pow_mul_exp

Exact expanded PA statement

forall a e f p x y z. p = e * f -> (exists ff_b_mul_base ff_c_mul_base. ((forall ff_i_mul_base_repeat. (exists ff_lt_mul_base_repeat_bound. ff_lt_mul_base_repeat_bound + S ff_i_mul_base_repeat = e) -> (((exists ff_h_mul_base_repeat_decoded. ff_h_mul_base_repeat_decoded + S (a) = S ((S (ff_i_mul_base_repeat)) * ff_c_mul_base)) /\ exists ff_q_mul_base_repeat_decoded. ff_b_mul_base = ff_q_mul_base_repeat_decoded * S ((S (ff_i_mul_base_repeat)) * ff_c_mul_base) + (a)))) /\ (exists ff_u_mul_base_product ff_v_mul_base_product. ((((exists ff_h_mul_base_product_start. ff_h_mul_base_product_start + S (1) = S ((S (0)) * ff_v_mul_base_product)) /\ exists ff_q_mul_base_product_start. ff_u_mul_base_product = ff_q_mul_base_product_start * S ((S (0)) * ff_v_mul_base_product) + (1))) /\ ((((exists ff_h_mul_base_product_terminal. ff_h_mul_base_product_terminal + S (x) = S ((S (e)) * ff_v_mul_base_product)) /\ exists ff_q_mul_base_product_terminal. ff_u_mul_base_product = ff_q_mul_base_product_terminal * S ((S (e)) * ff_v_mul_base_product) + (x))) /\ forall ff_i_mul_base_product. (exists ff_lt_mul_base_product_bound. ff_lt_mul_base_product_bound + S ff_i_mul_base_product = e) -> exists ff_p_mul_base_product ff_r_mul_base_product ff_s_mul_base_product. ((((exists ff_h_mul_base_product_factor. ff_h_mul_base_product_factor + S (ff_p_mul_base_product) = S ((S (ff_i_mul_base_product)) * ff_c_mul_base)) /\ exists ff_q_mul_base_product_factor. ff_b_mul_base = ff_q_mul_base_product_factor * S ((S (ff_i_mul_base_product)) * ff_c_mul_base) + (ff_p_mul_base_product))) /\ ((((exists ff_h_mul_base_product_partial. ff_h_mul_base_product_partial + S (ff_r_mul_base_product) = S ((S (ff_i_mul_base_product)) * ff_v_mul_base_product)) /\ exists ff_q_mul_base_product_partial. ff_u_mul_base_product = ff_q_mul_base_product_partial * S ((S (ff_i_mul_base_product)) * ff_v_mul_base_product) + (ff_r_mul_base_product))) /\ ((((exists ff_h_mul_base_product_successor. ff_h_mul_base_product_successor + S (ff_s_mul_base_product) = S ((S (S ff_i_mul_base_product)) * ff_v_mul_base_product)) /\ exists ff_q_mul_base_product_successor. ff_u_mul_base_product = ff_q_mul_base_product_successor * S ((S (S ff_i_mul_base_product)) * ff_v_mul_base_product) + (ff_s_mul_base_product))) /\ ff_s_mul_base_product = ff_r_mul_base_product * ff_p_mul_base_product)))))))) -> (exists ff_b_mul_outer ff_c_mul_outer. ((forall ff_i_mul_outer_repeat. (exists ff_lt_mul_outer_repeat_bound. ff_lt_mul_outer_repeat_bound + S ff_i_mul_outer_repeat = f) -> (((exists ff_h_mul_outer_repeat_decoded. ff_h_mul_outer_repeat_decoded + S (x) = S ((S (ff_i_mul_outer_repeat)) * ff_c_mul_outer)) /\ exists ff_q_mul_outer_repeat_decoded. ff_b_mul_outer = ff_q_mul_outer_repeat_decoded * S ((S (ff_i_mul_outer_repeat)) * ff_c_mul_outer) + (x)))) /\ (exists ff_u_mul_outer_product ff_v_mul_outer_product. ((((exists ff_h_mul_outer_product_start. ff_h_mul_outer_product_start + S (1) = S ((S (0)) * ff_v_mul_outer_product)) /\ exists ff_q_mul_outer_product_start. ff_u_mul_outer_product = ff_q_mul_outer_product_start * S ((S (0)) * ff_v_mul_outer_product) + (1))) /\ ((((exists ff_h_mul_outer_product_terminal. ff_h_mul_outer_product_terminal + S (y) = S ((S (f)) * ff_v_mul_outer_product)) /\ exists ff_q_mul_outer_product_terminal. ff_u_mul_outer_product = ff_q_mul_outer_product_terminal * S ((S (f)) * ff_v_mul_outer_product) + (y))) /\ forall ff_i_mul_outer_product. (exists ff_lt_mul_outer_product_bound. ff_lt_mul_outer_product_bound + S ff_i_mul_outer_product = f) -> exists ff_p_mul_outer_product ff_r_mul_outer_product ff_s_mul_outer_product. ((((exists ff_h_mul_outer_product_factor. ff_h_mul_outer_product_factor + S (ff_p_mul_outer_product) = S ((S (ff_i_mul_outer_product)) * ff_c_mul_outer)) /\ exists ff_q_mul_outer_product_factor. ff_b_mul_outer = ff_q_mul_outer_product_factor * S ((S (ff_i_mul_outer_product)) * ff_c_mul_outer) + (ff_p_mul_outer_product))) /\ ((((exists ff_h_mul_outer_product_partial. ff_h_mul_outer_product_partial + S (ff_r_mul_outer_product) = S ((S (ff_i_mul_outer_product)) * ff_v_mul_outer_product)) /\ exists ff_q_mul_outer_product_partial. ff_u_mul_outer_product = ff_q_mul_outer_product_partial * S ((S (ff_i_mul_outer_product)) * ff_v_mul_outer_product) + (ff_r_mul_outer_product))) /\ ((((exists ff_h_mul_outer_product_successor. ff_h_mul_outer_product_successor + S (ff_s_mul_outer_product) = S ((S (S ff_i_mul_outer_product)) * ff_v_mul_outer_product)) /\ exists ff_q_mul_outer_product_successor. ff_u_mul_outer_product = ff_q_mul_outer_product_successor * S ((S (S ff_i_mul_outer_product)) * ff_v_mul_outer_product) + (ff_s_mul_outer_product))) /\ ff_s_mul_outer_product = ff_r_mul_outer_product * ff_p_mul_outer_product)))))))) -> (exists ff_b_mul_total ff_c_mul_total. ((forall ff_i_mul_total_repeat. (exists ff_lt_mul_total_repeat_bound. ff_lt_mul_total_repeat_bound + S ff_i_mul_total_repeat = p) -> (((exists ff_h_mul_total_repeat_decoded. ff_h_mul_total_repeat_decoded + S (a) = S ((S (ff_i_mul_total_repeat)) * ff_c_mul_total)) /\ exists ff_q_mul_total_repeat_decoded. ff_b_mul_total = ff_q_mul_total_repeat_decoded * S ((S (ff_i_mul_total_repeat)) * ff_c_mul_total) + (a)))) /\ (exists ff_u_mul_total_product ff_v_mul_total_product. ((((exists ff_h_mul_total_product_start. ff_h_mul_total_product_start + S (1) = S ((S (0)) * ff_v_mul_total_product)) /\ exists ff_q_mul_total_product_start. ff_u_mul_total_product = ff_q_mul_total_product_start * S ((S (0)) * ff_v_mul_total_product) + (1))) /\ ((((exists ff_h_mul_total_product_terminal. ff_h_mul_total_product_terminal + S (z) = S ((S (p)) * ff_v_mul_total_product)) /\ exists ff_q_mul_total_product_terminal. ff_u_mul_total_product = ff_q_mul_total_product_terminal * S ((S (p)) * ff_v_mul_total_product) + (z))) /\ forall ff_i_mul_total_product. (exists ff_lt_mul_total_product_bound. ff_lt_mul_total_product_bound + S ff_i_mul_total_product = p) -> exists ff_p_mul_total_product ff_r_mul_total_product ff_s_mul_total_product. ((((exists ff_h_mul_total_product_factor. ff_h_mul_total_product_factor + S (ff_p_mul_total_product) = S ((S (ff_i_mul_total_product)) * ff_c_mul_total)) /\ exists ff_q_mul_total_product_factor. ff_b_mul_total = ff_q_mul_total_product_factor * S ((S (ff_i_mul_total_product)) * ff_c_mul_total) + (ff_p_mul_total_product))) /\ ((((exists ff_h_mul_total_product_partial. ff_h_mul_total_product_partial + S (ff_r_mul_total_product) = S ((S (ff_i_mul_total_product)) * ff_v_mul_total_product)) /\ exists ff_q_mul_total_product_partial. ff_u_mul_total_product = ff_q_mul_total_product_partial * S ((S (ff_i_mul_total_product)) * ff_v_mul_total_product) + (ff_r_mul_total_product))) /\ ((((exists ff_h_mul_total_product_successor. ff_h_mul_total_product_successor + S (ff_s_mul_total_product) = S ((S (S ff_i_mul_total_product)) * ff_v_mul_total_product)) /\ exists ff_q_mul_total_product_successor. ff_u_mul_total_product = ff_q_mul_total_product_successor * S ((S (S ff_i_mul_total_product)) * ff_v_mul_total_product) + (ff_s_mul_total_product))) /\ ff_s_mul_total_product = ff_r_mul_total_product * ff_p_mul_total_product)))))))) -> y = z