valuation_threshold_bit_decides_power_divides

Exact expanded PA statement

forall p a f i bit. ((~(p = 1) /\ forall frm_prime_left_legendre_successor_prime frm_prime_right_legendre_successor_prime. p = frm_prime_left_legendre_successor_prime * frm_prime_right_legendre_successor_prime -> frm_prime_left_legendre_successor_prime = 1 \/ frm_prime_right_legendre_successor_prime = 1)) -> ~(a = 0) -> (((exists bpv_gap_legendre_successor_threshold_valuation_exponent_bound. bpv_gap_legendre_successor_threshold_valuation_exponent_bound + f = a) /\ (exists bpv_result_legendre_successor_threshold_valuation_selected. ((exists ff_b_legendre_successor_threshold_valuation_selected_power ff_c_legendre_successor_threshold_valuation_selected_power. ((forall ff_i_legendre_successor_threshold_valuation_selected_power_repeat. (exists ff_lt_legendre_successor_threshold_valuation_selected_power_repeat_bound. ff_lt_legendre_successor_threshold_valuation_selected_power_repeat_bound + S ff_i_legendre_successor_threshold_valuation_selected_power_repeat = f) -> (((exists ff_h_legendre_successor_threshold_valuation_selected_power_repeat_decoded. ff_h_legendre_successor_threshold_valuation_selected_power_repeat_decoded + S (p) = S ((S (ff_i_legendre_successor_threshold_valuation_selected_power_repeat)) * ff_c_legendre_successor_threshold_valuation_selected_power)) /\ exists ff_q_legendre_successor_threshold_valuation_selected_power_repeat_decoded. ff_b_legendre_successor_threshold_valuation_selected_power = ff_q_legendre_successor_threshold_valuation_selected_power_repeat_decoded * S ((S (ff_i_legendre_successor_threshold_valuation_selected_power_repeat)) * ff_c_legendre_successor_threshold_valuation_selected_power) + (p)))) /\ (exists ff_u_legendre_successor_threshold_valuation_selected_power_product ff_v_legendre_successor_threshold_valuation_selected_power_product. ((((exists ff_h_legendre_successor_threshold_valuation_selected_power_product_start. ff_h_legendre_successor_threshold_valuation_selected_power_product_start + S (1) = S ((S (0)) * ff_v_legendre_successor_threshold_valuation_selected_power_product)) /\ exists ff_q_legendre_successor_threshold_valuation_selected_power_product_start. ff_u_legendre_successor_threshold_valuation_selected_power_product = ff_q_legendre_successor_threshold_valuation_selected_power_product_start * S ((S (0)) * ff_v_legendre_successor_threshold_valuation_selected_power_product) + (1))) /\ ((((exists ff_h_legendre_successor_threshold_valuation_selected_power_product_terminal. ff_h_legendre_successor_threshold_valuation_selected_power_product_terminal + S (bpv_result_legendre_successor_threshold_valuation_selected) = S ((S (f)) * ff_v_legendre_successor_threshold_valuation_selected_power_product)) /\ exists ff_q_legendre_successor_threshold_valuation_selected_power_product_terminal. ff_u_legendre_successor_threshold_valuation_selected_power_product = ff_q_legendre_successor_threshold_valuation_selected_power_product_terminal * S ((S (f)) * ff_v_legendre_successor_threshold_valuation_selected_power_product) + (bpv_result_legendre_successor_threshold_valuation_selected))) /\ forall ff_i_legendre_successor_threshold_valuation_selected_power_product. (exists ff_lt_legendre_successor_threshold_valuation_selected_power_product_bound. ff_lt_legendre_successor_threshold_valuation_selected_power_product_bound + S ff_i_legendre_successor_threshold_valuation_selected_power_product = f) -> exists ff_p_legendre_successor_threshold_valuation_selected_power_product ff_r_legendre_successor_threshold_valuation_selected_power_product ff_s_legendre_successor_threshold_valuation_selected_power_product. ((((exists ff_h_legendre_successor_threshold_valuation_selected_power_product_factor. ff_h_legendre_successor_threshold_valuation_selected_power_product_factor + S (ff_p_legendre_successor_threshold_valuation_selected_power_product) = S ((S (ff_i_legendre_successor_threshold_valuation_selected_power_product)) * ff_c_legendre_successor_threshold_valuation_selected_power)) /\ exists ff_q_legendre_successor_threshold_valuation_selected_power_product_factor. ff_b_legendre_successor_threshold_valuation_selected_power = ff_q_legendre_successor_threshold_valuation_selected_power_product_factor * S ((S (ff_i_legendre_successor_threshold_valuation_selected_power_product)) * ff_c_legendre_successor_threshold_valuation_selected_power) + (ff_p_legendre_successor_threshold_valuation_selected_power_product))) /\ ((((exists ff_h_legendre_successor_threshold_valuation_selected_power_product_partial. ff_h_legendre_successor_threshold_valuation_selected_power_product_partial + S (ff_r_legendre_successor_threshold_valuation_selected_power_product) = S ((S (ff_i_legendre_successor_threshold_valuation_selected_power_product)) * ff_v_legendre_successor_threshold_valuation_selected_power_product)) /\ exists ff_q_legendre_successor_threshold_valuation_selected_power_product_partial. ff_u_legendre_successor_threshold_valuation_selected_power_product = ff_q_legendre_successor_threshold_valuation_selected_power_product_partial * S ((S (ff_i_legendre_successor_threshold_valuation_selected_power_product)) * ff_v_legendre_successor_threshold_valuation_selected_power_product) + (ff_r_legendre_successor_threshold_valuation_selected_power_product))) /\ ((((exists ff_h_legendre_successor_threshold_valuation_selected_power_product_successor. ff_h_legendre_successor_threshold_valuation_selected_power_product_successor + S (ff_s_legendre_successor_threshold_valuation_selected_power_product) = S ((S (S ff_i_legendre_successor_threshold_valuation_selected_power_product)) * ff_v_legendre_successor_threshold_valuation_selected_power_product)) /\ exists ff_q_legendre_successor_threshold_valuation_selected_power_product_successor. ff_u_legendre_successor_threshold_valuation_selected_power_product = ff_q_legendre_successor_threshold_valuation_selected_power_product_successor * S ((S (S ff_i_legendre_successor_threshold_valuation_selected_power_product)) * ff_v_legendre_successor_threshold_valuation_selected_power_product) + (ff_s_legendre_successor_threshold_valuation_selected_power_product))) /\ ff_s_legendre_successor_threshold_valuation_selected_power_product = ff_r_legendre_successor_threshold_valuation_selected_power_product * ff_p_legendre_successor_threshold_valuation_selected_power_product)))))))) /\ (exists bpv_factor_legendre_successor_threshold_valuation_selected_divides. a = bpv_result_legendre_successor_threshold_valuation_selected * bpv_factor_legendre_successor_threshold_valuation_selected_divides)))) /\ forall bpv_candidate_legendre_successor_threshold_valuation. (exists bpv_gap_legendre_successor_threshold_valuation_candidate_bound. bpv_gap_legendre_successor_threshold_valuation_candidate_bound + bpv_candidate_legendre_successor_threshold_valuation = a) -> (exists bpv_result_legendre_successor_threshold_valuation_candidate. ((exists ff_b_legendre_successor_threshold_valuation_candidate_power ff_c_legendre_successor_threshold_valuation_candidate_power. ((forall ff_i_legendre_successor_threshold_valuation_candidate_power_repeat. (exists ff_lt_legendre_successor_threshold_valuation_candidate_power_repeat_bound. ff_lt_legendre_successor_threshold_valuation_candidate_power_repeat_bound + S ff_i_legendre_successor_threshold_valuation_candidate_power_repeat = bpv_candidate_legendre_successor_threshold_valuation) -> (((exists ff_h_legendre_successor_threshold_valuation_candidate_power_repeat_decoded. ff_h_legendre_successor_threshold_valuation_candidate_power_repeat_decoded + S (p) = S ((S (ff_i_legendre_successor_threshold_valuation_candidate_power_repeat)) * ff_c_legendre_successor_threshold_valuation_candidate_power)) /\ exists ff_q_legendre_successor_threshold_valuation_candidate_power_repeat_decoded. ff_b_legendre_successor_threshold_valuation_candidate_power = ff_q_legendre_successor_threshold_valuation_candidate_power_repeat_decoded * S ((S (ff_i_legendre_successor_threshold_valuation_candidate_power_repeat)) * ff_c_legendre_successor_threshold_valuation_candidate_power) + (p)))) /\ (exists ff_u_legendre_successor_threshold_valuation_candidate_power_product ff_v_legendre_successor_threshold_valuation_candidate_power_product. ((((exists ff_h_legendre_successor_threshold_valuation_candidate_power_product_start. ff_h_legendre_successor_threshold_valuation_candidate_power_product_start + S (1) = S ((S (0)) * ff_v_legendre_successor_threshold_valuation_candidate_power_product)) /\ exists ff_q_legendre_successor_threshold_valuation_candidate_power_product_start. ff_u_legendre_successor_threshold_valuation_candidate_power_product = ff_q_legendre_successor_threshold_valuation_candidate_power_product_start * S ((S (0)) * ff_v_legendre_successor_threshold_valuation_candidate_power_product) + (1))) /\ ((((exists ff_h_legendre_successor_threshold_valuation_candidate_power_product_terminal. ff_h_legendre_successor_threshold_valuation_candidate_power_product_terminal + S (bpv_result_legendre_successor_threshold_valuation_candidate) = S ((S (bpv_candidate_legendre_successor_threshold_valuation)) * ff_v_legendre_successor_threshold_valuation_candidate_power_product)) /\ exists ff_q_legendre_successor_threshold_valuation_candidate_power_product_terminal. ff_u_legendre_successor_threshold_valuation_candidate_power_product = ff_q_legendre_successor_threshold_valuation_candidate_power_product_terminal * S ((S (bpv_candidate_legendre_successor_threshold_valuation)) * ff_v_legendre_successor_threshold_valuation_candidate_power_product) + (bpv_result_legendre_successor_threshold_valuation_candidate))) /\ forall ff_i_legendre_successor_threshold_valuation_candidate_power_product. (exists ff_lt_legendre_successor_threshold_valuation_candidate_power_product_bound. ff_lt_legendre_successor_threshold_valuation_candidate_power_product_bound + S ff_i_legendre_successor_threshold_valuation_candidate_power_product = bpv_candidate_legendre_successor_threshold_valuation) -> exists ff_p_legendre_successor_threshold_valuation_candidate_power_product ff_r_legendre_successor_threshold_valuation_candidate_power_product ff_s_legendre_successor_threshold_valuation_candidate_power_product. ((((exists ff_h_legendre_successor_threshold_valuation_candidate_power_product_factor. ff_h_legendre_successor_threshold_valuation_candidate_power_product_factor + S (ff_p_legendre_successor_threshold_valuation_candidate_power_product) = S ((S (ff_i_legendre_successor_threshold_valuation_candidate_power_product)) * ff_c_legendre_successor_threshold_valuation_candidate_power)) /\ exists ff_q_legendre_successor_threshold_valuation_candidate_power_product_factor. ff_b_legendre_successor_threshold_valuation_candidate_power = ff_q_legendre_successor_threshold_valuation_candidate_power_product_factor * S ((S (ff_i_legendre_successor_threshold_valuation_candidate_power_product)) * ff_c_legendre_successor_threshold_valuation_candidate_power) + (ff_p_legendre_successor_threshold_valuation_candidate_power_product))) /\ ((((exists ff_h_legendre_successor_threshold_valuation_candidate_power_product_partial. ff_h_legendre_successor_threshold_valuation_candidate_power_product_partial + S (ff_r_legendre_successor_threshold_valuation_candidate_power_product) = S ((S (ff_i_legendre_successor_threshold_valuation_candidate_power_product)) * ff_v_legendre_successor_threshold_valuation_candidate_power_product)) /\ exists ff_q_legendre_successor_threshold_valuation_candidate_power_product_partial. ff_u_legendre_successor_threshold_valuation_candidate_power_product = ff_q_legendre_successor_threshold_valuation_candidate_power_product_partial * S ((S (ff_i_legendre_successor_threshold_valuation_candidate_power_product)) * ff_v_legendre_successor_threshold_valuation_candidate_power_product) + (ff_r_legendre_successor_threshold_valuation_candidate_power_product))) /\ ((((exists ff_h_legendre_successor_threshold_valuation_candidate_power_product_successor. ff_h_legendre_successor_threshold_valuation_candidate_power_product_successor + S (ff_s_legendre_successor_threshold_valuation_candidate_power_product) = S ((S (S ff_i_legendre_successor_threshold_valuation_candidate_power_product)) * ff_v_legendre_successor_threshold_valuation_candidate_power_product)) /\ exists ff_q_legendre_successor_threshold_valuation_candidate_power_product_successor. ff_u_legendre_successor_threshold_valuation_candidate_power_product = ff_q_legendre_successor_threshold_valuation_candidate_power_product_successor * S ((S (S ff_i_legendre_successor_threshold_valuation_candidate_power_product)) * ff_v_legendre_successor_threshold_valuation_candidate_power_product) + (ff_s_legendre_successor_threshold_valuation_candidate_power_product))) /\ ff_s_legendre_successor_threshold_valuation_candidate_power_product = ff_r_legendre_successor_threshold_valuation_candidate_power_product * ff_p_legendre_successor_threshold_valuation_candidate_power_product)))))))) /\ (exists bpv_factor_legendre_successor_threshold_valuation_candidate_divides. a = bpv_result_legendre_successor_threshold_valuation_candidate * bpv_factor_legendre_successor_threshold_valuation_candidate_divides))) -> (exists bpv_gap_legendre_successor_threshold_valuation_maximal. bpv_gap_legendre_successor_threshold_valuation_maximal + bpv_candidate_legendre_successor_threshold_valuation = f)) -> ((bit = 1 /\ (exists blsr_le_gap_legendre_successor_threshold_inside. blsr_le_gap_legendre_successor_threshold_inside + (S i) = (f))) \/ (bit = 0 /\ (exists blsr_lt_gap_legendre_successor_threshold_outside. blsr_lt_gap_legendre_successor_threshold_outside + S (f) = (S i)))) -> ((bit = 1 /\ (exists bpvi_result_legendre_successor_threshold_divides. ((exists bpvi_b_legendre_successor_threshold_divides_power bpvi_c_legendre_successor_threshold_divides_power. ((forall bpvi_i_legendre_successor_threshold_divides_power. (exists bpvi_repeat_gap_legendre_successor_threshold_divides_power. bpvi_repeat_gap_legendre_successor_threshold_divides_power + S bpvi_i_legendre_successor_threshold_divides_power = S i) -> (((exists bpvi_h_legendre_successor_threshold_divides_power_repeat. bpvi_h_legendre_successor_threshold_divides_power_repeat + S (p) = S ((S (bpvi_i_legendre_successor_threshold_divides_power)) * bpvi_c_legendre_successor_threshold_divides_power)) /\ exists bpvi_q_legendre_successor_threshold_divides_power_repeat. bpvi_b_legendre_successor_threshold_divides_power = bpvi_q_legendre_successor_threshold_divides_power_repeat * S ((S (bpvi_i_legendre_successor_threshold_divides_power)) * bpvi_c_legendre_successor_threshold_divides_power) + (p)))) /\ (exists bpvi_u_legendre_successor_threshold_divides_power bpvi_v_legendre_successor_threshold_divides_power. ((((exists bpvi_h_legendre_successor_threshold_divides_power_start. bpvi_h_legendre_successor_threshold_divides_power_start + S (1) = S ((S (0)) * bpvi_v_legendre_successor_threshold_divides_power)) /\ exists bpvi_q_legendre_successor_threshold_divides_power_start. bpvi_u_legendre_successor_threshold_divides_power = bpvi_q_legendre_successor_threshold_divides_power_start * S ((S (0)) * bpvi_v_legendre_successor_threshold_divides_power) + (1))) /\ ((((exists bpvi_h_legendre_successor_threshold_divides_power_terminal. bpvi_h_legendre_successor_threshold_divides_power_terminal + S (bpvi_result_legendre_successor_threshold_divides) = S ((S (S i)) * bpvi_v_legendre_successor_threshold_divides_power)) /\ exists bpvi_q_legendre_successor_threshold_divides_power_terminal. bpvi_u_legendre_successor_threshold_divides_power = bpvi_q_legendre_successor_threshold_divides_power_terminal * S ((S (S i)) * bpvi_v_legendre_successor_threshold_divides_power) + (bpvi_result_legendre_successor_threshold_divides))) /\ forall bpvi_j_legendre_successor_threshold_divides_power. (exists bpvi_product_gap_legendre_successor_threshold_divides_power. bpvi_product_gap_legendre_successor_threshold_divides_power + S bpvi_j_legendre_successor_threshold_divides_power = S i) -> exists bpvi_factor_legendre_successor_threshold_divides_power bpvi_partial_legendre_successor_threshold_divides_power bpvi_successor_legendre_successor_threshold_divides_power. ((((exists bpvi_h_legendre_successor_threshold_divides_power_factor. bpvi_h_legendre_successor_threshold_divides_power_factor + S (bpvi_factor_legendre_successor_threshold_divides_power) = S ((S (bpvi_j_legendre_successor_threshold_divides_power)) * bpvi_c_legendre_successor_threshold_divides_power)) /\ exists bpvi_q_legendre_successor_threshold_divides_power_factor. bpvi_b_legendre_successor_threshold_divides_power = bpvi_q_legendre_successor_threshold_divides_power_factor * S ((S (bpvi_j_legendre_successor_threshold_divides_power)) * bpvi_c_legendre_successor_threshold_divides_power) + (bpvi_factor_legendre_successor_threshold_divides_power))) /\ ((((exists bpvi_h_legendre_successor_threshold_divides_power_partial. bpvi_h_legendre_successor_threshold_divides_power_partial + S (bpvi_partial_legendre_successor_threshold_divides_power) = S ((S (bpvi_j_legendre_successor_threshold_divides_power)) * bpvi_v_legendre_successor_threshold_divides_power)) /\ exists bpvi_q_legendre_successor_threshold_divides_power_partial. bpvi_u_legendre_successor_threshold_divides_power = bpvi_q_legendre_successor_threshold_divides_power_partial * S ((S (bpvi_j_legendre_successor_threshold_divides_power)) * bpvi_v_legendre_successor_threshold_divides_power) + (bpvi_partial_legendre_successor_threshold_divides_power))) /\ ((((exists bpvi_h_legendre_successor_threshold_divides_power_successor. bpvi_h_legendre_successor_threshold_divides_power_successor + S (bpvi_successor_legendre_successor_threshold_divides_power) = S ((S (S bpvi_j_legendre_successor_threshold_divides_power)) * bpvi_v_legendre_successor_threshold_divides_power)) /\ exists bpvi_q_legendre_successor_threshold_divides_power_successor. bpvi_u_legendre_successor_threshold_divides_power = bpvi_q_legendre_successor_threshold_divides_power_successor * S ((S (S bpvi_j_legendre_successor_threshold_divides_power)) * bpvi_v_legendre_successor_threshold_divides_power) + (bpvi_successor_legendre_successor_threshold_divides_power))) /\ bpvi_successor_legendre_successor_threshold_divides_power = bpvi_partial_legendre_successor_threshold_divides_power * bpvi_factor_legendre_successor_threshold_divides_power)))))))) /\ exists bpvi_divisor_factor_legendre_successor_threshold_divides. a = bpvi_result_legendre_successor_threshold_divides * bpvi_divisor_factor_legendre_successor_threshold_divides))) \/ (bit = 0 /\ ~(exists bpvi_result_legendre_successor_threshold_divides. ((exists bpvi_b_legendre_successor_threshold_divides_power bpvi_c_legendre_successor_threshold_divides_power. ((forall bpvi_i_legendre_successor_threshold_divides_power. (exists bpvi_repeat_gap_legendre_successor_threshold_divides_power. bpvi_repeat_gap_legendre_successor_threshold_divides_power + S bpvi_i_legendre_successor_threshold_divides_power = S i) -> (((exists bpvi_h_legendre_successor_threshold_divides_power_repeat. bpvi_h_legendre_successor_threshold_divides_power_repeat + S (p) = S ((S (bpvi_i_legendre_successor_threshold_divides_power)) * bpvi_c_legendre_successor_threshold_divides_power)) /\ exists bpvi_q_legendre_successor_threshold_divides_power_repeat. bpvi_b_legendre_successor_threshold_divides_power = bpvi_q_legendre_successor_threshold_divides_power_repeat * S ((S (bpvi_i_legendre_successor_threshold_divides_power)) * bpvi_c_legendre_successor_threshold_divides_power) + (p)))) /\ (exists bpvi_u_legendre_successor_threshold_divides_power bpvi_v_legendre_successor_threshold_divides_power. ((((exists bpvi_h_legendre_successor_threshold_divides_power_start. bpvi_h_legendre_successor_threshold_divides_power_start + S (1) = S ((S (0)) * bpvi_v_legendre_successor_threshold_divides_power)) /\ exists bpvi_q_legendre_successor_threshold_divides_power_start. bpvi_u_legendre_successor_threshold_divides_power = bpvi_q_legendre_successor_threshold_divides_power_start * S ((S (0)) * bpvi_v_legendre_successor_threshold_divides_power) + (1))) /\ ((((exists bpvi_h_legendre_successor_threshold_divides_power_terminal. bpvi_h_legendre_successor_threshold_divides_power_terminal + S (bpvi_result_legendre_successor_threshold_divides) = S ((S (S i)) * bpvi_v_legendre_successor_threshold_divides_power)) /\ exists bpvi_q_legendre_successor_threshold_divides_power_terminal. bpvi_u_legendre_successor_threshold_divides_power = bpvi_q_legendre_successor_threshold_divides_power_terminal * S ((S (S i)) * bpvi_v_legendre_successor_threshold_divides_power) + (bpvi_result_legendre_successor_threshold_divides))) /\ forall bpvi_j_legendre_successor_threshold_divides_power. (exists bpvi_product_gap_legendre_successor_threshold_divides_power. bpvi_product_gap_legendre_successor_threshold_divides_power + S bpvi_j_legendre_successor_threshold_divides_power = S i) -> exists bpvi_factor_legendre_successor_threshold_divides_power bpvi_partial_legendre_successor_threshold_divides_power bpvi_successor_legendre_successor_threshold_divides_power. ((((exists bpvi_h_legendre_successor_threshold_divides_power_factor. bpvi_h_legendre_successor_threshold_divides_power_factor + S (bpvi_factor_legendre_successor_threshold_divides_power) = S ((S (bpvi_j_legendre_successor_threshold_divides_power)) * bpvi_c_legendre_successor_threshold_divides_power)) /\ exists bpvi_q_legendre_successor_threshold_divides_power_factor. bpvi_b_legendre_successor_threshold_divides_power = bpvi_q_legendre_successor_threshold_divides_power_factor * S ((S (bpvi_j_legendre_successor_threshold_divides_power)) * bpvi_c_legendre_successor_threshold_divides_power) + (bpvi_factor_legendre_successor_threshold_divides_power))) /\ ((((exists bpvi_h_legendre_successor_threshold_divides_power_partial. bpvi_h_legendre_successor_threshold_divides_power_partial + S (bpvi_partial_legendre_successor_threshold_divides_power) = S ((S (bpvi_j_legendre_successor_threshold_divides_power)) * bpvi_v_legendre_successor_threshold_divides_power)) /\ exists bpvi_q_legendre_successor_threshold_divides_power_partial. bpvi_u_legendre_successor_threshold_divides_power = bpvi_q_legendre_successor_threshold_divides_power_partial * S ((S (bpvi_j_legendre_successor_threshold_divides_power)) * bpvi_v_legendre_successor_threshold_divides_power) + (bpvi_partial_legendre_successor_threshold_divides_power))) /\ ((((exists bpvi_h_legendre_successor_threshold_divides_power_successor. bpvi_h_legendre_successor_threshold_divides_power_successor + S (bpvi_successor_legendre_successor_threshold_divides_power) = S ((S (S bpvi_j_legendre_successor_threshold_divides_power)) * bpvi_v_legendre_successor_threshold_divides_power)) /\ exists bpvi_q_legendre_successor_threshold_divides_power_successor. bpvi_u_legendre_successor_threshold_divides_power = bpvi_q_legendre_successor_threshold_divides_power_successor * S ((S (S bpvi_j_legendre_successor_threshold_divides_power)) * bpvi_v_legendre_successor_threshold_divides_power) + (bpvi_successor_legendre_successor_threshold_divides_power))) /\ bpvi_successor_legendre_successor_threshold_divides_power = bpvi_partial_legendre_successor_threshold_divides_power * bpvi_factor_legendre_successor_threshold_divides_power)))))))) /\ exists bpvi_divisor_factor_legendre_successor_threshold_divides. a = bpvi_result_legendre_successor_threshold_divides * bpvi_divisor_factor_legendre_successor_threshold_divides))))