beta_signed_horner_lift_preserves_simplicity

Exact expanded first-order arithmetic statement

forall pb pc nb nc a l vp dp vn dn m p s M r. (((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 sph_inverse_unit. ((exists hpl_gap_unit. hpl_gap_unit + S (sph_inverse_unit) = (p)) /\ (exists hgcrt_mod_left_hpl_unit hgcrt_mod_right_hpl_unit. (dp * sph_inverse_unit) + p * hgcrt_mod_left_hpl_unit = (1 + dn * sph_inverse_unit) + p * hgcrt_mod_right_hpl_unit))) -> m = p * s -> (((exists hpl_gap_lift. hpl_gap_lift + S (r) = (M)) /\ ((exists hgcrt_mod_left_hpl_lift hgcrt_mod_right_hpl_lift. r + m * hgcrt_mod_left_hpl_lift = a + m * hgcrt_mod_right_hpl_lift) /\ (exists sph_positive_lift sph_negative_lift. ((exists ff_u_ph_hpl_sph_lift_positive ff_v_ph_hpl_sph_lift_positive. ((((exists fs_h_ph_hpl_sph_lift_positive_body_start. fs_h_ph_hpl_sph_lift_positive_body_start + S (0) = S ((S (0)) * ff_v_ph_hpl_sph_lift_positive)) /\ exists fs_q_ph_hpl_sph_lift_positive_body_start. ff_u_ph_hpl_sph_lift_positive = fs_q_ph_hpl_sph_lift_positive_body_start * S ((S (0)) * ff_v_ph_hpl_sph_lift_positive) + (0))) /\ ((((exists fs_h_ph_hpl_sph_lift_positive_body_terminal. fs_h_ph_hpl_sph_lift_positive_body_terminal + S (sph_positive_lift) = S ((S (l)) * ff_v_ph_hpl_sph_lift_positive)) /\ exists fs_q_ph_hpl_sph_lift_positive_body_terminal. ff_u_ph_hpl_sph_lift_positive = fs_q_ph_hpl_sph_lift_positive_body_terminal * S ((S (l)) * ff_v_ph_hpl_sph_lift_positive) + (sph_positive_lift))) /\ forall ff_i_ph_hpl_sph_lift_positive_body_steps. (exists ph_bound_hpl_sph_lift_positive_body_steps. ph_bound_hpl_sph_lift_positive_body_steps + S ff_i_ph_hpl_sph_lift_positive_body_steps = l) -> exists ff_coefficient_ph_hpl_sph_lift_positive_body_steps ff_previous_ph_hpl_sph_lift_positive_body_steps ff_current_ph_hpl_sph_lift_positive_body_steps. ((((exists fs_h_ph_hpl_sph_lift_positive_body_steps_coefficient. fs_h_ph_hpl_sph_lift_positive_body_steps_coefficient + S (ff_coefficient_ph_hpl_sph_lift_positive_body_steps) = S ((S (ff_i_ph_hpl_sph_lift_positive_body_steps)) * pc)) /\ exists fs_q_ph_hpl_sph_lift_positive_body_steps_coefficient. pb = fs_q_ph_hpl_sph_lift_positive_body_steps_coefficient * S ((S (ff_i_ph_hpl_sph_lift_positive_body_steps)) * pc) + (ff_coefficient_ph_hpl_sph_lift_positive_body_steps))) /\ ((((exists fs_h_ph_hpl_sph_lift_positive_body_steps_before. fs_h_ph_hpl_sph_lift_positive_body_steps_before + S (ff_previous_ph_hpl_sph_lift_positive_body_steps) = S ((S (ff_i_ph_hpl_sph_lift_positive_body_steps)) * ff_v_ph_hpl_sph_lift_positive)) /\ exists fs_q_ph_hpl_sph_lift_positive_body_steps_before. ff_u_ph_hpl_sph_lift_positive = fs_q_ph_hpl_sph_lift_positive_body_steps_before * S ((S (ff_i_ph_hpl_sph_lift_positive_body_steps)) * ff_v_ph_hpl_sph_lift_positive) + (ff_previous_ph_hpl_sph_lift_positive_body_steps))) /\ ((((exists fs_h_ph_hpl_sph_lift_positive_body_steps_after. fs_h_ph_hpl_sph_lift_positive_body_steps_after + S (ff_current_ph_hpl_sph_lift_positive_body_steps) = S ((S (S ff_i_ph_hpl_sph_lift_positive_body_steps)) * ff_v_ph_hpl_sph_lift_positive)) /\ exists fs_q_ph_hpl_sph_lift_positive_body_steps_after. ff_u_ph_hpl_sph_lift_positive = fs_q_ph_hpl_sph_lift_positive_body_steps_after * S ((S (S ff_i_ph_hpl_sph_lift_positive_body_steps)) * ff_v_ph_hpl_sph_lift_positive) + (ff_current_ph_hpl_sph_lift_positive_body_steps))) /\ ff_current_ph_hpl_sph_lift_positive_body_steps = ff_previous_ph_hpl_sph_lift_positive_body_steps * r + ff_coefficient_ph_hpl_sph_lift_positive_body_steps)))))) /\ ((exists ff_u_ph_hpl_sph_lift_negative ff_v_ph_hpl_sph_lift_negative. ((((exists fs_h_ph_hpl_sph_lift_negative_body_start. fs_h_ph_hpl_sph_lift_negative_body_start + S (0) = S ((S (0)) * ff_v_ph_hpl_sph_lift_negative)) /\ exists fs_q_ph_hpl_sph_lift_negative_body_start. ff_u_ph_hpl_sph_lift_negative = fs_q_ph_hpl_sph_lift_negative_body_start * S ((S (0)) * ff_v_ph_hpl_sph_lift_negative) + (0))) /\ ((((exists fs_h_ph_hpl_sph_lift_negative_body_terminal. fs_h_ph_hpl_sph_lift_negative_body_terminal + S (sph_negative_lift) = S ((S (l)) * ff_v_ph_hpl_sph_lift_negative)) /\ exists fs_q_ph_hpl_sph_lift_negative_body_terminal. ff_u_ph_hpl_sph_lift_negative = fs_q_ph_hpl_sph_lift_negative_body_terminal * S ((S (l)) * ff_v_ph_hpl_sph_lift_negative) + (sph_negative_lift))) /\ forall ff_i_ph_hpl_sph_lift_negative_body_steps. (exists ph_bound_hpl_sph_lift_negative_body_steps. ph_bound_hpl_sph_lift_negative_body_steps + S ff_i_ph_hpl_sph_lift_negative_body_steps = l) -> exists ff_coefficient_ph_hpl_sph_lift_negative_body_steps ff_previous_ph_hpl_sph_lift_negative_body_steps ff_current_ph_hpl_sph_lift_negative_body_steps. ((((exists fs_h_ph_hpl_sph_lift_negative_body_steps_coefficient. fs_h_ph_hpl_sph_lift_negative_body_steps_coefficient + S (ff_coefficient_ph_hpl_sph_lift_negative_body_steps) = S ((S (ff_i_ph_hpl_sph_lift_negative_body_steps)) * nc)) /\ exists fs_q_ph_hpl_sph_lift_negative_body_steps_coefficient. nb = fs_q_ph_hpl_sph_lift_negative_body_steps_coefficient * S ((S (ff_i_ph_hpl_sph_lift_negative_body_steps)) * nc) + (ff_coefficient_ph_hpl_sph_lift_negative_body_steps))) /\ ((((exists fs_h_ph_hpl_sph_lift_negative_body_steps_before. fs_h_ph_hpl_sph_lift_negative_body_steps_before + S (ff_previous_ph_hpl_sph_lift_negative_body_steps) = S ((S (ff_i_ph_hpl_sph_lift_negative_body_steps)) * ff_v_ph_hpl_sph_lift_negative)) /\ exists fs_q_ph_hpl_sph_lift_negative_body_steps_before. ff_u_ph_hpl_sph_lift_negative = fs_q_ph_hpl_sph_lift_negative_body_steps_before * S ((S (ff_i_ph_hpl_sph_lift_negative_body_steps)) * ff_v_ph_hpl_sph_lift_negative) + (ff_previous_ph_hpl_sph_lift_negative_body_steps))) /\ ((((exists fs_h_ph_hpl_sph_lift_negative_body_steps_after. fs_h_ph_hpl_sph_lift_negative_body_steps_after + S (ff_current_ph_hpl_sph_lift_negative_body_steps) = S ((S (S ff_i_ph_hpl_sph_lift_negative_body_steps)) * ff_v_ph_hpl_sph_lift_negative)) /\ exists fs_q_ph_hpl_sph_lift_negative_body_steps_after. ff_u_ph_hpl_sph_lift_negative = fs_q_ph_hpl_sph_lift_negative_body_steps_after * S ((S (S ff_i_ph_hpl_sph_lift_negative_body_steps)) * ff_v_ph_hpl_sph_lift_negative) + (ff_current_ph_hpl_sph_lift_negative_body_steps))) /\ ff_current_ph_hpl_sph_lift_negative_body_steps = ff_previous_ph_hpl_sph_lift_negative_body_steps * r + ff_coefficient_ph_hpl_sph_lift_negative_body_steps)))))) /\ (exists hgcrt_mod_left_hpl_lift hgcrt_mod_right_hpl_lift. sph_positive_lift + M * hgcrt_mod_left_hpl_lift = sph_negative_lift + M * hgcrt_mod_right_hpl_lift))))))) -> (exists sph_vp_simple sph_dp_simple sph_vn_simple sph_dn_simple. ((((exists ff_u_hd_hpl_sph_simple_positive ff_v_hd_hpl_sph_simple_positive ff_d_hd_hpl_sph_simple_positive ff_e_hd_hpl_sph_simple_positive. ((((((exists fs_h_ph_hd_hpl_sph_simple_positive_body_value_start. fs_h_ph_hd_hpl_sph_simple_positive_body_value_start + S (0) = S ((S (0)) * ff_v_hd_hpl_sph_simple_positive)) /\ exists fs_q_ph_hd_hpl_sph_simple_positive_body_value_start. ff_u_hd_hpl_sph_simple_positive = fs_q_ph_hd_hpl_sph_simple_positive_body_value_start * S ((S (0)) * ff_v_hd_hpl_sph_simple_positive) + (0))) /\ ((((exists fs_h_ph_hd_hpl_sph_simple_positive_body_value_terminal. fs_h_ph_hd_hpl_sph_simple_positive_body_value_terminal + S (sph_vp_simple) = S ((S (l)) * ff_v_hd_hpl_sph_simple_positive)) /\ exists fs_q_ph_hd_hpl_sph_simple_positive_body_value_terminal. ff_u_hd_hpl_sph_simple_positive = fs_q_ph_hd_hpl_sph_simple_positive_body_value_terminal * S ((S (l)) * ff_v_hd_hpl_sph_simple_positive) + (sph_vp_simple))) /\ forall ff_i_ph_hd_hpl_sph_simple_positive_body_value_steps. (exists ph_bound_hd_hpl_sph_simple_positive_body_value_steps. ph_bound_hd_hpl_sph_simple_positive_body_value_steps + S ff_i_ph_hd_hpl_sph_simple_positive_body_value_steps = l) -> exists ff_coefficient_ph_hd_hpl_sph_simple_positive_body_value_steps ff_previous_ph_hd_hpl_sph_simple_positive_body_value_steps ff_current_ph_hd_hpl_sph_simple_positive_body_value_steps. ((((exists fs_h_ph_hd_hpl_sph_simple_positive_body_value_steps_coefficient. fs_h_ph_hd_hpl_sph_simple_positive_body_value_steps_coefficient + S (ff_coefficient_ph_hd_hpl_sph_simple_positive_body_value_steps) = S ((S (ff_i_ph_hd_hpl_sph_simple_positive_body_value_steps)) * pc)) /\ exists fs_q_ph_hd_hpl_sph_simple_positive_body_value_steps_coefficient. pb = fs_q_ph_hd_hpl_sph_simple_positive_body_value_steps_coefficient * S ((S (ff_i_ph_hd_hpl_sph_simple_positive_body_value_steps)) * pc) + (ff_coefficient_ph_hd_hpl_sph_simple_positive_body_value_steps))) /\ ((((exists fs_h_ph_hd_hpl_sph_simple_positive_body_value_steps_before. fs_h_ph_hd_hpl_sph_simple_positive_body_value_steps_before + S (ff_previous_ph_hd_hpl_sph_simple_positive_body_value_steps) = S ((S (ff_i_ph_hd_hpl_sph_simple_positive_body_value_steps)) * ff_v_hd_hpl_sph_simple_positive)) /\ exists fs_q_ph_hd_hpl_sph_simple_positive_body_value_steps_before. ff_u_hd_hpl_sph_simple_positive = fs_q_ph_hd_hpl_sph_simple_positive_body_value_steps_before * S ((S (ff_i_ph_hd_hpl_sph_simple_positive_body_value_steps)) * ff_v_hd_hpl_sph_simple_positive) + (ff_previous_ph_hd_hpl_sph_simple_positive_body_value_steps))) /\ ((((exists fs_h_ph_hd_hpl_sph_simple_positive_body_value_steps_after. fs_h_ph_hd_hpl_sph_simple_positive_body_value_steps_after + S (ff_current_ph_hd_hpl_sph_simple_positive_body_value_steps) = S ((S (S ff_i_ph_hd_hpl_sph_simple_positive_body_value_steps)) * ff_v_hd_hpl_sph_simple_positive)) /\ exists fs_q_ph_hd_hpl_sph_simple_positive_body_value_steps_after. ff_u_hd_hpl_sph_simple_positive = fs_q_ph_hd_hpl_sph_simple_positive_body_value_steps_after * S ((S (S ff_i_ph_hd_hpl_sph_simple_positive_body_value_steps)) * ff_v_hd_hpl_sph_simple_positive) + (ff_current_ph_hd_hpl_sph_simple_positive_body_value_steps))) /\ ff_current_ph_hd_hpl_sph_simple_positive_body_value_steps = ff_previous_ph_hd_hpl_sph_simple_positive_body_value_steps * r + ff_coefficient_ph_hd_hpl_sph_simple_positive_body_value_steps)))))) /\ (((((exists fs_h_ph_hd_hpl_sph_simple_positive_body_derivative_start. fs_h_ph_hd_hpl_sph_simple_positive_body_derivative_start + S (0) = S ((S (0)) * ff_e_hd_hpl_sph_simple_positive)) /\ exists fs_q_ph_hd_hpl_sph_simple_positive_body_derivative_start. ff_d_hd_hpl_sph_simple_positive = fs_q_ph_hd_hpl_sph_simple_positive_body_derivative_start * S ((S (0)) * ff_e_hd_hpl_sph_simple_positive) + (0))) /\ ((((exists fs_h_ph_hd_hpl_sph_simple_positive_body_derivative_terminal. fs_h_ph_hd_hpl_sph_simple_positive_body_derivative_terminal + S (sph_dp_simple) = S ((S (l)) * ff_e_hd_hpl_sph_simple_positive)) /\ exists fs_q_ph_hd_hpl_sph_simple_positive_body_derivative_terminal. ff_d_hd_hpl_sph_simple_positive = fs_q_ph_hd_hpl_sph_simple_positive_body_derivative_terminal * S ((S (l)) * ff_e_hd_hpl_sph_simple_positive) + (sph_dp_simple))) /\ forall ff_i_ph_hd_hpl_sph_simple_positive_body_derivative_steps. (exists ph_bound_hd_hpl_sph_simple_positive_body_derivative_steps. ph_bound_hd_hpl_sph_simple_positive_body_derivative_steps + S ff_i_ph_hd_hpl_sph_simple_positive_body_derivative_steps = l) -> exists ff_coefficient_ph_hd_hpl_sph_simple_positive_body_derivative_steps ff_previous_ph_hd_hpl_sph_simple_positive_body_derivative_steps ff_current_ph_hd_hpl_sph_simple_positive_body_derivative_steps. ((((exists fs_h_ph_hd_hpl_sph_simple_positive_body_derivative_steps_coefficient. fs_h_ph_hd_hpl_sph_simple_positive_body_derivative_steps_coefficient + S (ff_coefficient_ph_hd_hpl_sph_simple_positive_body_derivative_steps) = S ((S (ff_i_ph_hd_hpl_sph_simple_positive_body_derivative_steps)) * ff_v_hd_hpl_sph_simple_positive)) /\ exists fs_q_ph_hd_hpl_sph_simple_positive_body_derivative_steps_coefficient. ff_u_hd_hpl_sph_simple_positive = fs_q_ph_hd_hpl_sph_simple_positive_body_derivative_steps_coefficient * S ((S (ff_i_ph_hd_hpl_sph_simple_positive_body_derivative_steps)) * ff_v_hd_hpl_sph_simple_positive) + (ff_coefficient_ph_hd_hpl_sph_simple_positive_body_derivative_steps))) /\ ((((exists fs_h_ph_hd_hpl_sph_simple_positive_body_derivative_steps_before. fs_h_ph_hd_hpl_sph_simple_positive_body_derivative_steps_before + S (ff_previous_ph_hd_hpl_sph_simple_positive_body_derivative_steps) = S ((S (ff_i_ph_hd_hpl_sph_simple_positive_body_derivative_steps)) * ff_e_hd_hpl_sph_simple_positive)) /\ exists fs_q_ph_hd_hpl_sph_simple_positive_body_derivative_steps_before. ff_d_hd_hpl_sph_simple_positive = fs_q_ph_hd_hpl_sph_simple_positive_body_derivative_steps_before * S ((S (ff_i_ph_hd_hpl_sph_simple_positive_body_derivative_steps)) * ff_e_hd_hpl_sph_simple_positive) + (ff_previous_ph_hd_hpl_sph_simple_positive_body_derivative_steps))) /\ ((((exists fs_h_ph_hd_hpl_sph_simple_positive_body_derivative_steps_after. fs_h_ph_hd_hpl_sph_simple_positive_body_derivative_steps_after + S (ff_current_ph_hd_hpl_sph_simple_positive_body_derivative_steps) = S ((S (S ff_i_ph_hd_hpl_sph_simple_positive_body_derivative_steps)) * ff_e_hd_hpl_sph_simple_positive)) /\ exists fs_q_ph_hd_hpl_sph_simple_positive_body_derivative_steps_after. ff_d_hd_hpl_sph_simple_positive = fs_q_ph_hd_hpl_sph_simple_positive_body_derivative_steps_after * S ((S (S ff_i_ph_hd_hpl_sph_simple_positive_body_derivative_steps)) * ff_e_hd_hpl_sph_simple_positive) + (ff_current_ph_hd_hpl_sph_simple_positive_body_derivative_steps))) /\ ff_current_ph_hd_hpl_sph_simple_positive_body_derivative_steps = ff_previous_ph_hd_hpl_sph_simple_positive_body_derivative_steps * r + ff_coefficient_ph_hd_hpl_sph_simple_positive_body_derivative_steps)))))))) /\ (exists ff_u_hd_hpl_sph_simple_negative ff_v_hd_hpl_sph_simple_negative ff_d_hd_hpl_sph_simple_negative ff_e_hd_hpl_sph_simple_negative. ((((((exists fs_h_ph_hd_hpl_sph_simple_negative_body_value_start. fs_h_ph_hd_hpl_sph_simple_negative_body_value_start + S (0) = S ((S (0)) * ff_v_hd_hpl_sph_simple_negative)) /\ exists fs_q_ph_hd_hpl_sph_simple_negative_body_value_start. ff_u_hd_hpl_sph_simple_negative = fs_q_ph_hd_hpl_sph_simple_negative_body_value_start * S ((S (0)) * ff_v_hd_hpl_sph_simple_negative) + (0))) /\ ((((exists fs_h_ph_hd_hpl_sph_simple_negative_body_value_terminal. fs_h_ph_hd_hpl_sph_simple_negative_body_value_terminal + S (sph_vn_simple) = S ((S (l)) * ff_v_hd_hpl_sph_simple_negative)) /\ exists fs_q_ph_hd_hpl_sph_simple_negative_body_value_terminal. ff_u_hd_hpl_sph_simple_negative = fs_q_ph_hd_hpl_sph_simple_negative_body_value_terminal * S ((S (l)) * ff_v_hd_hpl_sph_simple_negative) + (sph_vn_simple))) /\ forall ff_i_ph_hd_hpl_sph_simple_negative_body_value_steps. (exists ph_bound_hd_hpl_sph_simple_negative_body_value_steps. ph_bound_hd_hpl_sph_simple_negative_body_value_steps + S ff_i_ph_hd_hpl_sph_simple_negative_body_value_steps = l) -> exists ff_coefficient_ph_hd_hpl_sph_simple_negative_body_value_steps ff_previous_ph_hd_hpl_sph_simple_negative_body_value_steps ff_current_ph_hd_hpl_sph_simple_negative_body_value_steps. ((((exists fs_h_ph_hd_hpl_sph_simple_negative_body_value_steps_coefficient. fs_h_ph_hd_hpl_sph_simple_negative_body_value_steps_coefficient + S (ff_coefficient_ph_hd_hpl_sph_simple_negative_body_value_steps) = S ((S (ff_i_ph_hd_hpl_sph_simple_negative_body_value_steps)) * nc)) /\ exists fs_q_ph_hd_hpl_sph_simple_negative_body_value_steps_coefficient. nb = fs_q_ph_hd_hpl_sph_simple_negative_body_value_steps_coefficient * S ((S (ff_i_ph_hd_hpl_sph_simple_negative_body_value_steps)) * nc) + (ff_coefficient_ph_hd_hpl_sph_simple_negative_body_value_steps))) /\ ((((exists fs_h_ph_hd_hpl_sph_simple_negative_body_value_steps_before. fs_h_ph_hd_hpl_sph_simple_negative_body_value_steps_before + S (ff_previous_ph_hd_hpl_sph_simple_negative_body_value_steps) = S ((S (ff_i_ph_hd_hpl_sph_simple_negative_body_value_steps)) * ff_v_hd_hpl_sph_simple_negative)) /\ exists fs_q_ph_hd_hpl_sph_simple_negative_body_value_steps_before. ff_u_hd_hpl_sph_simple_negative = fs_q_ph_hd_hpl_sph_simple_negative_body_value_steps_before * S ((S (ff_i_ph_hd_hpl_sph_simple_negative_body_value_steps)) * ff_v_hd_hpl_sph_simple_negative) + (ff_previous_ph_hd_hpl_sph_simple_negative_body_value_steps))) /\ ((((exists fs_h_ph_hd_hpl_sph_simple_negative_body_value_steps_after. fs_h_ph_hd_hpl_sph_simple_negative_body_value_steps_after + S (ff_current_ph_hd_hpl_sph_simple_negative_body_value_steps) = S ((S (S ff_i_ph_hd_hpl_sph_simple_negative_body_value_steps)) * ff_v_hd_hpl_sph_simple_negative)) /\ exists fs_q_ph_hd_hpl_sph_simple_negative_body_value_steps_after. ff_u_hd_hpl_sph_simple_negative = fs_q_ph_hd_hpl_sph_simple_negative_body_value_steps_after * S ((S (S ff_i_ph_hd_hpl_sph_simple_negative_body_value_steps)) * ff_v_hd_hpl_sph_simple_negative) + (ff_current_ph_hd_hpl_sph_simple_negative_body_value_steps))) /\ ff_current_ph_hd_hpl_sph_simple_negative_body_value_steps = ff_previous_ph_hd_hpl_sph_simple_negative_body_value_steps * r + ff_coefficient_ph_hd_hpl_sph_simple_negative_body_value_steps)))))) /\ (((((exists fs_h_ph_hd_hpl_sph_simple_negative_body_derivative_start. fs_h_ph_hd_hpl_sph_simple_negative_body_derivative_start + S (0) = S ((S (0)) * ff_e_hd_hpl_sph_simple_negative)) /\ exists fs_q_ph_hd_hpl_sph_simple_negative_body_derivative_start. ff_d_hd_hpl_sph_simple_negative = fs_q_ph_hd_hpl_sph_simple_negative_body_derivative_start * S ((S (0)) * ff_e_hd_hpl_sph_simple_negative) + (0))) /\ ((((exists fs_h_ph_hd_hpl_sph_simple_negative_body_derivative_terminal. fs_h_ph_hd_hpl_sph_simple_negative_body_derivative_terminal + S (sph_dn_simple) = S ((S (l)) * ff_e_hd_hpl_sph_simple_negative)) /\ exists fs_q_ph_hd_hpl_sph_simple_negative_body_derivative_terminal. ff_d_hd_hpl_sph_simple_negative = fs_q_ph_hd_hpl_sph_simple_negative_body_derivative_terminal * S ((S (l)) * ff_e_hd_hpl_sph_simple_negative) + (sph_dn_simple))) /\ forall ff_i_ph_hd_hpl_sph_simple_negative_body_derivative_steps. (exists ph_bound_hd_hpl_sph_simple_negative_body_derivative_steps. ph_bound_hd_hpl_sph_simple_negative_body_derivative_steps + S ff_i_ph_hd_hpl_sph_simple_negative_body_derivative_steps = l) -> exists ff_coefficient_ph_hd_hpl_sph_simple_negative_body_derivative_steps ff_previous_ph_hd_hpl_sph_simple_negative_body_derivative_steps ff_current_ph_hd_hpl_sph_simple_negative_body_derivative_steps. ((((exists fs_h_ph_hd_hpl_sph_simple_negative_body_derivative_steps_coefficient. fs_h_ph_hd_hpl_sph_simple_negative_body_derivative_steps_coefficient + S (ff_coefficient_ph_hd_hpl_sph_simple_negative_body_derivative_steps) = S ((S (ff_i_ph_hd_hpl_sph_simple_negative_body_derivative_steps)) * ff_v_hd_hpl_sph_simple_negative)) /\ exists fs_q_ph_hd_hpl_sph_simple_negative_body_derivative_steps_coefficient. ff_u_hd_hpl_sph_simple_negative = fs_q_ph_hd_hpl_sph_simple_negative_body_derivative_steps_coefficient * S ((S (ff_i_ph_hd_hpl_sph_simple_negative_body_derivative_steps)) * ff_v_hd_hpl_sph_simple_negative) + (ff_coefficient_ph_hd_hpl_sph_simple_negative_body_derivative_steps))) /\ ((((exists fs_h_ph_hd_hpl_sph_simple_negative_body_derivative_steps_before. fs_h_ph_hd_hpl_sph_simple_negative_body_derivative_steps_before + S (ff_previous_ph_hd_hpl_sph_simple_negative_body_derivative_steps) = S ((S (ff_i_ph_hd_hpl_sph_simple_negative_body_derivative_steps)) * ff_e_hd_hpl_sph_simple_negative)) /\ exists fs_q_ph_hd_hpl_sph_simple_negative_body_derivative_steps_before. ff_d_hd_hpl_sph_simple_negative = fs_q_ph_hd_hpl_sph_simple_negative_body_derivative_steps_before * S ((S (ff_i_ph_hd_hpl_sph_simple_negative_body_derivative_steps)) * ff_e_hd_hpl_sph_simple_negative) + (ff_previous_ph_hd_hpl_sph_simple_negative_body_derivative_steps))) /\ ((((exists fs_h_ph_hd_hpl_sph_simple_negative_body_derivative_steps_after. fs_h_ph_hd_hpl_sph_simple_negative_body_derivative_steps_after + S (ff_current_ph_hd_hpl_sph_simple_negative_body_derivative_steps) = S ((S (S ff_i_ph_hd_hpl_sph_simple_negative_body_derivative_steps)) * ff_e_hd_hpl_sph_simple_negative)) /\ exists fs_q_ph_hd_hpl_sph_simple_negative_body_derivative_steps_after. ff_d_hd_hpl_sph_simple_negative = fs_q_ph_hd_hpl_sph_simple_negative_body_derivative_steps_after * S ((S (S ff_i_ph_hd_hpl_sph_simple_negative_body_derivative_steps)) * ff_e_hd_hpl_sph_simple_negative) + (ff_current_ph_hd_hpl_sph_simple_negative_body_derivative_steps))) /\ ff_current_ph_hd_hpl_sph_simple_negative_body_derivative_steps = ff_previous_ph_hd_hpl_sph_simple_negative_body_derivative_steps * r + ff_coefficient_ph_hd_hpl_sph_simple_negative_body_derivative_steps)))))))))) /\ ((exists hgcrt_mod_left_hpl_simple hgcrt_mod_right_hpl_simple. sph_vp_simple + M * hgcrt_mod_left_hpl_simple = sph_vn_simple + M * hgcrt_mod_right_hpl_simple) /\ (exists sph_inverse_simple. ((exists hpl_gap_simple. hpl_gap_simple + S (sph_inverse_simple) = (p)) /\ (exists hgcrt_mod_left_hpl_simple hgcrt_mod_right_hpl_simple. (sph_dp_simple * sph_inverse_simple) + p * hgcrt_mod_left_hpl_simple = (1 + sph_dn_simple * sph_inverse_simple) + p * hgcrt_mod_right_hpl_simple))))))