factorial_exists

Exact expanded PA statement

forall n. exists z. (exists ff_b_exists ff_c_exists. ((forall ff_i_exists_range. (exists ff_lt_exists_range_bound. ff_lt_exists_range_bound + S ff_i_exists_range = n) -> (((exists ff_h_exists_range_decoded. ff_h_exists_range_decoded + S (1 + ff_i_exists_range) = S ((S (ff_i_exists_range)) * ff_c_exists)) /\ exists ff_q_exists_range_decoded. ff_b_exists = ff_q_exists_range_decoded * S ((S (ff_i_exists_range)) * ff_c_exists) + (1 + ff_i_exists_range)))) /\ (exists ff_u_exists_product ff_v_exists_product. ((((exists ff_h_exists_product_start. ff_h_exists_product_start + S (1) = S ((S (0)) * ff_v_exists_product)) /\ exists ff_q_exists_product_start. ff_u_exists_product = ff_q_exists_product_start * S ((S (0)) * ff_v_exists_product) + (1))) /\ ((((exists ff_h_exists_product_terminal. ff_h_exists_product_terminal + S (z) = S ((S (n)) * ff_v_exists_product)) /\ exists ff_q_exists_product_terminal. ff_u_exists_product = ff_q_exists_product_terminal * S ((S (n)) * ff_v_exists_product) + (z))) /\ forall ff_i_exists_product. (exists ff_lt_exists_product_bound. ff_lt_exists_product_bound + S ff_i_exists_product = n) -> exists ff_p_exists_product ff_r_exists_product ff_s_exists_product. ((((exists ff_h_exists_product_factor. ff_h_exists_product_factor + S (ff_p_exists_product) = S ((S (ff_i_exists_product)) * ff_c_exists)) /\ exists ff_q_exists_product_factor. ff_b_exists = ff_q_exists_product_factor * S ((S (ff_i_exists_product)) * ff_c_exists) + (ff_p_exists_product))) /\ ((((exists ff_h_exists_product_partial. ff_h_exists_product_partial + S (ff_r_exists_product) = S ((S (ff_i_exists_product)) * ff_v_exists_product)) /\ exists ff_q_exists_product_partial. ff_u_exists_product = ff_q_exists_product_partial * S ((S (ff_i_exists_product)) * ff_v_exists_product) + (ff_r_exists_product))) /\ ((((exists ff_h_exists_product_successor. ff_h_exists_product_successor + S (ff_s_exists_product) = S ((S (S ff_i_exists_product)) * ff_v_exists_product)) /\ exists ff_q_exists_product_successor. ff_u_exists_product = ff_q_exists_product_successor * S ((S (S ff_i_exists_product)) * ff_v_exists_product) + (ff_s_exists_product))) /\ ff_s_exists_product = ff_r_exists_product * ff_p_exists_product))))))))