legendre_sum_functional

Exact expanded PA statement

forall p n e f. (exists bls_code_bls_functional_left bls_scale_bls_functional_left. ((forall bls_index_bls_functional_left_prefix. (exists bls_gap_bls_functional_left_prefix_bound. bls_gap_bls_functional_left_prefix_bound + S (bls_index_bls_functional_left_prefix) = (n)) -> exists bls_power_bls_functional_left_prefix bls_quotient_bls_functional_left_prefix bls_remainder_bls_functional_left_prefix. ((exists bpvi_b_bls_bls_functional_left_prefix_power bpvi_c_bls_bls_functional_left_prefix_power. ((forall bpvi_i_bls_bls_functional_left_prefix_power. (exists bpvi_repeat_gap_bls_bls_functional_left_prefix_power. bpvi_repeat_gap_bls_bls_functional_left_prefix_power + S bpvi_i_bls_bls_functional_left_prefix_power = S bls_index_bls_functional_left_prefix) -> (((exists bpvi_h_bls_bls_functional_left_prefix_power_repeat. bpvi_h_bls_bls_functional_left_prefix_power_repeat + S (p) = S ((S (bpvi_i_bls_bls_functional_left_prefix_power)) * bpvi_c_bls_bls_functional_left_prefix_power)) /\ exists bpvi_q_bls_bls_functional_left_prefix_power_repeat. bpvi_b_bls_bls_functional_left_prefix_power = bpvi_q_bls_bls_functional_left_prefix_power_repeat * S ((S (bpvi_i_bls_bls_functional_left_prefix_power)) * bpvi_c_bls_bls_functional_left_prefix_power) + (p)))) /\ (exists bpvi_u_bls_bls_functional_left_prefix_power bpvi_v_bls_bls_functional_left_prefix_power. ((((exists bpvi_h_bls_bls_functional_left_prefix_power_start. bpvi_h_bls_bls_functional_left_prefix_power_start + S (1) = S ((S (0)) * bpvi_v_bls_bls_functional_left_prefix_power)) /\ exists bpvi_q_bls_bls_functional_left_prefix_power_start. bpvi_u_bls_bls_functional_left_prefix_power = bpvi_q_bls_bls_functional_left_prefix_power_start * S ((S (0)) * bpvi_v_bls_bls_functional_left_prefix_power) + (1))) /\ ((((exists bpvi_h_bls_bls_functional_left_prefix_power_terminal. bpvi_h_bls_bls_functional_left_prefix_power_terminal + S (bls_power_bls_functional_left_prefix) = S ((S (S bls_index_bls_functional_left_prefix)) * bpvi_v_bls_bls_functional_left_prefix_power)) /\ exists bpvi_q_bls_bls_functional_left_prefix_power_terminal. bpvi_u_bls_bls_functional_left_prefix_power = bpvi_q_bls_bls_functional_left_prefix_power_terminal * S ((S (S bls_index_bls_functional_left_prefix)) * bpvi_v_bls_bls_functional_left_prefix_power) + (bls_power_bls_functional_left_prefix))) /\ forall bpvi_j_bls_bls_functional_left_prefix_power. (exists bpvi_product_gap_bls_bls_functional_left_prefix_power. bpvi_product_gap_bls_bls_functional_left_prefix_power + S bpvi_j_bls_bls_functional_left_prefix_power = S bls_index_bls_functional_left_prefix) -> exists bpvi_factor_bls_bls_functional_left_prefix_power bpvi_partial_bls_bls_functional_left_prefix_power bpvi_successor_bls_bls_functional_left_prefix_power. ((((exists bpvi_h_bls_bls_functional_left_prefix_power_factor. bpvi_h_bls_bls_functional_left_prefix_power_factor + S (bpvi_factor_bls_bls_functional_left_prefix_power) = S ((S (bpvi_j_bls_bls_functional_left_prefix_power)) * bpvi_c_bls_bls_functional_left_prefix_power)) /\ exists bpvi_q_bls_bls_functional_left_prefix_power_factor. bpvi_b_bls_bls_functional_left_prefix_power = bpvi_q_bls_bls_functional_left_prefix_power_factor * S ((S (bpvi_j_bls_bls_functional_left_prefix_power)) * bpvi_c_bls_bls_functional_left_prefix_power) + (bpvi_factor_bls_bls_functional_left_prefix_power))) /\ ((((exists bpvi_h_bls_bls_functional_left_prefix_power_partial. bpvi_h_bls_bls_functional_left_prefix_power_partial + S (bpvi_partial_bls_bls_functional_left_prefix_power) = S ((S (bpvi_j_bls_bls_functional_left_prefix_power)) * bpvi_v_bls_bls_functional_left_prefix_power)) /\ exists bpvi_q_bls_bls_functional_left_prefix_power_partial. bpvi_u_bls_bls_functional_left_prefix_power = bpvi_q_bls_bls_functional_left_prefix_power_partial * S ((S (bpvi_j_bls_bls_functional_left_prefix_power)) * bpvi_v_bls_bls_functional_left_prefix_power) + (bpvi_partial_bls_bls_functional_left_prefix_power))) /\ ((((exists bpvi_h_bls_bls_functional_left_prefix_power_successor. bpvi_h_bls_bls_functional_left_prefix_power_successor + S (bpvi_successor_bls_bls_functional_left_prefix_power) = S ((S (S bpvi_j_bls_bls_functional_left_prefix_power)) * bpvi_v_bls_bls_functional_left_prefix_power)) /\ exists bpvi_q_bls_bls_functional_left_prefix_power_successor. bpvi_u_bls_bls_functional_left_prefix_power = bpvi_q_bls_bls_functional_left_prefix_power_successor * S ((S (S bpvi_j_bls_bls_functional_left_prefix_power)) * bpvi_v_bls_bls_functional_left_prefix_power) + (bpvi_successor_bls_bls_functional_left_prefix_power))) /\ bpvi_successor_bls_bls_functional_left_prefix_power = bpvi_partial_bls_bls_functional_left_prefix_power * bpvi_factor_bls_bls_functional_left_prefix_power)))))))) /\ ((((exists ff_h_bls_bls_functional_left_prefix_quotient_entry. ff_h_bls_bls_functional_left_prefix_quotient_entry + S (bls_quotient_bls_functional_left_prefix) = S ((S (bls_index_bls_functional_left_prefix)) * bls_scale_bls_functional_left)) /\ exists ff_q_bls_bls_functional_left_prefix_quotient_entry. bls_code_bls_functional_left = ff_q_bls_bls_functional_left_prefix_quotient_entry * S ((S (bls_index_bls_functional_left_prefix)) * bls_scale_bls_functional_left) + (bls_quotient_bls_functional_left_prefix))) /\ ((n = bls_power_bls_functional_left_prefix * bls_quotient_bls_functional_left_prefix + bls_remainder_bls_functional_left_prefix /\ exists bls_remainder_gap_bls_functional_left_prefix_division. bls_remainder_gap_bls_functional_left_prefix_division + S (bls_remainder_bls_functional_left_prefix) = bls_power_bls_functional_left_prefix))))) /\ (exists ff_u_bls_bls_functional_left_sum ff_v_bls_bls_functional_left_sum. ((((exists ff_h_bls_bls_functional_left_sum_start. ff_h_bls_bls_functional_left_sum_start + S (0) = S ((S (0)) * ff_v_bls_bls_functional_left_sum)) /\ exists ff_q_bls_bls_functional_left_sum_start. ff_u_bls_bls_functional_left_sum = ff_q_bls_bls_functional_left_sum_start * S ((S (0)) * ff_v_bls_bls_functional_left_sum) + (0))) /\ ((((exists ff_h_bls_bls_functional_left_sum_terminal. ff_h_bls_bls_functional_left_sum_terminal + S (e) = S ((S (n)) * ff_v_bls_bls_functional_left_sum)) /\ exists ff_q_bls_bls_functional_left_sum_terminal. ff_u_bls_bls_functional_left_sum = ff_q_bls_bls_functional_left_sum_terminal * S ((S (n)) * ff_v_bls_bls_functional_left_sum) + (e))) /\ forall ff_i_bls_bls_functional_left_sum. (exists ff_lt_bls_bls_functional_left_sum_bound. ff_lt_bls_bls_functional_left_sum_bound + S ff_i_bls_bls_functional_left_sum = n) -> exists ff_a_bls_bls_functional_left_sum ff_r_bls_bls_functional_left_sum ff_s_bls_bls_functional_left_sum. ((((exists ff_h_bls_bls_functional_left_sum_summand. ff_h_bls_bls_functional_left_sum_summand + S (ff_a_bls_bls_functional_left_sum) = S ((S (ff_i_bls_bls_functional_left_sum)) * bls_scale_bls_functional_left)) /\ exists ff_q_bls_bls_functional_left_sum_summand. bls_code_bls_functional_left = ff_q_bls_bls_functional_left_sum_summand * S ((S (ff_i_bls_bls_functional_left_sum)) * bls_scale_bls_functional_left) + (ff_a_bls_bls_functional_left_sum))) /\ ((((exists ff_h_bls_bls_functional_left_sum_partial. ff_h_bls_bls_functional_left_sum_partial + S (ff_r_bls_bls_functional_left_sum) = S ((S (ff_i_bls_bls_functional_left_sum)) * ff_v_bls_bls_functional_left_sum)) /\ exists ff_q_bls_bls_functional_left_sum_partial. ff_u_bls_bls_functional_left_sum = ff_q_bls_bls_functional_left_sum_partial * S ((S (ff_i_bls_bls_functional_left_sum)) * ff_v_bls_bls_functional_left_sum) + (ff_r_bls_bls_functional_left_sum))) /\ ((((exists ff_h_bls_bls_functional_left_sum_successor. ff_h_bls_bls_functional_left_sum_successor + S (ff_s_bls_bls_functional_left_sum) = S ((S (S ff_i_bls_bls_functional_left_sum)) * ff_v_bls_bls_functional_left_sum)) /\ exists ff_q_bls_bls_functional_left_sum_successor. ff_u_bls_bls_functional_left_sum = ff_q_bls_bls_functional_left_sum_successor * S ((S (S ff_i_bls_bls_functional_left_sum)) * ff_v_bls_bls_functional_left_sum) + (ff_s_bls_bls_functional_left_sum))) /\ ff_s_bls_bls_functional_left_sum = ff_r_bls_bls_functional_left_sum + ff_a_bls_bls_functional_left_sum)))))))) -> (exists bls_code_bls_functional_right bls_scale_bls_functional_right. ((forall bls_index_bls_functional_right_prefix. (exists bls_gap_bls_functional_right_prefix_bound. bls_gap_bls_functional_right_prefix_bound + S (bls_index_bls_functional_right_prefix) = (n)) -> exists bls_power_bls_functional_right_prefix bls_quotient_bls_functional_right_prefix bls_remainder_bls_functional_right_prefix. ((exists bpvi_b_bls_bls_functional_right_prefix_power bpvi_c_bls_bls_functional_right_prefix_power. ((forall bpvi_i_bls_bls_functional_right_prefix_power. (exists bpvi_repeat_gap_bls_bls_functional_right_prefix_power. bpvi_repeat_gap_bls_bls_functional_right_prefix_power + S bpvi_i_bls_bls_functional_right_prefix_power = S bls_index_bls_functional_right_prefix) -> (((exists bpvi_h_bls_bls_functional_right_prefix_power_repeat. bpvi_h_bls_bls_functional_right_prefix_power_repeat + S (p) = S ((S (bpvi_i_bls_bls_functional_right_prefix_power)) * bpvi_c_bls_bls_functional_right_prefix_power)) /\ exists bpvi_q_bls_bls_functional_right_prefix_power_repeat. bpvi_b_bls_bls_functional_right_prefix_power = bpvi_q_bls_bls_functional_right_prefix_power_repeat * S ((S (bpvi_i_bls_bls_functional_right_prefix_power)) * bpvi_c_bls_bls_functional_right_prefix_power) + (p)))) /\ (exists bpvi_u_bls_bls_functional_right_prefix_power bpvi_v_bls_bls_functional_right_prefix_power. ((((exists bpvi_h_bls_bls_functional_right_prefix_power_start. bpvi_h_bls_bls_functional_right_prefix_power_start + S (1) = S ((S (0)) * bpvi_v_bls_bls_functional_right_prefix_power)) /\ exists bpvi_q_bls_bls_functional_right_prefix_power_start. bpvi_u_bls_bls_functional_right_prefix_power = bpvi_q_bls_bls_functional_right_prefix_power_start * S ((S (0)) * bpvi_v_bls_bls_functional_right_prefix_power) + (1))) /\ ((((exists bpvi_h_bls_bls_functional_right_prefix_power_terminal. bpvi_h_bls_bls_functional_right_prefix_power_terminal + S (bls_power_bls_functional_right_prefix) = S ((S (S bls_index_bls_functional_right_prefix)) * bpvi_v_bls_bls_functional_right_prefix_power)) /\ exists bpvi_q_bls_bls_functional_right_prefix_power_terminal. bpvi_u_bls_bls_functional_right_prefix_power = bpvi_q_bls_bls_functional_right_prefix_power_terminal * S ((S (S bls_index_bls_functional_right_prefix)) * bpvi_v_bls_bls_functional_right_prefix_power) + (bls_power_bls_functional_right_prefix))) /\ forall bpvi_j_bls_bls_functional_right_prefix_power. (exists bpvi_product_gap_bls_bls_functional_right_prefix_power. bpvi_product_gap_bls_bls_functional_right_prefix_power + S bpvi_j_bls_bls_functional_right_prefix_power = S bls_index_bls_functional_right_prefix) -> exists bpvi_factor_bls_bls_functional_right_prefix_power bpvi_partial_bls_bls_functional_right_prefix_power bpvi_successor_bls_bls_functional_right_prefix_power. ((((exists bpvi_h_bls_bls_functional_right_prefix_power_factor. bpvi_h_bls_bls_functional_right_prefix_power_factor + S (bpvi_factor_bls_bls_functional_right_prefix_power) = S ((S (bpvi_j_bls_bls_functional_right_prefix_power)) * bpvi_c_bls_bls_functional_right_prefix_power)) /\ exists bpvi_q_bls_bls_functional_right_prefix_power_factor. bpvi_b_bls_bls_functional_right_prefix_power = bpvi_q_bls_bls_functional_right_prefix_power_factor * S ((S (bpvi_j_bls_bls_functional_right_prefix_power)) * bpvi_c_bls_bls_functional_right_prefix_power) + (bpvi_factor_bls_bls_functional_right_prefix_power))) /\ ((((exists bpvi_h_bls_bls_functional_right_prefix_power_partial. bpvi_h_bls_bls_functional_right_prefix_power_partial + S (bpvi_partial_bls_bls_functional_right_prefix_power) = S ((S (bpvi_j_bls_bls_functional_right_prefix_power)) * bpvi_v_bls_bls_functional_right_prefix_power)) /\ exists bpvi_q_bls_bls_functional_right_prefix_power_partial. bpvi_u_bls_bls_functional_right_prefix_power = bpvi_q_bls_bls_functional_right_prefix_power_partial * S ((S (bpvi_j_bls_bls_functional_right_prefix_power)) * bpvi_v_bls_bls_functional_right_prefix_power) + (bpvi_partial_bls_bls_functional_right_prefix_power))) /\ ((((exists bpvi_h_bls_bls_functional_right_prefix_power_successor. bpvi_h_bls_bls_functional_right_prefix_power_successor + S (bpvi_successor_bls_bls_functional_right_prefix_power) = S ((S (S bpvi_j_bls_bls_functional_right_prefix_power)) * bpvi_v_bls_bls_functional_right_prefix_power)) /\ exists bpvi_q_bls_bls_functional_right_prefix_power_successor. bpvi_u_bls_bls_functional_right_prefix_power = bpvi_q_bls_bls_functional_right_prefix_power_successor * S ((S (S bpvi_j_bls_bls_functional_right_prefix_power)) * bpvi_v_bls_bls_functional_right_prefix_power) + (bpvi_successor_bls_bls_functional_right_prefix_power))) /\ bpvi_successor_bls_bls_functional_right_prefix_power = bpvi_partial_bls_bls_functional_right_prefix_power * bpvi_factor_bls_bls_functional_right_prefix_power)))))))) /\ ((((exists ff_h_bls_bls_functional_right_prefix_quotient_entry. ff_h_bls_bls_functional_right_prefix_quotient_entry + S (bls_quotient_bls_functional_right_prefix) = S ((S (bls_index_bls_functional_right_prefix)) * bls_scale_bls_functional_right)) /\ exists ff_q_bls_bls_functional_right_prefix_quotient_entry. bls_code_bls_functional_right = ff_q_bls_bls_functional_right_prefix_quotient_entry * S ((S (bls_index_bls_functional_right_prefix)) * bls_scale_bls_functional_right) + (bls_quotient_bls_functional_right_prefix))) /\ ((n = bls_power_bls_functional_right_prefix * bls_quotient_bls_functional_right_prefix + bls_remainder_bls_functional_right_prefix /\ exists bls_remainder_gap_bls_functional_right_prefix_division. bls_remainder_gap_bls_functional_right_prefix_division + S (bls_remainder_bls_functional_right_prefix) = bls_power_bls_functional_right_prefix))))) /\ (exists ff_u_bls_bls_functional_right_sum ff_v_bls_bls_functional_right_sum. ((((exists ff_h_bls_bls_functional_right_sum_start. ff_h_bls_bls_functional_right_sum_start + S (0) = S ((S (0)) * ff_v_bls_bls_functional_right_sum)) /\ exists ff_q_bls_bls_functional_right_sum_start. ff_u_bls_bls_functional_right_sum = ff_q_bls_bls_functional_right_sum_start * S ((S (0)) * ff_v_bls_bls_functional_right_sum) + (0))) /\ ((((exists ff_h_bls_bls_functional_right_sum_terminal. ff_h_bls_bls_functional_right_sum_terminal + S (f) = S ((S (n)) * ff_v_bls_bls_functional_right_sum)) /\ exists ff_q_bls_bls_functional_right_sum_terminal. ff_u_bls_bls_functional_right_sum = ff_q_bls_bls_functional_right_sum_terminal * S ((S (n)) * ff_v_bls_bls_functional_right_sum) + (f))) /\ forall ff_i_bls_bls_functional_right_sum. (exists ff_lt_bls_bls_functional_right_sum_bound. ff_lt_bls_bls_functional_right_sum_bound + S ff_i_bls_bls_functional_right_sum = n) -> exists ff_a_bls_bls_functional_right_sum ff_r_bls_bls_functional_right_sum ff_s_bls_bls_functional_right_sum. ((((exists ff_h_bls_bls_functional_right_sum_summand. ff_h_bls_bls_functional_right_sum_summand + S (ff_a_bls_bls_functional_right_sum) = S ((S (ff_i_bls_bls_functional_right_sum)) * bls_scale_bls_functional_right)) /\ exists ff_q_bls_bls_functional_right_sum_summand. bls_code_bls_functional_right = ff_q_bls_bls_functional_right_sum_summand * S ((S (ff_i_bls_bls_functional_right_sum)) * bls_scale_bls_functional_right) + (ff_a_bls_bls_functional_right_sum))) /\ ((((exists ff_h_bls_bls_functional_right_sum_partial. ff_h_bls_bls_functional_right_sum_partial + S (ff_r_bls_bls_functional_right_sum) = S ((S (ff_i_bls_bls_functional_right_sum)) * ff_v_bls_bls_functional_right_sum)) /\ exists ff_q_bls_bls_functional_right_sum_partial. ff_u_bls_bls_functional_right_sum = ff_q_bls_bls_functional_right_sum_partial * S ((S (ff_i_bls_bls_functional_right_sum)) * ff_v_bls_bls_functional_right_sum) + (ff_r_bls_bls_functional_right_sum))) /\ ((((exists ff_h_bls_bls_functional_right_sum_successor. ff_h_bls_bls_functional_right_sum_successor + S (ff_s_bls_bls_functional_right_sum) = S ((S (S ff_i_bls_bls_functional_right_sum)) * ff_v_bls_bls_functional_right_sum)) /\ exists ff_q_bls_bls_functional_right_sum_successor. ff_u_bls_bls_functional_right_sum = ff_q_bls_bls_functional_right_sum_successor * S ((S (S ff_i_bls_bls_functional_right_sum)) * ff_v_bls_bls_functional_right_sum) + (ff_s_bls_bls_functional_right_sum))) /\ ff_s_bls_bls_functional_right_sum = ff_r_bls_bls_functional_right_sum + ff_a_bls_bls_functional_right_sum)))))))) -> e = f