four_pow_central_seed_package

Exact expanded PA statement

(forall n a b. (((exists bcf_lt_gap_bcb4we_recurrence_predecessor_out_of_range. bcf_lt_gap_bcb4we_recurrence_predecessor_out_of_range + S (n + n) = n) /\ a = 0) \/ ((exists bcf_le_gap_bcb4we_recurrence_predecessor_in_range. bcf_le_gap_bcb4we_recurrence_predecessor_in_range + (n) = n + n) /\ (exists bcf_row_code_code_bcb4we_recurrence_predecessor bcf_row_code_scale_bcb4we_recurrence_predecessor bcf_row_scale_code_bcb4we_recurrence_predecessor bcf_row_scale_scale_bcb4we_recurrence_predecessor bcf_row_code_bcb4we_recurrence_predecessor bcf_row_scale_bcb4we_recurrence_predecessor. ((forall bcf_row_index_bcb4we_recurrence_predecessor_table. (exists bcf_lt_gap_bcb4we_recurrence_predecessor_table_row_bound. bcf_lt_gap_bcb4we_recurrence_predecessor_table_row_bound + S (bcf_row_index_bcb4we_recurrence_predecessor_table) = S (n + n)) -> exists bcf_row_code_bcb4we_recurrence_predecessor_table bcf_row_scale_bcb4we_recurrence_predecessor_table. ((((exists bcf_height_bcb4we_recurrence_predecessor_table_decoded_row_code. bcf_height_bcb4we_recurrence_predecessor_table_decoded_row_code + S (bcf_row_code_bcb4we_recurrence_predecessor_table) = S ((S (bcf_row_index_bcb4we_recurrence_predecessor_table)) * bcf_row_code_scale_bcb4we_recurrence_predecessor)) /\ exists bcf_quotient_bcb4we_recurrence_predecessor_table_decoded_row_code. bcf_row_code_code_bcb4we_recurrence_predecessor = bcf_quotient_bcb4we_recurrence_predecessor_table_decoded_row_code * S ((S (bcf_row_index_bcb4we_recurrence_predecessor_table)) * bcf_row_code_scale_bcb4we_recurrence_predecessor) + (bcf_row_code_bcb4we_recurrence_predecessor_table))) /\ ((((exists bcf_height_bcb4we_recurrence_predecessor_table_decoded_row_scale. bcf_height_bcb4we_recurrence_predecessor_table_decoded_row_scale + S (bcf_row_scale_bcb4we_recurrence_predecessor_table) = S ((S (bcf_row_index_bcb4we_recurrence_predecessor_table)) * bcf_row_scale_scale_bcb4we_recurrence_predecessor)) /\ exists bcf_quotient_bcb4we_recurrence_predecessor_table_decoded_row_scale. bcf_row_scale_code_bcb4we_recurrence_predecessor = bcf_quotient_bcb4we_recurrence_predecessor_table_decoded_row_scale * S ((S (bcf_row_index_bcb4we_recurrence_predecessor_table)) * bcf_row_scale_scale_bcb4we_recurrence_predecessor) + (bcf_row_scale_bcb4we_recurrence_predecessor_table))) /\ ((bcf_row_index_bcb4we_recurrence_predecessor_table = 0 /\ (forall bcf_index_bcb4we_recurrence_predecessor_table_zero_row. (exists bcf_lt_gap_bcb4we_recurrence_predecessor_table_zero_row_bound. bcf_lt_gap_bcb4we_recurrence_predecessor_table_zero_row_bound + S (bcf_index_bcb4we_recurrence_predecessor_table_zero_row) = S (n + n)) -> exists bcf_value_bcb4we_recurrence_predecessor_table_zero_row. ((((exists bcf_height_bcb4we_recurrence_predecessor_table_zero_row_entry. bcf_height_bcb4we_recurrence_predecessor_table_zero_row_entry + S (bcf_value_bcb4we_recurrence_predecessor_table_zero_row) = S ((S (bcf_index_bcb4we_recurrence_predecessor_table_zero_row)) * bcf_row_scale_bcb4we_recurrence_predecessor_table)) /\ exists bcf_quotient_bcb4we_recurrence_predecessor_table_zero_row_entry. bcf_row_code_bcb4we_recurrence_predecessor_table = bcf_quotient_bcb4we_recurrence_predecessor_table_zero_row_entry * S ((S (bcf_index_bcb4we_recurrence_predecessor_table_zero_row)) * bcf_row_scale_bcb4we_recurrence_predecessor_table) + (bcf_value_bcb4we_recurrence_predecessor_table_zero_row))) /\ ((bcf_index_bcb4we_recurrence_predecessor_table_zero_row = 0 /\ bcf_value_bcb4we_recurrence_predecessor_table_zero_row = 1) \/ exists bcf_predecessor_bcb4we_recurrence_predecessor_table_zero_row. bcf_index_bcb4we_recurrence_predecessor_table_zero_row = S bcf_predecessor_bcb4we_recurrence_predecessor_table_zero_row /\ bcf_value_bcb4we_recurrence_predecessor_table_zero_row = 0)))) \/ exists bcf_predecessor_bcb4we_recurrence_predecessor_table bcf_previous_code_bcb4we_recurrence_predecessor_table bcf_previous_scale_bcb4we_recurrence_predecessor_table. bcf_row_index_bcb4we_recurrence_predecessor_table = S bcf_predecessor_bcb4we_recurrence_predecessor_table /\ ((((exists bcf_height_bcb4we_recurrence_predecessor_table_decoded_previous_code. bcf_height_bcb4we_recurrence_predecessor_table_decoded_previous_code + S (bcf_previous_code_bcb4we_recurrence_predecessor_table) = S ((S (bcf_predecessor_bcb4we_recurrence_predecessor_table)) * bcf_row_code_scale_bcb4we_recurrence_predecessor)) /\ exists bcf_quotient_bcb4we_recurrence_predecessor_table_decoded_previous_code. bcf_row_code_code_bcb4we_recurrence_predecessor = bcf_quotient_bcb4we_recurrence_predecessor_table_decoded_previous_code * S ((S (bcf_predecessor_bcb4we_recurrence_predecessor_table)) * bcf_row_code_scale_bcb4we_recurrence_predecessor) + (bcf_previous_code_bcb4we_recurrence_predecessor_table))) /\ ((((exists bcf_height_bcb4we_recurrence_predecessor_table_decoded_previous_scale. bcf_height_bcb4we_recurrence_predecessor_table_decoded_previous_scale + S (bcf_previous_scale_bcb4we_recurrence_predecessor_table) = S ((S (bcf_predecessor_bcb4we_recurrence_predecessor_table)) * bcf_row_scale_scale_bcb4we_recurrence_predecessor)) /\ exists bcf_quotient_bcb4we_recurrence_predecessor_table_decoded_previous_scale. bcf_row_scale_code_bcb4we_recurrence_predecessor = bcf_quotient_bcb4we_recurrence_predecessor_table_decoded_previous_scale * S ((S (bcf_predecessor_bcb4we_recurrence_predecessor_table)) * bcf_row_scale_scale_bcb4we_recurrence_predecessor) + (bcf_previous_scale_bcb4we_recurrence_predecessor_table))) /\ (forall bcf_index_bcb4we_recurrence_predecessor_table_row_step. (exists bcf_lt_gap_bcb4we_recurrence_predecessor_table_row_step_bound. bcf_lt_gap_bcb4we_recurrence_predecessor_table_row_step_bound + S (bcf_index_bcb4we_recurrence_predecessor_table_row_step) = S (n + n)) -> exists bcf_value_bcb4we_recurrence_predecessor_table_row_step. ((((exists bcf_height_bcb4we_recurrence_predecessor_table_row_step_entry. bcf_height_bcb4we_recurrence_predecessor_table_row_step_entry + S (bcf_value_bcb4we_recurrence_predecessor_table_row_step) = S ((S (bcf_index_bcb4we_recurrence_predecessor_table_row_step)) * bcf_row_scale_bcb4we_recurrence_predecessor_table)) /\ exists bcf_quotient_bcb4we_recurrence_predecessor_table_row_step_entry. bcf_row_code_bcb4we_recurrence_predecessor_table = bcf_quotient_bcb4we_recurrence_predecessor_table_row_step_entry * S ((S (bcf_index_bcb4we_recurrence_predecessor_table_row_step)) * bcf_row_scale_bcb4we_recurrence_predecessor_table) + (bcf_value_bcb4we_recurrence_predecessor_table_row_step))) /\ ((bcf_index_bcb4we_recurrence_predecessor_table_row_step = 0 /\ bcf_value_bcb4we_recurrence_predecessor_table_row_step = 1) \/ exists bcf_predecessor_bcb4we_recurrence_predecessor_table_row_step bcf_left_bcb4we_recurrence_predecessor_table_row_step bcf_right_bcb4we_recurrence_predecessor_table_row_step. bcf_index_bcb4we_recurrence_predecessor_table_row_step = S bcf_predecessor_bcb4we_recurrence_predecessor_table_row_step /\ ((((exists bcf_height_bcb4we_recurrence_predecessor_table_row_step_previous_left. bcf_height_bcb4we_recurrence_predecessor_table_row_step_previous_left + S (bcf_left_bcb4we_recurrence_predecessor_table_row_step) = S ((S (bcf_predecessor_bcb4we_recurrence_predecessor_table_row_step)) * bcf_previous_scale_bcb4we_recurrence_predecessor_table)) /\ exists bcf_quotient_bcb4we_recurrence_predecessor_table_row_step_previous_left. bcf_previous_code_bcb4we_recurrence_predecessor_table = bcf_quotient_bcb4we_recurrence_predecessor_table_row_step_previous_left * S ((S (bcf_predecessor_bcb4we_recurrence_predecessor_table_row_step)) * bcf_previous_scale_bcb4we_recurrence_predecessor_table) + (bcf_left_bcb4we_recurrence_predecessor_table_row_step))) /\ ((((exists bcf_height_bcb4we_recurrence_predecessor_table_row_step_previous_right. bcf_height_bcb4we_recurrence_predecessor_table_row_step_previous_right + S (bcf_right_bcb4we_recurrence_predecessor_table_row_step) = S ((S (S (bcf_predecessor_bcb4we_recurrence_predecessor_table_row_step))) * bcf_previous_scale_bcb4we_recurrence_predecessor_table)) /\ exists bcf_quotient_bcb4we_recurrence_predecessor_table_row_step_previous_right. bcf_previous_code_bcb4we_recurrence_predecessor_table = bcf_quotient_bcb4we_recurrence_predecessor_table_row_step_previous_right * S ((S (S (bcf_predecessor_bcb4we_recurrence_predecessor_table_row_step))) * bcf_previous_scale_bcb4we_recurrence_predecessor_table) + (bcf_right_bcb4we_recurrence_predecessor_table_row_step))) /\ bcf_value_bcb4we_recurrence_predecessor_table_row_step = bcf_left_bcb4we_recurrence_predecessor_table_row_step + bcf_right_bcb4we_recurrence_predecessor_table_row_step))))))))))) /\ ((((exists bcf_height_bcb4we_recurrence_predecessor_decoded_row_code. bcf_height_bcb4we_recurrence_predecessor_decoded_row_code + S (bcf_row_code_bcb4we_recurrence_predecessor) = S ((S (n + n)) * bcf_row_code_scale_bcb4we_recurrence_predecessor)) /\ exists bcf_quotient_bcb4we_recurrence_predecessor_decoded_row_code. bcf_row_code_code_bcb4we_recurrence_predecessor = bcf_quotient_bcb4we_recurrence_predecessor_decoded_row_code * S ((S (n + n)) * bcf_row_code_scale_bcb4we_recurrence_predecessor) + (bcf_row_code_bcb4we_recurrence_predecessor))) /\ ((((exists bcf_height_bcb4we_recurrence_predecessor_decoded_row_scale. bcf_height_bcb4we_recurrence_predecessor_decoded_row_scale + S (bcf_row_scale_bcb4we_recurrence_predecessor) = S ((S (n + n)) * bcf_row_scale_scale_bcb4we_recurrence_predecessor)) /\ exists bcf_quotient_bcb4we_recurrence_predecessor_decoded_row_scale. bcf_row_scale_code_bcb4we_recurrence_predecessor = bcf_quotient_bcb4we_recurrence_predecessor_decoded_row_scale * S ((S (n + n)) * bcf_row_scale_scale_bcb4we_recurrence_predecessor) + (bcf_row_scale_bcb4we_recurrence_predecessor))) /\ (((exists bcf_height_bcb4we_recurrence_predecessor_decoded_value. bcf_height_bcb4we_recurrence_predecessor_decoded_value + S (a) = S ((S (n)) * bcf_row_scale_bcb4we_recurrence_predecessor)) /\ exists bcf_quotient_bcb4we_recurrence_predecessor_decoded_value. bcf_row_code_bcb4we_recurrence_predecessor = bcf_quotient_bcb4we_recurrence_predecessor_decoded_value * S ((S (n)) * bcf_row_scale_bcb4we_recurrence_predecessor) + (a))))))))) -> (((exists bcf_lt_gap_bcb4we_recurrence_successor_out_of_range. bcf_lt_gap_bcb4we_recurrence_successor_out_of_range + S (S n + S n) = S n) /\ b = 0) \/ ((exists bcf_le_gap_bcb4we_recurrence_successor_in_range. bcf_le_gap_bcb4we_recurrence_successor_in_range + (S n) = S n + S n) /\ (exists bcf_row_code_code_bcb4we_recurrence_successor bcf_row_code_scale_bcb4we_recurrence_successor bcf_row_scale_code_bcb4we_recurrence_successor bcf_row_scale_scale_bcb4we_recurrence_successor bcf_row_code_bcb4we_recurrence_successor bcf_row_scale_bcb4we_recurrence_successor. ((forall bcf_row_index_bcb4we_recurrence_successor_table. (exists bcf_lt_gap_bcb4we_recurrence_successor_table_row_bound. bcf_lt_gap_bcb4we_recurrence_successor_table_row_bound + S (bcf_row_index_bcb4we_recurrence_successor_table) = S (S n + S n)) -> exists bcf_row_code_bcb4we_recurrence_successor_table bcf_row_scale_bcb4we_recurrence_successor_table. ((((exists bcf_height_bcb4we_recurrence_successor_table_decoded_row_code. bcf_height_bcb4we_recurrence_successor_table_decoded_row_code + S (bcf_row_code_bcb4we_recurrence_successor_table) = S ((S (bcf_row_index_bcb4we_recurrence_successor_table)) * bcf_row_code_scale_bcb4we_recurrence_successor)) /\ exists bcf_quotient_bcb4we_recurrence_successor_table_decoded_row_code. bcf_row_code_code_bcb4we_recurrence_successor = bcf_quotient_bcb4we_recurrence_successor_table_decoded_row_code * S ((S (bcf_row_index_bcb4we_recurrence_successor_table)) * bcf_row_code_scale_bcb4we_recurrence_successor) + (bcf_row_code_bcb4we_recurrence_successor_table))) /\ ((((exists bcf_height_bcb4we_recurrence_successor_table_decoded_row_scale. bcf_height_bcb4we_recurrence_successor_table_decoded_row_scale + S (bcf_row_scale_bcb4we_recurrence_successor_table) = S ((S (bcf_row_index_bcb4we_recurrence_successor_table)) * bcf_row_scale_scale_bcb4we_recurrence_successor)) /\ exists bcf_quotient_bcb4we_recurrence_successor_table_decoded_row_scale. bcf_row_scale_code_bcb4we_recurrence_successor = bcf_quotient_bcb4we_recurrence_successor_table_decoded_row_scale * S ((S (bcf_row_index_bcb4we_recurrence_successor_table)) * bcf_row_scale_scale_bcb4we_recurrence_successor) + (bcf_row_scale_bcb4we_recurrence_successor_table))) /\ ((bcf_row_index_bcb4we_recurrence_successor_table = 0 /\ (forall bcf_index_bcb4we_recurrence_successor_table_zero_row. (exists bcf_lt_gap_bcb4we_recurrence_successor_table_zero_row_bound. bcf_lt_gap_bcb4we_recurrence_successor_table_zero_row_bound + S (bcf_index_bcb4we_recurrence_successor_table_zero_row) = S (S n + S n)) -> exists bcf_value_bcb4we_recurrence_successor_table_zero_row. ((((exists bcf_height_bcb4we_recurrence_successor_table_zero_row_entry. bcf_height_bcb4we_recurrence_successor_table_zero_row_entry + S (bcf_value_bcb4we_recurrence_successor_table_zero_row) = S ((S (bcf_index_bcb4we_recurrence_successor_table_zero_row)) * bcf_row_scale_bcb4we_recurrence_successor_table)) /\ exists bcf_quotient_bcb4we_recurrence_successor_table_zero_row_entry. bcf_row_code_bcb4we_recurrence_successor_table = bcf_quotient_bcb4we_recurrence_successor_table_zero_row_entry * S ((S (bcf_index_bcb4we_recurrence_successor_table_zero_row)) * bcf_row_scale_bcb4we_recurrence_successor_table) + (bcf_value_bcb4we_recurrence_successor_table_zero_row))) /\ ((bcf_index_bcb4we_recurrence_successor_table_zero_row = 0 /\ bcf_value_bcb4we_recurrence_successor_table_zero_row = 1) \/ exists bcf_predecessor_bcb4we_recurrence_successor_table_zero_row. bcf_index_bcb4we_recurrence_successor_table_zero_row = S bcf_predecessor_bcb4we_recurrence_successor_table_zero_row /\ bcf_value_bcb4we_recurrence_successor_table_zero_row = 0)))) \/ exists bcf_predecessor_bcb4we_recurrence_successor_table bcf_previous_code_bcb4we_recurrence_successor_table bcf_previous_scale_bcb4we_recurrence_successor_table. bcf_row_index_bcb4we_recurrence_successor_table = S bcf_predecessor_bcb4we_recurrence_successor_table /\ ((((exists bcf_height_bcb4we_recurrence_successor_table_decoded_previous_code. bcf_height_bcb4we_recurrence_successor_table_decoded_previous_code + S (bcf_previous_code_bcb4we_recurrence_successor_table) = S ((S (bcf_predecessor_bcb4we_recurrence_successor_table)) * bcf_row_code_scale_bcb4we_recurrence_successor)) /\ exists bcf_quotient_bcb4we_recurrence_successor_table_decoded_previous_code. bcf_row_code_code_bcb4we_recurrence_successor = bcf_quotient_bcb4we_recurrence_successor_table_decoded_previous_code * S ((S (bcf_predecessor_bcb4we_recurrence_successor_table)) * bcf_row_code_scale_bcb4we_recurrence_successor) + (bcf_previous_code_bcb4we_recurrence_successor_table))) /\ ((((exists bcf_height_bcb4we_recurrence_successor_table_decoded_previous_scale. bcf_height_bcb4we_recurrence_successor_table_decoded_previous_scale + S (bcf_previous_scale_bcb4we_recurrence_successor_table) = S ((S (bcf_predecessor_bcb4we_recurrence_successor_table)) * bcf_row_scale_scale_bcb4we_recurrence_successor)) /\ exists bcf_quotient_bcb4we_recurrence_successor_table_decoded_previous_scale. bcf_row_scale_code_bcb4we_recurrence_successor = bcf_quotient_bcb4we_recurrence_successor_table_decoded_previous_scale * S ((S (bcf_predecessor_bcb4we_recurrence_successor_table)) * bcf_row_scale_scale_bcb4we_recurrence_successor) + (bcf_previous_scale_bcb4we_recurrence_successor_table))) /\ (forall bcf_index_bcb4we_recurrence_successor_table_row_step. (exists bcf_lt_gap_bcb4we_recurrence_successor_table_row_step_bound. bcf_lt_gap_bcb4we_recurrence_successor_table_row_step_bound + S (bcf_index_bcb4we_recurrence_successor_table_row_step) = S (S n + S n)) -> exists bcf_value_bcb4we_recurrence_successor_table_row_step. ((((exists bcf_height_bcb4we_recurrence_successor_table_row_step_entry. bcf_height_bcb4we_recurrence_successor_table_row_step_entry + S (bcf_value_bcb4we_recurrence_successor_table_row_step) = S ((S (bcf_index_bcb4we_recurrence_successor_table_row_step)) * bcf_row_scale_bcb4we_recurrence_successor_table)) /\ exists bcf_quotient_bcb4we_recurrence_successor_table_row_step_entry. bcf_row_code_bcb4we_recurrence_successor_table = bcf_quotient_bcb4we_recurrence_successor_table_row_step_entry * S ((S (bcf_index_bcb4we_recurrence_successor_table_row_step)) * bcf_row_scale_bcb4we_recurrence_successor_table) + (bcf_value_bcb4we_recurrence_successor_table_row_step))) /\ ((bcf_index_bcb4we_recurrence_successor_table_row_step = 0 /\ bcf_value_bcb4we_recurrence_successor_table_row_step = 1) \/ exists bcf_predecessor_bcb4we_recurrence_successor_table_row_step bcf_left_bcb4we_recurrence_successor_table_row_step bcf_right_bcb4we_recurrence_successor_table_row_step. bcf_index_bcb4we_recurrence_successor_table_row_step = S bcf_predecessor_bcb4we_recurrence_successor_table_row_step /\ ((((exists bcf_height_bcb4we_recurrence_successor_table_row_step_previous_left. bcf_height_bcb4we_recurrence_successor_table_row_step_previous_left + S (bcf_left_bcb4we_recurrence_successor_table_row_step) = S ((S (bcf_predecessor_bcb4we_recurrence_successor_table_row_step)) * bcf_previous_scale_bcb4we_recurrence_successor_table)) /\ exists bcf_quotient_bcb4we_recurrence_successor_table_row_step_previous_left. bcf_previous_code_bcb4we_recurrence_successor_table = bcf_quotient_bcb4we_recurrence_successor_table_row_step_previous_left * S ((S (bcf_predecessor_bcb4we_recurrence_successor_table_row_step)) * bcf_previous_scale_bcb4we_recurrence_successor_table) + (bcf_left_bcb4we_recurrence_successor_table_row_step))) /\ ((((exists bcf_height_bcb4we_recurrence_successor_table_row_step_previous_right. bcf_height_bcb4we_recurrence_successor_table_row_step_previous_right + S (bcf_right_bcb4we_recurrence_successor_table_row_step) = S ((S (S (bcf_predecessor_bcb4we_recurrence_successor_table_row_step))) * bcf_previous_scale_bcb4we_recurrence_successor_table)) /\ exists bcf_quotient_bcb4we_recurrence_successor_table_row_step_previous_right. bcf_previous_code_bcb4we_recurrence_successor_table = bcf_quotient_bcb4we_recurrence_successor_table_row_step_previous_right * S ((S (S (bcf_predecessor_bcb4we_recurrence_successor_table_row_step))) * bcf_previous_scale_bcb4we_recurrence_successor_table) + (bcf_right_bcb4we_recurrence_successor_table_row_step))) /\ bcf_value_bcb4we_recurrence_successor_table_row_step = bcf_left_bcb4we_recurrence_successor_table_row_step + bcf_right_bcb4we_recurrence_successor_table_row_step))))))))))) /\ ((((exists bcf_height_bcb4we_recurrence_successor_decoded_row_code. bcf_height_bcb4we_recurrence_successor_decoded_row_code + S (bcf_row_code_bcb4we_recurrence_successor) = S ((S (S n + S n)) * bcf_row_code_scale_bcb4we_recurrence_successor)) /\ exists bcf_quotient_bcb4we_recurrence_successor_decoded_row_code. bcf_row_code_code_bcb4we_recurrence_successor = bcf_quotient_bcb4we_recurrence_successor_decoded_row_code * S ((S (S n + S n)) * bcf_row_code_scale_bcb4we_recurrence_successor) + (bcf_row_code_bcb4we_recurrence_successor))) /\ ((((exists bcf_height_bcb4we_recurrence_successor_decoded_row_scale. bcf_height_bcb4we_recurrence_successor_decoded_row_scale + S (bcf_row_scale_bcb4we_recurrence_successor) = S ((S (S n + S n)) * bcf_row_scale_scale_bcb4we_recurrence_successor)) /\ exists bcf_quotient_bcb4we_recurrence_successor_decoded_row_scale. bcf_row_scale_code_bcb4we_recurrence_successor = bcf_quotient_bcb4we_recurrence_successor_decoded_row_scale * S ((S (S n + S n)) * bcf_row_scale_scale_bcb4we_recurrence_successor) + (bcf_row_scale_bcb4we_recurrence_successor))) /\ (((exists bcf_height_bcb4we_recurrence_successor_decoded_value. bcf_height_bcb4we_recurrence_successor_decoded_value + S (b) = S ((S (S n)) * bcf_row_scale_bcb4we_recurrence_successor)) /\ exists bcf_quotient_bcb4we_recurrence_successor_decoded_value. bcf_row_code_bcb4we_recurrence_successor = bcf_quotient_bcb4we_recurrence_successor_decoded_value * S ((S (S n)) * bcf_row_scale_bcb4we_recurrence_successor) + (b))))))))) -> S n * b = (2 * S (n + n)) * a) -> ((forall n. exists z. (((exists bcf_lt_gap_bcb4we_exists_out_of_range. bcf_lt_gap_bcb4we_exists_out_of_range + S (n + n) = n) /\ z = 0) \/ ((exists bcf_le_gap_bcb4we_exists_in_range. bcf_le_gap_bcb4we_exists_in_range + (n) = n + n) /\ (exists bcf_row_code_code_bcb4we_exists bcf_row_code_scale_bcb4we_exists bcf_row_scale_code_bcb4we_exists bcf_row_scale_scale_bcb4we_exists bcf_row_code_bcb4we_exists bcf_row_scale_bcb4we_exists. ((forall bcf_row_index_bcb4we_exists_table. (exists bcf_lt_gap_bcb4we_exists_table_row_bound. bcf_lt_gap_bcb4we_exists_table_row_bound + S (bcf_row_index_bcb4we_exists_table) = S (n + n)) -> exists bcf_row_code_bcb4we_exists_table bcf_row_scale_bcb4we_exists_table. ((((exists bcf_height_bcb4we_exists_table_decoded_row_code. bcf_height_bcb4we_exists_table_decoded_row_code + S (bcf_row_code_bcb4we_exists_table) = S ((S (bcf_row_index_bcb4we_exists_table)) * bcf_row_code_scale_bcb4we_exists)) /\ exists bcf_quotient_bcb4we_exists_table_decoded_row_code. bcf_row_code_code_bcb4we_exists = bcf_quotient_bcb4we_exists_table_decoded_row_code * S ((S (bcf_row_index_bcb4we_exists_table)) * bcf_row_code_scale_bcb4we_exists) + (bcf_row_code_bcb4we_exists_table))) /\ ((((exists bcf_height_bcb4we_exists_table_decoded_row_scale. bcf_height_bcb4we_exists_table_decoded_row_scale + S (bcf_row_scale_bcb4we_exists_table) = S ((S (bcf_row_index_bcb4we_exists_table)) * bcf_row_scale_scale_bcb4we_exists)) /\ exists bcf_quotient_bcb4we_exists_table_decoded_row_scale. bcf_row_scale_code_bcb4we_exists = bcf_quotient_bcb4we_exists_table_decoded_row_scale * S ((S (bcf_row_index_bcb4we_exists_table)) * bcf_row_scale_scale_bcb4we_exists) + (bcf_row_scale_bcb4we_exists_table))) /\ ((bcf_row_index_bcb4we_exists_table = 0 /\ (forall bcf_index_bcb4we_exists_table_zero_row. (exists bcf_lt_gap_bcb4we_exists_table_zero_row_bound. bcf_lt_gap_bcb4we_exists_table_zero_row_bound + S (bcf_index_bcb4we_exists_table_zero_row) = S (n + n)) -> exists bcf_value_bcb4we_exists_table_zero_row. ((((exists bcf_height_bcb4we_exists_table_zero_row_entry. bcf_height_bcb4we_exists_table_zero_row_entry + S (bcf_value_bcb4we_exists_table_zero_row) = S ((S (bcf_index_bcb4we_exists_table_zero_row)) * bcf_row_scale_bcb4we_exists_table)) /\ exists bcf_quotient_bcb4we_exists_table_zero_row_entry. bcf_row_code_bcb4we_exists_table = bcf_quotient_bcb4we_exists_table_zero_row_entry * S ((S (bcf_index_bcb4we_exists_table_zero_row)) * bcf_row_scale_bcb4we_exists_table) + (bcf_value_bcb4we_exists_table_zero_row))) /\ ((bcf_index_bcb4we_exists_table_zero_row = 0 /\ bcf_value_bcb4we_exists_table_zero_row = 1) \/ exists bcf_predecessor_bcb4we_exists_table_zero_row. bcf_index_bcb4we_exists_table_zero_row = S bcf_predecessor_bcb4we_exists_table_zero_row /\ bcf_value_bcb4we_exists_table_zero_row = 0)))) \/ exists bcf_predecessor_bcb4we_exists_table bcf_previous_code_bcb4we_exists_table bcf_previous_scale_bcb4we_exists_table. bcf_row_index_bcb4we_exists_table = S bcf_predecessor_bcb4we_exists_table /\ ((((exists bcf_height_bcb4we_exists_table_decoded_previous_code. bcf_height_bcb4we_exists_table_decoded_previous_code + S (bcf_previous_code_bcb4we_exists_table) = S ((S (bcf_predecessor_bcb4we_exists_table)) * bcf_row_code_scale_bcb4we_exists)) /\ exists bcf_quotient_bcb4we_exists_table_decoded_previous_code. bcf_row_code_code_bcb4we_exists = bcf_quotient_bcb4we_exists_table_decoded_previous_code * S ((S (bcf_predecessor_bcb4we_exists_table)) * bcf_row_code_scale_bcb4we_exists) + (bcf_previous_code_bcb4we_exists_table))) /\ ((((exists bcf_height_bcb4we_exists_table_decoded_previous_scale. bcf_height_bcb4we_exists_table_decoded_previous_scale + S (bcf_previous_scale_bcb4we_exists_table) = S ((S (bcf_predecessor_bcb4we_exists_table)) * bcf_row_scale_scale_bcb4we_exists)) /\ exists bcf_quotient_bcb4we_exists_table_decoded_previous_scale. bcf_row_scale_code_bcb4we_exists = bcf_quotient_bcb4we_exists_table_decoded_previous_scale * S ((S (bcf_predecessor_bcb4we_exists_table)) * bcf_row_scale_scale_bcb4we_exists) + (bcf_previous_scale_bcb4we_exists_table))) /\ (forall bcf_index_bcb4we_exists_table_row_step. (exists bcf_lt_gap_bcb4we_exists_table_row_step_bound. bcf_lt_gap_bcb4we_exists_table_row_step_bound + S (bcf_index_bcb4we_exists_table_row_step) = S (n + n)) -> exists bcf_value_bcb4we_exists_table_row_step. ((((exists bcf_height_bcb4we_exists_table_row_step_entry. bcf_height_bcb4we_exists_table_row_step_entry + S (bcf_value_bcb4we_exists_table_row_step) = S ((S (bcf_index_bcb4we_exists_table_row_step)) * bcf_row_scale_bcb4we_exists_table)) /\ exists bcf_quotient_bcb4we_exists_table_row_step_entry. bcf_row_code_bcb4we_exists_table = bcf_quotient_bcb4we_exists_table_row_step_entry * S ((S (bcf_index_bcb4we_exists_table_row_step)) * bcf_row_scale_bcb4we_exists_table) + (bcf_value_bcb4we_exists_table_row_step))) /\ ((bcf_index_bcb4we_exists_table_row_step = 0 /\ bcf_value_bcb4we_exists_table_row_step = 1) \/ exists bcf_predecessor_bcb4we_exists_table_row_step bcf_left_bcb4we_exists_table_row_step bcf_right_bcb4we_exists_table_row_step. bcf_index_bcb4we_exists_table_row_step = S bcf_predecessor_bcb4we_exists_table_row_step /\ ((((exists bcf_height_bcb4we_exists_table_row_step_previous_left. bcf_height_bcb4we_exists_table_row_step_previous_left + S (bcf_left_bcb4we_exists_table_row_step) = S ((S (bcf_predecessor_bcb4we_exists_table_row_step)) * bcf_previous_scale_bcb4we_exists_table)) /\ exists bcf_quotient_bcb4we_exists_table_row_step_previous_left. bcf_previous_code_bcb4we_exists_table = bcf_quotient_bcb4we_exists_table_row_step_previous_left * S ((S (bcf_predecessor_bcb4we_exists_table_row_step)) * bcf_previous_scale_bcb4we_exists_table) + (bcf_left_bcb4we_exists_table_row_step))) /\ ((((exists bcf_height_bcb4we_exists_table_row_step_previous_right. bcf_height_bcb4we_exists_table_row_step_previous_right + S (bcf_right_bcb4we_exists_table_row_step) = S ((S (S (bcf_predecessor_bcb4we_exists_table_row_step))) * bcf_previous_scale_bcb4we_exists_table)) /\ exists bcf_quotient_bcb4we_exists_table_row_step_previous_right. bcf_previous_code_bcb4we_exists_table = bcf_quotient_bcb4we_exists_table_row_step_previous_right * S ((S (S (bcf_predecessor_bcb4we_exists_table_row_step))) * bcf_previous_scale_bcb4we_exists_table) + (bcf_right_bcb4we_exists_table_row_step))) /\ bcf_value_bcb4we_exists_table_row_step = bcf_left_bcb4we_exists_table_row_step + bcf_right_bcb4we_exists_table_row_step))))))))))) /\ ((((exists bcf_height_bcb4we_exists_decoded_row_code. bcf_height_bcb4we_exists_decoded_row_code + S (bcf_row_code_bcb4we_exists) = S ((S (n + n)) * bcf_row_code_scale_bcb4we_exists)) /\ exists bcf_quotient_bcb4we_exists_decoded_row_code. bcf_row_code_code_bcb4we_exists = bcf_quotient_bcb4we_exists_decoded_row_code * S ((S (n + n)) * bcf_row_code_scale_bcb4we_exists) + (bcf_row_code_bcb4we_exists))) /\ ((((exists bcf_height_bcb4we_exists_decoded_row_scale. bcf_height_bcb4we_exists_decoded_row_scale + S (bcf_row_scale_bcb4we_exists) = S ((S (n + n)) * bcf_row_scale_scale_bcb4we_exists)) /\ exists bcf_quotient_bcb4we_exists_decoded_row_scale. bcf_row_scale_code_bcb4we_exists = bcf_quotient_bcb4we_exists_decoded_row_scale * S ((S (n + n)) * bcf_row_scale_scale_bcb4we_exists) + (bcf_row_scale_bcb4we_exists))) /\ (((exists bcf_height_bcb4we_exists_decoded_value. bcf_height_bcb4we_exists_decoded_value + S (z) = S ((S (n)) * bcf_row_scale_bcb4we_exists)) /\ exists bcf_quotient_bcb4we_exists_decoded_value. bcf_row_code_bcb4we_exists = bcf_quotient_bcb4we_exists_decoded_value * S ((S (n)) * bcf_row_scale_bcb4we_exists) + (z)))))))))) /\ (forall p c. (exists pa_b_bfplcb4_power pa_c_bfplcb4_power. ((forall pa_i_bfplcb4_power_repeat. (exists pa_lt_bfplcb4_power_repeat_bound. pa_lt_bfplcb4_power_repeat_bound + S pa_i_bfplcb4_power_repeat = 4) -> (((exists pa_h_bfplcb4_power_repeat_decoded. pa_h_bfplcb4_power_repeat_decoded + S (4) = S ((S (pa_i_bfplcb4_power_repeat)) * pa_c_bfplcb4_power)) /\ exists pa_q_bfplcb4_power_repeat_decoded. pa_b_bfplcb4_power = pa_q_bfplcb4_power_repeat_decoded * S ((S (pa_i_bfplcb4_power_repeat)) * pa_c_bfplcb4_power) + (4)))) /\ (exists pa_u_bfplcb4_power_product pa_v_bfplcb4_power_product. ((((exists pa_h_bfplcb4_power_product_start. pa_h_bfplcb4_power_product_start + S (1) = S ((S (0)) * pa_v_bfplcb4_power_product)) /\ exists pa_q_bfplcb4_power_product_start. pa_u_bfplcb4_power_product = pa_q_bfplcb4_power_product_start * S ((S (0)) * pa_v_bfplcb4_power_product) + (1))) /\ ((((exists pa_h_bfplcb4_power_product_terminal. pa_h_bfplcb4_power_product_terminal + S (p) = S ((S (4)) * pa_v_bfplcb4_power_product)) /\ exists pa_q_bfplcb4_power_product_terminal. pa_u_bfplcb4_power_product = pa_q_bfplcb4_power_product_terminal * S ((S (4)) * pa_v_bfplcb4_power_product) + (p))) /\ forall pa_i_bfplcb4_power_product. (exists pa_lt_bfplcb4_power_product_bound. pa_lt_bfplcb4_power_product_bound + S pa_i_bfplcb4_power_product = 4) -> exists pa_p_bfplcb4_power_product pa_r_bfplcb4_power_product pa_s_bfplcb4_power_product. ((((exists pa_h_bfplcb4_power_product_factor. pa_h_bfplcb4_power_product_factor + S (pa_p_bfplcb4_power_product) = S ((S (pa_i_bfplcb4_power_product)) * pa_c_bfplcb4_power)) /\ exists pa_q_bfplcb4_power_product_factor. pa_b_bfplcb4_power = pa_q_bfplcb4_power_product_factor * S ((S (pa_i_bfplcb4_power_product)) * pa_c_bfplcb4_power) + (pa_p_bfplcb4_power_product))) /\ ((((exists pa_h_bfplcb4_power_product_partial. pa_h_bfplcb4_power_product_partial + S (pa_r_bfplcb4_power_product) = S ((S (pa_i_bfplcb4_power_product)) * pa_v_bfplcb4_power_product)) /\ exists pa_q_bfplcb4_power_product_partial. pa_u_bfplcb4_power_product = pa_q_bfplcb4_power_product_partial * S ((S (pa_i_bfplcb4_power_product)) * pa_v_bfplcb4_power_product) + (pa_r_bfplcb4_power_product))) /\ ((((exists pa_h_bfplcb4_power_product_successor. pa_h_bfplcb4_power_product_successor + S (pa_s_bfplcb4_power_product) = S ((S (S pa_i_bfplcb4_power_product)) * pa_v_bfplcb4_power_product)) /\ exists pa_q_bfplcb4_power_product_successor. pa_u_bfplcb4_power_product = pa_q_bfplcb4_power_product_successor * S ((S (S pa_i_bfplcb4_power_product)) * pa_v_bfplcb4_power_product) + (pa_s_bfplcb4_power_product))) /\ pa_s_bfplcb4_power_product = pa_r_bfplcb4_power_product * pa_p_bfplcb4_power_product)))))))) -> (((exists bcf_lt_gap_bfplcb4_central_out_of_range. bcf_lt_gap_bfplcb4_central_out_of_range + S (4 + 4) = 4) /\ c = 0) \/ ((exists bcf_le_gap_bfplcb4_central_in_range. bcf_le_gap_bfplcb4_central_in_range + (4) = 4 + 4) /\ (exists bcf_row_code_code_bfplcb4_central bcf_row_code_scale_bfplcb4_central bcf_row_scale_code_bfplcb4_central bcf_row_scale_scale_bfplcb4_central bcf_row_code_bfplcb4_central bcf_row_scale_bfplcb4_central. ((forall bcf_row_index_bfplcb4_central_table. (exists bcf_lt_gap_bfplcb4_central_table_row_bound. bcf_lt_gap_bfplcb4_central_table_row_bound + S (bcf_row_index_bfplcb4_central_table) = S (4 + 4)) -> exists bcf_row_code_bfplcb4_central_table bcf_row_scale_bfplcb4_central_table. ((((exists bcf_height_bfplcb4_central_table_decoded_row_code. bcf_height_bfplcb4_central_table_decoded_row_code + S (bcf_row_code_bfplcb4_central_table) = S ((S (bcf_row_index_bfplcb4_central_table)) * bcf_row_code_scale_bfplcb4_central)) /\ exists bcf_quotient_bfplcb4_central_table_decoded_row_code. bcf_row_code_code_bfplcb4_central = bcf_quotient_bfplcb4_central_table_decoded_row_code * S ((S (bcf_row_index_bfplcb4_central_table)) * bcf_row_code_scale_bfplcb4_central) + (bcf_row_code_bfplcb4_central_table))) /\ ((((exists bcf_height_bfplcb4_central_table_decoded_row_scale. bcf_height_bfplcb4_central_table_decoded_row_scale + S (bcf_row_scale_bfplcb4_central_table) = S ((S (bcf_row_index_bfplcb4_central_table)) * bcf_row_scale_scale_bfplcb4_central)) /\ exists bcf_quotient_bfplcb4_central_table_decoded_row_scale. bcf_row_scale_code_bfplcb4_central = bcf_quotient_bfplcb4_central_table_decoded_row_scale * S ((S (bcf_row_index_bfplcb4_central_table)) * bcf_row_scale_scale_bfplcb4_central) + (bcf_row_scale_bfplcb4_central_table))) /\ ((bcf_row_index_bfplcb4_central_table = 0 /\ (forall bcf_index_bfplcb4_central_table_zero_row. (exists bcf_lt_gap_bfplcb4_central_table_zero_row_bound. bcf_lt_gap_bfplcb4_central_table_zero_row_bound + S (bcf_index_bfplcb4_central_table_zero_row) = S (4 + 4)) -> exists bcf_value_bfplcb4_central_table_zero_row. ((((exists bcf_height_bfplcb4_central_table_zero_row_entry. bcf_height_bfplcb4_central_table_zero_row_entry + S (bcf_value_bfplcb4_central_table_zero_row) = S ((S (bcf_index_bfplcb4_central_table_zero_row)) * bcf_row_scale_bfplcb4_central_table)) /\ exists bcf_quotient_bfplcb4_central_table_zero_row_entry. bcf_row_code_bfplcb4_central_table = bcf_quotient_bfplcb4_central_table_zero_row_entry * S ((S (bcf_index_bfplcb4_central_table_zero_row)) * bcf_row_scale_bfplcb4_central_table) + (bcf_value_bfplcb4_central_table_zero_row))) /\ ((bcf_index_bfplcb4_central_table_zero_row = 0 /\ bcf_value_bfplcb4_central_table_zero_row = 1) \/ exists bcf_predecessor_bfplcb4_central_table_zero_row. bcf_index_bfplcb4_central_table_zero_row = S bcf_predecessor_bfplcb4_central_table_zero_row /\ bcf_value_bfplcb4_central_table_zero_row = 0)))) \/ exists bcf_predecessor_bfplcb4_central_table bcf_previous_code_bfplcb4_central_table bcf_previous_scale_bfplcb4_central_table. bcf_row_index_bfplcb4_central_table = S bcf_predecessor_bfplcb4_central_table /\ ((((exists bcf_height_bfplcb4_central_table_decoded_previous_code. bcf_height_bfplcb4_central_table_decoded_previous_code + S (bcf_previous_code_bfplcb4_central_table) = S ((S (bcf_predecessor_bfplcb4_central_table)) * bcf_row_code_scale_bfplcb4_central)) /\ exists bcf_quotient_bfplcb4_central_table_decoded_previous_code. bcf_row_code_code_bfplcb4_central = bcf_quotient_bfplcb4_central_table_decoded_previous_code * S ((S (bcf_predecessor_bfplcb4_central_table)) * bcf_row_code_scale_bfplcb4_central) + (bcf_previous_code_bfplcb4_central_table))) /\ ((((exists bcf_height_bfplcb4_central_table_decoded_previous_scale. bcf_height_bfplcb4_central_table_decoded_previous_scale + S (bcf_previous_scale_bfplcb4_central_table) = S ((S (bcf_predecessor_bfplcb4_central_table)) * bcf_row_scale_scale_bfplcb4_central)) /\ exists bcf_quotient_bfplcb4_central_table_decoded_previous_scale. bcf_row_scale_code_bfplcb4_central = bcf_quotient_bfplcb4_central_table_decoded_previous_scale * S ((S (bcf_predecessor_bfplcb4_central_table)) * bcf_row_scale_scale_bfplcb4_central) + (bcf_previous_scale_bfplcb4_central_table))) /\ (forall bcf_index_bfplcb4_central_table_row_step. (exists bcf_lt_gap_bfplcb4_central_table_row_step_bound. bcf_lt_gap_bfplcb4_central_table_row_step_bound + S (bcf_index_bfplcb4_central_table_row_step) = S (4 + 4)) -> exists bcf_value_bfplcb4_central_table_row_step. ((((exists bcf_height_bfplcb4_central_table_row_step_entry. bcf_height_bfplcb4_central_table_row_step_entry + S (bcf_value_bfplcb4_central_table_row_step) = S ((S (bcf_index_bfplcb4_central_table_row_step)) * bcf_row_scale_bfplcb4_central_table)) /\ exists bcf_quotient_bfplcb4_central_table_row_step_entry. bcf_row_code_bfplcb4_central_table = bcf_quotient_bfplcb4_central_table_row_step_entry * S ((S (bcf_index_bfplcb4_central_table_row_step)) * bcf_row_scale_bfplcb4_central_table) + (bcf_value_bfplcb4_central_table_row_step))) /\ ((bcf_index_bfplcb4_central_table_row_step = 0 /\ bcf_value_bfplcb4_central_table_row_step = 1) \/ exists bcf_predecessor_bfplcb4_central_table_row_step bcf_left_bfplcb4_central_table_row_step bcf_right_bfplcb4_central_table_row_step. bcf_index_bfplcb4_central_table_row_step = S bcf_predecessor_bfplcb4_central_table_row_step /\ ((((exists bcf_height_bfplcb4_central_table_row_step_previous_left. bcf_height_bfplcb4_central_table_row_step_previous_left + S (bcf_left_bfplcb4_central_table_row_step) = S ((S (bcf_predecessor_bfplcb4_central_table_row_step)) * bcf_previous_scale_bfplcb4_central_table)) /\ exists bcf_quotient_bfplcb4_central_table_row_step_previous_left. bcf_previous_code_bfplcb4_central_table = bcf_quotient_bfplcb4_central_table_row_step_previous_left * S ((S (bcf_predecessor_bfplcb4_central_table_row_step)) * bcf_previous_scale_bfplcb4_central_table) + (bcf_left_bfplcb4_central_table_row_step))) /\ ((((exists bcf_height_bfplcb4_central_table_row_step_previous_right. bcf_height_bfplcb4_central_table_row_step_previous_right + S (bcf_right_bfplcb4_central_table_row_step) = S ((S (S (bcf_predecessor_bfplcb4_central_table_row_step))) * bcf_previous_scale_bfplcb4_central_table)) /\ exists bcf_quotient_bfplcb4_central_table_row_step_previous_right. bcf_previous_code_bfplcb4_central_table = bcf_quotient_bfplcb4_central_table_row_step_previous_right * S ((S (S (bcf_predecessor_bfplcb4_central_table_row_step))) * bcf_previous_scale_bfplcb4_central_table) + (bcf_right_bfplcb4_central_table_row_step))) /\ bcf_value_bfplcb4_central_table_row_step = bcf_left_bfplcb4_central_table_row_step + bcf_right_bfplcb4_central_table_row_step))))))))))) /\ ((((exists bcf_height_bfplcb4_central_decoded_row_code. bcf_height_bfplcb4_central_decoded_row_code + S (bcf_row_code_bfplcb4_central) = S ((S (4 + 4)) * bcf_row_code_scale_bfplcb4_central)) /\ exists bcf_quotient_bfplcb4_central_decoded_row_code. bcf_row_code_code_bfplcb4_central = bcf_quotient_bfplcb4_central_decoded_row_code * S ((S (4 + 4)) * bcf_row_code_scale_bfplcb4_central) + (bcf_row_code_bfplcb4_central))) /\ ((((exists bcf_height_bfplcb4_central_decoded_row_scale. bcf_height_bfplcb4_central_decoded_row_scale + S (bcf_row_scale_bfplcb4_central) = S ((S (4 + 4)) * bcf_row_scale_scale_bfplcb4_central)) /\ exists bcf_quotient_bfplcb4_central_decoded_row_scale. bcf_row_scale_code_bfplcb4_central = bcf_quotient_bfplcb4_central_decoded_row_scale * S ((S (4 + 4)) * bcf_row_scale_scale_bfplcb4_central) + (bcf_row_scale_bfplcb4_central))) /\ (((exists bcf_height_bfplcb4_central_decoded_value. bcf_height_bfplcb4_central_decoded_value + S (c) = S ((S (4)) * bcf_row_scale_bfplcb4_central)) /\ exists bcf_quotient_bfplcb4_central_decoded_value. bcf_row_code_bfplcb4_central = bcf_quotient_bfplcb4_central_decoded_value * S ((S (4)) * bcf_row_scale_bfplcb4_central) + (c))))))))) -> (exists bcf_lt_gap_bfplcb4_result. bcf_lt_gap_bfplcb4_result + S (p) = 4 * c)))