pow_two

Exact expanded PA statement

forall a e n. e = 2 -> (exists ff_b_two ff_c_two. ((forall ff_i_two_repeat. (exists ff_lt_two_repeat_bound. ff_lt_two_repeat_bound + S ff_i_two_repeat = e) -> (((exists ff_h_two_repeat_decoded. ff_h_two_repeat_decoded + S (a) = S ((S (ff_i_two_repeat)) * ff_c_two)) /\ exists ff_q_two_repeat_decoded. ff_b_two = ff_q_two_repeat_decoded * S ((S (ff_i_two_repeat)) * ff_c_two) + (a)))) /\ (exists ff_u_two_product ff_v_two_product. ((((exists ff_h_two_product_start. ff_h_two_product_start + S (1) = S ((S (0)) * ff_v_two_product)) /\ exists ff_q_two_product_start. ff_u_two_product = ff_q_two_product_start * S ((S (0)) * ff_v_two_product) + (1))) /\ ((((exists ff_h_two_product_terminal. ff_h_two_product_terminal + S (n) = S ((S (e)) * ff_v_two_product)) /\ exists ff_q_two_product_terminal. ff_u_two_product = ff_q_two_product_terminal * S ((S (e)) * ff_v_two_product) + (n))) /\ forall ff_i_two_product. (exists ff_lt_two_product_bound. ff_lt_two_product_bound + S ff_i_two_product = e) -> exists ff_p_two_product ff_r_two_product ff_s_two_product. ((((exists ff_h_two_product_factor. ff_h_two_product_factor + S (ff_p_two_product) = S ((S (ff_i_two_product)) * ff_c_two)) /\ exists ff_q_two_product_factor. ff_b_two = ff_q_two_product_factor * S ((S (ff_i_two_product)) * ff_c_two) + (ff_p_two_product))) /\ ((((exists ff_h_two_product_partial. ff_h_two_product_partial + S (ff_r_two_product) = S ((S (ff_i_two_product)) * ff_v_two_product)) /\ exists ff_q_two_product_partial. ff_u_two_product = ff_q_two_product_partial * S ((S (ff_i_two_product)) * ff_v_two_product) + (ff_r_two_product))) /\ ((((exists ff_h_two_product_successor. ff_h_two_product_successor + S (ff_s_two_product) = S ((S (S ff_i_two_product)) * ff_v_two_product)) /\ exists ff_q_two_product_successor. ff_u_two_product = ff_q_two_product_successor * S ((S (S ff_i_two_product)) * ff_v_two_product) + (ff_s_two_product))) /\ ff_s_two_product = ff_r_two_product * ff_p_two_product)))))))) -> n = a * a