central_binom_prime_contribution_product_exists

Exact expanded PA statement

forall n C. (((exists bcf_lt_gap_bcbpcpe_central_out_of_range. bcf_lt_gap_bcbpcpe_central_out_of_range + S (n + n) = n) /\ C = 0) \/ ((exists bcf_le_gap_bcbpcpe_central_in_range. bcf_le_gap_bcbpcpe_central_in_range + (n) = n + n) /\ (exists bcf_row_code_code_bcbpcpe_central bcf_row_code_scale_bcbpcpe_central bcf_row_scale_code_bcbpcpe_central bcf_row_scale_scale_bcbpcpe_central bcf_row_code_bcbpcpe_central bcf_row_scale_bcbpcpe_central. ((forall bcf_row_index_bcbpcpe_central_table. (exists bcf_lt_gap_bcbpcpe_central_table_row_bound. bcf_lt_gap_bcbpcpe_central_table_row_bound + S (bcf_row_index_bcbpcpe_central_table) = S (n + n)) -> exists bcf_row_code_bcbpcpe_central_table bcf_row_scale_bcbpcpe_central_table. ((((exists bcf_height_bcbpcpe_central_table_decoded_row_code. bcf_height_bcbpcpe_central_table_decoded_row_code + S (bcf_row_code_bcbpcpe_central_table) = S ((S (bcf_row_index_bcbpcpe_central_table)) * bcf_row_code_scale_bcbpcpe_central)) /\ exists bcf_quotient_bcbpcpe_central_table_decoded_row_code. bcf_row_code_code_bcbpcpe_central = bcf_quotient_bcbpcpe_central_table_decoded_row_code * S ((S (bcf_row_index_bcbpcpe_central_table)) * bcf_row_code_scale_bcbpcpe_central) + (bcf_row_code_bcbpcpe_central_table))) /\ ((((exists bcf_height_bcbpcpe_central_table_decoded_row_scale. bcf_height_bcbpcpe_central_table_decoded_row_scale + S (bcf_row_scale_bcbpcpe_central_table) = S ((S (bcf_row_index_bcbpcpe_central_table)) * bcf_row_scale_scale_bcbpcpe_central)) /\ exists bcf_quotient_bcbpcpe_central_table_decoded_row_scale. bcf_row_scale_code_bcbpcpe_central = bcf_quotient_bcbpcpe_central_table_decoded_row_scale * S ((S (bcf_row_index_bcbpcpe_central_table)) * bcf_row_scale_scale_bcbpcpe_central) + (bcf_row_scale_bcbpcpe_central_table))) /\ ((bcf_row_index_bcbpcpe_central_table = 0 /\ (forall bcf_index_bcbpcpe_central_table_zero_row. (exists bcf_lt_gap_bcbpcpe_central_table_zero_row_bound. bcf_lt_gap_bcbpcpe_central_table_zero_row_bound + S (bcf_index_bcbpcpe_central_table_zero_row) = S (n + n)) -> exists bcf_value_bcbpcpe_central_table_zero_row. ((((exists bcf_height_bcbpcpe_central_table_zero_row_entry. bcf_height_bcbpcpe_central_table_zero_row_entry + S (bcf_value_bcbpcpe_central_table_zero_row) = S ((S (bcf_index_bcbpcpe_central_table_zero_row)) * bcf_row_scale_bcbpcpe_central_table)) /\ exists bcf_quotient_bcbpcpe_central_table_zero_row_entry. bcf_row_code_bcbpcpe_central_table = bcf_quotient_bcbpcpe_central_table_zero_row_entry * S ((S (bcf_index_bcbpcpe_central_table_zero_row)) * bcf_row_scale_bcbpcpe_central_table) + (bcf_value_bcbpcpe_central_table_zero_row))) /\ ((bcf_index_bcbpcpe_central_table_zero_row = 0 /\ bcf_value_bcbpcpe_central_table_zero_row = 1) \/ exists bcf_predecessor_bcbpcpe_central_table_zero_row. bcf_index_bcbpcpe_central_table_zero_row = S bcf_predecessor_bcbpcpe_central_table_zero_row /\ bcf_value_bcbpcpe_central_table_zero_row = 0)))) \/ exists bcf_predecessor_bcbpcpe_central_table bcf_previous_code_bcbpcpe_central_table bcf_previous_scale_bcbpcpe_central_table. bcf_row_index_bcbpcpe_central_table = S bcf_predecessor_bcbpcpe_central_table /\ ((((exists bcf_height_bcbpcpe_central_table_decoded_previous_code. bcf_height_bcbpcpe_central_table_decoded_previous_code + S (bcf_previous_code_bcbpcpe_central_table) = S ((S (bcf_predecessor_bcbpcpe_central_table)) * bcf_row_code_scale_bcbpcpe_central)) /\ exists bcf_quotient_bcbpcpe_central_table_decoded_previous_code. bcf_row_code_code_bcbpcpe_central = bcf_quotient_bcbpcpe_central_table_decoded_previous_code * S ((S (bcf_predecessor_bcbpcpe_central_table)) * bcf_row_code_scale_bcbpcpe_central) + (bcf_previous_code_bcbpcpe_central_table))) /\ ((((exists bcf_height_bcbpcpe_central_table_decoded_previous_scale. bcf_height_bcbpcpe_central_table_decoded_previous_scale + S (bcf_previous_scale_bcbpcpe_central_table) = S ((S (bcf_predecessor_bcbpcpe_central_table)) * bcf_row_scale_scale_bcbpcpe_central)) /\ exists bcf_quotient_bcbpcpe_central_table_decoded_previous_scale. bcf_row_scale_code_bcbpcpe_central = bcf_quotient_bcbpcpe_central_table_decoded_previous_scale * S ((S (bcf_predecessor_bcbpcpe_central_table)) * bcf_row_scale_scale_bcbpcpe_central) + (bcf_previous_scale_bcbpcpe_central_table))) /\ (forall bcf_index_bcbpcpe_central_table_row_step. (exists bcf_lt_gap_bcbpcpe_central_table_row_step_bound. bcf_lt_gap_bcbpcpe_central_table_row_step_bound + S (bcf_index_bcbpcpe_central_table_row_step) = S (n + n)) -> exists bcf_value_bcbpcpe_central_table_row_step. ((((exists bcf_height_bcbpcpe_central_table_row_step_entry. bcf_height_bcbpcpe_central_table_row_step_entry + S (bcf_value_bcbpcpe_central_table_row_step) = S ((S (bcf_index_bcbpcpe_central_table_row_step)) * bcf_row_scale_bcbpcpe_central_table)) /\ exists bcf_quotient_bcbpcpe_central_table_row_step_entry. bcf_row_code_bcbpcpe_central_table = bcf_quotient_bcbpcpe_central_table_row_step_entry * S ((S (bcf_index_bcbpcpe_central_table_row_step)) * bcf_row_scale_bcbpcpe_central_table) + (bcf_value_bcbpcpe_central_table_row_step))) /\ ((bcf_index_bcbpcpe_central_table_row_step = 0 /\ bcf_value_bcbpcpe_central_table_row_step = 1) \/ exists bcf_predecessor_bcbpcpe_central_table_row_step bcf_left_bcbpcpe_central_table_row_step bcf_right_bcbpcpe_central_table_row_step. bcf_index_bcbpcpe_central_table_row_step = S bcf_predecessor_bcbpcpe_central_table_row_step /\ ((((exists bcf_height_bcbpcpe_central_table_row_step_previous_left. bcf_height_bcbpcpe_central_table_row_step_previous_left + S (bcf_left_bcbpcpe_central_table_row_step) = S ((S (bcf_predecessor_bcbpcpe_central_table_row_step)) * bcf_previous_scale_bcbpcpe_central_table)) /\ exists bcf_quotient_bcbpcpe_central_table_row_step_previous_left. bcf_previous_code_bcbpcpe_central_table = bcf_quotient_bcbpcpe_central_table_row_step_previous_left * S ((S (bcf_predecessor_bcbpcpe_central_table_row_step)) * bcf_previous_scale_bcbpcpe_central_table) + (bcf_left_bcbpcpe_central_table_row_step))) /\ ((((exists bcf_height_bcbpcpe_central_table_row_step_previous_right. bcf_height_bcbpcpe_central_table_row_step_previous_right + S (bcf_right_bcbpcpe_central_table_row_step) = S ((S (S (bcf_predecessor_bcbpcpe_central_table_row_step))) * bcf_previous_scale_bcbpcpe_central_table)) /\ exists bcf_quotient_bcbpcpe_central_table_row_step_previous_right. bcf_previous_code_bcbpcpe_central_table = bcf_quotient_bcbpcpe_central_table_row_step_previous_right * S ((S (S (bcf_predecessor_bcbpcpe_central_table_row_step))) * bcf_previous_scale_bcbpcpe_central_table) + (bcf_right_bcbpcpe_central_table_row_step))) /\ bcf_value_bcbpcpe_central_table_row_step = bcf_left_bcbpcpe_central_table_row_step + bcf_right_bcbpcpe_central_table_row_step))))))))))) /\ ((((exists bcf_height_bcbpcpe_central_decoded_row_code. bcf_height_bcbpcpe_central_decoded_row_code + S (bcf_row_code_bcbpcpe_central) = S ((S (n + n)) * bcf_row_code_scale_bcbpcpe_central)) /\ exists bcf_quotient_bcbpcpe_central_decoded_row_code. bcf_row_code_code_bcbpcpe_central = bcf_quotient_bcbpcpe_central_decoded_row_code * S ((S (n + n)) * bcf_row_code_scale_bcbpcpe_central) + (bcf_row_code_bcbpcpe_central))) /\ ((((exists bcf_height_bcbpcpe_central_decoded_row_scale. bcf_height_bcbpcpe_central_decoded_row_scale + S (bcf_row_scale_bcbpcpe_central) = S ((S (n + n)) * bcf_row_scale_scale_bcbpcpe_central)) /\ exists bcf_quotient_bcbpcpe_central_decoded_row_scale. bcf_row_scale_code_bcbpcpe_central = bcf_quotient_bcbpcpe_central_decoded_row_scale * S ((S (n + n)) * bcf_row_scale_scale_bcbpcpe_central) + (bcf_row_scale_bcbpcpe_central))) /\ (((exists bcf_height_bcbpcpe_central_decoded_value. bcf_height_bcbpcpe_central_decoded_value + S (C) = S ((S (n)) * bcf_row_scale_bcbpcpe_central)) /\ exists bcf_quotient_bcbpcpe_central_decoded_value. bcf_row_code_bcbpcpe_central = bcf_quotient_bcbpcpe_central_decoded_value * S ((S (n)) * bcf_row_scale_bcbpcpe_central) + (C))))))))) -> exists z. (exists bpr_product_code_bcbpcpe_product bpr_product_scale_bcbpcpe_product. ((forall bpr_prefix_index_bcbpcpe_product_prefix. (exists bpr_gap_bcbpcpe_product_prefix_bound. bpr_gap_bcbpcpe_product_prefix_bound + S (bpr_prefix_index_bcbpcpe_product_prefix) = n + n) -> exists bpr_prefix_value_bcbpcpe_product_prefix. ((((exists bpr_height_bcbpcpe_product_prefix_decoded. bpr_height_bcbpcpe_product_prefix_decoded + S (bpr_prefix_value_bcbpcpe_product_prefix) = S ((S (bpr_prefix_index_bcbpcpe_product_prefix)) * bpr_product_scale_bcbpcpe_product)) /\ exists bpr_quotient_bcbpcpe_product_prefix_decoded. bpr_product_code_bcbpcpe_product = bpr_quotient_bcbpcpe_product_prefix_decoded * S ((S (bpr_prefix_index_bcbpcpe_product_prefix)) * bpr_product_scale_bcbpcpe_product) + (bpr_prefix_value_bcbpcpe_product_prefix))) /\ (((((~(S (bpr_prefix_index_bcbpcpe_product_prefix) = 1) /\ forall bpr_left_bcbpcpe_product_prefix_choice_prime bpr_right_bcbpcpe_product_prefix_choice_prime. S (bpr_prefix_index_bcbpcpe_product_prefix) = bpr_left_bcbpcpe_product_prefix_choice_prime * bpr_right_bcbpcpe_product_prefix_choice_prime -> bpr_left_bcbpcpe_product_prefix_choice_prime = 1 \/ bpr_right_bcbpcpe_product_prefix_choice_prime = 1)) /\ exists bpr_choice_exponent_bcbpcpe_product_prefix_choice. ((((exists bpr_le_gap_bcbpcpe_product_prefix_choice_valuation_selected_bound. bpr_le_gap_bcbpcpe_product_prefix_choice_valuation_selected_bound + (bpr_choice_exponent_bcbpcpe_product_prefix_choice) = (C)) /\ (exists bpr_power_value_bcbpcpe_product_prefix_choice_valuation_selected. ((exists bpr_power_code_bcbpcpe_product_prefix_choice_valuation_selected_power bpr_power_scale_bcbpcpe_product_prefix_choice_valuation_selected_power. ((forall bpr_power_index_bcbpcpe_product_prefix_choice_valuation_selected_power. (exists bpr_gap_bcbpcpe_product_prefix_choice_valuation_selected_power_repeat_bound. bpr_gap_bcbpcpe_product_prefix_choice_valuation_selected_power_repeat_bound + S (bpr_power_index_bcbpcpe_product_prefix_choice_valuation_selected_power) = bpr_choice_exponent_bcbpcpe_product_prefix_choice) -> (((exists bpr_height_bcbpcpe_product_prefix_choice_valuation_selected_power_repeat_entry. bpr_height_bcbpcpe_product_prefix_choice_valuation_selected_power_repeat_entry + S (S (bpr_prefix_index_bcbpcpe_product_prefix)) = S ((S (bpr_power_index_bcbpcpe_product_prefix_choice_valuation_selected_power)) * bpr_power_scale_bcbpcpe_product_prefix_choice_valuation_selected_power)) /\ exists bpr_quotient_bcbpcpe_product_prefix_choice_valuation_selected_power_repeat_entry. bpr_power_code_bcbpcpe_product_prefix_choice_valuation_selected_power = bpr_quotient_bcbpcpe_product_prefix_choice_valuation_selected_power_repeat_entry * S ((S (bpr_power_index_bcbpcpe_product_prefix_choice_valuation_selected_power)) * bpr_power_scale_bcbpcpe_product_prefix_choice_valuation_selected_power) + (S (bpr_prefix_index_bcbpcpe_product_prefix))))) /\ (exists ff_u_bcbpcpe_product_prefix_choice_valuation_selected_power_product ff_v_bcbpcpe_product_prefix_choice_valuation_selected_power_product. ((((exists ff_h_bcbpcpe_product_prefix_choice_valuation_selected_power_product_start. ff_h_bcbpcpe_product_prefix_choice_valuation_selected_power_product_start + S (1) = S ((S (0)) * ff_v_bcbpcpe_product_prefix_choice_valuation_selected_power_product)) /\ exists ff_q_bcbpcpe_product_prefix_choice_valuation_selected_power_product_start. ff_u_bcbpcpe_product_prefix_choice_valuation_selected_power_product = ff_q_bcbpcpe_product_prefix_choice_valuation_selected_power_product_start * S ((S (0)) * ff_v_bcbpcpe_product_prefix_choice_valuation_selected_power_product) + (1))) /\ ((((exists ff_h_bcbpcpe_product_prefix_choice_valuation_selected_power_product_terminal. ff_h_bcbpcpe_product_prefix_choice_valuation_selected_power_product_terminal + S (bpr_power_value_bcbpcpe_product_prefix_choice_valuation_selected) = S ((S (bpr_choice_exponent_bcbpcpe_product_prefix_choice)) * ff_v_bcbpcpe_product_prefix_choice_valuation_selected_power_product)) /\ exists ff_q_bcbpcpe_product_prefix_choice_valuation_selected_power_product_terminal. ff_u_bcbpcpe_product_prefix_choice_valuation_selected_power_product = ff_q_bcbpcpe_product_prefix_choice_valuation_selected_power_product_terminal * S ((S (bpr_choice_exponent_bcbpcpe_product_prefix_choice)) * ff_v_bcbpcpe_product_prefix_choice_valuation_selected_power_product) + (bpr_power_value_bcbpcpe_product_prefix_choice_valuation_selected))) /\ forall ff_i_bcbpcpe_product_prefix_choice_valuation_selected_power_product. (exists ff_lt_bcbpcpe_product_prefix_choice_valuation_selected_power_product_bound. ff_lt_bcbpcpe_product_prefix_choice_valuation_selected_power_product_bound + S ff_i_bcbpcpe_product_prefix_choice_valuation_selected_power_product = bpr_choice_exponent_bcbpcpe_product_prefix_choice) -> exists ff_p_bcbpcpe_product_prefix_choice_valuation_selected_power_product ff_r_bcbpcpe_product_prefix_choice_valuation_selected_power_product ff_s_bcbpcpe_product_prefix_choice_valuation_selected_power_product. ((((exists ff_h_bcbpcpe_product_prefix_choice_valuation_selected_power_product_factor. ff_h_bcbpcpe_product_prefix_choice_valuation_selected_power_product_factor + S (ff_p_bcbpcpe_product_prefix_choice_valuation_selected_power_product) = S ((S (ff_i_bcbpcpe_product_prefix_choice_valuation_selected_power_product)) * bpr_power_scale_bcbpcpe_product_prefix_choice_valuation_selected_power)) /\ exists ff_q_bcbpcpe_product_prefix_choice_valuation_selected_power_product_factor. bpr_power_code_bcbpcpe_product_prefix_choice_valuation_selected_power = ff_q_bcbpcpe_product_prefix_choice_valuation_selected_power_product_factor * S ((S (ff_i_bcbpcpe_product_prefix_choice_valuation_selected_power_product)) * bpr_power_scale_bcbpcpe_product_prefix_choice_valuation_selected_power) + (ff_p_bcbpcpe_product_prefix_choice_valuation_selected_power_product))) /\ ((((exists ff_h_bcbpcpe_product_prefix_choice_valuation_selected_power_product_partial. ff_h_bcbpcpe_product_prefix_choice_valuation_selected_power_product_partial + S (ff_r_bcbpcpe_product_prefix_choice_valuation_selected_power_product) = S ((S (ff_i_bcbpcpe_product_prefix_choice_valuation_selected_power_product)) * ff_v_bcbpcpe_product_prefix_choice_valuation_selected_power_product)) /\ exists ff_q_bcbpcpe_product_prefix_choice_valuation_selected_power_product_partial. ff_u_bcbpcpe_product_prefix_choice_valuation_selected_power_product = ff_q_bcbpcpe_product_prefix_choice_valuation_selected_power_product_partial * S ((S (ff_i_bcbpcpe_product_prefix_choice_valuation_selected_power_product)) * ff_v_bcbpcpe_product_prefix_choice_valuation_selected_power_product) + (ff_r_bcbpcpe_product_prefix_choice_valuation_selected_power_product))) /\ ((((exists ff_h_bcbpcpe_product_prefix_choice_valuation_selected_power_product_successor. ff_h_bcbpcpe_product_prefix_choice_valuation_selected_power_product_successor + S (ff_s_bcbpcpe_product_prefix_choice_valuation_selected_power_product) = S ((S (S ff_i_bcbpcpe_product_prefix_choice_valuation_selected_power_product)) * ff_v_bcbpcpe_product_prefix_choice_valuation_selected_power_product)) /\ exists ff_q_bcbpcpe_product_prefix_choice_valuation_selected_power_product_successor. ff_u_bcbpcpe_product_prefix_choice_valuation_selected_power_product = ff_q_bcbpcpe_product_prefix_choice_valuation_selected_power_product_successor * S ((S (S ff_i_bcbpcpe_product_prefix_choice_valuation_selected_power_product)) * ff_v_bcbpcpe_product_prefix_choice_valuation_selected_power_product) + (ff_s_bcbpcpe_product_prefix_choice_valuation_selected_power_product))) /\ ff_s_bcbpcpe_product_prefix_choice_valuation_selected_power_product = ff_r_bcbpcpe_product_prefix_choice_valuation_selected_power_product * ff_p_bcbpcpe_product_prefix_choice_valuation_selected_power_product)))))))) /\ (exists bpr_divides_quotient_bcbpcpe_product_prefix_choice_valuation_selected_divides. C = (bpr_power_value_bcbpcpe_product_prefix_choice_valuation_selected) * bpr_divides_quotient_bcbpcpe_product_prefix_choice_valuation_selected_divides)))) /\ forall bpr_valuation_candidate_bcbpcpe_product_prefix_choice_valuation. (exists bpr_le_gap_bcbpcpe_product_prefix_choice_valuation_candidate_bound. bpr_le_gap_bcbpcpe_product_prefix_choice_valuation_candidate_bound + (bpr_valuation_candidate_bcbpcpe_product_prefix_choice_valuation) = (C)) -> (exists bpr_power_value_bcbpcpe_product_prefix_choice_valuation_candidate. ((exists bpr_power_code_bcbpcpe_product_prefix_choice_valuation_candidate_power bpr_power_scale_bcbpcpe_product_prefix_choice_valuation_candidate_power. ((forall bpr_power_index_bcbpcpe_product_prefix_choice_valuation_candidate_power. (exists bpr_gap_bcbpcpe_product_prefix_choice_valuation_candidate_power_repeat_bound. bpr_gap_bcbpcpe_product_prefix_choice_valuation_candidate_power_repeat_bound + S (bpr_power_index_bcbpcpe_product_prefix_choice_valuation_candidate_power) = bpr_valuation_candidate_bcbpcpe_product_prefix_choice_valuation) -> (((exists bpr_height_bcbpcpe_product_prefix_choice_valuation_candidate_power_repeat_entry. bpr_height_bcbpcpe_product_prefix_choice_valuation_candidate_power_repeat_entry + S (S (bpr_prefix_index_bcbpcpe_product_prefix)) = S ((S (bpr_power_index_bcbpcpe_product_prefix_choice_valuation_candidate_power)) * bpr_power_scale_bcbpcpe_product_prefix_choice_valuation_candidate_power)) /\ exists bpr_quotient_bcbpcpe_product_prefix_choice_valuation_candidate_power_repeat_entry. bpr_power_code_bcbpcpe_product_prefix_choice_valuation_candidate_power = bpr_quotient_bcbpcpe_product_prefix_choice_valuation_candidate_power_repeat_entry * S ((S (bpr_power_index_bcbpcpe_product_prefix_choice_valuation_candidate_power)) * bpr_power_scale_bcbpcpe_product_prefix_choice_valuation_candidate_power) + (S (bpr_prefix_index_bcbpcpe_product_prefix))))) /\ (exists ff_u_bcbpcpe_product_prefix_choice_valuation_candidate_power_product ff_v_bcbpcpe_product_prefix_choice_valuation_candidate_power_product. ((((exists ff_h_bcbpcpe_product_prefix_choice_valuation_candidate_power_product_start. ff_h_bcbpcpe_product_prefix_choice_valuation_candidate_power_product_start + S (1) = S ((S (0)) * ff_v_bcbpcpe_product_prefix_choice_valuation_candidate_power_product)) /\ exists ff_q_bcbpcpe_product_prefix_choice_valuation_candidate_power_product_start. ff_u_bcbpcpe_product_prefix_choice_valuation_candidate_power_product = ff_q_bcbpcpe_product_prefix_choice_valuation_candidate_power_product_start * S ((S (0)) * ff_v_bcbpcpe_product_prefix_choice_valuation_candidate_power_product) + (1))) /\ ((((exists ff_h_bcbpcpe_product_prefix_choice_valuation_candidate_power_product_terminal. ff_h_bcbpcpe_product_prefix_choice_valuation_candidate_power_product_terminal + S (bpr_power_value_bcbpcpe_product_prefix_choice_valuation_candidate) = S ((S (bpr_valuation_candidate_bcbpcpe_product_prefix_choice_valuation)) * ff_v_bcbpcpe_product_prefix_choice_valuation_candidate_power_product)) /\ exists ff_q_bcbpcpe_product_prefix_choice_valuation_candidate_power_product_terminal. ff_u_bcbpcpe_product_prefix_choice_valuation_candidate_power_product = ff_q_bcbpcpe_product_prefix_choice_valuation_candidate_power_product_terminal * S ((S (bpr_valuation_candidate_bcbpcpe_product_prefix_choice_valuation)) * ff_v_bcbpcpe_product_prefix_choice_valuation_candidate_power_product) + (bpr_power_value_bcbpcpe_product_prefix_choice_valuation_candidate))) /\ forall ff_i_bcbpcpe_product_prefix_choice_valuation_candidate_power_product. (exists ff_lt_bcbpcpe_product_prefix_choice_valuation_candidate_power_product_bound. ff_lt_bcbpcpe_product_prefix_choice_valuation_candidate_power_product_bound + S ff_i_bcbpcpe_product_prefix_choice_valuation_candidate_power_product = bpr_valuation_candidate_bcbpcpe_product_prefix_choice_valuation) -> exists ff_p_bcbpcpe_product_prefix_choice_valuation_candidate_power_product ff_r_bcbpcpe_product_prefix_choice_valuation_candidate_power_product ff_s_bcbpcpe_product_prefix_choice_valuation_candidate_power_product. ((((exists ff_h_bcbpcpe_product_prefix_choice_valuation_candidate_power_product_factor. ff_h_bcbpcpe_product_prefix_choice_valuation_candidate_power_product_factor + S (ff_p_bcbpcpe_product_prefix_choice_valuation_candidate_power_product) = S ((S (ff_i_bcbpcpe_product_prefix_choice_valuation_candidate_power_product)) * bpr_power_scale_bcbpcpe_product_prefix_choice_valuation_candidate_power)) /\ exists ff_q_bcbpcpe_product_prefix_choice_valuation_candidate_power_product_factor. bpr_power_code_bcbpcpe_product_prefix_choice_valuation_candidate_power = ff_q_bcbpcpe_product_prefix_choice_valuation_candidate_power_product_factor * S ((S (ff_i_bcbpcpe_product_prefix_choice_valuation_candidate_power_product)) * bpr_power_scale_bcbpcpe_product_prefix_choice_valuation_candidate_power) + (ff_p_bcbpcpe_product_prefix_choice_valuation_candidate_power_product))) /\ ((((exists ff_h_bcbpcpe_product_prefix_choice_valuation_candidate_power_product_partial. ff_h_bcbpcpe_product_prefix_choice_valuation_candidate_power_product_partial + S (ff_r_bcbpcpe_product_prefix_choice_valuation_candidate_power_product) = S ((S (ff_i_bcbpcpe_product_prefix_choice_valuation_candidate_power_product)) * ff_v_bcbpcpe_product_prefix_choice_valuation_candidate_power_product)) /\ exists ff_q_bcbpcpe_product_prefix_choice_valuation_candidate_power_product_partial. ff_u_bcbpcpe_product_prefix_choice_valuation_candidate_power_product = ff_q_bcbpcpe_product_prefix_choice_valuation_candidate_power_product_partial * S ((S (ff_i_bcbpcpe_product_prefix_choice_valuation_candidate_power_product)) * ff_v_bcbpcpe_product_prefix_choice_valuation_candidate_power_product) + (ff_r_bcbpcpe_product_prefix_choice_valuation_candidate_power_product))) /\ ((((exists ff_h_bcbpcpe_product_prefix_choice_valuation_candidate_power_product_successor. ff_h_bcbpcpe_product_prefix_choice_valuation_candidate_power_product_successor + S (ff_s_bcbpcpe_product_prefix_choice_valuation_candidate_power_product) = S ((S (S ff_i_bcbpcpe_product_prefix_choice_valuation_candidate_power_product)) * ff_v_bcbpcpe_product_prefix_choice_valuation_candidate_power_product)) /\ exists ff_q_bcbpcpe_product_prefix_choice_valuation_candidate_power_product_successor. ff_u_bcbpcpe_product_prefix_choice_valuation_candidate_power_product = ff_q_bcbpcpe_product_prefix_choice_valuation_candidate_power_product_successor * S ((S (S ff_i_bcbpcpe_product_prefix_choice_valuation_candidate_power_product)) * ff_v_bcbpcpe_product_prefix_choice_valuation_candidate_power_product) + (ff_s_bcbpcpe_product_prefix_choice_valuation_candidate_power_product))) /\ ff_s_bcbpcpe_product_prefix_choice_valuation_candidate_power_product = ff_r_bcbpcpe_product_prefix_choice_valuation_candidate_power_product * ff_p_bcbpcpe_product_prefix_choice_valuation_candidate_power_product)))))))) /\ (exists bpr_divides_quotient_bcbpcpe_product_prefix_choice_valuation_candidate_divides. C = (bpr_power_value_bcbpcpe_product_prefix_choice_valuation_candidate) * bpr_divides_quotient_bcbpcpe_product_prefix_choice_valuation_candidate_divides))) -> (exists bpr_le_gap_bcbpcpe_product_prefix_choice_valuation_candidate_below. bpr_le_gap_bcbpcpe_product_prefix_choice_valuation_candidate_below + (bpr_valuation_candidate_bcbpcpe_product_prefix_choice_valuation) = (bpr_choice_exponent_bcbpcpe_product_prefix_choice))) /\ (exists bpr_power_code_bcbpcpe_product_prefix_choice_power bpr_power_scale_bcbpcpe_product_prefix_choice_power. ((forall bpr_power_index_bcbpcpe_product_prefix_choice_power. (exists bpr_gap_bcbpcpe_product_prefix_choice_power_repeat_bound. bpr_gap_bcbpcpe_product_prefix_choice_power_repeat_bound + S (bpr_power_index_bcbpcpe_product_prefix_choice_power) = bpr_choice_exponent_bcbpcpe_product_prefix_choice) -> (((exists bpr_height_bcbpcpe_product_prefix_choice_power_repeat_entry. bpr_height_bcbpcpe_product_prefix_choice_power_repeat_entry + S (S (bpr_prefix_index_bcbpcpe_product_prefix)) = S ((S (bpr_power_index_bcbpcpe_product_prefix_choice_power)) * bpr_power_scale_bcbpcpe_product_prefix_choice_power)) /\ exists bpr_quotient_bcbpcpe_product_prefix_choice_power_repeat_entry. bpr_power_code_bcbpcpe_product_prefix_choice_power = bpr_quotient_bcbpcpe_product_prefix_choice_power_repeat_entry * S ((S (bpr_power_index_bcbpcpe_product_prefix_choice_power)) * bpr_power_scale_bcbpcpe_product_prefix_choice_power) + (S (bpr_prefix_index_bcbpcpe_product_prefix))))) /\ (exists ff_u_bcbpcpe_product_prefix_choice_power_product ff_v_bcbpcpe_product_prefix_choice_power_product. ((((exists ff_h_bcbpcpe_product_prefix_choice_power_product_start. ff_h_bcbpcpe_product_prefix_choice_power_product_start + S (1) = S ((S (0)) * ff_v_bcbpcpe_product_prefix_choice_power_product)) /\ exists ff_q_bcbpcpe_product_prefix_choice_power_product_start. ff_u_bcbpcpe_product_prefix_choice_power_product = ff_q_bcbpcpe_product_prefix_choice_power_product_start * S ((S (0)) * ff_v_bcbpcpe_product_prefix_choice_power_product) + (1))) /\ ((((exists ff_h_bcbpcpe_product_prefix_choice_power_product_terminal. ff_h_bcbpcpe_product_prefix_choice_power_product_terminal + S (bpr_prefix_value_bcbpcpe_product_prefix) = S ((S (bpr_choice_exponent_bcbpcpe_product_prefix_choice)) * ff_v_bcbpcpe_product_prefix_choice_power_product)) /\ exists ff_q_bcbpcpe_product_prefix_choice_power_product_terminal. ff_u_bcbpcpe_product_prefix_choice_power_product = ff_q_bcbpcpe_product_prefix_choice_power_product_terminal * S ((S (bpr_choice_exponent_bcbpcpe_product_prefix_choice)) * ff_v_bcbpcpe_product_prefix_choice_power_product) + (bpr_prefix_value_bcbpcpe_product_prefix))) /\ forall ff_i_bcbpcpe_product_prefix_choice_power_product. (exists ff_lt_bcbpcpe_product_prefix_choice_power_product_bound. ff_lt_bcbpcpe_product_prefix_choice_power_product_bound + S ff_i_bcbpcpe_product_prefix_choice_power_product = bpr_choice_exponent_bcbpcpe_product_prefix_choice) -> exists ff_p_bcbpcpe_product_prefix_choice_power_product ff_r_bcbpcpe_product_prefix_choice_power_product ff_s_bcbpcpe_product_prefix_choice_power_product. ((((exists ff_h_bcbpcpe_product_prefix_choice_power_product_factor. ff_h_bcbpcpe_product_prefix_choice_power_product_factor + S (ff_p_bcbpcpe_product_prefix_choice_power_product) = S ((S (ff_i_bcbpcpe_product_prefix_choice_power_product)) * bpr_power_scale_bcbpcpe_product_prefix_choice_power)) /\ exists ff_q_bcbpcpe_product_prefix_choice_power_product_factor. bpr_power_code_bcbpcpe_product_prefix_choice_power = ff_q_bcbpcpe_product_prefix_choice_power_product_factor * S ((S (ff_i_bcbpcpe_product_prefix_choice_power_product)) * bpr_power_scale_bcbpcpe_product_prefix_choice_power) + (ff_p_bcbpcpe_product_prefix_choice_power_product))) /\ ((((exists ff_h_bcbpcpe_product_prefix_choice_power_product_partial. ff_h_bcbpcpe_product_prefix_choice_power_product_partial + S (ff_r_bcbpcpe_product_prefix_choice_power_product) = S ((S (ff_i_bcbpcpe_product_prefix_choice_power_product)) * ff_v_bcbpcpe_product_prefix_choice_power_product)) /\ exists ff_q_bcbpcpe_product_prefix_choice_power_product_partial. ff_u_bcbpcpe_product_prefix_choice_power_product = ff_q_bcbpcpe_product_prefix_choice_power_product_partial * S ((S (ff_i_bcbpcpe_product_prefix_choice_power_product)) * ff_v_bcbpcpe_product_prefix_choice_power_product) + (ff_r_bcbpcpe_product_prefix_choice_power_product))) /\ ((((exists ff_h_bcbpcpe_product_prefix_choice_power_product_successor. ff_h_bcbpcpe_product_prefix_choice_power_product_successor + S (ff_s_bcbpcpe_product_prefix_choice_power_product) = S ((S (S ff_i_bcbpcpe_product_prefix_choice_power_product)) * ff_v_bcbpcpe_product_prefix_choice_power_product)) /\ exists ff_q_bcbpcpe_product_prefix_choice_power_product_successor. ff_u_bcbpcpe_product_prefix_choice_power_product = ff_q_bcbpcpe_product_prefix_choice_power_product_successor * S ((S (S ff_i_bcbpcpe_product_prefix_choice_power_product)) * ff_v_bcbpcpe_product_prefix_choice_power_product) + (ff_s_bcbpcpe_product_prefix_choice_power_product))) /\ ff_s_bcbpcpe_product_prefix_choice_power_product = ff_r_bcbpcpe_product_prefix_choice_power_product * ff_p_bcbpcpe_product_prefix_choice_power_product)))))))))) \/ (~((~(S (bpr_prefix_index_bcbpcpe_product_prefix) = 1) /\ forall bpr_left_bcbpcpe_product_prefix_choice_prime bpr_right_bcbpcpe_product_prefix_choice_prime. S (bpr_prefix_index_bcbpcpe_product_prefix) = bpr_left_bcbpcpe_product_prefix_choice_prime * bpr_right_bcbpcpe_product_prefix_choice_prime -> bpr_left_bcbpcpe_product_prefix_choice_prime = 1 \/ bpr_right_bcbpcpe_product_prefix_choice_prime = 1)) /\ bpr_prefix_value_bcbpcpe_product_prefix = 1))))) /\ (exists ff_u_bcbpcpe_product_product ff_v_bcbpcpe_product_product. ((((exists ff_h_bcbpcpe_product_product_start. ff_h_bcbpcpe_product_product_start + S (1) = S ((S (0)) * ff_v_bcbpcpe_product_product)) /\ exists ff_q_bcbpcpe_product_product_start. ff_u_bcbpcpe_product_product = ff_q_bcbpcpe_product_product_start * S ((S (0)) * ff_v_bcbpcpe_product_product) + (1))) /\ ((((exists ff_h_bcbpcpe_product_product_terminal. ff_h_bcbpcpe_product_product_terminal + S (z) = S ((S (n + n)) * ff_v_bcbpcpe_product_product)) /\ exists ff_q_bcbpcpe_product_product_terminal. ff_u_bcbpcpe_product_product = ff_q_bcbpcpe_product_product_terminal * S ((S (n + n)) * ff_v_bcbpcpe_product_product) + (z))) /\ forall ff_i_bcbpcpe_product_product. (exists ff_lt_bcbpcpe_product_product_bound. ff_lt_bcbpcpe_product_product_bound + S ff_i_bcbpcpe_product_product = n + n) -> exists ff_p_bcbpcpe_product_product ff_r_bcbpcpe_product_product ff_s_bcbpcpe_product_product. ((((exists ff_h_bcbpcpe_product_product_factor. ff_h_bcbpcpe_product_product_factor + S (ff_p_bcbpcpe_product_product) = S ((S (ff_i_bcbpcpe_product_product)) * bpr_product_scale_bcbpcpe_product)) /\ exists ff_q_bcbpcpe_product_product_factor. bpr_product_code_bcbpcpe_product = ff_q_bcbpcpe_product_product_factor * S ((S (ff_i_bcbpcpe_product_product)) * bpr_product_scale_bcbpcpe_product) + (ff_p_bcbpcpe_product_product))) /\ ((((exists ff_h_bcbpcpe_product_product_partial. ff_h_bcbpcpe_product_product_partial + S (ff_r_bcbpcpe_product_product) = S ((S (ff_i_bcbpcpe_product_product)) * ff_v_bcbpcpe_product_product)) /\ exists ff_q_bcbpcpe_product_product_partial. ff_u_bcbpcpe_product_product = ff_q_bcbpcpe_product_product_partial * S ((S (ff_i_bcbpcpe_product_product)) * ff_v_bcbpcpe_product_product) + (ff_r_bcbpcpe_product_product))) /\ ((((exists ff_h_bcbpcpe_product_product_successor. ff_h_bcbpcpe_product_product_successor + S (ff_s_bcbpcpe_product_product) = S ((S (S ff_i_bcbpcpe_product_product)) * ff_v_bcbpcpe_product_product)) /\ exists ff_q_bcbpcpe_product_product_successor. ff_u_bcbpcpe_product_product = ff_q_bcbpcpe_product_product_successor * S ((S (S ff_i_bcbpcpe_product_product)) * ff_v_bcbpcpe_product_product) + (ff_s_bcbpcpe_product_product))) /\ ff_s_bcbpcpe_product_product = ff_r_bcbpcpe_product_product * ff_p_bcbpcpe_product_product)))))))) /\ C = z