prime_support_common_divisor_implies_all_valuations

Exact expanded first-order arithmetic statement

forall n pb pc eb ec vb vc l k. (((~((n) = 0)) /\ (((forall pfp_i_pvs_common_supportdistinct pfp_j_pvs_common_supportdistinct pfp_a_pvs_common_supportdistinct. (exists pfp_gap_pvs_common_supportdistinctfirst. pfp_gap_pvs_common_supportdistinctfirst + S (pfp_i_pvs_common_supportdistinct) = (l)) -> (exists pfp_gap_pvs_common_supportdistinctsecond. pfp_gap_pvs_common_supportdistinctsecond + S (pfp_j_pvs_common_supportdistinct) = (l)) -> (((exists ff_h_pfp_pvs_common_supportdistinctleft. ff_h_pfp_pvs_common_supportdistinctleft + S (pfp_a_pvs_common_supportdistinct) = S ((S (pfp_i_pvs_common_supportdistinct)) * pc)) /\ exists ff_q_pfp_pvs_common_supportdistinctleft. pb = ff_q_pfp_pvs_common_supportdistinctleft * S ((S (pfp_i_pvs_common_supportdistinct)) * pc) + (pfp_a_pvs_common_supportdistinct))) -> (((exists ff_h_pfp_pvs_common_supportdistinctright. ff_h_pfp_pvs_common_supportdistinctright + S (pfp_a_pvs_common_supportdistinct) = S ((S (pfp_j_pvs_common_supportdistinct)) * pc)) /\ exists ff_q_pfp_pvs_common_supportdistinctright. pb = ff_q_pfp_pvs_common_supportdistinctright * S ((S (pfp_j_pvs_common_supportdistinct)) * pc) + (pfp_a_pvs_common_supportdistinct))) -> pfp_i_pvs_common_supportdistinct = pfp_j_pvs_common_supportdistinct) /\ (((forall pvs_index_common_supportentries. (exists pvs_gap_common_supportentriesindex. pvs_gap_common_supportentriesindex + S (pvs_index_common_supportentries) = (l)) -> exists pvs_prime_common_supportentries pvs_exponent_common_supportentries pvs_power_common_supportentries. (((((exists ff_h_pvs_common_supportentriesprime. ff_h_pvs_common_supportentriesprime + S (pvs_prime_common_supportentries) = S ((S (pvs_index_common_supportentries)) * pc)) /\ exists ff_q_pvs_common_supportentriesprime. pb = ff_q_pvs_common_supportentriesprime * S ((S (pvs_index_common_supportentries)) * pc) + (pvs_prime_common_supportentries))) /\ (((((exists ff_h_pvs_common_supportentriesexponent. ff_h_pvs_common_supportentriesexponent + S (pvs_exponent_common_supportentries) = S ((S (pvs_index_common_supportentries)) * ec)) /\ exists ff_q_pvs_common_supportentriesexponent. eb = ff_q_pvs_common_supportentriesexponent * S ((S (pvs_index_common_supportentries)) * ec) + (pvs_exponent_common_supportentries))) /\ (((((exists ff_h_pvs_common_supportentriespower. ff_h_pvs_common_supportentriespower + S (pvs_power_common_supportentries) = S ((S (pvs_index_common_supportentries)) * vc)) /\ exists ff_q_pvs_common_supportentriespower. vb = ff_q_pvs_common_supportentriespower * S ((S (pvs_index_common_supportentries)) * vc) + (pvs_power_common_supportentries))) /\ (((~((pvs_prime_common_supportentries) = 1) /\ forall pvs_left_common_supportentriesdomain pvs_right_common_supportentriesdomain. (pvs_prime_common_supportentries) = pvs_left_common_supportentriesdomain * pvs_right_common_supportentriesdomain -> pvs_left_common_supportentriesdomain = 1 \/ pvs_right_common_supportentriesdomain = 1) /\ (((~(pvs_exponent_common_supportentries = 0)) /\ (((((exists bpd_gap_pvs_common_supportentriesvaluation_selected_bound. bpd_gap_pvs_common_supportentriesvaluation_selected_bound + (pvs_exponent_common_supportentries) = (n)) /\ (exists bpvi_result_pvs_common_supportentriesvaluation_selected. ((exists bpvi_b_pvs_common_supportentriesvaluation_selected_power bpvi_c_pvs_common_supportentriesvaluation_selected_power. ((forall bpvi_i_pvs_common_supportentriesvaluation_selected_power. (exists bpvi_repeat_gap_pvs_common_supportentriesvaluation_selected_power. bpvi_repeat_gap_pvs_common_supportentriesvaluation_selected_power + S bpvi_i_pvs_common_supportentriesvaluation_selected_power = pvs_exponent_common_supportentries) -> (((exists bpvi_h_pvs_common_supportentriesvaluation_selected_power_repeat. bpvi_h_pvs_common_supportentriesvaluation_selected_power_repeat + S (pvs_prime_common_supportentries) = S ((S (bpvi_i_pvs_common_supportentriesvaluation_selected_power)) * bpvi_c_pvs_common_supportentriesvaluation_selected_power)) /\ exists bpvi_q_pvs_common_supportentriesvaluation_selected_power_repeat. bpvi_b_pvs_common_supportentriesvaluation_selected_power = bpvi_q_pvs_common_supportentriesvaluation_selected_power_repeat * S ((S (bpvi_i_pvs_common_supportentriesvaluation_selected_power)) * bpvi_c_pvs_common_supportentriesvaluation_selected_power) + (pvs_prime_common_supportentries)))) /\ (exists bpvi_u_pvs_common_supportentriesvaluation_selected_power bpvi_v_pvs_common_supportentriesvaluation_selected_power. ((((exists bpvi_h_pvs_common_supportentriesvaluation_selected_power_start. bpvi_h_pvs_common_supportentriesvaluation_selected_power_start + S (1) = S ((S (0)) * bpvi_v_pvs_common_supportentriesvaluation_selected_power)) /\ exists bpvi_q_pvs_common_supportentriesvaluation_selected_power_start. bpvi_u_pvs_common_supportentriesvaluation_selected_power = bpvi_q_pvs_common_supportentriesvaluation_selected_power_start * S ((S (0)) * bpvi_v_pvs_common_supportentriesvaluation_selected_power) + (1))) /\ ((((exists bpvi_h_pvs_common_supportentriesvaluation_selected_power_terminal. bpvi_h_pvs_common_supportentriesvaluation_selected_power_terminal + S (bpvi_result_pvs_common_supportentriesvaluation_selected) = S ((S (pvs_exponent_common_supportentries)) * bpvi_v_pvs_common_supportentriesvaluation_selected_power)) /\ exists bpvi_q_pvs_common_supportentriesvaluation_selected_power_terminal. bpvi_u_pvs_common_supportentriesvaluation_selected_power = bpvi_q_pvs_common_supportentriesvaluation_selected_power_terminal * S ((S (pvs_exponent_common_supportentries)) * bpvi_v_pvs_common_supportentriesvaluation_selected_power) + (bpvi_result_pvs_common_supportentriesvaluation_selected))) /\ forall bpvi_j_pvs_common_supportentriesvaluation_selected_power. (exists bpvi_product_gap_pvs_common_supportentriesvaluation_selected_power. bpvi_product_gap_pvs_common_supportentriesvaluation_selected_power + S bpvi_j_pvs_common_supportentriesvaluation_selected_power = pvs_exponent_common_supportentries) -> exists bpvi_factor_pvs_common_supportentriesvaluation_selected_power bpvi_partial_pvs_common_supportentriesvaluation_selected_power bpvi_successor_pvs_common_supportentriesvaluation_selected_power. ((((exists bpvi_h_pvs_common_supportentriesvaluation_selected_power_factor. bpvi_h_pvs_common_supportentriesvaluation_selected_power_factor + S (bpvi_factor_pvs_common_supportentriesvaluation_selected_power) = S ((S (bpvi_j_pvs_common_supportentriesvaluation_selected_power)) * bpvi_c_pvs_common_supportentriesvaluation_selected_power)) /\ exists bpvi_q_pvs_common_supportentriesvaluation_selected_power_factor. bpvi_b_pvs_common_supportentriesvaluation_selected_power = bpvi_q_pvs_common_supportentriesvaluation_selected_power_factor * S ((S (bpvi_j_pvs_common_supportentriesvaluation_selected_power)) * bpvi_c_pvs_common_supportentriesvaluation_selected_power) + (bpvi_factor_pvs_common_supportentriesvaluation_selected_power))) /\ ((((exists bpvi_h_pvs_common_supportentriesvaluation_selected_power_partial. bpvi_h_pvs_common_supportentriesvaluation_selected_power_partial + S (bpvi_partial_pvs_common_supportentriesvaluation_selected_power) = S ((S (bpvi_j_pvs_common_supportentriesvaluation_selected_power)) * bpvi_v_pvs_common_supportentriesvaluation_selected_power)) /\ exists bpvi_q_pvs_common_supportentriesvaluation_selected_power_partial. bpvi_u_pvs_common_supportentriesvaluation_selected_power = bpvi_q_pvs_common_supportentriesvaluation_selected_power_partial * S ((S (bpvi_j_pvs_common_supportentriesvaluation_selected_power)) * bpvi_v_pvs_common_supportentriesvaluation_selected_power) + (bpvi_partial_pvs_common_supportentriesvaluation_selected_power))) /\ ((((exists bpvi_h_pvs_common_supportentriesvaluation_selected_power_successor. bpvi_h_pvs_common_supportentriesvaluation_selected_power_successor + S (bpvi_successor_pvs_common_supportentriesvaluation_selected_power) = S ((S (S bpvi_j_pvs_common_supportentriesvaluation_selected_power)) * bpvi_v_pvs_common_supportentriesvaluation_selected_power)) /\ exists bpvi_q_pvs_common_supportentriesvaluation_selected_power_successor. bpvi_u_pvs_common_supportentriesvaluation_selected_power = bpvi_q_pvs_common_supportentriesvaluation_selected_power_successor * S ((S (S bpvi_j_pvs_common_supportentriesvaluation_selected_power)) * bpvi_v_pvs_common_supportentriesvaluation_selected_power) + (bpvi_successor_pvs_common_supportentriesvaluation_selected_power))) /\ bpvi_successor_pvs_common_supportentriesvaluation_selected_power = bpvi_partial_pvs_common_supportentriesvaluation_selected_power * bpvi_factor_pvs_common_supportentriesvaluation_selected_power)))))))) /\ exists bpvi_divisor_factor_pvs_common_supportentriesvaluation_selected. n = bpvi_result_pvs_common_supportentriesvaluation_selected * bpvi_divisor_factor_pvs_common_supportentriesvaluation_selected))) /\ forall bpd_candidate_pvs_common_supportentriesvaluation. (exists bpd_gap_pvs_common_supportentriesvaluation_candidate_bound. bpd_gap_pvs_common_supportentriesvaluation_candidate_bound + (bpd_candidate_pvs_common_supportentriesvaluation) = (n)) -> (exists bpvi_result_pvs_common_supportentriesvaluation_candidate. ((exists bpvi_b_pvs_common_supportentriesvaluation_candidate_power bpvi_c_pvs_common_supportentriesvaluation_candidate_power. ((forall bpvi_i_pvs_common_supportentriesvaluation_candidate_power. (exists bpvi_repeat_gap_pvs_common_supportentriesvaluation_candidate_power. bpvi_repeat_gap_pvs_common_supportentriesvaluation_candidate_power + S bpvi_i_pvs_common_supportentriesvaluation_candidate_power = bpd_candidate_pvs_common_supportentriesvaluation) -> (((exists bpvi_h_pvs_common_supportentriesvaluation_candidate_power_repeat. bpvi_h_pvs_common_supportentriesvaluation_candidate_power_repeat + S (pvs_prime_common_supportentries) = S ((S (bpvi_i_pvs_common_supportentriesvaluation_candidate_power)) * bpvi_c_pvs_common_supportentriesvaluation_candidate_power)) /\ exists bpvi_q_pvs_common_supportentriesvaluation_candidate_power_repeat. bpvi_b_pvs_common_supportentriesvaluation_candidate_power = bpvi_q_pvs_common_supportentriesvaluation_candidate_power_repeat * S ((S (bpvi_i_pvs_common_supportentriesvaluation_candidate_power)) * bpvi_c_pvs_common_supportentriesvaluation_candidate_power) + (pvs_prime_common_supportentries)))) /\ (exists bpvi_u_pvs_common_supportentriesvaluation_candidate_power bpvi_v_pvs_common_supportentriesvaluation_candidate_power. ((((exists bpvi_h_pvs_common_supportentriesvaluation_candidate_power_start. bpvi_h_pvs_common_supportentriesvaluation_candidate_power_start + S (1) = S ((S (0)) * bpvi_v_pvs_common_supportentriesvaluation_candidate_power)) /\ exists bpvi_q_pvs_common_supportentriesvaluation_candidate_power_start. bpvi_u_pvs_common_supportentriesvaluation_candidate_power = bpvi_q_pvs_common_supportentriesvaluation_candidate_power_start * S ((S (0)) * bpvi_v_pvs_common_supportentriesvaluation_candidate_power) + (1))) /\ ((((exists bpvi_h_pvs_common_supportentriesvaluation_candidate_power_terminal. bpvi_h_pvs_common_supportentriesvaluation_candidate_power_terminal + S (bpvi_result_pvs_common_supportentriesvaluation_candidate) = S ((S (bpd_candidate_pvs_common_supportentriesvaluation)) * bpvi_v_pvs_common_supportentriesvaluation_candidate_power)) /\ exists bpvi_q_pvs_common_supportentriesvaluation_candidate_power_terminal. bpvi_u_pvs_common_supportentriesvaluation_candidate_power = bpvi_q_pvs_common_supportentriesvaluation_candidate_power_terminal * S ((S (bpd_candidate_pvs_common_supportentriesvaluation)) * bpvi_v_pvs_common_supportentriesvaluation_candidate_power) + (bpvi_result_pvs_common_supportentriesvaluation_candidate))) /\ forall bpvi_j_pvs_common_supportentriesvaluation_candidate_power. (exists bpvi_product_gap_pvs_common_supportentriesvaluation_candidate_power. bpvi_product_gap_pvs_common_supportentriesvaluation_candidate_power + S bpvi_j_pvs_common_supportentriesvaluation_candidate_power = bpd_candidate_pvs_common_supportentriesvaluation) -> exists bpvi_factor_pvs_common_supportentriesvaluation_candidate_power bpvi_partial_pvs_common_supportentriesvaluation_candidate_power bpvi_successor_pvs_common_supportentriesvaluation_candidate_power. ((((exists bpvi_h_pvs_common_supportentriesvaluation_candidate_power_factor. bpvi_h_pvs_common_supportentriesvaluation_candidate_power_factor + S (bpvi_factor_pvs_common_supportentriesvaluation_candidate_power) = S ((S (bpvi_j_pvs_common_supportentriesvaluation_candidate_power)) * bpvi_c_pvs_common_supportentriesvaluation_candidate_power)) /\ exists bpvi_q_pvs_common_supportentriesvaluation_candidate_power_factor. bpvi_b_pvs_common_supportentriesvaluation_candidate_power = bpvi_q_pvs_common_supportentriesvaluation_candidate_power_factor * S ((S (bpvi_j_pvs_common_supportentriesvaluation_candidate_power)) * bpvi_c_pvs_common_supportentriesvaluation_candidate_power) + (bpvi_factor_pvs_common_supportentriesvaluation_candidate_power))) /\ ((((exists bpvi_h_pvs_common_supportentriesvaluation_candidate_power_partial. bpvi_h_pvs_common_supportentriesvaluation_candidate_power_partial + S (bpvi_partial_pvs_common_supportentriesvaluation_candidate_power) = S ((S (bpvi_j_pvs_common_supportentriesvaluation_candidate_power)) * bpvi_v_pvs_common_supportentriesvaluation_candidate_power)) /\ exists bpvi_q_pvs_common_supportentriesvaluation_candidate_power_partial. bpvi_u_pvs_common_supportentriesvaluation_candidate_power = bpvi_q_pvs_common_supportentriesvaluation_candidate_power_partial * S ((S (bpvi_j_pvs_common_supportentriesvaluation_candidate_power)) * bpvi_v_pvs_common_supportentriesvaluation_candidate_power) + (bpvi_partial_pvs_common_supportentriesvaluation_candidate_power))) /\ ((((exists bpvi_h_pvs_common_supportentriesvaluation_candidate_power_successor. bpvi_h_pvs_common_supportentriesvaluation_candidate_power_successor + S (bpvi_successor_pvs_common_supportentriesvaluation_candidate_power) = S ((S (S bpvi_j_pvs_common_supportentriesvaluation_candidate_power)) * bpvi_v_pvs_common_supportentriesvaluation_candidate_power)) /\ exists bpvi_q_pvs_common_supportentriesvaluation_candidate_power_successor. bpvi_u_pvs_common_supportentriesvaluation_candidate_power = bpvi_q_pvs_common_supportentriesvaluation_candidate_power_successor * S ((S (S bpvi_j_pvs_common_supportentriesvaluation_candidate_power)) * bpvi_v_pvs_common_supportentriesvaluation_candidate_power) + (bpvi_successor_pvs_common_supportentriesvaluation_candidate_power))) /\ bpvi_successor_pvs_common_supportentriesvaluation_candidate_power = bpvi_partial_pvs_common_supportentriesvaluation_candidate_power * bpvi_factor_pvs_common_supportentriesvaluation_candidate_power)))))))) /\ exists bpvi_divisor_factor_pvs_common_supportentriesvaluation_candidate. n = bpvi_result_pvs_common_supportentriesvaluation_candidate * bpvi_divisor_factor_pvs_common_supportentriesvaluation_candidate)) -> (exists bpd_gap_pvs_common_supportentriesvaluation_maximal. bpd_gap_pvs_common_supportentriesvaluation_maximal + (bpd_candidate_pvs_common_supportentriesvaluation) = (pvs_exponent_common_supportentries))) /\ (exists pa_b_pvs_common_supportentriesvalue pa_c_pvs_common_supportentriesvalue. ((forall pa_i_pvs_common_supportentriesvalue_repeat. (exists pa_lt_pvs_common_supportentriesvalue_repeat_bound. pa_lt_pvs_common_supportentriesvalue_repeat_bound + S pa_i_pvs_common_supportentriesvalue_repeat = pvs_exponent_common_supportentries) -> (((exists pa_h_pvs_common_supportentriesvalue_repeat_decoded. pa_h_pvs_common_supportentriesvalue_repeat_decoded + S (pvs_prime_common_supportentries) = S ((S (pa_i_pvs_common_supportentriesvalue_repeat)) * pa_c_pvs_common_supportentriesvalue)) /\ exists pa_q_pvs_common_supportentriesvalue_repeat_decoded. pa_b_pvs_common_supportentriesvalue = pa_q_pvs_common_supportentriesvalue_repeat_decoded * S ((S (pa_i_pvs_common_supportentriesvalue_repeat)) * pa_c_pvs_common_supportentriesvalue) + (pvs_prime_common_supportentries)))) /\ (exists pa_u_pvs_common_supportentriesvalue_product pa_v_pvs_common_supportentriesvalue_product. ((((exists pa_h_pvs_common_supportentriesvalue_product_start. pa_h_pvs_common_supportentriesvalue_product_start + S (1) = S ((S (0)) * pa_v_pvs_common_supportentriesvalue_product)) /\ exists pa_q_pvs_common_supportentriesvalue_product_start. pa_u_pvs_common_supportentriesvalue_product = pa_q_pvs_common_supportentriesvalue_product_start * S ((S (0)) * pa_v_pvs_common_supportentriesvalue_product) + (1))) /\ ((((exists pa_h_pvs_common_supportentriesvalue_product_terminal. pa_h_pvs_common_supportentriesvalue_product_terminal + S (pvs_power_common_supportentries) = S ((S (pvs_exponent_common_supportentries)) * pa_v_pvs_common_supportentriesvalue_product)) /\ exists pa_q_pvs_common_supportentriesvalue_product_terminal. pa_u_pvs_common_supportentriesvalue_product = pa_q_pvs_common_supportentriesvalue_product_terminal * S ((S (pvs_exponent_common_supportentries)) * pa_v_pvs_common_supportentriesvalue_product) + (pvs_power_common_supportentries))) /\ forall pa_i_pvs_common_supportentriesvalue_product. (exists pa_lt_pvs_common_supportentriesvalue_product_bound. pa_lt_pvs_common_supportentriesvalue_product_bound + S pa_i_pvs_common_supportentriesvalue_product = pvs_exponent_common_supportentries) -> exists pa_p_pvs_common_supportentriesvalue_product pa_r_pvs_common_supportentriesvalue_product pa_s_pvs_common_supportentriesvalue_product. ((((exists pa_h_pvs_common_supportentriesvalue_product_factor. pa_h_pvs_common_supportentriesvalue_product_factor + S (pa_p_pvs_common_supportentriesvalue_product) = S ((S (pa_i_pvs_common_supportentriesvalue_product)) * pa_c_pvs_common_supportentriesvalue)) /\ exists pa_q_pvs_common_supportentriesvalue_product_factor. pa_b_pvs_common_supportentriesvalue = pa_q_pvs_common_supportentriesvalue_product_factor * S ((S (pa_i_pvs_common_supportentriesvalue_product)) * pa_c_pvs_common_supportentriesvalue) + (pa_p_pvs_common_supportentriesvalue_product))) /\ ((((exists pa_h_pvs_common_supportentriesvalue_product_partial. pa_h_pvs_common_supportentriesvalue_product_partial + S (pa_r_pvs_common_supportentriesvalue_product) = S ((S (pa_i_pvs_common_supportentriesvalue_product)) * pa_v_pvs_common_supportentriesvalue_product)) /\ exists pa_q_pvs_common_supportentriesvalue_product_partial. pa_u_pvs_common_supportentriesvalue_product = pa_q_pvs_common_supportentriesvalue_product_partial * S ((S (pa_i_pvs_common_supportentriesvalue_product)) * pa_v_pvs_common_supportentriesvalue_product) + (pa_r_pvs_common_supportentriesvalue_product))) /\ ((((exists pa_h_pvs_common_supportentriesvalue_product_successor. pa_h_pvs_common_supportentriesvalue_product_successor + S (pa_s_pvs_common_supportentriesvalue_product) = S ((S (S pa_i_pvs_common_supportentriesvalue_product)) * pa_v_pvs_common_supportentriesvalue_product)) /\ exists pa_q_pvs_common_supportentriesvalue_product_successor. pa_u_pvs_common_supportentriesvalue_product = pa_q_pvs_common_supportentriesvalue_product_successor * S ((S (S pa_i_pvs_common_supportentriesvalue_product)) * pa_v_pvs_common_supportentriesvalue_product) + (pa_s_pvs_common_supportentriesvalue_product))) /\ pa_s_pvs_common_supportentriesvalue_product = pa_r_pvs_common_supportentriesvalue_product * pa_p_pvs_common_supportentriesvalue_product))))))))))))))))))))) /\ (((forall pvs_divisor_common_supportcover. (~((pvs_divisor_common_supportcover) = 1) /\ forall pvs_left_common_supportcoverprime pvs_right_common_supportcoverprime. (pvs_divisor_common_supportcover) = pvs_left_common_supportcoverprime * pvs_right_common_supportcoverprime -> pvs_left_common_supportcoverprime = 1 \/ pvs_right_common_supportcoverprime = 1) -> (exists pvs_factor_common_supportcoverdivides. (n) = (pvs_divisor_common_supportcover) * pvs_factor_common_supportcoverdivides) -> exists pvs_position_common_supportcover. (exists pvs_gap_common_supportcoverbound. pvs_gap_common_supportcoverbound + S (pvs_position_common_supportcover) = (l)) /\ (((exists ff_h_pvs_common_supportcoverentry. ff_h_pvs_common_supportcoverentry + S (pvs_divisor_common_supportcover) = S ((S (pvs_position_common_supportcover)) * pc)) /\ exists ff_q_pvs_common_supportcoverentry. pb = ff_q_pvs_common_supportcoverentry * S ((S (pvs_position_common_supportcover)) * pc) + (pvs_divisor_common_supportcover)))) /\ (exists ff_u_pvs_common_supportproduct ff_v_pvs_common_supportproduct. ((((exists ff_h_pvs_common_supportproduct_start. ff_h_pvs_common_supportproduct_start + S (1) = S ((S (0)) * ff_v_pvs_common_supportproduct)) /\ exists ff_q_pvs_common_supportproduct_start. ff_u_pvs_common_supportproduct = ff_q_pvs_common_supportproduct_start * S ((S (0)) * ff_v_pvs_common_supportproduct) + (1))) /\ ((((exists ff_h_pvs_common_supportproduct_terminal. ff_h_pvs_common_supportproduct_terminal + S (n) = S ((S (l)) * ff_v_pvs_common_supportproduct)) /\ exists ff_q_pvs_common_supportproduct_terminal. ff_u_pvs_common_supportproduct = ff_q_pvs_common_supportproduct_terminal * S ((S (l)) * ff_v_pvs_common_supportproduct) + (n))) /\ forall ff_i_pvs_common_supportproduct. (exists ff_lt_pvs_common_supportproduct_bound. ff_lt_pvs_common_supportproduct_bound + S ff_i_pvs_common_supportproduct = l) -> exists ff_p_pvs_common_supportproduct ff_r_pvs_common_supportproduct ff_s_pvs_common_supportproduct. ((((exists ff_h_pvs_common_supportproduct_factor. ff_h_pvs_common_supportproduct_factor + S (ff_p_pvs_common_supportproduct) = S ((S (ff_i_pvs_common_supportproduct)) * vc)) /\ exists ff_q_pvs_common_supportproduct_factor. vb = ff_q_pvs_common_supportproduct_factor * S ((S (ff_i_pvs_common_supportproduct)) * vc) + (ff_p_pvs_common_supportproduct))) /\ ((((exists ff_h_pvs_common_supportproduct_partial. ff_h_pvs_common_supportproduct_partial + S (ff_r_pvs_common_supportproduct) = S ((S (ff_i_pvs_common_supportproduct)) * ff_v_pvs_common_supportproduct)) /\ exists ff_q_pvs_common_supportproduct_partial. ff_u_pvs_common_supportproduct = ff_q_pvs_common_supportproduct_partial * S ((S (ff_i_pvs_common_supportproduct)) * ff_v_pvs_common_supportproduct) + (ff_r_pvs_common_supportproduct))) /\ ((((exists ff_h_pvs_common_supportproduct_successor. ff_h_pvs_common_supportproduct_successor + S (ff_s_pvs_common_supportproduct) = S ((S (S ff_i_pvs_common_supportproduct)) * ff_v_pvs_common_supportproduct)) /\ exists ff_q_pvs_common_supportproduct_successor. ff_u_pvs_common_supportproduct = ff_q_pvs_common_supportproduct_successor * S ((S (S ff_i_pvs_common_supportproduct)) * ff_v_pvs_common_supportproduct) + (ff_s_pvs_common_supportproduct))) /\ ff_s_pvs_common_supportproduct = ff_r_pvs_common_supportproduct * ff_p_pvs_common_supportproduct)))))))))))))) -> (forall ppf_index_common_exponents ppf_entry_common_exponents. (exists pvs_gap_common_exponentsbound. pvs_gap_common_exponentsbound + S (ppf_index_common_exponents) = (l)) -> (((exists ff_h_pvs_common_exponentsentry. ff_h_pvs_common_exponentsentry + S (ppf_entry_common_exponents) = S ((S (ppf_index_common_exponents)) * ec)) /\ exists ff_q_pvs_common_exponentsentry. eb = ff_q_pvs_common_exponentsentry * S ((S (ppf_index_common_exponents)) * ec) + (ppf_entry_common_exponents))) -> (exists pvs_factor_common_exponentsdivisor. (ppf_entry_common_exponents) = (k) * pvs_factor_common_exponentsdivisor)) -> (forall ppf_prime_common_all_primes ppf_exponent_common_all_primes. (~((ppf_prime_common_all_primes) = 1) /\ forall pvs_left_common_all_primesdomain pvs_right_common_all_primesdomain. (ppf_prime_common_all_primes) = pvs_left_common_all_primesdomain * pvs_right_common_all_primesdomain -> pvs_left_common_all_primesdomain = 1 \/ pvs_right_common_all_primesdomain = 1) -> (((exists bpd_gap_pvs_common_all_primesvaluation_selected_bound. bpd_gap_pvs_common_all_primesvaluation_selected_bound + (ppf_exponent_common_all_primes) = (n)) /\ (exists bpvi_result_pvs_common_all_primesvaluation_selected. ((exists bpvi_b_pvs_common_all_primesvaluation_selected_power bpvi_c_pvs_common_all_primesvaluation_selected_power. ((forall bpvi_i_pvs_common_all_primesvaluation_selected_power. (exists bpvi_repeat_gap_pvs_common_all_primesvaluation_selected_power. bpvi_repeat_gap_pvs_common_all_primesvaluation_selected_power + S bpvi_i_pvs_common_all_primesvaluation_selected_power = ppf_exponent_common_all_primes) -> (((exists bpvi_h_pvs_common_all_primesvaluation_selected_power_repeat. bpvi_h_pvs_common_all_primesvaluation_selected_power_repeat + S (ppf_prime_common_all_primes) = S ((S (bpvi_i_pvs_common_all_primesvaluation_selected_power)) * bpvi_c_pvs_common_all_primesvaluation_selected_power)) /\ exists bpvi_q_pvs_common_all_primesvaluation_selected_power_repeat. bpvi_b_pvs_common_all_primesvaluation_selected_power = bpvi_q_pvs_common_all_primesvaluation_selected_power_repeat * S ((S (bpvi_i_pvs_common_all_primesvaluation_selected_power)) * bpvi_c_pvs_common_all_primesvaluation_selected_power) + (ppf_prime_common_all_primes)))) /\ (exists bpvi_u_pvs_common_all_primesvaluation_selected_power bpvi_v_pvs_common_all_primesvaluation_selected_power. ((((exists bpvi_h_pvs_common_all_primesvaluation_selected_power_start. bpvi_h_pvs_common_all_primesvaluation_selected_power_start + S (1) = S ((S (0)) * bpvi_v_pvs_common_all_primesvaluation_selected_power)) /\ exists bpvi_q_pvs_common_all_primesvaluation_selected_power_start. bpvi_u_pvs_common_all_primesvaluation_selected_power = bpvi_q_pvs_common_all_primesvaluation_selected_power_start * S ((S (0)) * bpvi_v_pvs_common_all_primesvaluation_selected_power) + (1))) /\ ((((exists bpvi_h_pvs_common_all_primesvaluation_selected_power_terminal. bpvi_h_pvs_common_all_primesvaluation_selected_power_terminal + S (bpvi_result_pvs_common_all_primesvaluation_selected) = S ((S (ppf_exponent_common_all_primes)) * bpvi_v_pvs_common_all_primesvaluation_selected_power)) /\ exists bpvi_q_pvs_common_all_primesvaluation_selected_power_terminal. bpvi_u_pvs_common_all_primesvaluation_selected_power = bpvi_q_pvs_common_all_primesvaluation_selected_power_terminal * S ((S (ppf_exponent_common_all_primes)) * bpvi_v_pvs_common_all_primesvaluation_selected_power) + (bpvi_result_pvs_common_all_primesvaluation_selected))) /\ forall bpvi_j_pvs_common_all_primesvaluation_selected_power. (exists bpvi_product_gap_pvs_common_all_primesvaluation_selected_power. bpvi_product_gap_pvs_common_all_primesvaluation_selected_power + S bpvi_j_pvs_common_all_primesvaluation_selected_power = ppf_exponent_common_all_primes) -> exists bpvi_factor_pvs_common_all_primesvaluation_selected_power bpvi_partial_pvs_common_all_primesvaluation_selected_power bpvi_successor_pvs_common_all_primesvaluation_selected_power. ((((exists bpvi_h_pvs_common_all_primesvaluation_selected_power_factor. bpvi_h_pvs_common_all_primesvaluation_selected_power_factor + S (bpvi_factor_pvs_common_all_primesvaluation_selected_power) = S ((S (bpvi_j_pvs_common_all_primesvaluation_selected_power)) * bpvi_c_pvs_common_all_primesvaluation_selected_power)) /\ exists bpvi_q_pvs_common_all_primesvaluation_selected_power_factor. bpvi_b_pvs_common_all_primesvaluation_selected_power = bpvi_q_pvs_common_all_primesvaluation_selected_power_factor * S ((S (bpvi_j_pvs_common_all_primesvaluation_selected_power)) * bpvi_c_pvs_common_all_primesvaluation_selected_power) + (bpvi_factor_pvs_common_all_primesvaluation_selected_power))) /\ ((((exists bpvi_h_pvs_common_all_primesvaluation_selected_power_partial. bpvi_h_pvs_common_all_primesvaluation_selected_power_partial + S (bpvi_partial_pvs_common_all_primesvaluation_selected_power) = S ((S (bpvi_j_pvs_common_all_primesvaluation_selected_power)) * bpvi_v_pvs_common_all_primesvaluation_selected_power)) /\ exists bpvi_q_pvs_common_all_primesvaluation_selected_power_partial. bpvi_u_pvs_common_all_primesvaluation_selected_power = bpvi_q_pvs_common_all_primesvaluation_selected_power_partial * S ((S (bpvi_j_pvs_common_all_primesvaluation_selected_power)) * bpvi_v_pvs_common_all_primesvaluation_selected_power) + (bpvi_partial_pvs_common_all_primesvaluation_selected_power))) /\ ((((exists bpvi_h_pvs_common_all_primesvaluation_selected_power_successor. bpvi_h_pvs_common_all_primesvaluation_selected_power_successor + S (bpvi_successor_pvs_common_all_primesvaluation_selected_power) = S ((S (S bpvi_j_pvs_common_all_primesvaluation_selected_power)) * bpvi_v_pvs_common_all_primesvaluation_selected_power)) /\ exists bpvi_q_pvs_common_all_primesvaluation_selected_power_successor. bpvi_u_pvs_common_all_primesvaluation_selected_power = bpvi_q_pvs_common_all_primesvaluation_selected_power_successor * S ((S (S bpvi_j_pvs_common_all_primesvaluation_selected_power)) * bpvi_v_pvs_common_all_primesvaluation_selected_power) + (bpvi_successor_pvs_common_all_primesvaluation_selected_power))) /\ bpvi_successor_pvs_common_all_primesvaluation_selected_power = bpvi_partial_pvs_common_all_primesvaluation_selected_power * bpvi_factor_pvs_common_all_primesvaluation_selected_power)))))))) /\ exists bpvi_divisor_factor_pvs_common_all_primesvaluation_selected. n = bpvi_result_pvs_common_all_primesvaluation_selected * bpvi_divisor_factor_pvs_common_all_primesvaluation_selected))) /\ forall bpd_candidate_pvs_common_all_primesvaluation. (exists bpd_gap_pvs_common_all_primesvaluation_candidate_bound. bpd_gap_pvs_common_all_primesvaluation_candidate_bound + (bpd_candidate_pvs_common_all_primesvaluation) = (n)) -> (exists bpvi_result_pvs_common_all_primesvaluation_candidate. ((exists bpvi_b_pvs_common_all_primesvaluation_candidate_power bpvi_c_pvs_common_all_primesvaluation_candidate_power. ((forall bpvi_i_pvs_common_all_primesvaluation_candidate_power. (exists bpvi_repeat_gap_pvs_common_all_primesvaluation_candidate_power. bpvi_repeat_gap_pvs_common_all_primesvaluation_candidate_power + S bpvi_i_pvs_common_all_primesvaluation_candidate_power = bpd_candidate_pvs_common_all_primesvaluation) -> (((exists bpvi_h_pvs_common_all_primesvaluation_candidate_power_repeat. bpvi_h_pvs_common_all_primesvaluation_candidate_power_repeat + S (ppf_prime_common_all_primes) = S ((S (bpvi_i_pvs_common_all_primesvaluation_candidate_power)) * bpvi_c_pvs_common_all_primesvaluation_candidate_power)) /\ exists bpvi_q_pvs_common_all_primesvaluation_candidate_power_repeat. bpvi_b_pvs_common_all_primesvaluation_candidate_power = bpvi_q_pvs_common_all_primesvaluation_candidate_power_repeat * S ((S (bpvi_i_pvs_common_all_primesvaluation_candidate_power)) * bpvi_c_pvs_common_all_primesvaluation_candidate_power) + (ppf_prime_common_all_primes)))) /\ (exists bpvi_u_pvs_common_all_primesvaluation_candidate_power bpvi_v_pvs_common_all_primesvaluation_candidate_power. ((((exists bpvi_h_pvs_common_all_primesvaluation_candidate_power_start. bpvi_h_pvs_common_all_primesvaluation_candidate_power_start + S (1) = S ((S (0)) * bpvi_v_pvs_common_all_primesvaluation_candidate_power)) /\ exists bpvi_q_pvs_common_all_primesvaluation_candidate_power_start. bpvi_u_pvs_common_all_primesvaluation_candidate_power = bpvi_q_pvs_common_all_primesvaluation_candidate_power_start * S ((S (0)) * bpvi_v_pvs_common_all_primesvaluation_candidate_power) + (1))) /\ ((((exists bpvi_h_pvs_common_all_primesvaluation_candidate_power_terminal. bpvi_h_pvs_common_all_primesvaluation_candidate_power_terminal + S (bpvi_result_pvs_common_all_primesvaluation_candidate) = S ((S (bpd_candidate_pvs_common_all_primesvaluation)) * bpvi_v_pvs_common_all_primesvaluation_candidate_power)) /\ exists bpvi_q_pvs_common_all_primesvaluation_candidate_power_terminal. bpvi_u_pvs_common_all_primesvaluation_candidate_power = bpvi_q_pvs_common_all_primesvaluation_candidate_power_terminal * S ((S (bpd_candidate_pvs_common_all_primesvaluation)) * bpvi_v_pvs_common_all_primesvaluation_candidate_power) + (bpvi_result_pvs_common_all_primesvaluation_candidate))) /\ forall bpvi_j_pvs_common_all_primesvaluation_candidate_power. (exists bpvi_product_gap_pvs_common_all_primesvaluation_candidate_power. bpvi_product_gap_pvs_common_all_primesvaluation_candidate_power + S bpvi_j_pvs_common_all_primesvaluation_candidate_power = bpd_candidate_pvs_common_all_primesvaluation) -> exists bpvi_factor_pvs_common_all_primesvaluation_candidate_power bpvi_partial_pvs_common_all_primesvaluation_candidate_power bpvi_successor_pvs_common_all_primesvaluation_candidate_power. ((((exists bpvi_h_pvs_common_all_primesvaluation_candidate_power_factor. bpvi_h_pvs_common_all_primesvaluation_candidate_power_factor + S (bpvi_factor_pvs_common_all_primesvaluation_candidate_power) = S ((S (bpvi_j_pvs_common_all_primesvaluation_candidate_power)) * bpvi_c_pvs_common_all_primesvaluation_candidate_power)) /\ exists bpvi_q_pvs_common_all_primesvaluation_candidate_power_factor. bpvi_b_pvs_common_all_primesvaluation_candidate_power = bpvi_q_pvs_common_all_primesvaluation_candidate_power_factor * S ((S (bpvi_j_pvs_common_all_primesvaluation_candidate_power)) * bpvi_c_pvs_common_all_primesvaluation_candidate_power) + (bpvi_factor_pvs_common_all_primesvaluation_candidate_power))) /\ ((((exists bpvi_h_pvs_common_all_primesvaluation_candidate_power_partial. bpvi_h_pvs_common_all_primesvaluation_candidate_power_partial + S (bpvi_partial_pvs_common_all_primesvaluation_candidate_power) = S ((S (bpvi_j_pvs_common_all_primesvaluation_candidate_power)) * bpvi_v_pvs_common_all_primesvaluation_candidate_power)) /\ exists bpvi_q_pvs_common_all_primesvaluation_candidate_power_partial. bpvi_u_pvs_common_all_primesvaluation_candidate_power = bpvi_q_pvs_common_all_primesvaluation_candidate_power_partial * S ((S (bpvi_j_pvs_common_all_primesvaluation_candidate_power)) * bpvi_v_pvs_common_all_primesvaluation_candidate_power) + (bpvi_partial_pvs_common_all_primesvaluation_candidate_power))) /\ ((((exists bpvi_h_pvs_common_all_primesvaluation_candidate_power_successor. bpvi_h_pvs_common_all_primesvaluation_candidate_power_successor + S (bpvi_successor_pvs_common_all_primesvaluation_candidate_power) = S ((S (S bpvi_j_pvs_common_all_primesvaluation_candidate_power)) * bpvi_v_pvs_common_all_primesvaluation_candidate_power)) /\ exists bpvi_q_pvs_common_all_primesvaluation_candidate_power_successor. bpvi_u_pvs_common_all_primesvaluation_candidate_power = bpvi_q_pvs_common_all_primesvaluation_candidate_power_successor * S ((S (S bpvi_j_pvs_common_all_primesvaluation_candidate_power)) * bpvi_v_pvs_common_all_primesvaluation_candidate_power) + (bpvi_successor_pvs_common_all_primesvaluation_candidate_power))) /\ bpvi_successor_pvs_common_all_primesvaluation_candidate_power = bpvi_partial_pvs_common_all_primesvaluation_candidate_power * bpvi_factor_pvs_common_all_primesvaluation_candidate_power)))))))) /\ exists bpvi_divisor_factor_pvs_common_all_primesvaluation_candidate. n = bpvi_result_pvs_common_all_primesvaluation_candidate * bpvi_divisor_factor_pvs_common_all_primesvaluation_candidate)) -> (exists bpd_gap_pvs_common_all_primesvaluation_maximal. bpd_gap_pvs_common_all_primesvaluation_maximal + (bpd_candidate_pvs_common_all_primesvaluation) = (ppf_exponent_common_all_primes))) -> (exists pvs_factor_common_all_primesdivides. (ppf_exponent_common_all_primes) = (k) * pvs_factor_common_all_primesdivides))