prime_legendre_sum_succ

forall p n f e g. ((~(p = 1) /\ forall frm_prime_left_blrr_recurrence_prime frm_prime_right_blrr_recurrence_prime. p = frm_prime_left_blrr_recurrence_prime * frm_prime_right_blrr_recurrence_prime -> frm_prime_left_blrr_recurrence_prime = 1 \/ frm_prime_right_blrr_recurrence_prime = 1)) -> ((((exists blsr_le_gap_blrr_recurrence_valuation_exponent_bound. blsr_le_gap_blrr_recurrence_valuation_exponent_bound + (f) = (S n)) /\ (exists bpvi_result_blrr_recurrence_valuation_selected. ((exists bpvi_b_blrr_recurrence_valuation_selected_power bpvi_c_blrr_recurrence_valuation_selected_power. ((forall bpvi_i_blrr_recurrence_valuation_selected_power. (exists bpvi_repeat_gap_blrr_recurrence_valuation_selected_power. bpvi_repeat_gap_blrr_recurrence_valuation_selected_power + S bpvi_i_blrr_recurrence_valuation_selected_power = f) -> (((exists bpvi_h_blrr_recurrence_valuation_selected_power_repeat. bpvi_h_blrr_recurrence_valuation_selected_power_repeat + S (p) = S ((S (bpvi_i_blrr_recurrence_valuation_selected_power)) * bpvi_c_blrr_recurrence_valuation_selected_power)) /\ exists bpvi_q_blrr_recurrence_valuation_selected_power_repeat. bpvi_b_blrr_recurrence_valuation_selected_power = bpvi_q_blrr_recurrence_valuation_selected_power_repeat * S ((S (bpvi_i_blrr_recurrence_valuation_selected_power)) * bpvi_c_blrr_recurrence_valuation_selected_power) + (p)))) /\ (exists bpvi_u_blrr_recurrence_valuation_selected_power bpvi_v_blrr_recurrence_valuation_selected_power. ((((exists bpvi_h_blrr_recurrence_valuation_selected_power_start. bpvi_h_blrr_recurrence_valuation_selected_power_start + S (1) = S ((S (0)) * bpvi_v_blrr_recurrence_valuation_selected_power)) /\ exists bpvi_q_blrr_recurrence_valuation_selected_power_start. bpvi_u_blrr_recurrence_valuation_selected_power = bpvi_q_blrr_recurrence_valuation_selected_power_start * S ((S (0)) * bpvi_v_blrr_recurrence_valuation_selected_power) + (1))) /\ ((((exists bpvi_h_blrr_recurrence_valuation_selected_power_terminal. bpvi_h_blrr_recurrence_valuation_selected_power_terminal + S (bpvi_result_blrr_recurrence_valuation_selected) = S ((S (f)) * bpvi_v_blrr_recurrence_valuation_selected_power)) /\ exists bpvi_q_blrr_recurrence_valuation_selected_power_terminal. bpvi_u_blrr_recurrence_valuation_selected_power = bpvi_q_blrr_recurrence_valuation_selected_power_terminal * S ((S (f)) * bpvi_v_blrr_recurrence_valuation_selected_power) + (bpvi_result_blrr_recurrence_valuation_selected))) /\ forall bpvi_j_blrr_recurrence_valuation_selected_power. (exists bpvi_product_gap_blrr_recurrence_valuation_selected_power. bpvi_product_gap_blrr_recurrence_valuation_selected_power + S bpvi_j_blrr_recurrence_valuation_selected_power = f) -> exists bpvi_factor_blrr_recurrence_valuation_selected_power bpvi_partial_blrr_recurrence_valuation_selected_power bpvi_successor_blrr_recurrence_valuation_selected_power. ((((exists bpvi_h_blrr_recurrence_valuation_selected_power_factor. bpvi_h_blrr_recurrence_valuation_selected_power_factor + S (bpvi_factor_blrr_recurrence_valuation_selected_power) = S ((S (bpvi_j_blrr_recurrence_valuation_selected_power)) * bpvi_c_blrr_recurrence_valuation_selected_power)) /\ exists bpvi_q_blrr_recurrence_valuation_selected_power_factor. bpvi_b_blrr_recurrence_valuation_selected_power = bpvi_q_blrr_recurrence_valuation_selected_power_factor * S ((S (bpvi_j_blrr_recurrence_valuation_selected_power)) * bpvi_c_blrr_recurrence_valuation_selected_power) + (bpvi_factor_blrr_recurrence_valuation_selected_power))) /\ ((((exists bpvi_h_blrr_recurrence_valuation_selected_power_partial. bpvi_h_blrr_recurrence_valuation_selected_power_partial + S (bpvi_partial_blrr_recurrence_valuation_selected_power) = S ((S (bpvi_j_blrr_recurrence_valuation_selected_power)) * bpvi_v_blrr_recurrence_valuation_selected_power)) /\ exists bpvi_q_blrr_recurrence_valuation_selected_power_partial. bpvi_u_blrr_recurrence_valuation_selected_power = bpvi_q_blrr_recurrence_valuation_selected_power_partial * S ((S (bpvi_j_blrr_recurrence_valuation_selected_power)) * bpvi_v_blrr_recurrence_valuation_selected_power) + (bpvi_partial_blrr_recurrence_valuation_selected_power))) /\ ((((exists bpvi_h_blrr_recurrence_valuation_selected_power_successor. bpvi_h_blrr_recurrence_valuation_selected_power_successor + S (bpvi_successor_blrr_recurrence_valuation_selected_power) = S ((S (S bpvi_j_blrr_recurrence_valuation_selected_power)) * bpvi_v_blrr_recurrence_valuation_selected_power)) /\ exists bpvi_q_blrr_recurrence_valuation_selected_power_successor. bpvi_u_blrr_recurrence_valuation_selected_power = bpvi_q_blrr_recurrence_valuation_selected_power_successor * S ((S (S bpvi_j_blrr_recurrence_valuation_selected_power)) * bpvi_v_blrr_recurrence_valuation_selected_power) + (bpvi_successor_blrr_recurrence_valuation_selected_power))) /\ bpvi_successor_blrr_recurrence_valuation_selected_power = bpvi_partial_blrr_recurrence_valuation_selected_power * bpvi_factor_blrr_recurrence_valuation_selected_power)))))))) /\ exists bpvi_divisor_factor_blrr_recurrence_valuation_selected. S n = bpvi_result_blrr_recurrence_valuation_selected * bpvi_divisor_factor_blrr_recurrence_valuation_selected))) /\ forall blsr_candidate_blrr_recurrence_valuation. (exists blsr_le_gap_blrr_recurrence_valuation_candidate_bound. blsr_le_gap_blrr_recurrence_valuation_candidate_bound + (blsr_candidate_blrr_recurrence_valuation) = (S n)) -> (exists bpvi_result_blrr_recurrence_valuation_candidate. ((exists bpvi_b_blrr_recurrence_valuation_candidate_power bpvi_c_blrr_recurrence_valuation_candidate_power. ((forall bpvi_i_blrr_recurrence_valuation_candidate_power. (exists bpvi_repeat_gap_blrr_recurrence_valuation_candidate_power. bpvi_repeat_gap_blrr_recurrence_valuation_candidate_power + S bpvi_i_blrr_recurrence_valuation_candidate_power = blsr_candidate_blrr_recurrence_valuation) -> (((exists bpvi_h_blrr_recurrence_valuation_candidate_power_repeat. bpvi_h_blrr_recurrence_valuation_candidate_power_repeat + S (p) = S ((S (bpvi_i_blrr_recurrence_valuation_candidate_power)) * bpvi_c_blrr_recurrence_valuation_candidate_power)) /\ exists bpvi_q_blrr_recurrence_valuation_candidate_power_repeat. bpvi_b_blrr_recurrence_valuation_candidate_power = bpvi_q_blrr_recurrence_valuation_candidate_power_repeat * S ((S (bpvi_i_blrr_recurrence_valuation_candidate_power)) * bpvi_c_blrr_recurrence_valuation_candidate_power) + (p)))) /\ (exists bpvi_u_blrr_recurrence_valuation_candidate_power bpvi_v_blrr_recurrence_valuation_candidate_power. ((((exists bpvi_h_blrr_recurrence_valuation_candidate_power_start. bpvi_h_blrr_recurrence_valuation_candidate_power_start + S (1) = S ((S (0)) * bpvi_v_blrr_recurrence_valuation_candidate_power)) /\ exists bpvi_q_blrr_recurrence_valuation_candidate_power_start. bpvi_u_blrr_recurrence_valuation_candidate_power = bpvi_q_blrr_recurrence_valuation_candidate_power_start * S ((S (0)) * bpvi_v_blrr_recurrence_valuation_candidate_power) + (1))) /\ ((((exists bpvi_h_blrr_recurrence_valuation_candidate_power_terminal. bpvi_h_blrr_recurrence_valuation_candidate_power_terminal + S (bpvi_result_blrr_recurrence_valuation_candidate) = S ((S (blsr_candidate_blrr_recurrence_valuation)) * bpvi_v_blrr_recurrence_valuation_candidate_power)) /\ exists bpvi_q_blrr_recurrence_valuation_candidate_power_terminal. bpvi_u_blrr_recurrence_valuation_candidate_power = bpvi_q_blrr_recurrence_valuation_candidate_power_terminal * S ((S (blsr_candidate_blrr_recurrence_valuation)) * bpvi_v_blrr_recurrence_valuation_candidate_power) + (bpvi_result_blrr_recurrence_valuation_candidate))) /\ forall bpvi_j_blrr_recurrence_valuation_candidate_power. (exists bpvi_product_gap_blrr_recurrence_valuation_candidate_power. bpvi_product_gap_blrr_recurrence_valuation_candidate_power + S bpvi_j_blrr_recurrence_valuation_candidate_power = blsr_candidate_blrr_recurrence_valuation) -> exists bpvi_factor_blrr_recurrence_valuation_candidate_power bpvi_partial_blrr_recurrence_valuation_candidate_power bpvi_successor_blrr_recurrence_valuation_candidate_power. ((((exists bpvi_h_blrr_recurrence_valuation_candidate_power_factor. bpvi_h_blrr_recurrence_valuation_candidate_power_factor + S (bpvi_factor_blrr_recurrence_valuation_candidate_power) = S ((S (bpvi_j_blrr_recurrence_valuation_candidate_power)) * bpvi_c_blrr_recurrence_valuation_candidate_power)) /\ exists bpvi_q_blrr_recurrence_valuation_candidate_power_factor. bpvi_b_blrr_recurrence_valuation_candidate_power = bpvi_q_blrr_recurrence_valuation_candidate_power_factor * S ((S (bpvi_j_blrr_recurrence_valuation_candidate_power)) * bpvi_c_blrr_recurrence_valuation_candidate_power) + (bpvi_factor_blrr_recurrence_valuation_candidate_power))) /\ ((((exists bpvi_h_blrr_recurrence_valuation_candidate_power_partial. bpvi_h_blrr_recurrence_valuation_candidate_power_partial + S (bpvi_partial_blrr_recurrence_valuation_candidate_power) = S ((S (bpvi_j_blrr_recurrence_valuation_candidate_power)) * bpvi_v_blrr_recurrence_valuation_candidate_power)) /\ exists bpvi_q_blrr_recurrence_valuation_candidate_power_partial. bpvi_u_blrr_recurrence_valuation_candidate_power = bpvi_q_blrr_recurrence_valuation_candidate_power_partial * S ((S (bpvi_j_blrr_recurrence_valuation_candidate_power)) * bpvi_v_blrr_recurrence_valuation_candidate_power) + (bpvi_partial_blrr_recurrence_valuation_candidate_power))) /\ ((((exists bpvi_h_blrr_recurrence_valuation_candidate_power_successor. bpvi_h_blrr_recurrence_valuation_candidate_power_successor + S (bpvi_successor_blrr_recurrence_valuation_candidate_power) = S ((S (S bpvi_j_blrr_recurrence_valuation_candidate_power)) * bpvi_v_blrr_recurrence_valuation_candidate_power)) /\ exists bpvi_q_blrr_recurrence_valuation_candidate_power_successor. bpvi_u_blrr_recurrence_valuation_candidate_power = bpvi_q_blrr_recurrence_valuation_candidate_power_successor * S ((S (S bpvi_j_blrr_recurrence_valuation_candidate_power)) * bpvi_v_blrr_recurrence_valuation_candidate_power) + (bpvi_successor_blrr_recurrence_valuation_candidate_power))) /\ bpvi_successor_blrr_recurrence_valuation_candidate_power = bpvi_partial_blrr_recurrence_valuation_candidate_power * bpvi_factor_blrr_recurrence_valuation_candidate_power)))))))) /\ exists bpvi_divisor_factor_blrr_recurrence_valuation_candidate. S n = bpvi_result_blrr_recurrence_valuation_candidate * bpvi_divisor_factor_blrr_recurrence_valuation_candidate)) -> (exists blsr_le_gap_blrr_recurrence_valuation_maximal. blsr_le_gap_blrr_recurrence_valuation_maximal + (blsr_candidate_blrr_recurrence_valuation) = (f)))) -> (exists bls_code_blrr_recurrence_old bls_scale_blrr_recurrence_old. ((forall bls_index_blrr_recurrence_old_prefix. (exists bls_gap_blrr_recurrence_old_prefix_bound. bls_gap_blrr_recurrence_old_prefix_bound + S (bls_index_blrr_recurrence_old_prefix) = (n)) -> exists bls_power_blrr_recurrence_old_prefix bls_quotient_blrr_recurrence_old_prefix bls_remainder_blrr_recurrence_old_prefix. ((exists bpvi_b_bls_blrr_recurrence_old_prefix_power bpvi_c_bls_blrr_recurrence_old_prefix_power. ((forall bpvi_i_bls_blrr_recurrence_old_prefix_power. (exists bpvi_repeat_gap_bls_blrr_recurrence_old_prefix_power. bpvi_repeat_gap_bls_blrr_recurrence_old_prefix_power + S bpvi_i_bls_blrr_recurrence_old_prefix_power = S bls_index_blrr_recurrence_old_prefix) -> (((exists bpvi_h_bls_blrr_recurrence_old_prefix_power_repeat. bpvi_h_bls_blrr_recurrence_old_prefix_power_repeat + S (p) = S ((S (bpvi_i_bls_blrr_recurrence_old_prefix_power)) * bpvi_c_bls_blrr_recurrence_old_prefix_power)) /\ exists bpvi_q_bls_blrr_recurrence_old_prefix_power_repeat. bpvi_b_bls_blrr_recurrence_old_prefix_power = bpvi_q_bls_blrr_recurrence_old_prefix_power_repeat * S ((S (bpvi_i_bls_blrr_recurrence_old_prefix_power)) * bpvi_c_bls_blrr_recurrence_old_prefix_power) + (p)))) /\ (exists bpvi_u_bls_blrr_recurrence_old_prefix_power bpvi_v_bls_blrr_recurrence_old_prefix_power. ((((exists bpvi_h_bls_blrr_recurrence_old_prefix_power_start. bpvi_h_bls_blrr_recurrence_old_prefix_power_start + S (1) = S ((S (0)) * bpvi_v_bls_blrr_recurrence_old_prefix_power)) /\ exists bpvi_q_bls_blrr_recurrence_old_prefix_power_start. bpvi_u_bls_blrr_recurrence_old_prefix_power = bpvi_q_bls_blrr_recurrence_old_prefix_power_start * S ((S (0)) * bpvi_v_bls_blrr_recurrence_old_prefix_power) + (1))) /\ ((((exists bpvi_h_bls_blrr_recurrence_old_prefix_power_terminal. bpvi_h_bls_blrr_recurrence_old_prefix_power_terminal + S (bls_power_blrr_recurrence_old_prefix) = S ((S (S bls_index_blrr_recurrence_old_prefix)) * bpvi_v_bls_blrr_recurrence_old_prefix_power)) /\ exists bpvi_q_bls_blrr_recurrence_old_prefix_power_terminal. bpvi_u_bls_blrr_recurrence_old_prefix_power = bpvi_q_bls_blrr_recurrence_old_prefix_power_terminal * S ((S (S bls_index_blrr_recurrence_old_prefix)) * bpvi_v_bls_blrr_recurrence_old_prefix_power) + (bls_power_blrr_recurrence_old_prefix))) /\ forall bpvi_j_bls_blrr_recurrence_old_prefix_power. (exists bpvi_product_gap_bls_blrr_recurrence_old_prefix_power. bpvi_product_gap_bls_blrr_recurrence_old_prefix_power + S bpvi_j_bls_blrr_recurrence_old_prefix_power = S bls_index_blrr_recurrence_old_prefix) -> exists bpvi_factor_bls_blrr_recurrence_old_prefix_power bpvi_partial_bls_blrr_recurrence_old_prefix_power bpvi_successor_bls_blrr_recurrence_old_prefix_power. ((((exists bpvi_h_bls_blrr_recurrence_old_prefix_power_factor. bpvi_h_bls_blrr_recurrence_old_prefix_power_factor + S (bpvi_factor_bls_blrr_recurrence_old_prefix_power) = S ((S (bpvi_j_bls_blrr_recurrence_old_prefix_power)) * bpvi_c_bls_blrr_recurrence_old_prefix_power)) /\ exists bpvi_q_bls_blrr_recurrence_old_prefix_power_factor. bpvi_b_bls_blrr_recurrence_old_prefix_power = bpvi_q_bls_blrr_recurrence_old_prefix_power_factor * S ((S (bpvi_j_bls_blrr_recurrence_old_prefix_power)) * bpvi_c_bls_blrr_recurrence_old_prefix_power) + (bpvi_factor_bls_blrr_recurrence_old_prefix_power))) /\ ((((exists bpvi_h_bls_blrr_recurrence_old_prefix_power_partial. bpvi_h_bls_blrr_recurrence_old_prefix_power_partial + S (bpvi_partial_bls_blrr_recurrence_old_prefix_power) = S ((S (bpvi_j_bls_blrr_recurrence_old_prefix_power)) * bpvi_v_bls_blrr_recurrence_old_prefix_power)) /\ exists bpvi_q_bls_blrr_recurrence_old_prefix_power_partial. bpvi_u_bls_blrr_recurrence_old_prefix_power = bpvi_q_bls_blrr_recurrence_old_prefix_power_partial * S ((S (bpvi_j_bls_blrr_recurrence_old_prefix_power)) * bpvi_v_bls_blrr_recurrence_old_prefix_power) + (bpvi_partial_bls_blrr_recurrence_old_prefix_power))) /\ ((((exists bpvi_h_bls_blrr_recurrence_old_prefix_power_successor. bpvi_h_bls_blrr_recurrence_old_prefix_power_successor + S (bpvi_successor_bls_blrr_recurrence_old_prefix_power) = S ((S (S bpvi_j_bls_blrr_recurrence_old_prefix_power)) * bpvi_v_bls_blrr_recurrence_old_prefix_power)) /\ exists bpvi_q_bls_blrr_recurrence_old_prefix_power_successor. bpvi_u_bls_blrr_recurrence_old_prefix_power = bpvi_q_bls_blrr_recurrence_old_prefix_power_successor * S ((S (S bpvi_j_bls_blrr_recurrence_old_prefix_power)) * bpvi_v_bls_blrr_recurrence_old_prefix_power) + (bpvi_successor_bls_blrr_recurrence_old_prefix_power))) /\ bpvi_successor_bls_blrr_recurrence_old_prefix_power = bpvi_partial_bls_blrr_recurrence_old_prefix_power * bpvi_factor_bls_blrr_recurrence_old_prefix_power)))))))) /\ ((((exists ff_h_bls_blrr_recurrence_old_prefix_quotient_entry. ff_h_bls_blrr_recurrence_old_prefix_quotient_entry + S (bls_quotient_blrr_recurrence_old_prefix) = S ((S (bls_index_blrr_recurrence_old_prefix)) * bls_scale_blrr_recurrence_old)) /\ exists ff_q_bls_blrr_recurrence_old_prefix_quotient_entry. bls_code_blrr_recurrence_old = ff_q_bls_blrr_recurrence_old_prefix_quotient_entry * S ((S (bls_index_blrr_recurrence_old_prefix)) * bls_scale_blrr_recurrence_old) + (bls_quotient_blrr_recurrence_old_prefix))) /\ ((n = bls_power_blrr_recurrence_old_prefix * bls_quotient_blrr_recurrence_old_prefix + bls_remainder_blrr_recurrence_old_prefix /\ exists bls_remainder_gap_blrr_recurrence_old_prefix_division. bls_remainder_gap_blrr_recurrence_old_prefix_division + S (bls_remainder_blrr_recurrence_old_prefix) = bls_power_blrr_recurrence_old_prefix))))) /\ (exists ff_u_bls_blrr_recurrence_old_sum ff_v_bls_blrr_recurrence_old_sum. ((((exists ff_h_bls_blrr_recurrence_old_sum_start. ff_h_bls_blrr_recurrence_old_sum_start + S (0) = S ((S (0)) * ff_v_bls_blrr_recurrence_old_sum)) /\ exists ff_q_bls_blrr_recurrence_old_sum_start. ff_u_bls_blrr_recurrence_old_sum = ff_q_bls_blrr_recurrence_old_sum_start * S ((S (0)) * ff_v_bls_blrr_recurrence_old_sum) + (0))) /\ ((((exists ff_h_bls_blrr_recurrence_old_sum_terminal. ff_h_bls_blrr_recurrence_old_sum_terminal + S (e) = S ((S (n)) * ff_v_bls_blrr_recurrence_old_sum)) /\ exists ff_q_bls_blrr_recurrence_old_sum_terminal. ff_u_bls_blrr_recurrence_old_sum = ff_q_bls_blrr_recurrence_old_sum_terminal * S ((S (n)) * ff_v_bls_blrr_recurrence_old_sum) + (e))) /\ forall ff_i_bls_blrr_recurrence_old_sum. (exists ff_lt_bls_blrr_recurrence_old_sum_bound. ff_lt_bls_blrr_recurrence_old_sum_bound + S ff_i_bls_blrr_recurrence_old_sum = n) -> exists ff_a_bls_blrr_recurrence_old_sum ff_r_bls_blrr_recurrence_old_sum ff_s_bls_blrr_recurrence_old_sum. ((((exists ff_h_bls_blrr_recurrence_old_sum_summand. ff_h_bls_blrr_recurrence_old_sum_summand + S (ff_a_bls_blrr_recurrence_old_sum) = S ((S (ff_i_bls_blrr_recurrence_old_sum)) * bls_scale_blrr_recurrence_old)) /\ exists ff_q_bls_blrr_recurrence_old_sum_summand. bls_code_blrr_recurrence_old = ff_q_bls_blrr_recurrence_old_sum_summand * S ((S (ff_i_bls_blrr_recurrence_old_sum)) * bls_scale_blrr_recurrence_old) + (ff_a_bls_blrr_recurrence_old_sum))) /\ ((((exists ff_h_bls_blrr_recurrence_old_sum_partial. ff_h_bls_blrr_recurrence_old_sum_partial + S (ff_r_bls_blrr_recurrence_old_sum) = S ((S (ff_i_bls_blrr_recurrence_old_sum)) * ff_v_bls_blrr_recurrence_old_sum)) /\ exists ff_q_bls_blrr_recurrence_old_sum_partial. ff_u_bls_blrr_recurrence_old_sum = ff_q_bls_blrr_recurrence_old_sum_partial * S ((S (ff_i_bls_blrr_recurrence_old_sum)) * ff_v_bls_blrr_recurrence_old_sum) + (ff_r_bls_blrr_recurrence_old_sum))) /\ ((((exists ff_h_bls_blrr_recurrence_old_sum_successor. ff_h_bls_blrr_recurrence_old_sum_successor + S (ff_s_bls_blrr_recurrence_old_sum) = S ((S (S ff_i_bls_blrr_recurrence_old_sum)) * ff_v_bls_blrr_recurrence_old_sum)) /\ exists ff_q_bls_blrr_recurrence_old_sum_successor. ff_u_bls_blrr_recurrence_old_sum = ff_q_bls_blrr_recurrence_old_sum_successor * S ((S (S ff_i_bls_blrr_recurrence_old_sum)) * ff_v_bls_blrr_recurrence_old_sum) + (ff_s_bls_blrr_recurrence_old_sum))) /\ ff_s_bls_blrr_recurrence_old_sum = ff_r_bls_blrr_recurrence_old_sum + ff_a_bls_blrr_recurrence_old_sum)))))))) -> (exists blrr_code_recurrence_new blrr_scale_recurrence_new. ((forall bls_index_recurrence_new_prefix. (exists bls_gap_recurrence_new_prefix_bound. bls_gap_recurrence_new_prefix_bound + S (bls_index_recurrence_new_prefix) = (S n)) -> exists bls_power_recurrence_new_prefix bls_quotient_recurrence_new_prefix bls_remainder_recurrence_new_prefix. ((exists bpvi_b_bls_recurrence_new_prefix_power bpvi_c_bls_recurrence_new_prefix_power. ((forall bpvi_i_bls_recurrence_new_prefix_power. (exists bpvi_repeat_gap_bls_recurrence_new_prefix_power. bpvi_repeat_gap_bls_recurrence_new_prefix_power + S bpvi_i_bls_recurrence_new_prefix_power = S bls_index_recurrence_new_prefix) -> (((exists bpvi_h_bls_recurrence_new_prefix_power_repeat. bpvi_h_bls_recurrence_new_prefix_power_repeat + S (p) = S ((S (bpvi_i_bls_recurrence_new_prefix_power)) * bpvi_c_bls_recurrence_new_prefix_power)) /\ exists bpvi_q_bls_recurrence_new_prefix_power_repeat. bpvi_b_bls_recurrence_new_prefix_power = bpvi_q_bls_recurrence_new_prefix_power_repeat * S ((S (bpvi_i_bls_recurrence_new_prefix_power)) * bpvi_c_bls_recurrence_new_prefix_power) + (p)))) /\ (exists bpvi_u_bls_recurrence_new_prefix_power bpvi_v_bls_recurrence_new_prefix_power. ((((exists bpvi_h_bls_recurrence_new_prefix_power_start. bpvi_h_bls_recurrence_new_prefix_power_start + S (1) = S ((S (0)) * bpvi_v_bls_recurrence_new_prefix_power)) /\ exists bpvi_q_bls_recurrence_new_prefix_power_start. bpvi_u_bls_recurrence_new_prefix_power = bpvi_q_bls_recurrence_new_prefix_power_start * S ((S (0)) * bpvi_v_bls_recurrence_new_prefix_power) + (1))) /\ ((((exists bpvi_h_bls_recurrence_new_prefix_power_terminal. bpvi_h_bls_recurrence_new_prefix_power_terminal + S (bls_power_recurrence_new_prefix) = S ((S (S bls_index_recurrence_new_prefix)) * bpvi_v_bls_recurrence_new_prefix_power)) /\ exists bpvi_q_bls_recurrence_new_prefix_power_terminal. bpvi_u_bls_recurrence_new_prefix_power = bpvi_q_bls_recurrence_new_prefix_power_terminal * S ((S (S bls_index_recurrence_new_prefix)) * bpvi_v_bls_recurrence_new_prefix_power) + (bls_power_recurrence_new_prefix))) /\ forall bpvi_j_bls_recurrence_new_prefix_power. (exists bpvi_product_gap_bls_recurrence_new_prefix_power. bpvi_product_gap_bls_recurrence_new_prefix_power + S bpvi_j_bls_recurrence_new_prefix_power = S bls_index_recurrence_new_prefix) -> exists bpvi_factor_bls_recurrence_new_prefix_power bpvi_partial_bls_recurrence_new_prefix_power bpvi_successor_bls_recurrence_new_prefix_power. ((((exists bpvi_h_bls_recurrence_new_prefix_power_factor. bpvi_h_bls_recurrence_new_prefix_power_factor + S (bpvi_factor_bls_recurrence_new_prefix_power) = S ((S (bpvi_j_bls_recurrence_new_prefix_power)) * bpvi_c_bls_recurrence_new_prefix_power)) /\ exists bpvi_q_bls_recurrence_new_prefix_power_factor. bpvi_b_bls_recurrence_new_prefix_power = bpvi_q_bls_recurrence_new_prefix_power_factor * S ((S (bpvi_j_bls_recurrence_new_prefix_power)) * bpvi_c_bls_recurrence_new_prefix_power) + (bpvi_factor_bls_recurrence_new_prefix_power))) /\ ((((exists bpvi_h_bls_recurrence_new_prefix_power_partial. bpvi_h_bls_recurrence_new_prefix_power_partial + S (bpvi_partial_bls_recurrence_new_prefix_power) = S ((S (bpvi_j_bls_recurrence_new_prefix_power)) * bpvi_v_bls_recurrence_new_prefix_power)) /\ exists bpvi_q_bls_recurrence_new_prefix_power_partial. bpvi_u_bls_recurrence_new_prefix_power = bpvi_q_bls_recurrence_new_prefix_power_partial * S ((S (bpvi_j_bls_recurrence_new_prefix_power)) * bpvi_v_bls_recurrence_new_prefix_power) + (bpvi_partial_bls_recurrence_new_prefix_power))) /\ ((((exists bpvi_h_bls_recurrence_new_prefix_power_successor. bpvi_h_bls_recurrence_new_prefix_power_successor + S (bpvi_successor_bls_recurrence_new_prefix_power) = S ((S (S bpvi_j_bls_recurrence_new_prefix_power)) * bpvi_v_bls_recurrence_new_prefix_power)) /\ exists bpvi_q_bls_recurrence_new_prefix_power_successor. bpvi_u_bls_recurrence_new_prefix_power = bpvi_q_bls_recurrence_new_prefix_power_successor * S ((S (S bpvi_j_bls_recurrence_new_prefix_power)) * bpvi_v_bls_recurrence_new_prefix_power) + (bpvi_successor_bls_recurrence_new_prefix_power))) /\ bpvi_successor_bls_recurrence_new_prefix_power = bpvi_partial_bls_recurrence_new_prefix_power * bpvi_factor_bls_recurrence_new_prefix_power)))))))) /\ ((((exists ff_h_bls_recurrence_new_prefix_quotient_entry. ff_h_bls_recurrence_new_prefix_quotient_entry + S (bls_quotient_recurrence_new_prefix) = S ((S (bls_index_recurrence_new_prefix)) * blrr_scale_recurrence_new)) /\ exists ff_q_bls_recurrence_new_prefix_quotient_entry. blrr_code_recurrence_new = ff_q_bls_recurrence_new_prefix_quotient_entry * S ((S (bls_index_recurrence_new_prefix)) * blrr_scale_recurrence_new) + (bls_quotient_recurrence_new_prefix))) /\ ((S n = bls_power_recurrence_new_prefix * bls_quotient_recurrence_new_prefix + bls_remainder_recurrence_new_prefix /\ exists bls_remainder_gap_recurrence_new_prefix_division. bls_remainder_gap_recurrence_new_prefix_division + S (bls_remainder_recurrence_new_prefix) = bls_power_recurrence_new_prefix))))) /\ (exists fs_u_blrr_recurrence_new_sum fs_v_blrr_recurrence_new_sum. ((((exists fs_h_blrr_recurrence_new_sum_body_start. fs_h_blrr_recurrence_new_sum_body_start + S (0) = S ((S (0)) * fs_v_blrr_recurrence_new_sum)) /\ exists fs_q_blrr_recurrence_new_sum_body_start. fs_u_blrr_recurrence_new_sum = fs_q_blrr_recurrence_new_sum_body_start * S ((S (0)) * fs_v_blrr_recurrence_new_sum) + (0))) /\ ((((exists fs_h_blrr_recurrence_new_sum_body_terminal. fs_h_blrr_recurrence_new_sum_body_terminal + S (g) = S ((S (S n)) * fs_v_blrr_recurrence_new_sum)) /\ exists fs_q_blrr_recurrence_new_sum_body_terminal. fs_u_blrr_recurrence_new_sum = fs_q_blrr_recurrence_new_sum_body_terminal * S ((S (S n)) * fs_v_blrr_recurrence_new_sum) + (g))) /\ forall fs_i_blrr_recurrence_new_sum_body_steps. (exists fs_lt_blrr_recurrence_new_sum_body_steps_bound. fs_lt_blrr_recurrence_new_sum_body_steps_bound + S fs_i_blrr_recurrence_new_sum_body_steps = S n) -> exists fs_a_blrr_recurrence_new_sum_body_steps fs_r_blrr_recurrence_new_sum_body_steps fs_s_blrr_recurrence_new_sum_body_steps. ((((exists fs_h_blrr_recurrence_new_sum_body_steps_summand. fs_h_blrr_recurrence_new_sum_body_steps_summand + S (fs_a_blrr_recurrence_new_sum_body_steps) = S ((S (fs_i_blrr_recurrence_new_sum_body_steps)) * blrr_scale_recurrence_new)) /\ exists fs_q_blrr_recurrence_new_sum_body_steps_summand. blrr_code_recurrence_new = fs_q_blrr_recurrence_new_sum_body_steps_summand * S ((S (fs_i_blrr_recurrence_new_sum_body_steps)) * blrr_scale_recurrence_new) + (fs_a_blrr_recurrence_new_sum_body_steps))) /\ ((((exists fs_h_blrr_recurrence_new_sum_body_steps_partial. fs_h_blrr_recurrence_new_sum_body_steps_partial + S (fs_r_blrr_recurrence_new_sum_body_steps) = S ((S (fs_i_blrr_recurrence_new_sum_body_steps)) * fs_v_blrr_recurrence_new_sum)) /\ exists fs_q_blrr_recurrence_new_sum_body_steps_partial. fs_u_blrr_recurrence_new_sum = fs_q_blrr_recurrence_new_sum_body_steps_partial * S ((S (fs_i_blrr_recurrence_new_sum_body_steps)) * fs_v_blrr_recurrence_new_sum) + (fs_r_blrr_recurrence_new_sum_body_steps))) /\ ((((exists fs_h_blrr_recurrence_new_sum_body_steps_successor. fs_h_blrr_recurrence_new_sum_body_steps_successor + S (fs_s_blrr_recurrence_new_sum_body_steps) = S ((S (S fs_i_blrr_recurrence_new_sum_body_steps)) * fs_v_blrr_recurrence_new_sum)) /\ exists fs_q_blrr_recurrence_new_sum_body_steps_successor. fs_u_blrr_recurrence_new_sum = fs_q_blrr_recurrence_new_sum_body_steps_successor * S ((S (S fs_i_blrr_recurrence_new_sum_body_steps)) * fs_v_blrr_recurrence_new_sum) + (fs_s_blrr_recurrence_new_sum_body_steps))) /\ fs_s_blrr_recurrence_new_sum_body_steps = fs_r_blrr_recurrence_new_sum_body_steps + fs_a_blrr_recurrence_new_sum_body_steps)))))))) -> g = e + f