prime_valuation_divisible_power_root_exists

Exact expanded first-order arithmetic statement

forall n k. ~(n = 0) -> ~(k = 0) -> (forall ppf_prime_root_unbounded_divisibility ppf_exponent_root_unbounded_divisibility. (~((ppf_prime_root_unbounded_divisibility) = 1) /\ forall pvs_left_root_unbounded_divisibilitydomain pvs_right_root_unbounded_divisibilitydomain. (ppf_prime_root_unbounded_divisibility) = pvs_left_root_unbounded_divisibilitydomain * pvs_right_root_unbounded_divisibilitydomain -> pvs_left_root_unbounded_divisibilitydomain = 1 \/ pvs_right_root_unbounded_divisibilitydomain = 1) -> (((exists bpd_gap_pvs_root_unbounded_divisibilityvaluation_selected_bound. bpd_gap_pvs_root_unbounded_divisibilityvaluation_selected_bound + (ppf_exponent_root_unbounded_divisibility) = (n)) /\ (exists bpvi_result_pvs_root_unbounded_divisibilityvaluation_selected. ((exists bpvi_b_pvs_root_unbounded_divisibilityvaluation_selected_power bpvi_c_pvs_root_unbounded_divisibilityvaluation_selected_power. ((forall bpvi_i_pvs_root_unbounded_divisibilityvaluation_selected_power. (exists bpvi_repeat_gap_pvs_root_unbounded_divisibilityvaluation_selected_power. bpvi_repeat_gap_pvs_root_unbounded_divisibilityvaluation_selected_power + S bpvi_i_pvs_root_unbounded_divisibilityvaluation_selected_power = ppf_exponent_root_unbounded_divisibility) -> (((exists bpvi_h_pvs_root_unbounded_divisibilityvaluation_selected_power_repeat. bpvi_h_pvs_root_unbounded_divisibilityvaluation_selected_power_repeat + S (ppf_prime_root_unbounded_divisibility) = S ((S (bpvi_i_pvs_root_unbounded_divisibilityvaluation_selected_power)) * bpvi_c_pvs_root_unbounded_divisibilityvaluation_selected_power)) /\ exists bpvi_q_pvs_root_unbounded_divisibilityvaluation_selected_power_repeat. bpvi_b_pvs_root_unbounded_divisibilityvaluation_selected_power = bpvi_q_pvs_root_unbounded_divisibilityvaluation_selected_power_repeat * S ((S (bpvi_i_pvs_root_unbounded_divisibilityvaluation_selected_power)) * bpvi_c_pvs_root_unbounded_divisibilityvaluation_selected_power) + (ppf_prime_root_unbounded_divisibility)))) /\ (exists bpvi_u_pvs_root_unbounded_divisibilityvaluation_selected_power bpvi_v_pvs_root_unbounded_divisibilityvaluation_selected_power. ((((exists bpvi_h_pvs_root_unbounded_divisibilityvaluation_selected_power_start. bpvi_h_pvs_root_unbounded_divisibilityvaluation_selected_power_start + S (1) = S ((S (0)) * bpvi_v_pvs_root_unbounded_divisibilityvaluation_selected_power)) /\ exists bpvi_q_pvs_root_unbounded_divisibilityvaluation_selected_power_start. bpvi_u_pvs_root_unbounded_divisibilityvaluation_selected_power = bpvi_q_pvs_root_unbounded_divisibilityvaluation_selected_power_start * S ((S (0)) * bpvi_v_pvs_root_unbounded_divisibilityvaluation_selected_power) + (1))) /\ ((((exists bpvi_h_pvs_root_unbounded_divisibilityvaluation_selected_power_terminal. bpvi_h_pvs_root_unbounded_divisibilityvaluation_selected_power_terminal + S (bpvi_result_pvs_root_unbounded_divisibilityvaluation_selected) = S ((S (ppf_exponent_root_unbounded_divisibility)) * bpvi_v_pvs_root_unbounded_divisibilityvaluation_selected_power)) /\ exists bpvi_q_pvs_root_unbounded_divisibilityvaluation_selected_power_terminal. bpvi_u_pvs_root_unbounded_divisibilityvaluation_selected_power = bpvi_q_pvs_root_unbounded_divisibilityvaluation_selected_power_terminal * S ((S (ppf_exponent_root_unbounded_divisibility)) * bpvi_v_pvs_root_unbounded_divisibilityvaluation_selected_power) + (bpvi_result_pvs_root_unbounded_divisibilityvaluation_selected))) /\ forall bpvi_j_pvs_root_unbounded_divisibilityvaluation_selected_power. (exists bpvi_product_gap_pvs_root_unbounded_divisibilityvaluation_selected_power. bpvi_product_gap_pvs_root_unbounded_divisibilityvaluation_selected_power + S bpvi_j_pvs_root_unbounded_divisibilityvaluation_selected_power = ppf_exponent_root_unbounded_divisibility) -> exists bpvi_factor_pvs_root_unbounded_divisibilityvaluation_selected_power bpvi_partial_pvs_root_unbounded_divisibilityvaluation_selected_power bpvi_successor_pvs_root_unbounded_divisibilityvaluation_selected_power. ((((exists bpvi_h_pvs_root_unbounded_divisibilityvaluation_selected_power_factor. bpvi_h_pvs_root_unbounded_divisibilityvaluation_selected_power_factor + S (bpvi_factor_pvs_root_unbounded_divisibilityvaluation_selected_power) = S ((S (bpvi_j_pvs_root_unbounded_divisibilityvaluation_selected_power)) * bpvi_c_pvs_root_unbounded_divisibilityvaluation_selected_power)) /\ exists bpvi_q_pvs_root_unbounded_divisibilityvaluation_selected_power_factor. bpvi_b_pvs_root_unbounded_divisibilityvaluation_selected_power = bpvi_q_pvs_root_unbounded_divisibilityvaluation_selected_power_factor * S ((S (bpvi_j_pvs_root_unbounded_divisibilityvaluation_selected_power)) * bpvi_c_pvs_root_unbounded_divisibilityvaluation_selected_power) + (bpvi_factor_pvs_root_unbounded_divisibilityvaluation_selected_power))) /\ ((((exists bpvi_h_pvs_root_unbounded_divisibilityvaluation_selected_power_partial. bpvi_h_pvs_root_unbounded_divisibilityvaluation_selected_power_partial + S (bpvi_partial_pvs_root_unbounded_divisibilityvaluation_selected_power) = S ((S (bpvi_j_pvs_root_unbounded_divisibilityvaluation_selected_power)) * bpvi_v_pvs_root_unbounded_divisibilityvaluation_selected_power)) /\ exists bpvi_q_pvs_root_unbounded_divisibilityvaluation_selected_power_partial. bpvi_u_pvs_root_unbounded_divisibilityvaluation_selected_power = bpvi_q_pvs_root_unbounded_divisibilityvaluation_selected_power_partial * S ((S (bpvi_j_pvs_root_unbounded_divisibilityvaluation_selected_power)) * bpvi_v_pvs_root_unbounded_divisibilityvaluation_selected_power) + (bpvi_partial_pvs_root_unbounded_divisibilityvaluation_selected_power))) /\ ((((exists bpvi_h_pvs_root_unbounded_divisibilityvaluation_selected_power_successor. bpvi_h_pvs_root_unbounded_divisibilityvaluation_selected_power_successor + S (bpvi_successor_pvs_root_unbounded_divisibilityvaluation_selected_power) = S ((S (S bpvi_j_pvs_root_unbounded_divisibilityvaluation_selected_power)) * bpvi_v_pvs_root_unbounded_divisibilityvaluation_selected_power)) /\ exists bpvi_q_pvs_root_unbounded_divisibilityvaluation_selected_power_successor. bpvi_u_pvs_root_unbounded_divisibilityvaluation_selected_power = bpvi_q_pvs_root_unbounded_divisibilityvaluation_selected_power_successor * S ((S (S bpvi_j_pvs_root_unbounded_divisibilityvaluation_selected_power)) * bpvi_v_pvs_root_unbounded_divisibilityvaluation_selected_power) + (bpvi_successor_pvs_root_unbounded_divisibilityvaluation_selected_power))) /\ bpvi_successor_pvs_root_unbounded_divisibilityvaluation_selected_power = bpvi_partial_pvs_root_unbounded_divisibilityvaluation_selected_power * bpvi_factor_pvs_root_unbounded_divisibilityvaluation_selected_power)))))))) /\ exists bpvi_divisor_factor_pvs_root_unbounded_divisibilityvaluation_selected. n = bpvi_result_pvs_root_unbounded_divisibilityvaluation_selected * bpvi_divisor_factor_pvs_root_unbounded_divisibilityvaluation_selected))) /\ forall bpd_candidate_pvs_root_unbounded_divisibilityvaluation. (exists bpd_gap_pvs_root_unbounded_divisibilityvaluation_candidate_bound. bpd_gap_pvs_root_unbounded_divisibilityvaluation_candidate_bound + (bpd_candidate_pvs_root_unbounded_divisibilityvaluation) = (n)) -> (exists bpvi_result_pvs_root_unbounded_divisibilityvaluation_candidate. ((exists bpvi_b_pvs_root_unbounded_divisibilityvaluation_candidate_power bpvi_c_pvs_root_unbounded_divisibilityvaluation_candidate_power. ((forall bpvi_i_pvs_root_unbounded_divisibilityvaluation_candidate_power. (exists bpvi_repeat_gap_pvs_root_unbounded_divisibilityvaluation_candidate_power. bpvi_repeat_gap_pvs_root_unbounded_divisibilityvaluation_candidate_power + S bpvi_i_pvs_root_unbounded_divisibilityvaluation_candidate_power = bpd_candidate_pvs_root_unbounded_divisibilityvaluation) -> (((exists bpvi_h_pvs_root_unbounded_divisibilityvaluation_candidate_power_repeat. bpvi_h_pvs_root_unbounded_divisibilityvaluation_candidate_power_repeat + S (ppf_prime_root_unbounded_divisibility) = S ((S (bpvi_i_pvs_root_unbounded_divisibilityvaluation_candidate_power)) * bpvi_c_pvs_root_unbounded_divisibilityvaluation_candidate_power)) /\ exists bpvi_q_pvs_root_unbounded_divisibilityvaluation_candidate_power_repeat. bpvi_b_pvs_root_unbounded_divisibilityvaluation_candidate_power = bpvi_q_pvs_root_unbounded_divisibilityvaluation_candidate_power_repeat * S ((S (bpvi_i_pvs_root_unbounded_divisibilityvaluation_candidate_power)) * bpvi_c_pvs_root_unbounded_divisibilityvaluation_candidate_power) + (ppf_prime_root_unbounded_divisibility)))) /\ (exists bpvi_u_pvs_root_unbounded_divisibilityvaluation_candidate_power bpvi_v_pvs_root_unbounded_divisibilityvaluation_candidate_power. ((((exists bpvi_h_pvs_root_unbounded_divisibilityvaluation_candidate_power_start. bpvi_h_pvs_root_unbounded_divisibilityvaluation_candidate_power_start + S (1) = S ((S (0)) * bpvi_v_pvs_root_unbounded_divisibilityvaluation_candidate_power)) /\ exists bpvi_q_pvs_root_unbounded_divisibilityvaluation_candidate_power_start. bpvi_u_pvs_root_unbounded_divisibilityvaluation_candidate_power = bpvi_q_pvs_root_unbounded_divisibilityvaluation_candidate_power_start * S ((S (0)) * bpvi_v_pvs_root_unbounded_divisibilityvaluation_candidate_power) + (1))) /\ ((((exists bpvi_h_pvs_root_unbounded_divisibilityvaluation_candidate_power_terminal. bpvi_h_pvs_root_unbounded_divisibilityvaluation_candidate_power_terminal + S (bpvi_result_pvs_root_unbounded_divisibilityvaluation_candidate) = S ((S (bpd_candidate_pvs_root_unbounded_divisibilityvaluation)) * bpvi_v_pvs_root_unbounded_divisibilityvaluation_candidate_power)) /\ exists bpvi_q_pvs_root_unbounded_divisibilityvaluation_candidate_power_terminal. bpvi_u_pvs_root_unbounded_divisibilityvaluation_candidate_power = bpvi_q_pvs_root_unbounded_divisibilityvaluation_candidate_power_terminal * S ((S (bpd_candidate_pvs_root_unbounded_divisibilityvaluation)) * bpvi_v_pvs_root_unbounded_divisibilityvaluation_candidate_power) + (bpvi_result_pvs_root_unbounded_divisibilityvaluation_candidate))) /\ forall bpvi_j_pvs_root_unbounded_divisibilityvaluation_candidate_power. (exists bpvi_product_gap_pvs_root_unbounded_divisibilityvaluation_candidate_power. bpvi_product_gap_pvs_root_unbounded_divisibilityvaluation_candidate_power + S bpvi_j_pvs_root_unbounded_divisibilityvaluation_candidate_power = bpd_candidate_pvs_root_unbounded_divisibilityvaluation) -> exists bpvi_factor_pvs_root_unbounded_divisibilityvaluation_candidate_power bpvi_partial_pvs_root_unbounded_divisibilityvaluation_candidate_power bpvi_successor_pvs_root_unbounded_divisibilityvaluation_candidate_power. ((((exists bpvi_h_pvs_root_unbounded_divisibilityvaluation_candidate_power_factor. bpvi_h_pvs_root_unbounded_divisibilityvaluation_candidate_power_factor + S (bpvi_factor_pvs_root_unbounded_divisibilityvaluation_candidate_power) = S ((S (bpvi_j_pvs_root_unbounded_divisibilityvaluation_candidate_power)) * bpvi_c_pvs_root_unbounded_divisibilityvaluation_candidate_power)) /\ exists bpvi_q_pvs_root_unbounded_divisibilityvaluation_candidate_power_factor. bpvi_b_pvs_root_unbounded_divisibilityvaluation_candidate_power = bpvi_q_pvs_root_unbounded_divisibilityvaluation_candidate_power_factor * S ((S (bpvi_j_pvs_root_unbounded_divisibilityvaluation_candidate_power)) * bpvi_c_pvs_root_unbounded_divisibilityvaluation_candidate_power) + (bpvi_factor_pvs_root_unbounded_divisibilityvaluation_candidate_power))) /\ ((((exists bpvi_h_pvs_root_unbounded_divisibilityvaluation_candidate_power_partial. bpvi_h_pvs_root_unbounded_divisibilityvaluation_candidate_power_partial + S (bpvi_partial_pvs_root_unbounded_divisibilityvaluation_candidate_power) = S ((S (bpvi_j_pvs_root_unbounded_divisibilityvaluation_candidate_power)) * bpvi_v_pvs_root_unbounded_divisibilityvaluation_candidate_power)) /\ exists bpvi_q_pvs_root_unbounded_divisibilityvaluation_candidate_power_partial. bpvi_u_pvs_root_unbounded_divisibilityvaluation_candidate_power = bpvi_q_pvs_root_unbounded_divisibilityvaluation_candidate_power_partial * S ((S (bpvi_j_pvs_root_unbounded_divisibilityvaluation_candidate_power)) * bpvi_v_pvs_root_unbounded_divisibilityvaluation_candidate_power) + (bpvi_partial_pvs_root_unbounded_divisibilityvaluation_candidate_power))) /\ ((((exists bpvi_h_pvs_root_unbounded_divisibilityvaluation_candidate_power_successor. bpvi_h_pvs_root_unbounded_divisibilityvaluation_candidate_power_successor + S (bpvi_successor_pvs_root_unbounded_divisibilityvaluation_candidate_power) = S ((S (S bpvi_j_pvs_root_unbounded_divisibilityvaluation_candidate_power)) * bpvi_v_pvs_root_unbounded_divisibilityvaluation_candidate_power)) /\ exists bpvi_q_pvs_root_unbounded_divisibilityvaluation_candidate_power_successor. bpvi_u_pvs_root_unbounded_divisibilityvaluation_candidate_power = bpvi_q_pvs_root_unbounded_divisibilityvaluation_candidate_power_successor * S ((S (S bpvi_j_pvs_root_unbounded_divisibilityvaluation_candidate_power)) * bpvi_v_pvs_root_unbounded_divisibilityvaluation_candidate_power) + (bpvi_successor_pvs_root_unbounded_divisibilityvaluation_candidate_power))) /\ bpvi_successor_pvs_root_unbounded_divisibilityvaluation_candidate_power = bpvi_partial_pvs_root_unbounded_divisibilityvaluation_candidate_power * bpvi_factor_pvs_root_unbounded_divisibilityvaluation_candidate_power)))))))) /\ exists bpvi_divisor_factor_pvs_root_unbounded_divisibilityvaluation_candidate. n = bpvi_result_pvs_root_unbounded_divisibilityvaluation_candidate * bpvi_divisor_factor_pvs_root_unbounded_divisibilityvaluation_candidate)) -> (exists bpd_gap_pvs_root_unbounded_divisibilityvaluation_maximal. bpd_gap_pvs_root_unbounded_divisibilityvaluation_maximal + (bpd_candidate_pvs_root_unbounded_divisibilityvaluation) = (ppf_exponent_root_unbounded_divisibility))) -> (exists pvs_factor_root_unbounded_divisibilitydivides. (ppf_exponent_root_unbounded_divisibility) = (k) * pvs_factor_root_unbounded_divisibilitydivides)) -> exists r. (exists pa_b_pvs_root_unbounded_result pa_c_pvs_root_unbounded_result. ((forall pa_i_pvs_root_unbounded_result_repeat. (exists pa_lt_pvs_root_unbounded_result_repeat_bound. pa_lt_pvs_root_unbounded_result_repeat_bound + S pa_i_pvs_root_unbounded_result_repeat = k) -> (((exists pa_h_pvs_root_unbounded_result_repeat_decoded. pa_h_pvs_root_unbounded_result_repeat_decoded + S (r) = S ((S (pa_i_pvs_root_unbounded_result_repeat)) * pa_c_pvs_root_unbounded_result)) /\ exists pa_q_pvs_root_unbounded_result_repeat_decoded. pa_b_pvs_root_unbounded_result = pa_q_pvs_root_unbounded_result_repeat_decoded * S ((S (pa_i_pvs_root_unbounded_result_repeat)) * pa_c_pvs_root_unbounded_result) + (r)))) /\ (exists pa_u_pvs_root_unbounded_result_product pa_v_pvs_root_unbounded_result_product. ((((exists pa_h_pvs_root_unbounded_result_product_start. pa_h_pvs_root_unbounded_result_product_start + S (1) = S ((S (0)) * pa_v_pvs_root_unbounded_result_product)) /\ exists pa_q_pvs_root_unbounded_result_product_start. pa_u_pvs_root_unbounded_result_product = pa_q_pvs_root_unbounded_result_product_start * S ((S (0)) * pa_v_pvs_root_unbounded_result_product) + (1))) /\ ((((exists pa_h_pvs_root_unbounded_result_product_terminal. pa_h_pvs_root_unbounded_result_product_terminal + S (n) = S ((S (k)) * pa_v_pvs_root_unbounded_result_product)) /\ exists pa_q_pvs_root_unbounded_result_product_terminal. pa_u_pvs_root_unbounded_result_product = pa_q_pvs_root_unbounded_result_product_terminal * S ((S (k)) * pa_v_pvs_root_unbounded_result_product) + (n))) /\ forall pa_i_pvs_root_unbounded_result_product. (exists pa_lt_pvs_root_unbounded_result_product_bound. pa_lt_pvs_root_unbounded_result_product_bound + S pa_i_pvs_root_unbounded_result_product = k) -> exists pa_p_pvs_root_unbounded_result_product pa_r_pvs_root_unbounded_result_product pa_s_pvs_root_unbounded_result_product. ((((exists pa_h_pvs_root_unbounded_result_product_factor. pa_h_pvs_root_unbounded_result_product_factor + S (pa_p_pvs_root_unbounded_result_product) = S ((S (pa_i_pvs_root_unbounded_result_product)) * pa_c_pvs_root_unbounded_result)) /\ exists pa_q_pvs_root_unbounded_result_product_factor. pa_b_pvs_root_unbounded_result = pa_q_pvs_root_unbounded_result_product_factor * S ((S (pa_i_pvs_root_unbounded_result_product)) * pa_c_pvs_root_unbounded_result) + (pa_p_pvs_root_unbounded_result_product))) /\ ((((exists pa_h_pvs_root_unbounded_result_product_partial. pa_h_pvs_root_unbounded_result_product_partial + S (pa_r_pvs_root_unbounded_result_product) = S ((S (pa_i_pvs_root_unbounded_result_product)) * pa_v_pvs_root_unbounded_result_product)) /\ exists pa_q_pvs_root_unbounded_result_product_partial. pa_u_pvs_root_unbounded_result_product = pa_q_pvs_root_unbounded_result_product_partial * S ((S (pa_i_pvs_root_unbounded_result_product)) * pa_v_pvs_root_unbounded_result_product) + (pa_r_pvs_root_unbounded_result_product))) /\ ((((exists pa_h_pvs_root_unbounded_result_product_successor. pa_h_pvs_root_unbounded_result_product_successor + S (pa_s_pvs_root_unbounded_result_product) = S ((S (S pa_i_pvs_root_unbounded_result_product)) * pa_v_pvs_root_unbounded_result_product)) /\ exists pa_q_pvs_root_unbounded_result_product_successor. pa_u_pvs_root_unbounded_result_product = pa_q_pvs_root_unbounded_result_product_successor * S ((S (S pa_i_pvs_root_unbounded_result_product)) * pa_v_pvs_root_unbounded_result_product) + (pa_s_pvs_root_unbounded_result_product))) /\ pa_s_pvs_root_unbounded_result_product = pa_r_pvs_root_unbounded_result_product * pa_p_pvs_root_unbounded_result_product))))))))