HornerDerivativeTrace(b,c,t,l,u,v,d,e)

((((((exists fs_h_hd_research_value_start. fs_h_hd_research_value_start + S (0) = S ((S (0)) * v)) /\ exists fs_q_hd_research_value_start. u = fs_q_hd_research_value_start * S ((S (0)) * v) + (0))) /\ (forall ff_i_ph_hd_research_value_steps. (exists ph_bound_hd_research_value_steps. ph_bound_hd_research_value_steps + S ff_i_ph_hd_research_value_steps = l) -> exists ff_coefficient_ph_hd_research_value_steps ff_previous_ph_hd_research_value_steps ff_current_ph_hd_research_value_steps. ((((exists fs_h_ph_hd_research_value_steps_coefficient. fs_h_ph_hd_research_value_steps_coefficient + S (ff_coefficient_ph_hd_research_value_steps) = S ((S (ff_i_ph_hd_research_value_steps)) * c)) /\ exists fs_q_ph_hd_research_value_steps_coefficient. b = fs_q_ph_hd_research_value_steps_coefficient * S ((S (ff_i_ph_hd_research_value_steps)) * c) + (ff_coefficient_ph_hd_research_value_steps))) /\ ((((exists fs_h_ph_hd_research_value_steps_before. fs_h_ph_hd_research_value_steps_before + S (ff_previous_ph_hd_research_value_steps) = S ((S (ff_i_ph_hd_research_value_steps)) * v)) /\ exists fs_q_ph_hd_research_value_steps_before. u = fs_q_ph_hd_research_value_steps_before * S ((S (ff_i_ph_hd_research_value_steps)) * v) + (ff_previous_ph_hd_research_value_steps))) /\ ((((exists fs_h_ph_hd_research_value_steps_after. fs_h_ph_hd_research_value_steps_after + S (ff_current_ph_hd_research_value_steps) = S ((S (S ff_i_ph_hd_research_value_steps)) * v)) /\ exists fs_q_ph_hd_research_value_steps_after. u = fs_q_ph_hd_research_value_steps_after * S ((S (S ff_i_ph_hd_research_value_steps)) * v) + (ff_current_ph_hd_research_value_steps))) /\ ff_current_ph_hd_research_value_steps = ff_previous_ph_hd_research_value_steps * t + ff_coefficient_ph_hd_research_value_steps)))))) /\ (((((exists fs_h_hd_research_derivative_start. fs_h_hd_research_derivative_start + S (0) = S ((S (0)) * e)) /\ exists fs_q_hd_research_derivative_start. d = fs_q_hd_research_derivative_start * S ((S (0)) * e) + (0))) /\ (forall ff_i_ph_hd_research_derivative_steps. (exists ph_bound_hd_research_derivative_steps. ph_bound_hd_research_derivative_steps + S ff_i_ph_hd_research_derivative_steps = l) -> exists ff_coefficient_ph_hd_research_derivative_steps ff_previous_ph_hd_research_derivative_steps ff_current_ph_hd_research_derivative_steps. ((((exists fs_h_ph_hd_research_derivative_steps_coefficient. fs_h_ph_hd_research_derivative_steps_coefficient + S (ff_coefficient_ph_hd_research_derivative_steps) = S ((S (ff_i_ph_hd_research_derivative_steps)) * v)) /\ exists fs_q_ph_hd_research_derivative_steps_coefficient. u = fs_q_ph_hd_research_derivative_steps_coefficient * S ((S (ff_i_ph_hd_research_derivative_steps)) * v) + (ff_coefficient_ph_hd_research_derivative_steps))) /\ ((((exists fs_h_ph_hd_research_derivative_steps_before. fs_h_ph_hd_research_derivative_steps_before + S (ff_previous_ph_hd_research_derivative_steps) = S ((S (ff_i_ph_hd_research_derivative_steps)) * e)) /\ exists fs_q_ph_hd_research_derivative_steps_before. d = fs_q_ph_hd_research_derivative_steps_before * S ((S (ff_i_ph_hd_research_derivative_steps)) * e) + (ff_previous_ph_hd_research_derivative_steps))) /\ ((((exists fs_h_ph_hd_research_derivative_steps_after. fs_h_ph_hd_research_derivative_steps_after + S (ff_current_ph_hd_research_derivative_steps) = S ((S (S ff_i_ph_hd_research_derivative_steps)) * e)) /\ exists fs_q_ph_hd_research_derivative_steps_after. d = fs_q_ph_hd_research_derivative_steps_after * S ((S (S ff_i_ph_hd_research_derivative_steps)) * e) + (ff_current_ph_hd_research_derivative_steps))) /\ ff_current_ph_hd_research_derivative_steps = ff_previous_ph_hd_research_derivative_steps * t + ff_coefficient_ph_hd_research_derivative_steps)))))))

The unchanged native kernel never receives this surface symbol. Binder-safe expansion produces only its existing first-order syntax.