beta_signed_horner_root_value_derivative_exists

Exact expanded first-order arithmetic statement

forall pb pc nb nc a l M. (exists sph_positive_root sph_negative_root. ((exists ff_u_ph_hpl_sph_root_positive ff_v_ph_hpl_sph_root_positive. ((((exists fs_h_ph_hpl_sph_root_positive_body_start. fs_h_ph_hpl_sph_root_positive_body_start + S (0) = S ((S (0)) * ff_v_ph_hpl_sph_root_positive)) /\ exists fs_q_ph_hpl_sph_root_positive_body_start. ff_u_ph_hpl_sph_root_positive = fs_q_ph_hpl_sph_root_positive_body_start * S ((S (0)) * ff_v_ph_hpl_sph_root_positive) + (0))) /\ ((((exists fs_h_ph_hpl_sph_root_positive_body_terminal. fs_h_ph_hpl_sph_root_positive_body_terminal + S (sph_positive_root) = S ((S (l)) * ff_v_ph_hpl_sph_root_positive)) /\ exists fs_q_ph_hpl_sph_root_positive_body_terminal. ff_u_ph_hpl_sph_root_positive = fs_q_ph_hpl_sph_root_positive_body_terminal * S ((S (l)) * ff_v_ph_hpl_sph_root_positive) + (sph_positive_root))) /\ forall ff_i_ph_hpl_sph_root_positive_body_steps. (exists ph_bound_hpl_sph_root_positive_body_steps. ph_bound_hpl_sph_root_positive_body_steps + S ff_i_ph_hpl_sph_root_positive_body_steps = l) -> exists ff_coefficient_ph_hpl_sph_root_positive_body_steps ff_previous_ph_hpl_sph_root_positive_body_steps ff_current_ph_hpl_sph_root_positive_body_steps. ((((exists fs_h_ph_hpl_sph_root_positive_body_steps_coefficient. fs_h_ph_hpl_sph_root_positive_body_steps_coefficient + S (ff_coefficient_ph_hpl_sph_root_positive_body_steps) = S ((S (ff_i_ph_hpl_sph_root_positive_body_steps)) * pc)) /\ exists fs_q_ph_hpl_sph_root_positive_body_steps_coefficient. pb = fs_q_ph_hpl_sph_root_positive_body_steps_coefficient * S ((S (ff_i_ph_hpl_sph_root_positive_body_steps)) * pc) + (ff_coefficient_ph_hpl_sph_root_positive_body_steps))) /\ ((((exists fs_h_ph_hpl_sph_root_positive_body_steps_before. fs_h_ph_hpl_sph_root_positive_body_steps_before + S (ff_previous_ph_hpl_sph_root_positive_body_steps) = S ((S (ff_i_ph_hpl_sph_root_positive_body_steps)) * ff_v_ph_hpl_sph_root_positive)) /\ exists fs_q_ph_hpl_sph_root_positive_body_steps_before. ff_u_ph_hpl_sph_root_positive = fs_q_ph_hpl_sph_root_positive_body_steps_before * S ((S (ff_i_ph_hpl_sph_root_positive_body_steps)) * ff_v_ph_hpl_sph_root_positive) + (ff_previous_ph_hpl_sph_root_positive_body_steps))) /\ ((((exists fs_h_ph_hpl_sph_root_positive_body_steps_after. fs_h_ph_hpl_sph_root_positive_body_steps_after + S (ff_current_ph_hpl_sph_root_positive_body_steps) = S ((S (S ff_i_ph_hpl_sph_root_positive_body_steps)) * ff_v_ph_hpl_sph_root_positive)) /\ exists fs_q_ph_hpl_sph_root_positive_body_steps_after. ff_u_ph_hpl_sph_root_positive = fs_q_ph_hpl_sph_root_positive_body_steps_after * S ((S (S ff_i_ph_hpl_sph_root_positive_body_steps)) * ff_v_ph_hpl_sph_root_positive) + (ff_current_ph_hpl_sph_root_positive_body_steps))) /\ ff_current_ph_hpl_sph_root_positive_body_steps = ff_previous_ph_hpl_sph_root_positive_body_steps * a + ff_coefficient_ph_hpl_sph_root_positive_body_steps)))))) /\ ((exists ff_u_ph_hpl_sph_root_negative ff_v_ph_hpl_sph_root_negative. ((((exists fs_h_ph_hpl_sph_root_negative_body_start. fs_h_ph_hpl_sph_root_negative_body_start + S (0) = S ((S (0)) * ff_v_ph_hpl_sph_root_negative)) /\ exists fs_q_ph_hpl_sph_root_negative_body_start. ff_u_ph_hpl_sph_root_negative = fs_q_ph_hpl_sph_root_negative_body_start * S ((S (0)) * ff_v_ph_hpl_sph_root_negative) + (0))) /\ ((((exists fs_h_ph_hpl_sph_root_negative_body_terminal. fs_h_ph_hpl_sph_root_negative_body_terminal + S (sph_negative_root) = S ((S (l)) * ff_v_ph_hpl_sph_root_negative)) /\ exists fs_q_ph_hpl_sph_root_negative_body_terminal. ff_u_ph_hpl_sph_root_negative = fs_q_ph_hpl_sph_root_negative_body_terminal * S ((S (l)) * ff_v_ph_hpl_sph_root_negative) + (sph_negative_root))) /\ forall ff_i_ph_hpl_sph_root_negative_body_steps. (exists ph_bound_hpl_sph_root_negative_body_steps. ph_bound_hpl_sph_root_negative_body_steps + S ff_i_ph_hpl_sph_root_negative_body_steps = l) -> exists ff_coefficient_ph_hpl_sph_root_negative_body_steps ff_previous_ph_hpl_sph_root_negative_body_steps ff_current_ph_hpl_sph_root_negative_body_steps. ((((exists fs_h_ph_hpl_sph_root_negative_body_steps_coefficient. fs_h_ph_hpl_sph_root_negative_body_steps_coefficient + S (ff_coefficient_ph_hpl_sph_root_negative_body_steps) = S ((S (ff_i_ph_hpl_sph_root_negative_body_steps)) * nc)) /\ exists fs_q_ph_hpl_sph_root_negative_body_steps_coefficient. nb = fs_q_ph_hpl_sph_root_negative_body_steps_coefficient * S ((S (ff_i_ph_hpl_sph_root_negative_body_steps)) * nc) + (ff_coefficient_ph_hpl_sph_root_negative_body_steps))) /\ ((((exists fs_h_ph_hpl_sph_root_negative_body_steps_before. fs_h_ph_hpl_sph_root_negative_body_steps_before + S (ff_previous_ph_hpl_sph_root_negative_body_steps) = S ((S (ff_i_ph_hpl_sph_root_negative_body_steps)) * ff_v_ph_hpl_sph_root_negative)) /\ exists fs_q_ph_hpl_sph_root_negative_body_steps_before. ff_u_ph_hpl_sph_root_negative = fs_q_ph_hpl_sph_root_negative_body_steps_before * S ((S (ff_i_ph_hpl_sph_root_negative_body_steps)) * ff_v_ph_hpl_sph_root_negative) + (ff_previous_ph_hpl_sph_root_negative_body_steps))) /\ ((((exists fs_h_ph_hpl_sph_root_negative_body_steps_after. fs_h_ph_hpl_sph_root_negative_body_steps_after + S (ff_current_ph_hpl_sph_root_negative_body_steps) = S ((S (S ff_i_ph_hpl_sph_root_negative_body_steps)) * ff_v_ph_hpl_sph_root_negative)) /\ exists fs_q_ph_hpl_sph_root_negative_body_steps_after. ff_u_ph_hpl_sph_root_negative = fs_q_ph_hpl_sph_root_negative_body_steps_after * S ((S (S ff_i_ph_hpl_sph_root_negative_body_steps)) * ff_v_ph_hpl_sph_root_negative) + (ff_current_ph_hpl_sph_root_negative_body_steps))) /\ ff_current_ph_hpl_sph_root_negative_body_steps = ff_previous_ph_hpl_sph_root_negative_body_steps * a + ff_coefficient_ph_hpl_sph_root_negative_body_steps)))))) /\ (exists hgcrt_mod_left_hpl_root hgcrt_mod_right_hpl_root. sph_positive_root + M * hgcrt_mod_left_hpl_root = sph_negative_root + M * hgcrt_mod_right_hpl_root)))) -> exists vp dp vn dn. ((((exists ff_u_hd_hpl_sph_pair_positive ff_v_hd_hpl_sph_pair_positive ff_d_hd_hpl_sph_pair_positive ff_e_hd_hpl_sph_pair_positive. ((((((exists fs_h_ph_hd_hpl_sph_pair_positive_body_value_start. fs_h_ph_hd_hpl_sph_pair_positive_body_value_start + S (0) = S ((S (0)) * ff_v_hd_hpl_sph_pair_positive)) /\ exists fs_q_ph_hd_hpl_sph_pair_positive_body_value_start. ff_u_hd_hpl_sph_pair_positive = fs_q_ph_hd_hpl_sph_pair_positive_body_value_start * S ((S (0)) * ff_v_hd_hpl_sph_pair_positive) + (0))) /\ ((((exists fs_h_ph_hd_hpl_sph_pair_positive_body_value_terminal. fs_h_ph_hd_hpl_sph_pair_positive_body_value_terminal + S (vp) = S ((S (l)) * ff_v_hd_hpl_sph_pair_positive)) /\ exists fs_q_ph_hd_hpl_sph_pair_positive_body_value_terminal. ff_u_hd_hpl_sph_pair_positive = fs_q_ph_hd_hpl_sph_pair_positive_body_value_terminal * S ((S (l)) * ff_v_hd_hpl_sph_pair_positive) + (vp))) /\ forall ff_i_ph_hd_hpl_sph_pair_positive_body_value_steps. (exists ph_bound_hd_hpl_sph_pair_positive_body_value_steps. ph_bound_hd_hpl_sph_pair_positive_body_value_steps + S ff_i_ph_hd_hpl_sph_pair_positive_body_value_steps = l) -> exists ff_coefficient_ph_hd_hpl_sph_pair_positive_body_value_steps ff_previous_ph_hd_hpl_sph_pair_positive_body_value_steps ff_current_ph_hd_hpl_sph_pair_positive_body_value_steps. ((((exists fs_h_ph_hd_hpl_sph_pair_positive_body_value_steps_coefficient. fs_h_ph_hd_hpl_sph_pair_positive_body_value_steps_coefficient + S (ff_coefficient_ph_hd_hpl_sph_pair_positive_body_value_steps) = S ((S (ff_i_ph_hd_hpl_sph_pair_positive_body_value_steps)) * pc)) /\ exists fs_q_ph_hd_hpl_sph_pair_positive_body_value_steps_coefficient. pb = fs_q_ph_hd_hpl_sph_pair_positive_body_value_steps_coefficient * S ((S (ff_i_ph_hd_hpl_sph_pair_positive_body_value_steps)) * pc) + (ff_coefficient_ph_hd_hpl_sph_pair_positive_body_value_steps))) /\ ((((exists fs_h_ph_hd_hpl_sph_pair_positive_body_value_steps_before. fs_h_ph_hd_hpl_sph_pair_positive_body_value_steps_before + S (ff_previous_ph_hd_hpl_sph_pair_positive_body_value_steps) = S ((S (ff_i_ph_hd_hpl_sph_pair_positive_body_value_steps)) * ff_v_hd_hpl_sph_pair_positive)) /\ exists fs_q_ph_hd_hpl_sph_pair_positive_body_value_steps_before. ff_u_hd_hpl_sph_pair_positive = fs_q_ph_hd_hpl_sph_pair_positive_body_value_steps_before * S ((S (ff_i_ph_hd_hpl_sph_pair_positive_body_value_steps)) * ff_v_hd_hpl_sph_pair_positive) + (ff_previous_ph_hd_hpl_sph_pair_positive_body_value_steps))) /\ ((((exists fs_h_ph_hd_hpl_sph_pair_positive_body_value_steps_after. fs_h_ph_hd_hpl_sph_pair_positive_body_value_steps_after + S (ff_current_ph_hd_hpl_sph_pair_positive_body_value_steps) = S ((S (S ff_i_ph_hd_hpl_sph_pair_positive_body_value_steps)) * ff_v_hd_hpl_sph_pair_positive)) /\ exists fs_q_ph_hd_hpl_sph_pair_positive_body_value_steps_after. ff_u_hd_hpl_sph_pair_positive = fs_q_ph_hd_hpl_sph_pair_positive_body_value_steps_after * S ((S (S ff_i_ph_hd_hpl_sph_pair_positive_body_value_steps)) * ff_v_hd_hpl_sph_pair_positive) + (ff_current_ph_hd_hpl_sph_pair_positive_body_value_steps))) /\ ff_current_ph_hd_hpl_sph_pair_positive_body_value_steps = ff_previous_ph_hd_hpl_sph_pair_positive_body_value_steps * a + ff_coefficient_ph_hd_hpl_sph_pair_positive_body_value_steps)))))) /\ (((((exists fs_h_ph_hd_hpl_sph_pair_positive_body_derivative_start. fs_h_ph_hd_hpl_sph_pair_positive_body_derivative_start + S (0) = S ((S (0)) * ff_e_hd_hpl_sph_pair_positive)) /\ exists fs_q_ph_hd_hpl_sph_pair_positive_body_derivative_start. ff_d_hd_hpl_sph_pair_positive = fs_q_ph_hd_hpl_sph_pair_positive_body_derivative_start * S ((S (0)) * ff_e_hd_hpl_sph_pair_positive) + (0))) /\ ((((exists fs_h_ph_hd_hpl_sph_pair_positive_body_derivative_terminal. fs_h_ph_hd_hpl_sph_pair_positive_body_derivative_terminal + S (dp) = S ((S (l)) * ff_e_hd_hpl_sph_pair_positive)) /\ exists fs_q_ph_hd_hpl_sph_pair_positive_body_derivative_terminal. ff_d_hd_hpl_sph_pair_positive = fs_q_ph_hd_hpl_sph_pair_positive_body_derivative_terminal * S ((S (l)) * ff_e_hd_hpl_sph_pair_positive) + (dp))) /\ forall ff_i_ph_hd_hpl_sph_pair_positive_body_derivative_steps. (exists ph_bound_hd_hpl_sph_pair_positive_body_derivative_steps. ph_bound_hd_hpl_sph_pair_positive_body_derivative_steps + S ff_i_ph_hd_hpl_sph_pair_positive_body_derivative_steps = l) -> exists ff_coefficient_ph_hd_hpl_sph_pair_positive_body_derivative_steps ff_previous_ph_hd_hpl_sph_pair_positive_body_derivative_steps ff_current_ph_hd_hpl_sph_pair_positive_body_derivative_steps. ((((exists fs_h_ph_hd_hpl_sph_pair_positive_body_derivative_steps_coefficient. fs_h_ph_hd_hpl_sph_pair_positive_body_derivative_steps_coefficient + S (ff_coefficient_ph_hd_hpl_sph_pair_positive_body_derivative_steps) = S ((S (ff_i_ph_hd_hpl_sph_pair_positive_body_derivative_steps)) * ff_v_hd_hpl_sph_pair_positive)) /\ exists fs_q_ph_hd_hpl_sph_pair_positive_body_derivative_steps_coefficient. ff_u_hd_hpl_sph_pair_positive = fs_q_ph_hd_hpl_sph_pair_positive_body_derivative_steps_coefficient * S ((S (ff_i_ph_hd_hpl_sph_pair_positive_body_derivative_steps)) * ff_v_hd_hpl_sph_pair_positive) + (ff_coefficient_ph_hd_hpl_sph_pair_positive_body_derivative_steps))) /\ ((((exists fs_h_ph_hd_hpl_sph_pair_positive_body_derivative_steps_before. fs_h_ph_hd_hpl_sph_pair_positive_body_derivative_steps_before + S (ff_previous_ph_hd_hpl_sph_pair_positive_body_derivative_steps) = S ((S (ff_i_ph_hd_hpl_sph_pair_positive_body_derivative_steps)) * ff_e_hd_hpl_sph_pair_positive)) /\ exists fs_q_ph_hd_hpl_sph_pair_positive_body_derivative_steps_before. ff_d_hd_hpl_sph_pair_positive = fs_q_ph_hd_hpl_sph_pair_positive_body_derivative_steps_before * S ((S (ff_i_ph_hd_hpl_sph_pair_positive_body_derivative_steps)) * ff_e_hd_hpl_sph_pair_positive) + (ff_previous_ph_hd_hpl_sph_pair_positive_body_derivative_steps))) /\ ((((exists fs_h_ph_hd_hpl_sph_pair_positive_body_derivative_steps_after. fs_h_ph_hd_hpl_sph_pair_positive_body_derivative_steps_after + S (ff_current_ph_hd_hpl_sph_pair_positive_body_derivative_steps) = S ((S (S ff_i_ph_hd_hpl_sph_pair_positive_body_derivative_steps)) * ff_e_hd_hpl_sph_pair_positive)) /\ exists fs_q_ph_hd_hpl_sph_pair_positive_body_derivative_steps_after. ff_d_hd_hpl_sph_pair_positive = fs_q_ph_hd_hpl_sph_pair_positive_body_derivative_steps_after * S ((S (S ff_i_ph_hd_hpl_sph_pair_positive_body_derivative_steps)) * ff_e_hd_hpl_sph_pair_positive) + (ff_current_ph_hd_hpl_sph_pair_positive_body_derivative_steps))) /\ ff_current_ph_hd_hpl_sph_pair_positive_body_derivative_steps = ff_previous_ph_hd_hpl_sph_pair_positive_body_derivative_steps * a + ff_coefficient_ph_hd_hpl_sph_pair_positive_body_derivative_steps)))))))) /\ (exists ff_u_hd_hpl_sph_pair_negative ff_v_hd_hpl_sph_pair_negative ff_d_hd_hpl_sph_pair_negative ff_e_hd_hpl_sph_pair_negative. ((((((exists fs_h_ph_hd_hpl_sph_pair_negative_body_value_start. fs_h_ph_hd_hpl_sph_pair_negative_body_value_start + S (0) = S ((S (0)) * ff_v_hd_hpl_sph_pair_negative)) /\ exists fs_q_ph_hd_hpl_sph_pair_negative_body_value_start. ff_u_hd_hpl_sph_pair_negative = fs_q_ph_hd_hpl_sph_pair_negative_body_value_start * S ((S (0)) * ff_v_hd_hpl_sph_pair_negative) + (0))) /\ ((((exists fs_h_ph_hd_hpl_sph_pair_negative_body_value_terminal. fs_h_ph_hd_hpl_sph_pair_negative_body_value_terminal + S (vn) = S ((S (l)) * ff_v_hd_hpl_sph_pair_negative)) /\ exists fs_q_ph_hd_hpl_sph_pair_negative_body_value_terminal. ff_u_hd_hpl_sph_pair_negative = fs_q_ph_hd_hpl_sph_pair_negative_body_value_terminal * S ((S (l)) * ff_v_hd_hpl_sph_pair_negative) + (vn))) /\ forall ff_i_ph_hd_hpl_sph_pair_negative_body_value_steps. (exists ph_bound_hd_hpl_sph_pair_negative_body_value_steps. ph_bound_hd_hpl_sph_pair_negative_body_value_steps + S ff_i_ph_hd_hpl_sph_pair_negative_body_value_steps = l) -> exists ff_coefficient_ph_hd_hpl_sph_pair_negative_body_value_steps ff_previous_ph_hd_hpl_sph_pair_negative_body_value_steps ff_current_ph_hd_hpl_sph_pair_negative_body_value_steps. ((((exists fs_h_ph_hd_hpl_sph_pair_negative_body_value_steps_coefficient. fs_h_ph_hd_hpl_sph_pair_negative_body_value_steps_coefficient + S (ff_coefficient_ph_hd_hpl_sph_pair_negative_body_value_steps) = S ((S (ff_i_ph_hd_hpl_sph_pair_negative_body_value_steps)) * nc)) /\ exists fs_q_ph_hd_hpl_sph_pair_negative_body_value_steps_coefficient. nb = fs_q_ph_hd_hpl_sph_pair_negative_body_value_steps_coefficient * S ((S (ff_i_ph_hd_hpl_sph_pair_negative_body_value_steps)) * nc) + (ff_coefficient_ph_hd_hpl_sph_pair_negative_body_value_steps))) /\ ((((exists fs_h_ph_hd_hpl_sph_pair_negative_body_value_steps_before. fs_h_ph_hd_hpl_sph_pair_negative_body_value_steps_before + S (ff_previous_ph_hd_hpl_sph_pair_negative_body_value_steps) = S ((S (ff_i_ph_hd_hpl_sph_pair_negative_body_value_steps)) * ff_v_hd_hpl_sph_pair_negative)) /\ exists fs_q_ph_hd_hpl_sph_pair_negative_body_value_steps_before. ff_u_hd_hpl_sph_pair_negative = fs_q_ph_hd_hpl_sph_pair_negative_body_value_steps_before * S ((S (ff_i_ph_hd_hpl_sph_pair_negative_body_value_steps)) * ff_v_hd_hpl_sph_pair_negative) + (ff_previous_ph_hd_hpl_sph_pair_negative_body_value_steps))) /\ ((((exists fs_h_ph_hd_hpl_sph_pair_negative_body_value_steps_after. fs_h_ph_hd_hpl_sph_pair_negative_body_value_steps_after + S (ff_current_ph_hd_hpl_sph_pair_negative_body_value_steps) = S ((S (S ff_i_ph_hd_hpl_sph_pair_negative_body_value_steps)) * ff_v_hd_hpl_sph_pair_negative)) /\ exists fs_q_ph_hd_hpl_sph_pair_negative_body_value_steps_after. ff_u_hd_hpl_sph_pair_negative = fs_q_ph_hd_hpl_sph_pair_negative_body_value_steps_after * S ((S (S ff_i_ph_hd_hpl_sph_pair_negative_body_value_steps)) * ff_v_hd_hpl_sph_pair_negative) + (ff_current_ph_hd_hpl_sph_pair_negative_body_value_steps))) /\ ff_current_ph_hd_hpl_sph_pair_negative_body_value_steps = ff_previous_ph_hd_hpl_sph_pair_negative_body_value_steps * a + ff_coefficient_ph_hd_hpl_sph_pair_negative_body_value_steps)))))) /\ (((((exists fs_h_ph_hd_hpl_sph_pair_negative_body_derivative_start. fs_h_ph_hd_hpl_sph_pair_negative_body_derivative_start + S (0) = S ((S (0)) * ff_e_hd_hpl_sph_pair_negative)) /\ exists fs_q_ph_hd_hpl_sph_pair_negative_body_derivative_start. ff_d_hd_hpl_sph_pair_negative = fs_q_ph_hd_hpl_sph_pair_negative_body_derivative_start * S ((S (0)) * ff_e_hd_hpl_sph_pair_negative) + (0))) /\ ((((exists fs_h_ph_hd_hpl_sph_pair_negative_body_derivative_terminal. fs_h_ph_hd_hpl_sph_pair_negative_body_derivative_terminal + S (dn) = S ((S (l)) * ff_e_hd_hpl_sph_pair_negative)) /\ exists fs_q_ph_hd_hpl_sph_pair_negative_body_derivative_terminal. ff_d_hd_hpl_sph_pair_negative = fs_q_ph_hd_hpl_sph_pair_negative_body_derivative_terminal * S ((S (l)) * ff_e_hd_hpl_sph_pair_negative) + (dn))) /\ forall ff_i_ph_hd_hpl_sph_pair_negative_body_derivative_steps. (exists ph_bound_hd_hpl_sph_pair_negative_body_derivative_steps. ph_bound_hd_hpl_sph_pair_negative_body_derivative_steps + S ff_i_ph_hd_hpl_sph_pair_negative_body_derivative_steps = l) -> exists ff_coefficient_ph_hd_hpl_sph_pair_negative_body_derivative_steps ff_previous_ph_hd_hpl_sph_pair_negative_body_derivative_steps ff_current_ph_hd_hpl_sph_pair_negative_body_derivative_steps. ((((exists fs_h_ph_hd_hpl_sph_pair_negative_body_derivative_steps_coefficient. fs_h_ph_hd_hpl_sph_pair_negative_body_derivative_steps_coefficient + S (ff_coefficient_ph_hd_hpl_sph_pair_negative_body_derivative_steps) = S ((S (ff_i_ph_hd_hpl_sph_pair_negative_body_derivative_steps)) * ff_v_hd_hpl_sph_pair_negative)) /\ exists fs_q_ph_hd_hpl_sph_pair_negative_body_derivative_steps_coefficient. ff_u_hd_hpl_sph_pair_negative = fs_q_ph_hd_hpl_sph_pair_negative_body_derivative_steps_coefficient * S ((S (ff_i_ph_hd_hpl_sph_pair_negative_body_derivative_steps)) * ff_v_hd_hpl_sph_pair_negative) + (ff_coefficient_ph_hd_hpl_sph_pair_negative_body_derivative_steps))) /\ ((((exists fs_h_ph_hd_hpl_sph_pair_negative_body_derivative_steps_before. fs_h_ph_hd_hpl_sph_pair_negative_body_derivative_steps_before + S (ff_previous_ph_hd_hpl_sph_pair_negative_body_derivative_steps) = S ((S (ff_i_ph_hd_hpl_sph_pair_negative_body_derivative_steps)) * ff_e_hd_hpl_sph_pair_negative)) /\ exists fs_q_ph_hd_hpl_sph_pair_negative_body_derivative_steps_before. ff_d_hd_hpl_sph_pair_negative = fs_q_ph_hd_hpl_sph_pair_negative_body_derivative_steps_before * S ((S (ff_i_ph_hd_hpl_sph_pair_negative_body_derivative_steps)) * ff_e_hd_hpl_sph_pair_negative) + (ff_previous_ph_hd_hpl_sph_pair_negative_body_derivative_steps))) /\ ((((exists fs_h_ph_hd_hpl_sph_pair_negative_body_derivative_steps_after. fs_h_ph_hd_hpl_sph_pair_negative_body_derivative_steps_after + S (ff_current_ph_hd_hpl_sph_pair_negative_body_derivative_steps) = S ((S (S ff_i_ph_hd_hpl_sph_pair_negative_body_derivative_steps)) * ff_e_hd_hpl_sph_pair_negative)) /\ exists fs_q_ph_hd_hpl_sph_pair_negative_body_derivative_steps_after. ff_d_hd_hpl_sph_pair_negative = fs_q_ph_hd_hpl_sph_pair_negative_body_derivative_steps_after * S ((S (S ff_i_ph_hd_hpl_sph_pair_negative_body_derivative_steps)) * ff_e_hd_hpl_sph_pair_negative) + (ff_current_ph_hd_hpl_sph_pair_negative_body_derivative_steps))) /\ ff_current_ph_hd_hpl_sph_pair_negative_body_derivative_steps = ff_previous_ph_hd_hpl_sph_pair_negative_body_derivative_steps * a + ff_coefficient_ph_hd_hpl_sph_pair_negative_body_derivative_steps)))))))))) /\ (exists hgcrt_mod_left_hpl_mod hgcrt_mod_right_hpl_mod. vp + M * hgcrt_mod_left_hpl_mod = vn + M * hgcrt_mod_right_hpl_mod))