beta_horner_prime_power_iterated_lifts_exists_unique

forall b c a l n d p k j m. ~(p = 0) -> ~(k = 0) -> (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 = k) -> (((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 (m) = S ((S (k)) * 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 (k)) * pa_v_hpl_power_product) + (m))) /\ 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 = k) -> 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)))))))) -> (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)))))))) -> (exists hgcrt_mod_left_hpl_mod hgcrt_mod_right_hpl_mod. n + m * hgcrt_mod_left_hpl_mod = 0 + m * hgcrt_mod_right_hpl_mod) -> (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 M. ((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 = k + j) -> (((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 (M) = S ((S (k + j)) * 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 (k + j)) * pa_v_hpl_power_product) + (M))) /\ 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 = k + j) -> 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)))))))) /\ exists r. ((((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 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 + M * hgcrt_mod_left_hpl_lift = 0 + M * hgcrt_mod_right_hpl_lift)))))) /\ forall z. (((exists hpl_gap_lift. hpl_gap_lift + S (z) = (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 + M * hgcrt_mod_left_hpl_lift = 0 + M * hgcrt_mod_right_hpl_lift)))))) -> z = r))