lucas_multidigit_congruence

Exact expanded first-order arithmetic statement

forall p n k l qb qc db dc ub uc vb vc z t s w P C T. ((~(p = 1) /\ forall frm_prime_left_lmd_full_prime frm_prime_right_lmd_full_prime. p = frm_prime_left_lmd_full_prime * frm_prime_right_lmd_full_prime -> frm_prime_left_lmd_full_prime = 1 \/ frm_prime_right_lmd_full_prime = 1)) -> (((((exists ff_h_lmd_full_n_initial. ff_h_lmd_full_n_initial + S (n) = S ((S (0)) * qc)) /\ exists ff_q_lmd_full_n_initial. qb = ff_q_lmd_full_n_initial * S ((S (0)) * qc) + (n))) /\ forall lmd_index_full_n. (exists lmd_gap_full_n_index. lmd_gap_full_n_index + S (lmd_index_full_n) = (l)) -> exists lmd_current_full_n lmd_successor_full_n lmd_digit_full_n. ((((exists ff_h_lmd_full_n_current. ff_h_lmd_full_n_current + S (lmd_current_full_n) = S ((S (lmd_index_full_n)) * qc)) /\ exists ff_q_lmd_full_n_current. qb = ff_q_lmd_full_n_current * S ((S (lmd_index_full_n)) * qc) + (lmd_current_full_n))) /\ ((((exists ff_h_lmd_full_n_successor. ff_h_lmd_full_n_successor + S (lmd_successor_full_n) = S ((S (S lmd_index_full_n)) * qc)) /\ exists ff_q_lmd_full_n_successor. qb = ff_q_lmd_full_n_successor * S ((S (S lmd_index_full_n)) * qc) + (lmd_successor_full_n))) /\ ((((exists ff_h_lmd_full_n_digit. ff_h_lmd_full_n_digit + S (lmd_digit_full_n) = S ((S (lmd_index_full_n)) * dc)) /\ exists ff_q_lmd_full_n_digit. db = ff_q_lmd_full_n_digit * S ((S (lmd_index_full_n)) * dc) + (lmd_digit_full_n))) /\ ((lmd_current_full_n = (p) * (lmd_successor_full_n) + (lmd_digit_full_n)) /\ (exists lmd_gap_full_n_digit_bound. lmd_gap_full_n_digit_bound + S (lmd_digit_full_n) = (p)))))))) -> (((((exists ff_h_lmd_full_k_initial. ff_h_lmd_full_k_initial + S (k) = S ((S (0)) * uc)) /\ exists ff_q_lmd_full_k_initial. ub = ff_q_lmd_full_k_initial * S ((S (0)) * uc) + (k))) /\ forall lmd_index_full_k. (exists lmd_gap_full_k_index. lmd_gap_full_k_index + S (lmd_index_full_k) = (l)) -> exists lmd_current_full_k lmd_successor_full_k lmd_digit_full_k. ((((exists ff_h_lmd_full_k_current. ff_h_lmd_full_k_current + S (lmd_current_full_k) = S ((S (lmd_index_full_k)) * uc)) /\ exists ff_q_lmd_full_k_current. ub = ff_q_lmd_full_k_current * S ((S (lmd_index_full_k)) * uc) + (lmd_current_full_k))) /\ ((((exists ff_h_lmd_full_k_successor. ff_h_lmd_full_k_successor + S (lmd_successor_full_k) = S ((S (S lmd_index_full_k)) * uc)) /\ exists ff_q_lmd_full_k_successor. ub = ff_q_lmd_full_k_successor * S ((S (S lmd_index_full_k)) * uc) + (lmd_successor_full_k))) /\ ((((exists ff_h_lmd_full_k_digit. ff_h_lmd_full_k_digit + S (lmd_digit_full_k) = S ((S (lmd_index_full_k)) * vc)) /\ exists ff_q_lmd_full_k_digit. vb = ff_q_lmd_full_k_digit * S ((S (lmd_index_full_k)) * vc) + (lmd_digit_full_k))) /\ ((lmd_current_full_k = (p) * (lmd_successor_full_k) + (lmd_digit_full_k)) /\ (exists lmd_gap_full_k_digit_bound. lmd_gap_full_k_digit_bound + S (lmd_digit_full_k) = (p)))))))) -> (forall lmd_choose_index_full_quotient_choose. (exists lmd_gap_full_quotient_choose_bound. lmd_gap_full_quotient_choose_bound + S (lmd_choose_index_full_quotient_choose) = (S l)) -> exists lmd_choose_upper_full_quotient_choose lmd_choose_lower_full_quotient_choose lmd_choose_value_full_quotient_choose. ((((exists ff_h_lmd_full_quotient_choose_upper. ff_h_lmd_full_quotient_choose_upper + S (lmd_choose_upper_full_quotient_choose) = S ((S (lmd_choose_index_full_quotient_choose)) * qc)) /\ exists ff_q_lmd_full_quotient_choose_upper. qb = ff_q_lmd_full_quotient_choose_upper * S ((S (lmd_choose_index_full_quotient_choose)) * qc) + (lmd_choose_upper_full_quotient_choose))) /\ ((((exists ff_h_lmd_full_quotient_choose_lower. ff_h_lmd_full_quotient_choose_lower + S (lmd_choose_lower_full_quotient_choose) = S ((S (lmd_choose_index_full_quotient_choose)) * uc)) /\ exists ff_q_lmd_full_quotient_choose_lower. ub = ff_q_lmd_full_quotient_choose_lower * S ((S (lmd_choose_index_full_quotient_choose)) * uc) + (lmd_choose_lower_full_quotient_choose))) /\ ((((exists ff_h_lmd_full_quotient_choose_result. ff_h_lmd_full_quotient_choose_result + S (lmd_choose_value_full_quotient_choose) = S ((S (lmd_choose_index_full_quotient_choose)) * t)) /\ exists ff_q_lmd_full_quotient_choose_result. z = ff_q_lmd_full_quotient_choose_result * S ((S (lmd_choose_index_full_quotient_choose)) * t) + (lmd_choose_value_full_quotient_choose))) /\ (((exists bcf_lt_gap_lmd_full_quotient_choose_choose_out_of_range. bcf_lt_gap_lmd_full_quotient_choose_choose_out_of_range + S (lmd_choose_upper_full_quotient_choose) = lmd_choose_lower_full_quotient_choose) /\ lmd_choose_value_full_quotient_choose = 0) \/ ((exists bcf_le_gap_lmd_full_quotient_choose_choose_in_range. bcf_le_gap_lmd_full_quotient_choose_choose_in_range + (lmd_choose_lower_full_quotient_choose) = lmd_choose_upper_full_quotient_choose) /\ (exists bcf_row_code_code_lmd_full_quotient_choose_choose bcf_row_code_scale_lmd_full_quotient_choose_choose bcf_row_scale_code_lmd_full_quotient_choose_choose bcf_row_scale_scale_lmd_full_quotient_choose_choose bcf_row_code_lmd_full_quotient_choose_choose bcf_row_scale_lmd_full_quotient_choose_choose. ((forall bcf_row_index_lmd_full_quotient_choose_choose_table. (exists bcf_lt_gap_lmd_full_quotient_choose_choose_table_row_bound. bcf_lt_gap_lmd_full_quotient_choose_choose_table_row_bound + S (bcf_row_index_lmd_full_quotient_choose_choose_table) = S (lmd_choose_upper_full_quotient_choose)) -> exists bcf_row_code_lmd_full_quotient_choose_choose_table bcf_row_scale_lmd_full_quotient_choose_choose_table. ((((exists bcf_height_lmd_full_quotient_choose_choose_table_decoded_row_code. bcf_height_lmd_full_quotient_choose_choose_table_decoded_row_code + S (bcf_row_code_lmd_full_quotient_choose_choose_table) = S ((S (bcf_row_index_lmd_full_quotient_choose_choose_table)) * bcf_row_code_scale_lmd_full_quotient_choose_choose)) /\ exists bcf_quotient_lmd_full_quotient_choose_choose_table_decoded_row_code. bcf_row_code_code_lmd_full_quotient_choose_choose = bcf_quotient_lmd_full_quotient_choose_choose_table_decoded_row_code * S ((S (bcf_row_index_lmd_full_quotient_choose_choose_table)) * bcf_row_code_scale_lmd_full_quotient_choose_choose) + (bcf_row_code_lmd_full_quotient_choose_choose_table))) /\ ((((exists bcf_height_lmd_full_quotient_choose_choose_table_decoded_row_scale. bcf_height_lmd_full_quotient_choose_choose_table_decoded_row_scale + S (bcf_row_scale_lmd_full_quotient_choose_choose_table) = S ((S (bcf_row_index_lmd_full_quotient_choose_choose_table)) * bcf_row_scale_scale_lmd_full_quotient_choose_choose)) /\ exists bcf_quotient_lmd_full_quotient_choose_choose_table_decoded_row_scale. bcf_row_scale_code_lmd_full_quotient_choose_choose = bcf_quotient_lmd_full_quotient_choose_choose_table_decoded_row_scale * S ((S (bcf_row_index_lmd_full_quotient_choose_choose_table)) * bcf_row_scale_scale_lmd_full_quotient_choose_choose) + (bcf_row_scale_lmd_full_quotient_choose_choose_table))) /\ ((bcf_row_index_lmd_full_quotient_choose_choose_table = 0 /\ (forall bcf_index_lmd_full_quotient_choose_choose_table_zero_row. (exists bcf_lt_gap_lmd_full_quotient_choose_choose_table_zero_row_bound. bcf_lt_gap_lmd_full_quotient_choose_choose_table_zero_row_bound + S (bcf_index_lmd_full_quotient_choose_choose_table_zero_row) = S (lmd_choose_upper_full_quotient_choose)) -> exists bcf_value_lmd_full_quotient_choose_choose_table_zero_row. ((((exists bcf_height_lmd_full_quotient_choose_choose_table_zero_row_entry. bcf_height_lmd_full_quotient_choose_choose_table_zero_row_entry + S (bcf_value_lmd_full_quotient_choose_choose_table_zero_row) = S ((S (bcf_index_lmd_full_quotient_choose_choose_table_zero_row)) * bcf_row_scale_lmd_full_quotient_choose_choose_table)) /\ exists bcf_quotient_lmd_full_quotient_choose_choose_table_zero_row_entry. bcf_row_code_lmd_full_quotient_choose_choose_table = bcf_quotient_lmd_full_quotient_choose_choose_table_zero_row_entry * S ((S (bcf_index_lmd_full_quotient_choose_choose_table_zero_row)) * bcf_row_scale_lmd_full_quotient_choose_choose_table) + (bcf_value_lmd_full_quotient_choose_choose_table_zero_row))) /\ ((bcf_index_lmd_full_quotient_choose_choose_table_zero_row = 0 /\ bcf_value_lmd_full_quotient_choose_choose_table_zero_row = 1) \/ exists bcf_predecessor_lmd_full_quotient_choose_choose_table_zero_row. bcf_index_lmd_full_quotient_choose_choose_table_zero_row = S bcf_predecessor_lmd_full_quotient_choose_choose_table_zero_row /\ bcf_value_lmd_full_quotient_choose_choose_table_zero_row = 0)))) \/ exists bcf_predecessor_lmd_full_quotient_choose_choose_table bcf_previous_code_lmd_full_quotient_choose_choose_table bcf_previous_scale_lmd_full_quotient_choose_choose_table. bcf_row_index_lmd_full_quotient_choose_choose_table = S bcf_predecessor_lmd_full_quotient_choose_choose_table /\ ((((exists bcf_height_lmd_full_quotient_choose_choose_table_decoded_previous_code. bcf_height_lmd_full_quotient_choose_choose_table_decoded_previous_code + S (bcf_previous_code_lmd_full_quotient_choose_choose_table) = S ((S (bcf_predecessor_lmd_full_quotient_choose_choose_table)) * bcf_row_code_scale_lmd_full_quotient_choose_choose)) /\ exists bcf_quotient_lmd_full_quotient_choose_choose_table_decoded_previous_code. bcf_row_code_code_lmd_full_quotient_choose_choose = bcf_quotient_lmd_full_quotient_choose_choose_table_decoded_previous_code * S ((S (bcf_predecessor_lmd_full_quotient_choose_choose_table)) * bcf_row_code_scale_lmd_full_quotient_choose_choose) + (bcf_previous_code_lmd_full_quotient_choose_choose_table))) /\ ((((exists bcf_height_lmd_full_quotient_choose_choose_table_decoded_previous_scale. bcf_height_lmd_full_quotient_choose_choose_table_decoded_previous_scale + S (bcf_previous_scale_lmd_full_quotient_choose_choose_table) = S ((S (bcf_predecessor_lmd_full_quotient_choose_choose_table)) * bcf_row_scale_scale_lmd_full_quotient_choose_choose)) /\ exists bcf_quotient_lmd_full_quotient_choose_choose_table_decoded_previous_scale. bcf_row_scale_code_lmd_full_quotient_choose_choose = bcf_quotient_lmd_full_quotient_choose_choose_table_decoded_previous_scale * S ((S (bcf_predecessor_lmd_full_quotient_choose_choose_table)) * bcf_row_scale_scale_lmd_full_quotient_choose_choose) + (bcf_previous_scale_lmd_full_quotient_choose_choose_table))) /\ (forall bcf_index_lmd_full_quotient_choose_choose_table_row_step. (exists bcf_lt_gap_lmd_full_quotient_choose_choose_table_row_step_bound. bcf_lt_gap_lmd_full_quotient_choose_choose_table_row_step_bound + S (bcf_index_lmd_full_quotient_choose_choose_table_row_step) = S (lmd_choose_upper_full_quotient_choose)) -> exists bcf_value_lmd_full_quotient_choose_choose_table_row_step. ((((exists bcf_height_lmd_full_quotient_choose_choose_table_row_step_entry. bcf_height_lmd_full_quotient_choose_choose_table_row_step_entry + S (bcf_value_lmd_full_quotient_choose_choose_table_row_step) = S ((S (bcf_index_lmd_full_quotient_choose_choose_table_row_step)) * bcf_row_scale_lmd_full_quotient_choose_choose_table)) /\ exists bcf_quotient_lmd_full_quotient_choose_choose_table_row_step_entry. bcf_row_code_lmd_full_quotient_choose_choose_table = bcf_quotient_lmd_full_quotient_choose_choose_table_row_step_entry * S ((S (bcf_index_lmd_full_quotient_choose_choose_table_row_step)) * bcf_row_scale_lmd_full_quotient_choose_choose_table) + (bcf_value_lmd_full_quotient_choose_choose_table_row_step))) /\ ((bcf_index_lmd_full_quotient_choose_choose_table_row_step = 0 /\ bcf_value_lmd_full_quotient_choose_choose_table_row_step = 1) \/ exists bcf_predecessor_lmd_full_quotient_choose_choose_table_row_step bcf_left_lmd_full_quotient_choose_choose_table_row_step bcf_right_lmd_full_quotient_choose_choose_table_row_step. bcf_index_lmd_full_quotient_choose_choose_table_row_step = S bcf_predecessor_lmd_full_quotient_choose_choose_table_row_step /\ ((((exists bcf_height_lmd_full_quotient_choose_choose_table_row_step_previous_left. bcf_height_lmd_full_quotient_choose_choose_table_row_step_previous_left + S (bcf_left_lmd_full_quotient_choose_choose_table_row_step) = S ((S (bcf_predecessor_lmd_full_quotient_choose_choose_table_row_step)) * bcf_previous_scale_lmd_full_quotient_choose_choose_table)) /\ exists bcf_quotient_lmd_full_quotient_choose_choose_table_row_step_previous_left. bcf_previous_code_lmd_full_quotient_choose_choose_table = bcf_quotient_lmd_full_quotient_choose_choose_table_row_step_previous_left * S ((S (bcf_predecessor_lmd_full_quotient_choose_choose_table_row_step)) * bcf_previous_scale_lmd_full_quotient_choose_choose_table) + (bcf_left_lmd_full_quotient_choose_choose_table_row_step))) /\ ((((exists bcf_height_lmd_full_quotient_choose_choose_table_row_step_previous_right. bcf_height_lmd_full_quotient_choose_choose_table_row_step_previous_right + S (bcf_right_lmd_full_quotient_choose_choose_table_row_step) = S ((S (S (bcf_predecessor_lmd_full_quotient_choose_choose_table_row_step))) * bcf_previous_scale_lmd_full_quotient_choose_choose_table)) /\ exists bcf_quotient_lmd_full_quotient_choose_choose_table_row_step_previous_right. bcf_previous_code_lmd_full_quotient_choose_choose_table = bcf_quotient_lmd_full_quotient_choose_choose_table_row_step_previous_right * S ((S (S (bcf_predecessor_lmd_full_quotient_choose_choose_table_row_step))) * bcf_previous_scale_lmd_full_quotient_choose_choose_table) + (bcf_right_lmd_full_quotient_choose_choose_table_row_step))) /\ bcf_value_lmd_full_quotient_choose_choose_table_row_step = bcf_left_lmd_full_quotient_choose_choose_table_row_step + bcf_right_lmd_full_quotient_choose_choose_table_row_step))))))))))) /\ ((((exists bcf_height_lmd_full_quotient_choose_choose_decoded_row_code. bcf_height_lmd_full_quotient_choose_choose_decoded_row_code + S (bcf_row_code_lmd_full_quotient_choose_choose) = S ((S (lmd_choose_upper_full_quotient_choose)) * bcf_row_code_scale_lmd_full_quotient_choose_choose)) /\ exists bcf_quotient_lmd_full_quotient_choose_choose_decoded_row_code. bcf_row_code_code_lmd_full_quotient_choose_choose = bcf_quotient_lmd_full_quotient_choose_choose_decoded_row_code * S ((S (lmd_choose_upper_full_quotient_choose)) * bcf_row_code_scale_lmd_full_quotient_choose_choose) + (bcf_row_code_lmd_full_quotient_choose_choose))) /\ ((((exists bcf_height_lmd_full_quotient_choose_choose_decoded_row_scale. bcf_height_lmd_full_quotient_choose_choose_decoded_row_scale + S (bcf_row_scale_lmd_full_quotient_choose_choose) = S ((S (lmd_choose_upper_full_quotient_choose)) * bcf_row_scale_scale_lmd_full_quotient_choose_choose)) /\ exists bcf_quotient_lmd_full_quotient_choose_choose_decoded_row_scale. bcf_row_scale_code_lmd_full_quotient_choose_choose = bcf_quotient_lmd_full_quotient_choose_choose_decoded_row_scale * S ((S (lmd_choose_upper_full_quotient_choose)) * bcf_row_scale_scale_lmd_full_quotient_choose_choose) + (bcf_row_scale_lmd_full_quotient_choose_choose))) /\ (((exists bcf_height_lmd_full_quotient_choose_choose_decoded_value. bcf_height_lmd_full_quotient_choose_choose_decoded_value + S (lmd_choose_value_full_quotient_choose) = S ((S (lmd_choose_lower_full_quotient_choose)) * bcf_row_scale_lmd_full_quotient_choose_choose)) /\ exists bcf_quotient_lmd_full_quotient_choose_choose_decoded_value. bcf_row_code_lmd_full_quotient_choose_choose = bcf_quotient_lmd_full_quotient_choose_choose_decoded_value * S ((S (lmd_choose_lower_full_quotient_choose)) * bcf_row_scale_lmd_full_quotient_choose_choose) + (lmd_choose_value_full_quotient_choose))))))))))))) -> (forall lmd_choose_index_full_digit_choose. (exists lmd_gap_full_digit_choose_bound. lmd_gap_full_digit_choose_bound + S (lmd_choose_index_full_digit_choose) = (l)) -> exists lmd_choose_upper_full_digit_choose lmd_choose_lower_full_digit_choose lmd_choose_value_full_digit_choose. ((((exists ff_h_lmd_full_digit_choose_upper. ff_h_lmd_full_digit_choose_upper + S (lmd_choose_upper_full_digit_choose) = S ((S (lmd_choose_index_full_digit_choose)) * dc)) /\ exists ff_q_lmd_full_digit_choose_upper. db = ff_q_lmd_full_digit_choose_upper * S ((S (lmd_choose_index_full_digit_choose)) * dc) + (lmd_choose_upper_full_digit_choose))) /\ ((((exists ff_h_lmd_full_digit_choose_lower. ff_h_lmd_full_digit_choose_lower + S (lmd_choose_lower_full_digit_choose) = S ((S (lmd_choose_index_full_digit_choose)) * vc)) /\ exists ff_q_lmd_full_digit_choose_lower. vb = ff_q_lmd_full_digit_choose_lower * S ((S (lmd_choose_index_full_digit_choose)) * vc) + (lmd_choose_lower_full_digit_choose))) /\ ((((exists ff_h_lmd_full_digit_choose_result. ff_h_lmd_full_digit_choose_result + S (lmd_choose_value_full_digit_choose) = S ((S (lmd_choose_index_full_digit_choose)) * w)) /\ exists ff_q_lmd_full_digit_choose_result. s = ff_q_lmd_full_digit_choose_result * S ((S (lmd_choose_index_full_digit_choose)) * w) + (lmd_choose_value_full_digit_choose))) /\ (((exists bcf_lt_gap_lmd_full_digit_choose_choose_out_of_range. bcf_lt_gap_lmd_full_digit_choose_choose_out_of_range + S (lmd_choose_upper_full_digit_choose) = lmd_choose_lower_full_digit_choose) /\ lmd_choose_value_full_digit_choose = 0) \/ ((exists bcf_le_gap_lmd_full_digit_choose_choose_in_range. bcf_le_gap_lmd_full_digit_choose_choose_in_range + (lmd_choose_lower_full_digit_choose) = lmd_choose_upper_full_digit_choose) /\ (exists bcf_row_code_code_lmd_full_digit_choose_choose bcf_row_code_scale_lmd_full_digit_choose_choose bcf_row_scale_code_lmd_full_digit_choose_choose bcf_row_scale_scale_lmd_full_digit_choose_choose bcf_row_code_lmd_full_digit_choose_choose bcf_row_scale_lmd_full_digit_choose_choose. ((forall bcf_row_index_lmd_full_digit_choose_choose_table. (exists bcf_lt_gap_lmd_full_digit_choose_choose_table_row_bound. bcf_lt_gap_lmd_full_digit_choose_choose_table_row_bound + S (bcf_row_index_lmd_full_digit_choose_choose_table) = S (lmd_choose_upper_full_digit_choose)) -> exists bcf_row_code_lmd_full_digit_choose_choose_table bcf_row_scale_lmd_full_digit_choose_choose_table. ((((exists bcf_height_lmd_full_digit_choose_choose_table_decoded_row_code. bcf_height_lmd_full_digit_choose_choose_table_decoded_row_code + S (bcf_row_code_lmd_full_digit_choose_choose_table) = S ((S (bcf_row_index_lmd_full_digit_choose_choose_table)) * bcf_row_code_scale_lmd_full_digit_choose_choose)) /\ exists bcf_quotient_lmd_full_digit_choose_choose_table_decoded_row_code. bcf_row_code_code_lmd_full_digit_choose_choose = bcf_quotient_lmd_full_digit_choose_choose_table_decoded_row_code * S ((S (bcf_row_index_lmd_full_digit_choose_choose_table)) * bcf_row_code_scale_lmd_full_digit_choose_choose) + (bcf_row_code_lmd_full_digit_choose_choose_table))) /\ ((((exists bcf_height_lmd_full_digit_choose_choose_table_decoded_row_scale. bcf_height_lmd_full_digit_choose_choose_table_decoded_row_scale + S (bcf_row_scale_lmd_full_digit_choose_choose_table) = S ((S (bcf_row_index_lmd_full_digit_choose_choose_table)) * bcf_row_scale_scale_lmd_full_digit_choose_choose)) /\ exists bcf_quotient_lmd_full_digit_choose_choose_table_decoded_row_scale. bcf_row_scale_code_lmd_full_digit_choose_choose = bcf_quotient_lmd_full_digit_choose_choose_table_decoded_row_scale * S ((S (bcf_row_index_lmd_full_digit_choose_choose_table)) * bcf_row_scale_scale_lmd_full_digit_choose_choose) + (bcf_row_scale_lmd_full_digit_choose_choose_table))) /\ ((bcf_row_index_lmd_full_digit_choose_choose_table = 0 /\ (forall bcf_index_lmd_full_digit_choose_choose_table_zero_row. (exists bcf_lt_gap_lmd_full_digit_choose_choose_table_zero_row_bound. bcf_lt_gap_lmd_full_digit_choose_choose_table_zero_row_bound + S (bcf_index_lmd_full_digit_choose_choose_table_zero_row) = S (lmd_choose_upper_full_digit_choose)) -> exists bcf_value_lmd_full_digit_choose_choose_table_zero_row. ((((exists bcf_height_lmd_full_digit_choose_choose_table_zero_row_entry. bcf_height_lmd_full_digit_choose_choose_table_zero_row_entry + S (bcf_value_lmd_full_digit_choose_choose_table_zero_row) = S ((S (bcf_index_lmd_full_digit_choose_choose_table_zero_row)) * bcf_row_scale_lmd_full_digit_choose_choose_table)) /\ exists bcf_quotient_lmd_full_digit_choose_choose_table_zero_row_entry. bcf_row_code_lmd_full_digit_choose_choose_table = bcf_quotient_lmd_full_digit_choose_choose_table_zero_row_entry * S ((S (bcf_index_lmd_full_digit_choose_choose_table_zero_row)) * bcf_row_scale_lmd_full_digit_choose_choose_table) + (bcf_value_lmd_full_digit_choose_choose_table_zero_row))) /\ ((bcf_index_lmd_full_digit_choose_choose_table_zero_row = 0 /\ bcf_value_lmd_full_digit_choose_choose_table_zero_row = 1) \/ exists bcf_predecessor_lmd_full_digit_choose_choose_table_zero_row. bcf_index_lmd_full_digit_choose_choose_table_zero_row = S bcf_predecessor_lmd_full_digit_choose_choose_table_zero_row /\ bcf_value_lmd_full_digit_choose_choose_table_zero_row = 0)))) \/ exists bcf_predecessor_lmd_full_digit_choose_choose_table bcf_previous_code_lmd_full_digit_choose_choose_table bcf_previous_scale_lmd_full_digit_choose_choose_table. bcf_row_index_lmd_full_digit_choose_choose_table = S bcf_predecessor_lmd_full_digit_choose_choose_table /\ ((((exists bcf_height_lmd_full_digit_choose_choose_table_decoded_previous_code. bcf_height_lmd_full_digit_choose_choose_table_decoded_previous_code + S (bcf_previous_code_lmd_full_digit_choose_choose_table) = S ((S (bcf_predecessor_lmd_full_digit_choose_choose_table)) * bcf_row_code_scale_lmd_full_digit_choose_choose)) /\ exists bcf_quotient_lmd_full_digit_choose_choose_table_decoded_previous_code. bcf_row_code_code_lmd_full_digit_choose_choose = bcf_quotient_lmd_full_digit_choose_choose_table_decoded_previous_code * S ((S (bcf_predecessor_lmd_full_digit_choose_choose_table)) * bcf_row_code_scale_lmd_full_digit_choose_choose) + (bcf_previous_code_lmd_full_digit_choose_choose_table))) /\ ((((exists bcf_height_lmd_full_digit_choose_choose_table_decoded_previous_scale. bcf_height_lmd_full_digit_choose_choose_table_decoded_previous_scale + S (bcf_previous_scale_lmd_full_digit_choose_choose_table) = S ((S (bcf_predecessor_lmd_full_digit_choose_choose_table)) * bcf_row_scale_scale_lmd_full_digit_choose_choose)) /\ exists bcf_quotient_lmd_full_digit_choose_choose_table_decoded_previous_scale. bcf_row_scale_code_lmd_full_digit_choose_choose = bcf_quotient_lmd_full_digit_choose_choose_table_decoded_previous_scale * S ((S (bcf_predecessor_lmd_full_digit_choose_choose_table)) * bcf_row_scale_scale_lmd_full_digit_choose_choose) + (bcf_previous_scale_lmd_full_digit_choose_choose_table))) /\ (forall bcf_index_lmd_full_digit_choose_choose_table_row_step. (exists bcf_lt_gap_lmd_full_digit_choose_choose_table_row_step_bound. bcf_lt_gap_lmd_full_digit_choose_choose_table_row_step_bound + S (bcf_index_lmd_full_digit_choose_choose_table_row_step) = S (lmd_choose_upper_full_digit_choose)) -> exists bcf_value_lmd_full_digit_choose_choose_table_row_step. ((((exists bcf_height_lmd_full_digit_choose_choose_table_row_step_entry. bcf_height_lmd_full_digit_choose_choose_table_row_step_entry + S (bcf_value_lmd_full_digit_choose_choose_table_row_step) = S ((S (bcf_index_lmd_full_digit_choose_choose_table_row_step)) * bcf_row_scale_lmd_full_digit_choose_choose_table)) /\ exists bcf_quotient_lmd_full_digit_choose_choose_table_row_step_entry. bcf_row_code_lmd_full_digit_choose_choose_table = bcf_quotient_lmd_full_digit_choose_choose_table_row_step_entry * S ((S (bcf_index_lmd_full_digit_choose_choose_table_row_step)) * bcf_row_scale_lmd_full_digit_choose_choose_table) + (bcf_value_lmd_full_digit_choose_choose_table_row_step))) /\ ((bcf_index_lmd_full_digit_choose_choose_table_row_step = 0 /\ bcf_value_lmd_full_digit_choose_choose_table_row_step = 1) \/ exists bcf_predecessor_lmd_full_digit_choose_choose_table_row_step bcf_left_lmd_full_digit_choose_choose_table_row_step bcf_right_lmd_full_digit_choose_choose_table_row_step. bcf_index_lmd_full_digit_choose_choose_table_row_step = S bcf_predecessor_lmd_full_digit_choose_choose_table_row_step /\ ((((exists bcf_height_lmd_full_digit_choose_choose_table_row_step_previous_left. bcf_height_lmd_full_digit_choose_choose_table_row_step_previous_left + S (bcf_left_lmd_full_digit_choose_choose_table_row_step) = S ((S (bcf_predecessor_lmd_full_digit_choose_choose_table_row_step)) * bcf_previous_scale_lmd_full_digit_choose_choose_table)) /\ exists bcf_quotient_lmd_full_digit_choose_choose_table_row_step_previous_left. bcf_previous_code_lmd_full_digit_choose_choose_table = bcf_quotient_lmd_full_digit_choose_choose_table_row_step_previous_left * S ((S (bcf_predecessor_lmd_full_digit_choose_choose_table_row_step)) * bcf_previous_scale_lmd_full_digit_choose_choose_table) + (bcf_left_lmd_full_digit_choose_choose_table_row_step))) /\ ((((exists bcf_height_lmd_full_digit_choose_choose_table_row_step_previous_right. bcf_height_lmd_full_digit_choose_choose_table_row_step_previous_right + S (bcf_right_lmd_full_digit_choose_choose_table_row_step) = S ((S (S (bcf_predecessor_lmd_full_digit_choose_choose_table_row_step))) * bcf_previous_scale_lmd_full_digit_choose_choose_table)) /\ exists bcf_quotient_lmd_full_digit_choose_choose_table_row_step_previous_right. bcf_previous_code_lmd_full_digit_choose_choose_table = bcf_quotient_lmd_full_digit_choose_choose_table_row_step_previous_right * S ((S (S (bcf_predecessor_lmd_full_digit_choose_choose_table_row_step))) * bcf_previous_scale_lmd_full_digit_choose_choose_table) + (bcf_right_lmd_full_digit_choose_choose_table_row_step))) /\ bcf_value_lmd_full_digit_choose_choose_table_row_step = bcf_left_lmd_full_digit_choose_choose_table_row_step + bcf_right_lmd_full_digit_choose_choose_table_row_step))))))))))) /\ ((((exists bcf_height_lmd_full_digit_choose_choose_decoded_row_code. bcf_height_lmd_full_digit_choose_choose_decoded_row_code + S (bcf_row_code_lmd_full_digit_choose_choose) = S ((S (lmd_choose_upper_full_digit_choose)) * bcf_row_code_scale_lmd_full_digit_choose_choose)) /\ exists bcf_quotient_lmd_full_digit_choose_choose_decoded_row_code. bcf_row_code_code_lmd_full_digit_choose_choose = bcf_quotient_lmd_full_digit_choose_choose_decoded_row_code * S ((S (lmd_choose_upper_full_digit_choose)) * bcf_row_code_scale_lmd_full_digit_choose_choose) + (bcf_row_code_lmd_full_digit_choose_choose))) /\ ((((exists bcf_height_lmd_full_digit_choose_choose_decoded_row_scale. bcf_height_lmd_full_digit_choose_choose_decoded_row_scale + S (bcf_row_scale_lmd_full_digit_choose_choose) = S ((S (lmd_choose_upper_full_digit_choose)) * bcf_row_scale_scale_lmd_full_digit_choose_choose)) /\ exists bcf_quotient_lmd_full_digit_choose_choose_decoded_row_scale. bcf_row_scale_code_lmd_full_digit_choose_choose = bcf_quotient_lmd_full_digit_choose_choose_decoded_row_scale * S ((S (lmd_choose_upper_full_digit_choose)) * bcf_row_scale_scale_lmd_full_digit_choose_choose) + (bcf_row_scale_lmd_full_digit_choose_choose))) /\ (((exists bcf_height_lmd_full_digit_choose_choose_decoded_value. bcf_height_lmd_full_digit_choose_choose_decoded_value + S (lmd_choose_value_full_digit_choose) = S ((S (lmd_choose_lower_full_digit_choose)) * bcf_row_scale_lmd_full_digit_choose_choose)) /\ exists bcf_quotient_lmd_full_digit_choose_choose_decoded_value. bcf_row_code_lmd_full_digit_choose_choose = bcf_quotient_lmd_full_digit_choose_choose_decoded_value * S ((S (lmd_choose_lower_full_digit_choose)) * bcf_row_scale_lmd_full_digit_choose_choose) + (lmd_choose_value_full_digit_choose))))))))))))) -> (exists ff_u_lmd_full_product ff_v_lmd_full_product. ((((exists ff_h_lmd_full_product_start. ff_h_lmd_full_product_start + S (1) = S ((S (0)) * ff_v_lmd_full_product)) /\ exists ff_q_lmd_full_product_start. ff_u_lmd_full_product = ff_q_lmd_full_product_start * S ((S (0)) * ff_v_lmd_full_product) + (1))) /\ ((((exists ff_h_lmd_full_product_terminal. ff_h_lmd_full_product_terminal + S (P) = S ((S (l)) * ff_v_lmd_full_product)) /\ exists ff_q_lmd_full_product_terminal. ff_u_lmd_full_product = ff_q_lmd_full_product_terminal * S ((S (l)) * ff_v_lmd_full_product) + (P))) /\ forall ff_i_lmd_full_product. (exists ff_lt_lmd_full_product_bound. ff_lt_lmd_full_product_bound + S ff_i_lmd_full_product = l) -> exists ff_p_lmd_full_product ff_r_lmd_full_product ff_s_lmd_full_product. ((((exists ff_h_lmd_full_product_factor. ff_h_lmd_full_product_factor + S (ff_p_lmd_full_product) = S ((S (ff_i_lmd_full_product)) * w)) /\ exists ff_q_lmd_full_product_factor. s = ff_q_lmd_full_product_factor * S ((S (ff_i_lmd_full_product)) * w) + (ff_p_lmd_full_product))) /\ ((((exists ff_h_lmd_full_product_partial. ff_h_lmd_full_product_partial + S (ff_r_lmd_full_product) = S ((S (ff_i_lmd_full_product)) * ff_v_lmd_full_product)) /\ exists ff_q_lmd_full_product_partial. ff_u_lmd_full_product = ff_q_lmd_full_product_partial * S ((S (ff_i_lmd_full_product)) * ff_v_lmd_full_product) + (ff_r_lmd_full_product))) /\ ((((exists ff_h_lmd_full_product_successor. ff_h_lmd_full_product_successor + S (ff_s_lmd_full_product) = S ((S (S ff_i_lmd_full_product)) * ff_v_lmd_full_product)) /\ exists ff_q_lmd_full_product_successor. ff_u_lmd_full_product = ff_q_lmd_full_product_successor * S ((S (S ff_i_lmd_full_product)) * ff_v_lmd_full_product) + (ff_s_lmd_full_product))) /\ ff_s_lmd_full_product = ff_r_lmd_full_product * ff_p_lmd_full_product)))))) -> (((exists ff_h_lmd_full_coefficient_start. ff_h_lmd_full_coefficient_start + S (C) = S ((S (0)) * t)) /\ exists ff_q_lmd_full_coefficient_start. z = ff_q_lmd_full_coefficient_start * S ((S (0)) * t) + (C))) -> (((exists ff_h_lmd_full_coefficient_end. ff_h_lmd_full_coefficient_end + S (T) = S ((S (l)) * t)) /\ exists ff_q_lmd_full_coefficient_end. z = ff_q_lmd_full_coefficient_end * S ((S (l)) * t) + (T))) -> (exists lmd_mod_left_full_result lmd_mod_right_full_result. (C) + (p) * lmd_mod_left_full_result = (T * P) + (p) * lmd_mod_right_full_result)