ND0108

SignedDeterminantHistory(b,c,l)

Every node in an actual finite history satisfies its genuine strict-child determinant rule; cyclic or supplied-value evaluations are excluded.

Conservative notation; not a theorem, primitive, or axiom.

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_i_secondwave. Lt(mdr_i_secondwave,l) → ∃ x. ∃ y. ∃ z. ∃ n. ∃ m. ∃ k. ∃ i. SignedDeterminantNodeAt(b,c,mdr_i_secondwave,x,y,z,n,m,k,i)SignedDeterminantLocalStep(b,c,mdr_i_secondwave,x,y,z,n,m,k,i)

Only definitions earlier in this acyclic notation graph are used here.

Hygienic expanded first-order definition
forall mdr_i_secondwave. (exists mdr_gap_secondwavei. mdr_gap_secondwavei + S (mdr_i_secondwave) = (l)) -> exists mdr_d_secondwave mdr_pb_secondwave mdr_pc_secondwave mdr_nb_secondwave mdr_nc_secondwave mdr_p_secondwave mdr_n_secondwave. ((exists mdr_z_secondwaver. ((exists mdr_a_secondwaverc mdr_b_secondwaverc mdr_c_secondwaverc mdr_e_secondwaverc mdr_f_secondwaverc. ((mdr_a_secondwaverc = ((mdr_d_secondwave) + (mdr_pb_secondwave)) * S ((mdr_d_secondwave) + (mdr_pb_secondwave)) + ((mdr_pb_secondwave) + (mdr_pb_secondwave))) /\ ((mdr_b_secondwaverc = ((mdr_pc_secondwave) + (mdr_nb_secondwave)) * S ((mdr_pc_secondwave) + (mdr_nb_secondwave)) + ((mdr_nb_secondwave) + (mdr_nb_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_nc_secondwave) + (mdr_e_secondwaverc)) * S ((mdr_nc_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))))) /\ (((((mdr_d_secondwave) = 0) /\ (((mdr_p_secondwave) = 1) /\ ((mdr_n_secondwave) = 0))) \/ exists mdr_q_secondwaves mdr_eb_secondwaves mdr_ec_secondwaves mdr_fb_secondwaves mdr_fc_secondwaves. (((mdr_d_secondwave) = S (mdr_q_secondwaves)) /\ ((forall mdr_j_secondwavesc. (exists mdr_gap_secondwavescj. mdr_gap_secondwavescj + S (mdr_j_secondwavesc) = (S (mdr_q_secondwaves))) -> exists mdr_i_secondwavesc mdr_up_secondwavesc mdr_us_secondwavesc mdr_un_secondwavesc mdr_ut_secondwavesc mdr_p_secondwavesc mdr_n_secondwavesc. ((exists mdr_gap_secondwavesci. mdr_gap_secondwavesci + S (mdr_i_secondwavesc) = (mdr_i_secondwave)) /\ ((exists mdr_z_secondwavescr. ((exists mdr_a_secondwavescrc mdr_b_secondwavescrc mdr_c_secondwavescrc mdr_e_secondwavescrc mdr_f_secondwavescrc. ((mdr_a_secondwavescrc = ((mdr_q_secondwaves) + (mdr_up_secondwavesc)) * S ((mdr_q_secondwaves) + (mdr_up_secondwavesc)) + ((mdr_up_secondwavesc) + (mdr_up_secondwavesc))) /\ ((mdr_b_secondwavescrc = ((mdr_us_secondwavesc) + (mdr_un_secondwavesc)) * S ((mdr_us_secondwavesc) + (mdr_un_secondwavesc)) + ((mdr_un_secondwavesc) + (mdr_un_secondwavesc))) /\ ((mdr_c_secondwavescrc = ((mdr_a_secondwavescrc) + (mdr_b_secondwavescrc)) * S ((mdr_a_secondwavescrc) + (mdr_b_secondwavescrc)) + ((mdr_b_secondwavescrc) + (mdr_b_secondwavescrc))) /\ ((mdr_e_secondwavescrc = ((mdr_p_secondwavesc) + (mdr_n_secondwavesc)) * S ((mdr_p_secondwavesc) + (mdr_n_secondwavesc)) + ((mdr_n_secondwavesc) + (mdr_n_secondwavesc))) /\ ((mdr_f_secondwavescrc = ((mdr_ut_secondwavesc) + (mdr_e_secondwavescrc)) * S ((mdr_ut_secondwavesc) + (mdr_e_secondwavescrc)) + ((mdr_e_secondwavescrc) + (mdr_e_secondwavescrc))) /\ ((mdr_z_secondwavescr) = ((mdr_c_secondwavescrc) + (mdr_f_secondwavescrc)) * S ((mdr_c_secondwavescrc) + (mdr_f_secondwavescrc)) + ((mdr_f_secondwavescrc) + (mdr_f_secondwavescrc))))))))) /\ (((exists ff_h_mdr_secondwavescrb. ff_h_mdr_secondwavescrb + S (mdr_z_secondwavescr) = S ((S (mdr_i_secondwavesc)) * c)) /\ exists ff_q_mdr_secondwavescrb. b = ff_q_mdr_secondwavescrb * S ((S (mdr_i_secondwavesc)) * c) + (mdr_z_secondwavescr))))) /\ ((((forall ff_index_mdm_prefix_mdr_secondwavescm_positive. (exists ff_gap_mdm_lt_mdr_secondwavescm_positive_index_bound. ff_gap_mdm_lt_mdr_secondwavescm_positive_index_bound + S (ff_index_mdm_prefix_mdr_secondwavescm_positive) = ((mdr_q_secondwaves) * (mdr_q_secondwaves))) -> exists ff_row_mdm_prefix_mdr_secondwavescm_positive ff_column_mdm_prefix_mdr_secondwavescm_positive ff_value_mdm_prefix_mdr_secondwavescm_positive. (ff_index_mdm_prefix_mdr_secondwavescm_positive = (mdr_q_secondwaves) * ff_row_mdm_prefix_mdr_secondwavescm_positive + ff_column_mdm_prefix_mdr_secondwavescm_positive /\ ((exists ff_gap_mdm_lt_mdr_secondwavescm_positive_column_bound. ff_gap_mdm_lt_mdr_secondwavescm_positive_column_bound + S (ff_column_mdm_prefix_mdr_secondwavescm_positive) = (mdr_q_secondwaves)) /\ ((exists ff_row_mdm_cell_mdr_secondwavescm_positive_cell ff_column_mdm_cell_mdr_secondwavescm_positive_cell. (((((exists ff_gap_mdm_lt_mdr_secondwavescm_positive_cell_row_before. ff_gap_mdm_lt_mdr_secondwavescm_positive_cell_row_before + S (ff_row_mdm_prefix_mdr_secondwavescm_positive) = (0)) /\ ff_row_mdm_cell_mdr_secondwavescm_positive_cell = ff_row_mdm_prefix_mdr_secondwavescm_positive) \/ ((exists ff_gap_mdm_le_mdr_secondwavescm_positive_cell_row_after. ff_gap_mdm_le_mdr_secondwavescm_positive_cell_row_after + (0) = (ff_row_mdm_prefix_mdr_secondwavescm_positive)) /\ ff_row_mdm_cell_mdr_secondwavescm_positive_cell = S ff_row_mdm_prefix_mdr_secondwavescm_positive))) /\ (((((exists ff_gap_mdm_lt_mdr_secondwavescm_positive_cell_column_before. ff_gap_mdm_lt_mdr_secondwavescm_positive_cell_column_before + S (ff_column_mdm_prefix_mdr_secondwavescm_positive) = (mdr_j_secondwavesc)) /\ ff_column_mdm_cell_mdr_secondwavescm_positive_cell = ff_column_mdm_prefix_mdr_secondwavescm_positive) \/ ((exists ff_gap_mdm_le_mdr_secondwavescm_positive_cell_column_after. ff_gap_mdm_le_mdr_secondwavescm_positive_cell_column_after + (mdr_j_secondwavesc) = (ff_column_mdm_prefix_mdr_secondwavescm_positive)) /\ ff_column_mdm_cell_mdr_secondwavescm_positive_cell = S ff_column_mdm_prefix_mdr_secondwavescm_positive))) /\ (((exists ff_h_mdm_mdr_secondwavescm_positive_cell_source. ff_h_mdm_mdr_secondwavescm_positive_cell_source + S (ff_value_mdm_prefix_mdr_secondwavescm_positive) = S ((S ((ff_row_mdm_cell_mdr_secondwavescm_positive_cell) * (S (mdr_q_secondwaves)) + (ff_column_mdm_cell_mdr_secondwavescm_positive_cell))) * mdr_pc_secondwave)) /\ exists ff_q_mdm_mdr_secondwavescm_positive_cell_source. mdr_pb_secondwave = ff_q_mdm_mdr_secondwavescm_positive_cell_source * S ((S ((ff_row_mdm_cell_mdr_secondwavescm_positive_cell) * (S (mdr_q_secondwaves)) + (ff_column_mdm_cell_mdr_secondwavescm_positive_cell))) * mdr_pc_secondwave) + (ff_value_mdm_prefix_mdr_secondwavescm_positive)))))) /\ (((exists ff_h_mdm_mdr_secondwavescm_positive_target. ff_h_mdm_mdr_secondwavescm_positive_target + S (ff_value_mdm_prefix_mdr_secondwavescm_positive) = S ((S (ff_index_mdm_prefix_mdr_secondwavescm_positive)) * mdr_us_secondwavesc)) /\ exists ff_q_mdm_mdr_secondwavescm_positive_target. mdr_up_secondwavesc = ff_q_mdm_mdr_secondwavescm_positive_target * S ((S (ff_index_mdm_prefix_mdr_secondwavescm_positive)) * mdr_us_secondwavesc) + (ff_value_mdm_prefix_mdr_secondwavescm_positive))))))) /\ (forall ff_index_mdm_prefix_mdr_secondwavescm_negative. (exists ff_gap_mdm_lt_mdr_secondwavescm_negative_index_bound. ff_gap_mdm_lt_mdr_secondwavescm_negative_index_bound + S (ff_index_mdm_prefix_mdr_secondwavescm_negative) = ((mdr_q_secondwaves) * (mdr_q_secondwaves))) -> exists ff_row_mdm_prefix_mdr_secondwavescm_negative ff_column_mdm_prefix_mdr_secondwavescm_negative ff_value_mdm_prefix_mdr_secondwavescm_negative. (ff_index_mdm_prefix_mdr_secondwavescm_negative = (mdr_q_secondwaves) * ff_row_mdm_prefix_mdr_secondwavescm_negative + ff_column_mdm_prefix_mdr_secondwavescm_negative /\ ((exists ff_gap_mdm_lt_mdr_secondwavescm_negative_column_bound. ff_gap_mdm_lt_mdr_secondwavescm_negative_column_bound + S (ff_column_mdm_prefix_mdr_secondwavescm_negative) = (mdr_q_secondwaves)) /\ ((exists ff_row_mdm_cell_mdr_secondwavescm_negative_cell ff_column_mdm_cell_mdr_secondwavescm_negative_cell. (((((exists ff_gap_mdm_lt_mdr_secondwavescm_negative_cell_row_before. ff_gap_mdm_lt_mdr_secondwavescm_negative_cell_row_before + S (ff_row_mdm_prefix_mdr_secondwavescm_negative) = (0)) /\ ff_row_mdm_cell_mdr_secondwavescm_negative_cell = ff_row_mdm_prefix_mdr_secondwavescm_negative) \/ ((exists ff_gap_mdm_le_mdr_secondwavescm_negative_cell_row_after. ff_gap_mdm_le_mdr_secondwavescm_negative_cell_row_after + (0) = (ff_row_mdm_prefix_mdr_secondwavescm_negative)) /\ ff_row_mdm_cell_mdr_secondwavescm_negative_cell = S ff_row_mdm_prefix_mdr_secondwavescm_negative))) /\ (((((exists ff_gap_mdm_lt_mdr_secondwavescm_negative_cell_column_before. ff_gap_mdm_lt_mdr_secondwavescm_negative_cell_column_before + S (ff_column_mdm_prefix_mdr_secondwavescm_negative) = (mdr_j_secondwavesc)) /\ ff_column_mdm_cell_mdr_secondwavescm_negative_cell = ff_column_mdm_prefix_mdr_secondwavescm_negative) \/ ((exists ff_gap_mdm_le_mdr_secondwavescm_negative_cell_column_after. ff_gap_mdm_le_mdr_secondwavescm_negative_cell_column_after + (mdr_j_secondwavesc) = (ff_column_mdm_prefix_mdr_secondwavescm_negative)) /\ ff_column_mdm_cell_mdr_secondwavescm_negative_cell = S ff_column_mdm_prefix_mdr_secondwavescm_negative))) /\ (((exists ff_h_mdm_mdr_secondwavescm_negative_cell_source. ff_h_mdm_mdr_secondwavescm_negative_cell_source + S (ff_value_mdm_prefix_mdr_secondwavescm_negative) = S ((S ((ff_row_mdm_cell_mdr_secondwavescm_negative_cell) * (S (mdr_q_secondwaves)) + (ff_column_mdm_cell_mdr_secondwavescm_negative_cell))) * mdr_nc_secondwave)) /\ exists ff_q_mdm_mdr_secondwavescm_negative_cell_source. mdr_nb_secondwave = ff_q_mdm_mdr_secondwavescm_negative_cell_source * S ((S ((ff_row_mdm_cell_mdr_secondwavescm_negative_cell) * (S (mdr_q_secondwaves)) + (ff_column_mdm_cell_mdr_secondwavescm_negative_cell))) * mdr_nc_secondwave) + (ff_value_mdm_prefix_mdr_secondwavescm_negative)))))) /\ (((exists ff_h_mdm_mdr_secondwavescm_negative_target. ff_h_mdm_mdr_secondwavescm_negative_target + S (ff_value_mdm_prefix_mdr_secondwavescm_negative) = S ((S (ff_index_mdm_prefix_mdr_secondwavescm_negative)) * mdr_ut_secondwavesc)) /\ exists ff_q_mdm_mdr_secondwavescm_negative_target. mdr_un_secondwavesc = ff_q_mdm_mdr_secondwavescm_negative_target * S ((S (ff_index_mdm_prefix_mdr_secondwavescm_negative)) * mdr_ut_secondwavesc) + (ff_value_mdm_prefix_mdr_secondwavescm_negative))))))))) /\ ((((exists ff_h_mdr_secondwavescp. ff_h_mdr_secondwavescp + S (mdr_p_secondwavesc) = S ((S (mdr_j_secondwavesc)) * mdr_ec_secondwaves)) /\ exists ff_q_mdr_secondwavescp. mdr_eb_secondwaves = ff_q_mdr_secondwavescp * S ((S (mdr_j_secondwavesc)) * mdr_ec_secondwaves) + (mdr_p_secondwavesc))) /\ (((exists ff_h_mdr_secondwavescn. ff_h_mdr_secondwavescn + S (mdr_n_secondwavesc) = S ((S (mdr_j_secondwavesc)) * mdr_fc_secondwaves)) /\ exists ff_q_mdr_secondwavescn. mdr_fb_secondwaves = ff_q_mdr_secondwavescn * S ((S (mdr_j_secondwavesc)) * mdr_fc_secondwaves) + (mdr_n_secondwavesc)))))))) /\ (exists ff_ub_mce_fold_mdr_secondwavesf ff_uc_mce_fold_mdr_secondwavesf ff_vb_mce_fold_mdr_secondwavesf ff_vc_mce_fold_mdr_secondwavesf. ((forall ff_index_mce_alternating_mdr_secondwavesf_prefix. (exists ff_gap_mce_mdr_secondwavesf_prefix_index. ff_gap_mce_mdr_secondwavesf_prefix_index + S (ff_index_mce_alternating_mdr_secondwavesf_prefix) = (S (mdr_q_secondwaves))) -> exists ff_ap_mce_alternating_mdr_secondwavesf_prefix ff_an_mce_alternating_mdr_secondwavesf_prefix ff_bp_mce_alternating_mdr_secondwavesf_prefix ff_bn_mce_alternating_mdr_secondwavesf_prefix ff_p_mce_alternating_mdr_secondwavesf_prefix ff_n_mce_alternating_mdr_secondwavesf_prefix. ((((exists ff_h_mce_mdr_secondwavesf_prefix_ap. ff_h_mce_mdr_secondwavesf_prefix_ap + S (ff_ap_mce_alternating_mdr_secondwavesf_prefix) = S ((S (ff_index_mce_alternating_mdr_secondwavesf_prefix)) * mdr_pc_secondwave)) /\ exists ff_q_mce_mdr_secondwavesf_prefix_ap. mdr_pb_secondwave = ff_q_mce_mdr_secondwavesf_prefix_ap * S ((S (ff_index_mce_alternating_mdr_secondwavesf_prefix)) * mdr_pc_secondwave) + (ff_ap_mce_alternating_mdr_secondwavesf_prefix))) /\ ((((exists ff_h_mce_mdr_secondwavesf_prefix_an. ff_h_mce_mdr_secondwavesf_prefix_an + S (ff_an_mce_alternating_mdr_secondwavesf_prefix) = S ((S (ff_index_mce_alternating_mdr_secondwavesf_prefix)) * mdr_nc_secondwave)) /\ exists ff_q_mce_mdr_secondwavesf_prefix_an. mdr_nb_secondwave = ff_q_mce_mdr_secondwavesf_prefix_an * S ((S (ff_index_mce_alternating_mdr_secondwavesf_prefix)) * mdr_nc_secondwave) + (ff_an_mce_alternating_mdr_secondwavesf_prefix))) /\ ((((exists ff_h_mce_mdr_secondwavesf_prefix_bp. ff_h_mce_mdr_secondwavesf_prefix_bp + S (ff_bp_mce_alternating_mdr_secondwavesf_prefix) = S ((S (ff_index_mce_alternating_mdr_secondwavesf_prefix)) * mdr_ec_secondwaves)) /\ exists ff_q_mce_mdr_secondwavesf_prefix_bp. mdr_eb_secondwaves = ff_q_mce_mdr_secondwavesf_prefix_bp * S ((S (ff_index_mce_alternating_mdr_secondwavesf_prefix)) * mdr_ec_secondwaves) + (ff_bp_mce_alternating_mdr_secondwavesf_prefix))) /\ ((((exists ff_h_mce_mdr_secondwavesf_prefix_bn. ff_h_mce_mdr_secondwavesf_prefix_bn + S (ff_bn_mce_alternating_mdr_secondwavesf_prefix) = S ((S (ff_index_mce_alternating_mdr_secondwavesf_prefix)) * mdr_fc_secondwaves)) /\ exists ff_q_mce_mdr_secondwavesf_prefix_bn. mdr_fb_secondwaves = ff_q_mce_mdr_secondwavesf_prefix_bn * S ((S (ff_index_mce_alternating_mdr_secondwavesf_prefix)) * mdr_fc_secondwaves) + (ff_bn_mce_alternating_mdr_secondwavesf_prefix))) /\ ((((exists ff_h_mce_mdr_secondwavesf_prefix_positive. ff_h_mce_mdr_secondwavesf_prefix_positive + S (ff_p_mce_alternating_mdr_secondwavesf_prefix) = S ((S (ff_index_mce_alternating_mdr_secondwavesf_prefix)) * ff_uc_mce_fold_mdr_secondwavesf)) /\ exists ff_q_mce_mdr_secondwavesf_prefix_positive. ff_ub_mce_fold_mdr_secondwavesf = ff_q_mce_mdr_secondwavesf_prefix_positive * S ((S (ff_index_mce_alternating_mdr_secondwavesf_prefix)) * ff_uc_mce_fold_mdr_secondwavesf) + (ff_p_mce_alternating_mdr_secondwavesf_prefix))) /\ ((((exists ff_h_mce_mdr_secondwavesf_prefix_negative. ff_h_mce_mdr_secondwavesf_prefix_negative + S (ff_n_mce_alternating_mdr_secondwavesf_prefix) = S ((S (ff_index_mce_alternating_mdr_secondwavesf_prefix)) * ff_vc_mce_fold_mdr_secondwavesf)) /\ exists ff_q_mce_mdr_secondwavesf_prefix_negative. ff_vb_mce_fold_mdr_secondwavesf = ff_q_mce_mdr_secondwavesf_prefix_negative * S ((S (ff_index_mce_alternating_mdr_secondwavesf_prefix)) * ff_vc_mce_fold_mdr_secondwavesf) + (ff_n_mce_alternating_mdr_secondwavesf_prefix))) /\ (((exists ff_even_mce_term_mdr_secondwavesf_prefix_term. ff_index_mce_alternating_mdr_secondwavesf_prefix = 2 * ff_even_mce_term_mdr_secondwavesf_prefix_term) /\ (ff_p_mce_alternating_mdr_secondwavesf_prefix = (ff_ap_mce_alternating_mdr_secondwavesf_prefix) * (ff_bp_mce_alternating_mdr_secondwavesf_prefix) + (ff_an_mce_alternating_mdr_secondwavesf_prefix) * (ff_bn_mce_alternating_mdr_secondwavesf_prefix) /\ ff_n_mce_alternating_mdr_secondwavesf_prefix = (ff_ap_mce_alternating_mdr_secondwavesf_prefix) * (ff_bn_mce_alternating_mdr_secondwavesf_prefix) + (ff_an_mce_alternating_mdr_secondwavesf_prefix) * (ff_bp_mce_alternating_mdr_secondwavesf_prefix))) \/ ((exists ff_odd_mce_term_mdr_secondwavesf_prefix_term. ff_index_mce_alternating_mdr_secondwavesf_prefix = 2 * ff_odd_mce_term_mdr_secondwavesf_prefix_term + 1) /\ (ff_p_mce_alternating_mdr_secondwavesf_prefix = (ff_ap_mce_alternating_mdr_secondwavesf_prefix) * (ff_bn_mce_alternating_mdr_secondwavesf_prefix) + (ff_an_mce_alternating_mdr_secondwavesf_prefix) * (ff_bp_mce_alternating_mdr_secondwavesf_prefix) /\ ff_n_mce_alternating_mdr_secondwavesf_prefix = (ff_ap_mce_alternating_mdr_secondwavesf_prefix) * (ff_bp_mce_alternating_mdr_secondwavesf_prefix) + (ff_an_mce_alternating_mdr_secondwavesf_prefix) * (ff_bn_mce_alternating_mdr_secondwavesf_prefix))))))))))) /\ ((exists ff_u_mce_mdr_secondwavesf_positive ff_v_mce_mdr_secondwavesf_positive. ((((exists ff_h_mce_mdr_secondwavesf_positive_start. ff_h_mce_mdr_secondwavesf_positive_start + S (0) = S ((S (0)) * ff_v_mce_mdr_secondwavesf_positive)) /\ exists ff_q_mce_mdr_secondwavesf_positive_start. ff_u_mce_mdr_secondwavesf_positive = ff_q_mce_mdr_secondwavesf_positive_start * S ((S (0)) * ff_v_mce_mdr_secondwavesf_positive) + (0))) /\ ((((exists ff_h_mce_mdr_secondwavesf_positive_terminal. ff_h_mce_mdr_secondwavesf_positive_terminal + S (mdr_p_secondwave) = S ((S ((S (mdr_q_secondwaves)))) * ff_v_mce_mdr_secondwavesf_positive)) /\ exists ff_q_mce_mdr_secondwavesf_positive_terminal. ff_u_mce_mdr_secondwavesf_positive = ff_q_mce_mdr_secondwavesf_positive_terminal * S ((S ((S (mdr_q_secondwaves)))) * ff_v_mce_mdr_secondwavesf_positive) + (mdr_p_secondwave))) /\ forall ff_i_mce_mdr_secondwavesf_positive. (exists ff_lt_mce_mdr_secondwavesf_positive_bound. ff_lt_mce_mdr_secondwavesf_positive_bound + S ff_i_mce_mdr_secondwavesf_positive = (S (mdr_q_secondwaves))) -> exists ff_a_mce_mdr_secondwavesf_positive ff_r_mce_mdr_secondwavesf_positive ff_s_mce_mdr_secondwavesf_positive. ((((exists ff_h_mce_mdr_secondwavesf_positive_summand. ff_h_mce_mdr_secondwavesf_positive_summand + S (ff_a_mce_mdr_secondwavesf_positive) = S ((S (ff_i_mce_mdr_secondwavesf_positive)) * ff_uc_mce_fold_mdr_secondwavesf)) /\ exists ff_q_mce_mdr_secondwavesf_positive_summand. ff_ub_mce_fold_mdr_secondwavesf = ff_q_mce_mdr_secondwavesf_positive_summand * S ((S (ff_i_mce_mdr_secondwavesf_positive)) * ff_uc_mce_fold_mdr_secondwavesf) + (ff_a_mce_mdr_secondwavesf_positive))) /\ ((((exists ff_h_mce_mdr_secondwavesf_positive_partial. ff_h_mce_mdr_secondwavesf_positive_partial + S (ff_r_mce_mdr_secondwavesf_positive) = S ((S (ff_i_mce_mdr_secondwavesf_positive)) * ff_v_mce_mdr_secondwavesf_positive)) /\ exists ff_q_mce_mdr_secondwavesf_positive_partial. ff_u_mce_mdr_secondwavesf_positive = ff_q_mce_mdr_secondwavesf_positive_partial * S ((S (ff_i_mce_mdr_secondwavesf_positive)) * ff_v_mce_mdr_secondwavesf_positive) + (ff_r_mce_mdr_secondwavesf_positive))) /\ ((((exists ff_h_mce_mdr_secondwavesf_positive_successor. ff_h_mce_mdr_secondwavesf_positive_successor + S (ff_s_mce_mdr_secondwavesf_positive) = S ((S (S ff_i_mce_mdr_secondwavesf_positive)) * ff_v_mce_mdr_secondwavesf_positive)) /\ exists ff_q_mce_mdr_secondwavesf_positive_successor. ff_u_mce_mdr_secondwavesf_positive = ff_q_mce_mdr_secondwavesf_positive_successor * S ((S (S ff_i_mce_mdr_secondwavesf_positive)) * ff_v_mce_mdr_secondwavesf_positive) + (ff_s_mce_mdr_secondwavesf_positive))) /\ ff_s_mce_mdr_secondwavesf_positive = ff_r_mce_mdr_secondwavesf_positive + ff_a_mce_mdr_secondwavesf_positive)))))) /\ (exists ff_u_mce_mdr_secondwavesf_negative ff_v_mce_mdr_secondwavesf_negative. ((((exists ff_h_mce_mdr_secondwavesf_negative_start. ff_h_mce_mdr_secondwavesf_negative_start + S (0) = S ((S (0)) * ff_v_mce_mdr_secondwavesf_negative)) /\ exists ff_q_mce_mdr_secondwavesf_negative_start. ff_u_mce_mdr_secondwavesf_negative = ff_q_mce_mdr_secondwavesf_negative_start * S ((S (0)) * ff_v_mce_mdr_secondwavesf_negative) + (0))) /\ ((((exists ff_h_mce_mdr_secondwavesf_negative_terminal. ff_h_mce_mdr_secondwavesf_negative_terminal + S (mdr_n_secondwave) = S ((S ((S (mdr_q_secondwaves)))) * ff_v_mce_mdr_secondwavesf_negative)) /\ exists ff_q_mce_mdr_secondwavesf_negative_terminal. ff_u_mce_mdr_secondwavesf_negative = ff_q_mce_mdr_secondwavesf_negative_terminal * S ((S ((S (mdr_q_secondwaves)))) * ff_v_mce_mdr_secondwavesf_negative) + (mdr_n_secondwave))) /\ forall ff_i_mce_mdr_secondwavesf_negative. (exists ff_lt_mce_mdr_secondwavesf_negative_bound. ff_lt_mce_mdr_secondwavesf_negative_bound + S ff_i_mce_mdr_secondwavesf_negative = (S (mdr_q_secondwaves))) -> exists ff_a_mce_mdr_secondwavesf_negative ff_r_mce_mdr_secondwavesf_negative ff_s_mce_mdr_secondwavesf_negative. ((((exists ff_h_mce_mdr_secondwavesf_negative_summand. ff_h_mce_mdr_secondwavesf_negative_summand + S (ff_a_mce_mdr_secondwavesf_negative) = S ((S (ff_i_mce_mdr_secondwavesf_negative)) * ff_vc_mce_fold_mdr_secondwavesf)) /\ exists ff_q_mce_mdr_secondwavesf_negative_summand. ff_vb_mce_fold_mdr_secondwavesf = ff_q_mce_mdr_secondwavesf_negative_summand * S ((S (ff_i_mce_mdr_secondwavesf_negative)) * ff_vc_mce_fold_mdr_secondwavesf) + (ff_a_mce_mdr_secondwavesf_negative))) /\ ((((exists ff_h_mce_mdr_secondwavesf_negative_partial. ff_h_mce_mdr_secondwavesf_negative_partial + S (ff_r_mce_mdr_secondwavesf_negative) = S ((S (ff_i_mce_mdr_secondwavesf_negative)) * ff_v_mce_mdr_secondwavesf_negative)) /\ exists ff_q_mce_mdr_secondwavesf_negative_partial. ff_u_mce_mdr_secondwavesf_negative = ff_q_mce_mdr_secondwavesf_negative_partial * S ((S (ff_i_mce_mdr_secondwavesf_negative)) * ff_v_mce_mdr_secondwavesf_negative) + (ff_r_mce_mdr_secondwavesf_negative))) /\ ((((exists ff_h_mce_mdr_secondwavesf_negative_successor. ff_h_mce_mdr_secondwavesf_negative_successor + S (ff_s_mce_mdr_secondwavesf_negative) = S ((S (S ff_i_mce_mdr_secondwavesf_negative)) * ff_v_mce_mdr_secondwavesf_negative)) /\ exists ff_q_mce_mdr_secondwavesf_negative_successor. ff_u_mce_mdr_secondwavesf_negative = ff_q_mce_mdr_secondwavesf_negative_successor * S ((S (S ff_i_mce_mdr_secondwavesf_negative)) * ff_v_mce_mdr_secondwavesf_negative) + (ff_s_mce_mdr_secondwavesf_negative))) /\ ff_s_mce_mdr_secondwavesf_negative = ff_r_mce_mdr_secondwavesf_negative + ff_a_mce_mdr_secondwavesf_negative))))))))))))))

The unchanged native kernel never receives this surface symbol. Binder-safe expansion produces only its existing first-order syntax.

Direct definition dependencies

Definitions depending on this notation

Checked theorems using this definition