central_binom_succ_recurrence

Exact expanded PA statement

forall n c d. (((exists bcf_lt_gap_bcbsr_predecessor_out_of_range. bcf_lt_gap_bcbsr_predecessor_out_of_range + S (n + n) = n) /\ c = 0) \/ ((exists bcf_le_gap_bcbsr_predecessor_in_range. bcf_le_gap_bcbsr_predecessor_in_range + (n) = n + n) /\ (exists bcf_row_code_code_bcbsr_predecessor bcf_row_code_scale_bcbsr_predecessor bcf_row_scale_code_bcbsr_predecessor bcf_row_scale_scale_bcbsr_predecessor bcf_row_code_bcbsr_predecessor bcf_row_scale_bcbsr_predecessor. ((forall bcf_row_index_bcbsr_predecessor_table. (exists bcf_lt_gap_bcbsr_predecessor_table_row_bound. bcf_lt_gap_bcbsr_predecessor_table_row_bound + S (bcf_row_index_bcbsr_predecessor_table) = S (n + n)) -> exists bcf_row_code_bcbsr_predecessor_table bcf_row_scale_bcbsr_predecessor_table. ((((exists bcf_height_bcbsr_predecessor_table_decoded_row_code. bcf_height_bcbsr_predecessor_table_decoded_row_code + S (bcf_row_code_bcbsr_predecessor_table) = S ((S (bcf_row_index_bcbsr_predecessor_table)) * bcf_row_code_scale_bcbsr_predecessor)) /\ exists bcf_quotient_bcbsr_predecessor_table_decoded_row_code. bcf_row_code_code_bcbsr_predecessor = bcf_quotient_bcbsr_predecessor_table_decoded_row_code * S ((S (bcf_row_index_bcbsr_predecessor_table)) * bcf_row_code_scale_bcbsr_predecessor) + (bcf_row_code_bcbsr_predecessor_table))) /\ ((((exists bcf_height_bcbsr_predecessor_table_decoded_row_scale. bcf_height_bcbsr_predecessor_table_decoded_row_scale + S (bcf_row_scale_bcbsr_predecessor_table) = S ((S (bcf_row_index_bcbsr_predecessor_table)) * bcf_row_scale_scale_bcbsr_predecessor)) /\ exists bcf_quotient_bcbsr_predecessor_table_decoded_row_scale. bcf_row_scale_code_bcbsr_predecessor = bcf_quotient_bcbsr_predecessor_table_decoded_row_scale * S ((S (bcf_row_index_bcbsr_predecessor_table)) * bcf_row_scale_scale_bcbsr_predecessor) + (bcf_row_scale_bcbsr_predecessor_table))) /\ ((bcf_row_index_bcbsr_predecessor_table = 0 /\ (forall bcf_index_bcbsr_predecessor_table_zero_row. (exists bcf_lt_gap_bcbsr_predecessor_table_zero_row_bound. bcf_lt_gap_bcbsr_predecessor_table_zero_row_bound + S (bcf_index_bcbsr_predecessor_table_zero_row) = S (n + n)) -> exists bcf_value_bcbsr_predecessor_table_zero_row. ((((exists bcf_height_bcbsr_predecessor_table_zero_row_entry. bcf_height_bcbsr_predecessor_table_zero_row_entry + S (bcf_value_bcbsr_predecessor_table_zero_row) = S ((S (bcf_index_bcbsr_predecessor_table_zero_row)) * bcf_row_scale_bcbsr_predecessor_table)) /\ exists bcf_quotient_bcbsr_predecessor_table_zero_row_entry. bcf_row_code_bcbsr_predecessor_table = bcf_quotient_bcbsr_predecessor_table_zero_row_entry * S ((S (bcf_index_bcbsr_predecessor_table_zero_row)) * bcf_row_scale_bcbsr_predecessor_table) + (bcf_value_bcbsr_predecessor_table_zero_row))) /\ ((bcf_index_bcbsr_predecessor_table_zero_row = 0 /\ bcf_value_bcbsr_predecessor_table_zero_row = 1) \/ exists bcf_predecessor_bcbsr_predecessor_table_zero_row. bcf_index_bcbsr_predecessor_table_zero_row = S bcf_predecessor_bcbsr_predecessor_table_zero_row /\ bcf_value_bcbsr_predecessor_table_zero_row = 0)))) \/ exists bcf_predecessor_bcbsr_predecessor_table bcf_previous_code_bcbsr_predecessor_table bcf_previous_scale_bcbsr_predecessor_table. bcf_row_index_bcbsr_predecessor_table = S bcf_predecessor_bcbsr_predecessor_table /\ ((((exists bcf_height_bcbsr_predecessor_table_decoded_previous_code. bcf_height_bcbsr_predecessor_table_decoded_previous_code + S (bcf_previous_code_bcbsr_predecessor_table) = S ((S (bcf_predecessor_bcbsr_predecessor_table)) * bcf_row_code_scale_bcbsr_predecessor)) /\ exists bcf_quotient_bcbsr_predecessor_table_decoded_previous_code. bcf_row_code_code_bcbsr_predecessor = bcf_quotient_bcbsr_predecessor_table_decoded_previous_code * S ((S (bcf_predecessor_bcbsr_predecessor_table)) * bcf_row_code_scale_bcbsr_predecessor) + (bcf_previous_code_bcbsr_predecessor_table))) /\ ((((exists bcf_height_bcbsr_predecessor_table_decoded_previous_scale. bcf_height_bcbsr_predecessor_table_decoded_previous_scale + S (bcf_previous_scale_bcbsr_predecessor_table) = S ((S (bcf_predecessor_bcbsr_predecessor_table)) * bcf_row_scale_scale_bcbsr_predecessor)) /\ exists bcf_quotient_bcbsr_predecessor_table_decoded_previous_scale. bcf_row_scale_code_bcbsr_predecessor = bcf_quotient_bcbsr_predecessor_table_decoded_previous_scale * S ((S (bcf_predecessor_bcbsr_predecessor_table)) * bcf_row_scale_scale_bcbsr_predecessor) + (bcf_previous_scale_bcbsr_predecessor_table))) /\ (forall bcf_index_bcbsr_predecessor_table_row_step. (exists bcf_lt_gap_bcbsr_predecessor_table_row_step_bound. bcf_lt_gap_bcbsr_predecessor_table_row_step_bound + S (bcf_index_bcbsr_predecessor_table_row_step) = S (n + n)) -> exists bcf_value_bcbsr_predecessor_table_row_step. ((((exists bcf_height_bcbsr_predecessor_table_row_step_entry. bcf_height_bcbsr_predecessor_table_row_step_entry + S (bcf_value_bcbsr_predecessor_table_row_step) = S ((S (bcf_index_bcbsr_predecessor_table_row_step)) * bcf_row_scale_bcbsr_predecessor_table)) /\ exists bcf_quotient_bcbsr_predecessor_table_row_step_entry. bcf_row_code_bcbsr_predecessor_table = bcf_quotient_bcbsr_predecessor_table_row_step_entry * S ((S (bcf_index_bcbsr_predecessor_table_row_step)) * bcf_row_scale_bcbsr_predecessor_table) + (bcf_value_bcbsr_predecessor_table_row_step))) /\ ((bcf_index_bcbsr_predecessor_table_row_step = 0 /\ bcf_value_bcbsr_predecessor_table_row_step = 1) \/ exists bcf_predecessor_bcbsr_predecessor_table_row_step bcf_left_bcbsr_predecessor_table_row_step bcf_right_bcbsr_predecessor_table_row_step. bcf_index_bcbsr_predecessor_table_row_step = S bcf_predecessor_bcbsr_predecessor_table_row_step /\ ((((exists bcf_height_bcbsr_predecessor_table_row_step_previous_left. bcf_height_bcbsr_predecessor_table_row_step_previous_left + S (bcf_left_bcbsr_predecessor_table_row_step) = S ((S (bcf_predecessor_bcbsr_predecessor_table_row_step)) * bcf_previous_scale_bcbsr_predecessor_table)) /\ exists bcf_quotient_bcbsr_predecessor_table_row_step_previous_left. bcf_previous_code_bcbsr_predecessor_table = bcf_quotient_bcbsr_predecessor_table_row_step_previous_left * S ((S (bcf_predecessor_bcbsr_predecessor_table_row_step)) * bcf_previous_scale_bcbsr_predecessor_table) + (bcf_left_bcbsr_predecessor_table_row_step))) /\ ((((exists bcf_height_bcbsr_predecessor_table_row_step_previous_right. bcf_height_bcbsr_predecessor_table_row_step_previous_right + S (bcf_right_bcbsr_predecessor_table_row_step) = S ((S (S (bcf_predecessor_bcbsr_predecessor_table_row_step))) * bcf_previous_scale_bcbsr_predecessor_table)) /\ exists bcf_quotient_bcbsr_predecessor_table_row_step_previous_right. bcf_previous_code_bcbsr_predecessor_table = bcf_quotient_bcbsr_predecessor_table_row_step_previous_right * S ((S (S (bcf_predecessor_bcbsr_predecessor_table_row_step))) * bcf_previous_scale_bcbsr_predecessor_table) + (bcf_right_bcbsr_predecessor_table_row_step))) /\ bcf_value_bcbsr_predecessor_table_row_step = bcf_left_bcbsr_predecessor_table_row_step + bcf_right_bcbsr_predecessor_table_row_step))))))))))) /\ ((((exists bcf_height_bcbsr_predecessor_decoded_row_code. bcf_height_bcbsr_predecessor_decoded_row_code + S (bcf_row_code_bcbsr_predecessor) = S ((S (n + n)) * bcf_row_code_scale_bcbsr_predecessor)) /\ exists bcf_quotient_bcbsr_predecessor_decoded_row_code. bcf_row_code_code_bcbsr_predecessor = bcf_quotient_bcbsr_predecessor_decoded_row_code * S ((S (n + n)) * bcf_row_code_scale_bcbsr_predecessor) + (bcf_row_code_bcbsr_predecessor))) /\ ((((exists bcf_height_bcbsr_predecessor_decoded_row_scale. bcf_height_bcbsr_predecessor_decoded_row_scale + S (bcf_row_scale_bcbsr_predecessor) = S ((S (n + n)) * bcf_row_scale_scale_bcbsr_predecessor)) /\ exists bcf_quotient_bcbsr_predecessor_decoded_row_scale. bcf_row_scale_code_bcbsr_predecessor = bcf_quotient_bcbsr_predecessor_decoded_row_scale * S ((S (n + n)) * bcf_row_scale_scale_bcbsr_predecessor) + (bcf_row_scale_bcbsr_predecessor))) /\ (((exists bcf_height_bcbsr_predecessor_decoded_value. bcf_height_bcbsr_predecessor_decoded_value + S (c) = S ((S (n)) * bcf_row_scale_bcbsr_predecessor)) /\ exists bcf_quotient_bcbsr_predecessor_decoded_value. bcf_row_code_bcbsr_predecessor = bcf_quotient_bcbsr_predecessor_decoded_value * S ((S (n)) * bcf_row_scale_bcbsr_predecessor) + (c))))))))) -> (((exists bcf_lt_gap_bcbsr_successor_out_of_range. bcf_lt_gap_bcbsr_successor_out_of_range + S (S n + S n) = S n) /\ d = 0) \/ ((exists bcf_le_gap_bcbsr_successor_in_range. bcf_le_gap_bcbsr_successor_in_range + (S n) = S n + S n) /\ (exists bcf_row_code_code_bcbsr_successor bcf_row_code_scale_bcbsr_successor bcf_row_scale_code_bcbsr_successor bcf_row_scale_scale_bcbsr_successor bcf_row_code_bcbsr_successor bcf_row_scale_bcbsr_successor. ((forall bcf_row_index_bcbsr_successor_table. (exists bcf_lt_gap_bcbsr_successor_table_row_bound. bcf_lt_gap_bcbsr_successor_table_row_bound + S (bcf_row_index_bcbsr_successor_table) = S (S n + S n)) -> exists bcf_row_code_bcbsr_successor_table bcf_row_scale_bcbsr_successor_table. ((((exists bcf_height_bcbsr_successor_table_decoded_row_code. bcf_height_bcbsr_successor_table_decoded_row_code + S (bcf_row_code_bcbsr_successor_table) = S ((S (bcf_row_index_bcbsr_successor_table)) * bcf_row_code_scale_bcbsr_successor)) /\ exists bcf_quotient_bcbsr_successor_table_decoded_row_code. bcf_row_code_code_bcbsr_successor = bcf_quotient_bcbsr_successor_table_decoded_row_code * S ((S (bcf_row_index_bcbsr_successor_table)) * bcf_row_code_scale_bcbsr_successor) + (bcf_row_code_bcbsr_successor_table))) /\ ((((exists bcf_height_bcbsr_successor_table_decoded_row_scale. bcf_height_bcbsr_successor_table_decoded_row_scale + S (bcf_row_scale_bcbsr_successor_table) = S ((S (bcf_row_index_bcbsr_successor_table)) * bcf_row_scale_scale_bcbsr_successor)) /\ exists bcf_quotient_bcbsr_successor_table_decoded_row_scale. bcf_row_scale_code_bcbsr_successor = bcf_quotient_bcbsr_successor_table_decoded_row_scale * S ((S (bcf_row_index_bcbsr_successor_table)) * bcf_row_scale_scale_bcbsr_successor) + (bcf_row_scale_bcbsr_successor_table))) /\ ((bcf_row_index_bcbsr_successor_table = 0 /\ (forall bcf_index_bcbsr_successor_table_zero_row. (exists bcf_lt_gap_bcbsr_successor_table_zero_row_bound. bcf_lt_gap_bcbsr_successor_table_zero_row_bound + S (bcf_index_bcbsr_successor_table_zero_row) = S (S n + S n)) -> exists bcf_value_bcbsr_successor_table_zero_row. ((((exists bcf_height_bcbsr_successor_table_zero_row_entry. bcf_height_bcbsr_successor_table_zero_row_entry + S (bcf_value_bcbsr_successor_table_zero_row) = S ((S (bcf_index_bcbsr_successor_table_zero_row)) * bcf_row_scale_bcbsr_successor_table)) /\ exists bcf_quotient_bcbsr_successor_table_zero_row_entry. bcf_row_code_bcbsr_successor_table = bcf_quotient_bcbsr_successor_table_zero_row_entry * S ((S (bcf_index_bcbsr_successor_table_zero_row)) * bcf_row_scale_bcbsr_successor_table) + (bcf_value_bcbsr_successor_table_zero_row))) /\ ((bcf_index_bcbsr_successor_table_zero_row = 0 /\ bcf_value_bcbsr_successor_table_zero_row = 1) \/ exists bcf_predecessor_bcbsr_successor_table_zero_row. bcf_index_bcbsr_successor_table_zero_row = S bcf_predecessor_bcbsr_successor_table_zero_row /\ bcf_value_bcbsr_successor_table_zero_row = 0)))) \/ exists bcf_predecessor_bcbsr_successor_table bcf_previous_code_bcbsr_successor_table bcf_previous_scale_bcbsr_successor_table. bcf_row_index_bcbsr_successor_table = S bcf_predecessor_bcbsr_successor_table /\ ((((exists bcf_height_bcbsr_successor_table_decoded_previous_code. bcf_height_bcbsr_successor_table_decoded_previous_code + S (bcf_previous_code_bcbsr_successor_table) = S ((S (bcf_predecessor_bcbsr_successor_table)) * bcf_row_code_scale_bcbsr_successor)) /\ exists bcf_quotient_bcbsr_successor_table_decoded_previous_code. bcf_row_code_code_bcbsr_successor = bcf_quotient_bcbsr_successor_table_decoded_previous_code * S ((S (bcf_predecessor_bcbsr_successor_table)) * bcf_row_code_scale_bcbsr_successor) + (bcf_previous_code_bcbsr_successor_table))) /\ ((((exists bcf_height_bcbsr_successor_table_decoded_previous_scale. bcf_height_bcbsr_successor_table_decoded_previous_scale + S (bcf_previous_scale_bcbsr_successor_table) = S ((S (bcf_predecessor_bcbsr_successor_table)) * bcf_row_scale_scale_bcbsr_successor)) /\ exists bcf_quotient_bcbsr_successor_table_decoded_previous_scale. bcf_row_scale_code_bcbsr_successor = bcf_quotient_bcbsr_successor_table_decoded_previous_scale * S ((S (bcf_predecessor_bcbsr_successor_table)) * bcf_row_scale_scale_bcbsr_successor) + (bcf_previous_scale_bcbsr_successor_table))) /\ (forall bcf_index_bcbsr_successor_table_row_step. (exists bcf_lt_gap_bcbsr_successor_table_row_step_bound. bcf_lt_gap_bcbsr_successor_table_row_step_bound + S (bcf_index_bcbsr_successor_table_row_step) = S (S n + S n)) -> exists bcf_value_bcbsr_successor_table_row_step. ((((exists bcf_height_bcbsr_successor_table_row_step_entry. bcf_height_bcbsr_successor_table_row_step_entry + S (bcf_value_bcbsr_successor_table_row_step) = S ((S (bcf_index_bcbsr_successor_table_row_step)) * bcf_row_scale_bcbsr_successor_table)) /\ exists bcf_quotient_bcbsr_successor_table_row_step_entry. bcf_row_code_bcbsr_successor_table = bcf_quotient_bcbsr_successor_table_row_step_entry * S ((S (bcf_index_bcbsr_successor_table_row_step)) * bcf_row_scale_bcbsr_successor_table) + (bcf_value_bcbsr_successor_table_row_step))) /\ ((bcf_index_bcbsr_successor_table_row_step = 0 /\ bcf_value_bcbsr_successor_table_row_step = 1) \/ exists bcf_predecessor_bcbsr_successor_table_row_step bcf_left_bcbsr_successor_table_row_step bcf_right_bcbsr_successor_table_row_step. bcf_index_bcbsr_successor_table_row_step = S bcf_predecessor_bcbsr_successor_table_row_step /\ ((((exists bcf_height_bcbsr_successor_table_row_step_previous_left. bcf_height_bcbsr_successor_table_row_step_previous_left + S (bcf_left_bcbsr_successor_table_row_step) = S ((S (bcf_predecessor_bcbsr_successor_table_row_step)) * bcf_previous_scale_bcbsr_successor_table)) /\ exists bcf_quotient_bcbsr_successor_table_row_step_previous_left. bcf_previous_code_bcbsr_successor_table = bcf_quotient_bcbsr_successor_table_row_step_previous_left * S ((S (bcf_predecessor_bcbsr_successor_table_row_step)) * bcf_previous_scale_bcbsr_successor_table) + (bcf_left_bcbsr_successor_table_row_step))) /\ ((((exists bcf_height_bcbsr_successor_table_row_step_previous_right. bcf_height_bcbsr_successor_table_row_step_previous_right + S (bcf_right_bcbsr_successor_table_row_step) = S ((S (S (bcf_predecessor_bcbsr_successor_table_row_step))) * bcf_previous_scale_bcbsr_successor_table)) /\ exists bcf_quotient_bcbsr_successor_table_row_step_previous_right. bcf_previous_code_bcbsr_successor_table = bcf_quotient_bcbsr_successor_table_row_step_previous_right * S ((S (S (bcf_predecessor_bcbsr_successor_table_row_step))) * bcf_previous_scale_bcbsr_successor_table) + (bcf_right_bcbsr_successor_table_row_step))) /\ bcf_value_bcbsr_successor_table_row_step = bcf_left_bcbsr_successor_table_row_step + bcf_right_bcbsr_successor_table_row_step))))))))))) /\ ((((exists bcf_height_bcbsr_successor_decoded_row_code. bcf_height_bcbsr_successor_decoded_row_code + S (bcf_row_code_bcbsr_successor) = S ((S (S n + S n)) * bcf_row_code_scale_bcbsr_successor)) /\ exists bcf_quotient_bcbsr_successor_decoded_row_code. bcf_row_code_code_bcbsr_successor = bcf_quotient_bcbsr_successor_decoded_row_code * S ((S (S n + S n)) * bcf_row_code_scale_bcbsr_successor) + (bcf_row_code_bcbsr_successor))) /\ ((((exists bcf_height_bcbsr_successor_decoded_row_scale. bcf_height_bcbsr_successor_decoded_row_scale + S (bcf_row_scale_bcbsr_successor) = S ((S (S n + S n)) * bcf_row_scale_scale_bcbsr_successor)) /\ exists bcf_quotient_bcbsr_successor_decoded_row_scale. bcf_row_scale_code_bcbsr_successor = bcf_quotient_bcbsr_successor_decoded_row_scale * S ((S (S n + S n)) * bcf_row_scale_scale_bcbsr_successor) + (bcf_row_scale_bcbsr_successor))) /\ (((exists bcf_height_bcbsr_successor_decoded_value. bcf_height_bcbsr_successor_decoded_value + S (d) = S ((S (S n)) * bcf_row_scale_bcbsr_successor)) /\ exists bcf_quotient_bcbsr_successor_decoded_value. bcf_row_code_bcbsr_successor = bcf_quotient_bcbsr_successor_decoded_value * S ((S (S n)) * bcf_row_scale_bcbsr_successor) + (d))))))))) -> S n * d = (2 * S (n + n)) * c