lucas_digit_carry_iff_prime_divides

forall p a b C. ((~(p = 1) /\ forall frm_prime_left_lucas_digit_prime frm_prime_right_lucas_digit_prime. p = frm_prime_left_lucas_digit_prime * frm_prime_right_lucas_digit_prime -> frm_prime_left_lucas_digit_prime = 1 \/ frm_prime_right_lucas_digit_prime = 1)) -> (exists ldc_lt_left. ldc_lt_left + S (a) = p) -> (exists ldc_lt_right. ldc_lt_right + S (b) = p) -> (((exists bcf_lt_gap_lucas_digit_choose_out_of_range. bcf_lt_gap_lucas_digit_choose_out_of_range + S (a + b) = a) /\ C = 0) \/ ((exists bcf_le_gap_lucas_digit_choose_in_range. bcf_le_gap_lucas_digit_choose_in_range + (a) = a + b) /\ (exists bcf_row_code_code_lucas_digit_choose bcf_row_code_scale_lucas_digit_choose bcf_row_scale_code_lucas_digit_choose bcf_row_scale_scale_lucas_digit_choose bcf_row_code_lucas_digit_choose bcf_row_scale_lucas_digit_choose. ((forall bcf_row_index_lucas_digit_choose_table. (exists bcf_lt_gap_lucas_digit_choose_table_row_bound. bcf_lt_gap_lucas_digit_choose_table_row_bound + S (bcf_row_index_lucas_digit_choose_table) = S (a + b)) -> exists bcf_row_code_lucas_digit_choose_table bcf_row_scale_lucas_digit_choose_table. ((((exists bcf_height_lucas_digit_choose_table_decoded_row_code. bcf_height_lucas_digit_choose_table_decoded_row_code + S (bcf_row_code_lucas_digit_choose_table) = S ((S (bcf_row_index_lucas_digit_choose_table)) * bcf_row_code_scale_lucas_digit_choose)) /\ exists bcf_quotient_lucas_digit_choose_table_decoded_row_code. bcf_row_code_code_lucas_digit_choose = bcf_quotient_lucas_digit_choose_table_decoded_row_code * S ((S (bcf_row_index_lucas_digit_choose_table)) * bcf_row_code_scale_lucas_digit_choose) + (bcf_row_code_lucas_digit_choose_table))) /\ ((((exists bcf_height_lucas_digit_choose_table_decoded_row_scale. bcf_height_lucas_digit_choose_table_decoded_row_scale + S (bcf_row_scale_lucas_digit_choose_table) = S ((S (bcf_row_index_lucas_digit_choose_table)) * bcf_row_scale_scale_lucas_digit_choose)) /\ exists bcf_quotient_lucas_digit_choose_table_decoded_row_scale. bcf_row_scale_code_lucas_digit_choose = bcf_quotient_lucas_digit_choose_table_decoded_row_scale * S ((S (bcf_row_index_lucas_digit_choose_table)) * bcf_row_scale_scale_lucas_digit_choose) + (bcf_row_scale_lucas_digit_choose_table))) /\ ((bcf_row_index_lucas_digit_choose_table = 0 /\ (forall bcf_index_lucas_digit_choose_table_zero_row. (exists bcf_lt_gap_lucas_digit_choose_table_zero_row_bound. bcf_lt_gap_lucas_digit_choose_table_zero_row_bound + S (bcf_index_lucas_digit_choose_table_zero_row) = S (a + b)) -> exists bcf_value_lucas_digit_choose_table_zero_row. ((((exists bcf_height_lucas_digit_choose_table_zero_row_entry. bcf_height_lucas_digit_choose_table_zero_row_entry + S (bcf_value_lucas_digit_choose_table_zero_row) = S ((S (bcf_index_lucas_digit_choose_table_zero_row)) * bcf_row_scale_lucas_digit_choose_table)) /\ exists bcf_quotient_lucas_digit_choose_table_zero_row_entry. bcf_row_code_lucas_digit_choose_table = bcf_quotient_lucas_digit_choose_table_zero_row_entry * S ((S (bcf_index_lucas_digit_choose_table_zero_row)) * bcf_row_scale_lucas_digit_choose_table) + (bcf_value_lucas_digit_choose_table_zero_row))) /\ ((bcf_index_lucas_digit_choose_table_zero_row = 0 /\ bcf_value_lucas_digit_choose_table_zero_row = 1) \/ exists bcf_predecessor_lucas_digit_choose_table_zero_row. bcf_index_lucas_digit_choose_table_zero_row = S bcf_predecessor_lucas_digit_choose_table_zero_row /\ bcf_value_lucas_digit_choose_table_zero_row = 0)))) \/ exists bcf_predecessor_lucas_digit_choose_table bcf_previous_code_lucas_digit_choose_table bcf_previous_scale_lucas_digit_choose_table. bcf_row_index_lucas_digit_choose_table = S bcf_predecessor_lucas_digit_choose_table /\ ((((exists bcf_height_lucas_digit_choose_table_decoded_previous_code. bcf_height_lucas_digit_choose_table_decoded_previous_code + S (bcf_previous_code_lucas_digit_choose_table) = S ((S (bcf_predecessor_lucas_digit_choose_table)) * bcf_row_code_scale_lucas_digit_choose)) /\ exists bcf_quotient_lucas_digit_choose_table_decoded_previous_code. bcf_row_code_code_lucas_digit_choose = bcf_quotient_lucas_digit_choose_table_decoded_previous_code * S ((S (bcf_predecessor_lucas_digit_choose_table)) * bcf_row_code_scale_lucas_digit_choose) + (bcf_previous_code_lucas_digit_choose_table))) /\ ((((exists bcf_height_lucas_digit_choose_table_decoded_previous_scale. bcf_height_lucas_digit_choose_table_decoded_previous_scale + S (bcf_previous_scale_lucas_digit_choose_table) = S ((S (bcf_predecessor_lucas_digit_choose_table)) * bcf_row_scale_scale_lucas_digit_choose)) /\ exists bcf_quotient_lucas_digit_choose_table_decoded_previous_scale. bcf_row_scale_code_lucas_digit_choose = bcf_quotient_lucas_digit_choose_table_decoded_previous_scale * S ((S (bcf_predecessor_lucas_digit_choose_table)) * bcf_row_scale_scale_lucas_digit_choose) + (bcf_previous_scale_lucas_digit_choose_table))) /\ (forall bcf_index_lucas_digit_choose_table_row_step. (exists bcf_lt_gap_lucas_digit_choose_table_row_step_bound. bcf_lt_gap_lucas_digit_choose_table_row_step_bound + S (bcf_index_lucas_digit_choose_table_row_step) = S (a + b)) -> exists bcf_value_lucas_digit_choose_table_row_step. ((((exists bcf_height_lucas_digit_choose_table_row_step_entry. bcf_height_lucas_digit_choose_table_row_step_entry + S (bcf_value_lucas_digit_choose_table_row_step) = S ((S (bcf_index_lucas_digit_choose_table_row_step)) * bcf_row_scale_lucas_digit_choose_table)) /\ exists bcf_quotient_lucas_digit_choose_table_row_step_entry. bcf_row_code_lucas_digit_choose_table = bcf_quotient_lucas_digit_choose_table_row_step_entry * S ((S (bcf_index_lucas_digit_choose_table_row_step)) * bcf_row_scale_lucas_digit_choose_table) + (bcf_value_lucas_digit_choose_table_row_step))) /\ ((bcf_index_lucas_digit_choose_table_row_step = 0 /\ bcf_value_lucas_digit_choose_table_row_step = 1) \/ exists bcf_predecessor_lucas_digit_choose_table_row_step bcf_left_lucas_digit_choose_table_row_step bcf_right_lucas_digit_choose_table_row_step. bcf_index_lucas_digit_choose_table_row_step = S bcf_predecessor_lucas_digit_choose_table_row_step /\ ((((exists bcf_height_lucas_digit_choose_table_row_step_previous_left. bcf_height_lucas_digit_choose_table_row_step_previous_left + S (bcf_left_lucas_digit_choose_table_row_step) = S ((S (bcf_predecessor_lucas_digit_choose_table_row_step)) * bcf_previous_scale_lucas_digit_choose_table)) /\ exists bcf_quotient_lucas_digit_choose_table_row_step_previous_left. bcf_previous_code_lucas_digit_choose_table = bcf_quotient_lucas_digit_choose_table_row_step_previous_left * S ((S (bcf_predecessor_lucas_digit_choose_table_row_step)) * bcf_previous_scale_lucas_digit_choose_table) + (bcf_left_lucas_digit_choose_table_row_step))) /\ ((((exists bcf_height_lucas_digit_choose_table_row_step_previous_right. bcf_height_lucas_digit_choose_table_row_step_previous_right + S (bcf_right_lucas_digit_choose_table_row_step) = S ((S (S (bcf_predecessor_lucas_digit_choose_table_row_step))) * bcf_previous_scale_lucas_digit_choose_table)) /\ exists bcf_quotient_lucas_digit_choose_table_row_step_previous_right. bcf_previous_code_lucas_digit_choose_table = bcf_quotient_lucas_digit_choose_table_row_step_previous_right * S ((S (S (bcf_predecessor_lucas_digit_choose_table_row_step))) * bcf_previous_scale_lucas_digit_choose_table) + (bcf_right_lucas_digit_choose_table_row_step))) /\ bcf_value_lucas_digit_choose_table_row_step = bcf_left_lucas_digit_choose_table_row_step + bcf_right_lucas_digit_choose_table_row_step))))))))))) /\ ((((exists bcf_height_lucas_digit_choose_decoded_row_code. bcf_height_lucas_digit_choose_decoded_row_code + S (bcf_row_code_lucas_digit_choose) = S ((S (a + b)) * bcf_row_code_scale_lucas_digit_choose)) /\ exists bcf_quotient_lucas_digit_choose_decoded_row_code. bcf_row_code_code_lucas_digit_choose = bcf_quotient_lucas_digit_choose_decoded_row_code * S ((S (a + b)) * bcf_row_code_scale_lucas_digit_choose) + (bcf_row_code_lucas_digit_choose))) /\ ((((exists bcf_height_lucas_digit_choose_decoded_row_scale. bcf_height_lucas_digit_choose_decoded_row_scale + S (bcf_row_scale_lucas_digit_choose) = S ((S (a + b)) * bcf_row_scale_scale_lucas_digit_choose)) /\ exists bcf_quotient_lucas_digit_choose_decoded_row_scale. bcf_row_scale_code_lucas_digit_choose = bcf_quotient_lucas_digit_choose_decoded_row_scale * S ((S (a + b)) * bcf_row_scale_scale_lucas_digit_choose) + (bcf_row_scale_lucas_digit_choose))) /\ (((exists bcf_height_lucas_digit_choose_decoded_value. bcf_height_lucas_digit_choose_decoded_value + S (C) = S ((S (a)) * bcf_row_scale_lucas_digit_choose)) /\ exists bcf_quotient_lucas_digit_choose_decoded_value. bcf_row_code_lucas_digit_choose = bcf_quotient_lucas_digit_choose_decoded_value * S ((S (a)) * bcf_row_scale_lucas_digit_choose) + (C))))))))) -> ((((exists ldc_le_carry. ldc_le_carry + (p) = a + b) -> (exists ldc_quotient_digit. C = p * ldc_quotient_digit)) /\ ((exists ldc_quotient_digit. C = p * ldc_quotient_digit) -> (exists ldc_le_carry. ldc_le_carry + (p) = a + b))))