lucas_prime_row_sparse_complete

Exact expanded first-order arithmetic statement

forall p k j C C0 Cp. ((~(p = 1) /\ forall frm_prime_left_lucas_row_prime frm_prime_right_lucas_row_prime. p = frm_prime_left_lucas_row_prime * frm_prime_right_lucas_row_prime -> frm_prime_left_lucas_row_prime = 1 \/ frm_prime_right_lucas_row_prime = 1)) -> k + j = p -> (exists ldc_lt_row_left. ldc_lt_row_left + S (k) = p) -> (exists ldc_lt_row_right. ldc_lt_row_right + S (j) = p) -> (((exists bcf_lt_gap_lucas_row_initial_out_of_range. bcf_lt_gap_lucas_row_initial_out_of_range + S (p) = 0) /\ C0 = 0) \/ ((exists bcf_le_gap_lucas_row_initial_in_range. bcf_le_gap_lucas_row_initial_in_range + (0) = p) /\ (exists bcf_row_code_code_lucas_row_initial bcf_row_code_scale_lucas_row_initial bcf_row_scale_code_lucas_row_initial bcf_row_scale_scale_lucas_row_initial bcf_row_code_lucas_row_initial bcf_row_scale_lucas_row_initial. ((forall bcf_row_index_lucas_row_initial_table. (exists bcf_lt_gap_lucas_row_initial_table_row_bound. bcf_lt_gap_lucas_row_initial_table_row_bound + S (bcf_row_index_lucas_row_initial_table) = S (p)) -> exists bcf_row_code_lucas_row_initial_table bcf_row_scale_lucas_row_initial_table. ((((exists bcf_height_lucas_row_initial_table_decoded_row_code. bcf_height_lucas_row_initial_table_decoded_row_code + S (bcf_row_code_lucas_row_initial_table) = S ((S (bcf_row_index_lucas_row_initial_table)) * bcf_row_code_scale_lucas_row_initial)) /\ exists bcf_quotient_lucas_row_initial_table_decoded_row_code. bcf_row_code_code_lucas_row_initial = bcf_quotient_lucas_row_initial_table_decoded_row_code * S ((S (bcf_row_index_lucas_row_initial_table)) * bcf_row_code_scale_lucas_row_initial) + (bcf_row_code_lucas_row_initial_table))) /\ ((((exists bcf_height_lucas_row_initial_table_decoded_row_scale. bcf_height_lucas_row_initial_table_decoded_row_scale + S (bcf_row_scale_lucas_row_initial_table) = S ((S (bcf_row_index_lucas_row_initial_table)) * bcf_row_scale_scale_lucas_row_initial)) /\ exists bcf_quotient_lucas_row_initial_table_decoded_row_scale. bcf_row_scale_code_lucas_row_initial = bcf_quotient_lucas_row_initial_table_decoded_row_scale * S ((S (bcf_row_index_lucas_row_initial_table)) * bcf_row_scale_scale_lucas_row_initial) + (bcf_row_scale_lucas_row_initial_table))) /\ ((bcf_row_index_lucas_row_initial_table = 0 /\ (forall bcf_index_lucas_row_initial_table_zero_row. (exists bcf_lt_gap_lucas_row_initial_table_zero_row_bound. bcf_lt_gap_lucas_row_initial_table_zero_row_bound + S (bcf_index_lucas_row_initial_table_zero_row) = S (p)) -> exists bcf_value_lucas_row_initial_table_zero_row. ((((exists bcf_height_lucas_row_initial_table_zero_row_entry. bcf_height_lucas_row_initial_table_zero_row_entry + S (bcf_value_lucas_row_initial_table_zero_row) = S ((S (bcf_index_lucas_row_initial_table_zero_row)) * bcf_row_scale_lucas_row_initial_table)) /\ exists bcf_quotient_lucas_row_initial_table_zero_row_entry. bcf_row_code_lucas_row_initial_table = bcf_quotient_lucas_row_initial_table_zero_row_entry * S ((S (bcf_index_lucas_row_initial_table_zero_row)) * bcf_row_scale_lucas_row_initial_table) + (bcf_value_lucas_row_initial_table_zero_row))) /\ ((bcf_index_lucas_row_initial_table_zero_row = 0 /\ bcf_value_lucas_row_initial_table_zero_row = 1) \/ exists bcf_predecessor_lucas_row_initial_table_zero_row. bcf_index_lucas_row_initial_table_zero_row = S bcf_predecessor_lucas_row_initial_table_zero_row /\ bcf_value_lucas_row_initial_table_zero_row = 0)))) \/ exists bcf_predecessor_lucas_row_initial_table bcf_previous_code_lucas_row_initial_table bcf_previous_scale_lucas_row_initial_table. bcf_row_index_lucas_row_initial_table = S bcf_predecessor_lucas_row_initial_table /\ ((((exists bcf_height_lucas_row_initial_table_decoded_previous_code. bcf_height_lucas_row_initial_table_decoded_previous_code + S (bcf_previous_code_lucas_row_initial_table) = S ((S (bcf_predecessor_lucas_row_initial_table)) * bcf_row_code_scale_lucas_row_initial)) /\ exists bcf_quotient_lucas_row_initial_table_decoded_previous_code. bcf_row_code_code_lucas_row_initial = bcf_quotient_lucas_row_initial_table_decoded_previous_code * S ((S (bcf_predecessor_lucas_row_initial_table)) * bcf_row_code_scale_lucas_row_initial) + (bcf_previous_code_lucas_row_initial_table))) /\ ((((exists bcf_height_lucas_row_initial_table_decoded_previous_scale. bcf_height_lucas_row_initial_table_decoded_previous_scale + S (bcf_previous_scale_lucas_row_initial_table) = S ((S (bcf_predecessor_lucas_row_initial_table)) * bcf_row_scale_scale_lucas_row_initial)) /\ exists bcf_quotient_lucas_row_initial_table_decoded_previous_scale. bcf_row_scale_code_lucas_row_initial = bcf_quotient_lucas_row_initial_table_decoded_previous_scale * S ((S (bcf_predecessor_lucas_row_initial_table)) * bcf_row_scale_scale_lucas_row_initial) + (bcf_previous_scale_lucas_row_initial_table))) /\ (forall bcf_index_lucas_row_initial_table_row_step. (exists bcf_lt_gap_lucas_row_initial_table_row_step_bound. bcf_lt_gap_lucas_row_initial_table_row_step_bound + S (bcf_index_lucas_row_initial_table_row_step) = S (p)) -> exists bcf_value_lucas_row_initial_table_row_step. ((((exists bcf_height_lucas_row_initial_table_row_step_entry. bcf_height_lucas_row_initial_table_row_step_entry + S (bcf_value_lucas_row_initial_table_row_step) = S ((S (bcf_index_lucas_row_initial_table_row_step)) * bcf_row_scale_lucas_row_initial_table)) /\ exists bcf_quotient_lucas_row_initial_table_row_step_entry. bcf_row_code_lucas_row_initial_table = bcf_quotient_lucas_row_initial_table_row_step_entry * S ((S (bcf_index_lucas_row_initial_table_row_step)) * bcf_row_scale_lucas_row_initial_table) + (bcf_value_lucas_row_initial_table_row_step))) /\ ((bcf_index_lucas_row_initial_table_row_step = 0 /\ bcf_value_lucas_row_initial_table_row_step = 1) \/ exists bcf_predecessor_lucas_row_initial_table_row_step bcf_left_lucas_row_initial_table_row_step bcf_right_lucas_row_initial_table_row_step. bcf_index_lucas_row_initial_table_row_step = S bcf_predecessor_lucas_row_initial_table_row_step /\ ((((exists bcf_height_lucas_row_initial_table_row_step_previous_left. bcf_height_lucas_row_initial_table_row_step_previous_left + S (bcf_left_lucas_row_initial_table_row_step) = S ((S (bcf_predecessor_lucas_row_initial_table_row_step)) * bcf_previous_scale_lucas_row_initial_table)) /\ exists bcf_quotient_lucas_row_initial_table_row_step_previous_left. bcf_previous_code_lucas_row_initial_table = bcf_quotient_lucas_row_initial_table_row_step_previous_left * S ((S (bcf_predecessor_lucas_row_initial_table_row_step)) * bcf_previous_scale_lucas_row_initial_table) + (bcf_left_lucas_row_initial_table_row_step))) /\ ((((exists bcf_height_lucas_row_initial_table_row_step_previous_right. bcf_height_lucas_row_initial_table_row_step_previous_right + S (bcf_right_lucas_row_initial_table_row_step) = S ((S (S (bcf_predecessor_lucas_row_initial_table_row_step))) * bcf_previous_scale_lucas_row_initial_table)) /\ exists bcf_quotient_lucas_row_initial_table_row_step_previous_right. bcf_previous_code_lucas_row_initial_table = bcf_quotient_lucas_row_initial_table_row_step_previous_right * S ((S (S (bcf_predecessor_lucas_row_initial_table_row_step))) * bcf_previous_scale_lucas_row_initial_table) + (bcf_right_lucas_row_initial_table_row_step))) /\ bcf_value_lucas_row_initial_table_row_step = bcf_left_lucas_row_initial_table_row_step + bcf_right_lucas_row_initial_table_row_step))))))))))) /\ ((((exists bcf_height_lucas_row_initial_decoded_row_code. bcf_height_lucas_row_initial_decoded_row_code + S (bcf_row_code_lucas_row_initial) = S ((S (p)) * bcf_row_code_scale_lucas_row_initial)) /\ exists bcf_quotient_lucas_row_initial_decoded_row_code. bcf_row_code_code_lucas_row_initial = bcf_quotient_lucas_row_initial_decoded_row_code * S ((S (p)) * bcf_row_code_scale_lucas_row_initial) + (bcf_row_code_lucas_row_initial))) /\ ((((exists bcf_height_lucas_row_initial_decoded_row_scale. bcf_height_lucas_row_initial_decoded_row_scale + S (bcf_row_scale_lucas_row_initial) = S ((S (p)) * bcf_row_scale_scale_lucas_row_initial)) /\ exists bcf_quotient_lucas_row_initial_decoded_row_scale. bcf_row_scale_code_lucas_row_initial = bcf_quotient_lucas_row_initial_decoded_row_scale * S ((S (p)) * bcf_row_scale_scale_lucas_row_initial) + (bcf_row_scale_lucas_row_initial))) /\ (((exists bcf_height_lucas_row_initial_decoded_value. bcf_height_lucas_row_initial_decoded_value + S (C0) = S ((S (0)) * bcf_row_scale_lucas_row_initial)) /\ exists bcf_quotient_lucas_row_initial_decoded_value. bcf_row_code_lucas_row_initial = bcf_quotient_lucas_row_initial_decoded_value * S ((S (0)) * bcf_row_scale_lucas_row_initial) + (C0))))))))) -> (((exists bcf_lt_gap_lucas_row_terminal_out_of_range. bcf_lt_gap_lucas_row_terminal_out_of_range + S (p) = p) /\ Cp = 0) \/ ((exists bcf_le_gap_lucas_row_terminal_in_range. bcf_le_gap_lucas_row_terminal_in_range + (p) = p) /\ (exists bcf_row_code_code_lucas_row_terminal bcf_row_code_scale_lucas_row_terminal bcf_row_scale_code_lucas_row_terminal bcf_row_scale_scale_lucas_row_terminal bcf_row_code_lucas_row_terminal bcf_row_scale_lucas_row_terminal. ((forall bcf_row_index_lucas_row_terminal_table. (exists bcf_lt_gap_lucas_row_terminal_table_row_bound. bcf_lt_gap_lucas_row_terminal_table_row_bound + S (bcf_row_index_lucas_row_terminal_table) = S (p)) -> exists bcf_row_code_lucas_row_terminal_table bcf_row_scale_lucas_row_terminal_table. ((((exists bcf_height_lucas_row_terminal_table_decoded_row_code. bcf_height_lucas_row_terminal_table_decoded_row_code + S (bcf_row_code_lucas_row_terminal_table) = S ((S (bcf_row_index_lucas_row_terminal_table)) * bcf_row_code_scale_lucas_row_terminal)) /\ exists bcf_quotient_lucas_row_terminal_table_decoded_row_code. bcf_row_code_code_lucas_row_terminal = bcf_quotient_lucas_row_terminal_table_decoded_row_code * S ((S (bcf_row_index_lucas_row_terminal_table)) * bcf_row_code_scale_lucas_row_terminal) + (bcf_row_code_lucas_row_terminal_table))) /\ ((((exists bcf_height_lucas_row_terminal_table_decoded_row_scale. bcf_height_lucas_row_terminal_table_decoded_row_scale + S (bcf_row_scale_lucas_row_terminal_table) = S ((S (bcf_row_index_lucas_row_terminal_table)) * bcf_row_scale_scale_lucas_row_terminal)) /\ exists bcf_quotient_lucas_row_terminal_table_decoded_row_scale. bcf_row_scale_code_lucas_row_terminal = bcf_quotient_lucas_row_terminal_table_decoded_row_scale * S ((S (bcf_row_index_lucas_row_terminal_table)) * bcf_row_scale_scale_lucas_row_terminal) + (bcf_row_scale_lucas_row_terminal_table))) /\ ((bcf_row_index_lucas_row_terminal_table = 0 /\ (forall bcf_index_lucas_row_terminal_table_zero_row. (exists bcf_lt_gap_lucas_row_terminal_table_zero_row_bound. bcf_lt_gap_lucas_row_terminal_table_zero_row_bound + S (bcf_index_lucas_row_terminal_table_zero_row) = S (p)) -> exists bcf_value_lucas_row_terminal_table_zero_row. ((((exists bcf_height_lucas_row_terminal_table_zero_row_entry. bcf_height_lucas_row_terminal_table_zero_row_entry + S (bcf_value_lucas_row_terminal_table_zero_row) = S ((S (bcf_index_lucas_row_terminal_table_zero_row)) * bcf_row_scale_lucas_row_terminal_table)) /\ exists bcf_quotient_lucas_row_terminal_table_zero_row_entry. bcf_row_code_lucas_row_terminal_table = bcf_quotient_lucas_row_terminal_table_zero_row_entry * S ((S (bcf_index_lucas_row_terminal_table_zero_row)) * bcf_row_scale_lucas_row_terminal_table) + (bcf_value_lucas_row_terminal_table_zero_row))) /\ ((bcf_index_lucas_row_terminal_table_zero_row = 0 /\ bcf_value_lucas_row_terminal_table_zero_row = 1) \/ exists bcf_predecessor_lucas_row_terminal_table_zero_row. bcf_index_lucas_row_terminal_table_zero_row = S bcf_predecessor_lucas_row_terminal_table_zero_row /\ bcf_value_lucas_row_terminal_table_zero_row = 0)))) \/ exists bcf_predecessor_lucas_row_terminal_table bcf_previous_code_lucas_row_terminal_table bcf_previous_scale_lucas_row_terminal_table. bcf_row_index_lucas_row_terminal_table = S bcf_predecessor_lucas_row_terminal_table /\ ((((exists bcf_height_lucas_row_terminal_table_decoded_previous_code. bcf_height_lucas_row_terminal_table_decoded_previous_code + S (bcf_previous_code_lucas_row_terminal_table) = S ((S (bcf_predecessor_lucas_row_terminal_table)) * bcf_row_code_scale_lucas_row_terminal)) /\ exists bcf_quotient_lucas_row_terminal_table_decoded_previous_code. bcf_row_code_code_lucas_row_terminal = bcf_quotient_lucas_row_terminal_table_decoded_previous_code * S ((S (bcf_predecessor_lucas_row_terminal_table)) * bcf_row_code_scale_lucas_row_terminal) + (bcf_previous_code_lucas_row_terminal_table))) /\ ((((exists bcf_height_lucas_row_terminal_table_decoded_previous_scale. bcf_height_lucas_row_terminal_table_decoded_previous_scale + S (bcf_previous_scale_lucas_row_terminal_table) = S ((S (bcf_predecessor_lucas_row_terminal_table)) * bcf_row_scale_scale_lucas_row_terminal)) /\ exists bcf_quotient_lucas_row_terminal_table_decoded_previous_scale. bcf_row_scale_code_lucas_row_terminal = bcf_quotient_lucas_row_terminal_table_decoded_previous_scale * S ((S (bcf_predecessor_lucas_row_terminal_table)) * bcf_row_scale_scale_lucas_row_terminal) + (bcf_previous_scale_lucas_row_terminal_table))) /\ (forall bcf_index_lucas_row_terminal_table_row_step. (exists bcf_lt_gap_lucas_row_terminal_table_row_step_bound. bcf_lt_gap_lucas_row_terminal_table_row_step_bound + S (bcf_index_lucas_row_terminal_table_row_step) = S (p)) -> exists bcf_value_lucas_row_terminal_table_row_step. ((((exists bcf_height_lucas_row_terminal_table_row_step_entry. bcf_height_lucas_row_terminal_table_row_step_entry + S (bcf_value_lucas_row_terminal_table_row_step) = S ((S (bcf_index_lucas_row_terminal_table_row_step)) * bcf_row_scale_lucas_row_terminal_table)) /\ exists bcf_quotient_lucas_row_terminal_table_row_step_entry. bcf_row_code_lucas_row_terminal_table = bcf_quotient_lucas_row_terminal_table_row_step_entry * S ((S (bcf_index_lucas_row_terminal_table_row_step)) * bcf_row_scale_lucas_row_terminal_table) + (bcf_value_lucas_row_terminal_table_row_step))) /\ ((bcf_index_lucas_row_terminal_table_row_step = 0 /\ bcf_value_lucas_row_terminal_table_row_step = 1) \/ exists bcf_predecessor_lucas_row_terminal_table_row_step bcf_left_lucas_row_terminal_table_row_step bcf_right_lucas_row_terminal_table_row_step. bcf_index_lucas_row_terminal_table_row_step = S bcf_predecessor_lucas_row_terminal_table_row_step /\ ((((exists bcf_height_lucas_row_terminal_table_row_step_previous_left. bcf_height_lucas_row_terminal_table_row_step_previous_left + S (bcf_left_lucas_row_terminal_table_row_step) = S ((S (bcf_predecessor_lucas_row_terminal_table_row_step)) * bcf_previous_scale_lucas_row_terminal_table)) /\ exists bcf_quotient_lucas_row_terminal_table_row_step_previous_left. bcf_previous_code_lucas_row_terminal_table = bcf_quotient_lucas_row_terminal_table_row_step_previous_left * S ((S (bcf_predecessor_lucas_row_terminal_table_row_step)) * bcf_previous_scale_lucas_row_terminal_table) + (bcf_left_lucas_row_terminal_table_row_step))) /\ ((((exists bcf_height_lucas_row_terminal_table_row_step_previous_right. bcf_height_lucas_row_terminal_table_row_step_previous_right + S (bcf_right_lucas_row_terminal_table_row_step) = S ((S (S (bcf_predecessor_lucas_row_terminal_table_row_step))) * bcf_previous_scale_lucas_row_terminal_table)) /\ exists bcf_quotient_lucas_row_terminal_table_row_step_previous_right. bcf_previous_code_lucas_row_terminal_table = bcf_quotient_lucas_row_terminal_table_row_step_previous_right * S ((S (S (bcf_predecessor_lucas_row_terminal_table_row_step))) * bcf_previous_scale_lucas_row_terminal_table) + (bcf_right_lucas_row_terminal_table_row_step))) /\ bcf_value_lucas_row_terminal_table_row_step = bcf_left_lucas_row_terminal_table_row_step + bcf_right_lucas_row_terminal_table_row_step))))))))))) /\ ((((exists bcf_height_lucas_row_terminal_decoded_row_code. bcf_height_lucas_row_terminal_decoded_row_code + S (bcf_row_code_lucas_row_terminal) = S ((S (p)) * bcf_row_code_scale_lucas_row_terminal)) /\ exists bcf_quotient_lucas_row_terminal_decoded_row_code. bcf_row_code_code_lucas_row_terminal = bcf_quotient_lucas_row_terminal_decoded_row_code * S ((S (p)) * bcf_row_code_scale_lucas_row_terminal) + (bcf_row_code_lucas_row_terminal))) /\ ((((exists bcf_height_lucas_row_terminal_decoded_row_scale. bcf_height_lucas_row_terminal_decoded_row_scale + S (bcf_row_scale_lucas_row_terminal) = S ((S (p)) * bcf_row_scale_scale_lucas_row_terminal)) /\ exists bcf_quotient_lucas_row_terminal_decoded_row_scale. bcf_row_scale_code_lucas_row_terminal = bcf_quotient_lucas_row_terminal_decoded_row_scale * S ((S (p)) * bcf_row_scale_scale_lucas_row_terminal) + (bcf_row_scale_lucas_row_terminal))) /\ (((exists bcf_height_lucas_row_terminal_decoded_value. bcf_height_lucas_row_terminal_decoded_value + S (Cp) = S ((S (p)) * bcf_row_scale_lucas_row_terminal)) /\ exists bcf_quotient_lucas_row_terminal_decoded_value. bcf_row_code_lucas_row_terminal = bcf_quotient_lucas_row_terminal_decoded_value * S ((S (p)) * bcf_row_scale_lucas_row_terminal) + (Cp))))))))) -> (((exists bcf_lt_gap_lucas_row_choose_out_of_range. bcf_lt_gap_lucas_row_choose_out_of_range + S (p) = k) /\ C = 0) \/ ((exists bcf_le_gap_lucas_row_choose_in_range. bcf_le_gap_lucas_row_choose_in_range + (k) = p) /\ (exists bcf_row_code_code_lucas_row_choose bcf_row_code_scale_lucas_row_choose bcf_row_scale_code_lucas_row_choose bcf_row_scale_scale_lucas_row_choose bcf_row_code_lucas_row_choose bcf_row_scale_lucas_row_choose. ((forall bcf_row_index_lucas_row_choose_table. (exists bcf_lt_gap_lucas_row_choose_table_row_bound. bcf_lt_gap_lucas_row_choose_table_row_bound + S (bcf_row_index_lucas_row_choose_table) = S (p)) -> exists bcf_row_code_lucas_row_choose_table bcf_row_scale_lucas_row_choose_table. ((((exists bcf_height_lucas_row_choose_table_decoded_row_code. bcf_height_lucas_row_choose_table_decoded_row_code + S (bcf_row_code_lucas_row_choose_table) = S ((S (bcf_row_index_lucas_row_choose_table)) * bcf_row_code_scale_lucas_row_choose)) /\ exists bcf_quotient_lucas_row_choose_table_decoded_row_code. bcf_row_code_code_lucas_row_choose = bcf_quotient_lucas_row_choose_table_decoded_row_code * S ((S (bcf_row_index_lucas_row_choose_table)) * bcf_row_code_scale_lucas_row_choose) + (bcf_row_code_lucas_row_choose_table))) /\ ((((exists bcf_height_lucas_row_choose_table_decoded_row_scale. bcf_height_lucas_row_choose_table_decoded_row_scale + S (bcf_row_scale_lucas_row_choose_table) = S ((S (bcf_row_index_lucas_row_choose_table)) * bcf_row_scale_scale_lucas_row_choose)) /\ exists bcf_quotient_lucas_row_choose_table_decoded_row_scale. bcf_row_scale_code_lucas_row_choose = bcf_quotient_lucas_row_choose_table_decoded_row_scale * S ((S (bcf_row_index_lucas_row_choose_table)) * bcf_row_scale_scale_lucas_row_choose) + (bcf_row_scale_lucas_row_choose_table))) /\ ((bcf_row_index_lucas_row_choose_table = 0 /\ (forall bcf_index_lucas_row_choose_table_zero_row. (exists bcf_lt_gap_lucas_row_choose_table_zero_row_bound. bcf_lt_gap_lucas_row_choose_table_zero_row_bound + S (bcf_index_lucas_row_choose_table_zero_row) = S (p)) -> exists bcf_value_lucas_row_choose_table_zero_row. ((((exists bcf_height_lucas_row_choose_table_zero_row_entry. bcf_height_lucas_row_choose_table_zero_row_entry + S (bcf_value_lucas_row_choose_table_zero_row) = S ((S (bcf_index_lucas_row_choose_table_zero_row)) * bcf_row_scale_lucas_row_choose_table)) /\ exists bcf_quotient_lucas_row_choose_table_zero_row_entry. bcf_row_code_lucas_row_choose_table = bcf_quotient_lucas_row_choose_table_zero_row_entry * S ((S (bcf_index_lucas_row_choose_table_zero_row)) * bcf_row_scale_lucas_row_choose_table) + (bcf_value_lucas_row_choose_table_zero_row))) /\ ((bcf_index_lucas_row_choose_table_zero_row = 0 /\ bcf_value_lucas_row_choose_table_zero_row = 1) \/ exists bcf_predecessor_lucas_row_choose_table_zero_row. bcf_index_lucas_row_choose_table_zero_row = S bcf_predecessor_lucas_row_choose_table_zero_row /\ bcf_value_lucas_row_choose_table_zero_row = 0)))) \/ exists bcf_predecessor_lucas_row_choose_table bcf_previous_code_lucas_row_choose_table bcf_previous_scale_lucas_row_choose_table. bcf_row_index_lucas_row_choose_table = S bcf_predecessor_lucas_row_choose_table /\ ((((exists bcf_height_lucas_row_choose_table_decoded_previous_code. bcf_height_lucas_row_choose_table_decoded_previous_code + S (bcf_previous_code_lucas_row_choose_table) = S ((S (bcf_predecessor_lucas_row_choose_table)) * bcf_row_code_scale_lucas_row_choose)) /\ exists bcf_quotient_lucas_row_choose_table_decoded_previous_code. bcf_row_code_code_lucas_row_choose = bcf_quotient_lucas_row_choose_table_decoded_previous_code * S ((S (bcf_predecessor_lucas_row_choose_table)) * bcf_row_code_scale_lucas_row_choose) + (bcf_previous_code_lucas_row_choose_table))) /\ ((((exists bcf_height_lucas_row_choose_table_decoded_previous_scale. bcf_height_lucas_row_choose_table_decoded_previous_scale + S (bcf_previous_scale_lucas_row_choose_table) = S ((S (bcf_predecessor_lucas_row_choose_table)) * bcf_row_scale_scale_lucas_row_choose)) /\ exists bcf_quotient_lucas_row_choose_table_decoded_previous_scale. bcf_row_scale_code_lucas_row_choose = bcf_quotient_lucas_row_choose_table_decoded_previous_scale * S ((S (bcf_predecessor_lucas_row_choose_table)) * bcf_row_scale_scale_lucas_row_choose) + (bcf_previous_scale_lucas_row_choose_table))) /\ (forall bcf_index_lucas_row_choose_table_row_step. (exists bcf_lt_gap_lucas_row_choose_table_row_step_bound. bcf_lt_gap_lucas_row_choose_table_row_step_bound + S (bcf_index_lucas_row_choose_table_row_step) = S (p)) -> exists bcf_value_lucas_row_choose_table_row_step. ((((exists bcf_height_lucas_row_choose_table_row_step_entry. bcf_height_lucas_row_choose_table_row_step_entry + S (bcf_value_lucas_row_choose_table_row_step) = S ((S (bcf_index_lucas_row_choose_table_row_step)) * bcf_row_scale_lucas_row_choose_table)) /\ exists bcf_quotient_lucas_row_choose_table_row_step_entry. bcf_row_code_lucas_row_choose_table = bcf_quotient_lucas_row_choose_table_row_step_entry * S ((S (bcf_index_lucas_row_choose_table_row_step)) * bcf_row_scale_lucas_row_choose_table) + (bcf_value_lucas_row_choose_table_row_step))) /\ ((bcf_index_lucas_row_choose_table_row_step = 0 /\ bcf_value_lucas_row_choose_table_row_step = 1) \/ exists bcf_predecessor_lucas_row_choose_table_row_step bcf_left_lucas_row_choose_table_row_step bcf_right_lucas_row_choose_table_row_step. bcf_index_lucas_row_choose_table_row_step = S bcf_predecessor_lucas_row_choose_table_row_step /\ ((((exists bcf_height_lucas_row_choose_table_row_step_previous_left. bcf_height_lucas_row_choose_table_row_step_previous_left + S (bcf_left_lucas_row_choose_table_row_step) = S ((S (bcf_predecessor_lucas_row_choose_table_row_step)) * bcf_previous_scale_lucas_row_choose_table)) /\ exists bcf_quotient_lucas_row_choose_table_row_step_previous_left. bcf_previous_code_lucas_row_choose_table = bcf_quotient_lucas_row_choose_table_row_step_previous_left * S ((S (bcf_predecessor_lucas_row_choose_table_row_step)) * bcf_previous_scale_lucas_row_choose_table) + (bcf_left_lucas_row_choose_table_row_step))) /\ ((((exists bcf_height_lucas_row_choose_table_row_step_previous_right. bcf_height_lucas_row_choose_table_row_step_previous_right + S (bcf_right_lucas_row_choose_table_row_step) = S ((S (S (bcf_predecessor_lucas_row_choose_table_row_step))) * bcf_previous_scale_lucas_row_choose_table)) /\ exists bcf_quotient_lucas_row_choose_table_row_step_previous_right. bcf_previous_code_lucas_row_choose_table = bcf_quotient_lucas_row_choose_table_row_step_previous_right * S ((S (S (bcf_predecessor_lucas_row_choose_table_row_step))) * bcf_previous_scale_lucas_row_choose_table) + (bcf_right_lucas_row_choose_table_row_step))) /\ bcf_value_lucas_row_choose_table_row_step = bcf_left_lucas_row_choose_table_row_step + bcf_right_lucas_row_choose_table_row_step))))))))))) /\ ((((exists bcf_height_lucas_row_choose_decoded_row_code. bcf_height_lucas_row_choose_decoded_row_code + S (bcf_row_code_lucas_row_choose) = S ((S (p)) * bcf_row_code_scale_lucas_row_choose)) /\ exists bcf_quotient_lucas_row_choose_decoded_row_code. bcf_row_code_code_lucas_row_choose = bcf_quotient_lucas_row_choose_decoded_row_code * S ((S (p)) * bcf_row_code_scale_lucas_row_choose) + (bcf_row_code_lucas_row_choose))) /\ ((((exists bcf_height_lucas_row_choose_decoded_row_scale. bcf_height_lucas_row_choose_decoded_row_scale + S (bcf_row_scale_lucas_row_choose) = S ((S (p)) * bcf_row_scale_scale_lucas_row_choose)) /\ exists bcf_quotient_lucas_row_choose_decoded_row_scale. bcf_row_scale_code_lucas_row_choose = bcf_quotient_lucas_row_choose_decoded_row_scale * S ((S (p)) * bcf_row_scale_scale_lucas_row_choose) + (bcf_row_scale_lucas_row_choose))) /\ (((exists bcf_height_lucas_row_choose_decoded_value. bcf_height_lucas_row_choose_decoded_value + S (C) = S ((S (k)) * bcf_row_scale_lucas_row_choose)) /\ exists bcf_quotient_lucas_row_choose_decoded_value. bcf_row_code_lucas_row_choose = bcf_quotient_lucas_row_choose_decoded_value * S ((S (k)) * bcf_row_scale_lucas_row_choose) + (C))))))))) -> ((C0 = S (0)) /\ ((Cp = S (0)) /\ (exists ldc_quotient_row. C = p * ldc_quotient_row)))