power_valuation_mul_lower

Exact expanded PA statement

forall p a b e f g. ((~(p = 1) /\ forall frm_prime_left_bpd_prime frm_prime_right_bpd_prime. p = frm_prime_left_bpd_prime * frm_prime_right_bpd_prime -> frm_prime_left_bpd_prime = 1 \/ frm_prime_right_bpd_prime = 1)) -> ~(a = 0) -> ~(b = 0) -> (((exists bpv_gap_bpd_valuation_a_exponent_bound. bpv_gap_bpd_valuation_a_exponent_bound + e = a) /\ (exists bpv_result_bpd_valuation_a_selected. ((exists ff_b_bpd_valuation_a_selected_power ff_c_bpd_valuation_a_selected_power. ((forall ff_i_bpd_valuation_a_selected_power_repeat. (exists ff_lt_bpd_valuation_a_selected_power_repeat_bound. ff_lt_bpd_valuation_a_selected_power_repeat_bound + S ff_i_bpd_valuation_a_selected_power_repeat = e) -> (((exists ff_h_bpd_valuation_a_selected_power_repeat_decoded. ff_h_bpd_valuation_a_selected_power_repeat_decoded + S (p) = S ((S (ff_i_bpd_valuation_a_selected_power_repeat)) * ff_c_bpd_valuation_a_selected_power)) /\ exists ff_q_bpd_valuation_a_selected_power_repeat_decoded. ff_b_bpd_valuation_a_selected_power = ff_q_bpd_valuation_a_selected_power_repeat_decoded * S ((S (ff_i_bpd_valuation_a_selected_power_repeat)) * ff_c_bpd_valuation_a_selected_power) + (p)))) /\ (exists ff_u_bpd_valuation_a_selected_power_product ff_v_bpd_valuation_a_selected_power_product. ((((exists ff_h_bpd_valuation_a_selected_power_product_start. ff_h_bpd_valuation_a_selected_power_product_start + S (1) = S ((S (0)) * ff_v_bpd_valuation_a_selected_power_product)) /\ exists ff_q_bpd_valuation_a_selected_power_product_start. ff_u_bpd_valuation_a_selected_power_product = ff_q_bpd_valuation_a_selected_power_product_start * S ((S (0)) * ff_v_bpd_valuation_a_selected_power_product) + (1))) /\ ((((exists ff_h_bpd_valuation_a_selected_power_product_terminal. ff_h_bpd_valuation_a_selected_power_product_terminal + S (bpv_result_bpd_valuation_a_selected) = S ((S (e)) * ff_v_bpd_valuation_a_selected_power_product)) /\ exists ff_q_bpd_valuation_a_selected_power_product_terminal. ff_u_bpd_valuation_a_selected_power_product = ff_q_bpd_valuation_a_selected_power_product_terminal * S ((S (e)) * ff_v_bpd_valuation_a_selected_power_product) + (bpv_result_bpd_valuation_a_selected))) /\ forall ff_i_bpd_valuation_a_selected_power_product. (exists ff_lt_bpd_valuation_a_selected_power_product_bound. ff_lt_bpd_valuation_a_selected_power_product_bound + S ff_i_bpd_valuation_a_selected_power_product = e) -> exists ff_p_bpd_valuation_a_selected_power_product ff_r_bpd_valuation_a_selected_power_product ff_s_bpd_valuation_a_selected_power_product. ((((exists ff_h_bpd_valuation_a_selected_power_product_factor. ff_h_bpd_valuation_a_selected_power_product_factor + S (ff_p_bpd_valuation_a_selected_power_product) = S ((S (ff_i_bpd_valuation_a_selected_power_product)) * ff_c_bpd_valuation_a_selected_power)) /\ exists ff_q_bpd_valuation_a_selected_power_product_factor. ff_b_bpd_valuation_a_selected_power = ff_q_bpd_valuation_a_selected_power_product_factor * S ((S (ff_i_bpd_valuation_a_selected_power_product)) * ff_c_bpd_valuation_a_selected_power) + (ff_p_bpd_valuation_a_selected_power_product))) /\ ((((exists ff_h_bpd_valuation_a_selected_power_product_partial. ff_h_bpd_valuation_a_selected_power_product_partial + S (ff_r_bpd_valuation_a_selected_power_product) = S ((S (ff_i_bpd_valuation_a_selected_power_product)) * ff_v_bpd_valuation_a_selected_power_product)) /\ exists ff_q_bpd_valuation_a_selected_power_product_partial. ff_u_bpd_valuation_a_selected_power_product = ff_q_bpd_valuation_a_selected_power_product_partial * S ((S (ff_i_bpd_valuation_a_selected_power_product)) * ff_v_bpd_valuation_a_selected_power_product) + (ff_r_bpd_valuation_a_selected_power_product))) /\ ((((exists ff_h_bpd_valuation_a_selected_power_product_successor. ff_h_bpd_valuation_a_selected_power_product_successor + S (ff_s_bpd_valuation_a_selected_power_product) = S ((S (S ff_i_bpd_valuation_a_selected_power_product)) * ff_v_bpd_valuation_a_selected_power_product)) /\ exists ff_q_bpd_valuation_a_selected_power_product_successor. ff_u_bpd_valuation_a_selected_power_product = ff_q_bpd_valuation_a_selected_power_product_successor * S ((S (S ff_i_bpd_valuation_a_selected_power_product)) * ff_v_bpd_valuation_a_selected_power_product) + (ff_s_bpd_valuation_a_selected_power_product))) /\ ff_s_bpd_valuation_a_selected_power_product = ff_r_bpd_valuation_a_selected_power_product * ff_p_bpd_valuation_a_selected_power_product)))))))) /\ (exists bpv_factor_bpd_valuation_a_selected_divides. a = bpv_result_bpd_valuation_a_selected * bpv_factor_bpd_valuation_a_selected_divides)))) /\ forall bpv_candidate_bpd_valuation_a. (exists bpv_gap_bpd_valuation_a_candidate_bound. bpv_gap_bpd_valuation_a_candidate_bound + bpv_candidate_bpd_valuation_a = a) -> (exists bpv_result_bpd_valuation_a_candidate. ((exists ff_b_bpd_valuation_a_candidate_power ff_c_bpd_valuation_a_candidate_power. ((forall ff_i_bpd_valuation_a_candidate_power_repeat. (exists ff_lt_bpd_valuation_a_candidate_power_repeat_bound. ff_lt_bpd_valuation_a_candidate_power_repeat_bound + S ff_i_bpd_valuation_a_candidate_power_repeat = bpv_candidate_bpd_valuation_a) -> (((exists ff_h_bpd_valuation_a_candidate_power_repeat_decoded. ff_h_bpd_valuation_a_candidate_power_repeat_decoded + S (p) = S ((S (ff_i_bpd_valuation_a_candidate_power_repeat)) * ff_c_bpd_valuation_a_candidate_power)) /\ exists ff_q_bpd_valuation_a_candidate_power_repeat_decoded. ff_b_bpd_valuation_a_candidate_power = ff_q_bpd_valuation_a_candidate_power_repeat_decoded * S ((S (ff_i_bpd_valuation_a_candidate_power_repeat)) * ff_c_bpd_valuation_a_candidate_power) + (p)))) /\ (exists ff_u_bpd_valuation_a_candidate_power_product ff_v_bpd_valuation_a_candidate_power_product. ((((exists ff_h_bpd_valuation_a_candidate_power_product_start. ff_h_bpd_valuation_a_candidate_power_product_start + S (1) = S ((S (0)) * ff_v_bpd_valuation_a_candidate_power_product)) /\ exists ff_q_bpd_valuation_a_candidate_power_product_start. ff_u_bpd_valuation_a_candidate_power_product = ff_q_bpd_valuation_a_candidate_power_product_start * S ((S (0)) * ff_v_bpd_valuation_a_candidate_power_product) + (1))) /\ ((((exists ff_h_bpd_valuation_a_candidate_power_product_terminal. ff_h_bpd_valuation_a_candidate_power_product_terminal + S (bpv_result_bpd_valuation_a_candidate) = S ((S (bpv_candidate_bpd_valuation_a)) * ff_v_bpd_valuation_a_candidate_power_product)) /\ exists ff_q_bpd_valuation_a_candidate_power_product_terminal. ff_u_bpd_valuation_a_candidate_power_product = ff_q_bpd_valuation_a_candidate_power_product_terminal * S ((S (bpv_candidate_bpd_valuation_a)) * ff_v_bpd_valuation_a_candidate_power_product) + (bpv_result_bpd_valuation_a_candidate))) /\ forall ff_i_bpd_valuation_a_candidate_power_product. (exists ff_lt_bpd_valuation_a_candidate_power_product_bound. ff_lt_bpd_valuation_a_candidate_power_product_bound + S ff_i_bpd_valuation_a_candidate_power_product = bpv_candidate_bpd_valuation_a) -> exists ff_p_bpd_valuation_a_candidate_power_product ff_r_bpd_valuation_a_candidate_power_product ff_s_bpd_valuation_a_candidate_power_product. ((((exists ff_h_bpd_valuation_a_candidate_power_product_factor. ff_h_bpd_valuation_a_candidate_power_product_factor + S (ff_p_bpd_valuation_a_candidate_power_product) = S ((S (ff_i_bpd_valuation_a_candidate_power_product)) * ff_c_bpd_valuation_a_candidate_power)) /\ exists ff_q_bpd_valuation_a_candidate_power_product_factor. ff_b_bpd_valuation_a_candidate_power = ff_q_bpd_valuation_a_candidate_power_product_factor * S ((S (ff_i_bpd_valuation_a_candidate_power_product)) * ff_c_bpd_valuation_a_candidate_power) + (ff_p_bpd_valuation_a_candidate_power_product))) /\ ((((exists ff_h_bpd_valuation_a_candidate_power_product_partial. ff_h_bpd_valuation_a_candidate_power_product_partial + S (ff_r_bpd_valuation_a_candidate_power_product) = S ((S (ff_i_bpd_valuation_a_candidate_power_product)) * ff_v_bpd_valuation_a_candidate_power_product)) /\ exists ff_q_bpd_valuation_a_candidate_power_product_partial. ff_u_bpd_valuation_a_candidate_power_product = ff_q_bpd_valuation_a_candidate_power_product_partial * S ((S (ff_i_bpd_valuation_a_candidate_power_product)) * ff_v_bpd_valuation_a_candidate_power_product) + (ff_r_bpd_valuation_a_candidate_power_product))) /\ ((((exists ff_h_bpd_valuation_a_candidate_power_product_successor. ff_h_bpd_valuation_a_candidate_power_product_successor + S (ff_s_bpd_valuation_a_candidate_power_product) = S ((S (S ff_i_bpd_valuation_a_candidate_power_product)) * ff_v_bpd_valuation_a_candidate_power_product)) /\ exists ff_q_bpd_valuation_a_candidate_power_product_successor. ff_u_bpd_valuation_a_candidate_power_product = ff_q_bpd_valuation_a_candidate_power_product_successor * S ((S (S ff_i_bpd_valuation_a_candidate_power_product)) * ff_v_bpd_valuation_a_candidate_power_product) + (ff_s_bpd_valuation_a_candidate_power_product))) /\ ff_s_bpd_valuation_a_candidate_power_product = ff_r_bpd_valuation_a_candidate_power_product * ff_p_bpd_valuation_a_candidate_power_product)))))))) /\ (exists bpv_factor_bpd_valuation_a_candidate_divides. a = bpv_result_bpd_valuation_a_candidate * bpv_factor_bpd_valuation_a_candidate_divides))) -> (exists bpv_gap_bpd_valuation_a_maximal. bpv_gap_bpd_valuation_a_maximal + bpv_candidate_bpd_valuation_a = e)) -> (((exists bpv_gap_bpd_valuation_b_exponent_bound. bpv_gap_bpd_valuation_b_exponent_bound + f = b) /\ (exists bpv_result_bpd_valuation_b_selected. ((exists ff_b_bpd_valuation_b_selected_power ff_c_bpd_valuation_b_selected_power. ((forall ff_i_bpd_valuation_b_selected_power_repeat. (exists ff_lt_bpd_valuation_b_selected_power_repeat_bound. ff_lt_bpd_valuation_b_selected_power_repeat_bound + S ff_i_bpd_valuation_b_selected_power_repeat = f) -> (((exists ff_h_bpd_valuation_b_selected_power_repeat_decoded. ff_h_bpd_valuation_b_selected_power_repeat_decoded + S (p) = S ((S (ff_i_bpd_valuation_b_selected_power_repeat)) * ff_c_bpd_valuation_b_selected_power)) /\ exists ff_q_bpd_valuation_b_selected_power_repeat_decoded. ff_b_bpd_valuation_b_selected_power = ff_q_bpd_valuation_b_selected_power_repeat_decoded * S ((S (ff_i_bpd_valuation_b_selected_power_repeat)) * ff_c_bpd_valuation_b_selected_power) + (p)))) /\ (exists ff_u_bpd_valuation_b_selected_power_product ff_v_bpd_valuation_b_selected_power_product. ((((exists ff_h_bpd_valuation_b_selected_power_product_start. ff_h_bpd_valuation_b_selected_power_product_start + S (1) = S ((S (0)) * ff_v_bpd_valuation_b_selected_power_product)) /\ exists ff_q_bpd_valuation_b_selected_power_product_start. ff_u_bpd_valuation_b_selected_power_product = ff_q_bpd_valuation_b_selected_power_product_start * S ((S (0)) * ff_v_bpd_valuation_b_selected_power_product) + (1))) /\ ((((exists ff_h_bpd_valuation_b_selected_power_product_terminal. ff_h_bpd_valuation_b_selected_power_product_terminal + S (bpv_result_bpd_valuation_b_selected) = S ((S (f)) * ff_v_bpd_valuation_b_selected_power_product)) /\ exists ff_q_bpd_valuation_b_selected_power_product_terminal. ff_u_bpd_valuation_b_selected_power_product = ff_q_bpd_valuation_b_selected_power_product_terminal * S ((S (f)) * ff_v_bpd_valuation_b_selected_power_product) + (bpv_result_bpd_valuation_b_selected))) /\ forall ff_i_bpd_valuation_b_selected_power_product. (exists ff_lt_bpd_valuation_b_selected_power_product_bound. ff_lt_bpd_valuation_b_selected_power_product_bound + S ff_i_bpd_valuation_b_selected_power_product = f) -> exists ff_p_bpd_valuation_b_selected_power_product ff_r_bpd_valuation_b_selected_power_product ff_s_bpd_valuation_b_selected_power_product. ((((exists ff_h_bpd_valuation_b_selected_power_product_factor. ff_h_bpd_valuation_b_selected_power_product_factor + S (ff_p_bpd_valuation_b_selected_power_product) = S ((S (ff_i_bpd_valuation_b_selected_power_product)) * ff_c_bpd_valuation_b_selected_power)) /\ exists ff_q_bpd_valuation_b_selected_power_product_factor. ff_b_bpd_valuation_b_selected_power = ff_q_bpd_valuation_b_selected_power_product_factor * S ((S (ff_i_bpd_valuation_b_selected_power_product)) * ff_c_bpd_valuation_b_selected_power) + (ff_p_bpd_valuation_b_selected_power_product))) /\ ((((exists ff_h_bpd_valuation_b_selected_power_product_partial. ff_h_bpd_valuation_b_selected_power_product_partial + S (ff_r_bpd_valuation_b_selected_power_product) = S ((S (ff_i_bpd_valuation_b_selected_power_product)) * ff_v_bpd_valuation_b_selected_power_product)) /\ exists ff_q_bpd_valuation_b_selected_power_product_partial. ff_u_bpd_valuation_b_selected_power_product = ff_q_bpd_valuation_b_selected_power_product_partial * S ((S (ff_i_bpd_valuation_b_selected_power_product)) * ff_v_bpd_valuation_b_selected_power_product) + (ff_r_bpd_valuation_b_selected_power_product))) /\ ((((exists ff_h_bpd_valuation_b_selected_power_product_successor. ff_h_bpd_valuation_b_selected_power_product_successor + S (ff_s_bpd_valuation_b_selected_power_product) = S ((S (S ff_i_bpd_valuation_b_selected_power_product)) * ff_v_bpd_valuation_b_selected_power_product)) /\ exists ff_q_bpd_valuation_b_selected_power_product_successor. ff_u_bpd_valuation_b_selected_power_product = ff_q_bpd_valuation_b_selected_power_product_successor * S ((S (S ff_i_bpd_valuation_b_selected_power_product)) * ff_v_bpd_valuation_b_selected_power_product) + (ff_s_bpd_valuation_b_selected_power_product))) /\ ff_s_bpd_valuation_b_selected_power_product = ff_r_bpd_valuation_b_selected_power_product * ff_p_bpd_valuation_b_selected_power_product)))))))) /\ (exists bpv_factor_bpd_valuation_b_selected_divides. b = bpv_result_bpd_valuation_b_selected * bpv_factor_bpd_valuation_b_selected_divides)))) /\ forall bpv_candidate_bpd_valuation_b. (exists bpv_gap_bpd_valuation_b_candidate_bound. bpv_gap_bpd_valuation_b_candidate_bound + bpv_candidate_bpd_valuation_b = b) -> (exists bpv_result_bpd_valuation_b_candidate. ((exists ff_b_bpd_valuation_b_candidate_power ff_c_bpd_valuation_b_candidate_power. ((forall ff_i_bpd_valuation_b_candidate_power_repeat. (exists ff_lt_bpd_valuation_b_candidate_power_repeat_bound. ff_lt_bpd_valuation_b_candidate_power_repeat_bound + S ff_i_bpd_valuation_b_candidate_power_repeat = bpv_candidate_bpd_valuation_b) -> (((exists ff_h_bpd_valuation_b_candidate_power_repeat_decoded. ff_h_bpd_valuation_b_candidate_power_repeat_decoded + S (p) = S ((S (ff_i_bpd_valuation_b_candidate_power_repeat)) * ff_c_bpd_valuation_b_candidate_power)) /\ exists ff_q_bpd_valuation_b_candidate_power_repeat_decoded. ff_b_bpd_valuation_b_candidate_power = ff_q_bpd_valuation_b_candidate_power_repeat_decoded * S ((S (ff_i_bpd_valuation_b_candidate_power_repeat)) * ff_c_bpd_valuation_b_candidate_power) + (p)))) /\ (exists ff_u_bpd_valuation_b_candidate_power_product ff_v_bpd_valuation_b_candidate_power_product. ((((exists ff_h_bpd_valuation_b_candidate_power_product_start. ff_h_bpd_valuation_b_candidate_power_product_start + S (1) = S ((S (0)) * ff_v_bpd_valuation_b_candidate_power_product)) /\ exists ff_q_bpd_valuation_b_candidate_power_product_start. ff_u_bpd_valuation_b_candidate_power_product = ff_q_bpd_valuation_b_candidate_power_product_start * S ((S (0)) * ff_v_bpd_valuation_b_candidate_power_product) + (1))) /\ ((((exists ff_h_bpd_valuation_b_candidate_power_product_terminal. ff_h_bpd_valuation_b_candidate_power_product_terminal + S (bpv_result_bpd_valuation_b_candidate) = S ((S (bpv_candidate_bpd_valuation_b)) * ff_v_bpd_valuation_b_candidate_power_product)) /\ exists ff_q_bpd_valuation_b_candidate_power_product_terminal. ff_u_bpd_valuation_b_candidate_power_product = ff_q_bpd_valuation_b_candidate_power_product_terminal * S ((S (bpv_candidate_bpd_valuation_b)) * ff_v_bpd_valuation_b_candidate_power_product) + (bpv_result_bpd_valuation_b_candidate))) /\ forall ff_i_bpd_valuation_b_candidate_power_product. (exists ff_lt_bpd_valuation_b_candidate_power_product_bound. ff_lt_bpd_valuation_b_candidate_power_product_bound + S ff_i_bpd_valuation_b_candidate_power_product = bpv_candidate_bpd_valuation_b) -> exists ff_p_bpd_valuation_b_candidate_power_product ff_r_bpd_valuation_b_candidate_power_product ff_s_bpd_valuation_b_candidate_power_product. ((((exists ff_h_bpd_valuation_b_candidate_power_product_factor. ff_h_bpd_valuation_b_candidate_power_product_factor + S (ff_p_bpd_valuation_b_candidate_power_product) = S ((S (ff_i_bpd_valuation_b_candidate_power_product)) * ff_c_bpd_valuation_b_candidate_power)) /\ exists ff_q_bpd_valuation_b_candidate_power_product_factor. ff_b_bpd_valuation_b_candidate_power = ff_q_bpd_valuation_b_candidate_power_product_factor * S ((S (ff_i_bpd_valuation_b_candidate_power_product)) * ff_c_bpd_valuation_b_candidate_power) + (ff_p_bpd_valuation_b_candidate_power_product))) /\ ((((exists ff_h_bpd_valuation_b_candidate_power_product_partial. ff_h_bpd_valuation_b_candidate_power_product_partial + S (ff_r_bpd_valuation_b_candidate_power_product) = S ((S (ff_i_bpd_valuation_b_candidate_power_product)) * ff_v_bpd_valuation_b_candidate_power_product)) /\ exists ff_q_bpd_valuation_b_candidate_power_product_partial. ff_u_bpd_valuation_b_candidate_power_product = ff_q_bpd_valuation_b_candidate_power_product_partial * S ((S (ff_i_bpd_valuation_b_candidate_power_product)) * ff_v_bpd_valuation_b_candidate_power_product) + (ff_r_bpd_valuation_b_candidate_power_product))) /\ ((((exists ff_h_bpd_valuation_b_candidate_power_product_successor. ff_h_bpd_valuation_b_candidate_power_product_successor + S (ff_s_bpd_valuation_b_candidate_power_product) = S ((S (S ff_i_bpd_valuation_b_candidate_power_product)) * ff_v_bpd_valuation_b_candidate_power_product)) /\ exists ff_q_bpd_valuation_b_candidate_power_product_successor. ff_u_bpd_valuation_b_candidate_power_product = ff_q_bpd_valuation_b_candidate_power_product_successor * S ((S (S ff_i_bpd_valuation_b_candidate_power_product)) * ff_v_bpd_valuation_b_candidate_power_product) + (ff_s_bpd_valuation_b_candidate_power_product))) /\ ff_s_bpd_valuation_b_candidate_power_product = ff_r_bpd_valuation_b_candidate_power_product * ff_p_bpd_valuation_b_candidate_power_product)))))))) /\ (exists bpv_factor_bpd_valuation_b_candidate_divides. b = bpv_result_bpd_valuation_b_candidate * bpv_factor_bpd_valuation_b_candidate_divides))) -> (exists bpv_gap_bpd_valuation_b_maximal. bpv_gap_bpd_valuation_b_maximal + bpv_candidate_bpd_valuation_b = f)) -> (((exists bpd_gap_bpd_valuation_product_selected_bound. bpd_gap_bpd_valuation_product_selected_bound + (g) = (a * b)) /\ (exists bpvi_result_bpd_valuation_product_selected. ((exists bpvi_b_bpd_valuation_product_selected_power bpvi_c_bpd_valuation_product_selected_power. ((forall bpvi_i_bpd_valuation_product_selected_power. (exists bpvi_repeat_gap_bpd_valuation_product_selected_power. bpvi_repeat_gap_bpd_valuation_product_selected_power + S bpvi_i_bpd_valuation_product_selected_power = g) -> (((exists bpvi_h_bpd_valuation_product_selected_power_repeat. bpvi_h_bpd_valuation_product_selected_power_repeat + S (p) = S ((S (bpvi_i_bpd_valuation_product_selected_power)) * bpvi_c_bpd_valuation_product_selected_power)) /\ exists bpvi_q_bpd_valuation_product_selected_power_repeat. bpvi_b_bpd_valuation_product_selected_power = bpvi_q_bpd_valuation_product_selected_power_repeat * S ((S (bpvi_i_bpd_valuation_product_selected_power)) * bpvi_c_bpd_valuation_product_selected_power) + (p)))) /\ (exists bpvi_u_bpd_valuation_product_selected_power bpvi_v_bpd_valuation_product_selected_power. ((((exists bpvi_h_bpd_valuation_product_selected_power_start. bpvi_h_bpd_valuation_product_selected_power_start + S (1) = S ((S (0)) * bpvi_v_bpd_valuation_product_selected_power)) /\ exists bpvi_q_bpd_valuation_product_selected_power_start. bpvi_u_bpd_valuation_product_selected_power = bpvi_q_bpd_valuation_product_selected_power_start * S ((S (0)) * bpvi_v_bpd_valuation_product_selected_power) + (1))) /\ ((((exists bpvi_h_bpd_valuation_product_selected_power_terminal. bpvi_h_bpd_valuation_product_selected_power_terminal + S (bpvi_result_bpd_valuation_product_selected) = S ((S (g)) * bpvi_v_bpd_valuation_product_selected_power)) /\ exists bpvi_q_bpd_valuation_product_selected_power_terminal. bpvi_u_bpd_valuation_product_selected_power = bpvi_q_bpd_valuation_product_selected_power_terminal * S ((S (g)) * bpvi_v_bpd_valuation_product_selected_power) + (bpvi_result_bpd_valuation_product_selected))) /\ forall bpvi_j_bpd_valuation_product_selected_power. (exists bpvi_product_gap_bpd_valuation_product_selected_power. bpvi_product_gap_bpd_valuation_product_selected_power + S bpvi_j_bpd_valuation_product_selected_power = g) -> exists bpvi_factor_bpd_valuation_product_selected_power bpvi_partial_bpd_valuation_product_selected_power bpvi_successor_bpd_valuation_product_selected_power. ((((exists bpvi_h_bpd_valuation_product_selected_power_factor. bpvi_h_bpd_valuation_product_selected_power_factor + S (bpvi_factor_bpd_valuation_product_selected_power) = S ((S (bpvi_j_bpd_valuation_product_selected_power)) * bpvi_c_bpd_valuation_product_selected_power)) /\ exists bpvi_q_bpd_valuation_product_selected_power_factor. bpvi_b_bpd_valuation_product_selected_power = bpvi_q_bpd_valuation_product_selected_power_factor * S ((S (bpvi_j_bpd_valuation_product_selected_power)) * bpvi_c_bpd_valuation_product_selected_power) + (bpvi_factor_bpd_valuation_product_selected_power))) /\ ((((exists bpvi_h_bpd_valuation_product_selected_power_partial. bpvi_h_bpd_valuation_product_selected_power_partial + S (bpvi_partial_bpd_valuation_product_selected_power) = S ((S (bpvi_j_bpd_valuation_product_selected_power)) * bpvi_v_bpd_valuation_product_selected_power)) /\ exists bpvi_q_bpd_valuation_product_selected_power_partial. bpvi_u_bpd_valuation_product_selected_power = bpvi_q_bpd_valuation_product_selected_power_partial * S ((S (bpvi_j_bpd_valuation_product_selected_power)) * bpvi_v_bpd_valuation_product_selected_power) + (bpvi_partial_bpd_valuation_product_selected_power))) /\ ((((exists bpvi_h_bpd_valuation_product_selected_power_successor. bpvi_h_bpd_valuation_product_selected_power_successor + S (bpvi_successor_bpd_valuation_product_selected_power) = S ((S (S bpvi_j_bpd_valuation_product_selected_power)) * bpvi_v_bpd_valuation_product_selected_power)) /\ exists bpvi_q_bpd_valuation_product_selected_power_successor. bpvi_u_bpd_valuation_product_selected_power = bpvi_q_bpd_valuation_product_selected_power_successor * S ((S (S bpvi_j_bpd_valuation_product_selected_power)) * bpvi_v_bpd_valuation_product_selected_power) + (bpvi_successor_bpd_valuation_product_selected_power))) /\ bpvi_successor_bpd_valuation_product_selected_power = bpvi_partial_bpd_valuation_product_selected_power * bpvi_factor_bpd_valuation_product_selected_power)))))))) /\ exists bpvi_divisor_factor_bpd_valuation_product_selected. a * b = bpvi_result_bpd_valuation_product_selected * bpvi_divisor_factor_bpd_valuation_product_selected))) /\ forall bpd_candidate_bpd_valuation_product. (exists bpd_gap_bpd_valuation_product_candidate_bound. bpd_gap_bpd_valuation_product_candidate_bound + (bpd_candidate_bpd_valuation_product) = (a * b)) -> (exists bpvi_result_bpd_valuation_product_candidate. ((exists bpvi_b_bpd_valuation_product_candidate_power bpvi_c_bpd_valuation_product_candidate_power. ((forall bpvi_i_bpd_valuation_product_candidate_power. (exists bpvi_repeat_gap_bpd_valuation_product_candidate_power. bpvi_repeat_gap_bpd_valuation_product_candidate_power + S bpvi_i_bpd_valuation_product_candidate_power = bpd_candidate_bpd_valuation_product) -> (((exists bpvi_h_bpd_valuation_product_candidate_power_repeat. bpvi_h_bpd_valuation_product_candidate_power_repeat + S (p) = S ((S (bpvi_i_bpd_valuation_product_candidate_power)) * bpvi_c_bpd_valuation_product_candidate_power)) /\ exists bpvi_q_bpd_valuation_product_candidate_power_repeat. bpvi_b_bpd_valuation_product_candidate_power = bpvi_q_bpd_valuation_product_candidate_power_repeat * S ((S (bpvi_i_bpd_valuation_product_candidate_power)) * bpvi_c_bpd_valuation_product_candidate_power) + (p)))) /\ (exists bpvi_u_bpd_valuation_product_candidate_power bpvi_v_bpd_valuation_product_candidate_power. ((((exists bpvi_h_bpd_valuation_product_candidate_power_start. bpvi_h_bpd_valuation_product_candidate_power_start + S (1) = S ((S (0)) * bpvi_v_bpd_valuation_product_candidate_power)) /\ exists bpvi_q_bpd_valuation_product_candidate_power_start. bpvi_u_bpd_valuation_product_candidate_power = bpvi_q_bpd_valuation_product_candidate_power_start * S ((S (0)) * bpvi_v_bpd_valuation_product_candidate_power) + (1))) /\ ((((exists bpvi_h_bpd_valuation_product_candidate_power_terminal. bpvi_h_bpd_valuation_product_candidate_power_terminal + S (bpvi_result_bpd_valuation_product_candidate) = S ((S (bpd_candidate_bpd_valuation_product)) * bpvi_v_bpd_valuation_product_candidate_power)) /\ exists bpvi_q_bpd_valuation_product_candidate_power_terminal. bpvi_u_bpd_valuation_product_candidate_power = bpvi_q_bpd_valuation_product_candidate_power_terminal * S ((S (bpd_candidate_bpd_valuation_product)) * bpvi_v_bpd_valuation_product_candidate_power) + (bpvi_result_bpd_valuation_product_candidate))) /\ forall bpvi_j_bpd_valuation_product_candidate_power. (exists bpvi_product_gap_bpd_valuation_product_candidate_power. bpvi_product_gap_bpd_valuation_product_candidate_power + S bpvi_j_bpd_valuation_product_candidate_power = bpd_candidate_bpd_valuation_product) -> exists bpvi_factor_bpd_valuation_product_candidate_power bpvi_partial_bpd_valuation_product_candidate_power bpvi_successor_bpd_valuation_product_candidate_power. ((((exists bpvi_h_bpd_valuation_product_candidate_power_factor. bpvi_h_bpd_valuation_product_candidate_power_factor + S (bpvi_factor_bpd_valuation_product_candidate_power) = S ((S (bpvi_j_bpd_valuation_product_candidate_power)) * bpvi_c_bpd_valuation_product_candidate_power)) /\ exists bpvi_q_bpd_valuation_product_candidate_power_factor. bpvi_b_bpd_valuation_product_candidate_power = bpvi_q_bpd_valuation_product_candidate_power_factor * S ((S (bpvi_j_bpd_valuation_product_candidate_power)) * bpvi_c_bpd_valuation_product_candidate_power) + (bpvi_factor_bpd_valuation_product_candidate_power))) /\ ((((exists bpvi_h_bpd_valuation_product_candidate_power_partial. bpvi_h_bpd_valuation_product_candidate_power_partial + S (bpvi_partial_bpd_valuation_product_candidate_power) = S ((S (bpvi_j_bpd_valuation_product_candidate_power)) * bpvi_v_bpd_valuation_product_candidate_power)) /\ exists bpvi_q_bpd_valuation_product_candidate_power_partial. bpvi_u_bpd_valuation_product_candidate_power = bpvi_q_bpd_valuation_product_candidate_power_partial * S ((S (bpvi_j_bpd_valuation_product_candidate_power)) * bpvi_v_bpd_valuation_product_candidate_power) + (bpvi_partial_bpd_valuation_product_candidate_power))) /\ ((((exists bpvi_h_bpd_valuation_product_candidate_power_successor. bpvi_h_bpd_valuation_product_candidate_power_successor + S (bpvi_successor_bpd_valuation_product_candidate_power) = S ((S (S bpvi_j_bpd_valuation_product_candidate_power)) * bpvi_v_bpd_valuation_product_candidate_power)) /\ exists bpvi_q_bpd_valuation_product_candidate_power_successor. bpvi_u_bpd_valuation_product_candidate_power = bpvi_q_bpd_valuation_product_candidate_power_successor * S ((S (S bpvi_j_bpd_valuation_product_candidate_power)) * bpvi_v_bpd_valuation_product_candidate_power) + (bpvi_successor_bpd_valuation_product_candidate_power))) /\ bpvi_successor_bpd_valuation_product_candidate_power = bpvi_partial_bpd_valuation_product_candidate_power * bpvi_factor_bpd_valuation_product_candidate_power)))))))) /\ exists bpvi_divisor_factor_bpd_valuation_product_candidate. a * b = bpvi_result_bpd_valuation_product_candidate * bpvi_divisor_factor_bpd_valuation_product_candidate)) -> (exists bpd_gap_bpd_valuation_product_maximal. bpd_gap_bpd_valuation_product_maximal + (bpd_candidate_bpd_valuation_product) = (g))) -> (exists bpd_gap_valuation_mul_lower. bpd_gap_valuation_mul_lower + (e + f) = (g))