pow_block_bound_from_total

Exact expanded PA statement

forall a b d e m x y X Y. (forall bpt_a_hj32_block bpt_e_hj32_block. exists bpt_x_hj32_block. (exists ff_b_bpt_value_hj32_block ff_c_bpt_value_hj32_block. ((forall ff_i_bpt_value_hj32_block_repeat. (exists ff_lt_bpt_value_hj32_block_repeat_bound. ff_lt_bpt_value_hj32_block_repeat_bound + S ff_i_bpt_value_hj32_block_repeat = bpt_e_hj32_block) -> (((exists ff_h_bpt_value_hj32_block_repeat_decoded. ff_h_bpt_value_hj32_block_repeat_decoded + S (bpt_a_hj32_block) = S ((S (ff_i_bpt_value_hj32_block_repeat)) * ff_c_bpt_value_hj32_block)) /\ exists ff_q_bpt_value_hj32_block_repeat_decoded. ff_b_bpt_value_hj32_block = ff_q_bpt_value_hj32_block_repeat_decoded * S ((S (ff_i_bpt_value_hj32_block_repeat)) * ff_c_bpt_value_hj32_block) + (bpt_a_hj32_block)))) /\ (exists ff_u_bpt_value_hj32_block_product ff_v_bpt_value_hj32_block_product. ((((exists ff_h_bpt_value_hj32_block_product_start. ff_h_bpt_value_hj32_block_product_start + S (1) = S ((S (0)) * ff_v_bpt_value_hj32_block_product)) /\ exists ff_q_bpt_value_hj32_block_product_start. ff_u_bpt_value_hj32_block_product = ff_q_bpt_value_hj32_block_product_start * S ((S (0)) * ff_v_bpt_value_hj32_block_product) + (1))) /\ ((((exists ff_h_bpt_value_hj32_block_product_terminal. ff_h_bpt_value_hj32_block_product_terminal + S (bpt_x_hj32_block) = S ((S (bpt_e_hj32_block)) * ff_v_bpt_value_hj32_block_product)) /\ exists ff_q_bpt_value_hj32_block_product_terminal. ff_u_bpt_value_hj32_block_product = ff_q_bpt_value_hj32_block_product_terminal * S ((S (bpt_e_hj32_block)) * ff_v_bpt_value_hj32_block_product) + (bpt_x_hj32_block))) /\ forall ff_i_bpt_value_hj32_block_product. (exists ff_lt_bpt_value_hj32_block_product_bound. ff_lt_bpt_value_hj32_block_product_bound + S ff_i_bpt_value_hj32_block_product = bpt_e_hj32_block) -> exists ff_p_bpt_value_hj32_block_product ff_r_bpt_value_hj32_block_product ff_s_bpt_value_hj32_block_product. ((((exists ff_h_bpt_value_hj32_block_product_factor. ff_h_bpt_value_hj32_block_product_factor + S (ff_p_bpt_value_hj32_block_product) = S ((S (ff_i_bpt_value_hj32_block_product)) * ff_c_bpt_value_hj32_block)) /\ exists ff_q_bpt_value_hj32_block_product_factor. ff_b_bpt_value_hj32_block = ff_q_bpt_value_hj32_block_product_factor * S ((S (ff_i_bpt_value_hj32_block_product)) * ff_c_bpt_value_hj32_block) + (ff_p_bpt_value_hj32_block_product))) /\ ((((exists ff_h_bpt_value_hj32_block_product_partial. ff_h_bpt_value_hj32_block_product_partial + S (ff_r_bpt_value_hj32_block_product) = S ((S (ff_i_bpt_value_hj32_block_product)) * ff_v_bpt_value_hj32_block_product)) /\ exists ff_q_bpt_value_hj32_block_product_partial. ff_u_bpt_value_hj32_block_product = ff_q_bpt_value_hj32_block_product_partial * S ((S (ff_i_bpt_value_hj32_block_product)) * ff_v_bpt_value_hj32_block_product) + (ff_r_bpt_value_hj32_block_product))) /\ ((((exists ff_h_bpt_value_hj32_block_product_successor. ff_h_bpt_value_hj32_block_product_successor + S (ff_s_bpt_value_hj32_block_product) = S ((S (S ff_i_bpt_value_hj32_block_product)) * ff_v_bpt_value_hj32_block_product)) /\ exists ff_q_bpt_value_hj32_block_product_successor. ff_u_bpt_value_hj32_block_product = ff_q_bpt_value_hj32_block_product_successor * S ((S (S ff_i_bpt_value_hj32_block_product)) * ff_v_bpt_value_hj32_block_product) + (ff_s_bpt_value_hj32_block_product))) /\ ff_s_bpt_value_hj32_block_product = ff_r_bpt_value_hj32_block_product * ff_p_bpt_value_hj32_block_product))))))))) -> (exists pa_b_hj32_block_left_seed pa_c_hj32_block_left_seed. ((forall pa_i_hj32_block_left_seed_repeat. (exists pa_lt_hj32_block_left_seed_repeat_bound. pa_lt_hj32_block_left_seed_repeat_bound + S pa_i_hj32_block_left_seed_repeat = d) -> (((exists pa_h_hj32_block_left_seed_repeat_decoded. pa_h_hj32_block_left_seed_repeat_decoded + S (a) = S ((S (pa_i_hj32_block_left_seed_repeat)) * pa_c_hj32_block_left_seed)) /\ exists pa_q_hj32_block_left_seed_repeat_decoded. pa_b_hj32_block_left_seed = pa_q_hj32_block_left_seed_repeat_decoded * S ((S (pa_i_hj32_block_left_seed_repeat)) * pa_c_hj32_block_left_seed) + (a)))) /\ (exists pa_u_hj32_block_left_seed_product pa_v_hj32_block_left_seed_product. ((((exists pa_h_hj32_block_left_seed_product_start. pa_h_hj32_block_left_seed_product_start + S (1) = S ((S (0)) * pa_v_hj32_block_left_seed_product)) /\ exists pa_q_hj32_block_left_seed_product_start. pa_u_hj32_block_left_seed_product = pa_q_hj32_block_left_seed_product_start * S ((S (0)) * pa_v_hj32_block_left_seed_product) + (1))) /\ ((((exists pa_h_hj32_block_left_seed_product_terminal. pa_h_hj32_block_left_seed_product_terminal + S (x) = S ((S (d)) * pa_v_hj32_block_left_seed_product)) /\ exists pa_q_hj32_block_left_seed_product_terminal. pa_u_hj32_block_left_seed_product = pa_q_hj32_block_left_seed_product_terminal * S ((S (d)) * pa_v_hj32_block_left_seed_product) + (x))) /\ forall pa_i_hj32_block_left_seed_product. (exists pa_lt_hj32_block_left_seed_product_bound. pa_lt_hj32_block_left_seed_product_bound + S pa_i_hj32_block_left_seed_product = d) -> exists pa_p_hj32_block_left_seed_product pa_r_hj32_block_left_seed_product pa_s_hj32_block_left_seed_product. ((((exists pa_h_hj32_block_left_seed_product_factor. pa_h_hj32_block_left_seed_product_factor + S (pa_p_hj32_block_left_seed_product) = S ((S (pa_i_hj32_block_left_seed_product)) * pa_c_hj32_block_left_seed)) /\ exists pa_q_hj32_block_left_seed_product_factor. pa_b_hj32_block_left_seed = pa_q_hj32_block_left_seed_product_factor * S ((S (pa_i_hj32_block_left_seed_product)) * pa_c_hj32_block_left_seed) + (pa_p_hj32_block_left_seed_product))) /\ ((((exists pa_h_hj32_block_left_seed_product_partial. pa_h_hj32_block_left_seed_product_partial + S (pa_r_hj32_block_left_seed_product) = S ((S (pa_i_hj32_block_left_seed_product)) * pa_v_hj32_block_left_seed_product)) /\ exists pa_q_hj32_block_left_seed_product_partial. pa_u_hj32_block_left_seed_product = pa_q_hj32_block_left_seed_product_partial * S ((S (pa_i_hj32_block_left_seed_product)) * pa_v_hj32_block_left_seed_product) + (pa_r_hj32_block_left_seed_product))) /\ ((((exists pa_h_hj32_block_left_seed_product_successor. pa_h_hj32_block_left_seed_product_successor + S (pa_s_hj32_block_left_seed_product) = S ((S (S pa_i_hj32_block_left_seed_product)) * pa_v_hj32_block_left_seed_product)) /\ exists pa_q_hj32_block_left_seed_product_successor. pa_u_hj32_block_left_seed_product = pa_q_hj32_block_left_seed_product_successor * S ((S (S pa_i_hj32_block_left_seed_product)) * pa_v_hj32_block_left_seed_product) + (pa_s_hj32_block_left_seed_product))) /\ pa_s_hj32_block_left_seed_product = pa_r_hj32_block_left_seed_product * pa_p_hj32_block_left_seed_product)))))))) -> (exists pa_b_hj32_block_right_seed pa_c_hj32_block_right_seed. ((forall pa_i_hj32_block_right_seed_repeat. (exists pa_lt_hj32_block_right_seed_repeat_bound. pa_lt_hj32_block_right_seed_repeat_bound + S pa_i_hj32_block_right_seed_repeat = e) -> (((exists pa_h_hj32_block_right_seed_repeat_decoded. pa_h_hj32_block_right_seed_repeat_decoded + S (b) = S ((S (pa_i_hj32_block_right_seed_repeat)) * pa_c_hj32_block_right_seed)) /\ exists pa_q_hj32_block_right_seed_repeat_decoded. pa_b_hj32_block_right_seed = pa_q_hj32_block_right_seed_repeat_decoded * S ((S (pa_i_hj32_block_right_seed_repeat)) * pa_c_hj32_block_right_seed) + (b)))) /\ (exists pa_u_hj32_block_right_seed_product pa_v_hj32_block_right_seed_product. ((((exists pa_h_hj32_block_right_seed_product_start. pa_h_hj32_block_right_seed_product_start + S (1) = S ((S (0)) * pa_v_hj32_block_right_seed_product)) /\ exists pa_q_hj32_block_right_seed_product_start. pa_u_hj32_block_right_seed_product = pa_q_hj32_block_right_seed_product_start * S ((S (0)) * pa_v_hj32_block_right_seed_product) + (1))) /\ ((((exists pa_h_hj32_block_right_seed_product_terminal. pa_h_hj32_block_right_seed_product_terminal + S (y) = S ((S (e)) * pa_v_hj32_block_right_seed_product)) /\ exists pa_q_hj32_block_right_seed_product_terminal. pa_u_hj32_block_right_seed_product = pa_q_hj32_block_right_seed_product_terminal * S ((S (e)) * pa_v_hj32_block_right_seed_product) + (y))) /\ forall pa_i_hj32_block_right_seed_product. (exists pa_lt_hj32_block_right_seed_product_bound. pa_lt_hj32_block_right_seed_product_bound + S pa_i_hj32_block_right_seed_product = e) -> exists pa_p_hj32_block_right_seed_product pa_r_hj32_block_right_seed_product pa_s_hj32_block_right_seed_product. ((((exists pa_h_hj32_block_right_seed_product_factor. pa_h_hj32_block_right_seed_product_factor + S (pa_p_hj32_block_right_seed_product) = S ((S (pa_i_hj32_block_right_seed_product)) * pa_c_hj32_block_right_seed)) /\ exists pa_q_hj32_block_right_seed_product_factor. pa_b_hj32_block_right_seed = pa_q_hj32_block_right_seed_product_factor * S ((S (pa_i_hj32_block_right_seed_product)) * pa_c_hj32_block_right_seed) + (pa_p_hj32_block_right_seed_product))) /\ ((((exists pa_h_hj32_block_right_seed_product_partial. pa_h_hj32_block_right_seed_product_partial + S (pa_r_hj32_block_right_seed_product) = S ((S (pa_i_hj32_block_right_seed_product)) * pa_v_hj32_block_right_seed_product)) /\ exists pa_q_hj32_block_right_seed_product_partial. pa_u_hj32_block_right_seed_product = pa_q_hj32_block_right_seed_product_partial * S ((S (pa_i_hj32_block_right_seed_product)) * pa_v_hj32_block_right_seed_product) + (pa_r_hj32_block_right_seed_product))) /\ ((((exists pa_h_hj32_block_right_seed_product_successor. pa_h_hj32_block_right_seed_product_successor + S (pa_s_hj32_block_right_seed_product) = S ((S (S pa_i_hj32_block_right_seed_product)) * pa_v_hj32_block_right_seed_product)) /\ exists pa_q_hj32_block_right_seed_product_successor. pa_u_hj32_block_right_seed_product = pa_q_hj32_block_right_seed_product_successor * S ((S (S pa_i_hj32_block_right_seed_product)) * pa_v_hj32_block_right_seed_product) + (pa_s_hj32_block_right_seed_product))) /\ pa_s_hj32_block_right_seed_product = pa_r_hj32_block_right_seed_product * pa_p_hj32_block_right_seed_product)))))))) -> (exists bqb_le_gap_hj32_block_seed_bound. bqb_le_gap_hj32_block_seed_bound + (x) = (y)) -> (exists pa_b_hj32_block_left pa_c_hj32_block_left. ((forall pa_i_hj32_block_left_repeat. (exists pa_lt_hj32_block_left_repeat_bound. pa_lt_hj32_block_left_repeat_bound + S pa_i_hj32_block_left_repeat = d * m) -> (((exists pa_h_hj32_block_left_repeat_decoded. pa_h_hj32_block_left_repeat_decoded + S (a) = S ((S (pa_i_hj32_block_left_repeat)) * pa_c_hj32_block_left)) /\ exists pa_q_hj32_block_left_repeat_decoded. pa_b_hj32_block_left = pa_q_hj32_block_left_repeat_decoded * S ((S (pa_i_hj32_block_left_repeat)) * pa_c_hj32_block_left) + (a)))) /\ (exists pa_u_hj32_block_left_product pa_v_hj32_block_left_product. ((((exists pa_h_hj32_block_left_product_start. pa_h_hj32_block_left_product_start + S (1) = S ((S (0)) * pa_v_hj32_block_left_product)) /\ exists pa_q_hj32_block_left_product_start. pa_u_hj32_block_left_product = pa_q_hj32_block_left_product_start * S ((S (0)) * pa_v_hj32_block_left_product) + (1))) /\ ((((exists pa_h_hj32_block_left_product_terminal. pa_h_hj32_block_left_product_terminal + S (X) = S ((S (d * m)) * pa_v_hj32_block_left_product)) /\ exists pa_q_hj32_block_left_product_terminal. pa_u_hj32_block_left_product = pa_q_hj32_block_left_product_terminal * S ((S (d * m)) * pa_v_hj32_block_left_product) + (X))) /\ forall pa_i_hj32_block_left_product. (exists pa_lt_hj32_block_left_product_bound. pa_lt_hj32_block_left_product_bound + S pa_i_hj32_block_left_product = d * m) -> exists pa_p_hj32_block_left_product pa_r_hj32_block_left_product pa_s_hj32_block_left_product. ((((exists pa_h_hj32_block_left_product_factor. pa_h_hj32_block_left_product_factor + S (pa_p_hj32_block_left_product) = S ((S (pa_i_hj32_block_left_product)) * pa_c_hj32_block_left)) /\ exists pa_q_hj32_block_left_product_factor. pa_b_hj32_block_left = pa_q_hj32_block_left_product_factor * S ((S (pa_i_hj32_block_left_product)) * pa_c_hj32_block_left) + (pa_p_hj32_block_left_product))) /\ ((((exists pa_h_hj32_block_left_product_partial. pa_h_hj32_block_left_product_partial + S (pa_r_hj32_block_left_product) = S ((S (pa_i_hj32_block_left_product)) * pa_v_hj32_block_left_product)) /\ exists pa_q_hj32_block_left_product_partial. pa_u_hj32_block_left_product = pa_q_hj32_block_left_product_partial * S ((S (pa_i_hj32_block_left_product)) * pa_v_hj32_block_left_product) + (pa_r_hj32_block_left_product))) /\ ((((exists pa_h_hj32_block_left_product_successor. pa_h_hj32_block_left_product_successor + S (pa_s_hj32_block_left_product) = S ((S (S pa_i_hj32_block_left_product)) * pa_v_hj32_block_left_product)) /\ exists pa_q_hj32_block_left_product_successor. pa_u_hj32_block_left_product = pa_q_hj32_block_left_product_successor * S ((S (S pa_i_hj32_block_left_product)) * pa_v_hj32_block_left_product) + (pa_s_hj32_block_left_product))) /\ pa_s_hj32_block_left_product = pa_r_hj32_block_left_product * pa_p_hj32_block_left_product)))))))) -> (exists pa_b_hj32_block_right pa_c_hj32_block_right. ((forall pa_i_hj32_block_right_repeat. (exists pa_lt_hj32_block_right_repeat_bound. pa_lt_hj32_block_right_repeat_bound + S pa_i_hj32_block_right_repeat = e * m) -> (((exists pa_h_hj32_block_right_repeat_decoded. pa_h_hj32_block_right_repeat_decoded + S (b) = S ((S (pa_i_hj32_block_right_repeat)) * pa_c_hj32_block_right)) /\ exists pa_q_hj32_block_right_repeat_decoded. pa_b_hj32_block_right = pa_q_hj32_block_right_repeat_decoded * S ((S (pa_i_hj32_block_right_repeat)) * pa_c_hj32_block_right) + (b)))) /\ (exists pa_u_hj32_block_right_product pa_v_hj32_block_right_product. ((((exists pa_h_hj32_block_right_product_start. pa_h_hj32_block_right_product_start + S (1) = S ((S (0)) * pa_v_hj32_block_right_product)) /\ exists pa_q_hj32_block_right_product_start. pa_u_hj32_block_right_product = pa_q_hj32_block_right_product_start * S ((S (0)) * pa_v_hj32_block_right_product) + (1))) /\ ((((exists pa_h_hj32_block_right_product_terminal. pa_h_hj32_block_right_product_terminal + S (Y) = S ((S (e * m)) * pa_v_hj32_block_right_product)) /\ exists pa_q_hj32_block_right_product_terminal. pa_u_hj32_block_right_product = pa_q_hj32_block_right_product_terminal * S ((S (e * m)) * pa_v_hj32_block_right_product) + (Y))) /\ forall pa_i_hj32_block_right_product. (exists pa_lt_hj32_block_right_product_bound. pa_lt_hj32_block_right_product_bound + S pa_i_hj32_block_right_product = e * m) -> exists pa_p_hj32_block_right_product pa_r_hj32_block_right_product pa_s_hj32_block_right_product. ((((exists pa_h_hj32_block_right_product_factor. pa_h_hj32_block_right_product_factor + S (pa_p_hj32_block_right_product) = S ((S (pa_i_hj32_block_right_product)) * pa_c_hj32_block_right)) /\ exists pa_q_hj32_block_right_product_factor. pa_b_hj32_block_right = pa_q_hj32_block_right_product_factor * S ((S (pa_i_hj32_block_right_product)) * pa_c_hj32_block_right) + (pa_p_hj32_block_right_product))) /\ ((((exists pa_h_hj32_block_right_product_partial. pa_h_hj32_block_right_product_partial + S (pa_r_hj32_block_right_product) = S ((S (pa_i_hj32_block_right_product)) * pa_v_hj32_block_right_product)) /\ exists pa_q_hj32_block_right_product_partial. pa_u_hj32_block_right_product = pa_q_hj32_block_right_product_partial * S ((S (pa_i_hj32_block_right_product)) * pa_v_hj32_block_right_product) + (pa_r_hj32_block_right_product))) /\ ((((exists pa_h_hj32_block_right_product_successor. pa_h_hj32_block_right_product_successor + S (pa_s_hj32_block_right_product) = S ((S (S pa_i_hj32_block_right_product)) * pa_v_hj32_block_right_product)) /\ exists pa_q_hj32_block_right_product_successor. pa_u_hj32_block_right_product = pa_q_hj32_block_right_product_successor * S ((S (S pa_i_hj32_block_right_product)) * pa_v_hj32_block_right_product) + (pa_s_hj32_block_right_product))) /\ pa_s_hj32_block_right_product = pa_r_hj32_block_right_product * pa_p_hj32_block_right_product)))))))) -> (exists bqb_le_gap_hj32_block_result. bqb_le_gap_hj32_block_result + (X) = (Y))