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.
Definition in prerequisite notation
∀ mdr_j_secondwave. Lt(mdr_j_secondwave,l) → ∃ x. ∃ y. ∃ z. ∃ n. ∃ m. ∃ k. ∃ i. Lt(x,limit) ∧ (SignedDeterminantNodeAt(b,c,x,q,y,z,n,m,k,i) ∧ (SignedMatrixMinor(pb,pc,nb,nc,S q,0,mdr_j_secondwave,q,y,z,n,m) ∧ (BetaAt(eb,ec,mdr_j_secondwave,k) ∧ BetaAt(fb,fc,mdr_j_secondwave,i))))Only definitions earlier in this acyclic notation graph are used here.
Hygienic expanded first-order definition
forall mdr_j_secondwave. (exists mdr_gap_secondwavej. mdr_gap_secondwavej + S (mdr_j_secondwave) = (l)) -> exists mdr_i_secondwave mdr_up_secondwave mdr_us_secondwave mdr_un_secondwave mdr_ut_secondwave mdr_p_secondwave mdr_n_secondwave. ((exists mdr_gap_secondwavei. mdr_gap_secondwavei + S (mdr_i_secondwave) = (limit)) /\ ((exists mdr_z_secondwaver. ((exists mdr_a_secondwaverc mdr_b_secondwaverc mdr_c_secondwaverc mdr_e_secondwaverc mdr_f_secondwaverc. ((mdr_a_secondwaverc = ((q) + (mdr_up_secondwave)) * S ((q) + (mdr_up_secondwave)) + ((mdr_up_secondwave) + (mdr_up_secondwave))) /\ ((mdr_b_secondwaverc = ((mdr_us_secondwave) + (mdr_un_secondwave)) * S ((mdr_us_secondwave) + (mdr_un_secondwave)) + ((mdr_un_secondwave) + (mdr_un_secondwave))) /\ ((mdr_c_secondwaverc = ((mdr_a_secondwaverc) + (mdr_b_secondwaverc)) * S ((mdr_a_secondwaverc) + (mdr_b_secondwaverc)) + ((mdr_b_secondwaverc) + (mdr_b_secondwaverc))) /\ ((mdr_e_secondwaverc = ((mdr_p_secondwave) + (mdr_n_secondwave)) * S ((mdr_p_secondwave) + (mdr_n_secondwave)) + ((mdr_n_secondwave) + (mdr_n_secondwave))) /\ ((mdr_f_secondwaverc = ((mdr_ut_secondwave) + (mdr_e_secondwaverc)) * S ((mdr_ut_secondwave) + (mdr_e_secondwaverc)) + ((mdr_e_secondwaverc) + (mdr_e_secondwaverc))) /\ ((mdr_z_secondwaver) = ((mdr_c_secondwaverc) + (mdr_f_secondwaverc)) * S ((mdr_c_secondwaverc) + (mdr_f_secondwaverc)) + ((mdr_f_secondwaverc) + (mdr_f_secondwaverc))))))))) /\ (((exists ff_h_mdr_secondwaverb. ff_h_mdr_secondwaverb + S (mdr_z_secondwaver) = S ((S (mdr_i_secondwave)) * c)) /\ exists ff_q_mdr_secondwaverb. b = ff_q_mdr_secondwaverb * S ((S (mdr_i_secondwave)) * c) + (mdr_z_secondwaver))))) /\ ((((forall ff_index_mdm_prefix_mdr_secondwavem_positive. (exists ff_gap_mdm_lt_mdr_secondwavem_positive_index_bound. ff_gap_mdm_lt_mdr_secondwavem_positive_index_bound + S (ff_index_mdm_prefix_mdr_secondwavem_positive) = ((q) * (q))) -> exists ff_row_mdm_prefix_mdr_secondwavem_positive ff_column_mdm_prefix_mdr_secondwavem_positive ff_value_mdm_prefix_mdr_secondwavem_positive. (ff_index_mdm_prefix_mdr_secondwavem_positive = (q) * ff_row_mdm_prefix_mdr_secondwavem_positive + ff_column_mdm_prefix_mdr_secondwavem_positive /\ ((exists ff_gap_mdm_lt_mdr_secondwavem_positive_column_bound. ff_gap_mdm_lt_mdr_secondwavem_positive_column_bound + S (ff_column_mdm_prefix_mdr_secondwavem_positive) = (q)) /\ ((exists ff_row_mdm_cell_mdr_secondwavem_positive_cell ff_column_mdm_cell_mdr_secondwavem_positive_cell. (((((exists ff_gap_mdm_lt_mdr_secondwavem_positive_cell_row_before. ff_gap_mdm_lt_mdr_secondwavem_positive_cell_row_before + S (ff_row_mdm_prefix_mdr_secondwavem_positive) = (0)) /\ ff_row_mdm_cell_mdr_secondwavem_positive_cell = ff_row_mdm_prefix_mdr_secondwavem_positive) \/ ((exists ff_gap_mdm_le_mdr_secondwavem_positive_cell_row_after. ff_gap_mdm_le_mdr_secondwavem_positive_cell_row_after + (0) = (ff_row_mdm_prefix_mdr_secondwavem_positive)) /\ ff_row_mdm_cell_mdr_secondwavem_positive_cell = S ff_row_mdm_prefix_mdr_secondwavem_positive))) /\ (((((exists ff_gap_mdm_lt_mdr_secondwavem_positive_cell_column_before. ff_gap_mdm_lt_mdr_secondwavem_positive_cell_column_before + S (ff_column_mdm_prefix_mdr_secondwavem_positive) = (mdr_j_secondwave)) /\ ff_column_mdm_cell_mdr_secondwavem_positive_cell = ff_column_mdm_prefix_mdr_secondwavem_positive) \/ ((exists ff_gap_mdm_le_mdr_secondwavem_positive_cell_column_after. ff_gap_mdm_le_mdr_secondwavem_positive_cell_column_after + (mdr_j_secondwave) = (ff_column_mdm_prefix_mdr_secondwavem_positive)) /\ ff_column_mdm_cell_mdr_secondwavem_positive_cell = S ff_column_mdm_prefix_mdr_secondwavem_positive))) /\ (((exists ff_h_mdm_mdr_secondwavem_positive_cell_source. ff_h_mdm_mdr_secondwavem_positive_cell_source + S (ff_value_mdm_prefix_mdr_secondwavem_positive) = S ((S ((ff_row_mdm_cell_mdr_secondwavem_positive_cell) * (S (q)) + (ff_column_mdm_cell_mdr_secondwavem_positive_cell))) * pc)) /\ exists ff_q_mdm_mdr_secondwavem_positive_cell_source. pb = ff_q_mdm_mdr_secondwavem_positive_cell_source * S ((S ((ff_row_mdm_cell_mdr_secondwavem_positive_cell) * (S (q)) + (ff_column_mdm_cell_mdr_secondwavem_positive_cell))) * pc) + (ff_value_mdm_prefix_mdr_secondwavem_positive)))))) /\ (((exists ff_h_mdm_mdr_secondwavem_positive_target. ff_h_mdm_mdr_secondwavem_positive_target + S (ff_value_mdm_prefix_mdr_secondwavem_positive) = S ((S (ff_index_mdm_prefix_mdr_secondwavem_positive)) * mdr_us_secondwave)) /\ exists ff_q_mdm_mdr_secondwavem_positive_target. mdr_up_secondwave = ff_q_mdm_mdr_secondwavem_positive_target * S ((S (ff_index_mdm_prefix_mdr_secondwavem_positive)) * mdr_us_secondwave) + (ff_value_mdm_prefix_mdr_secondwavem_positive))))))) /\ (forall ff_index_mdm_prefix_mdr_secondwavem_negative. (exists ff_gap_mdm_lt_mdr_secondwavem_negative_index_bound. ff_gap_mdm_lt_mdr_secondwavem_negative_index_bound + S (ff_index_mdm_prefix_mdr_secondwavem_negative) = ((q) * (q))) -> exists ff_row_mdm_prefix_mdr_secondwavem_negative ff_column_mdm_prefix_mdr_secondwavem_negative ff_value_mdm_prefix_mdr_secondwavem_negative. (ff_index_mdm_prefix_mdr_secondwavem_negative = (q) * ff_row_mdm_prefix_mdr_secondwavem_negative + ff_column_mdm_prefix_mdr_secondwavem_negative /\ ((exists ff_gap_mdm_lt_mdr_secondwavem_negative_column_bound. ff_gap_mdm_lt_mdr_secondwavem_negative_column_bound + S (ff_column_mdm_prefix_mdr_secondwavem_negative) = (q)) /\ ((exists ff_row_mdm_cell_mdr_secondwavem_negative_cell ff_column_mdm_cell_mdr_secondwavem_negative_cell. (((((exists ff_gap_mdm_lt_mdr_secondwavem_negative_cell_row_before. ff_gap_mdm_lt_mdr_secondwavem_negative_cell_row_before + S (ff_row_mdm_prefix_mdr_secondwavem_negative) = (0)) /\ ff_row_mdm_cell_mdr_secondwavem_negative_cell = ff_row_mdm_prefix_mdr_secondwavem_negative) \/ ((exists ff_gap_mdm_le_mdr_secondwavem_negative_cell_row_after. ff_gap_mdm_le_mdr_secondwavem_negative_cell_row_after + (0) = (ff_row_mdm_prefix_mdr_secondwavem_negative)) /\ ff_row_mdm_cell_mdr_secondwavem_negative_cell = S ff_row_mdm_prefix_mdr_secondwavem_negative))) /\ (((((exists ff_gap_mdm_lt_mdr_secondwavem_negative_cell_column_before. ff_gap_mdm_lt_mdr_secondwavem_negative_cell_column_before + S (ff_column_mdm_prefix_mdr_secondwavem_negative) = (mdr_j_secondwave)) /\ ff_column_mdm_cell_mdr_secondwavem_negative_cell = ff_column_mdm_prefix_mdr_secondwavem_negative) \/ ((exists ff_gap_mdm_le_mdr_secondwavem_negative_cell_column_after. ff_gap_mdm_le_mdr_secondwavem_negative_cell_column_after + (mdr_j_secondwave) = (ff_column_mdm_prefix_mdr_secondwavem_negative)) /\ ff_column_mdm_cell_mdr_secondwavem_negative_cell = S ff_column_mdm_prefix_mdr_secondwavem_negative))) /\ (((exists ff_h_mdm_mdr_secondwavem_negative_cell_source. ff_h_mdm_mdr_secondwavem_negative_cell_source + S (ff_value_mdm_prefix_mdr_secondwavem_negative) = S ((S ((ff_row_mdm_cell_mdr_secondwavem_negative_cell) * (S (q)) + (ff_column_mdm_cell_mdr_secondwavem_negative_cell))) * nc)) /\ exists ff_q_mdm_mdr_secondwavem_negative_cell_source. nb = ff_q_mdm_mdr_secondwavem_negative_cell_source * S ((S ((ff_row_mdm_cell_mdr_secondwavem_negative_cell) * (S (q)) + (ff_column_mdm_cell_mdr_secondwavem_negative_cell))) * nc) + (ff_value_mdm_prefix_mdr_secondwavem_negative)))))) /\ (((exists ff_h_mdm_mdr_secondwavem_negative_target. ff_h_mdm_mdr_secondwavem_negative_target + S (ff_value_mdm_prefix_mdr_secondwavem_negative) = S ((S (ff_index_mdm_prefix_mdr_secondwavem_negative)) * mdr_ut_secondwave)) /\ exists ff_q_mdm_mdr_secondwavem_negative_target. mdr_un_secondwave = ff_q_mdm_mdr_secondwavem_negative_target * S ((S (ff_index_mdm_prefix_mdr_secondwavem_negative)) * mdr_ut_secondwave) + (ff_value_mdm_prefix_mdr_secondwavem_negative))))))))) /\ ((((exists ff_h_mdr_secondwavep. ff_h_mdr_secondwavep + S (mdr_p_secondwave) = S ((S (mdr_j_secondwave)) * ec)) /\ exists ff_q_mdr_secondwavep. eb = ff_q_mdr_secondwavep * S ((S (mdr_j_secondwave)) * ec) + (mdr_p_secondwave))) /\ (((exists ff_h_mdr_secondwaven. ff_h_mdr_secondwaven + S (mdr_n_secondwave) = S ((S (mdr_j_secondwave)) * fc)) /\ exists ff_q_mdr_secondwaven. fb = ff_q_mdr_secondwaven * S ((S (mdr_j_secondwave)) * fc) + (mdr_n_secondwave)))))))The unchanged native kernel never receives this surface symbol. Binder-safe expansion produces only its existing first-order syntax.