central_binom_nonzero_strong_upper

Exact expanded PA statement

forall n c q. ~(n = 0) -> (((exists bcf_lt_gap_bcnzsu_central_out_of_range. bcf_lt_gap_bcnzsu_central_out_of_range + S (n + n) = n) /\ c = 0) \/ ((exists bcf_le_gap_bcnzsu_central_in_range. bcf_le_gap_bcnzsu_central_in_range + (n) = n + n) /\ (exists bcf_row_code_code_bcnzsu_central bcf_row_code_scale_bcnzsu_central bcf_row_scale_code_bcnzsu_central bcf_row_scale_scale_bcnzsu_central bcf_row_code_bcnzsu_central bcf_row_scale_bcnzsu_central. ((forall bcf_row_index_bcnzsu_central_table. (exists bcf_lt_gap_bcnzsu_central_table_row_bound. bcf_lt_gap_bcnzsu_central_table_row_bound + S (bcf_row_index_bcnzsu_central_table) = S (n + n)) -> exists bcf_row_code_bcnzsu_central_table bcf_row_scale_bcnzsu_central_table. ((((exists bcf_height_bcnzsu_central_table_decoded_row_code. bcf_height_bcnzsu_central_table_decoded_row_code + S (bcf_row_code_bcnzsu_central_table) = S ((S (bcf_row_index_bcnzsu_central_table)) * bcf_row_code_scale_bcnzsu_central)) /\ exists bcf_quotient_bcnzsu_central_table_decoded_row_code. bcf_row_code_code_bcnzsu_central = bcf_quotient_bcnzsu_central_table_decoded_row_code * S ((S (bcf_row_index_bcnzsu_central_table)) * bcf_row_code_scale_bcnzsu_central) + (bcf_row_code_bcnzsu_central_table))) /\ ((((exists bcf_height_bcnzsu_central_table_decoded_row_scale. bcf_height_bcnzsu_central_table_decoded_row_scale + S (bcf_row_scale_bcnzsu_central_table) = S ((S (bcf_row_index_bcnzsu_central_table)) * bcf_row_scale_scale_bcnzsu_central)) /\ exists bcf_quotient_bcnzsu_central_table_decoded_row_scale. bcf_row_scale_code_bcnzsu_central = bcf_quotient_bcnzsu_central_table_decoded_row_scale * S ((S (bcf_row_index_bcnzsu_central_table)) * bcf_row_scale_scale_bcnzsu_central) + (bcf_row_scale_bcnzsu_central_table))) /\ ((bcf_row_index_bcnzsu_central_table = 0 /\ (forall bcf_index_bcnzsu_central_table_zero_row. (exists bcf_lt_gap_bcnzsu_central_table_zero_row_bound. bcf_lt_gap_bcnzsu_central_table_zero_row_bound + S (bcf_index_bcnzsu_central_table_zero_row) = S (n + n)) -> exists bcf_value_bcnzsu_central_table_zero_row. ((((exists bcf_height_bcnzsu_central_table_zero_row_entry. bcf_height_bcnzsu_central_table_zero_row_entry + S (bcf_value_bcnzsu_central_table_zero_row) = S ((S (bcf_index_bcnzsu_central_table_zero_row)) * bcf_row_scale_bcnzsu_central_table)) /\ exists bcf_quotient_bcnzsu_central_table_zero_row_entry. bcf_row_code_bcnzsu_central_table = bcf_quotient_bcnzsu_central_table_zero_row_entry * S ((S (bcf_index_bcnzsu_central_table_zero_row)) * bcf_row_scale_bcnzsu_central_table) + (bcf_value_bcnzsu_central_table_zero_row))) /\ ((bcf_index_bcnzsu_central_table_zero_row = 0 /\ bcf_value_bcnzsu_central_table_zero_row = 1) \/ exists bcf_predecessor_bcnzsu_central_table_zero_row. bcf_index_bcnzsu_central_table_zero_row = S bcf_predecessor_bcnzsu_central_table_zero_row /\ bcf_value_bcnzsu_central_table_zero_row = 0)))) \/ exists bcf_predecessor_bcnzsu_central_table bcf_previous_code_bcnzsu_central_table bcf_previous_scale_bcnzsu_central_table. bcf_row_index_bcnzsu_central_table = S bcf_predecessor_bcnzsu_central_table /\ ((((exists bcf_height_bcnzsu_central_table_decoded_previous_code. bcf_height_bcnzsu_central_table_decoded_previous_code + S (bcf_previous_code_bcnzsu_central_table) = S ((S (bcf_predecessor_bcnzsu_central_table)) * bcf_row_code_scale_bcnzsu_central)) /\ exists bcf_quotient_bcnzsu_central_table_decoded_previous_code. bcf_row_code_code_bcnzsu_central = bcf_quotient_bcnzsu_central_table_decoded_previous_code * S ((S (bcf_predecessor_bcnzsu_central_table)) * bcf_row_code_scale_bcnzsu_central) + (bcf_previous_code_bcnzsu_central_table))) /\ ((((exists bcf_height_bcnzsu_central_table_decoded_previous_scale. bcf_height_bcnzsu_central_table_decoded_previous_scale + S (bcf_previous_scale_bcnzsu_central_table) = S ((S (bcf_predecessor_bcnzsu_central_table)) * bcf_row_scale_scale_bcnzsu_central)) /\ exists bcf_quotient_bcnzsu_central_table_decoded_previous_scale. bcf_row_scale_code_bcnzsu_central = bcf_quotient_bcnzsu_central_table_decoded_previous_scale * S ((S (bcf_predecessor_bcnzsu_central_table)) * bcf_row_scale_scale_bcnzsu_central) + (bcf_previous_scale_bcnzsu_central_table))) /\ (forall bcf_index_bcnzsu_central_table_row_step. (exists bcf_lt_gap_bcnzsu_central_table_row_step_bound. bcf_lt_gap_bcnzsu_central_table_row_step_bound + S (bcf_index_bcnzsu_central_table_row_step) = S (n + n)) -> exists bcf_value_bcnzsu_central_table_row_step. ((((exists bcf_height_bcnzsu_central_table_row_step_entry. bcf_height_bcnzsu_central_table_row_step_entry + S (bcf_value_bcnzsu_central_table_row_step) = S ((S (bcf_index_bcnzsu_central_table_row_step)) * bcf_row_scale_bcnzsu_central_table)) /\ exists bcf_quotient_bcnzsu_central_table_row_step_entry. bcf_row_code_bcnzsu_central_table = bcf_quotient_bcnzsu_central_table_row_step_entry * S ((S (bcf_index_bcnzsu_central_table_row_step)) * bcf_row_scale_bcnzsu_central_table) + (bcf_value_bcnzsu_central_table_row_step))) /\ ((bcf_index_bcnzsu_central_table_row_step = 0 /\ bcf_value_bcnzsu_central_table_row_step = 1) \/ exists bcf_predecessor_bcnzsu_central_table_row_step bcf_left_bcnzsu_central_table_row_step bcf_right_bcnzsu_central_table_row_step. bcf_index_bcnzsu_central_table_row_step = S bcf_predecessor_bcnzsu_central_table_row_step /\ ((((exists bcf_height_bcnzsu_central_table_row_step_previous_left. bcf_height_bcnzsu_central_table_row_step_previous_left + S (bcf_left_bcnzsu_central_table_row_step) = S ((S (bcf_predecessor_bcnzsu_central_table_row_step)) * bcf_previous_scale_bcnzsu_central_table)) /\ exists bcf_quotient_bcnzsu_central_table_row_step_previous_left. bcf_previous_code_bcnzsu_central_table = bcf_quotient_bcnzsu_central_table_row_step_previous_left * S ((S (bcf_predecessor_bcnzsu_central_table_row_step)) * bcf_previous_scale_bcnzsu_central_table) + (bcf_left_bcnzsu_central_table_row_step))) /\ ((((exists bcf_height_bcnzsu_central_table_row_step_previous_right. bcf_height_bcnzsu_central_table_row_step_previous_right + S (bcf_right_bcnzsu_central_table_row_step) = S ((S (S (bcf_predecessor_bcnzsu_central_table_row_step))) * bcf_previous_scale_bcnzsu_central_table)) /\ exists bcf_quotient_bcnzsu_central_table_row_step_previous_right. bcf_previous_code_bcnzsu_central_table = bcf_quotient_bcnzsu_central_table_row_step_previous_right * S ((S (S (bcf_predecessor_bcnzsu_central_table_row_step))) * bcf_previous_scale_bcnzsu_central_table) + (bcf_right_bcnzsu_central_table_row_step))) /\ bcf_value_bcnzsu_central_table_row_step = bcf_left_bcnzsu_central_table_row_step + bcf_right_bcnzsu_central_table_row_step))))))))))) /\ ((((exists bcf_height_bcnzsu_central_decoded_row_code. bcf_height_bcnzsu_central_decoded_row_code + S (bcf_row_code_bcnzsu_central) = S ((S (n + n)) * bcf_row_code_scale_bcnzsu_central)) /\ exists bcf_quotient_bcnzsu_central_decoded_row_code. bcf_row_code_code_bcnzsu_central = bcf_quotient_bcnzsu_central_decoded_row_code * S ((S (n + n)) * bcf_row_code_scale_bcnzsu_central) + (bcf_row_code_bcnzsu_central))) /\ ((((exists bcf_height_bcnzsu_central_decoded_row_scale. bcf_height_bcnzsu_central_decoded_row_scale + S (bcf_row_scale_bcnzsu_central) = S ((S (n + n)) * bcf_row_scale_scale_bcnzsu_central)) /\ exists bcf_quotient_bcnzsu_central_decoded_row_scale. bcf_row_scale_code_bcnzsu_central = bcf_quotient_bcnzsu_central_decoded_row_scale * S ((S (n + n)) * bcf_row_scale_scale_bcnzsu_central) + (bcf_row_scale_bcnzsu_central))) /\ (((exists bcf_height_bcnzsu_central_decoded_value. bcf_height_bcnzsu_central_decoded_value + S (c) = S ((S (n)) * bcf_row_scale_bcnzsu_central)) /\ exists bcf_quotient_bcnzsu_central_decoded_value. bcf_row_code_bcnzsu_central = bcf_quotient_bcnzsu_central_decoded_value * S ((S (n)) * bcf_row_scale_bcnzsu_central) + (c))))))))) -> (exists pa_b_bcnzsu_power pa_c_bcnzsu_power. ((forall pa_i_bcnzsu_power_repeat. (exists pa_lt_bcnzsu_power_repeat_bound. pa_lt_bcnzsu_power_repeat_bound + S pa_i_bcnzsu_power_repeat = n) -> (((exists pa_h_bcnzsu_power_repeat_decoded. pa_h_bcnzsu_power_repeat_decoded + S (4) = S ((S (pa_i_bcnzsu_power_repeat)) * pa_c_bcnzsu_power)) /\ exists pa_q_bcnzsu_power_repeat_decoded. pa_b_bcnzsu_power = pa_q_bcnzsu_power_repeat_decoded * S ((S (pa_i_bcnzsu_power_repeat)) * pa_c_bcnzsu_power) + (4)))) /\ (exists pa_u_bcnzsu_power_product pa_v_bcnzsu_power_product. ((((exists pa_h_bcnzsu_power_product_start. pa_h_bcnzsu_power_product_start + S (1) = S ((S (0)) * pa_v_bcnzsu_power_product)) /\ exists pa_q_bcnzsu_power_product_start. pa_u_bcnzsu_power_product = pa_q_bcnzsu_power_product_start * S ((S (0)) * pa_v_bcnzsu_power_product) + (1))) /\ ((((exists pa_h_bcnzsu_power_product_terminal. pa_h_bcnzsu_power_product_terminal + S (q) = S ((S (n)) * pa_v_bcnzsu_power_product)) /\ exists pa_q_bcnzsu_power_product_terminal. pa_u_bcnzsu_power_product = pa_q_bcnzsu_power_product_terminal * S ((S (n)) * pa_v_bcnzsu_power_product) + (q))) /\ forall pa_i_bcnzsu_power_product. (exists pa_lt_bcnzsu_power_product_bound. pa_lt_bcnzsu_power_product_bound + S pa_i_bcnzsu_power_product = n) -> exists pa_p_bcnzsu_power_product pa_r_bcnzsu_power_product pa_s_bcnzsu_power_product. ((((exists pa_h_bcnzsu_power_product_factor. pa_h_bcnzsu_power_product_factor + S (pa_p_bcnzsu_power_product) = S ((S (pa_i_bcnzsu_power_product)) * pa_c_bcnzsu_power)) /\ exists pa_q_bcnzsu_power_product_factor. pa_b_bcnzsu_power = pa_q_bcnzsu_power_product_factor * S ((S (pa_i_bcnzsu_power_product)) * pa_c_bcnzsu_power) + (pa_p_bcnzsu_power_product))) /\ ((((exists pa_h_bcnzsu_power_product_partial. pa_h_bcnzsu_power_product_partial + S (pa_r_bcnzsu_power_product) = S ((S (pa_i_bcnzsu_power_product)) * pa_v_bcnzsu_power_product)) /\ exists pa_q_bcnzsu_power_product_partial. pa_u_bcnzsu_power_product = pa_q_bcnzsu_power_product_partial * S ((S (pa_i_bcnzsu_power_product)) * pa_v_bcnzsu_power_product) + (pa_r_bcnzsu_power_product))) /\ ((((exists pa_h_bcnzsu_power_product_successor. pa_h_bcnzsu_power_product_successor + S (pa_s_bcnzsu_power_product) = S ((S (S pa_i_bcnzsu_power_product)) * pa_v_bcnzsu_power_product)) /\ exists pa_q_bcnzsu_power_product_successor. pa_u_bcnzsu_power_product = pa_q_bcnzsu_power_product_successor * S ((S (S pa_i_bcnzsu_power_product)) * pa_v_bcnzsu_power_product) + (pa_s_bcnzsu_power_product))) /\ pa_s_bcnzsu_power_product = pa_r_bcnzsu_power_product * pa_p_bcnzsu_power_product)))))))) -> (exists bcf_le_gap_bcnzsu_result. bcf_le_gap_bcnzsu_result + (2 * c) = q)