_mdr_rank_integer_secondzeroevaluationdeterminanthsf) + (ff_p_mce_alternating_mdr_rank_integer_secondzeroevaluationdeterminanthsf_prefix))) /\ ((((exists ff_h_mce_mdr_rank_integer_secondzeroevaluationdeterminanthsf_prefix_negative. ff_h_mce_mdr_rank_integer_secondzeroevaluationdeterminanthsf_prefix_negative + S (ff_n_mce_alternating_mdr_rank_integer_secondzeroevaluationdeterminanthsf_prefix) = S ((S (ff_index_mce_alternating_mdr_rank_integer_secondzeroevaluationdeterminanthsf_prefix)) * ff_vc_mce_fold_mdr_rank_integer_secondzeroevaluationdeterminanthsf)) /\ exists ff_q_mce_mdr_rank_integer_secondzeroevaluationdeterminanthsf_prefix_negative. ff_vb_mce_fold_mdr_rank_integer_secondzeroevaluationdeterminanthsf = ff_q_mce_mdr_rank_integer_secondzeroevaluationdeterminanthsf_prefix_negative * S ((S (ff_index_mce_alternating_mdr_rank_integer_secondzeroevaluationdeterminanthsf_prefix)) * ff_vc_mce_fold_mdr_rank_integer_secondzeroevaluationdeterminanthsf) + (ff_n_mce_alternating_mdr_rank_integer_secondzeroevaluationdeterminanthsf_prefix))) /\ (((exists ff_even_mce_term_mdr_rank_integer_secondzeroevaluationdeterminanthsf_prefix_term. ff_index_mce_alternating_mdr_rank_integer_secondzeroevaluationdeterminanthsf_prefix = 2 * ff_even_mce_term_mdr_rank_integer_secondzeroevaluationdeterminanthsf_prefix_term) /\ (ff_p_mce_alternating_mdr_rank_integer_secondzeroevaluationdeterminanthsf_prefix = (ff_ap_mce_alternating_mdr_rank_integer_secondzeroevaluationdeterminanthsf_prefix) * (ff_bp_mce_alternating_mdr_rank_integer_secondzeroevaluationdeterminanthsf_prefix) + (ff_an_mce_alternating_mdr_rank_integer_secondzeroevaluationdeterminanthsf_prefix) * (ff_bn_mce_alternating_mdr_rank_integer_secondzeroevaluationdeterminanthsf_prefix) /\ ff_n_mce_alternating_mdr_rank_integer_secondzeroevaluationdeterminanthsf_prefix = (ff_ap_mce_alternating_mdr_rank_integer_secondzeroevaluationdeterminanthsf_prefix) * (ff_bn_mce_alternating_mdr_rank_integer_secondzeroevaluationdeterminanthsf_prefix) + (ff_an_mce_alternating_mdr_rank_integer_secondzeroevaluationdeterminanthsf_prefix) * (ff_bp_mce_alternating_mdr_rank_integer_secondzeroevaluationdeterminanthsf_prefix))) \/ ((exists ff_odd_mce_term_mdr_rank_integer_secondzeroevaluationdeterminanthsf_prefix_term. ff_index_mce_alternating_mdr_rank_integer_secondzeroevaluationdeterminanthsf_prefix = 2 * ff_odd_mce_term_mdr_rank_integer_secondzeroevaluationdeterminanthsf_prefix_term + 1) /\ (ff_p_mce_alternating_mdr_rank_integer_secondzeroevaluationdeterminanthsf_prefix = (ff_ap_mce_alternating_mdr_rank_integer_secondzeroevaluationdeterminanthsf_prefix) * (ff_bn_mce_alternating_mdr_rank_integer_secondzeroevaluationdeterminanthsf_prefix) + (ff_an_mce_alternating_mdr_rank_integer_secondzeroevaluationdeterminanthsf_prefix) * (ff_bp_mce_alternating_mdr_rank_integer_secondzeroevaluationdeterminanthsf_prefix) /\ ff_n_mce_alternating_mdr_rank_integer_secondzeroevaluationdeterminanthsf_prefix = (ff_ap_mce_alternating_mdr_rank_integer_secondzeroevaluationdeterminanthsf_prefix) * (ff_bp_mce_alternating_mdr_rank_integer_secondzeroevaluationdeterminanthsf_prefix) + (ff_an_mce_alternating_mdr_rank_integer_secondzeroevaluationdeterminanthsf_prefix) * (ff_bn_mce_alternating_mdr_rank_integer_secondzeroevaluationdeterminanthsf_prefix))))))))))) /\ ((exists ff_u_mce_mdr_rank_integer_secondzeroevaluationdeterminanthsf_positive ff_v_mce_mdr_rank_integer_secondzeroevaluationdeterminanthsf_positive. ((((exists ff_h_mce_mdr_rank_integer_secondzeroevaluationdeterminanthsf_positive_start. ff_h_mce_mdr_rank_integer_secondzeroevaluationdeterminanthsf_positive_start + S (0) = S ((S (0)) * ff_v_mce_mdr_rank_integer_secondzeroevaluationdeterminanthsf_positive)) /\ exists ff_q_mce_mdr_rank_integer_secondzeroevaluationdeterminanthsf_positive_start. ff_u_mce_mdr_rank_integer_secondzeroevaluationdeterminanthsf_positive = ff_q_mce_mdr_rank_integer_secondzeroevaluationdeterminanthsf_positive_start * S ((S (0)) * ff_v_mce_mdr_rank_integer_secondzeroevaluationdeterminanthsf_positive) + (0))) /\ ((((exists ff_h_mce_mdr_rank_integer_secondzeroevaluationdeterminanthsf_positive_terminal. ff_h_mce_mdr_rank_integer_secondzeroevaluationdeterminanthsf_positive_terminal + S (mdr_p_rank_integer_secondzeroevaluationdeterminanth) = S ((S ((S (mdr_q_rank_integer_secondzeroevaluationdeterminanths)))) * ff_v_mce_mdr_rank_integer_secondzeroevaluationdeterminanthsf_positive)) /\ exists ff_q_mce_mdr_rank_integer_secondzeroevaluationdeterminanthsf_positive_terminal. ff_u_mce_mdr_rank_integer_secondzeroevaluationdeterminanthsf_positive = ff_q_mce_mdr_rank_integer_secondzeroevaluationdeterminanthsf_positive_terminal * S ((S ((S (mdr_q_rank_integer_secondzeroevaluationdeterminanths)))) * ff_v_mce_mdr_rank_integer_secondzeroevaluationdeterminanthsf_positive) + (mdr_p_rank_integer_secondzeroevaluationdeterminanth))) /\ forall ff_i_mce_mdr_rank_integer_secondzeroevaluationdeterminanthsf_positive. (exists ff_lt_mce_mdr_rank_integer_secondzeroevaluationdeterminanthsf_positive_bound. ff_lt_mce_mdr_rank_integer_secondzeroevaluationdeterminanthsf_positive_bound + S ff_i_mce_mdr_rank_integer_secondzeroevaluationdeterminanthsf_positive = (S (mdr_q_rank_integer_secondzeroevaluationdeterminanths))) -> exists ff_a_mce_mdr_rank_integer_secondzeroevaluationdeterminanthsf_positive ff_r_mce_mdr_rank_integer_secondzeroevaluationdeterminanthsf_positive ff_s_mce_mdr_rank_integer_secondzeroevaluationdeterminanthsf_positive. ((((exists ff_h_mce_mdr_rank_integer_secondzeroevaluationdeterminanthsf_positive_summand. ff_h_mce_mdr_rank_integer_secondzeroevaluationdeterminanthsf_positive_summand + S (ff_a_mce_mdr_rank_integer_secondzeroevaluationdeterminanthsf_positive) = S ((S (ff_i_mce_mdr_rank_integer_secondzeroevaluationdeterminanthsf_positive)) * ff_uc_mce_fold_mdr_rank_integer_secondzeroevaluationdeterminanthsf)) /\ exists ff_q_mce_mdr_rank_integer_secondzeroevaluationdeterminanthsf_positive_summand. ff_ub_mce_fold_mdr_rank_integer_secondzeroevaluationdeterminanthsf = ff_q_mce_mdr_rank_integer_secondzeroevaluationdeterminanthsf_positive_summand * S ((S (ff_i_mce_mdr_rank_integer_secondzeroevaluationdeterminanthsf_positive)) * ff_uc_mce_fold_mdr_rank_integer_secondzeroevaluationdeterminanthsf) + (ff_a_mce_mdr_rank_integer_secondzeroevaluationdeterminanthsf_positive))) /\ ((((exists ff_h_mce_mdr_rank_integer_secondzeroevaluationdeterminanthsf_positive_partial. ff_h_mce_mdr_rank_integer_secondzeroevaluationdeterminanthsf_positive_partial + S (ff_r_mce_mdr_rank_integer_secondzeroevaluationdeterminanthsf_positive) = S ((S (ff_i_mce_mdr_rank_integer_secondzeroevaluationdeterminanthsf_positive)) * ff_v_mce_mdr_rank_integer_secondzeroevaluationdeterminanthsf_positive)) /\ exists ff_q_mce_mdr_rank_integer_secondzeroevaluationdeterminanthsf_positive_partial. ff_u_mce_mdr_rank_integer_secondzeroevaluationdeterminanthsf_positive = ff_q_mce_mdr_rank_integer_secondzeroevaluationdeterminanthsf_positive_partial * S ((S (ff_i_mce_mdr_rank_integer_secondzeroevaluationdeterminanthsf_positive)) * ff_v_mce_mdr_rank_integer_secondzeroevaluationdeterminanthsf_positive) + (ff_r_mce_mdr_rank_integer_secondzeroevaluationdeterminanthsf_positive))) /\ ((((exists ff_h_mce_mdr_rank_integer_secondzeroevaluationdeterminanthsf_positive_successor. ff_h_mce_mdr_rank_integer_secondzeroevaluationdeterminanthsf_positive_successor + S (ff_s_mce_mdr_rank_integer_secondzeroevaluationdeterminanthsf_positive) = S ((S (S ff_i_mce_mdr_rank_integer_secondzeroevaluationdeterminanthsf_positive)) * ff_v_mce_mdr_rank_integer_secondzeroevaluationdeterminanthsf_positive)) /\ exists ff_q_mce_mdr_rank_integer_secondzeroevaluationdeterminanthsf_positive_successor. ff_u_mce_mdr_rank_integer_secondzeroevaluationdeterminanthsf_positive = ff_q_mce_mdr_rank_integer_secondzeroevaluationdeterminanthsf_positive_successor * S ((S (S ff_i_mce_mdr_rank_integer_secondzeroevaluationdeterminanthsf_positive)) * ff_v_mce_mdr_rank_integer_secondzeroevaluationdeterminanthsf_positive) + (ff_s_mce_mdr_rank_integer_secondzeroevaluationdeterminanthsf_positive))) /\ ff_s_mce_mdr_rank_integer_secondzeroevaluationdeterminanthsf_positive = ff_r_mce_mdr_rank_integer_secondzeroevaluationdeterminanthsf_positive + ff_a_mce_mdr_rank_integer_secondzeroevaluationdeterminanthsf_positive)))))) /\ (exists ff_u_mce_mdr_rank_integer_secondzeroevaluationdeterminanthsf_negative ff_v_mce_mdr_rank_integer_secondzeroevaluationdeterminanthsf_negative. ((((exists ff_h_mce_mdr_rank_integer_secondzeroevaluationdeterminanthsf_negative_start. ff_h_mce_mdr_rank_integer_secondzeroevaluationdeterminanthsf_negative_start + S (0) = S ((S (0)) * ff_v_mce_mdr_rank_integer_secondzeroevaluationdeterminanthsf_negative)) /\ exists ff_q_mce_mdr_rank_integer_secondzeroevaluationdeterminanthsf_negative_start. ff_u_mce_mdr_rank_integer_secondzeroevaluationdeterminanthsf_negative = ff_q_mce_mdr_rank_integer_secondzeroevaluationdeterminanthsf_negative_start * S ((S (0)) * ff_v_mce_mdr_rank_integer_secondzeroevaluationdeterminanthsf_negative) + (0))) /\ ((((exists ff_h_mce_mdr_rank_integer_secondzeroevaluationdeterminanthsf_negative_terminal. ff_h_mce_mdr_rank_integer_secondzeroevaluationdeterminanthsf_negative_terminal + S (mdr_n_rank_integer_secondzeroevaluationdeterminanth) = S ((S ((S (mdr_q_rank_integer_secondzeroevaluationdeterminanths)))) * ff_v_mce_mdr_rank_integer_secondzeroevaluationdeterminanthsf_negative)) /\ exists ff_q_mce_mdr_rank_integer_secondzeroevaluationdeterminanthsf_negative_terminal. ff_u_mce_mdr_rank_integer_secondzeroevaluationdeterminanthsf_negative = ff_q_mce_mdr_rank_integer_secondzeroevaluationdeterminanthsf_negative_terminal * S ((S ((S (mdr_q_rank_integer_secondzeroevaluationdeterminanths)))) * ff_v_mce_mdr_rank_integer_secondzeroevaluationdeterminanthsf_negative) + (mdr_n_rank_integer_secondzeroevaluationdeterminanth))) /\ forall ff_i_mce_mdr_rank_integer_secondzeroevaluationdeterminanthsf_negative. (exists ff_lt_mce_mdr_rank_integer_secondzeroevaluationdeterminanthsf_negative_bound. ff_lt_mce_mdr_rank_integer_secondzeroevaluationdeterminanthsf_negative_bound + S ff_i_mce_mdr_rank_integer_secondzeroevaluationdeterminanthsf_negative = (S (mdr_q_rank_integer_secondzeroevaluationdeterminanths))) -> exists ff_a_mce_mdr_rank_integer_secondzeroevaluationdeterminanthsf_negative ff_r_mce_mdr_rank_integer_secondzeroevaluationdeterminanthsf_negative ff_s_mce_mdr_rank_integer_secondzeroevaluationdeterminanthsf_negative. ((((exists ff_h_mce_mdr_rank_integer_secondzeroevaluationdeterminanthsf_negative_summand. ff_h_mce_mdr_rank_integer_secondzeroevaluationdeterminanthsf_negative_summand + S (ff_a_mce_mdr_rank_integer_secondzeroevaluationdeterminanthsf_negative) = S ((S (ff_i_mce_mdr_rank_integer_secondzeroevaluationdeterminanthsf_negative)) * ff_vc_mce_fold_mdr_rank_integer_secondzeroevaluationdeterminanthsf)) /\ exists ff_q_mce_mdr_rank_integer_secondzeroevaluationdeterminanthsf_negative_summand. ff_vb_mce_fold_mdr_rank_integer_secondzeroevaluationdeterminanthsf = ff_q_mce_mdr_rank_integer_secondzeroevaluationdeterminanthsf_negative_summand * S ((S (ff_i_mce_mdr_rank_integer_secondzeroevaluationdeterminanthsf_negative)) * ff_vc_mce_fold_mdr_rank_integer_secondzeroevaluationdeterminanthsf) + (ff_a_mce_mdr_rank_integer_secondzeroevaluationdeterminanthsf_negative))) /\ ((((exists ff_h_mce_mdr_rank_integer_secondzeroevaluationdeterminanthsf_negative_partial. ff_h_mce_mdr_rank_integer_secondzeroevaluationdeterminanthsf_negative_partial + S (ff_r_mce_mdr_rank_integer_secondzeroevaluationdeterminanthsf_negative) = S ((S (ff_i_mce_mdr_rank_integer_secondzeroevaluationdeterminanthsf_negative)) * ff_v_mce_mdr_rank_integer_secondzeroevaluationdeterminanthsf_negative)) /\ exists ff_q_mce_mdr_rank_integer_secondzeroevaluationdeterminanthsf_negative_partial. ff_u_mce_mdr_rank_integer_secondzeroevaluationdeterminanthsf_negative = ff_q_mce_mdr_rank_integer_secondzeroevaluationdeterminanthsf_negative_partial * S ((S (ff_i_mce_mdr_rank_integer_secondzeroevaluationdeterminanthsf_negative)) * ff_v_mce_mdr_rank_integer_secondzeroevaluationdeterminanthsf_negative) + (ff_r_mce_mdr_rank_integer_secondzeroevaluationdeterminanthsf_negative))) /\ ((((exists ff_h_mce_mdr_rank_integer_secondzeroevaluationdeterminanthsf_negative_successor. ff_h_mce_mdr_rank_integer_secondzeroevaluationdeterminanthsf_negative_successor + S (ff_s_mce_mdr_rank_integer_secondzeroevaluationdeterminanthsf_negative) = S ((S (S ff_i_mce_mdr_rank_integer_secondzeroevaluationdeterminanthsf_negative)) * ff_v_mce_mdr_rank_integer_secondzeroevaluationdeterminanthsf_negative)) /\ exists ff_q_mce_mdr_rank_integer_secondzeroevaluationdeterminanthsf_negative_successor. ff_u_mce_mdr_rank_integer_secondzeroevaluationdeterminanthsf_negative = ff_q_mce_mdr_rank_integer_secondzeroevaluationdeterminanthsf_negative_successor * S ((S (S ff_i_mce_mdr_rank_integer_secondzeroevaluationdeterminanthsf_negative)) * ff_v_mce_mdr_rank_integer_secondzeroevaluationdeterminanthsf_negative) + (ff_s_mce_mdr_rank_integer_secondzeroevaluationdeterminanthsf_negative))) /\ ff_s_mce_mdr_rank_integer_secondzeroevaluationdeterminanthsf_negative = ff_r_mce_mdr_rank_integer_secondzeroevaluationdeterminanthsf_negative + ff_a_mce_mdr_rank_integer_secondzeroevaluationdeterminanthsf_negative))))))))))))))) /\ ((exists mdr_gap_rank_integer_secondzeroevaluationdeterminanti. mdr_gap_rank_integer_secondzeroevaluationdeterminanti + S (mdr_i_rank_integer_secondzeroevaluationdeterminant) = (mdr_l_rank_integer_secondzeroevaluationdeterminant)) /\ (exists mdr_z_rank_integer_secondzeroevaluationdeterminantr. ((exists mdr_a_rank_integer_secondzeroevaluationdeterminantrc mdr_b_rank_integer_secondzeroevaluationdeterminantrc mdr_c_rank_integer_secondzeroevaluationdeterminantrc mdr_e_rank_integer_secondzeroevaluationdeterminantrc mdr_f_rank_integer_secondzeroevaluationdeterminantrc. ((mdr_a_rank_integer_secondzeroevaluationdeterminantrc = ((mdr_q_rank_integer_second) + (mdr_ub_rank_integer_secondzeroevaluation)) * S ((mdr_q_rank_integer_second) + (mdr_ub_rank_integer_secondzeroevaluation)) + ((mdr_ub_rank_integer_secondzeroevaluation) + (mdr_ub_rank_integer_secondzeroevaluation))) /\ ((mdr_b_rank_integer_secondzeroevaluationdeterminantrc = ((mdr_uc_rank_integer_secondzeroevaluation) + (mdr_vb_rank_integer_secondzeroevaluation)) * S ((mdr_uc_rank_integer_secondzeroevaluation) + (mdr_vb_rank_integer_secondzeroevaluation)) + ((mdr_vb_rank_integer_secondzeroevaluation) + (mdr_vb_rank_integer_secondzeroevaluation))) /\ ((mdr_c_rank_integer_secondzeroevaluationdeterminantrc = ((mdr_a_rank_integer_secondzeroevaluationdeterminantrc) + (mdr_b_rank_integer_secondzeroevaluationdeterminantrc)) * S ((mdr_a_rank_integer_secondzeroevaluationdeterminantrc) + (mdr_b_rank_integer_secondzeroevaluationdeterminantrc)) + ((mdr_b_rank_integer_secondzeroevaluationdeterminantrc) + (mdr_b_rank_integer_secondzeroevaluationdeterminantrc))) /\ ((mdr_e_rank_integer_secondzeroevaluationdeterminantrc = ((mdr_p_rank_integer_secondzero) + (mdr_n_rank_integer_secondzero)) * S ((mdr_p_rank_integer_secondzero) + (mdr_n_rank_integer_secondzero)) + ((mdr_n_rank_integer_secondzero) + (mdr_n_rank_integer_secondzero))) /\ ((mdr_f_rank_integer_secondzeroevaluationdeterminantrc = ((mdr_vc_rank_integer_secondzeroevaluation) + (mdr_e_rank_integer_secondzeroevaluationdeterminantrc)) * S ((mdr_vc_rank_integer_secondzeroevaluation) + (mdr_e_rank_integer_secondzeroevaluationdeterminantrc)) + ((mdr_e_rank_integer_secondzeroevaluationdeterminantrc) + (mdr_e_rank_integer_secondzeroevaluationdeterminantrc))) /\ ((mdr_z_rank_integer_secondzeroevaluationdeterminantr) = ((mdr_c_rank_integer_secondzeroevaluationdeterminantrc) + (mdr_f_rank_integer_secondzeroevaluationdeterminantrc)) * S ((mdr_c_rank_integer_secondzeroevaluationdeterminantrc) + (mdr_f_rank_integer_secondzeroevaluationdeterminantrc)) + ((mdr_f_rank_integer_secondzeroevaluationdeterminantrc) + (mdr_f_rank_integer_secondzeroevaluationdeterminantrc))))))))) /\ (((exists ff_h_mdr_rank_integer_secondzeroevaluationdeterminantrb. ff_h_mdr_rank_integer_secondzeroevaluationdeterminantrb + S (mdr_z_rank_integer_secondzeroevaluationdeterminantr) = S ((S (mdr_i_rank_integer_secondzeroevaluationdeterminant)) * mdr_c_rank_integer_secondzeroevaluationdeterminant)) /\ exists ff_q_mdr_rank_integer_secondzeroevaluationdeterminantrb. mdr_b_rank_integer_secondzeroevaluationdeterminant = ff_q_mdr_rank_integer_secondzeroevaluationdeterminantrb * S ((S (mdr_i_rank_integer_secondzeroevaluationdeterminant)) * mdr_c_rank_integer_secondzeroevaluationdeterminant) + (mdr_z_rank_integer_secondzeroevaluationdeterminantr)))))))))) -> mdr_p_rank_integer_secondzero = mdr_n_rank_integer_secondzero)))))) -> rank = otherComplete tactic proof in conservative notation
All 39 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
39 script commands · 5 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 assumptionsL11–15
03Use earlier factsL16–25
Instantiate or apply named facts and discharge the corresponding proof obligations.
- L16
specialize rectangular_matrix_rank_functional (eb) - L17
specialize rectangular_matrix_rank_functional (ec) - L18
specialize rectangular_matrix_rank_functional (fb) - L19
specialize rectangular_matrix_rank_functional (fc) - L20
specialize rectangular_matrix_rank_functional (r) - L21
specialize rectangular_matrix_rank_functional (w) - L22
specialize rectangular_matrix_rank_functional (rank) - L23
specialize rectangular_matrix_rank_functional (other) - L24
apply rectangular_matrix_rank_functional - L25
specialize rectangular_matrix_rank_integer_transport (ab)
04Use earlier factsL26–35
Instantiate or apply named facts and discharge the corresponding proof obligations.
- L26
specialize rectangular_matrix_rank_integer_transport (ac) - L27
specialize rectangular_matrix_rank_integer_transport (bb) - L28
specialize rectangular_matrix_rank_integer_transport (bc) - L29
specialize rectangular_matrix_rank_integer_transport (eb) - L30
specialize rectangular_matrix_rank_integer_transport (ec) - L31
specialize rectangular_matrix_rank_integer_transport (fb) - L32
specialize rectangular_matrix_rank_integer_transport (fc) - L33
specialize rectangular_matrix_rank_integer_transport (r) - L34
specialize rectangular_matrix_rank_integer_transport (w) - L35
specialize rectangular_matrix_rank_integer_transport (rank)
Original defined command ledger · 39 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 other - 0013
intro hequal - 0014
intro hfirst - 0015
intro hsecond - 0016
specialize rectangular_matrix_rank_functional (eb) - 0017
specialize rectangular_matrix_rank_functional (ec) - 0018
specialize rectangular_matrix_rank_functional (fb) - 0019
specialize rectangular_matrix_rank_functional (fc) - 0020
specialize rectangular_matrix_rank_functional (r) - 0021
specialize rectangular_matrix_rank_functional (w) - 0022
specialize rectangular_matrix_rank_functional (rank) - 0023
specialize rectangular_matrix_rank_functional (other) - 0024
apply rectangular_matrix_rank_functional - 0025
specialize rectangular_matrix_rank_integer_transport (ab) - 0026
specialize rectangular_matrix_rank_integer_transport (ac) - 0027
specialize rectangular_matrix_rank_integer_transport (bb) - 0028
specialize rectangular_matrix_rank_integer_transport (bc) - 0029
specialize rectangular_matrix_rank_integer_transport (eb) - 0030
specialize rectangular_matrix_rank_integer_transport (ec) - 0031
specialize rectangular_matrix_rank_integer_transport (fb) - 0032
specialize rectangular_matrix_rank_integer_transport (fc) - 0033
specialize rectangular_matrix_rank_integer_transport (r) - 0034
specialize rectangular_matrix_rank_integer_transport (w) - 0035
specialize rectangular_matrix_rank_integer_transport (rank) - 0036
apply rectangular_matrix_rank_integer_transport - 0037
exact hequal - 0038
exact hfirst - 0039
exact hsecond