central_binom_four_weighted_of_recurrence

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 c. (((exists bcf_lt_gap_bcb4we_source_out_of_range. bcf_lt_gap_bcb4we_source_out_of_range + S (4 + 4) = 4) /\ c = 0) \/ ((exists bcf_le_gap_bcb4we_source_in_range. bcf_le_gap_bcb4we_source_in_range + (4) = 4 + 4) /\ (exists bcf_row_code_code_bcb4we_source bcf_row_code_scale_bcb4we_source bcf_row_scale_code_bcb4we_source bcf_row_scale_scale_bcb4we_source bcf_row_code_bcb4we_source bcf_row_scale_bcb4we_source. ((forall bcf_row_index_bcb4we_source_table. (exists bcf_lt_gap_bcb4we_source_table_row_bound. bcf_lt_gap_bcb4we_source_table_row_bound + S (bcf_row_index_bcb4we_source_table) = S (4 + 4)) -> exists bcf_row_code_bcb4we_source_table bcf_row_scale_bcb4we_source_table. ((((exists bcf_height_bcb4we_source_table_decoded_row_code. bcf_height_bcb4we_source_table_decoded_row_code + S (bcf_row_code_bcb4we_source_table) = S ((S (bcf_row_index_bcb4we_source_table)) * bcf_row_code_scale_bcb4we_source)) /\ exists bcf_quotient_bcb4we_source_table_decoded_row_code. bcf_row_code_code_bcb4we_source = bcf_quotient_bcb4we_source_table_decoded_row_code * S ((S (bcf_row_index_bcb4we_source_table)) * bcf_row_code_scale_bcb4we_source) + (bcf_row_code_bcb4we_source_table))) /\ ((((exists bcf_height_bcb4we_source_table_decoded_row_scale. bcf_height_bcb4we_source_table_decoded_row_scale + S (bcf_row_scale_bcb4we_source_table) = S ((S (bcf_row_index_bcb4we_source_table)) * bcf_row_scale_scale_bcb4we_source)) /\ exists bcf_quotient_bcb4we_source_table_decoded_row_scale. bcf_row_scale_code_bcb4we_source = bcf_quotient_bcb4we_source_table_decoded_row_scale * S ((S (bcf_row_index_bcb4we_source_table)) * bcf_row_scale_scale_bcb4we_source) + (bcf_row_scale_bcb4we_source_table))) /\ ((bcf_row_index_bcb4we_source_table = 0 /\ (forall bcf_index_bcb4we_source_table_zero_row. (exists bcf_lt_gap_bcb4we_source_table_zero_row_bound. bcf_lt_gap_bcb4we_source_table_zero_row_bound + S (bcf_index_bcb4we_source_table_zero_row) = S (4 + 4)) -> exists bcf_value_bcb4we_source_table_zero_row. ((((exists bcf_height_bcb4we_source_table_zero_row_entry. bcf_height_bcb4we_source_table_zero_row_entry + S (bcf_value_bcb4we_source_table_zero_row) = S ((S (bcf_index_bcb4we_source_table_zero_row)) * bcf_row_scale_bcb4we_source_table)) /\ exists bcf_quotient_bcb4we_source_table_zero_row_entry. bcf_row_code_bcb4we_source_table = bcf_quotient_bcb4we_source_table_zero_row_entry * S ((S (bcf_index_bcb4we_source_table_zero_row)) * bcf_row_scale_bcb4we_source_table) + (bcf_value_bcb4we_source_table_zero_row))) /\ ((bcf_index_bcb4we_source_table_zero_row = 0 /\ bcf_value_bcb4we_source_table_zero_row = 1) \/ exists bcf_predecessor_bcb4we_source_table_zero_row. bcf_index_bcb4we_source_table_zero_row = S bcf_predecessor_bcb4we_source_table_zero_row /\ bcf_value_bcb4we_source_table_zero_row = 0)))) \/ exists bcf_predecessor_bcb4we_source_table bcf_previous_code_bcb4we_source_table bcf_previous_scale_bcb4we_source_table. bcf_row_index_bcb4we_source_table = S bcf_predecessor_bcb4we_source_table /\ ((((exists bcf_height_bcb4we_source_table_decoded_previous_code. bcf_height_bcb4we_source_table_decoded_previous_code + S (bcf_previous_code_bcb4we_source_table) = S ((S (bcf_predecessor_bcb4we_source_table)) * bcf_row_code_scale_bcb4we_source)) /\ exists bcf_quotient_bcb4we_source_table_decoded_previous_code. bcf_row_code_code_bcb4we_source = bcf_quotient_bcb4we_source_table_decoded_previous_code * S ((S (bcf_predecessor_bcb4we_source_table)) * bcf_row_code_scale_bcb4we_source) + (bcf_previous_code_bcb4we_source_table))) /\ ((((exists bcf_height_bcb4we_source_table_decoded_previous_scale. bcf_height_bcb4we_source_table_decoded_previous_scale + S (bcf_previous_scale_bcb4we_source_table) = S ((S (bcf_predecessor_bcb4we_source_table)) * bcf_row_scale_scale_bcb4we_source)) /\ exists bcf_quotient_bcb4we_source_table_decoded_previous_scale. bcf_row_scale_code_bcb4we_source = bcf_quotient_bcb4we_source_table_decoded_previous_scale * S ((S (bcf_predecessor_bcb4we_source_table)) * bcf_row_scale_scale_bcb4we_source) + (bcf_previous_scale_bcb4we_source_table))) /\ (forall bcf_index_bcb4we_source_table_row_step. (exists bcf_lt_gap_bcb4we_source_table_row_step_bound. bcf_lt_gap_bcb4we_source_table_row_step_bound + S (bcf_index_bcb4we_source_table_row_step) = S (4 + 4)) -> exists bcf_value_bcb4we_source_table_row_step. ((((exists bcf_height_bcb4we_source_table_row_step_entry. bcf_height_bcb4we_source_table_row_step_entry + S (bcf_value_bcb4we_source_table_row_step) = S ((S (bcf_index_bcb4we_source_table_row_step)) * bcf_row_scale_bcb4we_source_table)) /\ exists bcf_quotient_bcb4we_source_table_row_step_entry. bcf_row_code_bcb4we_source_table = bcf_quotient_bcb4we_source_table_row_step_entry * S ((S (bcf_index_bcb4we_source_table_row_step)) * bcf_row_scale_bcb4we_source_table) + (bcf_value_bcb4we_source_table_row_step))) /\ ((bcf_index_bcb4we_source_table_row_step = 0 /\ bcf_value_bcb4we_source_table_row_step = 1) \/ exists bcf_predecessor_bcb4we_source_table_row_step bcf_left_bcb4we_source_table_row_step bcf_right_bcb4we_source_table_row_step. bcf_index_bcb4we_source_table_row_step = S bcf_predecessor_bcb4we_source_table_row_step /\ ((((exists bcf_height_bcb4we_source_table_row_step_previous_left. bcf_height_bcb4we_source_table_row_step_previous_left + S (bcf_left_bcb4we_source_table_row_step) = S ((S (bcf_predecessor_bcb4we_source_table_row_step)) * bcf_previous_scale_bcb4we_source_table)) /\ exists bcf_quotient_bcb4we_source_table_row_step_previous_left. bcf_previous_code_bcb4we_source_table = bcf_quotient_bcb4we_source_table_row_step_previous_left * S ((S (bcf_predecessor_bcb4we_source_table_row_step)) * bcf_previous_scale_bcb4we_source_table) + (bcf_left_bcb4we_source_table_row_step))) /\ ((((exists bcf_height_bcb4we_source_table_row_step_previous_right. bcf_height_bcb4we_source_table_row_step_previous_right + S (bcf_right_bcb4we_source_table_row_step) = S ((S (S (bcf_predecessor_bcb4we_source_table_row_step))) * bcf_previous_scale_bcb4we_source_table)) /\ exists bcf_quotient_bcb4we_source_table_row_step_previous_right. bcf_previous_code_bcb4we_source_table = bcf_quotient_bcb4we_source_table_row_step_previous_right * S ((S (S (bcf_predecessor_bcb4we_source_table_row_step))) * bcf_previous_scale_bcb4we_source_table) + (bcf_right_bcb4we_source_table_row_step))) /\ bcf_value_bcb4we_source_table_row_step = bcf_left_bcb4we_source_table_row_step + bcf_right_bcb4we_source_table_row_step))))))))))) /\ ((((exists bcf_height_bcb4we_source_decoded_row_code. bcf_height_bcb4we_source_decoded_row_code + S (bcf_row_code_bcb4we_source) = S ((S (4 + 4)) * bcf_row_code_scale_bcb4we_source)) /\ exists bcf_quotient_bcb4we_source_decoded_row_code. bcf_row_code_code_bcb4we_source = bcf_quotient_bcb4we_source_decoded_row_code * S ((S (4 + 4)) * bcf_row_code_scale_bcb4we_source) + (bcf_row_code_bcb4we_source))) /\ ((((exists bcf_height_bcb4we_source_decoded_row_scale. bcf_height_bcb4we_source_decoded_row_scale + S (bcf_row_scale_bcb4we_source) = S ((S (4 + 4)) * bcf_row_scale_scale_bcb4we_source)) /\ exists bcf_quotient_bcb4we_source_decoded_row_scale. bcf_row_scale_code_bcb4we_source = bcf_quotient_bcb4we_source_decoded_row_scale * S ((S (4 + 4)) * bcf_row_scale_scale_bcb4we_source) + (bcf_row_scale_bcb4we_source))) /\ (((exists bcf_height_bcb4we_source_decoded_value. bcf_height_bcb4we_source_decoded_value + S (c) = S ((S (4)) * bcf_row_scale_bcb4we_source)) /\ exists bcf_quotient_bcb4we_source_decoded_value. bcf_row_code_bcb4we_source = bcf_quotient_bcb4we_source_decoded_value * S ((S (4)) * bcf_row_scale_bcb4we_source) + (c))))))))) -> 4 * c = (2 * S (3 + 3)) * 20