signed_cofactor_minor_record_projects_minor

Exact expanded first-order arithmetic statement

forall pb pc nb nc q j z. (exists ff_up_mce_record_result ff_us_mce_record_result ff_un_mce_record_result ff_ut_mce_record_result. ((z = ((((ff_up_mce_record_result) + (ff_us_mce_record_result)) * S ((ff_up_mce_record_result) + (ff_us_mce_record_result)) + ((ff_us_mce_record_result) + (ff_us_mce_record_result))) + (((ff_un_mce_record_result) + (ff_ut_mce_record_result)) * S ((ff_un_mce_record_result) + (ff_ut_mce_record_result)) + ((ff_ut_mce_record_result) + (ff_ut_mce_record_result)))) * S ((((ff_up_mce_record_result) + (ff_us_mce_record_result)) * S ((ff_up_mce_record_result) + (ff_us_mce_record_result)) + ((ff_us_mce_record_result) + (ff_us_mce_record_result))) + (((ff_un_mce_record_result) + (ff_ut_mce_record_result)) * S ((ff_un_mce_record_result) + (ff_ut_mce_record_result)) + ((ff_ut_mce_record_result) + (ff_ut_mce_record_result)))) + ((((ff_un_mce_record_result) + (ff_ut_mce_record_result)) * S ((ff_un_mce_record_result) + (ff_ut_mce_record_result)) + ((ff_ut_mce_record_result) + (ff_ut_mce_record_result))) + (((ff_un_mce_record_result) + (ff_ut_mce_record_result)) * S ((ff_un_mce_record_result) + (ff_ut_mce_record_result)) + ((ff_ut_mce_record_result) + (ff_ut_mce_record_result))))) /\ (((forall ff_index_mdm_prefix_mce_result_minor_positive. (exists ff_gap_mdm_lt_mce_result_minor_positive_index_bound. ff_gap_mdm_lt_mce_result_minor_positive_index_bound + S (ff_index_mdm_prefix_mce_result_minor_positive) = ((q) * (q))) -> exists ff_row_mdm_prefix_mce_result_minor_positive ff_column_mdm_prefix_mce_result_minor_positive ff_value_mdm_prefix_mce_result_minor_positive. (ff_index_mdm_prefix_mce_result_minor_positive = (q) * ff_row_mdm_prefix_mce_result_minor_positive + ff_column_mdm_prefix_mce_result_minor_positive /\ ((exists ff_gap_mdm_lt_mce_result_minor_positive_column_bound. ff_gap_mdm_lt_mce_result_minor_positive_column_bound + S (ff_column_mdm_prefix_mce_result_minor_positive) = (q)) /\ ((exists ff_row_mdm_cell_mce_result_minor_positive_cell ff_column_mdm_cell_mce_result_minor_positive_cell. (((((exists ff_gap_mdm_lt_mce_result_minor_positive_cell_row_before. ff_gap_mdm_lt_mce_result_minor_positive_cell_row_before + S (ff_row_mdm_prefix_mce_result_minor_positive) = (0)) /\ ff_row_mdm_cell_mce_result_minor_positive_cell = ff_row_mdm_prefix_mce_result_minor_positive) \/ ((exists ff_gap_mdm_le_mce_result_minor_positive_cell_row_after. ff_gap_mdm_le_mce_result_minor_positive_cell_row_after + (0) = (ff_row_mdm_prefix_mce_result_minor_positive)) /\ ff_row_mdm_cell_mce_result_minor_positive_cell = S ff_row_mdm_prefix_mce_result_minor_positive))) /\ (((((exists ff_gap_mdm_lt_mce_result_minor_positive_cell_column_before. ff_gap_mdm_lt_mce_result_minor_positive_cell_column_before + S (ff_column_mdm_prefix_mce_result_minor_positive) = (j)) /\ ff_column_mdm_cell_mce_result_minor_positive_cell = ff_column_mdm_prefix_mce_result_minor_positive) \/ ((exists ff_gap_mdm_le_mce_result_minor_positive_cell_column_after. ff_gap_mdm_le_mce_result_minor_positive_cell_column_after + (j) = (ff_column_mdm_prefix_mce_result_minor_positive)) /\ ff_column_mdm_cell_mce_result_minor_positive_cell = S ff_column_mdm_prefix_mce_result_minor_positive))) /\ (((exists ff_h_mdm_mce_result_minor_positive_cell_source. ff_h_mdm_mce_result_minor_positive_cell_source + S (ff_value_mdm_prefix_mce_result_minor_positive) = S ((S ((ff_row_mdm_cell_mce_result_minor_positive_cell) * (S (q)) + (ff_column_mdm_cell_mce_result_minor_positive_cell))) * pc)) /\ exists ff_q_mdm_mce_result_minor_positive_cell_source. pb = ff_q_mdm_mce_result_minor_positive_cell_source * S ((S ((ff_row_mdm_cell_mce_result_minor_positive_cell) * (S (q)) + (ff_column_mdm_cell_mce_result_minor_positive_cell))) * pc) + (ff_value_mdm_prefix_mce_result_minor_positive)))))) /\ (((exists ff_h_mdm_mce_result_minor_positive_target. ff_h_mdm_mce_result_minor_positive_target + S (ff_value_mdm_prefix_mce_result_minor_positive) = S ((S (ff_index_mdm_prefix_mce_result_minor_positive)) * ff_us_mce_record_result)) /\ exists ff_q_mdm_mce_result_minor_positive_target. ff_up_mce_record_result = ff_q_mdm_mce_result_minor_positive_target * S ((S (ff_index_mdm_prefix_mce_result_minor_positive)) * ff_us_mce_record_result) + (ff_value_mdm_prefix_mce_result_minor_positive))))))) /\ (forall ff_index_mdm_prefix_mce_result_minor_negative. (exists ff_gap_mdm_lt_mce_result_minor_negative_index_bound. ff_gap_mdm_lt_mce_result_minor_negative_index_bound + S (ff_index_mdm_prefix_mce_result_minor_negative) = ((q) * (q))) -> exists ff_row_mdm_prefix_mce_result_minor_negative ff_column_mdm_prefix_mce_result_minor_negative ff_value_mdm_prefix_mce_result_minor_negative. (ff_index_mdm_prefix_mce_result_minor_negative = (q) * ff_row_mdm_prefix_mce_result_minor_negative + ff_column_mdm_prefix_mce_result_minor_negative /\ ((exists ff_gap_mdm_lt_mce_result_minor_negative_column_bound. ff_gap_mdm_lt_mce_result_minor_negative_column_bound + S (ff_column_mdm_prefix_mce_result_minor_negative) = (q)) /\ ((exists ff_row_mdm_cell_mce_result_minor_negative_cell ff_column_mdm_cell_mce_result_minor_negative_cell. (((((exists ff_gap_mdm_lt_mce_result_minor_negative_cell_row_before. ff_gap_mdm_lt_mce_result_minor_negative_cell_row_before + S (ff_row_mdm_prefix_mce_result_minor_negative) = (0)) /\ ff_row_mdm_cell_mce_result_minor_negative_cell = ff_row_mdm_prefix_mce_result_minor_negative) \/ ((exists ff_gap_mdm_le_mce_result_minor_negative_cell_row_after. ff_gap_mdm_le_mce_result_minor_negative_cell_row_after + (0) = (ff_row_mdm_prefix_mce_result_minor_negative)) /\ ff_row_mdm_cell_mce_result_minor_negative_cell = S ff_row_mdm_prefix_mce_result_minor_negative))) /\ (((((exists ff_gap_mdm_lt_mce_result_minor_negative_cell_column_before. ff_gap_mdm_lt_mce_result_minor_negative_cell_column_before + S (ff_column_mdm_prefix_mce_result_minor_negative) = (j)) /\ ff_column_mdm_cell_mce_result_minor_negative_cell = ff_column_mdm_prefix_mce_result_minor_negative) \/ ((exists ff_gap_mdm_le_mce_result_minor_negative_cell_column_after. ff_gap_mdm_le_mce_result_minor_negative_cell_column_after + (j) = (ff_column_mdm_prefix_mce_result_minor_negative)) /\ ff_column_mdm_cell_mce_result_minor_negative_cell = S ff_column_mdm_prefix_mce_result_minor_negative))) /\ (((exists ff_h_mdm_mce_result_minor_negative_cell_source. ff_h_mdm_mce_result_minor_negative_cell_source + S (ff_value_mdm_prefix_mce_result_minor_negative) = S ((S ((ff_row_mdm_cell_mce_result_minor_negative_cell) * (S (q)) + (ff_column_mdm_cell_mce_result_minor_negative_cell))) * nc)) /\ exists ff_q_mdm_mce_result_minor_negative_cell_source. nb = ff_q_mdm_mce_result_minor_negative_cell_source * S ((S ((ff_row_mdm_cell_mce_result_minor_negative_cell) * (S (q)) + (ff_column_mdm_cell_mce_result_minor_negative_cell))) * nc) + (ff_value_mdm_prefix_mce_result_minor_negative)))))) /\ (((exists ff_h_mdm_mce_result_minor_negative_target. ff_h_mdm_mce_result_minor_negative_target + S (ff_value_mdm_prefix_mce_result_minor_negative) = S ((S (ff_index_mdm_prefix_mce_result_minor_negative)) * ff_ut_mce_record_result)) /\ exists ff_q_mdm_mce_result_minor_negative_target. ff_un_mce_record_result = ff_q_mdm_mce_result_minor_negative_target * S ((S (ff_index_mdm_prefix_mce_result_minor_negative)) * ff_ut_mce_record_result) + (ff_value_mdm_prefix_mce_result_minor_negative))))))))))) -> exists up us un ut. (((forall ff_index_mdm_prefix_mce_record_projection_positive. (exists ff_gap_mdm_lt_mce_record_projection_positive_index_bound. ff_gap_mdm_lt_mce_record_projection_positive_index_bound + S (ff_index_mdm_prefix_mce_record_projection_positive) = ((q) * (q))) -> exists ff_row_mdm_prefix_mce_record_projection_positive ff_column_mdm_prefix_mce_record_projection_positive ff_value_mdm_prefix_mce_record_projection_positive. (ff_index_mdm_prefix_mce_record_projection_positive = (q) * ff_row_mdm_prefix_mce_record_projection_positive + ff_column_mdm_prefix_mce_record_projection_positive /\ ((exists ff_gap_mdm_lt_mce_record_projection_positive_column_bound. ff_gap_mdm_lt_mce_record_projection_positive_column_bound + S (ff_column_mdm_prefix_mce_record_projection_positive) = (q)) /\ ((exists ff_row_mdm_cell_mce_record_projection_positive_cell ff_column_mdm_cell_mce_record_projection_positive_cell. (((((exists ff_gap_mdm_lt_mce_record_projection_positive_cell_row_before. ff_gap_mdm_lt_mce_record_projection_positive_cell_row_before + S (ff_row_mdm_prefix_mce_record_projection_positive) = (0)) /\ ff_row_mdm_cell_mce_record_projection_positive_cell = ff_row_mdm_prefix_mce_record_projection_positive) \/ ((exists ff_gap_mdm_le_mce_record_projection_positive_cell_row_after. ff_gap_mdm_le_mce_record_projection_positive_cell_row_after + (0) = (ff_row_mdm_prefix_mce_record_projection_positive)) /\ ff_row_mdm_cell_mce_record_projection_positive_cell = S ff_row_mdm_prefix_mce_record_projection_positive))) /\ (((((exists ff_gap_mdm_lt_mce_record_projection_positive_cell_column_before. ff_gap_mdm_lt_mce_record_projection_positive_cell_column_before + S (ff_column_mdm_prefix_mce_record_projection_positive) = (j)) /\ ff_column_mdm_cell_mce_record_projection_positive_cell = ff_column_mdm_prefix_mce_record_projection_positive) \/ ((exists ff_gap_mdm_le_mce_record_projection_positive_cell_column_after. ff_gap_mdm_le_mce_record_projection_positive_cell_column_after + (j) = (ff_column_mdm_prefix_mce_record_projection_positive)) /\ ff_column_mdm_cell_mce_record_projection_positive_cell = S ff_column_mdm_prefix_mce_record_projection_positive))) /\ (((exists ff_h_mdm_mce_record_projection_positive_cell_source. ff_h_mdm_mce_record_projection_positive_cell_source + S (ff_value_mdm_prefix_mce_record_projection_positive) = S ((S ((ff_row_mdm_cell_mce_record_projection_positive_cell) * (S q) + (ff_column_mdm_cell_mce_record_projection_positive_cell))) * pc)) /\ exists ff_q_mdm_mce_record_projection_positive_cell_source. pb = ff_q_mdm_mce_record_projection_positive_cell_source * S ((S ((ff_row_mdm_cell_mce_record_projection_positive_cell) * (S q) + (ff_column_mdm_cell_mce_record_projection_positive_cell))) * pc) + (ff_value_mdm_prefix_mce_record_projection_positive)))))) /\ (((exists ff_h_mdm_mce_record_projection_positive_target. ff_h_mdm_mce_record_projection_positive_target + S (ff_value_mdm_prefix_mce_record_projection_positive) = S ((S (ff_index_mdm_prefix_mce_record_projection_positive)) * us)) /\ exists ff_q_mdm_mce_record_projection_positive_target. up = ff_q_mdm_mce_record_projection_positive_target * S ((S (ff_index_mdm_prefix_mce_record_projection_positive)) * us) + (ff_value_mdm_prefix_mce_record_projection_positive))))))) /\ (forall ff_index_mdm_prefix_mce_record_projection_negative. (exists ff_gap_mdm_lt_mce_record_projection_negative_index_bound. ff_gap_mdm_lt_mce_record_projection_negative_index_bound + S (ff_index_mdm_prefix_mce_record_projection_negative) = ((q) * (q))) -> exists ff_row_mdm_prefix_mce_record_projection_negative ff_column_mdm_prefix_mce_record_projection_negative ff_value_mdm_prefix_mce_record_projection_negative. (ff_index_mdm_prefix_mce_record_projection_negative = (q) * ff_row_mdm_prefix_mce_record_projection_negative + ff_column_mdm_prefix_mce_record_projection_negative /\ ((exists ff_gap_mdm_lt_mce_record_projection_negative_column_bound. ff_gap_mdm_lt_mce_record_projection_negative_column_bound + S (ff_column_mdm_prefix_mce_record_projection_negative) = (q)) /\ ((exists ff_row_mdm_cell_mce_record_projection_negative_cell ff_column_mdm_cell_mce_record_projection_negative_cell. (((((exists ff_gap_mdm_lt_mce_record_projection_negative_cell_row_before. ff_gap_mdm_lt_mce_record_projection_negative_cell_row_before + S (ff_row_mdm_prefix_mce_record_projection_negative) = (0)) /\ ff_row_mdm_cell_mce_record_projection_negative_cell = ff_row_mdm_prefix_mce_record_projection_negative) \/ ((exists ff_gap_mdm_le_mce_record_projection_negative_cell_row_after. ff_gap_mdm_le_mce_record_projection_negative_cell_row_after + (0) = (ff_row_mdm_prefix_mce_record_projection_negative)) /\ ff_row_mdm_cell_mce_record_projection_negative_cell = S ff_row_mdm_prefix_mce_record_projection_negative))) /\ (((((exists ff_gap_mdm_lt_mce_record_projection_negative_cell_column_before. ff_gap_mdm_lt_mce_record_projection_negative_cell_column_before + S (ff_column_mdm_prefix_mce_record_projection_negative) = (j)) /\ ff_column_mdm_cell_mce_record_projection_negative_cell = ff_column_mdm_prefix_mce_record_projection_negative) \/ ((exists ff_gap_mdm_le_mce_record_projection_negative_cell_column_after. ff_gap_mdm_le_mce_record_projection_negative_cell_column_after + (j) = (ff_column_mdm_prefix_mce_record_projection_negative)) /\ ff_column_mdm_cell_mce_record_projection_negative_cell = S ff_column_mdm_prefix_mce_record_projection_negative))) /\ (((exists ff_h_mdm_mce_record_projection_negative_cell_source. ff_h_mdm_mce_record_projection_negative_cell_source + S (ff_value_mdm_prefix_mce_record_projection_negative) = S ((S ((ff_row_mdm_cell_mce_record_projection_negative_cell) * (S q) + (ff_column_mdm_cell_mce_record_projection_negative_cell))) * nc)) /\ exists ff_q_mdm_mce_record_projection_negative_cell_source. nb = ff_q_mdm_mce_record_projection_negative_cell_source * S ((S ((ff_row_mdm_cell_mce_record_projection_negative_cell) * (S q) + (ff_column_mdm_cell_mce_record_projection_negative_cell))) * nc) + (ff_value_mdm_prefix_mce_record_projection_negative)))))) /\ (((exists ff_h_mdm_mce_record_projection_negative_target. ff_h_mdm_mce_record_projection_negative_target + S (ff_value_mdm_prefix_mce_record_projection_negative) = S ((S (ff_index_mdm_prefix_mce_record_projection_negative)) * ut)) /\ exists ff_q_mdm_mce_record_projection_negative_target. un = ff_q_mdm_mce_record_projection_negative_target * S ((S (ff_index_mdm_prefix_mce_record_projection_negative)) * ut) + (ff_value_mdm_prefix_mce_record_projection_negative)))))))))