beta_prime_product_valuation_from_sum

Exact expanded first-order arithmetic statement

forall p b c vb vc l z e. ((~(p = 1) /\ forall frm_prime_left_mkm_construct_prime frm_prime_right_mkm_construct_prime. p = frm_prime_left_mkm_construct_prime * frm_prime_right_mkm_construct_prime -> frm_prime_left_mkm_construct_prime = 1 \/ frm_prime_right_mkm_construct_prime = 1)) -> (forall gcrt_positive_index_mkm_construct_nonzero gcrt_positive_value_mkm_construct_nonzero. (exists ff_lt_gcrt_mkm_construct_nonzero_bound. ff_lt_gcrt_mkm_construct_nonzero_bound + S gcrt_positive_index_mkm_construct_nonzero = l) -> (((exists ff_h_gcrt_mkm_construct_nonzero_entry. ff_h_gcrt_mkm_construct_nonzero_entry + S (gcrt_positive_value_mkm_construct_nonzero) = S ((S (gcrt_positive_index_mkm_construct_nonzero)) * c)) /\ exists ff_q_gcrt_mkm_construct_nonzero_entry. b = ff_q_gcrt_mkm_construct_nonzero_entry * S ((S (gcrt_positive_index_mkm_construct_nonzero)) * c) + (gcrt_positive_value_mkm_construct_nonzero))) -> ~(gcrt_positive_value_mkm_construct_nonzero = 0)) -> (forall mkm_index_construct_valuations. (exists mkm_lt_construct_valuations_bound. mkm_lt_construct_valuations_bound + S (mkm_index_construct_valuations) = (l)) -> (exists mkm_value_construct_valuations_point mkm_exponent_construct_valuations_point. (((exists fs_h_mkm_construct_valuations_point_source. fs_h_mkm_construct_valuations_point_source + S (mkm_value_construct_valuations_point) = S ((S (mkm_index_construct_valuations)) * c)) /\ exists fs_q_mkm_construct_valuations_point_source. b = fs_q_mkm_construct_valuations_point_source * S ((S (mkm_index_construct_valuations)) * c) + (mkm_value_construct_valuations_point))) /\ ((((exists fs_h_mkm_construct_valuations_point_decoded. fs_h_mkm_construct_valuations_point_decoded + S (mkm_exponent_construct_valuations_point) = S ((S (mkm_index_construct_valuations)) * vc)) /\ exists fs_q_mkm_construct_valuations_point_decoded. vb = fs_q_mkm_construct_valuations_point_decoded * S ((S (mkm_index_construct_valuations)) * vc) + (mkm_exponent_construct_valuations_point))) /\ (((exists bpv_gap_mkm_construct_valuations_point_valuation_exponent_bound. bpv_gap_mkm_construct_valuations_point_valuation_exponent_bound + mkm_exponent_construct_valuations_point = (mkm_value_construct_valuations_point)) /\ (exists bpv_result_mkm_construct_valuations_point_valuation_selected. ((exists ff_b_mkm_construct_valuations_point_valuation_selected_power ff_c_mkm_construct_valuations_point_valuation_selected_power. ((forall ff_i_mkm_construct_valuations_point_valuation_selected_power_repeat. (exists ff_lt_mkm_construct_valuations_point_valuation_selected_power_repeat_bound. ff_lt_mkm_construct_valuations_point_valuation_selected_power_repeat_bound + S ff_i_mkm_construct_valuations_point_valuation_selected_power_repeat = mkm_exponent_construct_valuations_point) -> (((exists ff_h_mkm_construct_valuations_point_valuation_selected_power_repeat_decoded. ff_h_mkm_construct_valuations_point_valuation_selected_power_repeat_decoded + S (p) = S ((S (ff_i_mkm_construct_valuations_point_valuation_selected_power_repeat)) * ff_c_mkm_construct_valuations_point_valuation_selected_power)) /\ exists ff_q_mkm_construct_valuations_point_valuation_selected_power_repeat_decoded. ff_b_mkm_construct_valuations_point_valuation_selected_power = ff_q_mkm_construct_valuations_point_valuation_selected_power_repeat_decoded * S ((S (ff_i_mkm_construct_valuations_point_valuation_selected_power_repeat)) * ff_c_mkm_construct_valuations_point_valuation_selected_power) + (p)))) /\ (exists ff_u_mkm_construct_valuations_point_valuation_selected_power_product ff_v_mkm_construct_valuations_point_valuation_selected_power_product. ((((exists ff_h_mkm_construct_valuations_point_valuation_selected_power_product_start. ff_h_mkm_construct_valuations_point_valuation_selected_power_product_start + S (1) = S ((S (0)) * ff_v_mkm_construct_valuations_point_valuation_selected_power_product)) /\ exists ff_q_mkm_construct_valuations_point_valuation_selected_power_product_start. ff_u_mkm_construct_valuations_point_valuation_selected_power_product = ff_q_mkm_construct_valuations_point_valuation_selected_power_product_start * S ((S (0)) * ff_v_mkm_construct_valuations_point_valuation_selected_power_product) + (1))) /\ ((((exists ff_h_mkm_construct_valuations_point_valuation_selected_power_product_terminal. ff_h_mkm_construct_valuations_point_valuation_selected_power_product_terminal + S (bpv_result_mkm_construct_valuations_point_valuation_selected) = S ((S (mkm_exponent_construct_valuations_point)) * ff_v_mkm_construct_valuations_point_valuation_selected_power_product)) /\ exists ff_q_mkm_construct_valuations_point_valuation_selected_power_product_terminal. ff_u_mkm_construct_valuations_point_valuation_selected_power_product = ff_q_mkm_construct_valuations_point_valuation_selected_power_product_terminal * S ((S (mkm_exponent_construct_valuations_point)) * ff_v_mkm_construct_valuations_point_valuation_selected_power_product) + (bpv_result_mkm_construct_valuations_point_valuation_selected))) /\ forall ff_i_mkm_construct_valuations_point_valuation_selected_power_product. (exists ff_lt_mkm_construct_valuations_point_valuation_selected_power_product_bound. ff_lt_mkm_construct_valuations_point_valuation_selected_power_product_bound + S ff_i_mkm_construct_valuations_point_valuation_selected_power_product = mkm_exponent_construct_valuations_point) -> exists ff_p_mkm_construct_valuations_point_valuation_selected_power_product ff_r_mkm_construct_valuations_point_valuation_selected_power_product ff_s_mkm_construct_valuations_point_valuation_selected_power_product. ((((exists ff_h_mkm_construct_valuations_point_valuation_selected_power_product_factor. ff_h_mkm_construct_valuations_point_valuation_selected_power_product_factor + S (ff_p_mkm_construct_valuations_point_valuation_selected_power_product) = S ((S (ff_i_mkm_construct_valuations_point_valuation_selected_power_product)) * ff_c_mkm_construct_valuations_point_valuation_selected_power)) /\ exists ff_q_mkm_construct_valuations_point_valuation_selected_power_product_factor. ff_b_mkm_construct_valuations_point_valuation_selected_power = ff_q_mkm_construct_valuations_point_valuation_selected_power_product_factor * S ((S (ff_i_mkm_construct_valuations_point_valuation_selected_power_product)) * ff_c_mkm_construct_valuations_point_valuation_selected_power) + (ff_p_mkm_construct_valuations_point_valuation_selected_power_product))) /\ ((((exists ff_h_mkm_construct_valuations_point_valuation_selected_power_product_partial. ff_h_mkm_construct_valuations_point_valuation_selected_power_product_partial + S (ff_r_mkm_construct_valuations_point_valuation_selected_power_product) = S ((S (ff_i_mkm_construct_valuations_point_valuation_selected_power_product)) * ff_v_mkm_construct_valuations_point_valuation_selected_power_product)) /\ exists ff_q_mkm_construct_valuations_point_valuation_selected_power_product_partial. ff_u_mkm_construct_valuations_point_valuation_selected_power_product = ff_q_mkm_construct_valuations_point_valuation_selected_power_product_partial * S ((S (ff_i_mkm_construct_valuations_point_valuation_selected_power_product)) * ff_v_mkm_construct_valuations_point_valuation_selected_power_product) + (ff_r_mkm_construct_valuations_point_valuation_selected_power_product))) /\ ((((exists ff_h_mkm_construct_valuations_point_valuation_selected_power_product_successor. ff_h_mkm_construct_valuations_point_valuation_selected_power_product_successor + S (ff_s_mkm_construct_valuations_point_valuation_selected_power_product) = S ((S (S ff_i_mkm_construct_valuations_point_valuation_selected_power_product)) * ff_v_mkm_construct_valuations_point_valuation_selected_power_product)) /\ exists ff_q_mkm_construct_valuations_point_valuation_selected_power_product_successor. ff_u_mkm_construct_valuations_point_valuation_selected_power_product = ff_q_mkm_construct_valuations_point_valuation_selected_power_product_successor * S ((S (S ff_i_mkm_construct_valuations_point_valuation_selected_power_product)) * ff_v_mkm_construct_valuations_point_valuation_selected_power_product) + (ff_s_mkm_construct_valuations_point_valuation_selected_power_product))) /\ ff_s_mkm_construct_valuations_point_valuation_selected_power_product = ff_r_mkm_construct_valuations_point_valuation_selected_power_product * ff_p_mkm_construct_valuations_point_valuation_selected_power_product)))))))) /\ (exists bpv_factor_mkm_construct_valuations_point_valuation_selected_divides. (mkm_value_construct_valuations_point) = bpv_result_mkm_construct_valuations_point_valuation_selected * bpv_factor_mkm_construct_valuations_point_valuation_selected_divides)))) /\ forall bpv_candidate_mkm_construct_valuations_point_valuation. (exists bpv_gap_mkm_construct_valuations_point_valuation_candidate_bound. bpv_gap_mkm_construct_valuations_point_valuation_candidate_bound + bpv_candidate_mkm_construct_valuations_point_valuation = (mkm_value_construct_valuations_point)) -> (exists bpv_result_mkm_construct_valuations_point_valuation_candidate. ((exists ff_b_mkm_construct_valuations_point_valuation_candidate_power ff_c_mkm_construct_valuations_point_valuation_candidate_power. ((forall ff_i_mkm_construct_valuations_point_valuation_candidate_power_repeat. (exists ff_lt_mkm_construct_valuations_point_valuation_candidate_power_repeat_bound. ff_lt_mkm_construct_valuations_point_valuation_candidate_power_repeat_bound + S ff_i_mkm_construct_valuations_point_valuation_candidate_power_repeat = bpv_candidate_mkm_construct_valuations_point_valuation) -> (((exists ff_h_mkm_construct_valuations_point_valuation_candidate_power_repeat_decoded. ff_h_mkm_construct_valuations_point_valuation_candidate_power_repeat_decoded + S (p) = S ((S (ff_i_mkm_construct_valuations_point_valuation_candidate_power_repeat)) * ff_c_mkm_construct_valuations_point_valuation_candidate_power)) /\ exists ff_q_mkm_construct_valuations_point_valuation_candidate_power_repeat_decoded. ff_b_mkm_construct_valuations_point_valuation_candidate_power = ff_q_mkm_construct_valuations_point_valuation_candidate_power_repeat_decoded * S ((S (ff_i_mkm_construct_valuations_point_valuation_candidate_power_repeat)) * ff_c_mkm_construct_valuations_point_valuation_candidate_power) + (p)))) /\ (exists ff_u_mkm_construct_valuations_point_valuation_candidate_power_product ff_v_mkm_construct_valuations_point_valuation_candidate_power_product. ((((exists ff_h_mkm_construct_valuations_point_valuation_candidate_power_product_start. ff_h_mkm_construct_valuations_point_valuation_candidate_power_product_start + S (1) = S ((S (0)) * ff_v_mkm_construct_valuations_point_valuation_candidate_power_product)) /\ exists ff_q_mkm_construct_valuations_point_valuation_candidate_power_product_start. ff_u_mkm_construct_valuations_point_valuation_candidate_power_product = ff_q_mkm_construct_valuations_point_valuation_candidate_power_product_start * S ((S (0)) * ff_v_mkm_construct_valuations_point_valuation_candidate_power_product) + (1))) /\ ((((exists ff_h_mkm_construct_valuations_point_valuation_candidate_power_product_terminal. ff_h_mkm_construct_valuations_point_valuation_candidate_power_product_terminal + S (bpv_result_mkm_construct_valuations_point_valuation_candidate) = S ((S (bpv_candidate_mkm_construct_valuations_point_valuation)) * ff_v_mkm_construct_valuations_point_valuation_candidate_power_product)) /\ exists ff_q_mkm_construct_valuations_point_valuation_candidate_power_product_terminal. ff_u_mkm_construct_valuations_point_valuation_candidate_power_product = ff_q_mkm_construct_valuations_point_valuation_candidate_power_product_terminal * S ((S (bpv_candidate_mkm_construct_valuations_point_valuation)) * ff_v_mkm_construct_valuations_point_valuation_candidate_power_product) + (bpv_result_mkm_construct_valuations_point_valuation_candidate))) /\ forall ff_i_mkm_construct_valuations_point_valuation_candidate_power_product. (exists ff_lt_mkm_construct_valuations_point_valuation_candidate_power_product_bound. ff_lt_mkm_construct_valuations_point_valuation_candidate_power_product_bound + S ff_i_mkm_construct_valuations_point_valuation_candidate_power_product = bpv_candidate_mkm_construct_valuations_point_valuation) -> exists ff_p_mkm_construct_valuations_point_valuation_candidate_power_product ff_r_mkm_construct_valuations_point_valuation_candidate_power_product ff_s_mkm_construct_valuations_point_valuation_candidate_power_product. ((((exists ff_h_mkm_construct_valuations_point_valuation_candidate_power_product_factor. ff_h_mkm_construct_valuations_point_valuation_candidate_power_product_factor + S (ff_p_mkm_construct_valuations_point_valuation_candidate_power_product) = S ((S (ff_i_mkm_construct_valuations_point_valuation_candidate_power_product)) * ff_c_mkm_construct_valuations_point_valuation_candidate_power)) /\ exists ff_q_mkm_construct_valuations_point_valuation_candidate_power_product_factor. ff_b_mkm_construct_valuations_point_valuation_candidate_power = ff_q_mkm_construct_valuations_point_valuation_candidate_power_product_factor * S ((S (ff_i_mkm_construct_valuations_point_valuation_candidate_power_product)) * ff_c_mkm_construct_valuations_point_valuation_candidate_power) + (ff_p_mkm_construct_valuations_point_valuation_candidate_power_product))) /\ ((((exists ff_h_mkm_construct_valuations_point_valuation_candidate_power_product_partial. ff_h_mkm_construct_valuations_point_valuation_candidate_power_product_partial + S (ff_r_mkm_construct_valuations_point_valuation_candidate_power_product) = S ((S (ff_i_mkm_construct_valuations_point_valuation_candidate_power_product)) * ff_v_mkm_construct_valuations_point_valuation_candidate_power_product)) /\ exists ff_q_mkm_construct_valuations_point_valuation_candidate_power_product_partial. ff_u_mkm_construct_valuations_point_valuation_candidate_power_product = ff_q_mkm_construct_valuations_point_valuation_candidate_power_product_partial * S ((S (ff_i_mkm_construct_valuations_point_valuation_candidate_power_product)) * ff_v_mkm_construct_valuations_point_valuation_candidate_power_product) + (ff_r_mkm_construct_valuations_point_valuation_candidate_power_product))) /\ ((((exists ff_h_mkm_construct_valuations_point_valuation_candidate_power_product_successor. ff_h_mkm_construct_valuations_point_valuation_candidate_power_product_successor + S (ff_s_mkm_construct_valuations_point_valuation_candidate_power_product) = S ((S (S ff_i_mkm_construct_valuations_point_valuation_candidate_power_product)) * ff_v_mkm_construct_valuations_point_valuation_candidate_power_product)) /\ exists ff_q_mkm_construct_valuations_point_valuation_candidate_power_product_successor. ff_u_mkm_construct_valuations_point_valuation_candidate_power_product = ff_q_mkm_construct_valuations_point_valuation_candidate_power_product_successor * S ((S (S ff_i_mkm_construct_valuations_point_valuation_candidate_power_product)) * ff_v_mkm_construct_valuations_point_valuation_candidate_power_product) + (ff_s_mkm_construct_valuations_point_valuation_candidate_power_product))) /\ ff_s_mkm_construct_valuations_point_valuation_candidate_power_product = ff_r_mkm_construct_valuations_point_valuation_candidate_power_product * ff_p_mkm_construct_valuations_point_valuation_candidate_power_product)))))))) /\ (exists bpv_factor_mkm_construct_valuations_point_valuation_candidate_divides. (mkm_value_construct_valuations_point) = bpv_result_mkm_construct_valuations_point_valuation_candidate * bpv_factor_mkm_construct_valuations_point_valuation_candidate_divides))) -> (exists bpv_gap_mkm_construct_valuations_point_valuation_maximal. bpv_gap_mkm_construct_valuations_point_valuation_maximal + bpv_candidate_mkm_construct_valuations_point_valuation = mkm_exponent_construct_valuations_point))))) -> (exists ff_u_mkm_construct_product ff_v_mkm_construct_product. ((((exists ff_h_mkm_construct_product_start. ff_h_mkm_construct_product_start + S (1) = S ((S (0)) * ff_v_mkm_construct_product)) /\ exists ff_q_mkm_construct_product_start. ff_u_mkm_construct_product = ff_q_mkm_construct_product_start * S ((S (0)) * ff_v_mkm_construct_product) + (1))) /\ ((((exists ff_h_mkm_construct_product_terminal. ff_h_mkm_construct_product_terminal + S (z) = S ((S (l)) * ff_v_mkm_construct_product)) /\ exists ff_q_mkm_construct_product_terminal. ff_u_mkm_construct_product = ff_q_mkm_construct_product_terminal * S ((S (l)) * ff_v_mkm_construct_product) + (z))) /\ forall ff_i_mkm_construct_product. (exists ff_lt_mkm_construct_product_bound. ff_lt_mkm_construct_product_bound + S ff_i_mkm_construct_product = l) -> exists ff_p_mkm_construct_product ff_r_mkm_construct_product ff_s_mkm_construct_product. ((((exists ff_h_mkm_construct_product_factor. ff_h_mkm_construct_product_factor + S (ff_p_mkm_construct_product) = S ((S (ff_i_mkm_construct_product)) * c)) /\ exists ff_q_mkm_construct_product_factor. b = ff_q_mkm_construct_product_factor * S ((S (ff_i_mkm_construct_product)) * c) + (ff_p_mkm_construct_product))) /\ ((((exists ff_h_mkm_construct_product_partial. ff_h_mkm_construct_product_partial + S (ff_r_mkm_construct_product) = S ((S (ff_i_mkm_construct_product)) * ff_v_mkm_construct_product)) /\ exists ff_q_mkm_construct_product_partial. ff_u_mkm_construct_product = ff_q_mkm_construct_product_partial * S ((S (ff_i_mkm_construct_product)) * ff_v_mkm_construct_product) + (ff_r_mkm_construct_product))) /\ ((((exists ff_h_mkm_construct_product_successor. ff_h_mkm_construct_product_successor + S (ff_s_mkm_construct_product) = S ((S (S ff_i_mkm_construct_product)) * ff_v_mkm_construct_product)) /\ exists ff_q_mkm_construct_product_successor. ff_u_mkm_construct_product = ff_q_mkm_construct_product_successor * S ((S (S ff_i_mkm_construct_product)) * ff_v_mkm_construct_product) + (ff_s_mkm_construct_product))) /\ ff_s_mkm_construct_product = ff_r_mkm_construct_product * ff_p_mkm_construct_product)))))) -> (exists fs_u_mkm_construct_sum fs_v_mkm_construct_sum. ((((exists fs_h_mkm_construct_sum_body_start. fs_h_mkm_construct_sum_body_start + S (0) = S ((S (0)) * fs_v_mkm_construct_sum)) /\ exists fs_q_mkm_construct_sum_body_start. fs_u_mkm_construct_sum = fs_q_mkm_construct_sum_body_start * S ((S (0)) * fs_v_mkm_construct_sum) + (0))) /\ ((((exists fs_h_mkm_construct_sum_body_terminal. fs_h_mkm_construct_sum_body_terminal + S (e) = S ((S (l)) * fs_v_mkm_construct_sum)) /\ exists fs_q_mkm_construct_sum_body_terminal. fs_u_mkm_construct_sum = fs_q_mkm_construct_sum_body_terminal * S ((S (l)) * fs_v_mkm_construct_sum) + (e))) /\ forall fs_i_mkm_construct_sum_body_steps. (exists fs_lt_mkm_construct_sum_body_steps_bound. fs_lt_mkm_construct_sum_body_steps_bound + S fs_i_mkm_construct_sum_body_steps = l) -> exists fs_a_mkm_construct_sum_body_steps fs_r_mkm_construct_sum_body_steps fs_s_mkm_construct_sum_body_steps. ((((exists fs_h_mkm_construct_sum_body_steps_summand. fs_h_mkm_construct_sum_body_steps_summand + S (fs_a_mkm_construct_sum_body_steps) = S ((S (fs_i_mkm_construct_sum_body_steps)) * vc)) /\ exists fs_q_mkm_construct_sum_body_steps_summand. vb = fs_q_mkm_construct_sum_body_steps_summand * S ((S (fs_i_mkm_construct_sum_body_steps)) * vc) + (fs_a_mkm_construct_sum_body_steps))) /\ ((((exists fs_h_mkm_construct_sum_body_steps_partial. fs_h_mkm_construct_sum_body_steps_partial + S (fs_r_mkm_construct_sum_body_steps) = S ((S (fs_i_mkm_construct_sum_body_steps)) * fs_v_mkm_construct_sum)) /\ exists fs_q_mkm_construct_sum_body_steps_partial. fs_u_mkm_construct_sum = fs_q_mkm_construct_sum_body_steps_partial * S ((S (fs_i_mkm_construct_sum_body_steps)) * fs_v_mkm_construct_sum) + (fs_r_mkm_construct_sum_body_steps))) /\ ((((exists fs_h_mkm_construct_sum_body_steps_successor. fs_h_mkm_construct_sum_body_steps_successor + S (fs_s_mkm_construct_sum_body_steps) = S ((S (S fs_i_mkm_construct_sum_body_steps)) * fs_v_mkm_construct_sum)) /\ exists fs_q_mkm_construct_sum_body_steps_successor. fs_u_mkm_construct_sum = fs_q_mkm_construct_sum_body_steps_successor * S ((S (S fs_i_mkm_construct_sum_body_steps)) * fs_v_mkm_construct_sum) + (fs_s_mkm_construct_sum_body_steps))) /\ fs_s_mkm_construct_sum_body_steps = fs_r_mkm_construct_sum_body_steps + fs_a_mkm_construct_sum_body_steps)))))) -> (((exists bpv_gap_mkm_construct_val_exponent_bound. bpv_gap_mkm_construct_val_exponent_bound + e = (z)) /\ (exists bpv_result_mkm_construct_val_selected. ((exists ff_b_mkm_construct_val_selected_power ff_c_mkm_construct_val_selected_power. ((forall ff_i_mkm_construct_val_selected_power_repeat. (exists ff_lt_mkm_construct_val_selected_power_repeat_bound. ff_lt_mkm_construct_val_selected_power_repeat_bound + S ff_i_mkm_construct_val_selected_power_repeat = e) -> (((exists ff_h_mkm_construct_val_selected_power_repeat_decoded. ff_h_mkm_construct_val_selected_power_repeat_decoded + S (p) = S ((S (ff_i_mkm_construct_val_selected_power_repeat)) * ff_c_mkm_construct_val_selected_power)) /\ exists ff_q_mkm_construct_val_selected_power_repeat_decoded. ff_b_mkm_construct_val_selected_power = ff_q_mkm_construct_val_selected_power_repeat_decoded * S ((S (ff_i_mkm_construct_val_selected_power_repeat)) * ff_c_mkm_construct_val_selected_power) + (p)))) /\ (exists ff_u_mkm_construct_val_selected_power_product ff_v_mkm_construct_val_selected_power_product. ((((exists ff_h_mkm_construct_val_selected_power_product_start. ff_h_mkm_construct_val_selected_power_product_start + S (1) = S ((S (0)) * ff_v_mkm_construct_val_selected_power_product)) /\ exists ff_q_mkm_construct_val_selected_power_product_start. ff_u_mkm_construct_val_selected_power_product = ff_q_mkm_construct_val_selected_power_product_start * S ((S (0)) * ff_v_mkm_construct_val_selected_power_product) + (1))) /\ ((((exists ff_h_mkm_construct_val_selected_power_product_terminal. ff_h_mkm_construct_val_selected_power_product_terminal + S (bpv_result_mkm_construct_val_selected) = S ((S (e)) * ff_v_mkm_construct_val_selected_power_product)) /\ exists ff_q_mkm_construct_val_selected_power_product_terminal. ff_u_mkm_construct_val_selected_power_product = ff_q_mkm_construct_val_selected_power_product_terminal * S ((S (e)) * ff_v_mkm_construct_val_selected_power_product) + (bpv_result_mkm_construct_val_selected))) /\ forall ff_i_mkm_construct_val_selected_power_product. (exists ff_lt_mkm_construct_val_selected_power_product_bound. ff_lt_mkm_construct_val_selected_power_product_bound + S ff_i_mkm_construct_val_selected_power_product = e) -> exists ff_p_mkm_construct_val_selected_power_product ff_r_mkm_construct_val_selected_power_product ff_s_mkm_construct_val_selected_power_product. ((((exists ff_h_mkm_construct_val_selected_power_product_factor. ff_h_mkm_construct_val_selected_power_product_factor + S (ff_p_mkm_construct_val_selected_power_product) = S ((S (ff_i_mkm_construct_val_selected_power_product)) * ff_c_mkm_construct_val_selected_power)) /\ exists ff_q_mkm_construct_val_selected_power_product_factor. ff_b_mkm_construct_val_selected_power = ff_q_mkm_construct_val_selected_power_product_factor * S ((S (ff_i_mkm_construct_val_selected_power_product)) * ff_c_mkm_construct_val_selected_power) + (ff_p_mkm_construct_val_selected_power_product))) /\ ((((exists ff_h_mkm_construct_val_selected_power_product_partial. ff_h_mkm_construct_val_selected_power_product_partial + S (ff_r_mkm_construct_val_selected_power_product) = S ((S (ff_i_mkm_construct_val_selected_power_product)) * ff_v_mkm_construct_val_selected_power_product)) /\ exists ff_q_mkm_construct_val_selected_power_product_partial. ff_u_mkm_construct_val_selected_power_product = ff_q_mkm_construct_val_selected_power_product_partial * S ((S (ff_i_mkm_construct_val_selected_power_product)) * ff_v_mkm_construct_val_selected_power_product) + (ff_r_mkm_construct_val_selected_power_product))) /\ ((((exists ff_h_mkm_construct_val_selected_power_product_successor. ff_h_mkm_construct_val_selected_power_product_successor + S (ff_s_mkm_construct_val_selected_power_product) = S ((S (S ff_i_mkm_construct_val_selected_power_product)) * ff_v_mkm_construct_val_selected_power_product)) /\ exists ff_q_mkm_construct_val_selected_power_product_successor. ff_u_mkm_construct_val_selected_power_product = ff_q_mkm_construct_val_selected_power_product_successor * S ((S (S ff_i_mkm_construct_val_selected_power_product)) * ff_v_mkm_construct_val_selected_power_product) + (ff_s_mkm_construct_val_selected_power_product))) /\ ff_s_mkm_construct_val_selected_power_product = ff_r_mkm_construct_val_selected_power_product * ff_p_mkm_construct_val_selected_power_product)))))))) /\ (exists bpv_factor_mkm_construct_val_selected_divides. (z) = bpv_result_mkm_construct_val_selected * bpv_factor_mkm_construct_val_selected_divides)))) /\ forall bpv_candidate_mkm_construct_val. (exists bpv_gap_mkm_construct_val_candidate_bound. bpv_gap_mkm_construct_val_candidate_bound + bpv_candidate_mkm_construct_val = (z)) -> (exists bpv_result_mkm_construct_val_candidate. ((exists ff_b_mkm_construct_val_candidate_power ff_c_mkm_construct_val_candidate_power. ((forall ff_i_mkm_construct_val_candidate_power_repeat. (exists ff_lt_mkm_construct_val_candidate_power_repeat_bound. ff_lt_mkm_construct_val_candidate_power_repeat_bound + S ff_i_mkm_construct_val_candidate_power_repeat = bpv_candidate_mkm_construct_val) -> (((exists ff_h_mkm_construct_val_candidate_power_repeat_decoded. ff_h_mkm_construct_val_candidate_power_repeat_decoded + S (p) = S ((S (ff_i_mkm_construct_val_candidate_power_repeat)) * ff_c_mkm_construct_val_candidate_power)) /\ exists ff_q_mkm_construct_val_candidate_power_repeat_decoded. ff_b_mkm_construct_val_candidate_power = ff_q_mkm_construct_val_candidate_power_repeat_decoded * S ((S (ff_i_mkm_construct_val_candidate_power_repeat)) * ff_c_mkm_construct_val_candidate_power) + (p)))) /\ (exists ff_u_mkm_construct_val_candidate_power_product ff_v_mkm_construct_val_candidate_power_product. ((((exists ff_h_mkm_construct_val_candidate_power_product_start. ff_h_mkm_construct_val_candidate_power_product_start + S (1) = S ((S (0)) * ff_v_mkm_construct_val_candidate_power_product)) /\ exists ff_q_mkm_construct_val_candidate_power_product_start. ff_u_mkm_construct_val_candidate_power_product = ff_q_mkm_construct_val_candidate_power_product_start * S ((S (0)) * ff_v_mkm_construct_val_candidate_power_product) + (1))) /\ ((((exists ff_h_mkm_construct_val_candidate_power_product_terminal. ff_h_mkm_construct_val_candidate_power_product_terminal + S (bpv_result_mkm_construct_val_candidate) = S ((S (bpv_candidate_mkm_construct_val)) * ff_v_mkm_construct_val_candidate_power_product)) /\ exists ff_q_mkm_construct_val_candidate_power_product_terminal. ff_u_mkm_construct_val_candidate_power_product = ff_q_mkm_construct_val_candidate_power_product_terminal * S ((S (bpv_candidate_mkm_construct_val)) * ff_v_mkm_construct_val_candidate_power_product) + (bpv_result_mkm_construct_val_candidate))) /\ forall ff_i_mkm_construct_val_candidate_power_product. (exists ff_lt_mkm_construct_val_candidate_power_product_bound. ff_lt_mkm_construct_val_candidate_power_product_bound + S ff_i_mkm_construct_val_candidate_power_product = bpv_candidate_mkm_construct_val) -> exists ff_p_mkm_construct_val_candidate_power_product ff_r_mkm_construct_val_candidate_power_product ff_s_mkm_construct_val_candidate_power_product. ((((exists ff_h_mkm_construct_val_candidate_power_product_factor. ff_h_mkm_construct_val_candidate_power_product_factor + S (ff_p_mkm_construct_val_candidate_power_product) = S ((S (ff_i_mkm_construct_val_candidate_power_product)) * ff_c_mkm_construct_val_candidate_power)) /\ exists ff_q_mkm_construct_val_candidate_power_product_factor. ff_b_mkm_construct_val_candidate_power = ff_q_mkm_construct_val_candidate_power_product_factor * S ((S (ff_i_mkm_construct_val_candidate_power_product)) * ff_c_mkm_construct_val_candidate_power) + (ff_p_mkm_construct_val_candidate_power_product))) /\ ((((exists ff_h_mkm_construct_val_candidate_power_product_partial. ff_h_mkm_construct_val_candidate_power_product_partial + S (ff_r_mkm_construct_val_candidate_power_product) = S ((S (ff_i_mkm_construct_val_candidate_power_product)) * ff_v_mkm_construct_val_candidate_power_product)) /\ exists ff_q_mkm_construct_val_candidate_power_product_partial. ff_u_mkm_construct_val_candidate_power_product = ff_q_mkm_construct_val_candidate_power_product_partial * S ((S (ff_i_mkm_construct_val_candidate_power_product)) * ff_v_mkm_construct_val_candidate_power_product) + (ff_r_mkm_construct_val_candidate_power_product))) /\ ((((exists ff_h_mkm_construct_val_candidate_power_product_successor. ff_h_mkm_construct_val_candidate_power_product_successor + S (ff_s_mkm_construct_val_candidate_power_product) = S ((S (S ff_i_mkm_construct_val_candidate_power_product)) * ff_v_mkm_construct_val_candidate_power_product)) /\ exists ff_q_mkm_construct_val_candidate_power_product_successor. ff_u_mkm_construct_val_candidate_power_product = ff_q_mkm_construct_val_candidate_power_product_successor * S ((S (S ff_i_mkm_construct_val_candidate_power_product)) * ff_v_mkm_construct_val_candidate_power_product) + (ff_s_mkm_construct_val_candidate_power_product))) /\ ff_s_mkm_construct_val_candidate_power_product = ff_r_mkm_construct_val_candidate_power_product * ff_p_mkm_construct_val_candidate_power_product)))))))) /\ (exists bpv_factor_mkm_construct_val_candidate_divides. (z) = bpv_result_mkm_construct_val_candidate * bpv_factor_mkm_construct_val_candidate_divides))) -> (exists bpv_gap_mkm_construct_val_maximal. bpv_gap_mkm_construct_val_maximal + bpv_candidate_mkm_construct_val = e))