lucas_low_digit_product_congruence

Exact expanded first-order arithmetic statement

forall p q d e C A B. ((~(p = 1) /\ forall frm_prime_left_lucas_low_digit_prime frm_prime_right_lucas_low_digit_prime. p = frm_prime_left_lucas_low_digit_prime * frm_prime_right_lucas_low_digit_prime -> frm_prime_left_lucas_low_digit_prime = 1 \/ frm_prime_right_lucas_low_digit_prime = 1)) -> (exists lld_gap_product_upper_bound. lld_gap_product_upper_bound + S (d) = (p)) -> (exists lld_gap_product_lower_bound. lld_gap_product_lower_bound + S (e) = (p)) -> (((exists bcf_lt_gap_lucas_low_digit_product_upper_out_of_range. bcf_lt_gap_lucas_low_digit_product_upper_out_of_range + S (p * q + d) = e) /\ C = 0) \/ ((exists bcf_le_gap_lucas_low_digit_product_upper_in_range. bcf_le_gap_lucas_low_digit_product_upper_in_range + (e) = p * q + d) /\ (exists bcf_row_code_code_lucas_low_digit_product_upper bcf_row_code_scale_lucas_low_digit_product_upper bcf_row_scale_code_lucas_low_digit_product_upper bcf_row_scale_scale_lucas_low_digit_product_upper bcf_row_code_lucas_low_digit_product_upper bcf_row_scale_lucas_low_digit_product_upper. ((forall bcf_row_index_lucas_low_digit_product_upper_table. (exists bcf_lt_gap_lucas_low_digit_product_upper_table_row_bound. bcf_lt_gap_lucas_low_digit_product_upper_table_row_bound + S (bcf_row_index_lucas_low_digit_product_upper_table) = S (p * q + d)) -> exists bcf_row_code_lucas_low_digit_product_upper_table bcf_row_scale_lucas_low_digit_product_upper_table. ((((exists bcf_height_lucas_low_digit_product_upper_table_decoded_row_code. bcf_height_lucas_low_digit_product_upper_table_decoded_row_code + S (bcf_row_code_lucas_low_digit_product_upper_table) = S ((S (bcf_row_index_lucas_low_digit_product_upper_table)) * bcf_row_code_scale_lucas_low_digit_product_upper)) /\ exists bcf_quotient_lucas_low_digit_product_upper_table_decoded_row_code. bcf_row_code_code_lucas_low_digit_product_upper = bcf_quotient_lucas_low_digit_product_upper_table_decoded_row_code * S ((S (bcf_row_index_lucas_low_digit_product_upper_table)) * bcf_row_code_scale_lucas_low_digit_product_upper) + (bcf_row_code_lucas_low_digit_product_upper_table))) /\ ((((exists bcf_height_lucas_low_digit_product_upper_table_decoded_row_scale. bcf_height_lucas_low_digit_product_upper_table_decoded_row_scale + S (bcf_row_scale_lucas_low_digit_product_upper_table) = S ((S (bcf_row_index_lucas_low_digit_product_upper_table)) * bcf_row_scale_scale_lucas_low_digit_product_upper)) /\ exists bcf_quotient_lucas_low_digit_product_upper_table_decoded_row_scale. bcf_row_scale_code_lucas_low_digit_product_upper = bcf_quotient_lucas_low_digit_product_upper_table_decoded_row_scale * S ((S (bcf_row_index_lucas_low_digit_product_upper_table)) * bcf_row_scale_scale_lucas_low_digit_product_upper) + (bcf_row_scale_lucas_low_digit_product_upper_table))) /\ ((bcf_row_index_lucas_low_digit_product_upper_table = 0 /\ (forall bcf_index_lucas_low_digit_product_upper_table_zero_row. (exists bcf_lt_gap_lucas_low_digit_product_upper_table_zero_row_bound. bcf_lt_gap_lucas_low_digit_product_upper_table_zero_row_bound + S (bcf_index_lucas_low_digit_product_upper_table_zero_row) = S (p * q + d)) -> exists bcf_value_lucas_low_digit_product_upper_table_zero_row. ((((exists bcf_height_lucas_low_digit_product_upper_table_zero_row_entry. bcf_height_lucas_low_digit_product_upper_table_zero_row_entry + S (bcf_value_lucas_low_digit_product_upper_table_zero_row) = S ((S (bcf_index_lucas_low_digit_product_upper_table_zero_row)) * bcf_row_scale_lucas_low_digit_product_upper_table)) /\ exists bcf_quotient_lucas_low_digit_product_upper_table_zero_row_entry. bcf_row_code_lucas_low_digit_product_upper_table = bcf_quotient_lucas_low_digit_product_upper_table_zero_row_entry * S ((S (bcf_index_lucas_low_digit_product_upper_table_zero_row)) * bcf_row_scale_lucas_low_digit_product_upper_table) + (bcf_value_lucas_low_digit_product_upper_table_zero_row))) /\ ((bcf_index_lucas_low_digit_product_upper_table_zero_row = 0 /\ bcf_value_lucas_low_digit_product_upper_table_zero_row = 1) \/ exists bcf_predecessor_lucas_low_digit_product_upper_table_zero_row. bcf_index_lucas_low_digit_product_upper_table_zero_row = S bcf_predecessor_lucas_low_digit_product_upper_table_zero_row /\ bcf_value_lucas_low_digit_product_upper_table_zero_row = 0)))) \/ exists bcf_predecessor_lucas_low_digit_product_upper_table bcf_previous_code_lucas_low_digit_product_upper_table bcf_previous_scale_lucas_low_digit_product_upper_table. bcf_row_index_lucas_low_digit_product_upper_table = S bcf_predecessor_lucas_low_digit_product_upper_table /\ ((((exists bcf_height_lucas_low_digit_product_upper_table_decoded_previous_code. bcf_height_lucas_low_digit_product_upper_table_decoded_previous_code + S (bcf_previous_code_lucas_low_digit_product_upper_table) = S ((S (bcf_predecessor_lucas_low_digit_product_upper_table)) * bcf_row_code_scale_lucas_low_digit_product_upper)) /\ exists bcf_quotient_lucas_low_digit_product_upper_table_decoded_previous_code. bcf_row_code_code_lucas_low_digit_product_upper = bcf_quotient_lucas_low_digit_product_upper_table_decoded_previous_code * S ((S (bcf_predecessor_lucas_low_digit_product_upper_table)) * bcf_row_code_scale_lucas_low_digit_product_upper) + (bcf_previous_code_lucas_low_digit_product_upper_table))) /\ ((((exists bcf_height_lucas_low_digit_product_upper_table_decoded_previous_scale. bcf_height_lucas_low_digit_product_upper_table_decoded_previous_scale + S (bcf_previous_scale_lucas_low_digit_product_upper_table) = S ((S (bcf_predecessor_lucas_low_digit_product_upper_table)) * bcf_row_scale_scale_lucas_low_digit_product_upper)) /\ exists bcf_quotient_lucas_low_digit_product_upper_table_decoded_previous_scale. bcf_row_scale_code_lucas_low_digit_product_upper = bcf_quotient_lucas_low_digit_product_upper_table_decoded_previous_scale * S ((S (bcf_predecessor_lucas_low_digit_product_upper_table)) * bcf_row_scale_scale_lucas_low_digit_product_upper) + (bcf_previous_scale_lucas_low_digit_product_upper_table))) /\ (forall bcf_index_lucas_low_digit_product_upper_table_row_step. (exists bcf_lt_gap_lucas_low_digit_product_upper_table_row_step_bound. bcf_lt_gap_lucas_low_digit_product_upper_table_row_step_bound + S (bcf_index_lucas_low_digit_product_upper_table_row_step) = S (p * q + d)) -> exists bcf_value_lucas_low_digit_product_upper_table_row_step. ((((exists bcf_height_lucas_low_digit_product_upper_table_row_step_entry. bcf_height_lucas_low_digit_product_upper_table_row_step_entry + S (bcf_value_lucas_low_digit_product_upper_table_row_step) = S ((S (bcf_index_lucas_low_digit_product_upper_table_row_step)) * bcf_row_scale_lucas_low_digit_product_upper_table)) /\ exists bcf_quotient_lucas_low_digit_product_upper_table_row_step_entry. bcf_row_code_lucas_low_digit_product_upper_table = bcf_quotient_lucas_low_digit_product_upper_table_row_step_entry * S ((S (bcf_index_lucas_low_digit_product_upper_table_row_step)) * bcf_row_scale_lucas_low_digit_product_upper_table) + (bcf_value_lucas_low_digit_product_upper_table_row_step))) /\ ((bcf_index_lucas_low_digit_product_upper_table_row_step = 0 /\ bcf_value_lucas_low_digit_product_upper_table_row_step = 1) \/ exists bcf_predecessor_lucas_low_digit_product_upper_table_row_step bcf_left_lucas_low_digit_product_upper_table_row_step bcf_right_lucas_low_digit_product_upper_table_row_step. bcf_index_lucas_low_digit_product_upper_table_row_step = S bcf_predecessor_lucas_low_digit_product_upper_table_row_step /\ ((((exists bcf_height_lucas_low_digit_product_upper_table_row_step_previous_left. bcf_height_lucas_low_digit_product_upper_table_row_step_previous_left + S (bcf_left_lucas_low_digit_product_upper_table_row_step) = S ((S (bcf_predecessor_lucas_low_digit_product_upper_table_row_step)) * bcf_previous_scale_lucas_low_digit_product_upper_table)) /\ exists bcf_quotient_lucas_low_digit_product_upper_table_row_step_previous_left. bcf_previous_code_lucas_low_digit_product_upper_table = bcf_quotient_lucas_low_digit_product_upper_table_row_step_previous_left * S ((S (bcf_predecessor_lucas_low_digit_product_upper_table_row_step)) * bcf_previous_scale_lucas_low_digit_product_upper_table) + (bcf_left_lucas_low_digit_product_upper_table_row_step))) /\ ((((exists bcf_height_lucas_low_digit_product_upper_table_row_step_previous_right. bcf_height_lucas_low_digit_product_upper_table_row_step_previous_right + S (bcf_right_lucas_low_digit_product_upper_table_row_step) = S ((S (S (bcf_predecessor_lucas_low_digit_product_upper_table_row_step))) * bcf_previous_scale_lucas_low_digit_product_upper_table)) /\ exists bcf_quotient_lucas_low_digit_product_upper_table_row_step_previous_right. bcf_previous_code_lucas_low_digit_product_upper_table = bcf_quotient_lucas_low_digit_product_upper_table_row_step_previous_right * S ((S (S (bcf_predecessor_lucas_low_digit_product_upper_table_row_step))) * bcf_previous_scale_lucas_low_digit_product_upper_table) + (bcf_right_lucas_low_digit_product_upper_table_row_step))) /\ bcf_value_lucas_low_digit_product_upper_table_row_step = bcf_left_lucas_low_digit_product_upper_table_row_step + bcf_right_lucas_low_digit_product_upper_table_row_step))))))))))) /\ ((((exists bcf_height_lucas_low_digit_product_upper_decoded_row_code. bcf_height_lucas_low_digit_product_upper_decoded_row_code + S (bcf_row_code_lucas_low_digit_product_upper) = S ((S (p * q + d)) * bcf_row_code_scale_lucas_low_digit_product_upper)) /\ exists bcf_quotient_lucas_low_digit_product_upper_decoded_row_code. bcf_row_code_code_lucas_low_digit_product_upper = bcf_quotient_lucas_low_digit_product_upper_decoded_row_code * S ((S (p * q + d)) * bcf_row_code_scale_lucas_low_digit_product_upper) + (bcf_row_code_lucas_low_digit_product_upper))) /\ ((((exists bcf_height_lucas_low_digit_product_upper_decoded_row_scale. bcf_height_lucas_low_digit_product_upper_decoded_row_scale + S (bcf_row_scale_lucas_low_digit_product_upper) = S ((S (p * q + d)) * bcf_row_scale_scale_lucas_low_digit_product_upper)) /\ exists bcf_quotient_lucas_low_digit_product_upper_decoded_row_scale. bcf_row_scale_code_lucas_low_digit_product_upper = bcf_quotient_lucas_low_digit_product_upper_decoded_row_scale * S ((S (p * q + d)) * bcf_row_scale_scale_lucas_low_digit_product_upper) + (bcf_row_scale_lucas_low_digit_product_upper))) /\ (((exists bcf_height_lucas_low_digit_product_upper_decoded_value. bcf_height_lucas_low_digit_product_upper_decoded_value + S (C) = S ((S (e)) * bcf_row_scale_lucas_low_digit_product_upper)) /\ exists bcf_quotient_lucas_low_digit_product_upper_decoded_value. bcf_row_code_lucas_low_digit_product_upper = bcf_quotient_lucas_low_digit_product_upper_decoded_value * S ((S (e)) * bcf_row_scale_lucas_low_digit_product_upper) + (C))))))))) -> (((exists bcf_lt_gap_lucas_low_digit_product_quotient_out_of_range. bcf_lt_gap_lucas_low_digit_product_quotient_out_of_range + S (q) = 0) /\ A = 0) \/ ((exists bcf_le_gap_lucas_low_digit_product_quotient_in_range. bcf_le_gap_lucas_low_digit_product_quotient_in_range + (0) = q) /\ (exists bcf_row_code_code_lucas_low_digit_product_quotient bcf_row_code_scale_lucas_low_digit_product_quotient bcf_row_scale_code_lucas_low_digit_product_quotient bcf_row_scale_scale_lucas_low_digit_product_quotient bcf_row_code_lucas_low_digit_product_quotient bcf_row_scale_lucas_low_digit_product_quotient. ((forall bcf_row_index_lucas_low_digit_product_quotient_table. (exists bcf_lt_gap_lucas_low_digit_product_quotient_table_row_bound. bcf_lt_gap_lucas_low_digit_product_quotient_table_row_bound + S (bcf_row_index_lucas_low_digit_product_quotient_table) = S (q)) -> exists bcf_row_code_lucas_low_digit_product_quotient_table bcf_row_scale_lucas_low_digit_product_quotient_table. ((((exists bcf_height_lucas_low_digit_product_quotient_table_decoded_row_code. bcf_height_lucas_low_digit_product_quotient_table_decoded_row_code + S (bcf_row_code_lucas_low_digit_product_quotient_table) = S ((S (bcf_row_index_lucas_low_digit_product_quotient_table)) * bcf_row_code_scale_lucas_low_digit_product_quotient)) /\ exists bcf_quotient_lucas_low_digit_product_quotient_table_decoded_row_code. bcf_row_code_code_lucas_low_digit_product_quotient = bcf_quotient_lucas_low_digit_product_quotient_table_decoded_row_code * S ((S (bcf_row_index_lucas_low_digit_product_quotient_table)) * bcf_row_code_scale_lucas_low_digit_product_quotient) + (bcf_row_code_lucas_low_digit_product_quotient_table))) /\ ((((exists bcf_height_lucas_low_digit_product_quotient_table_decoded_row_scale. bcf_height_lucas_low_digit_product_quotient_table_decoded_row_scale + S (bcf_row_scale_lucas_low_digit_product_quotient_table) = S ((S (bcf_row_index_lucas_low_digit_product_quotient_table)) * bcf_row_scale_scale_lucas_low_digit_product_quotient)) /\ exists bcf_quotient_lucas_low_digit_product_quotient_table_decoded_row_scale. bcf_row_scale_code_lucas_low_digit_product_quotient = bcf_quotient_lucas_low_digit_product_quotient_table_decoded_row_scale * S ((S (bcf_row_index_lucas_low_digit_product_quotient_table)) * bcf_row_scale_scale_lucas_low_digit_product_quotient) + (bcf_row_scale_lucas_low_digit_product_quotient_table))) /\ ((bcf_row_index_lucas_low_digit_product_quotient_table = 0 /\ (forall bcf_index_lucas_low_digit_product_quotient_table_zero_row. (exists bcf_lt_gap_lucas_low_digit_product_quotient_table_zero_row_bound. bcf_lt_gap_lucas_low_digit_product_quotient_table_zero_row_bound + S (bcf_index_lucas_low_digit_product_quotient_table_zero_row) = S (q)) -> exists bcf_value_lucas_low_digit_product_quotient_table_zero_row. ((((exists bcf_height_lucas_low_digit_product_quotient_table_zero_row_entry. bcf_height_lucas_low_digit_product_quotient_table_zero_row_entry + S (bcf_value_lucas_low_digit_product_quotient_table_zero_row) = S ((S (bcf_index_lucas_low_digit_product_quotient_table_zero_row)) * bcf_row_scale_lucas_low_digit_product_quotient_table)) /\ exists bcf_quotient_lucas_low_digit_product_quotient_table_zero_row_entry. bcf_row_code_lucas_low_digit_product_quotient_table = bcf_quotient_lucas_low_digit_product_quotient_table_zero_row_entry * S ((S (bcf_index_lucas_low_digit_product_quotient_table_zero_row)) * bcf_row_scale_lucas_low_digit_product_quotient_table) + (bcf_value_lucas_low_digit_product_quotient_table_zero_row))) /\ ((bcf_index_lucas_low_digit_product_quotient_table_zero_row = 0 /\ bcf_value_lucas_low_digit_product_quotient_table_zero_row = 1) \/ exists bcf_predecessor_lucas_low_digit_product_quotient_table_zero_row. bcf_index_lucas_low_digit_product_quotient_table_zero_row = S bcf_predecessor_lucas_low_digit_product_quotient_table_zero_row /\ bcf_value_lucas_low_digit_product_quotient_table_zero_row = 0)))) \/ exists bcf_predecessor_lucas_low_digit_product_quotient_table bcf_previous_code_lucas_low_digit_product_quotient_table bcf_previous_scale_lucas_low_digit_product_quotient_table. bcf_row_index_lucas_low_digit_product_quotient_table = S bcf_predecessor_lucas_low_digit_product_quotient_table /\ ((((exists bcf_height_lucas_low_digit_product_quotient_table_decoded_previous_code. bcf_height_lucas_low_digit_product_quotient_table_decoded_previous_code + S (bcf_previous_code_lucas_low_digit_product_quotient_table) = S ((S (bcf_predecessor_lucas_low_digit_product_quotient_table)) * bcf_row_code_scale_lucas_low_digit_product_quotient)) /\ exists bcf_quotient_lucas_low_digit_product_quotient_table_decoded_previous_code. bcf_row_code_code_lucas_low_digit_product_quotient = bcf_quotient_lucas_low_digit_product_quotient_table_decoded_previous_code * S ((S (bcf_predecessor_lucas_low_digit_product_quotient_table)) * bcf_row_code_scale_lucas_low_digit_product_quotient) + (bcf_previous_code_lucas_low_digit_product_quotient_table))) /\ ((((exists bcf_height_lucas_low_digit_product_quotient_table_decoded_previous_scale. bcf_height_lucas_low_digit_product_quotient_table_decoded_previous_scale + S (bcf_previous_scale_lucas_low_digit_product_quotient_table) = S ((S (bcf_predecessor_lucas_low_digit_product_quotient_table)) * bcf_row_scale_scale_lucas_low_digit_product_quotient)) /\ exists bcf_quotient_lucas_low_digit_product_quotient_table_decoded_previous_scale. bcf_row_scale_code_lucas_low_digit_product_quotient = bcf_quotient_lucas_low_digit_product_quotient_table_decoded_previous_scale * S ((S (bcf_predecessor_lucas_low_digit_product_quotient_table)) * bcf_row_scale_scale_lucas_low_digit_product_quotient) + (bcf_previous_scale_lucas_low_digit_product_quotient_table))) /\ (forall bcf_index_lucas_low_digit_product_quotient_table_row_step. (exists bcf_lt_gap_lucas_low_digit_product_quotient_table_row_step_bound. bcf_lt_gap_lucas_low_digit_product_quotient_table_row_step_bound + S (bcf_index_lucas_low_digit_product_quotient_table_row_step) = S (q)) -> exists bcf_value_lucas_low_digit_product_quotient_table_row_step. ((((exists bcf_height_lucas_low_digit_product_quotient_table_row_step_entry. bcf_height_lucas_low_digit_product_quotient_table_row_step_entry + S (bcf_value_lucas_low_digit_product_quotient_table_row_step) = S ((S (bcf_index_lucas_low_digit_product_quotient_table_row_step)) * bcf_row_scale_lucas_low_digit_product_quotient_table)) /\ exists bcf_quotient_lucas_low_digit_product_quotient_table_row_step_entry. bcf_row_code_lucas_low_digit_product_quotient_table = bcf_quotient_lucas_low_digit_product_quotient_table_row_step_entry * S ((S (bcf_index_lucas_low_digit_product_quotient_table_row_step)) * bcf_row_scale_lucas_low_digit_product_quotient_table) + (bcf_value_lucas_low_digit_product_quotient_table_row_step))) /\ ((bcf_index_lucas_low_digit_product_quotient_table_row_step = 0 /\ bcf_value_lucas_low_digit_product_quotient_table_row_step = 1) \/ exists bcf_predecessor_lucas_low_digit_product_quotient_table_row_step bcf_left_lucas_low_digit_product_quotient_table_row_step bcf_right_lucas_low_digit_product_quotient_table_row_step. bcf_index_lucas_low_digit_product_quotient_table_row_step = S bcf_predecessor_lucas_low_digit_product_quotient_table_row_step /\ ((((exists bcf_height_lucas_low_digit_product_quotient_table_row_step_previous_left. bcf_height_lucas_low_digit_product_quotient_table_row_step_previous_left + S (bcf_left_lucas_low_digit_product_quotient_table_row_step) = S ((S (bcf_predecessor_lucas_low_digit_product_quotient_table_row_step)) * bcf_previous_scale_lucas_low_digit_product_quotient_table)) /\ exists bcf_quotient_lucas_low_digit_product_quotient_table_row_step_previous_left. bcf_previous_code_lucas_low_digit_product_quotient_table = bcf_quotient_lucas_low_digit_product_quotient_table_row_step_previous_left * S ((S (bcf_predecessor_lucas_low_digit_product_quotient_table_row_step)) * bcf_previous_scale_lucas_low_digit_product_quotient_table) + (bcf_left_lucas_low_digit_product_quotient_table_row_step))) /\ ((((exists bcf_height_lucas_low_digit_product_quotient_table_row_step_previous_right. bcf_height_lucas_low_digit_product_quotient_table_row_step_previous_right + S (bcf_right_lucas_low_digit_product_quotient_table_row_step) = S ((S (S (bcf_predecessor_lucas_low_digit_product_quotient_table_row_step))) * bcf_previous_scale_lucas_low_digit_product_quotient_table)) /\ exists bcf_quotient_lucas_low_digit_product_quotient_table_row_step_previous_right. bcf_previous_code_lucas_low_digit_product_quotient_table = bcf_quotient_lucas_low_digit_product_quotient_table_row_step_previous_right * S ((S (S (bcf_predecessor_lucas_low_digit_product_quotient_table_row_step))) * bcf_previous_scale_lucas_low_digit_product_quotient_table) + (bcf_right_lucas_low_digit_product_quotient_table_row_step))) /\ bcf_value_lucas_low_digit_product_quotient_table_row_step = bcf_left_lucas_low_digit_product_quotient_table_row_step + bcf_right_lucas_low_digit_product_quotient_table_row_step))))))))))) /\ ((((exists bcf_height_lucas_low_digit_product_quotient_decoded_row_code. bcf_height_lucas_low_digit_product_quotient_decoded_row_code + S (bcf_row_code_lucas_low_digit_product_quotient) = S ((S (q)) * bcf_row_code_scale_lucas_low_digit_product_quotient)) /\ exists bcf_quotient_lucas_low_digit_product_quotient_decoded_row_code. bcf_row_code_code_lucas_low_digit_product_quotient = bcf_quotient_lucas_low_digit_product_quotient_decoded_row_code * S ((S (q)) * bcf_row_code_scale_lucas_low_digit_product_quotient) + (bcf_row_code_lucas_low_digit_product_quotient))) /\ ((((exists bcf_height_lucas_low_digit_product_quotient_decoded_row_scale. bcf_height_lucas_low_digit_product_quotient_decoded_row_scale + S (bcf_row_scale_lucas_low_digit_product_quotient) = S ((S (q)) * bcf_row_scale_scale_lucas_low_digit_product_quotient)) /\ exists bcf_quotient_lucas_low_digit_product_quotient_decoded_row_scale. bcf_row_scale_code_lucas_low_digit_product_quotient = bcf_quotient_lucas_low_digit_product_quotient_decoded_row_scale * S ((S (q)) * bcf_row_scale_scale_lucas_low_digit_product_quotient) + (bcf_row_scale_lucas_low_digit_product_quotient))) /\ (((exists bcf_height_lucas_low_digit_product_quotient_decoded_value. bcf_height_lucas_low_digit_product_quotient_decoded_value + S (A) = S ((S (0)) * bcf_row_scale_lucas_low_digit_product_quotient)) /\ exists bcf_quotient_lucas_low_digit_product_quotient_decoded_value. bcf_row_code_lucas_low_digit_product_quotient = bcf_quotient_lucas_low_digit_product_quotient_decoded_value * S ((S (0)) * bcf_row_scale_lucas_low_digit_product_quotient) + (A))))))))) -> (((exists bcf_lt_gap_lucas_low_digit_product_digit_out_of_range. bcf_lt_gap_lucas_low_digit_product_digit_out_of_range + S (d) = e) /\ B = 0) \/ ((exists bcf_le_gap_lucas_low_digit_product_digit_in_range. bcf_le_gap_lucas_low_digit_product_digit_in_range + (e) = d) /\ (exists bcf_row_code_code_lucas_low_digit_product_digit bcf_row_code_scale_lucas_low_digit_product_digit bcf_row_scale_code_lucas_low_digit_product_digit bcf_row_scale_scale_lucas_low_digit_product_digit bcf_row_code_lucas_low_digit_product_digit bcf_row_scale_lucas_low_digit_product_digit. ((forall bcf_row_index_lucas_low_digit_product_digit_table. (exists bcf_lt_gap_lucas_low_digit_product_digit_table_row_bound. bcf_lt_gap_lucas_low_digit_product_digit_table_row_bound + S (bcf_row_index_lucas_low_digit_product_digit_table) = S (d)) -> exists bcf_row_code_lucas_low_digit_product_digit_table bcf_row_scale_lucas_low_digit_product_digit_table. ((((exists bcf_height_lucas_low_digit_product_digit_table_decoded_row_code. bcf_height_lucas_low_digit_product_digit_table_decoded_row_code + S (bcf_row_code_lucas_low_digit_product_digit_table) = S ((S (bcf_row_index_lucas_low_digit_product_digit_table)) * bcf_row_code_scale_lucas_low_digit_product_digit)) /\ exists bcf_quotient_lucas_low_digit_product_digit_table_decoded_row_code. bcf_row_code_code_lucas_low_digit_product_digit = bcf_quotient_lucas_low_digit_product_digit_table_decoded_row_code * S ((S (bcf_row_index_lucas_low_digit_product_digit_table)) * bcf_row_code_scale_lucas_low_digit_product_digit) + (bcf_row_code_lucas_low_digit_product_digit_table))) /\ ((((exists bcf_height_lucas_low_digit_product_digit_table_decoded_row_scale. bcf_height_lucas_low_digit_product_digit_table_decoded_row_scale + S (bcf_row_scale_lucas_low_digit_product_digit_table) = S ((S (bcf_row_index_lucas_low_digit_product_digit_table)) * bcf_row_scale_scale_lucas_low_digit_product_digit)) /\ exists bcf_quotient_lucas_low_digit_product_digit_table_decoded_row_scale. bcf_row_scale_code_lucas_low_digit_product_digit = bcf_quotient_lucas_low_digit_product_digit_table_decoded_row_scale * S ((S (bcf_row_index_lucas_low_digit_product_digit_table)) * bcf_row_scale_scale_lucas_low_digit_product_digit) + (bcf_row_scale_lucas_low_digit_product_digit_table))) /\ ((bcf_row_index_lucas_low_digit_product_digit_table = 0 /\ (forall bcf_index_lucas_low_digit_product_digit_table_zero_row. (exists bcf_lt_gap_lucas_low_digit_product_digit_table_zero_row_bound. bcf_lt_gap_lucas_low_digit_product_digit_table_zero_row_bound + S (bcf_index_lucas_low_digit_product_digit_table_zero_row) = S (d)) -> exists bcf_value_lucas_low_digit_product_digit_table_zero_row. ((((exists bcf_height_lucas_low_digit_product_digit_table_zero_row_entry. bcf_height_lucas_low_digit_product_digit_table_zero_row_entry + S (bcf_value_lucas_low_digit_product_digit_table_zero_row) = S ((S (bcf_index_lucas_low_digit_product_digit_table_zero_row)) * bcf_row_scale_lucas_low_digit_product_digit_table)) /\ exists bcf_quotient_lucas_low_digit_product_digit_table_zero_row_entry. bcf_row_code_lucas_low_digit_product_digit_table = bcf_quotient_lucas_low_digit_product_digit_table_zero_row_entry * S ((S (bcf_index_lucas_low_digit_product_digit_table_zero_row)) * bcf_row_scale_lucas_low_digit_product_digit_table) + (bcf_value_lucas_low_digit_product_digit_table_zero_row))) /\ ((bcf_index_lucas_low_digit_product_digit_table_zero_row = 0 /\ bcf_value_lucas_low_digit_product_digit_table_zero_row = 1) \/ exists bcf_predecessor_lucas_low_digit_product_digit_table_zero_row. bcf_index_lucas_low_digit_product_digit_table_zero_row = S bcf_predecessor_lucas_low_digit_product_digit_table_zero_row /\ bcf_value_lucas_low_digit_product_digit_table_zero_row = 0)))) \/ exists bcf_predecessor_lucas_low_digit_product_digit_table bcf_previous_code_lucas_low_digit_product_digit_table bcf_previous_scale_lucas_low_digit_product_digit_table. bcf_row_index_lucas_low_digit_product_digit_table = S bcf_predecessor_lucas_low_digit_product_digit_table /\ ((((exists bcf_height_lucas_low_digit_product_digit_table_decoded_previous_code. bcf_height_lucas_low_digit_product_digit_table_decoded_previous_code + S (bcf_previous_code_lucas_low_digit_product_digit_table) = S ((S (bcf_predecessor_lucas_low_digit_product_digit_table)) * bcf_row_code_scale_lucas_low_digit_product_digit)) /\ exists bcf_quotient_lucas_low_digit_product_digit_table_decoded_previous_code. bcf_row_code_code_lucas_low_digit_product_digit = bcf_quotient_lucas_low_digit_product_digit_table_decoded_previous_code * S ((S (bcf_predecessor_lucas_low_digit_product_digit_table)) * bcf_row_code_scale_lucas_low_digit_product_digit) + (bcf_previous_code_lucas_low_digit_product_digit_table))) /\ ((((exists bcf_height_lucas_low_digit_product_digit_table_decoded_previous_scale. bcf_height_lucas_low_digit_product_digit_table_decoded_previous_scale + S (bcf_previous_scale_lucas_low_digit_product_digit_table) = S ((S (bcf_predecessor_lucas_low_digit_product_digit_table)) * bcf_row_scale_scale_lucas_low_digit_product_digit)) /\ exists bcf_quotient_lucas_low_digit_product_digit_table_decoded_previous_scale. bcf_row_scale_code_lucas_low_digit_product_digit = bcf_quotient_lucas_low_digit_product_digit_table_decoded_previous_scale * S ((S (bcf_predecessor_lucas_low_digit_product_digit_table)) * bcf_row_scale_scale_lucas_low_digit_product_digit) + (bcf_previous_scale_lucas_low_digit_product_digit_table))) /\ (forall bcf_index_lucas_low_digit_product_digit_table_row_step. (exists bcf_lt_gap_lucas_low_digit_product_digit_table_row_step_bound. bcf_lt_gap_lucas_low_digit_product_digit_table_row_step_bound + S (bcf_index_lucas_low_digit_product_digit_table_row_step) = S (d)) -> exists bcf_value_lucas_low_digit_product_digit_table_row_step. ((((exists bcf_height_lucas_low_digit_product_digit_table_row_step_entry. bcf_height_lucas_low_digit_product_digit_table_row_step_entry + S (bcf_value_lucas_low_digit_product_digit_table_row_step) = S ((S (bcf_index_lucas_low_digit_product_digit_table_row_step)) * bcf_row_scale_lucas_low_digit_product_digit_table)) /\ exists bcf_quotient_lucas_low_digit_product_digit_table_row_step_entry. bcf_row_code_lucas_low_digit_product_digit_table = bcf_quotient_lucas_low_digit_product_digit_table_row_step_entry * S ((S (bcf_index_lucas_low_digit_product_digit_table_row_step)) * bcf_row_scale_lucas_low_digit_product_digit_table) + (bcf_value_lucas_low_digit_product_digit_table_row_step))) /\ ((bcf_index_lucas_low_digit_product_digit_table_row_step = 0 /\ bcf_value_lucas_low_digit_product_digit_table_row_step = 1) \/ exists bcf_predecessor_lucas_low_digit_product_digit_table_row_step bcf_left_lucas_low_digit_product_digit_table_row_step bcf_right_lucas_low_digit_product_digit_table_row_step. bcf_index_lucas_low_digit_product_digit_table_row_step = S bcf_predecessor_lucas_low_digit_product_digit_table_row_step /\ ((((exists bcf_height_lucas_low_digit_product_digit_table_row_step_previous_left. bcf_height_lucas_low_digit_product_digit_table_row_step_previous_left + S (bcf_left_lucas_low_digit_product_digit_table_row_step) = S ((S (bcf_predecessor_lucas_low_digit_product_digit_table_row_step)) * bcf_previous_scale_lucas_low_digit_product_digit_table)) /\ exists bcf_quotient_lucas_low_digit_product_digit_table_row_step_previous_left. bcf_previous_code_lucas_low_digit_product_digit_table = bcf_quotient_lucas_low_digit_product_digit_table_row_step_previous_left * S ((S (bcf_predecessor_lucas_low_digit_product_digit_table_row_step)) * bcf_previous_scale_lucas_low_digit_product_digit_table) + (bcf_left_lucas_low_digit_product_digit_table_row_step))) /\ ((((exists bcf_height_lucas_low_digit_product_digit_table_row_step_previous_right. bcf_height_lucas_low_digit_product_digit_table_row_step_previous_right + S (bcf_right_lucas_low_digit_product_digit_table_row_step) = S ((S (S (bcf_predecessor_lucas_low_digit_product_digit_table_row_step))) * bcf_previous_scale_lucas_low_digit_product_digit_table)) /\ exists bcf_quotient_lucas_low_digit_product_digit_table_row_step_previous_right. bcf_previous_code_lucas_low_digit_product_digit_table = bcf_quotient_lucas_low_digit_product_digit_table_row_step_previous_right * S ((S (S (bcf_predecessor_lucas_low_digit_product_digit_table_row_step))) * bcf_previous_scale_lucas_low_digit_product_digit_table) + (bcf_right_lucas_low_digit_product_digit_table_row_step))) /\ bcf_value_lucas_low_digit_product_digit_table_row_step = bcf_left_lucas_low_digit_product_digit_table_row_step + bcf_right_lucas_low_digit_product_digit_table_row_step))))))))))) /\ ((((exists bcf_height_lucas_low_digit_product_digit_decoded_row_code. bcf_height_lucas_low_digit_product_digit_decoded_row_code + S (bcf_row_code_lucas_low_digit_product_digit) = S ((S (d)) * bcf_row_code_scale_lucas_low_digit_product_digit)) /\ exists bcf_quotient_lucas_low_digit_product_digit_decoded_row_code. bcf_row_code_code_lucas_low_digit_product_digit = bcf_quotient_lucas_low_digit_product_digit_decoded_row_code * S ((S (d)) * bcf_row_code_scale_lucas_low_digit_product_digit) + (bcf_row_code_lucas_low_digit_product_digit))) /\ ((((exists bcf_height_lucas_low_digit_product_digit_decoded_row_scale. bcf_height_lucas_low_digit_product_digit_decoded_row_scale + S (bcf_row_scale_lucas_low_digit_product_digit) = S ((S (d)) * bcf_row_scale_scale_lucas_low_digit_product_digit)) /\ exists bcf_quotient_lucas_low_digit_product_digit_decoded_row_scale. bcf_row_scale_code_lucas_low_digit_product_digit = bcf_quotient_lucas_low_digit_product_digit_decoded_row_scale * S ((S (d)) * bcf_row_scale_scale_lucas_low_digit_product_digit) + (bcf_row_scale_lucas_low_digit_product_digit))) /\ (((exists bcf_height_lucas_low_digit_product_digit_decoded_value. bcf_height_lucas_low_digit_product_digit_decoded_value + S (B) = S ((S (e)) * bcf_row_scale_lucas_low_digit_product_digit)) /\ exists bcf_quotient_lucas_low_digit_product_digit_decoded_value. bcf_row_code_lucas_low_digit_product_digit = bcf_quotient_lucas_low_digit_product_digit_decoded_value * S ((S (e)) * bcf_row_scale_lucas_low_digit_product_digit) + (B))))))))) -> (exists lld_left_product_result lld_right_product_result. (C) + (p) * lld_left_product_result = (A * B) + (p) * lld_right_product_result)