prime_contribution_prefix_restrict_add

Exact expanded PA statement

forall n a b c l. (forall bpr_prefix_index_bpcpra_source. (exists bpr_gap_bpcpra_source_bound. bpr_gap_bpcpra_source_bound + S (bpr_prefix_index_bpcpra_source) = a + l) -> exists bpr_prefix_value_bpcpra_source. ((((exists bpr_height_bpcpra_source_decoded. bpr_height_bpcpra_source_decoded + S (bpr_prefix_value_bpcpra_source) = S ((S (bpr_prefix_index_bpcpra_source)) * c)) /\ exists bpr_quotient_bpcpra_source_decoded. b = bpr_quotient_bpcpra_source_decoded * S ((S (bpr_prefix_index_bpcpra_source)) * c) + (bpr_prefix_value_bpcpra_source))) /\ (((((~(S (bpr_prefix_index_bpcpra_source) = 1) /\ forall bpr_left_bpcpra_source_choice_prime bpr_right_bpcpra_source_choice_prime. S (bpr_prefix_index_bpcpra_source) = bpr_left_bpcpra_source_choice_prime * bpr_right_bpcpra_source_choice_prime -> bpr_left_bpcpra_source_choice_prime = 1 \/ bpr_right_bpcpra_source_choice_prime = 1)) /\ exists bpr_choice_exponent_bpcpra_source_choice. ((((exists bpr_le_gap_bpcpra_source_choice_valuation_selected_bound. bpr_le_gap_bpcpra_source_choice_valuation_selected_bound + (bpr_choice_exponent_bpcpra_source_choice) = (n)) /\ (exists bpr_power_value_bpcpra_source_choice_valuation_selected. ((exists bpr_power_code_bpcpra_source_choice_valuation_selected_power bpr_power_scale_bpcpra_source_choice_valuation_selected_power. ((forall bpr_power_index_bpcpra_source_choice_valuation_selected_power. (exists bpr_gap_bpcpra_source_choice_valuation_selected_power_repeat_bound. bpr_gap_bpcpra_source_choice_valuation_selected_power_repeat_bound + S (bpr_power_index_bpcpra_source_choice_valuation_selected_power) = bpr_choice_exponent_bpcpra_source_choice) -> (((exists bpr_height_bpcpra_source_choice_valuation_selected_power_repeat_entry. bpr_height_bpcpra_source_choice_valuation_selected_power_repeat_entry + S (S (bpr_prefix_index_bpcpra_source)) = S ((S (bpr_power_index_bpcpra_source_choice_valuation_selected_power)) * bpr_power_scale_bpcpra_source_choice_valuation_selected_power)) /\ exists bpr_quotient_bpcpra_source_choice_valuation_selected_power_repeat_entry. bpr_power_code_bpcpra_source_choice_valuation_selected_power = bpr_quotient_bpcpra_source_choice_valuation_selected_power_repeat_entry * S ((S (bpr_power_index_bpcpra_source_choice_valuation_selected_power)) * bpr_power_scale_bpcpra_source_choice_valuation_selected_power) + (S (bpr_prefix_index_bpcpra_source))))) /\ (exists ff_u_bpcpra_source_choice_valuation_selected_power_product ff_v_bpcpra_source_choice_valuation_selected_power_product. ((((exists ff_h_bpcpra_source_choice_valuation_selected_power_product_start. ff_h_bpcpra_source_choice_valuation_selected_power_product_start + S (1) = S ((S (0)) * ff_v_bpcpra_source_choice_valuation_selected_power_product)) /\ exists ff_q_bpcpra_source_choice_valuation_selected_power_product_start. ff_u_bpcpra_source_choice_valuation_selected_power_product = ff_q_bpcpra_source_choice_valuation_selected_power_product_start * S ((S (0)) * ff_v_bpcpra_source_choice_valuation_selected_power_product) + (1))) /\ ((((exists ff_h_bpcpra_source_choice_valuation_selected_power_product_terminal. ff_h_bpcpra_source_choice_valuation_selected_power_product_terminal + S (bpr_power_value_bpcpra_source_choice_valuation_selected) = S ((S (bpr_choice_exponent_bpcpra_source_choice)) * ff_v_bpcpra_source_choice_valuation_selected_power_product)) /\ exists ff_q_bpcpra_source_choice_valuation_selected_power_product_terminal. ff_u_bpcpra_source_choice_valuation_selected_power_product = ff_q_bpcpra_source_choice_valuation_selected_power_product_terminal * S ((S (bpr_choice_exponent_bpcpra_source_choice)) * ff_v_bpcpra_source_choice_valuation_selected_power_product) + (bpr_power_value_bpcpra_source_choice_valuation_selected))) /\ forall ff_i_bpcpra_source_choice_valuation_selected_power_product. (exists ff_lt_bpcpra_source_choice_valuation_selected_power_product_bound. ff_lt_bpcpra_source_choice_valuation_selected_power_product_bound + S ff_i_bpcpra_source_choice_valuation_selected_power_product = bpr_choice_exponent_bpcpra_source_choice) -> exists ff_p_bpcpra_source_choice_valuation_selected_power_product ff_r_bpcpra_source_choice_valuation_selected_power_product ff_s_bpcpra_source_choice_valuation_selected_power_product. ((((exists ff_h_bpcpra_source_choice_valuation_selected_power_product_factor. ff_h_bpcpra_source_choice_valuation_selected_power_product_factor + S (ff_p_bpcpra_source_choice_valuation_selected_power_product) = S ((S (ff_i_bpcpra_source_choice_valuation_selected_power_product)) * bpr_power_scale_bpcpra_source_choice_valuation_selected_power)) /\ exists ff_q_bpcpra_source_choice_valuation_selected_power_product_factor. bpr_power_code_bpcpra_source_choice_valuation_selected_power = ff_q_bpcpra_source_choice_valuation_selected_power_product_factor * S ((S (ff_i_bpcpra_source_choice_valuation_selected_power_product)) * bpr_power_scale_bpcpra_source_choice_valuation_selected_power) + (ff_p_bpcpra_source_choice_valuation_selected_power_product))) /\ ((((exists ff_h_bpcpra_source_choice_valuation_selected_power_product_partial. ff_h_bpcpra_source_choice_valuation_selected_power_product_partial + S (ff_r_bpcpra_source_choice_valuation_selected_power_product) = S ((S (ff_i_bpcpra_source_choice_valuation_selected_power_product)) * ff_v_bpcpra_source_choice_valuation_selected_power_product)) /\ exists ff_q_bpcpra_source_choice_valuation_selected_power_product_partial. ff_u_bpcpra_source_choice_valuation_selected_power_product = ff_q_bpcpra_source_choice_valuation_selected_power_product_partial * S ((S (ff_i_bpcpra_source_choice_valuation_selected_power_product)) * ff_v_bpcpra_source_choice_valuation_selected_power_product) + (ff_r_bpcpra_source_choice_valuation_selected_power_product))) /\ ((((exists ff_h_bpcpra_source_choice_valuation_selected_power_product_successor. ff_h_bpcpra_source_choice_valuation_selected_power_product_successor + S (ff_s_bpcpra_source_choice_valuation_selected_power_product) = S ((S (S ff_i_bpcpra_source_choice_valuation_selected_power_product)) * ff_v_bpcpra_source_choice_valuation_selected_power_product)) /\ exists ff_q_bpcpra_source_choice_valuation_selected_power_product_successor. ff_u_bpcpra_source_choice_valuation_selected_power_product = ff_q_bpcpra_source_choice_valuation_selected_power_product_successor * S ((S (S ff_i_bpcpra_source_choice_valuation_selected_power_product)) * ff_v_bpcpra_source_choice_valuation_selected_power_product) + (ff_s_bpcpra_source_choice_valuation_selected_power_product))) /\ ff_s_bpcpra_source_choice_valuation_selected_power_product = ff_r_bpcpra_source_choice_valuation_selected_power_product * ff_p_bpcpra_source_choice_valuation_selected_power_product)))))))) /\ (exists bpr_divides_quotient_bpcpra_source_choice_valuation_selected_divides. n = (bpr_power_value_bpcpra_source_choice_valuation_selected) * bpr_divides_quotient_bpcpra_source_choice_valuation_selected_divides)))) /\ forall bpr_valuation_candidate_bpcpra_source_choice_valuation. (exists bpr_le_gap_bpcpra_source_choice_valuation_candidate_bound. bpr_le_gap_bpcpra_source_choice_valuation_candidate_bound + (bpr_valuation_candidate_bpcpra_source_choice_valuation) = (n)) -> (exists bpr_power_value_bpcpra_source_choice_valuation_candidate. ((exists bpr_power_code_bpcpra_source_choice_valuation_candidate_power bpr_power_scale_bpcpra_source_choice_valuation_candidate_power. ((forall bpr_power_index_bpcpra_source_choice_valuation_candidate_power. (exists bpr_gap_bpcpra_source_choice_valuation_candidate_power_repeat_bound. bpr_gap_bpcpra_source_choice_valuation_candidate_power_repeat_bound + S (bpr_power_index_bpcpra_source_choice_valuation_candidate_power) = bpr_valuation_candidate_bpcpra_source_choice_valuation) -> (((exists bpr_height_bpcpra_source_choice_valuation_candidate_power_repeat_entry. bpr_height_bpcpra_source_choice_valuation_candidate_power_repeat_entry + S (S (bpr_prefix_index_bpcpra_source)) = S ((S (bpr_power_index_bpcpra_source_choice_valuation_candidate_power)) * bpr_power_scale_bpcpra_source_choice_valuation_candidate_power)) /\ exists bpr_quotient_bpcpra_source_choice_valuation_candidate_power_repeat_entry. bpr_power_code_bpcpra_source_choice_valuation_candidate_power = bpr_quotient_bpcpra_source_choice_valuation_candidate_power_repeat_entry * S ((S (bpr_power_index_bpcpra_source_choice_valuation_candidate_power)) * bpr_power_scale_bpcpra_source_choice_valuation_candidate_power) + (S (bpr_prefix_index_bpcpra_source))))) /\ (exists ff_u_bpcpra_source_choice_valuation_candidate_power_product ff_v_bpcpra_source_choice_valuation_candidate_power_product. ((((exists ff_h_bpcpra_source_choice_valuation_candidate_power_product_start. ff_h_bpcpra_source_choice_valuation_candidate_power_product_start + S (1) = S ((S (0)) * ff_v_bpcpra_source_choice_valuation_candidate_power_product)) /\ exists ff_q_bpcpra_source_choice_valuation_candidate_power_product_start. ff_u_bpcpra_source_choice_valuation_candidate_power_product = ff_q_bpcpra_source_choice_valuation_candidate_power_product_start * S ((S (0)) * ff_v_bpcpra_source_choice_valuation_candidate_power_product) + (1))) /\ ((((exists ff_h_bpcpra_source_choice_valuation_candidate_power_product_terminal. ff_h_bpcpra_source_choice_valuation_candidate_power_product_terminal + S (bpr_power_value_bpcpra_source_choice_valuation_candidate) = S ((S (bpr_valuation_candidate_bpcpra_source_choice_valuation)) * ff_v_bpcpra_source_choice_valuation_candidate_power_product)) /\ exists ff_q_bpcpra_source_choice_valuation_candidate_power_product_terminal. ff_u_bpcpra_source_choice_valuation_candidate_power_product = ff_q_bpcpra_source_choice_valuation_candidate_power_product_terminal * S ((S (bpr_valuation_candidate_bpcpra_source_choice_valuation)) * ff_v_bpcpra_source_choice_valuation_candidate_power_product) + (bpr_power_value_bpcpra_source_choice_valuation_candidate))) /\ forall ff_i_bpcpra_source_choice_valuation_candidate_power_product. (exists ff_lt_bpcpra_source_choice_valuation_candidate_power_product_bound. ff_lt_bpcpra_source_choice_valuation_candidate_power_product_bound + S ff_i_bpcpra_source_choice_valuation_candidate_power_product = bpr_valuation_candidate_bpcpra_source_choice_valuation) -> exists ff_p_bpcpra_source_choice_valuation_candidate_power_product ff_r_bpcpra_source_choice_valuation_candidate_power_product ff_s_bpcpra_source_choice_valuation_candidate_power_product. ((((exists ff_h_bpcpra_source_choice_valuation_candidate_power_product_factor. ff_h_bpcpra_source_choice_valuation_candidate_power_product_factor + S (ff_p_bpcpra_source_choice_valuation_candidate_power_product) = S ((S (ff_i_bpcpra_source_choice_valuation_candidate_power_product)) * bpr_power_scale_bpcpra_source_choice_valuation_candidate_power)) /\ exists ff_q_bpcpra_source_choice_valuation_candidate_power_product_factor. bpr_power_code_bpcpra_source_choice_valuation_candidate_power = ff_q_bpcpra_source_choice_valuation_candidate_power_product_factor * S ((S (ff_i_bpcpra_source_choice_valuation_candidate_power_product)) * bpr_power_scale_bpcpra_source_choice_valuation_candidate_power) + (ff_p_bpcpra_source_choice_valuation_candidate_power_product))) /\ ((((exists ff_h_bpcpra_source_choice_valuation_candidate_power_product_partial. ff_h_bpcpra_source_choice_valuation_candidate_power_product_partial + S (ff_r_bpcpra_source_choice_valuation_candidate_power_product) = S ((S (ff_i_bpcpra_source_choice_valuation_candidate_power_product)) * ff_v_bpcpra_source_choice_valuation_candidate_power_product)) /\ exists ff_q_bpcpra_source_choice_valuation_candidate_power_product_partial. ff_u_bpcpra_source_choice_valuation_candidate_power_product = ff_q_bpcpra_source_choice_valuation_candidate_power_product_partial * S ((S (ff_i_bpcpra_source_choice_valuation_candidate_power_product)) * ff_v_bpcpra_source_choice_valuation_candidate_power_product) + (ff_r_bpcpra_source_choice_valuation_candidate_power_product))) /\ ((((exists ff_h_bpcpra_source_choice_valuation_candidate_power_product_successor. ff_h_bpcpra_source_choice_valuation_candidate_power_product_successor + S (ff_s_bpcpra_source_choice_valuation_candidate_power_product) = S ((S (S ff_i_bpcpra_source_choice_valuation_candidate_power_product)) * ff_v_bpcpra_source_choice_valuation_candidate_power_product)) /\ exists ff_q_bpcpra_source_choice_valuation_candidate_power_product_successor. ff_u_bpcpra_source_choice_valuation_candidate_power_product = ff_q_bpcpra_source_choice_valuation_candidate_power_product_successor * S ((S (S ff_i_bpcpra_source_choice_valuation_candidate_power_product)) * ff_v_bpcpra_source_choice_valuation_candidate_power_product) + (ff_s_bpcpra_source_choice_valuation_candidate_power_product))) /\ ff_s_bpcpra_source_choice_valuation_candidate_power_product = ff_r_bpcpra_source_choice_valuation_candidate_power_product * ff_p_bpcpra_source_choice_valuation_candidate_power_product)))))))) /\ (exists bpr_divides_quotient_bpcpra_source_choice_valuation_candidate_divides. n = (bpr_power_value_bpcpra_source_choice_valuation_candidate) * bpr_divides_quotient_bpcpra_source_choice_valuation_candidate_divides))) -> (exists bpr_le_gap_bpcpra_source_choice_valuation_candidate_below. bpr_le_gap_bpcpra_source_choice_valuation_candidate_below + (bpr_valuation_candidate_bpcpra_source_choice_valuation) = (bpr_choice_exponent_bpcpra_source_choice))) /\ (exists bpr_power_code_bpcpra_source_choice_power bpr_power_scale_bpcpra_source_choice_power. ((forall bpr_power_index_bpcpra_source_choice_power. (exists bpr_gap_bpcpra_source_choice_power_repeat_bound. bpr_gap_bpcpra_source_choice_power_repeat_bound + S (bpr_power_index_bpcpra_source_choice_power) = bpr_choice_exponent_bpcpra_source_choice) -> (((exists bpr_height_bpcpra_source_choice_power_repeat_entry. bpr_height_bpcpra_source_choice_power_repeat_entry + S (S (bpr_prefix_index_bpcpra_source)) = S ((S (bpr_power_index_bpcpra_source_choice_power)) * bpr_power_scale_bpcpra_source_choice_power)) /\ exists bpr_quotient_bpcpra_source_choice_power_repeat_entry. bpr_power_code_bpcpra_source_choice_power = bpr_quotient_bpcpra_source_choice_power_repeat_entry * S ((S (bpr_power_index_bpcpra_source_choice_power)) * bpr_power_scale_bpcpra_source_choice_power) + (S (bpr_prefix_index_bpcpra_source))))) /\ (exists ff_u_bpcpra_source_choice_power_product ff_v_bpcpra_source_choice_power_product. ((((exists ff_h_bpcpra_source_choice_power_product_start. ff_h_bpcpra_source_choice_power_product_start + S (1) = S ((S (0)) * ff_v_bpcpra_source_choice_power_product)) /\ exists ff_q_bpcpra_source_choice_power_product_start. ff_u_bpcpra_source_choice_power_product = ff_q_bpcpra_source_choice_power_product_start * S ((S (0)) * ff_v_bpcpra_source_choice_power_product) + (1))) /\ ((((exists ff_h_bpcpra_source_choice_power_product_terminal. ff_h_bpcpra_source_choice_power_product_terminal + S (bpr_prefix_value_bpcpra_source) = S ((S (bpr_choice_exponent_bpcpra_source_choice)) * ff_v_bpcpra_source_choice_power_product)) /\ exists ff_q_bpcpra_source_choice_power_product_terminal. ff_u_bpcpra_source_choice_power_product = ff_q_bpcpra_source_choice_power_product_terminal * S ((S (bpr_choice_exponent_bpcpra_source_choice)) * ff_v_bpcpra_source_choice_power_product) + (bpr_prefix_value_bpcpra_source))) /\ forall ff_i_bpcpra_source_choice_power_product. (exists ff_lt_bpcpra_source_choice_power_product_bound. ff_lt_bpcpra_source_choice_power_product_bound + S ff_i_bpcpra_source_choice_power_product = bpr_choice_exponent_bpcpra_source_choice) -> exists ff_p_bpcpra_source_choice_power_product ff_r_bpcpra_source_choice_power_product ff_s_bpcpra_source_choice_power_product. ((((exists ff_h_bpcpra_source_choice_power_product_factor. ff_h_bpcpra_source_choice_power_product_factor + S (ff_p_bpcpra_source_choice_power_product) = S ((S (ff_i_bpcpra_source_choice_power_product)) * bpr_power_scale_bpcpra_source_choice_power)) /\ exists ff_q_bpcpra_source_choice_power_product_factor. bpr_power_code_bpcpra_source_choice_power = ff_q_bpcpra_source_choice_power_product_factor * S ((S (ff_i_bpcpra_source_choice_power_product)) * bpr_power_scale_bpcpra_source_choice_power) + (ff_p_bpcpra_source_choice_power_product))) /\ ((((exists ff_h_bpcpra_source_choice_power_product_partial. ff_h_bpcpra_source_choice_power_product_partial + S (ff_r_bpcpra_source_choice_power_product) = S ((S (ff_i_bpcpra_source_choice_power_product)) * ff_v_bpcpra_source_choice_power_product)) /\ exists ff_q_bpcpra_source_choice_power_product_partial. ff_u_bpcpra_source_choice_power_product = ff_q_bpcpra_source_choice_power_product_partial * S ((S (ff_i_bpcpra_source_choice_power_product)) * ff_v_bpcpra_source_choice_power_product) + (ff_r_bpcpra_source_choice_power_product))) /\ ((((exists ff_h_bpcpra_source_choice_power_product_successor. ff_h_bpcpra_source_choice_power_product_successor + S (ff_s_bpcpra_source_choice_power_product) = S ((S (S ff_i_bpcpra_source_choice_power_product)) * ff_v_bpcpra_source_choice_power_product)) /\ exists ff_q_bpcpra_source_choice_power_product_successor. ff_u_bpcpra_source_choice_power_product = ff_q_bpcpra_source_choice_power_product_successor * S ((S (S ff_i_bpcpra_source_choice_power_product)) * ff_v_bpcpra_source_choice_power_product) + (ff_s_bpcpra_source_choice_power_product))) /\ ff_s_bpcpra_source_choice_power_product = ff_r_bpcpra_source_choice_power_product * ff_p_bpcpra_source_choice_power_product)))))))))) \/ (~((~(S (bpr_prefix_index_bpcpra_source) = 1) /\ forall bpr_left_bpcpra_source_choice_prime bpr_right_bpcpra_source_choice_prime. S (bpr_prefix_index_bpcpra_source) = bpr_left_bpcpra_source_choice_prime * bpr_right_bpcpra_source_choice_prime -> bpr_left_bpcpra_source_choice_prime = 1 \/ bpr_right_bpcpra_source_choice_prime = 1)) /\ bpr_prefix_value_bpcpra_source = 1))))) -> (forall bpr_prefix_index_bpcpra_target. (exists bpr_gap_bpcpra_target_bound. bpr_gap_bpcpra_target_bound + S (bpr_prefix_index_bpcpra_target) = a) -> exists bpr_prefix_value_bpcpra_target. ((((exists bpr_height_bpcpra_target_decoded. bpr_height_bpcpra_target_decoded + S (bpr_prefix_value_bpcpra_target) = S ((S (bpr_prefix_index_bpcpra_target)) * c)) /\ exists bpr_quotient_bpcpra_target_decoded. b = bpr_quotient_bpcpra_target_decoded * S ((S (bpr_prefix_index_bpcpra_target)) * c) + (bpr_prefix_value_bpcpra_target))) /\ (((((~(S (bpr_prefix_index_bpcpra_target) = 1) /\ forall bpr_left_bpcpra_target_choice_prime bpr_right_bpcpra_target_choice_prime. S (bpr_prefix_index_bpcpra_target) = bpr_left_bpcpra_target_choice_prime * bpr_right_bpcpra_target_choice_prime -> bpr_left_bpcpra_target_choice_prime = 1 \/ bpr_right_bpcpra_target_choice_prime = 1)) /\ exists bpr_choice_exponent_bpcpra_target_choice. ((((exists bpr_le_gap_bpcpra_target_choice_valuation_selected_bound. bpr_le_gap_bpcpra_target_choice_valuation_selected_bound + (bpr_choice_exponent_bpcpra_target_choice) = (n)) /\ (exists bpr_power_value_bpcpra_target_choice_valuation_selected. ((exists bpr_power_code_bpcpra_target_choice_valuation_selected_power bpr_power_scale_bpcpra_target_choice_valuation_selected_power. ((forall bpr_power_index_bpcpra_target_choice_valuation_selected_power. (exists bpr_gap_bpcpra_target_choice_valuation_selected_power_repeat_bound. bpr_gap_bpcpra_target_choice_valuation_selected_power_repeat_bound + S (bpr_power_index_bpcpra_target_choice_valuation_selected_power) = bpr_choice_exponent_bpcpra_target_choice) -> (((exists bpr_height_bpcpra_target_choice_valuation_selected_power_repeat_entry. bpr_height_bpcpra_target_choice_valuation_selected_power_repeat_entry + S (S (bpr_prefix_index_bpcpra_target)) = S ((S (bpr_power_index_bpcpra_target_choice_valuation_selected_power)) * bpr_power_scale_bpcpra_target_choice_valuation_selected_power)) /\ exists bpr_quotient_bpcpra_target_choice_valuation_selected_power_repeat_entry. bpr_power_code_bpcpra_target_choice_valuation_selected_power = bpr_quotient_bpcpra_target_choice_valuation_selected_power_repeat_entry * S ((S (bpr_power_index_bpcpra_target_choice_valuation_selected_power)) * bpr_power_scale_bpcpra_target_choice_valuation_selected_power) + (S (bpr_prefix_index_bpcpra_target))))) /\ (exists ff_u_bpcpra_target_choice_valuation_selected_power_product ff_v_bpcpra_target_choice_valuation_selected_power_product. ((((exists ff_h_bpcpra_target_choice_valuation_selected_power_product_start. ff_h_bpcpra_target_choice_valuation_selected_power_product_start + S (1) = S ((S (0)) * ff_v_bpcpra_target_choice_valuation_selected_power_product)) /\ exists ff_q_bpcpra_target_choice_valuation_selected_power_product_start. ff_u_bpcpra_target_choice_valuation_selected_power_product = ff_q_bpcpra_target_choice_valuation_selected_power_product_start * S ((S (0)) * ff_v_bpcpra_target_choice_valuation_selected_power_product) + (1))) /\ ((((exists ff_h_bpcpra_target_choice_valuation_selected_power_product_terminal. ff_h_bpcpra_target_choice_valuation_selected_power_product_terminal + S (bpr_power_value_bpcpra_target_choice_valuation_selected) = S ((S (bpr_choice_exponent_bpcpra_target_choice)) * ff_v_bpcpra_target_choice_valuation_selected_power_product)) /\ exists ff_q_bpcpra_target_choice_valuation_selected_power_product_terminal. ff_u_bpcpra_target_choice_valuation_selected_power_product = ff_q_bpcpra_target_choice_valuation_selected_power_product_terminal * S ((S (bpr_choice_exponent_bpcpra_target_choice)) * ff_v_bpcpra_target_choice_valuation_selected_power_product) + (bpr_power_value_bpcpra_target_choice_valuation_selected))) /\ forall ff_i_bpcpra_target_choice_valuation_selected_power_product. (exists ff_lt_bpcpra_target_choice_valuation_selected_power_product_bound. ff_lt_bpcpra_target_choice_valuation_selected_power_product_bound + S ff_i_bpcpra_target_choice_valuation_selected_power_product = bpr_choice_exponent_bpcpra_target_choice) -> exists ff_p_bpcpra_target_choice_valuation_selected_power_product ff_r_bpcpra_target_choice_valuation_selected_power_product ff_s_bpcpra_target_choice_valuation_selected_power_product. ((((exists ff_h_bpcpra_target_choice_valuation_selected_power_product_factor. ff_h_bpcpra_target_choice_valuation_selected_power_product_factor + S (ff_p_bpcpra_target_choice_valuation_selected_power_product) = S ((S (ff_i_bpcpra_target_choice_valuation_selected_power_product)) * bpr_power_scale_bpcpra_target_choice_valuation_selected_power)) /\ exists ff_q_bpcpra_target_choice_valuation_selected_power_product_factor. bpr_power_code_bpcpra_target_choice_valuation_selected_power = ff_q_bpcpra_target_choice_valuation_selected_power_product_factor * S ((S (ff_i_bpcpra_target_choice_valuation_selected_power_product)) * bpr_power_scale_bpcpra_target_choice_valuation_selected_power) + (ff_p_bpcpra_target_choice_valuation_selected_power_product))) /\ ((((exists ff_h_bpcpra_target_choice_valuation_selected_power_product_partial. ff_h_bpcpra_target_choice_valuation_selected_power_product_partial + S (ff_r_bpcpra_target_choice_valuation_selected_power_product) = S ((S (ff_i_bpcpra_target_choice_valuation_selected_power_product)) * ff_v_bpcpra_target_choice_valuation_selected_power_product)) /\ exists ff_q_bpcpra_target_choice_valuation_selected_power_product_partial. ff_u_bpcpra_target_choice_valuation_selected_power_product = ff_q_bpcpra_target_choice_valuation_selected_power_product_partial * S ((S (ff_i_bpcpra_target_choice_valuation_selected_power_product)) * ff_v_bpcpra_target_choice_valuation_selected_power_product) + (ff_r_bpcpra_target_choice_valuation_selected_power_product))) /\ ((((exists ff_h_bpcpra_target_choice_valuation_selected_power_product_successor. ff_h_bpcpra_target_choice_valuation_selected_power_product_successor + S (ff_s_bpcpra_target_choice_valuation_selected_power_product) = S ((S (S ff_i_bpcpra_target_choice_valuation_selected_power_product)) * ff_v_bpcpra_target_choice_valuation_selected_power_product)) /\ exists ff_q_bpcpra_target_choice_valuation_selected_power_product_successor. ff_u_bpcpra_target_choice_valuation_selected_power_product = ff_q_bpcpra_target_choice_valuation_selected_power_product_successor * S ((S (S ff_i_bpcpra_target_choice_valuation_selected_power_product)) * ff_v_bpcpra_target_choice_valuation_selected_power_product) + (ff_s_bpcpra_target_choice_valuation_selected_power_product))) /\ ff_s_bpcpra_target_choice_valuation_selected_power_product = ff_r_bpcpra_target_choice_valuation_selected_power_product * ff_p_bpcpra_target_choice_valuation_selected_power_product)))))))) /\ (exists bpr_divides_quotient_bpcpra_target_choice_valuation_selected_divides. n = (bpr_power_value_bpcpra_target_choice_valuation_selected) * bpr_divides_quotient_bpcpra_target_choice_valuation_selected_divides)))) /\ forall bpr_valuation_candidate_bpcpra_target_choice_valuation. (exists bpr_le_gap_bpcpra_target_choice_valuation_candidate_bound. bpr_le_gap_bpcpra_target_choice_valuation_candidate_bound + (bpr_valuation_candidate_bpcpra_target_choice_valuation) = (n)) -> (exists bpr_power_value_bpcpra_target_choice_valuation_candidate. ((exists bpr_power_code_bpcpra_target_choice_valuation_candidate_power bpr_power_scale_bpcpra_target_choice_valuation_candidate_power. ((forall bpr_power_index_bpcpra_target_choice_valuation_candidate_power. (exists bpr_gap_bpcpra_target_choice_valuation_candidate_power_repeat_bound. bpr_gap_bpcpra_target_choice_valuation_candidate_power_repeat_bound + S (bpr_power_index_bpcpra_target_choice_valuation_candidate_power) = bpr_valuation_candidate_bpcpra_target_choice_valuation) -> (((exists bpr_height_bpcpra_target_choice_valuation_candidate_power_repeat_entry. bpr_height_bpcpra_target_choice_valuation_candidate_power_repeat_entry + S (S (bpr_prefix_index_bpcpra_target)) = S ((S (bpr_power_index_bpcpra_target_choice_valuation_candidate_power)) * bpr_power_scale_bpcpra_target_choice_valuation_candidate_power)) /\ exists bpr_quotient_bpcpra_target_choice_valuation_candidate_power_repeat_entry. bpr_power_code_bpcpra_target_choice_valuation_candidate_power = bpr_quotient_bpcpra_target_choice_valuation_candidate_power_repeat_entry * S ((S (bpr_power_index_bpcpra_target_choice_valuation_candidate_power)) * bpr_power_scale_bpcpra_target_choice_valuation_candidate_power) + (S (bpr_prefix_index_bpcpra_target))))) /\ (exists ff_u_bpcpra_target_choice_valuation_candidate_power_product ff_v_bpcpra_target_choice_valuation_candidate_power_product. ((((exists ff_h_bpcpra_target_choice_valuation_candidate_power_product_start. ff_h_bpcpra_target_choice_valuation_candidate_power_product_start + S (1) = S ((S (0)) * ff_v_bpcpra_target_choice_valuation_candidate_power_product)) /\ exists ff_q_bpcpra_target_choice_valuation_candidate_power_product_start. ff_u_bpcpra_target_choice_valuation_candidate_power_product = ff_q_bpcpra_target_choice_valuation_candidate_power_product_start * S ((S (0)) * ff_v_bpcpra_target_choice_valuation_candidate_power_product) + (1))) /\ ((((exists ff_h_bpcpra_target_choice_valuation_candidate_power_product_terminal. ff_h_bpcpra_target_choice_valuation_candidate_power_product_terminal + S (bpr_power_value_bpcpra_target_choice_valuation_candidate) = S ((S (bpr_valuation_candidate_bpcpra_target_choice_valuation)) * ff_v_bpcpra_target_choice_valuation_candidate_power_product)) /\ exists ff_q_bpcpra_target_choice_valuation_candidate_power_product_terminal. ff_u_bpcpra_target_choice_valuation_candidate_power_product = ff_q_bpcpra_target_choice_valuation_candidate_power_product_terminal * S ((S (bpr_valuation_candidate_bpcpra_target_choice_valuation)) * ff_v_bpcpra_target_choice_valuation_candidate_power_product) + (bpr_power_value_bpcpra_target_choice_valuation_candidate))) /\ forall ff_i_bpcpra_target_choice_valuation_candidate_power_product. (exists ff_lt_bpcpra_target_choice_valuation_candidate_power_product_bound. ff_lt_bpcpra_target_choice_valuation_candidate_power_product_bound + S ff_i_bpcpra_target_choice_valuation_candidate_power_product = bpr_valuation_candidate_bpcpra_target_choice_valuation) -> exists ff_p_bpcpra_target_choice_valuation_candidate_power_product ff_r_bpcpra_target_choice_valuation_candidate_power_product ff_s_bpcpra_target_choice_valuation_candidate_power_product. ((((exists ff_h_bpcpra_target_choice_valuation_candidate_power_product_factor. ff_h_bpcpra_target_choice_valuation_candidate_power_product_factor + S (ff_p_bpcpra_target_choice_valuation_candidate_power_product) = S ((S (ff_i_bpcpra_target_choice_valuation_candidate_power_product)) * bpr_power_scale_bpcpra_target_choice_valuation_candidate_power)) /\ exists ff_q_bpcpra_target_choice_valuation_candidate_power_product_factor. bpr_power_code_bpcpra_target_choice_valuation_candidate_power = ff_q_bpcpra_target_choice_valuation_candidate_power_product_factor * S ((S (ff_i_bpcpra_target_choice_valuation_candidate_power_product)) * bpr_power_scale_bpcpra_target_choice_valuation_candidate_power) + (ff_p_bpcpra_target_choice_valuation_candidate_power_product))) /\ ((((exists ff_h_bpcpra_target_choice_valuation_candidate_power_product_partial. ff_h_bpcpra_target_choice_valuation_candidate_power_product_partial + S (ff_r_bpcpra_target_choice_valuation_candidate_power_product) = S ((S (ff_i_bpcpra_target_choice_valuation_candidate_power_product)) * ff_v_bpcpra_target_choice_valuation_candidate_power_product)) /\ exists ff_q_bpcpra_target_choice_valuation_candidate_power_product_partial. ff_u_bpcpra_target_choice_valuation_candidate_power_product = ff_q_bpcpra_target_choice_valuation_candidate_power_product_partial * S ((S (ff_i_bpcpra_target_choice_valuation_candidate_power_product)) * ff_v_bpcpra_target_choice_valuation_candidate_power_product) + (ff_r_bpcpra_target_choice_valuation_candidate_power_product))) /\ ((((exists ff_h_bpcpra_target_choice_valuation_candidate_power_product_successor. ff_h_bpcpra_target_choice_valuation_candidate_power_product_successor + S (ff_s_bpcpra_target_choice_valuation_candidate_power_product) = S ((S (S ff_i_bpcpra_target_choice_valuation_candidate_power_product)) * ff_v_bpcpra_target_choice_valuation_candidate_power_product)) /\ exists ff_q_bpcpra_target_choice_valuation_candidate_power_product_successor. ff_u_bpcpra_target_choice_valuation_candidate_power_product = ff_q_bpcpra_target_choice_valuation_candidate_power_product_successor * S ((S (S ff_i_bpcpra_target_choice_valuation_candidate_power_product)) * ff_v_bpcpra_target_choice_valuation_candidate_power_product) + (ff_s_bpcpra_target_choice_valuation_candidate_power_product))) /\ ff_s_bpcpra_target_choice_valuation_candidate_power_product = ff_r_bpcpra_target_choice_valuation_candidate_power_product * ff_p_bpcpra_target_choice_valuation_candidate_power_product)))))))) /\ (exists bpr_divides_quotient_bpcpra_target_choice_valuation_candidate_divides. n = (bpr_power_value_bpcpra_target_choice_valuation_candidate) * bpr_divides_quotient_bpcpra_target_choice_valuation_candidate_divides))) -> (exists bpr_le_gap_bpcpra_target_choice_valuation_candidate_below. bpr_le_gap_bpcpra_target_choice_valuation_candidate_below + (bpr_valuation_candidate_bpcpra_target_choice_valuation) = (bpr_choice_exponent_bpcpra_target_choice))) /\ (exists bpr_power_code_bpcpra_target_choice_power bpr_power_scale_bpcpra_target_choice_power. ((forall bpr_power_index_bpcpra_target_choice_power. (exists bpr_gap_bpcpra_target_choice_power_repeat_bound. bpr_gap_bpcpra_target_choice_power_repeat_bound + S (bpr_power_index_bpcpra_target_choice_power) = bpr_choice_exponent_bpcpra_target_choice) -> (((exists bpr_height_bpcpra_target_choice_power_repeat_entry. bpr_height_bpcpra_target_choice_power_repeat_entry + S (S (bpr_prefix_index_bpcpra_target)) = S ((S (bpr_power_index_bpcpra_target_choice_power)) * bpr_power_scale_bpcpra_target_choice_power)) /\ exists bpr_quotient_bpcpra_target_choice_power_repeat_entry. bpr_power_code_bpcpra_target_choice_power = bpr_quotient_bpcpra_target_choice_power_repeat_entry * S ((S (bpr_power_index_bpcpra_target_choice_power)) * bpr_power_scale_bpcpra_target_choice_power) + (S (bpr_prefix_index_bpcpra_target))))) /\ (exists ff_u_bpcpra_target_choice_power_product ff_v_bpcpra_target_choice_power_product. ((((exists ff_h_bpcpra_target_choice_power_product_start. ff_h_bpcpra_target_choice_power_product_start + S (1) = S ((S (0)) * ff_v_bpcpra_target_choice_power_product)) /\ exists ff_q_bpcpra_target_choice_power_product_start. ff_u_bpcpra_target_choice_power_product = ff_q_bpcpra_target_choice_power_product_start * S ((S (0)) * ff_v_bpcpra_target_choice_power_product) + (1))) /\ ((((exists ff_h_bpcpra_target_choice_power_product_terminal. ff_h_bpcpra_target_choice_power_product_terminal + S (bpr_prefix_value_bpcpra_target) = S ((S (bpr_choice_exponent_bpcpra_target_choice)) * ff_v_bpcpra_target_choice_power_product)) /\ exists ff_q_bpcpra_target_choice_power_product_terminal. ff_u_bpcpra_target_choice_power_product = ff_q_bpcpra_target_choice_power_product_terminal * S ((S (bpr_choice_exponent_bpcpra_target_choice)) * ff_v_bpcpra_target_choice_power_product) + (bpr_prefix_value_bpcpra_target))) /\ forall ff_i_bpcpra_target_choice_power_product. (exists ff_lt_bpcpra_target_choice_power_product_bound. ff_lt_bpcpra_target_choice_power_product_bound + S ff_i_bpcpra_target_choice_power_product = bpr_choice_exponent_bpcpra_target_choice) -> exists ff_p_bpcpra_target_choice_power_product ff_r_bpcpra_target_choice_power_product ff_s_bpcpra_target_choice_power_product. ((((exists ff_h_bpcpra_target_choice_power_product_factor. ff_h_bpcpra_target_choice_power_product_factor + S (ff_p_bpcpra_target_choice_power_product) = S ((S (ff_i_bpcpra_target_choice_power_product)) * bpr_power_scale_bpcpra_target_choice_power)) /\ exists ff_q_bpcpra_target_choice_power_product_factor. bpr_power_code_bpcpra_target_choice_power = ff_q_bpcpra_target_choice_power_product_factor * S ((S (ff_i_bpcpra_target_choice_power_product)) * bpr_power_scale_bpcpra_target_choice_power) + (ff_p_bpcpra_target_choice_power_product))) /\ ((((exists ff_h_bpcpra_target_choice_power_product_partial. ff_h_bpcpra_target_choice_power_product_partial + S (ff_r_bpcpra_target_choice_power_product) = S ((S (ff_i_bpcpra_target_choice_power_product)) * ff_v_bpcpra_target_choice_power_product)) /\ exists ff_q_bpcpra_target_choice_power_product_partial. ff_u_bpcpra_target_choice_power_product = ff_q_bpcpra_target_choice_power_product_partial * S ((S (ff_i_bpcpra_target_choice_power_product)) * ff_v_bpcpra_target_choice_power_product) + (ff_r_bpcpra_target_choice_power_product))) /\ ((((exists ff_h_bpcpra_target_choice_power_product_successor. ff_h_bpcpra_target_choice_power_product_successor + S (ff_s_bpcpra_target_choice_power_product) = S ((S (S ff_i_bpcpra_target_choice_power_product)) * ff_v_bpcpra_target_choice_power_product)) /\ exists ff_q_bpcpra_target_choice_power_product_successor. ff_u_bpcpra_target_choice_power_product = ff_q_bpcpra_target_choice_power_product_successor * S ((S (S ff_i_bpcpra_target_choice_power_product)) * ff_v_bpcpra_target_choice_power_product) + (ff_s_bpcpra_target_choice_power_product))) /\ ff_s_bpcpra_target_choice_power_product = ff_r_bpcpra_target_choice_power_product * ff_p_bpcpra_target_choice_power_product)))))))))) \/ (~((~(S (bpr_prefix_index_bpcpra_target) = 1) /\ forall bpr_left_bpcpra_target_choice_prime bpr_right_bpcpra_target_choice_prime. S (bpr_prefix_index_bpcpra_target) = bpr_left_bpcpra_target_choice_prime * bpr_right_bpcpra_target_choice_prime -> bpr_left_bpcpra_target_choice_prime = 1 \/ bpr_right_bpcpra_target_choice_prime = 1)) /\ bpr_prefix_value_bpcpra_target = 1)))))