prime_contribution_interval_prefix_shift

Exact expanded PA statement

forall n a b c d e l. (forall bpr_prefix_index_bpcips_source. (exists bpr_gap_bpcips_source_bound. bpr_gap_bpcips_source_bound + S (bpr_prefix_index_bpcips_source) = a + l) -> exists bpr_prefix_value_bpcips_source. ((((exists bpr_height_bpcips_source_decoded. bpr_height_bpcips_source_decoded + S (bpr_prefix_value_bpcips_source) = S ((S (bpr_prefix_index_bpcips_source)) * c)) /\ exists bpr_quotient_bpcips_source_decoded. b = bpr_quotient_bpcips_source_decoded * S ((S (bpr_prefix_index_bpcips_source)) * c) + (bpr_prefix_value_bpcips_source))) /\ (((((~(S (bpr_prefix_index_bpcips_source) = 1) /\ forall bpr_left_bpcips_source_choice_prime bpr_right_bpcips_source_choice_prime. S (bpr_prefix_index_bpcips_source) = bpr_left_bpcips_source_choice_prime * bpr_right_bpcips_source_choice_prime -> bpr_left_bpcips_source_choice_prime = 1 \/ bpr_right_bpcips_source_choice_prime = 1)) /\ exists bpr_choice_exponent_bpcips_source_choice. ((((exists bpr_le_gap_bpcips_source_choice_valuation_selected_bound. bpr_le_gap_bpcips_source_choice_valuation_selected_bound + (bpr_choice_exponent_bpcips_source_choice) = (n)) /\ (exists bpr_power_value_bpcips_source_choice_valuation_selected. ((exists bpr_power_code_bpcips_source_choice_valuation_selected_power bpr_power_scale_bpcips_source_choice_valuation_selected_power. ((forall bpr_power_index_bpcips_source_choice_valuation_selected_power. (exists bpr_gap_bpcips_source_choice_valuation_selected_power_repeat_bound. bpr_gap_bpcips_source_choice_valuation_selected_power_repeat_bound + S (bpr_power_index_bpcips_source_choice_valuation_selected_power) = bpr_choice_exponent_bpcips_source_choice) -> (((exists bpr_height_bpcips_source_choice_valuation_selected_power_repeat_entry. bpr_height_bpcips_source_choice_valuation_selected_power_repeat_entry + S (S (bpr_prefix_index_bpcips_source)) = S ((S (bpr_power_index_bpcips_source_choice_valuation_selected_power)) * bpr_power_scale_bpcips_source_choice_valuation_selected_power)) /\ exists bpr_quotient_bpcips_source_choice_valuation_selected_power_repeat_entry. bpr_power_code_bpcips_source_choice_valuation_selected_power = bpr_quotient_bpcips_source_choice_valuation_selected_power_repeat_entry * S ((S (bpr_power_index_bpcips_source_choice_valuation_selected_power)) * bpr_power_scale_bpcips_source_choice_valuation_selected_power) + (S (bpr_prefix_index_bpcips_source))))) /\ (exists ff_u_bpcips_source_choice_valuation_selected_power_product ff_v_bpcips_source_choice_valuation_selected_power_product. ((((exists ff_h_bpcips_source_choice_valuation_selected_power_product_start. ff_h_bpcips_source_choice_valuation_selected_power_product_start + S (1) = S ((S (0)) * ff_v_bpcips_source_choice_valuation_selected_power_product)) /\ exists ff_q_bpcips_source_choice_valuation_selected_power_product_start. ff_u_bpcips_source_choice_valuation_selected_power_product = ff_q_bpcips_source_choice_valuation_selected_power_product_start * S ((S (0)) * ff_v_bpcips_source_choice_valuation_selected_power_product) + (1))) /\ ((((exists ff_h_bpcips_source_choice_valuation_selected_power_product_terminal. ff_h_bpcips_source_choice_valuation_selected_power_product_terminal + S (bpr_power_value_bpcips_source_choice_valuation_selected) = S ((S (bpr_choice_exponent_bpcips_source_choice)) * ff_v_bpcips_source_choice_valuation_selected_power_product)) /\ exists ff_q_bpcips_source_choice_valuation_selected_power_product_terminal. ff_u_bpcips_source_choice_valuation_selected_power_product = ff_q_bpcips_source_choice_valuation_selected_power_product_terminal * S ((S (bpr_choice_exponent_bpcips_source_choice)) * ff_v_bpcips_source_choice_valuation_selected_power_product) + (bpr_power_value_bpcips_source_choice_valuation_selected))) /\ forall ff_i_bpcips_source_choice_valuation_selected_power_product. (exists ff_lt_bpcips_source_choice_valuation_selected_power_product_bound. ff_lt_bpcips_source_choice_valuation_selected_power_product_bound + S ff_i_bpcips_source_choice_valuation_selected_power_product = bpr_choice_exponent_bpcips_source_choice) -> exists ff_p_bpcips_source_choice_valuation_selected_power_product ff_r_bpcips_source_choice_valuation_selected_power_product ff_s_bpcips_source_choice_valuation_selected_power_product. ((((exists ff_h_bpcips_source_choice_valuation_selected_power_product_factor. ff_h_bpcips_source_choice_valuation_selected_power_product_factor + S (ff_p_bpcips_source_choice_valuation_selected_power_product) = S ((S (ff_i_bpcips_source_choice_valuation_selected_power_product)) * bpr_power_scale_bpcips_source_choice_valuation_selected_power)) /\ exists ff_q_bpcips_source_choice_valuation_selected_power_product_factor. bpr_power_code_bpcips_source_choice_valuation_selected_power = ff_q_bpcips_source_choice_valuation_selected_power_product_factor * S ((S (ff_i_bpcips_source_choice_valuation_selected_power_product)) * bpr_power_scale_bpcips_source_choice_valuation_selected_power) + (ff_p_bpcips_source_choice_valuation_selected_power_product))) /\ ((((exists ff_h_bpcips_source_choice_valuation_selected_power_product_partial. ff_h_bpcips_source_choice_valuation_selected_power_product_partial + S (ff_r_bpcips_source_choice_valuation_selected_power_product) = S ((S (ff_i_bpcips_source_choice_valuation_selected_power_product)) * ff_v_bpcips_source_choice_valuation_selected_power_product)) /\ exists ff_q_bpcips_source_choice_valuation_selected_power_product_partial. ff_u_bpcips_source_choice_valuation_selected_power_product = ff_q_bpcips_source_choice_valuation_selected_power_product_partial * S ((S (ff_i_bpcips_source_choice_valuation_selected_power_product)) * ff_v_bpcips_source_choice_valuation_selected_power_product) + (ff_r_bpcips_source_choice_valuation_selected_power_product))) /\ ((((exists ff_h_bpcips_source_choice_valuation_selected_power_product_successor. ff_h_bpcips_source_choice_valuation_selected_power_product_successor + S (ff_s_bpcips_source_choice_valuation_selected_power_product) = S ((S (S ff_i_bpcips_source_choice_valuation_selected_power_product)) * ff_v_bpcips_source_choice_valuation_selected_power_product)) /\ exists ff_q_bpcips_source_choice_valuation_selected_power_product_successor. ff_u_bpcips_source_choice_valuation_selected_power_product = ff_q_bpcips_source_choice_valuation_selected_power_product_successor * S ((S (S ff_i_bpcips_source_choice_valuation_selected_power_product)) * ff_v_bpcips_source_choice_valuation_selected_power_product) + (ff_s_bpcips_source_choice_valuation_selected_power_product))) /\ ff_s_bpcips_source_choice_valuation_selected_power_product = ff_r_bpcips_source_choice_valuation_selected_power_product * ff_p_bpcips_source_choice_valuation_selected_power_product)))))))) /\ (exists bpr_divides_quotient_bpcips_source_choice_valuation_selected_divides. n = (bpr_power_value_bpcips_source_choice_valuation_selected) * bpr_divides_quotient_bpcips_source_choice_valuation_selected_divides)))) /\ forall bpr_valuation_candidate_bpcips_source_choice_valuation. (exists bpr_le_gap_bpcips_source_choice_valuation_candidate_bound. bpr_le_gap_bpcips_source_choice_valuation_candidate_bound + (bpr_valuation_candidate_bpcips_source_choice_valuation) = (n)) -> (exists bpr_power_value_bpcips_source_choice_valuation_candidate. ((exists bpr_power_code_bpcips_source_choice_valuation_candidate_power bpr_power_scale_bpcips_source_choice_valuation_candidate_power. ((forall bpr_power_index_bpcips_source_choice_valuation_candidate_power. (exists bpr_gap_bpcips_source_choice_valuation_candidate_power_repeat_bound. bpr_gap_bpcips_source_choice_valuation_candidate_power_repeat_bound + S (bpr_power_index_bpcips_source_choice_valuation_candidate_power) = bpr_valuation_candidate_bpcips_source_choice_valuation) -> (((exists bpr_height_bpcips_source_choice_valuation_candidate_power_repeat_entry. bpr_height_bpcips_source_choice_valuation_candidate_power_repeat_entry + S (S (bpr_prefix_index_bpcips_source)) = S ((S (bpr_power_index_bpcips_source_choice_valuation_candidate_power)) * bpr_power_scale_bpcips_source_choice_valuation_candidate_power)) /\ exists bpr_quotient_bpcips_source_choice_valuation_candidate_power_repeat_entry. bpr_power_code_bpcips_source_choice_valuation_candidate_power = bpr_quotient_bpcips_source_choice_valuation_candidate_power_repeat_entry * S ((S (bpr_power_index_bpcips_source_choice_valuation_candidate_power)) * bpr_power_scale_bpcips_source_choice_valuation_candidate_power) + (S (bpr_prefix_index_bpcips_source))))) /\ (exists ff_u_bpcips_source_choice_valuation_candidate_power_product ff_v_bpcips_source_choice_valuation_candidate_power_product. ((((exists ff_h_bpcips_source_choice_valuation_candidate_power_product_start. ff_h_bpcips_source_choice_valuation_candidate_power_product_start + S (1) = S ((S (0)) * ff_v_bpcips_source_choice_valuation_candidate_power_product)) /\ exists ff_q_bpcips_source_choice_valuation_candidate_power_product_start. ff_u_bpcips_source_choice_valuation_candidate_power_product = ff_q_bpcips_source_choice_valuation_candidate_power_product_start * S ((S (0)) * ff_v_bpcips_source_choice_valuation_candidate_power_product) + (1))) /\ ((((exists ff_h_bpcips_source_choice_valuation_candidate_power_product_terminal. ff_h_bpcips_source_choice_valuation_candidate_power_product_terminal + S (bpr_power_value_bpcips_source_choice_valuation_candidate) = S ((S (bpr_valuation_candidate_bpcips_source_choice_valuation)) * ff_v_bpcips_source_choice_valuation_candidate_power_product)) /\ exists ff_q_bpcips_source_choice_valuation_candidate_power_product_terminal. ff_u_bpcips_source_choice_valuation_candidate_power_product = ff_q_bpcips_source_choice_valuation_candidate_power_product_terminal * S ((S (bpr_valuation_candidate_bpcips_source_choice_valuation)) * ff_v_bpcips_source_choice_valuation_candidate_power_product) + (bpr_power_value_bpcips_source_choice_valuation_candidate))) /\ forall ff_i_bpcips_source_choice_valuation_candidate_power_product. (exists ff_lt_bpcips_source_choice_valuation_candidate_power_product_bound. ff_lt_bpcips_source_choice_valuation_candidate_power_product_bound + S ff_i_bpcips_source_choice_valuation_candidate_power_product = bpr_valuation_candidate_bpcips_source_choice_valuation) -> exists ff_p_bpcips_source_choice_valuation_candidate_power_product ff_r_bpcips_source_choice_valuation_candidate_power_product ff_s_bpcips_source_choice_valuation_candidate_power_product. ((((exists ff_h_bpcips_source_choice_valuation_candidate_power_product_factor. ff_h_bpcips_source_choice_valuation_candidate_power_product_factor + S (ff_p_bpcips_source_choice_valuation_candidate_power_product) = S ((S (ff_i_bpcips_source_choice_valuation_candidate_power_product)) * bpr_power_scale_bpcips_source_choice_valuation_candidate_power)) /\ exists ff_q_bpcips_source_choice_valuation_candidate_power_product_factor. bpr_power_code_bpcips_source_choice_valuation_candidate_power = ff_q_bpcips_source_choice_valuation_candidate_power_product_factor * S ((S (ff_i_bpcips_source_choice_valuation_candidate_power_product)) * bpr_power_scale_bpcips_source_choice_valuation_candidate_power) + (ff_p_bpcips_source_choice_valuation_candidate_power_product))) /\ ((((exists ff_h_bpcips_source_choice_valuation_candidate_power_product_partial. ff_h_bpcips_source_choice_valuation_candidate_power_product_partial + S (ff_r_bpcips_source_choice_valuation_candidate_power_product) = S ((S (ff_i_bpcips_source_choice_valuation_candidate_power_product)) * ff_v_bpcips_source_choice_valuation_candidate_power_product)) /\ exists ff_q_bpcips_source_choice_valuation_candidate_power_product_partial. ff_u_bpcips_source_choice_valuation_candidate_power_product = ff_q_bpcips_source_choice_valuation_candidate_power_product_partial * S ((S (ff_i_bpcips_source_choice_valuation_candidate_power_product)) * ff_v_bpcips_source_choice_valuation_candidate_power_product) + (ff_r_bpcips_source_choice_valuation_candidate_power_product))) /\ ((((exists ff_h_bpcips_source_choice_valuation_candidate_power_product_successor. ff_h_bpcips_source_choice_valuation_candidate_power_product_successor + S (ff_s_bpcips_source_choice_valuation_candidate_power_product) = S ((S (S ff_i_bpcips_source_choice_valuation_candidate_power_product)) * ff_v_bpcips_source_choice_valuation_candidate_power_product)) /\ exists ff_q_bpcips_source_choice_valuation_candidate_power_product_successor. ff_u_bpcips_source_choice_valuation_candidate_power_product = ff_q_bpcips_source_choice_valuation_candidate_power_product_successor * S ((S (S ff_i_bpcips_source_choice_valuation_candidate_power_product)) * ff_v_bpcips_source_choice_valuation_candidate_power_product) + (ff_s_bpcips_source_choice_valuation_candidate_power_product))) /\ ff_s_bpcips_source_choice_valuation_candidate_power_product = ff_r_bpcips_source_choice_valuation_candidate_power_product * ff_p_bpcips_source_choice_valuation_candidate_power_product)))))))) /\ (exists bpr_divides_quotient_bpcips_source_choice_valuation_candidate_divides. n = (bpr_power_value_bpcips_source_choice_valuation_candidate) * bpr_divides_quotient_bpcips_source_choice_valuation_candidate_divides))) -> (exists bpr_le_gap_bpcips_source_choice_valuation_candidate_below. bpr_le_gap_bpcips_source_choice_valuation_candidate_below + (bpr_valuation_candidate_bpcips_source_choice_valuation) = (bpr_choice_exponent_bpcips_source_choice))) /\ (exists bpr_power_code_bpcips_source_choice_power bpr_power_scale_bpcips_source_choice_power. ((forall bpr_power_index_bpcips_source_choice_power. (exists bpr_gap_bpcips_source_choice_power_repeat_bound. bpr_gap_bpcips_source_choice_power_repeat_bound + S (bpr_power_index_bpcips_source_choice_power) = bpr_choice_exponent_bpcips_source_choice) -> (((exists bpr_height_bpcips_source_choice_power_repeat_entry. bpr_height_bpcips_source_choice_power_repeat_entry + S (S (bpr_prefix_index_bpcips_source)) = S ((S (bpr_power_index_bpcips_source_choice_power)) * bpr_power_scale_bpcips_source_choice_power)) /\ exists bpr_quotient_bpcips_source_choice_power_repeat_entry. bpr_power_code_bpcips_source_choice_power = bpr_quotient_bpcips_source_choice_power_repeat_entry * S ((S (bpr_power_index_bpcips_source_choice_power)) * bpr_power_scale_bpcips_source_choice_power) + (S (bpr_prefix_index_bpcips_source))))) /\ (exists ff_u_bpcips_source_choice_power_product ff_v_bpcips_source_choice_power_product. ((((exists ff_h_bpcips_source_choice_power_product_start. ff_h_bpcips_source_choice_power_product_start + S (1) = S ((S (0)) * ff_v_bpcips_source_choice_power_product)) /\ exists ff_q_bpcips_source_choice_power_product_start. ff_u_bpcips_source_choice_power_product = ff_q_bpcips_source_choice_power_product_start * S ((S (0)) * ff_v_bpcips_source_choice_power_product) + (1))) /\ ((((exists ff_h_bpcips_source_choice_power_product_terminal. ff_h_bpcips_source_choice_power_product_terminal + S (bpr_prefix_value_bpcips_source) = S ((S (bpr_choice_exponent_bpcips_source_choice)) * ff_v_bpcips_source_choice_power_product)) /\ exists ff_q_bpcips_source_choice_power_product_terminal. ff_u_bpcips_source_choice_power_product = ff_q_bpcips_source_choice_power_product_terminal * S ((S (bpr_choice_exponent_bpcips_source_choice)) * ff_v_bpcips_source_choice_power_product) + (bpr_prefix_value_bpcips_source))) /\ forall ff_i_bpcips_source_choice_power_product. (exists ff_lt_bpcips_source_choice_power_product_bound. ff_lt_bpcips_source_choice_power_product_bound + S ff_i_bpcips_source_choice_power_product = bpr_choice_exponent_bpcips_source_choice) -> exists ff_p_bpcips_source_choice_power_product ff_r_bpcips_source_choice_power_product ff_s_bpcips_source_choice_power_product. ((((exists ff_h_bpcips_source_choice_power_product_factor. ff_h_bpcips_source_choice_power_product_factor + S (ff_p_bpcips_source_choice_power_product) = S ((S (ff_i_bpcips_source_choice_power_product)) * bpr_power_scale_bpcips_source_choice_power)) /\ exists ff_q_bpcips_source_choice_power_product_factor. bpr_power_code_bpcips_source_choice_power = ff_q_bpcips_source_choice_power_product_factor * S ((S (ff_i_bpcips_source_choice_power_product)) * bpr_power_scale_bpcips_source_choice_power) + (ff_p_bpcips_source_choice_power_product))) /\ ((((exists ff_h_bpcips_source_choice_power_product_partial. ff_h_bpcips_source_choice_power_product_partial + S (ff_r_bpcips_source_choice_power_product) = S ((S (ff_i_bpcips_source_choice_power_product)) * ff_v_bpcips_source_choice_power_product)) /\ exists ff_q_bpcips_source_choice_power_product_partial. ff_u_bpcips_source_choice_power_product = ff_q_bpcips_source_choice_power_product_partial * S ((S (ff_i_bpcips_source_choice_power_product)) * ff_v_bpcips_source_choice_power_product) + (ff_r_bpcips_source_choice_power_product))) /\ ((((exists ff_h_bpcips_source_choice_power_product_successor. ff_h_bpcips_source_choice_power_product_successor + S (ff_s_bpcips_source_choice_power_product) = S ((S (S ff_i_bpcips_source_choice_power_product)) * ff_v_bpcips_source_choice_power_product)) /\ exists ff_q_bpcips_source_choice_power_product_successor. ff_u_bpcips_source_choice_power_product = ff_q_bpcips_source_choice_power_product_successor * S ((S (S ff_i_bpcips_source_choice_power_product)) * ff_v_bpcips_source_choice_power_product) + (ff_s_bpcips_source_choice_power_product))) /\ ff_s_bpcips_source_choice_power_product = ff_r_bpcips_source_choice_power_product * ff_p_bpcips_source_choice_power_product)))))))))) \/ (~((~(S (bpr_prefix_index_bpcips_source) = 1) /\ forall bpr_left_bpcips_source_choice_prime bpr_right_bpcips_source_choice_prime. S (bpr_prefix_index_bpcips_source) = bpr_left_bpcips_source_choice_prime * bpr_right_bpcips_source_choice_prime -> bpr_left_bpcips_source_choice_prime = 1 \/ bpr_right_bpcips_source_choice_prime = 1)) /\ bpr_prefix_value_bpcips_source = 1))))) -> (forall bpr_index_bpcips_interval. (exists bpr_gap_bpcips_interval_bound. bpr_gap_bpcips_interval_bound + S (bpr_index_bpcips_interval) = l) -> exists bpr_value_bpcips_interval. ((((exists bpr_height_bpcips_interval_decoded. bpr_height_bpcips_interval_decoded + S (bpr_value_bpcips_interval) = S ((S (bpr_index_bpcips_interval)) * e)) /\ exists bpr_quotient_bpcips_interval_decoded. d = bpr_quotient_bpcips_interval_decoded * S ((S (bpr_index_bpcips_interval)) * e) + (bpr_value_bpcips_interval))) /\ (((((~(S (a + bpr_index_bpcips_interval) = 1) /\ forall bpr_left_bpcips_interval_choice_prime bpr_right_bpcips_interval_choice_prime. S (a + bpr_index_bpcips_interval) = bpr_left_bpcips_interval_choice_prime * bpr_right_bpcips_interval_choice_prime -> bpr_left_bpcips_interval_choice_prime = 1 \/ bpr_right_bpcips_interval_choice_prime = 1)) /\ exists bpr_choice_exponent_bpcips_interval_choice. ((((exists bpr_le_gap_bpcips_interval_choice_valuation_selected_bound. bpr_le_gap_bpcips_interval_choice_valuation_selected_bound + (bpr_choice_exponent_bpcips_interval_choice) = (n)) /\ (exists bpr_power_value_bpcips_interval_choice_valuation_selected. ((exists bpr_power_code_bpcips_interval_choice_valuation_selected_power bpr_power_scale_bpcips_interval_choice_valuation_selected_power. ((forall bpr_power_index_bpcips_interval_choice_valuation_selected_power. (exists bpr_gap_bpcips_interval_choice_valuation_selected_power_repeat_bound. bpr_gap_bpcips_interval_choice_valuation_selected_power_repeat_bound + S (bpr_power_index_bpcips_interval_choice_valuation_selected_power) = bpr_choice_exponent_bpcips_interval_choice) -> (((exists bpr_height_bpcips_interval_choice_valuation_selected_power_repeat_entry. bpr_height_bpcips_interval_choice_valuation_selected_power_repeat_entry + S (S (a + bpr_index_bpcips_interval)) = S ((S (bpr_power_index_bpcips_interval_choice_valuation_selected_power)) * bpr_power_scale_bpcips_interval_choice_valuation_selected_power)) /\ exists bpr_quotient_bpcips_interval_choice_valuation_selected_power_repeat_entry. bpr_power_code_bpcips_interval_choice_valuation_selected_power = bpr_quotient_bpcips_interval_choice_valuation_selected_power_repeat_entry * S ((S (bpr_power_index_bpcips_interval_choice_valuation_selected_power)) * bpr_power_scale_bpcips_interval_choice_valuation_selected_power) + (S (a + bpr_index_bpcips_interval))))) /\ (exists ff_u_bpcips_interval_choice_valuation_selected_power_product ff_v_bpcips_interval_choice_valuation_selected_power_product. ((((exists ff_h_bpcips_interval_choice_valuation_selected_power_product_start. ff_h_bpcips_interval_choice_valuation_selected_power_product_start + S (1) = S ((S (0)) * ff_v_bpcips_interval_choice_valuation_selected_power_product)) /\ exists ff_q_bpcips_interval_choice_valuation_selected_power_product_start. ff_u_bpcips_interval_choice_valuation_selected_power_product = ff_q_bpcips_interval_choice_valuation_selected_power_product_start * S ((S (0)) * ff_v_bpcips_interval_choice_valuation_selected_power_product) + (1))) /\ ((((exists ff_h_bpcips_interval_choice_valuation_selected_power_product_terminal. ff_h_bpcips_interval_choice_valuation_selected_power_product_terminal + S (bpr_power_value_bpcips_interval_choice_valuation_selected) = S ((S (bpr_choice_exponent_bpcips_interval_choice)) * ff_v_bpcips_interval_choice_valuation_selected_power_product)) /\ exists ff_q_bpcips_interval_choice_valuation_selected_power_product_terminal. ff_u_bpcips_interval_choice_valuation_selected_power_product = ff_q_bpcips_interval_choice_valuation_selected_power_product_terminal * S ((S (bpr_choice_exponent_bpcips_interval_choice)) * ff_v_bpcips_interval_choice_valuation_selected_power_product) + (bpr_power_value_bpcips_interval_choice_valuation_selected))) /\ forall ff_i_bpcips_interval_choice_valuation_selected_power_product. (exists ff_lt_bpcips_interval_choice_valuation_selected_power_product_bound. ff_lt_bpcips_interval_choice_valuation_selected_power_product_bound + S ff_i_bpcips_interval_choice_valuation_selected_power_product = bpr_choice_exponent_bpcips_interval_choice) -> exists ff_p_bpcips_interval_choice_valuation_selected_power_product ff_r_bpcips_interval_choice_valuation_selected_power_product ff_s_bpcips_interval_choice_valuation_selected_power_product. ((((exists ff_h_bpcips_interval_choice_valuation_selected_power_product_factor. ff_h_bpcips_interval_choice_valuation_selected_power_product_factor + S (ff_p_bpcips_interval_choice_valuation_selected_power_product) = S ((S (ff_i_bpcips_interval_choice_valuation_selected_power_product)) * bpr_power_scale_bpcips_interval_choice_valuation_selected_power)) /\ exists ff_q_bpcips_interval_choice_valuation_selected_power_product_factor. bpr_power_code_bpcips_interval_choice_valuation_selected_power = ff_q_bpcips_interval_choice_valuation_selected_power_product_factor * S ((S (ff_i_bpcips_interval_choice_valuation_selected_power_product)) * bpr_power_scale_bpcips_interval_choice_valuation_selected_power) + (ff_p_bpcips_interval_choice_valuation_selected_power_product))) /\ ((((exists ff_h_bpcips_interval_choice_valuation_selected_power_product_partial. ff_h_bpcips_interval_choice_valuation_selected_power_product_partial + S (ff_r_bpcips_interval_choice_valuation_selected_power_product) = S ((S (ff_i_bpcips_interval_choice_valuation_selected_power_product)) * ff_v_bpcips_interval_choice_valuation_selected_power_product)) /\ exists ff_q_bpcips_interval_choice_valuation_selected_power_product_partial. ff_u_bpcips_interval_choice_valuation_selected_power_product = ff_q_bpcips_interval_choice_valuation_selected_power_product_partial * S ((S (ff_i_bpcips_interval_choice_valuation_selected_power_product)) * ff_v_bpcips_interval_choice_valuation_selected_power_product) + (ff_r_bpcips_interval_choice_valuation_selected_power_product))) /\ ((((exists ff_h_bpcips_interval_choice_valuation_selected_power_product_successor. ff_h_bpcips_interval_choice_valuation_selected_power_product_successor + S (ff_s_bpcips_interval_choice_valuation_selected_power_product) = S ((S (S ff_i_bpcips_interval_choice_valuation_selected_power_product)) * ff_v_bpcips_interval_choice_valuation_selected_power_product)) /\ exists ff_q_bpcips_interval_choice_valuation_selected_power_product_successor. ff_u_bpcips_interval_choice_valuation_selected_power_product = ff_q_bpcips_interval_choice_valuation_selected_power_product_successor * S ((S (S ff_i_bpcips_interval_choice_valuation_selected_power_product)) * ff_v_bpcips_interval_choice_valuation_selected_power_product) + (ff_s_bpcips_interval_choice_valuation_selected_power_product))) /\ ff_s_bpcips_interval_choice_valuation_selected_power_product = ff_r_bpcips_interval_choice_valuation_selected_power_product * ff_p_bpcips_interval_choice_valuation_selected_power_product)))))))) /\ (exists bpr_divides_quotient_bpcips_interval_choice_valuation_selected_divides. n = (bpr_power_value_bpcips_interval_choice_valuation_selected) * bpr_divides_quotient_bpcips_interval_choice_valuation_selected_divides)))) /\ forall bpr_valuation_candidate_bpcips_interval_choice_valuation. (exists bpr_le_gap_bpcips_interval_choice_valuation_candidate_bound. bpr_le_gap_bpcips_interval_choice_valuation_candidate_bound + (bpr_valuation_candidate_bpcips_interval_choice_valuation) = (n)) -> (exists bpr_power_value_bpcips_interval_choice_valuation_candidate. ((exists bpr_power_code_bpcips_interval_choice_valuation_candidate_power bpr_power_scale_bpcips_interval_choice_valuation_candidate_power. ((forall bpr_power_index_bpcips_interval_choice_valuation_candidate_power. (exists bpr_gap_bpcips_interval_choice_valuation_candidate_power_repeat_bound. bpr_gap_bpcips_interval_choice_valuation_candidate_power_repeat_bound + S (bpr_power_index_bpcips_interval_choice_valuation_candidate_power) = bpr_valuation_candidate_bpcips_interval_choice_valuation) -> (((exists bpr_height_bpcips_interval_choice_valuation_candidate_power_repeat_entry. bpr_height_bpcips_interval_choice_valuation_candidate_power_repeat_entry + S (S (a + bpr_index_bpcips_interval)) = S ((S (bpr_power_index_bpcips_interval_choice_valuation_candidate_power)) * bpr_power_scale_bpcips_interval_choice_valuation_candidate_power)) /\ exists bpr_quotient_bpcips_interval_choice_valuation_candidate_power_repeat_entry. bpr_power_code_bpcips_interval_choice_valuation_candidate_power = bpr_quotient_bpcips_interval_choice_valuation_candidate_power_repeat_entry * S ((S (bpr_power_index_bpcips_interval_choice_valuation_candidate_power)) * bpr_power_scale_bpcips_interval_choice_valuation_candidate_power) + (S (a + bpr_index_bpcips_interval))))) /\ (exists ff_u_bpcips_interval_choice_valuation_candidate_power_product ff_v_bpcips_interval_choice_valuation_candidate_power_product. ((((exists ff_h_bpcips_interval_choice_valuation_candidate_power_product_start. ff_h_bpcips_interval_choice_valuation_candidate_power_product_start + S (1) = S ((S (0)) * ff_v_bpcips_interval_choice_valuation_candidate_power_product)) /\ exists ff_q_bpcips_interval_choice_valuation_candidate_power_product_start. ff_u_bpcips_interval_choice_valuation_candidate_power_product = ff_q_bpcips_interval_choice_valuation_candidate_power_product_start * S ((S (0)) * ff_v_bpcips_interval_choice_valuation_candidate_power_product) + (1))) /\ ((((exists ff_h_bpcips_interval_choice_valuation_candidate_power_product_terminal. ff_h_bpcips_interval_choice_valuation_candidate_power_product_terminal + S (bpr_power_value_bpcips_interval_choice_valuation_candidate) = S ((S (bpr_valuation_candidate_bpcips_interval_choice_valuation)) * ff_v_bpcips_interval_choice_valuation_candidate_power_product)) /\ exists ff_q_bpcips_interval_choice_valuation_candidate_power_product_terminal. ff_u_bpcips_interval_choice_valuation_candidate_power_product = ff_q_bpcips_interval_choice_valuation_candidate_power_product_terminal * S ((S (bpr_valuation_candidate_bpcips_interval_choice_valuation)) * ff_v_bpcips_interval_choice_valuation_candidate_power_product) + (bpr_power_value_bpcips_interval_choice_valuation_candidate))) /\ forall ff_i_bpcips_interval_choice_valuation_candidate_power_product. (exists ff_lt_bpcips_interval_choice_valuation_candidate_power_product_bound. ff_lt_bpcips_interval_choice_valuation_candidate_power_product_bound + S ff_i_bpcips_interval_choice_valuation_candidate_power_product = bpr_valuation_candidate_bpcips_interval_choice_valuation) -> exists ff_p_bpcips_interval_choice_valuation_candidate_power_product ff_r_bpcips_interval_choice_valuation_candidate_power_product ff_s_bpcips_interval_choice_valuation_candidate_power_product. ((((exists ff_h_bpcips_interval_choice_valuation_candidate_power_product_factor. ff_h_bpcips_interval_choice_valuation_candidate_power_product_factor + S (ff_p_bpcips_interval_choice_valuation_candidate_power_product) = S ((S (ff_i_bpcips_interval_choice_valuation_candidate_power_product)) * bpr_power_scale_bpcips_interval_choice_valuation_candidate_power)) /\ exists ff_q_bpcips_interval_choice_valuation_candidate_power_product_factor. bpr_power_code_bpcips_interval_choice_valuation_candidate_power = ff_q_bpcips_interval_choice_valuation_candidate_power_product_factor * S ((S (ff_i_bpcips_interval_choice_valuation_candidate_power_product)) * bpr_power_scale_bpcips_interval_choice_valuation_candidate_power) + (ff_p_bpcips_interval_choice_valuation_candidate_power_product))) /\ ((((exists ff_h_bpcips_interval_choice_valuation_candidate_power_product_partial. ff_h_bpcips_interval_choice_valuation_candidate_power_product_partial + S (ff_r_bpcips_interval_choice_valuation_candidate_power_product) = S ((S (ff_i_bpcips_interval_choice_valuation_candidate_power_product)) * ff_v_bpcips_interval_choice_valuation_candidate_power_product)) /\ exists ff_q_bpcips_interval_choice_valuation_candidate_power_product_partial. ff_u_bpcips_interval_choice_valuation_candidate_power_product = ff_q_bpcips_interval_choice_valuation_candidate_power_product_partial * S ((S (ff_i_bpcips_interval_choice_valuation_candidate_power_product)) * ff_v_bpcips_interval_choice_valuation_candidate_power_product) + (ff_r_bpcips_interval_choice_valuation_candidate_power_product))) /\ ((((exists ff_h_bpcips_interval_choice_valuation_candidate_power_product_successor. ff_h_bpcips_interval_choice_valuation_candidate_power_product_successor + S (ff_s_bpcips_interval_choice_valuation_candidate_power_product) = S ((S (S ff_i_bpcips_interval_choice_valuation_candidate_power_product)) * ff_v_bpcips_interval_choice_valuation_candidate_power_product)) /\ exists ff_q_bpcips_interval_choice_valuation_candidate_power_product_successor. ff_u_bpcips_interval_choice_valuation_candidate_power_product = ff_q_bpcips_interval_choice_valuation_candidate_power_product_successor * S ((S (S ff_i_bpcips_interval_choice_valuation_candidate_power_product)) * ff_v_bpcips_interval_choice_valuation_candidate_power_product) + (ff_s_bpcips_interval_choice_valuation_candidate_power_product))) /\ ff_s_bpcips_interval_choice_valuation_candidate_power_product = ff_r_bpcips_interval_choice_valuation_candidate_power_product * ff_p_bpcips_interval_choice_valuation_candidate_power_product)))))))) /\ (exists bpr_divides_quotient_bpcips_interval_choice_valuation_candidate_divides. n = (bpr_power_value_bpcips_interval_choice_valuation_candidate) * bpr_divides_quotient_bpcips_interval_choice_valuation_candidate_divides))) -> (exists bpr_le_gap_bpcips_interval_choice_valuation_candidate_below. bpr_le_gap_bpcips_interval_choice_valuation_candidate_below + (bpr_valuation_candidate_bpcips_interval_choice_valuation) = (bpr_choice_exponent_bpcips_interval_choice))) /\ (exists bpr_power_code_bpcips_interval_choice_power bpr_power_scale_bpcips_interval_choice_power. ((forall bpr_power_index_bpcips_interval_choice_power. (exists bpr_gap_bpcips_interval_choice_power_repeat_bound. bpr_gap_bpcips_interval_choice_power_repeat_bound + S (bpr_power_index_bpcips_interval_choice_power) = bpr_choice_exponent_bpcips_interval_choice) -> (((exists bpr_height_bpcips_interval_choice_power_repeat_entry. bpr_height_bpcips_interval_choice_power_repeat_entry + S (S (a + bpr_index_bpcips_interval)) = S ((S (bpr_power_index_bpcips_interval_choice_power)) * bpr_power_scale_bpcips_interval_choice_power)) /\ exists bpr_quotient_bpcips_interval_choice_power_repeat_entry. bpr_power_code_bpcips_interval_choice_power = bpr_quotient_bpcips_interval_choice_power_repeat_entry * S ((S (bpr_power_index_bpcips_interval_choice_power)) * bpr_power_scale_bpcips_interval_choice_power) + (S (a + bpr_index_bpcips_interval))))) /\ (exists ff_u_bpcips_interval_choice_power_product ff_v_bpcips_interval_choice_power_product. ((((exists ff_h_bpcips_interval_choice_power_product_start. ff_h_bpcips_interval_choice_power_product_start + S (1) = S ((S (0)) * ff_v_bpcips_interval_choice_power_product)) /\ exists ff_q_bpcips_interval_choice_power_product_start. ff_u_bpcips_interval_choice_power_product = ff_q_bpcips_interval_choice_power_product_start * S ((S (0)) * ff_v_bpcips_interval_choice_power_product) + (1))) /\ ((((exists ff_h_bpcips_interval_choice_power_product_terminal. ff_h_bpcips_interval_choice_power_product_terminal + S (bpr_value_bpcips_interval) = S ((S (bpr_choice_exponent_bpcips_interval_choice)) * ff_v_bpcips_interval_choice_power_product)) /\ exists ff_q_bpcips_interval_choice_power_product_terminal. ff_u_bpcips_interval_choice_power_product = ff_q_bpcips_interval_choice_power_product_terminal * S ((S (bpr_choice_exponent_bpcips_interval_choice)) * ff_v_bpcips_interval_choice_power_product) + (bpr_value_bpcips_interval))) /\ forall ff_i_bpcips_interval_choice_power_product. (exists ff_lt_bpcips_interval_choice_power_product_bound. ff_lt_bpcips_interval_choice_power_product_bound + S ff_i_bpcips_interval_choice_power_product = bpr_choice_exponent_bpcips_interval_choice) -> exists ff_p_bpcips_interval_choice_power_product ff_r_bpcips_interval_choice_power_product ff_s_bpcips_interval_choice_power_product. ((((exists ff_h_bpcips_interval_choice_power_product_factor. ff_h_bpcips_interval_choice_power_product_factor + S (ff_p_bpcips_interval_choice_power_product) = S ((S (ff_i_bpcips_interval_choice_power_product)) * bpr_power_scale_bpcips_interval_choice_power)) /\ exists ff_q_bpcips_interval_choice_power_product_factor. bpr_power_code_bpcips_interval_choice_power = ff_q_bpcips_interval_choice_power_product_factor * S ((S (ff_i_bpcips_interval_choice_power_product)) * bpr_power_scale_bpcips_interval_choice_power) + (ff_p_bpcips_interval_choice_power_product))) /\ ((((exists ff_h_bpcips_interval_choice_power_product_partial. ff_h_bpcips_interval_choice_power_product_partial + S (ff_r_bpcips_interval_choice_power_product) = S ((S (ff_i_bpcips_interval_choice_power_product)) * ff_v_bpcips_interval_choice_power_product)) /\ exists ff_q_bpcips_interval_choice_power_product_partial. ff_u_bpcips_interval_choice_power_product = ff_q_bpcips_interval_choice_power_product_partial * S ((S (ff_i_bpcips_interval_choice_power_product)) * ff_v_bpcips_interval_choice_power_product) + (ff_r_bpcips_interval_choice_power_product))) /\ ((((exists ff_h_bpcips_interval_choice_power_product_successor. ff_h_bpcips_interval_choice_power_product_successor + S (ff_s_bpcips_interval_choice_power_product) = S ((S (S ff_i_bpcips_interval_choice_power_product)) * ff_v_bpcips_interval_choice_power_product)) /\ exists ff_q_bpcips_interval_choice_power_product_successor. ff_u_bpcips_interval_choice_power_product = ff_q_bpcips_interval_choice_power_product_successor * S ((S (S ff_i_bpcips_interval_choice_power_product)) * ff_v_bpcips_interval_choice_power_product) + (ff_s_bpcips_interval_choice_power_product))) /\ ff_s_bpcips_interval_choice_power_product = ff_r_bpcips_interval_choice_power_product * ff_p_bpcips_interval_choice_power_product)))))))))) \/ (~((~(S (a + bpr_index_bpcips_interval) = 1) /\ forall bpr_left_bpcips_interval_choice_prime bpr_right_bpcips_interval_choice_prime. S (a + bpr_index_bpcips_interval) = bpr_left_bpcips_interval_choice_prime * bpr_right_bpcips_interval_choice_prime -> bpr_left_bpcips_interval_choice_prime = 1 \/ bpr_right_bpcips_interval_choice_prime = 1)) /\ bpr_value_bpcips_interval = 1))))) -> forall i p. (exists bpr_gap_bpcips_bound. bpr_gap_bpcips_bound + S (i) = l) -> (((exists bpr_height_bpcips_source_entry. bpr_height_bpcips_source_entry + S (p) = S ((S (a + i)) * c)) /\ exists bpr_quotient_bpcips_source_entry. b = bpr_quotient_bpcips_source_entry * S ((S (a + i)) * c) + (p))) -> (((exists bpr_height_bpcips_target_entry. bpr_height_bpcips_target_entry + S (p) = S ((S (i)) * e)) /\ exists bpr_quotient_bpcips_target_entry. d = bpr_quotient_bpcips_target_entry * S ((S (i)) * e) + (p)))