legendre_sum_extended_prefix_exists

forall p n e g. ((~(p = 1) /\ forall frm_prime_left_b5cvlsepe_prime frm_prime_right_b5cvlsepe_prime. p = frm_prime_left_b5cvlsepe_prime * frm_prime_right_b5cvlsepe_prime -> frm_prime_left_b5cvlsepe_prime = 1 \/ frm_prime_right_b5cvlsepe_prime = 1)) -> (exists bls_code_b5cvlsepe_legendre bls_scale_b5cvlsepe_legendre. ((forall bls_index_b5cvlsepe_legendre_prefix. (exists bls_gap_b5cvlsepe_legendre_prefix_bound. bls_gap_b5cvlsepe_legendre_prefix_bound + S (bls_index_b5cvlsepe_legendre_prefix) = (n)) -> exists bls_power_b5cvlsepe_legendre_prefix bls_quotient_b5cvlsepe_legendre_prefix bls_remainder_b5cvlsepe_legendre_prefix. ((exists bpvi_b_bls_b5cvlsepe_legendre_prefix_power bpvi_c_bls_b5cvlsepe_legendre_prefix_power. ((forall bpvi_i_bls_b5cvlsepe_legendre_prefix_power. (exists bpvi_repeat_gap_bls_b5cvlsepe_legendre_prefix_power. bpvi_repeat_gap_bls_b5cvlsepe_legendre_prefix_power + S bpvi_i_bls_b5cvlsepe_legendre_prefix_power = S bls_index_b5cvlsepe_legendre_prefix) -> (((exists bpvi_h_bls_b5cvlsepe_legendre_prefix_power_repeat. bpvi_h_bls_b5cvlsepe_legendre_prefix_power_repeat + S (p) = S ((S (bpvi_i_bls_b5cvlsepe_legendre_prefix_power)) * bpvi_c_bls_b5cvlsepe_legendre_prefix_power)) /\ exists bpvi_q_bls_b5cvlsepe_legendre_prefix_power_repeat. bpvi_b_bls_b5cvlsepe_legendre_prefix_power = bpvi_q_bls_b5cvlsepe_legendre_prefix_power_repeat * S ((S (bpvi_i_bls_b5cvlsepe_legendre_prefix_power)) * bpvi_c_bls_b5cvlsepe_legendre_prefix_power) + (p)))) /\ (exists bpvi_u_bls_b5cvlsepe_legendre_prefix_power bpvi_v_bls_b5cvlsepe_legendre_prefix_power. ((((exists bpvi_h_bls_b5cvlsepe_legendre_prefix_power_start. bpvi_h_bls_b5cvlsepe_legendre_prefix_power_start + S (1) = S ((S (0)) * bpvi_v_bls_b5cvlsepe_legendre_prefix_power)) /\ exists bpvi_q_bls_b5cvlsepe_legendre_prefix_power_start. bpvi_u_bls_b5cvlsepe_legendre_prefix_power = bpvi_q_bls_b5cvlsepe_legendre_prefix_power_start * S ((S (0)) * bpvi_v_bls_b5cvlsepe_legendre_prefix_power) + (1))) /\ ((((exists bpvi_h_bls_b5cvlsepe_legendre_prefix_power_terminal. bpvi_h_bls_b5cvlsepe_legendre_prefix_power_terminal + S (bls_power_b5cvlsepe_legendre_prefix) = S ((S (S bls_index_b5cvlsepe_legendre_prefix)) * bpvi_v_bls_b5cvlsepe_legendre_prefix_power)) /\ exists bpvi_q_bls_b5cvlsepe_legendre_prefix_power_terminal. bpvi_u_bls_b5cvlsepe_legendre_prefix_power = bpvi_q_bls_b5cvlsepe_legendre_prefix_power_terminal * S ((S (S bls_index_b5cvlsepe_legendre_prefix)) * bpvi_v_bls_b5cvlsepe_legendre_prefix_power) + (bls_power_b5cvlsepe_legendre_prefix))) /\ forall bpvi_j_bls_b5cvlsepe_legendre_prefix_power. (exists bpvi_product_gap_bls_b5cvlsepe_legendre_prefix_power. bpvi_product_gap_bls_b5cvlsepe_legendre_prefix_power + S bpvi_j_bls_b5cvlsepe_legendre_prefix_power = S bls_index_b5cvlsepe_legendre_prefix) -> exists bpvi_factor_bls_b5cvlsepe_legendre_prefix_power bpvi_partial_bls_b5cvlsepe_legendre_prefix_power bpvi_successor_bls_b5cvlsepe_legendre_prefix_power. ((((exists bpvi_h_bls_b5cvlsepe_legendre_prefix_power_factor. bpvi_h_bls_b5cvlsepe_legendre_prefix_power_factor + S (bpvi_factor_bls_b5cvlsepe_legendre_prefix_power) = S ((S (bpvi_j_bls_b5cvlsepe_legendre_prefix_power)) * bpvi_c_bls_b5cvlsepe_legendre_prefix_power)) /\ exists bpvi_q_bls_b5cvlsepe_legendre_prefix_power_factor. bpvi_b_bls_b5cvlsepe_legendre_prefix_power = bpvi_q_bls_b5cvlsepe_legendre_prefix_power_factor * S ((S (bpvi_j_bls_b5cvlsepe_legendre_prefix_power)) * bpvi_c_bls_b5cvlsepe_legendre_prefix_power) + (bpvi_factor_bls_b5cvlsepe_legendre_prefix_power))) /\ ((((exists bpvi_h_bls_b5cvlsepe_legendre_prefix_power_partial. bpvi_h_bls_b5cvlsepe_legendre_prefix_power_partial + S (bpvi_partial_bls_b5cvlsepe_legendre_prefix_power) = S ((S (bpvi_j_bls_b5cvlsepe_legendre_prefix_power)) * bpvi_v_bls_b5cvlsepe_legendre_prefix_power)) /\ exists bpvi_q_bls_b5cvlsepe_legendre_prefix_power_partial. bpvi_u_bls_b5cvlsepe_legendre_prefix_power = bpvi_q_bls_b5cvlsepe_legendre_prefix_power_partial * S ((S (bpvi_j_bls_b5cvlsepe_legendre_prefix_power)) * bpvi_v_bls_b5cvlsepe_legendre_prefix_power) + (bpvi_partial_bls_b5cvlsepe_legendre_prefix_power))) /\ ((((exists bpvi_h_bls_b5cvlsepe_legendre_prefix_power_successor. bpvi_h_bls_b5cvlsepe_legendre_prefix_power_successor + S (bpvi_successor_bls_b5cvlsepe_legendre_prefix_power) = S ((S (S bpvi_j_bls_b5cvlsepe_legendre_prefix_power)) * bpvi_v_bls_b5cvlsepe_legendre_prefix_power)) /\ exists bpvi_q_bls_b5cvlsepe_legendre_prefix_power_successor. bpvi_u_bls_b5cvlsepe_legendre_prefix_power = bpvi_q_bls_b5cvlsepe_legendre_prefix_power_successor * S ((S (S bpvi_j_bls_b5cvlsepe_legendre_prefix_power)) * bpvi_v_bls_b5cvlsepe_legendre_prefix_power) + (bpvi_successor_bls_b5cvlsepe_legendre_prefix_power))) /\ bpvi_successor_bls_b5cvlsepe_legendre_prefix_power = bpvi_partial_bls_b5cvlsepe_legendre_prefix_power * bpvi_factor_bls_b5cvlsepe_legendre_prefix_power)))))))) /\ ((((exists ff_h_bls_b5cvlsepe_legendre_prefix_quotient_entry. ff_h_bls_b5cvlsepe_legendre_prefix_quotient_entry + S (bls_quotient_b5cvlsepe_legendre_prefix) = S ((S (bls_index_b5cvlsepe_legendre_prefix)) * bls_scale_b5cvlsepe_legendre)) /\ exists ff_q_bls_b5cvlsepe_legendre_prefix_quotient_entry. bls_code_b5cvlsepe_legendre = ff_q_bls_b5cvlsepe_legendre_prefix_quotient_entry * S ((S (bls_index_b5cvlsepe_legendre_prefix)) * bls_scale_b5cvlsepe_legendre) + (bls_quotient_b5cvlsepe_legendre_prefix))) /\ ((n = bls_power_b5cvlsepe_legendre_prefix * bls_quotient_b5cvlsepe_legendre_prefix + bls_remainder_b5cvlsepe_legendre_prefix /\ exists bls_remainder_gap_b5cvlsepe_legendre_prefix_division. bls_remainder_gap_b5cvlsepe_legendre_prefix_division + S (bls_remainder_b5cvlsepe_legendre_prefix) = bls_power_b5cvlsepe_legendre_prefix))))) /\ (exists ff_u_bls_b5cvlsepe_legendre_sum ff_v_bls_b5cvlsepe_legendre_sum. ((((exists ff_h_bls_b5cvlsepe_legendre_sum_start. ff_h_bls_b5cvlsepe_legendre_sum_start + S (0) = S ((S (0)) * ff_v_bls_b5cvlsepe_legendre_sum)) /\ exists ff_q_bls_b5cvlsepe_legendre_sum_start. ff_u_bls_b5cvlsepe_legendre_sum = ff_q_bls_b5cvlsepe_legendre_sum_start * S ((S (0)) * ff_v_bls_b5cvlsepe_legendre_sum) + (0))) /\ ((((exists ff_h_bls_b5cvlsepe_legendre_sum_terminal. ff_h_bls_b5cvlsepe_legendre_sum_terminal + S (e) = S ((S (n)) * ff_v_bls_b5cvlsepe_legendre_sum)) /\ exists ff_q_bls_b5cvlsepe_legendre_sum_terminal. ff_u_bls_b5cvlsepe_legendre_sum = ff_q_bls_b5cvlsepe_legendre_sum_terminal * S ((S (n)) * ff_v_bls_b5cvlsepe_legendre_sum) + (e))) /\ forall ff_i_bls_b5cvlsepe_legendre_sum. (exists ff_lt_bls_b5cvlsepe_legendre_sum_bound. ff_lt_bls_b5cvlsepe_legendre_sum_bound + S ff_i_bls_b5cvlsepe_legendre_sum = n) -> exists ff_a_bls_b5cvlsepe_legendre_sum ff_r_bls_b5cvlsepe_legendre_sum ff_s_bls_b5cvlsepe_legendre_sum. ((((exists ff_h_bls_b5cvlsepe_legendre_sum_summand. ff_h_bls_b5cvlsepe_legendre_sum_summand + S (ff_a_bls_b5cvlsepe_legendre_sum) = S ((S (ff_i_bls_b5cvlsepe_legendre_sum)) * bls_scale_b5cvlsepe_legendre)) /\ exists ff_q_bls_b5cvlsepe_legendre_sum_summand. bls_code_b5cvlsepe_legendre = ff_q_bls_b5cvlsepe_legendre_sum_summand * S ((S (ff_i_bls_b5cvlsepe_legendre_sum)) * bls_scale_b5cvlsepe_legendre) + (ff_a_bls_b5cvlsepe_legendre_sum))) /\ ((((exists ff_h_bls_b5cvlsepe_legendre_sum_partial. ff_h_bls_b5cvlsepe_legendre_sum_partial + S (ff_r_bls_b5cvlsepe_legendre_sum) = S ((S (ff_i_bls_b5cvlsepe_legendre_sum)) * ff_v_bls_b5cvlsepe_legendre_sum)) /\ exists ff_q_bls_b5cvlsepe_legendre_sum_partial. ff_u_bls_b5cvlsepe_legendre_sum = ff_q_bls_b5cvlsepe_legendre_sum_partial * S ((S (ff_i_bls_b5cvlsepe_legendre_sum)) * ff_v_bls_b5cvlsepe_legendre_sum) + (ff_r_bls_b5cvlsepe_legendre_sum))) /\ ((((exists ff_h_bls_b5cvlsepe_legendre_sum_successor. ff_h_bls_b5cvlsepe_legendre_sum_successor + S (ff_s_bls_b5cvlsepe_legendre_sum) = S ((S (S ff_i_bls_b5cvlsepe_legendre_sum)) * ff_v_bls_b5cvlsepe_legendre_sum)) /\ exists ff_q_bls_b5cvlsepe_legendre_sum_successor. ff_u_bls_b5cvlsepe_legendre_sum = ff_q_bls_b5cvlsepe_legendre_sum_successor * S ((S (S ff_i_bls_b5cvlsepe_legendre_sum)) * ff_v_bls_b5cvlsepe_legendre_sum) + (ff_s_bls_b5cvlsepe_legendre_sum))) /\ ff_s_bls_b5cvlsepe_legendre_sum = ff_r_bls_b5cvlsepe_legendre_sum + ff_a_bls_b5cvlsepe_legendre_sum)))))))) -> exists b c. ((forall bls_index_b5cvlsepe_prefix. (exists bls_gap_b5cvlsepe_prefix_bound. bls_gap_b5cvlsepe_prefix_bound + S (bls_index_b5cvlsepe_prefix) = (n + g)) -> exists bls_power_b5cvlsepe_prefix bls_quotient_b5cvlsepe_prefix bls_remainder_b5cvlsepe_prefix. ((exists bpvi_b_bls_b5cvlsepe_prefix_power bpvi_c_bls_b5cvlsepe_prefix_power. ((forall bpvi_i_bls_b5cvlsepe_prefix_power. (exists bpvi_repeat_gap_bls_b5cvlsepe_prefix_power. bpvi_repeat_gap_bls_b5cvlsepe_prefix_power + S bpvi_i_bls_b5cvlsepe_prefix_power = S bls_index_b5cvlsepe_prefix) -> (((exists bpvi_h_bls_b5cvlsepe_prefix_power_repeat. bpvi_h_bls_b5cvlsepe_prefix_power_repeat + S (p) = S ((S (bpvi_i_bls_b5cvlsepe_prefix_power)) * bpvi_c_bls_b5cvlsepe_prefix_power)) /\ exists bpvi_q_bls_b5cvlsepe_prefix_power_repeat. bpvi_b_bls_b5cvlsepe_prefix_power = bpvi_q_bls_b5cvlsepe_prefix_power_repeat * S ((S (bpvi_i_bls_b5cvlsepe_prefix_power)) * bpvi_c_bls_b5cvlsepe_prefix_power) + (p)))) /\ (exists bpvi_u_bls_b5cvlsepe_prefix_power bpvi_v_bls_b5cvlsepe_prefix_power. ((((exists bpvi_h_bls_b5cvlsepe_prefix_power_start. bpvi_h_bls_b5cvlsepe_prefix_power_start + S (1) = S ((S (0)) * bpvi_v_bls_b5cvlsepe_prefix_power)) /\ exists bpvi_q_bls_b5cvlsepe_prefix_power_start. bpvi_u_bls_b5cvlsepe_prefix_power = bpvi_q_bls_b5cvlsepe_prefix_power_start * S ((S (0)) * bpvi_v_bls_b5cvlsepe_prefix_power) + (1))) /\ ((((exists bpvi_h_bls_b5cvlsepe_prefix_power_terminal. bpvi_h_bls_b5cvlsepe_prefix_power_terminal + S (bls_power_b5cvlsepe_prefix) = S ((S (S bls_index_b5cvlsepe_prefix)) * bpvi_v_bls_b5cvlsepe_prefix_power)) /\ exists bpvi_q_bls_b5cvlsepe_prefix_power_terminal. bpvi_u_bls_b5cvlsepe_prefix_power = bpvi_q_bls_b5cvlsepe_prefix_power_terminal * S ((S (S bls_index_b5cvlsepe_prefix)) * bpvi_v_bls_b5cvlsepe_prefix_power) + (bls_power_b5cvlsepe_prefix))) /\ forall bpvi_j_bls_b5cvlsepe_prefix_power. (exists bpvi_product_gap_bls_b5cvlsepe_prefix_power. bpvi_product_gap_bls_b5cvlsepe_prefix_power + S bpvi_j_bls_b5cvlsepe_prefix_power = S bls_index_b5cvlsepe_prefix) -> exists bpvi_factor_bls_b5cvlsepe_prefix_power bpvi_partial_bls_b5cvlsepe_prefix_power bpvi_successor_bls_b5cvlsepe_prefix_power. ((((exists bpvi_h_bls_b5cvlsepe_prefix_power_factor. bpvi_h_bls_b5cvlsepe_prefix_power_factor + S (bpvi_factor_bls_b5cvlsepe_prefix_power) = S ((S (bpvi_j_bls_b5cvlsepe_prefix_power)) * bpvi_c_bls_b5cvlsepe_prefix_power)) /\ exists bpvi_q_bls_b5cvlsepe_prefix_power_factor. bpvi_b_bls_b5cvlsepe_prefix_power = bpvi_q_bls_b5cvlsepe_prefix_power_factor * S ((S (bpvi_j_bls_b5cvlsepe_prefix_power)) * bpvi_c_bls_b5cvlsepe_prefix_power) + (bpvi_factor_bls_b5cvlsepe_prefix_power))) /\ ((((exists bpvi_h_bls_b5cvlsepe_prefix_power_partial. bpvi_h_bls_b5cvlsepe_prefix_power_partial + S (bpvi_partial_bls_b5cvlsepe_prefix_power) = S ((S (bpvi_j_bls_b5cvlsepe_prefix_power)) * bpvi_v_bls_b5cvlsepe_prefix_power)) /\ exists bpvi_q_bls_b5cvlsepe_prefix_power_partial. bpvi_u_bls_b5cvlsepe_prefix_power = bpvi_q_bls_b5cvlsepe_prefix_power_partial * S ((S (bpvi_j_bls_b5cvlsepe_prefix_power)) * bpvi_v_bls_b5cvlsepe_prefix_power) + (bpvi_partial_bls_b5cvlsepe_prefix_power))) /\ ((((exists bpvi_h_bls_b5cvlsepe_prefix_power_successor. bpvi_h_bls_b5cvlsepe_prefix_power_successor + S (bpvi_successor_bls_b5cvlsepe_prefix_power) = S ((S (S bpvi_j_bls_b5cvlsepe_prefix_power)) * bpvi_v_bls_b5cvlsepe_prefix_power)) /\ exists bpvi_q_bls_b5cvlsepe_prefix_power_successor. bpvi_u_bls_b5cvlsepe_prefix_power = bpvi_q_bls_b5cvlsepe_prefix_power_successor * S ((S (S bpvi_j_bls_b5cvlsepe_prefix_power)) * bpvi_v_bls_b5cvlsepe_prefix_power) + (bpvi_successor_bls_b5cvlsepe_prefix_power))) /\ bpvi_successor_bls_b5cvlsepe_prefix_power = bpvi_partial_bls_b5cvlsepe_prefix_power * bpvi_factor_bls_b5cvlsepe_prefix_power)))))))) /\ ((((exists ff_h_bls_b5cvlsepe_prefix_quotient_entry. ff_h_bls_b5cvlsepe_prefix_quotient_entry + S (bls_quotient_b5cvlsepe_prefix) = S ((S (bls_index_b5cvlsepe_prefix)) * c)) /\ exists ff_q_bls_b5cvlsepe_prefix_quotient_entry. b = ff_q_bls_b5cvlsepe_prefix_quotient_entry * S ((S (bls_index_b5cvlsepe_prefix)) * c) + (bls_quotient_b5cvlsepe_prefix))) /\ ((n = bls_power_b5cvlsepe_prefix * bls_quotient_b5cvlsepe_prefix + bls_remainder_b5cvlsepe_prefix /\ exists bls_remainder_gap_b5cvlsepe_prefix_division. bls_remainder_gap_b5cvlsepe_prefix_division + S (bls_remainder_b5cvlsepe_prefix) = bls_power_b5cvlsepe_prefix))))) /\ (exists fs_u_b5cvlsepe_sum fs_v_b5cvlsepe_sum. ((((exists fs_h_b5cvlsepe_sum_body_start. fs_h_b5cvlsepe_sum_body_start + S (0) = S ((S (0)) * fs_v_b5cvlsepe_sum)) /\ exists fs_q_b5cvlsepe_sum_body_start. fs_u_b5cvlsepe_sum = fs_q_b5cvlsepe_sum_body_start * S ((S (0)) * fs_v_b5cvlsepe_sum) + (0))) /\ ((((exists fs_h_b5cvlsepe_sum_body_terminal. fs_h_b5cvlsepe_sum_body_terminal + S (e) = S ((S (n + g)) * fs_v_b5cvlsepe_sum)) /\ exists fs_q_b5cvlsepe_sum_body_terminal. fs_u_b5cvlsepe_sum = fs_q_b5cvlsepe_sum_body_terminal * S ((S (n + g)) * fs_v_b5cvlsepe_sum) + (e))) /\ forall fs_i_b5cvlsepe_sum_body_steps. (exists fs_lt_b5cvlsepe_sum_body_steps_bound. fs_lt_b5cvlsepe_sum_body_steps_bound + S fs_i_b5cvlsepe_sum_body_steps = n + g) -> exists fs_a_b5cvlsepe_sum_body_steps fs_r_b5cvlsepe_sum_body_steps fs_s_b5cvlsepe_sum_body_steps. ((((exists fs_h_b5cvlsepe_sum_body_steps_summand. fs_h_b5cvlsepe_sum_body_steps_summand + S (fs_a_b5cvlsepe_sum_body_steps) = S ((S (fs_i_b5cvlsepe_sum_body_steps)) * c)) /\ exists fs_q_b5cvlsepe_sum_body_steps_summand. b = fs_q_b5cvlsepe_sum_body_steps_summand * S ((S (fs_i_b5cvlsepe_sum_body_steps)) * c) + (fs_a_b5cvlsepe_sum_body_steps))) /\ ((((exists fs_h_b5cvlsepe_sum_body_steps_partial. fs_h_b5cvlsepe_sum_body_steps_partial + S (fs_r_b5cvlsepe_sum_body_steps) = S ((S (fs_i_b5cvlsepe_sum_body_steps)) * fs_v_b5cvlsepe_sum)) /\ exists fs_q_b5cvlsepe_sum_body_steps_partial. fs_u_b5cvlsepe_sum = fs_q_b5cvlsepe_sum_body_steps_partial * S ((S (fs_i_b5cvlsepe_sum_body_steps)) * fs_v_b5cvlsepe_sum) + (fs_r_b5cvlsepe_sum_body_steps))) /\ ((((exists fs_h_b5cvlsepe_sum_body_steps_successor. fs_h_b5cvlsepe_sum_body_steps_successor + S (fs_s_b5cvlsepe_sum_body_steps) = S ((S (S fs_i_b5cvlsepe_sum_body_steps)) * fs_v_b5cvlsepe_sum)) /\ exists fs_q_b5cvlsepe_sum_body_steps_successor. fs_u_b5cvlsepe_sum = fs_q_b5cvlsepe_sum_body_steps_successor * S ((S (S fs_i_b5cvlsepe_sum_body_steps)) * fs_v_b5cvlsepe_sum) + (fs_s_b5cvlsepe_sum_body_steps))) /\ fs_s_b5cvlsepe_sum_body_steps = fs_r_b5cvlsepe_sum_body_steps + fs_a_b5cvlsepe_sum_body_steps)))))))