beta_product_bit_weighted_lower_power

forall b c d f B l z k Q. (forall pc_index_weight_lower_source pc_factor_weight_lower_source pc_bit_weight_lower_source. (exists pc_lt_weight_lower_source_index. pc_lt_weight_lower_source_index + S (pc_index_weight_lower_source) = (l)) -> (((exists fs_h_pc_weight_lower_source_factor. fs_h_pc_weight_lower_source_factor + S (pc_factor_weight_lower_source) = S ((S (pc_index_weight_lower_source)) * c)) /\ exists fs_q_pc_weight_lower_source_factor. b = fs_q_pc_weight_lower_source_factor * S ((S (pc_index_weight_lower_source)) * c) + (pc_factor_weight_lower_source))) -> (((exists fs_h_pc_weight_lower_source_bit. fs_h_pc_weight_lower_source_bit + S (pc_bit_weight_lower_source) = S ((S (pc_index_weight_lower_source)) * f)) /\ exists fs_q_pc_weight_lower_source_bit. d = fs_q_pc_weight_lower_source_bit * S ((S (pc_index_weight_lower_source)) * f) + (pc_bit_weight_lower_source))) -> ((pc_bit_weight_lower_source = 0 /\ (exists pc_le_weight_lower_source_zero. pc_le_weight_lower_source_zero + (1) = (pc_factor_weight_lower_source))) \/ (pc_bit_weight_lower_source = 1 /\ (exists pc_le_weight_lower_source_one. pc_le_weight_lower_source_one + (B) = (pc_factor_weight_lower_source))))) -> (exists ff_u_pc_weight_lower_product ff_v_pc_weight_lower_product. ((((exists ff_h_pc_weight_lower_product_start. ff_h_pc_weight_lower_product_start + S (1) = S ((S (0)) * ff_v_pc_weight_lower_product)) /\ exists ff_q_pc_weight_lower_product_start. ff_u_pc_weight_lower_product = ff_q_pc_weight_lower_product_start * S ((S (0)) * ff_v_pc_weight_lower_product) + (1))) /\ ((((exists ff_h_pc_weight_lower_product_terminal. ff_h_pc_weight_lower_product_terminal + S (z) = S ((S (l)) * ff_v_pc_weight_lower_product)) /\ exists ff_q_pc_weight_lower_product_terminal. ff_u_pc_weight_lower_product = ff_q_pc_weight_lower_product_terminal * S ((S (l)) * ff_v_pc_weight_lower_product) + (z))) /\ forall ff_i_pc_weight_lower_product. (exists ff_lt_pc_weight_lower_product_bound. ff_lt_pc_weight_lower_product_bound + S ff_i_pc_weight_lower_product = l) -> exists ff_p_pc_weight_lower_product ff_r_pc_weight_lower_product ff_s_pc_weight_lower_product. ((((exists ff_h_pc_weight_lower_product_factor. ff_h_pc_weight_lower_product_factor + S (ff_p_pc_weight_lower_product) = S ((S (ff_i_pc_weight_lower_product)) * c)) /\ exists ff_q_pc_weight_lower_product_factor. b = ff_q_pc_weight_lower_product_factor * S ((S (ff_i_pc_weight_lower_product)) * c) + (ff_p_pc_weight_lower_product))) /\ ((((exists ff_h_pc_weight_lower_product_partial. ff_h_pc_weight_lower_product_partial + S (ff_r_pc_weight_lower_product) = S ((S (ff_i_pc_weight_lower_product)) * ff_v_pc_weight_lower_product)) /\ exists ff_q_pc_weight_lower_product_partial. ff_u_pc_weight_lower_product = ff_q_pc_weight_lower_product_partial * S ((S (ff_i_pc_weight_lower_product)) * ff_v_pc_weight_lower_product) + (ff_r_pc_weight_lower_product))) /\ ((((exists ff_h_pc_weight_lower_product_successor. ff_h_pc_weight_lower_product_successor + S (ff_s_pc_weight_lower_product) = S ((S (S ff_i_pc_weight_lower_product)) * ff_v_pc_weight_lower_product)) /\ exists ff_q_pc_weight_lower_product_successor. ff_u_pc_weight_lower_product = ff_q_pc_weight_lower_product_successor * S ((S (S ff_i_pc_weight_lower_product)) * ff_v_pc_weight_lower_product) + (ff_s_pc_weight_lower_product))) /\ ff_s_pc_weight_lower_product = ff_r_pc_weight_lower_product * ff_p_pc_weight_lower_product)))))) -> (exists fs_u_pc_weight_lower_sum fs_v_pc_weight_lower_sum. ((((exists fs_h_pc_weight_lower_sum_body_start. fs_h_pc_weight_lower_sum_body_start + S (0) = S ((S (0)) * fs_v_pc_weight_lower_sum)) /\ exists fs_q_pc_weight_lower_sum_body_start. fs_u_pc_weight_lower_sum = fs_q_pc_weight_lower_sum_body_start * S ((S (0)) * fs_v_pc_weight_lower_sum) + (0))) /\ ((((exists fs_h_pc_weight_lower_sum_body_terminal. fs_h_pc_weight_lower_sum_body_terminal + S (k) = S ((S (l)) * fs_v_pc_weight_lower_sum)) /\ exists fs_q_pc_weight_lower_sum_body_terminal. fs_u_pc_weight_lower_sum = fs_q_pc_weight_lower_sum_body_terminal * S ((S (l)) * fs_v_pc_weight_lower_sum) + (k))) /\ forall fs_i_pc_weight_lower_sum_body_steps. (exists fs_lt_pc_weight_lower_sum_body_steps_bound. fs_lt_pc_weight_lower_sum_body_steps_bound + S fs_i_pc_weight_lower_sum_body_steps = l) -> exists fs_a_pc_weight_lower_sum_body_steps fs_r_pc_weight_lower_sum_body_steps fs_s_pc_weight_lower_sum_body_steps. ((((exists fs_h_pc_weight_lower_sum_body_steps_summand. fs_h_pc_weight_lower_sum_body_steps_summand + S (fs_a_pc_weight_lower_sum_body_steps) = S ((S (fs_i_pc_weight_lower_sum_body_steps)) * f)) /\ exists fs_q_pc_weight_lower_sum_body_steps_summand. d = fs_q_pc_weight_lower_sum_body_steps_summand * S ((S (fs_i_pc_weight_lower_sum_body_steps)) * f) + (fs_a_pc_weight_lower_sum_body_steps))) /\ ((((exists fs_h_pc_weight_lower_sum_body_steps_partial. fs_h_pc_weight_lower_sum_body_steps_partial + S (fs_r_pc_weight_lower_sum_body_steps) = S ((S (fs_i_pc_weight_lower_sum_body_steps)) * fs_v_pc_weight_lower_sum)) /\ exists fs_q_pc_weight_lower_sum_body_steps_partial. fs_u_pc_weight_lower_sum = fs_q_pc_weight_lower_sum_body_steps_partial * S ((S (fs_i_pc_weight_lower_sum_body_steps)) * fs_v_pc_weight_lower_sum) + (fs_r_pc_weight_lower_sum_body_steps))) /\ ((((exists fs_h_pc_weight_lower_sum_body_steps_successor. fs_h_pc_weight_lower_sum_body_steps_successor + S (fs_s_pc_weight_lower_sum_body_steps) = S ((S (S fs_i_pc_weight_lower_sum_body_steps)) * fs_v_pc_weight_lower_sum)) /\ exists fs_q_pc_weight_lower_sum_body_steps_successor. fs_u_pc_weight_lower_sum = fs_q_pc_weight_lower_sum_body_steps_successor * S ((S (S fs_i_pc_weight_lower_sum_body_steps)) * fs_v_pc_weight_lower_sum) + (fs_s_pc_weight_lower_sum_body_steps))) /\ fs_s_pc_weight_lower_sum_body_steps = fs_r_pc_weight_lower_sum_body_steps + fs_a_pc_weight_lower_sum_body_steps)))))) -> (exists pa_b_pc_weight_lower_power pa_c_pc_weight_lower_power. ((forall pa_i_pc_weight_lower_power_repeat. (exists pa_lt_pc_weight_lower_power_repeat_bound. pa_lt_pc_weight_lower_power_repeat_bound + S pa_i_pc_weight_lower_power_repeat = k) -> (((exists pa_h_pc_weight_lower_power_repeat_decoded. pa_h_pc_weight_lower_power_repeat_decoded + S (B) = S ((S (pa_i_pc_weight_lower_power_repeat)) * pa_c_pc_weight_lower_power)) /\ exists pa_q_pc_weight_lower_power_repeat_decoded. pa_b_pc_weight_lower_power = pa_q_pc_weight_lower_power_repeat_decoded * S ((S (pa_i_pc_weight_lower_power_repeat)) * pa_c_pc_weight_lower_power) + (B)))) /\ (exists pa_u_pc_weight_lower_power_product pa_v_pc_weight_lower_power_product. ((((exists pa_h_pc_weight_lower_power_product_start. pa_h_pc_weight_lower_power_product_start + S (1) = S ((S (0)) * pa_v_pc_weight_lower_power_product)) /\ exists pa_q_pc_weight_lower_power_product_start. pa_u_pc_weight_lower_power_product = pa_q_pc_weight_lower_power_product_start * S ((S (0)) * pa_v_pc_weight_lower_power_product) + (1))) /\ ((((exists pa_h_pc_weight_lower_power_product_terminal. pa_h_pc_weight_lower_power_product_terminal + S (Q) = S ((S (k)) * pa_v_pc_weight_lower_power_product)) /\ exists pa_q_pc_weight_lower_power_product_terminal. pa_u_pc_weight_lower_power_product = pa_q_pc_weight_lower_power_product_terminal * S ((S (k)) * pa_v_pc_weight_lower_power_product) + (Q))) /\ forall pa_i_pc_weight_lower_power_product. (exists pa_lt_pc_weight_lower_power_product_bound. pa_lt_pc_weight_lower_power_product_bound + S pa_i_pc_weight_lower_power_product = k) -> exists pa_p_pc_weight_lower_power_product pa_r_pc_weight_lower_power_product pa_s_pc_weight_lower_power_product. ((((exists pa_h_pc_weight_lower_power_product_factor. pa_h_pc_weight_lower_power_product_factor + S (pa_p_pc_weight_lower_power_product) = S ((S (pa_i_pc_weight_lower_power_product)) * pa_c_pc_weight_lower_power)) /\ exists pa_q_pc_weight_lower_power_product_factor. pa_b_pc_weight_lower_power = pa_q_pc_weight_lower_power_product_factor * S ((S (pa_i_pc_weight_lower_power_product)) * pa_c_pc_weight_lower_power) + (pa_p_pc_weight_lower_power_product))) /\ ((((exists pa_h_pc_weight_lower_power_product_partial. pa_h_pc_weight_lower_power_product_partial + S (pa_r_pc_weight_lower_power_product) = S ((S (pa_i_pc_weight_lower_power_product)) * pa_v_pc_weight_lower_power_product)) /\ exists pa_q_pc_weight_lower_power_product_partial. pa_u_pc_weight_lower_power_product = pa_q_pc_weight_lower_power_product_partial * S ((S (pa_i_pc_weight_lower_power_product)) * pa_v_pc_weight_lower_power_product) + (pa_r_pc_weight_lower_power_product))) /\ ((((exists pa_h_pc_weight_lower_power_product_successor. pa_h_pc_weight_lower_power_product_successor + S (pa_s_pc_weight_lower_power_product) = S ((S (S pa_i_pc_weight_lower_power_product)) * pa_v_pc_weight_lower_power_product)) /\ exists pa_q_pc_weight_lower_power_product_successor. pa_u_pc_weight_lower_power_product = pa_q_pc_weight_lower_power_product_successor * S ((S (S pa_i_pc_weight_lower_power_product)) * pa_v_pc_weight_lower_power_product) + (pa_s_pc_weight_lower_power_product))) /\ pa_s_pc_weight_lower_power_product = pa_r_pc_weight_lower_power_product * pa_p_pc_weight_lower_power_product)))))))) -> (exists pc_le_weight_lower_result. pc_le_weight_lower_result + (Q) = (z))