HL0020

beta_signed_horner_simple_root_hensel_lift_exists_unique

Alpha v34 independently verified · alpha_closed; checked-use authorized; not Stable

An unrestricted root of any finite integer-coefficient polynomial has exactly one canonical next-modulus lift whenever its actual integer derivative is a unit.

Current library: Alpha v34, 4,223 checked-use theorems; Stable remains 432. Historical first admissions, original proof editions, and non-admitted aliases are preserved. Exact original first-admission records.

Exact expanded first-order arithmetic statement

forall pb pc nb nc a l vp dp vn dn m p s. ~(p = 0) -> ~(m = 0) -> m = p * s -> (((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) -> (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))) -> exists r. ((((exists hpl_gap_lift. hpl_gap_lift + S (r) = (m * p)) /\ ((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 * p) * hgcrt_mod_left_hpl_lift = sph_negative_lift + (m * p) * hgcrt_mod_right_hpl_lift))))))) /\ forall z. (((exists hpl_gap_lift. hpl_gap_lift + S (z) = (m * p)) /\ ((exists hgcrt_mod_left_hpl_lift hgcrt_mod_right_hpl_lift. z + 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 * z + 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 * z + ff_coefficient_ph_hpl_sph_lift_negative_body_steps)))))) /\ (exists hgcrt_mod_left_hpl_lift hgcrt_mod_right_hpl_lift. sph_positive_lift + (m * p) * hgcrt_mod_left_hpl_lift = sph_negative_lift + (m * p) * hgcrt_mod_right_hpl_lift))))))) -> z = r)

Constructive proof overview

Generated structural guide

An unrestricted root of any finite integer-coefficient polynomial has exactly one canonical next-modulus lift whenever its actual integer derivative is a unit.

The unchanged tactic script uses 3 declared prerequisites and contains 58 exact native proof lines.

Alpha v34 checked-use · first admitted v27 · independently kernel and Lean verified; not Stable

Proof neighborhood

Direct dependencies

pow_exists Stable theorem; checked-use authorized pow_one Stable theorem; checked-use authorized HL001F beta_signed_horner_hensel_iterated_exists_unique

Direct dependents

HL0021 beta_signed_horner_prime_power_hensel_lift_exists_unique

Formal native tactic body

Dependencies are introduced as named hypotheses before line 1. Local theorem links identify exact declared prerequisites. This exact body belongs to a complete independently kernel-checked constructive proof bundle and has Alpha checked-use authority; it does not imply Stable membership.

Read the argument

Proof checkpoints

58 script commands · 9 reading checkpoints · 3 local claims

This is a reading aid, not a new proof or a proof-tree certificate. Checkpoint groups are consecutive commands, not inferred branch boundaries. Every step links to the preserved script.

Named ingredients (1)
beta signed horner hensel iterated exists unique

Long local formulas use this family’s existing definitions. Each new abbreviation was expanded back to the identical native formula, including its free-variable context. The original edition is preserved below.

01Fix variables and assumptionsL1–10

Work with arbitrary variables or the premises of the current implication.

  1. L1
    intro pb
  2. L2
    intro pc
  3. L3
    intro nb
  4. L4
    intro nc
  5. L5
    intro a
  6. L6
    intro l
  7. L7
    intro vp
  8. L8
    intro dp
  9. L9
    intro vn
  10. L10
    intro dn
02Fix variables and assumptionsL11–19

Work with arbitrary variables or the premises of the current implication.

  1. L11
    intro m
  2. L12
    intro p
  3. L13
    intro s
  4. L14
    intro hp
  5. L15
    intro hm
  6. L16
    intro hfactor
  7. L17
    intro hpair
  8. L18
    intro hroot
  9. L19
    intro hunit
03Establish hpowerL20–20

Establish this local claim before using it. It is not an additional assumption.

  1. L20
    have hpower : Pow(p,1,p)Definitions: Pow
04Establish hexistsL21–24

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply pow exists.

  1. L21
    have hexists : ∃ q. Pow(p,1,q)Definitions: Pow
  2. L22
    specialize pow_exists p
  3. L23
    specialize pow_exists 1
  4. L24
    apply pow_exists
05Separate the logical casesL25–25

Follow the explicit conjunction, disjunction, witness, or contradiction step recorded below.

  1. L25
    cases hexists
06Establish heqL26–35

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply pow one.

  1. L26
    have heq : x = p
  2. L27
    specialize pow_one p
  3. L28
    specialize pow_one 1
  4. L29
    specialize pow_one x
  5. L30
    apply pow_one
  6. L31
    refl
  7. L32
    exact hexists_witness
  8. L33
    rewrite heq at hexists_witness
  9. L34
    rewrite heq at hexists_witness
  10. L35
    exact hexists_witness
07Use earlier factsL36–45

Instantiate or apply named facts and discharge the corresponding proof obligations.

  1. L36
    specialize beta_signed_horner_hensel_iterated_exists_unique pb
  2. L37
    specialize beta_signed_horner_hensel_iterated_exists_unique pc
  3. L38
    specialize beta_signed_horner_hensel_iterated_exists_unique nb
  4. L39
    specialize beta_signed_horner_hensel_iterated_exists_unique nc
  5. L40
    specialize beta_signed_horner_hensel_iterated_exists_unique a
  6. L41
    specialize beta_signed_horner_hensel_iterated_exists_unique l
  7. L42
    specialize beta_signed_horner_hensel_iterated_exists_unique vp
  8. L43
    specialize beta_signed_horner_hensel_iterated_exists_unique dp
  9. L44
    specialize beta_signed_horner_hensel_iterated_exists_unique vn
  10. L45
    specialize beta_signed_horner_hensel_iterated_exists_unique dn
08Use earlier factsL46–55

Instantiate or apply named facts and discharge the corresponding proof obligations.

  1. L46
    specialize beta_signed_horner_hensel_iterated_exists_unique m
  2. L47
    specialize beta_signed_horner_hensel_iterated_exists_unique p
  3. L48
    specialize beta_signed_horner_hensel_iterated_exists_unique s
  4. L49
    specialize beta_signed_horner_hensel_iterated_exists_unique 1
  5. L50
    specialize beta_signed_horner_hensel_iterated_exists_unique p
  6. L51
    apply beta_signed_horner_hensel_iterated_exists_unique
  7. L52
    exact hp
  8. L53
    exact hm
  9. L54
    exact hfactor
  10. L55
    exact hpair
09Use earlier factsL56–58

Instantiate or apply named facts and discharge the corresponding proof obligations.

  1. L56
    exact hroot
  2. L57
    exact hunit
  3. L58
    exact hpower

Library-wide reading audit

Original exact command ledger · 58 lines
  1. 0001intro pb
  2. 0002intro pc
  3. 0003intro nb
  4. 0004intro nc
  5. 0005intro a
  6. 0006intro l
  7. 0007intro vp
  8. 0008intro dp
  9. 0009intro vn
  10. 0010intro dn
  11. 0011intro m
  12. 0012intro p
  13. 0013intro s
  14. 0014intro hp
  15. 0015intro hm
  16. 0016intro hfactor
  17. 0017intro hpair
  18. 0018intro hroot
  19. 0019intro hunit
  20. 0020have hpower : exists pa_b_hpl_power pa_c_hpl_power. ((forall pa_i_hpl_power_repeat. (exists pa_lt_hpl_power_repeat_bound. pa_lt_hpl_power_repeat_bound + S pa_i_hpl_power_repeat = 1) -> (((exists pa_h_hpl_power_repeat_decoded. pa_h_hpl_power_repeat_decoded + S (p) = S ((S (pa_i_hpl_power_repeat)) * pa_c_hpl_power)) /\ exists pa_q_hpl_power_repeat_decoded. pa_b_hpl_power = pa_q_hpl_power_repeat_decoded * S ((S (pa_i_hpl_power_repeat)) * pa_c_hpl_power) + (p)))) /\ (exists pa_u_hpl_power_product pa_v_hpl_power_product. ((((exists pa_h_hpl_power_product_start. pa_h_hpl_power_product_start + S (1) = S ((S (0)) * pa_v_hpl_power_product)) /\ exists pa_q_hpl_power_product_start. pa_u_hpl_power_product = pa_q_hpl_power_product_start * S ((S (0)) * pa_v_hpl_power_product) + (1))) /\ ((((exists pa_h_hpl_power_product_terminal. pa_h_hpl_power_product_terminal + S (p) = S ((S (1)) * pa_v_hpl_power_product)) /\ exists pa_q_hpl_power_product_terminal. pa_u_hpl_power_product = pa_q_hpl_power_product_terminal * S ((S (1)) * pa_v_hpl_power_product) + (p))) /\ forall pa_i_hpl_power_product. (exists pa_lt_hpl_power_product_bound. pa_lt_hpl_power_product_bound + S pa_i_hpl_power_product = 1) -> exists pa_p_hpl_power_product pa_r_hpl_power_product pa_s_hpl_power_product. ((((exists pa_h_hpl_power_product_factor. pa_h_hpl_power_product_factor + S (pa_p_hpl_power_product) = S ((S (pa_i_hpl_power_product)) * pa_c_hpl_power)) /\ exists pa_q_hpl_power_product_factor. pa_b_hpl_power = pa_q_hpl_power_product_factor * S ((S (pa_i_hpl_power_product)) * pa_c_hpl_power) + (pa_p_hpl_power_product))) /\ ((((exists pa_h_hpl_power_product_partial. pa_h_hpl_power_product_partial + S (pa_r_hpl_power_product) = S ((S (pa_i_hpl_power_product)) * pa_v_hpl_power_product)) /\ exists pa_q_hpl_power_product_partial. pa_u_hpl_power_product = pa_q_hpl_power_product_partial * S ((S (pa_i_hpl_power_product)) * pa_v_hpl_power_product) + (pa_r_hpl_power_product))) /\ ((((exists pa_h_hpl_power_product_successor. pa_h_hpl_power_product_successor + S (pa_s_hpl_power_product) = S ((S (S pa_i_hpl_power_product)) * pa_v_hpl_power_product)) /\ exists pa_q_hpl_power_product_successor. pa_u_hpl_power_product = pa_q_hpl_power_product_successor * S ((S (S pa_i_hpl_power_product)) * pa_v_hpl_power_product) + (pa_s_hpl_power_product))) /\ pa_s_hpl_power_product = pa_r_hpl_power_product * pa_p_hpl_power_product)))))))
  21. 0021have hexists : exists q. (exists pa_b_hpl_power pa_c_hpl_power. ((forall pa_i_hpl_power_repeat. (exists pa_lt_hpl_power_repeat_bound. pa_lt_hpl_power_repeat_bound + S pa_i_hpl_power_repeat = 1) -> (((exists pa_h_hpl_power_repeat_decoded. pa_h_hpl_power_repeat_decoded + S (p) = S ((S (pa_i_hpl_power_repeat)) * pa_c_hpl_power)) /\ exists pa_q_hpl_power_repeat_decoded. pa_b_hpl_power = pa_q_hpl_power_repeat_decoded * S ((S (pa_i_hpl_power_repeat)) * pa_c_hpl_power) + (p)))) /\ (exists pa_u_hpl_power_product pa_v_hpl_power_product. ((((exists pa_h_hpl_power_product_start. pa_h_hpl_power_product_start + S (1) = S ((S (0)) * pa_v_hpl_power_product)) /\ exists pa_q_hpl_power_product_start. pa_u_hpl_power_product = pa_q_hpl_power_product_start * S ((S (0)) * pa_v_hpl_power_product) + (1))) /\ ((((exists pa_h_hpl_power_product_terminal. pa_h_hpl_power_product_terminal + S (q) = S ((S (1)) * pa_v_hpl_power_product)) /\ exists pa_q_hpl_power_product_terminal. pa_u_hpl_power_product = pa_q_hpl_power_product_terminal * S ((S (1)) * pa_v_hpl_power_product) + (q))) /\ forall pa_i_hpl_power_product. (exists pa_lt_hpl_power_product_bound. pa_lt_hpl_power_product_bound + S pa_i_hpl_power_product = 1) -> exists pa_p_hpl_power_product pa_r_hpl_power_product pa_s_hpl_power_product. ((((exists pa_h_hpl_power_product_factor. pa_h_hpl_power_product_factor + S (pa_p_hpl_power_product) = S ((S (pa_i_hpl_power_product)) * pa_c_hpl_power)) /\ exists pa_q_hpl_power_product_factor. pa_b_hpl_power = pa_q_hpl_power_product_factor * S ((S (pa_i_hpl_power_product)) * pa_c_hpl_power) + (pa_p_hpl_power_product))) /\ ((((exists pa_h_hpl_power_product_partial. pa_h_hpl_power_product_partial + S (pa_r_hpl_power_product) = S ((S (pa_i_hpl_power_product)) * pa_v_hpl_power_product)) /\ exists pa_q_hpl_power_product_partial. pa_u_hpl_power_product = pa_q_hpl_power_product_partial * S ((S (pa_i_hpl_power_product)) * pa_v_hpl_power_product) + (pa_r_hpl_power_product))) /\ ((((exists pa_h_hpl_power_product_successor. pa_h_hpl_power_product_successor + S (pa_s_hpl_power_product) = S ((S (S pa_i_hpl_power_product)) * pa_v_hpl_power_product)) /\ exists pa_q_hpl_power_product_successor. pa_u_hpl_power_product = pa_q_hpl_power_product_successor * S ((S (S pa_i_hpl_power_product)) * pa_v_hpl_power_product) + (pa_s_hpl_power_product))) /\ pa_s_hpl_power_product = pa_r_hpl_power_product * pa_p_hpl_power_product))))))))
  22. 0022specialize pow_exists p
  23. 0023specialize pow_exists 1
  24. 0024apply pow_exists
  25. 0025cases hexists
  26. 0026have heq : x = p
  27. 0027specialize pow_one p
  28. 0028specialize pow_one 1
  29. 0029specialize pow_one x
  30. 0030apply pow_one
  31. 0031refl
  32. 0032exact hexists_witness
  33. 0033rewrite heq at hexists_witness
  34. 0034rewrite heq at hexists_witness
  35. 0035exact hexists_witness
  36. 0036specialize beta_signed_horner_hensel_iterated_exists_unique pb
  37. 0037specialize beta_signed_horner_hensel_iterated_exists_unique pc
  38. 0038specialize beta_signed_horner_hensel_iterated_exists_unique nb
  39. 0039specialize beta_signed_horner_hensel_iterated_exists_unique nc
  40. 0040specialize beta_signed_horner_hensel_iterated_exists_unique a
  41. 0041specialize beta_signed_horner_hensel_iterated_exists_unique l
  42. 0042specialize beta_signed_horner_hensel_iterated_exists_unique vp
  43. 0043specialize beta_signed_horner_hensel_iterated_exists_unique dp
  44. 0044specialize beta_signed_horner_hensel_iterated_exists_unique vn
  45. 0045specialize beta_signed_horner_hensel_iterated_exists_unique dn
  46. 0046specialize beta_signed_horner_hensel_iterated_exists_unique m
  47. 0047specialize beta_signed_horner_hensel_iterated_exists_unique p
  48. 0048specialize beta_signed_horner_hensel_iterated_exists_unique s
  49. 0049specialize beta_signed_horner_hensel_iterated_exists_unique 1
  50. 0050specialize beta_signed_horner_hensel_iterated_exists_unique p
  51. 0051apply beta_signed_horner_hensel_iterated_exists_unique
  52. 0052exact hp
  53. 0053exact hm
  54. 0054exact hfactor
  55. 0055exact hpair
  56. 0056exact hroot
  57. 0057exact hunit
  58. 0058exact hpower