prime_contribution_prefix_extend

Exact expanded PA statement

forall n b c m. (forall bpr_prefix_index_bpcpe_before. (exists bpr_gap_bpcpe_before_bound. bpr_gap_bpcpe_before_bound + S (bpr_prefix_index_bpcpe_before) = m) -> exists bpr_prefix_value_bpcpe_before. ((((exists bpr_height_bpcpe_before_decoded. bpr_height_bpcpe_before_decoded + S (bpr_prefix_value_bpcpe_before) = S ((S (bpr_prefix_index_bpcpe_before)) * c)) /\ exists bpr_quotient_bpcpe_before_decoded. b = bpr_quotient_bpcpe_before_decoded * S ((S (bpr_prefix_index_bpcpe_before)) * c) + (bpr_prefix_value_bpcpe_before))) /\ (((((~(S (bpr_prefix_index_bpcpe_before) = 1) /\ forall bpr_left_bpcpe_before_choice_prime bpr_right_bpcpe_before_choice_prime. S (bpr_prefix_index_bpcpe_before) = bpr_left_bpcpe_before_choice_prime * bpr_right_bpcpe_before_choice_prime -> bpr_left_bpcpe_before_choice_prime = 1 \/ bpr_right_bpcpe_before_choice_prime = 1)) /\ exists bpr_choice_exponent_bpcpe_before_choice. ((((exists bpr_le_gap_bpcpe_before_choice_valuation_selected_bound. bpr_le_gap_bpcpe_before_choice_valuation_selected_bound + (bpr_choice_exponent_bpcpe_before_choice) = (n)) /\ (exists bpr_power_value_bpcpe_before_choice_valuation_selected. ((exists bpr_power_code_bpcpe_before_choice_valuation_selected_power bpr_power_scale_bpcpe_before_choice_valuation_selected_power. ((forall bpr_power_index_bpcpe_before_choice_valuation_selected_power. (exists bpr_gap_bpcpe_before_choice_valuation_selected_power_repeat_bound. bpr_gap_bpcpe_before_choice_valuation_selected_power_repeat_bound + S (bpr_power_index_bpcpe_before_choice_valuation_selected_power) = bpr_choice_exponent_bpcpe_before_choice) -> (((exists bpr_height_bpcpe_before_choice_valuation_selected_power_repeat_entry. bpr_height_bpcpe_before_choice_valuation_selected_power_repeat_entry + S (S (bpr_prefix_index_bpcpe_before)) = S ((S (bpr_power_index_bpcpe_before_choice_valuation_selected_power)) * bpr_power_scale_bpcpe_before_choice_valuation_selected_power)) /\ exists bpr_quotient_bpcpe_before_choice_valuation_selected_power_repeat_entry. bpr_power_code_bpcpe_before_choice_valuation_selected_power = bpr_quotient_bpcpe_before_choice_valuation_selected_power_repeat_entry * S ((S (bpr_power_index_bpcpe_before_choice_valuation_selected_power)) * bpr_power_scale_bpcpe_before_choice_valuation_selected_power) + (S (bpr_prefix_index_bpcpe_before))))) /\ (exists ff_u_bpcpe_before_choice_valuation_selected_power_product ff_v_bpcpe_before_choice_valuation_selected_power_product. ((((exists ff_h_bpcpe_before_choice_valuation_selected_power_product_start. ff_h_bpcpe_before_choice_valuation_selected_power_product_start + S (1) = S ((S (0)) * ff_v_bpcpe_before_choice_valuation_selected_power_product)) /\ exists ff_q_bpcpe_before_choice_valuation_selected_power_product_start. ff_u_bpcpe_before_choice_valuation_selected_power_product = ff_q_bpcpe_before_choice_valuation_selected_power_product_start * S ((S (0)) * ff_v_bpcpe_before_choice_valuation_selected_power_product) + (1))) /\ ((((exists ff_h_bpcpe_before_choice_valuation_selected_power_product_terminal. ff_h_bpcpe_before_choice_valuation_selected_power_product_terminal + S (bpr_power_value_bpcpe_before_choice_valuation_selected) = S ((S (bpr_choice_exponent_bpcpe_before_choice)) * ff_v_bpcpe_before_choice_valuation_selected_power_product)) /\ exists ff_q_bpcpe_before_choice_valuation_selected_power_product_terminal. ff_u_bpcpe_before_choice_valuation_selected_power_product = ff_q_bpcpe_before_choice_valuation_selected_power_product_terminal * S ((S (bpr_choice_exponent_bpcpe_before_choice)) * ff_v_bpcpe_before_choice_valuation_selected_power_product) + (bpr_power_value_bpcpe_before_choice_valuation_selected))) /\ forall ff_i_bpcpe_before_choice_valuation_selected_power_product. (exists ff_lt_bpcpe_before_choice_valuation_selected_power_product_bound. ff_lt_bpcpe_before_choice_valuation_selected_power_product_bound + S ff_i_bpcpe_before_choice_valuation_selected_power_product = bpr_choice_exponent_bpcpe_before_choice) -> exists ff_p_bpcpe_before_choice_valuation_selected_power_product ff_r_bpcpe_before_choice_valuation_selected_power_product ff_s_bpcpe_before_choice_valuation_selected_power_product. ((((exists ff_h_bpcpe_before_choice_valuation_selected_power_product_factor. ff_h_bpcpe_before_choice_valuation_selected_power_product_factor + S (ff_p_bpcpe_before_choice_valuation_selected_power_product) = S ((S (ff_i_bpcpe_before_choice_valuation_selected_power_product)) * bpr_power_scale_bpcpe_before_choice_valuation_selected_power)) /\ exists ff_q_bpcpe_before_choice_valuation_selected_power_product_factor. bpr_power_code_bpcpe_before_choice_valuation_selected_power = ff_q_bpcpe_before_choice_valuation_selected_power_product_factor * S ((S (ff_i_bpcpe_before_choice_valuation_selected_power_product)) * bpr_power_scale_bpcpe_before_choice_valuation_selected_power) + (ff_p_bpcpe_before_choice_valuation_selected_power_product))) /\ ((((exists ff_h_bpcpe_before_choice_valuation_selected_power_product_partial. ff_h_bpcpe_before_choice_valuation_selected_power_product_partial + S (ff_r_bpcpe_before_choice_valuation_selected_power_product) = S ((S (ff_i_bpcpe_before_choice_valuation_selected_power_product)) * ff_v_bpcpe_before_choice_valuation_selected_power_product)) /\ exists ff_q_bpcpe_before_choice_valuation_selected_power_product_partial. ff_u_bpcpe_before_choice_valuation_selected_power_product = ff_q_bpcpe_before_choice_valuation_selected_power_product_partial * S ((S (ff_i_bpcpe_before_choice_valuation_selected_power_product)) * ff_v_bpcpe_before_choice_valuation_selected_power_product) + (ff_r_bpcpe_before_choice_valuation_selected_power_product))) /\ ((((exists ff_h_bpcpe_before_choice_valuation_selected_power_product_successor. ff_h_bpcpe_before_choice_valuation_selected_power_product_successor + S (ff_s_bpcpe_before_choice_valuation_selected_power_product) = S ((S (S ff_i_bpcpe_before_choice_valuation_selected_power_product)) * ff_v_bpcpe_before_choice_valuation_selected_power_product)) /\ exists ff_q_bpcpe_before_choice_valuation_selected_power_product_successor. ff_u_bpcpe_before_choice_valuation_selected_power_product = ff_q_bpcpe_before_choice_valuation_selected_power_product_successor * S ((S (S ff_i_bpcpe_before_choice_valuation_selected_power_product)) * ff_v_bpcpe_before_choice_valuation_selected_power_product) + (ff_s_bpcpe_before_choice_valuation_selected_power_product))) /\ ff_s_bpcpe_before_choice_valuation_selected_power_product = ff_r_bpcpe_before_choice_valuation_selected_power_product * ff_p_bpcpe_before_choice_valuation_selected_power_product)))))))) /\ (exists bpr_divides_quotient_bpcpe_before_choice_valuation_selected_divides. n = (bpr_power_value_bpcpe_before_choice_valuation_selected) * bpr_divides_quotient_bpcpe_before_choice_valuation_selected_divides)))) /\ forall bpr_valuation_candidate_bpcpe_before_choice_valuation. (exists bpr_le_gap_bpcpe_before_choice_valuation_candidate_bound. bpr_le_gap_bpcpe_before_choice_valuation_candidate_bound + (bpr_valuation_candidate_bpcpe_before_choice_valuation) = (n)) -> (exists bpr_power_value_bpcpe_before_choice_valuation_candidate. ((exists bpr_power_code_bpcpe_before_choice_valuation_candidate_power bpr_power_scale_bpcpe_before_choice_valuation_candidate_power. ((forall bpr_power_index_bpcpe_before_choice_valuation_candidate_power. (exists bpr_gap_bpcpe_before_choice_valuation_candidate_power_repeat_bound. bpr_gap_bpcpe_before_choice_valuation_candidate_power_repeat_bound + S (bpr_power_index_bpcpe_before_choice_valuation_candidate_power) = bpr_valuation_candidate_bpcpe_before_choice_valuation) -> (((exists bpr_height_bpcpe_before_choice_valuation_candidate_power_repeat_entry. bpr_height_bpcpe_before_choice_valuation_candidate_power_repeat_entry + S (S (bpr_prefix_index_bpcpe_before)) = S ((S (bpr_power_index_bpcpe_before_choice_valuation_candidate_power)) * bpr_power_scale_bpcpe_before_choice_valuation_candidate_power)) /\ exists bpr_quotient_bpcpe_before_choice_valuation_candidate_power_repeat_entry. bpr_power_code_bpcpe_before_choice_valuation_candidate_power = bpr_quotient_bpcpe_before_choice_valuation_candidate_power_repeat_entry * S ((S (bpr_power_index_bpcpe_before_choice_valuation_candidate_power)) * bpr_power_scale_bpcpe_before_choice_valuation_candidate_power) + (S (bpr_prefix_index_bpcpe_before))))) /\ (exists ff_u_bpcpe_before_choice_valuation_candidate_power_product ff_v_bpcpe_before_choice_valuation_candidate_power_product. ((((exists ff_h_bpcpe_before_choice_valuation_candidate_power_product_start. ff_h_bpcpe_before_choice_valuation_candidate_power_product_start + S (1) = S ((S (0)) * ff_v_bpcpe_before_choice_valuation_candidate_power_product)) /\ exists ff_q_bpcpe_before_choice_valuation_candidate_power_product_start. ff_u_bpcpe_before_choice_valuation_candidate_power_product = ff_q_bpcpe_before_choice_valuation_candidate_power_product_start * S ((S (0)) * ff_v_bpcpe_before_choice_valuation_candidate_power_product) + (1))) /\ ((((exists ff_h_bpcpe_before_choice_valuation_candidate_power_product_terminal. ff_h_bpcpe_before_choice_valuation_candidate_power_product_terminal + S (bpr_power_value_bpcpe_before_choice_valuation_candidate) = S ((S (bpr_valuation_candidate_bpcpe_before_choice_valuation)) * ff_v_bpcpe_before_choice_valuation_candidate_power_product)) /\ exists ff_q_bpcpe_before_choice_valuation_candidate_power_product_terminal. ff_u_bpcpe_before_choice_valuation_candidate_power_product = ff_q_bpcpe_before_choice_valuation_candidate_power_product_terminal * S ((S (bpr_valuation_candidate_bpcpe_before_choice_valuation)) * ff_v_bpcpe_before_choice_valuation_candidate_power_product) + (bpr_power_value_bpcpe_before_choice_valuation_candidate))) /\ forall ff_i_bpcpe_before_choice_valuation_candidate_power_product. (exists ff_lt_bpcpe_before_choice_valuation_candidate_power_product_bound. ff_lt_bpcpe_before_choice_valuation_candidate_power_product_bound + S ff_i_bpcpe_before_choice_valuation_candidate_power_product = bpr_valuation_candidate_bpcpe_before_choice_valuation) -> exists ff_p_bpcpe_before_choice_valuation_candidate_power_product ff_r_bpcpe_before_choice_valuation_candidate_power_product ff_s_bpcpe_before_choice_valuation_candidate_power_product. ((((exists ff_h_bpcpe_before_choice_valuation_candidate_power_product_factor. ff_h_bpcpe_before_choice_valuation_candidate_power_product_factor + S (ff_p_bpcpe_before_choice_valuation_candidate_power_product) = S ((S (ff_i_bpcpe_before_choice_valuation_candidate_power_product)) * bpr_power_scale_bpcpe_before_choice_valuation_candidate_power)) /\ exists ff_q_bpcpe_before_choice_valuation_candidate_power_product_factor. bpr_power_code_bpcpe_before_choice_valuation_candidate_power = ff_q_bpcpe_before_choice_valuation_candidate_power_product_factor * S ((S (ff_i_bpcpe_before_choice_valuation_candidate_power_product)) * bpr_power_scale_bpcpe_before_choice_valuation_candidate_power) + (ff_p_bpcpe_before_choice_valuation_candidate_power_product))) /\ ((((exists ff_h_bpcpe_before_choice_valuation_candidate_power_product_partial. ff_h_bpcpe_before_choice_valuation_candidate_power_product_partial + S (ff_r_bpcpe_before_choice_valuation_candidate_power_product) = S ((S (ff_i_bpcpe_before_choice_valuation_candidate_power_product)) * ff_v_bpcpe_before_choice_valuation_candidate_power_product)) /\ exists ff_q_bpcpe_before_choice_valuation_candidate_power_product_partial. ff_u_bpcpe_before_choice_valuation_candidate_power_product = ff_q_bpcpe_before_choice_valuation_candidate_power_product_partial * S ((S (ff_i_bpcpe_before_choice_valuation_candidate_power_product)) * ff_v_bpcpe_before_choice_valuation_candidate_power_product) + (ff_r_bpcpe_before_choice_valuation_candidate_power_product))) /\ ((((exists ff_h_bpcpe_before_choice_valuation_candidate_power_product_successor. ff_h_bpcpe_before_choice_valuation_candidate_power_product_successor + S (ff_s_bpcpe_before_choice_valuation_candidate_power_product) = S ((S (S ff_i_bpcpe_before_choice_valuation_candidate_power_product)) * ff_v_bpcpe_before_choice_valuation_candidate_power_product)) /\ exists ff_q_bpcpe_before_choice_valuation_candidate_power_product_successor. ff_u_bpcpe_before_choice_valuation_candidate_power_product = ff_q_bpcpe_before_choice_valuation_candidate_power_product_successor * S ((S (S ff_i_bpcpe_before_choice_valuation_candidate_power_product)) * ff_v_bpcpe_before_choice_valuation_candidate_power_product) + (ff_s_bpcpe_before_choice_valuation_candidate_power_product))) /\ ff_s_bpcpe_before_choice_valuation_candidate_power_product = ff_r_bpcpe_before_choice_valuation_candidate_power_product * ff_p_bpcpe_before_choice_valuation_candidate_power_product)))))))) /\ (exists bpr_divides_quotient_bpcpe_before_choice_valuation_candidate_divides. n = (bpr_power_value_bpcpe_before_choice_valuation_candidate) * bpr_divides_quotient_bpcpe_before_choice_valuation_candidate_divides))) -> (exists bpr_le_gap_bpcpe_before_choice_valuation_candidate_below. bpr_le_gap_bpcpe_before_choice_valuation_candidate_below + (bpr_valuation_candidate_bpcpe_before_choice_valuation) = (bpr_choice_exponent_bpcpe_before_choice))) /\ (exists bpr_power_code_bpcpe_before_choice_power bpr_power_scale_bpcpe_before_choice_power. ((forall bpr_power_index_bpcpe_before_choice_power. (exists bpr_gap_bpcpe_before_choice_power_repeat_bound. bpr_gap_bpcpe_before_choice_power_repeat_bound + S (bpr_power_index_bpcpe_before_choice_power) = bpr_choice_exponent_bpcpe_before_choice) -> (((exists bpr_height_bpcpe_before_choice_power_repeat_entry. bpr_height_bpcpe_before_choice_power_repeat_entry + S (S (bpr_prefix_index_bpcpe_before)) = S ((S (bpr_power_index_bpcpe_before_choice_power)) * bpr_power_scale_bpcpe_before_choice_power)) /\ exists bpr_quotient_bpcpe_before_choice_power_repeat_entry. bpr_power_code_bpcpe_before_choice_power = bpr_quotient_bpcpe_before_choice_power_repeat_entry * S ((S (bpr_power_index_bpcpe_before_choice_power)) * bpr_power_scale_bpcpe_before_choice_power) + (S (bpr_prefix_index_bpcpe_before))))) /\ (exists ff_u_bpcpe_before_choice_power_product ff_v_bpcpe_before_choice_power_product. ((((exists ff_h_bpcpe_before_choice_power_product_start. ff_h_bpcpe_before_choice_power_product_start + S (1) = S ((S (0)) * ff_v_bpcpe_before_choice_power_product)) /\ exists ff_q_bpcpe_before_choice_power_product_start. ff_u_bpcpe_before_choice_power_product = ff_q_bpcpe_before_choice_power_product_start * S ((S (0)) * ff_v_bpcpe_before_choice_power_product) + (1))) /\ ((((exists ff_h_bpcpe_before_choice_power_product_terminal. ff_h_bpcpe_before_choice_power_product_terminal + S (bpr_prefix_value_bpcpe_before) = S ((S (bpr_choice_exponent_bpcpe_before_choice)) * ff_v_bpcpe_before_choice_power_product)) /\ exists ff_q_bpcpe_before_choice_power_product_terminal. ff_u_bpcpe_before_choice_power_product = ff_q_bpcpe_before_choice_power_product_terminal * S ((S (bpr_choice_exponent_bpcpe_before_choice)) * ff_v_bpcpe_before_choice_power_product) + (bpr_prefix_value_bpcpe_before))) /\ forall ff_i_bpcpe_before_choice_power_product. (exists ff_lt_bpcpe_before_choice_power_product_bound. ff_lt_bpcpe_before_choice_power_product_bound + S ff_i_bpcpe_before_choice_power_product = bpr_choice_exponent_bpcpe_before_choice) -> exists ff_p_bpcpe_before_choice_power_product ff_r_bpcpe_before_choice_power_product ff_s_bpcpe_before_choice_power_product. ((((exists ff_h_bpcpe_before_choice_power_product_factor. ff_h_bpcpe_before_choice_power_product_factor + S (ff_p_bpcpe_before_choice_power_product) = S ((S (ff_i_bpcpe_before_choice_power_product)) * bpr_power_scale_bpcpe_before_choice_power)) /\ exists ff_q_bpcpe_before_choice_power_product_factor. bpr_power_code_bpcpe_before_choice_power = ff_q_bpcpe_before_choice_power_product_factor * S ((S (ff_i_bpcpe_before_choice_power_product)) * bpr_power_scale_bpcpe_before_choice_power) + (ff_p_bpcpe_before_choice_power_product))) /\ ((((exists ff_h_bpcpe_before_choice_power_product_partial. ff_h_bpcpe_before_choice_power_product_partial + S (ff_r_bpcpe_before_choice_power_product) = S ((S (ff_i_bpcpe_before_choice_power_product)) * ff_v_bpcpe_before_choice_power_product)) /\ exists ff_q_bpcpe_before_choice_power_product_partial. ff_u_bpcpe_before_choice_power_product = ff_q_bpcpe_before_choice_power_product_partial * S ((S (ff_i_bpcpe_before_choice_power_product)) * ff_v_bpcpe_before_choice_power_product) + (ff_r_bpcpe_before_choice_power_product))) /\ ((((exists ff_h_bpcpe_before_choice_power_product_successor. ff_h_bpcpe_before_choice_power_product_successor + S (ff_s_bpcpe_before_choice_power_product) = S ((S (S ff_i_bpcpe_before_choice_power_product)) * ff_v_bpcpe_before_choice_power_product)) /\ exists ff_q_bpcpe_before_choice_power_product_successor. ff_u_bpcpe_before_choice_power_product = ff_q_bpcpe_before_choice_power_product_successor * S ((S (S ff_i_bpcpe_before_choice_power_product)) * ff_v_bpcpe_before_choice_power_product) + (ff_s_bpcpe_before_choice_power_product))) /\ ff_s_bpcpe_before_choice_power_product = ff_r_bpcpe_before_choice_power_product * ff_p_bpcpe_before_choice_power_product)))))))))) \/ (~((~(S (bpr_prefix_index_bpcpe_before) = 1) /\ forall bpr_left_bpcpe_before_choice_prime bpr_right_bpcpe_before_choice_prime. S (bpr_prefix_index_bpcpe_before) = bpr_left_bpcpe_before_choice_prime * bpr_right_bpcpe_before_choice_prime -> bpr_left_bpcpe_before_choice_prime = 1 \/ bpr_right_bpcpe_before_choice_prime = 1)) /\ bpr_prefix_value_bpcpe_before = 1))))) -> exists d e. (forall bpr_prefix_index_bpcpe_after. (exists bpr_gap_bpcpe_after_bound. bpr_gap_bpcpe_after_bound + S (bpr_prefix_index_bpcpe_after) = S m) -> exists bpr_prefix_value_bpcpe_after. ((((exists bpr_height_bpcpe_after_decoded. bpr_height_bpcpe_after_decoded + S (bpr_prefix_value_bpcpe_after) = S ((S (bpr_prefix_index_bpcpe_after)) * e)) /\ exists bpr_quotient_bpcpe_after_decoded. d = bpr_quotient_bpcpe_after_decoded * S ((S (bpr_prefix_index_bpcpe_after)) * e) + (bpr_prefix_value_bpcpe_after))) /\ (((((~(S (bpr_prefix_index_bpcpe_after) = 1) /\ forall bpr_left_bpcpe_after_choice_prime bpr_right_bpcpe_after_choice_prime. S (bpr_prefix_index_bpcpe_after) = bpr_left_bpcpe_after_choice_prime * bpr_right_bpcpe_after_choice_prime -> bpr_left_bpcpe_after_choice_prime = 1 \/ bpr_right_bpcpe_after_choice_prime = 1)) /\ exists bpr_choice_exponent_bpcpe_after_choice. ((((exists bpr_le_gap_bpcpe_after_choice_valuation_selected_bound. bpr_le_gap_bpcpe_after_choice_valuation_selected_bound + (bpr_choice_exponent_bpcpe_after_choice) = (n)) /\ (exists bpr_power_value_bpcpe_after_choice_valuation_selected. ((exists bpr_power_code_bpcpe_after_choice_valuation_selected_power bpr_power_scale_bpcpe_after_choice_valuation_selected_power. ((forall bpr_power_index_bpcpe_after_choice_valuation_selected_power. (exists bpr_gap_bpcpe_after_choice_valuation_selected_power_repeat_bound. bpr_gap_bpcpe_after_choice_valuation_selected_power_repeat_bound + S (bpr_power_index_bpcpe_after_choice_valuation_selected_power) = bpr_choice_exponent_bpcpe_after_choice) -> (((exists bpr_height_bpcpe_after_choice_valuation_selected_power_repeat_entry. bpr_height_bpcpe_after_choice_valuation_selected_power_repeat_entry + S (S (bpr_prefix_index_bpcpe_after)) = S ((S (bpr_power_index_bpcpe_after_choice_valuation_selected_power)) * bpr_power_scale_bpcpe_after_choice_valuation_selected_power)) /\ exists bpr_quotient_bpcpe_after_choice_valuation_selected_power_repeat_entry. bpr_power_code_bpcpe_after_choice_valuation_selected_power = bpr_quotient_bpcpe_after_choice_valuation_selected_power_repeat_entry * S ((S (bpr_power_index_bpcpe_after_choice_valuation_selected_power)) * bpr_power_scale_bpcpe_after_choice_valuation_selected_power) + (S (bpr_prefix_index_bpcpe_after))))) /\ (exists ff_u_bpcpe_after_choice_valuation_selected_power_product ff_v_bpcpe_after_choice_valuation_selected_power_product. ((((exists ff_h_bpcpe_after_choice_valuation_selected_power_product_start. ff_h_bpcpe_after_choice_valuation_selected_power_product_start + S (1) = S ((S (0)) * ff_v_bpcpe_after_choice_valuation_selected_power_product)) /\ exists ff_q_bpcpe_after_choice_valuation_selected_power_product_start. ff_u_bpcpe_after_choice_valuation_selected_power_product = ff_q_bpcpe_after_choice_valuation_selected_power_product_start * S ((S (0)) * ff_v_bpcpe_after_choice_valuation_selected_power_product) + (1))) /\ ((((exists ff_h_bpcpe_after_choice_valuation_selected_power_product_terminal. ff_h_bpcpe_after_choice_valuation_selected_power_product_terminal + S (bpr_power_value_bpcpe_after_choice_valuation_selected) = S ((S (bpr_choice_exponent_bpcpe_after_choice)) * ff_v_bpcpe_after_choice_valuation_selected_power_product)) /\ exists ff_q_bpcpe_after_choice_valuation_selected_power_product_terminal. ff_u_bpcpe_after_choice_valuation_selected_power_product = ff_q_bpcpe_after_choice_valuation_selected_power_product_terminal * S ((S (bpr_choice_exponent_bpcpe_after_choice)) * ff_v_bpcpe_after_choice_valuation_selected_power_product) + (bpr_power_value_bpcpe_after_choice_valuation_selected))) /\ forall ff_i_bpcpe_after_choice_valuation_selected_power_product. (exists ff_lt_bpcpe_after_choice_valuation_selected_power_product_bound. ff_lt_bpcpe_after_choice_valuation_selected_power_product_bound + S ff_i_bpcpe_after_choice_valuation_selected_power_product = bpr_choice_exponent_bpcpe_after_choice) -> exists ff_p_bpcpe_after_choice_valuation_selected_power_product ff_r_bpcpe_after_choice_valuation_selected_power_product ff_s_bpcpe_after_choice_valuation_selected_power_product. ((((exists ff_h_bpcpe_after_choice_valuation_selected_power_product_factor. ff_h_bpcpe_after_choice_valuation_selected_power_product_factor + S (ff_p_bpcpe_after_choice_valuation_selected_power_product) = S ((S (ff_i_bpcpe_after_choice_valuation_selected_power_product)) * bpr_power_scale_bpcpe_after_choice_valuation_selected_power)) /\ exists ff_q_bpcpe_after_choice_valuation_selected_power_product_factor. bpr_power_code_bpcpe_after_choice_valuation_selected_power = ff_q_bpcpe_after_choice_valuation_selected_power_product_factor * S ((S (ff_i_bpcpe_after_choice_valuation_selected_power_product)) * bpr_power_scale_bpcpe_after_choice_valuation_selected_power) + (ff_p_bpcpe_after_choice_valuation_selected_power_product))) /\ ((((exists ff_h_bpcpe_after_choice_valuation_selected_power_product_partial. ff_h_bpcpe_after_choice_valuation_selected_power_product_partial + S (ff_r_bpcpe_after_choice_valuation_selected_power_product) = S ((S (ff_i_bpcpe_after_choice_valuation_selected_power_product)) * ff_v_bpcpe_after_choice_valuation_selected_power_product)) /\ exists ff_q_bpcpe_after_choice_valuation_selected_power_product_partial. ff_u_bpcpe_after_choice_valuation_selected_power_product = ff_q_bpcpe_after_choice_valuation_selected_power_product_partial * S ((S (ff_i_bpcpe_after_choice_valuation_selected_power_product)) * ff_v_bpcpe_after_choice_valuation_selected_power_product) + (ff_r_bpcpe_after_choice_valuation_selected_power_product))) /\ ((((exists ff_h_bpcpe_after_choice_valuation_selected_power_product_successor. ff_h_bpcpe_after_choice_valuation_selected_power_product_successor + S (ff_s_bpcpe_after_choice_valuation_selected_power_product) = S ((S (S ff_i_bpcpe_after_choice_valuation_selected_power_product)) * ff_v_bpcpe_after_choice_valuation_selected_power_product)) /\ exists ff_q_bpcpe_after_choice_valuation_selected_power_product_successor. ff_u_bpcpe_after_choice_valuation_selected_power_product = ff_q_bpcpe_after_choice_valuation_selected_power_product_successor * S ((S (S ff_i_bpcpe_after_choice_valuation_selected_power_product)) * ff_v_bpcpe_after_choice_valuation_selected_power_product) + (ff_s_bpcpe_after_choice_valuation_selected_power_product))) /\ ff_s_bpcpe_after_choice_valuation_selected_power_product = ff_r_bpcpe_after_choice_valuation_selected_power_product * ff_p_bpcpe_after_choice_valuation_selected_power_product)))))))) /\ (exists bpr_divides_quotient_bpcpe_after_choice_valuation_selected_divides. n = (bpr_power_value_bpcpe_after_choice_valuation_selected) * bpr_divides_quotient_bpcpe_after_choice_valuation_selected_divides)))) /\ forall bpr_valuation_candidate_bpcpe_after_choice_valuation. (exists bpr_le_gap_bpcpe_after_choice_valuation_candidate_bound. bpr_le_gap_bpcpe_after_choice_valuation_candidate_bound + (bpr_valuation_candidate_bpcpe_after_choice_valuation) = (n)) -> (exists bpr_power_value_bpcpe_after_choice_valuation_candidate. ((exists bpr_power_code_bpcpe_after_choice_valuation_candidate_power bpr_power_scale_bpcpe_after_choice_valuation_candidate_power. ((forall bpr_power_index_bpcpe_after_choice_valuation_candidate_power. (exists bpr_gap_bpcpe_after_choice_valuation_candidate_power_repeat_bound. bpr_gap_bpcpe_after_choice_valuation_candidate_power_repeat_bound + S (bpr_power_index_bpcpe_after_choice_valuation_candidate_power) = bpr_valuation_candidate_bpcpe_after_choice_valuation) -> (((exists bpr_height_bpcpe_after_choice_valuation_candidate_power_repeat_entry. bpr_height_bpcpe_after_choice_valuation_candidate_power_repeat_entry + S (S (bpr_prefix_index_bpcpe_after)) = S ((S (bpr_power_index_bpcpe_after_choice_valuation_candidate_power)) * bpr_power_scale_bpcpe_after_choice_valuation_candidate_power)) /\ exists bpr_quotient_bpcpe_after_choice_valuation_candidate_power_repeat_entry. bpr_power_code_bpcpe_after_choice_valuation_candidate_power = bpr_quotient_bpcpe_after_choice_valuation_candidate_power_repeat_entry * S ((S (bpr_power_index_bpcpe_after_choice_valuation_candidate_power)) * bpr_power_scale_bpcpe_after_choice_valuation_candidate_power) + (S (bpr_prefix_index_bpcpe_after))))) /\ (exists ff_u_bpcpe_after_choice_valuation_candidate_power_product ff_v_bpcpe_after_choice_valuation_candidate_power_product. ((((exists ff_h_bpcpe_after_choice_valuation_candidate_power_product_start. ff_h_bpcpe_after_choice_valuation_candidate_power_product_start + S (1) = S ((S (0)) * ff_v_bpcpe_after_choice_valuation_candidate_power_product)) /\ exists ff_q_bpcpe_after_choice_valuation_candidate_power_product_start. ff_u_bpcpe_after_choice_valuation_candidate_power_product = ff_q_bpcpe_after_choice_valuation_candidate_power_product_start * S ((S (0)) * ff_v_bpcpe_after_choice_valuation_candidate_power_product) + (1))) /\ ((((exists ff_h_bpcpe_after_choice_valuation_candidate_power_product_terminal. ff_h_bpcpe_after_choice_valuation_candidate_power_product_terminal + S (bpr_power_value_bpcpe_after_choice_valuation_candidate) = S ((S (bpr_valuation_candidate_bpcpe_after_choice_valuation)) * ff_v_bpcpe_after_choice_valuation_candidate_power_product)) /\ exists ff_q_bpcpe_after_choice_valuation_candidate_power_product_terminal. ff_u_bpcpe_after_choice_valuation_candidate_power_product = ff_q_bpcpe_after_choice_valuation_candidate_power_product_terminal * S ((S (bpr_valuation_candidate_bpcpe_after_choice_valuation)) * ff_v_bpcpe_after_choice_valuation_candidate_power_product) + (bpr_power_value_bpcpe_after_choice_valuation_candidate))) /\ forall ff_i_bpcpe_after_choice_valuation_candidate_power_product. (exists ff_lt_bpcpe_after_choice_valuation_candidate_power_product_bound. ff_lt_bpcpe_after_choice_valuation_candidate_power_product_bound + S ff_i_bpcpe_after_choice_valuation_candidate_power_product = bpr_valuation_candidate_bpcpe_after_choice_valuation) -> exists ff_p_bpcpe_after_choice_valuation_candidate_power_product ff_r_bpcpe_after_choice_valuation_candidate_power_product ff_s_bpcpe_after_choice_valuation_candidate_power_product. ((((exists ff_h_bpcpe_after_choice_valuation_candidate_power_product_factor. ff_h_bpcpe_after_choice_valuation_candidate_power_product_factor + S (ff_p_bpcpe_after_choice_valuation_candidate_power_product) = S ((S (ff_i_bpcpe_after_choice_valuation_candidate_power_product)) * bpr_power_scale_bpcpe_after_choice_valuation_candidate_power)) /\ exists ff_q_bpcpe_after_choice_valuation_candidate_power_product_factor. bpr_power_code_bpcpe_after_choice_valuation_candidate_power = ff_q_bpcpe_after_choice_valuation_candidate_power_product_factor * S ((S (ff_i_bpcpe_after_choice_valuation_candidate_power_product)) * bpr_power_scale_bpcpe_after_choice_valuation_candidate_power) + (ff_p_bpcpe_after_choice_valuation_candidate_power_product))) /\ ((((exists ff_h_bpcpe_after_choice_valuation_candidate_power_product_partial. ff_h_bpcpe_after_choice_valuation_candidate_power_product_partial + S (ff_r_bpcpe_after_choice_valuation_candidate_power_product) = S ((S (ff_i_bpcpe_after_choice_valuation_candidate_power_product)) * ff_v_bpcpe_after_choice_valuation_candidate_power_product)) /\ exists ff_q_bpcpe_after_choice_valuation_candidate_power_product_partial. ff_u_bpcpe_after_choice_valuation_candidate_power_product = ff_q_bpcpe_after_choice_valuation_candidate_power_product_partial * S ((S (ff_i_bpcpe_after_choice_valuation_candidate_power_product)) * ff_v_bpcpe_after_choice_valuation_candidate_power_product) + (ff_r_bpcpe_after_choice_valuation_candidate_power_product))) /\ ((((exists ff_h_bpcpe_after_choice_valuation_candidate_power_product_successor. ff_h_bpcpe_after_choice_valuation_candidate_power_product_successor + S (ff_s_bpcpe_after_choice_valuation_candidate_power_product) = S ((S (S ff_i_bpcpe_after_choice_valuation_candidate_power_product)) * ff_v_bpcpe_after_choice_valuation_candidate_power_product)) /\ exists ff_q_bpcpe_after_choice_valuation_candidate_power_product_successor. ff_u_bpcpe_after_choice_valuation_candidate_power_product = ff_q_bpcpe_after_choice_valuation_candidate_power_product_successor * S ((S (S ff_i_bpcpe_after_choice_valuation_candidate_power_product)) * ff_v_bpcpe_after_choice_valuation_candidate_power_product) + (ff_s_bpcpe_after_choice_valuation_candidate_power_product))) /\ ff_s_bpcpe_after_choice_valuation_candidate_power_product = ff_r_bpcpe_after_choice_valuation_candidate_power_product * ff_p_bpcpe_after_choice_valuation_candidate_power_product)))))))) /\ (exists bpr_divides_quotient_bpcpe_after_choice_valuation_candidate_divides. n = (bpr_power_value_bpcpe_after_choice_valuation_candidate) * bpr_divides_quotient_bpcpe_after_choice_valuation_candidate_divides))) -> (exists bpr_le_gap_bpcpe_after_choice_valuation_candidate_below. bpr_le_gap_bpcpe_after_choice_valuation_candidate_below + (bpr_valuation_candidate_bpcpe_after_choice_valuation) = (bpr_choice_exponent_bpcpe_after_choice))) /\ (exists bpr_power_code_bpcpe_after_choice_power bpr_power_scale_bpcpe_after_choice_power. ((forall bpr_power_index_bpcpe_after_choice_power. (exists bpr_gap_bpcpe_after_choice_power_repeat_bound. bpr_gap_bpcpe_after_choice_power_repeat_bound + S (bpr_power_index_bpcpe_after_choice_power) = bpr_choice_exponent_bpcpe_after_choice) -> (((exists bpr_height_bpcpe_after_choice_power_repeat_entry. bpr_height_bpcpe_after_choice_power_repeat_entry + S (S (bpr_prefix_index_bpcpe_after)) = S ((S (bpr_power_index_bpcpe_after_choice_power)) * bpr_power_scale_bpcpe_after_choice_power)) /\ exists bpr_quotient_bpcpe_after_choice_power_repeat_entry. bpr_power_code_bpcpe_after_choice_power = bpr_quotient_bpcpe_after_choice_power_repeat_entry * S ((S (bpr_power_index_bpcpe_after_choice_power)) * bpr_power_scale_bpcpe_after_choice_power) + (S (bpr_prefix_index_bpcpe_after))))) /\ (exists ff_u_bpcpe_after_choice_power_product ff_v_bpcpe_after_choice_power_product. ((((exists ff_h_bpcpe_after_choice_power_product_start. ff_h_bpcpe_after_choice_power_product_start + S (1) = S ((S (0)) * ff_v_bpcpe_after_choice_power_product)) /\ exists ff_q_bpcpe_after_choice_power_product_start. ff_u_bpcpe_after_choice_power_product = ff_q_bpcpe_after_choice_power_product_start * S ((S (0)) * ff_v_bpcpe_after_choice_power_product) + (1))) /\ ((((exists ff_h_bpcpe_after_choice_power_product_terminal. ff_h_bpcpe_after_choice_power_product_terminal + S (bpr_prefix_value_bpcpe_after) = S ((S (bpr_choice_exponent_bpcpe_after_choice)) * ff_v_bpcpe_after_choice_power_product)) /\ exists ff_q_bpcpe_after_choice_power_product_terminal. ff_u_bpcpe_after_choice_power_product = ff_q_bpcpe_after_choice_power_product_terminal * S ((S (bpr_choice_exponent_bpcpe_after_choice)) * ff_v_bpcpe_after_choice_power_product) + (bpr_prefix_value_bpcpe_after))) /\ forall ff_i_bpcpe_after_choice_power_product. (exists ff_lt_bpcpe_after_choice_power_product_bound. ff_lt_bpcpe_after_choice_power_product_bound + S ff_i_bpcpe_after_choice_power_product = bpr_choice_exponent_bpcpe_after_choice) -> exists ff_p_bpcpe_after_choice_power_product ff_r_bpcpe_after_choice_power_product ff_s_bpcpe_after_choice_power_product. ((((exists ff_h_bpcpe_after_choice_power_product_factor. ff_h_bpcpe_after_choice_power_product_factor + S (ff_p_bpcpe_after_choice_power_product) = S ((S (ff_i_bpcpe_after_choice_power_product)) * bpr_power_scale_bpcpe_after_choice_power)) /\ exists ff_q_bpcpe_after_choice_power_product_factor. bpr_power_code_bpcpe_after_choice_power = ff_q_bpcpe_after_choice_power_product_factor * S ((S (ff_i_bpcpe_after_choice_power_product)) * bpr_power_scale_bpcpe_after_choice_power) + (ff_p_bpcpe_after_choice_power_product))) /\ ((((exists ff_h_bpcpe_after_choice_power_product_partial. ff_h_bpcpe_after_choice_power_product_partial + S (ff_r_bpcpe_after_choice_power_product) = S ((S (ff_i_bpcpe_after_choice_power_product)) * ff_v_bpcpe_after_choice_power_product)) /\ exists ff_q_bpcpe_after_choice_power_product_partial. ff_u_bpcpe_after_choice_power_product = ff_q_bpcpe_after_choice_power_product_partial * S ((S (ff_i_bpcpe_after_choice_power_product)) * ff_v_bpcpe_after_choice_power_product) + (ff_r_bpcpe_after_choice_power_product))) /\ ((((exists ff_h_bpcpe_after_choice_power_product_successor. ff_h_bpcpe_after_choice_power_product_successor + S (ff_s_bpcpe_after_choice_power_product) = S ((S (S ff_i_bpcpe_after_choice_power_product)) * ff_v_bpcpe_after_choice_power_product)) /\ exists ff_q_bpcpe_after_choice_power_product_successor. ff_u_bpcpe_after_choice_power_product = ff_q_bpcpe_after_choice_power_product_successor * S ((S (S ff_i_bpcpe_after_choice_power_product)) * ff_v_bpcpe_after_choice_power_product) + (ff_s_bpcpe_after_choice_power_product))) /\ ff_s_bpcpe_after_choice_power_product = ff_r_bpcpe_after_choice_power_product * ff_p_bpcpe_after_choice_power_product)))))))))) \/ (~((~(S (bpr_prefix_index_bpcpe_after) = 1) /\ forall bpr_left_bpcpe_after_choice_prime bpr_right_bpcpe_after_choice_prime. S (bpr_prefix_index_bpcpe_after) = bpr_left_bpcpe_after_choice_prime * bpr_right_bpcpe_after_choice_prime -> bpr_left_bpcpe_after_choice_prime = 1 \/ bpr_right_bpcpe_after_choice_prime = 1)) /\ bpr_prefix_value_bpcpe_after = 1)))))