central_binom_odd_middle_le_four_pow

Proof neighborhood

Direct dependencies

Direct dependents

Formal native tactic body

Dependencies are hypotheses of this body receipt. The focused endpoint audits separately check the complete empty-context certificates.

1. 0001intro n
2. 0002intro m
3. 0003intro q
4. 0004intro hmiddle
5. 0005intro hpower
6. 0006have hpackage : ((((forall n c d. (((exists bcf_lt_gap_bcbrdb_predecessor_out_of_range. bcf_lt_gap_bcbrdb_predecessor_out_of_range + S (n + n) = n) /\ c = 0) \/ ((exists bcf_le_gap_bcbrdb_predecessor_in_range. bcf_le_gap_bcbrdb_predecessor_in_range + (n) = n + n) /\ (exists bcf_row_code_code_bcbrdb_predecessor bcf_row_code_scale_bcbrdb_predecessor bcf_row_scale_code_bcbrdb_predecessor bcf_row_scale_scale_bcbrdb_predecessor bcf_row_code_bcbrdb_predecessor bcf_row_scale_bcbrdb_predecessor. ((forall bcf_row_index_bcbrdb_predecessor_table. (exists bcf_lt_gap_bcbrdb_predecessor_table_row_bound. bcf_lt_gap_bcbrdb_predecessor_table_row_bound + S (bcf_row_index_bcbrdb_predecessor_table) = S (n + n)) -> exists bcf_row_code_bcbrdb_predecessor_table bcf_row_scale_bcbrdb_predecessor_table. ((((exists bcf_height_bcbrdb_predecessor_table_decoded_row_code. bcf_height_bcbrdb_predecessor_table_decoded_row_code + S (bcf_row_code_bcbrdb_predecessor_table) = S ((S (bcf_row_index_bcbrdb_predecessor_table)) * bcf_row_code_scale_bcbrdb_predecessor)) /\ exists bcf_quotient_bcbrdb_predecessor_table_decoded_row_code. bcf_row_code_code_bcbrdb_predecessor = bcf_quotient_bcbrdb_predecessor_table_decoded_row_code * S ((S (bcf_row_index_bcbrdb_predecessor_table)) * bcf_row_code_scale_bcbrdb_predecessor) + (bcf_row_code_bcbrdb_predecessor_table))) /\ ((((exists bcf_height_bcbrdb_predecessor_table_decoded_row_scale. bcf_height_bcbrdb_predecessor_table_decoded_row_scale + S (bcf_row_scale_bcbrdb_predecessor_table) = S ((S (bcf_row_index_bcbrdb_predecessor_table)) * bcf_row_scale_scale_bcbrdb_predecessor)) /\ exists bcf_quotient_bcbrdb_predecessor_table_decoded_row_scale. bcf_row_scale_code_bcbrdb_predecessor = bcf_quotient_bcbrdb_predecessor_table_decoded_row_scale * S ((S (bcf_row_index_bcbrdb_predecessor_table)) * bcf_row_scale_scale_bcbrdb_predecessor) + (bcf_row_scale_bcbrdb_predecessor_table))) /\ ((bcf_row_index_bcbrdb_predecessor_table = 0 /\ (forall bcf_index_bcbrdb_predecessor_table_zero_row. (exists bcf_lt_gap_bcbrdb_predecessor_table_zero_row_bound. bcf_lt_gap_bcbrdb_predecessor_table_zero_row_bound + S (bcf_index_bcbrdb_predecessor_table_zero_row) = S (n + n)) -> exists bcf_value_bcbrdb_predecessor_table_zero_row. ((((exists bcf_height_bcbrdb_predecessor_table_zero_row_entry. bcf_height_bcbrdb_predecessor_table_zero_row_entry + S (bcf_value_bcbrdb_predecessor_table_zero_row) = S ((S (bcf_index_bcbrdb_predecessor_table_zero_row)) * bcf_row_scale_bcbrdb_predecessor_table)) /\ exists bcf_quotient_bcbrdb_predecessor_table_zero_row_entry. bcf_row_code_bcbrdb_predecessor_table = bcf_quotient_bcbrdb_predecessor_table_zero_row_entry * S ((S (bcf_index_bcbrdb_predecessor_table_zero_row)) * bcf_row_scale_bcbrdb_predecessor_table) + (bcf_value_bcbrdb_predecessor_table_zero_row))) /\ ((bcf_index_bcbrdb_predecessor_table_zero_row = 0 /\ bcf_value_bcbrdb_predecessor_table_zero_row = 1) \/ exists bcf_predecessor_bcbrdb_predecessor_table_zero_row. bcf_index_bcbrdb_predecessor_table_zero_row = S bcf_predecessor_bcbrdb_predecessor_table_zero_row /\ bcf_value_bcbrdb_predecessor_table_zero_row = 0)))) \/ exists bcf_predecessor_bcbrdb_predecessor_table bcf_previous_code_bcbrdb_predecessor_table bcf_previous_scale_bcbrdb_predecessor_table. bcf_row_index_bcbrdb_predecessor_table = S bcf_predecessor_bcbrdb_predecessor_table /\ ((((exists bcf_height_bcbrdb_predecessor_table_decoded_previous_code. bcf_height_bcbrdb_predecessor_table_decoded_previous_code + S (bcf_previous_code_bcbrdb_predecessor_table) = S ((S (bcf_predecessor_bcbrdb_predecessor_table)) * bcf_row_code_scale_bcbrdb_predecessor)) /\ exists bcf_quotient_bcbrdb_predecessor_table_decoded_previous_code. bcf_row_code_code_bcbrdb_predecessor = bcf_quotient_bcbrdb_predecessor_table_decoded_previous_code * S ((S (bcf_predecessor_bcbrdb_predecessor_table)) * bcf_row_code_scale_bcbrdb_predecessor) + (bcf_previous_code_bcbrdb_predecessor_table))) /\ ((((exists bcf_height_bcbrdb_predecessor_table_decoded_previous_scale. bcf_height_bcbrdb_predecessor_table_decoded_previous_scale + S (bcf_previous_scale_bcbrdb_predecessor_table) = S ((S (bcf_predecessor_bcbrdb_predecessor_table)) * bcf_row_scale_scale_bcbrdb_predecessor)) /\ exists bcf_quotient_bcbrdb_predecessor_table_decoded_previous_scale. bcf_row_scale_code_bcbrdb_predecessor = bcf_quotient_bcbrdb_predecessor_table_decoded_previous_scale * S ((S (bcf_predecessor_bcbrdb_predecessor_table)) * bcf_row_scale_scale_bcbrdb_predecessor) + (bcf_previous_scale_bcbrdb_predecessor_table))) /\ (forall bcf_index_bcbrdb_predecessor_table_row_step. (exists bcf_lt_gap_bcbrdb_predecessor_table_row_step_bound. bcf_lt_gap_bcbrdb_predecessor_table_row_step_bound + S (bcf_index_bcbrdb_predecessor_table_row_step) = S (n + n)) -> exists bcf_value_bcbrdb_predecessor_table_row_step. ((((exists bcf_height_bcbrdb_predecessor_table_row_step_entry. bcf_height_bcbrdb_predecessor_table_row_step_entry + S (bcf_value_bcbrdb_predecessor_table_row_step) = S ((S (bcf_index_bcbrdb_predecessor_table_row_step)) * bcf_row_scale_bcbrdb_predecessor_table)) /\ exists bcf_quotient_bcbrdb_predecessor_table_row_step_entry. bcf_row_code_bcbrdb_predecessor_table = bcf_quotient_bcbrdb_predecessor_table_row_step_entry * S ((S (bcf_index_bcbrdb_predecessor_table_row_step)) * bcf_row_scale_bcbrdb_predecessor_table) + (bcf_value_bcbrdb_predecessor_table_row_step))) /\ ((bcf_index_bcbrdb_predecessor_table_row_step = 0 /\ bcf_value_bcbrdb_predecessor_table_row_step = 1) \/ exists bcf_predecessor_bcbrdb_predecessor_table_row_step bcf_left_bcbrdb_predecessor_table_row_step bcf_right_bcbrdb_predecessor_table_row_step. bcf_index_bcbrdb_predecessor_table_row_step = S bcf_predecessor_bcbrdb_predecessor_table_row_step /\ ((((exists bcf_height_bcbrdb_predecessor_table_row_step_previous_left. bcf_height_bcbrdb_predecessor_table_row_step_previous_left + S (bcf_left_bcbrdb_predecessor_table_row_step) = S ((S (bcf_predecessor_bcbrdb_predecessor_table_row_step)) * bcf_previous_scale_bcbrdb_predecessor_table)) /\ exists bcf_quotient_bcbrdb_predecessor_table_row_step_previous_left. bcf_previous_code_bcbrdb_predecessor_table = bcf_quotient_bcbrdb_predecessor_table_row_step_previous_left * S ((S (bcf_predecessor_bcbrdb_predecessor_table_row_step)) * bcf_previous_scale_bcbrdb_predecessor_table) + (bcf_left_bcbrdb_predecessor_table_row_step))) /\ ((((exists bcf_height_bcbrdb_predecessor_table_row_step_previous_right. bcf_height_bcbrdb_predecessor_table_row_step_previous_right + S (bcf_right_bcbrdb_predecessor_table_row_step) = S ((S (S (bcf_predecessor_bcbrdb_predecessor_table_row_step))) * bcf_previous_scale_bcbrdb_predecessor_table)) /\ exists bcf_quotient_bcbrdb_predecessor_table_row_step_previous_right. bcf_previous_code_bcbrdb_predecessor_table = bcf_quotient_bcbrdb_predecessor_table_row_step_previous_right * S ((S (S (bcf_predecessor_bcbrdb_predecessor_table_row_step))) * bcf_previous_scale_bcbrdb_predecessor_table) + (bcf_right_bcbrdb_predecessor_table_row_step))) /\ bcf_value_bcbrdb_predecessor_table_row_step = bcf_left_bcbrdb_predecessor_table_row_step + bcf_right_bcbrdb_predecessor_table_row_step))))))))))) /\ ((((exists bcf_height_bcbrdb_predecessor_decoded_row_code. bcf_height_bcbrdb_predecessor_decoded_row_code + S (bcf_row_code_bcbrdb_predecessor) = S ((S (n + n)) * bcf_row_code_scale_bcbrdb_predecessor)) /\ exists bcf_quotient_bcbrdb_predecessor_decoded_row_code. bcf_row_code_code_bcbrdb_predecessor = bcf_quotient_bcbrdb_predecessor_decoded_row_code * S ((S (n + n)) * bcf_row_code_scale_bcbrdb_predecessor) + (bcf_row_code_bcbrdb_predecessor))) /\ ((((exists bcf_height_bcbrdb_predecessor_decoded_row_scale. bcf_height_bcbrdb_predecessor_decoded_row_scale + S (bcf_row_scale_bcbrdb_predecessor) = S ((S (n + n)) * bcf_row_scale_scale_bcbrdb_predecessor)) /\ exists bcf_quotient_bcbrdb_predecessor_decoded_row_scale. bcf_row_scale_code_bcbrdb_predecessor = bcf_quotient_bcbrdb_predecessor_decoded_row_scale * S ((S (n + n)) * bcf_row_scale_scale_bcbrdb_predecessor) + (bcf_row_scale_bcbrdb_predecessor))) /\ (((exists bcf_height_bcbrdb_predecessor_decoded_value. bcf_height_bcbrdb_predecessor_decoded_value + S (c) = S ((S (n)) * bcf_row_scale_bcbrdb_predecessor)) /\ exists bcf_quotient_bcbrdb_predecessor_decoded_value. bcf_row_code_bcbrdb_predecessor = bcf_quotient_bcbrdb_predecessor_decoded_value * S ((S (n)) * bcf_row_scale_bcbrdb_predecessor) + (c))))))))) -> (((exists bcf_lt_gap_bcbrdb_successor_out_of_range. bcf_lt_gap_bcbrdb_successor_out_of_range + S (S n + S n) = S n) /\ d = 0) \/ ((exists bcf_le_gap_bcbrdb_successor_in_range. bcf_le_gap_bcbrdb_successor_in_range + (S n) = S n + S n) /\ (exists bcf_row_code_code_bcbrdb_successor bcf_row_code_scale_bcbrdb_successor bcf_row_scale_code_bcbrdb_successor bcf_row_scale_scale_bcbrdb_successor bcf_row_code_bcbrdb_successor bcf_row_scale_bcbrdb_successor. ((forall bcf_row_index_bcbrdb_successor_table. (exists bcf_lt_gap_bcbrdb_successor_table_row_bound. bcf_lt_gap_bcbrdb_successor_table_row_bound + S (bcf_row_index_bcbrdb_successor_table) = S (S n + S n)) -> exists bcf_row_code_bcbrdb_successor_table bcf_row_scale_bcbrdb_successor_table. ((((exists bcf_height_bcbrdb_successor_table_decoded_row_code. bcf_height_bcbrdb_successor_table_decoded_row_code + S (bcf_row_code_bcbrdb_successor_table) = S ((S (bcf_row_index_bcbrdb_successor_table)) * bcf_row_code_scale_bcbrdb_successor)) /\ exists bcf_quotient_bcbrdb_successor_table_decoded_row_code. bcf_row_code_code_bcbrdb_successor = bcf_quotient_bcbrdb_successor_table_decoded_row_code * S ((S (bcf_row_index_bcbrdb_successor_table)) * bcf_row_code_scale_bcbrdb_successor) + (bcf_row_code_bcbrdb_successor_table))) /\ ((((exists bcf_height_bcbrdb_successor_table_decoded_row_scale. bcf_height_bcbrdb_successor_table_decoded_row_scale + S (bcf_row_scale_bcbrdb_successor_table) = S ((S (bcf_row_index_bcbrdb_successor_table)) * bcf_row_scale_scale_bcbrdb_successor)) /\ exists bcf_quotient_bcbrdb_successor_table_decoded_row_scale. bcf_row_scale_code_bcbrdb_successor = bcf_quotient_bcbrdb_successor_table_decoded_row_scale * S ((S (bcf_row_index_bcbrdb_successor_table)) * bcf_row_scale_scale_bcbrdb_successor) + (bcf_row_scale_bcbrdb_successor_table))) /\ ((bcf_row_index_bcbrdb_successor_table = 0 /\ (forall bcf_index_bcbrdb_successor_table_zero_row. (exists bcf_lt_gap_bcbrdb_successor_table_zero_row_bound. bcf_lt_gap_bcbrdb_successor_table_zero_row_bound + S (bcf_index_bcbrdb_successor_table_zero_row) = S (S n + S n)) -> exists bcf_value_bcbrdb_successor_table_zero_row. ((((exists bcf_height_bcbrdb_successor_table_zero_row_entry. bcf_height_bcbrdb_successor_table_zero_row_entry + S (bcf_value_bcbrdb_successor_table_zero_row) = S ((S (bcf_index_bcbrdb_successor_table_zero_row)) * bcf_row_scale_bcbrdb_successor_table)) /\ exists bcf_quotient_bcbrdb_successor_table_zero_row_entry. bcf_row_code_bcbrdb_successor_table = bcf_quotient_bcbrdb_successor_table_zero_row_entry * S ((S (bcf_index_bcbrdb_successor_table_zero_row)) * bcf_row_scale_bcbrdb_successor_table) + (bcf_value_bcbrdb_successor_table_zero_row))) /\ ((bcf_index_bcbrdb_successor_table_zero_row = 0 /\ bcf_value_bcbrdb_successor_table_zero_row = 1) \/ exists bcf_predecessor_bcbrdb_successor_table_zero_row. bcf_index_bcbrdb_successor_table_zero_row = S bcf_predecessor_bcbrdb_successor_table_zero_row /\ bcf_value_bcbrdb_successor_table_zero_row = 0)))) \/ exists bcf_predecessor_bcbrdb_successor_table bcf_previous_code_bcbrdb_successor_table bcf_previous_scale_bcbrdb_successor_table. bcf_row_index_bcbrdb_successor_table = S bcf_predecessor_bcbrdb_successor_table /\ ((((exists bcf_height_bcbrdb_successor_table_decoded_previous_code. bcf_height_bcbrdb_successor_table_decoded_previous_code + S (bcf_previous_code_bcbrdb_successor_table) = S ((S (bcf_predecessor_bcbrdb_successor_table)) * bcf_row_code_scale_bcbrdb_successor)) /\ exists bcf_quotient_bcbrdb_successor_table_decoded_previous_code. bcf_row_code_code_bcbrdb_successor = bcf_quotient_bcbrdb_successor_table_decoded_previous_code * S ((S (bcf_predecessor_bcbrdb_successor_table)) * bcf_row_code_scale_bcbrdb_successor) + (bcf_previous_code_bcbrdb_successor_table))) /\ ((((exists bcf_height_bcbrdb_successor_table_decoded_previous_scale. bcf_height_bcbrdb_successor_table_decoded_previous_scale + S (bcf_previous_scale_bcbrdb_successor_table) = S ((S (bcf_predecessor_bcbrdb_successor_table)) * bcf_row_scale_scale_bcbrdb_successor)) /\ exists bcf_quotient_bcbrdb_successor_table_decoded_previous_scale. bcf_row_scale_code_bcbrdb_successor = bcf_quotient_bcbrdb_successor_table_decoded_previous_scale * S ((S (bcf_predecessor_bcbrdb_successor_table)) * bcf_row_scale_scale_bcbrdb_successor) + (bcf_previous_scale_bcbrdb_successor_table))) /\ (forall bcf_index_bcbrdb_successor_table_row_step. (exists bcf_lt_gap_bcbrdb_successor_table_row_step_bound. bcf_lt_gap_bcbrdb_successor_table_row_step_bound + S (bcf_index_bcbrdb_successor_table_row_step) = S (S n + S n)) -> exists bcf_value_bcbrdb_successor_table_row_step. ((((exists bcf_height_bcbrdb_successor_table_row_step_entry. bcf_height_bcbrdb_successor_table_row_step_entry + S (bcf_value_bcbrdb_successor_table_row_step) = S ((S (bcf_index_bcbrdb_successor_table_row_step)) * bcf_row_scale_bcbrdb_successor_table)) /\ exists bcf_quotient_bcbrdb_successor_table_row_step_entry. bcf_row_code_bcbrdb_successor_table = bcf_quotient_bcbrdb_successor_table_row_step_entry * S ((S (bcf_index_bcbrdb_successor_table_row_step)) * bcf_row_scale_bcbrdb_successor_table) + (bcf_value_bcbrdb_successor_table_row_step))) /\ ((bcf_index_bcbrdb_successor_table_row_step = 0 /\ bcf_value_bcbrdb_successor_table_row_step = 1) \/ exists bcf_predecessor_bcbrdb_successor_table_row_step bcf_left_bcbrdb_successor_table_row_step bcf_right_bcbrdb_successor_table_row_step. bcf_index_bcbrdb_successor_table_row_step = S bcf_predecessor_bcbrdb_successor_table_row_step /\ ((((exists bcf_height_bcbrdb_successor_table_row_step_previous_left. bcf_height_bcbrdb_successor_table_row_step_previous_left + S (bcf_left_bcbrdb_successor_table_row_step) = S ((S (bcf_predecessor_bcbrdb_successor_table_row_step)) * bcf_previous_scale_bcbrdb_successor_table)) /\ exists bcf_quotient_bcbrdb_successor_table_row_step_previous_left. bcf_previous_code_bcbrdb_successor_table = bcf_quotient_bcbrdb_successor_table_row_step_previous_left * S ((S (bcf_predecessor_bcbrdb_successor_table_row_step)) * bcf_previous_scale_bcbrdb_successor_table) + (bcf_left_bcbrdb_successor_table_row_step))) /\ ((((exists bcf_height_bcbrdb_successor_table_row_step_previous_right. bcf_height_bcbrdb_successor_table_row_step_previous_right + S (bcf_right_bcbrdb_successor_table_row_step) = S ((S (S (bcf_predecessor_bcbrdb_successor_table_row_step))) * bcf_previous_scale_bcbrdb_successor_table)) /\ exists bcf_quotient_bcbrdb_successor_table_row_step_previous_right. bcf_previous_code_bcbrdb_successor_table = bcf_quotient_bcbrdb_successor_table_row_step_previous_right * S ((S (S (bcf_predecessor_bcbrdb_successor_table_row_step))) * bcf_previous_scale_bcbrdb_successor_table) + (bcf_right_bcbrdb_successor_table_row_step))) /\ bcf_value_bcbrdb_successor_table_row_step = bcf_left_bcbrdb_successor_table_row_step + bcf_right_bcbrdb_successor_table_row_step))))))))))) /\ ((((exists bcf_height_bcbrdb_successor_decoded_row_code. bcf_height_bcbrdb_successor_decoded_row_code + S (bcf_row_code_bcbrdb_successor) = S ((S (S n + S n)) * bcf_row_code_scale_bcbrdb_successor)) /\ exists bcf_quotient_bcbrdb_successor_decoded_row_code. bcf_row_code_code_bcbrdb_successor = bcf_quotient_bcbrdb_successor_decoded_row_code * S ((S (S n + S n)) * bcf_row_code_scale_bcbrdb_successor) + (bcf_row_code_bcbrdb_successor))) /\ ((((exists bcf_height_bcbrdb_successor_decoded_row_scale. bcf_height_bcbrdb_successor_decoded_row_scale + S (bcf_row_scale_bcbrdb_successor) = S ((S (S n + S n)) * bcf_row_scale_scale_bcbrdb_successor)) /\ exists bcf_quotient_bcbrdb_successor_decoded_row_scale. bcf_row_scale_code_bcbrdb_successor = bcf_quotient_bcbrdb_successor_decoded_row_scale * S ((S (S n + S n)) * bcf_row_scale_scale_bcbrdb_successor) + (bcf_row_scale_bcbrdb_successor))) /\ (((exists bcf_height_bcbrdb_successor_decoded_value. bcf_height_bcbrdb_successor_decoded_value + S (d) = S ((S (S n)) * bcf_row_scale_bcbrdb_successor)) /\ exists bcf_quotient_bcbrdb_successor_decoded_value. bcf_row_code_bcbrdb_successor = bcf_quotient_bcbrdb_successor_decoded_value * S ((S (S n)) * bcf_row_scale_bcbrdb_successor) + (d))))))))) -> S n * d = (2 * S (n + n)) * c) /\ (forall n d m. (((exists bcf_lt_gap_bcbrdb_successor_out_of_range. bcf_lt_gap_bcbrdb_successor_out_of_range + S (S n + S n) = S n) /\ d = 0) \/ ((exists bcf_le_gap_bcbrdb_successor_in_range. bcf_le_gap_bcbrdb_successor_in_range + (S n) = S n + S n) /\ (exists bcf_row_code_code_bcbrdb_successor bcf_row_code_scale_bcbrdb_successor bcf_row_scale_code_bcbrdb_successor bcf_row_scale_scale_bcbrdb_successor bcf_row_code_bcbrdb_successor bcf_row_scale_bcbrdb_successor. ((forall bcf_row_index_bcbrdb_successor_table. (exists bcf_lt_gap_bcbrdb_successor_table_row_bound. bcf_lt_gap_bcbrdb_successor_table_row_bound + S (bcf_row_index_bcbrdb_successor_table) = S (S n + S n)) -> exists bcf_row_code_bcbrdb_successor_table bcf_row_scale_bcbrdb_successor_table. ((((exists bcf_height_bcbrdb_successor_table_decoded_row_code. bcf_height_bcbrdb_successor_table_decoded_row_code + S (bcf_row_code_bcbrdb_successor_table) = S ((S (bcf_row_index_bcbrdb_successor_table)) * bcf_row_code_scale_bcbrdb_successor)) /\ exists bcf_quotient_bcbrdb_successor_table_decoded_row_code. bcf_row_code_code_bcbrdb_successor = bcf_quotient_bcbrdb_successor_table_decoded_row_code * S ((S (bcf_row_index_bcbrdb_successor_table)) * bcf_row_code_scale_bcbrdb_successor) + (bcf_row_code_bcbrdb_successor_table))) /\ ((((exists bcf_height_bcbrdb_successor_table_decoded_row_scale. bcf_height_bcbrdb_successor_table_decoded_row_scale + S (bcf_row_scale_bcbrdb_successor_table) = S ((S (bcf_row_index_bcbrdb_successor_table)) * bcf_row_scale_scale_bcbrdb_successor)) /\ exists bcf_quotient_bcbrdb_successor_table_decoded_row_scale. bcf_row_scale_code_bcbrdb_successor = bcf_quotient_bcbrdb_successor_table_decoded_row_scale * S ((S (bcf_row_index_bcbrdb_successor_table)) * bcf_row_scale_scale_bcbrdb_successor) + (bcf_row_scale_bcbrdb_successor_table))) /\ ((bcf_row_index_bcbrdb_successor_table = 0 /\ (forall bcf_index_bcbrdb_successor_table_zero_row. (exists bcf_lt_gap_bcbrdb_successor_table_zero_row_bound. bcf_lt_gap_bcbrdb_successor_table_zero_row_bound + S (bcf_index_bcbrdb_successor_table_zero_row) = S (S n + S n)) -> exists bcf_value_bcbrdb_successor_table_zero_row. ((((exists bcf_height_bcbrdb_successor_table_zero_row_entry. bcf_height_bcbrdb_successor_table_zero_row_entry + S (bcf_value_bcbrdb_successor_table_zero_row) = S ((S (bcf_index_bcbrdb_successor_table_zero_row)) * bcf_row_scale_bcbrdb_successor_table)) /\ exists bcf_quotient_bcbrdb_successor_table_zero_row_entry. bcf_row_code_bcbrdb_successor_table = bcf_quotient_bcbrdb_successor_table_zero_row_entry * S ((S (bcf_index_bcbrdb_successor_table_zero_row)) * bcf_row_scale_bcbrdb_successor_table) + (bcf_value_bcbrdb_successor_table_zero_row))) /\ ((bcf_index_bcbrdb_successor_table_zero_row = 0 /\ bcf_value_bcbrdb_successor_table_zero_row = 1) \/ exists bcf_predecessor_bcbrdb_successor_table_zero_row. bcf_index_bcbrdb_successor_table_zero_row = S bcf_predecessor_bcbrdb_successor_table_zero_row /\ bcf_value_bcbrdb_successor_table_zero_row = 0)))) \/ exists bcf_predecessor_bcbrdb_successor_table bcf_previous_code_bcbrdb_successor_table bcf_previous_scale_bcbrdb_successor_table. bcf_row_index_bcbrdb_successor_table = S bcf_predecessor_bcbrdb_successor_table /\ ((((exists bcf_height_bcbrdb_successor_table_decoded_previous_code. bcf_height_bcbrdb_successor_table_decoded_previous_code + S (bcf_previous_code_bcbrdb_successor_table) = S ((S (bcf_predecessor_bcbrdb_successor_table)) * bcf_row_code_scale_bcbrdb_successor)) /\ exists bcf_quotient_bcbrdb_successor_table_decoded_previous_code. bcf_row_code_code_bcbrdb_successor = bcf_quotient_bcbrdb_successor_table_decoded_previous_code * S ((S (bcf_predecessor_bcbrdb_successor_table)) * bcf_row_code_scale_bcbrdb_successor) + (bcf_previous_code_bcbrdb_successor_table))) /\ ((((exists bcf_height_bcbrdb_successor_table_decoded_previous_scale. bcf_height_bcbrdb_successor_table_decoded_previous_scale + S (bcf_previous_scale_bcbrdb_successor_table) = S ((S (bcf_predecessor_bcbrdb_successor_table)) * bcf_row_scale_scale_bcbrdb_successor)) /\ exists bcf_quotient_bcbrdb_successor_table_decoded_previous_scale. bcf_row_scale_code_bcbrdb_successor = bcf_quotient_bcbrdb_successor_table_decoded_previous_scale * S ((S (bcf_predecessor_bcbrdb_successor_table)) * bcf_row_scale_scale_bcbrdb_successor) + (bcf_previous_scale_bcbrdb_successor_table))) /\ (forall bcf_index_bcbrdb_successor_table_row_step. (exists bcf_lt_gap_bcbrdb_successor_table_row_step_bound. bcf_lt_gap_bcbrdb_successor_table_row_step_bound + S (bcf_index_bcbrdb_successor_table_row_step) = S (S n + S n)) -> exists bcf_value_bcbrdb_successor_table_row_step. ((((exists bcf_height_bcbrdb_successor_table_row_step_entry. bcf_height_bcbrdb_successor_table_row_step_entry + S (bcf_value_bcbrdb_successor_table_row_step) = S ((S (bcf_index_bcbrdb_successor_table_row_step)) * bcf_row_scale_bcbrdb_successor_table)) /\ exists bcf_quotient_bcbrdb_successor_table_row_step_entry. bcf_row_code_bcbrdb_successor_table = bcf_quotient_bcbrdb_successor_table_row_step_entry * S ((S (bcf_index_bcbrdb_successor_table_row_step)) * bcf_row_scale_bcbrdb_successor_table) + (bcf_value_bcbrdb_successor_table_row_step))) /\ ((bcf_index_bcbrdb_successor_table_row_step = 0 /\ bcf_value_bcbrdb_successor_table_row_step = 1) \/ exists bcf_predecessor_bcbrdb_successor_table_row_step bcf_left_bcbrdb_successor_table_row_step bcf_right_bcbrdb_successor_table_row_step. bcf_index_bcbrdb_successor_table_row_step = S bcf_predecessor_bcbrdb_successor_table_row_step /\ ((((exists bcf_height_bcbrdb_successor_table_row_step_previous_left. bcf_height_bcbrdb_successor_table_row_step_previous_left + S (bcf_left_bcbrdb_successor_table_row_step) = S ((S (bcf_predecessor_bcbrdb_successor_table_row_step)) * bcf_previous_scale_bcbrdb_successor_table)) /\ exists bcf_quotient_bcbrdb_successor_table_row_step_previous_left. bcf_previous_code_bcbrdb_successor_table = bcf_quotient_bcbrdb_successor_table_row_step_previous_left * S ((S (bcf_predecessor_bcbrdb_successor_table_row_step)) * bcf_previous_scale_bcbrdb_successor_table) + (bcf_left_bcbrdb_successor_table_row_step))) /\ ((((exists bcf_height_bcbrdb_successor_table_row_step_previous_right. bcf_height_bcbrdb_successor_table_row_step_previous_right + S (bcf_right_bcbrdb_successor_table_row_step) = S ((S (S (bcf_predecessor_bcbrdb_successor_table_row_step))) * bcf_previous_scale_bcbrdb_successor_table)) /\ exists bcf_quotient_bcbrdb_successor_table_row_step_previous_right. bcf_previous_code_bcbrdb_successor_table = bcf_quotient_bcbrdb_successor_table_row_step_previous_right * S ((S (S (bcf_predecessor_bcbrdb_successor_table_row_step))) * bcf_previous_scale_bcbrdb_successor_table) + (bcf_right_bcbrdb_successor_table_row_step))) /\ bcf_value_bcbrdb_successor_table_row_step = bcf_left_bcbrdb_successor_table_row_step + bcf_right_bcbrdb_successor_table_row_step))))))))))) /\ ((((exists bcf_height_bcbrdb_successor_decoded_row_code. bcf_height_bcbrdb_successor_decoded_row_code + S (bcf_row_code_bcbrdb_successor) = S ((S (S n + S n)) * bcf_row_code_scale_bcbrdb_successor)) /\ exists bcf_quotient_bcbrdb_successor_decoded_row_code. bcf_row_code_code_bcbrdb_successor = bcf_quotient_bcbrdb_successor_decoded_row_code * S ((S (S n + S n)) * bcf_row_code_scale_bcbrdb_successor) + (bcf_row_code_bcbrdb_successor))) /\ ((((exists bcf_height_bcbrdb_successor_decoded_row_scale. bcf_height_bcbrdb_successor_decoded_row_scale + S (bcf_row_scale_bcbrdb_successor) = S ((S (S n + S n)) * bcf_row_scale_scale_bcbrdb_successor)) /\ exists bcf_quotient_bcbrdb_successor_decoded_row_scale. bcf_row_scale_code_bcbrdb_successor = bcf_quotient_bcbrdb_successor_decoded_row_scale * S ((S (S n + S n)) * bcf_row_scale_scale_bcbrdb_successor) + (bcf_row_scale_bcbrdb_successor))) /\ (((exists bcf_height_bcbrdb_successor_decoded_value. bcf_height_bcbrdb_successor_decoded_value + S (d) = S ((S (S n)) * bcf_row_scale_bcbrdb_successor)) /\ exists bcf_quotient_bcbrdb_successor_decoded_value. bcf_row_code_bcbrdb_successor = bcf_quotient_bcbrdb_successor_decoded_value * S ((S (S n)) * bcf_row_scale_bcbrdb_successor) + (d))))))))) -> (((exists bcf_lt_gap_bcbrdb_middle_out_of_range. bcf_lt_gap_bcbrdb_middle_out_of_range + S (S (n + n)) = n) /\ m = 0) \/ ((exists bcf_le_gap_bcbrdb_middle_in_range. bcf_le_gap_bcbrdb_middle_in_range + (n) = S (n + n)) /\ (exists bcf_row_code_code_bcbrdb_middle bcf_row_code_scale_bcbrdb_middle bcf_row_scale_code_bcbrdb_middle bcf_row_scale_scale_bcbrdb_middle bcf_row_code_bcbrdb_middle bcf_row_scale_bcbrdb_middle. ((forall bcf_row_index_bcbrdb_middle_table. (exists bcf_lt_gap_bcbrdb_middle_table_row_bound. bcf_lt_gap_bcbrdb_middle_table_row_bound + S (bcf_row_index_bcbrdb_middle_table) = S (S (n + n))) -> exists bcf_row_code_bcbrdb_middle_table bcf_row_scale_bcbrdb_middle_table. ((((exists bcf_height_bcbrdb_middle_table_decoded_row_code. bcf_height_bcbrdb_middle_table_decoded_row_code + S (bcf_row_code_bcbrdb_middle_table) = S ((S (bcf_row_index_bcbrdb_middle_table)) * bcf_row_code_scale_bcbrdb_middle)) /\ exists bcf_quotient_bcbrdb_middle_table_decoded_row_code. bcf_row_code_code_bcbrdb_middle = bcf_quotient_bcbrdb_middle_table_decoded_row_code * S ((S (bcf_row_index_bcbrdb_middle_table)) * bcf_row_code_scale_bcbrdb_middle) + (bcf_row_code_bcbrdb_middle_table))) /\ ((((exists bcf_height_bcbrdb_middle_table_decoded_row_scale. bcf_height_bcbrdb_middle_table_decoded_row_scale + S (bcf_row_scale_bcbrdb_middle_table) = S ((S (bcf_row_index_bcbrdb_middle_table)) * bcf_row_scale_scale_bcbrdb_middle)) /\ exists bcf_quotient_bcbrdb_middle_table_decoded_row_scale. bcf_row_scale_code_bcbrdb_middle = bcf_quotient_bcbrdb_middle_table_decoded_row_scale * S ((S (bcf_row_index_bcbrdb_middle_table)) * bcf_row_scale_scale_bcbrdb_middle) + (bcf_row_scale_bcbrdb_middle_table))) /\ ((bcf_row_index_bcbrdb_middle_table = 0 /\ (forall bcf_index_bcbrdb_middle_table_zero_row. (exists bcf_lt_gap_bcbrdb_middle_table_zero_row_bound. bcf_lt_gap_bcbrdb_middle_table_zero_row_bound + S (bcf_index_bcbrdb_middle_table_zero_row) = S (S (n + n))) -> exists bcf_value_bcbrdb_middle_table_zero_row. ((((exists bcf_height_bcbrdb_middle_table_zero_row_entry. bcf_height_bcbrdb_middle_table_zero_row_entry + S (bcf_value_bcbrdb_middle_table_zero_row) = S ((S (bcf_index_bcbrdb_middle_table_zero_row)) * bcf_row_scale_bcbrdb_middle_table)) /\ exists bcf_quotient_bcbrdb_middle_table_zero_row_entry. bcf_row_code_bcbrdb_middle_table = bcf_quotient_bcbrdb_middle_table_zero_row_entry * S ((S (bcf_index_bcbrdb_middle_table_zero_row)) * bcf_row_scale_bcbrdb_middle_table) + (bcf_value_bcbrdb_middle_table_zero_row))) /\ ((bcf_index_bcbrdb_middle_table_zero_row = 0 /\ bcf_value_bcbrdb_middle_table_zero_row = 1) \/ exists bcf_predecessor_bcbrdb_middle_table_zero_row. bcf_index_bcbrdb_middle_table_zero_row = S bcf_predecessor_bcbrdb_middle_table_zero_row /\ bcf_value_bcbrdb_middle_table_zero_row = 0)))) \/ exists bcf_predecessor_bcbrdb_middle_table bcf_previous_code_bcbrdb_middle_table bcf_previous_scale_bcbrdb_middle_table. bcf_row_index_bcbrdb_middle_table = S bcf_predecessor_bcbrdb_middle_table /\ ((((exists bcf_height_bcbrdb_middle_table_decoded_previous_code. bcf_height_bcbrdb_middle_table_decoded_previous_code + S (bcf_previous_code_bcbrdb_middle_table) = S ((S (bcf_predecessor_bcbrdb_middle_table)) * bcf_row_code_scale_bcbrdb_middle)) /\ exists bcf_quotient_bcbrdb_middle_table_decoded_previous_code. bcf_row_code_code_bcbrdb_middle = bcf_quotient_bcbrdb_middle_table_decoded_previous_code * S ((S (bcf_predecessor_bcbrdb_middle_table)) * bcf_row_code_scale_bcbrdb_middle) + (bcf_previous_code_bcbrdb_middle_table))) /\ ((((exists bcf_height_bcbrdb_middle_table_decoded_previous_scale. bcf_height_bcbrdb_middle_table_decoded_previous_scale + S (bcf_previous_scale_bcbrdb_middle_table) = S ((S (bcf_predecessor_bcbrdb_middle_table)) * bcf_row_scale_scale_bcbrdb_middle)) /\ exists bcf_quotient_bcbrdb_middle_table_decoded_previous_scale. bcf_row_scale_code_bcbrdb_middle = bcf_quotient_bcbrdb_middle_table_decoded_previous_scale * S ((S (bcf_predecessor_bcbrdb_middle_table)) * bcf_row_scale_scale_bcbrdb_middle) + (bcf_previous_scale_bcbrdb_middle_table))) /\ (forall bcf_index_bcbrdb_middle_table_row_step. (exists bcf_lt_gap_bcbrdb_middle_table_row_step_bound. bcf_lt_gap_bcbrdb_middle_table_row_step_bound + S (bcf_index_bcbrdb_middle_table_row_step) = S (S (n + n))) -> exists bcf_value_bcbrdb_middle_table_row_step. ((((exists bcf_height_bcbrdb_middle_table_row_step_entry. bcf_height_bcbrdb_middle_table_row_step_entry + S (bcf_value_bcbrdb_middle_table_row_step) = S ((S (bcf_index_bcbrdb_middle_table_row_step)) * bcf_row_scale_bcbrdb_middle_table)) /\ exists bcf_quotient_bcbrdb_middle_table_row_step_entry. bcf_row_code_bcbrdb_middle_table = bcf_quotient_bcbrdb_middle_table_row_step_entry * S ((S (bcf_index_bcbrdb_middle_table_row_step)) * bcf_row_scale_bcbrdb_middle_table) + (bcf_value_bcbrdb_middle_table_row_step))) /\ ((bcf_index_bcbrdb_middle_table_row_step = 0 /\ bcf_value_bcbrdb_middle_table_row_step = 1) \/ exists bcf_predecessor_bcbrdb_middle_table_row_step bcf_left_bcbrdb_middle_table_row_step bcf_right_bcbrdb_middle_table_row_step. bcf_index_bcbrdb_middle_table_row_step = S bcf_predecessor_bcbrdb_middle_table_row_step /\ ((((exists bcf_height_bcbrdb_middle_table_row_step_previous_left. bcf_height_bcbrdb_middle_table_row_step_previous_left + S (bcf_left_bcbrdb_middle_table_row_step) = S ((S (bcf_predecessor_bcbrdb_middle_table_row_step)) * bcf_previous_scale_bcbrdb_middle_table)) /\ exists bcf_quotient_bcbrdb_middle_table_row_step_previous_left. bcf_previous_code_bcbrdb_middle_table = bcf_quotient_bcbrdb_middle_table_row_step_previous_left * S ((S (bcf_predecessor_bcbrdb_middle_table_row_step)) * bcf_previous_scale_bcbrdb_middle_table) + (bcf_left_bcbrdb_middle_table_row_step))) /\ ((((exists bcf_height_bcbrdb_middle_table_row_step_previous_right. bcf_height_bcbrdb_middle_table_row_step_previous_right + S (bcf_right_bcbrdb_middle_table_row_step) = S ((S (S (bcf_predecessor_bcbrdb_middle_table_row_step))) * bcf_previous_scale_bcbrdb_middle_table)) /\ exists bcf_quotient_bcbrdb_middle_table_row_step_previous_right. bcf_previous_code_bcbrdb_middle_table = bcf_quotient_bcbrdb_middle_table_row_step_previous_right * S ((S (S (bcf_predecessor_bcbrdb_middle_table_row_step))) * bcf_previous_scale_bcbrdb_middle_table) + (bcf_right_bcbrdb_middle_table_row_step))) /\ bcf_value_bcbrdb_middle_table_row_step = bcf_left_bcbrdb_middle_table_row_step + bcf_right_bcbrdb_middle_table_row_step))))))))))) /\ ((((exists bcf_height_bcbrdb_middle_decoded_row_code. bcf_height_bcbrdb_middle_decoded_row_code + S (bcf_row_code_bcbrdb_middle) = S ((S (S (n + n))) * bcf_row_code_scale_bcbrdb_middle)) /\ exists bcf_quotient_bcbrdb_middle_decoded_row_code. bcf_row_code_code_bcbrdb_middle = bcf_quotient_bcbrdb_middle_decoded_row_code * S ((S (S (n + n))) * bcf_row_code_scale_bcbrdb_middle) + (bcf_row_code_bcbrdb_middle))) /\ ((((exists bcf_height_bcbrdb_middle_decoded_row_scale. bcf_height_bcbrdb_middle_decoded_row_scale + S (bcf_row_scale_bcbrdb_middle) = S ((S (S (n + n))) * bcf_row_scale_scale_bcbrdb_middle)) /\ exists bcf_quotient_bcbrdb_middle_decoded_row_scale. bcf_row_scale_code_bcbrdb_middle = bcf_quotient_bcbrdb_middle_decoded_row_scale * S ((S (S (n + n))) * bcf_row_scale_scale_bcbrdb_middle) + (bcf_row_scale_bcbrdb_middle))) /\ (((exists bcf_height_bcbrdb_middle_decoded_value. bcf_height_bcbrdb_middle_decoded_value + S (m) = S ((S (n)) * bcf_row_scale_bcbrdb_middle)) /\ exists bcf_quotient_bcbrdb_middle_decoded_value. bcf_row_code_bcbrdb_middle = bcf_quotient_bcbrdb_middle_decoded_value * S ((S (n)) * bcf_row_scale_bcbrdb_middle) + (m))))))))) -> d = m + m))) /\ (forall n. exists z. (((exists bcf_lt_gap_bcbsuo_exists_out_of_range. bcf_lt_gap_bcbsuo_exists_out_of_range + S (n + n) = n) /\ z = 0) \/ ((exists bcf_le_gap_bcbsuo_exists_in_range. bcf_le_gap_bcbsuo_exists_in_range + (n) = n + n) /\ (exists bcf_row_code_code_bcbsuo_exists bcf_row_code_scale_bcbsuo_exists bcf_row_scale_code_bcbsuo_exists bcf_row_scale_scale_bcbsuo_exists bcf_row_code_bcbsuo_exists bcf_row_scale_bcbsuo_exists. ((forall bcf_row_index_bcbsuo_exists_table. (exists bcf_lt_gap_bcbsuo_exists_table_row_bound. bcf_lt_gap_bcbsuo_exists_table_row_bound + S (bcf_row_index_bcbsuo_exists_table) = S (n + n)) -> exists bcf_row_code_bcbsuo_exists_table bcf_row_scale_bcbsuo_exists_table. ((((exists bcf_height_bcbsuo_exists_table_decoded_row_code. bcf_height_bcbsuo_exists_table_decoded_row_code + S (bcf_row_code_bcbsuo_exists_table) = S ((S (bcf_row_index_bcbsuo_exists_table)) * bcf_row_code_scale_bcbsuo_exists)) /\ exists bcf_quotient_bcbsuo_exists_table_decoded_row_code. bcf_row_code_code_bcbsuo_exists = bcf_quotient_bcbsuo_exists_table_decoded_row_code * S ((S (bcf_row_index_bcbsuo_exists_table)) * bcf_row_code_scale_bcbsuo_exists) + (bcf_row_code_bcbsuo_exists_table))) /\ ((((exists bcf_height_bcbsuo_exists_table_decoded_row_scale. bcf_height_bcbsuo_exists_table_decoded_row_scale + S (bcf_row_scale_bcbsuo_exists_table) = S ((S (bcf_row_index_bcbsuo_exists_table)) * bcf_row_scale_scale_bcbsuo_exists)) /\ exists bcf_quotient_bcbsuo_exists_table_decoded_row_scale. bcf_row_scale_code_bcbsuo_exists = bcf_quotient_bcbsuo_exists_table_decoded_row_scale * S ((S (bcf_row_index_bcbsuo_exists_table)) * bcf_row_scale_scale_bcbsuo_exists) + (bcf_row_scale_bcbsuo_exists_table))) /\ ((bcf_row_index_bcbsuo_exists_table = 0 /\ (forall bcf_index_bcbsuo_exists_table_zero_row. (exists bcf_lt_gap_bcbsuo_exists_table_zero_row_bound. bcf_lt_gap_bcbsuo_exists_table_zero_row_bound + S (bcf_index_bcbsuo_exists_table_zero_row) = S (n + n)) -> exists bcf_value_bcbsuo_exists_table_zero_row. ((((exists bcf_height_bcbsuo_exists_table_zero_row_entry. bcf_height_bcbsuo_exists_table_zero_row_entry + S (bcf_value_bcbsuo_exists_table_zero_row) = S ((S (bcf_index_bcbsuo_exists_table_zero_row)) * bcf_row_scale_bcbsuo_exists_table)) /\ exists bcf_quotient_bcbsuo_exists_table_zero_row_entry. bcf_row_code_bcbsuo_exists_table = bcf_quotient_bcbsuo_exists_table_zero_row_entry * S ((S (bcf_index_bcbsuo_exists_table_zero_row)) * bcf_row_scale_bcbsuo_exists_table) + (bcf_value_bcbsuo_exists_table_zero_row))) /\ ((bcf_index_bcbsuo_exists_table_zero_row = 0 /\ bcf_value_bcbsuo_exists_table_zero_row = 1) \/ exists bcf_predecessor_bcbsuo_exists_table_zero_row. bcf_index_bcbsuo_exists_table_zero_row = S bcf_predecessor_bcbsuo_exists_table_zero_row /\ bcf_value_bcbsuo_exists_table_zero_row = 0)))) \/ exists bcf_predecessor_bcbsuo_exists_table bcf_previous_code_bcbsuo_exists_table bcf_previous_scale_bcbsuo_exists_table. bcf_row_index_bcbsuo_exists_table = S bcf_predecessor_bcbsuo_exists_table /\ ((((exists bcf_height_bcbsuo_exists_table_decoded_previous_code. bcf_height_bcbsuo_exists_table_decoded_previous_code + S (bcf_previous_code_bcbsuo_exists_table) = S ((S (bcf_predecessor_bcbsuo_exists_table)) * bcf_row_code_scale_bcbsuo_exists)) /\ exists bcf_quotient_bcbsuo_exists_table_decoded_previous_code. bcf_row_code_code_bcbsuo_exists = bcf_quotient_bcbsuo_exists_table_decoded_previous_code * S ((S (bcf_predecessor_bcbsuo_exists_table)) * bcf_row_code_scale_bcbsuo_exists) + (bcf_previous_code_bcbsuo_exists_table))) /\ ((((exists bcf_height_bcbsuo_exists_table_decoded_previous_scale. bcf_height_bcbsuo_exists_table_decoded_previous_scale + S (bcf_previous_scale_bcbsuo_exists_table) = S ((S (bcf_predecessor_bcbsuo_exists_table)) * bcf_row_scale_scale_bcbsuo_exists)) /\ exists bcf_quotient_bcbsuo_exists_table_decoded_previous_scale. bcf_row_scale_code_bcbsuo_exists = bcf_quotient_bcbsuo_exists_table_decoded_previous_scale * S ((S (bcf_predecessor_bcbsuo_exists_table)) * bcf_row_scale_scale_bcbsuo_exists) + (bcf_previous_scale_bcbsuo_exists_table))) /\ (forall bcf_index_bcbsuo_exists_table_row_step. (exists bcf_lt_gap_bcbsuo_exists_table_row_step_bound. bcf_lt_gap_bcbsuo_exists_table_row_step_bound + S (bcf_index_bcbsuo_exists_table_row_step) = S (n + n)) -> exists bcf_value_bcbsuo_exists_table_row_step. ((((exists bcf_height_bcbsuo_exists_table_row_step_entry. bcf_height_bcbsuo_exists_table_row_step_entry + S (bcf_value_bcbsuo_exists_table_row_step) = S ((S (bcf_index_bcbsuo_exists_table_row_step)) * bcf_row_scale_bcbsuo_exists_table)) /\ exists bcf_quotient_bcbsuo_exists_table_row_step_entry. bcf_row_code_bcbsuo_exists_table = bcf_quotient_bcbsuo_exists_table_row_step_entry * S ((S (bcf_index_bcbsuo_exists_table_row_step)) * bcf_row_scale_bcbsuo_exists_table) + (bcf_value_bcbsuo_exists_table_row_step))) /\ ((bcf_index_bcbsuo_exists_table_row_step = 0 /\ bcf_value_bcbsuo_exists_table_row_step = 1) \/ exists bcf_predecessor_bcbsuo_exists_table_row_step bcf_left_bcbsuo_exists_table_row_step bcf_right_bcbsuo_exists_table_row_step. bcf_index_bcbsuo_exists_table_row_step = S bcf_predecessor_bcbsuo_exists_table_row_step /\ ((((exists bcf_height_bcbsuo_exists_table_row_step_previous_left. bcf_height_bcbsuo_exists_table_row_step_previous_left + S (bcf_left_bcbsuo_exists_table_row_step) = S ((S (bcf_predecessor_bcbsuo_exists_table_row_step)) * bcf_previous_scale_bcbsuo_exists_table)) /\ exists bcf_quotient_bcbsuo_exists_table_row_step_previous_left. bcf_previous_code_bcbsuo_exists_table = bcf_quotient_bcbsuo_exists_table_row_step_previous_left * S ((S (bcf_predecessor_bcbsuo_exists_table_row_step)) * bcf_previous_scale_bcbsuo_exists_table) + (bcf_left_bcbsuo_exists_table_row_step))) /\ ((((exists bcf_height_bcbsuo_exists_table_row_step_previous_right. bcf_height_bcbsuo_exists_table_row_step_previous_right + S (bcf_right_bcbsuo_exists_table_row_step) = S ((S (S (bcf_predecessor_bcbsuo_exists_table_row_step))) * bcf_previous_scale_bcbsuo_exists_table)) /\ exists bcf_quotient_bcbsuo_exists_table_row_step_previous_right. bcf_previous_code_bcbsuo_exists_table = bcf_quotient_bcbsuo_exists_table_row_step_previous_right * S ((S (S (bcf_predecessor_bcbsuo_exists_table_row_step))) * bcf_previous_scale_bcbsuo_exists_table) + (bcf_right_bcbsuo_exists_table_row_step))) /\ bcf_value_bcbsuo_exists_table_row_step = bcf_left_bcbsuo_exists_table_row_step + bcf_right_bcbsuo_exists_table_row_step))))))))))) /\ ((((exists bcf_height_bcbsuo_exists_decoded_row_code. bcf_height_bcbsuo_exists_decoded_row_code + S (bcf_row_code_bcbsuo_exists) = S ((S (n + n)) * bcf_row_code_scale_bcbsuo_exists)) /\ exists bcf_quotient_bcbsuo_exists_decoded_row_code. bcf_row_code_code_bcbsuo_exists = bcf_quotient_bcbsuo_exists_decoded_row_code * S ((S (n + n)) * bcf_row_code_scale_bcbsuo_exists) + (bcf_row_code_bcbsuo_exists))) /\ ((((exists bcf_height_bcbsuo_exists_decoded_row_scale. bcf_height_bcbsuo_exists_decoded_row_scale + S (bcf_row_scale_bcbsuo_exists) = S ((S (n + n)) * bcf_row_scale_scale_bcbsuo_exists)) /\ exists bcf_quotient_bcbsuo_exists_decoded_row_scale. bcf_row_scale_code_bcbsuo_exists = bcf_quotient_bcbsuo_exists_decoded_row_scale * S ((S (n + n)) * bcf_row_scale_scale_bcbsuo_exists) + (bcf_row_scale_bcbsuo_exists))) /\ (((exists bcf_height_bcbsuo_exists_decoded_value. bcf_height_bcbsuo_exists_decoded_value + S (z) = S ((S (n)) * bcf_row_scale_bcbsuo_exists)) /\ exists bcf_quotient_bcbsuo_exists_decoded_value. bcf_row_code_bcbsuo_exists = bcf_quotient_bcbsuo_exists_decoded_value * S ((S (n)) * bcf_row_scale_bcbsuo_exists) + (z)))))))))))
7. 0007exact central_binom_upper_support_package
8. 0008cases hpackage
9. 0009cases hpackage_left
10. 0010have hcentral_exists : exists d. (((exists bcf_lt_gap_bcomlfp_central_out_of_range. bcf_lt_gap_bcomlfp_central_out_of_range + S (S n + S n) = S n) /\ d = 0) \/ ((exists bcf_le_gap_bcomlfp_central_in_range. bcf_le_gap_bcomlfp_central_in_range + (S n) = S n + S n) /\ (exists bcf_row_code_code_bcomlfp_central bcf_row_code_scale_bcomlfp_central bcf_row_scale_code_bcomlfp_central bcf_row_scale_scale_bcomlfp_central bcf_row_code_bcomlfp_central bcf_row_scale_bcomlfp_central. ((forall bcf_row_index_bcomlfp_central_table. (exists bcf_lt_gap_bcomlfp_central_table_row_bound. bcf_lt_gap_bcomlfp_central_table_row_bound + S (bcf_row_index_bcomlfp_central_table) = S (S n + S n)) -> exists bcf_row_code_bcomlfp_central_table bcf_row_scale_bcomlfp_central_table. ((((exists bcf_height_bcomlfp_central_table_decoded_row_code. bcf_height_bcomlfp_central_table_decoded_row_code + S (bcf_row_code_bcomlfp_central_table) = S ((S (bcf_row_index_bcomlfp_central_table)) * bcf_row_code_scale_bcomlfp_central)) /\ exists bcf_quotient_bcomlfp_central_table_decoded_row_code. bcf_row_code_code_bcomlfp_central = bcf_quotient_bcomlfp_central_table_decoded_row_code * S ((S (bcf_row_index_bcomlfp_central_table)) * bcf_row_code_scale_bcomlfp_central) + (bcf_row_code_bcomlfp_central_table))) /\ ((((exists bcf_height_bcomlfp_central_table_decoded_row_scale. bcf_height_bcomlfp_central_table_decoded_row_scale + S (bcf_row_scale_bcomlfp_central_table) = S ((S (bcf_row_index_bcomlfp_central_table)) * bcf_row_scale_scale_bcomlfp_central)) /\ exists bcf_quotient_bcomlfp_central_table_decoded_row_scale. bcf_row_scale_code_bcomlfp_central = bcf_quotient_bcomlfp_central_table_decoded_row_scale * S ((S (bcf_row_index_bcomlfp_central_table)) * bcf_row_scale_scale_bcomlfp_central) + (bcf_row_scale_bcomlfp_central_table))) /\ ((bcf_row_index_bcomlfp_central_table = 0 /\ (forall bcf_index_bcomlfp_central_table_zero_row. (exists bcf_lt_gap_bcomlfp_central_table_zero_row_bound. bcf_lt_gap_bcomlfp_central_table_zero_row_bound + S (bcf_index_bcomlfp_central_table_zero_row) = S (S n + S n)) -> exists bcf_value_bcomlfp_central_table_zero_row. ((((exists bcf_height_bcomlfp_central_table_zero_row_entry. bcf_height_bcomlfp_central_table_zero_row_entry + S (bcf_value_bcomlfp_central_table_zero_row) = S ((S (bcf_index_bcomlfp_central_table_zero_row)) * bcf_row_scale_bcomlfp_central_table)) /\ exists bcf_quotient_bcomlfp_central_table_zero_row_entry. bcf_row_code_bcomlfp_central_table = bcf_quotient_bcomlfp_central_table_zero_row_entry * S ((S (bcf_index_bcomlfp_central_table_zero_row)) * bcf_row_scale_bcomlfp_central_table) + (bcf_value_bcomlfp_central_table_zero_row))) /\ ((bcf_index_bcomlfp_central_table_zero_row = 0 /\ bcf_value_bcomlfp_central_table_zero_row = 1) \/ exists bcf_predecessor_bcomlfp_central_table_zero_row. bcf_index_bcomlfp_central_table_zero_row = S bcf_predecessor_bcomlfp_central_table_zero_row /\ bcf_value_bcomlfp_central_table_zero_row = 0)))) \/ exists bcf_predecessor_bcomlfp_central_table bcf_previous_code_bcomlfp_central_table bcf_previous_scale_bcomlfp_central_table. bcf_row_index_bcomlfp_central_table = S bcf_predecessor_bcomlfp_central_table /\ ((((exists bcf_height_bcomlfp_central_table_decoded_previous_code. bcf_height_bcomlfp_central_table_decoded_previous_code + S (bcf_previous_code_bcomlfp_central_table) = S ((S (bcf_predecessor_bcomlfp_central_table)) * bcf_row_code_scale_bcomlfp_central)) /\ exists bcf_quotient_bcomlfp_central_table_decoded_previous_code. bcf_row_code_code_bcomlfp_central = bcf_quotient_bcomlfp_central_table_decoded_previous_code * S ((S (bcf_predecessor_bcomlfp_central_table)) * bcf_row_code_scale_bcomlfp_central) + (bcf_previous_code_bcomlfp_central_table))) /\ ((((exists bcf_height_bcomlfp_central_table_decoded_previous_scale. bcf_height_bcomlfp_central_table_decoded_previous_scale + S (bcf_previous_scale_bcomlfp_central_table) = S ((S (bcf_predecessor_bcomlfp_central_table)) * bcf_row_scale_scale_bcomlfp_central)) /\ exists bcf_quotient_bcomlfp_central_table_decoded_previous_scale. bcf_row_scale_code_bcomlfp_central = bcf_quotient_bcomlfp_central_table_decoded_previous_scale * S ((S (bcf_predecessor_bcomlfp_central_table)) * bcf_row_scale_scale_bcomlfp_central) + (bcf_previous_scale_bcomlfp_central_table))) /\ (forall bcf_index_bcomlfp_central_table_row_step. (exists bcf_lt_gap_bcomlfp_central_table_row_step_bound. bcf_lt_gap_bcomlfp_central_table_row_step_bound + S (bcf_index_bcomlfp_central_table_row_step) = S (S n + S n)) -> exists bcf_value_bcomlfp_central_table_row_step. ((((exists bcf_height_bcomlfp_central_table_row_step_entry. bcf_height_bcomlfp_central_table_row_step_entry + S (bcf_value_bcomlfp_central_table_row_step) = S ((S (bcf_index_bcomlfp_central_table_row_step)) * bcf_row_scale_bcomlfp_central_table)) /\ exists bcf_quotient_bcomlfp_central_table_row_step_entry. bcf_row_code_bcomlfp_central_table = bcf_quotient_bcomlfp_central_table_row_step_entry * S ((S (bcf_index_bcomlfp_central_table_row_step)) * bcf_row_scale_bcomlfp_central_table) + (bcf_value_bcomlfp_central_table_row_step))) /\ ((bcf_index_bcomlfp_central_table_row_step = 0 /\ bcf_value_bcomlfp_central_table_row_step = 1) \/ exists bcf_predecessor_bcomlfp_central_table_row_step bcf_left_bcomlfp_central_table_row_step bcf_right_bcomlfp_central_table_row_step. bcf_index_bcomlfp_central_table_row_step = S bcf_predecessor_bcomlfp_central_table_row_step /\ ((((exists bcf_height_bcomlfp_central_table_row_step_previous_left. bcf_height_bcomlfp_central_table_row_step_previous_left + S (bcf_left_bcomlfp_central_table_row_step) = S ((S (bcf_predecessor_bcomlfp_central_table_row_step)) * bcf_previous_scale_bcomlfp_central_table)) /\ exists bcf_quotient_bcomlfp_central_table_row_step_previous_left. bcf_previous_code_bcomlfp_central_table = bcf_quotient_bcomlfp_central_table_row_step_previous_left * S ((S (bcf_predecessor_bcomlfp_central_table_row_step)) * bcf_previous_scale_bcomlfp_central_table) + (bcf_left_bcomlfp_central_table_row_step))) /\ ((((exists bcf_height_bcomlfp_central_table_row_step_previous_right. bcf_height_bcomlfp_central_table_row_step_previous_right + S (bcf_right_bcomlfp_central_table_row_step) = S ((S (S (bcf_predecessor_bcomlfp_central_table_row_step))) * bcf_previous_scale_bcomlfp_central_table)) /\ exists bcf_quotient_bcomlfp_central_table_row_step_previous_right. bcf_previous_code_bcomlfp_central_table = bcf_quotient_bcomlfp_central_table_row_step_previous_right * S ((S (S (bcf_predecessor_bcomlfp_central_table_row_step))) * bcf_previous_scale_bcomlfp_central_table) + (bcf_right_bcomlfp_central_table_row_step))) /\ bcf_value_bcomlfp_central_table_row_step = bcf_left_bcomlfp_central_table_row_step + bcf_right_bcomlfp_central_table_row_step))))))))))) /\ ((((exists bcf_height_bcomlfp_central_decoded_row_code. bcf_height_bcomlfp_central_decoded_row_code + S (bcf_row_code_bcomlfp_central) = S ((S (S n + S n)) * bcf_row_code_scale_bcomlfp_central)) /\ exists bcf_quotient_bcomlfp_central_decoded_row_code. bcf_row_code_code_bcomlfp_central = bcf_quotient_bcomlfp_central_decoded_row_code * S ((S (S n + S n)) * bcf_row_code_scale_bcomlfp_central) + (bcf_row_code_bcomlfp_central))) /\ ((((exists bcf_height_bcomlfp_central_decoded_row_scale. bcf_height_bcomlfp_central_decoded_row_scale + S (bcf_row_scale_bcomlfp_central) = S ((S (S n + S n)) * bcf_row_scale_scale_bcomlfp_central)) /\ exists bcf_quotient_bcomlfp_central_decoded_row_scale. bcf_row_scale_code_bcomlfp_central = bcf_quotient_bcomlfp_central_decoded_row_scale * S ((S (S n + S n)) * bcf_row_scale_scale_bcomlfp_central) + (bcf_row_scale_bcomlfp_central))) /\ (((exists bcf_height_bcomlfp_central_decoded_value. bcf_height_bcomlfp_central_decoded_value + S (d) = S ((S (S n)) * bcf_row_scale_bcomlfp_central)) /\ exists bcf_quotient_bcomlfp_central_decoded_value. bcf_row_code_bcomlfp_central = bcf_quotient_bcomlfp_central_decoded_value * S ((S (S n)) * bcf_row_scale_bcomlfp_central) + (d)))))))))
11. 0011apply hpackage_right
12. 0012cases hcentral_exists
13. 0013have hdouble : x = m + m
14. 0014apply hpackage_left_right
15. 0015exact hcentral_exists_witness
16. 0016exact hmiddle
17. 0017have hsuccessor_power : exists pa_b_bcomlfp_successor_power pa_c_bcomlfp_successor_power. ((forall pa_i_bcomlfp_successor_power_repeat. (exists pa_lt_bcomlfp_successor_power_repeat_bound. pa_lt_bcomlfp_successor_power_repeat_bound + S pa_i_bcomlfp_successor_power_repeat = S n) -> (((exists pa_h_bcomlfp_successor_power_repeat_decoded. pa_h_bcomlfp_successor_power_repeat_decoded + S (4) = S ((S (pa_i_bcomlfp_successor_power_repeat)) * pa_c_bcomlfp_successor_power)) /\ exists pa_q_bcomlfp_successor_power_repeat_decoded. pa_b_bcomlfp_successor_power = pa_q_bcomlfp_successor_power_repeat_decoded * S ((S (pa_i_bcomlfp_successor_power_repeat)) * pa_c_bcomlfp_successor_power) + (4)))) /\ (exists pa_u_bcomlfp_successor_power_product pa_v_bcomlfp_successor_power_product. ((((exists pa_h_bcomlfp_successor_power_product_start. pa_h_bcomlfp_successor_power_product_start + S (1) = S ((S (0)) * pa_v_bcomlfp_successor_power_product)) /\ exists pa_q_bcomlfp_successor_power_product_start. pa_u_bcomlfp_successor_power_product = pa_q_bcomlfp_successor_power_product_start * S ((S (0)) * pa_v_bcomlfp_successor_power_product) + (1))) /\ ((((exists pa_h_bcomlfp_successor_power_product_terminal. pa_h_bcomlfp_successor_power_product_terminal + S (q * 4) = S ((S (S n)) * pa_v_bcomlfp_successor_power_product)) /\ exists pa_q_bcomlfp_successor_power_product_terminal. pa_u_bcomlfp_successor_power_product = pa_q_bcomlfp_successor_power_product_terminal * S ((S (S n)) * pa_v_bcomlfp_successor_power_product) + (q * 4))) /\ forall pa_i_bcomlfp_successor_power_product. (exists pa_lt_bcomlfp_successor_power_product_bound. pa_lt_bcomlfp_successor_power_product_bound + S pa_i_bcomlfp_successor_power_product = S n) -> exists pa_p_bcomlfp_successor_power_product pa_r_bcomlfp_successor_power_product pa_s_bcomlfp_successor_power_product. ((((exists pa_h_bcomlfp_successor_power_product_factor. pa_h_bcomlfp_successor_power_product_factor + S (pa_p_bcomlfp_successor_power_product) = S ((S (pa_i_bcomlfp_successor_power_product)) * pa_c_bcomlfp_successor_power)) /\ exists pa_q_bcomlfp_successor_power_product_factor. pa_b_bcomlfp_successor_power = pa_q_bcomlfp_successor_power_product_factor * S ((S (pa_i_bcomlfp_successor_power_product)) * pa_c_bcomlfp_successor_power) + (pa_p_bcomlfp_successor_power_product))) /\ ((((exists pa_h_bcomlfp_successor_power_product_partial. pa_h_bcomlfp_successor_power_product_partial + S (pa_r_bcomlfp_successor_power_product) = S ((S (pa_i_bcomlfp_successor_power_product)) * pa_v_bcomlfp_successor_power_product)) /\ exists pa_q_bcomlfp_successor_power_product_partial. pa_u_bcomlfp_successor_power_product = pa_q_bcomlfp_successor_power_product_partial * S ((S (pa_i_bcomlfp_successor_power_product)) * pa_v_bcomlfp_successor_power_product) + (pa_r_bcomlfp_successor_power_product))) /\ ((((exists pa_h_bcomlfp_successor_power_product_successor. pa_h_bcomlfp_successor_power_product_successor + S (pa_s_bcomlfp_successor_power_product) = S ((S (S pa_i_bcomlfp_successor_power_product)) * pa_v_bcomlfp_successor_power_product)) /\ exists pa_q_bcomlfp_successor_power_product_successor. pa_u_bcomlfp_successor_power_product = pa_q_bcomlfp_successor_power_product_successor * S ((S (S pa_i_bcomlfp_successor_power_product)) * pa_v_bcomlfp_successor_power_product) + (pa_s_bcomlfp_successor_power_product))) /\ pa_s_bcomlfp_successor_power_product = pa_r_bcomlfp_successor_power_product * pa_p_bcomlfp_successor_power_product)))))))
18. 0018specialize pow_successor_compose 4
19. 0019specialize pow_successor_compose n
20. 0020specialize pow_successor_compose q
21. 0021specialize pow_successor_compose (q * 4)
22. 0022apply pow_successor_compose
23. 0023exact hpower
24. 0024refl
25. 0025have hstrong_all : forall n c q. (((exists bcf_lt_gap_bcbsuo_central_out_of_range. bcf_lt_gap_bcbsuo_central_out_of_range + S (S n + S n) = S n) /\ c = 0) \/ ((exists bcf_le_gap_bcbsuo_central_in_range. bcf_le_gap_bcbsuo_central_in_range + (S n) = S n + S n) /\ (exists bcf_row_code_code_bcbsuo_central bcf_row_code_scale_bcbsuo_central bcf_row_scale_code_bcbsuo_central bcf_row_scale_scale_bcbsuo_central bcf_row_code_bcbsuo_central bcf_row_scale_bcbsuo_central. ((forall bcf_row_index_bcbsuo_central_table. (exists bcf_lt_gap_bcbsuo_central_table_row_bound. bcf_lt_gap_bcbsuo_central_table_row_bound + S (bcf_row_index_bcbsuo_central_table) = S (S n + S n)) -> exists bcf_row_code_bcbsuo_central_table bcf_row_scale_bcbsuo_central_table. ((((exists bcf_height_bcbsuo_central_table_decoded_row_code. bcf_height_bcbsuo_central_table_decoded_row_code + S (bcf_row_code_bcbsuo_central_table) = S ((S (bcf_row_index_bcbsuo_central_table)) * bcf_row_code_scale_bcbsuo_central)) /\ exists bcf_quotient_bcbsuo_central_table_decoded_row_code. bcf_row_code_code_bcbsuo_central = bcf_quotient_bcbsuo_central_table_decoded_row_code * S ((S (bcf_row_index_bcbsuo_central_table)) * bcf_row_code_scale_bcbsuo_central) + (bcf_row_code_bcbsuo_central_table))) /\ ((((exists bcf_height_bcbsuo_central_table_decoded_row_scale. bcf_height_bcbsuo_central_table_decoded_row_scale + S (bcf_row_scale_bcbsuo_central_table) = S ((S (bcf_row_index_bcbsuo_central_table)) * bcf_row_scale_scale_bcbsuo_central)) /\ exists bcf_quotient_bcbsuo_central_table_decoded_row_scale. bcf_row_scale_code_bcbsuo_central = bcf_quotient_bcbsuo_central_table_decoded_row_scale * S ((S (bcf_row_index_bcbsuo_central_table)) * bcf_row_scale_scale_bcbsuo_central) + (bcf_row_scale_bcbsuo_central_table))) /\ ((bcf_row_index_bcbsuo_central_table = 0 /\ (forall bcf_index_bcbsuo_central_table_zero_row. (exists bcf_lt_gap_bcbsuo_central_table_zero_row_bound. bcf_lt_gap_bcbsuo_central_table_zero_row_bound + S (bcf_index_bcbsuo_central_table_zero_row) = S (S n + S n)) -> exists bcf_value_bcbsuo_central_table_zero_row. ((((exists bcf_height_bcbsuo_central_table_zero_row_entry. bcf_height_bcbsuo_central_table_zero_row_entry + S (bcf_value_bcbsuo_central_table_zero_row) = S ((S (bcf_index_bcbsuo_central_table_zero_row)) * bcf_row_scale_bcbsuo_central_table)) /\ exists bcf_quotient_bcbsuo_central_table_zero_row_entry. bcf_row_code_bcbsuo_central_table = bcf_quotient_bcbsuo_central_table_zero_row_entry * S ((S (bcf_index_bcbsuo_central_table_zero_row)) * bcf_row_scale_bcbsuo_central_table) + (bcf_value_bcbsuo_central_table_zero_row))) /\ ((bcf_index_bcbsuo_central_table_zero_row = 0 /\ bcf_value_bcbsuo_central_table_zero_row = 1) \/ exists bcf_predecessor_bcbsuo_central_table_zero_row. bcf_index_bcbsuo_central_table_zero_row = S bcf_predecessor_bcbsuo_central_table_zero_row /\ bcf_value_bcbsuo_central_table_zero_row = 0)))) \/ exists bcf_predecessor_bcbsuo_central_table bcf_previous_code_bcbsuo_central_table bcf_previous_scale_bcbsuo_central_table. bcf_row_index_bcbsuo_central_table = S bcf_predecessor_bcbsuo_central_table /\ ((((exists bcf_height_bcbsuo_central_table_decoded_previous_code. bcf_height_bcbsuo_central_table_decoded_previous_code + S (bcf_previous_code_bcbsuo_central_table) = S ((S (bcf_predecessor_bcbsuo_central_table)) * bcf_row_code_scale_bcbsuo_central)) /\ exists bcf_quotient_bcbsuo_central_table_decoded_previous_code. bcf_row_code_code_bcbsuo_central = bcf_quotient_bcbsuo_central_table_decoded_previous_code * S ((S (bcf_predecessor_bcbsuo_central_table)) * bcf_row_code_scale_bcbsuo_central) + (bcf_previous_code_bcbsuo_central_table))) /\ ((((exists bcf_height_bcbsuo_central_table_decoded_previous_scale. bcf_height_bcbsuo_central_table_decoded_previous_scale + S (bcf_previous_scale_bcbsuo_central_table) = S ((S (bcf_predecessor_bcbsuo_central_table)) * bcf_row_scale_scale_bcbsuo_central)) /\ exists bcf_quotient_bcbsuo_central_table_decoded_previous_scale. bcf_row_scale_code_bcbsuo_central = bcf_quotient_bcbsuo_central_table_decoded_previous_scale * S ((S (bcf_predecessor_bcbsuo_central_table)) * bcf_row_scale_scale_bcbsuo_central) + (bcf_previous_scale_bcbsuo_central_table))) /\ (forall bcf_index_bcbsuo_central_table_row_step. (exists bcf_lt_gap_bcbsuo_central_table_row_step_bound. bcf_lt_gap_bcbsuo_central_table_row_step_bound + S (bcf_index_bcbsuo_central_table_row_step) = S (S n + S n)) -> exists bcf_value_bcbsuo_central_table_row_step. ((((exists bcf_height_bcbsuo_central_table_row_step_entry. bcf_height_bcbsuo_central_table_row_step_entry + S (bcf_value_bcbsuo_central_table_row_step) = S ((S (bcf_index_bcbsuo_central_table_row_step)) * bcf_row_scale_bcbsuo_central_table)) /\ exists bcf_quotient_bcbsuo_central_table_row_step_entry. bcf_row_code_bcbsuo_central_table = bcf_quotient_bcbsuo_central_table_row_step_entry * S ((S (bcf_index_bcbsuo_central_table_row_step)) * bcf_row_scale_bcbsuo_central_table) + (bcf_value_bcbsuo_central_table_row_step))) /\ ((bcf_index_bcbsuo_central_table_row_step = 0 /\ bcf_value_bcbsuo_central_table_row_step = 1) \/ exists bcf_predecessor_bcbsuo_central_table_row_step bcf_left_bcbsuo_central_table_row_step bcf_right_bcbsuo_central_table_row_step. bcf_index_bcbsuo_central_table_row_step = S bcf_predecessor_bcbsuo_central_table_row_step /\ ((((exists bcf_height_bcbsuo_central_table_row_step_previous_left. bcf_height_bcbsuo_central_table_row_step_previous_left + S (bcf_left_bcbsuo_central_table_row_step) = S ((S (bcf_predecessor_bcbsuo_central_table_row_step)) * bcf_previous_scale_bcbsuo_central_table)) /\ exists bcf_quotient_bcbsuo_central_table_row_step_previous_left. bcf_previous_code_bcbsuo_central_table = bcf_quotient_bcbsuo_central_table_row_step_previous_left * S ((S (bcf_predecessor_bcbsuo_central_table_row_step)) * bcf_previous_scale_bcbsuo_central_table) + (bcf_left_bcbsuo_central_table_row_step))) /\ ((((exists bcf_height_bcbsuo_central_table_row_step_previous_right. bcf_height_bcbsuo_central_table_row_step_previous_right + S (bcf_right_bcbsuo_central_table_row_step) = S ((S (S (bcf_predecessor_bcbsuo_central_table_row_step))) * bcf_previous_scale_bcbsuo_central_table)) /\ exists bcf_quotient_bcbsuo_central_table_row_step_previous_right. bcf_previous_code_bcbsuo_central_table = bcf_quotient_bcbsuo_central_table_row_step_previous_right * S ((S (S (bcf_predecessor_bcbsuo_central_table_row_step))) * bcf_previous_scale_bcbsuo_central_table) + (bcf_right_bcbsuo_central_table_row_step))) /\ bcf_value_bcbsuo_central_table_row_step = bcf_left_bcbsuo_central_table_row_step + bcf_right_bcbsuo_central_table_row_step))))))))))) /\ ((((exists bcf_height_bcbsuo_central_decoded_row_code. bcf_height_bcbsuo_central_decoded_row_code + S (bcf_row_code_bcbsuo_central) = S ((S (S n + S n)) * bcf_row_code_scale_bcbsuo_central)) /\ exists bcf_quotient_bcbsuo_central_decoded_row_code. bcf_row_code_code_bcbsuo_central = bcf_quotient_bcbsuo_central_decoded_row_code * S ((S (S n + S n)) * bcf_row_code_scale_bcbsuo_central) + (bcf_row_code_bcbsuo_central))) /\ ((((exists bcf_height_bcbsuo_central_decoded_row_scale. bcf_height_bcbsuo_central_decoded_row_scale + S (bcf_row_scale_bcbsuo_central) = S ((S (S n + S n)) * bcf_row_scale_scale_bcbsuo_central)) /\ exists bcf_quotient_bcbsuo_central_decoded_row_scale. bcf_row_scale_code_bcbsuo_central = bcf_quotient_bcbsuo_central_decoded_row_scale * S ((S (S n + S n)) * bcf_row_scale_scale_bcbsuo_central) + (bcf_row_scale_bcbsuo_central))) /\ (((exists bcf_height_bcbsuo_central_decoded_value. bcf_height_bcbsuo_central_decoded_value + S (c) = S ((S (S n)) * bcf_row_scale_bcbsuo_central)) /\ exists bcf_quotient_bcbsuo_central_decoded_value. bcf_row_code_bcbsuo_central = bcf_quotient_bcbsuo_central_decoded_value * S ((S (S n)) * bcf_row_scale_bcbsuo_central) + (c))))))))) -> (exists pa_b_bcbsuo_power pa_c_bcbsuo_power. ((forall pa_i_bcbsuo_power_repeat. (exists pa_lt_bcbsuo_power_repeat_bound. pa_lt_bcbsuo_power_repeat_bound + S pa_i_bcbsuo_power_repeat = S n) -> (((exists pa_h_bcbsuo_power_repeat_decoded. pa_h_bcbsuo_power_repeat_decoded + S (4) = S ((S (pa_i_bcbsuo_power_repeat)) * pa_c_bcbsuo_power)) /\ exists pa_q_bcbsuo_power_repeat_decoded. pa_b_bcbsuo_power = pa_q_bcbsuo_power_repeat_decoded * S ((S (pa_i_bcbsuo_power_repeat)) * pa_c_bcbsuo_power) + (4)))) /\ (exists pa_u_bcbsuo_power_product pa_v_bcbsuo_power_product. ((((exists pa_h_bcbsuo_power_product_start. pa_h_bcbsuo_power_product_start + S (1) = S ((S (0)) * pa_v_bcbsuo_power_product)) /\ exists pa_q_bcbsuo_power_product_start. pa_u_bcbsuo_power_product = pa_q_bcbsuo_power_product_start * S ((S (0)) * pa_v_bcbsuo_power_product) + (1))) /\ ((((exists pa_h_bcbsuo_power_product_terminal. pa_h_bcbsuo_power_product_terminal + S (q) = S ((S (S n)) * pa_v_bcbsuo_power_product)) /\ exists pa_q_bcbsuo_power_product_terminal. pa_u_bcbsuo_power_product = pa_q_bcbsuo_power_product_terminal * S ((S (S n)) * pa_v_bcbsuo_power_product) + (q))) /\ forall pa_i_bcbsuo_power_product. (exists pa_lt_bcbsuo_power_product_bound. pa_lt_bcbsuo_power_product_bound + S pa_i_bcbsuo_power_product = S n) -> exists pa_p_bcbsuo_power_product pa_r_bcbsuo_power_product pa_s_bcbsuo_power_product. ((((exists pa_h_bcbsuo_power_product_factor. pa_h_bcbsuo_power_product_factor + S (pa_p_bcbsuo_power_product) = S ((S (pa_i_bcbsuo_power_product)) * pa_c_bcbsuo_power)) /\ exists pa_q_bcbsuo_power_product_factor. pa_b_bcbsuo_power = pa_q_bcbsuo_power_product_factor * S ((S (pa_i_bcbsuo_power_product)) * pa_c_bcbsuo_power) + (pa_p_bcbsuo_power_product))) /\ ((((exists pa_h_bcbsuo_power_product_partial. pa_h_bcbsuo_power_product_partial + S (pa_r_bcbsuo_power_product) = S ((S (pa_i_bcbsuo_power_product)) * pa_v_bcbsuo_power_product)) /\ exists pa_q_bcbsuo_power_product_partial. pa_u_bcbsuo_power_product = pa_q_bcbsuo_power_product_partial * S ((S (pa_i_bcbsuo_power_product)) * pa_v_bcbsuo_power_product) + (pa_r_bcbsuo_power_product))) /\ ((((exists pa_h_bcbsuo_power_product_successor. pa_h_bcbsuo_power_product_successor + S (pa_s_bcbsuo_power_product) = S ((S (S pa_i_bcbsuo_power_product)) * pa_v_bcbsuo_power_product)) /\ exists pa_q_bcbsuo_power_product_successor. pa_u_bcbsuo_power_product = pa_q_bcbsuo_power_product_successor * S ((S (S pa_i_bcbsuo_power_product)) * pa_v_bcbsuo_power_product) + (pa_s_bcbsuo_power_product))) /\ pa_s_bcbsuo_power_product = pa_r_bcbsuo_power_product * pa_p_bcbsuo_power_product)))))))) -> (exists bcf_le_gap_bcbsuo_result. bcf_le_gap_bcbsuo_result + (2 * c) = q)
26. 0026apply central_binom_strong_upper_of_laws
27. 0027exact hpackage_left_left
28. 0028exact hpackage_right
29. 0029have hstrong : exists bcf_le_gap_bcomlfp_strong. bcf_le_gap_bcomlfp_strong + (2 * x) = q * 4
30. 0030specialize hstrong_all n
31. 0031specialize hstrong_all x
32. 0032specialize hstrong_all (q * 4)
33. 0033apply hstrong_all
34. 0034exact hcentral_exists_witness
35. 0035exact hsuccessor_power
36. 0036have hleft : 2 * (m + m) = 4 * m
37. 0037trans 2 * m + 2 * m
38. 0038apply mul_add
39. 0039trans 2 * (2 * m)
40. 0040specialize two_mul_eq_add_self (2 * m)
41. 0041symm
42. 0042exact two_mul_eq_add_self
43. 0043trans (2 * 2) * m
44. 0044symm
45. 0045apply mul_assoc
46. 0046have htwo_two : 2 * 2 = 4
47. 0047norm_num
48. 0048rewrite htwo_two
49. 0049refl
50. 0050have hright : q * 4 = 4 * q
51. 0051apply mul_comm
52. 0052rewrite hdouble at hstrong
53. 0053rewrite hleft at hstrong
54. 0054rewrite hright at hstrong
55. 0055specialize mul_le_cancel_left_nonzero 4
56. 0056specialize mul_le_cancel_left_nonzero m
57. 0057specialize mul_le_cancel_left_nonzero q
58. 0058apply mul_le_cancel_left_nonzero
59. 0059intro hfour_zero
60. 0060apply PA1
61. 0061exact hfour_zero
62. 0062exact hstrong