prime_contribution_prefix_interval_split

Exact expanded PA statement

forall n a l z. (exists bpr_product_code_bpcpis_source bpr_product_scale_bpcpis_source. ((forall bpr_prefix_index_bpcpis_source_prefix. (exists bpr_gap_bpcpis_source_prefix_bound. bpr_gap_bpcpis_source_prefix_bound + S (bpr_prefix_index_bpcpis_source_prefix) = a + l) -> exists bpr_prefix_value_bpcpis_source_prefix. ((((exists bpr_height_bpcpis_source_prefix_decoded. bpr_height_bpcpis_source_prefix_decoded + S (bpr_prefix_value_bpcpis_source_prefix) = S ((S (bpr_prefix_index_bpcpis_source_prefix)) * bpr_product_scale_bpcpis_source)) /\ exists bpr_quotient_bpcpis_source_prefix_decoded. bpr_product_code_bpcpis_source = bpr_quotient_bpcpis_source_prefix_decoded * S ((S (bpr_prefix_index_bpcpis_source_prefix)) * bpr_product_scale_bpcpis_source) + (bpr_prefix_value_bpcpis_source_prefix))) /\ (((((~(S (bpr_prefix_index_bpcpis_source_prefix) = 1) /\ forall bpr_left_bpcpis_source_prefix_choice_prime bpr_right_bpcpis_source_prefix_choice_prime. S (bpr_prefix_index_bpcpis_source_prefix) = bpr_left_bpcpis_source_prefix_choice_prime * bpr_right_bpcpis_source_prefix_choice_prime -> bpr_left_bpcpis_source_prefix_choice_prime = 1 \/ bpr_right_bpcpis_source_prefix_choice_prime = 1)) /\ exists bpr_choice_exponent_bpcpis_source_prefix_choice. ((((exists bpr_le_gap_bpcpis_source_prefix_choice_valuation_selected_bound. bpr_le_gap_bpcpis_source_prefix_choice_valuation_selected_bound + (bpr_choice_exponent_bpcpis_source_prefix_choice) = (n)) /\ (exists bpr_power_value_bpcpis_source_prefix_choice_valuation_selected. ((exists bpr_power_code_bpcpis_source_prefix_choice_valuation_selected_power bpr_power_scale_bpcpis_source_prefix_choice_valuation_selected_power. ((forall bpr_power_index_bpcpis_source_prefix_choice_valuation_selected_power. (exists bpr_gap_bpcpis_source_prefix_choice_valuation_selected_power_repeat_bound. bpr_gap_bpcpis_source_prefix_choice_valuation_selected_power_repeat_bound + S (bpr_power_index_bpcpis_source_prefix_choice_valuation_selected_power) = bpr_choice_exponent_bpcpis_source_prefix_choice) -> (((exists bpr_height_bpcpis_source_prefix_choice_valuation_selected_power_repeat_entry. bpr_height_bpcpis_source_prefix_choice_valuation_selected_power_repeat_entry + S (S (bpr_prefix_index_bpcpis_source_prefix)) = S ((S (bpr_power_index_bpcpis_source_prefix_choice_valuation_selected_power)) * bpr_power_scale_bpcpis_source_prefix_choice_valuation_selected_power)) /\ exists bpr_quotient_bpcpis_source_prefix_choice_valuation_selected_power_repeat_entry. bpr_power_code_bpcpis_source_prefix_choice_valuation_selected_power = bpr_quotient_bpcpis_source_prefix_choice_valuation_selected_power_repeat_entry * S ((S (bpr_power_index_bpcpis_source_prefix_choice_valuation_selected_power)) * bpr_power_scale_bpcpis_source_prefix_choice_valuation_selected_power) + (S (bpr_prefix_index_bpcpis_source_prefix))))) /\ (exists ff_u_bpcpis_source_prefix_choice_valuation_selected_power_product ff_v_bpcpis_source_prefix_choice_valuation_selected_power_product. ((((exists ff_h_bpcpis_source_prefix_choice_valuation_selected_power_product_start. ff_h_bpcpis_source_prefix_choice_valuation_selected_power_product_start + S (1) = S ((S (0)) * ff_v_bpcpis_source_prefix_choice_valuation_selected_power_product)) /\ exists ff_q_bpcpis_source_prefix_choice_valuation_selected_power_product_start. ff_u_bpcpis_source_prefix_choice_valuation_selected_power_product = ff_q_bpcpis_source_prefix_choice_valuation_selected_power_product_start * S ((S (0)) * ff_v_bpcpis_source_prefix_choice_valuation_selected_power_product) + (1))) /\ ((((exists ff_h_bpcpis_source_prefix_choice_valuation_selected_power_product_terminal. ff_h_bpcpis_source_prefix_choice_valuation_selected_power_product_terminal + S (bpr_power_value_bpcpis_source_prefix_choice_valuation_selected) = S ((S (bpr_choice_exponent_bpcpis_source_prefix_choice)) * ff_v_bpcpis_source_prefix_choice_valuation_selected_power_product)) /\ exists ff_q_bpcpis_source_prefix_choice_valuation_selected_power_product_terminal. ff_u_bpcpis_source_prefix_choice_valuation_selected_power_product = ff_q_bpcpis_source_prefix_choice_valuation_selected_power_product_terminal * S ((S (bpr_choice_exponent_bpcpis_source_prefix_choice)) * ff_v_bpcpis_source_prefix_choice_valuation_selected_power_product) + (bpr_power_value_bpcpis_source_prefix_choice_valuation_selected))) /\ forall ff_i_bpcpis_source_prefix_choice_valuation_selected_power_product. (exists ff_lt_bpcpis_source_prefix_choice_valuation_selected_power_product_bound. ff_lt_bpcpis_source_prefix_choice_valuation_selected_power_product_bound + S ff_i_bpcpis_source_prefix_choice_valuation_selected_power_product = bpr_choice_exponent_bpcpis_source_prefix_choice) -> exists ff_p_bpcpis_source_prefix_choice_valuation_selected_power_product ff_r_bpcpis_source_prefix_choice_valuation_selected_power_product ff_s_bpcpis_source_prefix_choice_valuation_selected_power_product. ((((exists ff_h_bpcpis_source_prefix_choice_valuation_selected_power_product_factor. ff_h_bpcpis_source_prefix_choice_valuation_selected_power_product_factor + S (ff_p_bpcpis_source_prefix_choice_valuation_selected_power_product) = S ((S (ff_i_bpcpis_source_prefix_choice_valuation_selected_power_product)) * bpr_power_scale_bpcpis_source_prefix_choice_valuation_selected_power)) /\ exists ff_q_bpcpis_source_prefix_choice_valuation_selected_power_product_factor. bpr_power_code_bpcpis_source_prefix_choice_valuation_selected_power = ff_q_bpcpis_source_prefix_choice_valuation_selected_power_product_factor * S ((S (ff_i_bpcpis_source_prefix_choice_valuation_selected_power_product)) * bpr_power_scale_bpcpis_source_prefix_choice_valuation_selected_power) + (ff_p_bpcpis_source_prefix_choice_valuation_selected_power_product))) /\ ((((exists ff_h_bpcpis_source_prefix_choice_valuation_selected_power_product_partial. ff_h_bpcpis_source_prefix_choice_valuation_selected_power_product_partial + S (ff_r_bpcpis_source_prefix_choice_valuation_selected_power_product) = S ((S (ff_i_bpcpis_source_prefix_choice_valuation_selected_power_product)) * ff_v_bpcpis_source_prefix_choice_valuation_selected_power_product)) /\ exists ff_q_bpcpis_source_prefix_choice_valuation_selected_power_product_partial. ff_u_bpcpis_source_prefix_choice_valuation_selected_power_product = ff_q_bpcpis_source_prefix_choice_valuation_selected_power_product_partial * S ((S (ff_i_bpcpis_source_prefix_choice_valuation_selected_power_product)) * ff_v_bpcpis_source_prefix_choice_valuation_selected_power_product) + (ff_r_bpcpis_source_prefix_choice_valuation_selected_power_product))) /\ ((((exists ff_h_bpcpis_source_prefix_choice_valuation_selected_power_product_successor. ff_h_bpcpis_source_prefix_choice_valuation_selected_power_product_successor + S (ff_s_bpcpis_source_prefix_choice_valuation_selected_power_product) = S ((S (S ff_i_bpcpis_source_prefix_choice_valuation_selected_power_product)) * ff_v_bpcpis_source_prefix_choice_valuation_selected_power_product)) /\ exists ff_q_bpcpis_source_prefix_choice_valuation_selected_power_product_successor. ff_u_bpcpis_source_prefix_choice_valuation_selected_power_product = ff_q_bpcpis_source_prefix_choice_valuation_selected_power_product_successor * S ((S (S ff_i_bpcpis_source_prefix_choice_valuation_selected_power_product)) * ff_v_bpcpis_source_prefix_choice_valuation_selected_power_product) + (ff_s_bpcpis_source_prefix_choice_valuation_selected_power_product))) /\ ff_s_bpcpis_source_prefix_choice_valuation_selected_power_product = ff_r_bpcpis_source_prefix_choice_valuation_selected_power_product * ff_p_bpcpis_source_prefix_choice_valuation_selected_power_product)))))))) /\ (exists bpr_divides_quotient_bpcpis_source_prefix_choice_valuation_selected_divides. n = (bpr_power_value_bpcpis_source_prefix_choice_valuation_selected) * bpr_divides_quotient_bpcpis_source_prefix_choice_valuation_selected_divides)))) /\ forall bpr_valuation_candidate_bpcpis_source_prefix_choice_valuation. (exists bpr_le_gap_bpcpis_source_prefix_choice_valuation_candidate_bound. bpr_le_gap_bpcpis_source_prefix_choice_valuation_candidate_bound + (bpr_valuation_candidate_bpcpis_source_prefix_choice_valuation) = (n)) -> (exists bpr_power_value_bpcpis_source_prefix_choice_valuation_candidate. ((exists bpr_power_code_bpcpis_source_prefix_choice_valuation_candidate_power bpr_power_scale_bpcpis_source_prefix_choice_valuation_candidate_power. ((forall bpr_power_index_bpcpis_source_prefix_choice_valuation_candidate_power. (exists bpr_gap_bpcpis_source_prefix_choice_valuation_candidate_power_repeat_bound. bpr_gap_bpcpis_source_prefix_choice_valuation_candidate_power_repeat_bound + S (bpr_power_index_bpcpis_source_prefix_choice_valuation_candidate_power) = bpr_valuation_candidate_bpcpis_source_prefix_choice_valuation) -> (((exists bpr_height_bpcpis_source_prefix_choice_valuation_candidate_power_repeat_entry. bpr_height_bpcpis_source_prefix_choice_valuation_candidate_power_repeat_entry + S (S (bpr_prefix_index_bpcpis_source_prefix)) = S ((S (bpr_power_index_bpcpis_source_prefix_choice_valuation_candidate_power)) * bpr_power_scale_bpcpis_source_prefix_choice_valuation_candidate_power)) /\ exists bpr_quotient_bpcpis_source_prefix_choice_valuation_candidate_power_repeat_entry. bpr_power_code_bpcpis_source_prefix_choice_valuation_candidate_power = bpr_quotient_bpcpis_source_prefix_choice_valuation_candidate_power_repeat_entry * S ((S (bpr_power_index_bpcpis_source_prefix_choice_valuation_candidate_power)) * bpr_power_scale_bpcpis_source_prefix_choice_valuation_candidate_power) + (S (bpr_prefix_index_bpcpis_source_prefix))))) /\ (exists ff_u_bpcpis_source_prefix_choice_valuation_candidate_power_product ff_v_bpcpis_source_prefix_choice_valuation_candidate_power_product. ((((exists ff_h_bpcpis_source_prefix_choice_valuation_candidate_power_product_start. ff_h_bpcpis_source_prefix_choice_valuation_candidate_power_product_start + S (1) = S ((S (0)) * ff_v_bpcpis_source_prefix_choice_valuation_candidate_power_product)) /\ exists ff_q_bpcpis_source_prefix_choice_valuation_candidate_power_product_start. ff_u_bpcpis_source_prefix_choice_valuation_candidate_power_product = ff_q_bpcpis_source_prefix_choice_valuation_candidate_power_product_start * S ((S (0)) * ff_v_bpcpis_source_prefix_choice_valuation_candidate_power_product) + (1))) /\ ((((exists ff_h_bpcpis_source_prefix_choice_valuation_candidate_power_product_terminal. ff_h_bpcpis_source_prefix_choice_valuation_candidate_power_product_terminal + S (bpr_power_value_bpcpis_source_prefix_choice_valuation_candidate) = S ((S (bpr_valuation_candidate_bpcpis_source_prefix_choice_valuation)) * ff_v_bpcpis_source_prefix_choice_valuation_candidate_power_product)) /\ exists ff_q_bpcpis_source_prefix_choice_valuation_candidate_power_product_terminal. ff_u_bpcpis_source_prefix_choice_valuation_candidate_power_product = ff_q_bpcpis_source_prefix_choice_valuation_candidate_power_product_terminal * S ((S (bpr_valuation_candidate_bpcpis_source_prefix_choice_valuation)) * ff_v_bpcpis_source_prefix_choice_valuation_candidate_power_product) + (bpr_power_value_bpcpis_source_prefix_choice_valuation_candidate))) /\ forall ff_i_bpcpis_source_prefix_choice_valuation_candidate_power_product. (exists ff_lt_bpcpis_source_prefix_choice_valuation_candidate_power_product_bound. ff_lt_bpcpis_source_prefix_choice_valuation_candidate_power_product_bound + S ff_i_bpcpis_source_prefix_choice_valuation_candidate_power_product = bpr_valuation_candidate_bpcpis_source_prefix_choice_valuation) -> exists ff_p_bpcpis_source_prefix_choice_valuation_candidate_power_product ff_r_bpcpis_source_prefix_choice_valuation_candidate_power_product ff_s_bpcpis_source_prefix_choice_valuation_candidate_power_product. ((((exists ff_h_bpcpis_source_prefix_choice_valuation_candidate_power_product_factor. ff_h_bpcpis_source_prefix_choice_valuation_candidate_power_product_factor + S (ff_p_bpcpis_source_prefix_choice_valuation_candidate_power_product) = S ((S (ff_i_bpcpis_source_prefix_choice_valuation_candidate_power_product)) * bpr_power_scale_bpcpis_source_prefix_choice_valuation_candidate_power)) /\ exists ff_q_bpcpis_source_prefix_choice_valuation_candidate_power_product_factor. bpr_power_code_bpcpis_source_prefix_choice_valuation_candidate_power = ff_q_bpcpis_source_prefix_choice_valuation_candidate_power_product_factor * S ((S (ff_i_bpcpis_source_prefix_choice_valuation_candidate_power_product)) * bpr_power_scale_bpcpis_source_prefix_choice_valuation_candidate_power) + (ff_p_bpcpis_source_prefix_choice_valuation_candidate_power_product))) /\ ((((exists ff_h_bpcpis_source_prefix_choice_valuation_candidate_power_product_partial. ff_h_bpcpis_source_prefix_choice_valuation_candidate_power_product_partial + S (ff_r_bpcpis_source_prefix_choice_valuation_candidate_power_product) = S ((S (ff_i_bpcpis_source_prefix_choice_valuation_candidate_power_product)) * ff_v_bpcpis_source_prefix_choice_valuation_candidate_power_product)) /\ exists ff_q_bpcpis_source_prefix_choice_valuation_candidate_power_product_partial. ff_u_bpcpis_source_prefix_choice_valuation_candidate_power_product = ff_q_bpcpis_source_prefix_choice_valuation_candidate_power_product_partial * S ((S (ff_i_bpcpis_source_prefix_choice_valuation_candidate_power_product)) * ff_v_bpcpis_source_prefix_choice_valuation_candidate_power_product) + (ff_r_bpcpis_source_prefix_choice_valuation_candidate_power_product))) /\ ((((exists ff_h_bpcpis_source_prefix_choice_valuation_candidate_power_product_successor. ff_h_bpcpis_source_prefix_choice_valuation_candidate_power_product_successor + S (ff_s_bpcpis_source_prefix_choice_valuation_candidate_power_product) = S ((S (S ff_i_bpcpis_source_prefix_choice_valuation_candidate_power_product)) * ff_v_bpcpis_source_prefix_choice_valuation_candidate_power_product)) /\ exists ff_q_bpcpis_source_prefix_choice_valuation_candidate_power_product_successor. ff_u_bpcpis_source_prefix_choice_valuation_candidate_power_product = ff_q_bpcpis_source_prefix_choice_valuation_candidate_power_product_successor * S ((S (S ff_i_bpcpis_source_prefix_choice_valuation_candidate_power_product)) * ff_v_bpcpis_source_prefix_choice_valuation_candidate_power_product) + (ff_s_bpcpis_source_prefix_choice_valuation_candidate_power_product))) /\ ff_s_bpcpis_source_prefix_choice_valuation_candidate_power_product = ff_r_bpcpis_source_prefix_choice_valuation_candidate_power_product * ff_p_bpcpis_source_prefix_choice_valuation_candidate_power_product)))))))) /\ (exists bpr_divides_quotient_bpcpis_source_prefix_choice_valuation_candidate_divides. n = (bpr_power_value_bpcpis_source_prefix_choice_valuation_candidate) * bpr_divides_quotient_bpcpis_source_prefix_choice_valuation_candidate_divides))) -> (exists bpr_le_gap_bpcpis_source_prefix_choice_valuation_candidate_below. bpr_le_gap_bpcpis_source_prefix_choice_valuation_candidate_below + (bpr_valuation_candidate_bpcpis_source_prefix_choice_valuation) = (bpr_choice_exponent_bpcpis_source_prefix_choice))) /\ (exists bpr_power_code_bpcpis_source_prefix_choice_power bpr_power_scale_bpcpis_source_prefix_choice_power. ((forall bpr_power_index_bpcpis_source_prefix_choice_power. (exists bpr_gap_bpcpis_source_prefix_choice_power_repeat_bound. bpr_gap_bpcpis_source_prefix_choice_power_repeat_bound + S (bpr_power_index_bpcpis_source_prefix_choice_power) = bpr_choice_exponent_bpcpis_source_prefix_choice) -> (((exists bpr_height_bpcpis_source_prefix_choice_power_repeat_entry. bpr_height_bpcpis_source_prefix_choice_power_repeat_entry + S (S (bpr_prefix_index_bpcpis_source_prefix)) = S ((S (bpr_power_index_bpcpis_source_prefix_choice_power)) * bpr_power_scale_bpcpis_source_prefix_choice_power)) /\ exists bpr_quotient_bpcpis_source_prefix_choice_power_repeat_entry. bpr_power_code_bpcpis_source_prefix_choice_power = bpr_quotient_bpcpis_source_prefix_choice_power_repeat_entry * S ((S (bpr_power_index_bpcpis_source_prefix_choice_power)) * bpr_power_scale_bpcpis_source_prefix_choice_power) + (S (bpr_prefix_index_bpcpis_source_prefix))))) /\ (exists ff_u_bpcpis_source_prefix_choice_power_product ff_v_bpcpis_source_prefix_choice_power_product. ((((exists ff_h_bpcpis_source_prefix_choice_power_product_start. ff_h_bpcpis_source_prefix_choice_power_product_start + S (1) = S ((S (0)) * ff_v_bpcpis_source_prefix_choice_power_product)) /\ exists ff_q_bpcpis_source_prefix_choice_power_product_start. ff_u_bpcpis_source_prefix_choice_power_product = ff_q_bpcpis_source_prefix_choice_power_product_start * S ((S (0)) * ff_v_bpcpis_source_prefix_choice_power_product) + (1))) /\ ((((exists ff_h_bpcpis_source_prefix_choice_power_product_terminal. ff_h_bpcpis_source_prefix_choice_power_product_terminal + S (bpr_prefix_value_bpcpis_source_prefix) = S ((S (bpr_choice_exponent_bpcpis_source_prefix_choice)) * ff_v_bpcpis_source_prefix_choice_power_product)) /\ exists ff_q_bpcpis_source_prefix_choice_power_product_terminal. ff_u_bpcpis_source_prefix_choice_power_product = ff_q_bpcpis_source_prefix_choice_power_product_terminal * S ((S (bpr_choice_exponent_bpcpis_source_prefix_choice)) * ff_v_bpcpis_source_prefix_choice_power_product) + (bpr_prefix_value_bpcpis_source_prefix))) /\ forall ff_i_bpcpis_source_prefix_choice_power_product. (exists ff_lt_bpcpis_source_prefix_choice_power_product_bound. ff_lt_bpcpis_source_prefix_choice_power_product_bound + S ff_i_bpcpis_source_prefix_choice_power_product = bpr_choice_exponent_bpcpis_source_prefix_choice) -> exists ff_p_bpcpis_source_prefix_choice_power_product ff_r_bpcpis_source_prefix_choice_power_product ff_s_bpcpis_source_prefix_choice_power_product. ((((exists ff_h_bpcpis_source_prefix_choice_power_product_factor. ff_h_bpcpis_source_prefix_choice_power_product_factor + S (ff_p_bpcpis_source_prefix_choice_power_product) = S ((S (ff_i_bpcpis_source_prefix_choice_power_product)) * bpr_power_scale_bpcpis_source_prefix_choice_power)) /\ exists ff_q_bpcpis_source_prefix_choice_power_product_factor. bpr_power_code_bpcpis_source_prefix_choice_power = ff_q_bpcpis_source_prefix_choice_power_product_factor * S ((S (ff_i_bpcpis_source_prefix_choice_power_product)) * bpr_power_scale_bpcpis_source_prefix_choice_power) + (ff_p_bpcpis_source_prefix_choice_power_product))) /\ ((((exists ff_h_bpcpis_source_prefix_choice_power_product_partial. ff_h_bpcpis_source_prefix_choice_power_product_partial + S (ff_r_bpcpis_source_prefix_choice_power_product) = S ((S (ff_i_bpcpis_source_prefix_choice_power_product)) * ff_v_bpcpis_source_prefix_choice_power_product)) /\ exists ff_q_bpcpis_source_prefix_choice_power_product_partial. ff_u_bpcpis_source_prefix_choice_power_product = ff_q_bpcpis_source_prefix_choice_power_product_partial * S ((S (ff_i_bpcpis_source_prefix_choice_power_product)) * ff_v_bpcpis_source_prefix_choice_power_product) + (ff_r_bpcpis_source_prefix_choice_power_product))) /\ ((((exists ff_h_bpcpis_source_prefix_choice_power_product_successor. ff_h_bpcpis_source_prefix_choice_power_product_successor + S (ff_s_bpcpis_source_prefix_choice_power_product) = S ((S (S ff_i_bpcpis_source_prefix_choice_power_product)) * ff_v_bpcpis_source_prefix_choice_power_product)) /\ exists ff_q_bpcpis_source_prefix_choice_power_product_successor. ff_u_bpcpis_source_prefix_choice_power_product = ff_q_bpcpis_source_prefix_choice_power_product_successor * S ((S (S ff_i_bpcpis_source_prefix_choice_power_product)) * ff_v_bpcpis_source_prefix_choice_power_product) + (ff_s_bpcpis_source_prefix_choice_power_product))) /\ ff_s_bpcpis_source_prefix_choice_power_product = ff_r_bpcpis_source_prefix_choice_power_product * ff_p_bpcpis_source_prefix_choice_power_product)))))))))) \/ (~((~(S (bpr_prefix_index_bpcpis_source_prefix) = 1) /\ forall bpr_left_bpcpis_source_prefix_choice_prime bpr_right_bpcpis_source_prefix_choice_prime. S (bpr_prefix_index_bpcpis_source_prefix) = bpr_left_bpcpis_source_prefix_choice_prime * bpr_right_bpcpis_source_prefix_choice_prime -> bpr_left_bpcpis_source_prefix_choice_prime = 1 \/ bpr_right_bpcpis_source_prefix_choice_prime = 1)) /\ bpr_prefix_value_bpcpis_source_prefix = 1))))) /\ (exists ff_u_bpcpis_source_product ff_v_bpcpis_source_product. ((((exists ff_h_bpcpis_source_product_start. ff_h_bpcpis_source_product_start + S (1) = S ((S (0)) * ff_v_bpcpis_source_product)) /\ exists ff_q_bpcpis_source_product_start. ff_u_bpcpis_source_product = ff_q_bpcpis_source_product_start * S ((S (0)) * ff_v_bpcpis_source_product) + (1))) /\ ((((exists ff_h_bpcpis_source_product_terminal. ff_h_bpcpis_source_product_terminal + S (z) = S ((S (a + l)) * ff_v_bpcpis_source_product)) /\ exists ff_q_bpcpis_source_product_terminal. ff_u_bpcpis_source_product = ff_q_bpcpis_source_product_terminal * S ((S (a + l)) * ff_v_bpcpis_source_product) + (z))) /\ forall ff_i_bpcpis_source_product. (exists ff_lt_bpcpis_source_product_bound. ff_lt_bpcpis_source_product_bound + S ff_i_bpcpis_source_product = a + l) -> exists ff_p_bpcpis_source_product ff_r_bpcpis_source_product ff_s_bpcpis_source_product. ((((exists ff_h_bpcpis_source_product_factor. ff_h_bpcpis_source_product_factor + S (ff_p_bpcpis_source_product) = S ((S (ff_i_bpcpis_source_product)) * bpr_product_scale_bpcpis_source)) /\ exists ff_q_bpcpis_source_product_factor. bpr_product_code_bpcpis_source = ff_q_bpcpis_source_product_factor * S ((S (ff_i_bpcpis_source_product)) * bpr_product_scale_bpcpis_source) + (ff_p_bpcpis_source_product))) /\ ((((exists ff_h_bpcpis_source_product_partial. ff_h_bpcpis_source_product_partial + S (ff_r_bpcpis_source_product) = S ((S (ff_i_bpcpis_source_product)) * ff_v_bpcpis_source_product)) /\ exists ff_q_bpcpis_source_product_partial. ff_u_bpcpis_source_product = ff_q_bpcpis_source_product_partial * S ((S (ff_i_bpcpis_source_product)) * ff_v_bpcpis_source_product) + (ff_r_bpcpis_source_product))) /\ ((((exists ff_h_bpcpis_source_product_successor. ff_h_bpcpis_source_product_successor + S (ff_s_bpcpis_source_product) = S ((S (S ff_i_bpcpis_source_product)) * ff_v_bpcpis_source_product)) /\ exists ff_q_bpcpis_source_product_successor. ff_u_bpcpis_source_product = ff_q_bpcpis_source_product_successor * S ((S (S ff_i_bpcpis_source_product)) * ff_v_bpcpis_source_product) + (ff_s_bpcpis_source_product))) /\ ff_s_bpcpis_source_product = ff_r_bpcpis_source_product * ff_p_bpcpis_source_product)))))))) -> (exists x y. (exists bpr_product_code_bpcpis_prefix bpr_product_scale_bpcpis_prefix. ((forall bpr_prefix_index_bpcpis_prefix_prefix. (exists bpr_gap_bpcpis_prefix_prefix_bound. bpr_gap_bpcpis_prefix_prefix_bound + S (bpr_prefix_index_bpcpis_prefix_prefix) = a) -> exists bpr_prefix_value_bpcpis_prefix_prefix. ((((exists bpr_height_bpcpis_prefix_prefix_decoded. bpr_height_bpcpis_prefix_prefix_decoded + S (bpr_prefix_value_bpcpis_prefix_prefix) = S ((S (bpr_prefix_index_bpcpis_prefix_prefix)) * bpr_product_scale_bpcpis_prefix)) /\ exists bpr_quotient_bpcpis_prefix_prefix_decoded. bpr_product_code_bpcpis_prefix = bpr_quotient_bpcpis_prefix_prefix_decoded * S ((S (bpr_prefix_index_bpcpis_prefix_prefix)) * bpr_product_scale_bpcpis_prefix) + (bpr_prefix_value_bpcpis_prefix_prefix))) /\ (((((~(S (bpr_prefix_index_bpcpis_prefix_prefix) = 1) /\ forall bpr_left_bpcpis_prefix_prefix_choice_prime bpr_right_bpcpis_prefix_prefix_choice_prime. S (bpr_prefix_index_bpcpis_prefix_prefix) = bpr_left_bpcpis_prefix_prefix_choice_prime * bpr_right_bpcpis_prefix_prefix_choice_prime -> bpr_left_bpcpis_prefix_prefix_choice_prime = 1 \/ bpr_right_bpcpis_prefix_prefix_choice_prime = 1)) /\ exists bpr_choice_exponent_bpcpis_prefix_prefix_choice. ((((exists bpr_le_gap_bpcpis_prefix_prefix_choice_valuation_selected_bound. bpr_le_gap_bpcpis_prefix_prefix_choice_valuation_selected_bound + (bpr_choice_exponent_bpcpis_prefix_prefix_choice) = (n)) /\ (exists bpr_power_value_bpcpis_prefix_prefix_choice_valuation_selected. ((exists bpr_power_code_bpcpis_prefix_prefix_choice_valuation_selected_power bpr_power_scale_bpcpis_prefix_prefix_choice_valuation_selected_power. ((forall bpr_power_index_bpcpis_prefix_prefix_choice_valuation_selected_power. (exists bpr_gap_bpcpis_prefix_prefix_choice_valuation_selected_power_repeat_bound. bpr_gap_bpcpis_prefix_prefix_choice_valuation_selected_power_repeat_bound + S (bpr_power_index_bpcpis_prefix_prefix_choice_valuation_selected_power) = bpr_choice_exponent_bpcpis_prefix_prefix_choice) -> (((exists bpr_height_bpcpis_prefix_prefix_choice_valuation_selected_power_repeat_entry. bpr_height_bpcpis_prefix_prefix_choice_valuation_selected_power_repeat_entry + S (S (bpr_prefix_index_bpcpis_prefix_prefix)) = S ((S (bpr_power_index_bpcpis_prefix_prefix_choice_valuation_selected_power)) * bpr_power_scale_bpcpis_prefix_prefix_choice_valuation_selected_power)) /\ exists bpr_quotient_bpcpis_prefix_prefix_choice_valuation_selected_power_repeat_entry. bpr_power_code_bpcpis_prefix_prefix_choice_valuation_selected_power = bpr_quotient_bpcpis_prefix_prefix_choice_valuation_selected_power_repeat_entry * S ((S (bpr_power_index_bpcpis_prefix_prefix_choice_valuation_selected_power)) * bpr_power_scale_bpcpis_prefix_prefix_choice_valuation_selected_power) + (S (bpr_prefix_index_bpcpis_prefix_prefix))))) /\ (exists ff_u_bpcpis_prefix_prefix_choice_valuation_selected_power_product ff_v_bpcpis_prefix_prefix_choice_valuation_selected_power_product. ((((exists ff_h_bpcpis_prefix_prefix_choice_valuation_selected_power_product_start. ff_h_bpcpis_prefix_prefix_choice_valuation_selected_power_product_start + S (1) = S ((S (0)) * ff_v_bpcpis_prefix_prefix_choice_valuation_selected_power_product)) /\ exists ff_q_bpcpis_prefix_prefix_choice_valuation_selected_power_product_start. ff_u_bpcpis_prefix_prefix_choice_valuation_selected_power_product = ff_q_bpcpis_prefix_prefix_choice_valuation_selected_power_product_start * S ((S (0)) * ff_v_bpcpis_prefix_prefix_choice_valuation_selected_power_product) + (1))) /\ ((((exists ff_h_bpcpis_prefix_prefix_choice_valuation_selected_power_product_terminal. ff_h_bpcpis_prefix_prefix_choice_valuation_selected_power_product_terminal + S (bpr_power_value_bpcpis_prefix_prefix_choice_valuation_selected) = S ((S (bpr_choice_exponent_bpcpis_prefix_prefix_choice)) * ff_v_bpcpis_prefix_prefix_choice_valuation_selected_power_product)) /\ exists ff_q_bpcpis_prefix_prefix_choice_valuation_selected_power_product_terminal. ff_u_bpcpis_prefix_prefix_choice_valuation_selected_power_product = ff_q_bpcpis_prefix_prefix_choice_valuation_selected_power_product_terminal * S ((S (bpr_choice_exponent_bpcpis_prefix_prefix_choice)) * ff_v_bpcpis_prefix_prefix_choice_valuation_selected_power_product) + (bpr_power_value_bpcpis_prefix_prefix_choice_valuation_selected))) /\ forall ff_i_bpcpis_prefix_prefix_choice_valuation_selected_power_product. (exists ff_lt_bpcpis_prefix_prefix_choice_valuation_selected_power_product_bound. ff_lt_bpcpis_prefix_prefix_choice_valuation_selected_power_product_bound + S ff_i_bpcpis_prefix_prefix_choice_valuation_selected_power_product = bpr_choice_exponent_bpcpis_prefix_prefix_choice) -> exists ff_p_bpcpis_prefix_prefix_choice_valuation_selected_power_product ff_r_bpcpis_prefix_prefix_choice_valuation_selected_power_product ff_s_bpcpis_prefix_prefix_choice_valuation_selected_power_product. ((((exists ff_h_bpcpis_prefix_prefix_choice_valuation_selected_power_product_factor. ff_h_bpcpis_prefix_prefix_choice_valuation_selected_power_product_factor + S (ff_p_bpcpis_prefix_prefix_choice_valuation_selected_power_product) = S ((S (ff_i_bpcpis_prefix_prefix_choice_valuation_selected_power_product)) * bpr_power_scale_bpcpis_prefix_prefix_choice_valuation_selected_power)) /\ exists ff_q_bpcpis_prefix_prefix_choice_valuation_selected_power_product_factor. bpr_power_code_bpcpis_prefix_prefix_choice_valuation_selected_power = ff_q_bpcpis_prefix_prefix_choice_valuation_selected_power_product_factor * S ((S (ff_i_bpcpis_prefix_prefix_choice_valuation_selected_power_product)) * bpr_power_scale_bpcpis_prefix_prefix_choice_valuation_selected_power) + (ff_p_bpcpis_prefix_prefix_choice_valuation_selected_power_product))) /\ ((((exists ff_h_bpcpis_prefix_prefix_choice_valuation_selected_power_product_partial. ff_h_bpcpis_prefix_prefix_choice_valuation_selected_power_product_partial + S (ff_r_bpcpis_prefix_prefix_choice_valuation_selected_power_product) = S ((S (ff_i_bpcpis_prefix_prefix_choice_valuation_selected_power_product)) * ff_v_bpcpis_prefix_prefix_choice_valuation_selected_power_product)) /\ exists ff_q_bpcpis_prefix_prefix_choice_valuation_selected_power_product_partial. ff_u_bpcpis_prefix_prefix_choice_valuation_selected_power_product = ff_q_bpcpis_prefix_prefix_choice_valuation_selected_power_product_partial * S ((S (ff_i_bpcpis_prefix_prefix_choice_valuation_selected_power_product)) * ff_v_bpcpis_prefix_prefix_choice_valuation_selected_power_product) + (ff_r_bpcpis_prefix_prefix_choice_valuation_selected_power_product))) /\ ((((exists ff_h_bpcpis_prefix_prefix_choice_valuation_selected_power_product_successor. ff_h_bpcpis_prefix_prefix_choice_valuation_selected_power_product_successor + S (ff_s_bpcpis_prefix_prefix_choice_valuation_selected_power_product) = S ((S (S ff_i_bpcpis_prefix_prefix_choice_valuation_selected_power_product)) * ff_v_bpcpis_prefix_prefix_choice_valuation_selected_power_product)) /\ exists ff_q_bpcpis_prefix_prefix_choice_valuation_selected_power_product_successor. ff_u_bpcpis_prefix_prefix_choice_valuation_selected_power_product = ff_q_bpcpis_prefix_prefix_choice_valuation_selected_power_product_successor * S ((S (S ff_i_bpcpis_prefix_prefix_choice_valuation_selected_power_product)) * ff_v_bpcpis_prefix_prefix_choice_valuation_selected_power_product) + (ff_s_bpcpis_prefix_prefix_choice_valuation_selected_power_product))) /\ ff_s_bpcpis_prefix_prefix_choice_valuation_selected_power_product = ff_r_bpcpis_prefix_prefix_choice_valuation_selected_power_product * ff_p_bpcpis_prefix_prefix_choice_valuation_selected_power_product)))))))) /\ (exists bpr_divides_quotient_bpcpis_prefix_prefix_choice_valuation_selected_divides. n = (bpr_power_value_bpcpis_prefix_prefix_choice_valuation_selected) * bpr_divides_quotient_bpcpis_prefix_prefix_choice_valuation_selected_divides)))) /\ forall bpr_valuation_candidate_bpcpis_prefix_prefix_choice_valuation. (exists bpr_le_gap_bpcpis_prefix_prefix_choice_valuation_candidate_bound. bpr_le_gap_bpcpis_prefix_prefix_choice_valuation_candidate_bound + (bpr_valuation_candidate_bpcpis_prefix_prefix_choice_valuation) = (n)) -> (exists bpr_power_value_bpcpis_prefix_prefix_choice_valuation_candidate. ((exists bpr_power_code_bpcpis_prefix_prefix_choice_valuation_candidate_power bpr_power_scale_bpcpis_prefix_prefix_choice_valuation_candidate_power. ((forall bpr_power_index_bpcpis_prefix_prefix_choice_valuation_candidate_power. (exists bpr_gap_bpcpis_prefix_prefix_choice_valuation_candidate_power_repeat_bound. bpr_gap_bpcpis_prefix_prefix_choice_valuation_candidate_power_repeat_bound + S (bpr_power_index_bpcpis_prefix_prefix_choice_valuation_candidate_power) = bpr_valuation_candidate_bpcpis_prefix_prefix_choice_valuation) -> (((exists bpr_height_bpcpis_prefix_prefix_choice_valuation_candidate_power_repeat_entry. bpr_height_bpcpis_prefix_prefix_choice_valuation_candidate_power_repeat_entry + S (S (bpr_prefix_index_bpcpis_prefix_prefix)) = S ((S (bpr_power_index_bpcpis_prefix_prefix_choice_valuation_candidate_power)) * bpr_power_scale_bpcpis_prefix_prefix_choice_valuation_candidate_power)) /\ exists bpr_quotient_bpcpis_prefix_prefix_choice_valuation_candidate_power_repeat_entry. bpr_power_code_bpcpis_prefix_prefix_choice_valuation_candidate_power = bpr_quotient_bpcpis_prefix_prefix_choice_valuation_candidate_power_repeat_entry * S ((S (bpr_power_index_bpcpis_prefix_prefix_choice_valuation_candidate_power)) * bpr_power_scale_bpcpis_prefix_prefix_choice_valuation_candidate_power) + (S (bpr_prefix_index_bpcpis_prefix_prefix))))) /\ (exists ff_u_bpcpis_prefix_prefix_choice_valuation_candidate_power_product ff_v_bpcpis_prefix_prefix_choice_valuation_candidate_power_product. ((((exists ff_h_bpcpis_prefix_prefix_choice_valuation_candidate_power_product_start. ff_h_bpcpis_prefix_prefix_choice_valuation_candidate_power_product_start + S (1) = S ((S (0)) * ff_v_bpcpis_prefix_prefix_choice_valuation_candidate_power_product)) /\ exists ff_q_bpcpis_prefix_prefix_choice_valuation_candidate_power_product_start. ff_u_bpcpis_prefix_prefix_choice_valuation_candidate_power_product = ff_q_bpcpis_prefix_prefix_choice_valuation_candidate_power_product_start * S ((S (0)) * ff_v_bpcpis_prefix_prefix_choice_valuation_candidate_power_product) + (1))) /\ ((((exists ff_h_bpcpis_prefix_prefix_choice_valuation_candidate_power_product_terminal. ff_h_bpcpis_prefix_prefix_choice_valuation_candidate_power_product_terminal + S (bpr_power_value_bpcpis_prefix_prefix_choice_valuation_candidate) = S ((S (bpr_valuation_candidate_bpcpis_prefix_prefix_choice_valuation)) * ff_v_bpcpis_prefix_prefix_choice_valuation_candidate_power_product)) /\ exists ff_q_bpcpis_prefix_prefix_choice_valuation_candidate_power_product_terminal. ff_u_bpcpis_prefix_prefix_choice_valuation_candidate_power_product = ff_q_bpcpis_prefix_prefix_choice_valuation_candidate_power_product_terminal * S ((S (bpr_valuation_candidate_bpcpis_prefix_prefix_choice_valuation)) * ff_v_bpcpis_prefix_prefix_choice_valuation_candidate_power_product) + (bpr_power_value_bpcpis_prefix_prefix_choice_valuation_candidate))) /\ forall ff_i_bpcpis_prefix_prefix_choice_valuation_candidate_power_product. (exists ff_lt_bpcpis_prefix_prefix_choice_valuation_candidate_power_product_bound. ff_lt_bpcpis_prefix_prefix_choice_valuation_candidate_power_product_bound + S ff_i_bpcpis_prefix_prefix_choice_valuation_candidate_power_product = bpr_valuation_candidate_bpcpis_prefix_prefix_choice_valuation) -> exists ff_p_bpcpis_prefix_prefix_choice_valuation_candidate_power_product ff_r_bpcpis_prefix_prefix_choice_valuation_candidate_power_product ff_s_bpcpis_prefix_prefix_choice_valuation_candidate_power_product. ((((exists ff_h_bpcpis_prefix_prefix_choice_valuation_candidate_power_product_factor. ff_h_bpcpis_prefix_prefix_choice_valuation_candidate_power_product_factor + S (ff_p_bpcpis_prefix_prefix_choice_valuation_candidate_power_product) = S ((S (ff_i_bpcpis_prefix_prefix_choice_valuation_candidate_power_product)) * bpr_power_scale_bpcpis_prefix_prefix_choice_valuation_candidate_power)) /\ exists ff_q_bpcpis_prefix_prefix_choice_valuation_candidate_power_product_factor. bpr_power_code_bpcpis_prefix_prefix_choice_valuation_candidate_power = ff_q_bpcpis_prefix_prefix_choice_valuation_candidate_power_product_factor * S ((S (ff_i_bpcpis_prefix_prefix_choice_valuation_candidate_power_product)) * bpr_power_scale_bpcpis_prefix_prefix_choice_valuation_candidate_power) + (ff_p_bpcpis_prefix_prefix_choice_valuation_candidate_power_product))) /\ ((((exists ff_h_bpcpis_prefix_prefix_choice_valuation_candidate_power_product_partial. ff_h_bpcpis_prefix_prefix_choice_valuation_candidate_power_product_partial + S (ff_r_bpcpis_prefix_prefix_choice_valuation_candidate_power_product) = S ((S (ff_i_bpcpis_prefix_prefix_choice_valuation_candidate_power_product)) * ff_v_bpcpis_prefix_prefix_choice_valuation_candidate_power_product)) /\ exists ff_q_bpcpis_prefix_prefix_choice_valuation_candidate_power_product_partial. ff_u_bpcpis_prefix_prefix_choice_valuation_candidate_power_product = ff_q_bpcpis_prefix_prefix_choice_valuation_candidate_power_product_partial * S ((S (ff_i_bpcpis_prefix_prefix_choice_valuation_candidate_power_product)) * ff_v_bpcpis_prefix_prefix_choice_valuation_candidate_power_product) + (ff_r_bpcpis_prefix_prefix_choice_valuation_candidate_power_product))) /\ ((((exists ff_h_bpcpis_prefix_prefix_choice_valuation_candidate_power_product_successor. ff_h_bpcpis_prefix_prefix_choice_valuation_candidate_power_product_successor + S (ff_s_bpcpis_prefix_prefix_choice_valuation_candidate_power_product) = S ((S (S ff_i_bpcpis_prefix_prefix_choice_valuation_candidate_power_product)) * ff_v_bpcpis_prefix_prefix_choice_valuation_candidate_power_product)) /\ exists ff_q_bpcpis_prefix_prefix_choice_valuation_candidate_power_product_successor. ff_u_bpcpis_prefix_prefix_choice_valuation_candidate_power_product = ff_q_bpcpis_prefix_prefix_choice_valuation_candidate_power_product_successor * S ((S (S ff_i_bpcpis_prefix_prefix_choice_valuation_candidate_power_product)) * ff_v_bpcpis_prefix_prefix_choice_valuation_candidate_power_product) + (ff_s_bpcpis_prefix_prefix_choice_valuation_candidate_power_product))) /\ ff_s_bpcpis_prefix_prefix_choice_valuation_candidate_power_product = ff_r_bpcpis_prefix_prefix_choice_valuation_candidate_power_product * ff_p_bpcpis_prefix_prefix_choice_valuation_candidate_power_product)))))))) /\ (exists bpr_divides_quotient_bpcpis_prefix_prefix_choice_valuation_candidate_divides. n = (bpr_power_value_bpcpis_prefix_prefix_choice_valuation_candidate) * bpr_divides_quotient_bpcpis_prefix_prefix_choice_valuation_candidate_divides))) -> (exists bpr_le_gap_bpcpis_prefix_prefix_choice_valuation_candidate_below. bpr_le_gap_bpcpis_prefix_prefix_choice_valuation_candidate_below + (bpr_valuation_candidate_bpcpis_prefix_prefix_choice_valuation) = (bpr_choice_exponent_bpcpis_prefix_prefix_choice))) /\ (exists bpr_power_code_bpcpis_prefix_prefix_choice_power bpr_power_scale_bpcpis_prefix_prefix_choice_power. ((forall bpr_power_index_bpcpis_prefix_prefix_choice_power. (exists bpr_gap_bpcpis_prefix_prefix_choice_power_repeat_bound. bpr_gap_bpcpis_prefix_prefix_choice_power_repeat_bound + S (bpr_power_index_bpcpis_prefix_prefix_choice_power) = bpr_choice_exponent_bpcpis_prefix_prefix_choice) -> (((exists bpr_height_bpcpis_prefix_prefix_choice_power_repeat_entry. bpr_height_bpcpis_prefix_prefix_choice_power_repeat_entry + S (S (bpr_prefix_index_bpcpis_prefix_prefix)) = S ((S (bpr_power_index_bpcpis_prefix_prefix_choice_power)) * bpr_power_scale_bpcpis_prefix_prefix_choice_power)) /\ exists bpr_quotient_bpcpis_prefix_prefix_choice_power_repeat_entry. bpr_power_code_bpcpis_prefix_prefix_choice_power = bpr_quotient_bpcpis_prefix_prefix_choice_power_repeat_entry * S ((S (bpr_power_index_bpcpis_prefix_prefix_choice_power)) * bpr_power_scale_bpcpis_prefix_prefix_choice_power) + (S (bpr_prefix_index_bpcpis_prefix_prefix))))) /\ (exists ff_u_bpcpis_prefix_prefix_choice_power_product ff_v_bpcpis_prefix_prefix_choice_power_product. ((((exists ff_h_bpcpis_prefix_prefix_choice_power_product_start. ff_h_bpcpis_prefix_prefix_choice_power_product_start + S (1) = S ((S (0)) * ff_v_bpcpis_prefix_prefix_choice_power_product)) /\ exists ff_q_bpcpis_prefix_prefix_choice_power_product_start. ff_u_bpcpis_prefix_prefix_choice_power_product = ff_q_bpcpis_prefix_prefix_choice_power_product_start * S ((S (0)) * ff_v_bpcpis_prefix_prefix_choice_power_product) + (1))) /\ ((((exists ff_h_bpcpis_prefix_prefix_choice_power_product_terminal. ff_h_bpcpis_prefix_prefix_choice_power_product_terminal + S (bpr_prefix_value_bpcpis_prefix_prefix) = S ((S (bpr_choice_exponent_bpcpis_prefix_prefix_choice)) * ff_v_bpcpis_prefix_prefix_choice_power_product)) /\ exists ff_q_bpcpis_prefix_prefix_choice_power_product_terminal. ff_u_bpcpis_prefix_prefix_choice_power_product = ff_q_bpcpis_prefix_prefix_choice_power_product_terminal * S ((S (bpr_choice_exponent_bpcpis_prefix_prefix_choice)) * ff_v_bpcpis_prefix_prefix_choice_power_product) + (bpr_prefix_value_bpcpis_prefix_prefix))) /\ forall ff_i_bpcpis_prefix_prefix_choice_power_product. (exists ff_lt_bpcpis_prefix_prefix_choice_power_product_bound. ff_lt_bpcpis_prefix_prefix_choice_power_product_bound + S ff_i_bpcpis_prefix_prefix_choice_power_product = bpr_choice_exponent_bpcpis_prefix_prefix_choice) -> exists ff_p_bpcpis_prefix_prefix_choice_power_product ff_r_bpcpis_prefix_prefix_choice_power_product ff_s_bpcpis_prefix_prefix_choice_power_product. ((((exists ff_h_bpcpis_prefix_prefix_choice_power_product_factor. ff_h_bpcpis_prefix_prefix_choice_power_product_factor + S (ff_p_bpcpis_prefix_prefix_choice_power_product) = S ((S (ff_i_bpcpis_prefix_prefix_choice_power_product)) * bpr_power_scale_bpcpis_prefix_prefix_choice_power)) /\ exists ff_q_bpcpis_prefix_prefix_choice_power_product_factor. bpr_power_code_bpcpis_prefix_prefix_choice_power = ff_q_bpcpis_prefix_prefix_choice_power_product_factor * S ((S (ff_i_bpcpis_prefix_prefix_choice_power_product)) * bpr_power_scale_bpcpis_prefix_prefix_choice_power) + (ff_p_bpcpis_prefix_prefix_choice_power_product))) /\ ((((exists ff_h_bpcpis_prefix_prefix_choice_power_product_partial. ff_h_bpcpis_prefix_prefix_choice_power_product_partial + S (ff_r_bpcpis_prefix_prefix_choice_power_product) = S ((S (ff_i_bpcpis_prefix_prefix_choice_power_product)) * ff_v_bpcpis_prefix_prefix_choice_power_product)) /\ exists ff_q_bpcpis_prefix_prefix_choice_power_product_partial. ff_u_bpcpis_prefix_prefix_choice_power_product = ff_q_bpcpis_prefix_prefix_choice_power_product_partial * S ((S (ff_i_bpcpis_prefix_prefix_choice_power_product)) * ff_v_bpcpis_prefix_prefix_choice_power_product) + (ff_r_bpcpis_prefix_prefix_choice_power_product))) /\ ((((exists ff_h_bpcpis_prefix_prefix_choice_power_product_successor. ff_h_bpcpis_prefix_prefix_choice_power_product_successor + S (ff_s_bpcpis_prefix_prefix_choice_power_product) = S ((S (S ff_i_bpcpis_prefix_prefix_choice_power_product)) * ff_v_bpcpis_prefix_prefix_choice_power_product)) /\ exists ff_q_bpcpis_prefix_prefix_choice_power_product_successor. ff_u_bpcpis_prefix_prefix_choice_power_product = ff_q_bpcpis_prefix_prefix_choice_power_product_successor * S ((S (S ff_i_bpcpis_prefix_prefix_choice_power_product)) * ff_v_bpcpis_prefix_prefix_choice_power_product) + (ff_s_bpcpis_prefix_prefix_choice_power_product))) /\ ff_s_bpcpis_prefix_prefix_choice_power_product = ff_r_bpcpis_prefix_prefix_choice_power_product * ff_p_bpcpis_prefix_prefix_choice_power_product)))))))))) \/ (~((~(S (bpr_prefix_index_bpcpis_prefix_prefix) = 1) /\ forall bpr_left_bpcpis_prefix_prefix_choice_prime bpr_right_bpcpis_prefix_prefix_choice_prime. S (bpr_prefix_index_bpcpis_prefix_prefix) = bpr_left_bpcpis_prefix_prefix_choice_prime * bpr_right_bpcpis_prefix_prefix_choice_prime -> bpr_left_bpcpis_prefix_prefix_choice_prime = 1 \/ bpr_right_bpcpis_prefix_prefix_choice_prime = 1)) /\ bpr_prefix_value_bpcpis_prefix_prefix = 1))))) /\ (exists ff_u_bpcpis_prefix_product ff_v_bpcpis_prefix_product. ((((exists ff_h_bpcpis_prefix_product_start. ff_h_bpcpis_prefix_product_start + S (1) = S ((S (0)) * ff_v_bpcpis_prefix_product)) /\ exists ff_q_bpcpis_prefix_product_start. ff_u_bpcpis_prefix_product = ff_q_bpcpis_prefix_product_start * S ((S (0)) * ff_v_bpcpis_prefix_product) + (1))) /\ ((((exists ff_h_bpcpis_prefix_product_terminal. ff_h_bpcpis_prefix_product_terminal + S (x) = S ((S (a)) * ff_v_bpcpis_prefix_product)) /\ exists ff_q_bpcpis_prefix_product_terminal. ff_u_bpcpis_prefix_product = ff_q_bpcpis_prefix_product_terminal * S ((S (a)) * ff_v_bpcpis_prefix_product) + (x))) /\ forall ff_i_bpcpis_prefix_product. (exists ff_lt_bpcpis_prefix_product_bound. ff_lt_bpcpis_prefix_product_bound + S ff_i_bpcpis_prefix_product = a) -> exists ff_p_bpcpis_prefix_product ff_r_bpcpis_prefix_product ff_s_bpcpis_prefix_product. ((((exists ff_h_bpcpis_prefix_product_factor. ff_h_bpcpis_prefix_product_factor + S (ff_p_bpcpis_prefix_product) = S ((S (ff_i_bpcpis_prefix_product)) * bpr_product_scale_bpcpis_prefix)) /\ exists ff_q_bpcpis_prefix_product_factor. bpr_product_code_bpcpis_prefix = ff_q_bpcpis_prefix_product_factor * S ((S (ff_i_bpcpis_prefix_product)) * bpr_product_scale_bpcpis_prefix) + (ff_p_bpcpis_prefix_product))) /\ ((((exists ff_h_bpcpis_prefix_product_partial. ff_h_bpcpis_prefix_product_partial + S (ff_r_bpcpis_prefix_product) = S ((S (ff_i_bpcpis_prefix_product)) * ff_v_bpcpis_prefix_product)) /\ exists ff_q_bpcpis_prefix_product_partial. ff_u_bpcpis_prefix_product = ff_q_bpcpis_prefix_product_partial * S ((S (ff_i_bpcpis_prefix_product)) * ff_v_bpcpis_prefix_product) + (ff_r_bpcpis_prefix_product))) /\ ((((exists ff_h_bpcpis_prefix_product_successor. ff_h_bpcpis_prefix_product_successor + S (ff_s_bpcpis_prefix_product) = S ((S (S ff_i_bpcpis_prefix_product)) * ff_v_bpcpis_prefix_product)) /\ exists ff_q_bpcpis_prefix_product_successor. ff_u_bpcpis_prefix_product = ff_q_bpcpis_prefix_product_successor * S ((S (S ff_i_bpcpis_prefix_product)) * ff_v_bpcpis_prefix_product) + (ff_s_bpcpis_prefix_product))) /\ ff_s_bpcpis_prefix_product = ff_r_bpcpis_prefix_product * ff_p_bpcpis_prefix_product)))))))) /\ ((exists bpr_code_bpcpis_interval bpr_scale_bpcpis_interval. ((forall bpr_index_bpcpis_interval_prefix. (exists bpr_gap_bpcpis_interval_prefix_bound. bpr_gap_bpcpis_interval_prefix_bound + S (bpr_index_bpcpis_interval_prefix) = l) -> exists bpr_value_bpcpis_interval_prefix. ((((exists bpr_height_bpcpis_interval_prefix_decoded. bpr_height_bpcpis_interval_prefix_decoded + S (bpr_value_bpcpis_interval_prefix) = S ((S (bpr_index_bpcpis_interval_prefix)) * bpr_scale_bpcpis_interval)) /\ exists bpr_quotient_bpcpis_interval_prefix_decoded. bpr_code_bpcpis_interval = bpr_quotient_bpcpis_interval_prefix_decoded * S ((S (bpr_index_bpcpis_interval_prefix)) * bpr_scale_bpcpis_interval) + (bpr_value_bpcpis_interval_prefix))) /\ (((((~(S (a + bpr_index_bpcpis_interval_prefix) = 1) /\ forall bpr_left_bpcpis_interval_prefix_choice_prime bpr_right_bpcpis_interval_prefix_choice_prime. S (a + bpr_index_bpcpis_interval_prefix) = bpr_left_bpcpis_interval_prefix_choice_prime * bpr_right_bpcpis_interval_prefix_choice_prime -> bpr_left_bpcpis_interval_prefix_choice_prime = 1 \/ bpr_right_bpcpis_interval_prefix_choice_prime = 1)) /\ exists bpr_choice_exponent_bpcpis_interval_prefix_choice. ((((exists bpr_le_gap_bpcpis_interval_prefix_choice_valuation_selected_bound. bpr_le_gap_bpcpis_interval_prefix_choice_valuation_selected_bound + (bpr_choice_exponent_bpcpis_interval_prefix_choice) = (n)) /\ (exists bpr_power_value_bpcpis_interval_prefix_choice_valuation_selected. ((exists bpr_power_code_bpcpis_interval_prefix_choice_valuation_selected_power bpr_power_scale_bpcpis_interval_prefix_choice_valuation_selected_power. ((forall bpr_power_index_bpcpis_interval_prefix_choice_valuation_selected_power. (exists bpr_gap_bpcpis_interval_prefix_choice_valuation_selected_power_repeat_bound. bpr_gap_bpcpis_interval_prefix_choice_valuation_selected_power_repeat_bound + S (bpr_power_index_bpcpis_interval_prefix_choice_valuation_selected_power) = bpr_choice_exponent_bpcpis_interval_prefix_choice) -> (((exists bpr_height_bpcpis_interval_prefix_choice_valuation_selected_power_repeat_entry. bpr_height_bpcpis_interval_prefix_choice_valuation_selected_power_repeat_entry + S (S (a + bpr_index_bpcpis_interval_prefix)) = S ((S (bpr_power_index_bpcpis_interval_prefix_choice_valuation_selected_power)) * bpr_power_scale_bpcpis_interval_prefix_choice_valuation_selected_power)) /\ exists bpr_quotient_bpcpis_interval_prefix_choice_valuation_selected_power_repeat_entry. bpr_power_code_bpcpis_interval_prefix_choice_valuation_selected_power = bpr_quotient_bpcpis_interval_prefix_choice_valuation_selected_power_repeat_entry * S ((S (bpr_power_index_bpcpis_interval_prefix_choice_valuation_selected_power)) * bpr_power_scale_bpcpis_interval_prefix_choice_valuation_selected_power) + (S (a + bpr_index_bpcpis_interval_prefix))))) /\ (exists ff_u_bpcpis_interval_prefix_choice_valuation_selected_power_product ff_v_bpcpis_interval_prefix_choice_valuation_selected_power_product. ((((exists ff_h_bpcpis_interval_prefix_choice_valuation_selected_power_product_start. ff_h_bpcpis_interval_prefix_choice_valuation_selected_power_product_start + S (1) = S ((S (0)) * ff_v_bpcpis_interval_prefix_choice_valuation_selected_power_product)) /\ exists ff_q_bpcpis_interval_prefix_choice_valuation_selected_power_product_start. ff_u_bpcpis_interval_prefix_choice_valuation_selected_power_product = ff_q_bpcpis_interval_prefix_choice_valuation_selected_power_product_start * S ((S (0)) * ff_v_bpcpis_interval_prefix_choice_valuation_selected_power_product) + (1))) /\ ((((exists ff_h_bpcpis_interval_prefix_choice_valuation_selected_power_product_terminal. ff_h_bpcpis_interval_prefix_choice_valuation_selected_power_product_terminal + S (bpr_power_value_bpcpis_interval_prefix_choice_valuation_selected) = S ((S (bpr_choice_exponent_bpcpis_interval_prefix_choice)) * ff_v_bpcpis_interval_prefix_choice_valuation_selected_power_product)) /\ exists ff_q_bpcpis_interval_prefix_choice_valuation_selected_power_product_terminal. ff_u_bpcpis_interval_prefix_choice_valuation_selected_power_product = ff_q_bpcpis_interval_prefix_choice_valuation_selected_power_product_terminal * S ((S (bpr_choice_exponent_bpcpis_interval_prefix_choice)) * ff_v_bpcpis_interval_prefix_choice_valuation_selected_power_product) + (bpr_power_value_bpcpis_interval_prefix_choice_valuation_selected))) /\ forall ff_i_bpcpis_interval_prefix_choice_valuation_selected_power_product. (exists ff_lt_bpcpis_interval_prefix_choice_valuation_selected_power_product_bound. ff_lt_bpcpis_interval_prefix_choice_valuation_selected_power_product_bound + S ff_i_bpcpis_interval_prefix_choice_valuation_selected_power_product = bpr_choice_exponent_bpcpis_interval_prefix_choice) -> exists ff_p_bpcpis_interval_prefix_choice_valuation_selected_power_product ff_r_bpcpis_interval_prefix_choice_valuation_selected_power_product ff_s_bpcpis_interval_prefix_choice_valuation_selected_power_product. ((((exists ff_h_bpcpis_interval_prefix_choice_valuation_selected_power_product_factor. ff_h_bpcpis_interval_prefix_choice_valuation_selected_power_product_factor + S (ff_p_bpcpis_interval_prefix_choice_valuation_selected_power_product) = S ((S (ff_i_bpcpis_interval_prefix_choice_valuation_selected_power_product)) * bpr_power_scale_bpcpis_interval_prefix_choice_valuation_selected_power)) /\ exists ff_q_bpcpis_interval_prefix_choice_valuation_selected_power_product_factor. bpr_power_code_bpcpis_interval_prefix_choice_valuation_selected_power = ff_q_bpcpis_interval_prefix_choice_valuation_selected_power_product_factor * S ((S (ff_i_bpcpis_interval_prefix_choice_valuation_selected_power_product)) * bpr_power_scale_bpcpis_interval_prefix_choice_valuation_selected_power) + (ff_p_bpcpis_interval_prefix_choice_valuation_selected_power_product))) /\ ((((exists ff_h_bpcpis_interval_prefix_choice_valuation_selected_power_product_partial. ff_h_bpcpis_interval_prefix_choice_valuation_selected_power_product_partial + S (ff_r_bpcpis_interval_prefix_choice_valuation_selected_power_product) = S ((S (ff_i_bpcpis_interval_prefix_choice_valuation_selected_power_product)) * ff_v_bpcpis_interval_prefix_choice_valuation_selected_power_product)) /\ exists ff_q_bpcpis_interval_prefix_choice_valuation_selected_power_product_partial. ff_u_bpcpis_interval_prefix_choice_valuation_selected_power_product = ff_q_bpcpis_interval_prefix_choice_valuation_selected_power_product_partial * S ((S (ff_i_bpcpis_interval_prefix_choice_valuation_selected_power_product)) * ff_v_bpcpis_interval_prefix_choice_valuation_selected_power_product) + (ff_r_bpcpis_interval_prefix_choice_valuation_selected_power_product))) /\ ((((exists ff_h_bpcpis_interval_prefix_choice_valuation_selected_power_product_successor. ff_h_bpcpis_interval_prefix_choice_valuation_selected_power_product_successor + S (ff_s_bpcpis_interval_prefix_choice_valuation_selected_power_product) = S ((S (S ff_i_bpcpis_interval_prefix_choice_valuation_selected_power_product)) * ff_v_bpcpis_interval_prefix_choice_valuation_selected_power_product)) /\ exists ff_q_bpcpis_interval_prefix_choice_valuation_selected_power_product_successor. ff_u_bpcpis_interval_prefix_choice_valuation_selected_power_product = ff_q_bpcpis_interval_prefix_choice_valuation_selected_power_product_successor * S ((S (S ff_i_bpcpis_interval_prefix_choice_valuation_selected_power_product)) * ff_v_bpcpis_interval_prefix_choice_valuation_selected_power_product) + (ff_s_bpcpis_interval_prefix_choice_valuation_selected_power_product))) /\ ff_s_bpcpis_interval_prefix_choice_valuation_selected_power_product = ff_r_bpcpis_interval_prefix_choice_valuation_selected_power_product * ff_p_bpcpis_interval_prefix_choice_valuation_selected_power_product)))))))) /\ (exists bpr_divides_quotient_bpcpis_interval_prefix_choice_valuation_selected_divides. n = (bpr_power_value_bpcpis_interval_prefix_choice_valuation_selected) * bpr_divides_quotient_bpcpis_interval_prefix_choice_valuation_selected_divides)))) /\ forall bpr_valuation_candidate_bpcpis_interval_prefix_choice_valuation. (exists bpr_le_gap_bpcpis_interval_prefix_choice_valuation_candidate_bound. bpr_le_gap_bpcpis_interval_prefix_choice_valuation_candidate_bound + (bpr_valuation_candidate_bpcpis_interval_prefix_choice_valuation) = (n)) -> (exists bpr_power_value_bpcpis_interval_prefix_choice_valuation_candidate. ((exists bpr_power_code_bpcpis_interval_prefix_choice_valuation_candidate_power bpr_power_scale_bpcpis_interval_prefix_choice_valuation_candidate_power. ((forall bpr_power_index_bpcpis_interval_prefix_choice_valuation_candidate_power. (exists bpr_gap_bpcpis_interval_prefix_choice_valuation_candidate_power_repeat_bound. bpr_gap_bpcpis_interval_prefix_choice_valuation_candidate_power_repeat_bound + S (bpr_power_index_bpcpis_interval_prefix_choice_valuation_candidate_power) = bpr_valuation_candidate_bpcpis_interval_prefix_choice_valuation) -> (((exists bpr_height_bpcpis_interval_prefix_choice_valuation_candidate_power_repeat_entry. bpr_height_bpcpis_interval_prefix_choice_valuation_candidate_power_repeat_entry + S (S (a + bpr_index_bpcpis_interval_prefix)) = S ((S (bpr_power_index_bpcpis_interval_prefix_choice_valuation_candidate_power)) * bpr_power_scale_bpcpis_interval_prefix_choice_valuation_candidate_power)) /\ exists bpr_quotient_bpcpis_interval_prefix_choice_valuation_candidate_power_repeat_entry. bpr_power_code_bpcpis_interval_prefix_choice_valuation_candidate_power = bpr_quotient_bpcpis_interval_prefix_choice_valuation_candidate_power_repeat_entry * S ((S (bpr_power_index_bpcpis_interval_prefix_choice_valuation_candidate_power)) * bpr_power_scale_bpcpis_interval_prefix_choice_valuation_candidate_power) + (S (a + bpr_index_bpcpis_interval_prefix))))) /\ (exists ff_u_bpcpis_interval_prefix_choice_valuation_candidate_power_product ff_v_bpcpis_interval_prefix_choice_valuation_candidate_power_product. ((((exists ff_h_bpcpis_interval_prefix_choice_valuation_candidate_power_product_start. ff_h_bpcpis_interval_prefix_choice_valuation_candidate_power_product_start + S (1) = S ((S (0)) * ff_v_bpcpis_interval_prefix_choice_valuation_candidate_power_product)) /\ exists ff_q_bpcpis_interval_prefix_choice_valuation_candidate_power_product_start. ff_u_bpcpis_interval_prefix_choice_valuation_candidate_power_product = ff_q_bpcpis_interval_prefix_choice_valuation_candidate_power_product_start * S ((S (0)) * ff_v_bpcpis_interval_prefix_choice_valuation_candidate_power_product) + (1))) /\ ((((exists ff_h_bpcpis_interval_prefix_choice_valuation_candidate_power_product_terminal. ff_h_bpcpis_interval_prefix_choice_valuation_candidate_power_product_terminal + S (bpr_power_value_bpcpis_interval_prefix_choice_valuation_candidate) = S ((S (bpr_valuation_candidate_bpcpis_interval_prefix_choice_valuation)) * ff_v_bpcpis_interval_prefix_choice_valuation_candidate_power_product)) /\ exists ff_q_bpcpis_interval_prefix_choice_valuation_candidate_power_product_terminal. ff_u_bpcpis_interval_prefix_choice_valuation_candidate_power_product = ff_q_bpcpis_interval_prefix_choice_valuation_candidate_power_product_terminal * S ((S (bpr_valuation_candidate_bpcpis_interval_prefix_choice_valuation)) * ff_v_bpcpis_interval_prefix_choice_valuation_candidate_power_product) + (bpr_power_value_bpcpis_interval_prefix_choice_valuation_candidate))) /\ forall ff_i_bpcpis_interval_prefix_choice_valuation_candidate_power_product. (exists ff_lt_bpcpis_interval_prefix_choice_valuation_candidate_power_product_bound. ff_lt_bpcpis_interval_prefix_choice_valuation_candidate_power_product_bound + S ff_i_bpcpis_interval_prefix_choice_valuation_candidate_power_product = bpr_valuation_candidate_bpcpis_interval_prefix_choice_valuation) -> exists ff_p_bpcpis_interval_prefix_choice_valuation_candidate_power_product ff_r_bpcpis_interval_prefix_choice_valuation_candidate_power_product ff_s_bpcpis_interval_prefix_choice_valuation_candidate_power_product. ((((exists ff_h_bpcpis_interval_prefix_choice_valuation_candidate_power_product_factor. ff_h_bpcpis_interval_prefix_choice_valuation_candidate_power_product_factor + S (ff_p_bpcpis_interval_prefix_choice_valuation_candidate_power_product) = S ((S (ff_i_bpcpis_interval_prefix_choice_valuation_candidate_power_product)) * bpr_power_scale_bpcpis_interval_prefix_choice_valuation_candidate_power)) /\ exists ff_q_bpcpis_interval_prefix_choice_valuation_candidate_power_product_factor. bpr_power_code_bpcpis_interval_prefix_choice_valuation_candidate_power = ff_q_bpcpis_interval_prefix_choice_valuation_candidate_power_product_factor * S ((S (ff_i_bpcpis_interval_prefix_choice_valuation_candidate_power_product)) * bpr_power_scale_bpcpis_interval_prefix_choice_valuation_candidate_power) + (ff_p_bpcpis_interval_prefix_choice_valuation_candidate_power_product))) /\ ((((exists ff_h_bpcpis_interval_prefix_choice_valuation_candidate_power_product_partial. ff_h_bpcpis_interval_prefix_choice_valuation_candidate_power_product_partial + S (ff_r_bpcpis_interval_prefix_choice_valuation_candidate_power_product) = S ((S (ff_i_bpcpis_interval_prefix_choice_valuation_candidate_power_product)) * ff_v_bpcpis_interval_prefix_choice_valuation_candidate_power_product)) /\ exists ff_q_bpcpis_interval_prefix_choice_valuation_candidate_power_product_partial. ff_u_bpcpis_interval_prefix_choice_valuation_candidate_power_product = ff_q_bpcpis_interval_prefix_choice_valuation_candidate_power_product_partial * S ((S (ff_i_bpcpis_interval_prefix_choice_valuation_candidate_power_product)) * ff_v_bpcpis_interval_prefix_choice_valuation_candidate_power_product) + (ff_r_bpcpis_interval_prefix_choice_valuation_candidate_power_product))) /\ ((((exists ff_h_bpcpis_interval_prefix_choice_valuation_candidate_power_product_successor. ff_h_bpcpis_interval_prefix_choice_valuation_candidate_power_product_successor + S (ff_s_bpcpis_interval_prefix_choice_valuation_candidate_power_product) = S ((S (S ff_i_bpcpis_interval_prefix_choice_valuation_candidate_power_product)) * ff_v_bpcpis_interval_prefix_choice_valuation_candidate_power_product)) /\ exists ff_q_bpcpis_interval_prefix_choice_valuation_candidate_power_product_successor. ff_u_bpcpis_interval_prefix_choice_valuation_candidate_power_product = ff_q_bpcpis_interval_prefix_choice_valuation_candidate_power_product_successor * S ((S (S ff_i_bpcpis_interval_prefix_choice_valuation_candidate_power_product)) * ff_v_bpcpis_interval_prefix_choice_valuation_candidate_power_product) + (ff_s_bpcpis_interval_prefix_choice_valuation_candidate_power_product))) /\ ff_s_bpcpis_interval_prefix_choice_valuation_candidate_power_product = ff_r_bpcpis_interval_prefix_choice_valuation_candidate_power_product * ff_p_bpcpis_interval_prefix_choice_valuation_candidate_power_product)))))))) /\ (exists bpr_divides_quotient_bpcpis_interval_prefix_choice_valuation_candidate_divides. n = (bpr_power_value_bpcpis_interval_prefix_choice_valuation_candidate) * bpr_divides_quotient_bpcpis_interval_prefix_choice_valuation_candidate_divides))) -> (exists bpr_le_gap_bpcpis_interval_prefix_choice_valuation_candidate_below. bpr_le_gap_bpcpis_interval_prefix_choice_valuation_candidate_below + (bpr_valuation_candidate_bpcpis_interval_prefix_choice_valuation) = (bpr_choice_exponent_bpcpis_interval_prefix_choice))) /\ (exists bpr_power_code_bpcpis_interval_prefix_choice_power bpr_power_scale_bpcpis_interval_prefix_choice_power. ((forall bpr_power_index_bpcpis_interval_prefix_choice_power. (exists bpr_gap_bpcpis_interval_prefix_choice_power_repeat_bound. bpr_gap_bpcpis_interval_prefix_choice_power_repeat_bound + S (bpr_power_index_bpcpis_interval_prefix_choice_power) = bpr_choice_exponent_bpcpis_interval_prefix_choice) -> (((exists bpr_height_bpcpis_interval_prefix_choice_power_repeat_entry. bpr_height_bpcpis_interval_prefix_choice_power_repeat_entry + S (S (a + bpr_index_bpcpis_interval_prefix)) = S ((S (bpr_power_index_bpcpis_interval_prefix_choice_power)) * bpr_power_scale_bpcpis_interval_prefix_choice_power)) /\ exists bpr_quotient_bpcpis_interval_prefix_choice_power_repeat_entry. bpr_power_code_bpcpis_interval_prefix_choice_power = bpr_quotient_bpcpis_interval_prefix_choice_power_repeat_entry * S ((S (bpr_power_index_bpcpis_interval_prefix_choice_power)) * bpr_power_scale_bpcpis_interval_prefix_choice_power) + (S (a + bpr_index_bpcpis_interval_prefix))))) /\ (exists ff_u_bpcpis_interval_prefix_choice_power_product ff_v_bpcpis_interval_prefix_choice_power_product. ((((exists ff_h_bpcpis_interval_prefix_choice_power_product_start. ff_h_bpcpis_interval_prefix_choice_power_product_start + S (1) = S ((S (0)) * ff_v_bpcpis_interval_prefix_choice_power_product)) /\ exists ff_q_bpcpis_interval_prefix_choice_power_product_start. ff_u_bpcpis_interval_prefix_choice_power_product = ff_q_bpcpis_interval_prefix_choice_power_product_start * S ((S (0)) * ff_v_bpcpis_interval_prefix_choice_power_product) + (1))) /\ ((((exists ff_h_bpcpis_interval_prefix_choice_power_product_terminal. ff_h_bpcpis_interval_prefix_choice_power_product_terminal + S (bpr_value_bpcpis_interval_prefix) = S ((S (bpr_choice_exponent_bpcpis_interval_prefix_choice)) * ff_v_bpcpis_interval_prefix_choice_power_product)) /\ exists ff_q_bpcpis_interval_prefix_choice_power_product_terminal. ff_u_bpcpis_interval_prefix_choice_power_product = ff_q_bpcpis_interval_prefix_choice_power_product_terminal * S ((S (bpr_choice_exponent_bpcpis_interval_prefix_choice)) * ff_v_bpcpis_interval_prefix_choice_power_product) + (bpr_value_bpcpis_interval_prefix))) /\ forall ff_i_bpcpis_interval_prefix_choice_power_product. (exists ff_lt_bpcpis_interval_prefix_choice_power_product_bound. ff_lt_bpcpis_interval_prefix_choice_power_product_bound + S ff_i_bpcpis_interval_prefix_choice_power_product = bpr_choice_exponent_bpcpis_interval_prefix_choice) -> exists ff_p_bpcpis_interval_prefix_choice_power_product ff_r_bpcpis_interval_prefix_choice_power_product ff_s_bpcpis_interval_prefix_choice_power_product. ((((exists ff_h_bpcpis_interval_prefix_choice_power_product_factor. ff_h_bpcpis_interval_prefix_choice_power_product_factor + S (ff_p_bpcpis_interval_prefix_choice_power_product) = S ((S (ff_i_bpcpis_interval_prefix_choice_power_product)) * bpr_power_scale_bpcpis_interval_prefix_choice_power)) /\ exists ff_q_bpcpis_interval_prefix_choice_power_product_factor. bpr_power_code_bpcpis_interval_prefix_choice_power = ff_q_bpcpis_interval_prefix_choice_power_product_factor * S ((S (ff_i_bpcpis_interval_prefix_choice_power_product)) * bpr_power_scale_bpcpis_interval_prefix_choice_power) + (ff_p_bpcpis_interval_prefix_choice_power_product))) /\ ((((exists ff_h_bpcpis_interval_prefix_choice_power_product_partial. ff_h_bpcpis_interval_prefix_choice_power_product_partial + S (ff_r_bpcpis_interval_prefix_choice_power_product) = S ((S (ff_i_bpcpis_interval_prefix_choice_power_product)) * ff_v_bpcpis_interval_prefix_choice_power_product)) /\ exists ff_q_bpcpis_interval_prefix_choice_power_product_partial. ff_u_bpcpis_interval_prefix_choice_power_product = ff_q_bpcpis_interval_prefix_choice_power_product_partial * S ((S (ff_i_bpcpis_interval_prefix_choice_power_product)) * ff_v_bpcpis_interval_prefix_choice_power_product) + (ff_r_bpcpis_interval_prefix_choice_power_product))) /\ ((((exists ff_h_bpcpis_interval_prefix_choice_power_product_successor. ff_h_bpcpis_interval_prefix_choice_power_product_successor + S (ff_s_bpcpis_interval_prefix_choice_power_product) = S ((S (S ff_i_bpcpis_interval_prefix_choice_power_product)) * ff_v_bpcpis_interval_prefix_choice_power_product)) /\ exists ff_q_bpcpis_interval_prefix_choice_power_product_successor. ff_u_bpcpis_interval_prefix_choice_power_product = ff_q_bpcpis_interval_prefix_choice_power_product_successor * S ((S (S ff_i_bpcpis_interval_prefix_choice_power_product)) * ff_v_bpcpis_interval_prefix_choice_power_product) + (ff_s_bpcpis_interval_prefix_choice_power_product))) /\ ff_s_bpcpis_interval_prefix_choice_power_product = ff_r_bpcpis_interval_prefix_choice_power_product * ff_p_bpcpis_interval_prefix_choice_power_product)))))))))) \/ (~((~(S (a + bpr_index_bpcpis_interval_prefix) = 1) /\ forall bpr_left_bpcpis_interval_prefix_choice_prime bpr_right_bpcpis_interval_prefix_choice_prime. S (a + bpr_index_bpcpis_interval_prefix) = bpr_left_bpcpis_interval_prefix_choice_prime * bpr_right_bpcpis_interval_prefix_choice_prime -> bpr_left_bpcpis_interval_prefix_choice_prime = 1 \/ bpr_right_bpcpis_interval_prefix_choice_prime = 1)) /\ bpr_value_bpcpis_interval_prefix = 1))))) /\ (exists ff_u_bpcpis_interval_product ff_v_bpcpis_interval_product. ((((exists ff_h_bpcpis_interval_product_start. ff_h_bpcpis_interval_product_start + S (1) = S ((S (0)) * ff_v_bpcpis_interval_product)) /\ exists ff_q_bpcpis_interval_product_start. ff_u_bpcpis_interval_product = ff_q_bpcpis_interval_product_start * S ((S (0)) * ff_v_bpcpis_interval_product) + (1))) /\ ((((exists ff_h_bpcpis_interval_product_terminal. ff_h_bpcpis_interval_product_terminal + S (y) = S ((S (l)) * ff_v_bpcpis_interval_product)) /\ exists ff_q_bpcpis_interval_product_terminal. ff_u_bpcpis_interval_product = ff_q_bpcpis_interval_product_terminal * S ((S (l)) * ff_v_bpcpis_interval_product) + (y))) /\ forall ff_i_bpcpis_interval_product. (exists ff_lt_bpcpis_interval_product_bound. ff_lt_bpcpis_interval_product_bound + S ff_i_bpcpis_interval_product = l) -> exists ff_p_bpcpis_interval_product ff_r_bpcpis_interval_product ff_s_bpcpis_interval_product. ((((exists ff_h_bpcpis_interval_product_factor. ff_h_bpcpis_interval_product_factor + S (ff_p_bpcpis_interval_product) = S ((S (ff_i_bpcpis_interval_product)) * bpr_scale_bpcpis_interval)) /\ exists ff_q_bpcpis_interval_product_factor. bpr_code_bpcpis_interval = ff_q_bpcpis_interval_product_factor * S ((S (ff_i_bpcpis_interval_product)) * bpr_scale_bpcpis_interval) + (ff_p_bpcpis_interval_product))) /\ ((((exists ff_h_bpcpis_interval_product_partial. ff_h_bpcpis_interval_product_partial + S (ff_r_bpcpis_interval_product) = S ((S (ff_i_bpcpis_interval_product)) * ff_v_bpcpis_interval_product)) /\ exists ff_q_bpcpis_interval_product_partial. ff_u_bpcpis_interval_product = ff_q_bpcpis_interval_product_partial * S ((S (ff_i_bpcpis_interval_product)) * ff_v_bpcpis_interval_product) + (ff_r_bpcpis_interval_product))) /\ ((((exists ff_h_bpcpis_interval_product_successor. ff_h_bpcpis_interval_product_successor + S (ff_s_bpcpis_interval_product) = S ((S (S ff_i_bpcpis_interval_product)) * ff_v_bpcpis_interval_product)) /\ exists ff_q_bpcpis_interval_product_successor. ff_u_bpcpis_interval_product = ff_q_bpcpis_interval_product_successor * S ((S (S ff_i_bpcpis_interval_product)) * ff_v_bpcpis_interval_product) + (ff_s_bpcpis_interval_product))) /\ ff_s_bpcpis_interval_product = ff_r_bpcpis_interval_product * ff_p_bpcpis_interval_product)))))))) /\ z = x * y))