lucas_one_step_division_congruence

Exact expanded first-order arithmetic statement

forall p n k q r d e C A B. ((~(p = 1) /\ forall frm_prime_left_lucas_block_digit_prime frm_prime_right_lucas_block_digit_prime. p = frm_prime_left_lucas_block_digit_prime * frm_prime_right_lucas_block_digit_prime -> frm_prime_left_lucas_block_digit_prime = 1 \/ frm_prime_right_lucas_block_digit_prime = 1)) -> n = p * q + d -> k = p * r + e -> (exists lbd_gap_division_upper_bound. lbd_gap_division_upper_bound + S (d) = (p)) -> (exists lbd_gap_division_lower_bound. lbd_gap_division_lower_bound + S (e) = (p)) -> (((exists bcf_lt_gap_lucas_block_digit_division_whole_out_of_range. bcf_lt_gap_lucas_block_digit_division_whole_out_of_range + S (n) = k) /\ C = 0) \/ ((exists bcf_le_gap_lucas_block_digit_division_whole_in_range. bcf_le_gap_lucas_block_digit_division_whole_in_range + (k) = n) /\ (exists bcf_row_code_code_lucas_block_digit_division_whole bcf_row_code_scale_lucas_block_digit_division_whole bcf_row_scale_code_lucas_block_digit_division_whole bcf_row_scale_scale_lucas_block_digit_division_whole bcf_row_code_lucas_block_digit_division_whole bcf_row_scale_lucas_block_digit_division_whole. ((forall bcf_row_index_lucas_block_digit_division_whole_table. (exists bcf_lt_gap_lucas_block_digit_division_whole_table_row_bound. bcf_lt_gap_lucas_block_digit_division_whole_table_row_bound + S (bcf_row_index_lucas_block_digit_division_whole_table) = S (n)) -> exists bcf_row_code_lucas_block_digit_division_whole_table bcf_row_scale_lucas_block_digit_division_whole_table. ((((exists bcf_height_lucas_block_digit_division_whole_table_decoded_row_code. bcf_height_lucas_block_digit_division_whole_table_decoded_row_code + S (bcf_row_code_lucas_block_digit_division_whole_table) = S ((S (bcf_row_index_lucas_block_digit_division_whole_table)) * bcf_row_code_scale_lucas_block_digit_division_whole)) /\ exists bcf_quotient_lucas_block_digit_division_whole_table_decoded_row_code. bcf_row_code_code_lucas_block_digit_division_whole = bcf_quotient_lucas_block_digit_division_whole_table_decoded_row_code * S ((S (bcf_row_index_lucas_block_digit_division_whole_table)) * bcf_row_code_scale_lucas_block_digit_division_whole) + (bcf_row_code_lucas_block_digit_division_whole_table))) /\ ((((exists bcf_height_lucas_block_digit_division_whole_table_decoded_row_scale. bcf_height_lucas_block_digit_division_whole_table_decoded_row_scale + S (bcf_row_scale_lucas_block_digit_division_whole_table) = S ((S (bcf_row_index_lucas_block_digit_division_whole_table)) * bcf_row_scale_scale_lucas_block_digit_division_whole)) /\ exists bcf_quotient_lucas_block_digit_division_whole_table_decoded_row_scale. bcf_row_scale_code_lucas_block_digit_division_whole = bcf_quotient_lucas_block_digit_division_whole_table_decoded_row_scale * S ((S (bcf_row_index_lucas_block_digit_division_whole_table)) * bcf_row_scale_scale_lucas_block_digit_division_whole) + (bcf_row_scale_lucas_block_digit_division_whole_table))) /\ ((bcf_row_index_lucas_block_digit_division_whole_table = 0 /\ (forall bcf_index_lucas_block_digit_division_whole_table_zero_row. (exists bcf_lt_gap_lucas_block_digit_division_whole_table_zero_row_bound. bcf_lt_gap_lucas_block_digit_division_whole_table_zero_row_bound + S (bcf_index_lucas_block_digit_division_whole_table_zero_row) = S (n)) -> exists bcf_value_lucas_block_digit_division_whole_table_zero_row. ((((exists bcf_height_lucas_block_digit_division_whole_table_zero_row_entry. bcf_height_lucas_block_digit_division_whole_table_zero_row_entry + S (bcf_value_lucas_block_digit_division_whole_table_zero_row) = S ((S (bcf_index_lucas_block_digit_division_whole_table_zero_row)) * bcf_row_scale_lucas_block_digit_division_whole_table)) /\ exists bcf_quotient_lucas_block_digit_division_whole_table_zero_row_entry. bcf_row_code_lucas_block_digit_division_whole_table = bcf_quotient_lucas_block_digit_division_whole_table_zero_row_entry * S ((S (bcf_index_lucas_block_digit_division_whole_table_zero_row)) * bcf_row_scale_lucas_block_digit_division_whole_table) + (bcf_value_lucas_block_digit_division_whole_table_zero_row))) /\ ((bcf_index_lucas_block_digit_division_whole_table_zero_row = 0 /\ bcf_value_lucas_block_digit_division_whole_table_zero_row = 1) \/ exists bcf_predecessor_lucas_block_digit_division_whole_table_zero_row. bcf_index_lucas_block_digit_division_whole_table_zero_row = S bcf_predecessor_lucas_block_digit_division_whole_table_zero_row /\ bcf_value_lucas_block_digit_division_whole_table_zero_row = 0)))) \/ exists bcf_predecessor_lucas_block_digit_division_whole_table bcf_previous_code_lucas_block_digit_division_whole_table bcf_previous_scale_lucas_block_digit_division_whole_table. bcf_row_index_lucas_block_digit_division_whole_table = S bcf_predecessor_lucas_block_digit_division_whole_table /\ ((((exists bcf_height_lucas_block_digit_division_whole_table_decoded_previous_code. bcf_height_lucas_block_digit_division_whole_table_decoded_previous_code + S (bcf_previous_code_lucas_block_digit_division_whole_table) = S ((S (bcf_predecessor_lucas_block_digit_division_whole_table)) * bcf_row_code_scale_lucas_block_digit_division_whole)) /\ exists bcf_quotient_lucas_block_digit_division_whole_table_decoded_previous_code. bcf_row_code_code_lucas_block_digit_division_whole = bcf_quotient_lucas_block_digit_division_whole_table_decoded_previous_code * S ((S (bcf_predecessor_lucas_block_digit_division_whole_table)) * bcf_row_code_scale_lucas_block_digit_division_whole) + (bcf_previous_code_lucas_block_digit_division_whole_table))) /\ ((((exists bcf_height_lucas_block_digit_division_whole_table_decoded_previous_scale. bcf_height_lucas_block_digit_division_whole_table_decoded_previous_scale + S (bcf_previous_scale_lucas_block_digit_division_whole_table) = S ((S (bcf_predecessor_lucas_block_digit_division_whole_table)) * bcf_row_scale_scale_lucas_block_digit_division_whole)) /\ exists bcf_quotient_lucas_block_digit_division_whole_table_decoded_previous_scale. bcf_row_scale_code_lucas_block_digit_division_whole = bcf_quotient_lucas_block_digit_division_whole_table_decoded_previous_scale * S ((S (bcf_predecessor_lucas_block_digit_division_whole_table)) * bcf_row_scale_scale_lucas_block_digit_division_whole) + (bcf_previous_scale_lucas_block_digit_division_whole_table))) /\ (forall bcf_index_lucas_block_digit_division_whole_table_row_step. (exists bcf_lt_gap_lucas_block_digit_division_whole_table_row_step_bound. bcf_lt_gap_lucas_block_digit_division_whole_table_row_step_bound + S (bcf_index_lucas_block_digit_division_whole_table_row_step) = S (n)) -> exists bcf_value_lucas_block_digit_division_whole_table_row_step. ((((exists bcf_height_lucas_block_digit_division_whole_table_row_step_entry. bcf_height_lucas_block_digit_division_whole_table_row_step_entry + S (bcf_value_lucas_block_digit_division_whole_table_row_step) = S ((S (bcf_index_lucas_block_digit_division_whole_table_row_step)) * bcf_row_scale_lucas_block_digit_division_whole_table)) /\ exists bcf_quotient_lucas_block_digit_division_whole_table_row_step_entry. bcf_row_code_lucas_block_digit_division_whole_table = bcf_quotient_lucas_block_digit_division_whole_table_row_step_entry * S ((S (bcf_index_lucas_block_digit_division_whole_table_row_step)) * bcf_row_scale_lucas_block_digit_division_whole_table) + (bcf_value_lucas_block_digit_division_whole_table_row_step))) /\ ((bcf_index_lucas_block_digit_division_whole_table_row_step = 0 /\ bcf_value_lucas_block_digit_division_whole_table_row_step = 1) \/ exists bcf_predecessor_lucas_block_digit_division_whole_table_row_step bcf_left_lucas_block_digit_division_whole_table_row_step bcf_right_lucas_block_digit_division_whole_table_row_step. bcf_index_lucas_block_digit_division_whole_table_row_step = S bcf_predecessor_lucas_block_digit_division_whole_table_row_step /\ ((((exists bcf_height_lucas_block_digit_division_whole_table_row_step_previous_left. bcf_height_lucas_block_digit_division_whole_table_row_step_previous_left + S (bcf_left_lucas_block_digit_division_whole_table_row_step) = S ((S (bcf_predecessor_lucas_block_digit_division_whole_table_row_step)) * bcf_previous_scale_lucas_block_digit_division_whole_table)) /\ exists bcf_quotient_lucas_block_digit_division_whole_table_row_step_previous_left. bcf_previous_code_lucas_block_digit_division_whole_table = bcf_quotient_lucas_block_digit_division_whole_table_row_step_previous_left * S ((S (bcf_predecessor_lucas_block_digit_division_whole_table_row_step)) * bcf_previous_scale_lucas_block_digit_division_whole_table) + (bcf_left_lucas_block_digit_division_whole_table_row_step))) /\ ((((exists bcf_height_lucas_block_digit_division_whole_table_row_step_previous_right. bcf_height_lucas_block_digit_division_whole_table_row_step_previous_right + S (bcf_right_lucas_block_digit_division_whole_table_row_step) = S ((S (S (bcf_predecessor_lucas_block_digit_division_whole_table_row_step))) * bcf_previous_scale_lucas_block_digit_division_whole_table)) /\ exists bcf_quotient_lucas_block_digit_division_whole_table_row_step_previous_right. bcf_previous_code_lucas_block_digit_division_whole_table = bcf_quotient_lucas_block_digit_division_whole_table_row_step_previous_right * S ((S (S (bcf_predecessor_lucas_block_digit_division_whole_table_row_step))) * bcf_previous_scale_lucas_block_digit_division_whole_table) + (bcf_right_lucas_block_digit_division_whole_table_row_step))) /\ bcf_value_lucas_block_digit_division_whole_table_row_step = bcf_left_lucas_block_digit_division_whole_table_row_step + bcf_right_lucas_block_digit_division_whole_table_row_step))))))))))) /\ ((((exists bcf_height_lucas_block_digit_division_whole_decoded_row_code. bcf_height_lucas_block_digit_division_whole_decoded_row_code + S (bcf_row_code_lucas_block_digit_division_whole) = S ((S (n)) * bcf_row_code_scale_lucas_block_digit_division_whole)) /\ exists bcf_quotient_lucas_block_digit_division_whole_decoded_row_code. bcf_row_code_code_lucas_block_digit_division_whole = bcf_quotient_lucas_block_digit_division_whole_decoded_row_code * S ((S (n)) * bcf_row_code_scale_lucas_block_digit_division_whole) + (bcf_row_code_lucas_block_digit_division_whole))) /\ ((((exists bcf_height_lucas_block_digit_division_whole_decoded_row_scale. bcf_height_lucas_block_digit_division_whole_decoded_row_scale + S (bcf_row_scale_lucas_block_digit_division_whole) = S ((S (n)) * bcf_row_scale_scale_lucas_block_digit_division_whole)) /\ exists bcf_quotient_lucas_block_digit_division_whole_decoded_row_scale. bcf_row_scale_code_lucas_block_digit_division_whole = bcf_quotient_lucas_block_digit_division_whole_decoded_row_scale * S ((S (n)) * bcf_row_scale_scale_lucas_block_digit_division_whole) + (bcf_row_scale_lucas_block_digit_division_whole))) /\ (((exists bcf_height_lucas_block_digit_division_whole_decoded_value. bcf_height_lucas_block_digit_division_whole_decoded_value + S (C) = S ((S (k)) * bcf_row_scale_lucas_block_digit_division_whole)) /\ exists bcf_quotient_lucas_block_digit_division_whole_decoded_value. bcf_row_code_lucas_block_digit_division_whole = bcf_quotient_lucas_block_digit_division_whole_decoded_value * S ((S (k)) * bcf_row_scale_lucas_block_digit_division_whole) + (C))))))))) -> (((exists bcf_lt_gap_lucas_block_digit_division_quotient_out_of_range. bcf_lt_gap_lucas_block_digit_division_quotient_out_of_range + S (q) = r) /\ A = 0) \/ ((exists bcf_le_gap_lucas_block_digit_division_quotient_in_range. bcf_le_gap_lucas_block_digit_division_quotient_in_range + (r) = q) /\ (exists bcf_row_code_code_lucas_block_digit_division_quotient bcf_row_code_scale_lucas_block_digit_division_quotient bcf_row_scale_code_lucas_block_digit_division_quotient bcf_row_scale_scale_lucas_block_digit_division_quotient bcf_row_code_lucas_block_digit_division_quotient bcf_row_scale_lucas_block_digit_division_quotient. ((forall bcf_row_index_lucas_block_digit_division_quotient_table. (exists bcf_lt_gap_lucas_block_digit_division_quotient_table_row_bound. bcf_lt_gap_lucas_block_digit_division_quotient_table_row_bound + S (bcf_row_index_lucas_block_digit_division_quotient_table) = S (q)) -> exists bcf_row_code_lucas_block_digit_division_quotient_table bcf_row_scale_lucas_block_digit_division_quotient_table. ((((exists bcf_height_lucas_block_digit_division_quotient_table_decoded_row_code. bcf_height_lucas_block_digit_division_quotient_table_decoded_row_code + S (bcf_row_code_lucas_block_digit_division_quotient_table) = S ((S (bcf_row_index_lucas_block_digit_division_quotient_table)) * bcf_row_code_scale_lucas_block_digit_division_quotient)) /\ exists bcf_quotient_lucas_block_digit_division_quotient_table_decoded_row_code. bcf_row_code_code_lucas_block_digit_division_quotient = bcf_quotient_lucas_block_digit_division_quotient_table_decoded_row_code * S ((S (bcf_row_index_lucas_block_digit_division_quotient_table)) * bcf_row_code_scale_lucas_block_digit_division_quotient) + (bcf_row_code_lucas_block_digit_division_quotient_table))) /\ ((((exists bcf_height_lucas_block_digit_division_quotient_table_decoded_row_scale. bcf_height_lucas_block_digit_division_quotient_table_decoded_row_scale + S (bcf_row_scale_lucas_block_digit_division_quotient_table) = S ((S (bcf_row_index_lucas_block_digit_division_quotient_table)) * bcf_row_scale_scale_lucas_block_digit_division_quotient)) /\ exists bcf_quotient_lucas_block_digit_division_quotient_table_decoded_row_scale. bcf_row_scale_code_lucas_block_digit_division_quotient = bcf_quotient_lucas_block_digit_division_quotient_table_decoded_row_scale * S ((S (bcf_row_index_lucas_block_digit_division_quotient_table)) * bcf_row_scale_scale_lucas_block_digit_division_quotient) + (bcf_row_scale_lucas_block_digit_division_quotient_table))) /\ ((bcf_row_index_lucas_block_digit_division_quotient_table = 0 /\ (forall bcf_index_lucas_block_digit_division_quotient_table_zero_row. (exists bcf_lt_gap_lucas_block_digit_division_quotient_table_zero_row_bound. bcf_lt_gap_lucas_block_digit_division_quotient_table_zero_row_bound + S (bcf_index_lucas_block_digit_division_quotient_table_zero_row) = S (q)) -> exists bcf_value_lucas_block_digit_division_quotient_table_zero_row. ((((exists bcf_height_lucas_block_digit_division_quotient_table_zero_row_entry. bcf_height_lucas_block_digit_division_quotient_table_zero_row_entry + S (bcf_value_lucas_block_digit_division_quotient_table_zero_row) = S ((S (bcf_index_lucas_block_digit_division_quotient_table_zero_row)) * bcf_row_scale_lucas_block_digit_division_quotient_table)) /\ exists bcf_quotient_lucas_block_digit_division_quotient_table_zero_row_entry. bcf_row_code_lucas_block_digit_division_quotient_table = bcf_quotient_lucas_block_digit_division_quotient_table_zero_row_entry * S ((S (bcf_index_lucas_block_digit_division_quotient_table_zero_row)) * bcf_row_scale_lucas_block_digit_division_quotient_table) + (bcf_value_lucas_block_digit_division_quotient_table_zero_row))) /\ ((bcf_index_lucas_block_digit_division_quotient_table_zero_row = 0 /\ bcf_value_lucas_block_digit_division_quotient_table_zero_row = 1) \/ exists bcf_predecessor_lucas_block_digit_division_quotient_table_zero_row. bcf_index_lucas_block_digit_division_quotient_table_zero_row = S bcf_predecessor_lucas_block_digit_division_quotient_table_zero_row /\ bcf_value_lucas_block_digit_division_quotient_table_zero_row = 0)))) \/ exists bcf_predecessor_lucas_block_digit_division_quotient_table bcf_previous_code_lucas_block_digit_division_quotient_table bcf_previous_scale_lucas_block_digit_division_quotient_table. bcf_row_index_lucas_block_digit_division_quotient_table = S bcf_predecessor_lucas_block_digit_division_quotient_table /\ ((((exists bcf_height_lucas_block_digit_division_quotient_table_decoded_previous_code. bcf_height_lucas_block_digit_division_quotient_table_decoded_previous_code + S (bcf_previous_code_lucas_block_digit_division_quotient_table) = S ((S (bcf_predecessor_lucas_block_digit_division_quotient_table)) * bcf_row_code_scale_lucas_block_digit_division_quotient)) /\ exists bcf_quotient_lucas_block_digit_division_quotient_table_decoded_previous_code. bcf_row_code_code_lucas_block_digit_division_quotient = bcf_quotient_lucas_block_digit_division_quotient_table_decoded_previous_code * S ((S (bcf_predecessor_lucas_block_digit_division_quotient_table)) * bcf_row_code_scale_lucas_block_digit_division_quotient) + (bcf_previous_code_lucas_block_digit_division_quotient_table))) /\ ((((exists bcf_height_lucas_block_digit_division_quotient_table_decoded_previous_scale. bcf_height_lucas_block_digit_division_quotient_table_decoded_previous_scale + S (bcf_previous_scale_lucas_block_digit_division_quotient_table) = S ((S (bcf_predecessor_lucas_block_digit_division_quotient_table)) * bcf_row_scale_scale_lucas_block_digit_division_quotient)) /\ exists bcf_quotient_lucas_block_digit_division_quotient_table_decoded_previous_scale. bcf_row_scale_code_lucas_block_digit_division_quotient = bcf_quotient_lucas_block_digit_division_quotient_table_decoded_previous_scale * S ((S (bcf_predecessor_lucas_block_digit_division_quotient_table)) * bcf_row_scale_scale_lucas_block_digit_division_quotient) + (bcf_previous_scale_lucas_block_digit_division_quotient_table))) /\ (forall bcf_index_lucas_block_digit_division_quotient_table_row_step. (exists bcf_lt_gap_lucas_block_digit_division_quotient_table_row_step_bound. bcf_lt_gap_lucas_block_digit_division_quotient_table_row_step_bound + S (bcf_index_lucas_block_digit_division_quotient_table_row_step) = S (q)) -> exists bcf_value_lucas_block_digit_division_quotient_table_row_step. ((((exists bcf_height_lucas_block_digit_division_quotient_table_row_step_entry. bcf_height_lucas_block_digit_division_quotient_table_row_step_entry + S (bcf_value_lucas_block_digit_division_quotient_table_row_step) = S ((S (bcf_index_lucas_block_digit_division_quotient_table_row_step)) * bcf_row_scale_lucas_block_digit_division_quotient_table)) /\ exists bcf_quotient_lucas_block_digit_division_quotient_table_row_step_entry. bcf_row_code_lucas_block_digit_division_quotient_table = bcf_quotient_lucas_block_digit_division_quotient_table_row_step_entry * S ((S (bcf_index_lucas_block_digit_division_quotient_table_row_step)) * bcf_row_scale_lucas_block_digit_division_quotient_table) + (bcf_value_lucas_block_digit_division_quotient_table_row_step))) /\ ((bcf_index_lucas_block_digit_division_quotient_table_row_step = 0 /\ bcf_value_lucas_block_digit_division_quotient_table_row_step = 1) \/ exists bcf_predecessor_lucas_block_digit_division_quotient_table_row_step bcf_left_lucas_block_digit_division_quotient_table_row_step bcf_right_lucas_block_digit_division_quotient_table_row_step. bcf_index_lucas_block_digit_division_quotient_table_row_step = S bcf_predecessor_lucas_block_digit_division_quotient_table_row_step /\ ((((exists bcf_height_lucas_block_digit_division_quotient_table_row_step_previous_left. bcf_height_lucas_block_digit_division_quotient_table_row_step_previous_left + S (bcf_left_lucas_block_digit_division_quotient_table_row_step) = S ((S (bcf_predecessor_lucas_block_digit_division_quotient_table_row_step)) * bcf_previous_scale_lucas_block_digit_division_quotient_table)) /\ exists bcf_quotient_lucas_block_digit_division_quotient_table_row_step_previous_left. bcf_previous_code_lucas_block_digit_division_quotient_table = bcf_quotient_lucas_block_digit_division_quotient_table_row_step_previous_left * S ((S (bcf_predecessor_lucas_block_digit_division_quotient_table_row_step)) * bcf_previous_scale_lucas_block_digit_division_quotient_table) + (bcf_left_lucas_block_digit_division_quotient_table_row_step))) /\ ((((exists bcf_height_lucas_block_digit_division_quotient_table_row_step_previous_right. bcf_height_lucas_block_digit_division_quotient_table_row_step_previous_right + S (bcf_right_lucas_block_digit_division_quotient_table_row_step) = S ((S (S (bcf_predecessor_lucas_block_digit_division_quotient_table_row_step))) * bcf_previous_scale_lucas_block_digit_division_quotient_table)) /\ exists bcf_quotient_lucas_block_digit_division_quotient_table_row_step_previous_right. bcf_previous_code_lucas_block_digit_division_quotient_table = bcf_quotient_lucas_block_digit_division_quotient_table_row_step_previous_right * S ((S (S (bcf_predecessor_lucas_block_digit_division_quotient_table_row_step))) * bcf_previous_scale_lucas_block_digit_division_quotient_table) + (bcf_right_lucas_block_digit_division_quotient_table_row_step))) /\ bcf_value_lucas_block_digit_division_quotient_table_row_step = bcf_left_lucas_block_digit_division_quotient_table_row_step + bcf_right_lucas_block_digit_division_quotient_table_row_step))))))))))) /\ ((((exists bcf_height_lucas_block_digit_division_quotient_decoded_row_code. bcf_height_lucas_block_digit_division_quotient_decoded_row_code + S (bcf_row_code_lucas_block_digit_division_quotient) = S ((S (q)) * bcf_row_code_scale_lucas_block_digit_division_quotient)) /\ exists bcf_quotient_lucas_block_digit_division_quotient_decoded_row_code. bcf_row_code_code_lucas_block_digit_division_quotient = bcf_quotient_lucas_block_digit_division_quotient_decoded_row_code * S ((S (q)) * bcf_row_code_scale_lucas_block_digit_division_quotient) + (bcf_row_code_lucas_block_digit_division_quotient))) /\ ((((exists bcf_height_lucas_block_digit_division_quotient_decoded_row_scale. bcf_height_lucas_block_digit_division_quotient_decoded_row_scale + S (bcf_row_scale_lucas_block_digit_division_quotient) = S ((S (q)) * bcf_row_scale_scale_lucas_block_digit_division_quotient)) /\ exists bcf_quotient_lucas_block_digit_division_quotient_decoded_row_scale. bcf_row_scale_code_lucas_block_digit_division_quotient = bcf_quotient_lucas_block_digit_division_quotient_decoded_row_scale * S ((S (q)) * bcf_row_scale_scale_lucas_block_digit_division_quotient) + (bcf_row_scale_lucas_block_digit_division_quotient))) /\ (((exists bcf_height_lucas_block_digit_division_quotient_decoded_value. bcf_height_lucas_block_digit_division_quotient_decoded_value + S (A) = S ((S (r)) * bcf_row_scale_lucas_block_digit_division_quotient)) /\ exists bcf_quotient_lucas_block_digit_division_quotient_decoded_value. bcf_row_code_lucas_block_digit_division_quotient = bcf_quotient_lucas_block_digit_division_quotient_decoded_value * S ((S (r)) * bcf_row_scale_lucas_block_digit_division_quotient) + (A))))))))) -> (((exists bcf_lt_gap_lucas_block_digit_division_digit_out_of_range. bcf_lt_gap_lucas_block_digit_division_digit_out_of_range + S (d) = e) /\ B = 0) \/ ((exists bcf_le_gap_lucas_block_digit_division_digit_in_range. bcf_le_gap_lucas_block_digit_division_digit_in_range + (e) = d) /\ (exists bcf_row_code_code_lucas_block_digit_division_digit bcf_row_code_scale_lucas_block_digit_division_digit bcf_row_scale_code_lucas_block_digit_division_digit bcf_row_scale_scale_lucas_block_digit_division_digit bcf_row_code_lucas_block_digit_division_digit bcf_row_scale_lucas_block_digit_division_digit. ((forall bcf_row_index_lucas_block_digit_division_digit_table. (exists bcf_lt_gap_lucas_block_digit_division_digit_table_row_bound. bcf_lt_gap_lucas_block_digit_division_digit_table_row_bound + S (bcf_row_index_lucas_block_digit_division_digit_table) = S (d)) -> exists bcf_row_code_lucas_block_digit_division_digit_table bcf_row_scale_lucas_block_digit_division_digit_table. ((((exists bcf_height_lucas_block_digit_division_digit_table_decoded_row_code. bcf_height_lucas_block_digit_division_digit_table_decoded_row_code + S (bcf_row_code_lucas_block_digit_division_digit_table) = S ((S (bcf_row_index_lucas_block_digit_division_digit_table)) * bcf_row_code_scale_lucas_block_digit_division_digit)) /\ exists bcf_quotient_lucas_block_digit_division_digit_table_decoded_row_code. bcf_row_code_code_lucas_block_digit_division_digit = bcf_quotient_lucas_block_digit_division_digit_table_decoded_row_code * S ((S (bcf_row_index_lucas_block_digit_division_digit_table)) * bcf_row_code_scale_lucas_block_digit_division_digit) + (bcf_row_code_lucas_block_digit_division_digit_table))) /\ ((((exists bcf_height_lucas_block_digit_division_digit_table_decoded_row_scale. bcf_height_lucas_block_digit_division_digit_table_decoded_row_scale + S (bcf_row_scale_lucas_block_digit_division_digit_table) = S ((S (bcf_row_index_lucas_block_digit_division_digit_table)) * bcf_row_scale_scale_lucas_block_digit_division_digit)) /\ exists bcf_quotient_lucas_block_digit_division_digit_table_decoded_row_scale. bcf_row_scale_code_lucas_block_digit_division_digit = bcf_quotient_lucas_block_digit_division_digit_table_decoded_row_scale * S ((S (bcf_row_index_lucas_block_digit_division_digit_table)) * bcf_row_scale_scale_lucas_block_digit_division_digit) + (bcf_row_scale_lucas_block_digit_division_digit_table))) /\ ((bcf_row_index_lucas_block_digit_division_digit_table = 0 /\ (forall bcf_index_lucas_block_digit_division_digit_table_zero_row. (exists bcf_lt_gap_lucas_block_digit_division_digit_table_zero_row_bound. bcf_lt_gap_lucas_block_digit_division_digit_table_zero_row_bound + S (bcf_index_lucas_block_digit_division_digit_table_zero_row) = S (d)) -> exists bcf_value_lucas_block_digit_division_digit_table_zero_row. ((((exists bcf_height_lucas_block_digit_division_digit_table_zero_row_entry. bcf_height_lucas_block_digit_division_digit_table_zero_row_entry + S (bcf_value_lucas_block_digit_division_digit_table_zero_row) = S ((S (bcf_index_lucas_block_digit_division_digit_table_zero_row)) * bcf_row_scale_lucas_block_digit_division_digit_table)) /\ exists bcf_quotient_lucas_block_digit_division_digit_table_zero_row_entry. bcf_row_code_lucas_block_digit_division_digit_table = bcf_quotient_lucas_block_digit_division_digit_table_zero_row_entry * S ((S (bcf_index_lucas_block_digit_division_digit_table_zero_row)) * bcf_row_scale_lucas_block_digit_division_digit_table) + (bcf_value_lucas_block_digit_division_digit_table_zero_row))) /\ ((bcf_index_lucas_block_digit_division_digit_table_zero_row = 0 /\ bcf_value_lucas_block_digit_division_digit_table_zero_row = 1) \/ exists bcf_predecessor_lucas_block_digit_division_digit_table_zero_row. bcf_index_lucas_block_digit_division_digit_table_zero_row = S bcf_predecessor_lucas_block_digit_division_digit_table_zero_row /\ bcf_value_lucas_block_digit_division_digit_table_zero_row = 0)))) \/ exists bcf_predecessor_lucas_block_digit_division_digit_table bcf_previous_code_lucas_block_digit_division_digit_table bcf_previous_scale_lucas_block_digit_division_digit_table. bcf_row_index_lucas_block_digit_division_digit_table = S bcf_predecessor_lucas_block_digit_division_digit_table /\ ((((exists bcf_height_lucas_block_digit_division_digit_table_decoded_previous_code. bcf_height_lucas_block_digit_division_digit_table_decoded_previous_code + S (bcf_previous_code_lucas_block_digit_division_digit_table) = S ((S (bcf_predecessor_lucas_block_digit_division_digit_table)) * bcf_row_code_scale_lucas_block_digit_division_digit)) /\ exists bcf_quotient_lucas_block_digit_division_digit_table_decoded_previous_code. bcf_row_code_code_lucas_block_digit_division_digit = bcf_quotient_lucas_block_digit_division_digit_table_decoded_previous_code * S ((S (bcf_predecessor_lucas_block_digit_division_digit_table)) * bcf_row_code_scale_lucas_block_digit_division_digit) + (bcf_previous_code_lucas_block_digit_division_digit_table))) /\ ((((exists bcf_height_lucas_block_digit_division_digit_table_decoded_previous_scale. bcf_height_lucas_block_digit_division_digit_table_decoded_previous_scale + S (bcf_previous_scale_lucas_block_digit_division_digit_table) = S ((S (bcf_predecessor_lucas_block_digit_division_digit_table)) * bcf_row_scale_scale_lucas_block_digit_division_digit)) /\ exists bcf_quotient_lucas_block_digit_division_digit_table_decoded_previous_scale. bcf_row_scale_code_lucas_block_digit_division_digit = bcf_quotient_lucas_block_digit_division_digit_table_decoded_previous_scale * S ((S (bcf_predecessor_lucas_block_digit_division_digit_table)) * bcf_row_scale_scale_lucas_block_digit_division_digit) + (bcf_previous_scale_lucas_block_digit_division_digit_table))) /\ (forall bcf_index_lucas_block_digit_division_digit_table_row_step. (exists bcf_lt_gap_lucas_block_digit_division_digit_table_row_step_bound. bcf_lt_gap_lucas_block_digit_division_digit_table_row_step_bound + S (bcf_index_lucas_block_digit_division_digit_table_row_step) = S (d)) -> exists bcf_value_lucas_block_digit_division_digit_table_row_step. ((((exists bcf_height_lucas_block_digit_division_digit_table_row_step_entry. bcf_height_lucas_block_digit_division_digit_table_row_step_entry + S (bcf_value_lucas_block_digit_division_digit_table_row_step) = S ((S (bcf_index_lucas_block_digit_division_digit_table_row_step)) * bcf_row_scale_lucas_block_digit_division_digit_table)) /\ exists bcf_quotient_lucas_block_digit_division_digit_table_row_step_entry. bcf_row_code_lucas_block_digit_division_digit_table = bcf_quotient_lucas_block_digit_division_digit_table_row_step_entry * S ((S (bcf_index_lucas_block_digit_division_digit_table_row_step)) * bcf_row_scale_lucas_block_digit_division_digit_table) + (bcf_value_lucas_block_digit_division_digit_table_row_step))) /\ ((bcf_index_lucas_block_digit_division_digit_table_row_step = 0 /\ bcf_value_lucas_block_digit_division_digit_table_row_step = 1) \/ exists bcf_predecessor_lucas_block_digit_division_digit_table_row_step bcf_left_lucas_block_digit_division_digit_table_row_step bcf_right_lucas_block_digit_division_digit_table_row_step. bcf_index_lucas_block_digit_division_digit_table_row_step = S bcf_predecessor_lucas_block_digit_division_digit_table_row_step /\ ((((exists bcf_height_lucas_block_digit_division_digit_table_row_step_previous_left. bcf_height_lucas_block_digit_division_digit_table_row_step_previous_left + S (bcf_left_lucas_block_digit_division_digit_table_row_step) = S ((S (bcf_predecessor_lucas_block_digit_division_digit_table_row_step)) * bcf_previous_scale_lucas_block_digit_division_digit_table)) /\ exists bcf_quotient_lucas_block_digit_division_digit_table_row_step_previous_left. bcf_previous_code_lucas_block_digit_division_digit_table = bcf_quotient_lucas_block_digit_division_digit_table_row_step_previous_left * S ((S (bcf_predecessor_lucas_block_digit_division_digit_table_row_step)) * bcf_previous_scale_lucas_block_digit_division_digit_table) + (bcf_left_lucas_block_digit_division_digit_table_row_step))) /\ ((((exists bcf_height_lucas_block_digit_division_digit_table_row_step_previous_right. bcf_height_lucas_block_digit_division_digit_table_row_step_previous_right + S (bcf_right_lucas_block_digit_division_digit_table_row_step) = S ((S (S (bcf_predecessor_lucas_block_digit_division_digit_table_row_step))) * bcf_previous_scale_lucas_block_digit_division_digit_table)) /\ exists bcf_quotient_lucas_block_digit_division_digit_table_row_step_previous_right. bcf_previous_code_lucas_block_digit_division_digit_table = bcf_quotient_lucas_block_digit_division_digit_table_row_step_previous_right * S ((S (S (bcf_predecessor_lucas_block_digit_division_digit_table_row_step))) * bcf_previous_scale_lucas_block_digit_division_digit_table) + (bcf_right_lucas_block_digit_division_digit_table_row_step))) /\ bcf_value_lucas_block_digit_division_digit_table_row_step = bcf_left_lucas_block_digit_division_digit_table_row_step + bcf_right_lucas_block_digit_division_digit_table_row_step))))))))))) /\ ((((exists bcf_height_lucas_block_digit_division_digit_decoded_row_code. bcf_height_lucas_block_digit_division_digit_decoded_row_code + S (bcf_row_code_lucas_block_digit_division_digit) = S ((S (d)) * bcf_row_code_scale_lucas_block_digit_division_digit)) /\ exists bcf_quotient_lucas_block_digit_division_digit_decoded_row_code. bcf_row_code_code_lucas_block_digit_division_digit = bcf_quotient_lucas_block_digit_division_digit_decoded_row_code * S ((S (d)) * bcf_row_code_scale_lucas_block_digit_division_digit) + (bcf_row_code_lucas_block_digit_division_digit))) /\ ((((exists bcf_height_lucas_block_digit_division_digit_decoded_row_scale. bcf_height_lucas_block_digit_division_digit_decoded_row_scale + S (bcf_row_scale_lucas_block_digit_division_digit) = S ((S (d)) * bcf_row_scale_scale_lucas_block_digit_division_digit)) /\ exists bcf_quotient_lucas_block_digit_division_digit_decoded_row_scale. bcf_row_scale_code_lucas_block_digit_division_digit = bcf_quotient_lucas_block_digit_division_digit_decoded_row_scale * S ((S (d)) * bcf_row_scale_scale_lucas_block_digit_division_digit) + (bcf_row_scale_lucas_block_digit_division_digit))) /\ (((exists bcf_height_lucas_block_digit_division_digit_decoded_value. bcf_height_lucas_block_digit_division_digit_decoded_value + S (B) = S ((S (e)) * bcf_row_scale_lucas_block_digit_division_digit)) /\ exists bcf_quotient_lucas_block_digit_division_digit_decoded_value. bcf_row_code_lucas_block_digit_division_digit = bcf_quotient_lucas_block_digit_division_digit_decoded_value * S ((S (e)) * bcf_row_scale_lucas_block_digit_division_digit) + (B))))))))) -> (exists lbd_left_division_result lbd_right_division_result. (C) + (p) * lbd_left_division_result = (A * B) + (p) * lbd_right_division_result)