power_valuation_selected_and_successor_not_divides

Exact expanded PA statement

forall p a e. ((~(p = 1) /\ forall frm_prime_left_bpvl_prime frm_prime_right_bpvl_prime. p = frm_prime_left_bpvl_prime * frm_prime_right_bpvl_prime -> frm_prime_left_bpvl_prime = 1 \/ frm_prime_right_bpvl_prime = 1)) -> ~(a = 0) -> (((exists bpv_gap_bpvl_valuation_exponent_bound. bpv_gap_bpvl_valuation_exponent_bound + e = a) /\ (exists bpv_result_bpvl_valuation_selected. ((exists ff_b_bpvl_valuation_selected_power ff_c_bpvl_valuation_selected_power. ((forall ff_i_bpvl_valuation_selected_power_repeat. (exists ff_lt_bpvl_valuation_selected_power_repeat_bound. ff_lt_bpvl_valuation_selected_power_repeat_bound + S ff_i_bpvl_valuation_selected_power_repeat = e) -> (((exists ff_h_bpvl_valuation_selected_power_repeat_decoded. ff_h_bpvl_valuation_selected_power_repeat_decoded + S (p) = S ((S (ff_i_bpvl_valuation_selected_power_repeat)) * ff_c_bpvl_valuation_selected_power)) /\ exists ff_q_bpvl_valuation_selected_power_repeat_decoded. ff_b_bpvl_valuation_selected_power = ff_q_bpvl_valuation_selected_power_repeat_decoded * S ((S (ff_i_bpvl_valuation_selected_power_repeat)) * ff_c_bpvl_valuation_selected_power) + (p)))) /\ (exists ff_u_bpvl_valuation_selected_power_product ff_v_bpvl_valuation_selected_power_product. ((((exists ff_h_bpvl_valuation_selected_power_product_start. ff_h_bpvl_valuation_selected_power_product_start + S (1) = S ((S (0)) * ff_v_bpvl_valuation_selected_power_product)) /\ exists ff_q_bpvl_valuation_selected_power_product_start. ff_u_bpvl_valuation_selected_power_product = ff_q_bpvl_valuation_selected_power_product_start * S ((S (0)) * ff_v_bpvl_valuation_selected_power_product) + (1))) /\ ((((exists ff_h_bpvl_valuation_selected_power_product_terminal. ff_h_bpvl_valuation_selected_power_product_terminal + S (bpv_result_bpvl_valuation_selected) = S ((S (e)) * ff_v_bpvl_valuation_selected_power_product)) /\ exists ff_q_bpvl_valuation_selected_power_product_terminal. ff_u_bpvl_valuation_selected_power_product = ff_q_bpvl_valuation_selected_power_product_terminal * S ((S (e)) * ff_v_bpvl_valuation_selected_power_product) + (bpv_result_bpvl_valuation_selected))) /\ forall ff_i_bpvl_valuation_selected_power_product. (exists ff_lt_bpvl_valuation_selected_power_product_bound. ff_lt_bpvl_valuation_selected_power_product_bound + S ff_i_bpvl_valuation_selected_power_product = e) -> exists ff_p_bpvl_valuation_selected_power_product ff_r_bpvl_valuation_selected_power_product ff_s_bpvl_valuation_selected_power_product. ((((exists ff_h_bpvl_valuation_selected_power_product_factor. ff_h_bpvl_valuation_selected_power_product_factor + S (ff_p_bpvl_valuation_selected_power_product) = S ((S (ff_i_bpvl_valuation_selected_power_product)) * ff_c_bpvl_valuation_selected_power)) /\ exists ff_q_bpvl_valuation_selected_power_product_factor. ff_b_bpvl_valuation_selected_power = ff_q_bpvl_valuation_selected_power_product_factor * S ((S (ff_i_bpvl_valuation_selected_power_product)) * ff_c_bpvl_valuation_selected_power) + (ff_p_bpvl_valuation_selected_power_product))) /\ ((((exists ff_h_bpvl_valuation_selected_power_product_partial. ff_h_bpvl_valuation_selected_power_product_partial + S (ff_r_bpvl_valuation_selected_power_product) = S ((S (ff_i_bpvl_valuation_selected_power_product)) * ff_v_bpvl_valuation_selected_power_product)) /\ exists ff_q_bpvl_valuation_selected_power_product_partial. ff_u_bpvl_valuation_selected_power_product = ff_q_bpvl_valuation_selected_power_product_partial * S ((S (ff_i_bpvl_valuation_selected_power_product)) * ff_v_bpvl_valuation_selected_power_product) + (ff_r_bpvl_valuation_selected_power_product))) /\ ((((exists ff_h_bpvl_valuation_selected_power_product_successor. ff_h_bpvl_valuation_selected_power_product_successor + S (ff_s_bpvl_valuation_selected_power_product) = S ((S (S ff_i_bpvl_valuation_selected_power_product)) * ff_v_bpvl_valuation_selected_power_product)) /\ exists ff_q_bpvl_valuation_selected_power_product_successor. ff_u_bpvl_valuation_selected_power_product = ff_q_bpvl_valuation_selected_power_product_successor * S ((S (S ff_i_bpvl_valuation_selected_power_product)) * ff_v_bpvl_valuation_selected_power_product) + (ff_s_bpvl_valuation_selected_power_product))) /\ ff_s_bpvl_valuation_selected_power_product = ff_r_bpvl_valuation_selected_power_product * ff_p_bpvl_valuation_selected_power_product)))))))) /\ (exists bpv_factor_bpvl_valuation_selected_divides. a = bpv_result_bpvl_valuation_selected * bpv_factor_bpvl_valuation_selected_divides)))) /\ forall bpv_candidate_bpvl_valuation. (exists bpv_gap_bpvl_valuation_candidate_bound. bpv_gap_bpvl_valuation_candidate_bound + bpv_candidate_bpvl_valuation = a) -> (exists bpv_result_bpvl_valuation_candidate. ((exists ff_b_bpvl_valuation_candidate_power ff_c_bpvl_valuation_candidate_power. ((forall ff_i_bpvl_valuation_candidate_power_repeat. (exists ff_lt_bpvl_valuation_candidate_power_repeat_bound. ff_lt_bpvl_valuation_candidate_power_repeat_bound + S ff_i_bpvl_valuation_candidate_power_repeat = bpv_candidate_bpvl_valuation) -> (((exists ff_h_bpvl_valuation_candidate_power_repeat_decoded. ff_h_bpvl_valuation_candidate_power_repeat_decoded + S (p) = S ((S (ff_i_bpvl_valuation_candidate_power_repeat)) * ff_c_bpvl_valuation_candidate_power)) /\ exists ff_q_bpvl_valuation_candidate_power_repeat_decoded. ff_b_bpvl_valuation_candidate_power = ff_q_bpvl_valuation_candidate_power_repeat_decoded * S ((S (ff_i_bpvl_valuation_candidate_power_repeat)) * ff_c_bpvl_valuation_candidate_power) + (p)))) /\ (exists ff_u_bpvl_valuation_candidate_power_product ff_v_bpvl_valuation_candidate_power_product. ((((exists ff_h_bpvl_valuation_candidate_power_product_start. ff_h_bpvl_valuation_candidate_power_product_start + S (1) = S ((S (0)) * ff_v_bpvl_valuation_candidate_power_product)) /\ exists ff_q_bpvl_valuation_candidate_power_product_start. ff_u_bpvl_valuation_candidate_power_product = ff_q_bpvl_valuation_candidate_power_product_start * S ((S (0)) * ff_v_bpvl_valuation_candidate_power_product) + (1))) /\ ((((exists ff_h_bpvl_valuation_candidate_power_product_terminal. ff_h_bpvl_valuation_candidate_power_product_terminal + S (bpv_result_bpvl_valuation_candidate) = S ((S (bpv_candidate_bpvl_valuation)) * ff_v_bpvl_valuation_candidate_power_product)) /\ exists ff_q_bpvl_valuation_candidate_power_product_terminal. ff_u_bpvl_valuation_candidate_power_product = ff_q_bpvl_valuation_candidate_power_product_terminal * S ((S (bpv_candidate_bpvl_valuation)) * ff_v_bpvl_valuation_candidate_power_product) + (bpv_result_bpvl_valuation_candidate))) /\ forall ff_i_bpvl_valuation_candidate_power_product. (exists ff_lt_bpvl_valuation_candidate_power_product_bound. ff_lt_bpvl_valuation_candidate_power_product_bound + S ff_i_bpvl_valuation_candidate_power_product = bpv_candidate_bpvl_valuation) -> exists ff_p_bpvl_valuation_candidate_power_product ff_r_bpvl_valuation_candidate_power_product ff_s_bpvl_valuation_candidate_power_product. ((((exists ff_h_bpvl_valuation_candidate_power_product_factor. ff_h_bpvl_valuation_candidate_power_product_factor + S (ff_p_bpvl_valuation_candidate_power_product) = S ((S (ff_i_bpvl_valuation_candidate_power_product)) * ff_c_bpvl_valuation_candidate_power)) /\ exists ff_q_bpvl_valuation_candidate_power_product_factor. ff_b_bpvl_valuation_candidate_power = ff_q_bpvl_valuation_candidate_power_product_factor * S ((S (ff_i_bpvl_valuation_candidate_power_product)) * ff_c_bpvl_valuation_candidate_power) + (ff_p_bpvl_valuation_candidate_power_product))) /\ ((((exists ff_h_bpvl_valuation_candidate_power_product_partial. ff_h_bpvl_valuation_candidate_power_product_partial + S (ff_r_bpvl_valuation_candidate_power_product) = S ((S (ff_i_bpvl_valuation_candidate_power_product)) * ff_v_bpvl_valuation_candidate_power_product)) /\ exists ff_q_bpvl_valuation_candidate_power_product_partial. ff_u_bpvl_valuation_candidate_power_product = ff_q_bpvl_valuation_candidate_power_product_partial * S ((S (ff_i_bpvl_valuation_candidate_power_product)) * ff_v_bpvl_valuation_candidate_power_product) + (ff_r_bpvl_valuation_candidate_power_product))) /\ ((((exists ff_h_bpvl_valuation_candidate_power_product_successor. ff_h_bpvl_valuation_candidate_power_product_successor + S (ff_s_bpvl_valuation_candidate_power_product) = S ((S (S ff_i_bpvl_valuation_candidate_power_product)) * ff_v_bpvl_valuation_candidate_power_product)) /\ exists ff_q_bpvl_valuation_candidate_power_product_successor. ff_u_bpvl_valuation_candidate_power_product = ff_q_bpvl_valuation_candidate_power_product_successor * S ((S (S ff_i_bpvl_valuation_candidate_power_product)) * ff_v_bpvl_valuation_candidate_power_product) + (ff_s_bpvl_valuation_candidate_power_product))) /\ ff_s_bpvl_valuation_candidate_power_product = ff_r_bpvl_valuation_candidate_power_product * ff_p_bpvl_valuation_candidate_power_product)))))))) /\ (exists bpv_factor_bpvl_valuation_candidate_divides. a = bpv_result_bpvl_valuation_candidate * bpv_factor_bpvl_valuation_candidate_divides))) -> (exists bpv_gap_bpvl_valuation_maximal. bpv_gap_bpvl_valuation_maximal + bpv_candidate_bpvl_valuation = e)) -> ((exists bpv_result_bpvl_selected_divides. ((exists ff_b_bpvl_selected_divides_power ff_c_bpvl_selected_divides_power. ((forall ff_i_bpvl_selected_divides_power_repeat. (exists ff_lt_bpvl_selected_divides_power_repeat_bound. ff_lt_bpvl_selected_divides_power_repeat_bound + S ff_i_bpvl_selected_divides_power_repeat = e) -> (((exists ff_h_bpvl_selected_divides_power_repeat_decoded. ff_h_bpvl_selected_divides_power_repeat_decoded + S (p) = S ((S (ff_i_bpvl_selected_divides_power_repeat)) * ff_c_bpvl_selected_divides_power)) /\ exists ff_q_bpvl_selected_divides_power_repeat_decoded. ff_b_bpvl_selected_divides_power = ff_q_bpvl_selected_divides_power_repeat_decoded * S ((S (ff_i_bpvl_selected_divides_power_repeat)) * ff_c_bpvl_selected_divides_power) + (p)))) /\ (exists ff_u_bpvl_selected_divides_power_product ff_v_bpvl_selected_divides_power_product. ((((exists ff_h_bpvl_selected_divides_power_product_start. ff_h_bpvl_selected_divides_power_product_start + S (1) = S ((S (0)) * ff_v_bpvl_selected_divides_power_product)) /\ exists ff_q_bpvl_selected_divides_power_product_start. ff_u_bpvl_selected_divides_power_product = ff_q_bpvl_selected_divides_power_product_start * S ((S (0)) * ff_v_bpvl_selected_divides_power_product) + (1))) /\ ((((exists ff_h_bpvl_selected_divides_power_product_terminal. ff_h_bpvl_selected_divides_power_product_terminal + S (bpv_result_bpvl_selected_divides) = S ((S (e)) * ff_v_bpvl_selected_divides_power_product)) /\ exists ff_q_bpvl_selected_divides_power_product_terminal. ff_u_bpvl_selected_divides_power_product = ff_q_bpvl_selected_divides_power_product_terminal * S ((S (e)) * ff_v_bpvl_selected_divides_power_product) + (bpv_result_bpvl_selected_divides))) /\ forall ff_i_bpvl_selected_divides_power_product. (exists ff_lt_bpvl_selected_divides_power_product_bound. ff_lt_bpvl_selected_divides_power_product_bound + S ff_i_bpvl_selected_divides_power_product = e) -> exists ff_p_bpvl_selected_divides_power_product ff_r_bpvl_selected_divides_power_product ff_s_bpvl_selected_divides_power_product. ((((exists ff_h_bpvl_selected_divides_power_product_factor. ff_h_bpvl_selected_divides_power_product_factor + S (ff_p_bpvl_selected_divides_power_product) = S ((S (ff_i_bpvl_selected_divides_power_product)) * ff_c_bpvl_selected_divides_power)) /\ exists ff_q_bpvl_selected_divides_power_product_factor. ff_b_bpvl_selected_divides_power = ff_q_bpvl_selected_divides_power_product_factor * S ((S (ff_i_bpvl_selected_divides_power_product)) * ff_c_bpvl_selected_divides_power) + (ff_p_bpvl_selected_divides_power_product))) /\ ((((exists ff_h_bpvl_selected_divides_power_product_partial. ff_h_bpvl_selected_divides_power_product_partial + S (ff_r_bpvl_selected_divides_power_product) = S ((S (ff_i_bpvl_selected_divides_power_product)) * ff_v_bpvl_selected_divides_power_product)) /\ exists ff_q_bpvl_selected_divides_power_product_partial. ff_u_bpvl_selected_divides_power_product = ff_q_bpvl_selected_divides_power_product_partial * S ((S (ff_i_bpvl_selected_divides_power_product)) * ff_v_bpvl_selected_divides_power_product) + (ff_r_bpvl_selected_divides_power_product))) /\ ((((exists ff_h_bpvl_selected_divides_power_product_successor. ff_h_bpvl_selected_divides_power_product_successor + S (ff_s_bpvl_selected_divides_power_product) = S ((S (S ff_i_bpvl_selected_divides_power_product)) * ff_v_bpvl_selected_divides_power_product)) /\ exists ff_q_bpvl_selected_divides_power_product_successor. ff_u_bpvl_selected_divides_power_product = ff_q_bpvl_selected_divides_power_product_successor * S ((S (S ff_i_bpvl_selected_divides_power_product)) * ff_v_bpvl_selected_divides_power_product) + (ff_s_bpvl_selected_divides_power_product))) /\ ff_s_bpvl_selected_divides_power_product = ff_r_bpvl_selected_divides_power_product * ff_p_bpvl_selected_divides_power_product)))))))) /\ (exists bpv_factor_bpvl_selected_divides_divides. a = bpv_result_bpvl_selected_divides * bpv_factor_bpvl_selected_divides_divides))) /\ ~(exists bpvi_result_bpvl_successor_divides. ((exists bpvi_b_bpvl_successor_divides_power bpvi_c_bpvl_successor_divides_power. ((forall bpvi_i_bpvl_successor_divides_power. (exists bpvi_repeat_gap_bpvl_successor_divides_power. bpvi_repeat_gap_bpvl_successor_divides_power + S bpvi_i_bpvl_successor_divides_power = S e) -> (((exists bpvi_h_bpvl_successor_divides_power_repeat. bpvi_h_bpvl_successor_divides_power_repeat + S (p) = S ((S (bpvi_i_bpvl_successor_divides_power)) * bpvi_c_bpvl_successor_divides_power)) /\ exists bpvi_q_bpvl_successor_divides_power_repeat. bpvi_b_bpvl_successor_divides_power = bpvi_q_bpvl_successor_divides_power_repeat * S ((S (bpvi_i_bpvl_successor_divides_power)) * bpvi_c_bpvl_successor_divides_power) + (p)))) /\ (exists bpvi_u_bpvl_successor_divides_power bpvi_v_bpvl_successor_divides_power. ((((exists bpvi_h_bpvl_successor_divides_power_start. bpvi_h_bpvl_successor_divides_power_start + S (1) = S ((S (0)) * bpvi_v_bpvl_successor_divides_power)) /\ exists bpvi_q_bpvl_successor_divides_power_start. bpvi_u_bpvl_successor_divides_power = bpvi_q_bpvl_successor_divides_power_start * S ((S (0)) * bpvi_v_bpvl_successor_divides_power) + (1))) /\ ((((exists bpvi_h_bpvl_successor_divides_power_terminal. bpvi_h_bpvl_successor_divides_power_terminal + S (bpvi_result_bpvl_successor_divides) = S ((S (S e)) * bpvi_v_bpvl_successor_divides_power)) /\ exists bpvi_q_bpvl_successor_divides_power_terminal. bpvi_u_bpvl_successor_divides_power = bpvi_q_bpvl_successor_divides_power_terminal * S ((S (S e)) * bpvi_v_bpvl_successor_divides_power) + (bpvi_result_bpvl_successor_divides))) /\ forall bpvi_j_bpvl_successor_divides_power. (exists bpvi_product_gap_bpvl_successor_divides_power. bpvi_product_gap_bpvl_successor_divides_power + S bpvi_j_bpvl_successor_divides_power = S e) -> exists bpvi_factor_bpvl_successor_divides_power bpvi_partial_bpvl_successor_divides_power bpvi_successor_bpvl_successor_divides_power. ((((exists bpvi_h_bpvl_successor_divides_power_factor. bpvi_h_bpvl_successor_divides_power_factor + S (bpvi_factor_bpvl_successor_divides_power) = S ((S (bpvi_j_bpvl_successor_divides_power)) * bpvi_c_bpvl_successor_divides_power)) /\ exists bpvi_q_bpvl_successor_divides_power_factor. bpvi_b_bpvl_successor_divides_power = bpvi_q_bpvl_successor_divides_power_factor * S ((S (bpvi_j_bpvl_successor_divides_power)) * bpvi_c_bpvl_successor_divides_power) + (bpvi_factor_bpvl_successor_divides_power))) /\ ((((exists bpvi_h_bpvl_successor_divides_power_partial. bpvi_h_bpvl_successor_divides_power_partial + S (bpvi_partial_bpvl_successor_divides_power) = S ((S (bpvi_j_bpvl_successor_divides_power)) * bpvi_v_bpvl_successor_divides_power)) /\ exists bpvi_q_bpvl_successor_divides_power_partial. bpvi_u_bpvl_successor_divides_power = bpvi_q_bpvl_successor_divides_power_partial * S ((S (bpvi_j_bpvl_successor_divides_power)) * bpvi_v_bpvl_successor_divides_power) + (bpvi_partial_bpvl_successor_divides_power))) /\ ((((exists bpvi_h_bpvl_successor_divides_power_successor. bpvi_h_bpvl_successor_divides_power_successor + S (bpvi_successor_bpvl_successor_divides_power) = S ((S (S bpvi_j_bpvl_successor_divides_power)) * bpvi_v_bpvl_successor_divides_power)) /\ exists bpvi_q_bpvl_successor_divides_power_successor. bpvi_u_bpvl_successor_divides_power = bpvi_q_bpvl_successor_divides_power_successor * S ((S (S bpvi_j_bpvl_successor_divides_power)) * bpvi_v_bpvl_successor_divides_power) + (bpvi_successor_bpvl_successor_divides_power))) /\ bpvi_successor_bpvl_successor_divides_power = bpvi_partial_bpvl_successor_divides_power * bpvi_factor_bpvl_successor_divides_power)))))))) /\ exists bpvi_divisor_factor_bpvl_successor_divides. a = bpvi_result_bpvl_successor_divides * bpvi_divisor_factor_bpvl_successor_divides)))