integer_polynomial_prime_simple_root_lifts_all_positive_powers

Exact expanded first-order arithmetic statement

forall pb pc nb nc a l p k. ((~(p = 1) /\ forall frm_prime_left_hsc_all_precision_prime frm_prime_right_hsc_all_precision_prime. p = frm_prime_left_hsc_all_precision_prime * frm_prime_right_hsc_all_precision_prime -> frm_prime_left_hsc_all_precision_prime = 1 \/ frm_prime_right_hsc_all_precision_prime = 1)) -> ~(k = 0) -> (exists hsc_vp_all_precision_source hsc_dp_all_precision_source hsc_vn_all_precision_source hsc_dn_all_precision_source. ((((exists ff_u_hd_hpl_sph_hsc_all_precision_source_pair_positive ff_v_hd_hpl_sph_hsc_all_precision_source_pair_positive ff_d_hd_hpl_sph_hsc_all_precision_source_pair_positive ff_e_hd_hpl_sph_hsc_all_precision_source_pair_positive. ((((((exists fs_h_ph_hd_hpl_sph_hsc_all_precision_source_pair_positive_body_value_start. fs_h_ph_hd_hpl_sph_hsc_all_precision_source_pair_positive_body_value_start + S (0) = S ((S (0)) * ff_v_hd_hpl_sph_hsc_all_precision_source_pair_positive)) /\ exists fs_q_ph_hd_hpl_sph_hsc_all_precision_source_pair_positive_body_value_start. ff_u_hd_hpl_sph_hsc_all_precision_source_pair_positive = fs_q_ph_hd_hpl_sph_hsc_all_precision_source_pair_positive_body_value_start * S ((S (0)) * ff_v_hd_hpl_sph_hsc_all_precision_source_pair_positive) + (0))) /\ ((((exists fs_h_ph_hd_hpl_sph_hsc_all_precision_source_pair_positive_body_value_terminal. fs_h_ph_hd_hpl_sph_hsc_all_precision_source_pair_positive_body_value_terminal + S (hsc_vp_all_precision_source) = S ((S (l)) * ff_v_hd_hpl_sph_hsc_all_precision_source_pair_positive)) /\ exists fs_q_ph_hd_hpl_sph_hsc_all_precision_source_pair_positive_body_value_terminal. ff_u_hd_hpl_sph_hsc_all_precision_source_pair_positive = fs_q_ph_hd_hpl_sph_hsc_all_precision_source_pair_positive_body_value_terminal * S ((S (l)) * ff_v_hd_hpl_sph_hsc_all_precision_source_pair_positive) + (hsc_vp_all_precision_source))) /\ forall ff_i_ph_hd_hpl_sph_hsc_all_precision_source_pair_positive_body_value_steps. (exists ph_bound_hd_hpl_sph_hsc_all_precision_source_pair_positive_body_value_steps. ph_bound_hd_hpl_sph_hsc_all_precision_source_pair_positive_body_value_steps + S ff_i_ph_hd_hpl_sph_hsc_all_precision_source_pair_positive_body_value_steps = l) -> exists ff_coefficient_ph_hd_hpl_sph_hsc_all_precision_source_pair_positive_body_value_steps ff_previous_ph_hd_hpl_sph_hsc_all_precision_source_pair_positive_body_value_steps ff_current_ph_hd_hpl_sph_hsc_all_precision_source_pair_positive_body_value_steps. ((((exists fs_h_ph_hd_hpl_sph_hsc_all_precision_source_pair_positive_body_value_steps_coefficient. fs_h_ph_hd_hpl_sph_hsc_all_precision_source_pair_positive_body_value_steps_coefficient + S (ff_coefficient_ph_hd_hpl_sph_hsc_all_precision_source_pair_positive_body_value_steps) = S ((S (ff_i_ph_hd_hpl_sph_hsc_all_precision_source_pair_positive_body_value_steps)) * pc)) /\ exists fs_q_ph_hd_hpl_sph_hsc_all_precision_source_pair_positive_body_value_steps_coefficient. pb = fs_q_ph_hd_hpl_sph_hsc_all_precision_source_pair_positive_body_value_steps_coefficient * S ((S (ff_i_ph_hd_hpl_sph_hsc_all_precision_source_pair_positive_body_value_steps)) * pc) + (ff_coefficient_ph_hd_hpl_sph_hsc_all_precision_source_pair_positive_body_value_steps))) /\ ((((exists fs_h_ph_hd_hpl_sph_hsc_all_precision_source_pair_positive_body_value_steps_before. fs_h_ph_hd_hpl_sph_hsc_all_precision_source_pair_positive_body_value_steps_before + S (ff_previous_ph_hd_hpl_sph_hsc_all_precision_source_pair_positive_body_value_steps) = S ((S (ff_i_ph_hd_hpl_sph_hsc_all_precision_source_pair_positive_body_value_steps)) * ff_v_hd_hpl_sph_hsc_all_precision_source_pair_positive)) /\ exists fs_q_ph_hd_hpl_sph_hsc_all_precision_source_pair_positive_body_value_steps_before. ff_u_hd_hpl_sph_hsc_all_precision_source_pair_positive = fs_q_ph_hd_hpl_sph_hsc_all_precision_source_pair_positive_body_value_steps_before * S ((S (ff_i_ph_hd_hpl_sph_hsc_all_precision_source_pair_positive_body_value_steps)) * ff_v_hd_hpl_sph_hsc_all_precision_source_pair_positive) + (ff_previous_ph_hd_hpl_sph_hsc_all_precision_source_pair_positive_body_value_steps))) /\ ((((exists fs_h_ph_hd_hpl_sph_hsc_all_precision_source_pair_positive_body_value_steps_after. fs_h_ph_hd_hpl_sph_hsc_all_precision_source_pair_positive_body_value_steps_after + S (ff_current_ph_hd_hpl_sph_hsc_all_precision_source_pair_positive_body_value_steps) = S ((S (S ff_i_ph_hd_hpl_sph_hsc_all_precision_source_pair_positive_body_value_steps)) * ff_v_hd_hpl_sph_hsc_all_precision_source_pair_positive)) /\ exists fs_q_ph_hd_hpl_sph_hsc_all_precision_source_pair_positive_body_value_steps_after. ff_u_hd_hpl_sph_hsc_all_precision_source_pair_positive = fs_q_ph_hd_hpl_sph_hsc_all_precision_source_pair_positive_body_value_steps_after * S ((S (S ff_i_ph_hd_hpl_sph_hsc_all_precision_source_pair_positive_body_value_steps)) * ff_v_hd_hpl_sph_hsc_all_precision_source_pair_positive) + (ff_current_ph_hd_hpl_sph_hsc_all_precision_source_pair_positive_body_value_steps))) /\ ff_current_ph_hd_hpl_sph_hsc_all_precision_source_pair_positive_body_value_steps = ff_previous_ph_hd_hpl_sph_hsc_all_precision_source_pair_positive_body_value_steps * a + ff_coefficient_ph_hd_hpl_sph_hsc_all_precision_source_pair_positive_body_value_steps)))))) /\ (((((exists fs_h_ph_hd_hpl_sph_hsc_all_precision_source_pair_positive_body_derivative_start. fs_h_ph_hd_hpl_sph_hsc_all_precision_source_pair_positive_body_derivative_start + S (0) = S ((S (0)) * ff_e_hd_hpl_sph_hsc_all_precision_source_pair_positive)) /\ exists fs_q_ph_hd_hpl_sph_hsc_all_precision_source_pair_positive_body_derivative_start. ff_d_hd_hpl_sph_hsc_all_precision_source_pair_positive = fs_q_ph_hd_hpl_sph_hsc_all_precision_source_pair_positive_body_derivative_start * S ((S (0)) * ff_e_hd_hpl_sph_hsc_all_precision_source_pair_positive) + (0))) /\ ((((exists fs_h_ph_hd_hpl_sph_hsc_all_precision_source_pair_positive_body_derivative_terminal. fs_h_ph_hd_hpl_sph_hsc_all_precision_source_pair_positive_body_derivative_terminal + S (hsc_dp_all_precision_source) = S ((S (l)) * ff_e_hd_hpl_sph_hsc_all_precision_source_pair_positive)) /\ exists fs_q_ph_hd_hpl_sph_hsc_all_precision_source_pair_positive_body_derivative_terminal. ff_d_hd_hpl_sph_hsc_all_precision_source_pair_positive = fs_q_ph_hd_hpl_sph_hsc_all_precision_source_pair_positive_body_derivative_terminal * S ((S (l)) * ff_e_hd_hpl_sph_hsc_all_precision_source_pair_positive) + (hsc_dp_all_precision_source))) /\ forall ff_i_ph_hd_hpl_sph_hsc_all_precision_source_pair_positive_body_derivative_steps. (exists ph_bound_hd_hpl_sph_hsc_all_precision_source_pair_positive_body_derivative_steps. ph_bound_hd_hpl_sph_hsc_all_precision_source_pair_positive_body_derivative_steps + S ff_i_ph_hd_hpl_sph_hsc_all_precision_source_pair_positive_body_derivative_steps = l) -> exists ff_coefficient_ph_hd_hpl_sph_hsc_all_precision_source_pair_positive_body_derivative_steps ff_previous_ph_hd_hpl_sph_hsc_all_precision_source_pair_positive_body_derivative_steps ff_current_ph_hd_hpl_sph_hsc_all_precision_source_pair_positive_body_derivative_steps. ((((exists fs_h_ph_hd_hpl_sph_hsc_all_precision_source_pair_positive_body_derivative_steps_coefficient. fs_h_ph_hd_hpl_sph_hsc_all_precision_source_pair_positive_body_derivative_steps_coefficient + S (ff_coefficient_ph_hd_hpl_sph_hsc_all_precision_source_pair_positive_body_derivative_steps) = S ((S (ff_i_ph_hd_hpl_sph_hsc_all_precision_source_pair_positive_body_derivative_steps)) * ff_v_hd_hpl_sph_hsc_all_precision_source_pair_positive)) /\ exists fs_q_ph_hd_hpl_sph_hsc_all_precision_source_pair_positive_body_derivative_steps_coefficient. ff_u_hd_hpl_sph_hsc_all_precision_source_pair_positive = fs_q_ph_hd_hpl_sph_hsc_all_precision_source_pair_positive_body_derivative_steps_coefficient * S ((S (ff_i_ph_hd_hpl_sph_hsc_all_precision_source_pair_positive_body_derivative_steps)) * ff_v_hd_hpl_sph_hsc_all_precision_source_pair_positive) + (ff_coefficient_ph_hd_hpl_sph_hsc_all_precision_source_pair_positive_body_derivative_steps))) /\ ((((exists fs_h_ph_hd_hpl_sph_hsc_all_precision_source_pair_positive_body_derivative_steps_before. fs_h_ph_hd_hpl_sph_hsc_all_precision_source_pair_positive_body_derivative_steps_before + S (ff_previous_ph_hd_hpl_sph_hsc_all_precision_source_pair_positive_body_derivative_steps) = S ((S (ff_i_ph_hd_hpl_sph_hsc_all_precision_source_pair_positive_body_derivative_steps)) * ff_e_hd_hpl_sph_hsc_all_precision_source_pair_positive)) /\ exists fs_q_ph_hd_hpl_sph_hsc_all_precision_source_pair_positive_body_derivative_steps_before. ff_d_hd_hpl_sph_hsc_all_precision_source_pair_positive = fs_q_ph_hd_hpl_sph_hsc_all_precision_source_pair_positive_body_derivative_steps_before * S ((S (ff_i_ph_hd_hpl_sph_hsc_all_precision_source_pair_positive_body_derivative_steps)) * ff_e_hd_hpl_sph_hsc_all_precision_source_pair_positive) + (ff_previous_ph_hd_hpl_sph_hsc_all_precision_source_pair_positive_body_derivative_steps))) /\ ((((exists fs_h_ph_hd_hpl_sph_hsc_all_precision_source_pair_positive_body_derivative_steps_after. fs_h_ph_hd_hpl_sph_hsc_all_precision_source_pair_positive_body_derivative_steps_after + S (ff_current_ph_hd_hpl_sph_hsc_all_precision_source_pair_positive_body_derivative_steps) = S ((S (S ff_i_ph_hd_hpl_sph_hsc_all_precision_source_pair_positive_body_derivative_steps)) * ff_e_hd_hpl_sph_hsc_all_precision_source_pair_positive)) /\ exists fs_q_ph_hd_hpl_sph_hsc_all_precision_source_pair_positive_body_derivative_steps_after. ff_d_hd_hpl_sph_hsc_all_precision_source_pair_positive = fs_q_ph_hd_hpl_sph_hsc_all_precision_source_pair_positive_body_derivative_steps_after * S ((S (S ff_i_ph_hd_hpl_sph_hsc_all_precision_source_pair_positive_body_derivative_steps)) * ff_e_hd_hpl_sph_hsc_all_precision_source_pair_positive) + (ff_current_ph_hd_hpl_sph_hsc_all_precision_source_pair_positive_body_derivative_steps))) /\ ff_current_ph_hd_hpl_sph_hsc_all_precision_source_pair_positive_body_derivative_steps = ff_previous_ph_hd_hpl_sph_hsc_all_precision_source_pair_positive_body_derivative_steps * a + ff_coefficient_ph_hd_hpl_sph_hsc_all_precision_source_pair_positive_body_derivative_steps)))))))) /\ (exists ff_u_hd_hpl_sph_hsc_all_precision_source_pair_negative ff_v_hd_hpl_sph_hsc_all_precision_source_pair_negative ff_d_hd_hpl_sph_hsc_all_precision_source_pair_negative ff_e_hd_hpl_sph_hsc_all_precision_source_pair_negative. ((((((exists fs_h_ph_hd_hpl_sph_hsc_all_precision_source_pair_negative_body_value_start. fs_h_ph_hd_hpl_sph_hsc_all_precision_source_pair_negative_body_value_start + S (0) = S ((S (0)) * ff_v_hd_hpl_sph_hsc_all_precision_source_pair_negative)) /\ exists fs_q_ph_hd_hpl_sph_hsc_all_precision_source_pair_negative_body_value_start. ff_u_hd_hpl_sph_hsc_all_precision_source_pair_negative = fs_q_ph_hd_hpl_sph_hsc_all_precision_source_pair_negative_body_value_start * S ((S (0)) * ff_v_hd_hpl_sph_hsc_all_precision_source_pair_negative) + (0))) /\ ((((exists fs_h_ph_hd_hpl_sph_hsc_all_precision_source_pair_negative_body_value_terminal. fs_h_ph_hd_hpl_sph_hsc_all_precision_source_pair_negative_body_value_terminal + S (hsc_vn_all_precision_source) = S ((S (l)) * ff_v_hd_hpl_sph_hsc_all_precision_source_pair_negative)) /\ exists fs_q_ph_hd_hpl_sph_hsc_all_precision_source_pair_negative_body_value_terminal. ff_u_hd_hpl_sph_hsc_all_precision_source_pair_negative = fs_q_ph_hd_hpl_sph_hsc_all_precision_source_pair_negative_body_value_terminal * S ((S (l)) * ff_v_hd_hpl_sph_hsc_all_precision_source_pair_negative) + (hsc_vn_all_precision_source))) /\ forall ff_i_ph_hd_hpl_sph_hsc_all_precision_source_pair_negative_body_value_steps. (exists ph_bound_hd_hpl_sph_hsc_all_precision_source_pair_negative_body_value_steps. ph_bound_hd_hpl_sph_hsc_all_precision_source_pair_negative_body_value_steps + S ff_i_ph_hd_hpl_sph_hsc_all_precision_source_pair_negative_body_value_steps = l) -> exists ff_coefficient_ph_hd_hpl_sph_hsc_all_precision_source_pair_negative_body_value_steps ff_previous_ph_hd_hpl_sph_hsc_all_precision_source_pair_negative_body_value_steps ff_current_ph_hd_hpl_sph_hsc_all_precision_source_pair_negative_body_value_steps. ((((exists fs_h_ph_hd_hpl_sph_hsc_all_precision_source_pair_negative_body_value_steps_coefficient. fs_h_ph_hd_hpl_sph_hsc_all_precision_source_pair_negative_body_value_steps_coefficient + S (ff_coefficient_ph_hd_hpl_sph_hsc_all_precision_source_pair_negative_body_value_steps) = S ((S (ff_i_ph_hd_hpl_sph_hsc_all_precision_source_pair_negative_body_value_steps)) * nc)) /\ exists fs_q_ph_hd_hpl_sph_hsc_all_precision_source_pair_negative_body_value_steps_coefficient. nb = fs_q_ph_hd_hpl_sph_hsc_all_precision_source_pair_negative_body_value_steps_coefficient * S ((S (ff_i_ph_hd_hpl_sph_hsc_all_precision_source_pair_negative_body_value_steps)) * nc) + (ff_coefficient_ph_hd_hpl_sph_hsc_all_precision_source_pair_negative_body_value_steps))) /\ ((((exists fs_h_ph_hd_hpl_sph_hsc_all_precision_source_pair_negative_body_value_steps_before. fs_h_ph_hd_hpl_sph_hsc_all_precision_source_pair_negative_body_value_steps_before + S (ff_previous_ph_hd_hpl_sph_hsc_all_precision_source_pair_negative_body_value_steps) = S ((S (ff_i_ph_hd_hpl_sph_hsc_all_precision_source_pair_negative_body_value_steps)) * ff_v_hd_hpl_sph_hsc_all_precision_source_pair_negative)) /\ exists fs_q_ph_hd_hpl_sph_hsc_all_precision_source_pair_negative_body_value_steps_before. ff_u_hd_hpl_sph_hsc_all_precision_source_pair_negative = fs_q_ph_hd_hpl_sph_hsc_all_precision_source_pair_negative_body_value_steps_before * S ((S (ff_i_ph_hd_hpl_sph_hsc_all_precision_source_pair_negative_body_value_steps)) * ff_v_hd_hpl_sph_hsc_all_precision_source_pair_negative) + (ff_previous_ph_hd_hpl_sph_hsc_all_precision_source_pair_negative_body_value_steps))) /\ ((((exists fs_h_ph_hd_hpl_sph_hsc_all_precision_source_pair_negative_body_value_steps_after. fs_h_ph_hd_hpl_sph_hsc_all_precision_source_pair_negative_body_value_steps_after + S (ff_current_ph_hd_hpl_sph_hsc_all_precision_source_pair_negative_body_value_steps) = S ((S (S ff_i_ph_hd_hpl_sph_hsc_all_precision_source_pair_negative_body_value_steps)) * ff_v_hd_hpl_sph_hsc_all_precision_source_pair_negative)) /\ exists fs_q_ph_hd_hpl_sph_hsc_all_precision_source_pair_negative_body_value_steps_after. ff_u_hd_hpl_sph_hsc_all_precision_source_pair_negative = fs_q_ph_hd_hpl_sph_hsc_all_precision_source_pair_negative_body_value_steps_after * S ((S (S ff_i_ph_hd_hpl_sph_hsc_all_precision_source_pair_negative_body_value_steps)) * ff_v_hd_hpl_sph_hsc_all_precision_source_pair_negative) + (ff_current_ph_hd_hpl_sph_hsc_all_precision_source_pair_negative_body_value_steps))) /\ ff_current_ph_hd_hpl_sph_hsc_all_precision_source_pair_negative_body_value_steps = ff_previous_ph_hd_hpl_sph_hsc_all_precision_source_pair_negative_body_value_steps * a + ff_coefficient_ph_hd_hpl_sph_hsc_all_precision_source_pair_negative_body_value_steps)))))) /\ (((((exists fs_h_ph_hd_hpl_sph_hsc_all_precision_source_pair_negative_body_derivative_start. fs_h_ph_hd_hpl_sph_hsc_all_precision_source_pair_negative_body_derivative_start + S (0) = S ((S (0)) * ff_e_hd_hpl_sph_hsc_all_precision_source_pair_negative)) /\ exists fs_q_ph_hd_hpl_sph_hsc_all_precision_source_pair_negative_body_derivative_start. ff_d_hd_hpl_sph_hsc_all_precision_source_pair_negative = fs_q_ph_hd_hpl_sph_hsc_all_precision_source_pair_negative_body_derivative_start * S ((S (0)) * ff_e_hd_hpl_sph_hsc_all_precision_source_pair_negative) + (0))) /\ ((((exists fs_h_ph_hd_hpl_sph_hsc_all_precision_source_pair_negative_body_derivative_terminal. fs_h_ph_hd_hpl_sph_hsc_all_precision_source_pair_negative_body_derivative_terminal + S (hsc_dn_all_precision_source) = S ((S (l)) * ff_e_hd_hpl_sph_hsc_all_precision_source_pair_negative)) /\ exists fs_q_ph_hd_hpl_sph_hsc_all_precision_source_pair_negative_body_derivative_terminal. ff_d_hd_hpl_sph_hsc_all_precision_source_pair_negative = fs_q_ph_hd_hpl_sph_hsc_all_precision_source_pair_negative_body_derivative_terminal * S ((S (l)) * ff_e_hd_hpl_sph_hsc_all_precision_source_pair_negative) + (hsc_dn_all_precision_source))) /\ forall ff_i_ph_hd_hpl_sph_hsc_all_precision_source_pair_negative_body_derivative_steps. (exists ph_bound_hd_hpl_sph_hsc_all_precision_source_pair_negative_body_derivative_steps. ph_bound_hd_hpl_sph_hsc_all_precision_source_pair_negative_body_derivative_steps + S ff_i_ph_hd_hpl_sph_hsc_all_precision_source_pair_negative_body_derivative_steps = l) -> exists ff_coefficient_ph_hd_hpl_sph_hsc_all_precision_source_pair_negative_body_derivative_steps ff_previous_ph_hd_hpl_sph_hsc_all_precision_source_pair_negative_body_derivative_steps ff_current_ph_hd_hpl_sph_hsc_all_precision_source_pair_negative_body_derivative_steps. ((((exists fs_h_ph_hd_hpl_sph_hsc_all_precision_source_pair_negative_body_derivative_steps_coefficient. fs_h_ph_hd_hpl_sph_hsc_all_precision_source_pair_negative_body_derivative_steps_coefficient + S (ff_coefficient_ph_hd_hpl_sph_hsc_all_precision_source_pair_negative_body_derivative_steps) = S ((S (ff_i_ph_hd_hpl_sph_hsc_all_precision_source_pair_negative_body_derivative_steps)) * ff_v_hd_hpl_sph_hsc_all_precision_source_pair_negative)) /\ exists fs_q_ph_hd_hpl_sph_hsc_all_precision_source_pair_negative_body_derivative_steps_coefficient. ff_u_hd_hpl_sph_hsc_all_precision_source_pair_negative = fs_q_ph_hd_hpl_sph_hsc_all_precision_source_pair_negative_body_derivative_steps_coefficient * S ((S (ff_i_ph_hd_hpl_sph_hsc_all_precision_source_pair_negative_body_derivative_steps)) * ff_v_hd_hpl_sph_hsc_all_precision_source_pair_negative) + (ff_coefficient_ph_hd_hpl_sph_hsc_all_precision_source_pair_negative_body_derivative_steps))) /\ ((((exists fs_h_ph_hd_hpl_sph_hsc_all_precision_source_pair_negative_body_derivative_steps_before. fs_h_ph_hd_hpl_sph_hsc_all_precision_source_pair_negative_body_derivative_steps_before + S (ff_previous_ph_hd_hpl_sph_hsc_all_precision_source_pair_negative_body_derivative_steps) = S ((S (ff_i_ph_hd_hpl_sph_hsc_all_precision_source_pair_negative_body_derivative_steps)) * ff_e_hd_hpl_sph_hsc_all_precision_source_pair_negative)) /\ exists fs_q_ph_hd_hpl_sph_hsc_all_precision_source_pair_negative_body_derivative_steps_before. ff_d_hd_hpl_sph_hsc_all_precision_source_pair_negative = fs_q_ph_hd_hpl_sph_hsc_all_precision_source_pair_negative_body_derivative_steps_before * S ((S (ff_i_ph_hd_hpl_sph_hsc_all_precision_source_pair_negative_body_derivative_steps)) * ff_e_hd_hpl_sph_hsc_all_precision_source_pair_negative) + (ff_previous_ph_hd_hpl_sph_hsc_all_precision_source_pair_negative_body_derivative_steps))) /\ ((((exists fs_h_ph_hd_hpl_sph_hsc_all_precision_source_pair_negative_body_derivative_steps_after. fs_h_ph_hd_hpl_sph_hsc_all_precision_source_pair_negative_body_derivative_steps_after + S (ff_current_ph_hd_hpl_sph_hsc_all_precision_source_pair_negative_body_derivative_steps) = S ((S (S ff_i_ph_hd_hpl_sph_hsc_all_precision_source_pair_negative_body_derivative_steps)) * ff_e_hd_hpl_sph_hsc_all_precision_source_pair_negative)) /\ exists fs_q_ph_hd_hpl_sph_hsc_all_precision_source_pair_negative_body_derivative_steps_after. ff_d_hd_hpl_sph_hsc_all_precision_source_pair_negative = fs_q_ph_hd_hpl_sph_hsc_all_precision_source_pair_negative_body_derivative_steps_after * S ((S (S ff_i_ph_hd_hpl_sph_hsc_all_precision_source_pair_negative_body_derivative_steps)) * ff_e_hd_hpl_sph_hsc_all_precision_source_pair_negative) + (ff_current_ph_hd_hpl_sph_hsc_all_precision_source_pair_negative_body_derivative_steps))) /\ ff_current_ph_hd_hpl_sph_hsc_all_precision_source_pair_negative_body_derivative_steps = ff_previous_ph_hd_hpl_sph_hsc_all_precision_source_pair_negative_body_derivative_steps * a + ff_coefficient_ph_hd_hpl_sph_hsc_all_precision_source_pair_negative_body_derivative_steps)))))))))) /\ ((exists hgcrt_mod_left_hpl_hsc_all_precision_source_root hgcrt_mod_right_hpl_hsc_all_precision_source_root. hsc_vp_all_precision_source + p * hgcrt_mod_left_hpl_hsc_all_precision_source_root = hsc_vn_all_precision_source + p * hgcrt_mod_right_hpl_hsc_all_precision_source_root) /\ (~(exists hgcrt_mod_left_hpl_hsc_all_precision_source hgcrt_mod_right_hpl_hsc_all_precision_source. hsc_dp_all_precision_source + p * hgcrt_mod_left_hpl_hsc_all_precision_source = hsc_dn_all_precision_source + p * hgcrt_mod_right_hpl_hsc_all_precision_source))))) -> (exists hsc_modulus_all_precision_result. ((exists pa_b_hpl_hsc_all_precision_result_power pa_c_hpl_hsc_all_precision_result_power. ((forall pa_i_hpl_hsc_all_precision_result_power_repeat. (exists pa_lt_hpl_hsc_all_precision_result_power_repeat_bound. pa_lt_hpl_hsc_all_precision_result_power_repeat_bound + S pa_i_hpl_hsc_all_precision_result_power_repeat = k) -> (((exists pa_h_hpl_hsc_all_precision_result_power_repeat_decoded. pa_h_hpl_hsc_all_precision_result_power_repeat_decoded + S (p) = S ((S (pa_i_hpl_hsc_all_precision_result_power_repeat)) * pa_c_hpl_hsc_all_precision_result_power)) /\ exists pa_q_hpl_hsc_all_precision_result_power_repeat_decoded. pa_b_hpl_hsc_all_precision_result_power = pa_q_hpl_hsc_all_precision_result_power_repeat_decoded * S ((S (pa_i_hpl_hsc_all_precision_result_power_repeat)) * pa_c_hpl_hsc_all_precision_result_power) + (p)))) /\ (exists pa_u_hpl_hsc_all_precision_result_power_product pa_v_hpl_hsc_all_precision_result_power_product. ((((exists pa_h_hpl_hsc_all_precision_result_power_product_start. pa_h_hpl_hsc_all_precision_result_power_product_start + S (1) = S ((S (0)) * pa_v_hpl_hsc_all_precision_result_power_product)) /\ exists pa_q_hpl_hsc_all_precision_result_power_product_start. pa_u_hpl_hsc_all_precision_result_power_product = pa_q_hpl_hsc_all_precision_result_power_product_start * S ((S (0)) * pa_v_hpl_hsc_all_precision_result_power_product) + (1))) /\ ((((exists pa_h_hpl_hsc_all_precision_result_power_product_terminal. pa_h_hpl_hsc_all_precision_result_power_product_terminal + S (hsc_modulus_all_precision_result) = S ((S (k)) * pa_v_hpl_hsc_all_precision_result_power_product)) /\ exists pa_q_hpl_hsc_all_precision_result_power_product_terminal. pa_u_hpl_hsc_all_precision_result_power_product = pa_q_hpl_hsc_all_precision_result_power_product_terminal * S ((S (k)) * pa_v_hpl_hsc_all_precision_result_power_product) + (hsc_modulus_all_precision_result))) /\ forall pa_i_hpl_hsc_all_precision_result_power_product. (exists pa_lt_hpl_hsc_all_precision_result_power_product_bound. pa_lt_hpl_hsc_all_precision_result_power_product_bound + S pa_i_hpl_hsc_all_precision_result_power_product = k) -> exists pa_p_hpl_hsc_all_precision_result_power_product pa_r_hpl_hsc_all_precision_result_power_product pa_s_hpl_hsc_all_precision_result_power_product. ((((exists pa_h_hpl_hsc_all_precision_result_power_product_factor. pa_h_hpl_hsc_all_precision_result_power_product_factor + S (pa_p_hpl_hsc_all_precision_result_power_product) = S ((S (pa_i_hpl_hsc_all_precision_result_power_product)) * pa_c_hpl_hsc_all_precision_result_power)) /\ exists pa_q_hpl_hsc_all_precision_result_power_product_factor. pa_b_hpl_hsc_all_precision_result_power = pa_q_hpl_hsc_all_precision_result_power_product_factor * S ((S (pa_i_hpl_hsc_all_precision_result_power_product)) * pa_c_hpl_hsc_all_precision_result_power) + (pa_p_hpl_hsc_all_precision_result_power_product))) /\ ((((exists pa_h_hpl_hsc_all_precision_result_power_product_partial. pa_h_hpl_hsc_all_precision_result_power_product_partial + S (pa_r_hpl_hsc_all_precision_result_power_product) = S ((S (pa_i_hpl_hsc_all_precision_result_power_product)) * pa_v_hpl_hsc_all_precision_result_power_product)) /\ exists pa_q_hpl_hsc_all_precision_result_power_product_partial. pa_u_hpl_hsc_all_precision_result_power_product = pa_q_hpl_hsc_all_precision_result_power_product_partial * S ((S (pa_i_hpl_hsc_all_precision_result_power_product)) * pa_v_hpl_hsc_all_precision_result_power_product) + (pa_r_hpl_hsc_all_precision_result_power_product))) /\ ((((exists pa_h_hpl_hsc_all_precision_result_power_product_successor. pa_h_hpl_hsc_all_precision_result_power_product_successor + S (pa_s_hpl_hsc_all_precision_result_power_product) = S ((S (S pa_i_hpl_hsc_all_precision_result_power_product)) * pa_v_hpl_hsc_all_precision_result_power_product)) /\ exists pa_q_hpl_hsc_all_precision_result_power_product_successor. pa_u_hpl_hsc_all_precision_result_power_product = pa_q_hpl_hsc_all_precision_result_power_product_successor * S ((S (S pa_i_hpl_hsc_all_precision_result_power_product)) * pa_v_hpl_hsc_all_precision_result_power_product) + (pa_s_hpl_hsc_all_precision_result_power_product))) /\ pa_s_hpl_hsc_all_precision_result_power_product = pa_r_hpl_hsc_all_precision_result_power_product * pa_p_hpl_hsc_all_precision_result_power_product)))))))) /\ exists hsc_root_all_precision_result. ((((exists hpl_gap_hsc_all_precision_result_chosen. hpl_gap_hsc_all_precision_result_chosen + S (hsc_root_all_precision_result) = (hsc_modulus_all_precision_result)) /\ ((exists hgcrt_mod_left_hpl_hsc_all_precision_result_chosen hgcrt_mod_right_hpl_hsc_all_precision_result_chosen. hsc_root_all_precision_result + p * hgcrt_mod_left_hpl_hsc_all_precision_result_chosen = a + p * hgcrt_mod_right_hpl_hsc_all_precision_result_chosen) /\ (exists sph_positive_hsc_all_precision_result_chosen sph_negative_hsc_all_precision_result_chosen. ((exists ff_u_ph_hpl_sph_hsc_all_precision_result_chosen_positive ff_v_ph_hpl_sph_hsc_all_precision_result_chosen_positive. ((((exists fs_h_ph_hpl_sph_hsc_all_precision_result_chosen_positive_body_start. fs_h_ph_hpl_sph_hsc_all_precision_result_chosen_positive_body_start + S (0) = S ((S (0)) * ff_v_ph_hpl_sph_hsc_all_precision_result_chosen_positive)) /\ exists fs_q_ph_hpl_sph_hsc_all_precision_result_chosen_positive_body_start. ff_u_ph_hpl_sph_hsc_all_precision_result_chosen_positive = fs_q_ph_hpl_sph_hsc_all_precision_result_chosen_positive_body_start * S ((S (0)) * ff_v_ph_hpl_sph_hsc_all_precision_result_chosen_positive) + (0))) /\ ((((exists fs_h_ph_hpl_sph_hsc_all_precision_result_chosen_positive_body_terminal. fs_h_ph_hpl_sph_hsc_all_precision_result_chosen_positive_body_terminal + S (sph_positive_hsc_all_precision_result_chosen) = S ((S (l)) * ff_v_ph_hpl_sph_hsc_all_precision_result_chosen_positive)) /\ exists fs_q_ph_hpl_sph_hsc_all_precision_result_chosen_positive_body_terminal. ff_u_ph_hpl_sph_hsc_all_precision_result_chosen_positive = fs_q_ph_hpl_sph_hsc_all_precision_result_chosen_positive_body_terminal * S ((S (l)) * ff_v_ph_hpl_sph_hsc_all_precision_result_chosen_positive) + (sph_positive_hsc_all_precision_result_chosen))) /\ forall ff_i_ph_hpl_sph_hsc_all_precision_result_chosen_positive_body_steps. (exists ph_bound_hpl_sph_hsc_all_precision_result_chosen_positive_body_steps. ph_bound_hpl_sph_hsc_all_precision_result_chosen_positive_body_steps + S ff_i_ph_hpl_sph_hsc_all_precision_result_chosen_positive_body_steps = l) -> exists ff_coefficient_ph_hpl_sph_hsc_all_precision_result_chosen_positive_body_steps ff_previous_ph_hpl_sph_hsc_all_precision_result_chosen_positive_body_steps ff_current_ph_hpl_sph_hsc_all_precision_result_chosen_positive_body_steps. ((((exists fs_h_ph_hpl_sph_hsc_all_precision_result_chosen_positive_body_steps_coefficient. fs_h_ph_hpl_sph_hsc_all_precision_result_chosen_positive_body_steps_coefficient + S (ff_coefficient_ph_hpl_sph_hsc_all_precision_result_chosen_positive_body_steps) = S ((S (ff_i_ph_hpl_sph_hsc_all_precision_result_chosen_positive_body_steps)) * pc)) /\ exists fs_q_ph_hpl_sph_hsc_all_precision_result_chosen_positive_body_steps_coefficient. pb = fs_q_ph_hpl_sph_hsc_all_precision_result_chosen_positive_body_steps_coefficient * S ((S (ff_i_ph_hpl_sph_hsc_all_precision_result_chosen_positive_body_steps)) * pc) + (ff_coefficient_ph_hpl_sph_hsc_all_precision_result_chosen_positive_body_steps))) /\ ((((exists fs_h_ph_hpl_sph_hsc_all_precision_result_chosen_positive_body_steps_before. fs_h_ph_hpl_sph_hsc_all_precision_result_chosen_positive_body_steps_before + S (ff_previous_ph_hpl_sph_hsc_all_precision_result_chosen_positive_body_steps) = S ((S (ff_i_ph_hpl_sph_hsc_all_precision_result_chosen_positive_body_steps)) * ff_v_ph_hpl_sph_hsc_all_precision_result_chosen_positive)) /\ exists fs_q_ph_hpl_sph_hsc_all_precision_result_chosen_positive_body_steps_before. ff_u_ph_hpl_sph_hsc_all_precision_result_chosen_positive = fs_q_ph_hpl_sph_hsc_all_precision_result_chosen_positive_body_steps_before * S ((S (ff_i_ph_hpl_sph_hsc_all_precision_result_chosen_positive_body_steps)) * ff_v_ph_hpl_sph_hsc_all_precision_result_chosen_positive) + (ff_previous_ph_hpl_sph_hsc_all_precision_result_chosen_positive_body_steps))) /\ ((((exists fs_h_ph_hpl_sph_hsc_all_precision_result_chosen_positive_body_steps_after. fs_h_ph_hpl_sph_hsc_all_precision_result_chosen_positive_body_steps_after + S (ff_current_ph_hpl_sph_hsc_all_precision_result_chosen_positive_body_steps) = S ((S (S ff_i_ph_hpl_sph_hsc_all_precision_result_chosen_positive_body_steps)) * ff_v_ph_hpl_sph_hsc_all_precision_result_chosen_positive)) /\ exists fs_q_ph_hpl_sph_hsc_all_precision_result_chosen_positive_body_steps_after. ff_u_ph_hpl_sph_hsc_all_precision_result_chosen_positive = fs_q_ph_hpl_sph_hsc_all_precision_result_chosen_positive_body_steps_after * S ((S (S ff_i_ph_hpl_sph_hsc_all_precision_result_chosen_positive_body_steps)) * ff_v_ph_hpl_sph_hsc_all_precision_result_chosen_positive) + (ff_current_ph_hpl_sph_hsc_all_precision_result_chosen_positive_body_steps))) /\ ff_current_ph_hpl_sph_hsc_all_precision_result_chosen_positive_body_steps = ff_previous_ph_hpl_sph_hsc_all_precision_result_chosen_positive_body_steps * hsc_root_all_precision_result + ff_coefficient_ph_hpl_sph_hsc_all_precision_result_chosen_positive_body_steps)))))) /\ ((exists ff_u_ph_hpl_sph_hsc_all_precision_result_chosen_negative ff_v_ph_hpl_sph_hsc_all_precision_result_chosen_negative. ((((exists fs_h_ph_hpl_sph_hsc_all_precision_result_chosen_negative_body_start. fs_h_ph_hpl_sph_hsc_all_precision_result_chosen_negative_body_start + S (0) = S ((S (0)) * ff_v_ph_hpl_sph_hsc_all_precision_result_chosen_negative)) /\ exists fs_q_ph_hpl_sph_hsc_all_precision_result_chosen_negative_body_start. ff_u_ph_hpl_sph_hsc_all_precision_result_chosen_negative = fs_q_ph_hpl_sph_hsc_all_precision_result_chosen_negative_body_start * S ((S (0)) * ff_v_ph_hpl_sph_hsc_all_precision_result_chosen_negative) + (0))) /\ ((((exists fs_h_ph_hpl_sph_hsc_all_precision_result_chosen_negative_body_terminal. fs_h_ph_hpl_sph_hsc_all_precision_result_chosen_negative_body_terminal + S (sph_negative_hsc_all_precision_result_chosen) = S ((S (l)) * ff_v_ph_hpl_sph_hsc_all_precision_result_chosen_negative)) /\ exists fs_q_ph_hpl_sph_hsc_all_precision_result_chosen_negative_body_terminal. ff_u_ph_hpl_sph_hsc_all_precision_result_chosen_negative = fs_q_ph_hpl_sph_hsc_all_precision_result_chosen_negative_body_terminal * S ((S (l)) * ff_v_ph_hpl_sph_hsc_all_precision_result_chosen_negative) + (sph_negative_hsc_all_precision_result_chosen))) /\ forall ff_i_ph_hpl_sph_hsc_all_precision_result_chosen_negative_body_steps. (exists ph_bound_hpl_sph_hsc_all_precision_result_chosen_negative_body_steps. ph_bound_hpl_sph_hsc_all_precision_result_chosen_negative_body_steps + S ff_i_ph_hpl_sph_hsc_all_precision_result_chosen_negative_body_steps = l) -> exists ff_coefficient_ph_hpl_sph_hsc_all_precision_result_chosen_negative_body_steps ff_previous_ph_hpl_sph_hsc_all_precision_result_chosen_negative_body_steps ff_current_ph_hpl_sph_hsc_all_precision_result_chosen_negative_body_steps. ((((exists fs_h_ph_hpl_sph_hsc_all_precision_result_chosen_negative_body_steps_coefficient. fs_h_ph_hpl_sph_hsc_all_precision_result_chosen_negative_body_steps_coefficient + S (ff_coefficient_ph_hpl_sph_hsc_all_precision_result_chosen_negative_body_steps) = S ((S (ff_i_ph_hpl_sph_hsc_all_precision_result_chosen_negative_body_steps)) * nc)) /\ exists fs_q_ph_hpl_sph_hsc_all_precision_result_chosen_negative_body_steps_coefficient. nb = fs_q_ph_hpl_sph_hsc_all_precision_result_chosen_negative_body_steps_coefficient * S ((S (ff_i_ph_hpl_sph_hsc_all_precision_result_chosen_negative_body_steps)) * nc) + (ff_coefficient_ph_hpl_sph_hsc_all_precision_result_chosen_negative_body_steps))) /\ ((((exists fs_h_ph_hpl_sph_hsc_all_precision_result_chosen_negative_body_steps_before. fs_h_ph_hpl_sph_hsc_all_precision_result_chosen_negative_body_steps_before + S (ff_previous_ph_hpl_sph_hsc_all_precision_result_chosen_negative_body_steps) = S ((S (ff_i_ph_hpl_sph_hsc_all_precision_result_chosen_negative_body_steps)) * ff_v_ph_hpl_sph_hsc_all_precision_result_chosen_negative)) /\ exists fs_q_ph_hpl_sph_hsc_all_precision_result_chosen_negative_body_steps_before. ff_u_ph_hpl_sph_hsc_all_precision_result_chosen_negative = fs_q_ph_hpl_sph_hsc_all_precision_result_chosen_negative_body_steps_before * S ((S (ff_i_ph_hpl_sph_hsc_all_precision_result_chosen_negative_body_steps)) * ff_v_ph_hpl_sph_hsc_all_precision_result_chosen_negative) + (ff_previous_ph_hpl_sph_hsc_all_precision_result_chosen_negative_body_steps))) /\ ((((exists fs_h_ph_hpl_sph_hsc_all_precision_result_chosen_negative_body_steps_after. fs_h_ph_hpl_sph_hsc_all_precision_result_chosen_negative_body_steps_after + S (ff_current_ph_hpl_sph_hsc_all_precision_result_chosen_negative_body_steps) = S ((S (S ff_i_ph_hpl_sph_hsc_all_precision_result_chosen_negative_body_steps)) * ff_v_ph_hpl_sph_hsc_all_precision_result_chosen_negative)) /\ exists fs_q_ph_hpl_sph_hsc_all_precision_result_chosen_negative_body_steps_after. ff_u_ph_hpl_sph_hsc_all_precision_result_chosen_negative = fs_q_ph_hpl_sph_hsc_all_precision_result_chosen_negative_body_steps_after * S ((S (S ff_i_ph_hpl_sph_hsc_all_precision_result_chosen_negative_body_steps)) * ff_v_ph_hpl_sph_hsc_all_precision_result_chosen_negative) + (ff_current_ph_hpl_sph_hsc_all_precision_result_chosen_negative_body_steps))) /\ ff_current_ph_hpl_sph_hsc_all_precision_result_chosen_negative_body_steps = ff_previous_ph_hpl_sph_hsc_all_precision_result_chosen_negative_body_steps * hsc_root_all_precision_result + ff_coefficient_ph_hpl_sph_hsc_all_precision_result_chosen_negative_body_steps)))))) /\ (exists hgcrt_mod_left_hpl_hsc_all_precision_result_chosen hgcrt_mod_right_hpl_hsc_all_precision_result_chosen. sph_positive_hsc_all_precision_result_chosen + hsc_modulus_all_precision_result * hgcrt_mod_left_hpl_hsc_all_precision_result_chosen = sph_negative_hsc_all_precision_result_chosen + hsc_modulus_all_precision_result * hgcrt_mod_right_hpl_hsc_all_precision_result_chosen))))))) /\ forall hsc_competitor_all_precision_result. (((exists hpl_gap_hsc_all_precision_result_other. hpl_gap_hsc_all_precision_result_other + S (hsc_competitor_all_precision_result) = (hsc_modulus_all_precision_result)) /\ ((exists hgcrt_mod_left_hpl_hsc_all_precision_result_other hgcrt_mod_right_hpl_hsc_all_precision_result_other. hsc_competitor_all_precision_result + p * hgcrt_mod_left_hpl_hsc_all_precision_result_other = a + p * hgcrt_mod_right_hpl_hsc_all_precision_result_other) /\ (exists sph_positive_hsc_all_precision_result_other sph_negative_hsc_all_precision_result_other. ((exists ff_u_ph_hpl_sph_hsc_all_precision_result_other_positive ff_v_ph_hpl_sph_hsc_all_precision_result_other_positive. ((((exists fs_h_ph_hpl_sph_hsc_all_precision_result_other_positive_body_start. fs_h_ph_hpl_sph_hsc_all_precision_result_other_positive_body_start + S (0) = S ((S (0)) * ff_v_ph_hpl_sph_hsc_all_precision_result_other_positive)) /\ exists fs_q_ph_hpl_sph_hsc_all_precision_result_other_positive_body_start. ff_u_ph_hpl_sph_hsc_all_precision_result_other_positive = fs_q_ph_hpl_sph_hsc_all_precision_result_other_positive_body_start * S ((S (0)) * ff_v_ph_hpl_sph_hsc_all_precision_result_other_positive) + (0))) /\ ((((exists fs_h_ph_hpl_sph_hsc_all_precision_result_other_positive_body_terminal. fs_h_ph_hpl_sph_hsc_all_precision_result_other_positive_body_terminal + S (sph_positive_hsc_all_precision_result_other) = S ((S (l)) * ff_v_ph_hpl_sph_hsc_all_precision_result_other_positive)) /\ exists fs_q_ph_hpl_sph_hsc_all_precision_result_other_positive_body_terminal. ff_u_ph_hpl_sph_hsc_all_precision_result_other_positive = fs_q_ph_hpl_sph_hsc_all_precision_result_other_positive_body_terminal * S ((S (l)) * ff_v_ph_hpl_sph_hsc_all_precision_result_other_positive) + (sph_positive_hsc_all_precision_result_other))) /\ forall ff_i_ph_hpl_sph_hsc_all_precision_result_other_positive_body_steps. (exists ph_bound_hpl_sph_hsc_all_precision_result_other_positive_body_steps. ph_bound_hpl_sph_hsc_all_precision_result_other_positive_body_steps + S ff_i_ph_hpl_sph_hsc_all_precision_result_other_positive_body_steps = l) -> exists ff_coefficient_ph_hpl_sph_hsc_all_precision_result_other_positive_body_steps ff_previous_ph_hpl_sph_hsc_all_precision_result_other_positive_body_steps ff_current_ph_hpl_sph_hsc_all_precision_result_other_positive_body_steps. ((((exists fs_h_ph_hpl_sph_hsc_all_precision_result_other_positive_body_steps_coefficient. fs_h_ph_hpl_sph_hsc_all_precision_result_other_positive_body_steps_coefficient + S (ff_coefficient_ph_hpl_sph_hsc_all_precision_result_other_positive_body_steps) = S ((S (ff_i_ph_hpl_sph_hsc_all_precision_result_other_positive_body_steps)) * pc)) /\ exists fs_q_ph_hpl_sph_hsc_all_precision_result_other_positive_body_steps_coefficient. pb = fs_q_ph_hpl_sph_hsc_all_precision_result_other_positive_body_steps_coefficient * S ((S (ff_i_ph_hpl_sph_hsc_all_precision_result_other_positive_body_steps)) * pc) + (ff_coefficient_ph_hpl_sph_hsc_all_precision_result_other_positive_body_steps))) /\ ((((exists fs_h_ph_hpl_sph_hsc_all_precision_result_other_positive_body_steps_before. fs_h_ph_hpl_sph_hsc_all_precision_result_other_positive_body_steps_before + S (ff_previous_ph_hpl_sph_hsc_all_precision_result_other_positive_body_steps) = S ((S (ff_i_ph_hpl_sph_hsc_all_precision_result_other_positive_body_steps)) * ff_v_ph_hpl_sph_hsc_all_precision_result_other_positive)) /\ exists fs_q_ph_hpl_sph_hsc_all_precision_result_other_positive_body_steps_before. ff_u_ph_hpl_sph_hsc_all_precision_result_other_positive = fs_q_ph_hpl_sph_hsc_all_precision_result_other_positive_body_steps_before * S ((S (ff_i_ph_hpl_sph_hsc_all_precision_result_other_positive_body_steps)) * ff_v_ph_hpl_sph_hsc_all_precision_result_other_positive) + (ff_previous_ph_hpl_sph_hsc_all_precision_result_other_positive_body_steps))) /\ ((((exists fs_h_ph_hpl_sph_hsc_all_precision_result_other_positive_body_steps_after. fs_h_ph_hpl_sph_hsc_all_precision_result_other_positive_body_steps_after + S (ff_current_ph_hpl_sph_hsc_all_precision_result_other_positive_body_steps) = S ((S (S ff_i_ph_hpl_sph_hsc_all_precision_result_other_positive_body_steps)) * ff_v_ph_hpl_sph_hsc_all_precision_result_other_positive)) /\ exists fs_q_ph_hpl_sph_hsc_all_precision_result_other_positive_body_steps_after. ff_u_ph_hpl_sph_hsc_all_precision_result_other_positive = fs_q_ph_hpl_sph_hsc_all_precision_result_other_positive_body_steps_after * S ((S (S ff_i_ph_hpl_sph_hsc_all_precision_result_other_positive_body_steps)) * ff_v_ph_hpl_sph_hsc_all_precision_result_other_positive) + (ff_current_ph_hpl_sph_hsc_all_precision_result_other_positive_body_steps))) /\ ff_current_ph_hpl_sph_hsc_all_precision_result_other_positive_body_steps = ff_previous_ph_hpl_sph_hsc_all_precision_result_other_positive_body_steps * hsc_competitor_all_precision_result + ff_coefficient_ph_hpl_sph_hsc_all_precision_result_other_positive_body_steps)))))) /\ ((exists ff_u_ph_hpl_sph_hsc_all_precision_result_other_negative ff_v_ph_hpl_sph_hsc_all_precision_result_other_negative. ((((exists fs_h_ph_hpl_sph_hsc_all_precision_result_other_negative_body_start. fs_h_ph_hpl_sph_hsc_all_precision_result_other_negative_body_start + S (0) = S ((S (0)) * ff_v_ph_hpl_sph_hsc_all_precision_result_other_negative)) /\ exists fs_q_ph_hpl_sph_hsc_all_precision_result_other_negative_body_start. ff_u_ph_hpl_sph_hsc_all_precision_result_other_negative = fs_q_ph_hpl_sph_hsc_all_precision_result_other_negative_body_start * S ((S (0)) * ff_v_ph_hpl_sph_hsc_all_precision_result_other_negative) + (0))) /\ ((((exists fs_h_ph_hpl_sph_hsc_all_precision_result_other_negative_body_terminal. fs_h_ph_hpl_sph_hsc_all_precision_result_other_negative_body_terminal + S (sph_negative_hsc_all_precision_result_other) = S ((S (l)) * ff_v_ph_hpl_sph_hsc_all_precision_result_other_negative)) /\ exists fs_q_ph_hpl_sph_hsc_all_precision_result_other_negative_body_terminal. ff_u_ph_hpl_sph_hsc_all_precision_result_other_negative = fs_q_ph_hpl_sph_hsc_all_precision_result_other_negative_body_terminal * S ((S (l)) * ff_v_ph_hpl_sph_hsc_all_precision_result_other_negative) + (sph_negative_hsc_all_precision_result_other))) /\ forall ff_i_ph_hpl_sph_hsc_all_precision_result_other_negative_body_steps. (exists ph_bound_hpl_sph_hsc_all_precision_result_other_negative_body_steps. ph_bound_hpl_sph_hsc_all_precision_result_other_negative_body_steps + S ff_i_ph_hpl_sph_hsc_all_precision_result_other_negative_body_steps = l) -> exists ff_coefficient_ph_hpl_sph_hsc_all_precision_result_other_negative_body_steps ff_previous_ph_hpl_sph_hsc_all_precision_result_other_negative_body_steps ff_current_ph_hpl_sph_hsc_all_precision_result_other_negative_body_steps. ((((exists fs_h_ph_hpl_sph_hsc_all_precision_result_other_negative_body_steps_coefficient. fs_h_ph_hpl_sph_hsc_all_precision_result_other_negative_body_steps_coefficient + S (ff_coefficient_ph_hpl_sph_hsc_all_precision_result_other_negative_body_steps) = S ((S (ff_i_ph_hpl_sph_hsc_all_precision_result_other_negative_body_steps)) * nc)) /\ exists fs_q_ph_hpl_sph_hsc_all_precision_result_other_negative_body_steps_coefficient. nb = fs_q_ph_hpl_sph_hsc_all_precision_result_other_negative_body_steps_coefficient * S ((S (ff_i_ph_hpl_sph_hsc_all_precision_result_other_negative_body_steps)) * nc) + (ff_coefficient_ph_hpl_sph_hsc_all_precision_result_other_negative_body_steps))) /\ ((((exists fs_h_ph_hpl_sph_hsc_all_precision_result_other_negative_body_steps_before. fs_h_ph_hpl_sph_hsc_all_precision_result_other_negative_body_steps_before + S (ff_previous_ph_hpl_sph_hsc_all_precision_result_other_negative_body_steps) = S ((S (ff_i_ph_hpl_sph_hsc_all_precision_result_other_negative_body_steps)) * ff_v_ph_hpl_sph_hsc_all_precision_result_other_negative)) /\ exists fs_q_ph_hpl_sph_hsc_all_precision_result_other_negative_body_steps_before. ff_u_ph_hpl_sph_hsc_all_precision_result_other_negative = fs_q_ph_hpl_sph_hsc_all_precision_result_other_negative_body_steps_before * S ((S (ff_i_ph_hpl_sph_hsc_all_precision_result_other_negative_body_steps)) * ff_v_ph_hpl_sph_hsc_all_precision_result_other_negative) + (ff_previous_ph_hpl_sph_hsc_all_precision_result_other_negative_body_steps))) /\ ((((exists fs_h_ph_hpl_sph_hsc_all_precision_result_other_negative_body_steps_after. fs_h_ph_hpl_sph_hsc_all_precision_result_other_negative_body_steps_after + S (ff_current_ph_hpl_sph_hsc_all_precision_result_other_negative_body_steps) = S ((S (S ff_i_ph_hpl_sph_hsc_all_precision_result_other_negative_body_steps)) * ff_v_ph_hpl_sph_hsc_all_precision_result_other_negative)) /\ exists fs_q_ph_hpl_sph_hsc_all_precision_result_other_negative_body_steps_after. ff_u_ph_hpl_sph_hsc_all_precision_result_other_negative = fs_q_ph_hpl_sph_hsc_all_precision_result_other_negative_body_steps_after * S ((S (S ff_i_ph_hpl_sph_hsc_all_precision_result_other_negative_body_steps)) * ff_v_ph_hpl_sph_hsc_all_precision_result_other_negative) + (ff_current_ph_hpl_sph_hsc_all_precision_result_other_negative_body_steps))) /\ ff_current_ph_hpl_sph_hsc_all_precision_result_other_negative_body_steps = ff_previous_ph_hpl_sph_hsc_all_precision_result_other_negative_body_steps * hsc_competitor_all_precision_result + ff_coefficient_ph_hpl_sph_hsc_all_precision_result_other_negative_body_steps)))))) /\ (exists hgcrt_mod_left_hpl_hsc_all_precision_result_other hgcrt_mod_right_hpl_hsc_all_precision_result_other. sph_positive_hsc_all_precision_result_other + hsc_modulus_all_precision_result * hgcrt_mod_left_hpl_hsc_all_precision_result_other = sph_negative_hsc_all_precision_result_other + hsc_modulus_all_precision_result * hgcrt_mod_right_hpl_hsc_all_precision_result_other))))))) -> hsc_competitor_all_precision_result = hsc_root_all_precision_result)))