beta_horner_taylor_square_congruence

Exact expanded first-order arithmetic statement

forall b c t h l n d z. (exists ff_u_hd_pth_taylor_pair ff_v_hd_pth_taylor_pair ff_d_hd_pth_taylor_pair ff_e_hd_pth_taylor_pair. ((((((exists fs_h_ph_hd_pth_taylor_pair_body_value_start. fs_h_ph_hd_pth_taylor_pair_body_value_start + S (0) = S ((S (0)) * ff_v_hd_pth_taylor_pair)) /\ exists fs_q_ph_hd_pth_taylor_pair_body_value_start. ff_u_hd_pth_taylor_pair = fs_q_ph_hd_pth_taylor_pair_body_value_start * S ((S (0)) * ff_v_hd_pth_taylor_pair) + (0))) /\ ((((exists fs_h_ph_hd_pth_taylor_pair_body_value_terminal. fs_h_ph_hd_pth_taylor_pair_body_value_terminal + S (n) = S ((S (l)) * ff_v_hd_pth_taylor_pair)) /\ exists fs_q_ph_hd_pth_taylor_pair_body_value_terminal. ff_u_hd_pth_taylor_pair = fs_q_ph_hd_pth_taylor_pair_body_value_terminal * S ((S (l)) * ff_v_hd_pth_taylor_pair) + (n))) /\ forall ff_i_ph_hd_pth_taylor_pair_body_value_steps. (exists ph_bound_hd_pth_taylor_pair_body_value_steps. ph_bound_hd_pth_taylor_pair_body_value_steps + S ff_i_ph_hd_pth_taylor_pair_body_value_steps = l) -> exists ff_coefficient_ph_hd_pth_taylor_pair_body_value_steps ff_previous_ph_hd_pth_taylor_pair_body_value_steps ff_current_ph_hd_pth_taylor_pair_body_value_steps. ((((exists fs_h_ph_hd_pth_taylor_pair_body_value_steps_coefficient. fs_h_ph_hd_pth_taylor_pair_body_value_steps_coefficient + S (ff_coefficient_ph_hd_pth_taylor_pair_body_value_steps) = S ((S (ff_i_ph_hd_pth_taylor_pair_body_value_steps)) * c)) /\ exists fs_q_ph_hd_pth_taylor_pair_body_value_steps_coefficient. b = fs_q_ph_hd_pth_taylor_pair_body_value_steps_coefficient * S ((S (ff_i_ph_hd_pth_taylor_pair_body_value_steps)) * c) + (ff_coefficient_ph_hd_pth_taylor_pair_body_value_steps))) /\ ((((exists fs_h_ph_hd_pth_taylor_pair_body_value_steps_before. fs_h_ph_hd_pth_taylor_pair_body_value_steps_before + S (ff_previous_ph_hd_pth_taylor_pair_body_value_steps) = S ((S (ff_i_ph_hd_pth_taylor_pair_body_value_steps)) * ff_v_hd_pth_taylor_pair)) /\ exists fs_q_ph_hd_pth_taylor_pair_body_value_steps_before. ff_u_hd_pth_taylor_pair = fs_q_ph_hd_pth_taylor_pair_body_value_steps_before * S ((S (ff_i_ph_hd_pth_taylor_pair_body_value_steps)) * ff_v_hd_pth_taylor_pair) + (ff_previous_ph_hd_pth_taylor_pair_body_value_steps))) /\ ((((exists fs_h_ph_hd_pth_taylor_pair_body_value_steps_after. fs_h_ph_hd_pth_taylor_pair_body_value_steps_after + S (ff_current_ph_hd_pth_taylor_pair_body_value_steps) = S ((S (S ff_i_ph_hd_pth_taylor_pair_body_value_steps)) * ff_v_hd_pth_taylor_pair)) /\ exists fs_q_ph_hd_pth_taylor_pair_body_value_steps_after. ff_u_hd_pth_taylor_pair = fs_q_ph_hd_pth_taylor_pair_body_value_steps_after * S ((S (S ff_i_ph_hd_pth_taylor_pair_body_value_steps)) * ff_v_hd_pth_taylor_pair) + (ff_current_ph_hd_pth_taylor_pair_body_value_steps))) /\ ff_current_ph_hd_pth_taylor_pair_body_value_steps = ff_previous_ph_hd_pth_taylor_pair_body_value_steps * t + ff_coefficient_ph_hd_pth_taylor_pair_body_value_steps)))))) /\ (((((exists fs_h_ph_hd_pth_taylor_pair_body_derivative_start. fs_h_ph_hd_pth_taylor_pair_body_derivative_start + S (0) = S ((S (0)) * ff_e_hd_pth_taylor_pair)) /\ exists fs_q_ph_hd_pth_taylor_pair_body_derivative_start. ff_d_hd_pth_taylor_pair = fs_q_ph_hd_pth_taylor_pair_body_derivative_start * S ((S (0)) * ff_e_hd_pth_taylor_pair) + (0))) /\ ((((exists fs_h_ph_hd_pth_taylor_pair_body_derivative_terminal. fs_h_ph_hd_pth_taylor_pair_body_derivative_terminal + S (d) = S ((S (l)) * ff_e_hd_pth_taylor_pair)) /\ exists fs_q_ph_hd_pth_taylor_pair_body_derivative_terminal. ff_d_hd_pth_taylor_pair = fs_q_ph_hd_pth_taylor_pair_body_derivative_terminal * S ((S (l)) * ff_e_hd_pth_taylor_pair) + (d))) /\ forall ff_i_ph_hd_pth_taylor_pair_body_derivative_steps. (exists ph_bound_hd_pth_taylor_pair_body_derivative_steps. ph_bound_hd_pth_taylor_pair_body_derivative_steps + S ff_i_ph_hd_pth_taylor_pair_body_derivative_steps = l) -> exists ff_coefficient_ph_hd_pth_taylor_pair_body_derivative_steps ff_previous_ph_hd_pth_taylor_pair_body_derivative_steps ff_current_ph_hd_pth_taylor_pair_body_derivative_steps. ((((exists fs_h_ph_hd_pth_taylor_pair_body_derivative_steps_coefficient. fs_h_ph_hd_pth_taylor_pair_body_derivative_steps_coefficient + S (ff_coefficient_ph_hd_pth_taylor_pair_body_derivative_steps) = S ((S (ff_i_ph_hd_pth_taylor_pair_body_derivative_steps)) * ff_v_hd_pth_taylor_pair)) /\ exists fs_q_ph_hd_pth_taylor_pair_body_derivative_steps_coefficient. ff_u_hd_pth_taylor_pair = fs_q_ph_hd_pth_taylor_pair_body_derivative_steps_coefficient * S ((S (ff_i_ph_hd_pth_taylor_pair_body_derivative_steps)) * ff_v_hd_pth_taylor_pair) + (ff_coefficient_ph_hd_pth_taylor_pair_body_derivative_steps))) /\ ((((exists fs_h_ph_hd_pth_taylor_pair_body_derivative_steps_before. fs_h_ph_hd_pth_taylor_pair_body_derivative_steps_before + S (ff_previous_ph_hd_pth_taylor_pair_body_derivative_steps) = S ((S (ff_i_ph_hd_pth_taylor_pair_body_derivative_steps)) * ff_e_hd_pth_taylor_pair)) /\ exists fs_q_ph_hd_pth_taylor_pair_body_derivative_steps_before. ff_d_hd_pth_taylor_pair = fs_q_ph_hd_pth_taylor_pair_body_derivative_steps_before * S ((S (ff_i_ph_hd_pth_taylor_pair_body_derivative_steps)) * ff_e_hd_pth_taylor_pair) + (ff_previous_ph_hd_pth_taylor_pair_body_derivative_steps))) /\ ((((exists fs_h_ph_hd_pth_taylor_pair_body_derivative_steps_after. fs_h_ph_hd_pth_taylor_pair_body_derivative_steps_after + S (ff_current_ph_hd_pth_taylor_pair_body_derivative_steps) = S ((S (S ff_i_ph_hd_pth_taylor_pair_body_derivative_steps)) * ff_e_hd_pth_taylor_pair)) /\ exists fs_q_ph_hd_pth_taylor_pair_body_derivative_steps_after. ff_d_hd_pth_taylor_pair = fs_q_ph_hd_pth_taylor_pair_body_derivative_steps_after * S ((S (S ff_i_ph_hd_pth_taylor_pair_body_derivative_steps)) * ff_e_hd_pth_taylor_pair) + (ff_current_ph_hd_pth_taylor_pair_body_derivative_steps))) /\ ff_current_ph_hd_pth_taylor_pair_body_derivative_steps = ff_previous_ph_hd_pth_taylor_pair_body_derivative_steps * t + ff_coefficient_ph_hd_pth_taylor_pair_body_derivative_steps)))))))) -> (exists ff_u_ph_pth_taylor_shifted ff_v_ph_pth_taylor_shifted. ((((exists fs_h_ph_pth_taylor_shifted_body_start. fs_h_ph_pth_taylor_shifted_body_start + S (0) = S ((S (0)) * ff_v_ph_pth_taylor_shifted)) /\ exists fs_q_ph_pth_taylor_shifted_body_start. ff_u_ph_pth_taylor_shifted = fs_q_ph_pth_taylor_shifted_body_start * S ((S (0)) * ff_v_ph_pth_taylor_shifted) + (0))) /\ ((((exists fs_h_ph_pth_taylor_shifted_body_terminal. fs_h_ph_pth_taylor_shifted_body_terminal + S (z) = S ((S (l)) * ff_v_ph_pth_taylor_shifted)) /\ exists fs_q_ph_pth_taylor_shifted_body_terminal. ff_u_ph_pth_taylor_shifted = fs_q_ph_pth_taylor_shifted_body_terminal * S ((S (l)) * ff_v_ph_pth_taylor_shifted) + (z))) /\ forall ff_i_ph_pth_taylor_shifted_body_steps. (exists ph_bound_pth_taylor_shifted_body_steps. ph_bound_pth_taylor_shifted_body_steps + S ff_i_ph_pth_taylor_shifted_body_steps = l) -> exists ff_coefficient_ph_pth_taylor_shifted_body_steps ff_previous_ph_pth_taylor_shifted_body_steps ff_current_ph_pth_taylor_shifted_body_steps. ((((exists fs_h_ph_pth_taylor_shifted_body_steps_coefficient. fs_h_ph_pth_taylor_shifted_body_steps_coefficient + S (ff_coefficient_ph_pth_taylor_shifted_body_steps) = S ((S (ff_i_ph_pth_taylor_shifted_body_steps)) * c)) /\ exists fs_q_ph_pth_taylor_shifted_body_steps_coefficient. b = fs_q_ph_pth_taylor_shifted_body_steps_coefficient * S ((S (ff_i_ph_pth_taylor_shifted_body_steps)) * c) + (ff_coefficient_ph_pth_taylor_shifted_body_steps))) /\ ((((exists fs_h_ph_pth_taylor_shifted_body_steps_before. fs_h_ph_pth_taylor_shifted_body_steps_before + S (ff_previous_ph_pth_taylor_shifted_body_steps) = S ((S (ff_i_ph_pth_taylor_shifted_body_steps)) * ff_v_ph_pth_taylor_shifted)) /\ exists fs_q_ph_pth_taylor_shifted_body_steps_before. ff_u_ph_pth_taylor_shifted = fs_q_ph_pth_taylor_shifted_body_steps_before * S ((S (ff_i_ph_pth_taylor_shifted_body_steps)) * ff_v_ph_pth_taylor_shifted) + (ff_previous_ph_pth_taylor_shifted_body_steps))) /\ ((((exists fs_h_ph_pth_taylor_shifted_body_steps_after. fs_h_ph_pth_taylor_shifted_body_steps_after + S (ff_current_ph_pth_taylor_shifted_body_steps) = S ((S (S ff_i_ph_pth_taylor_shifted_body_steps)) * ff_v_ph_pth_taylor_shifted)) /\ exists fs_q_ph_pth_taylor_shifted_body_steps_after. ff_u_ph_pth_taylor_shifted = fs_q_ph_pth_taylor_shifted_body_steps_after * S ((S (S ff_i_ph_pth_taylor_shifted_body_steps)) * ff_v_ph_pth_taylor_shifted) + (ff_current_ph_pth_taylor_shifted_body_steps))) /\ ff_current_ph_pth_taylor_shifted_body_steps = ff_previous_ph_pth_taylor_shifted_body_steps * (t + h) + ff_coefficient_ph_pth_taylor_shifted_body_steps)))))) -> (exists hgcrt_mod_left_pth_taylor_square hgcrt_mod_right_pth_taylor_square. z + (h * h) * hgcrt_mod_left_pth_taylor_square = (n + h * d) + (h * h) * hgcrt_mod_right_pth_taylor_square)