lucas_low_digit_congruence

Exact expanded first-order arithmetic statement

forall p q d e C D. ((~(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_low_upper_bound. lld_gap_low_upper_bound + S (d) = (p)) -> (exists lld_gap_low_lower_bound. lld_gap_low_lower_bound + S (e) = (p)) -> (((exists bcf_lt_gap_lucas_low_digit_low_upper_out_of_range. bcf_lt_gap_lucas_low_digit_low_upper_out_of_range + S (p * q + d) = e) /\ C = 0) \/ ((exists bcf_le_gap_lucas_low_digit_low_upper_in_range. bcf_le_gap_lucas_low_digit_low_upper_in_range + (e) = p * q + d) /\ (exists bcf_row_code_code_lucas_low_digit_low_upper bcf_row_code_scale_lucas_low_digit_low_upper bcf_row_scale_code_lucas_low_digit_low_upper bcf_row_scale_scale_lucas_low_digit_low_upper bcf_row_code_lucas_low_digit_low_upper bcf_row_scale_lucas_low_digit_low_upper. ((forall bcf_row_index_lucas_low_digit_low_upper_table. (exists bcf_lt_gap_lucas_low_digit_low_upper_table_row_bound. bcf_lt_gap_lucas_low_digit_low_upper_table_row_bound + S (bcf_row_index_lucas_low_digit_low_upper_table) = S (p * q + d)) -> exists bcf_row_code_lucas_low_digit_low_upper_table bcf_row_scale_lucas_low_digit_low_upper_table. ((((exists bcf_height_lucas_low_digit_low_upper_table_decoded_row_code. bcf_height_lucas_low_digit_low_upper_table_decoded_row_code + S (bcf_row_code_lucas_low_digit_low_upper_table) = S ((S (bcf_row_index_lucas_low_digit_low_upper_table)) * bcf_row_code_scale_lucas_low_digit_low_upper)) /\ exists bcf_quotient_lucas_low_digit_low_upper_table_decoded_row_code. bcf_row_code_code_lucas_low_digit_low_upper = bcf_quotient_lucas_low_digit_low_upper_table_decoded_row_code * S ((S (bcf_row_index_lucas_low_digit_low_upper_table)) * bcf_row_code_scale_lucas_low_digit_low_upper) + (bcf_row_code_lucas_low_digit_low_upper_table))) /\ ((((exists bcf_height_lucas_low_digit_low_upper_table_decoded_row_scale. bcf_height_lucas_low_digit_low_upper_table_decoded_row_scale + S (bcf_row_scale_lucas_low_digit_low_upper_table) = S ((S (bcf_row_index_lucas_low_digit_low_upper_table)) * bcf_row_scale_scale_lucas_low_digit_low_upper)) /\ exists bcf_quotient_lucas_low_digit_low_upper_table_decoded_row_scale. bcf_row_scale_code_lucas_low_digit_low_upper = bcf_quotient_lucas_low_digit_low_upper_table_decoded_row_scale * S ((S (bcf_row_index_lucas_low_digit_low_upper_table)) * bcf_row_scale_scale_lucas_low_digit_low_upper) + (bcf_row_scale_lucas_low_digit_low_upper_table))) /\ ((bcf_row_index_lucas_low_digit_low_upper_table = 0 /\ (forall bcf_index_lucas_low_digit_low_upper_table_zero_row. (exists bcf_lt_gap_lucas_low_digit_low_upper_table_zero_row_bound. bcf_lt_gap_lucas_low_digit_low_upper_table_zero_row_bound + S (bcf_index_lucas_low_digit_low_upper_table_zero_row) = S (p * q + d)) -> exists bcf_value_lucas_low_digit_low_upper_table_zero_row. ((((exists bcf_height_lucas_low_digit_low_upper_table_zero_row_entry. bcf_height_lucas_low_digit_low_upper_table_zero_row_entry + S (bcf_value_lucas_low_digit_low_upper_table_zero_row) = S ((S (bcf_index_lucas_low_digit_low_upper_table_zero_row)) * bcf_row_scale_lucas_low_digit_low_upper_table)) /\ exists bcf_quotient_lucas_low_digit_low_upper_table_zero_row_entry. bcf_row_code_lucas_low_digit_low_upper_table = bcf_quotient_lucas_low_digit_low_upper_table_zero_row_entry * S ((S (bcf_index_lucas_low_digit_low_upper_table_zero_row)) * bcf_row_scale_lucas_low_digit_low_upper_table) + (bcf_value_lucas_low_digit_low_upper_table_zero_row))) /\ ((bcf_index_lucas_low_digit_low_upper_table_zero_row = 0 /\ bcf_value_lucas_low_digit_low_upper_table_zero_row = 1) \/ exists bcf_predecessor_lucas_low_digit_low_upper_table_zero_row. bcf_index_lucas_low_digit_low_upper_table_zero_row = S bcf_predecessor_lucas_low_digit_low_upper_table_zero_row /\ bcf_value_lucas_low_digit_low_upper_table_zero_row = 0)))) \/ exists bcf_predecessor_lucas_low_digit_low_upper_table bcf_previous_code_lucas_low_digit_low_upper_table bcf_previous_scale_lucas_low_digit_low_upper_table. bcf_row_index_lucas_low_digit_low_upper_table = S bcf_predecessor_lucas_low_digit_low_upper_table /\ ((((exists bcf_height_lucas_low_digit_low_upper_table_decoded_previous_code. bcf_height_lucas_low_digit_low_upper_table_decoded_previous_code + S (bcf_previous_code_lucas_low_digit_low_upper_table) = S ((S (bcf_predecessor_lucas_low_digit_low_upper_table)) * bcf_row_code_scale_lucas_low_digit_low_upper)) /\ exists bcf_quotient_lucas_low_digit_low_upper_table_decoded_previous_code. bcf_row_code_code_lucas_low_digit_low_upper = bcf_quotient_lucas_low_digit_low_upper_table_decoded_previous_code * S ((S (bcf_predecessor_lucas_low_digit_low_upper_table)) * bcf_row_code_scale_lucas_low_digit_low_upper) + (bcf_previous_code_lucas_low_digit_low_upper_table))) /\ ((((exists bcf_height_lucas_low_digit_low_upper_table_decoded_previous_scale. bcf_height_lucas_low_digit_low_upper_table_decoded_previous_scale + S (bcf_previous_scale_lucas_low_digit_low_upper_table) = S ((S (bcf_predecessor_lucas_low_digit_low_upper_table)) * bcf_row_scale_scale_lucas_low_digit_low_upper)) /\ exists bcf_quotient_lucas_low_digit_low_upper_table_decoded_previous_scale. bcf_row_scale_code_lucas_low_digit_low_upper = bcf_quotient_lucas_low_digit_low_upper_table_decoded_previous_scale * S ((S (bcf_predecessor_lucas_low_digit_low_upper_table)) * bcf_row_scale_scale_lucas_low_digit_low_upper) + (bcf_previous_scale_lucas_low_digit_low_upper_table))) /\ (forall bcf_index_lucas_low_digit_low_upper_table_row_step. (exists bcf_lt_gap_lucas_low_digit_low_upper_table_row_step_bound. bcf_lt_gap_lucas_low_digit_low_upper_table_row_step_bound + S (bcf_index_lucas_low_digit_low_upper_table_row_step) = S (p * q + d)) -> exists bcf_value_lucas_low_digit_low_upper_table_row_step. ((((exists bcf_height_lucas_low_digit_low_upper_table_row_step_entry. bcf_height_lucas_low_digit_low_upper_table_row_step_entry + S (bcf_value_lucas_low_digit_low_upper_table_row_step) = S ((S (bcf_index_lucas_low_digit_low_upper_table_row_step)) * bcf_row_scale_lucas_low_digit_low_upper_table)) /\ exists bcf_quotient_lucas_low_digit_low_upper_table_row_step_entry. bcf_row_code_lucas_low_digit_low_upper_table = bcf_quotient_lucas_low_digit_low_upper_table_row_step_entry * S ((S (bcf_index_lucas_low_digit_low_upper_table_row_step)) * bcf_row_scale_lucas_low_digit_low_upper_table) + (bcf_value_lucas_low_digit_low_upper_table_row_step))) /\ ((bcf_index_lucas_low_digit_low_upper_table_row_step = 0 /\ bcf_value_lucas_low_digit_low_upper_table_row_step = 1) \/ exists bcf_predecessor_lucas_low_digit_low_upper_table_row_step bcf_left_lucas_low_digit_low_upper_table_row_step bcf_right_lucas_low_digit_low_upper_table_row_step. bcf_index_lucas_low_digit_low_upper_table_row_step = S bcf_predecessor_lucas_low_digit_low_upper_table_row_step /\ ((((exists bcf_height_lucas_low_digit_low_upper_table_row_step_previous_left. bcf_height_lucas_low_digit_low_upper_table_row_step_previous_left + S (bcf_left_lucas_low_digit_low_upper_table_row_step) = S ((S (bcf_predecessor_lucas_low_digit_low_upper_table_row_step)) * bcf_previous_scale_lucas_low_digit_low_upper_table)) /\ exists bcf_quotient_lucas_low_digit_low_upper_table_row_step_previous_left. bcf_previous_code_lucas_low_digit_low_upper_table = bcf_quotient_lucas_low_digit_low_upper_table_row_step_previous_left * S ((S (bcf_predecessor_lucas_low_digit_low_upper_table_row_step)) * bcf_previous_scale_lucas_low_digit_low_upper_table) + (bcf_left_lucas_low_digit_low_upper_table_row_step))) /\ ((((exists bcf_height_lucas_low_digit_low_upper_table_row_step_previous_right. bcf_height_lucas_low_digit_low_upper_table_row_step_previous_right + S (bcf_right_lucas_low_digit_low_upper_table_row_step) = S ((S (S (bcf_predecessor_lucas_low_digit_low_upper_table_row_step))) * bcf_previous_scale_lucas_low_digit_low_upper_table)) /\ exists bcf_quotient_lucas_low_digit_low_upper_table_row_step_previous_right. bcf_previous_code_lucas_low_digit_low_upper_table = bcf_quotient_lucas_low_digit_low_upper_table_row_step_previous_right * S ((S (S (bcf_predecessor_lucas_low_digit_low_upper_table_row_step))) * bcf_previous_scale_lucas_low_digit_low_upper_table) + (bcf_right_lucas_low_digit_low_upper_table_row_step))) /\ bcf_value_lucas_low_digit_low_upper_table_row_step = bcf_left_lucas_low_digit_low_upper_table_row_step + bcf_right_lucas_low_digit_low_upper_table_row_step))))))))))) /\ ((((exists bcf_height_lucas_low_digit_low_upper_decoded_row_code. bcf_height_lucas_low_digit_low_upper_decoded_row_code + S (bcf_row_code_lucas_low_digit_low_upper) = S ((S (p * q + d)) * bcf_row_code_scale_lucas_low_digit_low_upper)) /\ exists bcf_quotient_lucas_low_digit_low_upper_decoded_row_code. bcf_row_code_code_lucas_low_digit_low_upper = bcf_quotient_lucas_low_digit_low_upper_decoded_row_code * S ((S (p * q + d)) * bcf_row_code_scale_lucas_low_digit_low_upper) + (bcf_row_code_lucas_low_digit_low_upper))) /\ ((((exists bcf_height_lucas_low_digit_low_upper_decoded_row_scale. bcf_height_lucas_low_digit_low_upper_decoded_row_scale + S (bcf_row_scale_lucas_low_digit_low_upper) = S ((S (p * q + d)) * bcf_row_scale_scale_lucas_low_digit_low_upper)) /\ exists bcf_quotient_lucas_low_digit_low_upper_decoded_row_scale. bcf_row_scale_code_lucas_low_digit_low_upper = bcf_quotient_lucas_low_digit_low_upper_decoded_row_scale * S ((S (p * q + d)) * bcf_row_scale_scale_lucas_low_digit_low_upper) + (bcf_row_scale_lucas_low_digit_low_upper))) /\ (((exists bcf_height_lucas_low_digit_low_upper_decoded_value. bcf_height_lucas_low_digit_low_upper_decoded_value + S (C) = S ((S (e)) * bcf_row_scale_lucas_low_digit_low_upper)) /\ exists bcf_quotient_lucas_low_digit_low_upper_decoded_value. bcf_row_code_lucas_low_digit_low_upper = bcf_quotient_lucas_low_digit_low_upper_decoded_value * S ((S (e)) * bcf_row_scale_lucas_low_digit_low_upper) + (C))))))))) -> (((exists bcf_lt_gap_lucas_low_digit_low_lower_out_of_range. bcf_lt_gap_lucas_low_digit_low_lower_out_of_range + S (d) = e) /\ D = 0) \/ ((exists bcf_le_gap_lucas_low_digit_low_lower_in_range. bcf_le_gap_lucas_low_digit_low_lower_in_range + (e) = d) /\ (exists bcf_row_code_code_lucas_low_digit_low_lower bcf_row_code_scale_lucas_low_digit_low_lower bcf_row_scale_code_lucas_low_digit_low_lower bcf_row_scale_scale_lucas_low_digit_low_lower bcf_row_code_lucas_low_digit_low_lower bcf_row_scale_lucas_low_digit_low_lower. ((forall bcf_row_index_lucas_low_digit_low_lower_table. (exists bcf_lt_gap_lucas_low_digit_low_lower_table_row_bound. bcf_lt_gap_lucas_low_digit_low_lower_table_row_bound + S (bcf_row_index_lucas_low_digit_low_lower_table) = S (d)) -> exists bcf_row_code_lucas_low_digit_low_lower_table bcf_row_scale_lucas_low_digit_low_lower_table. ((((exists bcf_height_lucas_low_digit_low_lower_table_decoded_row_code. bcf_height_lucas_low_digit_low_lower_table_decoded_row_code + S (bcf_row_code_lucas_low_digit_low_lower_table) = S ((S (bcf_row_index_lucas_low_digit_low_lower_table)) * bcf_row_code_scale_lucas_low_digit_low_lower)) /\ exists bcf_quotient_lucas_low_digit_low_lower_table_decoded_row_code. bcf_row_code_code_lucas_low_digit_low_lower = bcf_quotient_lucas_low_digit_low_lower_table_decoded_row_code * S ((S (bcf_row_index_lucas_low_digit_low_lower_table)) * bcf_row_code_scale_lucas_low_digit_low_lower) + (bcf_row_code_lucas_low_digit_low_lower_table))) /\ ((((exists bcf_height_lucas_low_digit_low_lower_table_decoded_row_scale. bcf_height_lucas_low_digit_low_lower_table_decoded_row_scale + S (bcf_row_scale_lucas_low_digit_low_lower_table) = S ((S (bcf_row_index_lucas_low_digit_low_lower_table)) * bcf_row_scale_scale_lucas_low_digit_low_lower)) /\ exists bcf_quotient_lucas_low_digit_low_lower_table_decoded_row_scale. bcf_row_scale_code_lucas_low_digit_low_lower = bcf_quotient_lucas_low_digit_low_lower_table_decoded_row_scale * S ((S (bcf_row_index_lucas_low_digit_low_lower_table)) * bcf_row_scale_scale_lucas_low_digit_low_lower) + (bcf_row_scale_lucas_low_digit_low_lower_table))) /\ ((bcf_row_index_lucas_low_digit_low_lower_table = 0 /\ (forall bcf_index_lucas_low_digit_low_lower_table_zero_row. (exists bcf_lt_gap_lucas_low_digit_low_lower_table_zero_row_bound. bcf_lt_gap_lucas_low_digit_low_lower_table_zero_row_bound + S (bcf_index_lucas_low_digit_low_lower_table_zero_row) = S (d)) -> exists bcf_value_lucas_low_digit_low_lower_table_zero_row. ((((exists bcf_height_lucas_low_digit_low_lower_table_zero_row_entry. bcf_height_lucas_low_digit_low_lower_table_zero_row_entry + S (bcf_value_lucas_low_digit_low_lower_table_zero_row) = S ((S (bcf_index_lucas_low_digit_low_lower_table_zero_row)) * bcf_row_scale_lucas_low_digit_low_lower_table)) /\ exists bcf_quotient_lucas_low_digit_low_lower_table_zero_row_entry. bcf_row_code_lucas_low_digit_low_lower_table = bcf_quotient_lucas_low_digit_low_lower_table_zero_row_entry * S ((S (bcf_index_lucas_low_digit_low_lower_table_zero_row)) * bcf_row_scale_lucas_low_digit_low_lower_table) + (bcf_value_lucas_low_digit_low_lower_table_zero_row))) /\ ((bcf_index_lucas_low_digit_low_lower_table_zero_row = 0 /\ bcf_value_lucas_low_digit_low_lower_table_zero_row = 1) \/ exists bcf_predecessor_lucas_low_digit_low_lower_table_zero_row. bcf_index_lucas_low_digit_low_lower_table_zero_row = S bcf_predecessor_lucas_low_digit_low_lower_table_zero_row /\ bcf_value_lucas_low_digit_low_lower_table_zero_row = 0)))) \/ exists bcf_predecessor_lucas_low_digit_low_lower_table bcf_previous_code_lucas_low_digit_low_lower_table bcf_previous_scale_lucas_low_digit_low_lower_table. bcf_row_index_lucas_low_digit_low_lower_table = S bcf_predecessor_lucas_low_digit_low_lower_table /\ ((((exists bcf_height_lucas_low_digit_low_lower_table_decoded_previous_code. bcf_height_lucas_low_digit_low_lower_table_decoded_previous_code + S (bcf_previous_code_lucas_low_digit_low_lower_table) = S ((S (bcf_predecessor_lucas_low_digit_low_lower_table)) * bcf_row_code_scale_lucas_low_digit_low_lower)) /\ exists bcf_quotient_lucas_low_digit_low_lower_table_decoded_previous_code. bcf_row_code_code_lucas_low_digit_low_lower = bcf_quotient_lucas_low_digit_low_lower_table_decoded_previous_code * S ((S (bcf_predecessor_lucas_low_digit_low_lower_table)) * bcf_row_code_scale_lucas_low_digit_low_lower) + (bcf_previous_code_lucas_low_digit_low_lower_table))) /\ ((((exists bcf_height_lucas_low_digit_low_lower_table_decoded_previous_scale. bcf_height_lucas_low_digit_low_lower_table_decoded_previous_scale + S (bcf_previous_scale_lucas_low_digit_low_lower_table) = S ((S (bcf_predecessor_lucas_low_digit_low_lower_table)) * bcf_row_scale_scale_lucas_low_digit_low_lower)) /\ exists bcf_quotient_lucas_low_digit_low_lower_table_decoded_previous_scale. bcf_row_scale_code_lucas_low_digit_low_lower = bcf_quotient_lucas_low_digit_low_lower_table_decoded_previous_scale * S ((S (bcf_predecessor_lucas_low_digit_low_lower_table)) * bcf_row_scale_scale_lucas_low_digit_low_lower) + (bcf_previous_scale_lucas_low_digit_low_lower_table))) /\ (forall bcf_index_lucas_low_digit_low_lower_table_row_step. (exists bcf_lt_gap_lucas_low_digit_low_lower_table_row_step_bound. bcf_lt_gap_lucas_low_digit_low_lower_table_row_step_bound + S (bcf_index_lucas_low_digit_low_lower_table_row_step) = S (d)) -> exists bcf_value_lucas_low_digit_low_lower_table_row_step. ((((exists bcf_height_lucas_low_digit_low_lower_table_row_step_entry. bcf_height_lucas_low_digit_low_lower_table_row_step_entry + S (bcf_value_lucas_low_digit_low_lower_table_row_step) = S ((S (bcf_index_lucas_low_digit_low_lower_table_row_step)) * bcf_row_scale_lucas_low_digit_low_lower_table)) /\ exists bcf_quotient_lucas_low_digit_low_lower_table_row_step_entry. bcf_row_code_lucas_low_digit_low_lower_table = bcf_quotient_lucas_low_digit_low_lower_table_row_step_entry * S ((S (bcf_index_lucas_low_digit_low_lower_table_row_step)) * bcf_row_scale_lucas_low_digit_low_lower_table) + (bcf_value_lucas_low_digit_low_lower_table_row_step))) /\ ((bcf_index_lucas_low_digit_low_lower_table_row_step = 0 /\ bcf_value_lucas_low_digit_low_lower_table_row_step = 1) \/ exists bcf_predecessor_lucas_low_digit_low_lower_table_row_step bcf_left_lucas_low_digit_low_lower_table_row_step bcf_right_lucas_low_digit_low_lower_table_row_step. bcf_index_lucas_low_digit_low_lower_table_row_step = S bcf_predecessor_lucas_low_digit_low_lower_table_row_step /\ ((((exists bcf_height_lucas_low_digit_low_lower_table_row_step_previous_left. bcf_height_lucas_low_digit_low_lower_table_row_step_previous_left + S (bcf_left_lucas_low_digit_low_lower_table_row_step) = S ((S (bcf_predecessor_lucas_low_digit_low_lower_table_row_step)) * bcf_previous_scale_lucas_low_digit_low_lower_table)) /\ exists bcf_quotient_lucas_low_digit_low_lower_table_row_step_previous_left. bcf_previous_code_lucas_low_digit_low_lower_table = bcf_quotient_lucas_low_digit_low_lower_table_row_step_previous_left * S ((S (bcf_predecessor_lucas_low_digit_low_lower_table_row_step)) * bcf_previous_scale_lucas_low_digit_low_lower_table) + (bcf_left_lucas_low_digit_low_lower_table_row_step))) /\ ((((exists bcf_height_lucas_low_digit_low_lower_table_row_step_previous_right. bcf_height_lucas_low_digit_low_lower_table_row_step_previous_right + S (bcf_right_lucas_low_digit_low_lower_table_row_step) = S ((S (S (bcf_predecessor_lucas_low_digit_low_lower_table_row_step))) * bcf_previous_scale_lucas_low_digit_low_lower_table)) /\ exists bcf_quotient_lucas_low_digit_low_lower_table_row_step_previous_right. bcf_previous_code_lucas_low_digit_low_lower_table = bcf_quotient_lucas_low_digit_low_lower_table_row_step_previous_right * S ((S (S (bcf_predecessor_lucas_low_digit_low_lower_table_row_step))) * bcf_previous_scale_lucas_low_digit_low_lower_table) + (bcf_right_lucas_low_digit_low_lower_table_row_step))) /\ bcf_value_lucas_low_digit_low_lower_table_row_step = bcf_left_lucas_low_digit_low_lower_table_row_step + bcf_right_lucas_low_digit_low_lower_table_row_step))))))))))) /\ ((((exists bcf_height_lucas_low_digit_low_lower_decoded_row_code. bcf_height_lucas_low_digit_low_lower_decoded_row_code + S (bcf_row_code_lucas_low_digit_low_lower) = S ((S (d)) * bcf_row_code_scale_lucas_low_digit_low_lower)) /\ exists bcf_quotient_lucas_low_digit_low_lower_decoded_row_code. bcf_row_code_code_lucas_low_digit_low_lower = bcf_quotient_lucas_low_digit_low_lower_decoded_row_code * S ((S (d)) * bcf_row_code_scale_lucas_low_digit_low_lower) + (bcf_row_code_lucas_low_digit_low_lower))) /\ ((((exists bcf_height_lucas_low_digit_low_lower_decoded_row_scale. bcf_height_lucas_low_digit_low_lower_decoded_row_scale + S (bcf_row_scale_lucas_low_digit_low_lower) = S ((S (d)) * bcf_row_scale_scale_lucas_low_digit_low_lower)) /\ exists bcf_quotient_lucas_low_digit_low_lower_decoded_row_scale. bcf_row_scale_code_lucas_low_digit_low_lower = bcf_quotient_lucas_low_digit_low_lower_decoded_row_scale * S ((S (d)) * bcf_row_scale_scale_lucas_low_digit_low_lower) + (bcf_row_scale_lucas_low_digit_low_lower))) /\ (((exists bcf_height_lucas_low_digit_low_lower_decoded_value. bcf_height_lucas_low_digit_low_lower_decoded_value + S (D) = S ((S (e)) * bcf_row_scale_lucas_low_digit_low_lower)) /\ exists bcf_quotient_lucas_low_digit_low_lower_decoded_value. bcf_row_code_lucas_low_digit_low_lower = bcf_quotient_lucas_low_digit_low_lower_decoded_value * S ((S (e)) * bcf_row_scale_lucas_low_digit_low_lower) + (D))))))))) -> (exists lld_left_low_result lld_right_low_result. (C) + (p) * lld_left_low_result = (D) + (p) * lld_right_low_result)