pow_successor_compose_from_total

Exact expanded PA statement

forall a e r n. (forall bpt_a_successor bpt_e_successor. exists bpt_x_successor. (exists ff_b_bpt_value_successor ff_c_bpt_value_successor. ((forall ff_i_bpt_value_successor_repeat. (exists ff_lt_bpt_value_successor_repeat_bound. ff_lt_bpt_value_successor_repeat_bound + S ff_i_bpt_value_successor_repeat = bpt_e_successor) -> (((exists ff_h_bpt_value_successor_repeat_decoded. ff_h_bpt_value_successor_repeat_decoded + S (bpt_a_successor) = S ((S (ff_i_bpt_value_successor_repeat)) * ff_c_bpt_value_successor)) /\ exists ff_q_bpt_value_successor_repeat_decoded. ff_b_bpt_value_successor = ff_q_bpt_value_successor_repeat_decoded * S ((S (ff_i_bpt_value_successor_repeat)) * ff_c_bpt_value_successor) + (bpt_a_successor)))) /\ (exists ff_u_bpt_value_successor_product ff_v_bpt_value_successor_product. ((((exists ff_h_bpt_value_successor_product_start. ff_h_bpt_value_successor_product_start + S (1) = S ((S (0)) * ff_v_bpt_value_successor_product)) /\ exists ff_q_bpt_value_successor_product_start. ff_u_bpt_value_successor_product = ff_q_bpt_value_successor_product_start * S ((S (0)) * ff_v_bpt_value_successor_product) + (1))) /\ ((((exists ff_h_bpt_value_successor_product_terminal. ff_h_bpt_value_successor_product_terminal + S (bpt_x_successor) = S ((S (bpt_e_successor)) * ff_v_bpt_value_successor_product)) /\ exists ff_q_bpt_value_successor_product_terminal. ff_u_bpt_value_successor_product = ff_q_bpt_value_successor_product_terminal * S ((S (bpt_e_successor)) * ff_v_bpt_value_successor_product) + (bpt_x_successor))) /\ forall ff_i_bpt_value_successor_product. (exists ff_lt_bpt_value_successor_product_bound. ff_lt_bpt_value_successor_product_bound + S ff_i_bpt_value_successor_product = bpt_e_successor) -> exists ff_p_bpt_value_successor_product ff_r_bpt_value_successor_product ff_s_bpt_value_successor_product. ((((exists ff_h_bpt_value_successor_product_factor. ff_h_bpt_value_successor_product_factor + S (ff_p_bpt_value_successor_product) = S ((S (ff_i_bpt_value_successor_product)) * ff_c_bpt_value_successor)) /\ exists ff_q_bpt_value_successor_product_factor. ff_b_bpt_value_successor = ff_q_bpt_value_successor_product_factor * S ((S (ff_i_bpt_value_successor_product)) * ff_c_bpt_value_successor) + (ff_p_bpt_value_successor_product))) /\ ((((exists ff_h_bpt_value_successor_product_partial. ff_h_bpt_value_successor_product_partial + S (ff_r_bpt_value_successor_product) = S ((S (ff_i_bpt_value_successor_product)) * ff_v_bpt_value_successor_product)) /\ exists ff_q_bpt_value_successor_product_partial. ff_u_bpt_value_successor_product = ff_q_bpt_value_successor_product_partial * S ((S (ff_i_bpt_value_successor_product)) * ff_v_bpt_value_successor_product) + (ff_r_bpt_value_successor_product))) /\ ((((exists ff_h_bpt_value_successor_product_successor. ff_h_bpt_value_successor_product_successor + S (ff_s_bpt_value_successor_product) = S ((S (S ff_i_bpt_value_successor_product)) * ff_v_bpt_value_successor_product)) /\ exists ff_q_bpt_value_successor_product_successor. ff_u_bpt_value_successor_product = ff_q_bpt_value_successor_product_successor * S ((S (S ff_i_bpt_value_successor_product)) * ff_v_bpt_value_successor_product) + (ff_s_bpt_value_successor_product))) /\ ff_s_bpt_value_successor_product = ff_r_bpt_value_successor_product * ff_p_bpt_value_successor_product))))))))) -> (exists ff_b_bpt_predecessor ff_c_bpt_predecessor. ((forall ff_i_bpt_predecessor_repeat. (exists ff_lt_bpt_predecessor_repeat_bound. ff_lt_bpt_predecessor_repeat_bound + S ff_i_bpt_predecessor_repeat = e) -> (((exists ff_h_bpt_predecessor_repeat_decoded. ff_h_bpt_predecessor_repeat_decoded + S (a) = S ((S (ff_i_bpt_predecessor_repeat)) * ff_c_bpt_predecessor)) /\ exists ff_q_bpt_predecessor_repeat_decoded. ff_b_bpt_predecessor = ff_q_bpt_predecessor_repeat_decoded * S ((S (ff_i_bpt_predecessor_repeat)) * ff_c_bpt_predecessor) + (a)))) /\ (exists ff_u_bpt_predecessor_product ff_v_bpt_predecessor_product. ((((exists ff_h_bpt_predecessor_product_start. ff_h_bpt_predecessor_product_start + S (1) = S ((S (0)) * ff_v_bpt_predecessor_product)) /\ exists ff_q_bpt_predecessor_product_start. ff_u_bpt_predecessor_product = ff_q_bpt_predecessor_product_start * S ((S (0)) * ff_v_bpt_predecessor_product) + (1))) /\ ((((exists ff_h_bpt_predecessor_product_terminal. ff_h_bpt_predecessor_product_terminal + S (r) = S ((S (e)) * ff_v_bpt_predecessor_product)) /\ exists ff_q_bpt_predecessor_product_terminal. ff_u_bpt_predecessor_product = ff_q_bpt_predecessor_product_terminal * S ((S (e)) * ff_v_bpt_predecessor_product) + (r))) /\ forall ff_i_bpt_predecessor_product. (exists ff_lt_bpt_predecessor_product_bound. ff_lt_bpt_predecessor_product_bound + S ff_i_bpt_predecessor_product = e) -> exists ff_p_bpt_predecessor_product ff_r_bpt_predecessor_product ff_s_bpt_predecessor_product. ((((exists ff_h_bpt_predecessor_product_factor. ff_h_bpt_predecessor_product_factor + S (ff_p_bpt_predecessor_product) = S ((S (ff_i_bpt_predecessor_product)) * ff_c_bpt_predecessor)) /\ exists ff_q_bpt_predecessor_product_factor. ff_b_bpt_predecessor = ff_q_bpt_predecessor_product_factor * S ((S (ff_i_bpt_predecessor_product)) * ff_c_bpt_predecessor) + (ff_p_bpt_predecessor_product))) /\ ((((exists ff_h_bpt_predecessor_product_partial. ff_h_bpt_predecessor_product_partial + S (ff_r_bpt_predecessor_product) = S ((S (ff_i_bpt_predecessor_product)) * ff_v_bpt_predecessor_product)) /\ exists ff_q_bpt_predecessor_product_partial. ff_u_bpt_predecessor_product = ff_q_bpt_predecessor_product_partial * S ((S (ff_i_bpt_predecessor_product)) * ff_v_bpt_predecessor_product) + (ff_r_bpt_predecessor_product))) /\ ((((exists ff_h_bpt_predecessor_product_successor. ff_h_bpt_predecessor_product_successor + S (ff_s_bpt_predecessor_product) = S ((S (S ff_i_bpt_predecessor_product)) * ff_v_bpt_predecessor_product)) /\ exists ff_q_bpt_predecessor_product_successor. ff_u_bpt_predecessor_product = ff_q_bpt_predecessor_product_successor * S ((S (S ff_i_bpt_predecessor_product)) * ff_v_bpt_predecessor_product) + (ff_s_bpt_predecessor_product))) /\ ff_s_bpt_predecessor_product = ff_r_bpt_predecessor_product * ff_p_bpt_predecessor_product)))))))) -> n = r * a -> (exists pa_b_bpt_successor pa_c_bpt_successor. ((forall pa_i_bpt_successor_repeat. (exists pa_lt_bpt_successor_repeat_bound. pa_lt_bpt_successor_repeat_bound + S pa_i_bpt_successor_repeat = S e) -> (((exists pa_h_bpt_successor_repeat_decoded. pa_h_bpt_successor_repeat_decoded + S (a) = S ((S (pa_i_bpt_successor_repeat)) * pa_c_bpt_successor)) /\ exists pa_q_bpt_successor_repeat_decoded. pa_b_bpt_successor = pa_q_bpt_successor_repeat_decoded * S ((S (pa_i_bpt_successor_repeat)) * pa_c_bpt_successor) + (a)))) /\ (exists pa_u_bpt_successor_product pa_v_bpt_successor_product. ((((exists pa_h_bpt_successor_product_start. pa_h_bpt_successor_product_start + S (1) = S ((S (0)) * pa_v_bpt_successor_product)) /\ exists pa_q_bpt_successor_product_start. pa_u_bpt_successor_product = pa_q_bpt_successor_product_start * S ((S (0)) * pa_v_bpt_successor_product) + (1))) /\ ((((exists pa_h_bpt_successor_product_terminal. pa_h_bpt_successor_product_terminal + S (n) = S ((S (S e)) * pa_v_bpt_successor_product)) /\ exists pa_q_bpt_successor_product_terminal. pa_u_bpt_successor_product = pa_q_bpt_successor_product_terminal * S ((S (S e)) * pa_v_bpt_successor_product) + (n))) /\ forall pa_i_bpt_successor_product. (exists pa_lt_bpt_successor_product_bound. pa_lt_bpt_successor_product_bound + S pa_i_bpt_successor_product = S e) -> exists pa_p_bpt_successor_product pa_r_bpt_successor_product pa_s_bpt_successor_product. ((((exists pa_h_bpt_successor_product_factor. pa_h_bpt_successor_product_factor + S (pa_p_bpt_successor_product) = S ((S (pa_i_bpt_successor_product)) * pa_c_bpt_successor)) /\ exists pa_q_bpt_successor_product_factor. pa_b_bpt_successor = pa_q_bpt_successor_product_factor * S ((S (pa_i_bpt_successor_product)) * pa_c_bpt_successor) + (pa_p_bpt_successor_product))) /\ ((((exists pa_h_bpt_successor_product_partial. pa_h_bpt_successor_product_partial + S (pa_r_bpt_successor_product) = S ((S (pa_i_bpt_successor_product)) * pa_v_bpt_successor_product)) /\ exists pa_q_bpt_successor_product_partial. pa_u_bpt_successor_product = pa_q_bpt_successor_product_partial * S ((S (pa_i_bpt_successor_product)) * pa_v_bpt_successor_product) + (pa_r_bpt_successor_product))) /\ ((((exists pa_h_bpt_successor_product_successor. pa_h_bpt_successor_product_successor + S (pa_s_bpt_successor_product) = S ((S (S pa_i_bpt_successor_product)) * pa_v_bpt_successor_product)) /\ exists pa_q_bpt_successor_product_successor. pa_u_bpt_successor_product = pa_q_bpt_successor_product_successor * S ((S (S pa_i_bpt_successor_product)) * pa_v_bpt_successor_product) + (pa_s_bpt_successor_product))) /\ pa_s_bpt_successor_product = pa_r_bpt_successor_product * pa_p_bpt_successor_product))))))))