hensel_canonical_horner_lift_exists_unique

forall b c a l n d m p s q. ~(p = 0) -> ~(m = 0) -> (exists ff_u_hd_hpl_pair ff_v_hd_hpl_pair ff_d_hd_hpl_pair ff_e_hd_hpl_pair. ((((((exists fs_h_ph_hd_hpl_pair_body_value_start. fs_h_ph_hd_hpl_pair_body_value_start + S (0) = S ((S (0)) * ff_v_hd_hpl_pair)) /\ exists fs_q_ph_hd_hpl_pair_body_value_start. ff_u_hd_hpl_pair = fs_q_ph_hd_hpl_pair_body_value_start * S ((S (0)) * ff_v_hd_hpl_pair) + (0))) /\ ((((exists fs_h_ph_hd_hpl_pair_body_value_terminal. fs_h_ph_hd_hpl_pair_body_value_terminal + S (n) = S ((S (l)) * ff_v_hd_hpl_pair)) /\ exists fs_q_ph_hd_hpl_pair_body_value_terminal. ff_u_hd_hpl_pair = fs_q_ph_hd_hpl_pair_body_value_terminal * S ((S (l)) * ff_v_hd_hpl_pair) + (n))) /\ forall ff_i_ph_hd_hpl_pair_body_value_steps. (exists ph_bound_hd_hpl_pair_body_value_steps. ph_bound_hd_hpl_pair_body_value_steps + S ff_i_ph_hd_hpl_pair_body_value_steps = l) -> exists ff_coefficient_ph_hd_hpl_pair_body_value_steps ff_previous_ph_hd_hpl_pair_body_value_steps ff_current_ph_hd_hpl_pair_body_value_steps. ((((exists fs_h_ph_hd_hpl_pair_body_value_steps_coefficient. fs_h_ph_hd_hpl_pair_body_value_steps_coefficient + S (ff_coefficient_ph_hd_hpl_pair_body_value_steps) = S ((S (ff_i_ph_hd_hpl_pair_body_value_steps)) * c)) /\ exists fs_q_ph_hd_hpl_pair_body_value_steps_coefficient. b = fs_q_ph_hd_hpl_pair_body_value_steps_coefficient * S ((S (ff_i_ph_hd_hpl_pair_body_value_steps)) * c) + (ff_coefficient_ph_hd_hpl_pair_body_value_steps))) /\ ((((exists fs_h_ph_hd_hpl_pair_body_value_steps_before. fs_h_ph_hd_hpl_pair_body_value_steps_before + S (ff_previous_ph_hd_hpl_pair_body_value_steps) = S ((S (ff_i_ph_hd_hpl_pair_body_value_steps)) * ff_v_hd_hpl_pair)) /\ exists fs_q_ph_hd_hpl_pair_body_value_steps_before. ff_u_hd_hpl_pair = fs_q_ph_hd_hpl_pair_body_value_steps_before * S ((S (ff_i_ph_hd_hpl_pair_body_value_steps)) * ff_v_hd_hpl_pair) + (ff_previous_ph_hd_hpl_pair_body_value_steps))) /\ ((((exists fs_h_ph_hd_hpl_pair_body_value_steps_after. fs_h_ph_hd_hpl_pair_body_value_steps_after + S (ff_current_ph_hd_hpl_pair_body_value_steps) = S ((S (S ff_i_ph_hd_hpl_pair_body_value_steps)) * ff_v_hd_hpl_pair)) /\ exists fs_q_ph_hd_hpl_pair_body_value_steps_after. ff_u_hd_hpl_pair = fs_q_ph_hd_hpl_pair_body_value_steps_after * S ((S (S ff_i_ph_hd_hpl_pair_body_value_steps)) * ff_v_hd_hpl_pair) + (ff_current_ph_hd_hpl_pair_body_value_steps))) /\ ff_current_ph_hd_hpl_pair_body_value_steps = ff_previous_ph_hd_hpl_pair_body_value_steps * a + ff_coefficient_ph_hd_hpl_pair_body_value_steps)))))) /\ (((((exists fs_h_ph_hd_hpl_pair_body_derivative_start. fs_h_ph_hd_hpl_pair_body_derivative_start + S (0) = S ((S (0)) * ff_e_hd_hpl_pair)) /\ exists fs_q_ph_hd_hpl_pair_body_derivative_start. ff_d_hd_hpl_pair = fs_q_ph_hd_hpl_pair_body_derivative_start * S ((S (0)) * ff_e_hd_hpl_pair) + (0))) /\ ((((exists fs_h_ph_hd_hpl_pair_body_derivative_terminal. fs_h_ph_hd_hpl_pair_body_derivative_terminal + S (d) = S ((S (l)) * ff_e_hd_hpl_pair)) /\ exists fs_q_ph_hd_hpl_pair_body_derivative_terminal. ff_d_hd_hpl_pair = fs_q_ph_hd_hpl_pair_body_derivative_terminal * S ((S (l)) * ff_e_hd_hpl_pair) + (d))) /\ forall ff_i_ph_hd_hpl_pair_body_derivative_steps. (exists ph_bound_hd_hpl_pair_body_derivative_steps. ph_bound_hd_hpl_pair_body_derivative_steps + S ff_i_ph_hd_hpl_pair_body_derivative_steps = l) -> exists ff_coefficient_ph_hd_hpl_pair_body_derivative_steps ff_previous_ph_hd_hpl_pair_body_derivative_steps ff_current_ph_hd_hpl_pair_body_derivative_steps. ((((exists fs_h_ph_hd_hpl_pair_body_derivative_steps_coefficient. fs_h_ph_hd_hpl_pair_body_derivative_steps_coefficient + S (ff_coefficient_ph_hd_hpl_pair_body_derivative_steps) = S ((S (ff_i_ph_hd_hpl_pair_body_derivative_steps)) * ff_v_hd_hpl_pair)) /\ exists fs_q_ph_hd_hpl_pair_body_derivative_steps_coefficient. ff_u_hd_hpl_pair = fs_q_ph_hd_hpl_pair_body_derivative_steps_coefficient * S ((S (ff_i_ph_hd_hpl_pair_body_derivative_steps)) * ff_v_hd_hpl_pair) + (ff_coefficient_ph_hd_hpl_pair_body_derivative_steps))) /\ ((((exists fs_h_ph_hd_hpl_pair_body_derivative_steps_before. fs_h_ph_hd_hpl_pair_body_derivative_steps_before + S (ff_previous_ph_hd_hpl_pair_body_derivative_steps) = S ((S (ff_i_ph_hd_hpl_pair_body_derivative_steps)) * ff_e_hd_hpl_pair)) /\ exists fs_q_ph_hd_hpl_pair_body_derivative_steps_before. ff_d_hd_hpl_pair = fs_q_ph_hd_hpl_pair_body_derivative_steps_before * S ((S (ff_i_ph_hd_hpl_pair_body_derivative_steps)) * ff_e_hd_hpl_pair) + (ff_previous_ph_hd_hpl_pair_body_derivative_steps))) /\ ((((exists fs_h_ph_hd_hpl_pair_body_derivative_steps_after. fs_h_ph_hd_hpl_pair_body_derivative_steps_after + S (ff_current_ph_hd_hpl_pair_body_derivative_steps) = S ((S (S ff_i_ph_hd_hpl_pair_body_derivative_steps)) * ff_e_hd_hpl_pair)) /\ exists fs_q_ph_hd_hpl_pair_body_derivative_steps_after. ff_d_hd_hpl_pair = fs_q_ph_hd_hpl_pair_body_derivative_steps_after * S ((S (S ff_i_ph_hd_hpl_pair_body_derivative_steps)) * ff_e_hd_hpl_pair) + (ff_current_ph_hd_hpl_pair_body_derivative_steps))) /\ ff_current_ph_hd_hpl_pair_body_derivative_steps = ff_previous_ph_hd_hpl_pair_body_derivative_steps * a + ff_coefficient_ph_hd_hpl_pair_body_derivative_steps)))))))) -> m = p * s -> n = m * q -> (forall hmi_divisor_hpl_coprime. (exists hmi_left_factor_hpl_coprime. d = hmi_divisor_hpl_coprime * hmi_left_factor_hpl_coprime) -> (exists hmi_right_factor_hpl_coprime. p = hmi_divisor_hpl_coprime * hmi_right_factor_hpl_coprime) -> hmi_divisor_hpl_coprime = 1) -> (exists hpl_gap_bound. hpl_gap_bound + S (a) = (m)) -> exists r. ((((exists hpl_gap_lift. hpl_gap_lift + S (r) = (p * 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 hpl_value_lift. ((exists ff_u_ph_hpl_lift ff_v_ph_hpl_lift. ((((exists fs_h_ph_hpl_lift_body_start. fs_h_ph_hpl_lift_body_start + S (0) = S ((S (0)) * ff_v_ph_hpl_lift)) /\ exists fs_q_ph_hpl_lift_body_start. ff_u_ph_hpl_lift = fs_q_ph_hpl_lift_body_start * S ((S (0)) * ff_v_ph_hpl_lift) + (0))) /\ ((((exists fs_h_ph_hpl_lift_body_terminal. fs_h_ph_hpl_lift_body_terminal + S (hpl_value_lift) = S ((S (l)) * ff_v_ph_hpl_lift)) /\ exists fs_q_ph_hpl_lift_body_terminal. ff_u_ph_hpl_lift = fs_q_ph_hpl_lift_body_terminal * S ((S (l)) * ff_v_ph_hpl_lift) + (hpl_value_lift))) /\ forall ff_i_ph_hpl_lift_body_steps. (exists ph_bound_hpl_lift_body_steps. ph_bound_hpl_lift_body_steps + S ff_i_ph_hpl_lift_body_steps = l) -> exists ff_coefficient_ph_hpl_lift_body_steps ff_previous_ph_hpl_lift_body_steps ff_current_ph_hpl_lift_body_steps. ((((exists fs_h_ph_hpl_lift_body_steps_coefficient. fs_h_ph_hpl_lift_body_steps_coefficient + S (ff_coefficient_ph_hpl_lift_body_steps) = S ((S (ff_i_ph_hpl_lift_body_steps)) * c)) /\ exists fs_q_ph_hpl_lift_body_steps_coefficient. b = fs_q_ph_hpl_lift_body_steps_coefficient * S ((S (ff_i_ph_hpl_lift_body_steps)) * c) + (ff_coefficient_ph_hpl_lift_body_steps))) /\ ((((exists fs_h_ph_hpl_lift_body_steps_before. fs_h_ph_hpl_lift_body_steps_before + S (ff_previous_ph_hpl_lift_body_steps) = S ((S (ff_i_ph_hpl_lift_body_steps)) * ff_v_ph_hpl_lift)) /\ exists fs_q_ph_hpl_lift_body_steps_before. ff_u_ph_hpl_lift = fs_q_ph_hpl_lift_body_steps_before * S ((S (ff_i_ph_hpl_lift_body_steps)) * ff_v_ph_hpl_lift) + (ff_previous_ph_hpl_lift_body_steps))) /\ ((((exists fs_h_ph_hpl_lift_body_steps_after. fs_h_ph_hpl_lift_body_steps_after + S (ff_current_ph_hpl_lift_body_steps) = S ((S (S ff_i_ph_hpl_lift_body_steps)) * ff_v_ph_hpl_lift)) /\ exists fs_q_ph_hpl_lift_body_steps_after. ff_u_ph_hpl_lift = fs_q_ph_hpl_lift_body_steps_after * S ((S (S ff_i_ph_hpl_lift_body_steps)) * ff_v_ph_hpl_lift) + (ff_current_ph_hpl_lift_body_steps))) /\ ff_current_ph_hpl_lift_body_steps = ff_previous_ph_hpl_lift_body_steps * r + ff_coefficient_ph_hpl_lift_body_steps)))))) /\ (exists hgcrt_mod_left_hpl_lift hgcrt_mod_right_hpl_lift. hpl_value_lift + (p * m) * hgcrt_mod_left_hpl_lift = 0 + (p * m) * hgcrt_mod_right_hpl_lift)))))) /\ forall z. (((exists hpl_gap_lift. hpl_gap_lift + S (z) = (p * m)) /\ ((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 hpl_value_lift. ((exists ff_u_ph_hpl_lift ff_v_ph_hpl_lift. ((((exists fs_h_ph_hpl_lift_body_start. fs_h_ph_hpl_lift_body_start + S (0) = S ((S (0)) * ff_v_ph_hpl_lift)) /\ exists fs_q_ph_hpl_lift_body_start. ff_u_ph_hpl_lift = fs_q_ph_hpl_lift_body_start * S ((S (0)) * ff_v_ph_hpl_lift) + (0))) /\ ((((exists fs_h_ph_hpl_lift_body_terminal. fs_h_ph_hpl_lift_body_terminal + S (hpl_value_lift) = S ((S (l)) * ff_v_ph_hpl_lift)) /\ exists fs_q_ph_hpl_lift_body_terminal. ff_u_ph_hpl_lift = fs_q_ph_hpl_lift_body_terminal * S ((S (l)) * ff_v_ph_hpl_lift) + (hpl_value_lift))) /\ forall ff_i_ph_hpl_lift_body_steps. (exists ph_bound_hpl_lift_body_steps. ph_bound_hpl_lift_body_steps + S ff_i_ph_hpl_lift_body_steps = l) -> exists ff_coefficient_ph_hpl_lift_body_steps ff_previous_ph_hpl_lift_body_steps ff_current_ph_hpl_lift_body_steps. ((((exists fs_h_ph_hpl_lift_body_steps_coefficient. fs_h_ph_hpl_lift_body_steps_coefficient + S (ff_coefficient_ph_hpl_lift_body_steps) = S ((S (ff_i_ph_hpl_lift_body_steps)) * c)) /\ exists fs_q_ph_hpl_lift_body_steps_coefficient. b = fs_q_ph_hpl_lift_body_steps_coefficient * S ((S (ff_i_ph_hpl_lift_body_steps)) * c) + (ff_coefficient_ph_hpl_lift_body_steps))) /\ ((((exists fs_h_ph_hpl_lift_body_steps_before. fs_h_ph_hpl_lift_body_steps_before + S (ff_previous_ph_hpl_lift_body_steps) = S ((S (ff_i_ph_hpl_lift_body_steps)) * ff_v_ph_hpl_lift)) /\ exists fs_q_ph_hpl_lift_body_steps_before. ff_u_ph_hpl_lift = fs_q_ph_hpl_lift_body_steps_before * S ((S (ff_i_ph_hpl_lift_body_steps)) * ff_v_ph_hpl_lift) + (ff_previous_ph_hpl_lift_body_steps))) /\ ((((exists fs_h_ph_hpl_lift_body_steps_after. fs_h_ph_hpl_lift_body_steps_after + S (ff_current_ph_hpl_lift_body_steps) = S ((S (S ff_i_ph_hpl_lift_body_steps)) * ff_v_ph_hpl_lift)) /\ exists fs_q_ph_hpl_lift_body_steps_after. ff_u_ph_hpl_lift = fs_q_ph_hpl_lift_body_steps_after * S ((S (S ff_i_ph_hpl_lift_body_steps)) * ff_v_ph_hpl_lift) + (ff_current_ph_hpl_lift_body_steps))) /\ ff_current_ph_hpl_lift_body_steps = ff_previous_ph_hpl_lift_body_steps * z + ff_coefficient_ph_hpl_lift_body_steps)))))) /\ (exists hgcrt_mod_left_hpl_lift hgcrt_mod_right_hpl_lift. hpl_value_lift + (p * m) * hgcrt_mod_left_hpl_lift = 0 + (p * m) * hgcrt_mod_right_hpl_lift)))))) -> z = r)