beta_horner_derivative_only_exists_unique

Exact expanded first-order arithmetic statement

forall b c t l. exists z. ((exists ff_value_hd_only. (exists ff_u_hd_only_pair ff_v_hd_only_pair ff_d_hd_only_pair ff_e_hd_only_pair. ((((((exists fs_h_ph_hd_only_pair_body_value_start. fs_h_ph_hd_only_pair_body_value_start + S (0) = S ((S (0)) * ff_v_hd_only_pair)) /\ exists fs_q_ph_hd_only_pair_body_value_start. ff_u_hd_only_pair = fs_q_ph_hd_only_pair_body_value_start * S ((S (0)) * ff_v_hd_only_pair) + (0))) /\ ((((exists fs_h_ph_hd_only_pair_body_value_terminal. fs_h_ph_hd_only_pair_body_value_terminal + S (ff_value_hd_only) = S ((S (l)) * ff_v_hd_only_pair)) /\ exists fs_q_ph_hd_only_pair_body_value_terminal. ff_u_hd_only_pair = fs_q_ph_hd_only_pair_body_value_terminal * S ((S (l)) * ff_v_hd_only_pair) + (ff_value_hd_only))) /\ forall ff_i_ph_hd_only_pair_body_value_steps. (exists ph_bound_hd_only_pair_body_value_steps. ph_bound_hd_only_pair_body_value_steps + S ff_i_ph_hd_only_pair_body_value_steps = l) -> exists ff_coefficient_ph_hd_only_pair_body_value_steps ff_previous_ph_hd_only_pair_body_value_steps ff_current_ph_hd_only_pair_body_value_steps. ((((exists fs_h_ph_hd_only_pair_body_value_steps_coefficient. fs_h_ph_hd_only_pair_body_value_steps_coefficient + S (ff_coefficient_ph_hd_only_pair_body_value_steps) = S ((S (ff_i_ph_hd_only_pair_body_value_steps)) * c)) /\ exists fs_q_ph_hd_only_pair_body_value_steps_coefficient. b = fs_q_ph_hd_only_pair_body_value_steps_coefficient * S ((S (ff_i_ph_hd_only_pair_body_value_steps)) * c) + (ff_coefficient_ph_hd_only_pair_body_value_steps))) /\ ((((exists fs_h_ph_hd_only_pair_body_value_steps_before. fs_h_ph_hd_only_pair_body_value_steps_before + S (ff_previous_ph_hd_only_pair_body_value_steps) = S ((S (ff_i_ph_hd_only_pair_body_value_steps)) * ff_v_hd_only_pair)) /\ exists fs_q_ph_hd_only_pair_body_value_steps_before. ff_u_hd_only_pair = fs_q_ph_hd_only_pair_body_value_steps_before * S ((S (ff_i_ph_hd_only_pair_body_value_steps)) * ff_v_hd_only_pair) + (ff_previous_ph_hd_only_pair_body_value_steps))) /\ ((((exists fs_h_ph_hd_only_pair_body_value_steps_after. fs_h_ph_hd_only_pair_body_value_steps_after + S (ff_current_ph_hd_only_pair_body_value_steps) = S ((S (S ff_i_ph_hd_only_pair_body_value_steps)) * ff_v_hd_only_pair)) /\ exists fs_q_ph_hd_only_pair_body_value_steps_after. ff_u_hd_only_pair = fs_q_ph_hd_only_pair_body_value_steps_after * S ((S (S ff_i_ph_hd_only_pair_body_value_steps)) * ff_v_hd_only_pair) + (ff_current_ph_hd_only_pair_body_value_steps))) /\ ff_current_ph_hd_only_pair_body_value_steps = ff_previous_ph_hd_only_pair_body_value_steps * t + ff_coefficient_ph_hd_only_pair_body_value_steps)))))) /\ (((((exists fs_h_ph_hd_only_pair_body_derivative_start. fs_h_ph_hd_only_pair_body_derivative_start + S (0) = S ((S (0)) * ff_e_hd_only_pair)) /\ exists fs_q_ph_hd_only_pair_body_derivative_start. ff_d_hd_only_pair = fs_q_ph_hd_only_pair_body_derivative_start * S ((S (0)) * ff_e_hd_only_pair) + (0))) /\ ((((exists fs_h_ph_hd_only_pair_body_derivative_terminal. fs_h_ph_hd_only_pair_body_derivative_terminal + S (z) = S ((S (l)) * ff_e_hd_only_pair)) /\ exists fs_q_ph_hd_only_pair_body_derivative_terminal. ff_d_hd_only_pair = fs_q_ph_hd_only_pair_body_derivative_terminal * S ((S (l)) * ff_e_hd_only_pair) + (z))) /\ forall ff_i_ph_hd_only_pair_body_derivative_steps. (exists ph_bound_hd_only_pair_body_derivative_steps. ph_bound_hd_only_pair_body_derivative_steps + S ff_i_ph_hd_only_pair_body_derivative_steps = l) -> exists ff_coefficient_ph_hd_only_pair_body_derivative_steps ff_previous_ph_hd_only_pair_body_derivative_steps ff_current_ph_hd_only_pair_body_derivative_steps. ((((exists fs_h_ph_hd_only_pair_body_derivative_steps_coefficient. fs_h_ph_hd_only_pair_body_derivative_steps_coefficient + S (ff_coefficient_ph_hd_only_pair_body_derivative_steps) = S ((S (ff_i_ph_hd_only_pair_body_derivative_steps)) * ff_v_hd_only_pair)) /\ exists fs_q_ph_hd_only_pair_body_derivative_steps_coefficient. ff_u_hd_only_pair = fs_q_ph_hd_only_pair_body_derivative_steps_coefficient * S ((S (ff_i_ph_hd_only_pair_body_derivative_steps)) * ff_v_hd_only_pair) + (ff_coefficient_ph_hd_only_pair_body_derivative_steps))) /\ ((((exists fs_h_ph_hd_only_pair_body_derivative_steps_before. fs_h_ph_hd_only_pair_body_derivative_steps_before + S (ff_previous_ph_hd_only_pair_body_derivative_steps) = S ((S (ff_i_ph_hd_only_pair_body_derivative_steps)) * ff_e_hd_only_pair)) /\ exists fs_q_ph_hd_only_pair_body_derivative_steps_before. ff_d_hd_only_pair = fs_q_ph_hd_only_pair_body_derivative_steps_before * S ((S (ff_i_ph_hd_only_pair_body_derivative_steps)) * ff_e_hd_only_pair) + (ff_previous_ph_hd_only_pair_body_derivative_steps))) /\ ((((exists fs_h_ph_hd_only_pair_body_derivative_steps_after. fs_h_ph_hd_only_pair_body_derivative_steps_after + S (ff_current_ph_hd_only_pair_body_derivative_steps) = S ((S (S ff_i_ph_hd_only_pair_body_derivative_steps)) * ff_e_hd_only_pair)) /\ exists fs_q_ph_hd_only_pair_body_derivative_steps_after. ff_d_hd_only_pair = fs_q_ph_hd_only_pair_body_derivative_steps_after * S ((S (S ff_i_ph_hd_only_pair_body_derivative_steps)) * ff_e_hd_only_pair) + (ff_current_ph_hd_only_pair_body_derivative_steps))) /\ ff_current_ph_hd_only_pair_body_derivative_steps = ff_previous_ph_hd_only_pair_body_derivative_steps * t + ff_coefficient_ph_hd_only_pair_body_derivative_steps))))))))) /\ forall w. (exists ff_value_hd_only_other. (exists ff_u_hd_only_other_pair ff_v_hd_only_other_pair ff_d_hd_only_other_pair ff_e_hd_only_other_pair. ((((((exists fs_h_ph_hd_only_other_pair_body_value_start. fs_h_ph_hd_only_other_pair_body_value_start + S (0) = S ((S (0)) * ff_v_hd_only_other_pair)) /\ exists fs_q_ph_hd_only_other_pair_body_value_start. ff_u_hd_only_other_pair = fs_q_ph_hd_only_other_pair_body_value_start * S ((S (0)) * ff_v_hd_only_other_pair) + (0))) /\ ((((exists fs_h_ph_hd_only_other_pair_body_value_terminal. fs_h_ph_hd_only_other_pair_body_value_terminal + S (ff_value_hd_only_other) = S ((S (l)) * ff_v_hd_only_other_pair)) /\ exists fs_q_ph_hd_only_other_pair_body_value_terminal. ff_u_hd_only_other_pair = fs_q_ph_hd_only_other_pair_body_value_terminal * S ((S (l)) * ff_v_hd_only_other_pair) + (ff_value_hd_only_other))) /\ forall ff_i_ph_hd_only_other_pair_body_value_steps. (exists ph_bound_hd_only_other_pair_body_value_steps. ph_bound_hd_only_other_pair_body_value_steps + S ff_i_ph_hd_only_other_pair_body_value_steps = l) -> exists ff_coefficient_ph_hd_only_other_pair_body_value_steps ff_previous_ph_hd_only_other_pair_body_value_steps ff_current_ph_hd_only_other_pair_body_value_steps. ((((exists fs_h_ph_hd_only_other_pair_body_value_steps_coefficient. fs_h_ph_hd_only_other_pair_body_value_steps_coefficient + S (ff_coefficient_ph_hd_only_other_pair_body_value_steps) = S ((S (ff_i_ph_hd_only_other_pair_body_value_steps)) * c)) /\ exists fs_q_ph_hd_only_other_pair_body_value_steps_coefficient. b = fs_q_ph_hd_only_other_pair_body_value_steps_coefficient * S ((S (ff_i_ph_hd_only_other_pair_body_value_steps)) * c) + (ff_coefficient_ph_hd_only_other_pair_body_value_steps))) /\ ((((exists fs_h_ph_hd_only_other_pair_body_value_steps_before. fs_h_ph_hd_only_other_pair_body_value_steps_before + S (ff_previous_ph_hd_only_other_pair_body_value_steps) = S ((S (ff_i_ph_hd_only_other_pair_body_value_steps)) * ff_v_hd_only_other_pair)) /\ exists fs_q_ph_hd_only_other_pair_body_value_steps_before. ff_u_hd_only_other_pair = fs_q_ph_hd_only_other_pair_body_value_steps_before * S ((S (ff_i_ph_hd_only_other_pair_body_value_steps)) * ff_v_hd_only_other_pair) + (ff_previous_ph_hd_only_other_pair_body_value_steps))) /\ ((((exists fs_h_ph_hd_only_other_pair_body_value_steps_after. fs_h_ph_hd_only_other_pair_body_value_steps_after + S (ff_current_ph_hd_only_other_pair_body_value_steps) = S ((S (S ff_i_ph_hd_only_other_pair_body_value_steps)) * ff_v_hd_only_other_pair)) /\ exists fs_q_ph_hd_only_other_pair_body_value_steps_after. ff_u_hd_only_other_pair = fs_q_ph_hd_only_other_pair_body_value_steps_after * S ((S (S ff_i_ph_hd_only_other_pair_body_value_steps)) * ff_v_hd_only_other_pair) + (ff_current_ph_hd_only_other_pair_body_value_steps))) /\ ff_current_ph_hd_only_other_pair_body_value_steps = ff_previous_ph_hd_only_other_pair_body_value_steps * t + ff_coefficient_ph_hd_only_other_pair_body_value_steps)))))) /\ (((((exists fs_h_ph_hd_only_other_pair_body_derivative_start. fs_h_ph_hd_only_other_pair_body_derivative_start + S (0) = S ((S (0)) * ff_e_hd_only_other_pair)) /\ exists fs_q_ph_hd_only_other_pair_body_derivative_start. ff_d_hd_only_other_pair = fs_q_ph_hd_only_other_pair_body_derivative_start * S ((S (0)) * ff_e_hd_only_other_pair) + (0))) /\ ((((exists fs_h_ph_hd_only_other_pair_body_derivative_terminal. fs_h_ph_hd_only_other_pair_body_derivative_terminal + S (w) = S ((S (l)) * ff_e_hd_only_other_pair)) /\ exists fs_q_ph_hd_only_other_pair_body_derivative_terminal. ff_d_hd_only_other_pair = fs_q_ph_hd_only_other_pair_body_derivative_terminal * S ((S (l)) * ff_e_hd_only_other_pair) + (w))) /\ forall ff_i_ph_hd_only_other_pair_body_derivative_steps. (exists ph_bound_hd_only_other_pair_body_derivative_steps. ph_bound_hd_only_other_pair_body_derivative_steps + S ff_i_ph_hd_only_other_pair_body_derivative_steps = l) -> exists ff_coefficient_ph_hd_only_other_pair_body_derivative_steps ff_previous_ph_hd_only_other_pair_body_derivative_steps ff_current_ph_hd_only_other_pair_body_derivative_steps. ((((exists fs_h_ph_hd_only_other_pair_body_derivative_steps_coefficient. fs_h_ph_hd_only_other_pair_body_derivative_steps_coefficient + S (ff_coefficient_ph_hd_only_other_pair_body_derivative_steps) = S ((S (ff_i_ph_hd_only_other_pair_body_derivative_steps)) * ff_v_hd_only_other_pair)) /\ exists fs_q_ph_hd_only_other_pair_body_derivative_steps_coefficient. ff_u_hd_only_other_pair = fs_q_ph_hd_only_other_pair_body_derivative_steps_coefficient * S ((S (ff_i_ph_hd_only_other_pair_body_derivative_steps)) * ff_v_hd_only_other_pair) + (ff_coefficient_ph_hd_only_other_pair_body_derivative_steps))) /\ ((((exists fs_h_ph_hd_only_other_pair_body_derivative_steps_before. fs_h_ph_hd_only_other_pair_body_derivative_steps_before + S (ff_previous_ph_hd_only_other_pair_body_derivative_steps) = S ((S (ff_i_ph_hd_only_other_pair_body_derivative_steps)) * ff_e_hd_only_other_pair)) /\ exists fs_q_ph_hd_only_other_pair_body_derivative_steps_before. ff_d_hd_only_other_pair = fs_q_ph_hd_only_other_pair_body_derivative_steps_before * S ((S (ff_i_ph_hd_only_other_pair_body_derivative_steps)) * ff_e_hd_only_other_pair) + (ff_previous_ph_hd_only_other_pair_body_derivative_steps))) /\ ((((exists fs_h_ph_hd_only_other_pair_body_derivative_steps_after. fs_h_ph_hd_only_other_pair_body_derivative_steps_after + S (ff_current_ph_hd_only_other_pair_body_derivative_steps) = S ((S (S ff_i_ph_hd_only_other_pair_body_derivative_steps)) * ff_e_hd_only_other_pair)) /\ exists fs_q_ph_hd_only_other_pair_body_derivative_steps_after. ff_d_hd_only_other_pair = fs_q_ph_hd_only_other_pair_body_derivative_steps_after * S ((S (S ff_i_ph_hd_only_other_pair_body_derivative_steps)) * ff_e_hd_only_other_pair) + (ff_current_ph_hd_only_other_pair_body_derivative_steps))) /\ ff_current_ph_hd_only_other_pair_body_derivative_steps = ff_previous_ph_hd_only_other_pair_body_derivative_steps * t + ff_coefficient_ph_hd_only_other_pair_body_derivative_steps))))))))) -> z = w)